Precise Predictions within the

Two-Higgs-Doublet Model

Lukas Altenkamp

Physikalisches Institut

Fakultat fur Mathematik und Physik

Albert-Ludwigs-Universitat Freiburg

Precise Predictions within the

Two-Higgs-Doublet Model

Dissertation

zur Erlangung des Doktorgrades

der

Fakultat fur Mathematik und Physik

der

Albert-Ludwigs-Universitat Freiburg im Breisgau

vorgelegt von

Lukas Altenkamp

Dezember 2016

Dekan: Prof. Dr. Gregor Herten

Referent: Prof. Dr. Stefan Dittmaier

Koreferent: JProf. Dr. Harald Ita

Tag der mundlichen Prufung: 21.02.2017

Contents

1. Introduction 1

2. Theoretical Preliminaries 5

2.1. The Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1. GSW Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.2. Quantum Chromodynamics . . . . . . . . . . . . . . . . . . . . . 11

2.2. The Two-Higgs-Doublet Model . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.1. The Higgs Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2.2. Reparameterizing the Potential . . . . . . . . . . . . . . . . . . . 17

2.2.3. Yukawa Couplings . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2.4. Gauge-Fixing and Faddeev-Popov Ghost Lagrangians . . . . . . . 24

3. Renormalization of the THDM 26

3.1. The Counterterm Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . 27

3.1.1. Higgs Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.1.1.1. The Reparameterized Bare Higgs Potential at NLO . . . 28

3.1.1.2. Renormalization of the Mixing Angles . . . . . . . . . . 32

3.1.1.3. Renormalization with a Diagonal Mass Matrix – version 1a 34

3.1.2. Fermionic and Gauge Parts . . . . . . . . . . . . . . . . . . . . . 36

3.1.3. The Higgs Kinetic Part . . . . . . . . . . . . . . . . . . . . . . . . 37

3.1.4. Yukawa Part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2. Renormalization Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.2.1. On-shell Renormalization Conditions . . . . . . . . . . . . . . . . 41

3.2.1.1. Higgs Sector . . . . . . . . . . . . . . . . . . . . . . . . 41

3.2.1.2. Electroweak Sector . . . . . . . . . . . . . . . . . . . . . 44

3.2.1.3. Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.2.2. Different Renormalization Schemes . . . . . . . . . . . . . . . . . 46

3.2.2.1. α MS Scheme . . . . . . . . . . . . . . . . . . . . . . . 46

3.2.2.2. λ3 MS Scheme . . . . . . . . . . . . . . . . . . . . . . . 49

3.2.2.3. The FJ Tadpole Scheme . . . . . . . . . . . . . . . . . . 51

3.2.2.4. The FJ λ3 Scheme . . . . . . . . . . . . . . . . . . . . . 54

3.2.3. Comparing Different Renormalization Schemes . . . . . . . . . . . 54

3.3. The Running of the Couplings . . . . . . . . . . . . . . . . . . . . . . . . 55

3.4. Implementation into a FeynArts Model File . . . . . . . . . . . . . . . 59

V

4. Higgs-Boson Decay into Four Fermions 60

4.1. Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.2. Theoretical Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.2.1. Tree Level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.2.2. EW Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.2.3. QCD Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.3. Amplitudes in the THDM . . . . . . . . . . . . . . . . . . . . . . . . . . 734.3.1. Tree Level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.3.2. EW Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.3.2.1. One-loop Diagrams . . . . . . . . . . . . . . . . . . . . . 744.3.2.2. Counterterm Amplitudes . . . . . . . . . . . . . . . . . . 764.3.2.3. Real Emission . . . . . . . . . . . . . . . . . . . . . . . . 78

4.3.3. QCD Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.3.3.1. One-loop Diagrams . . . . . . . . . . . . . . . . . . . . . 784.3.3.2. Counterterm Amplitudes . . . . . . . . . . . . . . . . . . 794.3.3.3. Real Emission . . . . . . . . . . . . . . . . . . . . . . . . 79

4.4. The Total Width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.5. Gµ Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.6. Complex-Mass Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5. Numerics 86

5.1. The Program Prophecy4f . . . . . . . . . . . . . . . . . . . . . . . . . 865.2. Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.2.1. Modifications in the Model File . . . . . . . . . . . . . . . . . . . 885.2.2. Generating the Amplitudes . . . . . . . . . . . . . . . . . . . . . 885.2.3. Embedding in the Prophecy4f program . . . . . . . . . . . . . 89

5.3. Input parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895.4. Scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 905.5. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.5.1. Low-Mass Scenario . . . . . . . . . . . . . . . . . . . . . . . . . . 965.5.1.1. Conversion of the Input Parameters . . . . . . . . . . . . 965.5.1.2. The Running of cβ−α . . . . . . . . . . . . . . . . . . . 985.5.1.3. Scale Variation of the Width . . . . . . . . . . . . . . . 995.5.1.4. Scan over cβ−α . . . . . . . . . . . . . . . . . . . . . . . 1025.5.1.5. Partial Widths . . . . . . . . . . . . . . . . . . . . . . . 1075.5.1.6. Differential Distributions . . . . . . . . . . . . . . . . . . 109

5.5.2. High-Mass Scenario B1 . . . . . . . . . . . . . . . . . . . . . . . . 1165.5.2.1. Conversion of the Input Parameters . . . . . . . . . . . . 1165.5.2.2. The Running of cβ−α . . . . . . . . . . . . . . . . . . . 1175.5.2.3. Scale Variation of the Width . . . . . . . . . . . . . . . 1185.5.2.4. Scan over cβ−α . . . . . . . . . . . . . . . . . . . . . . . 1205.5.2.5. Partial Widths . . . . . . . . . . . . . . . . . . . . . . . 124

5.5.3. High-Mass Scenario B2 . . . . . . . . . . . . . . . . . . . . . . . . 1245.5.3.1. Conversion of the Input Parameters . . . . . . . . . . . . 124

VI

5.5.3.2. The running of cβ−α . . . . . . . . . . . . . . . . . . . . 1255.5.3.3. Scale Variation of the Width . . . . . . . . . . . . . . . 1265.5.3.4. Scan over cβ−α . . . . . . . . . . . . . . . . . . . . . . . 1285.5.3.5. Partial Widths . . . . . . . . . . . . . . . . . . . . . . . 131

5.5.4. Different THDM Types . . . . . . . . . . . . . . . . . . . . . . . . 1325.5.5. Benchmark Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . 1335.5.6. Baryogenesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1345.5.7. Fermiophobic Heavy Higgs . . . . . . . . . . . . . . . . . . . . . . 134

6. Summary and Outlook 135

Appendix 137

A. Feynman Rules 139

B. Other Renormalization Prescriptions 150

B.1. Renormalization of the Basic Parameters – version 2a . . . . . . . . . . . 150B.2. A modified prescription – version 2b . . . . . . . . . . . . . . . . . . . . 153B.3. Field Rotation after Renormalization – version 1b . . . . . . . . . . . . . 153

C. Further Results 156

C.1. Scale Variation in the High-Mass Scenario . . . . . . . . . . . . . . . . . 156C.2. Differential Distributions of the High-Mass Scenario . . . . . . . . . . . . 159

Acknowledgements 173

VII

1. Introduction

The discovery of a Higgs boson at the Large Hadron Collider (LHC) in the year 2012[1, 2] marks a milestone of elementary particle physics. It was rewarded with a NobelPrize for Peter Higgs and Francois Englert who predicted this particle together withRobert Brout in 1964 [3–7]. Now, after almost 50 years of search, the last missing par-ticle of the Standard Model of particle physics (SM) seems to be found and the Higgsboson puzzle looks like to be solved. Beside this extraordinary accomplishment, mea-surements of the properties of this new particle to highest precision are necessary toverify whether this particle is truly the Higgs boson of the SM, or another particle withsimilar, but slightly different properties. This would either confirm the SM or give hintsto new physics beyond the Standard Model (BSM). The ATLAS and CMS collaborationswork on this challenge and a better understanding of the experiment, an improvementof experimental techniques, but also the new data collected in the Run II will improvemass, spin, parity, and coupling measurements in the next months and years. So far, allresults are compatible with the SM expectations [8], however, the uncertainties are stillwithin some 10%, leaving room for deviations and new effects not explained by the SM.

Nevertheless, the SM cannot be the ultimate theory, as it does not solve several fun-damental problems. For example, the matter–anti-matter asymmetry and the origin ofdark matter are not explained, and the fundamental force of gravity is not included atall. This indicates that the SM is a low-energy limit of a more fundamental theory. Forsuch a theory, there are several concepts ranging from String Theory [9, 10] over Su-persymmetry (SUSY) (see e.g. Refs. [11, 12]) to composite Higgs (see e.g. Ref. [13] andreferences therein) and high-dimensional Higgs (HEIDI) [14, 15] models. In the searchfor new physics, it is also possible to employ model-independent effective field theories(see e.g. Refs. [16, 17]) and smaller extensions of the SM, where only specific parts aremodified. One of these extensions is the Two-Higgs-Doublet Model (THDM), wherea second scalar doublet is added to the one already included in the SM. This simpleSM extension is unlikely to be the fundamental theory of nature, but it is embeddedin other models e.g. SUSY and Axion models, and allows for a comprehensive generalsearch for an extended Higgs sector without making further assumptions on fundamentalsymmetries and other parts of the theory. It contains five Higgs bosons, one chargedand its anti-particle, one pseudoscalar and two neutral Higgs bosons. Together with 12additional real parameters in the Higgs potential, it provides interesting phenomenologywhich possibly could solve some of the well-known problems of the SM, like the baryonasymmetry [18–21], or the origin of dark matter [22, 23]. In this thesis we considerspecial THDMs, which exclude flavor changing neutral currents at tree-level and arecharge and parity (CP) conserving so that the parameter space reduces to five addi-

1

tional real parameters with respect to the SM. The former requirement is motivated byexperiments, while the latter is a valid approximation as long as one is not interestedin specific observables of flavor physics, or in solving the problem of baryon asymmetry.In addition, in our THDM models the recently discovered particle is identified with thelight neutral CP-even Higgs boson. Since no sign of additional Higgs bosons have beenfound in direct searches [8], this thesis aims to improve predictions of the THDM inorder to quantify and find deviations of the properties of the discovered Higgs bosonfrom the SM expectations.To be able to measure small deviations from the SM, not only an excellent experimentalaccuracy is necessary, but also a high theoretical precision. Theoretical predictions forobservables are usually computed in a perturbative power series in terms of couplingconstants of the strong (QCD) and the electroweak (EW) forces. The first contributionof the series is called the Leading Order (LO). However, the current experimental accu-racy demands at least next-to-leading order (NLO) precision in theoretical predictions.An NLO calculation does not only improve the absolute values of the predictions, butalso reduces the dependence on an unphysical renormalization scale and therefore thetheoretical uncertainty. The computation of higher-order predictions becomes more andmore challenging with each order, and already NLO requires the use of sophisticatedtechniques as divergences and infinities usually occur in the calculation. The NLO con-tributions can be separated into the virtual corrections, which include an integrationover an undetermined, internal momentum, and the real corrections, which contain anadditional particle in the final state. Both contributions cannot be treated separately,because their final states become degenerate for neutral, massless particles with arbi-trarily small energies, which a detector is unable to resolve. Physically, this leads toinfrared (IR) divergences, which appear in the virtual corrections when the integrationover the internal momentum diverges for small momenta and in the real corrections inthe integration over the phase space. These divergences cancel against each other ac-cording to the Kinoshita–Lee–Nauenberg theorem [24, 25], although the cancellation isnon-trivial as the two contributions are defined on different phase spaces, demanding asophisticated subtraction or cutting procedure.In the virtual corrections, the integration over the internal momentum can become di-vergent for large momenta as well, leading to ultraviolet (UV) divergences. To removethem, a procedure called renormalization is necessary. There, the parameters of the La-grangian are redefined in terms of divergent bare parameters that can be expressed byfinite renormalized ones, and counterterms, which contain the divergent part and form acounterterm Lagrangian. The divergences arising from the loop integrals cancel exactlyagainst the ones from the counterterms rendering the observables UV-finite. However,the finite parts of the counterterms are not determined, and renormalization conditionsare needed to fix these parts. These conditions define a renormalization scheme as wellas the connection of renormalized parameters to observables.The renormalization of the EW sector of the THDM is one of the central parts of thiswork. It had been tackled already in the past [26–31], and we propose a full and con-sistent renormalization prescription including two new renormalization schemes. Therenormalization of the quantum chromodynamical part is performed as well, however,

2

the additional Higgs doublet does not interact strongly so that the renormalization ofthis part is similar to the SM.Recent developments in computational techniques allow for more and more automationin the calculation of NLO corrections, and we contribute to this process by implementingthe complete model including the counterterm Lagrangian in FeynRules [32] and gen-erating a FeynArts [33] model file where all renormalization conditions are included.Using the amplitude generator FeynArts and FormCalc [34], NLO QCD and EWamplitudes for arbitrary processes can be calculated fully automatically and evaluatednumerically in Fortran with LoopTools [34] or Collier [35]. The integration overthe phase space to obtain scattering cross sections and particle decay widths is processdependent and has to be implemented separately. However, the integration routinescan be adapted from SM calculations straightforwardly since the THDM has a similarstructure of infrared divergences.

In a first application, we compute the decay observables of the SM-like neutral, light,CP-even Higgs boson decaying into four fermions,

h→WW/ZZ → 4f, (1.1)

in the THDM at NLO. The decay of the heavy neutral CP-even Higgs boson H→4f isless relevant and not considered in this thesis as the heavy Higgs boson widely decouplesfrom the vector bosons, and an on-shell decay into vector-boson pairs is possible for mostparts of the parameter space, so that a full calculation of the decay into four fermions isnot needed. The fermions in the final state of our process can either be quarks or lep-tons, and especially the latter can be resolved very well in the detector. This signaturewas crucial in the discovery of the Higgs boson and the fully leptonic ZZ channels con-tributed to it with the best sensitivity of all observables [8]. This channel also providesa window to BSM physics, as, due to its high precision, small deviations from the SMcan be measured and differential distributions can be investigated and tested against theSM. The program Prophecy4f [36–38] performs the calculation of the full EW andQCD corrections with a SM Higgs boson in the complex-mass scheme [39–41] with twodifferent methods for the handling of IR singularities, and provides differential distribu-tions as well as unweighted event files. We extend its functionality to the computationin the THDM in such a way that the usage of the program and its applicability as eventgenerator basically remains the same. We investigate several popular scenarios [42–45]which fit the experimental results and obey constraints from vacuum stability, unitarity,and perturbativity. The results are compared with the SM expectation and deviationsare quantified. In addition, we implement four different renormalization schemes andperform a variation of the renormalization scale to investigate the perturbative stabilityof the results. Similar to what has already been found in the Minimal Supersymmet-ric SM [46], the different renormalization schemes can contain problems such as gaugedependence, singularities or numerical instabilities. The comparison of the results ob-tained with different renormalization schemes allows us to carve out the strengths andweaknesses of the different schemes, and to determine for which parameter regions theyprovide reliable predictions.

3

This thesis is structured as follows:

• We begin with the theoretical basics in Chapt. 2 where we summarize the SMand introduce conventions. The THDM is presented in a general way, and theparameter constraints we apply are explained in detail.

• In Chapter 3 we extend the discussion of the THDM to NLO, perform the renor-malization in different ways, i.e. we give a consistent set of renormalization condi-tions and explain the different renormalization schemes. The implementation intoa model file is described.

• In Chapter 4 a detailed overview over the process of a light CP-even Higgs bo-son decaying into four fermions is given. The calculation of this decay in theTHDM with a SM like light Higgs boson is presented and all relevant diagramsand amplitudes are stated.

• Chapter 5 is devoted to the numerical results of the Higgs boson decaying into fourfermions. We analyze the scale dependence and compare the different renormal-ization schemes. The corresponding decay widths are presented for several THDMscenarios, and we give differential distributions for specific parameter sets.

• We conclude the thesis in Chapter 6 with a summary and an outlook.

4

2. Theoretical Preliminaries

2.1. The Standard Model

The Standard Model (SM) of particle physics is a gauge theory based on the gauge group

SU(3)C × SU(2)W × U(1)Y. (2.1)

It describes three out of the four known fundamental forces, viz. the electromagnetic,the weak, and the strong force. A quantum theory for the fourth force, gravity, hasnot yet been found. The theoretical predictions of the SM have been extensively testedduring the past decades and have shown excellent agreement with results from colliderexperiments. With the recent discovery of a Higgs-like boson at the LHC [1, 2, 8], thelast missing piece of the SM puzzle seems to be found, even though the analysis ofthe properties of the newly found particle is not yet completely finished. The discus-sion of the SM in this section is, with some changes, taken from Sect. 1.2 of my owndiploma thesis [47] which follows the treatment of Ref. [48]. The gauge group SU(3)Cleads to quantum chromodynamics (QCD) [49–51], the theory that describes the stronginteraction between quarks and gluons, which will be discussed in Section 2.1.2. TheSU(2)W ×U(1)Y part, also known as the Glashow–Salam–Weinberg model (GSW) [52–55], combines the electromagnetic and weak interactions into a unified electroweak force(EW) and will be covered in Section 2.1.1.The fermion content of the SM is shown in Table 2.1 and can be divided into four types:neutrinos, charged leptons, up-type quarks, and down-type quarks. The latter two in-teract strongly and appear only in bound states, so-called hadrons. All fermions exist inthree generations, where the particles differ by their masses only. Particles of the secondand third generations are not stable and decay into lighter particles. Therefore, stablematter is built up from the light fermions of the first generation. A special case arethe neutrinos, which all are considered massless in the SM. Recent experimental resultsshow, however, that neutrinos have a mass different from zero, which indicates that theSM is incomplete. Nevertheless, an extension of the SM to include neutrino masses canbe accomplished in a straightforward manner, e.g. by introducing an analogon to theCKM matrix in the lepton sector. A new analysis by van der Bij [56] indicates that sucha matrix seems to violate unitarity in three generations, giving hints to sterile neutrinos.However, the SM is a consistent quantum field theory which is renormalizable [57], andmeasurable predictions can be computed within perturbation theory. Even though theagreement between the predictions and experimental results is impressive, we know thatthe SM cannot be the most fundamental theory, since it provides no answer to questionslike the origin of dark matter or the baryon asymmetry and does not include gravity at

5

leptons quarksi νi ei ui di1 νe e u d2 νµ µ c s3 ντ τ t b

Table 2.1.: All fermions exist in three identical copies, called generations, which are labeled bythe index i. The only difference are the larger masses of the particles with higher generationindex. However, all neutrinos are considered massless in the SM.

all. In addition, it suffers from fine-tuning problems like the hierarchy problem or thestrong CP problem. Therefore, the search for physics beyond SM is crucial for a furtherunderstanding of the universe.

2.1.1. GSW Model

The GSW model unifies the weak and the electromagnetic forces in a non-abelian gaugetheory, based on the gauge group SU(2)W × U(1)Y. The SU(2)W group describes theweak isospin and the generators in the fundamental representation, relevant for left-handed fermions, are connected to the Pauli matrices τa via IaW = τa/2,

I1W =1

2

(0 11 0

)

, I2W =1

2

(0 −ii 0

)

, I3W =1

2

(1 00 −1

)

, (2.2)

whereas the generator YW2

of the U(1)Y group is a real number called weak hypercharge.With each of the four generators above a gauge boson is associated. Experiments showthat the W± and Z bosons are massive, and as a plain mass term in the Lagrangianwould violate gauge invariance, the masses are implemented via the Higgs–Kibble mech-anism [3–7]. There, a scalar field, called the Higgs field, with non-vanishing vacuumexpectation value is introduced in order to break the symmetry of the gauge groupspontaneously. This is done in such a way that the symmetry of the electromagneticsubgroup U(1)em is preserved, so that the photon stays massless while the other gaugebosons acquire masses.In the GSW model left- and right-handed fermions transform differently under theSU(2)W group. The left-handed fermions are assembled into two-dimensional funda-mental representations whereas right-handed fermions transform trivially. This rendersthe weak interaction maximally parity violating. In a chiral gauge theory, however, naivemass terms of fermions are forbidden in the Lagrangian. This problem can be solvedusing the Higgs field, since Yukawa couplings between fermions and the Higgs bosonare gauge invariant and preserve renormalizability. For the sake of clarity we split theLagrangian into four parts,

LEW = LGauge + LHiggs + LFermion + LYukawa, (2.3)

which will be discussed successively in the following.

6

Gauge Part

The gauge part

LGauge = − 1

4W aµν(x)W

a,µν(x)− 1

4Bµν(x)B

µν(x)

= − 1

4(∂µW

aν − ∂νW

aµ + g2ǫ

abcW bµW

cν )

2 − 1

4(∂µBν − ∂νBµ)

2 (2.4)

consists of the kinetic terms of the four EW gauge fields. The isotriplet W aµ , a = 1, 2, 3,

and the isosinglet Bµ are associated to the three generators of the SU(2)W group and thegenerator YW of the U(1)Y group, respectively. Since the two subgroups commute, twoindependent coupling constants g1 and g2 are introduced and the covariant derivative isgiven by

Dµ = ∂µ ∓ ig2IaWW

aµ + ig1

YW2Bµ. (2.5)

where the sign in the second term is negative in the conventions of Bohm, Hollik andSpiesberger (BHS) [58] and positive in the convention of Haber and Kane (HK)[59].Throughout this thesis we will state all formulas in the BHS convention. For fields inthe trivial representation the respective contribution to the covariant derivative vanishes.

Higgs Part

For the Higgs sector, we introduce a complex doublet scalar field

Φ =

(φ+

φ0

)

(2.6)

with hypercharge YW = 1. The Higgs-boson part to the Lagrangian reads

LHiggs = (DµΦ)†(DµΦ)− V (Φ†Φ), (2.7)

with the covariant derivative from Eq. (2.5), and a potential of the form

V (Φ†Φ) =λ

4(Φ†Φ)2 − µ2Φ†Φ. (2.8)

The parameters λ and µ are chosen to be positive, so that the field develops a non-vanishing vacuum expectation, given by the condition |Φmin| = 2µ2

λ≡ v2

26= 0. From this

degenerate set of vacuum states one can choose

〈Φ〉 = 〈0|Φ|0〉 = Φ0 =

(0v√2

)

. (2.9)

This breaks the symmetry of the gauge group which can be seen from IaW〈Φ〉 6= 0. Theonly generator that annihilates the vacuum is the combination

(

I3W +YW2

)

〈Φ〉 = 0, (2.10)

7

and

Q ≡ I3W +YW2

(2.11)

can be identified with the electric charge operator. This relation is called Gell-Mann–Nishijima relation. The scalar field can now be expanded around the vacuum expectationvalue

Φ =

(φ+(x)

1√2(v + h(x) + iχ(x))

)

, (2.12)

which can also be rewritten in polar coordinates as

Φ = exp

(

iαa(x)IaW

v

)(0

1√2(v + h(x))

)

. (2.13)

The three degrees of freedom associated with αa are unphysical, and it is evident that agauge exists in which those fields are removed by appropriate gauge terms. This gaugeis called the “unitary” gauge.The three unphysical degrees of freedom represent would-be Goldstone bosons and man-ifest themselves, after elimination, in the longitudinal polarizations of the gauge bosons.The field h represents the Higgs boson. Inserting Eq. (2.13) in the unitary gauge intoEq. (2.7) yields

LHiggs =1

2(∂µh)(∂

µh)− µ2h2

+1

4

v2

2

(g22W

1µW

1,µ + g22W2µW

2,µ + (g2W3µ + g1Bµ)(g2W

3,µ + g1Bµ))

+ “interaction terms”, (2.14)

where the second and the third terms have the form of mass terms for the Higgs andthe vector bosons, respectively. Transforming the vector fields into fields correspondingto mass and electric-charge eigenstates using the transformations

(AµZµ

)

=

(cW −sWsW cW

)(Bµ

W 3µ

)

, W±µ =

1√2(W 1

µ ∓ iW 2µ ), (2.15)

where

cW = cos θW ≡ g2√

g21 + g22, sW = sin θW, (2.16)

leads to the following masses,

MW = g2v

2, MZ =

v

2

√

g21 + g22, MA =Mγ = 0, MH =√2µ. (2.17)

8

Rewriting the covariant derivative in terms of the fields above, we obtain

Dµ = ∂µ − ig2√2(W+

µ I+W +W−

µ I−W)

− i1

√

g21 + g22Zµ(g

22I

3W − g21YW)− i

g1g2√

g21 + g22Aµ(I

3W +

YW2

), (2.18)

where I± = I1W ± I2W. Due to the properties of the vector field Aµ, it can be identifiedwith the photon and the electromagnetic coupling with the elementary charge, yielding

e =g1g2

√

g21 + g22. (2.19)

All formulas stated can be transformed to the HK convention by changing the sign ofsW,MW, and MZ everywhere in the Lagrangian.

Fermionic Part

As already mentioned, the SU(2)W gauge group is maximally parity violating, becauseleft- and right-handed fermions fall into different representations. Left-handed fermions(both leptons (L) and quarks (Q)) are grouped into weak isospin doublets

L′Lj = ω−L

′j =

(ν ′Ljl′Lj

)

, Q′Lj = ω−Q

′j =

(u′Ljd′Lj

)

, (2.20)

with ω± = 1±γ52

denoting the chirality projector and j running over the three generations.The particles ν, l, u, and d stand for neutrinos, charged leptons, up-type, and down-typequarks, respectively. The right-handed fermions transform as isosinglets

l′Rj = ω+l′j , u′Rj = ω+u

′j, d′Rj = ω+d

′j. (2.21)

Since quarks interact strongly as well, they carry a color index in addition, which we,however, suppressed for simplicity in this notation. The weak hypercharge is chosenin such a way that the electrical charges are reproduced correctly by the Gell-Mann–Nishijima relation (Eq. (2.11)). In the SM, the right-handed neutrinos are omitted asthey would not take part in any interaction.The kinetic terms of the fermions in the Lagrangian is

LFermion =∑

i

(

L′Li i /DL

′Li + Q′L

i i /DQ′Li

)

+∑

i

(

l′Ri i /Dl

′Ri + u′

Ri i /Du

′Ri + d′

Ri i /Dd

′Ri

)

, (2.22)

where the covariant derivative for left-handed fermions is given in Eq. (2.18). For theright-handed fermions terms proportional to g2 vanish and for quarks a term proportionalto gs occurs as detailed described in Sect. 2.1.2.

9

Yukawa Couplings

The masses of the fermions have to be introduced via Yukawa couplings to the Higgsfield, since a direct implementation of mass terms in the chiral fermionic Lagrangianwould violate gauge invariance:

LYukawa =−∑

i,j

(L′Li G

lijl

′Rj Φ+ Q′L

i Guiju

′Rj Φ + Q′L

i Gdijd

′Rj Φ + h.c.), (2.23)

The field Φ = (φ0∗,−φ−)T is the charge-conjugate Higgs field with φ− = (φ+)∗. Thematrices Gij are 3 × 3 matrices in generation space. There is no need for them to bediagonal, since the fields with prime correspond to eigenstates of the weak interactionand they do not have to coincide with the mass eigenstates. To write the Lagrangian interms of fields corresponding to mass eigenstates, the mass matrices Gij are diagonalizedwith the transformations

fLi = Uf,Lik f ′L

k , fRi = Uf,Rik f ′R

k , (2.24)

where f denotes the type of fermion f=(ν, l, u, d). Then the masses of the fermions are

mf,i = Uf,Lik Gf

kmUf,R†mi

v√2. (2.25)

The transformation of the fermions introduces a non-diagonal matrix into the quark–W-boson coupling, since the latter is mutually connecting up- and down-type fermions.The different transformations Uu,L and Ud,L are multiplied with each other in generationspace and give rise to the Cabibbo–Kobayashi–Maskawa (CKM) matrix [60, 61]

Vij = Uu,Lik Ud,L†

kj , (2.26)

which leads to a mixing between the generations of quarks in quark–W-boson interac-tions. However, the quark–Z-boson coupling connects fermions of the same type andthus stays diagonal. Therefore flavor-changing neutral currents (FCNC) are absent attree level in the SM. In the leptonic sector such a matrix does not exist, as long as allgenerations of neutrinos are considered massless. Every transformation of the neutrinofields leads to mass eigenstates with zero mass, thus Uν,L is arbitrary and can be chosento be U l,L. This argument holds also when the masses of the neutrinos are non-vanishingbut different generations are degenerate.

Quantization

The quantization of the EW theory requires a fixing of the gauge which is usually doneby adding a gauge-fixing term to the Lagrangian. Spontaneous symmetry breaking in-troduces mixing terms of the form Vµ∂

µφ between gauge bosons and would-be Goldstoneboson fields φ±, χ which leads to non-diagonal propagators. The gauge-fixing terms arein BHS convention chosen as

F± = ∂µW±µ ∓ iMWξ

W2 φ

±, (2.27)

10

FZ = ∂µZµ −MZξZ2 χ, (2.28)

F γ = ∂µAµ, (2.29)

and lead to the gauge-fixing Lagrangian

Lfix = − 1

2ξA(F γ)2 − 1

2ξZ1(FZ)2 − 1

ξW1F+F−. (2.30)

By choosing ξZ2 = ξZ1 and ξW2 = ξW1 all quadratic mixing terms drop out (except forirrelevant total derivatives). This gauge is called the ’t Hooft gauge and is used in allour calculations. In the standard Faddeev–Popov quantization procedure, the unphysicaldegrees of freedom of the gauge group are separated from the physical degrees of freedomby the gauge fixing. In this procedure a compensation term occurs, which correspondsto an unphysical, fermionic, spin one field, named ghost field. These ghost fields appearonly as virtual particles and the ghost Lagrangian reads

Lghost = −gαuα(x) δF α

δθβ(x)uβ(x) (2.31)

with α, β = ±, Z, γ. The matrix δFα

δθβ(x)is the variation of the gauge-fixing operators F

under the infinitesimal gauge transformation defined by the local gauge group parametersθβ(x).A very common gauge is the ’t Hooft-Feynman gauge, ξα = 1, where the propagators ofthe gauge bosons are particularly simple and become

∆V (p) = −igµν

p2 −M2 + i0. (2.32)

The effective EW Lagrangian is the sum of Eqs. (2.3), (2.30), and (2.31):

LEWeff = LEW + Lfix + Lghost. (2.33)

2.1.2. Quantum Chromodynamics

QCD is based on the SU(3)C group of the SM. For simplicity we treat the strong in-teractions completely independent from the electroweak interactions and neglect thelatter in this section. The inclusion of QCD into the EW theory is straightforward andcan be done by adding the Lagrangians and omitting the overlapping part, describingfree fermions, in the QCD Lagrangian. Corresponding to the dimension of the gaugegroup there are eight self-interacting gauge fields, called gluons, which mediate the forcebetween the quarks. The interaction is flavor-conserving, and no mixing between thegenerations occurs. The quarks transform as three-dimensional fundamental represen-tations of SU(3)C and are written in a triplet with three different colors (r, g, b),

q =

qrqgqb

. (2.34)

11

The eight generators T a of the group can be represented by Gell-Mann matrices λa, viaT a = λa/2

T 1 =1

2

0 1 01 0 00 0 0

, T 2 =1

2

0 −i 0i 0 00 0 0

, T 3 =1

2

1 0 00 −1 00 0 0

,

T 4 =1

2

0 0 10 0 01 0 0

, T 5 =1

2

0 0 −i0 0 0i 0 0

, T 6 =1

2

0 0 00 0 10 1 0

,

T 7 =1

2

0 0 00 0 −i0 i 0

, T 8 =1

2√3

1 0 00 1 00 0 2

. (2.35)

The Lagrangian of QCD reads

LQCD = −1

4GaµνG

a,µν +∑

f

qf(i /D −mf )qf , (2.36)

with the gluon field Gaµ, its field-strength tensor T aGa

µν , and the covariant derivative

Dµ = ∂µ − igsGaµ(x)T

a, (2.37)

where gs is the strong coupling constant. From this Lagrangian, special properties ofQCD such as the asymptotic freedom [62, 63] of quarks at high energies or the quarkconfinement, which forces quarks to form bound states of hadrons, can be derived. Amore detailed prescription can be found, e.g., in Ref. [64].

Quantization

The quantization of QCD causes no additional complications since QCD is a pure Yang–Mills gauge theory. The gauge-fixing Lagrangian is chosen as

Lfix = − 1

2ξ(∂µGa

µ)(∂νGa

ν), (2.38)

which leads to the ghost Lagrangian

Lghost = −uα∂µDαβµ uβ, α, β = 1, . . . , 8, (2.39)

where the covariant derivative is the one in the adjoint representation,

Dαβµ = ∂µ − igs(T

aadj)

αβGaµ(x). (2.40)

As in the EW case, a common gauge is the ’t Hooft-Feynman gauge (ξ = 1).

12

2.2. The Two-Higgs-Doublet Model

The Two-Higgs-Doublet Model (THDM) is an extension of the Standard Model (SM)with a different Higgs sector, but based on the identical gauge group SU(3)C×SU(2)W×U(1)Y and with the same fermion content. The implementation of the Higgs–Kibblemechanism in the SM is minimal, since it contains only one weak-isospin Higgs doubletwith weak hypercharge YW = 1. The THDM is one of the simplest extensions andcontains two Higgs isospin doublets with weak hypercharge YW = 1,

Φ1 =

(φ+1

φ01

)

, Φ2 =

(φ+2

φ02

)

, (2.41)

with 8 real degrees of freedom. Hence, after spontaneous symmetry breaking, there are 5physical Higgs bosons in addition to the three Goldstone bosons. This gives possibilitiesto new effects which could solve some of the unanswered questions of the SM. Forexample, CP-violation in the Higgs sector could provide solutions to the problem ofbaryogenesis [18–21] and inert THDMs contain a dark matter candidate [22, 23]. An evenlarger motivation comes from the embedding of the THDM into more complex models,such as axion [65, 66] or supersymmetric models [59]. Some of the latter are promisingcandidates for a fundamental theory and supersymmetric Higgs sectors contain a THDM(in which, however, the doublets have opposite hypercharges). Even though the THDMis unlikely to be the fundamental theory of nature, it provides a rich phenomenology,which can be used in the search for a non-minimal Higgs sector without being limitedby constraints from a specific underlying model.The Lagrangian of the THDM is, as the SM one, composed of several parts:

LTHDM = LGauge + LFermion + LHiggs + LYukawa + LFix + LGhost. (2.42)

In the following we will discuss each part in detail. A very elaborate and completediscussion of the THDM Higgs and Yukawa Lagrangians, including general and specificcases, can be found in Ref. [67] and this paper is used as a primary source for our sum-mary. The discussion in this chapter will be, however, restricted to lowest order, theissues and difficulties appearing in higher orders will be treated in Chap. 3.Since the extension of the SM by a second Higgs doublet does not affect the gauge andthe fermionic parts, they are identical to its SM counterparts, given in Eqs. (2.4,2.36)and (2.22), respectively. The gauge-fixing Lagrangian in the ’t Hooft gauge and theghost Lagrangian are discussed in Sect. 2.2.4. However, the second Higgs doublet doesnot change the structure of these parts drastically, hence they can be obtained straight-forwardly from their SM counterparts. The Higgs Lagrangian, treated in Sect.2.2.1, andthe Yukawa couplings to the fermions, covered in Sect.2.2.3, are mostly affected by theadditional degrees of freedom in this model and are responsible for the rich phenomenol-ogy of the THDM.

13

2.2.1. The Higgs Lagrangian

The Higgs Lagrangian is more complicated than in the SM, since it contains kineticterms and a potential involving both doublets,

LHiggs = (DµΦ1)†(DµΦ1) + (DµΦ2)

†(DµΦ2)− V (Φ†1Φ1,Φ

†2Φ2,Φ

†2Φ1,Φ

†1Φ2), (2.43)

with the covariant derivative as stated in Eq. (2.5). The potential can involve all her-mitian functions of the two doublets up to dimension four and can be parameterized inthe most general case as follows [67, 68],

V =m211Φ

†1Φ1 +m2

22Φ†2Φ2 − [m2

12Φ†1Φ2 + h.c.] (2.44)

+1

2λ1(Φ

†1Φ1)

2 +1

2λ2(Φ

†2Φ2)

2 + λ3(Φ†1Φ1)(Φ

†2Φ2) + λ4(Φ

†1Φ2)(Φ

†2Φ1)

+

[1

2λ5(Φ

†1Φ2)

2 + (λ6(Φ†1Φ1) + λ7Φ

†2Φ2)Φ

†1Φ2 + h.c.

]

.

The parameters m211, m

222, λ1, λ2, λ3, λ4 are real, while the parameters m2

12, λ5, λ6, λ7 arecomplex, yielding a total number of 14 real degrees of freedom for the potential. Foreach Higgs doublet we demand that the fields develop a (complex) vacuum expectationvalue in the neutral component,

〈Φ1〉 = 〈0|Φ1|0〉 =(

0v1√2eiξ1

)

, 〈Φ2〉 = 〈0|Φ2|0〉 =(

0v2√2eiξ2

)

. (2.45)

It is non-trivial that such a minimum of the potential exists and is stable, and thisrestricts the allowed parameter space already strongly, see Ref. [69]. The complex phasesof the vacua introduce the possibility to break the CP-symmetry spontaneously. TheHiggs doublets can be decomposed as follows,

Φ1 =

(φ+1

1√2(η1 + ic01 + v1e

iξ1)

)

, Φ2 =

(φ+2

1√2(η2 + ic02 + v2e

iξ2)

)

, (2.46)

with the charged fields φ+1 , φ

+2 , the neutral CP-even fields η1, η2, and the neutral CP-odd

fields c01, c02.

Change of basis

The component fields of the two Higgs doublets Φ1 and Φ2 do not correspond to masseigenstates. This fact motivates a redefinition of these doublets without changing thephysics. It becomes clear if one writes the Higgs doublets itself in a vector (Φ1,Φ2) onwhich arbitrary U(2) transformations Uab can be carried out:

Φ′a =

2∑

b=1

UabΦb. (2.47)

14

This change of basis leaves the kinetic term of Eq. (2.43) invariant, but the coefficientsof the potential change. This means that different sets of coefficients describe the samephysics, so that not all of the 14 real parameters are physically measurable. The groupof basis change transformations is U(2) ∼ U(1)× SU(2), where a global U(1) transfor-mation has no effect on the parameters of the potential as can be seen from Eq. (2.44).However, the 3 parameters of the SU(2) part of the transformations can be used toeliminate 3 parameters in the potential, resulting in 11 physical degrees of freedom. Anappropriate basis choice can lead to a particularly simple description, especially in theYukawa Lagrangian (see Sect. 2.2.3). For the investigation of basis changes, special no-tations of the potential have been proposed. For example the one of Ref. [70] which issuited particularly for the study of invariants, i.e. symmetries of the potential underthese transformations. We do not want to cover a detailed prescription here and justpresent the so-called Higgs basis, in which only one of the two Higgs doublets obtains avacuum expectation value. This can be achieved by redefining the Higgs doublets,

H1 =v1e

−iξ1Φ1 + v2e−iξ2Φ2

vH1 =

−v2eiξ2Φ1 + v1eiξ1Φ2

v(2.48)

with v2 = v21 + v22 which yields 〈H1〉 = v/√2 and 〈H2〉 = 0. This basis is uniquely

defined up to a complex phase of H2. The potential in this basis has a similar form, butwith different coefficients which we denote with Y and Z,

V =Y 211Φ

†1Φ1 + Y 2

22Φ†2Φ2 − [Y 2

12Φ†1Φ2 + h.c.] (2.49)

+1

2Z1(Φ

†1Φ1)

2 +1

2Z2(Φ

†2Φ2)

2 + Z3(Φ†1Φ1)(Φ

†2Φ2) + Z4(Φ

†1Φ2)(Φ

†2Φ1)

+

[1

2Z5(Φ

†1Φ2)

2 + (Z6(Φ†1Φ1) + Z7Φ

†2Φ2)Φ

†1Φ2 + h.c.

]

.

The calculation of these coefficients is straightforward and can be found together withfurther details of this basis in Refs. [42, 67].

Additional constraints

Since the 11-dimensional parameter space of the potential is too large for early experi-mental analyses, we want to restrict the model in our analysis by imposing two additionalconditions, motivated by experimental results:

• absence of flavor-changing neutral currents at tree level,

• CP conservation in the Higgs sector (even though this holds only approximately).

The former requirement can be ensured by adding a discrete Z2 symmetry Φ1 → −Φ1

(see Sect. 2.2.3). This condition implies that the parameters λ6, λ7 vanish if we allowsoft breaking terms proportional to m12. Concerning the second condition, the potentialis CP-conserving if and only if there exists a basis of the Higgs doublets in which allparameters and the vacuum expectation values are real [71]. For our description we

15

assume that such a transformation has been done already (if the parameters or vac-uum expectation values were initially complex), so that we only have to deal with realparameters. This renders m12 and λ5 real and implies for the phases of the vacua

ξ1 = ξ2 = 0. (2.50)

However, at higher order in perturbation theory CP breaking terms and complex phasesin the Higgs sector are generated radiatively through loop contributions. For our analysisthis does not present a problem as they appear only beyond NLO. In addition we assumethat a basis of the doublets is chosen in which v1, v2 > 0 (this is always possible as aredefinition Φi → −Φi changes the sign of the vacuum expectation value). The potentialof Eq. (2.44) has then the following form,

V =m211Φ

†1Φ1 +m2

22Φ†2Φ2 − [m2

12Φ†1Φ2 + h.c.] (2.51)

+1

2λ1(Φ

†1Φ1)

2 +1

2λ2(Φ

†2Φ2)

2 + λ3(Φ†1Φ1)(Φ

†2Φ2) + λ4(Φ

†1Φ2)(Φ

†2Φ1)

+1

2λ5[(Φ

†1Φ2)

2 + (Φ†2Φ1)

2].

This parameterization is, for example, used in Ref. [67], and we use it as a standardparameterization, since it is derived from the general one in Eq.(2.44).A different parameterization, called the “a parameterization”, which is used in literature(e.g. in the “Higgs Hunters Guide” [72]), is

V =a1

(

Φ†1Φ1 −

v212

)2

+ a2

(

Φ†2Φ2 −

v222

)2

+ a3

[(

Φ†1Φ1 −

v212

)

+(

Φ†2Φ2 −

v222

)]2

(2.52)

+a4

[

(Φ†1Φ1)(Φ

†2Φ2)− (Φ†

1Φ2)(Φ†2Φ1)

]

+ a5

[

Re(Φ†1Φ2)−

v1v22

]2

+ a6Im(Φ†1Φ2),

where the vacuum expectation values appear already in the definition of the potential.The two parameterizations, Eqs. (2.51,2.52), are equivalent at leading order (up to irrel-evant constant terms) and can be transformed into each other via the transformations,

λ1 = 2(a1 + a3), λ2 = 2(a2 + a3), λ3 = 2a3 + a4, (2.53)

λ4 = −a4 +1

2(a5 + a6), λ5 =

1

2(a5 − a6), m2

12 = a5v1v2/2.

and

a1 =m2

12

v1v2+

1

2

(λ1 − λ3 − λ4 − λ5

), a2 =

m12

v1v2+

1

2

(λ2 − λ3 − λ4 − λ5

), (2.54)

a3 = −m212

v1v2+

1

2

(λ3 + λ4 + λ5

), a4 = 2

m212

v1v2− λ4 − λ5,

a5 = 2m2

12

v1v2, a6 = 2

m212

v1v2+ 2λ5.

16

Expanding the potential of Eq.(2.51) using the decomposition of Eq.(2.46) and orderingthe terms into powers of the fields, leads to the form

V = −tη1η1 − tη2η2 (2.55)

+1

2(η1, η2)Mη

(η1η2

)

+1

2(c1, c2)Mc

(c1c2

)

+1

2(φ+

1 , φ+2 )Mφ

(φ−1

φ−2

)

+ . . .

with the tadpole terms linear in the fields and the mass terms quadratic. Terms cubic orquartic in the fields are suppressed in the notation here. Only the neutral CP-even scalarfields can have tadpole terms, since they carry the quantum numbers of the vacuum.Further, in the mass terms, only particles with the same quantum numbers can mix, sothat the three different scalars (neutral CP-even, neutral CP-odd, and charged) do notmix with one another. Of course, through the cubic and quartic terms, which are notshown here, these particles interact with each other. The tadpole terms are

tη1 = −m211v1 − λ1v

31/2 + v2(m

212 − λ345v1v2/2), (2.56a)

tη2 = −m222v2 − λ2v

32/2 + v1(m

212 − λ345v1v2/2), (2.56b)

where we introduced the abbreviations λij... = λi + λj + . . . and the mass matrices are

Mη =

(m2

11 + 3λ1v21/2 + λ345v

22/2 −m2

12 + λ345v1v2−m2

12 + λ345v1v2 m222 + 3λ2v

22/2 + λ345v

21/2

)

, (2.57a)

Mc =

(m2

11 + λ1v21/2 + (λ34 − λ5)v

22/2 −m2

12 + λ5v1v2−m2

12 + λ5v1v2 m222 + λ2v

22/2 + (λ34 − λ5)v

21/2

)

, (2.57b)

Mφ =

(2m2

11 + λ1v21 + λ3v

22 −2m2

12 + λ45v1v2−2m2

12 + λ45v1v2 2m222 + λ2v

22 + λ3v

21

)

. (2.57c)

2.2.2. Reparameterizing the Potential

The parameterization used above to define the potential is intuitive to obtain the mostgeneral potential in a symmetric form, but the connection of parameters to observablesis non-trivial. Therefore it is convenient to reparameterize the potential in terms ofparameters that are closer to physical observables.

Tadpole terms

As tadpole contributions destroy the simple relation between propagators and irreducibletwo-point functions and lead to disconnected diagrams, it is convenient to remove thesefrom the Lagrangian. Thus, we demand that terms in the Lagrangian linear in thefields vanish as such terms are the only source of tadpole contributions at LO. Thisrequirement corresponds to an expansion of the Higgs fields around the minimum of thepotential. By setting tη1 = tη3 = 0, Eqs. (2.56) simplify, and the vacuum expectationvalues v1 and v2 can be directly related to the mass parameters m2

11 and m222,

m211 =

v2v1m2

12 −1

2λ1v

21 −

1

2λ345v

22, (2.58a)

17

m222 =

v1v2m2

12 −1

2λ2v

22 −

1

2λ345v

21 . (2.58b)

In the “a-parameterization” the tadpole terms vanish by construction, as can be seenby applying the decomposition of Eq. (2.46) to the potential of Eq. (2.52). This pa-rameterization comes also with two parameters less than the standard one, which com-pensates for the fact that parameters are related. However, this argumentation is notvalid at NLO where also loop diagrams can generate tadpole terms. Therefore the“a-parameterization” is not suited for a higher-order treatment of the THDM.

Masses

Even though the physics is not dependent on the field basis, the mass eigenstate basis isparticularly simple and intuitive as the eigenfields of this basis are free fields. One canobtain this basis through an appropriate rotation of the fields,

(c1c2

)

= Rc

(G0

A0

)

=

(cos βn − sin βnsin βn cos βn

)(G0

A0

)

, (2.59a)

(φ+1

φ+2

)

= R±φ

(G+

H+

)

=

(cos βc − sin βcsin βc cos βc

)(G+

H+

)

, (2.59b)

(η1η2

)

= Rη

(Hh

)

=

(cosα − sinαsinα cosα

)(Hh

)

, (2.59c)

which diagonalizes the mass matrices of Eq. (2.57c). The angles βn, βc, α are determinedfrom the conditions

RT

cMcRc = diag(M2

G0,M2

A0), (2.60)

RT

φ±Mφ±Rφ± = diag(M2G±,M2

H±), (2.61)

RT

η MηRη = diag(M2H ,M

2h). (2.62)

The off-diagonal terms of the first two matrices, which have to vanish, are

0!=

∂2V

∂A0∂G

∣∣∣∣fields=0

= −M2A0

tan (β − βn), (2.63)

0!=

∂2V

∂H+∂G−

∣∣∣∣fields=0

= −M2H± tan (β − βc). (2.64)

Here the angle β is introduced and defined by the ratio of the vacuum expectation values,

tan β ≡ v2/v1, (2.65)

with β ∈ [0, π/2], as v1, v2 are positive. That means the mass mixing terms vanish forβn = βc = β. It is important to mention that this relation holds only at LO, the NLO

18

case is more delicate and explained in Chap. 3. This leads to the following masses inthe two parameterizations,

M2G = 0, (2.66a)

M2A0

= m212/(cβsβ)− λ5v

2 = a6v2/2, (2.66b)

M2H± = m2

12/(cβsβ)− (λ4 + λ5)v2/2 = a4v

2/2, (2.66c)

M2G± = 0, (2.66d)

with the abbreviations sα ≡ sinα, cα ≡ cosα, tα ≡ tanα (analogously for the angle β)which are used throughout this thesis. We obtain three massless Goldstone bosons, oneCP-odd (A0) and a pair of charged (H±).The neutral CP-even degrees of freedom correspond to two physical Higgs bosons, oneheavy (H) and one light (h), and the condition for the angle α reads

0!=

∂2V

∂H∂h

∣∣∣∣fields=0

=s2(α−β)s2β

m212 +

v2

2

(s2α(−c2βλ1 + s2βλ2) + c2αs2βλ345

). (2.67)

The rotation angle can be restricted to α ∈ (−π/2, π/2], since an additional shift of πjust changes the signs of H and h and does not affect observables. In addition, withthe mass assignment MH ≥ Mh the angle α is fixed uniquely. This yields the squaredneutral Higgs-boson masses

M2H =

s2α−βcβsβ

m212 + v2

(λ1c

2βc

2α + λ2s

2αs

2β + s2αs2βλ345/2

), (2.68a)

M2h =

c2α−βcβsβ

m212 + v2

(λ1c

2βs

2α + λ2c

2αs

2β − s2αs2βλ345/2

). (2.68b)

With the above equations, m12, λ1, λ2, λ3, λ4 can be traded for the masses of the phys-ical bosons MH , Mh, MA, MH± , and the mixing angle α. The parameter λ5 cannot bereplaced by a mass or a mixing angle as it appears only in cubic and quartic Higgs cou-plings and acts like an additional coupling constant. Explicit relations can be obtainedby inverting Eqs.(2.66), (2.67), and (2.68), with the result

λ1 =1

c2βv2

(c2αM

2H + s2αM

2h − s2β(M

2A0

+ λ5v2)), (2.69a)

λ2 =1

s2βv2

(s2αM

2H + c2αM

2h − c2β(M

2A0

+ λ5v2)), (2.69b)

λ3 =s2αs2βv2

(M2H −M2

h )−1

v2(M2

A0− 2M2

H±)− λ5, (2.69c)

λ4 =2M2

A0− 2M2

H±

v2+ λ5, (2.69d)

m212 = cβsβ(M

2A0

+ λ5v2). (2.69e)

Of course, for the a-parameterization the argumentation of this LO section is completelyanalogous, since the two parameterizations can be transformed into each other.

19

Parameters of the gauge sector

Mass terms of the gauge bosons arise through the interaction of the gauge bosons withthe vacuum expectation values, analogous to the SM. After a rotation into fields corre-sponding to mass eigenstates one obtains relations similar to the SM ones:

MW = g2v

2, MZ =

v

2

√

g21 + g22, e =g1g2

√

g21 + g22, (2.70)

where the electric charge is identified with the coupling constant of the photon fieldAµ in the covariant derivative analogous to Eq. (2.18). Inverting these relations andintroducing the weak mixing angle θW as in Eq. (2.16), one can replace these parametersby

v =2MWsW

e, g1 =

e

cW, g2 =

e

sW. (2.71)

Mass parameterization

Equations (2.58) , (2.69), and (2.71) can be used to eliminate two parameters and tochange the parameter set from the defining one,

{pbasic} = {λ1, . . . , λ5, m211, m

222, m

212, v1, v2, g1, g2}, (2.72)

in favor of

{pmass} = {MH,Mh,MA0 ,MH±,MW,MZ, e, λ5, α, β}, (2.73)

in which the masses (and angles) are better accessible by experiments.The kinetic term of the Higgs Lagrangian is only dependent on the electroweak parame-ters, as in the SM. But the potential additionally depends on 4 Higgs-boson masses, twoangles, and one Higgs coupling constant λ5. The latter parameter acts like an additionalcoupling constant and therefore cannot be traded for a parameter which is connected tothe propagator of a mass eigenstate. Using this set of parameters and fields correspond-ing to mass eigenstates, the Higgs Lagrangian of Eq. (2.43) has a familiar form

LHiggs =1

2(∂µH)(∂µH) +

1

2(∂µh)(∂

µh) +1

2(∂µA0)(∂

µA0) + (∂µH+)(∂µH−) (2.74)

+1

2(∂µG0)(∂

µG0) + (∂µG+)(∂µG−)

+MZZµ(∂µG0)− iMW(W+

µ (∂µG−)−W−

µ (∂µG+))

− 1

2M2

HH2 − 1

2M2

hh2 − 1

2M2

A0A2

0 −M2H±H+H−

+M2WW

+W− +1

2M2

ZZ2

+ interaction terms,

20

Z WξH cos (β − α) cos (β − α)ξh sin (β − α) sin (β − α)

Table 2.2.: The couplings ξH,h of the neutral CP-odd Higgs bosons to the vector bosonsrelative to the SM Higgs–vector-boson coupling. The couplings of the Goldstone bosons tothe vector bosons are as in the SM.

where the interaction terms are completely determined, but very lengthy. The Feynmanrules resulting from this Lagrangian (see [73]) are stated in App. A and the couplings ofthe Higgs bosons to the gauge bosons are shown in Tab. 2.2.As the angle α has not such a clear physical meaning, one can also deal with the couplingparameter λ3 instead as the defining parameter. We denote the resulting set with anadditional prime,

{p′mass} = {MH,Mh,MA0,MH± ,MW,MZ, e, λ5, λ3, β}. (2.75)

The hybrid basis

The hybrid basis of parameters is a mixture of parameters from the mass parameteri-zation and parameters of the Higgs basis. It has been recently employed in a study ofthe parameter space, where theoretical (unitary and vacuum stability) and experimen-tal (new LHC data) constraints were taken into account [42]. In this paper benchmarkscenarios and parameter scans of the allowed parameter regions are proposed, providingsetups for further theoretical and experimental analysis. The hybrid basis is defined bythe parameter set,

{phybrid} = {MH,Mh, Z4, Z5, Z7,MZ, e, λ5, cos (α− β), β}, (2.76)

with the definitions of the Z parameters in the Higgs basis according to Eq. (2.49). Basedon this parameterization, one can obtain the parameters of the {pmass} basis analogouslyas done above in the basic parameterization and which yields [42],

M2A0

=M2Hsβ−α +M2

h − Z5v2, (2.77a)

M2H± =M2

A0− 1

2(Z4 − Z5)v

2, (2.77b)

λ5 =Z5 +(M2

h −M2H)sβ−αcβ−αt2β2v2

− Z7

2t2β. (2.77c)

With these equations, one can use the hybrid parameterization as an input and derivethe values in the mass parameterization therefrom, while the main calculations remainthe same.

2.2.3. Yukawa Couplings

The Higgs mechanism does not only give rise to the gauge-boson mass terms (which aredetermined by the vacuum expectation values), but via Yukawa couplings it introduces

21

a possibility to give masses to chiral fermions. Since both Higgs doublets can couple tofermions, the general Yukawa couplings are more complicated than in the SM shown inEq. (2.23) and, in the most general case, have the form

LYukawa =−∑

i,j

(L′Li ζ

l,1ij l

′Rj Φ1 + Q′L

i ζu,1ij u

′Rj Φ1 + Q′L

i ζd,1ij d

′Rj Φ1 + h.c.)

−∑

i,j

(L′Li ζ

l,2ij l

′Rj Φ2 + Q′L

i ζl,2ij u

′Rj Φ2 + Q′L

i ζl,2ij d

′Rj Φ2 + h.c.), (2.78)

with the mixing matrices ζf,k, k = 1, 2, in generation space for the interaction with Φ1

and Φ2, respectively. As there are two mixing matrices for each type f of fermions,FCNC can occur at tree level. These currents are experimentally strongly suppressed,and additional suppression mechanisms are necessary, as described in the next para-graph. Depending on the choice of the coupling parameters, several types of models aredistinguished.

Models which conserve flavor at tree-level

In the SM, diagonalizing the fermion mass matrices, automatically diagonalizes Yukawainteractions with the Z boson (Sect. 2.1), so that no FCNC occur at tree level. However,in the THDM this is not the case, since there are two mass mixing matrices for each typeof fermions. The Paschos–Glashow–Weinberg theorem [74, 75] states in a general formthat FCNC are absent at tree level if all the fermions with the same quantum numbers(which are able to mix) transform according to the same irreducible representation ofSU(2), correspond to the same eigenvalue T3 and that, in an appropriate basis, theyobtain their contributions in the mass mixing matrix from a single source. This isfulfilled if each type of fermion couples only to one of the Higgs doublets which canbe achieved by imposing an additional discrete Z2 symmetry. The Yukawa Lagrangianreduces then to

LYukawa = −∑

i,j

(L′Li ζ

lijl

′Rj Φn1 + Q′L

i ζuiju

′Rj Φn2 + Q′L

i ζdijd

′Rj Φn3 + h.c.), (2.79)

with ni = 1, 2. For each of the fermion types a redefinition of the fields can be performedin order to get diagonal mass matrices, analogously to the SM case (see Sect. 2.1.1,Yukawa Couplings). Similar to the SM the coupling of fermions to the Z boson is flavorconserving, and a CKM matrix appears in the coupling to the charged gauge bosons. Inour investigation the CKM matrix can be approximated by the identity matrix with anegligible error as long as we do not deal with flavor-sensitive observables. This rendersthe Yukawa interaction particularly simple,

LYukawa = −∑

i

(

√2ml,i

vn1

LLi Φn1lRi +

√2mf,u,i

vn3

LLi Φn2uRi +

√2md,i

vn2

QLi Φn3d

Ri + h.c.).

(2.80)

22

ui di ei Z2 SymmetryType I Φ2 Φ2 Φ2 Φ1 → −Φ1

Type II Φ2 Φ1 Φ1 (Φ1, di, ei) → −(Φ1, di, ei)Lepton-specific Φ2 Φ2 Φ1 (Φ1, ei) → −(Φ1, ei)

Flipped Φ2 Φ1 Φ2 (Φ1, di) → −(Φ1, di)

Table 2.3.: Different flavor-conserving THDM which have in common that only one of theHiggs doublet couples to each type of fermions. This can be achieved by imposing appro-priate Z2 symmetry charges to the fields.

Without introducing FCNC it is possible to break the Z2 symmetry softly in quadraticterms in the potential by the term proportional to m12. Depending on the exact form ofthe symmetry one distinguishes four types of THDMs. In Type I models, all fermionscouple to one Higgs doublet (conventionally Φ2, but this is equivalent to Φ1 due tobasis changes) which can be ensured by demanding a Φ1 → −Φ1 symmetry. In TypeII models down-type fermions couple to the other doublet, which can be enforced bythe symmetry Φ1 → −Φ1, d

′Rj → −d′Rj , l′R → −l′R . The other two possibilities are

called “lepton-specific” or Type X and “flipped” or Type Y models. An overview overthe couplings and symmetries of the different models is given in Tab. 2.3. By specifyingthe model type, the Higgs–fermion interaction is determined, and one can write them,in the notation of Ref. [67], as

Lint =−∑

f=u,d,l

mf

v

(

ξfh f fh+ ξfH f fH − iξfA fγ5fA0 − 2iI3W,f fγ5fG0

)

(2.81)

−{√2Vijv

ui(miξuAω− +mjξ

dAω+) djH

+ +

√2mlξ

lA

vνLlRH

+ + h.c.}

−{√2Vijv

ui(miω− +mjω+) djG+ +

√2ml

vνLlRG

+ + h.c.}

.

The coupling coefficients ξfH,h,A are defined as the couplings relative to the canonical SMvalue of mf/v and are shown in Tab. 2.4. Through basis changes, one could introducecouplings to both Higgs doublets, paying the price of giving up a simple Yukawa inter-action and the discrete symmetry. However, in any basis, the mixing matrices ζf,1ij are

proportional to ζf,2ij for all fermions f , and the proportionality factors for the three typesof fermions follow a pattern determined by the rotation of the Higgs doublets.There is another proposal of Pich and Tuzon [76, 77] that satisfies the Paschos–Glashow–Weinberg theorem. They assume that the basis in which there is only one source forthe mass mixing matrix is different for l, u, and d type fermions. This means that themixing matrices ζf,1ij and ζf,2ij are proportional to each other, but without relations ofthe proportionality factors between the different types of fermions, in contrast to themodels described above. In our project we focus on Type I and II models which havegained the most attention in the past. In fact, Percei–Quinn axion models [66] as well assupersymmetric models [59] contain a Type II THDM, which in these cases is demandedby the underlying symmetries and not by imposing additional discrete ones.

23

Type I Type II Lepton-specific FlippedξlH sinα/ sinβ cosα/ cosβ cosα/ cos β sinα/ sin βξuH sinα/ sinβ sinα/ sinβ sinα/ sin β sinα/ sin βξdH sinα/ sinβ cosα/ cosβ sinα/ sin β cosα/ cosβξlh cosα/ sin β − sinα/ cosβ − sinα/ cosβ cosα/ sinβξuh cosα/ sin β cosα/ sin β cosα/ sinβ cosα/ sinβξdh cosα/ sin β − sinα/ cosβ cosα/ sinβ -sinα/ cosβξlA − cot β tan β tanβ − cot βξuA cot β cotβ cot β cotβξdA − cot β tan β − cot β tan β

Table 2.4.: The coupling strengths ξf of H,h,A to the fermions relative to the SM value ofmf/v, see Eq. (2.81).

Models with FCNC at tree level: Type III

We do not deal with Type III models in our project, but for the sake of completenesswe discuss them very briefly here. In those models no discrete symmetry is demanded,so that FCNC occur at tree level. The Yukawa Lagrangian contains all the terms ofEq. (2.78) and it is best to discuss them in the Higgs basis in which only Φ1 developsa vev. Then, the interaction with Φ1 generates the fermion masses, the mixing matrixζ1ij can be diagonalized and does not lead to FCNC. However, the interaction withΦ2 cannot be diagonalized simultaneously and introduces FCNC. To be consistent withexperimental constraints, other mechanisms to suppress FCNC are needed. More detailson those types of THDM can be found in Ref. [67].

2.2.4. Gauge-Fixing and Faddeev-Popov Ghost Lagrangians

As in the SM, the quantization of the theory requires a gauge-fixing term in the La-grangian. From the kinetic term (DµΦ

k)†(DµΦk) of the Higgs bosons, mixing betweengauge bosons and would-be Goldstone bosons appear in the Lagrangian. Inserting thegeneral covariant derivative,

Dµ,ij = δij∂µ + igaAaµT

aij, (2.82)

the mixing terms are

LSV = (∂µΦki )†(igaA

aµT

aijΦ

k0,j) + h.c. = −2gaA

aµIm

[

(∂µΦki )†F a,k

i

]

, (2.83)

with the general gauge fields Aa, the corresponding generators T a, and the coupling1 tothis particular gauge field ga. In this notation we introduce the matrix F a,k

i = T aijΦk0,j

which connects the spontaneously broken gauge generators with the Goldstone boson

1For non-simple gauge groups such as U(1) × SU(2), the coupling constants do not have to be thesame for all fields.

24

degrees of freedom. In addition we assume that the signs in the definition of the covariantderivative are included in the couplings ga. In the GSW theory we have three generatorsof the SU(2) gauge group and one from the U(1) with the corresponding couplings g2and ±g1, respectively. One has to sum over the all Higgs doublets which are labeledwith k. To ensure the cancellation of the mixings, the gauge-fixing terms are chosen as

F a = ∂µAaµ + 2gaξaIm[

(Φki )†F a,k

i

]

, (2.84)

and the gauge-fixing Lagrangian in the ’t Hooft gauge must be

Lfix = − 1

2ξa(F a)2 (2.85)

for each gauge field. Inserting all fields of the THDM in the mixing term of Eq. (2.83)results in

LH,SV =MZZµ∂µG0 − iMW(W+

µ ∂µG− −W−

µ ∂µG+), (2.86)

which is identical to the one of the SM, so that a gauge-fixing Lagrangian in the sameform as in the SM (Eq. (2.30)) can be used. The matrix δFα

δθβ(x)in the definition of the

Fadeev–Popov ghost Lagrangian of Eq. (2.31) differs from the one in the SM as bothHiggs doublets contribute. The matrix can be calculated using the transformation ofthe fields under the gauge symmetry,

δΦki =iδθa(x)T aijΦkj (x), (2.87)

δAaµ =1

ga∂µδθ

a(x)− fabcθbAcµ =1

gaDabδθb, (2.88)

which results in

δF α

δθβ(x)=δF α

δAγδAγ

δθβ+δF α

δΦki

δΦkiδθβ

+δF α

δΦki†δΦki

†

δθβ(2.89)

=1

gα∂µD

αβ − 2Re[gβFa,ki

†(T βijΦ

kj )].

The former term is the pure gauge part, determining the interactions between gaugebosons and ghosts, the latter is the scalar part, determining the interactions of theghosts and the Higgs bosons. The factor 2 compensates for the fact that all degrees offreedom in Φ are normalized to 1/

√2.

25

3. Renormalization of the THDM

After we have discussed each part of the Lagrangian of the THDM at tree level, the nextstep in calculating higher-order corrections is the renormalization of the theory. In thissection we focus on electroweak corrections of O(αem). The QCD renormalization of theTHDM is straightforward and completely analogous to the SM case (c.f. QFT textbooks,e.g. Ref. [73]), since all scalar degrees of freedom are color singlets and do not interactstrongly. Renormalization is necessary in order to guarantee a cancellation of the UVdivergences originating from high-momentum regions in loop integrals. The renormal-ization is performed in dimensional regularization [78, 79] in which the dimension ofspace-time is shifted to

D = 4− 2ǫ, (3.1)

where ǫ is complex. In the procedure of renormalization, all parameters and fields ap-pearing in the Lagrangian are considered to be potentially divergent bare parameters andfields, which are not accessible in experiment. They are represented with an additional0 as index. The bare parameters and fields are related to renormalized ones (denotedwithout an additional index) via a multiplicative renormalization transformation,

c0 = Zcc = c(1 + δZc +O(α2

em))= c+ δc+O(α2

em), (3.2)

φ0 = Z12φ φ = φ

(1 +

1

2δZφ +O(α2

em)),

with generic parameters c, fields φ, and counterterms δZ, where δZ = O(αem) andcontains all divergent parts, so that the renormalized parameters and fields are finite.In the case that there are several fields with the same quantum numbers, e.g. φ and ψ,we allow for matrix valued field renormalization constants

(φ0

ψ0

)

=

(

Z12φ Z

12φψ

Z12ψφ Z

12ψ

)(φψ

)

=

(1 + 1

2δZφ

12δZφψ

12δZψφ 1 + 1

2δZψ

)(φψ

)

. (3.3)

Inserting these renormalization transformations into the Lagrangian and separating allcounterterms leads to

L0 = L+ δL, (3.4)

with L having the same functional dependence on the renormalized fields and parametersas L0 has for the bare quantities, thus leading to the same Feynman rules with bare

26

quantities replaced by renormalized ones. The counterterm Lagrangian δL can be splitinto several parts,

δL = δLGauge + δLFermion + δLHiggs,kin − δVHiggs + δLYukawa, (3.5)

contains the full dependence on renormalization constants, and leads to additional coun-terterm Feynman rules. Since the gauge fixing is applied after renormalization, nogauge-fixing counterterm occurs, and since ghost particles occur only in loop diagrams,a renormalization of the ghost sector is not necessary at NLO for the calculation of S-matrix elements. Though, for analyzing e.g. Slavnov–Taylor or Ward identities a com-plete renormalization procedure would be necessary. The derivation of the countertermLagrangian is performed in Sect. 3.1. Afterwards it is necessary to fix the renormal-ization constants with renormalization conditions (Sect. 3.2). The on-shell conditions,where the renormalized parameters correspond to measurable quantities are describedand applied in Sect. 3.2.1. The renormalization constants of parameters which do notdirectly correspond to physical quantities are fixed in the “modified minimal subtractionscheme” (MS) so that they contain only the UV-divergences and no finite terms. Wedescribe different renormalization schemes based on different definitions of the param-eters renormalized in MS in Sect. 3.2.2. Parameters renormalized in MS depend onan unphysical renormalization scale and their running is governed by the renormaliza-tion group equations (RGE). These equations are derived and their numerical solutionis sketched in Sect. 3.3. The description of the implementation of the results into anautomated matrix element generator can be found in Sect. 3.4.The renormalization of the THDM has been tackled in several papers: First by Santosand Barroso [26] where the fields and masses were renormalized in the on-shell scheme,but no prescription was given for the other parameters α, β, λ5. Later Kanemura etal. [27] as well as Lopez Val and Sola [28] used for the calculation of their processes arenormalization where the fields are not renormalized on-shell, but the usual on-shellrenormalization conditions for the field renormalization constants were used to fix theother parameters. A tool for an automated renormalization was written by Degrande[29], where all finite (R1, R2 terms) and divergent terms are computed using on-shellor MS conditions. However, this method is only in principle applicable to four-point-functions as the code is too slow for a practical use. A complete renormalization schemeemploying a special but gauge invariant treatment of the tadpole terms was describedrecently in Refs. [30, 31]. We propose within this chapter a complete and consistentrenormalization prescription and two new renormalization schemes where tadpole dia-grams do not appear in higher-order calculations.

3.1. The Counterterm Lagrangian

In this section we derive successively the different parts of the counterterm Lagrangianof Eq. (3.5), beginning in Sect. 3.1.1 with the most complicated part, the countertermHiggs potential. The structure of the gauge and the fermionic parts look similar to theSM case, so that we apply the well-known renormalization procedure of the SM [48] in

27

Sect. 3.1.2. For the Higgs kinetic and the Yukawa parts one can straightforwardly applythe transformations, as done in Sect. 3.1.3 and Sect. 3.1.4, respectively.

3.1.1. Higgs Potential

The renormalization of the Higgs potential involves some difficulties, since the Higgspotential (with our restrictions) contains 7 independent parameters. To avoid mistakesand to allow for checks, the renormalization has been carried out independently with 3different methods:

1. Renormalization of the Lagrangian in the mass parameterization

a) with renormalization of the mixing angles.

b) where the mixing angles are not taken as free parameters and the field trans-formation is applied after renormalization,

2. Renormalization of the basic parameters and a transformation to renormalizationconstants and parameters of the mass parameter set

a) analogously to Dabelstein in the MSSM [80],

b) using a modified method which allows for a renormalization of the mixingangles and a field renormalization of each field.

3. Independent implementation by H. Rzehak a la method 1b.

The results obtained independently with each method cannot directly be comparedwith each other, as the methods differ by the parameterization and also the number ofrenormalization constants. However, these differences can be quantified and calculatedso that we can compare and check the results. Physical observables are independentof a specific parameterization of the potential. In the following, we give a detaileddescription of method 1a which we also use throughout the rest of this thesis while theother methods are presented in App. B.

3.1.1.1. The Reparameterized Bare Higgs Potential at NLO

In the method 1a, the renormalization procedure is applied to the potential of Eq. (2.74)written in terms of the mass parameter set. However, at NLO two crucial requirementsmade in the derivation at LO do not hold, because higher-order terms of O(αem) appearin the counterterm potential and through one-loop contributions. First, at LO, wedemanded that tadpole terms in the potential vanish, but one-loop tadpole diagramsappear as well at NLO. Thus, a linear term in the counterterm potential could be used tocancel the one-loop contribution (shown in Sect. 3.2.1). Secondly, we defined the anglesα, βn, βc in such a way that the mass matrices are diagonal and fields correspondingto mass eigenstates are achieved. However, such a definition does not exist at NLOas momentum-dependent loop and counterterm contributions appear which cannot becanceled generally by a constant redefinition of the fields. Therefore, different definitions

28

of the higher-order parts of these parameters can be formulated. For a proper treatmentof the bare potential in terms of mass parameters, the relations derived in Sec. 2.2.2have to be recalculated without the requirements made at LO. As all quantities are bareones, we suppress the index 0 for bare quantities in this section.After a rotation of the fields using Eq. (2.59) (the angles are bare quantities as well),the potential has the following form

V =− tHH − thh (3.6)

+1

2(H, h)

(M2

H M2Hh

M2Hh M2

h

)(Hh

)

+1

2(G0, A0)

(M2

G M2A0G

M2A0G

M2A0

)(G0

A0

)

+ (G+, H+)

(M2

G± M2HG±

M2HG± M2

H±

)(G−

H−

)

+ interaction terms

with the non-vanishing tadpole terms

tH = cαtη1 + sαtη2 = cα(−m211v1 − λ1v

31/2 + v2(m

212 − λ345v1v2/2)) (3.7a)

+ sα(−m222v2 − λ2v

32/2 + v1(m

212 − λ345v1v2/2)),

th = −sαtη1 + cαtη2 = −sα(−m211v1 − λ1v

31/2 + v2(m

212 − λ345v1v2/2)) (3.7b)

+ cα(−m22v2 − λ2v32/2 + v1(m

212 − λ345v1v2/2)).

The coefficients of the mass matrices contain tadpole terms as well when m11, m22 areeliminated using the above equations,

M2H =

s2α−βcβsβ

m212 + v2

(λ1c

2βc

2α + λ2s

2αs

2β + s2αs2βλ345/2

)(3.8a)

− ths2αsα−β2vs2β

− tHs3αcβ + c3αsβ

vs2β,

M2h =

c2α−βs2β

m212 + v2

(λ1c

2βs

2α + λ2c

2αs

2β − s2αs2βλ345/2

)(3.8b)

− thc3αcβ − s3αsβ

vs2β− tH

s2αcα−β2vs2β

,

M2Hh =

s2(α−β)s2β

m212 +

v2

2

(s2α(−c2βλ1 + s2βλ2) + s2βc2αλ345

)(3.8c)

− ths2αcα−βvs2β

− tHs2αsα−βvs2β

,

and the entries of the other mass mixing matrices are

M2A0

= c2β−βn

(m212

sβcβ− λ5v

2)

− tHc2βncβsα + s2βnsβcα

vcβsβ− th

c2βncβcα − s2βnsβsα

vcβsβ, (3.9a)

M2G = s2β−βn

(m212

sβcβ− λ5v

2)

− tHs2βncβsα + c2βnsβcα

vcβsβ− th

s2βncβcα − c2βnsβsα

vcβsβ, (3.9b)

29

M2AG = cβ−βnsβ−βn

(m212

cβsβ− λ5v

2)

− tHs2βnsα−βvs2β

− ths2βncα−βvs2β

, (3.9c)

and

M2H± = c2β−βc

(m212

cβsβ− v2

2(λ4 + λ5)

)

− tHc2βccβsα + s2βcsβcα

vcβsβ− th

c2βccβcα − s2βcsβsα

vcβsβ,

(3.10a)

M2G± = s2β−βc

(m212

cβsβ− v2

2(λ4 + λ5)

)

− tHs2βccβsα + c2βcsβcα

vcβsβ− th

s2βccβcα − c2βcsβsα

vcβsβ,

(3.10b)

M2HG± = sβ−βccβ−βc

(m212

cβsβ− v2

2(λ4 + λ5)

)

− tHs2βcsα−βvs2β

− ths2βccα−βvs2β

. (3.10c)

At tree level we demand diagonal propagators, so that the mixing terms M2Hh, M

2AG,

M2HG± vanish at LO. As we want to obtain the tree-level relations also for renormalized

quantities, we demandM2Hh,ren =M2

AG,ren =M2HG±,ren = 0 which defines the renormalized

mixing angles to equal their LO values. At NLO such a diagonalization is, however, notpossible, as the propagators receive also mixing terms from the field renormalizationand from one-loop diagrams so that there is no distinct condition to define the mixingangles. Therefore we keep the mass mixing terms M2

Hh, M2AG, M

2HG± in this section,

and specify defining conditions for the bare parameters α, βn, βc in Sect. 3.1.1.3 for themethod 1a and in App. B.3 for the method 1b. However, as long as α, βn, βc are onlyused in the field transformation and not to replace a free parameter, different definitionsdo not affect physics, as proved in Sect. 3.1.1.2.With the Eqs. (3.7), (3.8), (3.9a), and (3.10a), the parameter set can be changed anal-ogously to the LO case in favor of the mass parameter set. Explicitly, the relations forthe tadpole terms are

m211 = tβm

212 −

1

2λ1v

2c2β −1

2λ345v

2s2β −tHcα − thsα

vcβ, (3.11a)

m222 =

1

tβm2

12 −1

2λ2v

2s2β −1

2λ345v

2c2β −tHsα + thcα

vsβ. (3.11b)

with v as defined in Eq. (2.71). The other parameters are related to the masses by

λ1 =1

c2βv2

(c2αM

2H + s2αM

2h −M2

Hhs2α − s2β(M2A0/c2β−βn + λ5v

2))

(3.12a)

tHcβn (2sβsβncα + cα+βcβn)

v3c2βc2β−βn

− thcβn (2sβsβnsα + sα+βcβn)

v3c2βc2β−βn

,

λ2 =1

s2βv2

(c2αM

2h + s2αM

2H +MHhs2α − c2β(M

2A0/c2β−βn + λ5v

2))

(3.12b)

+ tHsβn(2cβcβnsα − cα+βsβn)

v3s2βc2β−βn

+ thsβn(2cβcβncα + sα+βsβn)

v3s2βc2β−βn

,

30

λ3 =1

v2s2β

(s2α(M

2H −M2

h) + 2c2αM2Hh

)− M2

A0

v2c2β−βn+

2M2H±

v2c2β−βc− λ5 (3.12c)

+th

v3cβsβ

(

cβcα

( 2c2βcc2β−βc

−c2βnc2β−βn

)

+ sβsα

( s2βnc2β−βn

−2s2βcc2β−βc

))

+tH

v3cβsβ

(

sβcα

( 2s2βcc2β−βc

−s2βnc2β−βn

)

+ cβsα

( 2c2βcc2β−βc

−c2βnc2β−βn

))

λ4 =λ5 +2M2

A0

v2c2β−βn− 2M2

H±

v2c2β−βc+

2tHsβc−βn (sα+β−βc−βn − sβ−αcβc−βn)

v3c2β−βcc2β−βn

(3.12d)

+2thsβc−βn (cα+β−βc−βn + cβ−αcβc−βn)

v3c2β−βcc2β−βn

,

m212 =λ5v

2cβsβ +M2

A0cβsβ

c2β−βn(3.12e)

+tH(sβcαs

2βn

+ cβsαc2βn

)

vc2β−βn+th(cβcαc

2βn

− sβsαs2βn

)

vc2β−βn.

Parameters of the gauge sector

For the parameters of the gauge sector, we demand that the relations (2.70) hold to allorders, yielding the same transformation (2.71) for parameters as in LO, but with bareparameters.

Mass parameterization

The relations (3.11), (3.12a,b,d,e) and (2.71) between the masses, angles, and the ba-sic parameters can be used to reparameterize the Higgs Lagrangian and change theparameters from

{pbasic} = {λ1, . . . , λ5, m211, m

222, m

212, v1, v2, g1, g2}, (3.13)

in favor of the bare mass parameters including λ3,

{p′mass} = {MH,Mh,MA0,MH± ,MW,MZ, e, λ5, λ3, β, tH, th}. (3.14)

Additionally, one has to keep in mind that the counterterms of each mixing parameterα, βn, and βc are free and need to be fixed by an additional constraint (which we willgive later). For some conditions on the mixing angle α, one can use Eq. (3.12c) to tradeλ3 for α, in which case the mixing angle becomes a free parameter of the theory. Then,one obtains the parameter set

{pmass} = {MH,Mh,MA0 ,MH±,MW,MZ, e, λ5, α, β, tH, th}. (3.15)

The tadpole terms of both sets are of O(αem) and therefore do not appear in the LOparameter set. With this set of bare parameters, we achieve a similar form of the Higgspotential as in LO, but with additional contributions from tadpoles and mixing angles,

31

VHiggs = −tHH − thh +M2Hh({pmass}, α)Hh+

1

2M2

HH2 +

1

2M2

hh2 +

1

2M2

A0A2

0 +M2H±H+H−

+1

2M2

G({pmass}, βn)G2 +M2GA({pmass}, βn)GA0 +M2

G±({pmass}, βc)G+G−

+M2HG±({pmass}, βc)(H+G− +H−G+) + interaction terms, (3.16)

where the mass mixing terms are of O(αem) and the interaction terms also contain thehigher oder parts of tH, th, α, βn, βc. Having a potential depending on the bare massparameter set, it is now possible to renormalize it. In a first step, we will discuss thefreedom in the definition of the higher-order part of the mixing angles and its implicationon the mass mixing terms.

3.1.1.2. Renormalization of the Mixing Angles

In this section we show that the counterterms of mixing angles that are not used toreplace another free parameter of the theory have no physical relevance and do notchange any observable, no matter how the angles are defined at NLO. We perform theproof for generic scalar fields η1,2 from which the Lagrangian is built up. They aretransformed into fields corresponding to mass eigenstate fields h1, h2 through a rotationwith an angle we name αr,

(η1η2

)

=

(cαr −sαr

sαr cαr

)(h1h2

)

. (3.17)

The general argument given in this section can be applied to the neutral CP-even, theCP-odd, and the charged Higgs fields of the THDM by setting h1, h2 to H, h or G,A0

or G+,H+, respectively. The mixing angle is treated as a bare quantity and the fieldscorresponding to mass eigenstates are renormalized using matrix-valued renormalizationconstants, so that the renormalization transformations read

hi,0 = hi +∑

j

1

2δZhihjhj , αr,0 = αr + δαr. (3.18)

Performing the renormalization of Eq. (3.17) leads for η1(η1,0η2,0

)

=

[1

2

(cαr −sαr

sαr cαr

)(2 + δZh1h1 δZh1h2

δZh2h1 2 + δZh2h2

)

(3.19)

+

(−sαr −cαr

cαr −sαr

)

δαr

](h1h2

)

=1

2

[(cαr −sαr

sαr cαr

)(2 + δZh1h1 0

0 2 + δZh2h2

)

+

(−sαr −cαr

cαr −sαr

)(δZh2h1 + 2δαr 0

0 −δZh1h2 + 2δαr

)](h1h2

)

32

One can easily remove the dependence on the mixing angle by redefining the mixingfield renormalization constants and introduce

δZh2h1 = δZh2h1 + 2δαr, (3.20)

δZh1h2 = δZh1h2 − 2δαr. (3.21)

Then, the bare Eq. (3.17) reads

(η1,0η2,0

)

=1

2

[(cαr −sαr

sαr cαr

)(2 + δZh1h1 0

0 2 + δZh2h2

)

(3.22)

+

(−sαr −cαr

cαr −sαr

)(δZh2h1 0

0 −δZh1h2

)](h1h2

)

The bare fields η1,0 and η2,0 are thus independent of δαr, and as the whole Lagrangian isbuilt up from η1 and η2 and independent parameters, it is obvious that the dependenceof δαr can always be removed from the Lagrangian by parameterizing it through theredefined field renormalization constants. As a simple shift of the mixing field renormal-ization constant is performing the task so that the renormalization of the mixing angleαr can be seen as an additional field renormalization (as it is done e.g. in Ref. [81]). Thisargumentation is general and holds for any renormalization condition as the actual def-inition of the mixing angle at higher order is not needed. Without loss of generality onecan even assume that such an redefinition has already been performed and set δαr = 0from the beginning, as done in method 1b in App. B.3. Of course, the bookkeeping ofcounterterms depends on the way δαr is treated.This can be seen when the mass term of the potential is considered. The general baremass term can be written using the rotation matrix Rη(αr,0) as

Vh1h2 =1

2

(h1 h2

)RT

η (αr,0)Mη,0Rη(αr,0)

(h1h2

)

(3.23)

=1

2

(h1 h2

)RT

η (δαr)RT

η (αr) (Mη + δMη)Rη(αr)Rη(δαr)

(h1h2

)

.

This expression can be expanded into renormalized and counterterm contributions andyields

Vh1h2 =1

2

(h1 h2

)(

M2h1

+ δM2h1

δα(M2h2

−M2h1) + fαr({δp})

δα(M2h2

−M2h1) + fαr({δp}) M2

h2+ δM2

h2

)(h1h2

)

,

where we obtain off-diagonal terms from the counterterm of the mixing angle, and fromthe renormalization of the mass matrix. The latter contribution depends on the inde-pendent counterterms {δp} and is abbreviated by the function fαr({δp}). The massmixing at NLO reads

δM2h1h2 = δαr(M

2h2 −Mh1) + fαr({δp}), (3.24)

33

and since the renormalization of the mixing angle can be chosen freely, countertermcontributions in the Lagrangian can be shifted arbitrarily from mixing terms to mixingangle counterterms.

However, if δαr obtains a meaning beyond the angle of the rotation of the fields, e.g.when a free parameter is eliminated in favor of the mixing angle, things get more in-volved. This case occurs in the mass parameterization {pmass} of neutral CP-even Higgssector of the THDM when α is used instead of λ3 as an independent parameter to param-eterize the potential. Through the elimination of the bare λ3,0, α and its countertermenters in the Lagrangian additionally to the terms originating from the rotation of thefields. For a unique relation between those two parameters at NLO, the mass mixingterm δM2

Hh needs to be specified in dependence of the free parameters. Different mixingterms lead to different definitions of δα which cannot easily be compared with each otheras they lead to different divergent and finite terms in δα.

3.1.1.3. Renormalization with a Diagonal Mass Matrix – version 1a

In this prescription, we use the angle α as an independent parameter instead of λ3. Todefine α at NLO, we demand that the Higgs potential is diagonal for the CP-even Higgsbosons at all orders even though the field renormalization and loop diagrams destroythe diagonality. Equation (3.12c) with

MHh,0 = 0 (3.25)

then defines the parameter α. The relation between α and λ3 of Eq. (3.12c) is the samefor bare and renormalized quantities (except for tadpoles) and can be used to eliminateλ3 from the theory. For each of the independent parameters of the mass parameter set(Eq. (3.15)) we apply the renormalization transformation:

M2H,0 =M2

H + δM2H, M2

h,0 =M2h + δM2

h , M2A0,0

=M2A0

+ δM2A0, (3.26)

M2H±,0 =M2

H± + δM2H±, β0 = β + δβ, α0 = α + δα,

λ5,0 = λ5 + δλ5 M2W,0 =M2

W + δM2W, M2

Z,0 =M2Z + δM2

Z, ,

e0 = e+ δe, tH,0 = 0 + δtH, th,0 = 0 + δth,

so that the 12 parameter renormalization constants are

{δpmass} = {δMH, δMh, δMA0 , δMH±, δMW, δMZ, δe, δλ5, δα, δβ, δtH, δth}, (3.27)

corresponding to {pmass} with the additional tadpole counterterms.

The higher-order corrections of the mixing angles βn and βc are irrelevant according toSect. 3.1.1.2 and we can choose

βn,0 = βc,0 = β0 (3.28)

34

for the bare parameters which defines the mixing terms uniquely and ensures that theangles βn, βc, and β do not have to be distinguished at any order. The mixing parametersobtain the following renormalization transformation,

βc,0 = β + δβ, βn,0 = β + δβ. (3.29)

From these conditions, we can compute the mass mixing terms from Eq. (3.9c) andEq. (3.10c) to

M2A0G,0

= 0 + δM2A0G

= −eδthcα−β + δtHsα−β2MWsW

, (3.30)

M2HG±,0 = 0 + δM2

HG± = −eδthcα−β + δtHsα−β2MWsW

.

The field renormalization is performed in addition for each field corresponding to masseigenstates

(H0

h0

)

=

(1 + 1

2δZH

12δZHh

12δZhH 1 + 1

2δZh

)(Hh

)

, (3.31)

(G0,0

A0,0

)

=

(1 + 1

2δZG

12δZGA

12δZAG 1 + 1

2δZA0

)(G0

A0

)

,

(G±

0

H±0

)

=

(1 + 1

2δZG+

12δZGH+

12δZHG+ 1 + 1

2δZH+

)(G±

H±

)

,

and we denote the complete set of parameter and field renormalization constants with{δRmass}. All renormalization constants δX are of O(αem) and all contributions ofO(α2

em) are omitted.Applying the renormalization transformation of Eqs. (3.26,3.31) to the bare potential ofEq. (3.16) and linearizing in the renormalization constants results in

V ({pmass}) + δV ({pmass}, {δRmass}) (3.32)

with the already known LO potential of Eq. (2.74) and the counterterm potential ofO(αem),

δV ({pmass}, {δRmass}) =− δtHH − δthh (3.33)

+1

2(δM2

H + δZHM2H)H

2 +1

2(δM2

h + δZhM2h)h

2

+1

2(δM2

A0+ δZA0M

2A0)A2

0 + (δM2H± + δZH±M2

H±)H+H−

+e

4MWsW(δthsα−β − δtHcα−β)(G

20 + 2G+G−)

+ (1

2M2

HδZHh +1

2M2

hδZhH)Hh

+1

2M2

A0δZAGA0G0 +

1

2M2

H±δZH±G±(H+G− +G+H−)

35

− eδthcα−β + δtHsα−β

2MWsW(A0G0 +H+G− +G+H−)

+ interaction terms,

The interaction terms are very lengthy and we do not want to show them here, butthey can be used straightforwardly for further processing, e.g. in the calculation of thecounterterm Feynman rules. Since the relations and definitions of the mixing angles havethe same functional dependence for bare and renormalized quantities, one can abbreviatethis derivation of the counterterm potential and generate it by taking the potential attree level with bare parameters and add only the tadpole contributions.The prescription for the field renormalization of Eq. (3.31) is non-minimal as a renor-malization of the doublets with two renormalization constants (see App. B.1),

Φ1 → Z1/2H1

Φ1 = Φ1(1 +1

2δZH1), (3.34)

Φ2 → Z1/2H2

Φ2 = Φ2(1 +1

2δZH2),

is sufficient to cancel the UV divergences. However, the prescription with matrix valuedrenormalization constants allows to renormalize each field on-shell. The UV-divergentparts of these renormalization constants cannot be independent and relations betweenthe UV-divergent parts of the two prescriptions exist. They can be obtained by com-puting the higher-order parts of the bare field rotations (see Eqs. (2.59)),

δZh

∣∣UV

= sin2 α δZH1

∣∣UV

+ cos2 α δZH2

∣∣UV, (3.35)

δZH

∣∣UV

= cos2 α δZH1

∣∣UV

+ sin2 α δZH2

∣∣UV,

δZHh

∣∣UV

= sinα cosα (−δZH1

∣∣UV

+ δZH2

∣∣UV

) + 2δα∣∣UV,

δZhH

∣∣UV

= sinα cosα (−δZH1

∣∣UV

+ δZH2

∣∣UV

)− 2δα∣∣UV,

δZA0

∣∣UV

= δZH+

∣∣UV

= sin2 β δZH1

∣∣UV

+ cos2 β δZH2

∣∣UV,

δZG

∣∣UV

= δZG+

∣∣UV

= cos2 β δZH1

∣∣UV

+ sin2 β δZH2

∣∣UV,

δZGA0

∣∣UV

= δZGH±

∣∣UV

= sin β cos β (−δZH1

∣∣UV

+ δZH2

∣∣UV

) + 2δβ∣∣UV,

δZA0G

∣∣UV

= δZHG±

∣∣UV

= sin β cos β (−δZH1

∣∣UV

+ δZH2

∣∣UV

)− 2δβ∣∣UV.

and we will use these relations to derive UV divergent parts for specific renormaliza-tion constants in Sect. 3.2.2. In App. B.3 we discuss a different choice of the mixingangles which is suited for λ3 as an independent parameter. In addition we present inthis appendix renormalization prescriptions which apply the transformation to the ba-sic parameter set and transform afterwards to the physical parameters. The resultsof the different prescription have been compared successfully and guarantee a correctcomputation of the counterterm potential.

3.1.2. Fermionic and Gauge Parts

Since the THDM extension of the SM does not affect the gauge and the fermion partsof the Lagrangian, also the renormalization is identical to the SM case. It is reviewed

36

in detail in Ref. [48] in the BHS convention. However, as we take the CKM matrix tobe diagonal throughout this thesis, the renormalization of the fermion sector simplifies.The transformation of the bosonic fields and the left- and right-handed fermions are

FLi,0 =

(1 + 1

2δZf,u,L

i 0

0 1 + 12δZf,d,L

i

)

FLi , F = L,Q, (3.36)

fR0 = fRi +1

2δZf,R

i fRi , f = l, u, d, (3.37)

W0 = W+1

2δZWW, (3.38)

(Z0

A0

)

=

(1 + 1

2δZZZ

12ZZA

12δZAZ 1 + 1

2δZAA

)(ZA

)

. (3.39)

Mixing between left-handed up- and down-type fermions does not occur, as the electriccharge is conserved. Inserting this into the Lagrangian directly delivers the renormalizedand the counterterm Lagrangian.

3.1.3. The Higgs Kinetic Part

After expressing the Higgs kinetic term

LH,kin = (DµΦ1)†(DµΦ1) + (DµΦ2)

†(DµΦ2) (3.40)

through the bare physical fields, mixing angles, and parameters, one can apply therenormalization transformation above (Eqs. (3.26), (3.31)) to obtain the countertermpart of the kinetic Lagrangian. The mixing angles are treated in a general way (seeEq. (3.18)) which introduces the dependence on terms including δβc,n and from thevacuum expectation values the counterterm δβ arises.Also scalar–vector mixing terms are introduced, which are canceled at LO by the gauge-fixing contribution. Since the gauge fixing is applied using renormalized fields, NLOmixing contributions still survive1. The mixing of gauge bosons and Higgs fields usingbare parameters is

LH,SV =MZcβ−βnZµ∂µG0 − iMWcβ−βc(W

+µ ∂

µG− −W−µ ∂

µG+) (3.41)

+MZsβ−βnZµ∂µA0 − iMWsβ−βc(W

+µ ∂

µH− −W−µ ∂

µH+).

Together with the renormalization transformation of Eqs. (3.26), (3.31) one obtains theSV mixing counterterms as

δLZG =MZZµ∂µG0(M

2ZδZZZ + δM2

Z)/(2MZ), (3.42a)

δLWG± = −i(W+µ ∂

µH− −W−µ ∂

µH+)(M2WδZW + δM2

W)/(2MW) (3.42b)

δLZA =MZZµ∂µA0(δZGA/2 + δβ − δβn), (3.42c)

1It is also possible to formulate the gauge fixing in terms of bare fields, however, one has to renormalizeand fix all appearing gauge constants separately, which has to be done carefully.

37

δLWH± = −iMW(W+µ ∂

µH− −W−µ ∂

µH+)(δZGH±/2 + δβ − δβc). (3.42d)

When the counterterm definition of Eq. (3.29) is inserted, the contributions from themixing angle vanish.

3.1.4. Yukawa Part

The renormalization of the Yukawa sector is straightforward in Type I–IV models andcan be done by taking the Lagrangian of Eq. (2.80), replacing the vevs and the fieldsthrough their physical counterparts and apply the renormalization transformations ofSect. 3.1.1.3, as well as a renormalization of the fermion mass,

mf,i,0 = mf,i + δmf,i. (3.43)

The coupling counterterms of the neutral CP-even and pseudoscalar Higgs fields to thefermions factorize from the LO, while the couplings to the charged Higgs bosons obtainadditional terms. The terms of the Lagrangian read,

δLff,mass =mf f f(1

2δZf,R +

1

2δZf,L +

δmf

mf

)

, (3.44)

δLf fh

Lffh

=δZe −δM2

W

2M2W

− δsWsW

+δmf

mf

+1

2δZf,R +

1

2δZf,L +

1

2δZh +

δξfhξfh

+δZHhξ

fH

ξfh,

δLffH

LffH=δZe −

δM2W

2M2W

− δsWsW

+δmf

mf+

1

2δZf,R +

1

2δZf,L +

1

2δZH +

δξfHξfH

+δZhHξ

fh

ξfH,

δLffA0

LffA0

=δZe −δM2

W

2M2W

− δsWsW

+δmf

mf+

1

2δZf,R +

1

2δZf,L +

1

2δZA0 +

δξfAξfA

+δZGA02I

3W,f

ξfA,

δLf fG0

LffG0

=δZe −δM2

W

2M2W

− δsWsW

+δmf

mf+

1

2δZf,R +

1

2δZf,L + δξfG2I

3W,f ,

δLffH+ =(

δZe −δM2

W

2M2W

− δsWsW

+1

2δZH±

)

Lf fH+ +1

2δZHG±Lf fG+

+e√

2MWsWH+fu

[

mufξ

uAω−

(δmuf

muf

+1

2δZf,d,L +

1

2δZf,u,R +

δξuH+

ξuA

)

+mdfξ

dAω+

(δmdf

mdf

+1

2δZf,u,L +

1

2δZf,d,R +

δξdH+

ξdA

)]

f d,

δLf fH− =δL†f fH+,

δLf fG+ =(

δZe −δM2

W

2M2W

− δsWsW

+ δξG+

)

Lf fG+

+e√

2MWsWG+

[

muf f

uw−f d(δmu

f

muf

+1

2δZf,d,L +

1

2δZf,u,R

)

+mdf f

uw+f d(δmd

f

mdf

+1

2δZf,u,L +

1

2δZf,d,R

)]

,

38

Type I Type II Lepton-specific Flipped

δξlHcαsβδα− sαcβ

s2β

δβ −sαcβδα+

cαsβc2β

δβ −sαcβδα+

cαsβc2β

δβ cαδαsβ

− sαcβs2β

δβ

δξuHcαsβδα− sαcβ

s2βδβ cα

sβδα− sαcβ

s2βδβ cα

sβδα− sαcβ

s2βδβ cα

sβδα− sαcβ

s2βδβ

δξdHcαsβδα− sαcβ

s2β

δβ −sαcβδα+

cαsβc2β

δβ cαsβδα− sαcβ

s2β

δβ −sαcβδα+

cαsβc2β

δβ

δξlh −sαsβδα− cαcβ

s2β

δβ − cαcβδα− sαsβ

c2β

δβ − cαcβδα− sαsβ

c2β

δβ −sαsβδα− cαcβ

s2β

δβ

δξuh −sαsβδα− cαcβ

s2β

δβ −sαsβδα− cαcβ

s2β

δβ −sαsβδα− cαcβ

s2β

δβ −sαsβδα− cαcβ

s2β

δβ

δξdh −sαsβδα− cαcβ

s2βδβ − cα

cβδα− sαsβ

c2βδβ −sα

sβδα− cαcβ

s2βδβ − cα

cβδα− sαsβ

c2βδtβ

δξlA,H+ δβn,c +c2βs2βδβ δβn,c +

s2βc2βδβ δβn,c +

s2βc2βδβ δβn,c +

c2βs2βδβ

δξuA,H+ −δβn,c −c2βs2β

δβ −δβn,c −c2βs2β

δβ −δβn,c −c2βs2β

δβ −δβn,c −c2βs2β

δβ

δξdA,H+ δβn,c +c2βs2β

δβ δβn,c +s2βc2β

δβ δβn,c +c2βs2β

δβ δβn,c +s2βc2β

δβ

δξlG,G+ − cβsβ(δβn,c − δβ)

sβcβ(δβn,c − δβ)

sβcβ(δβn,c − δβ) − cβ

sβ(δβn,c − δβ)

δξuG,G+

cβsβ(δβn,c − δβ)

cβsβ(δβn,c − δβ)

cβsβ(δβn,c − δβ)

cβsβ(δβn,c − δβ)

δξdG,G+ − cβsβ(δβn,c − δβ)

sβcβ(δβn,c − δβ) − cβ

sβ(δβn,c − δβ)

sβcβ(δβn,c − δβ)

Table 3.1.: The dependence of the angular counterterms δξ for the different types of models.

δLffG− =δL†ffG+ .

In contrast to the SM case, the counterterms in the Higgs–fermion interaction involvealso the renormalization constants δβ (as vevs appear in the coupling constants), δβn,c,and δα (through the general renormalization of the mixing angles) which are hidden inthe δξ factors. The general values for the different types of THDM are summarized inTab. 3.1.

3.2. Renormalization Conditions

After applying the renormalization transformation to the Lagrangian it is necessaryto fix the renormalization constants introduced in the previous sections, so that thecounterterm diagrams cancel the divergences of the respective loop diagrams and theS-matrix elements become finite. To ensure this, a renormalization of the fields is notnecessary, however, such a procedure is very useful and simplifies calculations, because itrenders also Green functions finite. Would-be-Goldstone bosons are not physical and donot appear as external particles in S-matrix elements so that we do not renormalize theirfields. Demanding finite Green functions determines the renormalization constants onlyup to finite terms. The choice of the finite terms is done via renormalization conditions,and the set of renormalization conditions defines a renormalization scheme. A verycommon scheme is the on-shell scheme, where the renormalized parameters correspondto physically measurable parameters, e.g. the electromagnetic coupling constant to theelectric charge in the Thomson limit, or the location of the poles of the propagators

39

ParametersEW(3): δM2

Z, δM2W, δe, (δcW, δsW)

fermion masses(9): δmf,i, f = l, u, d, i = 1, 2, 3Higgs masses(4): δM2

H, δM2h , δM

2A0, δM2

H±

Higgs potential(3): δλ3 or δα, δλ5, δβtadpoles(2): δtH, δth

FieldsEW(5): δZW, δZZZ, δZZA, δZAZ, δZAA

left-handed fermions(12): δZf,Li , f = ν, l, u, d, i = 1, 2, 3

right-handed fermions(9): δZf,Ri , f = l, u, d, i = 1, 2, 3

Higgs(12): δZH, δZHh, δZhH, δZh

δZA0 , δZA0G, δZGA0, δZG

δZH±, δZHG±, δZGH±, δZG±

Table 3.2.: The renormalization constants used to describe the THDM, separated into sectorsof parameter and field renormalization. The renormalization constants in parentheses arenot independent, but useful for a better bookkeeping. The numbers in parentheses arethe numbers of independent renormalization constants. In total there are 38 field and 20parameter renormalization constants to fix.

to the measured masses. Such conditions can be translated into conditions for specificGreen functions. For the fields, one demands that on-mass-shell external fields do neitherobtain corrections due to loop diagrams nor mix with other particles. An alternativescheme, suited for parameters that are not directly accessible in experiments (such asthe quark masses), is the ”modified minimal subtraction scheme” (MS), in which therenormalization constants only contain the standard UV divergence

∆UV =2

4−D− γE + ln 4π =

1

ǫ− γE + ln 4π, (3.45)

where γE is the Euler–Mascheroni constant (γE = 0.57721 . . .). In this case, the renor-malization constant can be fixed by demanding an arbitrary Green function in whichthe desired renormalization constant occurs to be finite.In the THDM we impose conditions on one-particle irreducible two-point functions andmake use of relations between the divergent parts of field renormalization constants.Renormalization constants corresponding to couplings such as δe are fixed by appro-priate vertices. An overview over the renormalization constants we introduced in theprevious section is shown in Tab. 3.2. In our approach, we will use the on-shell schemewherever the parameter might be accessible in experiment and we state the on-shellrenormalization conditions in Sect. 3.2.1, beginning with the Higgs sector and followedby the electroweak and the fermion sectors. The latter two are identical to the SM, sothat we skip the derivation, which can be found in detail in Ref. [48]. However, notall parameters of the THDM correspond to a measurable observable, so that we renor-malize three parameters of the Higgs sector in MS. This is performed in Sect. 3.2.2 infour different ways, resulting in four different renormalization schemes. The treatment

40

H, hF

H, hS

H, hU

H, hV

Figure 3.1.: Generic tadpole diagrams. There is one diagram for each massive fermion, scalar,gauge-boson, and ghost field.

of renormalization constants within the complex mass scheme won’t be covered in thissection but in Sect. 4.6. In addition, we adapt the notation introduced in Ref. [48], sothat we use for the renormalized and the corresponding unrenormalized quantities thesame symbols, but give the former a superscript ˆ.

3.2.1. On-shell Renormalization Conditions

3.2.1.1. Higgs Sector

Tadpoles

We start with the renormalization of the one-point functions, the tadpoles

TH,h =H, h

. (3.46)

At NLO the renormalized tadpole consists of a counterterm contribution δt and anunrenormalized one-loop tadpole term T resulting from the diagrams shown in Fig. 3.1.We demand that these two contributions cancel each other,

TH = δtH + TH = 0, Th = δth + Th = 0, (3.47)

which means that explicit tadpole diagrams can be omitted from the one-loop diagramsfor any process. However, as a remnant of the tadpole diagrams the tadpole counter-terms appear also in various couplings and need to be calculated properly. We note thatthe condition on the tadpoles does affect physical observables, but shifts contributionsbetween Green functions and counterterms and changes the bookkeeping.

Self-energies

The irreducible two-point functions will be intensively used in the formulation of renor-malization conditions. For the scalars with momentum k they are

Γab(k) = a bk

= iδab(k2 −M2

a ) + iΣab(k), (3.48)

where both fields a, b are incoming and a, b = H, h,A0, G0, H±, G±. The first term is

the LO two-point function, while the functions Σab are the renormalized self-energiescontaining loop diagrams and counterterms. Generic diagrams contributing to the self-energies are shown in Fig. 3.2. A mixing between particles is only possible if both

41

H H

S

H H

V

H

H

F

F

H

H

S

S

H

H

U

U

H

H

V

V

H

H

S

V

Figure 3.2.: Generic self-energy diagrams for the neutral Higgs self-energy, for other scalarself-energies the diagrams are analogous. Only massive particles contribute.

external particles have the same quantum numbers, i.e. there is mixing between H andh or A0 and G0 or H± and G±. For the neutral CP-even fields we obtain

Σhh(k2) = Σhh(k2) + δZh(k2 −M2

h)− δM2h , (3.49a)

ΣHH(k2) = ΣHH(k2) + δZH(k2 −M2

H)− δM2H, (3.49b)

ΣHh(k2) = ΣHh(k2) +1

2δZHh(k

2 −M2H) +

1

2δZhH(k

2 −M2h)− δM2

Hh, (3.49c)

and for the CP-odd fields

ΣA0A0(k2) = ΣA0A0(k2) + δZA0(k2 −M2

A0)− δM2

A0, (3.50a)

ΣGG(k2) = ΣGG(k2) + δZGk2 − δM2

G, (3.50b)

ΣA0G(k2) = ΣA0G(k2) +1

2δZA0G(k

2 −M2A0) +

1

2δZGA0 k

2 − δM2A0G

. (3.50c)

The charged sector has the following self-energies

ΣH+H−

(k2) = ΣH+H−

(k2) + δZH±(k2 −M2H±)− δM2

H± , (3.51a)

ΣG+G−

(k2) = ΣG+G−

(k2) + δZG± k2 − δM2G± , (3.51b)

ΣH+G−

(k2) = ΣH+G−

(k2) +1

2δZHG±(k2 −M2

H±) +1

2δZGH± k2 − δM2

HG± , (3.51c)

ΣG+H−

(k2) = ΣH+G−

(k2), (3.51d)

with the particular values of Eq. (3.30) and δM2Hh = 0 for the mass mixing terms (the

other prescription for the mixing angles leads to the terms stated in Eq. (B.26)). Onthese two-point functions we can now impose our renormalization conditions. First, wewant to fix the renormalized mass parameters to the physical values, so that the zerosof the real parts of the one-particle-irreducible two-point functions are located at thesquares of the physical masses:

Re ΓHH(M2H) = Re ΣHH(M2

H) = 0, Re Γhh(M2h) = Re Σhh(M2

h) = 0, (3.52)

42

Re ΓA0A0(M2A0) = Re ΣA0A0(M2

A0) = 0, Re ΓH+H−

(M2H±) = Re ΣH+H−

(M2H±) = 0.

Using Eqs. (3.49), (3.50), and (3.51) this straightforwardly leads to a decomposition interms of counterterm contributions and unrenormalized self-energies. We obtain for themass renormalization constants

δM2H = ReΣHH(M2

H), δM2h = ReΣhh(M2

h ), (3.53)

δM2A0

= ReΣA0A0(M2A0), δM2

H± = ReΣH+H−

(M2H+).

For the propagators of the fields, we demand that the residues are not changed byhigher-order corrections. This will determine the field renormalization constants for thenon-mixing cases. This means for the one-particle-irreducible two-point functions

limk2→M2

H

ReΓHH(k2)

k2 −M2H

= i, limk2→M2

h

ReΓhh(k2)

k2 −M2h

= i, (3.54)

limk2→M2

A0

ReΓA0A0(k2)

k2 −M2A0

= i, limk2→M2

H±

ReΓH+H−

(k2)

k2 −M2H±

= i.

Inserting Eq. (3.48) and using Eq. (3.52) yields for the first condition

ΣHH(k2)

k2 −M2H

∣∣∣∣∣k2→M2

H

= Re∂ΣHH(k2)

∂k2

∣∣∣∣∣k2=M2

H

= Re Σ′HH(M2H) = 0, (3.55)

and similar results for the other conditions. Here we introduced Σ′(k2) as the derivativew.r.t to the argument k2. Rewriting the equations with unrenormalized quantities andinserting the counterterms finally determines the renormalization constants:

δZH = −ReΣ′HH(M2H), δZh = −ReΣ′hh(M2

h), (3.56)

δZA0 = −ReΣ′A0A0(M2A0), δZH+ = −ReΣ′H+H−

(M2H±).

The mixing renormalization constants still remain to be fixed. There we enforce thecondition that on-mass-shell fields do not mix, i.e.

ΓHh(M2H) = ΣHh(M2

H) = 0, ΓHh(M2h ) = ΣHh(M2

h) = 0, (3.57)

ΓA0G(M2A0) = ΣA0G(M2

A0) = 0, ΓH+G−

(M2H±) = ΣH+G−

(M2H±) = 0.

After inserting the renormalized self-energies we obtain

δZHh = −2−δM2

Hh + ReΣHh(M2h)

M2h −M2

H

, δZhH = −2−δM2

Hh + ReΣhH(M2H)

M2H −M2

h

, (3.58)

δZGA0 = −2−δM2

AG + ReΣA0G(M2A0)

M2A0

, δZGH+ = −2−δM2

HG± + ReΣH+G−(M2

H±)

M2H±

.

43

Since on-shell Goldstone bosons do not exist, we do not render Green functions withexternal Goldstone bosons finite, so that we can set the corresponding renormalizationconstants to zero:

δZG = 0, δZA0G = 0, δZG± = 0, δZHG± = 0. (3.59)

The possible ZA0 and W∓H± mixing vanishes for physical on-shell gauge bosons due tothe Lorenz structure of the two point function, and the fact that polarization vectorsare orthogonal to the corresponding momentum,

ǫµZΓZA0µ (k2)

∣∣∣k2=M2

Z

= ǫµZkµΣZA0(k2)

∣∣∣k2=M2

Z

= 0, (3.60)

ǫµWΓH±W∓

µ (k2)∣∣∣k2=M2

W

= ǫµWkµΣH±W∓

(k2)∣∣∣k2=M2

W

= 0. (3.61)

The mixing self-energies on the other on-shell pointsM2A0

andM2H± , respectively, are not

independent and can be calculated from a BRST symmetry [82]. The BRST variationof the Green functions of one anti-ghost and a Higgs particle

δBRST〈0|T uZ(x)A0(y)|0〉 = 0, δBRST〈0|T u±(x)H∓(y)|0〉 = 0, (3.62)

implies Slavnov–Taylor identities. While the variation of the anti-ghost fields yields thegauge-fixing term, the variation of the Higgs fields introduces ghost contributions whichshould not contribute when putting the momentum on shell, so that we get

[

q2ΣZA0(q2) +MZΣA0G(q2)

]

q2=M2A0

= 0, (3.63)

[

q2ΣH±W∓

(q2)± iMWΣH±G∓

(q2)]

q2=M2H±

= 0. (3.64)

We have checked these identities successfully numerically. Together with the renormal-ization condition of Eqs. (3.57) we conclude that ΣZA0(M2

A0) = 0 and ΣH±W∓

(M2H±) = 0.

This set of renormalization conditions on the two-point functions ensures that all on-shell particles do not obtain any one-loop corrections or mix with other particles andthe corresponding self-energy Feynman diagrams do not have to be taken into accountin any calculation.

3.2.1.2. Electroweak Sector

The fixing of the renormalization constants of the electroweak sector is identical to theSM case. The mass renormalization constants are fixed in such a way that the polesof the gauge-boson propagators lie at their physical (experimental) locations. The fieldrenormalization constants are fixed by the condition that on-shell propagators of thegauge bosons do not obtain higher-order corrections and that on-shell bosons do not

44

mix. For a better bookkeeping we also keep the dependent renormalization constantsδcW and δsW in our calculation since they occur very often. This results in

δM2W = ReΣWT (M2

W), δZW = −ReΣ′WT (M2

W), (3.65)

δM2Z = ReΣZZT (M2

Z),

δZZZ = −ReΣ′ZZT (M2

Z), δZAA = −ReΣ′AAT (0),

δZAZ = −2ReΣAZT (M2

Z)

M2Z

, δZZA = 2ReΣAZT (0)

M2Z

,

δcW =cW2

(δM2

W

M2W

− δM2Z

M2Z

)

, δsW = −cWsWδcW.

A detailed derivation of these equations in BHS convention can be found in Ref. [48]whose notation we adopted. The electric charge e is defined as the eeγ coupling in theThomson limit of on-shell external electrons and zero momentum transfer to the photon,

γ

e+, k′

e−, k

∣∣∣∣∣∣∣∣∣∣∣∣k=k′,k2=k′2=m2

e

= ieu(k)γµu(k), (3.66)

where the momentum flow is in the direction of the fermion flow. In on-shell renormal-ization, the corrections to the external legs vanish, so that we obtain

u(k)Γeeγ(k, k′)u(k′)|k=k′ = ieu(k)γµu(k). (3.67)

This yields in BHS convention [48]

δZe∣∣α0

= −1

2(δZAA +

sWcWδZZA). (3.68)

3.2.1.3. Fermions

The renormalization conditions for the fermions are identical to the ones in the SM,described in detail in Ref. [48] (in BHS convention). We demand that the propagatorpoles are at their measured masses, and that on-shell fermion propagators do not obtainloop corrections. As we do not take the CKM matrix into account, our results simplifyto

δmf,i =mf,i

2Re(Σf,Li (m2

f,i) + Σf,Ri (m2f,i) + 2Σf,S(m2

f,i)), (3.69)

δZf,Li = −ReΣf,Li (m2

f,i)−m2f,i

∂

∂k2Re[Σf,Li (k2) + Σf,Ri (k2) + 2Σf,Si (k2)

]∣∣∣k2=m2

f,i

,

δZf,Ri = −ReΣf,Ri (m2

f,i)−m2f,i

∂

∂k2Re[Σf,Li (k2) + Σf,Ri (k2) + 2Σf,Si (k2)

]∣∣∣k2=m2

f,i

.

45

3.2.2. Different Renormalization Schemes

A formulation of an on-shell condition for β, λ5 and the renormalization constant appear-ing in the neutral Higgs mixing, δα or δλ3 is not that clear. Of course, one could relatethe renormalization to some physical process (e.g. Higgs-boson decays), by demandingthat this process does not obtain higher-order corrections. As no measured observableindicates the realization of a THDM so far, there is no distinguished process and sucha prescription could introduce artificially large corrections to these parameters, whichwould spread to many other observables [30, 46]. That is why we chose to renormalizethese parameters within the MS scheme. However, different variables (such as α orλ3) can be chosen to parameterize the model. Imposing an MS condition on either ofthe parameters leads to differences in the calculation of observables. In addition, gaugedependent definitions of parameters which are renormalized in MS spoil the gauge in-variance of the theory. However, small gauge dependencies can be acceptable if therenormalization scheme yields stable results and a good convergence of the perturbationseries. The price to pay is that subsequent calculations should be done in the samegauge. We will discuss different renormalization schemes based on a different treatmentof δβ, the neutral Higgs mixing parameter and the Higgs coupling constant δλ5 in thefollowing. We begin with the so-called α MS scheme.

3.2.2.1. α MS Scheme

In this scheme the independent parameter set is {pmass} of Eq. (2.73) so that the param-eters β, α, and λ5 are renormalized in MS. The corresponding counterterm Lagrangianwas derived in Sect. 3.1.1.3.

The renormalization constant δβ

In the derivation of the UV-divergent part of δβ, one can employ a relation between thecounterterms of the vevs of the two doublets which, however, requires a parameteriza-tion of the counterterm potential as in App. B.1 using the basic parameter set. Whenrenormalizing the vevs of the two doublets (see App. B.1 and Ref. [80]) with

v1,0 = Z1/2H1

(v1 − δv1), v2,0 = Z1/2H2

(v2 − δv2), (3.70)

the counterterms are connected, as shown in Ref. [83], via

δv1/v1 − δv2/v2 = finite. (3.71)

The renormalization constant δβ can be obtained from the renormalization of the ratioof the vacuum expectation values (Eq. (2.65)), and using the previous relation yields inthe MS scheme

δβ =δv1v2v2

− δv2v1v2

− δZH1v1v22v2

+δZH2v1v2

2v2

∣∣∣UV

(3.72)

=cβsβ2

(−δZH1 + δZH2)∣∣∣UV

=cβsβ2

δZh − δZH

c2α − s2α

∣∣∣UV,

46

where |UV denotes that we take only the UV-divergent parts. The explicit calculation ofthe UV-divergent terms of δZh, δZH according to Eqs. (3.56) reveals that only diagramswith closed fermion loops contribute to the counterterm:

δβ = ∆UVe2cβsβ

64π2M2Ws

2W

∑

f

(±cf )m2f (1 + ξfA

2) (3.73)

with a positive (negative) sign for a Yukawa coupling of the respective fermion to Φ1

(Φ2), the color factors cf , the coupling coefficients ξfA as defined in Tab. 2.4 and thestandard divergence ∆UV. In Rξ gauges this result is gauge-invariant at one-loop [31].

Neutral Higgs mixing

In the neutral Higgs sector, relations between field renormalization constants can alsobe used to determine another parameter in MS. The first three equations of Eqs. (3.35)can be solved for δα, yielding

δα∣∣UV

=1

2δZHh

∣∣∣UV

+cαsα

2(c2α − s2α)(δZH − δZh)

∣∣∣UV. (3.74)

The field renormalization constants can be inserted according to Eqs. (3.56),(3.58), witha vanishing mixing counterterm δM2

Hh, thus

δα = −ReΣHh(M2h)

M2h −M2

H

∣∣∣∣UV

− cαsαcβsβ

δβ. (3.75)

An explicit calculation of the counterterm in ’t Hooft-Feynman gauge yields for thefermionic contribution,

δα∣∣ferm

= ∆UVe2s2α

128π2M2Ws

2W(M2

H −M2h)

∑

f

(±cf )(12m4f −m2

f (M2h +M2

H))(1 + ξfA2),

(3.76)

and for the bosonic contribution

δα∣∣bos

= ∆UV

λ25M2Ws

2W

(s2α (23c2β + c6β) + 4s32βc2α

)

8π2e2s22β(3.77)

+ ∆UVλ5

128π2s22β

[

s2αc2β(177M2A0

− 2(54M2h + 54M2

H + 7M2H± +M2

W)−M2Z)

+ c2α(s2β(5M

2A0

− 38M2H± + 6M2

W + 3M2Z)− s6β(7M

2A0

− 2M2H± + 2M2

W +M2Z))

+ s2αc6β(7M2A0

− 2M2H± + 2M2

W +M2Z)− 4s2α(M

2h −M2

H)(2c2(α−2β) + 13c2α

) ]

+∆UV1

1024π2M2Ws

2W

[

2s2αc2β

(

83M4A0

−M2A0(2(52M2

h + 52M2H + 7M2

H± +M2W) +M2

Z)

+ 5c4α(M2h −M2

H)2 + 31M4

h + 8M2H±(M2

h +M2H)

47

A0

A0

HF

F

F

A0

A0

HS

S

S

A0

A0

HS

S

V

A0

A0

HS

V

S

A0

A0

HV

S

S

A0

A0

HS

V

V

A0

A0

HS

S

A0

A0

H

SS

A0

A0

HS

S

A0

A0

HV

V

Figure 3.3.: Generic diagrams contributing to the renormalization of λ5.

+ 58M2hM

2H + 31M4

H − 4M4H± − 6

(2M4

W +M4Z

) )

+ c2α

(

s2β(2M2

A0(8M2

h + 8M2H − 38M2

H± + 6M2W + 3M2

Z)− 2M4A0

+ 32M2H±(M2

h +M2H))

+ s2β(5(M2

h −M2H)

2 + 24M4H± + 36

(2M4

W +M4Z

))

− 2s6β(5M4

A0+M2

A0(−2M2

H± + 2M2W +M2

Z) + 4M4H± + 12M4

W + 6M4Z

) )

+ 2s2αc6β(5M4

A0+M2

A0(−2M2

H± + 2M2W +M2

Z) + 4M4H± + 12M4

W + 6M4Z

)

− 5c6αs2β(M2h −M2

H)2 + s4α(M

2h −M2

H)(−c4β(7M2

A0− 2M2

H± + 2M2W +M2

Z)− 49M2A0

+ 36M2h + 36M2

H + 6M2H± + 2M2

W +M2Z

)

+ s4β(M2h −M2

H)(c4α(7M

2A0

− 2M2H± + 2M2

W +M2Z) +M2

A0+ 18M2

H± − 2M2W −M2

Z

) ]

.

Higgs self-coupling

The Higgs self-coupling parameter δλ5 has to be fixed using a vertex function. Also thisrenormalization constant is defined in MS, as there is no distinguished process to fixit on-shell. Any 3- or 4-point vertex function with external Higgs bosons is suitable tocalculate the divergent terms. After analyzing all different vertices, we conclude thatthe HA0A0 vertex correction involves fewest diagrams, and is the preferred choice. The

48

condition is

ΓHA0A0∣∣UV

=

A0

A0H

∣∣∣∣∣∣∣∣UV

= 0. (3.78)

Solving this equation for δλ5 fixes this renormalization constant. The generic one-loopdiagrams appearing in this vertex correction are shown in Fig. 3.3, the contribution ofthe diagrams involving closed fermion loops is

δλ5,ferm = ∆UVe2λ5

32π2M2Ws

2W

∑

f

(1 + ξfA2)cfm

2f . (3.79)

The diagrams containing only bosons lead to

δλ5,bos =∆UVλ516π2

(2λ1 + 2λ2 + 8λ3 + 12λ4 − 9g22 − 3g21)

)(3.80)

=−∆UV

λ25c22β

4π2s22β+∆UV

λ5e2

256π2M2Ws

2W

[

8M2A0

− 16M2H± − 24(2M2

W +M2Z)

+4

s22β

(2(M2

h +M2H)− 4M2

A0c4β − (M2

h −M2H)(c2(α−β) − 3c2(α+β))

) ]

.

3.2.2.2. λ3 MS Scheme

In this scheme, the independent parameter set is {p′mass} defined in Eq. (2.75). Therenormalization of β and λ5 is identical to the previous renormalization scheme and notstated again, but in this scheme the parameter λ3 is an independent parameter beingrenormalized inMS. This has the advantage that this parameter is gauge invariant as itis a defining parameter of the basic parameterization. The parameter β breaks generallygauge invariance, however, in Rξ gauges the gauge dependence cancels at one-loop [31]so that this scheme yields gauge-invariant results at NLO. We take the countertermpotential of Sect. 3.1.1.3 but treat δα as a dependent counterterm. As α is a puremixing angle, we choose to apply the renormalization prescription of Sect. 3.1.1.3 wherethe mixing angle diagonalizes the potential to all orders. The relation between δα andthe independent constants is given in Eq. (3.24) with δM2

Hh = 0,

δα = −fα({δp′mass})

M2h −M2

H

, (3.81)

where fα({δp′mass}) can be obtained from Eq. (3.8c) by applying the renormalizationtransformation of Eq. (B.22) (which is identical to the renormalization transformationof Sect. 3.1.1.3, but renormalizing λ3 instead of α) which yields

fα({δp′mass}) =1

2t2α(δM2

h − δM2H

)+s2β(δM2

A0− 2δM2

H±

)

2c2α(3.82)

49

+δβc2β (M

2H −M2

h) t2αs2β

+4M2

Wcβsβ(δλ3 + δλ5)s2W

e2c2α

−(s2β(M2

A0− 2M2

H±

)+ (M2

h −M2H)s2α

)(δM2

WsW + 2M2W (δsW − δZesW))

2M2Wc2αsW

− e (δth (cα−3β + 3cα+β) + δtH (sα−3β + 3sα+β))

8MWc2αsW.

The UV-divergent term of α has been calculated in Eq. (3.75) and by renormalizing δλ3in MS scheme, it is clear that the dependent δα must have a finite part in addition. Wechose this finite term in such a way that the finite part in δλ3 vanishes and we obtain

δα∣∣λ3MS

= −ReΣHh(M2h)

M2h −M2

H

∣∣∣∣UV

− cαsαcβsβ

δβ − fα({δp′mass})M2

h −M2H

∣∣∣∣finite

, (3.83)

where δλ3 drops out as it has no finite part. The divergent part of δλ3 can be calculatedby solving Eq. (3.81) and is

δλ3 =[ e2c2α4M2

Ws2Wcβsβ

ReΣHh(M2h )− δλ5 −

e2s2α8M2

Ws2Wcβsβ

(δM2

h − δM2H

)(3.84)

− s2βe2(δM2

A0− 2δM2

H±

)

8M2Ws

2Wcβsβ

− δβe2c2β(c2β + s2α) (M2H −M2

h)

2M2Ws

2Ws

22β

+e2(s2β(M2

A0− 2M2

H±

)+ (M2

h −M2H)s2α

)(δM2

WsW + 2M2W (δsW − δZesW))

8M4Ws

3Wsβcβ

+e3 (δth (cα−3β + 3cα+β) + δtH (sα−3β + 3sα+β))

32M3Ws

3Wcβsβ

]

UV,

although this explicit dependence is not needed in the calculation, as δλ3 enters onlyimplicitly via δα|λ3MS

which we already calculated. The computation of the renormal-ization constant gives for the fermionic contribution

δλ3

∣∣∣UV,ferm

=∆UVe4

256π2M4Ws

4W

[

− 8∑

F

cF (muFm

dF )

2(ξuA + ξdA)2 (3.85)

+∑

f

m2f

cf (1 + ξfA2)

sβcβ

((M2

H −M2h)s2α +

((2M2

H± −M2A0)− 4M2

Ws2Wλ5/e

2)s2β)]

,

with the fermions f and the fermion doublets F . For the bosonic contribution we obtain

δλ3

∣∣∣UV,bos

= ∆UV1

16π2

((λ1 + λ2)(6λ3 + 2λ4) + 4λ23 + 2λ24 + 2λ25 (3.86)

+ 3/4(3g42 + g41 − 2g22g21)− 3λ3(3g

22 + g21)

)

= ∆UVλ25

2π2s22β+∆UV

1

128π2M2Ws

32βs

2W

[

2s2α (c4β + 11) (M2h −M2

H)

50

+ 4c2αs4β(M2H −M2

h)− s6β(M2A0

− 2M2H± + 6M2

W + 3M2Z)

+ s2β(27M2A0

− 8M2h − 8M2

H − 38M2H± + 18M2

W + 9M2Z)]

+∆UVe4

256π2M4Ws

4W

[

1

s22β

(

(M2h −M2

H)(c4α(M

2H −M2

h)− 2c2αc2β(M2A0

− 4M2H±))

+ 4M4A0

− 2M2A0(M2

h +M2H + 8M2

H±) + 8M2H±(M2

h +M2H) + (M2

h −M2H)

2)

+M2

h −M2H

s32β

(

3s4αc2β(M2H −M2

h ) + s2α

(c4β(M

2A0

+ 4M2H± − 6M2

W − 3M2Z) + 11M2

A0− 6M2

h − 6M2H − 4M2

H± + 6M2W + 3M2

Z

) )

+ 4(M4

A0− 2M2

A0M2

H± + 3M2A0M2

W + 3M4H± − 6M2

H±M2W + 9M4

W

)

+ 6M2Z(M

2A0

− 2(M2H± + 2M2

W)) + 6M4Z

]

.

3.2.2.3. The FJ Tadpole Scheme

Since tadpole loop diagrams are gauge dependent as shown for the SM in Ref. [84], pa-rameters that contain such terms in their definition potentially become gauge dependentas well. This holds for the standard definitions of mass parameters and mixing angles.However, for renormalization conditions that are formulated on-shell via a physical quan-tity, the gauge dependence of the bare parameter cancels in all renormalized quantitiesand observables become gauge independent. On the other hand, for parameters renor-malized in MS, the gauge dependence is manifest at NLO which renders the connectionbetween renormalized parameters and observables gauge dependent. In the α MS andthe λ3 MS renormalization schemes, the bare definition of β and α contains implicit andexplicit tadpole terms so that these schemes do not preserve gauge invariance (althoughthe λ3 MS scheme is gauge invariant at NLO in Rξ gauges [31]).Fleischer and Jegerlehner [85] proposed a renormalization scheme for the SM, referredto as the FJ scheme in the following, which preserves gauge invariance for all bare pa-rameters, including the masses. In this scheme, the parameters are defined in such away that tadpole terms do not enter in the definition of any parameter so that they aregauge independent. This can be easily achieved by demanding the bare tadpole termsto vanish at all orders,

t0,i = 0, (3.87)

for all fields i with the quantum numbers of the vacuum. Since tadpole conditions haveno effect on physical observables and change only the bookkeeping such a procedureis possible. The disadvantage is that tadpole diagrams have to be taken explicitly intoaccount for all higher-order calculations. In particular, the one-particle reducible tadpolecontributions destroy the simple relation of propagators being the inverse two-pointfunction. In the SM, the FJ scheme does not affect observables, as all parameters are

51

renormalized using on-shell conditions. A gauge-invariant renormalization scheme for theTHDM can be defined by applying the FJ prescription and imposing the MS conditionon gauge-invariant mixing angles [30, 31]. The bare physical parameters defined in theFJ scheme (we omit the index 0 here),

{pFJ,mass} = {MH,FJ,Mh,FJ,MA0 ,FJ,MH± ,FJ,MW,FJ,MZ,FJ, eFJ, λ5,FJ, αFJ, βFJ}, (3.88)

differ by divergent and finite NLO tadpole contributions from the gauge-dependent def-inition of these bare parameters {pmass} given in Eq. (3.15). The only exception is theparameter λ5 which is a parameter of the basic potential and therefore gauge indepen-dent by construction. The renormalization of this parameter in MS is identical to theone in the previous schemes. In the FJ scheme of Ref. [31] a Lagrangian in terms of{pFJ,mass} is used, and additionally the tadpole counterterms are reintroduced by shift-ing the Higgs fields H → H + ∆vH and h → h + ∆vh. This affects the form of thecounterterm Lagrangian and the definition of the renormalization constants so that theformulae given in Eq. (3.33) and Sect.3.2.1 cannot be applied.In contrast, we keep the dependence of the Lagrangian in terms of gauge-dependentmasses and couplings in our implementation of the FJ scheme, as they are renormalizedusing on-shell conditions. In addition we employ the tadpole renormalization conditionof Eq. (3.47) so that the definitions of the renormalization constants of the on-shellparameters according to Sect. 3.2.1 remain valid. The mixing angles, which are renor-malized using an MS condition, are defined in a gauge-independent way using the FJprescription. Thus our bare parameters (without the index 0) describing the Lagrangianare

{pFJ′,mass} = {MH,Mh,MA0 ,MH±,MW,MZ, e, λ5, αFJ, βFJ, tH, th}. (3.89)

With such a parameterization we can take the counterterm Lagrangian of Eq. (3.33),and use the relation between the gauge-dependent and -independent definitions of theparameters,

δα = δαFJ + a(TH, Th), (3.90)

δβ = δβFJ + b(TH, Th), (3.91)

to replace α and β. The functions a and b depend on the one-loop tadpole contribu-tions TH and Th. In the following, we derive the explicit functional dependence of aand b. To this end, we compare the definition of the renormalization constants in the αMS scheme to the gauge-independent ones obtained via the FJ prescriptions. There, alltadpole counterterms vanish but explicit tadpole diagrams have to be taken into account.

The renormalization constant δβFJ

We impose an MS condition on the independent renormalization constant δβFJ. Toobtain the dependence of the counterterm Lagrangian on this renormalization constantwe compute b(TH, Th) by comparing analytically the UV-divergent parts of δβ and δβFJ.We begin with the computation of δβ in the α MS scheme. Relation (3.71) cannot

52

be applied in the FJ scheme and therefore we calculate the counterterm in the α MSscheme from the field renormalization constant by employing the last two equations ofEqs. (3.35). This gives

δβ|UV =δZGA0|UV − δZA0G|UV

4= −ReΣA0G(M2

A0) + ReΣA0G(0)− 2δM2

AG

2M2A0

∣∣∣∣UV

, (3.92)

with δM2AG as in Eq. (3.30), and δZGA0 from Eq. (3.58). The non-vanishing tadpole

counterterms in the α MS scheme are δti = −Ti, and

δZA0G = 2−δM2

AG + ReΣA0G(0)

M2A0

. (3.93)

In the FJ scheme the tadpole counterterms vanish and the same computation yields

δβFJ|UV =δZGA0 |UV − δZA0G|UV

4= −ReΣt,A0G(M2

A0) + ReΣt,A0G(0)

2M2A0

∣∣∣∣UV

, (3.94)

where the superscript “t” indicates that one-particle-reducible tadpole diagrams areincluded in the self-energies in addition to the one-particle-irreducible diagrams. Thedifference of Eq. (3.92) and Eq. (3.94) gives the functional dependence b(TH, Th) on thetadpole terms, which is identical for UV-divergent and finite parts. We obtain

b(TH, Th) =− ReΣA0G(M2A0) + ReΣA0G(0)− 2δM2

AG

2M2A0

+ReΣt,A0G(M2

A0) + ReΣt,A0G(0)

2M2A0

=δM2

AG

M2A0

+1

M2A0

(

hA0 G0

+ HA0 G0

)

=1

M2A0

(

eThcα−β + THsα−β

2MWsW+ eTh

(M2

A0−M2

h

)cα−β

2sWMWM2h

+ eTH

(M2

A0−M2

H

)sα−β

2sWMWM2H

)

=e

2sWMW

(

Thcα−βM2

h

+ THsα−βM2

H

)

(3.95)

which is in agreement with Ref. [31]. The gauge-dependent β can be calculated to

δβ∣∣βFJMS

=cβsβ2

δZh − δZH

c2α − s2α

∣∣∣∣UV

+e

2sWMW

(

Thcα−βM2

h

+ THsα−βM2

H

)∣∣∣∣finite

. (3.96)

The renormalization constant δαFJ

We apply the same method to the renormalization constant δαFJ, impose an MS con-dition, and compute a(TH, Th) by comparing the UV-divergent parts of δα in the α MSscheme of Eq. (3.74) with δαFJ. The latter can be derived from the same renormalizationcondition, but with explicit tadpoles taken into account,

δαFJ

∣∣UV

= −ReΣt,Hh(M2h)

M2h −M2

H

∣∣∣UV

+cαsα

2(c2α − s2α)(δZH − δZh)

∣∣∣UV

53

The renormalization constants δZH and δZh are identical to the ones of the α MSrenormalization scheme as they are defined as derivatives and do not contain tadpoleterms. Therefore we can omit the superscript in the following. The difference of thisresult and Eq. (3.74) gives the functional dependence of a on the tadpoles,

a(TH, Th) =− ReΣHh(M2h)

M2h −M2

H

+cαsα

2(c2α − s2α)(δZH − δZh) (3.97)

+ReΣt,Hh(M2

h)

M2h −M2

H

− cαsα2(c2α − s2α)

(δZH − δZh)

=1

M2h −M2

H

(

Hh H

+ hh H

)

=1

M2h −M2

H

(eThChhH

M2h

+eTHChHH

M2H

)

with the coupling factors Cijk defined in App. A. This result is again in agreement withRef. [31]. We can absorb the net-effect of the FJ prescription into a finite part of thegauge-dependent δα, and get

δα∣∣αFJMS

= − ReΣHh(M2h)

M2h −M2

H

∣∣∣∣UV

− cαsαcβsβ

δβ

∣∣∣∣UV

+1

M2h −M2

H

(eThChhH

M2h

+eTHChHH

M2H

)∣∣∣∣finite

.

(3.98)

Using the gauge-dependent parameters δα and δβ as an abbreviation, one can keep theparameterization of the counterterm Lagrangian of Eq. (3.33). With this implementa-tion of the FJ scheme, all calculations can be performed in the same way as for otherrenormalization schemes, except that the finite contributions of δα and δβ according toEqs. (3.98) and (3.96) need to be taken into account.

3.2.2.4. The FJ λ3 Scheme

In the λ3 MS scheme, the parameters λ3,5 are defining parameters of the basic para-meterization and gauge invariant by construction. Therefore, the condition on δβ is theonly renormalization condition violating gauge invariance. To provide a gauge-invariantrenormalization scheme where λ3 is an independent quantity, we apply the FJ schemeto the parameter β and keep the renormalization of λ3,5 as in the λ3 MS to obtain theFJ λ3 scheme. The renormalization of the parameters read:

δβ as in Eq. (3.96),

δλ3 as in Eq. (3.84).

3.2.3. Comparing Different Renormalization Schemes

In the previous section, we presented four different renormalization schemes, which treatthe mixing parameters differently. When observables calculated with different renormal-ization schemes are compared, particular care has to be taken that the input values corre-spond to each other. The bare values of identical (and in some renormalization schemes

54

dependent) parameters are equal and independent of the renormalization scheme. Ex-emplarily, for a parameter c in two different renormalization schemes 1 and 2,

c0 = c|1 + δc|1(c|1) = c|2 + δc|2(c|2) (3.99)

must hold for the considered order. For the UV-divergent terms this equation holds byconstruction, so that only the finite terms are different. If c is a dependent parameterin one or both schemes, it must be calculated from the independent renormalized para-meters and their counterterms from the tree-level and NLO relations. For converting aninput value from one scheme to another, one can solve for one renormalized quantity

c|1 = c|2 + δc|2(c|2)− δc|1(c|1). (3.100)

This equation can be solved approximately by inserting the input value of c2 in thecomputation of the last counterterm. The differences to a exact solution are of higherorder and beyond our desired accuracy. However, large counterterms or small tree-levelvalues can spoil the approximation so that in this case a proper solution using numericaltechniques could improve results.We derived the Higgs mixing angle α and its counterterm in all schemes and obtainedfor the finite terms:

δα∣∣αMS ,finite

=0, (3.101a)

δα∣∣λ3MS

,finite=− fα{δR}

M2h −M2

H

∣∣∣finite

, (3.101b)

δα∣∣αFJMS ,finite

=a(TH, Th)∣∣finite

=1

M2h −M2

H

(eThChhH

M2h

+eTHChHH

M2H

)∣∣∣∣finite

. (3.101c)

For the angle β we obtained in the MS and the FJ scheme the following finite terms,

δβ∣∣βMS ,finite

=0, (3.102a)

δβ∣∣βFJMS

,finite=b(TH, Th)

∣∣finite

=e

2sWMW

(

Thcα−βM2

h

+ THsα−βM2

H

)∣∣∣∣finite

. (3.102b)

With these formulae we can transform the input variables easily into each other, and wesummarize the transformations in Tab. 3.3, based on an input in the α MS scheme. Theconverted input value of the independent parameter can be obtained from renormalizeddependent input parameters by tree-level relations. Defining the input values in each ofthe four renormalization schemes corresponds to a different physical scenario and resultsin four different input prescriptions.

3.3. The Running of the Couplings

Parameters renormalized in the MS scheme depend on an unphysical renormalizationscale. In the following, we derive the renormalization group equations, which govern

55

Scheme i α|i β|i

λ3 MS α|αMS+ fα{δp′mass}

M2h−M2

H

∣∣∣finite

β|βMS

FJ α|αMS− a(TH, Th)

∣∣finite

β|βMS− b(TH, Th)

∣∣finite

FJ λ3 α|αMS+ fα{δp′mass}

M2h−M2

H

∣∣∣finite

β|βMS− b(TH, Th)

∣∣finite

Table 3.3.: The conversion of the input values of α, β in the different renormalization schemes,based on an input defined in the α MS scheme.

the scale dependence of those parameters, and sketch their solution for the differentrenormalization schemes.In order to keep the coupling constants dimensionless in a D-dimensional calculation,one introduces a correction factor which keeps the dimension of all one-loop integralsindependent of D. This leads to the substitution

∫d4q

(2π)4I → (2πµ)4−D

∫dDq

(2π)4I, (3.103)

of the momentum integral with the integrand I in dimensional regularization. Theparameter µ is an arbitrary constant with the dimension of a mass. At the one-looplevel, the result of the integration can be expanded in powers of ǫ and is generally of theform

(2πµ)4−D∫

dDq

(2π)4I =(2πµ)2ǫ

[1

ǫA+B +O(ǫ)

]

(3.104)

=(1 + ǫ ln (4π2µ2) +O(ǫ2)

)[1

ǫA +B +O(ǫ)

]

=

[1

ǫ+ ln 4π2 + lnµ2

]

A+B +O(ǫ).

Thus, the scale µ appears always in combination with the divergence 1/ǫ. It is possibleto define all renormalization constants by means of the combination 1/ǫ + lnµ2 whichensures that not only the divergences cancel, but also the scale µ drops out of thecalculation. Its numerical value is therefore completely arbitrary and does not affectobservables. However, as logarithms with a dimensionful argument cannot be evaluatedit is convenient to introduce another scale, the renormalization scale µR, so that anyloop integral has the form

[1

ǫ+ ln 4π2 + ln

µ2

µ2R

]

A+BR(µ2R) +O(ǫ). (3.105)

The explicit dependence of µR in the first term is compensated by a similar term withan opposite sign in BR (canceling there the mass dimension of another logarithm) so

56

that the full integrals are independent of the renormalization scale. The renormalizationconstants defined in theMS scheme are designed in such a way that the divergences andthe µ dependent terms cancel, but do not contain other finite terms. Thus, the generalMS counterterm of a parameter p0 = p+ δp is

δp ∝[1

ǫ+ ln

µ2

µ2R

]

=

[1

ǫ+ ln 4π − γe + ln

µ2

µ2R

]

=

[

∆UV + lnµ2

µ2R

]

, (3.106)

where the proportionality constant is the coefficient of the divergence. The renormal-ization scale (with the convention µ2

R = eγeµ2R/(4π)) is introduced to keep the logarithm

dimensionless. In all practical calculations the arbitrary scale µ is set to µR so that thelogarithm in the definition of the renormalization constants vanishes, but for the deriva-tion of the scale dependence it is more transparent to distinguish between those scales.With an MS definition of the counterterms the dependence of the renormalization con-stants on the renormalization scale becomes explicit as the second contribution of µRappearing in loop integrals is omitted. However, observables and bare quantities cannotdepend on an artificial scale which we introduced by hand, no matter which renormal-ization scheme is chosen. Therefore, renormalized quantities must compensate the scaledependence of the counterterms, leading to the RGE for a parameter renormalized inMS. It reads for a parameter p

d

d lnµ2R

p0 =∂

∂ lnµ2R

p+∂

∂ lnµ2R

δp = 0, (3.107)

from which follows

∂

∂ lnµ2R

p = − ∂

∂ lnµ2R

δp. (3.108)

As the renormalized parameter must compensate the dependence of the countertermon the renormalization scale, it becomes itself scale dependent and a running parame-ter. The derivative with respect to the renormalization scale can be substituted by aderivative w.r.t. the standard UV divergence yielding the β-function

βp(µ2R) =

∂

∂ lnµ2R

p(µ2R) =

∂

∂∆UV

δp. (3.109)

Since the renormalization constants are computed in a perturbative manner, the β-functions have also a perturbative expansion in the coupling parameters.Parameters renormalized in the on-shell scheme are not dependent on the scale as therenormalization constants are defined via a physical process and by means of Green func-tions. These contain complete loop integrals and not only their divergent contribution,so that the residual dependence on the renormalization scale vanishes. In THDM, theratio of the vevs, one Higgs mixing parameter, and one Higgs self-coupling are renor-malized in the MS scheme. However, in Sect. 3.2.2 we described four renormalizationschemes which take different parameters as independent variables renormalized in MS.

57

Scheme α MS λ3 MS FJ FJ λ3

indep. variables α, β, λ5 λ3, β, λ5 αFJ, βFJ, λ5 λ3, βFJ, λ5

β-functions βα, ββ, βλ5 βλ3 , ββ, βλ5 βαFJ, ββFJ

, βλ5 βλ3 , ββFJ, βλ5

compute indep. λ3 = λ3(α) βFJ = β, λ3 = λ3(α),

parameters αFJ = α βFJ = β

Solve the RGEs for the independent variables

α(µ2), α(µ2) = α(λ3(µ2)), α(µ2) = αFJ(µ

2), α(µ2) = α(λ3(µ2)),

input at µ2 β(µ2), β(µ2), β(µ2) = βFJ(µ2), β(µ2) = βFJ(µ

2),

λ5(µ2) λ5(µ

2) λ5(µ2) λ5(µ

2)

Table 3.4.: The independent variables, the respective β-functions, and the algorithmic stepsto solve the RGEs and obtain values for the (dependent) input parameters α, β, λ5 at thedesired scale for the different renormalization schemes.

For each renormalization scheme, one obtains a set of coupled RGEs involving the β-functions of the independent parameters. Therefore the scale dependence varies whendifferent schemes are applied. In the perturbative expansion of the beta function weconsider only the one-loop term, being second order in the coupling constants,

βp(µ2R) = Aα2

em +Bαemλ5 + Cλ25. (3.110)

The dependence on the strong coupling constant vanishes at one-loop as the parametersrenormalized inMS appear only in couplings of particles which do not interact strongly.The coefficients A,B,C of the respective renormalized parameter can be easily read offfrom the divergent terms which are derived in the previous section and we checked themagainst the β-functions given for λ3,5 in Ref. [67] and for β (supersymmetric contribu-tions need to be omitted) in Ref. [83].Since the RGEs are complicated coupled differential equations whose analytic solutionis not possible, except for the one for the parameter β in Type 1 models in the α MSscheme, as in this case ββ decouples from the other scale dependent parameters. Oth-erwise numerical techniques, e.g. a Runge–Kutta method need to be employed to solvethe RGEs and to compute the value of the parameters at a desired scale. However, weparameterized the Lagrangian in all renormalization schemes by the (dependent) param-eters α and β and computed their finite parts (see Sect.3.2.2) so that the independentparameters like λ3, αFJ, and βFJ do not appear in the Lagrangian and in the relevantβ-functions. Before the RGEs can be solved, the independent renormalized parametersmust be computed from the renormalized α, β via their tree-level relations. To obtainthe (dependent) α, β at the desired scale after the RGEs are solved, one can computethem from the independent variables at the respective scale again via tree–level relations.The steps to solve the RGEs for the input parameters α, β, λ5 are schematically shownin Tab. 3.4 for the different renormalization schemes. For the λ3 MS scheme Eq. (2.69c)can be used for the parameter transformations, while the angles in the FJ scheme differ

58

from α and β only at NLO, so that no transformation for the renormalized parameteris necessary.

3.4. Implementation into a FeynArts Model File

The tree-level and the counterterm Lagrangian as well as the renormalization conditionsin the α MS and λ3 MS schemes have been implemented into a model file for the am-plitude generator FeynArts (FA) [33]. With such a model file, NLO amplitudes forprocesses can be generated in an automatized way, with a guaranteed cancellation of theUV divergences. The Mathematica [86] package FeynRules (FR) [32] is a tool toproduce such a model file, since it calculates Feynman rules from a given Lagrangian andprovides a possibility for FA model files as output format. FR provides also possibilitiesto create model files for many other amplitude generators of which we, however, do notmake use in this thesis. With a new feature, also counterterm Lagrangians can be treatedin FR in the correct way by separating tree and counterterm vertices in the description,as FA demands, and by treating the counterterm two-point functions as additional ver-tices and not as a part of the propagator. Beside the Lagrangian, parameters, gaugegroups, fields, and the transformation properties under gauge transformations have tobe specified. The Lagrangian, except for the term of the Higgs potential, is directly in-serted into FR in the internal notation, and the corresponding counterterm Lagrangianis obtained by the renormalization transformations. The Higgs potential and the corre-sponding counterterm potential of Eq. (3.33) depending on physical fields, parameters,and renormalization constants are derived from the general one (Eq. (2.51)) with Math-

ematica and the result is given to FR as a part of the Lagrangian. We use the formof the counterterm potential from method 1a (see Sect. 3.1.1), in which all field renor-malization constants are independent, so that on-shell renormalization is possible for allfields.As an output of FR, we get a complete NLO model file for the THDM with the followingfeatures:

• Type I, II, flipped or lepton specific THDM;

• all tree-level and counterterm Feynman rules including four-point Higgs self-inter-actions;

• renormalization conditions according to the α MS and λ3 MS schemes;

• all renormalization constants are implemented additionally in MS scheme as well,which allows for fast checks of UV-finiteness;

• BHS and HK conventions.

This model file has been tested intensively, and we have also checked UV-finiteness forseveral processes, both numerically and analytically. This allows for the generation (andprocessing with FormCalc [34]) of amplitudes for any process at one-loop level, at anyparameter point of the THDM.

59

4. Higgs-Boson Decay into Four

Fermions

In a first application of our renormalization procedure, we compute the decay of thelight CP-even Higgs boson of the THDM via gauge bosons into four fermions. Withinthe THDM, this Higgs boson can be identified with the discovered scalar resonance at125 GeV, and deviations from the expectation of corresponding decay observables in theSM are of particular interest for the identification of the nature of the Higgs boson. Atfirst in Sect. 4.1, the experimental setup and the relevance of this process is discussed,followed by a theoretical description (Sect. 4.2). A survey of the relevant diagrams andthe differences to the SM Higgs-boson decay are given in Sect. 4.3, and the formulafor the total h→4f width is derived in Sect. 4.4. The Gµ scheme and the complex-mass scheme, which are needed in the calculation to provide accurate predictions, arepresented in Sects. 4.5 and 4.6.

4.1. Experimental Setup

The Higgs-boson decay into electroweak gauge-boson pairs which subsequently decayinto four fermions was a crucial process in the discovery of the Higgs boson, plays animportant part in the determination of the properties of the discovered particle, andmay provide a window to BSM physics. Measurements of Higgs physics related observ-ables in these decay channels provide significant contributions to the current precision,and deviations from the SM expectations could give hints to new physics and underlying

BR [%]

bb 57.7 ± 1.90

WW 21.5 ± 0.91

gg 8.57 ± 0.86

τ τ 6.32 ± 0.36

cc 2.91 ± 0.35

ZZ 2.64 ± 0.11

γγ 0.23 ± 0.011

Zγ 0.15 ± 0.014

Table 4.1.: Predicted SM branching ratios and their total uncertainties for the Higgs decaysat MH = 125 GeV as given in Ref. [87].

60

structures. The predicted branching ratios (BR) of a SM Higgs boson with a mass of 125GeV are shown in Tab. 4.1. Not only a high signal rate is important for experimental ob-servations, but also the signal selection efficiency and the achieved signal-to-backgroundratio.At hadron colliders, it is easier to identify leptonic final states than hadronic ones as theformer leave a clear track in the detector and deposit their energy in the electromag-netic or muon calorimeter. In contrast, the quarks in the final states hardonize and formjets consisting of several hadrons, leaving multiple tracks and energy deposits in thedetector. Therefore the trigger efficiency and the energy resolution for leptons is better.Furthermore, one can suppress multi-jet background easily in leptonic final states andit is not surprising that the h→ZZ→4e/2e2µ/4µ decay cascade with a fully leptonicfinal state was the “golden channel” for the Higgs-boson discovery. The results of themost recent analyses using only these final states were the most significant Higgs bosonobservations, even though the BR of both, h→ZZ, and Z→ll is small. Based on thedata of the first run of the LHC, a background-only hypothesis could be excluded by6.8σ (CMS [88]) and 8.1σ (ATLAS [89]) which is already enough to claim the discoveryof a new particle.Searching for the Higgs boson in the decay into a W-boson pair involves more difficulties.It has a larger rate, but the reconstruction of the final state is incomplete due to themissing transverse energy of the neutrino pair and no mass resonance can be seen ontop of the W-pair continuum distribution. Therefore, the separation of the signal fromthe W-pair production is more delicate and the significance is not as large as for the ZZdecay. The most recent results from Run I are 4.3σ (CMS [90]) and 6.1σ (ATLAS [91]).In contrast, the final state with the largest BR, bb, suffers from the overwhelming back-ground from QCD processes which occur in a large rate at hadron colliders like theLHC. This makes the separation of the signal from the background very challenging anddifficult. Experimental analyses need an additional signature from the Higgs productionmode such as the remnants of a W and Z boson or a top quark for the identification andthe separation from pure QCD processes. In addition, many analyses focus on boostedscenarios to increase the signal-to-background ratio. Nevertheless, they achieve muchsmaller sensitivity and the evidence for a Higgs boson in Run I bb analyses is only abouttwo sigma or less [92–96]. From the other final states, γγ achieves a significance almostcomparable to the ZZ→4l final state, a detailed overview over the results from the firstrun of the LHC is presented in Ref. [8].A Higgs decay into vector-boson pairs plays also an important role in the measurementof Higgs properties. Mass [98], width [99–102], couplings [89, 103], and spin/parity mea-surements [104, 105] are based on WW/ZZ decay measurements and partially on theirdifferential distributions [106].In experiment, the production and the decay of the Higgs boson cannot be separated, sothat instead of the partial widths and couplings, signal strengths for the different decaymodes can be measured. They offer a quick overview whether the experimental resultsare in agreement with the SM predictions. The signal strength is defined as the totalcollected signal rate normalized to the SM expectation. The decay signal strength iscalculated under the assumption that all production modes are equal to the SM while

61

the production signal strength is defined analogously. The combined results of the decaysignal strengths from the first run are shown in Fig. 4.1 revealing a slight excess of mea-sured events, especially for the vector-boson decay channels. This could originate fromBSM physics, or from statistical fluctuations as the SM prediction deviates from theexperimental measurement by less than one sigma. With the data collected currently atRun II, the experimental collaborations will provide a clearer picture in the near future.Interpreting deviations in the signal strengths in terms of coupling factors is non-trivial,as the SM predicts the Higgs couplings and manual changes of the coupling factorsviolate the consistency of the quantum field theory. However, a phenomenologically mo-tivated analysis is based on a rescaling of all appearing SM couplings with the couplingfactors κj in the production cross section σi, the decay width Γf and the total widthΓH ,

σ(i→ H → f) =σi(κj)Γf (κj)

ΓH(κj). (4.1)

Such a method can be used in the search for relatively small deviations from the SM, in-dependent of the (unknown) structure of the underlying theory. With the above formulaone can calculate an expectation for a particular set of couplings and these expectationscan be fitted to the data with a maximum likelihood method. The current fit of theATLAS collaboration is shown in Fig. 4.2 where the set of rescaling factors has beenreduced to one for the coupling to vector bosons κV and one for the coupling to fermions

Figure 4.1.: The combination of the ATLAS and CMS measurements of the signal strengthsfor the different Higgs decays [97]. The signal strengths show the total collected signal ratesnormalized to the SM expectation.

62

Figure 4.2.: The current fit of the Higgs coupling to fermions and vector bosons [107]. TheSM prediction agrees with the experimental measurement within the 68% CL, but a slightlylarger coupling to the vector bosons is favored.

Figure 4.3.: Constraints on different Higgs coupling rescaling factors (relative to the SMvalues) under two different physics assumptions [97]: First, no additional Higgs decay ispossible, as in our THDM scenario (yellow). Second, BSM decays are possible, but couplingsto vector bosons cannot be enhanced w.r.t to the SM (black).

63

κF [107]. The SM lies within the best fit region, however, deviations up to 20% in thevector-boson couplings cannot be excluded with more than one sigma.A result involving more Higgs couplings, additional assumptions on the physics, anddata from ATLAS and CMS are displayed in Fig. 4.3. The yellow bands correspondto physics where no additional decay channel of the Higgs boson is allowed and ourinvestigated THDM scenario, where all other Higgs bosons are heavier, matches theseassumptions. The black bands are not relevant for our THDM scenarios as these bandsare obtained under the assumption that the couplings to vector bosons are not enhancedand additional decay channels exist. In neither of the analyses a significant deviationfrom the SM expectation or an indication for new physics could be found [97].To summarize, the decay of the Higgs boson into four fermions is not only an importantchannel in the test of the SM, but is also a window for the search for BSM physics asnew contributions could change the partial widths, the branching ratios and the shape ofthe distributions. The current experimental results mainly agree with the SM predictionwithin some 10%. Therefore the impact of BSM physics must be sufficiently small, butthe measured deviations from the SM leave still space for BSM effects. With the datataken currently at Run II, all measurements will be improved in the near future andstatistical fluctuations will be reduced.

4.2. Theoretical Considerations

The precise measurements at the LHC, described in the previous section, demand forhigh precision in the theoretical predictions in order to compare the experimental resultswith the theoretical predictions. Beside the (partial) widths as the main (theoretical)observables of the Higgs decay into four fermions, also differential distributions of theHiggs decay into four fermions are of interest. In particular the measurement of theCP properties rely on angular correlations [108–114] and recent analyses make use ofthem [104, 105]. In addition, distributions could be used to investigate the Lorentzstructure of the HV V coupling as momentum dependent BSM couplings can change theshapes w.r.t. the SM. As the differential distributions can be distorted significantly byhigher-order effects such as photon radiation, higher-order corrections are mandatory toprovide reliable predictions. For the SM, several calculations including off-shell effectshave been made [115–126], however the first full next-to-leading order prediction waspresented in Refs. [36–38] including a program called Prophecy4f 1. But not only forSM predictions theoretical accuracy is needed, a similar precision is also demanded forthe interpretation of small deviations from the SM in the context of BSM models. Inthis section we outline the calculation of the decay h→4f at tree level and at NLO. Wekeep the structure generic in order to cover the SM and the THDM case and follow thedescription of the SM calculation [36]. For simplicity, all formulae in this section aregiven with real gauge-boson masses, even though an application of the complex massscheme is mandatory and performed in Sect. 4.6.

1The code is available at http://omnibus.uni-freiburg.de/sd565/programs/prophecy4f/prophecy4f.html

64

W+, Z

W−, Z

h

f4(k4, σ4)

f3(k3, σ3)

f2(k2, σ2)

f1(k1, σ1)

(a)

Z

Z

h

f4(k4, σ4)

f3(k3, σ3)

f2(k2, σ2)

f1(k1, σ1)

(b)

Figure 4.4.: Tree level diagrams of the decay h→4f . The diagram on the right hand sideexists only if the fermion pairs are quarks or leptons belonging to the same generation. Theordering of the fermions is in such a way, that the W+ is always decaying into the first twofermions and the crossed diagram involves always Z bosons.

4.2.1. Tree Level

The generic leading-order diagrams are shown in Fig. 4.4 and consist of a decay ofthe Higgs boson into a pair of gauge bosons (off-shell) which subsequently decay intofermions. A second diagram exists only if both fermion pairs are quarks or leptonsbelonging to the same generation and it can be obtained from the first diagram byexchanging the anti-fermions. As we are interested in the decay of the discovered bosonat 125 GeV (whatever kind of Higgs boson it is), all fermions but a top quark can occurin the final state. The small fermion masses of all the external fermions are neglected asthe energy scales appearing in this process are much larger. Using this approximation,the diagrams involving a direct coupling of the Higgs to the external fermions vanishand do not have to be taken into account. Therefore the diagrams shown in Fig. 4.4 arethe only ones at tree level.The possible final states can be separated into four classes by the vector bosons appearingand the possible interference diagrams. For this classification, we denote fermions withf , their isospin partner with f ′, and different fermions with F 6= f as in the notation ofRef. [36]. We then distinguish between:

neutral current

H → f fF F : As all four particles in the final state are different and as they are com-posed of two pairs of fermion–anti-fermions only the left diagram of Fig. 4.4 withan intermediate Z boson contributes,

Mσ1,σ2,σ3,σ4(k1, k2, k3, k4) = MZZ,σ1,σ2,σ3,σ4(k1, k2, k3, k4). (4.2)

neutral current with interference

H → f ff f : Both pairs of fermions are identical so that both diagrams contribute.As the second diagram can be obtained from the first one by interchanging two

65

leptonic semi-leptonic hadronic

neutral current

νeνeνµνµ(3) νeνeuu(6) uucc(1)

e−e+µ−µ+(3) νeνedd(6) ddss(1)

νeνeµ−µ+(6) e−e+uu(6) uuss(2)

e−e+dd(6)

neutral current with interferencee−e+e−e+(3) uuuu(2)

νeνeνeνe(3) dddd(2)

charged current νee+µ−νµ(6) νee

+du(12) udsc(2)

charged and neutral current νee+e−νe(3) uddu(2)

Table 4.2.: The possible final states for the decay H → WW/ZZ → 4f . They can bedistinguished by the intermediate gauge boson, and the number of lepton pairs. Finalstates which differ only by generation indices, but have the same diagrams have an identicalcalculation and are only stated once. The multiplicity of a final state obtained by changingthe generation indices is given in parenthesis.

external anti-fermion lines, it receives a sign change,

Mσ1,σ2,σ3,σ4(k1, k2, k3, k4) =MZZ,σ1,σ2,σ3,σ4(k1, k2, k3, k4) (4.3)

−MZZ,σ1,σ4,σ3,σ2(k1, k4, k3, k2)

charged current

H → f f ′F ′F : The fermion pairs have a non-vanishing charge, but all four fermions aredifferent so that only the left diagram of Fig. 4.4 with an intermediate W bosoncontributes,

Mσ1,σ2,σ3,σ4(k1, k2, k3, k4) = MWW,σ1,σ2,σ3,σ4(k1, k2, k3, k4). (4.4)

charged and neutral current

H → f f ′f ′f : The final state consists of a fermion, its isospin partner and their anti-particles so that both Z and W boson exchange is possible,

Mσ1,σ2,σ3,σ4(k1, k2, k3, k4) =MWW,σ1,σ2,σ3,σ4(k1, k2, k3, k4) (4.5)

−MZZ,σ1,σ4,σ3,σ2(k1, k4, k3, k2).

In addition one can separate between leptonic, semi-leptonic, and hadronic final states,depending on the number of lepton pairs. This results in 19 different final states, listedin Tab. 4.2. In this counting, final states which differ only by generation indices but haveidentical diagrams, such as a µ+µ−µ+µ− and e+e−e+e− are only counted once as thecalculation is identical due to the negligence of the masses. The fermion pairs are orderedin a canonical way so that the W+ is always decaying into the first two fermions. If

66

δc1c2

δc3c4

δc1c2

δc3c4

(a)

δc1c2

δc3c4

δc1c4

δc2c3

(b)

Figure 4.5.: The color structure of squared amplitudes. The squared diagram on the l.h.sobtains a color factor of 3 (1) for each closed quark (lepton) line, while for fully hadronicfinal states the interference term on the r.h.s gets a color factor of 3 in total as there is onlyone closed quark line.

possible, up-type fermions are ordered before their anti-fermion or corresponding down-type partner. Except for final states consisting of two fermion–anti-fermion pairs, whichhave the same isospin, this determines the numbering of the fermions uniquely (in thelatter cases we have chosen the ordering freely). Using the Feynman rules one cancalculate the generic Born matrix element belonging to Fig. 4.4(a) easily to

MV V,σ1,σ2,σ3,σ4born (k1, k2, k3, k4) =ie3ChV V

1

(k1 + k2)2 −M2V

1

(k3 + k4)2 −M2V

(4.6)

u(k1, σ1)γµ(C+V f1f2

ω+ + C−V f1f2

ω−)v(k2, σ2)

u(k3, σ3)γµ(C+

V f3f4ω+ + C−

V f3f4ω−)v(k4, σ4),

with the generic Higgs–vector-boson–vector-boson coupling ChV V and the generic cou-plings of vector boson to chiral fermions C±

V ff given in Ref. [48] (SM) and App. A(THDM). For quarks in the final state the matrix elements depend on the color indicesci of the external fermions,

MV V,σ1,σ2,σ3,σ4c1,...,c4,born

(k1, k2, k3, k4) = MV V,σ1,σ2,σ3,σ4born (k1, k2, k3, k4)δc1c2δc3c4 . (4.7)

The second diagram, Fig. 4.4(b), is a crossed diagram which can be obtained from thefirst diagram by an exchange of the particle numbers 2 and 4 which introduces a signchange relative to the first diagram. From the matrix elements, the decay width can becalculated as

ΓB =1

2Mh

∫

dΦ4

∑

σi

∑

cj

|Mσ1,σ2,σ3,σ4c1,...,c4,born

|2 ≡ 1

2Mh

∫

dΦ4|Mborn|2, (4.8)

with the color and helicity summed squared matrix element |Mborn|2 and the phase-spacedensity of a four-particle phase space

dΦ4 =1

(2π)8

[4∏

f=1

d4kf δ(k2f −m2

f )Θ(k0f)

]

δ(4)(∑

kf − ph

)

. (4.9)

67

In the case that both fermions and both anti-fermions are identical, the phase-spacedensity has to be multiplied with a symmetry factor of 1/4. Color factors play a rolein semi-leptonic and hadronic final states. By squaring the amplitudes and summingover the color indices, one obtains a factor of

∑

ciδc1c2δc1c2 = 3 for each qq pair in

the final state for the squared diagrams as can be seen from Fig. 4.5(a). The inter-ference term (Fig. 4.5(b)) in the qqqq and qq′q′q final state obtains a color factor of3 =

∑

ciδc1c2δc3c4δc1c4δc3c2 and is thus suppressed by a factor of 3 w.r.t. to the squared

diagram.

4.2.2. EW Corrections

The amplitudes of the next-to-leading order EW corrections to this process are of O(e5),and they can be separated into the virtual and real corrections. The former contain one-loop and counterterm diagrams while the real corrections have a photon in the finalstate. The full NLO width is

ΓNLO =

∫

4

dΓB +

∫

4

dΓV +

∫

5

dΓR (4.10)

where the number of the particles in the phase-space integrations is written under theintegral and

dΓV = 2dΦ4ReMvirtM∗born, (4.11)

dΓR = dΦ5|Mreal|2.

The generic one-loop diagrams for the virtual corrections are shown in Fig. 4.6 wherethe blob stands for one-particle-irreducible loops. There are self-energies of the gaugebosons, vertex corrections to the HV V , V ff , and Hff couplings (even though thelatter does not exist at tree level for massless fermions), box and pentagon loops. Inthese loops all particles can appear. This introduces diagrams with the other Higgsbosons in the THDM calculation, described in detail in Sec. 4.3. The loop integralscan become divergent for large loop momenta so that a regularization by means of aD-dimensional space-time is mandatory. The computation of one-loop integrals is non-trivial, however, during the past 40 years sophisticated methods have been developedwhich can be found in many textbooks (e.g. [73]) and several numerical implementationslike the Collier library by Denner and Dittmaier [35] are available, so that nowadayssuch integrals do not pose a challenge anymore. Together with the counterterm diagramsof the renormalization procedure (Sec. 3) the UV divergences cancel, and the virtualcorrections become UV-finite.The diagrams for the real corrections can be obtained easily by taking the born diagramand let a charged particle emit a photon into the final state. Exemplary diagrams areshown in Fig. 4.7. However, the real and the virtual corrections cannot be treatedseparately as physical final states where the photon is absent as well as final stateswith an arbitrarily soft and/or collinear photon are degenerate. Also the detector isunable to detect particles with arbitrarily small energies and is unable to resolve collinear

68

V

V

V

(a)

V

V

V

(b)

V

V

(c)

V

V

(d)

V

V

(e)

V

(f)

V

(g)

V

(h)

V

(i)

V

(j)

V

(k) (l)

Figure 4.6.: Generic diagrams for the one-loop virtual corrections (V = W,Z, γ), where theblobs stand for one-particle-irreducible loops. There are gauge-boson self-energies (a,b),vertex corrections (c-i), box (j,k) and pentagon (e) diagrams.

γ

W

W

H

(a)

γ

Z,W

Z,W

H

(b)

Figure 4.7.: Exemplary diagrams for the real corrections. Either the W bosons or chargedfermions can emit a photon. Only one diagram for each case is shown.

particles (the muon-photon pair is an exception for the latter case, as they deposit theirenergies in different calorimeters). In the amplitudes this manifests itself by infrared (IR)divergences, which appear in two types, the soft singularities which arise from a masslessparticle emitting a photon with arbitrarily small energy, and the collinear singularities,which occur when a photon becomes collinear to a massless final state fermion. The

69

loop integrals of the virtual corrections become singular for such configurations, whilein the real corrections the integration over the photon phase space is divergent. TheKinoshita-Lee-Nauenberg theorem [24, 25] states that these divergences cancel each otherfor sufficiently inclusive observables. However, the additional particle in the final statefor the real corrections changes the phase space so that the cancellation happens betweendifferent processes in perturbation theory which demands a cancellation after the phase-space integration.Several methods to cure the divergent terms were proposed in literature, where weconsider the dipole subtraction [127–129] and the phase-space slicing [130–132] method.In the former, an auxiliary function dΓA, called local counterterm, which has the samepoint-wise divergent behavior as the real corrections, is subtracted from them and rendersthe integration finite in four dimensions,

ΓNLO =

∫

5

[(dΓR)ǫ=0 − (dΓA)ǫ=0

]+

∫

4

[

dΓV +

∫

1

dΓA]

ǫ=0

. (4.12)

The local counterterm is built from the Born matrix elements and so-called dipole termsin such a way that the integration over the additional one-particle phase space can becarried out analytically,

dΓA =∑

dipoles

dΓB ⊗ dVdipole, (4.13)

where the dipole factors are process independent, but depend on color and helicity,and include the IR divergences. They can be found in Ref. [127] for massless and inRef. [128, 129] for massive particles emitting the photon. The divergences from theintegrated local counterterm are explicit and cancel against the ones from the virtualcorrections. After the cancellation a numerical integration in four dimensions can beperformed also for the virtual corrections.The idea of phase-space slicing is to separate the regions of phase space with an emissionof a hard photon from the region with soft and/or collinear photons. In our calculationthe so-called two-cutoff phase-space slicing [133, 134] is applied, where the regions of softand collinear photons are defined by the cut-off parameters ∆E ≪ ΓW and ∆θ ≪ 1. Thematrix elements can be evaluated and integrated in the non-singular region numericallywithout the need of regulators. In the soft region, k0γ < ∆E, they can be calculatedwith the soft-photon approximation (c.f. Ref [48]) and in the collinear region, θfγ < ∆θ,k0γ > ∆E using collinear factorization [135–137]. After the integration of the singularregions, the divergences are manifest in terms of logarithms of the photon, (gluon) orfermion regulator masses. As they match the IR singular structure of the virtual correc-tions, they cancel each other. However, a residual dependence of O(∆E) and O(∆θ) ofboth contributions on the cut-off parameters is introduced. For sufficient small values of∆E and ∆θ, the result has to be stable (within the numerical errors) against a variationof these parameters, but only in a certain range, because the numerical integration be-comes unstable for too small cut-off parameters. A plateau in the dependence signalizesthe correct result. Both techniques described here are implemented in the integratorProphecy4f.

70

4.2.3. QCD Corrections

V

V

(a)

V

V

(b)

f

(c)

V

V

g

(d)

V

V

(e)

Figure 4.8.: Exemplary diagrams for the one-loop virtual QCD corrections. In the semi-leptonic case, only the first diagram type exists. In the hadronic cases a similar diagraminvolving the other vertex occurs. In the second diagram, the gluon can connect also otherquarks. Only the interference of the last four diagrams with the crossed LO diagram havea non-vanishing color structure, demanding the quarks to be identical.

QCD corrections occur only if quarks are in the final state, and the diagrams are muchsimpler than in the EW case as the quarks must exchange a gluon. The additional Higgsbosons in the THDM do not interact strongly so that the calculations in the THDMand the SM involve the same diagrams of which exemplary ones are shown in Fig. 4.8.In the semi-leptonic case, only the first diagram exists. By analyzing the color factorsof the interference of these diagrams (the color structure of Fig. 4.8(b),(d), and (e) areidentical) with the LO diagram, shown in Fig. 4.9, some diagrams vanish already. Foreach of the full hadronic squared diagrams 4.9(a)-(f) respectively, the color factor reads

∑

ci,X

TXc4cnTXcnc3δc3c4 · δc2c1δc1c2 = 3

∑

X

Tr(TXTX) = 12, (4.14a)

∑

ci,X

δc2c1δc1c4TXc4cn

TXcnc3δc3c2 =∑

X

Tr(TXTX) = 4, (4.14b)

∑

ci,X

δc1c2TXc2c1δc3c4T

Xc4c3 =

∑

X

Tr(TX)Tr(TX) = 0, (4.14c)

∑

ci,X

TXc2c1δc1c4TXc4c3

δc3c2 =∑

X

Tr(TXTX) = 4, (4.14d)

∑

ci,X,Y,cti

TXc4c3TYc2c1T

Xct1ct2

T Yct2ct1δc1c2δc3c4 =∑

X,Y

Tr(TX)Tr(T Y )Tr(TXT Y ) = 0, (4.14e)

71

δc2c1 δc1c2

δc3c4ΓXc4cn

ΓX

cnc3

(a)

δc2c1 δc1c4

δc3c2ΓXc4cn

ΓX

cnc3

(b)

δc1c2

δc3c4

ΓXc2c1

ΓX

c4c3

(c)

δc1c4

δc3c2

ΓXc2c1

ΓX

c4c3

(d)

f

δc1c2

δc3c4

ΓX

c2c1

ΓY

c4c3

ΓX

cf1cf2

ΓYcf2

cf1

(e)

f

δc1c4

δc3c2

ΓX

c2c1

ΓY

c4c3

ΓX

cf1cf2

ΓYcf2

cf1

(f)

Figure 4.9.: The color structure of the QCD corrections. Diagrams (c) and (e) vanish as thegluon cannot be a color singlet.

∑

ci,X,Y,cti

TXc4c3TYc2c1

TXct1ct2TYct2ct1

δc1c4δc3c2 =∑

X,Y

(Tr(TXT Y

))2 =

∑

X,Y

1

4δXY = 2. (4.14f)

Therefore, the virtual diagrams, Fig. 4.8(b)-(e) contribute only in the interference withthe crossed LO diagram, if all quarks are of the same generation. In the semi-leptoniccase only the first virtual diagram exists, and the color factor is the one of Eq. (4.14a)divided by 3. In the real corrections a final-state quark radiates a gluon as representedin Fig. 4.10, and the color structure looks similar to the ones of Fig. 4.9(a)-(d), exceptthat the exchanged gluon connects via the final state to the interfered diagram.The computation of the NLO QCD correction to the width is analogous to EW case (see

72

g

Z,W

Z,W

H

(a)

Figure 4.10.: Exemplary diagram for the real QCD correction. Also the other quarks canradiate a gluon.

Eq. (4.10)), the IR divergences contain the same complications and can be controlled bydipole subtraction or slicing, but the implementation is simpler than in the EW case.With the methods described in this section, the NLO corrections to the Higgs decay canbe calculated. The amplitudes can be generated from the Feynman diagrams and thephase-space integration can be done numerically. As the structure of the calculationsin the SM and the THDM are similar, it is our goal to extend the functionality of theprogram Prophecy4f to the THDM.

4.3. Amplitudes in the THDM

In this section we show the diagrams appearing in the calculation in the THDM andcompute the corresponding amplitudes. We assume that one can identify the light CPeven Higgs boson of the THDM with the discovered 125 GeV resonance and that itsproperties are close to the SM Higgs boson.

4.3.1. Tree Level

In leading order, the same diagrams as in the SM occur (Fig. 4.4) and they have thegeneric amplitude of Eq. (4.6). The second Higgs doublet does not affect the gauge-boson coupling to the fermions so that the coupling coefficients C±

V ff remain as in theSM. The couplings of Higgs bosons to gauge bosons differ from the SM and are shown inTab. 2.2. One can see that the relative factor to the SM is sin (β − α), and this rendersthe LO matrix element

MV VTHDM,born = sin (β − α)MV V

SM,born. (4.15)

4.3.2. EW Corrections

The electroweak corrections impose the largest modifications with respect to the SM asall additional Higgs bosons are involved in the interactions. We will first treat the one-

73

loop legs fermionic bosonic SM bosonic THDM2 a,b a,b a,b3 c c-i c4 j,k5 l

Table 4.3.: Classification of the occurring one-loop diagrams in the THDM according to thegeneric diagrams of Fig. 4.6.

loop diagrams, then the counterterms, and finish with the real corrections. The genericstructure of the loop diagrams was already shown in Fig. 4.6 and we will concretize thesegeneric diagrams.

4.3.2.1. One-loop Diagrams

It is useful to separate the one-loop EW diagrams into three classes:

fermionic: These diagrams contain a closed fermion loop and appear also in the SMcalculation. The full mass dependence of the loop fermions for those diagrams iskept, as it is done for the SM in Prophecy4f. Therefore also the Hff coupling isnon-vanishing (but different w.r.t. the SM) and diagrams including such a couplingare kept.

bosonic SM like: These diagrams contain at least one boson in the loop and a cor-responding diagram exists in the SM. However, the couplings of the light Higgsboson are different from the SM case.

bosonic THDM: Diagrams with at least one of the H,A0, H± Higgs bosons in the loop.

They do not exist in the SM and yield new contributions.

The separation of the first class from the other diagrams is also performed in theProphecy4f program for the SM, and the treatment of the fermion masses appearingin closed fermion loops is different from the external fermions which are treated massless.This can be done as the diagrams with closed fermion loops form a gauge-invariant andfinite sub-contribution to the process and this allows for a straightforward extension ofthe calculation to include a heavy fourth fermion generation. By separating the remain-ing diagrams into a class which has corresponding diagrams in the SM and diagramswhich contain additional Higgs bosons, differences to the SM calculation are much moremanifest. All new contributions of the THDM are collected in the last class, while theonly difference to the SM calculation in the other classes are the coupling factors of thelight Higgs boson. Especially for parameters close to the the decoupling limit (cβ−α = 0,MH,MA0 ,MH±→∞), where the THDM phenomenology deviates only little from the SM,the origins of the deviations can be better understood. In the decoupling limit itself,the first two classes must become identical to the SM and the last class must vanish. Anumerical check of this limit in each class is simple to perform, but important for the

74

W,Z

W,Z

W,Z, γ

(a)

W,Z, γ

f

W,Z, γ

(b)

Figure 4.11.: Exemplary diagrams for the fermionic self-energy (l.h.s) and vertex correctiondiagrams. A change of the Z boson to a photon via the fermion loop is possible. As the fullmass dependence is kept, the triangle diagrams contain a hff coupling where f are leptonsand quarks.

consistency of the calculation. In each class the diagrams can be sub-classified by theexternal legs of the one-particle irreducible parts into self-energies, triangle, box, andpentagon diagrams as already shown in Fig. 4.6, a detailed list of the occurring types ofdiagrams for the different classes is given in Tab. 4.3.The fermionic diagrams are composed of self-energy diagrams of Figs. 4.6(a), (b) andvertex corrections to the HV V vertex of Fig. 4.6(c), and all diagrams exist also in theSM. One exemplary diagram for each class is shown in Fig. 4.11. Box and pentagondiagrams involve only external fermion lines and do not belong to this class. The self-energy diagrams are closely related to the SM by the tree-level factor sin (β − α), whilethe vertex corrections involve a hff coupling. These couplings are given in Tab. 2.4 anddiffer from their SM counterparts, so that we cannot factor out a single modificationfactor.The bosonic SM-like diagrams have a corresponding diagram in the SM calculationwhere the light CP-even neutral Higgs boson, h, corresponds to the Higgs boson in theSM. The SM-like diagrams are shown and discussed in Ref. [36]. The diagrams with-out a Higgs boson in the loop are identical in the SM up to the hV V coupling factorof sin (β − α) and involve diagrams of all types of Fig. 4.6, in particular also box andpentagon diagrams. Diagrams involving a Higgs boson in the loop differ by more thana single Higgs coupling from the SM, as all coupling factors of the SM Higgs boson aredifferent from the ones of the light Higgs boson in the THDM. However, the Higgs bosoncouples only to massive particles so that such diagrams are either self-energy diagramsor vertex corrections to the hV V vertex.The bosonic THDM diagrams contain at least one of the extra Higgs bosons, H,A0, H

±

in the loop. As they couple only to massive particles, their coupling to the externalfermions vanish in our setup of massless external fermions and only self-energy and tri-angle diagrams remain. They are shown in Fig. 4.12. These new diagrams cannot befactored from the LO and need to be computed with our amplitude generator.

75

W,Z

W,Z

W,Z, γ

(a)

W,Z

W,Z

W,Z, γ

(b)

W,Z

W,Z

(c)

W,Z, γ

W,Z, γ

(d)

W,Z, γ

W,Z, γ

(e)

W,Z, γ

W,Z, γ

(f)

W,Z, γ

W,Z, γ

(g)

W,Z, γ

W,Z, γ

(h)

W,ZW,Z

(i)

Figure 4.12.: Exemplary generic one-loop diagrams with additional Higgs bosons. For any ofthe dashed lines, one of the scalars can be chosen in such a way that at least one of themis H,A0,H

±. The gauge bosons can be W,Z, γ, depending on the charge and final state.

4.3.2.2. Counterterm Amplitudes

The counterterm diagrams are generated from the tree-level diagrams by counterterminsertions in all possible ways and are shown in Fig. 4.13. For the W -exchange diagramsthe Lorentz structure factors completely from the LO, rendering the CT amplitudeparticularly simple,

MWW,CT =MWW,LO[δZh

2+ 3δZe − δα

cβ−αsβ−α

+ δZHhcβ−α2sβ−α

+ δβcβ−αsβ−α

(4.16)

+ δM2W

( 1

2M2W

+1

2k1 · k2 −M2W

+1

2k3 · k4 −M2W

)− 3

sWδsW

+1

2

(δZf,L

f1+ δZf,L

f2+ δZf,L

f3+ δZf,L

f4

)]

.

As the Z boson has different couplings to left- and right-handed fermions, the Lorentzstructure of the neutral current counterterm diagrams factorizes only partially from thetree-level amplitude,

MZZ,CT = MZZ,LO[δZh

2+ 3δZe − δα

cβ−αsβ−α

+ δZHhcβ−α2sβ−α

+ δβcβ−αsβ−α

(4.17)

76

W,Z, γ

W,Z, γ

(a)

W,Z

W,Z

(b)

W,Z

W,Z

(c)

W,Z

W,Z

W,Z, γ

(d)

W,Z

W,Z

W,Z, γ

(e)

Figure 4.13.: The generic counterterm diagrams. The cross denotes the counterterm insertion.

+ δM2Z

( 1

2M2Z

+1

2k1 · k2 −M2Z

+1

2k3 · k4 −M2Z

)− 3M2

Z − 2M2W

M2WsW

δsW

]

+ ie3MWsβ−αsWc2W

1

(k1 + k2)2 −M2V

1

(k3 + k4)2 −M2V

[

u(k1, σ1)γµ(δC+f1ω+ + δC−

f1ω−)v(k2, σ2) u(k3, σ3)γ

µ(C+Zff

ω+ + C−Zff

ω−)v(k4, σ4)

+ f1, k1, σ1, k2, σ2 ↔ f3, k3, σ3, k4, σ4

]

,

with the abbreviations

δC+fi= δZf,R

fi, (4.18)

δC−fi= δZf,L

fi−

2I3W,fi

s2WcW

δsW. (4.19)

The full counterterm contribution for the four different classes of final states is put to-gether from the two CT amplitudes of Eqs. (4.16),(4.17) analogously to the tree levelmatrix elements discussed in Sect. 4.2.1. Using Mathematica, FeynArts, and Form-

Calc the one-loop and counterterm amplitudes can be generated as described in detailin Sect. 5.2.2.In order to have a similar structure as the one-loop diagrams and to treat the closedfermion loops consistent, we also split the contributions to the renormalization constantsinto the three classes. Therefore, each counterterm is composed of a fermionic, bosonicSM-like and bosonic THDM contribution,

δc =δcf + δcbosSM + δcbosTHDM, (4.20)

δZφ =δZfφ + δZbosSM

φ + δZbosTHDMφ . (4.21)

77

The definition of the counterterms remains as described in Sect. 3.2, however, in thedefining self-energies and vertex functions, only diagrams of the respective class aretaken into account. In the fermionic contribution, the masses of fermions belonging toclosed loops are kept, which is necessary to be consistent with the loop diagrams. Wewrite the counterterm matrix elements also with three contributions,

MCT = MCTf +MCT

bosSM +MCTbosTHDM, (4.22)

where the fermionic contribution cancels the UV divergences from the respective one-loop diagrams.

4.3.2.3. Real Emission

The real emission is very simple as at LO no charged Higgs boson occurs and the couplingof the other particles to the photon is identical to the SM. Therefore, we have the samediagrams of Fig. 4.7 with the tree-level factor sin (β − α), and obtain

ΓRTHDM = sin2 (β − α)ΓRSM. (4.23)

4.3.3. QCD Corrections

As the THDM does not change the strongly interacting part of the theory, the computa-tion of the QCD corrections is much simpler than for the EW corrections. The diagramsof the QCD correction are the same as the diagrams shown in Figs. 4.8 and 4.10.

4.3.3.1. One-loop Diagrams

In the one-loop diagrams of Figs. 4.8(a),(b),(d),(e) the only coupling that changes w.r.t.to the SM is the HV V coupling with an additional factor of sin (β − α). In the diagramsrepresented by Fig. 4.8(c) Hqq couplings appear instead of the HV V . For the THDMthese are shown in Tab. 2.4. For up-type quarks, the factor is independent of the specifictype of the THDM, so that

Mu−loop,QCDTHDM =

cosα

sin βMu−loop,QCD

SM . (4.24)

For the down-type quark, in Type II and flipped models the factor is instead − sinαcos β

.

Therefore the following modification of the Hqq coupling (defined in App. A) and thewidths lead to a correct treatment of the QCD corrections in the THDM:

Chuu,THDM =cosα

sin β sin (β − α)Chuu,SM, (4.25a)

Chdd,THDM =

{cosα

sinβ sin (β−α)Chdd,SM for Type I and lepton-specific models,

− sinαcos β sin (β−α)Chdd,SM for Type II and flipped models,

(4.25b)

M1−loop,QCDTHDM =sin (β − α)M1−loop,QCD

SM . (4.25c)

78

W,Z

W,Z

(a)

W,Z

W,Z

(b)

Figure 4.14.: The generic QCD counterterm diagrams. For semi-leptonic final states onlyone diagram exists. Crossed diagrams exist for the case where the fermions are equal.

The coupling factors of the SM can be found in Ref. [48]. Note that the replacementsreproduce the correct amplitudes only for our process and are not generally valid in theTHDM. In particular, the global factor of the amplitude (Eq. (4.25c)) needs to cancelagainst a part of the replacement factors of the Hff couplings ((4.25a),(b)) to yieldthe correct amplitudes for the fermion loop diagrams. The quark loop diagrams do notcontain IR singularities, so that the singular structure of the one-loop diagrams matchesthe one from the SM calculation multiplied by sin (β − α). As the Born amplitude ob-tains the same factor, the SM subtraction and slicing algorithms can be applied withoutmodification.

4.3.3.2. Counterterm Amplitudes

The counterterm amplitudes for the Z and W-boson diagrams are formally identicalto the ones in the EW case stated in Eqs. (4.17) and (4.16). However, only the fieldrenormalization constants of the fermions δZf,L

f , δZf,Rf obtain O(αs) corrections while all

other renormalization constants vanish at NLO QCD. Therefore, only the countertermof the V qq vertex contributes (see Fig. 4.14). Except for a factor sin (β − α) the QCDcounterterm contributions are identical to the SM case, so that

MCT,QCDTHDM =sin (β − α)MCT,QCD

SM (4.26)

holds. With these prescriptions, the virtual QCD corrections can be calculated directlyfrom the SM amplitudes, without generating them again in the THDM with our ma-chinery.

4.3.3.3. Real Emission

The THDM does not affect the real QCD corrections at all, except from the HV Vcoupling, thus

ΓR,QCDTHDM = sin2 (β − α)ΓR,QCDSM . (4.27)

With the calculations described in this section and the amplitudes of the SM calculation,one can extend the calculation of the decay to the THDM.

79

4.4. The Total Width

The total width is the sum of all partial widths which can be calculated with the aboveformulae. There are only 19 independent final states which are listed in Tab. 4.2, all otherfinal states differ only by generation indices and yield the same result since the externalfermion masses are neglected. One can weight these independent final states with theirmultiplicity (given in parenthesis in Tab. 4.2) instead of computing partial widths for allexisting final states. However, it is also of interest, to separate the contributions fromZZ, WW and ZZ/WW interference diagrams in the total width, meaning

Γh→4f = Γh→W ∗W ∗→4f + Γh→Z∗Z∗→4f + ΓWW/ZZ−int. (4.28)

The assignment of the different final states to the three contributions can be done bythe occurring diagrams and only for the final states composed of both, WW and ZZexchange diagrams, care needs to be taken, as they contribute to the WW, ZZ andZZ/WW interference partial width. Nevertheless, a separation is still possible as thereare three contributions to the squared amplitudes, one which contains only virtual Z-bosons, one with only virtual W-bosons, and an interference term which contains both.The first two are identical to the full squared amplitudes with a modified final statewhere one pair of fermions is replaced by the corresponding from a different generation.For the modified final state, the second diagram is absent so that the contributions tothe WW or ZZ partial width can be computed. The interference contribution is theremainder of the total squared amplitude of the charged and neutral current final state.Exemplarily for the νee

+e−νe final state this reads

Γh→W ∗W ∗→νee+e−νe = Γh→νee+µ−νµ , (4.29)

Γh→Z∗Z∗→νee+e−νe = Γh→νeνeµ−µ+ , (4.30)

ΓWW/ZZ−int→νee+e−νe = Γh→νee+e−νe − Γh→νee+µ−νµ − Γh→νeνeµ−µ+ . (4.31)

With this procedure the contribution of all final states to the ZZ, WW and ZZ/WWinterference partial width can be computed. This yields, including the multiplicities ofeach final state,

Γh→W ∗W ∗→4f =9Γh→e+νeνµµ− + 12Γh→e+νeud + 4Γh→udsc, (4.32)

Γh→Z∗Z∗→4f =3Γh→νeνeνµνµ + 3Γh→e+e−µ+µ− + 9Γh→νeνeµ+µ− + 3Γh→e+e−e+e− (4.33)

+ 3Γh→νeνeνeνe + 6Γh→νeνeuu + 9Γh→νeνedd + 6Γh→e+e−uu + 9Γh→e+e−dd

+ Γh→uucc + 3Γh→ddss + 6Γh→uuss + 2Γh→uuuu + 3Γh→dddd,

ΓWW/ZZ−int =3Γh→νee+e−νe − 3Γh→νeνeµ+µ− − 3Γh→e+νeνµµ− (4.34)

+ 2Γh→uddu − 2Γh→uuss − 2Γh→udsc.

One can verify that every final state contributes with the respective weight to the totalwidth, by inserting these formulae into Eq. (4.28).

80

4.5. Gµ Scheme

In the Gµ scheme, the electromagnetic coupling is defined at the electroweak scale. It isderived from the Fermi constant, Gµ, via

αGµ =

√2GµM

2W sin2 θWπ

. (4.35)

With this definition at the electroweak scale, large corrections proportional to α lnmf/sare re-summed and incorporated into the definition of α. In addition, large universalcorrections induced by the ρ parameter are absorbed by the lowest-order cross section.However, at NLO one has to avoid a double counting of the O(αem) contributions,i.e. the parts absorbed in the lowest order have to be subtracted in the countertermdefinition. These parts are the pure weak corrections to the muon decay [48, 138] (inBHS convention), quantified by the finite constant ∆r,

∆r =Σ′AAT (0)− c2

W

s2W

(ΣZZT (M2

Z)

M2Z

− ΣWT (M2W)

M2W

)

+ΣWT (0)− ΣWT (M2

W)

M2W

(4.36)

+ 2cWsW

ΣAZT (0)

M2Z

+α

4πs2W

(

6 +7− 4s2

W

2s2Wlog c2

W

)

.

Therefore, the charge renormalization constant in the Gµ scheme is

δZe∣∣Gµ

= δZe∣∣α0

− 1

2∆r1−loop. (4.37)

4.6. Complex-Mass Scheme

The description of resonances in processes, such as the Higgs-boson decay into fourfermions, requires a special treatment, since the propagators of the vector bosons inFig. 4.4 are singular at the on-shell poles. Therefore at least a partial summation ofself-energy diagrams is necessary to avoid singularities in the amplitudes. However,this mixes orders of perturbation theory and can quickly lead to inconsistencies. Forlowest-order predictions, there are several treatments to cure this issue, for NLO, thecomplex-mass scheme provides a gauge-invariant solution. This scheme is explained ingreat detail in Refs. [39–41], and in the following we state the main features. The majoridea is that the masses of unstable particles are complex, where the mass correspondsto the pole of the propagator in the complex k2 plane,

PV(k2) =

1

k2 −M2V + iMVΓV

:=1

k2 − µ2V

, (4.38)

defining the complex mass to be directly connected to the decay width ΓV ,

µ2V =M2

V − iMVΓV, (4.39)

81

with V = Z,W . At the NLO EW h decay, it is sufficient to treat only the W and Z bosonin the complex-mass scheme even though the other scalar particles are not stable. Weassume that in the THDM, the light Higgs boson has properties similar to the SM Higgsboson, and its width is very small, O(Γh/Mh) < O(10−4). Effects of this order can beneglected, as they are smaller than the contributions from NLO and have the same sizeas the uncertainties from separating h production and decay. The other unstable Higgsbosons of the THDM enter only in loop diagrams, and the corrections from the complexmasses are negligible as ΓS ≪ MS. However, an extension to unstable scalars andfermions is straightforward an can be found in Ref. [40]. A fully consistent replacementof the real masses by its complex counterparts includes also a complex definition of theweak mixing angle θW ,

cos2 θW =µ2W

µ2Z

=M2

W − iMWΓW

M2Z − iMZΓZ

. (4.40)

In this prescription an analytic continuation of the masses into the complex plane hasbeen done, and gauge invariance as well as all algebraic relations between amplitudesor Green functions that do not involve complex conjugation (such as Ward and SlavnovTaylor identities) remain valid. Except from replacing real by complex parameters, thecalculations described in Sects. 4.2 and 4.3 are still performed in the same way. However,via the introduction of complex parameters also spurious additional terms from the weakmixing angle enter, which are of O(ΓV/MV) = O(αem) and can be seen as higher-ordereffects that are taken into account.The generalization of this prescription to one-loop involves a splitting of the real baremasses into complex renormalized masses and complex renormalization constants. Asthe bare Lagrangian remains unchanged, terms are only rearranged and a double count-ing is avoided. Algebraic relations such as Ward or Slavnov-Taylor identities are notaffected by this reordering and thus stay intact. The perturbative expansion can beperformed as usual, only complex parameters and counterterms appear in several placesin the amplitudes. Due to this, loop integrals involving complex internal masses andmomenta can occur. The singular ones have been calculated in Refs. [139, 140] and thenon-singular ones can be obtained from the standard loop integrals by analytic conti-nuation [141].The introduction of complex renormalization constants demands also a proper renor-malization prescription. To this end, the on-shell scheme, reviewed in Sect. 3.2, canbe generalized to complex parameters. The bare real masses are split into complexrenormalized masses and complex counterterms,

M2W,0 =µ

2W + δµ2

W, M2Z,0 =µ

2Z + δµ2

Z. (4.41)

The fields are treated in a similar way,

W0 = W+1

2δZWW, (4.42)

(Z0

A0

)

=

(1 + 1

2δZZZ

12ZZA

12δZAZ 1 + 1

2δZAA

)(ZA

)

, (4.43)

82

with the complex renormalization constants δZ. Note that both W+ and W− aredefined with the same renormalization constant so that the counterterm Lagrangianis not hermitian. Nevertheless, the bare Lagrangian is. The transverse self-energies forthe vector bosons with complex masses, as defined in Ref. [48], are

ΣWWT (k2) =ΣW

T (k2)− δµW + (k2 − µ2W)δZW, (4.44)

ΣZZT (k2) =ΣZZ

T (k2)− δµZ + (k2 − µ2Z)δZZZ,

ΣAAT (k2) =ΣAA

T (k2) + k2δZAA,

ΣAZT (k2) =ΣAZ

T (k2) + k21

2δZAZ + (k2 − µ2

Z)1

2δZZA.

In analogy to the real case, the renormalization conditions for the mass counterterms,the field counterterms and the field mixing counterterms are

ΣWWT (µ2

W) = 0, ΣZZT (µ2

Z) = 0, (4.45)

Σ′WWT (µ2

W) = 0, Σ′ZZT (µ2

Z) = 0, Σ′AAT (0) = 0,

ΣAZT (µ2

Z) = 0, ΣAZT (0) = 0.

Here we do not take the real parts of the self-energies rendering also the field renormal-ization constants complex. Solving these equations for the counterterms analogously toSec. 3.2 yields

δµ2W = ΣWW

T (µ2W ), δµ2

Z = ΣZZT (µ2

Z), (4.46)

δZW = −Σ′WWT (µ2

W), δZZZ = −Σ′ZZT (µ2

Z), δZAA = −Σ′AAT (0),

δZZA =2

µ2Z

ΣAZT (0), δZAZ = − 2

µ2Z

ΣAZT (µ2

Z).

As a consequence the renormalization constants of the weak mixing angle are

δsWsW

= −c2W

s2W

δcWcW

= − c2W

2s2W

(δµ2

W

µ2W

− δµ2Z

µ2Z

)

. (4.47)

From those definitions, the width of the unstable particle can be obtained by taking theimaginary part of M2

V,0 = µ2V + δµ2

V yielding

MV ΓV = Im ΣVT (M2V − iMV ΓV ), (4.48)

which can be recursively solved. Of course, the Cutkosky cutting equations [142] relatethe decay process to the self energies, so that one of the two can be used to calculatethe width. However, as the introduction of complex parameters violates unitarity (onlyin higher orders as the bare Lagrangian is unchanged), the Cutkosky equations hold notexactly, but obtain higher-order corrections [143].Also the renormalization of the electric charge is modified in the complex-mass schemeand reads

δe

e= −1

2(δZAA + sW/cWδZZA). (4.49)

83

Since the bare charge is real, the imaginary part of the renormalized electric charge ispurely determined by the imaginary part of the self-energies. In one-loop calculations,this imaginary part drops out when calculating the squared matrix element, as α fac-torizes from the loop matrix element. Only in two-loop calculations the complex natureof α has to be taken into account.The field renormalization constants of the stable fermions and the scalars are also af-fected by treating Z and W boson in the complex-mass scheme as we do not take the realparts of the self-energies. Due to the appearing complex parameters, the self-energiesand also the renormalization constants become complex. However, the field renormal-ization constants of internal fields drop out and those of external fields factorize from theLO so that the imaginary part drops out after squaring the matrix element at NLO. Forexternal fields which can mix, this factorization does not hold and a complex calculationof the renormalization is obligatory.The calculation of the renormalization constants in the complex-mass scheme requiresthe evaluation of self-energies for complex squared momenta. This can be done byexpanding the self-energies around the real parts of the masses:

ΣVT (µ2V ) = ΣVT (M

2V ) + (µ2

V −M2V )Σ

′VT (M2

V ) +O(α3em) (4.50)

where Σ′VT (M2

V )(µ2V −M2

V ) = O(α2em). However, the expansion breaks down when dia-

grams including a photon exchange are involved, because contributions with a branchcut at k2 =M2

W appear. The difference between the exact calculation and the expansioncan be quantified [40] by

cWT =iα

πMWΓW +O(Γ2

W ln ΓW). (4.51)

Thus, we obtain for the mass counterterms

δµ2W = ΣWW

T (M2W) + (µ2

W −M2W)Σ′WW

T (M2W) +

iα

πMWΓW +O(α3

em), (4.52)

δµ2Z = ΣZZ

T (M2Z) + (µ2

Z −M2Z)Σ

′ZZT (M2

Z) +O(α3em),

and for the field renormalization constants

δZW = −Σ′WWT (M2

W) +O(α2em), δZZZ = −Σ′ZZ

T (M2Z) +O(α2

em), (4.53)

δZZA =2

µ2Z

ΣAZT (0) +O(α2

em), δZAZ = − 2

M2Z

ΣAZT (M2

Z) +

(µ2Z

M2Z

− 1

)

δZZA +O(α2em).

The missing O(α2em) terms in the field renormalization constants do not matter as there

are no external gauge bosons in our process so that the contributions from these fieldrenormalization constants drop out, and the missing O(α3

em) terms in the mass renor-malization constants are beyond the desired accuracy.With these definitions a clear renormalization scheme for unstable particles at NLO is de-fined, and we use this scheme for the calculation of the decay of the light CP-even Higgsboson into four fermions in the THDM. The definition of all renormalization constantswithin the complex-mass scheme needed in the calculation of our process is summarizedin Tab. 4.4.

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Higgs Sector

δZh =−Σ′hh(M2h ), δZHh =−2

ΣHh(M2h )

M2h−M2

H,

in addition, depending on renormalization scheme

• α MS (Sect. 3.2.2.1)

δβ =cβsβ2

δZh−δZH

c2α−s2α

∣∣∣UV

, δα =−ΣHh(M2h )

M2h−M2

H

∣∣∣UV

− cαsαcβsβ

δβ.

• λ3 MS (Sect. 3.2.2.2)

δβ =cβsβ2

δZh−δZH

c2α−s2α

∣∣∣UV

, δα = (3.83)∣∣MW→µW,MZ→µZ

.

For the calculation of Eq. (3.83):

δM2H =ΣHH(M2

H), δM2h =Σhh(M2

h),

δM2A0

=ΣA0A0(M2A0), δM2

H± =ΣH+H−(M2

H+),

δtH =−TH, δth =−Th,δλ3 = (3.86)+(3.85)

∣∣MW→µW,MZ→µZ

, δλ5 = (3.80)+(3.79)∣∣MW→µW,MZ→µZ

.

• FJ (Sect. 3.2.2.3)

δβ = (3.96)∣∣MW→µW,MZ→µZ

, δα = (3.98)∣∣MW→µW,MZ→µZ

.

• FJ λ3 (Sect. 3.2.2.4)

δβ = (3.96)∣∣MW→µW,MZ→µZ

, δα = (3.83)∣∣MW→µW,MZ→µZ

.

EW Sector

δµ2W =ΣWW

T (M2W) + (µ2

W −M2W)Σ′WW

T (M2W) + iα

πMWΓW,

δµ2Z =ΣZZ

T (M2Z) + (µ2

Z −M2Z)Σ

′ZZT (M2

Z),

δsW =− cWsWδcW, δcW = 1

2

(δµ2Wµ2W

− δµ2Zµ2Z

)

,

δZAA =−Σ′AAT (0), δZZA = 2

µ2ZΣAZT (0),

∆r =−δZAA − 2 δsWsW

+ΣW

T (0)−δµ2Wµ2W

+ 2 cWsW

ΣAZT (0)

µ2Z+ α

4πs2W

(

6 +7−4s2W2s2W

log c2W

)

δee= −1

2(δZAA + sW/cWδZZA +∆r).

Fermion Sector

δZf,Li =−Σf,Li (m2

f,i)−m2f,i

[Σ′f,Li (m2

f,i) + Σ′f,Ri (m2

f,i) + 2Σ′f,Si (m2

f,i)],

δZf,Ri =−Σf,Ri (m2

f,i)−m2f,i

[Σ′f,Li (m2

f,i) + Σ′f,Ri (m2

f,i) + 2Σ′f,Si (m2

f,i)].

Table 4.4.: The definition of the renormalization constants needed in the calculation of thedecay H → 4f in the complex-mass scheme.

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5. Numerics

In this chapter we discuss the numerical implementation, present the setup, and showthe results for the Higgs decay into four fermions in the THDM. At first, the programProphecy4f, on which our implementation is based, is introduced in detail with an ex-planation of its features (Sect. 5.1). Afterwards, in Sect. 5.2, we describe the generationof the amplitudes and the implementation of our code into Prophecy4f. We state thenumerical setup in Sect. 5.3 and present the investigated THDM scenarios in Sect. 5.4.The numerical results are shown and discussed in detail in Sect. 5.5.

5.1. The Program Prophecy4f

The computer program Prophecy4f provides a “PROPer description of the HiggsdECaY into 4 Fermions” and calculates the decay observables of the decay processh→WW/ZZ→4f at NLO EW+QCD in the SM as published in Refs. [36–38]. It is ourgoal to extend this program to the calculation of the corresponding decay in the THDMwhile keeping the features. A description of the calculation in the SM and THDM wasgiven in the previous chapter.The original code contains the matrix elements of all 19 final states listed in Tab. 4.2 ina generic way, and includes also improvements beyond NLO, like two-loop contributionsin the heavy-Higgs-mass limit. It takes into account the full off-shell effects of theintermediate W and Z bosons and evaluates the matrix elements in the complex-massscheme (Sect. 4.6). The loop integrals in the virtual corrections are evaluated using anearly version of theCollier library [35]. The UV-divergences are treated in dimensionalregularization while the soft and collinear divergences of the loop integrals are regularizedby introducing a small photon or gluon mass. The final-state fermions are consideredmassless, although the masses are used as regulators for IR divergences. However, indiagrams with a closed fermion loop the full mass dependence of those fermions is keptwhich allows to extend the calculation to include a heavy fourth fermion generation.The cancellation of the infrared divergences can be performed via phase space slicing ordipole subtraction which are described in Sect. 4.2.2.The integration over the phase space is done using an adaptive Monte Carlo integrator,where the integrand is evaluated at pseudo-random phase-space points and the densityof the points is adapted iteratively to the integrand to provide a better accuracy. TheMonte-Carlo generator can also be used to generate samples of unweighted events, whichis particularly interesting for generating Monte Carlo samples for experimental analysis.Prophecy4f automatically provides distributions for leptonic and semi-leptonic finalstates. For fully hadronic final states distributions are not implemented, since this should

86

be done in the hadronic production framework.The invariant mass for a lepton–anti-lepton pair is defined as

M2fafb

= (ka + kb)2 (5.1)

with the momentum ka of the fermion and kb of the anti-fermion. In the case that bothlepton pairs are of the same generation, the assignment of the pair belonging to one Zboson is not unique. For this situation, the pair whose invariant mass is closest to theZ-boson mass is assigned to be number one.Angular distributions are defined for specific leptonic and semi-leptonic final states. Forthe neutral-current final state with two pairs of charged leptons belonging to differentgenerations (i.e. e+e−µ+µ−), the angle φ between the decay planes of the two pairs isgiven by

cosφ =((k1 + k2)× k1) ((k1 + k2)× k3)

|(k1 + k2)× k1||(k1 + k2)× k3|, (5.2)

with the sign convention

sgn(sinφ) = sgn{(k1 + k2) ((k1 + k2)× k1)× ((k1 + k2)× k3)}. (5.3)

In the semi-leptonic final state, the signature contains of two indistinguishable jets, sothat any observable must be invariant under the permutation of the two jets, thereforethe angular variable is defined as [38]

| cosφ| =((kjet

1+ kjet

2

)× k1

) (kjet

1× kjet

2

)

|(kjet

1+ kjet

2

)× k1||kjet

1× kjet

2| . (5.4)

The decay involving the charged current has neutrinos in the final state so that theinformation about the momenta is incomplete so that the angular observables as definedabove cannot be used. However, in this case one can define a transverse angle betweenthe two charged leptons as [36]

cos φT =k2,T · k3,T

|k2,T||k3,T|, (5.5)

sgn(sin φT ) = sgn{ez · (k2,T × k3,T)}, (5.6)

with the transverse momentum kT, and ez the unit vector pointing in beam directionof the Higgs-boson production process. All angular distributions are defined in the restframe of the Higgs boson.To provide collinear-safe observables, a photon recombination can be performed in thereal corrections. This means that the photon momentum is added to the fermion inthe histogram if the invariant mass of a photon and a charged fermion is smaller than5 GeV. When this is possible with more than one fermion, the photon is added to thefermion which yields smallest invariant mass. We apply the photon recombination in allour calculations and further details to its impact are discussed in Ref. [36].

87

5.2. Implementation

The implementation of our calculation is performed in three steps. In the first one, weadapt the generated FA model file to the specific demands of the Prophecy4f calcula-tion. Afterwards, the squared electroweak amplitudes for all final states are computedand written as Fortran code, so that they can be implemented in the Prophecy4f

program, and last but not least, the program itself is modified and extended.

5.2.1. Modifications in the Model File

The THDM was implemented in FeynRules and a FeynArts model file was produced(see Sect.3.4). However, for the process H→4f specific changes are necessary, in orderto produce the matrix elements in the structure of the Prophecy4f code.First of all, the fermion masses have to be implemented in such a way that fermionsbelonging to closed loops are treated with the full mass dependence while for externalfermions the masses are neglected (except for mass-singular logarithms). In FeynArts

model files one can distinguish between the masses in different propagators – loop,internal, external – and the mass inserted outside of propagators (in couplings or ason-shell momenta). We use this function to distinguish between the types of masses andset them to appropriate values after the generation of the diagrams.In addition, to treat the resonance regions of the vector-boson propagator properly, thecomplex-mass scheme is implemented in the model file. To this end, real and complexvector-boson masses are implemented and the counterterms are defined according toTab. (4.4). Additionally, we implement the Yukawa couplings in a model-independentway in the Lagrangian so that amplitudes have only generated once for all four Typesof THDM, and the type can be specified in the numerical input file.

5.2.2. Generating the Amplitudes

The virtual electroweak amplitudes are generated in Mathematica with FeynArts

and FormCalc using the modified FA model file. As output we obtain Fortran codewhich is, after linking to the powerful and numerical stable Collier library for theevaluation of one-loop integrals, able to compute the squared matrix elements for agiven phase-space point. A script, which generates automatically the amplitudes andthe code for all final states is written in Mathematica. This scripts provides an optionwith which the model can be switched between the THDM and the SM so that theextension of the program can be checked against the inbuilt Prophecy4f calculation.In the generation of the code, we select the diagrams manually and split the contributionsinto the three classes described in Sect. 4.3.2. For each class the general fermion massesof the model file are then specified to the needed ones. In the first class, we treat theloop fermion masses and the insertion mass exact and all the others fermion massesas small, while for the other classes all fermion masses are treated as small and areneglected whenever they appear in nominators. We also adapt to the naming conventionof Prophecy4f and Fortran. The counterterm contribution is as well split into three

88

parts, as described in Sect. 4.3.2, where each renormalization constants is composed ofa fermionic, bosonic SM-like and bosonic THDM contribution. The sum of the threecontributions add up to full counterterm contribution, however, the fermion masses ineach of the contributions are set to the values of the respective one-loop diagrams.In a very last step, the generated code is modified in such a way that all calls to theloop library are complex, which is necessary in the complex-mass scheme.

5.2.3. Embedding in the Prophecy4f program

The modification of the Prophecy4f program in a such way that the calculation inthe THDM is possible while keeping all the features of the original program, was one ofthe major parts of this project. To this end, we keep the original program flow intactand add branches to it, whenever a calculation specific for the THDM has to be carriedout. The input file and the readout of the input values of Prophecy4f is extendedto the additional parameters of the THDM. There are two possible ways to specify theparameters, either using the {pmass} set defined in Eq. (2.73) or the hybrid basis withthe parameter set {phybrid} defined in Eq. (2.76). As the parameter cβ−α is standard inliterature, we also use this quantity as input in the {pmass} set. In the main program,three additional routines carry out the major part of the computation in the THDM:

• An initialization routine which sets the THDM parameters, modifies the hV V (forthe LO, real, and QCD corrections) and hff (for the QCD corrections) couplingsof the main program, and computes the renormalization constants of the THDM.

• A routine which sets for every phase-space point the kinematics for the FormCalc

amplitudes.

• A routine which evaluates for each phase-space point the electroweak squaredmatrix element for the respective final state.

The virtual QCD corrections use the amplitudes of Prophecy4f, with the simple mod-ification factors stated in Eq. (4.25). With this implementation the program structureis not changed at all and this allows us to keep all features of the original program.In particular for the end-user no changes occur, except from the specification of theadditional parameters of the THDM in the input.

5.3. Input parameters

For the values of the SM like input parameters we take the values recommended by theLHC Higgs Cross Section Working Group [45] which essentially follow the Particle DataGroup [144]:

Gµ =0.11663787 · 10−4 GeV−2, αs =0.118 (5.7)

MZ =91.1876 GeV, MW =80.385 GeV

89

ΓZ =2.4952 GeV, ΓW =2.085 GeV,

me =510.998928 keV, mµ =105.6583715 MeV, mτ =1.77682 GeV,

mu =100 MeV, mc =1.51 GeV, mt =172.5 GeV,

md =100 MeV, ms =100 MeV, mb =4.92 GeV,

Mh =125 GeV.

Prophecy4f performs its calculation in the complex-mass scheme and automaticallyconverts the experimental measured on-shell boson masses MOS

V given in the input topole masses Mpole

V of the propagators. This is done by [36]

MpoleV =MOS

V /√

1 + (ΓOSV /MOS

V )2, ΓpoleV = ΓOS

V /√

1 + (ΓOSV /MOS

V )2. (5.8)

From these measured input values, the program recalculates the width of the vectorbosons in O(αem) in the SM using real mass parameters everywhere. This recalculationensures that the branching ratios of the vector bosons are correctly normalized and addup to one for the SM. In the THDM, the heavy Higgs bosons enter the width in themass counterterms, however, as we are close to the alignment limit (cβ−α = 0) the effectsare negligible. The fermion masses are only inserted as regulator masses in soft and/orcollinear divergent terms and in mass terms of closed fermion loops. The final-statefermions are considered massless and the mass values are not used. The strong couplingconstant appears only in the QCD corrections (see Sect. 4.2.3), and we take the valueat the Z-boson mass.For the central renormalization scale we use the average mass of all scalar degrees offreedom,

µ0 =1

5(Mh +MH +MA0 + 2MH±). (5.9)

This scale choice might seem unintuitive at first glance, since the light Higgs-boson massis the center of mass energy of our process. However, the loop diagrams including heavyscalar particles with mass M introduce potentially large terms of

lnM2

µ2, (5.10)

in the amplitudes. Therefore we adapt the choice of the scale to the arithmetic meanof the Higgs-boson masses and the scale variations performed in Sect. 5.5 confirms thischoice. The input values of the additional parameters of the THDM depend on theinvestigated scenario and are given in the following. Thereby are the scale dependentinput parameters cβ−α, tβ , λ5 defined at the central scale.

5.4. Scenarios

It is our goal to analyze the Higgs decay in the context of the most relevant THDMscenarios. To compute phenomenologically relevant results, we need to take into account

90

current constraints which also restrict the large parameter space. The constraints cancome from direct LHC searches for heavy Higgs bosons, but also from theoretical aspectslike vacuum stability, perturbative unitarity, or perturbativity of the couplings, which isrequired for a meaningful perturbative theory. The new report of the LHC Higgs CrossSection Working Group [45] summarizes a selection of relevant benchmark scenariosproposed in other papers. These scenarios are only partially interesting for us as wedemand CP conservation and the absence of FCNC, and identify the light Higgs bosonwith the discovered resonance at 125 GeV. In addition, many listed scenarios are designedto provide interesting phenomenology for charged or pseudoscalar Higgs bosons, or tacklethe questions of baryogenesis and dark matter while the light Higgs boson behaves as inthe SM. Most relevant for our work is the benchmark scenario BP1A which was originallyproposed as scenario A in Ref. [42] in the hybrid basis. It provides a SM-like light Higgsboson but deviates from the alignment limit where cβ−α = 0.

For this benchmark scenario experimental constraints from direct LHC searches, shownin Fig. 5.1, as well as theoretical constraints from vacuum stability and perturbativeunitarity, illustrated in Fig. 5.2, are taken into account. Additionally, we employ per-turbativity constraints to improve this scenario. Large coupling factors spoil the con-vergence of the expansion in coupling constants and perturbation theory breaks down,so that we demand sufficiently small coupling factors. To this end, we compute thesize of each coupling factor of the four point Higgs-boson vertices and use the largestcoupling factor, λ/(4π), as a measure. As the masses and mixing angles appear in thecouplings (see Feynman Rules in App. A), the perturbativity criterion can be used tofurther reduce the parameter space. Since the convergence of the perturbation seriesbecomes worse with increasing coupling factors a clear discrimination of perturbativeand non-perturbative parameter points is impossible. However, values larger than one

Figure 5.1.: Direct constraints from LHC Higgs searches on the parameter space for the2HDM Type-I with MH = 300 GeV (left) and MH = 600 GeV (right). In both casesMh = 125 GeV, Z4 = Z5 = −2 and Z7 = 0 are given in the hybrid basis (c.f. Sec. 2.2.2).The colors indicate compatibility with the observed Higgs signal at 1σ (green), 2σ (yellow)and 3σ (blue). Exclusion bounds at 95% C.L. from the non-observation of the additionalHiggs states are overlaid in gray. The graphics and description are taken from Ref. [42].

91

Figure 5.2.: Example THDM parameter regions respecting perturbative unitarity and stabil-ity constraints (green) for MH = 300 GeV (left) and MH = 600 GeV (right), Z4 = Z5 = −2and Z7 = 0. The graphics and description are taken from Ref. [42].

-0.4 -0.2 0 0.2 0.4

cos (β − α)

1

10

tanβ

-0.4 -0.2 0 0.2 0.4

cos (β − α)

1

10tanβ

Figure 5.3.: The perturbativity measure for an example THDM parameter region with Z4 =Z5 = −2 and Z7 = 0 and MH = 300 GeV (left) and MH = 600 GeV (right). Gray areas areruled out while blue and yellow show coupling strengths λ/(4π) between 0.5 and 1, and 0.3and 0.5, respectively. Parameter sets with values smaller than 0.3 do not occur.

indicate that higher-order corrections do not systematically become smaller and pertur-bativity is not given anymore which rules out such parameter points. Values between 0.5and 1 usually still yield large higher-order corrections and need to be taken with care.The result of the perturbativity analysis, which was performed using Mathematica

and our FeynArts model file in the hybrid basis, is given in Fig. 5.3 forMH = 300 GeV(left) andMH = 600 (right). Excluded areas are gray while blue (0.5 < λ/(4π) < 1) andyellow (0.3 < λ/(4π) < 0.5) indicate different sizes of the coupling factors. Parameterpoints where all couplings are smaller than 0.3 do not appear. The excluded trench attanβ = 1 appears in the hybrid basis as the relation of Eq. (2.77c) becomes singularand coupling factors diverge at this point. Overlaying these results with the previousexperimental and theoretical constraints shows a significant reduction of the allowedparameter region. Nevertheless, we need to modify the scenarios proposed by Ref. [42]

92

-0.4 -0.2 0 0.2 0.4

cos (β − α)

1

10

tanβ

Figure 5.4.: The perturbativity measure for a heavy Higgs boson of 1 TeV. Large areas areexcluded by coupling factors λ/(4π) > 1 (gray) whereas values between 0.5 (0.3) and 1 (0.5)are colored blue (yellow).

only slightly to obtain the low- and high-mass as well as the benchmark plane scenariodescribed later. We also take the masses and λ5 as independent and constant parame-ters instead of the ones of the hybrid basis (c.f. Sec. 2.2.2) as originally proposed. Thisapproach is practical because the physical meaning of mass parameters is obvious andwe renormalize them on-shell. We do not consider heavy Higgs masses in the TeV rangebecause the allowed parameter space is dramatically reduced in this region which canbe seen in Fig. 5.4. Only parameters close to the alignment limit and with tβ ≈ 2 andtβ ≈ 0.5 remain allowed.

Low-mass scenario

The low-mass scenario consists of a heavy neutral CP-even Higgs boson of 300 GeV.It is inspired by the first benchmark scenario of Ref. [42]. However, we transform theparameters of the hybrid basis (Z4 = Z5 = −2, Z7 = 0) to the massesMA0 ,MH± and thecoupling λ5. We do this at cβ−α = 0.1 because this value is proposed for the benchmarkpoint in the original scenario. Since cβ−α is the only parameter of the THDM appearingat LO, our process is most sensitive w.r.t. to this parameter, and we vary cβ−α from −0.2to 0.2 in scenario A. The range of the scan is limited by constraints from experiments

Scenario cos (β − α) MH [GeV] MA0 [GeV] MH± [GeV] λ5 tanβ Type

A −0.2 . . . 0.2 300 460 460 −1.9 2 1

Aa 0.1 300 460 460 −1.9 2 1

Ab −0.1 300 460 460 −1.9 2 1

Table 5.1.: The input values for the low-mass scenario which is based on a benchmark scenarioproposed in Ref. [42]. The first one (A) is a scan in cβ−α in the mass parameterization, Aaand Ab are points of the scan region used to analyze the scale dependence.

93

and perturbative unitarity. These constraints indicate that values of |cβ−α| exceeding0.1 are phenomenologically disfavored [42], however, we perform our analysis with lessstringent bounds to get a more complete picture. Out of this scenario, we take twodistinguished points named Aa and Ab with cβ−α = ±0.1 to perform scale variations.The setup is shown in Tab. 5.1.

High-mass scenario

The high-mass scenario is similar to the low-mass scenario, however, with a heavy Higgsmass of 600 GeV. The parametersMA0 ,MH± , and λ5 are computed from the hybrid basis(Z4 = Z5 = −2, Z7 = 0) at cβ−α = 0.1. Constraints from stability and perturbativeunitarity (Fig. 5.2) reveal that positive and negative values of cβ−α are only allowed indifferent regions of tan β. Therefore we define two parameter scans (B1, B2) which areapplicable for positive (B1) and negative (B2) values of cβ−α. Analogously to the low-mass scenario, two parameter points with specific values of cβ−α (B1a, B2b) are usedto analyze the renormalization scale dependence. The input values of the high-massscenario are shown in Tab. 5.2.

Scenario cos (β − α) MH [GeV] MA0 [GeV] MH± [GeV] λ5 tanβ Type

B1 0 . . . 0.15 600 690 690 −1.9 4.5 1

B2 −0.15 . . . 0 600 690 690 −2.4 1.5 1

B1a 0.1 600 690 690 −1.9 4.5 1

B2b −0.1 600 690 690 −2.4 1.5 1

Table 5.2.: The input values for the high-mass scenario which is based on a benchmark scenarioproposed in Ref. [42]. The first ones (B1, B2) are scans in cβ−α in the mass parameterization,where B1 and B2 are due to constraints only valid for positive or negative values of cβ−α.There are also two benchmark points of the scan region used for scale variations.

Different THDM types

In this scenario, we compare different types of THDM. Yukawa couplings appear in ourprocess only in closed fermion loops so that the top-quark contribution is dominant.The couplings to up-type quarks is in all types of THDM identical so that we expectnegligible effects from changing the type. The comparison is performed for Higgs massesaccording to the low- and high-mass scenarios, and the setup is shown in Tab. 5.3.

Scenario cos (β − α) MH [GeV] MA0 [GeV] MH± [GeV] λ5 tan β Type

Aa 0.1 300 460 460 −1.9 2 1 . . . 4

B1a 0.1 600 690 690 −1.9 4.5 1 . . . 4

Table 5.3.: We take the low- and high-mass scenarios Aa and B1a for the comparison ofdifferent types of THDM.

94

Benchmark plane

For this scenario, we analyze a large area of theMH−tan β plane. The input parametersare based on the non-alignment scenario of Ref. [42], are given in the hybrid basis(c.f. Sect. 2.2.2), and are shown in Tab. 5.4. The scale is set dynamically to µ =(Mh +MH +MA0 + 2MH±)/5.

Scenario cos (β − α) MH [GeV] Z4 Z5 Z7 tan β Type

plane 0.1 300. . .750 −2 −2 0 1. . .50 1

Table 5.4.: Input parameters for the NLO corrections in a MH−tβ plane.

Baryogenesis

The BP3B scenario of Ref. [45] was initially proposed in Ref. [43]. With a second Higgsdoublet a first-order electroweak phase transition is possible, which could explain thebaryon asymmetry in the universe. The main signature for this model is the decay of apseudoscalar Higgs boson. Nevertheless, the non-alignment of the the benchmark pointsBP3B render these scenarios also interesting for our study. The parameterization in theoriginal form uses m12 for the Higgs self-coupling parameter from which we computeλ5 using Eq. (2.69e). The input parameters are shown in Tab. 5.5, and we use againconsistently µ = (Mh +MH +MA0 + 2MH±)/5.

Scenario cos (β − α) MH [GeV] MA0 [GeV] MH± [GeV] λ5 tan β Type

BP3B1 0.3 200 420 420 -2.58 3 1

BP3B2 0.5 200 420 420 -2.58 3 2

Table 5.5.: Input parameters for the BP3B benchmark scenario of Ref. [45]. The scenario wasproposed in Ref. [43] as a strong EW phase transition could possibly explain baryogenesis.

Fermiophobic heavy Higgs

By choosing a Type I THDM as well as a vanishing mixing angle α, the heavy Higgsboson decouples from the fermions. Such a scenario was proposed in Ref. [44] witha direct detection of the heavy Higgs bosons as the leading signature. However, thealignment limit (cβ−α = 0) cannot be reached in this model as this would require largevalues of tan β which are ruled out by stability constraints. This gives rise to possiblysizable effects on the light Higgs-boson decay. The input parameter m12 used originallyis strongly constrained by the perturbativity constraints and needs to be adapted fordifferent values of tan β although this is not mentioned in Ref. [45]. We transform theinput parameter m12 to our convention using Eq. (2.69e) and end up with a constant λ5for all tan β, the setup is shown in Tab. 5.6. We use a central scale of µ = (Mh +MH +MA0 + 2MH±)/5.

95

Scenario MH [GeV] MA0 [GeV] MH± [GeV] sα λ5 tan β Type

BP6a 200 500 500 0 -3.46 40 1

BP6b 200 500 500 0 -3.46 20 1

BP6c 200 500 500 0 -3.46 10 1

Table 5.6.: Input parameters for the fermiophobic heavy Higgs THDM model based onRef. [44] and the BP6 benchmark scenario of Ref. [45]. Since the model is Type 1 and theneutral mixing vanishes, the heavy Higgs boson decouples completely from the fermions.With larger tan β the alignment limit is approached.

5.5. Results

The numerical results for the light Higgs boson of the THDM decaying into four fermionsat NLO are presented in this section. Subsequently, we state the results for the differentscenarios described in the previous section, beginning with the low-mass scenario. There,we investigate at first the conversion of the renormalized input parameters between dif-ferent renormalization schemes and the running of the couplings. Afterwards we discussthe scale dependence of the h→4f width and show the scan in the cβ−α parameter.Finally, we study the partial widths and differential distributions in order to identify de-viations from the SM expectations. The same procedure is performed for the high-massscenario (split into two regions with positive or negative cβ−α), while we do not performsuch an extended analysis for the other scenarios. When we compare the NLO with theLO, we compute the latter as the contribution of it to the NLO calculation, e.g. usingNLO parameter conversion and NLO W/Z-boson width definitions.

5.5.1. Low-Mass Scenario

5.5.1.1. Conversion of the Input Parameters

As described in Sec. 3.2.3, input values for parameters in one renormalization schemecorrespond to different input values in other renormalization schemes due to differentfinite higher-order contributions to the bare parameter. We compute the differences atNLO up to O(αem), and calculate the conversion of the input parameters. This givesindications on the size of higher-order terms and on the convergence of the perturbationseries which transfers to the predictivity of the full calculation of the decay. The relationbetween the input values of one parameter in different renormalization schemes is givenin Eq. (3.100), and we employ the α MS scheme as one of the two schemes so that weonly have to deal with one counterterm contribution at a time. The conversion into theα MS scheme is straightforward as the last term of Eq. (3.100) vanishes, while for theother direction a non-linear equation needs to be solved. For consistency reasons, bothconversions should be inverse to each other, and this can be checked graphically by areflection on the diagonal. The non-linear equation is solved with a numerical fixed-point iteration, but also approximately by evaluating the counterterm with the knowninput parameters instead of the converted ones. The error of this approximation is of

96

cβ−α|αMS

λ3MS

FJ

FJ λ3−0.4

−0.2

0

0.2

0.4

−0.4 −0.2 0 0.2 0.4

µ = 361 GeV

cβ−α|i

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(a)

cβ−α|i

λ3MS

FJ

FJ λ3−0.4

−0.2

0

0.2

0.4

−0.4 −0.2 0 0.2 0.4

µ = 361 GeV

cβ−α|αMS

Scenario A

(b)

Figure 5.5.: Conversion of the value of cβ−α from α MS to the λ3 MS (green), FJ (pink) andFJ λ3 scheme (turquoise) for scenario A (a). Panel (b) shows the conversion to the α MSscheme using the same color coding. The gray dashed line is obtained using a fixed-pointiteration. The phenomenologically relevant region is highlighted in the center.

higher order and beyond our desired accuracy as long as the perturbation series behaveswell and higher-order terms are small. The comparison of both methods allows for aconsistency check of the computation. We extend the range of cβ−α values in scenario Ato −0.4 . . . 0.4 so that we get a larger picture even though the regions with large |cβ−α|are ruled out by phenomenology.The results are shown in Fig. 5.5 with a conversion from (l.h.s.) and to the α MS scheme(r.h.s.), while the λ3 MS (green), FJ (pink), and FJ λ3 MS (turquoise) schemes areemployed as the second scheme. All other conversions can be seen as a combination ofthe presented ones. On the left-hand side, the gray dashed lines are the result obtainedusing the fixed-point iteration. In both plots, we highlighted the phenomenologicallyrelevant region in the center.All curves show only minor changes in the parameter values and the fixed-point iterationagrees well with the approximate conversion, affirming that the higher-order functionsa, b, and fα of Tab. 3.3 are small, and perturbation theory is applicable. However, wewould like to mention that a parameter set in the alignment limit does not persist inthe conversion to other renormalization schemes. This reveals, that the alignment limitdepends on the definition of the parameters at NLO, and is not a distinguished pointof the theory. In addition, the curves for the transformation involving the schemes withλ3 as an independent parameter have a small region where large effects occur. Thisis an artifact of the 1/c2α proportionality constant in the relation (3.81) between δλ3and the dependent δα. In the vicinity of the singularity c2α = 0 (corresponding here tocβ−α ≈ −0.32), theMS renormalization of λ3 introduces large finite contributions to the

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µ0 = 361 GeV

Scenario A

λ3MS

αMS

FJ

FJ λ3

(b)

Figure 5.6.: The running of cβ−α for the benchmarks Aa (a) and Ab (b) in the α MS (blue),λ3 MS (green), FJ (pink), and FJ λ3 (turquoise) scheme.

conversion equation resulting in a breakdown of perturbation theory. This is a part ofthe λ3 MS and the FJ λ3 renormalization schemes which limits their use. Nevertheless,the phenomenologically relevant region is not affect by this artifact so that this does notpresent a problem in our calculation.

5.5.1.2. The Running of cβ−α

We have shown in Sect. 3.3 that parameters renormalized in MS are scale dependentand that the dependence is governed by the RGEs. For each renormalization scheme wesolve the RGEs of the independent parameters according to Tab. 3.4 using a classicalRunge–Kutta algorithm. We isolate the effects of the running from the conversion byconsidering each renormalization scheme separately and do not convert the input values,with the drawback that different schemes cannot be compared here. The scale depen-dence of cβ−α from 100 to 900 GeV is plotted in Fig. 5.6, for the benchmark points Aa(l.h.s) and Ab (r.h.s) and input values given at the central scale µ0. It shows that thechoice of the renormalization scheme has a large impact on the scale dependence. Whilethe α MS scheme introduces only a small running, the other schemes show a muchstronger scale dependence so that excluded and unphysical values of input parametersare reached quickly. A similar observation has also been made in supersymmetric modelsfor the parameter β (the ratio of the vacuum expectation values of the Higgs doubletsin SUSY models). Gauge dependent MS schemes have a small scale dependence whilereplacing the parameters by gauge independent ones introduces additional terms in theβ-functions, which arrange for a stronger scale dependence of such schemes [46]. In ad-dition, it is remarkable that the sign of the slope differs for the different renormalization

98

schemes. This is another consequence of the additional terms in the β-functions andshows that considering different parameters as independent has large effects. As somecurves hit the cβ−α = 0 axis and therefore run into the alignment limit, we concludethat this limit is only valid for specific scale choices and thus cannot be a fundamentalphysical point of the theory. In Fig 5.6(b) one can also see that the curves for the λ3MS and the FJ λ3 scheme terminate around 250 GeV. At this scale, the running ofλ3 yields unphysical values for which Eq. (2.69c) with the given Higgs masses becomesover-constrained and no solution with |s2α| < 1 exists. This is unique to the λ3 runningas only there an equation needs to be solved to obtain the input parameter α. For theother cases we prevent the angles from running out of their domain of definition bysolving the running for the tangent of the angles.

5.5.1.3. Scale Variation of the Width

Due to the appearance of heavy Higgs bosons in the loop diagrams multiple scales oc-cur in the calculation. Therefore, a naive choice of the central renormalization scale ofµ0 =

√s = Mh might not be appropriate. To choose and to justify our central scale

of Eq. (5.9), and to estimate the theoretical uncertainties, we compute the total widthaccording to Eq. (4.28) while the scale is varied as in the previous section from 100to 900 GeV. As a definition of the input parameters in each of the four renormaliza-tion schemes represents a physical scenario on its own, we have four input prescriptions(λ3 MS, α MS, FJ, FJ λ3), and we compute for each of them the result in all renor-malization schemes. In a first step, the input is converted, thereafter the scale is varied,the RGEs solved, and the width computed using the respective renormalization scheme.The results are shown in Figs. 5.7 and 5.8 at LO (dashed) and NLO EW (solid) forthe benchmark points Aa and Ab for each of the input prescriptions. The QCD cor-rections are not part of the EW scale variation and therefore omitted in these results.The benchmark point Aa shows textbook-like behavior, and the results are similar forall input prescriptions so that we discuss all of them simultaneously. First of all, the LOcomputation shows a strong scale dependence for all renormalization schemes, resultingin sizable differences between the curves. However, each of the NLO curve shows awide extremum with a large plateau, reducing the scale dependence drastically, as it isexpected for NLO calculations. The central scale µ0 = (Mh+MH+MA0 +2MH±)/5 liesperfectly in the middle of the plateau regions motivating this scale choice. In contrast,the naive scale choice µ0 =Mh is not within the plateau region, leads to large, unphys-ical corrections, and should not be chosen. The breakdown of the FJ curve for smallscales can be explained by the running which becomes unstable for these values (seeFig. 5.6(a)). For all renormalization schemes, the plateaus coincide and the agreementbetween the renormalization schemes is improved. This is expected since results ob-tained with different target renormalization schemes should be equal up to higher-orderterms, if the input parameters were converted. The relative renormalization scheme

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Γh→4f [MeV]

Scenario Aa

µ0

λ3MS

αMS

FJ

FJ λ3

(d)

Figure 5.7.: The h→4f width at LO with NLO conversion (dashed) and NLO EW (solid)for the scenario Aa in dependence of the renormalization scale. The panels (a), (b), (c),and (d) correspond to input values defined in the λ3 MS, α MS, FJ, and FJ λ3 scheme,respectively. For each of them, the result is computed in all four different renormalizationschemes and displayed using the usual color code.

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µ0

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FJ

FJ λ3

(c)

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1

100 300 900

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Γh→4f [MeV]

Scenario Ab

µ0

λ3MS

αMS

FJ

FJ λ3

(d)

Figure 5.8.: The h→4f width at LO with NLO conversion (dashed) and NLO EW (solid)for the scenario Ab in dependence of the renormalization scale. The panels (a), (b), (c),and (d) correspond to input values defined in the λ3 MS, α MS, FJ, and FJ λ3 scheme,respectively. For each of them, the result is computed in all four different renormalizationschemes and displayed using the usual color code.

101

Input Prescription λ3 MS α MS FJ FJ λ3

Scenario Aa

∆LOren[%] 0.81 0.73 1.25 0.85

∆NLOren [%] 0.30 0.30 0.27 0.29

Scenario Ab

∆LOren[%] 0.98 1.10 1.32 0.81

∆NLOren [%] 0.65 0.62 0.51 0.66

Table 5.7.: The variation of the h→4f width using different renormalization schemes ∆ren

for different input prescriptions.

dependence at the central scale,

∆ren = 2Γh→4fmax (µ0)− Γh→4f

min (µ0)

Γh→4fmax (µ0) + Γh→4f

min (µ0), (5.11)

expresses the dependence of the result on the renormalization scheme. It can be com-puted for a specific input prescription from the difference of the smallest and largestwidth of the four renormalization schemes normalized to their average. In Tab. 5.7,∆ren is given at LO and NLO for each of the input variants and confirms the reductionof the scheme dependence in the NLO calculation. In addition, as already perceivedwhen the running was analyzed, the α MS scheme shows the smallest dependence onthe renormalization scale and this attests a good absorption of further corrections intothe NLO prediction.The situation for the benchmark point Ab is more subtle. For negative values of cβ−α thetruncation of the schemes involving λ3 at 250−300 GeV as well as the breakdown of therunning of the FJ scheme, which both were observed in the running in Fig. 5.6(b), arealso manifest in the computation of the h→4f width. Therefore the results vary muchmore and the extrema with the plateau regions are not as distinct as for the previousbenchmark point, and for the truncated curves even missing. Nevertheless, the situationimproves at NLO and the relative renormalization scheme dependence reduces as shownin Tab. 5.7. Also the central scale choice of µ0 seems to be appropriate in contrast to anaive choice of Mh.For both benchmark points the estimation of the theoretical uncertainties by varyingthe scale by a factor of 2 from the central value is not generally appropriate or evenpossible as unphysical parameters can be reached or a strong running can occur. Thiswould render the uncertainty estimation unreasonable and wrong. We leave a properdefinition of the scale uncertainty to a future project and won’t compute numbers inthis thesis.

5.5.1.4. Scan over cβ−α

The scan in the cβ−α parameter space is one of the central results of our analysis, as thedecay observables of the Higgs boson into four fermions in the THDM are most sensi-

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Γh→4f [MeV]

Scenario A

λ3MS

αMS

FJ

FJ λ3

SM

(d)

Figure 5.9.: The h→4f width at LO with NLO conversion (dashed) and full NLO EW+QCD(solid) for the scenario A in dependence of cβ−α. The panels (a), (b), (c), and (d) corre-spond to input values defined in the λ3 MS, α MS, FJ, and FJ λ3 scheme, respectively.The different target schemes are displayed with the usual colors, and the SM is shown forcomparison in red.

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1

2

3

4

5

−0.2 −0.1 0 0.1 0.2

cβ−α|i

∆ren [%]

Scenario A λ3MS

αMS

FJ

FJ λ3

Figure 5.10.: The relative dependence of the h→4f width on the renormalization schemes asdefined in Eq. (5.11) for the LO with NLO conversion (dashed) and NLO EW+QCD (solid)calculation. The different colors correspond to calculations with input values defined in thedifferent renormalization schemes.

tive to this THDM parameter. The h→4f widths in dependence of cβ−α in scenario Aare pictured in Figs. 5.9(a)-(d) for the four different input prescriptions. The LO withNLO conversion (dashed) and the full NLO EW+QCD total width (solid) are computedin the different renormalization schemes after the NLO input conversion and using theconstant default scale µ0 of Eq. (5.9). The SM values are illustrated in red. The resultsare similar for all input prescriptions so that we discuss them at the same time. At treelevel they show the suppression w.r.t. to the SM with the factor s2β−α. The differencesbetween the renormalization schemes are due to the conversion of the input. A pure LOcomputation is identical for all renormalization schemes as the conversion vanishes. Thisis represented in each plot by the LO curve for which the used renormalization schemeis the one in which the input values are defined. The suppression w.r.t. to the SM com-putation does not change at NLO, while the shape becomes slightly asymmetric and theNLO shows a significantly better agreement between the renormalization schemes. Thisis also confirmed by the relative renormalization scheme dependence shown in Fig. 5.10.We look also into the relative corrections of the width, which are displayed in Fig. 5.11for input parameters defined in the λ3 MS scheme. For input parameters defined in theother schemes we obtain similar results, which are not shown. The different plots showthe full EW+QCD, the QCD, and the EW corrections where the first is just the sum ofthe two individual contributions. The relative QCD corrections lie practically on top ofeach other, so that only one line is visible even though the calculation was made in all

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3

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0

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Scenario A EW

SM λ3MS αMS FJ FJ λ3

Figure 5.11.: The relative NLO corrections of the full EW+QCD, the QCD, and the EWcalculation. The input is defined in the λ3 MS scheme, and the corrections are computedin all four schemes, which are displayed together with the SM corrections using the usualcolor code.

−2

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cβ−α|λ3MS

δNLO [%]

Scenario A SM

full

Hh-mix

THDM-virt

SM-like

Figure 5.12.: The relative NLO corrections spit into different sub-contributions. The SM-likecontribution consists of all diagrams which have a SM equivalent (brown), the THDM-virtcontribution includes all one-loop and CT contributions which have explicit heavy Higgsbosons (orange), and Hh-mixing contribution is displayed in yellow.

105

renormalization schemes. The corrections are almost identical to the SM, which is notsurprising as the interference of the diagram involving a closed fermion loop (Fig. 4.8(c))is the only contribution which factors not from the SM calculation with sβ−α. Thosediagrams contribute only little to the h→4f width, so that the relative QCD correctionsbecome similar to the SM. The EW corrections with the heavy Higgs bosons in the loopshow a small asymmetry w.r.t. to the sign of cβ−α and are between 0 and 3 %, in theregions of large cβ−α even exceeding the relative corrections in the SM.We also investigate the origin of the relative EW corrections and look at the differentcontributions. The first one contains all diagrams which have a SM correspondence,namely the fermionic and the bosonic SM like class of Sect.4.3.2.1 (and their countert-erm parts) as well as the real corrections of Sect. 4.3.2.3. The second part contains thecontributions of the renormalization constants δZHh and δα and therefore the mixingof the heavy and light Higgs boson. The third contribution consists of the remainingbosonic THDM diagrams (and counterterms) of Sect. 4.3.2.1 with a heavy Higgs bosonin the loop. Although these contributions do not form a gauge-invariant subset, theorigin of the differences w.r.t. the SM can be investigated. For the λ3 MS renormal-ization scheme, the single contributions are plotted in Fig. 5.12. The SM-like diagramsalmost reproduce the SM result showing that the modification of the coupling factorsin the THDM is small, but grows when the alignment limit is left. The THDM bosoniccontribution amounts only up to 0.5% and does not account for large deviations fromthe SM, except for the alignment limit, where only these diagrams introduce differencesw.r.t. the SM. The major deviation from the SM and the shape of the EW correctionsare for values of |cβ−α| > 0.05 mainly due to the Hh-Higgs mixing and these termsfactorize from the LO, and are universal for all final states.Deviations of the THDM results from the SM can be investigated when the SM Higgsboson mass is identified with the mass of light Higgs boson of the THDM. The relativedeviation of the full width from the SM is then

∆SM =ΓTHDM − ΓSM

ΓSM

, (5.12)

and we show it in Fig. 5.13 at LO (dashed) and NLO (solid) in percent for parametersdefined in the the λ3 MS scheme (other input definitions yield similar results). The SMexceeds the THDM widths at LO and NLO and at LO shows, as expected, the shapeof c2β−α with modifications due to input conversion. This shape is slightly distorted atNLO through the asymmetry of the EW corrections, and a small offset of −0.5% isvisible even in the alignment limit where the diagrams including heavy Higgs bosonsstill contribute. The NLO computations show larger negative deviations and this couldbe used to improve current exclusion bounds or increase their significance. Nevertheless,in the whole scan region the deviation from the SM is within 6% and for the parameterregion with |cβ−α| < 0.1 even less than 2%, which is challenging for experiments tomeasure.

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∆SM [%]

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λ3MS

αMS

FJ

FJ λ3

Figure 5.13.: The relative difference of the h→4f decay width in the THDM w.r.t. the SMprediction at LO with NLO conversion (dashed) and NLO EW+QCD (solid). The inputscheme is λ3 MS, and the corrections are computed in all four ren. schemes which aredisplayed using the usual color code.

5.5.1.5. Partial Widths

The composition of the h→4f width from the pure charged- and neutral-current pro-cesses and the interference contributions as well as the partial widths (as defined inSect. 4.4) can be used to improve the understanding of the results and provides a pos-sibility to single out final states with enhanced deviations w.r.t. the SM. The partialwidths are automatically computed in all our calculations and shown for the benchmarkpoints Aa and Ab in Tab. 5.8 and Tab. 5.9, respectively. They are produced using theλ3 MS input prescription and renormalization scheme. For other schemes, the numbersdiffer slightly, but show the same pattern so that we do not show them here. In thetables, we do not only state the full NLO QCD+EW partial widths, but also the rela-tive EW and QCD corrections. The qualitative picture is, however, for both benchmarkpoints similar. The WW contribution originating from charged-current final states yieldthe largest contribution, while the ZZ contribution is minor and the interference termyields a small negative contribution. The EW corrections are strongly dependent onthe final states with about 2−3% for final states including charged currents, less forneutral-current final states, and even negative for fully hadronic neutral-current finalstates. The QCD corrections are essentially the strong corrections to W/Z → qq andtherefore amount to αs/π for each pair of quarks in the final state. For uuuu and ddddfinal states the negative interference terms reduce the amount of the QCD correction to

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final state ΓNLO [MeV] δEW [%] δQCD [%] ∆NLOSM [%] ∆LO

SM [%]

Total 0.967297 2.71 4.96 −1.05 −1.00

ZZ 0.106126 0.34 4.88 −1.12 −1.00

WW 0.866304 3.00 5.01 −1.04 −1.00

WW/ZZ int. −0.005134 1.28 11.99 −1.39 −1.34

νee+µ−νµ 0.010201 3.03 0.00 −1.04 −1.00

νee+ud 0.031719 3.02 3.76 −1.05 −1.00

udsc 0.098465 2.97 7.52 −1.04 −1.00

νee+e−νe 0.010197 3.12 0.00 −1.04 −0.99

uddu 0.100473 2.85 7.35 −1.04 −0.99

νeνeνµνµ 0.000949 3.01 0.00 −1.14 −1.00

e−e+µ−µ+ 0.000239 1.30 0.00 −1.15 −1.00

νeνeµ−µ+ 0.000477 2.45 0.00 −1.13 −1.00

νeνeνeνe 0.000569 2.90 0.00 −1.15 −1.00

e−e+e−e+ 0.000132 1.12 0.00 −1.13 −1.00

νeνeuu 0.001679 0.60 3.76 −1.11 −1.00

νeνedd 0.002177 1.69 3.76 −1.12 −1.00

e−e+uu 0.000845 0.11 3.76 −1.12 −1.00

e−e+dd 0.001088 0.47 3.76 −1.12 −1.00

uucc 0.002971 −1.80 7.51 −1.11 −1.00

dddd 0.002556 −0.38 4.38 −1.19 −1.00

ddss 0.004956 −0.36 7.51 −1.11 −1.00

uuss 0.003852 −0.66 7.51 −1.10 −1.00

uuuu 0.001506 −1.92 4.06 −1.23 −1.00

Table 5.8.: Partial widths for the benchmark point Aa in the λ3 MS renormalization scheme.

around 4%. The deviation from the SM expectation ∆SM are shown at NLO and LO inthe last two columns as well. The LO deviation is due to the suppression factor of theHV V coupling s2β−α−1 = −c2β−α and therefore identical for all final states. At NLO thedeviation is slightly larger even though still within only 1.3% (2%) for Aa (Ab) bench-mark point. No single final state deviates with a particularly large value, even thoughsmall differences, way below a percent, between the charged-current and neutral-currentfinal states can be observed. Within a reasonable approximation the deviation of eachfinal state can be expressed by the one of the h→4f width.

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final state ΓNLO [MeV] δEW [%] δQCD [%] ∆NLOSM [%] ∆LO

SM [%]

Total 0.959800 1.87 4.97 −1.82 −1.00

ZZ 0.105464 −0.34 4.90 −1.74 −1.00

WW 0.859376 2.14 5.01 −1.84 −1.00

WW/ZZ int. −0.005040 0.51 10.70 −3.19 −1.34

νee+µ−νµ 0.010116 2.17 0.00 −1.87 −1.00

νee+ud 0.031463 2.16 3.76 −1.85 −1.00

udsc 0.097695 2.11 7.52 −1.82 −1.00

νee+e−νe 0.010112 2.27 0.00 −1.86 −0.99

uddu 0.099720 1.99 7.38 −1.78 −0.99

νeνeνµνµ 0.000943 2.34 0.00 −1.78 −1.00

e−e+µ−µ+ 0.000237 0.62 0.00 −1.81 −1.00

νeνeµ−µ+ 0.000474 1.78 0.00 −1.78 −1.00

νeνeνeνe 0.000565 2.23 0.00 −1.79 −1.00

e−e+e−e+ 0.000131 0.45 0.00 −1.78 −1.00

νeνeuu 0.001668 −0.08 3.76 −1.76 −1.00

νeνedd 0.002163 1.02 3.76 −1.75 −1.00

e−e+uu 0.000840 −0.57 3.76 −1.77 −1.00

e−e+dd 0.001081 −0.21 3.76 −1.76 −1.00

uucc 0.002952 −2.48 7.51 −1.75 −1.00

dddd 0.002545 −1.06 4.57 −1.65 −1.00

ddss 0.004925 −1.04 7.51 −1.73 −1.00

uuss 0.003828 −1.35 7.51 −1.74 −1.00

uuuu 0.001500 −2.60 4.31 −1.64 −1.00

Table 5.9.: Partial widths for the benchmark point Ab in the λ3 MS renormalization scheme.

5.5.1.6. Differential Distributions

The differential decay widths are another window to observe effects of the THDM as theshape of distributions might be distorted significantly by new and additional couplingstructures. This can occur even if the partial widths do not change significantly, andtherefore, the differential distributions of leptonic and semi-leptonic final states (seeSect. 5.1) are important observables for us. In the following, we study them for bothcharged- and neutral-current processes, e.g. the fully leptonic final states e−e+µ−µ+

(nc), νee+µ−νµ (cc), and the semi-leptonic final states e−e+qq (nc), νee

+du (cc). Mostlikely, differential distributions for fully hadronic final states are not experimentallyaccessible, because only jets and not the primary partons are measured in the hadroniccalorimeter. A detailed discussion of the SM distributions including the effects of the

109

photon recombination and final-state radiation can be found for the fully leptonic finalstates in Refs. [36, 37] and for semi-leptonic final states in Ref. [38]. In contrast, in ourstudy we emphasize the differences between the SM and the THDM results, since thefeatures of Prophecy4f for photon recombination work completely in the same wayin the THDM. The distributions are calculated using λ3 MS renormalization scheme;other renormalization schemes yield similar results.

Leptonic final states

We begin with the leptonic final state e−e+µ−µ+ which is a decay mediated by Z bosons.The NLO invariant-mass distributions within the SM and THDM (benchmark points Aaand Ab) are displayed in the first panel of Fig. 5.14(a) and show the Z-boson resonancewith a large tail. The relative corrections w.r.t. the LO, illustrated in the second panel,show the effects of final-state radiation after photon recombination is applied:

Photons radiated from a final-state fermion lower the invariant mass of the fermion pairand lead to positive corrections for invariant masses below the Z-boson threshold andnegative corrections above. These corrections contain a logarithm of the fermion mass sothat they are particularly large for electrons in the final state. The photon recombinationeases this large effect, as the photon momentum is added to one of the fermion pairsif its energy is sufficiently small. This shifts events back to larger invariant masses,but also cancels the mass logarithms, restores the applicability of the Kinoshita–Lee–Nauenberg theorem [24, 25], and leads to an inclusive observable which becomes identicalfor electrons and muons [36]. Yet, the photons with energies above a threshold are notrecombined and still account for a sizable net effect which is observed in the relativecorrections. Nevertheless, the shape is identical for SM and THDM computations sothat the impact of new Lorentz structures in the higher-order diagrams are negligible.Together with the fact that the LO THDM distributions are the SM ones re-weighted bythe factor sβ−α, it is not surprising, that the NLO deviations given in the lowest panelof Fig. 5.14(a) are constant with a value compatible with the deviation of the partialwidth of −1.14% for the benchmark point Aa and −1.80% for Ab. The differential decaywidth with respect to the angle between the µ+µ− and e−e+ decay planes is defined inEq. (5.2) and the distribution is shown in the first panel of Fig. 5.14(b). One observesa cos (2φ) pattern which can be used to determine the parity (see Refs. [108–114]), e.g.the distribution of a pseudo-scalar Higgs boson would have a different sign . Also herethe SM shape is not distorted by THDM effects, and the deviation is constant andsimilar to the one of the invariant-mass distribution. We also looked at the invariant-mass and angular distributions of the e−e+e−e+ final state, for which also interferenceterms appear. There, the assignment of the lepton pairs is not unique and the electronand positron whose invariant mass is closest to MZ is combined to a pair. However,all distributions are similar to the ones of e−e+µ−µ+, both in shape and also in thedifferences between THDM and the SM and therefore not shown.For the W-boson-mediated νee

+µ−νµ final state, similar plots are shown in Fig. 5.15.The invariant-mass distribution of Mνµµ is not experimentally accessible, but shown fortheoretical interest. The peak is here around MW, in the relative corrections the effects

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Figure 5.14.: Invariant-mass (a) and angular distributions (b) of the leptonic neutral-currentdecay h → µ−µ+e−e+ for the SM and the THDM benchmark points Aa and Ab. Therelative NLO corrections to the distributions are plotted in the middle panels. The lowerpanels illustrate the relative deviation of the THDM calculation w.r.t. to the SM at NLO.

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Figure 5.15.: Invariant-mass (a) and angular distributions (b) of the leptonic charged-currentdecay h → νµµ

+e−νe for the SM and the THDM benchmark points Aa and Ab. The relativeNLO corrections to the distributions are plotted in the middle panels. The lower panelsillustrate the relative deviation of the THDM calculation w.r.t. to the SM at NLO.

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of photon radiation are as discussed above, and the shape distortion in the THDM isalso completely negligible. As the neutrinos cannot be detected, neither the Higgs northe W boson can be reconstructed. Though, the angle between the charged leptonsin the plane perpendicular to the beam axis defined in Eq. (5.6) can be studied afterincluding the production process, or if the transverse momentum of the Higgs boson issmall or reconstructed. In the latter case it is possible to boost into the transverse restframe of the Higgs boson, in which the distribution can be studied. It is shown in thefirst panel of Fig. 5.15(b), and no shape distortion can be seen in the relative correctionsof the second panel. The constant deviation w.r.t. SM is also identical to the partialwidth deviation of −1.05% (Aa) and −1.88% (Ab). Other fully leptonic final statesshow similar patterns, so that their distributions are not separately shown here.

Semi-leptonic final states

The approach for the semi-leptonic final states is similar to the leptonic ones. Theinvariant-mass distributions of the quark pair of the neutral-current final state e−e+qqare displayed in Fig. 5.16 (l.h.s) and show similar characteristics as the ones of thecorresponding leptonic final state. SM and THDM show no significant shape difference,and photon radiation shifts the peak positions while the photon recombination ensurescollinear-safe observables. However, the effect of the non-recombined photons is lesspronounced as the charge factors are smaller than for leptons. Similar effects existalso for gluon radiation, however, all gluons are recombined with the quark pair sothat only a flat correction remains which accounts for the major portion of the relativecorrections [38]. As the jets cannot be distinguished, the angle between the decay planescannot be fully resolved so that | cosφ| as defined in Eq. (5.4) is used instead as angularobservable, and the results are shown on the r.h.s. of Fig. 5.16. The distribution is similarto the one of the leptonic case, however, the different parameterization accounts for thedifferent shape and taking the absolute value of cosφ flattens the relative corrections.Nevertheless, in both cases the THDM contributions do not affect the shape, so thatthe e−e+qq distributions deviate only by a flat shift which is in accordance with thedeviation of the partial width, −1.11% (Aa) and −1.76% (Ab).The invariant-mass distribution of the quark pair of the semi-leptonic W-boson-mediatedfinal state νee

+du is pictured in Fig. 5.17 (l.h.s) and shows the well known characteristics.No shape difference between the SM and the THDM is observed, the gluon radiationand recombination accounts for a constant correction, and the photon radiation andrecombination shifts the peaks. Yet, the shift is small as the neutrino does not radiatephotons. The angle between the electron and the hadronically decaying W boson, φeW,in the rest frame of the Higgs boson is the angular observable shown in Fig. 5.17 (r.h.s).The electron is predominantly produced in the opposite direction to W boson, whilethe EW corrections distort this shape slightly in contrast to the THDM contributions.The difference between SM and THDM is described well by the deviation of the partialwidth of −1.05% for the Aa and −1.85% for the Ab benchmark point.To summarize, the effects of the THDM on the shape of the distributions are negligible,and distributions cannot be effectively used in the search for the THDM.

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Figure 5.16.: Invariant-mass (a) and angular distributions (b) of the charged-current semi-leptonic decay h → qqe−νe for the SM and the THDM benchmark points Aa and Ab. Therelative NLO corrections to the distributions are plotted in the middle panels. The lowerpanels illustrate the relative deviation of the THDM calculation w.r.t. to the SM at NLO.

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Figure 5.17.: Invariant-mass (a) and angular distributions (b) of the charged-current semi-leptonic decay h → νee

+du for the SM and the THDM benchmark points Aa and Ab. Therelative NLO corrections to the distributions are plotted in the middle panels. The lowerpanels illustrate the relative deviation of the THDM calculation w.r.t. to the SM at NLO.

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5.5.2. High-Mass Scenario B1

The high-mass scenario is divided into two branches which are valid for positive ornegative cβ−α and have different tanβ. In this section we cover scenario B1 with positivevalues of cβ−α and refer to the next section for the discussion of the scenario B2 withnegative values. The perturbativity measure increases with rising MH as can be seen inFig. 5.3, which restricts the range in cβ−α and possibly affects the quality of the resultsin the high-mass scenario in a negative way. The discussion of the numerical results isstructured in the same way as for the previous scenario, beginning with the conversionof the input parameters for different renormalization schemes.

5.5.2.1. Conversion of the Input Parameters

We compute the conversion between the input values in different renormalization schemesfor the enlarged range of cβ−α = −0.3 . . . 0.3 and use α MS either as input or as targetscheme. For the latter, the conversion equation (3.100) is trivially solved, as the lastcounterterm vanishes, and the result is shown in Fig. 5.18(b). Using input parametersdefined in the FJ scheme leads to particularly large changes in the cβ−α which indicatesthat the NLO terms a, b stated in Tab. 3.3 are large and that the perturbative expan-sion converges poorly. But also for the FJ λ3 scheme sizable shifts are observed, but tosmaller values. The methods used in Sect. 5.5.1.1 to solve the non-linear equation in theconversion from the α MS to other schemes fail in this case, as the large higher-orderterms drive the error of the approximate solution, and the fixed-point iteration fails

cβ−α|αMS

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(b)

Figure 5.18.: Panel (a): Conversion of the value of cβ−α from α MS to the other schemeswith the usual color coding for scenario B1. Panel (b) shows the conversion to the α MSscheme. The solution of (b) is mirrored at the diagonal and displayed in (a) with orangedashed lines. The phenomenologically relevant region is highlighted in the center.

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to converge. Therefore we employ a two-dimensional Newton Method instead, yieldingFig. 5.18(a). Both conversions of Fig. 5.18 should be inverse to each other and we checkthe Newton Method by plotting the curves of (b) mirrored at the diagonal with orangedotted lines. The deviations are due to the projection of the conversion from two dimen-sions (α, β) to a single one and in addition, the Newton Method converges partially todifferent, unphysical solutions which are not shown. Because of the absence of bijectivityin the conversion involving the FJ scheme, a conversion of negative cβ−α input valuesinto the FJ scheme is impossible, albeit this parameter region is not relevant for thisscenario. It seems that the explicit subtraction of the tadpole terms in renormalizationschemes involving the FJ prescription contains logarithms which are enhanced by rela-tively large coupling factors in this scenario. As this affects the conversion strongly wealso expect to see remnants of these large terms in the Higgs widths.

5.5.2.2. The Running of cβ−α

The running of cβ−α is investigated analogously to the low-mass scenario for each renor-malization scheme independently, so that a conversion is not necessary here. The scaledependence of cβ−α(µ0) = 0.1 is computed from 300 to 1500 GeV using a Runge–Kuttamethod and shown in Fig. 5.19 for the benchmark point B1a. Regions where pertur-bativity is not valid (λ/(4π) > 1) are plotted using dotted lines. In comparison tothe low-mass scenario scenario (Fig. 5.6(a)) the scale dependence for the FJ λ3 schemeincreases. For scales above the central scale we obtain large values of cβ−α for whichpredictions become unreliable. But also the FJ scheme shows a remarkable behavior asthe alignment limit is approached for both, low and high scales.

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λ3MS

αMS

FJ

FJ λ3

Figure 5.19.: The running of cβ−α for the benchmark B1 in the α MS (blue), λ3 MS (green),FJ (pink), and FJ λ3 scheme. The breakdown of perturbativity (λ/(4π) > 1) is indicatedby changing the NLO curve to dotted lines.

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5.5.2.3. Scale Variation of the Width

We now turn to the calculation of the h→4f width where we perform a scale variationsimilarly to the previous section in order to estimate theoretical uncertainties and tojustify the central scale choice for the scenario B1a. The scale is varied from 300 GeV to1500 GeV, and the results are shown in Fig. 5.20 with one plot for each input prescription.At first, the input values are converted to the target scheme, and afterwards the scaleis changed. In regions where perturbativity is not valid (λ/(4π) > 1) the NLO resultis plotted using dotted lines. The results do not show such a clear picture as for thelow-mass scenario:

• The λ3 MS, Fig. 5.20(a), and the α MS input prescription, Fig. 5.20(b), yieldsimilar results. In both cases, these schemes as target schemes show very goodagreement, an extremum and a distinct plateau region in which the central scalefits perfectly. They begin to deviate when perturbativity breaks down. The otherrenormalization schemes do not support this as the FJ scheme has a significantoffset and drops dramatically for lower scales, until perturbativity breaks down atabout 400 GeV. The FJ λ3 scheme suffers from the strong running and diverges asexpected, while it shows relatively good (but not stable) agreement with the otherschemes for lower scales.

• For input values defined in the FJ scheme (Fig. 5.20(c)), the conversion transportsthe large NLO corrections to all other schemes, so that perturbativity is not givenat all, and all curves disagree. Together with the behavior of the FJ scheme in theother plots, we conclude that the perturbative predictions using the FJ scheme arenot trustworthy for this benchmark point.

• The FJ λ3 input prescription (Fig. 5.20(d)) seems to yield the best agreementbetween the schemes, however, the conversion to other renormalization schemesresults in particularly small values for cβ−α and therefore corresponds in the otherrenormalization schemes to a scenario closer to the alignment limit. Such scenar-ios have smaller couplings and are perturbatively more stable, so that a betteragreement is not surprising. We even computed the scale dependence of the h→4fwidth for a high-mass scenario B1 with cβ−α = 0.05 in App. C.1, where the reduc-tion of the scale dependence, the development of plateau regions for all schemes,and an overlap of the different schemes can be observed.

The discussion of the B1a benchmark can be quantified by computing the relative renor-malization scheme dependence at the central scale ∆ren which is shown in Tab. 5.10.Although many problems arise for this benchmark point, we observe a reduction of thescale dependence except for the FJ scheme, and especially the renormalization schemesλ3 MS and α MS yield very good results and reliable NLO predictions. If only thosetwo schemes are taken into account in the relative renormalization scheme dependence,the numbers of Tab. 5.10 would be significantly better.

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µ0

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FJ

FJ λ3

(d)

Figure 5.20.: The h→4f width at LO (dashed) and NLO EW (solid) for the scenario B1ain dependence of the renormalization scale. The panels (a), (b), (c), and (d) correspond toinput values defined in the λ3 MS, α MS, FJ, and FJ λ3 scheme, respectively. For eachof them, the result is computed in all four different renormalization schemes and displayedusing the usual color code. The breakdown of perturbativity (λ/(4π) > 1) is indicated bychanging the NLO curve to dotted lines.

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Input Prescription λ3 MS α MS FJ FJ λ3

Scenario B1a

∆LOren[%] 8.04 7.78 50.49 0.98

∆NLOren [%] 6.78 6.53 59.74 0.43

Table 5.10.: The variation of the h→4f width at the central scale using all four renormaliza-tion schemes in the calculation at the benchmark B1a. We show the result for each inputprescription.

5.5.2.4. Scan over cβ−α

The h→4f width of the Higgs decay as a function of positive cβ−α is shown for allcombinations of input prescriptions and renormalization schemes in Fig. 5.21 at thescale µ0. The results obtained using different input definitions differ significantly, butpatterns observed in the investigation of the scale dependence recur. The dashed linesrepresent the LO result with an NLO input conversion, while at pure LO, the conversionbecomes trivial and all renormalization schemes are equal. The respective curve is theone where no conversion is necessary. The well known s2β−α pattern can be observed atLO while the conversion into the FJ λ3 scheme introduces large corrections leading to abreakdown (see Fig. 5.18(a)) so that this scheme is only applicable for very low valuesof cβ−α. The NLO results are more complicated:

• The α MS and the λ3 MS input prescriptions (Figs. 5.21(a),(b)) have similarcharacteristics which is due to the small shifts of the parameters in the conversion.The width in the α MS and the λ3 MS renormalization scheme agrees very welland the agreement improves from LO to NLO, as it is desired. The FJ schemeshows differences which also can be explained by large higher-order terms shiftingthe input values towards the alignment limit (see Fig. 5.18(b)). These shifts areresummed in angular functions and, if large, cannot be compensated by the NLOcalculation. This indicates that further corrections in the FJ scheme are sizableand should be taken into account.

• Using input values defined in the FJ scheme, Fig. 5.21(c), expresses this problemmore clearly. The large corrections spread to the other renormalization schemesand affects perturbativity in a negative way. But also within the FJ scheme thecorrections are large and differ from the s2β−α shape seen for other input variants.This confirms the conclusion of the previous section that predictions obtained usingthe FJ are not reliable for this scenario.

• The good agreement of the renormalization schemes in the FJ λ3 input prescription(Fig. 5.21(d)) is based on the shift of the input values towards the alignment limitin the conversion. This shrinks the range effectively to cβ−α = 0 . . . 0.05 for theother target schemes after the conversion and pushes the results together.

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SM

(d)

Figure 5.21.: The h→4f width at LO (dashed) and NLO EW+QCD (solid) for the scenarioB1 in dependence of cβ−α. The panels (a), (b), (c), and (d) correspond to input valuesdefined in the λ3 MS, α MS, FJ, and FJ λ3 scheme, respectively. The result is displayed inall four schemes and for the SM using the usual color code. The breakdown of perturbativity(λ/(4π) > 1) is visualized by using dotted lines for the NLO curve.

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Figure 5.22.: The relative NLO corrections of the full EW+QCD, the QCD, and the EWcalculation in scenario B1. The input is defined in the λ3 MS scheme, and the correctionsare computed in all four schemes which are displayed using the usual color code. Forcomparison the SM corrections are shown as well.

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Figure 5.23.: The h→4f width at LO with NLO conversion (dashed) and NLO EW+QCD(solid) in the scenario B1 normalized to the respective SM values. The input is defined inthe λ3 MS scheme, and the results in the four schemes are displayed using the usual colorcode.

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For the input defined in the λ3 MS scheme (and similar for α MS), the relative cor-rections separated in EW, QCD, and EW+QCD are shown in Fig. 5.22. The QCDcorrections are similar for all schemes because only the Hff couplings yield differentfactors w.r.t. the SM than the sβ−α factor of the LO. In contrast to the low-mass sce-nario, the EW corrections decrease with increasing cβ−α so that the deviation from theSM shown in Fig. 5.23 are larger than in the low-mass case although it remains below 2%in for cβ−α < 0.1. This deviation could possibly be used to put stronger constraints onthe high-mass scenario with positive cβ−α. The relative renormalization scheme depen-dence can only be applied using the α MS and λ3 MS schemes as the results obtainedusing the two schemes involving FJ prescriptions are only partially reliable. From theplots 5.20(a),(b), one can see that the differences between the α MS and λ3 MS schemesdecrease from LO to NLO and the scheme dependence reduces.

final state ΓNLO [MeV] δEW [%] δQCD [%] ∆NLOSM [%] ∆LO

SM [%]

Total 0.959759 1.88 4.96 −1.82 −1.00

ZZ 0.105308 −0.48 4.88 −1.89 −1.00

WW 0.859588 2.16 5.02 −1.81 −1.00

WW/ZZ int. −0.005136 0.34 12.61 −1.34 −1.00

νee+µ−νµ 0.010118 2.19 0.00 −1.85 −1.00

νee+ud 0.031471 2.18 3.77 −1.82 −1.00

udsc 0.097719 2.14 7.52 −1.79 −1.00

νee+e−νe 0.010113 2.29 0.00 −1.85 −1.00

uddu 0.099692 2.02 7.34 −1.81 −1.00

νeνeνµνµ 0.000941 2.19 0.00 −1.92 −1.00

e−e+µ−µ+ 0.000237 0.49 0.00 −1.94 −1.00

νeνeµ−µ+ 0.000474 1.63 0.00 −1.91 −1.00

νeνeνeνe 0.000564 2.09 0.00 −1.93 −1.00

e−e+e−e+ 0.000131 0.31 0.00 −1.92 −1.00

νeνeuu 0.001666 −0.22 3.75 −1.89 −1.00

νeνedd 0.002160 0.88 3.75 −1.89 −1.00

e−e+uu 0.000839 −0.70 3.76 −1.89 −1.00

e−e+dd 0.001080 −0.35 3.76 −1.89 −1.00

uucc 0.002948 −2.61 7.51 −1.86 −1.00

dddd 0.002537 −1.20 4.42 −1.93 −1.00

ddss 0.004918 −1.17 7.50 −1.86 −1.00

uuss 0.003823 −1.48 7.51 −1.86 −1.00

uuuu 0.001495 −2.73 4.12 −1.95 −1.00

Table 5.11.: Partial widths for the benchmark point B1a within the λ3 MS scheme.

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5.5.2.5. Partial Widths

The partial NLO widths, the relative corrections δEW/QCD, and the deviation from the

SM, ∆LO/NLOSM , are shown in Tab. 5.11 for the benchmark point B1a using the λ3 MS

renormalization scheme. The α MS scheme yields similar results which differ only atthe permille level whereas the other schemes are not reliable at this benchmark point.The widths are slightly smaller than in the low-mass scenario (see Sec. 5.5.1.5) and thenegative deviation from the SM rises to almost 2%, however, no final state accounts fordistinctively large THDM effects that could be exploited in experiments.In addition, the differential distributions of the scenario B1 as defined in Sect. 5.1 donot change the shape w.r.t. to the SM significantly and are shown in App. C.2, togetherwith the SM ones and the ones from the scenario B2b. The deviation of the total width,or more accurate, the one of the respective final state resemble the constant shift in allof the distributions very well.

5.5.3. High-Mass Scenario B2

To complete the discussion of the high-mass scenario, we turn to the negative values ofcβ−α for which the parameter space is strongly reduced by perturbativity, stability, andunitarity constraints, leaving only a small branch around tanβ = 1.5 and leading tothe scenario B2. Being at the vicinity of excluded parameter sets could possibly affectthe conversion, the scale dependence, and the quality of the full results. We discuss theresults in the same manner as the scenario for positive cβ−α above.

5.5.3.1. Conversion of the Input Parameters

The conversion of the input parameter cβ−α between different renormalization schemesis shown in Fig. 5.24 for an enlarged range with α MS as either input or target schemewhich ensures in both cases that only one counterterm in the conversion equation (3.100)is non-vanishing. The conversion into the α MS scheme shows several ominous features.First of all, a divergence for the λ3 MS and FJ λ3 schemes occur at cβ−α ≈ −0.19. Wesaw such a divergence already in the low-mass scenario outside of the relevant region (c.f.Sect. 5.5.1.1), and it is caused by the relation between α and λ3 which becomes singularat c2α = 0. Since the ratio of the vacuum expectation values is lower in this scenario, thedivergence moves towards the alignment limit and closer to the relevant region. It affectsthe conversion for cβ−α|(FJ)λ3MS

< −0.1 and such values should be taken with care. Ifexperimental observations favor this region of parameter space, it becomes necessary toredefine the renormalization scheme and choose a different Higgs self-coupling parameter(e.g. λ1) or a combination (e.g. λ3+λ4) as independent parameter renormalized in MS.The singularity then appears in other parameter regions and allows for predictions in thevicinity of cβ−α = −0.2. Not only the schemes involving λ3 are problematic, but also theconversion from the FJ scheme, as large shifts indicate problems with the perturbativeexpansion analogously to the scenario B1.The “inverse” conversion from the α MS scheme involves the solution of a non-linear

124

cβ−α|αMS

λ3MS

FJ

FJ λ3−0.3

−0.2

−0.1

0

0.1

0.2

0.3

−0.3 −0.2 −0.1 0 0.1 0.2 0.3

µ = 559 GeV

cβ−α|i

Scenario B2

(a)

cβ−α|i

λ3MS

FJ

FJ λ3−0.3

−0.2

−0.1

0

0.1

0.2

0.3

−0.3 −0.2 −0.1 0 0.1 0.2 0.3

µ = 559 GeV

cβ−α|αMS

Scenario B2

(b)

Figure 5.24.: Conversion of the value of cβ−α from α MS to the other renormalizationschemes (a) for scenario B2. (b) shows the conversion to the α MS scheme with the samecolor coding. The orange dashed line is the solution of (b) mirrored at the diagonal. Thephenomenological relevant region is highlighted in the center.

equation for which the approximate solution encounters large errors and the fixed-pointiteration does not converge (more details to the methods are given in Sect. 5.5.1.1).Therefore we employ, as for the scenario B1, a Newton Method and show the results inFig. 5.24(a), together with the inverse of (b) obtained graphically by mirroring the curvesat the diagonal (dashed orange). Although the comparison of these curves involves areduction of the conversion to one dimension and is therefore not exact, it gives a quickoverview over the convergence of the Newton Method. As expected, the singularity inthe λ3−α relation reduces the domain of definition for the conversion in Fig. 5.24(a) tocβ−α|αMS

> −0.1 for the λ3 MS scheme and cβ−α|αMS> −0.05 for the FJ λ3 scheme.

Values outside this domain cannot be converted into these schemes and predictionscannot be made.

5.5.3.2. The running of cβ−α

The running of cβ−α(µ0) = −0.1 is computed from 300 GeV to 1500 GeV with a Runge–Kutta method for the benchmark point B2b. The result is shown in Fig. 5.25 for eachrenormalization scheme independently and no conversion is performed. The curves looksimilar to the one of the low-mass scenario pictured in Fig. 5.6(b) (although the rangeof µ is different) and we see the same effects: the truncation of the λ3 MS and FJ λ3schemes, but also the strong scale dependence of the FJ scheme and the good stabilityof the α MS scheme.

125

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

300 600 1200

µr [GeV]

tβ(µ0) = 1.5

cβ−α|i

µ0 = 559 GeV

Scenario B2b

λ3MS

αMS

FJ

FJ λ3

Figure 5.25.: The running of cβ−α for the benchmark B2 for the different schemes in theusual color code.

5.5.3.3. Scale Variation of the Width

For the h→4f width we perform as well a scale variation in order to estimate theoreticaluncertainties and to motivate the central scale choice. The method is as described inSect. 5.5.1.3, and the results are shown in Fig. 5.26. The FJ λ3 renormalization schemedoes not appear, as it is not possible to convert input values to it for cβ−α = −0.1 (seeFig. 5.24(a)), however, it can be used when the input values are defined in it. Theobservations correspond in the most cases to the ones of the scenario B1:

• The first two plots using parameters defined in the λ3 MS (Fig. 5.26(a)) and α MS(Fig. 5.26(b)) scheme show, as in the previous scenarios, similar characteristics.The result obtained with the α MS renormalization scheme shows almost no scaledependence, and its value agrees with the extremum of the the λ3 MS renormal-ization scheme which lies at the central scale. However, through the truncation ofthe running a clear plateau region cannot be observed for the latter scheme. TheFJ scheme has an offset at the plateau and decreases strongly for scales below µ0,as expected from the running of cβ−α.

• The results using the FJ input prescription (Fig. 5.26(c)) cannot be compared aslarge corrections from the conversion spread to all other schemes.

• The scale variation of the FJ λ3 input prescription (Fig. 5.26(d)) corresponds againto an aligned scenario for other renormalization schemes. Closer to the alignment,the agreement improves, which can also be seen from the separate scale variationof a more aligned scenario with cβ−α = −0.05 given in App. C.1.

However, the relative scale dependence reduces from LO to NLO as shown in Tab. 5.12,verifying that our computation improves the predictions and scale uncertainties reduce.

126

0.8

0.85

0.9

0.95

1

300 600 1200

µr [GeV]

cβ−α|λ3MS(µ0) = −0.1

Γh→4f [MeV]

Scenario B2b

µ0

λ3MS

αMS

FJ

(a)

0.8

0.85

0.9

0.95

1

300 600 1200

µr [GeV]

cβ−α|αMS

(µ0) = −0.1

Γh→4f [MeV]

Scenario B2b

µ0

λ3MS

αMS

FJ

(b)

0.8

0.85

0.9

0.95

1

300 600 1200

µr [GeV]

cβ−α|FJ(µ0) = −0.1

Γh→4f [MeV]

Scenario B2b

µ0

λ3MS

αMS

FJ

(c)

0.8

0.85

0.9

0.95

1

300 600 1200

µr [GeV]

cβ−α|FJλ3(µ0) = −0.1

Γh→4f [MeV]

Scenario B2b

µ0

λ3MS

αMS

FJ

FJ λ3

(d)

Figure 5.26.: The h→4f width at LO (dashed) and NLO EW (solid) for the scenario B2bin dependence of the renormalization scale. The panels (a), (b), (c), and (d) correspond toinput values defined in the λ3 MS, α MS, FJ, and FJ λ3 scheme, respectively. For eachof them, the result is computed in all four different renormalization schemes and displayedusing the usual color code. The FJ λ3 scheme is not defined as target schemes due tothe singular relation between α and λ3. The breakdown of perturbativity (λ/(4π) > 1) isindicated by changing the NLO curve to dotted lines.

127

Input Prescription λ3 MS α MS FJ FJ λ3

Scenario B1

∆LOren[%] 0.81 1.40 3.35 0.99

∆NLOren [%] 0.18 0.46 3.80 0.25

Table 5.12.: The variation of the h→4f width at the central scale using different renormal-ization schemes in the calculation at the benchmark B2b. We show the result for each inputprescription.

We obtain a better improvement compared to the benchmark point B1a, which probablyoriginates from smaller perturbativity measures (see Fig. 5.3). The central scale µ0 =(Mh+MH+MA0 +2MH±)/5 is a justifiable choice and shows that this scale is applicablegenerally to the THDM Higgs decay into four fermions. The renormalization schemesα MS and λ3 MS yield trustworthy and comparable results, even though one shouldrespect the domain of definition of the latter. Results in the FJ scheme should be takenwith care, while the FJ λ3 scheme cannot be applied for this parameter set.

5.5.3.4. Scan over cβ−α

The h→4f width in dependence of cβ−α is shown for the different input prescriptions inthe four Figures 5.27, for all renormalization schemes. The plots deviate significantly,demanding an individual discussion:

• The curves obtained using the λ3 MS and the FJ λ3 input prescription, Fig. 5.27(a)and Fig. 5.27(d), are affected by the singular region. For small values an increasecan be observed, which originates from the rise of cβ−α in the conversion. Asthese effects are unphysical they show that the applicability breaks down whenthe singularity is approached.

• Using input values defined in the α MS yields the smooth curves of Fig. 5.27(b)which have the expected s2β−α shape. The relative renormalization scheme depen-

dence reduces from LO to NLO, while the breakdown of the λ3 MS and FJ λ3schemes is manifest.

• The FJ input prescription shows largest deviations from the SM as large NLOcontributions spread to the other schemes through the conversion, shifting thevalues away from the alignment limit and increasing the deviations w.r.t. SM.

However, all results show a significantly better agreement between the renormalizationschemes at NLO for all regions including the problematic ones, suggesting that theperturbative expansion works for this scenario in despite of partially large NLO terms.In addition, we do not observe the complete breakdown of perturbativity (λ/(4π) < 1)in this scenario which supports the previous statement.As the λ3 MS scheme has a limited region of applicability, we show in Fig. 5.28 therelative corrections using the α MS scheme which is reliable over the whole scan range.

128

0.8

0.85

0.9

0.95

1

−0.15 −0.1 −0.05 0

cβ−α|λ3MS

Γh→4f [MeV]

Scenario B2

λ3MS

αMS

FJ

FJ λ3

SM

(a)

0.8

0.85

0.9

0.95

1

−0.15 −0.1 −0.05 0

cβ−α|αMS

Γh→4f [MeV]

Scenario B2

λ3MS

αMS

FJ

FJ λ3

SM

(b)

0.8

0.85

0.9

0.95

1

−0.15 −0.1 −0.05 0

cβ−α|FJ

Γh→4f [MeV]

Scenario B2

λ3MS

αMS

FJ

FJ λ3

SM

(c)

0.8

0.85

0.9

0.95

1

−0.15 −0.1 −0.05 0

cβ−α|FJλ3

Γh→4f [MeV]

Scenario B2

λ3MS

αMS

FJ

FJ λ3

SM

(d)

Figure 5.27.: The h→4f width at LO (dashed) and NLO EW+QCD (solid) for the scenarioB2 in dependence of cβ−α. The panels (a), (b), (c), and (d) correspond to input valuesdefined in the λ3 MS, α MS, FJ, and FJ λ3 scheme, respectively. The input valuesare converted to the desired target scheme (usual color code) in which the calculation isperformed. The SM is shown for comparison.

129

4

5

6

7

8

9

δNLO [%]

Scenario B2 EW+QCD

3

4

5

6

7Scenario B2 QCD

−1

0

1

2

3

4

−0.15 −0.1 −0.05 0

cβ−α|αMS

Scenario B2 EW

SM λ3MS αMS FJ FJ λ3

Figure 5.28.: The relative NLO corrections of the full EW+QCD, the QCD, and the EWcalculation in the B2 scenario. The input is defined in the α MS scheme and the correctionsare computed in all four schemes which are displayed together with the SM corrections usingthe usual color code.

−6

−5

−4

−3

−2

−1

0

1

−0.15 −0.1 −0.05 0

cβ−α|αMS

∆SM [%]

Scenario B2

λ3MS

αMS

FJ

FJ λ3

Figure 5.29.: The h→4f width at LO (dashed) and NLO EW+QCD (solid) in the THDMscenario B2 normalized to the respective SM values. The input is defined in the α MSscheme and the corrections are computed in all four schemes which are displayed using theusual color code.

130

final state ΓNLO [MeV] δEW [%] δQCD [%] ∆NLOSM [%] ∆LO

SM [%]

Total 0.960854 1.99 4.97 −1.71 −1.00

ZZ 0.105584 −0.22 4.90 −1.63 −1.00

WW 0.860363 2.26 5.02 −1.72 −1.00

WW/ZZ int. −0.005093 0.52 11.46 −2.19 −1.00

νee+µ−νµ 0.010128 2.29 0.00 −1.76 −1.00

νee+ud 0.031499 2.28 3.77 −1.73 −1.00

udsc 0.097805 2.23 7.52 −1.70 −1.00

νee+e−νe 0.010123 2.39 0.00 −1.75 −1.00

uddu 0.099810 2.12 7.37 −1.69 −1.00

νeνeνµνµ 0.000944 2.46 0.00 −1.67 −1.00

e−e+µ−µ+ 0.000237 0.74 0.00 −1.69 −1.00

νeνeµ−µ+ 0.000475 1.89 0.00 −1.66 −1.00

νeνeνeνe 0.000566 2.35 0.00 −1.68 −1.00

e−e+e−e+ 0.000131 0.57 0.00 −1.66 −1.00

νeνeuu 0.001670 0.04 3.75 −1.65 −1.00

νeνedd 0.002165 1.13 3.75 −1.65 −1.00

e−e+uu 0.000841 −0.45 3.76 −1.65 −1.00

e−e+dd 0.001082 −0.09 3.76 −1.65 −1.00

uucc 0.002955 −2.36 7.51 −1.63 −1.00

dddd 0.002548 −0.94 4.59 −1.52 −1.00

ddss 0.004930 −0.92 7.50 −1.63 −1.00

uuss 0.003832 −1.23 7.51 −1.62 −1.00

uuuu 0.001502 −2.48 4.35 −1.49 −1.00

Table 5.13.: Partial widths for the benchmark point B2b in the αMS renormalization scheme.

The QCD corrections are similar to the SM and renormalization scheme independent,while the EW corrections show the breakdown of the λ3 MS and FJ λ3 schemes. Thedifference between the FJ and the α MS scheme is slightly larger than in the low-mass case, however, the size of the corrections are almost equal. This results in similardeviations from the SM as can be seen by comparison of Fig. 5.29 with Fig. 5.13, so thatit is difficult to distinguish these scenarios using the Higgs decay into four fermions.

5.5.3.5. Partial Widths

We give the partial widths in Tab. 5.13 for the benchmark point B2b and in the α MSscheme, as this scheme provides reliable results for cβ−α = −0.1. All of the total andpartial widths are similar to the ones of the low-mass benchmark Ab (Tab. 5.9) in size,

131

but also in their relative EW and QCD corrections as well as in their deviation fromthe SM. Again, there is no final state particularly sensitive to the THDM contributions.The differential distributions analogous to Sect. 5.5.1.6 are shown together with the dis-tributions of the high-mass benchmark B1 in App. C.2 and yield no significant shapedistortion w.r.t. the SM, but only constant shifts that match the deviation of the re-spective partial widths.

5.5.4. Different THDM Types

In this section, we compare the h→4f decay widths of the Type I, Type II, leptonspecific, and flipped models for the two benchmark points Aa and B1a using the λ3MS renormalization scheme. The results are shown in Tab. 5.14 with the numericalerrors in parenthesis, and they confirm our expectation: The differences originatingfrom the different types in the higher-order corrections are below a permille, and theones of the complete width are even less. They are an order of magnitude smaller thanthe numerical error, although we employ large statistics with 190 million phase-spacepoints. The difference between Type I and the lepton specific model is only due to themodification hll couplings and this effect cannot even be resolved. We therefore concludethat our decay observable is insensitive with respect to the types of THDM, so that ourpredictions can be used universally for all types.

Scenario Aa

Model Γh→4fNLO [MeV] Γh→4f

EW [MeV] Γh→4fQCD [MeV]

Type 1 0.967297(68) 0.024352(13) 0.044581(46)

Type 2 0.967288(68) 0.024351(13) 0.044573(46)

Lepton-specific 0.967297(68) 0.024352(13) 0.044581(46)

Flipped 0.967287(68) 0.024350(13) 0.044573(46)

Scenario B1a

Model Γh→4fNLO [MeV] Γh→4f

EW [MeV] Γh→4fQCD [MeV]

Type 1 0.958118(66) 0.016809(11) 0.044602(46)

Type 2 0.958111(67) 0.016818(11) 0.044587(46)

Lepton-specific 0.958117(66) 0.016809(11) 0.044602(46)

Flipped 0.958111(67) 0.016818(11) 0.044587(46)

Table 5.14.: The h→4f widths for the different types of THDM for the scenarios Aa and B1awith the numerical errors in parenthesis.

132

300 350 400 450 500 550 600 650 700 750

MH

1

10

tanβ

−10

−8

−6

−4

−2

0

2cβ−α|λ3MS

= 0.1Z4 = Z5 = −2

Z7 = 0

Figure 5.30.: The deviation w.r.t. the SM at NLO for the benchmark plane scenario in the λ3

MS scheme. Gray areas are excluded by perturbativity while the corrections are indicatedby the color. We interpolate linearly between computed points to obtain a smooth result.

5.5.5. Benchmark Plane

For the benchmark plane scenario defined in Sect. 5.4 we analyze only the relativedeviation with respect to the SM in the λ3 MS scheme. At LO, this is −1% as cβ−α = 0.1is kept constant. The NLO corrections differentiate this picture as they are dependenton both, the heavy Higgs boson masses and tan β. We show the result for the wholeplane in Fig. 5.30 where the color of the parameter point indicates the deviation and grayareas are excluded by perturbativity constraints (λ/(4π) > 1). We interpolate betweenthe computed parameter points to obtain a smooth result, however, the original grid canbe seen at the border between the computed and the perturbative excluded area whereno meaningful interpolation is possible. The major deviation is negative and between0 and 5% and grows with increasing tanβ. For very large values of this parameter andclose to the perturbative exclusion, also values up to 8% occur. Very interesting is alsothe region with a small tanβ, as very small enhancements with respect to the SM canbe found around MH = 300 GeV (displayed in green). However, this region has a strongmass dependence because for large masses the negative corrections become −5%. Wenote that this effect is also visible using the α MS scheme and therefore cannot be anartifact of the singularity of the λ3 MS scheme.

133

5.5.6. Baryogenesis

In this section we discuss the results of the benchmark sets BP3 which were proposed asa possible solution to the problem of baryogenesis and defined in Sect. 5.4. We computethe results shown in Tab. 5.15 in the λ3 MS scheme without considering the otherschemes. In despite of the large distance to the alignment limit, the small heavy-Higgs-boson masses render both scenarios perturbatively relatively stable with perturbativitymeasures of 0.38 and 0.40. Already at tree level we observe a large negative deviationfrom the SM caused by the large values for cβ−α suppressing the hV V coupling. Theseeffects are enhanced at NLO for which we observe an increase of the negative deviationby 3%. This should be used in experiments measuring the Higgs decay into four fermionsto put stronger bounds on these scenarios.

ΓNLOTot [MeV] ΓEW [MeV] ΓQCD [MeV] ∆NLO

SM ∆LOSM

BP3B1 0.860421(81) −0.006236(7) 0.040929(59) −9.00 −11.98

BP3B2 0.702399(66) −0.011780(7) 0.033633(49) −25.00 −28.15

Table 5.15.: The h→4f widths in the λ3 MS scheme including the EW and QCD correctionsof the benchmarks BP3B1,B2 with the numerical errors in parenthesis. The last columnsshow the deviation w.r.t. the SM at LO and NLO.

5.5.7. Fermiophobic Heavy Higgs

The results for the fermiophobic heavy Higgs scenario defined in Sect. 5.4 are shown inTab. 5.16 for the λ3 MS scheme. The three scenarios have a perturbativity measureof 0.6 and differ by their value of tanβ and their deviation from the alignment limit.However, all scenarios are quite close to this limit, so that the SM width is almostreached at LO. The NLO correction shows, however, an increase of the deviation by1.25% for all scenarios. Although this does not seem to be large, it opens up windowsfor precision experiments to find these deviations or to rule out scenarios which wouldbe impossible at LO.

ΓNLOTot [MeV] ΓEW [MeV] ΓQCD [MeV] ∆NLO

SM ∆LOSM

BP6a 0.964564(92) 0.012704(13) 0.045033(64) −0.07 −1.33

BP6b 0.963041(92) 0.012963(13) 0.044947(64) −0.25 −1.49

BP6c 0.957009(91) 0.013992(13) 0.044607(64) −0.99 −2.10

Table 5.16.: The h→4f widths the λ3 MS scheme including the EW and QCD corrections ofthe benchmarks BP6a−c with the numerical errors in parenthesis. The last columns showthe deviation w.r.t. the SM at LO and NLO.

134

6. Summary and Outlook

Since no BSM particles have been directly found in experiments so far, precision observ-ables play a crucial role in the search for hints of new physics. They allow to quantifydeviations from the SM predictions, and BSM models can be tested against those mea-surements. However, a high experimental accuracy demands at least NLO precision inthe SM as well as for BSM predictions. The THDM extends the scalar sector of the SMand allows for a comprehensive study of the impact of new scalar degrees of freedomwithout introducing new fundamental symmetries. In this thesis we have renormalizedthe Type I, II, lepton specific, and flipped THDM and have built the foundation to anautomated computation of NLO cross sections and decay widths within these models.In the renormalization procedure we computed the counterterm Lagrangian in several in-dependent ways and defined two new, complete, and consistent renormalization schemes,namely the λ3 MS and the all-order gauge-invariant FJ λ3 scheme. In addition, we havedescribed and implemented the existing α MS and FJ schemes rendering a comparisonof popular renormalization schemes possible. Since renormalization conditions differ be-tween the renormalization schemes, also the meaning of renormalized parameters andtheir connection to observables are renormalization scheme dependent. The values ofthe parameters are related by a conversion which takes the NLO differences in theirdefinitions into account. The counterterm Lagrangian and the renormalization condi-tions have been implemented in FeynRules, which allows us to create a FeynArts

model file with which NLO amplitudes and Fortran code for the numerical evaluation ofsquared matrix elements can be fully automatically generated for arbitrary processes.As an application, we have considered the decay process

h→WW/ZZ → 4f (6.1)

in the THDM, where we identify the light neutral CP-even Higgs boson with the dis-covered particle. This signature contributed to the discovery of the Higgs boson andis important for the measurement of the Higgs boson’s properties. Furthermore, it iscrucial in the search for deviations from SM predictions. The calculation of the NLOEW+QCD corrections in the THDM is important to quantify deviations from the SM.To this end, we have extended the program Prophecy4f from the SM calculation tothe THDM one, and the code will be made publicly available soon. Experimental physi-cists can use the program exactly in the same way as before, so that an inclusion of ourresults in experimental analyses is straightforward.We have computed results for several benchmark scenarios proposed by the LHC HiggsCross Section Working Group. For all investigated scenarios, we observe that the THDMwidths are bounded by the SM ones and that the deviations from the SM increase at

135

NLO, which can be used to improve exclusion limits in the parameter space. The par-tial widths have similar deviations for all final states, and the shapes of the differentialdistributions are not distorted by THDM contributions, so that they cannot be usedeffectively in the search for the THDM.Furthermore, a significant change in the conversion of the values of the scheme depen-dent renormalized parameters to other renormalization schemes, indicate problems withrespect to perturbativity, stability, and the scale dependence. The alignment limit is alsoaffected by the definition of the parameters as it does not persist when the scale or therenormalization scheme is changed. This shows that a proper definition of parametersat NLO is mandatory and compulsory in the future, as so far a consistent treatmentcannot be found in literature.The comparison of the different renormalization schemes reveals that the gauge-depen-dent α MS scheme has a minimal scale dependence which reflects the perturbativestability. The λ3 MS scheme deviates only slightly from the former, yields reliable re-sults and is in addition gauge invariant at one loop in Rξ gauges. However, a singularregion exists in which the scheme is not defined and artificially large corrections are in-troduced. If this region is experimentally favored, it is necessary to redefine the schemein such a way that the singularity is avoided. The gauge-invariant FJ (λ3) schemespartially suffer from large corrections, and can only be applied for parameter pointswith sufficiently small coupling factors. Since the different schemes do not yield reliableresults for all scenarios, self-consistency checks should be performed for every scenariobefore higher-order corrections are computed.In the low-mass scenario with a heavy CP-even Higgs boson of 300 GeV we obtaintextbook-like results for the scale dependence, i.e. an improvement of the scale uncer-tainty and a reduction of differences between all four renormalization schemes at NLO,which indicates that perturbation theory works well. The deviations from the SM are,depending on the parameter parameter set, between 0% and −6%, of which the NLOcalculation is responsible for 1−2%. In the high-mass scenario, the coupling factors arelarger, resulting in less predictive results and larger differences between the renormal-ization schemes. While the α MS and the λ3 MS (in its domain of definition) schemesstill yield trustworthy results, the FJ and the FJ λ3 scheme suffer from large correctionsand their results should be taken with care. The deviations from the SM are similarto the low-mass case, and the NLO corrections similarly account for 1−2%. The otherinvestigated scenarios support the described picture as they yield similar results.

Outlook:

As all results show a constant, parameter-dependent factor w.r.t. the SM at NLO, ananalytic computation of this factor is an important continuation of this work. Thiswould result in a better understanding of the THDM corrections, but also in a fasterevaluation of the THDM corrections. With the foundation of automated computationsof NLO correction being laid, it is planned in future projects to compute precisionobservables to further processes in the THDM. For this also other amplitude generatorsmight be employed, since FeynRules provides interfaces with many more amplitudegenerators.

136

Appendix

137

A. Feynman Rules

In this appendix we list the generic tree-level Feynman Rules for the THDM. Couplingswhich do not involve a scalar particle are identical to the SM and not stated here, theycan be found together with the rules for the propagators in Ref. [48].

FFS-coupling:

S

F2

F1

= ie(C−ω− + C+ω+) (A.1)

with the values (the ξ-factors are listed in Tab. 2.4)

H, f , f : C− = C+ = −ξfHmf

2MWsW

h, f , f : C− = C+ = −ξfhmf

2MWsW

A0, f , f : C− = −C+ = −ξfA0

mf

2MWsW

G, f , f : C− = −C+ = −I3Wmf

MWsW

H+, u, d :

{

C− = ξuA0

mfu√2MWsW

C+ = −ξdA0

mfd√2MWsW

H+, ν, l :

{C− = 0C+ = −ξlA0

mfl√2MWsW

H−, d, u :

{

C− = −ξdA0

mfd√2MWsW

C+ = ξuA0

mfu√2MWsW

H−, l, ν :

{C− = −ξlA0

mfl√2MWsW

C+ = 0

G+, u, d :

{

C− =mfu√

2MWsW

C+ = − mfd√2MWsW

G+, ν, l :

{C− = 0C+ = − mfl√

2MWsW

139

G−, d, u :

{

C− = − mfd√2MWsW

C+ =mfu√

2MWsW

G−, l, ν :

{C− = − mfl√

2MWsW

C+ = 0

SSS-coupling:

S1

S3

S2

= ieC (A.2)

with the values

h,A0,A0 : C =4M2

A0c2βcα−β + cα+β (16λ5s

2WM2

W/e2 − 3M2

h)−M2hcα−3β

4s2βsWMW

h,G,A0 : C =

(M2

A0−M2

h

)cα−β

2sWMW

h,G,G : C =M2

hsα−β2sWMW

h,G±,G∓ : C =M2

hsα−β2sWMW

h, h, h : C =−3(cα+β

(3M2

h − 4c2α−β(M2

A0+ 4λ5s

2WM

2W/e

2))

+M2hc3α−β

)

4sWs2βMW

H,A0,A0 : C =2λ5sWMWsα+β

e2cβsβ−(M2

H − 2M2A0

)sα−3β +

(3M2

H − 2M2A0

)sα+β

4sWs2βMW

H,G,A0 : C =

(M2

A0−M2

H

)sα−β

2sWMW

H,G,G : C =− M2Hcα−β

2sWMW

H,G±,G∓ : C =− M2Hcα−β

2sWMW

h, h,H : C =− s2αcα−β (2M2h +M2

H)

2sWs2βMW− cα−β (s2β − 3s2α)

(e2M2

A0+ 4λ5s

2WM2

W

)

2e2sWMWs2β

h,H,H : C =− s2αsα−β (M2h + 2M2

H)

2sWs2βMW+sα−β (3s2α + s2β)

(e2M2

A0+ 4λ5s

2WM

2W

)

2e2sWMWs2β

H,H,H : C =3(sα+β

(4s2α−β

(e2M2

A0+ 4λ5s

2WM

2W

)− 3e2M2

H

)+ e2M2

Hs3α−β)

4e2sWs2βMW

140

A0,H±,G∓ : C =± i

(M2

A0−M2

H±

)

2sWMW

h,H±,G∓ : C =cα−β

(M2

H± −M2h

)

2sWMW

H,H±,G∓ : C =sα−β

(M2

H± −M2H

)

2sWMW

h,H+,H− : C =cα+β

(4M2

A0− 3M2

h − 2M2H±

)− cα−3β

(M2

h − 2M2H±

)

4sWs2βMW

+4λ5sWMWcα+β

e2s2β

H,H+,H− : C =sα+β

(4M2

A0− 3M2

H − 2M2H±

)− sα−3β

(M2

H − 2M2H±

)

4sWs2βMW

+4λ5sWMWsα+β

e2s2β

SSSS-coupling:

S2

S1

S4

S3

= ie2C (A.3)

A0,A0,A0,A0 : C =3e

16M2Ws

2Ws22β

(64λ5M2Ws

2Wc22β

e2+ 16M2

A0c22β

−M2h (3cα+β + cα−3β)

2 −M2H (3sα+β + sα−3β)

2)

G,A0,A0,A0 : C =3e

16M2Ws

2Ws2β

(32λ5M2Ws

2Wc2β

e2+ 8M2

A0c2β

+M2h

(−c2(α−2β) − 3c2α − 4c2β

)+M2

H

(c2(α−2β) + 3c2α − 4c2β

) )

G,G,A0,A0 : C =e

16M2Ws

2Ws2β

(32λ5M2Ws

2Ws2β

e2+ 8M2

A0s2β

+M2h

(3s2(α−2β) + s2α − 6s2β

)−M2

H

(3s2(α−2β) + 6s2β + s2α

) )

G,G,G,A0 : C =3e(M2

h −M2H)s2(α−β)

8M2Ws

2W

G,G,G,G : C =−3e(M2

Hc2α−β +M2

hs2α−β)

4M2Ws

2W

141

G+,G−,A0,A0 : C =e

16M2Ws

2Ws2β

(32λ5M2Ws

2Ws2β

e2+ 8M2

A0s2β − 8M2

H±s2β

+M2h

(s2(α−2β) + 3s2α − 2s2β

)−M2

H

(s2(α−2β) −+s2α + 2s2β

) )

G,G+,G−,A0 : C =e(M2

h −M2H)s2(α−β)

8M2Ws

2W

G,G,G+,G− : C =−e(M2

Hc2α−β +M2

hs2α−β)

4M2Ws

2W

G+,G+,G−,G− : C =−e(M2

Hc2α−β +M2

hs2α−β)

2M2Ws

2W

h, h,A0,A0 : C =e

16M2Ws

2Ws22β

(

−M2h (3cα+β + cα−3β) (c3α−β + 3cα+β)

−M2Hs2α

(s2(α−2β) + 3s2α + 2s2β

)

+4λ5M

2Ws

2W

(2(c2(α−β) + c4β + 3

)+ 5c2(α+β) + c2(α−3β)

)

e2

+ 8M2A0c2β (c2α + c2β)

)

h, h,G,A0 : C =e

16M2Ws

2Ws2β

(32λ5M2Ws

2Wc2βc

2α−β

e2+ 4M2

A0(c2α + c2β)

+M2h (−c4α−2β − 4c2α − 3c2β)− 2M2

Hs2αs2(α−β)

)

h, h,G,G : C =e

16M2Ws

2Ws2β

(32λ5M2Ws

2Ws2βc

2α−β

e2− 4M2

Hs2αc2α−β

+M2h (s4α−2β + 2s2α − 3s2β)

)

h, h,G+,G− : C =e

16M2Ws

2Ws

22β

(32λ5M2Ws

2Ws

22βc

2α−β

e2+ 8M2

A0s22βc

2α−β

− 4M2Hs2αs2βc

2α−β − 8M2

H±s22βc2α−β

+M2hs2β (s4α−2β + 2s2α − 3s2β)

)

h, h, h, h : C =3e

16M2Ws

2Ws22β

(16λ5M2Ws

2W(c2α + c2β)

2

e2+ 4M2

A0(c2α + c2β)

2

−M2h (c3α−β + 3cα+β)

2 − 4M2Hs

22αc

2α−β

)

h,H,A0,A0 : C =e

16M2Ws

2Ws22β

(

8M2A0s2αc2β −M2

hs2α(c2(α−2β) + 3c2α + 4c2β

)

+M2Hs2α

(c2(α−2β) + 3c2α − 4c2β

)

+4λ5M

2Ws

2W

(s2(α−3β) + 2s2(α−β) + 5s2(α+β)

)

e2

)

h,G,H,A0 : C =e

4M2Ws

2Ws2β

(4λ5M2Ws

2Wc2βs2(α−β)e2

+M2A0s2α

142

−M2hs2αc

2α−β −M2

Hs2αs2α−β

)

h,G,G,H : C =es2α(M

2h −M2

H)s2(α−β)8M2

Ws2Ws2β

+λ5s2(α−β)

e

h,G+,G−,H : C =e

8M2Ws

2Ws2β

(8λ5M2Ws

2Ws2βs2(α−β)e2

+ 2M2A0s2βs2(α−β)

+M2hs2αs2(α−β) −M2

Hs2αs2(α−β) − 2M2H±s2βs2(α−β)

)

h, h, h,H : C =3e

16M2Ws

2Ws22β

(16λ5M2Ws

2Ws2α (c2α + c2β)

e2+ 4M2

A0s2α (c2α + c2β)

−M2hs2α (c4α−2β + 4c2α + 3c2β)− 2M2

Hs22αs2(α−β)

)

H,H,A0,A0 : C =e

16M2Ws

2Ws22β

(

M2H (s3α−β − 3sα+β) (sα−3β + 3sα+β)

+ 8M2A0c2β (c2β − c2α)−M2

hs2α(s2(α−2β) + 3s2α − 2s2β

)

− 4λ5M

2Ws

2W

e2(c2(α−3β) + 2c2(α−β) + 5c2(α+β) − 2c4β − 6

) )

G,H,H,A0 : C =e

16M2Ws

2Ws2β

(32λ5M2Ws

2Wc2βs

2α−β

e2+ 4M2

A0(c2β − c2α)

+M2H (−c4α−2β + 4c2α − 3c2β)− 2M2

hs2αs2(α−β)

)

G,G,H,H : C =e

16M2Ws

2Ws2β

(32λ5M2Ws

2Ws2βs

2α−β

e2

+ 4M2hs2αs

2α−β +M2

H (s4α−2β − 2s2α − 3s2β))

G+,G−,H,H : C =e

16M2Ws

2Ws2β

(32λ5M2Ws

2Ws2βs

2α−β

e2+ 8M2

A0s2βs

2α−β

+ 4M2hs2αs

2α−β +M2

H (s4α−2β − 2s2α − 3s2β)− 8M2H±s2βs

2α−β

)

h, h,H,H : C =e

16M2Ws

2Ws

22β

(8λ5M2Ws

2W (−3c4α + c4β + 2)

e2

+M2A0

(−6c4α + 2c4β + 4) +M2hs2α (−3s4α−2β − 6s2α + s2β)

−M2Hs2α (−3s4α−2β + 6s2α + s2β)

)

h,H,H,H : C =3e

16M2Ws

2Ws22β

(16λ5M2Ws

2Ws2α (c2β − c2α)

e2+ 4M2

A0s2α (c2β − c2α)

−M2Hs2α (c4α−2β − 4c2α + 3c2β)− 2M2

hs22αs2(α−β)

)

H,H,H,H : C =3e

16M2Ws

2Ws

22β

(16λ5M2Ws

2W(c2α − c2β)

2

e2+ 4M2

A0(c2α − c2β)

2

− 4M2hs

22αs

2α−β −M2

H (s3α−β − 3sα+β)2)

143

G±,A0,A0,H∓ : C =

e

16M2Ws

2Ws2β

(32λ5M2Ws

2Wc2β

e2+ 8M2

A0c2β

+M2h

(−c2(α−2β) − 3c2α − 4c2β

)+M2

H

(c2(α−2β) + 3c2α − 4c2β

) )

G,G±,A0,H∓ : C =

e(−M2

hc2α−β −M2

Hs2α−β +M2

H±

)

4M2Ws

2W

G,G,G±,H∓ : C =e(M2

h −M2H)s2(α−β)

8M2Ws

2W

G+,G−,G±,H∓ : C =e(M2

h −M2H)s2(α−β)

4M2Ws

2W

h,G±,A0,H∓ : C =± ie(M2

A0−M2

H±)sα−β

4M2Ws

2W

h,G,G±,H∓ : C =± ie(M2A0

−M2H±)cα−β

4M2Ws

2W

h, h,G±,H∓ : C =ecα−β

16M2Ws

2Wcβsβ

(cα+β(2M

2A0

− 3M2h −M2

H + 2M2H±)

+ 2(M2A0

−M2H±)cα−3β + (M2

H −M2h)c3α−β

)+λ5c2βc

2α−β

ecβsβ

G±,H,A0,H∓ : C =± ie(M2

H± −M2A0)cα−β

4M2Ws

2W

G,G±,H,H∓ : C =± ie(M2A0

−M2H±)sα−β

4M2Ws

2W

h,G±,H,H∓ : C =e

4M2Ws

2Ws2β

(4λ5M2Ws

2Wc2βs2(α−β)e2

+M2A0c2βs2(α−β) −M2

hs2αc2α−β +M2

H±s2βc2(α−β) −M2Hs2αs

2α−β

)

G±,H,H,H∓ : C =λ5c2βs

2α−β

ecβsβ− esα−β

8M2Ws

2Ws2β

(

sα+β(−2M2A0

+M2h + 3M2

H − 2M2H±)

+ 2(M2H± −M2

A0)sα−3β + (M2

h −M2H)s3α−β

)

G±,G±,H∓,H∓ : C =e(−M2

hc2α−β −M2

Hs2α−β +M2

A0

)

2M2Ws

2W

A0,A0,H+,H− : C =

e

16M2Ws

2Ws22β

(64λ5M2Ws

2Wc22β

e2+ 8M2

A0(c4β + 1)

−M2h (cα−3β + 3cα+β)

2 −M2H (sα−3β + 3sα+β)

2)

G,A0,H+,H− : C =

e

16M2Ws

2Ws22β

(16λ5M2Ws

2Ws4β

e2+ 4M2

A0s4β

+ (M2H −M2

h)s2β(c2(α−2β) + 3c2α

)− 2(M2

H +M2h)s4β

)

144

G,G,H+,H− : C =e(

(M2h −M2

H)s2(α−2β)+3s2α

s2β+ 8M2

A0− 2(M2

h +M2H + 4M2

H±))

16M2Ws

2W

+2λ5e

G+,G−,H+,H− : C =e(

(M2h −M2

H)c2βs2(α−β)

s2β+M2

A0−M2

h −M2H

)

4M2Ws

2W

+2λ5e

h, h,H+,H− : C =e

32M2Ws

2Ws22β

(

2M2A0

(c2(α−3β) + 2

(c2(α−β) + c4β + 3

)+ 5c2(α+β)

)

+8λ5M

2Ws

2W

(c2(α−3β) + 2

(c2(α−β) + c4β + 3

)+ 5c2(α+β)

)

e2

+M2h

(−6c2(α−β) − c4(α−β) − 10c2(α+β) − 3c4α − 3c4β − 9

)

− 2M2Hs2α

(s2(α−2β) + 3s2α + 2s2β

)− 16M2

H±s22βs2α−β

)

h,H,H+,H− : C =e

16M2Ws

2Ws22β

(4λ5M2Ws

2W

(s2(α−3β) + 2s2(α−β) + 5s2(α+β)

)

e2

+M2A0

(s2(α−3β) + 2s2(α−β) + 5s2(α+β)

)

−M2hs2α

(c2(α−2β) + 3c2α + 4c2β

)

+M2Hs2α

(c2(α−2β) + 3c2α − 4c2β

)+ 4M2

H±s22βs2(α−β)

)

H,H,H+,H− : C =e

32M2Ws

2Ws

22β

(

− 2M2A0

(c2α−6β + 2c2(α−β) + 5c2(α+β) − 2c4β − 6

)

− 8λ5M2Ws

2W

(c2(α−3β) + 2c2(α−β) + 5c2(α+β) − 2c4β − 6

)

e2

+M2H

(6c2(α−β) − c4(α−β) + 10c2(α+β) − 3c4α − 3c4β − 9

)

− 16M2H±s22βc

2α−β − 2M2

hs2α(s2(α−2β) + 3s2α − 2s2β

) )

G±,H∓,H+,H− : C =4λ5c2βes2β

− e

8M2Ws

2Ws2β

(

(M2h −M2

H)(c2(α−2β) + 3c2α

)

+ 4c2β(−2M2A0

+M2h +M2

H))

H+,H+,H−,H− : C =e

8M2Ws

2Ws22β

(64λ5M2Ws

2Wc22β

e2+ 8M2

A0(c4β + 1)

−M2h (cα−3β + 3cα+β)

2 −M2H (sα−3β + 3sα+β)

2)

145

SVV-coupling:

S

V2,ν

V1,µ

= iegµνC (A.4)

with the values

G±,W∓, γ : C = −MW

H,W+,W− : C =MWcα−βsW

h,W+,W− : C = −MWsα−βsW

G±,W∓,Z : C = −sWMW

cW

H,Z,Z : C =MWcα−βc2WsW

h,Z,Z : C = −MWsα−βc2WsW

VSS-coupling:

Vµ

S2, k2

S1, k1

= ieC(k1 − k2)µ (A.5)

with the values

G+,G−, γ : C = 1

H+,H−, γ : C = 1

h,G±,W∓ : C = ±sα−β2sW

h,H±,W∓ : C = ∓cα−β2sW

H,G±,W∓ : C = ∓cα−β2sW

H,H±,W∓ : C = ∓sα−β2sW

A0,H±,W∓ : C =

i

2sW

146

G,G±,W∓ : C =i

2sW

h,A0,Z : C =icα−β2cWsW

h,G,Z : C = − isα−β2cWsW

H,A0,Z : C =isα−β2cWsW

H,G,Z : C =icα−β2cWsW

G−,G+,Z : C =s2W − c2W2cWsW

H−,H+,Z : C =s2W − c2W2cWsW

SSVV-coupling:

S2

S1

V2,ν

V1,µ

= ie2gµνC (A.6)

with the values

G+,G−, γ, γ : C = 2

H+,H−, γ, γ : C = 2

G,G±, γ,W∓ : C = ± i

2sW

H,G±, γ,W∓ : C = −cα−β2sW

A0,H±, γ,W∓ : C = ± i

2sW

h,H±, γ,W∓ : C = −cα−β2sW

h,G±, γ,W∓ : C =sα−β2sW

H,H±, γ,W∓ : C = −sα−β2sW

A0,A0,W+,W− : C =

1

2s2W

G,G,W+,W− : C =1

2s2W

147

G+,G−,W+,W− : C =1

2s2W

h, h,W+,W− : C =1

2s2W

H,H,W+,W− : C =1

2s2W

H+,H−,W+,W− : C =1

2s2W

G+,G−, γ,Z : C =s2W − c2WcWsW

H+,H−, γ,Z : C =s2W − c2WcWsW

G,G±,Z,W∓ : C = ± i

2cW

h,G±,Z,W∓ : C =sα−β2cW

H,G±,Z,W∓ : C = −cα−β2cW

A0,H±,Z,W∓ : C = ± i

2cW

h,H±,Z,W∓ : C = −cα−β2cW

H,H±,Z,W∓ : C = −sα−β2cW

A0,A0,Z,Z : C =1

2c2Ws2W

G,G,Z,Z : C =1

2c2Ws2W

G+,G−,Z,Z : C =(s2

W− c2

W)2

2c2Ws2W

h, h,Z,Z : C =1

2c2Ws2W

H,H,Z,Z : C =1

2c2Ws2W

H+,H−,Z,Z : C =(s2

W− c2

W)2

2c2Ws2W

148

SUU-coupling:

S

U2

U1

= ieC (A.7)

with the values

G±, u±, uγ : C =MW

G, u±, u± : C = ∓iMW

2sW

H, u±, u± : C = −MWcα−β2sW

h, u±, u± : C =MWsα−β

2sW

G±, u±, uZ : C =MW(s2

W− c2

W)

2cWsW

G±, uZ , u∓ : C =MW

2cWsW

H, uZ , uZ : C = −MWcα−β2c2

WsW

h, uZ , uZ : C =MWsα−β2c2WsW

149

B. Other Renormalization

Prescriptions

In this part, we present the other renormalization procedures we implemented to obtaina counterterm Lagrangian. The second method of the list in Sect. 3.1.1 is based ona renormalization of the basic parameters from which afterwards the renormalizationconstants and the counterterm potential of the mass parameter set is derived. First, wepresent in Sect. B.1 a prescription of this method in which the mixing angles are notrenormalized and afterwards in Sect. B.2 a modification which allows for a renormaliza-tion of the mixing angles and a matrix valued field renormalization. A method similarthan the one presented in Sect. 3.1.1.3 which is based on a renormalization of the barepotential of Eq. (3.16) is given in Sect. B.3, however, in contrast to Sect. 3.1.1.3 themixing angles are not renormalized and λ3 is an independent parameter.

B.1. Renormalization of the Basic Parameters – version

2a

In this renormalization prescription we proceed analogously to Dabelstein in the MSSM[80], since the renormalization transformation is applied to the basic parameters {pbasic}and the potential of Eq. (2.51) while the Higgs field renormalization is performed foreach Higgs doublet and not for each component field. This corresponds to the followingrenormalization transformations of the parameters

m211,0 = m2

11 + δm211, m2

22,0 = m222 + δm2

22, m212,0 = m2

12 + δm212, (B.1)

λ1,0 = λ1 + δλ1, λ2,0 = λ2 + δλ2, λ3,0 = λ3 + δλ3,

λ4,0 = λ4 + δλ4, λ5,0 = λ5 + δλ5,

g1,0 = g1 + δg1, g2,0 = g2 + δg2,

and for the fields

Φ1,0 = Z1/2H1

Φ1 = Φ1(1 +1

2δZH1), (B.2)

Φ2,0 = Z1/2H2

Φ2 = Φ2(1 +1

2δZH2).

Even though the vacuum expectation values are not defining parameters, we introducerenormalization constants for them,

v1,0 = v1 − δv1 = Z1/2H1

(v1 − δv1), v2,0 = v2 − δv2 = Z1/2H2

(v2 − δv2). (B.3)

150

As the vevs are part of the Higgs field one can separate the contributions from thefield and the parameter renormalization as done in the second definition. This canbe useful, since a consistency relation between δv1 and δv2 exists [83], which we useas renormalization condition in Sect. 3.2. By a linearization in the renormalizationconstants, one obtains the full NLO potential V + δV (αem). For a better bookkeeping,we write for the counterterm potential

δV = cT · δRbasic, (B.4)

where

δRbasic =(

δλ1, . . . , δλ5, δm211, δm

222, δm

212, δv1, δv2, δg1, δg2

︸ ︷︷ ︸

{δpbasic}

, δZH1, δZH2

)T

(B.5)

is the 14-component set of renormalization constants consisting of 12 parameter renor-malization constants {δpbasic} and two field renormalization constants, as defined inEqs. (B.1), and (B.2). The dependence on the basic parameters {pbasic} and the fieldsdefined in Eq. (2.46) are put into the coefficients c. This minimal set of renormaliza-tion constants is sufficient to render all Green functions finite. However, for calculatingprocesses at NLO, we are interested in the counterterm Feynman rules as functions ofthe mass parameter set and their corresponding renormalization constants and in de-pendence of fields corresponding to mass eigenstates. Thus, it is necessary to expressthe counterterm potential through such parameters and fields.In the special convention of Dabelstein [80], which we denote with δD, the countertermsof the masses are defined by

δDM2H = 2

∂2δV

∂H2

∣∣∣∣fields=0

, (B.6)

δDM2h = 2

∂2δV

∂h2

∣∣∣∣fields=0

, (B.7)

δDM2A0

= 2∂2δV

∂A2

∣∣∣∣fields=0

, (B.8)

δDM2H± =

∂2δV

∂H+∂H−

∣∣∣∣fields=0

. (B.9)

As δV contains the terms from the field renormalization, their contributions are partiallyabsorbed into the definitions of the mass renormalization constants δDM . This is incontrast to the more common definitions of the mass renormalization constants whichdo not include terms arising from the field redefinition.For both, bare and renormalized quantities, the angle β is defined via the ratio of thevacuum expectation values so that renormalization of Eq. (2.65) yields

δβ = cos β sin β

(δv1v1

− δv2v2

)

=δv1v2v2

− δv2v1v2

− δZH1v1v22v2

+δZH2v1v2

2v2. (B.10)

151

The tadpole counterterms are obtained by

δtH =∂δV

∂H

∣∣∣∣fields=0

, δth =∂δV

∂h

∣∣∣∣fields=0

. (B.11)

Finally, as the Eqs. (2.70) hold for bare and renormalized quantities, the W/Z-bosonmass and electric charge renormalization constants are

δM2W

M2W

= 2δg2g2

+ c2β

(

δZH1 − 2δv1v1

)

+ s2β

(

δZH2 − 2δv2v2

)

, (B.12)

δM2Z

M2Z

= 2g1δg1 + g2δg2

g21 + g22+ c2β

(

δZH1 − 2δv1v1

)

+ s2β

(

δZH2 − 2δv2v2

)

, (B.13)

δe

e= δZe =

g32δg1 + g31δg2(g31g2 + g32g1)

. (B.14)

The rotations stated in Eqs. (2.59) are not subject to renormalization and are appliedafter the renormalization, i.e. applied to the renormalized fields. Thus, the notion ofbare angles is not meaningful and the angles cannot represent free parameters of thetheory which must be defined as bare quantities. For each rotation we define the angleto equal the tree-level angle, viz α as defined in Eq. (2.67) and βn = βc = β, where β isdefined from the renormalized ratio of the vacuum expectation values. This definitionof the angles diagonalizes the potential only at LO, while counterterm contributionsintroduce mass mixing terms. Furthermore, one cannot trade λ3 and δλ3 for the angleα as the latter cannot represent a free parameter. The 10 linear equations (B.6)–(B.14)relate the renormalization constants from the potential, δRbasic, to

δDR′mass

(B.15)

=(δDM

2H, δDM

2h , δDM

2A0, δDM

2H±, δM2

W, δM2Z, δe, δλ5, δλ3, δβ, δtH, δth

︸ ︷︷ ︸

{δDp′mass}

, δZH1, δZH2

)T.

The coupling δλ5 remains unchanged. There is no simple corresponding physical pa-rameter, since λ5 originates from the Higgs self-interactions and has to be determinedfrom the Higgs self-couplings. The parameter renormalization constants in Eq. (B.15)correspond to the parameters from the set {p′mass}. By inverting the relations, we obtain

δDRbasic = KδDR′mass

. (B.16)

The transformation can be used to eliminate the dependence on basic renormalizationconstants from the counterterm potential resulting in

δV = c({pbasic})TK({pbasic})δDR′mass

. (B.17)

The parameters and fields appearing in the coefficients c and in the transformationmatrix K are renormalized quantities, for which we can use the LO transformationdiscussed in Sect. 2.2.2 resulting in

δV ({p′mass}) = c′({p′mass})T · δDR′mass

. (B.18)

152

B.2. A modified prescription – version 2b

In order to compare the counterterm potential with the method 1a, the above prescrip-tion was modified. In the modified version, the rotation of the fields of Eqs. (2.59) isperformed before the renormalization using bare angles. They transform in the renor-malization as

βc,0 = βn,0 = β0 = β + δβ, α0 = α+ δα. (B.19)

The parameter renormalization transformation is applied as in Eqs. (B.1) and (B.3). Allrelations to the renormalization constants in the mass parameterization are derived with-out applying a field renormalization, which is in accordance with the usual definitionsof mass counterterms. The counterterm δα can be defined by demanding

∂2δV

∂H∂h

∣∣∣∣fields=0

= 0. (B.20)

Together with the tree-level definition, one can eliminate the parameter λ3 and its coun-terterm in favor of the parameter α and its counterterm, so that the 12 parameter renor-malization constants are the ones of Eq. (3.27). The field renormalization is performedin addition for each field corresponding to mass eigenstates according to Eq. (3.31). Theresulting counterterm potential can be directly compared to the result of Sect. 3.1.1.3and we obtain full agreement.

B.3. Field Rotation after Renormalization – version 1b

This prescription is similar to the one of Sect. 3.1.1.3, however, the rotations of the fieldsare applied to the renormalized fields after the renormalization procedure. Therefore, α,βn, and βc are pure mixing angles and λ3 must be chosen to parameterize the potential(corresponding to the set {p′mass}). As no counterterms to the mixing angles exist, wecan write their behavior in the renormalization transformation schematically as

α0 = α+ 0, βc,0 = β + 0, βn,0 = β + 0. (B.21)

This is analogous to the renormalization of the MSSM suggested in Ref. [145], wherethe additional angle does not obtain any higher-order corrections. Each parameter ofEq. (3.14) has to be renormalized

M2H,0 =M2

H + δM2H, M2

h,0 =M2h + δM2

h , M2A0,0 =M2

A0+ δM2

A0, (B.22)

M2H±,0 =M2

H± + δM2H± , β0 = β + δβ, λ3,0 = λ3 + δλ3,

λ5,0 = λ5 + δλ5 M2W,0 =M2

W + δM2W, M2

Z,0 =M2Z + δM2

Z, ,

e0 = e + δe, tH,0 = 0 + δtH, th,0 = 0 + δth,

so that the parameter renormalization constants are

{δp′mass} = {δM2H, δM

2h , δM

2A0, δM2

H±, δM2W, δM

2Z, δe, δλ5 δλ3, δβ, δtH, δth}. (B.23)

153

In addition we renormalize each field according to Eq. (3.31). Applying the renormal-ization transformation of Eqs.(B.22,3.31) results in

V ({p′mass}) + δV ({p′mass}, {δR′mass}) (B.24)

with the already known LO potential and the counterterm potential up to quadraticterms

δV ({p′mass}, {δR′mass}) =− δtHH − δthh (B.25)

+1

2(δM2

H + δZHM2H)H

2 +1

2(δM2

h + δZhM2h)h

2

+1

2(δM2

A0+ δZAM

2A0)A2

0 + (δM2H± + δZH±M2

H±)H+H−

+e

4MWsW(δth sin (α− β)− δtH cos (α− β))(G2

0 + 2G+G−)

+ (δM2Hh +

1

2M2

HδZHh +1

2M2

hδZhH)Hh

+ (δM2AG +

1

2M2

A0δZAG)A0G0

+ (δM2H± +

1

2M2

H±δZH±G±)(H+G− +G+H−),

with the mixing terms

δM2Hh =

1

2t2α(δM2

h − δM2H

)+s2β(δM2

A0− 2δM2

H±

)

2c2α(B.26)

+δβc2β (M

2H −M2

h ) t2αs2β

+4M2

Wcβsβ(δλ3 + δλ5)s2W

e2c2α

−(s2β(M2

A0− 2M2

H±

)+ (M2

h −M2H)s2α

)(δM2

WsW + 2M2W (δsW − δZesW))

2M2Wc2αsW

− e (δth (cα−3β + 3cα+β) + δtH (sα−3β + 3sα+β))

8MWc2αsW,

δM2A0G = −M2

A0δβ − e

δthcα−β + δtHsα−β2MWsW

,

δM2HG± = −M2

H±δβ − eδthcα−β + δtHsα−β

2MWsW.

Comparison of the different computations

To compare the result with the renormalization of the basic parameters (method 2a,Sect. B.1), one has to adapt the conventions for the mass renormalization constants,and put the treatment of the field renormalization constants on equal footing. For theformer, a redefinition of the mass counterterms is necessary, so that they are in theDabelstein convention and include the field renormalization constants:

δDM2H = 2

∂2δV

∂H2

∣∣∣∣fields=0

, (B.27)

154

δDM2h = 2

∂2δV

∂h2

∣∣∣∣fields=0

, (B.28)

δDM2A0

= 2∂2δV

∂A2

∣∣∣∣fields=0

, (B.29)

δDMH± =∂2δV

∂H+∂H−

∣∣∣∣fields=0

. (B.30)

In addition, in the Dabelstein approach, there are only two independent field renormal-ization constants originating from each Higgs doublet, which are expressed through δZH1 ,δZH2 . By choosing a renormalization scheme in which Eq. (3.35) with δα = δβ = 0 holdsalso for the finite parts, one can express the 12 field renormalization constants intro-duced in Eq. (3.31) by δZH1 and δZH2 . The resulting counterterm potential is comparedto the one of method 2a, order by order in interactions of the fields. We find completeanalytic agreement in all terms. In addition the counterterm potential of Eq. (B.25) wasdirectly compared to a computation from H. Rzehak (method 3), with full agreement.

155

C. Further Results

In this section, we show additional results for the scale variation and differential distri-butions of the high-mass scenario. All the diagrams are similar to ones discussed alreadyin Sect. 5.5, and they do not change our major conclusions. However, in order to providethe reader with all details, we show them here.

C.1. Scale Variation in the High-Mass Scenario

As we stated in Sect. 5.5.3.3, the comparison of the scale dependence in different renor-malization schemes agree better for benchmark points closer to the alignment limit.Therefore we perform a scale variation using the benchmark scenarios B1 and B2 withcβ−α = ±0.05. These points are closer to the alignment limit and the scale dependence ofthe total width using these benchmark points (Figs. C.1,C.2) shows significant improve-ment w.r.t. to the ones of B1a and B2b (Figs. 5.20, 5.26). In addition the conversioninto the FJ λ3 is possible when the alignment limit is approached so that this scheme isincluded in the comparison of Fig. C.1.

156

0.8

0.85

0.9

0.95

1

300 600 1200

µr [GeV]

cβ−α|λ3MS(µ0) = 0.05

Γh→4f [MeV]

Scenario B1

µ0

λ3MS

αMS

FJ

FJ λ3

(a)

0.8

0.85

0.9

0.95

1

300 600 1200

µr [GeV]

cβ−α|αMS

(µ0) = 0.05

Γh→4f [MeV]

Scenario B1

µ0

λ3MS

αMS

FJ

FJ λ3

(b)

0.8

0.85

0.9

0.95

1

300 600 1200

µr [GeV]

cβ−α|FJ(µ0) = 0.05

Γh→4f [MeV]

Scenario B1

µ0

λ3MS

αMS

FJ

FJ λ3

(c)

0.8

0.85

0.9

0.95

1

300 600 1200

µr [GeV]

cβ−α|FJλ3(µ0) = 0.05

Γh→4f [MeV]

Scenario B1

µ0

λ3MS

αMS

FJ

FJ λ3

(d)

Figure C.1.: The h → 4f cross section at LO (dashed) and NLO EW (solid) for the scenarioB1 with cβ−α = 0.05 in dependence of the renormalization scale. The panels (a), (b),(c), and (d) correspond to input values defined in the λ3 MS, α MS, FJ, and FJ λ3

scheme, respectively. The result are shown in the different schemes with usual color code.The breakdown of perturbativity (λ/(4π) > 1) is indicated by changing the NLO curve todotted lines.

157

0.8

0.85

0.9

0.95

1

300 600 1200

µr [GeV]

cβ−α|λ3MS(µ0) = −0.05

Γh→4f [MeV]

Scenario B2

µ0

λ3MS

αMS

FJ

FJ λ3

(a)

0.8

0.85

0.9

0.95

1

300 600 1200

µr [GeV]

cβ−α|αMS

(µ0) = −0.05

Γh→4f [MeV]

Scenario B2

µ0

λ3MS

αMS

FJ

FJ λ3

(b)

0.8

0.85

0.9

0.95

1

300 600 1200

µr [GeV]

cβ−α|FJ(µ0) = −0.05

Γh→4f [MeV]

Scenario B2

µ0

λ3MS

αMS

FJ

FJ λ3

(c)

0.8

0.85

0.9

0.95

1

300 600 1200

µr [GeV]

cβ−α|FJλ3(µ0) = −0.05

Γh→4f [MeV]

Scenario B2

µ0

λ3MS

αMS

FJ

FJ λ3

(d)

Figure C.2.: The h → 4f cross section at LO (dashed) and NLO EW (solid) for the scenarioB2 with cβ−α = −0.05 in dependence of the renormalization scale. The panels(a), (b), (c),and (d) correspond to input values defined in the λ3 MS, α MS, FJ, and FJ λ3 scheme,respectively. The result are shown in the different schemes with usual color code and for suchsmall cβ−α the FJ λ3 scheme is defined as target scheme. The breakdown of perturbativity(λ/(4π) > 1) is indicated by changing the NLO curve to dotted lines.

158

C.2. Differential Distributions of the High-Mass

Scenario

All the investigated distributions do not change the shape, but only differ by a constantoffset w.r.t. the SM. The distributions of the low-mass scenario are shown in Sect. 5.5.1.6,while the ones for the high-mass scenario are similar. For the benchmark points B1aand B2b we compare them to the SM for leptonic final states in Figs. C.3 and C.4, andfor semi-leptonic ones in C.5 and C.6.

159

10−7

10−6

10−5

h → µ−µ+e−e+dΓdMµµ

−10

−5

0

5

10

15

δNLO [%]

−3

−2

−1

0

1

60 70 80 90 100

Mµµ[GeV]

∆SM [%]

SM

B1a

B2b

(a)

5.5

6.0

6.5

7.0

7.5

h → µ−µ+e−e+dΓdφ

[

10−7 MeVdeg

]

−6

−4

−2

0

2

4

6

δNLO [%]

−3

−2

−1

0

1

0 90 180 270 360

φ[deg]

∆SM [%]

SM

B1a

B2b

(b)

Figure C.3.: Invariant-mass (a) and angular distributions (b) of the leptonic neutral-currentdecay h → µ−µ+e−e+ for the SM and the THDM benchmark points B1b and B2b. Therelative NLO corrections to the distributions are plotted in the middle panels. The lowerpanels illustrate the relative deviation of the THDM calculation w.r.t. to the SM at NLO.

160

10−5

10−4

10−3

h → νµµ+e−νe

dΓdMνµµ

−10

−5

0

5

10

15

δNLO [%]

−3

−2

−1

0

1

50 60 70 80 90

Mνµµ[GeV]

∆SM [%]

SM

B1a

B2b

(a)

1

2

3

4

h → νµµ+e−νedΓ

dφµe,T

[

10−5 MeVdeg

]

0

2

4

6

δNLO [%]

−3

−2

−1

0

1

0 90 180 270 360

φµe,T[deg]

∆SM [%]

SM

B1a

B2b

(b)

Figure C.4.: Invariant-mass (a) and angular distributions (b) of the leptonic charged-currentdecay h → νµµ

+e−νe for the SM and the THDM benchmark points B1a and B2b. Therelative NLO corrections to the distributions are plotted in the middle panels. The lowerpanels illustrate the relative deviation of the THDM calculation w.r.t. to the SM at NLO.

161

10−6

10−5

10−4

h → qqe−e+dΓdMqq

0

2

4

6

8

10

δNLO [%]

−3

−2

−1

0

1

60 70 80 90 100

Mqq[GeV]

∆SM [%]

SM

B1a

B2b

(a)

0

0.005

0.01

h → qqe−e+dΓd| cos φ|[MeV]

0

2

4

6

8

10

δNLO [%]

−3

−2

−1

0

1

0 0.2 0.4 0.6 0.8 1

| cos φ|

∆SM [%]

SM

B1a

B2b

(b)

Figure C.5.: Invariant-mass (a) and angular distributions (b) of the charged-current semi-leptonic decay h → qqe−νe for the SM and the THDM benchmark points B1a and B2b.The relative NLO corrections to the distributions are plotted in the middle panels. Thelower panels illustrate the relative deviation of the THDM calculation w.r.t. to the SM atNLO.

162

10−5

10−4

10−3

h → νee+dudΓ

dMqq

0

2

4

6

8

10

δNLO [%]

−3

−2

−1

0

1

50 60 70 80 90

Mqq[GeV]

∆SM [%]

SM

B1a

B2b

(a)

0

0.01

0.02

0.03

0.04

0.05

0.06

h → νee+dudΓ

d cosφeW[MeV]

0

2

4

6

8

10

δNLO [%]

−3

−2

−1

0

1

−1 −0.5 0 0.5 1

cosφeW

∆SM [%]

SM

B1a

B2b

(b)

Figure C.6.: Invariant-mass (a) and angular distributions (b) of the charged-current semi-leptonic decay h → νee

+du for the SM and the THDM benchmark points B1a and B2b.The relative NLO corrections to the distributions are plotted in the middle panels. Thelower panels illustrate the relative deviation of the THDM calculation w.r.t. to the SM atNLO.

163

Bibliography

[1] ATLAS, G. Aad et al., Phys. Lett. B716 (2012) 1, arXiv:1207.7214 [hep-ex].

[2] CMS, S. Chatrchyan et al., Phys. Lett. B716 (2012) 30,arXiv:1207.7235 [hep-ex].

[3] P. W. Higgs, Phys.Lett. 12 (1964) 132.

[4] P. W. Higgs, Phys.Rev.Lett. 13 (1964) 508.

[5] F. Englert and R. Brout, Phys.Rev.Lett. 13 (1964) 321.

[6] P. W. Higgs, Phys.Rev. 145 (1966) 1156.

[7] T. Kibble, Phys.Rev. 155 (1967) 1554.

[8] ATLAS, CMS, M. Flechl, J. Phys. Conf. Ser. 631 (2015) 1, 012028,arXiv:1503.00632 [hep-ex].

[9] J. Polchinksi, String Theory: Volume 1, An Introduction to the Bosonic String.Cambridge University Press.

[10] B. Zwiebach, A First Course in String Theory. Cambridge University Press.

[11] A. Martin, W. Stirling, R. Thorne, and G. Watt, Eur.Phys.J. C63 (2009) 189,arXiv:0901.0002 [hep-ph].

[12] Particle Data Group, C. Patrignani et al., Chin. Phys. C40 (2016) 10, 1682.

[13] G. Cacciapaglia and F. Sannino, JHEP 04 (2014) 111,arXiv:1402.0233 [hep-ph].

[14] J. J. van der Bij, Phys. Lett. B636 (2006) 56,arXiv:hep-ph/0603082 [hep-ph].

[15] J. J. van der Bij and S. Dilcher, Phys. Lett. B638 (2006) 234,arXiv:hep-ph/0605008 [hep-ph].

[16] H. Georgi, Ann. Rev. Nucl. Part. Sci. 43 (1993) 209.

[17] C. P. Burgess, Ann. Rev. Nucl. Part. Sci. 57 (2007) 329,arXiv:hep-th/0701053 [hep-th].

164

Bibliography 165

[18] N. Turok and J. Zadrozny, Nucl. Phys. B358 (1991) 471.

[19] A. T. Davies, C. D. froggatt, G. Jenkins, and R. G. Moorhouse,Phys. Lett. B336 (1994) 464.

[20] J. M. Cline, K. Kainulainen, and A. P. Vischer, Phys. Rev. D54 (1996) 2451,arXiv:hep-ph/9506284 [hep-ph].

[21] L. Fromme, S. J. Huber, and M. Seniuch, JHEP 11 (2006) 038,arXiv:hep-ph/0605242 [hep-ph].

[22] N. G. Deshpande and E. MaPhys. Rev. D 18 (Oct, 1978) 2574.http://link.aps.org/doi/10.1103/PhysRevD.18.2574.

[23] E. M. Dolle and S. Su, Phys. Rev. D80 (2009) 055012,arXiv:0906.1609 [hep-ph].

[24] T. Kinoshita, J. Math. Phys. 3 (1962) 650.

[25] T. D. Lee and M. Nauenberg, Phys. Rev. 133 (1964) B1549.

[26] R. Santos and A. Barroso, Phys. Rev. D56 (1997) 5366,arXiv:hep-ph/9701257 [hep-ph].

[27] S. Kanemura, Y. Okada, E. Senaha, and C. P. Yuan,Phys. Rev. D70 (2004) 115002, arXiv:hep-ph/0408364 [hep-ph].

[28] D. Lopez-Val and J. Sola, Phys. Rev. D81 (2010) 033003,arXiv:0908.2898 [hep-ph].

[29] C. Degrande, Comput. Phys. Commun. 197 (2015) 239,arXiv:1406.3030 [hep-ph].

[30] M. Krause, R. Lorenz, M. Muhlleitner, R. Santos, and H. Ziesche,arXiv:1605.04853 [hep-ph].

[31] A. Denner, L. Jenniches, J.-N. Lang, and C. Sturm,arXiv:1607.07352 [hep-ph].

[32] N. D. Christensen and C. Duhr, Comput.Phys.Commun. 180 (2009) 1614,arXiv:0806.4194 [hep-ph].

[33] T. Hahn, Comput. Phys. Commun. 140 (2001) 418, arXiv:hep-ph/0012260.

[34] T. Hahn and M. Perez-Victoria, Comput. Phys. Commun. 118 (1999) 153,arXiv:hep-ph/9807565.

[35] A. Denner, S. Dittmaier, and L. Hofer, arXiv:1604.06792 [hep-ph].

166 Bibliography

[36] A. Bredenstein, A. Denner, S. Dittmaier, and M. M. Weber,Phys. Rev. D74 (2006) 013004, arXiv:hep-ph/0604011 [hep-ph].

[37] A. Bredenstein, A. Denner, S. Dittmaier, and M. M. Weber,Nucl. Phys. Proc. Suppl. 160 (2006) 131, arXiv:hep-ph/0607060 [hep-ph].

[38] A. Bredenstein, A. Denner, S. Dittmaier, and M. M. Weber,JHEP 02 (2007) 080, arXiv:hep-ph/0611234 [hep-ph].

[39] A. Denner, S. Dittmaier, M. Roth, and D. Wackeroth,Nucl. Phys. B560 (1999) 33, arXiv:hep-ph/9904472 [hep-ph].

[40] A. Denner, S. Dittmaier, M. Roth, and L. H. Wieders,Nucl. Phys. B724 (2005) 247, arXiv:hep-ph/0505042 [hep-ph]. [Erratum:Nucl. Phys.B854,504(2012)].

[41] A. Denner and S. Dittmaier, Nucl. Phys. Proc. Suppl. 160 (2006) 22,arXiv:hep-ph/0605312 [hep-ph].

[42] H. E. Haber and O. Stal, Eur. Phys. J. C75 (2015) 10, 491,arXiv:1507.04281 [hep-ph].

[43] G. C. Dorsch, S. J. Huber, K. Mimasu, and J. M. No,Phys. Rev. Lett. 113 (2014) 21, 211802, arXiv:1405.5537 [hep-ph].

[44] B. Hespel, D. Lopez-Val, and E. Vryonidou, JHEP 09 (2014) 124,arXiv:1407.0281 [hep-ph].

[45] LHC Higgs Cross Section Working Group, D. de Florian et al.,arXiv:1610.07922 [hep-ph].

[46] A. Freitas and D. Stockinger, Phys. Rev. D66 (2002) 095014,arXiv:hep-ph/0205281 [hep-ph].

[47] L. Altenkamp, QCD Corrections to HZ Production via Gluon Fusion at the LHC,Diploma Thesis, Universitat Stuttgart, 2012.

[48] A. Denner, Fortsch.Phys. 41 (1993) 307, arXiv:0709.1075 [hep-ph].

[49] O. W. Greenberg, Phys. Rev. Lett. 13 (1964) 598.

[50] M. Y. Han and Y. Nambu, Phys. Rev. 139 (1965) B1006.

[51] H. Fritzsch, M. Gell-Mann, and H. Leutwyler, Phys. Lett. B47 (1973) 365.

[52] S. Glashow, Nucl.Phys. 22 (1961) 579.

[53] S. Weinberg, Phys.Rev.Lett. 19 (1967) 1264.

Bibliography 167

[54] A. Salam, Svartholm: Elementary Particle Theory, Proceedings Of The NobelSymposium Held 1968 At Lerum, Sweden (1968) 367.

[55] S. Glashow, J. Iliopoulos, and L. Maiani, Phys.Rev. D2 (1970) 1285.

[56] L. Basso, O. Fischer, and J. J. van der Bij, Europhys. Lett. 105 (2014) 1, 11001,arXiv:1310.2057 [hep-ph].

[57] G. ’t Hooft, Nucl.Phys. B35 (1971) 167.

[58] M. Bohm, H. Spiesberger, and W. Hollik, Fortsch. Phys. 34 (1986) 687.

[59] H. E. Haber and G. L. Kane, Phys. Rept. 117 (1985) 75.

[60] N. Cabibbo, Phys. Rev. Lett. 10 (1963) 531.

[61] M. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49 (1973) 652.

[62] D. J. Gross and F. Wilczek, Phys. Rev. Lett. 30 (1973) 1343.

[63] H. D. Politzer, Phys. Rev. Lett. 30 (1973) 1346.

[64] R.K. Ellis, W.J. Sterling and B.R. Webber, QCD and Collider Physics.Cambride University Press, 1996.

[65] J. E. Kim, Phys. Rept. 150 (1987) 1.

[66] R. D. Peccei and H. R. Quinn, Phys. Rev. Lett. 38 (1977) 1440.

[67] G. Branco, P. Ferreira, L. Lavoura, M. Rebelo, M. Sher, et al.,Phys.Rept. 516 (2012) 1, arXiv:1106.0034 [hep-ph].

[68] Y. Wu and L. Wolfenstein, Phys.Rev.Lett. 73 (1994) 1762,arXiv:hep-ph/9409421 [hep-ph].

[69] P. M. Ferreira, R. Santos, and A. Barroso, Phys. Lett. B603 (2004) 219,arXiv:hep-ph/0406231 [hep-ph]. [Erratum: Phys. Lett.B629,114(2005)].

[70] F. Botella and J. P. Silva, Phys.Rev. D51 (1995) 3870,arXiv:hep-ph/9411288 [hep-ph].

[71] J. F. Gunion and H. E. Haber, Phys. Rev. D72 (2005) 095002,arXiv:hep-ph/0506227 [hep-ph].

[72] J. F. Gunion, H. E. Haber, G. L. Kane, and S. Dawson, The Higgs Hunter’s

Guide. Westview, 1900.

[73] M. Bohm, A. Denner, H. Joos, Gauge Theories of the Strong and Electroweak

Interaction. B.G. Teubner.

168 Bibliography

[74] S. L. Glashow and S. Weinberg, Phys.Rev. D15 (1977) 1958.

[75] E. Paschos, Phys.Rev. D15 (1977) 1966.

[76] A. Pich and P. Tuzon, Phys.Rev. D80 (2009) 091702,arXiv:0908.1554 [hep-ph].

[77] P. Tuzon and A. Pich, Acta Phys.Polon.Supp. 3 (2010) 215,arXiv:1001.0293 [hep-ph].

[78] G. ’t Hooft and M. Veltman, Nucl.Phys. B44 (1972) 189.

[79] C. G. Bollini and J. J. Giambiagi, Nuovo Cim. B12 (1972) 20.

[80] A. Dabelstein, Z.Phys. C67 (1995) 495, arXiv:hep-ph/9409375 [hep-ph].

[81] M. Aoki, S. Kanemura, K. Tsumura, and K. Yagyu,Phys.Rev. D80 (2009) 015017, arXiv:0902.4665 [hep-ph].

[82] C. Becchi, A. Rouet, and R. Stora, Annals Phys. 98 (1976) 287.

[83] M. Sperling, D. Stockinger, and A. Voigt, JHEP 07 (2013) 132,arXiv:1305.1548 [hep-ph].

[84] G. Degrassi and A. Sirlin, Nucl. Phys. B383 (1992) 73.

[85] J. Fleischer and F. JegerlehnerPhys. Rev. D 23 (Aug, 1980) 2001.http://cds.cern.ch/record/125234.

[86] S. Wolfram, Mathematica: A System for Doing Mathematics by Computer.Addison-Wesley, Redwood City, CA, 1988.

[87] LHC Higgs Cross Section Working Group, J. R. Andersen et al.,arXiv:1307.1347 [hep-ph].

[88] CMS, S. Chatrchyan et al., Phys. Rev. D89 (2014) 9, 092007,arXiv:1312.5353 [hep-ex].

[89] ATLAS, G. Aad et al., Phys. Rev. D91 (2015) 1, 012006,arXiv:1408.5191 [hep-ex].

[90] CMS, S. Chatrchyan et al., JHEP 01 (2014) 096, arXiv:1312.1129 [hep-ex].

[91] ATLAS, G. Aad et al., Phys. Rev. D92 (2015) 1, 012006,arXiv:1412.2641 [hep-ex].

[92] ATLAS, G. Aad et al., JHEP 01 (2015) 069, arXiv:1409.6212 [hep-ex].

[93] CMS, S. Chatrchyan et al., Phys. Rev. D89 (2014) 1, 012003,arXiv:1310.3687 [hep-ex].

Bibliography 169

[94] ATLAS, G. Aad et al., Eur. Phys. J. C75 (2015) 7, 349,arXiv:1503.05066 [hep-ex].

[95] CMS, V. Khachatryan et al., Phys. Rev. D92 (2015) 3, 032008,arXiv:1506.01010 [hep-ex].

[96] ATLAS, M. Aaboud et al., arXiv:1606.02181 [hep-ex].

[97] The ATLAS and CMS Collaborations, Tech. Rep. ATLAS-CONF-2015-044,CERN, Geneva, Sep, 2015. http://cds.cern.ch/record/2052552.

[98] ATLAS, CMS, G. Aad et al., Phys. Rev. Lett. 114 (2015) 191803,arXiv:1503.07589 [hep-ex].

[99] ATLAS, G. Aad et al., Phys. Rev. D90 (2014) 5, 052004,arXiv:1406.3827 [hep-ex].

[100] ATLAS Collaboration, Tech. Rep. ATLAS-CONF-2014-042, CERN, Geneva, Jul,2014. http://cds.cern.ch/record/1740973.

[101] CMS, V. Khachatryan et al., Phys. Rev. D92 (2015) 7, 072010,arXiv:1507.06656 [hep-ex].

[102] CMS, V. Khachatryan et al., Phys. Lett. B736 (2014) 64,arXiv:1405.3455 [hep-ex].

[103] ATLAS, CMS, G. Aad et al., JHEP 08 (2016) 045,arXiv:1606.02266 [hep-ex].

[104] ATLAS, G. Aad et al., Eur. Phys. J. C75 (2015) 10, 476,arXiv:1506.05669 [hep-ex].

[105] CMS, V. Khachatryan et al., Phys. Rev. D92 (2015) 1, 012004,arXiv:1411.3441 [hep-ex].

[106] ATLAS, G. Aad et al., Phys. Lett. B738 (2014) 234,arXiv:1408.3226 [hep-ex].

[107] ATLAS, G. Aad et al., arXiv:1507.04548 [hep-ex].

[108] C. A. Nelson, Phys. Rev. D37 (1988) 1220.

[109] A. Soni and R. M. Xu, Phys. Rev. D48 (1993) 5259,arXiv:hep-ph/9301225 [hep-ph].

[110] D. Chang, W.-Y. Keung, and I. Phillips, Phys. Rev. D48 (1993) 3225,arXiv:hep-ph/9303226 [hep-ph].

[111] A. Skjold and P. Osland, Phys. Lett. B311 (1993) 261,arXiv:hep-ph/9303294 [hep-ph].

170 Bibliography

[112] C. P. Buszello, I. Fleck, P. Marquard, and J. J. van der Bij,Eur. Phys. J. C32 (2004) 209, arXiv:hep-ph/0212396 [hep-ph].

[113] T. Arens and L. M. Sehgal, Z. Phys. C66 (1995) 89,arXiv:hep-ph/9409396 [hep-ph].

[114] S. Y. Choi, D. J. Miller, M. M. Muhlleitner, and P. M. Zerwas,Phys. Lett. B553 (2003) 61, arXiv:hep-ph/0210077 [hep-ph].

[115] G. Pocsik and T. Torma, Z. Phys. C6 (1980) 1.

[116] T. G. Rizzo, Phys. Rev. D22 (1980) 722.

[117] W.-Y. Keung and W. J. Marciano, Phys. Rev. D30 (1984) 248.

[118] R. N. Cahn, Rept. Prog. Phys. 52 (1989) 389.

[119] B. A. Kniehl, Phys. Lett. B244 (1990) 537.

[120] A. Grau, G. Panchieri, and R. J. N. Phillips, Phys. Lett. B251 (1990) 293.

[121] E. Gross, G. Wolf, and B. A. Kniehl, Z. Phys. C63 (1994) 417,arXiv:hep-ph/9404220 [hep-ph]. [Erratum: Z. Phys.C66,321(1995)].

[122] A. Djouadi, J. Kalinowski, and M. Spira,Comput. Phys. Commun. 108 (1998) 56, arXiv:hep-ph/9704448 [hep-ph].

[123] J. Fleischer and F. Jegerlehner, Phys. Rev. D23 (1981) 2001.

[124] B. A. Kniehl, Nucl. Phys. B352 (1991) 1.

[125] D. Yu. Bardin, P. K. Khristova, and B. M. Vilensky, Sov. J. Nucl. Phys. 54(1991) 833. [Yad. Fiz.54,1366(1991)].

[126] B. A. Kniehl, Nucl. Phys. B357 (1991) 439.

[127] S. Catani and M. H. Seymour, Nucl. Phys. B485 (1997) 291,arXiv:hep-ph/9605323.

[128] S. Dittmaier, Nucl. Phys. B565 (2000) 69, arXiv:hep-ph/9904440 [hep-ph].

[129] A. Bredenstein, S. Dittmaier, and M. Roth, Eur. Phys. J. C44 (2005) 27,arXiv:hep-ph/0506005 [hep-ph].

[130] M. Bohm and S. Dittmaier, Nucl. Phys. B409 (1993) 3.

[131] S. Dittmaier and M. Bohm, Nucl. Phys. B412 (1994) 39.

[132] U. Baur, S. Keller, and D. Wackeroth, Phys. Rev. D59 (1999) 013002,arXiv:hep-ph/9807417 [hep-ph].

Bibliography 171

[133] B. W. Harris and J. F. Owens, Phys. Rev. D65 (2002) 094032,arXiv:hep-ph/0102128 [hep-ph].

[134] B. W. Harris, E. Laenen, L. Phaf, Z. Sullivan, and S. Weinzierl,Phys. Rev. D66 (2002) 054024, arXiv:hep-ph/0207055 [hep-ph].

[135] F. A. Berends, R. Kleiss, and S. Jadach, Nucl. Phys. B202 (1982) 63.

[136] R. Kleiss, Z. Phys. C33 (1987) 433.

[137] V. N. Baier, V. S. Fadin, and V. A. Khoze, Nucl. Phys. B65 (1973) 381.

[138] A. Sirlin, Phys. Rev. D22 (1980) 971.

[139] W. Beenakker and A. Denner, Nucl. Phys. B338 (1990) 349.

[140] A. Denner, U. Nierste, and R. Scharf, Nucl. Phys. B367 (1991) 637.

[141] G. ’t Hooft and M. J. G. Veltman, Nucl. Phys. B153 (1979) 365.

[142] R. E. Cutkosky, J. Math. Phys. 1 (1960) 429.

[143] A. Denner and J.-N. Lang, Eur. Phys. J. C75 (2015) 8, 377,arXiv:1406.6280 [hep-ph].

[144] Particle Data Group, K. A. Olive et al., Chin. Phys. C38 (2014) 090001.

[145] M. Frank, T. Hahn, S. Heinemeyer, W. Hollik, H. Rzehak, and G. Weiglein,JHEP 02 (2007) 047, arXiv:hep-ph/0611326 [hep-ph].

Acknowledgements

An dieser Stelle mochte ich mich bei allen Personen bedanken, welche mich wahrendmeiner Doktorandenzeit unterstutzt und zum Gelingen dieser Arbeit beigetragen haben.

Allen voran mochte ich Prof. Dr. Stefan Dittmaier dafur danken, dass ich die Moglichkeitbekommen habe unter seiner Betreuung eine solche Forschungsarbeit zu realisieren. Vonseiner außergewohnlichen Expertise konnte ich sehr viel lernen, und seine Tipps undIdeen brachten mich auch in festgefahrenen Situationen immer wieder weiter.

Ein weiterer besonderer Dank gebuhrt Dr. Heidi Rzehak, die sich immer Zeit genommenhat, um mit mir Details zu diskutieren und Unklarheiten zu beseitigen sowie große Teiledieser Arbeit korrekturgelesen hat.

Zudem mochte ich mich bei der gesamten Arbeitsgruppe und ihren ehemaligen Mit-gliedern fur das Korrekturlesen dieser Arbeit, die vielen interessanten Diskussionen(auch abseits der Physik), die Mitgliedschaft im ~-racing Team, die schonen Abendebeim Fußball schauen, die Betriebsausfluge und Weihnachtsfeiern bedanken. Es hat im-mer viel Spaß gemacht.

Ein Dankeschon gebuhrt auch der DFG, die große Teile meiner Finanzierung stellte,sowie der HPC Abteilung im Rechenzentrum dafur, dass ich auf bis zu 13689 Prozes-soren gleichzeitig rechnen konnte.

Nicht zuletzt mochte ich mich bei meiner Familie, meinen Freunden und insbesonderebei Franziska dafur bedanken, mich in allen leichten und schwierigen Phasen der letztenJahre unterstutzt zu haben.