Radboud University

Master's thesis

Quantum Corrections and Renormalization SchemesImplications for the hierarchy problem

Author:

Joris van der Ven

Supervisor:

prof. dr. Wim Beenakker

Department of Theoretical High Energy Physics (THEP)Institute for Mathematics, Astrophysics and Particle Physics (IMAPP)

August 24, 2018

Contents

Introduction 20.1 Some important conventions, identities and properties . . . . . . . . . . . . . . . . 4

1 The Standard Model 61.1 Quantum electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2 Non-abelian gauge theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3 The Higgs mechanism and symmetry breaking . . . . . . . . . . . . . . . . . . . . 91.4 The Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Regularization and renormalization 142.1 Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.1.1 Cuto regulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.1.2 Dimensional regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.1.3 Pauli-Villars regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2 Renormalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2.1 Introduction to renormalization . . . . . . . . . . . . . . . . . . . . . . . . . 182.2.2 Renormalizability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.2.3 Renormalized perturbation theory . . . . . . . . . . . . . . . . . . . . . . . 212.2.4 Renormalization schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.2.5 Renormalization group equations . . . . . . . . . . . . . . . . . . . . . . . . 252.2.6 Renormalization scheme dependence . . . . . . . . . . . . . . . . . . . . . . 26

3 Gauge anomalies 273.1 Noether's theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2 The Ward identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.3 Anomalies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.4 Anomaly cancellation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.5 Anomalies for massive fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4 The importance of a regulator in the calculation of H → γγ 34

5 Two-loop corrections to the Higgs mass in the MSSM 395.1 A very brief introduction to the MSSM . . . . . . . . . . . . . . . . . . . . . . . . . 395.2 Denition of the elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405.3 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.4 Field mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.4.1 Input parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.5 The Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5.5.1 The Lagrangian for the Higgs sector . . . . . . . . . . . . . . . . . . . . . . 445.5.2 The interaction Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.6 Renormalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475.7 The counterterm Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.8 Feynman rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525.9 Calculating the diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.9.1 Renormalization conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 555.9.2 Notation of the loop integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 585.9.3 Steps in the calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

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5.9.4 The p2 = 0 approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.9.5 The results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.10 The pole masses of the physical Higgs elds . . . . . . . . . . . . . . . . . . . . . . 65

Conclusions and outlook 67

A Feynman rules for Chapter 5 68

Bibliography 72

2

Introduction

The main topic of this thesis concerns quantum corrections in quantum eld theories (QFT's).The calculation of quantum corrections is an important part of the theoretical work in the eldof high-energy physics. It is crucial that these calculations are done correctly. In particular, theprocess of renormalization requires great care if one wants to make accurate predictions from agiven eld theory. In the rst part of this thesis we will therefore review the important pointson gauge theories and how to renormalize them. In the second part we will take a look at somecomplications that may arise when calculating loop diagrams. In the third and nal part we willwork out an explicit calculation of two-loop corrections in the context of supersymmetry.

Let's rst go into more detail on the contents of this thesis. Chapter 1 provides a review of theStandard Model (SM). The gauge principle is explained and both abelian and non-abelian gaugetheories will be discussed. Of course, the workings of the Higgs mechanism are also explained. Thesection ends with an overview of the Standard Model, its gauge group and its particle elds.

The second Chapter is all about renormalization. We start from the concept that a physicalprocess can be described by a perturbative expansion of Feynman diagrams that increase in thenumber of loops they contain. This brings with it a technical problem, namely the appearanceof divergences when integrating over the loop momenta. These divergences can be handled usingdierent regularization methods. In the process of renormalization these divergences are then ab-sorbed into parameters of the theory in an appropriate way. As a result, all physical observablesthat can be calculated will be nite. Particular attention is given to the existence of dierentrenormalization schemes, as this will be important later on in Chapter 5.

Chapter 3 goes into some more detail on gauge symmetries. It starts o with an explanationof Noether's theorem, which relates a (gauge) symmetry to a conserved current. In a quantumtheory this current conservation is expressed in the form of the Ward identity. An important issuethat arises when calculating quantum corrections in a gauge theory is that this Ward identity maybe violated by a certain type of loop diagrams. This is called a gauge anomaly. It results in anunacceptable form of symmetry breaking. The solution is a full cancellation of all contributions tothe anomaly.

Chapter 4 then provides a sort of intermezzo on an interesting class of integrals that appearswhen calculating the one-loop diagram of a Higgs decaying into two photons. These integralsproduce a nite answer, but nevertheless require a regulator to get the correct result. An attemptto solve these integrals without regularization will lead to incorrect results. The reason for thiswill be explained in this chapter; it basically serves as a warning that one should always thinkvery carefully about what the correct way to calculate loop corrections is, because there are manypotential pitfalls in the calculations.

The fth and nal Chapter should be seen as the main content of this thesis. It concerns acalculation of the two-loop corrections to the Higgs mass in the Minimal Supersymmetric StandardModel (MSSM). The MSSM is the most commonly used model for supersymmetry, the symmetrythat associates a fermion to every boson and vice versa. In particular we will look at the impactof the gluino, the supersymmetric partner to the gluon, on the Higgs mass. Every step in thecalculation is worked out in detail, from dening the Lagrangian to calculating the relevant Feyn-man diagrams. Importantly, we will explore the impact of dierent renormalization schemes onthe results.

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This calculation can also be seen in relation to the hierarchy problem; we calculate quantumcorrections to the Higgs mass (at 125 GeV) originating from particles that live at a higher energyscale. In this case, we have a heavy particle, the gluino, that does not couple directly to the Higgseld, but only indirectly through top and stop loops. Thus, this calculation provides an interestingcase study of a model where we can probe if there is a hierarchy problem. After all, there is stilla lot of discussion in the literature about the hierarchy problem, when it appears and whether itis actually a serious problem at all.

0.1 Some important conventions, identities and properties

We use the summation convention: this means that repeated indices are summed over unless oth-erwise indicated.

For the metric of spacetime we use the convention

gµν = diag(1,−1,−1,−1), (µ, ν = 0, 1, 2, 3) (1)

For the 2-dimensional antisymmetric Levi-Civita symbol we use the convention

ε12 = −ε12 = +1 (2)

and similarlyε0123 = −ε0123 = +1 (3)

for its 4-dimensional counterpart.

In gauge theories one needs to choose a convention for the sign of the covariant derivative. Wewill use the plus sign

Dµ = ∂µ + igAaµTa (4)

where g is the coupling constant and

∂µ =∂

∂xµ(5)

the derivative with respect to the spacetime coordinates.

If the gauge group is SU(N) the generators T a are hermitian ((T a)† = T a) and traceless(Tr(T a) = 0) N × N matrices. These two conditions imply that SU(N) has N2 − 1 independentgenerators, so the index a runs from 1 to N2 − 1. The SU(N) generators satisfy the followingproperty:

T aikTakj = CF δij , (i, j = 1, . . . , N) (6)

where

CF =1

2

N2 − 1

N(7)

In (6) the sum over the index a runs from 1 to N2 − 1, while the sum over the index k runs from1 to N .

When calculating diagrams with fermions the Dirac gamma matrices make an appearance.While it is of course not necessary to choose a representation for the gamma matrices, it can makecalculations easier. A convenient choice that we will use is the Weyl representation

γµ =

(0 σµ

σµ 0

)(8)

withσµ = (1, ~σ), σµ = (1,−~σ) (9)

where ~σ = (σ1, σ2, σ3) are the Pauli matrices in the standard representation (1.11). The matrixγ5 is dened as

γ5 = iγ0γ1γ2γ3 (10)

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The chiral projection operators are given by

PL =1

2(1− γ5), PR =

1

2(1 + γ5) (11)

For the dimensional regularization of loop integrals the number of spacetime dimensions iscontinued from 4 to an arbitrary number d. We will consistently use the convention

ε = 4− d (12)

for the dierence between 4 and d dimensions.

5

Chapter 1

The Standard Model

1.1 Quantum electrodynamics

The simplest example of a gauge theory is quantum electrodynamics (QED), the theory describingthe interactions between photons and (electrically) charged particles. Consider the Lagrangian ofa free fermion eld ψ with mass m:

L = ψ(i/∂ −m)ψ (1.1)

where the slash notation has been used to indicate a contraction with the gamma matrices: /∂ =∂µγ

µ. This Lagrangian is invariant under a global U(1) transformation of the fermion eld, whichis essentially just a multiplication with a complex phase:

ψ → ψ′ = eiΛψ, Λ ∈ R (1.2)

Since ψ will transform with the opposite phase, the full Lagrangian will clearly be unchanged.Equivalently, we can say that the Lagrangian has a U(1) symmetry. This transformation is knownas a gauge transformation: it allows us to transform the eld without changing the physics. Thisis a manifestation of gauge freedom.

What the gauge principle now postulates is that the Lagrangian should also be invariant undera local U(1) transformation. What this means is that the phase can now be dierent for everyspacetime point x, so we have to replace the global phase Λ by a phase Λ(x) that is a functionof the spacetime coordinates. Clearly, if we enter this into the Lagrangian (1.1), it will not beinvariant under this local gauge transformation, since there will be an extra term originating fromwhen the derivative acts on the phase Λ(x). The way this is solved in gauge theories is by replacingthe normal derivative by a covariant derivative:

∂µ → Dµ = ∂µ + igAµ (1.3)

In this expression we have introduced g, a number which is called the coupling constant, and thegauge eld Aµ. To make the Lagrangian invariant under the local gauge transformation, the gaugeeld also has to transform. It turns out that it has to transform in the following way:

Aµ → A′µ = Aµ −1

g∂µΛ(x) (1.4)

The second term here will of course cancel against the extra term we just discussed, so that theLagrangian is now locally gauge invariant.

To complete the QED Lagrangian we also have to add a kinetic term for the gauge eld. Toget the correct equation of motion for the gauge eld (for the photon eld this should give theMaxwell equations!) this term has to be:

− 1

4FµνF

µν (1.5)

whereFµν = ∂µAν − ∂νAµ (1.6)

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is called the eld strength tensor of the gauge eld. So the full QED Lagrangian we have nowobtained is:

LQED = −1

4FµνF

µν + ψ(i /D −m)ψ = −1

4FµνF

µν + ψ(i/∂ −m)ψ − gψγµAµψ (1.7)

Note that we cannot give the gauge eld a mass term of the formM2AµAµ, since this would break

gauge invariance. So the gauge eld has to be massless. For QED this is okay, since we know thatthe photon is massless anyway. But we also know that the Standard Model has massive gaugeelds, namely the ones corresponding to the W and Z bosons. These will get their mass from theHiggs mechanism as we will see later.

1.2 Non-abelian gauge theory

We have just seen that QED is a U(1) gauge theory. The group U(1) is an abelian group, whichmeans that any two elements of the group will commute. This is easily seen for U(1) since theelements are complex phases, which will obviously commute.

The idea of gauge theories can also be extended to non-abelian groups, i.e. groups where theelements do not always commute. To do this we have to introduce some concepts from group theory.

First, we will have a look at the group axioms. A group is a set that is equipped with anoperation • that combines two elements g1 and g2 of the group to form a new element g1 • g2.We will call this operation group multiplication, although it does not necessarily have to be atraditional form of multiplication. The set and operation form a group G if they satisfy the fourgroup axioms.

(G1) Closure: if g1 and g2 are in G, then g1 • g2 is also in G.

(G2) Associativity: if g1, g2 and g3 are in G, then g1 • (g2 • g3) = (g1 • g2) • g3.

(G3) Identity: G contains an identity e, satisfying e • g = g • e = g for any g in G.

(G4) Inverse: for any g in G there is an inverse g−1 in G satisfying g • g−1 = g−1 • g = e.

From now on we will omit the multiplication symbol, i.e. g1g2 = g1 • g2.

An abelian group is a group where g1g2 = g2g1 for any pair of elements of the group, so that thegroup elements always commute. In a non-abelian group there are at least some pairs of elementsthat do not commute.

In particle physics the only groups that are relevant are Lie groups. These are groups that canbe parameterized by one or more continuous parameters. For U(1) for example, we only need oneparameter: all elements of the group are generated by exp(iα) with the continuous parameter α.

Every Lie group has a corresponding Lie algebra. The relation between these two is determinedby the following: if we have some element g of the Lie group G we can expand it around theidentity

g = 1+ iεaT a +O(ε2) (1.8)

where the εa are some real innitesimals and the T a are hermitian operators which are called thegenerators of the Lie algebra. These generators provide a basis for the Lie algebra. The closureaxiom for the group elements (G1) translates into a commutation rule for the generators

[T a, T b] = ifabcT c (1.9)

where the fabc form a set of real numbers called the structure constants of the group. These arespecic to a group and determine the commutation rules that the generators of that group haveto satisfy.

7

In Eq. (1.8) we obtained the generators by expanding an element of the group. This can alsobe reversed: if we have the generators we can obtain elements of the group with the exponentialmap

g(α) = eiαaTa (1.10)

where the αa are a set of real numbers specic to the group element.

To a mathematician the group elements and generators of the Lie algebra are just abstractoperators. But in a particle physics context, these operators will always be matrices and moreoverwe will always work with specic representations of these groups and Lie algebras. For example arepresentation of the generators of SU(2) is given by the Pauli matrices

σ1 =

(0 11 0

), σ2 =

(0 −ii 0

), σ3 =

(1 00 −1

)(1.11)

This is not the only set of matrices that satises the commutation rules for the SU(2) Lie algebra.There are many equivalent representations and while the Pauli matrices are 2-dimensional thereare also representations of higher dimensions.

The most important Lie groups in particle physics are the SO (special orthogonal) groups andthe SU (special unitary) groups. The group SO(N) is dened as the set of N × N matrices thatare orthogonal (OTO = OOT = 1) and have unit determinant (det(O) = 1). These are strictlyreal matrices. The group SU(N) is basically the complex version of SO(N). It is dened as the setof N ×N matrices that are unitary (U†U = UU† = 1) and have unit determinant (det(U) = 1).Note that this denes the N -dimensional fundamental representations of these groups and again,other representations are also possible.

We now have the necessary tools to treat non-abelian gauge theories. Suppose we again havea Lagrangian with a free fermion eld. A representation of a non-abelian group is at least two-dimensional (since a one-dimensional representation would consist of just numbers, which alwayscommute). The fermion eld therefore has to have multiple components to reect the fact thatit `lives in' that representation. If the representation is N -dimensional we say that the eld is anN -component multiplet. That means that the representation matrices of the Lie group can act onthe fermion eld and mix the components of that eld. We will indicate this by attaching an indexto the fermion eld. The most common multiplets are the doublet (2 components) and the triplet(3 components).

Suppose the Lagrangian has a symmetry under the transformation:

ψi → ψ′i = Mijψj , i, j = 1, 2, ..., N (1.12)

where Mij is an N -dimensional representation matrix of a non-abelian Lie group. Using the expo-nential map we can also write this matrix asM = exp(iαaT a), where the T a form anN -dimensionalrepresentation of the generators of the Lie algebra. For simplicity let's assume that we are workingwith an SO group or an SU group, so that M† = M−1.

According to the gauge principle we now have to make this symmetry local again. This proceedsin a very similar way as in QED earlier, so we replace Mij → Mij(x), or in the exponential mapαa → αa(x). The covariant derivative now becomes

Dijµ = ∂µδ

ij + igAaµ(T a)ij (1.13)

The indices i and j are the indices of the representation space. The index a sums over the gener-ators, so we see that the number of gauge elds that we have to introduce is equal to the numberof generators of the Lie algebra.

The transformation rule for the gauge elds is now modied by an extra term when comparedto Eq. (1.4):

Aaµ → A′aµ = Aaµ −

1

g∂µα

a(x)− fabcαbAcµ (1.14)

8

Note that this equation only holds for an innitesimal transformation. The extra term is obviouslyabsent for an abelian group; since the group elements of an abelian group commute, the generatorsalso commute, causing the structure constants to vanish.

If we want to perform a non-innitesimal transformation of the gauge elds we have to look atthe combination Aµ ≡ AaµT a. This has to transform as:

Aµ → A′µ = MAµM−1 +

i

g(∂µM)M−1 (1.15)

Finally, the eld strength tensor also gets an extra term when compared to Eq. (1.6)

Fµν = ∂µAν − ∂νAµ + ig[Aµ, Aν ] (1.16)

or, in terms of the components Fµν ≡ F aµνT a

F aµν = ∂µAaν − ∂νAaµ − gfabcAbµAcν (1.17)

Usually, we choose the generators such that they satisfy the normalization condition Tr(T aT b) =12δab. With this choice the kinetic term for the gauge elds is given by

− 1

2Tr(FµνF

µν) = −1

2F aµνF

b,µνTr(T aT b) = −1

4F aµνF

a,µν (1.18)

So every gauge eld has its own kinetic term with prefactor −1/4 as in Eq. (1.5).

1.3 The Higgs mechanism and symmetry breaking

Figure 1.1: The potential of Eq. (1.20). On the left is the case where the parameters m2 and λare both positive, on the right is the case where m2 is negative and λ is positive.

Consider the following Lagrangian for a complex scalar eld φ

L = (∂µφ)∗(∂µφ)−m2φ∗φ− λ

4(φ∗φ)2 (1.19)

The rst term is the kinetic term, while the second and third terms form the potential. We seethat this Lagrangian has a global U(1) symmetry. We also see that the potential depends only onthe absolute value of the eld

V (φ) = m2|φ|2 +λ

4|φ|4 (1.20)

If m2 and λ are both positive the potential has a minimum at |φ| = 0 (see the left graph in Fig.1.1). This is the ground state of the eld. This means that the vacuum expectation value (vev) ofthe eld is also 0: 〈0|φ|0〉 = 0.

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However, if m2 becomes negative the situation changes; |φ| = 0 now becomes a local maximum

of the potential, while the minimum of the potential is at |φ| =√−2m2

λ , as can be seen in the right

graph of Fig. 1.1. We now typically dene v as

1

2v2 =

−2m2

λ(1.21)

This system now acts as you would expect from classical mechanics intuition: since the localmaximum is unstable, any slight perturbation will cause the eld to `roll down' the potential andinto the minimum. This minimum is in fact a ring of minima if we take the complex phase of theeld into account. Then we can make the picture in Fig. 1.1 3-dimensional by adding an extraaxis with the imaginary part of φ and rotating the graph around the y-axis. This produces thefamous `Mexican hat' potential.

The eld will fall into one particular point on the ring of minima, with all points being equallylikely. The symmetry of the Lagrangian was reected in the ring of equivalent minima, but byexplicitly picking one point on this ring this symmetry is broken. Since the local maximum isunstable, this process can occur spontaneously, which is why one speaks of spontaneous symmetrybreaking. We can always dene the eld φ such that it falls into the minimum that has phase zero.So the ground state of the eld has changed and the eld has now obtained a non-zero vacuumexpectation value

〈0|φ|0〉 =v√2

(1.22)

To describe the physics of this system we should expand the eld around this new ground state.Since φ is a complex scalar eld it has two real degrees of freedom, which means it can be writtenas a combination of two real scalar elds. We will use the following expansion of the eld

φ(x) =1√2

(v + η(x))eiξ(x)/v (1.23)

where η and ξ are the two real scalar elds. The eld ξ describes the uctuations of the eld inthe direction of the ring of minima (called the `at direction', since the potential remains constantin this direction), while the eld η describes the uctuations in the direction away from the ringof minima. If we plug this expansion in the Lagrangian (Eq. (1.19)) and use the relation (1.21)we obtain

L =1

2(∂µη)(∂µη) +

1

2

(1 +

η

v

)2

(∂µξ)(∂µξ)− 1

4m2v2 +m2η2 − λ

4vη3 − λ

16η4 (1.24)

The rst two terms contain the kinetic terms for η and ξ. The second term also contains interac-tions between η and ξ. The third term is a constant and does not contribute to the physics1. Thefourth term is now a mass term for η with the correct sign. The fth and sixth terms describetriple and quartic self-interactions of the η-eld.

So what we now have is a massive eld η with mass√−2m2 > 0 (the factor 2 comes from the

normalization of the mass term for real scalar elds). At rst we expanded around |φ| = 0. At thispoint the second derivative of the potential is negative. This is the reason that the eld had a massterm with the wrong sign. Around the minimum however, the second derivative of the potentialis positive. That is why the eld η, which describes the uctuations around the minimum, has theright sign for the mass term.

We also have a massless eld ξ. We saw that this eld corresponded to the phase of the oldeld φ. In the new ground state the U(1) symmetry was spontaneously broken by choosing aphase; multiplying the new ground state by a phase does not leave it invariant. What we see here

1A constant term does not contribute to the eld theory. It can however be interpreted as a contribution to thevacuum energy. Current observations indicate that our universe has a small positive vacuum energy. This energy isalso known as `dark energy'.

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is a manifestation of Goldstone's theorem: for every generator of a continuous symmetry that isspontaneously broken one gets a massless boson called a Goldstone boson. The Goldstone bosonis a remnant of the symmetry after it is broken.

In a gauge theory we have to turn the global U(1) symmetry of Eq. (1.19) into a local one. Todo this we just have to replace the derivatives by covariant derivatives that couple the U(1) gaugeeld to the scalar eld φ:

∂µ → Dµ = ∂µ + igAµ (1.25)

We also have to add the kinetic term for the gauge eld, so the new Lagrangian is

L = −1

4FµνF

µν + (Dµφ)∗(Dµφ)−m2φ∗φ− λ

4(φ∗φ)2 (1.26)

This Lagrangian is now invariant under local U(1) gauge transformations

φ(x)→ φ′(x) = eiΛ(x)φ(x), Aµ(x)→ A′µ(x) = Aµ(x)− 1

g∂µΛ(x) (1.27)

We can then repeat the process we just went through. Again we use the expansion of φ in Eq.(1.23). If we plug this into the locally invariant Lagrangian (1.26) and work this out, the physicalcontent of the Lagrangian will be somewhat tricky to interpret. This becomes easier if we rst use

the gauge transformation (1.27) to go to the unitary gauge. This is done by taking Λ(x) = −ξ(x)v .

The elds in the unitary gauge are

φ′(x) =1√2

(v + η(x)), A′µ(x) = Aµ(x) +1

gv∂µξ(x) (1.28)

Plugging these elds into the Lagrangian (1.26) and working out the expression results in

L = −1

4FµνF

µν+1

2(∂µη)(∂µη)+m2η2+

1

2g2v2A′µA

′µ+1

2g2A′µA

′µ(2vη+η2)−λ4vη3− λ

16η4− 1

4m2v2

(1.29)From this expression we can easily read o the physical content of the theory. First we have thekinetic terms for the gauge eld A′µ and η. Note that the eld strength tensor does not have aprime since it is invariant under gauge transformations. Then we have the mass term for η thatwe had before and now we also have a mass term for the gauge eld A′µ. We see that the gaugeeld has obtained a mass MA = gv that depends on the vacuum expectation value of the eld φand the coupling constant. Then we have terms describing interactions between the gauge eldand the eld η. Finally, we have terms describing the self-interactions of η and a constant term,which we already had in Eq. (1.24).

In the unitary gauge the massless Goldstone boson has disappeared from the Lagrangian. How-ever the degree of freedom that the Goldstone boson carried is not gone. A massless vector bosonhas two degrees of freedom while a massive vector boson has three. Clearly, we need one extradegree of freedom to give the gauge eld a mass. This is supplied by the Goldstone boson. Peopleoften say that the gauge eld `eats' the Goldstone boson to become massive.

This process where a eld breaks symmetry by obtaining a non-zero vacuum expectation valueand then gives mass to gauge elds is called the Higgs mechanism. In this section we have workedout an example with a U(1) symmetry, but it can be done with other Lie groups as well.

1.4 The Standard Model

With the theory from the last sections we have the necessary ingredients to build the StandardModel (SM). The only thing we have not discussed yet is how to treat a gauge theory where thegauge group is a product group. This is a fairly straightforward generalization. We just have toadd multiple indices to the elds corresponding to the groups they transform under. We also haveto add extra terms to the covariant derivatives: every group gets its own term in the covariant

11

derivative with its own coupling constant and its own generators.

The SM was constructed throughout the years by taking a lot of input from observations. Thegauge group of the SM and the fermionic particle content cannot be determined from theory alone,but have to be determined by experiments. Of course the constraints of the theoretical frameworkdid allow quite a few particles to be predicted before they were discovered, most famously theHiggs boson.

The gauge group of the SM is SU(3)C × SU(2)L × U(1)Y . The rst group acts on fermionsthat have color. Color is a quantum number that can take on the three values red, green and blue.The colored fermions in the SM are the quarks. SU(3) has eight generators so there are also eightassociated gauge bosons, which are called gluons.

The second group acts only on left-handed fermions. `Left' refers to the chirality of a fermioneld. We can project a spinor onto the left-handed and right-handed chirality states by using theprojection operators PL and PR

ψL ≡ PLψ =1

2(1− γ5)ψ

ψR ≡ PRψ =1

2(1+ γ5)ψ

(1.30)

where γ5 ≡ iγ0γ1γ2γ3 in terms of the Dirac gamma matrices. Since this group acts only on left-handed elds, elds with left-handed chirality are treated dierently from elds with right-handedchirality in the SM. This makes the SM a so-called chiral theory. The quantum number associatedto this group is the weak isospin I. A eld with weak isospin I is a (2I + 1)-dimensional multiplet.The third component of isospin I3 runs from −I to +I in integer steps for the dierent componentsin the multiplet. The group SU(2) has three generators. The three associated gauge bosons aretypically called W 1, W 2 and W 3.

The third group acts on the quantum number Y , which is called hypercharge. U(1) only hasone generator, so this group only brings one gauge eld with it, which is usually called B.

However this is not the full story yet. We will now look at how the Higgs mechanism worksin the SM. This will only be a short explanation without working out the equations. The Higgsmechanism in the SM takes place in the groups SU(2)L × U(1)Y , which is called the electroweaksector of the SM. This sector has the four gauge bosons W 1, W 2, W 3 and B. The Higgs eld is acomplex eld that transforms as a doublet under SU(2) and has a hypercharge Y = 1. This meansthat it has 4 degrees of freedom. This eld then obtains a non-zero vacuum expectation value inthe way that was described in the last section. This breaks 3 of the 4 independent generators ofSU(2)L ×U(1)Y . Consequently, 3 gauge bosons acquire a mass. Two of these are the W± bosons,obtained from linear combinations of the W 1 and W 2 elds. Linear combinations of the W 3 andB elds give the massive Z boson and the massless photon eld A. The symmetry that remains isU(1)EM , the symmetry of electromagnetic charge. The relation between electromagnetic charge Q,hypercharge Y and the third component of isospin I3 is given by the Gell-Mann-Nishijima formula

Q = I3 +1

2Y (1.31)

Now we can list the complete eld content of the SM. This has been done in Table 1.1. We givethe representation of the elds under SU(3)C × SU(2)L × U(1)Y . For the rst two groups we listthe dimension of the representation, while for the last group we simply list the value of Y . Notethat the trivial representation of any group has dimension 1. For example (3,1,1) would indicate aeld that lives in the 3-dimensional SU(3) representation, the trivial SU(2) representation (i.e. itdoes not transform under SU(2)), and has hypercharge Y = 1.

It turns out that there are three families or generations of matter fermions. For the chargedleptons these comprise the electron e−, muon µ− and tau τ−. Similarly there are three types of

12

Name Field RepresentationLeft-handed leptons LL = (νL, eL) (1,2,-1)Right-handed neutrinos νR (1,1,0)Right-handed charged leptons eR (1,1,-2)Left-handed quarks QL = (uL, dL) (3,2, 13 )Right-handed up-type quarks uR (3,1, 43 )Right-handed down-type quarks dR (3,1,− 2

3 )Gluons G (8,1,0)W elds W i = (W 1,W 2,W 3) (1,3,0)B eld B (1,1,0)Higgs eld Φ = (φ+, φ0) (1,2,1)

Table 1.1: The eld content of the standard model

neutrinos (neutral leptons): electron neutrinos νe, muon neutrinos νµ and tau neutrinos ντ . Forthe up-type quarks we know the up quark u, the charm quark c and the top quark t, while for thedown-type quarks there is the down quark d, the strange quark s and the bottom quark b. Thequantum number that refers to these families is called avor. The existence of the three familiescannot be predicted from the theory of the SM, but is something that one might hope to explainfrom a more complete, underlying theory.

Fermion masses in the Standard ModelThere is one nal issue in the Standard Model that needs to be discussed here and that is how wecan give masses to the fermions. Writing down a (Dirac) mass term of the form

−mψψ = −m(ψLψR + ψRψL) (1.32)

does not work in a chiral theory, since such a term is not gauge invariant. Luckily there is a solutionin the form of the Higgs eld; since the Higgs eld is an SU(2) doublet, it can be combined withthe left-handed SU(2) doublets LL and QL (see Table 1.1) to create gauge invariant terms.

Let's take the quark doublet QL as an example. If we want to give the down quark a mass wecan write down a gauge invariant term of the form:

− YdQLΦdR + h.c. (1.33)

This term is called a Yukawa term and Yd is the Yukawa coupling of the down quark. To give amass to the up quark, we use the Higgs eld as well:

− YuQLΦuR + h.c. (1.34)

where Φ = iσ2Φ∗ and Yu is the Yukawa coupling of the up quark. Now consider what happensafter symmetry breaking: inserting an expansion of the form (1.23) for the Higgs eld into theYukawa terms (1.33) and (1.34), we can see that they produce mass terms for the quarks and alsoHiggs-quark-quark interactions. The quark masses will be ∼ Yqv, (q = u, d). Fermion massesgenerated by Yukawa terms are thus proportional to the vacuum expectation value of the Higgseld. We can give the leptons and neutrinos a mass in the same way by writing down their Yukawaterms.

13

Chapter 2

Regularization and renormalization

After setting up a theory with the right Lagrangian, we want to be able to calculate quantitiesthat we can measure. One of the general principles of any quantum theory is that when doing acomputation one has to sum over all possible ways that a process can occur. In QFT we almostalways do this using perturbation theory. This means that we assume that the coupling constantsare small so that we can make an expansion in the coupling constants. We can then do a calcula-tion order by order in perturbation theory. Because we assume the coupling constants to be small,each following order should give a smaller contribution. We can then truncate the perturbationseries once we reach the order we need for our desired calculational accuracy. These perturbationtheory expansions can be graphically represented by Feynman diagrams: the more vertices thereare in the diagram, the more factors of coupling constants the expression contains, which meansthe diagram represents a higher order term in the perturbation series.

As a simple example consider the massive φ4-theory which has the Lagrangian

L =1

2(∂µφ)(∂µφ)− 1

2m2φ2 − λ

4!φ4 (2.1)

where m is the mass and λ the coupling constant of the quartic interaction. This Lagrangian leadsto two Feynman rules, one for the propagator and one for the vertex:

Figure 2.1: The Feynman rules for massive φ4-theory. Picture adapted from [1].

Now look at the expansion belonging to the scattering process φ(p1)φ(p2)→ φ(p3)φ(p4):

Figure 2.2: The rst few terms in the perturbation expansion of the 4-point function in φ4-theory.Picture adapted from [1].

The rst term on the right-hand side is the lowest order term, also called the tree-level diagram

14

since it contains no loops. It has one vertex, so it represents anO(λ) term in the perturbation series.The next three diagrams contain two vertices, so they represent O(λ2) contributions to the pertur-bation series. Obviously the series goes on indenitely and in general at higher orders there willbe more possible diagrams. Also, the number of loops gets higher the further you go into the series.

Once we try to do the computation these loops will cause problems. Since the momentumgoing around the loop is undetermined, we had better integrate over all possible momenta. Sincewe have no reason to assume any restriction on the loop momentum, we should integrate overall momentum values from zero to innity. This will generally give rise to divergent integrals inthe computation. In the early days of QFT this was thought to be a major problem. However,over time techniques have been developed to handle these divergences. It will turn out that thesedivergences are not physical, but an artifact of the way we do our calculation.

The two main techniques needed to handle the divergences are called regularization and renor-malization. In the next sections I will explain what these terms mean and show how they work.

2.1 Regularization

The term regularization refers to a set of techniques where an extra parameter, the regulator,is introduced to make the divergent integrals nite. This allows us to handle the integrals anddo manipulations with them. At the end of our calculations we should be able to `remove' theregulator again without encountering divergences. Also, the nal result should be independentof the type of regulator we use, so we can usually choose a regulator that is convenient for thecalculation we are doing. We will now take a look at some of the most commonly used regulators.

2.1.1 Cuto regulator

The simplest and most intuitive type of regulator is the cuto regulator. The use of this regulatoris based on the fact that most divergences in QFT originate from trying to integrate to innitemomentum. These are called UV-divergences, because high momentum corresponds to a shortwavelength. To remove this type of divergence we can simply place an upper limit (a cuto) onthe momentum instead of integrating to innity.

As an example, we might encounter the following integral in a computation:∫d4p

1

p2 −m2(2.2)

After a Wick rotation this becomes (apart from numerical factors) the following Euclidean integralover spherical coordinates: ∫

dΩ3

∫ ∞0

dpEp3E

p2E +m2

(2.3)

where the rst integral is over the angles of the 3-sphere and the second integral is over the absolutevalue of the Euclidean momentum pE . The second integral is divergent, but we now make it niteby replacing the upper limit by a cuto value Λ:∫ Λ

0

dpEp3E

p2E +m2

=

[1

2p2E −

1

2m2 log(p2

E +m2)

]Λ

0

=1

2Λ2 +

1

2log

(m2

Λ2 +m2

)(2.4)

So we see that we have now obtained a nite answer that depends on the value of the cuto Λ. Wealso see that the leading term is quadratic in Λ, which shows that we have regularized a quadraticdivergence. Now we can continue our calculations and at the end we should be able to `remove'the regulator again by sending Λ to innity.

The reason I said that cuto regularization is the most intuitive type of regularization is be-cause it has a clear physical interpretation: we can interpret the cuto Λ as the energy scale whereour current theory is no longer valid and where new physics kicks in. Because large energy scalescorrespond to short length scales, integrating to innite momentum basically means that we aresaying that QFT is valid on arbitrarily small length scales. This is clearly absurd, because certainly

15

at the Planck scale we would expect our theory to break down due to gravitational eects. Butit is very much possible that even at much lower energy scales new physics will appear. So thecuto Λ parametrizes our ignorance of these high-scale physics. However, our QFT results shouldobviously be insensitive to the exact value of this arbitrary cuto, so at the end we should be ableto remove the regulator again.

Power counting

In the example above we saw that a quadratic divergence was turned into a quadratic dependenceon the cuto after regularization. This is a general feature of cuto regulators. We can determinethe divergence of an integral using power counting : this is simply adding the powers of the loopmomentum in the integration measure to the powers in the numerator and subtracting the powersin the denominator. The resulting number is called the degree of divergence D. D = 2 indicatesthat the leading divergence is quadratic, D = 1 means the leading divergence is linear, while D = 0is a logarithmic divergence. D = −1 or lower indicates a convergent integral.

Of course we can also extract the degree of divergence from the Feynman diagrams, since weknow what each part of a Feynman diagram contributes in terms of powers of the momentum.For example a bosonic propagator contributes −2 to the degree of divergence, while a fermionicpropagator contributes −1.

It is important to stress, however, that this simple way of power counting does not always givethe correct answer. Dimensionful coupling constants can `absorb' powers of momenta. Similarly,the Ward identities can cause the true degree of divergence to be dierent from the one computedusing power counting.

2.1.2 Dimensional regularization

In practice doing computations with cuto regularization can be very awkward since the cutobreaks translational invariance in momentum space. This is why a dierent type of regularization,called dimensional regularization (often shortened to DREG), is more commonly used. In thismethod the extra parameter that is introduced is the number of spacetime dimensions d. StandardQFT is formulated in 4-dimensional spacetime. This means that while doing power counting weget 4 powers of the momentum from the integration measure. In d dimensions we get d powersof momentum from the integration measure, so when d is low enough the integral will converge.Since we still integrate over the entire momentum space this does not break translational invariance.

This regularization method is based on the fact that all Feynman integrals in d dimensions canbe expressed in terms of the gamma function Γ(z). For example, consider the following type ofintegrals:

µ4−d∫

ddp

(2π)d1

(p2 −∆)n= µ4−d (−1)ni

(4π)d/2Γ(n− d

2

)Γ(n)

∆d/2−n (2.5)

where ∆ is a quantity with dimension [mass]2 and µ is an arbitrary energy scale; the factor µ4−d

has been added in front of the integral so that the total expression has the same mass dimensionas its counterpart in 4 dimensions. In the integral p is now a momentum vector with 1 time com-ponent and d− 1 spatial components. The gamma function Γ(z) is dened for all complex valuesof z except for its poles at z = 0,−1,−2, ... . We use this fact to dene the value of the Feynmanintegrals for any (complex!) number of spacetime dimensions d. In the example above we haveΓ(n − d/2) in the numerator. So when d = 4 we encounter the poles of this gamma function atn = 1 and n = 2. These poles reect the divergence of the Feynman integrals.

So whenever we encounter an integral that is divergent in 4 spacetime dimensions we can usedimensional regularization to assign a nite value to it. Once again we should be able to removethe regulator by setting d = 4 at the end of our calculation.

Often we do not use the d-dimensional result, but we perform an expansion around d = 4dimensions. There are dierent conventions to do this, but we will use d = 4− ε. This changes the

16

regularization parameter from d to ε. Let's apply this to the example of (2.5). For simplicity let'stake n = 2. Then we need the following expansions

µ4−d = µε ≈ 1 + ε logµ+O(ε2) (2.6)

(4π)−d/2 = (4π)ε/2−2 ≈ (4π)−2(

1 +ε

2log(4π) +O(ε2)

)(2.7)

∆d/2−2 = ∆−ε/2 ≈ 1− ε

2log ∆ +O(ε2) (2.8)

Γ

(2− d

2

)= Γ

( ε2

)≈ 2

ε− γE +O(ε) (2.9)

Putting this together we get

µ4−dΓ(2− d

2

)(4π)d/2

∆d/2−2 ≈ 1

(4π)2

(2

ε− γE + log(4π)− log(

∆

µ2) +O(ε)

)(2.10)

Here γE = 0.577... is the Euler-Mascheroni constant. Now we can explicitly see the pole in ε whichreects the divergence of the 4-dimensional integral. The constants γE and log(4π) are typical indimensional regularization. Of course, now removing the regulator means sending ε to zero.

A downside to dimensional regularization (at least depending on ones point of view) is that itdoes not have a clear physical interpretation. Varying the dimensionality of spacetime seems to bemore of a mathematical trick. A successful trick, certainly, but not something that can easily beunderstood by physical arguments.

One more thing to mention is how we can read o the degree of divergence in dimensional reg-ularization. The integral in Eq. (2.5) is quadratically divergent in 4 dimensions if n = 1. Then theright-hand side contains the factor Γ(1− d/2), which has a pole at d = 4 dimensions as expected,but also another pole at d = 2 dimensions. On the other hand, the integral is logarithmicallydivergent in 4 dimensions if n = 2. In this case the right-hand side contains Γ(2− d/2), which hasa pole at d = 4 dimensions, but not at d = 2 dimensions. So in dimensional regularization thelowest singular dimension is an indication of the degree of divergence.

Finally I would like to mention a limitation of dimensional regularization. This is when the γ5

matrix appears in the calculation. After taking the trace this produces the 4-dimensional Levi-Civita symbol εµνρσ. Both γ5 and εµνρσ are intrinsically 4-dimensional objects and we cannotanalytically continue them into d dimensions. There are ways to get around this problem, but wewill not go into them here.

2.1.3 Pauli-Villars regularization

The nal type of regularization I will mention here is Pauli-Villars regularization. This methodconsists of adding to a loop diagram another loop diagram with a heavy particle in the loop. Saywe have a diagram where a particle with mass m goes around in a loop. Then we add anotherdiagram where the particle in the loop has a large mass M and opposite spin statistics to theparticle in the original loop.

Here is an example of a Feynman integral and the expression we get after doing Pauli-Villarsregularization:∫

d4p

(2π)4

1

(p2 −m2)((p+ q)2 −m2)

P-V reg.−−−−−→∫

d4p

(2π)4

(1

(p2 −m2)((p+ q)2 −m2)− 1

(p2 −M2)((p+ q)2 −M2)

)(2.11)

Obviously in the large momentum limit (p → ∞) the two terms cancel due to the opposite spinstatistics. This means that the UV-divergence is canceled and the full integral, containing bothterms, produces a nite answer.

Clearly the regularization parameter is the mass M of the new, auxiliary particle. At the endof the calculation we should be able to remove the regulator by lettingM go to innity. Eectively,

17

this decouples the auxiliary particle.

This regularization procedure, just like the cuto method, is one that is based on physicalarguments: we are temporarily adding an auxiliary particle to help us do the calculation. Sure,this is not a physical particle that can be detected, but its particle interpretation does help usunderstand a bit more about what we are doing.

These are just a few of the regularization methods that are in use. There are many, many more,although a lot of those are just variations or extensions of these three basic methods. Dimensionalregularization is the most widely used method, since it is quite simple to apply from a calculationalpoint of view and it does not break most of the important symmetries.

2.2 Renormalization

2.2.1 Introduction to renormalization

After performing a suitable regularization procedure, we have turned all divergences originatingfrom loop diagrams into nite quantities. However, our expressions now contain an arbitrary reg-ularization parameter. This means that we do not have any predictive power yet. What do weneed to do so that our calculations can make predictions for actual measurable processes?

To nd the answer to this question we should go back to our original Lagrangian. The La-grangian contains parameters such as coupling constants and masses. These are called the bareparameters. When we originally made our perturbation expansion we did an expansion in the barecoupling constants. The example in Fig. 2.2 represents the following mathematical expression ifwe use cuto regularization for the loop diagrams:

iM = −iλ+iλ2

32π2

[log

(Λ2

s

)+ log

(Λ2

−t

)+ log

(Λ2

−u

)+ 3 + iπ

]+O(λ3) (2.12)

In this calculation the particles were taken to be massless. This simplies the expression and is agood approximation if the involved momenta are much larger than the particle mass (i.e. p2 m2).In turn the cuto scale was taken to be much larger than the involved momenta (Λ2 p2). Thevariables s, t and u are the usual Mandelstam variables:

s = (p1 + p2)2

t = (p1 − p3)2

u = (p1 − p4)2

(2.13)

While the matrix element itself is still not a measurable quantity, its absolute value squared|M|2 is closely related to the cross section of the process, which is measurable. Our expression(2.12) for the matrix element shows us that we have no chance of measuring the bare couplingconstant λ. We can only measure |M|2, which is an expansion in λ. This is true in general: noneof the bare parameters in the Lagrangian are measurable, because we have to take all quantumcorrections due to loops into account.

So what we measure is more like an `eective' interaction. We also see from (2.12) that thequantum corrections are momentum-dependent due to the creation and annihilation of virtualparticles in the loops. This idea of eective interactions is the key to understanding renormalization.Let's say we measure the strength of this eective interaction at some common energy scale s =−2t = −2u = q2 ≥ 0 and we use this to dene the physical coupling constant λph:

iM∣∣∣s=−2t=−2u=q2

≡ −iλph = −iλ+iλ2

32π2

[3 log

(Λ2

q2

)+ 2 log(2) + 3 + iπ

]+O(λ3) (2.14)

18

Now that we have measured λph at the energy scale q2, we can express (2.12) in terms of λph:

iM = −iλph +iλ2ph

32π2

[log

(q2

s

)+ log

(q2

−2t

)+ log

(q2

−2u

)]+O(λ3

ph) (2.15)

The dependence on the cuto Λ is now gone! We have now obtained an expression for the matrixelement in terms of strictly nite quantities and no reference to the regulator. The dependence onΛ has in fact been traded for a dependence on the energy scale q2. This is the energy scale at whichwe performed the measurement of the physical coupling constant λph and which now serves as areference scale. If we had chosen a dierent reference scale, we would have measured a dierentvalue for λph, such that the value ofM stays the same.

This is the principle of renormalization: by re-expressing our equations in terms of physicallymeasurable quantities the divergences have been removed. The reason the divergences appeared inthe rst place was because we used a non-physical bare parameter to set up the theory. This showsthat in fact the bare parameters must be innite in such a way that they cancel the divergencesoriginating from loop diagrams. One can also say that the divergence has been `absorbed' into thebare coupling constant. As already noted earlier, the renormalization procedure leaves a trace inthe form of the momentum-dependence of the eective coupling strength λph = λph(q2) due to thecreation and annihilation of virtual particles in loops. So the physical coupling `constant' λph is infact not constant.

It is sometimes stated that renormalization is needed to `remove innities'. This, however, isbased on a misconception; even if all loop diagrams were nite, we would still have to performrenormalization to express all physical observables in terms of physically measurable quantities. Itis therefore a consequence of our use of perturbation theory that renormalization is needed. Theexistence of innities in loop diagrams just makes renormalization all the more necessary.

The example we have worked out above is an example of coupling constant renormalization;at lowest order the 4-point function is simply the tree-level coupling and then higher orders addcorrections to this coupling. In φ4-theory there is also the 2-point function, which at lowest or-der is just the bare propagator and then higher orders add corrections to the propagator. Thesecorrections come from the so-called self-energy diagrams. Self-energy diagrams are 2-point one-particle irreducible (1PI) diagrams; a 1PI diagram is a diagram that cannot be split in two bycutting one internal line. In φ4-theory there is one self-energy diagram at one-loop order: Fig.2.3. The contribution from scalar self-energy diagrams is usually denoted by Σ(p2). In fact, we de-ne the self-energy diagrams to be iΣ(p2) (the sign is arbitrary, so it is just a matter of convention).

Figure 2.3: The one-loop self-energy diagram in φ4-theory.

We will not work out the calculation in detail here, but simply state the result; the renormal-ization of the 2-point function leads to the following modication of the propagator:

i

p2 −m2 + iδ→ i

p2 −m2 + Σ(p2) + iδ(2.16)

So the bare propagator on the left is replaced by the full propagator on the right1. There are twochanges here: rst, the position of the pole2 of the propagator has been shifted. Physically this

1The full propagator can also be called the renormalized propagator.2If we regard p2 as a variable the propagator has one pole. If we write out the components of the momentum,

p2 = (p0)2 − ~p2, there are two poles in the complex p0 plane.

19

means that the mass of the particle has changed. This is called the mass renormalization. Thelocation of the pole and thus the physical mass of the particle mph is dened by:

m2ph −m2 + Σ(m2

ph) = 0 (2.17)

Secondly, also the residue of the pole has changed. This can be seen if we expand the full propagatoraround its pole

i

p2 −m2 + Σ(p2) + iδ=

iZ

p2 −m2ph + iδ

+ regular terms (2.18)

The terms that are regular at p2 = m2ph can be ignored, since only the pole structure of the

propagator is of importance. The factor Z is called the wave function renormalization factor. Ifwe do the expansion explicitly we can see that it is in fact

Z−1 = 1 +dΣ(p2)

d(p2)

∣∣∣p2=m2

ph

(2.19)

So the residue of the pole has changed from i in the bare propagator to iZ in the full propagator.The factor Z reects a rescaling of the elds themselves rather than a shift in bare Lagrangian pa-rameters: φ =

√ZφR, where φ is the original (bare) eld and φR the renormalized eld. Therefore

this is also known as eld (strength) renormalization and Z can also be called the eld (strength)renormalization factor. This rescaling of elds is not an observable eect and does not aect anyphysical observables. It is however an important part of the renormalization program.

2.2.2 Renormalizability

In the previous section we have seen three forms of renormalization: coupling constant renormal-ization, mass renormalization and wave function renormalization. These are in fact all forms ofrenormalization there are. So when we set up a theory we immediately know how many quantitieswe have to renormalize: in φ4 theory we had one coupling constant (λ), one mass (m) and one typeof eld (φ). This means that these three quantities, the bare coupling constant, bare mass andbare eld, have to be able to `absorb' all the innities that can occur in loop diagrams in order forrenormalization to be successful. This may seem unlikely at rst because one can draw innitelymany loop diagrams. However if a theory is renormalizable, these innities will always appear inthe same few forms, making it possible to absorb them into the very limited number of quantitiesat our disposal.

In fact, this is the denition of the class of renormalizable theories: if all innities that occurin a theory can be absorbed into a nite number of parameters of the theory it is renormalizable.If we would need an innite number of parameters to absorb all innities, the theory is called non-renormalizable. As a fundamental theory of nature a nonrenormalizable theory has no predictivepower since we would have to perform an innite number of measurements before we would beable to make any predictions. However, nonrenormalizable interactions can be used in so-calledeective eld theories (EFTs), which as the name implies are not fundamental theories but arelimited to use at low energies only.

Of course we would like to know which theories are renormalizable and which ones are not.There is a simple criterion that allows us to do just that: if a theory contains coupling constantswith negative mass dimension it is nonrenormalizable, whereas if there are only coupling constantswith mass dimension zero or positive the theory is renormalizable. Note that this is not a lawand there are a few exceptions to this rule, but it works for the majority of theories. We canheuristically understand this rule by considering the following: if we use cuto regularization acoupling constant λ with negative mass dimension −n < 0 would appear in our expressions in thedimensionless combination λΛn. If we try to remove the regulator this diverges horribly. Thisis the basic origin of nonrenormalizability. With coupling constants that have a mass dimension+n ≥ 0 we do not run into such a problem when trying to remove the regulator.

20

2.2.3 Renormalized perturbation theory

The renormalization process we followed at the start of this section led us to replace the bareparameters λ and m by the physical parameters λph and mph. It turned out that the way we setup our perturbation theory based on the bare Lagrangian was not very convenient and forced usthrough this whole process of renormalization. It seems obvious to ask whether we could haveworked with the physical parameters in the rst place and saved ourselves a lot of work. It turnsout that this is indeed possible. Let's take φ4 theory again and consider the bare Lagrangian,

L =1

2(∂µφ0)(∂µφ0)− 1

2m2

0φ20 −

λ0

4!φ4

0 (2.20)

where the subscript 0 is used to indicate bare quantities. The rst step is now to rescale the eldswith the wave function renormalization factor as we saw in Section 2.2.1. So φ0 =

√ZφR, where

φR indicates the rescaled, renormalized eld. Plugging this into (2.20) we get

L =1

2Z(∂µφR)(∂µφR)− 1

2m2

0Zφ2R −

λ0

4!Z2φ4

R (2.21)

It does not seem like we have accomplished much with this step. But now in the second step werewrite our Lagrangian in terms of λph and mph:

L =1

2(∂µφR)(∂µφR)− 1

2m2phφ

2R −

λph4!φ4R +

1

2δZ(∂µφR)(∂µφR)− 1

2δmφ

2R −

δλ4!φ4R (2.22)

whereδZ = Z − 1, δm = m2

0Z −m2ph, δλ = λ0Z

2 − λph (2.23)

In our rewritten Lagrangian (2.22) the rst three terms are the same as the terms in the bareLagrangian, but now written entirely in terms of the renormalized quantities. The other threeterms form the counterterm Lagrangian Lct. In fact we have traded the three parameters Z, m0

and λ0 for three new parameters δZ , δm and δλ called the counterterms (for the eld strength,mass and coupling constant, respectively). We will only work with these counterterm parametersfrom now on.

What we have done here is called renormalized perturbation theory : we rewrite the Lagrangianand split it into a part that contains only renormalized parameters and a counterterm part. Thiscan be done for any renormalizable theory. For ease of notation we will now drop all subscripts inthe Lagrangian:

L =1

2(∂µφ)(∂µφ)− 1

2m2φ2 − λ

4!φ4 +

1

2δZ(∂µφ)(∂µφ)− 1

2δmφ

2 − δλ4!φ4 (2.24)

However, it is absolutely crucial to keep in mind that from now on m and λ refer to the physicalparameters and not the bare parameters!

Basically all we have done here is rewriting the Lagrangian and redening the parameters. How-ever, this changes the way we do our calculations. The innities that were previously absorbedinto Z, m0 and λ0 now have to be absorbed into the three counterterms δZ , δm and δλ. But sincethese counterterms have their own terms in the Lagrangian they lead to new Feynman rules. TheFeynman rules for the Lagrangian (2.24) can be seen in Fig. 2.4.

Now we have new Feynman rules for a counterterm propagator and for a counterterm vertex,indicated in the diagrams by a circle with a cross inside. So when we write all possible diagramsfor a process we now have to include diagrams that contain these new elements. The idea is thatthe innities arising from loop diagrams are canceled by the counterterms. Order by order inperturbation theory we have to adjust the counterterms such that this cancellation takes place. Ifthe theory is renormalizable it is indeed possible to cancel all divergences in loop diagrams withjust a nite number of counterterms. In this sense the process is very similar to the one we followedin Section 2.2.1, just with dierent parameters.

21

Figure 2.4: The Feynman rules for φ4 theory in renormalized perturbation theory. Picture adaptedfrom [1].

2.2.4 Renormalization schemes

The conclusion of the previous section is that in renormalized perturbation theory we have tochoose the counterterms such that they cancel the divergences from loop diagrams. This is in factthe only requirement on the counterterms. So only the innite/divergent part of the countertermsis xed by this requirement. The counterterms can also have a nite part, which is ambiguous.There are dierent ways for xing the nite parts of the counterterms. The dierent ways arecalled renormalization schemes and we will look at two very common renormalization schemes inthis section: MS renormalization and on-shell (OS) renormalization.

MS (and MS) renormalization

The most straightforward renormalization scheme is the minimal subtraction (MS) scheme. Inthis scheme one simply sets the nite parts of all counterterms equal to zero. Calculations areusually simplest to do when working in this scheme. It is practically always used in combinationwith dimensional regularization. The divergent part is then a pole in the regularization parameterε as we saw in Section 2.1.2. The counterterm then has to be the same pole with the oppositenumerical prefactor to cancel the divergence from the original diagram.

As an example we will again look at the 4-point function in φ4 theory. Up to one-loop orderwe have the same diagrams we had in Fig. 2.2, but since we are now working in renormalizedperturbation theory we also have a counterterm diagram:

Figure 2.5: The rst few terms in the perturbation expansion of the 4-point function in φ4-theory.Because we are now working in renormalized perturbation theory we have to include one morediagram on the right-hand side, the counterterm diagram. Picture adapted from [1].

In Section 2.2.1 we used a cuto regulator to do this calculation. Now we will use dimensionalregularization to regularize the divergences. One aspect of dimensional regularization that wasalready mentioned in Section 2.1.2 is the fact that we have to introduce a new arbitrary energyscale µ to keep the mass dimension of the expression unchanged and ensure that our calculationsare dimensionally correct. For example look at the following integral and its replacement afterdimensional regularization (again we use d = 4− ε):∫

d4p

(2π)4

1

(p2 −m2)2

DREG−−−−→ µε∫

ddp

(2π)d1

(p2 −m2)2(2.25)

22

This integral was dimensionless in 4 spacetime dimensions and its replacement is now also dimen-sionless. The scale µ is called the renormalization scale of dimensional regularization.

Then the diagrams in Fig. 2.5 represent the following expression:

iM = −iλ− iλ2

32π2

[−6

ε+ 3γE − 3 log(4π) +W (s) +W (t) +W (u)

]− iδλ +O(λ3) (2.26)

where3

W (p2) =

∫ 1

0

dx log

(m2 − x(1− x)p2

µ2

)(2.27)

Clearly, in the MS renormalization scheme the counterterm δλ has to be xed at the followingvalue to cancel the pole in Eq. (2.26):

δλ =3λ2

16π2

1

ε(2.28)

We have now completed the calculation of this counterterm in the MS scheme at O(λ2). Howeverthis is not the end of the story: order by order in perturbation theory we have to keep `updating'the counterterms to cancel new divergences that appear at higher orders.

In the modied minimal subtraction (MS) scheme we do not only cancel the divergent part ofthe amplitude, but we also cancel the numerical factors γE and log(4π) that appear when doingdimensional regularization. This simplies our expressions and is no less arbitrary than the MSscheme. With our convention that d = 4 − ε the divergent part will almost always occur in thefollowing combination:

Nε =2

ε− γE + log(4π) (2.29)

So in the MS scheme the counterterm in Eq. (2.26) has to be

δλ =3λ2

32π2Nε (2.30)

If we plug this into Eq. (2.26) we get

iM = −iλ− iλ2

32π2

∫ 1

0

dx log

([m2 − x(1− x)s][m2 − x(1− x)t][m2 − x(1− x)u]

µ6

)+O(λ3) (2.31)

The counterterm has canceled the divergence, so we are left with a nite expression, as an-ticipated. However, our expression now contains the new energy scale µ. We will look into themeaning of this energy scale later. First we will take a look at a dierent renormalization scheme.

On-shell renormalization

A dierent type of renormalization scheme is the on-shell (OS) scheme. In this type of scheme,instead of simply subtracting the divergence, we impose certain renormalization conditions. Theseconditions x the value of certain amplitudes at a specic energy. This in turn xes the values ofthe counterterms. In φ4 theory we can impose renormalization conditions on the 2-point amplitudeand the 4-point amplitude. Fig. 2.6 is an example of these two renormalization conditions.

At this point we should introduce a new notation. In renormalized perturbation theory we candistinguish between the renormalized self-energy Σ(p2), which includes all counterterm diagrams,and the unrenormalized self-energy Σ(p2), which does not include the counterterms. In Eq. (2.16)we wrote down the general form of the renormalized propagator; in renormalized perturbationtheory it is the renormalized self-energy Σ(p2) that appears in the denominator. So the fullrenormalized propagator, which we will now name Π(p2), looks like this:

Π(p2) =i

p2 −m2 + Σ(p2) + iδ(2.32)

3In this calculation we have ignored the +iδ prescription for the propagators. Strictly this prescription needs tobe taken into account to make sure that Eq. (2.27) has the correct analytic behavior in the complex plane. Butthis is unimportant for the points we want to illustrate regarding renormalization. Therefore, the +iδ prescriptionhas been ignored here.

23

Figure 2.6: Possible renormalization conditions for the 2-point and 4-point amplitude in φ4 theory.Picture from [1].

Thinking back to our discussion of Eq. (2.16) the rst diagram in Fig. 2.6 actually imposestwo conditions, because it xes both the location and the residue of the pole of the propagator. Ifwe translate these conditions in terms of the renormalized self-energy Σ(p2) we need to impose:

Σ(p2 = m2) = 0 anddΣ(p2)

d(p2)

∣∣∣p2=m2

= 0 (2.33)

So we see that we have to specify the momentum at which to impose these conditions.

Let's consider these conditions at the one-loop level: at this level the renormalized self-energyΣ(p2) consists of the one-loop 1PI diagram of Fig. 2.3 and the counterterm propagator given inFig. 2.4. We now dene −iΣ(1)(p2) to be the expression assigned to the one-loop 1PI diagram.Then at the one-loop level Eq. (2.33) explicitly leads to the two conditions

−Σ(1)(m2) +m2δZ − δm = 0

−dΣ(1)(p2)

d(p2)

∣∣∣p2=m2

+ δZ = 0(2.34)

from which the counterterms δZ and δm can easily be determined.

For the second diagram in Fig. 2.6 we also have to specify the momentum. The choice wasmade to dene the coupling constant λ as the coupling strength of the s-channel process with theminimum center-of-mass energy s = 4m2.

Let's work out the renormalization condition for the 4-point function. We will again do thecalculation at O(λ2). Taking the expression from Eq. (2.26) and imposing the condition from Fig.2.6:

− iλ− iλ2

32π2

[−3Nε +W (4m2) + 2W (0)

]− iδλ = −iλ (2.35)

Obviously, the counterterm is now xed at

δλ =λ2

32π2

[3Nε −W (4m2)− 2W (0)

](2.36)

Plugging this into Eq. (2.26) we get

iM = −iλ− iλ2

32π2

∫ 1

0

dx log

([m2 − x(1− x)s][m2 − x(1− x)t][m2 − x(1− x)u]

m4[m2 − 4x(1− x)m2]

)+O(λ3) (2.37)

This expression is quite similar to Eq. (2.31); the counterterm has again canceled the divergence,producing a nite expression, but now the dependence on µ has also canceled out.

Comparison between the MS and OS schemes

The on-shell scheme has the advantage that the renormalized parameters m and λ have a physi-cal interpretation: m is the physical (pole) mass of the particle, while λ is the physical coupling

24

strength measured at a predened energy scale. This is exactly the point of using renormalizationconditions. In the MS or MS scheme we cannot give such an interpretation to the parameters; inthat case m and λ are simply (renormalized) parameters that are needed to do the calculations.So while the parameters may have the same names their meaning is dierent depending on therenormalization scheme that is being used. This also shows that it was actually misleading to talkabout physical parameters when we set up our renormalized perturbation theory in Eq. (2.22);it would have been better to call them renormalized parameters, because whether they are alsophysical or not depends on the scheme, as we have now seen.

2.2.5 Renormalization group equations

Our expression in Eq. (2.31) contains the energy scale µ, which we called the renormalization scaleearlier. This is an arbitrary scale, so clearly any physical observable cannot depend on it. And yetour matrix element (2.31), which is closely related to a physical observable, does seem to dependon µ. The only way out of this is if, besides the explicit dependence on µ, there is also an implicitdependence on µ through the renormalized parameters, i.e. λ = λ(µ) and m = m(µ). Let's imposethe requirement that our matrix element does not depend on µ4

µdMdµ

= 0 (2.38)

Writing this out in terms of partial derivatives we have5[µ∂

∂µ+ µ

∂λ

∂µ

∂

∂λ+ µ

∂m

∂µ

∂

∂m

]M = 0 (2.39)

The rst term in this equation species the explicit dependence on µ, the second term the depen-dence on µ through λ and the third term the dependence on µ through m.

For the µ-dependence of λ we typically get:

β(λ) = µ∂λ

∂µ(2.40)

The β-function on the left-hand side can be computed from perturbation theory and is a polynomialin λ. Eq. (2.40) is called the renormalization group equation (RGE) for the coupling constant λ.It denes how λ evolves with the energy. This is usually referred to as the running of the couplingconstant. As we already saw in Section 2.2.1 this energy-dependence of the coupling constant is aremnant of the renormalization process, because we have absorbed the eects of virtual particlesinto our parameters. Similarly, there is an RGE for the mass parameter, which has also become arunning parameter:

γm(λ) =µ

m

∂m

∂µ(2.41)

γm is called the anomalous mass dimension.

The RGEs ensure that it does not matter which reference scale µ we choose, if all orders inperturbation theory are taken into account. Dierent values of µ just label dierent points onthe trajectories λ(µ) and m(µ). So when we evolve the set (µ, λ(µ),m(µ)) to a dierent scale(µ′, λ(µ′),m(µ′)) all physical observables are left invariant. However, when doing actual calcu-lations it is preferable to choose a value of µ such that the logarithms in which µ appears aresuppressed. The advantage of choosing the `right' renormalization scale is that fewer orders inperturbation theory are needed to get an accurate answer.

Note that in the on-shell renormalization scheme we followed in Section 2.2.4 the parametersdo not run because we xed them at a certain scale. We xed the coupling constant at a scaleO(m) (λ ≈ λ(m)), and we xed the mass parameter to be the physical mass (m = mph).

4Multiplying by µ is a convention since µ usually appears in logarithms and dd(log µ)

= µ ddµ

5It is more common to do this for actual observables instead of matrix elements. Eq. (2.39) will then appear ina slightly modied form. However, to illustrate the point, working with the matrix element is ne and we will notgo into more detail here.

25

2.2.6 Renormalization scheme dependence

In principle it should not matter which renormalization scheme we decide to use. In practicehowever, we are limited by our use of perturbation theory; we are usually only able to calculatethe rst few orders in a perturbation series. Imagine that we want to calculate the physical quantityf . In (renormalized) perturbation theory we can write the expansion of f as

f(g) = f0 + f1g + f2g2 + f3g

3 + ... (2.42)

where g is some generic coupling constant. As we have seen in Section 2.2.4 the denition andthus the value of g depends on the renormalization scheme that we use in our calculation. Butf(g) is a physical quantity, so it cannot depend on the renormalization scheme. That means thatthe coecients fi also have to be dierent in dierent renormalization schemes to make sure thatthe perturbation series converges to the same value for f(g).

Let's say that we are only able to calculate the rst three coecients f0, f1 and f2. We truncatethe series after the third term:

f (3)(g) = f0 + f1g + f2g2 (2.43)

Now we have only got an approximation f (3)(g) to the actual physical quantity f(g). This ap-proximation is in general renormalization scheme dependent. So when performing a calculation inperturbation theory to nite order it does matter which renormalization scheme is used.

26

Chapter 3

Gauge anomalies

3.1 Noether's theorem

We have seen in Chapter 1 that we can set up theories based on dierent symmetry groups. Theimportance of symmetries in the Lagrangian is explained by Noether's theorem. In short, the state-ment of this theorem is the following: for every global, continuous symmetry in the Lagrangianthere is a conserved current.

Let's work out what this statement looks like mathematically. If we have a classical LagrangianL depending on one eld φ its equation of motion is determined by the Euler-Lagrange equation:

∂L∂φ

= ∂µ

(∂L

∂(∂µφ)

)(3.1)

Now we perform an innitesimal global transformation of the eld:

φ→ φ+ α∆φ (3.2)

where α is an innitesimal parameter and ∆φ is determined by the type of transformation. Thistransformation is a symmetry of the Lagrangian if the Lagrangian is invariant under this transfor-mation or if it only changes by a 4-divergence1

L → L+ α∂µGµ (3.3)

where Gµ is some function of the eld φ that again depends on the type of transformation. Aftera simple derivation one nds that

∂µ

(∂L

∂(∂µφ)∆φ−Gµ

)= 0 (3.4)

i.e.

jµ =∂L

∂(∂µφ)∆φ−Gµ (3.5)

is a conserved current. This works for any global symmetry. It can also easily be extended totheories with multiple elds or elds with multiple components; in the rst term of Eq. (3.5) onethen simply sums over all elds and all components:

jµ =∂L

∂(∂µφa)∆φa −Gµ (3.6)

The index a runs over all elds and all components.

1We can allow the Lagrangian to change by a 4-divergence because the action S is the integral over the Lagrangian.In the integral the 4-divergence term turns into a boundary term due to the divergence theorem. Boundary termsdo not change the equation of motion (3.1).

27

3.2 The Ward identity

Noether's theorem holds in classical eld theories with a global symmetry. Still, the Noether cur-rent implies that charge2 is conserved locally. We have seen in Section 1.1 how we can turn aglobal symmetry into a gauge symmetry. Due to the idea of the gauge principle the symmetry isnow local. This has an eect on current conservation as well: the current belonging to the gaugesymmetry is now coupled to the gauge eld. After creating a gauge theory this way we can thenquantize the elds, so that we can derive the Feynman rules and calculate quantum mechanicalamplitudes. These amplitudes are represented by Feynman diagrams. In the language of Feynmandiagrams the consequence of current conservation is that charge is conserved at every vertex of aFeynman diagram.

The analog to Noether's theorem in the quantum theory is the Ward identity. This identity isusually written down in momentum space. If we have a matrix element (calculated in momentumspace) with an external massless gauge boson carrying a 4-momentum k we can write it as

M(k) = εµ(k)Mµ(k) (3.7)

where εµ(k) is the polarization vector belonging to the external gauge boson. The Ward identitynow states that if we replace the polarization vector by the momentum of the external gauge bosonthe expression vanishes

kµMµ(k) = 0 (3.8)

Physically, this equation is not only related to current conservation, but also to the fact that amassless gauge boson cannot have a longitudinal polarization. A longitudinal polarization vectorwould be proportional to the momentum vector and thus never contribute to any process accordingto theWard identity (3.8). This means that the longitudinal mode is unphysical and can be removedfrom the theory. Of course, by considering the number of degrees of freedom we could have alreadyconcluded that a massless gauge boson has only two physical modes, which, as it turns out, arethe two transverse polarizations.

3.3 Anomalies

The Ward identity is the result of a symmetry in the classical Lagrangian. It is not guaranteed thatthe classical symmetry is also a valid symmetry of the quantum theory, so we could ask ourselveswhether quantum eects can break the symmetry. It turns out that this can indeed happen. Sincethis symmetry breaking is a purely quantum eect, it can only occur in loop diagrams. The typeof diagram that does this is seen in Fig. 3.1: a triangle diagram with a fermion loop and threeexternal gauge bosons.

Figure 3.1: A diagram that can break the classical symmetry and violate the Ward identity.

In such a diagram one has to consider three Ward identities, one for each external gauge boson.It turns out that if the fermion trace contains a γ5-matrix it is not possible to satisfy all three

2The Noether charge is simply the Noether current integrated over a spatial volume.

28

Ward identities simultaneously.

Consider the diagram above, where the top gauge boson has a purely axial (γµγ5) coupling tothe fermions and the bottom two gauge bosons have a purely vector (γµ) coupling to the fermions.We will use the same coupling constant g for all three couplings. The three momenta are allincoming. We take the fermions to be massless. We will consider the non-abelian case, so at everyvertex there is also a generator T i of the non-abelian group. At the end of the calculation we canthen easily reduce to the abelian case. Besides the diagram in Fig. 3.1 we also have to considera second diagram where the fermion loop goes around in the other direction. One can easily seethat this is just equivalent to performing the simultaneous interchanges p↔ q, ν ↔ ρ and b↔ c.So the expression we are considering is

iM = εµ(k)εν(p)ερ(q) (iV µνρabc (p, q)) (3.9)

where

iV µνρabc (p, q) = −g3

∫d4l

(2π)4

Tr(γµγ5(/l + /p)γν/lγρ(/l − /q)

)l2(l + p)2(l − q)2

Tr(T aT bT c) + (p↔ q, ν ↔ ρ, b↔ c)

(3.10)Note that the two separate traces are over dierent spaces: the rst is over spinor space, the secondover the representation space of the non-abelian generators.

By power counting we see that the integral is linearly divergent, so we have to regularize it.The γ5-matrix is tough to deal with in dimensional regularization, so we will use Pauli-Villarsregularization instead (see Section 2.1.3). The virtual Pauli-Villars particle has a mass M , so theregularized integral takes the form

[iV µνρabc (p, q)]Reg = g3

∫d4l

(2π)4[Iµνρ0 (l, p, q)− IµνρM (l, p, q)]Tr(T aT bT c) + (p↔ q, ν ↔ ρ, b↔ c)

(3.11)with

Iµνρm (l, p, q) =Tr(γ5γµ(/l + /p+m)γν(/l +m)γρ(/l − /q +m)

)(l −m)2((l + p)2 −m2)((l − q)2 −m2)

(3.12)

We don't have to compute the whole expression, because we just want to check the Ward identities.First, let's check the Ward identity for the top gauge boson. That means we have to replace εµ(k) bykµ. In other words we have to work out the contraction of kµ with [iV µνρabc (p, q)]Reg. By momentumconservation kµ = −pµ − qµ so we would like to compute

− i(pµ + qµ)[V µνρabc (p, q)]Reg (3.13)

The rst step is to work out the fermion traces. This is easier now that we have contracted withthe momentum of the gauge boson, because we can use

γ5(/p+ /q) = γ5(/l + /p−m) + (/l − /q −m)γ5 + 2mγ5 (3.14)

The rst two terms on the right-hand side lead to signicantly simpler expressions; if we look atthe rst term we get

(/l + /p−m)(/l + /p+m) = (l + p)2 −m2 (3.15)

which cancels one of the factors in the denominator. The leftover trace is easier to evaluate

Tr(γ5γν(/l +m)γρ(/l − /q +m)) = −4iενλρκlλ(lκ − qκ) = 4iενλρκlλqκ (3.16)

In the second step of this equation we used the antisymmetry of the Levi-Civita symbol. Theintegral that we are now left with is

ενλρκqκ

∫d4l

(2π)4

[lλ

l2(l − q)2− lλ

(l2 −M2)((l − q)2 −M2)

](3.17)

We don't have to explicitly evaluate this integral. We know that the outcome of the integral has tobe a 4-vector and the only 4-vector available in the integrand is qλ. This leads us to conclude that

29

the integral has to be proportional to qλ. That means the expression in Eq. (3.17) is proportionalto ενλρκqκqλ, which vanishes. Note that this argument only holds because the integral (3.17) isconvergent due to the regularization.

For the second term in (3.14) the story is much the same as for the rst term. Now the factor((l − q)2 −m2) cancels in the denominator. The leftover trace is now

Tr(γ5(/l + /p+m)γν(/l +m)γρ) = −4iελνκρ(lλ + pλ)lκ = −4iελνκρpλlκ (3.18)

leaving the integral

ελνκρpλ

∫d4l

(2π)4

[lκ

l2(l + p)2− lκ

(l2 −M2)((l + p)2 −M2)

](3.19)

This integral is proportional to pκ by the same argument as before. Therefore the expression inEq. (3.19) also vanishes.

Finally, we have to look at the third term in Eq. (3.14). Obviously, for the m = 0 part ofthe integrand we don't get a contribution from this term. However, for the regulator part in theintegrand m = M and we could get a contribution. The trace for this term evaluates to

2MTr(γ5(/l + /p+M)γν(/l +M)γρ(/l − /q +M)) = 8iM2ελνρκpλqκ (3.20)

Now there is no loop momentum in the numerator anymore and the integral we have to look at is∫d4l

(2π)4

1

(l2 −M2)((l + p)2 −M2)((l − q)2 −M2)(3.21)

Because we are going to take the limit M →∞ at the end of our calculation we can already takethat into consideration at this step. In the large M limit we can ignore the dependence on p andq in Eq. (3.21). This simplies the integral and makes it easy to evaluate∫

d4l

(2π)4

1

(l2 −M2)3=

−i32π2M2

(3.22)

Taking everything together we obtain the nal result

−i(pµ + qµ)[V µνρabc (p, q)]Reg = g3(8iM2ελνρκpλqκ)

(−i

32π2M2

)Tr(T aT bT c) + (p↔ q, ν ↔ ρ, b↔ c)

=g3

4π2ελνρκpλqκTr(T

aT b, T c)(3.23)

where the curly brackets denote the anti-commutator. The dependence on the regularization pa-rameter M has disappeared, so there is no issue in taking the limit M → ∞. This result is ingeneral nonzero (depending on the group), which means the Ward identity is not satised. Inother words the gauge symmetry has been broken by this diagram. This is called an anomaly : aclassical symmetry that turns out to be broken by quantum eects. This particular anomaly wehave encountered here is usually called a gauge anomaly, since it breaks a gauge symmetry of thetheory. The expression on the right-hand side of Eq. (3.23) is called the anomaly contribution andthe diagram in Fig. 3.1 is called an anomaly diagram. Breaking a gauge symmetry is not accept-able if we want to have a consistent quantum theory. We will discuss how this issue is resolved inthe next section.

For now let's take a look at some other aspects of this calculation. If the group is abelian thematrices T i are just factors, namely the U(1) charge Q of the fermion. In that case the result is

− i(pµ + qµ)[V µνρ(p, q)]Reg =(Qg)3

2π2ελνρκpλqκ (3.24)

For completeness we can also take a look at the Ward identities for the other two gauge bosons.If we look at the Ward identity for the left gauge boson we need to compute

ipν [V µνρabc (p, q)]Reg (3.25)

30

The rst step is again to work out the fermion trace. In the trace we can again rewrite theexpression to cancel factors in the denominator

(/l + /p+m)/p(/l +m) = (/l + /p+m)[(/l + /p−m)− (/l −m)](/l +m)

= ((l + p)2 −m2)(/l +m)− (l2 −m2)(/l + /p+m)(3.26)

This step is similar to what we did in Eq. (3.14). The crucial dierence with Eq. (3.14) is theabsence of the third term, where no factor in the denominator is canceled.

For the rst term in Eq. (3.26) the leftover trace is

Tr(γ5γµ(/l +m)γρ(/l − /q +m)) = 4iεµλρκlλqκ (3.27)

The integral that remains is exactly the same as the one in Eq. (3.17), so combined with theLevi-Civita symbol and the qκ vector its contribution vanishes. For the second term in Eq. (3.26)the leftover trace is

Tr(γ5γµ(/l + /p+m)γρ(/l − /q +m)) = −4iεµλρκ(lλ + pλ)(lκ − qκ) (3.28)

The term where the two loop momenta are contracted with the Levi-Civita symbol will vanish dueto symmetry arguments. Now the integral we have to deal with requires a bit more care

εµλρκ∫

d4l

(2π)4(lλ + pλ)(lκ − qκ)

[1

(l + p)2(l − q)2− 1

((l + p)2 −M2)((l − q)2 −M2)

](3.29)

The way to handle this integral is to perform a shift in the loop momentum lµ → lµ + qµ. Sincethe regularized integral is convergent this shift is well-dened. Eq. (3.29) then becomes

εµλρκ(pλ + qλ)

∫d4l

(2π)4

[lκ

l2(l + p+ q)2− lκ

(l2 −M2)((l + p+ q)2 −M2)

](3.30)

Now we realize that pλ + qλ = −kλ, so the integral simplies to

− εµλρκkλ∫

d4l

(2π)4

[lκ

l2(l − k)2− lκ

(l2 −M2)((l − k)2 −M2)

](3.31)

By the same arguments as we used previously in this section the result of the integral has to beproportional to kκ, causing the expression to vanish.

We conclude that the Ward identity for the left gauge boson in Fig. 3.1 does hold. With almostthe same calculation we can show that the Ward identity for the right gauge boson also holds. Thereason for this is that we set up the calculation so that only the top gauge boson coupled to thefermions with a γ5. That is why the anomaly only shows up for this boson's Ward identity.

3.4 Anomaly cancellation

The fact that anomaly diagrams can violate the Ward identity is a serious problem; the gaugesymmetry is broken at the loop level. This means that renormalizability is lost, since it becomesimpossible to consistently calculate loop corrections. As a result the theory loses its predictivepower.

So it is clear that a consistent gauge theory should not have any gauge anomalies. Obviously atheory that has no couplings involving γ5 will automatically be anomaly-free. To see how theoriesthat do have couplings with a γ5 can be anomaly-free we have to take a look at the group-theoreticalpart of the anomaly contribution (3.23):

Tr(T aT b, T c) (3.32)

For some non-abelian groups this trace of generators is automatically zero, which makes the theoryanomaly-free. If the gauge group is such that this trace is not always zero, then there is a dierentway that we can have an anomaly-free theory; the particle content of the theory then has to be

31

such that the anomaly contributions from all the dierent particles that can propagate in the loopof the anomaly diagram cancel against each other.

This is what happens in the Standard Model. In the SM we have chiral couplings betweenfermions and gauge bosons involving the projection operators PL and PR. These operators containγ5 and can thus give rise to anomalies. The calculation of the anomaly diagrams in the SM is verysimilar to the calculation in the previous section; we nd that the anomaly contributions in the SMhave the same form as in Eq. (3.23), but the prefactor is slightly dierent, because the couplingsare chiral instead of purely axial or purely vector. It is also important to note that left-handedand right-handed elds contribute to the anomaly with opposite sign.

As an example let's check the cancellation of the hypercharge anomaly of the SM in the unbrokenphase. The hypercharge Y is the quantum number associated with a U(1) gauge group. The gaugeboson Bµ belongs to this gauge group, so if we want to calculate the hypercharge anomaly we haveto consider the diagram of Fig. 3.1 where the three external gauge bosons are B bosons. We thenhave to add up the anomaly contributions of all the dierent SM fermions that can propagate inthe loop. Since this is an abelian group these contributions will be proportional to Y 3 as in Eq.(3.24). Ignoring the prefactors which are the same for every eld we can then add up the anomalycontributions for all the elds in one family using the hypercharge values in Table 1.1

2Y 3L−Y 3

ν −Y 3e +3·2Y 3

Q−3Y 3u−3Y 3

d = 2(−1)3−03−(−2)3+6

(1

3

)3

−3

(4

3

)3

−3

(−2

3

)3

= 0 (3.33)

A factor of 2 is added for the left-handed fermion doublets, because both components of the dou-blet have to be considered separately. A factor of 3 is added for colored particles, because all threecolors contribute to the anomaly. We see that the hypercharge anomaly cancels in the unbrokenSM, as it should.

If the gauge group of a theory consists of multiple factors like in the Standard Model it is alsopossible to have `mixed' anomalies; for example in the SM we can consider an anomaly diagramwhere two of the gauge bosons are SU(2) bosons and one is a U(1) boson. Then Eq. (3.32) wouldtake the form

2Y Tr(T aT b) (3.34)

where Y is the U(1) charge of a fermion and T a and T b are SU(2) generators. Of course thesemixed anomalies also have to cancel. Again we can check that this happens in the SM; the onlyfermions in the SM that couple both to the SU(2) gauge bosons and to the U(1) gauge bosonare the left-handed leptons and left-handed quarks. Again, taking the colors of the quarks intoaccount, we get from (3.34)

(2YL + 3 · 2YQ)Tr(T aT b) = (−2 + 2)Tr(T aT b) = 0 (3.35)

where T a and T b are now explicitly the 2-dimensional SU(2) generators, since the left-handedleptons and left-handed quarks are both in an SU(2) doublet. So this mixed SU(2)-U(1) anomalyalso cancels in the SM.

3.5 Anomalies for massive fermions

In Section 3.3 we considered the fermions in the loop of the anomaly diagram to be massless. Wecan now discuss what happens if we try to give the fermions a mass. The steps in the calculationare still the same as in Section 3.3; the dierence is that for a fermion with mass mf the I0integrand in Eq. (3.11) is replaced by Imf . In Section 3.3 we saw that all contributions from I0vanished and that the anomaly contribution actually originated from the regulator part IM . Formassive fermions we will also get a non-zero contribution from the Imf part of the integral. Thisis not that important however; what is crucial is that the group-theoretical part of the expressionremains identical and is still given by:

Tr(T aT b, T c) (3.36)

32

Consider the case where we generate the mass of the fermion by putting a Dirac mass term inthe Lagrangian

mf ψψ = mf (ψLψR + ψRψL) (3.37)

Such a mass term mixes the left- and right-handed fermion elds. If this term is to be gauge-invariant, that means that ψL and ψR should be in the same representation of the gauge group,i.e. the theory should be non-chiral. Since left- and right-handed elds contribute to the anomalywith opposite sign as stated in the previous section, the contributions from ψL and ψR cancel eachother. So fermions with a Dirac mass do not contribute to a gauge anomaly.

The other option for a fermion mass term is a Majorana mass term. A Majorana mass termuses the charge-conjugate spinor ψC dened in Eq. (5.12) and has the following form

1

2mf (ψCLψL + ψLψ

CL ) (3.38)

The factor 1/2 is there because it is needed to give the Majorana mass term the correct normal-ization. It turns out that for such a mass term to be allowed, the eld ψL has to be in a realrepresentation of the gauge group3. A real representation is a representation whose generatorssatisfy

T a = −(T a)∗ (3.39)

Then we nd for the group-theoretical factor (3.36)

Tr(T aT b, T c) = −Tr((T a)∗(T b)∗, (T c)∗) = −Tr((T a)T (T b)T , (T c)T ) = −Tr(T aT b, T c)(3.40)

using the hermiticity of the generators. This shows that the anomaly contribution for a fermion ina real representation of the gauge group is zero.

To conclude, we see that if a fermion is in a representation of the gauge group that allows adirect mass term (either Dirac or Majorana) that fermion cannot contribute to a gauge anomaly;only massless particles can contribute to an anomaly [2].

Some extra discussion is warranted in the case of the Standard Model, a chiral theory wherethe fermion masses are generated by the Higgs mechanism. In such a theory it is very importantto consider the eects of spontaneous symmetry breaking and the dierence between the brokenand unbroken phase. The gauge symmetry of the unbroken SM is SU(3)C × SU(2)L × U(1)Y . Inthe unbroken phase the Higgs mechanism has not acted yet and all fermions are to be consideredmassless. This means that anomaly cancellation with respect to SU(3)C ×SU(2)L×U(1)Y is non-trivial; it has to be checked that all gauge anomalies cancel as we did for the pure U(1) anomalyand the mixed SU(2)-U(1) anomaly in the previous section.

Now we move to the broken phase: in this phase we will only look at the anomalies withrespect to the remaining gauge symmetry of the broken SM, which is SU(3)C × U(1)EM . In thebroken phase, the Higgs mechanism has generated Dirac mass terms from the Yukawa terms inthe Lagrangian and, as discussed earlier in this section, the anomaly cancellation for the massivefermions with respect to SU(3)C ×U(1)EM therefore becomes trivial.

3Proof: Consider the gauge transformation of ψL: ψL → eiαaTaψL. Using ψCL = CψTL we can then derive the

gauge transformation of the charge conjugate spinor and nd: ψCL → e−iαa(Ta)∗ψCL . Thus, the Majorana mass

term transforms as ψCLψL+ ψLψCL → eiα

a(Ta+(Ta)∗)ψCLψL+e−iαa(Ta+(Ta)∗)ψLψ

CL . Demanding gauge invariance

then leads to the requirement Ta = −(Ta)∗.

33

Chapter 4

The importance of a regulator in the

calculation of H → γγ

Figure 4.1: The diagram for H → γγ through a fermion loop. The second diagram that has to betaken into account is obtained by interchanging q1 ↔ q2 and µ↔ ν. Picture taken from [3].

In this section we will look at a peculiar feature in the calculation of the process H → γγ,namely the use of a regulator in a certain integral that appears in this calculation. It may seemthat this integral can be done in 4 dimensions without encountering any problems, but this pro-duces an incorrect result; a regulator is needed to get the correct result. We will only look at thefermion loop contribution to this process. The calculation can also be done for scalar loops and Wboson loops, giving similar results [3].

For the fermion loop contribution there are two diagrams: the one in Fig. 4.1 and the diagramobtained from that one by interchanging the two photons, q1 ↔ q2 and µ ↔ ν. The fermion f inthe loop has mass m and electromagnetic charge ef , while the loop momentum is called p. Thefermion couples to the Higgs with a Yukawa coupling λf/

√21. The photons have on-shell momenta

q1 and q2 (i.e. q21 = q2

2 = 0) and polarization vectors ε1 and ε2. Implementing all the Feynmanrules the full expression for the two diagrams in d dimensions is:

iMµνε∗µ1 ε∗ν2 = −ε∗µ1 ε∗ν2

λf√2e2f

∫ddp

(2π)d

[Tr[(/p+ /q1

+ /q2+m)γν(/p+ /q1

+m)γµ(/p+m)]

(p2 −m2)((p+ q1)2 −m2)((p+ q1 + q2)2 −m2)

+Tr[(/p+ /q1

+ /q2+m)γµ(/p+ /q2

+m)γν(/p+m)]

(p2 −m2)((p+ q2)2 −m2)((p+ q1 + q2)2 −m2)

] (4.1)

1The Feynman rule for the Higgs-fermion-fermion coupling is −iλf/√

2

34

Now we are going to check the Ward identity for both photons simultaneously, so we replaceboth polarization vectors by their respective momenta. If we do this and then work out the traceswe obtain the following result:

iMµνqµ1 qν2 = −4m

λf√2e2f

∫ddp

(2π)d

[4(p · q1)(p · q2)− (q1 · q2)p2 + 2(p · q1)(q1 · q2) +m2(q1 · q2)

(p2 −m2)((p+ q1)2 −m2)((p+ q1 + q2)2 −m2)

+4(p · q1)(p · q2)− (q1 · q2)p2 + 2(p · q2)(q1 · q2) +m2(q1 · q2)

(p2 −m2)((p+ q2)2 −m2)((p+ q1 + q2)2 −m2)

](4.2)

This expression can be worked out further by dividing out the factors in the denominators asmuch as possible. After performing the necessary algebra this gives:

iMµνqµ1 qν2 = −4m

λf√2e2f

∫ddp

(2π)d

[1

p2 −m2− 1

(p+ q1)2 −m2− 1

(p+ q2)2 −m2+

1

(p+ q1 + q2)2 −m2

](4.3)

Figure 4.2: The diagram for a Higgs tadpole with a fermion loop. Picture taken from [3]

Eq. (4.3) we recognize as a combination of tadpole diagrams, since in d dimensions a Higgstadpole with a fermion loop (Fig. 4.2) is given by:

iMtadpole =−λf√

2

∫ddp

(2π)dTr(/p+m)

p2 −m2= −4m

λf√2

∫ddp

(2π)d1

p2 −m2(4.4)

So Eq. (4.3) tells us that applying the Ward identity to both photons in Fig. 4.1 gives a combinationof four tadpoles with dierent momenta (p, p+ q1, p+ q2 and p+ q1 + q2) propagating through theloop, multiplied by the fermion charge squared e2

f . A tadpole by itself is quadratically divergent in4 dimensions, but since we are dealing with dierences between tadpoles the degree of divergenceis lower. This can easily be seen by combining the rst and second term in the integral in Eq.(4.3) and doing the same with the third and fourth term:

∫ddp

(2π)d

[2p · q1

(p2 −m2)((p+ q1)2 −m2)− 2(p+ q2) · q1

((p+ q2)2 −m2)((p+ q1 + q2)2 −m2)

](4.5)

Now let's try to see if the Ward identity is satised: we have two terms that are linearly divergentin d = 4 dimensions and that dier only by a momentum shift p→ p+ q2. In d = 4− ε dimensionsthe integral is less-than-linearly divergent. This means that there will not be a contribution from asurface term when performing a momentum shift in the integral, i.e. the integral is shift-invariant[4]. This means that the two terms in Eq. (4.5) cancel in d = 4 − ε dimensions and the Wardidentity is satised. However, if we try to follow this argument in d = 4 dimensions, we startrunning into problems: in 4 dimensions the integral is linearly divergent and the surface term willcontribute when performing a momentum shift, i.e. the integral is not shift-invariant. This means

35

that in 4 dimensions the expression in Eq. (4.5) is not zero, and the Ward identity is violated.This already signals that there is a problem with the calculation in d = 4 dimensions.

This dierence between d = 4 and d = 4− ε dimensions can also be seen by manipulating theintegral in Eq. (4.3) in a dierent way, namely by performing a Taylor expansion. We will expandthe terms in the integral around the point where q1 = q2 = 0. This is similar to the expansion ofa function f(x) = f(x0 + a) around a = 0 that is given by

f(x) = f(x0 + a) = f(x0) + f ′(x0)a+1

2f ′′(x0)a2 + . . . (4.6)

Where f ′(x) = df(x)dx . However, since we are now expanding not in a one-dimensional variable, but

in a 4-momentum, this equation has to modied in the following obvious way for an expansionaround the 4-momentum q = 02:

f(p+ q) = f(p) +∂f(p)

∂pµqµ +

1

2

∂2f(p)

∂pµ∂pνqµqν + . . . (4.7)

The rst term in Eq. (4.3) does not contain q1 or q2 so it only has a zeroth order contribution.The other three terms have the following expansions to second order, where we have used that∂pµ∂pν = gµν and that q1 and q2 are on-shell:

1

(p+ q1)2 −m2≈ 1

p2 −m2− 2p · q1

(p2 −m2)2+

4(p · q1)2

(p2 −m2)3

1

(p+ q2)2 −m2≈ 1

p2 −m2− 2p · q2

(p2 −m2)2+

4(p · q2)2

(p2 −m2)3

1

(p+ q1 + q2)2 −m2≈ 1

p2 −m2− 2p · (q1 + q2)

(p2 −m2)2+

(4pµpν

(p2 −m2)3− gµν

(p2 −m2)2

)(q1 + q2)µ(q1 + q2)ν

(4.8)

So we see that at zeroth and rst order all terms cancel. At second order, the terms where the twoloop momenta are both contracted with q1 or both with q2 cancel. This leaves only the two termsoriginating from the last term where the expression between the brackets is contracted with bothq1 and q2. Since the expression between the brackets is symmetric in µ and ν this gives twice thesame contribution. After combining the expression between the brackets into one fraction we areleft with

2qµ1 qν2

4pµpν − gµν(p2 −m2)

(p2 −m2)3(4.9)

We can also show that all higher order terms in the Taylor expansion do not contribute to theintegral. This is related to the fact that every time we take the derivative with respect to p thedegree of divergence is lowered by one; in 4 dimensions the zeroth order terms are quadraticallydivergent, the rst order terms are linearly divergent, the second order terms are logarithmicallydivergent and after that the terms become convergent. We will now show that all these convergentterms vanish.

To do this we make use of the fact that the higher order terms in the Taylor expansion containderivatives. The integral over the divergence of a vector eld can be expressed as a surface integralusing the divergence theorem. The general form of the divergence theorem for some vector eld Fis ∫

Ω

dV (∇ · F ) =

∮∂Ω

dS(F · n) (4.10)

where Ω is a volume and ∂Ω is the closed surface of the volume Ω, dV and dS are the volumeand surface integration elements, respectively, and n is the outward-pointing normal vector of thesurface ∂Ω. In Minkowski space `outward-pointing' is hard to dene due to the timelike direction.We can solve this issue by rst turning the integral into a Euclidean one with a Wick rotation ofthe timelike component and then applying the divergence theorem. We will use pE to denote the

2Note that∂f(p+q)∂qµ

∣∣∣q=0

=∂f(p+q)∂pµ

∣∣∣q=0

=∂f(p)∂pµ

allows us to simplify the derivatives in the expansion.

36

Euclidean momentum.

In our calculation the divergence operator is the derivative with respect to the momentum p,i.e. ∇ · F = ∂Fµ

∂pµ . Now we note that any higher order term in the Taylor series (4.7) can easily beexpressed as a divergence. For example, let's do this for the third order term

1

6

∂3f(p)

∂pµ∂pν∂pρqµqνqρ =

∂

∂pµ

[1

6

∂2f(p)

∂pν∂pρqµqνqρ

](4.11)

If we introduce a cuto on the momentum integral we can look at how the integral scales with thiscuto. Let's say we place the cuto at p2

E = R2. Applying the divergence theorem (4.10) to theterm (4.11) above then gives∫

p2E≤R2

ddpE∂

∂pµ

[1

6

∂2f(p)

∂pν∂pρqµqνqρ

]=

∮p2E=R2

dS

[1

6

∂2f(p)

∂pν∂pρqµqνqρ

]nµ (4.12)

Then the surface area in d dimensions will scale as Rd−1, while the integrand in the surface integralscales as R−4 for the example of the third order term. In general the integrand of the surface termscales as R−1−k where k is the order of the term in the Taylor expansion. So the total integralscales with Rd−k−2. Then we see that in d = 4 dimensions the terms will start scaling with inversepowers of R from the third order terms onwards. Removing the cuto R → ∞ we can concludethat the contribution from the terms of third order and higher in the Taylor expansion will go tozero.

We have now shown that the integral in Eq. (4.3) is equal to the integral of the expression inEq. (4.9). The Ward identity applied to both photons is therefore also equal to

iMµνqµ1 qν2 = −8m

λf√2e2fqµ1 qν2

∫ddp

(2π)d4pµpν − gµν(p2 −m2)

(p2 −m2)3(4.13)

We will now do the calculation of this integral in d = 4 and in d = 4 − ε dimensions. We willsee that the results do not agree. Afterwards we will argue why the result in d = 4 dimensions isincorrect.

In d = 4 dimensions the spherical symmetry in momentum space allows us to make the replace-ment

4pµpν → p2gµν (4.14)

in the numerator of the integrand, causing the rst two terms in the numerator to cancel. Theremaining integral is convergent in 4 dimensions and can be computed or looked up in an integraltable: ∫

d4p

(2π)4

1

(p2 −m2)3=−i

(4π)2

1

2m2(4.15)

So in d = 4 dimensions the Ward identity now states

iMµνqµ1 qν2

∣∣∣d=4

=i

(2π)2

λf√2me2

f (q1 · q2) (4.16)

which is, in general, non-zero. This means that if we do our computation in d = 4 dimensions theWard identity is violated. This violation of the Ward identity in d = 4 dimensions is what we alsosaw earlier in the discussion about the combination of tadpole diagrams.

Now let's compute the integral in Eq. (4.13) in d = 4− ε dimensions. The replacement due tospherical symmetry is now

4pµpν →4

dp2gµν ≈

(1 +

ε

4

)p2gµν (4.17)

We will only need this expression up to the rst order contribution in ε. The rst term in (4.17)will again cancel against the other p2-term in the numerator. However, the term proportional toε will now give an extra contribution. The ε-term is multiplied by the following integral:∫

ddp

(2π)dp2

(p2 −m2)3=

i

(4π)2

(2

ε+ nite terms

)(4.18)

37

where we are only interested in the ε−1 contribution, since we will take ε → 0 at the end ofthe calculation. After multiplying the ε−1-term with the ε from the expansion we get a nitecontribution which cancels exactly against the contribution (4.15) from the m2-term. So the totalintegral of Eq. (4.13) is zero if we use dimensional regularization:

iMµνqµ1 qν2

∣∣∣d=4−ε

= 0 (4.19)

This means that the Ward identity is explicitly satised if the computation is performed in d = 4−εdimensions, in contrast to what we saw in the d = 4 result (4.16).

The reason for this dierence is a mistake in the d = 4 calculation: as pointed out by Weinzierlin [5] the integral (4.13) produces a nite result because it has divergences in dierent parts ofthe integration region that cancel against each other. Weinzierl shows this by using generalizedd-dimensional spherical coordinates for the integral; the integral then factorizes into a radial partand an angular part. The radial part is UV divergent so its Taylor expansion starts at order ε−1.The angular part is nite and has the additional property that it vanishes for d = 4. This meansthat the Taylor expansion for the angular integral starts at order ε1. Multiplying the radial andangular parts together then produces a result for the full integral that starts at order ε0. However,Weinzierl also points out that the integration domain for one of the angular variables (called θ1

in [5]) can be split into regions; if we then integrate this angular variable over one region whileintegrating the radial variable and other angular variables over their full range, we obtain a UVdivergent result. When adding the results obtained by integrating over the other regions of the θ1

domain these divergences cancel, producing a nite result.

Because the result is nite, this may lead to the mistaken conclusion that one may do thecalculation in d = 4 dimensions, without using a regulator. However, for the correct treatment ofthe dierent divergences that exist in the integration region a regulator (in this case dimensionalregularization) is nevertheless needed. We were not allowed to do the replacement (4.14) in thed = 4 calculation since the integrand is not strictly integrable in 4 dimensions. That is why weobtained an incorrect result in (4.16), where it seemed like the Ward identity was violated. If thecalculation is done correctly with a regulator the Ward identity is satised (Eq. (4.19)) as it shouldbe.

38

Chapter 5

Two-loop corrections to the Higgs

mass in the MSSM

In this chapter we will extensively look at the calculation of corrections to the Higgs mass inthe Minimally Supersymmetric Standard Model (MSSM). In particular, we are interested in thecorrections originating from the gluino, the fermionic superpartner of the gluon. Since the gluinodoes not couple directly to the Higgs elds, these corrections will rst appear at the two-loop level.To be more precise, the corrections we will consider are of order O(αsαt) where αs = g2

s/(4π)and αt = Y 2

t /(4π), with gs and Yt being the strong coupling constant and the top quark Yukawacoupling, respectively.

The reason that we want to investigate these corrections is the following: it is often stated inthe context of the hierarchy problem1 that the Higgs mass is very sensitive to higher energy scalesthat the Higgs eld couples to. This is often generically called the `scale of new physics' ΛNP . Thegluino corrections to the Higgs mass provide a nice case study of a situation where the Higgs eldcouples indirectly to a new physics scale, namely the gluino mass.

5.1 A very brief introduction to the MSSM

Supersymmetry is a symmetry that links fermions and bosons. In a supersymmetric theory thenumber of bosonic and fermionic degrees of freedom is equal. The most commonly used super-symmetric extension of the standard model is the Minimally Supersymmetric Standard Model(MSSM). In this model every Standard Model particle has a new superpartner. Standard Modelfermions have scalar superpartners whose names start with s- (squarks, sleptons, etc.), while Stan-dard Model bosons have fermionic superpartners whose names end in -ino (gluino, Higgsino, etc.).We will follow the widely used convention in the MSSM to denote the SM elds as usual, whileadding a tilde to denote the superpartners of the SM elds (t, g, H, etc.).

If supersymmetry were an exact symmetry the masses of the Standard Model particles and theirsuperpartners would be equal. In that case we should have already observed the superpartners,but since we haven't we must conclude that supersymmetry is at best a broken symmetry. In theMSSM this issue is dealt with by introducing soft supersymmetry-breaking terms. These are termsthat explicitly break supersymmetry such as mass terms for sfermions of the form

m2ff∗f (5.1)

and trilinear scalar couplings of the form

AfHf∗1 f2 (5.2)

1The hierarchy problem refers to the issue that scalar masses like the Higgs mass receive quadratically divergentloop corrections. A very high level of ne-tuning between the bare Higgs mass and the loop corrections is thenrequired to get a physical Higgs mass at a low energy scale.

39

involving a Higgs eld H and two sfermions. These terms are called `soft' because they only containmasses and couplings with a positive mass dimension. From the point of view of renormalizabilitythis means that their contribution becomes smaller at higher energy scales.

5.2 Denition of the elds

In this section we will take a look at the elds that are needed in the calculation. Of course wewill need the Higgs elds. At the one-loop level the Higgs elds couple to the fermion and sfermionelds. The dominant contributions are expected to come from loops involving top and stop elds,since the top quark is by far the heaviest quark. To simplify the calculation we will thereforeconsider a reduced version of the MSSM containing only the top and stop elds and not the otheravors. Then, at the two-loop level we have the gluon and gluino elds which couple to the topand stop elds.

Top elds: The left-handed top eld is part of an SU(2) doublet that also contains the left-handedbottom eld: (

tb

)L

(5.3)

This doublet has hypercharge Y = 1/3, so that the left-handed top eld has the correct electriccharge Q = 2/3 using the Gell-Mann-Nishijima relation (1.31).

The right-handed top eld tR is an SU(2) singlet eld with hypercharge Y = 4/3. Both tL andtR are SU(3) color triplets. After SU(2) symmetry breaking we can put them into one 4-componentDirac spinor2:

t =

(tLtR

)(5.4)

Stop elds: In the MSSM the top elds have scalar superpartners called the stop elds. Theseare complex scalar elds with the same quantum numbers as the top elds. So there is a `left'3

stop eld tL with Y = 1/3 which is the upper component of an SU(2) doublet(tLbL

)(5.5)

and a `right' stop eld tR with Y = 4/3. Just as the top elds, the stop elds are color triplets.In general, after SU(2) symmetry breaking, the two elds tL and tR will mix to two new masseigenstates, which we will see later on.

Higgs elds: In the MSSM there are two complex Higgs doublets. There is a Higgs doublet H1

with hypercharge Y = −1 and a Higgs doublet H2 with hypercharge Y = +1. After electroweaksymmetry breaking both Higgs elds take on a non-zero vacuum expectation value. We will denethe Higgs elds as follows

H1 =

(H0

1

H−1

)=

( 1√2(v1 + φ1 − iχ1)

H−1

)(5.6)

H2 =

(H+

2

H02

)=

(H+

21√2(v2 + φ2 + iχ2)

)(5.7)

where v1 and v2 are the vacuum expectation values of the Higgs elds. The elds φ1 and φ2 are realscalar elds. They are electrically neutral and will mix to the CP-even Higgs eigenstates h0 andH0. The elds χ1 and χ2 are also neutral real scalars. They mix to the CP-odd Higgs eigenstatesA0 and G0. The lower component of H1 and the upper component of H2 contain the charged

2Here we make explicit use of the Weyl representation for the Dirac matrices.3Note that the `left' and `right' labels have no real meaning for scalar particles, since there are no chiral projections

for scalars.

40

Higgs elds H−1 and H+2 . These will not contribute in our calculation.

Finally, we follow the usual MSSM conventions and use the ratio of the vacuum expectationvalues to dene a new parameter β:

tanβ =v2

v1(5.8)

Gluon elds: The gluon elds are the gauge elds of SU(3). They are vector elds in the adjointrepresentation of SU(3). We will denote them

Aaµ (5.9)

with a = 1, ..., 8.

Gluino elds: The gluino elds are the fermionic superpartners of the gluon elds. These areMajorana fermions. At this point it might be useful to give a short explanation of what Majoranafermions are.

To this end we need to talk about charge conjugation. One can introduce a charge conjugationoperator C that is unitary (C† = C−1) and antisymmetric (CT = −C). This operator has theproperty that

CΓTi C−1 = ηiΓi (5.10)

with

ηi =

+1 for Γi = 1, iγ5, γµγ5

−1 for Γi = γµ, σµν ≡ i2 [γµ, γν ]

(5.11)

In the Weyl representation for the gamma matrices the charge conjugation operator is given byC = −iγ2γ0. A charge-conjugated spinor ψC is dened by

ψC = CψT (5.12)

with ψ = ψ†γ0 as usual. A Majorana spinor ψM is a spinor that is its own charge conjugate, i.e.

ψCM = ψM (5.13)

This condition reduces the number of degrees of freedom from 4 to 2.

The Majorana spinor for the gluinos is dened as

ga =

(−iλag+iλag

)=

(−iλag−σ2λa∗g

)(5.14)

with a = 1, ..., 8 again and λag denotes the 2-component gluino spinors. Note that the bar for

2-component spinors has a dierent meaning than the one for 4-component spinors: λ = iσ2λ∗ fora 2-component spinor λ. The factors of i in (5.14) are a conventional choice.

5.3 Parameters

Now that we have dened the necessary elds let's take a look at the reduced set of MSSM param-eters in our calculation and their meaning. We will also go into the relations between parameters.

First, we have the U(1) gauge coupling g1 and the SU(2) gauge coupling g2. In the MSSMthe tree-level masses of the Z and W boson are related to the gauge couplings and the vacuumexpectation values v1 and v2

M2Z =

1

4(g2

1 + g22)(v2

1 + v22)

M2W =

1

4g2

2(v21 + v2

2)

(5.15)

41

The weak mixing angle θW is dened in terms of the ratio of these masses

sin2 θW = 1− M2W

M2Z

(5.16)

After electroweak symmetry breaking the electric charge e makes its appearance. It is related tothe gauge couplings by

e = g2sW = g1cW (5.17)

where we have used the shorthand notations sW = sin θW and cW = cos θW . In the future we willuse this notation for dierent angles as well.

Of course there is also the SU(3) (`strong') coupling constant gs. The parameter tanβ as de-ned in (5.8) will also be used. The gluino mass parameter, which in the MSSM is conventionallycalled M3, is a soft-supersymmetry breaking parameter. We will rename it to mg = M3 to makeit more clear that it is the gluino mass.

The mass of the top quark is related to the top Yukawa coupling by

mt =v2Yt√

2(5.18)

There are a few more important parameters that enter the calculation, but they will be explainedas we encounter them in the next sections.

5.4 Field mixing

As already mentioned in Section 5.2 there will be mixing between certain elds. We will now takea closer look at this issue.

Mixing between stops: Including the soft supersymmetry-breaking terms the mass matrix inthe stop sector is [6]

Lmass,t = −(t∗L t∗R

)(M2tL

+m2t +M2

Z cos(2β)(I3t − s2WQt) −mt(At + µ cotβ)

−mt(At + µ cotβ) M2tR

+m2t +M2

Z cos(2β)s2WQt

)(tLtR

)(5.19)

where I3t = 1/2 is the third isospin component of the left-handed top quark and Qt = 2/3 is theelectric charge of the top quark. Furthermore, MtL

and MtRare soft SUSY-breaking masses of

the type (5.1) and At is a soft trilinear coupling of the type (5.2). The parameter µ is a SUSYparameter that originates from the MSSM Higgs sector. It is basically the mixing between the twoHiggs elds. Here we have imposed a simplication by assuming that both At and µ are real4.

The mixing parameter Xt = At +µ cotβ determines how large the mixing in the stop sector is.If Xt = 0 there is no mixing. If there is mixing we obtain the mass eigenstates by a rotation(

t1t2

)=

(cos θt sin θt− sin θt cos θt

)(tLtR

)(5.20)

In the t1, t2-basis the mass matrix is diagonal

Lmass,t = −(t∗1 t∗2

)(m2t1

0

0 m2t2

)(t1t2

)(5.21)

where m2t1and m2

t2are the mass eigenvalues. To avoid issues with the degeneracy of the stop mass

matrix we will say that the two mass eigenvalues are dierent; for deniteness we will take t1 to

4This means that we are working in the real MSSM (rMSSM). In the complex MSSM (cMSSM) parameters suchas At are allowed to be complex.

42

be the lighter eigenstate, so m2t1< m2

t2. Of course we could express θt, m

2t1

and m2t2

in terms of

the parameters in (5.19), but that is not needed for our purposes here.

However, since the interactions are more easily given in the tL, tR-basis, we now use (5.20) to`rotate back' to the tL, tR-basis. Eq. (5.21) then takes the form

Lmass,t = −(t∗L t∗R

)( c2θtm2t1

+ s2θtm2t2

sθtcθt(m2t1−m2

t2)

sθtcθt(m2t1−m2

t2) s2

θtm2t1

+ c2θtm2t2

)(tLtR

)(5.22)

This has the advantage that we can now work in the tL, tR-basis while using the three parametersθt, m

2t1

and m2t2

as the input parameters. Since these are the mass eigenvalues and the mixing

angle they have a very clear physical meaning. By comparing the o-diagonal elements of (5.19)and (5.22) we can derive a relation that expresses At in terms of our input parameters:

At =sin θt cos θt(m

2t2−m2

t1)

mt− µ cotβ (5.23)

Mixing in the Higgs sector: To discuss mixing in the Higgs sector we need to take a look atthe MSSM Higgs potential [6]

LHiggs =−m21H†1H1 −m2

2H†2H2 +m2

12(εabHa1H

b2 + h.c.)

− 1

8(g2

1 + g22)(H†1H1 −H†2H2)2 − 1

2g2

2 |H†1H2|2

(5.24)

In this expression m2i = µ2 + m2

i (i = 1, 2). The parameters mi2 and m2

12 are soft masses. If weinsert the decompositions (5.6) and (5.7) of the Higgs elds we get the following mass matrices forthe φ- and χ-elds:

Lmass,H =− 1

2

(φ1 φ2

)(m21 + 1

4 (g21 + g2

2)( 32v

21 − 1

2v22) m2

12 − 14 (g2

1 + g22)v1v2

m212 − 1

4 (g21 + g2

2)v1v2 m22 + 1

4 (g21 + g2

2)( 32v

22 − 1

2v21)

)(φ1

φ2

)− 1

2

(χ1 χ2

)(m21 + 1

8 (g21 + g2

2)(v21 − v2

2) m212

m212 m2

2 + 18 (g2

1 + g22)(v2

2 − v21)

)(χ1

χ2

)(5.25)

This shows that there will be mixing between φ1 and φ2 and mixing between χ1 and χ2.

The mass eigenstates of the φ-elds are usually called h0 and H0. These are neutral CP-evenHiggs states. By convention h0 is taken to be the lightest of the two. This Higgs mode should bethe one that corresponds to the Higgs found at the LHC, which means its mass should be around125 GeV. The elds h0 and H0 are related to the elds φ1 and φ2 by a rotation over an angle wecall α: (

H0

h0

)=

(cosα sinα− sinα cosα

)(φ1

φ2

)(5.26)

Again we could work out what the angle α is in terms of the other parameters, but that is notneeded.

The χ-elds will rotate into mass eigenstates called A0 and G0. A0 is a CP-odd neutral Higgsscalar, while G0 is a neutral Goldstone boson. G0 will be used to give a mass to the Z boson, justas in the Standard Model. In this sector the rotation angle that diagonalizes the mass matrix iscalled βn: (

G0

A0

)=

(cosβn sinβn− sinβn cosβn

)(χ1

χ2

)(5.27)

If we plug this into (5.24) we nd that the mass of the A0-boson is

M2A = m2

1 sin2 βn +m22 cos2 βn −m2

12 sin(2βn) +1

8(g2

1 + g22)(v2

2 − v21) cos(2βn) (5.28)

The parameter MA is important, since it is the mass of a physical particle, unlike parameters suchas m12. Therefore in the following sections we will rewrite the Higgs sector in terms of MA (anda few other parameters).

43

5.4.1 Input parameters

Now we have enough information about the parameters of the MSSM to choose a suitable set ofindependent input parameters. We will formulate our reduced version of the MSSM in terms ofthese 12 parameters:

e, gs,MW ,MZ ,MA, tanβ,mg, µ,mt1,mt2

, θt,mt (5.29)

These input parameters have been chosen so that most of them have a close link to physical observ-ables. Seven of them are masses of physical particles. Then there are the two coupling constantse and gs with a clear physical interpretation. The mixing angle θt also has a clear physical inter-pretation. The parameter µ is not directly observable but is closely related to the neutralino andchargino masses. The nal parameter tanβ is the only one that does not have a simple and clearconnection to a physical quantity; it has to be seen as a somewhat generic parameter of the Higgssector.

All other parameters can be determined from these input parameters using relations that havebeen given in the previous sections. In other words once the input parameters have been set themodel is completely determined.

5.5 The Lagrangian

It is now time to look at all the relevant parts of the MSSM Lagrangian. To write this Lagrangiandown we have used [6] and [7, 8] as references. First we have the kinetic terms for all the elds:

Lkin =it/∂t+ (∂µtL)∗(∂µtL) + (∂µtR)∗(∂µtR) + (∂µH1)∗(∂µH1) + (∂µH2)∗(∂µH2)

+i

2¯ga /∂ga −

1

4F aµνF

µν,a(5.30)

whereF aµν = ∂µA

aν − ∂νAaµ (5.31)

is the linear part of the gluon eld strength tensor. The factor 1/2 in front of the kinetic term forthe gluino is due to it being a Majorana particle.

Next, we have the mass terms, where we express the stop masses as in Eq. (5.22):

Lmass =−mttt− (c2θtm2t1

+ s2θtm2t2

)t∗LtL − (s2θtm2t1

+ c2θtm2t2

)t∗RtR

− [sθtcθt(m2t1−m2

t2)t∗LtR + h.c.]− 1

2mg

¯gaga(5.32)

The factor 1/2 in front of the gluino mass term is again because it is a Majorana fermion. Lmassdoes not include the mass terms for the Higgs boson, because those are part of the terms originat-ing from the Higgs potential, which we will now study in more detail.

5.5.1 The Lagrangian for the Higgs sector

For our calculation the most important part of the MSSM Lagrangian is the Lagrangian of theHiggs sector. The kinetic terms for the Higgs doublets are included in (5.30) and the Higgs poten-tial is given in (5.24). The Higgs potential in particular requires a careful treatment.

First of all it is important that the Higgs potential has the right characteristics with regardto symmetry breaking. As in the Standard Model the symmetry group SU(2)L × U(1)Y needs tobe broken to U(1)Q. To accomplish this the Higgs potential should be bounded from below andit needs to have a local minimum away from zero. The requirement that the Higgs potential isbounded from below leads to the following condition

m21 +m2

2 ≥ −2m212 (5.33)

44

The second requirement, that the origin is not a local minimum of the potential, leads to anothercondition on the Higgs parameters

m21m

22 < m4

12 (5.34)

If these conditions are satised then electroweak symmetry breaking will take place.

The next step is to perform the minimization of the potential. This equates to nding thevacuum state of the Higgs elds. Since the MSSM Higgs potential involves two Higgs doublets, weneed to minimize the potential with respect to both of them. Because the U(1) electromagneticsymmetry needs to stay intact, the charged components of the Higgs elds (H−1 and H+

2 ) have tovanish in the vacuum state. To nd the values of H0

1 and H02 in the vacuum state we dene the

vevs v1 and v2 as in (5.6) and (5.7). We then need to evaluate the two minimization conditionsfollowing from

∂V

∂|H1|=

∂V

∂|H2|= 0 (5.35)

with |Hi| =√H†iHi. Using the denition of tanβ (5.8) and the expression for the Z boson mass

(5.15) this produces these two relations between the Higgs parameters:

m21 +m2

12 tanβ +1

2M2Z cos(2β) = 0

m22 +m2

12 cotβ − 1

2M2Z cos(2β) = 0

(5.36)

Now note that these relations are only satised at the minimum of the potential. The usualprocedure is to rewrite the Higgs modes in terms of an expansion around the minimum as in (5.6)and (5.7). We could then use the minimization conditions (5.36) to simplify the Higgs potential byeliminating two parameters. However, we will not do that yet because we have to keep in mind thatwe need to renormalize the Higgs sector later on. In the process of renormalization the minimum ofthe potential may get shifted. Therefore the correct procedure is to renormalize the parameters ofthe Higgs sector and afterwards apply the minimization conditions to the renormalized parametersto nd the correct `new' minimum.

Another issue of the Higgs sector that is closely related to the minimization of the potentialis that of the Higgs tadpoles. From the Higgs potential (5.24) we get a large amount of terms.Besides the quadratic mass terms of (5.25) we also get interaction terms between 3 and 4 Higgselds and a constant term. Finally there are terms linear in the Higgs elds which are called thetadpole terms. To work out these terms we plug Eqs. (5.6) and (5.7) into the Higgs potential(5.24) again. Looking at the terms linear in the Higgs elds we obtain

LtadpoleHiggs = −T1φ1 − T2φ2,

T1 = m21v1 +m2

12v2 +1

8(g2

1 + g22)v1(v2

1 − v22)

T2 = m22v2 +m2

12v1 +1

8(g2

1 + g22)v2(v2

2 − v21)

(5.37)

where we have dened the tadpole parameters T1 and T2.

Consider what happens at the minimum of the potential; if we use the minimization conditions(5.36) we see that T1 = T2 = 0. This is exactly what you would expect: at the minimum ofthe potential the rst derivative with respect to the elds vanishes by denition, so an expansionaround the minimum should have no terms linear in the elds.

But as discussed shortly before, keeping in mind the renormalization process, we should notapply the minimization conditions yet. This means that the tadpole parameters do not vanish.In fact, what we will now do is rewrite the Higgs sector in terms of (among others) the tadpoleparameters. The Higgs sector originally contained 5 parameters in addition to the gauge couplingsg1 and g2

m1,m2,m12, v1, v2. (5.38)

45

Using Eqs. (5.8), (5.15), (5.28) and (5.37) we can rewrite the Higgs sector in terms of the 5 newparameters

MZ ,MA, tanβ, T1, T2 (5.39)

plus some auxiliary parameters such as the charge e and the mixing angle βn. The advantage isthat this set of parameters is more physical and the minimization conditions are much simpler toimplement. In this step we perform the rotation (5.27) from the χ-elds to A0 and G0, but we donot rotate the φ-elds. This is because it is easier to calculate the diagrams in the φ1, φ2-basis andthen perform the rotation (5.26) to the elds h0 and H0 at the end. So we rewrite the Lagrangianin the following form:

LHiggs =− T1φ1 − T2φ2 −1

2m2φ1φ2

1 −1

2m2φ2φ2

2 −m2φ1φ2

φ1φ2 −1

2M2A(A0)2

+ (Terms involving G0) + (3H − int.) + (4H − int.) + const.(5.40)

with [9]

m2φ1

= M2Z cos2 β +

1

cos2(β − βn)

[M2A sin2 β − e

4MW sWT1 cosβn (cos(β + βn)− 3 cos(β − βn))

− e

2MW sWT2 sinβ cos2 βn

]m2φ2

= M2Z sin2 β +

1

cos2(β − βn)

[M2A cos2 β − e

2MW sWT1 cosβ sin2 βn

+e

4MW sWT2 sinβn (cos(β + βn) + 3 cos(β − βn))

]m2φ1φ2

= −M2Z sinβ cosβ +

1

cos2(β − βn)

[−M2

A sinβ cosβ +e

2MW sW(T1 sinβ sin2 βn + T2 cosβ cos2 βn)

](5.41)

5.5.2 The interaction Lagrangian

The nal part of the Lagrangian that we need to consider is the interaction Lagrangian. We willonly consider the interactions that will appear in the diagrams that we have to calculate. Forexample, in Eq. (5.40) we did not specify the interactions between 3 Higgs elds, because theywill not appear in the diagrams.

There is a further simplication that we can apply to the Higgs-stop interactions. Consider forexample the φ2 − t∗L − tL interaction. Its interaction term is(

e2v2

2c2W s2W

[I3t − s2WQt]− v2Y

2t

)t∗LtLφ2 (5.42)

This interaction term consists of two parts: the rst part is the `gauge' part and the second partis the `Yukawa' part. In the nal result we are only interested in terms that originate from theYukawa part of the interaction because those are going to give the O(αtαs) corrections. Thereforewe can ignore the rst part of the interaction term.

To make this a bit more formal, what we will actually do is take the `gaugeless' limit. Thismeans that we let the gauge couplings g1 and g2 go to zero. As a result, also e, MW and MZ

will go to zero (see (5.15), (5.17)). This means that the rst term in Eq. (5.42) vanishes in thegaugeless limit and we will be left with just the Yukawa part of the interaction. One thing shouldbe kept in mind here; while taking the gaugeless limit we let ratios of parameters that both go tozero stay constant, such as the ratio e/MW that appears in Eq. (5.41) or the ratio MW /MZ thatappears in the denition of the weak mixing angle (5.16).

The interaction Lagrangian containing only the necessary interactions and after taking thegaugeless limit then consists of the following parts

Lgaugelessint = LHtt + LHtt + LHHtt + Lgtt + Lgtt + Lgtt + Ltttt (5.43)

46

where we will now give explicit expressions for the types of interactions that appear.

Higgs-top-top:

LHtt = − emt

2MW sW sinβttφ2 + i

emt cosβn2MW sW sinβ

tγ5tA0 (5.44)

Higgs-stop-stop:

LHtt =− em2t

MW sW sinβ(t∗LtLφ2 + t∗RtRφ2) +

[emt

2MW sW sinβµt∗RtLφ1 + h.c.

]+

[emt

2MW sW sinβAtt∗RtLφ2 + h.c.

]− i emt

2MW sW sinβ(At cosβn − µ sinβn)(t∗LtRA

0 − t∗RtLA0)

(5.45)

Higgs-Higgs-stop-stop:

LHHtt = − e2m2t

4M2W s

2W sin2 β

(t∗LtLφ22 + t∗RtRφ

22)− e2m2

t cos2 βn

4M2W s

2W sin2 β

(t∗LtL(A0)2 + t∗RtR(A0)2)

(5.46)

Gluino-top-stop:Lgtt = −[

√2gsT

aαβ(tαPRgat

βL − t

αPLgatβR) + h.c.] (5.47)

Gluon-top-top:Lgtt = −gsT aαβ tαγµtβAaµ (5.48)

Gluon-stop-stop:

Lgtt = −igsT aαβ(tα∗L←→∂ µtβL + tα∗R

←→∂ µtβR)Aaµ (5.49)

with the use of the notation

α←→∂ µβ = α(∂µβ)− β(∂µα) (5.50)

Quartic stop interactions:

Ltttt =− g2s

12(δαβδγδ + δαδδγβ)tα∗L tβLt

γ∗L t

δL −

g2s

12(δαβδγδ + δαδδγβ)tα∗R tβRt

γ∗R t

δR

+ g2s

[−1

6δαβδγδ +

1

2δαδδγβ

]tα∗L tβLt

γ∗R t

δR

(5.51)

Now we have completely specied the Lagrangian that we need. It contains all the necessaryparts of the MSSM Lagrangian and everything has been expressed in terms of our chosen eldsand input parameters. To conclude this section, what we have is this Lagrangian

L = Lkin + Lmass + LHiggs + Lint (5.52)

with Lkin specied in (5.30), Lmass in (5.32), LHiggs in (5.40) and (5.41) and Lint in (5.43)-(5.51)

5.6 Renormalization

The next important step is to renormalize the Lagrangian. We will follow the principles of renor-malized perturbation theory explained in Section 2.2.3 and introduce appropriate counterterms forthe parameters and elds in the Lagrangian. We will stay close to the approach of [10].

While introducing the counterterms we can already take into account the structure of the dia-grams that we are going to calculate; in the diagrams the Higgs sector appears at the tree level,the top and stop sectors come into play at the one-loop level and the gluon and gluino (the `strong'sector, if you will) only play a role at the two-loop level. The result of this is that we do nothave to renormalize everything to the two-loop level: parameters and elds that only appear atthe two-loop level in the diagrams do not have to be renormalized at all, those that appear at the

47

one-loop level have to be renormalized only up to the one-loop level and nally parameters andelds appearing at tree-level have to be renormalized to the two-loop level.

So let's start with the Higgs sector, where we have to perform a two-loop renormalization. Forthe Higgs elds we introduce one wave function renormalization (WFR) factor for each doublet:

H1 → Z1/2H1

H1

H2 → Z1/2H2

H2

(5.53)

The way this statement should be understood is that on the left-hand side we have the original(bare) elds and we replace them by the renormalized elds on the right-hand side which aremultiplied by the appropriate WFR factors. The WFR factors consist of a one-loop countertermand a two-loop counterterm:

ZHi = 1 + δZ(1)Hi

+ δZ(2)Hi, (i = 1, 2) (5.54)

The numbers in parentheses in superscript indicate the loop level. As an obvious consequence ofEq. (5.53) the renormalization of the φ-elds is given by

φi → Z1/2Hi

φi, (i = 1, 2) (5.55)

The renormalization of the parameters in the Higgs sector proceeds as follows

M2A →M2

A + δM2(1)A + δM

2(2)A

tanβ → (1 + δ tanβ(1) + δ tanβ(2)) tanβ

T1 → T1 + δT(1)1 + δT

(2)1

T2 → T2 + δT(1)2 + δT

(2)2

(5.56)

On the left-hand side are the original (bare) parameters which are replaced by the renormalizedparameters and counterterms on the right-hand side.

Next is the renormalization of the top and stop elds and the parameters that appear in the(s)top sector. All of these should be renormalized to the O(αs) one-loop level. The elds arerenormalized as follows

tL → Z1/2tL tL =

(1 +

1

2δZtL

)tL

tR → Z1/2tR tR =

(1 +

1

2δZtR

)tR

tL → Z1/2

tLtL =

(1 +

1

2δZtL

)tL

tR → Z1/2

tRtR =

(1 +

1

2δZtR

)tR

(5.57)

Since the MSSM is a chiral theory, the left- and right-handed top elds get their own WFRfactors. Because this is a one-loop renormalization there is no need to indicate the loop level ofthe counterterms. The top mass is renormalized by the replacement

mt → mt + δmt (5.58)

The parameters of the stop mass matrix also receive a straightforward renormalization

m2t1→ m2

t1+ δm2

t1

m2t2→ m2

t2+ δm2

t2

θt → θt + δθt

(5.59)

48

As mentioned earlier in this section the gluon and gluino elds do not have to be renormalized,nor do the parameters gs and mg related to them.

For convenience we dene a counterterm for the coupling At by taking the relation (5.23) andreplacing the parameters on the right-hand side according to (5.58) and (5.59)5. Then At →At + δAt and we nd

δAt =sθtcθt(m

2t2−m2

t1)

mt

[c2θt − s

2θt

sθtcθtδθt +

δm2t2− δm2

t1

m2t2−m2

t1

− δmt

mt

](5.60)

where we have taken into account that only O(αs) corrections to At are relevant to us, whereasthe parameters µ and tanβ appearing in (5.23) do not receive O(αs) corrections.

Finally, it is useful to discuss how the parameter βn is treated in the renormalization process.As a reminder, this is the mixing angle that diagonalizes the mass matrix of the χ-elds (seeEq. (5.27)). It is not an independent parameter, rather, it is a function of the parameters of theHiggs sector (5.38). This is why it does not get renormalized with its own counterterm; instead,we simply dene βn in the renormalized theory to be the angle that diagonalizes the mass matrixof the renormalized χ-elds. After renormalization we apply the minimization conditions of theHiggs potential to the renormalized Higgs parameters. It can be worked out that this gives:

βn

∣∣∣min. Higgs

= β (5.61)

Note the dierence in treatment between this mixing angle and the stop mixing angle θt. The stopmixing angle was used as an independent parameter to parametrize the stop sector. This is whyθt gets its own counterterm upon renormalization (5.59), in contrast to βn.

5.7 The counterterm Lagrangian

With the counterterms we have introduced in the previous section we can determine the coun-terterm Lagrangian. This is achieved by taking the Lagrangian of Section 5.5 and performing thereplacements outlined in the previous section. As a result the Lagrangian changes as follows

L → L+ δL (5.62)

where δL indicates the counterterm Lagrangian.

It should be stressed that we will only include the counterterms that will be relevantfor the calculation of the Higgs masses at O(αtαs). This means that certain countertermcontributions can be ignored because they are not relevant at that order. For example, in the Higgssector we get terms in the counterterm Lagrangian of the form (one-loop counterterm)×(one-loopcounterterm) such as

δT(1)1 δZ

(1)H1

(5.63)

At one-loop the parameters of the Higgs sector receive corrections of O(αt) but not O(αs) sincethe gluon and gluino do not directly couple to the Higgs elds. So the counterterm contribution(5.63) will contribute at O(α2

t ) and not O(αtαs). Therefore, it can be ignored in our calculation.

The same argument holds for the parameters e and µ: we do not have to consider their renor-malization since their counterterms will not produce relevant contributions at O(αtαs). Let's takeµ as an example: it appears in the Higgs-stop-stop interactions (5.45). The counterterm for thisinteraction will contribute to the calculation of the Higgs mass in the form of diagram (d) inFig. 5.2. This diagram contains an O(αt) stop loop, so to bring the total contribution to O(αtαs)

5We can treat the counterterm as a small quantity so that it works very similar to how derivatives work: forexample sin(θ + δθ) = sin θ + δθ cos θ

49

the vertex counterterm would have to be of O(αs). However, since µ is a Higgs sector parameter itwill not receive O(αs) corrections. So the renormalization of µ in the Higgs-stop-stop interactioncan be ignored.

We also do not have to consider the renormalization of the parameters MW and MZ ; the Wand Z boson do not couple to the (s)top with a Yukawa coupling, but with the gauge couplings g1

and g2; since we work in the gaugeless limit (see Section 5.5.2) these contributions are ignored inour approach.

As one might suspect after this discussion, whether the counterterm of a certain parameter willproduce relevant contributions also depends on the interactions in which the parameter appearsand at which order it can contribute. A good example of this is given by the parameter tanβ whichappears in multiple places. In the Higgs-top-top interactions (5.44) for instance, it would have tobe renormalized to one-loop order. However, δ tanβ(1) does not receive O(αs) corrections sinceit is purely a parameter of the Higgs sector. So, by the same reasoning as for µ in the previoussection, the contribution of δ tanβ(1) to the Higgs-top-top interactions can be ignored. On theother hand, tanβ also appears in the expressions for the φ-eld masses (5.41). Here it has to berenormalized up to two-loop order. The one-loop counterterm δ tanβ(1) again does not lead to anyrelevant contributions, but the two-loop counterterm δ tanβ(2) does.

We continue to apply this type of reasoning to all terms in the Lagrangian to get a countertermLagrangian that only takes into account the counterterms that will produce contributions to theHiggs mass at O(αtαs). We will now explicitly write down the counterterm Lagrangian. We willsplit the counterterm Lagrangian into the same 4 parts as the original Lagrangian in (5.52)

δL = δLkin + δLmass + δLHiggs + δLint (5.64)

To simplify the expressions we immediately apply the minimization conditions of the Higgs poten-tial after renormalizing. This means that the tadpole parameters vanish (T1 = T2 = 0) and thatβn = β (5.61).

The kinetic counterterms are as follows:

δLkin =iδZtL t/∂PLt+ iδZtR t/∂PRt+ δZtL(∂µtL)∗(∂µtL) + δZtR(∂µtR)∗(∂µtR)

+1

2δZ

(2)H1

(∂µφ1)(∂µφ1) +1

2δZ

(2)H2

(∂µφ2)(∂µφ2)

+1

2δZ

(2)AA(∂µA

0)(∂µA0) +1

2δZ

(2)GG(∂µG

0)(∂µG0) + δZ(2)AG(∂µG

0)(∂µA0)

(5.65)

In this equation we have not performed the rotation to the physical elds for the φ-elds. Theχ-elds have been rotated to the physical eld A0 and the Goldstone eld G0. The wave functionrenormalization factors for the mixed elds can be derived by performing the renormalization ofthe original Higgs doublets H1 and H2 (5.53) and then applying the rotation (5.27) (with nowβn = β). Explicitly, these WFR's are given by

δZ(i)AA = sin2 βδZ

(i)H1

+ cos2 βδZ(i)H2

δZ(i)GG = cos2 βδZ

(i)H1

+ sin2 βδZ(i)H2

δZ(i)AG = cosβ sinβ(δZ

(i)H2− δZ(i)

H1)

(5.66)

The counterterm Lagrangian for the mass terms looks like this

δLmass = −(

1

2mt (δZtL + δZtR) + δmt

)tt− δmLL

t t∗LtL− δmRRt t∗RtR− [δmLR

t t∗LtR +h.c.] (5.67)

where

δmLLt = (c2θtm

2t1

+ s2θtm2t2

)δZtL + s2θt(m2t2−m2

t1)δθt + c2θtδm

2t1

+ s2θtδm2

t2

δmRRt = (s2

θtm2t1

+ c2θtm2t2

)δZtR + s2θt(m2t1−m2

t2)δθt + s2

θtδm2

t1+ c2θtδm

2t2

δmLRt =

1

2cθtsθt(m

2t1−m2

t2)(δZtL + δZtR) + c2θt(m

2t1−m2

t2)δθt + cθtsθt(δm

2t1− δm2

t2)

(5.68)

50

are the explicit mass counterterms for the stop masses.

Importantly, we have the Higgs counterterm Lagrangian:

δLHiggs = −δT (2)1 φ1−δT (2)

2 φ2−1

2δm

2(2)φ1

φ21−

1

2δm

2(2)φ2

φ22−δm

2(2)φ1φ2

φ1φ2−1

2[M2

AδZ(2)AA+δM

2(2)A ](A0)2

(5.69)where

δm2(2)φ1

=(M2Z cos2 β +M2

A sin2 β)δZ(2)H1

+1

2sin2(2β)(M2

A −M2Z)δ tanβ(2) + sin2 βδM

2(2)A

+e

2MW sWcosβ(1 + sin2 β)δT

(2)1 − e

2MW sWsinβ cos2 βδT

(2)2

δm2(2)φ2

=(M2Z sin2 β +M2

A cos2 β)δZ(2)H2

+1

2sin2(2β)(M2

Z −M2A)δ tanβ(2) + cos2 βδM

2(2)A

− e

2MW sWcosβ sin2 βδT

(2)1 +

e

2MW sWsinβ(1 + cos2 β)δT

(2)2

δm2(2)φ1φ2

=− 1

2(M2

Z +M2A) sinβ cosβ(δZ

(2)H1

+ δZ(2)H2

)− 1

2(M2

Z +M2A) sin(2β) cos(2β)δ tanβ(2)

− sinβ cosβδM2(2)A +

e

2MW sWsin3 βδT

(2)1 +

e

2MW sWcos3 βδT

(2)2

(5.70)

are the explicit mass counterterms for the φ-elds [10].

Finally, we have the counterterm Lagrangian for the interactions. Again we will split it by thedierent types of interaction

δLint = δLHtt + δLHtt + δLHHtt (5.71)

Note that we only have to consider the renormalization of the couplings between the Higgs eldsand the top and stop elds as these are the only couplings with counterterm contributions relevantto our calculations.

Higgs-top-top

δLHtt = − emt

2MW sW sinβ

(δmt

mt+

1

2(δZtL + δZtR)

)ttφ2+i

emt

2MW sW tanβ

(δmt

mt+

1

2(δZtL + δZtR)

)tγ5tA0

(5.72)

Higgs-stop-stop

δLHtt =− em2t

MW sW sinβ

[(2δmt

mt+ δZtL

)t∗LtLφ2 +

(2δmt

mt+ δZtR

)t∗RtRφ2

]+

[emt

2MW sW sinβAt

(δmt

mt+δAtAt

+1

2(δZtL + δZtR)

)t∗RtLφ2 + h.c.

]+

[emt

2MW sW sinβµ

(δmt

mt+

1

2(δZtL + δZtR)

)t∗RtLφ1 + h.c.

]− i emt

2MW sW tanβ

[(δmt

mt+

1

2(δZtL + δZtR)

)(At − µ tanβ) + δAt

](t∗LtRA

0 − t∗RtLA0)

(5.73)

Higgs-Higgs-stop-stop

δLHHtt =− e2m2t

4M2W s

2W sin2 β

[(2δmt

mt+ δZtL

)t∗LtLφ

22 +

(2δmt

mt+ δZtR

)t∗RtRφ

22

]− e2m2

t

4M2W s

2W tan2 β

[(2δmt

mt+ δZtL

)t∗LtL(A0)2 +

(2δmt

mt+ δZtR

)t∗RtR(A0)2

](5.74)

51

5.8 Feynman rules

Now that we have all the terms in the Lagrangian that are needed for our calculation, includingall the counterterms, it is quite straightforward to derive the accompanying Feynman rules. Thesimplest way to calculate diagrams is to apply the Feynman rules, so it is useful to make a list ofthe Feynman rules. All Feynman rules originating from the Lagrangian of the previous sectionshave therefore been listed in Appendix A.

5.9 Calculating the diagrams

Now we are nally ready to calculate the two-loop O(αtαs) corrections to the Higgs mass in theMSSM. As a rst step we are going to write down all the diagrams that contribute. First we havethe genuine two-loop Higgs self-energy diagrams in Fig. 5.1. We can divide these diagrams intothree groups: the gluino corrections (a)-(d), the gluon corrections (e)-(i) and the pure scalar cor-rections (j)-(l). We will be mostly focusing on the gluino corrections later on since we are interestedin the dependence of these corrections on the gluino mass.

Figure 5.1: The genuine two-loop O(αtαs) Higgs self-energy diagrams. H can be either φ1, φ2

or A.

52

Of course we also need to take into account the counterterm diagrams that will contribute tothe Higgs self-energy at O(αtαs). These are seen in Fig. 5.2. Diagrams (a)-(f) are one-loop dia-grams with a one-loop counterterm insertion and diagram (g) is the two-loop counterterm diagram.

Figure 5.2: The counterterm diagrams contributing to the Higgs self-energy. H can be either φ1,φ2 or A.

One thing to note is that these are not all possible diagrams. First we have omitted all diagramswith a closed 1-point gluon loop. An example of such a diagram can be seen in Fig. 5.3.

Figure 5.3: A diagram with a closed gluon loop.

If we try to calculate such a diagram the closed gluon loop produces an integral of the form∫ddk

(2π)d1

k2(5.75)

53

Due to the masslessness of the gluon this integral contains no mass scale, so by dimensional argu-ments it must be zero. Therefore we can ignore all diagrams with closed 1-point gluon loops.

The second type of diagram that does not appear in Figs. 5.1 and 5.2 are `partner' diagramsof the original diagrams. For example, in diagram (b) of Fig. 5.1 we can reverse the direction ofthe fermion loop to produce a very similar, but dierent diagram, seen in Fig. 5.4. This diagramalso has to be added in the calculation; it will give the same contribution as the original diagram.The same type of `partner diagram' also exists for diagrams (c), (f), (i) and (j) in Fig. 5.1 anddiagrams (a) and (c) in Fig. 5.2.

Some diagrams have a dierent type of partner diagram. For example, looking at diagram (b)of Fig. 5.2 we can see that instead of placing the counterterm insertion on the left vertex we couldhave placed it on the right vertex. Again this produces a very similar, but dierent diagram. Thistype of partner diagram where left and right are ipped also exists for diagram (a) of Fig. 5.1 anddiagram (d) of Fig. 5.2.

Figure 5.4: The `partner diagram' to diagram (b) in Fig. 5.1

Now we know the complete set of Higgs self-energy diagrams that contribute at O(αtαs), butthis is not the full story. To complete the calculation we also need the explicit expressions for thecounterterms that appear in the diagrams of Fig. 5.2. This requires us to calculate an additionalset of diagrams. Of course, the exact explicit form of the counterterms depends on the renormal-ization scheme that is used, but the diagrams are always the same.

Figure 5.5: The one-loop top self-energy diagrams.

To calculate the (one-loop) top mass counterterm δmt we have to compute the self-energy di-agrams for the top eld seen in Fig. 5.5. For the stop mass counterterms δm2

tiand stop mixing

angle counterterm δθt we have to compute the self-energy diagrams for the stop elds in Fig. 5.6.These diagrams also allow us to compute the WFR factors for the top and stop elds but thesewill turn out to be irrelevant, as we will see later on. Finally, for the two-loop counterterms in theHiggs sector we have to compute two types of diagrams. First, there are the self-energy diagrams

of Fig. 5.1; these allow us to obtain the WFR factors δZ(2)Hi

if we compute the self-energies of the

φ elds and the A boson mass counterterm δM2(2)A if we compute the self-energy of the A boson.

Second, there are the tadpole diagrams of Fig. 5.7 that allow us to obtain the tadpole counterterms

54

Figure 5.6: The one-loop stop self-energy diagrams.

δT(2)i .

Now that we have all the diagrams, we can start calculating them. To simplify the calculationwe will set the mixing between the stops to zero from now on. So θt = 0 and the stop mass matrix(5.22) takes the form

Lmass,t = −(t∗L t∗R

)(m2t1

0

0 m2t2

)(tLtR

)(5.76)

So t1 = tL and t2 = tR are the mass eigenstates now, which greatly simplies the calculation.

5.9.1 Renormalization conditions

Before we move ahead with the calculation of the Feynman diagrams a word about the renormal-ization conditions is in order. We will look at two renormalization schemes, the on-shell schemeand the DR scheme. The DR scheme is closely related to another scheme called the DR scheme,just like the MS and MS schemes explained in Section 2.2.4. The abbreviation DR stands fordimensional reduction. The DR scheme is also a minimal subtraction scheme, but instead of usingconventional dimensional regularization, it uses a slightly dierent technique, which is dimensionalreduction. In this technique the loop momenta are continued to d spacetime dimensions, but thegauge elds are still kept in 4 dimensions. This is done so that the number of bosonic and fermionicdegrees of freedom keep matching, thereby preserving supersymmetry. Conventional dimensionalregularization breaks supersymmetry by changing the gauge eld degrees of freedom and is there-fore less suited for MSSM calculations6. The only dierence between the DR and DR schemeis that in the DR scheme the numerical constants γE and log(4π) are also subtracted. A morein-depth explanation of dimensional reduction can be found in [11].

Still, for us this is mostly an unimportant technical detail; since we are interested in the gluinodiagrams of Fig. 5.1 and not the gluon diagrams, we do not have to deal with gauge elds inour calculations. This means that for the gluino contributions there is no dierence betweenconventional dimensional regularization and dimensional reduction. As a nal point here, wewill re-use the notation of Eq. (2.29) for the typical combination of the divergent term and thenumerical constants in dimensionally regularized loop integrals:

Nε =2

ε− γE + log(4π) (5.77)

where γE is again the Euler-Mascheroni constant.

To be denite about the signs of the diagrams, we will dene the expressions belonging totadpole diagrams as iT and the expressions belonging to self-energy diagrams as iΣ.

To x the tadpole counterterms we impose the on-shell conditions:

Ti = 0, (i = 1, 2) (5.78)

6One would then have to add supersymmetry-preserving terms to ensure the correctness of the calculations.

55

Figure 5.7: The two-loop Higgs tadpole diagrams including the counterterm diagrams. H can beφ1 or φ2.

The hat is used to indicate a renormalized quantity: this means that it includes the countertermsneeded to make it nite, i.e. at the two-loop level the quantity T includes not just the loop

diagrams (a)-(i) of Fig. 5.7 (which we call the unrenormalized two-loop tadpoles T(2)i ) but also

the counterterm (j) which makes T nite. The conditions (5.78) are used to x the tadpolecounterterms:

δT(2)i = T

(2)i (5.79)

By imposing the condition (5.78) the tadpole loop diagrams are removed, so that the minimumof the Higgs potential does not shift. This means that the renormalized theory is still expandedaround the true minimum of the potential. This is the only sensible way to renormalize the tad-poles; if we impose renormalization conditions that shift the minimum of the potential, we will notbe able to extract the correct particle interpretation from the theory.

For the self-energy of the A-boson we impose the on-shell condition

ReΣA(M2A) = 0 (5.80)

56

which xes the mass counterterm for the A boson

δM(2)A = ReΣ

(2)A (M2

A) (5.81)

Again, note the distinction between the renormalized self-energy Σ which includes diagram (g) ofFig. 5.2 and the unrenormalized self-energy Σ which does not include this diagram.

For the eld renormalization of the Higgs doublets we impose DR renormalization conditions:

[ReΣ′φi(p2 = 0)]Nε = 0, (i = 1, 2) (5.82)

The prime indicates a derivative with respect to p2 and the subscriptNε is used to indicate that onlythe divergent part of the expression (terms with inverse powers of ε) and the numerical constantsγE and log(4π) will be subtracted. This xes the WFR factors of the Higgs doublets:

δZ(2)Hi

= −[ReΣ′(2)φi

(p2 = 0)]Nε (5.83)

It is hard to nd an appropriate renormalization condition for the counterterm δ tanβ, since itis not directly relatable to something like a self-energy. In the literature one can nd a lot ofdiscussion about how to renormalize tanβ; it is shown in [12, 13] that

δ tanβ(2) =1

2(δZ

(2)H2− δZ(2)

H1) (5.84)

gives a correct DR renormalization of tanβ.

Next, let's consider the renormalization conditions in the stop sector. For an OS renormalizationwe can impose the following conditions on the diagonal stop self-energies:

ReΣti(m2ti

) = 0, (i = 1, 2) (5.85)

Then the stop mass counterterms are xed as

δm2ti

= ReΣti(m2ti

) (5.86)

The mixing angle counterterm δθt can be xed by imposing a condition on the o-diagonal stopself-energy. One possibility for an on-shell renormalization is

ReΣt1 t2(m2t1

) = 0 (5.87)

which determines the mixing angle counterterm to be

δθt =1

m2t1−m2

t2

ReΣt1 t2(m2t1

) (5.88)

Alternatively we could have achieved an OS renormalization of δθt by imposing the condition (5.87)at p2 = m2

t2. A third option is to go for the more symmetric condition:

Re(Σt1 t2(m2t1

) + Σt1 t2(m2t2

)) = 0 (5.89)

resulting in

δθt =1

m2t1−m2

t2

1

2Re(Σt1 t2(m2

t1) + Σt1 t2(m2

t2)) (5.90)

If we want to change to a DR renormalization in the stop sector we just have to impose the condi-tions (5.85) and (5.87)/(5.89) on the divergent parts of the expressions and then also subtract theappropriate numerical constants.

Finally, the top mass counterterm in the OS scheme can be xed by imposing the condition

ReΣt(/p = mt) = 0 (5.91)

57

If we use the following decomposition for the top self-energy

Σt(p2) = /pPLΣLt (p2) + /pPRΣRt (p2) +mtPLΣSLt (p2) +mtPRΣSRt (p2) (5.92)

then the condition (5.91) xes the top mass counterterm as

δmt

mt=

1

2Re(ΣLt (m2

t ) + ΣRt (m2t ) + ΣSLt (m2

t ) + ΣSRt (m2t )) (5.93)

For a DR renormalization of δmt, again, we just have to impose condition (5.91) on the divergentpart of the expression and subtract the appropriate numerical constants.

5.9.2 Notation of the loop integrals

To help write down the expressions for the loop diagrams it is useful to introduce some notationfor the loop integrals. The one-loop tadpole integral is dened as

A0(m2) =(2πµ)ε

iπ2

∫ddl

1

l2 −m2(5.94)

The one-loop two-point scalar integral is dened as

B0(p2;m20,m

21) =

(2πµ)ε

iπ2

∫ddl

1

(l2 −m20)((l + p)2 −m2

1)(5.95)

Finally the one-loop three-point scalar integral is dened as

C0(p21, p

22, p

212;m2

0,m21,m

22) =

(2πµ)ε

iπ2

∫ddl

1

(l2 −m20)((l + p1)2 −m2

1)((l + p1 + p2)2 −m22)

(5.96)

with p212 = (p1 + p2)2.

For the two-loop integrals we follow the conventions of [14]. In two-loop two-point diagramsthere are three momenta: the external momentum p and the two loop momenta l1 and l2. Weorder them in the following ve combinations:

k1 = l1, k2 = l1 + p, k3 = l2 − l1, k4 = l2, k5 = l2 + p (5.97)

Fig. 5.8 gives an example of a two-loop self-energy diagram where the momenta have been labeledusing this notation.

Every two-loop two-point integral can be written in the form of a T-integral

Ti1i2···in(p2;m21,m

22, · · · ,m2

n) =

((2πµ)ε

iπ2

)2 ∫ddl1

∫ddl2

1

(k2i1−m2

1)(k2i2−m2

2) · · · (k2in−m2

n)(5.98)

or a Y-integral

Y j1···i1i2···in(p2;m21,m

22, · · · ,m2

n) =

((2πµ)ε

iπ2

)2 ∫ddl1

∫ddl2

k2j1· · ·

(k2i1−m2

1)(k2i2−m2

2) · · · (k2in−m2

n)(5.99)

where all indices range from 1 to 5 and j1 6= i1, i2, · · · , in, etc.

5.9.3 Steps in the calculation

To illustrate the steps involved in the calculation, we will work out one of the two-loop diagramsin detail. As an example we will choose diagram (b) in Fig. 5.1 and go through the calculationstep by step. In Fig. 5.8 this diagram is shown with all its momenta and color indices.

58

Figure 5.8: Diagram (b) of Fig. 5.1 where all momenta and color indices are explicitly indicated.

1. Write down the expression for the diagram that results from applying the Feynman rules.We will take the Higgs elds to be H = φ2 on both ends of the diagram. The externalmomentum will be called p and we will make use of the momenta k1 through k5 that weredened in (5.97). Finally, in every diagram we should consider all combinations of the stopelds that can propagate inside the diagram; in our chosen diagram there is only one stoppropagator, so we should do the calculation for both t1 and t2 propagating through. Fort1 = tL applying the Feynman rules gives

iΣ(b)φ2

= −µ2ε

∫ddl1

(2π)d

∫ddl2

(2π)dTr

[i

/k2 −mt

(−iemt

2MW sW sinβ

)i

/k1 −mt

(−√

2igsTaαβPR

)× i

−/k3 −mg

(−√

2igsTaβαPL

) i

/k1 −mt

(−iemt

2MW sW sinβ

)]i

k24 −m2

t1

(5.100)

The minus sign on the right-hand side is due to the closed fermion loop.

2. Simplify the expression: calculate the color factor and rewrite the prefactor for the integralto agree with the prefactor for the standard two-loop integrals dened in (5.98) and (5.99).In our example this gives

iΣ(b)φ2

=ie2mt2αs

32π3M2W s

2W sin2 β

((2πµ)ε

iπ2

)2 ∫ddl1

∫ddl2

Tr[( /k2 +mt)( /k1 +mt)PR(−/k3 +mg)PL( /k1 +mt)]

(k21 −m2

t )2(k2

2 −m2t )(k

23 −m2

g)(k24 −m2

t1)

(5.101)

3. Calculate the fermion trace. This can be done by hand or by using a computer algebraprogram such as FORM [15].

Tr[( /k2+mt)( /k1+mt)PR(−/k3+mg)PL( /k1+mt)] = −4(k1·k2)(k1·k3)+2k21(k2·k3)−2m2

t (2k1·k3+k2·k3)(5.102)

4. Perform a tensor reduction of the integral [14]. After taking the trace we are left with termsin the numerator of the integral that have a tensor structure. In this case we have a bunchof inner products involving the loop momenta. By doing a tensor reduction we can alwayswrite a two-loop integral in terms of the scalar T -integrals in (5.98) and the Y -integrals in(5.99). The procedure is as follows: every inner product of the form ki · kj can be rewrittenas a linear combination of squares of the momenta ki and the external momentum p2. Forexample

k1 · k2 = l21 + l1 · p =1

2(k2

1 + k22 − p2) (5.103)

The p2-term simply results in a scalar integral. If a k2i on the right-hand side also appears

in the denominator we can use it to cancel a factor in the denominator. In our example k21

59

can be used to cancel a factor in the denominator by applying the obvious identity

k21 = (k2

1 −m2t ) +m2

t (5.104)

The term between brackets exactly cancels a factor in the denominator to produce a scalarintegral and the m2

t term also gives a scalar integral.

Of course this cancellation cannot occur when there is a k2i in the numerator that does not

appear in the denominator. Then the best we can do at this point is write the integral asa Y -integral. It is possible to further reduce these Y -integrals to scalar integrals but thatrequires dierent kinds of tricks that we will not go into here.

Thus, reducing the expression for every diagram to a set of T -integrals and Y -integrals is apretty straightforward algebraic process. It can be done quite easily with computer algebra.For our example diagram we get the following set of integrals

iΣ(b)φ2

=ie2m2

tαs

32π3M2W s

2W sin2 β

[(4m2t − p2)(m2

t +m2g −m2

t1)T11234(p2;m2

t ,m2t ,m

2t ,m

2g,m

2t1

)+

(4m2t − p2)(T1124 − T1123 + T1234)−m2

t1T1234+

(m2t +m2

g −m2t1

)T1134 + T135 − T123 − T113 + T114 + Y 51234]

(5.105)

Only for the rst integral the arguments have been listed explicitly. This establishes thecorrespondence between the numbers in subscript and the masses, i.e. if an integral has thesubscript 3 this always corresponds to the argument m2

g, so there will be a factor (k23 −m2

g)in the denominator of that integral.

After following these steps we have brought the expression for the diagram into the form (5.105).This is a process that can be used for every diagram, so from now on we will not give a full calcu-lation for each diagram but simply list the result.

5.9.4 The p2 = 0 approximation

There is another very important approximation that we can make in our calculation. Consider thegeneric form of a full renormalized Higgs propagator

ΠH(p2) =i

p2 −m2H + ΣH(p2) + iδ

=iZH

p2 −M2H,phys + iδ

+ regular terms (5.106)

The physical Higgs mass is determined by the (real part of) the pole of the propagator:

M2H,phys −m2

H + ΣH(M2H,phys) = 0 (5.107)

Note that in the on-shell renormalization scheme ΣH(M2H,phys) = 0 so that m2

H = M2H,phys. How-

ever, this condition does not hold in general for other schemes, so that the value of the renormalizedLagrangian parameter m2

H changes depending on the scheme.

Our main interest is how the (renormalized) Higgs self-energy ΣH(p2) and, in turn, the physicalHiggs mass MH,phys, scales with the gluino mass mg.

Our goal in this section is to simplify ΣH(p2) by performing an expansion in p2. To do this wenow introduce the dimensionless quantity x:

x =p2

m2g

(5.108)

Close to the pole in the propagator p2 ≈ M2H,phys. Theoretically, we do not know what M2

H,phys

is, since it is what we are trying to calculate. That is why we need to use some extra input here:

60

we know that the (physical) mass of the light Higgs boson should be around 125 GeV even aftercorrections, while the gluino mass is going to be at least in the TeV range. As a result x shouldbe very small in our region of interest: x 1. This means that we can perform an expansion ofΣH(p2) in x.

To see if this small-momentum expansion in x is still valid in the theoretical limit where thegluino mass goes to innity we should consider how MH,phys scales with mg. If the physical Higgsmass depends logarithmically on the gluino mass

M2H,phys ∼ log(mg) (5.109)

then in the limitmg →∞ the value of x in the neighborhood of the pole will decrease asymptoticallyto 0. This means that the small x approximation remains justied. If the physical Higgs massscales proportionally with the gluino mass

M2H,phys ∼ m2

g (5.110)

then in the limit mg →∞ the value of x in the neighborhood of the pole remains constant. So ifour expansion was initially justied it will remain justied when the gluino mass is increased. Agood argument for why the expansion should be initially justied is that any mg-dependence ofthe Higgs mass will come with a two-loop numerical prefactor. This factor will suppress the gluinoeects by about 2 or 3 orders of magnitude.

The expansion in x will only fail if the physical Higgs mass scales with the gluino mass as

M2H,phys ∼ m3

g (5.111)

or worse. Then close to the pole x will increase while the gluino mass is increased, until a pointwhere the expansion for small x is no longer justied. However, it will turn out that this does nothappen in our calculation. The scaling behavior changes depending on the renormalization schemebut does not get worse than in (5.110). That means that we can always use the small x expansion.

In generic form this expansion of the self-energy will look like this

ΣH(p2) = c0 + c1x+ c2x2 + . . . (5.112)

Note that the expansion contains no terms with inverse powers or logarithms of x since the self-energy of a massive scalar particle is generally nite at p2 = 0. Since we are interested in theleading eects of the gluino mass on the Higgs mass the c0 term is what we are after. It is easy tosee that c0 = ΣH(0). Instead of doing the full calculation of the self-energy and then expanding,the leading term in the expansion can more easily be obtained by calculating the Higgs self-energywhile setting p2 = 0 from the start.

This has a few consequences that simplify the calculation. One of them is that instead ofcalculating the exact full propagator of (5.106) we get the approximate form

ΠH(p2) =i

p2 −m2H + ΣH(0) + iδ

=i

p2 −M2H,phys + iδ

(5.113)

By neglecting the momentum dependence of the self-energy the WFR factor Z is now automati-cally 1 and M2

H,phys = m2H − ΣH(0).

Another consequence of evaluating the self-energy at p2 = 0 is that it greatly reduces thecomplexity of the integrals that appear; all integrals that appear will be vacuum integrals, i.e.integrals from diagrams without external legs. Since there is no other external momentum for pµ

to be contracted with, letting p2 → 0 eectively means letting p → 0. This means that of the kiin Eq. (5.97) only k1, k3 and k4 will occur. For this class of integrals the explicit expressions areknown even for arbitrary masses.

One other change that has to be made in the p2 = 0 approximation is in the renormalization of

δM2(2)A . In (5.80) we imposed a renormalization condition on the A boson self-energy at p2 = M2

A.

61

Since we will now compute all Higgs self-energies, both for the A boson and the φ elds, at p2 = 0this renormalization condition has to be modied to

ReΣA(p2 = 0) = 0 (5.114)

which means the A boson mass counterterm will be

δM2(2)A = ReΣ

(2)A (0) (5.115)

5.9.5 The results

Now we can compute the results. There is one nal simplication we will make at this point;we already set the stop mixing parameter Xt = At + µ cotβ to zero, but now we set µ and Atseparately to zero7. So now the only coupling of the top/stop sector that we get contributionsfrom is the actual (dimensionless) Yukawa coupling Yt. This also means that the eld φ1 whichalready didn't couple to the top now also doesn't couple to the stop. As a result all self-energiesand tadpoles with an external φ1 vanish and we only receive contributions from diagrams withexternal φ2 elds. So the two-loop tadpole counterterm for the eld φ1 vanishes:

δT(2)1 = 0 (5.116)

As explained in the previous section, because of the p2 = 0 approximation the WFR factors forthe Higgs elds will be ZH = 1 so that

δZ(2)H1

= δZ(2)H2

= 0 (5.117)

Due to our denition of δ tanβ (5.84) this counterterm will also vanish

δ tanβ(2) = 0 (5.118)

Let's continue by listing the results for the other two-loop counterterms or more specically,the gluino contributions to the two-loop counterterms. It is helpful to introduce the followingshorthand notation

c2i = m2t +m2

g −m2ti

(5.119)

for a combination of masses that appears a lot.

Gluino contributions to the two-loop tadpole counterterm δT(2)2 come from two places: two-

loop tadpole diagrams with a gluino propagator (diagrams (a) and (b) in Fig. 5.7) and one-looptadpole diagrams with a one-loop top or stop mass counterterm (diagrams (f)-(i) in Fig. 5.7).Calculating these diagrams without inserting explicit expressions for the one-loop top and stopmass counterterms we get:

δgT(2)2 =

em2tαs

16π3MW sW sinβ

2∑i=1

[2c2i (T1134(m2t ,m

2t ,m

2g,m

2ti

) + T1134(m2ti,m2

ti,m2

g,m2t ))

− 2T113(m2t ,m

2t ,m

2g) + 2T113(m2

ti,m2

ti,m2

g)

+ 2T114(m2t ,m

2t ,m

2ti

) + 2T114(m2ti,m2

ti,m2

t )

− 6π

αs

δmt

mt(A0(m2

t )−A0(m2ti

) +m2tB0(0;m2

t ,m2t ))

+ 3π

αsδm2

tiB0(0;m2

ti,m2

ti)]

(5.120)

The T114 integrals on the third line do not depend on mg so they are not relevant to us, but sincethey originate from the gluino diagrams they have been included here for completeness.

7In terms of the input parameters (5.29) what we actually do is set θt = µ = 0. Then At is also automaticallyzero according to the relation (5.23)

62

The gluino contributions to the self-energies come from diagrams (a)-(d) in Fig. 5.1 and dia-grams (a)-(f) in Fig. 5.2. If we calculate these diagrams for the A boson we nd that its mass

counterterm is equal to the tadpole counterterm δT(2)2 aside from the prefactor:

δgM2(2)A =

e sinβ

2MW sW tan2 βδgT

(2)2 (5.121)

If we plug these results into the expressions for the φ mass counterterms in (5.70) we nd forthe gluino contributions to these counterterms:

δgm2(2)φ1

= 0

δgm2(2)φ2

=e

2MW sW sinβδgT

(2)2

δgm2(2)φ1φ2

= 0

(5.122)

Since the coupling of φ1 to the stop has been set to zero, as stated in the beginning of thissection, the unrenormalized self-energies involving φ1 automatically vanish:

Σ(2)φ1

(0) = Σ(2)φ1φ2

(0) = 0 (5.123)

Because the two-loop gluino counterterm contributions (5.122) to these self-energies also vanish,we get

Σg(2)φ1

(0) = Σg(2)φ1φ2

(0) = 0 (5.124)

for the gluino part of the renormalized self-energies.

For the gluino contribution to the φ2 self-energy we again calculate the diagrams (a)-(d) in Fig.5.1 and (a)-(f) in Fig. 5.2 to which we add the two-loop counterterm contribution (diagram (g) inFig. 5.2, (5.122)):

Σg(2)φ2

(0) =e2m4

tαs

4π3M2W s

2W sin2 β

2∑i=1

[c2i (T11134(m2ti,m2

ti,m2

ti,m2

g,m2t ) + T11134(m2

t ,m2t ,m

2t ,m

2g,m

2ti

)

+ T11344(m2t ,m

2t ,m

2g,m

2ti,m2

ti)) + T1113(m2

ti,m2

ti,m2

ti,m2

g)

− T1113(m2t ,m

2t ,m

2t ,m

2g) + T1114(m2

ti,m2

ti,m2

ti,m2

t )

+ T1114(m2t ,m

2t ,m

2t ,m

2ti

) + T1144(m2t ,m

2t ,m

2ti,mti

)

− 3π

αs

δmt

mt(B0(0;m2

t ,m2t )−B0(0;m2

ti,m2

ti) +m2

tC0(0, 0, 0;m2t ,m

2t ,m

2t ))

+3

2

π

αsδm2

tiC0(0, 0, 0;m2

ti,m2

ti,m2

ti)]

(5.125)

Again there are integrals (T1114 and T1144) that do not depend on mg but have been included sincethey originate from the gluino diagrams.

Now the nal step is to insert the top and stop mass counterterms in the dierent renormal-ization schemes. The gluino contribution to the top mass counterterm is determined by diagram(a) of Fig. 5.5. In the OS scheme it produces the following expression:

δgmOSt

mt=−αs

6πm2t

2∑i=1

[−c2iB0(m2t ;m

2g,m

2ti

) +A0(m2g)−A0(m2

ti)] (5.126)

In the DR scheme only the part of this expression proportional to Nε is included in the counterterm

δgmDRt

mt=αs3πNε (5.127)

63

For the gluino contribution to the stop mass counterterms we have to compute diagram (a) ofFig. 5.6, which gives the following result in the OS renormalization scheme

δgm2,OS

ti=−2αs

3π[c2iB0(m2

ti;m2

g,m2t ) +A0(m2

g) +A0(m2t )] (5.128)

In the DR scheme the result is

δgm2,DR

ti=−2αs

3πNε(c

2i +m2

g +m2t ) (5.129)

Now we can insert the top and stop mass counterterms in the dierent renormalization schemesinto Eq. (5.125) and look at the leading mg behavior. All the vacuum integrals appearing in theexpression can be reduced to a combination of one-loop integrals and the two-loop vacuum inte-gral T134(m2

1,m22,m

23), which is known explicitly for three dierent masses. The reduction of the

vacuum integrals can be found for example in Appendix C.2 of [16]. For the one-loop integralswe need the explicit expansion in ε up to O(ε1), while for the two-loop integral T134 we need theexplicit expansion in ε up to O(ε0). After inserting the explicit expressions for the countertermsand the one- and two-loop integrals we can then perform the expansion in mg. All of this caneasily be done with an algebraic program such as Mathematica.

The results are as follows:OS top+OS stop

Σ(2)φ2

(0) =e2m4

tαs

4π3M2W s

2W sin2 β

[−2 log2(m2g) + 2 log(m2

g)(log(m2t1

) + log(m2t2

) + 3)] +O(m0g) (5.130)

DR top+OS stop

Σ(2)φ2

(0) =e2m4

tαs

4π3M2W s

2W sin2 β

[−2 log2(m2g) + log(m2

g)(log(m2t1

) + log(m2t2

) + 2 log(m2t ) + 7)] +O(m0

g)

(5.131)OS top+DR stop

Σ(2)φ2

(0) =e2m4

tαs

4π3M2W s

2W sin2 β

[m2g

(1

m2t1

+1

m2t2

)(log

(m2g

µ2

)− 1)− 2 log2(m2

g)

+ log(m2g)(2 log(m2

t1) + 2 log(m2

t2) +m2

t

(1

m2t1

+1

m2t2

)+ 5)] +O(m0

g)

(5.132)

DR top+DR stop

Σ(2)φ2

(0) =e2m4

tαs

4π3M2W s

2W sin2 β

[m2g

(1

m2t1

+1

m2t2

)(log

(m2g

µ2

)− 1)− 2 log2(m2

g)

+ log(m2g)(log(m2

t1) + log(m2

t2) + 2 log(m2

t ) +m2t

(1

m2t1

+1

m2t2

)+ 6)] +O(m0

g)

(5.133)

Note that in all these results there is a still a 1/ε pole term. These pole terms should notdepend on mg since we have taken all gluino contributions into account; the absence of an mg

dependence in the pole term thus acts as a check on our calculation. These pole terms are presentsince we have not taken all diagrams of Fig. 5.1 into account, but only the gluino diagrams. Thesediagrams by themselves are not fully nite, a fact that was already established in the literature [17].

64

From these results it can be seen that if the stop masses are renormalized in the DR scheme theleading term goes with m2

g, while with an OS renormalization of the stop masses the leading term

in the expansion is a log2(m2g) term. This dierence can be traced back to a cancellation between

diagram (c) of Fig. 5.1 and diagrams (c)+(d) of Fig. 5.2; in the OS scheme the terms quadraticin mg are canceled exactly, while in the DR scheme this cancellation does not occur.

The results for the full on-shell calculation (5.130) and the full DR calculation (5.133) agreewith [18]. The full on-shell result also agrees with [19]. In [10] the full on-shell result is posted as

Σ(2)φ2

= CFα

π

αsπ

3m2t

2s2W sin2 β

[− log2(m2g) + log(mg)

2] +O(m0g) (5.134)

which is incorrect.

5.10 The pole masses of the physical Higgs elds

To close this chapter we will look at how we can go from the renormalized self-energies calculatedin the previous section to the pole masses of the physical Higgs elds. This involves rst translatingthe self-energies of the φ-elds into self-energies for the physical Higgs elds h0 and H0. Then wecan determine the masses of h0 and H0 by nding the poles of their propagators.

The rst step is to look at the propagators for the φ-elds. Since the φ-elds mix, we don'tsimply have a propagator for φ1 and a propagator for φ2. Instead, we should consider the inversepropagator matrix Π−1(p2). In the bare theory, this matrix has the following form:

Π−1φ,0(p2) = −i

(p2 −m2

φ1−m2

φ1φ2

−m2φ1φ2

p2 −m2φ2

)(5.135)

The subscript φ is used to indicate the φ-basis and the subscript 0 indicates that it applies to thebare theory.

Using the rotation (5.26) we can nd the inverse propagator matrix in the (h0, H0)-basis. Todo this, let's introduce the following notation for the rotation matrix:

R(α) =

(cosα sinα− sinα cosα

)(5.136)

Since this is a rotation matrix, it satises R−1(α) = RT (α). The inverse propagator matrix in the(H0, h0)-basis is given by

Π−1H,0(p2) = R(α)Π−1

φ,0(p2)R−1(α) = −i(p2 −m2

H 00 p2 −m2

h

)(5.137)

where the subscript H is used to indicate the (H0, h0)-basis and where

m2H = cos2(α)m2

φ1+ sin2(α)m2

φ2+ 2 sin(α) cos(α)m2

φ1φ2

m2h = sin2(α)m2

φ1+ cos2(α)m2

φ2− 2 sin(α) cos(α)m2

φ1φ2

(5.138)

are the masses in the (H0, h0)-basis. Naturally, the o-diagonal elements of Π−1H,0(p2) vanish. It is

then easy to invert this matrix and nd the propagators of the H0 and h0 elds:

ΠH,0(p2) = i

((p2 −m2

H)−1 00 (p2 −m2

h)−1

)(5.139)

Now we can repeat this story for the full renormalized Higgs propagators. In the φ-basis theinverse propagator matrix including loop corrections takes the following form:

Π−1φ (p2) = −i

(p2 −m2

φ1+ Σφ1(p2) −m2

φ1φ2+ Σφ1φ2(p2)

−m2φ1φ2

+ Σφ1φ2(p2) p2 −m2

φ2+ Σφ2

(p2)

)(5.140)

65

The rotation to the (H0, h0)-basis proceeds in exactly the same way as in Eq. (5.137):

Π−1H (p2) = R(α)Π−1

φ (p2)R−1(α) = −i(p2 −m2

H + ΣH(p2) ΣHh(p2)

ΣHh(p2) p2 −m2h + Σh(p2)

)(5.141)

where

ΣH(p2) = cos2(α)Σφ1(p2) + sin2(α)Σφ2

(p2) + 2 sin(α) cos(α)Σφ1φ2(p2)

Σh(p2) = sin2(α)Σφ1(p2) + cos2(α)Σφ2

(p2)− 2 sin(α) cos(α)Σφ1φ2(p2)

ΣHh(p2) = sin(α) cos(α)(Σφ2(p2)− Σφ1

(p2)) + (cos2(α)− sin2(α))Σφ1φ2(p2)

(5.142)

are the Higgs self-energies in the (H0, h0)-basis. We can then invert Π−1H (p2) to nd the full Higgs

propagator matrix:

ΠH(p2) =i

(p2 −m2H + ΣH(p2))(p2 −m2

h + Σh(p2))− (ΣHh(p2))2

(p2 −m2

h + Σh(p2) −ΣHh(p2)

−ΣHh(p2) p2 −m2H + ΣH(p2)

)(5.143)

The physical Higgs masses m2h,phys and m

2H,phys are determined by the poles of the expression in

front of the matrix in Eq. (5.143). Phrased dierently, the physical Higgs masses are given by thezeros of the determinant of the inverse propagator matrix.

The determination of the poles becomes quite simple in the case where ΣHh(p2) vanishes orbecomes negligibly small. Then the physical mass of H0 just gets shifted by ΣH(p2) and thephysical mass of h0 gets shifted by Σh(p2). So then the gluino dependence of the self-energies thatwe calculated in the previous section appear in the same way in the physical Higgs masses.

66

Conclusions and outlook

In this thesis we have come across many of the intricacies involved in calculating loop corrections;choosing a suitable regulator, choosing a set of input parameters, renormalizing the theory andpicking the renormalization scheme are all steps that require careful thought. Never mind the factthat issues like gauge anomalies have to be kept in mind when setting up a theory. This wholeprocess quickly becomes very complex in any realistic theory. Even in our (very) simplied MSSMmodel of Chapter 5 this was quite challenging.

In the end, we were able to calculate the two-loop gluino corrections to the Higgs masses ofthe MSSM under some simplifying assumptions: no stop mixing, the p2 = 0 approximation andµ = At = 0. The most important conclusion to be taken from the results is that a DR renormaliza-tion of the stop masses leads to a quadratic dependence of the Higgs mass corrections on the gluinomass, whereas an on-shell renormalization of the stop masses leads to a logarithmic dependenceon the gluino mass. The former case presents a problem: if the gluino mass becomes very largethe corrections to the Higgs mass can become unphysically large. In the latter case this is not aproblem, so it seems like the on-shell scheme should be preferred here.

The stronger dependence on the gluino mass in the DR scheme could be related to the sen-sitivity of the Higgs mass to scales of new physics, an argument that is often used to explainthe hierarchy problem. At rst this may seem paradoxical, since supersymmetry is supposed tosolve the hierarchy problem. But that is only true for exact supersymmetry; for a theory withbroken supersymmetry like the MSSM there may be new ne-tuning problems if the scale ofsupersymmetry-breaking ΛSUSY is suciently far removed from the electroweak scale ΛEW . Inthis case there is another interesting complication because the mass scale mg couples only indi-rectly to the Higgs sector. The renormalization scheme dependence is also interesting, because itwould mean that there is a ne-tuning problem if the DR scheme is used, whereas in the on-shellscheme there would be no such problem. These points require more exploration: should the DRcalculation just be discarded because it is less `physical' than the on-shell scheme? Is that thereason a possible ne-tuning problem shows up?

Another thing that could be explored is the more general case where µ and At are not set tozero and there is an arbitrary non-zero mixing in the stop sector. This calculation has also beendone in the literature [19], but it could still be looked at in more detail in this context where weare interested in the dependence on the renormalization scheme and the possible appearance of ane-tuning problem. Does the generalization signicantly change this picture or not? Of coursethis does complicate the calculation and makes obtaining analytical results more dicult. In theeven more general case where we also drop the p2 = 0 approximation an analytical approach ispretty much not viable any more [20].

67

Appendix A

Feynman rules for Chapter 5

In this Appendix all Feynman rules originating from the Lagrangian of Section 5.5 and the coun-terterm Lagrangian of Section 5.7 are listed.

Let's start with the Feynman rules for the tadpole counterterms:

−iδT (2)1

−iδT (2)2

68

Next are the Feynman rules for all the propagators. The gluon propagator has been written downin the 't Hooft-Feynman gauge. This particular gauge choice doesn't really matter as we will focuson the gluino diagrams and not the gluon diagrams.

i

/p−mt

i

[/p(δZtLPL + δZtRPR)− 1

2mt(δZtL + δZtR)− δmt

]i

p2 −m2t1

i

p2 −m2t2

i[p2δZtL − δmLLt ]

i[p2δZtR − δmRRt ]

−iδmLRt

i

p2 −m2h0

i

p2 −m2H0

i[p2δZ(2)H1− δm2(2)

φ1]

i[p2δZ(2)H2− δm2(2)

φ2]

−iδm2(2)φ1φ2

i

p2 −M2A

i[(p2 −M2A)δZ

(2)AA − δM

2(2)A ]

i

/p−mg

−igµνp2

A cross denotes a one-loop counterterm and a circled cross denotes a two-loop counterterm.

69

Next, we have the Feynman rules for 3-particle interactions between a Higgs eld and the top orstop elds:

−iemt

2MW sW sinβ

−iemt

2MW sW sinβ

(δmt

mt+

1

2(δZtL + δZtR)

)

−emt

2MW sW tanβγ5

−emt

2MW sW tanβγ5

(δmt

mt+

1

2(δZtL + δZtR)

)

−iem2t

MW sW sinβ

−iem2t

MW sW sinβ

(2δmt

mt+ δZtL/R

)

iemtAt2MW sW sinβ

iemtAt2MW sW sinβ

(δmt

mt+δAtAt

+1

2(δZtL + δZtR)

)

iemtµ

2MW sW sinβ

iemtµ

2MW sW sinβ

(δmt

mt+

1

2(δZtL + δZtR)

)

−/+emt

2MW sW tanβ(At − µ tanβ)

−/+emt

2MW sW tanβ

[(δmt

mt+

1

2(δZtL + δZtR)

)(At − µ tanβ) + δAt

]For the nal two Feynman rules the upper sign is used for a tL coming into the vertex and thelower sign for a tR coming into the vertex.

70

Then we have the 3-particle interactions between a gluon or gluino and the top and stop elds:

−/+

√2igsT

aαβPL/R

−/+

√2igsT

aαβPR/L

−igsT aαβγµ

−igsT aαβ(p+ p′)µ

For the rst two of these Feynman rules the upper sign is used when tL is involved and the lowersign when tR is involved.

71

Finally, we have the Feynman rules for the 4-particle interactions between two Higgs elds andtwo stop elds as well as the quartic stop interactions:

−ie2m2t

2M2W s

2W sin2 β

−ie2m2t

2M2W s

2W sin2 β

(2δmt

mt+ δZtL/R

)

−ie2m2t

2M2W s

2W tan2 β

−ie2m2t

2M2W s

2W tan2 β

(2δmt

mt+ δZtL/R

)

− i3g2s(δαβδρσ + δασδρβ)

−ig2s

(1

6δαβδρσ −

1

2δασδρβ

)

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