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Electromagnetic Interactions and Chiral Multi-Pion Dynamics Diplomarbeit von Maximilian Duell ur August, 2012 Betreuer: Prof. Dr. Norbert Kaiser Technische Universit¨ at M¨ unchen Physik-Department T39 (Prof. Dr. Wolfram Weise)

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Page 1: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

Electromagnetic Interactionsand Chiral Multi-Pion Dynamics

Diplomarbeitvon

Maximilian Duell

fur August, 2012

Betreuer:

Prof. Dr. Norbert Kaiser

Technische Universitat Munchen

Physik-Department

T39 (Prof. Dr. Wolfram Weise)

Page 2: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as
Page 3: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

Mehr Licht!

Johann Wolfgang von Goethe

Page 4: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as
Page 5: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

Contents

Introduction 7

1 Scalar Electrodynamics 9

2 Strong Interactions 112.1 Quantum Chromodynamics . . . . . . . . . . . . . . . . . . . . . 112.2 Low-Energy Effective Field Theory . . . . . . . . . . . . . . . . . 132.3 QCD in the Presence of External Fields . . . . . . . . . . . . . . 15

3 Chiral Perturbation Theory at Order p4 173.1 Contact Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 Pion Propagator at Order p4 . . . . . . . . . . . . . . . . . . . . 183.3 Renormalization of the Self-Energy . . . . . . . . . . . . . . . . . 193.4 Pion-Photon Coupling . . . . . . . . . . . . . . . . . . . . . . . . 20

3.4.1 Calculation using Feynman-Parameters . . . . . . . . . . 213.4.2 Calculation by Tensor Reduction . . . . . . . . . . . . . . 223.4.3 Renormalized γππ Vertex Function . . . . . . . . . . . . . 25

3.5 Pion-Photon-Photon Coupling . . . . . . . . . . . . . . . . . . . 263.5.1 Two-Propagator Contribution . . . . . . . . . . . . . . . . 263.5.2 Three-Propagator Contribution . . . . . . . . . . . . . . . 273.5.3 Renormalized γγππ Vertex Functions . . . . . . . . . . . 32

3.6 Contractions with Polarization Vectors . . . . . . . . . . . . . . . 343.7 Further Many-Particle Vertex Functions . . . . . . . . . . . . . . 35

4 Scattering Observables and Phase Space 374.1 Cross Sections and Amplitudes . . . . . . . . . . . . . . . . . . . 374.2 Kinematics of 2→ n Scattering . . . . . . . . . . . . . . . . . . . 394.3 Direct Parametrization of Two-Body Phase Space . . . . . . . . 394.4 Direct Parametrization of Three-Body Phase Space . . . . . . . . 414.5 2-Body Decompositions of n-Body Phase Space . . . . . . . . . . 444.6 Lorentz Invariance and Lorentz Transformations . . . . . . . . . 464.7 2-Body Parametrization of n-Body Phase Space . . . . . . . . . . 48

5 Hadronic Photon-Photon Fusion 515.1 Pion-Pair Production γγ −→ π+π− . . . . . . . . . . . . . . . . . 51

5.1.1 Kinematics and Mandelstam Variables . . . . . . . . . . . 515.1.2 Amplitude at Leading Order . . . . . . . . . . . . . . . . 525.1.3 Transversality-Preserving Simplification . . . . . . . . . . 52

5

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6 CONTENTS

5.1.4 Polarization-Averaged Amplitude . . . . . . . . . . . . . . 535.1.5 Skeleton Expansion at Next-to-Leading Order . . . . . . . 545.1.6 Differential and Total Cross Section . . . . . . . . . . . . 55

5.2 Radiative Process γγ −→ π+π−γ . . . . . . . . . . . . . . . . . . 575.2.1 LO Amplitude and Skeleton Expansion . . . . . . . . . . 575.2.2 Partial NLO Corrections . . . . . . . . . . . . . . . . . . . 585.2.3 Photon Spectrum and Partial Cross Section . . . . . . . . 595.2.4 Soft-Photon Approximation . . . . . . . . . . . . . . . . . 60

5.3 Four-Pion Final States γγ −→ ππππ . . . . . . . . . . . . . . . . 645.4 Primakoff Scattering γπ− −→ π−π+π−π+π− . . . . . . . . . . . 67

6 Photon Fusion and Experiment 716.1 Inclusion of the f2(1270) Resonance . . . . . . . . . . . . . . . . 716.2 The Equivalent Photon Method . . . . . . . . . . . . . . . . . . . 74

7 Outlook and Summary 817.1 Resonances and Chiral Symmetry . . . . . . . . . . . . . . . . . . 817.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

A Feynman Rules 85A.1 Cartesian Isospin Basis . . . . . . . . . . . . . . . . . . . . . . . . 85A.2 Charge Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

B Dimensional Regularization 89

C Veltman-Passarino Scalar Loop Functions 91C.1 A0(m2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91C.2 B0(k2;m2,m2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92C.3 C0(0, s, 0;m2,m2,m2) . . . . . . . . . . . . . . . . . . . . . . . . 93

D Phase Space Integrations 95

E Details on Results 101E.1 Kinematical Inequality . . . . . . . . . . . . . . . . . . . . . . . . 101E.2 Amplitude γγ −→ π+π−γ . . . . . . . . . . . . . . . . . . . . . . 102E.3 Amplitude γγ −→ π+π−π0π0 . . . . . . . . . . . . . . . . . . . . 103E.4 Amplitude γγ −→ π+π−π+π− . . . . . . . . . . . . . . . . . . . 104

Page 7: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

Introduction

The electromagnetic structure of pions is still an active area of research. Exper-imentally accessible energies are constantly increasing and high-statistics dataallows to study higher-order processes, with smaller experimental signatures.

New exclusive experimental data involving multi-pion final-states are con-stantly produced. Concerning e+e−-annihilation into pions, the reaction involv-ing a four-pion final state, e+e− −→ π+π−π+π−, has been recently studied ininitial-state radiation measurements at BaBar [L+12]. Theoretical predictionsfor such reactions at low energies can be obtained in the framework of chiralperturbation theory. For the above process this was successfully done by Eckerand Unterdorfer [EU02].

In this spirit, we study two-photon processes at low energy by first revisitingthe chiral perturbation theory calculation of γγ −→ π+π− at next-to-leadingorder. The result is used to calculate the total- and differential cross sectionof the photon-photon fusion process γγ −→ π+π−. We generalize the cal-culation to the radiative process γγ −→ π+π−γ, considering the correctionsdue to pion loops and the pion polarizability. The cases of soft, as well ashard bremsstrahlung are investigated. Furthermore we study photon-photonfusion processes with four pions in the final state, γγ −→ π+π−π+π− andγγ −→ π+π−π0π0.

In the light of the recently discovered new particle with mass around 125 GeV,we would also like to mention the recent proposal [BEL+12] to further study itsconsistency with the Higgs particle of the Standard Model using a photon-photoncollider.

This diploma thesis is organized as follows. In chapter 1, we give a brief treat-ment of electromagnetic pion interactions in scalar quantum electrodynamics.This is put into an extended framework in chapter 2, where we shortly reviewthe foundations of quantum chromodynamics and its low-energy effective fieldtheory, chiral perturbation theory. In chapter 3, we study higher-order chiralcorrections to interactions of pions with the electromagnetic field. Chapter 4treats many-particle phase-space integrals as required for calculating total anddifferential cross sections. Hadronic photon-photon fusion processes are studiedin chapter 5. In chapter 6, we establish a connection with currently available ex-perimental data, featuring the inclusion of resonances and possible applicationsto ultraperipheral heavy-ion scattering. Chapter 7 gives an outlook on treatingresonances in chiral perturbation theory by resumming to all orders.

7

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Chapter 1

Scalar Electrodynamics

The charged pions are massive spin-0 bosons. It is therefore natural to describetheir electromagnetic coupling by a complex scalar field ϕ(x). Its free dynamicsare described by the Klein-Gordon Lagrangian density

LKG = (∂µϕ)†∂µϕ−m2ϕ†ϕ.

We can incorporate photons into this theory by introducing a vector poten-tial Aµ and adding the Maxwell Lagrangian density to LKG,

L = LKG + LMaxwell,

LMaxwell = −1

4FµνFµν , Fµν := ∂µAν − ∂νAµ.

It is invariant under the local gauge transformations Aµ(x)→ Aµ(x) + ∂µχ(x).The combined theory can only be quantized in a consistent manner, if all

interactions are formulated respecting local gauge invariance1. If renormalizabil-ity is used as a guiding principle, there is a unique gauge-invariant interaction.This interaction can be introduced using the gauge-covariant derivative

∂µ → Dµ = ∂µ − ieAµ, Fµν =i

e[Dµ,Dν ] .

A simple calculation shows

(Dµϕ)†Dµϕ = (∂µ + ieAµ)ϕ†(∂µ − ieAµ)ϕ

= (∂µϕ)†∂µϕ+ ie[Aµϕ

†∂µϕ− (∂µϕ†)Aµϕ

]+ e2AµA

µϕ†ϕ.

For the perturbative treatment, the Lagrangian density is separated into freeand interaction parts as

L = (∂µϕ)†∂µϕ−m2ϕ†ϕ− 1

4FµνFµν + LI,

LI = ieAµ(ϕ†∂µϕ− ϕ∂µϕ†) + e2AµAµϕ†ϕ.

1This point of view on the gauge principle is advocated in [Wei05], Ch. 8.1.

9

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10 CHAPTER 1. SCALAR ELECTRODYNAMICS

One obtains the corresponding momentum-space Feynman-rules as

k, µ

p

p′

k1, µ

k2, ν

ie(p+ p′)µ, 2ie2gµν .

We will find in section 2, that this simple description is adequate as longas pion-pion interactions can be neglected. In principle one might attempt tomodel pion interactions in a renormalizable way, through a quartic term (ϕ†ϕ)2.

A proper description should take into account that pions are not elementaryparticles, but posess a substructure of quarks and gluons. By this argument,the pion field can only be considered a low-energy effective degree-of-freedomand the renormalizability of the theory governing its dynamics is not an issue.A more detailed analysis of the nature of the pion is necessary, in order toconstrain the possible interactions of pions and photons.

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Chapter 2

Theoretical Descriptions ofStrong Interactions

All currently available experimental data on non-gravitational scales are welldescribed within the standard model of particle physics. It is the fundamentaltheory of electro-weak and strong interactions. The modern point of view is,that the standard model is certainly a correct approximation up to scales of a fewhundred GeV. It is assumed to be incomplete at the interface with gravitationand at the unification scale of the three forces, where new physics is expectedto appear.

The explanation of the success of the standard model has shaped the idea ofeffective field theories, which states that effects due to physics at higher energyscales can be absorbed into non-renormalizable effective interactions, resultingonly in small corrections to the physics of the effective degrees of freedom atlower energy scales.

In the following, we will restrict ourselves to Quantum Chromodynamics,which is the part of the standard model responsible for strong interactions.Further we will take into account only the lightest quark fields.1

2.1 Quantum Chromodynamics

Quantum Chromodynamics (QCD) is defined by the Yang-Mills Lagrangian den-sity with gauge group SU(3), written as

LQCD = q(i /D −m)q − 1

4GAµνG

µνA. (2.1)

There are Nf copies or flavours of the quark fields

q = (qa)f=1,...,Nf , a = 1, 2, 3 .

Each of them is an element of a three-dimensional vector space acted on by thegauge group (colour -)SU(3). The quarks interact by the exchange of the gauge

1For this introduction we will loosely follow [MW07]. For literature on chiral perturbationtheory see [Leu01] and [Sch03]. The physics of the standard model including QCD is exposedin [Lan09]. An introduction to effective field theories is given by Pich [Pic97].

11

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12 CHAPTER 2. STRONG INTERACTIONS

bosons represented by the gluon fields GAµ , with field strengths

GAµν = ∂µGAν − ∂νGAµ − gsfABCGBµGCν .

The gauge invariance fixes the structure of the interaction through the gauge-covariant derivative

Dµ = ∂µ + igsGAµT

A.

The gauge group can be parametrized in terms of the conventional generatorsTA, A = 1, . . . , 8 , defining the structure constants fABCTC := [TA, TB ].

Instead of the strong coupling strength gs one often uses the QCD fine-structure constant,

αs :=g2s

4π.

We can extract αs from high energy scattering reactions, where the quarksand gluons are identified with jets of many strongly interacting particles. Thevalue of αs can be determined by matching perturbative calculations and ex-perimental data [Bet07] depending on the renormalization scale µ,

αs(µ) = 0.1189± 0.0010, for µ = MZ0 ≈ 91.2 GeV. (2.2)

The parameter µ may be used to “optimize” the predictions obtained inperturbative calculations involving quantum corrections. Choosing a value of µwhich minimizes the quantum corrections to the observable under investigation,we can also minimize higher-order corrections, that we do not calculate.

The change or running of the coupling constant, which comes with a differentchoice of µ, can be shown to satisfy the equation

αs(µ) =12π

(33− 2Nf ) ln µ2

Λ2QCD

, ΛQCD := µ exp

[− 6π

(33− 2Nf )αs(µ)

]

at one-loop order. This equation has positive and negative implications. On onehand, αs becomes small for large momentum scales, so that the short distancebehaviour is that of a theory of free particles and therefore well-defined. Thisis the paradigm of high-energy physics called asymptotic freedom [Wil05].

On the other hand, for momentum scales on the order of the lightest observedhadron masses, the coupling αs becomes large. As a consequence of this, theperturbative expansion of QCD is no longer suitable to examine the propertiesand interactions of such particles. It is assumed that QCD exhibits confinement,which means that there are no asymptotic states containing quarks or gluons,which are not bound together in colour singlets. Experimentally, the scale fornon-perturbative strong interaction effects is set by ΛQCD ∼ 200 MeV.

Alternative calculation methods have been successfully applied in this regime.Lattice QCD is based on numerically solving approximated versions of the Feyn-man path integrals on supercomputers with space-time discretized [DD06].

The approach we use is chiral perturbation theory. It replaces quarks andgluons by effective hadronic degrees of freedom and determines their interactionsfrom a low-energy expansion of the QCD Green’s functions, constrained by thesymmetry properties of the Lagrangian density of QCD.

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2.2. LOW-ENERGY EFFECTIVE FIELD THEORY 13

2.2 Effective Field Theory for Low Energies:Chiral Perturbation Theory

The masses of the three lightest quarks are small compared to the scale of QCD,

mu < md < ms ΛQCD mc < mb < mt.

To determine the low-energy particle spectrum of QCD, we may in the spiritof effective field theory restrict ourselves to the light quark flavours up, downand strange. Chiral perturbation theory relies on the smallness of their masses,when compared to the scale of strong interactions ΛQCD. Assuming the light-quark masses play only a quantitative role for the low-energy particle spectrum,we may at first set mu = md = ms = 0, and only later include effects due tonon-vanishing quark masses as a perturbation.2

In this chiral limit of QCD the approximate global SU(3)-flavour symmetryof LQCD becomes not only exact, but it is enlarged to a more restrictive chiralglobal symmetry group acting independently left- and right-handed quarks,

qL/R := PL/R q, PL/R :=1

2

(1∓ γ5

).

For an element of the symmetry group (L,R) ∈ SU(3)L × SU(3)R we define itsaction on the quark fields by

qL −→ LqL, qR −→ RqR.

qi /Dq = qLi /DqL + qRi /DqR

−→ q′i /Dq′ = qLL†i /DLqL + qRR

†i /DRqR

= qLL†Li /DqL + qRR

†Ri /DqR = qi /Dq,

so that (LχQCD)′

= LχQCD possesses a chiral symmetry for vanishing quark masses.It is not yet theoretically understood from first principles, why this symmetry

is spontaneously broken by a non-vanishing ground-state expectation value ofthe quark bilinears [Leu01],⟨

Ω∣∣∣qjRqkL∣∣∣Ω⟩ = v δkj , k, j ∈ u, d, s.

where v ∼ Λ3QCD is the order parameter of the symmetry breaking, the chiral

condensate. The diagonal structure can be seen as a consequence of flavoursymmetry in the chiral limit.

Subjecting the quark fields to a chiral transformation, we obtain for thechiral condensate,⟨

Ω∣∣∣qjRqkL∣∣∣Ω⟩ −→ ⟨

Ω∣∣∣q′jRq′kL ∣∣∣Ω⟩ =

⟨Ω∣∣(qRR†)j(LqL)k

∣∣Ω⟩ = v (LR†)kj .

Therefore, the non-perturbative dynamics generating the condensate break downthe chiral symmetry of the theory to its diagonal subgroup

SU(3)V :=

(M,M)∣∣M ∈ SU(3)

⊂ SU(3)L × SU(3)R.

2The same derivation may also be carried out keeping only u and d, when effects due tostrange particles are not important.

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14 CHAPTER 2. STRONG INTERACTIONS

By Goldstone’s theorem, every broken symmetry generator leads to a masslessGoldstone boson. Their dynamics may be described by a unitary matrix fieldU(x) ∈ SU(3). Identifying the flavour indices vU(x)kj ∼ qjR(x)qkL(x) allows usto demand 〈Ω|U |Ω〉 = 1 and the behaviour under chiral transformations,

U −→ LUR†. (2.3)

We will now change our description from fundamental to effective degrees-of-freedom, by assuming, that the dynamics of U(x) may be described in terms ofan effective Lagrangian density L, invariant under the symmetry transformation(2.3). At low energies we may expand L in the number of occurring derivativeterms,

L = L(2) + L(4) + . . . .

Analyzing possible invariant terms, one finds that the most general Lagrangianat lowest order is

L(2) = 14f

2 Tr[DµU(DµU)†],

with f a free constant.To take care of the constraint U ∈ SU(3), one may use the exponential

parametrization, with the meson fields canonically identified as

U = exp

[iϕ

f

], ϕ =

π0 + 1√

√2π+

√2K+

√2π− −π0 + 1√

3η√

2K0

√2K−

√2K0 − 2√

= λAϕA.

Inserting DµU = ∂µU = if λA(∂µϕA) + o(ϕ2

A) and Tr[λAλB ] = 2δAB we find

that L(2) indeed describes the propagation of eight massless particles ϕA,

L(2) =1

2∂µϕA∂

µϕA + . . . .

To conclude this section, we examine the influence of the explicit chiralsymmetry breaking by the quark mass term in LQCD,

qjmjkqk = qjRmjkqkL + h.c., m =

mu 0 00 md 00 0 ms

.

We would like to treat m as a perturbation, so we keep it as an external field.The behaviour of this field under chiral transformations can be fixed by de-manding invariance of the Lagrangian density,

qLmqR + h.c. −→ q′Lm′q′R + h.c. = qLL

†m′RqR + h.c.,

so that m −→ m′ = LmR†. This shows how the external field m may beincluded into the effective theory preserving chiral symmetry,

L(2) = 14f

2 Tr[DµU(DµU)†] + vTr[mU† + Um†].

This leads, amongst others, to mass-terms for the pseudo-Goldstone bosons, e.g.

Tr

[m exp

iϕ1,2λ1,2

f+ h.c.

]= Tr

[m

1−

ϕ21,2

f2λ2

1,2 + . . .

]

= const.− 1

f2(mu +md)ϕ

21,2 + o(ϕ3

1,2),

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2.3. QCD IN THE PRESENCE OF EXTERNAL FIELDS 15

m2π± = m2

ϕ1/2 =2v

f2(mu +md).

In the following we will exclude strangeness and only consider SU(2) chiralperturbation theory. For our purposes of calculating electrodynamic interactionsof the pseudo-Goldstone bosons we will use the special parametrization of SU(2)given by

U(x) =1

f

[σ(x) + i~τ · ~π(x)

], σ(x) =

√f2 − ~π2(x),

where the 3-vector ~π = (π1, π2, π3) represents the pseudo-Goldstone bosons inthe cartesian isospin basis given by the Pauli matrices ~τ = (τ1, τ2, τ3). Thisσ-gauge is very convenient, as it eliminates all LO electromagnetic interactionsother than the scalar QED vertices known from chapter 1.

For a more rigorous derivation of the low-energy equivalence of QCD andchiral perturbation theory relying on Ward identities see [Leu94].

2.3 QCD in the Presence of External Fields

The inclusion of electromagnetic interactions in SU(2) chiral perturbation theorycan be accomplished by an analysis similar to the inclusion of the mass terms.The vector potential Aµ couples to the up and down quarks by means of thegauge-covariant derivative,

qγµDµq = qγµ [∂µ + ieqAµ] q.

The charge matrix of the quarks eq can be written as

eq = e

(23 00 − 1

3

)= e

(1

6+

1

2τ3

),

so that we have generated further terms in LQCD with well-defined transforma-tion properties under chiral symmetry.

A direct analysis of the covariant derivative [Sch03, pp. 46–54 and pp. 134–136] shows, that the vector potential may be incorporated into SU(2) chiralperturbation theory at leading order by the gauge-covariant derivative

Dµ = ∂µ +ie

2Aµ [τ3, · ] . (2.4)

A heuristic argument can be given by considering that[τ3 ,

(π0

√2π+

√2π− −π0

)]=

(0 2

√2π+

−2√

2π− 0

),

so that e/2 [τ3, · ] is just the operator assigning to the pions the correct chargeeigenvalues and (2.4) amounts to the standard prescription of QED.

We will now briefly go back to showing that the σ-gauge is indeed convenientfor calculations involving electrodynamics, as there occur no photon-pion ver-tices with more than two photons or pions. We obtain for the gauge-covariantderivative using the σ-gauge,

U(x) =1

f

[σ(x) + i~τ · ~π(x)

], σ(x) =

√f2 − ~π2(x),

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16 CHAPTER 2. STRONG INTERACTIONS

DµU = ∂µU −ie

2fAµ[τ3, U ] = ∂µU +

e

fAµτaεa3bπb,

∂µU = − 1

Fσπa∂µπa +

i

fτa∂µπa =

1

f

(−πaσ

+ iτa

)∂µπa.

Inserting this into the Lagrangian density we obtain the interaction parts,

L(2) = 14f

2 Tr[DµU(DµU)†] + vTr[mU† + Um†],

Lπ,γI = Lγππ + Lγγππ.

Lγππ =e

4Tr[(−πaσ

+ iτa

)(∂µπa)Aµτcεc3bπb −Aµτcεc3bπb

(−πaσ− iτa

)(∂µπ

a)]

=ie

4Tr [τa, τc] (∂µπa)Aµεc3bπb

= ieεa3bAµπb∂µπa

Lγγππ =e2

4Tr [τaεa3bπbτcεc3dπd]AµA

µ =e2

2πbπdεa3bδacεc3dAµA

µ

=e2

2(δbd − δb3δd3)πbπdAµA

µ

The Feynman rules can be determined from all possible contractions in

〈πb(p′)|−iLγππ|πa(p)γ(k, µ)〉 and 〈πb(p′)|−iLγγππ|πa(p)γ(k1, µ)γ(k2, ν)〉 ,

resulting in

eεa3b(p+ p′)µ, 2ie2(δab − δa3δb3) gµν .

We have thus explicitly confirmed the non-occurrence of other πγ vertices inthe σ-gauge. Translating the cartesian pion fields into the charge basis, we findthat the leading-order description in the σ-gauge is equivalent to scalar quantumelectrodynamics.

The equivalence of on-shell matrix-elements for different realizations of theGoldstone-boson fields is established by a theorem [KOS61] [Chi61].

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Chapter 3

Pion-Structure Effects:Chiral Perturbation Theoryat Order p4

3.1 Contact Terms

The divergences generated by pion loops have to be absorbed into counter-terms, which are introduced at each order in chiral perturbation theory. For thecounter terms at order p4, we use the Gasser-Leutwyler Lagrangian density forSU(2) chiral perturbation theory [GL84]. It can be written as [Sch03, p. 271]

LGL4 = 1

4 l1

Tr[DµU(DµU)†]2

+ 14 l2Tr[DµU(DνU)†]Tr[DµU(DνU)†]

+ 116 l3

Tr(χU† + Uχ†)

2+ 1

4 l4Tr[DµU(Dµχ)† +Dµχ(DµU)†]

+ l5

Tr[fRµνUfµνL U†]− 1

2Tr[fLµνfµνL + fRµνf

µνR ]

+ i2 l6Tr[fRµνD

µU(DνU)† + fLµν(DµU)†DνU ]− 116 l7

Tr(χU† − Uχ†)

2

+ 14 (h1 + h3)Tr[χχ†]− 2h2Tr(fLµνf

µνL + fRµνf

µνR )

+ 116 (h1 − h3)

Tr[χU† + Uχ†]2 +Tr[χU† − Uχ†]2

−2Tr[χU†χU† + Uχ†Uχ†].

The tree-level contributions of L(4) to the scattering of two on-shell photonscan be conveniently summarized by the polarizability vertex [GIS06],

k1, µ

k2, ν 8πiβπmπ(k1 · k2 gµν − kµ2 kν1 ),

βπ = − α(l6 − l5)

48π2f2mπ.

17

Page 18: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

18 CHAPTER 3. CHIRAL PERTURBATION THEORY AT ORDER P 4

Using the current value extracted from radiative pion decays [BT97] [GHW04]

l6 − l5 = 3.0± 0.3,

we obtain the numerical value of the polarizability at next-to-leading order

βπ ≈ −3.0 · 10−4fm3.

3.2 Pion Propagator at Order p4

At the order p4 there is only one pion-loop correction to the pion propagatorgiven by the diagram

pµ, a pµ, b

.

In this entire chapter, we work in the cartesian isospin basis. Using the Feynmanrules of appendix A, the amputated contribution from this diagram, includingthe symmetry factor 1/2, is given by

1

2

i

f2

∫d4l

(2π)4

iδcdMabcd

l2 −m2 + iε,

Mabcd = δabδcd((p− p)2 −m2) + δacδbd((p+ l)2 −m2) + δadδbc((p− l)2 −m2).

Calculating the isospin contractions, we find

δcdMabcd = −3m2δab + ((p+ l)2 −m2)δab + ((p− l)2 −m2)δab

= δab((p+ l)2 + (p− l)2 − 5m2

),

− δab

2f2

∫d4l

(2π)4

(p+ l)2 + (p− l)2 − 5m2

l2 −m2 + iε.

And we have

(p+ l)2 + (p− l)2 − 5m2 = 2p2 + 2l2 − 5m2

= 2(l2 −m2 + iε) + 2p2 − 3m2 − 2iε.

Assuming only linearity of the integral, we get the regularization-scheme inde-pendent intermediate result for the pion-loop contribution Σ1 to the self-energyat next-to-leading order,

−iΣ1 = − 1

2f2

[2

(∫d4l

(2π)4

)+ (2p2 − 3m2 − 2iε)

∫d4l

(2π)4

1

l2 −m2 + iε

].

At this point some regularization method has to be applied to make sense of thedivergent integrals. We use the framework of dimensional regularization1 withthe space-time dimension set to d := 4− 2δ,

d4l

(2π)4−→ ddl

(2π)d,

1see appendix B

Page 19: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

3.3. RENORMALIZATION OF THE SELF-ENERGY 19

we have further

RegD

∫ddl

(2π)d= 0,

RegD

∫ddl

(2π)d1

l2 −m2=−im2

16π2Γ(δ − 1)

(4πµ2

m2

)δ=−im2

16π2

(−1

δ− 1 + Γ′(0) + o(δ)

)(1 + δ ln

(4πµ2

m2

)+ o(δ2)

)=

im2

16π2

(1

δ+ 1− γE + ln

(4πµ2

m2

)+ o(δ)

).

Defining ξUV = 1δ − γE + ln(4πµ2/m2), we get

RegD

∫ddl

(2π)d1

l2 −m2=

im2

16π2(ξUV + 1 + o(δ)) .

The final result is

Σ1 =m2

32π2f2(2p2 − 3m2) (ξUV + 1) ,

where the terms linear in iε and δ have been dropped.

3.3 Renormalization of the Self-Energy

The full propagator can be expanded as a geometric series in one-particle irre-ducible diagrams, which can be explicitely summed to give

G(p) =i

p2 −m2 − Σ(p2) + iε.

The self energy up to order p2 is renormalized by the counter-terms l3 and l4,which introduce shifts in the bare mass and the pion decay constant,

fπ = f

[1 +

m2

f2

(lr4 −

1

16π2lnm2

µ2

)],

m2π = m2

[1 +

m2

f2

(2lr3 +

1

32π2lnm2

µ2

)].

The infinite part of the counter-terms l3 is determined by the renormalizationconditions on the propagator G(p),

(I) G(p) has a pole at the physical pion mass mπ,m2 + Σ(m2) = 0

(II) the residue at this pole is the wave-function renormalization factor,Zπ = 1 + Σ′(m2).

The pion propagator then takes the final form [BKM95]

iZπq2 −m2

π

.

Page 20: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

20 CHAPTER 3. CHIRAL PERTURBATION THEORY AT ORDER P 4

The mass shift is taken into account by using the physical pion mass for calcu-lations. The Z factor is

Zπ = 1 +m2

f2

[1

16π2(ξUV + 1)− 2l4

].

There are also contributions of the l4 counter-term to the γππ and γγππvertex functions, which are cancelled by the Z factor. For brevity we will omitdiscussing those contributions in the following, using the Z factor without thel4 contribution.

3.4 Pion-Photon Coupling

Contributing to the pion-pion-photon amputated Green’s function at order p4,we have only one pion-loop diagram, given by

(l + k), d

l, c

k, µ

p, a

p′, b

.

It includes a symmetry factor 1/2 due to the symmetry under the exchange of thetwo loop propagators. The corresponding amplitude of the vector-pseudoscalar-pseudoscalar correlation function reads in the cartesian isospin basis

V PPµNLO =i i2

2f2e εc3d

∫ddl

(2π)d(2l + k)µMabcd

[l2 −m2 + iε] [(l + k)2 −m2 + iε],

Mabcd = δabδcd(k2 −m2) + δacδbd((p− l)2 −m2)

+ δadδbc((p+ l + k)2 −m2).

We calculate the overall isospin structure of this amplitude as

εc3dMabcd = εc3d(((((((((δabδcd(k

2 −m2) + δacδbd((p− l)2 −m2)

+ δadδbc((p+ l + k)2 −m2))

= εa3b

[(p− l)2 −m2 −

((p+ l + k)2 −m2

)]= −εa3b

[(4p+ 2k) · l + 2k · p+ k2

],

where we used δcdεc3d = 0, εb3a = −εa3b. The pion-loop correction to the pion-photon vertex has an isospin structure identical to the leading order amplitudeof e εa3b(p+ p′)µ.

V PPµNLO =i

2f2e εa3b

∫d4l

(2π)4

(2l + k)µ((4p+ 2k) · l + 2k · p+ k2

)[l2 −m2 + iε] [(l + k)2 −m2 + iε]

=:i

2f2eεa3bT

µ

Page 21: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

3.4. PION-PHOTON COUPLING 21

Whether the Lorentz-structure of Tµ also possesses this property can be deter-mined by carrying out the loop integration.

3.4.1 Calculation using Feynman-Parameters

We employ the Feynman-formula for two factors2,

1

AB=

∫ 1

0

dx1

(xA+ (1− x)B)2=

∫ 1

0

dx1

(B + (A−B)x)2.

Choosing B = l2−m2 +iε, A = (l+k)2−m2 +iε, we see that B+(A−B)x 6= 0.This assures that the integrals exist. Now we can recombine the propagatorsin isotopic form by exchanging the order of integration and applying the linearsubstitution lµ → lµ − xkµ,

Tµ =

∫d4l

(2π)4

∫ 1

0

dx(2l + k)µ

((4p+ 2k) · l + 2k · p+ k2

)(l2 −m2 + iε+ x (2l · k + k2))

2

=

∫ 1

0

dx

∫d4l

(2π)4

(2l + (1− 2x)k)µ((4p+ 2k) · (l − xk) + 2k · p+ k2

)(l2 −m2 + iε+ x(1− x) k2)

2 .

With the denominator in this form we can apply dimensional regularization3,

RegD

∫ddl

(2π)d1

(l2 − a)2=

i

16π2

(ξUV − ln

a

m2

),

RegD

∫ddl

(2π)dlµ

(l2 − a)n= 0,

RegD

∫ddl

(2π)dlµlν

(l2 − a)2=

iagµν

32π2

(ξUV + 1− ln

a

m2

).

To simplify the calculation we consider immediately the on-shell limit k2 → 0

Tµ =

∫ 1

0

dx

∫d4l

(2π)4

(2l + (1− 2x)k)µ ((4p+ 2k) · l + 2(1− 2x)k · p)(l2 −m2 + iε)

2

=

∫ 1

0

dx

(∫d4l

(2π)4

lµlν(8p+ 4k)ν

(l2 −m2 + iε)2 + kµ

∫d4l

(2π)4

2(1− 2x)2k · p(l2 −m2 + iε)

2

).

After dimensional regularization we are left with the Feynman-parameter inte-grals ∫ 1

0

dx (1− 2x)2 = 1/3,

∫ 1

0

dx = 1.

These yield the on-shell-photon result for the pion-loop correction to the photon-pion-pion interaction

Tµ = 4(2p+ k)ν

[im2gµν

32π2(ξUV + 1)

]+

2

3k · p kµ

[i

16π2ξUV

],

Tµ = (4p+ 2k)µ A0(m2) +2k · p3m2

kµ A0(m2)− i

24π2k · p kµ.

2see appendix B3see appendix B

Page 22: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

22 CHAPTER 3. CHIRAL PERTURBATION THEORY AT ORDER P 4

3.4.2 Calculation by Tensor Reduction

This calculation can be performed in a different way. This derivation is alsoshown here, because it illustrates the idea of Veltman-Passarino tensor reduction[PV79]. Using this method, it is possible to reduce tensor (one-)loop-functionsto linear combinations of scalar one-loop integrals by solving a system of linearequations.

To carry out the decomposition, we first split Tµ into tensor components V µ

and Wµν , which only depend on the photon momentum kµ,

Tµ =

∫d4l

(2π)4

(2l + k)µ(2k · p+ k2)

[l2 −m2 + iε] [(l + k)2 −m2 + iε]

+ (4p+ 2k)ν

∫d4l

(2π)4

(2l + k)µ lν

[l2 −m2 + iε] [(l + k)2 −m2 + iε]

= V µ + (4p+ 2k)νWµν .

The key observation is here, that Lorentz-symmetry permits to write

V µ = kµV, Wµν = kµkνW1 + gµνW2.

Assuming that the photon momentum is off-shell, we get

V =1

k2kµT

µ,

V =1

k2

∫d4l

(2π)4

[k · (2l + k)][2k · p+ k2

][l2 −m2 + iε] [(l + k)2 −m2 + iε]

,

k · (2l + k) = (l + k)2 −m2 − (l2 −m2),

V =1

k2

(∫d4l

(2π)4

2k · p+ k2

[l2 −m2 + iε]−∫

d4l

(2π)4

2k · p+ k2

[(l + k)2 −m2 + iε]

)= 0.

The last equality follows from translation invariance l → l − k of the secondintegral. Terms linear in ε will generally be dropped, because there occur no1/ε singularities in scalar loop integrals. Applying a continuity argument4 wecan also extend this to on-shell photon momenta

V µ = 0.

Similarily we can calculate Wµν

Wµν = W1kµkν +W2 g

µν .

Now we have to solve a linear system of two equations. The coefficients of thissystem can be determined by contracting with kµkν and gµν ,

kµkνWµν = W1k

4 +W2k2,

gµνWµν = W1k

2 + dW2. (3.1)

Here we have used gµνgµν = d := 4 − 2δ for the dimension of space-time, asrequired by dimensional regularization.

4If in doubt, an explicit check using a Feynman-parametrization can be used as confirma-tion.

Page 23: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

3.4. PION-PHOTON COUPLING 23

The linear system on the right-hand side of equation 3.1 is invertible fork2 6= 0 and we can obtain the solution for small δ,

W1 =12 + 2δ

9k4kµkνW

µν − 3 + 2δ

9k2gµνW

µν ,

W2 = −3 + 2δ

9k2kµkνW

µν +3 + 2δ

9gµνW

µν . (3.2)

It seems as if not much has been gained by this intermediate result, as wehave only rewritten our amplitude in terms of contractions. We will see now,that the contractions can be simplified considerably in a simple way, so that nomore tensors occur in the numerators.

gµνWµν =

∫d4l

(2π)4

(2l + k) · l[l2 −m2 + iε] [(l + k)2 −m2 + iε]

(2l + k) · l =3

2(l2 −m2) +

1

2((l + k)2 −m2) + 2m2 − 1

2k2

gµνWµν =

3

2

∫d4l

(2π)4

1

(l + k)2 −m2 + iε+

1

2

∫d4l

(2π)4

1

l2 −m2 + iε

+

(2m2 − 1

2k2

)∫d4l

(2π)4

1

[l2 −m2 + iε] [(l + k)2 −m2 + iε]

After shifting the loop momentum l + k → l in the first integral we obtain interms of the Veltman-Passarino Loop functions5

A0(m2) :=

∫d4l

(2π)4

1

l2 −m2 + iε,

B0(k2,m21,m

22) :=

∫d4l

(2π)4

1

[l2 −m21 + iε] [(l + k)2 −m2

2 + iε].

The result, written in terms of scalar one-loop integrals is

gµνWµν = 2 A0(m2) +

(2m2 − k2

2

)B0(k2;m2,m2).

Now we consider the second contraction:

kµkνWµν =

∫d4l

(2π)4

(2l · k + k2) (l · k)

[l2 −m2 + iε] [(l + k)2 −m2 + iε],

2l · k + k2 = (l + k)2 − l2 =((l + k)2 −m2

)− (l2 −m2),

kµkνWµν =

∫d4l

(2π)4

l · kl2 −m2 + iε

−∫

d4l

(2π)4

l · k(l + k)2 −m2 + iε

= −∫

d4l

(2π)4

(l − k) · kl2 −m2 + iε

= k2 A0(m2),

5Note that our normalization has been choosen different for notational convenience:

A0 =i

16π2A0, . . . (as compared to the original work [PV79])

Page 24: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

24 CHAPTER 3. CHIRAL PERTURBATION THEORY AT ORDER P 4

where we have again shifted l −→ l − k.

Together we havekµkνW

µν = k2 A0(m2),

gµνWµν = 2 A0(m2) +

(2m2 − k2

2

)B0(k2;m2,m2),

W1 =12 + 2δ

9k2A0(m2)− 3 + 2δ

9k2

[2 A0(m2) +

(2m2 − k2

2

)B0(k2;m2,m2)

],

W2 = −3 + 2δ

9A0(m2) +

3 + 2δ

9

[2 A0(m2) +

(2m2 − k2

2

)B0(k2;m2,m2)

].

Performing the limit δ → 0 at the level of the coefficient functions will generateadditional terms due to the divergences in the loop functions A0 and B0.

12 + 2δ

9k2A0(m2) =

12 + 2δ

9k2

[im2

16π2(ξUV + 1)

]

ξUV =1

δ− γe + ln

4πµ2

m2

12 + 2δ

9k2A0(m2) =

4

3k2A0(m2) +

2im2

144π2k2

Proceeding similarly for B0, we obtain the result with the limit δ → 0 shiftedto be implicitly included in the loop-functions

W1 =2

3k2A0(m2) +

(1

6− 2m2

3k2

)B0(k2;m2,m2) +

i

144π2k2(k2 − 6m2),

W2 =1

3A0(m2) +

1

6(4m2 − k2) B0(k2;m2,m2)− i

144π2(k2 − 6m2),

Wµν = W1kµkν +W2 g

µν .

Finally we arrived at a form-factor representation of Tµ,

Tµ = (4p+ 2k)νWµν = [(4p+ 2k) · k W1 + 2W2] kµ + 4W2p

µ.

Due to 1/k2 terms it is not obvious that this expression is non-singular in thelimit of on-shell photons. Inserting the asymtotic form of B0 for small k2 (seeappendix C)

m2 B0(k2;m2,m2) = A0(m2)− im2

16π2+

ik2

96π2+ o(k4),

we obtain for the limit k2 → 0,

Tµ = (4p+ 2k)µ A0(m2) +2k · p3m2

kµ A0(m2)− i

24π2k · p kµ,

V PPµNLO =i

2f2eεa3bT

µ. (3.3)

Page 25: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

3.4. PION-PHOTON COUPLING 25

This agrees with the result of the Feynman parametrization. The above pro-cedure is sufficiently general to work with arbitrary one-loop tensor integrals.Its advantage is, that only the scalar loop functions have to be carefully exam-ined for integrand cuts in the ε −→ 0 limit and the tensor integral is written asa linear combination of scalar loop functions.

If the integrand is required for special values of the momenta which result in asingular determinant of the coefficient matrix of (3.1), the computation of such alimit might be more laborious, when compared with a Feynman parametrization.

3.4.3 Renormalized γππ Vertex Function

To determine whether the pion-loop correction influences the scattering of on-shell photons, we calculate the renormalized 1-PI amputated Green’s Function.The field renormalization was given by

π = Z−1/2π πR, Zπ = 1 +

m2

16π2f2(ξUV + 1) = 1 + A0(m2).

We omitted the cancelling term containing l4. We can now calculate the renor-malized γππ vertex function,

V PPµR = ZπV PPµ = Zπ(V PPµLO + V PPµNLO)

= V PPµLO + V PPµNLO + (Zπ − 1)V PPµLO + higher order terms

= V PPµLO + V PPµNLO + A0(m2)V PPµLO + higher order terms.

This is equivalent to the diagrammatic representation

V PPµR = + +

+ + .

Using the result from the last section, we get

V PPµR = eεa3b(p+ p′)µ − eεa3bk · p48π2f2

ξUV kµ.

While this still contains infinities, observable on-shell amplitudes only involvecontractions of the amplitude tensors with polarization vectors which satisfy

ε(kµ, λ = ±1) · k = 0.

Therefore all observable amplitudes are already free of divergences.To obtain the full Green’s function, it is necessary to also consider the tree-

level contributions involving the NLO Lagrangian L(4). They may be calculatedas [Sch03, p.137],

el6εa3b

f2

(k2ε · (p′ + p) + ε · k(p2 − p′2)

).

Page 26: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

26 CHAPTER 3. CHIRAL PERTURBATION THEORY AT ORDER P 4

This vanishes in the case of an on-shell photon k2 = 0, ε · k = 0. The infinitepart of l6 can be used to absorb the divergence ∼ kµ for off-shell photons.

The vanishing of the NLO corrections to the scattering of pions with oneon-shell photon indicates that the relevant terms have to be sought in the pion-diphoton correlator.

3.5 Pion-Photon-Photon Coupling

The pion-loop contributions to the VVPP 1-PI Green’s function are given bytwo diagrams

V V PPµνNLO = + + .

3.5.1 Two-Propagator Contribution

Consider the first diagram,

l + k1 + k2, d

l, c

k2,ν

k1,µ

p, a

p′, b

.

Noting the symmetry factor 1/2, it corresponds to the expression

i2i

2f2

∫d4l

(2π)4

2ie2gµνP±cdM(1)abcd

[l2 −m2 + iε][(l + k1 + k2)2 −m2 + iε],

M(1)abcd = δabδcd((k1+k2)2−m2)+δacδbd((p−l)2−m2)+δadδbc((p+l+k1+k2)2−m2).

We simplify the corresponding expression by calculating the isospin contrac-tions, using the abbreviations P±ab := δab − δa3δb3 for the projector onto thecharged states and k12 := k1 + k2.

P±cdM(1)abcd = P±ab

[(p− l)2 + (p+ l + k12)2 + 2k2

12 − 4m2]− 2δa3δb3

[k2

12 −m2]

= P±ab[l2 −m2 + (l + k12)2 −m2 + p2 + (p+ k12)2 + k2

12 − 2m2]

− 2δa3δb3[k2

12 −m2]

The factor due to interactions can therefore be seperated into two isospin parts,responsible for charged states (a, b ∈ 1, 2) and neutral states (a = b = 3)respectively. It was rewritten to eliminate their tensorial structure and the

Page 27: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

3.5. PION-PHOTON-PHOTON COUPLING 27

corresponding amplitude can be immediately expressed in terms of scalar loopfunctions. For charged pions we obtain

e2

f2gµν

∫d4l

(2π)4

l2 −m2 + (l + k12)2 −m2 + p2 + (p+ k12)2 + k212 − 2m2

[l2 −m2 + iε][(l + k12)2 −m2 + iε]

=e2

f2gµν

[∫d4l

(2π)4

1

(l + k12)2 −m2 + iε+

∫d4l

(2π)4

1

l2 −m2 + iε

+

∫d4l

(2π)4

p2 + (p+ k12)2 + k212 − 2m2

[l2 −m2 + iε][(l + k12)2 −m2 + iε]

]=e2

f2gµν

[2 A0(m2) + (p2 + (p+ k12)2 + k2

12 − 2m2) B0(k212;m2,m2)

].

For completeness we also give the term relevant for neutral pion scattering,

−2e2

f2gµνδa3δb3

∫d4l

(2π)4

k212 −m2

[l2 −m2 + iε][(l + k12)2 −m2 + iε]

=2e2

f2gµνδa3δb3(m2 − k2

12) B0(k212;m2,m2),

where δa3δb3 projects onto the neutral pion states, a = b = 3.

This is an interesting feature: a neutral particle can interact electromagneti-cally by quantum-correction terms. In the underlying theory of QCD this is ob-vious, because neutral pions are composite particles with charged constituents.The situation is analogous to neutral atoms, which possess non-vanishing highermultipole moments.

3.5.2 Three-Propagator Contribution

The second diagram involves three pion propagators and has a symmetry factorof unity.

l + k2, e

l, d

l − k1, c

k2,ν

k1,µ

p, a

p′, b

i3i

f2

∫d4l

(2π)4

eεc3d(2l − k1)µeεd3e(2l + k2)νM(2)abce

[(l − k1)2 −m2 + iε][l2 −m2 + iε][(l + k2)2 −m2 + iε],

where we write again for the four-pion interaction

M(2)abce = δabδce(k

212−m2)+δacδbe((p− l+k1)2−m2)+δaeδbc((p+ l+k2)2−m2).

Page 28: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

28 CHAPTER 3. CHIRAL PERTURBATION THEORY AT ORDER P 4

Note that we have εc3dεd3e = −P±ce, P±ceP±ce = P±ceδce = 2. We split the isospin

contraction into parts responsible for charged and neutral pions,

− P±ceM(2)abce

= −P±ab[(p+ l + k2)2 + (p− l + k1)2 + 2k2

12 − 4m2]− 2δa3δb3

[k2

12 −m2]

= −P±ab[(l − k1)2 −m2 + (l + k2)2 −m2 + 2p2 + 2p · k12 + 2k2

12 − 2m2]

− 2δa3δb3[k2

12 −m2].

Considering only charged external pions (a, b ∈ 1, 2, so P±ab = 1) we are leftwith

− e2

f2

∫d4l

(2π)4

(2l − k1)µ(2l + k2)ν

[l2 −m2 + iε][(l + k2)2 −m2 + iε]

+

∫d4l

(2π)4

(2l − k1)µ(2l + k2)ν

[(l − k1)2 −m2 + iε][l2 −m2 + iε]

+

∫d4l

(2π)4

(2p2 + 2p · k12 + 2k2

12 − 2m2)

(2l − k1)µ(2l + k2)ν

[(l − k1)2 −m2 + iε][l2 −m2 + iε][(l + k2)2 −m2 + iε]

.

(3.4)

A tensor reduction would involve more possible tensors for this diagram. There-fore it would be more tedious as compared to the example of the correction tothe photon-pion-pion correlator. The available tensors are given by

gµν , kµ1 kν2 , kµ2 k

ν1 , kµ1 k

ν1 , kµ2 k

ν2 .

If we are only interested in contractions with polarization vectors ε1 and ε2, onlythe first two tensors in this list are relevant. The coefficients of the last two arerelated by the symmetry of the amplitude under (l, k1, µ)↔ (−l, k2, ν).

For simplicity we will employ a Feynman-parametrization. Consider nowthe first term,∫

d4l

(2π)4

(2l − k1)µ(2l + k2)ν

[l2 −m2 + iε][(l + k2)2 −m2 + iε]

=

∫ 1

0

dx

∫d4l

(2π)4

(2l − k1)µ(2l + k2)ν

[l2 −m2 + iε+ x(2l · k2 + 2x2k22)]2

=

∫ 1

0

dx

∫d4l

(2π)4

(2l − k1 − 2xk2)µ(2l + (1− 2x)k2)ν

[l2 −m2 + iε+ xk22]2

,

where we shift the integral l → l − xk to eliminate the inhomogeneity in thedenominator. Performing the on-shell photon limit k2

1/2 → 0, we can carry outthe Feynman-parameter integral∫ 1

0

dx (2l − k1 − 2xk2)µ(2l + (1− 2x)k2)ν =1

3kµ2 k

ν2 − 2kµ12l

ν + 4lµlν .

After dimensional regularization6

RegD

∫ddl

(2π)d1

(l2 −m2)2=

i

16π2ξUV =

1

m2A0(m2)− i

16π2,

6see appendix B.

Page 29: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

3.5. PION-PHOTON-PHOTON COUPLING 29

RegD

∫ddl

(2π)dlµ

(l2 −m2)n= 0,

RegD

∫ddl

(2π)dlµlν

(l2 −m2)2=

im2gµν

32π2(ξUV + 1) =

1

2gµν A0(m2),

we are left with∫d4l

(2π)4

(2l − k1)µ(2l + k2)ν

[l2 −m2 + iε][(l + k2)2 −m2 + iε]

=1

3m2kµ2 k

ν2

(A0(m2)− im2

16π2

)+ 2gµν A0(m2).

The second term results, if we exchange k1 ↔ −k2, µ↔ ν,∫d4l

(2π)4

(2l − k1)µ(2l + k2)ν

[(l − k1)2 −m2 + iε][l2 −m2 + iε]

=1

3m2kµ1 k

ν1

(A0(m2)− im2

16π2

)+ 2gµν A0(m2).

Now only the third integral remains to be worked out,∫d4l

(2π)4

(2l − k1)µ(2l + k2)ν

[(l − k1)2 −m2 + iε][l2 −m2 + iε][(l + k2)2 −m2 + iε].

In this case we need a Feynman-formula for 3 factors,

1

ABC= 2

∫ 1

0

dx

∫ 1−x

0

dy1

[xA+ yB + (1− x− y)C]3

= 2

∫ 1

0

dx

∫ 1−x

0

dy1

[C + x(A− C) + y(B − C)]3,

C = l2 −m2 + iε, A = (l − k1)2 −m2 + iε, B = (l + k2)2 −m2 + iε.

It has to be assured that we do not integrate over a pole in the denominator.This is the case, as we have ImC = ε 6= 0, Im(A− C) = Im(B − C) = 0. Thisallows us to rewrite the second term as∫

d4l

(2π)4

∫ 1

0

dx

∫ 1−x

0

dy2(2l − k1)µ(2l + k2)ν

[l2 −m2 + iε+ x(k21 − 2l · k1) + y(k2

2 + 2l · k2)]3 .

The denominator is homogeneous after the transformation l→ l + xk1 − yk2,∫ 1

0

dx

∫ 1−x

0

dy

∫d4l

(2π)4

2(2l − (1− 2x)k1 − 2yk2)µ(2l + xk1 + (1− 2y)k2)ν

[l2 −m2 + iε+ 2xy(k1 · k2)− x(x− 1)k21 − y(y − 1)k2

2]3 .

We take the limit of on-shell photons k21/2 → 0 and introduce the dimensionless

variable 2k1 · k2 =: sm2,∫ 1

0

dx

∫ 1−x

0

dy

∫d4l

(2π)4

2(2l − (1− 2x)k1 − 2yk2)µ(2l + xk1 + (1− 2y)k2)ν

[l2 −m2 + iε+ xysm2]3 .

Page 30: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

30 CHAPTER 3. CHIRAL PERTURBATION THEORY AT ORDER P 4

Dimensional regularization gives

RegD

∫ddl

(2π)d1

(l2 − a)3= − i

32π2

1

a,

RegD

∫ddl

(2π)dlµ

(l2 − a)3= 0,

RegD

∫ddl

(2π)dlµlν

(l2 − a)3=

igµν

64π2

(ξUV − ln

a

m2

).

We are left with the two contributions

8igµν

64π2

∫ 1

0

dx

∫ 1−x

0

dy [ξUV − ln(1− xys− iε)]

=igµν

16π2ξUV −

igµν

8π2

∫ 1

0

dx

∫ 1−x

0

dy ln(1− xys− iε),

2i

32π2

∫ 1

0

dx

∫ 1−x

0

dy((1− 2x)k1 + 2yk2)µ(xk1 + (1− 2y)k2)ν

1− xys− iε.

It remains to calculate the two finite loop-functions,

L1(s) :=

∫ 1

0

dx

∫ 1−x

0

dy ln(1− xys− iε),

Lµν2 :=

∫ 1

0

dx

∫ 1−x

0

dy((1− 2x)k1 + 2yk2)µ(xk1 + (1− 2y)k2)ν

1− xys.

We calculate the loop function for the case s < 0, where we do not need toworry about the branch cut of the logarithm. In the end we can analyticallycontinue our result to the relevant case s > 4, if needed letting s → s + iε torestore the imaginary part.∫ y

0

dy ln(1− xys) =1− xys−xs

ln(1− xys)− y

∫ 1

0

dx1− x(1− x)s

−xsln [1− x(1− x)s]− (1− x)

= −√

4− s4√−s

ln

√4− s−

√−s√

4− s+√−s

+

√4− s

2√−s

tanh−1

√−s

4− s

+1

sLi2

[2s

s+√−s(4− s)

]+

1

sLi2

[2s

s−√−s(4− s)

]− 3

2

This can be simplified using the dilogarithm identity

Li2(x) + Li2

(x

x− 1

)= −1

2ln2(1− x), (for x < 1).

Choosing x =2s

s−√−s(4− s)

< 1,x

x− 1=

2s

s+√−s(4− s)

, we get

Page 31: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

3.5. PION-PHOTON-PHOTON COUPLING 31

1

sLi2

[2s

s+√−s(4− s)

]+

1

sLi2

[2s

s−√−s(4− s)

]= − 1

2sln2 (√

4− s−√−s)2

4.

We observe further, that

tanh−1

√−s

4− s=

1

2ln

√4− s+

√−s√

4− s−√−s

= ln

√4− s+

√−s

2,

and we get the simplified result for s < 0∫ 1

0

dx1− xyssx

ln(1− xys)− y = − 1

2sln2 (√

4− s−√−s)2

4− 3

2

+

√4− s√−s

ln

√4− s+

√−s

2.

In this representation it can be analytically continued7 to s > 4,

L1(s) = − 1

2s

[ln

(√s+√s− 4)2

4− iπ

2

]2

− 3

2+

√s− 4√s

[ln

√s− 4 +

√s

2+

2

].

Now we turn to the second loop function. Again for s < 0,

Lµν2 :=

∫ 1

0

dx

∫ 1−x

0

dy((1− 2x)k1 + 2yk2)µ(xk1 + (1− 2y)k2)ν

1− xys

= −4kµ2 kν1 + (4 + s)kµ1 k

ν2

s2

[1

2ln2 (√

4− s−√−s)2

4+s

2

]+ kµ1 k

ν2

[s− 16

2s− 8√

4− s√−s3

ln

√4− s−

√−s

2

].

And by analytic continuation to s > 4,

Lµν2 = −4kµ2 kν1 + (4 + s)kµ1 k

ν2

s2

[1

2

(ln

(√s+√s− 4)2

4− iπ

2

)2

+s

2

]

+ kµ1 kν2

[s− 16

2s− 8√s− 4√s3

(ln

√s−√s− 4

2− iπ

2

) ].

We arrived now at the solution of the third term of (3.4),∫d4l

(2π)4

(2p2 + 2p · k12 + 2k2

12 − 2m2)

(2l − k1)µ(2l + k2)ν

[(l − k1)2 −m2 + iε][l2 −m2 + iε][(l + k2)2 −m2 + iε]

=i k2

0

16π2

(gµνξUV − 2gµνL1(s) + Lµν2 (s)

),

where we defined k20 := 2p2 + 2p · k12 + 2k2

12 − 2m2.We collect the final result

V V PPµν(loop 2) = − e2

f2P±ab

1

3m2(kµ2 k

ν2 + kµ1 k

ν1 )

(A0(m2)− im2

16π2

)+ 4gµν A0(m2)

− k2

0e2

f2P±ab

igµν

16π2ξUV −

igµν

8π2L1(s) +

i

16π2Lµν2

,

7see also appendix C.

Page 32: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

32 CHAPTER 3. CHIRAL PERTURBATION THEORY AT ORDER P 4

where

A0(m2) =im2

16π2(ξUV + 1) ,

k20 := 2p2 + 2p · k12 + 2k2

12 − 2m2 = k212 + p2 + p′ 2 − 2m2.

For pions and photons both on-shell, we have k20 = sm2. For the purpose of

renormalization we rewrite in terms of ξUV,

V V PPµν(loop 2) = − ie2

16π2f2P±ab

1

3(kµ2 k

ν2 + kµ1 k

ν1 ) ξUV + 4m2gµν(ξUV + 1)

+ k20 [gµν ξUV − 2gµνL1(s) + Lµν2 ]

.

Note that we have also calculated the necessary ingredients for externalneutral pions,

−2e2

f2δa3δb3

∫d4l

(2π)4

(k2

12 −m2)

(2l − k1)µ(2l + k2)ν

[(l − k1)2 −m2 + iε][l2 −m2 + iε][(l + k2)2 −m2 + iε]

= − ie2δa3δb38π2f2

(k2

12 −m2) (gµνξUV − 2gµνL1(s) + Lµν2 (s)

).

3.5.3 Renormalized γγππ Vertex Functions

We summarize the results. For neutral pions there occurs no wave functionrenormalization, because VVPP has no LO terms with neutral pions. Theresult is indeed finite,

V V P 0P 0µνNLO = V V P 0P 0µν

(loop 1) + V V P 0P 0µν(loop 2)

= − ie2

8π2f2

(k2

12 −m2) −gµν B0 +gµνξUV − 2gµνL1(s) + Lµν2 (s)

= − ie2

8π2f2

(k2

12 −m2) gµνL3(s) + Lµν2 (s)

,

where we inserted the definition of B0,

B0 = ξUV + B0f = ξUV − 2

√s− 4√s

[ln

√s+√s− 4

2+

2

]+ 2,

to cancel the divergence. We further summarized8

L3(s) := −2L1(s)− Bf0 (s) =4

s

[ln

√s−√s− 4

2+

2

]2

+ 1.

8In terms of Passarino-Veltman functions this cancellation of logarithms is easier seen. Itoccurs as a cancellation of the B0 from the first loop diagram with an equal B0 from thetensor decomposition of the second loop diagram.

Page 33: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

3.5. PION-PHOTON-PHOTON COUPLING 33

For charged pions and on-shell photons we have

V V PPµνLO = 2ie2gµν ,

V V PPµν(Pol.) = 8πiβπmπ(kµ2 kν1 − k1 · k2g

µν),

V V PPµν(loop 1) =e2

f2gµν

[2 A0 +k2

0 B0(sm2)],

V V PPµν(loop 2) = − ie2

16π2f2

1

3(kµ1 k

ν1 + kµ2 k

ν2 ) ξUV + 4m2gµν(ξUV + 1) ,

+ k20 [gµν ξUV − 2gµνL1(s) + Lµν2 ]

.

V V PPµνR = Zπ(V V PPµνLO + V V PPµνNLO1 + V V PPµνNLO2 + V V PPµνPol )

Zπ = 1 +m2

16π2f2(ξUV + 1)

V V PPµνR = + +

+ + +

We find that the divergence is lifted from the physically relevant terms, whichdo not contain kµ1 or kν2 :

V V PPµνR = 2ie2gµν + 8πiβπmπP±ab(k

µ2 k

ν1 − k1 · k2g

µν) +e2

f2gµνk2

0 B0(sm2)

− ie2

16π2f2

1

3(kµ1 k

ν1 + kµ2 k

ν2 ) ξUV + k2

0 [gµν ξUV − 2gµνL1(s) + Lµν2 ]

.

Similar to the neutral case, the divergence proportional to gµν is cancelled if weinsert the definition of B0,

B0(k2;m21 = m2

2 = m2) =i

16π2

(ξUV + Bf0 (k2)

).

V V PPµνR = 2ie2gµν + 8πiβπmπP±ab(k

µ2 k

ν1 − k1 · k2g

µν)

− ie2

16π2f2P±ab

1

3(kµ1 k

ν1 + kµ2 k

ν2 ) ξUV + k2

0

[Lµν2 + gµνL3(s)

]Observable amplitudes are again finite. For real photons the contraction withtwo polarization vectors will eliminate the divergence proportional to

kµ1 kν1 + kµ2 k

ν2 .

Page 34: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

34 CHAPTER 3. CHIRAL PERTURBATION THEORY AT ORDER P 4

3.6 Relevant Results for Contractions with Pho-ton Polarization Vectors

For on-shell pions and real photons we have k20 = 2k1 · k2 and ε · k = 0. Further

we have now all ingredients to calculate the amplitude of γγ → π+π− at next-to-leading order.

V PPµR = eεa3b(p+ p′)µ + eεa3bm2 −A0(m2)

48π2f2m2k · p kµ

V V PPµνR = 2ie2P±abgµν + 8πiβπmπP

±ab(k

µ2 k

ν1 − k1 · k2g

µν)

− ie2

16π2f2P±ab

1

3(kµ1 k

ν1 + kµ2 k

ν2 ) ξUV + k2

0 [Lµν2 + gµνL3]

− ie2δa3δb3

8π2f2

(k2

12 −m2) gµνL3 + Lµν2

k2

0 := 2p2 + 2p · k12 + 2k212 − 2m2 = k2

12 + p2 + p′ 2 − 2m2

L3(s) = 2m2C0

[0, sm2, 0;m2,m2,m2

]+ 1

Lµν2 = −4kµ2 kν1 + (4 + s)kµ1 k

ν2

s2

[2

(ln

√s+√s− 4

2− iπ

2

)2

+s

2

]

+ kµ1 kν2

[s− 16

2s− 8√s− 4√s3

(ln

√s−√s− 4

2− iπ

2

) ]= −4kµ2 k

ν1 + (4 + s)kµ1 k

ν2

2sL3 + kµ1 k

ν2

[1

2− 4

sB0

f

]When calculating amplitudes, Lorentz indices associated with external photonswill be contracted with the respective polarization vectors ε(k, λ). Due to thegauge invariance of QED we have

k · ε(k, λ) = 0.

Therefore when later contracting with polarization vectors we may then dropterms involving the photon momentum and the corresponding Lorentz index,

Lµν2 + gµνL3 = (k1 · k2gµν − kµ2 kν1 )

L3(s)

k1 · k2.

V PPµR = eεa3b(p+ p′)µ

V V PPµνR = 2ie2P±abgµν + P±ab(k

µ2 k

ν1 − k1 · k2g

µν)

[8πiβπmπ +

ie2k20L3

16π2 k1 · k2

]+

ie2δa3δb38π2f2

π

(kµ2 kν1 − k1 · k2g

µν)

(k2

12 −m2π

)L3

k1 · k2

It should be said that this simplification comes at the price of ruining Wardidentities if not carefully applied.9

9see sections 5.1.3 and 5.2.2

Page 35: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

3.7. FURTHER MANY-PARTICLE VERTEX FUNCTIONS 35

a) b)

Figure 3.1: Pion-loop contributions to the γγγππ vertex function

3.7 Further Many-Particle Vertex Functions

To obtain the NLO prediction of chiral perturbation theory for processes involv-ing many particles, the corresponding vertex functions need to be calculated.

Considering the process γγ −→ π+π−γ, we require at NLO the γγγππ vertexfunction, which involves the diagrams shown in figure 3.1.

The expression for the amplitude corresponding to a) involves a tensor ofrank four and four propagators. Introducing the abbreviation k12 := k1 + k2, itis given by

ie3εa3b

f2

∫d4l

(2π)4

(2l + k1)µ(2l + k1 + k12)ν(2l − k3)ρ

[(l − k3)2 −m2] [l2 −m2] [(l + k1)2 −m2] [(l + k12)2 −m2]

×

2p · (2l + k12 − k3)− (l − k3)2 + (l + k12)2.

These diagrams are in principle tractable by automatized tensor reductionmethods, which were carried out by hand in this chapter. The resulting expres-sions become very large and involve the scalar loop functions C0 and D0, whichfor general momenta can only be treated numerically.

Page 36: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as
Page 37: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

Chapter 4

Scattering Observables andPhase Space

4.1 Cross Sections and Amplitudes

To derive the formulas relating experimentally observable cross-sections to theGreen’s functions, we use the finite volume approach to quantum mechanicalscattering theory.1

Consider the situation of two initial particles scattering and producing nparticles in the final state,

i1, i2 −→ f1, . . . , fn.

The particles are on the mass shell and the total four momentum is conserved,

k2ij = m2

ij , k2fj = m2

fj and ki = kf ,

ki := ki1 + ki2, kf := kf1 + · · ·+ kfn.

To handle the square of the overall energy-conservation delta function inthe transition probability P , we restrict the Fourier representation of the four-dimensional delta function to a finite four volume of extent V T = L3T ,

V T =

∫T×V

d4l ei0·l = (2π)4δ(4)(0).

The transition probability can be obtained from matrix elements by

P =|〈f |i〉|2

〈f |f〉 〈i|i〉=

(2π)4δ(4)(kf − ki) · V T · |Tfi|2

(2V Ef1 . . . 2V Efn)(4V 2Ei1Ei2).

Here we used the definition of the T matrix,

(2π)4δ(4)(kf − ki)Tfi := 〈f |i〉 ,

1In this chapter we loosely follow [Sre07], ch. 11. For the more rigorous approach toscattering theory using wave packets, see [PS95], pp. 99–115.

37

Page 38: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

38 CHAPTER 4. SCATTERING OBSERVABLES AND PHASE SPACE

and the relativistic normalization of free-particle states

〈p|p〉 = (2π)3 2E δ(3)(p1 − p1) = 2E

∫d3k eik·(p1−p1) = 2 EV,

〈p1 . . . pn|p1 . . . pn〉 =∏j

〈pj |pj〉 = 2V E1 · · · 2V En.

Due to the quantization of momenta in finite volumes, the differential crosssection in a finite volume is not a continuous, but a discrete distribution on thespace of possible momenta. The differential cross section ∆σ in the laboratoryframe is now defined as the transition probability per time divided by the fluxof particles,

∆σf←i =(viV

)−1 P

T.

By vi we denote the relative velocity of the initial particles in the lab frame.To take the infinite volume limit, it is necessary to make the transition from

a discrete distribution ∆σ to a continuous density dσ/dΦn on the phase spaceΦn of all possible momenta.2

It is given by the manifold depending on the initial momentum ki = (k0i ,ki),

Φn(ki) :=

(kf1, . . . ,kfn) ∈ R3n ;∑j

kfj = ki,∑j

√k2fj −m2

fj = k0i

.

For convenience we would like to use a Lorentz-invariant integration measure3

on Φn, with respect to which we can define the differential cross-section,

dΦn = dΦn (kf1, . . . , kfn; ki)

=d4kf1

(2π)4. . .

d4kfn(2π)4

n∏j=1

2πδ(k2fj −m2

fj)θ(k0fj)

(2π)4δ(4)(ki −

∑kfj

)=

d3kf1

(2π)3 2E1. . .

d3kfn(2π)3 2En

(2π)4δ(4)(ki −

∑kfj

), (4.1)

Ej :=√k2fj −m2

fj .

This measure is called the lorentz-invariant n-particle phase space.To obtain a measurable cross-section in the infinite-volume limit, it is nec-

essary to sum over a small range of momenta. In the continuum limit this sumis replaced by an integral, ∑

kfj

−→ V

(2π)3

∫d3kfj .

2Mathematically speaking, the differential cross section dσ/dΦn is the Radon-Nikodymderivative of the scattering distribution with respect to the measure dΦn.

3It is given by the standard construction

d4p

(2π)42πδ(p2 −m2)Θ(p0) =

d4p

(2π)42p02πδ(p0 −

√p2 +m2) =

d3p

(2π)32p0.

When using the d3p -measure, we understand the integral over p0 carried out using the deltafunction, so that all occurences of p0 in the integrand have been set to p0 =

√p2 +m2.

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4.2. KINEMATICS OF 2→ N SCATTERING 39

For the scattering into some small region O ∈ Φn, we have now

σO :=

∫O

dΦn

(dσ

dΦn

)=

∫O

dΦn1

vi· |Tfi|

2

4Ei1Ei2.

The limit of infinite volume can now be manifestly carried out and we obtainthe formula for the differential cross section

dσ =1

4viEi1Ei2|Tfi|2 dΦn(pf1, . . . , pfn).

The fraction 1/(4viEi1Ei2) is called the flux factor. An important observablewe will compute is given by the total cross-section σtot,

σtot =

∫dσ.

4.2 Kinematics of 2→ n Scattering

The computation of the flux factor in the center of mass system requires a moredetailed study of the kinematics. For simplicity we assume that we are in thelaboratory frame, where one particle is at rest and use v = k/E,

1

4viEi1Ei2=

1

4|ki1|Ei2=

1

4|ki1|mi2.

Using the squared center-of-mass energy s = (ki1 + ki2)2,

s =

(√k2i +m2

i1 +mi2

)2

− k2i = m2

i1 +m2i2 + 2

√m2i2k

2i +m2

i1m2i2,

it becomes clear that the flux factor can be written in terms of scalar quantities,

mi2|ki| =1

2

√(s−m2

i1 −m2i2)2 − 4m2

i1m2i2.

The special cases relevant for γγ and γπ scattering are

limm→0

1

4m|ki|γγ=

1

2s,

1

4mπ|ki|γπ=

1

2· 1

s−m2π

.

4.3 Direct Parametrization of Two-Body PhaseSpace

To compute the total cross section, we require a coordinate atlas of the phasespace. For low-dimensional phase spaces in the center-of-mass system, such aparametrization can be obtained by directly imposing the constraints of thedelta functions.

For a collision of two particles the invariant square s of the total four mo-mentum is always non-negative,

s := k2i = (ki1 + ki2)2 ≥ 0.

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40 CHAPTER 4. SCATTERING OBSERVABLES AND PHASE SPACE

Therefore there always exists a Lorentz frame, the Center-of-Momentum orCenter-of-Mass frame4 (CMS), in which the total three-momentum vanishes,

ki = (√s,0).

The phase-space measure is defined by

dΦ2(p1, p2; pi) =d3p1

(2π)32E1

d3p2

(2π)32E2· (2π)4δ(4)(p1 + p2 − pi).

To obtain a parametrization of the manifold Φ2, we eliminate d3p2-integrationwith spatial part of the four-momentum-conservation delta function.

dΦ2(p1, p2; pi) =d3p1

(2π)34E1E2· 2π δ(E1 + E2 −

√s)∣∣p2=−p1

E2 =√p2

2 +m22 =

√p2

1 +m22, E1 =

√p2

1 +m21

One has to take care of the nonlinear argument of the delta function,

δ(E1 − E2 −√s) =

(∂

∂|p1|(E1 + E2 −

√s))−1

δ(|p1| − p1(s)),

∂|p1|(E1 + E2 −

√s)

=1

2E12|p1|+

1

2E22|p1| =

|p1|√s

E1E2.

Applying polar coordinates d3p = p2dp dΩ, with dΩ = dϕ dϑ sinϑ, we obtainthe final result

dΦ2

[p1, p2; pi = (

√s,0)

]=

|p1|16π2

√s

∫dΩ1|M(θ, φ)|2.

The magnitude of p1 is determined by the positive solution of

p1 = −p2, E1 + E2 =√s,

|p1| = p1(s) :=1

2√s

√s2 − 2s

(m2f1 +m2

f2

)+(m2f1 −m2

f2

)2

.

During the derivation we have ignored one subtlety now coming to light. Thenonlinear system above might not have a positive solution, which we requirefor the modulus of p1. For the two-body phase space this yields the physicallyobvious kinematical restriction

√s ≥ mf1 +mf2.

For higher dimensional phase spaces we will also have to restrict the integrationregion to respect such constraints.

Now that the integration measure has been parametrized, it remains to ex-press the integrand by the chosen integration parameters. To determine theindependent scalar products of particle momenta, we rewrite them in terms ofMandelstam variables

s = (ki1 + ki2)2, t = (ki1 − kf1)2, u = (ki1 − kf2)2.

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4.4. DIRECT PARAMETRIZATION OF THREE-BODY PHASE SPACE 41

2ki · kj ki1 ki2 kf1 kf2

ki1 m2i1 s−m2

i1 −m2i2 m2

i1 +m2f1 − t m2

i1 +m2f2 − u

ki2 m2i2 m2

i2 +m2f2 − u m2

i2 +m2f2 − t

kf1 m2f1 s−m2

f1 −m2f2

kf2 m2f2

Table 4.1: Scalar products of four-momenta in terms of Mandelstam variables,e.g. ki2 · ki1 = ki1 · ki2 = 1

2

(s−m2

i1 −m2i2

).

The relations between all possible scalar products of the momenta and theMandelstam variables are summarized in table 4.1. There is a linear dependencebetween the Mandelstam variables due to four-momentum conservation,

ki1 + ki2 = kf1 + kf2,

s+ t+ u = k2i1 + k2

i2 + k2f1 + k2

f2, k2j =: m2

j .

The Mandelstam variable s can be expressed in terms of the initial fourmomenta and therefore does not depend on the phase-space integrations. Tofind the relation between t and the angular integration dΩ, we need to analyzethe geometry of the scattering problem.

t := (ki1 − kf1)2, ki1 = (Ei1,k), kf1 = (Ef1,p)

t = (Ei1 − Ef1)2 −(k2 − 2k · p + p2

)= m2

i1 +m2f1 − 2Ei1Ef1 + 2|k||p| cosϑ

Here we have taken the liberty to choose the coordinate system of the angularintegration dΩ accordingly, so that the polar angle ϑ is the angle between thethree-momenta k and p. The Mandelstam variable t depends linearly on thevariable z := cosϑ and we may write

kf = ki = (√s,0), kf1 = (E1,p), kf2 = (E2,−p),∫

dΩ |M|2 =

∫ 2π

0

∫ π

0

dϑ sinϑ|M(ϕ, cosϑ)|2 =

∫ 2π

0

∫ 1

−1

dz |M(ϕ, z)|2.

4.4 Direct Parametrization of Three-Body PhaseSpace

The three-body phase space in the center-of-mass system can be treated on asimilar footing as the two-body phase space in the previous section,

dΦ3(p1, p2, p3; pi) =d3p1

(2π)32E1

d3p2

(2π)32E2

d3p3

(2π)32E3· (2π)4δ(4)(p1 + p2 + p3 − pi)

=1

(2π)5

d3p1 d3p2

8E1E2E3· δ(E1 + E2 + E3 −

√s)∣∣p3=−p1−p2

=1

(2π)5dE1 dΩ1 dE2 dΩ2

|p1| |p2|8E3

δ(E1 + E2 + E3 −√s).

4We will explicitly construct a suitable Lorentz transformation, which transforms from ageneral frame to the CMS and vice versa in section 4.6.

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42 CHAPTER 4. SCATTERING OBSERVABLES AND PHASE SPACE

We usedd3p = d|p| p2 dΩ = dE |p|E · dΩ .

To proceed we define the geometry of the angular integrations ,

dΩ1 = dcosϑ dϕ, dΩ2 = dcos γ dδ,

as shown in figure 4.1. The polar coordinates for p1 are chosen around the rayaxis p as zenith. The p2 coordinates are set up with zenith direction p1 asillustrated.

Using the remaining energy-conservation delta function we can eliminate onemore integration variable, which we choose to be x = cos γ. Setting zero thedelta function argument, E1 +E2 +E3 =

√s and using p3 = −p1−p2, we can

calculate

2|p1||p2|x = 2p1 · p2 = (p1 + p2)2 − p21 − p2

2 = p23 − p2

1 − p22

= E23 −m2

3 − p21 − p2

2

= (√s− E1 − E2)2 − E2

1 +m21 − E2

2 +m22 −m2

3

= s− 2√s(E1 + E2) + 2E1E2 +m2

1 +m22 −m2

3,

x =s− 2

√s(E1 + E2) + 2E1E2 +m2

1 +m22 −m2

3

2√E2

1 −m21

√E2

2 −m22

. (4.2)

The linearization of the delta-function argument yields a factor(∂

∂x(E1 + E2 + E3 −

√s)

)−1

=E3

|p1||p2|.

Requiring that the zero of the argument of the energy-conservation delta func-tion can be found within the integration region of x yields

|x| ≤ 1⇐⇒ x2 ≤ 1.

The solution of this nonlinear inequality determines the integration region ofE1 and E2. We solve the inequality with respect to E1,

(s− 2√s(E1 + E2) + 2E1E2 +m2

1 +m22 −m2

3)2 ≤ 4(E21 −m2

1)(E22 −m2

2)

⇔ 0 ≤ −(s− 2E2

√s+m2

2)E21 + c(

√s− E2)E1 −

1

4c2 − (E2

2 −m22)m2

1

⇔ 0 ≤ −(c−m21 +m2

3)

(E1 −

c

2·√s− E2

c−m21 +m2

3

)2

+c2

4· (√s− E2)2

c−m21 +m2

3

− 1

4c2 − (E2 −m2

2)m21 (4.3)

where we introduced the abbreviation c := s− 2√sE2 +m2

1 +m22 −m2

3. Fromfour-momentum conservation we can derive the sharp kinematical inequation5

E2 ≤s+m2

2 − (m1 +m3)2

2√s

=: E+2 . (4.4)

5For a proof see appendix E.1.

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4.4. DIRECT PARAMETRIZATION OF THREE-BODY PHASE SPACE 43

px

p1

ϑϕ

p2

γ

δ

Figure 4.1: Scattering geometry with three particles

0.5 1.0 1.5 2.0 2.5E2 @ mD

1.5

2.0

2.5

3.0

E1 @ mD

E1-

E1+

Figure 4.2: Integration region of E1 depending on the value of E3 in the directparametrization of dΦ3 for m1 = m3 = m, m2 = 0,

√s = 6m. Due to kinemat-

ical restrictions we have m ≤ E1 ≤√s

2 = 3m, 0 ≤ E2 ≤ s−4m2

2√s

= 83m.

As a consequence, we find that c −m21 + m2

3 > 0. Therefore the coefficient ofthe quadratic term in (4.3) is negative and the integration region is the intervalbetween the roots of the corresponding equality,

E±1 =(√s− E2) c± |p2|

√(c− 2m2

1)2 − 4m2

1m23

2(c−m21 +m2

3). (4.5)

The inequality (4.4) also shows that the solutions are always real, so that theE1 integral covers the interval

[E−1 , E

+1

]. Figure 4.2 shows the shape of this

integration region for the masses m1 = m3 = m, m2 = 0 at√s = 6m.

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44 CHAPTER 4. SCATTERING OBSERVABLES AND PHASE SPACE

k1 k2 k3 p1 p2

k1 0 12s

12 (s+ t1 + t2 − 2m2) 1

2 (m2 − t1) 12 (m2 − t2)

k2 0 12 (s1 + s2 − s− t1 − t2) 1

2 (s− s2 + t1) 12 (s− s1 + t2)

k3 0 12 (s1 −m2) 1

2 (s2 −m2)p1 m2 1

2 (s− s1 − s2)p2 m2

Table 4.2: Mandelstam variables for dΦ3. k2j = 0, p2

j = m2.

Altogether we can write a dΦ3 phase-space integral as∫dΦ3 |M|2 =

1

256π5

∫ E+2

m2

dE2

∫ E+1

E−1

dE1

∫dΩ1

∫ 2π

0

dδ |M|2∣∣∣∣∣x(E1,E2)

.

The scalar products can again be reduced to independent Mandelstam vari-ables. We will now restrict ourselves to the case relevant for

γ(k1)γ(k2) −→ π+(p1) γ(k3)π−(p2),

so that m1 = m2 = mπ and m3=0. We use Mandelstam variables defined by

s := (k1 + k2)2, s1 := (p1 + k3)2, s2 := (p2 + k3)2,

t1 := (k1 − p1)2, t2 := (k1 − p2)2.

The relations between all scalar products of the momenta and these Mandelstamvariables are summarized in table 4.2.

By analyzing the scattering geometry we obtain

s1 = s+m2 − 2√sE2,

s2 = s+m2 − 2√sE1,

t1 = m2 − 2√sE1 +

√s√E2

1 −m2 y ,

t2 = m2 − 2√sE2 +

√s√E2

2 −m2(xy −

√1− x2

√1− y2 cos δ

),

where we defined dΩ1 = dϕ dy and x is given by equation (4.2). The equationfor t2 can be obtained from the spherical version of the law of cosines.

4.5 2-Body Decompositions of n-Body Phase Space

For higher-dimensional phase spaces Φn, n ≥ 4, it is no longer advantageousto use Mandelstam variables.6 Further, the determination of the integrationregion for n ≥ 4 becomes even more difficult than in the case of 3 particles. Theinequality corresponding to (4.3) also involves angles and terms non-polynomialin the energies for such n.

We will therefore use a more elegant method to implement phase-space in-tegrals based on a 2-particle decomposition, which exploits Φ2 as elementary

6 A procedure to express all scalar products by 3n− 4 independent Mandelstam variablesis given for n = 4 in appendix D of [Kum69].

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4.5. 2-BODY DECOMPOSITIONS OF N -BODY PHASE SPACE 45

building block to construct higher-dimensional phase spaces.7 Consider an n-particle phase-space integral

∫dΦn(pµfj) |M(pfj)|2 =

∫d4pf1

(2π)4. . .

d4pfn(2π)4

n∏j=1

2πδ(p2fj −m2

fj)θ(p0fj)

× (2π)4δ(4)

(pi −

∑pfj

)|M|2.

The conceptional idea consists of transforming the integration over individualmomenta into an integration over total momenta of k-particle subsystems. Thissimplifies the elimination of the energy-conservation delta function.

Consequently, we define integration variables

p12 := p1 + p2, m212 : = p2

12,

.... . .

...

p1···n := p1 + . . .+ pn, m21···n := p2

1···n.

The associated decomposition of dΦn is illustrated schematically in figure 4.3.Mathematically, the transformation can be accomplished by using the identities

1 =

∫d4p1···k

(2π)4(2π)

4δ(4)(p1 + . . .+ pk − p1···k),

1 =

∫dm2

1···k2π

2πδ(p21···k −m2

1···k) θ(p01···k).

To achieve an iterative decompsition of dΦn we will insert the above equa-tions and regroup the integrals to get∫

dΦn(pj; p0) |M(pj)|2

=

∫d4p1

(2π)4. . .

d4pn(2π)4

n∏j=1

2πδ(p2j −m2

j )θ(p0j )

(2π)4δ(4)(p0 −

∑pj

)|M|2

=

∫ dm21···(n−1)

2π2πδ(p2

1···(n−1) −m21···(n−1)) θ(p

01···(n−1))

×∫

d4p1···(n−1)

(2π)4(2π)

4δ(4)(p1 + . . .+ p(n−1) − p1···(n−1)

)×∫

d4p1

(2π)4. . .

d4pn(2π)4

n∏j=1

2πδ(p2j −m2

j )θ(p0j )

(2π)4δ(4)(p0 −

∑pj

)|M|2

7The decomposition method can be found in [BJ80]. A pedagogical introduction is [Mur07].Early publications like [Blo56] were mostly concerned with constant matrix elements and onlydetermined the volume of the phase space.

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46 CHAPTER 4. SCATTERING OBSERVABLES AND PHASE SPACE

p1

p2

p12

p3

. . .

pn−1

p1...(n−1)

pn

pi

Figure 4.3: 2-Particle Decomposition of n-Particle Phase Space

=

∫ dm21···(n−1)

∫d4p1···(n−1)

(2π)4

d4pn(2π)4

2πδ(p21···(n−1) −m

21···(n−1)) θ(p

01···(n−1))

×2πδ(p2n −m2

n)θ(p0n)(2π)4δ(4)

(p0 − p1···(n−1) − pn

)×∫

d4p1

(2π)4. . .

d4pn−1

(2π)4

n−1∏j=1

2πδ(p2j −m2

j )θ(p0j )

(2π)4δ(4)

p0 −n−1∑j=1

pj

|M|2=

∫ dm21···(n−1)

∫dΦ2(p1···(n−1), pn; p0)

∫dΦn−1(p1, . . . , pn−1; p1···(n−1)) |M|2,

where we also used the definitions of dΦn−1 and dΦ2.Requiring that the arguments of the four-momentum conservation delta

functions of dΦ2 and dΦn−1 possess solutions within the

dm21···(n−1) = 2m1···(n−1)dm1···(n−1)

integration regions, shows, that we may restrict the integration to the interval

m1···(n−1) ∈[m1 + . . .+mn−1,

√s−mn

],

which is also just what one would expect on physical grounds.A full 2-body decomposition, as shown in figure 4.3, is accomplished by

iterating the above decomposition step,∫dΦn(pj; p0) |M|2 =

∫ dm21···(n−1)

2π. . .

dm212

∫dΦ2(p1···(n−1), pn; p0)

×∫

dΦ2(p1···(n−2), pn−1; p1···(n−1)) . . . dΦ2(p1, p2; p12) |M|2.

In the decomposition dΦ2 occurs in arbitrary frames, whereas in section 4.3 weonly considered the center-of-mass system. A connection can be established bystudying the Lorentz-transformation properties of dΦ2.

4.6 Lorentz Invariance and Lorentz Transforma-tions

In this section we discuss the definition and properties of Lorentz transfor-mations and their relation to the phase-space measure (4.1). Lastly, we willcomment upon a special Lorentz transformation needed in the numerical imple-mentation of phase-space integrals.

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4.6. LORENTZ INVARIANCE AND LORENTZ TRANSFORMATIONS 47

Consider the phase-space integration, defined as in equation (4.1). It isdefined in a manifestly Lorentz invariant way,

dΦn = dΦn (kf1, . . . , kfn; ki)

=d4kf1

(2π)4. . .

d4kfn(2π)4

n∏j=1

2πδ(k2fj −m2

fj)θ(k0fj)

(2π)4δ(4)(ki −

∑kfj

)= dΦ′n = dΦn (Λkf1, . . . ,Λkfn; Λki) .

Λ ∈ L↑ is an orthochronous Lorentz transformation, so that it leaves the signof the timelike component invariant,8

sgn (Λk)0 = sgn k0.

The invariance follows from the invariance of scalar products and squares,

k · l = (Λk) · (Λl), and consequently k2 = (Λk)2,

which is the defining property of a Lorentz transformation.9 By writing out thescalar product explicitly and inserting arbitrary four vectors k, l, the equivalentdefinition of Lorentz transformations as a matrix equation results,

kµgµν lν = k · l = (Λk) · (Λl) =

(Λµρkµ

)gρσ (Λνσlν) ,

⇐⇒ gµν = ΛµρgρσΛνσ = ΛρµgρσΛσν .

Or written as ordinary matrix product (using gµν = gµν)

g = Λ · g · Λ.

Taking the determinant it follows that

(det Λ)2 = 1⇒ det Λ = ±1.

Consequently, four-vector integrations are also invariant under Lorentz trans-formations, as can be seen by applying the integral transformation formula,∫

d4k f(k) =

∫d4l

|det Λ|f(Λl) =

∫d4l f(Λl),

δ(4)(k) =1

|det Λ|δ(4)(Λk) = δ(4)(Λk).

Therefore we may perform substitutions in phase-space integrals,∫dΦn (kj ; ki) f (k1, . . . , kn) =

∫dΦn (Λkj ; Λki) f

(Λ−1k1, . . . ,Λ

−1kn).

8If the integrand is, like the T -matrix, invariant under the transformation k0 → −k0, therestriction Λ ∈ L↑ can be dropped.

9Note that the two invariance properties are equivalent. This can be seen by the polariza-tion identity, which relates scalar products and quadratic forms,

k · l =1

4

((k + l)2 − (k − l)2

).

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48 CHAPTER 4. SCATTERING OBSERVABLES AND PHASE SPACE

To numerically implement the 2-body decomposition of Φn we require the2-body phase-space integration for general frames. By using the above integraltransformation, we can reduce the general case to the center of mass integration.

Given an arbitrary timelike four vector k, we need to construct a Lorentztransformation Λk which satisfies

Λkk = (√k2,0).

Defining

β := k/k0, γ :=1√

1− β2,

we can write for this general boost, that up to three-dimensional rotations

Λµνk =

γ −βtγ

−βγ 1 + (γ − 1)β · βt

β2

.

It is easy to verify that Λk indeed has the required property,

Λνµkν =

γ −βtγ

βγ (1− γ)β · βt

β2 − 1

· [k0

(1

β

)]= k0

(γ(1− β2)

0

)=

(±√k2

0

),

where the sign has to be chosen accordingly to match the sign of k0. The inversetransformation is defined by

(Λ−1k

)µν

(Λk)νρ = δµρ .

It can be obtained by replacing β → −β,

(Λ−1k

)µν=

γ βtγ

βγ 1 + (γ − 1)β · βt

β2 ,

.

4.7 2-Body Parametrization of n-Body Phase Space

Applying the Lorentz-transformation from the previous section, we can reducethe dΦ2-integrations from the two-body decomposition of dΦn of section 4.5 tocenter-of-mass dΦ2-integrations, which we studied in section 4.3.

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4.7. 2-BODY PARAMETRIZATION OF N-BODY PHASE SPACE 49

p1

p2

p12

p3

p123

p4

pi

Figure 4.4: 2-Particle Decomposition of 4-Particle Phase Space

∫dΦn(pj; p0) |M|2 =

∫ dm21···(n−1)

2π. . .

dm212

∫dΦ2(p1···(n−1), pn; p0)

×∫

dΦ2(p1···(n−2), pn−1; p1···(n−1)) . . . dΦ2(p1, p2; p12) |M(p1, . . . , pn) |2

=

∫ dm21···(n−1)

2π. . .

dm212

∫dΦc2(Λnp1···(n−1),Λnpn; p2

0)

×∫

dΦc2(Λn−1p1···(n−2),Λn−1pn−1; p21···(n−1)) . . . dΦc2(Λ2p1,Λ2p2; p2

12)

× |M(p1, . . . , pn) |2

=

∫ dm21···(n−1)

2π. . .

dm212

∫dΦc2(p1···(n−1), pn; p2

0)

×∫

dΦc2(p1···(n−2), pn−1;m21···(n−1)) . . . dΦc2(p1, p2;m2

12)

×∣∣M(Λ−1

2 p1,Λ−12 p2,Λ

−13 p3 . . . ,Λ

−1n pn

)∣∣2 ,where Λj := Λp1···j is a Lorentz transformation with Λjp1···j =

(√p2

1···j ,0)

and

Φc2(k1, k2; s) := Φ2(k1, k2; (√s,0)) is the two-body center-of-mass phase-space

measure.

Note that the Λn recursively depend on the new integration variables pj ,

Λn = Λp1···n p1···n = p1 + . . .+ pn−1 + pn = p0

Λn−1 = Λp1···(n−1)p1···(n−1) = p1 + . . .+ pn−1 = p0 − pn = Λ−1

n (p0 − pn)

......

Λ3 = Λp123 p123 = p1 + p2 + p3 = p1234 − p4 = Λ−14 (p1234 − p4)

Λ2 = Λp12 p12 = p1 + p2 = p123 − p3 = Λ−13 (p123 − p3) .

To give a concrete example we will conclude this section by writing down acomplete 4-body phase-space integration in the center-of-mass frame, in a waythat it can be implemented numerically. The 2-body decomposition we appliedis visualized in figure 4.4. A sample implementation can be found in appendix D.

Page 50: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

50 CHAPTER 4. SCATTERING OBSERVABLES AND PHASE SPACE

∫dΦc4 |M|

2=

∫ √s−m4

m1+m2+m3

dm123

πm123

∫ m123−m3

m1+m2

dm12

πm12∫

dΦc2(p1, p2;m212) dΦc2(p12, p3;m2

123) dΦc2(p123, p4; s)∣∣M(Λ−12 p1,Λ

−12 p2,Λ

−13 p3, p4

)∣∣2The two-body phase-space integrals dΦc2 are realized by polar coordinate

integrations,

dΦc2 (p1, p2; s) =|p1|

16π2√s

dz dϕ , z ∈ [−1, 1], ϕ ∈ [0, 2π],

|p1| =1

2√s

√s2 − 2s

(m2f1 +m2

f2

)+(m2f1 −m2

f2

)2

,

p1 = |p1|

√1− z2 cosϕ√1− z2 sinϕ

z

, p2 = −p1,

and Λ is defined as in section 4.6.Phase space integrals can be marginally reduced in dimension, if we take into

account that we work with Lorentz invariant and therefore especially rotation-ally invariant theories. Then the azimuth integration around the ray axis, givenby dϕp4 in our parametrization, is trivial and gives a factor of 2π. Further, thedzp4 integration is redundant in [−1, 0] and [0, 1].

Such redundancies have little effect when using Monte Carlo methods. Theywill notably increase the computation time when using deterministic integrators.

Page 51: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

Chapter 5

Hadronic Photon-PhotonFusion

5.1 Pion-Pair Production γγ −→ π+π−

Pion-pair production by photon-photon fusion has been studied within theframework of chiral perturbation theory at next-to-leading order by Bijnensand Cornet [BC88]. The two-loop corrections have been worked out by Bel-lucci et al. for neutral pions [BGS94] and by Burgi for charged pions [Bu96],where they have been found to be small. See also the publications by Gasser etal. [GIS05] [GIS06].

5.1.1 Kinematics and Mandelstam Variables

We consider the reaction

γ(k1, ε1) γ(k2, ε2) −→ π+(p1)π−(p2).

The momenta are denoted by ki, pi, and εi are the photon polarization vectors.We define Mandelstam variables as in section 4.3,

s := (k1 + k2)2, t := (k1 − p1)2, u := (k1 − p2)2.

The relations between these Mandelstam variables and the scalar products aresummarized in table 5.1. The Mandelstam variables are related by

s+ t+ u = 2m2π.

2kx · ky k1 k2 p1 p2

k1 0 s m2π − t m2

π − uk2 0 m2

π − u m2π − t

p1 m2π s− 2m2

π

p2 m2π

Table 5.1: Mandelstam variable substitutions for the process γγ −→ ππ,e.g. p2 · p1 = p1 · p2 = 1

2

(s− 2m2

π

).

51

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52 CHAPTER 5. HADRONIC PHOTON-PHOTON FUSION

The differential polarization-averaged cross section is given by1

dΩcm

=1

2s

√s− 4m2

π

32π2√s·⟨|T |2

⟩pol.

.

5.1.2 Amplitude at Leading Order

The leading contribution to the amplitude in chiral perturbation theory is oforder o(p0). In the charge basis it is given by the sum of diagrams

iT µν = + + .

For processes involving a larger number of particles it is convenient to exploitthe symmetry properties of the T -matrix. To illustrate the idea we will rewritethe T -matrix of this process as

iT µν = +

(k1 ↔ k2

µ↔ ν

)+ .

Applying the Feynman rules of appendix A.2 we get

T µν =e2(2p1 − k1)µ(2p2 − k2)ν

(k1 − p1)2 −m2π + iε

+

(k1 ↔ k2

µ↔ ν

)+ 2e2gµν .

For later discussions of processes involving more particles we also note, thatwe may just symmetrize the entire amplitude with respect to the photons, ifwe multiply by the corresponding symmetry factors of the diagrams under theaction of the permutation group,

iT µν =∑Sγ

+1

2

.

Here Sγ denotes the permutation group acting on the photons.

5.1.3 Transversality-Preserving Simplification

Due to the gauge symmetry in the theory, we know that currents have to beconserved. This condition is interpreted in Fourier space as

kµJµ = 0,

where Jµ denotes the operator of the electromagnetic current. Using the LSZreduction formalism, the T -matrix can also be written as

T µν = ie2

∫d4x e−i(q1x+ q2y)

⟨π+(p1)π−(p2)out

∣∣TJµ(x)Jν(y)∣∣0⟩ .

1see section 4.2 and 4.3

Page 53: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

5.1. PION-PAIR PRODUCTION γγ −→ π+π− 53

By contracting with the respective momenta, we find that the Compton-tensoris transverse2,

kµ1 Tµν = 0, Tµνkν2 = 0.

Given this identity we could assume, that any terms proportional to kµ1 andkν2 can be dropped to simplify the amplitude. Remembering that εµk

µ = 0, sucha modification will not change fixed-spin amplitudes. Concerning transversalityhowever, it will in general only preserve the weaker statement, that the simul-taneous contraction with kµ1 and kν2 vanishes,

kµ1 Tµνkν2 = 0.

It is possible to eliminate such terms without violating transversality. Con-sider the above T µν . The problem lies in the linear dependence of the momenta,in this case k1 + k2 = p1 + p2. After choosing a linear independent set of themomenta containing the photon momenta k1 and k2, we found that droppingall terms proportional to kµ1 and kν2 results in a transverse tensor.

Explicitly,

T µν =e2(2p1 − k1)µ(2p2 − k2)ν

(k1 − p1)2 −m2π + iε

+

(k1 ↔ k2

µ↔ ν

)+ 2e2gµν

=e2(2p1 − k1)µ(2k1 + k2 − 2p1)ν

(k1 − p1)2 −m2π + iε

+

(k1 ↔ k2

µ↔ ν

)+ 2e2gµν

−→ 4e2pµ1 (k1 − p1)ν

(k1 − p1)2 −m2π + iε

+

(k1 ↔ k2

µ↔ ν

)+ 2e2gµν .

This procedure has been verified for the leading-order amplitudes of theprocesses γγ −→ π+π− and γγ −→ π+π−γ. It can be applied to obtain fullyfinite and transverse amplitude tensors in the skeleton expansion for higherorders. This relies on the transversality of the vertex functions themselves.

5.1.4 Polarization-Averaged Amplitude

As a consequence of the transversality of the amplitude, polarization averagescan be calculated by taking the absolute square and contracting with −gµν ,which amounts to the replacement3

∑λ=±1

εµλε∗νλ −→ −gµν .

We average over polarizations of the initial state particles, so⟨|T |2

⟩polarizations

=1

22

∑λ1/2=±1

|T |2

=1

4T ∗µνTρσ

∑µ=±1

εµµε∗ρµ

∑ν=±1

εννε∗σν =

1

4T ∗µνT µν .

2This is not a derivation. A derivation using Ward Identities is given in [Sre07], ch. 59.3see [Sre07], ch.59

Page 54: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

54 CHAPTER 5. HADRONIC PHOTON-PHOTON FUSION

Carrying out the contractions gives⟨|T |2

⟩= 32π2α2 5m8 − 4m6(t+ u) +m4(t2 + u2) + t2u2

(m2π − t)2(m2

π − u)2,

or written in terms of the independent variable s and t,⟨|T |2

32π2α2

⟩=

5m8 − 4m6t+m4(s2 + 2st+ 6t2)−m2πt

2(s+ t) + t2(s+ t)2

(m2π − t)2(m2

π − s− t)2.

From this point on, we will use dimensionless Mandelstam variables,

s :=s

m2π

, t :=t

m2π

, u :=u

m2π

,⟨|T |2

32π2α2

⟩=

5− 4t+ (s2 + 2st+ 6t2)− t2(s+ t) + t2(s+ t)2

(1− t)2(1− s− t)2.

5.1.5 Skeleton Expansion at Next-to-Leading Order

To perform higher-order calculations it is convenient to do a skeleton expansionof the amplitude in terms of full propagators and vertex functions. For pairproduction, there are no additional vertex functions created at higher ordersnot already occuring at the tree-level. Therefore we may directly promote theleading-order diagrams to the skeleton expansion

iT = + + .

Inserting the vertex functions calculated in chapter 3, expanding in the chi-ral power counting and applying the simplification procedure of section 5.1.3leads to the prediction at next-to-leading order

T µν =4e2

m2π

(kµ2 p

ν1

1− s− t− kν1p

µ1

1− t− spµ1p

ν1

(1− t) (1− s− t)

)+ 2e2gµν

+ (kµ2 kν1 − k1 · k2 g

µν)

(2e2

(4πfπ)2(2C0(s) + 1) + 8πβπmπ

). (5.1)

The contact term contributions at next-to-leading order are summarized bythe charged pion polarizabilities, which at NLO are given by

απ = −βπ u 3.0 · 10−4fm3.

We use the physical value of the pion decay constant

fπ = 92.4 MeV.

The loop function is

C0(s) :=1

2s

(2 ln

√s− 4 +

√s

2− iπ

)2

.

Page 55: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

5.1. PION-PAIR PRODUCTION γγ −→ π+π− 55

Both lines in equation (5.1) constitute independent transverse tensors. The firstpart is due to the leading order T -matrix. The second part, proportional to

kµ2 kν1

k1 · k2− gµν =

2

m2π skµ2 k

ν1 − gµν ,

contains all of the NLO contributions. We may investigate the influence of indi-vidual terms due to pion-loops and contact terms by defining its dimensionlesscoefficient

C(s) :=e2sm2

π

(4πfπ)2(2C0(s) + 1) + 4πβπm

3π s

= Cloops + Cpolarizability.

It remains to calculate the average over photon polarizations,⟨|T |2

⟩=

1

4

(T µν(LO)T (LO)

µν + 2 T µν(LO) Re T (NLO)

µν + T ∗ µν(NLO)T (NLO)

µν

),

T µν(NLO) := (2kµ2 kν1/s− gµν)C(s), T µν = T µν(LO) + T µν(NLO).

The contractions are

T ∗µνNLO T NLO

µν =1

2|C(s)|2 ,

2T µν Re T NLO

µν =4e2s

(1− t)(1− s− t)ReC(s).

5.1.6 Differential and Total Cross Section

The polarization-averaged differential cross section is given by

dσ =

√s(s− 4)

64π2m2π s

2dΩcm |Tfi|2 .

The angular distribution is symmetric around cosϑcm = x = 0. This is dueto the symmetry under exchange of the photons. Figure 5.1 shows the angulardependency of dσ with ϕ integrated out. We also show the leading-order result,as well as the next-to-leading order result without pion loops. The latter canbe used as indication for the rate of convergence of the chiral expansion.

To calculate the total cross section we use the results from section 4.3,

dΩ1 = dϕ dx, x := cosϑ, t = 1− s

2+

1

2

√s(s− 4)x.

The total cross section σ is given by the analytic expression

σ =2πα2

s3m2π

[4 + s+ s|C|2

]√s(s− 4) + 8

[2− s+ sRe C

]ln

√s+√s− 4

2

,

C(s) = −βπm3π

2αs− m2

π

(4πfπ)2

s

2+ 2

[ln

√s+√s− 4

2− iπ

2

]2.

The result is shown in figure 5.2.

Page 56: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

56 CHAPTER 5. HADRONIC PHOTON-PHOTON FUSION

√s = 2.05mπ

√s = 2.2mπ

Π

43 Π

2

J

0.05

0.10

0.15

0.20

0.25

0.30

0.35Μb

full NLOTree + PolTree

Π

43 Π

2

J

0.05

0.10

0.15

0.20

0.25

0.30

0.35Μb

full NLOTree + PolTree

√s = 2.5mπ

√s = 3mπ

Π

43 Π

2

J

0.05

0.10

0.15

0.20

0.25

0.30

0.35Μb

full NLOTree + PolTree

Π

43 Π

2

J

0.05

0.10

0.15

0.20

0.25

0.30

0.35Μb

full NLOTree + PolTree

√s = 3.5mπ

√s = 5mπ

Π

43 Π

2

J

0.05

0.10

0.15

0.20

0.25

0.30

0.35Μb

full NLOTree + PolTree

Π

43 Π

2

J

0.05

0.10

0.15

0.20

0.25

0.30

0.35Μb

full NLOTree + PolTree

Figure 5.1: Angular distribution of the differential cross section γγ −→ π+π−

from NLO χ-PT. At next to leading order there is no qualitative change in theangular distributions at energies tractable in χ-PT.

Page 57: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

5.2. RADIATIVE PROCESS γγ −→ π+π−γ 57

2.5 3.0 3.5 4.0 4.5 5.0s @mΠ D

0.1

0.2

0.3

0.4

0.5

Σ tot @ ΜbD

full NLO

Tree+ Pol .

Tree

Figure 5.2: Total cross section γγ −→ π+π− from NLO χ-PT

5.2 Radiative Process γγ −→ π+π−γ

Now we consider a radiative process related to pair production,

γ(k1, λ1) γ(k2, λ2) −→ π+(p1)π−(p2) γ(k3, λ3).

5.2.1 LO Amplitude and Skeleton Expansion

We factor out the polarization vectors ε(ki, λi) and write the T -matrix in ten-sorial form,

T = T µνρ ε∗µ(k1, λ1) ε∗ν(k2, λ2) ερ(k3, λ3)

The average over the initial-photon polarizations and the sum over the final-photon polarization can be calculated by contracting with −gµν⟨

|T |2⟩

= −1

4T ∗µνρT µνρ.

At leading order the T -matrix is given by the twelve diagrams

iT =

+ +

+ +

+

(k1 ↔ k2

µ↔ ν

)

+ + .

Page 58: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

58 CHAPTER 5. HADRONIC PHOTON-PHOTON FUSION

The resulting T -matrix can be found in appendix E.2.We will now describe how to obtain a simpler representation of the T -matrix

by exploiting symmetries. When not thinking in terms of initial- and final statephotons, there are only two distinct diagram topologies,

iT =∑

S3×S2

1

2+

1

2

.

Here we symmetrize the amplitude in the contributions due to photon- andpseudoscalar operators. We sum over all 3! = 6 permutations of the photons inS3(γ1, γ2, γ3) and the two permutations of the pions in S2(π+, π−).

The symmetry factor 1/2 of the first diagram arises from simultaneouslyexchanging the pions and the outer photons. The second diagram is symmetricunder the exchange of the two photons on the right.

When we perform this symmetrization in the charge basis, we need to takecare of the relative directions of the momentum flow and the charge flow. Thisis done by writing the amplitude contribution corresponding to the diagrams as

iq31 (k1 + 2p)

µ1 (2(k1 + p) + k2)µ2 (2(k1 + k2 + p) + k3)

µ3

[(k1 + p)2 −m2π] [(k1 + k2 + p)2 −m2

π],

− 2ie2q1 (k1 + 2p)µ1 gµ2µ3

(k1 + p)2 −m2π

,

where q1 = ±e is the charge of pion with momentum p and all momenta arethought of flowing into the diagram. The amplitude tensor is now equivalentlygiven by the above terms, summed over all combinations

(p, q1) ∈

(−p1,+e) , (−p2,−e), (ki, µi) ∈

(k1, µ), (k2, ν), (−k3, ρ)

.

If we would now like to calculate higher order contribution we can promotethe tree-level diagrams to skeleton diagrams. For this process there occursat next to leading order also a new diagram representing the VVVPP vertexfunction,

iT =∑

S3×S2

1

2+

1

2+

1

12

.

5.2.2 Partial NLO Corrections

Considering that a numerical implementation of the VVVPP vertex function isvery cumbersome and high-precision experimental data are not yet available,4

we will only give a partial NLO result, based on neglecting the effects of theVVVPP vertex function. This might be justified for NLO contact terms basedon the symmetry factors in the symmetrized skeleton expansion, which countthe relative number of diagrams with the respective correlators inserted. The

4see section 3.7

Page 59: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

5.2. RADIATIVE PROCESS γγ −→ π+π−γ 59

pion-loop contribution to the VVVPP vertex function might be seen as the sumof all tree-level diagrams with the pion lines closed by a four-pion vertex, sothat such a diagram counting is not valid.

We will construct a finite, transverse amplitude by applying the simplifica-tion procedure of section 5.1.3 on the level of renormalized vertex functions. Toget a transverse amplitude we are forced to take the vertex functions on-shell.The restoration of the transversality of the off-shell part would require terms ofthe VVVPP vertex function.

We are left with the simple expressions for the renormalized vertex functionsalso found in the pair production amplitude,

V PPµR = γππµ(LO) = ie(p+ p′)µ,

V V PPµνR = 2ie2gµν

+ (kµ2 kν1 − k1 · k2g

µν)

2ie2

(4πfπ)2

(2m2

π C0 (s) + 1)

+ 8πiβπmπ

.

This equation shows, that the partial NLO calculation can be written as twocontributions. One is given by inserting the leading-order vertices ie(p + p′)µ

and 2ie2gµν , leading again to the LO result T µνρLO . The second part T µνρNLO hasthe γγππ-vertex function given by the transverse tensor

(kµ2 kν1 − k1 · k2g

µν)

2ie2

(4πfπ)2

(2m2

π C0

((k1 + k2)2

m2π

)+ 1

)+ 8πiβπmπ

.

The combined tensor is transverse and thereby gauge-invariant and physicallymeaningful,

kµ1 Tµνρ = 0, kν1Tµνρ = 0, kρ3Tµνρ = 0,

T µνρ = T µνρLO + T µνρNLO .

5.2.3 Photon Spectrum and Partial Cross Section

Using the results of section 4.4, we can write for the scattering cross-section intoa region of the three-body phase space O(E1, E2) ⊂ Φ3

σO =1

2s

∫O

dΦ3

⟨|T |2

⟩=

1

2s

∫O

1

256π5dE2 dE1 dΩ1 dδ

⟨|T |2

⟩. (5.2)

Due to the infrared problem of QED, the result for the total cross section σΦ3

(with O the full 3-body phase space) is divergent. Physically this divergenceis compensated by diagrams with virtual photons for the total cross sectionγγ −→ π+π−, as experimentally these processes cannot be distinguished forsufficiently soft photons.

Observables, which do not depend on T for soft photon energies are notaffected and may be calculated from the above expression. We will calculatethe invariant mass spectrum of the two pion system, or equivalently the photonenergy spectrum dσ/dω3.

The polarization-averaged photon energy spectrum dσ/dω3 can be conve-niently found in the parametrization of (5.2),

ω3 =√s− E1 − E2, dω3 = dE2,

Page 60: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

60 CHAPTER 5. HADRONIC PHOTON-PHOTON FUSION

dω3=

1

128π4s

∫ E+1

E−1

dE1

∫ 1

−1

dy

∫ π

0

dδ⟨|T |2

⟩,

where we eliminated redundant integrations.

The maximum photon energy is kinematically determined5 as

ω(max)3 =

s− 4m2π

2√s

.

The photon spectrum resulting from the partial implementation of NLO cor-rections of section 5.2.2 is shown in figure 5.3. The influence of polarizabilitycontact terms and the pion loops on the photon spectrum can be found in figure5.4. The s dependence of the photon spectrum is studied in figure 5.5, where thephoton energy has been fixed and s is varied. A partially integrated cross-sectioncan be found in figure 5.6, where a low-energy cutoff has been applied.

5.2.4 Soft-Photon Approximation

For small photon energies ω3 mπ, the process γγ −→ π+π−γ may be treatedby the soft-photon approximation.6 If the soft final state photon couples to anexternal pion, as in

,

the pion will experience only a small momentum transfer, so that it remainsapproximately on-shell. The associated inner propagator will be evaluated closeto its singularity, so that such diagrams become dominant. We may then neglectdiagrams, where this is not the case.

iT ≈

+

+

(k1 ↔ k2

µ↔ ν

)

+ +

= +

5see appendix E.16see [PS95], pp. 176-184

Page 61: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

5.2. RADIATIVE PROCESS γγ −→ π+π−γ 61

0.0 0.5 1.0 1.5 2.0Ω3 @mΠ D

1

2

3

4

dΩ3

@ nb

D

s 5 mΠ

s 3.5 mΠ

s 3 mΠ

s 2.5 mΠ

Figure 5.3: Photon spectra of γγ −→ π+π−γ including partial NLO corrections

0.5 1.0 1.5 2.0Ω3 @mΠ D

- 5

5

10

15

20

correction @%D

Figure 5.4: Magnitude of chiral correction in γγ −→ π+π−γDotted/solid lines show the relative magnitude of polarizability and partial-NLOcontributions with respect to leading order for the photon spectrum. Energiesare chosen as in figure 5.3.

Page 62: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

62 CHAPTER 5. HADRONIC PHOTON-PHOTON FUSION

Ω3 = 0.15 mΠ

Ω3 = 0.5 mΠ

Ω3 = 1.5 mΠ

Ω3 = 0.3 mΠ

3 4 5 6 7 8s @mΠ D0

2

4

6

8

10

dΩ3

@ nb

D

Figure 5.5:√s behaviour of the γγ −→ π+π−γ Photon Spectrum

Dashed/dotted/solid lines correspond to tree level, tree level + polarizabilityand partial-nlo photon spectrum.

3 4 5 6 7 8s @mΠ D0

1

2

3

4

5

6

Σ part @ nbD

full NLO

Tree+ Pol .

Tree

Figure 5.6: (Partial) total cross section σtot(γγ −→ π+π−γ)An infrared cutoff has been imposed on the photon energies, ω3 ≥ 4 MeV.

Page 63: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

5.2. RADIATIVE PROCESS γγ −→ π+π−γ 63

Here the approximated amplitude is reorganized in terms of the parent pro-cess without soft photon, which is the pair production amplitude

= + + .

Denoting the soft-photon momentum by l, the soft-photon amplitude is now

T µνρsoft = T µνpair

(i2e(−2p1 − l)ρ

(p1 + l)2 −m2π

+i2e(2p2 + l)ρ

(p2 + l)2 −m2π

)= −eT µνpair

((2p1 + l)ρ

2p1 · l− (2p2 + l)ρ

2p2 · l

),

where we set the external particles on-shell.The soft-photon tensor is transverse, lρT µνρsoft = 0. The spin sum can thus be

calculated by contracting with −gρρ′ ,⟨|Tsoft|2

⟩= −1

4T ∗µνρTµνρ

= −e2

4

⟨|Tpair|2

⟩·(

m2π

(p1 · l)2+

m2π

(p2 · l)2− 2p1 · p2

(p1 · l)(p2 · l)

). (5.3)

To have a completely transverse tensor, Tpair also has to be taken with theexternal pions on-shell, which amounts to setting l = 0.

At this point, the photon spectrum can be calculated as in section 5.2.3.By also assuming that the kinematical effects due to the soft photon may beneglected, we can reduce the photon spectrum to the total pair-production crosssection. Assuming, that

p1 + p2 + k3 ≈ p1 + p2, s ≈ (p1 + p2)2,

we may simplify the three-particle phase space,

dΦ3(p1, p2, p3; pi) =d3p1

(2π)32E1

d3p2

(2π)32E2

d3k3

(2π)32ω3· (2π)4δ(4)(p1 + p2 + k3 − pi)

≈ d3p1

(2π)32E1

d3p2

(2π)32E2

d3k3

(2π)32ω3· (2π)4δ(4)(p1 + p2 − pi)

=d3k3

(2π)32ω3dΦ2 (p1, p2; pi).

Choosing the k3 polar coordinate system with azimuth p1, we may separatethe integration containing k3 from the two-particle phase space. We obtain thesoft-photon spectrum in terms of the soft-photon correction factor δsoft.

dσsoft

dω3= e2ω3

2

∫dΩ3

(2π)3

(2p1 · p2

(p1 · l)(p2 · l)− m2

π

(p1 · l)2− m2

π

(p2 · l)2−)· σpair(s)

=: δsoft · σpair(s).

The soft-photon correction factor δsoft can be calculated analytically by in-serting the approximate kinematical identities following from the above,

pi · l = Eiω3 − pi · l = Eiω3 − |pi|ω3x, x = cosϑ,

Page 64: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

64 CHAPTER 5. HADRONIC PHOTON-PHOTON FUSION

δsoft = e2ω3

2

∫dΩ3

(2π)3

(2p1 · p2

(p1 · l)(p2 · l)− m2

π

(p1 · l)2− m2

π

(p2 · l)2

)=

e2

8π2ω3

∫ 1

−1

dx

(2p1 · p2ω

23

(p1 · l)(p2 · l)− m2

π

(E1 − |p1|x)2− m2

π

(E2 − |p2|x)2

),

∫ 1

−1

dxm2π

(Ei − |pi|x)2=

2m2π

E2i − p2

i

= 2.

The first term can be treated using a Feynman parameter z,∫ 1

−1

dxω2

3

(p1 · l)(p2 · l)=

∫ 1

−1

dx

∫ 1

0

dzω2

3

[l · (p1 + z(p2 − p1))]2

=

∫ 1

0

dz

∫ 1

−1

dx1[

E1 − x |p1 + z(p2 − p1)|]2

=

∫ 1

0

dz2

E21 − (p1 + z(p2 − p1))

2 ,

p2 = −p1, 2p1 · p2 = (s− 2)m2π, E1 = E2 =

√s/2,

E21 − |p1 + z(p2 − p1)|2 = E2

1 − (1− 2z)2p21 = m2

π + 4z(1− z)p21

= m2π

(1 + z(1− z) (s− 4)

),∫ 1

0

dz1

1 + z(1− z) (s− 4)=

4√(s− 4)s

ln

√s+√s− 4

2.

Summarizing, we have calculated the soft-photon correction factor

δsoft =2α

πω3

(2s− 4√(s− 4)s

ln

√s+√s− 4

2− 1

).

The resulting soft photon spectrum is shown in figure 5.7 in comparison withthe full result of section 5.2.2. When requiring an error below 10 % the soft-photon approximation should not be used at photon energies ω3 > 0.02mπ, 0.1mπ

when√s = 2.5mπ, 5mπ respectively.

A large part of the error of the soft-photon approximation is due to theapproximate phase-space. We may write the spin averages of the soft-photoncorrection factor and the pair-production amplitude in terms of the 3-bodyMandelstam variables of section 4.4. Then it is possible to separately judge theerrors involved in the approximated soft-photon amplitude and in the approxi-mated soft-photon phase space. The quality of the different approximations iscompared in figure 5.8 for the leading-order predictions. The larger error forhigher photon energies is due to the approximation of the phase space, whichmakes a compromise concerning energy-momentum conservation.

5.3 Four-Pion Final States γγ −→ ππππ

In this section we calculate the scattering amplitude and the total cross sectionof the process

γ(k1, µ)γ(k2, ν) −→ π(p1, a)π(p2, b)π(p3, c)π(p4, d).

Page 65: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

5.3. FOUR-PION FINAL STATES γγ −→ ππππ 65

soft approx .

s 5 mΠ

soft approx .

s 2.5 mΠ

0.0 0.1 0.2 0.3 0.4 0.5 0.6Ω3 @mΠ D0

5

10

15

20

dΩ3

@ nb

D

Figure 5.7: NLO soft-photon approximation for γγ −→ π+π−γ

0.0 0.5 1.0 1.5 2.0Ω 3 @ m Π D0

2

4

6

8

10

12

14

dΩ 3

@nb

m Π

D

soft F 3 & ampl .

soft ampl .

s 5 m Π

soft F 3 & ampl .

soft ampl .

s 2.5 m Π

Figure 5.8: Improved LO soft-photon approximation for γγ −→ π+π−γ withexact phase space.

Page 66: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

66 CHAPTER 5. HADRONIC PHOTON-PHOTON FUSION

The leading-order contributions are given by the diagrams

iT µνabcd =∑S4

1

6+

(k1 ↔ k2

µ↔ ν

)

1

6+

.

In this section we use the cartesian isospin basis. Hereby there is a largersymmetry of the amplitude, allowing a symmetrization over all pions in thefinal state as used above.

The T matrices for the different channels in the charge basis are obtainedby contracting the isospin indices with the corresponding vectors, for the initialstate

~η+ =1√2

1i0

, ~η− =1√2

1−i0

, ~η0 =

001

.

Exploiting the transversality of the tensors T µν+−00 and T µν+−+−, we calculate thepolarization-average by contracting,⟨

|T |2⟩

=1

22T ∗µνT µν ,

or by explicitly performing the contractions with the corresponding photon po-larization vectors.

For γγ → π+π−π0π0 we obtain that the T is independent on the π0 angles,

T µν+−00 = T µνab33η+∗a η−∗b =

e2

f2π

(µ20 −m2

π)tµν ,

where the η’s are complex conjugated, as they belong to final-state particles.The four-particle phase space may in this case be written as an integral overthe invariant mass square of the neutral pions, µ2

0 = (p3 + p4)2, going over athree-particle phase-space integral,

σtot(s) =α2

8sf2π

∫ √s−mπ2mπ

dµ0

õ2

0 − 4m2π · (µ2

0 −m2π)2

∫dΦ3 tµνt

µν .

For γγ → π+π−π+π− a full treatment of dΦ4 is required,

T µν+−+− = T µνabcdη+∗a η−∗b η+∗

c η−∗d .

All other channels vanish, because at leading order, the photon only couples tocharged pions. The cross-section integrals have been treated numerically using

Page 67: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

5.4. PRIMAKOFF SCATTERING γπ− −→ π−π+π−π+π− 67

5 6 7 8Ecm @mΠ D

0.5

1.0

1.5

2.0

2.5

3.0

Σ @ nbD

Figure 5.9: Total cross section γγ −→ π+π−π+π− and γγ −→ π+π−π0π0 fromLO χ-PT. The production of neutral pions is suppressed by a factor of ∼ 8 inthe region shown. Errors are given by 3 standard deviations of the Monte Carlointegration.

the CERN TGenPhaseSpace routine based on [Jam68] and independently usingthe routines found in appendix D with agreeing results.

After the phase-space integration, we have to divide by a factor of S = 2, 22

respectively for two or four charged pions, to account for identical particles inthe final state,

σ =1

2s· 1

S

∫dΦ4

⟨|T |2

⟩.

The result for the total cross section is shown in figure 5.9. At leading order,the production of neutral pions is suppressed. Because photons do not couple toneutral pions at leading order, this reduces the number of contributing diagramsin the charge basis. Effects due to pion structure and higher-order pion-pioninteractions can be incorporated similarly as for γγ −→ π+π−γ. One mightexpect, that the chiral NLO corrections to the pion-pion interaction dominatethe results for four-pion final states.

5.4 Primakoff Scattering γπ− −→ π−π+π−π+π−

Although not a photon-photon fusion reaction, the Primakoff scattering pro-cess γπ− −→ π−π+π−π+π− is currently under investigation in COMPASS atCERN. Its threshold energy is already close to the resonance region. If the totalcross section at low energies is dominated by the phase space behaviour, theprediction of leading-order chiral perturbation theory might possibly be extra-polated to the near-threshold region of this reaction.

Page 68: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

68 CHAPTER 5. HADRONIC PHOTON-PHOTON FUSION

5.5 6.0 6.5 7.0 7.5 8.0Ecm @mΠ D

0.1

0.2

0.3

0.4

Σ @ nbD

Figure 5.10: Total cross section γπ− −→ π−π+π−π+π− from LO χ-PT. Errorsare given by 3 standard deviations of the Monte Carlo Integration.

The diagrams contributing at leading order in the cartesian isospin basis aregiven by the complete symmetrization of

1

12+

1

36

+1

5!.

They correspond to 59 diagrams in the charge-basis contributing at tree-level.We showed, that the part consisting of the first two terms, which only involve

the four-pion interaction vertex, is transverse, i.e.

kµTµ(4π) = 0.

The full amplitude has been calculated by Kaiser [Kai12]. The total crosssection may be calculated along similar lines as outlined in section 5.3. It isgiven by the five-body phase-space integral over the polarization-averaged cross

Page 69: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

5.4. PRIMAKOFF SCATTERING γπ− −→ π−π+π−π+π− 69

section,

σtot =1

2(s−m2)

1

12

∫dΦ5

1

2(T 2x + T 2

y ),

where we carried out the average and the contraction with the polarizationvector of the photon in the center-of-mass system. The factor of 1/12 is dueto identical particles in the final state. The result for the total cross section isshown in figure 5.10.

Kaiser and Friedrich have also investigated the similar pion-production pro-cesses γπ− −→ π−π+π− and γπ− −→ π−π0π0 [KF08], studying chiral correc-tions [Kai10a], as well as radiative corrections [Kai10b]. The predictions are inagreement with the results from COMPASS [A+12].

Page 70: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as
Page 71: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

Chapter 6

Photon Fusion andExperiment

Photon-photon fusion is not directly accessible by experiment. Experimentaldata have to be extrapolated from the fusion of nearly-real photons, e.g. fromultraperipheral collisions of charged particles. Using charged hadrons, there willbe contaminations by strong interactions. Thus mostly lepton colliders are usedfor such measurements. The mechanism is illustrated in figure 6.1.

Experimental measurements of the photon-photon-fusion cross section intoa charged pion pair have been carried out, among others at Mark II [BBG+90],Cello [B+92] and Belle [M+07]. An overview of the current experimental data isgiven in the work of Hoferichter [HPS11]. The total cross section for charged andneutral pion-pair production is shown in figure 6.2. Experimentally, for energiesabove 1 GeV the cross section is dominated by the f2(1280) tensor resonance.

6.1 Inclusion of the f2(1270) Resonance

To investigate the influence of f2(1270) on electromagnetic pion pair production,we include it as an explicit degree of freedom into our theory. For the inclusionof the f2 we follow [DGPV99], appendix C. See also [EZ07].

General Photon-Photon-Fusion Pair Production

Figure 6.1: Leading-order diagrams for the equivalent-photon method

71

Page 72: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

72 CHAPTER 6. PHOTON FUSION AND EXPERIMENT

M. Hoferichter, D. R. Phillips, C. Schat: Roy–Steiner equations for γγ → ππ 17

Fig. 5. Total cross section for γγ → π0π0 [5,10] and γγ →π+π− [6,7,8] for | cos θ| ≤ 0.8 and | cos θ| ≤ 0.6, respectively.

Fig. 6. Total cross section for γγ → π0π0 for | cos θ| ≤ 0.8 inthe low-energy region.

Fig. 7. Dependence of |hI=02,− (t)| on the number of subtrac-

tions. The grey bands indicate the difference between CCLand GKPRY phases.

with the ππ phase below the matching point. However,if Cπ

f2Cγ

f2is chosen to be negative, the mismatch of the

phases is very small: we find a correction of δcorr = −0.09and δcorr = −0.04 in order to obtain agreement with theCCL and GKPRY phases, respectively.

Finally, we comment on the analyticity properties ofthe partial waves at the matching point. As shown inthe appendix of [48], the solutions in terms of Omnesfunctions automatically fulfill continuity at the matchingpoint, but the derivative at tm is not determined. There-fore, in general, strong cusps can occur at the matchingpoint. For example, if the background in the charged reac-tion is dropped, the neutral cross section above tm is stillcorrectly reproduced, but the input for the I = 0 compo-nent changes, which affects the neutral cross section belowtm: the result for |hI=0

2,− (t)| exhibits a strong cusp below tm,which translates into an (unphysical) sharp peak of about15 nb in the neutral cross section directly below tm. Thefact that this effect is much smaller in the full solution pro-vides evidence that our model for the high-energy regionis reasonably accurate, because only a specific input of ππphases, polarizabilities, and imaginary parts above tm willyield a smooth solution for hI=0

2− (t) around t = tm. In thelanguage of [56] such a solution corresponds to an “ana-lytic input”. If the input above the matching point weresufficiently well known, one could thus derive constraintson the polarizabilities by requiring a no-cusp condition.These constraints would be similar to those derived in[45,48] for ππ and πK scattering lengths. However, theinput above the matching point is not very well known inγγ → ππ, so we content ourself with requiring that thecusp at tm is not too large, such that the input we areusing is reasonably close to being “analytic”.

8.3 Two-photon coupling of the σ

We now present our results for the two-photon width Γσγγ

as a function of the pion polarizabilities. (α1 + β1) and

a)M. Hoferichter, D. R. Phillips, C. Schat: Roy–Steiner equations for γγ → ππ 17

Fig. 5. Total cross section for γγ → π0π0 [5,10] and γγ →π+π− [6,7,8] for | cos θ| ≤ 0.8 and | cos θ| ≤ 0.6, respectively.

Fig. 6. Total cross section for γγ → π0π0 for | cos θ| ≤ 0.8 inthe low-energy region.

Fig. 7. Dependence of |hI=02,− (t)| on the number of subtrac-

tions. The grey bands indicate the difference between CCLand GKPRY phases.

with the ππ phase below the matching point. However,if Cπ

f2Cγ

f2is chosen to be negative, the mismatch of the

phases is very small: we find a correction of δcorr = −0.09and δcorr = −0.04 in order to obtain agreement with theCCL and GKPRY phases, respectively.

Finally, we comment on the analyticity properties ofthe partial waves at the matching point. As shown inthe appendix of [48], the solutions in terms of Omnesfunctions automatically fulfill continuity at the matchingpoint, but the derivative at tm is not determined. There-fore, in general, strong cusps can occur at the matchingpoint. For example, if the background in the charged reac-tion is dropped, the neutral cross section above tm is stillcorrectly reproduced, but the input for the I = 0 compo-nent changes, which affects the neutral cross section belowtm: the result for |hI=0

2,− (t)| exhibits a strong cusp below tm,which translates into an (unphysical) sharp peak of about15 nb in the neutral cross section directly below tm. Thefact that this effect is much smaller in the full solution pro-vides evidence that our model for the high-energy regionis reasonably accurate, because only a specific input of ππphases, polarizabilities, and imaginary parts above tm willyield a smooth solution for hI=0

2− (t) around t = tm. In thelanguage of [56] such a solution corresponds to an “ana-lytic input”. If the input above the matching point weresufficiently well known, one could thus derive constraintson the polarizabilities by requiring a no-cusp condition.These constraints would be similar to those derived in[45,48] for ππ and πK scattering lengths. However, theinput above the matching point is not very well known inγγ → ππ, so we content ourself with requiring that thecusp at tm is not too large, such that the input we areusing is reasonably close to being “analytic”.

8.3 Two-photon coupling of the σ

We now present our results for the two-photon width Γσγγ

as a function of the pion polarizabilities. (α1 + β1) and

b)

Figure 6.2: Experimental data on photon-photon fusion γγ → π+π− a) andγγ → π0π0 b) and a prediction using Roy-Steiner equations [HPS11]. Thedata only covers the angles from |cosϑ| < 0.6 and |cosϑ| < 0.8 respectively,relatively enhancing the f2(1270) resonance.

√t denotes the center-of-mass

energy emphasizing the relation to the crossed process of Compton scattering.

Page 73: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

6.1. INCLUSION OF THE F2(1270) RESONANCE 73

gf2ππ 23.64gf2γγ 0.239mf2 1275.1 MeVΓf2 185 MeV

Table 6.1: Properties of the f2(1270) resonance [DGPV99].

The tensor meson propagator is given by

i∆µνρσ(p) =i

p2 −m2T + imTΓ

∑λ

εµν(p, λ)ε∗ρσ(p, λ),

∑λ

εµν(p, λ)ε∗ρσ(p, λ) =1

2

(KµρKνσ +KνρKµσ − 2

3KµνKρσ

),

Kµν := −gµν +pµpν

m2.

The couplings of the tensor meson to the pion field and the electromagnetic fieldare

M (fρσ2 → π(p, a)π(p′, b)) =gf2ππmf2

δabp′ρpσ,

M (fρσ2 → γ(k1, µ)γ(k2, ν)) = −i2e2 gf2γγmf2

Fρδ(k1, µ)Fδσ(k2, ν),

Fρδ(k, µ) := kρgδµ − kδgρµ. (6.1)

The contribution to the amplitude of γγ −→ π+π− reads

T µνf2 = −ie2gf2ππgf2γγ

m2f2

(p2 −m2

f2 + imf2Γf2

)× p1ρp2σ

(KµρKνσ +KνρKµσ − 2

3KµνKρσ

)Fρδ(k1, µ)Fδσ(k2, ν).

The magnitudes of the coupling constants gf2ππ and gf2γγ can be deter-mined by calculating the decay widths associated with the amplitudes aboveand matching to the measured partial widths. The resulting constants givenin [DGPV99] are shown in table 6.1.

The polarization-averaged amplitude can be calculated by contracting thetotal amplitude tensor

T µν = T µνNLO + T µνf2 ,⟨|T |2

⟩=

1

4T µνT ∗µν .

The total cross section is shown in figure 6.3. The interference term 2 Re T µνNLOT ∗ f2µν

is sensitive to the relative sign of the coupling constants gf2ππ and gf2γγ , whichcannot be determined from the partial widths. In figure 6.4 the cross section hasonly been integrated for angles | cosϑ| ≤ 0.6 to compare to experimental data.The result of restricting the angular integration in the direction parallel to theray axis reduces the total cross section in the near-threshold region. D-wavecontributions, as from the f2, are nearly unaffected.

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74 CHAPTER 6. PHOTON FUSION AND EXPERIMENT

Motivated by the large-Nc limit of QCD, the above procedure may be gen-eralized and systematized. This approach has been designated as ResonanceChiral Theory [EGPdR89]. It has been explicitly carried out for spin one reso-nances [EGL+89] and more recently for spin two resonances [EZ07].

6.2 The Equivalent Photon Method for Ultra-peripheral Heavy-Ion Collisions

The equivalent-photon method due to Fermi1 is a convenient formalism to cal-culate the photon-photon fusion contribution to scattering processes of chargedparticles at leading order in the electromagnetic interaction, as represented bythe diagrams shown in figure 6.1. It directly relates cross sections of scatter-ing reactions of charged particles to the cross sections of photon-photon fusionprocesses.

We will use the variant of the equivalent-photon method given in the reportby Bertulani and Baur [BB88]. For heavy ion collisions Z1 +Z2 −→ Z1 +Z2 +Xthe relevant relation is [BB88, sec. 4.2],

σ =

∫ ω(max)1

0

dω1

ω1

∫ ω(max)2

0

dω2

ω2σγγ→X(4ω1ω2)n(ω1)n(ω2), (6.2)

where σγγ→X(s) is the total cross section of the corresponding photon-photonfusion process γγ → X, in terms of the squared center-of-mass energy s. Theintegrals are calculated up to the adiabatic cutoff

ω(max)i =

γ

Rion

.

The equivalent-photon spectrum n(ω) is given in terms of modified Besselfunctions of the second kind [BB88, sec. 1],

n(ω) =2

πZ2

1α1

β

[ξK0(ξ)K1(ξ)− β2ξ2

2

(K2

1 (ξ)−K20 (ξ)

)]≈ 2

πZ2

1α ln1

ξfor γ 1, ωR 6≪ 1, ξ :=

ωRion

βγ,

where

γ =1√

1− β2=Eion

m.

By substituting x := ω1ω2 in equation (6.2) we may write

σ =

∫ ω2max

0

dx

xσ(4x)

∫ ωmax

0

dω2

ω2n1

(x

ω2

)n2(ω2)

=:

∫ ω2max

0

dx

xσ(4x)I(x). (6.3)

Here we defined the equivalent photon spectrum for heavy-ion collisions,

I(x) =

∫ ωmax

0

dω2

ω2n1

(x

ω2

)n2(ω2) . (6.4)

1see the original work [Fer24]

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6.2. THE EQUIVALENT PHOTON METHOD 75

4 6 8 10Ecm @mΠ D

0.1

0.2

0.3

0.4

0.5

Μb

Figure 6.3: Total cross section γγ −→ π+π− including the resonance f2(1270).The dashed curve shows the χ-PT NLO result.

4 6 8 10Ecm @mΠ D

0.05

0.10

0.15

0.20

0.25

0.30

Μb

Figure 6.4: Total cross section γγ −→ π+π− including the resonance f2(1270)integrated over the angular region | cosϑ| ≤ 0.6 to compare with experiment(figure 6.2). The dashed curve shows the χ-PT NLO result.

Page 76: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

76 CHAPTER 6. PHOTON FUSION AND EXPERIMENT

2 4 6 8 10Ecm@mΠ D

- 20

- 10

10

20

30

40

relative Error @% D

Figure 6.5: Relative error of the photon spectrum approximation (6.5), shownfor Pb83+ CMS-energies of Ecm = 13, 20, 35, 50, 100, 500, 25000mPb for pho-ton invariant masses relevant for pion pair production.

Bertulani and Baur also give an approximated version for large γ 1,

I∼(x) =

(Z1Z2α

π

)216

3x

(ln

γ√xR1R2

)3

. (6.5)

The relative error of this approximation is shown in figure 6.5 up to ion energiesof the LHC Pb runs. It remains on the level of about 10 %, giving a robustestimate. For our purposes we use the full equivalent-photon spectrum as givenby equation (6.4).

When using the scattering cross section σ(γγ → π+π−) as predicted bychiral perturbation theory we obtain a total cross section as shown in figure 6.6.If we include the f2(1270) resonance as discussed in section 6.1 a similar totalcross section is obtained, which can be seen in 6.7.

The error band has been estimated by calculating the cross-section integralfrom a cutoff energy to the maximum energy ωmax, chosen at

√s = 6.5mπ and√

s = 11.5mπ, for the calculation without and with inclusion of the f2(1270)resonance respectively. The upper error band has been determined by multiply-ing the above contribution from higher pion energies by the maximum relativedifference in the two photon spectra with and without the f2 resonance2, toaccount for contributions of possible additional resonances. The lower errorband was determined by dropping the unknown contribution from higher pionenergies.

If we drop the integrations we get a prediction for the equivalent mass spectraof the pion pair as in figure 6.8. Up to LHC energies effect of the f2 on the massspectrum is only marginal, due to the strong damping of the equivalent-photonspectrum at higher energies. The difference in the spectra is shown in figure6.10.

2 The maximum relative difference in the two photon spectra with and without includingthe f2 resonance is shown in figure 6.11.

Page 77: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

6.2. THE EQUIVALENT PHOTON METHOD 77

The radiative process γγ −→ π+π−γ, studied in section 5.2, plays the role ofa radiative correction to the pion-pair mass spectrum, if the final-state photonis not measured.

Obtaining a prediction for the total cross section of electromagnetic pionpair production in ultraperipheral heavy-ion collisions would require the correctasymptotic form of the total cross section at high energies. As QCD becomesperturbative at such energies, a direct calculation using pQCD seems possible.Such calculations have been done using models for the valence quark structureof the pion, see the original work by Brodsky and Lepage [BL81], and morerecent publications, e.g. [KGS11] and [HYKP+97].

Pion-pair production in heavy-ion collisions has been observed in ALICEat CERN [Sch11]. Their result for the pion transverse-momentum distributionis shown in figure 6.9. The photon-photon fusion processes investigated mightplay a role in the theoretical analysis of such measurements.

Page 78: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

78 CHAPTER 6. PHOTON FUSION AND EXPERIMENT

0 10 20 30 40EPb@mPbD0

2

4

6

8Σ tot @ mbD

Figure 6.6: Total cross section for electromagnetic pion pair production inPb83+

208 Pb83+208 collisions at low energies using the photon-photon fusion total cross

section predicted from NLO χ-PT. The error band estimates possible contribu-tions due to resonances above

√sπ = 6.5mπ.

0 10 20 30 40EPb@mPbD0

2

4

6

8Σ tot @ mbD

Figure 6.7: Total cross section for electromagnetic pion-pair production inPb83+

208 Pb83+208 collisions with inclusion of the f2(1270) resonance. The large error

band at higher energies is due to the influence of the momentum-dependent f2coupling for pion energies above

√sπ = 11.5mπ.

Page 79: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

6.2. THE EQUIVALENT PHOTON METHOD 79

4 6 8 10Ecm@mΠ D

2

4

6

8

dΣ dmΠ2 @ mb mΠ

2 D

Figure 6.8: Predictions for mass spectra of electromagnetically produced pionpairs in Pb83+

208 Pb83+208 collisions for

√sPb = 2, 12.5, 50 · (103mPb). Increasing

mass spectra correspond to increasing Pb center-of-mass energies.

XIV International Conference on Hadron Spectroscopy (hadron2011), 13-17 June 2011, Munich, Germany

5 Central meson production in PbPb-collisions

Diffractive and electromagnetic processes in PbPb-collisions show a variety of intriguingfeatures [4, 5]. First, photoabsorption can lead to giant dipole resonance excitation withsubsequent neutron decay. In such decays, the charge to mass ratio is modified. In addition,bound-free pair production also leads to a modified charge to mass ratio of one or both ofthe nuclei involved. Both processes contribute to the beam life time. Second, photon-photonprocesses result in electromagnetic production of pseudoscalars π0, η, η

′and of pairs of

bosons π+π−, K+K− and fermions e+e−, µ+µ−, τ+τ−. Third, photon-Pomeron processescan, for example, result in diffractive photoproduction of vector mesons ρ0, φ, J/Ψ, Υ.

The first heavy ion run at the LHC took place in Nov-Dec 2010. In this period, about 12× 106

minimum bias PbPb-collisions were recorded with the ALICE detectors. In addition tothe minimum bias trigger, data were taken with two dedicated triggers for investigatingmeson production in the ALICE central barrel. First, a trigger OM2 based on number ofTOF hits ≥ 2 was running. Second, a trigger CCUP2 defined by the logic combination:(TOF hits ≥ 2) AND (ITS pixel) AND (double gap condition) was defined.

(GeV/c)T

p0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

(Y

ield

pe

r 1

0 M

eV

/c)

TdN

/dp

0

20

40

60

80

100

120

140

160 = 2.76 TeV

NNsUltra­peripheral Pb+Pb events at

unlike sign track pairs

like sign track pairs

Triggered by SPD & TOF & not(V0)

Performance

23/02/2011

Figure 5: Transverse momentum of unlike and like sign CCUP2 triggered pairs in PbPb-collisions at

√sNN = 2.76 TeV.

The OM2 and CCUP2 triggered events with exactly two tracks in the central barrel wereselected, and the pair transverse momentum was calculated. Both of these triggers result insimilar pair pT distributions. The pair pT of CCUP2 triggered events is shown in Figure 5for like and unlike sign pairs. The unlike sign distribution clearly shows a strong peak atpT ≤ 100 MeV/c consistent with coherent production off a Pb nucleus.

Figure 6 shows the invariant mass for the unlike sign CCUP2 triggered pairs with pairpT ≤ 150 MeV/c. This distribution is not corrected for finite detector acceptance and fordetector resolution. The shape of the distribution shown in Figure 6 is consistent withproduction of the ρ0-meson with JPC = 1−−. As for the pT-distribution, the OM2 and CCUP2triggers result in similar pair invariant mass distributions. Hence these results indicate that

5

Figure 6.9: Transverse momentum distribution of pion-pairs measured by theALICE experiment at CERN [Sch11],

√s = 2.76 TeV/nucleon, corresponding

to√sPb208 ≈ 3 · 103mPb.

Page 80: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

80 CHAPTER 6. PHOTON FUSION AND EXPERIMENT

6 7 8 9 10Ecm@mΠ D0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

DdΣ dmΠ2 @ mb mΠ

2 D

Figure 6.10: Absolute difference of the pion pair mass spectrum after inclusionof the f2(1270) resonance for

√sPb = 2, 12.5, 50 · (103mPb). Increasing mass

spectra correspond to increasing Pb center-of-mass energies.

4 6 8 10Ecm@m

ΠD

0.5

1.0

1.5

2.0

2.5

3.0

3.5

relative difference

Figure 6.11: Relative difference of the pion-pair mass spectrum after inclusionof the f2(1270) resonance. The maximum relative contribution in this regiongiven by a factor of ∼ 4 has been used to estimate the error of the total crosssections in figures 6.6 and 6.7 due to resonance contributions.

Page 81: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

Chapter 7

Outlook and Summary

7.1 Resonances and Chiral Symmetry

We have seen in chapter 6, that we had to augment the chiral description atintermediate energies to account for the occurrence of resonances. In this sectionwe will briefly comment on the role of resonances in QCD and chiral perturbationtheory. The idea is to consider infinite sums of rescattering diagrams

+ + + . . . . (7.1)

We consider now pion-pion scattering,

π(p+Q, a)π(−Q, b) −→ π(p+ q, c)π(−q, d). (7.2)

The above infinite series can be rewritten as a Bethe-Salpeter equation,

= + .

By iteratively inserting the equation into itself we see that the right-hand sidewill generate the whole tower of diagrams.

Calculating the first few diagrams of the series (7.1), one finds that they allpossess the general structure ∑

λ∈Λ×Ξ

λfλ(p,Q), (7.3)

where Λ only contains the Lorentz structures required for calculating the dia-gram with another loop to the right of the blob diagram,

Λ =

1, p · q, q2.

81

Page 82: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

82 CHAPTER 7. OUTLOOK AND SUMMARY

qQ

pa

b d

c

Figure 7.1: Definition of the momentum flow and isospin indices

Ξ contains the isospin structures

Ξ = δabδcd, δacδbd, δadδbc ,

and the fλ are unknown coefficient functions.When calculating the general loop integral in the Bethe-Salpeter equation

using the parametrization of the blob vertex (7.3), it turns out that the loopintegration preserves the above structure.

We used an automatically calculated solution of the loop integrals1. Follow-ing Nieves and Ruiz Arriola [NRA00], we ignore the required renormalizationprocedure and use the divergence ξUV as a subtraction constant. By the regularrenormalization procedure, the divergences ξUV of the loop integrals would havebeen absorbed into higher order counter-terms, which in turn may possess finiteparts. The parametrization by ξUV may be seen as an attempt to simulate themissing contributions of higher-order counter terms.

After this procedure, the Bethe-Salpeter equation reduces to a finite linearsystem of equations. Its numerical solution is shown in figure 7.2. For largeregions of values of the subtraction constant, the T -matrix exhibits a behaviourwhich may be identified with a resonance. In figure 7.2, the subtraction constanthas been used for the peak to occur at the ρ mass.

Nieves and Ruiz Arriola [NRA00] discuss many subtleties that we haveignored in our simple treatment. Further details can be found in Lutz andKolomeitsev [LK04], or the original works of Oller and Oset [OO97] and [OO98].

The f2(1270) tensor resonance discussed in section 6.1 has been interpretedas a dynamically-generated bound state of vector mesons in the work of Oset,Geng et al. [OGMN10]. Using the result of resummed Goldstone-boson scatter-ing amplitudes it is possible to also calculate interactions of the resonanceswith external fields. This was carried out for the f2 by Branz, Geng andOset [BGO10]. Schematically this is achieved by using the resummed resultto calculate loop diagrams as shown in figure 7.3.

1see [MBD91]

Page 83: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

7.1. RESONANCES AND CHIRAL SYMMETRY 83

p-wave π+π− −→ π+π−

3 4 5 6 7 8s @m

ΠD

T ¤ 2

Figure 7.2: Result of a SU(2) Model Resummation. Shown is the p-wave pro-jection of the scattering amplitude of the channel π+π− → π+π−. The resultexhibits a resonance-like behaviour. The peak was fitted to mρ = 770 MeV us-ing the subtraction constant. The lower dashed curve shows the leading orderχ-PT result.

Figure 7.3: Contribution of final state interactions to γπ-vertex form factors

Page 84: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

84 CHAPTER 7. OUTLOOK AND SUMMARY

7.2 Summary

In this diploma thesis we have investigated hadronic photon-photon fusion pro-cesses at low energies. Predictions were calculated for total- and differen-tial cross sections for the processes γγ −→ π+π−, γγ −→ π+π−γ, γγ −→π+π−π0π0 and γγ −→ π+π−π+π− using the framework of chiral perturbationtheory.

The total cross sections for γγ −→ π+π−π0π0 and γγ −→ π+π−π+π− havebeen calculated using established Monte Carlo methods and by an independentoriginal implementation of the four-body phase-space integration. Our imple-mentation can be used to compute predictions for further observables, e.g. massspectra or angular distributions.

We analyzed the magnitude of chiral corrections to photon-pion interactionsnear the threshold, concerning their influence on the above reactions.

Comparing to experimental results, we included the f2(1270) tensor res-onance as an explicit degree of freedom. The result for the cross section ofγγ −→ π+π−, with f2 included, was used to obtain a prediction for chargedpion-pair production in ultraperipheral heavy-ion reactions by means of theequivalent photon method.

In the last chapter we showed how to resum pion-rescattering diagrams, toinclude effects due to bound-states of Goldstone bosons.

If experimental data reaches sufficient precision, the methods applied in thisthesis may be used to obtain a full NLO prediction for pion-photon interactionsin the radiative process γγ −→ π+π−γ. This could establish a new link betweenexperimental pion-pion-scattering data, connected to NLO contact terms, andthe interface of strong and electromagnetic interactions, probing the electro-magnetic structure of the pion with three photons.

Similarly the calculation can be generalized by considering pion-pion scat-tering at next-to-leading order in chiral perturbation theory to study associatedfinal-state effects for the reactions γγ −→ π+π−π0π0, γγ −→ π+π−π+π− andγπ− −→ π−π+π−π+π−, the latter being currently measured by the COMPASSexperiment at CERN.

The resummation method of the last chapter should be investigated for thepossibility of describing tensor resonances directly from chiral meson-meson in-teractions.

Page 85: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

Appendix A

Feynman Rules

A.1 Cartesian Isospin Basis

In the square-root parametrization of the special-unitary matrix field U(x) ∈SU(2) we have

U(x) =1

F0

[√F 2

0 − ~π2(x) + i~τ · ~π(x)

],

where ~τ = (τ1, τ2, τ3) are the hermitian and traceless Pauli matrices

τ1 =

(0 11 0

), τ2 =

(0 i−i 0

), τ3 =

(1 00 −1

).

The components of the pion field ~π = (π1, π2, π3) correspond to a cartesianisospin basis. The orientations of the momenta are indicated by the arrows.Photon momenta go into the diagram.

πa(p) πb(p)iδab

p2 −m2 + iε

γ(k, µ)

πa(p)

πb(p′)

eεa3b(p+ p′)µ

γ(k1, µ)

γ(k2, ν)

πa(p)

πb(p′)

2ie2(δab − δa3δb3) gµν

85

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86 APPENDIX A. FEYNMAN RULES

p1, a p2, b

p3, c p4, di

f2

[δabδcd(s−m2)

+δacδbd(t−m2)

+δadδbc(u−m2)]

s = (p1 + p2)2, t = (p1 − p3)2,

u = (p1 − p4)2

A.2 Charge Basis

Experimentally we see neutral and charged pions. Therefore we are interested,which combinations of the cartesian π1, π2 and π3 correspond to states withdefinite charge. Introducing the covariant derivative

DµU = ∂µU +i

2eAµ[τ3, U ],

we find the normalized charge eigenstates as

∣∣π+⟩

=1√2

1+i0

,∣∣π−⟩ =

1√2

1−i0

,∣∣π0⟩

=

001

.

For the Feynman rules we assume that the charge flow goes in the same directionas the momentum, from the bottom of the diagram to the top. Photon momentago into the diagram.

π±(p) π±(p)i

p2 −m2 + iε

π0(p) π0(p)i

p2 −m2 + iε

γ(k, µ)

π+(p)

π+(p′)

ie(p+ p′)µ

γ(k1, µ)

γ(k2, ν)

π+(p)

π+(p′)

2ie2gµν

Page 87: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

A.2. CHARGE BASIS 87

π+(p1) π+(p2)

π+(p3) π+(p4)

i

f2

[t+ u− 2m2

]

π+(p1) π0(p2)

π+(p3) π0(p4)

i

f2

[t−m2

]

π0(p1) π0(p2)

π0(p3) π0(p4)

i

f2

[s+ t+ u− 3m2

]

γ(k1, µ)

γ(k2, ν)

π(p)

π(p′)

8πiβπmπ(kµ2 kν1 − k1 · k2g

µν)

Where the Mandelstam variables are defined by

s = (p1 + p2)2, t = (p1 − p3)2, u = (p1 − p4)2.

Page 88: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as
Page 89: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

Appendix B

Dimensional Regularization

All denominators of loop integrands can be simplified to an isotropic form usingthe Feynman trick:

(n− 1)!

∫ 1

0

dx1 . . . dxnδ(x1 + · · ·+ xn − 1)

(x1A1 + · · ·+ xnAn)n=

1

A1A2 . . . An.

This formula is valid, as long as the Feynman-parameter integration does notencounter a pole of the integrand. For all Ai real, this is equivalent to all Aihaving the same sign.

Define d = 4 − 2δ as the space-time dimension in which the only radiallydependent integrals have to be performed. Then we have [Tic99],

RegD

∫d4l

(2π)4

1

(l2 − a)n=

i

16π2

Γ(n− 2 + δ)

Γ(n)

(−1

a

)n−2(4πµ2

a

)δ. (B.1)

By exploiting Lorentz symmetry, inhomogenities in the numerator become tractable,

RegD

∫d4l

(2π)4

(l2 − a)n= 0,

RegD

∫d4l

(2π)4

l2

(l2 − a)n=

i(2− δ)16π2

Γ(n− 3 + δ)

Γ(n)

(−1

a

)n−3(4πµ2

a

)δ,

RegD

∫d4l

(2π)4

lµlν

(l2 − a)n=

igµν

32π2

Γ(n− 3 + δ)

Γ(n)

(−1

a

)n−3(4πµ2

a

)δ.

Proofs can be found in [Sre07], ch. 14.

89

Page 90: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as
Page 91: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

Appendix C

Veltman-Passarino ScalarLoop Functions

The four Veltman-Passarino scalar loop functions A0, B0, C0 and D0 are definedas in [PV79]. They provide a convenient basis to express loop amplitudes. Weexpress them in terms of available Lorentz scalars,

A0[m2] :=16π2

i

∫ddl

(2π)d1

l2 −m2,

B0[k2;m21,m

22] :=

16π2

i

∫ddl

(2π)d1

[l2 −m21] [(l + k)2 −m2

2],

C0[k21, (k1 − k2)2, k2

2;m21,m

22,m

23]

:=16π2

i

∫ddl

(2π)d1

[l2 −m21] [(l + k1)2 −m2

2] [(l + k2)2 −m23],

and similarly for D0. We understand +iε added to each propagator denominator.Note that A0 and B0 contain divergences, C0 and D0 are finite.

The normalization is chosen in a way that the coefficient of the divergence

ξUV :=1

δ− γE + ln

4πµ2

m2

is unity. For convenience we also introduced differently normalized versions

A0[m2] :=

∫d4l

(2π)4

1

l2 −m2=

i

16π2A0[m2], . . . .

For our calculations we are only interested in A0, B0 and C0, with all massesset equal.

C.1 A0(m2)

For A0 we can immediately apply dimensional regularization:

A0[m2] =16π2

i

∫ddl

(2π)d1

l2 −m2= m2

(1

δ+ 1− γE + ln(4π) + ln

µ2

m2

)= m2(ξUV + 1),

91

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92 APPENDIX C. VELTMAN-PASSARINO SCALAR LOOP FUNCTIONS

where we applied the formula from appendix B and the definition of ξUV.

C.2 B0(k2;m2,m2)

To obtain B0 we use the standard Feynman trick,

1

AB=

∫ 1

0

dx1

(xA+ (1− x)B)2=

∫ 1

0

dx1

(B + (A−B)x)2

and the translation invariance of the dimensionally regularized integral. Forthe Feynman-parameter integration to be well defined, it should not hit a pole.This is the case if A and B have an identical, nonvanishing imaginary part.

B0 =16π2

i

∫ddl

(2π)d1

[l2 −m2 + iε] [(l + k)2 −m2 + iε]

=16π2

i

∫ddl

(2π)d

∫ 1

0

dx1

[l2 −m2 + x(2l · k + k2)]2

=

∫ 1

0

dx16π2

i

∫ddl

(2π)d1

[l2 −m2 + x(2l · k + k2)]2

=

∫ 1

0

dx16π2

i

∫ddl

(2π)d1

[l2 −m2 + x(1− x)k2]2

In the last equality we shifted the integration variable l→ l − xk.The equality of changing the order of integration can in principle also be

understood as a definition of dimensionally regularized integrals with inhomo-geneities of the integration variable in the denominator.

Dimensional regularization gives

RegD16π2

i

∫ddl

(2π)d1

(l2 − a)2=(ξUV − ln

a

m2

).

We can already calculate the divergent part:

B0[k2;m21 = m2

2 = m2] = ξUV −∫ 1

0

dx lnm2 − x(1− x)k2 − iε

m2.

The main work lies in calculating the finite loop function. We introduce the

dimensionless variable s = k2

m2 and calculate∫ 1

0

dx ln (1− x(1− x)s− iε) .

The simplest variant, which obtains the correct imaginary parts, is to cal-culate the loop function for s < 0, where the real part of the argument of thelogarithm remains positive. Then we do not have to worry about the branchesof the holomorphic logarithm and we can analytically continue to s > 4 after-wards.1

0 ≤ x ≤ 1, s < 0 =⇒ 1− x(1− x)s > 0

1A more tedious method would be to keep all iε’s. Then the antiderivative exists globally.The integration result then has to be examined for selecting the correct cuts.

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C.3. C0(0, S, 0;M2,M2,M2) 93

We obtain for s < 0∫ 1

0

dx ln (1− x(1− x)s) = −2

√4− s√−s

ln

√4− s−

√−s

2− 2,

and analytically continued for s > 4∫ 1

0

dx ln (1− x(1− x)s) = −2

√s− 4√s

[ln

√s−√s− 4

2− iπ

2

]− 2,

where we used ln(−ai) = ln a+ ln(−i) = ln a− iπ/2.The final result for s > 4 is

B0[k2;m21 = m2

2 = m2] = ξUV + 2

√s− 4√s

[ln

√s−√s− 4

2− iπ

2

]+ 2.

The finite part will be sometimes referred to as Bf0 (k2).The limiting cases k2 = 0 or k2 small are also often used. There is a useful

relation between A0 and B0,

m2B0[k2;m2,m2] = A0[m2]−m2 +k2

6+ o(k4).

C.3 C0(0, s, 0;m2,m2,m2)

C0 is defined as the scalar three-point integral

C0[k21, (k1 − k2)2, k2

2;m21,m

22,m

23] :=

16π2

i

∫ddl

(2π)d1

[l2 −m21] [(l + k1)2 −m2

2] [(l + k2)2 −m23].

The argument dependence is chosen to make it symmetric in the first threeparameters for equal masses. We require the special case

C0(s) := C0[0, s, 0;m2,m2,m2].

Introducing Feynman parameters x, y

1

ABC= 2

∫ 1

0

dx

∫ 1−x

0

dy1

[xA+ yB + (1− x− y)C]3

= 2

∫ 1

0

dx

∫ 1−x

0

dy1

[C + x(A− C) + y(B − C)]3 ,

we obtain for C0 (with iε’s supressed)

32π2

i

∫ddl

(2π)d

∫ 1

0

dx

∫ 1−x

0

dy1

[l2 −m2 + x(2k1 · l + k21) + y(2k2 · l + k2

2)]3

=32π2

i

∫ 1

0

dx

∫ 1−x

0

dy

∫ddl

(2π)d1

[l2 −m2 + x(1− x)k21 − 2xyk1 · k2 + y(1− y)k2

2)]3

=32π2

i

∫ 1

0

dx

∫ 1−x

0

dy

∫ddl

(2π)d1

[l2 −m2 + (1− x− y)(xk21 + yk2

2) + xy(k1 − k2)2]3 .

Page 94: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

94 APPENDIX C. VELTMAN-PASSARINO SCALAR LOOP FUNCTIONS

In the second equality we shifted the integration l→ l−xk1−yk2 and exchangedthe order of integration. We set k2

1 = k22 = 0, (k1 − k2)2 := sm2 and perform

the finite integral over the loop momentum2

32π2

i

∫ddl

(2π)d1

[l2 − a]3= −1

a.

We are left with the Feynman parameter integrations, which we perform againfor the simpler case s < 0. We will later analytically continue to s > 4

C0(s) = − 1

m2

∫ 1

0

dx

∫ 1−x

0

dy1

1− xys

=1

m2

∫ 1

0

dx1

sxln(1− x(1− x)s)

= − 1

sm2

[Li2

s−√−s(4− s)2

+ Li2s+

√−s(4− s)2

]

This can be simplified using the dilogarithm identity

Li2(x) + Li2

(x

x− 1

)= −1

2ln2(1− x), (for x < 1).

We choose

x :=s−

√−s(4− s)2

< 0,x

x− 1=s+

√−s(4− s)2

,

so that

C0(s) =1

2sm2ln2 (√

4− s+√−s)2

4.

We can now analytically continue to s > 4,

C0(s) =1

sm2

[ln−

(√s− 4 +

√s)2

4

]2

=1

sm2

[ln

(√s− 4 +

√s)2

4− iπ

]2

=2

sm2

[ln

√s− 4 +

√s

2− iπ

2

]2

=2

sm2

[ln

√s−√s− 4

2+

2

]2

,

where the lower branch of the logarithm has to be selected, because

Im[(√

s− 4 +√s)2]

> 0

when substituting s→ s+ iε.

2Applying the dimensional regularization formula (B.1) of appendix B gives the same resultby construction.

Page 95: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

Appendix D

Implementation of 4-BodyPhase Space Integrations

1 (∗ Phase Space I n t e g r a l ∗)

3 (∗ Convert an array o f 4− to 3−Vectors and Metric ∗)FTTs [ v s ] := vs [ [ All , 2 ; ; 4 ] ] ;

5 g=DiagonalMatrix [1 ,−1 ,−1 ,−1 ] ;

7 (∗ Unit Vector in Polar Coordinates ∗)Polar3 [ Phi , z ] :=Sqrt [1−z ˆ2 ]Cos [ Phi ] , Sqrt [1−z ˆ2 ]Sin [ Phi ] , z

9

(∗ Lorentz TF with LAMBDA −1 p0=Sqr t [ p0 ˆ2] ,011 f o r g iven FV p0 ∗)

LorentzTFtoP=Function [ p0 ,Module [gamma, beta ,13 beta=p0 [ [ 2 ; ; 4 ] ] / p0 [ [ 1 ] ] ; gamma=1/Sqrt [1−beta . beta ] ;

Flatten [Flatten [gamma,gamma beta ] ,15 Transpose [ Flatten [gamma beta ,

(Transpose [ beta ] . beta ) / beta . beta (gamma−1)17 +IdentityMatrix [ 3 ] , 1 ] ] , 1 ]

] ] ;19

(∗ 2−Par t i c l e CMS Phase−Space ,21 Parametrized by dOmega(Theta , Phi ) ∗)

23 (∗ Energy and 3−Momentum Modulus o f Pa r t i c l e 1 ∗)EOne [ s , m1 , m2 ] :=( s+m1ˆ2−m2ˆ2) /(2Sqrt [ s ] ) ;

25 POne [ s , m1 , m2 ] :=Sqrt [ EOne [ s ,m1,m2]ˆ2−m1ˆ 2 ] ;

27 (∗ Weight wrt . dOmega , i . e . dPHI2=PHI2 dOmega(Theta , Phi ) ∗)PHI2 [ s , m1 , m2 ] :=1/(4Pi ) ˆ2 POne [ s ,m1,m2] /Sqrt [ s ]

29

(∗ 4−Momenta o f Pa r t i c l e s 1 ,2 in t h i s parametr i za t ion ∗)31 pfOne [ E1 , m1 , Phi , z ] :=Flatten [E1 , Sqrt [ E1ˆ2−m1ˆ2 ] Polar3 [ Phi , z ] ] ;

pfTwo [ s , E1 , m1 , Phi , z ] := Flatten [Sqrt [ s ]−E1,−Sqrt [ E1ˆ2−m1ˆ2 ]Polar3 [ Phi , z ] ]

33

(∗Conf igurat ion ∗)35 m1=1;m2=1;m3=1;m4=1; ;

37 (∗ Phase Space Volume ,i . e . ang le i n t e g r a t i o n s car r i ed out , dOmega−>4Pi ∗)

39 PV4n [ s ] :=NIntegrate [ m12/Pi m123/Pi

95

Page 96: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

96 APPENDIX D. PHASE SPACE INTEGRATIONS

(4Pi ) ˆ3 PHI2 [ m12ˆ2 ,m1,m2] PHI2 [ m123ˆ2 ,m12 ,m3] PHI2 [ s , m123 ,m4] ,41 m123 ,m1+m2+m3, Sqrt [ s ]−m4 ,m12 ,m1+m2, m123−m3 ,

WorkingPrecision−>1543 ]

45 (∗ Ca lcu la t e the 4−Momenta in the Angle/ Invar iant−MassParametr izat ion ∗)

FinalMomenta [ s , m123 , m12 , t4 , f 4 , t3 , f 3 , t2 , f 2 ] :=47 Module [ p123 , p12 , p1 , p2 , p3 , p4 ,

(∗ Subsystem Momenta∗)49 p123=pfTwo [ s , EOne [ s ,m4, m123 ] ,m4, f4 , t4 ] ;

p12=LorentzTFtoP [ p123 ] .51 pfTwo [ m123ˆ2 ,EOne [ m123ˆ2 ,m3, m12 ] ,m3, f3 , t3 ] ;

53 p1=LorentzTFtoP [ p12 ] . pfTwo [ m12ˆ2 ,EOne [ m12ˆ2 ,m2,m1] ,m2, f2 , t2 ] ;p2=LorentzTFtoP [ p12 ] . pfOne [ EOne [ m12ˆ2 ,m2,m1] ,m2, f2 , t2 ] ;

55 p3=LorentzTFtoP [ p123 ] . pfOne [ EOne [ m123ˆ2 ,m3, m12 ] ,m3, f3 , t3 ] ;p4=pfOne [ EOne [ s ,m4, m123 ] ,m4, f4 , t4 ] ;

57

p1 , p2 , p3 , p459 ]

61 (∗ Only use numeric va lues , don ’ t e va l ua t e s ymbo l i c a l l y ∗)GetAmp [ Ampl2 , s , m123 , m12 , t4 , f 4 , t3 , f 3 , t 2 ?NumericQ, f 2 ] :=

63 Ampl2 [ s , FTTs [ FinalMomenta [ s , m123 , m12 , t4 , f4 , t3 , f3 , t2 , f 2 ] ] ]

65 (∗ Fu l l Phase Space I n t e g r a l ∗)PSI4 [ s , Ampl2 ] :=NIntegrate [

67 m12/Pi m123/PiPHI2 [ m12ˆ2 ,m1,m2] PHI2 [ m123ˆ2 ,m12 ,m3] PHI2 [ s , m123 ,m4]

69 GetAmp [ Ampl2 , s , m123 , m12 , t4 , f4 , t3 , f3 , t2 , f 2 ] ,m123 ,m1+m2+m3, Sqrt [ s ]−m4 ,m12 ,m1+m2, m123−m3 ,

71 f4 , 0 , 2Pi , t4 ,−1 ,1 , f3 , 0 , 2Pi , t3 ,−1 ,1 , f2 , 0 , 2Pi , t2 ,−1 ,1 ,Method−>”AdaptiveMonteCarlo” , ” Symbol icProcess ing ”−>0,

73 PrecisionGoal−>2]

75

(∗ Phase Space I n t e g r a l wi thout Numerical Redundancies ,77 f 2 in t e g ra t i on−>2Pi , f i x f2 value , h a l f t2 In t e g ra t i on ∗)

PSI4a [ s , Ampl2 , pg : 2 ] : = 2 (2 Pi ) NIntegrate [79 m12/Pi m123/Pi

PHI2 [ m12ˆ2 ,m1,m2] PHI2 [ m123ˆ2 ,m12 ,m3] PHI2 [ s , m123 ,m4]81 GetAmp [ Ampl2 , s , m123 , m12 , t4 , f4 , t3 , f3 , t2 , 0 ] ,

m123 ,m1+m2+m3, Sqrt [ s ]−m4 ,m12 ,m1+m2, m123−m3 ,83 f4 , 0 , 2Pi , t4 ,−1 ,1 , f3 , 0 , 2Pi , t3 ,−1 ,1 , t2 , 0 , 1 ,

Method−>”AdaptiveMonteCarlo” , ”MaxPoints”−>10ˆ7,85 PrecisionGoal−>pg

]87

(∗ Cross Sec t ion I n t e g r a l89 Flux f o r Mass less i n i t i a l Pa r t i c l e s 1/(2 s ) . ∗)

XSI4 [ s , Ampl2 ] :=1/(2 s ) PSI4a [ s , Ampl2 ] ;91 XSI4 [ s , Ampl2 , pg ] :=1/(2 s ) PSI4a [ s , Ampl2 , pg ] ;

psi4.m

Page 97: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

97

(∗ Ca lcu la t e Total Cross Sect ion ∗)2

(∗ Load Phase Space I n t e g r a l ∗)4 <<ps i 4 .m

6 (∗ Load Amplitude ∗)<<Ampl0c .m;

8 CalcAmplSquare0 [ s , ps ] := CalcAmplc [ s ,Table [ ps [ [ i ] ] , i ,Length [ ps ] ] ]

10 <<Phys ica lConstants .mf a c t o r=MassPionˆ2 ElementaryCharge ˆ4/FPionˆ4 HBarCˆ2

ConversionFmToBarn / . barn−>1//N12

SqrtS =5.014 r e s u l t = f a c t o r 10ˆ9 1/2 XSI4 [ SqrtS ˆ2 , CalcAmplSquare0 , 2 ]

(∗ 1/2 to account f o r i d e n t i c a l p a r t i c l e s ∗)

xs-4p0.m

1 (∗ Phys ica l Constants ∗)BeginPackage [ ” Phys ica lConstants ‘ ” ]

3

MassPion =139.57 MeV;5 FPion=92.4MeV;

ElementaryCharge=Sqrt [ 4Pi FineStructure \ [ Alpha ] ] ;7 FineStructure \ [ Alpha ]=1/137;

HBarC=197.036 MeV fm ;9 ConversionFmToBarn=10 10ˆ−3 barn/fm ˆ2 ;

11 EndPackage [ ] ;

PhysicalConstants.m

1 (∗ Sample Ampltude Fi le , gamma gamma −> pi− pi− pi0 pi0 ∗)CalcAmplc=Compile [ s , ps , Real , 2 , Module [

3 SPs=ConstantArray [ 0 . , 6 , 6 ] ,

5 Masses = 0 . , 0 . , 1 . , 1 . , 1 . , 1 . ,MyEn=ConstantArray [ 0 . , 6 ] ,

7 MyPs=ConstantArray [ 0 . , 6 , 3 ] ,

9

MyPs [ [ 1 ] ] = 0 . , 0 . , Sqrt [ s ] / 2 ;11 MyPs[ [ 2 ] ] = −0 . , 0 . , Sqrt [ s ] / 2 ;

Do[MyPs[ [2+ i ] ]= ps [ [ i ] ] , i , 4 ] ;13

(∗ Ca lcu la t e Energies and Sca lar Products ∗)15 MyEn=Table [ Sqrt [Sum[MyPs [ [ i , k ] ] MyPs [ [ i , k ] ] , k ,3 ]+ Masses [ [ i

] ] ˆ 2 ] , i , 6 ] ;SPs=Table [MyEn [ [ i ] ] MyEn [ [ j ] ]−Sum[MyPs [ [ i , k ] ] MyPs [ [ j , k ] ] , k , 3 ] ,

i , 6 , j , 6 ] ;17

(∗ Ca lcu la t e Spin−Averaged Amplitude ,19 i n c l ude s the averag ing f a c t o r 1/4 ∗)

( (16/(2∗ SPs [ [ 1 , 2 ] ] − 2∗SPs [ [ 1 , 3 ] ] − 2∗SPs [ [ 2 , 3 ] ] ) ˆ2 +21 ( (4 − 4∗SPs [ [ 1 , 3 ] ] ) ∗(4 + 4∗SPs [ [ 1 , 2 ] ] − 8∗SPs [ [ 1 , 3 ] ] − 4∗SPs

[ [ 2 , 3 ] ] ) ) /(4∗SPs [ [ 1 , 3 ] ] ˆ 2 ∗ ( 2 ∗ SPs [ [ 1 , 2 ] ] − 2∗SPs [ [ 1 , 3 ] ] − 2∗SPs [ [ 2 , 3 ] ] ) ˆ2)

+23 (2∗ (4 + SPs [ [ 1 , 2 ] ] − 6∗SPs [ [ 1 , 3 ] ] − 2∗SPs [ [ 2 , 3 ] ] ) ) /

( SPs [ [ 1 , 3 ] ] ∗ ( 2 ∗ SPs [ [ 1 , 2 ] ] − 2∗SPs [ [ 1 , 3 ] ] − 2∗SPs [ [ 2 , 3 ] ] ) ˆ2) +

Page 98: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

98 APPENDIX D. PHASE SPACE INTEGRATIONS

25 ( (4 − 4∗SPs [ [ 1 , 4 ] ] ) ∗(4 − 4∗SPs [ [ 2 , 3 ] ] ) ) /(16∗SPs [ [ 1 , 4 ] ] ˆ 2 ∗ SPs[ [ 2 , 3 ] ] ˆ 2 ) +

((4 + 4∗SPs [ [ 1 , 2 ] ] − 4∗SPs [ [ 1 , 3 ] ] − 8∗SPs [ [ 2 , 3 ] ] ) ∗(4 − 4∗SPs[ [ 2 , 3 ] ] ) ) /

27 (4∗ (2∗SPs [ [ 1 , 2 ] ] − 2∗SPs [ [ 1 , 3 ] ] − 2∗SPs [ [ 2 , 3 ] ] ) ˆ2∗SPs [ [ 2 , 3 ] ] ˆ 2 )+

(2∗ (4 + SPs [ [ 1 , 2 ] ] − 2∗SPs [ [ 1 , 3 ] ] − 6∗SPs [ [ 2 , 3 ] ] ) ) /29 ( (2∗ SPs [ [ 1 , 2 ] ] − 2∗SPs [ [ 1 , 3 ] ] − 2∗SPs [ [ 2 , 3 ] ] ) ˆ2∗SPs [ [ 2 , 3 ] ] ) +

(4 + 2∗SPs [ [ 1 , 2 ] ] − 4∗SPs [ [ 1 , 3 ] ] − 4∗SPs [ [ 2 , 3 ] ] ) ˆ2/31 (2∗SPs [ [ 1 , 3 ] ] ∗ ( 2 ∗ SPs [ [ 1 , 2 ] ] − 2∗SPs [ [ 1 , 3 ] ] − 2∗SPs [ [ 2 , 3 ] ] ) ˆ2∗

SPs [ [ 2 , 3 ] ] ) + 16/(2∗SPs [ [ 1 , 2 ] ] − 2∗SPs [ [ 1 , 4 ] ] − 2∗SPs [ [ 2 , 4 ] ] )ˆ2 +

33 ( (4 − 4∗SPs [ [ 1 , 4 ] ] ) ∗(4 + 4∗SPs [ [ 1 , 2 ] ] − 8∗SPs [ [ 1 , 4 ] ] − 4∗SPs[ [ 2 , 4 ] ] ) ) /

(4∗SPs [ [ 1 , 4 ] ] ˆ 2 ∗ ( 2 ∗ SPs [ [ 1 , 2 ] ] − 2∗SPs [ [ 1 , 4 ] ] − 2∗SPs [ [ 2 , 4 ] ] ) ˆ2)+

35 (2∗ (4 + SPs [ [ 1 , 2 ] ] − 6∗SPs [ [ 1 , 4 ] ] − 2∗SPs [ [ 2 , 4 ] ] ) ) /( SPs [ [ 1 , 4 ] ] ∗ ( 2 ∗ SPs [ [ 1 , 2 ] ] − 2∗SPs [ [ 1 , 4 ] ] − 2∗SPs [ [ 2 , 4 ] ] ) ˆ2) +

37 32/((2∗SPs [ [ 1 , 2 ] ] − 2∗SPs [ [ 1 , 3 ] ] − 2∗SPs [ [ 2 , 3 ] ] ) ∗(2∗SPs [ [ 1 , 2 ] ] − 2∗SPs [ [ 1 , 4 ] ] − 2∗SPs [ [ 2 , 4 ] ] ) ) +

39 (2∗ (4 + SPs [ [ 1 , 2 ] ] − 6∗SPs [ [ 1 , 3 ] ] − 2∗SPs [ [ 2 , 3 ] ] ) ) /( SPs [ [ 1 , 3 ] ] ∗ ( 2 ∗ SPs [ [ 1 , 2 ] ] − 2∗SPs [ [ 1 , 3 ] ] − 2∗SPs [ [ 2 , 3 ] ] ) ∗

41 (2∗SPs [ [ 1 , 2 ] ] − 2∗SPs [ [ 1 , 4 ] ] − 2∗SPs [ [ 2 , 4 ] ] ) ) +(2∗ (4 + SPs [ [ 1 , 2 ] ] − 2∗SPs [ [ 1 , 3 ] ] − 6∗SPs [ [ 2 , 3 ] ] ) ) /

43 ( (2∗ SPs [ [ 1 , 2 ] ] − 2∗SPs [ [ 1 , 3 ] ] − 2∗SPs [ [ 2 , 3 ] ] ) ∗SPs [ [ 2 , 3 ] ] ∗(2∗SPs [ [ 1 , 2 ] ] − 2∗SPs [ [ 1 , 4 ] ] − 2∗SPs [ [ 2 , 4 ] ] ) ) +

45 (2∗ (4 + SPs [ [ 1 , 2 ] ] − 6∗SPs [ [ 1 , 4 ] ] − 2∗SPs [ [ 2 , 4 ] ] ) ) /( SPs [ [ 1 , 4 ] ] ∗ ( 2 ∗ SPs [ [ 1 , 2 ] ] − 2∗SPs [ [ 1 , 3 ] ] − 2∗SPs [ [ 2 , 3 ] ] ) ∗

47 (2∗SPs [ [ 1 , 2 ] ] − 2∗SPs [ [ 1 , 4 ] ] − 2∗SPs [ [ 2 , 4 ] ] ) ) +( (4 − 4∗SPs [ [ 1 , 3 ] ] ) ∗(4 − 4∗SPs [ [ 2 , 4 ] ] ) ) /(16∗SPs [ [ 1 , 3 ] ] ˆ 2 ∗ SPs

[ [ 2 , 4 ] ] ˆ 2 ) +49 ( (4 + 4∗SPs [ [ 1 , 2 ] ] − 4∗SPs [ [ 1 , 4 ] ] − 8∗SPs [ [ 2 , 4 ] ] ) ∗(4 − 4∗SPs

[ [ 2 , 4 ] ] ) ) /(4∗ (2∗SPs [ [ 1 , 2 ] ] − 2∗SPs [ [ 1 , 4 ] ] − 2∗SPs [ [ 2 , 4 ] ] ) ˆ2∗SPs [ [ 2 , 4 ] ] ˆ 2 )

+51 (2∗ (4 + SPs [ [ 1 , 2 ] ] − 2∗SPs [ [ 1 , 4 ] ] − 6∗SPs [ [ 2 , 4 ] ] ) ) /

( (2∗ SPs [ [ 1 , 2 ] ] − 2∗SPs [ [ 1 , 4 ] ] − 2∗SPs [ [ 2 , 4 ] ] ) ˆ2∗SPs [ [ 2 , 4 ] ] ) +53 (4 + 2∗SPs [ [ 1 , 2 ] ] − 4∗SPs [ [ 1 , 4 ] ] − 4∗SPs [ [ 2 , 4 ] ] ) ˆ2/

(2∗SPs [ [ 1 , 4 ] ] ∗ ( 2 ∗ SPs [ [ 1 , 2 ] ] − 2∗SPs [ [ 1 , 4 ] ] − 2∗SPs [ [ 2 , 4 ] ] ) ˆ2∗55 SPs [ [ 2 , 4 ] ] ) + (2∗ (4 + SPs [ [ 1 , 2 ] ] − 2∗SPs [ [ 1 , 4 ] ] − 6∗SPs

[ [ 2 , 4 ] ] ) ) /( (2∗ SPs [ [ 1 , 2 ] ] − 2∗SPs [ [ 1 , 3 ] ] − 2∗SPs [ [ 2 , 3 ] ] ) ∗

57 (2∗SPs [ [ 1 , 2 ] ] − 2∗SPs [ [ 1 , 4 ] ] − 2∗SPs [ [ 2 , 4 ] ] ) ∗SPs [ [ 2 , 4 ] ] ) +( (4 + 2∗SPs [ [ 1 , 2 ] ] − 4∗SPs [ [ 1 , 3 ] ] − 4∗SPs [ [ 2 , 3 ] ] ) ∗

59 (−2∗SPs [ [ 1 , 3 ] ] − 2∗SPs [ [ 1 , 4 ] ] + 4∗SPs [ [ 3 , 4 ] ] ) ) /(4∗SPs [ [ 1 , 3 ] ] ∗ SPs [ [ 1 , 4 ] ] ∗ ( 2 ∗ SPs [ [ 1 , 2 ] ] − 2∗SPs [ [ 1 , 3 ] ] − 2∗SPs

[ [ 2 , 3 ] ] ) ∗61 SPs [ [ 2 , 3 ] ] ) + ( (4 + 2∗SPs [ [ 1 , 2 ] ] − 4∗SPs [ [ 1 , 4 ] ] − 4∗SPs

[ [ 2 , 4 ] ] ) ∗(−2∗SPs [ [ 1 , 3 ] ] − 2∗SPs [ [ 1 , 4 ] ] + 4∗SPs [ [ 3 , 4 ] ] ) ) /

63 (4∗SPs [ [ 1 , 3 ] ] ∗ SPs [ [ 1 , 4 ] ] ∗ ( 2 ∗ SPs [ [ 1 , 2 ] ] − 2∗SPs [ [ 1 , 4 ] ] − 2∗SPs[ [ 2 , 4 ] ] ) ∗

SPs [ [ 2 , 4 ] ] ) + ( (4 − 4∗SPs [ [ 2 , 4 ] ] ) ∗(2∗SPs [ [ 1 , 2 ] ] − 2∗SPs [ [ 1 , 3 ] ]−

65 2∗SPs [ [ 1 , 4 ] ] − 4∗SPs [ [ 2 , 3 ] ] + 4∗SPs [ [ 3 , 4 ] ] ) ) /(4∗SPs [ [ 1 , 3 ] ] ∗ ( 2 ∗ SPs [ [ 1 , 2 ] ] − 2∗SPs [ [ 1 , 4 ] ] − 2∗SPs [ [ 2 , 4 ] ] ) ∗

67 SPs [ [ 2 , 4 ] ] ˆ 2 ) + ( SPs [ [ 1 , 2 ] ] − 2∗SPs [ [ 1 , 4 ] ] − 2∗SPs [ [ 2 , 3 ] ] +4∗SPs [ [ 3 , 4 ] ] ) /( SPs [ [ 1 , 3 ] ] ∗ ( 2 ∗ SPs [ [ 1 , 2 ] ] − 2∗SPs [ [ 1 , 3 ] ] − 2∗SPs

[ [ 2 , 3 ] ] ) ∗69 SPs [ [ 2 , 4 ] ] ) + ( SPs [ [ 1 , 2 ] ] − 2∗SPs [ [ 1 , 4 ] ] − 2∗SPs [ [ 2 , 3 ] ] + 4∗

SPs [ [ 3 , 4 ] ] ) /

Page 99: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

99

( SPs [ [ 1 , 3 ] ] ∗ ( 2 ∗ SPs [ [ 1 , 2 ] ] − 2∗SPs [ [ 1 , 4 ] ] − 2∗SPs [ [ 2 , 4 ] ] ) ∗SPs[ [ 2 , 4 ] ] ) +

71 ( (4 − 4∗SPs [ [ 2 , 3 ] ] ) ∗(2∗SPs [ [ 1 , 2 ] ] − 2∗SPs [ [ 1 , 3 ] ] − 2∗SPs [ [ 1 , 4 ] ]−

4∗SPs [ [ 2 , 4 ] ] + 4∗SPs [ [ 3 , 4 ] ] ) ) /(4∗SPs [ [ 1 , 4 ] ] ∗73 (2∗SPs [ [ 1 , 2 ] ] − 2∗SPs [ [ 1 , 3 ] ] − 2∗SPs [ [ 2 , 3 ] ] ) ∗SPs [ [ 2 , 3 ] ] ˆ 2 ) +

( SPs [ [ 1 , 2 ] ] − 2∗SPs [ [ 1 , 3 ] ] − 2∗SPs [ [ 2 , 4 ] ] + 4∗SPs [ [ 3 , 4 ] ] ) /75 ( SPs [ [ 1 , 4 ] ] ∗ ( 2 ∗ SPs [ [ 1 , 2 ] ] − 2∗SPs [ [ 1 , 3 ] ] − 2∗SPs [ [ 2 , 3 ] ] ) ∗SPs

[ [ 2 , 3 ] ] ) +( SPs [ [ 1 , 2 ] ] − 2∗SPs [ [ 1 , 3 ] ] − 2∗SPs [ [ 2 , 4 ] ] + 4∗SPs [ [ 3 , 4 ] ] ) /

77 ( SPs [ [ 1 , 4 ] ] ∗ SPs [ [ 2 , 3 ] ] ∗ ( 2 ∗ SPs [ [ 1 , 2 ] ] − 2∗SPs [ [ 1 , 4 ] ] − 2∗SPs[ [ 2 , 4 ] ] ) ) +

( (4 + 2∗SPs [ [ 1 , 2 ] ] − 4∗SPs [ [ 1 , 3 ] ] − 4∗SPs [ [ 2 , 3 ] ] ) ∗79 (−2∗SPs [ [ 2 , 3 ] ] − 2∗SPs [ [ 2 , 4 ] ] + 4∗SPs [ [ 3 , 4 ] ] ) ) /

(4∗SPs [ [ 1 , 3 ] ] ∗ ( 2 ∗ SPs [ [ 1 , 2 ] ] − 2∗SPs [ [ 1 , 3 ] ] − 2∗SPs [ [ 2 , 3 ] ] ) ∗SPs[ [ 2 , 3 ] ] ∗

81 SPs [ [ 2 , 4 ] ] ) + ( (4 + 2∗SPs [ [ 1 , 2 ] ] − 4∗SPs [ [ 1 , 4 ] ] − 4∗SPs[ [ 2 , 4 ] ] ) ∗

(−2∗SPs [ [ 2 , 3 ] ] − 2∗SPs [ [ 2 , 4 ] ] + 4∗SPs [ [ 3 , 4 ] ] ) ) /83 (4∗SPs [ [ 1 , 4 ] ] ∗ SPs [ [ 2 , 3 ] ] ∗ ( 2 ∗ SPs [ [ 1 , 2 ] ] − 2∗SPs [ [ 1 , 4 ] ] − 2∗SPs

[ [ 2 , 4 ] ] ) ∗SPs [ [ 2 , 4 ] ] ) + ((−2∗SPs [ [ 1 , 3 ] ] − 2∗SPs [ [ 1 , 4 ] ] + 4∗SPs [ [ 3 , 4 ] ] ) ∗

85 (−2∗SPs [ [ 2 , 3 ] ] − 2∗SPs [ [ 2 , 4 ] ] + 4∗SPs [ [ 3 , 4 ] ] ) ) /(8∗SPs [ [ 1 , 3 ] ] ∗ SPs [ [ 1 , 4 ] ] ∗ SPs [ [ 2 , 3 ] ] ∗ SPs [ [ 2 , 4 ] ] ) +

87 ( (4∗ SPs [ [ 1 , 2 ] ] − 2∗SPs [ [ 1 , 3 ] ] − 2∗SPs [ [ 1 , 4 ] ] − 4∗SPs [ [ 2 , 3 ] ] −4∗SPs [ [ 2 , 4 ] ] + 4∗SPs [ [ 3 , 4 ] ] ) ∗(−2∗SPs [ [ 2 , 3 ] ] − 2∗SPs [ [ 2 , 4 ] ] +

89 4∗SPs [ [ 3 , 4 ] ] ) ) /(2∗(2∗SPs [ [ 1 , 2 ] ] − 2∗SPs [ [ 1 , 3 ] ] − 2∗SPs[ [ 2 , 3 ] ] ) ∗

SPs [ [ 2 , 3 ] ] ∗ ( 2 ∗ SPs [ [ 1 , 2 ] ] − 2∗SPs [ [ 1 , 4 ] ] − 2∗SPs [ [ 2 , 4 ] ] ) ∗SPs[ [ 2 , 4 ] ] ) +

91 ( (4 − 4∗SPs [ [ 1 , 4 ] ] ) ∗(2∗SPs [ [ 1 , 2 ] ] − 4∗SPs [ [ 1 , 3 ] ] − 2∗SPs [ [ 2 , 3 ] ]−

2∗SPs [ [ 2 , 4 ] ] + 4∗SPs [ [ 3 , 4 ] ] ) ) /(4∗SPs [ [ 1 , 4 ] ] ˆ 2 ∗ SPs [ [ 2 , 3 ] ] ∗93 (2∗SPs [ [ 1 , 2 ] ] − 2∗SPs [ [ 1 , 4 ] ] − 2∗SPs [ [ 2 , 4 ] ] ) ) +

((2∗ SPs [ [ 1 , 2 ] ] − 2∗SPs [ [ 1 , 3 ] ] − 2∗SPs [ [ 1 , 4 ] ] − 4∗SPs [ [ 2 , 4 ] ] +95 4∗SPs [ [ 3 , 4 ] ] ) ∗(2∗SPs [ [ 1 , 2 ] ] − 4∗SPs [ [ 1 , 3 ] ] − 2∗SPs [ [ 2 , 3 ] ] −

2∗SPs [ [ 2 , 4 ] ] + 4∗SPs [ [ 3 , 4 ] ] ) ) /(2∗SPs [ [ 1 , 4 ] ] ∗97 (2∗SPs [ [ 1 , 2 ] ] − 2∗SPs [ [ 1 , 3 ] ] − 2∗SPs [ [ 2 , 3 ] ] ) ∗SPs [ [ 2 , 3 ] ] ∗

(2∗SPs [ [ 1 , 2 ] ] − 2∗SPs [ [ 1 , 4 ] ] − 2∗SPs [ [ 2 , 4 ] ] ) ) +99 ( (4 − 4∗SPs [ [ 1 , 3 ] ] ) ∗(2∗SPs [ [ 1 , 2 ] ] − 4∗SPs [ [ 1 , 4 ] ] − 2∗SPs [ [ 2 , 3 ] ]

−2∗SPs [ [ 2 , 4 ] ] + 4∗SPs [ [ 3 , 4 ] ] ) ) /(4∗SPs [ [ 1 , 3 ] ] ˆ 2 ∗

101 (2∗SPs [ [ 1 , 2 ] ] − 2∗SPs [ [ 1 , 3 ] ] − 2∗SPs [ [ 2 , 3 ] ] ) ∗SPs [ [ 2 , 4 ] ] ) +((2∗ SPs [ [ 1 , 2 ] ] − 2∗SPs [ [ 1 , 3 ] ] − 2∗SPs [ [ 1 , 4 ] ] − 4∗SPs [ [ 2 , 3 ] ] +

103 4∗SPs [ [ 3 , 4 ] ] ) ∗(2∗SPs [ [ 1 , 2 ] ] − 4∗SPs [ [ 1 , 4 ] ] − 2∗SPs [ [ 2 , 3 ] ] −2∗SPs [ [ 2 , 4 ] ] + 4∗SPs [ [ 3 , 4 ] ] ) ) /(2∗SPs [ [ 1 , 3 ] ] ∗

105 (2∗SPs [ [ 1 , 2 ] ] − 2∗SPs [ [ 1 , 3 ] ] − 2∗SPs [ [ 2 , 3 ] ] ) ∗(2∗SPs [ [ 1 , 2 ] ] − 2∗SPs [ [ 1 , 4 ] ] − 2∗SPs [ [ 2 , 4 ] ] ) ∗SPs [ [ 2 , 4 ] ] ) +

107 ((−2∗SPs [ [ 1 , 3 ] ] − 2∗SPs [ [ 1 , 4 ] ] + 4∗SPs [ [ 3 , 4 ] ] ) ∗(4∗SPs [ [ 1 , 2 ] ] − 4∗SPs [ [ 1 , 3 ] ] − 4∗SPs [ [ 1 , 4 ] ] − 2∗SPs [ [ 2 , 3 ] ] −

109 2∗SPs [ [ 2 , 4 ] ] + 4∗SPs [ [ 3 , 4 ] ] ) ) /(2∗SPs [ [ 1 , 3 ] ] ∗ SPs [ [ 1 , 4 ] ] ∗(2∗SPs [ [ 1 , 2 ] ] − 2∗SPs [ [ 1 , 3 ] ] − 2∗SPs [ [ 2 , 3 ] ] ) ∗

111 (2∗SPs [ [ 1 , 2 ] ] − 2∗SPs [ [ 1 , 4 ] ] − 2∗SPs [ [ 2 , 4 ] ] ) ) ) ∗(1 + 2∗SPs[ [ 5 , 6 ] ] ) ˆ2) /4

113 ] ]

Ampl4c0.m

Page 100: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as
Page 101: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

Appendix E

Details on Results

We use natural units in which ~ = c = 1 and the Einstein summation convention.The metric g is written in the mainly-minus form,

gµν = gµν =

1 0 0 00 −1 0 00 0 −1 00 0 0 −1

.

E.1 Kinematical Inequality

In this section we will show how to derive the kinematical constraint on particleenergies in the center-of-mass system required for the parametrization of thethree-particle phase space. The sharp restriction is given by

E2 ≤s+m2

2 − (m1 +m3)2

2√s

.

In the center-of-mass system we have p1 + p2 + p3 = 0, thus

p22 = (p1 + p3)

2= p2

1 + p23 + 2p1 · p3 ≤ p1

2 + p23 + 2|p1||p3|,

by the Cauchy-Schwarz inequality, which becomes sharp for p2 = λp3, λ ≥ 0.Physically this means, that E2 becomes maximal, when p1 and p3 possess amomentum in the opposite direction to that of p2.

By applying a sharp (for |p1| = |p2|) binomial inequality we find

|p1|2|p3|2 = E21E

23 +m2

1m23 −m2

1E23 − E2

1m23

≤ E21E

23 +m2

1m23 − 2m1m2E1E2 = (E1E3 −m1m3)2.

Therefore, employing energy conservation,

p22 ≤ E2

1 + E23 + 2E1E3 −m2

1 −m23 − 2m1m3 = (E1 + E3)2 − (m1 +m3)2

= (√s− E2)2 − (m1 +m3)2.

Solving for E2 we obtain

E2 ≤s+m2

2 − (m1 +m3)2

2√s

.

101

Page 102: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

102 APPENDIX E. DETAILS ON RESULTS

This may be generalized to higher n-particle phase spaces,

Ej ≤s+m2

j − (mj −∑nk=1mk)2

2√s

,

where∑k 6=jmj is the lower bound on the invariant mass of the system consisting

of the all particles other than j.

E.2 Amplitude γγ −→ π+π−γ

The leading order amplitude tensor is given by

T µνρ

e3=

(k1 − 2p2)µ

(2p1 − k2)ν

(k1 − k2 + p1 − p2)ρ[

(p1 − k2)2 −m2

] [(k1 − p2)

2 −m2]

+(k1 − 2p2)

µ(k1 + k3 + p1 − p2)

ν(k3 + 2p1)

ρ[(k3 + p1)

2 −m2] [

(k1 − p2)2 −m2

]+

(2p1 − k1)µ

(k2 − 2p2)ν

(−k1 + k2 + p1 − p2)ρ[

(p1 − k1)2 −m2

] [(k2 − p2)

2 −m2]

+(k2 + k3 + p1 − p2)

µ(k2 − 2p2)

ν(k3 + 2p1)

ρ[(k3 + p1)

2 −m2] [

(k2 − p2)2 −m2

]+

(2p1 − k1)µ

(−k1 − k3 + p1 − p2)ν

(−k3 − 2p2)ρ[

(p1 − k1)2 −m2

] [(k3 + p2)

2 −m2]

+(−k2 − k3 + p1 − p2)

µ(2p1 − k2)

ν(−k3 − 2p2)

ρ[(p1 − k2)

2 −m2] [

(k3 + p2)2 −m2

]− 2gνρ (2p1 − k1)

µ

(p1 − k1)2 −m2

− 2gµρ (2p1 − k2)ν

(p1 − k2)2 −m2

− 2gµν (k3 + 2p1)ρ

(k3 + p1)2 −m2

− 2gνρ (k1 − 2p2)µ

(k1 − p2)2 −m2

− 2gµρ (k2 − 2p2)ν

(k2 − p2)2 −m2

− 2gµν (−k3 − 2p2)ρ

(k3 + p2)2 −m2

.

The partial on-shell NLO contributions are

T µνρNLO /e =C[(k1 + k2)

2]

(k3 + 2p1)ρ

(kµ2 kν1 − k1 · k2g

µν)

m2 − (k3 + p1)2

+C[(k1 + k2)

2]

(−k3 − 2p2)ρ

(kµ2 kν1 − k1 · k2g

µν)

m2 − (k3 + p2)2

−C[(k2 − k3)

2]

(2p1 − k1)µ

(kν3kρ2 − k2 · k3g

νρ)

m2 − (p1 − k1)2

−C[(k1 − k3)

2]

(2p1 − k2)ν

(kµ3 kρ1 − k1 · k3g

µρ)

m2 − (p1 − k2)2

Page 103: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

E.3. AMPLITUDE γγ −→ π+π−π0π0 103

−C[(k2 − k3)

2]

(k1 − 2p2)µ

(kν3kρ2 − k2 · k3g

νρ)

m2 − (k1 − p2)2

−C[(k1 − k3)

2]

(k2 − 2p2)ν

(kµ3 kρ1 − k1 · k3g

µρ)

m2 − (k2 − p2)2

The form factor C contains terms due to NLO contact terms and pion loops,

C(s) :=e2m2s

(4πfπ)2

(2m2

πC0(s) + 1)

+ 4πβπm3πs =: CNLO-loops + CNLO-tree.

The polarization-average/-sum of leading order soft-photon amplitude (5.3) isgiven in terms of dimensionless versions of the Mandelstam variables of section4.4 by ⟨

|T |2⟩

=⟨|Tpair|2

⟩· δsoft,

⟨|Tpair|2

⟩=

1

4e4

(−2 (3s− 2 (2s1 + s2 + t1 − t2 + 1))

1− t1

− 4 (t2 + 1) (s− s2 + t1 − 2)

(1− t2) 2− 4 (t1 + 1) (s− s1 + t2 − 2)

(1− t1) 2

− 2 (2s− 2s1 − 2s2 + t1 + t2 − 2) (s1 + s2 + t1 + t2)

(1− t1) (1− t2)

−2 (3s− 2 (s1 + 2s2 − t1 + t2 + 1))

1− t2+ 16

),

δsoft = 4e2 s(s1− 1)(s2− 1) + s12(−s2)− s1s22 + 2s1s2 + s1 + s2− 2

(s1− 1)2(s2− 1)2.

E.3 Amplitude γγ −→ π+π−π0π0

The LO amplitude tensor reads

T µν =e2

F 2

(− 2gµν

(−k1− k2 + p1)2 −m2− 2gµν

(−k1− k2 + p2)2 −m2

+(2p1− k2)

ν(−k1− 2k2 + 2p1)

µ

(m2 − (p1− k2)2) (m2 − (−k1− k2 + p1)2)

− (2p1− k1)µ

(−2k1− k2 + 2p1)ν

((p1− k1)2 −m2) (m2 − (−k1− k2 + p1)2)

− (2p2− k1)µ

(2p1− k2)ν

(m2 − (p2− k1)2) (m2 − (p1− k2)2)

− (2p1− k1)µ

(2p2− k2)ν

((p1− k1)2 −m2) ((p2− k2)2 −m2)

+(2p2− k1)

µ(−2k1− k2 + 2p2)

ν

(m2 − (p2− k1)2) (m2 − (−k1− k2 + p2)2)

+(2p2− k2)

ν(−k1− 2k2 + 2p2)

µ

(m2 − (p2− k2)2) (m2 − (−k1− k2 + p2)2)

)×((p3 + p4)2 −m2

).

Page 104: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

104 APPENDIX E. DETAILS ON RESULTS

E.4 Amplitude γγ −→ π+π−π+π−

T µν/

e2

F 2= −

2gµν((p1 + p2)2 −m2

)(−k1 − k2 + p3)2 −m2

−2gµν

((p1 + p2)2 −m2

)(−k1 − k2 + p4)2 −m2

−2gµν

((p2 + p3)2 −m2

)(−k1 − k2 + p1)2 −m2

−2gµν

((p1 + p4)2 −m2

)(−k1 − k2 + p2)2 −m2

−2gµν

((p1 + p4)2 −m2

)(−k1 − k2 + p3)2 −m2

−2gµν

((p3 + p4)2 −m2

)(−k1 − k2 + p1)2 −m2

−2gµν

((p3 + p4)2 −m2

)(−k1 − k2 + p2)2 −m2

−2gµν

((p2 + p3)2 −m2

)(−k1 − k2 + p4)2 −m2

+(2p3 − k1)

µ(−2k1 − k2 + 2p3)

ν ((p1 + p2)2 −m2

)((p3 − k1)2 −m2) ((−k1 − k2 + p3)2 −m2)

+(−k1 − 2k2 + 2p3)

µ(2p3 − k2)

ν ((p1 + p2)2 −m2

)((p3 − k2)2 −m2) ((−k1 − k2 + p3)2 −m2)

−(2p3 − k2)

ν(2p4 − k1)

µ ((p1 + p2)2 −m2

)((p3 − k2)2 −m2) ((p4 − k1)2 −m2)

−(2p3 − k1)

µ(2p4 − k2)

ν ((p1 + p2)2 −m2

)((p3 − k1)2 −m2) ((p4 − k2)2 −m2)

+(2p4 − k1)

µ(−2k1 − k2 + 2p4)

ν ((p1 + p2)2 −m2

)((p4 − k1)2 −m2) ((−k1 − k2 + p4)2 −m2)

+(−k1 − 2k2 + 2p4)

µ(2p4 − k2)

ν ((p1 + p2)2 −m2

)((p4 − k2)2 −m2) ((−k1 − k2 + p4)2 −m2)

+(2p1 − k1)

µ(−2k1 − k2 + 2p1)

ν ((p2 + p3)2 −m2

)((p1 − k1)2 −m2) ((−k1 − k2 + p1)2 −m2)

+(−k1 − 2k2 + 2p1)

µ(2p1 − k2)

ν ((p2 + p3)2 −m2

)((p1 − k2)2 −m2) ((−k1 − k2 + p1)2 −m2)

−(2p1 − k2)

ν(2p2 − k1)

µ ((−k1 + p2 + p3)2 −m2

)((p1 − k2)2 −m2) ((p2 − k1)2 −m2)

+(2p1 − k2)

ν(2p3 − k1)

µ ((−k1 + p2 + p3)2 −m2

)((p1 − k2)2 −m2) ((p3 − k1)2 −m2)

+(2p2 − k1)

µ(−2k1 − k2 + 2p2)

ν ((p1 + p4)2 −m2

)((p2 − k1)2 −m2) ((−k1 − k2 + p2)2 −m2)

+(−k1 − 2k2 + 2p2)

µ(2p2 − k2)

ν ((p1 + p4)2 −m2

)((p2 − k2)2 −m2) ((−k1 − k2 + p2)2 −m2)

−(2p2 − k2)

ν(2p3 − k1)

µ ((p1 + p4)2 −m2

)((p2 − k2)2 −m2) ((p3 − k1)2 −m2)

−(2p2 − k1)

µ(2p3 − k2)

ν ((p1 + p4)2 −m2

)((p2 − k1)2 −m2) ((p3 − k2)2 −m2)

+(2p3 − k1)

µ(−2k1 − k2 + 2p3)

ν ((p1 + p4)2 −m2

)((p3 − k1)2 −m2) ((−k1 − k2 + p3)2 −m2)

Page 105: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

E.4. AMPLITUDE γγ −→ π+π−π+π− 105

+(−k1 − 2k2 + 2p3)

µ(2p3 − k2)

ν ((p1 + p4)2 −m2

)((p3 − k2)2 −m2) ((−k1 − k2 + p3)2 −m2)

−(2p1 − k1)

µ(2p2 − k2)

ν ((−k1 + p1 + p4)2 −m2

)((p1 − k1)2 −m2) ((p2 − k2)2 −m2)

+(2p1 − k1)

µ(2p3 − k2)

ν ((−k1 + p1 + p4)2 −m2

)((p1 − k1)2 −m2) ((p3 − k2)2 −m2)

+(2p2 − k2)

ν(2p4 − k1)

µ ((−k1 + p1 + p4)2 −m2

)((p2 − k2)2 −m2) ((p4 − k1)2 −m2)

−(2p3 − k2)

ν(2p4 − k1)

µ ((−k1 + p1 + p4)2 −m2

)((p3 − k2)2 −m2) ((p4 − k1)2 −m2)

+(2p1 − k1)

µ(−2k1 − k2 + 2p1)

ν ((p3 + p4)2 −m2

)((p1 − k1)2 −m2) ((−k1 − k2 + p1)2 −m2)

+(−k1 − 2k2 + 2p1)

µ(2p1 − k2)

ν ((p3 + p4)2 −m2

)((p1 − k2)2 −m2) ((−k1 − k2 + p1)2 −m2)

−(2p1 − k2)

ν(2p2 − k1)

µ ((p3 + p4)2 −m2

)((p1 − k2)2 −m2) ((p2 − k1)2 −m2)

−(2p1 − k1)

µ(2p2 − k2)

ν ((p3 + p4)2 −m2

)((p1 − k1)2 −m2) ((p2 − k2)2 −m2)

+(2p2 − k1)

µ(−2k1 − k2 + 2p2)

ν ((p3 + p4)2 −m2

)((p2 − k1)2 −m2) ((−k1 − k2 + p2)2 −m2)

+(−k1 − 2k2 + 2p2)

µ(2p2 − k2)

ν ((p3 + p4)2 −m2

)((p2 − k2)2 −m2) ((−k1 − k2 + p2)2 −m2)

+(2p1 − k2)

ν(2p3 − k1)

µ ((−k1 + p3 + p4)2 −m2

)((p1 − k2)2 −m2) ((p3 − k1)2 −m2)

−(2p2 − k2)

ν(2p3 − k1)

µ ((−k1 + p3 + p4)2 −m2

)((p2 − k2)2 −m2) ((p3 − k1)2 −m2)

−(2p1 − k2)

ν(2p4 − k1)

µ ((−k1 + p3 + p4)2 −m2

)((p1 − k2)2 −m2) ((p4 − k1)2 −m2)

+(2p2 − k2)

ν(2p4 − k1)

µ ((−k1 + p3 + p4)2 −m2

)((p2 − k2)2 −m2) ((p4 − k1)2 −m2)

+(2p1 − k1)

µ(2p3 − k2)

ν ((−k1 + p1 + p2)2 −m2

)((p1 − k1)2 −m2) ((p3 − k2)2 −m2)

−(2p2 − k1)

µ(2p3 − k2)

ν ((−k1 + p1 + p2)2 −m2

)((p2 − k1)2 −m2) ((p3 − k2)2 −m2)

−(2p1 − k2)

ν(2p4 − k1)

µ ((p2 + p3)2 −m2

)((p1 − k2)2 −m2) ((p4 − k1)2 −m2)

−(2p1 − k1)

µ(2p4 − k2)

ν ((−k1 + p1 + p2)2 −m2

)((p1 − k1)2 −m2) ((p4 − k2)2 −m2)

+(2p2 − k1)

µ(2p4 − k2)

ν ((−k1 + p1 + p2)2 −m2

)((p2 − k1)2 −m2) ((p4 − k2)2 −m2)

Page 106: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

106 APPENDIX E. DETAILS ON RESULTS

−(2p1 − k1)

µ(2p4 − k2)

ν ((p2 + p3)2 −m2

)((p1 − k1)2 −m2) ((p4 − k2)2 −m2)

+(2p2 − k1)

µ(2p4 − k2)

ν ((−k1 + p2 + p3)2 −m2

)((p2 − k1)2 −m2) ((p4 − k2)2 −m2)

−(2p3 − k1)

µ(2p4 − k2)

ν ((−k1 + p2 + p3)2 −m2

)((p3 − k1)2 −m2) ((p4 − k2)2 −m2)

+(2p4 − k1)

µ(−2k1 − k2 + 2p4)

ν ((p2 + p3)2 −m2

)((p4 − k1)2 −m2) ((−k1 − k2 + p4)2 −m2)

+(−k1 − 2k2 + 2p4)

µ(2p4 − k2)

ν ((p2 + p3)2 −m2

)((p4 − k2)2 −m2) ((−k1 − k2 + p4)2 −m2)

Page 107: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

List of Figures

3.1 Pion-loop contributions to the γγγππ vertex function . . . . . . 35

4.1 Scattering geometry with three particles . . . . . . . . . . . . . . 434.2 Integration region of E1 in the direct parametrization of dΦ3 . . 434.3 2-Particle Decomposition of n-Particle Phase Space . . . . . . . . 464.4 2-Particle Decomposition of 4-Particle Phase Space . . . . . . . . 49

5.1 Angular distribution dσ/dϑ(γγ −→ π+π−) . . . . . . . . . . . . 565.2 Total cross section γγ −→ π+π− from NLO χ-PT . . . . . . . . . 575.3 Photon spectra of γγ −→ π+π−γ including partial NLO corrections 615.4 Magnitude of chiral correction in γγ −→ π+π−γ . . . . . . . . . 615.5

√s behaviour of the γγ −→ π+π−γ Photon Spectrum . . . . . . 62

5.6 (Partial) total cross section σtot(γγ −→ π+π−γ) . . . . . . . . . 625.7 NLO soft-photon approximation for γγ −→ π+π−γ . . . . . . . . 655.8 Soft-photon approximation for γγ −→ π+π−γ, exact phase space 655.9 Total cross section γγ −→ π+π−π+π− and γγ −→ π+π−π0π0 . . 675.10 Total cross section γπ− −→ π−π+π−π+π− from LO χ-PT. . . . 68

6.1 Leading-order diagrams for the equivalent-photon method . . . . 716.2 Experimental data on photon-photon fusion . . . . . . . . . . . . 726.3 Total cross section γγ −→ π+π− including f2(1270) . . . . . . . 756.4 Cross section γγ −→ π+π−, | cosϑ| ≤ 0.6|, including f2(1270) . . 756.5 Relative error of the photon spectrum approximation . . . . . . . 766.6 Total electromagnetic Pb83+

208 cross section for pion production . . 786.7 Total electromagnetic Pb83+

208 cross section including the f2(1270) 786.8 Mass Spectra of pion pairs in Pb83+

208 Pb83+208 collisions . . . . . . . 79

6.9 Transverse momentum distribution of pion-pairs from ALICE . . 796.10 Absolute contribution to pion-pair mass spectra of the f2(1270) . 806.11 Relative contribution to pion-pair mass spectra of the f2(1270). 80

7.1 Definition of the momentum flow and isospin indices . . . . . . . 827.2 Result of the model resummation . . . . . . . . . . . . . . . . . . 837.3 Contribution of final state interactions to γπ-vertex form factors 83

107

Page 108: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as
Page 109: Electromagnetic Interactions and Chiral Multi-Pion Dynamicscorrections to interactions of pions with the electromagnetic eld. Chapter4 treats many-particle phase-space integrals as

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Acknowledgements

I would like to utilize this last page to express my gratitude towards everyonesupporting me during the writing of this diploma thesis. Without their help, itwould not have been possible for me.

I have to thank one of the world-experts on chiral perturbation theory, Prof.Dr. Norbert Kaiser, for patiently guiding me through this glimpse of chiralperturbation theory (the quote is by Prof. Dr. Andrzej Buras).

I thank Prof. Dr. Wolfram Weise for providing me with the opportunityto write this diploma thesis at his chair, where he has managed to create avery friendly and productive working climate. I further have to thank him formaking possible my attendence at the DPG Fruhjahrstagung 2012 to presentmy results.

I have to say thank-you to Prof. Dr. Hiroshi Toki for sharing his deeppersonal insights into physics while having coffee-breaks with us.

I thank Boris Grube and Dimitri Ryabchikov for useful discussions about thephase space generator software TGenPhaseSpace and GENBOD. I further thankJan Friedrich for explaining experimental details about Primakoff scattering andultraperipheral heavy-ion reactions.

I would like to thank Michael Altenbuchinger not only for uncountably manydiscussions on physics, but also for fostering my knowledge of classical musicand my coffee addiction.

Salvatore Fiorilla has shared with me his appreciation for nature and coffee.We have enjoyed many walks along the Isar, which helped to clear my thoughts.

I would like to thank Robert Lang, with whom I have a common interest inquestions of mathematics and mathematical physics.

I thank Thomas Hell for brushing up my knowledge of theoretical mechanicswith very time-intensive tutorial work sheets, which he prepared for us withunmatchable appreciation to detail.

I have to thank Corbinian Wellenhofer, equally enthusiastic about music andphysics.

I thank Nino Bratovic, Florian Dandl, Matthias Drews, Jeremy Holt, Alexan-der Laschka, Stefan Petschauer and Sebastian Schulteß. Sadly our interactionswere suppressed by an increased space-like distance during the second half ofthe writing of this diploma thesis.

I sincerely thank Elke for continuously supporting and motivating me. Shehas been putting up with me during the entire writing of this diploma thesisand much longer.

Finally, I have to thank my family, my mother for always supporting meeven in difficult times, and Herbert for supporting my mother.

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Mehr nicht.Thomas Bernhard