Electromagnetic Interactions and Chiral Multi-Pion Dynamics

Transcription

1 Electromagnetic Interactions and Chiral Multi-Pion Dynamics Diplomarbeit von Maximilian Duell für August, 1 Betreuer: Prof. Dr. Norbert Kaiser Technische Universität München Physik-Department T39 (Prof. Dr. Wolfram Weise)

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3 Mehr Licht! Johann Wolfgang von Goethe

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5 Contents Introduction 7 1 Scalar Electrodynamics 9 Strong Interactions 11.1 Quantum Chromodynamics Low-Energy Effective Field Theory QCD in the Presence of External Fields Chiral Perturbation Theory at Order p Contact Terms Pion Propagator at Order p Renormalization of the Self-Energy Pion-Photon Coupling Calculation using Feynman-Parameters Calculation by Tensor Reduction Renormalized γππ Vertex Function Pion-Photon-Photon Coupling Two-Propagator Contribution Three-Propagator Contribution Renormalized γγππ Vertex Functions Contractions with Polarization Vectors Further Many-Particle Vertex Functions Scattering Observables and Phase Space Cross Sections and Amplitudes Kinematics of n Scattering Direct Parametrization of Two-Body Phase Space Direct Parametrization of Three-Body Phase Space Body Decompositions of n-body Phase Space Lorentz Invariance and Lorentz Transformations Body Parametrization of n-body Phase Space Hadronic Photon-Photon Fusion Pion-Pair Production γγ π + π Kinematics and Mandelstam Variables Amplitude at Leading Order Transversality-Preserving Simplification

6 6 CONTENTS Polarization-Averaged Amplitude Skeleton Expansion at Next-to-Leading Order Differential and Total Cross Section Radiative Process γγ π + π γ LO Amplitude and Skeleton Expansion Partial NLO Corrections Photon Spectrum and Partial Cross Section Soft-Photon Approximation Four-Pion Final States γγ ππππ Primakoff Scattering γπ π π + π π + π Photon Fusion and Experiment Inclusion of the f (17) Resonance The Equivalent Photon Method Outlook and Summary Resonances and Chiral Symmetry Summary A Feynman Rules 85 A.1 Cartesian Isospin Basis A. Charge Basis B Dimensional Regularization 89 C Veltman-Passarino Scalar Loop Functions 91 C.1 A (m ) C. B (k ; m, m ) C.3 C (, s, ; m, m, m ) D Phase Space Integrations 95 E Details on Results 11 E.1 Kinematical Inequality E. Amplitude γγ π + π γ E.3 Amplitude γγ π + π π π E.4 Amplitude γγ π + π π + π

7 Introduction The electromagnetic structure of pions is still an active area of research. Experimentally accessible energies are constantly increasing and high-statistics data allows to study higher-order processes, with smaller experimental signatures. New exclusive experimental data involving multi-pion final-states are constantly produced. Concerning e + e -annihilation into pions, the reaction involving a four-pion final state, e + e π + π π + π, has been recently studied in initial-state radiation measurements at BaBar [L + 1]. Theoretical predictions for such reactions at low energies can be obtained in the framework of chiral perturbation theory. For the above process this was successfully done by Ecker and Unterdorfer [EU]. In this spirit, we study two-photon processes at low energy by first revisiting the chiral perturbation theory calculation of γγ π + π at next-to-leading order. The result is used to calculate the total- and differential cross section of the photon-photon fusion process γγ π + π. We generalize the calculation to the radiative process γγ π + π γ, considering the corrections due to pion loops and the pion polarizability. The cases of soft, as well as hard bremsstrahlung are investigated. Furthermore we study photon-photon fusion processes with four pions in the final state, γγ π + π π + π and γγ π + π π π. In the light of the recently discovered new particle with mass around 15 GeV, we would also like to mention the recent proposal [BEL + 1] to further study its consistency with the Higgs particle of the Standard Model using a photon-photon collider. This diploma thesis is organized as follows. In chapter 1, we give a brief treatment of electromagnetic pion interactions in scalar quantum electrodynamics. This is put into an extended framework in chapter, where we shortly review the foundations of quantum chromodynamics and its low-energy effective field theory, chiral perturbation theory. In chapter 3, we study higher-order chiral corrections to interactions of pions with the electromagnetic field. Chapter 4 treats many-particle phase-space integrals as required for calculating total and differential cross sections. Hadronic photon-photon fusion processes are studied in chapter 5. In chapter 6, we establish a connection with currently available experimental data, featuring the inclusion of resonances and possible applications to ultraperipheral heavy-ion scattering. Chapter 7 gives an outlook on treating resonances in chiral perturbation theory by resumming to all orders. 7

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9 Chapter 1 Scalar Electrodynamics The charged pions are massive spin- bosons. It is therefore natural to describe their electromagnetic coupling by a complex scalar field ϕ(x). Its free dynamics are described by the Klein-Gordon Lagrangian density L KG = ( µ ϕ) µ ϕ m ϕ ϕ. We can incorporate photons into this theory by introducing a vector potential A µ and adding the Maxwell Lagrangian density to L KG, L = L KG + L Maxwell, L Maxwell = 1 4 F µν F µν, F µν := µ A ν ν A µ. It is invariant under the local gauge transformations A µ (x) A µ (x) + µ χ(x). The combined theory can only be quantized in a consistent manner, if all interactions are formulated respecting local gauge invariance 1. If renormalizability is used as a guiding principle, there is a unique gauge-invariant interaction. This interaction can be introduced using the gauge-covariant derivative A simple calculation shows µ D µ = µ iea µ, F µν = i e [Dµ, D ν ]. (D µ ϕ) D µ ϕ = ( µ + iea µ )ϕ ( µ iea µ )ϕ = ( µ ϕ) µ ϕ + ie [ A µ ϕ µ ϕ ( µ ϕ )A µ ϕ ] + e A µ A µ ϕ ϕ. For the perturbative treatment, the Lagrangian density is separated into free and interaction parts as L = ( µ ϕ) µ ϕ m ϕ ϕ 1 4 F µν F µν + L I, L I = iea µ (ϕ µ ϕ ϕ µ ϕ ) + e A µ A µ ϕ ϕ. 1 This point of view on the gauge principle is advocated in [Wei5], Ch

10 1 CHAPTER 1. SCALAR ELECTRODYNAMICS One obtains the corresponding momentum-space Feynman-rules as p k, ν k, µ p k 1, µ ie(p + p ) µ, ie g µν. We will find in section, that this simple description is adequate as long as pion-pion interactions can be neglected. In principle one might attempt to model pion interactions in a renormalizable way, through a quartic term (ϕ ϕ). A proper description should take into account that pions are not elementary particles, but posess a substructure of quarks and gluons. By this argument, the pion field can only be considered a low-energy effective degree-of-freedom and 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 to constrain the possible interactions of pions and photons.

11 Chapter Theoretical Descriptions of Strong Interactions All currently available experimental data on non-gravitational scales are well described within the standard model of particle physics. It is the fundamental theory 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 few hundred GeV. It is assumed to be incomplete at the interface with gravitation and at the unification scale of the three forces, where new physics is expected to appear. The explanation of the success of the standard model has shaped the idea of effective field theories, which states that effects due to physics at higher energy scales can be absorbed into non-renormalizable effective interactions, resulting only in small corrections to the physics of the effective degrees of freedom at lower 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. 1.1 Quantum Chromodynamics Quantum Chromodynamics (QCD) is defined by the Yang-Mills Lagrangian density with gauge group SU(3), written as L QCD = q(i /D m)q 1 4 GA µνg µνa. (.1) There are N f copies or flavours of the quark fields q = (q a ) f=1,...,nf, a = 1,, 3. Each of them is an element of a three-dimensional vector space acted on by the gauge group (colour-)su(3). The quarks interact by the exchange of the gauge 1 For this introduction we will loosely follow [MW7]. For literature on chiral perturbation theory see [Leu1] and [Sch3]. The physics of the standard model including QCD is exposed in [Lan9]. An introduction to effective field theories is given by Pich [Pic97]. 11

12 1 CHAPTER. STRONG INTERACTIONS bosons represented by the gluon fields G A µ, with field strengths G A µν = µ G A ν ν G A µ g s f ABC G B µ G C ν. The gauge invariance fixes the structure of the interaction through the gaugecovariant derivative D µ = µ + ig s G A µ T A. The gauge group can be parametrized in terms of the conventional generators T A, A = 1,..., 8, defining the structure constants f ABC T C := [T A, T B ]. Instead of the strong coupling strength g s one often uses the QCD finestructure constant, α s := g s 4π. We can extract α s from high energy scattering reactions, where the quarks and gluons are identified with jets of many strongly interacting particles. The value of α s can be determined by matching perturbative calculations and experimental data [Bet7] depending on the renormalization scale µ, α s (µ) =.1189 ±.1, for µ = M Z 91. GeV. (.) The parameter µ may be used to optimize the predictions obtained in perturbative calculations involving quantum corrections. Choosing a value of µ which 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 different choice of µ, can be shown to satisfy the equation α s (µ) = 1π (33 N f ) ln µ Λ QCD [, Λ QCD := µ exp ] 6π (33 N f )α s (µ) at one-loop order. This equation has positive and negative implications. On one hand, α s becomes small for large momentum scales, so that the short distance behaviour is that of a theory of free particles and therefore well-defined. This is the paradigm of high-energy physics called asymptotic freedom [Wil5]. On the other hand, for momentum scales on the order of the lightest observed hadron masses, the coupling α s becomes large. As a consequence of this, the perturbative expansion of QCD is no longer suitable to examine the properties and 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 for non-perturbative strong interaction effects is set by Λ QCD MeV. Alternative calculation methods have been successfully applied in this regime. Lattice QCD is based on numerically solving approximated versions of the Feynman path integrals on supercomputers with space-time discretized [DD6]. The approach we use is chiral perturbation theory. It replaces quarks and gluons by effective hadronic degrees of freedom and determines their interactions from a low-energy expansion of the QCD Green s functions, constrained by the symmetry properties of the Lagrangian density of QCD.

13 .. LOW-ENERGY EFFECTIVE FIELD THEORY 13. Effective Field Theory for Low Energies: Chiral Perturbation Theory The masses of the three lightest quarks are small compared to the scale of QCD, m u < m d < m s Λ QCD m c < m b < m t. To determine the low-energy particle spectrum of QCD, we may in the spirit of effective field theory restrict ourselves to the light quark flavours up, down and strange. Chiral perturbation theory relies on the smallness of their masses, when compared to the scale of strong interactions Λ QCD. Assuming the lightquark masses play only a quantitative role for the low-energy particle spectrum, we may at first set m u = m d = m s =, and only later include effects due to non-vanishing quark masses as a perturbation. In this chiral limit of QCD the approximate global SU(3)-flavour symmetry of L QCD becomes not only exact, but it is enlarged to a more restrictive chiral global symmetry group acting independently left- and right-handed quarks, q L/R := P L/R q, P L/R := 1 ( 1 γ5 ). For an element of the symmetry group (L, R) SU(3) L SU(3) R we define its action on the quark fields by q L Lq L, q R Rq R. qi /Dq = q L i /Dq L + q R i /Dq R q i /Dq = q L L i /DLq L + q R R i /DRq R = q L L Li /Dq L + q R R Ri /Dq R = qi /Dq, so that (L χ QCD) = L χ QCD 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 of the quark bilinears [Leu1], Ω Ω = v δ kj, k, j {u, d, s}. q j R qk L where v Λ 3 QCD is the order parameter of the symmetry breaking, the chiral condensate. The diagonal structure can be seen as a consequence of flavour symmetry in the chiral limit. Subjecting the quark fields to a chiral transformation, we obtain for the chiral condensate, Ω q j R qk L Ω Ω q j Ω L = Ω ( q R R ) j (Lq L ) k Ω = v (LR ) kj. R q k Therefore, the non-perturbative dynamics generating the condensate break down the chiral symmetry of the theory to its diagonal subgroup SU(3) V := { (M, M) M SU(3) } SU(3)L SU(3) R. The same derivation may also be carried out keeping only u and d, when effects due to strange particles are not important.

14 14 CHAPTER. STRONG INTERACTIONS By Goldstone s theorem, every broken symmetry generator leads to a massless Goldstone boson. Their dynamics may be described by a unitary matrix field U(x) SU(3). Identifying the flavour indices vu(x) kj q j R (x)qk L (x) allows us to demand Ω U Ω = 1 and the behaviour under chiral transformations, U LUR. (.3) We will now change our description from fundamental to effective degrees-offreedom, by assuming, that the dynamics of U(x) may be described in terms of an effective Lagrangian density L, invariant under the symmetry transformation (.3). At low energies we may expand L in the number of occurring derivative terms, L = L () + L (4) Analyzing possible invariant terms, one finds that the most general Lagrangian at lowest order is L () = 1 4 f Tr[D µ U(D µ U) ], 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 π + K + U = exp [ ] iϕ, ϕ = f π + 1 η 3 π K π η K K = λ A ϕ A. 3 η Inserting D µ U = µ U = i f λ A( µ ϕ A ) + o(ϕ A ) and Tr[λ Aλ B ] = δ AB we find that L () indeed describes the propagation of eight massless particles ϕ A, L () = 1 µϕ A µ ϕ A To conclude this section, we examine the influence of the explicit chiral symmetry breaking by the quark mass term in L QCD, q j m jk q k = q j R mjk ql k + h.c., m = m u m d. m s 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 demanding invariance of the Lagrangian density, q L mq R + h.c. q Lm q R + h.c. = q L L m Rq R + h.c., so that m m = LmR. This shows how the external field m may be included into the effective theory preserving chiral symmetry, L () = 1 4 f Tr[D µ U(D µ U) ] + v Tr[mU + Um ]. This leads, amongst others, to mass-terms for the pseudo-goldstone bosons, e.g. [ Tr m exp iϕ ] [ { }] 1,λ 1, + h.c. = Tr m 1 ϕ 1, f f λ 1, +... = const. 1 f (m u + m d )ϕ 1, + o(ϕ 3 1,),

15 .3. QCD IN THE PRESENCE OF EXTERNAL FIELDS 15 m π ± = m ϕ 1/ = v f (m u + m d ). In the following we will exclude strangeness and only consider SU() chiral perturbation theory. For our purposes of calculating electrodynamic interactions of the pseudo-goldstone bosons we will use the special parametrization of SU() given by U(x) = 1 [ ] σ(x) + i τ π(x), σ(x) = f f π (x), where the 3-vector π = (π 1, π, π 3 ) represents the pseudo-goldstone bosons in the cartesian isospin basis given by the Pauli matrices τ = (τ 1, τ, τ 3 ). This σ-gauge is very convenient, as it eliminates all LO electromagnetic interactions other than the scalar QED vertices known from chapter 1. For a more rigorous derivation of the low-energy equivalence of QCD and chiral perturbation theory relying on Ward identities see [Leu94]..3 QCD in the Presence of External Fields The inclusion of electromagnetic interactions in SU() chiral perturbation theory can be accomplished by an analysis similar to the inclusion of the mass terms. The vector potential A µ couples to the up and down quarks by means of the gauge-covariant derivative, qγ µ D µ q = qγ µ [ µ + ie q A µ ] q. The charge matrix of the quarks e q can be written as ( ) ( e q = e = e ) τ 3, so that we have generated further terms in L QCD with well-defined transformation properties under chiral symmetry. A direct analysis of the covariant derivative [Sch3, pp and pp ] shows, that the vector potential may be incorporated into SU() chiral perturbation theory at leading order by the gauge-covariant derivative D µ = µ + ie Aµ [τ 3, ]. (.4) A heuristic argument can be given by considering that [ ( τ 3, ( π π + π π )] = π + π so that e/ [τ 3, ] is just the operator assigning to the pions the correct charge eigenvalues and (.4) amounts to the standard prescription of QED. We will now briefly go back to showing that the σ-gauge is indeed convenient for calculations involving electrodynamics, as there occur no photon-pion vertices with more than two photons or pions. We obtain for the gauge-covariant derivative using the σ-gauge, U(x) = 1 [ ] σ(x) + i τ π(x), σ(x) = f f π (x), ),

16 16 CHAPTER. STRONG INTERACTIONS D µ U = µ U ie f A µ[τ 3, U] = µ U + e f A µτ a ɛ a3b π b, µ U = 1 F σ π a µ π a + i f τ a µ π a = 1 f ( π ) a σ + iτ a µ π a. Inserting this into the Lagrangian density we obtain the interaction parts, L () = 1 4 f Tr[D µ U(D µ U) ] + v Tr[mU + Um ], L π,γ I = L γππ + L γγππ. L γππ = e [( 4 Tr π ) ( a σ + iτ a ( µ π a )A µ τ c ɛ c3b π b A µ τ c ɛ c3b π b π ) ] a σ iτ a ( µ π a ) = ie 4 Tr [{τ a, τ c }] ( µ π a )A µ ɛ c3b π b = ieɛ a3b A µ π b µ π a L γγππ = e 4 Tr [τ aɛ a3b π b τ c ɛ c3d π d ] A µ A µ = e π bπ d ɛ a3b δ ac ɛ c3d A µ A µ = e (δ bd δ b3 δ d3 )π b π d A µ A µ The Feynman rules can be determined from all possible contractions in π b (p ) il γππ π a (p)γ(k, µ) and π b (p ) il γγππ π a (p)γ(k 1, µ)γ(k, ν), resulting in eɛ a3b (p + p ) µ, ie (δ ab δ a3 δ b3 ) g µν. We have thus explicitly confirmed the non-occurrence of other πγ vertices in the σ-gauge. Translating the cartesian pion fields into the charge basis, we find that the leading-order description in the σ-gauge is equivalent to scalar quantum electrodynamics. The equivalence of on-shell matrix-elements for different realizations of the Goldstone-boson fields is established by a theorem [KOS61] [Chi61].

17 Chapter 3 Pion-Structure Effects: Chiral Perturbation Theory at Order p Contact Terms The divergences generated by pion loops have to be absorbed into counterterms, which are introduced at each order in chiral perturbation theory. For the counter terms at order p 4, we use the Gasser-Leutwyler Lagrangian density for SU() chiral perturbation theory [GL84]. It can be written as [Sch3, p. 71] L GL 4 = 1 4 l { 1 Tr[Dµ U(D µ U) ] } l Tr[D µ U(D ν U) ]Tr[D µ U(D ν U) ] l { 3 Tr(χU + Uχ ) } l 4Tr[D µ U(D µ χ) + D µ χ(d µ U) ] { + l 5 Tr[f R µν Uf µν L U ] 1 Tr[f µνf L µν L + f µνf R µν R ]} + i l 6Tr[fµνD R µ U(D ν U) + fµν(d L µ U) D ν U] 1 16 l { 7 Tr(χU Uχ ) } (h 1 + h 3 )Tr[χχ ] h Tr(f L µνf µν L + f R µνf µν R ) (h 1 h 3 ) { Tr[χU + Uχ ] +Tr[χU Uχ ] Tr[χU χu + Uχ Uχ ] }. The tree-level contributions of L (4) to the scattering of two on-shell photons can be conveniently summarized by the polarizability vertex [GIS6], k, ν k 1, µ 8πiβ π m π (k 1 k g µν k µ kν 1 ), β π = α( l 6 l 5 ) 48π f m π. 17

18 18 CHAPTER 3. CHIRAL PERTURBATION THEORY AT ORDER P 4 Using the current value extracted from radiative pion decays [BT97] [GHW4] l6 l 5 = 3. ±.3, we obtain the numerical value of the polarizability at next-to-leading order β π fm Pion Propagator at Order p 4 At the order p 4 there is only one pion-loop correction to the pion propagator given by the diagram l µ p µ, a p µ, b. In this entire chapter, we work in the cartesian isospin basis. Using the Feynman rules of appendix A, the amputated contribution from this diagram, including the symmetry factor 1/, is given by 1 i f d 4 l iδ cd M abcd (π) 4 l m + iɛ, M abcd = δ ab δ cd ((p p) m ) + δ ac δ bd ((p + l) m ) + δ ad δ bc ((p l) m ). Calculating the isospin contractions, we find And we have δ cd M abcd = 3m δ ab + ((p + l) m )δ ab + ((p l) m )δ ab = δ ab ( (p + l) + (p l) 5m ), δab d 4 l (p + l) + (p l) 5m f (π) 4 l m. + iɛ (p + l) + (p l) 5m = p + l 5m = (l m + iɛ) + p 3m iɛ. Assuming only linearity of the integral, we get the regularization-scheme independent intermediate result for the pion-loop contribution Σ 1 to the self-energy at next-to-leading order, iσ 1 = 1 f [ ( d 4 l (π) 4 ) + (p 3m iɛ) d 4 ] l 1 (π) 4 l m. + iɛ At this point some regularization method has to be applied to make sense of the divergent integrals. We use the framework of dimensional regularization 1 with the space-time dimension set to d := 4 δ, 1 see appendix B d 4 l (π) 4 dd l (π) d,

19 3.3. RENORMALIZATION OF THE SELF-ENERGY 19 we have further d d l Reg D (π) d =, d d ( ) l 1 Reg D (π) d l m = im 4πµ δ Γ(δ 1) 16π m ( = im 1 ) ( δ 1 + Γ () + o(δ) 1 + δ ln 16π = im 16π ( 1 δ + 1 γ E + ln ( 4πµ m Defining ξ UV = 1 δ γ E + ln(4πµ /m ), we get The final result is Reg D ) ) + o(δ). ( 4πµ m d d l 1 (π) d l m = im 16π (ξ UV o(δ)). Σ 1 = m 3π f (p 3m ) (ξ UV + 1), where the terms linear in iɛ and δ have been dropped. 3.3 Renormalization of the Self-Energy ) ) + o(δ ) The full propagator can be expanded as a geometric series in one-particle irreducible diagrams, which can be explicitely summed to give G(p) = i p m Σ(p ) + iɛ. The self energy up to order p is renormalized by the counter-terms l 3 and l 4, which introduce shifts in the bare mass and the pion decay constant, ( f π = f [1 + m f l4 r 1 )] m ln 16π µ, [ ( m π = m 1 + m f l3 r + 1 )] m ln 3π µ. The infinite part of the counter-terms l 3 is determined by the renormalization conditions on the propagator G(p), (I) G(p) has a pole at the physical pion mass m π, m + Σ(m ) = (II) the residue at this pole is the wave-function renormalization factor, Z π = 1 + Σ (m ). The pion propagator then takes the final form [BKM95] iz π q m. π

20 CHAPTER 3. CHIRAL PERTURBATION THEORY AT ORDER P 4 The mass shift is taken into account by using the physical pion mass for calculations. The Z factor is [ ] Z π = 1 + m 1 f 16π (ξ UV + 1) l 4. There are also contributions of the l 4 counter-term to the γππ and γγππ vertex functions, which are cancelled by the Z factor. For brevity we will omit discussing those contributions in the following, using the Z factor without the l 4 contribution. 3.4 Pion-Photon Coupling Contributing to the pion-pion-photon amputated Green s function at order p 4, we have only one pion-loop diagram, given by p, b l, c k, µ. (l + k), d It includes a symmetry factor 1/ due to the symmetry under the exchange of the two loop propagators. The corresponding amplitude of the vector-pseudoscalarpseudoscalar correlation function reads in the cartesian isospin basis V P P µ NLO = i i f e ɛ c3d d d l (π) d p, a (l + k) µ M abcd [l m + iɛ] [(l + k) m + iɛ], M abcd = δ ab δ cd (k m ) + δ ac δ bd ((p l) m ) + δ ad δ bc ((p + l + k) m ). We calculate the overall isospin structure of this amplitude as ɛ c3d M abcd = ɛ c3d ( δ ab δ cd (k m ) + δ ac δ bd ((p l) m ) + δ ad δ bc ((p + l + k) m )) = ɛ a3b [ (p l) m ( (p + l + k) m )] = ɛ a3b [ (4p + k) l + k p + k ], where we used δ cd ɛ c3d =, ɛ b3a = ɛ a3b. The pion-loop correction to the pionphoton vertex has an isospin structure identical to the leading order amplitude of e ɛ a3b (p + p ) µ. ( (4p + k) l + k p + k ) V P P µ NLO = i d 4 f e ɛ l (l + k) µ a3b (π) 4 [l m + iɛ] [(l + k) m + iɛ] i =: f eɛ a3bt µ

21 3.4. PION-PHOTON COUPLING 1 Whether the Lorentz-structure of T µ also possesses this property can be determined by carrying out the loop integration Calculation using Feynman-Parameters We employ the Feynman-formula for two factors, 1 AB = 1 dx 1 (xa + (1 x)b) = 1 dx 1 (B + (A B)x). Choosing B = l m +iɛ, A = (l +k) m +iɛ, we see that B +(A B)x. This assures that the integrals exist. Now we can recombine the propagators in isotopic form by exchanging the order of integration and applying the linear substitution l µ l µ xk µ, d T µ 4 l 1 = (π) 4 dx (l + ( k)µ (4p + k) l + k p + k ) (l m + iɛ + x (l k + k )) 1 ( d 4 l (l + (1 x)k) ) µ (4p + k) (l xk) + k p + k = dx (π) 4 (l m + iɛ + x(1 x) k ). With the denominator in this form we can apply dimensional regularization 3, d d l 1 Reg D (π) d (l a) = i ( 16π ξ UV ln a ) m, d d l l µ Reg D (π) d (l a) n =, d d l l µ l ν ( Reg D (π) d (l a) = iagµν 3π ξ UV + 1 ln a ) m. To simplify the calculation we consider immediately the on-shell limit k 1 d T µ 4 l (l + (1 x)k) µ ((4p + k) l + (1 x)k p) = dx (π) 4 (l m + iɛ) ( 1 d 4 l l µ l ν ) (8p + 4k) ν d 4 = dx (π) 4 (l m + iɛ) + l (1 x) k p kµ (π) 4 (l m + iɛ). After dimensional regularization we are left with the Feynman-parameter integrals 1 dx (1 x) = 1/3, 1 dx = 1. These yield the on-shell-photon result for the pion-loop correction to the photonpion-pion interaction [ im T µ g µν ] = 4(p + k) ν 3π (ξ UV + 1) + [ ] i 3 k p kµ 16π ξ UV, T µ = (4p + k) µ Ã (m ) + k p 3m kµ Ã (m ) i 4π k p kµ. see appendix B 3 see appendix B

22 CHAPTER 3. CHIRAL PERTURBATION THEORY AT ORDER P Calculation by Tensor Reduction This calculation can be performed in a different way. This derivation is also shown here, because it illustrates the idea of Veltman-Passarino tensor reduction [PV79]. Using this method, it is possible to reduce tensor (one-)loop-functions to linear combinations of scalar one-loop integrals by solving a system of linear equations. To carry out the decomposition, we first split T µ into tensor components V µ and W µν, which only depend on the photon momentum k µ, T µ = d 4 l (l + k) µ (k p + k ) (π) 4 [l m + iɛ] [(l + k) m + iɛ] + (4p + k) ν d 4 l (π) 4 = V µ + (4p + k) ν W µν. (l + k) µ l ν [l m + iɛ] [(l + k) m + iɛ] The key observation is here, that Lorentz-symmetry permits to write V µ = k µ V, W µν = k µ k ν W 1 + g µν W. Assuming that the photon momentum is off-shell, we get V = 1 k ( V = 1 k d 4 l (π) 4 V = 1 k k µt µ, [ [k (l + k)] ] k p + k [l m + iɛ] [(l + k) m + iɛ], k (l + k) = (l + k) m (l m ), d 4 l k p + k d 4 (π) 4 [l m + iɛ] l k p + k ) (π) 4 [(l + k) m + iɛ] =. The last equality follows from translation invariance l l k of the second integral. Terms linear in ɛ will generally be dropped, because there occur no 1/ɛ singularities in scalar loop integrals. Applying a continuity argument 4 we can also extend this to on-shell photon momenta Similarily we can calculate W µν V µ =. W µν = W 1 k µ k ν + W g µν. Now we have to solve a linear system of two equations. The coefficients of this system can be determined by contracting with k µ k ν and g µν, k µ k ν W µν = W 1 k 4 + W k, g µν W µν = W 1 k + dw. (3.1) Here we have used g µν g µν = d := 4 δ for the dimension of space-time, as required by dimensional regularization. 4 If in doubt, an explicit check using a Feynman-parametrization can be used as confirmation.

23 3.4. PION-PHOTON COUPLING 3 The linear system on the right-hand side of equation 3.1 is invertible for k and we can obtain the solution for small δ, W 1 = 1 + δ 9k 4 k µ k ν W µν 3 + δ 9k g µνw µν, W = 3 + δ 9k k µk ν W µν δ g µν W µν. (3.) 9 It seems as if not much has been gained by this intermediate result, as we have 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 no more tensors occur in the numerators. g µν W µν = 3 g µν W µν = d 4 l (π) 4 (l + k) l [l m + iɛ] [(l + k) m + iɛ] (l + k) l = 3 (l m ) + 1 ((l + k) m ) + m 1 k d 4 l 1 (π) 4 (l + k) m + iɛ + 1 ( + m 1 ) k d 4 l 1 (π) 4 l m + iɛ d 4 l 1 (π) 4 [l m + iɛ] [(l + k) m + iɛ] After shifting the loop momentum l + k l in the first integral we obtain in terms of the Veltman-Passarino Loop functions 5 B (k, m 1, m ) := à (m ) := d 4 l 1 (π) 4 l m + iɛ, d 4 l (π) 4 1 [l m 1 + iɛ] [(l + k) m + iɛ]. The result, written in terms of scalar one-loop integrals is ) g µν W µν = Ã(m ) + (m k B (k ; m, m ). Now we consider the second contraction: d k µ k ν W µν 4 l (l k + k ) (l k) = (π) 4 [l m + iɛ] [(l + k) m + iɛ], l k + k = (l + k) l = ( (l + k) m ) (l m ), d k µ k ν W µν 4 l = (π) 4 = d 4 l (π) 4 l k l m + iɛ d 4 l (π) 4 (l k) k l m + iɛ = k à (m ), l k (l + k) m + iɛ 5 Note that our normalization has been choosen different for notational convenience: à = i 16π A,... (as compared to the original work [PV79])

24 4 CHAPTER 3. CHIRAL PERTURBATION THEORY AT ORDER P 4 where we have again shifted l l k. Together we have k µ k ν W µν = k à (m ), ) g µν W µν = Ã(m ) + (m k B (k ; m, m ), 1 + δ W 1 = 9k à (m ) 3 + δ [ ) 9k Ã(m ) + (m k W = 3 + δ à (m ) δ [ ) 9 9 Ã(m ) + (m k ] B (k ; m, m ), ] B (k ; m, m ). Performing the limit δ at the level of the coefficient functions will generate additional terms due to the divergences in the loop functions à and B. [ ] 1 + δ 9k à (m 1 + δ im ) = 9k 16π (ξ UV + 1) ξ UV = 1 δ γ e + ln 4πµ m 1 + δ 9k à (m ) = 4 3k Ã(m ) + im 144π k Proceeding similarly for B, we obtain the result with the limit δ shifted to be implicitly included in the loop-functions W 1 = ( ) 1 3k Ã(m ) + 6 m B 3k (k ; m, m i ) + 144π k (k 6m ), W = 1 3 Ã(m ) (4m k ) B (k ; m, m ) W µν = W 1 k µ k ν + W g µν. Finally we arrived at a form-factor representation of T µ, i 144π (k 6m ), T µ = (4p + k) ν W µν = [(4p + k) k W 1 + W ] k µ + 4W p µ. Due to 1/k terms it is not obvious that this expression is non-singular in the limit of on-shell photons. Inserting the asymtotic form of B for small k (see appendix C) m B (k ; m, m ) = Ã(m ) im 16π + ik 96π + o(k4 ), we obtain for the limit k, T µ = (4p + k) µ à (m ) + k p 3m kµ à (m ) i 4π k p kµ, V P P µ NLO = i f eɛ a3bt µ. (3.3)

25 3.4. PION-PHOTON COUPLING 5 This agrees with the result of the Feynman parametrization. The above procedure is sufficiently general to work with arbitrary one-loop tensor integrals. Its advantage is, that only the scalar loop functions have to be carefully examined for integrand cuts in the ɛ limit and the tensor integral is written as a linear combination of scalar loop functions. If the integrand is required for special values of the momenta which result in a singular determinant of the coefficient matrix of (3.1), the computation of such a limit might be more laborious, when compared with a Feynman parametrization Renormalized γππ Vertex Function To determine whether the pion-loop correction influences the scattering of onshell photons, we calculate the renormalized 1-PI amputated Green s Function. The field renormalization was given by π = Z 1/ π π R, Z π = 1 + m 16π f (ξ UV + 1) = 1 + Ã(m ). We omitted the cancelling term containing l 4. We can now calculate the renormalized γππ vertex function, V P P µ R = Z πv P P µ = Z π (V P P µ LO + V P P µ NLO ) = V P P µ LO + V P P µ NLO + (Z π 1) V P P µ LO + higher order terms = V P P µ LO + V P P µ NLO + Ã(m ) V P P µ LO + higher order terms. This is equivalent to the diagrammatic representation V P P µ R = Using the result from the last section, we get V P P µ R = eɛ a3b(p + p ) µ eɛ a3bk p 48π f ξ UV k µ. While this still contains infinities, observable on-shell amplitudes only involve contractions of the amplitude tensors with polarization vectors which satisfy ɛ(k µ, λ = ±1) k =. Therefore all observable amplitudes are already free of divergences. To obtain the full Green s function, it is necessary to also consider the treelevel contributions involving the NLO Lagrangian L (4). They may be calculated as [Sch3, p.137], el 6 ɛ a3b f ( k ɛ (p + p) + ɛ k(p p ) ).

26 6 CHAPTER 3. CHIRAL PERTURBATION THEORY AT ORDER P 4 This vanishes in the case of an on-shell photon k =, ɛ k =. The infinite part of l 6 can be used to absorb the divergence k µ for off-shell photons. The vanishing of the NLO corrections to the scattering of pions with one on-shell photon indicates that the relevant terms have to be sought in the piondiphoton correlator. 3.5 Pion-Photon-Photon Coupling The pion-loop contributions to the VVPP 1-PI Green s function are given by two diagrams V V P P µν NLO = Two-Propagator Contribution Consider the first diagram, k 1,µ p, b l, c. k,ν l + k 1 + k, d p, a Noting the symmetry factor 1/, it corresponds to the expression i i f d 4 l (π) 4 ie g µν P ± cd M(1) abcd [l m + iɛ][(l + k1 + k) m + iɛ], M (1) abcd = δ abδ cd ((k 1 +k ) m )+δ ac δ bd ((p l) m )+δ ad δ bc ((p+l+k 1 +k ) m ). We simplify the corresponding expression by calculating the isospin contractions, using the abbreviations P ± ab := δ ab δ a3 δ b3 for the projector onto the charged states and k 1 := k 1 + k. P ± cd M(1) abcd = P ± [ ab (p l) + (p + l + k 1 ) + k1 4m ] [ δ a3 δ b3 k 1 m ] = P ± [ ab l m + (l + k 1 ) m + p + (p + k 1 ) + k1 m ] [ δ a3 δ b3 k 1 m ] The factor due to interactions can therefore be seperated into two isospin parts, responsible for charged states (a, b {1, }) and neutral states (a = b = 3) respectively. It was rewritten to eliminate their tensorial structure and the

27 3.5. PION-PHOTON-PHOTON COUPLING 7 corresponding amplitude can be immediately expressed in terms of scalar loop functions. For charged pions we obtain e d 4 l l m + (l + k 1 ) m + p + (p + k 1 ) + k f gµν 1 m (π) 4 [l m + iɛ][(l + k 1 ) m + iɛ] [ = e d 4 l 1 d 4 f gµν (π) 4 (l + k 1 ) m + iɛ + l 1 (π) 4 l m + iɛ d 4 l p + (p + k 1 ) + k1 m ] + (π) 4 [l m + iɛ][(l + k 1 ) m + iɛ] [ = e f gµν Ã(m ) + (p + (p + k 1 ) + k1 m ) B ] (k1; m, m ). For completeness we also give the term relevant for neutral pion scattering, e f gµν δ a3 δ b3 d 4 l k1 m (π) 4 [l m + iɛ][(l + k 1 ) m + iɛ] = e f gµν δ a3 δ b3 (m k 1) B (k 1; m, m ), where δ a3 δ b3 projects onto the neutral pion states, a = b = 3. This is an interesting feature: a neutral particle can interact electromagnetically by quantum-correction terms. In the underlying theory of QCD this is obvious, because neutral pions are composite particles with charged constituents. The situation is analogous to neutral atoms, which possess non-vanishing higher multipole moments Three-Propagator Contribution The second diagram involves three pion propagators and has a symmetry factor of unity. k 1,µ p, b l k 1, c l, d l + k, e i 3 i f k,ν p, a d 4 l eɛ c3d (l k 1 ) µ eɛ d3e (l + k ) ν M () abce (π) 4 [(l k 1 ) m + iɛ][l m + iɛ][(l + k ) m + iɛ], where we write again for the four-pion interaction M () abce = δ abδ ce (k 1 m )+δ ac δ be ((p l+k 1 ) m )+δ ae δ bc ((p+l+k ) m ).

28 8 CHAPTER 3. CHIRAL PERTURBATION THEORY AT ORDER P 4 Note that we have ɛ c3d ɛ d3e = P ce, ± P cep ± ce ± = P ceδ ± ce =. We split the isospin contraction into parts responsible for charged and neutral pions, P cem ± () abce = P ± [ ab (p + l + k ) + (p l + k 1 ) + k1 4m ] [ δ a3 δ b3 k 1 m ] = P ± [ ab (l k1 ) m + (l + k ) m + p + p k 1 + k1 m ] δ a3 δ b3 [ k 1 m ]. Considering only charged external pions (a, b {1, }, so P ± ab = 1) we are left with { e d 4 l (l k 1 ) µ (l + k ) ν f (π) 4 [l m + iɛ][(l + k ) m + iɛ] d 4 l (l k 1 ) µ (l + k ) ν + + d 4 l (π) 4 (π) 4 [(l k 1 ) m + iɛ][l m + iɛ] ( p + p k 1 + k1 m ) (l k 1 ) µ (l + k ) ν [(l k 1 ) m + iɛ][l m + iɛ][(l + k ) m + iɛ] }. (3.4) A tensor reduction would involve more possible tensors for this diagram. Therefore it would be more tedious as compared to the example of the correction to the photon-pion-pion correlator. The available tensors are given by g µν, k µ 1 kν, k µ kν 1, k µ 1 kν 1, k µ kν. If we are only interested in contractions with polarization vectors ɛ 1 and ɛ, only the first two tensors in this list are relevant. The coefficients of the last two are related by the symmetry of the amplitude under (l, k 1, µ) ( l, k, ν). For simplicity we will employ a Feynman-parametrization. the first term, d 4 l (l k 1 ) µ (l + k ) ν (π) 4 [l m + iɛ][(l + k ) m + iɛ] 1 d 4 l = dx (π) 4 = 1 dx d 4 l (π) 4 (l k 1 ) µ (l + k ) ν [l m + iɛ + x(l k + x k )] (l k 1 xk ) µ (l + (1 x)k ) ν [l m + iɛ + xk ], Consider now where we shift the integral l l xk to eliminate the inhomogeneity in the denominator. Performing the on-shell photon limit k1/, we can carry out the Feynman-parameter integral 1 dx (l k 1 xk ) µ (l + (1 x)k ) ν = 1 3 kµ kν k µ 1 lν + 4l µ l ν. After dimensional regularization 6 Reg D 6 see appendix B. d d l (π) d 1 (l m ) = i 16π ξ UV = 1 m Ã(m ) i 16π,

29 3.5. PION-PHOTON-PHOTON COUPLING 9 Reg D Reg D d d l l µ (π) d (l m ) n =, d d l (π) d l µ l ν (l m ) = im g µν 3π (ξ UV + 1) = 1 gµν Ã (m ), we are left with d 4 l (π) 4 (l k 1 ) µ (l + k ) ν [l m + iɛ][(l + k ) m + iɛ] = 1 3m kµ kν (Ã (m ) im 16π ) + g µν Ã (m ). The second term results, if we exchange k 1 k, µ ν, (l k 1 ) µ (l + k ) ν [(l k 1 ) m + iɛ][l m + iɛ] = 1 3m kµ 1 kν 1 (Ã (m ) im 16π d 4 l (π) 4 ) + g µν Ã (m ). Now only the third integral remains to be worked out, d 4 l (π) 4 (l k 1 ) µ (l + k ) ν [(l k 1 ) m + iɛ][l m + iɛ][(l + k ) m + iɛ]. In this case we need a Feynman-formula for 3 factors, 1 ABC = = 1 1 dx dx 1 x 1 x dy dy 1 [xa + yb + (1 x y)c] 3 1 [C + x(a C) + y(b C)] 3, C = l m + iɛ, A = (l k 1 ) m + iɛ, B = (l + k ) m + iɛ. It has to be assured that we do not integrate over a pole in the denominator. This is the case, as we have Im C = ɛ, Im(A C) = Im(B C) =. This allows us to rewrite the second term as d 4 l (π) 4 1 dx 1 x dy (l k 1 ) µ (l + k ) ν [l m + iɛ + x(k 1 l k 1) + y(k + l k )] 3. The denominator is homogeneous after the transformation l l + xk 1 yk, 1 dx 1 x dy d 4 l (π) 4 (l (1 x)k 1 yk ) µ (l + xk 1 + (1 y)k ) ν [l m + iɛ + xy(k 1 k ) x(x 1)k 1 y(y 1)k ]3. We take the limit of on-shell photons k1/ and introduce the dimensionless variable k 1 k =: sm, 1 dx 1 x dy d 4 l (π) 4 (l (1 x)k 1 yk ) µ (l + xk 1 + (1 y)k ) ν [l m + iɛ + xysm ] 3.

30 3 CHAPTER 3. CHIRAL PERTURBATION THEORY AT ORDER P 4 Dimensional regularization gives d d l 1 Reg D (π) d (l a) 3 = i 1 3π a, d d l l µ Reg D (π) d (l a) 3 =, d d l l µ l ν ( Reg D (π) d (l a) 3 = igµν 64π ξ UV ln a ) m. We are left with the two contributions 8ig µν 64π 1 dx 1 x = igµν 16π ξ UV igµν 8π dy [ξ UV ln(1 xys iɛ)] 1 dx 1 x dy ln(1 xys iɛ), i 3π 1 dx 1 x dy ((1 x)k 1 + yk ) µ (xk 1 + (1 y)k ) ν. 1 xys iɛ It remains to calculate the two finite loop-functions, L 1 (s) := L µν := 1 1 dx dx 1 x 1 x dy ln(1 xys iɛ), dy ((1 x)k 1 + yk ) µ (xk 1 + (1 y)k ) ν. 1 xys We calculate the loop function for the case s <, where we do not need to worry about the branch cut of the logarithm. In the end we can analytically continue our result to the relevant case s > 4, if needed letting s s + iɛ to restore the imaginary part. y dy ln(1 xys) = 1 xys xs ln(1 xys) y 1 1 x(1 x)s dx ln [1 x(1 x)s] (1 x) xs 4 s 4 s s 4 s s = 4 s ln + 4 s + s s tanh 1 4 s [ ] [ ] + 1 s Li s s s(4 s) s Li s s 3 s(4 s) This can be simplified using the dilogarithm identity ( ) x Li (x) + Li = 1 x 1 ln (1 x), (for x < 1). Choosing x = s s s(4 s) < 1, x x 1 = s s +, we get s(4 s)

31 3.5. PION-PHOTON-PHOTON COUPLING 31 [ ] [ ] 1 s Li s s s(4 s) s Li s s s(4 s) = 1 ( 4 s s) s ln. 4 We observe further, that s tanh 1 4 s = 1 4 s + s 4 s + s ln = ln, 4 s s and we get the simplified result for s < 1 dx 1 xys sx ln(1 xys) y = 1 ( 4 s s) s ln s 4 s + s + ln. s In this representation it can be analytically continued 7 to s > 4, L 1 (s) = 1 [ln ( s + s 4) iπ ] 3 s 4 s 4 + s s 4 + [ln + iπ ]. s Now we turn to the second loop function. Again for s <, 1 1 x L µν := dx dy ((1 x)k 1 + yk ) µ (xk 1 + (1 y)k ) ν 1 xys = 4kµ kν 1 + (4 + s)k µ [ 1 kν 1 ( 4 s s) s ln + s ] 4 [ + k µ s 16 1 kν 8 ] 4 s 4 s s ln. s s 3 And by analytic continuation to s > 4, L µν = 4kµ kν 1 + (4 + s)k µ 1 kν + k µ 1 kν s [ s 16 s [ 1 (ln ( s + s 4) 4 s s 4 8 s 4 s 3 ( ln iπ ) ] + s ) ]. iπ We arrived now at the solution of the third term of (3.4), d 4 l (π) 4 ( p + p k 1 + k1 m ) (l k 1 ) µ (l + k ) ν [(l k 1 ) m + iɛ][l m + iɛ][(l + k ) m + iɛ] = i k 16π ( g µν ξ UV g µν L 1 (s) + L µν (s)), where we defined k := p + p k 1 + k1 m. We collect the final result { V V P P µν (loop ) = e 1 f P ± ab 3m (kµ kν + k µ 1 kν 1 ) k e { ig f P ± µν ab 16π ξ UV igµν 8π L 1(s) + 7 see also appendix C. ) } (Ã (m ) im 16π + 4g µν Ã (m ) i 16π Lµν },

32 3 CHAPTER 3. CHIRAL PERTURBATION THEORY AT ORDER P 4 where à (m ) = im 16π (ξ UV + 1), k := p + p k 1 + k 1 m = k 1 + p + p m. For pions and photons both on-shell, we have k = sm. For the purpose of renormalization we rewrite in terms of ξ UV, V V P P µν (loop ) = ie 16π f P ± ab { 1 3 (kµ kν + k µ 1 kν 1 ) ξ UV + 4m g µν (ξ UV + 1) } + k [g µν ξ UV g µν L 1 (s) + L µν ]. Note that we have also calculated the necessary ingredients for external neutral pions, e f δ a3δ b3 d 4 l (π) 4 ( k 1 m ) (l k 1 ) µ (l + k ) ν [(l k 1 ) m + iɛ][l m + iɛ][(l + k ) m + iɛ] = ie δ a3 δ b3 8π f ( k 1 m ) ( g µν ξ UV g µν L 1 (s) + L µν (s)) Renormalized γγππ Vertex Functions We summarize the results. For neutral pions there occurs no wave function renormalization, because VVPP has no LO terms with neutral pions. The result is indeed finite, V V P P µν NLO = V V P P µν (loop 1) + V V P P µν (loop ) = ie 8π f ( k 1 m ) { g µν B +g µν ξ UV g µν L 1 (s) + L µν (s)} = ie 8π f ( k 1 m ) { g µν L 3 (s) + L µν (s)}, where we inserted the definition of B, s 4 s + s 4 B = ξ UV + B f = ξ UV [ln + iπ ] +, s to cancel the divergence. We further summarized 8 L 3 (s) := L 1 (s) B f (s) = 4 s [ s s 4 ln + iπ ] In terms of Passarino-Veltman functions this cancellation of logarithms is easier seen. It occurs as a cancellation of the B from the first loop diagram with an equal B from the tensor decomposition of the second loop diagram.

33 3.5. PION-PHOTON-PHOTON COUPLING 33 For charged pions and on-shell photons we have V V P P µν LO = ie g µν, V V P P µν (Pol.) = 8πiβ π m π (k µ kν 1 k 1 k g µν ), V V P P µν (loop 1) = e V V P P µν (loop ) = ie 16π f [ f gµν Ã +k B ] (sm ), { 1 3 (kµ 1 kν 1 + k µ kν ) ξ UV + 4m g µν (ξ UV + 1), } + k [g µν ξ UV g µν L 1 (s) + L µν ]. V V P P µν R = Z π(v V P P µν LO + V V P P µν NLO1 + V V P P µν NLO + V V P P µν Pol ) Z π = 1 + m 16π f (ξ UV + 1) V V P P µν R = We find that the divergence is lifted from the physically relevant terms, which do not contain k µ 1 or kν : V V P P µν R = ie g µν + 8πiβ π m π P ± ab (kµ kν 1 k 1 k g µν ) + e f gµν k B (sm ) { } 1 3 (kµ 1 kν 1 + k µ kν ) ξ UV + k [g µν ξ UV g µν L 1 (s) + L µν ]. ie 16π f Similar to the neutral case, the divergence proportional to g µν is cancelled if we insert the definition of B, B (k ; m 1 = m = m ) = i ( ) 16π ξ UV + B f (k ). V V P P µν R = ie g µν + 8πiβ π m π P ± ab (kµ kν 1 k 1 k g µν ) { ie 1 16π f P ± ab 3 (kµ 1 kν 1 + k µ kν ) ξ UV + k [ L µν + g µν L 3 (s) ]} Observable amplitudes are again finite. For real photons the contraction with two polarization vectors will eliminate the divergence proportional to k µ 1 kν 1 + k µ kν.

34 34 CHAPTER 3. CHIRAL PERTURBATION THEORY AT ORDER P Relevant Results for Contractions with Photon Polarization Vectors For on-shell pions and real photons we have k = k 1 k and ɛ k =. Further we have now all ingredients to calculate the amplitude of γγ π + π at nextto-leading order. V P P µ R = eɛ a3b(p + p ) µ + eɛ a3b m A (m ) 48π f m k p k µ V V P P µν R = ie P ± ab gµν + 8πiβ π m π P ± ab (kµ kν 1 k 1 k g µν ) ie 16π f P ± ab { 1 3 (kµ 1 kν 1 + k µ kν ) ξ UV + k [L µν + g µν L 3 ] ie δ a3 δ b3 ( k 8π f 1 m ) { g µν L 3 + L µν } k := p + p k 1 + k 1 m = k 1 + p + p m L 3 (s) = m C [, sm, ; m, m, m ] + 1 L µν = 4kµ kν 1 + (4 + s)k µ 1 kν + k µ 1 kν s [ s 16 s [ 8 s 4 s 3 ( s + s 4 ln ( ln = 4kµ kν 1 + (4 + s)k µ 1 kν L 3 + k µ 1 s kν iπ ) ] + s s s 4 iπ ) ] [ 1 4 ] s B f When calculating amplitudes, Lorentz indices associated with external photons will be contracted with the respective polarization vectors ɛ(k, λ). Due to the gauge invariance of QED we have k ɛ(k, λ) =. Therefore when later contracting with polarization vectors we may then drop terms involving the photon momentum and the corresponding Lorentz index, L µν + g µν L 3 = (k 1 k g µν k µ kν 1 ) L 3(s) k 1 k. } V P P µ R = eɛ a3b(p + p ) µ [ V V P P µν R = ie P ± ab gµν + P ± ab (kµ kν 1 k 1 k g µν ) 8πiβ π m π + ie kl ] 3 16π k 1 k ( ) + ie δ a3 δ b3 k 8π fπ (k µ kν 1 k 1 k g µν ) 1 m π L3 k 1 k It should be said that this simplification comes at the price of ruining Ward identities if not carefully applied. 9 9 see sections and 5..

35 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 involving many particles, the corresponding vertex functions need to be calculated. Considering the process γγ π + π γ, we require at NLO the γγγππ vertex function, which involves the diagrams shown in figure 3.1. The expression for the amplitude corresponding to a) involves a tensor of rank four and four propagators. Introducing the abbreviation k 1 := k 1 + k, it is given by ie 3 ɛ a3b f d 4 l (π) 4 (l + k 1 ) µ (l + k 1 + k 1 ) ν (l k 3 ) ρ [(l k 3 ) m ] [l m ] [(l + k 1 ) m ] [(l + k 1 ) m ] { p (l + k 1 k 3 ) (l k 3 ) + (l + k 1 ) }. These diagrams are in principle tractable by automatized tensor reduction methods, which were carried out by hand in this chapter. The resulting expressions become very large and involve the scalar loop functions C and D, which for general momenta can only be treated numerically.

36

37 Chapter 4 Scattering Observables and Phase Space 4.1 Cross Sections and Amplitudes To derive the formulas relating experimentally observable cross-sections to the Green s functions, we use the finite volume approach to quantum mechanical scattering theory. 1 Consider the situation of two initial particles scattering and producing n particles in the final state, i 1, i f 1,..., f n. The particles are on the mass shell and the total four momentum is conserved, k ij = m ij, k fj = m fj and k i = k f, k i := k i1 + k i, k f := k f1 + + k fn. To handle the square of the overall energy-conservation delta function in the transition probability P, we restrict the Fourier representation of the fourdimensional delta function to a finite four volume of extent V T = L 3 T, V T = d 4 l e i l = (π) 4 δ (4) (). T V The transition probability can be obtained from matrix elements by P = f i f f i i = (π)4 δ (4) (k f k i ) V T T fi (V E f1... V E fn )(4V E i1 E i ). Here we used the definition of the T matrix, (π) 4 δ (4) (k f k i )T fi := f i, 1 In this chapter we loosely follow [Sre7], ch. 11. For the more rigorous approach to scattering theory using wave packets, see [PS95], pp

38 38 CHAPTER 4. SCATTERING OBSERVABLES AND PHASE SPACE and the relativistic normalization of free-particle states p p = (π) 3 E δ (3) (p 1 p 1 ) = E d 3 k e ik (p1 p1) = EV, p 1... p n p 1... p n = j p j p j = V E 1 V E n. Due to the quantization of momenta in finite volumes, the differential cross section in a finite volume is not a continuous, but a discrete distribution on the space of possible momenta. The differential cross section σ in the laboratory frame is now defined as the transition probability per time divided by the flux of particles, ( vi ) 1 P σ f i = V T. By v i 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/dφ n on the phase space Φ n of all possible momenta. It is given by the manifold depending on the initial momentum k i = (ki, k i), Φ n (k i ) := (k f1,..., k fn ) R 3n ; k fj = k i, k fj m fj = k i. j j For convenience we would like to use a Lorentz-invariant integration measure 3 on Φ n, with respect to which we can define the differential cross-section, dφ n = dφ n (k f1,..., k fn ; k i ) = d4 k f1 (π) 4... d4 k fn n (π) 4 πδ(kfj m fj)θ(kfj) ( (π) 4 δ (4) k i ) k fj j=1 = d3 k f1 d 3 k ( fn (π) 3... E 1 (π) 3 (π) 4 δ (4) k i ) k fj, (4.1) E n E j := k fj m 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 necessary to sum over a small range of momenta. In the continuum limit this sum is replaced by an integral, V (π) 3 d 3 k fj. k fj Mathematically speaking, the differential cross section dσ/dφ n is the Radon-Nikodym derivative of the scattering distribution with respect to the measure dφ n. 3 It is given by the standard construction d 4 p (π) 4 πδ(p m )Θ(p ) = d4 p (π) 4 p πδ(p p + m ) = d3 p (π) 3 p. When using the d 3 p -measure, we understand the integral over p carried out using the delta function, so that all occurences of p in the integrand have been set to p = p + m.