Techniques for the calculation of electroweak radiative corrections at the one-loop level and results for W-physics at LEP200
aa r X i v : . [ h e p - ph ] S e p Techniques for the calculation of electroweakradiative corrections at the one-loop leveland results for W -physics at LEP200 A. Denner
Physikalisches Institut, Universit¨at W¨urzburg8700 W¨urzburg, Germany
Abstract:
We review the techniques necessary for the calculation of virtual electroweak and softphotonic corrections at the one-loop level. In particular we describe renormalization,calculation of one-loop integrals and evaluation of one-loop Feynman amplitudes. Wesummarize many explicit results of general relevance. We give the Feynman rules and theexplicit form of the counter terms of the electroweak standard model, we list analyticalexpressions for scalar one-loop integrals and reduction of tensor integrals, we present thedecomposition of the invariant matrix element for processes with two external fermionsand we give the analytic form of soft photonic corrections. These techniques are appliedto physical processes with external W -bosons. We present the full set of analytical for-mulae and the corresponding numerical results for the decay width of the W -boson andthe top quark. We discuss the cross section for the production of W -bosons in e + e − -annihilaton including all O ( α ) radiative corrections and finite width effects. ImprovedBorn approximations for these processes are given. ontents N ≤ S → F ¯ F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.2.2 V → F ¯ F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.2.3 F ¯ F → SS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365.2.4 F ¯ F → SV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365.2.5 F ¯ F → V V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365.3 Calculation of standard matrix elements . . . . . . . . . . . . . . . . . . . 37 W → f i ¯ f ′ j for massless fermions . . . . . . . . . . . . . . . . . . 436.4 Computeralgebraic calculation of one-loop diagrams . . . . . . . . . . . . . 46 Soft photon bremsstrahlung 47 m t contributions . . . . . . . . . . . . . . . . . . . . . . . 558.2.3 Recipes for leading universal corrections . . . . . . . . . . . . . . . 56 W -boson 57
10 The top width 70 W -pair production 76
12 Conclusion 95A Feynman rules 97B Self energies 105C Vertex formfactors 109D Bremsstrahlung integrals 113
Introduction
All known experimental facts about the electroweak interaction are in agreement withthe Glashow-Salam-Weinberg (GSW) model [1, 2, 3, 4]. Therefore, this theory is calledthe standard model (SM) of electroweak physics. Despite its extraordinary experimentalsuccess it is by no means tested in its full scope. Many more experimental and theoreticalefforts are needed for its further confirmation.An important step in this direction is provided by the e + e − colliders SLC and LEP100which started a new era of precision experiments. The first important results from theseexperiments were the determination of the number of light neutrinos and the precisemeasurement of the mass of the neutral weak gauge boson, the Z -boson [5]. Furthermorethe total and partial widths of the Z -boson and various on-resonance asymmetries havebeen determined and will be measured with increasing accuracy. These experiments willuniquely allow to study in great detail all the properties of the Z -boson and its couplingsto fermions.There are, however, ingredients of the electroweak SM, which are not directly accessibleat SLC and LEP100. The most important one is probably the gauge boson self-interactionwhich is crucial for the nonabelian structure of the GSW model. It will be directly testedfor the first time at LEP200, the upgraded version of LEP. There the center of massenergy will be high enough to produce pairs of charged weak gauge bosons, the W -bosons,such that one can study the reaction e + e − → W + W − in great detail. It will allow theinvestigation of the nonabelian three-gauge boson interactions γW + W − and ZW + W − atthe classical level of the theory. Moreover, all the properties of the W -boson, like its massand its total and partial widths can be measured directly there. The statistics will not beas good as on the Z -peak. One expects of the order of 10 W -pairs and thus an accuracyat the percent level. The examination of several independent methods indicates that anerror of about 0.1% for the W -mass determination can be reached [6].Theoretical predictions should have an accuracy comparable to or even better thanthe experimental errors. If the experimental precision is of the order of one percent theclassical level of the theory is no longer sufficient. One is forced to take into account quan-tum corrections: the radiative corrections. In the case of the electroweak SM these canreach several percent. For the high precision experiments at LEP100 even the first ordercorrections are inadequate, one has to take into account leading higher order corrections,too.Radiative corrections are not only compelling for the precise comparison betweenthe theoretical predictions and the experimental results, but offer the possibility to getinformations about sectors of the theory that are not directly observable. While the directinvestigation of certain objects may not be possible because the available energy is toosmall to produce them they may affect the radiative corrections noticeably.In the electroweak SM there are at least two such objects. The top quark, the stillundiscovered constituent of the third fermion generation, and the Higgs boson, the phys-ical remnant of the Higgs-Kibble mechanism of spontaneous symmetry breaking. Bothparticles seem to be too massive to be produced directly in the existing colliders. How-ever, the high precision experiments performed so far together with the precise knowledgeof the radiative corrections of the electroweak SM already allow to derive limits on themass of the top quark within the SM [7, 5]. Since the sensitivity of radiative corrections1o the mass of the Higgs boson is weaker, the restrictions on this parameter are at presentonly marginal [8]. The situation may improve with increasing experimental accuracy.While direct determinations of physical parameters are in general to a large extent modelindependent, the information extracted from radiative corrections depends on the entirestructure of the underlying theory.Finally there is a third important issue concerning radiative corrections. It is likelythat the electroweak SM, despite its experimental success, is only an effective theory,the low-energy approximation of a more general structure. This would manifest itselftypically in small deviations from the SM predictions. Furthermore most of the presentlydiscussed new physics is connected with scales bigger than the experimentally accessibleenergies. Therefore new phenomena will show up predominantly via indirect effects ratherthan via direct production of new particles. In order to disentangle these small effectsone has to know once again the predictions of the SM accurately and thus needs radiativecorrections.The actual evaluation of the radiative corrections is a tedious and time consuming task.It requires extensive calculations involving many different techniques, like renormalization,evaluation of loop integrals, Dirac algebra calculations, phase space integrations and soon. Fortunately the whole procedure can be organized into different independent steps.Furthermore many steps can be facilitated with the help of computer algebra [9, 10, 11, 12].For the interesting processes at LEP100 radiative corrections have been calculated bymany authors [13]. Their structure is relatively simple since the masses of the externalfermions can be neglected. Calculations for gauge boson production processes at LEP200are already more complicated because the masses of the external gauge bosons are non-negligible. Such calculations have been performed by several groups and we will givethe most important results in the second part of this review. The whole complexity ofone-loop corrections will show up when considering reactions where all external particlesare massive like e.g. gauge boson scattering processes which may be investigated at theLHC or SSC. The calculation of radiative corrections to these processes has just started.In the first part of this review we collect the relevant formulae and techniques necessaryfor the calculation of electroweak one-loop radiative corrections. Although we discusseverything in the context of the SM the presented material is – apart from the explicitform of the renormalization constants – applicable to extended models as well. In thesecond part these methods are applied to physical processes with external W -bosons. Thispart not only gives examples for the calculation of one-loop electroweak corrections, butalso provides a survey on the status of radiative corrections for the production and thedecay of W -pairs in e + e − annihilation. The corresponding experiments will be carriedthrough in a few years at LEP200.The general techniques described in this paper are restricted to the virtual part of theelectroweak corrections and soft photon bremsstrahlung. We do not consider the methodsappropriate for hard photon bremsstrahlung. This can be efficiently treated using spinortechniques [14] and Monte Carlo simulations [15]. Furthermore we do not touch themethods developed for calculating higher order QCD corrections.This paper is organized as follows:In chapter 2 we specify the Lagrangian of the electroweak SM. Chapter 3 outlinesthe on-shell renormalization for the physical sector of the electroweak SM and providesexplicit expressions for the counter terms. All relevant formulae for the calculation of one-2oop Feynman integrals are collected in chapter 4. In chapter 5 we introduce the standardmatrix elements, a concept which allows to represent the results for one-loop diagrams ina systematic and simple way. In chapter 6 we show how everything is put together in theactual calculation of one-loop amplitudes and provide first simple examples. The relevantformulae for the calculation of the soft photon corrections are summarized in chapter 7.Chapter 8 serves to define our input parameters and the way of resumming higher ordercorrections.The remaining chapters are devoted to applications. In chapter 9 we give results forthe width of the W -boson, in chapter 10 for the width of the top quark. Finally theradiative corrections to the production of W -pairs in e + e − annihilation are discussed inchapter 11.The appendices contain the Feynman rules of the electroweak SM, the explicit expres-sions for the self energies of the physical particles and the vertex functions as well as thebremsstrahlung integrals relevant for the W -boson and top quark decay width.3 The Glashow-Salam-Weinberg Model
The Glashow-Salam-Weinberg (GSW) model of the electroweak interaction has beenproposed by Glashow [1], Weinberg [2], and Salam [3] for leptons and extended to thehadronic degrees of freedom by Glashow, Iliopoulos and Maiani [4]. It is the presently mostcomprehensive formulation of a theory of the unified electroweak interaction: theoreticallyconsistent and in agreement with all experimentally known phenomena of electroweak ori-gin. For energies that are small compared to the electroweak scale it reproduces quantumelectrodynamics and the Fermi model, which already accomplished a good description ofthe electromagnetic and weak interactions at low energies. It is minimal in the sense thatit contains the smallest number of degrees of freedom necessary to describe the knownexperimental facts.The electroweak standard model (SM) is a nonabelian gauge theory based on the non-simple group SU (2) W × U (1) Y . From experiment we know that three out of the fourassociated gauge bosons have to be massive. This is implemented via the Higgs-Kibblemechanism [16]. By introducing a scalar field with nonvanishing vacuum expectationvalue the SU (2) W × U (1) Y gauge symmetry is spontaneously broken in such a way thatinvariance under the electromagnetic subgroup U (1) em is preserved. The SM is chiralsince right- and left-handed fermions transform according to different representations ofthe gauge group. Consequently fermion masses are forbidden in the symmetric theory.They are generated through spontaneous symmetry breaking from the Yukawa couplings.Diagonalization of the fermion mass matrices introduces the quark mixing matrix in thequark sector. This can give rise to CP-violation. Fermions appear in generations. Themodel does not fix their number, but from experiment we know that there are exactlythree with light neutrinos [5].The SM is a consistent quantum field theory. It is renormalizable, as was proven by’t Hooft [17], and free of anomalies. Therefore it allows to calculate unique quantumcorrections. Given a finite set of input parameters measurable quantities can be predictedorder by order in perturbation theory.The classical Lagrangian L C of the SM is composed of a Yang-Mills, a Higgs and afermion part L C = L Y M + L H + L F . (2.1)Each of them is separately gauge invariant. They are specified as follows: The gauge fields are four vector fields transforming according to the adjoint represen-tation of the gauge group SU (2) W × U (1) Y . The isotriplet W aµ , a = 1 , , I aW of the weak isospin group SU (2) W , the isosinglet B µ with theweak hypercharge Y W of the group U (1) Y . The pure gauge field Lagrangian reads L Y M = − (cid:16) ∂ µ W aν − ∂ ν W aµ + g ε abc W bµ W cν (cid:17) −
14 ( ∂ µ B ν − ∂ ν B µ ) , (2.2)4here ε abc are the totally antisymmetric structure constants of SU (2). Since the gaugegroup is non-simple there are two gauge coupling constants, the SU (2) W gauge coupling g and the U (1) Y gauge coupling g . The covariant derivative is given by D µ = ∂ µ − ig I aW W aµ + ig Y W B µ . (2.3)The electric charge operator Q is composed of the weak isospin generator I W and theweak hypercharge according to the Gell-Mann Nishijima relation Q = I W + Y W . (2.4) The minimal Higgs sector consists of a single complex scalar SU (2) W doublet fieldwith hypercharge Y W = 1 Φ( x ) = φ + ( x ) φ ( x ) . (2.5)It is coupled to the gauge fields with the covariant derivative (2.3) and has a self couplingresulting in the Lagrangian L H = ( D µ Φ) † ( D µ Φ) − V (Φ) . (2.6)The Higgs potential V (Φ) = λ (cid:16) Φ † Φ (cid:17) − µ Φ † Φ (2.7)is constructed in such a way that it gives rise to spontaneous symmetry breaking. Thismeans that the parameters λ and µ are chosen such that the potential V (Φ) takes itsminimum for a nonvanishing Higgs field, i.e. the vacuum expectation value h Φ i of theHiggs field is nonzero. The left-handed fermions of each lepton ( L ) and quark ( Q ) generation are groupedinto SU (2) W doublets (we suppress the colour index) L ′ Lj = ω − L ′ j = ν ′ Lj l ′ Lj , Q ′ Lj = ω − Q ′ j = u ′ Lj d ′ Lj , (2.8)the right-handed fermions into singlets l ′ Rj = ω + l ′ j , u ′ Rj = ω + u ′ j , d ′ Rj = ω + d ′ j , (2.9)where ω ± = ± γ is the projector on right- and left-handed fields, respectively, j is thegeneration index and ν , l , u and d stand for neutrinos, charged leptons, up-type quarksand down-type quarks, respectively. The weak hypercharge of the right- and left-handedmultiplets is chosen such that the known electromagnetic charges of the fermions are5eproduced by the Gell-Mann-Nishijima relation (2.4). There are no right-handed neutri-nos. These could be easily added, but they would induce nonvanishing neutrino masses,which have not been observed experimentally so far.The fermionic part of the Lagrangian reads L F = X i (cid:16) L ′ Li iγ µ D µ L ′ Li + Q ′ Li iγ µ D µ Q ′ Li (cid:17) + X i (cid:16) l ′ Ri iγ µ D µ l ′ Ri + u ′ Ri iγ µ D µ u ′ Ri + d ′ Ri iγ µ D µ d ′ Ri (cid:17) − X ij (cid:16) L ′ Li G lij l ′ Rj Φ + Q ′ Li G uij u ′ Rj ˜Φ + Q ′ Li G dij d ′ Rj Φ + h.c. (cid:17) . (2.10)Note that in the covariant derivative D µ acting on right-handed fermions the term in-volving g is absent, since they are SU (2) W singlets. The primed fermion fields are bydefinition eigenstates of the electroweak gauge interaction, i.e. the covariant derivativesare diagonal in this basis with respect to the generation indices. G lij , G uij and G dij arethe Yukawa coupling matrices, ˜Φ = ( φ ∗ , − φ − ) T is the charge conjugated Higgs field and φ − = ( φ + ) ∗ . The SU (2) W × U (1) Y symmetry forbids explicit mass terms for the fermions.The masses of the fermions are generated through the Yukawa couplings via spontaneoussymmetry breaking. The theory is constructed such that the classical ground state of the scalar field satisfies |h Φ i| = 2 µ λ = v = 0 . (2.11)In perturbation theory one has to expand around the ground state. Its phase is chosensuch that the electromagnetic gauge invariance U (1) em is preserved and the Higgs field iswritten as Φ( x ) = φ + ( x ) √ ( v + H ( x ) + iχ ( x )) , (2.12)where the components φ + , H and χ have zero vacuum expectation values. φ + , φ − and χ are unphysical degrees of freedom and can be eliminated by a suitable gauge transforma-tion. The gauge in which they are absent is called unitary. The field H is the physicalHiggs field with mass M H = √ µ. (2.13)Inserting (2.12) into L C the vacuum expectation value v introduces couplings with massdimension and mass terms for the gauge bosons and fermions.6he physical gauge boson and fermion fields are obtained by diagonalizing the corre-sponding mass matrices W ± µ = 1 √ (cid:16) W µ ∓ iW µ (cid:17) , Z µ A µ = c W s W − s W c W W µ B µ ,f Li = U f,Lik f ′ Lk ,f Ri = U f,Rik f ′ Rk , (2.14)where c W = cos θ W = g q g + g , s W = sin θ W , (2.15)with the weak mixing angle θ W and f stands for ν , l , u or d . The resulting masses are M W = 12 g v, M Z = 12 q g + g v,M γ = 0 , m f,i = U f,Lik G fkm U f,R † mi v √ . (2.16)The neutrinos remain massless since the absence of the right-handed neutrinos forbids theYukawa couplings which would generate their masses. With (2.16) we find for the weakmixing angle c W = M W M Z . (2.17)Identifying the coupling of the photon field A µ to the electron with the electrical charge e = √ πα yields e = g g q g + g , (2.18)or g = ec W , g = es W . (2.19)The diagonalization of the fermion mass matrices introduces a matrix into the quark-W-boson couplings, the unitary quark mixing matrix V ij = U u,Lik U d,L † kj . (2.20)There is no corresponding matrix in the lepton sector. Since there is no neutrino massmatrix, U ν,L is completely arbitrary and can be chosen such that it cancels U l,L in thelepton-W-boson couplings. The same would also be true for the quark sector if all up-type or down-type quarks would be degenerate in masses. For degenerate masses one canchoose U L = U R † arbitrary without destroying the diagonality of the corresponding massmatrix and thus eliminate V ij .The above relations (2.13, 2.16, 2.18, 2.20) allow to replace the original set ofparameters g , g , λ, µ , G l , G u , G d (2.21)7y the parameters e, M W , M Z , M H , m f,i , V ij (2.22)which have a direct physical meaning. Thus we can express the Lagrangian (2.1) in termsof physical parameters and fields.Inserting (2.12) into L C generates a term linear in the Higgs field H which we denoteby tH ( x ) with t = v ( µ − λ v ) . (2.23)The tadpole t vanishes at lowest order due to the choice of v . We use t instead of v inthe following. Choosing v as the correct vacuum expectation value of the Higgs field Φ isequivalent to the vanishing of t . Quantization of L C and higher order calculations require the specification of a gauge.We choose a renormalizable ’t Hooft gauge with the following linear gauge fixings F ± = ( ξ W ) − ∂ µ W ± µ ∓ iM W ( ξ W ) φ ± ,F Z = ( ξ Z ) − ∂ µ Z µ − M Z ( ξ Z ) χ,F γ = ( ξ γ ) − ∂ µ A µ , (2.24)leading to the following gauge fixing Lagrangian L fix = − h ( F γ ) + ( F Z ) + 2 F + F − i . (2.25) L fix involves the unphysical components of the gauge fields. In order to compensatetheir effects one introduces Faddeev Popov ghosts u α ( x ), ¯ u α ( x ) ( α = ± , γ, Z ) with theLagrangian L F P = ¯ u α ( x ) δF α δθ β ( x ) u β ( x ) . (2.26) δF α δθ β ( x ) is the variation of the gauge fixing operators F α under infinitesimal gauge trans-formations characterized by θ β ( x ).The ’t Hooft Feynman gauge ξ α = 1 is particularly simple. At lowest order the polesof the ghost fields, unphysical Higgs fields and longitudinal gauge fields coincide withthe poles of the corresponding transverse gauge fields. Furthermore no gauge-field-Higgsmixing occurs.With L fix and L F P the complete renormalizable Lagrangian for the electroweak SMreads L GSW = L C + L fix + L F P . (2.27)The corresponding Feynman rules are given in App. A.8 Renormalization
The Lagrangian (2.1) of the minimal SU (2) W × U (1) Y model involves a certain num-ber of free parameters (2.22) which have to be determined experimentally. These arechosen such that they have an intuitive physical meaning at tree level (physical masses,couplings), i.e. they are directly related to experimental quantities. This direct relationis destroyed through higher order corrections. Moreover the parameters of the originalLagrangian, the so-called bare parameters, differ from the corresponding physical quanti-ties by UV-divergent contributions. However, in renormalizable theories these divergenciescancel in relations between physical quantities, thus allowing meaningful predictions. Therenormalizability of nonabelian gauge theories with spontaneous symmetry breaking andthus of the SM was proven by ’t Hooft [17].One possibility to evaluate predictions of a renormalizable model is the following: • Calculate physical quantities in terms of the bare parameters. • Use as many of the resulting relations as bare parameters exist to express these interms of physical observables. • Insert the resulting expressions into the remaining relations.Thus one arrives at predictions for physical observables in terms of other physical quan-tities, which have to be determined from experiment. In these predictions all UV-divergencies cancel in any order of perturbation theory. The predictions obtained fromdifferent input parameters differ in finite orders of perturbation theory by higher ordercontributions. This treatment of renormalization has been pioneered by Passarino, Velt-man and Consoli [18] and is the basis of the so-called ’star’ scheme of Kennedy and Lynn[19].We use the counterterm approach. Here the UV-divergent bare parameters are ex-pressed by finite renormalized parameters and divergent renormalization constants (coun-terterms). In addition the bare fields may be replaced by renormalized fields. The coun-terterms are fixed through renormalization conditions. These can be chosen arbitrarily,but determine the relation between renormalized and physical parameters. Further eval-uation proceeds like described above. The results depend in finite orders of perturbationtheory not only on the choice of the input parameters but also on the choice of therenormalized parameters. Clearly the physical results are unambiguous up to the orderswhich have been taken into account completely. The renormalization procedure can besummarized as follows: • Choose a set of independent parameters (e.g. (2.22) in the SM). • Separate the bare parameters (and fields) into renormalized parameters (fields) andrenormalization constants (see Sect. 3.1). • Choose renormalization conditions to fix the counterterms (see Sect. 3.2). • Express physical quantities in terms of the renormalized parameters. • Choose input data in order to fix the values of the renormalized parameters. • Evaluate predictions for physical quantities as functions of the input data.9he first three items in this list specify a renormalization scheme.Putting the counterterms equal to zero, the renormalized parameters equal the bareparameters and we recover the first approach.However, we can choose the counterterms such that the finite renormalized parametersare equal to physical parameters in all orders of perturbation theory. This is the so-called on-shell renormalization scheme. In the SM one uses the masses of the physicalparticles M W , M Z , M H , m f , the charge of the electron e and the quark mixing matrix V ij as renormalized parameters. This scheme was proposed by Ross and Taylor [20]and is widely used in the electroweak theory. The advantage of the on-shell schemeis that all parameters have a clear physical meaning and can be measured directly insuitable experiments . Furthermore the Thomson cross section from which e is obtainedis exact to all orders of perturbation theory. However, not all of the particle masses areknown experimentally with good accuracy. Therefore other schemes may sometimes beadvantageous.Renormalization of the parameters is sufficient to obtain finite S-matrix elements,but it leaves Green functions divergent. This is due to the fact that radiative correc-tions change the normalization of the fields by an infinite amount. In order to get finitepropagators and vertex functions the fields have to be renormalized, too. Furthermoreradiative corrections provide nondiagonal corrections to the mass matrices so that thebare fields are no longer mass eigenstates. In order to rediagonalize the mass matricesone has to introduce matrix valued field renormalization constants. These allow to definethe renormalized fields in such a way that they are the correct physical mass eigenstatesin all orders of perturbation theory. If one does not renormalize the fields in this way,one needs a nontrivial wave function renormalization for the external particles. This isrequired in going from Green functions to S-matrix elements in order to obtain a properlynormalized S-matrix.The results for physical S-matrix elements are independent of the specific choice of fieldrenormalization. There exist many different treatments in the literature [21, 22, 23, 24, 25].Calculations without field renormalization were performed by [26]. In the following we specify the on-shell renormalization scheme for the electroweak SMquantitatively. As independent parameters we choose the physical parameters specified in(2.22). The renormalized quantities and renormalization constants are defined as follows(we denote bare quantities by an index 0) e = Z e e = (1 + δZ e ) e,M W, = M W + δM W ,M Z, = M Z + δM Z , (3.1) M H, = M H + δM H ,m f,i, = m f,i + δm f,i , This is not the case for the quark masses, due to the presence of the strong interaction. In practicethese are replaced by suitable experimental input parameters (see Sect. 8.1). ij, = ( U V U † ) ij = V ij + δV ij .U and U are unitary matrices since V ij, and V ij are both unitary.Radiative corrections affect the Higgs potential in such a way that its minimum isshifted. In order to correct for this shift one introduces a counterterm to the vacuumexpectation value of the Higgs field, which is determined such that the renormalized v isgiven by the actual minimum of the effective Higgs potential. Since we have replaced v by t (2.23) we must introduce a counterterm δt . This is fixed such that it cancels all tadpolediagrams, i.e. that the effective potential contains no term linear in the Higgs field H .The counterterms defined above are sufficient to render all S-matrix elements finite.In order to have finite Green functions we must renormalize the fields, too. As explainedabove we need field renormalization matrices in order to be able to define renormalizedfields which are mass eigenstates W ± = Z / W W ± = (1 + δZ W ) W ± , Z A = Z / ZZ Z / ZA Z / AZ Z / AA ZA = δZ ZZ δZ ZA δZ AZ δZ AA ZA ,H = Z / H H = (1 + δZ H ) H,f
Li, = Z / ,f,Lij f Lj = ( δ ij + δZ f,Lij ) f Lj ,f Ri, = Z / ,f,Rij f Rj = ( δ ij + δZ f,Rij ) f Rj . (3.2)We do not discuss the renormalization constants of the unphysical ghost and Higgsfields. They do not affect Green functions of physical particles and are not relevant forthe calculation of physical one-loop amplitudes. Furthermore the renormalization of theunphysical sector decouples from the one of the physical sector. It is governed by theSlavnov-Taylor identities. A discussion of this subject can be found e.g. in [24, 25].In writing Z = 1 + δZ for the multiplicative renormalization constants (matrices)we can split the bare Lagrangian L into the basic Lagrangian L and the countertermLagrangian δ L L = L + δ L . (3.3) L has the same form as L but depends on renormalized parameters and fields insteadof unrenormalized ones. δ L yields the counterterms. The corresponding Feynman rulesare listed in App. A. They give rise to counterterm diagrams which have to be added tothe loop graphs. Since we are only interested in one-loop corrections, we neglect terms oforder ( δZ ) everywhere. The renormalization constants introduced in the previous section are fixed by imposingrenormalization conditions. These decompose into two sets. The conditions which definethe renormalized parameters and the ones which define the renormalized fields. Whilethe choice of the first affects physical predictions to finite orders of perturbation theory,11he second are only relevant for Green functions and drop out when calculating S-matrixelements. Nevertheless their use is very convenient in the on-shell scheme. They not onlyallow to eliminate the explicit wave function renormalization of the external particles,but also simplify the explicit form of the renormalization conditions for the physicalparameters considerably.In the on-shell scheme all renormalization conditions are formulated for on mass shellexternal fields. The field renormalization constants, the mass renormalization constantand the renormalization constant of the quark mixing matrix are fixed using the one-particle irreducible two-point functions. For the charge renormalization we need onethree-point function. For this we choose the eeγ -vertex function. In the following renor-malized quantities are denoted by the same symbols as the corresponding unrenormalizedquantities, but with the superscriptˆ.As discussed above the first renormalization condition involves the tadpole T , theHiggs field one-point amputated renormalized Green functionˆ T = H (cid:30)(cid:29)(cid:31)(cid:28) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) , (3.4)and simply states ˆ T = T + δt = 0 . (3.5)As a consequence of this condition no tadpoles need to be considered in actual calculations.Next we need the renormalized one-particle irreducible two-point functions. These aredefined as follows (we are using the ’t Hooft-Feynman gauge) W µ k (cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0)(cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1)(cid:30)(cid:29)(cid:31)(cid:28) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) (cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0)(cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) W ν = ˆΓ Wµν ( k )= − ig µν ( k − M W ) − i g µν − k µ k ν k ! ˆΣ WT ( k ) − i k µ k ν k ˆΣ WL ( k ) ,a, µk (cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0)(cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1)(cid:30)(cid:29)(cid:31)(cid:28) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) (cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0)(cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) b, ν = ˆΓ abµν ( k )= − ig µν ( k − M a ) δ ab − i g µν − k µ k ν k ! ˆΣ abT ( k ) − i k µ k ν k ˆΣ abL ( k ) , where a, b = A, Z, M A = 0 , (3.6) Hk (cid:30)(cid:29)(cid:31)(cid:28) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) H = ˆΓ H ( k ) = i ( k − M H ) + i ˆΣ H ( k ) , j p - (cid:30)(cid:29)(cid:31)(cid:28) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) - f i = ˆΓ fij ( p )= iδ ij ( p/ − m i ) + i h p/ω − ˆΣ f,Lij ( p ) + p/ω + ˆΣ f,Rij ( p ) + ( m f,i ω − + m f,j ω + ) ˆΣ f,Sij ( p ) i . The corresponding propagators are obtained as the inverse of these two-point functions.Note that we have to invert matrices for the neutral gauge bosons and for the fermions.The renormalized mass parameters of the physical particles are fixed by the require-ment that they are equal to the physical masses, i.e. to the real parts of the poles of thecorresponding propagators which are equivalent to the zeros of the one-particle irreducibletwo-point functions. In case of mass matrices these conditions have to be fulfilled by thecorresponding eigenvalues resulting in complicated expressions. These can be considerablysimplified by requiring simultaneously the on-shell conditions for the field renormalizationmatrices. These state that the renormalized one-particle irreducible two-point functionsare diagonal if the external lines are on their mass shell. This determines the nondiagonalelements of the field renormalization matrices. The diagonal elements are fixed such thatthe renormalized fields are properly normalized, i.e. that the residues of the renormal-ized propagators are equal to one. This choice of field renormalization implies that therenormalization conditions for the mass parameters (in all orders of perturbation theory)involve only the corresponding diagonal self energies. Thus we arrive at the followingrenormalization conditions for the two-point functions for on-shell external physical fields f Re ˆΓ
Wµν ( k ) ε ν ( k ) (cid:12)(cid:12)(cid:12) k = M W = 0 , Re ˆΓ
ZZµν ( k ) ε ν ( k ) (cid:12)(cid:12)(cid:12) k = M Z = 0 , Re ˆΓ
AZµν ( k ) ε ν ( k ) (cid:12)(cid:12)(cid:12) k = M Z = 0 , ˆΓ AZµν ( k ) ε ν ( k ) (cid:12)(cid:12)(cid:12) k =0 = 0 , ˆΓ AAµν ( k ) ε ν ( k ) (cid:12)(cid:12)(cid:12) k =0 = 0 , lim k → M W k − M W f Re ˆΓ
Wµν ( k ) ε ν ( k ) = − iε µ ( k ) , lim k → M Z k − M Z Re ˆΓ
ZZµν ( k ) ε ν ( k ) = − iε µ ( k ) , lim k → k Re ˆΓ
AAµν ( k ) ε ν ( k ) = − iε µ ( k ) , Re ˆΓ H ( k ) (cid:12)(cid:12)(cid:12) k = M H = 0 , lim k → M H k − M H Re ˆΓ H ( k ) = i, f Re ˆΓ fij ( p ) u j ( p ) (cid:12)(cid:12)(cid:12) p = m f,j = 0 , f Re ¯ u i ( p ′ )ˆΓ fij ( p ′ ) (cid:12)(cid:12)(cid:12) p ′ = m f,i = 0 , lim p → m f,i p/ + m f,i p − m f,i f Re ˆΓ fii ( p ) u i ( p ) = iu i ( p ) , lim p ′ → m f,i ¯ u i ( p ′ ) f Re ˆΓ fii ( p ′ ) p/ ′ + m f,i p ′ − m f,i = i ¯ u i ( p ′ ) . (3.7) ε ( k ), u ( p ) and ¯ u ( p ′ ) are the polarization vectors and spinors of the external fields. f Retakes the real part of the loop integrals appearing in the self energies but not of the quarkmixing matrix elements appearing there. Since we restrict ourselves to the one-loop orderwe apply it only to those quantities which depend on the quark mixing matrix at one loop.13n higher orders Re must be replaced by f Re everywhere. Re and f Re are only relevantabove thresholds and have no effect for the two-point functions of on-shell stable particles.If the quark mixing matrix is real f Re can be replaced by Re. This holds in particular fora unit quark mixing matrix which is often used.From the above equations we obtain the conditions for the self energy functions. f Re ˆΣ WT ( M W ) = 0 , Re ˆΣ
ZZT ( M Z ) = 0 , Re ˆΣ
AZT ( M Z ) = 0 , ˆΣ AZT (0) = 0 , ˆΣ AAT (0) = 0 , f Re ∂ ˆΣ WT ( k ) ∂k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k = M W = 0 , Re ∂ ˆΣ ZZT ( k ) ∂k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k = M Z = 0 , Re ∂ ˆΣ AAT ( k ) ∂k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k =0 = 0 , (3.8)Re ˆΣ H ( M H ) = 0 , Re ∂ ˆΣ H ( k ) ∂k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k = M H = 0 , (3.9) m f,j f Re ˆΣ f,Lij ( m f,j ) + m f,j f Re ˆΣ f,Sij ( m f,j ) = 0 ,m f,j f Re ˆΣ f,Rij ( m f,j ) + m f,i f Re ˆΣ f,Sij ( m f,j ) = 0 , f Re ˆΣ f,Rii ( m f,i ) + f Re ˆΣ f,Lii ( m f,i )+ 2 m f,i ∂∂p (cid:16) f Re ˆΣ f,Rii ( p ) + f Re ˆΣ f,Lii ( p ) + 2 f Re ˆΣ f,Sii ( p ) (cid:17)(cid:12)(cid:12)(cid:12) p = m f,i = 0 . (3.10)Note that the (unphysical) longitudinal part of the gauge boson self energies drops outfor on-shell external gauge bosons.Our choice for the renormalization condition of the quark mixing matrix V ij can bemotivated as follows. To lowest order V ij is given by (see eq. 2.20) V ,ij = U u,Lik U d,L † kj , (3.11)where the matrices U f,L transform the weak interaction eigenstates f ′ to the lowest ordermass eigenstates f U f,L † ij f Lj, = f ′ Li, . (3.12)In the on-shell renormalization scheme the higher order mass eigenstates are related tothe bare mass eigenstates through the field renormalization constants of the fermions f Li = Z / ,f,Lij f Lj, . (3.13)We define the renormalized quark mixing matrix in analogy to the unrenormalized onethrough the rotation from the weak interaction eigenstates to the renormalized mass This condition is automatically fulfilled due to a Ward identity. δZ L is simply given by the anti-Hermitean part δZ AH of δZ L δZ f,AHij = 12 ( δZ f,Lij − δZ f,L † ij ) . (3.14)Thus we are lead to define the renormalized quark mixing matrix as V ij = ( δ ik + δZ u,AH † ik ) U u,Lkm U d,L † mn ( δ nj + δZ d,AHnj )= ( δ ik + δZ u,AH † ik ) V ,kn ( δ nj + δZ d,AHnj ) . (3.15)It has been shown that this condition correctly cancels all one-loop divergencies and that V ij = V ,ij in the limit of degenerate up- or down-type quark masses [27].Finally the electrical charge is defined as the full eeγ -coupling for on-shell externalparticles in the Thomson limit. This means that all corrections to this vertex vanishon-shell and for zero momentum transfer A µ (cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0)(cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1)(cid:26)(cid:25)(cid:27)(cid:24) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) (cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0)(cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1)(cid:30)(cid:29)(cid:31)(cid:28) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) (cid:8)(cid:8)*(cid:8)(cid:8) (cid:26)(cid:25)(cid:27)(cid:24) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) (cid:8)(cid:8)*(cid:8)(cid:8)HHHYHH (cid:26)(cid:25)(cid:27)(cid:24) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) HHHYHH e + , p ′ e − , p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p = p ′ , p = p ′ = m e = ie ¯ u ( p ) γ µ u ( p ) . (3.16)The momenta p , p ′ flow in the direction of the fermion arrows. Due to our choice forthe field renormalization the corrections in the external legs vanish and we obtain thecondition ¯ u ( p )Γ eeγµ ( p, p ) u ( p ) (cid:12)(cid:12)(cid:12) p = m e = ie ¯ u ( p ) γ µ u ( p ) , (3.17)for the (amputated) vertex functionˆΓ eeγµ ( p, p ′ ) = A µ (cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0)(cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1)(cid:30)(cid:29)(cid:31)(cid:28) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) (cid:8)(cid:8)*(cid:8)(cid:8)HHHYHH e + , p ′ e − , p . (3.18) The renormalized quantities defined in Sect. 3.2 consist of the unrenormalized onesand the counterterms as specified by the Feynman rules in App. A. The renormalizationconditions allow to express the counterterms by the unrenormalized self energies at specialexternal momenta. This is evident for all renormalization constants apart from the one Due to the wave function renormalization of the external particles the self energy corrections in theexternal legs contribute only with a factor 1 / δt = − T,δM W = f Re Σ WT ( M W ) , δZ W = − Re ∂ Σ WT ( k ) ∂k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k = M W ,δM Z = Re Σ ZZT ( M Z ) , δZ ZZ = − Re ∂ Σ ZZT ( k ) ∂k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k = M Z ,δZ AZ = − AZT ( M Z ) M Z , δZ ZA = 2 Σ AZT (0) M Z ,δZ AA = − ∂ Σ AAT ( k ) ∂k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k =0 ,δM H = Re Σ H ( M H ) , δZ H = − Re ∂ Σ H ( k ) ∂k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k = M H . (3.19)In the fermion sector (3.10) yields δm f,i = m f,i f Re (cid:16) Σ f,Lii ( m f,i ) + Σ f,Rii ( m f,i ) + 2Σ f,Sii ( m f,i ) (cid:17) ,δZ f,Lij = 2 m f,i − m f,j f Re h m f,j Σ f,Lij ( m f,j ) + m f,i m f,j Σ f,Rij ( m f,j )+( m f,i + m f,j )Σ f,Sij ( m f,j ) i , i = j,δZ f,Rij = 2 m f,i − m f,j f Re h m f,j Σ f,Rij ( m f,j ) + m f,i m f,j Σ f,Lij ( m f,j ) (3.20)+2 m f,i m f,j Σ f,Sij ( m f,j ) i , i = j,δZ f,Lii = − f Re Σ f,Lii ( m f,i ) − m f,i ∂∂p f Re h Σ f,Lii ( p ) + Σ f,Rii ( p ) + 2Σ f,Sii ( p ) i(cid:12)(cid:12)(cid:12) p = m f,i ,δZ f,Rii = − f Re Σ f,Rii ( m f,i ) − m f,i ∂∂p f Re h Σ f,Lii ( p ) + Σ f,Rii ( p ) + 2Σ f,Sii ( p ) i(cid:12)(cid:12)(cid:12) p = m f,i . The use of f Re ensures reality of the renormalized Lagrangian. Furthermore it yields δZ † ij = δZ ij ( m i ↔ m j ) , (3.21)and in particular δZ † ii = δZ ii . (3.22)16n the lepton sector we have V ij = δ ij . Consequently all lepton self energies arediagonal and the off-diagonal lepton wave function renormalization constants are zero.The same holds for the quark sector if one replaces the quark mixing matrix by a unitmatrix as is usually done in calculations of radiative corrections for high energy processes.The renormalization constant for the quark mixing matrix V ij can be directly read offfrom (3.15) δV ij = 14 h ( δZ u,Lik − δZ u,L † ik ) V kj − V ik ( δZ d,Lkj − δZ d,L † kj ) i . (3.23)Inserting the fermion field renormalization constants (3.20) yields δV ij = 12 f Re (cid:26) m u,i − m u,k (cid:20) m u,k Σ u,Lik ( m u,k ) + m u,i Σ u,Lik ( m u,i )+ m u,i m u,k (cid:16) Σ u,Rik ( m u,k ) + Σ u,Rik ( m u,i ) (cid:17) +( m u,k + m u,i ) (cid:16) Σ u,Sik ( m u,k ) + Σ u,Sik ( m u,i ) (cid:17)(cid:21) V kj − V ik m d,k − m d,j (cid:20) m d,j Σ d,Lkj ( m d,j ) + m d,k Σ d,Lkj ( m d,k )+ m d,k m d,j (cid:16) Σ d,Rkj ( m d,j ) + Σ d,Rkj ( m d,k ) (cid:17) +( m d,k + m d,j ) (cid:16) Σ d,Skj ( m d,k ) + Σ d,Skj ( m d,j ) (cid:17)(cid:21)(cid:27) . (3.24)It remains to fix the charge renormalization constant δZ e . This is determined from the eeγ -vertex. To be more general we investigate the f f γ -vertex for arbitrary fermions f .The renormalized vertex function readsˆΓ γffij,µ ( p, p ′ ) = − ieQ f δ ij γ µ + ie ˆΛ γffij,µ ( p, p ′ ) . (3.25)For on-shell external fermions it can be decomposed as ( k = p ′ − p )ˆΛ γffij,µ ( p, p ′ ) = δ ij γ µ ˆΛ fV ( k ) − γ µ γ ˆΛ fA ( k ) + ( p + p ′ ) µ m f ˆΛ fS ( k ) + ( p ′ − p ) µ m f γ ˆΛ fP ( k ) ! . (3.26)Expressing the renormalized quantities by the unrenormalized ones and the countertermsand inserting this in the analogue of the renormalization condition (3.17) for arbitraryfermions we find, using the Gordon identities,0 = ¯ u ( p ) ˆΛ γffii,µ ( p, p ) u ( p )= ¯ u ( p ) γ µ u ( p ) h − Q f ( δZ e + δZ f,Vii + δZ AA ) + Λ fV (0) + Λ fS (0) + v f δZ ZA i − ¯ u ( p ) γ µ γ u ( p ) h − Q f δZ f,Aii + Λ fA (0) + a f δZ ZA i , (3.27)where δZ f,Vii = 12 ( δZ f,Lii + δZ f,Rii ) , δZ f,Aii = 12 ( δZ f,Lii − δZ f,Rii ) , (3.28)17nd v f , a f are the vector and axialvector couplings of the Z -boson to the fermion f , givenexplicitly in (A.15). This yields in fact two conditions, namely0 = − Q f ( δZ e + δZ f,Vii + 12 δZ AA ) + Λ fV (0) + Λ fS (0) + v f δZ ZA , (3.29)0 = − Q f δZ f,Aii + Λ fA (0) + a f δZ ZA . (3.30)The first one (3.29) for f = e fixes the charge renormalization constant. The second(3.30) is automatically fulfilled due to a Ward identity which can be derived from thegauge invariance of the theory. The same Ward identity moreover yieldsΛ fV (0) + Λ fS (0) − Q f δZ f,Vii + a f δZ ZA = 0 . (3.31)Inserting this in (3.29) we finally find (using v f − a f = − Q f s W /c W ) δZ e = − δZ AA − s W c W δZ ZA = 12 ∂ Σ AAT ( k ) ∂k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k =0 − s W c W Σ AZT (0) M Z . (3.32)This result is independent of the fermion species, reflecting electric charge universality.Clearly it does not depend on a specific choice of field renormalization. Consequently theanalogue of (3.17) holds for arbitrary fermions f .In the on-shell scheme the weak mixing angle is a derived quantity. Following Sirlin[26] we define it as sin θ W = s W = 1 − M W M Z , (3.33)using the renormalized gauge boson masses. This definition is independent of a specificprocess and valid to all orders of perturbation theory.Since the dependent parameters s W and c W frequently appear, it is useful to introducethe corresponding counterterms c W, = c W + δc W , s W, = s W + δs W . (3.34)Because of (3.33) these are directly related to the counterterms to the gauge boson masses.To one-loop order we obtain δc W c W = 12 δM W M W − δM Z M Z ! = 12 f Re Σ WT ( M W ) M W − Σ ZZT ( M Z ) M Z ! ,δs W s W = − c W s W δc W c W = − c W s W f Re Σ WT ( M W ) M W − Σ ZZT ( M Z ) M Z ! . (3.35)We have now determined all renormalization constants in terms of unrenormalized selfenergies. In the next sections we will describe the methods to calculate these self energiesand more general diagrams at the one-loop level.18 One-loop integrals
Perturbative calculations at one-loop order involve integrals over the loop momentum.In this chapter we discuss their classification and techniques for their calculation. Themethods described here are to a large extent based on the work of Passarino and Veltman[18], ’t Hooft and Veltman [28], and Melrose [29].
The one-loop integrals in D dimensions are classified according to the number N ofpropagator factors in the denominator and the number P of integration momenta in thenumerator. For P + D − N ≥ D = 4 (dimensional regu-larization). The UV-divergencies drop out in renormalized quantities. For renormalizabletheories we have P ≤ N and thus a finite number of divergent integrals.We define the general one-loop tensor integral (see Fig. 4.1) as T Nµ ...µ P ( p , . . . , p N − , m , . . . , m N − ) = (2 πµ ) − D iπ Z d D q q µ · · · q µ P D D · · · D N − (4.1)with the denominator factors D = q − m + iε, D i = ( q + p i ) − m i + iε, i = 1 , . . . , N − , (4.2)originating from the propagators in the Feynman diagram. Furthermore we introduce p i = p i and p ij = p i − p j . (4.3)Evidently the tensor integrals are invariant under arbitrary permutations of the prop-agators D i , i = 0 and totally symmetric in the Lorentz indices µ k . iε is an infinitesimalimaginary part which is needed to regulate singularities of the integrand. Its specificchoice ensures causality. After integration it determines the correct imaginary parts ofthe logarithms and dilogarithms. The parameter µ has mass dimension and serves to keep p p NN − q q + p N − q + p p N − N − p q + p N − q + p j HHHH YHHHH j HHHH*(cid:8)(cid:8)(cid:8)(cid:8) *(cid:8)(cid:8)(cid:8)(cid:8) (cid:25)(cid:8)(cid:8)(cid:8)(cid:8)6 ?6 qqqqqqq Figure 4.1: Conventions for the N-point integral.19he dimension of the integrals fixed for varying D . Conventionally T N is denoted by the N th character of the alphabet, i.e. T ≡ A , T ≡ B , . . . , and the scalar integrals carryan index 0.Lorentz covariance of the integrals allows to decompose the tensor integrals into ten-sors constructed from the external momenta p i , and the metric tensor g µν with totallysymmetric coefficient functions T Ni ...i P . We formally introduce an artificial momentum p in order to write the terms containing g µν in a compact way T Nµ ...µ P ( p , . . . , p N − , m , . . . , m N − ) = N − X i ,...,i P =0 T Ni ...i P p i µ · · · p i P µ P . (4.4)From this formula the correct g µν terms are recovered by omitting all terms containing anodd number of p ’s and replacing products of even numbers of p ’s by the correspondingtotally symmetric tensor constructed from the g µν , e.g. p µ p µ → g µ µ ,p µ p µ p µ p µ → g µ µ g µ µ + g µ µ g µ µ + g µ µ g µ µ . (4.5)The explicit Lorentz decompositions for the lowest order integrals read B µ = p µ B ,B µν = g µν B + p µ p ν B , (4.6) C µ = p µ C + p µ C = X i =1 p iµ C i ,C µν = g µν C + p µ p ν C + p µ p ν C + ( p µ p ν + p µ p ν ) C = g µν C + X i,j =1 p iµ p jν C ij ,C µνρ = ( g µν p ρ + g νρ p µ + g µρ p ν ) C + ( g µν p ρ + g νρ p µ + g µρ p ν ) C + p µ p ν p ρ C + p µ p ν p ρ C + ( p µ p ν p ρ + p µ p ν p ρ + p µ p ν p ρ ) C + ( p µ p ν p ρ + p µ p ν p ρ + p µ p ν p ρ ) C = X i =1 ( g µν p iρ + g νρ p iµ + g µρ p iν ) C i + X i,j,k =1 p iµ p jν p kρ C ijk , (4.7)20 µ = X i =1 p iµ D i ,D µν = g µν D + X i,j =1 p iµ p jν D ij ,D µνρ = X i =1 ( g µν p iρ + g νρ p iµ + g µρ p iν ) D i + X i,j,k =1 p iµ p jν p kρ D ijk ,D µνρσ = ( g µν g ρσ + g µρ g νσ + g µσ g νρ ) D + X i,j =1 ( g µν p iρ p jσ + g νρ p iµ p jσ + g µρ p iν p jσ + g µσ p iν p jρ + g νσ p iµ p jρ + g ρσ p iµ p jν ) D ij + X i,j,k,l =1 p iµ p jν p kρ p lσ D ijkl . (4.8)Since the four dimensional space is spanned by four Lorentz vectors the terms involving g µν should be omitted for N ≥ N ≥ T Nµ ...µ P ( p , . . . , p N − , m , . . . , m N − ) = X i ,...,i P =1 T Ni ...i P p i µ · · · p i P µ P , (4.9)where p , . . . , p is any set of four linear independent Lorentz vectors out of p , . . . , p N − .The symmetry of the tensor integrals under exchange of the propagators yields relationsbetween the scalar coefficient functions. Exchanging the arguments ( p i , m i ) ↔ ( p j , m j )together with the corresponding indices i ↔ j leaves the scalar coefficient functions in-variant T N... i...i |{z} n ... j...j |{z} m ... ( p , . . . , p i , . . . , p j , . . . , p N − , m , . . . , m i , . . . , m j , . . . , m N − )= T N... i...i |{z} m ... j...j |{z} n ... ( p , . . . , p j , . . . , p i , . . . , p N − , m , . . . , m j , . . . , m i , . . . , m N − ) , (4.10)e.g. C ( p , p , m , m , m ) = C ( p , p , m , m , m ) ,C ( p , p , m , m , m ) = C ( p , p , m , m , m ) ,C ( p , p , m , m , m ) = C ( p , p , m , m , m ) . (4.11)All one-loop tensor integrals can be reduced to the scalar ones T N . This is done inSect. 4.2. General analytical results for the scalar integrals A , B , C are D are listedin Sect. 4.3. The scalar integrals for N > D ’s in fourdimensions. The relevant formulae are given in Sect. 4.4. They apply as well to the21ensor integrals with N ≤ Using the Lorentz decomposition of the tensor integrals (4.4) the invariant functions T Ni ...i P can be iteratively reduced to the scalar integrals T N [18]. We derive the relevantformulae for the general tensor integral.The product of the integration momentum q µ with an external momentum can beexpressed in terms of the denominators qp k = 12 [ D k − D − f k ] , f k = p k − m k + m . (4.12)Multiplying (4.1) with p k and substituting (4.12) yields R N,kµ ...µ P − = T Nµ ...µ P p µ P k = 12 (2 πµ ) − D iπ Z d D q " q µ . . . q µ P − D . . . D k − D k +1 . . . D N − − q µ . . . q µ P − D . . . D N − − f k q µ . . . q µ P − D . . . D N − = h T N − µ ...µ P − ( k ) − T N − µ ...µ P − (0) − f k T Nµ ...µ P − i , (4.13)where the argument k of the tensor integrals in the last line indicates that the propagator D k was cancelled. Note that T N − µ ...µ P − (0) has an external momentum in its first propaga-tor. Therefore a shift of the integration momentum has to be performed in this integralin order to bring it to the form (4.1). All tensor integrals on the right-hand side of eq.(4.13) have one Lorentz index less than the original tensor integral. In two of them alsoone propagator is eliminated.For P ≥ g µν and using g µν q µ q ν = q = D + m . (4.14)This gives R N, µ ...µ P − = T Nµ ...µ P g µ P − µ P = (2 πµ ) − D iπ Z d D q (cid:20) q µ . . . q µ P − D . . . D N + m q µ . . . q µ P − D . . . D N (cid:21) = h T N − µ ...µ P − (0) + m T Nµ ...µ P − i . (4.15)Inserting the Lorentz decomposition (4.4) for the tensor integrals T into (4.13) and (4.15)we obtain set of linear equations for the corresponding coefficient functions. This set22ecomposes naturally into disjoint sets of N − X N − = p p p . . . p p N − p p p . . . p p N − ... ... . . . ... p N − p p N − p . . . p N − (4.16)exists, these can be solved for the invariant functions T Ni ...i p yielding them in terms ofinvariant functions of tensor integrals with fewer indices (see eqs. (4.18) and (4.19) below).In this way all tensor integrals are expressed iteratively in terms of scalar integrals T L with L ≤ N .If the matrix X N − becomes singular, the reduction algorithm breaks down. If thisis due to the linear dependence of the momenta we can leave out the linear dependentvectors of the set p , . . . , p N − in the Lorentz decomposition resulting in a smaller matrix X M . If X M is nonsingular the reduction algorithm works again. This happens usually atthe edge of phase space where some of the momenta p i become collinear.If the determinant of X N − , the Gram determinant, is zero but the momenta are notlinear dependent one has to use a different reduction algorithm [29, 30]. This will bediscussed in Sect. 4.4.Here we give the results for the reduction of arbitrary N-point integrals dependingon M ≤ N − D dimensions for nonsingular X M .Inserting the Lorentz decomposition of T N and R N,k as well as R N, R N,kµ ...µ P − = T Nµ ...µ P p µ P k = M X i ,...,i P − =0 R N,ki ...i P − p i µ · · · p i P − µ P − ,R N, µ ...µ P − = T Nµ ...µ P g µ P − µ P = M X i ,...,i P − =0 R N, i ...i P − p i µ · · · p i P − µ P − , (4.17)into the first lines of (4.13) and (4.15) these equations can be solved for the T Ni ...i p : T N i ...i P − = 1 D + P − − M " R N, i ...i P − − M X k =1 R N,kki ...i P − ,T Nki ...i P − = (cid:16) X − M (cid:17) kk ′ " R N,k ′ i ...i P − − P − X r =1 δ k ′ i r T N i ...i r − i r +1 ...i P − . (4.18)Note that the numerator of the prefactor in the first equation is always positve in therelevant cases P ≥ D > M . Using the third lines of (4.13) and (4.15) the R ’s canbe expressed in terms of T Ni ...i P − , T Ni ...i P − , and T N − i ...i q , with q < P as follows R N, i ...i q M...M | {z } P − − q = m T Ni ...i q M...M | {z } P − − q + ( − P − q ˜ T N − i ...i q (0) + (cid:18) P − − q (cid:19) M − X k =1 ˜ T N − i ...i q k (0) This can happen, because of the indefinite metric of space time. (cid:18) P − − q (cid:19) M − X k ,k =1 ˜ T N − i ...i q k k (0) + . . . + P − − qP − − q ! M − X k ,...,k P − − q =1 ˜ T N − i ...i q k ...k P − − q (0) ,R N,ki ...i q M...M | {z } P − − q = 12 ( T N − i ... ˜ i q ˜ M... ˜ M | {z } P − − q ( k ) θ ( k | i , . . . , i q , M, . . . , M | {z } P − − q ) − f k T Ni ...i q M...M | {z } P − − q (4.19) − ( − P − − q ˜ T N − i ...i q (0) + (cid:18) P − − q (cid:19) M − X k =1 ˜ T N − i ...i q k (0)+ (cid:18) P − − q (cid:19) M − X k ,k =1 ˜ T N − i ...i q k k (0) + . . . + P − − qP − − q ! M − X k ,...,k P − − q =1 ˜ T N − i ...i q k ...k P − − q (0) ) where i , . . . , i q = M and θ ( k | i , . . . , i P − ) = i r = k, r = 1 , . . . , P − , . (4.20)The indices ˜ i refer to the i -th momentum of the corresponding N -point function T N butto the ( i − N − T N − ( k ) if i > k . Again thearguments of the T ’s indicate the cancelled propagators. The tilde in ˜ T (0) means that ashift of the integration variable q → q − p M has been performed in order to obtain thestandard form of these integrals. This shift generates the terms in the square bracketsof (4.19). It is also the reason for the unsymmetric appearance of the index M in theabove equations. A different shift would result in similar results. An explicit exampleillustrating the use of these reduction formulae is given in App. C.The recursion formulae above determine the coefficients T i ...i P regardless of their sym-metries. Consequently coefficients whose indices are not all equal are obtained in differentways. This allows for checks on the analytical results as well as on numerical stability.If the number M of linear independent momenta equals the dimension D of space-timethen the terms containing g µν in the Lorentz decomposition have to be omitted, since g µν can be built up from the D momenta. In this case the coefficients T Ni ...i P are obtainedfrom the second equations in (4.18) and (4.19) with T N i ...i P − = 0. N ≤ T N provided the matrices X M are nonsingular. General analytical results24or A , B , C and D were derived in [28]. Algorithms for the numerical calculation of thescalar one-loop integrals based on these results have been presented in [31]. Here we givea new formula [32] for D involving only 16 dilogarithms compared to 24 of the solutionof [28]. For completeness we first list the results for A , B and C . The scalar one-point function reads A ( m ) = − m ( m πµ ) D − Γ(1 − D m (∆ − log m µ + 1) + O ( D − , (4.21)with the UV-divergence contained in∆ = 24 − D − γ E + log 4 π (4.22)and γ E is Euler’s constant. The terms of order O ( D −
4) are only relevant for two- orhigher-loop calculations.
The two-point function is given by B ( p , m , m ) = ∆ − Z dx log [ p x − x ( p − m + m ) + m − iε ] µ + O ( D − − log m m µ + m − m p log m m − m m p ( 1 r − r ) log r + O ( D − , (4.23)where r and r are determined from x + m + m − p − iεm m x + 1 = ( x + r )( x + 1 r ) . (4.24)The variable r never crosses the negative real axis even for complex physical masses ( m has a negative imaginary part!). For r < iε prescription yields Im r = ε sgn( r − r ).Consequently the result (4.23) is valid for arbitrary physical parameters.For the field renormalization constants we need the derivative of B with respect to p . This is easily obtained by differentiating the above result ∂∂p B ( p , m , m ) = − m − m p log m m + m m p ( 1 r − r ) log r − p (cid:16) r + 1 r − r (cid:17) + O ( D − . (4.25)25 .3.3 Scalar three-point function The general result for the scalar three-point function valid for all real momenta andphysical masses was calculated by [28]. It can be brought into the symmetric form C ( p , p , m , m , m ) = − Z dx Z x dy [ p x + p y + ( p − p − p ) xy + ( m − m − p ) x + ( m − m + p − p ) y + m − iε ] − (4.26)= 1 α X i =0 ( X σ = ± (cid:20) Li (cid:16) y i − y iσ (cid:17) − Li (cid:16) y i y iσ (cid:17) + η (cid:16) − x iσ , y iσ (cid:17) log y i − y iσ − η (cid:16) − x iσ , y iσ (cid:17) log y i y iσ (cid:21) − (cid:20) η ( − x i + , − x i − ) − η ( y i + , y i − ) − πiθ ( − p jk ) θ ( − Im( y i + y i − )) (cid:21) log 1 − y i − y i ) , with ( i, j, k = 0 , , y i = 12 αp jk [ p jk ( p jk − p ki − p ij + 2 m i − m j − m k ) − ( p ki − p ij )( m j − m k ) + α ( p jk − m j + m k )] ,x i ± = 12 p jk [ p jk − m j + m k ± α i ] ,y i ± = y i − x i ± , (4.27) α = κ ( p , p , p ) ,α i = κ ( p jk , m j , m k ) (1 + iεp jk ) , and κ is the K¨all´en function κ ( x, y, z ) = q x + y + z − xy + yz + zx ) . (4.28)The dilogarithm or Spence function Li ( x ) is defined asLi ( x ) = − Z dtt log(1 − xt ) , | arg (1 − x ) | < π. (4.29)The η -function compensates for cut crossings on the Riemann-sheet of the logarithms anddilogarithms. For a , b on the first Riemann sheet it is defined bylog( ab ) = log( a ) + log( b ) + η ( a, b ) . (4.30)All η -functions in (4.26) vanish if α and all the masses m i are real. Note that α is real inparticular for all on-shell decay and scattering processes.26 .3.4 Scalar four-point function The scalar four-point function D ( p , p , p , m , m , m , m ) can be expressed interms of 16 dilogarithms [32].Before we give the result we first introduce some useful variables and functions. Wedefine k ij = m i + m j − p ij m i m j , i, j = 0 , , , , (4.31)and r ij and ˜ r ij by x + k ij x + 1 = ( x + r ij )( x + 1 /r ij ) , (4.32)and x + ( k ij − iε ) x + 1 = ( x + ˜ r ij )( x + 1 / ˜ r ij ) . (4.33)Note that for real k ij the r ij ’s lie either on the real axis or on the complex unit circle.Furthermore P ( y , y , y , y ) = X ≤ i
6, because any five momenta are linear dependent in four dimensions.
Here we assume that the four external momenta appearing in the five-point functionspan the whole four-dimensional space . Then the integration momentum q dependslinearly on these four external momenta and the following equations holds0 = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q qp . . . qp p q p . . . p p ... ... . . . ...2 p q p p . . . p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) D + Y qp . . . qp D − D + Y − Y p . . . p p ... ... . . . ... D − D + Y − Y p p . . . p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (4.45)with Y ij = m i + m j − ( p i − p j ) . (4.46)and D i as defined in (4.2). Thus we have1 iπ Z d D q D D · · · D (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) D + Y qp . . . qp D − D + Y − Y p . . . p p ... ... . . . ... D − D + Y − Y p p . . . p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 . (4.47) The exceptional case, when they are linear dependent will be covered in the next section. h T (0) + Y T i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p p . . . p p ... . . . ...2 p p . . . p p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + X k =1 ( − k h T µ ( k ) − T µ (0) − p µ T (0)+ p µ T (0) + ( Y k − Y ) T µ i × (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p µ . . . p µ p p . . . p p ... . . . ...2 p k − p . . . p k − p p k +1 p . . . p k +1 p ... . . . ...2 p p . . . p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (4.48)where the arguments of the functions T denote again the cancelled propagators.The Lorentz decomposition of the vector integrals in (4.48) involves only the momenta p , . . . , p . T µµ ...µ P ( k ) does not depend on p k , consequently each term in its Lorentzdecomposition contains a factor p iµ , i = k and its contraction with the correspondingdeterminant in (4.48) vanishes. Similarly all terms in the tensor integral decomposition of T µ (0) + p µ T (0) involve a factor p iµ − p µ , i = 1 , ,
3, if one performs the shift q → q − p to bring the tensor integral to the standard form. Multiplying with the determinantsand performing the sum in (4.48) these terms drop out. Finally in the term p µ T (0)the determinant is nonzero only for k = 4 where it can be combined with the first termin (4.48). Rewriting the resulting equation as a determinant and reinserting the explicitform of the tensor integrals we find1 iπ Z d D q D D · · · D (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) D + Y qp . . . qp Y − Y p . . . p p ... ... . . . ... Y − Y p p . . . p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 . (4.49)Using 2 p i p j = Y ij − Y i − Y j + Y , qp j = D j − D + Y j − Y , (4.50)30dding the first column to each of the other columns and then enlarging the determinantby one column and one row this can be written as (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Y . . . Y T (0) + Y T . . . T (4) + Y T Y − Y . . . Y − Y ... ... . . . ...0 Y − Y . . . Y − Y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 (4.51)This is equivalent to (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T − T (0) − T (1) − T (2) − T (3) − T (4)1 Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 , (4.52)which can be solved for T if the determinant of the matrix Y ij , i, j = 0 , . . . , T (0) the momenta have not been shifted. In particular (4.52)yields the scalar five-point function T in terms of five scalar four-point functions. For vanishing Gram determinant | X N − | the following relation holds, if the Lorentzdecomposition of the appearing tensor integrals contains only momenta and no metrictensors, which is the case for N ≥ P = 0 (scalar integrals)(2 πµ ) − D iπ Z d D q q µ · · · q µ P D D · · · D N − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) D + Y qp . . . qp N − Y − Y p . . . p p N − ... ... . . . ... Y N − − Y p N − p . . . p N − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 . (4.53)Performing the same manipulations of the determinant as in (4.49) to (4.52) above thisresults in (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T Nµ ...µ P − T N − µ ...µ P (0) − T N − µ ...µ P (1) . . . − T N − µ ...µ P ( N − Y Y . . . Y N − Y Y . . . Y N − ... ... ... . . . ...1 Y N − Y N − . . . Y N − N − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 , (4.54)valid for | X N − | = 0 and N ≥ P = 0. We stress again that in the tensor integral T Nµ ...µ P (0) appearing in (4.54) the momenta have not been shifted. Eq. (4.54) determines T Nµ ...µ P in terms of T N − µ ...µ P ( i ), i = 0 , . . . , N −
1, if the determinant of the matrix Y ij is31onzero. The vanishing of this determinant corresponds to the leading Landau singularityof T N which is clearly not contained in T N − . In this case one has to calculate T N directly[30].Eq. (4.54) in particular expresses T N by T N − . For N = 5 and P = 0 (4.54) coincideswith (4.52), which is thus valid for arbitrary momenta. For N > N ≤
4, where(4.54) is only valid for scalar integrals, the Gram determinant is singular at the edge ofphase space where some of the momenta p i become collinear, i.e. for forward or backwardscattering or at the threshold of a certain process. Because in this special situations allintegrals can be reduced to lower ones one can obtain considerably simpler formulae thanin the general case (see e.g. [33]).With the methods described in this section all tensor integrals with N ≥ N . Note that this may yield tensorintegrals with P > N because P is not reduced simultaneously as in the reduction methoddescribed in Sect. 4.2. These tensor integrals are not directly present in renormalizabletheories. Nevertheless their reduction to scalar integrals can be done with the formulaegiven in Sect. 4.2. For practical calculations it is useful to know the UV-divergent parts of the tensorintegrals explicitly. We give directly the products of D − O ( D − D − A ( m ) = − m , ( D − B ( p , m , m ) = − , ( D − B ( p , m , m ) = 1 , ( D − B ( p , m , m ) = ( p − m − m ) , ( D − B ( p , m , m ) = − , ( D − C ( p , p , m , m , m ) = − , ( D − C i ( p , p , m , m , m ) = , ( D − D ( p , p , p , m , m , m , m ) = − . (4.55)All other scalar coefficients defined in (4.7) and (4.8) are UV-finite.32 Standard matrix elements
The invariant matrix elements for scattering and decay processes involving externalfermions and/or vector bosons depend on the polarizations σ i , σ ′ i and λ i of these particles.This dependence is completely contained in the polarization vectors ε µ i ( k i , λ i ) and spinors¯ v α i ( p ′ i , σ ′ i ) and u α i ( p i , σ i ). k i , p ′ i and p i denote the incoming momenta of the vector bosons,antifermions and fermions, respectively. For outgoing fermions one has to replace p , p ′ by − p , − p ′ and one must use u ( − p ) = v ( p ). If we split off the polarization vectors andspinors from the invariant matrix element M we are left with a tensor involving Lorentzand Dirac indices in the general case M = ¯ v α ( p ′ , σ ′ ) . . . ¯ v α n ( p ′ n , σ ′ n ) M µ ...µ m α ...α n β ...β n u β ( p , σ ) . . . u β n ( p n , σ n ) × ε µ ( k , λ ) . . . ε µ m ( k m , λ m ) . (5.1)To be definite we choose m external vector bosons and n external fermion-antifermionpairs. The tensor M µ ...µ m α ...β n can be decomposed into a set of covariant operators togetherwith the corresponding scalar formfactors F i M µ ...µ m α ...β n = X i M µ ...µ m i,α ...β n F i . (5.2)We call the covariant operators M µ ...µ n i,α ...β n multiplied by the corresponding polarizationvectors and spinors standard matrix elements M i M i = ¯ v α ( p ′ , σ ′ ) . . . ¯ v α n ( p ′ n , σ ′ n ) M µ ...µ m i,α ...β n u β ( p , σ ) . . . u β n ( p n , σ n ) × ε µ ( k , λ ) . . . ε µ m ( k m , λ m ) . (5.3)In this way the invariant amplitude M is decomposed into polarization independentformfactors F i and the standard matrix elements M i M = X i M i F i . (5.4)The formfactors F i are complicated model dependent functions involving in general theinvariant integrals T N and the counterterms. The standard matrix elements in contrastare simple model independent expressions which depend on the external particles onlybut contain the whole information on their polarization. They are purely kinematicalobjects. All of the dynamical information is contained in the formfactors.The covariant tensor operators forming the standard matrix elements can be con-structed from the external four-momenta p i , p ′ i and k i , the Lorentz tensors g µν and ε µνρσ and the Dirac matrices γ µ , γ . In general one thus obtains an overcomplete set. Diracalgebra and momentum conservation are used to eliminate superfluous operators. Sincethe external particles are on-shell, the Dirac equation for the fermion spinors p/ i u ( p i , σ i ) = m i u ( p i , σ i ) , ¯ v ( p ′ i , σ ′ i ) p/ ′ i = − m ′ i ¯ v ( p ′ i , σ ′ i ) (5.5)33nd the transversality condition for the polarization vectors k µ i i ε µ i ( k i , λ i ) = 0 (5.6)reduce the number of independent standard matrix elements further.The number of independent standard matrix elements cannot be larger than the num-ber of independent polarization combinations of the external particles. In four dimensionsthere are only four linear independent four-vectors. Expressing all four-vectors in a def-inite basis allows to derive the missing relations between the remaining standard matrixelements. Thus a minimal set of standard matrix elements can be constructed.If there are only few external particles there may be less independent standard ma-trix elements than different polarization combinations, since there are only few momentaavailable for their construction. In this case some of the polarized amplitudes are related.The number of standard matrix elements can be reduced further if the model underconsideration exhibits certain symmetries. These evidently also apply to the relevantstandard matrix elements.For many applications it is not essential to minimize the number of standard matrixelements. All one needs is a complete set.Furthermore the choice of the standard matrix elements is not unique. This allows toarrange for the most convenient set according to simplicity, the structure of the lowestorder amplitudes and, if present, symmetries. At least some of the formfactors can bechosen as generalizations of the lowest order couplings. This is useful in establishingimproved Born approximations.The concept of standard matrix elements is not indispensable for the calculation ofamplitudes in higher orders. It is, however, extremely helpful in organizing lengthy cal-culations, which often are inevitable. All complicated expressions are cast into the form-factors which are polarization independent and thus have to be evaluated only once.An alternative method would be to calculate directly the polarized amplitudes. Thisrequires a definite representation for the spinors and/or polarization vectors from thestart. The whole calculation has to be done for each polarization separately. A closerlook shows that this method can be represented as a particular case of the standardmatrix element approach. The corresponding covariant operators are constructed fromthe polarization vectors and spinors instead of the momenta, Lorentz tensors and Diracmatrices. Their explicit form is M µ ...µ m i,α ...α n β ...β n = ( − n v α ( p ′ ,σ ′ )2 m ′ . . . v αn ( p ′ n ,σ ′ n )2 m ′ n ¯ u β ( p ,σ )2 m . . . ¯ u βn ( p n ,σ n )2 m n × ε ∗ µ ( k , λ ) . . . ε ∗ µ m ( k m , λ m ) , (5.7)where m , m ′ are the masses of the external spinors. The indices i correspond to differentpolarization combinations. Consequently the number of different standard matrix ele-ments equals the number of polarizations of the external particles. For each polarizationonly one standard matrix element is nonzero. In this sense the set of standard matrixelements (5.7) is orthogonal. The formfactors equal the polarization amplitudes and aredirectly obtained by inserting explicitly the polarization vectors and spinors in the in-variant matrix element. Unlike in the approach outlined above these formfactors are nodirect generalizations of the lowest order couplings.In the following we list complete sets of standard matrix elements relevant for theproduction of bosons in fermion-antifermion annihilation.34 .2 Standard matrix elements for processes with two external fermions In this section we will give the standard matrix elements for processes involving twoexternal fermions ( F ¯ F ) and one [or two] scalar ( S ) or vector ( V ) bosons. The momentaand spinors of the fermions are denoted by p , p and ¯ v ( p ) = ¯ v ( p , σ ), u ( p ) = u ( p , σ ),the momenta and polarization vectors of the bosons by k , ε = ε ( k , λ ) [and k , ε = ε ( k , λ )]. The numbers of different polarizations for each scalar, fermion and vector bosonare 1, 2 and 3, respectively. If we use momentum conservation to eliminate k [or k + k ]the standard matrix elements are constructed from the momenta p and p [and k − k ],the polarization vectors of the vector bosons, the totally antisymmetric tensor ε µνρσ andDirac matrices between the spinors. If there are products of ε -tensors, pairs of them canbe eliminated using ε µνρσ ε αβγδ = − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) g µα g µβ g µγ g µδ g να g νβ g νγ g νδ g ρα g ρβ g ργ g ρδ g σα g σβ g σγ g σδ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (5.8)If any of the left over ε -tensors are contracted with four four-momenta, we write for oneof these momenta p α = { p/, γ α } between the spinors. Now all remaining ε -tensors arecontracted with one γ -matrix at least and can be eliminated using the Chisholm identity ε µνρσ γ σ = − i [ γ µ γ ν γ ρ − g µν γ ρ + g µρ γ ν − g νρ γ µ ] γ . (5.9)All Dirac matrices contracted with p or p can be eliminated using Dirac algebra andthe Dirac equation. Consequently the only remaining Dirac matrices are contracted withpolarization vectors [and k/ − k/ ] and there is at most one of each type. Finally in thescalar products involving the polarization vectors only one [or two] independent momentamay appear because of transversality and momentum conservation.Thus we arrive at the following sets of standard matrix elements (we suppress polar-ization indices in the following): S → F ¯ F There are 2 × M σ = ¯ u ( p ) ω σ v ( p ) , (5.10)where σ = ± and ω ± = ± γ and the fermions are outgoing. V → F ¯ F Replacing the scalar by a vector results in 3 × × M σ = ¯ u ( p ) ε/ ω σ v ( p ) , M σ = ¯ u ( p ) ω σ v ( p ) ε p . (5.11) Eq. (5.8) and (5.9) can be applied because the standard matrix elements involve only external vectorsand spinors which remain four-dimensional also in dimensional regularization. .2.3 F ¯ F → SS Here the number of independent polarizations four equals the number of standardmatrix elements M σ = ¯ v ( p ) ( k/ − k/ ) ω σ u ( p ) , M σ = ¯ v ( p ) ω σ u ( p ) . (5.12) F ¯ F → SV In this case we find twelve standard matrix elements for 2 × × M σ = ¯ v ( p ) ε/ ω σ u ( p ) , M σ = ¯ v ( p ) ( k/ − k/ ) ω σ u ( p ) ε p , M σ = ¯ v ( p ) ( k/ − k/ ) ω σ u ( p ) ε p , M σ = ¯ v ( p ) ε/ k/ ω σ u ( p ) , M σ = ¯ v ( p ) ω σ u ( p ) ε p , M σ = ¯ v ( p ) ω σ u ( p ) ε p . (5.13) F ¯ F → V V
There are 2 × × × M σ = ¯ v ( p ) ε/ ( k/ − p/ ) ε/ ω σ u ( p ) , M σ = ¯ v ( p ) ( k/ − k/ ) ( ε ε ) ω σ u ( p ) , M σ , = ¯ v ( p ) ε/ ω σ u ( p ) ( ε k ) , M σ , = − ¯ v ( p ) ε/ ω σ u ( p ) ( ε k ) , M σ , = ¯ v ( p ) ε/ ω σ u ( p ) ( ε p ) , M σ , = − ¯ v ( p ) ε/ ω σ u ( p ) ( ε p ) , M σ = ¯ v ( p ) ( k/ − k/ ) ω σ u ( p ) ( ε k ) ( ε k ) , M σ = ¯ v ( p ) ( k/ − k/ ) ω σ u ( p ) ( ε p ) ( ε p ) , M σ , = ¯ v ( p ) ( k/ − k/ ) ω σ u ( p ) ( ε k ) ( ε p ) , M σ , = ¯ v ( p ) ( k/ − k/ ) ω σ u ( p ) ( ε p ) ( ε k ) , (5.14) M σ = ¯ v ( p ) ε/ ε/ ω σ u ( p ) , M σ = ¯ v ( p ) ω σ u ( p ) ( ε ε ) , M σ , = ¯ v ( p ) ε/ k/ ω σ u ( p ) ( ε k ) , M σ , = ¯ v ( p ) k/ ε/ ω σ u ( p ) ( ε k ) , M σ , = ¯ v ( p ) ε/ k/ ω σ u ( p ) ( ε p ) , σ , = ¯ v ( p ) k/ ε/ ω σ u ( p ) ( ε p ) , M σ = ¯ v ( p ) ω σ u ( p ) ( ε k ) ( ε k ) , M σ = ¯ v ( p ) ω σ u ( p ) ( ε p ) ( ε p ) , M σ , = ¯ v ( p ) ω σ u ( p ) ( ε k ) ( ε p ) , M σ , = ¯ v ( p ) ω σ u ( p ) ( ε p ) ( ε k ) . We have obtained more than 36 standard matrix elements because we have not yet usedthe four dimensionality of space time, i.e. the fact that the five vectors p , p , k − k , ε and ε are linear dependent. The relations between the 40 standard matrix elements canbe found by representing these vectors in a certain basis using for example v = p + p , v = p − p , v = k − k , v ,µ = ε µνρσ v ν v ρ v σ . In this way one can derive the relation0 = 2( p p − m m ) ( M σ + M σ ) − p k − m − m m ) M σ , − p k − m − m m ) M σ , − k + ( k k )) M σ , − k + ( k k )) M σ , − M σ + 2( M σ , + M σ , ) − m ( m − p k ) + m ( m − p k ))( M σ − M σ ) (5.15)+ ( m + m )( p p − m m ) M σ − m M σ , − m M σ , + ( m + m ) (2 M σ , + 2 M σ , − M σ − M σ )+ (3 m + m ) M σ , + (3 m + m ) M σ , and a similar independent one allowing to eliminate four of the 40 standard matrix ele-ments (5.14).The construction of complete sets of standard matrix elements described above isstraightforward. The reduction of general structures to these standard matrix elementsis therefore easy to implement into computer algebra programs. In practical applicationssome of the standard matrix elements may not contribute due to the presence of symme-tries and/or the neglection of fermion masses. These aspects will be discussed togetherwith the applications in the following chapters. For the calculation of the standard matrix elements one has to choose a certain rep-resentation for the polarization vectors and spinors. This has to be done only once foreach process and not in the calculation of individual Feynman diagrams. If there areat least four external particles the polarization vectors can be constructed from theirfour-momenta respecting k i · ε i ( k i , λ i ) = 0 ,ε i ( k i , λ i ) ε i ( k i , λ ′ i ) = − δ λ i ,λ ′ i . (5.16)We thus obtain for ε ε µ ( k , k ) = 1 q [ p p (2 p k p k − k p p ) + p p k − p ( p k ) − p ( p k ) ]37 q [( p k + p k ) − k ( p + p ) ] × (cid:20) p µ (cid:16) p ( p + p ) k − p k ( p k + p k ) (cid:17) − p µ (cid:16) p ( p + p ) k − p k ( p k + p k ) (cid:17) + k µ (cid:16) p p ( p k − p k ) + p p k − p p k (cid:17)(cid:21) = (0 , cos ϑ, , − sin ϑ ) (5.17) ε µ ( k , ⊥ ) = 1 q p p (2 p k p k − k p p ) + p p k − p ( p k ) − p ( p k ) ǫ µνρσ p ν p ρ k σ = (0 , , ,
0) (5.18) ε µ ( k , L ) = 1 q k [( p k + p k ) − k ( p + p ) ] (cid:20) k µ ( p k + p k ) − ( p + p ) µ k (cid:21) = ( k, E sin ϑ, , E cos ϑ ) / q k . (5.19)where we have also given the simple expressions in the CMS-system of the fermions andbosons. In this system the four-momenta of the external particles read p , = ( ˜ E , , , , ∓| p | ) ,k , = ( E , , ∓| k | sin ϑ, , ∓| k | cos ϑ ) . (5.20)˜ E , are the energies and p the three-momentum of the fermions and E , the energiesand k the three-momentum of the bosons. ϑ is the angle between the spatial vectors p and k ,From the polarization vectors given above the ones for helicity states are obtained as ε µ ( k , ± ) = 1 √ h ε µ ( k , k ) ± iε µ ( k , ⊥ ) i , ε µ ( k ,
0) = ε µ ( k , L ) . (5.21)The polarization vector ε can be obtained by interchanging 1 ↔ V → F ¯ F , and to(5.12) for the annihilation processes F ¯ F → V V , V S . To calculate these remaining Diracmatrix elements one either inserts a definite representation for the spinors or evaluates the38uantities M σ ∗ i M σ ′ j via traces and reconstructs M σi from those if needed. Note that forthe calculation of |M| to one-loop order one only has to evalulate the products M σ ∗ i M σ ′ j for those values of i , where F σi is nonzero in lowest order |M| = |M + δ M | ≈ |M | + 2Re {M ∗ δ M } = X i,j F σ ∗ i, ( F σj, + 2 δF σj, )Re {M σ ∗ i M σj } . (5.22)Here M , F σi, denote the lowest order quantities and δ M , δF σi, the one-loop quantities.For massless external fermions the Dirac matrix elements (5.10) and (5.12) are equiva-lent to the helicity matrix elements. They do not interfere and can thus easily be obtainedfrom |M σi | as¯ v ( p ) ω σ u ( p ) = √ p p , ¯ v ( p ) ( k/ − k/ ) ω σ u ( p ) = q p ( k − k ) p ( k − k ) − p p ( k − k ) . (5.23)If one is only interested in unpolarized quantities it suffices to calculate P pol M σ ∗ i M σj using the polarization sums for vector bosons and spinors.39 Calculation of one-loop amplitudes
We have described all the ingredients necessary for the calculation of one-loop radiativecorrections. This chapter shows how one-loop amplitudes are evaluated in practice.First one has to specify a Lagrangian and to derive the corresponding Feynman rules.Then renormalization has to be carried out and the counterterms have to be determined.Both were done in Chap. 2 and 3 for the SM. Once the Feynman rules and the countertermsare fixed, the following steps apply to any renormalizable model.To calculate the amplitude of a certain process at the one-loop level one has to con-struct all tree and one-loop Feynman diagrams with the given external particles allowedby the specified model. Next each Feynman diagram has to be reduced algebraically toa form suitable for numerical evaluation. This procedure is explained in more detail inSect. 6.1. Finally the expressions for all diagrams have to be put together into a numeri-cal program which calculates the amplitude and the corresponding cross section or decayrate.
The algorithm for the reduction of one-loop diagrams is the following. The loop inte-gral obtained from the Feynman rules contains a product of propagators as denominatorand a numerator composed of Lorentz vectors and tensors, Dirac matrices and spinors andpolarization vectors of the external particles. The numerator is simplified using tensorand Dirac algebra, the mass shell conditions for the external particles and momentumconservation. One can also try to separate terms proportional to one or more of thedenominators. Cancelling these yields N-point functions of lower degree. Next the loopintegral is organized into the tensor integrals defined in Sect. 4.1. The Lorentz decompo-sition of these integrals is inserted and the whole Dirac and Lorentz structure is separatedoff from the integrals. Using again Dirac algebra and mass shell conditions it can bereduced to the appropriate standard matrix elements as discussed in Chap. 5.We thus arrive at an expression of the form δ M = X i M i δF i (6.1)for each one-loop Feynman diagram. The formfactors are linear combinations of theinvariant coefficient functions of the tensor integrals with coefficients being functions ofthe kinematical invariants.The formfactors can be further evaluated by applying the reduction scheme for theinvariant integrals described in Sects. 4.2 and 4.4. Finally they are obtained as linearcombinations of the scalar one-loop integrals A , B , C and D which are given explicitlyin Sect. 4.3. This last step may lead to very lengthy expressions. Their algebraic evalu-ation needs a lot of time and space. This can be avoided by performing the reduction toscalar integrals numerically.The evaluation of the counterterm diagrams and the Born diagrams is done in a similarway. Since no integrations have to be performed their calculation is much easier. In mostcases the counterterm diagrams can be obtained from the Born diagrams by replacing thelowest order couplings by the corresponding counterterm.40 − e + W − W + WWν e Z (cid:27)- 6 (cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0)(cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1)(cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0)(cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1)(cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0)(cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1)(cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0)(cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1)(cid:3)(cid:2)(cid:3)(cid:2)(cid:3)(cid:2)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) Figure 6.1: Box diagram contributing to e + e − → W + W − .As an illustration of the reduction method we present the explicit calculation of a boxdiagram contributing to e + e − → W + W − (Fig. 6.1). According to the Feynman rules thecorresponding contribution to the invariant matrix element δ M is given by (we include aglobal factor i in the Feynman rules in order to obtain real amplitudes) δ M = − i e c W s W µ − D Z d D q (2 π ) D ¯ v ( p ) γ µ ( q/ + k/ − p/ ) γ ν ω − u ( p )Γ λµρ Γ λσν ε ρ ε σ [ q − M Z ] [( q + k ) − M W ] ( q + k − p ) [( q − k ) − M W ] , (6.2)with Γ λµρ = g λµ (2 q + k ) ρ + g µρ ( − q − k ) λ + g ρλ ( k − q ) µ , Γ λσν = g λσ ( − k − q ) ν + g σν (2 k − q ) λ + g νλ (2 q − k ) σ . (6.3)Evaluating the numerator and introducing the tensor integrals D and C we arrive at δ M = α c W s W ¯ v ( p ) n − γ µ ε ν ε ρ D µνρ + D µν h ε/ γ µ ε/ k − k ) ν + ε/ (cid:16) − ε µ k ν − p µ ε ν + 2 p µ ε ν (cid:17) + ε/ γ µ k/ ε ν − ε/ (cid:16) − ε µ k ν − p µ ε ν + 2 p µ ε ν (cid:17) − k/ γ µ ε/ ε ν + k/ ε µ ε ν + γ µ (cid:16) − ε ν p ε + 16 ε ν p ε + 2( k − p ) ν ε ε (cid:17)i + D µ h ε/ ( k/ − p/ ) ε/ k − k ) µ + ε/ γ µ ε/ ( M Z − k k )+ ε/ (cid:16) − ε µ (3 M Z + 3 M W + 2 p k ) − k µ ε k + 4 p µ ε k (cid:17) − ε/ (cid:16) ε µ (3 M Z + 3 M W + 2 p k ) + 4 k µ ε k − p µ ε k (cid:17) + k/ (cid:16) k − p ) µ ε ε − ε µ ε p + 8 ε µ ε p (cid:17) (6.4)41 γ µ ε ε ( M Z − M W + 6 p k ) + k/ γ µ ε/ ε k + 4 ε/ γ µ k/ ε k i + D h ε/ ( k/ − p/ ) ε/ ( M Z − k k ) + k/ ε ε ( M W − p k − M Z )+ ε/ ε k (4 p k − M W + 2 M Z ) − ε/ ε k (4 p k − M W + 2 M Z ) i + C µ h ε/ γ µ ε/ − ε/ ε µ − ε/ ε µ + γ µ ε ε i + C h ε/ ε k − ε/ ε k + ε/ ε p − ε/ ε p − k/ ε ε io ω − u ( p ) . The three-point integrals arise from q terms in the numerator by writing q =( q − M Z )+ M Z and cancelling the first denominator factor. After that the shift q → q − k + p was performed in the three-point integrals (this shift conserves the manifest CP symmetryof the diagram). The arguments of the C and D functions are as follows D = D ( k , k − p , − k , M Z , M W , , M W ) ,C = C ( p , − p , , M W , M W ) . (6.5)Inserting the tensor integral decomposition eqs. (4.7, 4.8) yields the final expression δ M = α c W s W n M − h D + 2(4 M W − s ) D + 2( M W + t ) D + (12 M W + 4 t − s ) D + 2(4 M W − s ) D + (16 M W − s + 2 M Z ) D + (2 t − s + 6 M W + M Z ) D + (4 M W − s + M Z ) D i + M − h − C − D − D + 10 D + 2 tD + 2( M W + t )( D + D )+ 2( M W + 3 t ) D + 2( M W − t + M Z ) D + ( M Z − t ) D + ( t − M Z ) D i + h M − , + M − , ih − C + 2 C − D − D + (4 s − M W + 5 t ) D − M W + t ) D − M W + t ) D + ( t − M W − M Z ) D + 2( M Z − t ) D i (6.6)+ h M − , + M − , ih C + 3 C − D − D + 2( s − M W )( D + D ) − t + 2 M W ) D + (4 s − t − M W ) D + (4 s − M W − M Z ) D + ( t − M W − M Z ) D i + M − h D + 8 D − D i + M − h D + 16 D + 24 D + 32 D + 16 D i h M − , + M − , ih D + 8 D + 8 D + 8 D + 16 D io , where we introduced the standard matrix elements (5.14), the Mandelstam variables s = ( p + p ) , t = ( p − k ) , (6.7)and put k = k = M W , p = p = 0 . (6.8)Furthermore we made use of the relations which follow from the symmetry of the diagramunder the exchange e + ↔ e − , W + ↔ W − (CP invariance) D = D , D = D , D = D ,D = D , D = D , D = D ,D = D , D = D , C = C . (6.9)These reduce the number of independent invariant integrals considerably. Note that notall of the 40 standard matrix elements of (5.14) appear in (6.6). This is due to theneglection of fermion masses and CP invariance of the box diagram.This example shows that the reduction method is straightforward and universallyapplicable to one-loop Feynman diagrams, since they all have a similar structure. The huge number of algebraic calculations makes the evaluation of each Feynmandiagram very lengthy and tedious. Furthermore there are a large number of diagramscontributing to each process. Fortunately many of the diagrams resemble each otherin their algebraic structure and can be considered as special cases of generic diagrams.These are the Feynman diagrams of a theory with only one generic scalar, fermion, vectorboson and Faddeev-Popov ghost each and arbitrary renormalizable couplings betweenthose fields (for more details see [34]). It suffices to do the algebraic calculations for thesegeneric diagrams only. All actual diagrams are obtained from those by substituting theactual fields together with their coupling constants and masses. This saves a lot of workespecially if there are many fields in the theory.Clearly the generic diagrams can be calculated with the methods described above.The efficiency of generic diagrams is illustrated in the next section using the decay of the W -boson into massless fermions as example. W → f i ¯ f ′ j for massless fermions We will now apply the methods described above by calculating the one-loop amplitudefor the decay of the W -boson into massless fermions W + ( k ) → f i ( p ) ¯ f ′ j ( p ) . (6.10)In lowest order there is only one Feynman diagram (Fig.6.2) leading to the amplitude M = − eV ij √ s W ¯ u ( p ) ε/ ( k ) ω − v ( p ) = − eV ij √ s W M − . (6.11)43 i W + ¯ f ′ j s(cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0)(cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:26)(cid:26)>(cid:26)(cid:26)ZZ}ZZ Figure 6.2: Born diagram to W → f i ¯ f ′ j . W γ, Zf i f ′ j f i ¯ f ′ j s ss(cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0)(cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:8)(cid:8)*(cid:8)(cid:8)(cid:8)(cid:8)*(cid:8)(cid:8)HHHYHH HHHYHH (cid:3)(cid:2)(cid:3)(cid:2)(cid:3)(cid:2)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) W f i γ, ZW f i ¯ f ′ j s ss(cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0)(cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3)(cid:1) (cid:1) (cid:1) (cid:1)(cid:2) (cid:2) (cid:2) (cid:2) (cid:2)(cid:0) (cid:0) (cid:0) (cid:0) W f ′ j Wγ, Z f i ¯ f ′ j s ss(cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0)(cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3)(cid:1) (cid:1) (cid:1) (cid:1)(cid:2) (cid:2) (cid:2) (cid:2) (cid:2)(cid:0) (cid:0) (cid:0) (cid:0) f i W + ¯ f ′ j (cid:19)(cid:19)SS (cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0)(cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:26)(cid:26)>(cid:26)(cid:26)ZZ}ZZ Figure 6.3: One-loop diagrams and corresponding counterterm diagram to W → f i ¯ f ′ j .Neglecting the fermion masses the amplitude (6.11) leads to the following lowest orderdecay width Γ = α M W s W | V ij | . (6.12)At one-loop order there are six loop diagrams and one counterterm diagram (Fig. 6.3; thecounterterm is indicated by a cross).The first two loop diagrams correspond to one generic diagram and the other four toanother generic one. We first calculate the two generic diagrams. The expression for thefirst reads δ M = iµ − D Z d D q (2 π ) D q − M )( q + p ) ( q − p ) ¯ u ( p ) γ ν ( g − ω − + g +1 ω + )( q/ + p/ ) ε/ ( g − ω − + g +3 ω + )( q/ − p/ ) γ ν ( g − ω − + g +2 ω + ) v ( p ) , (6.13)44here g ± denote the generic left- and right-handed fermion-fermion-vector couplings. Sim-plification and decomposition into tensor integrals yields δ M = iµ − D Z d D q (2 π ) D q − M )( q + p ) ( q − p ) ¯ u ( p )[ − q/ − p/ ) ε/ ( q/ + p/ ) + (4 − D )( q/ ε/ q/ )]( g − g − g − ω − + g +1 g +2 g +3 ω + ) v ( p )= − π ¯ u ( p )[(2 − D ) C µν γ µ ε/ γ ν + 2 C µ ( p/ ε/ γ µ − γ µ ε/ p/ ) + 2 C p/ ε/ p/ ]( g − g − g − ω − + g +1 g +2 g +3 ω + ) v ( p ) . (6.14)Insertion of the Lorentz decomposition and further simplification gives δ M = − π ( g − g − g − M − + g +1 g +3 g +2 M +1 )[(2 − D ) C − k ( C + C + C + C )] . (6.15)Finally the reduction of the invariant integrals and the use of (4.55) leads to δ M = − π ( g − g − g − M − + g +1 g +3 g +2 M +1 )[ − k C (0 , k , , M, , M k ) − B ( k , , M k ) + 2 B (0 , M, M k ) − − π ( g − g − g − M − + g +1 g +3 g +2 M +1 ) V a (0 , k , , M, , , (6.16)where we introduced the generic vertex function V a which is defined in the general casein App. C.Similarly we obtain for the second generic diagram δ M = − iµ − D Z d D q (2 π ) D ¯ u ( p ) γ ν ( g − ω − + g +1 ω + )( − q/ ) γ ρ ( g − ω − + g +2 ω + ) v ( p ) q [( q + p ) − M ][( q − p ) − M ] g h g ρµ ( p + 2 p − q ) ν − g µν (2 p + p + q ) ρ + g νρ (2 q + p − p ) µ ] ε µ = π g ( g − g − M − + g +1 g +2 M +1 ) h D − C − k ( C + C + C ) i = π g ( g − g − M − + g +1 g +2 M +1 ) h M + M + M M k ) C (0 , k , , , M , M ) − (1 + M + M k ) B ( k , M , M )+(2 + M k ) B (0 , , M ) + (2 + M k ) B (0 , , M ) i = π g ( g − g − M − + g +1 g +2 M +1 ) V − b (0 , k , , , M , M ) . (6.17)The general definition of V − b can again be found in App. C.Inserting the actual couplings and masses of the six actual diagrams into the genericdiagrams and adding the counterterm diagram, which can be easily obtained from the45eynman rules or the Born diagram, we find for the virtual one-loop corrections to theinvariant amplitude for W → f i ¯ f ′ j δ M = − e √ s W α π V ij M − { Q f Q f ′ V a (0 , M W , , λ, , g − f g − f ′ V a (0 , M W , , M Z , , Q f V − b (0 , M W , , , λ, M W ) − Q f ′ V − b (0 , M W , , , M W , λ )+ c W s W g − f V − b (0 , M W , , , M Z , M W ) − c W s W g − f ′ V − b (0 , M W , , , M W , M Z )+ δZ f,L † ii + δZ f ′ ,Ljj + δZ W + δZ e − δs W s W } . (6.18)The left- and right-handed couplings g σf of the fermions to the Z -boson are defined in(A.14). Note that only one out of the four standard matrix elements (5.11) is contributingthere and that we need no counterterm to the quark mixing matrix for massless fermions.The counterterms are expressed in terms of the self energies in Sect. 3.3. These have tobe calculated to one-loop order to determine δ M completely. δ M contains infrared divergencies. These are regularized with a photon mass λ . Theydrop out in the decay width if the contribution from the decay W → f i ¯ f ′ j γ is added. Thiswill be discussed in more detail in Chap. 7.The example above was rather simple. If we keep the fermion masses finite or considerprocesses with more external particles the number and complexity of Feynman diagramsraises considerably. The procedure of generation and algebraic reduction of Feynman diagrams as describedabove is algorithmic and can be implemented in symbolic computation systems. Thereare several attempts to create such systems for high energy physics calculations [35]. Inaddition there exist special packages written in general purpose languages [9, 10, 11, 12].In particular the
MATHEMATICA packages
FEYN ARTS [11] and
FEYN CALC [12]have been developed for the automatic calculation of one-loop diagrams following theapproach outlined in this paper.
FEYN ARTS generates all graphs to a given process in a specified model together withtheir combinatorial factors (weights). It yields both analytical expressions and drawingsof the graphs. There is a version under development which uses the concept of genericdiagrams. It creates all relevant generic graphs together with a list of all possible substi-tutions yielding the actual graphs.
FEYN CALC performs the algebraic evaluation of Feynman diagrams. It starts fromthe output of
FEYN ARTS and uses exactly the reduction algorithm described above. Itcan deal with generic diagrams. The
FEYN CALC output can easily be translated into
FORTRAN code. 46
Soft photon bremsstrahlung
As mentioned in the last chapter the virtual one-loop corrections to the decay matrixelement W → f i ¯ f ′ j are infrared divergent. These divergencies originate from photoniccorrections and show up in any process with charged external particles.However, these processes are not of direct physical relevance since they cannot bedistinguished experimentally from those involving additional soft external photons. Sincethe photons are massless their energies can be arbitrarily small and thus less than theresolution of any detector. Therefore in observable processes in addition to the basicprocess those with arbitrary numbers of soft photons are included.For these observable processes one obtains theoretically satisfactory results. Addingincoherently the cross sections of all the different processes with arbitrary numbers ofphotons, all infrared divergencies cancel [36]. This cancellation takes place between thevirtual photonic corrections and the real bremsstrahlung corrections order by order inperturbation theory. To one-loop order one only needs to consider single photon radiation.For the cancellations only the soft photons, i.e. photons with energy k ≤ ∆ E , are relevant,where ∆ E is a cutoff parameter, which should be small compared to all relevant energyscales. Photons with energies k > ∆ E are called hard. They can also yield sizablecontributions especially arising from photon emission collinear to the external chargedparticles.In Sect. 7.1 we introduce the soft photon approximation and show that in this ap-proximation the bremsstrahlung diagrams are proportional to the Born diagrams. Thecorresponding soft photon cross section for arbitrary Born diagrams is given in Sect. 7.2. Attaching soft photons to a charged external particle line of an arbitrary Feynmandiagram yields diagrams which become singular for vanishing momentum of the soft pho-ton. This divergence arises from the propagator of the charged particle generated by theinclusion of the radiated photon line. In the soft photon approximation the momentaof the radiated photons are neglected everywhere but in this singular propagator. Thisapproximation is valid if the matrix element of the basic process does not change muchif a photon with energy ∆ E is emitted, i.e. the basic matrix element is a slowly varyingfunction of the photon energy for k < ∆ E . This is not the case if the basic processcontains a narrow resonance as e.g. in e + e − → µ + µ − . Then one must either choose ∆ E small compared to the width of the resonance or take into account the strong variationexactly [37, 38].We now extract the soft photon matrix elements for external fermions, scalars and vec-tor bosons. The general renormalizable couplings of these particles to the photon allowedby electromagnetic gauge invariance are (momenta are considered as incoming):¯ FA µ F s(cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0)(cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:26)(cid:26)>(cid:26)(cid:26)ZZ}ZZ = − ieQ F γ µ , (7.1)47 ∗ , p ′ A µ S, p s(cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0)(cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:26) (cid:26) (cid:26)Z Z Z}> = − ieQ S ( p − p ′ ) µ , (7.2) V ∗ ν , p ′ A µ , k V ρ , p s(cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0)(cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1)(cid:3) (cid:3) (cid:3) (cid:3)(cid:1) (cid:1) (cid:1) (cid:1)(cid:2) (cid:2) (cid:2) (cid:2)(cid:0) (cid:0) (cid:0) (cid:0) }> = + ieQ V g νρ ( p − p ′ ) µ − ieκ V [ k ρ g µν − k ν g µρ ] . (7.3)Quartic boson couplings do not give rise to IR-singularities and are thus not relevant inthe soft photon approximation. The terms involving the charges Q are obtained directlyfrom the covariant derivative with respect to QED. The term proportional to κ V , whichcontributes only to the magnetic moment, is gauge invariant by itself. Further termspresent in the γW W coupling in the SM do not contribute for physical vector bosons.Since we will use the unitary gauge in this section they drop out. In renormalizable gaugestheir contributions are cancelled by those of the corresponding unphysical Higgs bosons.Consider first radiation from a fermion line. Let the basic matrix element without softphotons be M = A ( p ) u ( p ) = F - (cid:30)(cid:29)(cid:31)(cid:28) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) , (7.4)where u ( p ) is the fermion spinor with momentum p , p = m and A ( p ) the remaining partof the matrix element. Inserting one photon (polarization vector ε , momentum k ) intothe fermion line yields M = F, p ε, k - - (cid:3) (cid:3) (cid:3) (cid:3)(cid:1) (cid:1) (cid:1) (cid:1) (cid:30)(cid:29)(cid:31)(cid:28) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) = A ( p − k ) i ( p/ − k/ + m )( p − k ) − m ( − ieQ F ε/ ) u ( p ) . (7.5)Anticommuting p/ − k/ with ε/ and using the Dirac equation this can be written as M = eQ F − pk A ( p − k )[2 εp − iε µ σ µν k ν ] u ( p ) , (7.6)where σ µν = i [ γ µ , γ ν ]. The denominator pk contains the IR-singularity. Neglecting allterms proportional k in the numerator we obtain the soft photon approximation M ,s = − eQ F εpkp A ( p ) u ( p ) = − eQ F εpkp M . (7.7)Note that the contributions of the magnetic moment term, the second term in the squarebracket in (7.6), are neglected in the soft photon approximation and that the soft photon48atrix element is proportional to the Born matrix element. For an outgoing fermion(¯ u ( p )) one finds in the same way M ,s = eQ F εpkp M . (7.8)This is equivalent to (7.7) apart from a minus sign originating from the different chargeflow.For an external vector line with polarization vector ε V ( p ) the basic matrix element is M = A σ ( p ) ε σV ( p ) = V σ (cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0)(cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) - (cid:30)(cid:29)(cid:31)(cid:28) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) . (7.9)The corresponding soft photon contribution is obtained from M = A σ ( p − k ) − i ( p − k ) − M ( g σν − ( p − k ) σ ( p − k ) ν M ) ε ρV ( p ) ε µ ( k ) ie [ Q V g νρ (2 p − k ) µ − κ V ( k ρ g µν − k ν g µρ )] , (7.10)using ε V · p = 0 (7.11)as M ,s = − eQ V pεpk M . (7.12)It is proportional to the Born matrix element and independent of the contribution of themagnetic moment κ V of the boson V . Again an outgoing vector yields an extra minussign.The soft photon matrix element for an external scalar line is derived analogously.Radiation from internal charged lines or quartic vertices does not lead to IR-singularities and is neglected in the soft photon approximation.Summarizing, the O ( α ) soft photon matrix element corresponding to an arbitrarymatrix element M can be written as M ,s = − e M X i ( ± Q i ) εp i kp i , (7.13)where p i , Q i are the momentum and the charge of the i -th external particle and k is theoutgoing photon momentum. The ± sign refers to charges flowing into or out of thediagram, respectively. The soft photon matrix element is always proportional to the Bornmatrix element. The proportionality factor depends only on the charges and momenta ofthe external particles. The soft photon cross section is obtained by squaring the soft photon matrix element(7.13), summing over the photon polarizations and integrating over the photon phasespace with | k | ≤ ∆ E dσd Ω ! s = − dσd Ω ! e (2 π ) Z | k |≤ ∆ E d k ω k X ij ± p i p j Q i Q j p i k p j k , (7.14)49here ω k = √ k + λ (7.15)and ± refers to the relative sign of the i -th and j -th term in (7.13). As in the virtualcorrections the IR-singularities are regularized by the photon mass λ . Note that theseintegrals are not Lorentz-invariant due to the integration region. The basic integrals I ij = Z | k |≤ ∆ E d k ω k p i p j p i k p j k (7.16)have been worked out e.g. by ’t Hooft and Veltman [28].The result is I ij = 4 π αp i p j ( αp i ) − p j (
12 log ( αp i ) p j log 4∆ E λ (7.17)+ (cid:20)
14 log u − | u | u + | u | + Li (cid:18) − u + | u | v (cid:19) +Li (cid:18) − u − | u | v (cid:19)(cid:21) u = αp i u = p j ) , with v = ( αp i ) − p j αp i − p j ) , (7.18)and α defined through α p i − αp i p j + p j = 0 , αp i − p j p j > . (7.19)For p i = p j this simplifies to I ii = 2 π ( log 4∆ E λ + p | p | log p − | p | p + | p | ) , (7.20)and for p i = − p j = p I ij = 2 π pq ( p + q ) | p | (
12 log p + | p | p − | p | log 4∆ E λ − Li (cid:18) | p | p + | p | (cid:19) −
14 log p + | p | p − | p | (7.21)+ 12 log q + | p | q − | p | log 4∆ E λ − Li (cid:18) | p | q + | p | (cid:19) −
14 log q + | p | q − | p | ) . Inserting the results for I ij into (7.14) yields the soft photon cross section. Adding it tothe one-loop corrected cross section for the corresponding basic process the IR-divergenciescancel and the limit λ → E/E necessary for the validity of the soft photonapproximation. Therefore also hard photons (with k > ∆ E ) are important. Theircontribution is UV-and IR-finite and can be treated separately. One merely has to makesure that the soft and hard part are properly adapted to each other.Hard photon corrections are treated with methods different from the ones presentedin this work. Their contribution depends sensitively on the experimental setup. They areusually incorporated by Monte Carlo simulations [15].50 Input parameters and leading higher order contributions
In order to complete all ingredients necessary for the calculation of radiative correc-tions we have to specify the input parameters. This is done in Sect. 8.1. The leadinghigher order corrections which become important for precision experiments are discussedin Sect. 8.2.
In its original symmetric version the SM depends on the parameters (2.21), which areessentially the couplings allowed by the SU (2) W × U (1) Y symmetry. These were replacedby the physical parameters (2.22), i.e. the particle masses, the electromagnetic couplingconstant, and the quark mixing matrix. In the on-shell renormalization scheme the renor-malized parameters are equal to these physical parameters in all orders of perturbationtheory.The numerical values of the physical parameters must be fixed through experimentalinput. However, this input may not necessarily consist of direct measurements of therenormalized parameters; it may be obtained from any suitable set of experimental results.In practice one uses those experiments which have the highest experimental accuracy andtheoretical reliability. This criterion is certainly fulfilled for the following set of parameterswhose numerical values are taken from [39]: • the fine structure constant α = 1 / . e = √ πα , • the masses of the charged leptons m e = 0 . , m µ = 105 . ,m τ = 1784 . +2 . − . MeV , • the mass of the Z -boson [5] M Z = 91 . , • and the Fermi constant G F = 1 . − GeV − , which is directly related to the muon lifetime.We do not use the W -mass as input parameter because it is experimentally not knownwith comparable accuracy.Besides the above listed well known parameters the still unknown masses of the topquark and the Higgs scalar are kept as free parameters. If the minimal SM is correct,the present experimental data restrict the top quark mass to the region 80 GeV < m t < < M H < m t = 140 GeV and M H = 100 GeV.The quark mixing matrix elements V ij are directly taken from experiment. We use theparametrization of Harari and Leurer [40] as advocated by the Particle Data Group andchoose the following numerical values for the parameters in agreement with [39] s = 0 . , s = 0 . , s = 0 .
007 (8.1)and δ = 0 for simplicity. This yields approximately the following numbers for the quarkmixing matrix elements: V ud = 0 . , V us = 0 . , V ub = 0 . ,V cd = − . , V cs = 0 . , V cb = 0 . ,V td = 0 . , V ts = − . , V tb = 0 . . (8.2)It remains to discuss the masses m q of the light quarks ( q = d, u, s, c, b ). In theelectroweak Lagrangian the quarks are treated as free particles with appropriate masses.This is not correct due to the presence of the strong interaction. Therefore the quarkmasses can at best be considered as somewhat effective parameters. Fortunately in typicalhigh energy experiments ( s ≫ m q ) theoretical predictions depend on the quark massesonly through universal quantities such as the hadronic vacuum polarization or the quarkstructure functions. These can be directly determined from experiment. Nonuniversalcontributions are suppressed as m q /s and thus negligible for sufficiently high energies.For processes without external quarks only the hadronic contribution to the vacuumpolarization Π AA ( s ) = Σ AAT ( s ) s (8.3)is relevant. In perturbation theory the contribution of light quarks is given byˆΠ AAhad ( s ) = 3 α π X d,u,s,c,b Q q − log − s − iεm q ! . (8.4)The large logarithmic terms contained in (8.4) constitute a dominant contribution tothe radiative corrections. They originate from the charge renormalization in the on-shellscheme at zero momentum transfer (see eq. 3.32) involvingΠ AA (0) = ∂ Σ AAT ( k ) ∂k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k =0 . (8.5)In this quantity nonperturbative strong interaction effects cannot be neglected. Since noreliable theoretical predictions are available one has to extract Π AAhad (0) from experimentaldata. Writing Π
AAhad (0) = Π
AAhad (0) − Re Π
AAhad ( s ) + Re Π AAhad ( s )= − Re ˆΠ
AAhad ( s ) + Re Π AAhad ( s ) , (8.6)52he unrenormalized hadronic vacuum polarization Π AAhad ( s ) can be evaluated perturbativelyfor s ≫ m q and the renormalized one Re ˆΠ AAhad ( s ) is given by the dispersion relationRe ˆΠ AAhad ( s ) = α π s Re Z ∞ m π ds ′ R AA ( s ′ ) s ′ ( s ′ − s − iε ) (8.7)with R AA ( s ′ ) = σ ( e + e − → γ ∗ → hadrons) σ ( e + e − → γ ∗ → µ + µ − ) . (8.8) R AA ( s ) can be taken from experiment up to some scale s , for larger s perturbative QCDis used. A recent analysis [41] involving data for the energy range below 40 GeV yieldsfor the contribution of the 5 light quarks in the energy region 50 GeV < s <
200 GeVRe ˆΠ AA (5) had ( s ) = − . ± . − . " log s (92 GeV) ! + 0 . s (92 GeV) − ! . For energies around the Z-boson mass this can be approximated by (8.4) using theeffective quark masses m u = 0 .
041 GeV , m c = 1 . ,m d = 0 .
041 GeV , m s = 0 .
15 GeV , m b = 4 . . (8.10)These quark masses, in particular the ones for the three lightest quarks, are effectiveparameters adjusted to fit the dispersion integral and have no further significance.In addition to the above parameters we need the strong coupling constant α s for theQCD corrections. Its value at the scale of the Z -boson mass is given by [5] α s = 0 . ± . . (8.11)For the numerical evaluation we use in the following α s = 0 . . (8.12)The W -mass is determined from the parameters given above through the relation M W (1 − M W M Z ) = πα √ G F [1 + ∆ r ] . (8.13)∆ r summarizes the radiative corrections to muon decay [26] apart from the QED correc-tions which coincide with those of the Fermi model. It depends on all parameters of theSM and is given by∆ r = Π AA (0) − c W s W Σ ZZT ( M Z ) M Z − Σ WT ( M W ) M W ! + Σ WT (0) − Σ WT ( M W ) M W +2 c W s W Σ AZT (0) M Z + α πs W − s W s W log c W ! . (8.14)The relation (8.13) can be improved by summing the leading higher order reducible cor-rections. This is done in the next section (eq. 8.23). For the set of parameters specifiedabove we obtain for the W -boson mass M W = 80 .
23 GeV . (8.15)53 .2 Leading higher order contributions The natural order of magnitude of one-loop radiative corrections is set by the loopexpansion parameter απ ∼ . The first type of enhanced corrections originates from the renormalization of α atzero momentum transfer where the relevant scale is set by the fermion masses. Theseare much smaller than the relevant scales in high energy experiments. The large ratio ofthese different scales leads to large logarithms which can be summarized in the universalquantity∆ α ( s ) = − Re ˆΠ AA ( s ) = α π X f N fC Q f log | s | m f + . . . = (∆ α ) LL + . . . , (8.16)where N fC is the colour factor of the fermions and the dots indicate nonleading contri-butions. Renormalization group arguments can be used to show [42] that the leadinglogarithms (∆ α ) LL are correctly summed to all orders in perturbation theory by the re-placement 1 + (∆ α ) LL → − (∆ α ) LL . (8.17)Since not only the leading logarithms but the whole fermionic contribution to ∆ α aregauge invariant we will sum the latter. The gauge dependent bosonic contribution cannot be summed, however, because this would violate gauge invariance in higher orders.Thus we arrive at1 + ∆ α = 1 + (∆ α ) ferm + (∆ α ) bos → − (∆ α ) ferm + (∆ α ) bos . (8.18)This corresponds to a resummation of the iterated one-loop fermionic vacuum polarizationto all orders.Since the leading logarithms contained in ∆ α originate from the charge renormalizationconstant δZ e they are universal, i.e. they appear everywhere where α appears in lowestorder. They can be taken into acount by replacing the lowest order α by a running α ( s )defined as α = α (0) → α ( s ) = α − (∆ α ( s )) ferm . (8.19)∆ α (0) = 0 due to the on-shell renormalization condition for α . Using α ( s ) instead of α effectively corresponds to renormalize α not at zero momentum transfer but at momentumtransfer s . Then the light fermion masses can be neglected everywhere and no largelogarithms appear.There are similar large logarithms associated with external fermion lines. These arerelated to collinear singularities, arising from the radiation of photons collinear to externalparticles. They can be consistently treated with the structure function method [43].54 .2.2 Leading m t contributions The second type of important corrections is also connected to the fermionic sector.The top quark gives rise to corrections ∝ m t /M W , which become large if the top quarkmass is large compared to the W -boson mass. For m t = 200 GeV they reach severalpercent. These terms arise from fermion loop contributions to the boson self energies andfrom the Yukawa couplings of the physical and unphysical Higgs fields, which show up invertex and fermionic self energy corrections. The latter are process dependent and aretherefore not discussed here. The former, however, are universal, they can be traced backto the renormalization of s W in the on-shell scheme δs W s W = − c W s W f Re Σ WT ( M W ) M W − Σ ZZT ( M Z ) M Z ! = 12 c W s W α π s W m t M W + . . . = 12 c W s W ∆ ρ + . . . (8.20)where the dots indicate again nonleading contributions.There is no general principle that determines the resummation of these corrections.The following recipe has been shown to yield the correct leading terms to O ( α ) [44] s W → s W + c W ∆ ¯ ρ = ¯ s W ,c W → c W (1 − ∆ ¯ ρ ) = ¯ c W , (8.21)where ∆ ¯ ρ = 3 G F m t √ π " G F m t √ π (19 − π ) (8.22)incorporates the result [45] from two-loop irreducible diagrams. Note that α has beenreplaced by G F in order to obtain the correct leading O ( α ) terms.In particular for the relation between M W and G F the correct resummation of theleading corrections is given by G F = πM W √ αs W − (∆ α ) ferm
11 + c W s W ∆ ¯ ρ " r − (∆ α ) ferm + c W s W ∆ ρ . (8.23)Note that the two types of leading corrections are summed separately. Inserting s W =1 − M W /M Z this relation can be used to determine the W -boson mass M W includingleading higher order contributions.Neglecting the nonleading contributions and using (8.19) and (8.21) eq. (8.23) can bewritten as πα ( M W )¯ s W ≈ √ G F M W . (8.24)With this relation the appearance of G F in ∆ ¯ ρ can be easily understood. All leadinguniversal corrections arise from the renormalization constants of α and s W . Consequentlythey can be absorbed by incorporating the leading finite parts of these renormalizationconstants into the effective parameters α ( s ) and ¯ s W (including the leading higher ordercorrections). Thus one obtains from the lowest order result directly the correspondingresult including the leading universal corrections. In particular (8.24) can be obtained inthis way. Applying this recipe to ∆ ρ and using (8.24) introduces naturally G F .55 .2.3 Recipes for leading universal corrections From the discussion above we infer that the universal leading higher order contribu-tions can be taken into account by the following replacements α → α ( s ) ,s W → ¯ s W , c W → ¯ c W ,e s W = 2 παs W → πα ( s )¯ s W ≈ √ G F M W ,e s W c W = παs W c W → πα ( s )¯ s W ¯ c W ≈ √ G F M Z − ∆ ¯ ρ . (8.25)Note that this does not include the nonuniversal corrections ∝ α m t /M W arising fromenhanced Yukawa couplings. These have to be evaluated for each process directly.56 The width of the W -boson In the coming years the upgrade of the LEP electron positron storage ring to 180 −
190 GeV CM-energy will allow to study the properties of the W -boson in detail in amodel independent way. Besides its mass M W also its total and partial decay widthsare of interest. While the leptonic partial widths allow to test lepton universality thehadronic partial widths can serve to determine the quark mixing matrix elements [46].The expected accuracy is δM W ≈
100 MeV and δ Γ W ≈
200 MeV [47].An accurate comparison between these experiments and theoretical predictions of theSM requires at least the inclusion of one-loop radiative corrections both for the productionand the decay of W -bosons. The W -bosons decay dominantly into fermion-antifermionpairs. The total and partial widths of the W -boson and the corresponding one-loopradiative corrections are discussed in this chapter.The electroweak and QCD corrections for decays into massless fermions ( m f ≪ M W )have been calculated in [48, 49, 50]. The hard bremsstrahlung contribution has beeninvestigated in [52, 53]. The QCD corrections for the decay into a massive top quarkand a massless bottom quark were given in [54]. The full one-loop electroweak andQCD corrections together with the complete photonic and gluonic bremsstrahlung wereevaluated for arbitrary finite fermion masses in [55].Since the top quark is probably heavier than the W -boson and since all other quarkmasses are small compared to the W -boson mass, the fermion mass effects are not ofgreat importance for the W -decay. However, the matrix element for the W -decay intotwo fermions is directly related via crossing to the one for the decay of the top quark intoa W -boson and a bottom quark. In this case the fermion masses are crucial. Since we willdiscuss top decay in Chap. 10 we give the results for W -decay for finite fermion masses. The Born amplitude for the decay W + → f i ¯ f ′ j was already given in (6.11). For finitefermion masses the following result is obtained for the corresponding partial decay widthΓ W f i f ′ j ( M W , m f,i , m f ′ ,j ) = N fC α
12 12 s W | V ij | κ ( M W , m f,i , m f ′ ,j ) M W G − , (9.1)where κ/M W originates from phase space and G − = X pol M −† M − = 2 M W − m f,i − m f ′ ,j − ( m f,i − m f ′ ,j ) M W (9.2)from the polarization sum of the matrix element squared. The K¨all´en function κ wasdefined in (4.28). The colour factor N fC is given by N fC = V ij = δ ij . The total width is obtained as a sum over thepartial fermionic decay widths with m f,i + m f ′ ,j < M W Γ W = X ij Γ W u i d j + X i Γ W ν i l i . (9.4)57e can write down another possible tree level representation for the partial decay widthby eliminating α/s W in favour of G F ¯Γ W f i f ′ j ( M W , m f,i , m f ′ ,j ) = N fC G F π √ | V ij | κ ( M W , m f,i , m f ′ ,j ) M W G − . (9.5) In our formulation of the on-shell renormalization scheme the invariant matrix elementto one-loop order has the following form M W f i f ′ j = − e √ s W n V ij M − [1 + δZ e − δs W s W ] + M − δV ij + V ij M − δZ W + M − X k [ δZ f,L † ik V kj + V ik δZ f ′ ,Lkj ]+ X a =1 X σ = ± M σa δF σa ( M W , m f,i , m f ′ ,j ) o . (9.6)The standard matrix elements M σa were defined in (5.11). The functions δF σa summarizethe loop corrections to the W f i f ′ j -vertex. There are no explicit self energy correctionsfrom the external lines. These are all absorbed into the field renormalization constants δZ W , δZ f,L and δZ f ′ ,L . These and the parameter renormalization constants were givenin terms of self energies in Sect. 3.3. The explicit forms of the self energies can be foundin App. B.Fig. 9.1 shows the Feynman diagrams contributing to the vertex corrections for massivefermions. They yield the following vertex form factors δF − ( M W , m , m ) = α π × ( Q f Q f ′ V a ( m , M W , m , λ, m , m ) + g − f g − f ′ V a ( m , M W , m , M Z , m , m )+ X σ = ± (cid:20) Q f V σb ( m , M W , m , m , λ, M W ) − Q f ′ V σb ( m , M W , m , m , λ, M W )+ c W s W g σf V σb ( m , M W , m , m , M Z , M W ) − c W s W g σf ′ V σb ( m , M W , m , m , M Z , M W ) (cid:21) + 14 s W V c ( m , M W , m , M H , m , m )+ 12 s W (cid:20) V d ( m , M W , m , m , M H , M W ) + V d ( m , M W , m , m , M H , M W ) (cid:21) (9.7)+ 12 s W (cid:20) V e ( m , M W , m , m , M H , M W ) + V e ( m , M W , m , m , M H , M W )58 ) W γ, Zf i f ′ j ¯ f ′ j f i s ss(cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0)(cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:8)(cid:8)*(cid:8)(cid:8)(cid:8)(cid:8)*(cid:8)(cid:8)HHHYHH HHHYHH (cid:3)(cid:2)(cid:3)(cid:2)(cid:3)(cid:2)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) b ) W f i γ, ZW ¯ f ′ j f i s ss(cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0)(cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3)(cid:1) (cid:1) (cid:1) (cid:1)(cid:2) (cid:2) (cid:2) (cid:2) (cid:2)(cid:0) (cid:0) (cid:0) (cid:0) W f ′ j Wγ, Z ¯ f ′ j f i s ss(cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0)(cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3)(cid:1) (cid:1) (cid:1) (cid:1)(cid:2) (cid:2) (cid:2) (cid:2) (cid:2)(cid:0) (cid:0) (cid:0) (cid:0) c ) W H, χf ′ j f i ¯ f ′ j f i s ss(cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0)(cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:8)(cid:8)*(cid:8)(cid:8)(cid:8)(cid:8)*(cid:8)(cid:8)HHHYHH HHHYHH d ) W f i HW ¯ f ′ j f i s ss(cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0)(cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2)(cid:0) (cid:0) (cid:0) (cid:0) (cid:8) (cid:8) (cid:8) W f ′ j WH ¯ f ′ j f i s ss(cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0)(cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3)(cid:1) (cid:1) (cid:1) (cid:1) e ) W f i H, χφ ¯ f ′ j f i s ss(cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0)(cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) W f ′ j φH, χ ¯ f ′ j f i s ss(cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0)(cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) f ) W f i γ, Zφ ¯ f ′ j f i s ss(cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0)(cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3)(cid:1) (cid:1) (cid:1) (cid:1) W f ′ j φγ, Z ¯ f ′ j f i s ss(cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0)(cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2)(cid:0) (cid:0) (cid:0) (cid:0) (cid:8) (cid:8) (cid:8) Figure 9.1: One-loop diagrams for W → f i ¯ f ′ j .59 V e ( m , M W , m , m , M Z , M W ) + V e ( m , M W , m , m , M Z , M W ) (cid:21) + X σ = ± (cid:20) Q f V σf ( m , M W , m , m , λ, M W ) − Q f ′ V σf ( m , M W , m , m , λ, M W ) − s W c W g σf V σf ( m , M W , m , m , M Z , M W ) + s W c W g σf ′ V σf ( m , M W , m , m , M Z , M W ) (cid:21)) ,δF +1 ( M W , m , m ) = α π m m × ( X σ = ± (cid:20) Q f Q f ′ W σa ( m , M W , m , λ, m , m ) + g σf g σf ′ W σa ( m , M W , m , M Z , m , m )+ Q f W σb ( m , M W , m , m , λ, M W ) − Q f ′ W σb ( m , M W , m , m , λ, M W )+ c W s W g σf W σb ( m , M W , m , m , M Z , M W ) − c W s W g σf ′ W σb ( m , M W , m , m , M Z , M W ) (cid:21) + 14 s W (cid:20) X σ = ± W σc ( m , M W , m , M H , m , m ) − W − c ( m , M W , m , M Z , m , m ) (cid:21) + 12 s W (cid:20) W d ( m , M W , m , m , M H , M W ) + W d ( m , M W , m , m , M H , M W ) (cid:21) (9.8)+ 12 s W (cid:20) W e ( m , M W , m , m , M H , M W ) + W e ( m , M W , m , m , M H , M W ) − W e ( m , M W , m , m , M Z , M W ) − W e ( m , M W , m , m , M Z , M W ) (cid:21) + X σ = ± (cid:20) Q f W σf ( m , M W , m , m , λ, M W ) − Q f ′ W σf ( m , M W , m , m , λ, M W ) − s W c W g σf W σf ( m , M W , m , m , M Z , M W ) + s W c W g σf ′ W σf ( m , M W , m , m , M Z , M W ) (cid:21)) ,δF − ( M W , m , m ) = α π m × ( X σ = ± (cid:20) Q f Q f ′ X σa ( m , M W , m , λ, m , m ) + g σf g − f ′ X σa ( m , M W , m , M Z , m , m )+ Q f X σb ( m , M W , m , m , λ, M W ) + c W s W g σf X σb ( m , M W , m , m , M Z , M W ) (cid:21) − Q f ′ X − b ( m , M W , m , m , M W , λ ) − c W s W g − f ′ X − b ( m , M W , m , m , M W , M Z )60 14 s W (cid:20) X σ = ± X − c ( m , M W , m , M H , m , m ) − X − c ( m , M W , m , M Z , m , m ) (cid:21) + 12 s W X d ( m , M W , m , m , M H , M W ) (9.9) − s W X σ = ± (cid:20) σ (cid:16) X σe ( m , M W , m , m , M H , M W ) − X σe ( m , M W , m , m , M H , M W ) (cid:17) − X σe ( m , M W , m , m , M Z , M W ) − X σe ( m , M W , m , m , M Z , M W ) (cid:21) − s W m M W (cid:20) X e ( m , M W , m , m , M H , M W ) − X e ( m , M W , m , m , M H , M W ) − X e ( m , M W , m , m , M Z , M W ) − X e ( m , M W , m , m , M Z , M W ) (cid:21) + Q f X f ( m , M W , m , m , λ, M W ) + Q f ′ X f ( m , M W , m , m , λ, M W ) − s W c W (cid:20) g + f X f ( m , M W , m , m , M Z , M W ) + g − f ′ X f ( m , M W , m , m , M Z , M W ) (cid:21)) . To obtain these results we had to evaluate six generic diagrams labeled by a, b, c, d, e andf in Fig. 9.1. The corresponding invariant functions V , W and X are listed in App. C. Theformfactor δF +2 can be obtained from δF − by the substitutions m ↔ m , Q f ↔ − Q f ′ , g f ↔ − g f ′ .Squaring the matrix element (9.6), summing over the polarizations of the externalparticles and multiplying the phase space factor yields the one-loop corrected widthΓ W f i f ′ j = N fC α
12 12 s W κ ( M W , m f,i , m f ′ ,j ) M W n | V ij | G − [1 + 2 δZ e − δs W s W + δZ W ]+ 12 G − X k [( δZ f,L † ik + δZ f,Lik ) V kj + V ik ( δZ f ′ ,L † kj + δZ f ′ ,Lkj )] V † ij + 2 | V ij | X a =1 X σ = ± G σa δF σa ( M W , m f,i , m f ′ ,j ) (9.10)= Γ W f i f ′ j (1 + δ ewvirt ) , where G +1 = X pol M −† M +1 = 6 m f,i m f ′ ,j , (9.11)61 γW f ′ j f i s s(cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0)(cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:8)(cid:8)*(cid:8)(cid:8)HHHYHH (cid:3) (cid:3) (cid:3) (cid:3)(cid:1) (cid:1) (cid:1) (cid:1)(cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0)(cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) W γφ f ′ j f i s s(cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0)(cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:8)(cid:8)*(cid:8)(cid:8)HHHYHH (cid:3) (cid:3) (cid:3) (cid:3)(cid:1) (cid:1) (cid:1) (cid:1) W f ′ j γ ¯ f ′ j f i s s(cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0)(cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:8)(cid:8)(cid:8)(cid:8)*(cid:8)(cid:8)(cid:8)(cid:8)HHHHYHHHH (cid:3) (cid:3) (cid:3) (cid:3) (cid:3)(cid:1) (cid:1) (cid:1) (cid:1) Y Y
W f i γ ¯ f ′ j f i s s(cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0)(cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:8)(cid:8)(cid:8)(cid:8)*(cid:8)(cid:8)(cid:8)(cid:8)HHHHYHHHH (cid:2) (cid:2) (cid:2) (cid:2) (cid:2)(cid:0) (cid:0) (cid:0) (cid:0) * * Figure 9.2: Bremsstrahlung Feynman diagrams for W → f i ¯ f ′ j γ . G − = X pol M −† M − = − m f,i κ ( M W , m f,i , m f ′ ,j ) M W ,G +2 = X pol M −† M +2 = − m f ′ ,j κ ( M W , m f,i , m f ′ ,j ) M W , and we have inserted δV ij from (3.23). Like any one-loop amplitude with external charged particles (9.6) and consequently(9.10) are IR-divergent due to virtual photonic corrections. These singularities are com-pensated by the the real bremsstrahlung corrections, i.e. the three-body decay W + ( k ) → f i ( p ) ¯ f ′ j ( p ) γ ( q ) . (9.12)The corresponding matrix element as given by the Feynman diagrams (Fig. 9.2) is M b = V ij e √ s W ¯ u ( p ) (cid:26) − Q f p q h p η / ǫ + / η / q / ǫ i + Q f ′ p q h p η / ǫ + / ǫ / q / η i + ( Q f − Q f ′ ) − kq h ( qη − kη ) / ǫ + 2 ǫη / q − qǫ / η i(cid:27) ω − v ( p ) , (9.13)where η denotes the polarization vector of the photon. Performing the polarization sumover the square of the amplitude gives X pol |M b | = α s W (64 π ) | V ij | ( Q f Q f ′ (2 p q ) (2 p q ) (cid:20) ( M W − m f,i − m f ′ ,j ) G − (cid:21) Q f (2 p q ) (cid:20) ( m f,i + 2 p q ) G − + (1 + m f,i + m f ′ ,j M W )(2 p q ) ( − kq ) + (2 p q ) (cid:21) − Q f ′ (2 p q ) (cid:20) ( m f ′ ,j + 2 p q ) G − + (1 + m f,i + m f ′ ,j M W )(2 p q ) ( − kq ) + (2 p q ) (cid:21) − ( Q f − Q f ′ ) ( − kq ) (cid:20) ( M W − kq ) G − + m f,i + m f ′ ,j M W ( − kq ) − p q )(2 p q ) (cid:21) − Q f ( Q f − Q f ′ ) − kq p q (cid:20) ( M W + m f,i − m f ′ ,j ) G − − p q )(2 p q ) (cid:21) (9.14)+ Q f ′ ( Q f − Q f ′ ) − kq p q (cid:20) ( M W − m f,i + m f ′ ,j ) G − − p q )(2 p q ) (cid:21)) . From this the complete bremsstrahlung cross section (including soft and hard photons) isobtained by integrating over the phase space of the photon and the two fermions asΓ
W f i f ′ j b ( M W , m f,i , m f ′ ,j ) = 1(2 π ) N fC M W Z d q q d p p d p p δ (4) ( p + p + q − k ) 13 X pol |M b | = Γ W f i f ′ j δ ewb ( M W , m f,i , m f ′ ,j ) (9.15)with δ ewb ( M W , m f,i , m f ′ ,j ) = (cid:18) − απ (cid:19) M W κ ( M W , m f,i , m f ′ ,j ) ( − Q f Q f ′ (cid:20) ( M W − m f,i − m f ′ ,j ) I (cid:21) + Q f (cid:20) ( m f,i I + I ) + (1 + m f,i + m f ′ ,j M W ) I G − + IG − (cid:21) + Q f ′ (cid:20) ( m f ′ ,j I + I ) + (1 + m f,i + m f ′ ,j M W ) I G − + IG − (cid:21) +( Q f − Q f ′ ) (cid:20) ( M W I + I ) + m f,i + m f ′ ,j M W IG − − I G − (cid:21) + Q f ( Q f − Q f ′ ) (cid:20) ( M W + m f,i − m f ′ ,j ) I − I G − (cid:21) − Q f ′ ( Q f − Q f ′ ) (cid:20) ( M W − m f,i + m f ′ ,j ) I − I G − (cid:21)) . (9.16)The bremsstrahlung phase space integrals I ······ = I ······ ( M W , m f,i , m f ′ ,j ) are given in App. D.The IR-singularities contained in the I kl are again regularized by a photon mass λ .63 .4 QCD corrections Like the electroweak corrections also the QCD corrections consist of virtual and realcontributions δ QCD ( M W , m f,i , m f ′ ,j ) = δ QCDvirt + δ QCDb , (9.17)which are individually IR-divergent. They are obtained from the electroweak results ofthe previous sections by keeping only the terms containing Q f , Q f ′ or Q f Q f ′ , setting Q f = Q f ′ = 1, replacing α by the strong coupling constant α s and multiplying an overall colourfactor C F = . In particular the virtual QCD-corrections arise only from the diagram ofFig. 9.1a with the photon replaced by a gluon and the corresponding corrections to thefermion wave function renormalization constants. Since this allows many simplificationswe give the explicit results δ QCDvirt = (cid:18) − α s π (cid:19) C F ( − B ( M W , m f,i , m f ′ ,j )+ 12 B (0 , m f,i , m f,i ) + 12 B (0 , m f ′ ,j , m f ′ ,j )+2( M W − m f,i − m f ′ ,j )( C + C + C ) − m f,i C − m f ′ ,j C (9.18) − m f,i m f ′ ,j λ ) − G +1 G − m f,i m f ′ ,j h C + C i +4 G − G − m f,i h C + C + C i +4 G +2 G − m f ′ ,j h C + C + C i) . The arguments of the three-point functions are C = C ( m f,i , M W , m f ′ ,j , λ, m f,i , m f ′ ,j ). Tothe gluonic bremsstrahlung δ QCDb only the first three terms of (9.16) contribute. For zerofermion masses the total QCD-correction reduces to δ QCD ( M W , ,
0) = α s π . (9.19) For numerical evaluation of the previous results we use the parameters listed inSect. 8.1 including the values for the quark mixing matrix as given by (8.1). The W -mass is determined from the relation (8.23).In the on-shell scheme the lowest order width is parametrized by α and the particlemasses (9.1). In this scheme large electroweak corrections arise due to fermion loopcontributions to the renormalization of α and s W . We can improve the results in thisscheme by resumming the corresponding one-loop contributions to all orders as discussedin Sect. 8.2. Thus we obtain for the corrected widthΓ = Γ h δ − (∆ α ) ferm + c W s W ∆ ρ i − (∆ α ) ferm
11 + c W s W ∆ ¯ ρ = Γ h δ i , (9.20)64igure 9.3: Electroweak radiative corrections δ ew and ¯ δ ew and QCD-corrections δ QCD tothe total W -width Γ W versus the top mass.where δ = δ ewvirt + δ ewb + δ QCDvirt + δ QCDb (9.21)is the proper one-loop correction without resummation. As in any charged current processwe can avoid the large corrections by parametrizing the lowest order decay width with G F and M W instead of α and s W (9.5). Using (8.23) we find the relation between thedecay width in both parametrizations¯Γ = Γ − (∆ α ) ferm
11 + c W s W ∆ ¯ ρ " r − (∆ α ) ferm + c W s W ∆ ρ , (9.22)¯Γ = ¯Γ h δ − ∆ r i = ¯Γ (1 + ¯ δ ) . The large fermionic contributions contained in δ are exactly cancelled by equal contri-butions in ∆ r and consequently the remaining corrections ¯ δ are small.The relative corrections to the total W -boson decay width are shown in Fig. 9.3.They are large and strongly m t -dependent in the on-shell scheme. This behaviour arisesfrom the fermionic contributions to the renormalization of the weak mixing angle in (9.6)( δs W /s W ) which contain terms ∝ α m t /M W . In contrast to this in the parametrizationwith G F the corrections depend only weakly on m t and remain below 0.6%. The QCDcorrections are practically constant and equal to 2 α s / (3 π ), their value for zero fermionmasses. 65 t M W Γ W ¯Γ W Γ Wew ¯Γ Wew . .
87 1 . . . . . .
99 1 . . . . . .
10 1 . . . . . .
23 1 . . . . . .
37 1 . . . . . .
52 2 . . . . . .
69 2 . . . . W parametrized by α and ¯Γ W parametrized by G F in lowestorder and including electroweak corrections. All values are given in GeV.Tab. 9.1 shows the lowest order width Γ and the width including electroweak cor-rections Γ ew in both parametrizations for various values of the top mass. The W -massobtained from (8.23) is also listed there. While the results for the lowest order width in thetwo parametrizations differ by several percent, the deviation of the first order expressionsis always less than 0.4%.The analytical results were presented above for finite external fermion masses andwith correct renormalization of the quark mixing matrix. However, since the top quarkis presumably heavier than the W -boson [57] all relevant actual fermion masses are smallcompared to the W -boson mass. Therefore in addition to the completely corrected numer-ical results Γ( M W , m f,i , m f ′ ,j ) for finite fermion masses we also give those for vanishingfermion masses Γ( M W , ,
0) in Tab. 9.2. The analytical results for vanishing fermionmasses were listed in Sect. 6.3. Finally we include an improved Born approximation con-sisting of the Born widths with zero fermion masses parametrized by G F and multipliedby the QCD correction factor for zero quark massesΓ W νlimp = G F M W √ π , Γ W udimp = G F M W √ π | V ij | (cid:18) α s π (cid:19) , (9.23)Γ Wimp = 3 G F M W √ π (cid:18) α s π (cid:19) , for the leptonic partial widths, the hadronic partial widhts and for the total width, re-spectively.The numerical values for the partial and total W -widths in these different approxi-mations are given in Tab. 9.2 assuming a top quark mass of 140 GeV and a Higgs bosonmass of 100 GeV. The improved Born approximation (9.23) reproduces the exact resultsup to 0.4% (0.6% for the decays into a b-quark). The effects of the fermion masses arebelow 0.3%. They are suppressed by m q /s . There are no mass singularities since the66orn complete m f = 0 improved Branchingwidth one-loop in Γ Born ratioΓ( W → eν e ) 0.2260 0.2252 0.2252 0.2260 0.1084Γ( W → µν µ ) 0.2259 0.2252 0.2252 0.2260 0.1084Γ( W → τ ν τ ) 0.2258 0.2250 0.2252 0.2260 0.1083Γ( W → lep. ) 0.6777 0.6754 0.6756 0.6778 0.3251Γ( W → ud ) 0.6450 0.6672 0.6672 0.6696 0.3211Γ( W → us ) ×
10 0.3281 0.3393 0.3393 0.3406 0.0163Γ( W → ub ) × W → cd ) ×
10 0.3281 0.3396 0.3396 0.3409 0.0163Γ( W → cs ) 0.6432 0.6656 0.6657 0.6682 0.3204Γ( W → cb ) × W → had. ) 1.3553 1.4022 1.4023 1.4075 0.6749Γ( W → all ) 2.0330 2.0776 2.0779 2.0853Table 9.2: Partial and total W-decay widths ¯Γ in different approximations for m t = 140 GeV, M H = 100 GeV and the corresponding W-mass M W = 80 .
23 GeV.width is obtained by integrating over the full phase space of the final state particles [56].Consequently the exact numerical values of the masses of the external fermions massesare irrelevant . The branching ratios derived from (9.23) BR ( W → lν ) = 19(1 + 2 α s / π ) ,BR ( W → leptons) = 13(1 + 2 α s / π ) ,BR ( W → u i d j ) = | V ij | (1 + α s /π )3(1 + 2 α s / π ) ,BR ( W → hadrons) = 2(1 + α s /π )3(1 + 2 α s / π ) (9.24)agree numerically within 0.1% with those obtained from the full one loop results. Theydepend only on α s and V ij .The dependence of the W -width on the unknown top and Higgs masses is shown inTab. 9.3 for Γ W ud and in Tab. 9.4 for Γ
W eν . A variation of m t between 80 and 200 GeVaffects the partial widths by ∼ M H between 50 and 1000 GeV by The values given in Sect. 8.1 are not appropriate for the external quarks. We only use them here todemonstrate the numerical irrelevance of the fermion mass effects. t M H = 50 M H = 100 M H = 300 M H = 100080.0 0.2212 0.2209 0.2202 0.2193100.0 0.2227 0.2224 0.2217 0.2208120.0 0.2239 0.2236 0.2229 0.2220140.0 0.2251 0.2248 0.2241 0.2232160.0 0.2264 0.2261 0.2254 0.2245180.0 0.2278 0.2276 0.2269 0.2260200.0 0.2294 0.2291 0.2284 0.2275Table 9.3: Partial W-decay width Γ W eν including first order QCD and electroweak cor-rections for different values of the top and Higgs masses. All values are given in GeV. m t M H = 50 M H = 100 M H = 300 M H = 100080.0 0.6539 0.6530 0.6509 0.6481100.0 0.6585 0.6575 0.6554 0.6526120.0 0.6622 0.6612 0.6591 0.6563140.0 0.6659 0.6650 0.6629 0.6601160.0 0.6700 0.6691 0.6670 0.6642180.0 0.6745 0.6736 0.6715 0.6686200.0 0.6793 0.6784 0.6763 0.6735Table 9.4: Partial W-decay width Γ W ud including first order QCD and electroweak cor-rections for different values of the top and Higgs masses. All values are given in GeV. ∼ α , G F and M Z . In this case the top mass dependence is mainly due to thevariation of M W with m t . Keeping instead M W , G F and M Z fixed the dependence on m t is considerably smaller. Remember, however, that the prediction for the decay width hasthe same uncertainty in this parametrization due to the uncertainty of the experimentalvalue for the W -boson mass.We have compared our results for the partial leptonic width for zero fermion massesto those of Jegerlehner [50] and Bardin et al. [49], who both use the parametrizationwith G F . Furthermore Jegerlehner includes two-loop QCD corrections into the boson selfenergies. If these are switched off the difference between his and our results is less than0 . . . t M H = 50 M H = 100 M H = 300 M H = 100080.0 2.0375 2.0347 2.0283 2.0196100.0 2.0517 2.0488 2.0423 2.0336120.0 2.0630 2.0602 2.0537 2.0449140.0 2.0747 2.0719 2.0654 2.0566160.0 2.0872 2.0844 2.0780 2.0692180.0 2.1009 2.0981 2.0917 2.0830200.0 2.1157 2.1129 2.1066 2.0980Table 9.5: Total W-decay width Γ W including first order QCD and electroweak correctionsfor different values of the top and Higgs masses. All values are given in GeV.69 The present lower limit from CDF data indicates that the top mass is at least 89 GeV[57]. Moreover, LEP data in combination with radiative correction calculations require m t = 137 ±
40 GeV within the minimal standard model [5, 8] at the 1 σ level. Thereforethe top mass lies presumably above the W b threshold and the dominant decay of the topquark is the one into a W -boson and a bottom quark ( t → W b ) and the total width ofthe top quark can be well described by the partial width Γ tW b = Γ( t → W b ).While the measurement of the top mass will provide a long missing input parameter,the measurement of its width will serve as a consistency check on the standard model.With the operation of LHC, SSC and/or a high energy e + e − collider one expects to obtaina sufficiently large number of tops so that both the mass and the width can be measuredwith good accuracy.The QCD corrections to the top decay t → W b were already evaluated in [58, 59].The first order electroweak corrections have been calculated by [60, 61].The electroweak corrections to this decay involve particularly interesting contributionsof O ( α m t /M W ) which are potentially large for large top masses. Those terms arise notonly from fermion loop contributions to the boson self energies but also from the Yukawacouplings of the Higgs fields, which show up in vertex and fermionic self energy corrections.As discussed in Sect. 8.2 contributions of the first type can be eliminated if the Bornapproximation is expressed by G F and M W . Surprisingly the effects from strong Yukawacouplings turn out to be small, as will be demonstrated in the following.We will only consider the decay of free top quarks and sum over the polarizations ofthe W -bosons. The results are obtained via crossing from the ones for W → t ¯ b . Because we want to use our results for the decay W + → t ¯ b we consider the decay ofan anti-top quark. The corresponding decay width is identical to the one of the top quarkbecause of the CPT theorem.The lowest order decay of an anti-top quark¯ t ( p ) → W − ( k )¯ b ( p ) (10.1)is described by the Feynman diagram of Fig. 10.1 yielding the amplitude M = − e √ s W V tb ¯ v ( p )/ ǫω − v ( p ) . (10.2) W − ¯ t ¯ b s (cid:27) (cid:3) (cid:3) (cid:3) (cid:3)(cid:1) (cid:1) (cid:1) (cid:1) ZZ}ZZ
Figure 10.1: Born diagram for the decay ¯ t → W − ¯ b
70t can be obtained from the Born amplitude (6.11) for the decay W + → t ¯ b by crossing.This amounts to change the signs of p and k and use u ( − p ) = v ( p ) and ε ( − k ) = ε ( k ).From (10.2) we get the lowest order widthΓ tW b ( m t , M W , m b ) = α s W | V tb | κ ( m t , M W , m b ) m t G − , (10.3)with G − = " m t + m b − M W + ( m t − m b ) M W . (10.4)Eq. (10.3) can directly be derived from (9.1) by substituting m i → m t and m j → m b in G − coming from the matrix element squared, exchanging M W and m t in the phase spacefactors, changing the spin average from 1/3 to 1/2 and supplying a minus sign originatingfrom the different signs of the momenta entering the matrix element squared. This minussign has been incorporated into the definition of G − which differs from the one in Chap. 9.Introducing G F instead of α the lowest order width reads¯Γ tW b ( m t , M W , m b ) = G F M W π √ | V tb | κ ( m t , M W , m b ) m t G − . (10.5) With the four Dirac matrix elements M − = ¯ v ( p )/ ǫω − v ( p ) , M +1 = ¯ v ( p )/ ǫω + v ( p ) , M − = ¯ v ( p ) ω − v ( p ) ǫ · p , M +2 = ¯ v ( p ) ω + v ( p ) ǫ · p (10.6)obtained from (5.11) by setting p → − p the virtual electroweak one-loop correctionstake the form of (9.6) with t and b instead of i and j . The corresponding decay widthΓ tW b follows from (9.10) using the substitutions specified after (10.4).The QCD corrections can be extracted from the electroweak ones in the same way asin Sect. 9.4. The real photonic contributions of O ( α ) to the top width arise from the radiativedecay ¯ t ( p ) → W − ( k )¯ b ( p ) γ ( q ) . (10.7)The corresponding amplitude squared, summed over all polarizations, can be derived from(9.14) by replacing the momenta k → − k , p → − p and multiplying an overall factor( − tW bb ( m t , M W , m b ) = 1(2 π ) m t Z d q q d k k d p p δ (4) ( p − p − q − k ) 12 X pol |M b |
71 Γ tW b δ ewb ( m t , M W , m b ) . (10.8)The correction factor reads δ ewb ( m t , M W , m b ) = (cid:18) − απ (cid:19) m t κ ( m t , M W , m b ) ( − Q t Q b (cid:20) ( M W − m t − m b ) I (cid:21) + Q t (cid:20) ( m t I + I ) − (1 + m t + m b M W ) I G − − IG − (cid:21) + Q b (cid:20) ( m b I + I ) − (1 + m t + m b M W ) I G − − IG − (cid:21) (10.9)+( Q t − Q b ) (cid:20) ( M W I + I ) − m t + m b M W IG − + 2 I G − (cid:21) + Q t ( Q t − Q b ) (cid:20) ( M W + m t − m b ) I + 2 I G − (cid:21) − Q b ( Q t − Q b ) (cid:20) ( M W − m t + m b ) I + 2 I G − (cid:21)) . The bremsstrahlung phase space integrals carry the arguments I ······ = I ······ ( m t , M W , m b ).They are given in App. D.From eq. (10.9) the gluonic bremsstrahlung corrections can be obtained by setting Q t = Q b = 1, replacing α by the strong coupling constant α s and multiplying with thecolour factor C F = . We again use the parameters listed in Sect. 8.1 as numerical input and calculate the W -mass from the relation (8.23). Unless stated otherwise, we choose for the Higgs mass M H = 100 GeV.We perform the same summation of the leading higher order corrections as discussedin Sect. 9.5, eq. (9.20), and introduce the parametrization with G F and M W as in (9.22).In this parametrization the large corrections arising from the renormalization of α and s W are absorbed into the lowest order expression. This is not the case for large contributionsproportional to α m t /M W arising from vertex and fermion self energy diagrams withenhanced Yukawa couplings.In Tab. 10.1 we give the lowest order width as well as the width including electroweakcorrections in both parametrizations for various values of the top mass together withthe W -mass obtained from (8.23). The results for the first order expressions of bothparametrizations agree within 0.05%.According to (10.3) the width increases with the top mass approximately like m t /M W .The corresponding relative corrections are shown in Fig. 10.2. The QCD corrections yieldabout −
10% with only a weak dependence on the top mass. In the on-shell scheme wefind sizable electroweak corrections which range from +7% at m t = 100 GeV to − t M W Γ tW b ¯Γ tW b Γ tW bew ¯Γ tW bew tW b parametrized by α and ¯Γ tW b by G F in lowest order andincluding electroweak corrections. All values are given in GeV.Figure 10.2: Electroweak radiative corrections δ ew and ¯ δ ew and QCD corrections δ QCD tothe top decay width versus the top mass. 73 t / GeV 200 300 500 1000leading term 0.023 0.036 0.051 -0.070next to -0.043 -0.065 -0.108 -0.217leading termTable 10.2: The two leading terms of the expansion in eq. (10.10) for various values ofthe top quark mass.at m t = 350 GeV. The large variation arises from terms ∝ α m t /M W in the first ordercorrections. Contrarily in the parametrization with G F the corrections are only ≈ +1%for m t = 100 GeV and remain almost constant at ≈ +1 .
7% for m t ≥
160 GeV. Thisfeature is independent of the Higgs boson mass. However, we expected large corrections ∝ m t to arise from the vertex diagrams containing large Yukawa couplings; in the similarcase of Z → b ¯ b the corresponding contributions are noticeable. The slow variation ofthe relative correction in this parametrization indicates that the strong Yukawa couplingshave no sizable effect on the top width. In order to demonstrate the absence of largecorrections the plot in Fig. 10.2 has been extended up to m t = 350 GeV, although this iswell above the present upper limits on the top quark mass.Some understanding of this surprising feature can be obtained from the expansion ofthe relative correction factor in the parametrization with G F and M W , ¯ δ ew , for large topquark masses ( m t ≫ M W , M Z , M H )¯ δ ew = δ ew − ∆ r ∼ α π s W m t M W (cid:26)h
174 + log( M H m t ) i + h − π M H m t i + O (cid:16) log ( m t M i ) (cid:17)(cid:27) (10.10)with M i = M W , M Z , M H . The two leading terms are evaluated in Tab. 10.2 for a widerange of values of the top quark mass and M W = 80 GeV, M H = 100 GeV. For m t < ∝ α m t /M W are not dominant unless the top mass has a value of several TeV. In the physicallyacceptable range of top quark masses, the quadratic terms are numerically compensatedby logarithmic contributions. This remains true also for large Higgs masses. The smallcorrections result from intricate cancellations between leading and nonleading terms.In Fig. 10.3 we show the lowest order width in both parametrizatons Γ , ˆΓ , theelectroweak corrections δ Γ ew , the QCD corrections δ Γ QCD as well as the fully correctedwidth Γ as a function of the top quark mass.The dependence of the total width on the Higgs mass is displayed in Tab. 10.3 wherethis parameter is varied from 50 to 1000 GeV. Although the influence of M H becomesstronger for large top masses it never exceeds 1%.We have compared the pure QCD corrections to those obtained in [58] and foundcomplete agreement. Our results for the electroweak corrections agree with those of [61].74igure 10.3: Top decay width Γ tW b versus the top mass in lowest order Γ , ¯Γ including allcorrections Γ and the contribution of the electroweak δ Γ ew and QCD corrections δ Γ QCD . m t M H = 50 M H = 100 M H = 300 M H = 1000100.0 0.0882 0.0883 0.0886 0.0891120.0 0.3022 0.3021 0.3020 0.3022140.0 0.6179 0.6174 0.6165 0.6157160.0 1.0385 1.0377 1.0356 1.0329180.0 1.5742 1.5734 1.5697 1.5641200.0 2.2373 2.2376 2.2321 2.2221220.0 3.0411 3.0438 3.0369 3.0207Table 10.3: Top decay width Γ tW b including first order QCD and electroweak correctionsfor different values of the Higgs mass. All values are given in GeV.75 W -pair production One of the predominant aims of LEP200 is the high precision investigation of theproperties of the W -boson, i.e. its mass, its total and partial widths and its couplings.Probably the most interesting aspect will be the study of the nonabelian gauge interactionwhich has no direct experimental evidence so far.The general properties of W -pair production at LEP200 have been studied in [62, 63].While the total cross section of e + e − → W + W − is extremely sensitive to deviationsfrom the triple gauge boson couplings of the SM at energies high above the productionthreshold, the sensitivity to variations of this coupling in the LEP200 energy range is onlyat the percent level. Consequently theoretical predictions should be better than 1% toobtain reasonable limits on the structure of the gauge boson self interaction.The W -pair production process allows an independent direct measurement of the W -boson mass with an expected accuracy of about 100 MeV [47]. Again this requires theknowledge of the total cross section with a precision better than 1%.Much effort has been made in recent years to obtain such precise theoretical predic-tions for W -pair production. The virtual electroweak and soft photonic corrections werecalculated by several authors [64, 65, 66, 67]. The complete analytical results for arbitrarypolarizations of the external particles were published in [66]. These will be used for ourevaluations. For the unpolarized case they numerically agree with those of [67] betterthan 0.3% and essentially also with [64]. The hard photon bremsstrahlung correctionshave been evaluated by [68, 14, 69, 70] for definite initial and final state polarizations.The effects arising from the finite width of the W-bosons have been studied in the Bornapproximation in [71] and including the leading weak corrections in [72]. Recently alsothe hard bremsstrahlung to the process e + e − → W + W − → W -pair production cross section. We discuss the process e + ( p , σ ) + e − ( p , σ ) → W + ( k , λ ) + W − ( k , λ ) . (11.1)The arguments indicate the momenta and helicities of the incoming fermions and outgoingbosons ( σ i = ± , λ i = 1 , , − s = ( p + p ) = ( k + k ) = 4 E ,t = ( p − k ) = ( p − k ) = M W − E + 2 E β cos ϑ. (11.2)Here E is the beam energy, ϑ the scattering angle between the e − and the W − and β = q − M W /E the velocity of the W -bosons in the center of mass frame. The electronmass has been consistently neglected. 76n the approximation of zero electron mass the invariant matrix element vanishes dueto chiral symmetry for equal helicities of the e + and e − . Consequently we can write M ( σ , σ , λ , λ , s, t ) = M ( σ, λ , λ , s, t ) (11.3)with σ = σ = − σ . If we neglect the CP-violating phase in the quark mixing matrix, CPis a symmetry of the process leading to the relation M ( σ, λ , λ , s, t ) = M ( σ, − λ , − λ , s, t ) . (11.4)Consequently there are only 12 independent helicity matrix elements instead of 36.As discussed in Chap. 5 the general matrix element M ( σ, λ , λ , s, t ) = ¯ v ( p , − σ ) M µν u ( p , σ ) ε µ ( k , λ ) ε ν ( k , λ ) (11.5)can be decomposed into formfactors and standard matrix elements. Due to the above-mentioned symmetries we do not need all of the (overcomplete) 40 standard matrix ele-ments given in (5.14) for a general process of vector pair production in fermion-antifermionannihilation, but only seven for each fermion helicity M ( σ, λ , λ , s, t ) = X i =1 M σi F σi ( s, t ) . (11.6)The standard matrix elements M σi are defined in (5.14) together with M σ = M σ , + M σ , , M σ = M σ , + M σ , , M σ = M σ , + M σ , . (11.7)Only six are linear independent. Using (5.15) we can express M by the others M σ = − s M σ + M σ ) + M W − t M σ + s M σ + M σ . (11.8)Therefore there are only twelve independent formfactors. The standard matrix elementscan be calculated with the methods described in Sect. 5.3. The explicit results can befound in [66, 74]. At the Born level three diagrams contribute to W -pair production (Fig. 11.1). Weomitted a Higgs-exchange diagram, which is suppressed by a factor m e /M W and thuscompletely negligible. The t -channel ν e -exchange diagram contributes only for left-handed77 + e − W + W − ν (cid:27)- 6 ss(cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0)(cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1)(cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0)(cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) e + e − W + W − γ, Z ZZ}ZZ(cid:26)(cid:26)>(cid:26)(cid:26) s s(cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0)(cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1)(cid:3) (cid:3) (cid:3) (cid:3)(cid:1) (cid:1) (cid:1) (cid:1)(cid:2) (cid:2) (cid:2) (cid:2)(cid:0) (cid:0) (cid:0) (cid:0)
Figure 11.1: Born diagrams for e + e − → W + W − .electrons whereas the s -channel diagrams containing the nonabelian gauge coupling con-tribute also for right handed electrons. The analytical expressions read M ( − , λ , λ , s, t ) = M − e s W t + 2( M − − M − ) e (cid:20) s − c W s W g − e s − M Z (cid:21) = e s W (cid:20) t M − + 2 s − M Z ( M − − M − ) (cid:21) + e (cid:20) s − s − M Z (cid:21) M − − M − ) , M (+ , λ , λ , s, t ) = 2( M +3 − M +2 ) e (cid:20) s − c W s W g + e s − M Z (cid:21) = e (cid:20) s − s − M Z (cid:21) M +3 − M +2 ) , (11.9)where we have inserted the explicit form of the Z -boson fermion couplings g − e , g + e (A.14).The corresponding cross section for arbitrary longitudinal polarizations of the leptons andbosons is given by dσd Ω ! = β π s X λ ,λ
14 (1 − σP + )(1 + 2 σP − ) |M ( σ, λ , λ , s, t ) | , (11.10)and P ± are the polarization degrees of the leptons ( P − = ± W -pair production. We firststudy the threshold behaviour [75, 76]. For small β the matrix elements behave as M σ , M σ ∝ β, M σ ∝ . (11.11)Consequently the s -channel diagrams vanish at threshold and the t -channel graph domi-nates in the threshold region. For β ≪ σ ( s ) ≈ πα s s W β + O ( β ) . (11.12)All terms ∝ β which are present in the differential cross section drop out in the total crosssection. s -channel diagrams yield contributions ∝ β . In the SM the coefficient of the β β term. As long as that coefficientis not enhanced drastically the β term is negligible in the threshold region, i.e. for 2 M W < √ s < M W + 10 GeV (for √ s = 2 M W + 10 GeV we have β = 0 .
33 and β = 0 . β and hence by kinematics alone. Any change in the couplings willonly affect the coefficient of the β term and thus the normalization of the cross section.Moreover many new physics effects such as anomalous gauge couplings contribute to thes-channel only and thus do not affect the leading term. The inclusion of the finite widthof the W -boson smears the threshold considerably (see Sect. 11.5). However, since thenext to leading β term becomes only sizable several Γ W above threshold only the leadingterm is relevant for the cross section in the region of the nominal threshold.This fact allows a model independent determination of the W -mass from the W -pair production threshold [75]. The measured cross section up to about 10 GeV abovethreshold is fitted with a three-parameter curve σ ( s ) = as + b σ SM ( M W , s ) (11.13)where a/s accounts for the background, b is a model dependent normalization factor and σ SM the W -pair production cross section in the SM depending on the W -mass. Eq. (11.13)is valid including radiative corrections and finite width effects.At high energies the W -pair production cross section is subject to large gauge cancel-lations arising from the contributions of longitudinally polarized W -bosons. For s ≫ M W the matrix elements behave as M σ t , M σ , s ∼ sM W , (11.14)but 1 t M σ + 1 s − M Z M σ − M σ ) ∼ M W t + O (1) , s − s − M Z ! M σ − M σ ) ∼ O (1) . (11.15)Consequently the Born matrix elements (11.9) have a good high energy behaviour. Whilethe cross sections corresponding to the s -channel or t -channel diagrams alone violateunitarity at high energies σ ,t ( s ) ≈ σ ,s ( s ) ≈ πα s s W s M W , (11.16)the SM cross section respects it σ ≈ πα s s W (cid:20) (cid:16) log sM W − (cid:17) − − c W + 524 c W (cid:21) . (11.17)The gauge cancellations are illustrated in Fig. 11.2. They reach one order of magnitudeat 400 GeV and two orders at 1 TeV. They only occur for longitudinal W -bosons. Afterthe gauge cancellations the t -channel again dominates the SM cross section. Compared to79igure 11.2: Gauge cancellations in the total cross section for W-pair production. Shownare the Born cross sections arising from the s -channel σ ,s and t -channel σ ,t diagramsalone and the SM cross section σ .the t -channel contribution all other contributions to the total cross section are suppressedby ∼
50 log( s/M W ). For example the cross section for right handed electrons is σ ( e − R e + → W + W − ) ≈ πα s c W , (11.18)and the cross section for longitudinal W -bosons σ ( e − e + → W + L W − L ) ≈ πα s c W s W + 1 ! . (11.19)We now consider the complete Born expressions for the W -pair production cross sec-tion. The differential cross section for the unpolarized case and for longitudinally polarized W -bosons is shown in Fig. 11.3 and 11.4, respectively. Due to the t -channel pole the un-polarized cross section is strongly peaked in forward direction at high energies and dropssmoothly with increasing scattering angle. In contrast the differential cross section forlongitudinal W -bosons has a minimum for a certain energy dependent finite scatteringangle = π .In order to show the importance of the separate contributions we give in Tab. 11.1 theintegrated cross section for different center of mass energies and different polarizations ofthe leptons and W -bosons.The gauge cancellations depend crucially on the values of the SM couplings. Anydeviations from these values can lead to sizable effects at higher energies since they areenhanced by a factor βs/M W . This fact concerns especially anomalous three gauge boson80igure 11.3: Lowest order differential cross section for the production of unpolarized W -pairs at different center of mass energies.Figure 11.4: Lowest order differential cross section for the production of longitudinal W -pairs at different center of mass energies.81 / pb √ s/ GeV unpolarized e − L e − R W + T W − T W + L W − L W + T W − L + W + L W − T M W = 80 .
23 GeV.couplings which have been studied by many authors [77]. The sensitivity to these effects isbest at high energies and large scattering angles where the t -channel pole is not dominant.Nevertheless one hopes to determine the anomalous couplings up to 20% at LEP200 [6].Using right-handed electrons one could study a pure triple gauge coupling process,but this would require longitudinally polarized electron beams. Furthermore the right-handed cross section is suppressed by two orders of magnitude compared to the dominantleft-handed mode, mainly because there is no t -channel contribution. On the other hand,nonstandard couplings or other new physics can enhance it drastically exactly for thisreason. The radiative corrections can be naturally divided into three classes, the virtualcorrections, the soft photonic, and the hard photonic corrections. Since the process e + e − → W + W − involves the charged current, the radiative corrections cannot be sepa-rated into electromagnetic and weak ones in a gauge invariant way. We first discuss thevirtual and soft photonic corrections. The virtual corrections get contributions from the ν e -, γ - and Z -self energies, the γ - Z -mixing energy, from the vertex corrections to the eeγ -, eeZ -, eν e W -, W W γ - and
W W Z -vertices and from box diagrams. The necessary counterterms involve in additionthe e - and W -self energies. Altogether one has to calculate more than 200 individualdiagrams. These can be treated using the methods described in the first part of thisreview. The number of generic diagrams to be evaluated is about 30. The results can beexpressed in terms of the formfactors defined in (11.6) δ M ( σ, λ , λ , s, t ) = X i =1 M σi δF σi . (11.20)82he formfactors δF σi can be evaluated for every CP-invariant set of diagrams separately.For CP-violating diagrams we need in addition the standard matrix elements M σ , −M σ , , M σ , − M σ , and M σ , − M σ , . These drop out in CP-invariant combinations. Theexplicit analytical results for the formfactors are given in terms of the scalar coefficientsof tensor integrals in [66, 74]. The reduction to scalar integrals and their evaluation isdone numerically using the formulae given in Chap. 4. The contribution of the virtualcorrections to the cross section is given by δ dσd Ω ! V = β π s X λ ,λ
14 (1 − σP + )(1 + 2 σP − )2Re ( M ∗ δ M ) . (11.21)The cancellations already present at the Born level occur as well at the level of radiativecorrections. These cancellations only work for gauge invariant quantities. Consequentlythe inclusion of the leading higher order contributions must be done such that gaugeinvariance is respected. Otherwise one may introduce sizable unphysical corrections. Thiswill be discussed in more detail in Sect. 11.3.3.The presence of these cancellations enforces very careful tests of the numerical stabilityof the computer programs. The reliability of the results is founded on agreement betweenindependent calculations [66, 67]. The soft photonic corrections can be easily obtained using the results of Chap. 7. Thesoft photon matrix element reads ( k is the photon momentum) M s = e M " εp kp − εp kp + εk kk − εk kk . (11.22)This yields the soft photon cross section as dσd Ω ! s = dσd Ω ! δ s (11.23)with δ s = − α π Z | k | < ∆ E d k ω k (cid:26) p ( p k ) + p ( p k ) − p p ( p k )( p k )+ k ( k k ) + k ( k k ) − k k ( k k )( k k ) − p k ( p k )( k k ) − p k ( p k )( k k ) + 2 p k ( p k )( k k ) + 2 p k ( p k )( k k ) (cid:27) = − απ ( Eλ − Eλ log sm e + 4 log 2∆ Eλ log M W − uM W − t + 1 + β β log 2∆ Eλ log − β β !
83 log m e s + 1 β log − β β ! + π m e s + 1 + β β " Li β β ! + 14 log − β β ! (11.24)+ 2 " Li − s (1 − β )2( M W − t ) ! + Li − s (1 + β )2( M W − t ) ! − Li − s (1 − β )2( M W − u ) ! − Li − s (1 + β )2( M W − u ) ! . Adding the soft photon cross section to the contribution of the virtual corrections(11.21) the IR-singularities cancel. Moreover also the large Sudakov double logarithmslog ( m e /s ) drop out. In order to set up improved Born approximations which are often very handy thefirst step is to extract the leading corrections [78]. The universal corrections involving∆ α and ∆ ρ can be easily obtained from (8.25) including the leading O ( α ) contributions.There are no nonuniversal corrections ∝ α m t /M W to the W -pair production cross sectionfor not too high energies, i.e. as long as the unitarity cancellations are not sizeable. Inthe LEP200 energy region also terms involving log m t or log M H may become important.These have been evaluated in the limit M H , m t ≫ s . In addition close to threshold apartfrom the large bremsstrahlung corrections which will be discussed in the next section thereis a sizable effect of the Coulomb singularity. This can be simply obtained from generalconsiderations or to O ( α ) directly from the loop diagrams involving photons exchangedbetween the final state W -bosons. Altogether this yields the following approximation M − a = e s W " t M − + 1 s − M Z M − − M − ) − ∆ α
11 + c W s W ∆ ρ + α π s W − c W s W ! log m t M W + α π
116 12 s W log M H M W + απ β + e s − s − M Z ! (cid:16) M − − M − (cid:17) " − ∆ α + απ β (11.25)+ e α π s − M Z M − − M − ) " s W − c W s W log m t M W − c W s W log M H M W , M + a = e s − s − M Z ! M +3 − M +2 ) " − ∆ α + απ β e α π s − M Z M +3 − M +2 ) 16 s W c W " log m t M W −
14 log M H M W . All terms in (11.25) respect the high energy cancellations apart from those involvinglog m t and log M H . However, these were obtained for M H , m t ≫ s , whereas the unitaritycancellations work for s ≫ M H , m t . In this limit the terms containing log m t and log M H are absent. These may, however, cause large effects for small energies and large top quarkor Higgs boson masses. This phenomenon was called delayed unitarity cancellation in[79]. Introducing G F instead of e /s W and the running α ( s ) we obtain M − a = 2 √ G F M W " t M − + 1 s − M Z M − − M − ) απ β + C − ( s, t ) + 4 πα ( s ) s − s − M Z ! (cid:16) M − − M − (cid:17) " απ β + C − ( s, t ) + e α π s − M Z M − − M − ) " s W − c W s W log m t M Z − c W s W log M H s , M + a = 4 πα ( s ) s − s − M Z ! M +3 − M +2 ) " απ β + C +2 ( s, t ) (11.26)+ 2 √ G F M W " t M +1 + 1 s − M Z M +3 − M +2 ) C +1 ( s, t )+ e α π s − M Z M +3 − M +2 ) 16 s W c W " log m t M Z −
14 log M H s . We have included four functions C i ( s, t ) , i = 1 , , σ = ± in this approximation. Theseare necessary to describe the angular dependence of the differential cross section. Thecomplete one-loop invariant matrix element for W -pair production involves 12 formfactors F σi . It turns out, however, that only four of them namely the C σi are relevant in theLEP200 energy region. For higher energies even C +1 can be omitted. The functions C σi have been determined such that they reproduce the corresponding exact one-loopformfactors sufficiently well in the LEP200 energy region [78].We want to stress that the naive summation of the Dyson series of the self energiesmay lead to incorrect results, i.e. a wrong high energy behaviour. This happens becausethe leading corrections are not only contained in the self energies but also in the vertexcorrections. The actual place of their appearance depends on the choice of the fieldrenormalization. We now present some numerical results for the radiative corrections in the soft photonapproximation. The numerical input parameters are defined in Sect. 8.1. The soft photoncutoff is chosen as ∆
E/E = 0 .
1. Different choices of ∆
E/E uniformly shift the absolute85igure 11.5: Radiative corrections to the differential cross section relative to lowest orderfor the unpolarized case at various center of mass energies.value of the corrections but do not influence their angular dependence very much. Fig. 11.5shows the relative correction factor δ defined through dσd Ω = dσd Ω ! (1 + δ s ) + δ dσd Ω ! V = dσd Ω ! (1 + δ ) (11.27)for the unpolarized case. While the variation with the scattering angle is relatively flatfor LEP200 energies it becomes stronger with increasing energy. In the forward directionwhere the Born cross section is dominated by the t -channel pole the energy dependence isvery weak. In the backward direction, however, the percentage correction varies stronglywith energy and reaches large negative values up to −
50% at 1 TeV. Nevertheless sincethe absolute cross section is small for large scattering angles (see Fig. 11.3), the relativecorrections to the integrated cross section stay below 20% up to 1 TeV. Note that theone-loop corrections are large exactly in that region where the sensitivity to new physicsis highest.The behaviour of the corrections for purely transverse W -bosons is similar to the unpo-larized case. In contrast to this the corrections for purely longitudinal bosons (Fig. 11.6)exhibit a strong angular dependence arising from the minima in the lowest order crosssection (Fig. 11.4).The sensitivity of the total unpolarized cross section on the unknown masses of theHiggs boson and top quark is illustrated in Tab. 11.2 and 11.3. A change of m t from80 to 200 GeV affects the cross section by less than 3% apart from the region very closeto threshold. The large effect close to threshold is due to the variation of M W and thus86igure 11.6: Radiative corrections to the differential cross section relative to lowest orderfor purely longitudinal W -bosons at various center of mass energies.the variation of the threshold with m t . A variation of M H between 50 and 1000 GeVinfluences the total cross section by less than 0.5%, again with the exception of thethreshold region. Note that this is valid for constant α , G F and M Z . Fixing instead G F , M W and M Z the dependence on m t is much weaker. This allows to determine M W fromthe cross section practically independently on m t and M H as pointed out by Jegerlehner[80]. These results for the top and Higgs mass dependence remain valid if we include hardphotonic corrections.Using the functions C σi given in [78] the improved Born approximation (11.26) agreeswith the full one-loop order result within 0.5% for √ s <
220 GeV and within 1% for √ s <
270 GeV in the case of the total cross section. For the differential cross sectionthe deviation is at most 1% for √ s <
210 GeV. The largest difference occurs for largescattering angles, i.e. where the cross section is small.
The complete hard photonic bremsstrahlung to e + e − → W + W − was determined in[68, 70]. The polarized amplitudes for the process e + e − → W + W − γ were calculatedusing three different methods. The first one, described in detail in [69] uses the Weylrepresentation for Dirac matrices and spinors and results in expressions for the amplitudesin terms of the components of momentum and polarization vectors in the center of massframe of the incoming leptons. The second method used in [14] is based on the Weyl-van87 t / GeV = 80 120 160 200 M W / GeV = 79.87 80.10 80.37 80.69 √ s/ GeV σ/ pb165.0 10.120 9.743 9.218 8.461180.0 15.521 15.620 15.646 15.626200.0 15.944 16.112 16.220 16.301500.0 5.689 5.760 5.807 5.8471000.0 2.064 2.088 2.103 2.113Table 11.2: Total unpolarized cross section for e + e − → W + W − including virtual and softphotonic corrections for different top quark masses at various center of mass energies. M H / GeV = 50 100 300 1000 M W /GeV = 80.26 80.23 80.16 80.06 √ s/ GeV σ/ pb165.0 9.488 9.503 9.614 9.793180.0 15.654 15.638 15.612 15.598200.0 16.168 16.168 16.143 16.105500.0 5.785 5.785 5.776 5.7641000.0 2.097 2.096 2.093 2.088Table 11.3: Total unpolarized cross section for e + e − → W + W − including virtual and softphotonic corrections for different Higgs boson masses at various center of mass energies.88er Waerden formalism. It yields concise analytical formulae for the amplitude whichare manifestly Lorentz invariant. The relative numerical difference between both resultswas found to be less than 10 − for the amplitude squared. In [70] the amplitudes werecalculated numerically.From this the total cross section is obtained as σ ( s ) = 1(2 π ) s Z d k k d k k d k k X pol |M| δ (4) ( p + p − k − k − k )= 12 s π ) Z d cos ϑ d cos ϑ dφ dk (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k | k | | k | ( √ s − k ) + k k c (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X pol |M| , (11.28)where ϑ , ϑ are the polar angles of the W − -boson and the photon, φ is the azimuthalangle of the photon with respect to the incoming electron and c = sin ϑ sin ϑ cos φ + cos ϑ cos ϑ. (11.29)The nontrivial phase space integrations are performed using Monte Carlo routines. Thusexperimental cuts can be easily implemented.Eq. (11.28) contains the soft photon poles. These are eliminated by a cut k > ∆ E onthe photon energy. After combining soft and hard photonic corrections the cut dependencedrops out. This has been checked numerically for ∆ E/E between 10 − and 10 − . The leading logarithmic (LL) QED corrections to the W -pair production cross sectionwere already calculated in [43]. The resulting cross section is given by [68] σ LL ( s ) = Z M W /s dzφ ( z )ˆ σ ( zs ) , (11.30)where ˆ σ ( zs ) denotes the (improved) Born cross section at the reduced CMS energysquared zs . The flux φ ( z ) reads φ ( z ) = δ (1 − z )+ απ ( L − (cid:20) δ (1 − z )2 log ε + θ (1 − ε − z ) 21 − z (cid:21) + απ L (cid:20) δ (1 − z ) 32 − θ (1 − ε − z )(1 + z ) (cid:21) (11.31)+ (cid:18) απ L (cid:19) (cid:26) δ (1 − z ) ε + 3 log ε + 98 − π ! + θ (1 − ε − z ) (cid:20) z − z (2 log(1 − z ) − log z + 32 ) + 12 (1 + z ) log z − (1 − z ) (cid:21)(cid:27) where L = log( Q /m e ) is the leading logarithm and ε = ∆ E/E the soft photon cutoff. φ ( z ) is given including O ( α ) LL -contributions. Furthermore some nonleading termsare incorporated taking into account the fact that the residue of the soft photon pole89s proportional to L − L for the initial state radiation. The scale Q is afree parameter. It can only be determined through higher order calculations. In [68] theintegral in (11.30) was performed numerically. Neglecting the O ( α ) leading logarithmsand the nonlogarithmic terms it has been evaluated analytically [78]. The results presented in this section were calculated by [68, 81]. The parameters arethe same as in Sect. 8.1. In the Born cross section ˆ σ , entering the LL approximation, α has been replaced by G F everywhere. Thus the large fermionic corrections are absorbedat least in the dominant t -channel contributions.The total cross section is plotted in Fig. 11.7 in the LEP200 energy range. Shownare the Born cross section with α replaced by G F , the cross section including the full O ( α ) corrections and including in addition the O ( α ) LL corrections. The correspondingnumbers are given in Tab. 11.4 for various CMS energies. The uncertainty of the full O ( α )result is due to the Monte Carlo integration of the hard bremsstrahlung corrections. Thiserror refers also to the last column of Tab. 11.4. The O ( α ) LL results were evaluated fortwo scale choices Q = sQ = − t min = − M W + s − β ) . (11.32)The second one is motivated by the fact that the total cross section is dominated by the t -channel pole. It reproduces the exact O ( α ) results better. The difference is found to beless than 2% for √ s >
165 GeV. Choosing the scale Q = s the deviation from the exact O ( α ) result is about 5% at 165 GeV. Also at higher energies the scale choice Q = − t min turns out to be preferable. It reproduces the complete O ( α ) result including hard photonbremsstrahlung within 1% for 170 GeV < √ s <
500 GeV. The effect of the O ( α ) LL contribution is demonstrated in the last column of Tab. 11.4. It reaches about 5% at165 GeV, decreases with increasing energy and is small for √ s >
190 GeV. A practicallyidentical result is obtained by soft photon exponentiation.The large deviation between O ( α ) LL and the exact result close to threshold is dueto the Coulomb singularity (see Sect. 11.3.3). It amounts to about 10% at 1 GeV abovethreshold and is not included in the O ( α ) LL result. Note also the large O ( α ) correctionclose to threshold (28% at 161 GeV). Realistic calculations for W -pair production must include the decays of the W -bosonsinto fermions. These are especially important around threshold.In real experiments one observes the reaction e + e − → W + W − → final states . (11.33)The W -bosons give rise to peaks in the invariant mass distributions of the final stateparticles. Therefore one has to calculate the cross section for e + e − → f ¯ f f ¯ f ( γ, g ).This task has been attacked but not completed so far [83]. Above the W -pair productionthreshold the dominant contributions come from Feynman diagrams containing resonant90igure 11.7: Total cross section for W -pair production including hard photonic correc-tions. incl. incl. incl. incl. √ s/GeV Born( G F ) O ( α ) LL O ( α ) LL O ( α ) exact O ( α ) exact Q = s Q = − t min + O ( α ) LL161.0 4.411 2.003 2.158 2.556 ± ± ± ± ± ± ± ± ± ± e + e − → W + W − in pb including hard photonic correc-tions. 91 + e − W + W − W + W − (cid:8)(cid:8)*(cid:8)(cid:8)HHHYHH &%'$(cid:3) (cid:3) (cid:3)(cid:1) (cid:1) (cid:1)(cid:2) (cid:2) (cid:2)(cid:0) (cid:0) (cid:0)(cid:26)(cid:25)(cid:27)(cid:24)(cid:26)(cid:25)(cid:27)(cid:24)(cid:3) (cid:3) (cid:3)(cid:1) (cid:1) (cid:1)(cid:2) (cid:2) (cid:2)(cid:0) (cid:0) (cid:0)(cid:30)(cid:29)(cid:31)(cid:28)(cid:30)(cid:29)(cid:31)(cid:28) (cid:16)(cid:16)(cid:16)(cid:16)PPPPP(cid:16)(cid:16)(cid:16)(cid:16)PPPPP ......Figure 11.8: General structure of diagrams containing two resonant W -propagators. W -propagators. The cross section for W -pair production is obtained by calculating alldiagrams containing two resonant W -propagators. Diagrams contributing to the samefinal state but without two resonant propagators are considered as background. They aresuppressed by a factor M W / Γ W ≈
40 if the full range of invariant masses √ s i of the finalstate particles is included. If √ s i is restricted by a cut ∆ M W − ∆ < √ s i < M W + ∆ , (11.34)the suppression is even M W / (Γ W ∆) ≈
300 for ∆ ≈
10 GeV. Explicit calculations showthat the background contributions are below 1% for √ s ≥ M W [83]. It becomes, however,more relevant below the nominal threshold. But even above threshold nonresonant Borncontributions to e + e − → f ¯ f f ¯ f ( γ, g ) must be taken into account to obtain an accuracyof better than 1%.There are three types of diagrams which may give resonant contributions. The mostimportant ones are factorizable diagrams with the structure shown in Fig. 11.8 whichevidently contain two resonant W -propagators. The corresponding cross section is givenby σ ( s ) = Z ( M W +∆) ( M W − ∆) ds ds σ ∗ ( s, s , s ) ρ ( s ) ρ ( s ) θ ( √ s − √ s − √ s ) , (11.35)where σ ∗ ( s, s , s ) is the ’cross section’ for the production of two off-shell W -bosons and ρ ( s ) = 1 π √ s Γ W ( s )( s − M W ) + s (Γ W ( s )) , (11.36)with the ’decay width’ Γ W ( s ) for an off-shell W -boson. Note that ρ ( s ) → δ ( s − M W ) for Γ W → . (11.37)The off-shell quantities σ ∗ ( s, s , s ) and Γ W ( s ) are not gauge invariant. However, theleading resonant contributions to σ ( s ) are. Eq. (11.35) closely resembles a Breit-Wignerapproximation for the unstable W -bosons. In the threshold region σ ∗ ( s, s , s ) dependsstrongly on s and s . Consequently σ ( s ) deviates considerably from σ ∗ ( s, M W , M W ),the cross section for on-shell stable W ’s. Fig. 11.9 [81] shows this effect in lowest orderand with the full O ( α ) corrections to σ ∗ and Γ W ( s ) included. This dependence is mainly92igure 11.9: Total cross section for W -pair production in lowest order and including thefull O ( α ) corrections with and without finite width effects.due to the threshold factor κ / ( s, s , s ) contained in σ ∗ ( s, s , s ). Extracting this factor, σ ∗ /κ / depends only weakly on s and s . This is also the case for Γ W ( s ) / √ s withrespect to s . Replacing these quantities by their on-shell values we find the followingapproximation σ ( s ) ≈ σ ∗ ( s, M W , M W ) κ / ( s, M W , M W ) Z ( M W +∆) ( M W − ∆) ds ds κ / ( s, s , s ) ˜ ρ ( s ) ˜ ρ ( s ) θ ( √ s − √ s − √ s )(11.38)with ˜ ρ ( s ) = 1 π sM W Γ W ( M W )( s − M W ) + s M W (Γ W ( M W )) . (11.39)This approximation is gauge invariant because σ ∗ ( s, M W , M W ) and Γ W ( M W ) are physicalon-shell quantities. It is particulary useful above the nominal threshold, whereas it getsworse below threshold because there at least one of the W -bosons has to be off-shell.For high energies ( s ≫ M W ) also κ ( s, s , s ) varies only weakly with s and s in theresonance region s ≈ s ≈ M W . Replacing it by its on-shell value we can perform theintegrations and obtain σ ( s ) ≈ σ ∗ ( s, M W , M W ) , for s ≫ M W and ∆ ≫ Γ W . (11.40)Eq. (11.35) incorporates all resonant lowest order contributions and all one-loop correc-tions associated either with the production of W -pairs or the decay of the W -bosons (givenin Chap. 9). This includes in particular all self energy and vertex corrections and thus all93) γ ss ss ss -(cid:27) 6 (cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0)(cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1)(cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0)(cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:16)(cid:16)1(cid:16)(cid:16)PPPPPi i(cid:16)(cid:16)(cid:16)(cid:16)1 1PPPiPP (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:3)(cid:2)(cid:3)(cid:2)(cid:3)(cid:2) b) γ ss sss -(cid:27) 6 (cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0)(cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1)(cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0)(cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:16)(cid:16)1(cid:16)(cid:16)PPPiPP(cid:16)(cid:16)1(cid:16)(cid:16)PPPiPP (cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3)(cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) Figure 11.10: Examples for additional diagrams leading to resonant contributions.leading corrections. A more thorough analysis of this kind has been carried through forthe case of Z -pair production [82].Feynman diagrams which do not fit into the structure shown in Fig. 11.8 give nonres-onant contributions and can thus be neglected with the exception of two classes. Bothoriginate from photonic corrections. The first type results from virtual photons exchangedbetween the external lines connected to different blobs in Fig. 11.8. An example is shownin Fig. 11.10a. These diagrams give rise to resonant contributions coming from photonswhich are nearly on-shell. From similar cases in µ -pair production we know that these arecancelled by the corresponding bremsstrahlung diagrams if one integrates over the wholephoton phase space. For stringent cuts, however, resonant contributions survive.The second type of diagrams consists of those where a real photon is emitted fromthe internal W -boson line (Fig. 11.10b). There are three W -propagators in the diagram.If the photon is hard the corresponding resonances appear in three different regions ofphase space. Therefore these diagrams seem not to fit into the simple Breit-Wigner-likepicture discussed above.In order to take into account these resonant contributions properly one has to calculatethe virtual and real photonic corrections to e + e − → With the electroweak standard model (SM) we have a theory that describes all knownexperimental facts about the electroweak interaction. It has succesfully survived all preci-sion experiments at low energies and at LEP100. The upcoming experiments at LEP200,HERA and the planned hadron colliders will allow to investigate sectors of the SM whichwere not directly accessible so far. For a conclusive confrontation of future experimentalresults with the SM precise predictions are mandatory.For the adequate theoretical description of the experiments at LEP100 the calculationof radiative corrections was inevitable. Although experiments outside the Z -region willnot profit from the presence of a resonance, the expected experimental accuracy will besuch that radiative corrections will be indispensable. Moreover radiative corrections allowto obtain information on otherwise not accessible quantities such as the mass of the topquark or the Higgs boson.One of the next important classes of experiments will be the investigation of the W -boson and its nonabelian couplings at LEP200. We have presented the relevant formulaenecessary for the corresponding higher order calculations. Together with [66, 14] thesecover the complete analytical expressions for processes with on-shell W -bosons. The one-loop virtual corrections are settled for the polarized differential and total cross section.Also hard bremsstrahlung has been calculated by several authors. We have given animproved Born approximation including the leading two-loop contributions for the to-tal and differential cross section. The effects of the finite width of the W -bosons havebeen discussed for the lowest order cross section and the cross section including radia-tive corrections. While the inclusion of the non-photonic corrections is simple the correctsimultaneous treatment of photonic corrections and finite width effects involves nonfac-torizable box diagrams. These contributions are under consideration.We have discussed the total and partial W -decay widths including all one-loop andleading two-loop corrections. Because the W -boson decays only into light fermions thewidths can be described by a very simple expression with an accuracy better than 0.6%.Furthermore we have given results on the t -quark decay width. Also in this case theelectroweak corrections can be incorporated into a simple approximation valid for a topmass below 250 GeV with an accuracy of about 1.7%.Apart from giving these explicit results we have discussed many techniques needed forthe calculation of one-loop corrections. We have compiled a comprehensive set of formulaewhich are relevant for the calculation of one-loop radiative corrections within and outsidethe standard model. We have listed the complete set of Feynman rules for the electroweakstandard model including the counter terms. These were expressed by the self energies ofthe physical particles in terms of two-point functions. We have given explicit results forthe scalar N-point functions for N = 1 , . . . , W + W − → W + W − ),electron photon reactions ( eγ → ν e W ), reactions with three or more final state particlesand so on.The methods described here have been implemented in the computer algebra package FEYN CALC . Many of the quoted formulae are included in this package. We hope thatthis compilation together with the packages
FEYN CALC and
FEYN ARTS can serveas a useful tool, facilitating future calculations of radiative corrections.
Acknowledgements
I would like to thank M. B¨ohm for his support and permanent encouragement duringthe past years. Many of the results presented here were calculated together with T. Sackwho also provided some of the figures and tables. The results on W -pair-production wereobtained in a fruitful collaboration with W. Beenakker, F. A. Berends, H. Kuijf, M. B¨ohmand T. Sack. I have profited from many stimulating and clarifying discussions with thepeople mentioned above and H. Spiesberger, W. Hollik, J. H. K¨uhn, F. Jegerlehner andW. L. van Neerven. I am grateful to R. Mertig, J. K¨ublbeck, R. Guth, H. Eck, R. Scharf,U. Nierste and S. Dittmaier for their assistance. I am indebted to my wife R. Denner fortyping parts of the manuscript and in particular for her patience.96 Feynman rules
In this appendix we list the Feynman rules of the SM in the ’t Hooft-Feynman gaugeincluding the counter terms in a way appropriate for the concept of generic diagrams. I.e.we write down generic Feynman rules and give the possible actual insertions. We omitany field renormalization constants for the unphysical fields. For brevity we introduce theshorthand notation c = c W , s = s W . (A.1)In the vertices all momenta are considered as incoming.Propagators:for gauge bosons V = γ, Z, W in the ’t Hooft Feynman gauge ( ξ i = 1) kV µ V ν (cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0)(cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1)s s = − ig µν k − M V , for Faddeev-Popov ghosts G = u γ , u Z , u W kG ¯ G - p p p p p p p p p p ps s = ik − M G , for scalar fields S = H, χ, φ kS S s s = ik − M S , and for fermion fields F = f i pF ¯ F - s s = i ( p/ + m F ) p − m F . In the ’t Hooft-Feynman gauge we have the following relations: M u γ = 0 , M u Z = M χ = M Z , M u ± = M φ = M W . (A.2)Tadpole: S (cid:19)(cid:19)SS = iδt. VV-counterterm: V ,ν V ,µ , k (cid:19)(cid:19)SS (cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0)(cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1)(cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0)(cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) = − ig µν h C k − C i with the actual values of V , V and C , C W + W − : C = δZ W , C = M W δZ W + δM W ,ZZ : C = δZ ZZ , C = M Z δZ ZZ + δM Z ,AZ : C = δZ AZ + δZ ZA , C = M Z δZ ZA ,AA : C = δZ AA , C = 0 . (A.3)97S-counterterm: S S , k (cid:19)(cid:19)SS = i h C k − C i with the actual values of S , S and C , C HH : C = δZ H , C = M H δZ H + δM H ,χχ : C = 0 , C = − e s δtM W + δM Z ,φ + φ − : C = 0 , C = − e s δtM W + δM W . (A.4)FF-counterterm: ¯ F F , p (cid:19)(cid:19)SS- - = i h C L p/ω − + C R p/ω + − C − S ω − − C + S ω + i with the actual values of F , ¯ F and C L , C R , C − S , C + S f j ¯ f i : C L = (cid:16) δZ f,Lij + δZ f,L † ij (cid:17) , C R = (cid:16) δZ f,Rij + δZ f,R † ij (cid:17) ,C − S = m f,i δZ f,Lij + m f,j δZ f,R † ij + δ ij δm f,i ,C + S = m f,i δZ f,Rij + m f,j δZ f,L † ij + δ ij δm f,i . (A.5)VVVV-coupling: V ,µ V ,ρ V ,ν V ,σ s(cid:3) (cid:3) (cid:3) (cid:3)(cid:1) (cid:1) (cid:1) (cid:1)(cid:2) (cid:2) (cid:2) (cid:2)(cid:0) (cid:0) (cid:0) (cid:0)(cid:3) (cid:3) (cid:3) (cid:3)(cid:1) (cid:1) (cid:1) (cid:1)(cid:2) (cid:2) (cid:2) (cid:2)(cid:0) (cid:0) (cid:0) (cid:0) = ie C h g µν g σρ − g νρ g µσ − g ρµ g νσ i with the actual values of V , V , V , V and CW + W + W − W − : C = s h δZ e − δss + 2 δZ W i ,W + W − ZZ : C = − c s h δZ e − c δss + δZ W + δZ ZZ i + cs δZ AZ ,W + W − AZ : C = cs h δZ e − c δss + δZ W + δZ ZZ + δZ AA i − δZ AZ − c s δZ ZA ,W + W − AA : C = − h δZ e + δZ W + δZ AA i + cs δZ ZA . (A.6)98VV-coupling: V ,ν , k V ,µ , k V ,ρ , k s(cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0)(cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1)(cid:3) (cid:3) (cid:3) (cid:3)(cid:1) (cid:1) (cid:1) (cid:1)(cid:2) (cid:2) (cid:2) (cid:2)(cid:0) (cid:0) (cid:0) (cid:0) = − ieC h g µν ( k − k ) ρ + g νρ ( k − k ) µ + g ρµ ( k − k ) ν i with the actual values of V , V , V and CAW + W − : C = 1 + δZ e + δZ W + δZ AA − cs δZ ZA ,ZW + W − : C = − cs (1 + δZ e − c δss + δZ W + δZ ZZ ) + δZ AZ . (A.7)SSSS-coupling: S S S S s (cid:26) (cid:26) (cid:26)Z Z Z(cid:26) (cid:26) (cid:26)Z Z Z = ie C with the actual values of S , S , S , S and CHHHH : C = − s M H M W h δZ e − δss + δM H M H + e s δtM W M H − δM W M W + 2 δZ H i ,HHχχHHφφ (cid:27) : C = − s M H M W h δZ e − δss + δM H M H + e s δtM W M H − δM W M W + δZ H i ,χχχχ : C = − s M H M W h δZ e − δss + δM H M H + e s δtM W M H − δM W M W i ,χχφφ : C = − s M H M W h δZ e − δss + δM H M H + e s δtM W M H − δM W M W i ,φφφφ : C = − s M H M W h δZ e − δss + δM H M H + e s δtM W M H − δM W M W i . (A.8)SSS-coupling: S S S s (cid:26) (cid:26) (cid:26)Z Z Z = ieC with the actual values of S , S , S and CHHH : C = − s M H M W h δZ e − δss + δM H M H + e s δtM W M H − δM W M W + δZ H i ,HχχHφφ (cid:27) : C = − s M H M W h δZ e − δss + δM H M H + e s δtM W M H − δM W M W + δZ H i . (A.9)99VSS-coupling: V ,µ V ,ν S S s(cid:3) (cid:3) (cid:3) (cid:3)(cid:1) (cid:1) (cid:1) (cid:1)(cid:2) (cid:2) (cid:2) (cid:2)(cid:0) (cid:0) (cid:0) (cid:0) (cid:26) (cid:26) (cid:26)Z Z Z = ie g µν C with the actual values of V , V , S , S and CW + W − HH : C = s h δZ e − δss + δZ W + δZ H i ,W + W − χχW + W − φ + φ − (cid:27) : C = s h δZ e − δss + δZ W i ,ZZφ + φ − : C = ( s − c ) s c h δZ e + s − c ) c δss + δZ ZZ i + s − c sc δZ AZ ,ZAφ + φ − : C = s − c sc h δZ e + s − c ) c δss + δZ ZZ + δZ AA i + ( s − c ) s c δZ ZA + δZ AZ ,AAφ + φ − : C = 2 h δZ e + δZ AA i + s − c sc δZ ZA ,ZZHH : C = s c h δZ e + 2 s − c c δss + δZ ZZ + δZ H i ,ZZχχ : C = s c h δZ e + 2 s − c c δss + δZ ZZ i ,ZAHHZAχχ (cid:27) : C = s c δZ ZA ,W ± Zφ ∓ H : C = − c h δZ e − δcc + δZ W + δZ H + δZ ZZ i − s δZ AZ ,W ± Aφ ∓ H : C = − s h δZ e − δss + δZ W + δZ H + δZ AA i − c δZ ZA ,W ± Zφ ∓ χ : C = ∓ i c h δZ e − δcc + δZ W + δZ ZZ i ∓ i s δZ AZ ,W ± Aφ ∓ χ : C = ∓ i s h δZ e − δss + δZ W + δZ AA i ∓ i c δZ ZA . (A.10)100SS-coupling: S , k V µ S , k s(cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0)(cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:26) (cid:26) (cid:26)Z Z Z = ieC ( k − k ) µ with the actual values of V, S , S and CAχH : C = − i cs δZ ZA ,ZχH : C = − i cs h δZ e + s − c c δss + δZ H + δZ ZZ i ,Aφ + φ − : C = − h δZ e + δZ AA + s − c sc δZ ZA i ,Zφ + φ − : C = − s − c sc h δZ e + s − c ) c δss + δZ ZZ i − δZ AZ ,W ± φ ∓ H : C = ∓ s h δZ e − δss + δZ W + δZ H i ,W ± φ ∓ χ : C = − i s h δZ e − δss + δZ W i . (A.11)SVV-coupling: V ,µ V ,ν S s(cid:3) (cid:3) (cid:3) (cid:3)(cid:1) (cid:1) (cid:1) (cid:1)(cid:2) (cid:2) (cid:2) (cid:2)(cid:0) (cid:0) (cid:0) (cid:0) = ieg µν C with the actual values of S, V , V and CHW + W − : C = M W s h δZ e − δss + δM W M W + δZ H + δZ W i ,HZZ : C = M W sc h δZ e + s − c c δss + δM W M W + δZ H + δZ ZZ i ,HZA : C = M W sc δZ ZA ,φ ± W ∓ Z : C = − M W sc h δZ e + c δss + δM W M W + δZ W + δZ ZZ i − M W δZ AZ ,φ ± W ∓ A : C = − M W h δZ e + δM W M W + δZ W + δZ AA i − M W sc δZ ZA . (A.12)101FF-coupling: ¯ F V µ F s(cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0)(cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:26)(cid:26)>(cid:26)(cid:26)ZZ}ZZ = ieγ µ ( C − ω − + C + ω + )with the actual values of V, ¯ F , F and C + , C − γ ¯ f i f j : C + = − Q f h δ ij (cid:16) δZ e + δZ AA (cid:17) + ( δZ f,Rij + δZ f,R † ij ) i + δ ij g + f δZ ZA ,C − = − Q f h δ ij (cid:16) δZ e + δZ AA (cid:17) + ( δZ f,Lij + δZ f,L † ij ) i + δ ij g − f δZ ZA ,Z ¯ f i f j : C + = g + f h δ ij (cid:16) δg + f g + f + δZ ZZ (cid:17) + ( δZ f,Rij + δZ f,R † ij ) i − δ ij Q f δZ AZ ,C − = g − f h δ ij (cid:16) δg − f g − f + δZ ZZ (cid:17) + ( δZ f,Lij + δZ f,L † ij ) i − δ ij Q f δZ AZ ,W + ¯ u i d j : C + = 0 , C − = √ s h V ij (cid:16) δZ e − δss + δZ W (cid:17) + δV ij + P k ( δZ u,L † ik V kj + V ik δZ d,Lkj ) i ,W − ¯ d j u i : C + = 0 , C − = √ s h V † ji (cid:16) δZ e − δss + δZ W (cid:17) + δV † ji + P k ( δZ d,L † jk V † ki + V † jk δZ u,Lki ) i ,W + ¯ ν i l j : C + = 0 , C − = √ s δ ij h δZ e − δss + δZ W + ( δZ ν,L † ii + δZ l,Lii ) i ,W − ¯ l j ν i : C + = 0 , C − = √ s δ ij h δZ e − δss + δZ W + ( δZ l,L † ii + δZ ν,Lii ) i , (A.13)where g + f = − sc Q f , δg + f = − sc Q f h δZ e + c δss i ,g − f = I W,f − s Q f sc , δg − f = I W,f sc h δZ e + s − c c δss i + δg + f . (A.14)The vector and axial vector couplings of the Z -boson are given by v f = ( g − f + g + f ) = I W,f − s Q f sc , a f = ( g − f − g + f ) = I W,f sc . (A.15)102FF-coupling: ¯ F S F s (cid:26)(cid:26)>(cid:26)(cid:26)ZZ}ZZ = ie ( C − ω − + C + ω + )with the actual values of S, ¯ F , F and C + , C − H ¯ f i f j : C + = − s M W h δ ij m f,i (cid:16) δZ e − δss + δm f,i m f,i − δM W M W + δZ H (cid:17) + ( m f,i δZ f,Rij + δZ f,L † ij m f,j ) i ,C − = − s M W h δ ij m f,i (cid:16) δZ e − δss + δm f,i m f,i − δM W M W + δZ H (cid:17) + ( m f,i δZ f,Lij + δZ f,R † ij m f,j ) i ,χ ¯ f i f j : C + = i s I W,f M W h δ ij m f,i (cid:16) δZ e − δss + δm f,i m f,i − δM W M W (cid:17) + ( m f,i δZ f,Rij + δZ f,L † ij m f,j ) i ,C − = − i s I W,f M W h δ ij m f,i (cid:16) δZ e − δss + δm f,i m f,i − δM W M W (cid:17) + ( m f,i δZ f,Lij + δZ f,R † ij m f,j ) i ,φ + ¯ u i d j : C + = − √ s M W h V ij m d,j (cid:16) δZ e − δss + δm d,j m d,j − δM W M W (cid:17) + δV ij m d,j + P k ( δZ u,L † ik V kj m d,j + V ik m d,k δZ d,Rkj ) i ,C − = √ s M W h m u,i V ij (cid:16) δZ e − δss + δm u,i m u,i − δM W M W (cid:17) + m u,i δV ij + P k ( δZ u,R † ik m u,k V kj + m u,i V ik δZ d,Lkj ) i ,φ − ¯ d j u i : C + = √ s M W h V † ji m u,i (cid:16) δZ e − δss + δm u,i m u,i − δM W M W (cid:17) + δV † ji m u,i + P k ( δZ d,L † jk V † ki m u,i + V † jk m u,k δZ u,Rki ) i ,C − = − √ s M W h m d,j V † ji (cid:16) δZ e − δss + δm d,j m d,j − δM W M W (cid:17) + m d,j δV † ji + P k ( δZ d,R † jk m d,k V † ki + m d,j V † jk δZ u,Lki ) i ,φ + ¯ ν i l j : C + = − √ s m l,i M W δ ij h δZ e − δss + δm l,i m l,i − δM W M W + ( δZ ν,L † ii + δZ l,Rii ) i ,C − = 0 ,φ − ¯ l j ν i : C + = 0 ,C − = − √ s m l,i M W δ ij h δZ e − δss + δm l,i m l,i − δM W M W + ( δZ l,R † ii + δZ ν,Lii ) i . (A.16)103GG-coupling: ¯ G , k V µ G s(cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0) (cid:3) (cid:0)(cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1)p p p p p p p p pp p p p p p p p p }> = iek ,µ C with the actual values of V, ¯ G , G and CA ¯ u ± u ± : C = ± h δZ e + δZ AA i ∓ cs δZ ZA ,Z ¯ u ± u ± : C = ∓ cs h δZ e − c δss + δZ ZZ i ± δZ AZ ,W ± ¯ u ± u Z : C = ± cs h δZ e − c δss + δZ W i ,W ± ¯ u Z u ∓ : C = ∓ cs h δZ e − c δss + δZ W i ,W ± ¯ u ± u γ : C = ∓ h δZ e + δZ W i ,W ± ¯ u γ u ∓ : C = ± h δZ e + δZ W i . (A.17)SGG-coupling: ¯ G S G sp p p p p p p p pp p p p p p p p p }> = ieC with the actual values of S, ¯ G , G and CH ¯ u Z u Z : C = − sc M W h δZ e + s − c c δss + δM W M W + δZ H i ,H ¯ u ± u ± : C = − s M W h δZ e − δss + δM W M W + δZ H i ,χ ¯ u ± u ± : C = ∓ i s M W h δZ e − δss + δM W M W i ,φ ± ¯ u Z u ± : C = sc M W h δZ e + s − c c δss + δM W M W i ,φ ± ¯ u ± u Z : C = s − c sc M W h δZ e + s − c ) c δss + δM W M W i ,φ ± ¯ u ± u γ : C = M W h δZ e + δM W M W i . (A.18)104 Self energies
In this appendix we list all self energies of the physical fields.The gauge boson self energies readΣ
AAT ( k ) = − α π ( X f,i N fC Q f h − ( k + 2 m f,i ) B ( k , m f,i , m f,i )+ 2 m f,i B (0 , m f,i , m f,i ) + 13 k i (B.1)+ (cid:26)h k + 4 M W i B ( k , M W , M W ) − M W B (0 , M W , M W ) (cid:27)) , Σ AZT ( k ) = − α π ( X f,i N fC ( − Q f ) (cid:16) g + f + g − f (cid:17)h − ( k + 2 m f,i ) B ( k , m f,i , m f,i )+ 2 m f,i B (0 , m f,i , m f,i ) + 13 k i − s W c W (cid:26)h (9 c W + 12 ) k + (12 c W + 4) M W i B ( k , M W , M W ) − (12 c W − M W B (0 , M W , M W ) + 13 k (cid:27)) , (B.2)Σ ZZT ( k ) = − α π ( X f,i N fC (cid:26)(cid:16) ( g + f ) + ( g − f ) (cid:17)h − ( k + 2 m f,i ) B ( k , m f,i , m f,i )+ 2 m f,i B (0 , m f,i , m f,i ) + 13 k i + 34 s W c W m f,i B ( k , m f,i , m f,i ) (cid:27) + 16 s W c W (cid:26)h (18 c W + 2 c W −
12 ) k + (24 c W + 16 c W − M W i B ( k , M W , M W ) − (24 c W − c W + 2) M W B (0 , M W , M W ) + (4 c W −
1) 13 k (cid:27) + 112 s W c W (cid:26)(cid:16) M H − M Z − k (cid:17) B ( k , M Z , M H ) − M Z B (0 , M Z , M Z ) − M H B (0 , M H , M H ) (B.3) − ( M Z − M H ) k (cid:16) B ( k , M Z , M H ) − B (0 , M Z , M H ) (cid:17) − k (cid:27)) , WT ( k ) = − α π (
23 12 s W X i (cid:20) − (cid:16) k − m l,i (cid:17) B ( k , , m l,i ) + 13 k + m l,i B (0 , m l,i , m l,i ) + m l,i k (cid:16) B ( k , , m l,i ) − B (0 , , m l,i ) (cid:17)(cid:21) + 23 12 s W X i,j | V ij | (cid:20) − (cid:16) k − m u,i + m d,j (cid:17) B ( k , m u,i , m d,j ) + 13 k + m u,i B (0 , m u,i , m u,i ) + m d,j B (0 , m d,j , m d,j )+ ( m u,i − m d,j ) k (cid:16) B ( k , m u,i , m d,j ) − B (0 , m u,i , m d,j ) (cid:17)(cid:21) + 23 (cid:26)(cid:16) M W + 5 k (cid:17) B ( k , M W , λ ) − M W B (0 , M W , M W ) − M W k (cid:16) B ( k , M W , λ ) − B (0 , M W , λ ) (cid:17) + 13 k (cid:27) + 112 s W (cid:26)h (40 c W − k + (16 c W + 54 − c − W ) M W i B ( k , M W , M Z ) − (16 c W + 2) h M W B (0 , M W , M W ) + M Z B (0 , M Z , M Z ) i + (4 c W −
1) 23 k − (8 c W + 1) ( M W − M Z ) k (cid:16) B ( k , M W , M Z ) − B (0 , M W , M Z ) (cid:17)(cid:27) + 112 s W (cid:26)(cid:16) M H − M W − k (cid:17) B ( k , M W , M H ) − M W B (0 , M W , M W ) − M H B (0 , M H , M H ) (B.4) − ( M W − M H ) k (cid:16) B ( k , M W , M H ) − B (0 , M W , M H ) (cid:17) − k (cid:27)) . For the self energy of the physical Higgs boson we obtainΣ HT ( k ) = − α π (X f,i N fC m f,i s W M W h A ( m f,i ) + (4 m f,i − k ) B ( k , m f,i , m f,i ) i − s W (cid:20)(cid:16) M W − k + M H M W (cid:17) B ( k , M W , M W )+ (cid:16) M H M W (cid:17) A ( M W ) − M W (cid:21) − s W c W (cid:20)(cid:16) M Z − k + M H M Z (cid:17) B ( k , M Z , M Z )+ (cid:16) M H M Z (cid:17) A ( M Z ) − M Z (cid:21) s W (cid:20) M H M W B ( k , M H , M H ) + M H M W A ( M H ) (cid:21)) . (B.5)The fermion self energies are given byΣ f,Lij ( p ) = − α π ( δ ij Q f (cid:20) B ( p , m f,i , λ ) + 1 (cid:21) + δ ij ( g − f ) (cid:20) B ( p , m f,i , M Z ) + 1 (cid:21) + δ ij s W m f,i M W (cid:20) B ( p , m f,i , M Z ) + B ( p , m f,i , M H ) (cid:21) + 12 s W X k V ik V † kj (cid:20)(cid:16) m f ′ ,k M W (cid:17) B ( p , m f ′ ,k , M W ) + 1 (cid:21)) , (B.6)Σ f,Rij ( p ) = − α π ( δ ij Q f (cid:20) B ( p , m f,i , λ ) + 1 (cid:21) + δ ij ( g + f ) (cid:20) B ( p , m f,i , M Z ) + 1 (cid:21) + δ ij s W m f,i M W (cid:20) B ( p , m f,i , M Z ) + B ( p , m f,i , M H ) (cid:21) + 12 s W m f,i m f,j M W X k V ik V † kj B ( p , m f ′ ,k , M W ) ) , (B.7)Σ f,Sij ( p ) = − α π ( δ ij Q f (cid:20) B ( p , m f,i , λ ) − (cid:21) + δ ij g + f g − f (cid:20) B ( p , m f,i , M Z ) − (cid:21) + δ ij s W m f,i M W (cid:20) B ( p , m f,i , M Z ) − B ( p , m f,i , M H ) (cid:21) + 12 s W X k V ik V † kj m f ′ ,k M W B ( p , m f ′ ,k , M W ) ) . (B.8) f ′ is the isospin partner of the fermion f and N fC the colour factor. i, j, k run over thefermion generations. For down-type quarks V ik V † kj has to be replaced by V † ik V kj .107he two-point function B was given in Sect. 4.3. For B we find B ( p , m , m ) = m − m p (cid:16) B ( p , m , m ) − B (0 , m , m ) (cid:17) − B ( p , m , m ) . (B.9)For the field renormalization constants one needs in addition the derivatives of theself energies with respect to k or p , respectively. These are easily obtained from theexpressions above. ∂B ∂p was given in Sect. 4.3, ∂B ∂p can be calculated from (B.9) as ∂∂p B ( p , m , m ) = − m − m p (cid:16) B ( p , m , m ) − B (0 , m , m ) (cid:17) + m − m − p p ∂∂p B ( p , m , m ) . (B.10)These derivatives become IR-singular for m = p and m = 0 or vice versa. This leadsto IR-singular contributions in the field renormalization constants of charged particlesarising from photonic corrections to the corresponding self energies. Because these reduceto very simple expressions we give the photonic contributions to the field renormalizationconstants of the W -boson and the charged fermions explicitly δZ W | photonic = − απ log λM W + α π
13 + 5 (cid:16) ∆ + 1 − log M W µ (cid:17)! , (B.11) δZ f,Lii | photonic = δZ f,Rii | photonic= − α π Q f " ∆ − log m f,i µ + 4 + 4 log λm f,i . (B.12)108 Vertex formfactors
The vertex formfactors V , W , X , can be expressed by the scalar one-loop integrals B ( m , M , M ), C ( m , m , m , M , M , M ) and the scalar coefficients of the vector andtensor integrals B ( m , M , M ), C i ( j ) ( m , m , m , M , M , M ) V a ( m , m , m , M , M , M ) = B ( m , M , M ) − − ( M − m − M ) C (C.1) − ( M − m − M ) C − m − m − m )( C + C + C ) , V − b ( m , m , m , M , M , M ) = 3 B ( m , M , M ) + 4 M C +(4 m + 2 m − m + M − M ) C (C.2)+(4 m + 2 m − m + M − M ) C , V + b ( m , m , m , M , M , M ) = 3 m C , (C.3) V c ( m , m , m , M , M , M ) = − m m M W ( C + C + 2 C ) , (C.4) V d ( m , m , m , M , M , M ) = m ( C − C ) , (C.5) V e ( m , m , m , M , M , M ) = m M W C , (C.6) V − f ( m , m , m , M , M , M ) = m C + m C , (C.7) V + f ( m , m , m , M , M , M ) = m C , (C.8) W − a ( m , m , m , M , M , M ) = 2( C + C + C ) , (C.9) W + a ( m , m , m , M , M , M ) = − C , (C.10) W − b ( m , m , m , M , M , M ) = 3( C + C ) , (C.11) W + b ( m , m , m , M , M , M ) = 3 C , (C.12) W − c ( m , m , m , M , M , M ) = 12 M W h B ( m , M , M ) − − M ( C + C ) i , (C.13) W + c ( m , m , m , M , M , M ) = 2 M W h m C + m C i , (C.14) W d ( m , m , m , M , M , M ) = − C , (C.15)109 e ( m , m , m , M , M , M ) = − M W C , (C.16) W − f ( m , m , m , M , M , M ) = − C , (C.17) W + f ( m , m , m , M , M , M ) = − C − C , (C.18) X − a ( m , m , m , M , M , M ) = − h C + C + 2 C + C + C i , (C.19) X + a ( m , m , m , M , M , M ) = 4 h C + C + C i , (C.20) X − b ( m , m , m , M , M , M ) = 2 h C + 2 C − C i , (C.21) X + b ( m , m , m , M , M , M ) = 6 h C + C i , (C.22) X − c ( m , m , m , M , M , M ) = − m M W h C + C i , (C.23) X + c ( m , m , m , M , M , M ) = − m M W C , (C.24) X d ( m , m , m , M , M , M ) = 2 C , (C.25) X − e ( m , m , m , M , M , M ) = m M W h C + C + C i , (C.26) X e ( m , m , m , M , M , M ) = h C + C + C i , (C.27) X + e ( m , m , m , M , M , M ) = m M W h C + C + C i , (C.28) X f ( m , m , m , M , M , M ) = − C . (C.29)Using the reduction methods decribed in Chap. 4 the vector and tensor coefficients canbe expressed by scalar integrals. For illustration we give the explicit reduction formulae.The vertex function is defined as C ··· = C ··· ( p , p , M , M , M ) = C ··· ( m , m , m , M , M , M )= (2 πµ ) − D iπ Z d D q · · · [ q − M ][( q + p ) − M ][( q + p ) − M ] (C.30)with p = m , p = m , p p = −
12 ( m − m − m ) . (C.31)110or the three-point vector functions (4.18) yields ( P = 1, M = N − C k = T k = ( X − ) kk ′ R ,k ′ (C.32)with k, k ′ = 1 , X = m
21 12 ( m + m − m ) ( m + m − m ) m . (C.33)Evaluating X − this gives C = − κ h m R , + 12 ( m − m − m ) R , i ,C = − κ h
12 ( m − m − m ) R , + m R , i , (C.34)where κ = κ ( m , m , m ) , (C.35)from (4.28). The R ’s are obtained from (4.19) as R , = 12 h B ( m , M , M ) − ( m − M + M ) C − B ( m , M , M ) i ,R , = 12 h B ( m , M , M ) − ( m − M + M ) C − B ( m , M , M ) i . (C.36)The tensor coefficients are evaluated analogously as ( P = 2, M = 2) C = 1 D − h R , − R , − R , i C ki = T ki = ( X − ) kk ′ [ R ,k ′ i − δ k ′ i C ] (C.37)or more explicitly C = 14 h B ( m , M , M ) + ( M − M + m ) C +( M − M + m ) C + 1 + 2 M C i ,C = − κ h m ( R , − C ) + 12 ( m − m − m ) R , i ,C = − κ h
12 ( m − m − m )( R , − C ) + m R , i = C = − κ h m R , + 12 ( m − m − m )( R , − C ) i ,C = − κ h
12 ( m − m − m ) R , + m ( R , − C ) i (C.38)111ith R , = M C + B ( m , M , M ) ,R , = 12 h − ( m − M + M ) C − B ( m , M , M ) i ,R , = 12 h B ( m , M , M ) − ( m − M + M ) C + ( B + B )( m , M , M ) i ,R , = 12 h B ( m , M , M ) − ( m − M + M ) C − B ( m , M , M ) i ,R , = 12 h − ( m − M + M ) C + ( B + B )( m , M , M ) i . (C.39)Note that C = C can be calculated in two different ways. In the evaluation of C weused (4.55) B was given in (B.9). The results for the scalar integrals can again be found inSect. 4.3. 112 Bremsstrahlung integrals
For the decay width of a massive particle with momentum p and mass m into twomassive particles with momenta p , p and masses m , m and a photon with momentum q and mass λ we need the following phase space integrals I j ,...,j m i ,...,i n ( m , m , m ) = 1 π Z d p p d p p d q q δ ( p − p − p − q ) ( ± qp j ) · · · ( ± qp j m )( ± qp i ) · · · ( ± qp i n ) . (D.1)Here j k , i l = 0 , , p , p , the minus signs to p .Introducing the abbreviations κ = κ ( m , m , m ) , (D.2)as defined in (4.28) and β = m − m − m + κ m m ,β = m − m + m − κ m m , β = m + m − m − κ m m , (D.3)with β β β = 1 , (D.4)we get compact expressions for the final results. From (D.1) it is evident that the integralswith the indices 1 and 2 interchanged are obtained by interchanging m and m . We listonly the independent integrals. The IR-singular ones are given by I = 14 m (cid:20) κ log (cid:16) κ λm m m (cid:17) − κ − ( m − m ) log (cid:16) β β (cid:17) − m log( β ) (cid:21) , (D.5) I = 14 m m (cid:20) κ log (cid:16) κ λm m m (cid:17) − κ − ( m − m ) log (cid:16) β β (cid:17) − m log( β ) (cid:21) , (D.6) I = 14 m (cid:20) − (cid:16) λm m m κ (cid:17) log( β ) + 2 log ( β ) − log ( β ) − log ( β )+2 Sp (1 − β ) − Sp (1 − β ) − Sp (1 − β ) (cid:21) , (D.7) I = − I − I = 14 m (cid:20) − (cid:16) λm m m κ (cid:17) log( β ) + 2 log ( β ) − log ( β ) − log ( β )+2 Sp (1 − β ) − Sp (1 − β ) − Sp (1 − β ) (cid:21) . (D.8)For the IR finite integrals we obtain I = 14 m (cid:20) κ m + m + m ) + 2 m m log( β ) + 2 m m log( β ) + 2 m m log( β ) (cid:21) , = 14 m (cid:20) − m log( β ) − m log( β ) − κ (cid:21) , (D.9) I = 14 m (cid:20) − m log( β ) − m log( β ) − κ (cid:21) , (D.10) I = 14 m (cid:20) m log( β ) − m (2 m − m + m ) log( β ) − κ m − m + 5 m ) (cid:21) , (D.11) I = 14 m (cid:20) m log( β ) − m (2 m − m + m ) log( β ) − κ m − m + 5 m ) (cid:21) , (D.12) I = − I − I = 14 m (cid:20) m log( β ) − m (2 m − m + m ) log( β ) − κ m − m + 5 m ) (cid:21) , (D.13) I = − m (cid:20) m log( β ) + m log( β ) + κ m + κ m + 3 m − m ) (cid:21) , (D.14) I = − m (cid:20) m log( β ) + m log( β ) + κ m + κ m + 3 m − m ) (cid:21) , (D.15) I = − I − I = 14 m (cid:20) m ( m + m − m ) log( β ) + κ m + 2 κm (cid:21) , (D.16) I = − I − I = 14 m (cid:20) m ( m + m − m ) log( β ) + κ m + 2 κm (cid:21) , (D.17) I = − I − I = 14 m (cid:20) m ( m + m − m ) log( β ) + κ m + 2 κm (cid:21) . (D.18)Note the symmetries in 0 ↔ ↔
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