1 Introduction

DESY 07-112

Electroweak corrections to hadronic production of bosons at large transverse momenta

Johann H. Kühn, A. Kulesza, S. Pozzorini, M. Schulze

Institut für Theoretische Teilchenphysik, Universität Karlsruhe,

D-76128 Karlsruhe, Germany

Deutsches Elektronen-Synchrotron DESY, Notkestrasse 85,

D–22607 Hamburg, Germany

Max-Planck-Institut für Physik, Föhringer Ring 6,

D–80805 Munich, Germany


To match the precision of present and future measurements of -boson production at hadron colliders electroweak radiative corrections must be included in the theory predictions. In this paper we consider their effect on the transverse momentum () distribution of bosons, with emphasis on large . We evaluate the full electroweak corrections to the processes and including virtual and real photonic contributions. We present the explicit expressions in analytical form for the virtual corrections and provide results for the real corrections, discussing in detail the treatment of soft and collinear singularities. We also provide compact approximate expressions which are valid in the high-energy region, where the electroweak corrections are strongly enhanced by logarithms of . These expressions describe the complete asymptotic behaviour at one loop as well as the leading and next-to-leading logarithms at two loops. Numerical results are presented for proton-proton collisions at and proton-antiproton collisions at . The corrections are negative and their size increases with . At the LHC, where transverse momenta of or more can be reached, the one- and two-loop corrections amount up to and , respectively, and will be important for a precise analysis of production. At the Tevatron, transverse momenta up to are within reach. In this case the electroweak corrections amount up to and are thus larger than the expected statistical error.

1 Introduction

After the startup of the Large Hadron Collider (LHC) hard scattering reactions will be explored with high event rates and momentum transfers up to several TeV. In order to identify new phenomena in this region, the predictions of the Standard Model have to be understood with adequate precision.

The study of gauge-boson production has been among the primary goals of hadron colliders, starting with the discovery of the and bosons more than two decades ago [1]. The investigation of the production dynamics, strictly predicted by the electroweak theory, constitutes one of the important tests of the Standard Model. Differential distributions of gauge bosons, in rapidity as well as in transverse momentum (), have always been the subject of theoretical and experimental studies. This allows to search for and set limits on anomalous gauge-boson couplings, measure the parton distribution functions and, if understood sufficiently well, use these reactions to calibrate the luminosity. For gauge-boson production at large the final state of the leading-order process consists of an electroweak gauge boson plus one recoiling jet. Being, in leading order, proportional to the strong coupling constant, these reactions could also lead to a determination of in the TeV region.

The high center-of-mass energy at the LHC in combination with its enormous luminosity will allow to produce gauge bosons with transverse momenta up to 2 TeV or even beyond. In this kinematic region the electroweak corrections are strongly enhanced, with the dominant terms in -loop approximation being leading logarithms (LL) of the form , next-to-leading logarithms (NLL) of the form , and so on. These corrections, also known as electroweak Sudakov logarithms, may well amount to several tens of percent [2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. (A recent survey of the literature on electroweak Sudakov logarithms can be found in Ref. [12].) Specifically, the electroweak corrections to the -distribution of photons and bosons at hadron colliders were studied in Refs. [7, 8, 9, 10]. In Refs. [8, 9, 10], it was found that at transverse momenta of the dominant two-loop contributions to these reactions amount to several percent and must be included to match the precision of the LHC experiments. This is quite different from the production of on-shell gauge bosons with small transverse momenta [13], where the electroweak corrections are not enhanced by Sudakov logarithms. With this motivation in mind we study the electroweak corrections to hadronic production of bosons in association with a jet, , at large .

As a consequence of the non-vanishing charge, QED corrections cannot be separated from the purely weak ones and will thus be included in our analysis. Thus, in comparison with -boson production, several new aspects arise. Real photon emission must be included to cancel the infrared divergencies from virtual photonic corrections. Collinear singularities, a consequence of radiation from massless quarks, must be isolated and absorbed in the parton distribution functions (PDFs) in the case of initial-state radiation. We regularize soft and collinear singularities in two different schemes: using small quark and photon masses which are set to zero at the end of the calculation and, alternatively, dimensional regularization. In events with real radiation, the of the boson is balanced both by the of the recoiling parton (quark or gluon) and the photon. Configurations involving a small- parton and a hard photon are better described as final states. We thus define the cross section imposing a lower limit on the jet transverse momentum, which is chosen independent of the -boson . In order to avoid final-state collinear singularities, we recombine collinear photon-quark final states.

The virtual EW corrections to production are formally connected with the real emission of and bosons, which leads to final states with . Both contributions are of . If integrated over the full phase space, the real emission of gauge bosons produces large Sudakov logarithms that partially cancel those resulting from virtual gauge bosons. However, in exclusive measurements of , the available phase space for gauge boson emission is strongly suppressed by the experimental cuts. We thus expect that real emission provides relatively small contributions while the bulk of electroweak effects originates from virtual corrections. In fact, for it was shown that, in presence of realistic (and relatively less exclusive) experimental cuts, the contribution of real emission is about five times smaller than the virtual corrections [11]. Moreover, real emission can be further reduced with a veto on additional jets, which suppresses multiple-jet events resulting from the hadronic decay of the radiated gauge bosons. Therefore we will restrict ourselves to the investigation of virtual electroweak corrections (and photon bremsstrahlung). The real emission of and bosons can be non-negligible and certainly deserves further detailed studies, however we do not expect a dramatic impact on our results.

The partonic reactions , and with are considered. All of them are, however, trivially related by CP- and crossing-symmetry relations. Quark-mass effects are neglected throughout, which allows to incorporate the effect of quark mixing through a simple redefinition of parton distribution functions (see Sect. 2.1). Our conventions for couplings, kinematics and two- as well as three-body phase space are also collected in Sect. 2. The calculation of the virtual corrections is described in Sect. 3. We present analytic expressions for the one-loop amplitude, specify the counterterms in the renormalization scheme and isolate the infrared singularities. The high-energy limit is studied in detail in Sect. 4. The analytic one-loop result is investigated in the limit , keeping quadratic and linear logarithms as well as constant terms. These results are compared to those derived in the NLL approximation [14]. In view of their numerical importance we also derive the dominant (NLL) two-loop terms, using the formalism of Refs. [15, 16]. The calculation of the real corrections is performed using the dipole subtraction formalism [17, 18, 19], as discussed in Sect. 5. The checks which we carried out in order to ensure the correctness of the results are described in Sect. 6.

The numerical results are presented in Sect. 7. After convolution with parton distribution functions, we obtain radiatively corrected predictions for -distributions of bosons at the LHC and the Tevatron. The quality of the one-loop NLL and NNLL approximations is investigated and the size of the dominant two-loop terms is compared with the expected statistical precision of the experiments. Concerning perturbative QCD, our predictions are based on the lowest order. To obtain realistic absolute cross sections, higher-order QCD corrections [20] must be included. However, the relative rates for , and production are expected to be more stable against QCD effects. Therefore, the impact of the electroweak corrections on these ratios is presented in Sect. 7. Our conclusions and a brief summary can be found in Sect. 8. Explicit analytic results are collected in the Appendices.

A short description of the method of calculation and the main results for LHC have been given in Ref. [21]. After completion of this work, Hollik, Kasprzik and Kniehl [22] reported results on hadronic -boson production at large qualitatively similar to those of Ref. [21] and the present paper.

2 Definitions and conventions

2.1 Hadronic cross section

The -distribution of bosons in the reaction is given by


where depends on the kinematic configuration of the final state and is specified at the end of Sect. 2.2. The indices denote initial-state partons and , are the corresponding parton distribution functions (PDFs). is the partonic cross section for the subprocess . The sum in (1) runs over all combinations corresponding to the subprocesses


for production, and similarly for production.

The dependence of the partonic cross sections on the family indices amounts to an overall factor . This factor can be easily absorbed by redefining the parton distribution functions as


The hadronic cross section (1) can be computed using the trivial CKM matrix and the redefined PDFs (2.1). Since we do not consider initial or final states involving (anti-)top quarks, only the contributions of the first two quark families () have to be included. The corresponding redefined PDFs ( with ) automatically include the (small) contributions associated with initial- and final-state bottom quarks.

2.2 Kinematics

For the subprocess the Mandelstam variables are defined in the standard way,


The momenta , , of the partons are assumed to be massless, whereas . In terms of and the collider energy we have


with corresponding to the cosine of the angle between the momenta and in the partonic center-of-mass frame.

The -distribution for the unpolarized partonic subprocess reads


where involves the sum over polarization and color as well as the average factor for initial-state polarization. The factor is given by


where , , with , account for the initial-state colour average. The phase-space measure is given by


while the function defines the observable of interest, i.e. the -boson -distribution in presence of a cut on the transverse momentum of the jet,


In the 2-particle phase space the jet is identified with the parton and momentum conservation implies . In practice, since we always consider the -distribution in the region , the cut on in (9) is irrelevant. The phase-space integral in (6) yields two contributions originating from kinematic configurations in the forward and backward hemispheres with opposite values of in the center-of-mass frame,




For the subprocess we define the following five independent invariants


and the four dependent invariants

The -distribution for this subprocess reads




In the 3-particle phase space, the -boson -distribution in production is defined by the observable function


The cut on the jet transverse momentum rejects events where the -boson is balanced by an isolated photon plus a parton with small transverse momentum. This observable is thus free from singularities associated with soft and collinear quarks or gluons. When applying the cut on the jet momentum in the 3-particle phase space, care must be taken that the definition of the jet is collinear-safe. In general the jet cannot be identified with the parton , since in presence of collinear photon radiation the transverse momentum of a charged parton is not a collinear-safe quantity. Thus we identify the jet with the parton only if is a quark well separated from the photon or a gluon. Otherwise, i.e. for collinear quark-photon configurations, the recombined momentum of the quark and photon is taken as momentum of the jet. In practice, we define the separation variable


where is the pseudo-rapidity and is the azimuthal angle of a particle . If , then the photon and quark momenta are recombined by simple four-vector addition into an effective momentum and then , otherwise . We note that, in the collinear region, lowest-order kinematics implies . This means that the recombination procedure effectively removes the cut on inside the collinear cone . For instance the recombined cross section is given by


In contrast, for the case of final-state gluons, we do not perform photon-gluon recombination and the cut on is imposed in the entire phase space.

This procedure has the advantage to avoid both collinear-photon and soft-gluon singularities. However it implies a different treatment of quark and gluon final states and can thus be regarded as an arbitrary cut-off prescription for the final-state collinear singularity. Moreover, the recombined cross section (18) has a logarithmic dependence on the cut-off parameter . These aspects are discussed in detail in Appendix A. There we compare the recombination procedure with a realistic experimental definition of exclusive production, where final-state quarks are subject to the same cut as final state gluons () within the entire phase space. Describing the exclusive cross section,


by means of quark fragmentation functions, we find that the quantitative difference between the two definitions (18) and (19) amounts to less than two permille. Moreover, we show that the recombined cross section is extremely stable with respect to variations of the parameter . This means that the recombination procedure used in our calculation provides a very good description of exclusive production.

Another treatment of the singularities, which does not require recombination and treats quark- and gluon-induced jets uniformly, has been proposed in Ref. [22]. There, contributions from production and production to a more inclusive observable, i.e. high- production, are both calculated. All soft and collinear singularities in the final state cancel in the approach of Ref. [22] as a result of the more inclusive observable definition than associated production of the boson together with a jet, considered in this work. The comparison of our results with those of Ref. [22] seems to indicate that these differences in the jet definitions have a quite small impact on the size of the electroweak corrections.

The quantity in (1) is related to the minimum partonic energy that is needed to produce final states with and ,


When we evaluate the contributions to the hadronic cross section (1), after analytic integration of the phase space in (6), we can set in (20).

2.3 Crossing symmetries

The unpolarized squared matrix elements for the processes in (2.1) are related by the crossing-symmetry relations


Moreover, due to CP symmetry, the unpolarized partonic cross section for the production of positively and negatively charged bosons are related by


Eqs. (2.3) and (22) permit to relate the six processes for production in (2.1) and the six charge conjugate ones to a single process. Hence the explicit computation of the unpolarized squared matrix element needs to be performed only once. In the following we will present explicit results for the process .

Similarly, for the unpolarized squared matrix elements for the processes in (2.1) we have




It is thus enough to perform calculations only for the subprocess.

2.4 Couplings and Born matrix element

For gauge couplings we adopt the conventions of Ref. [23]. With this notation the vertex and the vertices with read


where are the chiral projectors


are the Gell-Mann matrices and are matrices in the weak isospin space. For diagonal matrices such as and we write . In terms of the weak isospin and the weak hypercharge we have


with the shorthands and for the weak mixing angle . The eigenvalues of isospin, hypercharge and SU(2) Casimir operators for left-handed fermions are


The only non-vanishing components of the generators associated with bosons are


The triple gauge-bosons vertices read


where the totally anti-symmetric tensor is defined through the commutation relations


and has components and .

To lowest order in and , the unpolarized squared matrix element for the process reads


where and are the electromagnetic and the strong coupling constants.

3 Virtual corrections

In this section we present the virtual electroweak corrections to the process. The algebraic reduction to gauge-coupling structures, standard matrix elements and one-loop scalar integrals is described in Sect. 3.2. The renormalization of ultraviolet divergences and the subtraction of infrared singularities originating from soft and collinear virtual photons are discussed in Sect. 3.3. and Sect. 3.4, respectively. In Sect. 3.5 we summarize the one-loop result for the unpolarized squared matrix element.

3.1 Preliminaries

As discussed in the previous section, the twelve different processes relevant for production are related by CP and crossing symmetries. It is thus sufficient to consider only one of these processes.

\ArrowLine(0,20)(-40,20)\Vertex(0,20)1.5\Photon(0,20)(40,20)2.56\ArrowLine(-40,-20)(0,-20)\Vertex(0,-20)1.5\Gluon(0,-20)(40,-20)-2.56\ArrowLine(0,-20)(0,20)\Text(0,-35)[t](t1) \ArrowLine(0,20)(-40,20)\Vertex(0,20)1.5\Gluon(0,20)(40,20)2.56\ArrowLine(-40,-20)(0,-20)\Vertex(0,-20)1.5\Photon(0,-20)(40,-20)-2.56\ArrowLine(0,-20)(0,20)\Text(0,-35)[t](t2)
Figure 1: Tree-level Feynman diagrams for the process .

In the following we derive the one-loop corrections for the process. The matrix element


is expressed as a function of the Mandelstam invariants


The Born contribution results from the - and -channel diagrams of Fig. 1. The loop and counterterm diagrams contributing to the corrections,


are depicted in Fig. 2 and Fig. 3, respectively.

\ArrowLine(0,20)(-40,20)\Vertex(0,20)1.5\Photon(0,20)(40,20)2.56\ArrowLine(-40,-20)(0,-20)\Vertex(0,-20)1.5\Gluon(0,-20)(40,-20)-2.56\PhotonArc(0,0)(10,90,270)-2.54\Text(-15,0)[r]\ArrowLine(0,-20)(0,-10)\Vertex(0,-10)1.5\ArrowLine(0,-10)(0,10)\Vertex(0,10)1.5\ArrowLine(0,10)(0,20) \Text(0,-35)[t](s1) \ArrowLine(0,20)(-40,20)\Vertex(0,20)1.5\Gluon(0,20)(40,20)2.56\ArrowLine(-40,-20)(0,-20)\Vertex(0,-20)1.5\Photon(0,-20)(40,-20)-2.56\PhotonArc(0,0)(10,90,270)-2.54\Text(-15,0)[r]\ArrowLine(0,-20)(0,-10)\Vertex(0,-10)1.5\ArrowLine(0,-10)(0,10)\Vertex(0,10)1.5\ArrowLine(0,10)(0,20) \Text(0,-35)[t](s2) \ArrowLine(0,20)(-40,20)\Vertex(0,20)1.5\Photon(0,20)(40,20)2.56\ArrowLine(-40,-20)(-15,-20)\Vertex(-15,-20)1.5\Vertex(15,-20)1.5\Gluon(15,-20)(40,-20)-2.53.5\ArrowLine(-15,-20)(15,-20)\Photon(-15,-20)(0,5)2.54.5\Text(-13,-5)[r]\ArrowLine(15,-20)(0,5)\Vertex(0,5)1.5\ArrowLine(0,5)(0,20) \Text(0,-35)[t](v1) \ArrowLine(-15,20)(-40,20)\Vertex(-15,20)1.5\Vertex(15,20)1.5\Gluon(15,20)(40,20)2.53.5\ArrowLine(15,20)(-15,20)\Photon(0,-5)(-15,20)2.54.5\Text(-13,5)[r]\ArrowLine(0,-5)(15,20)\Vertex(0,-5)1.5\ArrowLine(0,-20)(0,-5) \ArrowLine(-40,-20)(0,-20)\Vertex(0,-20)1.5\Photon(0,-20)(40,-20)-2.56\Text(0,-35)[t](v2) \ArrowLine(-15,20)(-40,20)\Vertex(-15,20)1.5\Vertex(15,20)1.5\Photon(15,20)(40,20)2.53.5\ArrowLine(15,20)(-15,20)\Photon(0,-5)(-15,20)2.54.5\Text(-13,5)[r]\ArrowLine(0,-5)(15,20)\Vertex(0,-5)1.5\ArrowLine(0,-20)(0,-5) \ArrowLine(-40,-20)(0,-20)\Vertex(0,-20)1.5\Gluon(0,-20)(40,-20)-2.56\Text(0,-35)[t](v3) \ArrowLine(0,20)(-40,20)\Vertex(0,20)1.5\Gluon(0,20)(40,20)2.56\ArrowLine(-40,-20)(-15,-20)\Vertex(-15,-20)1.5\Vertex(15,-20)1.5\Photon(15,-20)(40,-20)-2.53.5\ArrowLine(-15,-20)(15,-20)\Photon(-15,-20)(0,5)2.54.5\Text(-13,-5)[r]\ArrowLine(15,-20)(0,5)\Vertex(0,5)1.5\ArrowLine(0,5)(0,20) \Text(0,-35)[t](v4) \ArrowLine(-15,20)(-40,20)\Vertex(-15,20)1.5\Vertex(15,20)1.5\Photon(15,20)(40,20)2.53.5\Photon(15,20)(-15,20)-2.54.5\Text(0,25)[b]\ArrowLine(0,-5)(-15,20)\Photon(0,-5)(15,20)2.54.5\Text(13,5)[l]\Vertex(0,-5)1.5\ArrowLine(0,-20)(0,-5) \ArrowLine(-40,-20)(0,-20)\Vertex(0,-20)1.5\Gluon(0,-20)(40,-20)-2.56\Text(0,-35)[t](v5) \ArrowLine(0,20)(-40,20)\Vertex(0,20)1.5\Gluon(0,20)(40,20)2.56\ArrowLine(-40,-20)(-15,-20)\Vertex(-15,-20)1.5\Vertex(15,-20)1.5\Photon(15,-20)(40,-20)-2.53.5\Photon(-15,-20)(15,-20)-2.54.5\Text(0,-25)[t]\ArrowLine(-15,-20)(0,5)\Photon(15,-20)(0,5)2.54.5\Text(13,-5)[l]\Vertex(0,5)1.5\ArrowLine(0,5)(0,20) \Text(0,-35)[t](v6) \ArrowLine(-15,20)(-40,20)\Vertex(-15,20)1.5\Vertex(15,20)1.5\Photon(15,20)(40,20)2.53.5\ArrowLine(15,20)(-15,20)\ArrowLine(-40,-20)(-15,-20)\Vertex(-15,-20)1.5\Vertex(15,-20)1.5\Gluon(15,-20)(40,-20)-2.53.5\ArrowLine(-15,-20)(15,-20)\Photon(-15,-20)(-15,20)2.55\Text(-20,0)[r]\ArrowLine(15,-20)(15,20)\Text(0,-35)[t](b1) \ArrowLine(-15,20)(-40,20)\Vertex(-15,20)1.5\Vertex(15,20)1.5\Gluon(15,20)(40,20)2.53.5\ArrowLine(15,20)(-15,20)\ArrowLine(-40,-20)(-15,-20)\Vertex(-15,-20)1.5\Vertex(15,-20)1.5\Photon(15,-20)(40,-20)-2.53.5\ArrowLine(-15,-20)(15,-20)\Photon(-15,-20)(-15,20)2.55\Text(-20,0)[r]\ArrowLine(15,-20)(15,20)\Text(0,-35)[t](b2) \ArrowLine(-15,20)(-40,20)\Vertex(-15,20)1.5\Vertex(15,20)1.5\Photon(15,20)(40,20)2.53.5\Photon(15,20)(-15,20)-2.54.5\Text(0,25)[b]\ArrowLine(-40,-20)(-15,-20)\Vertex(-15,-20)1.5\Vertex(15,-20)1.5\Gluon(15,-20)(40,-20)-2.53.5\ArrowLine(-15,-20)(15,-20)\Photon(-15,-20)(15,20)2.57\Text(13,0)[lb]\ArrowLine(15,-20)(-15,20)\Text(0,-35)[t](b3)
Figure 2: One-loop Feynman diagrams for the process . The diagrams s1, s2, v1 and v2 receive contributions from neutral and charged gauge bosons, . The diagrams v3, v4, b1 and b2 involve only neutral gauge bosons, . The remaining diagrams, v5, v6 and b3 involve two contributions with one charged and one neutral gauge boson: and with .
\ArrowLine(0,20)(-40,20)\Photon(0,20)(40,20)2.56\ArrowLine(-40,-20)(0,-20)\Vertex(0,-20)1.5\Gluon(0,-20)(40,-20)-2.56\ArrowLine(0,-20)(0,0)\ArrowLine(0,0)(0,20)\Text(0,-35)[t](c1) \BCirc(0,0)4\Line(-2.83,-2.83)(2.83,2.83)\Line(-2.83,2.83)(2.83,-2.83)\ArrowLine(0,20)(-40,20)\Vertex(0,20)1.5\Gluon(0,20)(40,20)2.56\ArrowLine(-40,-20)(0,-20)\Vertex(0,-20)1.5\Photon(0,-20)(40,-20)-2.56\ArrowLine(0,-20)(0,0)\ArrowLine(0,0)(0,20)\Text(0,-35)[t](c2) \BCirc(0,0)4\Line(-2.83,-2.83)(2.83,2.83)\Line(-2.83,2.83)(2.83,-2.83)\ArrowLine(0,20)(-40,20)\Photon(0,20)(40,20)2.56\ArrowLine(-40,-20)(0,-20)\Vertex(0,-20)1.5\Gluon(0,-20)(40,-20)-2.56\ArrowLine(0,-20)(0,20)\Text(0,-35)[t](c3) \BCirc(0,-20)4\Line(-2.83,-22.83)(2.83,-17.18)\Line(-2.83,-17.18)(2.83,-22.83)\ArrowLine(0,20)(-40,20)\Vertex(0,20)1.5\Gluon(0,20)(40,20)2.56\ArrowLine(-40,-20)(0,-20)\Vertex(0,-20)1.5\Photon(0,-20)(40,-20)-2.56\ArrowLine(0,-20)(0,20)\Text(0,-35)[t](c4) \BCirc(0,20)4\Line(-2.82,17.18)(2.83,22.82)\Line(-2.83,22.83)(2.83,17.18)\ArrowLine(0,20)(-40,20)\Photon(0,20)(40,20)2.56\ArrowLine(-40,-20)(0,-20)\Vertex(0,-20)1.5\Gluon(0,-20)(40,-20)-2.56\ArrowLine(0,-20)(0,20)\Text(0,-35)[t](c5) \BCirc(0,20)4\Line(-2.82,17.18)(2.83,22.82)\Line(-2.83,22.83)(2.83,17.18)\ArrowLine(0,20)(-40,20)\Vertex(0,20)1.5\Gluon(0,20)(40,20)2.56\ArrowLine(-40,-20)(0,-20)\Vertex(0,-20)1.5\Photon(0,-20)(40,-20)-2.56\ArrowLine(0,-20)(0,20)\Text(0,-35)[t](c6) \BCirc(0,-20)4\Line(-2.83,-22.83)(2.83,-17.18)\Line(-2.83,-17.18)(2.83,-22.83)
Figure 3: Counterterm diagrams for the process .

The quarks that are present in the loop diagrams of Fig. 2 are treated as massless, and the regularization of the collinear singularities that arise in this limit is discussed in Sect. 3.4. The only quark-mass effects that we take into account are the -terms that enter the counterterms through gauge-boson self-energies.

Our calculation has been performed at the matrix-element level and provides full control over polarization effects. However, at this level, the analytical expressions are too large to be published. Explicit results will thus be presented only for the unpolarized squared matrix element.

3.2 Algebraic reduction

The matrix element (33) has the general form


Since we neglect quark masses, consists of terms involving an odd number of matrices with . The -terms are isolated in the chiral projector defined in (26). The polarization dependence of the quark spinors and gauge-boson polarization vectors is implicitly understood. In analogy to (33) and (35) we write


Following the approach adopted in Ref. [9], we isolate the SU(2)U(1) couplings that appear in the Feynman diagrams and reduce the one-loop amplitude to a sum of contributions associated with independent coupling structures. As we will see, besides an abelian and a non-abelian contribution that are related to the ones found for production [9], for production we have two additional coupling structures.

The coupling structure of the Born amplitude is trivial and consists simply of the component of the SU(2) generator,


The contribution of the loop diagrams of Fig. 2 can be written as


In the following, treating the electroweak gauge couplings as isospin matrices and using group-theoretical identities (see App. B of Ref. [23]), we express the above amplitude in terms of the eigenvalues of isospin, hypercharge and SU(2) Casimir operators for left-handed fermions (28).

The tensors and in (39) describe the contributions of the diagrams s1, v1 and s2, v2, respectively. These diagrams may involve charged or neutral virtual bosons. In the former case , the corresponding couplings read111The following identities have to be understood as matrix identities, where the indices of the SU(2) generators are implicitly understood.


In the latter case () the coupling factors read