ARTICLE IN PRESS. Nuclear Physics B ( ) 17 Received 5 October 2004; accepted 11 November Introduction

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1 S ) /FLA AID:977 Vol. ) [DTD5] P ) NUPHB:m1 v 1.30 Prn:6/11/004; 9:17 npb977 by:is p. 1 Nuclear Physics B ) Logarithmic electroweak corrections to gauge-boson pair production at the LHC 14 a Dipartimento di Fisica Teorica Università di Torino Via P. Giuria Torino Italy b Paul Scherrer Institut CH-53 Villigen PSI Switzerland c Institute of Theoretical Physics University of Zürich CH-8057 Zürich Switzerland Received 5 October 004; accepted 11 November Abstract E. Accomando a A.Denner b A. Kaiser bc We have studied the effects of the complete logarithmic electroweak Oα) corrections on the production of vector-boson pairs WZ ZZ andww at the LHC. These corrections are implemented into a Monte Carlo program for pp 4f+γ)with final states involving four or two leptons using the double-pole approximation. We numerically investigate purely leptonic final states and find that electroweak corrections lower the predictions by 5 30% in the physically interesting region of large di-boson invariant mass and large angle of the produced vector bosons. 004 Published by Elsevier B.V. 30 PACS: 1.15.Lk; 1.15.Ji; e; t Introduction The production of gauge-boson pairs provides an excellent opportunity to test the non- Abelian structure of the Standard Model SM). Gauge-boson-pair-production amplitudes involve trilinear gauge-boson couplings. Therefore the corresponding cross sections depend very sensitively on the non-abelian structure of the underlying theory. For this reason vector-boson pair production has found continuous interest in the literature. In the last few 43 address: ansgar.denner@psi.ch A. Denner) /$ see front matter 004 Published by Elsevier B.V doi: /j.nuclphysb

2 S ) /FLA AID:977 Vol. ) [DTD5] P. 1-47) NUPHB:m1 v 1.30 Prn:6/11/004; 9:17 npb977 by:is p. E. Accomando et al. / Nuclear Physics B ) years gauge-boson self-interactions were directly measured at LEP and Tevatron. Still up to now these couplings have not been determined with the same precision as other boson properties such as their masses and couplings to fermions. Despite of the high statistics reached at LEP in producing W + W pairs the resulting limits on possible anomalous couplings which parametrize deviations from SM predictions due to new physics occurring at energy scales of order of tens of TeV are not very stringent. The weakness of the LEP measurement is the rather modest centre-of-mass CM) energy of the produced W -boson pairs. On the other hand anomalous gauge-boson couplings cause strong enhancements in the gauge-boson-pair-production cross section especially at large values of the di-boson invariant mass M VV VV = WZ). A significant improvement in the bounds on triple-gauge-boson couplings is expected from measurements at future colliders operating at high energies such as the Large Hadron Collider LHC). Therefore in order to achieve a better precision in the determination of these couplings it will be useful to analyse di-boson production at hadron colliders at the highest possible CM energies. Vector-boson pairs also constitute a background to other kinds of new-physics searches. One of the gold-plated signals for supersymmetry at hadron colliders is chargino neutralino pair production which would give rise to final states with three charged leptons and missing transverse momentum; the primary background to this signature is given by WZ production. Also final states coming from ZZ production could fake that supersymmetry signature if one of the leptons is lost in the beam pipe. Finally W ± W can dirty the measurements of chargino and slepton pair production which both give rise to two leptons and missing energy. Leptonic final states coming from pp VV VV = WZ) could also fake ZZ WZandWW vector-boson scattering signals which are again expected to be enhanced at high CM energies. In the near future the LHC will be the main source of vector-boson pairs with large invariant mass M VV. The machine will collect thousands of events the exact statistics depending on the particular process and luminosity [1]. With LHC approaching its goal of an integrated luminosity of 100 fb 1 a large data sample will be available to start a detailed investigation of the trilinear vertices. In order to match the experimental precision theoretical predictions need to have an accuracy of the order of a few per cent to allow for a decent analysis of the data. At lowest order this demands taking into account all spin correlations and finite-width effects. The easiest way to fulfill this requirement is to go beyond the production decay approach by computing the full processes pp 4f. The next step consists in a full understanding and control of higher-order QCD and electroweak EW) corrections. In the past years a large effort has gone into accurate calculations of hadronic di-boson production for a review on the subject see Ref. [1]). The Oα s ) QCD corrections to massive gauge-boson pair production and decay have been extensively analysed by many authors [ 8]. Several NLO Monte Carlo programs have been constructed and cross checked so that complete Oα s ) corrections are now available [78]. QCD corrections turn out to be quite significant at LHC energies. They can increase the lowest-order cross section by a factor of two if no cuts are applied and by one order of magnitude for large transverse momentum or large invariant mass of the vector bosons [3]. By including a jet veto their effects can be drastically reduced to the order of tens of per cent [67] but in any case they have to be considered to get realistic and reliable estimates of total cross sections and distributions

3 S ) /FLA AID:977 Vol. ) [DTD5] P ) NUPHB:m1 v 1.30 Prn:6/11/004; 9:17 npb977 by:is p. 3 E. Accomando et al. / Nuclear Physics B ) 3 In view of the envisaged precision of a few per cent at the LHC also a discussion of EW corrections is in order. For single W -andz-boson production Oα) corrections have been computed taking into account the full electromagnetic and weak contributions [9]. One-loop weak corrections have been also investigated for t t production [10] b b production [11] γ/z + jet hadro-production [1] WH and ZH production [13] aswellasfor γz production [14]. By contrast gauge-boson pair production at hadron colliders is commonly treated by including only universal radiative corrections such as the running of the electromagnetic coupling and corrections to the ρ parameter. This approach is based on the belief that the remaining EW corrections dominated by double-logarithmic contributions) are not relevant at the LHC just because physical cross sections decrease strongly with increasing invariant mass of the gauge-boson pairs i.e. where EW corrections can be non-negligible. However a first analysis of the effect of one-loop logarithmic EW corrections on WZ and Wγ production processes at the LHC [15] has instead demonstrated that Oα) corrections are of the same order or bigger than the statistical error when exploring the large di-boson invariant-mass and small rapidity region. The fact that Oα) EW corrections grow with increasing energy is well known since long time. EW corrections are in fact dominated by double and single logarithms of the ratio of the energy to the EW scale. Analyses of the general high-energy behaviour of EW corrections have been extensively performed see for instance Refs. [1617]). A processindependent recipe for the calculation of logarithmic EW corrections is given in Refs. [18 0] where it has been shown that the logarithmic one-loop corrections to arbitrary EW processes factorize into tree-level amplitudes times universal correction factors. Using the method of Refs. [18 0] we investigate in this paper the effect of logarithmic EW corrections to the hadronic production of W ± Z ZZ andw ± W pairs in the large-invariant-mass region of the hard process at the LHC. Going beyond the analysis of Ref. [15] which addressed only logarithmic contributions originating from above the EW scale here we consider also the effect of the complete logarithmic electromagnetic corrections. Since the aim of the paper is to analyse the structure of the Oα) EW corrections and to give an estimate of their size we have not included QCD corrections. The simplest experimental analyses of gauge-boson pair production will rely on purely leptonic final states. Semi-leptonic channels where one of the vector bosons decays hadronically have been analysed at the Tevatron [1] showing that these events suffer from the background due to the production of one vector boson plus jets via gluon exchange. For this reason we study only di-boson production where both gauge bosons decay leptonically into e or µ. The paper is organized as follows: the strategy of our calculation is presented in Section. In Section 3 we describe the calculation of the lowest-order matrix elements and in Section 4 the analytical results for the virtual logarithmic EW one-loop corrections are summarized. The treatment of soft and collinear singularities is discussed in Section 5. While the general setup of our numerical calculation is given in Section 6 Section 7 contains a numerical discussion for processes mediated by WZ ZZ andww production. Our findings are summarized in Section 8. Appendices A and B contain results for nonfactorizable corrections to a general class of processes and the corresponding integrals. Some coupling factors are listed in Appendix C

4 S ) /FLA AID:977 Vol. ) [DTD5] P ) NUPHB:m1 v 1.30 Prn:6/11/004; 9:17 npb977 by:is p. 4 4 E. Accomando et al. / Nuclear Physics B ). Strategy of the calculation We consider the production of massive gauge-boson pairs in hadron hadron collisions. The generic process can be written as h 1 + h V 1 + V + X f 3 + f 4 + f 5 + f 6 + X ) where h 1 and h denote the incoming hadrons V 1 and V two arbitrary massive gauge bosons e.g. W or Z bosons f 3 f 5 the outgoing fermions f 4 f 6 the outgoing antifermions and X the remnants of the hadrons. In the parton model the corresponding cross sections are obtained from a convolution as dσ h 1h P 1 P p f ) = ij dx 1 dx Φ ih1 x1 Q ) Φ jh x Q ) d ˆσ ij x 1 P 1 x P p f ).) where p f summarizes the final-state momenta Φ ih1 and Φ jh are the distribution functions of the partons i and j in the incoming hadrons h 1 and h with momenta P 1 and P respectively Q is the factorization scale and d ˆσ ij represent the differential cross sections for the partonic processes averaged over colours and spins of the partons. The sum ij runs over all possible quarks and antiquarks of flavour u d c ands. The relevant parton processes are of the form q 1 p 1 σ 1 ) + q p σ ) V 1 k 1 λ 1 ) + V k λ ) f 3 p 3 σ 3 ) + f 4 p 4 σ 4 ) + f 5 p 5 σ 5 ) + f 6 p 6 σ 6 )..3) The arguments label the momenta p i k l and helicities σ i =±1/λ l = 0 ±1 ofthecorresponding incoming partons outgoing fermions and virtual gauge bosons. We often use only the signs to denote the helicities. The momenta of the incoming partons are related to the momenta of the hadrons by p 1 = x 1 P 1 and p = x P if i is an antiquark and j a quark and by p = x 1 P 1 and p 1 = x P in the opposite case. The corresponding lowest-order partonic cross sections are calculated using the complete matrix elements. This means that we include the full set of Feynman diagrams in this way accounting for all irreducible background coming from non-doubly resonant contributions. The calculation of the matrix elements for the complete process q 1 p 1 σ 1 ) + q p σ ) f 3 p 3 σ 3 ) + f 4 p 4 σ 4 ) + f 5 p 5 σ 5 ) + f 6 p 6 σ 6 ).4) is described in Section 3. The electroweak radiative corrections to.4) consist of virtual corrections resulting from loop diagrams as well as of real corrections originating from the processes q 1 p 1 σ 1 ) + q p σ ) f 3 p 3 σ 3 ) + f 4 p 4 σ 4 ) + f 5 p 5 σ 5 ) + f 6 p 6 σ 6 ) + γkλ γ ).5)

5 S ) /FLA AID:977 Vol. ) [DTD5] P ) NUPHB:m1 v 1.30 Prn:6/11/004; 9:17 npb977 by:is p. 5 E. Accomando et al. / Nuclear Physics B ) 5 14 Fig. 1. Structure of the process pp V 1 V + X 4f + X. 14 with an additional photon with momentum k and helicity λ γ =±1. Both have to be combined properly in order to ensure the cancellations of soft and collinear singularities cf. Section 5). For the calculation of the radiative corrections we follow the approach used for the process e + e W + W 4f in Ref. []. The virtual corrections are calculated in the double-pole approximation DPA) i.e. we take only those terms into account that are enhanced by two resonant massive gauge-boson propagators. The real corrections are calculated from the full matrix elements for the processes.5)..1. Double-pole approximation for virtual corrections In DPA the processes q 1 q V 1 V 4f are divided into the production of on-shell gauge bosons and their decay into fermion antifermion pairs see Fig. 1). At tree level the matrix elements in DPA for the partonic processes q 1 q V 1 V 4f factorize into those for the production of two on-shell bosons M q 1q V 1λ1 V λ the prop- agators of these bosons and the matrix elements for their on-shell decays M V 1λ 1 f 3 f and M V λ f 5 f M q 1q V 1 V 4f DPA ) ) 37 = P.6) 38 V1 k 1 PV k M q1q V1λ1 V λ M V 1λ 1 f 3 f 4 M V λ f 5 f λ 39 1 λ 39 The sum runs over the physical helicities λ 1 λ = 0 ±1 of the on-shell projected gauge bosons V 1 and V with momenta k 1 and k respectively. The propagators of the massive gauge bosons P V p ) 1 = p MV + θp )im V Γ V V = WZ 44.7)

6 S ) /FLA AID:977 Vol. ) [DTD5] P ) NUPHB:m1 v 1.30 Prn:6/11/004; 9:17 npb977 by:is p. 6 6 E. Accomando et al. / Nuclear Physics B ) involve besides the masses of the gauge bosons also their widths which we consider as constant and finite for time-like momenta. The on-shell matrix elements are calculated using on-shell projected momenta as defined in Appendix A of Ref. []. Of course the momenta in the resonant propagators are not projected on shell. In DPA there are two types of corrections factorizable and non-factorizable ones. The former are those that can be associated to one of the production or decay subprocesses the latter are those that connect these subprocesses. The factorizable corrections can be expressed in terms of the corrections to the on-shell gauge-boson-pair-production and -decay subprocesses. The matrix elements for the virtual factorizable corrections to the processes q 1 q V 1 V 4f can be written as δm q 1q V 1 V 4f virtdpafact ) ) { 14 = P 14 V1 k 1 PV k q 1 q V 1λ1 V λ δm virt M V 1λ 1 f 3 f 4 M V λ f 5 f 6 15 λ 1 λ M q 1q V 1λ1 V λ 17 δm V 1λ 1 f 3 f 4 virt M V λ f 5 f M q 1q V 1λ1 V λ M V 1λ 1 f 3 f 4 δm V λ f 5 f 6 } virt.8) 19 1 where δm q 1q V 1λ1 V λ virt δm V 1λ 1 f 3 f 4 virt andδm V λ f 5 f 6 virt denote the virtual corrections 1 to the on-shell matrix elements for the gauge-boson production and decay processes. The non-factorizable corrections yield a simple correction factor δ virt to the lowestorder cross section. Its explicit form is given in Section 4.3. The contribution of the complete virtual corrections in DPA to the cross section reads 3 nfdpa 3 dσ q 1q V 1 V 4f 8 virtdpa 8 = 1 ŝ [ q dφ 4f Re M 1 q V 1 V 4f ) δm q 1 q V 1 V 4f ) 30 DPA virtdpafact q M 1 q V 1 V 4f.9) 3 DPA δnfdpa] virt 33 where dφ 4f denotes the four-particle phase-space element and ŝ = p 1 + p ) the square of the CM energy in the partonic system. For some four-fermion final states resonant massive gauge bosons can either be formed from the pairs f 3 f 4 ) and f 5 f 6 ) or from the pairs f 3 f 6 ) and f 5 f 4 ). Denoting the isospin partner of f by f this is the case for f 3 = f 4 = f 5 = f 6 and f 3 = f 4 = f 5 = f 6 which allow for two different sets of ZZ and WZ pairs respectively. In all these cases a DPA has to be defined for each of the two sets of resonant gauge bosons separately and the cross sections from these two cases have to be summed In our numerical calculation we actually do not add the cross sections but the matrix elements for the different resonant sets. Since the interference between these different contributions is non-doubly resonant this is equivalent within DPA accuracy. 45

7 S ) /FLA AID:977 Vol. ) [DTD5] P ) NUPHB:m1 v 1.30 Prn:6/11/004; 9:17 npb977 by:is p. 7 E. Accomando et al. / Nuclear Physics B ) 7 Finally we have to take care of the proper matching of the infrared IR) and collinear singularities. In general the total cross section is composed as dσ q1q 4fγ. 4 q dσ = dσ 1 q 4f q + dσ 1 q 4f virt +.10) 4 5 Φ 4f Φ 4f Φ 4fγ 5 Here dσ q 1q 4f is the full differential lowest-order cross section for q 1 q 4f that is to be integrated over the four-particle phase space Φ 4f i.e dσ q 1q 4f.11) 10 = 1 q M 1 q 4f dφ 4f. ŝ Similarly dσ q 1q 4fγ which describes the real corrections is the full lowest-order cross section for q 1 q 4fγ to be integrated over the five-particle phase space Φ 4fγ and 14 dσ q 1q 4f virt denotes the virtual one-loop corrections. 14 Both dσ q 1q 4fγ and dσ q 1q 4f virt involve soft and collinear singularities that cancel in their sum. Taking the DPA for dσ q 1q 4f but not for dσ q 1q 4fγ spoils this cancellation. Therefore we subtract the singular contributions dσ q 1q 4f before we impose the DPA and replace.10) by q dσ 1 q 4f ) 17 virt virtsing 18 q 1 dσ = dσ 1 q 4f + virtdpa dσ q 1q 4f virtsingdpa 1 Φ 4f Φ 4f 3 3 q 4 + dσ 1 q 4f virtsing +.1) 4 5 Φ 4f 5 Φ 4fγ dσ q1q 4fγ. Since the finite non-logarithmic terms of dσ q 1q 4f are not uniquely defined this procedure leads to an ambiguity which is however of the order of the uncertainty of the DPA. Since the IR- and fermion-mass-singular part is not treated in DPA the logarithmic photonic corrections are not affected by this ambiguity. We use the definition of dσ q 1q 4f as given in Section virtsing 7 30 virtsing 30.. High-energy approximation 33 In contrast to Ref. [] we do not calculate the EW Oα) corrections completely but we only calculate the logarithmic corrections in the high-energy limit. To this end we consider the limit where all kinematical invariants s ij = p i + p j ) are large compared with the weak-boson mass scale s ij MW and take into account all contributions proportional to α ln s ij /MW ) or α ln s ij /MW ). Note that this approximation is not applicable to the full processes q 1 q V 1 V 4f because of the presence of the resonances with invariant masses of the order of M W. But it is perfectly applicable to the subprocesses q 1 q V 1 V V 1 f 3 f 4 andv f 5 f 6 appearing in the DPA. Since we assume that all kinematical invariants are of the same order of magnitude we can write all large energy-dependent EW logarithms in terms of lnŝ/mw )whereŝ is the CM energy squared of the partonic process. In Ref. [15] we have taken into account all 44

8 S ) /FLA AID:977 Vol. ) [DTD5] P ) NUPHB:m1 v 1.30 Prn:6/11/004; 9:17 npb977 by:is p. 8 8 E. Accomando et al. / Nuclear Physics B ) contributions involving logarithms of the form lnŝ/mw ). These arise from scales larger than M W and can be written in an SU) U1) symmetric form. We did not include logarithmic corrections of purely electromagnetic origin arising from scales smaller than M W. These involve logarithms of the form lnmw /m f ) or lnm W /λ )whereλ is the photon mass regulator. In the present paper all large logarithms of electromagnetic origin are included as well. Since the decay processes involve no large-energy variable the corresponding virtual corrections involve no large EW logarithms. However they give rise to large electromagnetic logarithms. In the high-energy approximation we omit all mass-suppressed terms i.e. terms of order MW /ŝ. Therefore we can omit the channels with one longitudinal and one transverse gauge boson that are mass suppressed for the di-boson production processes q 1 q V 1 V 4f and take into account only the corrections to the dominating channels involving two transverse TT) or two longitudinal LL) gauge bosons. On the other hand we take into account the exact kinematics by evaluating the complete four-fermion phase space and use the exact values of the kinematical invariants in all formulas. The logarithmic virtual EW corrections to the dominating channels of q 1 q V 1 V are calculated using the general results for a high-energy approximation given in Refs. [18 0]. The validity of these results relies on the assumption that all kinematical variables ŝ ˆt and û are large compared with MW and approximately of the same size ŝ ˆt û MW..13) This implies that the produced gauge bosons have to be emitted at sufficiently large angles with respect to the beam. Hence the validity range of the high-energy logarithmic approximation for the radiative corrections corresponds to the central region of the boson scattering angle in the di-boson rest frame. The t-channel pole in the matrix element gives rise to additional enhanced logarithms when integrated over the full kinematical range. Since these terms are not included in our Oα) analysis we have to take care that we do not get sizeable contributions from small scattering angles with respect to the beam. On the other hand our formulas do not fake spurious contributions as long as ŝ ˆt û MW since the large logarithms become small for ŝ ˆt û MW. The logarithmic approximation yields the dominant corrections for large kinematical invariants s ij MW but neglects finite non-logarithmic process-dependent Oα) contributions. For e + e W + W where complete Oα) corrections and their high-energy limit are available [16] the latter turn out to be of order of a few per cent. We assume that this holds as well for similar processes like hadronic di-boson production. Neglecting non-logarithmic terms can therefore be considered a reasonable approximation at the LHC where the experimental accuracy in the high-energy regime is at the few-per-cent level The matrix elements for q 1 q 4f Our calculation involves two independent sets of matrix elements. The first set consists of the complete lowest-order matrix elements for the processes q 1 q 4f and q 1 q 4fγ. This set is based on the generic matrix elements for e + e 4 fermions + γ given in Ref. [3]. All Feynman diagrams contributing to a f 4f process can be constructed 44

9 S ) /FLA AID:977 Vol. ) [DTD5] P ) NUPHB:m1 v 1.30 Prn:6/11/004; 9:17 npb977 by:is p. 9 E. Accomando et al. / Nuclear Physics B ) 9 9 Fig.. The fundamental topologies for processes with six external fermions. 9 from only two fundamental topologies see Fig. ) by permuting the external particles f a...f f. The second set of matrix elements is used for the calculation of the corrections in the DPA. It includes the matrix elements M q 1q V 1 V for the production of a pair of transverse or longitudinal gauge bosons and the matrix elements M V f f for their decay Matrix elements for four-fermion production We need the amplitudes for the parton processes.4) and.5). To this end we consider a generic process with three incoming antifermions f 1 f 3 f 5 and three incoming fermions f f 4 f 6 : f 1 p 1 σ 1 ) + f p σ ) + f 3 p 3 σ 3 ) + f 4 p 4 σ 4 ) + f 5 p 5 σ 5 ) + f 6 p 6 σ 6 ) V 34 1 f a f C σ σ b V C σ cσ d f f b V 1 C σ eσ f f c f d V f e f f = 4e 4 C V3 W + W Cσ aσ b V 40 3 f C σ cσ d a f b W + f C σ eσ f c f d W 44 V 3.1) The arguments denote the incoming) momenta and helicities of the incoming fermions and antifermions. Each Feynman diagram for the process 3.1) corresponds to one of the two generic diagrams in Fig.. These generic diagrams are given by M av 1V f a f b f c f d f e f f ) ) = 4e 4 C σ aσ f P V 1 pc + p d ) ) P V pe + p f ) ) p b + p e + p f ) A σ aσ c σ e p a p b p c p d p e p f ) M bv 3 f a f b f c f d f e f f ) P f e f V3 pa + p b ) ) f P W pc + p d ) ) P W pe + p f ) ) A σ a 3 p ap b p c p d p e p f ) 3.) and the auxiliary functions A and A 3 can be found in Ref. [3]. The gauge-boson propagators P V are defined by.7)ande is the electric charge of the positron. For the photon V = A) the Z boson and the W boson the generic couplings C σ aσ b are listed in C.5) f a f b

10 S ) /FLA AID:977 Vol. ) [DTD5] P ) NUPHB:m1 v 1.30 Prn:6/11/004; 9:17 npb977 by:is p E. Accomando et al. / Nuclear Physics B ) with C.1). The coupling C V3 W + W is given by C AW + W = 1 C ZW + W = c W s W C σ aσ b 3.4) 9 g 9 M weak {i 1 i 3 i 5 } {i i 4 i 6 } 18 V 1 =W ± Zγ V =W ± Zγ 18 1 V 3 =Zγ 1 M gluon 5 5 {i 1 i 3 i 5 } {i i 4 i 6 } 9 V =W ± Zγ 9 3 V =W ± Zγ is the sum of all diagrams with a positive signature of all permutations and M weak is the M weak 3.7) ) where c W = M W /M Z and s W are the cosine and sine of the electroweak mixing angle respectively. If gluons are present 3.) corresponds to the matrix element with colour matrices omitted and the generic fermionic couplings read g s = δ f a f σa σ b b e with the strong gauge coupling g s = 4πα s. The matrix elements for outgoing particles are simply obtained by inverting the helicities and momenta. It is convenient to define the objects ± = sign{i 1 i 3 i 5 }) sign{i i 4 i 6 }) ± 1 and [ + ± = [ + M bv 3 f i1 f i f i3 f i4 f i5 f i6 ) M av 1V f i1 f i f i3 f i4 f i5 f i6 ) sign{i 1 i 3 i 5 }) sign{i i 4 i 6 }) ± 1 M agv f i1 f i f i3 f i4 f i5 f i6 ) M av g f i1 f i f i3 f i4 f i5 f i6 ) ] ] 3.5) 3.6) where the two sums run over the permutations of the fermions and antifermions and sign{i 1 i 3 i 5 }) and sign{i i 4 i 6 }) give the signs of these permutations. Note that M weak sum of all diagrams with a negative signature. All diagrams that are not present in the SM e.g. diagrams including a Zūe coupling or a W + du coupling drop out because the corresponding values of the generic couplings vanish. For a process that involves just one quark antiquark pair the matrix element squared and summed over colours and spins of the fermions and antifermions reads M quarks = N colour M weak σ 1...σ 6 with the colour factor N colour = 3. For a specific f 4f process this has to be divided by 4N av N sym in order to average over the polarizations and colours of the initial state

11 S ) /FLA AID:977 Vol. ) [DTD5] P ) NUPHB:m1 v 1.30 Prn:6/11/004; 9:17 npb977 by:is p. 11 E. Accomando et al. / Nuclear Physics B ) 11 and to take into account identical particles in the final state. For initial state quarks we have N av = 9 and the symmetry factor is given by N sym = N id for N id pairs of identical particles in the final state. If there are four quarks two additional complications must be taken into account. First the exchange of gluons between two pairs of quarks becomes possible giving rise to additional Feynman diagrams with gluon exchange and secondly we get different colour structures. For the squared matrix element with four quarks summed over polarizations and colours we find M 4 quarks = M weak + gluon M + + gluon M 10 σ 1...σ 6 [ 9 M weak ) M weak ) + 8Re M weak gluon) ) M + 6Re M weak Re M weak gluon) ) 4 M 3 Re M gluon gluon) )] M f ) We do not consider processes with six quarks. The matrix elements for four-fermion-plus-photon production are constructed in complete analogy to the matrix elements for four-fermion production from the generic diagrams given in Ref. [3]. 3.. Matrix elements for q 1 q V 1 V and V f In DPA we need matrix elements for the processes q 1 p 1 σ 1 ) + q p σ ) V 1 k 1 λ 1 ) + V k λ ) V 1 k 1 λ 1 ) f 3 p 3 σ 3 ) + f 4 p 4 σ 4 ) V k λ ) f 5 p 5 σ 5 ) + f 6 p 6 σ 6 ). 3.9) We take all pairs of massive gauge bosons i.e. W + W W ± ZandZZ into account. Owing to the mixing of the Z boson with the photon also matrix elements for W ± γ and Zγ production occur in the results for the logarithmic EW radiative corrections. The logarithmic corrections for matrix elements involving longitudinal gauge bosons are calculated with the Goldstone-boson equivalence theorem. As a consequence of the mixing of the would-be Goldstone bosons with the Higgs boson also the matrix elements for the production of a gauge boson and a Higgs boson appear. All these matrix elements have been calculated with the Weyl van der Waerden spinor formalism. The results can be found in Ref. [4]. 4. Logarithmic EW corrections In DPA the Oα) contributions consist of factorizable corrections to gauge-boson production and decay as well as non-factorizable corrections as summarized in.9). Inthe following we list these corrections in the high-energy approximation.

12 S ) /FLA AID:977 Vol. ) [DTD5] P ) NUPHB:m1 v 1.30 Prn:6/11/004; 9:17 npb977 by:is p. 1 1 E. Accomando et al. / Nuclear Physics B ) 4.1. Corrections to gauge-boson production In this section we present the analytical formulas for the logarithmic EW corrections to the polarized partonic subprocesses q 1 p 1 σ 1 ) + q p σ ) V 1 k 1 λ 1 ) + V k λ ) 33 M q 1q Φ 1 Φ = i) Q V +1) 1 i) Q V +1) M q 1q V 1L V L 41 M q 1q HΦ = i) Q V +1) M q 1q HV L 4.4) We denote the would-be Goldstone bosons corresponding to the Z and W bosons by χ and φ respectively ) which can be derived from the general results given in Ref. [18]. The photon field is denoted by A. The Mandelstam variables read ŝ = p 1 + p ) ˆt = p 1 k 1 ) û = p 1 k ) 4.) where the momenta of the initial and final states are incoming and outgoing respectively. The one-loop corrections are evaluated in the limit.13) and we neglect combinations of gauge-boson helicities that are mass-suppressed compared with ŝ in this limit. Thus we do not consider corrections to the case of mixed longitudinal and transversely polarized gauge bosons. We calculate corrections to the non-suppressed purely longitudinal final state λ 1 λ ) = 0 0) which we denote by λ 1 λ ) = L L) and to the purely transverse final states denoted by λ 1 λ ) = T T) which includes the non-suppressed opposite-helicity final states λ 1 λ ) = ± ) and the suppressed equal-helicity final states λ 1 λ ) = ± ±). The leading and next-to-leading logarithms depend only on tree-level amplitudes and quantum numbers of the external particles and thus are universal. Following Ref. [18]the logarithmic EW corrections can be written in the form δm q 1q V 1λ1 V λ = δ LSC M + δ SSC M + δ C M + δ PR M. 4.3) Here δ LSC M denotes the contribution of leading soft-collinear corrections and δ SSC M the contribution of the next-to-leading soft-collinear corrections which are angular dependent. The term δ C M contains the collinear logarithms and the logarithms related to the renormalization of the incoming and outgoing fields. Finally δ PR M are the logarithmic corrections that arise from parameter renormalization. The following results are directly based on the formulas of Ref. [18] and written in a generic way for all processes q 1 q V 1 V. In this way they have been directly implemented in our Monte Carlo program. The amplitudes involving longitudinal gauge bosons are evaluated using the Goldstone-boson equivalence theorem. Therefore it is convenient to write the results in terms of matrix elements with external would-be Goldstone bosons. These matrix elements have to be understood as shorthands for matrix elements with external longitudinal gauge bosons according to where Q V are the charges of the outgoing gauge bosons.

13 S ) /FLA AID:977 Vol. ) [DTD5] P ) NUPHB:m1 v 1.30 Prn:6/11/004; 9:17 npb977 by:is p. 13 E. Accomando et al. / Nuclear Physics B ) Leading soft-collinear corrections According to formulas 3.6) and 3.7) of Ref. [18] the leading soft-collinear corrections read δ LSC M q 1q V 1T V T = δ LSC q 1 q 1 + δq LSC q + δv LSC 1 V 1 + δv LSC ) q V M 1 q V 1T V T + δ 4.5) 8 V1 ZδAV LSC 1 M q 1q AV T + δ V ZδAV LSC M q 1q V 1T A 8 for transverse gauge bosons and δ LSC M q 1q V 1L V L = δ LSC q 1 q 1 + δq LSC q + δφ LSC 1 Φ 1 + δφ LSC ) q Φ M 1 q Φ 1 Φ ϕ ϕ 14 δ 16 ϕ LSC ϕ 16 8π Cew M + δ ϕ ϕ 4π W M ln W MW 6 4π MW λ λ λ 6 C 30 qq ew = Cew q q = 1 + c W ) + 4Q q I q 3Q q)sw 30 4s W c W C ew φ ± φ ± = C ew 33 4s C ew W c W s W CAA ew = Cew AZ = Cew ZA = c W CZZ ew 4.9) 35 s = c W W s 36 W sw c W sw 40 4s 43 W c W 4sW c W ) for longitudinal gauge bosons where Φ 1 and Φ denote the would-be Goldstone bosons corresponding to V 1L and V L respectively. The factors δ LSC are defined as = α ϕ ϕ ln ŝ ) 1 Q ϕ Lem ŝλ M ϕ) } { α I Z ϕ ) ln ŝ ) M ) Z 4.7) where M ϕ and Q ϕ are the mass and relative charge respectively of the field ϕ = q qw ± Zφ ± χandλ is the photon mass regulator. The term L em contains all leading soft-collinear logarithms of pure electromagnetic origin: L em { ) ŝλ Mϕ ) α ŝ M ) = ln ln W M + ln ) W M )} ln ϕ. 4.8) The relevant non-vanishing components of the EW Casimir operator C ew read χχ = 1 + c W W ± W ± = and the relevant squared Z-boson couplings are given by ) I Z ) Q q sw q = I Z q I q 3) = ) I Z c φ = W s W ) I Z χ ) = 1 ) I Z c W = W. 4.10) Finally Q q and Iq 3 denote the relative charge and the third component of the weak isospin of the quark q.

14 S ) /FLA AID:977 Vol. ) [DTD5] P ) NUPHB:m1 v 1.30 Prn:6/11/004; 9:17 npb977 by:is p E. Accomando et al. / Nuclear Physics B ) Subleading soft-collinear corrections The angular-dependent subleading soft-collinear corrections are obtained from formula 3.1) of Ref. [18]. For the production of transverse gauge bosons we get 5 δneutral SSC M q 1q V 1T V T 5 = α [ ŝ ln ) M )] + δ VA ln W π M λ 8 V =AZ W 8 { ) ) ˆt [I V ln q ŝ 1 q 1 I V V + I V 1 V q 1 q I V V ] û [I V + ln V q ŝ 1 q 1 I V V + I V V q q I V V ] } 1 V 1 M q 1q V 1T V T 1 1 from the exchange of neutral virtual gauge bosons and 15 δcharged SSC M q 1q V 1T V T ) 16 = α ŝ ln π M 18 V =W ± V =AZW ± q W 18 { ) ˆt [I V ln q ŝ q 1 I V V V M q q V T V T + I V 1 q q I V V V M q 1q V 1T V ] T ) û [I V + ln q q 1 I V V V M q q V 1T V T + I V q q I V V V M q 1q V T V ] } T ŝ 3 31 δneutral SSC 1q V 1L V L 31 = i 1+Q V 1 )+1+Q V )) δ 33 neutral SSC 1q S 1 S π 35 M λ V =AZ S 36 =χhφ ± W 36 ln 38 q 1 q 1 I V S S M q 1q S S δ 44 charged SSC M q 1q V 1L V L 44 charged M q 1q S 1 S 4.11) 4.1) from the exchange of charged virtual gauge bosons. The charge conjugated of the gauge boson V is denoted by V. The couplings I V 1 are defined in C.3). The couplings I V V V 3 q q given in C.1) involve the quark-mixing matrix and quark mixing requires the sum over q in 4.1). After using the unitarity of the quark-mixing matrix the EW logarithmic corrections have exactly the same dependence on its matrix elements as the lowest order. For the production of longitudinal gauge bosons we find = i 1+Q V 1 )+1+Q V )) α [ ŝ ln ) M )] + δ VA ln W { ) ˆt [I V + I V ŝ q q I V S S M q 1q S 1 S ] ) û [I V + ln q ŝ 1 q 1 I V S S M q 1q S 1 S + I V q q I V S S M q 1q S S ] } 1 from the exchange of neutral virtual gauge bosons and = i 1+Q V 1 )+1+Q V )) δ SSC 4.13)

15 S ) /FLA AID:977 Vol. ) [DTD5] P ) NUPHB:m1 v 1.30 Prn:6/11/004; 9:17 npb977 by:is p. 15 E. Accomando et al. / Nuclear Physics B ) 15 = i 1+Q V 1 )+1+Q V )) α ŝ ln π V =W ± S =χhφ ± q MW { ) ˆt [I V ln q q 1 I V S S M q q S S 1 + I V q q I V S S M q 1q S 1 S ] ) û [I V q ŝ q 1 I V S S M q q S 1 S ] } 5 ŝ 5 + ln 4.14) 7 + I V q q I V S S M q 1q S S 1 7 from the exchange of charged virtual gauge bosons where the couplings I V S S are defined in C.4) Collinear logarithms The single collinear logarithms can be read off from formulas 4.) 4.6) 4.10) 4.) and 4.33) of Ref. [18]. Their contribution to the gauge-boson-production matrix element reads δ C M q 1q V 1T V T = δ C q 18 1 q 1 + δq C q + δv C 1T V 1T + δ C ) q V T V T M 1 q V 1T V T 18 + δ V1 ZδAV C 1T M q 1q AV T + δ V ZδAV C T M q 1q V 1T A in the case of transverse gauge bosons and δ C M q 1q V 1L V L = δ C q 1 q 1 + δ C q q + δ C V 1L V 1L + δ C V L V L ) M q 1 q V 1L V L 4π Cew M 9 W m λ f 9 δ C 31 W 31 T ± W T ± 4π 1s ln W M + ln λ W 33 3π 1sW c W MW ) 35 4π 6s W c W M 37 W 37 δ C 39 W L ± W ± = α [ 1 + c ) W ŝ L 4π s ln 39 W c W M 3 m ) t ŝ M )] W 4sW MW ln m + ln W t λ 41 δ 41 Z C L Z L = α [ 1 + c ) W ŝ 4.17) 4 4π sw ln c W MW 3 m )] t ŝ 4sW MW ln m. t 4 44 ) 4.15) 4.16) in the case of longitudinal gauge bosons. The collinear correction factors for the different particles read δq C σ q σ = δ C q σ q σ = α [ 3 ŝ q σ q σ ln = α δ C Z T Z T = α δ C AZ T = α [ 19 ŝ ) 1 M + Q ) q σ ln W M ))] + ln W ) 19 38sW s4 W ŝ ln 19 + sw ŝ ln M )] W While in Ref. [18] all masses of the order of the EW scale were replaced by M W in the arguments of the large logarithms we keep m t in the logarithm resulting from top-quark loops. )

16 S ) /FLA AID:977 Vol. ) [DTD5] P ) NUPHB:m1 v 1.30 Prn:6/11/004; 9:17 npb977 by:is p E. Accomando et al. / Nuclear Physics B ) Logarithms from parameter renormalization The parameter renormalization gives rise to the so-called counter-term contributions which result from 5 δ PR M q 1q V 1 V = M q 1q V 1 V δe + M q 1q V 1 V δc 4.18) 5 W 6 e c W µ =ŝ 6 where δe and δc W are the counter terms to the electric charge e and the cosine of the weak mixing angle c W = M W /M Z respectively. The mass parameter µ of dimensional regularization is set to ŝ in order not to introduce spurious large logarithms. The counter terms depend on the explicit renormalization conditions. We fix them in the on-shell scheme and obtain in logarithmic approximation: [ ) δe α 11 ŝ 4.19) 3 ln + α MW ) ] δc W = α 19 + s W ŝ ln 4.0) 14 e = 1 4π MW ) 15 c W 4π 1c 17 W MW 17 where α MW ) α = N f M ) C Q f ln W 3π m 1 f t f 1 describes the running of α from zero to the EW scale. 4.. Corrections to gauge-boson decay 3 δm V λ f f = δ Vf f MV λ f f. 4.) δ Zf 36 4π Q f ln m f λ ln M Z λ + ln M Z m + ln M Z λ f + 1 α MZ) δ 39 Wf 4π Q f ln m f λ ln M W λ + ln M W 39 m + ln M W λ f 4 λ ln M W λ + ln M W m + ln M W λ 4 f λ ) Since the decay matrix elements are independent of a large energy scale no energydependent logarithms appear in the corresponding corrections. The only large logarithms result from electromagnetic corrections i.e. from diagrams with photon exchange. For massless fermions these corrections turn out to be proportional to the lowest-order matrix element In the logarithmic approximation the correction factors for V = Z and V = W ± read f = α [ ] f = α { 1 [ + 1 [ Q f ln m f } + Q f Q f ) ln M W + 1 α MW ). ] ] 4.3)

17 S ) /FLA AID:977 Vol. ) [DTD5] P ) NUPHB:m1 v 1.30 Prn:6/11/004; 9:17 npb977 by:is p. 17 E. Accomando et al. / Nuclear Physics B ) Non-factorizable corrections The non-factorizable corrections for a general class of processes are evaluated in Appendix A. For the processes.4) the corresponding correction factor to the lowest-order cross section reads 4 6 Q i Q j θ d i)θ d j) Re { 1 k 1 p i ; k p j ) } δnfdpa virt = α 8 π 8 i=3 j=5 + 4 Q k Q i θ d k)θ d i) Re { p k ; k 1 p i ) } k=1 i=3 + 6 Q k Q j θ d k)θ d j) Re { p k ; k p j ) }) 15 k=1 j=5 15 k 3 1 M 1 )M s 1j ) in DPA where Q i are the relative charges of the fermions corresponding to the external legs and θ d is defined by { +1 for incoming fermions and outgoing antifermions θ d i) = 4.5) 1 for incoming antifermions and outgoing fermions and accounts for the sign difference of the charges of fermions and antifermions. In the high-energy limit we assume MW r for all kinematical invariants r that are not fixed to a certain mass value like s 34 = p 3 + p 4 ) = MV after on-shell projection and keep only logarithmic terms. If we apply this approximation to the non-factorizable corrections we find rather simple expressions for the quantities 1 and : 1 k 1 p i ; k p j ) = 1 s ij s s 1j s i )D0 he k + p j k 1 + p j p i + p j m j M M 1 m i ) k + ln M )M ) 1 ln s i [ + + ln s ][ ] ij λm λm 1 ln s M + ln k M 1 k 1 [ λm l p k ; k l p j ) = ln M l kl ln t ] kl 1. t kj 4.6) 4.7) The invariants are defined as s lj = k l +p j ) t kl = p k k l ) s ij = p i +p j ) t ij = p i p j ) and s = s 1 = p 1 + p ) and have to be calculated using the appropriate on-shellprojected momenta. Note that the invariants s lj t kl and s differ from the corresponding invariants defined in B.3) appearing in A.16) and A.17). It it crucial for the cancellation of the IR singularities that we use here the same definitions as in 4.). In the high-energy limit we are interested in the differences between the definitions disappear. In the D he function which is given in B.30) also the on-shell-projected momenta enter. The original set of momenta enters the non-factorizable corrections only in the terms kl M l where

18 S ) /FLA AID:977 Vol. ) [DTD5] P ) NUPHB:m1 v 1.30 Prn:6/11/004; 9:17 npb977 by:is p E. Accomando et al. / Nuclear Physics B ) M l = Ml im l Γ l are the complex masses of the gauge bosons. We note that we could omit the first two lines of 4.6) in the logarithmic approximation since they do not contain large logarithms. 5. Treatment of soft and collinear photon emission In this section we describe the treatment of soft and collinear photon emission. Soft and collinear singularities are regularized by an infinitesimal photon mass λ and small fermion masses respectively. The masses of the external fermions are denoted by m i pi = m i 0) Phase-space slicing For the evaluation of the real corrections we use the phase-space slicing method where the phase space is divided into singular and non-singular regions. The singular regions are integrated analytically thus allowing the explicit cancellation of the singularities against their counterparts in the virtual corrections. The finite reminder is evaluated by using Monte Carlo techniques. For the actual implementation of this well-known procedure see e.g. Ref. [5]) we closely follow the approach of Ref. []. We divide the five-particle phase space into soft collinear and finite regions by introducing the cut parameters ŝ/ δ s and δ c respectively. The soft region contains photons with energies E γ <δ s = E in the CM frame of the incoming partons. The collinear region contains all photons with E γ > Ebut collinear to any charged fermion i.e. with 1 δ c < cos θ γf < 1 where θ γf is the angle between the charged fermion and the emitted photon in the partonic CM frame. The finite region contains all photons with E γ > Eand 1 < cos θ γf < 1 δ c for all charged fermions. In the soft and collinear regions the squared matrix element M q 1q 4fγ factorizes into the leading-order squared matrix element M q 1q 4f and a soft or collinear factor as long as δ s and δ c are sufficiently small. Also the five-particle phase space factorizes into a four-particle phase space and a soft or collinear part. As a consequence the contribution of the real corrections can be written as q 35 dσ 1 q 4fγ q real = dσ 1 q 4fγ finite + dσ soft + dσ coll. 5.1) Φ 4fγ Φ 4fγ Φ 4f Φ 4f In the soft-photon region we use the soft-photon approximation i.e. the photon fourmomentum k is omitted everywhere but in the IR-singular propagators. Since we neglect k also in the resonant gauge-boson propagators we have to assume E γ < E Γ V V = WZ. In this region dσ q 1q 4fγ can be written as [67] dσ soft = dσ q 1q 4f α 6 6 Q i Q j θ d i)θ d j)i ij π 45 i=1 j= )

19 S ) /FLA AID:977 Vol. ) [DTD5] P ) NUPHB:m1 v 1.30 Prn:6/11/004; 9:17 npb977 by:is p. 19 E. Accomando et al. / Nuclear Physics B ) 19 where I ij = 1 d 3 k p i p j kp i )kp j ). π E 4 γ 4 E γ < E 5 k =E 5 γ λ 1 1 π λ 13 i=1 j=i+1 13 m 16 i m m j i m j m i 16 m s 19 j ij 19 4 apartdσcoll initial originating from initial-state radiation and a part dσcoll final originating from 4 7 dσ coll = dσcoll initial + dσcoll final 5.5) 7 3 dσ initial α coll 3 π Q i 33 i= δ s z 35 dz i δ c i ln 5.6) 36 1 z i m z i dσ q 1q 4f z i p i ) 1 z 36 i i 0 4 dσ final α 4 coll = π Q i + ln δ c 1 ln E 43 i m i=3 i dσ q 1q 4f. 5.7) ) The explicit expression for the integrals I ij can be found in Refs. [78]. Since we only investigate high energies we can assume that the energies E i of the external fermions in the CM frame are large compared with their masses E i m i and keep the fermion masses m i only as regulators so that dσ soft can be written as dσ soft = dσ q 1q 4f α 5 6 { ) E Q i Q j θ d i)θ d j) ln [ s ij ln )] ) 4Ei E j ln + 1 4E ) i ln + 1 4E ) j ln + π 3 + Li 1 4E )} ie j 5.4) where s ij = p i + p j ). In the collinear region we use the collinear limit i.e. the components of the photon four-momentum k perpendicular to the momentum of the collinear fermion are omitted everywhere but in the singular propagators. The collinear cross section is divided into final-state radiation While the emission of photons from the final state does not change the kinematics of the subprocess the initial-state radiation causes a loss of energy of the incoming partons. In the latter case assuming unpolarized incoming partons the cross section reads = ŝ where z i denotes the fraction of energy of the incoming parton i that is left after emission of the collinear photon and ŝ is defined in 4.). The differential cross section for final-state radiation reads 6 [ )][ 3 E 4E )] i + 3 π ) 3 ) )

20 S ) /FLA AID:977 Vol. ) [DTD5] P ) NUPHB:m1 v 1.30 Prn:6/11/004; 9:17 npb977 by:is p. 0 0 E. Accomando et al. / Nuclear Physics B ) 14 Fig. 3. Dependence of the cross section for ν e e + µ ν µ production in the scenario 7.6) on the phase-space slicing cuts. Left: dependence on δ s for δ c = Right: dependence on δ c for δ s = Note that this procedure implicitly assumes that photons within small cones collinear to charged final-state fermions will never be separated from those collinear fermions. Subtracting the soft and collinear cross sections 5.4) and 5.5) from the cross section of the process q 1 q 4fγ 5.1) yields the finite part dσ q 1q 4fγ contributions depend on the cut parameters δ s and δ c. The dependence on these technical cuts cancels in the sum when the cut parameters are chosen to be small enough so that the soft-photon and collinear approximations apply. The variation of the cross section for ν e e + µ ν µ production in the scenario 7.6) with the parameters δ s and δ c is shown in Fig. 3. While the numerical integration becomes unstable for very small cuts and the soft and collinear approximations fail for too large cuts the cross section is independent of the cuts within integration errors for 10 5 δ s 10 3 and 10 6 δ c 10 3.Forthe numerical analysis we have chosen δ s = 10 4 and δ c = finite. The different Definition of finite virtual corrections We fix the finite parts of dσ q 1q 4f entering.1) by adopting the convention of Ref. [] 33 virtsing 33 α 5 dσ q 1q 4f 37 virtsingdpa) = dσ q 1q 4f DPA) Q i Q j θ d i)θ d j) π 37 i=1 j=i [ L s ij m ) i + L sij m ) ] j + Cij + C ji 5.8) with the invariants s ij = p i + p j ) the masses m i of the external fermions their relative charges Q i θ d given by 4.5) L s ij m ) m i = ln i ln λ + ln λ 1 s ij s ij s ij ln m i + 1 s ij ln m i s ij 5.9)

21 S ) /FLA AID:977 Vol. ) [DTD5] P ) NUPHB:m1 v 1.30 Prn:6/11/004; 9:17 npb977 by:is p. 1 E. Accomando et al. / Nuclear Physics B ) 1 and the constant terms C ij defined as C ij = π + if i and j are incoming 3 C ij = π 1 if i is incoming and j is outgoing 6 C ij = π + 3 if i is outgoing and j is incoming C ij = π 5.10) if i and j are outgoing. Of course in logarithmic approximation the constants C ij could be omitted Absorption of mass singularities in parton distributions After combining real and virtual corrections the Oα)-corrected partonic cross section still contains mass-singular terms of the form α ln m qk involving the masses m qk of the incoming partons. These terms arise from collinear emission of photons in the initial state. In analogy to the MS factorization scheme for next-to-leading-order QCD corrections we absorb these collinear singularities into the quark distributions. To this end we replace the parton-distribution functions in.) as 3 3 Φ qh xq ) Φ qh xq ) α 1 dz 4 π Q q z Φ qh z Q 4 x [ 1 + z ln Q 5.11) 7 1 z m ln1 z) 1 q + 7 where the usual [ ] + prescription is defined by dz [ fz) ] + gz) = 1 dz f z)gz) 3 x x The replacement 5.11) amounts to a contribution 36 dσ pdf = α Q 36 i π 37 i= [ z )] 39 dz ln Q 5.13) 40 1 z m ln1 z) 1 dσ q 1q 4f z i p i ) 40 i dσ q 1q 4f virtsing all IR and collinear singularities i.e. all lnλ ) and lnm i ) terms cancel x dz f z)g1). )] ) 5.1) that has to be added to the partonic cross section. When adding dσ pdf to dσ soft dσ coll and The absorption of the collinear Oα) singularities into the parton distributions requires also the inclusion of the corresponding corrections into the DGLAP evolution of these

22 S ) /FLA AID:977 Vol. ) [DTD5] P. 1-47) NUPHB:m1 v 1.30 Prn:6/11/004; 9:17 npb977 by:is p. E. Accomando et al. / Nuclear Physics B ) distributions and into their fit to experimental data. At present time there exist no published PDFs in which the photonic Oα) corrections are consistently included. An approximative inclusion of the Oα) corrections to the DGLAP evolution shows [9] that the impact of these corrections is below about 1%. Therefore these effects are below our aimed accuracy of a few per cent and can be neglected. 6. Setup of the numerical analysis We consider three classes of processes: i) pp lν l l l +γ) ii) pp l ll l +γ) iii) pp l ν l ν l l +γ) where ll = e or µ. In our notation lν l indicates both l ν l and l + ν l. The first class is characterized by three isolated charged leptons plus missing energy in the final state. This channel includes WZ production as intermediate state. The second class is purely mediated by ZZ production while the third class is related to W ± W production. When there is a unique flavor in the final state l = l the third process receives also a ZZ contribution. All above-mentioned processes are described by.). Since the two incoming hadrons are protons and we sum over final states with opposite charges we find dσ pp P 1 P p f ) 1 = dx 1 dx [ Φ Dp x1 Q ) Φ Up x Q ) d ˆσ DU x 1 P 1 x P p f ) 8 0 U=uc D=ds 8 + ΦŪp x1 Q ) Φ Dp x Q ) d ˆσ ŪD x 1 P 1 x P p f ) + Φ Dp x Q ) Φ Up x1 Q ) d ˆσ DU x P x 1 P 1 p f ) + ΦŪp x Q ) Φ Dp x1 Q ) d ˆσ ŪD x P x 1 P 1 p f ) ] 33 for WZ production and dσ pp P 1 P p f ) = dx 1 dx [ Φ qp x1 Q ) Φ qp x Q ) d ˆσ qq x 1 P 1 x P p f ) 40 0 q=udcs 40 + Φ qp x Q ) Φ qp x1 Q ) d ˆσ qq x P x 1 P 1 p f ) ] ) 6.) for ZZ and WW production in leading order of QCD. Since the initial state is forward backward symmetric for two incoming protons this cross section is forward backward symmetric.