We determine the accuracy of various approximations to the O(alpha)
corrections for on-shell W-pair production. While an approximation based on the
universal corrections arising from initial-state radiation, from the running of
alpha, and from corrections proportional to m_t^2 fails in the Linear-Collider
energy range, a high-energy approximation improved by the exact universal
corrections is sufficiently good above about 500GeV. These results indicate
that in Monte Carlo event generators for off-shell W-pair production the
incorporation of the universal corrections is not sufficient and more
corrections should be included.Comment: 7 pages, late

Denner, A.

Dittmaier, S.

English

arXiv

We determine the accuracy of various approximations to the O(alpha) corrections for on-shell W-pair production. While an approximation based on the universal corrections arising from initial-state radiation, from the running of alpha, and from corrections proportional to m_t^2 fails in the Linear-Collider energy range, a high-energy approximation improved by the exact universal corrections is sufficiently good above about 500GeV. These results indicate that in Monte Carlo event generators for off-shell W-pair production the incorporation of the universal corrections is not sufficient and more corrections should be included.We determine the accuracy of various approximations to the O(alpha) corrections for on-shell W-pair production. While an approximation based on the universal corrections arising from initial-state radiation, from the running of alpha, and from corrections proportional to m_t^2 fails in the Linear-Collider energy range, a high-energy approximation improved by the exact universal corrections is sufficiently good above about 500GeV. These results indicate that in Monte Carlo event generators for off-shell W-pair production the incorporation of the universal corrections is not sufficient and more corrections should be included

Denner, Ansgar

CERN Document Server

CERN-TH/97-128Approximations for W-Pair Productionat Linear-Collider EnergiesyA. Denner1 and S. Dittmaier21 PSI, Wu¨renlingen und Villigen, Switzerland2 CERN, Theory Division, Geneva, SwitzerlandAbstractWe determine the accuracy of various approximations to the O() corrections for on-shell W-pair production. While an approximation based on the universal corrections aris-ing from initial-state radiation, from the running of , and from corrections proportionalto m2t fails in the Linear-Collider energy range, a high-energy approximation improvedby the exact universal corrections is suciently good above about 500 GeV. These re-sults indicate that in Monte Carlo event generators for o-shell W-pair production theincorporation of the universal corrections is not sucient and more corrections should beincluded.CERN-TH/97-128June 1997yContribution to the Proceedings of the Joint ECFA/DESY Study: Physics and Detectors for a LinearCollider, Frascati, London, Munich, Hamburg, February 5 to November 22, 1996.Approximations for W-Pair Productionat Linear-Collider EnergiesA. Denner2 and S. Dittmaier21 PSI, Wu¨renlingen und Villigen, Switzerland2 CERN, Theory Division, Geneva, Switzerland1 IntroductionOne of the most important processes for testing the Minimal Standard Model (MSM)in the future is the production of W pairs [1{3]. It allows a direct study of the non-Abeliantriple gauge couplings in the clean environment of e+e− collisions. The sensitivity of thisprocess to anomalous gauge-boson couplings grows strongly with energy owing to the factthat these couplings in general spoil the unitarity cancellations that are present in theMSM for longitudinal gauge bosons. Consequently, more stringent limits on non-standardcouplings can be obtained at higher energies.In order to test for anomalous couplings, the cross-section for W-pair production hasto be known with an accuracy of 1% or better. At this level, the inclusion of radiativecorrections is mandatory. However, so far the complete electroweak O() corrections too-shell W-pair production are not available and the present Monte Carlo event generatorsinclude only the known leading universal corrections.In order to assess the theoretical uncertainty inherent in these event generators, on-shell W-pair production can be used as guideline. The corresponding cross-section inthe MSM is known, including the complete set of electroweak O() corrections [4]. Thesize of the non-leading O() corrections in the on-shell case should provide a reasonableestimate for the corresponding left-out non-leading corrections in the o-shell case.In this short article we discuss the quality of an improved Born approximation (IBA)for on-shell W-pair production that is based on the known leading universal corrections.We concentrate on the Linear-Collider (LC) energy range and emphasize the dierencesto the LEP2 case studied in Ref. [5]. In addition, we compare the full one-loop resultswith a form-factor approximation (FFA), which corresponds to the best possible improvedBorn approximation, and with a consistent high-energy approximation (HEA). We alsoinclude some remarks on the radiative corrections to W-pair productions in photon{photon collisions.2 Form-factor approximationWhereas in the lowest-order matrix element for e+e− ! W+W− only three dierenttensor structures occur, at O() twelve independent tensor structures are required, each1of which is associated with an independent invariant function. The dominant radiativecorrections, such as those that are related to UV, IR or mass singularities, in general havefactorization properties and are atO() restricted to those invariant functions that appearalready at lowest order. Therefore, the contributions of the other invariant functionsshould be relatively small. Indeed, a numerical analysis reveals that in a suitably chosenrepresentation for the basic set of independent matrix elements only the three Born-likeinvariant functions plus one extra right-handed piece are relevant for a suciently goodapproximation [6,7]. This suggests an approximation for the matrix element of the formMapp =MIFI +MQFQ; (1)where|following the notation of Ref. [1]|MI andMQ denote the tensor structures asso-ciated with the charged-current coupling and the electromagnetic coupling, respectively,in the lowest-order matrix element, and  is the electron helicity. The term involvingF+I is only needed for right-handed electrons in the LEP2 energy region. Neglecting theother invariant functions in the basis of Ref. [6] denes the FFA. The form factors F I andF Q are independent of the W polarizations and include all corrections that are relatedto the Born structure. However, they are both energy- and angular-dependent and theirevaluation is as complicated as the exact one-loop calculation.3 Improved Born approximationIn order to construct an IBA one has to specify simple expressions that reproduce theinvariant functions F I;Q with sucient accuracy. In the LEP2 energy region the followingexpressions provide a reasonable ansatz [6]F I;IBA ="2p2GM2W +42s2W4(1− 2)2#−;F Q;IBA ="4(s) + 44(1− 2)2#: (2)As usual, s, t, and u denote the Mandelstam variables, and  =q1− 4M2W=s is theW-boson velocity in the centre-of-mass system.The terms containing G and (s) incorporate all leading universal corrections as-sociated with the running of  and the corrections / m2t=M2W associated with the parameter. As these are linked to the renormalization of the electric charge at zero mo-mentum transfer and of the weak mixing angle, they only contribute to the structuresalready present at lowest order. The 1= term describes the eect of the Coulomb singu-larity, which is only relevant close to threshold and can be omitted at high energies. Thefactor (1−2)2 is introduced by hand to restrict the 1= contribution to the threshold re-gion. The ansatz (2) includes all leading logarithms originating from light fermions and allcorrections / m2t=M2W. As has been studied in Ref. [6], contributions /  log(mt=MW)and /  log(MH=MW) that occur when the masses are heavy and are not covered by (2)have a negligible numerical impact.In addition to the contributions described so far, one has to include the leading log-arithmic QED corrections. These can be calculated using the structure-function method2[8]. They comprise all contributions /  log(me=MW). Since we are interested in approx-imations for the virtual corrections, we restrict ourselves to the inclusion of real photoniccorrections in the soft-photon approximation. In this approximation the leading logarith-mic QED corrections to the cross-section are given by the factorγ = 2lnsm2eln2Eps+34lnQ2m2e− ln2Eps2− 2 lnM2W − tM2W − u−s− 2M2Wsln1 + 1−  ; (3)where E is the soft-photon cuto energy, and Q2 is a scale that is not determined inthe leading-log approximation and set to Q2 = s for the numerics.We note that all present event generators for o-shell W-pair production include onlythose radiative corrections that are included in our IBA.4 High-energy approximationAt high energies the full O() corrections can be approximated by exploiting the factthat s; jtj; juj  M2W;Z  me;;;d;u;s;c;b. Such an approximation has been constructed forthe process e+e− !W+W− by a systematic expansion of the exact one-loop correctionsfor arbitrary mt and MH [9,10]. For intermediate energies (500 GeV to 2 TeV) the high-energy approximation is improved [9] by exactly taking into account the leading universalcorrections of the IBA, dened in Section 3, apart from the Coulomb singularity. As theHEA has already been discussed in Ref. [11], we summarize here only its most importantfeatures.The dominant corrections at high energies result from terms proportional to thehigh-energy leading logarithms  log(q21=M2W) log(q22=M2W) with q2i = s; jtj; juj. Thesecorrections are non-universal and arise from vertex and box diagrams. For longitudi-nal gauge bosons, additional sizeable corrections appear, depending on the values ofthe top-quark and Higgs-boson masses. In the limit s  m2t ;M2H  M2W, terms con-taining m2t=M2W log(mt=MW) or M2H=M2W arise as a consequence of incomplete screen-ing. In the limit m2t ;M2H  s  M2W, corrections of the form s=M2W log(mt=MW) ands=M2W log(MH=MW) result from delayed unitarity eects.5 Numerical resultsFor the numerical evaluation we used the parameters given in Ref. [5]. The correspond-ing values for the W-boson, Higgs-boson, and top-quark masses are MW = 80:26 GeV,MH = 300 GeV, and mt = 165:3 GeV.The unpolarized lowest-order cross-sections integrated over the angular region 10   170 and the dierential cross section for three angles are listed in Table 1 at variousenergies. The quality app is dened asapp =app − exactBornexact ; (4)where exact is the full virtual and soft-photonic O() correction, app the correspond-ing approximation and Bornexact the exact lowest-order cross-section. When calculating the3ps=GeV  B= fb IBA=% FFA=% HEA=%161 (10; 170) 3753:2 1:5 0:00 3710 367:0 1:6 0:00 3690 300:7 1:4 0:00 37170 250:0 1:3 0:00 37175 (10; 170) 15591 1:3 0:03 1210 3380 1:7 0:00 1090 1001 1:0 0:05 12170 439 0:59 0:00 12200 (10; 170) 17107 1:5 0:01 3:710 6463 1:8 0:00 2:390 812 1:4 0:02 4:7170 255 1:3 0:00 3:8500 (10; 170) 4413:1 4:7 −0:06 −0:8510 11604:4 1:9 0:00 −0:6790 75:4 10 −0:29 −0:05170 6:5 14 −0:19 3:51000 (10; 170) 1084:3 11 0:06 0:2110 3292:3 3:9 0:00 1:190 16:7 23 0:08 0:54170 0:6 28 −0:77 6:42000 (10; 170) 267:57 22 0:12 0:1710 823:35 9:7 0:02 0:6490 4:03 39 −0:16 0:34170 0:09 46 −2:3 5:4Table 1: Quality of the approximations for various energiesdierence app − exact, care has to be taken that the leading higher-order correctionsare treated in the same way in the two expressions.The FFA is excellent; the corresponding error is well below the per-cent level, wheneverthe cross-section is sizeable. At high energies and large scattering angles, where the crosssection is extremely small, larger dierences appear. This signals a strong dominance ofthe Born structure and demonstrates that improved Born approximations are possible.Above LEP2 energies the IBA is reasonable only for extremely small scattering angles,and deviates for most angles by several per cent easily. This shows that approximationsbased on the leading universal corrections are certainly not sucient for a comparisonwith experiment at high energies. The HEA on the other hand is good at energies above4about 500 GeV for sizeable cross-sections, i.e. its deviation from the full O() correctionsis below 1%.6 Remarks on γγ !W+W−Another important process for the study of anomalous gauge-boson couplings is W-pair production in photon{photon collisions. The corresponding cross-section approaches80 pb at high energies and is thus much larger than the one for W-pair production ine+e− collisions. The corrections to the corresponding on-shell process have been studiedin Refs. [12] and briefly summarized in Ref. [13]. Almost all leading universal correctionsare absent. There are no eects from the running of , no corrections proportional to m2t ,and no enhanced logarithms arising from soft or collinear photons. Not even corrections/  log(MH=MW) arise in the limit of a large Higgs-boson mass. The Coulomb singularityis unimportant at LC energies. Thus, an IBA that is equivalent to the one discussed fore+e− ! W+W− at high energies becomes trivial, i.e. equal to the Born approximation.All dominating corrections are of non-universal origin and their size, which reaches severalper cent at 500 GeV and increases with energy, indicates that they cannot be neglected.Unlike for e+e− ! W+W−, no form-factor and high-energy approximations have beenworked out for γγ !W+W− so far.7 ConclusionsAt high energies the O() corrections to on-shell W-pair production in e+e− collisionsare large [O(10{40%)] and cannot be neglected. They are not dominated by universalcorrections from initial-state radiation, from the running of , and from corrections pro-portional to m2t , but by contributions like  log2(s=M2W), which originate from the bosonic(vertex and box) corrections. For on-shell W-pair production they are well approximatedby a consistent high-energy expansion that is improved by the exact leading universalcorrections in order to yield a sucient accuracy at intermediate energies.Present event generators for o-shell W-pair production include only the universal cor-rections. The theoretical uncertainty from neglecting non-universal corrections stronglyincreases with energy. While this uncertainty is only 1{2% at LEP2 energies it amounts toseveral tens of per cent in the TeV range. Thus, the non-universal corrections are requiredfor adequate theoretical predictions. This applies in the same way to γγ !W+W− whereessentially all non-universal corrections are absent.A rst step towards the inclusion of non-universal corrections into Monte Carlo eventgenerators for e+e− ! 4 fermions is provided by the fermion-loop scheme [14], wheregauge-boson widths are consistently introduced by the resummation of the fermion-loopcorrections. The incorporation of the bosonic corrections, which dominate the non-universal corrections at high energies, is more complicated owing to their complexityand problems with gauge invariance. The most promising approach consists in an ex-pansion according to the degree of resonance. In this approach the corrections are to alarge extent given by the known corrections for on-shell W bosons or, at LC energies, bythe corresponding high-energy approximation. For a genuine on-shell approach to W-pairproduction and the subsequent W decay, the O() corrections have already been includedin an event generator (see e.g. Ref. [15]). The combination of o-shell eects and O()corrections is still under investigation.5References[1] W. Beenakker and A. Denner, Int. J. Mod. Phys. A9 (1994) 4837.[2] Physics at LEP2, eds. G. Altarelli, T. Sjo¨strand and F. Zwirner (CERN/96-01,Geneva, 1996).[3] E. Accomando et al., DESY-97/100, hep-ph/9705442.[4] M. Bo¨hm et al., Nucl. Phys. B304 (1988) 463;J. Fleischer, F. Jegerlehner and M. Zra lek, Z. Phys. C42 (1989) 409;W. Beenakker, K. Ko lodziej and T. Sack, Phys. Lett. B258 (1991) 469;W. Beenakker, F.A. Berends and T. Sack, Nucl. Phys. B367 (1991) 287;K. Ko lodziej and M. Zra lek, Phys. Rev. 43 (1991) 3619;J. Fleischer, F. Jegerlehner and K. Ko lodziej, Phys. Rev. D47 (1993) 830.[5] W. Beenakker et al., hep-ph/9602351, in Ref. [2], p. 79.[6] M. Bo¨hm, A. Denner and S. Dittmaier, Nucl. Phys. B376 (1992) 29; E: Nucl. Phys.B391 (1993) 483.[7] J. Fleischer et al., Nucl. Phys. B378 (1992) 443; E: Nucl. Phys. B426 (1994) 246.[8] E.A. Kuraev and V.S. Fadin, Sov. J. Nucl. Phys. 41 (1985) 466;G. Altarelli and G. Martinelli, in Physics at LEP, eds. J. Ellis and R. Peccei(CERN/86-02, Geneva, 1986), Vol. 1, p. 47;W. Beenakker, F.A. Berends and W.L. van Neerven, in Radiative Corrections fore+e− Collisions, ed. J.H.Ku¨hn (Springer, Berlin, 1989), p. 3.[9] W. Beenakker et al., Nucl. Phys. B410 (1993) 245.[10] W. Beenakker et al., Phys. Lett. B317 (1993) 622.[11] W. Beenakker et al., in: e+e− Collisions at 500 GeV: The Physics Potential, ed.P. Zerwas (DESY 93-123C, Hamburg, 1993), p. 153.[12] A. Denner, S. Dittmaier and R. Schuster, Phys. Rev. D51 (1995) 4738 and Nucl.Phys. B452 (1995) 80;G. Jikia, FREIBURG-THEP-96-23, hep-ph/9612380.[13] A. Denner, S. Dittmaier and R. Schuster, hep-ph/9601355, in: e+e− Collisions at TeVEnergies: The Physics Potential, ed. P. Zerwas (DESY 96-123D, Hamburg, 1996),p. 233.[14] W. Beenakker et al., NIKHEF 96-031, PSI-PR-96-41, hep-ph/9612260, to appear inNucl. Phys. B.[15] S. Jadach et al., UTHEP-96-1101, hep-ph/9705429.6

Approximations for W-Pair Production at Linear-Collider Energies

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