We discuss the simultaneous resummation of threshold and recoil enhancements
to partonic cross sections due to soft radiation. Our method is based on a
refactorization of the parton cross section near its partonic threshold. It
avoids double counting, conserves the flow of partonic energy, and reproduces
either threshold or recoil resummation when the other enhancements are
neglected.Comment: 4 pages, LaTeX, 2 figs. Contributed to 30th International Conference
  on High-Energy Physics (ICHEP 2000), Osaka, Japan, 27 Jul - 2 Aug 200

Laenen, Eric

Sterman, George

Vogelsang, Werner

English

arXiv

We discuss the simultaneous resummation of threshold and recoil enhancements to partonic cross sections due to soft radiation. Our method is based on a refactorization of the parton cross section near its partonic threshold. It avoids double counting, conserves the flow of partonic energy, and reproduces either threshold or recoil resummation when the other enhancements are neglected

Utrecht University Repository

arXiv:hep-ph/0010184v1  17 Oct 2000NIKHEF/00-30YITP-SB-00-65BNL-HET-00/39RBREC 145COMBINED RECOIL AND THRESHOLD RESUMMATION FOR HARDSCATTERING CROSS SECTIONSERIC LAENENNIKHEF Theory Group, Kruislaan 4091098 SJ Amsterdam, The NetherlandsGEORGE STERMANC.N. Yang Institute for Theoretical Physics, SUNY Stony BrookStony Brook, New York 11794 – 3840, U.S.A.Physics Department, Brookhaven National Laboratory,Upton, NY 11973, U.S.A.WERNER VOGELSANGRIKEN-BNL Research Center, Brookhaven National Laboratory,Upton, NY 11973, U.S.A.We discuss the simultaneous resummation of threshold and recoil enhancements to partonic crosssections due to soft radiation. Our method is based on a refactorization of the parton cross sectionnear its partonic threshold. It avoids double counting, conserves the flow of partonic energy, andreproduces either threshold or recoil resummation when the other enhancements are neglected.1 IntroductionA large class of hard-scattering cross sec-tions in QCD may be factorized into convo-lutions of parton distributions and fragmen-tation functions with hard-scattering func-tions. 1 The latter are computed in per-turbation theory. Intermediate infinities as-sociated with virtual and emitted soft glu-ons cancel in their higher-order corrections,but finite remnants, assuming the form ofplus-distributions and delta-functions, maylead to large enhancements when integratedagainst the smooth functions in the convo-lutions. In physical terms, soft gluon radi-ation affects the hard-scattering process byreweighting the relative importance of near-threshold partonic subprocesses to the phys-ical cross section, and by providing the finalstate with overall transverse momentum QTthrough its recoil. When the conservationof energy is taken into account, the sign ofthe cumulative effect is a priori unclear: en-hancement effects of recoil may be offset bythe suppression due to the extra energy re-quired to produce the desired final state plusrecoil radiation. A combined and consistentanalysis is required 2 and in what follows wesketch our approach to this problem 3,4.aaContributed to the 30th International Conference onHigh-Energy Physics (ICHEP 2000), Osaka, Japan,27 Jul - 2 Aug 200012 Electroweak annihilationThese processes are characterized at lowestorder by ab → V , with a, b partons andV an electroweak final state of mass Q andtransverse momentum QT . Because of theirrelative simplicity and phenomenological in-terest, these processes have been studied in-tensely for resummation purposes. Partonicthreshold is at z ≡ Q2/ŝ = 1 and QT = 0,and the singular functions to be resummedare plus-distributions in 1 − z and QT /Q.Our method generalizes that of Ref. 5, andorganizes these distributions in the cross sec-tion. This is done by refactorizing partoniccross sections into short-distance functions,σ(H)ab , sensitive only to the hardest scale, Q,and matrix elements of appropriate operatorsthat absorb all dependence on (1 − z)Q andQT , to leading power in 1− z.dσab→VdQ2d2QT=1Sσ(H)ab→V (Q2, αs(Q2))×∫dxad2ka Ra/a(xa,ka, Q)×∫dxbd2kb Rb/b(xb,kb, Q)×∫dwsd2ks Uab(ws, Q,ks)×δ(1−Q2/S − (1− xa)− (1 − xb)− ws)× δ2 (QT − ka − kb − ks) + Yj , (1)whereRf/f (x,k, 2p0) =14√2Nc∫dλ2πd2b(2π)2e−iλxp0+ib·k×〈f(p)|q̄f(λ+,b, 0−)γ+qf (0)|f(p)〉 (2)is a partonic quark density at fixed energyand transverse momentum, and Uab a purelyeikonal function, depending on parton veloc-ities β, and on the soft radiation’s energyws Q. The x’s and k’s are defined by (2).The delta functions in the triple convolutionof (1) relate the singular behavior of the var-ious functions, and Yj represents the match-ing to finite order. The all-order behavior ofthe above functions and the cross section canbe analyzed through their eikonalized equiv-alents. The eikonal cross section can be ex-pressed as an exponent, Eab, an integral overdiagrammatically-defined functions referredto as “webs”, 6 wab,Eab = 2∫ Q d4−2ǫkΩ1−2ǫ× wab(k2,k · βk · β′β · β′ , µ2, αs(µ2), ǫ)×(e−N(k0/Q)−ik·b − 1). (3)The integral is over the energy and transversemomentum contributed by each web to thefinal state. The k-dependence of wab followsfrom the invariance of the eikonal cross sec-tion under rescalings of β and β′.3 Single-particle inclusiveFor definiteness we consider prompt photonproduction, but the method sketched belowis more general 3. A similar refactorization asin (1) holds for single-particle inclusive crosssections at high pT . In particular, it containsthe same Rf/f functions. The argumentssupporting this refactorization are somewhatmore involved than for (1), but reveal thatonly initial state radiation contributes to QT(the transverse momentum of the 2 → 2 par-ton cms frame) 7. In the inclusive cross sec-tion, final-state interactions require thresholdresummation only. In contrast to electroweakannihilation, QT is an unobserved variable,akin to 1 − z. Thus, the jointly resummedprompt photon pT spectrum may be writtenas an integral over QT of a “profile function”3Pij(N,QT , Q) =∫d2be−ib·QT eEij→γk (4)where the exponential exhibits the joint re-summation, asp3Tdσ(resum)AB→γdpT=2×∑ijp4T8πS2∫CdN2πiφ̃i/A(N)φ̃j/B(N)×∫ 10dx̃2T(x̃2T)N |Mij(x̃2T )|2√1− x̃2T×C(ij→γk)δ (αs, x̃2T )∫d2QT(2π)2Θ(µ̄−QT )×(S4p′T2)N+1Pij(N,QT , Q) (5)with x̃2T = 4|pT −QT /2|2/ŝ, φ̃i/A(N) Mellinmoments of the parton distributions, |Mij |2the Born amplitudes, and µ̄ a cut-off restrict-ing QT to sufficiently small values for re-summation to be relevant. The C(ij→γk)δ areinfrared safe coefficient functions, which in-clude short-distance dynamics at the scaleQ. For notational simplicity we have sup-pressed all factorization and renormalizationscale dependence. Cross sections computedon the basis of (5) are shown in Fig. 1 asfunctions of QT at fixed pT . The kinematicsare those of the E706 experiment; 9 see 3 fordetails of the calculation, in particular thoseregarding the evaluation of the b-integral in(4). The dashed lines are dσ(resum)pN→γX/dQTdpT ,with recoil neglected by fixing p′T = pT , thusshowing how each QT contributes to thresh-old enhancement. The solid lines show thesame, but now including the true recoil factor(S/4p′T2)N+1. The resulting enhancement isclearly substantial. For small pT , the en-hancement simply grows with QT , while forpT above 5 GeV it has a dip at about QT = 5GeV, which remains substantially above zero.This makes it difficult to determine µ̄ on thisbasis alone.So far, the numerical results given in 3are primarily to be regarded as illustrations,rather than quantitative predictions. Thisapplies in particular to the resummed QT -integrated cross section, shown in Fig. 2 forpT ≥ 3.5 GeV and µ̄ = 5 GeV. These figuresdemonstrate, however, the size of the addi-tional enhancement that recoil can produce,and its potential phenomenological impact.Figure 1. dσpN→γX/dQT dpT at√s = 31.5 GeV,as a function of QT for various values of photon pT .Dashed lines are computed without recoil (p′T= pTin (5)), solid lines are with recoil.Figure 2. Ed3σpN→γX/dp3 for pN collisions at√s =31.5 GeV. The dotted line represents the full NLOcalculation, while the dashed and solid lines respec-tively incorporate pure threshold resummation 8 andthe joint resummation described in this paper. Datahave been taken from 9.3AcknowledgementsThe work of E.L. is part of the research pro-gram of the Foundation for Fundamental Re-search of Matter (FOM) and the NationalOrganization for Scientific Research (NWO).The work of G.S. was supported in partby the National Science Foundation, grantPHY9722101. G.S. thanks Brookhaven Na-tional Laboratory for its hospitality. W.V.is grateful to RIKEN, Brookhaven Na-tional Laboratory and the U.S. Depart-ment of Energy (contract number DE-AC02-98CH10886) for providing the facilities essen-tial for the completion of this work.References1. J.C. Collins, D.E. Soper and G. Ster-man, in Perturbative Quantum Chromo-dynamics, ed. A.H. Mueller (World Sci-entific, Singapore, 1989).2. H.-n. Li, Phys. Lett. B454, 328 (1999),hep-ph/9812363.3. E. Laenen, G. Sterman and W. Vogel-sang, Phys. Rev. Lett.84, 4296 (2000),hep-ph/0002078.4. E. Laenen, G. Sterman and W. Vogel-sang, hep-ph/0010080.5. G. Sterman, Nucl. Phys. B281, 310(1987); S. Catani and L. Trentadue,Nucl. Phys. B327, 323 (1989), B353,183 (1991).6. J.G.M. Gatheral, Phys. Lett. B133, 9(1983); G. Sterman, in: Perturbativequantum chromodynamics, proc. of Tal-lahassee conference (1981), AIP Confer-ence Proceedings, eds. D. W. Duke andJ. F. Owens, p.22; J. Frenkel and J.C.Taylor, Nucl. Phys. B246, 231 (1984).7. H.-L. Lai and H.-n. Li, Phys. Rev. D58,114020 (1998), hep-ph/9802414.8. S. Catani, M.L. Mangano, P. Nason,C. Oleari andW. Vogelsang, JHEP 9903(1999) 025, hep-ph/9903436.9. L. Apanasevich et al., E706 Collab.,Phys. Rev. Lett. 81, 2642 (1998), hep-ex/9711017.4

Combined recoil and threshold resummation for hard scattering cross sections

Laenen, E.

Sterman, G.

Vogelsang, W.

UNT Digital Library

11YJ-00RBRC-145BNL-67842NIKHEF/oo-30YITP-SB-OO-65BNL-HET-oo/39COMBINED RECOIL AND THRESHOLD RESUMMATION FOR HARDSCATTERING CROSS SECTIONSERIC LAENENNIKHEF Theory Group, Kruislaan 4091098 SJ Amsterdam, The NetherlandsGEORGE STERMANC.N. Yang Institute -for Theoretical Physics, SUNY Stony BrookStony Brook, New York 11794-3840, U.S.A.Physics Department, Brookhaven National LaboratoryjUpton, NY 11973, U.S.A.WERNER VOGELSANGRIKEN-BNL Research Center, Brookhaven National Laboratory,Upton, NY 11973, U.S.A.We discuss the simultaneous resummation of threshold and recoil enhancements to partonic crosssections due to soft radiation. Our method is based on a refactorization of the parton cross sectionnear its partonic threshold. It avoids double counting, conserves the flow of partonic energy, andreproduces either threshold or recoil resummation when the other enhancements are neglected.1 IntroductionA large class of hard-scattering cross sec-tions in QCD may be factorized into convo-lutions of parton distributions and fragmen-tation functions with hard-scattering func-tions. 1 The latter are computed in per-turbation theory. Intermediate infinities as-sociated with virtual and emitted soft glu-ons cancel in their higher-order corrections,but finite remnants, assuming the form ofplus-distributions and delta-functions, maylead to large enhancements when integratedagainst the smooth functions in the conv~lutions. In physical terms, soft gluon radi-ation affects the hard-scattering process byreweighing the relative importance of near-threshold partonic subprocesses to the phys-ical cross section, and by providing the finalstate with overall transverse momentum Q~through its recoil. When the conservationof energy is taken into account, the sign ofthe cumulative effect is a p,riori unclear: en-hancement effects of recoil may be offset bythe suppression due to the extra energy re-quired to produce the desired final state plusrecoil radiation. A combined and consistentanalysis is required 2 and in what follows wesketch our approach to this problem 3’4.“aContributed to the 30th International Conference onHigh-Energy Physics (ICHEP 2000), Osaka, Japan,27 Jrd -2 Aug 2000HJx 4p;2 pij(N, QT, Q)(5)with i; = 41p~ – QT/’212/~, &iA (N) Mellinmoments of the parton distributions, lA4ij 12the Born amplitudes, and ji a cut-off restrict-ing QT to sufficiently small values for re-summation to be relevant. The C~i~+Tk) areinfrared safe coefficient functions, which in-clude short-distance dynamics at the scaleQ. For notational simplicity we have suppressed all factorization and renormalizationscale dependence. Cross sections computedon the basis of (5) are shown in Fig. 1 asfunctions of QT d fixed ~. The kinematicsare those of the E706 experiment; 9 see 3 fordetails of the calculation, in particular thoseregarding the evaluation of the b-integral in(4). The dashed lines are du~~~~~/dQTd~,with recoil neglected by fixing p~ = PT, thusshowing how each QT contributes to thresh-old enhancement. The solid lines show thesame, but now including the true recoil factor(S/4p~2)N+1. The resulting enhancement isclearly substantial. For small ~, the en-hancement simply grows with QT, while for~ above 5 GeV it haa a dip at about QT = 5GeV, which remains substantially above zero.This makes it difficult to determine ji on thisbasis alone.So far, the numerical results given in 3are primarily to be regarded as illustrations,rather than quantitative predictions. Thisapplies in particular to the resummed QT-integrated cross section, shown in Fig. 2 forpT >3.5 GeV and p = 5 GeV. These figuresdemonstrate, however, the size of the addi-tional enhancement that recoil can produce,and its potential phenomenological impact.3mh-nc+vz -\\\\‘.\00 *46‘;mim0246 024Q, (WV) ‘Figure 1. dup~+yx/dQTdpT at 6 = 31.5 GeV,as a function of QT for various values of photon pT.Dashed lines are computed without recoil (Pi = pTin (5)), solid lines are with recoil.1 ,1),11),11,14 6 8 10 12PT (Gev)Figure 2, Ed3upN+7x/dp3 for pN collisions at W =31.5 GeV. The dotted line represents the full NLOcalculation, while the dmhed and solid lines respec-tively incorporate pure threshold resummation 8 andthe joint resummation described in this paper. Datahave been taken from 9.2 Electroweak annihilationThese processes are characterized at lowestorder by ab + V, with a, b partons andV an electroweak final state of mass Q andtransverse momentum Q~. Because of theirrelative simplicity and phenomenological in-terest, these processes have been studied in-tensely for resummation purposes. Partonicthreshold is at z s Q2/s = 1 and Q~ = O,and the singular functions to be resummedare plus-distributions in 1 – z and QT /Q.Our method generalizes that of Ref. 5, andorganizes these distributions in the cross sec-tion. This is done by refactorizing partoniccross sections into short-dist ante functions,a~fl), sensitive only to the hardest scale, Q,and matrix elements of appropriate operatorsthat absorb all dependence on (1 – z)Q andQT, to leading power in I – z.du.b+v 1 (H)dQ2d2QT = ~ U.b+v(Q2, ~s(Q2))x/dq,d2kb Rb/b(*6, kb, Q)/X dw,d2k, ~.b(~,, Q;k,)xd(l –Q2/S– (1 –Za) –(1 –z~) –w. )xJ2(Q~—k~—kb —ks)+- Yj, “(l)whereRf/f(z, k, 2PO) =1[d~ d2b -i AzpO+ib.k——4&NC 27r (27r)2 eis a partonic quark density at fixed energyand transverse momentum, and Ucb a purelyeikonal function, depending on parton veloc-ities ~, and on the soft radiation’s energyw, Q. The x’s and k’s are defined by (2).The delta functions in the triple convolutionof ( 1) relate the singular behavior of the var-ious functions, and Yj represents the match-ing to finite order. The all-order behavior ofthe above functions and the cross section canbe analyzed through their eikonalized equiv-alents. The eikonal cross section can be ex-pressed as an exponent, J??=b,an integral overdiagrammatically-defined functions referredto as “webs”, 6 UI.b,/Q d+zfkEab=2 —(2]-2.(~ e–N(kO/Q)-ikb -l). (3)The integral is over the energy and transversemomentum contributed by each web to thefinal state. The k-dependence of w.b followsfrom the invariance of the eikonal cross sec-tion under resealings of ~ and /3’.3 Single-particle inclusiveFor definiteness we consider prompt photonproduction, but the method sketched belowis more general 3. A similar refactorization asin (1) holds for single-particle inclusive crosssections at high pl’. In particular, it containsthe same Rjfj functions. The argumentssupporting this refactorization are somewhatmore involved than for (1), but reveal thatonly initial state radiation contributes to QT(the transverse momentum of the 2 + 2 par-ton cms frame) 7. In the inclusive cross sec-tion, final-state interactions require thresholdresummation only. In contrast to electroweakannihilation, Q~ is an unobserved variable,akin to 1 – z. Thus, the jointly resummedprompt photon pT spectrum may be writtenas an integral over QT of a “profile function”3~ij(N,QT,Q) =/#be-ibQ~eE,,+,, (4)where the exponential exhibits the joint re-summation, as(resum)P;dUAB+y=dpT●Acknowledgements ex/9711017.The work of E.L. is part of the research pro-gram of the Foundation for Fundamental Re-search of Matter (FOM) and the NationalOrganization for Scientific Research (NWO).The work of G .S. was supported in partby the National Science Foundation, grantPHY9722101. G .S. thanks Brookhaven Na-tional Laboratory for its hospitality. W .V.is grateful to RIKEN, Brookhaven Na-tional Laboratory and the U.S. Depart-ment of Energy (contract number DEAC02-98CH10886) for providing the facilities essen-tial for the completion of this work.References1. J.C. Collins, D.E. Soper and G. Ster-man, in Perturbative Quantum Chromo-dynamics, ed. A.H. Mueller (World Sci-entific, Singapore, 1989).2. H.-n. Li, Phys. Lett. B454, 328 (1999),hepph/9812363.3. E. Laenen, G. Sterman and W. Vogel-sang, Phys. Rev. Lett.84, 4296 (2000),hepph/0002078.4. E. Laenen, G. Sterman and W. Vogel-sang, hepph/0010080.5. G. Sterman, Nucl. Phys. B281, 310(1987); S. Catani and L. Trentadue,Nucl. Phys. B327, 323 (1989), B353,183 (1991).6. J.G.M. Gatheral, Phys. Lett. B133, 9(1983); G. Sterman, in: Perturbativequantum chromodynamics, proc. of Tal-lahassee conference (198 1), AIP Confer-ence Proceedings, eds. D. W. Duke andJ. F. Owens, p.22; J. Frenkel and J.C.Taylor, Nucl. Phys. B246, 231 (1984).7. H.-L. Lai and H.-n. Li, Phys. Rev. D58,114020 (1998), hep-ph/9802414.8. S. Catani, M.L. Mangano, P. Nason,C. Oleari and W. Vogelsang, JHEP 9903(1999) 025, hepph/9903436.9. L. ‘Apanasevich et al., E706 Collab.,Phys. Rev. Lett. 81, 2642 (1998), hep

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Combined recoil and threshold resummation for hard scattering cross sections

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