We give a brief account of recent theoretical developments in prompt photon
production.Comment: Contribution to the proceedings of the ``8th International Workshop
  on Deep-Inelastic Scattering'' (DIS2000), April 2000, Liverpool, UK; 7 Pages,
  LaTeX, three figures as eps file

Laenen, Eric

Sterman, George

Vogelsang, Werner

English

arXiv

The authors give a brief account of recent theoretical developments in prompt photon production

Laenen, E.

Sterman, G.

Vogelsang, W.

UNT Digital Library

& .RBRC-111. BNL-68325o00N0mNIKHEF/oo-14YITP-00-33RIKEN-BNL preprintJune 2000Current Issues inPrompt Photon Production~Eric Laenena, George Sterman b, Werner Vogelsangc’t“ NIKHEF Theory Group, Kruislaan 4091098 SJ Amsterdam, The Netherlandsb C .N. Yang Institute for Theoretical Physics, SUNY Stony BrookStony Brook, New York 11794-3840, U.S.A.c RIKEN-BNL Research Center, Brookhaven National LaboratoryUpton, NY 11973, U.S.A.AbstractWe give a brief account of recent theoretical developments in prompt photonproduction.t Invited talk presented by W. Vogelsang at the “8th International Workshop on Deep-Inelastic Scattering” (DIS2000), 25-30 April 2000, Liverpool, UKi WV is grateful to RI~EN, Brookhaven National Laboratory and the U.S. Department ofEnergy (contract number DE-ACO2-98CH1O886) for providing the facilities essential for thecompletion of this workCURRENT ISSUES IN PROMPT PHOTON PRODUCTIONE. LAENENNIKHEFTheory Group, Kruislaan J09, 1098 SJ,4msterdam, The NetherlandsG. STERMANC.N. Yang Institute for Theoretical Physics, SUNY Stony BrookStony Brook, Neu> York 11794-.3840, U.S.A.W. VOGELSANGRIKEN-BNL Research Center, Brookhaven National Laboratory,Upton, NY 11973, U.S.A.We give a brief account of recent theoretical developments in prompt photon pro-duction.Prompt-photon production at high transverse momentum 1, pp, p~, pN -+7X, has been a classical tool for constraining the nucleon’s gluon density, sinceat leading order a photon can be produced in the Compt,on reaction qg + -yq.The ‘point-like’ coupling of the photon to the quark provides a supposedly‘clean’ electromagnetic probe of QCD hard scattering. In the framework ofQCD perturbation theory, the inclusive cross section is written in a factorizedform:For simplicity, we have integrated over the rapidity q of the prompt photon.In (1), f#I@/A(z., p) denotes the parton density for species a in hadron A, atfactorization scale p, Z. being the parton’s momentumfraction. d8.b+T./dpTare the partonic hard-scattering functions, which have been calculated to next-to-leading order 2. Hadronic and partonic scaling variables are Z; ❑ 4p~/Sand 2$ ❑ 4p~/.$, respectively, with ; = $a%bS.Unfortunately, in experiment, one has to deal with a substantial back-ground of photons from no decay. In addition, high-pT photons can be pro-duced in jets, when a pa.rton, resulting from a pure QCD reaction, fragmentsinto a photon plus a number of hadrons. This inevitably introduces dependenceon non-perturbative (parton-to-photon) fragment ation functions. So far, thelatter are insufficiently known. Theoretical studies 3,4,5~G, based on predk-tions 3’7’8 for the photon fragmentation functions, indicate that the fragmen-tation component is in practice a subdominant, albeit non-negligible, effect.In the fixed-target regime, fragmentation photons are believed 5 to contribute2at most 20% to the direct photon cross section. At collider energies, the frag-mentation mechanism can easily produce about half or more of the observed93’5 Here an ‘isolation’ cut is imposed on the photon signal in exper-photons 1 .iment, in order to suppress the m“ background. Isolation is usually realized bydrawing a cone of fixed aperture in p–q space around the photon, restrictingthe hadronic transverse energy allowed in this cone to a certain small fractions of the photon transverse energy. In this way, the fragmentation contribution,resulting from an essentially collinear process, is diminished 10, probably to alevel of 15–20Y0, or less 4’5.We mention that subtleties were observed 1112113 in the past concerningthe factorizability of the isolated prompt photon cross section. While in 13factorization was eventually proved to hold, it was also shown 131211 thatwith isolation the NLO partonic hard scattering function has a Sudakov-typesingular behavior at a certain point inside the physical region, requiring soft-gluon resummations in order to obtain a reliable theoretical result. It was alsopointed out 10 that the (in practice 14) small size of ~ can give rise to potentiallylarge logarithms of c, associated with soft-gluon emission into the isolationcone. Further work on the fragmentation component is clearly needed. Arecent suggestion 15 to refine isolation by allowing less and less hadronic energythe closer to the photon it is deposited, will, at least theoretically, eliminatethe fragmentation’ component altogether and could potentially avoid some ofthe problems just mentioned. This isolation prescription has been applied instudies for prompt photon production at RHIC and LHC 16.A pattern of disagreement between theoretical predictions and experimen-tal dat a for prompt photon production has been observed in recent years 14’17’18,not globally curable by changing factorization and renormalization scales or by5 19J6. The most serious problems relate to the‘fine-tuning’ the gluon density Jfixed-target data, where NLO theory dramatically underpredicts some datasets 17’1s. At collider energies, there is less reason for concern, but also herethe agreement is not fully satisfactory. The mutual consistency of the variousfixed-target data sets has been questioned in 6. On the other hand, variousimprovements of the theoretical framework have been developed.‘Threshold’ resumma.tions for the inclusive prompt photon cross sectionhave been performed in aozl and applied phenomenologically in 22’23. As s ap-proaches its minimum value at 2; = 1, corresponding to ‘partonic threshold’when the initial partons have just enough energy to produce the high-pT pho-ton and the recoiling jet, the phase space available for gluon bremsstrahlungvanishes, resulting in corrections to db/dpT as large as c$ lnzk (1 – ~~ ) &*ornat order cr~ in perturbation theory. Threshold resummation 24’20’21 organizesthis singular, but integrable, behavior of d&/dpT to all orders in as. It is310 I 1 1 I: IIPN, Eb_=530 GeV, lql<O.75GRV–989 !+=tir=PT8i ][-4 -$]1 [][]I / 5a ------------- ----“2 4 8 12PT (GeV) 8 ‘0F@re 1: Threshold-resummed prompt photon cross section based on the formalismsof20, (25J (solid) and 21, (Z~) dashed), divided by the pJLO cross section. parton densities are(from 26. The E706 prompt photon data17 are also shown in the same normalization.carried out in Mellin-iV moment space, where the logarithms are of the forma: in’k (N) &BOrn(N). Its application is particularly interesting in the fixed-target regime, since here the highest xT are attained in the data, and sincehere the discrepancy between data and theory is largest. As seen from Fig. 1,one obtains a significant, albeit not sufficient, enhancement of the theory pre-diction. Also, a dramatic reduction of scale dependence is found z’. One noticesthat the formalisms of ‘“”l, which differ in so far as the one of 20 resurns atfixed photon rapidity, whereas the one of 21 applies to the cross section fullyintegrated over rapidity, yield very similar predictions in practice.Valuable insight was gained in studies of direct photon production inwhich transverse smearing of the momenta of the initial partons was incorpo-rated 19,27J28J2gJ30.If, say, in the Compton process qg + yq the initial partonshave a non-zero kT, the -yq pair in the final state will acquire a net transversemomentum QT, which may make the process softer than it would be other-wise and result in an enhancement in the photon ~T spectrum. The first ap-proacheslg’27’28 assumed Gaussian dependence on kT and enj eyed phenomeno-logical success in that they were able to bridge the large gaps between data andtheory for appropriate choices of average (kT), guided by values of dimuon, di-jet and diphoton pair transverse momenta. measured in hadronic reactions. Onthe other hand, it was clear that a more developed theoretical framework was27 the physical origin forrequired. For inst ante, as acknowledged already in ,the parton kT dependence should be thought, of as initial-state gluon emission,so that QT takes the role of pair recoil against unobserved radiation. Ideally,the phenomenologically required kT smearing would have to be understoodin terms of a QT-resummation calculation, with perturbative as well as non-perturbative components, as familiar from the well-explored case of Drell-Yan4dimuon production, where soft-gluon radiation gives rise to powers of largelogarithms at small Q~, which can be resummed to all orders. In the inclusivepartonic prompt photon cross section d&ab+vc /dpT, such Q~-logarithms willnot be visible at any given order of perturbation theory; however, they willshow up when considering d6ah+7c/d2QT dpT and can be resummed prior toQT integration. Attempts in this direction were first made in 29’30, where par-ton densities, unintegrated over parton transverse momentum, but constructedfrom the usual densities, were used. The ‘double-logarithmic approximation’was employed in 29’30. In 30, only small recoil effects were found when a strongordering constraint was imposed.In 31, a simultaneousresummation in both threshold and transverse mo-mentum logarithms was achieved. The possibility of doing this had previouslybeen pointed out in 32. Contributions to the hard scattering function associ-ated with threshold resummation are redistributed over soft gluon transversemoment a, simultaneously conserving energy and transverse momentum. Largelogarithmic corrections in QT to d&ab+7c/d2QT dpT arising in the thresholdregion are resummed jointly with threshold logarithms. A further crucial anddistinct feature of 31 is that it remains within the formalism of collinear fac-torization, which implies use of ordinary parton densities.The possibility of joint resummation for singular behavior in QT and 1– ~$is ensured by the factorization properties of the partonic cross section nearthreshold 33. Singular QT behavior is organized in impact parameter h-space,where the logarithms exponentiate. Defining ‘profile functions’ in QT asP;j (N, QT, Q,p) =/(2)dzb e–ib” QT exp [Eij-+vk (~, b, Q, P)]>where the resummed exponents Eij+7k can be found in31 to NLL in both b andN, the resulting resummed cross section is given in terms of Mellin momentsof the parton distributions and of the squared Born amplitudes IJ4i,i ]2 asthe recoil effect entering through p$ = pT – QT/2. Eq. (3) reverts to thethreshold-resummed cross section for p! + pT. QT is limited by the scale Z,to avoid going outside the region where the singularities in QT dominate.5Q’sm%-B*Y, .-., \ \ \ .. .“.,4.,! l!, Al,,,,,,,,,,,,!z PN , Ebem = 530 GeV[7]<0.75‘ .. ..>\.>.\, .\. ,\, ...>.\‘,. ,\.\‘. \\‘. \1,, ,,l!!, ,1, ,,,4 6 8 10 12PT (Gev)Figure 2: Left: prompt photon cross section doPN+yx /dQq-dpT at fi = 31.5 GeV. Dashedlines are computed without recoil, solid lines are with recoil. Right: Ed3 LTPN+TX /dp3. Thedotted line represents the full NLO calculation, while the dashed and solid lines respectivelyincorporate pure threshold resummation (see21 and Fig. 1) and the joint resummation of 31.Cross sections computed on the basis of (3) are shown in Fig. 2 as func-tions of QT at fixed pT. The kinematics are those of the E706 experiment 17;see 31 for details of the calculation, in particular those regarding the evaluation(r=um) d ~dpT, with recoilof the b-integral in (2). The dashed lines are drpN+Tx / Qneglected by fixing p~ = pT, thus showing how each QT contributes to thresh-old enhancement. The solid lines show the same, but now including the true1 Z ~+1 The resulting enhancenlent is clearly subst a.ntial.recoil factor (S/4p~ ) .For small pT, the enhancement simply grows with QT, while for p~ above 5GeV it has a dip at about QT = 5 GeV, which remains substantially abovezero. This makes it difficult to confident y determine ~.So far, the numerical results given in 31 are primarily to be regarded asillustrations, rather than quantitative predict ions. This applies in particularto the resummed QT-integrated cross section, also shown in Fig. 2 for ~T >3.5GeV and ji = 5 GeV. Comparison with Fig. 1 demonstrates the size of the ad-ditional enhancement that recoil can produce and its potential phenomenolog-ical impact. Further work on the implementation of practical nonperturbat iveestimates and of matching procedures is required.References1.2.see: W. Vogelsa.ng, M. Whalley, J. Phys. G 23, Al (1997).P. Aurenche et al., Phys. Lett. B 140, 87 (1984); Nucl. Phys. B 297,661 (1988); H. Baer, J. Ohnemus, J.F. Owens, Phys. Lett. B 234, 127(1990); Phys. Rev. D 42,61 (1990); L.E. Gordon, W. Vogelsang, Phys.6“4v3.4.5.6.7.8.9.10.11.12.13.14.15.16.17.18.19.20.21.22.23.24.25.26.27,28.29.30.31.32.Rev. D 48, 3136 (1993); Phys. Rev, D 50, 1901 (1994).P. Aurenche et al., Ntici. Phys. B 399, 34 (1993).M. Gliick et al., Phys. Rev. Lett. 73, 388 (1994).W. Vogelsang, A. Vogt, Nucl. Phys. B 453, 334 (1995).P. Aurenche et al., Eur. J. Phys. C 9, 107 (1999).M. Gluck, E. Reya, A. Vogt, Phys, Rev. D 48, 116 (1993).L. Bourhis, M. Fontannazl J. Ph. Guillet, Eur. 7. Phys. C 2, 529 (1998).P. Aurenche, R. Baier, M. Fontannaz, Phys. Rev. D 42, 1440 (1990).E.L. Berger, J. Qiu, Phys. Rev. D 44, 2002 (1991).E.L. Berger, X. Guo, J. Qiu, Phys. Rev. Lett. 76, 2234 (1996); Phys,Rev. D 54, 5470 (1996).P. Aurenche et cd., Phys. Rev. D 55, 1124 (1997).S. Catani, M. Fontannaz, E. Pilon, Phys. Rev. D 58, 094025 (1998).F. Abe et al., C!DF Collab., Phys. Rev. Lett. 73, 2662 (1!394);S. Kuhlmann, et cd., CDF Collab., IVaci. Phys. Proc. Suppl. 79, 241(1999); B. Abbott et cd., DO Ccdlab., Phys. Rev. Lett. 84,2786 (2000).S. Frixione, Phys. Lett. B 429, 369 (1998).S. Frixione, W. Vogelsang, Nucl, Phys. B 568, 60 (2000); in: Pro-ceedings of “Workshop on Physics at TeV Colliders”, Les Houches,France, June 1999, QCD and Standard Model Working Group, convenersS. Cata.ni et al., hep-ph/0005114.L. Apanasevich et al., E706 Collab., Phys. Rev. LetL 81, 2642 (1998).G. Ballocchi et al., UA6 Collab., Phys. Lett. B 436, 222 (1998).J. Huston et aL, Phys. Rev. D D51, 6139 (199.5).E. Laenen, G. Oderda, G. Sterman, Phys. Lett. B 438, 173 (1998).S. Catani, M.L. Mangano, P. Nason, JHEP 9807, 024 (1998).S. Catani et al., JHEP 9903,025 (1999).N. Kidonakis, J.F. Owens, Phys. Rev. D 61, 094004 (2000).G. Sterman, Nucl. Phys. B 281, 310 (1987); S. Catani, L. Trentadue,Nucl. Phys. B B327, 323 (1989), B 353, 183 (1991); N. Kidona.kis, G.Sterman, Nucl. Ph.ys. B 505, 321 (1997); R. Bonciani et al., Nucl. Phys.B 529,424 (1998).G. Sterman, W. Vogelsang, in preparation.M. Gluck, E. Reya, A. Vogt, Eur. J. Phys.C5, 461 (1998).L. Apanasevich et al., Phys. Rev. D 59, 074007 (1999).A.D. Martin et aL, Eur. J. Phys. C 4, 463 (1998).H.-L. Lai, H.-n. Li, Phgs. Rev. D 58, 114020 (1998).M.A. Kimber, A.D. Martin, M. Ryskin, Eur.E. Laenen, G. Sterman, W. Vogelsang, Phys.H.-n. Li, Phys. Lett. B 454, 328 (1999).Phys. J. C 12, 655 (2000).Rev. Lett. 84,4296 (2000).733. S. Catani et al,, Nucl. Phys. B 407, 3 (1993); H. Contopanagos, E.Laenen, G. Sterman, Nucl. Phys. B 484, 303 (1997).8+’?●

Current issues in prompt photon production

We give a brief account of recent theoretical developments in prompt photon production

Utrecht University Repository

arXiv:hep-ph/0006352v1  30 Jun 2000NIKHEF/00-14YITP-00-33RIKEN-BNL preprintJune 2000Current Issues inPrompt Photon Production†Eric Laenena, George Stermanb, Werner Vogelsangc,‡a NIKHEF Theory Group, Kruislaan 4091098 SJ Amsterdam, The Netherlandsb C.N. Yang Institute for Theoretical Physics, SUNY Stony BrookStony Brook, New York 11794 – 3840, U.S.A.c RIKEN-BNL Research Center, Brookhaven National LaboratoryUpton, NY 11973, U.S.A.AbstractWe give a brief account of recent theoretical developments in prompt photonproduction.† Invited talk presented by W. Vogelsang at the “8th International Workshop on Deep-Inelastic Scattering” (DIS2000), 25-30 April 2000, Liverpool, UK‡ WV is grateful to RIKEN, Brookhaven National Laboratory and the U.S. Department ofEnergy (contract number DE-AC02-98CH10886) for providing the facilities essential for thecompletion of this workCURRENT ISSUES IN PROMPT PHOTON PRODUCTIONE. LAENENNIKHEF Theory Group, Kruislaan 409, 1098 SJ Amsterdam, The NetherlandsG. STERMANC.N. Yang Institute for Theoretical Physics, SUNY Stony BrookStony Brook, New York 11794 – 3840, U.S.A.W. VOGELSANGRIKEN-BNL Research Center, Brookhaven National Laboratory,Upton, NY 11973, U.S.A.We give a brief account of recent theoretical developments in prompt photon pro-duction.Prompt-photon production at high transverse momentum 1, pp, pp̄, pN →γX , has been a classical tool for constraining the nucleon’s gluon density, sinceat leading order a photon can be produced in the Compton reaction qg → γq.The ‘point-like’ coupling of the photon to the quark provides a supposedly‘clean’ electromagnetic probe of QCD hard scattering. In the framework ofQCD perturbation theory, the inclusive cross section is written in a factorizedform:dσAB→γX(x2T )dpT=∑abφa/A(xa, µ) ⊗ φb/B(xb, µ) ⊗dσ̂ab→γc(x̂2T , µ)dpT. (1)For simplicity, we have integrated over the rapidity η of the prompt photon.In (1), φa/A(xa, µ) denotes the parton density for species a in hadron A, atfactorization scale µ, xa being the parton’s momentum fraction. dσ̂ab→γc/dpTare the partonic hard-scattering functions, which have been calculated to next-to-leading order 2. Hadronic and partonic scaling variables are x2T ≡ 4p2T /Sand x̂2T ≡ 4p2T/ŝ, respectively, with ŝ = xaxbS.Unfortunately, in experiment, one has to deal with a substantial back-ground of photons from π0 decay. In addition, high-pT photons can be pro-duced in jets, when a parton, resulting from a pure QCD reaction, fragmentsinto a photon plus a number of hadrons. This inevitably introduces dependenceon non-perturbative (parton-to-photon) fragmentation functions. So far, thelatter are insufficiently known. Theoretical studies 3,4,5,6, based on predic-tions 3,7,8 for the photon fragmentation functions, indicate that the fragmen-tation component is in practice a subdominant, albeit non-negligible, effect.In the fixed-target regime, fragmentation photons are believed 5 to contribute2at most 20% to the direct photon cross section. At collider energies, the frag-mentation mechanism can easily produce about half or more of the observedphotons 9,3,5. Here an ‘isolation’ cut is imposed on the photon signal in exper-iment, in order to suppress the π0 background. Isolation is usually realized bydrawing a cone of fixed aperture in ϕ–η space around the photon, restrictingthe hadronic transverse energy allowed in this cone to a certain small fractionε of the photon transverse energy. In this way, the fragmentation contribution,resulting from an essentially collinear process, is diminished 10, probably to alevel of 15–20%, or less 4,5.We mention that subtleties were observed 11,12,13 in the past concerningthe factorizability of the isolated prompt photon cross section. While in 13factorization was eventually proved to hold, it was also shown 13,12,11 thatwith isolation the NLO partonic hard scattering function has a Sudakov-typesingular behavior at a certain point inside the physical region, requiring soft-gluon resummations in order to obtain a reliable theoretical result. It was alsopointed out10 that the (in practice14) small size of ε can give rise to potentiallylarge logarithms of ε, associated with soft-gluon emission into the isolationcone. Further work on the fragmentation component is clearly needed. Arecent suggestion15 to refine isolation by allowing less and less hadronic energythe closer to the photon it is deposited, will, at least theoretically, eliminatethe fragmentation component altogether and could potentially avoid some ofthe problems just mentioned. This isolation prescription has been applied instudies for prompt photon production at RHIC and LHC 16.A pattern of disagreement between theoretical predictions and experimen-tal data for prompt photon production has been observed in recent years14,17,18,not globally curable by changing factorization and renormalization scales or by‘fine-tuning’ the gluon density 5,19,6. The most serious problems relate to thefixed-target data, where NLO theory dramatically underpredicts some datasets 17,18. At collider energies, there is less reason for concern, but also herethe agreement is not fully satisfactory. The mutual consistency of the variousfixed-target data sets has been questioned in 6. On the other hand, variousimprovements of the theoretical framework have been developed.‘Threshold’ resummations for the inclusive prompt photon cross sectionhave been performed in 20,21 and applied phenomenologically in 22,23. As ŝ ap-proaches its minimum value at x̂2T = 1, corresponding to ‘partonic threshold’when the initial partons have just enough energy to produce the high-pT pho-ton and the recoiling jet, the phase space available for gluon bremsstrahlungvanishes, resulting in corrections to dσ̂/dpT as large as αks ln2k(1 − x̂2T ) σ̂Bornat order αks in perturbation theory. Threshold resummation24,20,21 organizesthis singular, but integrable, behavior of dσ̂/dpT to all orders in αs. It is3Figure 1: Threshold-resummed prompt photon cross section based on the formalismsof 20,(25) (solid) and21,(22) (dashed), divided by the NLO cross section. Parton densities arefrom 26. The E706 prompt photon data 17 are also shown in the same normalization.carried out in Mellin-N moment space, where the logarithms are of the formαks ln2k(N) σ̂Born(N). Its application is particularly interesting in the fixed-target regime, since here the highest xT are attained in the data, and sincehere the discrepancy between data and theory is largest. As seen from Fig. 1,one obtains a significant, albeit not sufficient, enhancement of the theory pre-diction. Also, a dramatic reduction of scale dependence is found22. One noticesthat the formalisms of 20,21, which differ in so far as the one of 20 resums atfixed photon rapidity, whereas the one of 21 applies to the cross section fullyintegrated over rapidity, yield very similar predictions in practice.Valuable insight was gained in studies of direct photon production inwhich transverse smearing of the momenta of the initial partons was incorpo-rated 19,27,28,29,30. If, say, in the Compton process qg → γq the initial partonshave a non-zero kT , the γ q pair in the final state will acquire a net transversemomentum QT , which may make the process softer than it would be other-wise and result in an enhancement in the photon pT spectrum. The first ap-proaches19,27,28 assumed Gaussian dependence on kT and enjoyed phenomeno-logical success in that they were able to bridge the large gaps between data andtheory for appropriate choices of average 〈kT 〉, guided by values of dimuon, di-jet and diphoton pair transverse momenta measured in hadronic reactions. Onthe other hand, it was clear that a more developed theoretical framework wasrequired. For instance, as acknowledged already in 27, the physical origin forthe parton kT dependence should be thought of as initial-state gluon emission,so that QT takes the role of pair recoil against unobserved radiation. Ideally,the phenomenologically required kT smearing would have to be understoodin terms of a QT -resummation calculation, with perturbative as well as non-perturbative components, as familiar from the well-explored case of Drell-Yan4dimuon production, where soft-gluon radiation gives rise to powers of largelogarithms at small QT , which can be resummed to all orders. In the inclusivepartonic prompt photon cross section dσ̂ab→γc/dpT , such QT -logarithms willnot be visible at any given order of perturbation theory; however, they willshow up when considering dσ̂ab→γc/d2QT dpT and can be resummed prior toQT integration. Attempts in this direction were first made in29,30, where par-ton densities, unintegrated over parton transverse momentum, but constructedfrom the usual densities, were used. The ‘double-logarithmic approximation’was employed in 29,30. In 30, only small recoil effects were found when a strongordering constraint was imposed.In 31, a simultaneous resummation in both threshold and transverse mo-mentum logarithms was achieved. The possibility of doing this had previouslybeen pointed out in 32. Contributions to the hard scattering function associ-ated with threshold resummation are redistributed over soft gluon transversemomenta, simultaneously conserving energy and transverse momentum. Largelogarithmic corrections in QT to dσ̂ab→γc/d2QT dpT arising in the thresholdregion are resummed jointly with threshold logarithms. A further crucial anddistinct feature of 31 is that it remains within the formalism of collinear fac-torization, which implies use of ordinary parton densities.The possibility of joint resummation for singular behavior in QT and 1−x̂2Tis ensured by the factorization properties of the partonic cross section nearthreshold 33. Singular QT behavior is organized in impact parameter b-space,where the logarithms exponentiate. Defining ‘profile functions’ in QT asPij (N,QT , Q, µ) =∫d2b e−ib·QT exp [Eij→γk (N, b,Q, µ)] , (2)where the resummed exponents Eij→γk can be found in31 to NLL in both b andN , the resulting resummed cross section is given in terms of Mellin momentsof the parton distributions and of the squared Born amplitudes |Mij |2 asp3Tdσ(resum)AB→γXdpT=∑ijp4T8πS2∫CdN2πiφ̃i/A(N,µ)φ̃j/B(N,µ)∫ 10dx̃2T(x̃2T)N×|Mij(x̃2T )|2√1− x̃2T∫d2QT(2π)2Θ(µ̄−QT )(S4p′T2)N+1Pij(N,QT ,2pTx̃T, µ), (3)the recoil effect entering through p′T = pT − QT /2. Eq. (3) reverts to thethreshold-resummed cross section for p′T → pT . QT is limited by the scale µ̄,to avoid going outside the region where the singularities in QT dominate.5Figure 2: Left: prompt photon cross section dσpN→γX/dQT dpT at√s = 31.5 GeV. Dashedlines are computed without recoil, solid lines are with recoil. Right: Ed3σpN→γX/dp3. Thedotted line represents the full NLO calculation, while the dashed and solid lines respectivelyincorporate pure threshold resummation (see21 and Fig. 1) and the joint resummation of 31.Cross sections computed on the basis of (3) are shown in Fig. 2 as func-tions of QT at fixed pT . The kinematics are those of the E706 experiment17;see 31 for details of the calculation, in particular those regarding the evaluationof the b-integral in (2). The dashed lines are dσ(resum)pN→γX/dQTdpT , with recoilneglected by fixing p′T = pT , thus showing how each QT contributes to thresh-old enhancement. The solid lines show the same, but now including the truerecoil factor (S/4p′T2)N+1. The resulting enhancement is clearly substantial.For small pT , the enhancement simply grows with QT , while for pT above 5GeV it has a dip at about QT = 5 GeV, which remains substantially abovezero. This makes it difficult to confidently determine µ̄.So far, the numerical results given in 31 are primarily to be regarded asillustrations, rather than quantitative predictions. This applies in particularto the resummed QT -integrated cross section, also shown in Fig. 2 for pT ≥ 3.5GeV and µ̄ = 5 GeV. Comparison with Fig. 1 demonstrates the size of the ad-ditional enhancement that recoil can produce and its potential phenomenolog-ical impact. Further work on the implementation of practical nonperturbativeestimates and of matching procedures is required.References1. see: W. Vogelsang, M. Whalley, J. Phys. G 23, A1 (1997).2. P. Aurenche et al., Phys. Lett. B 140, 87 (1984); Nucl. Phys. B 297,661 (1988); H. Baer, J. Ohnemus, J.F. Owens, Phys. Lett. B 234, 127(1990); Phys. Rev. D 42, 61 (1990); L.E. Gordon, W. Vogelsang, Phys.6Rev. D 48, 3136 (1993); Phys. Rev. D 50, 1901 (1994).3. P. Aurenche et al., Nucl. Phys. B 399, 34 (1993).4. M. Glück et al., Phys. Rev. Lett. 73, 388 (1994).5. W. 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Current Issues in Prompt Photon Production

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