Charmonium production in two-photon collisions at next-to-leading order
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Charmonium production in two-photon collisions at next-to-leading order Bernd A.Kniehl II.Institut für Theoretische Physik,Universit?t Hamburg,Luruper Chaussee 149,22761Hamburg,Germany Abstract.We review recent results on the production of prompt charmonium in association with a hadron jet or a prompt photon in two-photon collisions at next-to-leading order in the factorization framework of nonrelativistic quantum chromodynamics.Keywords:Nonrelativistic quantum chromodynamics,radiative corrections,charmonium,two-photon scattering PACS:12.38.Bx,12.39.St,13.66.Bc,14.40.Gx 1.INTRODUCTION The factorization formalism of nonrelativistic quantum chromodynamics (NRQCD)[1]provides a rigorous theoretical framework for the description of heavy-quarkonium production and decay that is renormalizable and predictive.Theoretical predictions are decomposed into sums over products of short-distance coef?cients,which can be calculated perturbatively as expansions in the strong-coupling constant αs ,and long-distance matrix elements (MEs),which are subject to relative-velocity (v )scaling rules and must be extracted from experiment,and are so organized as double expansions in αs and v .This formalism takes into account the complete structure of the Q
malization and factorization scale dependences,sizeable effects,e.g.due to the opening of new partonic production channels,are expected at NLO.On the other hand,the ex-perimental errors are still rather sizeable.The latter are being signi?cantly reduced by HERA II and run II at the Tevatron,and will be dramatically more so by the LHC and hopefully a future e+e?linear collider such as the TeV-Energy Superconducting Linear Accelerator(TESLA),which is presently being designed and planned at DESY. Recently,2→2processes of heavy-quarkonium production were for the?rst time studied at NLO in the NRQCD factorization formalism[2,3].Speci?cally,the produc-tion of prompt charmonium,which is produced either directly or through the decay of heavier charmonia,with?nite transverse momentum(p T)in association with a hadron jet[2]or a prompt photon[3]via direct photoproduction in two-photon collisions was considered.In this presentation,we review the most important conceptional issues and phenomenological results of Refs.[2,3],in Sections2and3,respectively.
2.CONCEPTIONAL ISSUES
We focus attention on the processγγ→J/ψ+X,where J/ψis promptly produced at ?nite value of p T and X is a purely hadronic remainder.Since the incoming photons can interact either directly with the quarks participating in the hard-scattering process(direct photoproduction)or via their quark and gluon content(resolved photoproduction),this process receives contributions from the direct,single-resolved,and double-resolved channels,which are formally of the same order in the perturbative expansion.At LO, the bulk of the cross section is due to single-resolved photoproduction,and the NRQCD prediction[4]based on the MEs determined from?ts[4]to Tevatron data nicely agrees with a recent measurement by the DELPHI Collaboration at LEP2[6].
Here,we consider direct photoproduction at NLO[2].At LO,there is only one partonic subprocess,namely
γ+γ→c
c[n]+g+g,n=3P(1)J,1S(8)0,3S(8)1,3P(8)J,(2)γ+γ→c u,n=3S(8)1,(3)
γ+γ→c q,n=1S(8)0,3S(8)1,3P(8)J,(4) where u and
O H[3S(8)1] also produce UV,IR,and Coulomb divergences.The UV and IR divergences are extracted using dimensional regularization in d=4?2εspace-time dimensions, leading to poles inεUV andεIR,respectively,while the Coulomb singularities are regu-larized by a small value of v.The UV divergences are removed by the renormalization of O H[3S(8)1] ,αs,the charm-quark mass and?eld,and the gluon?eld,which is performed in the modi?ed minimal-subtraction(
q quarks inside the resolved photon.The M dependence thus introduced is approxi-mately compensated by the LO single-resolved contribution.
Combining the contributions arising from the virtual corrections(vi),the parameter and wave-function renormalization(ct),the operator rede?nition(op),the initial-state (is)and?nal-state(fs)collinear con?gurations,the soft-gluon radiation(so),and the hard-parton emission(ha)as
dσ(μ,λ,M)=dσ0(μ,λ)[1+δvi(μ;εUV,εIR,v)+δct(μ;εUV,εIR)+δop(μ,λ;εIR,v) +δfs(μ;εIR,δf)]+dσis(μ,λ,M;δi)+dσso(μ,λ;εIR,δf)+dσha(μ,λ;δi,δf),(5) the regulatorsεUV,εIR,v,δi,andδf drop out and theμandλdependences formally cancel up to terms beyond NLO,while the M dependence is unscreened at NLO.
3.PHENOMENOLOGICAL RESULTS
We consider two-photon collisions at TESLA operating at a center-of-mass energy of 500GeV,where the photons arise from electromagnetic initial-state bremsstrahlung, with antitagging angleθmax=25mrad,and beamstrahlung,with effective beam-strahlung parameter?=0.053.The J/ψ,χcJ,andψ′MEs are adopted from Ref.[5] and the photon PDFs from Ref.[7].
In Fig.1,we study d2σ/d p T dy(a)for rapidity y=0as a function of p T and (b)for p T=5GeV as a function of y,comparing the LO(dashed lines)and NLO (solid lines)results of direct photoproduction with the LO result of single-resolved photoproduction(dotted lines).From Fig.1(a),we observe that,with increasing value of p T,the NLO result of direct photoproduction falls of considerably more slowly than the LO one.In fact,the QCD correction(K)factor,de?ned as the NLO to LO
ratio,rapidly increases with p T,exceeding10for p T～>10GeV.This feature may be understood by observing that so-called fragmentation-prone[8]partonic subprocesses start to contribute to direct photoproduction at NLO,while they are absent at LO.Such subprocesses contain a gluon with small virtuality,q2=4m2c,that splits into a c