Timelike and spacelike virtual Compton scattering in the generalized Bjorken scaling regime are complementary tools to access generalized parton distributions. We stress that the gluonic contributions are by no means negligible, even in the medium energy range which will be studied intensely at JLab12 and in the COMPASS-II experiment at CERN. Ultraperipheral collisions with proton or ion beams may also be used at RHIC and at collider or fixed target experiments at LHC.
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Manuscript submitted on 15-6-2013 |
Original Manuscript | Timelike vs Spacelike DVCS from JLab, Compass to Ultraperipheral Collisions and AFTER@LHC§ |
In the collinear factorization framework the scattering amplitudes for exclusive processes such as deeply virtual Compton scattering (DVCS) [1(a) Müller D, Robaschik D, Geyer B, Dittes FM, Horejši J. Wave functions, evolution equations and evolution kernels from light-ray operators of QCD Fortsch Phys 1994; 42: 101-41.; (b) Radyushkin AV. Scaling limit of deeply virtual compton scattering Phys Lett 1996; B380: 417-25.; (c) Ji XD, Osborne J. One loop corrections and all order factorization in deeply virtual compton scattering Phys Rev 1996; B380: 417-25.; (d) Collins J, Freund A. Proof of Factorization for Deeply Virtual Compton Scattering in QCD Phys Rev 1999; D59: 074009-.]
and its crossed reaction, timelike Compton scattering (TCS) [2Berger ER, Diehl M, Pire B. Timelike Compton scattering Exclusive photoproduction of lepton pairs Eur Phys J 2002; C23: 675-89.]
have been shown to factorize in specific kinematical regions, provided a large scale controls the separation of short distance dominated partonic subprocesses and long distance hadronic matrix elements, the generalized quark and gluon distributions (GPDs) [3(a) Goeke K, Polyakov MV, Vanderhaeghen M. Hard exclusive reactions and the structure of hadrons Prog Part Nucl Phys 2001; 47: 401-515.; (b) Diehl M. Generalized parton distributions Phys Rep 2003; 388: 41-227.; (c) Belitsky AV, Radyushkin AV. Unraveling hadron structure with generalized parton distributions Phys Rep 2005; 418: 1-378.] which encode much information on the partonic content of nucleons. After factorization, the DVCS (and similarly TCS) amplitudes are written in terms of Compton form factors (CFF) , and
with the CFFs defined as:
We report in Sect. 2 on a recent NLO analysis [4(a) Pire B, Szymanowski L, Wagner J. NLO corrections to timelike, spacelike and double deeply virtual Compton scattering Phys Rev 2011; D83: 034009-.; (b) Pire B, Szymanowski L, Wagner J. On timelike compton scattering at medium and high energies Few Body Syst 2012; 53: 125-7., 5Moutarde H, Pire B, Sabatie F, Szymanowski L, Wagner J. On timelike and spacelike deeply virtual Compton scattering at next to leading order Phys Rev D 2013; 87: 054029.] of DVCS and TCS amplitudes, and make a few remarks on the study of TCS in ultraperipheral collisions at hadron colliders (Sect. 3) and at fixed target experiments at LHC (Sect. 4).
TCS and DVCS amplitudes are identical (up to a complex conjugation) at lowest order in α_{s} but differ at next to leading order, in particular because of the quite different analytic structure of the scattering amplitudes of these reactions. Indeed, the production of a timelike photon enables the production of intermediate states in some channels which are kinematically forbidden in the DVCS case. This opens the way to new absorptive parts of the amplitude. To estimate Compton Form Factors (CFF), we use the NLO calculations of the coefficient functions which have been calculated in the DVCS case in the early days of GPD studies and more recently for the TCS case [4(a) Pire B, Szymanowski L, Wagner J. NLO corrections to timelike, spacelike and double deeply virtual Compton scattering Phys Rev 2011; D83: 034009-.; (b) Pire B, Szymanowski L, Wagner J. On timelike compton scattering at medium and high energies Few Body Syst 2012; 53: 125-7.], the two results being simply related thanks to the analyticity (in Q^{2}) properties of the amplitude [6Mueller D, Pire B, Szymanowski L, Wagner J. Timelike and spacelike hard exclusive reactions Phys Rev D 2012; 86: 031502-.]:
where + (–) sign corresponds to vector (axial) case.
Using two GPD models based on Double Distributions (DDs), as discussed in detail in [5Moutarde H, Pire B, Sabatie F, Szymanowski L, Wagner J. On timelike and spacelike deeply virtual Compton scattering at next to leading order Phys Rev D 2013; 87: 054029.]: the Goloskokov-Kroll (GK) model [7Goloskokov SV, Kroll P. The role of the quark and gluon GPDs in hard vector-meson electroproduction Eur Phys J 2008; C53: 367-84.] and a model based on the MSTW08 PDF parametrization [8Martin AD, Stirling WJ, Thorne RS, Watt G. Parton distributions for the LHC Eur Phys J 2009; C63: 189-285.], we get the results shown in Fig. (1) for the real and imaginary parts of the spacelike and timelike dominant CFF H(ξ,t) and H(η,t). Comparing dashed and solid lines in the upper panels, one sees that gluonic contributions are so important that they even change the sign of the real part of the CFF, and are dominant for almost all values of the skewness parameter. A moderate result arises from a similar comparison of the lower panels; the gluonic contribution to the imaginary part of the CFF remains sizeable for values of the skewness parameter up to 0.3.
The effects of NLO contributions to some of the DVCS observables at moderate energies are exemplified in Figs. (2, 3) which show specific observables to be measured at JLab and COMPASS. The difference between the dotted and solid lines demonstrates that NLO contributions are important, whereas the difference between the dashed and solid lines shows that gluon contributions should not be forgotten even at low energy when a precise data set is analyzed.
Timelike Compton scattering in ultraperipheral collisions at hadron colliders opens a new way to measure generalized parton distributions, in particular for very small values of the skewness parameter.
We estimated [9(a) Pire B, Szymanowski L, Wagner J. Can one measure timelike Compton scattering at LHC? Phys Rev D 2009; 79: 014010.; (b) Pire B, Szymanowski L, Wagner J. Timelike compton scattering at LHC Acta Phys Polon Supp 2009; 2: 373-8.; (c) Pire B, Szymanowski L, Wagner J. Exclusive photoproduction of lepton pairs at LHC Nucl Phys Proc Suppl 2008; 232: 179-80.] the different contributions to the lepton pair cross section for ultraperipheral collisions at the LHC. Since the cross sections decrease rapidly with Q^{2}, we are interested in the kinematics of moderate Q^{2}, say a few GeV^{2}, and large energy, thus very small values of η. Note however that for a given proton energy the photon flux is higher at smaller photon energy.
The Bethe-Heitler amplitude grows much when small θ angles are allowed. In the following we will use the limits [π/4,3π/4] where the Bethe-Heitler cross section is sufficiently big but does not dominate too much over the Compton process. The Bethe-Heitler cross section integrated over θ∈[π/4,3π/4],ϕ∈[0,2π], Q^{2}∈[4.5,5.5]Gev^{2},|t|∈[0.05,0.25]Gev^{2}, as a function of γp energy squared s is in the limit of large s constant and equals 28.4 pb.
Since the amplitudes for the Compton and Bethe-Heitler processes transform with opposite signs under reversal of the lepton charge, the interference term between TCS and BH is odd under exchange of the ℓ^{+} and ℓ^{-} momenta. It is thus possible to project out the interference term through a clever use of the angular distribution of the lepton pair. The interference part of the cross-section for
with
where
The lepton pair production differential cross sections (solid lines) for
In Fig. (4) we show the interference contribution to the cross section in comparison to the Bethe-Heitler and Compton processes, for various values of photon proton energy squared
Fig.(4) The lepton pair production differential cross sections (solid lines) for t = –0.2GeV^{2}, Q^{2} = 5GeV^{2} integrated over θ = [π/4; 3 π/4(a) Pire B, Szymanowski L, Wagner J. NLO corrections to timelike, spacelike and double deeply virtual Compton scattering Phys Rev 2011; D83: 034009-.; (b) Pire B, Szymanowski L, Wagner J. On timelike compton scattering at medium and high energies Few Body Syst 2012; 53: 125-7.], as a function of φ, for s = 107GeV^{2} (a) s = 105 GeV^{2} (b) with μ_{F}^{2} = 5GeV^{2}. We also display the Compton (dotted), Bethe-Heitler (dash-dotted) and Interference (dashed) contributions. |
The main idea of the multi-purpose project AFTER@LHC is to extract the halo of the LHC proton or ion beams by means of a bent-cristal and to use it as a beam in the fixed-target experiments [10(a) Brodsky SJ, Fleuret F, Hadjidakis C, Lansberg JP. Physics opportunities of a fixed-target experiment using LHC beams Phys Rep 2013; 522: 239-55.; (b) Lansberg JP, Arnaldi R, Brodsky SJ et al, et al. AFTER@LHC a precision machine to study the interface between particle and nuclear physics EPJ Web Conf 2014; 66: 11023-0.]. The extracted beams will have a sufficiently high energy to produce on fixed target in ultraperipheral scattering a lepton pair with high invariant mass or heavy mesons. In these experiments a nucleus projectile or a nucleus target is treated as a high-energy photon source which allows study of photon-hadron collisions. Figs. (5 and 6, respectively) shows preliminary estimates of the Bethe-Heitler, TCS, and Interference contributions to the cross section, as functions of the CMS rapidity y, after integration over
We did not discuss here the rich phenomenology of DVCS and TCS processes which electron-ion colliders [11(a) Boer D, Diehl M, Milner R et al, et al. Gluons and the quark sea at high energies distributions, polarization, tomography [arXiv: 1108 1713 [nucl-th]] ; (b) Abelleira Fernandez JL, Adolphsen C, Akay AN et al, et al. [LHeC Study Group Collaboration] A large hadron electron collider at CERN Report on the physics and design concepts for machine and detector J Phys G 2012; 39: 075001-.] will allow to study. Neither did we comment on recent progresses in higher twist contributions [12(a) Braun VM, Manashov AN, Pirnay B. Finite-t and target-mass corrections to deeply virtual compton scattering Phys Rev Lett 2012; 109: 242001.; (b) Pire B, Szymanowski L, Wallon S. Chiral-odd pion generalized parton distributions beyond leading twist [arXiv: 13090083 [hep-ph]] ] nor on the effect of resummation of higher order QCD corrections [13Altinoluk T, Pire B, Szymanowski L, Wallon S. Resumming soft and collinear contributions in deeply virtual Compton scattering J High Energy Phys 2012; 1210: 049.]. The physics of generalized parton distributions is definitely a domain of work in progress, both on the theory and on the experimental side.
The authors confirm that this article content has no conflicts of interest.
This work is supported by the Polish Grant NCN No DEC-2011/01/B/ST2/03915, the French-Polish collaboration agreement Polonium, the ANR project "Partons", the COPIN-IN2P3 Agreement and the Joint Research Activity "Study of Strongly Interacting Matter" (HadronPhysics3, Grant Agreement no. 283286) under the 7^{th} Framework Programme of the European Community.