Virtual Compton Scattering (VCS) at low transferred momenta to the proton
($t$) and sufficiently high c.m. energies ($s$) may be used to a) study
$Q^2$--dependence of leading $t$--channel exchanges and b) look for onset of
scaling behavior with increasing $Q^2$. I discuss the implications for
perturbative and nonperturbative QCD and suggest possible experiments.Comment: LaTeX, 7 pages, 3 figures, requires sprocl.sty, psfig.st

Afanasev, Andrei

English

arXiv

Virtual Compton Scattering (VCS) at low transferred momenta to the proton (t) and sufficiently high c.m. energies (s) may be used to (a) study Q{sup 2}-dependence of leading t-channel exchanges and (b) look for onset of scaling behavior with increasing Q{sup 2}. The author discusses the implications for perturbative and nonperturbative QCD and suggests possible experiments

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arXiv:hep-ph/9608305   12 Aug 1996JLAB–THY–96–01August 6, 1996VIRTUAL COMPTON SCATTERING ON THE PROTONAT HIGH s AND LOW t aAndrei Afanasev bJefferson Lab, 12000 Jefferson Ave, Newport News, VA 23606, USAandDepartment of Physics, Hampton University, Hampton, VA 23668, USAVirtual Compton Scattering (VCS) at low transferred momenta to the proton (t)and sufficiently high c.m. energies (s) may be used to a) study Q2–dependenceof leading t–channel exchanges and b) look for onset of scaling behavior withincreasing Q2. I discuss the implications for perturbative and nonperturbativeQCD and suggest possible experiments.IntroductionAnalysing various kinematic domains of Virtual Compton Scattering (VCS),one may obtain clues to various problems of strong interaction dynamics.If s −t, and Q2 is fixed, the amplitude of VSC may be described (Fig.1a)by the sum of t–channel Reggeon (R) exchanges 1,M =∑RsαR(t)βR(t, Q2)e−12 iπαR(t), (1)where the sum is taken over all possible Reggeons with positive charge parity,i.e. Pomeron (α(0) ≈ 1), f − a2 (α(0) ≈ 1/2) and pion (α(0) ≈ 0) trajectories.It implies that at asymptotically high energies (s →∞), the energy behaviorof Compton amplitude is governed by the Pomeron exchange.However, as I demonstrate below, if s and Q2 are in the range of a few(GeV/c)2, and t ≈ −m2π , where mπ is a pion mass, a relative contribution fromthe π0–exchange becomes large, exceeding even the diffractive (Pomeron) con-tribution at low t. Taking advantage of different quantum numbers and phasesof t–channel exchanges, it is possible to separate these contributions2 and mea-sure the form factor of γ∗π0 → γ transition (Fγ∗γπ0(Q2)) as a function of Q2.aInvited talk at the Workshop on Virtual Compton Scattering, Clermont–Ferrand, France,June 26–29, 1996bAlso at Theory Division, Kharkov Institute of Physics and Technology, Kharkov, Ukraine1γ∗γe e’p’pRγγM∗qqqa) b)Figure 1: a) VSC amplitude in terms of t–channel Reggeon exchanges, where R stands forthe Pomeron, f − a2, and pion trajectories, and the dashed blob denotes the two–photontransition form factor; b) Quark triangle diagram for γ∗M → γ transition, where M is apseudoscalar meson.Theoretically, the leading–twist QCD contribution to the Fγ∗γπ0 transitionform factor is given by the quark triangle diagram (Fig.1b) with no hard gluonexchange making this process 0th–order in QCD running coupling constantαs with non–perturbative dynamics being contained in the pion distributionamplitude ϕπ(x). 3 The function ϕπ(x) cannot be predicted by perturbativeQCD (except for it asymptotic behavior) and it is crucial for understandingwhether or not one may apply a perturbative QCD description to exclusiveprocesses at given energies. The data on Fγ∗γπ0(Q2) obtained at e+e− collid-ers 4,5 currently available for Q2 up to 8 (GeV/c)2 indicate that ϕπ(x) is closeto its asymptotic form, and therefore ‘soft’, nonperturbative mechanisms aredominant in this energy range, in agreement with theoretical predictions basedon QCD Sum Rules.6This conclusion is so important that it is desirable to have an independentmeasurement of Fγ∗γπ0(Q2), and such a measurement via VCS was proposedearlier at Jefferson Lab,7 for transferred momenta Q2 = 1.0 ÷ 3.5 (GeV/c)2.Higher values of Q2 are also possible for the Jefferson Lab energy upgrade.As Q2 increases, and s stays large, in the low–t limit one may observescaling behavior of the VCS amplitude predicted recently by X. Ji 8 andA. Radyushkin. 9Transition Form Factors of Leading t–channel ExchangesRegge phenomenology proved very successful at describing energy dependenceof hadronic total cross sections and differential cross–sections at low t andhigh s. For VCS, it predicts the slope of the Regge trajectory αR(t) to beindependent of the photon virtuality Q2. This is an important result of Reggetheory which needs to be tested experimentally, as was indicated earlier (see,2e.g., Ref. 10).As far as form factors of γ∗Reggeon → γ transitions are concerned, oneneeds a microscopic theory of Reggeon exchanges to be able to predict their Q2–dependence. This task is challenging for the ‘soft’ Pomeron exchange limitedto a few GeV energy scale, and this problem is still far from being solved inQCD.However, QCD predictions for the two–photon transition form factors re-lated to the t–channel of VCS are available for the case of π0 exchange. Forthe review of theoretical approaches, see Ref.6 I would also like to mentionhere effective quark models for Fγ∗γπ0 based on the extended NJL–model 11and the model of dynamical dressing of propagators and vertices,12 the latteralso addressed the off-shell behavior of Fγ∗γπ0 for the proposed Jefferson Labexperiment. 7The lightest meson which can be exchanged in the t–channel of VCS is π0.It may be possible to extract the corresponding form factor doing a Chew–Lowextrapolation like for the charged pion form factor measurements 13 scheduledat Jefferson Lab.Exchange of π0In the real Compton scattering on the proton at high s and low t, the maincontribution to the cross section is known to be diffractive, due to the Pomeronexchange. I will demonstrate here that the situation is different for the case ofvirtual photons, because of strong enhancement of the π0–exchange term.At small t, the π–trajectory is determined by the pion Born term. Thematrix element of the corresponding transition isM(π0)γ∗p→γp = e2Fγ∗γπ0(Q2)gπNNFπNN (t)Dπ(t)εµναβεµε′∗ν qαq′βū′γ5u, (2)where ε(ε′) is the polarization 4–vector of initial (final) photon, and q(q′) isits momentum (Q2 ≡ −q2), and u(u′) is a bispinor of the initial (final) proton.The pion propagator has the form iDπ(t) = (t−m2π)−1, and I also assumed aconventional monopole form for the cut–off form factor FπNN (t) = Λ2/(Λ2−t).Define four coincidence structure functions (SF) for the (unpolarized)p(e, e′γ)p cross section asd5σdE′dΩedΩp=α316π3E′E|p|c.m.mW1Q211− ε [σT + εσL + ε cos(2ϕ)σTT+√2ε(1 + ε) cos(ϕ)σLT ], (3)ε−1 ≡ 1− 2q2labq2tan2θe2, (4)3where E(E′) is the initial (final) electron energy, θe is the electron scatteringangle, ϕ is the azimuthal angle, and m is the proton mass. The matrix elementgiven by eq.(2) yields the following contributions to SF:σ(π0)T = [(|q| − q0 cos θ)2 + (|q| cos θ − q0)2]X, (5)σ(π0)L = 2Q2 sin2(θ)X, (6)σ(π0)LT = 2√Q2(|q| − q0 cos θ) sin(θ)X, (7)σ(π0)TT = Q2 sin2(θ)X, (8)X =−t(t−m2π)2[q′0FπNN (t)gπNNFγ∗γπ0(Q2)]2, (9)where the c.m. energy of the (initial) final photon is given by q0 = q2+W2−m22W(q′0 =W2−m22W ), and θ is the c.m. angle of outgoing photon.The π0 pole contributes the most to transverse photoabsorption in VCS;the corresponding SF is shown in Fig. 2. Contributions to the other structurefunctions are suppressed at small t by the factor of θ for σLT and θ2 for σLand σTT . The overall factor −t/(t − m2π)2 from Eq.(9) has a pole in theunphysical region, t = m2π , turns to zero at t = 0, and has a maximum inthe physical region at t = −m2π . As can be seen from Fig. 2, dependence ofthe π0 contribution on θ changes dramatically when going from real to virtualphotons. When Q2 = 0, it is suppressed at forward angles (i.e., low t); incontrast, for Q2 6= 0, it is peaked at forward angles. This result is due to theLorentz structure of the γ∗γπ0–vertex defined by Eq.(2): εµναβεµε′∗ν qαq′β.Enhancement of π0 exchange makes it large enough to reach, and evenexceed, the magnitude of diffractive term. For instance, assuming the VMD–model for the γ∗Pomeron→ γ transition form factor, π0 contribution to σT isevaluated to be 25% higher than from the Pomeron at Q2 = 2 (GeV/c)2 and,respectively, three times higher at Q2 = 3 (GeV/c)2. (This result is obtainedat W = 2.5 GeV, t = −m2π).It creates favorable conditions for extracting the form factor Fγ∗γπ0 fromVCS experiments.Separation of the t–channel exchangesOne can attempt to disentangle various t–channel exchanges in VCS.Indeed, the Pomeron a) has quantum numbers of vacuum (except for itsspin), b) contributes almost purely to the imaginary part of VCS amplitudeat low t, and c) does not flip the nucleon spin. On the other hand, the pion a)4050100150012340510Q2, GeV2θc.m.σTFigure 2: Contribution of the π0–exchange Born diagram to the (dimensionless) structurefunction σT of transverse virtual photoabsorption; the invariant mass of final γp state istaken W= 2.5 GeV.is isovector and pseudoscalar b) contributes to the real part of VCS amplitudeat low t, and c) flips the nucleon spin.These circumstances may be used to design the experiments in order toseparate these mechanisms, especially if nuclear targets are used. For instance,coherent VCS on 4He excludes pseudoscalar t–channel exchanges (π0, η, η′),thus providing useful information on Q2 evolution of the diffractive (Pomeron)term. Coherent VCS on deuterium target would rule out the π0–exchange, butkeep exchange of other pseudoscalars with zero isospin. On the other hand, ifthe hadronic target undergoes an isovector transition of any kind (e.g. thresholddeuteron dissociation), it would be completely due to the π0 exchange.In addition, both diffractive and pseudoscalar t–channel exchanges arestrongly suppressed in interference between VCS and the Bethe–Heitler ampli-tudes for the case of unpolarized particles. If, however, spin effects are included,the asymmetry and/or recoil polarization due to the proton polarized normalto the reaction plane would be caused by interference between diffractive andπ0 terms (in which case the electron beam polarization is not required), whilethe sideways (in–plane) asymmetry/polarization with longitudinally polarizedelectrons would be mainly caused by interference between π0–exchange andthe Bethe–Heitler amplitudes.Summary• Exchanges of π0 and the Pomeron in t–channel provide the largest contribu-tions to the amplitude of VCS on the proton at small t and s in the region of5a few GeV2.• It is possible to separate these contributions and study Q2–dependenceof the corresponding form factors.• For the form factor Fγ∗γπ0 , it gives information about the pion distribu-tion amplitude and QCD corrections. It also discriminates between predictionsof effective quark models.• Further increasing energies and momentum transfers, and keeping tsmall, one may observe a transition to the scaling limit of VCS predictedand studied theoretically by X. Ji 8 and A. Radyushkin.9AcknowledgmentsI would like to thank Vincent Breton and the other members of the organizingcommittee for the perfectly organized workshop at Clermont–Ferrand. Discus-sions with A. Radyushkin, N. Isgur, S. Brodsky, and P. Bertin are gratefullyacknowledged. This work was supported by the US Department of Energyunder contract DE–AC05–84ER40150.References1. M. Damashek, F.J. Gilman, Phys. Rev. D 1, 1319 (1970).2. A. Afanasev in Workshop on CEBAF at Higher Energies, Eds. N. Isgurand P. Stoler (CEBAF, Newport News, 1994), p.185.3. S. Brodsky, P. Lepage, Phys. Rev. D 24, 1808 (1981).4. H.J. Behrend et al. (CELLO Collab.), Z. Phys. C 49, 401 (1991).5. V. Savinov (for CLEO Collab.), hep-ex/9507005 (1995); D.M. Asner etal., Preprint CLEO CONF95–24.6. A.V. Radyushkin, R. Ruskov, Phys. Lett. B 374, 173 (1996); for thedetailed version, see Preprint CEBAF–TH–95–18 and hep–ph/9603408.7. CEBAF Letter of Intent #LOI–94/005. Co–Spokesmen: A. Afanasev,J. Gomez, and S. Nanda.8. X. Ji, Preprint MIT–CTP–2517, Cambridge (1996); hep-ph/9603249.9. A.V. Radyushkin, Phys. Lett. B 380, 417 (1996).10. S. Brodsky in Workshop on CEBAF at Higher Energies, Eds. N. Isgurand P. Stoler (CEBAF, Newport News, 1994), p.153.11. H. Ito, W.W. Buck, F. Gross, Phys. Lett. B 287, 23 (1992).12. M.R. Frank, K.L. Mitchell, C.D. Roberts, and P.C. Tandy, Phys. Lett.B 359, 17 (1995).13. CEBAF Experiment #E–93–021. Spokesman: D.J. Mack.6

Virtual Compton Scattering on the proton at high s and low t

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Virtual Compton Scattering on the Proton at High s and Low t

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