Virtual Compton scattering (VCS) γ ∗p → γp at low CM energy gives access to the generalized polarizabilities of the nucleon. These observables generalize the concept of electromagnetic polarizabilities to the case of a virtual photon. Dedicated VCS experiments have been performed at MAMI, Jefferson Lab and MIT-Bates. The experimental status is reviewed, including methods of analysis and physical results. The measurement of absolute (ep → epγ) cross sections allows the extraction of the two unpolarized VCS structure functions PLL − PTT/ǫ and PLT, which are combinations of the generalized polarizabilities of the proton. Future prospects in the field of VCS at low energy are also presented.Virtualno Comptonovo raspršenje (VCR) γ ∗p → γp na niskim energijama u CMS pruža mogućnost određivanja poopćenih polarizivosti nukleona. Te opservable poopćuju pojam elektromagnetske polarizivosti za slučaj virtualnih fotona. Usmjerena mjerenja VCR načinjena su u MAMI, Jefferson Labu i u MIT-Batesu. Dajemo pregled mjerenja, te metoda analize podataka i ishode. Mjerenja apsolutnih udarnih presjeka reakcije (ep→epγ) dozvoljava izvođenje dviju strukturnih funkcija VCR bez polarizacije PLL − PTT/ǫ i PLT, koje su kombinacije poopćenih polarizivosti protona. Predstavljaju se izgledi budućih istraživanja VCR a niskim energijama

Fonvieille (for the Jefferson Lab Hall A Collaboration and the VCS Collaboration), H.

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ISSN1330–0016CODEN FIZBE7VIRTUAL COMPTON SCATTERING AT LOW ENERGY AND THEGENERALIZED POLARIZABILITIES OF THE NUCLEONH. FONVIEILLEfor the Jefferson Lab Hall A Collaboration and the VCS CollaborationLaboratoire de Physique Corpusculaire IN2P3/CNRS,Université Blaise Pascal - Clermont II, 63170 AUBIERE Cedex, FRANCEReceived 12 September 2003; Accepted 26 April 2004Online 10 October 2004Virtual Compton scattering (VCS) γ∗p → γp at low CM energy gives access to thegeneralized polarizabilities of the nucleon. These observables generalize the conceptof electromagnetic polarizabilities to the case of a virtual photon. Dedicated VCSexperiments have been performed at MAMI, Jefferson Lab and MIT-Bates. The ex-perimental status is reviewed, including methods of analysis and physical results.The measurement of absolute (ep → epγ) cross sections allows the extraction ofthe two unpolarized VCS structure functions PLL − PTT/ǫ and PLT, which arecombinations of the generalized polarizabilities of the proton. Future prospects inthe field of VCS at low energy are also presented.PACS numbers: 13.60.Fz UDC 539.12Keywords: virtual Compton scattering, cross sections, nucleon generalized polarizabilities1. Real and virtual Compton scattering and the concept ofgeneralized polarizabilitiesWe recall that the electric and magnetic polarizabilities αE and βM that arestudied in real Compton scattering (RCS) [1] are a measure of the deformation ofthe nucleon structure under an applied electromagnetic (EM) field. Their smallnessindicates that the proton is a very rigid object. The smallness of βM relative to αEis usually interpreted in terms of a cancellation between a diamagnetic (pion cloud)and a paramagnetic (quark core) contribution.The concept of generalized polarizabilities (GP) was first established by P. Gui-chon et al. in 1995 [2], introducing observables that are defined not only at Q2 = 0but at any photon virtuality Q2. Thus the GPs measure the EM deformation locallyinside the nucleon, with a scale given by Q2, and they can be seen as “deformedFIZIKA B 13 (2004) 1, 209–216 209fonvieille: virtual compton scattering at low energy and the . . .form factors”. The full formalism [3, 4] leads to six independent GPs in the low-est order: two scalar ones αE(Q2) and βM(Q2) (giving the above mentioned RCSpolarizabilities at Q2 = 0) and four spin GPs.Virtual Compton scattering (VCS) can be accessed via photon electroproductionep → epγ. The amplitude of this process is decomposed into the coherent sum ofthe Bethe-Heitler (BH), the VCS Born, and the VCS non-Born amplitudes. The(BH+Born) cross section is known and depends only on the proton EM form factors.The VCS non-Born term contains the unknown physics, i.e. the GPs which are thegoal of the measurements.2. Methods of analysisUp to now, our knowledge of GPs comes from the measurement of cross sectionfor the (unpolarized) ep → epγ process. The main kinematic variables for physicalanalysis are the four-momentum transfer squared Q2, the available energy√s inthe (γp) centre-of-mass (CM) and the CM variables of the outgoing photon: itsmomentum q′ and its spherical angles θ and ϕ with respect to the virtual photonmomentum vector ~q. There are presently two methods to extract GPs, dependingon the range of CM energy. A short description of these methods is given below.2.1. The Low energy theoremBelow the pion production threshold (√s ≤ (mp + mπ)), one can use the Lowenergy theorem (LET), first applied to VCS by P. Guichon et al. [2]. In this context,one obtains the following expression for the (ep→epγ) cross section:d5σEXP = d5σBH+Born + (φq′) ×[v1(PLL −1ǫPTT)+ v2PLT]+ O(q′2) (1)where d5σBH+Born represents the Bethe-Heitler+Born cross section, (φq′) a phasespace factor, and v1 and v2 are known kinematic factors [3]. PLL − 1ǫ PTT and PLTare the structure functions to be measured, and ǫ is the virtual photon polarization.These structure functions are combinations of the GPs (see Ref. [3] for details); theycontain the electric and magnetic generalized polarizabilities αE(Q2) and βM(Q2)in the formPLL ∼ αE(Q2)PLT ∼ βM(Q2) + spin GPs. (2)To extract the structure functions from the data at fixed |~q| and fixed ǫ, one basicallymakes a fit of the quantity (d5σEXP − d5σBH+Born)/(φq′) to the expression of thebracketed term in Eq. (1), thus adjusting two free parameters. The main difficulty ofthe method is that the effect of GPs is small below the pion threshold; it amounts tono more than 10-15% of the cross section. Hence, the systematics of the experimentshave to be well under control.210 FIZIKA B 13 (2004) 1, 209–216fonvieille: virtual compton scattering at low energy and the . . .2.2. The approach of dispersion relationsBoth below and above pion threshold and up to the ππN threshold, one canuse the dispersion relation (DR) formalism developed by B. Pasquini et al. for realand virtual Compton scattering [5]. This allows VCS measurements in the ∆(1232)resonance region to be interpreted in terms of GPs. In this model, the GPs αE(Q2)and βM(Q2) contain free parameters; experimentally, we fit their values at a givenQ2. The advantage of this method is that the (epγ) cross section is more sensitiveto the GPs in the resonance region than below pion threshold.3. Experiments and resultsTable 1 summarizes the experiments performed as of today in the (ep→epγ)channel to investigate the GPs. All of them use magnetic spectrometers to detectthe final electron and proton, and they identify the single photon electroproductionreaction by the missing mass technique. These difficult experiments usually have tomeet the following requirements: i) a good spectrometer resolution to separate thenearby missing mass peak due to (ep→epπ0); ii) measurements of small absolutecross sections with good accuracy, implying a precise calculation of the solid angle;iii) a careful study of experimental cuts (data are not free of background).TABLE 1. Status of performed VCS experiments.Ref. Lab Q2 (GeV2) CM energy Data taking Status[6] MAMI (I) 0.33 < (mN + mπ) 1995+97 published[7] JLab E93-050 0.9, 1.8 up to 1.9 GeV 1998 final stage[8] Bates E97-03 0.05 < (mN + mπ) 2000 analysis[9] MAMI, ~e (II) 0.33 at ∆(1232) 2002-2003 analysisFigure 1 shows the fit of the structure functions using the LET approach, in theexperiments performed at MAMI [6] and in the Hall A of the Thomas Jefferson Na-tional Accelerator Facility (JLab) [10]. The reasonable goodness of the fit confirmsthe validity of the LET even at high Q2 (∼ 0.9 and 1.8 GeV2 at JLab). Figure 2shows (ep→epγ) cross sections from the JLab experiment [11], scanning the reso-nance region for the first time. The data are well reproduced by the DR model inthe region of the ∆(1232) resonance. These data are used to fit the free parametersof the model and subsequently extract the structure functions PLL−(1/ǫ)PTT andPLT at Q2 = 1 GeV2.Figure 3 gives a representation of the structure functions measured by the men-tioned experiments as a function of Q2. On the left plot, the structure functionPLL is obtained from the measurement of (PLL − PTT/ǫ) by adding the value ofFIZIKA B 13 (2004) 1, 209–216 211fonvieille: virtual compton scattering at low energy and the . . .-10-7.5-5-2.502.557.510-0.1 0 0.1 0.2 0.3 0.4 0.5v1/v2Ψ0 /v2 (GeV-2)JLab/Q2=0.92 GeV2MAMI/Q2=0.33 GeV2 JLab/Q2=1.76 GeV2Fig. 1. A graphical representation of the polarizability fit by the LET method. Theordinate is the quantity (d5σEXP −d5σBH+Born)/(φq′)/v2 extrapolated to q′ = 0 ineach bin in (θ, ϕ). The abscissa is the ratio of the kinematic factors v1 and v2. Thefitted slope yields PLL−PTT/ǫ and the intercept yields the PLT structure function.W (GeV)d5σ (pb/MeV/sr/sr)Fig. 2. Photon electroproduction cross section d5σ/dk′labd[Ωe]labd[Ωp]CM measuredin the JLab experiment [11] for six bins in ϕ, at Q2 = 1 GeV2 and cos θγ∗γCM =−0.975. The solid curve is the DR model prediction, and the dashed curve is the(BH+Born) calculation.212 FIZIKA B 13 (2004) 1, 209–216fonvieille: virtual compton scattering at low energy and the . . .Fig. 3. World results on VCS structure functions (statistical errors only). Thedata points at Q2 = 0 are deduced from the TAPS experiment [1], the ones atQ2 = 0.33 GeV2 are from MAMI [6] and the other ones are from the JLab VCSE93-050 experiment including both analysis methods [10, 11]. Some points havebeen shifted in abscissa for clarity. The dash-dotted curve is the calculation ofthe chiral perturbation theory [13], the solid and dashed curves are the DR modelpredictions for different values of the free parameters (Λα and Λβ , related to theGPs αE(Q2) and βM(Q2); see Ref. [5] for more details).PTT/ǫ calculated in the DR model, at the value of Q2 and ǫ of each experiment1.After dividing by the proton electric form factor GpE, one gets the Q2-behaviourof the generalized polarizability αE(Q2), up to a constant factor. On the rightplot, PLT/GpE is proportional to the generalized polarizability βM(Q2), neglectinga small term containing the spin GPs. The JLab data [12] are in good agreementwith the DR model (and parameter values of the dashed curve). At lower Q2,the chiral perturbation theory calculation to order p3 [13] is in good agreementwith the MAMI data for PLT and PLL − PTT/ǫ. Measurements tend to confirm1This is a model-dependent operation, since PTT has not been measured yet.FIZIKA B 13 (2004) 1, 209–216 213fonvieille: virtual compton scattering at low energy and the . . .the turn-over of PLT that is predicted by most models in the low Q2 region. Thisturn-over reflects the competing effects of para- and dia-magnetism in βM(Q2), andthe expected Bates measurement at Q2 = 0.05 GeV2 [8] is of great interest in thisregard.4. Future prospects and conclusionsTable 2 summarizes the foreseen experiments in the field of VCS at low energy.Most projects aim at a deeper disentangling of the individual GPs, by polarizationmeasurements or Rosenbluth-type separations.TABLE 2. Experimental prospects of VCS at low energies. SSA(DSA)= single(double) spin asymmetry.Type of Lab Meas. → structure functions √s Q2experiment (GeV2)Double MAMI DSA → separate the six GPs < πN 0.33polarization & Bates DSA < πN 0.20Polarized JLab d5σ → PLL − PTT/ǫ and PLT ∆ ≤ 4beam SSA → test inputs to DR modelUnpolarized MAMI d5σ → ǫ separation → PTT, PLL < πN 0.33& Bates < πN 0.20Virtual Compton scattering is an active field of research; generalized polarizabil-ities are new observables providing an original way to study nucleon structure, andthere is an ongoing effort to learn more about them both experimentally and theo-retically. Experiments make use of low- and high-energy machines (Bates, MAMI,JLab), of polarization degrees of freedom, and they exploit the versatility of meth-ods to extract GPs at low and high Q2. These observables are also predicted bymany theoretical approaches: chiral perturbation theory [13, 14], quark models [15],linear σ-model [16], dispersion relations [5], effective lagrangian model [17], ..., mostcalculations being valid at rather low Q2. In that view, the results of the JLab VCSE93-050 experiment should stimulate new calculations of the GPs at high Q2.AcknowledgementsThis work was supported by DOE, NSF, by contract DE-AC05-84ER40150 un-der which the Southeastern Universities Research Association (SURA) operatesthe Thomas Jefferson National Accelerator Facility for DOE, by the CEA, theCNRS-IN2P3 and the Conseil Régional d’Auvergne (France), the FWO-Flandersand the BOF-Gent University (Belgium) and by the European Commission ERBFMRX-CT96-0008. I thank all my VCS colleagues for stimulating work.214 FIZIKA B 13 (2004) 1, 209–216fonvieille: virtual compton scattering at low energy and the . . .References[1] V. Olmos de León et al., Eur. Phys. J. A 10 (2001) 207 and references therein.[2] P. A. M. Guichon, G. Q. Liu and A. W. Thomas, Nucl. Phys. A 591 (1995) 606.[3] P. A. M. Guichon and M. Vanderhaeghen, Prog. Part. Nucl. Phys. 41 (1998) 125.[4] D. Drechsel et al., Phys. Rev. C 55 (1997) 424; Phys. Rev. C 57 (1998) 941.[5] B. Pasquini et al., Eur. Phys. J. A 11 (2001) 185; D. Drechsel, B. Pasquini and M.Vanderhaeghen, Phys. Rept. 378 (2003) 99.[6] J. Roche et al., Phys. Rev. Lett. 85 (2000) 708.[7] P. Y. Bertin, C. Hyde-Wright, P. Guichon et al., CEBAF Proposal PR93050 (1993);http://hallaweb.jlab.org/experiment/E93-050/vcs.html[8] J. Shaw, R. Miskimen et al., MIT-Bates Proposal E97-03 (1997).[9] H. Merkel and N. D’Hose, spokespersons, MAMI Proposal (2000).[10] S. Jaminion et al., in preparation.[11] G. Laveissière et al., in preparation.[12] H. Fonvieille, JLab E93-050, preliminary results, in Proc. Baryons 2002 Conf., WorldScientific (2003), hep-ex/0206035.[13] T. Hemmert et al., Phys. Rev. D 62 (2000) 014013.[14] C. W. Kao and M. Vanderhaeghen, Phys. Rev. Lett. 89 (2002) 272002.[15] G. Q. Liu, A. W. Thomas and P. A. M. Guichon, Austral. J. Phys. 49 (1996) 905;B. Pasquini, S. Scherer and D. Drechsel, Phys. Rev. C 63 (2001) 025205.[16] A. Metz and D. Drechsel, Z. Phys. A 356 (1996) 351; Z. Phys. A 359 (1997) 165.[17] M. Vanderhaeghen, Phys. Lett. B 368 (1996) 13.FIZIKA B 13 (2004) 1, 209–216 215fonvieille: virtual compton scattering at low energy and the . . .VIRTUALNO COMPTONOVO RASPRŠENJE NA NISKIM ENERGIJAMA IPOOPĆENE POLARIZIVOSTI NUKLEONAVirtualno Comptonovo raspšenje (VCR) γ∗p → γp na niskim energijama u CMSpruža mogućnost odred–ivanja poopćenih polarizivosti nukleona. Te opservable po-općuju pojam elektromagnetske polarizivosti za slučaj virtualnih fotona. Usmjerenamjerenja VCR načinjena su u MAMI, Jefferson Labu i u MIT-Batesu. Dajemopregled mjerenja, te metoda analize podataka i ishode. Mjerenja apsolutnih udarnihpresjeka reakcije (ep→epγ) dozvoljava izvod–enje dviju strukturnih funkcija VCRbez polarizacije PLL − PTT/ǫ i PLT, koje su kombinacije poopćenih polarizivostiprotona. Predstavljaju se izgledi budućih istraživanja VCR a niskim energijama.216 FIZIKA B 13 (2004) 1, 209–216

Virtualno Comptonovo raspršenje na niskim energijama i poopćene polarizivosti nukleona

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