The low energy virtual Compton scattering process $eN\to e'N\gamma$ offers a
new and potentially high resolution window on nucleon sturcture via measurement
of so-called generalized polarizabilities (GPs). We present calculations of GPs
within heavy baryon chiral perturbation theory and discuss present experimental
efforts.Comment: Talk given at Workshop on "Physics of Mesons, Baryons and Nuclei,"
  Cracow, May 1998, in celebration of 60th birthday of Prof. Josef Spet

Holstein, Barry R.

English

arXiv

The low energy virtual Compton scattering process eN\rightarrow e\u27N\gamma offers a new and potentially high resolution window on nucleon structure via measurement of so-called generalized polarizabilities (GPs). We present calculations of GPs within heavy baryon chiral perturbation theory and discuss present experimental efforts

Holstein, BR

ScholarWorks@UMass Amherst

University of Massachusetts AmherstScholarWorks@UMass AmherstPhysics Department Faculty Publication Series Physics1998Low energy Compton scattering and nucleonstructureBR Holsteinholstein@physics.umass.eduFollow this and additional works at: https://scholarworks.umass.edu/physics_faculty_pubsPart of the Physical Sciences and Mathematics CommonsThis Article is brought to you for free and open access by the Physics at ScholarWorks@UMass Amherst. It has been accepted for inclusion in PhysicsDepartment Faculty Publication Series by an authorized administrator of ScholarWorks@UMass Amherst. For more information, please contactscholarworks@library.umass.edu.Recommended CitationHolstein, BR, "Low energy Compton scattering and nucleon structure" (1998). ACTA PHYSICA POLONICA B. 330.Retrieved from https://scholarworks.umass.edu/physics_faculty_pubs/330arXiv:nucl-th/9806035v1  12 Jun 1998Low Energy Compton Scattering and NucleonStructureBarry R. HolsteinDepartment of Physics and AstronomyUniversity of MassachusettsAmherst, MA 01002 USAandInstitut fu¨r KernphysikForschungszentrum Ju¨lichD-52425 Ju¨lich, GermanyFebruary 9, 2008AbstractThe low energy virtual Compton scattering process eN → e′Nγ of-fers a new and potentially high resolution window on nucleon structurevia measurement of so-called generalized polarizabilities (GPs). Wepresent calculations of GPs within heavy baryon chiral perturbationtheory and discuss present experimental efforts.01 (En)Lightning Real Compton ReviewThe physics of (real) Compton scattering has received a good deal of recentattention and it is useful, before plunging into the virtual case, to have aquick review of some of the interesting issues in RCS. One of the primarygoals of contemporary particle/nuclear physics is to understand the structureof the nucleon. Indeed this is being pursued at the very highest energy ma-chines such as SLAC and HERMES, wherein one probes the quark/partonsubstructure, as well as at lower energy accelerators such as MAMI andBATES, wherein one studies behavior of the nucleon in terms of a collectivethree quark mode. In recent years one of the important low energy probeshas been Compton scattering, by which one can study the deformation of thenucleon under the influence of quasi-static electric and/or magnetic fields.[1]For example, in the presence of an external electric field ~E the quark dis-tribution of the nucleon becomes distorted, leading to an induced electricdipole moment~p = 4παE ~E (1)in the direction of the applied field, where αE is the electric polarizability.The interaction of this dipole moment with the field leads to a correspondinginteraction energyU = −124παE ~E2 (2)Similarly in the presence of an applied magnetizing field ~H there will be ininduced magnetic dipole moment and interaction energy~µ = 4πβM ~H, U = −124πβM ~H2 (3)For wavelengths large compared to the size of the system, the effective Hamil-tonian for the interaction of a system of charge e and mass m with an elec-tromagnetic field is, of course, given by the simple formH(0) =(~p− e ~A)22m+ eφ (4)As the energy increases, however, one must also take into account polariz-ability effects and the effective Hamiltonian becomesHeff = H(0) −124π(αE ~E2 + βM ~H2) (5)1The Compton scattering cross section from such a system (taken, for sim-plicity, to be spinless) is then given bydσdΩ=(αemm)2 ( ωω′)2[12(1 + cos2 θ)−mωω′αem[12(αE + βM)(1 + cos θ)2+12(αE − βM)(1− cos θ)2 + . . .] (6)where αem is the fine structure constant and ω, ω′ are the initial, final photonenergies respectively. It is clear from Eq. 6 that careful measurement of thedifferential scattering cross section allows extraction of these structure de-pendent polarizability terms provided that i) the energy is large enough thatsuch terms are significant compared to the leading Thomson piece and ii)that the energy is not too large that higher order corrections become impor-tant. In this way the measurement of electric and magnetic polarizabilitiesfor the proton has recently been accomplished using photons in the energyrange 50 MeV < ω < 100 MeV, yielding[2]αpE = (12.1± 0.8± 0.5)× 10−4fm3, βpM = (2.1∓ 0.8∓ 0.5)× 10−4fm3 (7)¿From these results, which say that the polarizabilities of the proton arenearly a factor of a thousand smaller than the corresponding nucleon volume,we learn that the nucleon is a rather rigid object when compared to thehydrogen atom, for example, wherein the electric polarizability and volumeare comparable.Additional structure probes are possible if we exploit the feature of nu-cleon spin.[3] Thus, for example, the presence of a time varying electric fieldin the plane of a rotating system of charges will lead to a charge separationand induced electric dipole moment~p = −γ1~S ×∂ ~E∂t(8)with corresponding interaction energyU1 = −~p · ~E = γ1 ~E · ~S × (~∇× ~B) (9)where we have used the Maxwell equations in writing this form. (Note thatthe ”extra” time or spatial derivative is required by time reversal invariancesince ~S is T-odd.) Similarly other possible structures areU2 = γ2 ~B · ~∇~S · ~E, U3 = γ3 ~E · ~∇~S · ~B, U4 = γ4 ~B · ~S × (~∇× ~E) (10)2and the measurement of these various ”spin-polarizabilities” γi via polar-ized Compton scattering provides a rather different sort of probe for nucleonstructure. Because of the requirement for polarization not much is known atpresent about such spin-polarizabilities, although from dispersion relationsthe combination[4]γp0 ≡ γp1 − γp2 − 2γp4 ≈ −1.34× 10−4fm4 (11)has been calculated and from a global analysis of unpolarized Compton data,to which it contributes in higher orders, one has determined the so-calledbackward polarizability to be[5]γpi = γ1 + γ2 + 2γ4 = (27.7± 2.3± 2.5)× 10−4fm4 (12)Clearly such measurements represent an important goal for the future.2 Virtual Compton Scattering: FormalismRecently a new frontier in Compton scattering has been opened (see, e.g.,[6, 7]) and is in the beginning of being explored: the study of the elec-tron scattering process ep→ e′p′γ in order to obtain information concerningthe virtual Compton scattering (VCS) process γ∗N → γN . As will be dis-cussed below, in addition to the two kinematical variables of real Comptonscattering—the scattering angle θ and the energy ω′ of the outgoing photon—the invariant structure functions for VCS [8],[9] depend on a third kinematicalvariable—the magnitude of the three–momentum transfer to the nucleon inthe hadronic c.m. frame, q¯ ≡ |~q|. The VCS amplitude can then, as shown by[9], be characterized in terms of structure coefficients having q¯ dependenceand are called “generalized polarizabilities” (GPs) of the nucleon in anal-ogy to the well-known polarizability coefficients in real Compton scattering.(However, due to the specific kinematic approximation chosen in [9] there isno one–to–one correspondence between all the real Compton polarizabilitiesand the GPs of Guichon et al. in VCS [9, 10, 11].)The advantage of VCS lies in the virtual nature of the initial state photonand the associated possibility of an independent variation of photon energyand momentum, thus rendering access to a much greater variety of struc-ture information than in the case of real Compton scattering. For example,3one can hope to identify the individual signatures of specific nucleon reso-nances in the various GPs, which cannot be obtained in other processes [6].In this regard, it should be noted that a great deal of theoretical work hastaken place and predictions for both spin-independent and spin-dependentGPs are available within a non–relativistic constituent quark model [9] anda one–loop calculation in the linear sigma model [12]. In addition, variousapproaches have been used to calculate the two spin–independent polarizabil-ities α¯E(q¯2) and β¯M(q¯2), namely, an effective Lagrangian approach includingnucleon resonance effects [13], our calculation of the leading q¯2 dependencein heavy–baryon ChPT (HBChPT) [14] and a calculation of α¯E(q¯2) in theSkyrme model [15]. For an overview of the status at higher energies and inthe deep inelastic regime we refer to [6].The GPs of the nucleon are defined in terms of electromagnetic multipolesas functions of the initial photon momentum q¯ [9] ,P (ρ′L′,ρL)S(q¯2) =[1ω′Lq¯LH(ρ′L′,ρL)S(ω′, q¯)]ω′=0,Pˆ (ρ′L′,L)S(q¯2) =[1ω′Lq¯L+1Hˆ(ρ′L′,L)S(ω′, q¯)]ω′=0, (13)where L (L′) denotes the initial (final) photon orbital angular momentum, ρ(ρ′) the type of multipole transition (0 = C (scalar, Coulomb), 1 =M (mag-netic), 2 = E (electric)), and S distinguishes between non–spin–flip (S = 0)and spin–flip (S = 1) transitions. Mixed–type polarizabilities, Pˆ (ρ′L′,L)S(q¯2),have been introduced, which are neither purely electric nor purely Coulombtype. It is important to note that the above definitions are based on the kine-matical approximation that the multipoles are expanded around ω′ = 0 andonly terms linear in ω′ are retained, which together with current conserva-tion yields selection rules for the possible combinations of quantum numbersof the GPs. In this approximation, ten GPs have been introduced in [9]as functions of q¯2: P (01,01)0, P (11,11)0 , P (01,01)1 , P (11,11)1 , P (01,12)1 , P (11,02)1 ,P (11,00)1 , Pˆ (01,1)0 , Pˆ (01,1)1 , Pˆ (11,2)1 .However, recently it has been proved [10, 11] using crossing symmetryand charge conjugation invariance that only six of the above ten GPs areindependent. Then in the scalar (i.e. spin–independent) sector it is conve-nient to eliminate the mixed polarizability Pˆ (01,1)0 in favor of P (01,01)0 andP (11,11)1, because the latter are generalizations of the electric and magnetic4polarizabilities in real Compton scattering:α¯E(q¯2) = −e24π√32P (01,01)0(q¯2) , β¯M(q¯2) = −e24π√38P (11,11)0(q¯2) . (14)However, in the spin–dependent sector it is not a priori clear which threeGPs should be eliminated with the help of the C-constraints.3 Chiral Calculation of Generalized Polariz-abilitiesAs stated above, there have been a number of theoretical approaches to cal-culation of the generalized polarizabilities in addition to the heavy baryonchiral perturbative study reported below. An advantage of the latter, how-ever, is that it is guaranteed to satisfy all field theory constraints such ascrossing symmetry, charge conjugation invariance, etc. In addition, the cal-culation at O(p3) of nucleon electric and magnetic polarizabilities for the caseof real Compton scattering is know to be in agreement with experiment, soone hopes that the same may hold for the GPs. Indeed the diagrams are thesame. Only the kinematics is different—instead of the usual RCS variablesω′, θ, there is an additional variable |~q|, the center of mass momentum of theincident virtual photon in the VCS case. We have evaluated the GPs usingthe standard formalism of HBχPT and have obtained closed form expressionsfor each.We analyze the VCS process using the standard chiral perturbation the-ory Lagrangian in the heavy baryon formulation to O (p3) in the nucleonsector[16, 17],LχpiN = L(1)piN + L(2)piN + L(3)piN , (15)withL(1)piN = N¯v(iv ·D + gAS · u)Nv ,L(2)piN = −12MN¯v{D ·D − (v ·D)2−12εµνρσvρSσ[fµν+ (1 + 4c6) + 2v(s),µν (1 + 2c7)]}Nv − [Sµ, Sν ][Dµ, Dν]5L(3)piN =12M2N¯v{[fµν+(c6 +18)+ v(s),µν(c7 +14)]× εµνρσSσiDρ + h.c.}Nv , (16)where ε0123 = 1. Here we keep those terms which contribute to a O (p3) VCScalculation. In particular terms linear in the photon fields, which vanish inour gauge, have been omitted. The definitions of symbols used in Eq. 16 arestandard and can be found, e.g. in ref. [16]Explicit forms for each of the GPs can be found in ref. [18] Here for spacereasons we quote only the generalized electric and magnetic polarizabilitiesα¯(3)E (q¯) =e2g2Ampi64π2F 2pi4 + 2 q¯2m2pi−(8− 2 q¯2m2pi− q¯4m4pi)mpiq¯arctan q¯2mpiq¯2(4 + q¯2m2pi) ,β¯(3)M (q¯) =e2g2Ampi128π2F 2pi−(4 + 2 q¯2m2pi)+(8 + 6 q¯2m2pi+ q¯4m4pi)mpiq¯arctan q¯2mpiq¯2(4 + q¯2m2pi) .(17)The meaning of these forms can be found by expandingα¯(3)E (q¯) =5e2g2A384π2F 2pimpi[1−750q¯2m2pi+812800q¯4m4pi+O(q¯6)],β¯(3)M (q¯) =e2g2A768π2F 2pimpi[1 +15q¯2m2pi−39560q¯4m4pi+O(q¯6)], (18)We see then that at the real photon point—q¯ = 0—one reproduces the usualchiral formsαE(0) = 10βM(0) =αg2A48πF 2pimpi= 12.2× 10−4fm3 (19)in good agreement with experiment. New are the predictions for the q¯ depen-dence. In the case of the electric polarizability there is nothing unexpected—one sees a gradual fall-off with momentum transfer corresponding to a size∼1 fm. However, in the magnetic case, there is a surprise—the generalizedmagnetic polarizability is predicted to rise before reaching a maximum atq¯ ∼ 100 MeV and then falling. This behavior is given only in chiral models,and indicates the presence of contributions to the local magnetic polarizabil-ity of opposite sign. However, it is not clear at present what the physical6origin of this effect might be. In any case it will be interesting to look forexperimentally, as it distinguishes chiral models from constituent quark pre-dictions.4 Experimental PossibilitiesOf course, calculation of the generalized polarizabilities is only really inter-esting to the extent that such quantities can be confronted with experimentaldata. The challenges here are great. The problem is firstly that such effectsare relatively small. In the case of RCS, for example, the interference of ofthe polarizability terms in the cross section gives at most ∼ 10% modifica-tions to the cross section at a photon energy of 100 MeV. However, at thisenergy one must already worry about modifications also coming from termsin the effective Lagrangian of order ω4, which are estimated using disper-sive methods. The same is true of generalized polarizabilities. These arenot large effects. However, the problem is much worse. In the case of RCS,the primary background comes from Thomson scattering. However, in thecase of VCS, the basic reaction is ep → e′pγ, which means that one is sen-sitive both to the sought-for e, e′ spectator-γ∗p → γp reaction as well as top, p′ spectator-γ∗e → γe, i.e. the Bethe-Heitler process, wherein the finalphoton is radiated from either the initial or final state electron. Because ofthe lightness of the electron this bremsstrahlung process is very importantand generally dominates the cross section unless one chooses the kinematicregion carefully. In addition, the entire process is quite sensitive to radiativecorrections, which must be calculated quite precisely.[7]Despite these difficulties, several groups have taken up the experimentalchallenge. In the case of an experiment mounted at MAMI data taking hasalready taken place.[19] and it looks as if the group will be able to extractvalues for the generalized polarizabilities. The basic problem in this regard isthat in-plane kinematics were employed, meaning that the (e,e’) and (p,p’)planes were parallel. In this case the Bethe-Heitler reaction produces twoblow-torch-like peaks in the differential scattering cross section correspond-ing to radiation from either the initial or final state electron and the desiredGP effects are small perturbations. The careful measurements of this grouphas been able to verify the basic correctness of the radiative correction cal-culations and can reproduce the data by a sum of Bethe-Heitler and nucleon7Born diagram terms. The effect of the GPs is calculated to be about 10%in the backward direction (i.e. when the photon is emitted oppositely to theelectron directions.)A different approach has been taken by an approved BATES experimentby the OOPS collaboration.[20] In this case the use of the movable OOPSspectrometers allow an experiment to be performed at perpendicular orien-tation of the electron and proton planes, whereby the influence of the Bethe-Heitler forward peaks is minimized. Theoretically one expects the Born andBethe-Heitler contributions to make roughly equal contributions so that anyadditional effect from generalized polarizabilites should be possible to see.Alternatively an approved CEBAF measurement expects to get around theproblem of Bethe-Heitler backgrounds by a different route.[21] Even thoughemploying parallel kinematics, the use of the higher CEBAF beam energymeans that the experiment can be performed at a larger value of longitudinalpolarization—ǫ ≈ 0.95. In this case, the larger virtual photon flux, whichscales as 1/(1 − ǫ) means that the VCS contribution will be correspondingmagnified and calculations reveal possible 20-30 % effects coming from GPs.In addition to these experiments a great deal of work is focussing also onhigher energies and momentum transfers where one may be able to sort outthe basic and angular momentum and spin structure of the nucleon itself.[6]At the present time then the VCS glass is not only full—it is overflowing!AcknowledgementIt is a pleasure to acknowledge the support of the Alexander von Hum-boldt Foundation as well as the hospitality of Forchungszentrum Ju¨lich. Thiswork was also supported by the National Science Foundation.References[1] See, e.g. A.. L’vov, Int. J. Mod. Phys. A8, 5267 (1993); B.R. Holstein,Comm. Nucl. Part. Phys. 20, 301 (1992).[2] F.J. Federspiel et al., Phys. Rev. Lett. 67, 1511 (1991); A.L. Hallin etal., Phys. Rev. C48, 1497 (19939; A. Zieger et al., Phys. Lett. B278,34 (1992); B.E. MacGibbon et al., Phys. Rev. C52, 2097 (1995).8[3] See, e.g. D. Babusci et al., hep-ph/9803347.[4] A.M. Sandorfi et al., Phys. Rev. D50, R6681 (1994).[5] A. Sandorfi et al., nucl-th/9801008.[6] See, e.g. Proc. Workshop on Virtual Compton Scattering VCS96,Clermont-Ferrand, 1996, Ed. V. Breton.[7] P.A.M. Guichon and M. Vandergaeghen, CEA report, to appear in Prog.Part. Nucl. Phys. (1998).[8] R.A. Berg and C.N. Lindner, Nucl. Phys. 26, 259 (1961); H. Arenhoeveland D. Drechsel, Nucl. Phys. A233, 153 (1974); S. Scherer, A. Yu.Korchin, and J.H. Koch, Phys. Rev. C54, 904 (1996)[9] P.A.M. Guichon, G.Q. Liu, and A.W. Thomas, Nucl. Phys. A591, 606(1995) and Aust. J. Phys. 49, 905 (1996).[10] D. Drechsel, G. Kno¨chlein, A. Metz, and S. Scherer, Phys. Rev. C55,424 (1997).[11] D. Drechsel, G. Kno¨chlein, A. Yu. Korchin, A. Metz, and S. Scherer,nucl-th/9704064[12] A. Metz and D. Drechsel, Z. Phys. A356, 351 (1996) and nucl-th/9705010.[13] M. Vanderhaeghen, Phys. Lett. B368, 13 (1996).[14] T.R. Hemmert, B.R. Holstein, G. Knoechlein, and S. Scherer, Phys.Rev. D55, 2630 (1997); Phys. Rev. 79, 22 (1997).[15] M. Kim and D.-P. Min, hep-ph/9704381.[16] V. Bernard, J. Kambor, N. Kaiser, and Ulf-G. Meißner, Nucl. Phys.B388, 315 (1992).[17] G. Ecker and M. Mojzis, Phys. Lett. B365, 312 (1996).[18] D. Drechsel, T.R. Hemmert, B.R. Holstein, and G. Knoechlein, in prepa-ration.9[19] N. d’Hose and Th. Walcher, spokespersons MAMI proposal (1995); N.d’Hose et al., in Proc. of the Conf. on Perspectives in Hadronic Physics(Trieste 1997), eds. S. Boffi, C. Ciofi degli Atti, and M. Giannini,World Scientific (1998); N d’Hose presentation at ECT∗ Workshop onN∗ Physics and Nonperturbative QCD, May 1998.[20] J. Shaw and R. Miskimen, spokespersons BATES proposal (1997).[21] P. Bertin, P.A.M. Guichon and C.E. Hyde-Wright, spokespersons TJ-NAF proposal E-93-05010

Low energy Compton scattering and nucleon structure

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Low Energy Compton Scattering and Nucleon Structure

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