Poincare invariance, gauge invariance, conservation of parity and time
reversal invariance are respected in an impulse approximation evaluation of the
handbag diagram. Proton wave functions, previously constrained by comparison
with measured form factors, that incorporate the influence of quark transverse
and orbital angular momentum (and the corresponding violation of proton
helicity conservation) are used. Computed cross sections are found to be in
reasonably good agreement with early measurements. The helicity correlation
between the incident photon and outgoing proton, $K_{LL}$, is both large and
positive at back angles. For photon laboratory energies of $\le$ 6 GeV, we find
that $K_{LL}\ne A_{LL}$, $D_{LL}\ne1$, and that the polarization $P$ can be
large.Comment: 9 pages, 6 figures. Replacement fixes some typos, improves references
  and figures. An error in Fig. 6 was corrected and related comments in the
  text change

Gerald A. Miller

H. Rollnik

S. Théberge

V. B. Berestetskii

English

arXiv

Poincare invariance, gauge invariance, conservation of parity and time reversal invariance are respected in an impulse approximation evaluation of the handbag diagram. Proton wave functions, previously constrained by comparison with measured form factors, that incorporate the influence of quark transverse and orbital angular momentum (and the corresponding violation of proton helicity conservation) are used. Computed cross sections are found to be in reasonably good agreement with early measurements. The helicity correlation between the incident photon and outgoing proton, K{sub LL}, is both large and positive at back angles. For photon laboratory energies of {le} 6 GeV, we find that K{sub LL} {ne} A{sub LL}, D{sub LL} {ne} 1, and that the polarization P can be large

Miller, Gerald A.

UNT (University of North Texas) Digital Library

arXiv:nucl-th/0402092 v1   25 Feb 2004NT@UW-04-002,ECT-03-034,LBNL-54622Handling the Handbag Diagram in Compton Scattering on theProtonGerald A. MillerDepartment of Physics, University of Washington, Seattle, WA 98195-1560(Dated: February 25, 2004)AbstractPoincaré invariance, gauge invariance, conservation of parity and time reversal invariance arerespected in an impulse approximation evaluation of the handbag diagram. Proton wave functions,previously constrained by comparison with measured form factors, that incorporate the influence ofquark transverse and orbital angular momentum (and the corresponding violation of proton helicityconservation) are used. Computed cross sections are found to be in reasonably good agreement withearly measurements. The helicity correlation between the incident photon and outgoing proton,KLL, is both large and positive at back angles. For photon laboratory energies of ≤ 6 GeV, wefind that KLL 6= ALL, DLL 6= 1, and that the polarization P can be large.PACS numbers: 13.60.FzKeywords: Compton Scattering, high momentum transfer, relativistic constituent quark models1Kq q’(a) (b)K’p p’FIG. 1: Direct (a) and crossed (b) handbag graphs for the Compton amplitude.The recent and planned experimental accessibility[1, 2] of Real Compton Scattering onthe proton at large momentum transfer make this reaction a promising new probe of shortdistance structure. The scattering amplitude depends on the square of the charge of a struckquark, and so provides a ground-state to ground-state matrix element that is different thanthe ones involved in the electromagnetic form factors[3].One primary goal has been to determine the dominant reaction mechanism that allows theproton to accommodate the large momentum transfer while remaining a proton. Accordingto perturbative QCD (pQCD)[4, 5] the three active valence quarks share the momentumtransfer via the exchange of two gluons that each carry a large momentum. In the pQCDtreatment, hard gluon exchanges are included in an effective current operator, and so canbe taken to occur within the time duration of the reaction. The most recent calculations[6] find that this mechanism yields cross sections that are about 10 times smaller than onesmeasured at photon energies of 6 GeV or less. Another approach uses overlaps of softnon-perturbative wave functions[7, 8]. As noted in the review[9], the invariant amplitudeis obtained by evaluating the so-called handbag diagrams of Fig. 1. The large momentumtransfer occurs on a single quark, with a probability amplitude determined by the overlap ofthe initial and final state wave functions. Here the high momentum transfer is accommodatedby the exchange of an uncountable number of gluons that occurs either before and after thereaction takes place. Such calculations have had reasonable success in reproducing themeasurements [10].Previous treatments of the handbag diagram[7, 8, 10] have provided a unifying relationbetween elastic form factors, real Compton scattering and virtual Compton scattering. Thisapproach has led to interesting predictions that are testable experimentally. Our concernhere is with understanding the limitations of the approximations. One involves assertingthat the longitudinal momentum of the proton is carried by a single quark[8, 10]. Anotherinvolves the neglect of the effects of hadron helicity flip[7, 8, 10] in high momentum transferexclusive reactions. A specific consequence is that the observables KLL (which involves thehelicity of the final proton) and ALL (which involves the helicity of the initial proton) arepredicted to be the same.The present intent is to use a model wave function that provides a reasonably gooddescription of all four nucleon electromagnetic form factors[11, 12, 13] to evaluate the graphsof Fig. 1 in a manner that avoids neglecting the effects of hadronic helicity non-conservation.The essential feature of this is that relativistic effects induce significant quark transverse andorbital angular momentum that causes violations of hadronic helicity conservation.2TABLE I: Different parameter sets, units in terms of fm. From Ref. [13]Set(legend) m β γ1 dash 1.8 3.65 4.12 dot-dash 1.8 3.65 3.93 solid 1.7 2.65 3.7We begin describing the formalism by reviewing the salient features of the wave functionsof Refs. [11, 12, 13]. This model starts with a wave function for 3 relativistic constituentquarks:Ψ(pi) = u(p1)u(p2)u(p3)ψ(p1, p2, p3), (1)where pi represents space, spin and isospin indices: pi = pisi, τi and repeated indices aresummed over. The spinors u are canonical Dirac spinors. The components of momenta areexpressed in terms of light cone notation: pi ≡ (p+,p⊥)i, with p−i = (p⊥2i +m2)/p+i . Thethree momenta pi of the quarks can be transformed to the total and relative momenta tofacilitate the separation of the center of mass motion as P =∑i pi, ξ = p+1 /(p+1 + p+2 ), η =p+1 + p+2 /P+,k⊥ = (1− ξ)p1⊥− ξp2⊥ ,K⊥ = (1− η)(p1⊥ +p2⊥)− ηp3⊥ . One may expressthe proton wave function in the center of mass frame in which the individual momenta aregiven by p1⊥ = k⊥ + ξK⊥, p2⊥ = −k⊥ + (1 − ξ)K⊥ ,p3⊥ = −K⊥. The structure of thecolor-spin-isospin wave function can be understood in a familiar form. This eigenstate ofspin[14, 15] is a product of an anti-symmetric color wave function with a symmetric flavor-spin-momentum wave function, given by Ψ = 1√2(φρ χρ + φλ χλ) Φ, where φρ represents amixed-antisymmetric and φλ a mixed-symmetric flavor wave function and, χρ,λ representsmixed symmetric or anti-symmetric spin wave functions in terms of Dirac spinors. Thelower components of these contain terms in which the spin of the quark is opposite to thatof the proton, with the difference accommodated by the orbital angular momentum. Suchterms are responsible for reproducing the experimental feature thatQF2/F1 is approximatelyconstant for Q2 ≥ 2 (GeV/c)2[11, 12, 13].The spin-independent momentum-space wave function is a function of the mass op-erator M0 of a non-interacting system of any Pµ: M20 = (K2⊥ +m2η)/η(1 − η) +(k2⊥ +m2)/(ηξ(1− ξ)),where m is the u, d quark mass. We take the S-state orbital functionΦ(M0) to be of a power law form: Φ(M0) = N/(M20 + β2)γ that depends on β, γ and theconstituent quark mass m. Parameters of the Light Front Cloudy Bag Model[13], whichmakes the Cloudy Bag Model[16] relativistic, are displayed in Table I. The effects of thepion cloud are unimportant at high momentum transfer and are ignored here.The use of light front dynamics enables one to relate the proton wave functions in differentreference frames using a kinematic boost. If the proton acquires a transverse momentum (isboosted) by the absorption of a quantity of momentum, ∆ ≡ (0,∆⊥) = q− q′ by the thirdquark, the effects of the boost are obtained merely by replacing the momenta k⊥,K⊥ thatappeat in the wave function with k⊥, K⊥ − η∆⊥.The evaluation of the Compton scattering amplitude MS′,S(ǫ′, ǫ) is made using an im-pulse approximation in which the Compton scattering, occurring through the Born andcrossed Born graph of Fig. 1. The antisymmetric nature of Ψ allows us to take the scat-tering to occur on the third quark of charge Q3. The incident (outgoing) photon has fourmomentum and polarization vector q, ǫ (q, ǫ′), evaluated using the γp center of mass frame.3MS′,S(ǫ′, ǫ) depends on the initial spin S and final spin S ′ of the proton, as well as on ǫ, ǫ′:MS′,S(ǫ′, ǫ) = 3Q23∫dηd2K⊥∑s,s′ρS′,s′;S,s(η,K′,k)ū(K ′, s′)Oǫ′,ǫ(K′, K)u(K, s), (2)where ρ(η,K′,k) ≡∫dξd2k⊥Ψ†S′,s′(ξ,k⊥, η,K′⊥)ΨS,s(ξ,k⊥, η,K⊥), and in which the repeatedindices(s, s′) represent both spin and isospin (charge) quantum numbers of the struck thirdquark. The quantities K,K ′ are four vectors given by K = ((1−η)M0,−K) and 6Ku(K, s) =mu(K, s), 6K ′u(K ′, s) = mu(K ′, s). The latter two relations are necessary in maintaininggauge invariance. The operator Oǫ′,ǫ(K′, K) represents Compton scattering on a quark:Oǫ′,ǫ(K′, K) = 6ǫ ′∗12(K + q +K ′ + q′) · γ +m(K + q)2 −m26ǫ+ 6ǫ12(K ′ − q +K − q′) · γ +m(K ′ − q)2 −m26ǫ ′∗. (3)The numerators appearing in Eq. (3) are displayed in a form symmetric with respect to theinitial and final states. This is necessary to maintain the time reversal invariance of theresulting amplitudes. Deriving Eq. (2) from the full four-dimensional formalism involvesintegration over the minus components of the wave function and neglecting modifications ofthe intermediate propagator of the struck quark caused by spectator quarks. Note also thatthe total momentum of the proton does not enter into the expression (2), so that our resultsare independent of frame.Each of s, s′, ǫ, ǫ′, has two possible values, so there are 16 amplitudes. The use of parityconservation and time reversal invariance reduces the number of independent amplitudes tosix[17]. We calculate the 16 amplitudes and demonstrate explicitly that there really are onlysix independent ones. It is not obvious that the impulse approximation used in obtainingEq. (2) will yield only six amplitudes. Applying parity invariance immediately reduces thenumber of independent amplitudes to eight. But the effects of time reversal invarianceare more difficult to satisfy. This is because the sum of the quark minus-momenta is notequal to the minus-momentum of the proton. Thus conservation of four-momentum occursonly at the hadronic level, but not in the γq scattering. However, numerical calculationsshow that a reasonably accurate approximation can be made that leads to the respect oftime reversal invariance. Examine Eq. (2) and shift the variable of integration according toK⊥ → K⊥+η∆/2. Then note that if the momentum transfer ∆ is large compared to typicalmomenta appearing in the wave function, one may ignore the component of K parallel to∆ in evaluating the matrix element ū(K ′, s′)O(ǫ′, ǫ)u(K, s). Numerical work shows thatusing this approximation doesn’t change the computed values of observables by significantamounts, but does reduce the number of independent amplitudes to exactly six, and alsomaintain gauge invariance.The relevant experimental observables involve photons and protons of a definite helicity.The γp-cm helicity amplitudes are determined by making a unitary transformation on thespin amplitudes MS′,S(ǫ′, ǫ):Φµ′λ′,µλ =∑S′,ST ∗(p′)S′λ′MS′,S(ǫ′µ, ǫµ)TS,λ(p), (4)where µ, λ represent the helicity of the initial photon and initial proton, and TS,λ(p) ≡ū(p, s)uH(p, λ)/(2Mp) in which u represents an ordinary Dirac spinor and uH represents ahelicity spinor. Then the differential cross section is expressed as:dσdt=164π(s−m2)2Σµ,µ′,λ,λ′|Φµ′,λ′,µλ|2. (5)4FIG. 2: Cross sections. The data are from Ref. [18]. Three curves are obtained with parametersdefined in Table 1 and three are obtained by increasing the quark mass m by 10%.Our results for this are shown in Fig. 2. Using the parameter set of Table I leads to crosssections that are a bit too small, but increasing the quark mass by 10%, leads to qualitativelygood agreement. The dependence of dσ/dt on photon laboratory energy in shown in Fig. 3.There is a 20% overall normalization uncertainty, and the data shown in Figs. 2 and 3 havebeen mul.ltiplied by by 0.8.The spin dependent observables are defined according to Ref. [10]. One set involves bothphoton and proton helicities. The correlation KLL between the helicities of the incomingphoton and the outgoing proton isKLLdσdt=dσ(µ = +, λ′ = +)dt−dσ(µ = +, λ′ = −)dt, (6)and is especially interesting because the experimental result[1] will be announced soon. Thecorrelation, ALL, between the incident photon and the proton in the initial state isALLdσdt=12[dσ(µ = +, λ = +)dt−dσ(µ = +, λ = −)dt]. (7)In the handbag approach of Refs.[8, 10], the amplitudes Φ−−++ ≡ Φ2 and Φ−++− ≡ Φ6 havethe same magnitude, and this leads to the prediction that KLL = ALL. Our results, for thephoton energies of immediate experimental interest (3.2 and 4.3 GeV)[1, 2], are displayedin Fig. 4, and the related cross sections can be found in Fig. 5. Examining Fig. 4 showsthat the predicted values of KLL and ALL do not depend on the quark mass, and have littlevariation with energy. The values of KLL are large and positive at large scattering angles.This is similar to the KLL of Refs.[8, 10], but their predicted equality between KLL and ALLdoes not hold in our approach– for backward scattering angles, we find KLL ≈ −ALL.This difference demands explanation. In Ref. [10] the equality between KLL and ALLarises from neglecting the variation of the quark’s minus and tranverse momenta, and the5FIG. 3: Energy dependence of cross sections. The data are from Ref. [18]. The curves are obtainedusing set 3 of Table I, with an increase in the value of m by 10%.effects of quark orbital angular momentum that lead to vanishing proton helicity flip matrixelements. In particular, the massless quarks are taken to move collinearly with the protonwhich consequently does not change helicity. This could be valid for very large energy andmomentum transfer, but perhaps not for photons of a few GeV. In our model, the role oforbital angular momentum and non-conservation of the proton helicity is the crucial aspectin reproducing the proton form factors[11].Let’s consider scattering by 180◦ (back-angle scattering) to illustrate how it is thatKLL 6= ALL. Examine Eq. (4). The transformation matrices T (p)Sλ reduce to over-laps between two-component spinors, in which σ · p̂|λ〉 = λ|λ〉. Then for back-anglescattering T ∗(p′)S′λ′ =1√2(δS′,+1/2 + λ′δS′,−1/2), T (p)Sλ =1√2(δS,+1/2 − λδS,−1/2), so thatΦ2 =12[M++ + M−− −M+− −M−+] , Φ6 =12[M++ + M−− + M+− + M−+] . The de-pendence of MS′S on the photon polarization vectors is suppressed, as these are the samefor Φ2,6. In our notation, equality of Φ2 and Φ6 can only occur if each of the proton spin flipmatrix elements M+−,M−+ vanish, or if their sum vanishes. A brief inspection of Eq. (3)shows that the spin-flip matrix elements do not vanish. A further inspection shows thatthe terms γ · (q + q′) = γ02q0 lead to operators of the form (1 + σx) evaluated betweenPauli spinors. The operator σx raises and lowers spins with exactly strength. Thus the sumM+− + M−+ does not vanish, Φ2 6= Φ6, and ALL 6= KLL.There are other polarization variables[10]. The helicity transfer from the incoming to theoutgoing photon is given by DLLdσ/dt = (dσ(µ = +, µ′ = +)/dt− dσ(µ = +, µ′ = −)/dt) .Ref. [10] finds thatDLL ≈ 1. The polarization of the incoming proton is defined by Pdσ/dt =12[dσ(↑)/dt− dσ(↓)/dt] . In Ref. [10] small corrections lead to estimating that P ≈ 3%. Ourpredictions, shown in Fig 6, are that DLL and P can have very significant values.One may also consider sideway proton spin directions. The correlation between thehelicity of the incoming photon and the sideways (S) polarization of the incoming proton,6FIG. 4: KLL, ALL obtained using the parameter sets of Fig. 2.FIG. 5: Computed values of dσdt the parameter sets of Fig. 2.parallel or anti-parallel to the S-direction is defined[10] as ALS, and the one for the sidewayspolarization of the outgoing proton is defined as KLS. We find KLS = 0 and ALS = 0. Asingle-spin observable for Compton scattering is the incoming photon asymmetry Σ [10],and our result is that Σ = 0.Let’s summarize. Poincaré invariance, gauge invariance, conservation of parity and timereversal invariance are respected in our impulse approximation evaluation of the handbag7FIG. 6: Computed values of DLL,P the parameter sets of Fig. 2.diagrams. Proton wave functions, previously constrained by comparison with measuredform factors, that incorporate the influence of quark orbital angular momentum (and thecorresponding violation of proton helicity conservation) are used. omputed cross sectionsare in reasonably good agreement with early measurements. The value of KLL is largeand positive for scattering at large angles. The use of detailed wave functions leads toKLL 6= ALL, DLL 6= 1 and large values of the polarization, P . With our model functions,photon laboratory energies of 6 GeV or less are too low for the simplifying assumptionsthat lead to proton helicity conservation to be valid. Future experiments that measureany of ALL, DLL and P can determine whether or not proton helicity conservation holds inCompton scattering at any given energy.AcknowledgmentsI thank B. Wojtsekhowski for suggesting this project. I am grateful to the the ECT*(Trento), the National Institute for Nuclear Theory at the University of Washington, theNuclear Science Theory Group at LBL and the CSSSM (Adelaide) for providing hospitalityduring the course of this work. The Guggenheim foundation, on the other hand, was of nohelp at all. This work is partially supported by the USDOE.[1] Jefferson Laboratory Experiment E99-114, C. Hyde-Wright, A. M. Nathan, and B. Wojt-sekhowski, spokespersons.[2] Jefferson Laboratory Experiment E03-003, A. M. Nathan and B. Wojtsekhowski, spokesper-sons.8[3] M. K. Jones et al. [Jefferson Lab Hall A Collaboration], Phys. Rev. Lett. 84, 1398 (2000);O. Gayou et al.Phys. Rev. Lett. 88, 092301 (2002)[4] A. S. Kronfeld and B. Nizic, Phys. Rev. D 44, 3445 (1991) [Erratum-ibid. D 46, 2272 (1992)].[5] G. R. Farrar and H. y. Zhang, Phys. Rev. Lett. 65, 1721 (1990); Phys. Rev. D 41, 3348 (1990)[Erratum-ibid. D 42, 2413 (1990)]. E. Maina and G. R. Farrar, Phys. Lett. B 206, 120 (1988).[6] T. C. Brooks and L. J. Dixon, Phys. Rev. D 62, 114021 (2000).[7] A. V. Radyushkin, Phys. Rev. D 58, 114008 (1998).[8] M. Diehl, T. Feldmann, R. Jakob and P. Kroll, Eur. Phys. J. C 8, 409 (1999); Phys. Lett.B460,204 (1999)[9] M. Diehl, Phys. Rept. 388, 41 (2003).[10] M. Diehl, T. Feldmann, R. Jakob and P. Kroll, Phys. Lett. B 460, 204 (1999); Eur. Phys. J.C 23, 301 (2002) [Erratum-ibid. C 31, 279 (2003)].[11] G. A. Miller and M. R. Frank, Phys. Rev. C 65, 065205 (2002)[12] M.R. Frank, B.K. Jennings and G.A. Miller, Phys. Rev. C 54, 920 (1996).[13] G. A. Miller, Phys. Rev. C 66, 032201 (2002).[14] V. B. Berestetskii and M. V. Terent’ev. Sov. J. Nucl. Phys. 25, 347 (1977).[15] P. L. Chung and F. Coester. Phys. Rev. D 44, 229, (1991); F. Coester, Prog. in Part. andNucl. Phys. 29, 1 (1992). F. Schlumpf, U. Zurich Ph. D. Thesis, hep-ph/9211255.[16] S. Théberge, A. W. Thomas and G. A. Miller, Phys. Rev. D22 (1980) 2838; (1981) 2106;A. W. Thomas, S. Théberge, and G. A. Miller, Phys. Rev. D24 (1981) 216; S. Théberge,G. A. Miller and A. W. Thomas, Can. J. Phys. 60, 59 (1982). G. A. Miller, A. W. Thomasand S. Théberge, Phys. Lett. B 91, 192 (1980).[17] H. Rollnik and P. Stichel, in E.Paul et al., Elementary Particle Physics, Springer TractsVol.79, Berlin 1976. ; C. Bourrely, J. Soffer and E. Leader, Phys. Rept. 59, 95 (1980). M. Diehl,Eur. Phys. J. C 19, 485 (2001).[18] M. A. Shupe et al., Phys. Rev. D 19, 1921 (1979).9

Handling the Handbag Diagram in Compton Scattering on the Proton

eScholarship - University of California

Lawrence Berkeley National LaboratoryLawrence Berkeley National LaboratoryTitleHandling the Handbag Diagram in Compton Scattering on the ProtonPermalinkhttps://escholarship.org/uc/item/72b3h9zfAuthorMiller, Gerald A.Publication Date2004-02-25 Peer reviewedeScholarship.org Powered by the California Digital LibraryUniversity of CaliforniaarXiv:nucl-th/0402092 v1   25 Feb 2004NT@UW-04-002,ECT-03-034,LBNL-54622Handling the Handbag Diagram in Compton Scattering on theProtonGerald A. MillerDepartment of Physics, University of Washington, Seattle, WA 98195-1560(Dated: February 25, 2004)AbstractPoincaré invariance, gauge invariance, conservation of parity and time reversal invariance arerespected in an impulse approximation evaluation of the handbag diagram. Proton wave functions,previously constrained by comparison with measured form factors, that incorporate the influence ofquark transverse and orbital angular momentum (and the corresponding violation of proton helicityconservation) are used. Computed cross sections are found to be in reasonably good agreement withearly measurements. The helicity correlation between the incident photon and outgoing proton,KLL, is both large and positive at back angles. For photon laboratory energies of ≤ 6 GeV, wefind that KLL 6= ALL, DLL 6= 1, and that the polarization P can be large.PACS numbers: 13.60.FzKeywords: Compton Scattering, high momentum transfer, relativistic constituent quark models1Kq q’(a) (b)K’p p’FIG. 1: Direct (a) and crossed (b) handbag graphs for the Compton amplitude.The recent and planned experimental accessibility[1, 2] of Real Compton Scattering onthe proton at large momentum transfer make this reaction a promising new probe of shortdistance structure. The scattering amplitude depends on the square of the charge of a struckquark, and so provides a ground-state to ground-state matrix element that is different thanthe ones involved in the electromagnetic form factors[3].One primary goal has been to determine the dominant reaction mechanism that allows theproton to accommodate the large momentum transfer while remaining a proton. Accordingto perturbative QCD (pQCD)[4, 5] the three active valence quarks share the momentumtransfer via the exchange of two gluons that each carry a large momentum. In the pQCDtreatment, hard gluon exchanges are included in an effective current operator, and so canbe taken to occur within the time duration of the reaction. The most recent calculations[6] find that this mechanism yields cross sections that are about 10 times smaller than onesmeasured at photon energies of 6 GeV or less. Another approach uses overlaps of softnon-perturbative wave functions[7, 8]. As noted in the review[9], the invariant amplitudeis obtained by evaluating the so-called handbag diagrams of Fig. 1. The large momentumtransfer occurs on a single quark, with a probability amplitude determined by the overlap ofthe initial and final state wave functions. Here the high momentum transfer is accommodatedby the exchange of an uncountable number of gluons that occurs either before and after thereaction takes place. Such calculations have had reasonable success in reproducing themeasurements [10].Previous treatments of the handbag diagram[7, 8, 10] have provided a unifying relationbetween elastic form factors, real Compton scattering and virtual Compton scattering. Thisapproach has led to interesting predictions that are testable experimentally. Our concernhere is with understanding the limitations of the approximations. One involves assertingthat the longitudinal momentum of the proton is carried by a single quark[8, 10]. Anotherinvolves the neglect of the effects of hadron helicity flip[7, 8, 10] in high momentum transferexclusive reactions. A specific consequence is that the observables KLL (which involves thehelicity of the final proton) and ALL (which involves the helicity of the initial proton) arepredicted to be the same.The present intent is to use a model wave function that provides a reasonably gooddescription of all four nucleon electromagnetic form factors[11, 12, 13] to evaluate the graphsof Fig. 1 in a manner that avoids neglecting the effects of hadronic helicity non-conservation.The essential feature of this is that relativistic effects induce significant quark transverse andorbital angular momentum that causes violations of hadronic helicity conservation.2TABLE I: Different parameter sets, units in terms of fm. From Ref. [13]Set(legend) m β γ1 dash 1.8 3.65 4.12 dot-dash 1.8 3.65 3.93 solid 1.7 2.65 3.7We begin describing the formalism by reviewing the salient features of the wave functionsof Refs. [11, 12, 13]. This model starts with a wave function for 3 relativistic constituentquarks:Ψ(pi) = u(p1)u(p2)u(p3)ψ(p1, p2, p3), (1)where pi represents space, spin and isospin indices: pi = pisi, τi and repeated indices aresummed over. The spinors u are canonical Dirac spinors. The components of momenta areexpressed in terms of light cone notation: pi ≡ (p+,p⊥)i, with p−i = (p⊥2i +m2)/p+i . Thethree momenta pi of the quarks can be transformed to the total and relative momenta tofacilitate the separation of the center of mass motion as P =∑i pi, ξ = p+1 /(p+1 + p+2 ), η =p+1 + p+2 /P+,k⊥ = (1− ξ)p1⊥− ξp2⊥ ,K⊥ = (1− η)(p1⊥ +p2⊥)− ηp3⊥ . One may expressthe proton wave function in the center of mass frame in which the individual momenta aregiven by p1⊥ = k⊥ + ξK⊥, p2⊥ = −k⊥ + (1 − ξ)K⊥ ,p3⊥ = −K⊥. The structure of thecolor-spin-isospin wave function can be understood in a familiar form. This eigenstate ofspin[14, 15] is a product of an anti-symmetric color wave function with a symmetric flavor-spin-momentum wave function, given by Ψ = 1√2(φρ χρ + φλ χλ) Φ, where φρ represents amixed-antisymmetric and φλ a mixed-symmetric flavor wave function and, χρ,λ representsmixed symmetric or anti-symmetric spin wave functions in terms of Dirac spinors. Thelower components of these contain terms in which the spin of the quark is opposite to thatof the proton, with the difference accommodated by the orbital angular momentum. Suchterms are responsible for reproducing the experimental feature thatQF2/F1 is approximatelyconstant for Q2 ≥ 2 (GeV/c)2[11, 12, 13].The spin-independent momentum-space wave function is a function of the mass op-erator M0 of a non-interacting system of any Pµ: M20 = (K2⊥ +m2η)/η(1 − η) +(k2⊥ +m2)/(ηξ(1− ξ)),where m is the u, d quark mass. We take the S-state orbital functionΦ(M0) to be of a power law form: Φ(M0) = N/(M20 + β2)γ that depends on β, γ and theconstituent quark mass m. Parameters of the Light Front Cloudy Bag Model[13], whichmakes the Cloudy Bag Model[16] relativistic, are displayed in Table I. The effects of thepion cloud are unimportant at high momentum transfer and are ignored here.The use of light front dynamics enables one to relate the proton wave functions in differentreference frames using a kinematic boost. If the proton acquires a transverse momentum (isboosted) by the absorption of a quantity of momentum, ∆ ≡ (0,∆⊥) = q− q′ by the thirdquark, the effects of the boost are obtained merely by replacing the momenta k⊥,K⊥ thatappeat in the wave function with k⊥, K⊥ − η∆⊥.The evaluation of the Compton scattering amplitude MS′,S(ǫ′, ǫ) is made using an im-pulse approximation in which the Compton scattering, occurring through the Born andcrossed Born graph of Fig. 1. The antisymmetric nature of Ψ allows us to take the scat-tering to occur on the third quark of charge Q3. The incident (outgoing) photon has fourmomentum and polarization vector q, ǫ (q, ǫ′), evaluated using the γp center of mass frame.3MS′,S(ǫ′, ǫ) depends on the initial spin S and final spin S ′ of the proton, as well as on ǫ, ǫ′:MS′,S(ǫ′, ǫ) = 3Q23∫dηd2K⊥∑s,s′ρS′,s′;S,s(η,K′,k)ū(K ′, s′)Oǫ′,ǫ(K′, K)u(K, s), (2)where ρ(η,K′,k) ≡∫dξd2k⊥Ψ†S′,s′(ξ,k⊥, η,K′⊥)ΨS,s(ξ,k⊥, η,K⊥), and in which the repeatedindices(s, s′) represent both spin and isospin (charge) quantum numbers of the struck thirdquark. The quantities K,K ′ are four vectors given by K = ((1−η)M0,−K) and 6Ku(K, s) =mu(K, s), 6K ′u(K ′, s) = mu(K ′, s). The latter two relations are necessary in maintaininggauge invariance. The operator Oǫ′,ǫ(K′, K) represents Compton scattering on a quark:Oǫ′,ǫ(K′, K) = 6ǫ ′∗12(K + q +K ′ + q′) · γ +m(K + q)2 −m26ǫ+ 6ǫ12(K ′ − q +K − q′) · γ +m(K ′ − q)2 −m26ǫ ′∗. (3)The numerators appearing in Eq. (3) are displayed in a form symmetric with respect to theinitial and final states. This is necessary to maintain the time reversal invariance of theresulting amplitudes. Deriving Eq. (2) from the full four-dimensional formalism involvesintegration over the minus components of the wave function and neglecting modifications ofthe intermediate propagator of the struck quark caused by spectator quarks. Note also thatthe total momentum of the proton does not enter into the expression (2), so that our resultsare independent of frame.Each of s, s′, ǫ, ǫ′, has two possible values, so there are 16 amplitudes. The use of parityconservation and time reversal invariance reduces the number of independent amplitudes tosix[17]. We calculate the 16 amplitudes and demonstrate explicitly that there really are onlysix independent ones. It is not obvious that the impulse approximation used in obtainingEq. (2) will yield only six amplitudes. Applying parity invariance immediately reduces thenumber of independent amplitudes to eight. But the effects of time reversal invarianceare more difficult to satisfy. This is because the sum of the quark minus-momenta is notequal to the minus-momentum of the proton. Thus conservation of four-momentum occursonly at the hadronic level, but not in the γq scattering. However, numerical calculationsshow that a reasonably accurate approximation can be made that leads to the respect oftime reversal invariance. Examine Eq. (2) and shift the variable of integration according toK⊥ → K⊥+η∆/2. Then note that if the momentum transfer ∆ is large compared to typicalmomenta appearing in the wave function, one may ignore the component of K parallel to∆ in evaluating the matrix element ū(K ′, s′)O(ǫ′, ǫ)u(K, s). Numerical work shows thatusing this approximation doesn’t change the computed values of observables by significantamounts, but does reduce the number of independent amplitudes to exactly six, and alsomaintain gauge invariance.The relevant experimental observables involve photons and protons of a definite helicity.The γp-cm helicity amplitudes are determined by making a unitary transformation on thespin amplitudes MS′,S(ǫ′, ǫ):Φµ′λ′,µλ =∑S′,ST ∗(p′)S′λ′MS′,S(ǫ′µ, ǫµ)TS,λ(p), (4)where µ, λ represent the helicity of the initial photon and initial proton, and TS,λ(p) ≡ū(p, s)uH(p, λ)/(2Mp) in which u represents an ordinary Dirac spinor and uH represents ahelicity spinor. Then the differential cross section is expressed as:dσdt=164π(s−m2)2Σµ,µ′,λ,λ′|Φµ′,λ′,µλ|2. (5)4FIG. 2: Cross sections. The data are from Ref. [18]. Three curves are obtained with parametersdefined in Table 1 and three are obtained by increasing the quark mass m by 10%.Our results for this are shown in Fig. 2. Using the parameter set of Table I leads to crosssections that are a bit too small, but increasing the quark mass by 10%, leads to qualitativelygood agreement. The dependence of dσ/dt on photon laboratory energy in shown in Fig. 3.There is a 20% overall normalization uncertainty, and the data shown in Figs. 2 and 3 havebeen mul.ltiplied by by 0.8.The spin dependent observables are defined according to Ref. [10]. One set involves bothphoton and proton helicities. The correlation KLL between the helicities of the incomingphoton and the outgoing proton isKLLdσdt=dσ(µ = +, λ′ = +)dt−dσ(µ = +, λ′ = −)dt, (6)and is especially interesting because the experimental result[1] will be announced soon. Thecorrelation, ALL, between the incident photon and the proton in the initial state isALLdσdt=12[dσ(µ = +, λ = +)dt−dσ(µ = +, λ = −)dt]. (7)In the handbag approach of Refs.[8, 10], the amplitudes Φ−−++ ≡ Φ2 and Φ−++− ≡ Φ6 havethe same magnitude, and this leads to the prediction that KLL = ALL. Our results, for thephoton energies of immediate experimental interest (3.2 and 4.3 GeV)[1, 2], are displayedin Fig. 4, and the related cross sections can be found in Fig. 5. Examining Fig. 4 showsthat the predicted values of KLL and ALL do not depend on the quark mass, and have littlevariation with energy. The values of KLL are large and positive at large scattering angles.This is similar to the KLL of Refs.[8, 10], but their predicted equality between KLL and ALLdoes not hold in our approach– for backward scattering angles, we find KLL ≈ −ALL.This difference demands explanation. In Ref. [10] the equality between KLL and ALLarises from neglecting the variation of the quark’s minus and tranverse momenta, and the5FIG. 3: Energy dependence of cross sections. The data are from Ref. [18]. The curves are obtainedusing set 3 of Table I, with an increase in the value of m by 10%.effects of quark orbital angular momentum that lead to vanishing proton helicity flip matrixelements. In particular, the massless quarks are taken to move collinearly with the protonwhich consequently does not change helicity. This could be valid for very large energy andmomentum transfer, but perhaps not for photons of a few GeV. In our model, the role oforbital angular momentum and non-conservation of the proton helicity is the crucial aspectin reproducing the proton form factors[11].Let’s consider scattering by 180◦ (back-angle scattering) to illustrate how it is thatKLL 6= ALL. Examine Eq. (4). The transformation matrices T (p)Sλ reduce to over-laps between two-component spinors, in which σ · p̂|λ〉 = λ|λ〉. Then for back-anglescattering T ∗(p′)S′λ′ =1√2(δS′,+1/2 + λ′δS′,−1/2), T (p)Sλ =1√2(δS,+1/2 − λδS,−1/2), so thatΦ2 =12[M++ + M−− −M+− −M−+] , Φ6 =12[M++ + M−− + M+− + M−+] . The de-pendence of MS′S on the photon polarization vectors is suppressed, as these are the samefor Φ2,6. In our notation, equality of Φ2 and Φ6 can only occur if each of the proton spin flipmatrix elements M+−,M−+ vanish, or if their sum vanishes. A brief inspection of Eq. (3)shows that the spin-flip matrix elements do not vanish. A further inspection shows thatthe terms γ · (q + q′) = γ02q0 lead to operators of the form (1 + σx) evaluated betweenPauli spinors. The operator σx raises and lowers spins with exactly strength. Thus the sumM+− + M−+ does not vanish, Φ2 6= Φ6, and ALL 6= KLL.There are other polarization variables[10]. The helicity transfer from the incoming to theoutgoing photon is given by DLLdσ/dt = (dσ(µ = +, µ′ = +)/dt− dσ(µ = +, µ′ = −)/dt) .Ref. [10] finds thatDLL ≈ 1. The polarization of the incoming proton is defined by Pdσ/dt =12[dσ(↑)/dt− dσ(↓)/dt] . In Ref. [10] small corrections lead to estimating that P ≈ 3%. Ourpredictions, shown in Fig 6, are that DLL and P can have very significant values.One may also consider sideway proton spin directions. The correlation between thehelicity of the incoming photon and the sideways (S) polarization of the incoming proton,6FIG. 4: KLL, ALL obtained using the parameter sets of Fig. 2.FIG. 5: Computed values of dσdt the parameter sets of Fig. 2.parallel or anti-parallel to the S-direction is defined[10] as ALS, and the one for the sidewayspolarization of the outgoing proton is defined as KLS. We find KLS = 0 and ALS = 0. Asingle-spin observable for Compton scattering is the incoming photon asymmetry Σ [10],and our result is that Σ = 0.Let’s summarize. Poincaré invariance, gauge invariance, conservation of parity and timereversal invariance are respected in our impulse approximation evaluation of the handbag7FIG. 6: Computed values of DLL,P the parameter sets of Fig. 2.diagrams. Proton wave functions, previously constrained by comparison with measuredform factors, that incorporate the influence of quark orbital angular momentum (and thecorresponding violation of proton helicity conservation) are used. omputed cross sectionsare in reasonably good agreement with early measurements. The value of KLL is largeand positive for scattering at large angles. The use of detailed wave functions leads toKLL 6= ALL, DLL 6= 1 and large values of the polarization, P . With our model functions,photon laboratory energies of 6 GeV or less are too low for the simplifying assumptionsthat lead to proton helicity conservation to be valid. Future experiments that measureany of ALL, DLL and P can determine whether or not proton helicity conservation holds inCompton scattering at any given energy.AcknowledgmentsI thank B. Wojtsekhowski for suggesting this project. I am grateful to the the ECT*(Trento), the National Institute for Nuclear Theory at the University of Washington, theNuclear Science Theory Group at LBL and the CSSSM (Adelaide) for providing hospitalityduring the course of this work. The Guggenheim foundation, on the other hand, was of nohelp at all. This work is partially supported by the USDOE.[1] Jefferson Laboratory Experiment E99-114, C. Hyde-Wright, A. M. Nathan, and B. Wojt-sekhowski, spokespersons.[2] Jefferson Laboratory Experiment E03-003, A. M. Nathan and B. Wojtsekhowski, spokesper-sons.8[3] M. K. Jones et al. [Jefferson Lab Hall A Collaboration], Phys. Rev. Lett. 84, 1398 (2000);O. Gayou et al.Phys. Rev. Lett. 88, 092301 (2002)[4] A. S. Kronfeld and B. Nizic, Phys. Rev. D 44, 3445 (1991) [Erratum-ibid. D 46, 2272 (1992)].[5] G. R. Farrar and H. y. Zhang, Phys. Rev. Lett. 65, 1721 (1990); Phys. Rev. D 41, 3348 (1990)[Erratum-ibid. D 42, 2413 (1990)]. E. Maina and G. R. Farrar, Phys. Lett. B 206, 120 (1988).[6] T. C. Brooks and L. J. Dixon, Phys. Rev. D 62, 114021 (2000).[7] A. V. Radyushkin, Phys. Rev. D 58, 114008 (1998).[8] M. Diehl, T. Feldmann, R. Jakob and P. Kroll, Eur. Phys. J. C 8, 409 (1999); Phys. Lett.B460,204 (1999)[9] M. Diehl, Phys. Rept. 388, 41 (2003).[10] M. Diehl, T. Feldmann, R. Jakob and P. Kroll, Phys. Lett. B 460, 204 (1999); Eur. Phys. J.C 23, 301 (2002) [Erratum-ibid. C 31, 279 (2003)].[11] G. A. Miller and M. R. Frank, Phys. Rev. C 65, 065205 (2002)[12] M.R. Frank, B.K. Jennings and G.A. Miller, Phys. Rev. C 54, 920 (1996).[13] G. A. Miller, Phys. Rev. C 66, 032201 (2002).[14] V. B. Berestetskii and M. V. Terent’ev. Sov. J. Nucl. Phys. 25, 347 (1977).[15] P. L. Chung and F. Coester. Phys. Rev. D 44, 229, (1991); F. Coester, Prog. in Part. andNucl. Phys. 29, 1 (1992). F. Schlumpf, U. Zurich Ph. D. Thesis, hep-ph/9211255.[16] S. Théberge, A. W. Thomas and G. A. Miller, Phys. Rev. D22 (1980) 2838; (1981) 2106;A. W. Thomas, S. Théberge, and G. A. Miller, Phys. Rev. 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Handling the handbag diagram in Compton scattering on the proton

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Handling the Handbag Diagram in Compton Scattering on the Proton

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