Alternative way to understand the unexpected results of the JLab polarization experiments to measure the Sachs form factors ratio
aa r X i v : . [ h e p - ph ] M a r Alternative way to understand the unexpected results of the JLab polarizationexperiments to measure the Sachs form factors ratio
M. V. Galynskii ∗ and E. A. Kuraev Joint Institute for Power and Nuclear Research-Sosny, BAS, 220109 Minsk, Belarus Joint Institute for Nuclear Research, Dubna, Moscow region, 141980 Russia
In the one-photon exchange approximation we discuss questions related to the interpretation ofunexpected results of the JLab polarization experiments to measure the Sachs form factors ratio G E /G M in the region 1 . ≤ Q ≤ . . For this purpose, we developed an approach whichessentially is a generalization of the constituent-counting rules of the perturbative QCD (pQCD)for the case of massive quarks. We assume that at the lower boundary of the considered region thehard-scattering mechanism of pQCD is realized. Within the framework of the developed approachwe calculated the hard kernel of the proton current matrix elements J ± δ,δp for the full set of spincombinations corresponding to the number of the spin-flipped quarks, which contribute to the protontransition without spin-flip ( J δ,δp ) and with the spin-flip ( J − δ,δp ). This allows us to state that (i)around the lower boundary of the considered region, the leading scaling behavior of the Sachs formfactors has the form G E , G M ∼ /Q , (ii) the dipole dependence ( G E , G M ∼ /Q ) is realized in theasymptotic regime of pQCD when τ ≫ τ = Q / M ) in the case when the quark transitions withspin-flip dominate, (iii) the asymptotic regime of pQCD in the JLab experiments has not yet beenachieved, and (iv) the linear decrease of the ratio G E /G M at τ < J δ,δp by spin-flip transitions of two quarks and an additional contribution to J − δ,δp by spin-fliptransitions of three quarks. PACS numbers: 11.80.Cr, 13.40.Gp, 13.88.+e, 25.30.Bf
I. INTRODUCTION
Experiments aimed at studying the proton form fac-tors (FFs), the electric ( G E ) and magnetic ( G M ) ones,which are frequently referred to as the Sachs FFs, havebeen performed since the mid 1950 s [1, 2] by using elas-tic electron-proton scattering. In the case of unpolar-ized electrons and protons, all experimental data on thebehavior of the proton FFs were obtained by using theRosenbluth formula [1] for the differential cross sectionfor the reaction ep → ep ; that is, dσd Ω e = α E cos ( θ e / E sin ( θ e /
2) 11 + τ (cid:16) G E + τε G M (cid:17) . (1)Here, τ = Q / M , Q = − q = 4 E E sin ( θ e /
2) is thesquare of the momentum transfer to the proton and M is the proton mass; E , E , and θ e are, respectively, theinitial-electron energy, the final-electron energy, and theelectron scattering angle in the rest frame of the initialproton; ε is the degree of the linear polarization of thevirtual photon [3–5], ε − = 1 + 2(1 + τ ) tan ( θ e / α = 1 /
137 is the fine-structure constant. Expression (1)was obtained in the one-photon exchange approximationand the electron mass was set to zero.With the aid of Rosenbluth’s technique, it was foundthat the experimental dependences of G E and G M on Q are well described up to 10 GeV by the dipole- ∗ [email protected] approximation expression G E = G M /µ = G D ( Q ) ≡ (1 + Q / . − , (2)where µ is the proton magnetic moment ( µ = 2 . . < Q < . , there was a linear decreasein the ratio R = µG E /G M with increasing Q , R = 1 − .
13 ( Q − . , (3)which indicates that G E falls faster than G M . This is incontradiction with data obtained with the aid of Rosen-bluth’s technique; according to those, the approximateequality R ≈ R using the polarization transfermethod [9, 10] and by Rosenbluth’s method [11] only con-firmed this contradiction. In order to resolve this contra-diction, it was assumed that the discrepancy in questionmay be caused by disregarding, in the respective analy-sis, the contribution of two-photon exchange (TPE) (seework [12], the reviews [13, 14], and references therein).At the present time, three experiments aimed at studyingthe contribution of TPE are known. It is an experiment An analogous prediction was made by D.V. Volkov in 1965 basedon SU(6) symmetry for baryon octet [8]. at the VEPP-3 storage ring in Novosibirsk, the OLYM-PUS experiment at the DORIS accelerator at DESY, andthe EG5 CLAS experiment at JLab.In [15], we proposed a new alternative to the method[4] for determining the Sachs FFs in the process e~p → e~p on the basis of measuring cross sections for spin-flip andnon-spin-flip transitions for protons.The aims of this paper are (i) the interpretation inthe one-photon exchange approximation unexpected re-sults of the JLab polarization experiments to measure theSachs FFs ratio as well as the explanation of the reasonfor the linear dependence in (3), and (ii) the determina-tion of the conditions for the realization of the Sachs FFsdipole dependence based on the use of the hard-scatteringmechanism (HSM) of perturbative QCD (pQCD) underthe assumption that the onset of pQCD starts aroundthe lower boundary of the considered region.It is, in general, admitted that the onset of the asymp-totic regime of pQCD starts around the J/ Ψ masssquared. It was first observed in work [16] that the pro-ton magnetic FF, G M , follows the asymptotic pQCD pre-dictions of [17, 18] and Q G M becomes nearly constant(with the logarithmic accuracy, modulo log( Q ) factors)starting at Q ≈ . The answer to the questionwhat is in general admitted at present on the onset ofpQCD can be found in [19–21]. In Refs. [19, 20], basedon using completely different approaches, it is shown thatthe point of transition from non-perturbative QCD topQCD correspond to a momentum scale Q ∼ Q ∼ Q ∼ Q within the APT approach is the disappear-ance of the nonphysical singularities of the perturbationtheory series. It should be noted that in the known workof Belitsky et al. [22] the authors have performed numer-ical calculations in the framework of pQCD in the regionof 0 . ≤ Q ≤ . ; therefore, they proceeded fromthe assumption that the onset of pQCD starts alreadyat Q = 0 . . It is very likely that the results ofRef. [22] are an indirect proof of the correctness resultsof Ref. [21] obtained in the framework of the APT.In order to achieve goals, we will use the formalism ofthe method for calculating the matrix elements of QEDprocesses in the diagonal spin basis (DSB) [23–25]. II. PHYSICAL MEANING OF THE SACHS FFS
It is well known that in the Breit frame of the ini-tial and the final proton, the Sachs FFs G E and G M describe the distributions of the proton charge and mag- netic moment, respectively, and their advantage is dueto the simplification of expression (1). The question ofwhether there is any physical meaning behind the decom-position of G E and G M in Rosenbluth’s cross section wasnot raised and not discussed either in textbooks or in sci-entific literature. Nevertheless, it was shown many yearsago in the work of Sikach [23] that the FFs G E and G M factorize in the DSB even at the level of amplitudes incalculating (in an arbitrary reference frame) the protoncurrent matrix elements in the cases of non-spin-flip andspin-flip transitions for the proton. A. Diagonal spin basis
In the DSB, the spin four-vectors s and s of fermionswith four-momenta q (before the interaction) and q (af-ter it) have the form [23] s = − ( v v ) v − v p ( v v ) − , s = ( v v ) v − v p ( v v ) − , (4)where v = q /M and v = q /M . They satisfy ordinaryconditions – that is, s q = s q = 0 and s = s = −
1– and are invariant under the transformations of a littlegroup of Lorentz group L q q common to particles with 4-momenta q and q : L q q q = q and L q q q = q . Thisgroup is isomorphic to the one-parameter subgroup of therotational group SO (3) with an axis whose direction isdetermined by the three-dimensional vector [26] a = q /q − q /q . (5)For the two particles in question, the spin projectionsonto the direction specified by the vector a in Eq. (5)simultaneously have specific values [26]. Let us consider the realization of the DSB in the ini-tial proton rest frame, where q = ( q , q ) = ( M, ). Inthis case for the vector a in Eq. (5) we have a = n = q / | q | ; that is, the direction of the final proton motionis a common direction onto which one projects the spinsin question. Therefore, in the rest frame of the initialproton the polarization state of the final proton is a he-licity state, while the spin four-vectors s and s in theDSB (4) have the form s = (0 , n ) , s = ( | v | , v n ) . (6)Note in the DSB the particles with the 4-momenta q (before interaction) and q (after interaction) have com-mon spin operators [24, 25]. This makes it possible to The vector a in Eq. (5) is the difference of two three-dimensionalvectors, and the geometric image of the difference of two three-vectors is a diagonal of the parallelogram. This is the reason whythe term “DSB” was introduced by academician F.I. Fedorov.Note the Breit frame, where q = − q is a particular case of theDSB. separate the interactions with and without change in thespin states of the particles involved in the reaction and,thus, to trace the dynamics of the spin interaction. B. Amplitudes of the proton current in DSB
The matrix elements of the proton current in the one-photon exchange approximation has the form( J ± δ,δp ) µ = u ± δ ( q , s ) Γ µ ( q ) u δ ( q , s ) , (7)Γ µ ( q ) = F γ µ + F M (ˆ qγ µ − γ µ ˆ q ) , (8)where u ( q , s ) and u ( q , s ) are the bispinors of theprotons with four-momenta q and q and spin four-vectors s and s ; accordingly, we have q i = M , and u ( q i ) u ( q i ) = 2 M ( i = 1 , q = q − q is the four-momentum transfer to the proton; γ µ and ˆ q are the Diracoperators, ˆ q = γ µ q µ ; F and F are, respectively, theDirac and Pauli FFs.The matrix elements of the proton current (7) cor-responding to the proton transitions without and withspin-flip calculated in the DSB (4) have the form [23, 25]( J δ,δp ) µ = 2 M G E ( b ) µ , (9)( J − δ,δp ) µ = − M δ √ τ G M ( b δ ) µ , (10)where G E and G M are the Sachs FFs G E = F − τ F , G M = F + F . (11)In expressions (9), (10) we used an orthonormalized basis(tetrad) of four-vectors b A ( A = 0 , , , b = q + / q q , b = q − / q − q − , ( b ) µ = ε µνκσ b ν b κ b σ , ( b ) µ = ε µνκσ q ν q κ p σ /ρ . (12)Here, q + = q + q , q − = q = q − q , ε µνκσ is the Levi-Civita tensor ( ε = − p is the four-momentum ofthe initial electron, and ρ is determined from the nor-malization conditions b = b = b = − b = −
1, where b ± δ = b ± iδb , b ∗ δ = b − δ , and b δ b ∗ δ = − δ = ± G E , and magnetic, G M , FFs, respectively. It is precisely because of thisfactorization of G E and G M that Rosenbluth’s formula isdecomposed for the sum of two terms containing only G E and G M , which are responsible for the contributions ofthe transitions without and with spin-flip of the proton,respectively. The spin states of massless particles in the DSB coincide up tosign with helical states [25]; in this case, the DSB formalism isequivalent to the CALKUL group method [27].
In the case of pointlike particles having a mass m q ,their current amplitudes have the form( J δ,δq ) µ = 2 m q ( b ) µ , (13)( J − δ,δq ) µ = − m q δ √ τ q ( b δ ) µ , τ q = Q q / m q . (14)In the ultrarelativistic (massless) case, only spin-flip tran-sitions contribute to the cross section for the process be-ing considered, since the amplitudes without spin-flip inEq. (9) and Eq. (13) vanish. At first glance, this con-clusion contradicts the well-known fact that in the mass-less limit, only amplitudes of the processes correspond-ing to helicity-conserving transitions do not vanish. Suchprocesses are frequently referred to as non-spin-flip pro-cesses. However, this terminology is highly conditionalsince the particles involved have different directions ofmotion before and after the interaction event. Moreover,it is erroneous since in helicity-conserving processes athigh energies the spins of the particles are in fact flipped.There is no contradiction here since in the DSB the ini-tial state for ultrarelativistic particles is a helicity state,while the final state has a negative helicity [25], with theresult M − δ,δ = M − ( − λ ) ,λ = M λ,λ , M δ,δ = M − λ,λ = 0 . (15)Along with the representation (8) for Γ µ ( q ), anotherequivalent representation is often used,Γ µ ( q ) = G M γ µ − ( q + q ) µ M F . (16)On the basis of the explicit form (8) and (16) for Γ µ ( q ),it is often stated (see e.g. [6, 7, 28]) that the pro-ton Dirac FF F (proton Pauli FF F ) corresponds tohelicity-conserving (helicity-flip) transitions of the pro-ton, respectively. In fact, it is G M ( G E ) rather than F ( F ) that is responsible for helicity-conserving (helicity-flip) transitions at high q and q [see Eqs. (9), (10),(15)].We note that in the literature sometimes there is noclear understanding of the physical meaning of the quan-tity ε in formula (1). So in [6, 10, 11, 14, 28] it is writtenthat the quantity ε is a degree of the longitudinal polar-ization of the virtual photon. In fact ε is the degree ofthe linear polarization of the virtual photon (see [3–5]). III. THE Q DEPENDENCE OF THE SACHSFFS G E AND G M Let us consider the Q dependence of the absolute val-ues of the matrix elements of the proton currents (9), (10)and pointlike-particle ones (13), (14). We note that thefactorization of 2 M and 2 m q in expressions (9), (10), and(13), (14) is due to normalizing the particle bispinors bythe condition ¯ u i u i = 2 m i . In performing further calcula-tions, it is more convenient to employ the normalizationconditions ¯ u i u i = 1. Since | b | = 1 and | b δ | = √ J ± δ,δp and pointlike-particle ones J ± δ,δq , we getthe following expressions J δ,δp = G E , J − δ,δp = √ τ G M , (17) J δ,δq = 1 , J − δ,δq = √ τ q . (18)In these expressions due to | b δ | = √ τ and τ q but τ ′ = 2 τ and τ ′ q = 2 τ q ,but below we shall omit the primes.Let us consider the HSM of pQCD [17] in the process ep → ep that is realized as we believe at Q ≥ . Inthis case the leading contribution to the proton current(7) can be presented as a sum of the hard gluon exchangeprocesses, where the proton is replaced by a set of threealmost on mass shell quarks as illustrated in Fig. 1. FIG. 1: Typical Born diagrams for the proton FFs.
Below we will suppose the masses of quarks m q to beequal to 1 / M and the fraction oftheir transfer momenta to be equal. So we have τ q = τ . (19)Under such simplifying assumptions it can easily be veri-fied that the matrix element corresponding to the sum oftwo gauge-invariant diagrams, shown in Fig. 1, has theform ( J ± δ,δp , ) µ ∼ ( J ± δ,δq ) ν ( J ± δ,δq ) ν ( J ± δ,δq ) µ /Q , (20)where Q in the denominator corresponds to the prod-uct of two gluon propagators, of an order of magnitude1 /Q and two quark propagators of an order 1 /Q . There-fore, the absolute magnitudes of the proton current ma-trix elements J ± δ,δp that correspond to the contribution ofthe full set of possible Feynman diagrams can be writtenas the product of three point-quark current amplitudes J ± δ,δq (18) divided by Q , J ± δ,δp ∼ J ± δ,δq J ± δ,δq J ± δ,δq /Q . (21)Relations (17), (18), (19), and (21) make it possible toshow how there arises the Q dependence of G E and G M in the HSM of pQCD and explain the results of polariza-tion experiments at JLab.There are two possibilities for a proton non-spin-fliptransition: (i) none of the three quarks undergoes a spin-flip transition and (ii) two quarks undergo a spin-fliptransition, while the third does not. We denote the num-ber of such ways as n − δ,δqE = [0 , n − δ,δqM = [1 , n − δ,δqE × n − δ,δqM = (0 , ⊕ (0 , ⊕ (2 , ⊕ (2 , . (22)Note due to Eqs. (18), (19) at τ ≪ τ ≫
1) the quarktransition without (with) spin-flip dominates. Therefore,the sets (0,1) and (2,3) with the minimal and maximalnumber of spin-flip quarks are realized at τ ≪ τ ≫
1, respectively.
A. The set (0,1), G E , G M ∼ /Q , G E / G M ∼ Let us consider the first (0,1) set corresponding to aproton non-spin-flip transition J δ,δp for the case wherethere is no spin-flip for any of the three quarks and cor-responding to the proton transition J − δ,δp where spin-flip occurs only for one quark. According to Eqs. (17),the matrix elements of the proton current J δ,δp and J − δ,δp must be proportional to G E and G M , respectively; as aresult, we have J δ,δp = G E ∼ × × / Q , (23) J − δ,δp = √ τ G M ∼ √ τ × × / Q , (24)where the factors of unity and √ τ on the right-hand sideof Eqs. (23) and (24) correspond to non-spin-flip transi-tions for three pointlike quarks and to the spin-flip tran-sition for one quark. As a result, we have G E ∼ Q , G M ∼ Q , G E G M ∼ . (25)Therefore, for the set (0,1) the FFs ratio G E /G M behavesin just the same way as in the dipole case. However, thedependencies G E , G M ∼ /Q are not dipole ones. B. The set (0,3), G E ∼ /Q , G M ∼ /Q Let us consider the (0,3) set. For this purpose we writeequalities similar to (23) and (24); that is, J δ,δp = G E ∼ × × / Q , (26) J − δ,δp = √ τ G M ∼ √ τ × √ τ × √ τ / Q . (27)From here, we obtain G E ∼ Q , G M ∼ τQ , G E G M ∼ τ ∼ M Q , (28) Q G E G M ∼ M = const. (29)Relation (29) is sometimes called in the literature theBrodsky saturation law; it really corresponds to a maxi-mal possible number of the quark spin-flip transitions. C. The set (2,1), G E ∼ /Q , G M ∼ /Q Let us consider the set (2,1) in Eq. (22). Following thesame line of reasoning as above, we have G E ∼ τQ , G M ∼ Q , G E G M ∼ τ ∼ Q M , (30) Q G M G E ∼ M = const. (31) D. The set (2,3), G E , G M ∼ /Q , G E / G M ∼ The set (2,3) is generated by spin-flip transitions fortwo quarks in the case of the contribution to J δ,δp and byspin-flip transitions for all three quarks in the case of thecontribution to J − δ,δp . For this case we have J δ,δp = G E ∼ √ τ × √ τ × / Q , (32) J − δ,δp = √ τ G M ∼ √ τ × √ τ × √ τ / Q . (33)Hence, we obtain G E ∼ Q , G M ∼ Q , G E G M ∼ . (34)Therefore, the dipole dependence in the behavior of theFFs G E and G M on Q occurs in the set (2,3) at τ ≫ IV. SPIN PARAMETRIZATION FOR G E / G M The non-spin-flip ( J δ,δp ) and spin-flip ( J − δ,δp ) proton-current amplitudes can be represented as the linear com-binations J δ,δp = α J δ,δq J − δ, − δq J δ,δq + α J − δ,δq J δ, − δq J δ,δq , (35) J − δ,δp = β J − δ,δq J δ,δq J − δ, − δq + β J − δ,δq J δ, − δq J − δ,δq , (36)where the coefficients α , α , β , and β have a clearphysical meaning that is determined by their indices.With the aid of Eqs. (35) and (36), one can readily obtaina general expression for the ratio G E /G M . The result is G E G M = α + α τβ + β τ . (37) This expression may serve as a basis for constructing spinparametrization and fits experimental data obtained bymeasuring the ratio G E /G M .We showed above that at τ ≪ G E /G M ∼
1, mustoccur. In this case the coefficients α and β in Eq. (37)must have the values close to unity. With allowance forthis comment, we expand the right-hand side of (37) in apower series for τ . As a result, we get the law of a lineardecrease in the ratio R = G E /G M as Q increases, R ≈ − ( β − α ) τ . (38) V. CONCLUSION
We have discussed in the one-photon exchange ap-proximation the questions related to the interpretationof the JLab polarization experiment’s unexpected resultsto measure the Sachs FFs ratio G E /G M in the region1 . ≤ Q ≤ . . For this purpose, in the case ofthe HSM of the pQCD, we calculated the hard kernelof the proton current matrix elements J ± δ,δp for the fullset of spin combinations corresponding to a number ofthe spin-flipped quarks, which contribute to the protontransition without spin-flip ( J δ,δp ) and with the spin-flip( J − δ,δp ). This allows us to state that (i) around the lowerboundary of the considered region the leading scaling be-havior of the Sachs FFs has the form G E , G M ∼ /Q ,(ii) the dipole dependence ( G E , G M ∼ /Q ) is realizedin the asymptotic regime of pQCD when τ ≫ G E occurs at higher values Q than for G M , (iv) and the linear decrease of the ra-tio G E /G M at τ < J δ,δp by spin-flip transitions of two quarks and an ad-ditional contribution to J − δ,δp by spin-flip transitions ofthree quarks.Thus, abandoning the massless quarks, we were ableto explain in the one-photon exchange approximation theunexpected results of measurements of the proton SachsFFs ratio and analytically derive the experimentally es-tablished formula of the linear decrease law for this ratioat τ <
1. We believe that the interpretation presentedabove can be considered as a possible way to solve the G E /G M problem. ACKNOWLEDGEMENTS
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