Beam-normal single-spin asymmetry in elastic scattering of electrons from a spin-0 nucleus
Oleksandr Koshchii, Mikhail Gorchtein, Xavier Roca-Maza, Hubert Spiesberger
MMITP/21-008
Beam-normal single-spin asymmetry inelastic scattering of electrons from a spin-0 nucleus
Oleksandr Koshchii a , Mikhail Gorchtein b , Xavier Roca-Maza c , Hubert Spiesberger a a PRISMA + Cluster of Excellence, Institut f¨ur Physik,Johannes Gutenberg-Universit¨at, D-55099 Mainz, Germany b PRISMA + Cluster of Excellence, Institut f¨ur Kernphysik,Johannes Gutenberg-Universit¨at, D-55099 Mainz, Germany c Dipartimento di Fisica, Universit´a degli Studi di Milano, Via Celoria 16, I-20133 Milano, Italyand INFN, Sezione di Milano, Via Celoria 16, I-20133 Milano, Italy (Dated: February 24, 2021)We study the beam-normal single-spin asymmetry (BNSSA) in high-energy elastic electron scat-tering from several spin-0 nuclei. Existing theoretical approaches work in the plane-wave formalismand predict the BNSSA to scale as ∼ A/Z with the atomic number Z and nuclear mass number A .While this prediction holds for light and intermediate nuclei, a striking disagreement in both thesign and the magnitude of BNSSA was observed by the PREX collaboration for Pb, coined the“PREX puzzle”. To shed light on this disagreement, we go beyond the plane-wave approach whichneglects Coulomb distortions known to be significant for heavy nuclei. We explicitly investigate thedependence of BNSSA on A and Z by i) including inelastic intermediate states’ contributions intothe Coulomb problem in the form of an optical potential, ii) by accounting for the experimental in-formation on the A -dependence of the Compton slope parameter, and iii) giving a thorough accountof the uncertainties of the calculation. Despite of these improvements, the PREX puzzle remainsunexplained. We discuss further strategies to resolve this riddle. I. INTRODUCTION
Corrections due to two-photon exchange (TPE) Feyn-man diagrams for electron scattering have received con-siderable interest in recent years [1–12], primarily in thecontext of the discrepancy of the experimental results[13, 14] for the electric-to-magnetic form factor ratio,which sometimes is referred to as the proton form fac-tor puzzle. There are strong indications [15] that thispuzzle can be resolved by a proper inclusion of TPE inthe experimental analysis. In view of the interest in TPE,beam- and target-normal single spin asymmetries (SSAs)in elastic electron-nucleus ( eN ) scattering regained atten-tion of theorists [16–27]. It has been known for severaldecades that these transverse asymmetries are T-odd ob-servables which, in the absence of CP violation, are sen-sitive to the imaginary part of the scattering amplitude[28]. The T-even one-photon exchange amplitude (inthe plane-wave Born approximation, PWBA) is purelyreal, and it is the T-odd imaginary (absorptive) part ofthe TPE amplitude that gives rise to nonzero transverseasymmetries.The measurement of the BNSSA (Mott asymmetry,Sherman function and analyzing power are alternativenames which are more common for low-energy electronscattering) has been part of the parity-violation programover the past two decades [29–38]. Parity violation is ob-served in eN scattering when the incoming electron beamis longitudinally polarized. Measurements of the respec-tive parity-violating (PV) asymmetry have far-reachingapplications, including precision tests of the StandardModel [39–41] and studies of the nuclear structure [42–45]. Typical values of the PV asymmetry range fromparts per million to parts per billion, several orders of magnitude below BNSSA, hence a thorough control ofthis source of a potentially significant systematical uncer-tainty associated with an unknown transverse componentof the electron beam polarization has become a must-doin the analyses of PV electron scattering. Thanks to thefact that these experiments are designed for measuringthe much smaller PV asymmetry, in the past decadesgood-quality data of the BNSSA have become availablein a variety of kinematic regions and for a variety of tar-gets.The general theoretical treatment of transverse asym-metries in high-energy ( E b (cid:38) eN scat-tering is a highly challenging task. The two approachesthat have been pursued in the literature in this energyrange are i) solving the Dirac equation for the electronmoving in the Coulomb field of an infinitely heavy nu-cleus in the distorted-wave Born approximation (DWBA)upon neglecting nuclear and hadronic excitations of theintermediate states [21]; and ii) including the latter onlyin the approximation of the two-photon exchange [24],disregarding multi-photon exchange effects. The formerapproach enables one to accurately account for Coulombdistortion effects which scale with the nuclear charge, Zα ,and thus are important for electron scattering from heavynuclei. While this mechanism dominates at electron en-ergies in the few-MeV range, its contribution to BNSSAdrops with the electron energy, and for GeV electrons theinelastic hadronic contribution exceeds the former by sev-eral orders of magnitude [24]. When compared with thecorresponding scattering data from Jefferson Lab [34] andMAMI [37, 38], the second approach has been quite suc-cessful for light and intermediate-mass nuclei, e.g. He, C, Si, and Zr, while a stark disagreement betweentheory and experiment for BNSSA on the
Pb target a r X i v : . [ nu c l - t h ] F e b [34], sometimes called the “PREX puzzle”, was observed.This disagreement indicates that the theoretical calcula-tion of Ref. [24] may miss some important nuclear contri-butions which become important for very heavy nuclei,while only playing a minor role otherwise. One such ef-fect might be the exchange of many soft Coulomb pho-tons on top of the two-photon exchange which may leadto a substantial modification of the leading-order result.In this article, we join the two aforementioned ap-proaches. We include the contribution of the inelastichadronic states as an optical potential entering the Diracequation and study the interplay of the Coulomb distor-tion and two-photon exchange within one formalism. Wealso improve the existing calculations by using a more ex-tensive database for experimental information on Comp-ton scattering on nuclei. We use this information to ex-tract the dependence of the optical potential on the nu-clear mass number. II. DIRAC COULOMB PROBLEM ATRELATIVISTIC ENERGIES
We consider elastic scattering of an electron of mass m by a spin-0 nucleus of mass M , e − ( k , S i ) + N ( p ) → e − ( k , S f ) + N ( p ) , (1)where k ( k ) and p ( p ) denote the four-momenta ofthe initial (final) electron and initial (final) nucleus, and S i ( S f ) describes the spin projection of the initial (final)electron along the considered axis.The beam-normal single-spin asymmetry is defined as B n ≡ σ ↑ − σ ↓ σ ↑ + σ ↓ , (2)where σ ↑ ( σ ↓ ) represents the eN scattering cross sectionfor electrons with spin parallel (anti-parallel) to the nor-mal vector ξ µ given by ξ µ = (0 , (cid:126)ξ ) , (cid:126)ξ ≡ (cid:126)k × (cid:126)k | (cid:126)k × (cid:126)k | . (3)In order to account for Coulomb distortion and inelas-tic intermediate excitations in the considered scatteringprocess, we solve the relativistic Dirac equation : (cid:16) − i(cid:126)α · (cid:126) ∇ + βm + V c + iβV abs (cid:17) Ψ( (cid:126)r ) = E Ψ( (cid:126)r ) , (4)where (cid:126)α = γ (cid:126)γ and β = γ are Dirac matrices, and E the electron energy in the center of mass reference framerelated (neglecting the electron mass) to the laboratoryenergy E b by E = E b / (cid:112) E b /M . We use natural units throughout this paper.
The Coulomb potential V c ( r ) corresponds to the nu-clear charge distribution which is known from electronscattering experiments [46]. The absorptive potential V abs ( r, E ) represents the contribution of the inelastichadronic excitations in the two-photon exchange dia-gram, as discussed in detail in Sec. III. The inclusion ofthe absorptive component of the potential in the Diracproblem is the main novel feature of this work. Note thatthe form of this potential, iβV abs , is specific to the prob-lem at hand: an absorptive potential of the form i ˜ V abs only contributes to B n at higher order in α , exceeding theprecision goal of this study. Spherically symmetric V c ( r )and V abs ( r, E ) should be expected for spin-0 nuclei, andwe use this assumption throughout this paper.For electron scattering in a central field, the solutionof the Dirac equation can be expanded in spherical waves[47], Ψ κ,m z ( (cid:126)r ) = 1 r (cid:18) P κ ( r ) Ω κ,m z ( θ, φ ) iQ κ ( r ) Ω − κ,m z ( θ, φ ) (cid:19) , (5)where Ω κ,m z ( θ, φ ) are 2-component spherical spinors.The relativistic quantum number κ takes values κ and κ given by (cid:40) κ = − ( j + 1 /
2) if j = l + 1 / ,κ = +( j + 1 /
2) if j = l − / , (6)where l, j , and m z are the orbital angular momentum,total angular momentum, and total angular momentumprojection quantum numbers, respectively.The radial functions P κ ( r ) and Q κ ( r ) satisfy the fol-lowing coupled system of differential equations: dP κ dr = − κr P κ + (cid:16) E − V c + iV abs + m (cid:17) Q κ ,dQ κ dr = − (cid:16) E − V c − iV abs − m (cid:17) P κ + κr Q κ . (7)We normalize the spherical waves such that the radialfunction P κ ( r ) oscillates asymptotically with unit ampli-tude, P κ ( r → ∞ ) = sin (cid:16) kr − l π − η ln 2 kr + δ κ (cid:17) , (8)where k is the electron’s wave number and η = − ZαE/k is the relativistic Sommerfeld parameter. The scatteringphase shift δ κ is obtained by requiring continuity of theradial function P κ ( r ) and its derivative at large distance r m (matching distance), at which the numerical solutionof Eq. (7) is matched to the known analytical solutionof the Dirac equation for a point-like Coulomb poten-tial, V pc ( r ) = − Zα/r . The matching at large distancesis justified by the fact that both the absorptive poten-tial and the short range part of the Coulomb potentialcan be neglected beyond r m . As a result, the solutionof the point-like Coulomb potential provides the properasymptotic behavior.The absorptive potential, while having a shorter rangethan the Coulomb one, turns out to extend to distances ofthe order of the inverse electron mass 1 /m ∼
400 fm, andthe respective computation becomes cumbersome (detailsare discussed in Sec. III C). To perform the numerical cal-culation, we use the ELSEPA package [47, 48], properlymodified to include the absorptive potential.Knowledge of the phase shift enables one to determinethe direct and spin-flip scattering amplitudes, f ( θ ) and g ( θ ), respectively, in terms of which the beam-normalSSA is given by B n = i f ( θ ) g ∗ ( θ ) − f ∗ ( θ ) g ( θ ) | f ( θ ) | + | g ( θ ) | = 2 Im [ f ∗ ( θ ) g ( θ )] | f ( θ ) | + | g ( θ ) | . (9)These amplitudes admit the following partial-wave ex-pansions: f ( θ ) = 12 ik ∞ (cid:88) l =0 (cid:2) ( l + 1) (cid:0) e iδ κ − (cid:1) + l (cid:0) e iδ κ − (cid:1) (cid:3) P l (cos θ ) ,g ( θ ) = 12 ik ∞ (cid:88) l =0 (cid:2) e iδ κ − e iδ κ (cid:3) P l (cos θ ) , (10)where P l (cos θ ) and P l (cos θ ) are Legendre and associ-ated Legendre polynomials. The series in Eqs. (10) issingular at θ = 0 leading to a slow convergence when ap-proaching that limit. The convergence of the series can beaccelerated by using the reduced series method suggestedby Yennie et al. in Ref. [49]. This method prescribes toreduce the degree of the singularity of the original se-ries by expanding (1 − cos θ ) n f ( θ ) and (1 − cos θ ) n g ( θ )into analogous sums over Legendre polynomials. Thenew sums converge more quickly, however the extractionof the original amplitudes requires to divide by a factor(1 − cos θ ) n . As a result, for forward scattering the useof too many reductions becomes unstable. We found theoptimal number of reductions to be n = 2. III. ABSORPTIVE POTENTIAL FROM THETWO-PHOTON EXCHANGEA. Elastic eN scattering We turn to a field-theoretical description of the eN scattering process to deduce the explicit form of the po-tentials in Eq. (4). In the absence of P- and CP-violation,the invariant amplitude describing the scattering processEq. (1) for a spin-0 nucleus has two terms, [24], T = e | t | ¯ u ( k ) (cid:104) mA ( s, t ) + ( /p + /p ) A ( s, t ) (cid:105) u ( k ) , (11)with the usual Mandelstam invariants t = ( k − k ) and s = ( k + p ) , and two scalar amplitudes A and A . e ( k ) e ( k ) N ( p ) N ( p ) q q q N ( p ) N ( p ) e ( k ) e ( k ) e ( K ) X ( P )( a ) ( b ) FIG. 1. (a) One- and (b) two-photon exchange diagrams forelastic electron-nucleus scattering.
The initial and final electron Dirac spinors are denotedby u ( k ) and u ( k ), respectively.In the static approximation, | t | (cid:28) s, M , E , relativis-tic electron-nucleus scattering reduces to the problem ofpotential scattering of a relativistic electron in the fieldof a static nucleus. In the static limit, p = p = ( M, T = 2 M u † ( k ) (cid:20) e | t | (cid:18) mβ M A + A (cid:19)(cid:21) u ( k ) . (12)To leading order in Zα , the electron-nucleon interac-tion proceeds via the exchange of a virtual photon, cf.Fig. 1(a), and only the amplitude A survives at thisorder, A γ = 0 , A γ = ZF ch ( t ) . (13)Here, F ch denotes the nuclear charge form factor, whichis related to the spatial distribution of the nuclear charge ρ ch ( r ) by a three-dimensional Fourier transform, F ch ( t ) = (cid:90) ρ ch ( r ) e − i(cid:126)q · (cid:126)r d (cid:126)r, with | (cid:126)q | ≡ (cid:112) | t | , (14)with the normalization (cid:82) ρ ch ( r ) d (cid:126)r = 1.The T-odd observable B n is determined by the imag-inary part of the interference of A and A [28]. Animaginary part, Im A , for elastic eN scattering, i.e. for s > M , t <
0, appears first at next-to-leading order in Zα in the two-photon exchange contribution depicted inFig. 1(b).The absorptive and Coulomb components of the totalpotential which enter the Dirac equation (4), V abs and V c ,are related to Im A and A through a three-dimensionalFourier transform of the first and second terms in squarebrackets of Eq. (12). Including the leading nonvanishingterms in the perturbative expansion of these amplitudeswe obtain V c ( r ) = − (cid:90) d (cid:126)q (2 π ) e (cid:126)q A γ e i(cid:126)q · (cid:126)r = − Zα (cid:90) d (cid:126)r (cid:48) ρ ch ( (cid:126)r (cid:48) ) | (cid:126)r − (cid:126)r (cid:48) | , (15) V abs ( r, E b ) = − (cid:90) d (cid:126)q (2 π ) e (cid:126)q m M Im A γ e i(cid:126)q · (cid:126)r . (16)Other contributions, e.g. recoil corrections and higher-order contributions to A , are neglected here. In the nextsection we are going to study the perturbative result forthe two-photon exchange diagram in order to determinean explicit ansatz for the absorptive potential. B. Imaginary part of the two-photon exchangeamplitude
The imaginary part of the two-photon exchange am-plitude, displayed in Fig. 1(b), is given byIm T γ = e (cid:90) d (cid:126)K (2 π ) E K πL αβ W αβ Q Q , (17)where the momenta are defined as shown in Fig. 1, with Q , = − q , = − ( k , − K ) , and E K and (cid:126)K the energyand three-momentum of the intermediate electron insidethe loop, respectively. The leptonic tensor L αβ reads L αβ = ¯ u ( k ) γ α ( /K + m ) γ β u ( k ) , (18)and the doubly virtual Compton scattering (VVCS) ten-sor W αβ is defined as W αβ ≡ π (cid:88) X (cid:10) p | J † α (0) | X ( P ) (cid:11) (cid:104) X ( P ) | J β (0) | p (cid:105) (19) × (2 π ) δ (cid:0) p + q − P (cid:1) = 14 π (cid:90) d xe iq x (cid:10) p | [ J α † ( x ) , J β (0)] | p (cid:11) , (20)where (cid:80) X in Eq. (19) includes the phase-space inte-gral (cid:82) d (cid:126)P / ((2 π ) E P ). We note that in order to getfrom Eq. (19) to Eq. (20) one can apply a translationto the current operator, J α ( x ) = e iP x J α (0) e − iP x , and(2 π ) δ ( p + q − P ) = (cid:82) d xe i ( p + q − P ) x . The matrixelement of the hadronic current for the elastic intermedi-ate state is given by (cid:104) P | J β (0) | p (cid:105) = ( P + p ) β ZF ch ( Q ).To compute the imaginary part of the TPE diagramand perform a systematic study of its uncertainties, wenote that the result of the contraction with the leptonictensor can be decomposed into two parts, L αβ W αβ = m ¯ u ( k ) u ( k ) A + ¯ u ( k ) (cid:0) /p + /p (cid:1) u ( k ) A , (21)where A and A are analytical functions of t , Q , Q , W = ( p + q ) and s . With this notation, a straight-forward connection to the amplitude A can be made,Im A γ = α | t | π (cid:90) (cid:126)K d | (cid:126)K | d Ω K E K Q Q A ( t, Q , Q , W , s ) . (22)In the following we will obtain the long-range (i.e. low- t )behavior of Im A γ , adequate for devising the form of theabsorptive potential via Eq. (16).To that end, we follow Refs. [16–18] which observedthat B n is logarithmically enhanced in the kinematical regime m (cid:28) | t | (cid:28) s due to the collinear photon singu-larity. The integrals over the solid angle that are proneto this enhancement read I = | t | (cid:126)K π (cid:90) d Ω K Q Q ≈ ln | t | m ,I = E | (cid:126)K | π (cid:90) d Ω K Q = E | (cid:126)K | π (cid:90) d Ω K Q ≈ ln 4 E m , (23)with the energies E = ( s − M ) / (2 √ s ) and E K =( s − W ) / (2 √ s ) defined in the center-of-mass frame, and | (cid:126)K | = (cid:112) E K − m . Here we have listed only the leadingbehavior in the limit where (cid:112) | t | and E are large com-pared with the electron mass m . The exact expressionsare given in the Appendix A. For the values we are in-terested in, | t | ≈ .
01 GeV , the first logarithm is oforder 10. The second logarithm is of order 25 for E inthe GeV range, but is suppressed by an explicit factor | t | with respect to the former. This hierarchy defines ourapproximation scheme:Im A γ = α (cid:90) d | (cid:126)K | E K (cid:104) A (0)1 ( t ) I + | t | E K E A (1)1 ( t ) I + . . . (cid:105) , (24)where terms denoted by dots are doubly suppressed: theycontain one power of t and no large logarithm. To arriveat this result we have used an expansion in small photonvirtualities, A ( t, Q , Q , W , s ) = A (0)1 ( t ) + Q + Q A (1)1 ( t ) + . . . , (25)where we show explicitly only the dependence of A (0)1 and A (1)1 on t , while their dependence on the other fourinvariant variables is implicitly assumed. For consistency,we will only keep the “strong” t -dependence in A (0)1 , A (1)1 ,e.g. an exponential or the nuclear charge form factor, butwill neglect power corrections ∼ t/M , t/s , t/E . In theliterature only A (0)1 ( t ) has been obtained in the near-forward limit. In this work we include the second termand use it to estimate the uncertainty induced by theapproximations used.Next we proceed to derive explicit expressions for A (0)1 and A (1)1 . The optical theorem relates them to the to-tal cross sections for virtual photoabsorption at the firststep. The t -dependence is reconstructed at the secondstep from the measured differential cross section for realCompton scattering. This two-step procedure requiresthat we operate with the Compton amplitudes which arewell-defined in botha) the forward scattering limit, described by t = 0 andfinite Q = Q ≡ Q , andb) the real Compton scattering limit, described by Q = Q = 0 and finite t .The general virtual Compton tensor W αβ for a spin-less target consists of five independent Lorentz struc-tures, τ αβi , ( i = 1 , . . . ,
5) [50–52] multiplied by respectivescalar amplitudes F i ( t, Q , Q , W ). In the approxima-tion scheme we work in, the number of structures thatcontribute is further reduced upon neglecting terms thatvanish in both the forward and the real Compton scat-tering limits. This restricts our consideration to just twostructures τ αβi (in the original enumeration of Ref. [50]): W αβ = τ αβ Im F + τ αβ Im F ,τ αβ =( q · q ) g αβ − q α q β ,τ αβ =(¯ p · ¯ q ) g αβ − (¯ p · ¯ q )(¯ p β q α + ¯ p α q β )+ ( q · q )¯ p α ¯ p β , (26)with ¯ p = ( p + p ) / q = ( q + q ) /
2. Other structures(explicitly provided in Ref. [50]) surviving in the forwardlimit for virtual photons can always be expressed as linearcombinations of τ , .By contracting the leptonic and virtual Compton ten-sors, we find an explicit expression for A , A ( t ) = Im F (cid:2) ( Q + Q )( λ − − t (cid:3) (27)+ Im F (cid:2) (cid:0) p · ¯ q )(¯ p · ¯ k ) − ¯ p ( q · q ) (cid:1) × ( λ − − p · ¯ q ) ( λ − (cid:3) , where ¯ k = ( k + k ) / λ = 2( s − M )( s − W ) − ( Q + Q )( s + M )2( s − M ) . The forward limit of the amplitudes F , is determinedby the usual structure functions F and F , Q Im F (0 , Q , Q , W ) = F − x Bj F ≡ F L ,Q Im F (0 , Q , Q , W ) = − p · ¯ q ) F , (28)where x Bj = Q / (2¯ p · ¯ q ). The structure functions arerelated to the transverse and longitudinal inelastic virtualphotoabsorption cross sections σ T and σ L via F ( W , Q ) = W − M π α σ T ( W , Q ) , (29) F ( W , Q ) = W − M π α Q (¯ p · ¯ q )(¯ p · ¯ q ) + Q M × (cid:16) σ T ( W , Q ) + σ L ( W , Q ) (cid:17) . Expanding σ T,L at Q = 0 we obtain the following ex-pressions for A (0)1 and A (1)1 : A (0)1 (0) = M π α E K ωE σ T ( ω ) , (30) A (1)1 (0) = M π α (cid:20) (cid:18) − ω + 32 E b − ω ( E b + M )2 M E (cid:19) σ T ( ω )+ E K ωE σ (cid:48) T ( ω ) + 2 ωσ (cid:48) L ( ω ) (cid:21) , (31) where E b = ( s − M ) / (2 M ) and instead of the variable W we used ω = ( W − M ) / (2 M ). Moreover, σ T ( ω ) ≡ σ T ( ω, Q = 0) and σ (cid:48) T,L ( ω ) ≡ dσ T,L /dQ ( ω, Q = 0).The t -dependence of the Compton amplitudes can beretrieved from experimental studies of the differentialcross section for Compton scattering. Measurements areavailable at high energies, E ∼ − − t , 0 . < − t < .
06 GeV , see Refs. [53, 54]. In thiskinematic range, dσdt ≈ πα M ω | Im F | (cid:0) R (cid:1) , (32)where the terms suppressed with powers of t were ne-glected, and we defined R = | Re F | / | Im F | . The datafollow an exponential fall-off, dσdt ( ω, t ) = ae − B | t | F ( t ) + σ inc . (33) F ch ( t ) is the nuclear charge form factor. Dependingon the nucleus, we adopt a two-parameter Fermi model( Au,
Ag, and Cu), a Fourier-Bessel model ( Ti, Al, and C), or a sum of Gaussians ( He) [46, 55, 56].A (small) incoherent contribution σ inc was added to im-prove the description of the data around the first diffrac-tion minimum and above. In the t -range of interest thiscontribution is a slowly-varying function of t which canbe approximated by a polynomial. In practice, we foundthat only the constant term is reliably constrained by thedata. This is related to the rather small range of t wheredata are available, as well as large uncertainties at thelargest values of the momentum transfer. We performthe fit using the 3 GeV and 5 GeV data of Ref. [54] with B and σ inc as free parameters, and fix the normalizationof the coherent contribution a such that the sum a + σ inc reproduces the values of dσ/dt (0) reported in Ref. [54].We find the incoherent contributions to be irrelevantfor the description of the data at 3 GeV and set σ inc to0 (cf. second column of Table I). For the 5 GeV data, in-stead, its inclusion greatly improves the overall fit dueto a larger measured t -range. Importantly, however,whether including or excluding the incoherent contribu-tion from the fit barely affects the extracted value of B ,as the latter is determined by low- t data. For the Hedata of Ref. [53], we treat a as a free parameter. Theextracted values for a , B , and σ inc are listed in Table I.We display the fit of the 5 GeV data of Ref. [54] in Fig. 2.In the literature, BNSSA measurements have been re-ported for the following spin-0 nuclei: He, C, Si, Ca, Ca, Zr and
Pb. To obtain the slope param-eter B for Si, Ca, Ca, Zr and
Pb for which nodirect data is available, we use the values obtained fornuclei with the closest atomic weight in Table I. Morespecifically, we use the values of B from Al for Si, Ti for , Ca,
Ag for Zr, and
Au for
Pb.Values of B obtained from the fit to Compton data at ω = 3 GeV and ω = 5 GeV are compatible with eachother (where a comparison is possible). We use the more TABLE I. Average values and corresponding theoretical uncertainties for the Compton slope parameter B extracted fromCompton scattering data of Refs. [53] and [54]. For nuclei with A ≥ dσ/dt (0) = a + σ inc is fixed to the value reported inRef. [54]. The 5 GeV results are used for all nuclei except for He. ω (cid:39) ω = 5 GeVTarget a [ µ b/GeV ] σ inc [ µ b/GeV ] B [GeV − ] a [ µ b/GeV ] σ inc [ µ b/GeV ] B [GeV − ] He 13 . ± . . . ± . C 111 . . . ± . . ∓ . . ± . . ± . Al 523 . . . ± . . ∓ . . ± . . ± . Ti — — — 1210 . ∓ . . ± . . ± . Cu 2664 . . . ± . . ∓ . . ± . . ± . Ag 8406 . . . ± . . ∓ . . ± . . ± . Au — — — 20589 . ∓ . . ± . . ± . precise values from the ω = 5 GeV fit for all nuclei exceptfor He where only data at ω = 3 . t -dependence of the imagi-nary part of the Compton amplitude using Eqs. (32, 33),Im F ( ω, t ) = Im F ( ω, e − B | t | F ch ( t ) (cid:115) R ( ω, R ( ω, t ) , (34)and estimate the ratio R ( ω, t ) in a Regge model. Forthe latter, we use a recent Regge fit [57] of the totalphotoabsorption cross section measured for several nuclei[58–60]. The total cross section was fitted by a sum of aPomeron and a Reggeon exchange, σ totγA ( ω ) = c AP ( ω/ω ) α P (0) − + c AR ( ω/ω ) α R (0) − , (35) FIG. 2. Compton scattering cross section data [54] used todetermine the Compton slope B compared with our fit. with ω = 1 GeV and linear Regge trajectories α i ( t ) = α i + α (cid:48) i t . The intercepts are α P = 1 . α R = 0 . α (cid:48) P = 0 .
25 GeV − , α (cid:48) R = 0 . − [57]. Thevalues of c AP,R for carbon, aluminum, copper and lead arelisted in Table I of Ref. [57]. From the optical theorem, σ totγA ( ω ) ∝ Im F γA → γA . For a given Regge exchange con-tribution to an amplitude, its real and imaginary partsfollow from the known phase of the Regge propagator, P R ∼ e iπα R ( t ) + ξ, ξ = ± . (36)For Compton scattering only the exchange with natu-ral parity, ξ = +1, contributes to the spin-independentchannel, and R i ( t ) = cot [ πα i ( t ) /
2] with i = P, R . Wefound that removing the effect of the real part of the am-plitude from the differential cross section to obtain the t -dependence of the imaginary part, Eq. (34), is equivalentto a change in the slope parameter B of ≈ . − .We include this effect as an uncertainty of B , in addi-tion to those of the fit listed in Table I, and use a simpleexponential times charge form factor ansatz for the t -dependence of the amplitude Im F .Finally, our ansatz for the t -dependence of the coeffi-cients A (0)1 ( t ) and A (1)1 ( t ) in front of the large logarithmsat small momentum transfer t reads A (0 , ( t ) = A (0 , (0) e − B | t | F ch ( t ) . (37)While the ansatz for A (1)1 ( t ) is motivated by the continu-ity of the function A ( t, Q , Q , W , s ) in t , Q near theforward limit and near the real photon point, the qualityof this approximation is hard to estimate. We thereforeassign a 100% uncertainty to the A (1)1 contribution. C. Absorptive potential
The results of Eqs. (24, 30, 31), and (37) can be usedto compute the absorptive potential given by Eq. (16).Using the hierarchy introduced in Eq. (24), we splitthe absorptive potential into two parts, V abs ( r, E b ) = V (0)abs ( r, E b ) + V (1)abs ( r, E b ), and find the leading and sub-leading contributions to be given by V (0)abs = c E b (cid:90) ω π dωω σ T ( ω ) ∞ (cid:90) dq j ( qr ) F ch ( q ) e − B q I ,V (1)abs = c ∞ (cid:90) dq q j ( qr ) F ch ( q ) e − B q E b (cid:90) ω π dωI × (cid:26) (cid:20) E b − ω − ω ( E b + M )2 M E (cid:21) σ T + ω ( E b − ω ) E b σ (cid:48) T (cid:27) , (38)with c = − αm/ (2 π E b ), q ≡ | (cid:126)q | , j ( qr ) the Bessel func-tion of order zero, and ω π = m π + m π / (2 M ) the labo-ratory frame photon energy at the pion photoproductionthreshold. The expressions for I and I are provided inthe Appendix A.We note that the ω -weighting ∼ ωσ T ( ω ) in V (0)abs , to-gether with the overall 1 /E -weighting in c , puts the em-phasis on the photoabsorption in the hadronic energyrange. Nuclear photoabsorption occurs at much lowerenergies and its contribution to the leading term is sup-pressed.In this article, we focus on the evaluation of contribu-tions to the absorptive potential coming from photoab-sorption in the hadronic region. In the nucleon reso-nance region and slightly above, the total nuclear pho-toabsorption cross section is assumed to approximatelyscale with the atomic weight A as σ T ( ω ) ≈ Aσ T,γp ( ω ),where σ T,γp ( ω ) is the real photoabsorption cross sectionof the proton. For the evaluation of σ T,γp and σ (cid:48) T,γp ,we use the parametrization of Ref. [61]. We point outthat σ (cid:48) L,γp ( ω ) is zero in this parametrization, hence thiscontribution was omitted in Eq. (38).The naive linear A -scaling disregards the shadowing athigher energies and anti-shadowing in the resonance re-gion (cf. Fig. 10 of Ref. [60]). Nevertheless, since Eq. (38)operates with the integrated cross section rather thanthe cross section itself, the two effects should largelycancel out justifying our approximation. A comprehen-sive study of specifically nuclear effects in photoabsorp-tion, from the giant resonance to shadowing and anti-shadowing at hadronic energies, is postponed to a futurework.In Fig. 3, we display the result of a numerical eval-uation of the two-fold integrals in Eq. (38) which de-termine the leading (black dashed curve) and subleading(orange dashed curve) contributions to the absorptive po-tential for Pb at E b = 1 .
063 GeV. We compare theresult with the Coulomb potential (blue solid curve) ofthe lead nuclear charge distribution of Ref. [46] and withthe Coulomb potential for a point-like charge (red dottedcurve). The potentials for the point-like and the empiri-cal charge distributions approach each other just outsidethe r.m.s. radius, which is ∼ . r | V c | r | V pc | r | V abs ( ) | r | V abs ( ) | r ( fm ) r | V | FIG. 3. The r -dependence of weighted potentials r | V ( r ) | for Pb as a function of r in units of Fermi. The point-chargeCoulomb potential is shown by a red dotted horizontal lineat r | V pc ( r ) | = Zα . The Coulomb potential of the empiricalcharge distribution corresponds to the blue solid curve. Theblack and orange dashed curves show the leading and sublead-ing contributions to the absorptive potential, correspondingly,for E b = 1 .
063 GeV. the scale factor 10 in front of the leading contributionto V abs ) and has a finite range ( rV abs ( r → ∞ ) → ∼ /m ∼
400 fm. This prop-erty results in a large matching distance needed for aprecise evaluation of B n . The matching distance r m is the distance at which the total interaction potential V ( r ) = V ch ( r ) ± iV abs ( r ) has reached (within a given pre-cision) its asymptotic value V ( r m ) = V pc ( r m ), beyondwhich V abs can be set equal to zero. The determinationof the matching distance is of crucial importance for ourcalculation since r m is the distance where the numericalsolution of the Dirac equation is matched to the knownanalytical solution with V ( r ) = V pc ( r ). We observe thatthe CPU time for the numerical calculation grows ap-proximately linearly with r m , and a proper balance be-tween precision and computing time had to be found.We studied the dependence of predictions for B n onthe matching distance for electron scattering from Pbat E beam = 1 .
063 GeV. In previous calculations for theCoulomb problem with a nuclear charge distribution of atypical radius (cid:46) r m ∼
15 fm hadbeen used [21]. However, we found that the relative un-certainty of our calculation for B n at θ = 5 ◦ can not beexpected to be better than 10 − if r m is chosen smallerthan 120 fm. In addition, the precision of the calculationbecomes worse as the scattering angle increases. The re-sults of our calculation for B n , which are presented inSec. IV, are obtained with r m = 606 fm. Such a match-ing distance represents a compromise between achievingthe necessary numerical precision and keeping the calcu-lation time under control. With r m = 606 fm, the relative C Si Ca Zr Pb - - - - - - t ( GeV ) · B n FIG. 4. BNSSA for elastic electron scattering from C (solidblue curve), Si (dashed orange curve), Ca (solid greencurve), Zr (dashed-dotted black curve), and
Pb (solidred curve) versus momentum transfer squared | t | at E b =953 MeV for the case when only elastic intermediate-statecontributions are taken into account. For the distribution ofthe nuclear charge, ρ ch ( r ), we use an experimental fit of theworld data on elastic electron-nucleus scattering parametrizedin the form of a sum of Gaussians ( C, Si, Ca,
Pb) orFourier-Bessel ( Zr) as reported in Ref. [46]. intrinsic numerical uncertainty of our prediction for B n is well below ∼
1% in the range of momentum transfersconsidered in this paper, independently of the target andbeam energy.
IV. RESULTS
In this section, we present results for B n for electronscattering at energies ranging from 570 MeV to 3 GeVfrom a variety of nuclear targets. We note here that whileour formalism was developed for spin-0 nuclei, for elasticscattering on an unpolarized nuclear target non-zero nu-clear spin will only induce corrections of the order of thenuclear recoil, ∼ t/M , which can be safely neglected.The results of the calculation including Coulombdistortion (distorted-wave calculation, for short) ofthe beam-normal SSA for the case when only elasticintermediate-state contributions in the scattering processare taken into account ( V abs = 0) are displayed in Fig. 4.This figure illustrates the dependence of the asymmetryon details of the nuclear charge distribution at a fixedenergy of the incoming beam, E b = 953 MeV. Becausethe nuclear charge density is roughly represented by anearly homogeneous sphere with a relatively sharp edge,the prediction for the beam-normal SSA features a typ-ical diffractive pattern. The location of the first diffrac-tion minimum gives an idea of the characteristic size ofthe target nucleus. One can see that for light nuclei thediffraction minima are prominent and deep, with the ab-solute value of the asymmetry changing by an order ofmagnitude in the vicinity of the minimum. For heavy nuclei, Coulomb distortions are stronger, and the asym-metry experiences a less drastic change around the min-imum. The predictions for B n presented in Fig. 4 are ingood agreement with those reported in Ref. [21].Next we discuss results of the distorted-wave calcula-tion of the beam-normal SSA for the case when inelasticintermediate-state contributions in the scattering processare taken into account by including the absorptive poten-tial into the Coulomb problem. We calculate a theoreticaluncertainty in several steps. First, we evaluate a relativeuncertainty (cid:15) of the asymmetry due to the uncertaintyof B . The uncertainty of B receives itself two contribu-tions: (i) the first component is the uncertainty from thefit to the Compton data and is provided in Table I; (ii)the second component is associated with neglecting theeffect of the real part of the amplitude F in the fit of theCompton data and was estimated to be 2 . − , asdiscussed in Sec. III B. These two parts are combined inquadrature. Second, we evaluate a contribution (cid:15) to therelative uncertainty of the asymmetry due to the specificchoice of an ansatz for the t -dependence of the coefficient A (1)1 ( t ). (cid:15) is obtained as the relative difference betweenpredictions for B n computed with and without the con-tribution from A (1)1 ( t ) to V abs , while the parameter B is kept fixed at its central value. This prescription isequivalent to assigning a 100% uncertainty to the contri-bution from A (1)1 . Finally, the two components are addedin quadrature, i.e. (cid:15) = (cid:112) (cid:15) + (cid:15) is used to calculate un-certainty bands shown in the following figures.In Figs. 5 and 6, we display results for the BNSSAin the distorted-wave calculation including inelasticintermediate-states. Each curve in these figures belongsto a specific energy of the incoming beam as specifiedin the figure captions and a specific target nucleus asindicated on the plots. The central dashed lines cor-respond to the absorptive potential given by Eq. (38)and the parameter B fixed at its central value as pro-vided in Table I (5 GeV data). The solid bands aroundthe central lines indicate the estimated theoretical un-certainty as described in the previous paragraph. Bycomparing the results presented in Fig. 4 with those dis-played on the left panel of Fig. 6 (both figures corre-spond to E b = 953 MeV), we conclude that the inelasticexcitations of the intermediate state provide the domi-nant contribution to B n at GeV beam energies. Thisis consistent with the results of Ref. [24], in which onlythe leading-order inelastic intermediate-state excitationswere considered.In Fig. 5, we compare our prediction for B n with themeasurements by the PREX-I and HAPPEX collabora-tions at JLab [34] (left plot) and a series of experimentsperformed at MAMI [37, 38] (right plot). We note thatour framework has been designed for high-energy elec-tron scattering; apart from lacking contributions fromthe nuclear range, it operates with a phenomenologi-cal t -dependence motivated by the high-energy Comp-ton scattering data. While the high-energy measurement ●●■■ ◆◆ E b = E b = He Pb C ● ◆ ■ - - - - - t ( GeV ) · B n PREX - I + HAPPEX ●● ●● ●●●● ●●■■ ■■ ◆◆◆◆ E b =
570 MeV C Si Zr ● ■ ◆ - - - - - - - t ( GeV ) · B n MAMI
FIG. 5. BNSSA versus momentum transfer squared | t | in the kinematical range where measurements are available. Leftpanel: Predictions for He obtained with E b = 2 .
750 GeV and for C and
Pb with E b = 1 .
063 GeV. Experimental datapoints are from the PREX-I and HAPPEX experiments [34]. Right panel: Predictions for C, Si, and Zr obtained with E b = 570 MeV. Experimental data points are from the experiments of Refs. [37, 38] at MAMI. on He by the HAPPEX collaboration at 2 .
75 GeV iswell described, and so is a somewhat lower one on Cat 1 .
063 GeV, the agreement at lower MAMI energies isworse even for light and intermediate nuclei. This factindicates that the t -dependence of the Compton crosssection in the resonance region is likely not to follow theexponential fall-off as deduced from high-energy data.The data point by the PREX-I collaboration on the Pb target clearly stands out: the measured value of B n ≈ +0 . B n reported here and obtained with the updatedvalue of the slope parameter B reduces the disagreementbetween theory and experiment for Pb somewhat, itis still unable to explain the origin of the sign differencebetween measurement and prediction.We note that the predictions displayed in Fig. 5 areobtained using different values of the parameter B (seeTable I for details) for different nuclei. These values werededuced from the Compton scattering data on 8 nuclei[53, 54]. In contrast, theoretical predictions presented inRefs. [34, 37, 38] were based on the calculation of Ref. [24]which assumed a universal parameter B = 8 ± − ,independent of the target nucleus. This value stems fromthe high-energy Compton data on the proton. In Ref. [24]this value was found consistent with that for He, therebyconjecturing that it remains constant across the nuclearchart. The present, more careful study addressed the va-lidity of this assumption explicitly, see Table I, and foundit to hold for light nuclei, from He to Al. For heav-ier nuclei it gradually breaks down and for the heaviestnucleus,
Au the actual value of the slope is 7 timeslarger.In Refs. [37, 38], light and intermediate nuclei hadbeen studied at lower energies. In those references, the slope B was taken universal and constant [24], but theuncertainty was assumed to be 10% (20%) of the fullslope of the Compton cross section, i.e. of B + R / R Ch . For carbon, one has R Ch ≈ . B = 8 ± ±
12) GeV − for10% (20%) uncertainty, respectively. This is a conserva-tive estimate in view of experimental data that allow usto reduce the uncertainty of B considerably, as shown inTable I.Another difference between the approach of Refs. [37,38] and the one used in the present work concernsthe treatment of corrections to the leading-order behav-ior of the potential V abs and related uncertainties. Inthose references, the approximate result for I shown inEq. (23) was used to obtain the central value, while the t -independent non-logarithmic term appearing in the fullresult of Eqs. (A4, A5), was only used to estimate the un-certainty. Here we argue that the full expression for V (0)abs and the subleading contribution V (1)abs are exactly calcula-ble and should therefore be included in the central value.The leading term is model-independent as it is the onlyterm that carries the long-range behavior ∼ ln( | t | /m ).All other corrections, including V (1)abs , are of short-rangenature. Among these, V (1)abs is the only term enhanced bythe collinear logarithm ∼ ln(4 E /m ), and its coefficientis exactly calculable, based on the low- Q expansion ofthe near-forward virtual Compton amplitude. This en-hancement justifies using 100% of this contribution as aconservative uncertainty estimate for all neglected short-range pieces.The numerical hierarchy of parts of V abs may breakdown upon including effects originating from low-energynuclear excitations and the quasielastic peak. Suchcontributions carry a new intermediate scale Λ Nucl ∼
15 MeV with m (cid:28) Λ Nucl (cid:28) E b ∼ (cid:38)
300 MeV)0 E b =
953 MeV C Ca Pb - - - - t ( GeV ) · B n PREX - II E b = C Ca Ca Pb - - - - - - - - t ( GeV ) · B n CREX E b = C Al - - - - - - - t ( GeV ) · B n Qweak
FIG. 6. BNSSA versus momentum transfer squared | t | for the kinematical conditions of soon-to-be published measurements.Upper left panel: Predictions for C, Ca, and
Pb obtained for E b = 953 MeV (the PREX-II measurement [62]). Upperright panel: Predictions for C, Ca, Ca, and
Pb obtained for E b = 2 .
183 GeV (the CREX measurement [63]). Lowerpanel: Predictions for C and Al obtained for E b = 1 .
158 GeV (the Qweak measurement). The dashed vertical lines indicateapproximate values of t of the considered experiments. the leading term V (0)abs is exempt from a substantialmodification by the contributions from such low ener-gies. To see this we may use the approximate scalingof the integrated nuclear cross section without energyweighting, (cid:82) σ Nucl ( ω ) dω = NZA αM [64], with N the num-ber of neutrons. The expected energy-weighted resultthen reads, assuming N = Z , A (cid:82)
30 MeV0 ωσ Nucl ( ω ) dω ∼ α Λ Nucl M (cid:46) − . This is to be compared with the energy-weighted integral over the hadronic range for which wefind (cid:82) E b ω π ωσ γp ( ω ) dω ∼ .
3, for E b = 1 GeV and usingthe parametrization of Ref. [61]. The effect of these nu-clear contributions on the subleading term V (1)abs remainsa question which we plan to address in a future work.In Fig. 6, we present the prediction for the kinemat-ical conditions of soon-to-be published measurements of B n by the PREX-II [65], CREX [66], and Qweak [67]collaborations at Jefferson Lab.Finally, we confront the results of our distorted-wavecalculation to those obtained in the plane-wave approx-imation as reported in Ref. [24]. Here, the BNSSA is given by B n = − m (cid:112) | t | (cid:112) ( s − M ) − s | t | Im A γ A γ (39)in terms of the invariant amplitudes A γ (Eq. (22)) and A γ (Eq. (13)). In order to perform a meaningful com-parison between the two calculations, we tuned the in-put parameters of the plane-wave approach to be identi-cal to the input we used for our distorted-wave calcula-tion, i.e. instead of assuming a flat A/Z dependence ofthe Compton slope parameter as in Ref. [24], we usedthe experimental information on its Z ( A ) dependence assummarized in Table I. In addition, instead of evaluatingthe asymmetry in the leading logarithm approximation,i.e. by considering only those contributions to B n com-ing from approximating A ( t ) with A (0)1 ( t ), Eq. (30), wealso took into account the A (1)1 ( t ) contribution to A ( t ),Eq. (31). The results of the comparison are displayedin Fig. 7. We observe that Coulomb distortion increasesthe absolute value of the asymmetry. While the effect1 C Ca Ca Pb - - - - - - - t ( GeV ) · B n CREX
FIG. 7. BNSSA versus momentum transfer squared | t | forthe kinematical conditions of the CREX measurement, i.e. for E b = 2 .
183 GeV. Dashed curves represent predictions for theasymmetry obtained in the plane-wave approximation whilethe solid curves correspond to the exact calculation includingCoulomb distortion. (the difference between the solid and dashed curves ofthe same color) may be significant, the correspondingpredictions are qualitatively similar to those obtained inthe plane-wave approximation.
V. CONCLUSIONS
We have computed the beam-normal single-spin asym-metry in the diffractive regime of elastic scattering ofelectrons from a variety of spin-0 nuclei. This asymme-try is generated by the imaginary part of the interfer-ence between the direct, f ( θ ), and spin-flip, g ( θ ), scat-tering amplitudes. We have evaluated these amplitudesby studying the asymptotic behavior of the solution ofthe relativistic Dirac equation at large distances. To re-alistically describe the effective interaction between theelectron and target nucleus, we have employed an opticalpotential method. Within this approach, the electron-nucleus interaction is represented by two components ofthe potential: the (real) Coulomb one and (imaginary)absorptive one. The Coulomb component accounts forthe contribution from elastic intermediate states in thescattering process, whereas the absorptive component de-scribes inelastic contributions. To parametrize the ab-sorptive component of the potential, we made use of theresult for the imaginary part of the general amplitude A calculated to order α in the electromagnetic coupling.The corresponding perturbative calculation has been per-formed for the kinematics of diffractive electron scatter-ing, where the optical theorem can be used to relate theimaginary part of the amplitude A to the total photoab-sorption cross section of the nucleus. To describe the t -dependence of the asymmetry near the forward scatteringlimit ( t = 0), we utilized information on the t -dependence of the differential cross section of Compton scattering offnuclei at low t . Using this approach, we have obtaineddistorted-wave predictions for BNSSA for various spin-0nuclei and presented the results in the kinematical rangeof several experiments that have been performed.Our calculation contains several improvements with re-spect to earlier calculations: a) we included contributionsfrom inelastic intermediate states into the Coulomb prob-lem; b) we went beyond the leading logarithm approx-imation in evaluating B n by considering contributionscoming from A (1)1 ( t ); c) we explicitly outlined the ap-proximation scheme used for the evaluation of B n andobtained a more realistic estimate of uncertainties. Wefound however that neither of these improvements seemsto be enough to explain the PREX puzzle. A small andpositive value of B n obtained on a Pb target is at vari-ance with large negative values of B n predicted by thetheory and backed by all other measurements on lightand intermediate nuclei. As a possible improvement, weplan to study contributions coming from the nuclear re-gion of the photoabsorption cross section. ACKNOWLEDGMENTS
The authors acknowledge useful discussions withChuck Horowitz, Jens Erler and Stefan Wezorke. Thework of M. G., O. K., and H. S. is supported bythe German-Mexican research collaboration grant No.278017 (CONACyT) and No. SP 778/4-1 (DFG). M. G.is supported by the EU Horizon 2020 research andinnovation programme, project STRONG-2020, grantagreement No. 824093. X. R.-M. acknowledges fundingfrom the EU Horizon 2020 research and innovation pro-gramme, grant agreement No. 654002.
Appendix A: Master Integrals
In this Appendix, we briefly summarize the details ofthe calculation of the master integrals appearing in theexpression for the beam-normal SSA, Eq. (23), I = | t | (cid:126)K π (cid:90) d Ω K Q Q , (A1) I = E | (cid:126)K | π (cid:90) d Ω K Q = ln 4 E E K m E γ . (A2)where E = ( s − M ) / √ s and E K = ( s − W ) / √ s arethe energies of the external and the intermediate elec-trons in the center-of-mass frame. The center-of-mass en-ergy of the collinear quasi-real photon is E γ = E − E K =( W − M ) / √ s . The angular integration in Eq. (A1)can be performed by using the Feynman trick, (cid:90) d Ω K Q Q = (cid:90) dx (cid:90) d cos θ x dϕ [ Q + ( Q − Q ) x ] , (A3)2and choosing the polar axis to be oriented in such a waythat θ x is the angle between (cid:126)K and (cid:126)K x = (cid:126)k + x ( (cid:126)k − (cid:126)k ).Here (cid:126)k and (cid:126)k are the three-momenta of the initial andfinal electron in the center-of-mass frame. 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