On the QED corrections to elastic electron scattering at high momentum transfer
OOn the QED corrections to elastic electron scatteringat high momentum transfer
D. H. Jakubassa-AmundsenMathematics Institute, University of Munich, Theresienstrasse 39,80333 Munich, Germany (Dated: February 17, 2021)Estimates of QED and dispersion effects on the cross section for elastic electron scattering froma C nucleus are provided for collision energies in the range of 120 −
450 MeV. While in generalsuch corrections are smoothly varying with energy or scattering angle, they show structures in thevicinity of diffraction minima which are very sensitive to details of the theoretical models. Thiscasts doubt on the assertion that the discrepancy between QED background-corrected experimentaldata and theory in these minima originates solely from dispersion.
1. INTRODUCTION
In order to determine nuclear charge distributions ornuclear charge radii from the measurements of elasticelectron scattering by means of a comparison with phase-shift or distorted-wave Born calculations [1], the experi-mental data are corrected for quantum electrodynami-cal (QED) effects. For such nuclear structure investiga-tions light targets ranging from protons to carbon areused, where Coulomb distortion effects are assumed tobe small.Usually these QED corrections rely on the plane-waveBorn approximation (PWBA) for the vacuum polariza-tion, the vertex and self-energy correction and the radi-ation of soft unobserved photons. For high collision en-ergies, simple formulae are available to account for theseeffects (see, e.g. [2–5]).In the past decades great efforts were made to under-stand the deviation of the so corrected experimental scat-tering cross section data from theoretical predictions inthe region of diffraction, arising from the charge distri-bution inside the target nucleus [6]. It is well-known,however, that the PWBA, an appropriate high-energytheory for weak fields at low momentum transfer, fails tocorrectly describe the scattering process when diffractioneffects modulate the electron intensity. The displacementbetween the calculated PWBA position of the cross sec-tion minima and experiment is conventionally handled byintroducing an effective momentum transfer q eff at whichthe PWBA theory has to be evaluated [7], but substantialdeviations in intensity remain.The consideration of the QED effects beyond first orderin the fine-structure constant α is non-trivial. Second-order radiative corrections pertaining to the electron wererecently reported for proton targets [4, 8]. The inclusionof higher-order interactions with the target potential ismore involved, and is only straightforward in case of thevacuum polarization by means of the Uehling potential[9, 10].The largest QED effect, the vertex and self-energy cor-rection plus the contribution from soft bremsstrahlung(in the following termed vsb correction), may amount up to 20% in first order for collision energies around100 MeV [3]. A calculation of the second-order (in Zα ,where Z is the nuclear charge number) vsb effects wasattempted in [11] and further literature is provided inthe review by Maximon [3], but no tractable formula isavailable. A full account of the electron-target field forthis effect has not yet been accomplished.The assumption why such higher-order corrections canbe disregarded in the data reduction is based on thefact that the first-order Born amplitude, which multipliesthe QED corrections, vanishes in the diffraction minima.Hence the remaining discrepancies between experimentand theory are attributed to dispersion, a second-orderBorn contribution to Coulomb scattering which allowsfor an intermediate excitation of the nucleus [1, 12–15].Such deviations are of the order of 5 −
10% at collisionenergies between 300 −
500 MeV (increasing with energy[16]). However, estimates for the dispersion are com-monly in the percent region [13] and cannot account forthese discrepancies.The present work demonstrates the sensitivity of theQED corrections to the choice of different theoreticalprescriptions. In particular, the PWBA result is setagainst an improved model where Coulomb distortion isaccounted for by replacing the PWBA potential scatter-ing amplitudes by the phase-shift results in all next-to-leading order terms in α , as suggested in [3].The C nucleus is chosen as target because of its im-portance as a reference nucleus for nuclear structure stud-ies [17], but it will also be used in future experiments onparity violation [18]. Its central field is weak enough sothat the PWBA is valid at small angles. Moreover, it isa spin-zero nucleus, for which any scattering from mag-netic moment distributions is absent. Being a p -shellnucleus, C can reasonably well be described within theharmonic oscillator shell model [13]. Even realistic many-body interactions can in principle be taken into accountfor this nucleus [19]. Finally, there exists a large numberof high-precision elastic electron scattering measurements[16, 17, 20, 21].The paper is organized as follows. Section 2 recapitu-lates the QED corrections within the PWBA in the high-energy approximation. Dispersion is considered using the a r X i v : . [ nu c l - t h ] F e b second-order Born theory with a closure approximation.The so corrected differential cross section is providedboth in the Born approximation and when Coulomb dis-tortion is taken into account by means of the phase-shiftanalysis. In section 3, numerical results for the differ-ential cross sections and their change with QED effectsand dispersion are provided. Comparison is made withexperimental data in the energy region 150 −
430 MeV.Concluding remarks are given in section 4. Atomic units( (cid:126) = m = e = 1) are used unless indicated otherwise.
2. QED AND DISPERSION CORRECTIONS
We start by providing the QED corrections to elasticelectron scattering from spin-zero nuclei, consisting ofthe vacuum polarization and the vertex, self-energy andsoft bremsstrahlung (vsb) contributions. Since the ex-perimental data under consideration are recorded witha high-resolution spectrometer, hard bremsstrahlung,where the photon momentum has to be fully taken intoaccount, does not contribute.
In lowest-order Born approximation, the transitionamplitude for vacuum polarization is given by [5] A vac fi = 13 πc (cid:20) ln( − q /c ) − (cid:21) A B fi , (2.1)Here, A B fi is the first-order Born amplitude for elasticscattering. The 4-momentum transfer q to the nucleusis defined by q = ( E i − E f ) /c − q where q = k i − k f , and E i , k i , respectively E f , k f are the total energiesand momenta of incoming and scattered electron. Thevalidity of (2.1) is restricted to high momentum transfer, − q /c (cid:29)
1, which covers our cases of interest.Alternatively, vacuum polarization can be calculatedto all orders in
Z/c with the help of the Uehling potential[9], U e ( r ) = − πc (cid:90) d r (cid:48) (cid:37) N ( r (cid:48) ) | r − r (cid:48) | χ (2 c | r − r (cid:48) | ) , (2.2)which for a spherical nuclear charge distribution (cid:37) N ( r (cid:48) ),normalized to Z , reduces to [10] U e ( r ) = − c r (cid:90) ∞ r (cid:48) dr (cid:48) (cid:37) N ( r (cid:48) ) × [ χ (2 c | r − r (cid:48) | ) − χ (2 c | r + r (cid:48) | )] ,χ n ( x ) = (cid:90) ∞ dt e − xt t − n (cid:18) t (cid:19) (cid:18) − t (cid:19) . (2.3) For numerical estimates one can use a parametrization of χ in terms of rational functions of polynomials [22].To include the effect of vacuum polarization in elasticscattering exactly, the Uehling potential has to be addedto the target nuclear potential V T ( r ) when performingthe phase-shift analysis. We have confirmed numericallythat the first-order Born approximation to the Uehlingpotential agrees with (2.1) within 0.01 %, which validatesthe comparison of the exact result with the one based on(2.1). The lowest-order Born amplitude for the vertex cor-rection, after eliminating the UV divergence by renor-malizing via the inclusion of the self energy, is given by[4] A vs fi = F ( − q ) A B fi − (cid:112) E i E f c Z q F L ( q ) F ( − q ) ( u ( σ f )+ k f γ ( αq ) u ( σ i ) k i ) , (2.4)where α and γ denote Dirac matrices. The initial, re-spectively final states of the electron (with spin polariza-tion σ i and σ f ) are represented by the free 4-spinors u ( σ i ) k i and u ( σ f ) k f . The longitudinal (Coulombic) form factor F L is calculated from the Fourier transform of the nuclearcharge distribution (cid:37) N , F L ( q ) = 1 Z (cid:90) d x N (cid:37) N ( x N ) e i qx N . (2.5)It is normalized to unity at | q | = 0. In the high-energyapproximation, − q /c (cid:29)
1, the electric ( F ) and mag-netic ( F ) electron form factors are given by F ( − q ) = 12 πc (cid:26)
12 ln( − q /c ) (cid:2) − ln( − q /c ) (cid:3) − π (cid:21) + IR , (2.6) F ( − q ) = − πc c q ln( − q /c ) , (2.7)and the infrared divergent term IR readsIR = 12 πc (cid:0) ln λ (cid:1) (cid:2) ln( − q /c ) − (cid:3) , (2.8)where λ is an auxiliary finite photon mass. Since the electron detector has a finite energy resolu-tion ∆ E , electrons which have lost an energy ω < ∆ E bymeans of soft photon emission cannot be distinguishedfrom the elastically scattered electrons. Therefore thissoft-photon bremsstrahlung has to be added incoherentlyto the cross section for elastic scattering as long as thephotons are not observed. To lowest order Born withinthe high-energy approximation, − q /c (cid:29)
1, the differ-ential cross section for the soft photon emission is givenby [5] dσ soft d Ω f = (cid:20) − πc (cid:26)(cid:0) ln( − q /c ) − (cid:1) ln ω E i E f + 12 (cid:0) ln( − q /c ) (cid:1) − (cid:18) ln E i E f (cid:19) + Li (cid:18) cos ϑ f (cid:19) − π (cid:27)(cid:21) (cid:12)(cid:12) A B fi (cid:12)(cid:12) , (2.9)where ϑ f is the scattering angle and ω is the upper limitof radiation. Li( x ) = − (cid:82) x dt ln | − t | t is the Spence func-tion [2]. In the derivation of this formula, the photonmomentum k is partly neglected in the propagators be-fore the integration over the photon degrees of freedomis carried out [2, 5]. The dispersion correction is calculated from the boxdiagram, which accounts for two virtual photon cou-plings between electron and nucleus. The corresponding S -matrix element is given by [13, 23] S box fi = − i (cid:16) ec (cid:17) (cid:90) d x e d y e ¯ ψ f ( y e ) γ µ × S F ( y e − x e ) γ ν ψ i ( x e ) A µν ( y e , x e ) , (2.10)where ψ i and ψ f are the plane-wave electronic scat-tering states, ψ n ( x e ) = (2 π ) − e − ik n x e u ( σ n ) k n , n = i, f . γ µ , µ = 0 , ..., S F is the electronpropagator, defined by S F ( y − x ) = (cid:90) d p (2 π ) e − ip ( y − x ) cp µ γ µ + mc p − m c + i(cid:15) , (2.11)and A µν is the photon field, which for the direct term iscalculated from A µν ( y e , x e ) = 4 πi ( Ze ) (cid:90) d x N d y N D ( y e − y N ) × D ( x e − x N ) ¯ φ f ( y N ) γ J µ S N ( y N − x N ) γ J ν φ i ( x N ) , (2.12)while in the exchange term, x N and y N are interchangedin the photon propagators D , D ( x − y ) = − (cid:90) d Q (2 π ) e − iQ ( x − y ) Q + i(cid:15) , Q = ( Q , Q ) , (2.13)and µ and ν are interchanged in the operators J µ and J ν for the 4-currents of the nucleus. The nuclear propagatoris represented by S N ( y N − x N ) = (cid:88) n (cid:90) d P n (2 π ) φ n ( y N ) ¯ φ n ( x N ) P n − E n /c + i(cid:15) , (2.14)where P n = ( P n , P n ) is the 4-momentum of an interme-diate nuclear state φ n and E n = (cid:113) P n c + M c + ω n itstotal energy with ω n the nuclear excitation energy, and M is the target mass number.We will restrict ourselves to a pure Coulombic excita-tion, µ = ν = 0, while neglecting the magnetic interac-tion. In this approximation, one has¯ φ n ( x N ) γ J ν φ i ( x N ) = e i ( P n − P i ) x N × (cid:104) n | J ( P n − P i ) | i (cid:105) δ ν, , (2.15)where | n (cid:105) comprises the internal quantum numbers ofthe nuclear state φ n . The evaluation of the sum (2.14)over a complete set of nuclear states by means of theGreens function method is very involved [19]. Therefore,following Friar and Rosen [13], we resort to the closureapproximation by fixing ω n = ¯ ω with an appropriatechoice ¯ ω = 15 MeV, which represents the mean exci-tation energy of the giant dipole resonance. Friar andRosen suggested some additional approximations for theevaluation of the S -matrix element. We adopt the fol-lowing ones: the exchange term to (2.12) is dropped, theenergy difference Q in the denominator of the photonpropagators is omitted, and the contribution from thepole of the electron propagator is neglected. With theseapproximations, the transition amplitude for the box di-agram, defined by S box fi = − i δ ( P f − P i + k f − k i ) A box fi , (2.16)is given by A box fi = (cid:18) Ze c (cid:19) π c (2 π ) (cid:90) d q q q − q ) C ( q , q − q ) × ( u ( σ f )+ k f [ E i − ¯ ω − q M + c α ( k i − q ) + mc γ ] u ( σ i ) k i ) (cid:16) E i − ¯ ω − q M (cid:17) − ( k i − q ) c − m c + i(cid:15) . (2.17)The dispersion correction comprises only inelastic in-termediate nuclear states, i.e. n (cid:54) = i in (2.14), whereasthe elastic contribution ( n = i ) can be included by re-placing the Born amplitude A B fi in the leading term withthe respective phase-shift result. Since elastic scatteringimplies | i (cid:105) = | f (cid:105) = | (cid:105) , where | (cid:105) denotes the nuclearground state, the elastic contribution is characterized by (cid:104) | J ( q ) | (cid:105) = F L ( q ) . (2.18)Subtracting this term, the resulting A box fi is identifiedwith the dispersion amplitude, and the respective cor-relation function C in (2.17) is, using closure, givenby C ( q , q ) = 1 Z (cid:104) | J ( q ) J ( q ) | (cid:105)− F L ( q ) F L ( q ) . (2.19)Within the harmonic oscillator model for C, the cor-relation function can be reduced to an expression con-taining F L and the longitudinal proton form factor. It isexplicitly provided in [13]. The leading term in the PWBA transition amplitudefor elastic scattering is given by A B fi = − c Z (cid:112) E i E f q c (cid:16) u ( σ f )+ k f u ( σ i ) k i (cid:17) F L ( q ) . (2.20)In this expression, the magnetic current-current interac-tion is omitted, since C is a spin-zero nucleus. Corre-spondingly, the factor ( − q ) in the denominator of thegeneral theory has been replaced by q [24]. This is thesame approximation as applied to the photon propaga-tors in the box diagram.The corresponding differential cross section reads dσ B d Ω f = | k f || k i | f rec (cid:88) σ i σ f (cid:12)(cid:12) A B fi (cid:12)(cid:12) . (2.21)The cross section is reduced by the recoil factor f rec be-cause of the finite momentum q of the recoiling nucleus[7], f rec = 1 − q E f M c k f (cid:18) − c M E f (cid:19) . (2.22)Therefore, E f is strictly less than E i . Since spin polar-ization of the electron is not considered, an average over σ i and a sum over σ f has to be included in (2.21).The total cross section, accounting for the QED correc-tions and for the second-order elastic ( A B fi ) and inelastic( A box fi ) amplitudes, is in PWBA calculated from [25] dσ Born d Ω f = | k f || k i | f rec (cid:88) σ i σ f (cid:104)(cid:12)(cid:12) A B fi (cid:12)(cid:12) + 2 Re (cid:110) A ∗ B fi (cid:0) A vac fi + A vs fi + A B fi + A box fi (cid:1)(cid:111) + dσ soft d Ω f (cid:21) , (2.23)such that the IR terms in (2.6) and (2.9) cancel. Recall-ing that A B fi ∼ Zα (with α = c ), the terms in (2.23)proportional to A vac fi and A vs fi are of order Z α as is dσ soft d Ω f , while those relating to A B fi and A box fi are of order Z α . Hence (2.23) is consistent to third order in α . Itshould be noted that quadratic terms like | A box fi | haveto be omitted in the total cross section, since they are ofhigher than third order in α . Occasionally it is arguedthat | A box fi | is important in the diffraction minima where A B fi is zero, and is therefore retained [13, 16]. However,there are further contributions of the same order ( α ),like | A B fi | or ( A ∗ B fi + A ∗ box fi )( A vac fi + A vs fi ), which con-tribute in the diffraction minimum and hence should beconsidered if | A box fi | is.In order to account for Coulomb distortion, the leadingterm in (2.23), dσ B d Ω f , is conventionally replaced by theexact result, dσ coul d Ω f = | k f || k i | f rec (cid:88) σ i σ f | f coul ( σ i σ f ) | , (2.24)where f coul is the scattering amplitude obtained from thephase-shift analysis. In terms of the spin-conserving (A)and spin-flip (B) amplitudes [25], the contributions fromthe electronic states with positive (+) or negative (-) he-licity are given by f coul (+ +) = f coul ( − − ) = A,f coul (+ − ) = − f coul ( − +) = i B, (2.25)and further12 (cid:88) σ i σ f | f coul ( σ i σ f ) | = | A | + | B | . (2.26)This leads to the Born-type cross section formula, dσ B − type d Ω f = | k f || k i | f rec (cid:88) σ i σ f (cid:2) | f coul | + 2 Re (cid:8) A ∗ B fi (cid:0) A vac fi + A vs fi + A box fi (cid:1)(cid:9) + dσ soft d Ω f (cid:21) . (2.27)The term proportional to A B fi does no longer occur, sinceit is already incorporated into the exact leading term.At higher collision energies or larger scattering angles,when the electron-nucleus distance (given approximatelyby the inverse momentum transfer) becomes comparableto the nuclear radius, there occur notable differences be-tween f coul and A B fi , even for light nuclei such as C.This is shown in Fig.1 where dσ coul d Ω f is compared to dσ B d Ω f -10 -9 -8 -7 -6 -5 -4 -3
50 60 70 80 90 100 110e + C 240.2 MeV D i ff e r en t i a l c r o ss s e c t i on ( f m / s r) Scattering angle (deg)
FIG. 1: Differential cross section for the elastic scatteringof 240.2 MeV electrons from C as a function of scatter-ing angle ϑ f . Shown is the PWBA result dσ B /d Ω f without( · · · · · · ) and with ( − · − · − ) consideration of q eff , as well asthe phase-shift results dσ coul /d Ω f from (2.24) with collisionenergy E i, kin ( − − −− ) and with ¯ E instead (———-). Theexperimental data ( (cid:7) ) are from Reuter et al [17]. for a collision energy of 240.2 MeV. It is seen that diffrac-tion affects the PWBA at higher momentum transferthan the phase-shift theory, because in the latter the elec-tron is allowed to accelerate in the attractive field of thenucleus, diminishing the distance to the target.In order to improve on the Born amplitude, distortionis conventionally accounted for by shifting the momen-tum | q | , occurring in (2.20), to a slightly higher value q eff , given by [17] q eff = | q | Z c (cid:126) c (cid:113) (cid:104) r (cid:105) E i, kin , (2.28)with (cid:126) c = 197 . E i, kin = E i − c in MeV. The quantity (cid:112) (cid:104) r (cid:105) is the root-mean-square charge radius of the nucleus, which for C is 2.47fm [17, 20]. It is seen in Fig.1 that with this prescription,the minimum of the Born theory basically coincides withthe experimental position.The deviation of dσ coul d Ω f from experiment [17] near theposition of the minimum, seen in Fig.1, can be ascribedto the omission of recoil in the phase-shift analysis. Thiscontrasts the PWBA result where the different energiesof initial and final electronic states can easily be takeninto account.Several ways to include recoil in the phase-shift anal-ysis are considered in the literature. A modification ofthe Dirac equation is suggested in [26], and a scaling ofthe nuclear potential is derived in [27]. Here we adoptthe prescription from [3, 28] which consists in replacingthe kinetic energy E i, kin by an average collision energy¯ E = (cid:112) E i, kin E f, kin (where E f, kin = E f − c ) when per- forming the phase-shift analysis. Modifying f coul in thisway leads to a good agreement with the data concerningthe position of the diffraction minimum. By inspectionof Fig.1 one can also see that at angles 60 ◦ (cid:46) θ (cid:46) ◦ this result is very close to the q eff -modified PWBA result.Note that at very small angles (respectively at very lowmomentum transfer) the q eff -prescription fails, and therecoil-modified f coul is closer to the unshifted A B fi from(2.20).However, even with the above modifications, the mis-match in intensity between the Born theory and thephase-shift analysis persists near and beyond the min-imum. As a consequence, the lowest-order QED correc-tion terms, which in fact are proportional to | A B fi | , willbe seriously in error in the vicinity of the diffraction min-imum where they are most likely to be observed. Follow-ing the suggestion of Maximon [3] to consider Coulombdistortion also in the next-to-leading-order terms, we re-place A B fi with f coul throughout. This implies that (2.27)is exchanged for dσ tot d Ω f = | k f || k i | f rec (cid:88) σ i σ f (cid:2) | f coul | + 2 Re (cid:110) f ∗ coul (cid:16) ˜ A vac fi + ˜ A vs fi + A box fi (cid:17)(cid:111) + d ˜ σ soft d Ω f (cid:21) (2.29)where the tilde indicates that in (2.1), (2.4) and (2.9), A B fi is replaced by f coul . This does not affect the cancel-lation of the IR terms in the vsb contribution. We notethat the replacement implied in (2.29) should not be donefor perpendicular spin asymmetry considerations, sincethis would suppress its changes by the vacuum polar-ization and by the vsb correction for which | f coul | is acommon factor.
3. RESULTS
In this section we provide datails of our numerical com-putations and define the relative changes of the differ-ential cross section due to the QED and the dispersioneffects. Both angular and energy distributions are con-sidered in comparison with experiment. If not statedotherwise, the reduced energy ¯ E will be used throughoutin f coul , and the effective momentum q eff in A B fi . Finallywe discuss the findings of the Jefferson Lab experimentfor 362 MeV impact energy at an observation angle of61 ◦ [16], which is in the region of the first diffractionminimum. For obtaining the phase-shift result, the electronic scat-tering state ψ i is decomposed into partial waves, and foreach partial wave the radial Dirac equations are solvedby means of the Fortran code RADIAL from Salvat etal [29]. The weighted summation of the correspondingphase shifts is carried out with the help of a threefoldconvergence acceleration [30]. The target potential V T isgenerated from the spherical nuclear charge distribution (cid:37) N which for C is available in terms of a Fourier-Besselexpansion, (cid:37) N ( r ) = N (cid:88) k =1 a k j (cid:18) kπrR (cid:19) , r ≤ R , r > R , (3.1)where j is a spherical Bessel function, and the parame-ters a k , R and N are tabulated for C in [31]. Whenexplicitly stated, the tabulation from Offerman et al [20]is used instead. This Fourier-Bessel expansion allows foran analytical formula for the target potential, V T ( r ) = − π N (cid:88) k =1 a k α k (cid:20) sin( α k r ) α k − ( − k (cid:21) , r ≤ R − Zr , r > R , (3.2)where α k = kπR . From the representation (2.5), theCoulombic form factor F L is obtained by means of F L ( q ) = 2 πZ q N (cid:88) k =1 a k α k (cid:20) sin[( q − α k ) R ] q − α k − sin[( q + α k ) R ] q + α k (cid:21) , (3.3)where the argument of F L is identified with | q | . The formfactor F L entering into the correlation function C forthe box diagram is, however, calculated from the sim-ple formula provided in [13] (unless stated otherwise),in order to be consistent with the oscillator model for C . The evaluation of the integral in (2.17) is describedin [13]; however, we have not introduced any further ap-proximations. In particular, the two values of the integra-tion variable | q | where the energy denominator may be-come zero, are determined with Newton’s method. Oneof them is close to ¯ ω/c , the other is of the order of k i .Near these singularities, the integration over the polarangle ϑ q has to be carried out analytically, while the ra-dial and the azimuthal-angle integrals can be performednumerically. More details are given in [32].The free 4-spinors u ( σ n ) k n , n = i, f , entering into A B fi , A box fi and A vs fi , pertain to the initial and final helic-ity eigenstates and are given by u ( ± ) k n = (cid:115) E n + c E n (cid:18) c σk n / ( E n + c ) (cid:19) χ ± (3.4)with χ = (cid:0) (cid:1) and χ − = (cid:0) (cid:1) . The vector σ consists ofthe Pauli matrices.The final momentum k f is determined from the four-momentum conservation, q = P f − P i = k i − k f . (3.5) Upon squaring this equation and using Eq.(10.43) andEq.(10.48) of [24], one obtains, defining the scatteringangle ϑ f = ∠ ( k i , k f ) , | k f | = 1 a (cid:16) b + (cid:112) b − ad (cid:17) ,a = ( M c + E i /c ) − ( | k i | cos ϑ f ) ,b = ( M c · E i /c + c ) | k i | cos ϑ f ,d = − (cid:0) ( M c ) − c (cid:1) (cid:0) ( E i /c ) − c (cid:1) , (3.6)where M = 12 × z -axis is determined by k i and the scat-tering plane coincides with the ( x, z )-plane, such that k f = | k f | (sin ϑ f , , cos ϑ f ) . A comparison of the cross section with experimentfrom Offermann et al [20] for 431.4 MeV electron im-pact is provided in Fig.2a. Like in Fig.1, the experimen-tal data are corrected for the QED effects in order toreproduce the phase-shift results away from the diffrac-tion minimum, if calculated from a suitably fitted nuclearcharge distribution. Again, the recoil-corrected phase-shift analysis agrees very well with the cross section mea-surements. Also shown are results where all correctionsare included. When the formula (2.29) is used, whereCoulomb distortion is fully accounted for, the cross sec-tion is lowered by about 40%. In contrast, if the Born-type prescription (2.27) is applied, the corrections reduceto a few percent near the minimum at 50 ◦ (since A B fi = 0at the minimum position).Fig.2b shows an enlarged region around 50 ◦ . In addi-tion to dσ coul d Ω f , displayed in Fig.2a, we have included thephase-shift result without consideration of recoil, whichclearly shows the mismatch with the data. Also providedis the modified cross section resulting from adding thedispersion correction to the phase-shift result. Clearly,its effect is too small to compensate the underpredictionof the data. In the minimum at ϑ f = 49 . ◦ the devia-tion of the dispersion-corrected theory from the data is5.6%.In order to quantify the QED and dispersion correc-tions it is of advantage to define the relative change ofthe cross section according to [2, 5]∆ σ = dσ/d Ω f dσ LO /d Ω f − , (3.7)where dσd Ω f denotes the cross section in which any QEDeffect or dispersion is included, while the denominator -8 -7 -6 -5 -4 -3 -2
30 40 50 60 70(a) e + C 431.4 MeV D i ff e r en t i a l c r o ss s e c t i on ( f m / s r) Scattering angle (deg) 10 -7 -6
48 48.5 49 49.5 50 50.5 51 51.5 52(b)e + C 431.4 MeV D i ff e r en t i a l c r o ss s e c t i on ( f m / s r) Scattering angle (deg)-0.04-0.02 0 0.02 0.04 0.06 0.08 0.1 30 40 50 60 70 80(c)431.4 MeV D i s pe r s i on c o rr e c t i on Scattering angle (deg) -0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 30 40 50 60 70 80 90 100 110 120300.5 MeV (d) D i s pe r s i on c o rr e c t i on Scattering angle (deg)
FIG. 2: Differential cross section from 431.4 MeV e + C collisions and the cross section change by dispersion for 431.4MeV and 300.5 MeV as a function of scattering angle ϑ f . Shown in (a) is dσ coul /d Ω f (———-) and dσ tot /d Ω f ( − − −− ).Included are results from the Born-type theory (2.27) ( · · · · · · ). The experimental data ( (cid:4) ) are from Offermann et al [20].∆ E/E = 2 × − . Shown in (b) is the enlarged region of the minimum. Included are results from the phase-shift theory usingthe energy E i, kin ( − · − · − ), and when the dispersion correction is included in the cross section according to (2.29)( − − −− ).Shown in (c) is ∆ σ from the dispersion effect for 431.4 MeV: ∆ σ box (———), when in addition F L in C is treated exactly( − · − · − ), and the Born-type theory ( · · · · · · ). The experimental data ( (cid:4) ) are from Offermann et al [20]. Shown in (d) is ∆ σ from the dispersion effect for 300.5 MeV: ∆ σ box (——–) and when in addition F L in C is treated exactly ( − · − · − ). Theexperimental data ( (cid:7) ) are from Reuter et al [17]. represents the leading-order (LO) cross section. Un-less stated otherwise, dσ LO /d Ω f will always be identifiedwith dσ coul /d Ω f from (2.24).The dispersion correction ∆ σ box is calculated from theCoulomb-distortion theory, (3.7) with the use of (2.29),by ignoring the contributions from ˜ A vac fi , ˜ A vs fi and fromthe soft bremsstrahlung. In Fig.2c, ∆ σ box is comparedto the experimental change ∆ σ exp for which dσ/d Ω f inthe numerator of (3.7) refers to the QED-corrected datapoints. As anticipated from Fig.2b, ∆ σ exp increases to8% in the vicinity of the diffraction minimum, set againstthe experimental accuracy of about 2%.In order to display the influence of different models wehave included two more estimates for the dispersion cor-rection: (i) the Born-type theory resulting from (2.27)and (ii) the Coulomb-distortion theory where in the cor-relation function C the exact form factor F L accordingto (3.3) is employed. The latter model may be viewed asan accuracy test of the Friar and Rosen harmonic oscil-lator model for C . In their work [13] they have testedtheir model in a different way by improving on the clo-sure approximation (for an energy of 375 MeV) with thehelp of an expansion in terms of ω n − ¯ ω . They have found(as we do for model (ii)) that the negative excursion of∆ σ box has disappeared at the expense of reducing thepositive excursion near the minimum.It is obvious from Fig.2c that below the onset of diffrac-tion, all models lead to the same result. Strong devia-tions occur in the vicinity of the minimum and persistat still larger angles. However, none of these model-dependent changes can explain the large values of ∆ σ exp in the diffraction region.Fig.2d shows the dispersion effect for an energy of 300.5MeV in comparison with the experiments from Reuter etal [17]. For this energy, the deviation of the various mod-els for dispersion from the QED-corrected data is evenhigher, above 10 percent in the maximum. The samesituation persists for even lower energies. We thereforeascribe the deviations between ∆ σ exp and ∆ σ box partlyto an incorrect account of the QED effects when extract-ing the plotted experimental cross sections from the mea-sured raw data. This conjecture is substantiated below.Fig.3a provides the angular dependence of the disper-sion correction, calculated with the Coulomb-distortiontheory (2.29), for three collision energies, 240.2 MeV,300.5 MeV and 431.4 MeV, for which experimental dataare existing. Clearly, the diffraction structures shift tosmaller angles at higher energies, in accordance with afixed momentum transfer in the form factor which de-termines the cross section minimum. However, | ∆ σ box | decreases strongly with energy in the region of the crosssection minimum. At still larger angles the dispersioneffect is most pronounced for the highest E i as expected,increasing up to 10% for 431.4 MeV at 120 ◦ .The difference between the Born approximation andan exact treatment can easily be shown in the case ofvacuum polarization. In Fig.3b we compare the crosssection change (3.7) as obtained from the exact theory according to dσ vac d Ω f = | k f || k i | f rec (cid:88) σ i σ f | f vac | , (3.8)where f vac is the scattering amplitude obtained from thephase-shift analysis with central potential V T + U e , withthe Coulomb-distorted PWBA result by using (2.29) andretaining in the next-to-leading-order terms only the con-tribution from ˜ A vac fi . Due to the proportionality of thedifferential cross section to | f coul | in the latter case, anydiffraction effect cancels out in (3.7), and the resultingchange ∆ σ B is smoothly increasing with scattering an-gle and with collision energy. On the other hand, ∆ σ vac obtained from the exact theory shows large excursions inthe region of the cross section minima, up to 3% inde-pendent of energy, their width decreasing with E i .The change induced by the vsb correction is shown inFig.3c. Here, the upper limit ω of the bremsstrahlungradiation is set equal to ω = E i, kin ∆ EE , where the exper-imental spectrometer resolution ∆ EE = 2 × − is takenfrom Offermann et al [20]. Like for vacuum polarization,the Coulomb distortion theory obtained from (2.29) byomitting ˜ A vac fi and A box fi , leads to a monotonous increaseof the magnitude of this QED correction with ϑ f and E i .In concord with the findings for vacuum polarization, itis expected that any exact consideration of the vsb effectwill show structures in the vicinity of the cross sectionminima. Therefore, the subtraction of the vsb effect as asmooth background is considered to be erroneous.The total changes by the QED effects and by disper-sion are depicted in Fig.3d for the three collision energies.Shown are the Coulomb distortion results by either treat-ing vacuum polarization in Born according to (2.29), orby considering it exactly by using instead dσ ex d Ω f = | k f || k i | f rec (cid:88) σ i σ f (cid:2) | f vac | + 2 Re (cid:110) f ∗ coul ( ˜ A vs fi + A box fi ) (cid:111) + d ˜ σ soft d Ω f (cid:21) . (3.9)The corrections ∆ σ tot relating to (2.29), like those fromthe use of (3.9), increase in magnitude with ϑ f for smallangles, being dominated by the vsb effect (see Fig.3cwhere ∆ σ tot is included for 300.5 MeV). The structuresin ∆ σ tot near the minima, but also the increase at verylarge angles, relate to the dispersion effect. The exactconsideration of vacuum polarization, ∆ σ ex , leads to ashift of the structures to smaller angles, combined witha filling of the minima and an enhancement of the max-ima. At the positions of the cross section minima, thedeviations between ∆ σ tot and ∆ σ ex amount to 3 − In this subsection the Born result from (2.23) is in-cluded in our calculations of the cross section changes. -0.04-0.02 0 0.02 0.04 0.06 0.08 0.1 30 40 50 60 70 80 90 100 110 120 D i s pe r s i on c o rr e c t i on Scattering angle (deg)(a) 431.4240.2300.5 0.01 0.015 0.02 0.025 0.03 0.035 0.04 30 40 50 60 70 80 90 100 110 120 V a c uu m po l a r i z a t i on Scattering angle (deg) (b)431.4 300.5 240.2-0.5-0.48-0.46-0.44-0.42-0.4-0.38-0.36-0.34 30 40 50 60 70 80 90 100 110 120 vs b c o rr e c t i on Scattering angle (deg)(c)431.4 300.5240.2300.5tot -0.46-0.44-0.42-0.4-0.38-0.36-0.34 30 40 50 60 70 80 90 100 110 120 T o t a l c r o ss s e c t i on c hange Scattering angle (deg)(d) 431.4 300.5 240.2
FIG. 3: Change ∆ σ of the differential cross section from elastic e + C collisions (a) by dispersion, (b) by vacuum polarization,(c) by the vsb correction and (d) including all effects, as a function of scattering angle ϑ f . In (a) and (c), at the collisionenergies 240.2 MeV ( − − −− ), 300.5 MeV (———) and 431.4 MeV ( − · − · − ), these results are derived from the Coulombdistortion theory (2.29). Shown in (b) is ∆ σ vac for 240.2 MeV ( − − −− ), 300.5 MeV (——–) and 431.4 MeV ( − · − · − , upperline). Included is ∆ σ B for 240.2 MeV ( − − −− ), 300.5 MeV ( · · · · · · ) and 431.4 MeV ( − · − · − , lower line). In (d), the totalchange ∆ σ ex from (3.9) is shown for 240.2 MeV ( − − −− ), 300.5 MeV (——–) and 431.4 MeV ( − · − · − ). The results for ∆ σ tot from (2.29) are also shown ( · · · · · · ). This latter result for 300.5 MeV is included in (c) ( · · · · · · ). Fig.4 displays the energy dependence of the correctionfrom vacuum polarization at a backward scattering an-gle, 150 ◦ . At this angle, the diffraction minimum of thecross section is near 175 MeV. The exact consideration of vacuum polarization, ∆ σ vac , shows structures similar tothose present in the angular distribution (see Fig.3b).The results from the Coulomb-distorted Born theory,∆ σ B , are monotonously increasing with E i and can be0 o exactPWBA V a c uu m po l a r i z a t i on Collision energy E i,kin (MeV)
FIG. 4: Change ∆ σ of the differential cross section from e + C collisions by vacuum polarization at ϑ f = 150 ◦ as afunction of collision energy E i, kin . Shown is ∆ σ vac (——–)and ∆ σ B from (2.29) ( − − −− ). Included are results fromthe Born-type theory (2.27) ( · · · · · · ). viewed in terms of a mean cross section change across thediffraction structure. If the PWBA theory (2.23), with dσ LO /d Ω f in (3.7) identified with the Born cross section dσ B d Ω f , were used instead, the identical smooth line wouldbe obtained. This is also true for the vsb correction fromFig.3c, where the PWBA theory reproduces the smoothlines. If, on the other hand, the Born-type theory (2.27)is applied, the structures persist, but they differ notablyfrom those of ∆ σ vac , most obviously by a change in sign.In Fig.5 the energy dependence of the cross sectionwith its QED and dispersion corrections is shown for anangle of 90 ◦ . The experimental data included in Fig.5aresult, when not measured at 90 ◦ , from a spline interpola-tion of adjacent data points. The recoil-modified phase-shift result explains the measurements well. When allcorrections are included, the cross section is lowered by36 −
42% if (3.9) is used, while the corrections tend tozero in the vicinity of the minimum if (2.27) is applied.Fig.5b displays the modifications of the cross sectionby vacuum polarization, treated exactly, and by the dis-persion effect ∆ σ box according to (2.29). The two effectstend to compensate each other in the region 240 − σ ex near 250 MeV results basically from dispersion,with some modification from the vacuum polarization. If,on the other hand, the Born-type theory (2.27) is used,not only the total but also the vsb correction shows adiffraction structure, which, however, decreases to zeroin its maximum. This experiment was carried out at 362 MeV for a sin-gle angle. By working in the effective momentum approx-imation where a modified q eff is defined, which considerssimultaneously recoil and distortion [33], it was foundthat the cross section minimum is obtained for 61 ◦ , suchthat this angle was selected for the measurement [16]. Incontrast to the reduced data tabulated by Reuter et al[17] or Offermann et al [20], the measured cross sectionis provided without any subtraction of QED corrections.This allows for a straightforward comparison with ourQED- and dispersion-corrected theory.Let us first investigate the precise position of the mini-mum at 362 MeV with our method of including distortionand recoil. We consider this method superior to the effec-tive momentum approximation, since it is based on thephase-shift analysis and not on the PWBA. Our previousresults have shown that the minimum position remainsbasically unchanged when the QED and dispersion cor-rections are added to the phase-shift result, both in theangular as well as in the energy distributions. There-fore we can use the recoil-corrected phase-shift theoryto determine for each energy E i, kin the scattering angle ϑ min which leads to the smallest cross section, and sub-sequently to compare these values with the experimentalones obtained by a spline interpolation of adjacent datapoints. This is done in Fig.6a, and it is seen that ourresults agree with the angular position derived from theexperiments by Offermann et al [20] and Reuter et al [17]within 0.2%. On the other hand, the Jefferson Lab datumpoint is well above theory, suggesting that at 362 MeV,the minimum is at 60 . ◦ and not at 61 ◦ . Fig.6b shows thecorresponding cross section dσ coul d Ω f ( ϑ min ) as a function ofenergy. It is seen that beyond 300 MeV, experiment isconsiderably underestimated by theory.It is well understood that diffraction and hence theminimum position is basically determined by the formfactor F L from (3.3). In turn, F L depends only on themomentum transfer, which is a function of E i and ϑ f .This feature is commonly used to determine ϑ min for agiven E i by assuming a fixed momentum transfer. Inorder to test this assertion, we have included in Fig.6a ϑ min as obtained from the fixed value (cid:112) − q = 1 . − , corresponding to the momentum transfer at 300.5MeV. The deviation of this approximation from the exactphase-shift result amounts up to 1% for ϑ min (Fig.6a) andup to 8% for the corresponding cross sections (Fig.6b) inthe energy region under consideration. The validity ofthe q -scaling for the position of the cross section minimawas also investigated experimentally [17], necessitating,however, shifts in the corresponding intensities.Fig.7a displays the energy distribution of the differ-ential cross section at the angle of 61 ◦ , and the experi-mental data from Reuter et al [17] and Offermann et al[20], spline-interpolated to 61 ◦ , are well described by therecoil-modified phase-shift analysis. Identifying ω withthe detector resolution from [16], ∆ E/E = 5 × − , the1 -9 -8 -7 -6 -5 -4 -3
150 200 250 300 350(a) e + C 90 o D i ff e r en t i a l c r o ss s e c t i on ( f m / s r) Collision energy E i,kin (MeV) -0.03-0.02-0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 150 200 250 300 35090 o box vac(b) C r o ss s e c t i on c hange Collision energy E i,kin (MeV)-0.5-0.4-0.3-0.2-0.1 0 150 200 250 300 350(c)90 o vsbtotPWBA C r o ss s e c t i on c hange Collision energy E i,kin (MeV)
FIG. 5: (a) Differential cross section from elastic e + C collisions at ϑ f = 90 ◦ and the cross section change ∆ σ (b) fromvacuum polarization and the dispersion effect and (c) from the vsb correction and from the combined effecs as a function ofcollision energy E i, kin . Shown in (a) is dσ coul /d Ω f (———-) and the total cross section from (3.9) ( − − −− ) and from theBorn-type theory (2.27) ( · · · · · · ). The experimental data are from Reuter et al ( (cid:7) [17]) and from Offermann et al ( (cid:4) [20]).Shown in (b) is ∆ σ vac (——–, thin line) and ∆ σ box ( − · − · − ) as well as their sum (——–, thick line). In (c) the vsb resultsfrom (2.29) ( − · − · − ) and ∆ σ ex (————-) are compared to the Born-type results for the vsb effect ( · · · · · · ) and for the totalcorrection ( − − −− ). ∆ E/E = 2 . × − , corresponding to the experimental resolution of [17]. QED- and dispersion-corrected cross section, calculatedfrom (2.29), is lower by about 35%. If the Born-type the-ory is used instead, the corrections are slightly smallerbut tend to zero in the diffraction minimum. A magni- fication of the minimum region is shown in Fig.7b, andit is seen that the predictions of (2.29) overestimate theJefferson Lab datum point by as much as 28% (whichincreases to 30% if vacuum polarization is treated ex-2
40 50 60 70 80 90 100 110 250 300 350 400 450(a) e + C M i n i m u m po s i t i on ( deg ) Collision energy E i,kin (MeV)10 -8 -7
250 300 350 400 450(b) D i ff e r en t i a l c r o ss s e c t i on ( f m / s r) Collision energy E i,kin (MeV)
FIG. 6: (a) Position ϑ min of the first diffraction minimumin elastic e + C collisions and (b) differential cross section inthis minimum as a function of collision energy E i, kin . Shownin (a) are results from the recoil-corrected phase-shift theory(————–), and from fixing the four-momentum transfer at1.8154 fm − ( · · · · · · ). Shown in (b) are dσ coul d Ω f ( ϑ min ) (——–) and dσ coul d Ω f ( ϑ (cid:48) ) ( · · · · · · ), where ϑ (cid:48) relates to the angle forfixed momentum transfer from (a). The experimental dataare from Reuter et al ( (cid:7) [17]) and from Offermann et al ( (cid:4) [20]). In (a) the Jefferson Lab datum point is included ( (cid:12) [16]). actly according to (3.9)). Were the experimental pointshifted to the minimum at 60 ◦ (keeping its cross sectionvalue unchanged), the overprediction would reduce to 3%(respectively 5%). Included in Fig.7b are the respectiveresults when the charge density provided by Offermannet al [20] is used. The modifications are small, about 5%in the minimum region. Fig.7c isolates the dispersion correction at 61 ◦ . Com-parison is made between ∆ σ box from the Coulomb distor-tion theory, calculated with two different charge densitiesor with the harmonic-oscillator based F L in C replacedby the exact form factor. The use of (cid:37) N from [20] deepensthe minimum and reduces the maximum, while the resultdue to the exact F L leads to some damping of the struc-tures. Included is also a colculation from the Born-typetheory. However, all these models do not substantiallyenhance the dispersion effect and can therefore not ac-count for the discrepancy between theory and experimentin the minimum region.The total cross section changes ∆ σ tot and ∆ σ ex aredisplayed in Fig.7d, together with the modified resultfor ∆ σ ex when F L is treated exactly in C . While thedifferences between these models are small, a consider-able change is introduced if the spectrometer resolution∆ E/E (affecting ω ) is reduced from 5 × − to 2 × − .This lowers the cross section at 362 MeV by 15%. Anagreement between the present theory and experimentwould, however, require that ∆ E/E ≈ × − .
4. CONCLUSION
We have estimated the QED corrections to the differ-ential cross section for elastic scattering, resulting fromvacuum polarization, from the vertex, self energy andsoft bremsstrahlung correction (the vsb effect), as wellas from the dispersion effect. While vacuum polarizationis treated to all orders in Zα , the vsb effect can onlybe estimated in the first-order Born approximation. Thedispersion effect is obtained from the second-order Borntheory with the help of a closure approximation.The sensitivity of the vacuum polarization and the dis-persion correction to various model modifications was in-vestigated in the region of the first diffraction minimumwhere numerous experimental data on the C target areavailable. Considerable differences were found, being ofthe order of the corrections themselves, which is in thepercent region. These model variations pass on to thetotal change of the cross section by the combined QEDand dispersion effects.We did not find it possible to reconcile the earlier ex-perimental data above 200 MeV with our theory, the un-derprediction of the QED-corrected measurements by thetheoretical cross sections (including dispersion) amount-ing up to 10% in the vicinity of the diffraction minimum,which is even higher than quoted in the literature.There is some reason to question the conventionallyused PWBA approach to the vsb correction, which con-tributes with 30 −
35% the biggest portion to the crosssection change. From the investigation of vacuum po-larization, both in PWBA and in an exact treatment, itfollows that the exact theory exhibits structures in thediffraction region while PWBA predicts a monotonousbehaviour with energy or angle. It is therefore conjec-tured that an exact treatment of the vsb effect will show3 -8 -7 -6 -5 -4 -3 -2
150 200 250 300 350 400 45061 o e + C(a) D i ff e r en t i a l c r o ss s e c t i on ( f m / s r) Collision energy E i,kin (MeV) 10 -8 -7 -6 -5
300 320 340 360 380 400(b) 61 o D i ff e r en t i a l c r o ss s e c t i on ( f m / s r) Collision energy E i,kin (MeV)-0.02-0.01 0 0.01 0.02 0.03 150 200 250 300 350 400 450(c) 61 o D i s pe r s i on c o rr e c t i on Collision energy E i,kin (MeV) -0.44-0.42-0.4-0.38-0.36-0.34-0.32-0.3 150 200 250 300 350 400 450(d) 61 o -4 -4 T o t a l c r o ss s e c t i on c hange Collision energy E i,kin (MeV)
FIG. 7: (a,b) Differential cross section, (c) dispersion correction and (d) total corrections for elastic electron scattering from C at ϑ f = 61 ◦ as a function of collision energy E i, kin . Shown in (a) are results from dσ coul /d Ω f (———–) and for the crosssection including QED and dispersion effects by means of (2.29) ( − − −− ) and using the Born-type theory (2.27) ( · · · · · · ).The experimental data are from Reuter et al ( (cid:7) [17]), from Offermann et al ( (cid:4) [20]) and from Jefferson Lab ( (cid:12) [16]). Thesymbol ( ♦ ) shows this datum point when shifted upwards by the difference between the results from (3.9) and (2.24), which isequal to 2 . × − fm /sr. The deviation of this shifted datum point from dσ coul /d Ω f is 18%. In (b) the minimum region isenlarged. In addition to the results from (a) the respective results are shown when the charge density from [20] is used instead: dσ coul /d Ω f ( − · − · − ) and total cross section from (2.29) ( · · · · · · ). Shown in (c) are ∆ σ box (——–), ∆ σ box but with (cid:37) N from[20] ( − · − · − ), ∆ σ box but with exact F L in C ( − − −− ) and dispersion calculated from the Born-type theory ( · · · · · · ). Shownin (d) are the results for ∆ σ ex (————), ∆ σ tot ( · · · · · · ) and ∆ σ ex but with exact F L in C ( − · − · − ). Included are resultsfor ∆ σ tot if the spectrometer resolution is set to ∆ E/E = 2 × − instead of 5 × − ( − − −− ). ω , which we have identifiedwith the experimental detector resolution. An improve-ment of this resolution, respectively a decrease of ω , willlead to an increase in magnitude of the vsb correctionwhich, however, is only weakly dependent on energy andscattering angle.We have also tried to interpret the recent Jefferson Labmeasurement at 362 MeV, where an uncorrected experi-mental value of the cross section is available. In contrastto the underprediction of the data from the other groupsin the diffraction minimum by our theoretical model, theopposite is true for the 362 MeV datum point. Also here,one might attribute the disagreement between experi-ment and theory to an inaccurate account of the vsb cor-rection. However, it remains unclear why its behaviourshould be so fundamentally different when proceeding to lower or higher collision energies.In the literature interpretation of the 362 MeV experi-ment, using a PWBA-based theoretical model, it is con-jectured that also in this case, the experimental cross sec-tion is above the theoretical one. In this context it maybe of importance that we cannot confirm the claim thatthe chosen scattering angle of 61 ◦ leads to the minimumcross section, taken into consideration that our recoil-corrected phase-shift approach is in this respect in perfectagreement with the measurements at different energies.Concerning the applicability of our model at higherimpact energies, one has to cope with the problem thatbeyond 500 MeV the closure approximation, facilitatingthe dispersion estimate, will start to break down. It wasshown in the context of spin asymmetries that for ener-gies in the GeV region an exact treatment of dispersion,feasible in the forward direction, leads to fundamentallyhigher results than obtained within the phase-shift the-ory. This enhancement can be traced back to the influ-ence of highly excited intermediate nuclear states [15, 34],which are suppressed in the closure approximation. It isexpected that such a dispersion behaviour will also influ-ence the differential cross section. [1] T.De Forest Jr. and J.D.Walecka, Adv. Phys. , 1(1966)[2] Y.-S.Tsai, Phys. Rev. , 1898 (1961)[3] L.C.Maximon, Rev. Mod. Phys. , 193 (1969)[4] R.-D.Bucoveanus and H.Spiesberger, Eur. Phys. J. A :57 (2019)[5] L.C.Maximon and J.A.Tjon, Phys. Rev. C , 054320(2000)[6] H. ¨Uberall, Electron Scattering from Complex Nuclei (Academic Press, New York, 1971), § , 461(1984)[8] A.B.Arbuzov and T.V.Kopylova, Eur. Phys. J. C : 603(2015)[9] E.A.Uehling, Phys. Rev. , 55 (1935)[10] S.Klarsfeld, Phys. Lett. B, 86 (1977)[11] H.Mitter and P.Urban, Acta Physica Austriaca , 356(1954)[12] R.R.Lewis Jr., Phys. Rev. , 544 (1956)[13] J.L.Friar and M.Rosen, Ann. Phys. , 289 (1974)[14] T.Herrmann and R.Rosenfelder, Eur. Phys. J. A , 29(1998)[15] M.Gorshteyn and C.J.Horowitz, Phys. Rev. C , 044606(2008)[16] P.Gu`eye et al (Jefferson Lab Collaboration), Eur. Phys.J .A : 126 (2020)[17] W.Reuter, G.Fricke, K.Merle and H.Miska, Phys. Rev. C , 806 (1982)[18] K.Aulenbacher, Hyperfine Interact. : 3 (2011)[19] A.Lovato, S.Gandolfi, J.Carlson, S.C.Pieper andR.Schiavilla, Phys. Rev. Lett. , 082501 (2016)[20] E.A.J.M.Offermann, L.S.Cardman, C.W.De Jager,H.Miska, C.De Vries and H.De Vries, Phys. Rev. C , 1096 (1991)[21] N.Kalantar-Nayestanaki et al, Phys. Rev. Lett. , 2032(1989)[22] L.W.Fullerton and G.A.Rinker Jr., Phys. Rev. A ,1283 (1976)[23] J.D.Bjorken and S.D.Drell, Relativistic Quantum Me-chanics (McGraw-Hill, New York, 1964)[24] F.Gross,
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