Coulomb corrections to the Delbrueck scattering amplitude at low energies
aa r X i v : . [ h e p - ph ] A ug Coulomb corrections to the Delbr¨uck scattering amplitude at low energies
G.G. Kirilin ∗ and I.S. Terekhov † Budker Institute of Nuclear Physics, SB RAS, Novosibirsk (Dated: October 31, 2018)In this article, we study the Coulomb corrections to the Delbr¨uck scattering amplitude. We con-sider the limit when the energy of the photon is much less than the electron mass. The calculationsare carried out in the coordinate representation using the exact relativistic Green function of anelectron in a Coulomb field. The resulting relative corrections are of the order of a few percent forscattering on for a large charge of the nucleus. We compare the corrections with the correspondingones calculated through the dispersion integral of the pair production cross section and also with themagnetic loop contribution to the g-factor of a bound electron. The last one is in a good agreementwith our results but the corrections calculated through the dispersion relation are not.
INTRODUCTION
The elastic scattering of photons by an externalCoulomb field (the so called Delbr¨uck scattering [1]) isone of nontrivial predictions of quantum electrodynam-ics. In the perturbation theory, the Delbr¨uck scatteringamplitude begins from the second order in Zα ( Z | e | isthe charge of the nucleus, α = e ≈ /
137 is the fine-structure constant, we put c = 1, ~ = 1). Therefore,significant efforts have been made to calculate the ampli-tude for the arbitrary scattering angles and energies evenin the lowest-order Born approximation. The results ofthese calculations and the detailed bibliography can befound in the report [2].To calculate the Delbr¨uck scattering amplitude for Z ≫ Zα has been derived in Ref. [3] without any additional as-sumptions, but numerical results have not yet been ob-tained because it is fairly cumbersome.Considerable progress in the calculation of theCoulomb corrections to the lowest-order Born approxi-mation has been achieved for the case of the photon en-ergy ω much large than the electron mass m e and eitherat small scattering angles ∆ /ω ≪ | k − k | , where k and k are the momenta of the photon in the initialand final states, correspondingly) Refs. [4, 5, 6, 7, 8] orat large momentum transfer ∆ /m e ≫ ω = 889 KeVRefs. [13, 14, 15, 16, 17] but the corresponding energyfor the Coulomb corrections is ω = 2754 KeV Ref. [18]only, i. e. above the electron-positron pair productionthreshold.We aim here to calculate the Coulomb corrections atlow energy ω ≪ m e . These corrections have not yetbeen investigated neither experimentally nor theoreti- cally. Nevertheless, they are closely connected with theCoulomb corrections to the pair production cross sectiondue to the dispersion relation Refs. [19, 20] and, as we willshow below, with the magnetic-loop contribution to the g -factor of a bound electron Ref. [21]. We think that theDelbr¨uck scattering amplitude at low energy is useful toestimate the Coulomb corrections for both phenomena.The amplitude is calculated in the coordinate represen-tation with the help of the Green function of an electronmoving in a Coulomb field. The structure of this paperis the following: in Section 2, we provide all necessaryinformation about the Green function and the generalparametrization of the Delbr¨uck scattering amplitude. InSection 3, we show that the calculation in the coordinaterepresentation reproduces the result of the lowest-orderBorn approximation derived in the momentum represen-tation. We point out the difficulties specific for a cal-culation in the coordinate representation also occurringduring the calculation of the Coulomb corrections. Theresults for the Coulomb corrections are given in Section4. We also provide the simple parametrization of theirdependence on Z . In Section 5 we compare our resultswith those obtained via the dispersion relation. The es-timated value for the magnetic-loop contribution to the g -factor of a bound electron is given in Section 6. DELBR ¨UCK SCATTERING AMPLITUDE
We parameterize the Delbr¨uck scattering amplitude asfollows: A = ǫ (1) µ ǫ ⋆ (2) ν Π µν ( ω, k , k , Z ) , (1)Π µν ( ω, k , k , Z ) == α ( Zα ) m e { f ( ω, k , k , Z ) [ g µν k · k − k µ k ν ]+ f ( ω, k , k , Z ) (cid:2) ω g µν − ω ( n µ k ν + n ν k µ )+ n µ n ν k · k ] } (2)where k = ( ω, k ) , k = ( ω, k ) are the 4-momenta ofthe photon in the initial and final states, correspond-ingly, ǫ (1 , are the polarization vectors, the 4-vector n is defined as k · n = k · n = ω , f and f are the formfactors to be calculated. The main purpose of this articleis to calculate lim | k |→ f , (0 , k , k , Z ).In a point-like charge approximation (Coulomb field),the polarization tensor Π µν has the following form:Π µν ( . . . , Z ) = ˜Π µν ( . . . , Z ) − ˜Π µν ( . . . , , (3)˜Π µν ( ω, k , k , Z ) = iα Z d r d r exp( i k r − i k r ) × Z C dǫ π Sp n γ µ ˆ G ( r , r | ǫ ) γ ν G ( r , r | ǫ − ω ) o , (4) where ˆ G ( r , r | ǫ ) is the Green function of an electron ina Coulomb field. The contour of integration over ǫ in theexpression (4) goes from −∞ to ∞ so that it is below thereal axis in the left half-plane and above the real axis inthe right half-plane. It is convenient to turn the contouralong the imaginary axis. In this case the Green functiontakes the following form (see Ref. [22]): G ( r , r | iǫ ) = − πr r p ∞ X l =1 Z ∞ ds exp(2 iZα s ( ǫ/p ) − p ( r + r ) coth s ) × nh R + y I ′ ν ( y ) B l + R − lI ν ( y ) A l i ( γ iǫ + m ) + Zαγ [ im ( ˆn + ˆn ) + p R + coth s ] I ν ( y ) B l − i (cid:20) p r − r s ( ˆn + ˆn ) B l + p coth s ( ˆn − ˆn ) lA l (cid:21) I ν ( y ) (cid:27) , (5)where R ± = 1 ± n n ± i Σ ( n × n ) , Σ k = iǫ ijk (cid:2) γ i , γ j (cid:3) / , ˆn (1 , = γ n (1 , ,A l = ddx ( P l ( x ) + P l − ( x )) , B l = ddx ( P l ( x ) − P l − ( x )) , x = n n ,ν = q l − ( Zα ) , y = 2 p √ r r / sinh s, p = p m + ǫ . (6)For the sake of convenience, we calculate the time-timecomponent of the polarization tensor and the trace of thespatial components separately:Π ( ii ) ( k , Z ) = ( n µ n ν − g µν ) Π µν (0 , k , k , Z ) , (7)Π (00) ( k , Z ) = n µ n ν Π µν ( k , Z ) . (8)Substituting the parametrization of Π µν (2) in the right-hand side of the Eqs. (7) and (8) yields the relation be-tween Π ( ii ) , Π (00) and the form factors f (1 , :Π ( ii ) ( k , Z ) = 2 α ( Zα ) m e k f (0 , k , k , Z ) (9)Π (00) ( k , Z ) = − α ( Zα ) m e k ( f (0 , k , k , Z )+ f (0 , k , k , Z )) . (10) k1 k2 k1 k2(a) (b) FIG. 1: lowest-order Born approximation
LOWEST-ORDER BORN APPROXIMATION
The diagrams of the second order of the perturbationtheory in Zα are depicted in Fig. 1. Their contributionwere calculated in Refs. [23, 24]. We aim here to demon-strate that the calculation of this diagrams in the coordi-nate representation reproduces the result of the lowest-order Born approximation.To calculate the contributions of the diagrams, seeFig. 1, we expand the exponent function in the expres-sion (4) up to the second order in | k | = | k | = | k | :2Π µν ( a ) + Π µν ( b ) = α k Z d r d r | r − r | Z ∞−∞ dǫ π × Sp n γ µ G (0) ( r , r | iǫ ) γ ν G (2) ( r , r | iǫ )+ γ µ G (1) ( r , r | iǫ ) γ ν G (1) ( r , r | iǫ ) o , (11)where G ( n ) is the contribution to the Green function (5) of the n -th order in Zα : G (0) = − (cid:20) ( γ iǫ + m ) + i ˆr − ˆr | r − r | ∂∂ | r − r | (cid:21) × exp ( − | r − r | )4 π | r − r | , (12) G (1) = − Zα π − t r r (cid:26) i ǫρp (cid:20) γ iǫ + m + ip ρ ( ˆr − ˆr ) ∂σ∂σ + ip ˆn − ˆn ) ∂∂ρ (cid:21) F ( ρ, σ )+ 11 − σ γ (cid:18) ˆ R + − i mp ( ˆn + ˆn ) ∂∂ρ (cid:19) (cid:18) σ e − ρ σ − e − ρ (cid:19)(cid:27) , (13) G (2) = ( Zα ) πr r Z ∞ ds exp( ip ( r + r ) coth s ) ( ∞ X l =1 (cid:20)(cid:18) ˆ R + B l y ∂ ∂y + ˆ R − lA l (cid:19) ( γ iǫ/p + m/p ) (14) − i p ( r − r )2 sinh s ( ˆn + ˆn ) B l − i coth s ( ˆn − ˆn ) lA l (cid:21) ∂ l ∂ν I ν ( y ) (cid:12)(cid:12)(cid:12)(cid:12) ν = l (15)+ ǫs p " ǫs p ˆ T + γ (cid:18) mp ( ˆn + ˆn ) − ˆ R + i coth s (cid:19) y cos φ I (cid:18) y cos φ (cid:19) , (16)where F ( ρ, σ ) = 12 ρσ (cid:20) e − ρ σ log 1 + σ − σ + e ρ σ Γ (0 , ρ (1 + σ )) − e − ρ σ Γ (0 , ρ (1 − σ )) (cid:21) , (17)ˆ T = " y ( γ iǫ/p + m/p ) − i y coth s ( ˆn − ˆn ) − i p ( r − r )sinh s ( ˆn + ˆn )cos φ y ∂∂y I (cid:18) y cos φ (cid:19) , (18)where Γ ( a, z ) = R ∞ z t a − exp ( − t ) dt is the incompletegamma function. The following notations have been alsointroduced in the expressions (12-18): ρ = p ( r + r ) , σ = | r − r | r + r , (19) t = r − r r + r , cos φ (cid:18) n n (cid:19) / . (20)The contribution to the Green function G (2) of the secondorder is split into two parts: the contribution correspond-ing to the expansion of a Bessel function over an index(the part of the expressions (14), (15) separated by thesquare brackets) and the contribution in which we sub-stitute the indices of the Bessel functions for 2 l and sumover l explicitly. The latter can be called conditionally quasiclassical contribution because it corresponds to thecontribution of the large angular momenta l ≫ Zα .There are some points to be made. First of all, the con-tribution of each diagram Fig. 1 is infrared divergent, i. e.it is divergent at large distances. Since the divergence is canceled between the diagrams Fig. 1(a) and Fig. 1(b),the contribution each of them depends on the regular-ization of this divergence. However, the contribution ofseparated terms (for example, the quasiclassical contri-bution or the contribution proportional to ∂ ν I ν (14),(15)) and even the presence of the divergence dependson the order of the iterated integration over the spatialvariables r i and the inner variables of the Green functions– ”proper times” s i . An iterated improper integral, thevalue of which depends on the order of integration, wecall a conditionally convergent iterated integral. The ex-ample of such integral, that arises during the calculationof the diagram Fig. 1(b), is given in Appendix. To avoidthe complications due to an explicit regularization anddifficulties during the calculation of separated terms, oneshould fix the order of integration for all the diagramsand sum the contribution of each one before the inte-gration with respect to the last variable. We chose thevariable t defined in (20) as the last integration variable.In this case, the contributions of the diagrams (Fig. 1) H P H L - P B o r n H L L (cid:144) P B o r n H L FIG. 2: The relative Coulomb corrections to the trace of thetime-time component of the polarization tensor. The dashedcurve corresponds to the fit a ( Zα ) , the solid line correspondsto a ( Zα ) + b ( Zα ) have the following form:Π ( ii )( a ) = α ( Zα ) m e k (cid:18) − − Z dtt (cid:19) , (21)Π ( ii )( b ) = α ( Zα ) m e k (cid:18) Z dtt (cid:19) , (22)Π ( ii )Born = 2Π ( ii )( a ) + Π ( ii )( b ) = α ( Zα ) m e k . (23)The time-time component is derived in a similar manner:Π (00)Born = α ( Zα ) m e k . (24)Substituting the expressions (23) and (24) in (9) and (10)yields the form factors (4) in the lowest-order Born ap-proximation f = 716 · , f = − · . (25)As noticed above, they coincide with the results derivedin [23, 24]. COULOMB CORRECTIONS
Analytical derivation of the Coulomb corrections to thelowest-order Born approximation (25) is a rather com-plicated problem. We have calculated these correctionsmostly numerically. To increase the accuracy of the nu-merical calculations we have subtracted the lowest-orderBorn approximation Eq. (11) from the general expressionEq. (4) before any transformations. Let us denote the an-gular momenta and the proper times that appear in the H P H ii L - P B o r n H ii L L (cid:144) P B o r n H ii L FIG. 3: The relative Coulomb corrections to the trace of thespatial components of the polarization tensor. The dashedcurve corresponds to the fit a ( Zα ) , the solid line correspondsto a ( Zα ) + b ( Zα ) Green functions in Eq. (4) as l , l and s , s . After sub-traction, it is convenient to change variables as follows:( r , r ) → ( η = √ r r , t ′ = r /r ). Then we integrateanalytically over ǫ , x = ( r r ) / | r || r | , t ′ one by one. Af-ter that we also perform the analytical summation over l . Thus, the expression (4) can be reduced to the sumover l and the iterated integral over s , s and η . Theexplicit expression for the integrand is omitted here asbulky. Further analytical integration is only possible forseparate terms. The term previously referred to as the quasiclassical (which contains I l instead of I ν ) can berepresented as an one-fold integral or as an infinite seriesover Zα . For example, the corresponding contribution toΠ (00) is the following :Π (00)quasicl . − Π (00)quasicl . Born = α ( Zα ) m e k × (cid:18) π − Zα ) − π (cid:0) π − (cid:1) Zα ) + π (cid:0) π − (cid:1) Zα ) − π (cid:0) π − (cid:1) Zα ) + π (cid:0) π − (cid:1) Zα ) + . . . ! . (26)This contribution is finite, i. e. the infrared divergency isabsent in Eq. (26). Nevertheless, the residual part mustbe integrated in the same order as that used to deriveEq. (26). We check explicitly that the contribution con-taining the subtraction from the single Bessel function,i. e. which is proportional to I ν − I l − ( Zα ) l ∂ ν I ν (cid:12)(cid:12)(cid:12)(cid:12) ν = l = O ( Z α ) , (27)is a conditionally convergent iterated integral.The results of our numerical calculations are presentedin Figs. 2, 3 and Tab. I. TABLE I: Relative Coulomb correctionsZ “ Π (00) − Π (00)Born ” / Π (00)Born “ Π ( ii ) − Π ( ii )Born ” / Π ( ii )Born
50 1 . · − . · −
60 2 . · − . · −
70 3 . · − . · −
80 5 . · − . · − It should be noted that the quasiclassical contributionis of the order of 60% of the lowest-order Born approxi-mation for Z = 80. The residual part containing the sub-straction from Bessel functions reduces the corrections upto a few percents. This cancellation adversely affects theaccuracy of the calculation. The authors carried out thecalculations of the corrections independently. The accu-racy of the results (Tab. I) is better than one percent.Now we consider the dependence of the Coulomb cor-rections on Z . The first two terms in Eq. (26) domi-nate and give 97 −
96% of the quasiclassical contributionin spite of the fact that we calculate complete series in Zα . It comes as a surprise that the complete results canbe adequately fitted by a biquadratic polynomial in Zα without a free term:Π (00) − Π (00)Born Π (00)Born = 230459 h (3 . ± . · − ( Zα ) + (1 . ± . · − ( Zα ) i , (28)Π ( ii ) − Π ( ii )Born Π ( ii )Born = 5767 h (6 . ± . · − ( Zα ) + (3 . ± . · − ( Zα ) i . (29)The results of the fitting with a quadratic function a ( Zα ) and also the functions in Eqs. (28), (29) areshown in Figs. 2, 3. One further comment is in order.The coefficients at ( Zα ) in Eqs. (28) and (29) have amagnitude one or two orders less than those at ( Zα ) in the lowest-order Born approximation Eqs. (23), (24).If one assumes the same hierarchy between the coeffi-cients at ( Zα ) and ( Zα ) , then the coefficients of ( Zα ) could not be distinguished from zero with our accuracy.In this case, the maximal difference between the dashedand solid curves in Figs. 2, 3 shows the actual accuracyof our calculations. Substituting Eqs. (28), (29) in therelations Eqs. (9), (10), we obtain the form factors f (1 , : f = 71152 + 3 . · − ( Zα ) + 1 . · − ( Zα ) , (30) f = − − . · − ( Zα ) − . · − ( Zα ) . (31) PAIR PRODUCTION VIA DISPERSIONRELATION
In the papers [20], [19], Gluckstern and Rohrlich havederived the relation between the pair production crosssection in a Coulomb field and the Delbr¨uck amplitudeaveraged over the polarizations: A ( ω ) = ω π Z ∞ m σ γ → e + e − ( ω ′ ) ω ′ − ω + i dω ′ . (32)The amplitude (1) averaged over the polarizations of thephoton has the form: A = 12 (cid:18) δ ij − k i k j ω (cid:19) Π ij = − α ( Zα ) m e f ω . (33)One can find the relation between the Coulomb correc-tions to the form factor f and the pair production crosssection in a Coulomb field by using the expressions (32),(33) (we put m e = 1 in this section): f = − π α ( Zα ) Z ∞ σ ( ω ′ ) ω ′ dω ′ . (34)Let us check the formula (34) in the Born approximation.Substituting Z = 82 (lead) yields:12 π α ( Zα ) f B Z ∞ − σ B ( ω ′ ) ω ′ dω ′ = 1 + 4 · − , (35)where σ B is replaced by the asymptotical formulae de-rived by Maximon in Ref. [25] for ω < . σ B ( ω ) = α ( Zα ) π (cid:18) ω − ω (cid:19) × (cid:18) ǫ ǫ + 1160 ǫ + 29960 ǫ + O ( ǫ ) (cid:19) , (36) ǫ = 2 ω −
42 + ω + 2 (2 ω ) / , (37)for ω > . σ B ( ω ) = α ( Zα ) (cid:26)
289 log 2 ω − (cid:18) ω (cid:19) (cid:20) ω −
72 + 23 log ω − log ω − π ω + π ζ (3) (cid:21) − (cid:18) ω (cid:19) (cid:20)
316 log 2 ω + 18 (cid:21) − (cid:18) ω (cid:19) (cid:20)
29 log 2 ω · − · (cid:21) + O (cid:18) ω (cid:19)) . (38)Now we discuss the Coulomb corrections to the form fac-tors. Using the Eq. (31) we have obtained the relativecorrection to the form factor in the Born approximation Ω - H Π Α Z f B o r n L - Σ C H Ω L (cid:144) Ω FIG. 4: Coulomb corrections to the pair production crosssection ( Z = 82) (here and below all the calculations are carried out for Z = 82): f − f B f B = 4 . · − . (39)However, if we use the Coulomb corrections to the pairproduction cross section σ C ( ω ) = σ ( ω ) − σ B ( ω ) derivedin Ref. [26] for the photon energy ω <
10 and the inter-polation equation derived in Ref. [27] for ω >
10, thenthe relative correction to the form factor in the Born ap-proximation is − π α ( Zα ) f B Z ∞ . σ C ( ω ) ω dω = 2 . · − . (40)This result is 20 times less than that in Eq. (39). Theintegrand (40) as a function of ω is shown in Fig. 4. Itvaries mainly in the region 2 < ω <
30 but there is a longnegative ”tail” for ω → ∞ . The total integral is a resultof the almost complete cancelation between the positivecontribution for ω .
10 and the negative one for ω & R ∞ σ T ( ω ) /ω dω R ∞ | σ T ( ω ) /ω | dω = 3 . · − . (41)For the integral (40) to be calculated with the sufficientaccuracy it is necessary to derive the Coulomb correc-tions to the cross section with an accuracy better thana few percents. It is quite possible that this cancelationcauses the discrepancy due to the lack of precision inthe calculations of the positive part of the integrand inRef.[26].The cause of the discrepancy could also be the inter-polation equation derived in Ref. [27] (the region 10 . ω . /ω TABLE II: Integration of the 1 /ω -corrections over the nega-tive ”tail” [see Fig. 4 and Eq. (42)]Contribution in σ C when ω → ∞ − Z ∞ ω dω σ C ( ω ) ω π α ( Zα ) f B O (1) − . O (1) + O (1 /ω ) 0 . O (1) + O (1 /ω ) + O (1 /ω ) − . and (1/ ω ) log ω/ σ C ( ω ≫
2) = α ( αZ ) (cid:26) − f ( Zα ) + 1 ω (cid:20) − π g ( Zα ) − π ( Zα ) f ( Zα ) (cid:3) + bω log ω (cid:27) , (42)where the functions f, g and f are derived analyti-cally but the coefficient b is obtained by an interpola-tion procedure from the experimental data for ω & ω >
25. The expression(42) is zero when ω = 8 .
95 (see Fig. 4, the correspond-ing value for the approximation formula of Ref. [27] is ω = 10 . − R ∞ ω dωσ C ( ω ) / ( ω π α ( Zα ) f B ) so that the terms ofhigher orders in 1 /ω are accounted for in σ C ( ω ) one af-ter another. The results are presented in the table (II).One can see that the successive terms from Eq. (42) thusintegrated give the contributions of the same order, i. e.the process does not converge to a certain value of theintegral.It is also quite possible that, in order to resolve thecontradiction between the results (39) and (40), the pairproduction in bound-free states should be taken into ac-count because of the strong cancellation Eq. (41) of thecontribution of free-free states.The expression (40) coincides with that calculatedin Ref. [32] (more precisely, − D /f B in the nota-tions of Ref. [32]). The comparison of our results, i.e.( f ( Z ) − f B ) /f B , and the results of Ref. [32] − D /f B is made in Fig. 5.It should be noted that our results and those of Ref. [32]are essentially different because the last one have a non-monotonic dependance on Z . g -FACTOR OF A BOUND ELECTRON The amplitude of virtual light-by-light scattering isknown to be a part of the so-called magnetic loop contri-bution to the g -factor of a bound electron Ref. [34].For the 1 S / electron state, this contribution reads H f - f B L (cid:144) f B FIG. 5: Our results (triangles) for ( f ( Z ) − f B ) /f B and theapproximation formula Eq. (31) (dashed line) in comparisonwith the results of Ref. [32] (squares).
40 50 60 70 80 90Z00.20.40.60.81 D g10 FIG. 6: The squares represent the difference ∆ g Ref.[21] − ∆ g Ref.[33] , the solid line corresponds to the function(16 / α ( Zα ) (3 . · − ( Zα ) ), the dashed line is the func-tion (16 / α ( Zα ) (3 . · − ( Zα ) + 1 . · − ( Zα ) ), cor-responding to the Coulomb corrections to the form factor f Eq. (30). (see Ref. [33]):∆ g = − α ( Zα ) πm Z ∞ dq f ( q/m ) × Z ∞ dr r ˜ f ( r ) ˜ f ( r ) (cid:18) sin qrqr − cos qr (cid:19) , (43)where ˜ f and ˜ f are the radial parts of the electron wavefunction in a Coulomb field: ψ ( r ) = ˜ f Ω − i ˜ f ( σ · n ) Ω ! , (44)where Ω is the spherical spinor (see for example Ref. [35]).Using the lowest-order Born approximation for the formfactor f and the nonrelativistic expressions for the com-ponents of the wave function ˜ f ( r ) = 2 exp( − r/a B ) and ˜ f = ˜ f ′ ( r ) / m yields the leading correction to the g -factor of a bound electron in 1 S / state Ref. [34]:∆ g = 7216 α ( Zα ) . (45)In the case of small Zα , one can expand the form factor f in power series of Zα : f (0 , , q , Zα ) = F (cid:16) qm (cid:17) + ( Zα ) F (1) (cid:16) qm (cid:17) + O (cid:0) Z α (cid:1) . (46)The contribution of F ( q/m ) was considered in Ref. [33]in detail. To calculate the correction in Zα , it is suffi-cient to use the functions ˜ f and ˜ f in the nonrelativisticlimit and the expression (30) f − / F (1) (0). Theresults of the numerical calculation of the magnetic-loopcontribution exact in Zα are presented in Ref. [21].The comparison of the contribution of the Coulombcorrections to the form factor f Eq. (30) and the dif-ference of the results obtained in Ref. [21] and Ref. [33]is depicted in Fig. 6. It is surprising that our correctioncoincides with this difference not only for the small Zα ,but for Zα ∼ CONCLUSIONS
The Coulomb corrections to the Delbr¨uck scatteringamplitude have been considered in this article. We havecalculated these corrections in the low-energy limit buttaking into account all orders of the parameter Zα . Theaccuracy of the calculation is of the order of one percentfor Z = 50 and increases with Z . Our result is in agood agreement with the corresponding contribution tothe g -factor of a bound electron calculated previouslyin Ref. [21]. However, there is a contradiction with thedispersion integral of the Coulomb corrections to the pairproduction cross section calculated in Ref. [32].We would like to thank A.I. Milstein and R.N. Lee fortheir helpful comments and discussions. EXAMPLE OF REARRANGEMENT OF ACONDITIONALLY CONVERGENT INTEGRAL
While calculating the contribution of the first order in Zα , we have expanded the expression (5) on Zα and in-tegrated over s . The equation (22) corresponding to thecontribution of the diagram [Fig. 1(b)] has been derivedby integration over ρ , σ and t in order [see Eq. (20)].However, one can calculate this contribution in an alter-native way – by substituting the expansion of Eq. (5) inEq. (11) and integrating over r and r before the inte-gration over s and s in the Green functions (5). Oneof the typical expressions appeared is Y ( t , t , z ) = 1 + t t (1 + 2 z ( t + t ) ) ( t + t ) × − t t )( t + t ) ! , (A.47)where t , = coth s , ∈ (1 , ∞ ) and z = (1+ x ) / ∈ (0 , t , z and t one after another (or z , t and t ) and obtain afinite result, that is Z ∞ Z Z ∞ Y dt dz dt = Z ∞ Z ∞ Z Y dz dt dt = 13360 − π √ √ √ . (A.48)However, if one integrates Eq. (A.47) over t and t atfirst then the result is the function of z ˜ Y ( z ) = Z ∞ (cid:18)Z ∞ Y ( t , t , z ) dt (cid:19) dt = 160 z (cid:26) z √ z (cid:16) arctan (cid:16) √ z (cid:17) − π (cid:17) +16 z (5 − z ) log (cid:18) z z (cid:19) + 96 z − (cid:27) , (A.49)which has a nonintegrable singularity at z = 0˜ Y ( z ) = − π √ z / + O (cid:16) z − / (cid:17) . (A.50) ∗ [email protected] † [email protected][1] L. Meitner, H. K¨usters, and M. Delbr¨uck), Zeitschrift f¨urPhysik A Hadrons and Nuclei , 137 (1933).[2] P. Papatzacos and K. Mork, Phys. Rept. , 81 (1975).[3] A. Scherdin, A. Schafer, W. Greiner, and G. Soff, Phys.Rev. D45 , 2982 (1992).[4] H. Cheng and T. T. Wu, Phys. Rev. , 1873 (1969).[5] H. Cheng and T. T. Wu, Phys. Rev. D2 , 2444 (1970).[6] H. Cheng and T. T. Wu, Phys. Rev. D5 , 3077 (1972).[7] A. I. Milstein and V. M. Strakhovenko, Sov. Phys. -JETP , 8 (1983).[8] A. I. Milstein and V. M. Strakhovenko, Phys. Lett. A95 ,135 (1983).[9] H. Cheng, E.-C. Tsai, and X.-q. Zhu, Phys. Rev.
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