Charge asymmetry in the differential cross section of high-energy bremsstrahlung in the field of a heavy atom

# Charge asymmetry in the differential cross section of high-energy bremsstrahlung in the field of a heavy atom

P.A. Krachkov Budker Institute of Nuclear Physics, 630090 Novosibirsk, Russia Novosibirsk State University, 630090 Novosibirsk, Russia    A. I. Milstein Budker Institute of Nuclear Physics, 630090 Novosibirsk, Russia
July 26, 2019
###### Abstract

The distinction between the charged particle and antiparticle differential cross sections of high-energy bremsstrahlung in the electric field of a heavy atom is investigated. The consideration is based on the quasiclassical approximation to the wave functions in the external field. The charge asymmetry (the ratio of the antisymmetric and symmetric parts of the differential cross section) arises due to the account for the first quasiclassical correction to the differential cross section. All evaluations are performed with the exact account of the atomic field. We consider in detail the charge asymmetry for electrons and muons. For electrons, the nuclear size effect is not important while for muons this effect should be taken into account. For the longitudinal polarization of the initial charged particle, the account for the first quasiclassical correction to the differential cross section leads to the asymmetry in the cross section with respect to the replacement , where is the azimuth angle between the photon momentum and the momentum of the final charged particle.

###### pacs:
12.20.Ds, 32.80.-t

## I Introduction

The theoretical investigation of high-energy bremsstrahlung and high-energy particle-antiparticle photoproduction in the electric field of a heavy nucleus or atom has a long history because of importance of these processes for various applications; for the latter process see reviews in Refs. HO1980 (); H2000 (). These processes should be taken into account when considering electromagnetic showers in detectors, they also give the significant part of the radiative corrections in many cases. Therefore, it is necessary to know the cross sections of these processes with high accuracy. In the Born approximation, the cross sections of both processes have been obtained for arbitrary energies of particles and for arbitrary atomic form factors BH1934 (); Racah1934 () (see also Ref. BLP1982 ()). The Coulomb corrections to the cross section, which are the difference between the exact in the parameter cross section and the Born cross section, are very important (here is the atomic charge number, is the fine-structure constant, ). There are formal expressions for the Coulomb corrections to the cross sections exact in and energies of particles Overbo1968 (). However, the numerical computations based on these expressions become more and more difficult when energies are increasing, and, for instance, the numerical results for photoproduction have been obtained so far only for the photon energy  MeV SudSharma2006 ().

At high energies of initial particles, the final particle momenta usually have small angles with respect to the incident direction. In this case typical angular momenta, which provide the main contribution to the cross section, are large (, where is energy and is the momentum transfer). This is why the quasiclassical approximation, based on the account of large angular momenta contributions, becomes applicable. In this approximation, the wave functions and the Green’s functions of the Dirac equation in the external field have very simple forms which drastically simplify their use in specific calculations. The wave functions, obtained in the leading quasiclassical approximation for the Coulomb field, are the famous Furry-Sommerfeld-Maue wave functions Fu (); ZM () (see also Ref. BLP1982 ()). The quasiclassical Green’s function have been derived in Ref.  MS1983 () for the case of a pure Coulomb field, in Ref. LM95A () for an arbitrary spherically symmetric field, in Ref. LMS00 () for a localized field which generally possesses no spherical symmetry, and in Ref. DM2012 () for combined strong laser and atomic fields.

In the leading quasiclassical approximation, the cross sections for pair photoproduction and bremsstrahlung have been obtained in BM1954 (); DBM1954 (); OlsenMW1957 (); O1955 (); OM1959 (). The first quasiclassical corrections to the spectra of both processes, as well as to the total cross section of pair photoproduction, have been obtained in Refs. LMS2004 (); LMSS2005 (); DM10 (); DM12 (). Recently, the first quasiclassical correction to the fully differential cross section was obtained in Ref. LMS2012 () for pair photoproduction and in Ref. DLMR2014 () for pair photoproduction. As a result, the charge asymmetry in these processes (the asymmetry of the cross section with respect to permutation of particle and antiparticle momenta) was predicted. This asymmetry is absent in the cross section calculated in the Born approximation and also in the cross section exact in the parameter but calculated in the leading quasiclassical approximation. Thus, the charge asymmetry appears solely due to the quasiclassical corrections to the Coulomb corrections. The difference between the atomic field and the Coulomb field of a nucleus results in the modification of the cross sections (effect os screening). The influence of screening on the Coulomb corrections to pair photoproduction is small for the differential cross section and for the total cross section DBM1954 (). However, screening is important for the Born term. The quantitative investigation of the effect of screening on the Coulomb corrections to the photoproduction cross section is performed in Ref. LMS2004 ().

The influence of screening on the bremsstrahlung cross section in an atomic field is more complicated. It is shown in Refs. OlsenMW1957 (); LMSS2005 () that the Coulomb corrections to the differential cross section are very susceptible to screening. However, the Coulomb corrections to the cross section integrated over the momentum of final charged particle (electron or muon) are independent of screening in the leading approximation over a small parameter , where is a screening radius and is the electron mass. The quantitative investigation of the effect of screening on the Coulomb corrections to the spectrum of bremsstrahlung is performed in Ref. LMSS2005 (). The differential cross section of bremsstrahlung, calculated in the leading quasiclassical approximation, is the same for and (for and ). Therefore, to predict the charge asymmetry (the difference between the bremsstrahlung differential cross section for particles and antiparticles in the atomic field), one should perform calculations in the next-to-leading quasiclassical approximation. This is the main goal of our paper. The result is obtained exactly in the parameter . Besides, for the case of muons the nuclear size effect is taken into account.

The bremsstrahlung differential cross section from high-energy charged particle in an atomic field, , can be written as

 dσ(p,q,k,η)=dσs(p,q,k,η)+dσa(p,q,k,η), dσs(p,q,k,η)=dσ(p,q,k,η)+dσ(p,q,k,−η)2, dσa(p,q,k,η)=dσ(p,q,k,η)−dσ(p,q,k,−η)2, (1)

where is the photon momentum, and are the initial and final charged particle momenta, respectively. Evidently, the bremsstrahlung differential cross section from high-energy antiparticle can be obtained from by the replacement , so that it is equal to . We show that the antisymmetric part of the differential cross section, , is independent of screening in the kinematical region which provides the main contribution to the antisymmetric part of the spectrum.

The paper is organized as follows. In Sec. II we derive the general expression for the quasiclassical matrix element of the process. In Sec. III we find in the quasiclassical approximation all structures of the Green’s function of the squared Dirac equation for a charged particle in arbitrary localized potential and the corresponding structures of the wave functions. We obtain the leading terms and the first quasiclassical corrections as well. Using these wave functions, we derive in Sec. IV the matrix element of the process and the corresponding differential cross section for arbitrary localized potential and in the particular case of the pure Coulomb field. In Sec. V we investigate in detail the charge asymmetry in high-energy bremsstrahlung from electrons. In this case the nuclear size effect is not important. In Sec. VI we investigate the charge asymmetry in high-energy bremsstrahlung from muons which is sensitive to the deviation at small distances of the nuclear atomic field from the pure Coulomb field. Finally, in Sec. VII the main conclusions of the paper are presented.

## Ii General discussion

The differential cross section of bremsstrahlung in the electric field of a heavy atom reads BLP1982 ()

 dσ=αωqεq(2π)4dΩkdΩqdω|M|2, (2)

where and are the solid angles corresponding to the photon momentum and the final charged particle momentum , is the photon energy, , , and is the particle mass. Below we assume that and . The matrix element reads

 M=∫dr¯u(−)q(r)γ⋅e∗u(+)p(r)exp(−ik⋅r), (3)

where are the Dirac matrices, and are the solutions of the Dirac equation in the external field, is the photon polarization vector. The superscripts and remind us that the asymptotic forms of and at large contain, in addition to the plane wave, the spherical convergent and divergent waves, respectively. The wave functions and have the form DLMR2014 ()

 ¯u(−)q(r)=¯uq[f0(r,q)−α⋅f1(r,q)−Σ⋅f2(r,q)], u(+)p(r)=[g0(r,p)−α⋅g1(r,p)−Σ⋅g2(r,p)]up, up=√εp+m2εp⎛⎜⎝ϕσ⋅pεp+mϕ⎞⎟⎠,uq=√εq+m2εq⎛⎜⎝χσ⋅qεq+mχ⎞⎟⎠, (4)

where and are spinors, , , and are the Pauli matrices. The following relations hold

 g0(r,q)=f0(r,−q),g1(r,q)=f1(r,−q),g2(r,q)=−f2(r,−q). (5)

The wave functions in the atomic field can be found from the Green’s function of the “squared” Dirac equation in this field using the relations DLMR2014 ()

 expipr14πr1u(+)p(r2)=−limr1→∞D(r2,r1|εp)up,p=−pn1, expiqr24πr2¯u(−)q(r1)=−limr2→∞¯uqD(r2,r1|εq),q=qn2,

where , , and

 D(r2,r1|ε)=⟨r2|1^P2−m2+i0|r1⟩ =⟨r2|[(ε−V(r))2+∇2−m2+iα⋅∇V(r)+i0]−1|r1⟩. (7)

Here , , and is the atomic potential. It follows from Eq. (II) that the Green’s function can be written as

 D(r2,r1|ε)=d0(r2,r1)+α⋅d1(r2,r1)+Σ⋅d2(r2,r1). (8)

It is convenient to calculate the matrix element for definite helicities of the particles. Let , , and be the signs of the helicities of initial charged particle, final charged particle, and photon, respectively. We fix the coordinate system so that is directed along -axis and lies in the plane with . Denoting helicities by the subscripts, we have

 ϕμp χμq eλ =1√2(ex+iλey), (9)

where and are the polar angles of the vectors and , respectively. The unit vectors and are directed along and , respectively, where the notation for any vector is used. We also introduce the vectors , , and the matrix , which can be written as

 F=18(aμpμq+Σ⋅bμpμq)[γ0(1+PQ)+γ0γ5(P+Q)+(1−PQ)−γ5(P−Q)], P=μppεp+m,Q=μqqεq+m, (10)

where and are defined from

 ϕμpχ†μq=12(aμpμq+σ⋅bμpμq). (11)

We obtain from Eq.(II)

 aμμ=1−θ2pq8−iμ4ν⋅[θp×θq], aμ¯μ=μ√2eμ⋅θpq, bμμ={μ[1−18(θp+θq)2]+i4ν⋅[θp×θq]}ν +μ2(θp+θq)+i2[θpq×ν], bμ¯μ=√2eμ−1√2(eμ,θp+θq)ν, (12)

where and . The matrix element , Eq. (3), can be written as follows

 M=∫dre−ik⋅rSp{(f0−α⋅f1−Σ⋅f2)γ⋅e∗λ(g0−α⋅g1−Σ⋅g2)F}. (13)

Note that only the terms with and in , Eq. (II), contribute to the matrix element (13) because it contains the odd number of the gamma-matrices.

In the quasiclassical approximation the relative magnitude of the functions , , , and is different, so that

 f0∼lcf1∼l2cf2,g0∼lcg1∼l2cg2,d0∼lcd1∼l2cd2, (14)

where is the characteristic value of the angular momentum in the process, is the momentum transfer. To find the distinction between the differential cross section of bremsstrahlung from particles and antiparticles, it is necessary to take into account the first quasiclassical corrections to the functions , , , and , while the functions and can be taken in the leading quasiclassical approximation. Let us introduce the quantities

 (A00,A01,A10,A02,A20)=∫drexp(−ik⋅r)(f0g0,f0g1,f1g0,f0g2,f2g0). (15)

In terms of these quantities, the matrix element has the form

 M=δμpμq[δλμp(e∗λ,−θqA00−2A10+2μpA20) +δλ¯μp(e∗λ,−θpA00+2A01+2μpA02)]−mμp(p−q)√2pqδμq¯μpδλμpA00. (16)

Below we calculate all quantities in (II) for arbitrary atomic potential which includes the effect of screening and the nuclear size effect as well.

## Iii Green’s functions and wave functions

Let us consider the case of arbitrary central localized potential . We expand the Green’s function , Eq. (II), up to the second order with respect to the correction :

 D(r2,r1|ε)=⟨r2|1H−1Hiα⋅∇V(r)1H+1Hiα⋅∇V(r)1Hiα⋅∇V(r)1H|r1⟩, H=ε2−m2−2εφ(r)+∇2+i0,φ(r)=V(r)−V2(r)2ε. (17)

The function is the Green’s function of the Klein-Gordon equation. This function was found in the quasiclassical approximation with the first correction taken into account LMS00 () :

 D(0)(r2,r1|ε)=ieiκr4π2r∫dQexp[iQ2−ir∫10dxV(Rx)] ×⎧⎪⎨⎪⎩1+ir32κ1∫0dxx∫0dy(x−y)∇⊥V(Rx)⋅∇⊥V(Ry)⎫⎪⎬⎪⎭ , r=r2−r1,Rx=r1+xr+Q√2r1r2κr, (18)

where is a two-dimensional vector perpendicular to and is the component of the gradient perpendicular to . Within the same accuracy, coincides with the contribution to the Green’s function , Eq. (8).

Using this formula and Eqs. (II), (II), and (8), we obtain the function ,

 f0(r,q)=−iπe−iq⋅r∫dQexp[iQ2−i∫∞0dxV(rx)] ×⎧⎪⎨⎪⎩1+i2εq∞∫0dxx∫0dy(x−y)∇⊥V(rx)⋅∇⊥V(ry)⎫⎪⎬⎪⎭ , rx=r+xnq+Q√2rεq,Q⋅nq=0, (19)

where is the component of the gradient perpendicular to . Then we use the relation

 i∇V(r)=12ε[p,H]+i2ε∇V2(r), (20)

and write the linear in term in Eq. (III) as , where

 d1(r2,r1)=−i2ε(∇1+∇2)D(0)(r2,r1|ε)+δd1(r2,r1), δd1(r2,r1)=−⟨r2|1Hi2ε∇V2(r)1H|r1⟩. (21)

If we replace by in the operator , where , then we obtain from Eq. (III)

 δd1(r2,r1)=−ieiκr16π2ε2∫dQexp[iQ2−ir∫10dxV(Rx)]1∫0dx∇V2(Rx), (22)

where is given in (III). Using Eqs. (II), (II), and (8), we find the function ,

 f1(r,q)=12ε(i∇−q)f0(r,q)+δf1(r,q), δf1(r,q)=−i4πε2e−iq⋅r∫dQexp[iQ2−i∫∞0dxV(rx)]∞∫0dx∇V2(rx). (23)

where is given in Eq. (III).

To transform the third term in (III), we replace by . Then it follows from Eqs. (II), (II), and (8) that the function is

 d2(r2,r1)=−i(2ε)2[∇2×∇1]D(0)(r2,r1|ε)+δd2(r2,r1), δd2(r2,r1)=l21⟨r2|1HV′(r)2εr1H|r1⟩, (24)

where and . In (III) we use the relation . If we replace by in the operator , where , then we obtain from Eq. (III)

 δd2(r2,r1)=l21eiκr16π2ε2∫dQexp[iQ2−ir∫10dxV(Rx)]1∫0dxV′(Rx)Rx. (25)

Substituting this expression in (III), we finally find ,

 d2(r2,r1)=−reiκr16π2ε2∫dQexp[iQ2−ir∫10dxV(Rx)] ×1∫0dxx∫0dy[∇V(Rx)×∇V(Ry)]. (26)

The corresponding function is

 f2(r,q)=−e−iq⋅r4πε2∫dQexp[iQ2−ir∫10dxV(rx)] ×∞∫0dxx∫0dy[∇V(rx)×∇V(ry)]. (27)

For the Coulomb field , we find from (III), (III), and (III)

 f0(r,q)=FA+(1+nq⋅n)FC, f1(r,q)=(nq+n)ηFB, f2(r,q)=−iΣ⋅[nq×n]FC, (28)

where

 FA(r,q,η)=exp(πη2−iq⋅r)[Γ(1−iη)F(iη,1,iz) +πη2eiπ42√2qrΓ(1/2−iη)F(1/2+iη,1,iz)], FB(r,q,η)=−i2exp(πη2−iq⋅r)[Γ(1−iη)F(1+iη,2,iz) +πη2eiπ42√2qrΓ(1/2−iη)F(3/2+iη,2,iz)], FC(r,q,η)=−exp(πη2−iq⋅r)πη2eiπ48√2qrΓ(1/2−iη)F(3/2+iη,2,iz), z=(1+n⋅nq)qr,n=rr. (29)

Here is the Euler Gamma function and is the confluent hypergeometric function. The results (III) and (III) are in agreement with that obtained in LMS2012 ().

## Iv Calculation of the matrix element

The calculation of the quantities , , , , and (15) is performed in the same way as in Ref.LMSS2005 (). We present details of this very tricky calculation in Appendix. We obtain

 A00=1ωm4∫drexp[−iΔ⋅r−iχ(ρ)][i2εpεqξpξq(p⊥+q⊥) +m2(εpξp−εqξq)∞∫0dxx∇⊥V(r−xν)]⋅∇⊥V(r), A01=εqξqωm2∫drexp[−iΔ⋅r−iχ(ρ)] ×[i∇⊥V(r)+Δ2εp∞∫0dxx∇⊥V(r−xν)⋅∇⊥V(r)+i2εp∇⊥V2(r)], A02=−εqξq2ωεpm2∫drexp[−iΔ⋅r−iχ(ρ)] ×∞∫0dx[∇V(r−xν)×∇V(r)], A10=−A01(εq↔εp,ξq↔ξp),A20=−A02(εq↔εp,ξq↔ξp), χ(ρ)=∫∞−∞V(z,ρ)dz,ξp=m2m2+p2⊥,ξq=m2m2+q2⊥. (30)

Substituting Eq. (IV) in Eq. (II), we find the matrix element ,

 M=−δμpμq(εpδλμp+εqδλ¯μp)[N0(e∗λ,ξpp⊥−ξqq⊥)+N1(e∗λ,εpξpp⊥−εqξqq⊥)] −1√2mμpδμp¯μqδλμp(εp−εq)[N0(ξp−ξq)+N1(εpξp−εqξq)], N0=2iωm2Δ2∫drexp[−iΔ⋅r−iχ(ρ)]Δ⋅∇⊥V(r), N1=1ωm2εpεq∫drexp[−iΔ⋅r−iχ(ρ)]∞∫0dxx∇⊥V(r−xν)⋅∇⊥V(r). (31)

Note that in Eq. (II) the contributions of and cancel out the contributions of the terms with in and (IV). The amplitude is exact in the potential . It contains the leading quasiclassical contribution and the first quasiclassical correction as well. For high-energy bremsstrahlung from electrons in the field of a heavy atom, it is necessary to take into account the effect of screening. For high-energy bremsstrahlung from muons it is necessary to take also into account the finite nuclear radius (nuclear size effect), because the muon Compton wavelength, , is smaller than , for gold and for lead, is the muon mass.

From (IV) we have:

 ∑λμq|M|2=S0+S1+S2, S0=m2|N0|22[Δ2m2(ε2p+ε2q)ξpξq−2εpεq(ξp−ξq)2], S1=m2ReN0N∗12{Δ2m2(ε2p+ε2q)(εp+εq)ξpξq +[(ε2p+ε2q)(εp−εq)−4εpεq(εpξp−εqξq)](ξp−ξq)}, S2=−μpImN0N∗1ω2(εp+εq)ξpξq[p⊥×q⊥]⋅ν. (32)

The quantity is the even function of , it contributes to the symmetric term of the cross section (I). The quantity is the odd function of , it contributes to the antisymmetric term of the cross section (I). The quantity is the even function of , it contributes to the symmetric term of the cross section (I) which vanishes after averaging over the helicity of the initial electron. Note that the contribution of to the cross section is responsible for the effect of asymmetry with respect to the replacement , where are the azimuth angles of the final particles in the frame where the z axis is directed along . Such asymmetry is absent in the cross section calculated in the leading quasiclassical approximation. We emphasize that the contributions and are nonzero due to accounting for the next-to-leading quasiclassical terms.

The coefficients and depend on the momenta , , and via the momentum transfer . Therefore, it is easy to find from (2) and (IV) the cross section . A simple integration gives