High-energy \mu^{+}\mu^{-} electroproduction

# High-energy μ+μ− electroproduction

P. A. Krachkov    A. I. Milstein Budker Institute of Nuclear Physics, 630090 Novosibirsk, Russia
July 30, 2019
###### Abstract

The cross sections of high-energy pair production and paradimuonium production by a relativistic electron in the atomic field are discussed. The calculation is performed exactly in the parameters of the atomic field. Though the Coulomb corrections to the cross sections, related to the multiple Coulomb exchange of and an atom, are negligible, the Coulomb corrections to the differential cross sections, related to the electron interaction with an atom, are large. However, the latter Coulomb corrections to the cross sections integrated over the final electron momentum are small. Apparently, this effect can easily be observed experimentally. Furthermore, it is shown that the asymmetry of the cross section with respect to the permutation of the momenta of and is large.

electroproduction, photoproduction, bremsstrahlung, Coulomb corrections, screening
###### pacs:
12.20.Ds, 32.80.-t

## I Introduction

A process of pair production by a high-energy electron in the atomic field is one of the most important QED processes. Because of its importance, this process has been investigated in many papers Bhabha2 (); Racah37 (); BMV13 () in the leading in the parameter approximation (in the Born approximation), where is the atomic charge number and is the fine-structure constant. A pair of and may be in an unbound state as well as in the bound state (dimuonium). Dimuonium is now widely discussed BS61 (); BL09 (); BS12 (); CZ12 (); EB15 (); Lamm16 (); Olsen85 (); MSS71 (); HO87 (); ACS00 (); GJKKSS98 (); KKSS99 (); BNNT () because it is one of the simplest hydrogen-like atoms. This atom is very convenient for testing fundamental laws. Several proposals are currently under consideration. In proposal of Budker Institute BDLMS17 (), the scheme of a new experiment for the production of dimuonium in annihilation is suggested. In Jefferson Laboratory, it is planned to produce dimuonium in collision of an electron with a tungsten target Hansson14 (). In Fermilab, it is planned REDTOP () to observe dimuonium in the decay , where -meson is produced in collisions of protons with a beryllium target. An experiment on the production of dimuonium using the low-energy muon beams is also under consideration ISS15 ().

The Coulomb corrections (the difference between the exact in result and the Born result) to the cross section of the process under consideration are originated from the interaction of created pair with the atomic field and from the interaction of an electron with the atomic field. The Coulomb corrections, related to the interaction of created pair with the atomic field, are strongly suppressed by the atomic form factor HKS07 (), as well as in the case of photoproduction IM98 (). The Coulomb corrections, related to the interaction of an electron with the atomic field, have not been discussed yet. However, the account for this contribution may significantly modify the differential cross sections of the process. We have found this effect in our recent investigation of electroproduction by a heavy charged particle KM2017 () and by an ultrarelativistic electron KM2016 (); KM2017Arxiv () in the atomic field. In both cases, the Coulomb corrections are significant and reveal the interesting properties. It has been shown in Ref. KM2017 () that the cross section, differential over the angles of a heavy outgoing particle, changes significantly due to the exact account for the interaction of a heavy particle with the atomic field. However, the cross section integrated over these angles is not affected by this interaction. The same statement is also valid for the process of electroproduction by an ultrarelativistic electron KM2017Arxiv ().

In the present paper we investigate the impact of the electron interaction with the atomic field on the cross section of high-energy electroproduction. We discuss the electroproduction of unbound pair in Sec. II and electroproduction of paradimuonium in Sec. III. It is shown that in both cases the account for the interaction of an electron with the atomic field results in the large Coulomb corrections to the cross section differential over the electron transverse momentum. However, the cross section integrated over these momentum coincides with the Born result.

## Ii Electroproduction of unbound μ+μ− pair.

The differential cross section of high-energy electroproduction by an electron in the atomic field reads

 dσ=α2(2π)8dε3dε4dp2⊥dp3⊥dp4⊥12∑μi|T|2, (1)

where and are initial and final electron momenta, and are momenta of and , is the energy of the incoming electron, , , , is the electron mass, is the muon mass, and is the fine-structure constant, . In Eq. (1) the notation for any vector is used, corresponds to the helicity of the particle with the momentum , . Below we assume that and .

To calculate the amplitude (see Fig. 1), we use the quasiclassical approximation KLM2016 () developed in our recent paper KM2016 () for the problem of pair production by a relativistic electron in the atomic field. This approximation is based on the smallness of angles between the momenta of the final particles and the momentum of the initial particle at high energies. In this case typical angular momenta, which provide the main contribution to the cross section, are large, , where is energy of initial particle and is the momentum transfer. As a result, the quasiclassical approximation, based on the account for the large angular momentum contribution, becomes applicable.

Following Ref. KM2016 (), we write the amplitude (1) in the form

 T=T⊥+T∥, T⊥=−4π∑λ=±∫dkjλJλ(2π)3(ω2−k2+i0), T∥=−4πω2∫dk(2π)3j∥J∥, jλ=j⋅s∗λ,Jλ=J⋅sλ,j∥=j⋅ν,J∥=J⋅ν. (2)

where , , and are two orthogonal unit vectors perpendicular to . The functions and correspond to the matrix elements of virtual photon bremsstrahlung and pair production by virtual photon, respectively. We perform the calculation of these functions in the same way as it has been done in Refs. LMSS2005 (); KM2015 () for real bremsstrahlung and in Ref. KLM2014 () for pair production by a real photon. We obtain

 jλ=−A(Δ)ε1ε2[δμ1μ2(ε1δλμ1+ε2δλ¯μ1)(s∗λ,p2D1+p1D2) +1√2δμ1¯μ2δλμ1meωμ1(1D1+1D2)], j∥=−A(Δ)δμ1μ2(1D1+1D2), A(Δ)=−iΔ2⊥∫drexp[−iΔ⋅r−iχ(ρ)]Δ⊥⋅∇⊥V(r), χ(ρ)=∫∞−∞dzV(√z2+ρ2), D1=Δ2⊥2ε1+n1⋅Δ−i0,D2=Δ2⊥2ε2−n2⋅Δ−i0, Δ=k+p2−p1,ni=pi/pi, (3)

where is the atomic potential. At , where , is a screening radius, and is the radius of a nucleus, the function is independent of the potential shape LMSS2005 () and has the following form

 Aas(Δ)=−4πη(LΔ⊥)2iηΓ(1−iη)Δ2⊥Γ(1+iη), (4)

where is the Euler function, a specific value of is irrelevant because the factor disappears in . At , the function strongly depends on and on the shape of the atomic potential LMSS2005 ().

Note that the main contribution to the Coulomb corrections to the cross section of photoproduction is given by the impact parameter , where . Thus, the Coulomb corrections to the cross section are strongly suppressed by the form factor, and below we use the Born result for the matrix element and of photoproduction by a virtual photon:

 Jλ=J(0)λ+J(1)λ,J∥=J(0)∥+J(1)∥, J(0)λ=(2π)3ε3ε4δ(p3+p4−k)[δμ3¯μ4(sλ,δλμ3ε3p4+δλμ4ε4p3)−1√2δμ3μ4δλμ3mμωμ3], J(1)λ=−8πηF(Δ2)ωΔ2m2μ{δμ3¯μ4[ε3δμ3λsμ3−ε4δμ3λsμ4]⋅(ξ1p3+ξ2p4)+δμ3μ4δμ3λmeωμ3√2(ξ1−ξ2)}, J(0)∥=(2π)3δ(p3+p4−k)δμ3¯μ4,J(1)∥=8πηε3ε4F(Δ2)ω2Δ2m2μ(ξ1−ξ2)δμ3¯μ4, ξ1=m2μm2μ+ε3ε4(ω2−k2)/ω2+ε23θ23k,ξ2=m2μm2μ+ε3ε4(ω2−k2)/ω2+ε24θ24k, Δ=p3+p4−k, (5)

where is the atomic form factor, and are the angles between the momenta , and the virtual photon momentum .

Then we substitute Eqs. (II) and (II) in Eq. (II), take the integral over , and write the amplitudes and in the form

 T⊥=T(0)⊥+T(1)⊥,T∥=T(0)∥+T(1)∥. (6)

 T(0)⊥=8πA(Δ0)δμ1μ2m2μ+ζ2{δμ3¯μ4[ε3ω(s∗μ3⋅X)(sμ3⋅ζ)(ε1δμ1μ3+ε2δμ1μ4) −ε4ω(s∗μ4⋅X)(sμ4⋅ζ)(ε1δμ1μ4+ε2δμ1μ3)]+mμμ3√2δμ3μ4(s∗μ3⋅X)(ε1δμ3μ1+ε2δμ3¯μ1)}, T(0)∥=16πA(Δ0)δμ1μ2δμ3¯μ4θ21⋅Δ0⊥D2, (7)

where

 X=Δ0⊥ε1ε2D−2(θ21⋅Δ0⊥)θ21D2, D=ω2ε3ε4(m2μ+ζ2)+m2eω2ε1ε2+ε1ε2p22⊥,ζ=ε3ε4ωθ34, Δ0=p2+p3+p4−p1,θij=pi⊥εi−pj⊥εj, Δ0∥=−12[m2μωε3ε4+m2eωε1ε2+ε2θ221+ε3θ231+ε4θ241]. (8)

The terms and correspond to the amplitudes of electroproduction of pair non-interacting with the atomic field.

We perform the calculation of and as in Ref. KM2016 (). Then we have

 T(1)⊥=8iηε1ω∫dΔ⊥A(Δ⊥)F(Q2)Q2(m2eω2+ε21Y2)M, M=−δμ1μ2δμ3¯μ4ω[ε1(ε3δμ1μ3−ε4δμ1μ4)(s∗μ1⋅Y)(sμ1⋅I1) +ε2(ε3δμ1¯μ3−ε4δμ1¯μ4)(sμ1⋅Y)(s∗μ1⋅I1)]+δμ1¯μ2δμ3¯μ4meωμ1√2ε1(ε3δμ1μ3−ε4δμ1μ4)(sμ1⋅I1) +δμ1μ2δμ3μ4mμμ3√2(ε1δμ1μ3+ε2δμ1¯μ3)(s∗μ3⋅Y)I0−memμω22ε1δμ1¯μ2δμ3μ4δμ1μ3I0, T(1)∥=−8iηε3ε4ω3∫dΔ⊥A(Δ⊥)F(Q2)Q2I0δμ1μ2δμ3¯μ4, (9)

Here

 A(Δ⊥)=i∫dρexp[−iΔ⊥⋅ρ−iχ(ρ)], (10)

is the function , see Eq. (II), at . The integration over in Eqs. (II) is performed over two-dimensional vectors perpendicular to -axis. The following notations are used in Eqs. (II)

 M2=m2μ+ε3ε4ε1ε2m2e+ε1ε3ε4ε2ω2Y2,Y=Δ⊥−ε2θ21,ζ=ε3ε4ωθ34 Q=Δ⊥−Δ0,I0=2(Q⊥⋅ζ)(M2+ζ2)2,I1=Q⊥M2+ζ2−I0ζ. (11)

Note that Eq. (II) is valid for the region (where is the nuclear radius), which gives the main contribution to the cross section integrated over and . In this region the Coulomb corrections to the amplitude of pair production by virtual photon are absent. Besides, screening is important only for very high energies,

 ε1≳m2μαZ1/3me∼1TeV% ,

and we neglect this effect in our consideration.

Then, for the sake of simplicity of calculations, we use the model potential

 V(r)=−η√r2+R2. (12)

For this potential, the form factor and the function have a simple forms

 F(Q2)=QRK1(QR),A(Δ⊥)=Aas(Δ⊥)(Δ⊥R)1−iηK1−iη(Δ⊥R)2−iηΓ(1−iη), (13)

where is a modified Bessel function of the second kind and is given in Eq. (4).

In fact, the difference between the results obtained by using the realistic form factor and the model one is about (see Ref. JS09 () where the Born case has been considered). A small influence of the potential shape is irrelevant for the qualitative analysis of the importance of the Coulomb corrections related to the electron-atom interaction.

Let us consider the dimensionless quantity ,

 Σ=dσSdp2⊥dε3dε4,S=η2ω2m2μme, (14)

which is the differential cross section, integrated over and , in units . This quantity is shown in Fig. 2 as the function of for , , , (gold).

It is seen that the impact of the electron interaction with the atomic field on the cross section, differential over , is significant. In the region , the exact cross section is essentially smaller than that obtained in the Born approximation (the deviation of the Born result from the exact one is about ). At , the exact cross section is lager than the Born one (the deviation is about ).

In our resent paper KM2017 (), the process of pair production by a heavy particle in the atomic field has been discussed. In that paper it is shown that the cross section, differential with respect to the momentum of a heavy particle, is strongly affected by the interaction of this particle with the atomic field. However, the cross section integrated over the final momentum of a heavy particle is independent of this interaction. It turns out that a similar statement is also true for the process of pair production by a high-energy electron in the atomic field, i.e., the Coulomb corrections to the quantity

 Σ1=1me∫∞0Σdp2⊥ (15)

are strongly suppressed. In Fig. 3, the solid curve shows the quantity as the function of . Due to the strong suppression of the Coulomb corrections, the exact result coincides with the Born one for all . It is interesting to consider the relative contribution of the amplitude to the cross section. Compared to the term , the term contains the suppression factor at . This is why the contribution of to the cross section is important only for . This statement is confirmed by Fig. 3 where , obtained by neglecting the contribution of , is shown as the dotted curve.

The account for the term results in the asymmetry of the differential cross section with respect to the permutation of and momenta, . Due to the relations

 T(0)μ1μ2μ3μ4(p2,p3,p4)=T(0)μ1μ2μ4μ3(p2,p4,p3), T(1)μ1μ2μ3μ4(p2,p3,p4)=−T(1)μ1μ2μ4μ3(p2,p4,p3), (16)

this asymmetry arises as a result of interference between and .

Let us consider the cross section integrated over , , and define the asymmetry as

 A=dσ(p3,p4)−dσ(p4,p3)dσ(p3,p4)+dσ(p4,p3). (17)

In Fig. 4, the asymmetry is shown as the function of for a few values of and . It is seen that the asymmetry may reach tens of percent at .

In Ref. DLMR14 (), the charge asymmetry of the cross section of photoproduction in the atomic field has been investigated. The asymmetry arises due to the account for the first quasiclassical correction to the amplitude of the process. The cross section of photoproduction calculated in the leading quasiclassical approximation does not possess such an asymmetry. The question arises whether it is possible to use a beam of ultra-relativistic electrons as a source of equivalent photons for observation of the charge asymmetry in photoproduction, related to the next-to-leading quasiclassical approximation. Since the charge asymmetry due to interference of the amplitudes and at may be large, the observation of the charge asymmetry in the process of electroproduction, which appears due to the account for the next-to-leading quasiclassical corrections to the amplitude of pair production by a virtual photon, becomes problematic.

In this section we consider the electroproduction of pair in the bound state by ultrarelativistic electron in the atomic field (dimuonium). In this process a dimuonium is mainly produced in the state with the total spin zero (paradimuonium with the positive C-parity) because in this case the amplitude is determined by one virtual photon exchange of pair with the atomic center. To produce an orthodimuonium (the total spin one and negative C-parity), it is necessary to have either two virtual photon exchange of a pair with the atomic center in the amplitude , which is suppressed by the atomic form factor, or to account for only the amplitude , which is small compared to the amplitude .

The cross section of high-energy paradimuonium electroproduction with the total angular momentum and the principal quantum number has the form MSS71 (); Olsen85 ()

 dσPM=α2ω(2π)5mμ|ψn(0)|2dωdp2⊥dP⊥12∑μ1μ2|˜Tμ1μ2|2, (18)

where and are the energy and the momentum of dimuonium, respectively, is the dimuonium wave function at the origin, and . The amplitude is expressed via the amplitude , see Eq. (II), as follows

 ˜Tμ1μ2(p1,p2,P)=T(1)μ1μ2+−(p1,p2,P/2,P/2). (19)

We have

 ˜T=4iηε1ω∫dΔ⊥A(Δ⊥)F(Q2)Q2M2(m2eω2+ε21Y2)M, M=−μ1δμ1μ2[ε1(s∗μ1⋅Y)(sμ1⋅Q⊥)−ε2(sμ1⋅Y)(s∗μ1⋅Q⊥)]+δμ1¯μ2meω2√2ε1(sμ1⋅Q⊥), M2=m2μ+ω2m2e4ε1ε2+ε14ε2Y2,Y=Δ⊥−ε2θ21,Q=Δ⊥+p1−p2−P. (20)

The cross section of paradimuonium pair production has similar properties as the cross section of unbound pair production. I.e., the cross section differential over the electron transverse momentum has the large Coulomb corrections, in contrast to the commonly accepted point of view ACS00 (); GJKKSS98 (); KKSS99 (); HKS07 (). To illustrate this statements we plot the dependence of the dimensionless quantity ,

 ΣPM=dσPMSPMdp2⊥dω,SPM=α3η2ζ(3)8πωm2μme, (21)

on for and . In this formula, the summation over the principle quantum number is performed.

It is seen that the exact result in the peak region is about less than the Born result. In the wide region the exact result is about larger than the Born one. Again, after integration over the momentum , the cross section coincides with the Born result, see Fig. 6, where the quantity

 Σ1PM=1me∫dp2⊥ΣPM

is shown as the function of for and .

The dependence of on is very similar to that shown in Fig. 3 for .

## Iv Conclusion

We have investigated the cross sections of pair production by a high-energy electron in the strong atomic field. The interaction of electron with the atomic field is taken into account exactly in the parameter . The cases of the bound (paradimuonium) and unbound produced pair are considered. For the cross sections differential over the transverse electron momenta , the Coulomb corrections, related to the electron interaction with the atomic field, turns out to be large, in contrast to the commonly accepted point of view. Apparently, this effect can be easily observed experimentally. However, the Coulomb corrections to the cross sections integrated over are small. The asymmetry of the cross section with respect to the permutation of the momenta of and is large. This asymmetry appears due to interference of the amplitudes and corresponding to production of a pair with the opposite C-parity. This effect makes problematic a possibility to use an electron beam as a source of equivalent photons for the charge asymmetry observation in photoproduction in an atomic field. The latter asymmetry appears due to account for the next-to leading quasiclassical corrections to the photoproduction amplitude.

## Acknowledgement

This work has been supported by Russian Science Foundation (Project No. 14-50-00080).

## Appendix

Here we present the Born amplitude for the process of high-energy electroproduction by an electron in the atomic field. In this case, the terms and are given by Eq. (II) with the replacement

 A(Δ0)→AB(Δ0)=−4πηΔ20F(Δ20).

To derive the terms and , we use the following relation

 limη→0A(Δ⊥)=i(2π)2δ(Δ⊥). (22)

Then we obtain

 T(1)B⊥=−32π2ηε1F(Δ20)ωΔ20(m2eω2+ε21p22⊥)MB, MB=δμ1μ2δμ3¯μ4ω[ε1(ε3δμ1μ3−ε4δμ1μ4)(s∗μ1⋅p2⊥)(sμ1⋅IB1) +ε2(ε3δμ1¯μ3−ε4δμ1¯μ4)(sμ1⋅p2⊥)(s∗μ1⋅IB1)]+δμ1¯μ2δμ3¯μ4meωμ1√2ε1(ε3δμ1μ3−ε4δμ1μ4)(sμ1⋅IB1) −δμ1μ2δμ3μ4mμμ3√2(ε1δμ1μ3+ε2δμ1¯μ3)(s∗μ3⋅p2⊥)IB0−memμω22ε1δμ1¯μ2δμ3μ4δμ1μ3IB0, T(1)B∥=32π2ηε3ε4F(Δ20)ω3Δ20IB0δμ1μ2δμ3¯μ4, IB0=−2(Δ0⊥⋅ζ)(M2B+ζ2)2,IB1=−Δ0⊥M2B+ζ2−IB0ζ,Δ0=p2+p3+p4−p1, M2B=m2μ+ε3ε4ε1ε2m2e+ε1ε3ε4ε2ω2p22⊥,p2⊥=ε2θ21,ζ=ε3ε4ωθ34. (23)

The Born amplitude of the paradimuonium electroproduction (see Eq.(Appendix)) can also be obtained by means of Eq. (22):

 ˜TB=−16π2ηε1F(Δ20)ωΔ20M2B(m2eω2+ε21p22⊥)MB, MB=μ1δμ1μ2[ε1(s∗μ1⋅p2⊥)(sμ1⋅Δ0⊥)−ε2(sμ1⋅p2⊥)(s∗μ1⋅Δ0⊥)]+δμ1¯μ2meω2√2ε1(sμ1⋅Δ0⊥), M2B=m