JHEP01(2021)144

Published for SISSA by Springer

Received: November 2, 2020Accepted: December 12, 2020Published: January 22, 2021

Total cross sections of eγ → eXX̄ processes withX = µ, γ, e via multiloop methods

Roman N. Lee,a Alexey A. Lyubyakina,b and Vyacheslav A. Stotskya,baBudker Institute of Nuclear Physics,630090, Novosibirsk, RussiabNovosibirsk State University,630090, Novosibirsk, RussiaE-mail: r.n.lee@inp.nsk.su, a.liubiakin@g.nsu.ru,stotsky.slava@gmail.com

Abstract: Using modern multiloop calculation methods, we derive the analytical expres-sions for the total cross sections of the processes e−γ → e−XX̄ with X = µ, γ or e atarbitrary energies. For the first two processes our results are expressed via classical poly-logarithms. The cross section of e−γ → e−e−e+ is represented as a one-fold integral ofcomplete elliptic integral K and logarithms. Using our results, we calculate the thresholdand high-energy asymptotics and compare them with available results.

Keywords: NLO Computations

ArXiv ePrint: 2010.15430

Open Access, c© The Authors.Article funded by SCOAP3. https://doi.org/10.1007/JHEP01(2021)144

mailto:r.n.lee@inp.nsk.sumailto:a.liubiakin@g.nsu.rumailto:stotsky.slava@gmail.comhttps://arxiv.org/abs/2010.15430https://doi.org/10.1007/JHEP01(2021)144

JHEP01(2021)144

Contents

1 Introduction 1

2 Total cross section of the process e−γ → e−µ+µ− 22.1 Results 42.2 Cross-section near the threshold 52.3 High-energy asymptotics 52.4 Total cross-section of e−γ → e−π+π− 6

3 Total cross section of the process e−γ → e−γγ 63.1 Calculation of the total cross section in dimensional regularization 73.2 Soft-photon contribution to e−γ → e−γγ 83.3 Results 93.4 Asymptotics 12

4 Total cross section of the process e−γ → e−e−e+ 134.1 Asymptotics 16

5 Conclusion 17

1 Introduction

Since the invention of quantum electrodynamics (QED), one of its first touchstones was thecalculation of the cross sections of elementary processes, like e+e− → µ+µ− or eγ → eγ.In particular, their total Born cross sections for arbitrary energies invariably appear inany QED textbook. In its vast majority, these results concern the processes with twoparticles in the initial state and two in the final (2 → 2 processes). In contrast, thetotal Born cross sections of the 2 → N processes with N > 2 have been paid much lessattention.1 This circumstance is not incidental. It appears that, when there are massiveparticles in the final state and/or massive propagators in the amplitude, the N -particlephase-space integrals are not so simple to be taken by brute force. From the viewpointof contemporary multiloop methods, this is no wonder, as the N -particle phase spaceintegral corresponds to the N − 1-loop momentum space integral with bipartite cut. Thus,N > 3 case corresponds to (L > 2)-loop integrals, which are known to bring much morecomplexity than one-loop integrals. However, the last 40 years of development of multiloopcalculations methods have passed for good reason, and we are now in perfect position topatch this omission. We should not expect too simple results though. It is well-known that

1Perhaps, the only remarkable exception is the Racah results [1, 2] for the processes γZ → e+e−Z ande−Z → e−γZ.

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JHEP01(2021)144

m1

p1

k1

p1

k1p2p3

p4

1

2

3

4

5

m2

p1

k1

p1

k1

p2

p3

p4

1

2

3

4

56

Figure 1. Diagrams contributing to the total cross section of the e−γ → e−µ+µ− process. Firstthree diagrams correspond to C-odd muon pair, while the last two diagrams — to C-even pair.

already two-loop integrals can be impossible to express via harmonic polylogarithms [3] andeven via generalized polylogarithms [4]. In the present paper we calculate the total crosssections of the processes e−γ → e−XX̄ with X = e−, µ−, γ. These processes have manyastrophysical applications, see, e.g., refs. [5–11], and have been considered in a number ofpapers [6, 9, 12–18]. Our approach is based on using the optical theorem and Cutcoskyrules to express the total cross section via cut diagrams in forward kinematics.2 The masterformula is

σe−γ→e−XX̄ =ImAe−γ→e−XX̄

s−m2e, (1.1)

where s = (p1 + k1)2 is the square of total c.m. energy, me is the electron mass, andImAe−γ→e−XX̄ is the sum of the propagator-type diagrams with cut e−, X, X̄ lines. Thus,in order to calculate the cross section, we apply to this sum the contemporary multiloopcalculation methods: the IBP reduction and the calculation of the master integrals via thedifferential equations method.

Our calculation features a few methods not so widely known in the multiloop commu-nity. First, given two differential systems, ∂xj = Mxj and ∂yj = Myj, we consider thesolution of the second system in the form of generalized power series j =

∑α∈S+Z cα(x)yα

and show how to systematically obtain the differential equations with respect to x forthe finite subset {cα(x)|α ∈ S}. Next, we demonstrate that the contribution of soft pho-tons, as defined by the soft-photon approximation, can be calculated by means of themultiloop techniques, although it does not directly correspond to a sum of conventionalFeynman diagrams. Finally, we successfully apply the recently introduced approach to thenon-polylogarithmic integrals, based on the construction of �-regular basis, ref. [20].

2 Total cross section of the process e−γ → e−µ+µ−

Let us first consider the process e−γ → e−µ+µ−. As the muon mass mµ is much largerthan me, we can treat me as a small parameter. Nevertheless, we can not simply putme = 0 because the cross section becomes infrared divergent in this limit. Therefore, wekeep the electron mass in “large” logarithms and omit the power corrections in me, so thatour final result has the following form:

σe−γ→e−µ+µ− = σ0 + σ1 ln(mµ/me). (2.1)

Below in this section we will use the units mµ = 1. In order to obtain the above form,we proceed in the following way. We define two LiteRed [21] bases m1 and m2 containing

2There is, of course, not much novelty in using multiloop methods to the calculation of the phase-spaceintegrals. In particular, this approach has been used already in ref. [19] in 2002.

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JHEP01(2021)144

the denominators of the corresponding boxed diagrams in figure 1. We add the irreduciblenumerators D6 = k1 · p3 and D7 = p1 · p3 to basis m1 and D7 = p1 · p3 to m2. We reveal 7master integrals

jᵀ = (j1, j2, . . . , j7) =(jm10111000, j

m1−1111000, j

m1−2111000, j

m10111100, j

m20111010, j

m20111020, j

m21111000

)and construct the differential systems with respect to m2e and s:

∂m2ej = Mm2ej , ∂sj = Msj . (2.2)

Since the point m2e = 0 is a singular point of the first system, we search for its solutionin the form of generalized power series using the Frobenius method, along the lines ofref. [22]. The result has the form

j = Ũc , (2.3)

where Ũ is a fundamental matrix with entries being the generalized power series in m2e,and c is a column of constants. Note that we can always redefine Ũ by multiplying fromthe right by any non-degenerate matrix independent of m2e. In particular, we can use thisfreedom to secure that the column of constants c consists of specific coefficients in theasymptotic expansion of master integrals near the point m2e = 0. Namely, we secure that

cᵀ =([j1]m0e , [j2]m0e , [j5]m0e , [j3]m2−2�e , [j4]m−2�e , [j6]m2−2�e , [j7]m−2�e

), (2.4)

where [jk]mµe denotes to coefficient in front of mµe in small-me asymptotic of jk. The matrix

Ũ is found in a routine way, along the lines of ref. [22].Note that constants c depend nontrivially on s. In order to find the differential system

for c with respect to s, we treat Ũ in eq. (2.3) as a transformation matrix for the secondsystem in eq. (2.2). Then we obtain

∂sc = M̃sc , (2.5)

whereM̃s = Ũ−1

[MsŨ − ∂sŨ

](2.6)

Three remarks are in place here. First, since c is independent of m2e, so is the matrixM̃s. Therefore, in order to establish the exact form of M̃s, it is sufficient to know onlyfirst few terms of generalized power series in Ũ . Second, instead of inverting the matrix Ũof truncated generalized power series, we calculate Ũ−1 independently from the equation∂m2e(Ũ

−1)ᵀ = −(Mm2e)ᵀ(Ũ−1)ᵀ. Finally, equations for constants, corresponding to different

fractional powers of m2e, decouple, as they should, so that M̃s has a block diagonal form

M̃s =(

3 × 3 00 4 × 4

).

In order to find the �-forms of the resulting systems, we pass to the variable v =√1− 4/s. This variable has a simple physical meaning as the maximum muon velocity in

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JHEP01(2021)144

c.m.f. at a given energy. We have

∂v

c1c2c3

=

v2(2�−1)+1v3−v −

3(�−1)2v 0

− 4v−v3 −

2(v2−3)(�−1)v(v2−1) 0

−2v2�+v2+12v−2v3

3(�−1)4v

2vv2−1

c1c2c3

, (2.7)

∂v

c4c5c6c7

=

−3v2+2�−1v(v2−1) 0 0 0

18(v

2 − 1)x(�− 1) 2v(v2−1) 0 0(v2−1)3(2�−1)

64v 02v

(v2−1) 018(v3 − v

)(�− 1) 0 0 −2(2v�−v)

v2−1

c4c5c6c7

. (2.8)

We reduce both systems to �-form [23, 24] using Libra, ref. [25]. The boundaryconditions are fixed by evaluating the small-v asymptotics of j1 and j3:

j1 ∼4π2Γ(2− 2�)

Γ(

32 − �

)Γ(72 − 3�)

× v5−6�m0e, j3 ∼16π3/2Γ(�− 1)

Γ(

32 − �

) × v1−2�m2−2�e . (2.9)Using these boundary conditions, we obtain all c in terms of harmonic polylogarithms.Then we substitute c in eq. (2.3) and obtain a sufficient number of terms in the generalizedpower series representation for master integrals j. So, our results for master integrals havethe form of generalized power series in m2e (truncated at some order), whose coefficientsare series in � expressed via polylogarithms depending on v.

2.1 Results

Expressing the cross section via master integrals and substituting our results for the latter,we obtain the total cross section.

σe−γ→e−µ+µ− =α3

s

{ln(

(s− 4)2sm2e

)((7s2 − 66s+ 50) v9s +

4(s2 + 1

)3s2 ln

1 + v1− v

− s− 2s

[4Li2

(1− v2

)+ ln s ln 1 + v1− v

])+ 8(s− 2)

s

[2Li3

( 2v1 + v

)− 3Li3

(1− v2

)− 4Li3(v)

]− 12 ln

1 + v1− v

[14s3 − 107s2 + 504s− 1109s2 +

4(s+ 3) ln s3s

+ s− 2s

(73 ln

2 1 + v1− v + 2π

2 − 3 ln2 s)]−(436s2 − 4111s+ 1302

)v

108s

− 32(s2 + 1

)Li2(v)

3s2 −8(s+ 3)Li2

(1−v

2

)3s − (v → −v)

}. (2.10)

Diagrams from figure 1 can be divided into two groups according to C-parity of themuon pair. We present for the reference the contribution of C-odd diagrams (the first three

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JHEP01(2021)144

diagrams in figure 1) separately:

σodde−γ→e−µ+µ− =α3

s

{ln(

(s− 4)2sm2e

)(s2 − 12

3s2 ln1 + v1− v −

(8s− 26)v9s

)

+(17s− 154)v36s +12 ln

1 + v1− v

((35s2 − 144s+ 198

)9s2 −

4(s2 − 12

)ln s

3s2

)

− 8(s2 − 12

)3s2

[Li2 (v) + Li2

(1− v2

)]− (v → −v)

}. (2.11)

2.2 Cross-section near the threshold

When the muon kinetic energy ∼ v2 is comparable with me the derived formulae areinapplicable. It becomes obvious from the small-v asymptotics of eq. (2.10) which reads[ln 8v2me −

83

]v3, and, therefore, turns negative at v2 . me. Although this narrow region

might be not very relevant for the experiment, let us derive the appropriate expression forthe cross section for the sake of completeness. For this purpose we introduce the variable τvia√s = 2+(1+τ)me. We obtain the differential systems with respect to τ and me. Then

we search for master integrals as a generalized power series in m2e, but this time, at fixedτ . We need to fix two nonzero coefficients cᵀ =

([j1]m7/2−3�e , [j1]m5/2−3�e

). After expressing

the cross-section via c we have established that �0 terms of c are sufficient for our purpose.Then we can put � = 0, and obtain the system ∂τc = Mc with

M = 13τ(τ + 2)

(21(τ + 1) 916

(11τ2 + 22τ + 12

)−28 −9(τ + 1)

). (2.12)

The solution of this system can be expressed via complete elliptic integrals K(−τ/2)and E(−τ/2). We finally get

σthresholde−γ→e−µ+µ− =√

2α3

3 (me)3/2(−8(τ + 1)E

(−τ2

)+ (τ + 2)(3τ + 4)K

(−τ2

)). (2.13)

We remind here that τ =√s−2−meme

and that the formula above is valid when 0 <√s− 2−

me � 1, while eq. (2.10) is valid when me �√s − 2 −me. The two formulae agree with

each other in the overlapping region me �√s− 2−me � 1.

In figure 2 we present our exact result (2.10) and the found threshold and high-energyasymptotics. As a cross-check, we present on the same figure also a few points obtainedby numerical integration of the differential cross section using the Cuba library [26].

2.3 High-energy asymptotics

For the high-energy asymptotics of the total cross-section of e−γ → e−µ+µ− we have

σe−γ→e−µ+µ− =α3([28

9 ln(s

me

)− 21827

]+ 43s

{π2 ln

(s2

me

)− ln2 s ln

(s2

m3e

)+ ln s ln

(s5

m4e

)+ 803 lnme−

37312 ln s−12ζ3−

53π

2+ 149324

}+O

(s−2

)).

(2.14)

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JHEP01(2021)144

5 10 50 100

0

10

20

30

Figure 2. Exact cross-section of the e−γ → e−µ+µ− (solid curve) and asymptotics. Black dashedcurve — leading asymptotic, dotted curve — eq. (2.14), and dash-dotted curve — the cross-sectionnear threshold, eq. (2.13). Points represent the data obtained by numerical Monte-Carlo integrationof the differential cross section using Cuba library.

The leading term of this asymptotics has been derived in ref. [15] using the equivalentphoton approximation. It coincides with the leading term of our result.

2.4 Total cross-section of e−γ → e−π+π−

For the sake of completeness, let us present also the total cross section of e−γ → e−π+π−,where we consider π-meson as a pointlike scalar particle. We have

σe−γ→e−π+π− =α3

s2

{ln(

(s− 4)2sm2e

)((4s2 + 87s− 118) v36 − 4Li2

(1− v2

)

− ln 1 + v1− v

[11s2 + 2012s + ln s

])+ ln 1 + v1− v

(5s2 + 6s+ 126s ln s−

8s3 − 155s2 − 756s− 5072s

− 76 ln2 1 + v

1− v − π2 + 32 ln

2 s

)+ 2

(11s2 + 20

)Li2(v)

3s

+2(5s2 + 6s+ 12

)Li2

(1−v

2

)3s − 24Li3

(1− v2

)− 32Li3(v)

+ 16Li3( 2v

1 + v

)−(208s2 + 7547s− 4278

)v

432 − (v → −v)}. (2.15)

Note that the leading high-energy asymptotics of this formula, α3[

49 ln(s/me)−

2627

], per-

fectly agrees with the result of ref. [15].

3 Total cross section of the process e−γ → e−γγ

In this section we present the calculation of the cross section of the process e−γ → e−γγ(double Compton scattering) at arbitrary energies. To avoid the infrared divergences, we

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JHEP01(2021)144

g1

7 6 3 5 4

12

g2

4 5 3 6 7

12

Figure 3. Diagrams contributing to the total cross section of the e−γ → e−γγ process.

restrict the integration region by the conditions ω2 > ω0 and ω3 > ω0, where ω2 and ω3 arethe energies of the outgoing photons, and ω0 is a small cut-off parameter. This restrictionis frame-dependent, and we consider two physically relevant frames: the center-of-massframe and the initial electron rest frame. From the technical point of view, we proceed asfollows. Using dimensional regularization, we first calculate the total cross section, whichcontains the term ∝ 1� . Then we calculate separately the contribution of soft-photon regionand subtract it from the total cross section to obtain the physically observed cross sectionσe−γ→e−γγ(ω0).

3.1 Calculation of the total cross section in dimensional regularization

Using the optical theorem (1.1), the total cross section e−γ → e−γγ can be expressed interms of cut diagrams shown in figure 3. We set up two LiteRed bases g1 and g2 containingpropagators of boxed diagrams in figure 3. We find 14 master integrals,

jg11110000, j

g1111000−1, j

g11110001, j

g11110002, j

g11110011, j

g11110110, j

g11111001,

jg11110120, j

g11111002, j

g11110111, j

g11110112, j

g21110110, j

g21110111, j

g21111111.

Introducing the column-vector j = (j1, . . . , j14)ᵀ = (jg11110000, . . . , jg21111111)ᵀ we obtain the

differential system∂sj = M (s, �) j, (3.1)

where M is the matrix, rationally depending on s and �. Introducing the new variabley =

√s−1s+3 , we reduce the system (3.1) to �-form using Libra [25]:

∂yJ (y) = �[1yS0 +

1y − 1S1 +

1y + 1S2 +

2yy2 + 1/3S3

]J (y) , (3.2)

where Si are some constant matrices. The canonical basis J is connected with j by rationaltransformation j = T (y, �)J . The general solution of the system (3.2), J =

∑n �

nJ (n), canbe easily written in terms of generalized polylogarithms. We fix the boundary conditionsby calculating the coefficients in the asymptotic expansion of the master integrals at thethreshold. The only nontrivial coefficient of the leading threshold asymptotic is

j2 ∼ −y8−8�25−8�πΓ2 (1− �)

Γ (4− 4�) . (3.3)

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JHEP01(2021)144

Using this boundary condition, we find an expression for the master integrals and get thetotal Born cross sections e−γ → e−γγ in the form

σe−γ→e−γγ =σ1�

+ σ2. (3.4)

where σ1, σ2 can be expressed via generalized polylogarithms with letters 0,±1,± i√3 . Inthis expression we omit the terms suppressed by �. The 1� term is quite anticipated, it isdue to the contribution of soft-photon region. In order to get rid of this term, we have tosubtract the contribution related to soft photons.

3.2 Soft-photon contribution to e−γ → e−γγ

In a soft-photon approximation the differential cross section factorizes as

dσsofte−γ→e−γγ = dσe−γ→e−γ · dWγ , (3.5)

where dWγ is a soft-photon emission probability

dWγ = −dd−1k3

(2π)d−1 2ω3e2(

p1p1 · k3

− p2p2 · k3

)2. (3.6)

Here k3 is the four-momentum of the soft photon. The contribution to the total crosssection can be written as

σsofte−γ→e−γγ =∫dσe−γ→e−γdWγ θ (ω0 − ω3) . (3.7)

Here ω0 is the maximal energy of soft photon. As we have already mentioned, the infraredcutoff introduces the frame dependence. We calculate the cross section in the center-of-mass frame (cmf) and in the rest frame of the initial electron (rf). Below we present somedetails of the calculation for the electron rest frame.

The soft-photon contribution (3.7) can formally be written as a two-loop integral withcut propagators. Inserting a identity 1 =

∫dωδ (ω − ω3) into (3.7) and rescaling k3 → ωk3

we get

σsofte−γ→e−γγ = −ω−2�02�

∫dσe−γ→e−γ

ddk3

(2π)d−1δ(k23

)( p1p1 · k3

− p2p2 · k3

)2δ (p1 · k3 − 1) .

(3.8)We calculate the integral on the right side using the differential equations method. Thisintegral should be considered up to the O(�1) term.

To determine the basis for IBP reduction, we include propagators from single Comptonscattering, scalar products in denominators in (3.7), and cut propagators from δ-functions.We obtain scalar integral

jsoftn1,...,n7 =∫

ddk2ddk3

(p2 · k3)n5(1− p1 · k3)−n4 (k1 · k3)−n7(

k22)n1 (k23)n2 (p22 − 1)n3 ((p1 − k2)2 − 1)n6 , (3.9)

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JHEP01(2021)144

where p2 = p1 + k1 − k2. We imply that the first four propagators in this integral are cut,and that the last one is the irreducible numerator. Using LiteRed package we reveals thefollowing master integrals

jsoft1111000, jsoft1111010, j

soft1111100, j

soft1111200, j

soft111100−1, j

soft1111110. (3.10)

Using Libra we find the canonical basis J related to jsoft by

jsoft1111000 = (s− 1) J1, jsoft1111010 =s(2�− 1)3(s− 1)�J2, j

soft1111100 =

2(2�− 1)2

9(s− 1)�2 J4

jsoft1111200 =2(s+ 1)2(2�− 1)2

9(s− 1)s2 J4 −2(s+ 1)(2�− 1)2

9s2 J3 −2(2�− 1)2

s2J1,

jsoft111100−1 =s(2�− 1)

3� J5 −2s(2�− 1)2

9(s− 1)�2 J4,

jsoft1111110 =16(2�− 1)2

9y(s− 1)(s+ 3)�2J6. (3.11)

The differential equations for J have the form

J ′1 =1− 3s

(s− 1)s�J1, J′2 =

3s�J1 −

2s�J2, J

′3 =

9(3− s)2(s− 1)s�J1 −

2s�J3,

J ′4 = −92s�J1 −

1s�J3 −

s− 3(s− 1)s�J4, J

′5 = −

3s�J1 −

2s�J5,

J ′6 =9�J1

8y(s+ 3) −3�J2

8y(s+ 3) +�J3

4y(s+ 3) −2(s− 3)�J6

(s− 1)s(s+ 3) . (3.12)

To fix the boundary conditions it suffices to calculate the leading asymptotics of jsoft1111000,

jsoft1111000 ∼ −y2−4�23−2�π2

Γ2(

32 − �

) . (3.13)Finally, we obtain the contribution of soft-photons in the form

σsofte−γ→e−γγ =σ1�

+ σ3 · lnω0 + σ4. (3.14)

Note that 1� term in this formula appears to be the same as in eq. (3.4), as it should be.Therefore, in the difference the terms, containing 1� , vanish.

3.3 Results

The total Born cross section of the process e−γ → e−γγ, integrated over the kinematicregion, in which ω2,3 > ω0, in the electron rest frame

σrfe−γ→e−γγ (ω0)

=α3{(3s5 + 3s4 + 62s3 + 222s2 + 15s− 465) yA2 (y)

(s− 1)4(s+ 3) +20B3 (y)(s− 1)2

+ 12(s4 − 3s3 − 27s2 − 33s+ 14

)yC3 (y)

(s− 1)4(s+ 3) +6(s+ 1)

(s3 − 7s2 − 29s− 13

)yD3 (y)

(s− 1)4(s+ 3)

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JHEP01(2021)144

+ 6(s2 − 4s− 9

)A3 (y)

(s− 1)3 +6(s4 − 2s3 − 24s2 − 30s+ 23

)yE3 (y)

(s− 1)4(s+ 3)

+ 51s3 + 2629s2 − 459s+ 99

24(s− 1)2s2 +(5s5 − 295s4 − 1061s3 − 161s2 + 108s− 36

)ln s

4(s− 1)3s2(s+ 3)

−(4s5 + s4 − 36s3 − 60s2 − 28s− 9

)ln2 s

2(s− 1)4(s+ 3)

−(s5 + 4s4 − 44s3 − 98s2 + 35s+ 6

)Li2 (1− s)

(s− 1)3s(s+ 3) +2(5s+ 1) ln sLi2 (1− s)

(s− 1)3

+ ln(s− 12ω0

)(12 (s4 − 2s3 − 24s2 − 30s+ 7) yA2 (y)(s− 1)4(s+ 3) −

5s3 + 45s2 − 5s+ 3(s− 1)2s2

− 2(s5 − 27s4 − 86s3 − 34s2 + 5s− 3

)ln s

(s− 1)3s2(s+ 3) −2(s2 − 6s− 1

)ln2 s

(s− 1)3)}

. (3.15)

For center-of-mass frame we have

σcmfe−γ→e−γγ

=σrfe−γ→e−γγ + α3{6 (s4 − 6s3 − 28s2 − 26s+ 11) yA2 (y)

(s− 1)4(s+ 3)

− 24(s4 − 2s3 − 24s2 − 30s+ 7

)yD3 (y)

(s− 1)4(s+ 3)

−(7s6 − 16s5 − 67s4 − 8s3 − 39s2 − 8s+ 3

)ln2 s

2(s− 1)4s2(s+ 3)

− ln s(

6(s4−2s3−24s2−30s+7

)yA2 (y)

(s− 1)4(s+ 3) −5s5+29s4+24s3−16s2−29s+3

(s− 1)3s2(s+ 3)

)

− 12(s4 − 2s3 − 24s2 − 30s+ 7

)yC3 (y)

(s− 1)4(s+ 3) −5s3 + 9s2 − 5s− 1

(s− 1)2s2}. (3.16)

Here A2, A3, B3, C3, D3, E3 are functions which can be expressed via classical polyloga-rithms. They are defined as follows

A2 (y) =<[Li2

(eiπ3 q)− Li2

(eiπ3

q

)],

A3 (y) =<[Li3

(eiπ3 q)

+ Li3

(eiπ3

q

)]− 23ζ3,

B3 (y) =π2

6 ln s+12 ln (s− 1) ln

2 s− 16 ln3 s+ Li2 (1− s) ln s+ Li3

(1s

)− ζ3,

C3 (y) =∫ s

1

ln (x) ln(s+3x+3

)√

(x− 1) (x+ 3)dx,

D3 (y) = −∫ s

1

ln2 (x)2√

(x− 1) (x+ 3)dx,

E3 (y) =∫ s

1

Li2 (1− x)√(x− 1) (x+ 3)

dx, (3.17)

– 10 –

JHEP01(2021)144

0.01 0.10 1 10 100

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

Figure 4. Dependence of the exact cross section on the infrared cutoff parameter ω0 at√s = 100.

The upper thick line shows the cross section in electron rest frame, the lower thin line — the crosssection in center-of-mass frame, points correspond to numerical integration of the differential crosssection with Cuba library.

where q = 1−y1+y , y =√

s−1s+3 . The integrals in C3, D3 and E3 can be taken in terms of Li3

and similar functions; the corresponding expressions are quite lengthy and given in thesupplementary material in the file CDEfunctions.m.

Let us discuss the limits of applicability of the soft photon approximation used in thepresent work. For the low-energy region a natural physical requirement is that the cut-offparameter ω0 should be much smaller than the initial photon energy ω1 ≈ (s − 1)/2. Interms of the low-energy asymptotics presented below in eq. (3.18) this restriction meansthat the logarithm in that formula should dominate. For the high-energy region, therequirements for ω0 is essentially different for the two reference frames we consider. Inthe center-of-mass frame the infrared cut-off parameter should be small compared to

√s.

Thus, it is natural to expect that in the electron rest frame the soft photon approximationbreaks down already at ω0 & me = 1. This is because the photon with energy ∼ me inthe electron rest frame can have energy ∼

√s when boosted to the center-of-mass frame.

This is demonstrated in figure 4, where the results of the present paper are compared withthose of numerical integration using the Cuba library [26].

Figure 5 shows the dependence of the cross section on s at ω0 = 0.01. On the samefigure we have shown a few points obtained by numerical integration of the differentialcross section using Cuba library. We have also performed the comparison of the obtainedresults with known numerical and/or approximate results. First, the numerical result ofref. [27] disagrees with our result by a factor of 2. The discrepancy is possibly due tothe overlooked Bose symmetry factor 12! in ref. [27]. Once this factor is recovered, we findperfect agreement. Numerical results of ref. [28] agree with our results up to s ≈ 30m2e. Thediscrepancy for higher energies is quite expected due to inapplicability of the soft-photonapproximation for the parameter ω0 = ω150 =

s−m2e100me chosen in ref. [28].

– 11 –

JHEP01(2021)144

1 5 10 50 100

0

1

2

3

4

5

Figure 5. Exact cross section as a function of√s/me at ω0 = 0.01me. The upper thick line —

the cross section in electron rest frame, the lower thin line — in center-of-mass frame, dashed lines— the high-energy asymptotics (3.19), (3.20), dash-dotted line — the threshold asymptotics (3.18),points — Cuba numerical integration of the differential cross section.

3.4 Asymptotics

Using the expressions for the exact cross sections (3.15) and (3.16) via polylogarithms, itis easy to calculate both the threshold asymptotics and the high-energy asymptotics.

Threshold asymptotics

σe−γ→e−γγ (ω0)s→1= 8α

3

9 (s− 1)2(

ln(s− 12ω0

)− 167120

)+O (s− 1)3 (3.18)

coincides with the non-relativistic asymptotic in ref. [17]. We remind that this formulaimplies that ω0 � s−1� 1. Note that this asymptotics holds both for electron rest frameand for the center-of-mass frame.

The high-energy asymptotics has the form

σrfe−γ→e−γγ (ω0)s→∞= α

3

s

{ln(√

s

2ω0

)(4 ln2 s− 2 ln s− 5− 2π

2

3

)

+ ln3 s− 3 ln2 s

2 − ln s(

5π2

3 −34

)+ 10ζ3 +

2π2

3 +338 +O

(s−1

)}(3.19)

in the electron rest frame, and

σcmfe−γ→e−γγ (ω0)s→∞= α

3

s

{ln(√

s

2ω0

)(4 ln2 s− 2 ln s− 5− 2π

2

3

)

− 2 ln2 s− ln s(

2π2

3 −234

)+ 2ζ3 +

π2

3 −78 +O

(s−1

)}(3.20)

in center-of-mass frame. We note that the high-energy asymptotics of the double Comptonscattering is obtained for the first time here to the best of our knowledge.

– 12 –

JHEP01(2021)144

e1

1

2

3

4

56

7

e2

1 2

345 6

7

e3

1

2

3

4

5

6

e4

1

23

4

5

67

Figure 6. Diagrams contributing to the total cross section of the e−γ → e−e−e+ process.

4 Total cross section of the process e−γ → e−e−e+

The total cross section of the process e−γ → e−e−e+ is not expressible via polylogarithms.This is easy to understand because among the relevant master integrals there is an equalmass two-loop sunrise integrals which is a classical example of non-polylogarithmic inte-grals. Therefore, we will rely here on the approach of ref. [20]. We construct an �-regularbasis sufficient for the calculation of the cross section.3

The cut diagrams which contribute to the cross section are shown in figure 6. Wedefine 4 LiteRed bases e1, e2, e3, e4, corresponding to the denominators of the frameddiagrams in the same figure. We find 18 master integrals which we choose as follows:

je10101010, je10101020, j

e10101011, j

e10101111, j

e11101011, j

e11101−111, j

e11111010, j

e11111011, j

e11111−11,

je11111110, je20101111, j

e21101011, j

e21101111, j

e20111111, j

e21111111, j

e31101011, j

e31101111, j

e31111111.

Note that the integrals je10101111 and je31101111 can, in principle, be expressed via the integralsof the lower sectors, and can be replaced by, e.g., je10101110 and je31101012, respectively. Howeverwe find our present choice to be convenient as we empirically observe that the IBP reductionof any other integral to our set of masters does not generate inverse powers of � in thecoefficients. In particular, the total cross section can be expressed as a linear combinationof these master integrals with regular coefficients. The differential system for these masterintegrals is also regular at � → 0. Therefore, we can safely put � = 0 starting fromthis point.

In order to further simplify the differential system, we use Libra [25] and pass to newmaster integrals J1, . . . , J18 related to our previous master integrals via

je10101010 = (s+ 3)J2 − 4J1, je10101020 = 2J2 − 2J1,

je10101011 =8J3s− 1 − 2J1, j

e10101111 =

4J4s− 1 ,

je11101011 = −8J4s− 1 +

4J6s− 1 −

4J7s− 1 ,

3Note that the cross section has no IR divergences.

– 13 –

JHEP01(2021)144

je11101−111 = −2(−s2 + 2s+ 3

)J3

s2 − s− 4J1 + 2J2 −

2(s− 1)J5s

,

je11111010 = −4J4s− 1 −

2J7s− 1 , j

e11111011 =

4yJ8(s− 1)2 ,

je11111−11 = −4J6s− 1 +

2J7s− 1 −

2J13s− 1 ,

je11111110 =4J4

(s− 1)2 +4J14

(s− 1)2 , je20101111 =

2zJ9s− 1 −

6J3s− 1 ,

je21101011 = 2zJ9 +8(3− s)J3s− 1 + 2J5 + 2J15, j

e21101111 =

2zJ10s2 − 6s+ 5 ,

je20111111 = −2J11

(s− 1)2 , je21111111 =

4zJ12(s− 5)(s− 1)2 ,

je31101011 =2J5s− 6J16

s,

je31101111 = −4J6

(s− 1)2 +2J7

(s− 1)2 −2J13

(s− 1)2 −4J17

(s− 1)2 ,

je31111111 =4yJ8

(s− 1)3 +8yJ18

(s− 1)3 . (4.1)

Here we have introduced the notations y =√

s−1s+3 and z =

√s−5s−1 .

The differential system for Jk has the form

J ′1 =8J1

(s− 9)(s− 1) +J2s− 1 , J

′2 =

2(s+ 3)J1 + (s− 9)J2s(s− 9)(s− 1) ,

J ′3 =2J1s− 9 , J

′4 =

J3s− 1 ,

J ′5 =J1s− 1 , J

′6 =

J52s ,

J ′7 = −3J32s , J

′8 =

y

s− 1(3J3 + J5),

J ′9 =(7s− 15)zJ1(s− 9)(s− 5) , J

′10 =

z

s− 5 (2J5 − 9J3 + 2J15) +J9s− 1 ,

J ′11 =zJ9s− 5 , J

′12 =

3zJ3 − zJ5 + J9 − zJ15s− 5 ,

J ′13 = 0, J ′14 = 0,

J ′15 = 0, J ′16 = 0,

J ′17 = −3J162s , J

′18 = −

3yJ162(s− 1) . (4.2)

The boundary conditions follow from the threshold asymptotics s → 9. Moreover, forintegrals J3−18 it is sufficient to know that they vanish in this limit. In particular, usingthe last line of the above formula we have J13 = J14 = J15 = J16 = J17 = J18 = 0. Thismeans that the J1−18 do not form a complete �-regular basis, but the subset J1−12 does,as far as the total cross section is concerned.

– 14 –

JHEP01(2021)144

Note that, apart from the first two equations, the differential system is strictly triangu-lar and the integrals J3−12 can be elementary expressed as the iterated integrals dependingon J1 by integrating the corresponding equation. Alternatively, similar to ref. [20], we canexpress them as one-fold integrals of J1 and (poly)logarithms.4 We have

J1(s) =s− 9

6 (√s− 1)3/2 (

√s+ 3)1/2

K(

(√s− 3) (

√s+ 1)3

(√s− 1)3 (

√s+ 3)

),

J2(s) =18s(√s− 1

)3/2 (√s+ 3

)1/2 E((√s− 3) (√s+ 1)3(√s− 1)3 (

√s+ 3)

)− 2J1(s)√

s(√s− 3) ,

J3(s) = 2∫ s

9ds̃J1(s̃)s̃− 9 ,

J4(s) = 2∫ s

9ds̃J1(s̃)s̃− 9 ln

(s− 1s̃− 1

),

J5(s) =∫ s

9ds̃J1(s̃)s̃− 1 ,

J6(s) =12

∫ s9ds̃J1(s̃)s̃− 1 ln

(s

s̃

), J7(s) = −3

∫ s9ds̃J1(s̃)s̃− 9 ln

(s

s̃

),

J8(s) = 2∫ s

9ds̃

(7s̃− 15)J1(s̃)(s̃− 9) (s̃− 1) ln

(√s− 1 +

√s+ 3√

s̃− 1 +√s̃+ 3

),

J9(s) =∫ s

9ds̃

7s̃−15s̃−9 J1(s̃)√

(s̃− 5)(s̃− 1),

J10(s) =∫ s

9ds̃

7s̃−15s̃−9 J1(s̃)√

(s̃− 5)(s̃− 1)ln(s− 1s̃− 1

)

− 32∫ s

9ds̃

s̃J1(s̃)(s̃− 9)(s̃− 1) ln

(√s− 5 +

√s− 1√

s̃− 5 +√s̃− 1

),

J11(s) =∫ s

9ds̃

J1(s̃)√(s̃− 5)(s̃− 1)

7s̃− 15s̃− 9 ln

(√s− 5 +

√s− 1√

s̃− 5 +√s̃− 1

),

J12(s) =∫ s

9ds̃

7s̃−15s̃−9 J1(s̃)√

(s̃− 5)(s̃− 1)ln(s− 5s̃− 5

)

+ 2∫ s

9ds̃

(5s̃+ 3)J1(s̃)(s̃− 9) (s̃− 1) ln

(√s− 5 +

√s− 1√

s̃− 5 +√s̃− 1

). (4.3)

Taking the traces over γ-matrices, performing the IBP reduction and expressing theresult via Jk, we obtain for the total cross section

σe−γ→e−e−e+ =α3

s− 1

[4 (168s4 − 2714s3 + 8697s2 + 1128s+ 81) J127(s− 1)2s

− 2(872s4 − 11289s3 + 29949s2 + 729s+ 27

)J2

27(s− 1)2s −32(s2 − 5s− 2

)J6

(s− 1)2

+ 4(24s6 − 38s5 − 228s4 + 507s3 + 1161s2 − 45s+ 27

)J3

3(s− 1)2s2(s+ 3)4Note that similar one-fold representations for non-polylogarithmic loop integrals have been obtained

earlier in refs. [29, 30], but within approach which differs from that of ref. [20] (and of the present paper).

– 15 –

JHEP01(2021)144

− 16(9s2 − 46s+ 17

)J4

(s− 1)2 −8(3s6−3s5−59s4−94s3−159s2+33s−9

)J5

3(s− 1)2s2(s+ 3)

+ 16(2s2 − 8s− 1

)J7

(s− 1)2 +32(s4 − s3 − 21s2 − 27s+ 16

)yJ8

(s− 1)3(s+ 3)

− 8(3s4 − 30s3 + 116s2 − 146s− 7

)zJ9

3(s− 5)(s− 1)2 −8(s+ 1)zJ10

s− 1

+ 8(3s2 − 6s− 5

)J11

(s− 1)2 −16(s3 − 3s2 − 9s+ 19

)zJ12

(s− 5)(s− 1)2]. (4.4)

The above formula, together with eq. (4.3), represents our exact result for the total crosssection.

4.1 Asymptotics

Let us now derive the asymptotics of the obtained results.

Threshold asymptotics. The threshold asymptotics can be easily obtained by expand-ing under the integral sign. Taking into account the position of singularities in the differ-ential system (4.2), we choose to expand in the variable r = s−9s−1 . This choice secures thefast convergence of power series for any s > 9.

We haveσe−γ→e−e−e+ =

π

3√

3

[4r2

27 +602r3

243 +4217r4

4374 +O(r5)]. (4.5)

The leading term is in agreement with the asymptotics found in ref. [12].

High-energy asymptotics. The high-energy asymptotics of J1 and J2 can be calculateddirectly:

J1(s)s→∞∼ ln s8 −

3 ln s+ 24s −

6 ln s− 54s2 +O

(s−3

), (4.6)

J2(s)s→∞∼ 18 −

2 ln s+ 38s −

3 ln s− 14s2 +O

(s−3

). (4.7)

However, the calculation of the asymptotics of J3−12 is more tricky. Let us explain ourapproach on the example of J3(s). We transform it in the following way:

J3(s) = 2∫ s

9ds̃J1(s̃)s̃− 9 = 2

∫ s9ds̃

[J1(s̃)s̃− 9 −

ln s̃8s̃

]+ ln

2 s− ln2 98 (4.8)

= ln2 s− ln2 9

8 + C3 − 2∫ ∞s

ds̃

[J1(s̃)s̃− 9 −

ln s̃8s̃

], (4.9)

where C3 = 2∫∞

9 ds̃[J1(s̃)s̃−9 −

ln s̃8s̃

]. The first transition in eq. (4.9) is the subtraction from

the integrand of its large-s̃ asymptotics so as to secure the convergence of the integral whenits upper limit goes to ∞. Now, in the last term we can substitute the asymptotics of theintegrand and integrate term-wise. The only nontrivial task left is to evaluate C3. Whilewe were not able to find an analytic way to perform this integration, we have successfullyapplied PSLQ algorithm [31] to find that C3 = 18 ln

2 9− π224 . Thus, we obtain

J3(s) =ln2 s

8 −π2

24 +1− 3 ln s

4s +37− 30 ln s

16s2 + . . . (4.10)

– 16 –

JHEP01(2021)144

Note that we could have obtained as many terms of the asymptotics as needed. Similarly,we have found the asymptotics of all integrals. Finally we obtain

σe−γ→e−e−e+ =α3[28 ln s

9 −21827 −

1s

(83 ln

3 s− 496 ln2 s−

(2π2 − 4099

)ln s

+ 12ζ3 +31π2

18 −108112

)+O

(s−2

) ](4.11)

The leading term α3(

28 ln s9 −

21827

)agrees with ref. [15]. The subleading term has been

considered in ref. [16]. In that paper the expansion has been performed with respect tophoton energy k = (s−1)/2 in the rest frame of the initial electron. If we express eq. (4.11)in terms of k, we obtain

σe−γ→e−e−e+ =α3[28

9 ln(2k)−21827 +

1k

(− 43 ln

3(2k) + 4912 ln2(2k) +

(π2 − 40918

)ln(2k)

− 6ζ3 −31π2

36 +335572

)+O

(k−2

) ](4.12)

≈α3[28

9 ln(2k)−21827 −

1k

(43 ln

3(2k)− 4.08 ln2(2k)

+ 12.85 ln(2k)− 30.89)

+O(k−2

) ]. (4.13)

This is to be compared with the result of ref. [16]:

σRef. [16]e−γ→e−e−e+ ≈ α

3[28

9 ln(2k)−21827 −

1k

(43 ln

3(2k)− 3.86 ln2(2k) + 11 ln(2k)− 27.9)]

.

(4.14)

Figure 7 shows the cross section as a function of energy. It is remarkable that the twofirst terms of high-energy asymptotics, eq. (4.11), provide a very good approximation ofthe total cross section with accuracy better than 1% for

√s > 8me. In the same figure we

have presented a few points of ref. [32] obtained by numerical integration. These pointsserve as a perfect cross check of our analytic results.

5 Conclusion

In the present paper we have calculated the total cross sections for the processes e−γ →e−XX̄ with X = µ−, γ, e− for arbitrary energies. Apart from the last case, our exactresults are expressed via classical polylogarithms. We have calculated the threshold andhigh-energy asymptotics of the obtained results and compared them with the availableresults finding some discrepancies in the latter.

Note that our method allows one to obtain exact expressions also for the integratedcross sections weighted by some kinematic invariants. As an example, we have calculatedthe cross sections weighted by the invariant mass of a pair of produced particles for each

– 17 –

JHEP01(2021)144

5 10 50 100

0

5

10

15

20

Figure 7. Exact cross section (solid curve) and asymptotics. The high-energy asymptotics (dottedline) corresponds to the leading term α3

( 28 ln s9 −

21827)only, while eq. (4.11), taking into account

the next-to-leading term, would lead to the curve hardly discernible from the exact result: e.g., at√s = 10me the difference is a tiny 0.4%. Points show the numerical results of ref. [32].

5 10 50 100

0

10

20

30

40

50

60

(a) Cross section averagedsquare of the µ+µ− pair in-variant mass, in units m2µ, asfunction of

√s/mµ.

1 5 10 50 100

0

20

40

60

80

100

120

(b) Double Compton scatteringcross section weighted by thesquare of invariant mass of pho-tons as function of

√s/mµ.

5 10 50 1000.0

0.1

0.2

0.3

0.4

0.5

0.6

(c) Cross section averagedsquare of e−e− invariant mass,divided by s, as function of√s/mµ. The low- and high-

energy asymptotics are 4/9 and1/2, respectively.

Figure 8. Weighted cross sections defined in eq. (5.1).

process. More precisely, we consider

〈µ2µ+µ−〉def=∫µ2µ+µ−dσe−γ→e−µ+µ− ,

〈µ22γ〉def=∫µ22γdσe−γ→e−γγ ,

〈µ22e−〉def=∫µ22e−dσe−γ→e−e−e+ . (5.1)

i.e., we consider the cross section weighted by the square of invariant mass of the muonpair, the photon pair, and the electron pair for e−γ → e−µ+µ−, e−γ → e−γγ, ande−γ → e−e−e+, respectively. Note that

∫µ2γγdσe−γ→e−γγ is infrared safe. The numeri-

cal results are presented in figure 8. We provide the explicit expressions for the weighted

– 18 –

JHEP01(2021)144

cross sections in eq. (5.1) in the supplementary material, see below. Let us note that theleading asymptotics of 〈µ22e−〉 is equal to that of σe−γ→e−e−e+ multiplied by s/2. This canbe easily understood on physical ground as follows. At high energies the electron-positronpair is mostly produced with small invariant mass by equivalent photon mechanism. Thenwe have p3 ≈ p4, p2 ≈ (

√s/2,n2

√s/2), and p3 + p4 ≈ 2p3 ≈ q(

√s/2,−n2

√s/2). Then

µ22e− = (p2 + p3)2 ≈ 2p2 · p3 ≈ s/2.

For reader convenience, we provide in the supplementary material the file CSall.mwhich contains evaluation-ready expressions for the exact cross sections defined ineqs. (2.10), (2.15), (3.15), (3.16), (4.4), and (5.1). When read in from Mathematica ses-sion with Get["CSall.m"] the file prints all necessary information about the functionsdefined therein.

The last but not the least, the calculation presented in this paper has demonstrated afew methods not so widely known in the multiloop community. First, in the calculation ofthe e−γ → e−µ+µ− cross section we have shown how to obtain the differential equationswith respect to one variable (s in our case) for the coefficients of generalized power serieswith respect to another variable (m2e in our case). Next, we have explicitly demonstratedthat the contribution of soft photons to the integrated cross section can be calculatedby means of the multiloop techniques. Finally, we have successfully applied the recentlyintroduced [20] approach to the non-polylogarithmic integrals, based on the construction of�-regular basis. These approaches can be applied in other physically relevant calculations.

Acknowledgments

This work has been supported by Russian Science Foundation, grant 20-12-00205.

Open Access. This article is distributed under the terms of the Creative CommonsAttribution License (CC-BY 4.0), which permits any use, distribution and reproduction inany medium, provided the original author(s) and source are credited.

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IntroductionTotal cross section of the process e- gamma -> e- mu+ mu-ResultsCross-section near the thresholdHigh-energy asymptoticsTotal cross-section of e- gamma -> e- pi+ pi-

Total cross section of the process e- gamma -> e- gamma gammaCalculation of the total cross section in dimensional regularizationSoft-photon contribution to e- gamma -> e- gamma gammaResultsAsymptotics

Total cross section of the process e- gamma -> e- e- e+Asymptotics

Conclusion