We apply the differential equation method to the calculation of the total
Born cross section of the process $Z_1Z_2\to Z_1Z_2e^+e^-$. We obtain explicit
expression for the cross section exact in the relative velocity of the nuclei.Comment: 5 pages, 2 figures, ancillary file MastersExpansion.m attached to
  submissio

Lee, Roman N.

Mingulov, Kirill T.

Physics Letters B

English

arXiv

AbstractWe apply the differential equation method to the calculation of the total Born cross section of the process Z1Z2→Z1Z2e+e−. We obtain explicit expression for the cross section exact in the relative velocity of the nuclei

Elsevier - Publisher Connector 

Total Born cross section of e+e−-pair production in relativistic ion collisions from differential equations 

Physics Letters B 757 (2016) 207–210Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Total Born cross section of e+e−-pair production in relativistic ion 
collisions from differential equations
Roman N. Lee a,∗, Kirill T. Mingulov a,b
a Budker Institute of Nuclear Physics, 630090, Novosibirsk, Russia
b Novosibirsk State University, 630090, Novosibirsk, Russia
a r t i c l e i n f o a b s t r a c t
Article history:
Received 15 February 2016
Accepted 29 March 2016
Available online 4 April 2016
Editor: A. Ringwald
We apply the differential equation method to the calculation of the total Born cross section of the process 
Z1 Z2 → Z1 Z2e+e−. We obtain explicit expression for the cross section exact in the relative velocity of 
the nuclei.
© 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license 
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.1. Introduction
Theoretical investigation of electromagnetic e+e− pair produc-
tion in relativistic heavy-ion collisions goes back to the paper [1]
where the Born cross section of the process at high energy was 
calculated in the leading logarithmic approximation. Racah, in his 
remarkable paper [2], has calculated the high-energy asymptotics 
of the Born cross section up to power-suppressed terms in 1/γ
(γ is a Lorentz factor of the colliding nuclei). Recently there was 
a certain rise of the interest to this process connected with the 
functioning of heavy ion colliders, like RHIC and LHC, see Ref. [3]. 
In particular, a great attention has been paid to the investigation 
of the Coulomb corrections to the cross section at high energies 
[4–8].
Speaking of the total Born cross section, the problem of its cal-
culation is of a three-loop complexity level. Probably, this is the 
main reason why this quantity was not calculated exactly at ar-
bitrary velocities of the colliding nuclei. This is in contrast to the 
Born cross section of pair photoproduction in the ﬁeld of an ion, 
where the total Born cross section is known exactly for any energy 
of the initial photon since Refs. [9,10]. Now that we have an essen-
tial progress in the multiloop calculations, we are in position to ﬁll 
this gap and to calculate the total Born cross section of e+e− pair 
production in relativistic ion collisions.
The consideration of the present paper is based on the fol-
lowing approach. Using the optical theorem we express the total 
cross section via the sum of cut three-loop integrals. Then we ap-
ply the standard approach to multiloop calculations, based on the 
* Corresponding author.
E-mail addresses: r.n.lee@inp.nsk.su (R.N. Lee), k.t.mingulov@gmail.com
(K.T. Mingulov).http://dx.doi.org/10.1016/j.physletb.2016.03.083
0370-2693/© 2016 The Authors. Published by Elsevier B.V. This is an open access article
SCOAP3.IBP reduction and differential equations for master integrals. The 
differential equations for the master integrals are ﬁrst reduced to 
-form [11] using the algorithm of Ref. [12], and then solved re-
cursively up to the required order in  . Thus, we obtain the total 
Born cross section exactly in the relative velocity β of the collid-
ing nuclei. Our result perfectly agrees with the celebrated result 
of Racah [2] in the limit of large relativistic factor. At small β we 
compare our result with estimate obtained in the recent paper [13]
and ﬁnd a complete disagreement. In order to ﬁnd the origin of 
the disagreement, we perform a straightforward calculation of the 
low-energy asymptotics of the cross section differential with re-
spect to the electron and positron momenta. The direct integration 
then reproduces our result obtained with the help of the differen-
tial equations.
2. Born cross section for the production of e+e− pair
Using optical theorem, the total cross section of the process 
Z1 Z2 → Z1 Z2e+e− can be written as
σ = 8 ImA
γ β
, (1)
where ImA is given by the sum of two cut diagrams depicted in 
Fig. 1, β is the relative velocity of the colliding nuclei, and γ =
[1 −β2]− 12 is the Lorentz factor. Contribution of both diagrams can 
be expresses in terms of the scalar integrals
I(n1, . . . ,n12) =
∫
ddl ddq1 ddq2
(2π)3d
θ(q01 − l0)θ(q02 + l0)
×
4∏
Im
1
(Dk + i0)nk
12∏ 1
(Dk + i0)nk ,k=1 k=5
 under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by 
208 R.N. Lee, K.T. Mingulov / Physics Letters B 757 (2016) 207–210Fig. 1. Cut diagrams for the calculation of the total cross section of e+e− pair 
production in the collisions of relativistic nuclei. Cut thin line denotes the cut prop-
agator −2π iδ(p2 − m2)(pˆ + m) of the electron, cut double line denotes the cut 
propagator −2π iδ(2u · q) of a heavy particle, interaction vertex with the heavy par-
ticle is −iuμ (u = P/M is a four-velocity of the heavy particle).
D1 = −2q1 · u1 , D2 = −2q2 · u2 , D3 = (l − q1)2 − 1 ,
D4 = (l + q2)2 − 1 , D5 = l2 − 1 , D6 = (l − q1 + q2)2 − 1 ,
D7 = q21 , D8 = q22 , D9 = −2l · u1 , D10 = −2l · u2 ,
D11 = −2q2 · u1 , D12 = −2q1 · u2 . (2)
Here u1 and u2 are the four-velocities of the nuclei, so that u1 ·
u2 = γ .
We proceed in the following way. First, we perform the IBP 
reduction of the cut integrals from the above topologies in d =
4 − 2 . For this step we use LiteRed, Refs. [14,15]. We end up 
with 8 master integrals
J1 = I(1,1,1,1,0,0,0,1,0,0,0,0),
J2 = I(1,1,1,1,0,1,0,0,0,0,0,0),
J3 = I(1,1,1,1,0,2,0,0,0,0,0,0),
J4 = I(1,1,1,1,−1,1,0,0,0,0,0,0),
J5 = I(1,1,1,1,0,0,1,1,0,0,0,0),
J6 = I(1,1,1,1,1,1,0,0,0,0,0,0),
J7 = I(1,1,1,1,0,0,1,1,−1,−1,0,0),
J8 = I(1,1,1,1,1,1,0,0,−1,0,−1,0) .
Introducing the column-vector J = ( J1, . . . , J8)T , we obtain the 
differential system
∂
∂γ
J = M(γ , )J , (3)
where M(γ , ) is a matrix with entries being rational functions 
of both γ and  . Passing to new variable, x = 1−β1+β , we apply 
the algorithm from Ref. [12] to reduce the differential system 
(3) to -form [11]. The differential system for the new basis ˜ J =
(˜ J1, . . . , ˜J8)T has the form
∂
∂ x˜
J = 
[
1
x
M0 + 1
x− 1M1 +
1
x+ 1M2
]
J˜ , (4)
M0 =
⎡
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
−1 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0
0 1 −1 0 0 0 0 0
0 0 0 −1 0 1 0 0
0 0 0 0 3 3 0 0
0 0 0 −1 −1 0 0 0
0 0 0 0 0 0 −1 −1
0 0 0 2 1 0 1 1
⎤
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦
, (5)M1 = diag(2,0,2,2,−6,0,2,0) , (6)
M2 = diag(0,0,0,0,0,0,0,−2) . (7)
We obtain -expansion of ˜J =∑n n˜J(n) term-by-term using the 
formula
J˜(n+1) =
∫
dx
[
1
x
M0 + 1
x− 1M1 +
1
x+ 1M2
]
J˜(n) + const (8)
and ﬁxing the constant from small-β asymptotics. In order to cal-
culate this asymptotics, we use the method of expansion by re-
gions [16]. The only nontrivial boundary conditions come from 
O (β2−1) term in small-β asymptotics of J1 and J2:
J1 ∼ J2 ∼ −2
8−16π3−5
()2
(2 − 1)
(3 − 2)

(4 − 1) β
2−1 . (9)
As a result, the -expansions of both ˜J and J are expressed in terms 
of HPLs. Plugging the obtained expansions in the cross section ex-
pressed via ˜J we observe the cancellation of the terms O (n) with 
n = −4, . . . , −1. The O (0) term gives us the result
σ = (Z1α)
2(Z2α)2
πm2
{
− 1− β
2
12β2
L4 + 2
(
23β2 − 37) S3a
9β2
+ 2
(
11β2 − 25) S3b
9β2
− 26S2
9β
−
(
β6 + 217β4 − 135β2 + 45) L2
54β6
+ 5
(
67β4 − 48β2 + 18) L
27β5
− 2
(
78β4 − 35β2 + 15)
9β4
}
, (10)
S3a = Li3
(
1− β
1+ β
)
+ L Li2
(
1− β
1+ β
)
− L
2
2
log
(
2β
1+ β
)
− L
3
12
− ζ3 ,
S3b = Li3
(
−1− β
1+ β
)
+ L
2
Li2
(
−1− β
1+ β
)
+ L
3
24
− π
2L
24
+ 3ζ3
4
,
S2 = Li2
(
−1− β
1+ β
)
+ L log
(
β + 1
2
)
− L
2
4
+ π
2
12
,
L = log
(
1+ β
1− β
)
.
2.1. Asymptotics
Given the expression (10), it is easy to calculate both high-
energy and low-energy asymptotics of the total cross section. For 
γ  1 we have
σ = (Z1α)
2(Z2α)2
πm2
{
28L30
27
− 178L
2
0
27
+
(
370
27
+ 7π
2
27
)
L0 + 7ζ3
9
− 13π
2
54
− 116
9
− 1
γ 2
[
4L40
3
− 98L
3
0
27
+ 188L
2
0
27
−
(
172
27
+ 25π
2
54
)
L0 − 73ζ3
18
+ 5π
2
27
+ 43
27
]
+ . . .
}
, (11)
where L0 = ln(2γ ). The ﬁrst line of Eq. (11) is the celebrated 
Racah’s result [2], and the second line is the ﬁrst correction to 
R.N. Lee, K.T. Mingulov / Physics Letters B 757 (2016) 207–210 209Fig. 2. Left: cross section σ , Eq. (10), in units of σ0 = (Z1α)2(Z2α)2/m2 as a function of u = γ β (solid curve). Dashed and dotted curves correspond to high- and low-energy 
asymptotics, respectively. Right: relative error δ = σh,l/σ − 1 of the high- and low-energy asymptotics.it. It is interesting to note that the correction is ampliﬁed by the 
fourth power of L0.
For β  1 there is a strong compensation between separate 
terms in Eq. (10), which leads to ∝ β8 suppression of the cross 
section. We have
σ = 296(Z1α)
2(Z2α)2β8
55125πm2
(
1+ 7708β
2
3663
+ . . .
)
. (12)
Recently, the small-β asymptotics of the total cross section was 
discussed in Ref. [13]. The estimate σ ∝ β5 given there is in clear 
contradiction with our result (12). In fact, the estimate σ ∝ β8 can 
be justiﬁed in the following way. Using the kinematic constraints
q1 · u1 = q2 · u2 = 0 , (q1 + q2)2 > 4m2 (13)
for momentum transfers q1,2, it is easy to understand that the 
main contribution to the cross section is given by the region 
where
|q1,2| ∼m/β , |q1 + q2| ∼ q01,2 ∼m . (14)
The characteristic momenta of the produced particles are of the 
order of their mass. Using these estimates, it is easy to count pow-
ers of β in the total cross section. We have β−3 from dq1dq2, β8
from the photon propagators, β4 from the denominator of electron 
propagator, and β−1 from the ﬂux of the colliding particles. As to 
the numerator of the electron propagator, one might check that it 
does not give β−1 factor in the sum of two diagrams due to the 
estimate uˆ1qˆ1,2uˆ2 − uˆ2qˆ1,2uˆ1 ∼ O (β0).
Using these estimates, we have derived the differential cross 
section at β  1 and then obtained the leading term of (12) by 
the direct integration. To avoid cluttering, we refrain from present-
ing fully differential cross section here. We only present the cross 
section, differential with respect to the energies of the produced 
particles:
dσ
dε+dε−
≈ 16(Z1α)
2(Z2α)2β8p−p+
45π (ε− + ε+)10
×
[
(33ε+ε− − 49m2)(ε2− + ε2+) − 14ε2+ε2−
+ 78ε+ε−m2 − 32m4
]
. (15)
Here p± =
√
ε2± −m2 and β  1 is the relative velocity of the nu-
clei. Integrating this cross section over ε± , we obtain the leading 
term of Eq. (12).
3. Discussion and conclusion
It is interesting to compare our result with the leading high-
energy (Racah) and low-energy asymptotics σh,l . These asymptotics 
are given by the ﬁrst line of Eq. (11) and the leading term of Eq. (12), respectively. Fig. 2 demonstrates this comparison. One can 
see that both low- and high-energy asymptotics essentially depart 
from the exact result in the region 0.3  γ β  10.
Our approach based on the IBP reduction and calculation of the 
master integrals allows us, without additional efforts, to calculate 
the total Born cross section of the production of a pair of point-like 
scalar charged particles. We have
σs = (Z1α)
2 (Z2α)2
πm2
{
4
(
β2 − 2) S3a
9β2
+ 4
(
β2 − 2) S3b
9β2
− 5S2
9β
+
(
β6 + β4 − 81β2 + 45) L2
108β6
−
(
10β4 − 66β2 + 45) L
27β5
− 17β
2 − 15
9β4
}
. (16)
The high- and the low-energy asymptotics of this cross section 
have the form
σs = (Z1α)
2 (Z2α)2
πm2
×
⎧⎨
⎩
4L30
27 −
19L20
27 + 22+π
2
27 L0 + ζ39 − 5π
2
108 − 29 + . . . at γ  1
4β4
135π
(
1+ 27β235 + 694β
4
1225 + . . .
)
at β  1
(17)
The leading term L30 in the high-energy asymptotics agrees with 
the result obtained within the equivalent photon approximation. 
Curiously, the low-energy asymptotics of the cross section for 
scalar particles scales differently (∝ β4) than that for spinor par-
ticles. Inspection of the contributions of separate diagrams shows 
that in this limit only the contribution of seagull diagram survives.
It is interesting to discuss the applicability region of our re-
sults. Of course, by using the Born approximation we assume 
that Z1,2α  1. In principle, at small β one may expect higher-
order corrections of the relative order Z1,2α/β . However, since 
at small β the velocities of the produced particles are not small 
(see, e.g., their spectrum (15)), we would guess that such correc-
tions are forbidden. One may also expect corrections of the order 
Z1 Z2α/β due to additional Coulomb exchanges between the nu-
clei, but they seem to be accompanied by the factor m/M1,2  1, 
where M are the masses of the nuclei. In fact a stronger condition 
m/(βM1,2)  1 is deﬁnitely required because otherwise the limit 
M1,2 → ∞ is no longer valid. Nevertheless, we must admit that 
the determination of the correct magnitude of the higher-order ef-
fects at small β requires a separate examination.
Acknowledgements
We are grateful to A.I. Milstein for the interest to our work and 
useful discussions. This work has been supported by Russian Sci-
210 R.N. Lee, K.T. Mingulov / Physics Letters B 757 (2016) 207–210ence Foundation (Project No. 14-50-00080). Partial support of RFBR 
through Grant No. 15-02-07893 is also acknowledged.
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