Physics Letters B 791 (2019) 206–209
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
The O (α
2
) initial state QED corrections to e
+
e
−
annihilation to a
neutral vector boson revisited
J. Blümlein
a,∗
, A. De Freitas
a
, C.G. Raab
b
, K. Schönwald
a
a
Deutsches Elektronen–Synchrotron, DESY, Platanenallee 6, D-15738 Zeuthen, Germany
b
Institute of Algebra, Johannes Kepler University, Altenbergerstraße 69, A-4040, Linz, Austria
a r t i c l e i n f o a b s t r a c t
Article history:
Received
28 January 2019
Accepted
20 February 2019
Available
online 26 February 2019
Editor:
A. Ringwald
Keywords:
Higher
order QED correction
e
+
e
−
annihilation
Precision
physics at ILC
FCC
Iterated
integrals
We calculate the non-singlet, the pure singlet contribution, and their interference term, at O (α
2
) due to
electron-pair initial state radiation to e
+
e
−
annihilation into a neutral vector boson in a direct analytic
computation without any approximation. The correction is represented in terms of iterated incomplete
elliptic integrals. Performing the limit s m
2
e
we find discrepancies with the earlier results of Ref. [1]
and
confirm results obtained in Ref. [2]where the effective method of massive operator matrix elements
has been used, which works for all but the power corrections in m
2
/s. In this way, we also confirm the
validity of the factorization of massive partons in the Drell–Yan process. We also add non-logarithmic
terms at O (α
2
) which have not been considered in [1]. The corrections are of central importance for
precision analyses in e
+
e
−
annihilation into γ
∗
/Z
∗
at high luminosity.
© 2019 The Author(s). 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 SCOAP
3
.
The initial state QED corrections (ISR) to e
+
e
−
annihilation are
of crucial importance for the experimental analyses at LEP [3] and
for planned projects like the ILC and CLIC [4], the FCC_ee [5],
muon colliders [6], and notably also at e
+
e
−
Higgs factories us-
ing
the process e
+
e
−
→ ZH
0
, as well as e
+
e
−
→ t
¯
t. The initial
state corrections have been carried out analytically to O
(
αL)
5
in the leading logarithmic series using the structure function
method [7]. A small z-resummation has been performed in [8].
In Ref. [1]the O (α
2
) corrections were calculated neglecting terms
of O
m
2
s
ln(
m
2
s
)
. Here s is the cms energy squared and m = m
e
.
These corrections are used in analysis codes such as TOPAZ0 [9]
and
ZFITTER [10]. The initial state QED corrections can be writ-
ten
in terms of the following functions
H
z, α,
s
m
2
= δ(
1 − z) +
∞
k=1
α
4π
k
C
k
z,
s
m
2
(1)
C
k
z,
s
m
2
=
k
l=0
ln
k−l
s
m
2
c
k,l
(z), (2)
*
Corresponding author.
E-mail
address: Johannes.Bluemlein@desy.de (J. Blümlein).
which yield the respective differential cross sections by
dσ
e
+
e
−
ds
=
1
s
σ
e
+
e
−
(s
)H
z, α,
s
m
2
,
(3)
with σ
e
+
e
−
(s
) the scattering cross section without the ISR correc-
tions,
α ≡ α(s) the fine structure constant and z = s
/s, where s
is
the invariant mass of the produced (off-shell) γ /Z boson.
In
Ref. [2]the method of massive operator matrix elements
(OMEs) [11] has been applied to calculate the O (α
2
) corrections,
factorizing the process into universal massive contributions and
the massless Wilson coefficients of the Drell–Yan process [12,13].
This method has been known to work in the case of external mass-
less
fields, including the non-logarithmic contributions in QCD,
cf. [11,14]. However, the results of Refs. [1] and [2]were found
to disagree.
One
way to find out the correct answer is to perform the di-
rect
analytic calculation of the corresponding contributions with-
out
doing any approximation. This is done in the present paper
for three of the subprocesses of the O (α
2
) corrections related to
fermion-pair production, i.e. for the processes II–IV of Ref. [1].
Like in Refs. [1,2]we will consider the case of pure vector cou-
plings
in the following. It is known already from the massless
case [12]that in some of the processes axialvector-vector terms
receive different corrections if compared to the pure vector or ax-
ialvector
case. These aspects will be presented in Ref. [15]. In the
calculation we used the packages FORM, Sigma, Harmonic-
https://doi.org/10.1016/j.physletb.2019.02.038
0370-2693/
© 2019 The Author(s). 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
SCOAP
3
.
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