Eur. Phys. J. C (2018) 78:644 Page 3 of 12 644
C
ab
(α
s
, z) = δ
ab
δ(1 − z) +
∞
n=1
α
s
2π
n
C
(n)
ab
(z). (8)
At present, the perturbative Sudakov form factor can be eval-
uated at next-to-next-to-leading logarithmic (NNLL) accu-
racy [11]. In the q ¯q annihilation channel pertinent to Drell–
Yan production, the evaluation of the Sudakov form factor at
next-to-leading logarithmic (NLL) accuracy, the one reached
in the present analysis, involves the coefficients
A
(1)
= 2C
F
B
(1)
=−3C
F
, (9)
which are the coefficient of the singular (1−z)
−1
and δ(1−z)
terms of the one-loop splitting function P
(0)
qq
(z) and
A
(2)
= KA
(1)
, K = C
A
67
18
−
π
2
6
− n
f
T
R
10
9
, (10)
which is the coefficient of the singular term of the two-loop
splitting function P
(1)
qq
(z) in the z → 1 limit [34]. The general
expression for C
(1)
ab
are given by [11,35]
C
(1)
qa
(z) = C
(1)
¯qb
(z)
= δ
qa
C
F
(1 − z) + δ
qa
δ(1 − z)C
F
−4 +
π
2
2
,
C
(1)
qg
(z) = C
(1)
¯qg
(z) = 2T
R
z(1 − z). (11)
Color factors in the previous equations are given by C
A
= 3,
C
F
= 4/3, T
R
= 1/2 with n
f
being the number of active
flavours. Together with the use of NLO pdfs, this guarantees
the evaluation of the cross section at small q
T
at NLL accu-
racy. The last ingredient in Eq. (2) is the non perturbative
form factor, S
h
1
h
2
NP
(b), which encodes the transverse struc-
ture of both the colliding hadrons. The latter is either fixed
by comparison with data or parametrized with the help of
hadronic models, as we shall do in this paper.
2.2 Proton structure
Predictions for the transverse momentum spectrum of DY
pairs produced in pion–proton collisions do rely on the
knowledge of the proton NP form factor. The latter is
extracted from the transverse momentum spectrum of DY
pairs produced in proton–proton (pp) and proton–nucleus
( pA) collisions. Quite recent analyses [15,16] have appeared
which address such an extraction. Since our aim here is to
establish the possibility of studying the pion transverse non
perturbative structure in pion–nucleus DY experiments, we
here intend to minimize the uncertainity coming from the
proton structure part of the calculation. We use the well
known and widely accepted results of Konychev and Nadol-
sky (KN05) [12] obtained within the CSS formalism [6]
where S
pp
NP
(b) is extracted from global fit to Z -boson and
low mass DY data, updating the results presented in Ref. [13].
The latter is parametrised as
S
pp
NP
(b) (12)
= exp{−[a
1
+ a
2
ln(M/(3.2GeV)) + a
3
ln(100x
1
x
2
)]b
2
}.
The a
i
parameters appearing in Eq. (12) are determined by a
minimisation procedure against data and are given by [12]
a
1
= 0.201 ±0.011, a
2
= 0.184 ±0.018,
a
3
=−0.026 ±0.007. (13)
The fit is fully specified once a prescription for the treatment
of the non perturbative, large-b, region both in the Sudakov
form factor, Eq. (6), and the parton distributions is given.
The authors of Ref. [12] adopt the so-called b
-prescription,
substituting b with
b
(b, b
max
) =
b
1 +
b
b
max
2
, (14)
and setting b
max
= 1.5GeV
−1
in the perturbative form fac-
tor. In principle, the same setting should be used in PDFs,
which are evaluated at the factorisation scale μ
F
= b
0
/b
∗
.
However this choice for b
max
may imply a call to a specific
PDFs parameterization below their lowest available scale,
Q
in
. Since in Ref. [12] cross sections are evaluated with
the NLO CTEQ6M PDFs [36], whose lowest Q accessible
is Q
in
= 1.3 GeV, the b
-prescription entering PDFs calls
is used with b
max
= b
0
/Q
in
0.86 GeV
−1
which always
guarantees μ
F
> Q
in
. It is important to remark that the non
perturbative form factor is determined not only by fitting the
parameters of the chosen functional form, but also by the
specific regularisation prescription and its associated param-
eters adopted to deal with the infrared region. In general all
these ingredients have been found to be highly correlated.
In order to present a benchmark of our code and to gauge
how theory performs in extrapolation regions, we compare
predictions from KN05 to the pA data of Ref. [37]. An addi-
tional ±25% normalisation error is assigned to the data [37].
In the original KN05 analysis, only the data at p
lab
= 400
GeV, q
T
< 1.4GeV,5< M/GeV < 9 were included in the
fit. In such a restricted region indeed the theory (solid lines)
performs well offering a good benchmark of our code, as
shown in the first row of Fig. 1. Since the π W data to be ana-
lyzed in the following are at p
lab
=252 GeV, it is important to
check how well the theory performs in extrapolation regions
at lower
√
s and higher DY rapidity. Therefore we present
in the second and third rows of Fig. 1 the KN05 benchmark
(dashed lines) versus data [37]at p
lab
= 200 and 300 GeV,
which were not included in the KN05 fit. By using Eq. (4)
and Eq. (5) and assuming the invariant mass values indicated
on the plots, the rapidity coverage of these data can be con-
123
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