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Vector-boson production at hadron colliders: hard-collinear coefficients at the NNLO

22 May 2012-European Physical Journal C (Springer-Verlag)-Vol. 72, Iss: 11, pp 2013

Abstract: We consider QCD radiative corrections to vector-boson production in hadron collisions. We present the next-to-next-to-leading order (NNLO) result of the hard-collinear coefficient function for the all-order resummation of logarithmically enhanced contributions at small transverse momenta. The coefficient function controls NNLO contributions in resummed calculations at full next-to-next-to-leading logarithmic accuracy. The same coefficient function is used in applications of the subtraction method to perform fully exclusive perturbative calculations up to NNLO.
Topics: Resummation (57%), Vector boson (51%)

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Eur. Phys. J. C (2012) 72:2195
DOI 10.1140/epjc/s10052-012-2195-7
Regular Article - Theoretical Physics
Vector-boson production at hadron colliders:
hard-collinear coefficients at the NNLO
Stefano Catani
1
, Leandro Cieri
1
, Daniel de Florian
2
, Giancarlo Ferrera
3
, Massimiliano Grazzini
4,a,b
1
INFN, Sezione di Firenze and Dipartimento di Fisica e Astronomia, Università di Firenze, 50019 Sesto Fiorentino, Florence, Italy
2
Departamento de Física, FCEYN, Universidad de Buenos Aires, (1428) Pabellón 1 Ciudad Universitaria, Capital Federal, Argentina
3
Dipartimento di Fisica, Università di Milano and INFN, Sezione di Milano, 20133 Milan, Italy
4
Institut für Theoretische Physik, Universität Zürich, 8057 Zürich, Switzerland
Received: 22 August 2012 / Published online: 3 November 2012
© Springer-Verlag Berlin Heidelberg and Società Italiana di Fisica 2012
Abstract We consider QCD radiative corrections to vector-
boson production in hadron collisions. We present the next-
to-next-to-leading order (NNLO) result of the hard-collinear
coefficient function for the all-order resummation of log-
arithmically enhanced contributions at small transverse
momenta. The coefficient function controls NNLO contri-
butions in resummed calculations at full next-to-next-to-
leading logarithmic accuracy. The same coefficient function
is used in applications of the subtraction method to perform
fully exclusive perturbative calculations up to NNLO.
The transverse-momentum (q
T
) distribution of systems with
high invariant mass M (Drell–Yan lepton pairs, vector bo-
son(s), Higgs boson(s) and so forth) produced in hadronic
collisions is important for physics studies within and beyond
the Standard Model (SM).
The computation of these distributions in perturbative
QCD is complicated by the presence of large logarithmic
contributions of the form ln(M
2
/q
2
T
) that need to be re-
summed to all perturbative orders in the QCD coupling α
S
.
The method to perform the resummation is known [18],
including recent developments on the discovered and re-
summed effects [9, 10] due to helicity and azimuthal cor-
relations in gluon fusion subprocesses. The structure of the
resummed calculation is organized in a process-independent
form that is controlled by a set of perturbative functions
with computable ‘resummation coefficients’. All the resum-
mation coefficients that are process independent are known
since some time [1118] up to the second order in α
S
,
and the third-order coefficient A
(3)
has been obtained in
a
e-mail: grazzini@physik.uzh.ch
b
On leave of absence from INFN, Sezione di Firenze, Sesto Fiorentino,
Florence, Italy.
Ref. [19]. The complete computations of the second-order
resummation coefficients have been carried out in Refs. [20]
and [21] for two benchmark processes, namely, the pro-
duction of the SM Higgs boson through gluon fusion and
vector-boson production through the Drell–Yan (DY) mech-
anism of quark–antiquark annihilation. The explicit ana-
lytic expressions for the O
2
S
) hard-collinear resummation
coefficients in the case of SM Higgs boson production in
the large-m
top
limit have been presented in Ref. [22]. This
paper parallels Ref. [22]: we concentrate on single vector-
boson production, and we present the corresponding ana-
lytic expressions of the second-order hard-collinear coeffi-
cient functions H
(2)
.
QCD predictions for vector-boson production at hadron
colliders are important for present and forthcoming stud-
ies at the Tevatron and the LHC. Resummed calculations
of the q
T
spectrum of vector bosons and of related observ-
ables are presented in Refs. [2339]. Calculations for vector-
boson production at the fully exclusive level with respect
to the accompanying QCD radiation have been carried out
in Refs. [21, 4043] up to the next-to-next-to-leading order
(NNLO) in perturbative QCD.
In this paper we compute the hard-collinear coefficient
function H
(2)
and, thus, the complete analytical expression
of the NNLO cross section for vector-boson production in
the small-q
T
region. These results have a twofold relevance,
in the context of both resummed and fixed-order calcula-
tions.
The knowledge of H
(2)
can be implemented in re-
summed calculations at full next-to-next-to-leading loga-
rithmic (NNLL) order to achieve uniform NNLO accuracy
in the small-q
T
region. In the case of vector-boson produc-
tion, this implementation has been carried out in Ref. [33]by
using the impact-parameter space resummation formalism
developed in Refs. [44, 45]. This formalism enforces a uni-
tarity constraint and thus it guarantees that (upon inclusion

Page 2 of 9 Eur. Phys. J. C (2012) 72:2195
of H
(2)
) the resummed q
T
spectrum returns the complete
NNLO total cross section after integration over q
T
.
The subtraction method of Ref. [20] exploits the knowl-
edge of transverse-momentum resummation coefficients at
O
2
S
) to perform NNLO calculations at the fully exclu-
sive level. The Higgs boson coefficient functions presented
in Ref. [22] were used in the numerical computations of
Refs. [20, 46]. The coefficient functions presented in this pa-
per are precisely those that are needed for the actual imple-
mentation of this subtraction method in DY-type processes:
theyareusedinRefs.[21] and [47] for the NNLO numerical
computations of vector-boson production and of associated
production of a Higgs boson and a W boson. The diphoton
NNLO calculation of Ref. [48] also uses part of the results
of the present paper to treat the quark-antiquark annihilation
subprocess q ¯q γγ.
This paper is organized as follows. We first introduce our
notation and illustrate the NNLO calculation of the vector-
boson cross section at small values of q
T
. Then we recall the
transverse-momentum resummation formalism. Finally, we
present our NNLO results in analytic form and the relation
with the q
T
resummation coefficients at O
2
S
).
We briefly introduce the theoretical framework and our
notation. We consider the production of a vector boson V
(V = W
±
,Z and/or γ
) in hadron–hadron collisions. We
use the narrow width approximation and we treat the vec-
tor boson as an on-shell particle with mass M.TheQCD
expression of the vector-boson transverse-momentum cross
section
1
is
dq
2
T
(q
T
,M,s)
=
a,b
1
0
dz
1
1
0
dz
2
f
a/h
1
z
1
,M
2
f
b/h
2
z
2
,M
2
×
d ˆσ
ab
dq
2
T
q
T
,M,ˆs =z
1
z
2
s;α
S
M
2

, (1)
where f
a/h
i
(x, μ
2
F
) (a = q
f
, ¯q
f
,g) are the parton densi-
ties of the colliding hadrons (h
1
and h
2
) at the factoriza-
tion scale μ
F
, and d ˆσ
ab
/dq
2
T
are the partonic cross sections.
The centre-of-mass energy of the two colliding hadrons is
denoted by s, and ˆs is the partonic centre-of-mass energy.
We use parton densities as defined in the
MS factoriza-
tion scheme, and α
S
2
R
) is the QCD running coupling at
the renormalization scale μ
R
in the MS renormalization
scheme. In Eq. (1) and throughout the paper, the arbitrary
factorization and renormalization scales, μ
F
and μ
R
,areset
to be equal to the vector-boson mass M.
1
If V = γ
or if the vector boson V is not an on-shell particle, the
transverse-momentum cross section dσ/dq
2
T
has to be replaced by the
doubly differential distribution M
2
dσ/dM
2
dq
2
T
,whereM is the in-
variant mass of V .
The partonic cross sections d ˆσ
ab
/dq
2
T
are computable
in QCD perturbation theory as power series expansions in
α
S
(M
2
). We are interested in the perturbative contributions
that are large in the small-q
T
region (q
T
M) and, even-
tually, singular in the limit q
T
0. To explicitly recall the
perturbative structure of these enhanced terms at small q
T
,
we follow Ref. [22] and we introduce the cumulative par-
tonic cross section:
2
Q
2
0
0
dq
2
T
d ˆσ
ab
dq
2
T
q
T
,M,ˆs =M
2
/z;α
S
M
2

c=q
f
, ¯q
f
(0)
c ¯c,V
ˆ
R
V
c ¯cab
z, M/Q
0
;α
S
M
2

, (2)
where the overall normalization of the function
ˆ
R
V
is de-
fined with respect to σ
(0)
q
f
¯q
f
,V
, which is the Born level
cross section for the quark–antiquark annihilation subpro-
cess q
f
¯q
f
V (the quark flavours f and f
are equal if
V = Z,γ
). The partonic function
ˆ
R
V
has the following
perturbative expansion:
ˆ
R
V
c ¯cab
(z, M/Q
0
;α
S
)
=δ
ca
δ
¯cb
δ(1 z) +
n=1
α
S
π
n
ˆ
R
V(n)
c ¯cab
(z, M/Q
0
). (3)
The next-to-leading order (NLO) and NNLO contributions
to the cumulative cross section in Eq. (2) are determined by
the functions
ˆ
R
V(1)
and
ˆ
R
V(2)
, respectively. The small-q
T
region of the cross section d ˆσ
ab
/dq
2
T
is probed by perform-
ing the limit Q
0
M in Eq. (2). In this limit, the NLO and
NNLO functions
ˆ
R
V(1)
and
ˆ
R
V(2)
have the following be-
haviour:
ˆ
R
V(1)
c ¯cab
(z, M/Q
0
) = l
2
0
ˆ
R
(1;2)
c ¯cab
(z) +l
0
ˆ
R
(1;1)
c ¯cab
(z)
+
ˆ
R
(1;0)
c ¯cab
(z) +O
Q
2
0
/M
2
, (4)
ˆ
R
V(2)
c ¯cab
(z, M/Q
0
) = l
4
0
ˆ
R
(2;4)
c ¯cab
(z) +l
3
0
ˆ
R
(2;3)
c ¯cab
(z)
+l
2
0
ˆ
R
(2;2)
c ¯cab
(z) +l
0
ˆ
R
(2;1)
c ¯cab
(z)
+
ˆ
R
(2;0)
c ¯cab
(z) +O
Q
2
0
/M
2
, (5)
where l
0
= ln(M
2
/Q
2
0
).InEqs.(4) and (5), the powers
of the large logarithm l
0
are produced by the singular
(though integrable) behaviour of d ˆσ
ab
/dq
2
T
at small values
of q
T
. The coefficients
ˆ
R
(1;m)
(with m 2) and
ˆ
R
(2;m)
(with
2
In our notation, the subscripts c and ¯c denote a quark and an anti-
quark (or vice versa) that do not necessarily have the same flavour. The
flavour structure depends on the produced vector boson V and it is (im-
plicitly) specified by the specific form of the Born level cross section
σ
(0)
c ¯c,V
.

Eur. Phys. J. C (2012) 72:2195 Page 3 of 9
m 4) of the large logarithms are independent of Q
0
; these
coefficients depend on the partonic centre-of-mass energy ˆs
and, more precisely, they are functions of the energy frac-
tion z = M
2
/ˆs. As is well known (see also Eq. (7)), the
logarithmic coefficients
ˆ
R
(n;m)
do not depend on the spe-
cific vector boson that is produced by q ¯q annihilation and,
therefore, we have removed the explicit superscript V (i.e.,
ˆ
R
V(n;m)
=
ˆ
R
(n;m)
).
In this paper we present the result of the computation of
the cumulative cross section in Eq. (2) up to NNLO. The par-
tonic calculation is performed in analytic form by neglect-
ing terms of O(Q
2
0
/M
2
) in the limit Q
0
M. Therefore,
we determine the coefficient functions
ˆ
R
(n;m)
(z) in Eqs. (4)
and (5).
To perform our calculation, we follow the same method
as used in Ref. [22] to evaluate the transverse-momentum
cross section for Higgs boson production. The q
T
integra-
tion in Eq. (2) is thus rewritten in the following form:
Q
2
0
0
dq
2
T
d ˆσ
ab
dq
2
T
(q
T
,M,ˆs;α
S
)
+∞
0
dq
2
T
d ˆσ
ab
dq
2
T
(q
T
,M,ˆs;α
S
)
+∞
Q
2
0
dq
2
T
d ˆσ
ab
dq
2
T
(q
T
,M,ˆs;α
S
)
σ
(tot)
ab
(M, ˆs;α
S
)
Q
2
0
dq
2
T
+∞
−∞
d ˆy
d ˆσ
ab
d ˆydq
2
T
( ˆy,q
T
,M,ˆs;α
S
), (6)
where ˆσ
(tot)
ab
is the vector-boson total (i.e. integrated over
q
T
) cross section and d ˆσ
ab
/d ˆydq
2
T
is the corresponding
doubly differential cross section with respect to the trans-
verse momentum and rapidity ( ˆy is the rapidity of V in the
centre-of-mass frame of the two colliding partons a and b)
of the vector boson. The total cross section ˆσ
(tot)
ab
(M, ˆs;α
S
)
is known [49, 50] in analytic form up to NNLO (i.e., up
to O
2
S
σ
(0)
V
)). In the region of large or, more precisely,
non-vanishing values of q
T
, the differential distribution
d ˆσ
ab
/d ˆydq
2
T
is also known [5153] in analytic form up
to O
2
S
σ
(0)
V
). Using these known results and exploiting
Eq. (6), we can compute the cumulative partonic cross sec-
tion up to the NNLO. Note that q
T
>Q
0
in the last term
on the right-hand side of Eq. (6). Therefore the correspond-
ing integration of the expression d ˆσ
ab
/d ˆydq
2
T
[5153] over
ˆy and q
2
T
is finite as long as Q
0
= 0: using the explicit
expression of d ˆσ
ab
/d ˆydq
2
T
from
3
Ref. [52], we carry out
the integration in analytic from in the limit Q
0
M (i.e.,
3
We list some typos that we have found and corrected in some formulae
of Ref. [52]. In Eq. (2.12), B
qG
2
has to be replaced by B
qG
2
+ C
qG
2
,
we neglect terms of O(Q
2
0
/M
2
) on the right-hand side of
Eq. (6)). The result of our calculation
4
confirms the logarith-
mic structure in Eqs. (4) and (5), and it allows us to deter-
mine the NLO and NNLO coefficients
ˆ
R
(1;m)
(with m 2)
and
ˆ
R
(2;m)
(with m 4) of the cumulative cross section in
Eq. (2).
The results of the coefficient functions
ˆ
R
(n;m)
(z) are con-
veniently expressed in terms of transverse-momentum re-
summation coefficients. Therefore, before presenting the re-
sults, we recall how these functions are related to the pertur-
bative coefficients of the transverse-momentum resumma-
tion formula for vector-boson production [7]. This relation
also shows that from the knowledge of Eq. (5) we can fully
determine the NNLO rapidity distribution of the vector bo-
soninthesmall-q
T
region.
To present the transverse-momentum resummation for-
mula, we first decompose the partonic cross section d ˆσ
ab
/
dq
2
T
in Eq. (1) in the form d ˆσ
ab
= d ˆσ
(sing)
ab
+ d ˆσ
(reg)
ab
.The
singular component, d ˆσ
(sing)
ab
, contains all the contributions
that are enhanced at small q
T
. These contributions are
proportional to δ(q
2
T
) or to large logarithms of the type
1/q
2
T
ln
m
(M
2
/q
2
T
). The remaining component, d ˆσ
(reg)
ab
,of
the partonic cross section is regular order-by-order in α
S
as q
T
0: the integration of d ˆσ
(reg)
ab
/dq
2
T
over the range
0 q
T
Q
0
leads to a result that, at each fixed order in α
S
,
it vanishes in the limit Q
0
0. Therefore, d ˆσ
(reg)
ab
only con-
tributes to the terms of O(Q
2
0
/M
2
) on the right-hand side of
Eqs. (4) and (5).
Inserting the decomposition d ˆσ
ab
= d ˆσ
(sing)
ab
+ d ˆσ
(reg)
ab
in Eq. (1), we obtain a corresponding decomposition,
=
(sing)
+
(reg)
, of the hadronic cross section. The
transverse-momentum resummation formula for the singular
component of the q
T
cross section at fixed value of the ra-
pidity y (the rapidity is defined in the centre-of-mass frame
of the two colliding hadrons) of the vector boson reads [7, 8]
(sing)
dy dq
2
T
(y, q
T
,M,s)
=
M
2
s
c=q
f
, ¯q
f
σ
(0)
c ¯c,V
+∞
0
db
b
2
J
0
(bq
T
)S
q
(M, b)
×
a
1
,a
2
1
x
1
dz
1
z
1
1
x
2
dz
2
z
2
H
F
C
1
C
2
c ¯c;a
1
a
2
×f
a
1
/h
1
x
1
/z
1
,b
2
0
/b
2
f
a
2
/h
2
x
2
/z
2
,b
2
0
/b
2
, (7)
and C
qG
2
has to be replaced by C
qG
3
. In Eq. (A.4), two signs have to be
changed: B
qG
1
has to be replaced by B
qG
1
,andA
qG
has to be replaced
by A
qG
. In the first line of Eq. (A.10), the term C
F
(f
u
f
s
f
t
)
has to be replaced by C
A
(f
u
f
s
f
t
).
4
Some technical details related to the limit Q
0
M are illustrated in
Ref. [22].

Page 4 of 9 Eur. Phys. J. C (2012) 72:2195
where the kinematical variables x
i
(i = 1, 2) are x
1
=
e
+y
M/
s and x
2
= e
y
M/
s. The integration variable
b is the impact parameter, J
0
(bq
T
) is the zeroth-order
Bessel function, and b
0
= 2e
γ
E
(γ
E
= 0.5772 ... is the
Euler number) is a numerical coefficient. The symbol
[H
F
C
1
C
2
]
c ¯c;a
1
a
2
briefly denotes the following function of
the longitudinal-momentum fractions z
1
and z
2
:
H
DY
C
1
C
2
c ¯c;a
1
a
2
=H
DY
q
α
S
M
2

C
ca
1
z
1
;α
S
b
2
0
/b
2

×C
¯ca
2
z
2
;α
S
b
2
0
/b
2

, (8)
where H
DY
q
S
) and C
ca
(z;α
S
) (c = q
f
, ¯q
f
) are perturba-
tive functions of α
S
(see Eqs. (12)–(13)).
The quark form factor S
q
(M, b) in Eq. (7) is a process-
independent quantity [7, 8, 1113]. Its functional depen-
dence on M and b is controlled by two perturbative func-
tions, which are usually denoted as A
q
S
) and B
q
S
) (see,
e.g., Ref. [10] that uses the same notation as in Eq. (7)).
Their corresponding nth order perturbative coefficients are
A
(n)
q
and B
(n)
q
. The coefficients A
(1)
q
, B
(1)
q
, A
(2)
q
[1113] and
B
(2)
q
[14, 15] are known: their knowledge fully determines
the perturbative expression of S
q
(M, b) up to O
2
S
).
The perturbative function H
DY
q
S
) in Eq. (8)ispro-
cess dependent, since it is directly related to the production
mechanism of the vector boson through quark–antiquark an-
nihilation. However, H
DY
q
is independent of the specific type
of vector boson V (V = W
±
,Z
), and we have intro-
duced the generic superscript DY.
The partonic functions C
q
f
a
and C
¯q
f
a
in Eq. (8)arein-
stead process independent, as a consequence of the univer-
sality features of QCD collinear radiation. Owing to their
process independence, these partonic functions fulfil the fol-
lowing relations:
C
q
f
q
f
(z;α
S
)
=C
¯q
f
¯q
f
(z;α
S
)
C
qq
(z;α
S
ff
+C
qq
(z;α
S
)(1 δ
ff
), (9)
C
q
f
¯q
f
(z;α
S
)
=C
¯q
f
q
f
(z;α
S
)
C
q ¯q
(z;α
S
ff
+C
q ¯q
(z;α
S
)(1 δ
ff
), (10)
C
q
f
g
(z;α
S
) =C
¯q
f
g
(z;α
S
) C
qg
(z;α
S
), (11)
which are a consequence of charge conjugation invariance
and flavour symmetry of QCD. The dependence of the ma-
trix C
ca
on the parton labels is thus fully specified by the
ve independent quark functions C
qq
, C
qq
, C
q ¯q
, C
q ¯q
and
C
qg
on the right-hand side of Eqs. (9)–(11).
We recall that the function H
DY
q
S
), the quark func-
tions C
qa
S
) and the perturbative function B
q
S
) of the
quark form factor are not separately computable in an un-
ambiguous way. Indeed, these three functions are related by
a renormalization-group symmetry [8] that follows from the
b-space factorization structure of Eq. (7). The unambiguous
definition of these three functions thus requires the specifi-
cation
5
of a resummation scheme [8]. Note, however, that
considering the perturbative expansion
6
of Eq. (7) (i.e., the
perturbative expansion of the singular component of the q
T
cross section), the resummation-scheme dependence exactly
cancels order-by-order in α
S
.
The perturbative expansion of the quark functions
C
qa
S
) and of the vector-boson function H
DY
q
S
) is de-
fined as follows:
C
qa
(z;α
S
) = δ
qa
δ(1 z) +
n=1
α
S
π
n
C
(n)
qa
(z),
a =g,q, ¯q,q
, ¯q
, (12)
H
DY
q
S
) = 1 +
n=1
α
S
π
n
H
DY(n)
q
. (13)
The first-order coefficient function C
(1)
qg
(z) is independent of
the resummation scheme; its expression is [14, 15]
C
(1)
qg
(z) =
1
2
z(1 z). (14)
The first-order coefficients C
(1)
qq
(z), C
(1)
q ¯q
(z) and C
(1)
q ¯q
(z)
vanish,
C
(1)
qq
(z) =C
(1)
q ¯q
(z) =C
(1)
q ¯q
(z) =0, (15)
while the coefficients C
(1)
qq
(z) and H
DY(1)
q
fulfil the follow-
ing relation [14, 15, 17, 18]:
C
(1)
qq
(z) +
1
2
H
DY(1)
q
δ(1 z)
=
C
F
2

π
2
2
4
δ(1 z) +1 z
. (16)
The separate determination of C
(1)
qq
(z) and H
DY(1)
q
requires
the specification of a resummation scheme. For instance,
considering the resummation scheme in which the coeffi-
cient H
DY(1)
q
vanishes, the right-hand side of Eq. (16)gives
the value of C
(1)
qq
(z), and the corresponding value of the
5
The reader who is not interested in issues related to the specifica-
tion of a resummation scheme can simply assume that H
DY
q
S
) 1
throughout this paper. The choice H
DY
q
S
) =1 is customarily used in
most of the literature on q
T
resummation for vector-boson production.
6
The resummation-scheme dependence also cancels by consistently
expanding Eq. (7) in terms of classes of resummed (leading, next-to-
leading and so forth) logarithmic contributions [44].

Eur. Phys. J. C (2012) 72:2195 Page 5 of 9
quark form factor coefficient B
(2)
q
is explicitly computed in
Refs. [14, 15]. The computation of the second-order coeffi-
cients C
(2)
qq
, C
(2)
qq
, C
(2)
q ¯q
, C
(2)
q ¯q
, C
(2)
qg
and H
DY(2)
q
is the aim of
the calculation described in this paper.
To the purpose of presenting the NNLO results for the
cumulative cross section in Eq. (2), we also define the fol-
lowing hard-collinear coefficient function:
H
DY
q ¯qab
(z;α
S
) H
DY
q
S
)
1
0
dz
1
1
0
dz
2
δ(z z
1
z
2
)
×C
qa
(z
1
;α
S
)C
¯qb
(z
2
;α
S
), (17)
which is directly related to the coefficient function in Eq. (8).
The function H
DY
depends only on the energy fraction z,
and it arises after integration of the resummation formula
(7) over the rapidity of the vector boson. Note that H
DY
is
independent of the resummation scheme [8]. The perturba-
tive expansion of the function H
DY
directly follows from
Eqs. (12)–(13). We have
H
DY
q ¯qab
(z;α
S
) = δ
qa
δ
¯qb
δ(1 z)
+
n=1
α
S
π
n
H
DY(n)
q ¯qab
(z), (18)
where the first-order and second-order contributions are
H
DY(1)
q ¯qab
(z) = δ
qa
δ
¯qb
δ(1 z)H
DY(1)
q
+δ
qa
C
(1)
¯qb
(z) +δ
¯qb
C
(1)
qa
(z), (19)
H
DY(2)
q ¯qab
(z) = δ
qa
δ
¯qb
δ(1 z)H
DY(2)
q
+δ
qa
C
(2)
¯qb
(z)
+δ
¯qb
C
(2)
qa
(z)
+H
DY(1)
q
δ
qa
C
(1)
¯qb
(z) +δ
¯qb
C
(1)
qa
(z)
+
C
(1)
qa
C
(1)
¯qb
(z). (20)
In Eq. (20) and in the following, the symbol denotes
the convolution integral (i.e., we define (g h)(z)
1
0
dz
1
1
0
dz
2
δ(z z
1
z
2
)g(z
1
)h(z
2
)).
In the limit Q
0
M, the perturbative expansion of the
cumulative partonic cross section in Eq. (2) can directly be
related to the resummation coefficients of Eq. (7). We refer
the reader to Ref. [22] for a concise illustration of this re-
lation and to Ref. [44] for more technical details. The NLO
and NNLO functions
ˆ
R
V(1)
and
ˆ
R
V(2)
in Eqs. (4) and (5)
have the following expressions:
ˆ
R
V(1)
q ¯qab
(z, M/Q
0
)
=l
2
0
Σ
DY(1;2)
q ¯qab
(z) +l
0
Σ
DY(1;1)
q ¯qab
(z)
+H
DY(1)
q ¯qab
(z) +O
Q
2
0
/M
2
, (21)
ˆ
R
V(2)
q ¯qab
(z, M/Q
0
)
=l
4
0
Σ
DY(2;4)
q ¯qab
(z) +l
3
0
Σ
DY(2;3)
q ¯qab
(z) +l
2
0
Σ
DY(2;2)
q ¯qab
(z)
+l
0
Σ
DY(2;1)
q ¯qab
(z) 16ζ
3
Σ
DY(2;4)
q ¯qab
(z)
+
H
DY(2)
q ¯qab
(z) 4ζ
3
Σ
DY(2;3)
q ¯qab
(z)
+O
Q
2
0
/M
2
,
(22)
where we have used the same notation as in Ref. [44]. The
explicit expressions of the coefficient functions Σ
DY(n;m)
q ¯qab
(z)
in terms of the resummation coefficients are given in
Eqs. (63), (64), (66)–(69) of Ref. [44](wehavetoset
μ
R
=μ
F
=Q =M, where μ
R
F
and Q are the auxiliary
scales of Ref. [44]) and are not reported here. The coeffi-
cients H
DY(1)
q ¯qab
and H
DY(2)
q ¯qab
are exactly those in Eqs. (19)
and (20) (they are also given in Eqs. (65) and (70) of
Ref. [44]) The first-order terms Σ
DY(1;2)
q ¯qab
and Σ
DY(1;1)
q ¯qab
de-
pend on the quark form factor S
q
(M, b). The second-order
terms Σ
DY(2;m)
q ¯qab
depend on H
DY(1)
q ¯qab
and on the quark form
factor S
q
(M, b) up to O
2
S
). The numerical coefficient
ζ
3
1.202 ... (ζ
k
is the Riemann ζ -function) on the right-
hand side of Eq. (22) originates from the Bessel transfor-
mations (see, e.g., Eqs. (B.18) and (B.30) in Appendix B of
Ref. [44]).
We now document our results of the NNLO computation
of the cumulative partonic cross section. Using Eqs. (21)
and (22), the results for
ˆ
R
V(1)
and
ˆ
R
V(2)
allow us to ex-
tract Σ
DY(n;m)
and H
DY(n)
up to O
2
S
). The explicit re-
sult of the NLO function
ˆ
R
V(1)
(z) confirms the expressions
of Σ
DY(1;2)
q ¯qab
(z), Σ
DY(1;1)
q ¯qab
(z) and H
DY(1)
q ¯qab
(z), as predicted
by the q
T
resummation coefficients at O
S
). At NNLO,
the present knowledge [1115]oftheq
T
resummation co-
efficients at O
2
S
) predicts the expressions of the terms
Σ
DY(2;m)
q ¯qab
(z), with m = 1, 2, 3, 4. Our result for the NNLO
function
ˆ
R
V(2)
(z) confirms this prediction, and it allows us
to extract the explicit expression of the second-order coeffi-
cient function H
DY(2)
q ¯qab
(z).
We obtain
H
DY(2)
q ¯qq ¯q
(z)
=C
A
C
F

7ζ
3
2
101
27

1
1 z
+
+
59ζ
3
18
1535
192
+
215π
2
216
π
4
240
δ(1 z)
+
1 +z
2
1 z
Li
3
(1 z)
2
+Li
3
(z)
Li
2
(z) log(z)
2
1
2
Li
2
(z) log(1 z)
1
24
log
3
(z)
1
2
log
2
(1 z) log(z) +
1
12
π
2
log(1 z)
π
2
8

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