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 [1–8],

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 [11–18] 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(α

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. [23–39]. Calculations for vector-

boson production at the fully exclusive level with respect

to the accompanying QCD radiation have been carried out

in Refs. [21, 40–43] 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(α

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(α

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

dσ

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

(μ

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

zσ

(0)

c ¯c,V

ˆ

R

V

c ¯c←ab

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 ¯c←ab

(z, M/Q

0

;α

S

)

=δ

ca

δ

¯cb

δ(1 −z) +

∞

n=1

α

S

π

n

ˆ

R

V(n)

c ¯c←ab

(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 ¯c←ab

(z, M/Q

0

) = l

2

0

ˆ

R

(1;2)

c ¯c←ab

(z) +l

0

ˆ

R

(1;1)

c ¯c←ab

(z)

+

ˆ

R

(1;0)

c ¯c←ab

(z) +O

Q

2

0

/M

2

, (4)

ˆ

R

V(2)

c ¯c←ab

(z, M/Q

0

) = l

4

0

ˆ

R

(2;4)

c ¯c←ab

(z) +l

3

0

ˆ

R

(2;3)

c ¯c←ab

(z)

+l

2

0

ˆ

R

(2;2)

c ¯c←ab

(z) +l

0

ˆ

R

(2;1)

c ¯c←ab

(z)

+

ˆ

R

(2;0)

c ¯c←ab

(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

)

=ˆσ

(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(α

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 [51–53] in analytic form up

to O(α

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

[51–53] 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,

dσ = dσ

(sing)

+dσ

(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]

dσ

(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

(α

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, 11–13]. Its functional depen-

dence on M and b is controlled by two perturbative func-

tions, which are usually denoted as A

q

(α

S

) and B

q

(α

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

[11–13] and

B

(2)

q

[14, 15] are known: their knowledge fully determines

the perturbative expression of S

q

(M, b) up to O(α

2

S

).

The perturbative function H

DY

q

(α

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,γ

∗

), 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

)δ

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

)δ

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

ﬁve 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

(α

S

), the quark func-

tions C

qa

(α

S

) and the perturbative function B

q

(α

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

(α

S

) and of the vector-boson function H

DY

q

(α

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

(α

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

(α

S

) ≡ 1

throughout this paper. The choice H

DY

q

(α

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 ¯q←ab

(z;α

S

) ≡ H

DY

q

(α

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 ¯q←ab

(z;α

S

) = δ

qa

δ

¯qb

δ(1 −z)

+

∞

n=1

α

S

π

n

H

DY(n)

q ¯q←ab

(z), (18)

where the ﬁrst-order and second-order contributions are

H

DY(1)

q ¯q←ab

(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 ¯q←ab

(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 ¯q←ab

(z, M/Q

0

)

=l

2

0

Σ

DY(1;2)

q ¯q←ab

(z) +l

0

Σ

DY(1;1)

q ¯q←ab

(z)

+H

DY(1)

q ¯q←ab

(z) +O

Q

2

0

/M

2

, (21)

ˆ

R

V(2)

q ¯q←ab

(z, M/Q

0

)

=l

4

0

Σ

DY(2;4)

q ¯q←ab

(z) +l

3

0

Σ

DY(2;3)

q ¯q←ab

(z) +l

2

0

Σ

DY(2;2)

q ¯q←ab

(z)

+l

0

Σ

DY(2;1)

q ¯q←ab

(z) −16ζ

3

Σ

DY(2;4)

q ¯q←ab

(z)

+

H

DY(2)

q ¯q←ab

(z) −4ζ

3

Σ

DY(2;3)

q ¯q←ab

(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 ¯q←ab

(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

,μ

F

and Q are the auxiliary

scales of Ref. [44]) and are not reported here. The coefﬁ-

cients H

DY(1)

q ¯q←ab

and H

DY(2)

q ¯q←ab

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 ¯q←ab

and Σ

DY(1;1)

q ¯q←ab

de-

pend on the quark form factor S

q

(M, b). The second-order

terms Σ

DY(2;m)

q ¯q←ab

depend on H

DY(1)

q ¯q←ab

and on the quark form

factor S

q

(M, b) up to O(α

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(α

2

S

). The explicit re-

sult of the NLO function

ˆ

R

V(1)

(z) conﬁrms the expressions

of Σ

DY(1;2)

q ¯q←ab

(z), Σ

DY(1;1)

q ¯q←ab

(z) and H

DY(1)

q ¯q←ab

(z), as predicted

by the q

T

resummation coefﬁcients at O(α

S

). At NNLO,

the present knowledge [11–15]oftheq

T

resummation co-

efﬁcients at O(α

2

S

) predicts the expressions of the terms

Σ

DY(2;m)

q ¯q←ab

(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 ¯q←ab

(z).

We obtain

H

DY(2)

q ¯q←q ¯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