#### Nested soft-collinear subtractions in NNLO QCD computations

Eur. Phys. J. C
Nested soft-collinear subtractions in NNLO QCD computations
Fabrizio Caola 1 2
Kirill Melnikov 0
Raoul Röntsch 0
0 Institute for Theoretical Particle Physics , KIT, Karlsruhe , Germany
1 IPPP, Durham University , Durham , UK
2 CERN, Theoretical Physics Department , Geneva , Switzerland
We discuss a modification of the next-to-nextto-leading order (NNLO) subtraction scheme based on the residue-improved sector decomposition that reduces the number of double-real emission sectors from five to four. In particular, a sector where energies and angles of unresolved particles vanish in a correlated fashion is redundant and can be discarded. This simple observation allows us to formulate a transparent iterative subtraction procedure for double-real emission contributions, to demonstrate the cancellation of soft and collinear singularities in an explicit and (almost) process-independent way and to write the result of a NNLO calculation in terms of quantities that can be computed in four space-time dimensions. We illustrate this procedure explicitly in the simple case of O(αs2) gluonic corrections to the Drell-Yan process of qq¯ annihilation into a lepton pair. We show that this framework leads to fast and numerically stable computation of QCD corrections.
1 Introduction . . . . . . . . . . . . . . . . . . . . . .
2 Preliminary remarks . . . . . . . . . . . . . . . . . .
3 The NLO calculation . . . . . . . . . . . . . . . . . .
4 The NNLO computation: general considerations . . .
5 The NNLO computation: ultraviolet and PDF
6 The NNLO computation: double-virtual corrections .
7 The NNLO computation: real-virtual corrections . . .
8 The NNLO computation: double-real emission . . . .
8.4.1 The double-collinear partitions . . . . . . .
8.4.2 The triple-collinear partition 14, 15 . . . .
1 Introduction
One of the most important recent advances in
perturbative QCD was the discovery of practical ways to
perform fully differential next-to-next-to-leading order (NNLO)
QCD computations for hadron collider processes. These
methods, which include antenna [1–9], residue-improved
sector-decomposition [10–12] (see also [13]) and projection
to Born [14] subtraction schemes, as well as q⊥ [15, 16] and
N -jettiness [17, 18] slicing methods, were used to perform an
impressive number of NNLO QCD computations relevant for
LHC phenomenology [14, 15, 18–62].1
1 We also note that recently the CoLoRFulNNLO scheme was fully
worked out for e+e− colliders [63,64].
However, in spite of these remarkable successes, it is
important to recognize that existing implementations of
subtraction schemes are complex, not transparent, and require
significant CPU time to produce stable results. On the other
hand, slicing methods, while conceptually simple, have to be
carefully controlled to avoid dependence of the final result on
the slicing parameter. Given these shortcomings, it is
important to study whether improvements to existing methods are
possible. In the context of the N -jettiness slicing method,
there has been recent progress toward a better control of the
soft-collinear region [65,66].
In this paper we study the residue-improved subtraction
scheme introduced in Refs. [10–12]. This scheme is
interesting because it is the only existing framework for NNLO
QCD computations that is fully local in multi-particle phase
space. As such, it should demonstrate exemplary
numerical stability, at least in theory. Although this scheme is well
understood and was applied to a large number of non-trivial
problems, we will argue in this paper that certain aspects
of it are redundant. Interestingly, once this redundancy is
recognized and removed, the residue-improved subtraction
scheme becomes very transparent and physical. In addition,
the technical simplifications that occur become so significant
that the cancellation of the divergent terms can be
demonstrated independently of the hard matrix element and almost
entirely analytically, and the final finite result for the NNLO
contribution to (in principle) any process can be written in
a compact form in terms of generic four-dimensional matrix
elements.2
Although the improvements that we just described hold
true for an arbitrary complicated process, in this paper, for
the sake of clarity, we restrict our discussion to the
production of a colorless final state in qq¯ annihilation. This allows us
to discuss all the relevant conceptual and technical aspects of
the computational framework, without cluttering the notation
and limiting the bookkeeping to a minimum. The
generalization of the framework described here to arbitrary processes
is – at least conceptually – straightforward.
Admittedly, compared to NNLO QCD problems studied
recently, the production of a colorless final state in qq¯
annihilation is a very simple process, which has been discussed
in the literature many times. However, we believe that the
simplicity of our approach and the structures that emerge
justify revisiting it one more time. Moreover, thanks to the
simplicity of this process, we will be able to describe our
approach in detail and demonstrate many intermediate steps
of the calculation. Hopefully this will allow us to make the
rather technical subject of NNLO subtractions accessible to
a broader part of the particle physics community.
2 A different improvement of the original scheme [10,11], which also
allows one to deal with four-dimensional matrix elements was presented
in Ref. [12].
The paper is organized as follows. We begin with
preliminary remarks in Sect. 2, where we also precisely define the
problem that we plan to address. In Sect. 3 we discuss the
next-to-leading order (NLO) QCD computation as a
prototype of the following NNLO QCD construction. In Sect. 4, we
describe how the NLO computation generalizes to the NNLO
case. We elaborate on this in Sect. 5, where we discuss
ultraviolet and collinear renormalization, and in Sects. 6, 7 and 8,
where we study two-loop virtual corrections, one-loop
corrections to single-real emission process, and the double-real
emission contributions, respectively. In Sect. 9, we combine
the different contributions and present the final result for the
NNLO QCD corrections to color singlet production in qq¯
annihilation. In Sect. 10, we show some numerical results
and a comparison with earlier analytic calculations. We
conclude in Sect. 11. A collection of useful formulas is provided
in the appendices.
2 Preliminary remarks
We consider the production of a colorless final state V in the
collision of two protons
We are interested in computing the differential cross section
for the process in Eq. (2.1)
where dσˆi j is the finite partonic scattering cross section, fi, j
are parton distribution functions and x1,2 are momenta
fractions of the incoming hadrons that are carried to a hard
collision by partons i and j , respectively. The dependence on the
renormalization and factorization scales and all other
parameters of the process is understood. The finite partonic
scattering cross section is obtained after the renormalization of
the strong coupling constant removes all ultraviolet
divergences, all soft and final state collinear divergences cancel in
the sum of cross sections with different partonic
multiplicities, and the initial state collinear divergences are subtracted
by redefining parton distribution functions.
Since the process under consideration is driven by a
conserved current that is independent of αs, the ultraviolet
renormalization reduces to the following (MS) redefinition of the
strong coupling constant:
1 −
where S = (4π ) e− γE , γE ≈ 0.577216 is the Euler–
Mascheroni constant, = (4 − d)/2 and
because of the ultraviolet and collinear renormalizations. The
latter are obtained with the procedure just described and read
1 ⊗ dσˆ NLO + dσˆ NLO ⊗ 1 −
1 ⊗ dσˆ LO ⊗ 1 + 2 ⊗ dσˆ LO + dσˆ LO ⊗ 2 ,
is the leading-order (LO) QCD β-function.
Collinear divergences associated with initial state QCD
radiation are removed by a redefinition of parton
distributions. In the MS scheme, this amounts to the replacement
+ O(αs3) ⊗ f j (μ),
where ⊗ stands for the convolution
g(z) = [ f1 ⊗ f2](z) =
(0,1) are the Altarelli–Parisi splitting functions.
and Pˆi j
As we already mentioned, we focus on gluonic corrections
to the qq¯ annihilation channel,
q + q¯ → V + ng.
This allows us to present all the features of the framework
while limiting the bookkeeping to a minimum and, therefore,
to keep the discussion relatively concise. All other partonic
channels relevant for the Drell–Yan process can be obtained
by a simple generalization of what we will describe.
The collinear-renormalized partonic cross section for qq¯
annihilation into a vector boson is expanded in series of αs.
We write
dσˆ ≡ dσˆqq¯ = dσˆ LO + dσˆ NLO + dσˆ NNLO,
dσˆ NLO = dσ V + dσ R
Pˆq(q0) ⊗ dσˆ LO + dσˆ LO ⊗ Pˆq(q0) ,
Various contributions in Eq. (2.9) refer to virtual and real
corrections, as well as to contributions to cross sections that arise
1 =
2 =
− 2
and the relevant splitting functions are provided in
Appendix A.
The cross section dσˆ (N)NLO is finite but all the
individual contributions in Eq. (2.9) are divergent. The well-known
problem is that these divergences are explicit in some of the
terms and implicit in the others. Indeed, soft and collinear
divergences appear as explicit 1/ poles in virtual corrections
but they only become evident in real corrections once
integration over gluon momenta is performed. However, since
we would like to keep the kinematics of all the final state
particles intact, we cannot integrate over momenta of any of
the final state particles if it is resolved. It is this point that
makes extraction of implicit singularities complicated and
requires us to devise a procedure to do it.
Depending on how these implicit singularities are
extracted, it may or may not be straightforward to recognize how
they combine and cancel, once all contributions to the
physical cross section are put together. At NNLO, this was done
for the antenna subtraction scheme and, in a less transparent
way, for the residue-improved sector decomposition. One
thing we would like to do, therefore, is to combine the
individual terms that contribute to partonic cross sections, and
cancel all the 1/ divergences explicitly, without any
reference to the matrix elements that contribute to the different
terms in Eq. (2.9). In the next section we show how to do that
at next-to-leading order in the perturbative expansion for the
Drell–Yan process. This will allow us to set up the formalism
and the notation that will be used for the NNLO analysis of
Sects. 4–8.
3 The NLO calculation We will illustrate our approach by studying the production of a lepton pair in quark–antiquark annihilation at next-toleading order in perturbative QCD. We note that, at this order,
the method that we would like to describe is identical to the
FKS subtraction scheme introduced in Refs. [67, 68].
However, we formulate the FKS method in a way that makes its
extension to next-to-next-to-leading order as straightforward
as possible.3 One point that we found helpful, especially for
bookkeeping, was to introduce soft and collinear subtraction
operators, and we show how to use them in the NLO
computation below.
We are interested in the calculation of the finite partonic
cross section dσˆ NLO defined in Eq. (2.9). It receives
contributions from the virtual and real corrections and the collinear
subtraction term. We will start the discussion with the
realemission contribution. It refers to the process
q( p1) + q¯ ( p2) → V + g( p4),
where V is a generic notation for all colorless particles in
the final state. We write the cross section for the process in
Eq. (3.1) as
[dg4]FL M (1, 2, 4),
where s is the partonic center-of-mass energy,
FL M (1, 2, 4) = dLipsV |M(1, 2, 4, V )|2 Fkin(1, 2, 4, V ).
(3.4)
In Eq. (3.4), dLipsV is the Lorentz-invariant phase space
for colorless particles, including the momentum-conserving
δ(d)( p1 + p2 − p4 − pV ), M(1, 2, 4, V ) is the matrix element
for the process in Eq. (3.1) and Fkin(1, 2, 4, V ) is an
(infrared safe) observable that depends on kinematic variables of
all particles in the process. Also, Emax is an arbitrary auxiliary
parameter that has to be large enough to accommodate all
possible kinematic configurations for qq¯ → V + g. The
need to introduce such a parameter is a consequence of our
construction, as explained in detail below.
We would like to isolate and extract soft and collinear
singularities that appear when the integration over [dg4] in
Eq. (3.2) is attempted. To this end, we introduce two operators
that define soft and collinear projections
Si A = Elii→m0 A,
3 For earlier efforts, see Ref. [69].
where ρi j = 1 − ni · n j and ni is a unit vector that describes
the direction of the momentum of the i th particle in (d −
1)dimensional space. By definition, operators in Eq. (3.5) act on
everything that appears to the right of them. The limit
operations, on the right hand side of Eq. (3.5), are to be understood
in the sense of extracting the most singular contribution
provided that limits in the conventional sense do not exist. We
will also use the averaging sign ... to represent integration
over momenta of final state particles. This integration is
supposed to be performed in the center-of-mass frame of
incoming partons. We emphasize that this remark is important since
our construction of the subtraction terms is frame-dependent
and not Lorentz-invariant.
We rewrite Eq. (3.2) in the following way:
[dg4]FL M (1, 2, 4)
=
= S4 FL M (1, 2, 4) + (C41 + C42)(I − S4)FL M (1, 2, 4)
Oˆ NLO FL M (1, 2, 4) ,
where I is the identity operator and Oˆ NLO is a short-hand
notation for a combination of soft and collinear projection
operators
Oˆ NLO = (I − C41 − C42)(I − S4).
Note that in Eq. (3.6) soft and collinear projection operators
act on FL M (1, 2, 4), which, according to Eq. (3.4), contains
the energy-momentum conserving δ-function; we stress that
the soft and collinear limits must be taken in that δ-function
as well.
The reason for rewriting dσ R as in Eq. (3.6) is that the last
term there is finite, thanks to the nested structure of
subtraction terms. This term can, therefore, be integrated
numerically in four dimensions. We emphasize again that the
subtraction terms, as formulated here, are not Lorentz-invariant.
This means that all the three terms in Eq. (3.6) should be
computed in the same reference frame that, as already
mentioned, is taken to be the center-of-mass reference frame of
the colliding partons.
We now consider the remaining two terms in Eq. (3.6).
Their common feature is either complete or partial
decoupling of the gluon g4 from the matrix element thanks to the
fact that they contain either soft or collinear projection
operators. Hence, those terms can be re-written in such a way that
all singularities are extracted and canceled, without
specifying the matrix elements for the hard process.
To see this explicitly, consider first two terms in Eq. (3.6)
and write them as
(I −C41−C42)S4 FL M (1, 2, 4) + (C41+C42)FL M (1, 2, 4) .
(3.8)
It is easy to see that the first term in Eq. (3.8) vanishes.4
Indeed, in the limit when the gluon g4 becomes soft, we find
S4 FL M (1, 2, 4) =
FL M (1, 2),
where gs,b is the bare QCD coupling, CF = 4/3 is the QCD
color factor, and FL M (1, 2) is closely related to the LO cross
section
FL M (1, 2) = 2s · dσˆ LO
dLipsV |M(1, 2, V )|2 Fkin(1, 2, V ).
ρ12 1 1
ρ14ρ24 = ρ14 + ρ24 ;
this implies (I − C41 − C42)S4 FL M (1, 2, 4) = 0.
Hence, the only term that we need to consider in Eq. (3.8)
is the collinear subtraction
(C41 + C42)FL M (1, 2, 4) .
We will consider the action of the operator C41 on FL M (1, 2, 4)
and then infer the result for the operator C42. First, we find
the collinear limit
To simplify C41 FL M (1, 2, 4) , we integrate over the
emission angle of the gluon g4, rewrite the integration over
its energy as an integral over z and express gs,b in terms
of the renormalized coupling αs(μ). After straightforward
manipulations we find
C41 FL M (1, 2, 4) = − [αs]
(2E1)−2
where zmin = 1− Emax/E1 and we introduced the short-hand
notation
(1 − )
We note that in Eq. (3.16) integration over z leads to
divergences caused by the soft z → 1 singularity in the splitting
functions. These singularities are regulated dimensionally in
Eq. (3.16). On the other hand, this equation has the form of
a convolution of a hard matrix element with a splitting
function, so we expect that divergences present there will cancel
against the collinear subtraction terms. However, collinear
subtractions employ regularization of soft singularities that
is based on the plus-prescription. Our goal, therefore, is to
rewrite Eq. (3.16) in such a way that all soft singularities are
regulated by the plus-prescription; once this is done,
combining this contribution with virtual corrections and collinear
subtractions becomes straightforward.
To simplify the notation, we denote FL M (z · 1, 2)/z =
G(z) and split Pqq (z) into a piece that is singular at z = 1
and a regular piece
We also note that we can extend the integral over z in
Eq. (3.16) to z = 0 since if E4 > Emax, FL M (z · 1, 2)
will vanish because there is not enough energy to produce
the final state. We will use this fact frequently in our NNLO
analysis. We write
We note that a new variable z = 1 − E4/E1 is introduced
in Eq. (3.14). The notation FL M (z · 1, 2) implies that in the
computation of FL M (1, 2), cf. Eq. (3.10), the momentum
p1 is replaced with zp1 everywhere, including the
energymomentum conserving δ-function. We also used Pqq (z) to
denote the splitting function
Pqq (z) = CF
4 This feature is particular to the process under consideration.
= −
+ (1 − z)−2 Pqreqg(z)G(z) .
The expression in Eq. (3.19) can be expanded in a power
series in to the required order and the plus-distributions
can be used to write the result in a compact form. Indeed, the
following equation holds:
∞ (−1)n(2 )n
n=0
where Dn(z) = [lnn(1 − z)/(1 − z)]+. It is now
straightforward to rewrite Eq. (3.16) in such a way that all soft, z → 1,
singularities are regulated using the plus-prescription. We
use the fact that we are in the center-of-mass frame of the
incoming qq¯ pair, so that 2E1 = 2E2 = √s. We find
C41 FL M (1, 2, 4) = −
× ⎣ −
FL M (1, 2)
The splitting function in Eq. (3.21) reads5
where Pˆq(q0)(z) is the LO Altarelli–Parisi splitting kernel, see
Eq. (A.1), and Pq(q),R is defined as
We note that in Eq. (3.24) all singularities of the real-emission
contribution are explicit and a straightforward expansion in
is, in principle, possible. However, such an expansion is
inconvenient since it involves higher-order terms of
lowermultiplicity amplitude. To avoid these contributions, it is
useful to combine Eq. (3.24) with virtual corrections and
collinear counterterms.
For the virtual corrections, all divergent parts can be
separated using Catani’s representation of renormalized one-loop
scattering amplitudes [70]. We obtain
2s · dσ V = FLV (1, 2)
dLipsV 2Re
M(1, 2)M1∗−loop(1, 2)
× Fkin(1, 2, V )
s−
× FL M (1, 2) + FLfiVn (1, 2) ,
where FLfiVn (1, 2) is free of singularities and μ-independent.
To arrive at the final result, we add virtual, real and
collinear subtraction terms, cf. Eq. (2.9), and obtain
+ FLfiVn (1, 2) −
dz Pˆq(q0)(z)
The result for C42 FL M (1, 2, 4) is obtained by a simple
replacement FL M (z · 1, 2) → FL M (1, z · 2) in Eq. (3.21).
Putting everything together, we find the following result for
the real-emission cross section:
S(1, 2) = 2s−
+ Oˆ NLO FL M (1, 2, 4)
FL M (1, 2)
5 The O( 2) contribution to Pqq,R, relevant for NNLO contributions,
is reported in Appendix A.
and the extra z terms in the denominator of the last line
of Eq. (3.26) appear because of the z-dependent flux
factor in the collinear counterterm cross section. Taking the
limit → 0 in Eq. (3.26), we find the final formula for
the NLO QCD contribution to the scattering cross section
for q( p1) + q¯ ( p2) → V + X . It reads
FLfiVn (1, 2) + αs(μ) 2 π 2CF FL M (1, 2)
2π 3
+ Oˆ NLO FL M (1, 2, 4)
5 The NNLO computation: ultraviolet and PDF renormalization
In this section we study the contributions to dσˆ NNLO coming
from the ultraviolet and the collinear renormalization,
beginning with the former. As mentioned previously, because the
process qq¯ → V is driven by a conserved current which is
independent of αs, the ultraviolet renormalization
contribution is very simple. Combining Eq. (2.10) and the first two
lines of Eq. (3.26), it is straightforward to obtain
Oˆ NLO FL M (1, 2, 4) + FLfiVn (1, 2)
(1 − 2 )
(1 − 2 )
× 1 − cos(π ) 2(1 − )
× FL M (1, 2) −
Pˆq(q0)(z) − Pq(q),R (z)
It follows that the NLO cross section is computed as a sum of
low-multiplicity terms, including those where FL M (z · 1, 2)
or FL M (1, z · 2) are convoluted with particular splitting
functions, and the subtracted real-emission term described by
ONLO FL M (1, 2, 4) . We note that terms that involve matrix
elements of different multiplicities, as well as terms that
involve different types of convolutions, are separately finite.
We will use this observation at NNLO, to check for the
cancellation of 1/ divergences in an efficient way.
4 The NNLO computation: general considerations
We would like to extend the above framework to NNLO in
QCD. Apart from the UV and collinear renormalization
discussed in Sect. 2, the NNLO cross section receives
contributions from two-loop virtual corrections to qq¯ → V (double
virtual), from one-loop corrections to the process with an
additional gluon in the final state qq¯ → V + g (real-virtual),
and from the tree-level process with two additional gluons in
the final state qq¯ → V + gg (double real).
The double-virtual corrections can be dealt with in a
straightforward way since all the divergences of the two-loop
matrix elements are explicit, universal and well understood
[70]. For our purposes, we only need to write them in a
convenient form. The real-virtual corrections are more tricky,
but do not require new conceptual developments. Indeed, the
kinematic regions that lead to singularities in one-loop
amplitudes with an additional gluon in the final state are identical to
those appearing in the NLO computations and, furthermore,
the limiting behavior of one-loop amplitudes with one
additional parton is well understood [71–73]. Hence, we can deal
with the real-virtual contribution by a simple generalization
of what we did at NLO.
The qualitatively new element of the NNLO computation
is the double-real emission process qq¯ → V + gg. The
methods that are applicable at next-to-leading order need to
be adjusted to become useful in the NNLO case. However,
somewhat surprisingly, these adjustments appear to be
relatively minor although, of course, the bookkeeping becomes
much more complex.
We begin by discussing the ultraviolet and PDF
renormalizations at NNLO, as well as the double-virtual and
the real-virtual contributions. We then move on to a more
involved analysis of the double-real emission contribution to
dσˆ NNLO.
⊗ Pˆq(q0)
dz dz¯ Pˆq(q0)(z)
× Pˆq(q0)(z¯) +
We proceed to the collinear subtraction. Rewriting Eq. (2.10)
to make the convolutions explicit, we obtain
Terms that involve convolutions of the various splitting
functions with FL M are, in principle, straightforward to deal with.
These terms are fully regulated and can be expanded in
powers of without further ado. In practice, we combine those
terms with other contributions in order to cancel the
singularities prior to integration over z.
It is less straightforward to rewrite Pˆ ⊗ dσ R+V and
dσ R+V ⊗ Pˆ in a form convenient for combining them with
other contributions to dσˆ NNLO. We focus on Pˆ ⊗dσ R+V, and
consider the effect of the convolution on the first two lines of
Eq. (3.26).
First, we consider the term proportional to S(1, 2)
FL M (1, 2) in Eq. (3.26). It receives contributions from the
divergent part of virtual corrections and from the soft
regularization of collinear subtraction terms. These terms scale
differently with z. The virtual correction depends on s−
which becomes (zs)− once the momentum p1 is changed to
zp1. On the other hand, the soft remainders of the collinear
subtracted terms scale as Ei−2 , with i = 1, 2. Hence, in the
calculation of dσ R+V(z · 1, 2), the corresponding
contribution scales with z either as ∼ z−2 or as ∼ 1. Therefore, we
have to compute
dz Pˆq(q0)(z)S(z · 1, 2) ×
S(z · 1, 2) = s−
= s− CF 23 π 2 + ln2 z +
For future convenience, we rewrite Eq. (5.3) as
1
dz Pˆq(q0)(z)S(z · 1, 2) ×
= s−
where the splitting function Pqq,NLOCV (z) is defined in
Eq. (A.6).
The other two terms that we need involve convolutions of
splitting function Pqq,R with FL M (1, 2), cf. Eq. (3.26). The
first term can be written as a double convolution
0
1
dz Pqq ⊗ Pqq NLOCV (z)
where the splitting function Pqq ⊗ Pqq NLOCV is defined in
Eq. (A.7). The second term is the left–right convolution,
s−
dx Pˆq(q0)(x )
Combining all these terms we find
2s · Pˆq(q0) ⊗ dσqRq+V + dσ R+V ⊗ Pˆq(q0) = − [αs]s−
dz dz¯ Pqq,R(z) FL M (z · 1, z¯ · 2) z+z FL M (z¯ · 1, z · 2) Pˆq(q0)(z¯)
¯
dz Pˆq(q0)(z) FLfiVn (z · 1, 2) +z FLfiVn (1, z · 2)
Each term that appears on the right hand side of Eq. (5.8)
is regularized and can be expanded in powers of
independently of the other terms in that equation.
6 The NNLO computation: double-virtual corrections
The calculation of double-virtual corrections proceeds in
the standard way. We start with the scattering amplitude for
qq¯ → V and write it as an expansion in the renormalized
strong coupling constant
M = Mtree +
M1−loop
where the dependence of the scattering amplitudes on the
momenta of the external particles is suppressed. By analogy
with what was done in Sect. 3, we write
dLips12→V 2Re{M2−loopMt∗ree}
+ |M1−loop|2
Fkin(1, 2, V ) +
≡ FLV V (1, 2) ,
where in the second line we removed the renormalization
contribution that is already accounted for in Eq. (5.1).
FLV V can be directly expanded in a Laurent series in
and integrated over the phase space of the final state
particles since this integration does not introduce soft or collinear
divergences. Before doing that, it is convenient to
explicitly extract the 1/ poles from FLV V . Soft and collinear
singularities of a generic scattering amplitude are given in
Ref. [70]. Using these results, we rewrite FLV V as
FLV V (1, 2) =
(1 − )
(1 − )
FLfiVn (1, 2)
+ FLfiVn 2 (1, 2) + FLfiVn V (1, 2) .
2(1 − )
FL M (1, 2)
11 83 π 2
− 12 3 − 18 2 + 12 2
7 The NNLO computation: real-virtual corrections
The kinematics of the real-virtual corrections is identical to
the NLO case described in Sect. 3. The procedure for making
these corrections expandable in is, therefore, the same. We
write
2s · dσ RV ≡ FL RV (1, 2, 4) = S4 FL RV (1, 2, 4)
I − S4 C41 + C42 FL RV (1, 2, 4)
We remind the reader that, according to our notation,
softand collinear-projection operators in Eq. (7.1) do not act on
the phase space of the gluon g4. It remains to compute the
corresponding limits in Eq. (7.1) and to rewrite them, where
appropriate, as convolutions of the hard matrix elements with
splitting functions.
The soft limit of a general one-loop amplitude is discussed
in Refs. [71,72]. Adapting those results to our case, we find
E42 S4 FL RV (1, 2, 4) = 2CF gs2,b ρ1ρ41ρ224 FLV (1, 2)
1 5(1 − ) 3(1 + )
− CA[αs] 2 2(1 − 2 ) (1 + 2 )
We need to integrate Eq. (7.2) over the phase space of the
gluon g4. This can easily be done, with the result
The sum of the last two terms in Eq. (6.3) is a finite remainder
of the O(αs2) contribution to the virtual corrections once its
divergent part is written in a form suggested in Ref. [70].
More specifically, FLfiVn 2 is the finite remainder of the
oneloop amplitude squared, while FLfiVn V is the genuine two-loop
finite remainder. Note that, contrary to FLfiVn (1, 2) and FLfiVn 2 ,
FLfiVn V is scale-dependent; the scale-dependent contribution
reads
44
FLfiVn V (μ2, s) − FLfiVn V (s, s) = 3 CFCA log
As follows from Eq. (6.3), the singularities of the
doublevirtual corrections are proportional to the leading-order
contribution FL M (1, 2) and to the finite part of the virtual
corrections to the NLO cross section FLfiVn (1, 2), given in Sect. 3.
Our goal is to rewrite the real-virtual and the double-real
emission contributions in a way that allows explicit
cancellation of the divergences in Eq. (6.3) without specifying hard
matrix elements.
S4 FL RV (1, 2, 4) = 2CF[αs]
2(1 − 2 ) (1 + 2 )
FLV (1, 2)
FL M (1, 2) .
Note that in order to obtain a meaningful result, it is
crucial that the integration over gluon energy is bounded from
above; as we already explained in the context of the NLO
computations, we use a parameter Emax for this purpose, cf.
Eq. (3.3).
The second term that we need to consider is the
softregulated collinear subtraction term
I − S4 C41 + C42 FL RV (1, 2, 4) .
We will only discuss the collinear projection operator C41;
the contribution of C42 is obtained along the same lines.
Collinear limits of loop amplitudes were studied in Refs. [71,
73]. Using these results and adapting them to our case, we
find
2
C41 FL RV (1, 2, 4) = Eg42sρ,b41 (1 − z) Pqq (z)
2CF FLV (1, 2) − 2CACF [α2s] 2−
Integrating over emission angles of the gluon g4 and rewriting
the result through plus-distributions, following the discussion
in Sect. 3, we obtain a convenient representation for the
softregulated collinear subtraction term. It reads
where the two splitting functions are defined in Eqs. (A.10)
and (A.11). We replace FL M (z · 1, 2) with FL M (1, z · 2) in
Eq. (7.7) to obtain the result for I − S4 C42 FL RV (1, 2, 4) .
We are now in a position to present the final result for the
real-virtual part. Collecting results shown in Eqs. (7.3) and
(7.7), we obtain
FLV (1, 2, 4)
= ONLO FLV (1, 2, 4) + 2CF[αs]s−
× FLV (1, 2) − CA [αs]s−
4 4
−2
FL M (1, 2)
I − S4 C41 FL RV (1, 2, 4)
1
We stress that each term on the r.h.s. in Eq. (7.8) can be
expanded in powers of ; we will make full use of this to
cancel 1/ singularities when combining Eq. (7.8) with other
contributions to dσˆ NNLO. To this end, it is useful to make all
the 1/ singularities explicit in Eq. (7.8) by writing FLV (1, 2)
in terms of FLfiVn (1, 2) and FL M (1, 2), cf. Eq. (3.25), and
FLV (1, 2, 4) as
(s1−4 + s2−4 )
× FL M (1, 2, 4) + FLfiVn (1, 2, 4).
We used si j = 2 pi · p j and denoted a finite remainder which
does not depend on the scale μ by FLfiVn (1, 2, 4).
8 The NNLO computation: double-real emission
In this section, we discuss the double-real emission
contributions to dσˆ NNLO. Similar to the NLO case, we need to
determine all kinematic configurations that may lead to
singularities and understand the factorization of the matrix element
that describes qq¯ → V + gg in these regions. In the case
of the two-gluon emission in qq¯ annihilation into a
colorless final state, the singular regions correspond to soft and/or
collinear emissions, with collinear directions being the
collision axis and the direction of either one of the two gluons.
The difficulty of the NNLO case is that each of these
kinematic limits can be approached in several different ways
and all of them have to be identified and regularized
separately. To do so, we introduce several soft and collinear
projection operators. They are defined as follows. Consider
a quantity A that depends on the four-momenta of some or
all of the particles in the process. The action of operators
SS, S4,5, CC1,2, C14, C15, C24, C25, C45 on A is described by
8.1 General considerations
the following formulas:
CCi A = ρ4i ,lρim5i →0 A, with non vanishing
To make use of these projection operators, we need to rewrite
the two-gluon phase space in a way that allows these limits
to be taken. It is convenient to order gluon energies as the
first step. We write
[dg4][dg5]θ (E4 − E5)|M(1, 2, 4, 5, V )|2
× dLips12−45→V Fkin(1, 2, 4, 5, V )
= FL M (1, 2, 4, 5) ,
where, as in Sect. 3, dLips is the phase space for the final
state V , including momentum-conserving delta-function.
The gluon phase space elements [dg4,5] are defined as in
Eq. (3.3)
As we already saw when considering the real-virtual
contribution, it is necessary to introduce the θ -function in order
to define integrals over gluon energies in the soft limits. We
work in the center-of-mass frame of the colliding quark and
antiquark; it is in this frame that all the energies in the above
formulas are defined.
We recall that, similar to the NLO case, soft and collinear
projection operators act on everything that appears to the
right of them. However, in the NNLO case we will find it
convenient, occasionally, to also simplify the phase space in
the collinear limits. If so, we will explicitly show the
corresponding part of the phase space to the right of the operator.
For example
O[dg4]FL M (1, 2, 4, 5)
[dg5]O[dg4]FL M (1, 2, 4, 5),
implies that the operator O acts on FL M (1, 2, 4, 5) and on
the phase space element [dg4].
We begin by extracting soft singularities from the
doublereal process, largely repeating what we have done at
next-toleading order.6 We write
FL M (1, 2, 4, 5)
+ (I − S5)(I − SS)FL M (1, 2, 4, 5) .
In Eq. (8.5), the last term is soft-regulated, in the second term
gluon g5 is soft and soft singularities associated with g4 are
regulated, and in the first term both g4 and g5 are soft.
All of these terms contain collinear singularities.
Regulating these is more difficult since collinear singularities
overlap. For this reason, we first need to split the phase space
into mutually exclusive partitions that, ideally, select a
single kinematic configuration that leads to singularities. We
write
1 =
where the label i runs through the elements of the set
i ∈ {14, 15; 24, 25; 14, 25; 15, 24}. We refer to the first
two elements of the set as triple-collinear and to the last two
elements of the set as double-collinear partitions. We
construct the weights wi in such a way that when they
multiply the matrix element M(1, 2, 4, 5) squared, the resulting
expression is only singular in a well-defined subset of limits.
For example, in the partition 14, 15 collinear singularities in
w14,15|M|2 only occur when gluons 4 and/or 5 are emitted
along the direction of the incoming quark q( p1) or when
their momenta are parallel to each other. Similarly, in the
partition 24, 25, the singularities occur when momenta of g4
and/or g5 are parallel to p2 or to each other. In the partitions
14, 25 and 15, 24, singularities only occur when momenta of
g4 and g5 are collinear to p1 and p2 or p2 and p1,
respectively. Apart from these requirements, the specific form of
wi is arbitrary. Weights used in this calculation are given in
Appendix B. In the following, we assume that weights do not
depend on gluon energies and, therefore, commute with soft
operators.
The triple-collinear partitions require further splitting
to factorize all the relevant singularities. The purpose of
this splitting is to establish a well-defined hierarchy for
the parameters ρ4i , ρ5i , ρ45, since different orderings
correspond to different limiting behavior. This splitting is not
6 The very possibility of regulating soft singularities independently of
collinear ones follows directly from QCD color coherence. It is the
primary reason why we do not need to consider sectors where soft and
collinear singularities are entangled. Note that this feature does not
apply to individual diagrams but to on-shell QCD scattering amplitudes
as a whole.
unique; a possible choice consistent with the phase space
parametrization that we employ later (cf. Appendix B)
reads
where, as usual, ηi j = ρi j /2 = (1 − cos θi j )/2. We will refer
to the four contributions shown in Eq. (8.7) as sectors a, b, c
and d. We note that only two of the sectors are qualitatively
different, since the other two are just obtained by the 4 ↔ 5
replacement. However, because of the energy ordering E5 <
E4, we no longer have the 4 ↔ 5 symmetry, and we have to
consider all the four sectors separately.
A suitable parametrization of all angular variables that
supports splitting of the angular phase space as shown in
Eq. (8.7) and allows for factorization of singularities in hard
amplitudes was provided in Refs. [10, 11] and is reviewed
in Appendix B. We will use this parametrization to carry
out integrations over sectors θ (a), θ (b), θ (c), θ (d) explicitly
in what follows.
Having introduced partitions and sectors as a tool to
identify singularities that may appear in the course of
integrating over the angles of the final state gluons, we are now in
a position to write the result for the double-real emission
cross section as a sum of terms that either can be integrated
in four dimensions, or that depend on hard matrix elements
of lower multiplicity. The latter contributions still diverge,
either explicitly or implicitly, and we will have to combine
them with double-virtual and real-virtual contributions to
arrive at the finite result.
We can thus rewrite the double-real emission cross section
FL M (1, 2, 4, 5)
= SS FL M (1, 2, 4, 5) +
where the soft-regulated, single-collinear term
FLsrMct (1, 2, 4, 5)
I − SS S5 FL M (1, 2, 4, 5)
i∈tc
the soft-regulated, triple-collinear terms FLsrMct reads
= −
I − SS
I − S5
(i j )∈dc
× C4i C5 j [dg4][dg5]wi4, j 5 FL M (1, 2, 4, 5)
I − SS
I − S5
× [dg4][dg5]wi4,i5 FL M (1, 2, 4, 5) ,
and the fully regulated term
(i j )∈dc
I − SS
I − S5
(I − C5 j )(I − C4i )
× [dg4][dg5]wi4, j 5 FL M (1, 2, 4, 5)
i∈tc
+ θ (b) I − CCi I − C45
I − SS
I − S5
θ (a) I − CCi I − C5i
× [dg4][dg5]wi4,i5 FL M (1, 2, 4, 5) .
For the process under consideration, dc = {(1, 2), (2, 1)}
and t c = {1, 2}. The above results are obtained by
combining soft-regulated expression for FL M (1, 2, 4, 5) with
multiple partitions of unity for the angular projections, and the
understanding of which collinear divergences can appear in
each partition and sector. One can easily check, starting from
Eq. (8.8), that the collinear projection operators add up to an
identity operator for each partition and each sector.
For example, the contribution of the double-collinear
sector i 4, j 5 follows from an expansion of an identity operator
written in the following form:
I = (I − C4i + C4i )(I − C5 j + C5 j )
The reason we restrict ourselves to the subtraction of the C4i
and C5 j collinear projection operators is that in the partition
i 4, j 5 no other collinear singularities appear, thanks to the
damping factor wi4, j 5. Similarly, taking e.g. the sector a of
the triple-collinear partition wi4,i5, we write
+(I − CCi )(I − C5i ),
I = (I − CCi + CCi )(I − C5i + C5i ) = C5i + CCi (I − C5i )
(8.13)
because in this case a singularity can only occur either in a
triple-collinear limit η4i ∼ η5i → 0 or if η5i → 0 at fixed
η4i .
It is worth pointing out a few things in connection with
Eq. (8.8).
• The procedure that we used to write Eq. (8.8) is, in
principle, process- and phase space
parametrizationindependent. We will use a particular parametrization of
the phase space to perform the required computation but
one should keep in mind that the freedom of changing the
parametrization exists and, perhaps, it is worth exploring
it in the future.
• The first term in Eq. (8.8), SS FL M (1, 2, 4, 5) , is the
double-soft subtraction term. It contains unregulated
soft and collinear singularities and cannot be directly
expanded in . However, it only involves the tree-level
matrix element FL M (1, 2) which means that emitted
gluons decouple from both the hard matrix element and
the phase space constraints. When integrated over gluon
energies and angles, this term gives rise to 1/ n poles,
n ≤ 4.
• The second term in Eq. (8.8), I − SS S5 FL M (1, 2, 4, 5) ,
is the double-soft regulated, single-soft subtraction term.
It contains FL M (1, 2, 4) and matrix elements of lower
multiplicity. This term still contains unregulated
singularities that occur when the momentum of the gluon g4
becomes collinear to the collision axis or to the direction
of g5. When integrated over gluon energies and angles,
this term gives rise to 1/ n poles, with n ≤ 3.
• The term FLsrMcs in Eq. (8.9) is the soft-regulated
singlecollinear subtraction term. Note that thanks to the
damping and θ factors only one kind of collinear
singularity per term is present. FLsrMcs involves FL M (1, 2, 4(5)),
depending on the partition and the sector and,
therefore, contains unregulated collinear singularities related
to gluon emissions along the collision axis. It gives rise
to 1/ 2 and 1/ poles.
• The term FLsrMct in Eq. (8.10) is the triple-collinear
subtraction, where all other singularities are regulated. As
we will see, contributions of the double-collinear
partitions to F sr ct have the “double-convolution” structure.
L M
The FLsrMct term contains 1/ poles in contributions of
triple-collinear partitions, and 1/ 2 poles in contributions
of double-collinear partitions.
• The term FLsrMcr in Eq. (8.11) is completely regulated,
thanks to the nested subtractions. It can be evaluated in
four dimensions. It is the only term that involves the full
hard matrix element for the process q( p1) + q¯ ( p2) →
V + g( p4) + g( p5).
Following our general strategy, we need to study the
first four terms in Eq. (8.8), which involve matrix elements
of reduced multiplicity, and rewrite them in terms of
integrable quantities that admit straightforward expansions in
the dimensional regularization parameter . We will discuss
how to do this in the following subsections.
8.2 The double-soft subtraction term
We begin with the discussion of the first term in Eq. (8.8),
SS FL M (1, 2, 4, 5) . It corresponds to the kinematic situation
where momenta of both gluons vanish at a comparable rate.
The corresponding limit for the amplitude squared is given
in Refs. [74, 75] and allows us to write
[dg4][dg5] Eik(1, 2, 4, 5) FL M (1, 2) ,
SS FL M (1, 2, 4, 5)
− S11(4, 5) − S22(4, 5) ,
Si j (q1, q2) = Sisjo(q1, q2) +
Si j (q) ≡ ( pi ·pqi )·( pp jj · q) = 2 siqsisjjq ,
pi · q1 p j · q2 + pi · q2 p j · q1
− 2
− pi · q1 p j · q1 pi · q2 p j · q2
At this point, we stress again that the hard matrix element
FL M (1, 2) corresponds to a tree-level process and that the
emitted gluons have no impact on the kinematic properties
of the final state V because they decouple from the
energymomentum conserving δ-function.
The goal now is to integrate the eikonal factor over the
momenta of the two gluons. We note that, at this point, unless
1.999995(8) CF2
0.499999(2) CACF
Table 1 Coefficients of the expansion of the double-soft projected
real-emission contribution. Full results are given in the first row. Results
for individual color factors are given in the second and third rows.
−11.73653(7)
−7.3253(7)
1.1077(3) CF2
−2.3236(1) CACF
−20.796(5)
1.522(1) CF2
−5.876(1) CACF
−54.65(7)
1.961(4) CF2
−14.52(1) CACF
Numerical errors are such that their contribution to the final result is
below the per mille level
put in by hand, the integration over gluon energies becomes
unconstrained since the energy-momentum conserving
δfunction becomes independent of the gluon momenta after
the double-soft limit is taken. It is for this reason that we need
to introduce Emax as in Eq. (8.3).
To satisfy constraints on gluon energies, we parametrize
them as
E4 = Emax x1,
E5 = E4 x2 = Emax x1 x2.
Written in these variables, the eikonal factor becomes
The important point is that the dependence on the overall
energy scale x1 factorizes and that the remaining
(complicated) function E is independent of energies of the
incoming partons. We also use the parametrization of energies
Eq. (8.18) in the phase space to obtain
[dg4][dg5]Eik(1, 2, 4, 5) = −
dx2 d 4 d 5
x 1+2 2(2π )d−1 2(2π )d−1 E (x2, n1, n2, n4, n5).
2
For the case of a color-singlet final state, this integral is just a
constant.7 The abelian contribution is simple to obtain since
it is just the product of NLO structures. In principle it should
be possible to compute the non-abelian contribution
analytically along the lines of e.g. Refs. [76–78]. However, it
is also straightforward to obtain it numerically. To do this,
we partition the phase space as in residue-improved sector
decomposition [10, 11]. The corresponding formulas for the
angular phase space are given in Appendix B. Performing the
required decomposition and integrating Eq. (8.20)
numerically, we obtain the -expansion of the double-soft
subtraction term,
SS FL M (1, 2, 4, 5) = [αs]2 Em−a4x FL M (1, 2)
the coefficients cSS are shown in Table 1. There we also report
numerical results for the abelian contribution, which are in
perfect agreement with the analytic calculation. The result
for the double-soft subtraction SS FL M (1, 2, 4, 5) does not
require any further regularization; we will later combine it
with other contributions with tree-level kinematics to cancel
the 1/ singularities explicitly.
8.3 The single-soft term We now consider the second term that contributes to Eq. (8.8). It is a double-soft regulated, single-soft singular expression that reads
I − SS S5 FL M (1, 2, 4, 5) .
Note that this contribution implicitly depends on FL M (1, 2, 4)
and FL M (1, 2), and the hard matrix element that appears in
FL M (1, 2, 4) still contains collinear singularities that arise
when the momentum of gluon g4 becomes parallel to the
momenta of the incoming partons or to the direction of g5.
These divergences will have to be extracted and regulated.
We start by computing the soft limit for the gluon g5. We
7 In a more general NNLO problem, this integral is a function of the
scalar product of the three-momenta of the two hard partons.
Since the gluon g5 decouples from the hard matrix element,
we can integrate over its momentum. We find
(I − SS)S5 FL M (1, 2, 4, 5) = J124 (I − S4)FL M (1, 2, 4) ,
(8.24)
where zmin = 1 − Emax/E4. The splitting function operator
Pˆq(−q) is defined by means of the following equation:
J124 =
Ki j =
+ CA (2ρ14)− K14 + (2ρ24)− K24 ,
(21(1−−2 )) ηi1j+ F21(1, 1, 1 − , 1 − ηi j )
= 1 +
We need to simplify Eq. (8.24) because it still contains
collinear singularities that appear when the momentum of
gluon g4 becomes parallel to the collision axis. To extract
them, we write
J124 I − S4 FL M (1, 2, 4) =
I − C41 − C42
× J124 I − S4 FL M (1, 2, 4)
C41 + C42 J124 I − S4 FL M (1, 2, 4) .
We reiterate that according to our notational conventions,
the collinear projection operators do not act on the phase
space element of the gluon g4 in Eq. (8.27). The first term
in Eq. (8.27) is explicitly regulated and can be expanded in
powers of ; for this reason, we will only be concerned with
the second term. We focus on the projection operator C41; the
contribution of the projection operator C42 is then obtained
by analogy.
First, we consider how C41 acts on J124. Using η12 = 1,
C41ρ24 = ρ12 and taking the ρ41 → 0 limit on K14, K24 we
obtain
C41 J124 = [α2s] E4−2
× 21−2 CF + CA (1 + ) (1 − )(2ρ14)− .
Integrating over the energy and angle of the gluon g4 we
arrive at
C41 J124 I − S4 FL M (1, 2, 4)
= −
Pˆq(−q) f (z) = Pqq (z) f (z) − 2CF f (1),
and the splitting function Pqq (z) is given in Eq. (3.15).
Note that C41 J124 I − S4 FL M (1, 2, 4) in Eq. (8.29) can
be directly expanded in powers of since all the singularities
are regulated. The only problem that needs to be addressed
is the fact that the integration over z does not start at z = 0,
as is the case for the convolutions. The lower integration
boundary zmin must be kept in Eq. (8.29) because of the
subtraction term 2CF FL M (1, 2). Indeed, if zmin is replaced
with zero, the integration over the gluon energy for this term
extends to the region E4 > Emax, in contradiction with the
original phase space parametrization. The extension of the
integration region in Eq. (8.29) is accomplished following
steps discussed in the context of the NLO QCD
computation in Sect. 3. Effectively, this leads to a redefinition of the
splitting function
where Pqq,R R1 (z) is given in Eq. (A.12). We note that the
contribution of the collinear operator C42 to the second term
in Eq. (8.27) is computed in a similar way; the computation
leads to the same result as in Eq. (8.29) up to an obvious
replacement FL M (z · 1, 2) → FL M (1, z · 2).
Putting everything together, we find the final result for
the double-soft regulated single-soft singular contribution to
FL M (1, 2, 4, 5) ,
I − SS S5 FL M (1, 2, 4, 5)
I − C41 − C42 I − S4 J124 FL M (1, 2, 4)
8.4 The single-collinear term
Next, we consider the soft-regulated, single-collinear
contribution to FL M (1, 2, 4, 5)
FLsrMcs =
I − SS I − S5
(i j)∈dc
i∈tc
C4i [dg4] + C5 j [dg5] wi4, j5 FL M (1, 2, 4, 5)
I − SS I − S5 θ (a)C5i + θ (b)C45 + θ (c)C4i
We need to rewrite Eq. (8.34) in such a way that extraction
of the remaining collinear singularities becomes
straightforward. We note that FLsrMcs contains contributions from
double- and triple-collinear partitions, which we will treat
separately. We will start with the double-collinear partitions
since they are somewhat simpler.
8.4.1 The double-collinear partitions
In this subsection, we will consider the contribution of the
double-collinear partitions to FLsrMcs . We begin with the
partition 14, 25. For the first term, we need to compute
where w˜ 41|4|,125 = limρ41→0 w14,25 does not depend on
the momentum of gluon g4 anymore. To further simplify
Eq. (8.35), note that collinear and soft projection operators
commute with each other and that
−SS S5 FL M (1 − 4, 2, 5) = 0.
This implies that we can drop the SS term in Eq. (8.35). We
use the collinear limit for C41 FL M (1, 2, 4, 5) obtained by a
straightforward generalization of Eq. (3.14). We define z =
1 − E4/E1 and obtain
x3(1 − x3) − dx3 θ (zmax(E5) − z),
x3
E5 2E5
zmax(E5) = 1 − E1 = 1 − √s .
In this parametrization, the action of C41 implies replacing
[x3(1 − x3)]− with x3− . Putting everything together, we
obtain
= −
Note that, similar to the NLO case, the lower boundary zmin
is not important when integrating FL M (z · 1, 2, 4) since for
z < zmin there is no sufficient energy in the incoming partons
to produce the required final state.
Next we consider the action of the C52 projection
operator. Following the preceding discussion and using z =
1 − E5/E2 = 1 − 2E5/√s, we obtain
= −
× FL M (1, 2 · z, 4) .
E4 2E4
zmin(E4) = 1 − E2 = 1 − √s .
The sum of Eqs. (8.40) and (8.41) gives the required result
for the collinear sector 14, 25.
The partition 15, 24 is obtained from the results for 14, 25
after a few obvious replacements. We find
Pqq (z) I − S5 FL M (z · 1, 2, 5).
The function Pqq (z) was introduced in Eq. (8.31).
We now consider the phase space. According to Eq. (8.34),
C41 acts on the phase space element [dg4]. We introduce
x3 = (1 − cos θ41)/2 to get
× w˜ 41|5|,224 I − S5 FL M (1, z · 2, 5)
= −
× FL M (1 · z, 2, 4) .
lations we find
We can now combine the contributions of the two
doublecollinear partitions. In doing so, it is convenient to always
denote the “resolved” (i.e. the non-collinear) gluon by g4.
Out of the four terms that we need to combine, two
correspond to the collinear emission along the direction of the
incoming quark p1 and two along the direction of the
incoming antiquark p2. We consider terms that belong to the former
category first.
When combining results, it is important to realize that
√s. We will denote it by
zz4mi>n( Ezm4 )in==z1m−ax(2EE4m) a=x/√1−s. A2Eft4e/r straightforward
manipudz 15,24
(1 − z)1+2 Pˆq(−q) w˜ 5||1
× [dg4][dg5]FL M (1, 2, 4, 5)
= −
(1 − z)1+2
Pˆq(−q) I − S4 FL M (z · 1, 2, 4)
+ θ (z4 − z) 2CF I − S4 FL M (1, 2, 4)
+ θ (z − z4) Pˆq(−q) S4 FL M (z · 1, 2, 4) .
Note that the lower integration boundary in this formula
should be z = zmin but we can extend the integration region
to z = 0, without making a mistake. This is so because every
time FL M (z · 1, . . .) appears in the integrand, the z > zmin
condition is automatically enforced by the requirement that
the initial state should have enough energy to produce the
final state. On the other hand, if FL M (z · 1, . . .) does not
appear, θ -functions require that z > z4 > zmin. We also note
that, thanks to explicit subtractions and constraints due to
θ -functions, each term in Eq. (8.45) vanishes if z → 1 or
E4 → 0. Finally, we stress that Pˆq(−q) and S4 commute since
they act on different variables.
We can write a similar equation for the sum of the two
terms where the collinear gluon is emitted along the direction
of the antiquark q¯ ( p2). Finally, putting everything together,
we obtain the contribution of the double-collinear partitions
to FLsrMcs . We find
× [dg4][dg5]FL M (1, 2, 4, 5)
= −
+ FL M (1, z · 2, 4)
Note that the second term in the curly bracket only depends
on z through the θ -function and so the z-integration of this
term can be performed explicitly.
8.4.2 The triple-collinear partition 14, 15
We consider the triple-collinear partition 14, 15 and study
the contribution of single-collinear limits in Eq. (8.34). We
begin with sector (a). The relevant expression reads
The calculation is identical to the case of the double-collinear
partition except that we need to account for the constraint
that defines sector (a) when integrating over the angle of
gluon g5. Writing ρ14 = 2x3 and ρ15 = 2x4, and taking
θa = θ (ρ14/2 − ρ15), we find
= −
(x3/2)−
= −
Using this result, we obtain
= −
× θ (z − z4)Pˆq(−q) FL M (1, 2, 4) .
A similar calculation for the sector (c) gives
I − SS I − S5 θ (c)C41w14,15[dg4]FL M (1, 2, 4, 5)
= −
(1 − z)1+2
× Pqq (z) I − S5 FL M (z · 1, 2, 5) .
In parallel to the case of the double-collinear partitions, it
is again convenient to always call the resolved gluon g4.
We combine contributions of sectors (a) and (c), renaming
g5 → g4 where appropriate, and we obtain
I − SS I − S5
× w14,15[dg4][dg5]FL M (1, 2, 4, 5)
= −
(1 − z)1+2
[I − θ (z4 − z)S4] × Pˆq(−q) FL M (z · 1, 2, 4)
+ θ (z4 − z)2CF I − S4 FL M (1, 2, 4) .
We now turn to sectors (b) and (d). These sectors are
different from the other triple-collinear sectors and from the
double-collinear partitions. Indeed, the single-collinear
limits that we consider in sectors (b) and (d) correspond to
gluons g4 and g5 becoming collinear to each other. We consider
and start with the discussion of how the collinear projection
operator C45 acts on FL M . We find
2
C45 FL M (1, 2, 4, 5) = Eg52sρ,b45 EE45 Pgg,μν (z)FLμMν (1, 2, 4 + 5)
d ≡
Pg(g0)(z)FL M (1, 2, 45)
+ Pg⊥g(z)κ⊥,μκ⊥,ν FLμMν (1, 2, 45) ,
z = E5/(E4 + E5).
where p4+5 = p45 = (E4 + E5)/E4 · p4, i.e. the hard matrix
element must be taken in the collinear limit. The splitting
functions are
Pg(g0)(z) = 2CA
Pg⊥g(z) = 4CA(1 − )z(1 − z),
1 − z
,
z
and z is the fraction of the total momentum p45 = p4 + p5
carried by gluon g5,
defined by the following decomposition:
where p¯4 = ( p4(0), − p4) and k⊥ · p4 = k⊥ · p¯4 = 0.
We now construct these vectors explicitly. For this, we
need the parametrization of the angular phase space of the two
gluons valid for sectors (b) and (d); it is given in Appendix B.
Here we repeat the relevant formulas and discuss
simplifications that occur in the limit where the momenta of g4 and g5
become collinear. We write the four-momenta of g4 and g5
as
where t μ = (1, 0), e3μ = (0, 0, 0, 1; 0 . . .), b · t = b · e3 = 0
and a · t = a · e3 = a · b = 0. Our goal is to parametrize the
phase space in such a way that explicit averaging over
directions of k⊥ can be performed. The phase space
parametrization for sectors (b) and (d) can be written as
[dg4][dg5] =
× θ (Emax − E4)θ (E4 − E5)d (4b5,d),
= N (b,d) d g4 1 (4π )
(2π )d−1 8π 2 (1 − )
d d−3,a
d−3
(1 + ) (1 − ) λ−1/2+ (1 − λ)−1/2−
Here, x4 → 0 corresponds to the 4||5 collinear limit, η45 =
(1 − cos θ45)/2, x3 = ρ41/2 and λ is related to the azimuthal
angle ϕ45. Further details about the parametrization as well
as expressions of scalar products in terms of x3,4 and λ can
be found in Appendix B. In this parametrization, the vector
κ⊥ reads8
Using this expression in Eq. (8.53) together with momenta
parametrization Eq. (8.58) and the phase space limit
Eq. (8.60), we observe that integrations over λ and the
directions of the vector aμ can be performed since neither λ nor
aμ appear in the hard matrix element. We define
d−3
d−3
d−3
= −
= 1,
1 + 2
,
2
= −
− g⊥μν,(d−3)a + r μr ν
It follows that averaging over transverse directions
introduces an -dependent leftover, as a consequence of the
chosen phase space parametrization [10,11].
To write the result of the integration over unresolved phase
space variables, it is convenient to define an additional
splitting function,
Pgg(z, ) = Pg(g0)(z) +
Pg⊥g(z)
= 2CA
Combining the results discussed above, we write an
expression for the contribution of the C45 collinear projection
operator in sector (b). We obtain
8 This expression is valid for sector (b). For sector (d), one should
replace r → −r˜ with r˜ = sin θ51 e3 − cos θ51 b in Eq. (8.62).
[dg4]w˜ 41|4|,515x3− (1 − x3)
I − SS I − S5 P45(1, 2, 4, 5),
FL M (1, 2, 4 + 5)
It follows from Eq. (8.68) that we need to know how an
operator (I −SS)(I −S5) acts on P45(1, 2, 4, 5). We recall that
the action of SS on energy variables implies that E4, E5 →
0 at fixed E4/E5. Computing the soft limits is simple and
standard except for the spin-correlated part that we address
below. In principle, we need to know three soft limits SS, S5
and SS S5. However, since
we only need to consider SSrμrν FLμMν (1, 2, 4 + 5). We find
it using the known soft limits for amplitudes and the explicit
form of the vector r given in Eq. (8.62). Indeed, since r · p4 =
0 and r 2 = −1, r μ is a valid polarization vector of the gluon
with momentum 4 + 5, in the collinear 4||5 approximation.
For this reason, the soft limit of rμrν FLμMν follows from the
standard soft limit of the amplitude for qq¯ → V + g, not
averaged over gluon polarizations. We obtain
FL M (1, 2)
= S4 FL M (1, 2, 4).
FL M (1, 2)
Collecting all the soft limits, we find
× I − S45 FLμMν (1, 2, 45) − 2CA I − S4
× FL M (1, 2, 4),
Fig. 1 Integration region for the (E4, E5) → (E45, z) change of
variables. The colored triangle is the allowed 0 < E5 < E4 < Emax
region. The blue region “A” is the “physical” one, i.e. the one which
is not removed by a phase space θ -function inside FL M (1, 2, 45). The
orange region “B” only contributes to the soft limit, since there no θ
function from FL M is preventing it. Lines of fixed E45 are shown in
solid red (for E45 = Emax) and dashed orange (for E45 > Emax). In
dot-dashed blue, lines of constant z are shown. In the “physical” region,
only the z < 1/2 condition is relevant. In the “B” region, we also have to
impose z > 1 − Emax/E45, to prevent the E45 integration to go outside
the triangle (see the intersection of blue and orange lines)
where z is defined in Eq. (8.55). This implies
E5 = z E45,
E4 = (1 − z)E45,
E45 = E4 + E5. (8.73)
We can now use Eqs. (8.72) in (8.68) and integrate over all
variables that are not present in the hard matrix elements. This
requires different variable transformations in the first and the
second terms in Eq. (8.72). To integrate the first term, we
change the integration variables from E4,5 to E45 and z.
We find that the integration region splits into two regions
(cf. Fig. 1)
d E5 =
E45 d E45
1−Emax/E45
E45 d E45
Following this separation, we split the integral into two parts
I − SS
I − S5 θ (b)C45w14,15[dg5]FL M (1, 2, 4, 5)
= I A + IB ,
where the integral I A corresponds to the region E45 < Emax
and the integral IB to the region E45 > Emax; see Fig. 1.
We obtain an integral representation for I A starting from
Eq. (8.72), changing variables (E4, E5) → (E45, z) in the
first term, and E5 = z E4 in the second term in Eq. (8.72)
and, finally, renaming E45 → E4. We obtain
I A = − 2
[dg4]w˜ 41|4|,515 x3− (1 − x3) E −2
4
z(1 − z)−2
× I − S4 FLμMν (1, 2, 4)θ (1/2 − z)
− 2CA I − S4 FL M (1, 2, 4) .
To compute IB , we notice that only the soft S45 FLμMν (1, 2, 45)
term of Eq. (8.72) contributes. Again renaming E45 → E4
and using Eq. (8.71), we obtain
IB =
4x3(1 − x3)
FL M (1, 2),
w˜ 41|4|,515 x3− (1 − x3)
Pgg (z, k⊥)
where z4 = 1 − E4/Emax and we expressed [dg4] as an
integral over energy E4 and the angular integration variable
x3 = ρ14/2. We have also integrated over d − 3 angular
variables that do not appear in the hard matrix element and
in the splitting function.
Finally, we consider sector (d). We need to compute
I − SS
I − S5 θ (d)C45w14,15[dg4]FL M (1, 2, 4, 5) .
(8.78)
The calculation is similar to what we just described for sector
(b), apart from the following modifications of the integration
boundaries:
Incorporating these changes in Eqs. (8.76) and (8.77)
provides us with the result for sector (d).
Significant simplifications occur if the results for the two
sectors are added; this happens because the z-integration
boundaries in sectors (b) and (d) complement each other.
Also, for both IA and IB the z-integration decouples from the
rest and can be performed independently of the hard matrix
element. In IA, it yields
[αs] w14,15 ρ14 −
˜ 4||5 2
× γ˜g( ) I − S4 FL M (1, 2, 4) + γ˜g( , k⊥)
× I − S4 rμrν FLμMν (1, 2, 4) ,
where we used x3 = ρ14/2 and the constants γ˜g( ), γ˜g( , k⊥)
are defined in Eq. (A.19). In the integral IB the hard matrix
element is that of the leading-order process which implies
that integration over all variables related to radiated gluons
can be explicitly performed. We find
4x3(1 − x3)
8.4.3 Summing double- and triple-collinear partitions
Summing up the above results, we obtain an intermediate
representation of FLsrMcs . We write it as a sum of four terms,
FLsrMcs =
(i j)∈dc
I − SS I − S5
C4i [dg4] + C5 j [dg5]
× wi4, j5 FL M (1, 2, 4, 5) +
I − SS I − S5
i∈tc
× [dg4][dg5]wi4,i5 FL M (1, 2, 4, 5)
= C1(z · 1, 2, 4) + C2(1, z · 2, 4)
+ C3(1, 2, 4) + C4(1, 2, 4) .
These terms are defined as
C1(z · 1, 2, 4) = −
(1 − z)1+2
(1 − z)1+2
[I − θ (z4 − z)S4] Pˆq(−q) FL M (1, z · 2, 4)
+ 2CFθ (z4 − z) I − S4 FL M (1, 2, 4) ,
C3(1, 2, 4) = [αs]
+ γ˜g( , k⊥)rμrν FLμMν (1, 2, 4) ,
C4(1, 2, 4) = [αs]2 Em−a4x δg( )
d (d−1),4
FL M (1, 2) ,
d−2
d d−1,4 = d cos θ sin2 θ
= 4x3(1 − x3)
8.5 The single-collinear term: extracting the last singularities
The four contributions to FLsrMcs described at the end of the
previous section require further manipulations because they
cannot be expanded in series of as they are. Indeed, all of
them exhibit collinear singularities in the limits 4||1 and 4||2,
which need to be extracted before expansion in becomes
possible. To deal with this issue, we again rewrite the identity
operator through collinear projections. For example, we write
+ (I − C41 − C42)C1(z · 1, 2, 4) .
The first two terms can be further simplified by
considering respective collinear limits; the last term is regulated and
can be Taylor-expanded in . The single-collinear
subtraction term can be analyzed in the same way as all the other
collinear limits discussed previously. The only new element
here is the action of the collinear projection operators on
the spin-correlated part. Using the explicit expression for the
vector r in Eq. (8.62) we find
E42ρ41C41rμrν FLμMν (1, 2, 4) = gs2,bCF
FL M (z·1, 2),
where z is defined in the usual way z = 1 − E4/E1. Taking
this into account, after tedious but straightforward
calculations we arrive at
FLsrMcs =
− gμν γ˜g + γ˜g⊥rμrν FLμMν (1, 2, 4)
× 2CFOˆ NLO
(Emax/E1)−2
− 1
Pqq ⊗ Pqq R R(z)
× γ˜gPqq,R R1 (z) + γ˜g⊥Pqq,R R5 (z)
dz dz¯ Pqq,R R2 (z)Pqq,R R2 (z¯)
5||1 =
5||2 =
4||5 =
Em−a4x FL M (1, 2) .
In the above equation, we used the following notation:
= 1 + O( ),
= 1 + O( ),
= 1 + O( ).
The relevant splitting functions are defined in Appendix A.
Also,
FLz M (1, 2) ≡
Oˆ C ≡ I − C41 − C42,
Oˆ NLO ≡ I − S4 I − C41 − C42 ,
5||2 FL M (1, z · 2, 4)
We note that in Eq. (8.90) the first curly bracket is fully
regulated, while the second contains subtraction terms. Note
also that since i|| j = 1 + O( ), if we are only interested in
the 1/ poles, we can substitute i|| j → 1 in Eq. (8.90). We
also note that, for the process of interest, terms that contain
OC S1(42),ρ can easily be integrated over the relative angles of
the gluon g4 with respect to the collision axis. We find
[dρ41]OC S1(42),ρ 5||1 =
= (1 + ln 2) + O( 2),
(i j)∈dc
I − SS I − S5
× C4i C5 j [dg4][dg5]wi4, j5 FL M (1, 2, 4, 5)
= −
I − SS I − S5 C41C52 + C51C42
× [dg4][dg5]FL M (1, 2, 4, 5) .
Note that, following our notational convention, the collinear
projection operators act on the phase space elements [dg4]
and [dg5].
We begin with the C41C52 term. Introducing
E4 = (1 − z)E1, E5 = (1 − z¯)E2,
and calculating collinear limits we obtain
E42 E52ρ14ρ25C41C52 FL M (1, 2, 4, 5)
4
= gs,bPqq (z)Pqq (z¯)FL M (z · 1, z¯ · 2).
Since the momenta of gluons g4 and g5 decouple from each
other, we find
FL M (1, 2).
As the result, the original expression simplifies
I − SS I − S5 C41C52 FL M (1, 2, 4, 5)
= I − S5 C41C52 FL M (1, 2, 4, 5).
We will use these results when presenting the final formula
for the double-real contribution.
We now turn to the term FLsrMct and begin by considering the
contribution of the double-collinear partitions. It reads
− I − SS I − S5 C41C52[dg4][dg5]FL M (1, 2, 4, 5)
1 1
= −
(1 − z)1+2
× Pqq (z)Pqq (z¯)FL M (z · 1, z¯ · 2)
− 2CFPqq (z)FL M (z · 1, 2) ,
where, as usual, the z integrals do not need a lower cut-off
whenever z is present in FL M .
The term with collinear operators C51C42 in Eq. (8.95)
can be simplified in a similar way. Combining the two
contributions, we obtain
Performing the angular integrations and accounting for the
hierarchy of energies E4 > E5, we obtain
In this section, we consider the contribution of the
triplecollinear partitions to FLsrMct . It reads
I − SS I − S5 C41C52 + C51C42
× [dg4][dg5]FL M (1, 2, 4, 5)
= −
− 2CF
+ FL M (1, z · 2) ×
We can rewrite Eq. (8.101) to ensure that all
singularities that appear in z and z¯ integrals are regulated with the
plus-prescription. This gives the final result for the
doublecollinear contribution
= −
dz dz¯ Pqq,R R2 (z)Pqq,R R2 (z¯)
The relevant splitting functions are given in Appendix A.
8.7 The double-unresolved collinear limit: triple collinear
I − SS I − S5 C41C52 + C51C42
× [dg4][dg5]FL M (1, 2, 4, 5)
i∈tc
I − SS I − S5 θ (a)CCi I − C5i
+ θ (c)CCi I − C4i + θ (d)CCi I − C45
× [dg4][dg5]wi4,i5 FL M (1, 2, 4, 5) .
This contribution always contains the triple-collinear
projection operator CCi that acts on the hard matrix elements. For
i = 1, this gives, schematically,
CC1 FL M (1, 2, 4, 5) =
Pggq (1, 4, 5)
× FL M (1 − 4 − 5, 2),
where s145 = ( p1 − p4 − p5)2 and Pggq (1, 4, 5) is the known
triple-collinear splitting function [75,79,80]. The reduced
matrix element in Eq. (8.104) has to be evaluated in the
exact collinear limit, i.e. p1−4−5 ≡ (E1 − E4 − E5)/E1 · p1.
Other projection operators that appear in Eq. (8.103)
provide subtractions that are needed to make the triple-collinear
splitting function integrable over the unresolved parts of the
(g4, g5) phase space. For definiteness, we focus here on the
triple-collinear partition where gluons are emitted along the
direction of the incoming quark with momentum p1; this
corresponds to taking i = 1 in Eq. (8.103).
To proceed, we note that the damping factors in Eq. (8.103)
can be removed since the collinear projection operator CC1
acting on them yields 1. Next, we need to study the
triplecollinear limit of the angular phase space. The generic phase
space parametrization is described in Appendix B and we
use it to compute the triple-collinear limits. We stress that
since the phase space parametrization changes from sector
to sector, we need to consider all the four sectors separately.
Without going into further detail of the angular integration,
it is clear that once this integration is performed, each sector
in Eq. (8.103) provides the following contribution to the final
integral over energies:
θ (k)CC1 I − Ci j d (4k5) FL M (1, 2, 4, 5)
≡ [αs]2TC(k)(E1, E4, E5)FL M (1 − 4 − 5, 2),
where the auxiliary function TC(k) in Eq. (8.105) is defined as
[αs]2TC(k)(E1, E4, E5) =
θ (k)CC1 I − Ci j d (4k5)
4 Pggq (z4, z5, z1, s45, s41, s51)
We note that the reason that CC1 is present in Eq. (8.106) is
that it still has to act on the phase space; its action on the
matrix element has already been accounted for and resulted
in the factorized form of Eq. (8.105) and the appearance of
the triple-collinear splitting function in Eq. (8.106). We use
Eq. (8.105) to write
I − SS I − S5 θ (k)CC1 I − Ci j [dg4][dg5]
× wi4,i5 FL M (1, 2, 4, 5)
d E4 E41−2
d E5 E51−2 I − SS
where 145 ≡ 1 − 4 − 5 in the collinear approximation.
In what follows, we discuss the integration over energies
in Eq. (8.107). Our goal is to change variables in such a way
that the argument of the hard matrix element becomes z · 1;
once this happens, Eq. (8.107) becomes a convolution of a
hard matrix element with a splitting function. Although, in
principle, changing variables in an integral is straightforward,
it turns out that it is beneficial to do so in different ways in
the four terms that appear in Eq. (8.107); for this reason,
we consider them separately. We emphasize that since the
individual contributions to Eq. (8.107) diverge, it is important
to keep dimensional regularization in place until the end of
the computation.
We begin with the term that contains the identity operator
I and change the variables as follows:
r r
E4 = E1(1 − z) 1 − 2 , E5 = E1(1 − z) 2 ,
with r ∈ (0, 1) and z ∈ (0, 1). We note that the lower
integration boundary for z can be taken to be z = 0 because
FL M (z · 1, 2) always appears in this contribution. As we
already discussed several times, this automatically cuts off
the integral over z at a proper minimal value. With this in
mind, we write9
d E4 E41−2
(1 − z)1+4 r 1+2
9 For ease of notation, we will drop the sector index in the intermediate
calculations and restore it at the end.
TC (E1, E1(1 − z)
FL M (z · 1, 2).
Next, we consider the S5 operator. It describes the limit
where E5 → 0 at fixed E4. Calculating this limit with the
parametrization in Eq. (8.108) mixes z and r and, therefore, is
inconvenient. A better way is to change the parametrization.
We choose
E4 = E1(1 − z), E5 = E1(1 − z)r,
where r ∈ (0, 1). In principle, we should use z > zmin but,
since z enters the hard matrix element, we can extend all the
integrals to z ∈ (0, 1). We find
d E4 E41−2
d E5 E51−2 S5TC (E1, E4, E5)
× FL M (1 − 4 − 5, 2) = E 4−4
1
Note that the r 2 prefactor ensures that the r → 0 limit of the
square bracket exists.
We can also use the change of variables in Eq. (8.110) for
terms with operators SS and SS S5. The only difference is that
since in those terms z does not appear in FL M , we have to
keep the lower integration boundary at z = zmin. We write
d E4 E41−2
d E5 E51−2 SSTC (E1, E4, E5)
× FL M (1 − 4 − 5, 2) = E 4−4
1
FL M (1, 2)
Also in this case, the (1 − z)4 prefactor ensures the existence
of the z → 1 limit of the term in the square bracket. The
term with an operator SS S5 in Eq. (8.107) is obtained from
Eq. (8.112) by taking the r → 0 limit in the expression in
square brackets.
− z
To proceed, it is convenient to define two z-dependent
functions and a constant
r r
× TC E1, E1(1 − z) 1 − 2 , E1(1 − z) 2
A3 ≡
(1 − z)4r2TC (E1, E1(1 − z), E1(1 − z)r)
− (1 − z)4r2TC (E1, E1(1 − z), E1(1 − z)r)
We can further simplify the function A2 if we realize that the
S5 limit of the triple-collinear splitting function is
homogeneous in E5. This implies that the r → 0 limit of the two TC
functions in the formula for A2 in Eq. (8.113) are related and
can be combined. Changing variables r → r/2 in the first
term on the r.h.s. of the integral for A2, we obtain
A2(z) = z
Since the term in the square bracket no longer depends on r
after the r → 0 limit is taken, we can perform the r
integrations in the first line to get
1 −
R(k)(z) =
This integral can be re-written in such a way that all the
z → 1 singularities are regulated by plus-prescriptions. We
have already discussed how this can be done several times;
for this reason, we do not repeat this discussion again and
only present the result. It reads
I − SS I − S5 θ (k)CC1 I − Ci j [dg4][dg5]
× wi4,i5 FL M (1, 2, 4, 5) = [αs]2 E 4−4
1
A(1k)(z) + A(2k)(z) − A(1k)(1) − A(2k)(1)
(1 − z)1+4
R(k)
+ = A(1k)(1) + A(2k)(1),
The functions A(1k,2)(z) and the constant A(k) are given in
3
Eqs. (8.113) and (8.115). We have also used the following
notation:
[(1 − z)1+4 ]+ =
n=0
10 At this point, we restore the sector label.
We can now write the result for the integral that we are
interested in using A1,2(z) and A3. We find10
We are now in a position to write the contribution of this term
in the final form
(1 − z)1+4
I − SS I − S5 θ (k)CC1 I − Ci j [dg4][dg5]
× wi4,i5 FL M (1, 2, 4, 5) = [αs]2 E 4−4
1
dz A(1k)(z) + A(2k)(z) FL M (z · 1, 2)
(1 − z)1+4 z
FL M (1, 2) .
I − SS I − S5 θ (k)CC1 I − Ci j
× [dg4][dg5]w14,15 FL M (1, 2, 4, 5) = [αs]2
All the terms in Eq. (8.120) can be expanded in power series
in the dimensional regularization parameter . The functions
R(k)(z) and the constants R(k) and Rδ(k) are calculated
numerically. +
9 Pole cancellation and finite remainders
We are now in position to discuss the final result for the
NNLO QCD contribution to the cross section. We consider
All the different contributions to Eq. (9.1) were considered
in the previous sections. It should be clear from these
discussions that the result for the NNLO cross section is given
by a linear combination of integrated matrix elements with
different multiplicities, which may or may not be convoluted
with generalized splitting functions. Since, for well-defined
observables, the cancellation of soft and collinear
divergences occurs point-by-point in the phase space,
contributions proportional to FL M (1, 2, 4, 5), FL M (1, 2, 4), FL M (z ·
1, 2, 4) etc. must be separately finite. For this reason, it is
convenient to present the result for the NNLO QCD
contribution to the cross section as a sum of seven terms
which are individually finite. Each of the individual terms in
Eq. (9.2) has a subscript that indicates the highest
multiplicity matrix element that it contains. Below we collect all the
different contributions to dσˆ NNLO and present finite
remainders for terms with different multiplicities. For simplicity, we
fix the arbitrary parameter Emax = √s/2.
9.1 Terms involving FL M (1, 2, 4, 5)
This contribution is the only one that involves the matrix
element for qq¯ → V + gg. We repeat here the result, already
given in Eq. (8.11)
FLsrMcr =
I − SS I − S5
(i j)∈dc
× (I − C5 j )(I − C4i )
× [dg4][dg5]wi4, j5 FL M (1, 2, 4, 5)
i∈tc
+ θ (b) I − CCi I − C45
I − SS I − S5 θ (a) I − CCi I − C5i
× [dg4][dg5]wi4,i5 FL M (1, 2, 4, 5) .
It follows that dσˆ FNLNML(O1,2,4,5) is expressed through a
combination of nested soft and collinear subtractions and can be
directly computed in four dimensions.
9.2 Terms involving Oˆ N L O FL M (1, 2, 4)
We continue with terms that involve FL M (1, 2, 4). They are
present in the double-real contribution, Eqs. (8.33) and (8.90)
and in the real-virtual contribution, Eq. (7.8); they are also
found in terms that appear due to ultraviolet Eq. (5.1) and
collinear Eq. (5.8) renormalizations of the next-to-leading
order cross section. Extracting these terms, we observe that
all the 1/ singularities cancel out. The finite remainder reads
dz 4CFD˜ 1(z)− Pˆq(q0)(z) ln
− Pˆq(q)(z)
+ CF Oˆ NLO 23 π 2 − 2 ln 2√Es4 ln ρ441 w˜ 51|4|,115
2E4 ln ρ442 w˜ 5||2
− 2 ln √s 24,25 FL M (1, 2, 4)
11 ln ρ14 w14,15
− 6 ρ24 ˜ 4||5 + ln ρρ2144 w˜ 42|4|,525
+ 23 ln 2√Es4 + ln2 2√Es4 + 43 ln ρ144ρ24
− C3A Oˆ NLOrμrν FLμMν (1, 2, 4)
FL M (1, 2, 4)
D˜ i (z) = Di (z) −
+ Oˆ NLO FLfiVn (1, 2, 4) ,
9.3 Terms involving FLfiVn (1, 2) and FLfiVn V (1, 2)
Next, we collect the finite remainders of the one-loop and
two-loop virtual contributions to the qq¯ → V process. These
contributions appear in the real-virtual, the double-virtual,
the collinear subtraction and the ultraviolet renormalization.
Upon combining them and expanding the resulting
contributions in , we obtain
+ FLfiVn 2 (1, 2) + FLfiVn V (1, 2)
dz 4CFD˜ 1(z) − ln
Pˆq(0q)(z) − Pq(q)(z)
9.4 Terms of the form P1 ⊗ dσ ⊗ P2
Terms of the type P1⊗dσ ⊗P2, where P1,2 are some splitting
functions, appear in the double-real contribution as well as
in the collinear renormalization. Combining all the relevant
terms, we find
−4D˜ 1(z) − (1 − z)
−4D˜ 1(z¯) − (1 − z¯) .
dz dz¯ 2D˜ 0(z) ln
2D˜ 0(z¯) ln
9.5 Terms of the form P ⊗ dσ
These terms appear in the double-real, real-virtual, collinear
subtraction and ultraviolet renormalization contributions. We
note that starting from O(1/ ), the part of the double-real
contribution related to the integral of the triple-collinear
splitting function is only known numerically; see the discussion
in Sect. 8.7.
Combining all the terms, we observe analytic cancellation
of the poles up to 1/ 2. For the 1/ poles and the finite part,
it is useful to split the contribution into a scale-independent
and a scale-dependent term
dσˆ FNLNML(Oz·1,2) ≡ dσˆ FNLNML(Oz·1,2)(μ2 = s) +
NNLO
We also introduce an expansion of the functions R(k)(z) and
the constants R(k), which were introduced in Section 8.7, in
+
powers of
R(k)(z) = R(0)(z) +
k∈sectors
k∈sectors
The scale-independent term reads
dσˆ FNLNML(Oz·1,2)(μ2 = s) =
dz CF2 8D˜ 3(z) + 4D˜ 1(z)(1 + ln 2)
ln z + (1 − 3z) ln 2 ln z
+ 2(1 − 3z)Li2(1 − z)
+ (1 − z) 43 π 2 + 27 ln2 2 − 2 ln2(1 − z)
+ ln 2 4 ln(1 − z) − 6 + ln2 z
+ Li2(1 − z) + (1 + z) − π32 ln z − 47 ln2 2 ln z
− 2 ln 2 ln(1 − z) ln z + 4 ln2(1 − z) ln z
+ 4 ln(1 − z) − 2 ln 2 Li2(1 − z)
ln(1 − z) 3Li2(1 − z) − 2 ln2 z
5 − 3z2 ln z
− 1 − z Li3(1 − z) + (1 − z)
× 12 ln(1 − z) −
802 11 2 ln 2
− 27 + 18 π + (2π 2 − 1) 3 + 11 ln2 2 + 16ζ3 D˜ 0(z)
37 − 28z 1 − 4z 61 161
9 + 3 ln 2 − 9 + 18 z ln(1 − z)
+ (1 + z) ln(1 − z)
− (1 − z)
π62 + Li2(1 − z) − 23(+1 −11zz)2 ln 2 ln z
The scale-dependent term reads
dz C2
F − 12D˜ 1(z) − 12D˜ 2(z) − 6 + 5z
+ 2(1 − z) ln(1 − z) − 2 ln z 1 + z + z2
1 − z
ln2 z
− (1 + z) 2 ln z ln(1 − z) + 2
+ 2Li2(1 − z)
ln(1 − z) ln z − Li2(1 − z)
3 D˜ 1(z) −
2 + 11z2 1 + z2
− 6(1 − z) ln z + 1 − z Li2(1 − z)
1 + 3z2
d z CF2 4D˜ 1(z) + 6D˜ 0(z) − (1 − z) − 2(1 − z) ln z
To arrive at these results, we check the cancellation of 1/
poles in dσˆ FNLNML(Oz·1,2) and then, assuming that the cancellation
is exact, deduce the analytic form of R(0)(z) and R(0). These
+
analytic results are then used in the scale-dependent term.
Thus the only numerical contributions needed for the finite
part are R( )(z) and R( ).
+
9.6 Terms involving FL M (1, 2)
All the different contributions to the NNLO cross section
produce terms proportional to FL M (1, 2). These include
constants R(k) originating from the triple-collinear splitting
funcδ
tion, which, as mentioned in the previous subsection, are only
known numerically. As before, we introduce an expansion in
for these constants.
k∈sectors
Furthermore, for the double-soft contribution we know the
abelian constants of Eq. (8.21) analytically, but only have
numerical results for the non-abelian constants, which are
reported in Table 1. Thus, for each order in 1/ , we check the
cancellation of terms numerically and then, assuming that the
cancellation is actually exact, we deduce an analytic form of
the triple-collinear splitting and double-soft constants at this
order. This form is then used in determining the cancellation
at lower orders in 1/ . Thus, the only numerical constants
appearing in our final formula are Rδ( ) and c0SS,CACF . The final
formula reads
403 2 17 4 671
+ 72 π − 48 π − 36 ζ3
22 16
+ 9 ln3 2 − 3 ln4 2 − 34ζ3 ln 2
FL M (1, 2)
FL M (1, 2) .
10 Numerical results
Having described the subtraction procedure in some detail,
we will now study how well it works in practice. We have
implemented it in a partonic Monte Carlo program to
compute NNLO QCD corrections to the production of a vector
boson γ ∗ in proton–proton collisions.11 The calculation is
11 We remind the reader that we only consider the qq¯ annihilation
channel and restrict ourselves to gluonic corrections in this paper.
10−5
Numerical result
Analytic result
Fig. 2 Comparison of the NNLO QCD contribution dσ NNLO/dQ
computed in this paper with the analytic results in Ref. [87]
fully differential; we consider decays of the virtual photons to
massless leptons and study NNLO QCD corrections to lepton
observables. We extracted the relevant matrix elements from
Refs. [81,82] as implemented in [83], and from Refs. [84,85].
For all computations reported below we employ the NNLO
parton distribution functions from the NNPDF3.0 set [86].
We begin by comparing the analytic result for the NNLO
QCD correction to the pp → γ ∗ → e+e− + X cross
section, which we extract from Ref. [87], and the result of the
numerical computation based on the formulas reported in
the previous section. We wish to emphasize that this
comparison is performed using the NNLO contribution to the
cross section, and not the full cross section at NNLO, which
would have included LO and NLO contributions as well.
We take 14 TeV as the center-of-mass collision energy. We
include lepton pairs with invariant masses Q in the range
50 GeV < Q < 350 GeV and take μ = 100 GeV for
the renormalization and factorization scales. We obtain the
NNLO corrections to the cross sections
where the first result is ours and the second is extracted from
Ref. [87]. The agreement between the two results is quite
impressive; it is significantly better than a permille. To
further illustrate the degree of agreement, we repeat the
comparison using the kinematic distribution dσ NNLO/d Q, shown in
Fig. 2. In the upper pane of Fig. 2, we see a perfect
agreement of analytic and numerical results for a range of Q-values
where the cross section changes by five orders of magnitude.
The ratio of numerical and analytic cross sections is shown
in the lower pane of Fig. 2. We see that the agreement is
between a fraction of permille and a few percent for all values
of Q considered. We reiterate that we plot the NNLO
correction to the differential cross section and not the full cross
section at NNLO. Given that the NNLO contribution changes the
NLO result by about 10%, the permille to percent precision
−5
−4
−3
−2
−1
−5
−4
−3
−2
−1
−4
−3
−2
−1
−4
−3
−2
−1
Fig. 3 Upper panes Rapidity distribution of the vector boson, rapidity
distribution of a lepton and pT distribution of a lepton at different orders
of perturbation theory. Lower panes the ratio of NNLO/NLO
prediction for a given observable. Plots on the left the runtime of O(10) CPU
on the NNLO correction leads to almost absolute precision
for physical cross sections and simple kinematic
distributions. We will further illustrate this point below. Before doing
so, we note that we found a similar level of agreement for
individual color structures and for individual contributions to
the final result. We also note that although we report results
for a single scale choice here, using the results in the
previous sections and the known amplitudes for qq¯ → e+e− + X ,
it is easy to check analytically the scale dependence of our
result against the one reported in Ref. [87]. Full agreement
is found.
hours; plots on the right the runtime of O(100) CPU hours. Note that the
dip in the ratio of NNLO/NLO lepton pT distribution at pT ∼ 25 GeV
is a physical feature and not a fluctuation
As we mentioned in the Introduction, one of the
important issues for current NNLO QCD computations is their
practicality. For example, with the increasing precision of
Drell–Yan measurements, one may require very accurate
theoretical predictions for fiducial volume cross sections. It is
then important to clarify whether a given implementation of
the NNLO QCD corrections can produce results that satisfy
advanced stability requirements and, if so, how much CPU
time is needed to achieve them.
To illustrate this aspect of our computational scheme, we
show the rapidity distribution of the dilepton pair, the rapidity
distribution of a lepton, and the lepton transverse momentum
distribution in Fig. 3. The plots on the left and on the right
provide identical information: the upper panes show
nextto-leading and next-to-next-to-leading order predictions for
the respective observable, and the lower panes the ratio of
the NNLO to NLO distributions. The difference between the
plots on the left and the plots on the right is the CPU time
required to obtain them; it changes from O(10) CPU hours
for the plots on the left, to O(100) CPU hours for the plots on
the right. The different run times are reflected in different
binto-bin fluctuations seen in both plots. The bin-to-bin
fluctuations for the two rapidity distributions are at the percent-level
or better in the plots on the left, and they become practically
unobservable in the plots on the right. The situation is slightly
worse for the transverse momentum of the lepton. However,
this observable is rather delicate in the γ ∗ case, as each bin
receives contributions from a large range of invariant masses.
The introduction of a Z boson propagator will localize the
bulk of the cross section in a much smaller invariant mass
window, and lead to improved stability in this case.12
Nevertheless, the results shown in Fig. 3 imply that the numerical
implementation of our subtraction scheme allows for high
precision computations, while also delivering results that are
acceptable for phenomenology even after relatively short run
times.
11 Conclusions
In this paper we described a modification of the
residueimproved subtraction scheme [10, 11] that allows us to
remove one of the five sectors that are traditionally used
to fully factorize singularities of the double-real emission
matrix elements squared. The redundant sector includes
correlated soft-collinear limits where energies of emitted
gluons and their angles vanish in a correlated fashion. Once this
sector is removed, the physical picture of independent soft
and collinear emissions leading to singularities in
scattering amplitudes is recovered and the bookkeeping simplifies
considerably.
Using these simplifications, we reformulated a NNLO
subtraction scheme, based on nested subtractions of soft and
collinear singularities that, in a straightforward way, leads to
an integrable remainder for the double-real emission cross
section. The subtraction terms are related to cross sections
of reduced multiplicity; they can be rewritten in a way that
allows us to prove the cancellation of 1/ singularities
independent of the hard matrix elements. Once singularities
cancel, the NNLO QCD corrected cross section is written in
terms of quantities that can be computed in four dimensions.
12 The state-of-the-art comparison of this and other observables in
Drell–Yan production between different NNLO codes was presented in
Ref. [88].
Although we believe that this framework is applicable for
generic NNLO QCD computations, in this paper, for the
sake of simplicity, we studied dilepton pair production in
quark–antiquark annihilation and computed gluonic
contributions to NNLO corrections. We implemented our
formulas in a numerical program and used it to calculate NNLO
QCD corrections to the production cross section of a
vector boson in hadron collisions with a sub-percent precision.
We also showed that kinematic distributions, including the
lepton rapidity and transverse momentum distributions, can
be computed precisely and efficiently. We look forward to
the application of the computational framework discussed in
this paper to more complex processes, relevant for the LHC
phenomenology.
Acknowledgements We are grateful to Bernhard Mistlberger for
providing a cross-check on the analytic formula for the NNLO Drell–Yan
coefficient function. We would like to thank KIT and CERN for
hospitality at various stages of this project. The work of F.C. was supported
in part by the ERC starting grant 637019 “MathAm” and by the ERC
advanced grant 291377 “LHCtheory”. K.M. and R.R. are supported
by the German Federal Ministry for Education and Research (BMBF)
under grant 05H15VKCCA.
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm
ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
Funded by SCOAP3.
Appendix A: Generalized splitting functions
In this appendix, we summarize all the various splitting
functions that appear in our calculation.
We start by presenting the Altarelli–Parisi splitting
functions that are relevant for the calculation. At NLO, we only
require the leading-order splitting function
3
Pˆq(q0)(z) = CF 2D0(z) − (1 + z) + 2 δ(1 − z) .
For NNLO computations, two more splitting functions are
needed; see Eqs. (2.10) and (2.11). The first one is the
convolution of two leading-order splitting functions. It reads
! Pˆq(q0) ⊗ Pˆq(q0) (z)
= CF2 8D1(z) + 6D0(z) +
1 − z
ln z − 4(1 + z) ln(1 − z) − 5 − z .
The second is the splitting function Pˆq(q1). We emphasize that,
for our case, we need the NLO splitting function for a
continuous quark line entering the hard matrix element after the
emission of two gluons. We then only have to consider the
non-singlet NLO splitting function from which the
contribution of identical quarks is subtracted. We extract the relevant
information from Ref. [87]. Pˆq(q1) in Eq. (2.11) must then be
identified with
2 + 11x 2
D0(x ) + 6(1 − x ) ln x
Li2(1 − x )
We now list the definitions of the various splitting functions
used in the text. In these formulas, we use the definition
(a) The tree-level splitting function used in the NLO
computation is defined as
2
Pqq ⊗ Pqq NLOCV (z) = CF
6D0(z) + 8D1(z)
+ 43 π 2D0(z) − 6D1(z) − 12D2(z) − 8ζ3δ(1 − z)
8 π 2D1(z) + 6D2(z) + 3 D3(z)
32
+ 16ζ3D0(z) − 3
− 5 − z − 4(1 + z) ln(1 − z) −
× 2Li2(z) − 3 ln2 z + 6 ln2(1 − z) − π 2
+ (1 − z) 3 ln(1 − z) − 2
− 2 ln z − 2(3 + z) ln2 z
16 ln3(1 − z) − ln2(1 − z) 3 + 2 ln z
+ (1 + z) − 3
× 2π 2 − 4Li2(z) − 4Li3(z) − 4Li3(1 − z) − 4ζ3
(d) The one-loop splitting functions used in the computation
of the real-virtual limits reads
Pqloqop,i(z) = −(1 − z)Pqq (z)
− 2 Li2(1 − z) ln(1 − z) CA
+ ln(1 − z)Li2(1 − z) − Li3(1 − z)
+ (CA − CF)PqRqV,new(z) + O( 2),
1 2(1 − ) 2(1 + ) 1
2 (1 − 2 ) (1 + 2 ) (1 − z)2 + 2Li2(1 − z)
− Li2(1 − z) − ln z ln(1 − z) +
ln2(1 − z) ln z
Pqq,R (z) = CF
− 4D1(z) + 4D2(z) 2
− (1 + z) + 2(1 + z) ln(1 − z) − (1 − z)
(b) The splitting function Pqq,NLOCV (z) reads
Pqq,NLOCV (z)
= CF Pˆq(0q) 2π3 2 + (π 2 − 4ζ3)
RV,new(z) = −CF(1 − z) z + (1 + z − z ln(1 − z))
Pqq
9 ln2(1 − z) ln z
+ 6 ln(1 − z)Li2(1 − z) − 2Li3(1 − z)
(e) The tree-level splitting function used in the real-virtual
contribution is defined as
Pqq,RV,1(z) = CF 2 δ(1 − z)L1 − D0(z)
+ 2 2D1(z) − δ(1 − z)L12
2 8
− 3 δ(1 − z)L14 + 3 D3(z)
+ (1 − z) − 2(1 + z) ln(1 − z)
+ 2 (1 + z) ln2(1 − z) − (1 − z) ln(1 − z)
(f) The one-loop splitting function used in the real-virtual
contribution reads
PRV,2(z) = −CACF
− 31 δ(1 − z)L1 π 2 − 8L21 + π32 D0(z) − 8D2(z)
+ 2 !2D1(z) − δ(1 − z)L12
+ 23 δ(1 − z)L12 × π 2 − 4L21
+ (1 + z) 4 ln2(1 − z) − 6
(1 − z) − 4(1 + z) × ln(1 − z)
(g) The splitting function Pqq,R R1 reads
Pqq,R R1 (z) = CF
2 D0(z) − L 1δ(1 − z)
− 4 2D1(z) − L 21δ(1 − z)
1
+ 16 D2(z) − 3 L 31δ(1 − z)
+ 4(1 − z) ln(1 − z) − 8(1 + z) ln2(1 − z) 2
32
+ 3 (1 + z) ln3(1 − z) − 8(1 − z) ln2(1 − z)
(h) The splitting function Pqq,R R2 (z) reads
Pqq,R R2 (z) = CF
2D0(z) − 4D1(z)
+ 2(1 − z) ln(1 − z) − 2(1 + z) ln2(1 − z) 2
4
+ −2(1 − z) ln2(1 − z) + 3 (1 + z) ln3(1 − z)
(i) The splitting function Pqq,R R3 (z) reads
− 3 + 4π 2 ln(1 − z) − 32 ln3(1 − z)
Pqq,R R3 (z) = CF !2L1D0(z) − 2D1(z) − δ(1 − z)L21
− CF2 −
× ln(1 − z)Li2(1 − z)
− (1 − z) 3 ln(1 − z) ln z + Li2(1 − z)
+ !6D2(z) − 4L1D1(z) − 2L21D0(z) + 2δ(1 − z)L31
+ − 3 D3(z) + 4L1D2(z) + 4L21D1(z) + 43 L13D0(z)
28
+ (1 − z)(ln(1 − z) − L1)
2 + [(1 + z)(ln(1 − z) − L1)]
+ (1 + z) L1 + 2L1 ln(1 − z) − 3 ln2(1 − z)
2
+ (1 − z) L1 + 2L1 ln(1 − z) − 3 ln2(1 − z) + (1 + z)
2
(j) The splitting function Pqq,R R4 (z) reads
Pqq,R R4 (z) = CF
!δ(1 − z)L12 − 2D1(z)
+ !−2δ(1 − z)L13 + 6D2(z)
7 28
+ 3 δ(1 − z)L14 − 3 D3(z)
+ !(1 − z) ln(1 − z) − 3(1 + z) ln2(1 − z)
+ 134 (1 + z) ln3(1 − z) − 3(1 − z) ln2(1 − z)
+ O( 3).
2 + [(1 + z) ln(1 − z)]
(k) The splitting function Pqq,R R5 (z) reads
Pqq,R R5 (z) = CF 2 D0(z) − L1δ(1 − z)
+ 4L21δ(1 − z) − 8D1(z)
− 3 +2 z + 2(3 + z) ln(1 − z)
(l) The splitting function Pqq,R R6 (z) reads
56
Pqq,R R6 (z) = CF2 4D1(z) − 12D2(z) + 3 D3(z) 2
− 2(1 + z) ln(1 − z)
− 2(1 − z) 2 ln2(1 − z) + 2L1 ln(1 − z) + L1 + 5
2
+ (1 + z) 4 ln2(1 − z)L1 + 4 ln(1 − z)L21
+ 43 L31 − 43 π 2 + 8π 2 ln(1 − z)
− 3 2 π 2 ln z
56 ln3(1 − z) − 5 ln z + 3
2 ln3 z + 8Li2(z)
− 8 ln2(1 − z) ln z − 2 ln2 z + 3
− 16 ln(1 − z)Li2(z) − 16Li3(1 − z)
We also require the following constants:
11 137 2π 2
γ˜g( ) = 6 CA + CA 18 − 3
+ CA 82273 − 1181 π 2 − 16ζ3
γ˜g( , k⊥) = − C3A − 7C9A + O( 2),
− 17321 + π62 + 161 ln 2
+ CA − 1251461 + 1118 π 2 − ln62 + 4ζ3
+ CA − 9362047 + 122156 π 2 + 475 π 4 + ln 2
+ 1181 π 2 ln 2 + 767 ζ3 2 + O( 3).
Appendix B: Phase space parametrization
and partitioning
We consider the phase space element of the two gluons
[dg4][dg5]θ (Emax − E4)θ (E4 − E5)
and discuss its parametrization.
We take E4 = Emax x1, E5 = Emax x1x2 with x1 ∈ (0, 1)
and x2 ∈ (0, 1) and write the phase space in Eq. (B.1) as
dx1 dx2
[dg4][dg5] = x 1+4 x 1+2 x14 x22 E m4−ax4 d 45,
1 2
d 45 = 2(2π )d−1 2(2π )d−1 .
d 4(d−1) d (5d−1)
We will now introduce the parametrization of the angular
phase space. This parametrization is tailored for the process
that we are interested in,
q¯ ( p1) + q( p2) → V ( pV ) + g( p4) + g( p5),
and, as we will see, allows us to expose all the collinear
singularities in a straightforward manner. We assume that
momenta of quarks in the initial state point along the z-axis
p1,2 = E1,2 t μ ± e3μ , t μ = (1, 0, 0, 0; . . .),
e3μ = (0, 0, 0, 1; 0, 0 . . .),
and parametrize the gluon momenta as
p4 = E4 t μ + cos θ41e3μ + sin θ41bμ ,
p5 = E5 t μ + cos θ51e3μ +sin θ51 cos ϕ45bμ +sin ϕ45aμ ,
(B.5)
The (d-dimensional) unit vectors bμ and aμ are chosen in
such a way that
Given this choice, the angular phase space is written as [10,
11]
where ηi = ηi1 and ηi j = ρi j /2, D = η4 + η5 − 2η4η5 +
2(2λ − 1)√η4η5(1 − η4)(1 − η5), and
Below we present the phase space for each of the four sectors
employing the above parametrization. To this end, we will
need the following function:
N (x3, x4, λ) = 1 + x4(1 − 2x3)
−2(1 − 2λ)$x4(1 − x3)(1 − x3x4). (B.9)
d (4a5,c) =
d (4b5,d) =
d (bd−2)d a(d−3) dx3 dx4
d (db−)2 d (da−)3
d−2
d−3
(256F )− 4F0 x32x4,
(1 − x3)(1 − x3x4/2)(1 − x4/2)2
F = 2N (x3, x4/2, λ)2
F0 = 2N((1x−3, xx44//22), λ) ,
d (bd−2)d a(d−3) dx3
d (db−)2 d (da−)3
d−2
d−3
dλ
× π(λ(1 − λ))1/2+
F = (1 − x3)(41N−(xx34,/12−)(1x4−/2x,3λ(1)2− x4/2)) ,
1
F0 = 4N (x3, 1 − x4/2, λ) .
(256F )− 4F0 x32x42,
It turns out that the angular phase spaces for sectors a and c
and for sectors b and d are identical. The results read
Furthermore, it is beneficial to remove singular dependence
on λ in Eqs. (B.10) and (B.11) by changing variable λ → y
as
To deal with one singularity at a time, we have to partition
the phase space. We write
w14,15 =
w24,25 =
F −
F −
ρ24ρ15ρ45 , w24,15
= d4d5d4512
= d4d5d4521
We have introduced the following notation:
d4512 = ρ45 + ρ41 + ρ52.
In Eq. (B.13), the term w14,15 corresponds to the
triplecollinear sector where singular radiation occurs along the
direction of the incoming quark with momentum p1, w24,25
to the triple-collinear sector where singular radiation occurs
along the direction of the incoming antiquark with
momentum p2, and w14,25 and w15,24 to the double-collinear sectors.
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