#### Boosted top production: factorization and resummation for single-particle inclusive distributions

Andrea Ferroglia
2
Simone Marzani
3
Ben D. Pecjak
3
Li Lin Yang
0
1
4
0
Collaborative Innovation Center of Quantum Matter
,
Beijing, China
1
Center for High Energy Physics, Peking University
,
Beijing 100871, China
2
New York City College of Technology
, 300 Jay Street, Brooklyn,
NY 11201, U.S.A
3
Institute for Particle Physics Phenomenology, University of Durham
, DH1 3LE Durham,
U.K
4
School of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University
,
Beijing 100871, China
We study single-particle inclusive (1PI) distributions in top-quark pair production at hadron colliders, working in the highly boosted regime where the top-quark pT is much larger than its mass. In particular, we derive a novel factorization formula valid in the small-mass and soft limits of the differential partonic cross section. This provides a framework for the simultaneous resummation of soft gluon corrections and small-mass logarithms, and also an efficient means of obtaining higher-order corrections to the differential cross section in this limit. The result involves five distinct one-scale functions, three of which arise through the subfactorization of soft real radiation in the small-mass limit. We list the NNLO corrections to each of these functions, building on results in the literature by performing a new calculation of a soft function involving four light-like Wilson lines to this order. We thus obtain a nearly complete description of the small-mass limit of the differential partonic cross section at NNLO near threshold, missing only terms involving closed top-quark loops in the virtual corrections.
Contents
1 Introduction 2 3 4
Kinematics and factorization
Factorizing soft real radiation in the small-mass limit
3.1 NLO phase space integrals and momentum regions
3.2 All-order factorization in the small-mass limit Fixed-order expansions and resummation Heavy-quark loops Conclusions
A The massless soft function to NNLO
Matching coefficients and anomalous dimensions
Introduction
Nowadays, top-quark production is of great interest in elementary particle phenomenology
at hadron colliders. This is due to the fact that top-quark physics is closely connected to
the study of the recently discovered Higgs boson [1, 2] and to the search for new particles.
Millions of top-quark pair events have already been produced at the Large Hadron Collider
(LHC). For this reason, the ATLAS and CMS collaborations were able to measure the
topquark pair production cross section with remarkable precision, e.g. [38]. On the theoretical
side, precise measurements require calculations of the measured observables which include
corrections beyond the next-to-leading-order (NLO) in QCD. As an example, the total cross
section, which can be measured with a relative error of approximately 5%, was recently
evaluated at next-to-next-to-leading order (NNLO) in perturbation theory [9].
Differential distributions, such as the pair invariant mass distribution, the top-quark
rapidity distribution, and the distributions with respect to the transverse momentum (both
of the individual top quark or of the tt system) are also of great interest, especially in the
search for new physics. The ATLAS and CMS collaborations already measured several
differential distributions [10, 11]. To date, the full set of NNLO QCD corrections to these
observables is not known. However, studies of the soft gluon emission corrections to the
toppair invariant mass distribution up to next-to-next-to-leading-logarithmic (NNLL) accuracy
were presented in [12, 13]. In those works, the resummation of the soft corrections was
carried out in momentum space by employing methods developed in [1416]. In the same
papers, approximate formulas including all of the terms proportional to logarithmic (plus
distribution) corrections up to NNLO were obtained starting from the NNLL resummation
formulas. A study of the top-quark transverse momentum and rapidity distributions within
the same approach was carried out in [17].1 The NNLL resummation of the transverse
momentum distribution of the tt system, which presents additional technical complications
with respect to the two distributions mentioned above, was considered in [20, 21].
A kinematic situation of special interest for new physics searches is the so-called
boosted regime, where the top quarks are produced with energies much larger than their
mass. Examples of boosted top production include the differential distribution at high
values of pair invariant mass M , or the high-pT tail of the top-quark transverse
momentum distribution. The presence of new heavy particles decaying into pairs of energetic top
quarks could generate bumps or more subtle distortions of differential distributions in this
kinematic region. The LHC at center-of-mass energies of 7 TeV and 8 TeV has started to
explore boosted top production experimentally, and more data will become available with
the future 14 TeV run. At the same time, highly-boosted production is characterized by
energy scales much larger than the top-quark mass, and QCD calculations of the differential
cross sections must take this into account.
A factorization formalism appropriate for describing QCD corrections to the pair
invariant mass distribution in the limit M mt was put forth in [22], opening up the
opportunity to resum simultaneously soft-gluon corrections and small-mass logarithms of
the form ln(mt/M ). This same formalism can be used as a way of simplifying the
calculation of higher-order corrections in the small-mass limit, a fact exploited in [23] to obtain an
NNLO soft plus virtual approximation to the invariant mass distribution in this limit. The
goal of the present work is to develop the framework necessary for describing the
highlyboosted limit of single-particle inclusive (1PI) distributions, for instance the pT mt
region of the top-quark transverse momentum distribution. To this end, we derive a
factorization formalism appropriate for describing the double soft and small-mass limit of the
differential partonic cross section.
Our results have some common ground with those for the pair invariant mass
distribution in [22], a fact which is particularly clear when deriving results in the double soft
and small-mass limit using those in the soft limit as a starting point. In the soft limit, the
partonic cross section for top-quark pair production factorizes into a hard function, related
to virtual corrections, and a soft function, related to real emission in the soft limit [13, 17
19, 2430]. The hard function is common to both cases, while the soft function depends on
the observable. The small-mass factorization of the virtual corrections to 1PI observables
can thus be taken directly from [22]. It involves the virtual corrections calculated with
mt = 0, in the form of a massless hard function, and a second function encoding all mt
dependence and related to collinear divergences in the small-mass limit.
On the other hand, the small-mass factorization of soft real radiation for 1PI
observables in top-quark pair production has not yet been discussed in the literature,2 and turns
1Approximate NNLO formulas for the same observables obtained by means of standard Mellin space
resummation methods can be found in [18, 19].
2Note, however, that the single-hadron inclusive cross section at large values of the transverse momentum
out to be rather different than that for the pair invariant mass distribution. A main
result of our paper is that such real radiation factorizes into three component functions, as
shown in (2.15) below. The physical interpretation is that soft radiation collinear to the
observed top quark, soft radiation collinear to the unobserved anti-top quark, and wide
angle soft emission are decoherent and factorize. We make a technical distinction between
these different kinds of soft radiation in two ways: diagrammatically, through the method
of regions, and at the operator level, in terms of Wilson loops. Our final results associate i )
wide-angle soft emission with a Wilson loop built out of four light-like Wilson lines and
involving a delta-function constraint particular to 1PI observables; ii ) soft radiation collinear
to the top quark with the Wilson loop defining the soft part of the heavy-quark
fragmentation function [32] (this is equivalent to the partonic shape-function from B meson decays);
and iii ) soft radiation collinear with the anti-top quark with the Wilson loop defining the
heavy-quark jet function introduced in [33]. While the massless soft function involving
four light-like Wilson lines is a matrix in color space, the two types of soft-collinear objects
are color diagonal.
With this factorization at hand, one can resum soft and small-mass logarithms at the
level of the differential partonic cross section by deriving and solving renormalization-group
(RG) equations for the five component functions, or else use it as a tool for simplifying the
calculation of higher-order corrections in this limit. In fact, of the five component functions
mentioned above, only the massless soft function has not yet been calculated to NNLO;
we build on the literature by performing this computation here. We thus achieve a nearly
complete NNLO soft plus virtual approximation to the differential partonic cross section
in the small-mass limit. The final missing piece is the NNLO virtual corrections involving
closed heavy-quark loops and proportional to powers of nh = 1 for the top quark. We leave
an analysis of these corrections to future work, emphasizing their potential complications
on the factorization formalism in the small-mass limit. Even in their absence, our results
represent the most complete fixed-order calculation of the pT distribution for boosted
production performed so far. They go beyond the approximate NNLO formulas derived
in [17, 18] by determining the non-logarithmic (delta-function) coefficient in addition to
the logarithmic plus distribution terms. They are also consistent with them, and the fact
that the NNLO logarithmic plus distribution contributions obtained with the two methods
are identical in the small-mass limit is a strong check on our factorization formalism.
The paper is organized as follows. In section 2 we introduce some notation and provide
the factorization formula for the partonic cross section in the double small-mass and soft
limit. We then devote section 3 to details of factorizing soft real radiation in the
smallmass limit. In section 3.1 we analyze NLO phase space integrals in this limit using the
method of regions, identifying three distinct momentum configurations which appear at
leading power. In section 3.2 we discuss the all-order factorization of the soft function into
a convolution of component parts related to these three momentum regions. In section 4
we give expressions needed for fixed-order expansions and present RG equations needed to
resum logarithmic corrections. We discuss subtleties related to closed top-quark loops in
of the produced hadron was recently studied in [31].
section 5, and conclude in section 6. Some details of the NNLO calculation of the massless
soft function for 1PI kinematics are given in appendix A, while explicit expressions for the
anomalous dimensions and matching coefficients are collected in appendix B.
Kinematics and factorization
We consider the scattering process
N1(P1) + N2(P2) t(p3) + t(p4) + X ,
where N1 and N2 indicate the incoming protons (at the LHC) or proton and anti-proton (at
the Tevatron), while X represents an inclusive hadronic final state. In the Born
approximation and also to leading order in the soft limit we will deal with later on, two different
production channels contribute to the partonic scattering process (2.1): the quark-antiquark
annihilation and gluon fusion channels. The partonic processes which we will analyze in
detail are thus
q(p1) + q(p2) t(p3) + t(p4) + X (k) ,
g(p1) + g(p2) t(p3) + t(p4) + X (k) ,
where X contains any number of emitted partons. The relations between the hadronic
momenta (Pi) and the momenta of the incoming partons (pi) are p1 = x1P1 and p2 = x2P2.
At the hadronic level, we define the Mandelstam variables as
s = (P1 + P2)2 , t1 = (P1 p3)2 mt2 ,
u1 = (P2 p3)2 mt2 ,
while the corresponding quantities at the partonic level are given by
It will also be useful to introduce the variable
s = x1x2s , t1 = x1t1 ,
u1 = x2u1 .
s4 = s + t1 + u1 = (p4 + k)2 mt2 .
Momentum conservation implies that s4 = 0 at Born level (k = 0).
We will be interested in the double differential distribution with respect to the
transverse momentum pT and rapidity y of the top quark in the laboratory frame. Such 1PI
observables are obtained by integrating over the phase space of the unobserved anti-top
quark, along with any extra real radiation. The pT and rapidity are related to the hadronic
invariants (2.3) according to
t1 = s m ey ,
u1 = s mey ,
where m = qp2T + mt2. Using (2.4) allows one to express the partonic Mandelstam
variables in terms of the pT , y, x1, x2. Then, assuming factorization in QCD3 and ignoring
3We should mention however potential subtleties in two-to-two processes pointed out in [3437].
fi/N1 (x1, f ) fj/N2 (x2, f ) Cij (s4, s, t1, u1, mt, f ) ,
where the fi/N are universal non-perturbative PDFs for the parton i in the hadron N
and the hard-scattering kernels Cij are related to the partonic cross section and can be
calculated perturbatively as series in the strong coupling constant. In addition, the lower
limits of integration are given by
The hard-scattering kernel is a function of the kinematic invariants needed to describe
the differential cross section. As long as these invariants are parametrically of the same
order, an expansion of the Cij in fixed orders of the strong coupling constant is appropriate.
An interesting situation arises when there is a large hierarchy among two or more of the
kinematic invariants. In that case it is often possible to factorize the hard-scattering kernel
into a product of simpler functions depending only on a single mass scale, up to corrections
in the small ratio of disparate scales. This factorization is useful for two reasons. First,
the component functions are typically easier to calculate than the full hard-scattering
kernels. Second, the factorization formula can be used as a starting point for resumming large
logarithmic corrections in the ratio of scales which appear in the higher-order
perturbative corrections.
An example often considered in the literature is the soft gluon emission limit, where the
partonic invariants satisfy the parametric relation s4 mt2, s, t1, u1. In this limit the Cij
factorize into a matrix product of a hard function Himj and a soft function Simj as follows:4
Such a factorization was first derived in [38]. The hard and soft functions are two-by-two
matrices for the qq channel, and three-by-three for the gg channel; the matrix structure is
related to the mixing of a basis of color-singlet amplitudes through soft gluon exchange.
The hard function is related to virtual corrections, and the soft function is related to real
emission in the soft limit. Real emission in the soft limit is considerably easier to calculate
than in the generic case. The eikonal factors related to soft gluon emissions exponentiate
into Wilson lines, and at the level of the squared matrix element form a gauge invariant
Wilson loop operator. Much is known about the perturbative properties of such Wilson loops.
In position, Laplace, or Mellin space they contain a series of double logarithmic corrections.
4The superscript m in (2.9) indicates that the hard and soft function are evaluated as exact functions
of mt, as opposed to the corresponding functions calculated with mt = 0 and used below.
In momentum space, these translate into logarithmic plus distribution and delta-function
corrections. In particular, defining expansion coefficients of the hard-scattering kernels as
Ci(j2)(s4, s, t1, u1, mt, f ) + . . . ,
the n-th order term contains a tower of logarithmic plus distributions, a delta function
term, and regular terms in the s4 0 limit. Consider for instance the NNLO coefficient,
which is currently not known. It has the form
Pn(s4)
where the plus distributions
are defined by
g(s4) =
Soft-gluon resummation at NNLL can be used to determine the coefficients Di of the
plus-distribution contributions [17, 18]. The delta-function coefficient, which is formally of
NNNLL order, is unknown, as is the term R, which is non-singular in the s4 0 limit and
is related to hard gluon emission.
In this paper we will discuss the application of the factorization formula (2.9) in the
soft limit to the high-pT region of the double differential cross section, where mt pT .
Producing the top quark with high transverse momentum requires that the partonic
centerof-mass energy be large, so in this regime the generic situation is that s4 mt2 s, t1, u1.
We will refer to such a hierarchy of scales as the double soft and small-mass limit. In this
limit it is possible to factorize the hard and soft functions themselves. We explain the form
of this factorization in the remainder of the section. To simplify the discussion we ignore
for the moment contributions from closed top-quark loops appearing in virtual corrections.
The factorization of the hard function in the small-mass limit was derived in [22].
It reads
where xt t1/s and we used momentum conservation u1/s = 1 xt (valid at Born
level and also in the soft limit) in order to simplify dependence on the Mandelstam
variables. This factorization can be thought of as a division of the virtual corrections into two
momentum regions. The hard matrix appearing on the right-hand side is related to the
virtual corrections evaluated with mt = 0 and receives contributions from loop momenta
whose virtuality is at the scale s, while the coefficient function CD contains all the collinear
singularities appearing in the limit mt 0 and receives contributions from loop momenta
with virtuality at the scale mt2. The factorization thus separates physics from the widely
separated scales mt2 s. Two different ways of deriving (2.14) were discussed in [22].
The first relied on the factorization of the heavy-quark fragmentation function in the soft
limit [3942], and the second used the factorization formula [43] (see also [44]) relating
massive amplitudes in the small-mass limit to their massless counterparts.
The factorization of the soft function in the small-mass limit is more subtle. Compared
to the factorization of the hard function and even the analogous factorization for the soft
function appearing in the top-pair invariant mass distribution [22], a complication here is
that the soft function in the small-mass limit is characterized by three distinct momentum
scales rather than two. In the next section, we derive the following result:
Simj (s4, s, t1, u1, mt, ) =
+ O(s4/mt2) + O(mt2/s) .
At leading order, each of the above functions is a delta function in its first argument. At
higher orders, the three functions on the right-hand side are characterized by logarithmic
corrections at the scale shown in their second argument and following from the
parametric relation i s4. Before moving on, we discuss the interpretation of each of these
component functions.
First, the massless soft function Sij is related to wide-angle soft real emission
corrections to the partonic processes (qq, gg) QQ, where q and Q are massless distinct quarks.
Such emissions are associated with a characteristic mass scale s s4/s. This massless
soft function is analogous to that entering the factorization formula for the invariant mass
distribution in the mt 0 limit and calculated to NNLO in [45]. In fact, we will be able
to construct results for the Sij to NNLO using calculations from that paper.
Second, the function SD describes soft emissions which are simultaneously collinear
to the observed top quark. The characteristic scale for such soft-collinear emissions is
d mts4/s. This function has to do with the soft part of the perturbative
heavyquark fragmentation function. Field theoretically, it is related to a Wilson-loop operator
closing at infinity and containing a finite segment with light-like separation. It is equivalent
to the partonic shape function familiar from inclusive B decays, and was calculated to
NNLO in [46].
Finally, the function SB describes soft emissions which are simultaneously collinear to
the unobserved anti-top quark. The characteristic scale for this function is b s4/mt
(note that s4 mt2 is important in this context). Our analysis shows that this function is
the so-called heavy-quark jet function introduced in [33] and calculated to NNLO in [47].
This function is very similar to Wilson-loop operator used in defining SD, the difference
being that it contains a finite segment with a time-like separation instead of a light-like one.
By combining (2.14) and (2.15) one arrives at the factorized form of the hard-scattering
kernel valid in the double soft and small-mass limit. The only subtlety is the treatment of
terms proportional to nh = 1 and related to top-quark loops. For the counting s4 mt2,
such contributions appear only in virtual corrections and modify (2.14). We discuss them
in more detail in section 5.
Factorizing soft real radiation in the small-mass limit
In this section we discuss the factorization of soft real radiation in the small-mass limit.
Such a factorization is equivalent to that of the massive soft function (2.15). We make this
clear in the preliminary discussion below, introducing some notations and definitions in
the process. We then approach the small-mass factorization in two steps. In section 3.1,
we perform a diagrammatic factorization at NLO using the method of regions. Then, in
section 3.2, we explain how to encode the all-order contributions from these regions in
terms of three distinct Wilson-loop operators.
The massive soft function can be defined as
where dR = Nc in the quark annihilation channel and dR = Nc2 1 in the gluon fusion
channel, with Nc = 3 colors in QCD. The final state X is built of soft gluons in the massive
theory (i.e. that relevant for the kinematic limit s4 mt2, s, t1, u1),5 and
Osm(x) = Svm1 Svm2 Svm3 Svm4 (x)
is a Wilson loop operator built out of soft Wilson lines
Z
Svmi (x) = P exp igs
where vi is the velocity vector associated with parton i. For i = 1, 2 we have vi2 = 0,
while for i = 3, 4 we have vi2 = 1. We have made use of the basis-independent color-space
formalism of [48] in our definitions. This allows us to deal simultaneously with the two
cases (qa1 qa2 , ga1 ga2 ) ta3 ta4 , where ai is the color index of the parton with velocity vi.
Details on how to convert products Ti Tj Pa TiaTja to the basis-dependent matrices
used in (2.9) can be found, for instance, in [45], and we do not repeat them here. For
now we just mention that the amplitude of the scattering process is represented by an
abstract color-space vector |Mi, and the generators Tia act on these vectors according to
rules specific to whether i is a quark or gluon, and in the initial or final state. For example,
5Here and in the remainder of the paper we avoid notational clutter by dropping the hat on the partonic
state X .
where we have used the identification of and to bring the definitions of the normal
Wilson lines to the convention in the literature. Other important properties of the color
generators are that Ti Tj = Tj Ti for i 6= j, and that Ti Ti = Ci, with Ci = CF for quarks
and antiquarks and Ci = CA for gluons. In addition, amplitudes satisfy color conservation,
While this is often expressed as a relation between the generators, Pi Tia = 0, it is
important to keep in mind that it holds only when acting on a color-singlet vector, as above.
The soft function takes into account real radiation in the soft limit. We illustrate
this by considering the structure of NLO phase-space integrals for single-particle inclusive
observables. The three-body phase space for a final state containing the top-quark pair
and a gluon with momentum k is
(PS)3 = Z
ddp3 ddp4 ddk
(2)d1 (2)d1 (2)d1 (2)d (d)(p1 + p2 p3 p4 k)
the Wilson lines Sv3 and Sv4 in (3.3) are converted to
igs
Svm3(x) ,
Svm4 (x) ,
We wish to integrate over the unobserved momenta p4 and k. To do so, we use a technique
introduced in [49]. The idea is to shift integration variables to p4k = p4 + k and then split
the phase space up into two Lorentz invariant pieces: that for the two-to-two production
p1 + p2 p3 + p4k, and that for a subsequent two-body decay p4k p4 + k. We thus write
(PS)3 = (2)12d3 Z
Z ddk ddp4 +(k2) +(p24 mt2) (d)(p4k p4 k) .
ddp3 ddp4k ds4 +(p23 mt2) +(p24k mt2 s4) (d)(p1 +p2 p3 p4k)
After a trivial integration the piece on the second line can be written as an integral over
the unobserved gluon momentum:
(PS)k = Z ddk +(k2) +(s4 2p4k k) .
The piece on the first line can be arranged into a form appropriate for describing the double
differential 1PI observables and is unimportant for what follows.
To evaluate the NLO real emission corrections to the differential cross section one
integrates the squared matrix element over the phase space (3.8). The structure of these
phase-space integrals simplifies in the soft limit k 0, in which case s4 mt2, s, t1, u1.
In this limit one can replace the squared matrix element by eikonal factors for a gluon
emission from each leg, approximate p4k p4 in the delta-function constraint, and drop
any k dependence in the matrix element arising from the shift p4 p4k k. One must
then evaluate integrals of the form6
Iimj = 1+eE 2 Z ddk+(k2) +(s4 2p4k k)
We have introduced factors convenient for the MS renormalization scheme, and absorbed
them into the integral measure [dk] defined on the second line. The quantity = (4
d)/2 is the dimensional regulator. These integrals are exactly those appearing in the
NLO corrections to the soft function (3.1), which shows explicitly its connection with real
radiation. In fact, the NLO bare soft function is calculated by associating a color factor
Ti Tj with each integral and summing over possible attachments to the partons i, j. A
first step to factorizing the soft function in the small-mass limit is thus to understand the
structure of the integrals (3.9). We turn to this problem in the following subsection, using
the method of regions as a tool for performing a diagrammatic factorization.
We end this section with some comments concerning the arguments of the massive soft
function (3.1). The Wilson lines entering its definition depend on the velocity vectors vi,
so the object on the left-hand side depends on invariants formed from the velocities and
p4 = mtv4. In order to keep contact with our physical picture of the soft function as
representing soft real radiation, we express these scalar products in terms of the Mandelstam
variables. However, in studying the properties of the integrals it is sometimes useful to
keep the structure of the scalar products explicit. For instance, by considering properties
of the integrals (3.9) under simultaneous rescalings of the different vectors and s4 (see, for
instance, [50]) one finds their general functional form is
Iimj =
NLO phase space integrals and momentum regions
The NLO integrals (3.9) were evaluated for arbitrary mt in [17]. Here we are interested in
the asymptotic expansion of those integrals in the small-mass limit, where mt2 s, t1, u1.
To leading order in mt2/s the results are
I1m2 =
2
6The integrals Iij in (3.9) are connected to the position space integrals Iij in eq. (20) of ref. [17] through
relation
Iij(x0) = 2 (4eE ) Z0 ds4 exp mteE se4(L0/2) Iimj (s4) .
I3m3 =
I3m4 =
I4m4 =
2
2
2
2
In the above equations, we have defined xt = 1 xt. These explicit results make clear that
some of the integrals are characterized by a single mass scale, while some of them depend
on more than one mass scale and contain logarithms of mt2/s.
We will now show how to reproduce these results using the method of regions [51].
This allows us to factorize the multiscale integrals into a sum of simpler, one-scale integrals.
While this method was originally developed to construct the asymptotic expansions of loop
integrals and is usually discussed in that context, it applies equally well to the phase-space
integrals considered here. At the technical level, the reason for this is that integrals such
as (3.9) are equivalent to loop integrals, since one can rewrite the delta-function constraint
as the discontinuity of propagators (see for example [52]). Rather than actually doing this,
one can simply apply the normal procedure for expanding loop diagrams by regions to
the phase-space integrals directly. This proceeds as follows. First, one defines a region by
associating a specific scaling to the components of the undetermined momentum k in terms
of the external expansion parameter (in our case mt2/s) . One then expands the integrand
as appropriate for the particular momentum region, and integrates over the whole phase
space. After finding all of the possible momentum regions which contribute at a given
power, one adds their contributions together to obtain the asymptotic expansion of the
full integral.
The exact scalings of the regions which contribute to the integrals (3.9) in the
smallmass limit are perhaps not obvious at first sight. However, physical intuition suggests three
possibilities: wide-angle soft emission, soft emission collinear to the observed top quark,
and soft emission collinear to the unobserved anti-top quark. The regions analysis below
shows that this is indeed correct, and moreover fixes the momentum scale associated with
each of these regions.
To discuss the momentum regions, let us first introduce four light-like vectors n1, n2,
n3 and n4, whose space components are aligned with the momenta p1, p2, p3 and p4k,
respectively. For convenience we normalize the vectors to satisfy n1 n2 = n3 n4 = 2. The
other scalar products are then fixed to n1 n3 = n2 n4 = 2xt and n1 n4 = n2 n3 = 2xt.
Picking two reference vectors ni and nj , we define the light-cone decomposition of an
In the following we drop the ij labels when there is no danger of confusion. A judicious
choice of the light-cone vectors for a given integral can significantly simplify calculations, as
will become evident in the examples below. For the discussion of regions, it is particularly
convenient to choose i = 3 and j = 4. The scaling of the momentum p4k in the limit
s4 mt2 s is then given by
p4k = (p4k+, p4k, p4k) s(2, 1, 0) ,
We can use the relations
v1 v2 (1, 1, 1) ,
arbitrary four-vector k as
k+ij =
kij =
Iisj =
ksc
ksc sm4pt24 s ms4t2 (2, 1, )
(soft, collinear to the top),
(soft, collinear to the anti-top).
However, not every region contributes to each integral. For example, it is clear from
powercounting that the sc (i.e. soft, collinear to the top) region only contributes to integrals
involving v3, while the sc (i.e. soft, collinear to the anti-top) region only contributes to
integrals involving v4. In the following, we structure our discussion by analyzing how the
three regions contribute to the list of integrals in (3.11).
Wide-angle soft emission. We first discuss the wide-angle soft region. In this region,
to leading power in , we can approximate 2p4k k sn4 k in the delta function and also
for the propagators. The contribution to the integral Iimj from the soft region is then given
by the integral
Iisj =
2
The above result has several important features. First, the mass scale on which it depends
is characterized by s s4/ s |ks|. The scaling of ks enforced under the integrand
determines the mass scale in the integral. Second, the dependence on the light-cone vectors
ni is of the form required by (3.10). Finally, the result is non-zero only if i 6= j, and if
i, j 6= 4, because otherwise one of the scalar products in (3.19) vanishes and the prefactor
is zero in dimensional regularization. We can see this also in intermediate results. An
explicit example is the following integral:
I1s4 =
= Z
The equality follows because the integral is scaleless, as one can verify by choosing n1 and
n4 as basis vectors for the light-cone coordinate decomposition (3.12) and then integrating
over the transverse and ks n4 components. The reason this happens is that when a gluon
connects partons i and j, the more precise definition of wide-angle soft is
ksij s4s (ni + nj ) .
If, say, nj = n4, one must drop its contribution inside the delta-function constraint, in
which case the square of the soft momentum vanishes and the integral is scaleless.
The total contribution from the wide-angle soft region to the NLO phase-space integrals
is obtained by associating a factor of Ti Tj with each integral Iisj and summing over legs.
The result is proportional to
Is = 2T1 T2 I1s2 + 2T1 T3 I1s3 + 2T2 T3 I1s2 .
The contributions above are derived from the general integrals after the replacement vi
ni. The time-like vectors are expanded out into light-like ones, which corresponds to
calculating real emission corrections with massless partons. We use this fact in the next
section to define the massless soft function, and calculate it to NNLO in appendix A.
The NLO result for the bare soft function is exactly that given in (3.22), showing the direct
correspondence between the operator definition and regions calculation.
7A step-by-step derivation is given in [50].
Soft emission collinear to the top quark. We next consider soft emission which is
simultaneously collinear to the top quark. We call this region soft-collinear or simply sc.
The scaling of soft-collinear momenta is ksc p3 s4/s. In contrast to the wide-angle soft
region, to expand the integrand in the soft-collinear region we must keep the mt-dependence
in the parameterization of p3, i.e, p3 = sn3 /2 + n4 mt2/2s, such that v32 = p32/mt2 = 1.
For all other velocities vi with i 6= 3, it is enough to know that vi v3 viv3+/2 and
vi ksc viks+c/2, no further specifications are needed.
We can now consider contributions from the soft-collinear region, starting with that to
I1m3. Using the scalings in (3.15) and (3.16b) to perform the expansion under the integrand,
we find to leading order in :
I1s3c = Z [dk] +(s4 sks+c) (v3+ksc +2vv3+3ks+c) ks+c
where the second equality follows after a straightforward integration. Note that the integral
is characterized by the single mass scale 2 ks2c, and that the scaling of ks+c is such that
the two terms in the delta-function are of the same order. Moreover, the integral contains
no information about the velocity v1. It is therefore easy to show that I1s3c = I2s3c = I3s4c. The
only other contribution from the soft-collinear region is to I3m3, which is in fact saturated
by that region:
I3s3c = Z
In the second equality we used color conservation (3.5), after which one sees that the
contribution is diagonal in color space. Furthermore, the expansion in the soft-collinear
region is such that the delta-function constraint has the form (s4 sn4 ksc), i.e. the
constraint vector n4 is light-like. We will see in the next section that both of these features
are important when identifying the contributions of this region with the soft part of the
heavy-quark fragmentation function SD defined in (3.49) below. In fact, one can check
that the NLO bare contributions to that function are exactly reproduced by (3.25), which
is especially obvious after writing down the NLO integrals using the Feynman rules for
Wilson lines and noting the correspondence with the integrands expanded in the
softcollinear region.
Soft emission collinear to the anti-top quark. Finally, we consider soft emission
which is simultaneously collinear to the unobserved anti-top quark. The scaling of such
sc momenta is ksc s4p4 /mt2. To perform an expansion in this region we parametrize
p4 = sn4/2 + n3mt2/2s, and then v42 = p42/mt2 = 1. For all other velocities vi with i 6= 4,
we can approximate vi v4 vi+v4/2 and vi ksc vi+ksc /2.
The analysis of the contributions from the sc region to the integrals is very similar to
that of the sc region. Using the scalings (3.15) and (3.16c), the contribution to I1m4 from
this region is:
I1s4c = Z
2
The final equality follows from direct integration using the standard techniques. One sees
that I1s4c = I1m4. The integrals show familiar features: the particular scaling of ksc (3.16c)
ensures that the two terms in the argument of the delta-function scale the same, and the
characteristic scale is 2 ks2c . Moreover, the integral depends only on quantities related
to parton 4. One can show that I2s4c = I3s4c = I1s4c . Furthermore, I4s4c = I4m4. The total
contribution of this region to the NLO soft function is thus proportional to
As was the case with the emissions collinear to the top, the total contribution is color
diagonal. The two regions do not, however, give identical contributions. The reason for
this is that while after expansion in the sc region the delta-function constraint involves
a light-like vector n4, in the sc region the delta-function constraint involves the
timelike vector v4. Instead of being related to the heavy-quark fragmentation function, these
contributions are related to a different object, the heavy-quark jet function SB defined
in (3.55) below. Here again one can check that the NLO contributions to SB are exactly
those arising from (3.27).
Comments. To summarize, we have found three distinct momentum regions: soft,
associated with the scale s s4/s; soft and collinear to p3, associated with the scale
sc mts4/s; and soft and collinear to p4, associated with the scale sc s4/mt.
Although all of these scales vanish in the limit s4 0, the method of regions provides a
technical way of separating out their contributions to the soft function, and identifying the
exact mass scale associated with them.
One could use the regions method to prove the factorization formula (2.15) to all
orders diagrammatically. It is convenient instead to use effective field theory to reorganize
contributions from the different regions into field-theoretical objects encoding their all-order
structure, a problem we turn to next. In either case, one might wonder if the three regions
identified here are sufficient also at higher orders. Our explicit checks on factorization
described below have shown that this is the case at least to NNLO. We have no proof
beyond that, yet also see no physical effect (other than complications from heavy-quark
loops we deal with later) that would give rise to other regions. This is an assumption in the
all-order analysis that follows, and is common to most proofs of factorization relying
on the regions method, effective field-theory based or not.
All-order factorization in the small-mass limit
Having identified the momentum regions which contribute to the phase-space integrals
in the double soft and small-mass limit, we are now in position to explore their all-order
structure. There are two possible routes to doing so. The first is to construct an appropriate
version of soft-collinear effective theory and apply it to double differential cross sections
for 1PI observables using a multistep matching procedure. Many of the steps of such a
construction can be taken over from [13], for the soft limit, and from [33], for the boosted
limit. A second, more direct route is to start from the definition of the soft function (3.1)
for arbitrary mt, factorize the QCD gluon field appearing in the single Wilson-loop operator
into a sum of fields whose Fourier components are restricted to certain regions, and then
see how the different component fields factorize into operators. We pursue this second
method here, and then comment on the alternate derivation at the end of the section.
Our aim is to decompose the operator definition of the massive soft function (3.1) into
component operators whose diagrammatic expansions encode the contributions of the three
distinct momentum regions. These operators are functions of gluon fields whose Fourier
components are restricted to the scalings appropriate for a particular region. We write the
decomposition as
The Wilson lines in the definition (3.3) then decompose into a product of three Wilson
lines containing gluons of the different scalings. This works as follows. Let us first define
Svi (x) = P exp igs
Yvi (x) = P exp igs
Yvi (x) = P exp igs
Z
By using the following identity for path-ordered exponentials:
dx (A(x) + B(x))
= P exp
it is easy to show that
Svmi (x) = Yvi (x) Svi (x) Yvi (x) ,
Svi (x) = P exp igs
Yvi (x) = P exp igs
Z
ds hvi Asac Svi Yvi TiaYvi Svi i (x + svi) .
We now need to perform a consistent power expansion in . This involves the expansion of
the gluon fields themselves as well as their momenta. In the soft-collinear effective theory,
the gluon fields scale the same as their momenta [53, 54]. It is then clear that only the +
component of the Asc field and its momentum needs to be kept when it interacts with the
As or the Asc field. The same is true for the As field when it interacts with the Asc field.
This is often called multipole expansion in the literature [54]. After this expansion, the
velocity vectors in the Wilson lines Yvi and Svi on the right side of (3.32) can be replaced
by their components along the plus direction, which in our reference system of choice is
n4, i.e. vi n4 in (3.32). We then redefine the fields as
Asa(x) Yn4,i(x)TiaYn4,i(x) Asa(x) Tia ,
Yn4,i(x) = P exp igs
ds n4 Asac(x + sn4) Tia
Z
Z
is a Wilson line along the direction of n4 but in the color representation of parton i, and
similarly for Sn4,i. To show that these field redefinitions are actually the same for each i,
we use the identity
Yn4,iTiaYn4,i = Ynb4a,adj Tib ,
which holds whether Ti is the color generator for a quark or a gluon. Here Yn4,adj is a
Wilson line in the adjoint representation along the direction of n4,
Yna4b,adj(x) = P exp igs
with (Tacdj)ab = if cab. It then follows that the field redefinition of, e.g. the soft gluon
field is
hYn4,adj(x)iab Abs(x) Asa(x) ,
which does not involve the particular color generator at all.
After the field redefinitions, the two functions in (3.32) are just usual Wilson lines
in terms of the new fields. We work with the redefined fields in the following, and drop
the tilde on the two functions. In soft-collinear effective theory, such field redefinitions
also remove the interactions among the various fields in the Lagrangian and are therefore
Sm(s4, s, t1, mt, ) =
ds dsc dsc (s4 s sc sc )
X h0|Os(0)|Xsi hXs|Os(0)|0i (s 2p4 pXs )
dR Xs
The above Wilson line operators (defined in (3.40), (3.53), and (3.56) below) arise after
a multipole expansion appropriate for the particular momentum region, according to the
rules which we explain below. This expansion ensures that the Feynman rules for the
Wilson-line attachments in the different regions are such that they produce the correct,
homogeneous expansion appropriate for the momentum region the gluon fields are restricted
to. The form of this expansion is very similar to what appeared in the regions analysis in
the previous subsection. We thus structure our discussion in a similar way, performing a
region-by-region analysis which leads to operator definitions of the objects in (3.38).
The wide-angle soft region. Let us first consider the wide-angle soft region. We
parametrize the external momenta as described in the previous subsection. The expansion
inside the delta-function and Wilson lines then reads
Note that for i = 3, 4, we have vi (s/2mt)ni . However, an important property of
Wilson lines is their invariance under rescalings of the reference vector ni ni, for
an arbitrary number , which can be verified immediately from the definition (3.3) after a
change of variables. We used this fact to eliminate factors of s/2mt. The expansion above
implies that in the wide-angle soft region we can treat all partons as massless, replacing
the time-like vectors v3 and v4 with light-like ones n3 and n4.
From the above discussion, we are led to define the massless soft function as
1
=
Os(x) = Sn1Sn2Sn3Sn4 (x) .
We calculate the massless soft function to NNLO in appendix A. Our explicit
calculations show that integrals involving parton 4 vanish to this order. It seems likely to us that
this is also true at higher-orders, but do not pursue a formal proof here.
1 n3 .
vi Asac(svi) 2 ni n3 n3 Asac sni n3 2
We again use the invariance of Wilson lines under the scaling ni ni. With an
appropriate choice of , the scalar product becomes
vi Asac(svi) n3 Asac(sn3)
irrespective of whether i = 1, 2, 4. It follows that we can replace the product of Wilson
lines as
[Yv1Yv2Yv4](x) = P exp igs
Z
Z
= P exp igs
ds n3 Asac(x + sn3) (T3a) = Yn3(x) ,
where we have used color conservation. To understand the appearance of anti-path ordering
in the second line, we note that color conservation Pi6=3 Tia = T3a only applies when acting
on the color singlet amplitude directly, as in (3.5), so one must replace, e.g.,
where we have used that Ti and Tj commute when i 6= j. On the other hand, when i = 3
we just have a standard soft-collinear Wilson line for particle 3, and no further expansion
is possible. Therefore, we can identify
Osc(x) = Yv3(x) Yn3(x) .
The squared matrix element involving soft collinear structure is then
Osc(x) = Yv3 (x) Yn3 (x) ,
with the Wilson line Yn defined as in (3.4) but with sc fields only.
We now use the Fourier representation of the delta function to write
X Z dt ei(sc/sn3pXsc )t h0|Osc(0)|Xsci hXsc|Osc(0)|0i ,
s Xsc 2
and we shift the argument of Osc:
Osc(0) = eitn3P Osc(tn3) eitn3P ,
where P is an operator acting on the external states to pick up their momenta. This
operator produces a term that cancels the pXsc dependence in the Fourier exponent, allowing
us to perform the sum over states to find
Soft emission collinear to the anti-top quark. Finally, we consider the sc region,
where pXsc p4s4/mt2. For the scalar products vi Asac , i 6= 4, we can perform exactly the
same arguments as for the sc region. The sc region then involves the operator
Osc (x) = Yn4 (x) Yv4 (x) ,
where n4 = n3 and the Wilson line Yn is defined as in (3.4) but with sc fields only. As
for the the delta-function constraint, there is no possible expansion. Therefore, the matrix
element for the sc region is
= X h0|Osc (0)|Xsc i hXsc |Osc (0)|0i (sc 2mtv4 pXsc )
Xsc
8See, e.g., appendix C of [16].
The difference between SB and SD is the time-like vector in the delta-function constraint,
as opposed to a light-like one. We can now go through the steps discussed above for SD to
arrive at the result
1 Z dt eisc t/(2mt) h0|T [Osc (tv4)] T [Osc (0)]|0i ,
which is consistent with the definition of the heavy-quark jet function in eq. (46) of ref. [47],
after making the adaptions necessary to describe a final-state antiquark.
Comments. After inserting the matrix elements (3.40), (3.53), and (3.56) into (3.38) we
arrive at the factorization formula (2.15) for the massive soft function in the small-mass
limit. We achieved this by studying the factorization of the Wilson-line definition of the
massive soft function in this limit.
Another option would have been to analyze the differential cross section in soft-collinear
effective theory through a multistep matching procedure, similarly to the analysis in [33],
where energetic top-pair production in e+e collisions was studied. In that case, after
integrating out virtualities of order s and mt, one is left with two copies of boosted HQET,
which interact only through soft cross talk. In our analysis, the sc- and sc-momenta
play the role of the residual momenta for the two copies of boosted HQET, and the soft
momenta the role of the soft cross talk. It is then evident that many steps of an
effectivetheory analysis could be carried over from [33] and lead to the same final result. We refer
the interested reader to that work for the set-up that could be used in such an
effectivetheory analysis.
Fixed-order expansions and resummation
The factorization formalism derived in this work can be used in different ways. The first is
to view it as a tool for reformulating the calculation of complicated, multiscale higher-order
corrections to the coefficient functions Cij in terms of much simpler one-scale calculations,
up to corrections to the soft and small-mass limit. In that case, we need only fixed-order
expansions of the component functions appearing in the factorization formula. However,
in the limit where the mass scales characterizing the component functions are widely
separated, for any choice of a common factorization scale f the fixed-order expansion of Cij
contains large logarithms of scale ratios which can be resummed by deriving and solving
RG equations for the component functions. In this section we collect results for the
fixedorder expansions of the component functions to NNLO, and then discuss the structure of
their RG equations.
It is simplest to discuss higher-order corrections and RG equations in Laplace space,
where the distribution-valued functions related to soft real emission become simple
functions, and convolutions reduce to multiplication. We define Laplace transforms of the
component functions as
Z
Z
Z
cij (N, s, t1, u1, mt, ) = s
We now discuss the NNLO corrections to the various component functions above. The
channel-independent functions sB, sD, and CD, all related to (soft) collinear emissions, are
particularly simple. For these, we can define coefficients as
sD(L, ) = 1 +
and similarly for sB and CD. Compact results for all of these functions to NNLO can be
extracted from the literature, and are gathered in appendix B.
For the channel-dependent, matrix-valued massless hard and soft functions, we define
perturbative expansion coefficients
s = s(0) +
Here and in the remainder of the section we suppress the subscript indicating the channel
dependence of the hard and soft functions, as well as their explicit arguments. While
the NNLO hard functions H(2) are unknown, the quantities Tr H(2)s(0) were recently
extracted in [23], using NNLO corrections from massless two-to-two scattering obtained
in [5660] along with a subtraction procedure. The rather lengthy expressions can be
found in electronic form with the arXiv version of that paper. The massless soft functions
are not available in the literature, but we construct results up to NNLO in appendix A.
Here again the results are lengthy, and are included in Mathematica files with the arXiv
submission of the present paper.
We can make use of these results to form approximations to the
Laplacetransformations of the expansion coefficients of the hard-scattering kernels defined in (2.10).
To leading order in the soft and small mass limits, we have
c(0) =
c(1) =
c(2) =
3 Tr hH(0)s(0)i ,
3 Tr hH(1)s(0) + H(0)s(1) + s(1) + s(D1) + 2CD(1)
3 Tr H(2)s(0) + H(0)s(2) + H(1)s(1)
H(0)s(1) + H(1)s(0)
CD(1) 2 + s(B1)s(D1) + 2CD(1) s(1) + s(1)
B D
The above result for the soft and small-mass limit of the NNLO coefficient c(2) is particularly
interesting because the exact coefficient in fixed-order perturbation theory is unknown.
Approximations to this coefficient based on soft -gluon resummation to NNLL order for
arbitrary mt were derived in [17, 18]. To explain how the results given here go beyond
those works, we define an explicit expansion of the coefficient function as
The NNLO approximations from [17, 18] determine the coefficients c(2,n) with n = 1 . . . 4,
as exact functions mt. From the viewpoint of a fixed-order expansion, the results for these
coefficients in the small-mass limit, determined from (4.6), do not offer an improvement.
However, it is a very non-trivial check on the factorization formalism that the coefficients
derived above agree with the small-mass limit of those from [17, 18], a fact which we have
confirmed.
The NNLO approximations from [17, 18] do not determine c(2,0), as this coefficient is
formally of NNNLL order. The expansion (4.6) determines it in the small-mass limit (up
to corrections involving heavy-quark loops, which we return to in the next section). It
will thus be interesting to study the numerical implications of our results for high-pT top
production, where corrections to the small-mass limit are negligible and the extra terms
calculated here offer a clear improvement on the NNLO approximations from [17, 18]. In
fact, our results form the basis for a full NNLO soft plus virtual approximation in the small
mass limit, meaning that they determine also the delta-function coefficient in (2.11).
The convergence of the fixed-order expansion discussed above can be invalidated when
the logarithms of scale ratios are large. In that case, one must resum the logarithms
by deriving and solving RG equations. The RG equations for the channel-independent
functions read
CF cusp(s) ln m2t2 + 2B(s) sB ln mt ,
CF cusp(s) ln m2t2 2S(s) sD ln mt ,
CF cusp(s) ln m2t2 + 2S(s) + 2q CD ln m2t2 ,
The coefficients As and s can be derived from the results above, along with the condition
that the -dependence of the partonic cross section cancels against that in the PDFs to
give a -independent hadronic cross section. We write the Altarelli-Parisi kernels in the
soft limit as
Pii(z, ) = 2Cicusp(s) + 2i(s)(1 z) .
(1 z)+
s(xt, s) = h(xt, s) + 2(s) + 2q (s)
+ B(s) + S(s) As(s) ln xt(1 xt) 1 .
The term proportional to ln xt(1 xt) is needed to cancel the -dependence of the PDFs
and follows from the derivation given in section 3.2 of [17].
These RG equations can be solved in the standard way, and in fact many of the
ingredients can be recycled directly from [22] after appropriate replacements. The perturbative
components gathered here form a starting point for an analysis of a simultaneous
smallmass and soft-gluon resummation to NNLL. We plan to return to a phenomenological
analysis of resummation effects, and a comparison with the fixed-order approximations
defined above, in future work.
A technical subtlety we now address is the treatment of closed heavy-quark loops. The
way in which they contribute to the factorization formula is determined by our parametric
counting s4 mt2. Since in that case the heavy-quark mass is much larger than any of
the scales characteristic of real radiation, it is not possible for a soft gluon to split into
on-shell top quarks. Therefore, heavy-quark loops do not contribute to any of the functions
in (2.15) related to soft real emission. Heavy-quark loops decouple from these functions,
and the correct prescription is to evaluate them in a theory with five massless flavors.
For the virtual corrections, on the other hand, there is no such decoupling of
heavyquark loops, and one must include them explicitly through diagrammatic computations.
This leads to a modification of (2.14). In general, we can write
The notation is such that the Chij contains any explicit dependence on nh = 1 and represents
the effects of closed top-quark loops, while the second and third factor in the r.h.s. of (5.1)
are the same as in (2.14) and are evaluated with nl = 5 light flavors.
A few words about the renormalization schemes are in order. The hard functions
are the Wilson coefficients arising when matching from the amplitudes in full QCD to
the ones in the effective theory. To obtain the finite hard functions, we need to perform
several renormalizations. These include the usual renormalization of the strong coupling
constant, the quark masses, the quark and gluon propagators in full QCD, as well as the
renormalization of the operators in the effective theory. We renormalize the masses and
the propagators in the on-shell scheme, and renormalize the effective operators in the MS
scheme. Furthermore, we renormalize the running coupling constant in the MS scheme with
five active flavors (in practice by first performing renormalization with six active flavors
and then applying a decoupling transformation).
An interesting question is whether the matching coefficient Ch in (5.1) can be factorized
into one-scale functions, whether it is diagonal in color space, and whether it depends on
the channel. In the absence of heavy-quark loops, the mt dependence is contained solely in
CD, and is related to regions of loop momenta collinear to p3 and p4. Top-quark loops can
introduce an mt dependence even in diagrams not involving p3 or p4, and can otherwise
change the regions analysis in such a way that not all mt dependence is related to collinear
regions. A full analysis of these higher-order corrections is beyond the scope of the paper.
However, we note that the two-loop corrections depending on nh were calculated in the
small-mass limit in [61, 62], and do not appear to factorize in a simple way, as pointed
out in [43]. The same is true of the analogous NNLO corrections calculated for Bhabha
scattering in [44].
In any case, the closed heavy-quark loops can be included in fixed-order perturbation
theory by calculating the contributions involving powers of nh to the massive hard function
in the small-mass limit. The NLO results can be extracted from [13]. To obtain the NNLO
results as a matrix would be rather involved. However, we hope to extract the contribution
of the nh-dependent part of NNLO hard function to the coefficient function (2.9) using
the the method followed in [23] in future work. This requires two main pieces related to
nh-dependent corrections. The first is the UV renormalized NNLO virtual corrections in
the small-mass limit. The two-loop contributions were given in [61, 62], but the one-loop
squared pieces are not readily available in the literature. In addition, one must determine
certain color-decomposed one-loop amplitudes to order 2. With these building blocks in
place, one can calculate the finite remainder of the nh-dependent terms and add them to
the results gathered in the previous section to achieve a full soft plus virtual approximation
at NNLO in the double soft and small-mass limit.
Conclusions
We have derived a novel factorization formula appropriate for the double soft and
smallmass limits of single-particle inclusive cross sections in top-quark pair production at hadron
colliders. This formula applies to double differential distributions in the rapidity and pT of
the heavy top (or anti-top) quark within this limit. In the absence of closed heavy-quark
loops in virtual corrections, we found that the partonic cross section factorizes into five
component functions, each depending on a single momentum scale.
Our method was to start from the factorization formula (2.9) for top-quark pair
production in the soft limit and then subfactorize the component parts as appropriate for
the small-mass limit. For virtual corrections contained in the so-called hard function the
method for doing this was already available in the literature. On the other hand, our
result (2.15) for the factorization of the soft function, related to real emission in the double
soft and small-mass limits, is new. We motivated the result by first performing a
diagrammatic factorization of real emission using the method of regions in section 3.1. Our
analysis revealed that three types of soft radiation contribute: radiation simultaneously
soft and collinear to the observed top-quark, radiation soft and collinear to the unobserved
anti-top quark, and wide angle soft emission. In section 3.2 we showed how to factorize the
general Wilson-loop operator definition of the soft function into three component
Wilsonloop operators related to these regions. We demonstrated explicitly that the two types of
collinear operators are diagonal in color space and connected the Wilson loop operators
with the soft part of the heavy-quark fragmentation function and the heavy-quark jet
function introduced in [33]. The wide-angle soft emission goes into a massless soft function
defined in (3.40). It involves a non-trivial matrix structure characteristic of soft emissions
in two-to-two scattering.
Most of the component functions entering our factorization formula could be extracted
to NNLO from results in the literature. We added to this literature by computing the
NNLO massless soft function. We showed that the anomalous dimensions appearing in the
renormalization-group equation for the NNLO function is consistent with the factorization
formalism to this order, providing a strong consistency check on the factorized formula
as well as our higher-order perturbative computation. An equivalent check, described
in section 4, is that the NNLO logarithmic corrections obtained by expanding out the
factorization formula agree with the small-mass limit of those determined by NNLL
softgluon resummation in [17, 18]. Our results provide nearly all elements for the construction
of an NNLO soft plus virtual approximation to the differential cross section in the
smallmass limit. The missing piece is the NNLO virtual corrections related to closed heavy-quark
loops, which we hope to calculate in future work.
We expect that the results obtained here will provide useful insight into the structure of
higher-order QCD corrections to 1PI observables in the boosted regime. On the one hand,
the NNLO soft plus virtual approximation can be used to study numerically to what extent
logarithmic soft gluon corrections dominate over non-logarithmic delta-function terms, thus
assessing the reliability of the NNLO approximations [17, 18] for boosted production. On
the other hand, our factorization formalism provides the starting point for a simultaneous
resummation of small-mass and soft logarithms in the partonic cross section to NNLL.
This work is supported in part by the National Natural Science Foundation of China under
Grant No. 11345001. The work of A.F. was supported in part by the PSC-CUNY Award
No. 66590-00-44 and by the National Science Foundation Grant No. PHY-1068317. S.M.
is supported the UKs STFC. B.P. is grateful to KITP Santa Barbara for hospitality and
support, and B.P. and S.M. would like to thank the ESI Vienna for hospitality and support.
The massless soft function to NNLO
The calculation of the massless soft function (3.40) proceeds similarly to [45]. The difference
in the present case is that the constraint vector is light-like instead of time-like, but since
the basic phase-space integrals were calculated for an arbitrary constraint vector, this
offers no additional complication. In fact, the end results are slightly simpler, and integrals
involving parton 4 vanish.
We first go through the NLO calculation as an example. We obtain the bare NLO soft
function through the following sum over legs:
Sb(1a)re =
s !2
s !2
legs 6= 4
w1(12) I1(a12) + w1(13) I1(a13) + w2(13) I1(a23) .
Explicit results for the matrices wi(j1) can be found in [45]. The integral I1 is defined as10
while the stripped integral I1 can be obtained from (A.2) through the relation
I1(aij ) =
Z ddk +(k2)( sn4 k)
10This integral is identical to the one found in eqs. (13-14) of [50].
The general result for the stripped integral resulting from gluon emissions associated with
partons ij is
I1(aij ) =
aij =
(Note that aij is different from the one defined in [45].) With this definition, we have
a12 =
a13 =
a23 =
The stripped integral can easily be expanded in (and shown to be equivalent to I12
from [17], which contrary to first appearance does not depend on mt when Laplace
transformed with respect to s4/s instead of s4/mt). We observe that the -expansion of the
integral I1(aij ) involves only logarithms of the argument, since the 2F1 function does not
depend on aij :
The NNLO contributions read (taking account that attachments to parton 4 vanish)
Sb(2a)re =
s !4
2w1(12) I2(a12) + CAI6(a12) + CAI7,1(a12) + CAI7,2(a12)
+ 2w1(13) I2(a13) + CAI6(a13) + CAI7,1(a13) + CAI7,2(a13)
+ 2w2(13) I2(a23) + CAI6(a23) + CAI7,1(a23) + CAI7,2(a23)
+ 2(w1(32)I3(a12) + w1(33)I3(a13) + w2(33)I3(a23))
+ w1(42) I4(a12)+2I5(a12) +w1(43) I4(a13) + 2I5(a13) +w2(43) I4(a23) + 2I5(a23)
+ 2w1(82)3I8(a12, a13) + 2w2(81)3I8(a12, a23) + 2w3(81)2I8(a13, a23) .
The integrals Ii (i {2, , 8}) are the same as the corresponding integrals in [45], except
that with respect to the results in that paper one should replace the integral argument a
according a 1 a in order to fit the notation of the present work, and then expand to
leading order for a 0.
The bare function has poles in which must be removed through a renormalization
procedure. This is most easily done in Laplace space. It is straightforward to obtain
the bare Laplace-transformed function by performing the integral in the definition (4.1).
where the renormalization factor Zs reads
A2s,20 +
We have defined expansion coefficients
As,1 + 20 (As,0L + 0s) + As,1L + 1s + .
82 4
and similarly for the other anomalous dimensions. Note that s depends on the h
through (4.15). An explicit expression for h(xt, s) can be found in appendix B.
By evaluating (A.9) using the above expressions, and by renormalizing the bare
coupling constant which appears in sbare in the MS scheme by replacing
s s 0 + O(s2) ,
bare = eE (4) 2s 1 4
The form of the RG equation (4.11) implies that the renormalized function can be found
through the relation
s = ZssbareZs ,
one finds that the renormalized soft function on the left-hand side is indeed finite. This
provides a strong check on the validity of the factorization formula used to derive the
RG equation, and also on our calculation of the bare NNLO function. Specifically, by
expanding everything in powers of s/4, at NLO and NNLO one finds
s(1)(L, xt) = s(b1a)re + Z(1)s(0) + s(0)Zs(1) ,
s
s(2)(L, xt) = s(b2a)re + Z(2)s(0) + s(0)Zs(2) + Z(1)s(1)
s s bare
+ s(b1a)reZs(1) + Zs(1)s(0)Zs(1) 0 s(b1a)re .
(1)
sqq,11 = 8L2 16L (ln(1 xt) + ln(xt)) + 8 (ln(1 xt) + ln(xt))2 + 42 ,
s(q1q),12 = 136 L (ln(xt) ln(1 xt)) ,
(1)
sgg,11 = 18L2 36L (ln(1 xt) + ln(xt)) + 18 (ln(1 xt) + ln(xt))2 + 92 ,
Matching coefficients and anomalous dimensions
Here we list the matching coefficients and anomalous dimensions appearing in section 4.
We first list results for the coefficients in (4.4) as well the corresponding coefficients in
the expansions of sB and CD. Using [46], we find (for simplicity, here and below we set
the number of colors to Nc = 3 in the NNLO coefficient)
s(D1)(L/2) = CF
1904 + 76272 + 2176 nl L2
723 + 1861 nl 16272 nl L
where nl indicates the number of active flavors. The matching coefficient CD can be
written as
CD(1)(L) = CF
+ 21165253 + 10732 7449054 + 26903 + 1692 ln 2
1254431 + 74812 + 102473 nl C 42CACF .
The NNLO coefficient was originally extracted in [42] using the result for sD along with the
NNLO result for the heavy-quark fragmentation function calculated in [63], and yields the
above equation with C = 0 in the last line. It was extracted directly using the relationship
between small-mass and massless amplitudes in the appendix of [22], which yields the above
result with C = 1. The discrepancy between the two methods of extracting the NNLO
coefficient remains unresolved.
Finally, the Laplace transformed heavy-quark jet function is easily derived from results
given in [47]. Explicitly,
s(B1)(L/2) = CF
496 282 64 2
9 27 27 nl L
+ 403 + 78512 nl L + 2488124 + 1932
4244936 10812 + 287 3 nl .
We next collect expansion coefficients for the anomalous dimensions appearing in the
RG equations in section 4. We define these as
and similarly for the other anomalous dimensions. One has [64]
532672 + 44454 + 838 3 + CATF nl 162772 + 162072
+ CF TF nl 2230 + 643 2674 TF2 nl2 .
167 + 2292 123 CF TF nl 23 + 892
qhq(xt, s) = 4q(s)1+Nccusp(s) (ln xt + i)
in the quark annihilation channel, while in the gluon fusion channel one finds
0 0 0
ghg(xt, s) = 2 (g(s) + q(s)) 1 + Nccusp(s) (ln xt + i) 0 1 0
0 0 1
+ 2cusp(s) ln 1 xtxt 1 N4c N4c2NN4cc4 . (B.13)
0 Nc
4
The anomalous dimensions q and g, entering in (B.12), (B.13) and, consequently in s
in (4.15), are [15, 65]
3 961
2 + 22 243 + CF CA 54
and [16, 65]
+ 4CF TF nl .
(s) = 2s 0 4s + . . .
3 CA 3 TF nl .
where we need only
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