Universality of nexttoleading power threshold effects for colourless final states in hadronic collisions
JHE
Universality of nexttoleading power threshold e ects for colourless nal states in hadronic collisions
V. Del Duca 0 2 4 7 8 10 11 12
E. Laenen 0 2 3 4 5 6 7 8 10 11 12
L. Magnea 0 1 2 4 7 8 10 11 12
L. Vernazzag 0 2 4 7 8 10 11 12
C.D. Whiteh 0 2 4 7 8 10 11 12
ETH Zurich 0 2 4 7 8 10 11 12
Institut fur theoretische Physik 0 2 4 7 8 10 11 12
0 and INFN , Sezione di Torino
1 Dipartimento di Fisica and ArnoldRegge Center, Universita di Torino
2 Science Park 904 , Amsterdam , The Netherlands
3 ITFA, University of Amsterdam
4 WolfgangPaulistr. 27, 8093, Zurich , Switzerland
5 ITF, Utrecht University , Leuvenlaan 4, Utrecht , The Netherlands
6 Nikhef , Science Park 105, NL
7 327 Mile End Road , London E1 4NS , U.K
8 Queen Mary University of London
9 1098 XG Amsterdam , The Netherlands
10 Edinburgh EH9 3JZ , Scotland , U.K
11 The University of Edinburgh
12 Via P. Giuria 1, I10125 Torino , Italy
We consider the production of an arbitrary number of coloursinglet particles near partonic threshold, and show that nexttoleading order cross sections for this class of processes have a simple universal form at nexttoleading power (NLP) in the energy of the emitted gluon radiation. Our analysis relies on a recently derived factorisation formula for NLP threshold e ects at amplitude level, and therefore applies both if the leadingorder process is treelevel and if it is loopinduced. It holds for di erential distributions as well. The results can furthermore be seen as applications of recently derived nexttosoft theorems for gauge theory amplitudes. We use our universal expression to rederive known results for the production of up to three Higgs bosons at NLO in the large top mass limit, and for the hadroproduction of a pair of electroweak gauge bosons. Finally, we present new analytic results for Higgs boson pair production at NLO and NLP, with exact topmass dependence.
NLO Computations; QCD Phenomenology

1 Introduction
2
3
4
5
6
7
8
NLP amplitude factorisation
Coloursinglet particle production in the gluon channel
Coloursinglet particle production in the quark channel
Single Higgs boson production via gluon fusion
Multiple Higgs boson production
Vector boson pair production
Conclusion
corrections has reached the stage where automated tools are being used and multijet
production rates are evaluated (see for example [
1
]).
Attention then focuses on QCD
corrections at NNLO or above, and on the inclusion of higherorder electroweak e ects.
In this context, processes which are induced by loop e ects present special di culties:
typically, the leading order contribution involves a loop containing heavy particles, such
as top quarks or electroweak vector bosons, and is often computed in the context of an
e ective eld theory. NLO QCD corrections with exact dependence on the heavy particle
masses then involve intricate twoloop calculations: for multiparticle
nal states, these
corrections are not known, and even for twoparticle
nal states they are typically only
to the incoming partons. In such cases, one commonly de nes a dimensionless
threshold variable , vanishing at the threshold, and it is well known that the corresponding
di erential cross section has the generic form
d
d
= Kew (4
s)n0 X
1
n=0
s n 2Xn1
m=0
c(nm1) logm
+
+ c(n ) ( )+ c(n0m) logm
+: : : ; (1.1)
where the overall factor is associated with the leadingorder cross section: for loopinduced
processes, this may be proportional to a power of the strong coupling, as indicated, while
Kew contains electroweak couplings. The rst two sets of terms on the righthand side of
eq. (1.1) originate from soft and collinear radiation (real or virtual), and correspond to the
leading power in the threshold variable, and to corrections localised at the threshold,
respectively. These contributions are known to have a universal (processindependent) form,
that permits their resummation to all orders in perturbation theory. This resummation is
well understood and widely applied, and can be performed within a variety of approaches
(see for example [2{11]). Even without a full resummation, a
xed order evaluation of
these terms can be useful in estimating higherorder corrections to the crosssection, when
these are not known (see for example [12] for a review).
The third set of terms on the righthand side of eq. (1.1) de nes nexttoleading power
(NLP) contributions in the threshold variable, roughly corresponding to gluon radiation
that can be nexttosoft or collinear. Although powersuppressed, these terms are still
singular as
! 0, and can be numerically signi cant: for example, they contribute signi
cantly to the theoretical uncertainty for Higgs boson production in the gluon fusion
channel [13], and their numerical impact has been explicitly con rmed by the recent calculation
of this process at N3LO [14{18]. A full, generally applicable resummation prescription for
NLP contributions is not presently known, even in the relatively simple case of parton
annihilation into electroweak
nal states: the problem has been intensively studied in recent
years, and partial progress has been made using a variety of methods [19{40], building
upon the earlier work of [41{43]. Even without a full resummation, however, knowledge
of NLP contributions at
xed order can provide useful analytic information where this
is missing, as well as furnishing improved approximations for unknown higherorder cross
sections. This is especially true if one can derive universal properties of NLP contributions,
applicable at a given order for a broad class of processes.
In this paper, we examine the universal properties of NLP radiation in the
hadroproduction of an arbitrary number of coloursinglet particles at NLO. We will prove that
the NLO crosssection for this class of processes can be written in terms of the leadingorder
crosssection with shifted kinematics, convolved with a simple universal Kfactor. More
precisely, we nd that the Kfactor does not depend on the spin of the emitting parton,
and depends on the color representation only through a trivial replacement of colour
factors. Our starting point will be a factorisation formula for NLP e ects at amplitude level
recently derived in refs. [23, 24], which expresses the e ect of adding an additional gluon
to an arbitrary hard process with an electroweak nal state in terms of universal functions.
Our results provide a useful testing ground for this formula, and an examination of its
simpler consequences, aside from its role as a basis for resummation. Furthermore, our
{ 2 {
results have more practical consequences, providing an analytic approximation to the NLO
cross section for a number of interesting loopinduced processes for which only limited
information is available. Interestingly, the universality of the result extends to di erential
distributions, provided the shift in LO kinematics is properly understood. From a
theoretical point of view, the universality and simplicity of our results at NLO can be seen
as a consequence of recently derived nexttosoft theorems [44{46] for radiative treelevel
gauge theory amplitudes: we note however that our factorisation formula is an allorder
result, and therefore will yield more general results, once the appropriate ingredients are
computed at the relevant orders.
From the point of view of phenomenology, the most interesting applications of our
results will concern the production of Higgs bosons in the gluon fusion channel, possibly
in association with electroweak gauge bosons. We will however show explicitly that the
formalism can be used also with (anti)quark initial states, and compare our results with
existing calculations. In the gluon fusion channel, we will begin by showing how known
properties of the single Higgs boson cross section emerge as a special case of our result.
We next move on to multiple Higgs boson production, which has been the focus of much
recent research. Beyond the leadingorder result [47], we note that analytic expressions for
the cross section are known only in the large top mass limit for Higgs pair production [48,
49] and for triple Higgs boson production [50]. In the case of Higgs pair production,
leading order results with full top mass dependence were obtained in refs. [51, 52], and
numerical results have recently been presented at NLO [53] (see also [54]); leading power
threshold corrections have been considered in ref. [55], and corrections to the large top
mass approximation in ref. [56]. In the case of triple Higgs boson production, numerical
results with full top mass dependence at leading order and for real radiation at NLO were
obtained in ref. [57] and in ref. [58], respectively. Associated production of electroweak
bosons and Higgs bosons in the gluon fusion channel, discussed in ref. [59], also falls within
the scope of our method, although we will not discuss it in detail here.
In this paper, we go beyond previous analytic results for Higgs boson pair production,
by providing NLO corrections, up to NLP in the threshold variable, with full top mass
dependence. As a further illustration and check, we demonstrate consistency with known
results for triple Higgs boson production in the large top mass limit [50], diphoton
production [60], and the production of W +W
pairs [61]. These results serve as an illustration
of the method: we postpone a detailed phenomenological analysis, as well as applications
to triple Higgs production with full top mass dependencce, and to associated production
of Higgs bosons with Z bosons, to future work. We note in passing that the present results
also provide a strong consistency check on all future NLO analytic computations of
loopinduced processes of coloursinglet particles, which of course must agree with the simple
factorised expressions we derive at NLP.
The structure of our paper is as follows. In section 2, we brie y review the nexttosoft
factorisation formula of refs. [23, 24], before describing how it can be extended for use
in gluoninduced processes. In section 3, we derive an explicit expression for the NLO
crosssection of a coloursinglet nal state in gluon fusion, valid up to NLP level. A similar
result for the quark channel is derived in section 4. In section 5, we show how known results
{ 3 {
in single Higgs boson production are reproduced, before examining multiple Higgs boson
production in section 6. Vector boson pair production is considered in section 7. Finally,
we discuss our results and future prospects in section 8.
2
NLP amplitude factorisation
In this section, we brie y review the results of refs. [23, 24], which derive a factorisation
formula for QCD radiation up to NLP level, and we provide a generalisation of these results
to the case of external incoming gluons. Consider an amplitude with two incoming partons
of momenta p1 and p2, and any number N of
nal state colour singlet particles, with
Here S~(p1; p2) is a nexttosoft function, dressing the hard interaction with virtual
exchanges of (nextto)soft gluons, which couple to the external partons through the
nexttoeikonal Feynman rules described in refs. [20, 21]. Associated with each hard parton is a jet
function J (pi; ni), which collects collinear singularities, and which depends on an auxiliary
vector ni. The nexttoeikonal jet J~(pi; ni) corrects for the double counting of
contributions from gluons which are both (nextto)soft and collinear, and
nally the hard function
H~ (fpig; n1; n2) is de ned by matching to the amplitude on the lefthand side of eq. (2.1), so
that all dependence on the auxiliary vectors fnig cancels out. If one ignores the presence of
nexttoeikonal Feynman rules, eq. (2.1) reduces to the wellknown softcollinear
factorisation formula, describing the dressing of a given hard interaction process with leadingpower
soft and collinear radiation (see, for example, [62]). The form of eq. (2.1), however, is a
crucial intermediate step in considering the emission of an additional gluon, of momentum
k and colour a. Up to NLP in this momentum, the resulting amplitude is given by [24]
log
J (pl; nl)
J~(pl; nl)
+
#
J a (pl; nl; k)
J (pl; nl)
A (fpig)
A
a; J~l (fpig; k) ;
where gs is the QCD coupling,1 Tia a colour generator on line i with adjoint index a, and
we have introduced the tensor [63]
G
l
=
(2pl
2pl k
k)
k2
k :
In addition to the functions already appearing in eq. (2.1), eq. (2.2) contains two more
universal functions. First, the radiative nexttosoft function S~a is a matrix element of
1We absorb a factor , where is the dimensional regularisation scale, into the coupling for simplicity.
{ 4 {
(2.2)
)
(2.3)
nexttoeikonal Wilson lines directed along the directions of the incoming partons, like
the virtual nexttosoft function S~, but with a single gluon present in the
Furthermore, eq. (2.2) includes a radiative jet function J a collecting all contributions
nal state.
associated with the emission of a gluon from the ith parton, and enhanced by virtual
collinear poles. This function was rst introduced in the context of abelian gauge theory
in ref. [43], and its de nition was recently generalised to nonabelian theories in ref. [24].
The radiative functions can be de ned in terms of operator matrix elements, but for our
present NLO analysis, where radiative functions enter only at tree level, a diagrammatic
de nition is su cient.
The nal term on the righthand side of eq. (2.2) is a subtraction term that removes
any double counting of contributions occuring in both the radiative nexttosoft emission
function, and in the radiative jet emission functions: it can be obtained simply by taking
the nexttosoft limit of the radiative jet function. As was done in ref. [24], eq. (2.2) can
be considerably simpli ed by using renormalisation group arguments and computing the
righthand side in the bare theory and with lightlike reference vectors ni2 = 0 for the jets.
With these choices, one can use the bare quantities
S~ (p1; p2) = J (pi; ni) = J~ (pi; ni) = 1 ;
ni2 = 0 ;
and the amplitude can be written as
simpli cation: indeed, the leadingorder term in the nexttosoft emission function, S~(1;)a
consists of single gluon emissions from the hard incoming partons, and these contributions
are completely cancelled [24] by the leadingorder subtraction term A ; a
(1);J~l , leaving
A(1;a) (fpig; k) =
2 "
X
l=1
(0) (fpig) ;
(2.6)
particle species in each jet.
the diagram gives2
external quark line.
which expresses the complete onegluon radiative amplitude at NLO and NLP in terms of
the Born amplitude. Note that in eq. (2.6) the nonradiative amplitude and jet emission
functions are understood as carrying implicit spin indices, depending on the identity of the
The quark radiative jet function at leading order is simply given by the emission of a
single gluon from the incoming (anti)quark [24, 43], as shown in
gure 1(a). Evaluating
J a(p; n; k) = gsT a (2p
k)
2p k
+
ik
p k
S
;
S
=
[ ; ] ;
(2.7)
2Note that, by de nition, the radiative jet function does not include the spinor wave function for the
{ 5 {
#
i
4
p
i
a
(p−k)
j
(a)
n
p
b
k
(p−k)
c
a
(b)
n
M
;
= i
where we have decomposed the result into spindependent and spinindependent parts,
introducing the generator S
of Lorentz transformations on spinors. Note that at leading
order the quark jet function is independent of the auxiliary vector n , consistently with
eq. (2.6), which represents a physical amplitude and cannot depend on n. For the gluon
radiative jet function, at leading order, we can simply use a diagrammatic de nition,
analogous to the radiative quark jet, and shown in gure 1(b). Restoring explicit spin indices
for the external gluon, we can write the result of this diagram as
J a; (p; n; k) = gsT a (2p
k)
2p k
M
;
;
where we have introduced the generator of Lorentz transformations acting on vector elds,
ik
p k
:
Once again, we have decomposed the kinematic part into its spindependent and
spinindependent parts (see for example [64]). The colour operator for the gluon case can be
explicitly interpreted as
[T a]bc = ifabc :
l
Turning now to the derivative contribution to eq. (2.6), one may note that the action of the
projector G
de ned in eq. (2.3), up to NLP order, can be recast in terms of the orbital
angular momentum of parton l. Indeed, to this accuracy
=
k
pl k
=
ik L(l)
pi k
;
where L(l) is the orbital angular momentum operator associated with the lth parton. Using
eqs. (2.8), (2.9), we can now rewrite eq. (2.6) in a uni ed notation for quarks and gluons, as
X gs Tl;a pl k
A
{ 7 {
(l) is the spin angular momentum operator for parton l, in the
relevant representation of the Lorentz group, acting as
for spin one, while J(l) is the total angular momentum operator. Furthermore, in the second
line, we have omitted the term proportional to k , which gives a vanishing contribution
S(l) for spin one half, and as M (l)
when contracted with a physical polarisation vector for the emitted gluon.
Equation (2.12) is recognisable as the recently derived nexttosoft theorem [44], which
mirrors a similar result derived in gravity [45, 46]. As noted, this formula encompasses both
the quark and gluon cases, provided the spin operator is interpreted appropriately,
validating our diagrammatic de nition for the leading order gluon radiative jet function. For
the NLO analysis performed in this paper, we could in fact have simply adopted eq. (2.12)
as the starting point for our following analysis; note, however, that eq. (2.2) and eq. (2.5)
are much more general results, applicable in principle to any order in perturbation theory.
3
Coloursinglet particle production in the gluon channel
In this section, we apply the result of eq. (2.12) to obtain a general expression for the NLO
crosssection for the production of N coloursinglet particles near threshold. We begin by
considering the gluoninduced process shown in gure 2, while we will turn to the
quarkinduced process in section 4. At Born level, the momenta introduced in gure 2 satisfy the
leadingorder momentum conservation condition
2
X pi =
i=1
N+2
X p
i=3
i
P ;
with the Bornlevel centreofmass energy squared given by s = P 2. Beyond Born level, we
may de ne the dimensionless variable
which represents the fraction of the partonic centreofmass energy carried into the nal
state by all colour singlet particles. At leading order obviously z = 1; beyond leading
order, additional real radiation may be emitted, in which case 0
z
1, and
1
z
is a dimensionless threshold variable of the kind introduced in eq. (1.1). In particular, at
(3.1)
(3.2)
NLO only a single gluon can be emitted, and all contributions up to NLP in the emitted
momentum k are captured by eq. (2.12). We can then use this to obtain a crosssection
formula that is correct up to the rst subleading order in . To this end, it is useful to
write the complete radiative amplitude (before contraction with external gluon polarisation
vectors) as
;
;
A NLP = A scal: + A spin + A orb: ;
;
;
where
is the Lorentz index of the emitted gluon, while
and
are the Lorentz indices
of the incoming gluons, and, for simplicity, we have suppressed momentum dependence,
colour indices and the superscript denoting the perturbative order. The three terms on the
righthand side correspond to the scalar, spindependent and orbital angular momentum
terms in eq. (2.12) respectively. The colour indices of the incoming gluons are displayed
in
gure 2, and we note that by colour conservation the leading order amplitude must be
proportional to bc. Following eqs. (2.10), (2.12), one may write the scalarlike contribution
to the amplitude as
;
A scal: = igsfabc
(2p1
k)
2p1 k
(2p2
k)
2p2 k
A
amplitude A
is given by
where, as above, we omitted the superscript denoting the perturbative order for the Born
. Using eqs. (2.9), (2.12), the spindependent contribution to the amplitude
(3.3)
(3.4)
(3.5)
(3.6)
(3.7)
(3.8)
;
A spin = igsfabc p1 k
(k
A
Finally, the orbital angular momentum contribution is
;
A orb: = igsfabc G1
k
)
G2
After including polarisation vectors for the two incoming gluons, the squared matrix
element, accurate to NLP level and summed over polarisations and colours, is
where we de ned the polarisation sum
P
(p; l)
X
( )(p) ( ) (p) =
+
p l + p l
p l
;
with l is an arbitrary lightlike reference vector used to de ne physical polarisation states,
whose dependence must cancel in the
nal result. Alternatively, one could sum over all
polarisations, using P
=
, and correct for this by including external ghost
contributions. Following this second approach, it is fairly easy to conclude that ghost contributions
vanish at NLP: indeed, nal state ghost emission is suppressed by a power of the energy at
amplitude level, and thus contributes at NNLP at cross section level; furthermore, diagrams
with a ghostantighost pair in the initial state do not couple directly to fermions or to the
Higgs boson, and are strongly suppressed. These expectations are borne out by a direct
calculation, showing that all terms proportional to the vector l in eq. (3.7) are beyond the
required accuracy. We conclude that we can perform a sum over all polarisation, so that
eq. (3.7) simpli es to
where in the second term we need to keep only those terms which are leading power in
the scalar part of the amplitude. It is straightforward to show that the rst term on the
HJEP1(207)5
righthand side yields
X
colours
;
A scal:
2 = 2gs2Nc Nc2
1
so that only the leading power term survives. For the gluoninitiated process we are
considering in this section, the spin term in eq. (3.9) can be shown to vanish upon summing
over all polarisations. For example, the spin contribution from the rst leg is
X
colours
h
;
2 Re A spin A scal: ;
i
=
2gs2CA Nc2
1
p1 k
k
k
p1
p1 k
Re A A
p2
The prefactor in the second line is antisymmetric under the interchange of
thus vanishes when contracted with the squared Born amplitude, which is symmetric; the
same argument applies to the second incoming gluon. Note that the argument applies also
when the Born amplitude is loop induced, and thus may acquire an imaginary part (as is
the case here). The orbital angular momentum contributions give
(3.9)
(3.10)
(3.11)
(3.12)
where we de ned
p1 =
1
2
p2 k
p1 p2 1
p
p1 k
p1 p2 p2 +k
; p2 =
1
2
p1 k
p1 p2 2
p
p2 k
p1 p2 p1 +k
:
(3.13)
Note that these shifts are proportional to the soft momentum k and transverse to their
respective momenta, pi
pi = 0. This second property follows from the fact that the
ith momentum shift is derived from the orbital angular momentum operator of eq. (2.11),
{ 9 {
G2
1 p1 p2 "
p
which generates an in nitesimal Lorentz transformation transverse to the momentum pi.
Combining eq. (3.12) with eq. (3.10), we can write
jANLPj2 =
Eq. (3.14) is a focal point of this paper: it shows that all NLP contributions to the NLO
squared matrix element for the production of an arbitrary coloursinglet nal state can be
absorbed into a shift in the kinematics of the Born contribution. Corrections to this shifting
procedure involve terms at least quadratic in pi, and thus beyond the NLP approximation.
In section 4, we will show that the same property is shared by quarkinitiated processes.
Note that eq. (3.14) is fully di erential in
nal state momenta, and can be applied to
generate distributions valid at NLO and to NLP accuracy. On the other hand, using
simple properties of phase space, one can also derive a similarly simple expression for the
inclusive cross section. In order to do so, note that the e ect of the required momentum
shifts on the partonic centreofmass energy is given by a simple rescaling. Indeed,
s ! (p1 + p2 + p1 + p2)2 = s + 2 ( p1 + p2) (p1 + p2) :
Substituting the de nitions of eq. (3.13) in eq. (3.15), and using eq. (3.2), together with
the NLO momentum conservation condition
it is easy to show that eq. (3.15) can be written simply as
p1 + p2 = k + P ;
s ! zs :
To construct the partonic crosssection, we must now introduce the appropriate factors to
average over initial state colours and spins, integrate over the (N + 1)body
nal state
phase space, and include the ux factor. We nd
^N(gLgP) =
(d
1
denotes the nbody Lorentzinvariant phase space for a process with total nal state
momentum Q = Pi qi, and qi = (Ei; qi) in a suitable frame. For the phase space, we may
use the wellknown result
d N+1 (P + k; p3; : : : pN+1; k) =
dP 2 d 2 (P + k; P; k) d N (P ; p3; : : : pN+2) ;
(3.15)
(3.16)
(3.17)
(3.18)
(3.19)
(3.20)
factorising the phase space of the N coloursinglet particles from a twobody phase space
involving the total momentum of the colourless system, and the additional gluon
momentum k. The latter can be written more explicitly by parametrising
Introducing the variable
one then nds (see for example ref. [21] for a recent derivation)
Using eq. (3.23) together with eqs. (3.2), (3.14) in eq. (3.18) yields
d 2 (P + k; P; k) =
4
s
8
1
(1
z)1 2 h
y(1
y)
i
dy :
Z
2)2 2 (Nc2
d (Nz)
A
1) (1
) s
(p1 + p1; p2 + p2) ;
d^N(gLgP) =
dz
dyhy(1
y)
i
1
(1
z) 1 2
where we reinstated the explicit dependence on the dimensional regularisation scale , and
we denoted by d (Nz) the phase space for N (coloursinglet) particles with a partonic
centreofmass energy shifted according to eq. (3.17). We may easily rewrite this result in terms
of the leadingorder cross section with shifted kinematics, which is given by
B(gogr)n (zs) =
2(d
1
This leads us to our second central result: a simple factorised expression for the inclusive
cross section, valid at NLO and NLP for the production of a generic coloursinglet system,
which can be written as
d^N(gLgP) = CA KNLP (z; ) ^B(gogr)n (zs) ;
where the nexttoleading power K factor is easily computed, with the result
KNLP (z; ) =
=
dz
s
s
4
s
2
s
2
Di(z)
In the second line of eq. (3.27) we expanded the result to NLP in (1
z) and to nite order
in , and we introduced the MS scale 2 =
2 e ln(4 ) E and the plus distributions
(3.21)
(3.22)
(3.23)
(3.24)
(3.26)
(3.28)
HJEP1(207)5
j
.
.
.
p
p
4
pair, without nal state QCD radiation.
Eqs. (3.26), (3.27) show explicitly that the NLO Kfactor for the production of N
coloursinglet particles in the gluon channel is simple and universal, up to nexttoleading power
in the threshold variable. This is a powerful constraint, and we will discuss some speci c
examples in the following sections. First, however, we consider an analogous formula in
the quark channel.
4
Coloursinglet particle production in the quark channel
In the previous section, we have derived an explicit universal Kfactor for multiple
coloursinglet particle production in the gluongluon channel. In this section, we consider the cross
section for quarkinduced production of coloursinglet particles, and show that an identical
result holds, up to a trivial replacement of colour factors. The universality of the result is
not obvious from the outset, and it comes about through an interesting reshu ing of the
contributions of spin and angular momentum operators, as compared to the gluoninduced
process. We take the leading order process shown in
gure 3, and consider the radiation
of an additional gluon from the incoming quark and antiquark lines. One may write the
LO amplitude as
Aij (fpig) = ij v(p2) A (fpig) u(p1) ;
where i, j are the colour indices of the incoming quark and antiquark (suppressed in what
follows for brevity), and the factor A(fpig), matrixvalued in spinor space, is the
quantity entering eq. (2.6), namely the leadingorder amplitude with external wave functions
removed. Following the procedure adopted in the gluon case, we may decompose the NLO
amplitude, before contraction with external spinors, according to
A NLP = A scal: + A spin + A orb: ;
where the three terms on the righthand side denote the scalarlike, spin, and orbital angular
momentum contributions, and we have suppressed spinor indices (as well as color labels)
for brevity. For the scalar and orbital angular momentum contributions, which do not
depend explicitly on the spin (apart from replacing the vector indices on the leadingorder
amplitude with spinor indices), the arguments of the previous section may be repeated,
(4.1)
(4.2)
and one may write3
A scal: + A orb: = gs
p
1
p1 k
p
2
p2 k
A (p1 + p1; p2 + p2) ;
(4.3)
where the momentum shifts are de ned in eq. (3.13). Including the (anti)quark wave
functions and performing color and spin sums, we then nd
2
jANLPj scal:+orb: = gs2NcCF p1 k p2 k
= gs2NcCF p1 k p2 k
where in the second line we have introduced the dimensionless vectors
ni = pi
p
s
i = 1; 2 :
By comparing eq. (4.4) with its LO counterpart,
jA(p1; p2)j2 = Nc s Tr h6 n1A(p1; p2)6 n2Ay(p1; p2)i ;
and using eq. (3.17) we may promote the momentum shift in eq. (4.4) to apply to the entire
squared amplitude. This leads to
2
jA NLPj scal: + orb: =
gs2CF
z
s
p1 k p2 k jA (p1 + p1; p2 + p2)j2 :
Note the close resemblance of eq. (4.7) and eq. (3.14): they di er only by the color factor
and a rescaling by a factor of z. We must still, however, add to eq. (4.7) the interference
between the spindependent part of the NLO amplitude, and the eikonal amplitude. In
the gluon case, this turned out to vanish in Feynman gauge, upon summing over all gluon
polarisations, which was allowed at NLP accuracy. For an incoming fermion, we
nd
that the spin contribution does not vanish, and indeed it precisely compensates for the z
rescaling observed in eq. (4.7), recovering the universality of the result.
The spindependent part of the NLO amplitude is given by the diagrams of gure 4,
which evaluate to
(k) tjai v(p2) A spin u(p1) =
i gs tjai k
(k) v(p2)
=
gs tjai
(k) v(p2)
A 6 k
2p1 k
A
p1 k
+
2p2 k
A
p2 k
6 k A
u(p1)
u(p1) :
In the second line of eq. (4.8), we anticommuted Dirac matrices and used the physical
polarisation condition k
(k) = 0 to write the result in a form which will be more
convenient in what follows. Up to NLP accuracy in the squared amplitude, we only need to
3Following the convention of eq. (4.1), we do not include the colour generator tjai in the fundamental
representation in the de nition of the stripped amplitude A.
(4.4)
(4.5)
(4.6)
(4.7)
(4.8)
p
1
p
2
i
j
k
a
...
p
4
...
denotes the magnetic moment coupling of the gluon to the spin of the quark.
consider the interference of eq. (4.8) with the (leading power) scalar part of the NLO
amplitude. Furthermore, since we are considering the emission of a single gluon, we can sum
over all polarisations, rather than restricting to physical polarisations only. The relevant
contribution to the squared matrix element is then
X
To simplify this further, we may expand the emitted gluon momentum in the Sudakov
decomposition
k =
p2 k
p1 p2
p1 +
p1 k
p1 p2
p2 + kT ;
kT p1 = kT p2 = 0 :
(4.10)
We then observe that, to linear order in k , the Dirac trace in eq. (4.10) cannot depend on
kT . Indeed, one easily nds
X
colours
2 Re hAyscal: A spin NLP
i
=
gs2Nc CF p1 k p2 k
k (p1 + p2)
p1 p2
jA(p1; p2)j2 : (4.11)
By comparing with the squared scalar part of the amplitude
X
colours
Ayscal: A scal: = gs2Nc CF p1 k p2 k jA(p1; p2)j2 ;
we see that the spindependent contribution to the squared amplitude can be obtained
from the part which is leading power in the gluon momentum, simply through rescaling by
the factor
k (p1 + p2)
=
(1
z) ;
where we have used the momentum parametrisation of eq. (3.21).
Combining eq. (4.7) with eq. (4.12), we see that the rescaling factors cancel at NLP in
(4.12)
(4.13)
HJEP1(207)5
(1
z). Indeed one may write
jA NLPj2 = gs2 CF
s
)
z) :
(4.14)
Expanding now in powers of (1
z), one gets to rst order
jANLPj2 = gs2 CF
s
and one observes that the second line is e ectively O(1
z)2. We nd then
s
jA NLPj2 = gs2 CF p1 k p2 k jA(p1 + p1; p2 + p2)j2 ;
+
jA(p1 + p1; p2 + p2)j2
z) ;
which is precisely analogous to eq. (3.14), except for the replacement of the colour factor,
which here is associated with the fundamental rather than adjoint representation of the
gauge group. Once again, at NLP, eq. (4.16) can be used in a fully di erential
implementation for the nal state kinematics.
Having obtained eq. (4.16), one may form the crosssection by integrating with the
(N + 1)body phase space, exactly as was done in the gluon case. One nds then
dz
d^(NqqL)P = CF KNLP (z; ) ^B(qoqr)n (zs) ;
(4.15)
(4.16)
(4.17)
(4.18)
(4.19)
(4.20)
(4.21)
so that eq. (3.26) becomes
d^qNqLP =
dz
=
4
s CF z
s CF
2
2D0(z)
8
8D0(z)
which precisely agrees with the wellknown results quoted for example in refs. [21, 65].
with the same factor KNLP (z; ), given in eq. (3.27).
production of a vector boson of invariant mass Q2, where one has
A
rst check on this result is that it reproduces the NLO Kfactor for DrellYan
z =
Q2
s
:
In this case, as for any 2 ! 1 process, the LO partonic cross section has support only on
the partonic threshold: for DrellYan production,
so that the LO cross section with shifted kinematics is
The delta function imposes the correct de nition of the threshold variable at NLO, while
the rest of the cross section is una ected by the shift in kinematics. To compare with
standard results, we must note that the MS scale
invariant mass Q2. To this end, one may write
2 is usually set equal to the nal state
B(qoqr)n(s) /
(Q2
s) =
B(qoqr)n(zs) /
Q
2
zs =
2
s
=
2
Q2
1
s
1
s
Q2
s
Q2
s
Q2
s
1 ;
z :
! z ;
(a)
(b)
(c)
Contact interaction in the large top mass limit; (c) Contact interaction for radiation of an extra
gluon.
5
Single Higgs boson production via gluon fusion
Having presented our results for both quark and gluoninduced coloursinglet particle
production, we now examine a rst signi cant application of the gluon result, eq. (3.27):
the single production of Higgs bosons in the gluon fusion channel. As is well known, this
is the principal production mode for Higgs bosons at the LHC, and higherorder QCD
corrections have been studied in great detail and with great e orts in recent years. In
the e ective
eld theory with the top quark integrated out, they have been calculated
up to N3LO in perturbation theory [14{18, 66{69]. Topmass e ects are know exactly at
NLO [70], and have been studied at NNLO as a power expansion in m2h=mt2 [71, 72]. Here
we will see how the intricate top mass dependence at NLO simpli es considerably in the
threshold region, including NLP corrections.
At leading order, the incoming gluons couple to the Higgs boson via a topquark loop, as
shown in gure 5(a). The leading order crosssection for this process (see for example [
66
])
can be written as
HJEP1(207)5
h
Born(s) =
2
s
57m62hv2 (1 + ) F ( ; ) (s
m2h) ;
where mh and v are the Higgs mass and vacuum expectation value respectively.4 The form
factor F ( ; ) depends on the dimensionless variable
2
z
+ O( ) ;
m2h
s
:
(5.1)
(5.2)
(5.3)
(5.4)
F ( ; ) =
and it is given by [
66
]
with a normalisation chosen so that F ( ; ) ! 1 as
shifted according to eq. (3.15) can then be written as
! 0. The cross section with kinematics
=
s
4mt2
;
4In eq. (5.1) we have omitted scale factors relating to the ddimensional coupling s, which amounts to
the choice
= mh.
Substituting this result into eq. (3.26) and expanding in powers of (1
z) and
one nds
d NhLP =
dz
s3CA
It is easy to check that eq. (5.5) agrees with the known analytic NLO result of ref. [
66
] in
the mt ! 1 limit. We note, however, that the result of eq. (5.5) is much more informative:
it includes the full dependence on the top quark mass up to NLP order, and can thus be
applied for arbitrary mt. This is a remarkable simpli cation of the intricate result of ref. [70]
for the full mt dependence: after shifting the kinematics of the leading order result, the
resulting Kfactor is entirely independent of the top quark mass, which makes the formula
HJEP1(207)5
especially simple to apply in practical applications.5
It is interesting to examine the anatomy of the result in eq. (5.5) in slightly more detail.
If one were to calculate the NLO cross section by starting manifestly in the large top mass
limit (i.e. by using an e ective
eld theory), the leading order graph would contain an
e ective pointlike interaction coupling the two incoming gluons to a Higgs, as shown in
gure 5(b). At NLO, one can radiate the extra gluon from either of the incoming gluons,
and one must also include the additional e ective coupling shown in
gure 5(c), namely
a pointlike interaction between three gluons and a Higgs boson. If one resolves the top
quark loop as in
gure 5(a), this extra interaction corresponds to emissions from inside
the top quark loop. In the NLP calculation, there is no need to include any additional
diagrams to capture these contributions: they are generated precisely by the orbital angular
momentum contributions in eq. (3.6): therefore, as the above analysis reveals, we can choose
to associate these terms with a shift in the kinematics of the leading order result, up to
corrections subleading in soft momentum. Seen from the point of view of the e ective eld
theory at large mt, it is highly nontrivial that such a shift captures the contribution of
higherorder operators in the e ective Lagrangian.
6
Multiple Higgs boson production
In the previous section we have tested our main result, given by eq. (3.26) for gluon
scattering, by reproducing known results in the cross section for single Higgs boson production
via gluon fusion. We now consider the case of Higgs boson pair production, a process of
ongoing interest at the LHC, due to its potential role in extracting the Higgs boson
selfcoupling. Analytic results for this process are known up to NNLO in the large top mass
limit [47{49], but only at leading order with full top mass dependence [51, 52]. Further
studies have looked at systematically improving the e ective eld theory results by
including leadingpower threshold e ects [55], or contributions suppressed by powers of the top
mass [56]. Recently, numerical results at NLO accuracy with full top mass dependence
have become available [53] (see also [54]). This, however, does not preclude the desire
for analytic results, which can serve to improve the e ciency of numerical computations,
whilst also providing clues regarding higherorder structures in perturbation theory. This
5Indeed, we have checked that eq. (5.5) reproduces the Kfactor reported in ref. [71], which features a
double expansion in threshold parameter and top mass.
where (in the Standard Model)
A
=
2 vs2 h (C4F4 + C F ) T0;
+ C G T2;
i
;
C
s
are form factors arising from the triangle and box graphs, as indicated by
the subscripts. They depend on the Higgs boson and top masses, as well as the partonic
centre of mass energy s and the other Mandelstam invariants
t =
u =
1
2
1
2
"
"
s
s
2m2h
2m2h + s 1
r
s 1
r
4m2h cos
4m2h cos
s
s
#
#
;
:
The basis tensors Ts; , with s = 0; 2, in eq. (6.1) are associated with the exchange of spin
0 and spin 2 in the s channel, respectively. Denoting the gluon momenta by p1 and p2 and
the Higgs boson momenta by p3 and p4, their explicit forms are
is especially true in Higgs boson pair production, given that the large top mass limit is
not a good approximation, unlike the case of single Higgs production. The leading order
diagrams for Higgs pair production are shown in gure 6, and the leading order amplitude
With these notations, the leadingorder distribution in the Mandelstam invariant t can be
2 + C G
This expression simpli es considerably in the large top mass limit, where
F
so that eq. (6.6) becomes
We observe that, in the large top mass limit, the leadingorder cross section vanishes at
threshold, as s ! 4m2h, due to the cancellation between the box and triangle contributions.
This property is one of the reasons that make the large top mass limit a poor approximation
in Higgs boson pair production, necessitating the calculation of higher order corrections
with full mt dependence. It also causes problems when trying to de ne a Kfactor as a
function of the variable z. Ordinarily, one would divide the NLO cross section by the LO
one, however this becomes illde ned in the threshold region z ! 1. In ref. [47] this problem
is circumvented by dividing by the LO cross section with kinematics shifted according to
eq. (3.17). The resulting Kfactor thus matches precisely the quantity de ned in eq. (3.26).
With this convention, the NLO cross section for Higgs boson pair production, up to
+ 12D1(z)
24 log(1
z)
hh
Born (zs) ;
(6.9)
NLP accuracy, can be written as
z
d NhhLP =
dz
3
where we have extracted an explicit factor of z on the lefthand side, to match the
conventions adopted in ref. [47]. In the large top mass limit, eq. (6.9) reproduces the results
of ref. [47]. As in the case of single Higgs production, however, the result is much more
powerful, in that it applies to the full top mass dependence. Eq. (6.9) thus provides an
explicit analytic form of the cross section, at NLO in perturbation theory, and up to NLP
in the threshold variable (1
z). This extends the results of ref. [55], which considered
supplementing the
xedorder cross section with threshold e ects at leading power only.
Ref. [56], on the other hand, studied the systematic improvement of the large top mass
limit, by including corrections expressed as a power series in m2h=mt2. The authors found
that the convergence of this expansion could be dramatically improved by factorising the
leading order crosssection with exact top mass dependence. The results of this paper
explain why this is so: indeed, we nd that, to rst subleading order in the threshold
expansion, the NLO cross section can be completely expressed in terms of the leadingorder
cross section with shifted kinematics, and with full top mass dependence.
Returning to the large top mass limit, we can go further and consider triple Higgs
production, a process for which analytic NLO corrections were presented recently in ref. [50].
If we denote the invariant mass of the triple Higgs system by M3h, and we set the MS scale
according to 2 = M32h, the NLO cross section in the gluon fusion channel can be written
as [50]
with
M32h ddMh3h2hh
=
2
s Bhhohrn (zs) (z) ;
(z) = 24D1(z)
24z
z + z2 + 2 log(1
z)
12(z2 + 1
z)2
1
z
log(z)
11(1
z)3 + C3(h) (1
z) ;
(6.10)
(6.11)
where C3(h) can be read o from ref. [50], and does not a ect our arguments. Expanding
to NLP in (1
z), we may rewrite eq. (6.10) as
d hhh
dz
=
4
s CA h16D1(z)
32 log(1
z) + 8 + O(1
z)
i Bhhohrn (zs) :
(6.12)
This result is indeed reproduced from eq. (3.26): implementing the scale choice as in
eq. (4.21), one can rewrite eq. (3.26) in the present case as
z
d hhh
dz
=
4
s CA
16
extracting the nite part in the MS scheme, one nds precise agreement with eq. (6.12).
7
Vector boson pair production
In the preceding two sections, we have illustrated the application of our general expression
for the NLO Kfactor in the gluon channel. In order to verify and illustrate the quark
result, eq. (4.17), in a nontrivial case, it is instructive to see how known matrix elements
in vector boson pair production can be reproduced. This calculation is also an important
illustration of the fact that our prediction, while based on power counting in the threshold
variable z, applies to the fully di erential squared amplitude, and not only to the integrated
cross section. As an example, we consider diboson production,
q(p1) + q(p2) ! V (p3) + V (p4) ;
where V is an electroweak gauge boson. Let us start with diphoton production, the
amplitudes for which can be found in ref. [60] up to NLO, in four spacetime dimensions.
The squared LO matrix element for this process, summed and averaged over colours and
spins, is given by
ABorn
2 =
2eq4 t2 + u2
Nc
tu
=
4eq4 1 + cos2
Nc 1
cos2
;
where eq is the electromagnetic charge of the quark. One may now consider the radiation
of an additional gluon, leading to the process
q(p1) + q(p2) ! V (p3) + V (p4) + g(k) :
Computing the diphoton cross section at NLO, one must include the squared Born matrix
element for the process in eq. (7.3), again summed and averaged over colours and spins. It
is given by [60]
ANLO
g 2
=
e
4
Nqc gs2CF
s Pi(p1 ki)(p2 ki) (p1 ki)2 +(p2 ki)2
Qi(p1 ki)(p2 ki)
;
ki 2 fp3; p4; kg :
(7.4)
Our aim is to show that, up to NLP accuracy, this matrix element can be obtained by
shifting kinematics in the LO squared amplitude, as dictated by eq. (4.16). To this end,
one may rescale the gluon momentum as k !
k in eq. (7.4), and expand to nexttoleading
(7.1)
(7.2)
(7.3)
where
p3 =
p4 =
2
2
ps2 1;
p01 =
s + tk
2ps2
cos
= 1
;
s
2p01p02
;
! 1. Next, one can parametrise the momenta in the centre of
mass frame of the V V system, as (see for example ref. [61])
;
cos 0 = 1 +
t
k
2p01k0
;
k0 =
z =
r
1
tk + uk
2ps2
;
4m2V ;
zs
and we have introduced the invariants
tk = (p1
k)2 ;
uk = (p2
k)2 ;
s2 = s + tk + uk :
Note that in these notations the threshold variable is
= 1
z = 1
s2=s. With these
de nitions, we can now expand to
rst subleading power in the gluon energy, and set
mV = 0 for the diphoton case. The result is
ANLP
g 2 = gs2CF p1 k p2 k Nc 1
s
4eq4 1 + cos2 1
8 sin 1 cos 1 cos 2
By performing the same procedure, one may easily show that the LO amplitude of eq. (7.2),
evaluated with the kinematic shifts de ned in eq. (3.13), yields
ABorn (p1 + p1; p2 + p2) =
2
4eq4 1+cos2 1
Nc 1 cos2 1
8 sin 1 cos 1 cos 2
We therefore see that eq. (7.8) explicitly con rms the expectations raised by eq. (4.16).
A similar exercise may be carried out for W +W
production: the NLO squared
amplitudes for this process (again in four spacetime dimensions) may be found in ref. [61].
The Bornlevel squared matrix element, summed and averaged over colours and spins, is
given by
ABWoWrn 2 =
4Nc
1 hcitt Fi(0)(s; t)
cits(s) Ji(0)(s; t) + ciss(s) Ki(0)(s; t) ;
i
(7.10)
where citt, ciss and cits are coe cients associated with tchannel, schannel and interference
graphs, respectively, for a quark of avour i, and can be found in ref. [61]. The remaining
functions of Mandelstam invariants are given by
p
p
s
s
Fi(0)(s; t) = 16
Ji(0)(s; t) = 16
Ki(0)(s; t) = 8
ut
m4W
ut
m4W
ut
m4W
1
1
1
1
4
s
4
2
s
4
+
m4W
t
2
m2W
2
+ 16
m4W
t
s
m2W ;
+ 16s
sm2W + 3m4W
+ 8s2
s
m2W
s
m2W
2 + 2
4 :
m2W
t
;
(7.11)
(7.5)
(7.6)
(7.7)
; (7.8)
:
(7.9)
Similarly, the NLO squared matrix element including the radiation of a gluon, for the qq
initial state, also summed and averaged over colours and spins, is given by
ANWLWOg 2
4
tkuk
s hcitt Xbi cits(zs) Ybi + ciss(zs) Zbii ;
(7.12)
where the functions Xbi, Ybi and Zbi have been obtained by rescaling the corresponding
functions Xi, Yi and Zi, given in appendix D of ref. [61], extracting the singular prefactor
4s=(tkuk). Parametrising momenta as in eq. (7.5), one may expand each function up to
NLP in
= (1
z), using the same procedure as outlined above for the diphoton matrix
element. Introducing the notation
= m2W =s, the results can be written as
HJEP1(207)5
Xbi NLP
3 cos2 1 32 3 32 2 +10
+ cos 1 96 4 112 3 +70 2 20 +2
+
1
1
2
Ybi NLP
32s
1 2 +(4
1)cos 1
2 cos 1( +2)+2 2 +3
2
cos 1(4
1)+5
+s cos2 1 96
ssin cos 2 sin 1
(4 1)cos 1+1 2 2 cos3 1
"
+ cos2 1 768 3 1088 2 +656
204+ 32
2
2
16+ 10 + 12 +
cos2 1 64
We have explicitly checked that the same results are obtained from eq. (4.16), where for
the righthand side one must use the LO squared amplitude given in eq. (7.10), with
momenta shifted according to eq. (3.13). This is a highly nontrivial crosscheck: comparing
eqs. (7.13){(7.15) with the diphoton case given in eq. (7.8), one sees that the W W case
involves a much more complicated dependence on the opening angle 1, and the partonic
centre of mass energy s. A similar analysis could be carried out for ZZ production [73], and
also to provide analytic information for triple vector boson production, numerical results
for which have been presented in refs. [
74, 75
].
8
In this paper, we have considered the hadroproduction of an arbitrary heavy colourless
system, in both the gluonfusion and quarkantiquarkannihilation channels, near partonic
threshold for the production of the selected
nal state. Our starting point is an allorder
factorisation formula for the relevant scattering amplitudes, introduced in refs. [23, 24],
given here in eq. (2.2), and valid to nexttoleading power in the threshold expansion.
Specialising this formula to NLO in the strong coupling, we have observed how the general
expression simpli es, and takes the form of a nexttosoft theorem, as derived for example
in [44{46]. This simple expression, in turn, leads to a universal form for the transition
probability, completely di erential in the nal state variables, and proportional to the
Bornlevel transition probability, computed with a speci c shift for the initial parton momenta.
The result is the same for quarks and gluons (up to a trivial substitution of color factors),
and is reported in eqs. (3.14), (4.16). When the transition probability is integrated over
nal state variables, one nds that the inclusive cross section for the selected process can
also be written in a simple and universal factorised form, given here in eqs. (3.26), (4.17).
More precisely, at NLO, and up to nexttoleading power in the threshold expansion, the
crosssections can be written as a universal Kfactor, multiplying the leading order
crosssection with a shifted partonic centreofmass energy. All these results apply regardless of
whether the leading order process is treelevel or loopinduced, and the resulting Kfactors
are independent of hard scales such as heavy quark masses.
We have checked our results by reproducing known expressions for the production of
up to three Higgs bosons at NLO, in the large top mass limit. Away from this limit, our
formula provides new analytic information in the case of Higgs pair production at NLO,
where only numerical results are presently known. Furthermore, we explain the observation,
made previously in ref. [56], that the convergence of the large top mass expansion can
be improved in this process by factoring out the LO crosssection with exact top mass
dependence. In the quark channel, we have shown how our formula is consistent with
previous results for the production of photon and W boson pairs, again at the level of
di erential distributions.
The results we have presented show that, to NLP accuracy, di erential and inclusive
NLO cross sections for coloursinglet nal states are dramatically simpler than exact results,
and we expect that they will be very easy to implement in numerical codes, providing checks
of existing calculations, and improved approximations for di erential distributions when
complete results are not available, as is the case for loop induced processes with
multiparticle electroweak
nal states. A detailed phenomenological analysis, including a study
of the accuracy of the NLP approximation in di erent processes and kinematic domains,
has been left to future work.
We emphasise that the simple universal expressions that we nd at NLO can be
systematically improved upon by relying on eq. (2.2): in particular, a NLO calculation of the
radiative soft function and of the radiative jet functions for quarks and gluons, which is
under way, will lead to NLP approximations for cross sections of the type studied in this
paper at NNLO level, exploring uncharted territory, in particular for processes which are
loop induced and have so far been studied predominantly in the context of e ective eld
theory approximations.
Beyond NLO, we expect the K factor obtained from eq. (2.5) to acquire a degree of
process dependence, and we expect the simple interpretation of the result as a kinematic
shift to receive corrections. First of all, the oneloop radiative jet functions for quarks
and gluons will di er, and the nexttosoft function will acquire spin dependence beyond
leading power. Furthermore, the exact cancellation between the nexttosoft function and
the softcollinear amplitude, leading from eq. (2.5) to eq. (2.6), will be lost at the
oneloop level.
We note however that these corrections are only sensitive to the nature of
the incoming particles, and do not depend on the hard interaction, preserving a degree of
universality of the result. What is likely to be lost is the simple geometric interpretation
in terms of a kinematic shift, displayed for example in eq. (3.14) and in eq. (4.16): this
is not surprising, given the close relation of these results to the existence of (nextto)soft
theorems for amplitudes [44, 46, 64], which are known to break down beyond tree level.
Furthermore, we note that, at NNLO and beyond, one must also consider multiple gluon
emissions from the jets and from the nexttosoft functions. There is good reason to believe
such contributions to be universal, in the sense of eq. (2.5): in the soft case, for example,
they are known to exponentiate independently of the hard process [20, 21] . Nevertheless,
to what extent universality persists for multiple emissions can only be checked by an
explicit computation.
One may also wonder whether the property of universality persist for contributions
suppressed by two or more powers of the threshold variable (i.e. at NNLP and beyond). This
seems very unlikely: indeed, as has been known since the pioneering work of refs. [41, 42],
emissions from inside the hard interaction, starting at NNLP, are no longer dictated by
gauge invariance, which is the key ingredient leading to eq. (2.2).
Note that in this paper we considered only processes involving two coloured particles
in the Born interaction, so that colour correlations are simple, given the inclusive nature
of the crosssection. One may wonder whether our formalism would also correctly capture
colour correlations in multiparton processes, which are much more intricate. We do not
expect additional colour structures to pose a serious problem, since their structure would
be fully encoded in the appropriate nexttosoft function, which can be formulated in terms
of webs [20, 21].
Finally, we note that the universal expressions we have derived for inclusive cross
sections are also applicable to colourless nal states arising beyond the Standard Model,
in which case our results can provide a controlled approximation to estimate the size of
higherorder corrections in selected models without the need to perform calculations that
would be expensive for loopinduced processes.
Acknowledgments
We thank Fabrizio Caola, Jort SinningheDamste, Falko Dulat, Bernhard Mistlberger and
Adrian Signer for discussions. We also thank Michael Spira for kindly providing us with
personal notes relevant to our results in section 3. This research was supported by the
Research Executive Agency (REA) of the European Union through the contract
PITNGA2012316704 (HIGGSTOOLS). EL was supported by the Netherlands Foundation for
Fundamental Research of Matter (FOM) programme 156, \Higgs as Probe and Portal",
and by the National Organisation for Scienti c Research (NWO). CDW is supported by
the U.K. Science and Technology Facilities Council (STFC), and thanks ETH, Zurich and
the Higgs Centre for Theoretical Physics, University of Edinburgh, for hospitality. LV is
supported by the People Programme (Marie Curie Actions) of the European Union Horizon
2020 Framework H2020MSCAIF2014, under REA grant N. 656463 (Soft Gluons).
Open Access.
This article is distributed under the terms of the Creative Commons Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
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