Preprint

Soft quark effects on H+jet production at NLP accuracy

Sourav Pal and Satyajit Seth

Theoretical Physics Division, Physical Research Laboratory,
Navrangpura, Ahmedabad 380009, India

E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>

Abstract: We define soft quark operators that enable construction of colour-
ordered helicity amplitudes out of the non-radiative ones. We explore these
operators for Higgs plus one jet production in a hadron collider and study the
next-to-leading power corrections on all partonic channels involving quark(s)
and/or anti-quark(s). We also investigate the effect of next-to-soft gluon
radiation in these channels employing the technique illustrated in [1].
Analytical expressions of next-to-leading power leading logarithms thus
obtained are simple and compact in nature. Further, we discuss the connection
of such logarithms with that of the pseudo-scalar Higgs plus one jet
production.

###### Contents

  1. 1 Introduction
  2. 2 Next-to-soft gluon radiation
  3. 3 Soft quark radiation
  4. 4 H+jet production
    1. 4.1 Colour-ordered amplitudes
    2. 4.2 Leading order amplitudes
    3. 4.3 NLP amplitudes for $q\bar{q}ggH$
      1. 4.3.1 Radiation of soft quark
      2. 4.3.2 Radiation of soft anti-quark
      3. 4.3.3 Radiation of soft gluons
    4. 4.4 NLP amplitudes for $q\bar{q}Q\bar{Q}$
  5. 5 NLP logarithms
    1. 5.1 NLP logarithms: $q\bar{q}Hgg$
      1. 5.1.1 $q\bar{q}$ initiated process
      2. 5.1.2 $gg$ initiated process
      3. 5.1.3 $qg$ initiated process
      4. 5.1.4 $\bar{q}g$ initiated process
    2. 5.2 NLP logarithms: $q\bar{q}HQ\bar{Q}$
  6. 6 Psuedo-scalar Higgs plus jet production
  7. 7 Conclusions

##  1 Introduction

Precise data obtained from the Large Hadron Collider, coupled with the absence
of compelling new physics indications necessitate a better understanding of
the Standard Model. Performing high precision perturbative Quantum
Chromodynamics (QCD) calculations, either by incorporating higher order
perturbative terms for fixed order corrections, or by performing resummation
that account for specific enhanced logarithms across all orders in the
perturbation series, are indispensable to understand the data obtained from
the collider experiments.

For all collider processes, one can define a threshold variable ($\xi$) that
vanishes in the threshold limit and the differential cross-section may take
the following form,

$\displaystyle\frac{d\sigma}{d\xi}\,\approx\,\sum_{n=0}^{\infty}\alpha_{s}^{n}\left\\{\sum_{m=0}^{2n-1}C_{nm}\left(\frac{\log^{m}\xi}{\xi}\right)_{+}+d_{n}\delta(\xi)+\sum_{m=0}^{2n-1}D_{nm}\,\log^{m}\xi\right\\}\,.$
(1)

The first set of logarithms with plus distributions and the delta functions
appear due to the leading power (LP) approximation, whereas the second set of
logarithms are the outcome of the next-to-leading power (NLP) approximation.
The LP logarithms exhibit universality due to the factorisation of soft and
collinear radiations from the hard interaction and as a consequence,
resummation of these logarithms is well established in the literature. The
pioneering works of refs. [2, 3, 4, 5, 6, 7, 8, 9] based on diagrammatics
helped in devising methods for LP resummation. Subsequently several
alternative methods of LP resummation were developed based on Wilson lines
[10, 11], renormalisation group (RG) [12] and Soft Collinear Effective Theory
(SCET) [13, 14, 15, 16]. A comprehensive comparison among different approaches
can be found in refs. [17, 18, 19].

In view of the sizeable numerical impacts of NLP logarithms [20, 21, 22, 23,
24, 25, 26, 27], several methods have been proposed during the past decade to
resum such logarithms [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41,
42, 43, 44, 45, 46, 47, 48, 49, 50, 27, 51, 52, 53, 54, 55, 56, 57, 58, 59,
60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75]. Colour
singlet production processes exhibit a universal pattern [38], however the
universality of such terms is yet to be recognised for processes with coloured
particles in the final state. To come up with a global resummation formulae,
it is necessary to study jet-associated processes in order to understand the
nature of NLP logarithms that originates from the emission of next-to-soft
gluons and soft (anti-)quarks. Very few processes with a single coloured
particle in the final state are studied so far [39, 76, 44, 77] at the NLP
accuracy and such scarcity of results arose due to the complexity in applying
the method of momenta shifting at the squared amplitude level [39]. Very
recently, we have reported the NLP results of H+jet production via gluon
fusion [1] and shown that the shifting of dipole spinors in the non-radiative
helicity amplitudes uniquely determines all colour-ordered radiative
amplitudes at the NLP accuracy by capturing the effect of a next-to-soft gluon
radiation. In fact, the method developed in [1] is generic in nature and can
be applied for processes with multiple jets in the final state.

As discussed in [1], a blended mechanism comprising of helicity amplitudes and
soft theorems of gauge theories can be used to obtain the NLP leading
logarithms by shifting relevant spinors of the colour dipoles in non-radiative
helicity amplitudes. However, as soft theorems are formulated specifically for
the emission of massless bosons, they do not account for the generation of NLP
logarithms resulting from the emission of soft (anti-)quarks. In this article,
we study the NLP logarithms that emerge due to the soft quark radiation. We
define soft quark operators that act on the colour-ordered non-radiative
helicity amplitudes, thereby making the calculation simple and tractable.
Using these operators, we look into the Higgs plus one jet production process
and compute NLP leading logarithms arising from different partonic channels
that involve a soft quark or anti-quark.

The structure of our paper is as follows. In section 2, we review the
radiation of soft and next-to-soft gluon in terms of spinor shifts. Next, we
construct operators for soft (anti-)quark radiation in section 3. In section
4, using these operators, we calculate NLP amplitudes for all the partonic
channels that take part in Higgs plus one jet production and contain colour
triplet particles in the initial and/or final states. Finally, we perform
phase space integration over the squared NLP amplitudes to obtain the NLP
leading logarithms for each of these channels in section 5. Throughout this
study, we have used a combination of in-house routines based on QGRAF [78],
FORM [79] and Mathematica [80] to calculate all amplitudes and to perform
phase space integrations. Feynman diagrams in this article are drawn using
FeynGame-2.1 [81].

##  2 Next-to-soft gluon radiation

In this section, we briefly review the emission of next-to-soft gluon in terms
of spinor shifts in the non-radiative helicity amplitudes.

The LP and NLP scattering amplitudes for $(n+1)$ particles with one next-to-
soft gluon carrying momentum $p_{s}$ and helicity ‘$\displaystyle+$’ can be
expressed in terms $n$-particle scattering amplitude as [82, 83, 1],

$\displaystyle\mathcal{A}_{\,n+1}^{\text{LP+NLP}}\bigg{(}\left\\{\lambda|s\rangle,|s]\right\\},\left\\{|1\rangle,|1]\right\\},\ldots,\left\\{|n\rangle,|n]\right\\}\bigg{)}$
$\displaystyle\,=\,\frac{1}{\lambda^{2}}\frac{\braket{1n}}{\braket{1s}\braket{ns}}\times\mathcal{A}_{\,n}\bigg{(}\left\\{|1\rangle,|1^{\prime}]\right\\},\ldots,\left\\{|n\rangle,|n^{\prime}]\right\\}\bigg{)}\,,$
(2)

where

$|1^{\prime}]\,=\,|1]+\Delta_{s}^{(1,n)}|s]\,,\\\
|n^{\prime}]\,=\,|n]+\Delta_{s}^{(n,1)}|s]\,,$

and,

$\Delta_{s}^{(i,j)}=\lambda\frac{\braket{js}}{\braket{ji}}\,.$ (3)

Here $|i\rangle$ and $|i]$ denote the holomorphic and anti-holomorphic spinors
associated with the particle $i$ carrying momentum $p_{i}$. $\lambda$ is the
scaling parameter by which the radiative gluon momentum is being scaled and
the exponent of this parameter determines the leading and sub-leading terms
that are of $\mathcal{O}(\lambda^{-2})$ and $\mathcal{O}(\lambda^{-1})$
respectively. To obtain the above formulae, we have used the holomorphic soft
limit [83, 82] i.e.,

$\displaystyle|s\rangle\rightarrow\lambda\,|s\rangle\,,\quad|s]\rightarrow|s]\,,$
(4)

and the BCFW deformation of $s$ and $n$ pair when particle $1$ forms a three
particle amplitude involving particle $s$ and the on-shell cut propagator.
Note that, in the colour-ordered amplitude of eq. (2), the soft gluon $s$ is
placed in between particles $1$ and $n$, and that can essentially be
calculated by shifting the anti-holomorphic spinors (i.e., $|1]$ and $|n]$) of
the colour dipole $\mathcal{D}_{1n}$. In fact, for a $n$-gluon amplitude one
can form ${}^{n}C_{2}=n(n-1)/2$ colour dipoles and in the next-to-soft limit,
shifting of corresponding anti-holomorphic spinors in each dipole produces one
independent colour-ordered amplitude that participates in the $(n+1)$-gluon
scattering. When the helicity of the next-to-soft gluon is ‘$\displaystyle-$’
one needs to replace all angle spinors with square spinors and vice versa in
the above set of formulae.

##  3 Soft quark radiation

We now turn our attention towards the emission of a soft quark or anti-quark
from the non-radiative amplitudes. Our aim here is to express the colour-
ordered $(n+1)$-particle amplitudes as a blend of soft quark operators and
$n$-particle amplitudes; akin to eq. (2). In order to construct soft quark
operators, we consider combining of a soft quark to the adjacent hard colour
particles, thereby forming new hard candidates to engage in non-radiative
amplitudes. Such merging can occur in two possible ways: ($i$) a soft quark
can get merged to an adjacent gluon and form a quark, as shown in fig. (1a),
($ii$) a soft quark can get merged to an adjacent anti-quark and form a gluon,
as depicted in fig. (1b).

(a)  (b)

Figure 1: Feynman diagrams showing the radiation of soft quarks. Diagram (a)
shows merging of a soft quark and a gluon; diagram (b) shows merging of a soft
quark with an antiquark.

We first discuss the scenario shown in fig. (1a) where a soft quark operator
is to be formed by melding a soft quark with gluon. Note that we work on the
colour ordered amplitude and consider one merging at a time which always
involves one power of strong coupling. Therefore, for simplicity, the colour
and coupling strength are stripped off the amplitude and then it is expressed
as,

$\displaystyle\mathcal{A}^{q_{s}g_{n}\to q}_{n+1}\,$
$\displaystyle=\,\bar{u}_{h_{s}}(p_{s})\gamma^{\mu}\frac{i(\not{p_{n}}+\not{p}_{s})}{(p_{n}+p_{s})^{2}}\frac{-i}{\sqrt{2}}\mathcal{E}^{\mu}(p_{n},h_{n})\,\mathcal{A}\,$
$\displaystyle=\,\frac{1}{\sqrt{2}}\bar{u}_{h_{s}}(p_{s})\not{\mathcal{E}}(p_{n},h_{n})\frac{\not{p}_{n}}{2p_{n}\cdot
p_{s}}\mathcal{A}\,.$ (5)

where the $n$-particle Born amplitude in the soft quark limit can be written
as,

$\displaystyle\mathcal{A}_{n}\,=\,\bar{u}_{h_{n}}(p_{n})\,\mathcal{A}\,.$ (6)

All the remaining $(n-1)$ particles are kept implicit throughout as they do
not intervene in the present calculation directly. We recall that the
dimension of $\bar{u}_{h_{s}}(p_{s})$ is $\mathcal{O}(\sqrt{p_{s}})$ and that
leaves the dimension of the overall amplitude presented in eq. (5) as
$\mathcal{O}(1/{\sqrt{p_{s}}})$ in the soft quark limit. Therefore, the
squared amplitude, although it does not generate any LP threshold
contribution, can produce NLP logarithms.

We replace $\not{p}_{n}$ in eq. (5) as
$\not{p}_{n}=\sum\limits_{h_{n}}u_{h_{n}}(p_{n})\bar{u}_{h_{n}}(p_{n})$ and
then it is straightforward to show that,

$\displaystyle\mathcal{A}^{q_{s}^{+}g_{n}^{+}\to
q^{+}}_{n+1}\,=\,-\frac{1}{\braket{ns}}\,\mathcal{A}_{n}\,,\qquad\mathcal{A}^{q_{s}^{-}g_{n}^{-}\to
q^{-}}_{n+1}\,=\,\frac{1}{[ns]}\,\mathcal{A}_{n}\,,$ (7)

and QCD being a helicity conserving theory, all the remaining combinations
that can arise due to other helicities of the mingling soft quark and gluon
become zero.

A similar calculation can be performed for fig. (1b), where a soft quark
merges with an anti-quark and produces a hard gluon to enter in
$\mathcal{A}_{n}$. In this case, the non-zero contributions can be represented
as,

$\displaystyle\mathcal{A}^{q_{s}^{+}\bar{q}_{n}^{-}\to
g^{-}}_{n+1}\,=\,\frac{1}{\braket{ns}}\,\mathcal{A}_{n}\,,\qquad\mathcal{A}^{q_{s}^{-}\bar{q}_{n}^{+}\to
g^{+}}_{n+1}\,=\,-\frac{1}{[ns]}\,\mathcal{A}_{n}\,.$ (8)

The effects of radiation of soft anti-quarks can easily be obtained in a
similar fashion as above.

We slightly alter the notation to be more explicit and represent such
factorisations in terms soft (anti-)quark operators ($\mathcal{Q}$) as
follows,

$\displaystyle\mathcal{A}^{s^{i}h^{j}}_{n+1}\,=\,\mathcal{Q}^{s^{i}h^{j}\to
c^{k}}\,\mathcal{A}^{c^{k}}_{n}\,,\qquad$ (9)

where $s^{i}$ denotes the “soft ” (anti-)quark with helicity $i$; $h^{j}$ is
the “hard ” particle with helicity $j$ and the “clubbed ” colour particle
$c^{k}$ with helicity $k$ is being formed by combining the hard and soft
particles of the $(n+1)$-particle amplitude. Note that the clubbed particle
carries the helicity and momentum of the hard particle, but not its colour and
therefore $j=k$ always. Two independent non-zero operators are listed in Table
(1) and all the rest of the non-zero operators, including the soft anti-quark
operators, are charge and/or parity conjugate of these two. Except $q\bar{q}$
merging, interchanging of the position of the soft and hard particles may
incur an additional negative sign in the soft (anti-)quark operators to take
care of a definite colour ordering.

Soft Quark Operator | Explicit Form
---|---
$\mathcal{Q}^{q_{s}^{+}\bar{q}_{h}^{-}\to g_{c}^{-}}$ | ${\displaystyle-\frac{1}{\braket{sh}}}$
$\mathcal{Q}^{g_{h}^{+}q_{s}^{+}\to q_{c}^{+}}$ | ${\displaystyle-\frac{1}{\braket{sh}}}$
Table 1: Two independent soft quark operators for two different types of
merging. All other non-zero combinations including the soft anti-quark
operators can be obtained by taking appropriate parity and/or charge
conjugations.

Note that the soft quark operators were first defined in [39], where the NLP
amplitudes were calculated using conventional Feynman diagrammatic approach
and they were also used in [84] to explain exponentiation of the soft quark
effects. Those operators exhibit intricate colour structures, thereby
rendering them challenging to employ in practical computations. However,
colour ordered helicity amplitudes together with the soft quark operators
discussed above remarkably streamline the computation of NLP amplitudes and
hence the derivation of NLP logarithms. We apply this technique to acquire NLP
logarithms that originate due to soft (anti-)quark radiations in the case of
Higgs plus one jet production at the LHC.

##  4 H+jet production

The most dominant production mechanism of Higgs boson at the LHC is through
gluon fusion. Though gluons do not interact directly with the Higgs boson, it
can be produced via a massive quark loop. Since top quark is the heaviest
among quarks, the dominant contribution comes from the top quark loop. In the
limit $m_{t}\rightarrow\infty$, the top quark effect can be integrated out to
obtain an effective Lagrangian [85, 86] given by,

$\displaystyle\mathcal{L}_{\,\text{eff}}\,=\,-\frac{1}{4}\,G\,\,H\,\text{Tr}(F_{\mu\nu}^{a}F^{\mu\nu,a})\,,$
(10)

where $F_{\mu\nu}^{a}$ is the QCD field strength tensor and the effective
coupling at the lowest order is given by $G=\alpha_{s}/3\pi v$. Here
$\alpha_{s}$ denotes the strong coupling constant and $v$ is the vacuum
expectation value of the Higgs field.

We stick to this effective Lagrangian and study the soft (anti-)quark and the
next-to-soft gluon radiation effects on Higgs plus one hard jet production at
the LHC. Therefore, the process that we are interested in involves Higgs plus
four partons among which one is either a soft (anti-)quark or a next-to-soft
gluon. NLP amplitudes that arise from Higgs plus 4-gluon channel are
thoroughly discussed in [1]. There are two other types of channels that
involve Higgs and four partons: $(i)$ one quark anti-quark pair with two
gluons and Higgs $\left(q\bar{q}ggH\right)$, and $(ii)$ two pairs of quark
anti-quark and a Higgs $\left(q\bar{q}Q\bar{Q}H\right)$. Scattering amplitudes
of these processes are given in [87].

NLP contributions of Higgs boson plus one jet production can come from four
different scattering processes that fall into the first category i.e.,
$q\bar{q}ggH$ and those four processes are given by,

$\displaystyle
q(p_{1})+\bar{q}(p_{2})+H(-p_{3})+g(-p_{4})+g(-p_{5})\rightarrow 0$
$\displaystyle\,\,,$ (11) $\displaystyle
q(p_{1})+g(p_{2})+H(-p_{3})+\bar{q}(-p_{4})+g(-p_{5})\rightarrow 0$
$\displaystyle\,,$ (12)
$\displaystyle\bar{q}(p_{1})+g(p_{2})+H(-p_{3})+q(-p_{4})+g(-p_{5})\rightarrow
0$ $\displaystyle\,,$ (13) $\displaystyle
g(p_{1})+g(p_{2})+H(-p_{3})+q(-p_{4})+\bar{q}(-p_{5})\rightarrow 0$
$\displaystyle\,.$ (14)

NLP logarithms can appear when either one of the partons with momentum $p_{4}$
and $p_{5}$ becomes (next-to-)soft. However, at the scattering amplitude
level, all these four processes are related to one another by simple
relabelling of their momenta. For simplicity, one can always consider all
external particles as incoming and calculate NLP amplitudes for any one out of
these four processes.

The second kind of scattering process that contributes to NLP logarithms
involves a Higgs boson and two pairs of quark and anti-quark,

$\displaystyle
q(p_{1})+\bar{q}(p_{2})+H(-p_{3})+Q(-p_{4})+\bar{Q}(-p_{5})\rightarrow 0$
$\displaystyle\,.$ (15)

Here $q$ and $Q$ denote different quark flavours. This process has only one
independent helicity amplitude which is given in [87] and when all
(anti-)quarks are of same flavour, one needs to add an additional contribution
by replacing $p_{2}\leftrightarrow p_{5}$.

###  4.1 Colour-ordered amplitudes

Scattering amplitude of $n$-gluons with one Higgs boson can be expressed as,

$\mathcal{A}_{nH}(\\{p_{i},h_{i},c_{i}\\})\,=\,-i\,\left(\frac{\alpha_{s}}{6\pi
v}\right)g_{s}^{n-2}\sum_{\sigma\in\mathcal{S}_{n^{\prime}}}\text{Tr}\left({\bf
T}^{c_{1}}{\bf T}^{c_{2}}{\bf T}^{c_{4}}\ldots{\bf
T}^{c_{n+1}}\right)\mathcal{A}^{h_{1}\,h_{2}\,h_{4}\,\ldots\,h_{n+1}}_{1_{g}\,2_{g}\,H\,4_{g}\,\ldots\,(n+1)_{g}}\,,$
(16)

where $\mathcal{S}_{n^{\prime}}$ denotes $(n-1)!$ non-cycling permutations of
$n$-gluons. ${\bf T}^{c_{i}}$ represents SU(3) colour matrix in the
fundamental representation and such generators are normalized as
$\text{Tr}({\bf T}^{c_{1}}{\bf T}^{c_{2}})=\delta^{c_{1}c_{2}}$. We can write
the full amplitude of 4-gluons and Higgs for a given helicity configuration in
terms of colour-ordered amplitudes as,

$\displaystyle\mathcal{A}_{4H}(\\{p_{i},h_{i},c_{i}\\})\,$
$\displaystyle=\,-i\,\left(\frac{\alpha_{s}}{6\pi
v}\right)\,g_{s}^{2}\Bigg{[}\left\\{\text{Tr}\left({\bf T}^{c_{1}}{\bf
T}^{c_{2}}{\bf T}^{c_{4}}{\bf T}^{c_{5}}\right)+\text{Tr}\left({\bf
T}^{c_{1}}{\bf T}^{c_{5}}{\bf T}^{c_{4}}{\bf
T}^{c_{2}}\right)\right\\}\mathcal{A}^{h_{1}\,h_{2}\,h_{4}\,h_{5}}_{1_{g}\,2_{g}\,4_{g}\,5_{g}}$
$\displaystyle+\left\\{\text{Tr}\left({\bf T}^{c_{1}}{\bf T}^{c_{4}}{\bf
T}^{c_{5}}{\bf T}^{c_{2}}\right)+\text{Tr}\left({\bf T}^{c_{1}}{\bf
T}^{c_{2}}{\bf T}^{c_{5}}{\bf
T}^{c_{4}}\right)\right\\}\mathcal{A}^{h_{1}\,h_{2}\,h_{4}\,h_{5}}_{1_{g}\,4_{g}\,5_{g}\,2_{g}\,}$
$\displaystyle+\left\\{\text{Tr}\left({\bf T}^{c_{1}}{\bf T}^{c_{5}}{\bf
T}^{c_{2}}{\bf T}^{c_{4}}\right)+\text{Tr}\left({\bf T}^{c_{1}}{\bf
T}^{c_{4}}{\bf T}^{c_{2}}{\bf
T}^{c_{5}}\right)\right\\}\mathcal{A}^{h_{1}\,h_{2}\,h_{4}\,h_{5}}_{1_{g}\,5_{g}\,2_{g}\,4_{g}}\Bigg{]}\,.$
(17)

For brevity, we avoid writing H explicitly in the colour-ordered amplitudes
here and in the rest of the paper. Squaring the above equation and summing
over colours we obtain,

$\displaystyle\sum_{\text{colours}}|\mathcal{A}_{4H}(\\{p_{i},h_{i},c_{i}\\})|^{2}$
$\displaystyle=\bigg{[}\left(\frac{\alpha_{s}}{6\pi
v}\right)g_{s}^{2}\bigg{]}^{2}(N^{2}-1)\bigg{\\{}2N^{2}\left(\left|\mathcal{A}^{h_{1}\,h_{2}\,h_{4}\,h_{5}}_{1_{g}\,2_{g}\,4_{g}\,5_{g}}\right|^{2}+\left|\mathcal{A}^{h_{1}\,h_{2}\,h_{4}\,h_{5}}_{1_{g}\,4_{g}\,5_{g}\,2_{g}}\right|^{2}+\left|\mathcal{A}^{h_{1}\,h_{2}\,h_{4}\,h_{5}}_{1_{g}\,5_{g}\,2_{g}\,4_{g}}\right|^{2}\right)$
$\displaystyle-4\frac{(N^{2}-3)}{N^{2}}\left|\mathcal{A}^{h_{1}\,h_{2}\,h_{4}\,h_{5}}_{1_{g}\,2_{g}\,4_{g}\,5_{g}}+\mathcal{A}^{h_{1}\,h_{2}\,h_{4}\,h_{5}}_{1_{g}\,4_{g}\,5_{g}\,2_{g}}+\mathcal{A}^{h_{1}\,h_{2}\,h_{4}\,h_{5}}_{1_{g}\,5_{g}\,2_{g}\,4_{g}}\right|^{2}\bigg{\\}}\,.$
(18)

in which the sub-leading colour term given in the second line of the above
equation does not contribute at all due to the dual Ward identity [88, 87].

The general form of scattering amplitude involving a quark antiquark pair, $n$
gluons and a Higgs boson is given by,

$\displaystyle\mathcal{A}^{q\bar{q}}_{nH}(\\{p_{i},h_{i},c_{i}\\})\,=\,-i\left(\frac{\alpha_{s}}{6\pi
v}\right)g_{s}^{n}\sum_{\sigma\in S_{n}}({\bf T}^{c_{4}}\ldots{\bf
T}^{c_{n+3}})_{j_{2}j_{1}}{}\mathcal{A}^{h_{1}\,h_{2}\,h_{4}\,\ldots\,h_{n+3}}_{1_{q}\,2_{\bar{q}}\,H\,4_{g}\,\ldots\,(n+3)_{g}}\,,$
(19)

where $S_{n}$ is the $n!$ possible non-cyclic permutations of $n$ gluons in
the presence of a $q\bar{q}$ pair.

Now, using the above equation, the amplitude for the process given in eq. (11)
can be expressed as,

$\displaystyle\mathcal{A}^{q\bar{q}}_{2H}(\\{p_{i},h_{i},c_{i}\\})\,=\,-i\left(\frac{\alpha_{s}}{6\pi
v}\right)g_{s}^{2}\left[\left({\bf T}^{c_{4}}{\bf
T}^{c_{5}}\right)_{j_{2}j_{1}}\mathcal{A}^{h_{1}\,h_{2}\,h_{4}\,h_{5}}_{1_{q}\,2_{\bar{q}}\,4_{g}\,5_{g}}+\left({\bf
T}^{c_{5}}{\bf
T}^{c_{4}}\right)_{j_{2}j_{1}}\mathcal{A}^{h_{1}\,h_{2}\,h_{4}\,h_{5}}_{1_{q}\,2_{\bar{q}}\,5_{g}\,4_{g}}\right]\,,$
(20)

and squaring the above amplitude and summing over colours we obtain,

$\displaystyle\sum_{\text{colours}}\left|\mathcal{A}^{q\bar{q}}_{2H}(\\{p_{i},h_{i},c_{i}\\})\right|^{2}\,$
$\displaystyle=\,\left(\frac{\alpha_{s}}{6\pi
v}\right)^{2}g_{s}^{4}\,(N^{2}-1)\bigg{\\{}N\left(\left|\mathcal{A}^{h_{1}\,h_{2}\,h_{4}\,h_{5}}_{1_{q}\,2_{\bar{q}}\,4_{g}\,5_{g}}\right|^{2}+\left|\mathcal{A}^{h_{1}\,h_{2}\,h_{4}\,h_{5}}_{1_{q}\,2_{\bar{q}}\,5_{g}\,4_{g}}\right|^{2}\right)$
$\displaystyle-\frac{1}{N}\left|\mathcal{A}^{h_{1}\,h_{2}\,h_{4}\,h_{5}}_{1_{q}\,2_{\bar{q}}\,4_{g}\,5_{g}}+\mathcal{A}^{h_{1}\,h_{2}\,h_{4}\,h_{5}}_{1_{q}\,2_{\bar{q}}\,5_{g}\,4_{g}}\right|^{2}\bigg{\\}}\,.$
(21)

The colour-ordered amplitude for Higgs plus two quarks and two anti-quarks is
given by,

$\displaystyle\mathcal{A}^{q\bar{q}Q\bar{Q}}_{H}(\\{p_{i},h_{i},c_{i}\\})\,=\,-i\left(\frac{\alpha_{s}}{6\pi
v}\right)\,g_{s}^{2}\,({\bf T}^{c_{1}})_{j_{2}j_{1}}({\bf
T}^{c_{1}})_{j_{4}j_{3}}\,\mathcal{A}^{h_{1}\,h_{2}\,h_{4}\,h_{5}}_{1_{q}\,2_{\bar{q}}\,4_{Q}\,5_{\bar{Q}}}$
(22)

where the helicities of the antiquarks are fixed by the helicities of the
quarks i.e., $h_{2}=-h_{1}$ and $h_{4}=-h_{5}$. Squaring the amplitude and
summing over colours we get,

$\displaystyle\sum_{\text{colours}}|\mathcal{A}^{q\bar{q}Q\bar{Q}}_{H}(\\{p_{i},h_{i},c_{i}\\})|^{2}\,=\,\left(\frac{\alpha_{s}}{6\pi
v}\right)^{2}g_{s}^{4}\,(N^{2}-1)\left|\mathcal{A}^{h_{1}\,h_{2}\,h_{4}\,h_{5}}_{1_{q}\,2_{\bar{q}}\,4_{Q}\,5_{\bar{Q}}}\right|^{2}\,,$
(23)

and here different two different quark flavours are considered. For the case
where quarks are of same flavours, we can write the squared amplitude as,

$\displaystyle\sum_{\text{colours}}|\mathcal{A}^{q\bar{q}Q\bar{Q}}_{H}(\\{p_{i},h_{i},c_{i}\\})|^{2}\,$
$\displaystyle=\,\left(\frac{\alpha_{s}}{6\pi
v}\right)^{2}g_{s}^{4}\,(N^{2}-1)\bigg{\\{}\left|\mathcal{A}^{h_{1}\,h_{2}\,h_{4}\,h_{5}}_{1_{q}\,2_{\bar{q}}\,4_{Q}\,5_{\bar{Q}}}\right|^{2}+\left|\mathcal{A}^{h_{1}\,h_{2}\,h_{4}\,h_{5}}_{1_{q}\,2_{\bar{q}}\,4_{q}\,5_{\bar{q}}}\right|^{2}$
$\displaystyle+\frac{\delta_{h_{1}h_{4}}}{N}\bigg{(}\mathcal{A}^{h_{1}\,h_{2}\,h_{4}\,h_{5}}_{1_{q}\,2_{\bar{q}}\,4_{Q}\,5_{\bar{Q}}}\mathcal{A}^{\dagger\,h_{1}\,h_{2}\,h_{4}\,h_{5}}_{1_{q}\,2_{\bar{q}}\,4_{q}\,5_{\bar{q}}}+\mathcal{A}^{h_{1}\,h_{2}\,h_{4}\,h_{5}}_{1_{q}\,2_{\bar{q}}\,4_{q}\,5_{\bar{q}}}\mathcal{A}^{\dagger\,h_{1}\,h_{2}\,h_{4}\,h_{5}}_{1_{q}\,2_{\bar{q}}\,4_{Q}\,5_{\bar{Q}}}\bigg{)}\bigg{\\}}\,,$
(24)

where
$\mathcal{A}^{h_{1}\,h_{2}\,h_{4}\,h_{5}}_{1_{q}\,2_{\bar{q}}\,4_{q}\,5_{\bar{q}}}=\mathcal{A}^{h_{1}\,h_{2}\,h_{4}\,h_{5}}_{1_{q}\,5_{\bar{q}}\,4_{Q}\,2_{\bar{Q}}}$,
and the sub-leading colour term exists only when the quarks carry the same
helicity.

###  4.2 Leading order amplitudes

The next-to-soft gluon effects can be obtained by shifting appropriate spinors
in the leading order amplitudes. On the other hand, the soft quark effects can
be obtained through soft quark operators which eventually result into spinor
factors multiplied with the leading order amplitudes. So the basic ingredients
to compute NLP amplitudes are the leading order helicity amplitudes that
contain Higgs plus three partons and we calculate them here under.

The leading order process with a quark anti-quark pair, a Higgs, and a gluon
is given by,

$\displaystyle q(p_{1})+\bar{q}(p_{2})+H(p_{3})+g(p_{4})\rightarrow 0\,.$ (25)

Note that, for simplicity, we consider all external legs as incoming
throughout this article unless mentioned otherwise. There is only one
independent colour-ordered helicity amplitude for this process, and that is
given by,

$\displaystyle\mathcal{A}^{+-+}_{1_{q}\,2_{\bar{q}}\,4_{g}}\,=\,\frac{\,\,[14]^{2}}{[12]}\,.$
(26)

The remaining helicity amplitudes for this process can be obtained by taking
either parity or charge conjugation of the above amplitude.

The other leading order process which involves Higgs boson and three gluons
can be represented as,

$\displaystyle g(p_{1})+g(p_{2})+H(p_{3})+g(p_{4})\rightarrow 0\,,$ (27)

and two independent colour-ordered helicity amplitudes for this process are,

$\displaystyle\mathcal{A}_{1_{g}\,2_{g}\,4_{g}}^{+++}$
$\displaystyle=\frac{m_{H}^{4}}{\braket{12}\braket{24}\braket{41}}\,,\qquad\mathcal{A}_{1_{g}\,2_{g}\,4_{g}}^{-++}=\frac{[24]^{3}}{[12][14]}\,.$
(28)

All other helicity combinations for this process can be constructed out of
these two helicity configurations. All these amplitudes agree with the results
given in [87].

###  4.3 NLP amplitudes for $q\bar{q}ggH$

We present here the NLP amplitudes for the scattering involving Higgs, a quark
antiquark pair, and two gluons. Throughout this section, we choose the
kinematic configuration given in eq. (11), except the fact that all external
particles are now incoming. Due to the fact that quark and anti-quark always
possess opposite helicities, there are total $\displaystyle 2^{3}=8$ helicity
configurations. Out of these eight, one needs to calculate only three
configurations, since the remaining configurations can be obtained by using
parity and charge conjugation. We choose to calculate NLP amplitudes for the
following three independent helicity configurations: ($i$)
$\mathcal{A}^{+-++}_{1_{q}\,2_{\bar{q}}\,4_{g}\,5_{g}}$, ($ii$)
$\mathcal{A}^{+-+-}_{1_{q}\,2_{\bar{q}}\,4_{g}\,5_{g}}$, ($iii$)
$\mathcal{A}^{+--+}_{1_{q}\,2_{\bar{q}}\,4_{g}\,5_{g}}$, and their full
results are given by [87],

$\displaystyle\mathcal{A}^{+-++}_{1_{q}\,2_{\bar{q}}\,4_{g}\,5_{g}}$
$\displaystyle=\frac{\langle 2|1+5|4]^{2}}{s_{125}}\frac{[15]}{\langle
25\rangle}\left(\frac{1}{s_{12}}+\frac{1}{s_{15}}\right)-\frac{\langle
2|1+4|5]^{2}}{s_{124}\,s_{12}}\frac{[14]}{\langle 24\rangle}+\frac{\langle
2|4+5|1]^{2}}{[12]\langle 24\rangle\langle 25\rangle\langle 45\rangle}\,,$
$\displaystyle\mathcal{A}^{+-+-}_{1_{q}\,2_{\bar{q}}\,4_{g}\,5_{g}}$
$\displaystyle=\frac{\langle 25\rangle^{3}}{\langle 12\rangle\langle
24\rangle\langle 45\rangle}-\frac{[14]^{3}}{[12][15][45]}\,,$
$\displaystyle\mathcal{A}^{+--+}_{1_{q}\,2_{\bar{q}}\,4_{g}\,5_{g}}$
$\displaystyle=\frac{\langle 24\rangle^{2}\langle 14\rangle}{\langle
12\rangle\langle 15\rangle\langle
45\rangle}-\frac{[15]^{2}[25]}{[12][24][45]}\,,$ (29)

where $s_{ijk}=(s_{ij}+s_{jk}+s_{ik})$ and $s_{ij}=2p_{i}.p_{j}$, when $p_{i}$
and $p_{j}$ denote momenta of massless particles.

####  4.3.1 Radiation of soft quark

Soft quark contribution can be obtained by expanding the $q\bar{q}ggH$
helicity amplitudes given in eq. (29) in powers of the holomorphic quark
spinor and then keeping the leading terms only. Alternatively, they can also
be obtained using the factorisation of amplitudes in terms of soft quark
operators, defined in section 3. In what follows, we take the latter approach
to express the NLP amplitudes due to soft quark radiation. In every case,
these results perfectly match with the ones derived using the former approach.

We choose to calculate NLP amplitudes for the process given in eq. (11),
provided all the external particles are now incoming. As discussed in section
3, in a colour-ordered helicity amplitude the soft quark can couple either to
an adjacent anti-quark or to an adjacent gluon, and can produce a clubbed
gluon or quark. Following that, the independent NLP amplitudes can be
expressed as,

$\displaystyle\left.\mathcal{A}^{+-++}_{1_{q(s)}\,2_{\bar{q}}\,4_{g}\,5_{g}}\right|_{\text{NLP}}\,=\,$
$\displaystyle\mathcal{Q}^{q_{1}^{+}\bar{q}_{2}^{-}\to
g_{2}^{-}}\mathcal{A}^{-++}_{2_{g}\,4_{g}\,5_{g}}+\mathcal{Q}^{g_{5}^{+}q_{1}^{+}\to
q_{5}^{+}}\mathcal{A}^{+-+}_{5_{q}\,2_{\bar{q}}\,4_{g}}\,,$
$\displaystyle\left.\mathcal{A}^{+-+-}_{1_{q(s)}\,2_{\bar{q}}\,4_{g}\,5_{g}}\right|_{\text{NLP}}\,=\,$
$\displaystyle\mathcal{Q}^{q_{1}^{+}\bar{q}_{2}^{-}\to
g_{2}^{-}}\mathcal{A}^{-+-}_{2_{g}\,4_{g}\,5_{g}}\,,$
$\displaystyle\left.\mathcal{A}^{+--+}_{1_{q(s)}\,2_{\bar{q}}\,4_{g}\,5_{g}}\right|_{\text{NLP}}\,=\,$
$\displaystyle\mathcal{Q}^{q_{1}^{+}\bar{q}_{2}^{-}\to
g_{2}^{-}}\mathcal{A}^{--+}_{2_{g}\,4_{g}\,5_{g}}+\mathcal{Q}^{g_{5}^{+}q_{1}^{+}\to
q_{5}^{+}}\mathcal{A}^{+--}_{5_{q}\,2_{\bar{q}}\,4_{g}}\,.$ (30)

Explicit forms of the soft quark operators that take part in the above
equations are provided in Table 1. Note that the second NLP amplitude contains
only one single operator as the operator of type
$\mathcal{Q}^{g_{h}^{-}q_{s}^{+}\to q_{c}^{-}}=0$.

####  4.3.2 Radiation of soft anti-quark

A soft anti-quark can either couple to a hard quark and form a clubbed gluon,
or it can be combined to a hard gluon to produce a clubbed anti-quark. The
three different helicity amplitudes in case of soft anti-quark radiation can
be written as,

$\displaystyle\left.\mathcal{A}^{+-++}_{1_{q}\,2_{\bar{q}(s)}\,4_{g}\,5_{g}}\right|_{\text{NLP}}\,=\,$
$\displaystyle\mathcal{Q}^{q_{1}^{+}\bar{q}_{2}^{-}\to
g_{1}^{+}}\mathcal{A}^{+++}_{1_{g}\,4_{g}\,5_{g}}\,,$
$\displaystyle\left.\mathcal{A}^{+-+-}_{1_{q}\,2_{\bar{q}(s)}\,4_{g}\,5_{g}}\right|_{\text{NLP}}\,=\,$
$\displaystyle\mathcal{Q}^{q_{1}^{+}\bar{q}_{2}^{-}\to
g_{1}^{+}}\mathcal{A}^{++-}_{1_{g}\,4_{g}\,5_{g}}\,,$
$\displaystyle\left.\mathcal{A}^{+--+}_{1_{q}\,2_{\bar{q}(s)}\,4_{g}\,5_{g}}\right|_{\text{NLP}}\,=\,$
$\displaystyle\mathcal{Q}^{q_{1}^{+}\bar{q}_{2}^{-}\to
g_{1}^{+}}\mathcal{A}^{+-+}_{1_{g}\,4_{g}\,5_{g}}+\mathcal{Q}^{\bar{q}_{2}^{-}g_{4}^{-}\to\bar{q}_{4}^{-}}\mathcal{A}^{+--}_{1_{q}\,4_{\bar{q}}\,5_{g}}\,.$
(31)

The generic forms of the two types of non-zero soft anti-quark operators that
play the part here are, $\mathcal{Q}^{q_{h}^{+}\bar{q}_{s}^{-}\to
g_{c}^{+}}=\frac{1}{[sh]}$ and
$\mathcal{Q}^{\bar{q}_{s}^{-}g_{h}^{-}\to\bar{q}_{c}^{-}}=-\frac{1}{[sh]}$,
and the same can be inferred once appropriate conjugations on the ingredients
given in Table 1 are applied. Again, we find perfect agreement when the same
set of results are produced by retaining only the leading terms in the
expansion of eq. (29) in powers of the anti-holomorphic anti-quark spinor.

####  4.3.3 Radiation of soft gluons

The colour-ordered amplitudes for soft gluon radiation can be obtained very
easily through formation of colour dipoles and shifting of appropriate spinors
in the leading order amplitudes[1]. The helicity of the emitted soft gluon can
be both ‘$\displaystyle+$’ or ‘$\displaystyle-$’ and therefore configurations
that are maximally helicity violating (MHV) may also appear. However, it is to
be noted that any MHV configuration does not contribute to the NLP amplitude.
We describe this by considering a simple example. Let us consider that a gluon
with ‘$\displaystyle-$’ helicity is being emitted from
$\mathcal{A}^{+-+}_{1_{q}\,2_{\bar{q}}\,4_{g}}$. Now following the arguments
illustrated in section 2, one needs to shift the holomorphic spinors depending
on where the soft gluon is going to be attached. However, the leading order
amplitude given in eq. (26) does not contain any holomorphic spinor at all and
hence NLP amplitudes of
$\mathcal{A}_{1_{q}^{+}2_{\bar{q}}^{-};4_{g}^{+}5_{g}^{-}}$ vanishes. The same
can be verified by expanding the full result given in eq. (29) in terms of
anti-holomorphic spinor $|5]$. We note in passing that no NMHV amplitudes
contribute at the NLP the case of Higgs plus one jet production via gluon
fusion [1]. Therefore vanishing of NLP amplitudes with alternative helicity
signatures, is independent of the production channel when next-to-soft gluon
radiation is considered.

In the absence of MHV amplitudes at the NLP, we are left with only one type of
non-vanishing amplitude
$\mathcal{A}^{+-++}_{1_{q}\,2_{\bar{q}}\,4_{g}\,5_{g}}$. Its various colour-
ordered components, as given in eq. (21), can be obtained by shifting
appropriate spinors of the colour dipoles formed at the leading order. The LP
and NLP contributions can be expressed as [1],

$\displaystyle\left.\mathcal{A}_{1_{q}\,2_{\bar{q}}\,4_{g}\,5_{g}}^{+-++}\right|_{\text{LP+NLP}}\,=\,\frac{\braket{14}}{\braket{15}\braket{45}}\mathcal{A}^{+-+}_{1^{{}^{\prime}}_{q}\,2_{\bar{q}}\,4^{{}^{\prime}}_{g}}\,,$
(32)

where the primes indicate the shifted spinors $|1^{{}^{\prime}}]$ and
$|4^{{}^{\prime}}]$ according to eq. (2). Similarly, the other colour-ordered
amplitude can be obtained by shifting the spinors $|2^{{}^{\prime}}]$ and
$|4^{{}^{\prime}}]$, and that is given by,

$\displaystyle\left.\mathcal{A}_{1_{q}\,2_{\bar{q}}\,5_{g}\,4_{g}}^{+-++}\right|_{\text{LP+NLP}}\,=\,\frac{\braket{24}}{\braket{25}\braket{54}}\mathcal{A}^{+-+}_{1_{q}\,2^{{}^{\prime}}_{\bar{q}}\,4^{{}^{\prime}}_{g}}\,.$
(33)

After placing the expressions of shifted anti-holomorphic spinors in eq. (32)
and extracting out the NLP portions we get,

$\displaystyle\left.\mathcal{A}^{+-++}_{1_{q}\,2_{\bar{q}}\,4_{g}\,5_{g}}\right|_{\text{NLP}}=$
$\displaystyle\frac{\braket{14}}{\braket{15}\braket{45}}\left(\frac{2s_{15}}{s_{14}}+\frac{2s_{45}}{s_{14}}-\frac{\braket{45}[25]}{\braket{14}[12]}\right)\mathcal{A}^{+-+}_{1_{q}\,2_{\bar{q}}\,4_{g}}\,.$
(34)

This amplitude agree with the sub-leading terms obtained by expanding the
first equality in eq. (29) around the gluon momentum $p_{5}$ in the
holomorphic limit. Similarly for eq. (33) we get,

$\displaystyle\left.\mathcal{A}^{+-++}_{1_{q}\,2_{\bar{q}}\,5_{g}\,4_{g}}\right|_{\text{NLP}}=$
$\displaystyle\frac{\braket{24}}{\braket{25}\braket{54}}\left(\frac{\braket{45}[15]}{\braket{24}[12]}+\frac{2\braket{25}[15]}{\braket{24}[14]}\right)\mathcal{A}^{+-+}_{1_{q}\,2_{\bar{q}}\,4_{g}}\,.$
(35)

once the exact form of the shifted spinors are inserted. We note in passing
that the addition of the two colour-ordered amplitudes given in eqs. (32) and
(33) can be obtained by using a shifted pair of spinors $|1^{{}^{\prime}}]$
and $|2^{{}^{\prime}}]$ in the leading order amplitude as,

$\displaystyle\left.\left(\mathcal{A}^{+-++}_{1_{q}\,2_{\bar{q}}\,4_{g}\,5_{g}}+\mathcal{A}^{+-++}_{1_{q}\,2_{\bar{q}}\,5_{g}\,4_{g}}\right)\right|_{\text{LP+NLP}}\,=\,\frac{\braket{12}}{\braket{15}\braket{25}}\mathcal{A}^{+-+}_{1^{{}^{\prime}}_{q}\,2^{{}^{\prime}}_{\bar{q}}\,4_{g}}\,.$
(36)

This essentially means that in the next-to-soft gluon limit, the sub-leading
colour term of eq. (21) has its genesis in the dipole $\mathcal{D}_{12}$ of
the leading order amplitude.

###  4.4 NLP amplitudes for $q\bar{q}Q\bar{Q}$

In this sub-section, we calculate the NLP amplitudes for the scattering
process involving Higgs and two pairs of quarks and anti-quarks. In this case,
out of the $2^{2}=4$ helicity configurations, only one is independent.
Considering all the particles to be incoming for the process given in eq. (21)
and the quark with momentum $p_{1}$ to be soft, the amplitude can be written
as,

$\displaystyle\left.\mathcal{A}^{+-+-}_{1_{q(s)}\,2_{\bar{q}}\,4_{Q}\,5_{\bar{Q}}}\right|_{\text{NLP}}\,=\,\mathcal{Q}^{q_{1}^{+}\bar{q}_{2}^{-}\to
g_{2}^{-}}\mathcal{A}^{-+-}_{2_{g}\,4_{Q}\,5_{\bar{Q}}}\,,$ (37)

where the explicit form of quark operator is provided in Table 1. In case the
anti-quark with momentum $p_{2}$ becomes soft, the amplitude takes the
following form,

$\displaystyle\left.\mathcal{A}^{+-+-}_{1_{q}\,2_{\bar{q}(s)}\,4_{Q}\,5_{\bar{Q}}}\right|_{\text{NLP}}\,=\,\mathcal{Q}^{q_{1}^{+}\bar{q}_{2}^{-}\to
g_{1}^{+}}\mathcal{A}^{++-}_{1_{g}\,4_{Q}\,5_{\bar{Q}}}\,,$ (38)

and as mentioned earlier in sec. 4.3.2,
$\mathcal{Q}^{q_{h}^{+}\bar{q}_{s}^{-}\to g_{c}^{+}}=\frac{1}{[sh]}$. In the
above, we have considered the scenario when flavours of the two $q\bar{q}$
pairs are different. For all (anti-)quarks to be of same flavour, one needs to
take into account additional clubbing of soft quark with another adjacent
anti-quark and vice versa. Needless to say, one can arrive at the same results
when (anti-)holomorphic expansion over the soft (anti-)quark spinor is taken
in the full result [87] and all sub-leading terms are discarded.

##  5 NLP logarithms

To get the NLP logarithms, we need to integrate the squared amplitudes over
the unobserved parton phase space in the rest frame of $p_{4}$ and $p_{5}$. We
factorise three-body phase space is into two two-body phase spaces in the
usual manner: (i) one containing two coloured particles with momenta $p_{4}$
and $p_{5}$, (ii) another one containing the Higgs ($p_{3}$) and the joint
contribution of the two coloured particles ($p_{4}+p_{5}$). We choose the
phase space parametrisation in $d=(4-2\epsilon)$ dimension [89, 90] as,

$\displaystyle p_{1}$ $\displaystyle=$
$\displaystyle(E_{1},0,\cdots,0,E_{1})\,,$ $\displaystyle p_{2}$
$\displaystyle=$ $\displaystyle(E_{2},0,\cdots,0,p_{3}\sin\psi,p_{3}\cos\psi-
E_{1})\,,$ $\displaystyle p_{3}$ $\displaystyle=$
$\displaystyle-(E_{3},0,\cdots,0,p_{3}\sin\psi,p_{3}\cos\psi)\,,$
$\displaystyle p_{4}$ $\displaystyle=$
$\displaystyle-\frac{\sqrt{s_{45}}}{2}(1,0,\cdots,0,\sin\theta_{1}\sin\theta_{2},\sin\theta_{1}\cos\theta_{2},\cos\theta_{1})\,,$
$\displaystyle p_{5}$ $\displaystyle=$
$\displaystyle-\frac{\sqrt{s_{45}}}{2}(1,0,\cdots,0,-\sin\theta_{1}\sin\theta_{2},-\sin\theta_{1}\cos\theta_{2},-\cos\theta_{1})\,.$
(39)

The helicity dependent differential cross-section at NLP can then be written
as,

$\displaystyle\left.s_{12}^{2}\frac{d^{2}\sigma^{h_{1}h_{2}h_{3}h_{4}}}{ds_{13}ds_{23}}\right|_{\text{NLP}}\,$
$\displaystyle=\,\mathcal{F}_{ab}\left(\frac{s_{45}}{\bar{\mu}^{2}}\right)^{-\epsilon}\,\overline{\mathcal{A}_{\text{NLP}}^{2}}\,,$
(40)

where

$\displaystyle\mathcal{F}_{ab}\,=\,\frac{1}{16}\kappa_{ab}\,(N^{2}-1)\,G^{2}\left(\frac{\alpha_{s}(\bar{\mu}^{2})}{4\pi}\right)^{2}\,\left(\frac{s_{13}\,s_{23}-m_{H}^{2}\,s_{45}}{\bar{\mu}^{2}\,s_{12}}\right)^{-\epsilon}\,.$
(41)

Here $\bar{\mu}^{2}=4\pi e^{-\gamma_{E}\epsilon}\mu_{r}^{2}$, and
$\kappa_{ab}$ denotes the factor due to the colour average over the initial
states i.e., $\kappa_{gg}=1/(N^{2}-1)^{2}$, $\kappa_{q\bar{q}}=1/N^{2}$,
$\kappa_{qg}=\kappa_{\bar{q}g}=1/(N(N^{2}-1))$. Angular integrated squared
amplitude $\overline{\mathcal{A}_{\text{NLP}}^{2}}$ is defined as,

$\displaystyle\overline{\mathcal{A}_{\text{NLP}}^{2}}\,=$
$\displaystyle\,\int_{0}^{\pi}d\theta_{1}\,(\sin\theta_{1})^{1-2\epsilon}\int_{0}^{\pi}d\theta_{2}\,(\sin\theta_{2})^{-2\epsilon}[\mathcal{\tilde{A}}^{2}]_{\,}|_{\text{NLP}}.$
(42)

where $[\mathcal{\tilde{A}}^{2}]|_{\text{NLP}}$ is the colour-summed squared
NLP amplitude for a given helicity configuration. It is important to take the
mass factorisation into account in order for the cancellation of singularities
that appear once the above angular integration is performed.

###  5.1 NLP logarithms: $q\bar{q}Hgg$

Differential cross-sections would be different for different initial states.
Squared NLP colour-ordered amplitudes
$\left[\mathcal{A}\right]^{2}_{\text{NLP}}$ can be obtained through the
exchange of momenta applied in the expressions of colour-ordered NLP
amplitudes and then squaring them accordingly. For next-to-soft gluon emission
$\left[\mathcal{A}\right]^{2}_{\text{NLP}}=2\,\text{Re}(\mathcal{A}_{\rm
NLP}\mathcal{A}^{\dagger}_{\rm LP})$, whereas
$\left[\mathcal{A}\right]^{2}_{\text{NLP}}=(\mathcal{A}_{\rm
NLP}\mathcal{A}^{\dagger}_{\rm NLP})$ for soft (anti-)quark emission. The
momentum carried by the (next-to-)soft particle, which can be identified via a
suffix $s$ within the bracket, is either $p_{4}$ or $p_{5}$, as we choose to
work in the $(p_{4}+p_{5})$ rest frame. In the following, we provide the NLP
leading logarithms for the independent helicity configurations, as parity
conjugate configurations produce same results and the other helicity
configurations can be obtained by re-arrenging the external momenta.

####  5.1.1 $q\bar{q}$ initiated process

We define the scattering process as follows,

$\displaystyle
q(p_{1})+\bar{q}(p_{2})+H(-p_{3})+g(-p_{4})+g(-p_{5})\rightarrow 0$
$\displaystyle\,\,,$ (43)

and consider the gluon with momentum $p_{5}$ as next-to-soft. As discussed in
sec. 4.3.3, there is only one independent helicity configuration that can
produce NLP contribution, because the rest are MHV amplitudes and they do not
participate at NLP.

The squared colour-ordered amplitudes that take part in the colour-summed
squared amplitude given in eq. (21) are,

$\displaystyle\left[\mathcal{A}^{+-++}_{1_{q}\,2_{\bar{q}}\,4_{g}\,5_{g(s)}}\right]^{2}_{\text{NLP}}\,=\,\left(\frac{3}{s_{15}}+\frac{4}{s_{45}}+\frac{s_{24}}{s_{12}s_{45}}-\frac{s_{14}s_{25}}{s_{12}s_{15}s_{45}}\right)\left|\mathcal{A}^{+-+}_{1_{q}\,2_{\bar{q}}\,4_{g}}\right|^{2}\,,$
$\displaystyle\left[\mathcal{A}^{+-++}_{1_{q}\,2_{\bar{q}}\,5_{g(s)}\,4_{g}}\right]^{2}_{\text{NLP}}\,=\,\left(-\frac{1}{s_{25}}+\frac{2}{s_{45}}+\frac{s_{14}}{s_{12}s_{45}}-\frac{2s_{12}}{s_{14}s_{25}}-\frac{s_{15}s_{24}}{s_{25}s_{45}s_{12}}+\frac{2s_{15}s_{24}}{s_{14}s_{25}s_{45}}\right)\left|\mathcal{A}^{+-+}_{1_{q}\,2_{\bar{q}}\,4_{g}}\right|^{2}\,,$
$\displaystyle\left[\mathcal{A}^{+-++}_{1_{q}\,2_{\bar{q}}\,4_{g}\,5_{g(s)}}+\mathcal{A}^{+-++}_{1_{q}\,2_{\bar{q}}\,5_{g(s)}\,4_{g}}\right]^{2}_{\text{NLP}}\,=\,\left(-\frac{2}{s_{25}}-\frac{2s_{24}}{s_{14}s_{25}}+\frac{2s_{12}s_{45}}{s_{14}s_{15}s_{25}}\right)\left|\mathcal{A}^{+-+}_{1_{q}\,2_{\bar{q}}\,4_{g}}\right|^{2}\,.$
(44)

Now using eq. (40), we calculate the helicity driven NLP leading logarithms
that contribute to the differential cross-section i.e.,

$\displaystyle\left.s_{12}^{2}\frac{d^{2}\sigma^{+-++}_{1_{q}\,2_{\bar{q}}\,4_{g}\,5_{g(s)}}}{ds_{13}\,ds_{23}}\right|_{\text{NLP-
LL}}\,$ $\displaystyle=\,\mathcal{F}_{q\bar{q}}\,\bigg{\\{}4\pi
N\left(\frac{1}{s_{23}}-\frac{1}{s_{13}}\right)\log\left(\frac{s_{45}}{\mu^{2}}\right)$
$\displaystyle-\frac{4\pi}{N}\left[\left(\frac{1}{s_{23}}-\frac{1}{s_{13}}\right)\log\left(\frac{s_{45}}{\mu^{2}}\right)+\frac{2}{s_{23}}\log\left(\frac{s_{12}s_{45}}{s_{13}s_{23}}\right)\right]\bigg{\\}}\left|\mathcal{A}^{+-+}_{1_{q}\,2_{\bar{q}}\,4_{g}}\right|^{2}.$
(45)

The coefficients of $\log\left(\frac{s_{45}}{\mu^{2}}\right)$ coming from the
leading and sub-leading colour terms are exactly same.

####  5.1.2 $gg$ initiated process

For this process, we assign the momenta of the external particles as follows,

$\displaystyle
g(p_{1})+g(p_{2})+H(-p_{3})+q(-p_{4})+\bar{q}(-p_{5})\rightarrow 0$
$\displaystyle\,.$ (46)

In this case, either the quark or the anti-quark in the final state can be
soft. When the final state quark is soft, the squared colour-ordered
amplitudes that are the ingredients of eq. (21) can be expressed for the two
independent helicity configurations as,

$\displaystyle\left[\mathcal{A}^{+++-}_{1_{g}\,2_{g}\,4_{q}\,5_{\bar{q}(s)}}\right]^{2}_{\text{NLP}}\,=\,\frac{1}{s_{45}}\left|\mathcal{A}^{+++}_{1_{g}\,2_{g}\,4_{g}}\right|^{2}\,,$
$\displaystyle\left[\mathcal{A}^{+++-}_{2_{g}\,1_{g}\,4_{q}\,5_{\bar{q}(s)}}\right]^{2}_{\text{NLP}}\,=\,\frac{1}{s_{45}}\left|\mathcal{A}^{+++}_{1_{g}\,2_{g}\,4_{g}}\right|^{2}\,,$
$\displaystyle\left[\mathcal{A}^{+++-}_{1_{g}\,2_{g}\,4_{q}\,5_{\bar{q}(s)}}+\mathcal{A}^{+++-}_{2_{g}\,1_{g}\,4_{q}\,5_{\bar{q}(s)}}\right]^{2}_{\text{NLP}}\,=\,0\,,$
(47)

and

$\displaystyle\left[\mathcal{A}^{+-+-}_{1_{g}\,2_{g}\,4_{q}\,5_{\bar{q}(s)}}\right]^{2}_{\text{NLP}}\,=\,\frac{1}{s_{45}}\left|\mathcal{A}^{+-+}_{1_{g}\,2_{g}\,4_{g}}\right|^{2}\,,$
$\displaystyle\left[\mathcal{A}^{+-+-}_{2_{g}\,1_{g}\,4_{q}\,5_{\bar{q}(s)}}\right]^{2}_{\text{NLP}}\,=\,\frac{s_{15}s_{24}}{s_{14}s_{25}s_{45}}\left|\mathcal{A}^{+-+}_{1_{g}\,2_{g}\,4_{g}}\right|^{2}\,,$
$\displaystyle\left[\mathcal{A}^{+-+-}_{1_{g}\,2_{g}\,4_{q}\,5_{\bar{q}(s)}}+\mathcal{A}^{+-+-}_{2_{g}\,1_{g}\,4_{q}\,5_{\bar{q}(s)}}\right]^{2}_{\text{NLP}}\,=\,\frac{s_{12}}{s_{14}s_{25}}\left|\mathcal{A}^{+-+}_{1_{g}\,2_{g}\,4_{g}}\right|^{2}\,.$
(48)

The NLP leading logarithms that appear in the differential cross-sections of
these two independent helicity configurations are given by,

$\displaystyle\left.s_{12}^{2}\frac{d^{2}\sigma^{+++-}_{1_{g}\,2_{g}\,4_{q}\,5_{\bar{q}(s)}}}{ds_{13}\,ds_{23}}\right|_{\text{NLP-
LL}}\,$ $\displaystyle=\,0\,,$
$\displaystyle\left.s_{12}^{2}\frac{d^{2}\sigma^{+-+-}_{1_{g}\,2_{g}\,4_{q}\,5_{\bar{q}(s)}}}{ds_{13}\,ds_{23}}\right|_{\text{NLP-
LL}}\,$ $\displaystyle=\,\mathcal{F}_{gg}\,\Bigg{\\{}2\pi
N\left(\frac{s_{12}}{s_{13}s_{23}}\right)\log\left(\frac{s_{45}}{\mu^{2}}\right)-\frac{2\pi}{N}\left(\frac{s_{12}}{s_{13}s_{23}}\right)\log\left(\frac{s_{45}}{\mu^{2}}\right)\Bigg{\\}}\left|\mathcal{A}^{+-+}_{1_{g}\,2_{g}\,4_{g}}\right|^{2}\,.$
(49)

Interestingly, no NLP leading logarithm appears for the former helicity
configuration and for the latter configuration, no logarithmic term of the
form $\log\left(\frac{s_{12}s_{45}}{s_{13}s_{23}}\right)$ is present in the
above equation.

A similar exercise can be performed when the final state anti-quark is soft.
In that case, the squared colour-ordered amplitudes read as,

$\displaystyle\left[\mathcal{A}^{+++-}_{1_{g}\,2_{g}\,4_{q(s)}\,5_{\bar{q}}}\right]^{2}_{\text{NLP}}\,=\,\left[\left(\frac{1}{s_{24}}+\frac{1}{s_{45}}\right)\left(1+\frac{s_{12}}{s_{15}}\right)-\frac{s_{14}s_{25}}{s_{15}s_{24}s_{45}}\right]\left|\mathcal{A}^{+-+}_{2_{q}\,5_{\bar{q}}\,1_{g}}\right|^{2}\,,$
$\displaystyle\left[\mathcal{A}^{+++-}_{2_{g}\,1_{g}\,4_{q(s)}\,5_{\bar{q}}}\right]^{2}_{\text{NLP}}\,=\,\frac{s_{25}}{s_{15}}\left[\left(\frac{1}{s_{14}}+\frac{1}{s_{45}}\right)\left(1+\frac{s_{12}}{s_{25}}\right)-\frac{s_{15}s_{24}}{s_{14}s_{25}s_{45}}\right]\left|\mathcal{A}^{+-+}_{2_{q}\,5_{\bar{q}}\,1_{g}}\right|^{2}\,,$
$\displaystyle\left[\mathcal{A}^{+++-}_{1_{g}\,2_{g}\,4_{q(s)}\,5_{\bar{q}}}+\mathcal{A}^{+++-}_{2_{g}\,1_{g}\,4_{q}\,5_{\bar{q}(s)}}\right]^{2}_{\text{NLP}}\,=\,\left[\left(\frac{1}{s_{14}}+\frac{1}{s_{24}}\right)\left(1+\frac{s_{25}}{s_{15}}\right)-\frac{s_{12}s_{45}}{s_{14}s_{15}s_{24}}\right]\left|\mathcal{A}^{+-+}_{2_{q}\,5_{\bar{q}}\,1_{g}}\right|^{2}\,,$
(50)
$\displaystyle\left[\mathcal{A}^{+-+-}_{1_{g}\,2_{g}\,4_{q(s)}\,5_{\bar{q}}}\right]^{2}_{\text{NLP}}\,=\frac{1}{s_{45}}\left|\mathcal{A}^{+--}_{1_{g}\,2_{g}\,5_{g}}\right|^{2}\,,$
$\displaystyle\left[\mathcal{A}^{+-+-}_{2_{g}\,1_{g}\,4_{q(s)}\,5_{\bar{q}}}\right]^{2}_{\text{NLP}}\,=\frac{s_{15}s_{24}}{s_{14}s_{25}s_{45}}\left|\mathcal{A}^{+--}_{1_{g}\,2_{g}\,5_{g}}\right|^{2}\,,$
$\displaystyle\left[\mathcal{A}^{+-+-}_{1_{g}\,2_{g}\,4_{q(s)}\,5_{\bar{q}}}+\mathcal{A}^{+-+-}_{2_{g}\,1_{g}\,4_{q}\,5_{\bar{q}(s)}}\right]^{2}_{\text{NLP}}\,=\,\frac{s_{12}}{s_{14}s_{25}}\left|\mathcal{A}^{+--}_{1_{g}\,2_{g}\,5_{g}}\right|^{2}\,,$
(51)

and the NLP leading logarithms of the differential cross-sections can be
expressed as,

$\displaystyle\left.s_{12}^{2}\frac{d^{2}\sigma^{+++-}_{1_{g}\,2_{g}\,4_{q(s)}\,5_{\bar{q}}}}{ds_{13}\,ds_{23}}\right|_{\text{NLP-
LL}}\,$ $\displaystyle=\,\mathcal{F}_{gg}\,\Bigg{\\{}2\pi
N\left(\frac{1}{s_{13}}+\frac{s_{13}}{s_{23}^{2}}\right)\log\left(\frac{s_{45}}{\mu^{2}}\right)$
$\displaystyle-\frac{2\pi}{N}\left[\left(\frac{1}{s_{13}}+\frac{s_{13}}{s_{23}^{2}}\right)\log\left(\frac{s_{45}}{\mu^{2}}\right)-\frac{2}{s_{23}}\log\left(\frac{s_{12}s_{45}}{s_{13}s_{23}}\right)\right]\Bigg{\\}}\left|\mathcal{A}^{+-+}_{2_{q}\,5_{\bar{q}}\,1_{g}}\right|^{2}\,,$
$\displaystyle\left.s_{12}^{2}\frac{d^{2}\sigma^{+-+-}_{1_{g}\,2_{g}\,4_{q(s)}\,5_{\bar{q}}}}{ds_{13}\,ds_{23}}\right|_{\text{NLP-
LL}}\,$ $\displaystyle=\,\mathcal{F}_{gg}\,\Bigg{\\{}2\pi
N\left(\frac{s_{12}}{s_{13}s_{23}}\right)\log\left(\frac{s_{45}}{\mu^{2}}\right)-\frac{2\pi}{N}\left(\frac{s_{12}}{s_{13}s_{23}}\right)\log\left(\frac{s_{45}}{\mu^{2}}\right)\Bigg{\\}}\left|\mathcal{A}^{+--}_{1_{g}\,2_{g}\,5_{g}}\right|^{2}\,.$
(52)

Unlike eq. (49), the first helicity configuration in the above produces a non-
zero result when the final state anti-quark is soft, whereas the result for
the latter configuration matches to the soft quark case except for the overall
leading order squared amplitude.

####  5.1.3 $qg$ initiated process

Here the process we are interested in is described as,

$\displaystyle
g(p_{1})+q(p_{2})+H(-p_{3})+\bar{q}(-p_{4})+g(-p_{5})\rightarrow 0$
$\displaystyle\,.$ (53)

Note that either of the final state gluon or quark can be (next-to-)soft and
they can contribute to the NLP threshold cross-section. We first consider the
case when the gluon is next-to-soft and as discussed earlier, no MHV
configuration would play any role here. There is only one independent helicity
configuration and to get the colour-summed squared amplitude of that
configuration, we calculate the following colour-ordered squared amplitudes,

$\displaystyle\left[\mathcal{A}^{++-+}_{1_{g}\,2_{q}\,4_{\bar{q}}\,5_{g(s)}}\right]^{2}_{\text{NLP}}\,=\,\left[\mathcal{A}^{+-++}_{1_{q}\,2_{\bar{q}}\,5_{g(s)}\,4_{g}}\right]^{2}_{\text{NLP}}\Big{\\{}1\to
2,2\to 4,4\to 1\Big{\\}}\,,$
$\displaystyle\left[\mathcal{A}^{++-+}_{5_{g}(s)\,2_{q}\,4_{\bar{q}}\,1_{g}}\right]^{2}_{\text{NLP}}\,=\,\left[\mathcal{A}^{+-++}_{1_{q}\,2_{\bar{q}}\,4_{g}\,5_{g(s)}}\right]^{2}_{\text{NLP}}\Big{\\{}1\to
2,2\to 4,4\to 1\Big{\\}}\,,$
$\displaystyle\left[\mathcal{A}^{++-+}_{1_{g}\,2_{q}\,4_{\bar{q}}\,5_{g(s)}}+\mathcal{A}^{++-+}_{5_{g}(s)\,2_{q}\,4_{\bar{q}}\,1_{g}}\right]^{2}_{\text{NLP}}\,=\,\left[\mathcal{A}^{+-++}_{1_{q}\,2_{\bar{q}}\,5_{g(s)}\,4_{g}}+\mathcal{A}^{+-++}_{1_{q}\,2_{\bar{q}}\,4_{g}\,5_{g(s)}}\right]^{2}_{\text{NLP}}\Big{\\{}1\to
2,2\to 4,4\to 1\Big{\\}}\,.$ (54)

The colour-summed squared amplitude thus obtained takes part in the
computation of the differential cross-section and the NLP leading logarithms
of that cross-section can be written as,

$\displaystyle\left.s_{12}^{2}\frac{d^{2}\sigma^{++-+}_{1_{g}\,2_{q}\,4_{\bar{q}}\,5_{g(s)}}}{ds_{13}\,ds_{23}}\right|_{\text{NLP-
LL}}\,=$ $\displaystyle\,\mathcal{F}_{qg}\,\bigg{\\{}4\pi
N\left[\left(\frac{1}{s_{13}}+\frac{4}{s_{23}}\right)\log\left(\frac{s_{45}}{\mu^{2}}\right)-\frac{1}{s_{13}}\log\left(\frac{s_{12}s_{45}}{s_{13}s_{23}}\right)\right]$
$\displaystyle-\frac{4\pi}{N}\left(\frac{1}{s_{13}}\right)\log\left(\frac{s_{45}}{\mu^{2}}\right)\bigg{\\}}\left|\mathcal{A}^{+-+}_{2_{q}\,4_{\bar{q}}\,1_{g}}\right|^{2}\,.$
(55)

Now we provide the results when the final state quark is soft. Following are
the expressions of the squared colour-ordered amplitudes for the two
independent helicity configurations,

$\displaystyle\left[\mathcal{A}^{++-+}_{1_{g}\,2_{q}\,4_{\bar{q}(s)}\,5_{g}}\right]^{2}_{\text{NLP}}\,=\,\left[\mathcal{A}^{+++-}_{1_{g}\,2_{g}\,4_{q}\,5_{\bar{q}(s)}}\right]^{2}_{\text{NLP}}\Big{\\{}1\to
5,2\to 1,4\to 2,5\to 4\Big{\\}}\,,$
$\displaystyle\left[\mathcal{A}^{++-+}_{1_{g}\,2_{q}\,4_{\bar{q}(s)}\,5_{g}}\right]^{2}_{\text{NLP}}\,=\,\left[\mathcal{A}^{+++-}_{2_{g}\,1_{g}\,4_{q}\,5_{\bar{q}(s)}}\right]^{2}_{\text{NLP}}\Big{\\{}1\to
5,2\to 1,4\to 2,5\to 4\Big{\\}}\,,$
$\displaystyle\left[\mathcal{A}^{++-+}_{1_{g}\,2_{q}\,4_{\bar{q}(s)}\,5_{g}}+\mathcal{A}^{++-+}_{1_{g}\,2_{q}\,4_{\bar{q}(s)}\,5_{g}}\right]^{2}_{\text{NLP}}\,=\,\left[\mathcal{A}^{+++-}_{1_{g}\,2_{g}\,4_{q}\,5_{\bar{q}(s)}}+\mathcal{A}^{+++-}_{2_{g}\,1_{g}\,4_{q}\,5_{\bar{q}(s)}}\right]^{2}_{\text{NLP}}\
\Big{\\{}1\to 5,2\to 1,4\to 2,5\to 4\Big{\\}}\,,$ (56)

and

$\displaystyle\left[\mathcal{A}^{-+-+}_{1_{g}\,2_{q}\,4_{\bar{q}(s)}\,5_{g}}\right]^{2}_{\text{NLP}}\,=\left[\mathcal{A}^{+-+-}_{1_{g}\,2_{g}\,4_{q}\,5_{\bar{q}(s)}}\right]^{2}_{\text{NLP}}\Big{\\{}1\to
5,2\to 1,4\to 2,5\to 4\Big{\\}}\,,$
$\displaystyle\left[\mathcal{A}^{-+-+}_{5_{g}\,2_{q}\,4_{\bar{q}(s)}\,1_{g}}\right]^{2}_{\text{NLP}}\,=\left[\mathcal{A}^{+-+-}_{2_{g}\,1_{g}\,4_{q}\,5_{\bar{q}(s)}}\right]^{2}_{\text{NLP}}\Big{\\{}1\to
5,2\to 1,4\to 2,5\to 4\Big{\\}}\,,$
$\displaystyle\left[\mathcal{A}^{-+-+}_{1_{g}\,2_{q}\,4_{\bar{q}(s)}\,5_{g}}+\mathcal{A}^{-+-+}_{5_{g}\,2_{q}\,4_{\bar{q}(s)}\,1_{g}}\right]^{2}_{\text{NLP}}\,=\,\left[\mathcal{A}^{+-+-}_{1_{g}\,2_{g}\,4_{q}\,5_{\bar{q}(s)}}+\mathcal{A}^{+-+-}_{2_{g}\,1_{g}\,4_{q}\,5_{\bar{q}(s)}}\right]^{2}_{\text{NLP}}\
\Big{\\{}1\to 5,2\to 1,4\to 2,5\to 4\Big{\\}}\,.$ (57)

The NLP leading logarithms are then given by,

$\displaystyle\left.s_{12}^{2}\frac{d^{2}\sigma^{++-+}_{1_{g}\,2_{q}\,4_{\bar{q}(s)}\,5_{g}}}{ds_{13}\,ds_{23}}\right|_{\text{NLP-
LL}}\,$ $\displaystyle=\,\mathcal{F}_{qg}\,\bigg{\\{}4\pi
N\left(\frac{1}{s_{13}}\right)\log\left(\frac{s_{45}}{\mu^{2}}\right)\bigg{\\}}\left|\mathcal{A}^{+++}_{1_{g}\,2_{g}\,5_{g}}\right|^{2}\,,$
$\displaystyle\left.s_{12}^{2}\frac{d^{2}\sigma^{-+-+}_{1_{g}\,2_{q}\,4_{\bar{q}(s)}\,5_{g}}}{ds_{13}\,ds_{23}}\right|_{\text{NLP-
LL}}\,$ $\displaystyle=\,\mathcal{F}_{qg}\,\bigg{\\{}2\pi
N\left[\left(\frac{3}{s_{13}}\right)\log\left(\frac{s_{45}}{\mu^{2}}\right)+\frac{2}{s_{13}}\log\left(\frac{s_{12}s_{45}}{s_{13}s_{23}}\right)\right]$
$\displaystyle-\frac{2\pi}{N}\left(\frac{1}{s_{13}}\right)\log\left(\frac{s_{45}}{\mu^{2}}\right)\bigg{\\}}\left|\mathcal{A}^{-++}_{1_{g}\,2_{g}\,5_{g}}\right|^{2}\,,$
(58)

where the sub-leading colour term does not contribute to the
‘$+\\!+\\!-\\!\;+$’ configuration.

####  5.1.4 $\bar{q}g$ initiated process

To describe this process we consider the following assignment of external
momenta,

$\displaystyle\bar{q}(p_{1})+g(p_{2})+H(-p_{3})+g(-p_{4})+q(-p_{5})\rightarrow
0$ $\displaystyle\,.$ (59)

As described in the previous sub-section, next-to-soft gluon emission leads to
only one independent helicity configuration, whereas soft anti-quark in the
final state causes two independent helicity structures to appear at the NLP.
The results for the squared colour-ordered amplitudes and the NLP leading
logarithms are first provided for the next-to-soft gluon and then for the soft
quark emission.

For the next-to-soft gluon emission,

$\displaystyle\left[\mathcal{A}^{-+++}_{1_{\bar{q}}\,2_{g}\,4_{g(s)}\,5_{q}}\right]^{2}_{\text{NLP}}\,=\,\left[\mathcal{A}^{+-++}_{1_{q}\,2_{\bar{q}}\,4_{g}\,5_{g(s)}}\right]^{2}_{\text{NLP}}\Big{\\{}1\to
5,2\to 1,4\to 2,5\to 4\Big{\\}}\,,$
$\displaystyle\left[\mathcal{A}^{-+++}_{1_{\bar{q}}\,4_{g(s)}\,2_{g}\,5_{q}}\right]^{2}_{\text{NLP}}\,=\,\left[\mathcal{A}^{+-++}_{1_{q}\,2_{\bar{q}}\,5_{g(s)\,4_{g}}}\right]^{2}_{\text{NLP}}\Big{\\{}1\to
5,2\to 1,4\to 2,5\to 4\Big{\\}}\,,$
$\displaystyle\left[\mathcal{A}^{-+++}_{1_{\bar{q}}\,2_{g}\,4_{g(s)}\,5_{q}}+\mathcal{A}^{-+++}_{1_{\bar{q}}\,4_{g(s)}\,2_{g}\,5_{q}}\right]^{2}_{\text{NLP}}\,=\,\left[\mathcal{A}^{+-++}_{1_{q}\,2_{\bar{q}}\,4_{g}\,5_{g(s)}}+\mathcal{A}^{+-++}_{1_{q}\,2_{\bar{q}}\,5_{g(s)}\,4_{g}}\right]^{2}_{\text{NLP}}\Big{\\{}1\to
5,2\to 1,4\to 2,5\to 4\Big{\\}}\ \,,$ (60)

and,

$\displaystyle\left.s_{12}^{2}\frac{d^{2}\sigma^{-+++}_{1_{\bar{q}}\,2_{g}\,4_{g(s)}\,5_{q}}}{ds_{13}\,ds_{23}}\right|_{\text{NLP-
LL}}\,=$ $\displaystyle\,\mathcal{F}_{\bar{q}g}\,\bigg{\\{}4\pi
N\left[\left(\frac{4}{s_{13}}-\frac{1}{s_{23}}\right)\log\left(\frac{s_{45}}{\mu^{2}}\right)+\left(\frac{2}{s_{13}}-\frac{1}{s_{23}}\right)\log\left(\frac{s_{12}s_{45}}{s_{13}s_{23}}\right)\right]$
$\displaystyle+\frac{4\pi}{N}\left(\frac{1}{s_{23}}\right)\log\left(\frac{s_{45}}{\mu^{2}}\right)\bigg{\\}}\left|\mathcal{A}^{+-+}_{5_{q}\,1_{\bar{q}}\,2_{g}}\right|^{2}\,.$
(61)

Such results for the final state soft quark read as,

$\displaystyle\left[\mathcal{A}^{-+++}_{1_{\bar{q}}\,2_{g}\,4_{g}\,5_{q(s)}}\right]^{2}_{\text{NLP}}\,=\,\left[\mathcal{A}^{+++-}_{1_{g}\,2_{g}\,4_{q(s)}\,5_{\bar{q}}}\right]^{2}_{\text{NLP}}\Big{\\{}1\to
2,2\to 4,4\to 5,5\to 1\Big{\\}}\,,$
$\displaystyle\left[\mathcal{A}^{-+++}_{1_{\bar{q}}\,4_{g}\,2_{g}\,5_{q(s)}}\right]^{2}_{\text{NLP}}\,=\,\left[\mathcal{A}^{+++-}_{2_{g}\,1_{g}\,4_{q(s)}\,5_{\bar{q}}}\right]^{2}_{\text{NLP}}\Big{\\{}1\to
2,2\to 4,4\to 5,5\to 1\Big{\\}}\,,$
$\displaystyle\left[\mathcal{A}^{-+++}_{1_{\bar{q}}\,2_{g}\,4_{g}\,5_{q(s)}}+\mathcal{A}^{-+++}_{1_{\bar{q}}\,4_{g}\,2_{g}\,5_{q(s)}}\right]^{2}_{\text{NLP}}\,=\,\left[\mathcal{A}^{+++-}_{1_{g}\,2_{g}\,4_{q(s)}\,5_{\bar{q}}}+\mathcal{A}^{+++-}_{2_{g}\,1_{g}\,4_{q(s)}\,5_{\bar{q}}}\right]^{2}_{\text{NLP}}\Big{\\{}1\to
2,2\to 4,4\to 5,5\to 1\Big{\\}}\,,$ (62)
$\displaystyle\left[\mathcal{A}^{-+-+}_{1_{\bar{q}}\,2_{g}\,4_{g}\,5_{q(s)}}\right]^{2}_{\text{NLP}}\,=\left[\mathcal{A}^{+-+-}_{1_{g}\,2_{g}\,4_{q(s)}\,5_{\bar{q}}}\right]^{2}_{\text{NLP}}\Big{\\{}1\to
2,2\to 4,4\to 5,5\to 1\Big{\\}}\,,$
$\displaystyle\left[\mathcal{A}^{-+-+}_{1_{\bar{q}}\,4_{g}\,2_{g}\,5_{q(s)}}\right]^{2}_{\text{NLP}}\,=\left[\mathcal{A}^{+-+-}_{2_{g}\,1_{g}\,4_{q(s)}\,5_{\bar{q}}}\right]^{2}_{\text{NLP}}\Big{\\{}1\to
2,2\to 4,4\to 5,5\to 1\Big{\\}}\,,$
$\displaystyle\left[\mathcal{A}^{-+-+}_{1_{\bar{q}}\,2_{g}\,4_{g}\,5_{q(s)}}+\mathcal{A}^{-+-+}_{1_{\bar{q}}\,4_{g}\,2_{g}\,5_{q(s)}}\right]^{2}_{\text{NLP}}\,=\,\left[\mathcal{A}^{+-+-}_{1_{g}\,2_{g}\,4_{q(s)}\,5_{\bar{q}}}+\mathcal{A}^{+-+-}_{2_{g}\,1_{g}\,4_{q(s)}\,5_{\bar{q}}}\right]^{2}_{\text{NLP}}\Big{\\{}1\to
2,2\to 4,4\to 5,5\to 1\Big{\\}}\,,$ (63)

and,

$\displaystyle\left.s_{12}^{2}\frac{d^{2}\sigma^{-+++}_{1_{\bar{q}}\,2_{g}\,4_{g}\,5_{q(s)}}}{ds_{13}\,ds_{23}}\right|_{\text{NLP-
LL}}\,$ $\displaystyle=\,\mathcal{F}_{\bar{q}g}\,\bigg{\\{}2\pi
N\left[\left(\frac{2s_{13}}{s_{12}s_{23}}+\frac{s_{23}}{s_{12}s_{13}}\right)\log\left(\frac{s_{45}}{\mu^{2}}\right)-\left(\frac{2}{s_{12}}\right)\log\left(\frac{s_{12}s_{45}}{s_{13}s_{23}}\right)\right]$
$\displaystyle-\frac{2\pi}{N}\left(\frac{s_{23}}{s_{12}s_{13}}\right)\log\left(\frac{s_{45}}{\mu^{2}}\right)\bigg{\\}}\left|\mathcal{A}^{+-+}_{4_{q}\,1_{\bar{q}}\,2_{g}}\right|^{2}\,,$
$\displaystyle\left.s_{12}^{2}\frac{d^{2}\sigma^{-+-+}_{1_{\bar{q}}\,2_{g}\,4_{g}\,5_{q(s)}}}{ds_{13}\,ds_{23}}\right|_{\text{NLP-
LL}}\,$ $\displaystyle=\,\mathcal{F}_{\bar{q}g}\,\bigg{\\{}2\pi
N\left[\left(\frac{3}{s_{23}}\right)\log\left(\frac{s_{45}}{\mu^{2}}\right)+\frac{2}{s_{23}}\log\left(\frac{s_{12}s_{45}}{s_{13}s_{23}}\right)\right]$
$\displaystyle-\frac{2\pi}{N}\left(\frac{1}{s_{23}}\right)\log\left(\frac{s_{45}}{\mu^{2}}\right)\bigg{\\}}\left|\mathcal{A}^{-+-}_{1_{g}\,2_{g}\,4_{g}}\right|^{2}\,.$
(64)

As compared to eq. (58), the results for the ‘$-\\!+\\!-\\!\;+$’ configuration
are related to each other by $s_{13}\leftrightarrow s_{23}$ interchange due to
their inherent association at the leading order in the soft limit, whereas
this is not the case for the other helicity configuration.

###  5.2 NLP logarithms: $q\bar{q}HQ\bar{Q}$

The $q\bar{q}$ initiated process that produces one Higgs boson and another
pair of quark and anti-quark can be written as,

$\displaystyle
q(p_{1})+\bar{q}(p_{2})+H(-p_{3})+Q(-p_{4})+\bar{Q}(-p_{5})\rightarrow 0\,,$
(65)

where the flavour of the final state pair can be same or different as compared
to the initial state $q\bar{q}$ pair. Here either of the final state quark or
anti-quark can be soft, thereby leading to NLP leading logarithms at the
differential cross-section level. There is only one independent helicity
configuration and the expression of the colour-summed squared amplitude is
given in eq. (24). It is evident that all the leading and sub-leading colour
terms exist only when the quarks are of same flavour. Considering the final
state quark to be soft, the colour-ordered squared amplitudes are as follows,

$\displaystyle\left[\mathcal{A}^{+-+-}_{1_{q}\,2_{\bar{q}}\,4_{Q}\,5_{\bar{Q}(s)}}\right]^{2}_{\text{NLP}}\,=\,\frac{1}{s_{45}}\left|\mathcal{A}^{+-+}_{1_{q}\,2_{\bar{q}}\,4_{g}}\right|^{2}\,,$
$\displaystyle\left[\mathcal{A}^{+-+-}_{1_{q}\,2_{\bar{q}}\,4_{q}\,5_{{\bar{q}}(s)}}\right]^{2}_{\text{NLP}}\,=\,\frac{s_{12}}{s_{15}s_{24}}\left|\mathcal{A}^{+-+}_{1_{q}\,2_{\bar{q}}\,4_{g}}\right|^{2}\,,$
$\displaystyle\left[\mathcal{A}^{+-+-}_{1_{q}\,2_{\bar{q}}\,4_{Q}\,5_{{\bar{Q}}(s)}}\mathcal{A}^{\dagger\,+-+-}_{1_{q}\,2_{\bar{q}}\,4_{q}\,5_{{\bar{q}}(s)}}+c.c.\right]_{\text{NLP}}\,=\,\left(\frac{1}{s_{45}}+\frac{s_{12}}{s_{15}s_{24}}-\frac{s_{14}s_{25}}{s_{15}s_{24}s_{45}}\right)\left|\mathcal{A}^{+-+}_{1_{q}\,2_{\bar{q}}\,4_{g}}\right|^{2}\,.$
(66)

The NLP leading logarithm results are given by,

$\displaystyle\left.s_{12}^{2}\frac{d^{2}\sigma^{+-+-}_{1_{q}\,2_{\bar{q}}\,4_{q}\,5_{{\bar{q}}(s)}}}{ds_{13}\,ds_{23}}\right|_{\text{NLP-
LL}}\,=\,\mathcal{F}_{q\bar{q}}\,\bigg{\\{}2\pi
N\bigg{(}\frac{s_{12}}{s_{13}s_{23}}\bigg{)}\log\left(\frac{s_{45}}{\mu^{2}}\right)\bigg{\\}}\left|\mathcal{A}^{+-+}_{1_{q}\,2_{\bar{q}}\,4_{g}}\right|^{2}\,.$
(67)

Similar calculations for the final state anti-quark to be soft result into,

$\displaystyle\left[\mathcal{A}^{+-+-}_{1_{q}\,2_{\bar{q}}\,4_{Q(s)}\,5_{\bar{Q}}}\right]^{2}_{\text{NLP}}\,=\,\frac{1}{s_{45}}\left|\mathcal{A}^{+--}_{1_{q}\,2_{\bar{q}}\,5_{g}}\right|^{2}\,,$
$\displaystyle\left[\mathcal{A}^{+-+-}_{1_{q}\,2_{\bar{q}}\,4_{q(s)}\,5_{\bar{q}}}\right]^{2}_{\text{NLP}}\,=\,\frac{s_{12}}{s_{15}s_{24}}\left|\mathcal{A}^{+--}_{1_{q}\,2_{\bar{q}}\,5_{g}}\right|^{2}\,,$
$\displaystyle\left[\mathcal{A}^{+-+-}_{1_{q}\,2_{\bar{q}}\,4_{Q(s)}\,5_{{\bar{Q}}}}\mathcal{A}^{\dagger\,+-+-}_{1_{q}\,2_{\bar{q}}\,4_{q(s)}\,5_{{\bar{q}}}}+c.c.\right]_{\text{NLP}}\,=\,\left(\frac{1}{s_{45}}+\frac{s_{12}}{s_{15}s_{24}}-\frac{s_{14}s_{25}}{s_{15}s_{24}s_{45}}\right)\left|\mathcal{A}^{+--}_{1_{q}\,2_{\bar{q}}\,5_{g}}\right|^{2}\,.$
(68)

and,

$\displaystyle\left.s_{12}^{2}\frac{d^{2}\sigma^{+-+-}_{1_{q}\,2_{\bar{q}}\,4_{q(s)}\,5_{\bar{q}}}}{ds_{13}\,ds_{23}}\right|_{\text{NLP-
LL}}\,=\,\mathcal{F}_{q\bar{q}}\,\bigg{\\{}2\pi
N\bigg{(}\frac{s_{12}}{s_{13}s_{23}}\bigg{)}\log\left(\frac{s_{45}}{\mu^{2}}\right)\bigg{\\}}\left|\mathcal{A}^{+--}_{1_{q}\,2_{\bar{q}}\,5_{g}}\right|^{2}\,.$
(69)

Note that, except the squared leading order amplitudes, all expressions both
in the soft quark and anti-quark cases are same. In the NLP leading
logarithms, no contribution comes from the sub-leading colour term and the
non-zero leading colour term exists only when all the quarks and anti-quarks
are of same flavour.

##  6 Psuedo-scalar Higgs plus jet production

To explore the universality of the NLP logarithms obtained in the previous
section, the natural and straightforward extension is to study the case when
the Higgs boson is replaced by a pseudo-scalar Higgs. In the large top mass
limit when the top quark effect is integrated out, the effective Lagrangian
for this case can be written as,

$\displaystyle\mathcal{L}_{\text{eff}}\,=\,G_{a}\,A\,\text{Tr}(\tilde{F}^{a}_{\mu\nu}F^{\mu\nu,a})\,$
(70)

where $A$ represents the pseudo-scalar Higgs, the coupling
$G_{a}=\alpha_{s}/(8\pi v)$ and
$\tilde{F}^{a}_{\mu\nu}=\frac{1}{2}\epsilon^{\mu\nu\lambda\rho}{F}^{a}_{\lambda\rho}$.
As the pseudo-scalar does not carry any colour charge, colour orderings of the
helicity amplitudes remain exactly same to the scalar Higgs plus jet
production process given in section 4.1.

Pseudo-scalar Higgs plus four gluon amplitudes are identical to that of Higgs
plus four gluon amplitudes for all non-NMHV helicity configurations, however
the NMHV amplitudes differ due to a relative sign change in one of the two
terms [91]. Following [1], it is already known that Higgs plus four gluon NMHV
amplitudes do not contribute at NLP leading logarithmic order and we find the
same holds true here as well for the pseudo-scalar Higgs case. As a
consequence, forms of all NLP leading logarithms for the pseudo-scalar Higgs
plus jet production match to the Higgs plus jet production process for each
and every helicity configuration [1].

Expressions of the scattering amplitudes that include a $q\bar{q}$ pair and
two gluons along with a Higgs or pseudo-scalar Higgs are same when the gluons
posses same helicities. When the two gluons carry opposite helicities, they
differ by a relative sign change between the two terms in the last two
identities given in eq. (29). Recall that NLP leading logarithms for next-to-
soft gluon radiation can only arise when the gluons have same helicities, as
MHV amplitudes do not play any role here. This feature ensures that the
structures of NLP leading logarithms for the psuedo-scalar Higgs plus jet
production are identical to that of Higgs plus jet production when the
radiated gluon is next-to-soft. For soft quark or anti-quark radiation, the
forms of NLP leading logarithms for the Higgs and pseudo-scalar Higgs cases
appear to be same because – ($i$) the first identity of eq. (29) remains same
in both the cases, ($ii$) any one of the two terms in the last two identities
of eq. (29) contributes at NLP, thereby diminishing the effect of relative
sign changes in them.

For the process involving a Higgs or pseudo-scalar Higgs in association with
two $q\bar{q}$ pairs, helicity amplitudes again differ by a relative sign
change in one of the two terms as given in [91]. However this change in sign
does not alter the forms of NLP leading logarithms, as one of the two types of
terms takes part in the NLP calculation at a time depending on whether the
quark or anti-quark is soft.

Through this study we establish the universality of NLP leading logarithms for
the case of Higgs and pseudo-scalar Higgs production in association with a
final state jet. Numerical estimations of these logarithms depend on the
values of coupling $G_{a}$ and mass of the pseudo-scalar Higgs $m_{A}$, which
can consistently be absorbed in $\mathcal{F}_{ab}$ of eq. (40). When masses of
Higgs and pseudo-scalar Higgs are considered to be equal, one can readily
scale the NLP results for the Higgs plus one jet production by a factor of
$(G_{a}^{2}/G^{2})$ and obtain them for the pseudo-scalar Higgs plus one jet
production.

##  7 Conclusions

Enormous amount of data from the LHC demands precise predictions from
perturbative QCD. The required precision is achievable by computing higher
order corrections and performing resummation originating from the singular
region of the phase space. NLP resummation is one of the many different ways
to achieve percent-level accuracy in the theoretical predictions. To build a
consistent resummation method for NLP logarithms, one needs to calculate many
process at this accuracy to test the validity of universality at this order.
The draught of NLP results for jet-associated processes in the literature
indicates the need for a better method to calculate them.

In this paper, we have focused on computing the NLP leading logarithms for all
possible channels involving (anti-)quarks for the Higgs plus one jet
production in a hadron collider. These NLP logarithms originate when either a
next-to-soft gluon or a soft (anti-)quark is emitted from the leading-order
process. To obtain the results, we have utilised a method involving shifted
spinors, as outlined in [1], to calculate the next-to-soft gluon corrections.
Additionally, we have introduced for the first time soft (anti-)quark
operators based on spinor variables to account for the NLP effects arising
from soft (anti-)quark radiation in Higgs plus one jet production. Our
analysis has revealed that MHV amplitudes for processes involving a $q\bar{q}$
pair alongside a Higgs and two gluons do not contribute at the NLP in the
next-to-soft gluon limit. Together with the findings in [1], our results
provide NLP leading logarithms for all the channels that take part in Higgs
plus one jet production. The absence of MHV amplitudes for the above mentioned
channel and NMHV amplitudes for Higgs plus four-gluons channel at NLP
indicates the need for further exploration, particularly from the generic
structural point of view in scattering amplitudes.

The universality of NLP logarithms and their resummation is an active area of
research and many different approaches have been proposed in the literature to
explore that direction. One of the ways to understand this universality is to
calculate many processes with increasing complexities and investigate the
structure of NLP logarithms. Although colour singlet production processes
exhibit a universal pattern, the universality of NLP terms for processes with
coloured final states is not yet in place. In this article, we have observed
the universal nature of NLP leading logarithms when production of a scalar or
a pseudo-scalar Higgs is considered in association with a final state jet. We
believe that the simplicity of the shifted dipole spinors technique of [1]
along with the soft (anti-)quark spinor operator technique developed in this
paper will speed up the calculation of NLP logarithms for many more jet-
associated processes relevant at the LHC.

## Acknowledgements

We thank V. Ravindran for useful discussions and gratefully acknowledge the
hospitality of IMSc during the preparation of this manuscript. We also thank
R. K. Ellis for carefully reading the manuscript and providing useful
suggestions. SS is supported in part by the SERB-MATRICS under Grant No.
MTR/2022/000135.

## References

  * [1] S. Pal and S. Seth, _On H+jet production at NLP accuracy_ , 2309.08343.
  * [2] G. Parisi, _Summing large perturbative corrections in QCD_ , _Phys. Lett._ B90 (1980) 295.
  * [3] G. Curci and M. Greco, _Large Infrared Corrections in QCD Processes_ , _Phys. Lett._ B92 (1980) 175.
  * [4] G. Sterman, _Summation of large corrections to short distance hadronic cross-sections_ , _Nucl. Phys._ B281 (1987) 310.
  * [5] S. Catani and L. Trentadue, _Resummation of the QCD Perturbative Series for Hard Processes_ , _Nucl. Phys._ B327 (1989) 323.
  * [6] S. Catani and L. Trentadue, _Comment on qcd exponentiation at large x_ , _Nucl. Phys._ B353 (1991) 183.
  * [7] J. G. M. Gatheral, _Exponentiation of eikonal cross-sections in nonabelian gauge theories_ , _Phys. Lett._ B133 (1983) 90.
  * [8] J. Frenkel and J. C. Taylor, _Nonabelian eikonal exponentiation_ , _Nucl. Phys._ B246 (1984) 231.
  * [9] G. Sterman, _Infrared divergences in perturbative QCD_ , .
  * [10] G. P. Korchemsky and G. Marchesini, _Structure function for large x and renormalization of wilson loop_ , _Nucl. Phys._ B406 (1993) 225 [hep-ph/9210281].
  * [11] G. P. Korchemsky and G. Marchesini, _Resummation of large infrared corrections using Wilson loops_ , _Phys. Lett._ B313 (1993) 433.
  * [12] S. Forte and G. Ridolfi, _Renormalization group approach to soft gluon resummation_ , _Nucl. Phys._ B650 (2003) 229 [hep-ph/0209154].
  * [13] T. Becher and M. Neubert, _Threshold resummation in momentum space from effective field theory_ , _Phys. Rev. Lett._ 97 (2006) 082001 [hep-ph/0605050].
  * [14] M. D. Schwartz, _Resummation and NLO matching of event shapes with effective field theory_ , _Phys.Rev._ D77 (2008) 014026 [0709.2709].
  * [15] C. W. Bauer, S. P. Fleming, C. Lee and G. F. Sterman, _Factorization of e+e- Event Shape Distributions with Hadronic Final States in Soft Collinear Effective Theory_ , _Phys.Rev._ D78 (2008) 034027 [0801.4569].
  * [16] J.-y. Chiu, A. Fuhrer, R. Kelley and A. V. Manohar, _Factorization Structure of Gauge Theory Amplitudes and Application to Hard Scattering Processes at the LHC_ , _Phys.Rev._ D80 (2009) 094013 [0909.0012].
  * [17] G. Luisoni and S. Marzani, _QCD resummation for hadronic final states_ , _J. Phys._ G42 (2015) 103101 [1505.04084].
  * [18] T. Becher, A. Broggio and A. Ferroglia, _Introduction to Soft-Collinear Effective Theory_ , _Lect. Notes Phys._ 896 (2015) pp.1 [1410.1892].
  * [19] J. Campbell, J. Huston and F. Krauss, _The Black Book of Quantum Chromodynamics_. Oxford University Press, 2017.
  * [20] M. Kramer, E. Laenen and M. Spira, _Soft gluon radiation in Higgs boson production at the LHC_ , _Nucl. Phys._ B511 (1998) 523 [hep-ph/9611272].
  * [21] R. D. Ball, M. Bonvini, S. Forte, S. Marzani and G. Ridolfi, _Higgs production in gluon fusion beyond NNLO_ , _Nucl.Phys._ B874 (2013) 746 [1303.3590].
  * [22] M. Bonvini, S. Forte, G. Ridolfi and L. Rottoli, _Resummation prescriptions and ambiguities in SCET vs. direct QCD: Higgs production as a case study_ , _JHEP_ 01 (2015) 046 [1409.0864].
  * [23] C. Anastasiou, C. Duhr, F. Dulat, F. Herzog and B. Mistlberger, _Higgs Boson Gluon-Fusion Production in QCD at Three Loops_ , _Phys. Rev. Lett._ 114 (2015) 212001 [1503.06056].
  * [24] C. Anastasiou, C. Duhr, F. Dulat, E. Furlan, T. Gehrmann, F. Herzog et al., _High precision determination of the gluon fusion Higgs boson cross-section at the LHC_ , _JHEP_ 05 (2016) 058 [1602.00695].
  * [25] M. van Beekveld, W. Beenakker, R. Basu, E. Laenen, A. Misra and P. Motylinski, _Next-to-leading power threshold effects for resummed prompt photon production_ , _Phys. Rev._ D100 (2019) 056009 [1905.11771].
  * [26] M. van Beekveld, E. Laenen, J. Sinninghe Damsté and L. Vernazza, _Next-to-leading power threshold corrections for finite order and resummed colour-singlet cross sections_ , _JHEP_ 05 (2021) 114 [2101.07270].
  * [27] A. H. Ajjath, P. Mukherjee, V. Ravindran, A. Sankar and S. Tiwari, _Next-to SV resummed Drell-Yan cross section beyond leading-logarithm_ , 2107.09717.
  * [28] G. Grunberg and V. Ravindran, _On threshold resummation beyond leading 1-x order_ , _JHEP_ 10 (2009) 055 [0902.2702].
  * [29] S. Moch and A. Vogt, _On non-singlet physical evolution kernels and large-x coefficient functions in perturbative QCD_ , _JHEP_ 11 (2009) 099 [0909.2124].
  * [30] S. Moch and A. Vogt, _Threshold Resummation of the Structure Function F(L)_ , _JHEP_ 04 (2009) 081 [0902.2342].
  * [31] E. Laenen, L. Magnea, G. Stavenga and C. D. White, _Next-to-eikonal corrections to soft gluon radiation: a diagrammatic approach_ , _JHEP_ 1101 (2011) 141 [1010.1860].
  * [32] E. Laenen, G. Stavenga and C. D. White, _Path integral approach to eikonal and next-to-eikonal exponentiation_ , _JHEP_ 03 (2009) 054 [0811.2067].
  * [33] D. de Florian, J. Mazzitelli, S. Moch and A. Vogt, _Approximate N 3LO Higgs-boson production cross section using physical-kernel constraints_, _JHEP_ 10 (2014) 176 [1408.6277].
  * [34] N. Lo Presti, A. Almasy and A. Vogt, _Leading large-x logarithms of the quark & gluon contributions to inclusive Higgs-boson and lepton-pair production_, _Phys. Lett._ B737 (2014) 120 [1407.1553].
  * [35] D. Bonocore, E. Laenen, L. Magnea, S. Melville, L. Vernazza and C. D. White, _A factorization approach to next-to-leading-power threshold logarithms_ , _JHEP_ 06 (2015) 008 [1503.05156].
  * [36] D. Bonocore, E. Laenen, L. Magnea, L. Vernazza and C. D. White, _Non-abelian factorisation for next-to-leading-power threshold logarithms_ , _JHEP_ 12 (2016) 121 [1610.06842].
  * [37] E. Laenen, J. Sinninghe Damsté, L. Vernazza, W. Waalewijn and L. Zoppi, _Towards all-order factorization of QED amplitudes at next-to-leading power_ , _Phys. Rev._ D103 (2021) 034022 [2008.01736].
  * [38] V. Del Duca, E. Laenen, L. Magnea, L. Vernazza and C. D. White, _Universality of next-to-leading power threshold effects for colourless final states in hadronic collisions_ , _JHEP_ 11 (2017) 057 [1706.04018].
  * [39] M. van Beekveld, W. Beenakker, E. Laenen and C. D. White, _Next-to-leading power threshold effects for inclusive and exclusive processes with final state jets_ , _JHEP_ 03 (2020) 106 [1905.08741].
  * [40] D. Bonocore, E. Laenen, L. Magnea, L. Vernazza and C. D. White, _The method of regions and next-to-soft corrections in Drell-Yan production_ , _Phys. Lett._ B742 (2015) 375 [1410.6406].
  * [41] N. Bahjat-Abbas, J. Sinninghe Damsté, L. Vernazza and C. D. White, _On next-to-leading power threshold corrections in Drell-Yan production at N 3LO_, _JHEP_ 10 (2018) 144 [1807.09246].
  * [42] M. A. Ebert, I. Moult, I. W. Stewart, F. J. Tackmann, G. Vita and H. X. Zhu, _Power Corrections for N-Jettiness Subtractions at ${\cal O}(\alpha_{s})$_, _JHEP_ 12 (2018) 084 [1807.10764].
  * [43] R. Boughezal, A. Isgrò and F. Petriello, _Next-to-leading-logarithmic power corrections for $N$-jettiness subtraction in color-singlet production_, _Phys. Rev._ D97 (2018) 076006 [1802.00456].
  * [44] R. Boughezal, A. Isgrò and F. Petriello, _Next-to-leading power corrections to $V+1$ jet production in $N$-jettiness subtraction_, _Phys. Rev._ D101 (2020) 016005 [1907.12213].
  * [45] N. Bahjat-Abbas, D. Bonocore, J. Sinninghe Damsté, E. Laenen, L. Magnea, L. Vernazza et al., _Diagrammatic resummation of leading-logarithmic threshold effects at next-to-leading power_ , _JHEP_ 11 (2019) 002 [1905.13710].
  * [46] A. H. Ajjath, P. Mukherjee and V. Ravindran, _On next to soft corrections to Drell-Yan and Higgs Boson productions_ , 2006.06726.
  * [47] A. H. Ajjath, P. Mukherjee, V. Ravindran, A. Sankar and S. Tiwari, _On next to soft threshold corrections to DIS and SIA processes_ , _JHEP_ 04 (2021) 131 [2007.12214].
  * [48] A. H. Ajjath, P. Mukherjee, V. Ravindran, A. Sankar and S. Tiwari, _On next to soft corrections for Drell-Yan and Higgs boson rapidity distributions beyond N 3LO_, _Phys. Rev._ D103 (2021) L111502 [2010.00079].
  * [49] T. Ahmed, A. A. H., P. Mukherjee, V. Ravindran and A. Sankar, _Soft-virtual correction and threshold resummation for $n$-colorless particles to fourth order in QCD: Part II_, 2010.02980.
  * [50] T. Ahmed, A. H. Ajjath, G. Das, P. Mukherjee, V. Ravindran and S. Tiwari, _Soft-virtual correction and threshold resummation for $n$-colorless particles to fourth order in QCD: Part I_, 2010.02979.
  * [51] D. W. Kolodrubetz, I. Moult and I. W. Stewart, _Building Blocks for Subleading Helicity Operators_ , _JHEP_ 05 (2016) 139 [1601.02607].
  * [52] I. Moult, L. Rothen, I. W. Stewart, F. J. Tackmann and H. X. Zhu, _Subleading Power Corrections for N-Jettiness Subtractions_ , _Phys. Rev._ D95 (2017) 074023 [1612.00450].
  * [53] I. Feige, D. W. Kolodrubetz, I. Moult and I. W. Stewart, _A Complete Basis of Helicity Operators for Subleading Factorization_ , _JHEP_ 11 (2017) 142 [1703.03411].
  * [54] M. Beneke, M. Garny, R. Szafron and J. Wang, _Anomalous dimension of subleading-power N-jet operators_ , _JHEP_ 03 (2018) 001 [1712.04416].
  * [55] M. Beneke, M. Garny, R. Szafron and J. Wang, _Anomalous dimension of subleading-power $N$-jet operators. Part II_, _JHEP_ 11 (2018) 112 [1808.04742].
  * [56] A. Bhattacharya, I. Moult, I. W. Stewart and G. Vita, _Helicity Methods for High Multiplicity Subleading Soft and Collinear Limits_ , _JHEP_ 05 (2019) 192 [1812.06950].
  * [57] M. Beneke, M. Garny, R. Szafron and J. Wang, _Violation of the Kluberg-Stern-Zuber theorem in SCET_ , _JHEP_ 09 (2019) 101 [1907.05463].
  * [58] G. T. Bodwin, J.-H. Ee, J. Lee and X.-P. Wang, _Renormalization of the radiative jet function_ , 2107.07941.
  * [59] I. Moult, I. W. Stewart and G. Vita, _Subleading Power Factorization with Radiative Functions_ , _JHEP_ 11 (2019) 153 [1905.07411].
  * [60] M. Beneke, A. Broggio, S. Jaskiewicz and L. Vernazza, _Threshold factorization of the Drell-Yan process at next-to-leading power_ , _JHEP_ 07 (2020) 078 [1912.01585].
  * [61] Z. L. Liu and M. Neubert, _Factorization at subleading power and endpoint-divergent convolutions in $h\to\gamma\gamma$ decay_, _JHEP_ 04 (2020) 033 [1912.08818].
  * [62] Z. L. Liu, B. Mecaj, M. Neubert and X. Wang, _Factorization at subleading power, Sudakov resummation, and endpoint divergences in soft-collinear effective theory_ , _Phys. Rev._ D104 (2021) 014004 [2009.04456].
  * [63] R. Boughezal, X. Liu and F. Petriello, _Power Corrections in the N-jettiness Subtraction Scheme_ , _JHEP_ 03 (2017) 160 [1612.02911].
  * [64] I. Moult, I. W. Stewart and G. Vita, _A subleading operator basis and matching for gg → H_ , _JHEP_ 07 (2017) 067 [1703.03408].
  * [65] C.-H. Chang, I. W. Stewart and G. Vita, _A Subleading Power Operator Basis for the Scalar Quark Current_ , _JHEP_ 04 (2018) 041 [1712.04343].
  * [66] I. Moult, I. W. Stewart, G. Vita and H. X. Zhu, _First Subleading Power Resummation for Event Shapes_ , _JHEP_ 08 (2018) 013 [1804.04665].
  * [67] M. Beneke, A. Broggio, M. Garny, S. Jaskiewicz, R. Szafron, L. Vernazza et al., _Leading-logarithmic threshold resummation of the Drell-Yan process at next-to-leading power_ , _JHEP_ 03 (2019) 043 [1809.10631].
  * [68] M. A. Ebert, I. Moult, I. W. Stewart, F. J. Tackmann, G. Vita and H. X. Zhu, _Subleading power rapidity divergences and power corrections for q T_, _JHEP_ 04 (2019) 123 [1812.08189].
  * [69] M. Beneke, M. Garny, S. Jaskiewicz, R. Szafron, L. Vernazza and J. Wang, _Leading-logarithmic threshold resummation of Higgs production in gluon fusion at next-to-leading power_ , _JHEP_ 01 (2020) 094 [1910.12685].
  * [70] I. Moult, I. W. Stewart, G. Vita and H. X. Zhu, _The Soft Quark Sudakov_ , _JHEP_ 05 (2020) 089 [1910.14038].
  * [71] Z. L. Liu and M. Neubert, _Two-Loop Radiative Jet Function for Exclusive $B$-Meson and Higgs Decays_, _JHEP_ 06 (2020) 060 [2003.03393].
  * [72] Z. L. Liu, B. Mecaj, M. Neubert, X. Wang and S. Fleming, _Renormalization and Scale Evolution of the Soft-Quark Soft Function_ , _JHEP_ 07 (2020) 104 [2005.03013].
  * [73] J. Wang, _Resummation of double logarithms in loop-induced processes with effective field theory_ , 1912.09920.
  * [74] M. Beneke, M. Garny, S. Jaskiewicz, R. Szafron, L. Vernazza and J. Wang, _Large-x resummation of off-diagonal deep-inelastic parton scattering from d-dimensional refactorization_ , _JHEP_ 10 (2020) 196 [2008.04943].
  * [75] M. van Beekveld, L. Vernazza and C. D. White, _Threshold resummation of new partonic channels at next-to-leading power_ , _JHEP_ 12 (2021) 087 [2109.09752].
  * [76] G. Sterman and W. Vogelsang, _Power corrections to electroweak boson production from threshold resummation_ , _Phys. Rev. D_ 107 (2023) 014009 [2208.00937].
  * [77] M. van Beekveld, A. Danish, E. Laenen, S. Pal, A. Tripathi and C. D. White, _Next-to-soft radiation from a different angle_ , 2308.12850.
  * [78] P. Nogueira, _Automatic Feynman graph generation_ , _J. Comput. Phys._ 105 (1993) 279.
  * [79] J. A. M. Vermaseren, _New features of form_ , math-ph/0010025.
  * [80] W. R. Inc., “Mathematica, Version 13.0.”
  * [81] R. Harlander, S. Y. Klein and M. C. Schaaf, _FeynGame-2.1 – Feynman diagrams made easy_ , _PoS_ EPS-HEP2023 (2024) 657 [2401.12778].
  * [82] H. Luo, P. Mastrolia and W. J. T. Bobadilla, _On the Subleading-Soft Behaviour of QCD Amplitudes_ , 1411.1669.
  * [83] E. Casali, _Soft sub-leading divergences in Yang-Mills amplitudes_ , _JHEP_ 08 (2014) 077 [1404.5551].
  * [84] M. van Beekveld, L. Vernazza and C. D. White, _Exponentiation of soft quark effects from the replica trick_ , 2312.11606.
  * [85] F. Wilczek, _Decays of Heavy Vector Mesons Into Higgs Particles_ , _Phys. Rev. Lett._ 39 (1977) 1304.
  * [86] M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, _Remarks on Higgs Boson Interactions with Nucleons_ , _Phys. Lett. B_ 78 (1978) 443.
  * [87] R. P. Kauffman, S. V. Desai and D. Risal, _Production of a Higgs boson plus two jets in hadronic collisions_ , _Phys. Rev. D_ 55 (1997) 4005 [hep-ph/9610541].
  * [88] L. J. Dixon, E. W. N. Glover and V. V. Khoze, _MHV rules for Higgs plus multi-gluon amplitudes_ , _JHEP_ 12 (2004) 015 [hep-th/0411092].
  * [89] V. Ravindran, J. Smith and W. L. Van Neerven, _Next-to-leading order QCD corrections to differential distributions of Higgs boson production in hadron hadron collisions_ , _Nucl. Phys. B_ 634 (2002) 247 [hep-ph/0201114].
  * [90] W. Beenakker, H. Kuijf, W. L. van Neerven and J. Smith, _QCD Corrections to Heavy Quark Production in p anti-p Collisions_ , _Phys. Rev. D_ 40 (1989) 54.
  * [91] R. P. Kauffman, S. V. Desai and D. Risal, _Amplitudes for Higgs bosons plus four partons_ , in _American Physical Society (APS) Meeting of the Division of Particles and Fields (DPF 99)_ , 1, 1999, hep-ph/9903330.