IFJPAN-IV-2022-20

One-loop gauge invariant amplitudes
with a space-like gluon

Etienne<EMAIL_ADDRESS>Alessandro
<EMAIL_ADDRESS>Andreas van
<EMAIL_ADDRESS>
Institute of Nuclear Physics Polisch Academy of Sciences,
PL-31342 Kraków, Poland
Piotr<EMAIL_ADDRESS>
AGH University Of Science and Technology, Physics Faculty,
S. Mickiewicza 30, 30-059 Kraków, Poland

Abstract
Nowadays the particle physics has entered an era where high precision
calculations are required in order to compare the theoretical predictions with
the experimental data. In this paper, we explicitly compute the virtual
contributions for the space-like one-jet processes, $\emptyset\to
g^{\star}gg,\emptyset\to g^{\star}q\bar{q},\emptyset\to g^{\star}gH$ and
$\emptyset\to g^{\star}\bar{q}qe^{+}e^{-}$ within the auxiliary parton method.
Our results, which are expected to play an important role in high precision
description of small $x$ physics, explicitly confirm the conjecture developed
in Ref. [1], thus helping to bridge the gap between lowest order calculations
and NLO corrections within hybrid $k_{T}$-factorization scheme.

###### Contents

  1. 1 Introduction
  2. 2 The auxiliary parton method
  3. 3 Results
  4. 4 Conclusions
  5. A Special functions
  6. B $\bm{\emptyset}\bm{\to}\bm{g}^{\bm{\star}}\bm{g}\bm{g}$ from auxiliary gluons
  7. C $\bm{\emptyset}\bm{\to}\bm{g}^{\bm{\star}}\bar{\bm{q}}\bm{q}$ from auxiliary quarks
  8. D $\bm{\emptyset}\bm{\to}\bm{g}^{\bm{\star}}\bar{\bm{q}}\bm{q}\bm{e}^{\bm{+}}\bm{e}^{\bm{-}}$ from auxiliary gluons
  9. E Spinor helicity method

##  1 Introduction

The fundamental idea that guides phenomenological investigations in high-
energy hadronic physics is factorization. In collinear factorization (see e.g.
[2]) it is assumed that the incoming partons carry only longitudinal momenta.
More precisely, the transverse momenta in the hard process are power-
suppressed, while in the soft part they are integrated over. In the small $x$
limit of perturbative QCD, the centre-of-mass energy $\sqrt{s}$ is much bigger
than any hard scales of the problem and typically the leading power collinear
approximation is not adequate. Instead, the so-called High Energy
Factorization (HEF), also called $k_{T}$-factorization, applies [3, 4]. The
key difference between the $k_{T}$-factorization and the collinear
factorization is that, in the former, both the hard part and the soft hadronic
part depend on the parton transverse momenta $k_{T}$. This implies that the
hard matrix elements include explicit higher powers.

In the small $x$ regime, the collinear Parton Distribution Functions (PDFs)
are replaced by unintegrated PDFs (with explicit transverse momentum
dependence) that follow the BFKL evolution equations [5, 6, 7, 8, 9, 10, 11].
When there is an asymmetry in the longitudinal fractions carried by the
colliding initial state partons (a situation present in forward jet
production) the HEF can be reduced to the so-called hybrid
$k_{T}$-factorization [12, 13], where only one initial-state parton has an
explicit dependence on the transverse momentum $k_{T}$. This factorization
scheme requires partonic hard matrix elements with one space-like (off-shell)
initial-state parton. Any physically relevant scattering amplitude must be
gauge invariant, i.e. satisfy freedom in the choice of gluon propagators and
Ward identity, but these two conditions only hold if all external particles
are on-shell. For this reason, the calculation of amplitudes with off-shell
intial state cannot be obtained by a naive application of the conventional QCD
Feynman rules. In the literature, this problem was solved at tree-level by the
Lipatov’s effective action [14, 15], and methods which manually restore gauge
invariance. The latter are based on Ward identities [16], matrix elements of
straight infinite Wilson lines [17], or auxiliary parton method [18], which is
the method chosen in the present work.

The idea of the auxiliary parton method relies on embedding the off-shell
process into a new process in which the off-shell gluon is replaced by an
auxiliary on-shell quark anti-quark pair or by two auxiliary gluons, whose
momenta are written as a function of a dimensionless parameter called
$\Lambda$ [18]. Being this new process on-shell, the constructed amplitude is
gauge invariant for any value of $\Lambda$. Most important, in the
$\Lambda\to\infty$ limit, the constructed amplitude is the desired off-shell
scattering amplitude of the original process. At Leading Order (LO) one can
use equivalently auxiliary quarks or auxiliary gluons, provided to properly
take care of the color factors. Moving to Next-to Leading Order (NLO)
inevitably leads to infrared divergences in real and virtual contributions,
which do not cancel for processes with colored partons and thus need to be
treated by a renormalization of proper operators. In collinear factorization,
this is rigorously established (partly through renormalization of the PDFs)
for all orders in perturbation theory [19] while, in hybrid
$k_{T}$-factorization, an equivalent rigorous all-order proof has still to be
developed. However, working order-by-order, there has been great progress in
$k_{T}$ factorization, both concerning the evolution of unintegrated PDFs and
the partonic processes. The NLO BFKL kernel is known [20, 21, 22] as well as
the NLO Balitsky-Kovchegov equation [23] and the whole B-JIMWLK equation [24,
25]. In the language of impact factors, several NLO results exist, for example
the $\gamma^{*}\rightarrow\bar{q}q$ impact factor [26, 27, 28, 29, 30], Higgs
plus a hadron [31]. In Color Glass Condensate theory, the NLO results include
the single inclusive jet production [32], dijet production in DIS [33, 34]. In
addition, there are NLO calculations in the context of the Lipatov’s effective
action [35, 36, 37, 38, 39, 40, 41, 42].

The motivation for the following work is provided by tremendous progress in
calculating hard matrix elements in collinear factorization, where, by now,
any tree or loop-level process can be computed automatically.

In $k_{T}$ factorization, on the other hand, so far only tree-level processes
are fully automated. In particular, the KaTie Monte Carlo [43] allows for
efficient computation of any Standard Model processes within the $k_{T}$
factorization at tree-level, based on the embedding method [18]. The
automation at loop-level is still not achieved, but there has been
considerable progress. In Ref. [44] the embedding method was used to obtain
all-plus helicity off-shell gauge invariant amplitudes at one loop. These
amplitudes are all finite and it turns out that there is “auxiliary parton
universality” property, similar to tree-level, namely, one obtains the same
result irrespectively whether the auxiliary parton line is a quark line or
gluon line. In [1] the authors made the first rigorous attempt to understand
the auxiliary parton approach to $k_{T}$ factorization at NLO in a general
setting. It was shown that at NLO the auxiliary parton universality is
violated, but these violations are process independent. Moreover, the authors
identified the explicit structure of the virtual and real contributions based
on a conjecture regarding the $\Lambda$-dependent virtual contributions.

Let us now summarize the key properties of our approach and outline how we
plan to use it in actual cross-section calculations. First, computations of
loop diagrams are easier using the auxiliary parton approach than explicitly
computing loop diagrams using, for example, the Lipatov’s effective action;
clearly the approach utilizes existing on-shell one-loop results. Furthermore,
although the present work provides analytic results, it seems that the
approach can be performed numerically, which would allow for full automation
of one-loop computations. Second, the auxiliary parton approach gives in
general different results for the divergent parts than the Lipatov’s effective
action approach. This fact should not be surprising as the auxiliary parton
approach uses an entirely different regularization scheme. Full cross-section
computation requires also real corrections and proper subtractions of infrared
sensitive terms. Since the auxiliary parton approach is already fully
understood at tree-level, it is possible to recognize all singular regions
with auxiliary partons and construct a general framework for computing the
real corrections, in the spirit of the dipole subtraction method [45] or the
antenna method [46, 47]. Once this is done, proper infrared subtraction terms,
consistent with the $k_{T}$ factorization can be constructed. Currently, there
is an ongoing research on the real corrections within the auxiliary parton
approach. Finally, it is important to stress that the ultimate goal is to
build a Monte Carlo code that can handle NLO computation of a wide range of
observables within the $k_{T}$ factorization approach and the auxiliary parton
approach seems well suited for that purpose.

In this paper, we explicitly calculate the virtual contributions for the
following processes: $\emptyset\to g^{\star}gg,\emptyset\to
g^{\star}q\bar{q},\emptyset\to g^{\star}gH$ and $\emptyset\to
g^{\star}\bar{q}qe^{+}e^{-}$, with explicit helicity projections using the
auxiliary parton approach. In each case we check their collinear limit, i.e.
the limit when the gluon has vanishing transverse momentum and becomes on-
shell. Our results explicitly confirm the conjecture developed in [1]
regarding the structure of the divergent auxiliary parton depended terms.
Originally, the auxiliary parton method was introduced just as a trick to
obtain LO helicity amplitudes involving space-like partons and there were not
any a priori theoretical justifications which guarantee its application to
compute NLO cross sections. Our explicit calculation of the virtual
contributions within the auxiliary parton method confirms for the considered
processes that the auxiliary parton universality is violated, but in a
universal way, thus opening the way for the establishment of a rigorous
formalism to calculate higher order corrections suitable for a numerical
approach.

##  2 The auxiliary parton method

###  2.1 Definitions

Our results are written in the language of the spinor helicity method (see
Appendix E) and $SU(N_{c})$ color decomposition. When presenting helicity
amplitudes, it is most convenient to imagine all momenta to be outgoing, and
have momentum conservation as

$\sum_{i=1}^{n}k_{i}^{\mu}=0~{},$ (1)

where $k_{i}^{\mu}$ for $i=1,\ldots,n$ are the external momenta. This implies
that the initial-state momenta have negative energy. The space-like initial-
state momentum will always be the first one. It has an explicit longitudinal
component and transverse components, and we find it most convenient to write
it as

$k_{1}^{\mu}=k^{\mu}=p^{\mu}+k_{T}^{\mu}~{},$ (2)

where $p^{\mu}$ is the light-like longitudinal component that will appear in
spinor products etc., and $p\\!\cdot\\!k_{T}=0$. It is eventually equal to

$p^{\mu}=-xP_{\mathrm{hadron}}^{\mu}~{},$ (3)

where $P_{\mathrm{hadron}}^{\mu}$ is the (positive-energy) hadron momentum and
$x$ its positive momentum fraction. The definition (2) is actually consistent
with the so-called quasi Regge kinematics in high energy scattering, see eg.
[15]. Since $k^{\mu}$ is space-like, its square is negative, which we express
with absolute value symbols as follows:

$k^{2}=-|k_{T}^{2}|~{}.$ (4)

In order to extract helicity amplitudes with the auxiliary parton method, we
would like to apply exact kinematics, and introduce auxiliary parton momenta

$\displaystyle p_{A}^{\mu}$ $\displaystyle=\hskip 23.2499pt\Lambda
p^{\mu}+\alpha q^{\mu}+\hskip 23.68048pt\beta k_{T}^{\mu}$ (5) $\displaystyle
p_{B}^{\mu}$ $\displaystyle=(1-\Lambda)p^{\mu}-\alpha
q^{\mu}+(1-\beta)k_{T}^{\mu}~{},$ (6)

where $q^{\mu}$ is light-like with $p\\!\cdot\\!q>0$ and
$q\\!\cdot\\!k_{T}=0$, and where

$\alpha=\frac{-\beta^{2}k_{T}^{2}}{\Lambda(p+q)^{2}}\quad,\quad\beta=\frac{1}{1+\sqrt{1-1/\Lambda}}\quad.$
(7)

With this choice, the momenta $p_{A}^{\mu},p_{B}^{\mu}$ satisfy the relations

$p_{A}^{2}=p_{B}^{2}=0\quad,\quad p_{A}^{\mu}+p_{B}^{\mu}=p^{\mu}+k_{T}^{\mu}$
(8)

for any value of the parameter $\Lambda$, which is why we call it “exact”.
Thus, the momenta $p_{A}$ and $p_{B}$ represent the momenta of external
auxiliary partons (a $q$-$\overline{q}$ pair or a pair of gluons) whose sum
gives the momentum of the off-shell gluon (2). We now consider an actual
amplitude with these external partons and _define_ the gauge invariant
amplitude with a space-like gluon as follows:

$\frac{1}{\Lambda}\,\mathcal{M}\Big{(}q\big{(}p_{A}\big{)},\bar{q}\big{(}p_{B}\big{)},\ldots\Big{)}\;\overset{\Lambda\to\infty}{\longrightarrow}\;\mathcal{M}\big{(}g^{\star}(p+k_{T}),\ldots\big{)}~{},$
(9)

where the ellipses stand for other particles and partons involved in the
scattering process. Above, we assumed that the auxiliary pair are quarks. In
earlier work, e.g. [44], there is a factor $x|k_{T}|/g_{\mathrm{s}}$ included
on the left-hand side. The factor $x$ is included implicitly here because we
chose to parametrize the auxiliary parton momenta in terms of $p$ instead of
$P_{\mathrm{hadron}}$. The factor $|k_{T}|$ is omitted to give the amplitudes
presented here a more natural analytic form. We also decided to omit an
imaginary unit from the amplitudes for convenience, and we summarize with the
statement that

a factor $\mathrm{i}|k_{T}|$ is omitted from the presented amplitudes and must
be included in the end.

The coupling $g_{\mathrm{s}}$ will be considered to be equal to $1$ in this
write-up. We must just imagine that in the end, tree-level amplitudes get a
factor $g_{\mathrm{s}}^{n-2}$, and one-loop amplitudes get a factor
$g_{\mathrm{s}}^{n}$, where $n$ is the number of partons involved in the
scattering process. The procedure in Eq. (9) of taking $\Lambda\to\infty$ on
auxiliary parton momenta and dividing by $\Lambda$ will also be abbreviated
with

$\mathcal{M}\big{(}A,B,\ldots\big{)}\;\overset{\Lambda}{\longrightarrow}\;\mathcal{M}\big{(}g^{\star},\ldots\big{)}~{}.$
(10)

The transverse momentum $k_{T}^{\mu}$ can be written in terms of polarization
vectors constructed with the help of $q^{\mu}$

$k_{T}^{\mu}=-\frac{\kappa^{*}}{\sqrt{2}}\,\frac{\langle
q|\gamma^{\mu}|p]}{\sqrt{2}\,\langle
qp\rangle}-\frac{\kappa}{\sqrt{2}}\,\frac{\langle
p|\gamma^{\mu}|q]}{\sqrt{2}\,[pq]}\quad\textrm{with}\quad\kappa=\frac{\langle
q|k/|p]}{\langle qp\rangle}\quad,\quad\kappa^{*}=\frac{\langle
p|k/|q]}{[pq]}~{}.$ (11)

Both $\kappa$ and $\kappa^{*}$ do not implicitly depend on $x$, since both
$|p\rangle$ and $|p]$ scale as $\sqrt{x}$ for $x>0$. Also, they do not depend
on $q^{\mu}$, and we have

$\kappa\kappa^{*}=|k_{T}^{2}|~{}.$ (12)

Realizing that $\sqrt{1-1/\Lambda}=1/\beta-1$, the Weyl spinors for
$p_{A}^{\mu},p_{B}^{\mu}$ can be written as

$\displaystyle|A\rangle=\sqrt{\Lambda}\,|p\rangle-\frac{\beta\kappa^{*}}{\sqrt{\Lambda}\,\langle
qp\rangle}\,|q\rangle\quad$
$\displaystyle,\quad|A]=\sqrt{\Lambda}\,|p]-\frac{\beta\kappa}{\sqrt{\Lambda}\,[pq]}\,|q]\quad,$
(13)
$\displaystyle|B\rangle=\sqrt{\Lambda-1}\,|p\rangle+\frac{\beta\kappa^{*}}{\sqrt{\Lambda}\,\langle
qp\rangle}\,|q\rangle\quad$
$\displaystyle,\quad|B]=-\sqrt{\Lambda-1}\,|p]-\frac{\beta\kappa}{\sqrt{\Lambda}\,[pq]}\,|q]\quad.$

Explicitly, the calculation of $\Lambda\to\infty$ limit of the left side of
Eq. (10) consists in performing the substitutions reported in Eq. 13, for
example $\langle A4\rangle\to\sqrt{\Lambda}\langle p4\rangle$, in a precise
order respecting the following sequence:

  * •

1)

$\langle AB\rangle\to-\kappa^{*}\quad,\quad[AB]\to-\kappa\quad,\quad
p_{A}^{\mu}+p_{B}^{\mu}\to k^{\mu}~{}.$ (14)

The latter implies also

$s_{AB}=(p_{A}+p_{B})^{2}\to k^{2}=-\kappa\kappa^{*}\quad,\quad
t_{ABi}=(p_{A}+p_{B}+p_{i})^{2}\to(k+p_{i})^{2}=s_{ki}$ (15)

etc.

  * •

2)

$p_{A}^{\mu}\to\Lambda p^{\mu}\quad,\quad p_{B}^{\mu}\to-\Lambda p^{\mu}$ (16)

so that

$s_{Ai}\to\Lambda s_{pi}\quad,\quad s_{Bi}\to-\Lambda s_{pi}~{},$ (17)

  * •

3)

$|A\rangle\to\sqrt{\Lambda}\,|p\rangle\quad,\quad|A]\to\sqrt{\Lambda}\,|p]\quad,\quad|B\rangle\to\sqrt{\Lambda}\,|p\rangle\quad,\quad|B]\to-\sqrt{\Lambda}\,|p]~{}.$
(18)

These operations give the correct result under the condition that all terms in
an expression to which the operations are applied exhibit at most the leading
power behavior of $\Lambda^{p}$ with $p=1$. We encountered only one expression
with higher-order terms that cancel among each other and require including the
$1/\sqrt{\Lambda}$ terms in Eq. (13). It will be mentioned in Appendix D.

We follow the conventions in literature, in particular [48, 49, 50, 51], to
decompose amplitudes into partial amplitudes. Consider for example multi-gluon
amplitudes. At tree-level, we have

$\mathcal{M}^{\mathrm{tree}}(1,\ldots,n)=\sum_{\sigma\in
S_{n}/Z_{n}}\mathrm{Tr}\big{(}T^{a_{\sigma(1)}}\cdots
T^{a_{\sigma(n)}}\big{)}\mathcal{A}^{\mathrm{tree}}(\sigma(1),\ldots,\sigma(n))~{}.$
(19)

The arguments of the amplitude on the left-hand side indicate that it depends
on $n$ gluon momenta, helicities, and colors. We note, that it is common to
include an index $n$ for an amplitude, but to make our notation simpler we
shall omit that index, as the number of external states is implied by the
argument list. On the right-hand side, only the color variables $a_{i}$ are
made explicit, elucidating the color decomposition. The partial amplitude
$\mathcal{A}^{\mathrm{tree}}$ does not depend on these variables. It is
however not symmetric in its arguments enumerating the gluon momenta and
helicities. The sum is over all cyclically nonequivalent permutations.
Regarding the color algebra, we follow the conventions with

$\mathrm{Tr}\big{(}T^{a}T^{b}\big{)}=\delta^{ab}\quad,\quad[T^{a},T^{b}]=\sqrt{2}\,\mathrm{i}f^{abc}T^{c}~{},$
(20)

and $f^{123}=1$. Tree-level amplitudes will carry the superscript as above,
while amplitudes without this superscript will always be implied to be at one
loop. The color decomposition for multi-gluon amplitudes becomes

$\mathcal{M}(1,\ldots,n)=\sum_{\sigma\in
S_{n}/Z_{n}}N_{\mathrm{c}}\mathrm{Tr}\big{(}T^{a_{\sigma(1)}}\cdots
T^{a_{\sigma(n)}}\big{)}\mathcal{A}_{1}(\sigma(1),\ldots,\sigma(n))\\\
+\sum_{c=2}^{\lfloor n/2\rfloor+1}\sum_{\sigma\in
S_{n;c}/Z_{n}}\mathrm{Tr}\big{(}T^{a_{\sigma(1)}}\cdots
T^{a_{\sigma(c-1)}}\big{)}\mathrm{Tr}\big{(}T^{a_{\sigma(c)}}\cdots
T^{a_{\sigma(n)}}\big{)}\mathcal{A}_{c}(\sigma(1),\ldots,\sigma(n))~{}.$ (21)

More color structures appear, and $S_{n;c}$ is the subset of $S_{n}$ that
leaves the corresponding double trace structure invariant.

The subscript $c$ on a partial one-loop amplitude $\mathcal{A}_{c}$ indicates
to which color structure it belongs, where $c=1$ always refers the “leading-
color” tree-level structure.

The one-loop partial amplitudes are typically decomposed further into more
basic building blocks called primitive amplitudes. They will be labelled with
superscripts. Just like all amplitudes mentioned so far, they are gauge
invariant.

The color decomposition examples above are for on-shell gluons, but also valid
in case there is a space-like one. As mentioned earlier, the space-like one
will always be labelled as number $1$. Amplitudes involving the space-like
gluons will not carry any special label, but will be implied to be as such by
having the argument $1^{\star}$. So for example in $\mathcal{M}(1,2,3,4)$ all
gluons are on-shell, while in $\mathcal{M}(1^{\star},2,3,4)$ number $1$ is
space-like. Helicities are indicated with superscripts to the arguments, while
the space-like gluon does not come with helicities and will always have
superscript $\star$ as in
$\mathcal{A}^{\mathrm{tree}}(1^{\star},2^{+},3^{-},4^{-})$. If arguments do
not refer to gluons, then this is indicated with subscripts. For example,
$\mathcal{M}(1^{\star},2,3_{\bar{q}},4_{q})$ refers to the process with a
space-like gluon, an on-shell gluon, and a quark-antiquark pair.

One particular “primitive” amplitude that will appear for many processes with
a space-like gluon in the following and deserves a dedicated symbol is

$\mathcal{A}^{\mathrm{Rt}}=c_{\Gamma}\bigg{(}\frac{\mu^{2}}{|k_{T}^{2}|}\bigg{)}^{\epsilon}\frac{1}{\epsilon}\,\mathcal{A}^{\mathrm{tree}}~{},$
(22)

giving us the opportunity to introduce the one-loop constant

$c_{\Gamma}=\frac{1}{(4\pi)^{2-\epsilon}}\frac{\Gamma(1+\epsilon)\Gamma^{2}(1-\epsilon)}{\Gamma(1-2\epsilon)}~{},$
(23)

the dimensional regularization parameter

$\epsilon=\frac{4-\mathrm{dim}}{2}~{},$ (24)

and the renormalization scale $\mu$. The amplitude $\mathcal{A}^{\mathrm{Rt}}$
will typically be accompanied by one of the functions

$\Theta(s)=2\,\mathrm{i}\,\pi\,\theta(s)\quad,\quad\Upsilon(s)=\mathrm{i}\,\pi\,\mathrm{sgn}(s)~{},$
(25)

where $s$ is one of the invariants in the process. Amplitudes coming with this
function represent imaginary parts that can be ignored at NLO.

In order to make the correspondence with real-radiation contributions in NLO
calculations, it is useful to mention the relation

$g_{\mathrm{s}}^{2}c_{\Gamma}=\frac{1}{2}\cdot\frac{\alpha_{\mathrm{s}}}{2\pi}\frac{(4\pi)^{\epsilon}}{\Gamma(1-\epsilon)}+\mathcal{O}\big{(}\epsilon^{3}\big{)}~{},$
(26)

where the explicit factor $1/2$ is compensated by the factor $2$ in the
virtual contribution being
$2\mathrm{Re}\big{(}\mathcal{M}^{\mathrm{tree}\dagger}\mathcal{M}\big{)}$.

###  2.2 Analytic continuation

Our starting point will be expressions from literature that are valid when all
invariants are negative, and are analytically continued by imagining those
invariants have a small positive imaginary part, e.g.

$\ln\bigg{(}\frac{-s_{1}}{-s_{2}}\bigg{)}=\ln(-s_{1}-\mathrm{i}\eta)-\ln(-s_{2}-\mathrm{i}\eta)\quad,\quad\eta>0~{}.$
(27)

In the auxiliary parton method outlined before, this imaginary part of
invariants involving parton $B$ may flip sign, that is

$s_{Bi}+\mathrm{i}\eta\to-\Lambda
s_{pi}+\mathrm{i}\eta=-\Lambda(s_{pi}-\mathrm{i}\eta/\Lambda)~{}.$ (28)

This will only be relevant for arguments of the $\ln$ function, and we will
present expressions brought to the standard form, with positive imaginary
parts for all invariants, using the relation

$\ln(s-\mathrm{i}\eta)=\ln(-s-\mathrm{i}\eta)+\Upsilon(s)~{},$ (29)

with $\Upsilon$ defined in Eq. (25).

We also mention here that ratios of invariants raised to a power of the
parameter $\epsilon$ from dimensional regularization (like in Eq. (22)) must
always be imagined to be expanded in that parameter as

$\frac{x^{\epsilon}}{\epsilon^{2}}=\frac{1}{\epsilon^{2}}+\frac{\ln
x}{\epsilon}+\frac{\ln^{2}x}{2}+\mathcal{O}(\epsilon)\;\;,\quad\frac{x^{\epsilon}}{\epsilon}=\frac{1}{\epsilon}+\ln
x+\mathcal{O}(\epsilon)\;\;,\quad x^{\epsilon}=1+\mathcal{O}(\epsilon)\;\;,$
(30)

where we ignore $\mathcal{O}(\epsilon)$.

###  2.3 Auxiliary parton type

Instead of amplitudes with an auxiliary quark pair, one can also use
amplitudes with an auxiliary gluon pair to extract the amplitude with a space-
like gluon. In Appendix B of [1] it is explained that the amplitudes obtained
follow the relations

$\displaystyle\mathcal{M}^{\mathrm{tree},\imath_{B}}_{\hskip
7.53471pti_{A}}(B_{\bar{q}},A_{q},\ldots)$
$\displaystyle\overset{\Lambda}{\longrightarrow}\big{(}T^{a_{1}}\big{)}^{\imath_{B}}_{i_{A}}\mathcal{M}^{\mathrm{tree},a_{1}}(1^{\star},\ldots)~{},$
(31) $\displaystyle\mathcal{M}^{\mathrm{tree},a_{B}a_{A}}(B,A,\ldots)$
$\displaystyle\overset{\Lambda}{\longrightarrow}\sqrt{2}\,\mathrm{i}\,f^{a_{B}a_{A}a_{1}}\mathcal{M}^{\mathrm{tree},a_{1}}(1^{\star},\ldots)~{},~{}.$
(32)

Here we made only relevant color indices explicit. Also realize that in [1]
the $T$ matrices are normalized differently. The relations can be inverted as

$\displaystyle\big{(}T^{a_{1}}\big{)}_{\imath_{B}}^{i_{A}}\mathcal{M}^{\mathrm{tree},\imath_{B}}_{\hskip
7.53471pti_{A}}(B_{\bar{q}},A_{q},\ldots)$
$\displaystyle\overset{\Lambda}{\longrightarrow}\mathcal{M}^{\mathrm{tree},a_{1}}(1^{\star},\ldots)~{},$
(33)
$\displaystyle\frac{-\mathrm{i}}{\sqrt{2}N_{\mathrm{c}}}\,f^{a_{B}a_{A}a_{1}}\mathcal{M}^{\mathrm{tree},a_{B}a_{A}}(B,A,\ldots)$
$\displaystyle\overset{\Lambda}{\longrightarrow}\mathcal{M}^{\mathrm{tree},a_{1}}(1^{\star},\ldots)~{}.$
(34)

At one loop, the color structure is richer than at tree-level, and the
relations generally only hold for the leading-color, tree-like, color
structures. However, in an NLO calculation, the one-loop structures are
contracted with tree-level structures, and we will see in the examples here
that only the leading-color structures survive.

We denote by $\mathcal{M}^{\mathrm{NLO}}$ those parts of the one-loop
amplitude that are relevant at NLO.

At tree-level, the limit of Eq. (9) is well-defined, but at one loop, it
produces terms proportional to $\ln\Lambda$. Furthermore, the result depends
on the type of auxiliary partons. It involves the leading-color structures,
and the form of the dependence was conjectured in [1] to be

$\displaystyle\mathcal{A}_{1}^{\mathrm{aux}\textrm{-}\mathrm{g}}$
$\displaystyle=\mathcal{A}_{1}+c_{\Gamma}\mathcal{A}^{\mathrm{tree}}\bigg{(}\frac{\mu^{2}}{|k_{T}^{2}|}\bigg{)}^{\epsilon}\bigg{[}\frac{1}{\epsilon}\big{(}2\ln\Lambda-\mathrm{i}\pi\big{)}-\frac{1}{\epsilon^{2}}+\frac{\pi^{2}}{3}\bigg{]}~{},$
(35) $\displaystyle\mathcal{A}_{1}^{\mathrm{aux}\textrm{-}\mathrm{q}}$
$\displaystyle=\mathcal{A}_{1}+c_{\Gamma}\mathcal{A}^{\mathrm{tree}}\bigg{(}\frac{\mu^{2}}{|k_{T}^{2}|}\bigg{)}^{\epsilon}\bigg{[}\frac{1}{\epsilon}\big{(}2\ln\Lambda-\mathrm{i}\pi\big{)}+\frac{1}{\epsilon}\frac{13}{6}+\frac{\pi^{2}}{3}+\frac{83-3\delta_{R}}{18}$
$\displaystyle\hskip
120.55518pt+\frac{1}{N_{\mathrm{c}}^{2}}\bigg{(}\frac{1}{\epsilon^{2}}+\frac{3}{2\epsilon}+\frac{7+\delta_{R}}{2}\bigg{)}-\frac{n_{f}}{N_{\mathrm{c}}}\bigg{(}\frac{2}{3\epsilon}+\frac{10}{9}\bigg{)}\bigg{]}~{}.$
(36)

The conjecture states that $\mathcal{A}_{1}$ indeed will be independent of the
auxiliary parton type, and produces the expected on-shell limit for
$|k_{T}|\to 0$. More specifically, the divergent part of $\mathcal{A}_{1}$ has
the on-shell form, but with the longitudinal momentum component $p^{\mu}$
instead of the momentum $k^{\mu}$, e.g. involves the invariants $s_{pi}$
instead of $s_{ki}$.

In the following, we directly present the expressions for $\mathcal{A}_{1}$.
Amplitudes coming with sub-leading color structures $\mathcal{A}_{i>1}$ do not
depend on the type of auxiliary parton or on $\Lambda$.

The subtractions of Eq. (35) and Eq. (36) to arrive at $\mathcal{A}_{1}$ will
be implied to be included in the operation
“$\overset{\Lambda}{\longrightarrow}$” of Eq. (10).

Finally, the amplitudes we present are not UV-subtracted, and include the
parameter $\delta_{R}$ to distinguish between the ’t Hooft-Veltman scheme of
dimensional regularization ($\delta_{R}=1$) and the four-dimensional helicity
scheme ($\delta_{R}=0$).

###  2.4 Symmetrization

We find that, despite the subtractions of Eq. (35) and Eq. (36), amplitudes
with extra quark pair still depend on the type of auxiliary partons. The
dependence is caused by terms proportional to $\Upsilon(s)$ originating from
Eq. (29) for some invariants. We argue that the difference vanishes upon
summation over helicities and symmetrization as follows.

In [1] it was argued that the auxiliary parton method must be considered to be
applied at the cross-section level, rather than just at the amplitude level,
in order to take the real radiation contribution into account correctly. This
means that also the auxiliary partons must be summed over their helicities
before taking the $\Lambda$ limit. At tree-level, already for the amplitudes,
only the opposite-helicity configurations for the auxiliary partons
contribute, and either configuration gives the same result. Consider auxiliary
quarks. For the virtual contribution we can then write

$\displaystyle\mathcal{V}_{q}$
$\displaystyle=2\mathrm{Re}\Big{\\{}\mathcal{M}^{\mathrm{tree}\,\dagger}(A_{\bar{q}}^{+},B_{q}^{-})\mathcal{M}(A_{\bar{q}}^{+},B_{q}^{-})+\mathcal{M}^{\mathrm{tree}\,\dagger}(A_{\bar{q}}^{-},B_{q}^{+})\mathcal{M}(A_{\bar{q}}^{-},B_{q}^{+})\Big{\\}}$
$\displaystyle=2\mathrm{Re}\Big{\\{}\mathcal{M}^{\mathrm{tree}\,\dagger}(A_{\bar{q}}^{+},B_{q}^{-})\big{[}\mathcal{M}(A_{\bar{q}}^{+},B_{q}^{-})+\mathcal{M}(A_{\bar{q}}^{-},B_{q}^{+})\big{]}\Big{\\}}~{}.$
(37)

Here we used the “$\dagger$” to indicate complex conjugation, and only wrote
the helicity sum for the auxiliary partons explicitly. Considering the
definition of the momenta $p_{A},p_{B}$, the anti-quark in the amplitude must
be seen as an initial-state quark, and one could argue that the contribution
from anti-quark scattering must be included too. For the amplitudes, this
simply means that the role of $A,B$ must be reversed. At tree-level, this just
amounts to an overall minus sign, and we can write

$\displaystyle\mathcal{V}_{q}+\mathcal{V}_{\bar{q}}$
$\displaystyle=2\mathrm{Re}\Big{\\{}\mathcal{M}^{\mathrm{tree}\,\dagger}(A_{\bar{q}}^{+},B_{q}^{-})\big{[}\mathcal{M}(A_{\bar{q}}^{+},B_{q}^{-})+\mathcal{M}(A_{\bar{q}}^{-},B_{q}^{+})$
$\displaystyle\hskip
103.33301pt-\mathcal{M}(B_{\bar{q}}^{+},A_{q}^{-})-\mathcal{M}(B_{\bar{q}}^{-},A_{q}^{+})\big{]}\Big{\\}}~{}.$
(38)

We will see that in this combination of one-loop amplitudes, the problematic
terms that cause the auxiliary parton dependence drop out.

Regarding auxiliary gluons, one can argue that the same symmetrization
regarding the role of $A,B$ must be applied. In a bigger scope, one could say
that the hard process is just a factorized part of the full process, the
latter including the diagrams with reversed roles. Thus, following the same
procedure as for auxiliary quarks, we should take the combination

$\mathcal{M}(A^{+},B^{-})+\mathcal{M}(A^{-},B^{+})-\mathcal{M}(B^{+},A^{-})-\mathcal{M}(B^{-},A^{+})~{},$
(39)

and include a factor $1/2$. At tree-level, this does not make any difference.
We will see that at one loop this causes the problematic terms to drop out
again.

##  3 Results

In this section we present our results for the processes, $\emptyset\to
g^{\star}gg,\emptyset\to g^{\star}q\bar{q},\emptyset\to g^{\star}gH$ and
$\emptyset\to g^{\star}\bar{q}qe^{+}e^{-}$, using the auxiliary parton method
described above. All of them carry one power of the strong coupling constant,
but are increasingly less trivial regarding the kinematics, with the last two
having a non-vanishing on-shell limit.

We only present one helicity configuration for each process. The opposite-
helicity configuration is obtained by exchanging

$\kappa\leftrightarrow\kappa^{*}\quad,\quad\langle
ab\rangle\leftrightarrow[ba]$ (40)

everywhere in the expression. The latter also implies $\langle
a|b|c]\leftrightarrow\langle c|b|a]$. For the process $\emptyset\to
g^{\star}\bar{q}qe^{+}e^{-}$, the above includes the electron-positron pair.
Configurations for which only the electron and positron have flipped helicity
are obtained by simply exchanging their labels $(4,5)$ everywhere in the
expression.

###  3.1 $\bm{\emptyset}\bm{\to}\bm{g}^{\bm{\star}}\bm{g}\bm{g}$

In order to obtain the scattering amplitude for $\emptyset\to g^{\star}gg$
process we apply the auxiliary quark prescription to the results from Appendix
II in [50]. This means to use $\emptyset\to\bar{q}qgg$ as the on-shell
embedding process, with quark-antiquark pair taking the role of auxiliary
partons. In Appendix B we show how the method works with auxiliary gluons.

At tree-level, the color decomposition is

$\displaystyle\mathcal{M}^{\mathrm{tree}}(B_{\bar{q}},A_{q},2,3)=\big{(}T^{a_{2}}T^{a_{3}}\big{)}^{\imath_{B}}_{i_{A}}\,\mathcal{A}^{\mathrm{tree}}(B_{\bar{q}},A_{q},2,3)+\big{(}T^{a_{3}}T^{a_{2}}\big{)}^{\imath_{B}}_{i_{A}}\,\mathcal{A}^{\mathrm{tree}}(B_{\bar{q}},A_{q},3,2)~{}.$
(41)

Following Eq. (33) we find

$\mathcal{M}^{\mathrm{tree}}(1^{\star},2,3)=\mathrm{Tr}\big{(}T^{a_{1}}T^{a_{2}}T^{a_{3}}\big{)}\mathcal{A}^{\mathrm{tree}}(1^{\star},2,3)+\mathrm{Tr}\big{(}T^{a_{1}}T^{a_{3}}T^{a_{2}}\big{)}\mathcal{A}^{\mathrm{tree}}(1^{\star},3,2)~{}.$
(42)

The helicity amplitudes are

$\mathcal{A}^{\mathrm{tree}}(1^{\star},2^{\pm},3^{\pm})=0\quad,\quad\mathcal{A}^{\mathrm{tree}}(1^{\star},2^{-},3^{+})=\frac{\langle
p2\rangle^{3}}{\kappa^{*}\langle 23\rangle\langle 3p\rangle}~{}.$ (43)

Realize that
$\mathcal{A}^{\mathrm{tree}}(1^{\star},3,2)=-\mathcal{A}^{\mathrm{tree}}(1^{\star},2,3)$
(multiply numerator and denominator by $\langle p2\rangle$, and then only
exchange $2\leftrightarrow 3$ in the denominator). Thus, we can also decompose
the amplitude as

$\mathcal{M}^{\mathrm{tree}}(1^{\star},2,3)=\sqrt{2}\,\mathrm{i}f^{a_{1}a_{2}a_{3}}\mathcal{A}^{\mathrm{tree}}(1^{\star},2,3)~{}.$
(44)

The auxiliary quark one-loop color decomposition is

$\displaystyle\mathcal{M}(B_{\bar{q}},A_{q},2,3)$
$\displaystyle=N_{\mathrm{c}}\big{(}T^{a_{2}}T^{a_{3}}\big{)}^{\imath_{B}}_{i_{A}}\,\mathcal{A}_{1}(B_{\bar{q}},A_{q},2,3)+N_{\mathrm{c}}\big{(}T^{a_{3}}T^{a_{2}}\big{)}^{\imath_{B}}_{i_{A}}\,\mathcal{A}_{1}(B_{\bar{q}},A_{q},3,2)$
$\displaystyle\hskip
17.22217pt+\mathrm{Tr}\big{(}T^{a_{2}}T^{a_{3}}\big{)}\delta^{\imath_{B}}_{i_{A}}\,\mathcal{A}_{3}(B_{\bar{q}},A_{q},2,3)$
$\displaystyle=\sqrt{2}\,\mathrm{i}f^{a_{2}a_{3}a_{1}}N_{\mathrm{c}}\big{(}T^{a_{1}}\big{)}^{\imath_{B}}_{i_{A}}\mathcal{A}_{1}(B_{\bar{q}},A_{q},2,3)$
(45) $\displaystyle\hskip
17.22217pt+N_{\mathrm{c}}\big{(}T^{a_{3}}T^{a_{2}}\big{)}^{\imath_{B}}_{i_{A}}\Big{[}\mathcal{A}_{1}(B_{\bar{q}},A_{q},2,3)+\mathcal{A}_{1}(B_{\bar{q}},A_{q},3,2)\Big{]}$
$\displaystyle\hskip
17.22217pt+\mathrm{Tr}\big{(}T^{a_{2}}T^{a_{3}}\big{)}\delta^{\imath_{B}}_{i_{A}}\,\mathcal{A}_{3}(B_{\bar{q}},A_{q},2,3)~{}.$

The decomposition of $\mathcal{A}_{1}$ into primitive amplitudes is given in
Eq. (4.2) of [50], and $\mathcal{A}_{3}$ in Eq.(4.6). Regarding the latter,
realize that the primitive amplitudes for which the auxiliary partons are not
adjacent do not contribute. The amplitude
$\mathcal{A}_{1}(1^{\star},2^{+},3^{+})$ does not vanish, while it corresponds
to a helicity configuration that vanishes at tree-level. It was already
calculated in [44], and can be written slightly more compactly as

$\mathcal{A}_{1}(1^{\star},2^{+},3^{+})=\bigg{(}1-\frac{n_{f}}{N_{\mathrm{c}}}\bigg{)}\frac{1}{48\pi^{2}}\,\frac{[2p][p3]}{\kappa\langle
23\rangle}~{}.$ (46)

For the other helicity amplitudes, upon taking the $\Lambda$-limit we find

$\displaystyle\mathcal{A}_{3}(1^{\star},2^{+},3^{+})$ $\displaystyle=0~{},$
(47) $\displaystyle\mathcal{A}_{1}(1^{\star},2^{-},3^{+})$
$\displaystyle=\mathcal{A}^{\mathrm{re}}_{1}(1^{\star},2^{-},3^{+})-\Upsilon(s_{p2})\mathcal{A}^{\mathrm{Rt}}(1^{\star},2^{-},3^{+})$
(48)
$\displaystyle\mathcal{A}_{1}(1^{\star},2^{-},3^{+})+\mathcal{A}_{1}(1^{\star},3^{+},2^{-})$
$\displaystyle={-}2\Upsilon(s_{p2})\mathcal{A}^{\mathrm{Rt}}(1^{\star},2^{-},3^{+})$
(49) $\displaystyle\mathcal{A}_{3}(1^{\star},2^{-},3^{+})$
$\displaystyle={-}2\Upsilon(s_{p2})\mathcal{A}^{\mathrm{Rt}}(1^{\star},2^{-},3^{+})~{},$
(50)

where $\mathcal{A}^{\mathrm{Rt}}$ and $\Upsilon$ are defined in Eq. (22) and
Eq. (25), and

$\displaystyle\mathcal{A}^{\mathrm{re}}_{1}(1^{\star},2,3)$
$\displaystyle=c_{\Gamma}\mathcal{A}^{\mathrm{tree}}(1^{\star},2,3)\bigg{(}\frac{\mu^{2}}{|k_{T}^{2}|}\bigg{)}^{\epsilon}$
$\displaystyle\hskip
17.22217pt\times\bigg{[}\frac{-3}{\epsilon^{2}}+\frac{2}{\epsilon}\ln\frac{|s_{p2}|}{|k_{T}^{2}|}+\frac{2\pi^{2}}{3}-\frac{64+3\delta_{R}}{9}-\frac{11-2n_{f}/N_{\mathrm{c}}}{3\epsilon}+\frac{n_{f}}{N_{\mathrm{c}}}\frac{10}{9}\bigg{]}~{}.$
(51)

There is no discontinuity problem for $|k_{T}|\to 0$ since
$|k_{T}|\mathcal{A}^{\mathrm{tree}}$ vanishes in that limit. The helicities of
the auxiliary partons were not mentioned, but either choice gives the same
result. Switching the role of $A,B$ results in an overall minus sign at tree-
level and also for $\mathcal{A}^{\mathrm{re}}_{1}$ and
$\mathcal{A}^{\mathrm{Rt}}$, while the argument of $\Upsilon$ switches from
$s_{p2}$ to $s_{p3}$. Due to the kinematic restrictions, we have
$s_{p3}=-s_{p2}$, and we see that the contributions proportional to $\Upsilon$
vanish in the combination of Eq. (38). We can, however, already ignore those
contributions because of the color content. We have

$\displaystyle\mathcal{M}(B_{\bar{q}},A_{q},2,3)\overset{\Lambda}{\longrightarrow}\sqrt{2}\,\mathrm{i}f^{a_{2}a_{3}a_{1}}N_{\mathrm{c}}\big{(}T^{a_{1}}\big{)}^{\imath_{B}}_{i_{A}}\,\mathcal{A}^{\mathrm{re}}_{1}(1^{\star},2,3)$
$\displaystyle\hskip
43.05542pt-\Big{[}N_{\mathrm{c}}\big{(}T^{a_{2}}T^{a_{3}}\big{)}^{\imath_{B}}_{i_{A}}+N_{\mathrm{c}}\big{(}T^{a_{3}}T^{a_{2}}\big{)}^{\imath_{B}}_{i_{A}}+2\mathrm{Tr}\big{(}T^{a_{2}}T^{a_{3}}\big{)}\delta^{\imath_{B}}_{i_{A}}\Big{]}\Upsilon(s_{p2})\mathcal{A}^{\mathrm{Rt}}(1^{\star},2,3)~{}.$
(52)

Since the second line is symmetric in $a_{2},a_{3}$, it vanishes when
contracted with tree-level amplitude, which is anti-symmetric. Thus, we
eventually find

$\mathcal{M}^{\mathrm{NLO}}(1^{\star},2,3)=\sqrt{2}\,\mathrm{i}\,f^{a_{1}a_{2}a_{3}}N_{\mathrm{c}}\,\mathcal{A}^{\mathrm{re}}_{1}(1^{\star},2,3)~{}.$
(53)

As mentioned before, this process vanishes in the on-shell limit $|k_{T}|\to
0$, which is manifest through the vanishing tree-level amplitudes.

###  3.2 $\bm{\emptyset}\bm{\to}\bm{g}^{\bm{\star}}\bar{\bm{q}}\bm{q}$

For $g^{*}q\bar{q}$ we can use the same on-shell embedding process as for
$g^{*}gg$, but considering this time the gluon pair to play the role of
auxiliary partons. At tree-level, the color decomposition is

$\displaystyle\mathcal{M}^{\mathrm{tree}}(2_{\bar{q}},3_{q},B,A)=\big{(}T^{a_{B}}T^{a_{A}}\big{)}^{\imath_{2}}_{i_{3}}\,\mathcal{A}^{\mathrm{tree}}(2_{\bar{q}},3_{q},B,A)+\big{(}T^{a_{A}}T^{a_{B}}\big{)}^{\imath_{2}}_{i_{3}}\,\mathcal{A}^{\mathrm{tree}}(2_{\bar{q}},3_{q},A,B)~{}.$
(54)

with

$\displaystyle\mathcal{A}^{\mathrm{tree}}(2_{\bar{q}},3_{q},B,A)$
$\displaystyle\overset{\Lambda}{\longrightarrow}\mathcal{A}^{\mathrm{tree}}(1^{\star},2_{\bar{q}},3_{q})$
(55) $\displaystyle\mathcal{A}^{\mathrm{tree}}(2_{\bar{q}},3_{q},A,B)$
$\displaystyle\overset{\Lambda}{\longrightarrow}-\mathcal{A}^{\mathrm{tree}}(1^{\star},2_{\bar{q}},3_{q})$
(56)

so

$\displaystyle\mathcal{M}^{\mathrm{tree}}(2_{\bar{q}},3_{q},B,A)\overset{\Lambda}{\longrightarrow}\Big{[}\big{(}T^{a_{B}}T^{a_{A}}\big{)}^{\imath_{2}}_{i_{3}}-\big{(}T^{a_{A}}T^{a_{B}}\big{)}^{\imath_{2}}_{i_{3}}\Big{]}\mathcal{A}^{\mathrm{tree}}(1^{\star},2_{\bar{q}},3_{q})$
$\displaystyle\hskip
51.6665pt=\sqrt{2}\,\mathrm{i}f^{a_{B}a_{A}a_{1}}\big{(}T^{a_{1}}\big{)}^{\imath_{2}}_{i_{3}}\,\mathcal{A}^{\mathrm{tree}}(1^{\star},2_{\bar{q}},3_{q})=\sqrt{2}\,\mathrm{i}f^{a_{B}a_{A}a_{1}}\mathcal{M}^{\mathrm{tree},a_{1}}(1^{\star},2_{\bar{q}},3_{q})~{},$
(57)

confirming Eq. (32). The helicity amplitudes are given by

$\mathcal{A}^{\mathrm{tree}}(1^{\star},2_{\bar{q}}^{+},3_{q}^{+})=0\quad,\quad\mathcal{A}^{\mathrm{tree}}(1^{\star},2_{\bar{q}}^{-},3_{q}^{+})=-\frac{\langle
2p\rangle^{2}}{\kappa^{*}\langle 23\rangle}~{}.$ (58)

The $\kappa^{*}$ in the denominator came from $\langle BA\rangle$, and gets a
minus sign when exchanging $A\leftrightarrow B$. The one-loop color
decomposition is

$\displaystyle\mathcal{M}(2_{\bar{q}},3_{q},B,A)$
$\displaystyle=N_{\mathrm{c}}\big{(}T^{a_{B}}T^{a_{A}}\big{)}^{\imath_{2}}_{i_{3}}\,\mathcal{A}_{1}(2_{\bar{q}},3_{q},B,A)+N_{\mathrm{c}}\big{(}T^{a_{A}}T^{a_{B}}\big{)}^{\imath_{2}}_{i_{3}}\,\mathcal{A}_{1}(2_{\bar{q}},3_{q},A,B)$
$\displaystyle\hskip
17.22217pt+\mathrm{Tr}\big{(}T^{a_{A}}T^{a_{B}}\big{)}\delta^{\imath_{2}}_{i_{3}}\,\mathcal{A}_{3}(2_{\bar{q}},3_{q},B,A)$
(59)
$\displaystyle=\sqrt{2}\,\mathrm{i}f^{a_{B}a_{A}a_{1}}N_{\mathrm{c}}\big{(}T^{a_{1}}\big{)}^{\imath_{2}}_{i_{3}}\mathcal{A}_{1}(2_{\bar{q}},3_{q},B,A)$
$\displaystyle\hskip
17.22217pt+N_{\mathrm{c}}\big{(}T^{a_{A}}T^{a_{B}}\big{)}^{\imath_{2}}_{i_{3}}\Big{[}\mathcal{A}_{1}(2_{\bar{q}},3_{q},B,A)+\mathcal{A}_{1}(2_{\bar{q}},3_{q},A,B)\Big{]}$
$\displaystyle\hskip
17.22217pt+\mathrm{Tr}\big{(}T^{a_{A}}T^{a_{B}}\big{)}\delta^{\imath_{2}}_{i_{3}}\,\mathcal{A}_{3}(2_{\bar{q}},3_{q},B,A)~{}.$
(60)

The expression of $\mathcal{A}_{1}$ into primitive amplitudes is given in Eq.
(4.2) of [50], and $\mathcal{A}_{3}$ in Eq.(4.6). Regarding the latter,
realize that the primitive amplitudes for which the auxiliary partons are not
adjacent do not contribute. We must choose opposite helicities for the
auxiliary gluons, and may choose $B^{-},A^{+}$. For the helicity amplitudes
after the $\Lambda$-limit we find

$\displaystyle\mathcal{A}_{1}(1^{\star},2_{\bar{q}}^{+},3_{q}^{+})$
$\displaystyle=\mathcal{A}_{3}(1^{\star},2_{\bar{q}}^{+},3_{q}^{+})=0~{},$
(61) $\displaystyle\mathcal{A}_{1}(1^{\star},2_{\bar{q}}^{-},3_{q}^{+})$
$\displaystyle=\mathcal{A}^{\mathrm{re}}_{1}(1^{\star},2_{\bar{q}}^{-},3_{q}^{+})-\Upsilon(s_{p2})\mathcal{A}^{\mathrm{Rt}}(1^{\star},2_{\bar{q}}^{-},3_{q}^{+})~{},$
(62) $\displaystyle\mathcal{A}_{3}(1^{\star},2_{\bar{q}}^{-},3_{q}^{+})$
$\displaystyle=-2\Upsilon(s_{p2})\mathcal{A}^{\mathrm{Rt}}(1^{\star},2_{\bar{q}}^{-},3_{q}^{+})$
(63)
$\displaystyle\mathcal{A}_{1}(2_{\bar{q}}^{-},3_{q}^{+},B^{-},A^{+})+\mathcal{A}_{1}(2_{\bar{q}}^{-},3_{q}^{+},A^{+},B^{-})$
$\displaystyle\overset{\Lambda}{\longrightarrow}-2\Upsilon(s_{p2})\mathcal{A}^{\mathrm{Rt}}(1^{\star},2_{\bar{q}}^{-},3_{q}^{+})~{},$
(64)

where $\mathcal{A}^{\mathrm{Rt}}$ and $\Upsilon$ are defined in Eq. (22) and
Eq. (25), and

$\displaystyle\mathcal{A}^{\mathrm{re}}_{1}(1^{\star},2_{\bar{q}},3_{q})=c_{\Gamma}\mathcal{A}^{\mathrm{tree}}(1^{\star},2_{\bar{q}},3_{q})\bigg{(}\frac{\mu^{2}}{|k_{T}^{2}|}\bigg{)}^{\epsilon}$
$\displaystyle\hskip
34.44434pt\times\bigg{[}\frac{-2}{\epsilon^{2}}+\frac{1}{\epsilon}\bigg{(}2\ln\frac{|s_{p2}|}{|k_{T}^{2}|}-\frac{3}{2}\bigg{)}+\frac{2\pi^{2}}{3}-\frac{5+\delta_{R}}{2}+\frac{1}{N_{\mathrm{c}}^{2}}\bigg{(}\frac{1}{\epsilon^{2}}+\frac{3}{2\epsilon}+\frac{7+\delta_{R}}{2}\bigg{)}\bigg{]}~{}.$
(65)

With the same reasoning as for $\emptyset\to g^{\star}q\bar{q}$, we see that
the contributions proportional to $\Upsilon$ vanish in the combination of Eq.
(39). Also again, those contributions are already irrelevant due to the color
content. Combining, we find

$\displaystyle\mathcal{M}(2_{\bar{q}},3_{q},B,A)\overset{\Lambda}{\longrightarrow}\sqrt{2}\,\mathrm{i}f^{a_{B}a_{A}a_{1}}N_{\mathrm{c}}\big{(}T^{a_{1}}\big{)}^{\imath_{2}}_{i_{3}}\,\mathcal{A}^{\mathrm{re}}_{1}(1^{\star},2_{\bar{q}},3_{q})$
$\displaystyle\hskip
34.44434pt-\bigg{[}N_{\mathrm{c}}\big{(}T^{a_{A}}T^{a_{B}}\big{)}^{\imath_{2}}_{i_{3}}+N_{\mathrm{c}}\big{(}T^{a_{B}}T^{a_{A}}\big{)}^{\imath_{2}}_{i_{3}}+2\mathrm{Tr}\big{(}T^{a_{A}}T^{a_{B}}\big{)}\delta^{\imath_{2}}_{i_{3}}\bigg{]}\Upsilon(s_{p2})\mathcal{A}^{\mathrm{Rt}}(1^{\star},2_{\bar{q}},3_{q})~{}.$
(66)

Since the second line is symmetric in $a_{2},a_{3}$, it vanishes when
contracted with tree-level amplitude, which is anti-symmetric. Thus, we find

$\mathcal{M}^{\mathrm{NLO}}(1^{\star},2_{\bar{q}},3_{q})=N_{\mathrm{c}}\big{(}T^{a_{1}}\big{)}^{\imath_{2}}_{i_{3}}\,\mathcal{A}^{\mathrm{re}}_{1}(1^{\star},2_{\bar{q}},3_{q})~{}.$
(67)

Also this process vanishes in the on-shell limit $|k_{T}|\to 0$, which is
manifest through the vanishing tree-level amplitudes.

###  3.3 $\bm{\emptyset}\bm{\to}\bm{g}^{\bm{\star}}\bm{g}\bm{H}$

For this process the color decomposition is trivial and there is only one
partial amplitude. Following [52], with auxiliary quarks we have both at tree-
level and one-loop

$\displaystyle\mathcal{M}^{\mathrm{tree}}(A_{q},B_{\bar{q}},2,H)$
$\displaystyle\overset{\Lambda}{\longrightarrow}-\frac{\alpha_{\mathrm{s}}}{3\pi
v}\frac{1}{2}\big{(}T^{a_{2}}\big{)}^{\imath_{B}}_{i_{A}}\,\mathcal{A}^{\mathrm{tree}}(1^{\star},2,H)~{},$
(68) $\displaystyle\mathcal{M}(A_{q},B_{\bar{q}},2,H)$
$\displaystyle\overset{\Lambda}{\longrightarrow}-\frac{\alpha_{\mathrm{s}}}{3\pi
v}\frac{1}{2}\,N_{\mathrm{c}}\big{(}T^{a_{2}}\big{)}^{\imath_{B}}_{i_{A}}\,\mathcal{A}(1^{\star},2,H)~{},$
(69)

that is

$\displaystyle\mathcal{M}^{\mathrm{tree}}(1^{\star},2,H)$
$\displaystyle=-\frac{\alpha_{\mathrm{s}}}{3\pi
v}\frac{1}{2}\,\delta^{a_{1}a_{2}}\,\mathcal{A}^{\mathrm{tree}}(1^{\star},2,H)~{},$
(70) $\displaystyle\mathcal{M}(1^{\star},2,H)$
$\displaystyle=-\frac{\alpha_{\mathrm{s}}}{3\pi
v}\frac{1}{2}\,N_{\mathrm{c}}\delta^{a_{1}a_{2}}\,\mathcal{A}(1^{\star},2,H)~{}.$
(71)

The factor $-\alpha_{\mathrm{s}}/(3\pi v)$ is related to the effective Higgs-
gluon coupling. At tree-level, we get

$\mathcal{A}^{\mathrm{tree}}(1^{\star},2^{+},H)=\frac{[p2]^{2}}{\kappa}$ (72)

and at one loop we get

$\displaystyle\mathcal{A}(1^{\star},2,H)$
$\displaystyle=c_{\Gamma}\mathcal{A}^{\mathrm{tree}}(1^{\star},2,H)\bigg{[}{-}\frac{2}{\epsilon^{2}}\left(\frac{\mu^{2}}{-M_{H}^{2}}\right)^{\epsilon}+\frac{2}{\epsilon}\bigg{(}\frac{\mu^{2}}{|k_{T}^{2}|}\bigg{)}^{\epsilon}\ln\frac{|s_{p2}|}{-M_{H}^{2}}$
$\displaystyle\hskip
163.6106pt+\frac{\pi^{2}}{3}-2\mathrm{Li}_{2}\left(1+\frac{|k_{T}^{2}|}{M_{H}^{2}}\right)\bigg{]}~{}.$
(73)

Here we must imagine that $M_{H}^{2}$ has a small positive imaginary part (so
it plays the role of an invariant rather than a squared mass, which would
naturally get a negative imaginary part). Notice that the first term in the
square brackets gives the on-shell result, while the other terms vanish for
$|k_{T}|\to 0$ (and negative $M_{H}^{2}$ implying $-M_{H}^{2}\to|s_{p2}|$).
The following organization of terms better matches the singularity structure
of the real radiation contribution in an NLO calculation:

$\displaystyle\mathcal{A}(1^{\star},2,H)$
$\displaystyle=c_{\Gamma}\mathcal{A}^{\mathrm{tree}}(1^{\star},2,H)\bigg{\\{}{-}\frac{2}{\epsilon^{2}}\left(\frac{\mu^{2}}{|s_{p2}|}\right)^{\epsilon}+\ln\bigg{(}\frac{|s_{p2}|}{|M_{H}^{2}|}\bigg{)}\ln\bigg{(}\frac{|s_{p2}||M_{H}^{2}|}{|k_{T}^{2}|^{2}}\bigg{)}$
$\displaystyle\hskip
43.05542pt+\frac{\pi^{2}}{3}-2\mathrm{Li}_{2}\left(1+\frac{|k_{T}^{2}|}{M_{H}^{2}}\right)+\theta\big{(}M_{H}^{2}\big{)}\bigg{[}\pi^{2}+2\mathrm{i}\pi\ln\bigg{(}\frac{|M_{H}^{2}|}{|k_{T}^{2}|}\bigg{)}\bigg{]}\bigg{\\}}~{}.$
(74)

###  3.4
$\bm{\emptyset}\bm{\to}\bm{g}^{\bm{\star}}\bar{\bm{q}}\bm{q}\bm{e}^{\bm{+}}\bm{e}^{\bm{-}}$

In this case, we use the results from [51], and consider auxiliary quarks. We
are after the amplitude in Eq. (2.11) of that publication, that is

$\displaystyle\mathcal{M}^{\mathrm{1-loop}}(1_{q},2_{\bar{Q}},3_{Q},4_{\bar{q}},4_{e^{+}},5_{e^{-}})$
$\displaystyle=2e^{2}\big{(}{-}Q^{q}+v^{e}_{L,R}v^{q}_{L,R}\mathcal{P}_{Z}(s_{45})\big{)}\bigg{[}N_{\mathrm{c}}\delta^{\imath_{A}}_{i_{3}}\delta^{\imath_{2}}_{i_{B}}\mathcal{A}_{1}(1_{q},2_{\bar{Q}},3_{Q},4_{\bar{q}})+\delta^{\imath_{2}}_{i_{3}}\delta^{\imath_{A}}_{i_{B}}\mathcal{A}_{2}(1_{q},2_{\bar{Q}},3_{Q},4_{\bar{q}})\bigg{]}$
$\displaystyle+2e^{2}\big{(}{-}Q^{Q}+v^{e}_{L,R}v^{Q}_{L,R}\mathcal{P}_{Z}(s_{45})\big{)}\bigg{[}N_{\mathrm{c}}\delta^{\imath_{A}}_{i_{3}}\delta^{\imath_{2}}_{i_{B}}\mathcal{A}_{1}(3_{Q},4_{\bar{q}},1_{q},2_{\bar{Q}})+\delta^{\imath_{2}}_{i_{3}}\delta^{\imath_{A}}_{i_{B}}\mathcal{A}_{2}(3_{Q},4_{\bar{q}},1_{q},2_{\bar{Q}})\bigg{]}$
$\displaystyle\hskip 77.49976pt+2e^{2}\frac{v^{e}_{L,R}}{\sin
2\theta_{W}}\,\mathcal{P}_{Z}(s_{45})\bigg{(}\delta^{\imath_{A}}_{i_{3}}\delta^{\imath_{2}}_{i_{B}}-\frac{1}{N_{\mathrm{c}}}\delta^{\imath_{2}}_{i_{3}}\delta^{\imath_{A}}_{i_{B}}\bigg{)}\mathcal{A}_{3}(1_{q},2_{\bar{Q}},3_{Q},4_{\bar{q}})~{}.$
(75)

On the left-hand side, the enumeration of the $e^{+},e^{-}$ pair is written
explicitly, while it is omitted in the partial amplitudes. We will write it
explicitly in the partial amplitudes obtained after the $\Lambda$ limit. One
issue to keep in mind is that in our prescription, the vector-boson must not
couple to the auxiliary quark line. This is achieved by including only the
contributions of Figure 3 in [51], while assigning $Q\bar{Q}$ as the auxiliary
quark-antiquark pair. This simply means that we must exclude the amplitudes
with ordering $\big{(}3_{Q},4_{\bar{q}},1_{q},2_{\bar{Q}}\big{)}$ in the
expression above. Thus, we need

$\displaystyle\mathcal{M}(3_{q},A_{\bar{Q}},B_{Q},2_{\bar{q}},4_{e^{+}},5_{e^{-}})$
$\displaystyle=c_{1}\bigg{[}N_{\mathrm{c}}\delta^{\imath_{A}}_{i_{3}}\delta^{\imath_{2}}_{i_{B}}\mathcal{A}_{1}(3_{q},A_{\bar{Q}},B_{Q},2_{\bar{q}})+\delta^{\imath_{2}}_{i_{3}}\delta^{\imath_{A}}_{i_{B}}\mathcal{A}_{2}(3_{q},A_{\bar{Q}},B_{Q},2_{\bar{q}})\bigg{]}$
$\displaystyle+c_{3}\bigg{(}\delta^{\imath_{A}}_{i_{3}}\delta^{\imath_{2}}_{i_{B}}-\frac{1}{N_{\mathrm{c}}}\delta^{\imath_{2}}_{i_{3}}\delta^{\imath_{A}}_{i_{B}}\bigg{)}\mathcal{A}_{3}(3_{q},A_{\bar{Q}},B_{Q},2_{\bar{q}})~{},$
(76)

where we abbreviate

$c_{1}=2e^{2}\big{(}{-}Q^{q}+v^{e}_{L,R}v^{q}_{L,R}\mathcal{P}_{Z}(s_{45})\big{)}\quad,\quad
c_{3}=2e^{2}\frac{v^{e}_{L,R}}{\sin 2\theta_{W}}\,\mathcal{P}_{Z}(s_{45})~{}.$
(77)

We copied the symbols for the constants, and refer to page 7 of [51] for their
definition. We only mention that the partial amplitudes include the photon
propagator, and the factor $\mathcal{P}_{Z}(s_{45})$ corrects it to the
$Z$-boson propagator, so both contributions are included depending on whether
the constants $v^{e,q}_{L,R}$ are switched on. The tree-level decomposition is
given by

$\mathcal{M}^{\mathrm{tree}}(3_{q},A_{\bar{Q}},B_{Q},2_{\bar{q}},4_{e^{+}},5_{e^{-}})=c_{1}\bigg{(}\delta^{\imath_{A}}_{i_{3}}\delta^{\imath_{2}}_{i_{B}}-\frac{1}{N_{\mathrm{c}}}\delta^{\imath_{2}}_{i_{3}}\delta^{\imath_{A}}_{i_{B}}\bigg{)}\mathcal{A}^{\mathrm{tree}}(3_{q},A_{\bar{Q}},B_{Q},2_{\bar{q}})~{}.$
(78)

Notice that the tree-level expression vanishes under contraction with
$\delta_{\imath_{A}}^{i_{B}}$. We can write the one-loop amplitude in terms of
the tree-level color structure, and one that vanishes if contracted with the
tree-level structure as

$\displaystyle\mathcal{M}(3_{q},A_{\bar{Q}},B_{Q},2_{\bar{q}},4_{e^{+}},5_{e^{-}})$
$\displaystyle=\bigg{(}\delta^{\imath_{A}}_{i_{3}}\delta^{\imath_{2}}_{i_{B}}-\frac{1}{N_{\mathrm{c}}}\delta^{\imath_{2}}_{i_{3}}\delta^{\imath_{A}}_{i_{B}}\bigg{)}\bigg{[}c_{1}N_{\mathrm{c}}\mathcal{A}_{1}(3_{q},A_{\bar{Q}},B_{Q},2_{\bar{q}})$
$\displaystyle\hskip
111.94409pt+c_{3}\mathcal{A}_{3}(3_{q},A_{\bar{Q}},B_{Q},2_{\bar{q}})\bigg{]}$
$\displaystyle+c_{1}\,\delta^{\imath_{2}}_{i_{3}}\delta^{\imath_{A}}_{i_{B}}\bigg{[}\mathcal{A}_{1}(3_{q},A_{\bar{Q}},B_{Q},2_{\bar{q}})+\mathcal{A}_{2}(3_{q},A_{\bar{Q}},B_{Q},2_{\bar{q}})\bigg{]}~{}.$
(79)

The contraction of color indices of the tree-level amplitude with the last
line vanishes. Omitting this term and contracting with
$(T^{a_{1}})_{\imath_{A}}^{i_{B}}$, we find

$\displaystyle\mathcal{M}^{\mathrm{tree}}(1^{\star},2_{\bar{q}},3_{q},4_{e^{+}},5_{e^{-}})$
$\displaystyle=(T^{a_{1}})^{i_{2}}_{i_{3}}\,c_{1}\,\mathcal{A}^{\mathrm{tree}}(1^{\star},2_{\bar{q}},3_{q},4_{e^{+}},5_{e^{-}})~{},$
(80)
$\displaystyle\mathcal{M}^{\mathrm{NLO}}(1^{\star},2_{\bar{q}},3_{q},4_{e^{+}},5_{e^{-}})$
$\displaystyle=(T^{a_{1}})^{i_{2}}_{i_{3}}\,\Big{[}c_{1}N_{\mathrm{c}}\mathcal{A}_{1}(1^{\star},2_{\bar{q}},3_{q},4_{e^{+}},5_{e^{-}})$
$\displaystyle\hskip
51.6665pt+c_{3}\mathcal{A}_{3}(1^{\star},2_{\bar{q}},3_{q},4_{e^{+}},5_{e^{-}})\Big{]}~{}.$
(81)

We only present expressions for the helicity amplitude
$\mathcal{M}(1^{\star},2_{\bar{q}}^{-},3_{q}^{+},4_{e^{+}}^{-},5_{e^{-}}^{+})$.
The others can be derived with the rules mentioned at the beginning of this
section.

The partial amplitudes are decomposed into primitive amplitudes in [51] as

$\displaystyle\mathcal{A}_{1}(3_{q}^{+},A_{\bar{Q}}^{+},B_{Q}^{-},2_{\bar{q}}^{-})$
$\displaystyle=\mathcal{A}^{++}(3,A,B,2)+\frac{1}{N_{\mathrm{c}}^{2}}\,\mathcal{A}^{\mathrm{sl}}(A,B,3,2)$
$\displaystyle-\frac{2}{N_{\mathrm{c}}^{2}}\big{[}\mathcal{A}^{++}(3,A,B,2)+\mathcal{A}^{+-}(3,B,A,2)\big{]}$
(82)
$\displaystyle-\frac{n_{f}}{N_{\mathrm{c}}}\big{[}\mathcal{A}^{f,++}(3,A,B,2)+\mathcal{A}^{s,++}(3,A,B,2)\big{]}+\frac{1}{N_{\mathrm{c}}}\,\mathcal{A}^{t,++}(3,A,B,2)~{},$

and

$\displaystyle\mathcal{A}_{1}(3_{q}^{+},A_{\bar{Q}}^{-},B_{Q}^{+},2_{\bar{q}}^{-})$
$\displaystyle=\mathcal{A}^{+-}(3,A,B,2)-\frac{1}{N_{\mathrm{c}}^{2}}\,\mathcal{A}^{\mathrm{sl}}(B,A,3,2)$
$\displaystyle-\frac{2}{N_{\mathrm{c}}^{2}}\big{[}\mathcal{A}^{+-}(3,A,B,2)+\mathcal{A}^{++}(3,B,A,2)\big{]}$
(83)
$\displaystyle-\frac{n_{f}}{N_{\mathrm{c}}}\big{[}\mathcal{A}^{f,+-}(3,A,B,2)+\mathcal{A}^{s,+-}(3,A,B,2)\big{]}+\frac{1}{N_{\mathrm{c}}}\,\mathcal{A}^{t,+-}(3,A,B,2)~{}.$

From now on, we omit the top quark vacuum polarization contribution
$\mathcal{A}^{t}$, since it is suppressed by a factor $|k_{T}^{2}|/m_{t}^{2}$
already at the amplitude level. Also, we will not mention the
$n_{f}/N_{\mathrm{c}}$ contribution from massless fermion loops any further,
because it turns out to be completely given by the one in Eq. (36) and does
not show up in
$\mathcal{A}_{1}(1^{\star},2_{\bar{q}},3_{q},4_{e^{+}},5_{e^{-}})$.

The second line Eq. (82) turns out to lead to a contribution that causes a
difference compared to auxiliary gluons. In the sum over helicities of the
auxiliary partons following Eq. (37), that is by and adding Eq. (83), this
$1/N_{\mathrm{c}}^{2}$ contribution becomes symmetric in $A,B$. Including then
the contribution with the role of $A,B$ switched as in Eq. (38), this
contribution vanishes.

The primitive amplitudes in [51] are decomposed further as (remember that we
took out an overall factor $\mathrm{i}$ compared to [51])

$\mathcal{A}=c_{\Gamma}\big{(}\mathcal{V}\,\mathcal{A}^{\mathrm{tree}}+\mathcal{F}\big{)}~{},$
(84)

where $\mathcal{V}$ contains all $1/\epsilon$ poles and $\mathcal{F}$ is
finite. This decomposition is not uniquely defined since $\mathcal{V}$ also
contains finite parts. We will also present the results in this type of
decomposition, but with a distribution of finite terms between
$\mathcal{V}\,\mathcal{A}^{\mathrm{tree}}$ and $\mathcal{F}$ that does not
directly correspond to the one in [51]. The cancellation of terms proportional
to $\ln\Lambda$ and $\ln^{2}\Lambda$ happens between
$\mathcal{V}\,\mathcal{A}^{\mathrm{tree}}$ and $\mathcal{F}$ from [51] and
does not stay within each of these contributions. The subtraction dictated by
Eq. (36), however, does apply only to the contribution
$\mathcal{V}\,\mathcal{A}^{\mathrm{tree}}$. We write

$\displaystyle c_{\Gamma}^{-1}\mathcal{A}^{++}(3,A,B,2)$
$\displaystyle\overset{\Lambda}{\longrightarrow}\mathcal{V}^{\,\star}\mathcal{A}^{\mathrm{tree},\star}+\mathcal{F}^{\,\mathrm{re},\star}+\Upsilon(s_{p2})\mathcal{F}^{\,\mathrm{im},\star}+\Theta(s_{p2})\mathcal{A}^{\mathrm{Rt},\star}~{},$
$\displaystyle c_{\Gamma}^{-1}\mathcal{A}^{\mathrm{sl}}(A,B,3,2)$
$\displaystyle\overset{\Lambda}{\longrightarrow}{-}\mathcal{V}^{\,\mathrm{sl},\star}\mathcal{A}^{\mathrm{tree},\star}+\mathcal{F}^{\,\mathrm{sl},\star}~{},$
(85)

where we use the single superscript $\star$ instead of the argument list
$(1^{\star},2_{\bar{q}}^{-},3_{q}^{+},4_{e^{+}}^{-},5_{e^{-}}^{+})$ for
brevety, and $\Theta,\Upsilon$ are defined in Eq. (25). In the following, we
present the expressions for the amplitudes on the right hand sides. At tree
level, we have

$\displaystyle\mathcal{A}^{\mathrm{tree},\star}$
$\displaystyle=\frac{-1}{s_{45}}\bigg{[}\frac{[3p]\langle
42\rangle[p5]}{\kappa s_{k3}}+\frac{\langle p2\rangle[53]\langle
4p\rangle}{\kappa^{*}s_{k2}}+\frac{\langle
24\rangle[53]}{\kappa\kappa^{*}}\bigg{(}\frac{s_{p3}}{s_{k3}}-\frac{s_{p2}}{s_{k2}}\bigg{)}\bigg{]}~{}.$
(86)

Realize that

$s_{ki}\;\overset{|k_{T}|\to
0}{\longrightarrow}\;s_{pi}\qquad\textrm{so}\qquad\frac{s_{p3}}{s_{k3}}-\frac{s_{p2}}{s_{k2}}\;\overset{|k_{T}|\to
0}{\longrightarrow}\;0$ (87)

and the amplitude behaves only as $|k_{T}|^{-1}$, not as $|k_{T}|^{-2}$.
Another form in which this is explicit is given by

$\displaystyle\mathcal{A}^{\mathrm{tree},\star}=-\frac{\mathcal{T}^{\,\flat}+\mathcal{T}^{\,\sharp}}{2}~{},$
(88)

where

$\mathcal{T}^{\,\flat}=\frac{[3p]^{2}\langle 24\rangle^{2}}{\kappa\langle
45\rangle s_{k3}\langle 2|k|3]}+\frac{\langle
p|(2+4)|5]^{2}}{\kappa^{*}[54]s_{k3}\langle 3|k|2]}~{},$ (89)

and where $\mathcal{T}^{\,\sharp}$ is obtained from $\mathcal{T}^{\,\flat}$
with the “flip” operation, defined through the rules

$\displaystyle\mathrm{flip}:\;\;2\leftrightarrow 3\;\;,\quad 4\leftrightarrow
5\;\;,\quad\langle
ab\rangle\leftrightarrow[ab]\;\;,\quad\kappa\leftrightarrow\kappa^{*}\;\;,\quad\rho_{\mathrm{flip}}=1\rightarrow-1\;\;.$
(90)

The third rule also implies $\langle
a|b|c]\leftrightarrow[a|b|c\rangle=\langle c|b|a]$, and the last rule will
show to be usefull at one loop. Notice that each ${-}\mathcal{T}^{\,\flat}$
and ${-}\mathcal{T}^{\,\sharp}$ produce the on-shell limit when $|k_{T}|\to
0$, with the $1/\kappa^{*}$ term giving the minus helicity, and the $1/\kappa$
term giving the plus helicity. For the one-loop amplitude, we find

$\mathcal{V}^{\,\star}=-\frac{1}{\epsilon^{2}}\bigg{[}\bigg{(}\frac{\mu^{2}}{-s_{p3}}\bigg{)}^{\epsilon}+\bigg{(}\frac{\mu^{2}}{-s_{p2}}\bigg{)}^{\epsilon}\bigg{]}-\frac{3}{2\epsilon}\bigg{(}\frac{\mu^{2}}{-s_{45}}\bigg{)}^{\epsilon}-\frac{7}{2}-\frac{\delta_{R}}{3}~{}.$
(91)

We included the leading-color contribution of

$-\frac{\delta_{R}}{2}\bigg{(}1-\frac{1}{N_{\mathrm{c}}^{2}}\bigg{)}$ (92)

to switch the amplitude between the ’t Hooft-Velman scheme ($\delta_{R}=1$)
and the four-dimensional helicity scheme ($\delta_{R}=0)$, and subtracted the
necessary leading color terms following Eq. (36). For the remaining finite
parts of the amplitude we find

$\displaystyle\mathcal{F}^{\,\mathrm{re},\star}$
$\displaystyle=\mathcal{T}^{\,\flat}\bigg{[}\ln\bigg{(}\frac{|k_{T}^{2}|}{-s_{k3}}\bigg{)}\ln\bigg{(}\frac{-s_{p3}}{-s_{k3}}\bigg{)}-\frac{1}{2}\ln^{2}\bigg{(}\frac{-s_{p2}}{-s_{k3}}\bigg{)}$
$\displaystyle\hskip
34.44434pt+\frac{1}{2}\ln\bigg{(}\frac{-s_{p2}}{-s_{45}}\bigg{)}\ln\bigg{(}\frac{-s_{p2}}{-s_{p3}}\bigg{)}+\mathrm{Li}_{2}\bigg{(}1+\frac{|k_{T}^{2}|}{s_{k3}}\bigg{)}-\frac{\pi^{2}}{4}-\mathrm{Li}_{2}\bigg{(}1-\frac{s_{45}}{s_{k3}}\bigg{)}\bigg{]}$
$\displaystyle+2\frac{\langle p|(2+4)|5]}{s_{k3}[45]\langle
3|k|2]}\bigg{[}\rho_{\mathrm{flip}}\,\frac{\langle
3|k|5][3p]}{s_{k3}}\,\mathrm{L}_{0}\bigg{(}\frac{|k_{T}^{2}|}{-s_{k3}}\bigg{)}-\frac{\langle
p|2|5]}{\kappa^{*}}\,\mathrm{L}_{0}\bigg{(}\frac{-s_{45}}{-s_{k3}}\bigg{)}\bigg{]}$
(93) $\displaystyle-\frac{1}{2}\frac{\langle
3|k|5]^{2}[3p]^{2}}{\kappa[45]s_{k3}^{2}\langle
3|k|2]}\,\bigg{[}\frac{|k_{T}^{2}|}{s_{k3}}\,\mathrm{L}_{1}\bigg{(}\frac{|k_{T}^{2}|}{-s_{k3}}\bigg{)}-1\bigg{]}+\frac{1}{2}\frac{\langle
p|2|5]^{2}}{\kappa^{*}[45]s_{k3}\langle
3|k|2]}\,\mathrm{L}_{1}\bigg{(}\frac{-s_{45}}{-s_{k3}}\bigg{)}$
$\displaystyle+\phantom{\bigg{[}}\mathrm{flip}~{},$

where $\mathrm{flip}$ acts on all preceding terms. For the other one, we find

$\mathcal{F}^{\,\mathrm{im},\star}=\frac{\mathcal{T}^{\,\flat}-\mathcal{T}^{\,\sharp}}{2}\ln\bigg{(}\frac{(-s_{p2})(-s_{p3})}{|k_{T}^{2}|(-s_{45})}\bigg{)}+\mathcal{T}^{\,\flat}\ln\bigg{(}\frac{-s_{k3}}{-s_{p3}}\bigg{)}-\mathcal{T}^{\,\sharp}\ln\bigg{(}\frac{-s_{k2}}{-s_{p2}}\bigg{)}~{}.$
(94)

While the quantities $\mathcal{F}^{\,\mathrm{re},\star}$ and
$\Upsilon(s_{p2})\mathcal{F}^{\,\mathrm{im},\star}$ are not strictly real and
imaginary, we decide to still use these labels to distinguish them from each
other. Notice that the arguments of the logarithms in
$\mathcal{F}^{\,\mathrm{im},\star}$ never become negative: first of all
$s_{ki}$ and $s_{pi}$ always have the same sign. For $e^{+},e^{-}$ in the
final state, one of $p_{2,3}$ must be in the initial state while the other is
in the final state and we have $s_{45}>0$ and $s_{p2}s_{p3}<0$. For DIS-type
kinematics we have $s_{45}<0$, $s_{p2}<0$, $s_{p3}<0$.

Also notice that $\mathcal{F}^{\,\mathrm{im},\star}$ vanishes in the on-shell
limit. The terms proportional to $1/\kappa$ in
$\mathcal{V}^{\,\star}\mathcal{A}^{\mathrm{tree},\star}+\mathcal{F}^{\,\mathrm{re},\star}$
exactly give the on-shell limit for $|k_{T}|\to 0$, as given in the formulas
in Appendix IV in [51], which involve a positive-helicity gluon. For the
combination of amplitudes in Eq. (38) we find

$4\big{(}\mathcal{V}^{\,\star}\mathcal{A}^{\mathrm{tree},\star}+\mathcal{F}^{\,\mathrm{re},\star}\big{)}+2\big{[}\Upsilon(s_{p2})-\Upsilon(s_{p3})\big{]}\mathcal{F}^{\,\mathrm{im},\star}+2\big{[}\Theta(s_{p2})+\Theta(s_{p3})\big{]}\mathcal{A}^{\mathrm{Rt},\star}~{}.$
(95)

Notice that the “imaginary” terms vanish for DIS-type kinematics. The term
with $\mathcal{A}^{\mathrm{Rt},\star}$ is truly imaginary, and never
contributes to the virtual contribution.

The $1/N_{\mathrm{c}}^{2}$ contribution does not contain such pieces, and we
find

$\displaystyle-\mathcal{V}^{\,\mathrm{sl},\star}$
$\displaystyle=\frac{1}{\epsilon^{2}}\bigg{(}\frac{\mu^{2}}{-s_{23}}\bigg{)}^{\epsilon}+\frac{3}{2\epsilon}\bigg{(}\frac{\mu^{2}}{-s_{23}}\bigg{)}^{\epsilon}+4~{},$
(96)

where we included the sub-leading color part of (92) and subtracted the
necessary sub-leading color terms following Eq. (36). For the finite
contribution, we find

$\displaystyle\mathcal{F}^{\,\mathrm{sl},\star}$
$\displaystyle=\bigg{[}{-}\frac{[p3]^{2}\langle 24\rangle^{2}}{\kappa\langle
45\rangle s_{k3}\langle 2|k|3]}+\frac{\langle 3|k|5]^{2}\langle
p|(4+5)|2]^{2}}{\kappa^{*}[45]s_{k3}\langle
3|k|2]^{3}}\bigg{]}\mathrm{Ls}_{-1}^{2mh}(s_{23},s_{k3},k_{T}^{2},s_{45})$
$\displaystyle+\bigg{[}\frac{1}{2}\frac{(3s_{23}\delta_{23}-\Delta)(s_{k2}-s_{k3})(s_{p3}+s_{p2})\langle
4|k|5]}{\langle
3|k|2]\Delta^{2}}+\frac{1}{2}\frac{s_{23}(s_{k2}-s_{k3})\langle
4p\rangle[p5]}{\langle 3|k|2]\Delta}$ $\displaystyle\hskip
17.22217pt+\frac{\kappa\langle 45\rangle\langle p|(2+3)|5]^{2}}{\langle
3|k|2]\Delta}+\frac{\langle p3\rangle[25]\delta_{23}(\kappa\langle
4p\rangle\delta_{k^{2}}-\langle 4|5|p]\delta_{45})}{\langle 3|k|2]^{2}\Delta}$
$\displaystyle\hskip 17.22217pt-\frac{\langle p3\rangle[25]\langle
4|(2+5)|p]}{\langle 3|k|2]^{2}}+2\frac{[p3]\langle 24\rangle\langle
p|(4+5)|2]\langle 3|k|5]}{s_{k3}^{2}\langle
3|k|2]}\bigg{]}I_{3}(k_{T}^{2},s_{23},s_{45})$
$\displaystyle{-}\frac{1}{2}\frac{[52]^{2}\langle
2p\rangle^{2}s_{k3}}{\kappa^{*}[45]\langle
3|k|2]}\frac{\mathrm{L}_{1}\big{(}\frac{-s_{45}}{-s_{k3}}\big{)}}{s_{k3}^{2}}-\frac{1}{2}\frac{[p2]^{2}\langle
24\rangle^{2}s_{k2}}{\kappa\langle 54\rangle\langle
3|k|2]}\frac{\mathrm{L}_{1}\big{(}\frac{-k_{T}^{2}}{-s_{k2}}\big{)}}{s_{k2}^{2}}$
$\displaystyle+\frac{\langle p2\rangle[25]}{\kappa^{*}[45]\langle
3|k|2]}\bigg{[}2\langle p|(2+4)|5]+\frac{\langle p3\rangle[25]s_{k3}}{\langle
3|k|2]}\bigg{]}\frac{\mathrm{L}_{0}\big{(}\frac{-s_{k3}}{-s_{45}}\big{)}}{s_{45}}$
$\displaystyle-\frac{\langle 42\rangle[2p]}{\kappa\langle 54\rangle\langle
3|k|2]}\bigg{[}2\langle 4|(5+3)|p]+\frac{\langle 43\rangle[2p]s_{k2}}{\langle
3|k|2]}\bigg{]}\frac{\mathrm{L}_{0}\big{(}\frac{-s_{k2}}{-k_{T}^{2}}\big{)}}{k_{T}^{2}}$
$\displaystyle+\frac{\langle p|(2+4)|5]}{\kappa^{*}[45]\langle
3|k|2]}\bigg{(}\frac{3}{4}\frac{\langle
p|(2+4)|5]}{s_{k3}}-\frac{1}{2}\frac{\langle p3\rangle[52]}{\langle
3|k|2]}\bigg{)}\ln\bigg{(}\frac{(-s_{k3})(-s_{23})}{(-s_{45})^{2}}\bigg{)}$
$\displaystyle+\frac{\langle 4|(5+3)|p]}{\kappa\langle 54\rangle\langle
3|k|2]}\bigg{(}\frac{3}{4}\frac{\langle
4|(5+3)|p]}{s_{k2}}-\frac{1}{2}\frac{\langle 43\rangle[p2]}{\langle
3|k|2]}\bigg{)}\ln\bigg{(}\frac{(-s_{k2})(-s_{23})}{(-k_{T}^{2})^{2}}\bigg{)}$
$\displaystyle+\bigg{[}\frac{3}{2}\frac{\delta_{45}(s_{k2}-s_{k3})(s_{p3}+s_{p2})\langle
4|k|5]}{\langle 3|k|2]\Delta}-\frac{\kappa\langle p3\rangle[25]}{\langle
3|k|2]^{2}\Delta}\big{(}\langle p4\rangle(s_{k2}-s_{k3})+2\kappa^{*}\langle
45\rangle[5p]\big{)}$ $\displaystyle\hskip 17.22217pt+\frac{\kappa\langle
p4\rangle\langle p|(2-3)|5]}{\langle 3|k|2]\Delta}+\frac{[p5]}{[45]\langle
3|k|2]\Delta}\bigg{(}\langle p|3|5]s_{k3}-\langle p|2|5]s_{k2}+\frac{\langle
p3\rangle[25]\delta_{23}s_{k3}}{\langle 3|k|2]}\bigg{)}\bigg{]}$
$\displaystyle\hskip
8.61108pt\times\ln\bigg{(}\frac{-k_{T}^{2}}{-s_{23}}\bigg{)}$
$\displaystyle+\bigg{[}\frac{3}{2}\frac{\delta_{k^{2}}(s_{k2}-s_{k3})(s_{p2}+s_{p3})\langle
4|k|5]}{\langle 3|k|2]\Delta^{2}}-\frac{[54]\langle 43\rangle[2p]}{\langle
3|k|2]^{2}\Delta}\big{(}\langle 4p\rangle(s_{k2}-s_{k3})+2\kappa^{*}\langle
45\rangle[5p]\big{)}$ $\displaystyle\hskip 17.22217pt-\frac{[54]\langle
p4\rangle\langle 4|(2-3)|p]}{\langle 3|k|2]\Delta}-\frac{[5p]}{\kappa\langle
3|k|2]\Delta}\bigg{(}\langle 4|3|p]s_{k2}-\langle 4|2|p]s_{k3}-\frac{\langle
43\rangle[2p]\delta_{23}s_{k2}}{\langle 3|k|2]}\bigg{)}\bigg{]}$
$\displaystyle\hskip
8.61108pt\times\ln\bigg{(}\frac{-s_{45}}{-s_{23}}\bigg{)}$
$\displaystyle+\frac{1}{2}\frac{[p5](s_{k3}-s_{k2})([p5]\delta_{23}+2\kappa\langle
p4\rangle[45])}{\kappa[45]\langle 3|k|2]\Delta}\;\;+\;\;\mathrm{flip}~{},$
(97)

where $\mathrm{flip}$ again acts on all preceding terms. Furthermore, we
denote

$\delta_{k^{2}}=k_{T}^{2}-s_{23}-s_{45}\quad,\quad\delta_{23}=s_{23}-s_{45}-k_{T}^{2}\quad,\quad\delta_{45}=s_{45}-k_{T}^{2}-s_{23}~{},$
(98)

and

$\Delta=(k_{T}^{2})^{2}+s_{23}^{2}+s_{45}^{2}-2k_{T}^{2}s_{23}-2s_{23}s_{45}-2s_{45}k_{T}^{2}~{}.$
(99)

Also for the terms proportional to $1/\kappa$ in
${-}\mathcal{V}^{\,\mathrm{sl},\star}\mathcal{A}^{\mathrm{tree},\star}+\mathcal{F}^{\,\mathrm{sl},\star}$
we find the correct on-shell limit of $|k_{T}|\to 0$, matching the expressions
in Appendix IV of [51].

The amplitude $\mathcal{A}_{3}$ in Eq. (81) is the axial vector quark triangle
contribution, and given by

$\displaystyle\mathcal{A}_{3}(1^{\star},2_{\bar{q}},3_{q},4_{e^{+}},5_{e^{-}})$
$\displaystyle\hskip
17.22217pt=\frac{2}{(4\pi)^{2}}\frac{f(m_{t};s_{23},k_{T}^{2},s_{45})-f(m_{b};s_{23},k_{T}^{2},s_{45})}{s_{45}}\bigg{(}\frac{[5p]\langle
p2\rangle\langle 24\rangle}{\langle 23\rangle}-\frac{[53][3p]\langle
p4\rangle}{[23]}\bigg{)}$ $\displaystyle\hskip
17.22217pt+\frac{2}{(4\pi)^{2}}\frac{f(m_{t};k_{T}^{2},s_{23},s_{45})-f(m_{b};k_{T}^{2},s_{23},s_{45})}{s_{45}}\bigg{(}\frac{[53]\langle
2p\rangle\langle p4\rangle}{\kappa^{*}}-\frac{[5p][p3]\langle
24\rangle}{\kappa}\bigg{)}~{},$ (100)

with

$\displaystyle
f(m;s,t,u)=\int_{0}^{1}da\int_{0}^{1}db\int_{0}^{1}dc\,\delta(1-a-b-c)\,\frac{bc}{m^{2}-sab-
tbc-uca}~{}.$ (101)

This function simplifies for $m=0$, and can be expanded in $1/m^{2}$ for
$m\to\infty$. The expressions can be found in [51]. We do mention that when
$|k_{T}|\to 0$, only the last line in Eq. (100) contributes, and we have

$f(m\to\infty;0,s_{23},s_{45})-f(0;0,s_{23},s_{45})=-\frac{1}{2s_{45}}\,\mathrm{L}_{1}\bigg{(}\frac{-s_{23}}{-s_{45}}\bigg{)}+\frac{1}{24m^{2}}+\mathcal{O}\big{(}1/m^{4}\big{)}~{}.$
(102)

##  4 Conclusions

In this paper, we explicitly calculate the virtual contributions for the one-
jet processes $\emptyset\to g^{\star}gg,\emptyset\to g^{\star}q\bar{q}$,
$\emptyset\to g^{\star}gH$ and $\emptyset\to g^{\star}\bar{q}qe^{+}e^{-}$, and
we check their collinear limit. The off-shell scattering amplitudes for these
processes have been systematically extracted from the on-shell embedding
processes available in the literature according to the auxiliary parton
method. The main result of this work, besides providing the explicit
expressions the off-shell scattering amplitudes corresponding to the mentioned
processes, is to prove, process by process, the conjecture developed in [1]
for the structure of the NLO virtual contribution in hybrid
$k_{T}$-factorization. Our explicit calculations of the NLO virtual
contributions to the parton level cross-sections of the mentioned processes
completely agree with the conjecture formulated in Ref. [1]. Furthermore, the
explicit results obtained here will be useful to the theoretical community for
the cross-section calculations at NLO within hybrid $k_{T}$-factorization
scheme and will play an important role toward the establishment of a higher
precision description of small $x$ phenomena. Our explicit calculations of the
virtual corrections to the $\emptyset\to g^{\star}gg,\emptyset\to
g^{\star}q\bar{q},\emptyset\to g^{\star}gH$ and $\emptyset\to
g^{\star}\bar{q}qe^{+}e^{-}$ LO cross sections show how to systematically
derive the one loop amplitudes from the on-shell helicity amplitudes. This is
the core of our investigation: to show with explicit examples that our method
can be successfully extended to NLO and to provide the guiding principles to
extract the virtual corrections from the on-shell amplitudes. The computation
of the real emission corrections still remains to be done and it is the
subject of our present investigation. Our calculation scheme for the virtual
corrections, which has been presented in this work, together with a general
framework for computing the real corrections, which is under investigation, is
an essential step forward toward the implementation of a numerical code for an
event simulation generator, based on $k_{T}$ factorization, which can handle
NLO calculations and which is therefore able to provide a significant
contribution in the rich field of small-x physics.

### Acknowledgments

EB acknowledge partial support by NCN grant No. DEC-2017/27/B/ST2/01985. AG
and AvH are supported by grant no. 2019/35/B/ST2/03531 of the Polish National
Science Centre. PK is supported by the Polish National Science Centre, grant
no. 2020/39/O/ST2/03011.

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##  Appendix A Special functions

We use the same functions as defined in for example [51]. There is

$\mathrm{Li}_{2}(x)=-\int_{0}^{x}dz\,\frac{\ln(1-z)}{z}~{},$ (103)

which satisfies the useful relation

$\mathrm{Li}_{2}\big{(}1-x^{-1}\big{)}=-\mathrm{Li}_{2}(1-x)-\frac{1}{2}\ln^{2}(x)~{}.$
(104)

Then there are

$\mathrm{L}_{0}(x)=\frac{\ln(x)}{1-x}\quad,\quad\mathrm{L}_{1}(x)=\frac{\ln(x)+1-x}{(1-x)^{2}}~{},$
(105)

which satisfy the useful relations

$\mathrm{L}_{0}\big{(}x^{-1}\big{)}=x\,\mathrm{L}_{0}(x)\quad,\quad\mathrm{L}_{1}\big{(}x^{-1}\big{)}=-x^{2}\,\mathrm{L}_{1}(x)-x~{}.$
(106)

The function

$\displaystyle\mathrm{Ls}_{-1}(x,y)$
$\displaystyle=\mathrm{Li}_{2}(1-x)+\mathrm{Li}_{2}(1-y)+\ln(x)\ln(y)-\frac{\pi^{2}}{6}$
(107)

does not show up in any of our expressions, but will appear in the limit of a
function we do use and present below. First, however, we present

$\mathrm{I}_{3}^{3m}(x,y,z)=\int_{0}^{1}da\int_{0}^{1}db\int_{0}^{1}dc\,\delta(1-a-b-c)\,\frac{-1}{xab+ybc+zca}~{},$
(108)

which is symmetric in its arguments, and satisfies the limit

$\mathrm{I}_{3}^{3m}(x,y,z\to
0)=\frac{1}{x-y}\bigg{[}\ln\frac{x}{y}\ln\frac{z}{y}+2\mathrm{Li}_{2}\bigg{(}1-\frac{x}{y}\bigg{)}\bigg{]}+\mathcal{O}(z)~{}.$
(109)

And finally, we use

$\displaystyle\mathrm{Ls}_{-1}^{2mh}(x_{1},x_{2},y_{1},y_{2})$
$\displaystyle=-\mathrm{Li}_{2}\bigg{(}1-\frac{y_{1}}{x_{2}}\bigg{)}-\mathrm{Li}_{2}\bigg{(}1-\frac{y_{2}}{x_{2}}\bigg{)}-\frac{1}{2}\ln^{2}\bigg{(}\frac{-x_{1}}{-x_{2}}\bigg{)}$
(110)
$\displaystyle+\frac{1}{2}\ln\bigg{(}\frac{-x_{1}}{-y_{1}}\bigg{)}\ln\bigg{(}\frac{-x_{1}}{-y_{2}}\bigg{)}+\frac{1}{2}\bigg{[}\frac{x_{1}-y_{1}-y_{2}}{2}+\frac{y_{1}y_{2}}{x_{2}}\bigg{]}\mathrm{I}_{3}^{3m}(x_{1},y_{1},y_{2})~{},$

which satisfies the limit

$\mathrm{Ls}_{-1}^{2mh}(x_{1},x_{2},y,0)=\mathrm{Ls}_{-1}\bigg{(}\frac{x_{1}}{y},\frac{x_{2}}{y}\bigg{)}~{}.$
(111)

##  Appendix B $\bm{\emptyset}\bm{\to}\bm{g}^{\bm{\star}}\bm{g}\bm{g}$ from
auxiliary gluons

Given the general color decomposition of Eq. (19) and the fact that partial
amplitudes with non-adjacent auxiliary gluons do not contribute, the tree-
level decomposition can be written in a compact notation as

$\mathcal{M}^{\mathrm{tree}}(A,2,3,B)=T_{A23B}\mathcal{A}^{\mathrm{tree}}_{A23B}+T_{A32B}\mathcal{A}^{\mathrm{tree}}_{A32B}+T_{B23A}\mathcal{A}^{\mathrm{tree}}_{B23A}+T_{B32A}\mathcal{A}^{\mathrm{tree}}_{B32A}~{},$
(112)

where

$T_{ijkl}=\mathrm{Tr}\big{(}T^{a_{i}}T^{a_{j}}T^{a_{k}}T^{a_{l}}\big{)}\quad\textrm{and}\quad\mathcal{A}^{\mathrm{tree}}_{ijkl}=\mathcal{A}^{\mathrm{tree}}(i,j,k,l)~{}.$
(113)

This can be reorganized as

$\displaystyle\mathcal{M}^{\mathrm{tree}}(A,2,3,B)$
$\displaystyle=\big{(}T_{A23B}-T_{A32B}-T_{B23A}+T_{B32A}\big{)}\mathcal{A}^{\mathrm{tree}}_{A23B}+T_{A32B}\big{(}\mathcal{A}^{\mathrm{tree}}_{A23B}+\mathcal{A}^{\mathrm{tree}}_{A32B}\big{)}$
$\displaystyle\hskip
68.88867pt+T_{B23A}\big{(}\mathcal{A}^{\mathrm{tree}}_{A23B}+\mathcal{A}^{\mathrm{tree}}_{B23A}\big{)}+T_{B32A}\big{(}\mathcal{A}^{\mathrm{tree}}_{B32A}-\mathcal{A}^{\mathrm{tree}}_{A23B}\big{)}$
$\displaystyle=\big{(}\sqrt{2}\,\mathrm{i}f^{BAc}\big{)}\big{(}\sqrt{2}\,\mathrm{i}f^{23c}\big{)}\mathcal{A}^{\mathrm{tree}}_{A23B}+T_{A32B}\big{(}\mathcal{A}^{\mathrm{tree}}_{A23B}+\mathcal{A}^{\mathrm{tree}}_{A32B}\big{)}$
$\displaystyle\hskip
68.88867pt+T_{B23A}\big{(}\mathcal{A}^{\mathrm{tree}}_{A23B}+\mathcal{A}^{\mathrm{tree}}_{B23A}\big{)}+T_{B32A}\big{(}\mathcal{A}^{\mathrm{tree}}_{B32A}-\mathcal{A}^{\mathrm{tree}}_{A23B}\big{)}~{}.$
(114)

The pairs of tree-level amplitudes appearing with the same trace structure
vanish in the $\Lambda$ limit, and we are left with the same result as with
auxiliary quarks. At one loop we have, following Eq. (21) and again including
only amplitudes for which the auxiliary partons are adjacent, the color
decomposition

$\displaystyle\mathcal{M}(A,2,3,B)$
$\displaystyle=N_{\mathrm{c}}\big{(}T_{A23B}-T_{A32B}-T_{B23A}+T_{B32A}\big{)}\mathcal{A}_{A23B}+N_{\mathrm{c}}T_{A32B}\big{(}\mathcal{A}_{A23B}+\mathcal{A}_{A32B}\big{)}$
$\displaystyle\hskip
34.44434pt+N_{\mathrm{c}}T_{B23A}\big{(}\mathcal{A}_{A23B}+\mathcal{A}_{B23A}\big{)}+N_{\mathrm{c}}T_{B32A}\big{(}\mathcal{A}_{B32A}-\mathcal{A}_{A23B}\big{)}$
$\displaystyle\hskip
68.88867pt+T_{AB}T_{23}\mathcal{A}_{AB;23}+T_{A2}T_{B3}\mathcal{A}_{A2;B3}+T_{A3}T_{2B}\mathcal{A}_{A3;2B}~{},$
(115)

with

$T_{ij}=\mathrm{Tr}\big{(}T^{a_{i}}T^{a_{j}}\big{)}=\delta^{a_{i}a_{j}}\quad\textrm{and}\quad\mathcal{A}_{ijkl}=\mathcal{A}_{1}(i,j,k,l)\quad\textrm{and}\quad\mathcal{A}_{ij;kl}=\mathcal{A}_{3}(i,j,k,l)~{}.$
(116)

The amplitudes $\mathcal{A}_{3}$ can be expressed in terms of
$\mathcal{A}_{1}$ following Eq. (6.6) in [48], and including only the ones
that survive the $\Lambda$-limit, we have

$\mathcal{A}_{AB;23}=\mathcal{A}_{A2;B3}=\mathcal{A}_{A3;2B}=\mathcal{A}_{A23B}+\mathcal{A}_{A32B}+\mathcal{A}_{B23A}+\mathcal{A}_{B32A}~{}.$
(117)

Notice that this would vanish at tree level, as expected. At one loop, the
combinations do not quite vanish, and we find

$\displaystyle\mathcal{A}_{A23B}+\mathcal{A}_{A32B}$
$\displaystyle\overset{\Lambda}{\longrightarrow}-2\Upsilon(s_{p2})\mathcal{A}^{\mathrm{Rt}}(1^{\star},2,3)$
(118) $\displaystyle\mathcal{A}_{A23B}+\mathcal{A}_{B23A}$
$\displaystyle\overset{\Lambda}{\longrightarrow}-2\Upsilon(s_{p2})\mathcal{A}^{\mathrm{Rt}}(1^{\star},2,3)$
(119) $\displaystyle\mathcal{A}_{B32A}-\mathcal{A}_{A23B}$
$\displaystyle\overset{\Lambda}{\longrightarrow}0~{}.$ (120)

Furthermore, we find

$\mathcal{A}_{A23B}\overset{\Lambda}{\longrightarrow}\mathcal{A}^{\mathrm{re}}(1^{\star},2,3)-\Upsilon(s_{p2})\mathcal{A}^{\mathrm{Rt}}(1^{\star},2,3)$
(121)

with $\mathcal{A}^{\mathrm{re}}$ from Eq. (51), and thus

$\displaystyle\mathcal{M}(A,2,3,B)$
$\displaystyle\overset{\Lambda}{\longrightarrow}N_{\mathrm{c}}\big{(}T_{A23B}-T_{A32B}-T_{B23A}+T_{B32A}\big{)}\mathcal{A}^{\mathrm{re}}(1^{\star},2,3)$
$\displaystyle\hskip
34.44434pt-\Big{[}N_{\mathrm{c}}\big{(}T_{A23B}-T_{A32B}-T_{B23A}+T_{B32A}\big{)}+2N_{\mathrm{c}}\big{(}T_{A32B}+T_{B23A}\big{)}$
$\displaystyle\hskip
86.11084pt+4\big{(}T_{AB}T_{23}+T_{A2}T_{B3}+T_{A3}T_{2B}\big{)}\Big{]}\Upsilon(s_{p2})\mathcal{A}^{\mathrm{Rt}}(1^{\star},2,3)$
$\displaystyle\hskip
8.61108pt=\big{(}\sqrt{2}\,\mathrm{i}f^{BAc}\big{)}\big{(}\sqrt{2}\,\mathrm{i}f^{23c}\big{)}N_{\mathrm{c}}\mathcal{A}^{\mathrm{re}}(1^{\star},2,3)$
$\displaystyle\hskip
34.44434pt-\Big{[}N_{\mathrm{c}}\big{(}T_{A23B}+T_{A32B}+T_{B23A}+T_{B32A}\big{)}$
$\displaystyle\hskip
68.88867pt+4\big{(}T_{AB}T_{23}+T_{A2}T_{B3}+T_{A3}T_{2B}\big{)}\Big{]}\Upsilon(s_{p2})\mathcal{A}^{\mathrm{Rt}}(1^{\star},2,3)~{}.$
(122)

The last two lines are symmetric in $A,B$ and $2,3$, and do not contribute at
NLO.

##  Appendix C $\bm{\emptyset}\bm{\to}\bm{g}^{\bm{\star}}\bar{\bm{q}}\bm{q}$
from auxiliary quarks

From [53], we have the formula

$\displaystyle\mathcal{M}(2_{\bar{q}},B_{\bar{Q}},A_{Q},3_{q})$
$\displaystyle=\bigg{(}\delta^{\imath_{B}}_{i_{3}}\delta^{\imath_{2}}_{i_{A}}-\frac{1}{N_{\mathrm{c}}}\delta^{\imath_{2}}_{i_{3}}\delta^{\imath_{B}}_{i_{A}}\bigg{)}a_{1}(2,B;A,3)+\delta^{\imath_{B}}_{i_{3}}\delta^{\imath_{2}}_{i_{A}}\,a_{2}(2,B;A,3)$
$\displaystyle=\bigg{(}\delta^{\imath_{B}}_{i_{3}}\delta^{\imath_{2}}_{i_{A}}-\frac{1}{N_{\mathrm{c}}}\delta^{\imath_{2}}_{i_{3}}\delta^{\imath_{B}}_{i_{A}}\bigg{)}\big{[}a_{1}(2,B;A,3)+a_{2}(2,B;A,3)\big{]}$
(123) $\displaystyle\hskip
51.6665pt+\frac{1}{N_{\mathrm{c}}}\delta^{\imath_{2}}_{i_{3}}\delta^{\imath_{B}}_{i_{A}}\,a_{2}(2,B;A,3)~{},$

where the last line vanishes upon contraction with the tree-level color
structure. For the partial amplitudes, we find, after “un-doing” the UV
subtraction by putting $\beta_{0}=0$,

$a_{1}(2,B;A,3)+a_{2}(2,B;A,3)\overset{\Lambda}{\longrightarrow}\mathcal{A}^{\mathrm{re}}(1^{\star},2_{\bar{q}},3_{q})-\bigg{(}1-\frac{4}{N_{\mathrm{c}}^{2}}\bigg{)}\Upsilon(s_{p2})\mathcal{A}^{\mathrm{Rt}}(1^{\star},2_{\bar{q}},3_{q})~{},$
(124)

with $\mathcal{A}^{\mathrm{re}}$ from Eq. (65). Again, with the same reasoning
as argued there, we see that the contributions proportional to $\Upsilon$
vanish in the combination of Eq. (38).

##  Appendix D
$\bm{\emptyset}\bm{\to}\bm{g}^{\bm{\star}}\bar{\bm{q}}\bm{q}\bm{e}^{\bm{+}}\bm{e}^{\bm{-}}$
from auxiliary gluons

We omit the arguments $(4_{e^{+}},5_{e^{-}})$ from all amplitudes here. The
tree-level partial amplitudes satisfy

$\displaystyle\mathcal{A}^{\mathrm{tree}}(3_{q},A,B,2_{\bar{q}})$
$\displaystyle\overset{\Lambda}{\longrightarrow}\mathcal{A}^{\mathrm{tree}}(1^{\star},2_{\bar{q}},3_{q})~{},$
(125) $\displaystyle\mathcal{A}^{\mathrm{tree}}(3_{q},B,A,2_{\bar{q}})$
$\displaystyle\overset{\Lambda}{\longrightarrow}-\mathcal{A}^{\mathrm{tree}}(1^{\star},2_{\bar{q}},3_{q})~{},$
(126)

so, from Eq. (2.6) in [51] it is clear that the tree-level amplitude is
proportional to
$f^{a_{A}a_{B}a_{1}}\big{(}T^{a_{1}}\big{)}_{i_{3}}^{\imath_{2}}$. Thus, the
only terms of the one-loop amplitude of Eq. (2.9) in [51] that are relevant
and survive contraction with the tree-level color structure are

$\displaystyle\mathcal{M}^{\mathrm{relevant}}(3_{q},A,B,2_{\bar{q}})$
$\displaystyle=c_{1}N_{\mathrm{c}}\Big{\\{}\big{(}T^{a_{A}}T^{a_{B}}\big{)}_{i_{3}}^{\imath_{2}}\mathcal{A}_{1}(3_{q},A,B,2_{\bar{q}})+\big{(}T^{a_{B}}T^{a_{A}}\big{)}_{i_{3}}^{\imath_{2}}\mathcal{A}_{1}(3_{q},B,A,2_{\bar{q}})\Big{\\}}$
$\displaystyle+c_{3}\Big{\\{}\big{(}T^{a_{A}}T^{a_{B}}\big{)}_{i_{3}}^{\imath_{2}}\mathcal{A}^{\mathrm{ax}}_{4}(3_{q},2_{\bar{q}};A,B)+\big{(}T^{a_{B}}T^{a_{A}}\big{)}_{i_{3}}^{\imath_{2}}\mathcal{A}^{\mathrm{ax}}_{4}(3_{q},2_{\bar{q}};B,A)\Big{\\}}~{}.$
(127)

This can be reorganized to

$\displaystyle\mathcal{M}^{\mathrm{relevant}}(3_{q},A,B,2_{\bar{q}})$
$\displaystyle=f^{a_{A}a_{B}a_{1}}\big{(}T^{a_{1}}\big{)}_{i_{3}}^{\imath_{2}}\bigg{\\{}c_{1}N_{\mathrm{c}}\mathcal{A}_{1}(3_{q},A,B,2_{\bar{q}})+c_{3}\mathcal{A}^{\mathrm{ax}}_{4}(3_{q},2_{\bar{q}};A,B)$
$\displaystyle\hskip
60.27759pt-\frac{c_{1}N_{\mathrm{c}}}{2}\Big{[}\mathcal{A}_{1}(3_{q},A,B,2_{\bar{q}})+\mathcal{A}_{1}(3_{q},B,A,2_{\bar{q}})\Big{]}$
(128) $\displaystyle\hskip
68.88867pt-\frac{c_{3}}{2}\Big{[}\mathcal{A}^{\mathrm{ax}}_{4}(3_{q},2_{\bar{q}};A,B)+\mathcal{A}^{\mathrm{ax}}_{4}(3_{q},2_{\bar{q}};B,A)\Big{]}\bigg{\\}}~{},$

where we omitted contributions proportional to
$\big{[}\big{(}T^{a_{A}}T^{a_{B}}\big{)}_{i_{3}}^{\imath_{2}}+\big{(}T^{a_{B}}T^{a_{A}}\big{)}_{i_{3}}^{\imath_{2}}\big{]}$,
because they vanish upon contraction with the tree-level color structure.

Now, it is the second line that would cause a difference with the result
obtained with auxiliary quarks. It is the equivalent of the second line of Eq.
(82), and one can immediately see that the factors $2/N_{\mathrm{c}}^{2}$ and
$N_{\mathrm{c}}/2$ do not match. This contribution above clearly becomes
symmetric in $A,B$ if summed over the auxiliary parton helicities, and then
vanishes in the combination of Eq. (39).

The decomposition of $\mathcal{A}_{1}$ into primitive amplitudes from [51] is

$\displaystyle\mathcal{A}_{1}(3_{q},A,B,2_{\bar{q}})$
$\displaystyle=\mathcal{A}(3_{q},A,B,2_{\bar{q}})-\frac{1}{N_{\mathrm{c}}^{2}}\mathcal{A}(3_{q},2_{\bar{q}},B,A)$
$\displaystyle\hskip
73.19421pt-\frac{n_{f}}{N_{\mathrm{c}}}\big{[}\mathcal{A}^{f}(3_{q},A,B,2_{\bar{q}})+\mathcal{A}^{s}(3_{q},A,B,2_{\bar{q}})\big{]}~{},$
(129)

where we omitted the top-loop contributions that behave as
$|k_{T}|^{2}/m_{t}^{2}$ again. We confirm that, referring to the amplitudes in
Eq. (85),

$\displaystyle c_{\Gamma}^{-1}$
$\displaystyle\mathcal{A}(3_{q}^{+},A^{+},B^{-},2_{\bar{q}}^{-})\overset{\Lambda}{\longrightarrow}\mathcal{V}^{\,\star}\mathcal{A}^{\mathrm{tree},\star}+\mathcal{F}^{\,\mathrm{re},\star}+\Theta(s_{p2})\mathcal{A}^{\mathrm{Rt},\star}+\Upsilon(s_{p2})\mathcal{F}^{\,\mathrm{im},\star}~{},$
(130) $\displaystyle c_{\Gamma}^{-1}$
$\displaystyle\mathcal{A}(3_{q}^{+},2_{\bar{q}}^{-},B^{-},A^{+})\overset{\Lambda}{\longrightarrow}\mathcal{V}^{\,\mathrm{sl},\star}\mathcal{A}^{\mathrm{tree},\star}-\mathcal{F}^{\,\mathrm{sl},\star}~{},$
(131)
$\displaystyle\mathcal{A}^{f}(3_{q}^{+},A^{+},B^{-},2_{\bar{q}}^{-})+\mathcal{A}^{s}(3_{q}^{+},A^{+},B^{-},2_{\bar{q}}^{-})\overset{\Lambda}{\longrightarrow}0~{}.$
(132)

We also confirm

$\mathcal{A}_{4}^{\mathrm{ax}}(3_{q}^{+},2_{\bar{q}}^{-};A^{+},B^{-})+\mathcal{A}_{4}^{\mathrm{ax}}(3_{q}^{+},2_{\bar{q}}^{-},B^{-};A^{+})\overset{\Lambda}{\longrightarrow}0~{},$
(133)

and we confirm

$\mathcal{A}_{4}^{\mathrm{ax}}(3_{q}^{+},2_{\bar{q}}^{-};A^{+},B^{-})\overset{\Lambda}{\longrightarrow}\mathcal{A}_{3}(1^{\star},2_{\bar{q}}^{-},3_{q}^{+})$
(134)

in the large $m_{t}$ and $m_{b}=0$ approximation. The latter involves the
expression for $C^{\mathrm{ax}}$ in Eq. (11.8) of [51], which is the only one
we encountered with individual terms that behave as $\Lambda^{2}$, while the
whole expressions behaves as $\Lambda$.

##  Appendix E Spinor helicity method

In presenting our results we use the well-known spinor helicity formalism,
which is the basis language of modern scattering amplitude calculations. Let
$u(p)$ and $v(p)$ respectively denote the positive and negative energy
solutions to the massless Dirac equation

$p/u(p)=p/v(p)=0\,,$ (135)

where the slash notation stands for a contraction with the gamma matrices,
$p/\equiv\gamma_{\mu}p^{\mu}$, the latter obeying to the Clifford algebra,
$\\{\gamma_{\mu},\gamma_{\nu}\\}=2g_{\mu\nu}$, and $g_{\mu\nu}$ is the
Minkowski metric with signature $(1,-1,-1,-1)$. When dealing with massless
spinors an explicit useful realization of the gamma matrices is the Weyl
representation:

$\gamma_{\mu}=\left(\begin{array}[]{cc}0&\sigma_{\mu}\\\
\bar{\sigma}_{\mu}&0\end{array}\right),\;\;\;\gamma^{5}=i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}=\left(\begin{array}[]{cc}-I_{2}&0\\\
0&I_{2}\end{array}\right)$ (136)

where $\sigma_{\mu}=(I_{2},\vec{\sigma})$ and
$\bar{\sigma}_{\mu}=(I_{2},-\vec{\sigma})$ and
$\vec{\sigma}=(\sigma_{1},\sigma_{2},\sigma_{3})$ are the standard Pauli
matrices, defined as:

$\sigma_{1}=\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right),\;\;\;\sigma_{2}=\left(\begin{array}[]{cc}0&-i\\\
i&0\end{array}\right),\;\;\;\sigma_{3}=\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right)\;.$ (137)

We can use the matrix $\gamma_{5}$ to construct the projection operators onto
the upper and lower parts the four-component Dirac spinors $u(p)$ and $v(p)$:

$u_{\pm}(p)=\frac{1\pm\gamma_{5}}{2}u(p),\;\;\;\text{and}\;\;\;v_{\pm}(p)=\frac{1\mp\gamma_{5}}{2}v(p)\,,$
(138)

where $u_{\pm}(p)$ are and $v_{\pm}(p)$ are the solutions of the massless
Dirac equation with definite helicity, which is the projection of the spin
along the particle momentum. Since in the massless limit $u_{+}(p)=v_{-}(p)$
and $u_{-}(p)=v_{+}(p)$ it is useful to introduce the following short-hand
notation:

$|p]\equiv u_{-}(p)=\left(\begin{array}[]{c}L(p)\\\
0\end{array}\right)\;\;\text{and}\;\;|p\rangle\equiv
u_{+}(p)=\left(\begin{array}[]{c}0\\\ R(p)\end{array}\right)$ (139)

where $L(p)$ and $R(p)$ are found by imposing that $u_{+}(p)$ and $u_{-}(p)$
satisfy the massless Dirac equation, Eq. 135, and they can be explicitly
written in the Weyl representation of the gamma matrices as follows [54]:

$L(p)=\frac{1}{\sqrt{|p_{0}+p_{3}|}}\left(\begin{array}[]{c}-p_{1}+ip_{2}\\\
p_{0}+p_{3}\end{array}\right),\;\;\;\;\;\;R(p)=\frac{\sqrt{|p_{0}+p_{3}|}}{p_{0}+p_{3}|}\left(\begin{array}[]{c}p_{0}+p_{3}\\\
p_{1}+ip_{2}\end{array}\right)\,.$ (140)

The “dual” Weyl spinors are defined as [54]:

$[p|=((\varepsilon L(p))^{T},0),\;\;\;\langle
p|=(0,(\varepsilon^{T}R(p))^{T}),\;\;\;\text{where}\;\;\;\varepsilon=\left(\begin{array}[]{cc}0&1\\\
-1&0\end{array}\right)\,.$ (141)

We use the following conventions for the spinor products:

$\langle kl\rangle=\bar{u}_{-}(k)u_{+}(l);\;\;[kl]=\bar{u}_{+}(k)u_{-}(l)\,.$
(142)

Observe that the spinor products are, up to a phase, square roots of Lorentz
products:

$\langle
kl\rangle=\sqrt{2kl}e^{i\eta(k,l)}\,,\;\;\;[lk]=\sqrt{2kl}e^{-i\eta(k,l)}$
(143)

which implies the very useful relation

$\langle kl\rangle[lk]=2kl=s_{kl}\,.$ (144)

Note that the mapping between a light-like vector and a Weyl spinor is non-
linear: the spinor $|p\rangle+|q\rangle$ is another massless spinor, but it
cannot be used to represent the four-vector $p+q$, which is not light-like. On
the other hand we have the linear maps:

$|p\rangle[p|=\frac{1+\gamma_{5}}{2}p/,\;\;\;\text{and}\;\;\;|p]\langle
p|=\frac{1-\gamma_{5}}{2}p/$ (145)

which implies another very useful relation

$|p\rangle[p|+|p]\langle p|=p/\,.$ (146)

The spinors products also satisfy the Gordon identity:

$\langle i|\gamma_{\mu}|i]=2p_{\mu},$ (147)

antisymmetry:

$\langle ij\rangle=-\langle ji\rangle,\;\;\;[ij]=-[ji],\;\;\;\langle
ii\rangle=[ii]=0,$ (148)

Fierz rearrangement:

$\langle i|\gamma_{\mu}|j]\langle k|\gamma^{\mu}|l]=\langle ik\rangle[jl],$
(149)

and the Schouten identities:

$\langle pq\rangle\langle kl\rangle=\langle pk\rangle\langle ql\rangle+\langle
pl\rangle\langle kq\rangle,\;\;\;[pq][kl]=[pk][ql]+[pl][kq]\,.$ (150)

Given two light-like vectors, $p$ (momentum of a massless particle) and $n$
(any external light-like vector), the polarisation vector for a gluon with a
given helicity, $\epsilon^{\mu}_{\lambda}(p)$, and momentum $p$, can be
expressed as spinor products as follows [54]:

$\epsilon_{+}^{\mu}(p)=\frac{\langle n|\gamma^{\mu}|p]}{\sqrt{2}\langle
np\rangle},\;\;\;\epsilon_{-}^{\mu}(p)=\frac{\langle
p|\gamma^{\mu}|n]}{\sqrt{2}[pn]}$ (151)

A good choice of the auxiliary vectors $n$ can simplify the calculations of
helicity amplitudes enormously.