aainstitutetext:  Dipartimento di Fisica and Arnold-Regge Center, Università
di Torino, and INFN, Sezione di Torino, Via P. Giuria 1, I-10125 Torino, Italy
bbinstitutetext:  Cavendish Laboratory, University of Cambridge, Cambridge CB3
0HE, United Kingdom ccinstitutetext:  Institute for Particle Physics
Phenomenology, Department of Physics, Durham University, Durham DH1 3LE,
United Kingdom

# Two-loop leading colour helicity amplitudes for $W^{\pm}\gamma+j$ production
at the LHC

Simon Badger b Heribertus Bayu Hartanto a,c Jakub Kryś a Simone Zoia
simondavid.badger@unito,it<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We present the two-loop leading colour QCD helicity amplitudes for the process
$pp\to W(\to l\nu)\gamma+j$. We implement a complete reduction of the
amplitudes, including the leptonic decay of the $W$-boson, using finite field
arithmetic, and extract the analytic finite remainders using a recently
identified basis of special functions. Simplified analytic expressions are
obtained after considering permutations of a rational kinematic
parametrisation and multivariate partial fractioning. We demonstrate efficient
numerical evaluation of the two-loop colour and helicity summed finite
remainders for physical kinematics, and hence the suitability for
phenomenological applications.

††preprint: CAVENDISH-HEP-22/01

##  1 Introduction

High-precision theoretical predictions are a high priority for the current
experiments at the Large Hadron Collider (LHC). Processes with a pair of
electroweak vector bosons $(W^{\pm},Z,\gamma)$ offer a wide range of
observables which can test electroweak couplings and probe the Higgs sector of
the Standard Model. In particular, the production of a $W$ boson in
association with a photon ($pp\to W\gamma$) is one of the processes observed
at the LHC with relatively large cross sections where clean signatures can be
acquired when the $W$ boson decays leptonically. $W\gamma$ production enables
direct access to the $WW\gamma$ triple gauge boson coupling, which can be
modified in certain new physics scenarios. Both ATLAS and CMS experiments have
measured the $W\gamma$ process CMS:2011myh ; ATLAS:2011nmx ; ATLAS:2012bpb ;
ATLAS:2013way ; CMS:2013ryd ; CMS:2021foa  and set the limit on the anomalous
$WW\gamma$ coupling.

Predictions for $pp\to W^{\pm}\gamma$ are available through to next-to-next-
to-leading order (NNLO) in QCD Gehrmann:2011ab ; Grazzini:2015nwa ;
Campbell:2021mlr  and NLO in the electroweak (EW) coupling Accomando:2005ra ;
Denner:2014bna  as well as combined NNLO in QCD and NLO in EW Grazzini:2019jkl
. The colourless final state makes the process well suited for the $q_{T}$
Catani:2007vq  and $N$-jettiness Boughezal:2015aha ; Gaunt:2015pea
subtraction methods as implemented within the Matrix Grazzini:2017mhc  and
MCFM Boughezal:2016wmq  Monte Carlo event generators, respectively. Resummed
predictions including parton shower effects are now available Cridge:2021hfr ,
making this one of the most precisely known theoretical predictions.
Experimental measurements are constantly improving and provide rich
opportunities for precision SM tests CMS:2021foa ; CMS:2021cxr . In order to
suppress different types of backgrounds in the experimental analysis, it is a
common practice to divide the measurement according to the jet multiplicities,
i.e. $W\gamma$+0 jet, $W\gamma$+1 jet, $W\gamma$+2 jets, etc. Increasing the
precision of the theoretical predictions for each of the jet bins amounts to
computing higher order corrections to $W\gamma$ production in association with
additional jets.

In this article we compute the two-loop helicity amplitudes for the process
$pp\to W^{\pm}(\to l^{\pm}\nu)\gamma+j$ for the first time. The amplitude-
level ingredients we provide will give useful information for future precision
measurements of anomalous couplings and potentially for complete global fits
of the Standard Model Effective Theory (SMEFT). A fully differential
computation of $W\gamma+j$ at NNLO in QCD would also open up the possibility
of N3LO QCD predictions for $W\gamma$ production.

The process has already been well studied at NLO in QCD Campanario:2009um ,
including the anomalous couplings Campanario:2010hv , and is easily within the
reach of standard automated tools including electroweak corrections. Compact
analytic formulae at one loop have been obtained recently Campbell:2021mlr .

Recent years have seen tremendous progress in the analytic computation of two-
loop amplitudes for processes involving five massless particles
Gehrmann:2015bfy ; Badger:2018enw ; Abreu:2018aqd ; Chicherin:2018yne ;
Chicherin:2019xeg ; Abreu:2019rpt ; Abreu:2018zmy ; Abreu:2018jgq ;
Abreu:2019odu ; Badger:2019djh ; Abreu:2020cwb ; Chawdhry:2020for ; Caron-
Huot:2020vlo ; Chicherin:2020oor ; DeLaurentis:2020qle ; Agarwal:2021grm ;
Abreu:2021oya ; Chawdhry:2021mkw ; Agarwal:2021vdh ; Badger:2021imn . The
methods are now reaching maturity also for two-loop five-particle amplitudes
with an external off-shell leg, with the first sets of analytic helicity
amplitudes in the planar limit appearing over the last 12 months
Badger:2021nhg ; Badger:2021ega ; Abreu:2021asb . This progress has been made
possible by the classification of the special functions appearing in the
finite remainders through the differential equations they satisfy
Papadopoulos:2015jft ; Papadopoulos:2019iam ; Abreu:2021smk ; Abreu:2020jxa ;
Canko:2020ylt ; Syrrakos:2020kba ; Chicherin:2021dyp . Furthermore, finite-
field arithmetic for scattering amplitudes vonManteuffel:2014ixa ;
Peraro:2016wsq ; Peraro:2019svx  allows us to avoid the large intermediate
expressions and provides efficient solutions to large systems of integration-
by-parts integral identities Tkachov:1981wb ; Chetyrkin:1981qh ;
Laporta:2000dsw , which are the main bottlenecks in the analytic computation
of multi-loop scattering amplitudes.

Our paper is organised as follows. In Section 2 we describe the structure of
the amplitudes for $pp\to W\gamma j$ up to two loops, paying particular
attention to the description of the decay of the $W$ boson. In Section 3 we
describe the finite field reduction setup used to extract the finite
remainders, and propose an approach to simplify dramatically the analytic
expressions of the latter based on a systematic search of a better
parameterisation of the kinematics in terms of momentum-twistor variables. We
describe a number of validation tests that have been performed on our results
in Section 4, and then present some numerical results for the colour and
helicity summed finite remainders in Section 5. We present our conclusions and
outlook for the future in Section 6. Complete analytic expressions are
provided in the associated ancillary files on the arXiv. In addition, we
include appendices describing the details of the renormalisation constants,
and giving explicit one-loop results to facilitate future cross-checks.

##  2 Structure of the Amplitudes

We compute the two-loop amplitudes for the production of a $W^{+}$ boson in
association with a photon and a jet at hadron colliders, where the $W^{+}$
boson decays to a positron and an electron neutrino ($pp\to\nu_{e}e^{+}\gamma
j$), in the leading colour approximation,

$0\rightarrow\gamma(p_{1},h_{1})+\bar{u}(p_{2},h_{2})+g(p_{3},h_{3})+d(p_{4},h_{4})+\nu_{e}(p_{5},h_{5})+e^{+}(p_{6},h_{6})\,.$
(1)

For simplicity we denote this process as $W^{+}\gamma j$ production
henceforth. Sample two-loop Feynman diagrams contributing at leading colour
are shown in Figures 1 and 2.

Figure 1: Sample two-loop Feynman diagrams for $W^{+}\gamma j$ production.
Figure 2: Sample two-loop Feynman diagrams for $W^{+}\gamma j$ production
containing closed fermion loop. $A_{6,q}^{(2),n_{f}}$ vanishes due to Furry’s
theorem.

The colour decomposition of the $W^{+}\gamma j$ $L$-loop amplitude is given by

$\mathcal{A}_{6}^{(L)}(1_{\gamma},2_{\bar{u}},3_{g},4_{d},5_{\nu},6_{\bar{e}})=\sqrt{2}eg_{W}^{2}g_{s}\,n^{L}\,(T^{a_{3}})_{i_{4}}^{\;\;\bar{i}_{2}}\,A_{6}^{(L)}(1_{\gamma},2_{\bar{u}},3_{g},4_{d},5_{\nu},6_{\bar{e}})\,,$
(2)

where $n=m_{\epsilon}\alpha_{s}/(4\pi)$, $\alpha_{s}=g_{s}^{2}/(4\pi)$,
$m_{\epsilon}=i(4\pi)^{\epsilon}e^{-\epsilon\gamma_{E}}$, $\epsilon=(4-d)/2$
is the dimensional regulator, and $T^{a}$ are the generators of $SU(N_{c})$ in
the fundamental representation, normalised according to $\operatorname{\rm
tr}(T^{a}T^{b})=\delta^{ab}$. We denote by $e$, $g_{W}$ and $g_{s}$ the
electron charge, the weak and the strong coupling constants, respectively. The
$W^{+}\gamma j$ amplitude can be further decomposed according to the source of
photon radiation,

$\displaystyle\begin{aligned} A_{6}^{(L)}=\
&\left[Q_{u}A^{(L)}_{6,u}+Q_{d}A^{(L)}_{6,d}+\left(\sum_{q}Q_{q}\right)A^{(L)}_{6,q}\right]P(s_{56})\\\
&+\left(Q_{u}-Q_{d}\right)\left[A^{(L)}_{6,e}+A^{(L)}_{6,W}P(s_{56})\right]P(s_{156})\,,\end{aligned}$
(3)

where $s_{ij}=(p_{i}+p_{j})^{2}$ and $s_{ijk}=(p_{i}+p_{j}+p_{k})^{2}$,
$Q_{u}$ and $Q_{d}$ are the up- and down-quark charges respectively, the sum
runs over the quark flavours $q$, while

$P(s)=\frac{1}{s-M_{W}^{2}+iM_{W}\Gamma_{W}}\,,$ (4)

is the denominator factor of $W$ boson propagator. $M_{W}$ and $\Gamma_{W}$
are the mass and decay width of the $W$ boson, respectively. The sub-
amplitudes $A_{6,i}^{(L)}$ in Eq. (3) are categorised as follows:

  * •

$A^{(L)}_{6,u}$: the photon is radiated off the $u$ quark;

  * •

$A^{(L)}_{6,d}$: the photon is radiated off the $d$ quark;

  * •

$A^{(L)}_{6,W}$: the photon is radiated off the $W$ boson;

  * •

$A^{(L)}_{6,e}$: the photon is radiated off the positron;

  * •

$A^{(L)}_{6,q}$: the photon is radiated off the internal quark loop.

We stress that the sub-amplitudes are not separately gauge invariant in the
electroweak sector. Using the relation Campbell:2021mlr

$P(s_{56})P(s_{156})=\frac{1}{s_{156}-s_{56}}\bigg{[}P(s_{56})-P(s_{156})\bigg{]}$
(5)

we can rewrite Eq. (3) as

$\displaystyle A_{6}^{(L)}$
$\displaystyle=Q_{u}\bigg{[}A^{(L)}_{6,u}+\frac{1}{s_{156}-s_{56}}A^{(L)}_{6,W}\bigg{]}P(s_{56})+Q_{d}\bigg{[}A^{(L)}_{6,d}-\frac{1}{s_{156}-s_{56}}A^{(L)}_{6,W}\bigg{]}P(s_{56})$
$\displaystyle\quad+(Q_{u}-Q_{d})\bigg{[}A^{(L)}_{6,e}-\frac{1}{s_{156}-s_{56}}A^{(L)}_{6,W}\bigg{]}P(s_{156})+\left(\sum_{q}Q_{q}\right)A^{(L)}_{6,q}P(s_{56})\,,$
(6)

such that the combinations of sub-amplitudes in the square brackets and
$A^{(L)}_{6,q}$ are the gauge invariant pieces. We further decompose the sub-
amplitudes according to their closed fermion loop contributions. At leading
colour we have

$\displaystyle\begin{aligned} &A^{(1)}_{6,i}=N_{c}A^{(1),1}_{6,i}\,,\\\
&A^{(2)}_{6,i}=N_{c}^{2}A^{(2),1}_{6,i}+N_{c}n_{f}A^{(2),n_{f}}_{6,i}\,,\\\
&A^{(2)}_{6,q}=N_{c}A^{(2),1}_{6,q}\,,\end{aligned}$ (7)

where $i=u,d,W,e$, and $n_{f}$ is the number of massless quark flavours
running in the loop. We note that $A^{(L)}_{6,q}$ vanishes at tree level and
one loop, while at two loops it includes non-planar contributions, and thus
will not be considered in this work.

The coupling of the $W$ boson to fermions involves vector and axial-vector
($V-A$) vertices in the form of $\gamma^{\mu}(1-\gamma_{5})/2$. The massless
fermion pairs that are coupled to the $W$ boson are connected to the external
states and the $V-A$ coupling fixes the helicity of the fermion pairs.
Therefore we only need to take into account the vector coupling of the $W$
boson to fermions when computing the helicity amplitudes. The contributing
helicity configurations are

$A_{6}^{(L)}(1^{\pm}_{\gamma},2^{+}_{\bar{u}},3^{\pm}_{g},4^{-}_{d},5^{-}_{\nu},6^{+}_{\bar{e}})\,.\\\
$

We choose $\scriptstyle+++--+$ and $\scriptstyle-++--+$ as the independent
helicity configurations and focus on them. We obtain the amplitudes in the
other helicity configurations from the independent ones by suitably permuting
the external momenta and conjugating space-time parity.

The sub-amplitudes $A^{(L)}_{6,u}$ and $A^{(L)}_{6,d}$ are related by

$A_{6,u}^{(L)}(1^{h_{1}}_{\gamma},2^{+}_{\bar{u}},3^{h_{3}}_{g},4^{-}_{d},5^{h_{5}}_{\nu},6^{h_{6}}_{\bar{e}})=A_{6,d}^{(L)}(1^{h_{1}}_{\gamma},4^{-}_{\bar{u}},3^{h_{3}}_{g},2^{+}_{d},5^{h_{5}}_{\nu},6^{h_{6}}_{\bar{e}})\,.$
(8)

As a result, we can limit ourselves to computing the $A^{(L)}_{6,d}$ sub-
amplitudes with the following helicity configurations,

$A_{6,d}^{(L)}(1^{\pm}_{\gamma},2^{\pm}_{\bar{u}},3^{\pm}_{g},4^{\mp}_{d},5^{-}_{\nu},6^{+}_{\bar{e}})\,.\\\
$

The independent helicity amplitudes for $A^{(L)}_{6,u}$ are then obtained
through

$A_{6,u}^{(L)}(1^{\pm}_{\gamma},2^{+}_{\bar{u}},3^{+}_{g},4^{-}_{d},5^{-}_{\nu},6^{+}_{\bar{e}})=A_{6,d}^{(L)}(1^{\pm}_{\gamma},4^{-}_{\bar{u}},3^{+}_{g},2^{+}_{d},5^{-}_{\nu},6^{+}_{\bar{e}})\,.$
(9)

The pole structure of the unrenormalised $W^{+}\gamma j$ amplitudes in the ’t
Hooft-Veltman (tHV) scheme at one and two loops is given by Catani:1998bh ;
Becher:2009qa ; Becher:2009cu ; Gardi:2009qi

$\displaystyle P_{6}^{(1)}$
$\displaystyle=2I_{1}(\epsilon)+\frac{\beta_{0}}{2\epsilon},$ (10)
$\displaystyle P_{6}^{(2)}$
$\displaystyle=2I_{1}(\epsilon)\bigg{(}\hat{A}^{(1)}_{6}-\frac{\beta_{0}}{2\epsilon}\bigg{)}+4I_{2}(\epsilon)+\frac{3\beta_{0}}{2\epsilon}\hat{A}^{(1)}_{6}-\frac{3\beta_{0}^{2}}{8\epsilon^{2}}+\frac{\beta_{1}}{4\epsilon}\,,$
(11)

where $\hat{A}^{(1)}_{6}$ is the unrenormalised one-loop amplitude divided by
the tree-level amplitude. The $I_{2}(\epsilon)$ operator is given by

$I_{2}(\epsilon)=-\frac{1}{2}I_{1}(\epsilon)\left[I_{1}(\epsilon)+\frac{\beta_{0}}{\epsilon}\right]+\frac{N(\epsilon)}{N(2\epsilon)}\left[\frac{\beta_{0}}{2\epsilon}+\frac{\gamma_{1}^{{\mathrm{cusp}}}}{8}\right]I_{1}(2\epsilon)+H^{(2)}(\epsilon)\,,$
(12)

while the $I_{1}(\epsilon)$ operator is given at leading colour by

$\displaystyle I_{1}(\epsilon)$
$\displaystyle=-N_{c}\frac{N(\epsilon)}{2}\bigg{(}\frac{1}{\epsilon^{2}}+\frac{3}{4\epsilon}+\frac{\beta_{0}}{4N_{c}\epsilon}\bigg{)}\big{[}\left(-s_{23}\right)^{-\epsilon}+\left(-s_{34}\right)^{-\epsilon}\big{]}\,,$
(13)

where $N(\epsilon)={e^{\epsilon\gamma_{E}}}/{\Gamma(1-\epsilon)}$ and

$\displaystyle H^{(2)}(\epsilon)$
$\displaystyle=\frac{1}{16\epsilon}\bigg{\\{}\big{(}2\gamma_{1}^{q}+\gamma_{1}^{g}\big{)}-\gamma_{1}^{\mathrm{cusp}}\left(\frac{\gamma_{0}^{q}}{2}+\frac{\gamma_{0}^{g}}{4}\right)+\frac{\pi^{2}}{8}\beta_{0}\gamma_{0}^{\mathrm{cusp}}\left(C_{F}+\frac{C_{A}}{2}\right)\bigg{\\}}\,.$
(14)

The $\beta$ function coefficients and anomalous dimensions are tabulated in
Appendix A. We stress that the pole terms in Eqs. (10) and (11) include both
the ultraviolet and infrared singularities. We then extract the $L$-loop
partial finite remainder by subtracting the poles $P_{6}^{(L)}$ from the
unrenormalised partial amplitude $A_{6}^{(L)}$ and sending $\epsilon$ to $0$,

$F_{6}^{(L)}=\lim_{\epsilon\to
0}\left[A_{6}^{(L)}-P_{6}^{(L)}A_{6}^{(0)}\right]\,.$ (15)

The finite remainder $F_{6}^{(L)}$ follows the same decomposition as the
unrenormalised partial amplitude $A_{6}^{(L)}$ (see Eqs. (3) and (7)),

$\displaystyle
F_{6}^{(L)}=\,\left[Q_{u}F^{(L)}_{6,u}+Q_{d}F^{(L)}_{6,d}\right]P(s_{56})+(Q_{u}-Q_{d})\left[F^{(L)}_{6,e}+F^{(L)}_{6,W}P(s_{56})\right]P(s_{156})\,,$
(16)

with

$\displaystyle\begin{aligned}
&F^{(1)}_{6,i}=N_{c}F^{(1),1}_{6,i}+n_{f}F^{(1),n_{f}}_{6,i}\,,\\\
&F^{(2)}_{6,i}=N_{c}^{2}F^{(2),1}_{6,i}+N_{c}n_{f}F^{(2),n_{f}}_{6,i}+n_{f}^{2}F^{(2),n_{f}^{2}}_{6,i}\,,\end{aligned}$
(17)

where $i=u,d,W,e$. We note that, although the bare sub-amplitudes
$A^{(1),n_{f}}_{6,i}$ and $A^{(2),n_{f}^{2}}_{6,i}$ vanish, there are finite
contributions to the finite remainders $F^{(1),n_{f}}_{6,i}$ and
$F^{(2),n_{f}^{2}}_{6,i}$ from the UV renormalisation and IR subtraction terms
specified in Eqs. (10) and (11). As discussed below Eqs. (7), we defer the
computation of $F^{(2)}_{6,q}$, as it involves the non-planar integrals.

For the charge-conjugated process, i.e. $pp\to\bar{\nu}_{e}e^{-}\gamma j$, we
consider the amplitudes for

$0\rightarrow\gamma(p_{1},h_{1})+\bar{d}(p_{2},h_{2})+g(p_{3},h_{3})+u(p_{4},h_{4})+e^{-}(p_{5},h_{5})+\bar{\nu}_{e}(p_{6},h_{6})\,,$

which we denote by $W^{-}\gamma j$ production. The amplitudes for $W^{-}\gamma
j$ production can be obtained from the $W^{+}\gamma j$ results through the
following relation,

$A_{6}^{(L)}(1^{-h_{1}}_{\gamma},2^{+}_{\bar{d}},3^{-h_{3}}_{g},4^{-}_{u},5^{-}_{e},6^{+}_{\bar{\nu}})=\bigg{[}A_{6}^{(L)}(1^{h_{1}}_{\gamma},4^{+}_{\bar{u}},3^{h_{3}}_{g},2^{-}_{d},6^{-}_{\nu},5^{+}_{\bar{e}})\bigg{]}_{\langle
ij\rangle\leftrightarrow[ij]}\,.$ (18)

##  3 Amplitude Computation

In this section we describe the computation of the two-loop $W^{+}\gamma j$
amplitude in the leading colour approximation. To derive the analytic form of
the amplitude we employ a framework that combines Feynman diagram input, the
four-dimensional projector method Peraro:2019cjj ; Peraro:2020sfm ,
integration-by-parts (IBP) reduction Tkachov:1981wb ; Chetyrkin:1981qh ;
Laporta:2000dsw , Laurent expansion onto a basis of special function,
numerical evaluations over finite fields, and analytic reconstruction
techniques vonManteuffel:2014ixa ; Peraro:2016wsq ; Peraro:2019svx .

Instead of computing the loop amplitudes using the full six-particle
kinematics, we detach the leptonic decay of the $W$ boson from the amplitude
and only compute the $W$-production amplitudes. For the $A^{(L)}_{6,u}$ and
$A^{(L)}_{6,d}$ sub-amplitudes the $W$-production amplitude is a five-point
amplitude with an off-shell leg (denoted by $A_{5,u/d}^{{}^{(L)}\mu}$), while
for the $A^{(L)}_{6,W}$ and $A^{(L)}_{6,e}$ sub-amplitudes the $W$-production
amplitude is a four-point amplitude with an off-shell leg (denoted by
$A_{4}^{{(L)}\mu}$),

$\displaystyle
A^{(L)}_{6,u/d}(p_{1},p_{2},p_{3},p_{4},p_{5},p_{6})=A_{5,u/d}^{(L)\mu}(p_{1},p_{2},p_{3},p_{4},p_{W})\;L_{A,\mu}(p_{5},p_{6})\,,$
(19) $\displaystyle
A^{(L)}_{6,e/W}(p_{1},p_{2},p_{3},p_{4},p_{5},p_{6})=A_{4}^{(L)\mu}(p_{2},p_{3},p_{4},\tilde{p}_{W})\;L^{e/W}_{B,\mu}(p_{1},p_{5},p_{6})\,,$
(20)

where $p_{W}=p_{5}+p_{6}$ and $\tilde{p}_{W}=p_{1}+p_{5}+p_{6}$,
$L^{e}_{B,\mu}$ ($L^{W}_{B,\mu}$) is the decay current where the photon is
emitted from the positron ($W$ boson), while $L_{A,\mu}$ is simply the
$W^{+}\to\nu e^{+}$ decay current. The QCD corrections affect only the
$W$-production amplitudes $A_{5,u/d}^{(L)\mu}$ and $A_{4}^{(L)\mu}$. We adopt
the same decomposition for the finite remainders,

$\displaystyle
F^{(L)}_{6,u/d}(p_{1},p_{2},p_{3},p_{4},p_{5},p_{6})=F_{5,u/d}^{(L)\mu}(p_{1},p_{2},p_{3},p_{4},p_{W})\;L_{A,\mu}(p_{5},p_{6})\,,$
(21) $\displaystyle
F^{(L)}_{6,e/W}(p_{1},p_{2},p_{3},p_{4},p_{5},p_{6})=F_{4}^{(L)\mu}(p_{2},p_{3},p_{4},\tilde{p}_{W})\;L^{e/W}_{B,\mu}(p_{1},p_{5},p_{6})\,.$
(22)

In the next subsections we discuss the computation of $F_{5,d}^{(L)\mu}$ and
$F_{4}^{(L)\mu}$. We recall that $F_{6,u}^{(L)}$ can be obtained from
$F_{6,d}^{(L)}$ through the amplitude-level relation given in Eq. (8), which
we rewrite here for the finite remainders,

$F_{6,u}^{(L)}(1^{h_{1}}_{\gamma},2^{+}_{\bar{u}},3^{h_{3}}_{g},4^{-}_{d},5^{h_{5}}_{\nu},6^{h_{6}}_{\bar{e}})=F_{6,d}^{(L)}(1^{h_{1}}_{\gamma},4^{-}_{\bar{u}},3^{h_{3}}_{g},2^{+}_{d},5^{h_{5}}_{\nu},6^{h_{6}}_{\bar{e}})\,.$
(23)

We begin by describing how we parameterise the kinematics. Next, we discuss
how we decompose the $W$-production five- and four-particle amplitudes, in
Sections 3.2 and 3.3 respectively, using the projector method. Section 3.4 is
devoted to the finite-field setup which we use to reconstruct the analytic
expressions of the finite remainders as linear combinations of rational
coefficients and monomials of independent special functions. In Section 3.5 we
present a strategy which allows us to simplify dramatically the expressions of
the rational coefficients. Finally, in Section 3.6 we discuss how our analytic
results for the minimal set of independent finite remainders can be used
efficiently to evaluate numerically all the contributions to the squared
finite remainder summed over helicity and colour.

###  3.1 Kinematics

In this section we describe the kinematics of the process $W^{+}\gamma j$ (1).
All the external momenta $p_{i}^{\mu}$ are massless,

$p_{i}^{2}=0\quad\forall\,i=1,\ldots,6\,,$ (24)

and taken to be outgoing, so that momentum conservation is

$\sum_{i=1}^{6}p_{i}=0\,.$ (25)

We consider the external momenta $p_{i}^{\mu}$ to live in a four-dimensional
Minkowski space. As a result there are eight independent scalar invariants,
which we choose as

$\vec{s}_{6}=\left\\{s_{12},s_{23},s_{34},s_{45},s_{56},s_{16},s_{123},s_{234}\right\\}\,.$
(26)

It is also possible to form pseudo-scalar invariants by contracting the Levi-
Civita symbol $\epsilon_{\mu\nu\rho\sigma}$ with any four external momenta.
The six-particle kinematics is therefore fully determined by the scalar
invariants $\vec{s}_{6}$ and by one pseudo-scalar invariant, which captures
all the space-time parity information of the phase space. In analogy with the
five-particle kinematics, we choose

$\displaystyle\operatorname{\rm
tr}_{5}=4i\epsilon_{\mu\nu\rho\sigma}p_{1}^{\mu}p_{2}^{\nu}p_{3}^{\rho}p_{4}^{\sigma}\,.$
(27)

The latter is related to the scalar invariants through

$\displaystyle\operatorname{\rm
tr}_{5}^{2}=\Delta_{5}:=\text{det}\left(2p_{i}\cdot
p_{j}\right)\bigl{|}_{i,j=1,\ldots,4}\,,$ (28)

where the right-hand side is a degree-four polynomial in the scalar
invariants.

Only a subset of these invariants are relevant for the computation of
$A_{5,d}^{(L)\mu}$, which has five-point kinematics with an external massive
particle. We choose the following independent five-point scalar invariants for
computing $A_{5,d}^{(L)\mu}$,

$\vec{s}_{5}=\left\\{s_{12},s_{23},s_{34},s_{123},s_{234},s_{56}\right\\}\,,$
(29)

together with $\operatorname{\rm tr}_{5}$. Even fewer variables are relevant
for $A_{4}^{(L)\mu}$. Since the latter has four-point kinematics with an
external massive particle, no pseudo-scalar invariant can be formed and it is
thus independent of $\operatorname{\rm tr}_{5}$. Moreover, it depends only on
three of the scalar invariants in $\vec{s}_{5}$. Nonetheless, we view it as a
function of $\vec{s}_{5}$ in order to have a homogeneous setup.

When attaching the $W$-boson decay currents ($L_{A,\mu}$ and
$L^{e/W}_{B,\mu}$) to $A_{5,d}^{(L)\mu}$ and $A_{4}^{(L)\mu}$ (see Eqs. (19)
and (20)) we find it convenient to describe the massless six-point kinematics
using a parameterisation based on momentum twistors Hodges:2009hk  (see e.g.
Refs. Badger:2013gxa ; Badger:2016uuq  for a thorough discussion of momentum
twistors in this context). We adopt the following momentum-twistor
parameterisation,

$\displaystyle Z=\begin{pmatrix}1&0&y_{1}&y_{2}&y_{3}&y_{4}\\\ 0&1&1&1&1&1\\\
0&0&0&\tfrac{x_{5}}{x_{2}}&x_{6}&1\\\
0&0&1&1&x_{7}&1-\tfrac{x_{8}}{x_{5}}\end{pmatrix}\,,$ (30)

where the columns give the four-component momentum twistors of the six
external particles, and we used the short-hand notation
$y_{i}=\sum_{j=1}^{i}\prod_{k=1}^{j}\frac{1}{x_{k}}$. The eight momentum-
twistor variables $x_{i}$ are related to the external momenta through

$\displaystyle x_{1}$ $\displaystyle=s_{12}\,,$ $\displaystyle x_{2}$
$\displaystyle=-\frac{\operatorname{\rm tr}_{+}(1234)}{s_{12}s_{34}}\,,$
$\displaystyle x_{3}$ $\displaystyle=-\frac{\operatorname{\rm
tr}_{+}(1345)}{s_{45}s_{13}}\,,$ $\displaystyle x_{4}$
$\displaystyle=-\frac{\operatorname{\rm tr}_{+}(1456)}{s_{14}s_{56}}\,,$
$\displaystyle x_{5}$ $\displaystyle=\frac{s_{23}}{s_{12}}\,,$ $\displaystyle
x_{6}$ $\displaystyle=-\frac{\operatorname{\rm
tr}_{+}(15(3+4)2)}{s_{12}s_{15}}\,,$ $\displaystyle x_{7}$
$\displaystyle=\frac{\operatorname{\rm
tr}_{+}(51(2+3)(2+3+4))}{s_{15}s_{23}}\,,$ $\displaystyle x_{8}$
$\displaystyle=\frac{s_{123}}{s_{12}}\,,$ (31)

with $\operatorname{\rm tr}_{\pm}(ij\cdots kl)=\operatorname{\rm
tr}((1\pm\gamma_{5})\not{p}_{i}\not{p}_{j}\cdots\not{p}_{k}\not{p}_{l})/2$.
Note that the momentum-twistor variables $x_{i}$ in general transform in a
non-trivial way under space-time parity. We implement the action of parity on
momentum-twistor expressions as a change of momentum-twistor variables which
leave unchanged the scalar invariants and flips the sign of $\operatorname{\rm
tr}_{5}$. The definition of the parity-flipped momentum-twistor variables can
be obtained by trading $\operatorname{\rm tr}_{+}$ for $\operatorname{\rm
tr}_{-}$ in Eqs. (31).

###  3.2 Structure of the five-particle $W$-production amplitudes

We decompose the five-point $W$-production amplitude $A_{5,d}^{(L)\mu}$ using
the external momenta $(p_{1},p_{2},p_{3},p_{4})$ as the spanning basis,

$\displaystyle
A_{5,d}^{(L)\mu}=p_{1}^{\mu}a^{(L)}_{1}+p_{2}^{\mu}a^{(L)}_{2}+p_{3}^{\mu}a^{(L)}_{3}+p_{4}^{\mu}a^{(L)}_{4}\,.$
(32)

The coefficients $a^{(L)}_{i}$ can be obtained by inverting the system of
equations

$\displaystyle
a^{(L)}_{i}=\sum_{j=1}^{4}\left(\Delta^{-1}\right)_{ij}\;\tilde{A}_{5,j}^{(L)}\,,$
(33)

where

$\displaystyle\Delta_{ij}=p_{i}\cdot p_{j}\,,$ (34)
$\displaystyle\tilde{A}_{5,i}^{(L)}=p_{i}\cdot A_{5,d}^{(L)}\,.$ (35)

The contracted amplitudes $\tilde{A}_{5,i}^{(L)}$ are computed by first
generating the five-point process with an on-shell $W$ boson, followed by
replacing the $W$-boson polarisation vector by the four external momenta in
the spanning basis, $(p_{1},p_{2},p_{3},p_{4})$. We then apply tensor
decomposition, taking into account the four-dimensional nature of the external
states as proposed in Refs. Peraro:2019cjj ; Peraro:2020sfm , to express each
contracted amplitude $\tilde{A}_{5,i}^{(L)}$ as a linear combination of 8
independent tensor structures $\\{T_{j}\\}_{j=1}^{8}$,

$\displaystyle\tilde{A}_{5,i}^{(L)}=\sum_{j=1}^{8}T_{j}\alpha_{i,j}^{(L)}\,,$
(36)

where

$\displaystyle T_{1}$
$\displaystyle=\bar{u}(p_{4})\not{p}_{1}v(p_{2})\;p_{2}\cdot\varepsilon(p_{1},q_{1})\;p_{2}\cdot\varepsilon(p_{3},q_{3})\,,$
$\displaystyle T_{2}$
$\displaystyle=\bar{u}(p_{4})\not{p}_{1}v(p_{2})\;p_{2}\cdot\varepsilon(p_{1},q_{1})\;p_{4}\cdot\varepsilon(p_{3},q_{3})\,,$
$\displaystyle T_{3}$
$\displaystyle=\bar{u}(p_{4})\not{p}_{1}v(p_{2})\;p_{4}\cdot\varepsilon(p_{1},q_{1})\;p_{2}\cdot\varepsilon(p_{3},q_{3})\,,$
$\displaystyle T_{4}$
$\displaystyle=\bar{u}(p_{4})\not{p}_{1}v(p_{2})\;p_{4}\cdot\varepsilon(p_{1},q_{1})\;p_{4}\cdot\varepsilon(p_{3},q_{3})\,,$
(37) $\displaystyle T_{5}$
$\displaystyle=\bar{u}(p_{4})\not{p}_{3}v(p_{2})\;p_{2}\cdot\varepsilon(p_{1},q_{1})\;p_{2}\cdot\varepsilon(p_{3},q_{3})\,,$
$\displaystyle T_{6}$
$\displaystyle=\bar{u}(p_{4})\not{p}_{3}v(p_{2})\;p_{2}\cdot\varepsilon(p_{1},q_{1})\;p_{4}\cdot\varepsilon(p_{3},q_{3})\,,$
$\displaystyle T_{7}$
$\displaystyle=\bar{u}(p_{4})\not{p}_{3}v(p_{2})\;p_{4}\cdot\varepsilon(p_{1},q_{1})\;p_{2}\cdot\varepsilon(p_{3},q_{3})\,,$
$\displaystyle T_{8}$
$\displaystyle=\bar{u}(p_{4})\not{p}_{3}v(p_{2})\;p_{4}\cdot\varepsilon(p_{1},q_{1})\;p_{4}\cdot\varepsilon(p_{3},q_{3})\,.$

Here, $q_{1}$ and $q_{3}$ are arbitrary reference vectors for the photon and
the gluon polarisation states, respectively. We set $q_{1}=p_{3}$ and
$q_{3}=p_{1}$ throughout our computation. The tensor coefficients
$\alpha^{(L)}_{i,j}$ can be obtained by

$\displaystyle\alpha^{(L)}_{i,j}=\sum_{k=1}^{8}\left(\Theta^{-1}\right)_{jk}\;\sum_{\text{pol}}{T}_{k}^{\dagger}\tilde{A}_{5,i}^{(L)}\,,$
(38)

where

$\displaystyle\Theta_{ij}=\sum_{\text{pol}}T_{i}^{\dagger}T_{j}\,,$ (39)

and the polarisation-vector sum for the photon and gluon is

$\displaystyle\sum_{\text{pol}}\varepsilon^{*}_{\mu}(p_{i},q_{i})\varepsilon_{\nu}(p_{i},q_{i})=-g_{\mu\nu}+\frac{p_{i\mu}q_{i\nu}+q_{i\mu}p_{i\nu}}{p_{i}\cdot
q_{i}},\qquad i=1,3\,.$ (40)

We further specify the helicity states of the spinors and polarisation vectors
in the tensor structures $\\{T_{j}\\}_{j=1}^{8}$,

$\displaystyle T^{h_{1}h_{2}h_{3}h_{4}}_{1}$
$\displaystyle=\bar{u}(p_{4},h_{4})\not{p}_{1}v(p_{2},h_{2})\;p_{2}\cdot\varepsilon(p_{1},q_{1},h_{1})\;p_{2}\cdot\varepsilon(p_{3},q_{3},h_{3})\,,$
$\displaystyle T^{h_{1}h_{2}h_{3}h_{4}}_{2}$
$\displaystyle=\bar{u}(p_{4},h_{4})\not{p}_{1}v(p_{2},h_{2})\;p_{2}\cdot\varepsilon(p_{1},q_{1},h_{1})\;p_{4}\cdot\varepsilon(p_{3},q_{3},h_{3})\,,$
$\displaystyle T^{h_{1}h_{2}h_{3}h_{4}}_{3}$
$\displaystyle=\bar{u}(p_{4},h_{4})\not{p}_{1}v(p_{2},h_{2})\;p_{4}\cdot\varepsilon(p_{1},q_{1},h_{1})\;p_{2}\cdot\varepsilon(p_{3},q_{3},h_{3})\,,$
$\displaystyle T^{h_{1}h_{2}h_{3}h_{4}}_{4}$
$\displaystyle=\bar{u}(p_{4},h_{4})\not{p}_{1}v(p_{2},h_{2})\;p_{4}\cdot\varepsilon(p_{1},q_{1},h_{1})\;p_{4}\cdot\varepsilon(p_{3},q_{3},h_{3})\,,$
(41) $\displaystyle T^{h_{1}h_{2}h_{3}h_{4}}_{5}$
$\displaystyle=\bar{u}(p_{4},h_{4})\not{p}_{3}v(p_{2},h_{2})\;p_{2}\cdot\varepsilon(p_{1},q_{1},h_{1})\;p_{2}\cdot\varepsilon(p_{3},q_{3},h_{3})\,,$
$\displaystyle T^{h_{1}h_{2}h_{3}h_{4}}_{6}$
$\displaystyle=\bar{u}(p_{4},h_{4})\not{p}_{3}v(p_{2},h_{2})\;p_{2}\cdot\varepsilon(p_{1},q_{1},h_{1})\;p_{4}\cdot\varepsilon(p_{3},q_{3},h_{3})\,,$
$\displaystyle T^{h_{1}h_{2}h_{3}h_{4}}_{7}$
$\displaystyle=\bar{u}(p_{4},h_{4})\not{p}_{3}v(p_{2},h_{2})\;p_{4}\cdot\varepsilon(p_{1},q_{1},h_{1})\;p_{2}\cdot\varepsilon(p_{3},q_{3},h_{3})\,,$
$\displaystyle T^{h_{1}h_{2}h_{3}h_{4}}_{8}$
$\displaystyle=\bar{u}(p_{4},h_{4})\not{p}_{3}v(p_{2},h_{2})\;p_{4}\cdot\varepsilon(p_{1},q_{1},h_{1})\;p_{4}\cdot\varepsilon(p_{3},q_{3},h_{3})\,,$

from which we obtain the contracted helicity amplitudes,

$\displaystyle\tilde{A}_{5,i}^{(L),h_{1}h_{2}h_{3}h_{4}}=\sum_{j,k=1}^{8}T_{j}^{h_{1}h_{2}h_{3}h_{4}}\left(\Theta^{-1}\right)_{jk}\tilde{\mathcal{A}}_{5,k\,i}^{(L)}\,,$
(42)

with

$\displaystyle\tilde{\mathcal{A}}_{5,k\,i}^{(L)}=\sum_{\text{pol}}{T}_{k}^{\dagger}\tilde{A}_{5,i}^{(L)}\,.$
(43)

We carry out the same decomposition for the five-particle finite remainder
$F_{5,d}^{(L)\mu}$, arriving at the following formula for the contracted
helicity finite remainders,

$\displaystyle\tilde{F}_{5,i}^{(L),h_{1}h_{2}h_{3}h_{4}}=\sum_{j,k=1}^{8}T_{j}^{h_{1}h_{2}h_{3}h_{4}}\left(\Theta^{-1}\right)_{jk}\,\tilde{\mathcal{F}}_{5,k\,i}^{(L)}\,,$
(44)

where

$\displaystyle\tilde{\mathcal{F}}_{5,k\,i}^{(L)}=\sum_{\text{pol}}{T}_{k}^{\dagger}\,p_{i\,\mu}\,F_{5,d}^{(L)\mu}\,.$
(45)

As discussed in Section 2, the independent helicity configurations which we
need to compute are

$\left\\{\tilde{F}_{5,i}^{(L),+++-},\tilde{F}_{5,i}^{(L),-++-},\tilde{F}_{5,i}^{(L),+-++},\tilde{F}_{5,i}^{(L),--++}\right\\}\,.$
(46)

We note that it is possible to compute directly the contracted finite
remainders $\tilde{F}^{(L)}_{5,i}$ without specifying the helicity states. In
our setup, however, such a computation would lead to more complicated analytic
expressions compared to the results obtained for the contracted helicity
amplitudes.

###  3.3 Structure of the four-particle $W$-production amplitudes

The four-particle $W$-production amplitude $A_{4}^{(L)\mu}$ has been computed
in the context of $W+1j$ production at the LHC ($q\bar{q}\to Wg$)
Gehrmann:2011ab , which is a crossing of the $e^{+}e^{-}\to q\bar{q}g$
amplitude Garland:2002ak . In our case it is convenient to express
$A_{4}^{(L)\mu}$ in terms of the same special function basis as
$A_{5,d}^{(L)\mu}$. This guarantees a uniform combination of the different
contributions to the full amplitude. We therefore re-derive the
$A_{4}^{(L)\mu}$ amplitude using our computational framework. We decompose the
$A_{4}^{(L)\mu}$ amplitude using the following tensor structures
Garland:2002ak ,

$A_{4}^{(L)\mu}(p_{2},p_{3},p_{4})=\sum_{i=1}^{7}b_{i}^{(L)}Y_{i}^{\mu}\,,$
(47)

where

$\displaystyle Y_{1}^{\mu}$
$\displaystyle=\bar{u}(p_{4})\not{p}_{3}v(p_{2})\,\varepsilon_{3}\cdot
p_{4}\,p_{4}^{\mu}-p_{3}\cdot
p_{4}\,\bar{u}(p_{4})\not{\varepsilon}_{3}v(p_{2})\,p_{4}^{\mu}$
$\displaystyle\quad-(p_{2}\cdot p_{4}+p_{3}\cdot
p_{4})\left[\bar{u}(p_{4})\not{p}_{3}v(p_{2})\,\varepsilon_{3}^{\mu}-\bar{u}(p_{4})\not{\varepsilon}_{3}v(p_{2})\,p_{3}^{\mu}\right]\,,$
$\displaystyle Y_{2}^{\mu}$
$\displaystyle=\bar{u}(p_{4})\not{p}_{3}v(p_{2})\,\varepsilon_{3}\cdot
p_{4}\,p_{3}^{\mu}-p_{3}\cdot
p_{4}\,\bar{u}(p_{4})\not{\varepsilon}_{3}v(p_{2})\,p_{3}^{\mu}$
$\displaystyle\quad-(p_{2}\cdot p_{3}+p_{3}\cdot
p_{4})\left[\bar{u}(p_{4})\not{p}_{3}v(p_{2})\,\varepsilon_{3}^{\mu}-\bar{u}(p_{4})\not{\varepsilon}_{3}v(p_{2})\,p_{3}^{\mu}\right]\,,$
$\displaystyle Y_{3}^{\mu}$
$\displaystyle=\bar{u}(p_{4})\not{p}_{3}v(p_{2})\,\varepsilon_{3}\cdot
p_{4}\,p_{2}^{\mu}-p_{3}\cdot
p_{4}\,\bar{u}(p_{4})\not{\varepsilon}_{3}v(p_{2})\,p_{2}^{\mu}$
$\displaystyle\quad-(p_{2}\cdot p_{3}+p_{2}\cdot
p_{4})\left[\bar{u}(p_{4})\not{p}_{3}v(p_{2})\,\varepsilon_{3}^{\mu}-\bar{u}(p_{4})\not{\varepsilon}_{3}v(p_{2})\,p_{3}^{\mu}\right]\,,$
(48) $\displaystyle Y_{4}^{\mu}$
$\displaystyle=\bar{u}(p_{4})\not{p}_{3}v(p_{2})\,\varepsilon_{3}\cdot
p_{2}\,p_{4}^{\mu}-p_{2}\cdot
p_{3}\,\bar{u}(p_{4})\not{\varepsilon}_{3}v(p_{2})\,p_{4}^{\mu}\,,$
$\displaystyle Y_{5}^{\mu}$
$\displaystyle=\bar{u}(p_{4})\not{p}_{3}v(p_{2})\,\varepsilon_{3}\cdot
p_{2}\,p_{3}^{\mu}-p_{2}\cdot
p_{3}\,\bar{u}(p_{4})\not{\varepsilon}_{3}v(p_{2})\,p_{3}^{\mu}\,,$
$\displaystyle Y_{6}^{\mu}$
$\displaystyle=\bar{u}(p_{4})\not{p}_{3}v(p_{2})\,\varepsilon_{3}\cdot
p_{2}\,p_{2}^{\mu}-p_{2}\cdot
p_{3}\,\bar{u}(p_{4})\not{\varepsilon}_{3}v(p_{2})\,p_{2}^{\mu}\,,$
$\displaystyle Y_{7}^{\mu}$ $\displaystyle=p_{3}\cdot
p_{4}\,\bar{u}(p_{4})\gamma^{\mu}v(p_{2})\,\varepsilon_{3}\cdot
p_{2}-p_{2}\cdot
p_{3}\,\bar{u}(p_{4})\gamma^{\mu}v(p_{2})\,\varepsilon_{3}\cdot p_{4}$
$\displaystyle\quad-(p_{2}\cdot
p_{3})\left[\bar{u}(p_{4})\not{p}_{3}v(p_{2})\,\varepsilon_{3}^{\mu}-\bar{u}(p_{4})\not{\varepsilon}_{3}v(p_{2})\,p_{3}^{\mu}\right]\,.$

The coefficients $b_{i}^{(L)}$ can be determined through

$b^{(L)}_{i}=\sum_{j}\left(\Omega^{-1}\right)_{ij}\tilde{A}^{(L)}_{4,j}\,,$
(49)

where

$\displaystyle\Omega_{ij}=\sum_{\text{pol}}Y^{\mu\dagger}_{i}Y_{j\mu}\,,$ (50)
$\displaystyle\tilde{A}^{(L)}_{4,i}=\sum_{\text{pol}}Y^{\mu\dagger}_{i}A^{(L)}_{4\,\mu}\,.$
(51)

The gluon polarisation-vector sum follows from Eq. (40). We note that the
tensor structures in Eq. (48) are different from the ones employed in Ref.
Garland:2002ak . Here, we start from 12 tensor structures that are linearly
independent in 4 dimensions Peraro:2019cjj ; Peraro:2020sfm  and reduce them
to 7 by imposing Ward identities. Since $A^{(L)\mu}_{4}$ is a four-point
amplitude, it does not depend on $\operatorname{\rm tr}_{5}$. For the sake of
uniformity, we express it in terms of the five-point Mandelstam invariants
$\vec{s}_{5}$ (29). In contrast to the computation of $A^{(L)\mu}_{5,d}$, here
we derive the contracted amplitudes $\tilde{A}^{(L)}_{4,i}$ directly without
specifying the helicity states, since the four-point computation is relatively
simple. The helicity states for the tensor structures $Y_{\mu}^{i}$ are
specified when the decay currents are attached, following Eq. (20).

Once again we perform the same decomposition on the corresponding four-
particle finite remainder $F^{(L)\mu}_{4}$. The resulting formula for the
contracted finite remainders is

$\displaystyle\tilde{F}^{(L)}_{4,i}=\sum_{\text{pol}}Y^{\mu\dagger}_{i}F^{(L)}_{4\,\mu}\,.$
(52)

###  3.4 Amplitude reduction and analytic reconstruction

In this section we present the analytic computation of the contracted five-
and four-particle finite remainders,
$\tilde{F}_{5,i}^{(L),h_{1}h_{2}h_{3}h_{4}}$ (44) and $\tilde{F}_{4,i}^{(L)}$
(52) respectively, at one and two loops. We adopt the framework used in Refs.
Badger:2021nhg ; Badger:2021owl ; Badger:2021imn ; Badger:2021ega , based on
Feynman diagrams and functional reconstruction from numerical sampling over
finite field. For the latter we employ the FiniteFlow framework Peraro:2019svx
.

In order to use the finite field technique, we need to have a rational
parameterisation of the kinematics. As discussed in Section 3.1, the five-
particle phase space is described by the six scalar invariants $\vec{s}_{5}$
(29) together with the pseudo-scalar invariant $\operatorname{\rm tr}_{5}$
(27). It follows from Eq. (28) that $\operatorname{\rm tr}_{5}$ is given by
the square root of $\Delta_{5}$, up to the overall sign which encodes the
parity information. This square root may be dealt with in two ways: either
parameterise the kinematics to explicitly rationalise $\sqrt{\Delta}_{5}$, or
parameterise the finite remainders
$\tilde{F}_{5,i}^{(L),h_{1}h_{2}h_{3}h_{4}}$ such that the dependence on
$\operatorname{\rm tr}_{5}$ is analytic. We choose the latter option in this
case. The pseudo-scalar invariant $\operatorname{\rm tr}_{5}$ can enter the
computation in three distinct ways. First, it can originate from the
$\gamma_{5}$ in the axial coupling of the $W$ boson. As we discussed in
Section 2, we set up the computation so that only the vector coupling of $W$
is used. Second, $\operatorname{\rm tr}_{5}$ is in general needed to capture
the parity-odd part of the spinor-helicity expressions. However, the pseudo-
scalar invariant enters the contracted helicity finite remainder in Eq. (44)
only through the spanning basis elements $T_{j}^{h_{1}h_{2}h_{3}h_{4}}$, which
are known analytically and do not need to be reconstructed over finite fields.
Finally, $\operatorname{\rm tr}_{5}$ is present in the definition of the
canonical master integrals, which also introduces three other square roots,
$\sqrt{\Delta_{3}^{(i)}}$ with $i=1,2,3$, relevant for the analytic structure
of the Feynman integrals Abreu:2020jxa ; Chicherin:2021dyp . Since the square
roots appear only as overall normalisation factors, we can re-absorb them in
the definition of the canonical master integrals, and thus reduce the finite
remainders to master integrals which are all manifestly scalar. With this
setup the only parts of the contracted finite remainders which need to be
reconstructed are rational functions of the scalar invariants $\vec{s}_{5}$
only, and can thus be sampled numerically over finite fields.

The starting point of our computation is the expression of the $W$-production
five- and four-particle amplitudes, $A_{5,d}^{(L)\mu}$ and $A_{4}^{(L)\mu}$,
in terms of Feynman diagrams, which we generate using Qgraf Nogueira:1991ex .
For those interested in counting Feynman diagrams, there are 20 diagrams for
$A_{5,d}^{(1),1\mu}$, 231 diagrams for $A_{5,d}^{(2),1\mu}$, 32 diagrams for
$A_{5,d}^{(2),n_{f}\mu}$, 7 diagrams for $A_{4}^{(1),1\mu}$, 74 diagrams for
$A_{4}^{(2),1\mu}$, and 13 diagrams for $A_{4}^{(2),n_{f}\mu}$. Clearly this
signifies nothing more than that the number of diagrams is not a good measure
of complexity. We want to obtain analytic, IBP-compatible expressions for the
contracted amplitudes, $\tilde{A}_{5,i}^{(L),h_{1}h_{2}h_{3}h_{4}}$ and
$\tilde{A}_{4,i}^{(L)}$. For the four-point amplitude $A^{(L)\mu}_{4}$ we
apply the projectors and sum over all polarisation states as in Eq. (51). For
the five-point amplitude $A^{(L)\mu}_{5,d}$ we contract by the external
momenta and apply the projectors, as in Eqs. (35) and (43), respectively. We
then rewrite the resulting expressions in terms of scalar Feynman integrals
belonging to the master topologies defined in Refs. Badger:2021nhg ;
Badger:2021ega . We carry out all these operations analytically using
Mathematica and Form Kuipers:2012rf ; Ruijl:2017dtg  scripts. As a result, we
obtain analytic expressions for $\tilde{\mathcal{A}}_{5,i\,k}^{(L)}$ (43) and
$\tilde{A}_{4,i}^{(L)}$ (51) as linear combinations of scalar Feynman
integrals with rational coefficients functions of $\vec{s}_{5}$. In order to
obtain the contracted helicity-amplitudes
$\tilde{A}_{5,i}^{(L),h_{1}h_{2}h_{3}h_{4}}$ from the
$\tilde{\mathcal{A}}_{5,i\,k}^{(L)}$’s we further need to multiply by the
spanning basis elements $T^{h_{1}h_{2}h_{3}h_{4}}_{j}$ and by the inverse of
$\Theta$, as shown in Eq. (42). We do these operations (including the
inversion of $\Theta$) numerically within the finite field framework.

We reduce the scalar integrals to the canonical master integrals identified in
Ref. Abreu:2020jxa , which we modified so as to re-absorb the square roots. We
generate the IBP relations Chetyrkin:1981qh  using LiteRed Lee:2012cn  in
Mathematica, and solve them numerically over finite fields using the Laporta
algorithm Laporta:2000dsw  through FiniteFlow’s linear solver. We then perform
a Laurent expansion of the rational coefficients around $\epsilon=0$, and map
the canonical master integrals onto square roots and the special function
basis of Ref. Badger:2021nhg  up to the required order in $\epsilon$. We label
the special function basis by $\\{f_{k}\\}$. We truncate the $\epsilon$
expansion at order $\epsilon^{2}$ at one loop and at order $\epsilon^{0}$ at
two loops. Finally, we subtract the UV/IR poles as in Eq. (15) and define the
contracted finite remainders, which we represent as

$\displaystyle\begin{aligned}
&\tilde{F}_{5,i}^{(L),h_{1}h_{2}h_{3}h_{4}}=\Phi^{h_{1}h_{2}h_{3}h_{4}}_{5}\sum_{j}\left[q_{i,j}^{h_{1}h_{2}h_{3}h_{4}}\left(\vec{s}_{5}\right)+\operatorname{\rm
tr}_{5}\,r_{i,j}^{h_{1}h_{2}h_{3}h_{4}}\left(\vec{s}_{5}\right)\right]\text{mon}_{j}\left(\operatorname{\rm
tr}_{5},\sqrt{\Delta_{3}^{(l)}},\\{f_{k}\\}\right)\,,\\\
&\tilde{F}_{4,i}^{(L)}=\sum_{j}t_{i,j}\left(\vec{s}_{5}\right)\text{mon}_{j}\left(\operatorname{\rm
tr}_{5},\sqrt{\Delta_{3}^{(l)}},\\{f_{k}\\}\right)\,,\\\ \end{aligned}$ (53)

where $\text{mon}_{j}(x,y,\ldots)$ denotes monomials in $x,y,\ldots$, while
$q_{i,j}^{h_{1}h_{2}h_{3}h_{4}}$, $r_{i,j}^{h_{1}h_{2}h_{3}h_{4}}$ and
$t_{i,j}$ are rational functions of $\vec{s}_{5}$. Note that we pull out from
the five-particle finite remainders an arbitrary phase factor
$\Phi^{h_{1}h_{2}h_{3}h_{4}}_{5}$ carrying all the helicity weights, so that
the coefficients $q_{i,j}^{h_{1}h_{2}h_{3}h_{4}}$ and
$r_{i,j}^{h_{1}h_{2}h_{3}h_{4}}$ are scalar and hence functions of
$\vec{s}_{5}$ only. We recall that the helicity configuration is assigned to
the four-particle finite remainders when attaching the decay current, as
discussed in Section 3.3. The cancellation of the poles at this stage provides
a robust check of the result prior to the rational reconstruction.
Furthermore, it typically leads to simplifications which make the finite
remainders easier to reconstruct than the bare amplitudes. This chain of
operations is implemented in the FiniteFlow framework, and ultimately amounts
to an algorithm which samples numerically over finite fields the rational
coefficients in the contracted finite remainders.

Finally, we need to reconstruct the rational coefficients of the contracted
finite remainders from their numerical values. Following Refs. Abreu:2018zmy ;
Badger:2021nhg ; Badger:2021imn ; Badger:2021ega , we perform a number of
optimisations to reduce the number of required sample points. We follow the
strategy outlined in Ref. Badger:2021imn . First of all, we set $s_{12}=1$. We
recover the analytic dependence on $s_{12}$ a posteriori through dimensional
analysis. Second, we fit the $\mathbb{Q}$-linear relations among the rational
coefficients, and solve them so as to express the most complicated
coefficients in terms of the simplest ones. Third, we reconstruct the
coefficients on a random univariate phase-space slice modulo a large prime
number, and match them with ansätze made of the following factors (with
$s_{12}=1$):

$\displaystyle\begin{aligned}
\biggl{\\{}&s_{12}\,,s_{23}\,,s_{34}\,,s_{23}+s_{34}\,,s_{23}-s_{234}\,,s_{234}-s_{34}\,,s_{123}-s_{56}\,,s_{234}-s_{56}\,,s_{12}-s_{123}+s_{23}\,,\\\
&s_{12}+s_{234}-s_{34}\,,s_{23}-s_{234}+s_{34}\,,s_{12}+s_{234}-s_{56}\,,s_{12}-s_{123}-s_{34}\,,s_{123}s_{234}-s_{23}s_{56}\,,\\\
&s_{12}-s_{123}+s_{23}-s_{34}\,,s_{123}-s_{23}+s_{234}-s_{56}\,,s_{12}+s_{234}-s_{34}-s_{56}\,,\\\
&s_{12}s_{234}-s_{123}s_{234}+s_{23}s_{234}-s_{234}s_{34}+s_{34}s_{56}\,,\\\
&s_{12}s_{234}-s_{123}s_{234}-s_{234}s_{34}+s_{23}s_{56}+s_{34}s_{56}\,,s_{12}s_{234}+s_{234}^{2}-s_{234}s_{34}-s_{234}s_{56}+s_{34}s_{56}\,,\\\
&s_{12}s_{123}+s_{123}s_{234}-s_{123}s_{34}-s_{12}s_{56}-s_{23}s_{56}\,,s_{12}s_{123}-s_{123}^{2}-s_{123}s_{34}-s_{12}s_{56}+s_{123}s_{56}\,,\\\
&s_{12}^{2}-s_{12}s_{123}+s_{12}s_{234}-s_{123}s_{234}-2s_{12}s_{34}+s_{123}s_{34}-s_{234}s_{34}+s_{34}^{2}+s_{23}s_{56}\,,\\\
&s_{12}s_{23}-s_{12}s_{234}+s_{23}s_{234}-s_{234}^{2}-s_{23}s_{34}+s_{234}s_{34}-s_{23}s_{56}+s_{234}s_{56}-s_{34}s_{56}\,,\\\
&\lambda\left(s_{12},s_{34},s_{56}\right)\,,\lambda\left(s_{23},s_{14},s_{56}\right)\,,\operatorname{\rm
tr}_{5}^{2}\biggr{\\}}\,,\end{aligned}$ (54)

where $\lambda$ is the Källen function,

$\displaystyle\lambda(a,b,c)=a^{2}+b^{2}+c^{2}-2ab-2bc-2ca\,.$ (55)

The factors in the list (54) are a subset of the letters of the (planar) one-
mass pentagon alphabet Abreu:2020jxa , namely of the arguments of the
logarithmic integration kernels appearing in the differential equations
satisfied by the master integrals. The letters govern the singularity
structure of the master integrals and hence of the special function basis
$\\{f_{k}\\}$. It is therefore natural to expect that the denominators of the
rational coefficients multiplying the special functions should factorise in
terms of letters, and indeed the previous experience has shown that this is
the case Abreu:2018zmy ; Badger:2021nhg ; Badger:2021imn ; Badger:2021ega . It
follows that we can determine entirely the denominators of the rational
coefficients by matching them against ansätze made of the factors in Eq. (54)
on a univariate slice. Part of the numerators may in general be caught by this
approach as well. In Table 1 we show the impact of this strategy on the
highest polynomial degrees of the rational coefficients which need to be
reconstructed for the five-particle contracted finite remainders. Note that we
process all helicity configurations of the five-particle finite remainders
simultaneously, but for the $n_{f}^{0}$ ones we separate the contractions by
the external momenta into two subsets, $\\{p_{1},p_{2}\\}$ and
$\\{p_{3},p_{4}\\}$, to reduce the memory usage. After this optimisation is
done, the rational coefficients are reconstructed using the multivariate
functional reconstruction algorithms implemented in FiniteFlow Peraro:2019svx
.

| $s_{12}=1$ | linear relations | factor matching
---|---|---|---
$\tilde{F}_{5,i}^{(2),1\,h_{1}h_{2}h_{3}h_{4}}$ with $i=1,2$ | $44/44$ | $41/40$ | $41/0$
$\tilde{F}_{5,i}^{(2),1\,h_{1}h_{2}h_{3}h_{4}}$ with $i=3,4$ | $48/47$ | $42/42$ | $42/0$
$\tilde{F}_{5,i}^{(2),n_{f}\,h_{1}h_{2}h_{3}h_{4}}$ with $i=1,2,3,4$ | $39/38$ | $26/24$ | $26/0$
Table 1: Maximal total polynomial degrees of the rational coefficients of the
contracted two-loop five-particle finite remainders at each stage of the
optimisation procedure for the finite-field reconstruction, in the form
numerator/denominator. The coefficients are functions of the five scalar
invariants $\\{s_{23},s_{34},s_{123},s_{234},s_{56}\\}$ ($s_{12}=1$). The
independent helicity configurations (46) are processed simultaneously, while
the contractions by the external momenta for
$\tilde{F}_{5,i}^{(2),1\,h_{1}h_{2}h_{3}h_{4}}$ are separated into two subsets
to reduce the memory usage.

###  3.5 Simplification of the rational coefficients

The resulting analytic expressions of the rational coefficients of the finite
remainders are rather bulky. The standard approach to simplify them relies on
partial fraction decomposition, either multivariate Leinartas:1978 ;
Raichev:2012 ; Abreu:2019odu ; Boehm:2020ijp ; Heller:2021qkz ; Bendle:2021ueg
or univariate with respect to a suitable variable Badger:2021nhg ;
Badger:2021imn ; Badger:2021ega ; Abreu:2021asb . For the rational
coefficients of the four-point finite remainders,
$t_{i,j}\left(\vec{s}_{5}\right)$ in Eq. (53), we achieve a satisfactory
simplification by performing a multivariate partial fraction decomposition
with the Mathematica package MultivariateApart Heller:2021qkz , enhanced by
Singular DGPS  for the computation of the Gröbner bases.

The rational coefficients of the five-particle finite remainders are instead
substantially more involved. In order to simplify them, we look for a
parameterisation of the five-particle kinematics leading to more compact
expressions than the scalar invariants $\vec{s}_{5}$ (29). We investigate how
the complexity of the expressions varies when using momentum-twistor
parameterisations Hodges:2009hk . The pseudo-scalar invariant
$\operatorname{\rm tr}_{5}$ is given by a rational function in terms of
momentum-twistor variables, and we can thus add up the two terms of the
coefficients of the special function monomials,

$\displaystyle\left(q_{i,j}^{h_{1}h_{2}h_{3}h_{4}}\left(\vec{s}_{5}\right)+\operatorname{\rm
tr}_{5}\,r_{i,j}^{h_{1}h_{2}h_{3}h_{4}}\left(\vec{s}_{5}\right)\right)\biggl{|}_{\vec{s}_{5}=\vec{s}_{5}(\vec{z})}=u_{i,j}^{h_{1}h_{2}h_{3}h_{4}}\left(\vec{z}\right)\,,$
(56)

where by $\vec{z}=\\{z_{i}\\}_{i=1,\ldots,6}$ we denote generally the
independent momentum-twistor variables. In particular, we consider the
parameterisation proposed in Ref. Badger:2021ega ,

$\displaystyle z_{1}=s_{12}\,,\qquad$ $\displaystyle
z_{2}=-\frac{\operatorname{\rm tr}_{+}(1234)}{s_{12}s_{34}}\,,$ (57)
$\displaystyle z_{3}=\frac{\operatorname{\rm
tr}_{+}(1341(5+6)2)}{s_{13}\operatorname{\rm tr}_{+}{(14(5+6)2)}}\,,\qquad$
$\displaystyle z_{4}=\frac{s_{23}}{s_{12}}\,,$ $\displaystyle
z_{5}=-\frac{\operatorname{\rm
tr}_{-}(1(2+3)(1+5+6)(5+6)23)}{s_{23}\operatorname{\rm
tr}_{-}(1(5+6)23)}\,,\qquad$ $\displaystyle z_{6}=\frac{s_{456}}{s_{12}}\,.$

In previous applications, such a parameterisation has been used globally, i.e.
in all amplitudes/finite remainders irrespective of their helicity
configuration. We find that this approach does not perform well in this case,
and does not lead to a major simplification in comparison with the expressions
in terms of scalar invariants $\vec{s}_{5}$ and $\operatorname{\rm tr}_{5}$.
The rational parameterisation has the effect of breaking some symmetries in
the kinematic quantities, which results in some configurations being simpler
than others. There is no reason for the parameterisation to be a global
choice, and in this case we exploit this fact and consider different
parameterisations for each helicity configuration.

In practice, we consider all parameterisations which are obtained by permuting
the massless momenta on the right-hand side of Eqs. (57). For each helicity
configuration we determine which permutations of the parameterisation lead to
the most compact expression of the finite remainder at one loop. We then use
them at two loops, and select the one which results in the simplest
expression. We perform the change of variables over finite fields within the
FiniteFlow framework, and measure the “simplicity” of the rational
coefficients in terms of their numerator/denominator polynomial degrees, which
can be determined without reconstructing the expression of the coefficients in
terms of the new variables. Once the “best” parameterisation
$\vec{s}_{5}=\vec{s}_{5}(\vec{z})$ is chosen for each helicity configuration,
we reconstruct the analytic expression of the coefficients in terms of the new
variables $\vec{z}$. For this purpose we make use of the finite field
algorithm for univariate partial fraction decomposition presented in Refs.
Badger:2021nhg ; Badger:2021imn . We choose the variable to partial fraction
with respect to so as to minimise the polynomial degrees of the separate terms
of the decomposition. Breaking down the coefficients into univariate partial
fractions simplifies the subsequent multivariate partial fraction
decomposition, which we perform using MultivariateApart Heller:2021qkz
enhanced with Singular DGPS . We apply it to each term of the univariate
partial fraction decomposition separately, which is convenient as each term is
by itself much simpler than the full coefficient. This is possible because
MultivariateApart’s algorithm commutes with summation by design. The spurious
poles introduced by the univariate partial fraction decomposition therefore
cancel out after the multivariate partial fraction decomposition. In summary,
our algorithm for the simplification of the rational coefficients of the five-
particle finite remainders is the following.

  1. 1.

Try all permutations of a given momentum-twistor parameterisation on the one-
loop expressions and select the ones which lead to the lowest polynomial
degrees.

  2. 2.

Apply the parameterisations selected at step 1 on the two-loop rational
coefficients and choose the one which leads to the lowest polynomial degree.

  3. 3.

Decompose the two-loop rational coefficients in terms of the new variables
into univariate partial fractions with respect to the variable which leads to
the lowest polynomial degrees in the separate terms.

  4. 4.

Decompose into multivariate partial fractions the separate terms of the
univariate partial fractions using the algorithm of Ref. Heller:2021qkz , and
sum them up cancelling the spurious poles.

In hindsight, the first three steps could have been implemented within the
original FiniteFlow setup. We did not attempt this approach because we did not
need any further optimisation to reconstruct the rational coefficients of the
$W$-production five-particle finite remainders. However, we believe that this
strategy may also help to improve the rational reconstruction.

We apply this procedure separately on each of the helicity configurations,
leading to different parameterisations for each of them. The resulting
expressions for the coefficients are remarkably more compact than the original
ones in terms of the scalar invariants $\vec{s}_{5}$. For the most complicated
finite remainder we achieved a compression in the file size of more than two
orders of magnitude. The evaluation time of the rational coefficients is
similarly improved.

###  3.6 Numerical evaluation and permutations of the amplitudes

In order to obtain the values of all the amplitudes in all the possible
scattering channels we need to evaluate the minimal set of independent objects
we reconstructed for different permutations of the external momenta (see e.g.
Eq. (8) for an explicit example). In this subsection we discuss how we
implement this operation in an efficient way at the level of the numerical
evaluation.

We denote a generic permutation of the external momenta by

$\displaystyle\sigma=\left(\sigma_{1}\sigma_{2}\sigma_{3}\sigma_{4}\sigma_{5}\sigma_{6}\right)\,,$
(58)

where the $\sigma_{i}$’s take distinct values in $\\{1,2,3,4,5,6\\}$, such
that the action of $\sigma$ on an external momentum is given by

$\displaystyle\sigma\circ p_{i}=p_{\sigma_{i}}\,.$ (59)

Not all $S_{6}$ permutations of $\\{1,2,3,4,5,6\\}$ are needed for this
application. The required permutations belong to the subset $S_{4}\times
Z_{2}$, i.e. they are obtained by composing an $S_{4}$ permutation of
$\\{p_{1},p_{2},p_{3},p_{4}\\}$ and a $Z_{2}$ exchange of $\\{p_{5},p_{6}\\}$.
In particular, $p_{5}$ and $p_{6}$ need to be exchanged in order to obtain the
$W^{-}\gamma j$ amplitudes according to Eq. (18). Only the $S_{4}$
permutations are relevant for the $W$-production amplitudes (and hence for the
special functions), since $p_{5}$ and $p_{6}$ enter them only in the sum
$p_{5}+p_{6}$ (see e.g. Eqs. (19) and (20)). The $Z_{2}$ exchange is relevant
only for the leptonic currents ($L_{A,\mu}$ and $L^{e/W}_{B,\mu}$), which are
rational functions.

Given a generic amplitude/finite remainder $A$, function of the external
momenta $\\{p_{i}\\}$, we define its permutation $\sigma$ as

$\displaystyle\left(\sigma\circ
A\right)\left(\\{p_{i}\\}\right)=A\left(\\{\sigma\circ p_{i}\\}\right)\,.$
(60)

In other words, we can obtain the value of the permuted amplitude by
evaluating the amplitude in the original orientation of the external momenta
at a permuted phase space point. While this operation is trivial for the
rational functions, it is in general very subtle for the special functions.
The reason is that a permutation in general maps the phase-space point to a
different scattering region. This would require a complicated analytic
continuation, since the special functions have a very intricate branch cut
structure.

One way to overcome this problem is to evaluate the special functions
numerically using the generalised series expansion method Moriello:2019yhu ,
implemented in the public Mathematica package DiffExp Hidding:2020ytt , as
done in Refs. Badger:2021nhg ; Badger:2021ega . Within this method the
analytic continuation can be carried out systematically. This approach however
requires that, for each phase-space point where we want to evaluate the
permuted amplitudes, we evaluate the special functions at as many points as
the number of needed permutations.

For phase-space points in the physical scattering region we can adopt a much
more efficient evaluation strategy: we use the C++ package PentagonFunctions++
Chicherin:2021dyp , which allows us to evaluate in the physical scattering
region a larger basis of special functions, named one-mass pentagon functions.
We denote them by $\\{g_{i}\\}$. For this purpose we translate the special
function basis $\\{f_{i}\\}$ of Ref. Badger:2021nhg  to the one-mass pentagon
function basis $\\{g_{i}\\}$ implemented in PentagonFunctions++. The
translation takes the form

$\displaystyle
f_{i}=\sum_{j}w_{ij}\,\text{mon}_{j}\left(\\{g_{k}\\}\right)\,,$ (61)

where $w_{ij}\in\mathbb{Q}$, and the sum runs over all the required monomials
of the one-mass pentagon functions $\\{g_{k}\\}$. We obtain the transformation
rules (61) by matching the expressions of the master integrals in terms of
special special functions given in Ref. Badger:2021nhg  with that of Ref.
Chicherin:2021dyp . The advantage of the one-mass pentagon functions with
respect to the function basis $\\{f_{i}\\}$ or Ref. Badger:2021nhg  is that
their evaluation through the package PentagonFunctions++ is extremely
efficient, and their design allows us to generate the values of all $S_{4}$
permutations of the functions from those at the un-permuted phase-space point.
The one-mass pentagon function basis $\\{g_{i}\\}$ is in fact closed under
$S_{4}$ permutations. This means that, for any $\sigma\in S_{4}$, we can
express the permuted one-mass pentagon functions evaluated at a given phase-
space point as a combination of un-permuted pentagon functions evaluated at
the same point,

$\displaystyle\left(\sigma\circ g_{i}\right)(\vec{s}_{5},\operatorname{\rm
tr}_{5})=\sum_{j}\Sigma_{ij}^{(\sigma)}\text{mon}_{j}\left[\\{g_{k}\left(\vec{s}_{5},\operatorname{\rm
tr}_{5}\right)\\}\right]\,,$ (62)

where $\Sigma_{ij}^{(\sigma)}\in\mathbb{Q}$, and we spelled out the dependence
on the kinematics for the sake of clarity. These transformation rules are
provided in Ref. Chicherin:2021dyp .111Note that Ref. Chicherin:2021dyp  has a
different labelling of the external momenta. Moreover, the package
PentagonFunctions++ works in a specific physical scattering region (the
$s_{45}$ channel using the notation of Ref. Chicherin:2021dyp ). A relabelling
and a further permutation of the momenta are required to use
PentagonFunctions++ in the scattering region relevant for our application. We
implemented these operations in the Mathematica evaluation script provided in
the ancillary files, and refer to the original work Chicherin:2021dyp  for a
discussion of how to use PentagonFunctions++ in a physical region different
from the default one. This strategy is advantageous because it minimises the
number of evaluations of the special functions, which is the most time-
consuming step in the numerical evaluation of the colour and helicity summed
squared amplitudes.

It is worth highlighting the special behaviour of the pseudo-scalar invariant
$\operatorname{\rm tr}_{5}$ in this chain of operations. In the physical
scattering regions the reality of the momenta implies that $\operatorname{\rm
tr}_{5}^{2}<0$. In other words, $\operatorname{\rm tr}_{5}$ is purely
imaginary. The library PentagonFunctions++ always assumes that
$\text{Im}\left[\operatorname{\rm tr}_{5}\right]>0$. The sign of
$\operatorname{\rm tr}_{5}$ however may change upon the action of an odd-
signature permutation,

$\displaystyle\sigma\circ\text{tr}_{5}=\text{sign}(\sigma)\,\text{tr}_{5}\,,$
(63)

or space-time parity. The values of the one-mass pentagon functions for a
negative imaginary part of $\operatorname{\rm tr}_{5}$ can be obtained by
flipping the sign of a subset of functions specified in Ref. Chicherin:2021dyp
. In our setup however we do not need to. As discussed in Section 3.4, we
reduce the amplitudes to manifestly scalar master integrals, and group
together the special functions and the square roots arising from the
definition of the canonical master integrals. As a result, the monomials of
special functions and square roots in the finite remainders (53) are scalar as
well. Any sign change in the pentagon functions due to permutations or space-
time parity is therefore compensated by that of the accompanying factor of
$\operatorname{\rm tr}_{5}$, and we can thus evaluate both with a value of
$\operatorname{\rm tr}_{5}$ such that $\text{Im}\left[\operatorname{\rm
tr}_{5}\right]>0$ — as by default in PentagonFunctions++ — regardless of the
permutations or space-time parity. We must only keep track of the sign of
$\operatorname{\rm tr}_{5}$ in the rational coefficients (see Eqs. (53)),
which enters our final expressions for the five-particle finite remainders
through the values of the momentum twistors, and is determined by the values
of the external momenta through its definition (27). The same holds for the
other square roots in the problem, $\sqrt{\Delta_{3}^{(i)}}$, which appear
only in the special function monomials. The polynomials $\Delta_{3}^{(i)}$ are
positive in the physical scattering regions. We adopt the convention of Ref.
Chicherin:2021dyp  that their square roots are positive,
$\sqrt{\Delta_{3}^{(i)}}>0$, as done in PentagonFunctions++.

In conclusion, we reconstruct the analytic expressions for the minimal set of
independent finite remainders, and generate the values of the remaining ones
by permuting the former at the numerical evaluation stage. We do this by
evaluating the rational coefficients at permuted points, whereas we obtain the
values of all permutations of the special functions from the values of the
functions at the original phase-space point only. This allows us to minimise
the amount of analytic data, whose size may otherwise become problematic, and
at the same time evaluate the results efficiently.

##  4 Validation

In this section we discuss a number of validations performed on the analytic
results derived in this work. First, let us remind the readers that the
quantities that we reconstructed analytically are the $L$-loop finite
remainders, where the UV and IR poles contained in the $L$-loop bare
amplitudes are cancelled by the pole terms according to Eq. (15). These pole
cancellations already provide a strong consistency check of our calculation.
In the following subsections we present further checks.

###  4.1 Comparison against full six-point computation

In order to verify the analytic expressions obtained by detaching the leptonic
decay current as described in Section 3, we cross-check them against the
helicity amplitudes obtained by computing the six-point process directly using
a framework that has been applied to the computation of several two-loop
amplitudes Hartanto:2019uvl ; Badger:2021owl ; Badger:2021imn ; Badger:2021ega
. We perform the full six-point computation numerically using the momentum
twistor parameterisation (31) by assigning rational values to the variables
$x_{1},\ldots,x_{8}$ in the rational coefficients and treating the special
functions symbolically. We derive numerical results for all the sub-amplitudes
— $A^{(L)}_{6,u}$, $A^{(L)}_{6,d}$, $A^{(L)}_{6,W}$ and $A^{(L)}_{6,e}$ — in
all four contributing helicity configurations. We find full numerical
agreement between the two approaches. This provides a further robust check of
our analytic computation, where we derived analytic expressions only for the
independent helicity configurations and obtained the remaining ones by complex
conjugation and permutation of the external momenta.

###  4.2 Gauge invariance

The gauge-invariance structure of the $W^{+}\gamma j$ amplitude is slightly
complicated by the different sources of photon emission, as discussed in
Section 2. The individual sub-amplitudes ($A^{(L)}_{6,i}$ with $i=u,d,W,e$)
are not separately gauge invariant in the electroweak (EW) sector. Only linear
combinations of them, defined in Eq. (6), are. We rewrite them here for
convenience,

$\left\\{A^{(L)}_{6,u}+\frac{1}{s_{156}-s_{56}}A^{(L)}_{6,W}\,,\quad
A^{(L)}_{6,d}-\frac{1}{s_{156}-s_{56}}A^{(L)}_{6,W}\,,\quad
A^{(L)}_{6,e}-\frac{1}{s_{156}-s_{56}}A^{(L)}_{6,W}\right\\}\,.$ (64)

We verify explicitly that these combinations satisfy the EW Ward identity by
replacing the photon polarisation vector with its momentum
($\varepsilon(p_{1})\to p_{1}$) and checking that the resulting expressions
vanish.

The QCD Ward identity (performed by replacing the gluon polarisation vector
with its momentum, $\varepsilon(p_{3})\to p_{3}$), instead, is already
satisfied by the individual sub-amplitudes. We checked this explicitly as
well.

We further demonstrate the gauge invariance by evaluating the helicity
amplitudes using two different sets of reference momenta for the photon and
gluon polarisation vectors ($q_{1}$ and $q_{3}$), finding perfect agreement.

###  4.3 Renormalisation scale dependence

The $L$-loop finite remainders depend on the renormalisation scale, $\mu$. In
deriving analytic results for the process $W^{+}\gamma j$ we set the
renormalisation scale to unity ($\mu=1$). The dependence on the
renormalisation scale of the finite remainders can be restored as follows,

$F^{(L),i}_{6}\left(\mu^{2}\right)=F^{(L),i}_{6}\left(\mu^{2}=1\right)+\delta
F^{(L),i}_{6}\left(\mu^{2}\right)\,,$ (65)

where we omitted the dependence on the external momenta to simplify the
notation. The $\mu$-restoring terms $\delta F_{6}^{(L),i}$ are built out of
the lower-loop finite remainders evaluated at $\mu^{2}=1$ and logarithms of
$\mu^{2}$. Explicitly, they are given by

$\displaystyle\delta
F^{(1),1}_{6}\left(\mu^{2}\right)=\frac{11}{6}A_{6}^{(0)}\log\left(\mu^{2}\right)\,,$
(66) $\displaystyle\delta
F^{(1),n_{f}}_{6}\left(\mu^{2}\right)=-\frac{1}{3}A_{6}^{(0)}\log\left(\mu^{2}\right)\,,$
(67) $\displaystyle\delta
F^{(2),1}_{6}\left(\mu^{2}\right)=\log(\mu^{2})\bigg{\\{}\bigg{(}\frac{1813}{216}-\frac{11}{36}\pi^{2}+8\zeta_{3}\bigg{)}A^{(0)}_{6}+\frac{11}{2}F^{(1),1}_{6}(1)\bigg{\\}}+\frac{121}{24}A^{(0)}_{6}\log^{2}(\mu^{2})\,,$
(68) $\displaystyle\delta
F^{(2),n_{f}}_{6}\left(\mu^{2}\right)=\log(\mu^{2})\bigg{\\{}\bigg{(}\frac{\pi^{2}}{18}-\frac{77}{18}\bigg{)}A^{(0)}_{6}-F^{(1),1}_{6}(1)+\frac{11}{2}F^{(1),n_{f}}_{6}(1)\bigg{\\}}-\frac{11}{6}A^{(0)}_{6}\log^{2}(\mu^{2})\,,$
(69) $\displaystyle\delta
F^{(2),n_{f}^{2}}_{6}\left(\mu^{2}\right)=\log(\mu^{2})\bigg{\\{}\frac{10}{27}A^{(0)}_{6}-F^{(1),n_{f}}_{6}(1)\bigg{\\}}+\frac{1}{6}A^{(0)}_{6}\log^{2}(\mu^{2})\,.$
(70)

The dependence on the external momenta is understood. We can then use Eq. (65)
to check that the finite remainders we computed have the correct scale
dependence. We do this by evaluating the finite remainders at two phase-space
points that are connected by a rescaling by some positive factor $a$,

$\displaystyle\vec{p}=(p_{1},p_{2},p_{3},p_{4},p_{5},p_{6})\,,$ (71)
$\displaystyle\vec{p}^{\,\prime}=a\,\vec{p}=(a\,p_{1},a\,p_{2},a\,p_{3},a\,p_{4},a\,p_{5},a\,p_{6})\,.$

These two evaluations allow us to confirm numerically that the finite
remainders exhibit the correct scaling behaviour, in the form

$\frac{F_{6}^{(L),i}\left(1,a\,\vec{p}\right)+\delta
F_{6}^{(L),i}\left(a^{2},a\,\vec{p}\right)}{A_{6}^{(0)}\left(a\,\vec{p}\right)}=\frac{F_{6}^{(L),i}\left(1,\vec{p}\right)}{A_{6}^{(0)}\left(\vec{p}\right)}\,,$
(72)

where we have made the dependence on the kinematic point explicit.

###  4.4 Tree-level and one-loop checks

We validated the tree-level and one-loop amplitudes derived in this paper
against the results available in the literature. For the tree-level amplitude
we compared our helicity amplitudes against the analytic results presented in
Ref. Campbell:2021mlr  and additionally, for the full colour tree-level
squared matrix elements, against Madgraph5 Alwall:2014hca  for both processes
$W^{+}\gamma j$ and $W^{-}\gamma j$. As for the one-loop amplitudes, we
compared our results against the leading colour contributions of the
$W^{+}\gamma j$ amplitudes presented in Ref. Campbell:2021mlr . In all cases
we find perfect agreement. We would like to point out that our choice of
reference vectors for the photon and the gluon is different from the one used
in Ref. Campbell:2021mlr . For this reason we compared the gauge invariant
combinations of sub-amplitudes shown in Eq. (64). This check therefore further
validates the gauge invariance of our result.

###  4.5 Four-point amplitude comparison

We performed a cross check of the four-point amplitudes $A_{4}^{(L)\mu}$ which
contribute to the sub-amplitudes $A^{(L)}_{6,W}$ and $A^{(L)}_{6,e}$ against
the results provided in Ref. Gehrmann:2011ab  for the scattering process
$q\bar{q}\to Vg$. In Ref. Gehrmann:2011ab  analytic results are presented for
the helicity coefficients which are linear combinations of the form factors
$b_{i}^{(L)}$ in Eq. (47), evaluated at $\mu^{2}=s_{234}$. In order to enable
a direct comparison for the one- and two-loop leading colour finite
remainders, we recomputed the $A_{4}^{(L)\mu}$ amplitudes in Eq. (47) using
the tensor structures employed in Ref. Gehrmann:2011ab . Since we compute the
finite remainders with $\mu^{2}=1$, we obtain the results at $\mu^{2}=s_{234}$
using the formulae to restore the dependence on $\mu$ shown in Section 4.3. We
obtain perfect numerical agreement for the helicity coefficients. We further
check that the four-particle finite remainders $F_{4}^{(L)\mu}$ computed using
the tensor structures of Ref. Gehrmann:2011ab  match the ones we derived using
the tensor structures defined in Section 3.3 after contracting them with the
decay currents according to Eq. (22).

##  5 Results

We provide analytic expressions for the five- and four-point contracted
amplitudes ($\tilde{A}^{(L)}_{5,i}$ and $\tilde{A}^{(L)}_{4,i}$), at one and
two loops, together with the decay currents ($L_{A,\mu}$, $L^{e}_{B,\mu}$,
$L^{W}_{B,\mu}$) and the relevant projection matrices ($\Delta$ (34) and
$\Omega$ (50)) in the ancillary files. The amplitudes are presented as linear
combinations of independent rational coefficients that multiply a monomial
basis of square roots and special functions.

We confirm the previous observations about the cancellation of the pentagon
functions involving certain letters Badger:2021nhg ; Badger:2021ega ;
Abreu:2021asb . We observe that the functions involving the letters
$\mathcal{Z}=\\{W_{18},W_{25},W_{34},W_{45},W_{46},W_{57}\\}$ (in the notation
of Refs. Abreu:2021smk ; Chicherin:2021dyp ) are present in the contributing
integrals but drop out from the amplitudes truncated at order
$\epsilon^{0}$,222This holds for the set of independent amplitudes we
reconstructed explicitly, which receive contributions only from the cyclic
permutations of the master integrals. Since the set of letters $\mathcal{Z}$
is not closed under all $S_{4}$ permutations, these letters are present in
some of the permuted amplitudes which contribute to the helicity and colour
summed squared finite remainders. and that the functions involving the letter
$W_{198}=\operatorname{\rm tr}_{5}$ are present in the amplitudes and drop out
from the finite remainders. The latter phenomenon is by now well established
in the case of fully massless scattering, where it has been linked to an
underlying cluster algebra structure of the letter alphabet Chicherin:2020umh
.

We include a Mathematica script to demonstrate the assembly of both the
$W^{+}\gamma j$ and $W^{-}\gamma j$ amplitudes, and to perform the numerical
evaluation of the finite remainders at a given kinematic point. We evaluate
the special functions in the physical scattering region using the package
PentagonFunctions++ Chicherin:2021dyp , as discussed in Section 3.6.

We use the following configuration of momenta,

$-p_{2}-p_{4}\to p_{1}+p_{3}+p_{5}+p_{6}\,,$ (73)

to define the six scattering channels for $pp\to W^{+}\gamma j$ production,

$\displaystyle\mathbf{u\bar{d}}:\quad
u(-p_{2})+\bar{d}(-p_{4})\to\gamma(p_{1})+g(p_{3})+\nu_{e}(p_{5})+e^{+}(p_{6})\,,$
(74)
$\displaystyle\mathbf{\bar{d}u}:\quad\bar{d}(-p_{2})+u(-p_{4})\to\gamma(p_{1})+g(p_{3})+\nu_{e}(p_{5})+e^{+}(p_{6})\,,$
$\displaystyle\mathbf{ug}:\quad
u(-p_{2})+g(-p_{4})\to\gamma(p_{1})+d(p_{3})+\nu_{e}(p_{5})+e^{+}(p_{6})\,,$
$\displaystyle\mathbf{gu}:\quad
g(-p_{2})+u(-p_{4})\to\gamma(p_{1})+d(p_{3})+\nu_{e}(p_{5})+e^{+}(p_{6})\,,$
$\displaystyle\mathbf{\bar{d}g}:\quad\bar{d}(-p_{2})+g(-p_{4})\to\gamma(p_{1})+\bar{u}(p_{3})+\nu_{e}(p_{5})+e^{+}(p_{6})\,,$
$\displaystyle\mathbf{g\bar{d}}:\quad
g(-p_{2})+\bar{d}(-p_{4})\to\gamma(p_{1})+\bar{u}(p_{3})+\nu_{e}(p_{5})+e^{+}(p_{6})\,,$

and similarly for $pp\to W^{-}\gamma j$ production,

$\displaystyle\mathbf{d\bar{u}}:\quad
d(-p_{2})+\bar{u}(-p_{4})\to\gamma(p_{1})+g(p_{3})+e^{-}(p_{5})+\bar{\nu}_{e}(p_{6})\,,$
(75)
$\displaystyle\mathbf{\bar{u}d}:\quad\bar{u}(-p_{2})+d(-p_{4})\to\gamma(p_{1})+g(p_{3})+e^{-}(p_{5})+\bar{\nu}_{e}(p_{6})\,,$
$\displaystyle\mathbf{dg}:\quad
d(-p_{2})+g(-p_{4})\to\gamma(p_{1})+u(p_{3})+e^{-}(p_{5})+\bar{\nu}_{e}(p_{6})\,,$
$\displaystyle\mathbf{gd}:\quad
g(-p_{2})+d(-p_{4})\to\gamma(p_{1})+u(p_{3})+e^{-}(p_{5})+\bar{\nu}_{e}(p_{6})\,,$
$\displaystyle\mathbf{\bar{u}g}:\quad\bar{u}(-p_{2})+g(-p_{4})\to\gamma(p_{1})+\bar{d}(p_{3})+e^{-}(p_{5})+\bar{\nu}_{e}(p_{6})\,,$
$\displaystyle\mathbf{g\bar{u}}:\quad
g(-p_{2})+\bar{u}(-p_{4})\to\gamma(p_{1})+\bar{d}(p_{3})+e^{-}(p_{5})+\bar{\nu}_{e}(p_{6})\,.$

The interference between the $L$-loop finite remainders and the tree-level
amplitudes summed over colour and helicity in the leading colour approximation
is given by

$\sum_{\mathrm{colour}}\sum_{\mathrm{helicity}}\mathcal{A}_{6}^{(0)*}\mathcal{F}_{6}^{(L)}=:2e^{2}g_{W}^{4}g_{s}^{2}n^{L}N_{c}^{2}\,\mathcal{H}^{(L)}\,,$
(76)

where the reduced squared finite remainder $\mathcal{H}^{(L)}$ is defined by

$\mathcal{H}^{(L)}=\sum_{\mathrm{helicity}}A_{6}^{(0)*}F_{6}^{(L)}\,,$ (77)

for all scattering channels given in Eqs. (74) and (75). The reduced squared
finite remainder obeys the same decomposition according to the closed fermion
loop contributions as $F_{6}^{(L)}$,

$\displaystyle\begin{aligned}
&\mathcal{H}^{(1)}=N_{c}\mathcal{H}^{(1),1}+n_{f}\mathcal{H}^{(1),n_{f}}\,,\\\
&\mathcal{H}^{(2)}=N_{c}^{2}\mathcal{H}^{(2),1}+N_{c}n_{f}\mathcal{H}^{(2),n_{f}}+n_{f}^{2}\mathcal{H}^{(2),n_{f}^{2}}\,.\end{aligned}$
(78)

We present a benchmark evaluation at the following phase-space point in the
physical scattering region specified by Eq. 73 (the momenta are given in units
of GeV),

$\displaystyle\begin{aligned}
p_{1}&=(88.551333054,-22.100690287,40.080353191,-75.805430956)\,,\\\
p_{2}&=(-500,0,0,-500)\,,\\\
p_{4}&=(328.32941922,-103.84961188,-301.93375538,76.494921387)\,,\\\
p_{2}&=(-500,0,0,500)\,,\\\
p_{5}&=(152.35810946,-105.88095966,-97.709638326,49.548385226)\,,\\\
p_{6}&=(430.76113825,231.83126183,359.56304052,-50.237875657)\,,\end{aligned}$
(79)

with $\operatorname{\rm tr}_{5}=2.167055i\cdot 10^{10}\;\mathrm{GeV}^{4}$. We
take the $W$ boson mass and width to be

$M_{W}=80.4109\;\mathrm{GeV}\,,\qquad\qquad\Gamma_{W}=2.0467\;\mathrm{GeV}\,.$
(80)

High precision values for the phase space point in Eq. (79) as well as the
input parameters in Eq. (80) are provided in the ancillary files. We present
in Tables 2 and 3 the values of the bare two-loop amplitudes normalised to the
tree-level amplitudes in the $\mathbf{u\bar{d}}$ scattering channel for each
individual sub-amplitude,

$\hat{A}_{6,i}^{(L),j}=\frac{A_{6,i}^{(L),j}}{A_{6,i}^{(0)}}\,,$ (81)

for $i=u,d,W,e$ and the two closed fermion loop contributions specified in Eq.
(7), namely $j=1,n_{f}$. The results are presented only for the two
independent helicity configurations ($\scriptstyle+++--+$ and
$\scriptstyle-++--+$). In Table 4 we show the values of the two-loop reduced
squared finite remainders normalised to the reduced squared tree-level
amplitudes,

$\hat{\mathcal{H}}^{(L)}=\frac{\mathcal{H}^{(L)}}{\mathcal{H}^{(0)}}\,,$ (82)

for all channels of both $pp\to W^{+}\gamma j$ and $pp\to W^{-}\gamma j$
production. We give analogous tables for the one-loop amplitudes in Appendix
B.

|  helicity |  $\epsilon^{-4}$ |  $\epsilon^{-3}$ |  $\epsilon^{-2}$ |  $\epsilon^{-1}$ |  $\epsilon^{0}$
---|---|---|---|---|---|---
$\hat{A}^{(2),1}_{6,u}$ |  $\scriptstyle+++--+$ |  2 |  -49.5288 |  $603.232+4.18740i$ |  $-4813.11-82.3401i$ |  $28289.7+713.980i$
|  $\scriptstyle-++--+$ |  2 |  -49.5288 |  $605.560+1.03233i$ |  $-4867.68-10.1740i$ |  $28904.1-84.4212i$
$\hat{A}^{(2),1}_{6,d}$ |  $\scriptstyle+++--+$ |  2 |  -49.5288 |  $606.017+4.37613i$ |  $-4883.27-87.6955i$ |  $29148.2+787.284i$
|  $\scriptstyle-++--+$ |  2 |  -49.5288 |  $604.589+4.36093i$ |  $-4848.83-90.4281i$ |  $28743.3+856.481i$
$\hat{A}^{(2),1}_{6,W}$ |  $\scriptstyle+++--+$ |  2 |  -49.5288 |  $605.100+3.07126i$ |  $-4859.29-58.7793i$ |  $28844.2+480.026i$
|  $\scriptstyle-++--+$ |  2 |  -49.5288 |  $605.637+2.40762i$ |  $-4871.59-43.1992i$ |  $28978.3+302.671i$
$\hat{A}^{(2),1}_{6,e}$ |  $\scriptstyle+++--+$ |  2 |  -49.5288 |  $605.140+2.93702i$ |  $-4860.19-55.6437i$ |  $28853.6+444.669i$
|  $\scriptstyle-++--+$ |  2 |  -49.5288 |  $606.606+2.97710i$ |  $-4894.35-56.2398i$ |  $29236.9+444.300i$
Table 2:  Bare two-loop helicity sub-amplitudes (normalised to the tree-level amplitudes as in Eq. (81)) without any closed fermion loop contribution for $W^{+}\gamma j$ production in the $\mathbf{u\bar{d}}$ scattering channel evaluated at the kinematic point given in Eq. (79). The results are shown for the two independent helicity configurations and obtained with $q_{1}=p_{3}$ and $q_{3}=p_{1}$ where $q_{1}$ ($q_{3}$) is the reference momentum for the photon (gluon) polarisation vector.  |  helicity |  $\epsilon^{-4}$ |  $\epsilon^{-3}$ |  $\epsilon^{-2}$ |  $\epsilon^{-1}$ |  $\epsilon^{0}$
---|---|---|---|---|---|---
$\hat{A}^{(2),n_{f}}_{6,u}$ |  $\scriptstyle+++--+$ |  0 |  0.333333 |  -7.39369 |  $79.8302+1.39580i$ |  $-556.215-14.3791i$
|  $\scriptstyle-++--+$ |  0 |  0.333333 |  -7.39369 |  $80.6063+0.34411i$ |  $-570.821+1.89741i$
$\hat{A}^{(2),n_{f}}_{6,d}$ |  $\scriptstyle+++--+$ |  0 |  0.333333 |  -7.39369 |  $80.7586+1.45871i$ |  $-576.048-17.8798i$
|  $\scriptstyle-++--+$ |  0 |  0.333333 |  -7.39369 |  $80.2827+1.45364i$ |  $-566.721-19.4318i$
$\hat{A}^{(2),n_{f}}_{6,W}$ |  $\scriptstyle+++--+$ |  0 |  0.333333 |  -7.39369 |  $80.4531+1.02375i$ |  $-569.113-9.88866i$
|  $\scriptstyle-++--+$ |  0 |  0.333333 |  -7.39369 |  $80.6321+0.802539i$ |  $-572.278-6.31448i$
$\hat{A}^{(2),n_{f}}_{6,e}$ |  $\scriptstyle+++--+$ |  0 |  0.333333 |  -7.39369 |  $80.4664+0.979007i$ |  $-569.368-9.14880i$
|  $\scriptstyle-++--+$ |  0 |  0.333333 |  -7.39369 |  $80.9551+0.992365i$ |  $-577.544-9.72557i$
Table 3:  Bare two-loop helicity sub-amplitudes (normalised to the tree-level amplitudes as in Eq. (81)) with one closed fermion loop for $W^{+}\gamma j$ production in the $\mathbf{u\bar{d}}$ scattering channel evaluated at the kinematic point given in Eq. (79). The results are shown for the two independent helicity configurations and obtained with $q_{1}=p_{3}$ and $q_{3}=p_{1}$ where $q_{1}$ ($q_{3}$) is the reference momentum for the photon (gluon) polarisation vector.  $W^{+}\gamma j$ |  $\mathrm{Re}\,\hat{\mathcal{H}}^{(2),1}$ |  $\mathrm{Re}\,\hat{\mathcal{H}}^{(2),n_{f}}$ |  $\mathrm{Re}\,\hat{\mathcal{H}}^{(2),n_{f}^{2}}$
---|---|---|---
$\mathbf{u\bar{d}}$ |  483.506205134 |  -222.568846475 |  22.1747738519
$\mathbf{\bar{d}u}$ |  462.732386147 |  -219.389809502 |  22.1747738519
$\mathbf{ug}$ |  894.669569294 |  -309.802310098 |  24.2425489305
$\mathbf{gu}$ |  796.031872994 |  -288.292629199 |  23.3127252902
$\mathbf{\bar{d}g}$ |  954.097242371 |  -317.336400774 |  24.2425489305
$\mathbf{g\bar{d}}$ |  898.961273740 |  -302.856612446 |  23.3127252902
$W^{-}\gamma j$ |  $\mathrm{Re}\,\hat{\mathcal{H}}^{(2),1}$ |  $\mathrm{Re}\,\hat{\mathcal{H}}^{(2),n_{f}}$ |  $\mathrm{Re}\,\hat{\mathcal{H}}^{(2),n_{f}^{2}}$
$\mathbf{d\bar{u}}$ |  498.332524932 |  -222.702160434 |  22.1747738519
$\mathbf{\bar{u}d}$ |  732.600496818 |  -268.121335492 |  22.1747738519
$\mathbf{dg}$ |  1786.14253164 |  -305.863467669 |  24.2425489305
$\mathbf{gd}$ |  1612.34790163 |  -407.732735568 |  23.3127252902
$\mathbf{\bar{u}g}$ |  320.710353060 |  -152.382317276 |  24.2425489305
$\mathbf{g\bar{u}}$ |  1300.37372328 |  -375.944229843 |  23.3127252902
Table 4:  Reduced squared finite remainders (normalised to the reduced squared
tree level amplitudes) for all closed fermion loop contributions and
scattering channels evaluated at the kinematic point given in Eq. (79) for
both $pp\to W^{+}\gamma j$ and $pp\to W^{-}\gamma j$ production.

In order to show the suitability and stability of our evaluation strategy, we
present in Figure 3 the evaluation of the reduced squared finite remainders on
a one-dimensional slice of the physical phase space for all channels of
$W^{+}\gamma j$ production. We begin by parameterising the momenta of the one-
mass five-particle process relevant for the $W$-production amplitudes as

$\displaystyle\begin{aligned}
&p_{1}^{\mu}=u_{1}\,\frac{\sqrt{s}}{2}\,\left(1,1,0,0\right)\,,\\\
&p_{2}^{\mu}=\frac{\sqrt{s}}{2}\,\left(-1,0,0,-1\right)\,,\\\
&p_{3}^{\mu}=u_{2}\,\frac{\sqrt{s}}{2}\,\left(1,\cos\theta,-\sin\phi\sin\theta,-\cos\phi\sin\theta\right)\,,\\\
&p_{4}^{\mu}=\frac{\sqrt{s}}{2}\,\left(-1,0,0,1\right)\,.\\\ \end{aligned}$
(83)

We fix the value of $\cos\theta$ by requiring that

$\displaystyle(p_{5}+p_{6})^{2}=M^{2}_{ll}\,,$ (84)

where $M^{2}_{ll}$ is the invariant mass of the leptonic pair. We then
parameterise the momenta of the leptonic pair,

$\displaystyle
p_{5}^{\mu}=u_{3}\,\frac{\sqrt{s}}{2}\left(1,\cos\theta_{ll},-\sin\phi_{ll}\sin\theta_{ll},-\cos\phi_{ll}\sin\theta_{ll}\right)\,,$
(85)

and $p_{6}$ follows from momentum conservation. We fix $u_{3}$ by requiring
that $p_{6}^{2}=0$. In order to define a univariate phase-space slice, we
choose

$\displaystyle s=10^{4}\,\text{GeV}^{2}\,,\qquad
M_{ll}=60\,\text{GeV}\,,\qquad\phi=\frac{1}{10}\,,\qquad
u_{1}=\frac{1}{7}\,,\qquad\theta_{ll}=\frac{\pi}{2}\,,\qquad\phi_{ll}=\frac{\pi}{3}\,.$
(86)

The remaining variable, $u_{2}$, is constrained to the interval
$[87/175,29/50]$. We chose these values arbitrarily so that the slice crosses
a number of spurious poles, i.e. points where the rational coefficients
diverge whereas the finite remainders stay finite. We checked explicitly that,
while approaching such spurious poles, the values of the rational coefficients
become larger and larger, while the finite remainders converge. This is a
robust check of the stability of the evaluation, since the convergence
requires large numerical cancellations among various terms of the finite
remainders. Figure 3 shows the plots of the reduced squared finite remainders
up to two-loop order for all channels of $W^{+}\gamma j$ production on the
univariate phase-space slice defined above.

(a) $\mathbf{u\bar{d}}$

(b) $\mathbf{\bar{d}u}$

(c) $\mathbf{ug}$

(d) $\mathbf{gu}$

(e) $\mathbf{\bar{d}g}$

(f) $\mathbf{g\bar{d}}$

Figure 3: Reduced squared finite remainders $\mathcal{H}^{(L)}$ at tree level,
one and two loops evaluated on the univariate phase-space slice defined by
Eqs. (83), (84) and (85), with the parameters given in Eq. (86), for all
channels of $W^{+}\gamma j$ production defined in Eq. (74).

##  6 Conclusions

In this article we have presented the two-loop leading colour QCD helicity
amplitudes for the process $W^{\pm}\gamma j$ for the first time. We have
obtained relatively compact analytic expressions that can be efficiently
evaluated across the full physical phase-space. We constructed the colour and
helicity summed finite remainders, and performed several validation tests.
This opens the path for precision phenomenological predictions at NNLO
accuracy in the strong coupling.

To obtain the best possible theoretical predictions it will be necessary to
improve upon the leading colour approximation taken in this article. While it
is expected that the leading colour contribution dominates, a quantitative
statement is not possible without explicit computation. Sub-leading colour
corrections require non-planar topologies to be taken into account, and
represent a considerable increase in analytic and algebraic complexity.
Progress on the relevant Feynman integrals has been made in this direction
quite recently Abreu:2021smk , although a few topologies contributing to the
full amplitudes are still missing. We note that the missing closed fermion
loop contributions, $A^{(2)}_{6,q}$, only require non-planar hexaboxes and
therefore could be considered on a shorter timescale.

We also hope that our approach to the simplification of the reconstructed
amplitudes will be of use in subsequent amplitude computations. An improved
understanding of how a rational parameterisation can be tuned to simplify a
particular rational coefficient would certainly be of great value. We expect
this to be of particular importance when dealing with sub-leading colour and
non-planar configurations, in which many different orderings appear
simultaneously. It would also be interesting to study the effect of this
method on the reconstruction of the amplitude, i.e. whether the reconstruction
is performed in terms of $s_{ij},\operatorname{\rm tr}_{5}$ variables or a
rational parametrisation.

###### Acknowledgements.

SZ wishes to thank Yang Zhang for useful discussions about partial fraction
decomposition. We would also like to thank Tiziano Peraro for useful
discussions and comments on the manuscript. This project received funding from
the European Union’s Horizon 2020 research and innovation programmes New level
of theoretical precision for LHC Run 2 and beyond (grant agreement No 683211),
and High precision multi-jet dynamics at the LHC (grant agreement No 772009).
HBH was partially supported by STFC consolidated HEP theory grant
ST/T000694/1. SZ gratefully acknowledges the computing resources provided by
the Max Planck Institute for Physics and by the Max Planck Computing & Data
Facility.

##  Appendix A Renormalisation Constants

In this appendix we list the values of the $\beta$ function coefficients and
anomalous dimensions relevant for the IR and UV structure of the amplitudes
discussed in Section 2 Becher:2009qa :

$\displaystyle\beta_{0}=$
$\displaystyle\;\frac{11}{3}C_{A}-\frac{4}{3}T_{F}n_{f}\,,$ (87)
$\displaystyle\beta_{1}=$
$\displaystyle\;\frac{34}{3}C_{A}^{2}-\frac{20}{3}C_{A}T_{F}n_{f}-4C_{F}T_{F}n_{f}\,,$
(88) $\displaystyle\gamma_{0}^{g}=$
$\displaystyle\;-\frac{11}{3}C_{A}+\frac{4}{3}T_{F}n_{f}\,,$ (89)
$\displaystyle\gamma_{1}^{g}=$
$\displaystyle\;C_{A}^{2}\left(-\frac{692}{27}+\frac{11\pi^{2}}{18}+2\zeta_{3}\right)+4C_{F}T_{F}n_{f}+C_{A}T_{F}n_{f}\left(\frac{256}{27}-\frac{2\pi^{2}}{9}\right)\,,$
(90) $\displaystyle\gamma_{0}^{q}=$ $\displaystyle\;-3C_{F}\,,$ (91)
$\displaystyle\gamma_{1}^{q}=$
$\displaystyle\;C_{F}^{2}\left(-\frac{3}{2}+2\pi^{2}-24\zeta_{3}\right)+C_{F}C_{A}\left(-\frac{961}{54}-\frac{11\pi^{2}}{6}+26\zeta_{3}\right)$
$\displaystyle\;+C_{F}T_{F}n_{f}\left(\frac{130}{27}+\frac{2\pi^{2}}{3}\right)\,,$
(92) $\displaystyle\gamma_{0}^{\mathrm{cusp}}=$ $\displaystyle\;4\,,$ (93)
$\displaystyle\gamma_{1}^{\mathrm{cusp}}=$
$\displaystyle\;\left(\frac{268}{9}-\frac{4\pi^{2}}{3}\right)C_{A}-\frac{80}{9}T_{F}n_{f}\,,$
(94)

where $C_{A}=N_{c}$, $C_{F}=(N_{c}^{2}-1)/(2N_{c})$, and $T_{F}=1/2$.

##  Appendix B One-Loop Results

We show in Table 5 the numerical values of the one-loop bare sub-amplitudes
normalised to the tree-level amplitudes ($\hat{A}^{(1),1}_{i}$ for
$i=u,d,W,e$) in the $\mathbf{u\bar{d}}$ scattering channel evaluated at the
kinematic point given in Eq. (79). The corresponding reduced squared tree-
level amplitudes $\mathcal{H}^{(0)}$ and reduced squared one-loop finite
remainders normalised to the reduced squared tree-level amplitudes
($\hat{\mathcal{H}}^{(1),j}$ with $j=1,n_{f}$) are shown in Table 6 for both
$W^{+}\gamma j$ and $W^{-}\gamma j$ production.

|  helicity |  $\epsilon^{-2}$ |  $\epsilon^{-1}$ |  $\epsilon^{0}$ |  $\epsilon^{1}$ |  $\epsilon^{2}$
---|---|---|---|---|---|---
$\hat{A}^{(1),1}_{u}$ |  $\scriptstyle+++--+$ |  -2 |  23.8477 |  $-138.615-2.09370i$ |  $523.949+12.3666i$ |  $-1448.23-21.7701i$
|  $\scriptstyle-++--+$ |  -2 |  23.8477 |  $-139.779-0.516164i$ |  $535.218-2.01397i$ |  $-1503.32+46.1044i$
$\hat{A}^{(1),1}_{d}$ |  $\scriptstyle+++--+$ |  -2 |  23.8477 |  $-140.008-2.18806i$ |  $539.871+13.7461i$ |  $-1538.48-30.4669i$
|  $\scriptstyle-++--+$ |  -2 |  23.8477 |  $-139.294-2.18046i$ |  $532.471+15.2170i$ |  $-1499.51-44.3866i$
$\hat{A}^{(1),1}_{W}$ |  $\scriptstyle+++--+$ |  -2 |  23.8477 |  $-139.550-1.53563i$ |  $534.185+8.26368i$ |  $-1503.99-6.62167i$
|  $\scriptstyle-++--+$ |  -2 |  23.8477 |  $-139.818-1.20381i$ |  $536.639+5.03856i$ |  $-1515.10+9.39205i$
$\hat{A}^{(1),1}_{e}$ |  $\scriptstyle+++--+$ |  -2 |  23.8477 |  $-139.570-1.46851i$ |  $534.360+7.61926i$ |  $-1504.71-3.48177i$
|  $\scriptstyle-++--+$ |  -2 |  23.8477 |  $-140.303-1.48855i$ |  $541.353+7.64165i$ |  $-1538.44-2.31498i$
Table 5:  Bare one-loop helicity sub-amplitudes (normalised to the tree-level amplitudes as in Eq. (81)) without any closed fermion loop contribution for $W^{+}\gamma j$ production in the $\mathbf{u\bar{d}}$ scattering channel evaluated at the kinematic point given in Eq. (79). The results are shown for the two independent helicity configurations and obtained with $q_{1}=p_{3}$ and $q_{3}=p_{1}$ where $q_{1}$ ($q_{3}$) is the reference momentum for the photon (gluon) polarisation vector.  $W^{+}\gamma j$ |  $\mathcal{H}^{(0)}\;[\times 10^{-10}\,\mathrm{GeV}^{-4}]$ |  $\mathrm{Re}\;\hat{\mathcal{H}}^{(1),1}$ |  $\mathrm{Re}\;\hat{\mathcal{H}}^{(1),n_{f}}$
---|---|---|---
$\mathbf{u\bar{d}}$ |  32.9224527109 |  -20.4269208141 |  4.22462265354
$\mathbf{\bar{d}u}$ |  35.8863373066 |  -20.0350027848 |  4.22462265354
$\mathbf{ug}$ |  4.84655650134 |  -26.9389515414 |  4.45445318051
$\mathbf{gu}$ |  15.2151742999 |  -25.3235043118 |  4.37533965902
$\mathbf{\bar{d}g}$ |  9.18270882925 |  -28.3542876136 |  4.45445318051
$\mathbf{g\bar{d}}$ |  26.4333120479 |  -27.3120879601 |  4.37533965902
$W^{-}\gamma j$ |  $\mathcal{H}^{(0)}\;[\times 10^{-10}\,\mathrm{GeV}^{-4}]$ |  $\mathrm{Re}\;\hat{\mathcal{H}}^{(1),1}$ |  $\mathrm{Re}\;\hat{\mathcal{H}}^{(1),n_{f}}$
$\mathbf{d\bar{u}}$ |  48.5521763841 |  -20.5759435967 |  4.22462265354
$\mathbf{\bar{u}d}$ |  5.60724308955 |  -25.0921274652 |  4.22462265354
$\mathbf{dg}$ |  0.161819754065 |  -53.2745933316 |  4.45445318051
$\mathbf{gd}$ |  2.59919214772 |  -35.7387232774 |  4.37533965902
$\mathbf{\bar{u}g}$ |  0.471356750696 |  -25.5067063821 |  4.45445318051
$\mathbf{g\bar{u}}$ |  27.6357549618 |  -32.8902240077 |  4.37533965902
Table 6:  Reduced squared tree-level amplitude (absolute) and reduced squared
one-loop finite remainders (normalised to the reduced squared tree amplitudes)
for the various closed fermion loop contributions and scattering channels of
both $pp\to W^{+}\gamma j$ and $pp\to W^{-}\gamma j$ production, evaluated at
the kinematic point given in Eq. (79).

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