# Electroweak corrections to Higgs boson production via W W fusion at the
future LHeC

Bowen Wang<EMAIL_ADDRESS>Zhejiang Institute of Modern Physics and School
of Physics, Zhejiang University, Hangzhou, Zhejiang 310027, China  School of
Physics, Hangzhou Normal University, Hangzhou, Zhejiang 311121, China  Kai
Wang<EMAIL_ADDRESS>Zhejiang Institute of Modern Physics and School of
Physics, Zhejiang University, Hangzhou, Zhejiang 310027, China  Hanying Xiong
<EMAIL_ADDRESS>Zhejiang Institute of Modern Physics and School of
Physics, Zhejiang University, Hangzhou, Zhejiang 310027, China

###### Abstract

Precision measurement of quark Yukawa couplings is a crucial aspect of Higgs
property study. Proposed as a future upgrade of the Large Hadron Collider
(LHC), the Large Hadron electron Collider (LHeC) provides opportunities to
probe quark Yukawa couplings with a high precision because of relatively low
rate from the QCD background as compared with that of the Higgs processes at
the LHC. For this purpose, it is important to have a precision prediction of
the Higgs production rate at the LHeC. The leading production channel of the
Higgs boson at the LHeC is via weak boson fusion (WBF) which has the unique
kinematic feature of forward tagging jets. As QCD corrections are suppressed
by the simple color structure of the process, electroweak radiative
corrections become particularly important. In this paper, we focus on the
electroweak radiative corrections at next-to-leading order for the Higgs-boson
production via charge current WBF processes at the LHeC. In the two
renormalization schemes we use, the loop corrections are respectively at the
level of $9\%$ and $18\%$ relative to the leading order result, for a center-
of-mass energy at $1.98$ TeV, and the next-to-leading order results in both
schemes agree up to a truncation error expected from the perturbative method.
The size of the EW corrections exceeds that from QCD radiations and is shown
to be important for the study of Higgs phenomenology.

##  I Introduction

Enormous efforts have been made to study the properties of the 125 GeV Higgs
boson at an increasing precision level since its discovery Aad:2012tfa ;
Chatrchyan:2012xdj . In particular, the measurement of Higgs couplings to
various particles Cadamuro:2019tcf ; Langford:2021osp ; Ordek:2022ppj  is an
important part of the endeavor. Recent experiments have achieved a percent
level accuracy in the extraction of Higgs couplings to gauge bosons and the
top quark ATLAS:2020qdt ; CMS:2020gsy , which are all in agreement with the
standard model (SM) predictions. These measurements are preferably carried out
in the leptonic decay channels of the Higg boson. For instance, in Ref.
CMS:2020gsy  the Higgs coupling to the $Z$ boson is extracted from
$H\rightarrow ZZ^{*}\rightarrow 4\,\text{leptons}$; the top quark Yukawa is
measured via the Higgs production associated with a top quark pair, with
$H\rightarrow\text{leptons}$.

In contrast, the hadronic final states from Higgs decays are much more
difficult to probe because of the large rate for the background jets (produced
not via intermediate Higgs bosons). The $H\rightarrow\tau^{+}\tau^{-}$ with
subsequent hadronic decays produces narrow $\tau$ jets, but the jet shape
alone is insufficient for identifying the process from the hadronic
background. Nonetheless, constraining of $H$-$\tau$ Yukawa coupling can be
carried out in a production mode commonly referred to as the “weak boson
fusion” (WBF), where a pair of intermediate W or Z bosons is radiated by the
colliding fermions and couples to a Higgs. The energetic incoming particles
evolve to a final state with a pair of fermions (at the parton level)
separated by a large rapidity gap, in association with the Higgs and other
particles produced in the central rapidity region. The transverse momenta of
the two fermions are of order the weak boson mass. This configuration of event
is successfully utilized for suppressing backgrounds in the search of
$H\rightarrow\tau\tau$ at the Large Hadron Collider (LHC), for both hadronic
and leptonic decays of $\tau$’s CMS:2017zyp ; ATLAS:2018ynr .

The identification of $H$-$b$ and $H$-$c$ couplings via WBF at hardron
colliders gets even more challenging CMS:2015ebl ; ATLAS:2016mzy  because of
the huge QCD multi-jet background. On the other hand, the Large Hadron
electron Collider (LHeC) LHeCStudyGroup:2012zhm  is proposed partly to measure
Higgs couplings such as $H$-$b$ and $H$-$c$. In this upgrade option for the
LHC, a beam of electrons will be aligned to collide with the 7 TeV protons.
The electron beam energy is considered to range in 50-200 GeV, making the
facility a factory to produce Higgs bosons mainly via WBF. Studies have shown
the prospect of making the measurement of Yukawa couplings at the LHeC, based
on the analysis using leading order (LO) matrix elements in the simulations
Han:2009pe ; LHeCStudyGroup:2012zhm ; LHeC:2020van ; Li:2019xwd . An account
for the next-to-leading order (NLO) corrections is needed both for a precise
knowledge of the signal cross section and for determination of various
selection criterions affected by the distortion of distributions at NLO.

Computation of NLO QCD and (part of) QED corrections for WBF production of the
Higgs has been performed by Blumlein Blumlein:1992eh , and Jager Jager:2010zm
. The QCD corrections to the total cross section are shown to be at the level
of a few percent, while their impact on differential distributions is found to
be more sizable, i.e. as large as $10\%\sim 20\%$. The dominant QED radiative
corrections are about -5%. As we will show, it is important to include all the
NLO EW processes, which give rise to substantial corrections in both
integrated cross section and differential distributions.

We note that part of the EW loop effect has been calculated in a study
Li:2019jba  to estimate the significance of the triple Higgs-self coupling at
the LHeC. However, the calculation only involves a small subset of the loop
diagrams associated with the coupling. We shall see that the background (with
no Higgs self coupling) cross section in this study can be reduced by
including all EW loop processes, which makes a negative contribution and
therefore yields a better constraint on the Higgs self coupling.

In this paper we present a calculation of the full NLO EW corrections to the
WBF cross section in $e$-$p$ collisions. To make the discussion general we
consider only the partonic scattering amplitude with an on-shell Higgs, and do
not simulate the final state to which the Higgs decays. For simplicity, the
focus will be on the charge current(CC) WBF, which is dominant over the
neutral current(NC) process.111The integrated cross section of CC WBF is about
5 times that of NC WBF at the energy regime of the LHeC. Moreover, the
outgoing electron of the NC process could be tagged in a broad kinematical
region where it could be distinguished from the CC process.

The past two decades have seen the fast development of automation programs for
the calculation of QCD and EW corrections for various scattering processes at
one loop accuracy Alwall:2014hca ; Buccioni:2019sur ; Sherpa:2019gpd ;
Bellm:2015jjp ; Alioli:2010xd ; Alioli:2012fc ; Kilian:2007gr . However, these
tools prioritize the processes in $p$-$p$ and $e^{+}$-$e^{-}$ scatterings, and
to our best knowledge, no implementation is made as yet in the publicly
available Monte-Carlo programs to fully automatize the NLO calculation for e-p
collisions. To treat the WBF process at the LHeC, we need to organize
different parts of the calculation on our own. The calculation proceeds in the
dipole subtraction formalism Catani:1996vz , first designed for a unified
treatment of various singularities encountered in scattering amplitudes with
real and virtual QCD radiations. The method was then adapted to the
calculation of QED radiative corrections and generalized to include the
radiative effect that is sensitive to the mass of the initial or final
fermions Dittmaier:1999mb ; Catani:2002hc ; Schonherr:2017qcj . We shall show
how to consistently include the fermion mass effect in our calculation of the
WBF process using the dipole method.

The rest of the paper is organized as follows. In Sec. II, we give the
description of the details for the analytical and numerical parts of the
calculation. In Sec. III, we discuss the numerical significance of the NLO EW
corrections in both total and differential cross sections of the WBF process.
Then we conclude the paper in Sec. IV.

##  II Detailed calculation of the process

###  II.1 LO contribution

The CC WBF $ep\rightarrow\nu_{e}jH$ is depicted at the leading order by the
partonic process $eq\rightarrow\nu_{e}q^{\prime}H$. An example diagram is
shown in Fig. 1. The cross section is given by

$\displaystyle\sigma^{LO}=\sum_{q}\int_{0}^{1}d\eta_{q}f_{q}(\eta_{q},\mu_{F})\frac{1}{2\hat{s}}d\Phi_{3}B(P_{A},p_{q}),$
(1)

where $B(P_{A},p_{q})$ denotes the born amplitude squared at the parton level
with the incoming electron and quark momenta $P_{A}$ and $p_{q}$. The partonic
center-of-mass (CM) energy squared is given by $\hat{s}=(P_{A}+p_{q})^{2}$,
and $d\Phi_{3}$ is the phase space element of the 3-body final state. In this
paper we assume no particular polarization for external particles, so that in
$B(P_{A},p_{q})$ and all other squared amplitudes below, we sum and average
over colors and spins of the initial and final states. $\eta_{q}$ is the
fraction of the proton momentum $P_{B}$ carried by the incoming quark $q$. The
hadronic cross section is factorized at the scale $\mu_{F}$ into a quark
distribution function $f_{q}(\eta_{b},\mu_{F})$, convolved with a squared hard
scattering amplitude depending implicitly on $\eta_{q}$. The evolution of
$f_{q}(\eta_{b},\mu_{F})$ introduces NLO QCD and QED terms to the LO cross
section, which is not a problem as long as we do not double count these terms
in our NLO calculation. The dominant contribution comes from the initial quark
flavors $u$, $c$, $\bar{d}$ and $\bar{s}$, while the effect of $\bar{b}$ and
$t$ is marginal and neglected.

Figure 1: A representative leading order diagram for CC WBF at the LHeC.

###  II.2 NLO EW corrections

The numerically dominant NLO EW corrections are from the loop diagrams of
various topologies, ranging from self energy graphs to pentagons. Soft and
collinear singularities may arise in some of these diagrams when photons are
present in the loop. Representative loop diagrams with and without these
singularities are shown in Fig. 3 and Fig. 2, respectively.

Figure 2: Representative 1-loop diagrams for CC WBF at the LHeC with no
soft/collinear divergences. Figure 3: Representative 1-loop diagrams for CC
WBF at the LHeC with soft/collinear divergences.

A consistent treatment of the singularities requires the inclusion of all
diagrams with real emission of photons , whose momenta can potentially be
either soft or collinear to the emittee. In properly defined observables,
cancellation of part of the singularities takes place between real-emission
and loop diagrams. Mass singularities associated with initial state radiations
are removed by the collinear counter terms provided by the factorization
procedure, to be discussed below. There are also real-emission processes
induced by photons that are treated as constituents of the initial state
protons. In this case, no corresponding loop diagrams are present and the
singular terms are cancelled solely by the collinear counter terms.

Within the energy range of the LHeC, we treat $u$, $d$, $s$, and $c$ (and
their anti-particles) as massless flavors, whose couplings with the Higgs are
also neglected. The contribution from $b$ and $t$ (and their anti-particles)
is marginal and not included. The role of the electron mass is somewhat
intricate and will be discussed later. With these simplifications at NLO,
there are in total 628 loop diagrams produced by the program MadGraph5_aMC@NLO
Alwall:2014hca , and 42 real emission diagrams by FeynArts Hahn:2000kx  with
quark, anti-quark, and photon initial states.

####  II.2.1 Renormalization

The calculation of one-loop diagrams has been performed in the t’Hooft-Feynman
gauge with the program MadLoop hirschi:2011pa . Several packages are linked by
MadLoop for the reduction of tensor integrals, each based on one of two
distinct procedures: the Tensor Integral Reduction Passarino:1978jh ;
Davydychev:1991va  and the Ossola-Papadopoulos-Pittau method Ossola:2006us .
For simplicity, we neglect quark mixing and do not renormalize the quark
mixing matrix.

The mass and field strength renormalization constants are determined within
the on-shell renormalization scheme, except for the field strength $Z_{AA}$ of
the photon that is closely related to the renormalization of the electric
charge. It can be obtained by imposing a renormalization condition at some
momentum scale, or equivalently, by specifying the form of the renormalized
fine structure constant. In this work we adopt two schemes. The $G_{\mu}$
scheme is defined by the choice Denner:2019vbn

$\displaystyle\alpha_{G_{\mu}}=\frac{\sqrt{2}G_{\mu}M_{W}^{2}}{\pi}\left(1-\frac{M_{W}^{2}}{M_{Z}^{2}}\right)\approx\frac{1}{132},$
(2)

where the Fermi constant $G_{\mu}$ is determined from the muon decay
experiment, while the $\alpha(M_{Z})$ scheme is obtained by evolving the
physical fine structure constant $\alpha(0)\approx 1/137$ from zero momentum
transfer to the scale $M_{Z}$, yielding $\alpha(M_{Z})\approx 1/129$. The
numerical value of $\alpha$ in both schemes differ from $\alpha(0)$, which is
obtained by requiring that all high order corrections to the electron-photon
3-point function vanish in the Thomson limit Denner:1991kt ; Denner:2019vbn .
The differences are sensitive to the light fermion masses on which the loop
contribution to the photon wave function depend. The sensitivity manifests as
logarithmic terms of the form $\ln m_{f}$ Ellis:2007qk ; Denner:2019vbn ,
indicating a collinear divergence in the on-shell photon wave function as the
fermion masses $m_{f}$ approach 0. It is straightforward to show that in the
$G_{\mu}$ and $\alpha(M_{Z})$ schemes, $Z_{AA}$, as well as the charge
renormalization constant $Z_{e}$, are free from the collinear divergence, such
that the definition of the physical charge on-shell is maintained. As long as
no external photons are present in the LO process, we are able to consistently
implement the renormalization constants and neglect the small mass $m_{f}$ in
the calculation of the loop diagrams with dimensional regularization.

####  II.2.2 Factorization

After factorizing the parton distribution functions (PDFs) of the incoming
quarks, their mass can be safely set to zero in the partonic scattering
amplitudes initiated by these flavors. The mass singularities from the initial
state photon-quark splitting signal our ignorance of the nonperturbative
interactions of the partons in the proton below some factorization scale
$\mu_{F}$, on which the QED evolution of the PDFs depends. The corresponding
poles in the hard scattering process are subtracted by adding a collinear
counter term to the squared amplitudes. In this calculation we use the PDF set
CT14qed_inc_proton Schmidt:2015zda  from the LHAPDF library Buckley:2014ana
with the NLO QED evolution. The collinear counter term is consistently
implemented in the $\overline{\mbox{MS}}$ factorization scheme.

On the other hand, the incoming electron has no internal structure to
accommodate for the singularity from collinear radiations. In principle, one
could keep $m_{e}$ throughout the calculation and retain the dependence of the
hard scattering amplitude on $\ln m_{e}$. This collinear singularity is
physical and would break the convergence of the perturbative calculation if
$m_{e}$ were too small. Fortunately, the size of the fine structure constant
at the electro-weak scale ensures the convergence of the perturbation series
in most of the kinematic regions we will consider (see Sec. III.3). However,
in reducing the tensor integrals we find that some of the pentagon diagrams
are sensitive to the electron mass, which leads to a significant disagreement
among different reduction programs when $m_{e}$ takes a small but non-zero
value. This numerical instability is cured if $m_{e}$ is neglected and the
mass singularity is regulated dimensionally.

With this observation, we factor out the mass singularity from the collinear
photon radiation off the incoming electron. The singular terms are then put in
a lepton distribution function, with the form to $\mathcal{O}(\alpha)$ in the
$\overline{\mbox{MS}}$ factorization scheme given by Liu:2021jfp  (Here and
below, $\alpha$ can be $\alpha_{G_{\mu}}$ or $\alpha(M_{Z})$, depending on the
renormalization scheme used)

$\displaystyle
f_{e}(x)=\delta(1-x)+\frac{\alpha}{2\pi}\Big{[}\frac{1+x^{2}}{1-x}\ln\frac{\mu_{F}^{2}}{(1-x)^{2}m_{e}^{2}}\Big{]}_{+}.$
(3)

This leaves a hard scattering part independent of the electron mass. Following
this path, we are able to avoid the numerical problem during the reduction
procedure and the calculation of the hard scattering cross section can be
greatly simplified.

####  II.2.3 4-particle final states

4-particle final states with an on-shell Higgs at the parton level can be
produced at NLO ($O(\alpha^{4})$) via
$e^{-}+q\rightarrow\nu_{e}+\gamma+q^{\prime}+H$ and
$e^{-}+\gamma\rightarrow\nu_{e}+q+q^{\prime}+H$. Example diagrams for these
processes are given in Fig. 4. There can be resonant $W$ propagators, as shown
by the last diagram in the figure. To handle such cases, we employ the
complex-mass scheme Denner:1999gp ; Denner:2005fg ; Denner:2006ic  in the
calculation of the amplitudes.

Figure 4: Representative real-emission diagrams for CC WBF at the LHeC. The
first and second lines correspond respectively to the processes
$e^{-}+q\rightarrow\nu_{e}+\gamma+q^{\prime}+H$ and
$e^{-}+\gamma\rightarrow\nu_{e}+q+q^{\prime}+H$, in which $q=u$,
$q^{\prime}=d$.

The subtraction procedure starts by constructing terms that asymptotically
approach the squared amplitude for the real emission process in the phase
space region where the radiated particle (the emittee) becomes soft or
collinear to the emitter. The dipole structure of the subtraction term is
encoded in the charge correlation between the splitting pair and a third
particle (spectator). Three types of dipoles are present for our processes and
the subtraction term takes the form Schonherr:2017qcj

$\displaystyle|\mathcal{M}_{sub}|^{2}=\sum_{i,j}\sum_{a}\mathcal{D}_{ij}^{a}+\sum_{a,j}\sum_{k\neq
j}\mathcal{D}_{j,k}^{a}+\sum_{a,j}\sum_{b\neq a}\mathcal{D}_{j}^{a,b}.$ (4)

The indices $i$, $j$, $k$ denote the emitter, emittee, and spectator,
respectively, in the final state. $a$ and $b$ are initial state partons, where
$a$ plays the role of the spectator in $\mathcal{D}_{ij}^{a}$, and the emitter
in $\mathcal{D}_{j,k}^{a}$ and $\mathcal{D}_{j}^{a,b}$, while $b$ is the
spectator. Here we extend the meaning of “parton” to include the initial state
electron and photon that undergo electromagnetic interactions. The outer sum
is over all possible choices of the splitting pairs, while the inner sum runs
over the spectators. The diagrammatic representation of the three dipole types
can be found in Ref. Catani:1996vz ; Schonherr:2017qcj . In each dipole term,
the singular behavior is extracted and encoded in a splitting function
factorized from the corresponding born process, whose final state coincides
with the kinematics of the radiative process in the soft or collinear limit.
To obtain the born kinematics one combines the emitter and emittee from the
corresponding splitting process to a single particle. The momenta of this
particle and of the spectator are then shifted in such a way as to respect
momentum conservation and on-shell conditions for the 3-particle final state
Catani:1996vz . The explicit expressions for the dipole terms and shifted
momenta are given in Appendix A

Now we subtract these terms from the real emission part to remove any
singularity in the phase space. The cross section is thereby expressed by

$\displaystyle\sigma_{4}^{NLO}=\sum_{b}\int_{0}^{1}d\eta_{b}f_{b}(\eta_{b},\mu_{F})\frac{1}{2\hat{s}}d\Phi_{4}\left\\{|\mathcal{M}_{R}^{b}|^{2}F^{(2)}(p_{1},p_{2},p_{3},p_{4};P_{A},p_{b})-|\mathcal{M}_{sub}^{b}|^{2}\right\\},$
(5)

where $b$ denotes the parton that collides with the electron; it can be a
quark, anti-quark, or photon, and plays the role of either an emitter or
spectator. It should not be confused with the “$b$” in Eq. 4, which only
denotes a spectator.

In order to construct collinear-safe observables, we assign a jet function
$F^{(2)}$ to the real emission process to combine quarks and photons according
to a certain choice of the jet algorithm. $F^{(2)}$ depends on the momenta of
the 4-particle final state. Correspondingly, for each dipole term in Eq. 4,
there is a function $F^{(1)}$ of the 3-particle phase space obtained from the
real emission process via the procedure described after Eq. 4. $F^{(1)}$’s are
implicitly included in Eqs. 4 and 5, as well as in Eq. 1. To ensure the
cancellation of singularities, $F^{(2)}$ must approach the corresponding
$F^{(1)}$ in various soft and collinear limits.

At this order, only the delta function term of the electron PDF in Eq. 3
contributes, which is trivially integrated out in Eq. 5 over the electron
momentum fraction (This is also true for the LO case in Eq. 1). Since the
integrand in Eq. 5 is finite, integration over the 4-particle phase space can
be safely done in four dimensions with numerical methods.

####  II.2.4 3-particle final states

Evidently, the subtracted dipole terms are auxiliary and need to be put back
to avoid any artifact in the calculation. The structure of these terms is
simple enough such that the singularities can be isolated in a factorized form
after an analytical integration over a one-particle phase space. These
“integrated” dipole terms have a phase space with one fewer particle, and are
added back to combine with the loop diagrams. The result contains only
collinear singularities from the initial state radiations, which are then
canceled by the collinear counter terms from the factorization procedure.

The dipole and collinear counter terms can be reorganized into a term $\bm{I}$
that contains all the soft and collinear divergences present in the loop
diagrams, and two other terms $\bm{K}$ and $\bm{P}$ that have finite
remainders after cancellation of singularities. In particular, $\bm{K}$ and
$\bm{P}$ include the finite terms from factorization of initial state
collinear singularities. The terms that depend on the choice of factorization
scheme are contained in $\bm{K}$, while $\bm{P}$ gives the dependence on the
factorization scale $\mu_{F}$. The cross section from the 3-particle final
states can therefore be expressed in terms of the contribution from the born
and loop processes together with $\bm{I}$,$\bm{K}$ and $\bm{P}$ Catani:1996vz
; Catani:2002hc ; Schonherr:2017qcj

$\displaystyle\sigma_{3}^{NLO}=$ $\displaystyle\sum_{b}\int
d\eta_{b}f_{b}(\eta_{b},\mu_{F})\bigg{\\{}\int\frac{1}{2\hat{s}}d\Phi_{3}^{(4)}\Big{[}V_{ab}(\Phi_{3},P_{A},p_{b})+B_{ab}(\Phi_{3},P_{A},p_{b})\bm{I}^{b}(\epsilon,\mu^{2})\Big{]}_{\epsilon=0}$
(6) $\displaystyle+\sum_{a^{\prime}}\int
dx_{a}\int\frac{1}{2\hat{s}}d\Phi_{3}^{(4)}B_{a^{\prime}b}(\Phi_{3}^{(4)},x_{a}P_{A},p_{b})\Big{[}\bm{K}^{b}_{aa^{\prime}}(x_{a})+\bm{P}^{b}_{aa^{\prime}}(x_{a};\mu_{F}^{2})\Big{]}$
$\displaystyle+\sum_{b^{\prime}}\int
dx_{b}\int\frac{1}{2\hat{s}}d\Phi_{3}^{(4)}B_{ab^{\prime}}(\Phi_{3}^{(4)},P_{A},x_{b}p_{b})\Big{[}\bm{K}^{a}_{bb^{\prime}}(x_{b})+\bm{P}^{a}_{bb^{\prime}}(x_{b};\mu_{F}^{2})\Big{]}\bigg{\\}}$
$\displaystyle+\sum_{b}\int
d\eta_{a}d\eta_{b}f^{\mathcal{O}(\alpha)}_{e}(\eta_{a},\mu_{F})f_{b}(\eta_{b},\mu_{F})\int\frac{1}{2\hat{s}}d\Phi_{3}^{(4)}B_{ab}(\Phi_{3}^{(4)},p_{a},p_{b}).$

Here $B$ is the squared born amplitude, and $V$ the interference between born
and loop processes. $a$ labels the electron and $b$ the parton from the
proton; $a^{\prime}$ ($b^{\prime}$) is the flavor from the splitting of $a$
($b$) that enters the LO hard scattering amplitude. The beam and initial state
parton momenta are related by $p_{a}=\eta_{a}P_{A}$ and $p_{b}=\eta_{b}P_{B}$.
In the last line of Eq. 6, we use the $\mathcal{O}(\alpha)$ term of the
electron PDF, as is indicated by its superscript. The first three lines are
from convolution with the delta term in Eq. 3, for which $\eta_{a}=1$. In each
term on the R.H.S of Eq. 6, the phase space $\Phi_{3}^{(4)}$ and flux factor
$1/2\hat{s}$ both depend on the CM energy of the corresponding born factor.
The subscripts of $B$ and $V$ in the first and last lines show that the
flavors entering these factors are the same as those from the incoming beams
(i.e., from the PDFs). In contrast, the subscripts of $\bm{K}$ and $\bm{P}$
show that the flavors may change before and after initial state splittings in
these terms. The superscript of $\bm{K}$ or $\bm{P}$ labels the other incoming
particle that does not go through the splitting. As in Eqs. 1, 4 and 5, we
implicitly include a jet function $F^{(1)}$ for the phase space of each term
in Eq. 6.

The loop integrals in $V$ are done in $d=4-2\epsilon$ dimensions and all
singularities after renormalization manifest as single and double poles, to be
canceled exactly by the poles from $\bm{I}$. Therefore, the phase space
$\Phi_{3}$ of $V$ and $B$ in the first line of Eq. 6 should be the same, i.e.,
both in either 4 or $d$ dimensions. After the poles are canceled, one takes
the limit $\epsilon\rightarrow 0$ and performs the phase space integration in
4 dimensions, as denoted by “$\epsilon=0$” and “$d\Phi_{3}^{(4)}$” in the
first line. While $\bm{I}$,$\bm{K}$ and $\bm{P}$ all contribute to the
quark/anti-quark induced processes, photon induced processes only receive
contribution from non-singular terms $\bm{K}$ and $\bm{P}$.

The expressions of $\bm{I}$,$\bm{K}$ and $\bm{P}$ for specific processes are
listed in Appendix B. The integrand of all terms in Eq. 6 are finite and the
phase space can be integrated over numerically in four dimensions. Once
$\sigma_{3}^{NLO}$ and $\sigma_{4}^{NLO}$ are obtained, we combine them to
give the full corrections to the cross section at NLO. Note that neither of
the two contributions alone is physical because of the auxiliary dipole terms
introduced.

##  III Numerical result

###  III.1 Setup

To make numerical predictions for the WBF cross section, we take the energies
of the incoming electron and proton to be

$\displaystyle E_{e}=140\;\mbox{GeV},\qquad E_{p}=7\;\mbox{TeV},$ (7)

corresponding to a CM energy $\sqrt{s}=2\sqrt{E_{p}E_{e}}\approx
1.98\;\mbox{TeV}$, which is the choice in the study of $H$-$b$ coupling at the
LHeC Han:2009pe . The renormalization and factorization scales are set to
$M_{W}$. In addition, we use the following set of parameters

$\displaystyle G_{\mu}=1.16639\times
10^{-5}\;\mbox{GeV}^{-2},\qquad\alpha_{G_{\mu}}=1/132.5,\qquad\alpha(M_{Z})=1/128.93,$
(8) $\displaystyle
M_{W}=80.419\;\mbox{GeV},\qquad\Gamma_{W}=2.09291\;\mbox{GeV},\qquad
M_{Z}=91.188\;\mbox{GeV},$ $\displaystyle\Gamma_{Z}=2.49877\;\mbox{GeV},\qquad
c_{W}^{2}=1-s_{W}^{2}=\frac{M_{W}^{2}}{M_{Z}^{2}},$ $\displaystyle
m_{e}=0.510998928\;\mbox{MeV},\qquad M_{H}=125\;\mbox{GeV}.$

As discussed in Sec. II.2.1 and II.2.2, the dominant fermion mass effect is
incorporated into the renormalized fine structure constant as well as the
electron distribution. Hence we do not specify the values of fermion masses
here, and set them to zero throughout the calculation of the hard scattering
cross section. The only exception is the explicit use of $m_{e}$ in the
electron distribution from Eq. 3. As stated before, we use a unit CKM matrix
in this work.

In this calculation, the tree level Feynman diagrams and amplitudes are
produced and evaluated by the FeynArts, FormCalc, and LoopTools package set
Hahn:2000kx ; Hahn:1998yk . The analytical calculation of the dipole terms are
carried out with the program Mathematica. The one-loop amplitudes are computed
using the program MadLoop hirschi:2011pa  implemented by MadGraph5_aMC@NLO
package Alwall:2014hca . We have developed our own code to generate the 3- and
4-particle phase space, over which the numerical integration is performed with
the Vegas Monte-Carlo program implemented by the Cuba library Hahn:2004fe .

###  III.2 Consistency checks

To verify the calculation we have done several checks at different levels.
First, the tree-level amplitudes for born and real emission processes computed
with FormCalc are compared with those obtained from MadGraph5.

In the 4-particle final state part, we have made sure that the real emission
contribution approach the corresponding dipole terms in various soft and
collinear limits, and that the integrand after subtraction be stable in a
Monte-Carlo integration. Also, the explicit form of the dipole terms for the
real emission processes is compared numerically with the dipoles generated by
MadDipole Frederix:2008hu ; Gehrmann:2010ry .

For the 3-particle final state terms, we have verified analytically in the
case of triangle graphs that all double and single poles produced in
dimensional regularization are canceled among the loop graphs (whose forms are
derived from the scalar integrals in Ref. Ellis:2007qk  after reduction),
dipoles, and the collinear counter terms. This verification is then carried
out for all loop diagrams (computed with MadLoop) numerically. In the finite
part, we checked that the dependence on the renormalization scale is canceled
between the loop and dipole terms.222Note that the factorization scale
independence could not be checked in this calculation because the PDF sets
undergo QCD evolution. The corresponding QCD corrections need to be included
in order to cancel the scale dependence at the order of evolution. There is no
running of the coupling and masses in the renormalization schemes we choose,
and the cancellation of the scale dependence is exact at the order we are
working with.

###  III.3 Phenomenology

First, we report the result of the integrated cross section computed in two
renormalization schemes. Table. 1 lists contribution from LO and NLO terms,
where all $W$ fusion processes that produce a Higgs boson are included.

Schemes | total | LO | 3-particle | 4-particle
---|---|---|---|---
$G_{\mu}$ | 223.27 | 245.48 | -20.48 | -1.73
$\alpha_{M_{Z}}$ | 216.74 | 266.48 | -47.82 | -1.93
Table 1: Integrated cross sections in fb for CC WBF at the LHeC at LO and NLO,
computed in two renormalization schemes $G_{\mu}$ and $\alpha_{M_{Z}}$. The
electron and proton beam energies are $140$ GeV and $7$ TeV, respectively.

There is a prominent difference in the LO cross sections obtained in the two
schemes. This is solely due to the difference in the renormalized fine
structure constants. In the $G_{\mu}$ scheme, the NLO terms reduces the LO
result by $\sim 9\%$, while the relative correction in the $\alpha(M_{Z})$
scheme is as large as $\sim 18\%$. The sum of LO and NLO result, however,
agree in both schemes, up to a difference at one higher order. This in fact
provides another consistency check for the calculation. In both schemes the
bulk of the NLO corrections come from the 3-particle final states. The
allocation of the contribution between 3- and 4- particle final states is an
artifact that relies on the choice of the subtraction terms away from the
divergent region of the phase space. But this will of course not affect the
sum of the two contributions. Notice that the coupling in the $G_{\mu}$ scheme
is smaller and leads to a much faster convergence of the perturbation series.
In the following discussion we shall stick with this scheme.

Next, we turn to the study of distributions. As remarked in the introduction,
a distinctive feature of the WBF event in $e$-$p$ collisions is an energetic
jet produced in the far backward direction with large transverse momentum (we
take the direction of the incoming electron to be forward). This is very
prominent at the LHeC because the final state is strongly boosted along the
direction of the proton. The decay product of the Higgs tends to be well
separated from this jet, with a rapidity near the central region. To make full
use of this event shape, we follow the strategy in Ref. Han:2009pe ;
Jager:2010zm  and identify the “tagging jet” as the one with the largest
transverse momentum, imposing the cuts

$\displaystyle p_{T}^{j}>30~{}\text{GeV},\qquad-5<\eta_{j}<-1,\qquad
M_{H,j}>250~{}\text{GeV},$ (9)

where $p^{j}_{T}$ and $\eta^{j}$ are the transverse momenta and rapidity of
the tagging jet; $M_{H,j}$ is the invariant mass of the Higgs-tagging jet
system. Further, the large momentum transfer to the neutrino from the hard
scattering is reflected by the requirement

$\displaystyle\not{E}_{T}>25~{}\text{GeV},$ (10)

with $\not{E}_{T}$ denoting the transverse energy of the neutrino. Normally
one also applies basic cuts on the Higgs decay products, which is apparently
not done in our case as we are not performing a full signal-background
analysis in this work.

In the following, we present distributions in various observables
$\mathcal{O}$ computed at LO and NLO (here “NLO” also includes the LO
contribution). Each jet function in Eqs. 5 and 6 plays the role of
constructing the observable with the momenta from its argument list.333Note
that the momenta for a dipole term are transformed from those of the
corresponding real emission process. Here we use $k_{T}$ algorithm
Catani:1992zp ; Ellis:1993tq ; Blazey:2000qt  with the parameter $D=0.8$ to
define jet observables. In order to show the relative correction from the EW
radiations with respect to the LO result, we define the $K$ factor

$\displaystyle
K(\mathcal{O})\equiv\frac{d\sigma^{NLO}/d\mathcal{O}}{d\sigma^{LO}/d\mathcal{O}},$
(11)

as a function of the observable $\mathcal{O}$.

(a) $p_{T}^{j}$ distribution (b) $K$ factor

Figure 5: Distribution in the transverse momentum $p_{T}^{j}$ of the tagging
jet (a), and the corresponding $K$ factors (b).

We first display the differential distribution of the tagging jet transverse
momentum $p_{T}^{j}$ in Fig. 5 (a), where the LO and NLO curves are in red and
blue respectively. The cross section peaks near small value of $p_{T}^{j}$ and
drops apace with increasing jet transverse momentum, with an average
$p_{T}^{j}$ of order the $W$ boson mass. The relative EW correction is about
-10% over the broad $p_{T}^{j}$ range, as shown by the blue curve on the
right. In addition, we show separately the $K$ factor of the two dominant
contributions at NLO. The short-dashed magenta line gives the correction from
the initial state radiation (ISR) of the photon off the electron, which is
about -5% and is consistent with the result obtained by Ref. Blumlein:1992eh .
This collinear radiation results in large logarithmic terms of the form
$\sim\ln m_{e}$ (See the last line of Eq. 6), and gives the main part of the
QED corrections. Another contribution in long-dashed brown is from the EW loop
diagrams, with IR singular terms removed by subtractions. In Fig. 5 it is of
the similar size as ISR, and both curves have a rather flat shape.

(a) $\eta_{j}$ distribution (b) $K$ factor

Figure 6: Distribution in the rapidity $\eta_{j}$ of the tagging jet (a), and
the corresponding $K$ factors (b).

The distribution in the rapidity $\eta_{j}$ of the tagging parton is shown in
Fig. 6, where the peak is clearly located in the backward region. The EW
corrections vary between -8% and -12%, with a slow increase as $\eta_{j}$
approaches the central region. The loop contribution increases more rapidly at
large $\eta_{j}$ than the full NLO curve, while the ISR terms display an
opposite trend in this region.

(a) $p_{T}^{H}$ distribution (b) $K$ factor

Figure 7: Distribution in the transverse momentum $p_{T}^{H}$ of the Higgs
boson (a), and the corresponding $K$ factors (b).

(a) $\eta_{H}$ distribution (b) $K$ factor

Figure 8: Distribution in the rapidity $\eta_{H}$ of the Higgs boson (a), and
the corresponding $K$ factors (b).

Figs. 7 and 8 show that the Higgs observables (transverse momentum $p_{T}^{H}$
and rapidity $\eta_{H}$) are more sensitive to the NLO terms. The size of the
corrections can be as large as -17% (for $p_{T}^{H}$) or -14% (for
$\eta_{H}$). There are also considerable changes of the $K$ factors in the
ranges of both variables. In particular, ISR terms play a prominant role in
the $\eta_{H}$ distribution. It is argued Denner:2003ri ; Denner:2004jy  that
the ISR contribution can be enhanced near the threshold for producing the
final state. This is already seen at large $p_{T}^{H}$ or positive $\eta_{H}$
in the plots. The analysis in Ref. Han:2009pe  shows that the WBF process
tends to produce the final state with a larger invariant mass than that of the
background. This may induce a heavier suppression of the WBF signal from the
ISR terms in the region where the signal dominates, potentially making the
probe of e.g. $H$-$b$ Yukawa more difficult. Of course, radiative corrections
for the background need also be accounted for in order to yield a full
phenomenological effect.

(a) $\Delta\phi_{MET-j}$ distribution (b) $K$ factor

Figure 9: Distribution in the azimuthal angle correlation between the tagging
jet and the missing transverse energy (a), and the corresponding $K$ factors
(b).

Finally, we present in Fig. 9 the distribution in the azimuthal angle between
the tagging jet and missing transverse energy. This distribution is sensitive
to the anomalous $HWW$ couplings Plehn:2001nj ; Biswal:2012mp , whereas in the
SM the differential cross section exhibits only a steady increase with the
increasing $\Delta\phi_{MET-j}$. The relative EW corrections also display a
mild dependence on $\Delta\phi_{MET-j}$ and change in the range between -14%
to -7%. This shape distortion at NLO is caused by the loop contribution that
is insensitive to the electron mass $m_{e}$. ISR in this case adds essentially
a constant correction of -5%. At this point, it is still hard to predict the
role of the loop terms in a systematic study of Higgs phenomenology at the
LHeC.

##  IV Conclusion

We have computed the NLO EW corrections for the Higgs production process via
CC WBF at the LHeC. This is the first calculation that takes into account the
full EW effect at one loop for this process. To handle the singularities of
various origins in the amplitudes, we have worked in the dipole subtraction
formalism and factorized the collinear radiation of the initial states. We
have developed our own program to implement the subtraction procedure and
carry out the numerical integration over the phase space. Checks at many
levels have been done to verify the consistency of the calculation.

For the CM energy at $1.98$ TeV, we have found a relative correction of 9%
(18%) in the total cross section with no cuts using the $G_{\mu}$
($\alpha_{M_{Z}}$) renormalization scheme. The agreement between the two
schemes are significantly improved when the LO and NLO-correction terms are
added. The differential cross sections for several observables have also been
computed in the $G_{\mu}$ scheme under the selection cuts for the WBF process.
The corrections for these variables are within -20%. Sizable distortion of the
distributions are also observed due to the ISR and loop corrections. Despite
the smallness of the EW coupling $\alpha$ as compared with the strong coupling
$\alpha_{S}$, EW corrections are considerably larger than QCD corrections at
NLO both because of the large number of diagrams (compared with the number of
the QCD loop diagrams), and because of the ISR effect associated with photon
radiations by the electron beam. The results show that the EW corrections is
very important for the study of Higgs phenomenology at the LHeC and should not
be neglected in a full analysis of the WBF process.

##  V Acknowledgement

The authors thank Lilin Yang, Ruibo Li, and Xiang Lv for helpful discussions.
B.W. would like to thank Zhuoni Qian for useful suggestions on histograming,
and Ted Rogers for providing a very important reference. This work was
supported by National Science Foundation of China (11875232, 12147103,
12105068). B.W. was also supported by Hangzhou Normal University Start-up
Funds.

##  Appendix A Expressions for the 4-particle final states

The specific form of the subtraction term for the quark- and anti-quark-
initiated process

$\displaystyle
e^{-}(p_{a})+q(p_{b})\rightarrow\nu_{e}(p_{1})+\gamma(p_{2})+q^{\prime}(p_{3})+H(p_{4})$

reads

$\displaystyle|\mathcal{M}_{sub}|^{2}=\mathcal{D}_{q^{\prime}\gamma}^{e}+\mathcal{D}_{\gamma,q^{\prime}}^{e}+\mathcal{D}_{\gamma}^{e,q}+\mathcal{D}_{q^{\prime}\gamma}^{q}+\mathcal{D}_{\gamma,q^{\prime}}^{q}+\mathcal{D}_{\gamma}^{q,e}$
(12)
$\displaystyle\qquad\quad=-\frac{1}{2p_{3}p_{2}}\frac{1}{x_{32,a}}8\pi\mu^{2\epsilon}\alpha\left[\frac{2}{2-x_{32,a}-z_{3a}}-1-z_{3a}-\epsilon(1-z_{3a})\right]\bm{Q}_{3a}^{2}|\mathcal{M}^{q}_{0}(x_{32,a}p_{a}+P_{3a};x_{32,a}p_{a})|^{2}$
$\displaystyle\qquad\qquad-\frac{1}{2p_{a}p_{2}}\frac{1}{x_{a2,3}}8\pi\mu^{2\epsilon}\alpha\left[\frac{2}{2-x_{a2,3}-z_{3a}}-1-x_{a2,3}-\epsilon(1-x_{a2,3})\right]\bm{Q}_{a3}^{2}|\mathcal{M}^{q}_{0}(x_{a2,3}p_{a}+P_{3a};x_{a2,3}p_{a})|^{2}$
$\displaystyle\qquad\qquad-\frac{1}{2p_{a}p_{2}}\frac{1}{x_{a2,b}}8\pi\mu^{2\epsilon}\alpha\left[\frac{2}{1-x_{a2,b}}-1-x_{a2,b}-\epsilon(1-x_{a2,b})\right]\bm{Q}_{ab}^{2}|\mathcal{M}^{q}_{0}(\tilde{p}_{i}(x_{a2,b});x_{a2,b}p_{a})|^{2}$
$\displaystyle\qquad\qquad-\frac{1}{2p_{3}p_{2}}\frac{1}{x_{32,b}}8\pi\mu^{2\epsilon}\alpha\left[\frac{2}{2-x_{32,b}-z_{3b}}-1-z_{3b}-\epsilon(1-z_{3b})\right]\bm{Q}_{3b}^{2}|\mathcal{M}^{q}_{0}(x_{32,b}p_{b}+P_{3b};x_{32,b}p_{b})|^{2}$
$\displaystyle\qquad\qquad-\frac{1}{2p_{b}p_{2}}\frac{1}{x_{b2,3}}8\pi\mu^{2\epsilon}\alpha\left[\frac{2}{2-x_{b2,3}-z_{3b}}-1-x_{b2,3}-\epsilon(1-x_{b2,3})\right]\bm{Q}_{b3}^{2}|\mathcal{M}^{q}_{0}(x_{b2,3}p_{b}+P_{3b};x_{b2,3}p_{b})|^{2}$
$\displaystyle\qquad\qquad-\frac{1}{2p_{b}p_{2}}\frac{1}{x_{b2,a}}8\pi\mu^{2\epsilon}\alpha\left[\frac{2}{1-x_{b2,a}}-1-x_{b2,a}-\epsilon(1-x_{b2,a})\right]\bm{Q}_{ba}^{2}|\mathcal{M}^{q}_{0}(\tilde{p}_{i}(x_{b2,a});x_{b2,a}p_{b})|^{2},$

where $\mathcal{M}^{q}_{0}$ is the amplitude for the born process
$eq\rightarrow\nu_{e}q^{\prime}H$. The momentum fractions are defined as

$\displaystyle
x_{32,a}=x_{a2,3}=1-\frac{p_{3}p_{2}}{(p_{3}+p_{2})p_{a}},\qquad
z_{3a}=\frac{p_{3}p_{a}}{(p_{3}+p_{2})p_{a}},$ (13) $\displaystyle
x_{32,b}=x_{b2,3}=1-\frac{p_{3}p_{2}}{(p_{3}+p_{2})p_{b}},\qquad
z_{3b}=\frac{p_{3}p_{b}}{(p_{3}+p_{2})p_{b}}$ $\displaystyle
x_{a2,b}=x_{b2,a}=1-\frac{p_{2}(p_{a}+p_{b})}{p_{a}p_{b}}.$

As remarked in Sec. II.2.3, the born amplitudes in Eqs. 12 depend on the
shifted momenta obtained from the corresponding radiative processes. These
momenta are shown as the arguments of the born amplitudes, where the momenta
from the initial/final state is on the right/left of the semicolon (Momenta
with no modifications are not shown). They are related to the momenta of the
real emission processes by

$\displaystyle
P_{3a}=p_{3}+p_{2}-p_{a}=p_{3}+p_{2}-(1-x_{32,a})p_{a}-x_{32,a}p_{a},$ (14)
$\displaystyle
P_{3b}=p_{3}+p_{2}-p_{b}=p_{3}+p_{2}-(1-x_{32,b})p_{b}-x_{32,b}p_{b},$
$\displaystyle
P_{ab}=p_{a}+p_{b}-p_{2},\qquad\tilde{P}_{ab}=x_{a2,b}p_{a}+p_{b},$
$\displaystyle\tilde{p}_{i}^{\mu}=\Lambda_{\nu}^{\mu}p_{i}^{\nu}=\bigg{(}g_{\nu}^{\mu}-\frac{(P_{ab}+\tilde{P}_{ab})^{\mu}(P_{ab}+\tilde{P}_{ab})_{\nu}}{P_{ab}^{2}+P_{ab}\tilde{P}_{ab}}+\frac{\tilde{P}_{ab}^{\mu}P_{ab,\nu}}{P_{ab}^{2}}\bigg{)}p_{i}^{\nu}.$

Note that $\tilde{p}_{i}$ for all final state particles in
$\mathcal{D}_{\gamma}^{e,q}$ and $\mathcal{D}_{\gamma}^{q,e}$ must transform
according to Eq. 14. The charge correlator $\bm{Q}_{ik}^{2}$ is defined by the
charges of the flavor $i$ and $k$ as

$\displaystyle\bm{Q}_{ik}^{2}=Q_{i}Q_{k}\theta_{i}\theta_{k},$ (15)

where $\theta_{i/k}$ is $1$ ($-1$) if it is in the final (initial) state. For
quark-initiated processes in our calculation, i.e., $q=u,\,c$, and
$q^{\prime}=d,\,s$, the charge correlators in Eq. 12 are

$\displaystyle\bm{Q}_{3a}^{2}=\bm{Q}_{a3}^{2}=Q_{3}(-Q_{a})=-\frac{1}{3},\qquad\bm{Q}_{3b}^{2}=\bm{Q}_{b3}^{2}=Q_{3}(-Q_{b})=\frac{2}{9},$
(16)
$\displaystyle\bm{Q}_{ab}^{2}=\bm{Q}_{ba}^{2}=(-Q_{a})(-Q_{b})=-\frac{2}{3},$

where each initial state fermion acquires a minus sign on its charge. The
anti-quark-initiated processes, with $q=\bar{d},\,\bar{s}$, and
$q^{\prime}=\bar{u},\,\bar{c}$, differ from the corresponding quark-initiated
processes only by the charge correlators, which are

$\displaystyle\bm{Q}_{3a}^{2}=\bm{Q}_{a3}^{2}=-\frac{2}{3},\qquad\bm{Q}_{3b}^{2}=\bm{Q}_{b3}^{2}=\frac{2}{9},\qquad\bm{Q}_{ab}^{2}=\bm{Q}_{ba}^{2}=-\frac{1}{3}.$
(17)

For the photon-induced processes

$\displaystyle
e^{-}(p_{a})+\gamma(p_{b})\rightarrow\nu_{e}(p_{1})+q(p_{2})+q^{\prime}(p_{3})+H(p_{4})$

the subtraction term takes the form

$\displaystyle|\mathcal{M}_{sub}|^{2}$
$\displaystyle=\mathcal{D}_{q,q^{\prime}}^{\gamma}+\mathcal{D}_{q}^{\gamma,e}+\mathcal{D}_{q^{\prime},q}^{\gamma}+\mathcal{D}_{q^{\prime}}^{\gamma,e}$
(18)
$\displaystyle=-\frac{1}{2p_{b}p_{2}}\frac{1}{x_{b2,3}}8\pi\mu^{2\epsilon}\alpha\Big{[}1-\epsilon-2x_{b2,3}(1-x_{b2,3})\Big{]}N_{C,f}\bm{Q}_{\widetilde{b2}3}^{2}|\mathcal{M}^{\gamma}_{0}(x_{b2,3}p_{b}+P_{3b},x_{b2,3}p_{b})|^{2}$
$\displaystyle\quad-\frac{1}{2p_{b}p_{2}}\frac{1}{x_{b2,a}}8\pi\mu^{2\epsilon}\alpha\Big{[}1-\epsilon-2x_{b2,a}(1-x_{b2,a})\Big{]}N_{C,f}\bm{Q}_{\widetilde{b2}a}^{2}|\mathcal{M}^{\gamma}_{0}(\tilde{p}_{i},x_{b2,a}p_{b})|^{2}$
$\displaystyle\quad-\frac{1}{2p_{b}p_{3}}\frac{1}{x_{b3,2}}8\pi\mu^{2\epsilon}\alpha\Big{[}1-\epsilon-2x_{b3,2}(1-x_{b3,2})\Big{]}N_{C,f}\bm{Q}_{\widetilde{b3}2}^{2}|\mathcal{M}^{\gamma}_{0}(x_{b3,2}p_{b}+P_{3b},x_{b3,2}p_{b})|^{2}$
$\displaystyle\quad-\frac{1}{2p_{b}p_{3}}\frac{1}{x_{b3,a}}8\pi\mu^{2\epsilon}\alpha\Big{[}1-\epsilon-2x_{b3,a}(1-x_{b3,a})\Big{]}N_{C,f}\bm{Q}_{\widetilde{b3}a}^{2}|\mathcal{M}^{\gamma}_{0}(\tilde{p}_{i},x_{b3,a}p_{b})|^{2},$

where $\mathcal{M}^{\gamma}_{0}$ is the amplitude for the born process
$e\bar{q}\rightarrow\nu_{e}q^{\prime}H$ (or equivalently,
$e\bar{q^{\prime}}\rightarrow\nu_{e}qH$). In addition to the definitions made
in Eq. 13, we have

$\displaystyle x_{b3,2}=1-\frac{p_{3}p_{2}}{(p_{3}+p_{2})p_{b}},\qquad
x_{b3,a}=1-\frac{p_{3}(p_{a}+p_{b})}{p_{a}p_{b}},\qquad N_{C,f}=3.$ (19)

The “tilde” symbol in the subscript of each charge correlator denotes the
flavor from the photon splitting that enters the corresponding born process,
namely, $\widetilde{b2}$ denotes $\bar{q}$ and $\widetilde{b3}$ denotes
$\bar{q^{\prime}}$. The charge correlators for $q=\bar{u},\,\bar{c}$, and
$q^{\prime}=d,\,s$ are

$\displaystyle\bm{Q}_{\widetilde{b2}3}^{2}=\frac{2}{9},\qquad\bm{Q}_{\widetilde{b3}2}^{2}=\frac{2}{9},\qquad\bm{Q}_{\widetilde{b2}a}^{2}=-\frac{2}{3},\qquad\bm{Q}_{\widetilde{b3}a}^{2}=-\frac{1}{3}.$
(20)

##  Appendix B Expressions for the 3-particle final states

In this appendix we list the specific form of the $\bm{I}$,$\bm{K}$ and
$\bm{P}$ terms that are factorized from the born process

$\displaystyle e^{-}+q\rightarrow\nu_{e}(p_{1})+q^{\prime}(p_{2})+H(p_{3}),$

where the momenta of the initial particles are not shown explicitly. For the
corresponding $\bm{I}$,$\bm{K}$ and $\bm{P}$ terms, they are given by the
arguments of the born factors in the first three lines of Eq. 6. Recall that
there $p_{a}=P_{A}$, and $p_{b}=\eta_{b}P_{B}$.

The generic expressions for the quark- and anti-quark-initiated processes are

$\displaystyle\bm{I}^{q}(\epsilon)$
$\displaystyle=-\frac{\alpha}{2\pi}\frac{(4\pi)^{\epsilon}}{\Gamma(1-\epsilon)}\left[\frac{1}{\epsilon^{2}}+\frac{3}{2\epsilon}+5-\frac{\pi^{2}}{2}\right]\Bigg{\\{}\bm{Q}_{2a}^{2}\left(\frac{\mu^{2}}{2p_{2}p_{a}}\right)^{\epsilon}+\bm{Q}_{a2}^{2}\left(\frac{\mu^{2}}{2p_{a}p_{2}}\right)^{\epsilon}$
(21)
$\displaystyle\qquad\qquad+\bm{Q}_{ab}^{2}\left(\frac{\mu^{2}}{2p_{a}p_{b}}\right)^{\epsilon}+\bm{Q}_{2b}^{2}\left(\frac{\mu^{2}}{2p_{2}p_{b}}\right)^{\epsilon}+\bm{Q}_{b2}^{2}\left(\frac{\mu^{2}}{2p_{b}p_{2}}\right)^{\epsilon}+\bm{Q}_{ba}^{2}\left(\frac{\mu^{2}}{2p_{b}p_{a}}\right)^{\epsilon}\Bigg{\\}},$
$\displaystyle\bm{K}^{c^{\prime}}_{ff}(x)$
$\displaystyle=\frac{\alpha}{2\pi}\Bigg{\\{}-Q_{f}^{2}(1+x)\log\frac{1-x}{x}+Q_{f}^{2}(1-x)+\bigg{[}Q_{f}^{2}\Big{(}\frac{2}{1-x}\log\frac{1-x}{x}\Big{)}_{+}-\delta(1-x)\Big{(}(5-\pi^{2})Q_{f}^{2}\Big{)}\bigg{]}$
(22)
$\displaystyle\quad+\frac{3}{2}\frac{\bm{Q}_{2f}^{2}}{Q_{f}^{2}}\bigg{[}\Big{(}\frac{1}{1-x}\Big{)}_{+}+\delta(1-x)\bigg{]}-\bm{Q}_{fc^{\prime}}^{2}\Bigg{[}-(1+x)\log(1-x)+\bigg{[}2\bigg{(}\frac{\log(1-x)}{1-x}\bigg{)}_{+}-\frac{\pi^{2}}{3}\delta(1-x)\bigg{]}\Bigg{]}\Bigg{\\}},$
$\displaystyle\bm{P}^{c^{\prime}}_{ff}(x,\mu_{F}^{2})$
$\displaystyle=\frac{\alpha}{2\pi}\bigg{\\{}-(1+x)+2\Big{(}\frac{1}{1-x}\Big{)}_{+}+\frac{3}{2}\delta(1-x)\bigg{\\}}\bigg{[}\bm{Q}_{f2}^{2}\log\frac{\mu_{F}^{2}}{2xp_{f}p_{2}}+\bm{Q}_{f{c^{\prime}}}^{2}\log\frac{\mu_{F}^{2}}{2xp_{f}p_{c^{\prime}}}\bigg{]},$
(23)

where “$q$” in $\bm{I}^{q}$ can be a quark or anti-quark. In $\bm{K}^{b}_{ff}$
and $\bm{P}^{b}_{ff}$, “$f$” can be a quark, anti-quark, or electron, while
“$c^{\prime}$” is the initial flavor not going through splitting (“$c$”
without prime is reserved for the charm quark, which is only a particular case
of $c^{\prime}$. See below). Inserting the charge correlators for specific
flavors, we find

$\displaystyle\bm{I}^{u/c}(\epsilon)$ $\displaystyle=$
$\displaystyle=-\frac{\alpha}{2\pi}\frac{(4\pi\mu^{2})^{\epsilon}}{\Gamma(1-\epsilon)}\left[\frac{1}{\epsilon^{2}}+\frac{3}{2\epsilon}+5-\frac{\pi^{2}}{2}\right]\Bigg{\\{}-\frac{2}{3}\left(\frac{1}{2p_{a}p_{2}}\right)^{\epsilon}-\frac{4}{3}\left(\frac{1}{2p_{a}p_{b}}\right)^{\epsilon}+\frac{4}{9}\left(\frac{1}{2p_{b}p_{2}}\right)^{\epsilon}\Bigg{\\}},$
(24)
$\displaystyle\bm{I}^{\bar{d}/\bar{s}}(\epsilon)=-\frac{\alpha}{2\pi}\frac{(4\pi\mu^{2})^{\epsilon}}{\Gamma(1-\epsilon)}\left[\frac{1}{\epsilon^{2}}+\frac{3}{2\epsilon}+5-\frac{\pi^{2}}{2}\right]\Bigg{\\{}-\frac{4}{3}\left(\frac{1}{2p_{a}p_{2}}\right)^{\epsilon}-\frac{2}{3}\left(\frac{1}{2p_{a}p_{b}}\right)^{\epsilon}+\frac{4}{9}\left(\frac{1}{2p_{b}p_{2}}\right)^{\epsilon}\Bigg{\\}},$
(25) $\displaystyle\bm{K}^{u/c}_{ee}(x)$
$\displaystyle=\frac{\alpha}{2\pi}\bigg{\\{}-(1+x)\log\frac{1-x}{x}+(1-x)+\Big{(}\frac{2}{1-x}\log\frac{1-x}{x}\Big{)}_{+}-\frac{9}{2}\Big{(}\frac{1}{1-x}\Big{)}_{+}-\frac{2}{3}(1+x)\log(1-x)$
(26)
$\displaystyle\qquad\qquad+\frac{4}{3}\Big{(}\frac{\log(1-x)}{1-x}\Big{)}_{+}+\Big{(}\frac{7}{9}\pi^{2}-\frac{19}{2}\Big{)}\delta(1-x)\bigg{\\}},$
$\displaystyle\bm{K}^{e}_{uu/cc}(x)$
$\displaystyle=\frac{\alpha}{2\pi}\bigg{\\{}-\frac{4}{9}(1+x)\log\frac{1-x}{x}+\frac{4}{9}(1-x)+\frac{4}{9}\Big{(}\frac{2}{1-x}\log\frac{1-x}{x}\Big{)}_{+}+3\Big{(}\frac{1}{1-x}\Big{)}_{+}-\frac{2}{3}(1+x)\log(1-x)$
(27)
$\displaystyle\qquad\qquad+\frac{4}{3}\Big{(}\frac{\log(1-x)}{1-x}\Big{)}_{+}+\Big{(}\frac{2}{9}\pi^{2}+\frac{7}{9}\Big{)}\delta(1-x)\bigg{\\}},$
$\displaystyle\bm{K}^{\bar{d}/\bar{s}}_{ee}(x)$
$\displaystyle=\frac{\alpha}{2\pi}\bigg{\\{}-(1+x)\log\frac{1-x}{x}+(1-x)+\Big{(}\frac{2}{1-x}\log\frac{1-x}{x}\Big{)}_{+}-\frac{9}{4}\Big{(}\frac{1}{1-x}\Big{)}_{+}-\frac{1}{3}(1+x)\log(1-x)$
(28)
$\displaystyle\qquad\qquad+\frac{2}{3}\Big{(}\frac{\log(1-x)}{1-x}\Big{)}_{+}+\Big{(}\frac{8}{9}\pi^{2}-\frac{29}{4}\Big{)}\delta(1-x)\bigg{\\}},$
$\displaystyle\bm{K}^{e}_{\bar{d}\bar{d}/\bar{s}\bar{s}}(x)$
$\displaystyle=\frac{\alpha}{2\pi}\bigg{\\{}-\frac{1}{9}(1+x)\log\frac{1-x}{x}+\frac{1}{9}(1-x)+\frac{1}{9}\Big{(}\frac{2}{1-x}\log\frac{1-x}{x}\Big{)}_{+}+\frac{3}{4}\Big{(}\frac{1}{1-x}\Big{)}_{+}-\frac{1}{3}(1+x)\log(1-x)$
(29)
$\displaystyle\qquad\qquad+\frac{2}{3}\bigg{(}\frac{\log(1-x)}{1-x}\bigg{)}_{+}+\frac{7}{36}\delta(1-x)\bigg{\\}},$
$\displaystyle\bm{P}^{u/c}_{ee}(x,\mu_{F}^{2})=\frac{\alpha}{2\pi}\bigg{\\{}-(1+x)+2\Big{(}\frac{1}{1-x}\Big{)}_{+}+\frac{3}{2}\delta(1-x)\bigg{\\}}\bigg{[}-\frac{1}{3}\log\frac{\mu_{F}^{2}}{2xp_{a}p_{2}}-\frac{2}{3}\log\frac{\mu_{F}^{2}}{2xp_{a}p_{b}}\bigg{]},$
(30)
$\displaystyle\bm{P}^{e}_{uu/cc}(x,\mu_{F}^{2})=\frac{\alpha}{2\pi}\bigg{\\{}-(1+x)+2\Big{(}\frac{1}{1-x}\Big{)}_{+}+\frac{3}{2}\delta(1-x)\bigg{\\}}\bigg{[}\frac{2}{9}\log\frac{\mu_{F}^{2}}{2xp_{b}p_{2}}-\frac{2}{3}\log\frac{\mu_{F}^{2}}{2xp_{a}p_{b}}\bigg{]},$
(31)
$\displaystyle\bm{P}^{\bar{d}/\bar{s}}_{ee}(x,\mu_{F}^{2})=\frac{\alpha}{2\pi}\bigg{\\{}-(1+x)+2\Big{(}\frac{1}{1-x}\Big{)}_{+}+\frac{3}{2}\delta(1-x)\bigg{\\}}\bigg{[}-\frac{2}{3}\log\frac{\mu_{F}^{2}}{2xp_{a}p_{2}}-\frac{1}{3}\log\frac{\mu_{F}^{2}}{2xp_{a}p_{b}}\bigg{]},$
(32)
$\displaystyle\bm{P}^{e}_{\bar{d}\bar{d}/\bar{s}\bar{s}}(x,\mu_{F}^{2})=\frac{\alpha}{2\pi}\bigg{\\{}-(1+x)+2\Big{(}\frac{1}{1-x}\Big{)}_{+}+\frac{3}{2}\delta(1-x)\bigg{\\}}\bigg{[}\frac{2}{9}\log\frac{\mu_{F}^{2}}{2xp_{b}p_{2}}-\frac{1}{3}\log\frac{\mu_{F}^{2}}{2xp_{a}p_{b}}\bigg{]}.$
(33)

For the photon-initiated processes, $\bm{I}$ terms do not contribute. Also the
electron splitting terms are of higher order in $\alpha$ and do not contribute
in this case either. $\bm{K}$ and $\bm{P}$ terms take the form

$\displaystyle\bm{K}^{a}_{\gamma f}(x)$
$\displaystyle=\frac{\alpha}{2\pi}\Bigg{\\{}N_{C,f}Q_{f}^{2}\frac{1+(1-x)^{2}}{x}\log\frac{1-x}{x}+N_{C,f}Q_{f}^{2}x(1-x)-\bm{Q}_{fa}^{2}N_{C,f}\frac{1+(1-x)^{2}}{x}\log(1-x)\Bigg{\\}},$
(34) $\displaystyle\bm{P}^{a}_{\gamma
f}(x,\mu_{F}^{2})=\frac{\alpha}{2\pi}\bigg{\\{}N_{C,f}\frac{1+(1-x)^{2}}{x}\bigg{\\}}\bigg{[}\bm{Q}_{f2}^{2}\log\frac{\mu_{F}^{2}}{2xp_{b}p_{2}}+\bm{Q}_{fa}^{2}\log\frac{\mu_{F}^{2}}{2xp_{a}p_{b}}\bigg{]}.$
(35)

The expressions for specific splitting processes are

$\displaystyle\bm{K}^{e}_{\gamma
u/c}(x)=\frac{\alpha}{2\pi}\Bigg{\\{}\frac{4}{3}\frac{1+(1-x)^{2}}{x}\log\frac{1-x}{x}+\frac{4}{3}x(1-x)+2\frac{1+(1-x)^{2}}{x}\log(1-x)\Bigg{\\}},$
(36)
$\displaystyle\bm{K}^{e}_{\gamma\bar{d}/\bar{s}}(x)=\frac{\alpha}{2\pi}\Bigg{\\{}\frac{1}{3}\frac{1+(1-x)^{2}}{x}\log\frac{1-x}{x}+\frac{1}{3}x(1-x)+\frac{1+(1-x)^{2}}{x}\log(1-x)\Bigg{\\}},$
(37) $\displaystyle\bm{P}^{e}_{\gamma
u/c}(x,\mu_{F}^{2})=\frac{\alpha}{2\pi}\bigg{\\{}\frac{4}{3}\frac{1+(1-x)^{2}}{x}\bigg{\\}}\bigg{[}\frac{1}{2}\log\frac{\mu_{F}^{2}}{2xp_{b}p_{2}}-\frac{3}{2}\log\frac{\mu_{F}^{2}}{2xp_{a}p_{b}}\bigg{]},$
(38)
$\displaystyle\bm{P}^{e}_{\gamma\bar{d}/\bar{s}}(x,\mu_{F}^{2})=\frac{\alpha}{2\pi}\bigg{\\{}\frac{1}{3}\frac{1+(1-x)^{2}}{x}\bigg{\\}}\bigg{[}2\log\frac{\mu_{F}^{2}}{2xp_{b}p_{2}}-3\log\frac{\mu_{F}^{2}}{2xp_{a}p_{b}}\bigg{]}.$
(39)

## References

  * (1) ATLAS, G. Aad et al., “Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC,” Phys. Lett. B716 (2012) 1–29, arXiv:1207.7214 [hep-ex].
  * (2) CMS, S. Chatrchyan et al., “Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC,” Phys. Lett. B716 (2012) 30–61, arXiv:1207.7235 [hep-ex].
  * (3) ATLAS, CMS, L. Cadamuro, “Higgs boson couplings and properties,” PoS LHCP2019 (2019) 101
  * (4) ATLAS, CMS, J. M. Langford, “Combination of Higgs measurements from ATLAS and CMS : couplings and $\mathcal{k}$\- framework,” PoS LHCP2020 (2021) 136
  * (5) ATLAS, CMS, S. Ordek, “Measurements of Higgs couplings to fermions and bosons at the LHC,” PoS PANIC2021 (2022) 418
  * (6) ATLAS, “A combination of measurements of Higgs boson production and decay using up to $139$ fb-1 of proton–proton collision data at $\sqrt{s}=$ 13 TeV collected with the ATLAS experiment,”
  * (7) CMS, “Combined Higgs boson production and decay measurements with up to 137 fb-1 of proton-proton collision data at $\sqrt{s}$ = 13 TeV,”
  * (8) CMS, A. M. Sirunyan et al., “Observation of the Higgs boson decay to a pair of $\tau$ leptons with the CMS detector,” Phys. Lett. B 779 (2018) 283–316, arXiv:1708.00373 [hep-ex]
  * (9) ATLAS, M. Aaboud et al., “Cross-section measurements of the Higgs boson decaying into a pair of $\tau$-leptons in proton-proton collisions at $\sqrt{s}=13$ TeV with the ATLAS detector,” Phys. Rev. D 99 (2019) 072001, arXiv:1811.08856 [hep-ex]
  * (10) CMS, V. Khachatryan et al., “Search for the standard model Higgs boson produced through vector boson fusion and decaying to $b\overline{b}$,”Phys. Rev. D 92 3, (2015) 032008, arXiv:1506.01010 [hep-ex]
  * (11) ATLAS, M. Aaboud et al., “Search for the Standard Model Higgs boson produced by vector-boson fusion and decaying to bottom quarks in $\sqrt{s}=8$ TeV pp collisions with the ATLAS detector,” JHEP 11 (2016) 112, arXiv:1606.02181 [hep-ex]
  * (12) LHeC Study Group, J. L. Abelleira Fernandez et al., “A Large Hadron Electron Collider at CERN: Report on the Physics and Design Concepts for Machine and Detector,” J. Phys. G 39 (2012) 075001, arXiv:1206.2913 [physics.acc-ph]
  * (13) T. Han and B. Mellado, “Higgs Boson Searches and the H b anti-b Coupling at the LHeC,” Phys. Rev. D82 (2010) 016009, arXiv:0909.2460 [hep-ph].
  * (14) LHeC, FCC-he Study Group, P. Agostini et al., “The Large Hadron-Electron Collider at the HL-LHC,”J. Phys. G 48 11, (2021) 110501, arXiv:2007.14491 [hep-ex]
  * (15) R. Li, B.-W. Wang, K. Wang, X. Zhang, and Z. Zhou, “Probing the charm Yukawa coupling at future $e^{-}p$ and $e^{+}e^{-}$ colliders,”Phys. Rev. D 100 5, (2019) 053008, arXiv:1905.09457 [hep-ph]
  * (16) J. Blumlein, G. J. van Oldenborgh, and R. Ruckl, “QCD and QED corrections to Higgs boson production in charged current e p scattering,” Nucl. Phys. B 395 (1993) 35–59, arXiv:hep-ph/9209219
  * (17) B. Jager, “Next-to-leading order QCD corrections to Higgs production at a future lepton-proton collider,” Phys. Rev. D 81 (2010) 054018, arXiv:1001.3789 [hep-ph]
  * (18) R. Li, X.-M. Shen, B.-W. Wang, K. Wang, and G. Zhu, “Probing the trilinear Higgs boson self-coupling via single Higgs production at the LHeC,”Phys. Rev. D 101 7, (2020) 075036, arXiv:1910.09424 [hep-ph]
  * (19) J. Alwall, R. Frederix, S. Frixione, V. Hirschi, F. Maltoni, O. Mattelaer, H. S. Shao, T. Stelzer, P. Torrielli, and M. Zaro, “The automated computation of tree-level and next-to-leading order differential cross sections, and their matching to parton shower simulations,” JHEP 07 (2014) 079, arXiv:1405.0301 [hep-ph]
  * (20) F. Buccioni, J.-N. Lang, J. M. Lindert, P. Maierhöfer, S. Pozzorini, H. Zhang, and M. F. Zoller, “OpenLoops 2,”Eur. Phys. J. C 79 10, (2019) 866, arXiv:1907.13071 [hep-ph]
  * (21) Sherpa, E. Bothmann et al., “Event Generation with Sherpa 2.2,”SciPost Phys. 7 3, (2019) 034, arXiv:1905.09127 [hep-ph]
  * (22) J. Bellm et al., “Herwig 7.0/Herwig++ 3.0 release note,”Eur. Phys. J. C 76 4, (2016) 196, arXiv:1512.01178 [hep-ph]
  * (23) S. Alioli, P. Nason, C. Oleari, and E. Re, “A general framework for implementing NLO calculations in shower Monte Carlo programs: the POWHEG BOX,” JHEP 06 (2010) 043, arXiv:1002.2581 [hep-ph]
  * (24) S. Alioli, C. W. Bauer, C. J. Berggren, A. Hornig, F. J. Tackmann, C. K. Vermilion, J. R. Walsh, and S. Zuberi, “Combining Higher-Order Resummation with Multiple NLO Calculations and Parton Showers in GENEVA,” JHEP 09 (2013) 120, arXiv:1211.7049 [hep-ph]
  * (25) W. Kilian, T. Ohl, and J. Reuter, “WHIZARD: Simulating Multi-Particle Processes at LHC and ILC,” Eur. Phys. J. C 71 (2011) 1742, arXiv:0708.4233 [hep-ph]
  * (26) S. Catani and M. H. Seymour, “A General algorithm for calculating jet cross-sections in NLO QCD,” Nucl. Phys. B 485 (1997) 291–419, arXiv:hep-ph/9605323. [Erratum: Nucl.Phys.B 510, 503–504 (1998)]
  * (27) S. Dittmaier, “A General approach to photon radiation off fermions,” Nucl. Phys. B 565 (2000) 69–122, arXiv:hep-ph/9904440
  * (28) S. Catani, S. Dittmaier, M. H. Seymour, and Z. Trocsanyi, “The Dipole formalism for next-to-leading order QCD calculations with massive partons,” Nucl. Phys. B 627 (2002) 189–265, arXiv:hep-ph/0201036
  * (29) M. Schönherr, “An automated subtraction of NLO EW infrared divergences,”Eur. Phys. J. C 78 2, (2018) 119, arXiv:1712.07975 [hep-ph]
  * (30) T. Hahn, “Generating Feynman diagrams and amplitudes with FeynArts 3,” Comput. Phys. Commun. 140 (2001) 418–431, arXiv:hep-ph/0012260
  * (31) V. Hirschi, R. Frederix, S. Frixione, M. V. Garzelli, F. Maltoni, and R. Pittau, “Automation of one-loop QCD corrections,” JHEP 05 (2011) 044, arXiv:1103.0621 [hep-ph]
  * (32) G. Passarino and M. J. G. Veltman, “One Loop Corrections for e+ e- Annihilation Into mu+ mu- in the Weinberg Model,” Nucl. Phys. B 160 (1979) 151–207
  * (33) A. I. Davydychev, “A Simple formula for reducing Feynman diagrams to scalar integrals,” Phys. Lett. B 263 (1991) 107–111
  * (34) G. Ossola, C. G. Papadopoulos, and R. Pittau, “Reducing full one-loop amplitudes to scalar integrals at the integrand level,” Nucl. Phys. B 763 (2007) 147–169, arXiv:hep-ph/0609007
  * (35) A. Denner and S. Dittmaier, “Electroweak Radiative Corrections for Collider Physics,” Phys. Rept. 864 (2020) 1–163, arXiv:1912.06823 [hep-ph]
  * (36) A. Denner, “Techniques for calculation of electroweak radiative corrections at the one loop level and results for W physics at LEP-200,” Fortsch. Phys. 41 (1993) 307–420, arXiv:0709.1075 [hep-ph]
  * (37) R. K. Ellis and G. Zanderighi, “Scalar one-loop integrals for QCD,” JHEP 02 (2008) 002, arXiv:0712.1851 [hep-ph]
  * (38) C. Schmidt, J. Pumplin, D. Stump, and C. P. Yuan, “CT14QED parton distribution functions from isolated photon production in deep inelastic scattering,”Phys. Rev. D93 11, (2016) 114015, arXiv:1509.02905 [hep-ph].
  * (39) A. Buckley, J. Ferrando, S. Lloyd, K. Nordström, B. Page, M. Rüfenacht, M. Schönherr, and G. Watt, “LHAPDF6: parton density access in the LHC precision era,” Eur. Phys. J. C75 (2015) 132, arXiv:1412.7420 [hep-ph].
  * (40) T. Liu, W. Melnitchouk, J.-W. Qiu, and N. Sato, “A new approach to semi-inclusive deep-inelastic scattering with QED and QCD factorization,” JHEP 11 (2021) 157, arXiv:2108.13371 [hep-ph]
  * (41) A. Denner, S. Dittmaier, M. Roth, and D. Wackeroth, “Predictions for all processes e+ e- —$>$ 4 fermions + gamma,” Nucl. Phys. B 560 (1999) 33–65, arXiv:hep-ph/9904472
  * (42) A. Denner, S. Dittmaier, M. Roth, and L. H. Wieders, “Electroweak corrections to charged-current e+ e- —$>$ 4 fermion processes: Technical details and further results,” Nucl. Phys. B 724 (2005) 247–294, arXiv:hep-ph/0505042. [Erratum: Nucl.Phys.B 854, 504–507 (2012)]
  * (43) A. Denner and S. Dittmaier, “The Complex-mass scheme for perturbative calculations with unstable particles,” Nucl. Phys. B Proc. Suppl. 160 (2006) 22–26, arXiv:hep-ph/0605312
  * (44) T. Hahn and M. Perez-Victoria, “Automatized one loop calculations in four-dimensions and D-dimensions,” Comput. Phys. Commun. 118 (1999) 153–165, arXiv:hep-ph/9807565
  * (45) T. Hahn, “CUBA: A Library for multidimensional numerical integration,” Comput. Phys. Commun. 168 (2005) 78–95, arXiv:hep-ph/0404043
  * (46) R. Frederix, T. Gehrmann, and N. Greiner, “Automation of the Dipole Subtraction Method in MadGraph/MadEvent,” JHEP 09 (2008) 122, arXiv:0808.2128 [hep-ph]
  * (47) T. Gehrmann and N. Greiner, “Photon Radiation with MadDipole,” JHEP 12 (2010) 050, arXiv:1011.0321 [hep-ph]
  * (48) S. Catani, Y. L. Dokshitzer, and B. R. Webber, “The $K^{-}$ perpendicular clustering algorithm for jets in deep inelastic scattering and hadron collisions,” Phys. Lett. B 285 (1992) 291–299
  * (49) S. D. Ellis and D. E. Soper, “Successive combination jet algorithm for hadron collisions,” Phys. Rev. D 48 (1993) 3160–3166, arXiv:hep-ph/9305266
  * (50) G. C. Blazey et al., “Run II jet physics,” in Physics at Run II: QCD and Weak Boson Physics Workshop: Final General Meeting,  pp. 47–77.  arXiv:hep-ex/0005012
  * (51) A. Denner, S. Dittmaier, M. Roth, and M. M. Weber, “Electroweak radiative corrections to $e^{+}e^{-}\to t\overline{t}$H,” Phys. Lett. B 575 (2003) 290–299, arXiv:hep-ph/0307193
  * (52) A. Denner, S. Dittmaier, M. Roth, and M. M. Weber, “Electroweak corrections to e+ e- —$>$ f anti-f H,” Nucl. Phys. B Proc. Suppl. 135 (2004) 88–91, arXiv:hep-ph/0406335
  * (53) T. Plehn, D. L. Rainwater, and D. Zeppenfeld, “Determining the Structure of Higgs Couplings at the LHC,” Phys. Rev. Lett. 88 (2002) 051801, arXiv:hep-ph/0105325
  * (54) S. S. Biswal, R. M. Godbole, B. Mellado, and S. Raychaudhuri, “Azimuthal Angle Probe of Anomalous $HWW$ Couplings at a High Energy $ep$ Collider,” Phys. Rev. Lett. 109 (2012) 261801, arXiv:1203.6285 [hep-ph].