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

Hanying Xiong<EMAIL_ADDRESS>School of Physics, Hangzhou Normal
University, Hangzhou, Zhejiang 311121, China  Hongsheng Hou<EMAIL_ADDRESS>School of Physics, Hangzhou Normal University, Hangzhou, Zhejiang 311121,
China  Zhuoni Qian<EMAIL_ADDRESS>School of Physics, Hangzhou Normal
University, Hangzhou, Zhejiang 311121, China  Qingjun Xu
<EMAIL_ADDRESS>School of Physics, Hangzhou Normal University, Hangzhou,
Zhejiang 311121, China  Bowen Wang<EMAIL_ADDRESS>School of Physics,
Hangzhou Normal University, Hangzhou, Zhejiang 311121, China

###### Abstract

An important mechanism for production of the Higgs boson at the prospective
Large Hadron-electron Collider (LHeC) is via neutral current (NC) weak boson
fusion (WBF) processes. Aside from its role in measurements of Higgs couplings
within the standard model, this production mode is particularly useful in
searchings of Higgs decays into invisble particles in various models for the
Higg portal dark matter. In this work we compute the electroweak corrections
for the NC WBF at the LHeC up to the 1-loop level. For a center-of-mass energy
of 1.98 TeV, the magnitudes of the relative corrections for the total cross
section at next-to-leading (NLO) order are respectively 8% and 17%, in the two
renormalization schemes we use. The NLO terms also distort various
distributions (notably, those for Higgs and electron observables) computed at
the leading order. Along with our previous treatment of the charge current
processes, this paper completes the calulation of the NLO EW effects for the
dominant Higgs production modes at the LHeC.

##  I Introduction

Various Higgs boson production mechanisims have been explored to study the
properties of the particle, especially over the decade after its discovery
Aad:2012tfa ; Chatrchyan:2012xdj . Among these, the production of the Higgs
via weak boson fusion (WBF) has unique kinematic characteristics that have
been succesfully utilized in the determination of Higgs couplings with other
particles. For instance, the measurement of Higgs-gauge boson couplings was
found to be sensitive to the WBF production mode (as well as to other modes
such as the gluon-gluon fusion, or ggH) at the LHC ATLAS:2020qdt ; CMS:2020gsy
. The combined analysis reaches an accuracy at the percent level. In addition,
this channel also plays an important role in probing the
$H\rightarrow\tau^{+}\tau^{-}$ decay CMS:2017zyp ; ATLAS:2018ynr

To avoid large QCD background, the measurement of Higgs Yukawa couplings with
light quarks is proposed to be performed at lepton-hadron colliders such as
the future LHeC LHeCStudyGroup:2012zhm 111Of course, there are many other
motivations for building the LHeC. For more details see e.g., Ref.
LHeCStudyGroup:2012zhm , which is planned to run at a center-of-mass energy of
$1\sim 2$ TeV with the current proton beam at the LHC scattering on an
additional beam of electrons. Studies have already shown the potential of
probing the $H$-$b$ Yukawa coupling at the LHeC Han:2009pe ;
LHeCStudyGroup:2012zhm ; LHeC:2020van , where Higgs bosons are predominantly
produced via WBF. Positive results are also obtained in restricting the
$H$-$c$ Yukawa and triple-higgs-self couplings Li:2019xwd ; Li:2019jba ;
LHeC:2020van . These advancements give rise to the need for including higher
order corrections of the cross sections in the simulations of the relevant
processes. In fact, the QCD Jager:2010zm  and part of the electro-weak
Blumlein:1992eh  corrections to the Higgs WBF production on e-p colliders at
1-loop level were computed even before many of these phenomenological studies.

In a previous paper Wang:2022awk , we222Two of the three authors of this
reference are in the author list of the current paper. treated the Higgs WBF
production via W boson fusion with the full EW corrections at 1 loop (next-to-
leading order, or NLO). At the tree level, the charge current (CC) processes
account for approximately 80% of the total cross section for Higgs production
at the energy regime of the LHeC. However, the neutral current (NC) processes
via Z Z fusion is not negligible, and moreover, has its own merit because the
final state electrons can be observed in about half of the scattering events
(in contrast to the missing energy carried by neutrinos in CC events). The NC
final state will typically contain two visible fermions with large rapidity
difference and invariant mass, as well as the products from the fusion
subprocesses near the central region of the rapidity gap. This configuration
not only provides a prospect of restricting the $H$-$b$ coupling with the NC
WBF Han:2009pe , but is also useful particularly when probing final states
with invisible fusion products, thanks to the kinematic handles provided by
the forward and backward fermions333The classification into “forward”,
“backward”, and “central” may be changed by boosts along the beams, but it is
possible to distinguish various final states with observables such as the
rapidities of the particles..

In fact, studies at the LHC are the first to make use of the WBF mode in dark
matter searches Bai:2011wz ; Ghosh:2012ep ; Dutta:2012xe ; Delannoy:2013ata ;
Bernaciak:2014pna , where the forward and backward jets can be tagged in both
CC and NC processes. In particular, Higgs decays to dark matter particles
allowed by various extentions Arkani-Hamed:1998wuz ; Eboli:2000ze ;
Djouadi:2011aa ; Englert:2011yb ; Mambrini:2011ik ; Baek:2012se ;
Belanger:2013kya ; Curtin:2013fra  of the standard model (SM) are investigated
experimentally by ATLAS ATLAS:2019cid  and CMS CMS:2016dhk  groups. In these
searches, the ggH mode with jets is not preferable since it suffers from large
background, e.g., $Z\rightarrow\nu\overline{\nu}$ \+ jets in the central
region. Due to lack of discriminative power from the event topology, strict
restrictions on the jets have to be imposed in order to suppress the
background, which substantially lowers the signal rate. The search in VH (V
denotes W or Z) production, where V decays leptonically, gives better result,
but WBF turns out to be the most sensitive channel in identifying invisible
decays of the Higgs, and sets the lowest upper limit (18% by CMS CMS:2022qva ;
CMS:2018yfx  and 28% by ATLAS ATLAS:2015gvj ; ATLAS:2021pdg , at 95%
confidence level) for the branching ratio of these decays.

On the other hand, it is shown that more stringent limit is to be expected at
the LHeC, where the background involves Z and W bosons produced via purely
electro-weak interactions Tang:2015uha . In contrast, the background for the
LHC analysis includes Z/W +jets, with jets produced by strong interactions,
which is less clean and controllable. However, the W W fusion processes cannot
be used in dark matter searches at the LHeC as the single jet signal is not
quite differentiable from the charge current deep inelastic scattering. The
study in Ref. Tang:2015uha  focuses then on the NC processes and reaches an
upper limit of 6% at $2\sigma$ level.

Encouraged by these results, we are to carry out in this paper the caculation
of the EW corrections for the NC WBF processes at the LHeC. We shall follow
closely the method and notations of our previous treatment of the CC processes
Wang:2022awk , and organize the rest of this paper as follows. Section II
gives the details of the calculation. Section III shows numerically the
significance of the NLO corrections for the NC processes. In Section IV we
discuss the result and conclude the paper.

##  II Calculation of the processes

In describing the treatment of the NC processes, we shall keep to the minimum
the discussion that repeats the CC calculation Wang:2022awk , and focus more
on the new features of the NC processes.

###  II.1 Leading order (LO) contribution

The LO cross section for the NC processes is computed according to

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

where the squared amplitude $B(P_{A},p_{q})$ for the born process
$eq\rightarrow eqH$, along with the corresponding flux factor, is convolved
with the proper parton distribution function. The result is then summed over
all quark (antiquark) flavors, except for $b$, $\bar{b}$, $t$, and $\bar{t}$,
whose contributions are marginal. A LO graph with an incoming u quark is shown
in Fig. 1.

Figure 1: A representative LO graph for NC WBF at the LHeC.

###  II.2 NLO EW corrections

The calculation of NLO EW corrections involves cancellation of various types
of singularities and must be treated with care. As with the CC case, here we
still work in the framework of the dipole subtraction method Catani:1996vz ;
Schonherr:2017qcj , and organize the calculation into numerical integrations
over 4- and 3-particle final states respectively.

####  II.2.1 4-particle final states

Neglecting the contribution from $t$, $b$, and their antiparticles gives 48
NLO real emission graphs with a photon in the initial or final state, some of
which are shown in Fig. 2. These tree graphs, as well as the LO graph, are
evaluated with the packages FeynArts, FormCalc, and LoopTools Hahn:2000kx ;
Hahn:1998yk . The photon induced graphs may involve resonant propagators and
are handled with the complex-mass-scheme Denner:1999gp ; Denner:2005fg ;
Denner:2006ic  in the calculation.

Figure 2: Representative real-emission graphs for NC WBF at the LHeC. The
first and second lines correspond respectively to the processes
$e^{-}+q\rightarrow e^{-}+\gamma+q+H$ and $e^{-}+\gamma\rightarrow
e^{-}+q+\bar{q}+H$, in which $q=u$.

The subtraction term of the real emission processes have the form

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

where the labeling of the emitter, emittee and spectator follows the
convention in Refs. Schonherr:2017qcj ; Wang:2022awk . Summing over all
contributing flavor combinations gives complete sets of dipoles for the quark
(photon) induced processes, whose detailed forms are collected in Appendix A.
Note that the first term in Eq. 2 represents dipoles with both the splitting
particles and the spectator from the final state. This type of dipoles is not
present in the CC processes.

The contribution from 4-particle final states is then obtained by subtracting
the singularities from the real emission cross section:

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

with a sum over the parton flavor $b$ that collides with the incoming
electron. A particular jet algorithm is implemented via the function $F^{(2)}$
as well as an implicit $F^{(1)}$ multiplied with the subtraction term in order
to ensure IR and collinear safety for jet observables.444One could also
consider collinear unsafe observables Dittmaier:2008md . Distributions in this
case are enhanced logarithmically by final state radiations in collinear
regions. For simplicity, however, this kind of observables is not explored in
this study.

####  II.2.2 3-particle final states

Figure 3: Representative 1-loop graphs for NC WBF at the LHeC.

The contribution of the 3-particle final states comes from loop-graph terms
combined with integrated dipoles as well as collinear counter terms in a
certain factorization scheme. After a reorganization of terms, one arrives at
an expression

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

The terms with $V_{ab}$ is from 1-loop graphs. In this calculation, 2280
1-loop graphs (see examples in Fig. 3) are produced and evaluated by the
program MadGraph5_aMC@NLO Alwall:2014hca . Renormalization of parameters is
done in both $G_{\mu}$ and $\alpha(M_{Z})$ schemes Denner:1991kt ;
Denner:2019vbn . The explicit form of the $\bm{I}$,$\bm{K}$ and $\bm{P}$
Catani:1996vz ; Catani:2002hc ; Schonherr:2017qcj  terms are given in Appendix
B. The last line of Eq. 4 is obtained by factorization of an electron
distribution function to simplify the treatment of the initial state
radiations, where the $\mathcal{O}(\alpha)$ electron distribution reads
Liu:2021jfp

$\displaystyle
f^{\mathcal{O}(\alpha)}_{e}(x,\mu_{F})=\frac{\alpha}{2\pi}\Big{[}\frac{1+x^{2}}{1-x}\ln\frac{\mu_{F}^{2}}{(1-x)^{2}m_{e}^{2}}\Big{]}_{+}.$
(5)

The factorization of both the electron and partons from the proton is carried
out in the $\overline{\mbox{MS}}$ scheme. A jet definition, as remarked at the
end of section II.2.1, is applied to the 3-particle final states by an
implicit function $F^{(1)}$ both at LO and NLO.

##  III Numerical result

The beam energies and SM parameters for this calculation are taken as

$\displaystyle E_{e}=140\;\mbox{GeV},\qquad E_{p}=7\;\mbox{TeV},$ (6)
$\displaystyle G_{\mu}=1.16639\times
10^{-5}\;\mbox{GeV}^{-2},\qquad\alpha_{G_{\mu}}=1/132.5,\qquad\alpha(M_{Z})=1/128.93,$
$\displaystyle
M_{W}=80.419\;\mbox{GeV},\qquad\Gamma_{W}=2.09291\;\mbox{GeV},\qquad
M_{Z}=91.188\;\mbox{GeV},$ $\displaystyle\Gamma_{Z}=2.49877\;\mbox{GeV},\qquad
c_{W}^{2}=1-s_{W}^{2}=\frac{M_{W}^{2}}{M_{Z}^{2}},$ $\displaystyle
m_{e}=0.510998928\;\mbox{MeV},\qquad M_{H}=125\;\mbox{GeV}.$

A unit CKM matrix is used for simplicity, and we take $M_{W}$ for the
renormalization and factorization scales. All quark and electron masses are
set to zero in the scattering amplitudes, which reduces substantially the
number of graphs to be evaluated. The 3- and 4- particle phase spaces are
generated in our own program that relies on the Vegas Monte-Carlo algorithm
from the Cuba library Hahn:2004fe  for numerical integration. Checks similar
to those in the CC case are done until a consistent and stable result is
obtained.

Now we show our numerical results of cross sections computed with the setup
above. The total integrated cross sections listed in Table 1 are about 20% of
those in CC case. However, the corrections from the NLO terms relative to the
LO ones are respectively -8% and -17% in $G_{\mu}$ and $\alpha(M_{Z})$
schemes, which are very close to the results for CC processes. Furthermore,
the decomposition of the total cross sections into various contributions at LO
and NLO in two schemes shows a pattern of convergence very similar to the CC
result, which reflects once again the renormalization scheme independence of
the physical result. At NLO, the ratio between the contributions from 3- and
4- particle final states increases by a factor of 4 as compared with the CC
case. This is rouphly proportional to the increase of the ratio between the
number of loop- and real-graphs in NC WBF.

Schemes | total | LO | 3-particle | 4-particle
---|---|---|---|---
$G_{\mu}$ | 45.02 | 48.78 | -3.85 | 0.09
$\alpha(M_{Z})$ | 43.81 | 52.95 | -9.24 | 0.10
Table 1: Integrated cross sections in fb for NC 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.

For differential distributions, we first construct quark jets and electron
observables according to the $k_{T}$ algorithm Catani:1992zp ; Ellis:1993tq ;
Blazey:2000qt  with the parameter $D=0.8$. Hence the electron observables we
shall discuss below receive contribution from the photon momentum in the soft
and collinear limits. Next, we apply selection cuts to the jet with the
largest transverse momentum (tagging jet), as well as to the electron
observables, requiring

$\displaystyle p_{T}^{j}>30~{}\text{GeV},\qquad p_{T}^{e}>25~{}\text{GeV},$
(7) $\displaystyle\qquad-5<\eta_{j},\,\eta_{e}<-5,$

where these quantities are defined with respect to the forward direction in
which the incoming electron is moving. Cuts on transverse momenta are
inherited from the CC calculation to maintain a typical event shape of WBF.
The requirement on rapidities is loosened as compared with the CC case,
because the NC signal is much smaller at the LHeC and could not afford a
substantial loss. For the same reason we no longer restrict the invant mass of
the Higgs-tagging jet (or of the Higgs-electron) system Han:2009pe .

To present the result, we compare various cross sections computed to LO and
NLO (LO + NLO corrections). We shall work in the $G_{\mu}$ scheme as it leads
to a faster convergence of the perturbative calculation. Fig. 4 shows the
transverse momentum distribution and the corresponding K factors of the
tagging jet. The curves on the left drop rapidly at large $p_{T}^{j}$, and the
NLO correction is negative throughout the entire spectrum shown. The
$p_{T}^{j}$ of a typical event is located around $\mathcal{O}(M_{Z})$, as is
expected for the WBF. The K factor plot on the right shows the distribution
computed up to NLO divided by the corresponding one at LO. Also displayed are
The K factors with solely the loop contribution and with solely the initial
state radiations (ISR) off the electron (i.e., the contribution from the
$\mathcal{O}(\alpha)$ electron distribution function), respectively, at NLO.
Corrections from both sources are about -5% and change slowly with increasing
$p_{T}^{j}$. However, their sum gives almost a constant K factor $\sim 0.93$,
as shown by the blue curve.

The tagging jet rapidity in Fig. 5 covers mostly the backward region and peaks
around $\eta_{j}=-3$. The relative corrections at NLO are still rather stable
($\sim 7\%$) for different $\eta_{j}$, even though the K factors of loop and
ISR contributions vary slowly in opposite trends as $\eta_{j}$ changes.

In searchings of Higgs invisible decays, it is important to make use of the
electron observables together with those of the tagging jet. This proves to be
useful in suppressing the background $p+e^{-}\rightarrow W^{-}(\rightarrow
e^{-}\bar{\nu}_{e})+j+\nu_{e}$ at the LHeC Tang:2015uha . The distribution of
$p_{T}^{e}$ in Fig. 6 behaves very similar to that of $p_{T}^{j}$, both at LO
and NLO, as to be expected from the characteristics of WBF events. In
contrast, the electron and tagging jet rapidities are very different. The
outgoing electron tends to be in the forward region with a more concentrated
distribution of $\eta_{e}$ (as compared with $\eta_{j}$) around 1 (see Fig.
7(a)). Furthermore, the NLO corrections in Fig. 7(b) become positive and
sizeable (as large as 35%) in going to the negative $\eta_{e}$, despite that
the cross sections in this region die out. It is interesting to observe that
both the loop and ISR corrections at NLO are positive in this region. While
the loop terms are very complicated to analyse, it is fairly straightforward
to explain the behavior of the ISR corrections. The negative corrections from
the virtual ISR become more important when the CM energy of the beams is near
the threshold of producing the final state Denner:2003ri ; Denner:2004jy . The
K factor of the ISR above 1 in the negative $\eta_{e}$ region, however,
suggests the converse scenario where the CM energy is much larger than the
invariant mass of the final state. This allows for a large phase space for the
positive contribution from the real photon radiations off the inital electron
that may compensate the negative corrections of the virtual ISR. The net NLO
corrections at large and positive $\eta_{e}$ reaches -15%. Over the entire
spectrum of $\eta_{e}$, the variation of loop corrections is milder than the
ISR.

The typical transverse momentum of the Higgs is much larger than those of the
final state fermions. Fig. 8 shows that $p_{T}^{H}$ peaks at 100 GeV and
extends over 300 GeV. The relative corrections at NLO changes between -5% to
-12% in this range, where the loop and ISR terms give comparable
contributions. The $\eta_{H}$ distribution in Fig. 9 centers near -1 and
appears fairly symmetric on the two sides. Within the range $-4<\eta_{H}<2$,
the NLO K factor varies between 0.97 to 0.9, and essentially interpolates
between the curves of the loop terms and of the ISR. The decrease of the ISR K
factor at large $p_{T}^{H}$ and positive $\eta_{H}$ is consistent with the
fact that these regions are reached by a large collision energy.

The last observable we shall explore is the azimuthal angle difference between
the tagging jet and the final state electron. This quantity has been shown to
be sensitive to Higgs anomalous couplings with gauge bosons, and are used to
distinguish these couplings from the SM ones Plehn:2001nj ; Biswal:2012mp ;
Sharma:2022epc . As with the CC case, the distribution of $\Delta\phi_{e-j}$
in Fig. 10 is going through a steady and monotonic increase within the range
$\Delta\phi_{e-j}\in[0,\pi]$. The K factor increases within the same range
from 0.9 to 0.95, with the trend dominated by loop corrections. The ISR terms
are not sensitive to the change of $\Delta\phi_{e-j}$ and contribute roughly a
constant relative correction about -4%. It is worth mentioning that the
azimuthal angle difference is an effective observable to reduce backgrounds
for WBF production of invisible Higgs bosons at both the LHC Eboli:2000ze  and
LHeC Tang:2015uha . These searches require the azimuthal angle difference to
be less than about $70^{\circ}$, which is in Fig. 10 (b) just the region where
the NLO corrections are most pronounced.

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

Figure 4: Distribution in the transverse momentum $p_{T}^{j}$ of the tagging
jet (a), and the corresponding $K$ factors (b).

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

Figure 5: Distribution in the rapidity $\eta_{j}$ of the tagging jet (a), and
the corresponding $K$ factors (b).

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

Figure 6: Distribution in the transverse momentum $p_{T}^{e}$ of the outgoing
electron (a), and the corresponding $K$ factors (b).

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

Figure 7: Distribution in the rapidity $\eta_{e}$ of the outgoing electron
(a), and the corresponding $K$ factors (b).

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

Figure 8: 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 9: Distribution in the rapidity $\eta_{H}$ of the Higgs boson (a), and
the corresponding $K$ factors (b).

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

Figure 10: Distribution in the azimuthal angle difference between the tagging
jet and the outgoing electron (a), and the corresponding $K$ factors (b).

##  IV Conclusion

In this work we have computed the NLO EW effects for the NC WBF processes at
the LHeC. The dipole subtraction formalism is adopted for organizing various
singularities encountered at 1-loop order. A factorization of the electron
distribution is done in order to simplify the electron mass dependence in the
hard scattering cross sections. The numerical result of the calculation is
obtained in both $G_{\mu}$ and $\alpha(M_{Z})$ renormalization schemes for the
CM energy of the electron-proton beams at 1.98 TeV. While the sizes of the NLO
terms computed in the two schemes differ by $\sim 50\%$, the results of LO +
NLO in two schemes agree up to higher order corrections. The $G_{\mu}$ scheme
leads to a faster convergence of the perturbative calculation, and is used for
computing several differential distributions. In the spectra of the considered
observables where most contributing events reside, the corrections are
negative and within the range [-12%, 0]. The only large positive corrections
are observed in the region where the rapidity of the final state electron
becomes negative, i.e., as it recoils against the proton beam. The Higgs
observables are more sensitive to the EW radiative effects. The roles of the
ISR and loop terms are quite different in distorting the LO distributions,
despite that they both give sizable corrections. Overall, the NLO corrections
are at the level of 10% and are significant in modifying both the
normalization and shape of the LO distributions; they should be included in a
full phenomenological analysis of the NC WBF production of the Higgs at the
LHeC.

##  V Acknowledgement

This work was supported by National Natural Science Foundation of China
(12105068). B.W. and H.X. were 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 processes

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

reads

$\displaystyle|\mathcal{M}_{sub}|^{2}=\mathcal{D}_{e\gamma,q}+\mathcal{D}_{e\gamma}^{e}+\mathcal{D}_{e\gamma}^{q}+\mathcal{D}_{q\gamma,e}+\mathcal{D}_{q\gamma}^{e}+\mathcal{D}_{q\gamma}^{q}+\mathcal{D}_{\gamma,e}^{e}+\mathcal{D}_{\gamma,q}^{e}+\mathcal{D}_{\gamma}^{e,q}+\mathcal{D}_{\gamma,e}^{q}+\mathcal{D}_{\gamma,q}^{q}+\mathcal{D}_{\gamma}^{q,e}$
(8)
$\displaystyle\qquad\quad=-\frac{1}{2p_{1}p_{2}}\frac{1}{1-y_{13}}8\pi\mu^{2\epsilon}\alpha\left[\frac{2}{1-z_{13}(1-y_{13})}-1-z_{13}-\epsilon(1-z_{13})\right]\bm{Q}_{13}^{2}|\mathcal{M}^{q}_{0}(\tilde{p}_{1},\tilde{p}_{3})|^{2}$
$\displaystyle\qquad\qquad-\frac{1}{2p_{1}p_{2}}\frac{1}{x_{12,a}}8\pi\mu^{2\epsilon}\alpha\left[\frac{2}{2-x_{12,a}-z_{1a}}-1-z_{1a}-\epsilon(1-z_{1a})\right]\bm{Q}_{1a}^{2}|\mathcal{M}^{q}_{0}(x_{12,a}p_{a}+P_{1a};x_{12,a}p_{a})|^{2}$
$\displaystyle\qquad\qquad-\frac{1}{2p_{1}p_{2}}\frac{1}{x_{12,b}}8\pi\mu^{2\epsilon}\alpha\left[\frac{2}{2-x_{12,b}-z_{1b}}-1-z_{1b}-\epsilon(1-z_{1b})\right]\bm{Q}_{1b}^{2}|\mathcal{M}^{q}_{0}(x_{12,b}p_{b}+P_{1b};x_{12,b}p_{b})|^{2}$
$\displaystyle\qquad\qquad-\frac{1}{2p_{3}p_{2}}\frac{1}{1-y_{31}}8\pi\mu^{2\epsilon}\alpha\left[\frac{2}{1-z_{31}(1-y_{31})}-1-z_{31}-\epsilon(1-z_{31})\right]\bm{Q}_{13}^{2}|\mathcal{M}^{q}_{0}(\tilde{p}_{1},\tilde{p}_{3})|^{2}$
$\displaystyle\qquad\qquad-\frac{1}{2p_{3}p_{2}}\frac{1}{x_{32,a}}8\pi\mu^{2\epsilon}\alpha\left[\frac{2}{2-x_{32,a}-z_{3a}}-1-z_{3a}-\epsilon(1-z_{3a})\right]\bm{Q}_{3a}^{2}|\mathcal{M}^{q}_{0}(x_{32,a}p_{a}+P_{3a};x_{32,a}p_{a})|^{2}$
$\displaystyle\qquad\qquad-\frac{1}{2p_{3}p_{2}}\frac{1}{x_{32,b}}8\pi\mu^{2\epsilon}\alpha\left[\frac{2}{2-x_{32,b}-z_{3b}}-1-z_{3b}-\epsilon(1-z_{3b})\right]\bm{Q}_{3b}^{2}|\mathcal{M}^{q}_{0}(x_{32,b}p_{b}+P_{3b};x_{32,b}p_{b})|^{2}$
$\displaystyle\qquad\qquad-\frac{1}{2p_{a}p_{2}}\frac{1}{x_{a2,1}}8\pi\mu^{2\epsilon}\alpha\left[\frac{2}{2-x_{a2,1}-z_{1a}}-1-x_{a2,1}-\epsilon(1-x_{a2,1})\right]\bm{Q}_{1a}^{2}|\mathcal{M}^{q}_{0}(x_{a2,1}p_{a}+P_{1a};x_{a2,1}p_{a})|^{2}$
$\displaystyle\qquad\qquad-\frac{1}{2p_{a}p_{2}}\frac{1}{x_{a2,3}}8\pi\mu^{2\epsilon}\alpha\left[\frac{2}{2-x_{a2,3}-z_{3a}}-1-x_{a2,3}-\epsilon(1-x_{a2,3})\right]\bm{Q}_{3a}^{2}|\mathcal{M}^{q}_{0}(x_{a2,3}p_{a}+P_{3a};x_{a2,3}p_{a})|^{2}$
$\displaystyle\qquad\qquad-\frac{1}{2p_{a}p_{2}}\frac{1}{x_{a2,b}}8\pi\mu^{2\epsilon}\alpha\left[\frac{2}{1-x_{a2,b}}-1-x_{a2,b}-\epsilon(1-x_{a2,b})\right]\bm{Q}_{ab}^{2}|\mathcal{M}^{q}_{0}(\tilde{k}_{i}(x_{a2,b});x_{a2,b}p_{a})|^{2}$
$\displaystyle\qquad\qquad-\frac{1}{2p_{b}p_{2}}\frac{1}{x_{b2,1}}8\pi\mu^{2\epsilon}\alpha\left[\frac{2}{2-x_{b2,1}-z_{1b}}-1-x_{b2,1}-\epsilon(1-x_{b2,1})\right]\bm{Q}_{1b}^{2}|\mathcal{M}^{q}_{0}(x_{b2,1}p_{b}+P_{1b};x_{b2,1}p_{b})|^{2}$
$\displaystyle\qquad\qquad-\frac{1}{2p_{b}p_{2}}\frac{1}{x_{b2,3}}8\pi\mu^{2\epsilon}\alpha\left[\frac{2}{2-x_{b2,3}-z_{3b}}-1-x_{b2,3}-\epsilon(1-x_{b2,3})\right]\bm{Q}_{3b}^{2}|\mathcal{M}^{q}_{0}(x_{b2,3}p_{b}+P_{3b};x_{b2,3}p_{b})|^{2}$
$\displaystyle\qquad\qquad-\frac{1}{2p_{b}p_{2}}\frac{1}{x_{b2,a}}8\pi\mu^{2\epsilon}\alpha\left[\frac{2}{1-x_{b2,a}}-1-x_{b2,a}-\epsilon(1-x_{b2,a})\right]\bm{Q}_{ab}^{2}|\mathcal{M}^{q}_{0}(\tilde{k}_{i}(x_{b2,a});x_{b2,a}p_{b})|^{2},$

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

$\displaystyle
y_{13}=\frac{p_{1}p_{2}}{p_{1}p_{2}+p_{1}p_{3}+p_{2}p_{3}},\qquad
z_{13}=\frac{p_{1}p_{3}}{p_{1}p_{3}+p_{2}p_{3}},$ (9) $\displaystyle
y_{31}=\frac{p_{3}p_{2}}{p_{1}p_{2}+p_{1}p_{3}+p_{2}p_{3}},\qquad
z_{31}=\frac{p_{1}p_{3}}{p_{1}p_{2}+p_{1}p_{3}},$ $\displaystyle
x_{12,a}=x_{a2,1}=1-\frac{p_{1}p_{2}}{(p_{1}+p_{2})p_{a}},\qquad
z_{1a}=\frac{p_{1}p_{a}}{(p_{1}+p_{2})p_{a}},$ $\displaystyle
x_{12,b}=x_{b2,1}=1-\frac{p_{1}p_{2}}{(p_{1}+p_{2})p_{b}},\qquad
z_{1b}=\frac{p_{1}p_{b}}{(p_{1}+p_{2})p_{b}},$ $\displaystyle
x_{32,a}=x_{a2,3}=1-\frac{p_{3}p_{2}}{(p_{3}+p_{2})p_{a}},\qquad
z_{3a}=\frac{p_{3}p_{a}}{(p_{3}+p_{2})p_{a}},$ $\displaystyle
x_{32,b}=x_{b2,3}=1-\frac{p_{3}p_{2}}{(p_{3}+p_{2})p_{b}},\qquad
z_{3b}=\frac{p_{3}p_{b}}{(p_{3}+p_{2})p_{b}}$ $\displaystyle
x_{a2,b}=x_{b2,a}=1-\frac{p_{2}(p_{a}+p_{b})}{p_{a}p_{b}}.$

As remarked in Sec. II.2.1, the born amplitudes in Eqs. 8 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 are 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

$\displaystyle\tilde{p}_{1}=p_{1}+p_{2}-\frac{y_{13}}{1-y_{13}}p_{3}\quad\text{and}\quad\tilde{p}_{3}=\frac{1}{1-y_{13}}p_{3}\quad\text{for}\quad\mathcal{D}_{e\gamma,q^{\prime}},$
(10)
$\displaystyle\tilde{p}_{1}=\frac{1}{1-y_{31}}p_{1}\quad\text{and}\quad\tilde{p}_{3}=p_{3}+p_{2}-\frac{y_{31}}{1-y_{31}}p_{1}\quad\text{for}\quad\mathcal{D}_{q^{\prime}\gamma,e},$
$\displaystyle
P_{1a}=p_{1}+p_{2}-p_{a}=p_{1}+p_{2}-(1-x_{12,a})p_{a}-x_{12,a}p_{a},$
$\displaystyle
P_{1b}=p_{1}+p_{2}-p_{b}=p_{1}+p_{2}-(1-x_{12,b})p_{b}-x_{12,b}p_{b},$
$\displaystyle
P_{3a}=p_{3}+p_{2}-p_{a}=p_{3}+p_{2}-(1-x_{32,a})p_{a}-x_{32,a}p_{a},$
$\displaystyle
P_{3b}=p_{3}+p_{2}-p_{b}=p_{3}+p_{2}-(1-x_{32,b})p_{b}-x_{32,b}p_{b},$
$\displaystyle\tilde{k}_{i}^{\mu}=\Lambda_{\nu}^{\mu}k_{i}^{\nu}=\bigg{(}g_{\nu}^{\mu}-\frac{(P_{ab}+\tilde{P}_{ab})^{\mu}(P_{ab}+\tilde{P}_{ab})_{\nu}}{P_{ab}^{2}+P_{ab}\tilde{P}_{ab}}+\frac{\tilde{P}_{ab}^{\mu}P_{ab,\nu}}{P_{ab}^{2}}\bigg{)}k_{i}^{\nu},$
$\displaystyle
P_{ab}=p_{a}+p_{b}-p_{2}\quad\text{for}\quad\tilde{k}_{i}(x_{a2,b})\quad\text{and}\quad\tilde{k}_{i}(x_{b2,a}),$
$\displaystyle\tilde{P}_{ab}=x_{a2,b}p_{a}+p_{b}\quad\text{for}\quad\tilde{k}_{i}(x_{a2,b})\quad\text{and}\quad\tilde{P}_{ab}=p_{a}+x_{b2,a}p_{b}\quad\text{for}\quad\tilde{k}_{i}(x_{b2,a}).$

Note that $\tilde{k}_{i}$ for all final state particles in
$\mathcal{D}_{\gamma}^{e,q}$ and $\mathcal{D}_{\gamma}^{q,e}$ must transform
according to Eq. 10. The charge correlator $\bm{Q}_{ik}^{2}$ is defined by the
charges of the flavor $i$ and $k$ as

$\displaystyle\bm{Q}_{ik}^{2}=Q_{i}Q_{k}\theta_{i}\theta_{k},$ (11)

where $\theta_{i/k}$ is $1$ ($-1$) if it is in the final (initial) state. For
processes initiated by various quark (antiquark) flavors $q$ that contribute
in our calculation, the charge correlators in Eq. 8 are summarized in table 2.

$q$ | $\bm{Q}_{13}^{2}$ | $\bm{Q}_{1a}^{2}$ | $\bm{Q}_{1b}^{2}$ | $\bm{Q}_{3a}^{2}$ | $\bm{Q}_{3b}^{2}$ | $\bm{Q}_{ab}^{2}$
---|---|---|---|---|---|---
$u,\,c$ | $-\frac{2}{3}$ | -1 | $\frac{2}{3}$ | $\frac{2}{3}$ | $-\frac{4}{9}$ | $-\frac{2}{3}$
$d,\,s$ | $\frac{1}{3}$ | -1 | $-\frac{1}{3}$ | $-\frac{1}{3}$ | $-\frac{1}{9}$ | $\frac{1}{3}$
$\bar{u},\,\bar{c}$ | $\frac{2}{3}$ | -1 | $-\frac{2}{3}$ | $-\frac{2}{3}$ | $-\frac{4}{9}$ | $\frac{2}{3}$
$\bar{d},\,\bar{s}$ | $-\frac{1}{3}$ | -1 | $\frac{1}{3}$ | $\frac{1}{3}$ | $-\frac{1}{9}$ | $-\frac{1}{3}$
Table 2: Charge correlators of the dipole terms for quark- and antiquark-
induced real emission processes.

For the photon-induced processes

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

the subtraction term takes the form

$\displaystyle|\mathcal{M}_{sub}|^{2}$
$\displaystyle=\mathcal{D}_{q,\bar{q}}^{\gamma}+\mathcal{D}_{q,e}^{\gamma}+\mathcal{D}_{q}^{\gamma,e}+\mathcal{D}_{\bar{q},q}^{\gamma}+\mathcal{D}_{\bar{q},e}^{\gamma}+\mathcal{D}_{\bar{q}}^{\gamma,e}$
(12)
$\displaystyle=-\frac{1}{2p_{b}p_{2}}\frac{1}{x_{b2,3}}8\pi\mu^{2\epsilon}\alpha\Big{[}1-\epsilon-2x_{b2,3}(1-x_{b2,3})\Big{]}N_{C,f}\bm{Q}_{\widetilde{b2}3}^{2}|\mathcal{M}^{\bar{q}}_{0}(x_{b2,3}p_{b}+P_{23};x_{b2,3}p_{b})|^{2}$
$\displaystyle\quad-\frac{1}{2p_{b}p_{2}}\frac{1}{x_{b2,1}}8\pi\mu^{2\epsilon}\alpha\Big{[}1-\epsilon-2x_{b2,1}(1-x_{b2,1})\Big{]}N_{C,f}\bm{Q}_{\widetilde{b2}1}^{2}|\mathcal{M}^{\bar{q}}_{0}(x_{b2,1}p_{b}+P_{21};x_{b2,1}p_{b})|^{2}$
$\displaystyle\quad-\frac{1}{2p_{b}p_{2}}\frac{1}{x_{b2,a}}8\pi\mu^{2\epsilon}\alpha\Big{[}1-\epsilon-2x_{b2,a}(1-x_{b2,a})\Big{]}N_{C,f}\bm{Q}_{\widetilde{b2}a}^{2}|\mathcal{M}^{\bar{q}}_{0}(\tilde{k}_{i}(x_{b2,a});x_{b2,a}p_{b})|^{2}$
$\displaystyle\quad-\frac{1}{2p_{b}p_{3}}\frac{1}{x_{b3,2}}8\pi\mu^{2\epsilon}\alpha\Big{[}1-\epsilon-2x_{b3,2}(1-x_{b3,2})\Big{]}N_{C,f}\bm{Q}_{\widetilde{b3}2}^{2}|\mathcal{M}^{q}_{0}(x_{b3,2}p_{b}+P_{32};x_{b3,2}p_{b})|^{2}$
$\displaystyle\quad-\frac{1}{2p_{b}p_{3}}\frac{1}{x_{b3,1}}8\pi\mu^{2\epsilon}\alpha\Big{[}1-\epsilon-2x_{b3,1}(1-x_{b3,1})\Big{]}N_{C,f}\bm{Q}_{\widetilde{b3}1}^{2}|\mathcal{M}^{q}_{0}(x_{b3,1}p_{b}+P_{31};x_{b3,1}p_{b})|^{2}$
$\displaystyle\quad-\frac{1}{2p_{b}p_{3}}\frac{1}{x_{b3,a}}8\pi\mu^{2\epsilon}\alpha\Big{[}1-\epsilon-2x_{b3,a}(1-x_{b3,a})\Big{]}N_{C,f}\bm{Q}_{\widetilde{b3}a}^{2}|\mathcal{M}^{q}_{0}(\tilde{k}_{i}(x_{b3,a});x_{b3,a}p_{b})|^{2},$

where $\mathcal{M}^{\bar{q}}_{0}$ and $\mathcal{M}^{q}_{0}$ are the amplitudes
for the born processes $e\bar{q}\rightarrow e\bar{q}H$ and $eq\rightarrow
eqH$, respectively. In addition to the definitions made in Eq. 9 and 10, we
have

$\displaystyle
x_{b2,3}=x_{b3,2}=1-\frac{p_{3}p_{2}}{(p_{3}+p_{2})p_{b}},\qquad
x_{b2,1}=1-\frac{p_{1}p_{2}}{(p_{1}+p_{2})p_{b}},\qquad
x_{b3,1}=1-\frac{p_{1}p_{3}}{(p_{1}+p_{3})p_{b}},$ (13) $\displaystyle
x_{b2,a}=1-\frac{p_{2}(p_{a}+p_{b})}{p_{a}p_{b}},\qquad
x_{b3,a}=1-\frac{p_{3}(p_{a}+p_{b})}{p_{a}p_{b}},\qquad N_{C,f}=3,$
$\displaystyle
P_{23}=P_{32}=p_{2}+p_{3}-p_{b}=p_{2}+p_{3}-(1-x_{b2,3})p_{b}-x_{b2,3}p_{b},$
$\displaystyle
P_{21}=p_{1}+p_{2}-p_{b}=p_{1}+p_{2}-(1-x_{b2,1})p_{b}-x_{b2,1}p_{b},$
$\displaystyle
P_{31}=p_{1}+p_{3}-p_{b}=p_{1}+p_{3}-(1-x_{b3,1})p_{b}-x_{b3,1}p_{b},$
$\displaystyle
P_{ab}=p_{a}+p_{b}-p_{3}\quad\text{and}\quad\tilde{P}_{ab}=p_{a}+x_{b3,a}p_{b}\quad\text{for}\quad\tilde{k}_{i}(x_{b3,a}).$

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 $q$.
The charge correlators for choices of $q$ in Eq. 12 are listed in table 3

$q$ | $\bm{Q}_{\widetilde{b2}3}^{2}$ | $\bm{Q}_{\widetilde{b2}1}^{2}$ | $\bm{Q}_{\widetilde{b2}a}^{2}$ | $\bm{Q}_{\widetilde{b3}2}^{2}$ | $\bm{Q}_{\widetilde{b3}1}^{2}$ | $\bm{Q}_{\widetilde{b3}a}^{2}$
---|---|---|---|---|---|---
$u,\,c$ | $-\frac{4}{9}$ | $-\frac{2}{3}$ | $\frac{2}{3}$ | $-\frac{4}{9}$ | $\frac{2}{3}$ | $-\frac{2}{3}$
$d,\,s$ | $-\frac{1}{9}$ | $\frac{1}{3}$ | $-\frac{1}{3}$ | $-\frac{1}{9}$ | $-\frac{1}{3}$ | $\frac{1}{3}$
Table 3: Charge correlators of the dipole terms for photon-induced real
emission processes.

##  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 processes

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

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. 4. 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

$\displaystyle\bm{I}^{q}(\epsilon)$
$\displaystyle=-\frac{\alpha}{2\pi}\frac{(4\pi)^{\epsilon}}{\Gamma(1-\epsilon)}\left[\frac{1}{\epsilon^{2}}+\frac{3}{2\epsilon}+5-\frac{\pi^{2}}{2}\right]\Bigg{\\{}2\bm{Q}_{12}^{2}\left(\frac{\mu^{2}}{2p_{1}p_{2}}\right)^{\epsilon}+2\bm{Q}_{1a}^{2}\left(\frac{\mu^{2}}{2p_{a}p_{1}}\right)^{\epsilon}$
(14)
$\displaystyle\qquad\qquad+2\bm{Q}_{1b}^{2}\left(\frac{\mu^{2}}{2p_{b}p_{1}}\right)^{\epsilon}+2\bm{Q}_{2a}^{2}\left(\frac{\mu^{2}}{2p_{a}p_{2}}\right)^{\epsilon}+2\bm{Q}_{2b}^{2}\left(\frac{\mu^{2}}{2p_{b}p_{2}}\right)^{\epsilon}+2\bm{Q}_{ab}^{2}\left(\frac{\mu^{2}}{2p_{a}p_{b}}\right)^{\epsilon}\Bigg{\\}},$
$\displaystyle\bm{K}^{c^{\prime}}_{ff}(x)$
$\displaystyle=\frac{\alpha}{2\pi}\Bigg{\\{}Q_{f}^{2}\bigg{[}-(1+x)\log\frac{1-x}{x}+(1-x)+\Big{(}\frac{2}{1-x}\log\frac{1-x}{x}\Big{)}_{+}-\delta(1-x)(5-\pi^{2})\bigg{]}$
(15)
$\displaystyle\quad+\frac{3}{2}\Big{(}\frac{\bm{Q}_{1f}^{2}}{Q_{1}^{2}}+\frac{\bm{Q}_{2f}^{2}}{Q_{2}^{2}}\Big{)}\bigg{[}\Big{(}\frac{1}{1-x}\Big{)}_{+}+\delta(1-x)\bigg{]}+\bm{Q}_{fc^{\prime}}^{2}\bigg{[}(1+x)\log(1-x)-2\bigg{(}\frac{\log(1-x)}{1-x}\bigg{)}_{+}+\frac{\pi^{2}}{3}\delta(1-x)\bigg{]}\Bigg{\\}},$
$\displaystyle\bm{P}^{c^{\prime}}_{ff}(x,\mu_{F}^{2})$
$\displaystyle=\frac{\alpha}{2\pi}\bigg{\\{}-(1+x)+2\Big{(}\frac{1}{1-x}\Big{)}_{+}+\frac{3}{2}\delta(1-x)\bigg{\\}}\bigg{[}\bm{Q}_{f1}^{2}\log\frac{\mu_{F}^{2}}{2xp_{f}p_{1}}+\bm{Q}_{f2}^{2}\log\frac{\mu_{F}^{2}}{2xp_{f}p_{2}}+\bm{Q}_{f{c^{\prime}}}^{2}\log\frac{\mu_{F}^{2}}{2xp_{f}p_{c^{\prime}}}\bigg{]},$
(16)

where “$q$” in $\bm{I}^{q}$ can be a quark or anti-quark. In
$\bm{K}^{c^{\prime}}_{ff}$ and $\bm{P}^{c^{\prime}}_{ff}$, “$f$” can be a
quark, anti-quark, or electron, while “$c^{\prime}$” is the initial particle
species not going through splitting (“$c$” without prime is reserved for the
charm quark, which is only a particular case of $c^{\prime}$. See below). The
specific choices of charge correlators for Eqs. 14, 15, and 16 are collected
in tables 4 and 5.

$\bm{I}^{q}$ | $\bm{Q}_{12}^{2}$ | $\bm{Q}_{1a}^{2}$ | $\bm{Q}_{1b}^{2}$ | $\bm{Q}_{2a}^{2}$ | $\bm{Q}_{2b}^{2}$ | $\bm{Q}_{ab}^{2}$
---|---|---|---|---|---|---
$\bm{I}^{u/c}(\epsilon)$ | $-\frac{2}{3}$ | -1 | $\frac{2}{3}$ | $\frac{2}{3}$ | $-\frac{4}{9}$ | $-\frac{2}{3}$
$\bm{I}^{d/s}(\epsilon)$ | $\frac{1}{3}$ | -1 | $-\frac{1}{3}$ | $-\frac{1}{3}$ | $-\frac{1}{9}$ | $\frac{1}{3}$
$\bm{I}^{\bar{u}/\bar{c}}(\epsilon)$ | $\frac{2}{3}$ | -1 | $-\frac{2}{3}$ | $-\frac{2}{3}$ | $-\frac{4}{9}$ | $\frac{2}{3}$
$\bm{I}^{\bar{d}/\bar{s}}(\epsilon)$ | $-\frac{1}{3}$ | -1 | $\frac{1}{3}$ | $\frac{1}{3}$ | $-\frac{1}{9}$ | $-\frac{1}{3}$
Table 4: Charge correlators of $\bm{I}$ terms for quark- and antiquark-induced processes. $\bm{R}^{c^{\prime}}_{ff}$ | $Q_{f}^{2}$ | $\bm{Q}_{1f}^{2}$ | $\bm{Q}_{2f}^{2}$ | $\bm{Q}_{1}^{2}$ | $\bm{Q}_{2}^{2}$ | $\bm{Q}_{fc^{\prime}}^{2}$
---|---|---|---|---|---|---
$\bm{R}^{u/c}_{ee}$ | 1 | -1 | $\frac{2}{3}$ | 1 | $\frac{4}{9}$ | $-\frac{2}{3}$
$\bm{R}^{e}_{uu/cc}$ | $\frac{4}{9}$ | $\frac{2}{3}$ | $-\frac{4}{9}$ | 1 | $\frac{4}{9}$ | $-\frac{2}{3}$
$\bm{R}^{d/s}_{ee}$ | 1 | -1 | $-\frac{1}{3}$ | 1 | $\frac{1}{9}$ | $\frac{1}{3}$
$\bm{R}^{e}_{dd/ss}$ | $\frac{1}{9}$ | $-\frac{1}{3}$ | $-\frac{1}{9}$ | 1 | $\frac{1}{9}$ | $\frac{1}{3}$
$\bm{R}^{\bar{u}/\bar{c}}_{ee}$ | 1 | -1 | $-\frac{2}{3}$ | 1 | $\frac{4}{9}$ | $\frac{2}{3}$
$\bm{R}^{e}_{\bar{u}\bar{u}/\bar{c}\bar{c}}$ | $\frac{4}{9}$ | $-\frac{2}{3}$ | $-\frac{4}{9}$ | 1 | $\frac{4}{9}$ | $\frac{2}{3}$
$\bm{R}^{\bar{d}/\bar{s}}_{ee}$ | 1 | -1 | $\frac{1}{3}$ | 1 | $\frac{1}{9}$ | $-\frac{1}{3}$
$\bm{R}^{e}_{\bar{d}\bar{d}/\bar{s}\bar{s}}$ | $\frac{1}{9}$ | $\frac{1}{3}$ | $-\frac{1}{9}$ | 1 | $\frac{1}{9}$ | $-\frac{1}{3}$
Table 5: Charge correlators of $\bm{R}$ terms for quark- and antiquark-induced
processes, where $\bm{R}$ stands for $\bm{K}$ or $\bm{P}$. Note that
$\bm{Q}_{f1}^{2}$ and $\bm{Q}_{f2}^{2}$ are not listed since
$\bm{Q}_{f1}^{2}=\bm{Q}_{1f}^{2}$, and $\bm{Q}_{f2}^{2}=\bm{Q}_{2f}^{2}$.

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

$\displaystyle\bm{K}^{a}_{\gamma f}(x)$
$\displaystyle=\frac{\alpha}{2\pi}\Bigg{\\{}N_{C,f}Q_{f}^{2}\frac{1+(1-x)^{2}}{x}\log\frac{1-x}{x}+N_{C,f}Q_{f}^{2}x(1-x)-\bm{Q}_{fa}^{2}N_{C,f}\frac{1+(1-x)^{2}}{x}\log(1-x)\Bigg{\\}},$
(17) $\displaystyle\bm{P}^{a}_{\gamma
f}(x,\mu_{F}^{2})=\frac{\alpha}{2\pi}\bigg{\\{}N_{C,f}\frac{1+(1-x)^{2}}{x}\bigg{\\}}\bigg{[}\bm{Q}_{f1}^{2}\log\frac{\mu_{F}^{2}}{2xp_{b}p_{1}}+\bm{Q}_{f2}^{2}\log\frac{\mu_{F}^{2}}{2xp_{b}p_{2}}+\bm{Q}_{fa}^{2}\log\frac{\mu_{F}^{2}}{2xp_{a}p_{b}}\bigg{]}.$
(18)

The charge correlators for Eqs. 17 and 18 are collected in table 6.

$\bm{R}^{a}_{\gamma f}$ | $Q_{f}^{2}$ | $\bm{Q}_{f1}^{2}$ | $\bm{Q}_{f2}^{2}$ | $\bm{Q}_{fa}^{2}$
---|---|---|---|---
$\bm{R}^{e}_{\gamma u/c}$ | $\frac{4}{9}$ | $\frac{2}{3}$ | $-\frac{4}{9}$ | $-\frac{2}{3}$
$\bm{R}^{e}_{\gamma d/s}$ | $\frac{1}{9}$ | $-\frac{1}{3}$ | $-\frac{1}{9}$ | $\frac{1}{3}$
$\bm{R}^{e}_{\gamma\bar{u}/\bar{c}}$ | $\frac{4}{9}$ | $-\frac{2}{3}$ | $-\frac{4}{9}$ | $\frac{2}{3}$
$\bm{R}^{e}_{\gamma\bar{d}/\bar{s}}$ | $\frac{1}{9}$ | $\frac{1}{3}$ | $-\frac{1}{9}$ | $-\frac{1}{3}$
Table 6: Charge correlators of $\bm{R}$ terms for photon-induced processes,
where $\bm{R}$ stands for $\bm{K}$ or $\bm{P}$.

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