#  Charged lepton flavor violation associated with heavy quark production in
deep inelastic lepton-nucleon scattering via scalar exchange

Yuichiro Kiyo<EMAIL_ADDRESS>Department of Physics, Juntendo
University, Inzai, 270-1695 Japan  Michihisa Takeuchi
<EMAIL_ADDRESS>Department of Physics, Osaka University,
Osaka 560-0043, Japan  Yuichi Uesaka<EMAIL_ADDRESS>Faculty of
Science and Engineering, Kyushu Sangyo University, 2-3-1 Matsukadai, Higashi-
ku, Fukuoka 813-8503, Japan  Masato Yamanaka<EMAIL_ADDRESS>Department of Mathematics and Physics, Osaka City University, Osaka 558-8585,
Japan  Nambu Yoichiro Institute of Theoretical and Experimental Physics
(NITEP), Osaka City University, Osaka 558-8585, Japan

###### Abstract

We study charged lepton flavor violation (CLFV) associated with heavy quark
pair production in lepton-nucleon deep-inelastic scattering
$\ell_{i}N\to\ell_{j}q\bar{q}X$. Here $\ell_{i}$ and $\ell_{j}$ denote the
initial and final leptons; $N$ and $X$ are respectively the initial nucleon
and arbitrary final hadronic system. We employ a model Lagrangian in which a
scalar and pseudoscalar mediator generates the CLFV. We derive heavy quark
structure functions for scalar and pseudoscalar currents and compute momentum
distributions of the final lepton for the process. Our focus is on the heavy
quark mass effects in the final lepton momentum distribution. We clarify the
necessity of inclusion of the heavy quark mass to obtain reliable theory
predictions for the CLFV signal searches in the deep-inelastic scattering.

††preprint: OCU-PHYS 545, NITEP 90, OU-HET-1102

##  I Introduction

A variety of new physics models provides contributions to the Charged Lepton
Flavor Violating (CLFV) observables by extra degrees of freedom, for instance
extensions of particle contents, additional space dimensions, etc. (See
Refs.Kuno and Okada (2001); Raidal et al. (2008) and references therein for
reviews on this subject.) It is often the case that the CLFV mediators couple
with not only leptons but also quarks. In that case, noticeable processes
which involve hadronic interactions, e.g., $\mu\to e$ conversion in nuclei,
$\tau\to\ell_{i}\pi\pi$ ($\ell_{i}=e,\mu$), different flavor di-lepton
$\ell_{i}\ell_{j}$ production at hadron collider experiments, would be
expected. No signal for such types of CLFV processes is discovered so far
although a lot of effort to search for them has been devoted in the various
experiments. Though these results are translated into the stringent limits on
the CLFV interactions, the limits are mainly on the interactions related to
light flavor quarks. This motivates us to revisit the scenarios where the CLFV
mediators dominantly couples with heavy quarks. Such scenarios are motivated
theoretically as well as experimentally, e.g. extra-dimension models Huber
(2003); Moreau and Silva-Marcos (2006); Agashe et al. (2006); Davidson et al.
(2008), two Higgs doublet models Kanemura et al. (2006); Davidson and Grenier
(2010); Crivellin et al. (2013); Botella et al. (2016), leptoquark models
Crivellin et al. (2021), models with flavor symmetry Tsumura and Velasco-
Sevilla (2010), and the next to minimal flavor violation scenarios Agashe et
al. (2005).

In such scenarios, CLFV processes via deep-inelastic scattering (DIS)
$\ell_{i}N\to\ell_{j}X$, where $N$ is a nucleon, $\ell_{i}$ and $\ell_{j}$ are
respectively the initial and final leptons, offer a good prospect for CLFV
searches. Such processes can be probed at fixed target experiments and lepton-
hadron colliders. In both experiments, the problems due to pile-up and QCD
background are better controlled than in the environments of the hadron-hadron
colliders. In this article, we will focus on the study at the fixed target
experiments. The event rates in such experiments increase with the beam
energy, beam intensity and target density. Typical beam energy $E_{\ell}$ in
the next-generation experiments is up to $10~{}{\rm TeV}$, which corresponds
to $\sqrt{s}\lesssim 100$ GeV. Although it seems not high, it is sufficient to
open production thresholds for the CLFV processes. Therefore it is expected to
observe enough signals in the CLFV searches at the fixed target experiments.

The DIS processes are studied with a variety of theoretical motivations in the
context of the CLFV Gninenko et al. (2002); Sher and Turan (2004); Kanemura et
al. (2005); Gonderinger and Ramsey-Musolf (2010); Bolanos et al. (2013); Liao
and Wu (2016); Abada et al. (2017); Takeuchi et al. (2017); Gninenko et al.
(2018); Antusch et al. (2020); Husek et al. (2021); Cirigliano et al. (2021);
Antusch et al. (2021), as well as a probe for the Standard Model (SM) and new
physics Cakir et al. (2009); Han and Mellado (2010); Liang et al. (2010);
Blaksley et al. (2011); Biswal et al. (2012); Dutta et al. (2015); Li et al.
(2018); Curtin et al. (2018); Li et al. (2019); Azuelos et al. (2020). The
HERA experiment searched for the CLFV DIS processes and put the bound on the
related parameters Aktas et al. (2007); Aaron et al. (2011). Searches for the
CLFV DIS processes are proposed at the upcoming experiments, and shown to
reach higher sensitivities than the current bounds by a few orders of
magnitude Accardi et al. (2016); Abelleira Fernandez et al. (2012).

In this article, we study the CLFV DIS processes, $\ell_{i}N\to\ell_{j}X_{H}$,
in the scenario where a (pseudo-)scalar CLFV mediator dominantly couples with
heavy flavor quarks, like the SM Higgs boson. Here $X_{H}$ denotes an
inclusive hadronic final state involving heavy quarks. It is worth
investigating the CLFV DIS processes associated with heavy flavor quarks,
since the CLFV operators involving heavy flavor quarks are usually difficult
to probe directly in the low-energy flavor experiments. Experimental signals
for the processes are characterized as the existence of a heavy charged lepton
$\ell_{j}$ and heavy quarks in the final state. Such signals seem distinctive,
but there is always a competition between the signals and the background.
Thus, precise understanding of the backgrounds and also the accurate theory
prediction for the signal processes would be required. In this article, we
focus on the latter point. One of differences between heavy quarks and light
quarks is the mass effect, and it is important to understand how the heavy
quark mass affects the CLFV DIS observables. At first glance, it looks simple
and straightforward, but it turns out rather complicated when the issue is
related to the problem of the large logarithmic resummation in the
perturbative QCD. A resolution to the problem was given in a series of seminal
papers by ACOT Aivazis et al. (1994a, b) at the leading order (LO), and the
result has been extended to include higher-order effect in a consistent manner
Buza et al. (1998); Cacciari et al. (1998); Collins (1998); Forte et al.
(2010). In the present article, we apply the ACOT method to the CLFV DIS
involving heavy quarks. Although we work at LO in QCD strong coupling
expansion, we include some of the important effects of heavy quark mass which
were obtained in the studies of QCD structure functions in the literatures
Krämer et al. (2000); Tung et al. (2002); Kretzer et al. (2004); Stavreva et
al. (2012). We aim for the construction of heavy quark structure functions of
(pseudo-)scalar exchange coming from CLFV interactions. With the constructed
heavy quark structure functions we analyze some distributions of the final
lepton momentum to investigate how the heavy quark mass effect modifies the
CLFV DIS observables. We focus on the analysis of the CLFV DIS processes
associated with heavy quark production in the present article, and a
comprehensive phenomenological study taking other modes will be reported in a
separate paper.

The present article is organized in the following way. In section II.A we will
describe the model Lagrangian of the CLFV (pseudo-)scalar mediator which
strongly couples with heavy quarks. With the model Lagrangian, we will compute
the cross section for CLFV DIS associated with heavy quark pair production. We
shall introduce a structure function of (pseudo-)scalar current. In section
II.B we will show the cross section formula and momentum distribution of the
final lepton, where an inverse moment of the structure function shall be
introduced. In section II.C we will compute the heavy quark contribution to
the structure function at the leading order in $\alpha_{s}$ with massive
quark. In sections II.D and E, the SACOT scheme and threshold improvement of
the heavy quark structure function are discussed. The issue here is the
unification of heavy quark mass effect and large logarithm resummation in a
consistent manner. In section III we will perform the numerical analysis of
the structure functions for the production of bottom and charm quarks. In
section IV we will present the numerical results of the CLFV cross section
associated with heavy quark production.

##  II CLFV DIS and Heavy quark production

###  II.1 CLFV DIS via scalar or pseudoscalar current

We start with an interaction Lagrangian for a neutral scalar or pseudoscalar
field $\phi\in\\{S,P\\}$ coupled with charged leptons $\ell_{i},\ell_{j}$ and
heavy flavor quarks $q=c,b,t$;

$\begin{split}\mathcal{L}_{\phi}&=-\sum_{ij}\left(\rho_{ij}^{\phi}\bar{\ell}_{j}P_{L}\ell_{i}\phi+h.c.\right)-\sum_{q}\rho_{qq}^{\phi}\bar{q}\,\Gamma^{\phi}q\phi,\end{split}$
(1)

where $i,j$ run over flavor indices of charged leptons, $q$ runs over heavy
flavor quarks, and $P_{L}=(1-\gamma_{5})/2$. The vertex factors $\Gamma^{S}=1$
and $\Gamma^{P}=i\gamma_{5}$ are the matrices in Dirac-space respectively for
scalar and pseudoscalar cases. The lepton and quark fields in the interaction
Lagrangians are mass eigenstates. We assume that the CLFV mediators interact
with quarks through flavor diagonal couplings $\rho_{qq}^{\phi}$, while in the
lepton sector the off-diagonal coupling $\rho_{ij}^{\phi}(i\neq j)$ induces
the CLFV.

Figure 1:  The CLFV lepton-nucleon scattering induced by $t$-channel exchange
of the (pseudo-)scalar mediator. The initial and final lepton momenta are
$k_{i}$ and $k_{j}$, respectively, and the nucleon momentum is $P$.

In Fig. 1 schematic diagram for the process $\ell_{i}N\to\ell_{j}X_{H}$ is
shown, where initial lepton $\ell_{i}$ with a momentum $k_{i}$ and a nucleon
$N$ with a momentum $P$ are scattered by exchanging the CLFV mediator $\phi$
with a momentum $q=k_{i}-k_{j}$. The final states are lepton $\ell_{j}$ with
momentum $k_{j}$ and an arbitrary hadronic system $X_{H}$ which contains heavy
quarks. The amplitude, where the CLFV mediator $\phi$ is exchanged in
$t$-channel, is factorized into leptonic and hadronic amplitudes. The cross
section $\sigma$ consists of the leptonic and hadronic parts, which are
respectively denoted by $L_{\phi}$ and $F_{\phi}$, and is written in the
following form:

$\displaystyle\frac{d\sigma}{dxdy}\left(\ell_{i}N\to\ell_{j}X\right)=\frac{y\left|\rho_{qq}^{\phi}\right|^{2}L_{\phi}(Q^{2})\,F_{\phi}(x,Q^{2})}{16\pi\left(Q^{2}+m_{\phi}^{2}\right)^{2}},$
(2)

where $m_{\phi}$ is a mediator mass, and the dimensionless variables $x,y$ are
defined by

$\displaystyle x=\frac{Q^{2}}{2P\cdot q},\hskip 28.45274pty=\frac{2P\cdot
q}{2P\cdot k_{i}},$ (3)

with $Q^{2}=-q^{2}=xy(s-m_{N}^{2})$. Here $s=(P+k_{i})^{2}$ is the collision
energy squared of the initial lepton-nucleon system, and $m_{N}$ is the
nucleon mass.

The leptonic part is given by

$\displaystyle L_{\phi}$ $\displaystyle=$
$\displaystyle\big{(}|\rho_{ij}^{\phi}|^{2}+|\rho_{ji}^{\phi}|^{2}\big{)}\big{(}Q^{2}+m_{i}^{2}+m_{j}^{2}\big{)}+4\,{\rm
Re}\left(\rho_{ij}^{\phi}\rho_{ji}^{\phi}\right)m_{i}m_{j},$ (4)

with $m_{i(j)}$ being the initial (final) lepton mass, and the hadronic part
is called structure function written in a convolution form as

$\displaystyle F_{\phi}(x,Q^{2})$ $\displaystyle=$
$\displaystyle\sum_{k}\int_{0}^{1}\frac{d\xi}{\xi}\
C_{\phi,\,k}\left(\frac{x}{\xi}\right)f_{k/N}(\xi,\mu_{f}^{2}),$ (5)

where $k\in\\{g,q,\bar{q}\\}$ is a parton which contributes to the process
$\phi k\rightarrow X$. The $C_{\phi,k}$ is a coefficient function calculable
in perturbative QCD, while the parton distribution function (PDF)
$f_{k/N}(x,\mu_{f}^{2})$ is a nonperturbative object which describes a
probability of parton $k$ having momentum fraction $\xi$ inside the nucleon
$N$ at a factorization scale $\mu_{f}^{2}$. The $\mu_{f}$-dependence is
governed by renormalization group equation, so called Dokshitzer-Gribov-
Lipatov-Altarelli-Parisi (DGLAP) equation Gribov and Lipatov (1972); Altarelli
and Parisi (1977) :

$\displaystyle\mu_{f}^{2}\frac{\partial~{}}{\partial\mu_{f}^{2}}f_{k/N}(\xi,\mu_{f}^{2})$
$\displaystyle=$
$\displaystyle\sum_{l}\frac{\alpha_{s}(\mu_{f}^{2})}{2\pi}\int_{\xi}^{1}\frac{d\eta}{\eta}{\cal
P}_{kl}\left(\frac{\xi}{\eta}\right)f_{l/N}(\xi,\mu_{f}^{2})+{\cal
O}(\alpha_{s}^{2}),$ (6)

where $l$ runs over possible quark flavors and gluon, and ${\cal P}_{kl}$ is a
splitting function at one-loop level. Conventionally the factorization scale
is chosen as $\mu_{f}^{2}\sim Q^{2}$ to be the same order as the hard scale in
the process. For the heavy quark production, there are two hard scales
$Q^{2},m_{q}^{2}$ in the process, and we will adopt a more refined scale
setting (see Eq. (26)).

###  II.2 Cross sections and distribution

In theory discussion it is sometimes convenient to use $x,Q^{2}$ as
independent variables instead of $x,y$. The conversion formula is given by

$\displaystyle\frac{d^{2}\sigma}{dxdQ^{2}}(\ell_{i}N\to\ell_{j}X)$
$\displaystyle=$
$\displaystyle\left(\frac{1}{xs}\right)\frac{d^{2}\sigma}{dxdy}(\ell_{i}N\to\ell_{j}X).$
(7)

Integrated over $x$ one obtains the differential cross section
$d\sigma/dQ^{2}$ as

$\displaystyle\frac{d\sigma}{dQ^{2}}$ $\displaystyle=$
$\displaystyle\frac{|\rho_{qq}^{\phi}|^{2}}{16\pi
s^{2}}\frac{Q^{2}L_{\phi}(Q^{2})M_{\phi}(s,Q^{2})}{(Q^{2}+m_{\phi}^{2})^{2}},$
(8)

where $M_{\phi}$ is the second inverse moment of the structure function
defined by

$\displaystyle M_{\phi}(s,Q^{2})$ $\displaystyle\equiv$
$\displaystyle\int_{x_{\rm min}(s,Q^{2})}^{x_{\rm
max}(Q^{2})}x^{-2}\,F_{\phi}(x,Q^{2})dx.$ (9)

The integration region $[x_{\rm min}(s,Q^{2}),x_{\rm max}(Q^{2})]$ depends on
partonic processes, and the $x_{\rm min}(s,Q^{2})$ depends on $s$ (see (58)),
which introduces the collision energy dependence in the inverse moment. A
derivation of the physical region for the CLFV DIS is given in the Appendix.

As a direct observable in collider experiments, we study the momentum
distribution of the final lepton. To make our discussion concrete we give a
cross section formula for $eN\to\tau X$ for fixed target experiment where the
initial nucleon is at the rest. The initial nucleon and electron momenta are
parametrized by $k_{e}=(E_{e},0,0,E_{e})$ and $P=(m_{N},0,0,0)$, respectively.
Here we ignored the electron mass, and the nucleon mass is set to be
$m_{N}=938\,{\rm MeV}$. The electron beam energy $E_{e}$ is related to the
collision energy by $E_{e}=\sqrt{(s-m_{N}^{2})/2m_{N}}$. The final
$\tau$-momentum at the nucleon rest frame is parametrized as
$k_{\tau}=\left(E_{\tau},p_{T},0,p_{Z}\right)$ where $p_{T},p_{Z}$ are related
to the dimensionless parameters $x,y$ as

$\displaystyle p_{Z}=(1-y)E_{e}-xym_{N}-\frac{m_{\tau}^{2}}{2E_{e}},\hskip
14.22636ptp_{T}=\sqrt{(1-y)^{2}E_{e}^{2}-m_{\tau}^{2}-p_{Z}^{2}},$ (10)

with $E_{\tau}=(1-y)E_{e}$. Then the $\tau$-momentum distribution at the
nucleon rest frame is given by

$\displaystyle\frac{d^{\,2}\sigma}{dp_{T}\,dp_{Z}}(\ell_{i}N\to\ell_{j}X)$
$\displaystyle=$
$\displaystyle\bigg{[}\frac{2p_{T}}{E_{\tau}y(s-m_{N}^{2})}\bigg{]}\frac{d^{2}\sigma}{dxdy}(\ell_{i}N\to\ell_{j}X),$
(11)

where a Jacobian factor is multiplied in the right-hand side of Eq. (11) to
convert the independent variables from $(x,y)$ to $(p_{T},p_{Z})$.

Figure 2:  The physical region of the $\tau$-momentum for the process
$eN\to\tau X$ in a fixed target experiment. The electron beam energy is
$E_{e}=200\,{\rm GeV}$(left panel)/$1\,{\rm TeV}$(right panel). The contour
lines of fixed $x$ (black lines) and fixed $Q$ (dashed blue lines) are plotted
in the momentum space $(p_{z},p_{T})$.

In Fig. 2 the physical region of the $\tau$-momentum $(p_{Z},p_{T})$ is
plotted for the beam energies $E_{e}=200\,{\rm GeV}$ and $1\,{\rm TeV}$
respectively in the left and right panels. The black lines are the contours
for $x=0.1,0.2,\cdots,1.0$ from smaller to larger arcs, and the blue dashed
lines are the contours for $Q$ values indicated in the plots. For CLFV signal
searches in the fixed target experiments it is of great importance to have a
reliable theory prediction that covers all the physical regions of Fig. 2.

###  II.3 Heavy quark contribution to structure function

In the following, we consider the heavy quark production via the CLFV scalar
and pseudoscalar interactions. To be concrete we assume that the heavy quark
$q$ is bottom or charm quark, which is much heavier than the nucleon, and
treat other lighter quarks as massless. Formulae derived here can be also
applied to top quark, but the threshold of top quark pair production is too
high, and we do not discuss its phenomenology in the present article. The
heavy quark mass is so large that its intrinsic partonic content inside the
nucleon is zero. Yet the heavy quark can be produced in pair with its anti-
quark via a gluon splitting $g\to q\bar{q}$ and subsequently $q$ or $\bar{q}$
is scattered by $\phi=S,P$ via the quark-mediator interaction of ${\cal
L}_{\phi}$. In Fig. 3 an example Feynman diagram is shown for such a process.

Figure 3: A Feynman diagram for the heavy quark pair production in the CLFV
lepton-nucleon scattering.

The corresponding hadronic part $F_{\phi}$ which contributes to the
$\ell_{i}N\to\ell_{j}q\bar{q}X$ is given by

$\displaystyle F_{\phi}^{{\rm M}}(x,Q^{2})$ $\displaystyle=$
$\displaystyle\int_{0}^{1}\frac{d\xi}{\xi}\ C_{\phi,g}^{{\rm
M}}\left(\frac{x}{\xi},\frac{Q^{2}}{m_{q}^{2}}\right)f_{g/N}(\xi,\mu_{f}^{2}),$
(12)

where $C_{\phi,g}^{{\rm M}}$ is a heavy quark contribution to the coefficient
function. The superscript M indicates that the quantity is computed in massive
scheme, where the heavy quark mass is retained in the computation:

$\displaystyle C_{S,g}^{{\rm
M}}\left(\frac{x}{\xi},\frac{Q^{2}}{m_{q}^{2}}\right)$ $\displaystyle=$
$\displaystyle\frac{\alpha_{s}}{2\pi}\frac{T_{F}}{(Q^{2}+w^{2})^{2}}\Theta\left(w^{2}-4m_{q}^{2}\right)\bigg{\\{}2Kw^{2}(Q^{2}+4m_{q}^{2})$
(13) $\displaystyle\hskip
28.45274pt+\bigg{[}Q^{2}(Q^{2}+4m_{q}^{2})+(w^{2}-4m_{q}^{2})^{2}\bigg{]}\ln\frac{1+K}{1-K}\bigg{\\}},$
$\displaystyle C_{P,g}^{{\rm
M}}\left(\frac{x}{\xi},\frac{Q^{2}}{m_{q}^{2}}\right)$ $\displaystyle=$
$\displaystyle\frac{\alpha_{s}}{2\pi}\frac{T_{F}}{(Q^{2}+w^{2})^{2}}\Theta\left(w^{2}-4m_{q}^{2}\right)\bigg{\\{}2Kw^{2}Q^{2}$
(14) $\displaystyle\hskip
28.45274pt+\bigg{[}Q^{2}(Q^{2}+4m_{q}^{2})+w^{4}\bigg{]}\ln\frac{1+K}{1-K}\bigg{\\}},$

with $T_{F}=1/2$ and $\Theta(x)$ being the Heaviside step function. The
$K=\sqrt{1-4m_{q}^{2}/w^{2}}$ is the speed of the heavy quark in the center-
of-mass frame of the produced heavy quark pair with $w^{2}=(q+\xi
P)^{2}=Q^{2}(\xi/x-1)$ being the invariant mass of $q\bar{q}$. Thus, the step
function can be rewritten as
$\Theta(w^{2}-4m_{q}^{2})=\Theta\left(\xi-\chi\right)$ with

$\displaystyle\chi\equiv\left(1+\frac{4m_{q}^{2}}{Q^{2}}\right)x.$ (15)

The appearance of $\chi$, not $x$, is an important mass effect due to the
threshold of pair production of $q\bar{q}$. This leads $\chi$-rescaling
prescription Tung et al. (2002); Kretzer et al. (2004) based on an idea of so-
called slow-rescaling Barnett (1976), which will be discussed later.

###  II.4 SACOT scheme

(a)  (b)

Figure 4: Feynman diagrams for (a) the heavy quark pair production in scalar-
gluon fusion, and (b) heavy quark excitation.

In Eqs.(13) and (14), the heavy quark contribution to the coefficient function
is calculated retaining the heavy quark mass, which corresponds to the heavy
quark pair production in scalar-gluon fusion (a) in Fig. 4, and its behavior
around the threshold $w^{2}\sim 4m_{q}^{2}$ is correctly described order by
order in perturbative expansion of $\alpha_{s}$. However when $Q^{2}\gg
m_{q}^{2}$ there appears a large logarithm $\ln(Q^{2}/m_{q}^{2})$ in
$C_{\phi,g}^{{\rm M}}$, and its appearance deteriorates the perturbative
computation for $C_{\phi,g}^{{\rm M}}$. It can be seen by expanding
$C_{\phi,g}^{{\rm M}}$ in the limit $(m_{q}^{2}/Q^{2})\to 0$:

$\displaystyle C_{\phi,g}^{{\rm M}}$ $\displaystyle\approx$
$\displaystyle\int_{0}^{1}\frac{d\eta}{\eta}\bigg{[}C_{\phi,q}^{{\rm
ZM}}\left(\frac{x}{\eta}\right)+C_{\phi,\bar{q}}^{{\rm
ZM}}\left(\frac{x}{\eta}\right)\bigg{]}\,\Theta(\xi-\eta)\bigg{[}\left(\frac{\alpha_{s}}{2\pi}\right){\cal
P}_{qg}\left(\frac{\eta}{\xi}\right)\ln\left(\frac{Q^{2}}{m_{q}^{2}}\right)\bigg{]},$
(16)

where ${\cal P}_{qg}(z)=T_{F}\big{[}z^{2}+(1-z)^{2}\big{]}$ is a gluon-quark
splitting function, and $C_{\phi,q}^{{\rm ZM}}(x),C_{\phi,\bar{q}}^{{\rm
ZM}}(x)$ are single heavy quark contributions to the coefficient function in
the massless limit, corresponding to the heavy quark excitation diagram (b) in
Fig. 4:

$\displaystyle C_{\phi,q}^{{\rm ZM}}\left(x\right)=C_{\phi,\bar{q}}^{{\rm
ZM}}\left(x\right)=\frac{1}{2}\delta\left(1-x\right).$ (17)

Therefore the high-energy limit of the structure function can be written as

$\displaystyle F_{\rm\phi}^{{\rm M0}}(x,Q^{2})$ $\displaystyle=$
$\displaystyle\int_{0}^{1}\frac{d\eta}{\eta}\ 2C_{\phi,q}^{{\rm
ZM}}\left(\frac{x}{\eta}\right)\frac{\alpha_{s}}{2\pi}\ln\left(\frac{\mu_{f}^{2}}{m_{q}^{2}}\right)\int_{\eta}^{1}\frac{d\xi}{\xi}{\cal
P}_{qg}\left(\frac{\eta}{\xi}\right)f_{g/N}(\xi,\mu_{f}^{2})$ (18)

where the superscript M0 denotes the leading contribution in the massless
limit of the massive scheme structure function. Namely the $F_{\phi}^{{\rm
M0}}$ contains the mass singularity of the massive structure function, and
$(F_{\phi}^{{\rm M}}-F_{\phi}^{{\rm M0}})$ is finite in the massless limit.
Here, the original collinear logarithm existing in Eq. (16) is separated as
$\alpha_{s}\ln(Q^{2}/m_{q}^{2})=\alpha_{s}\ln(Q^{2}/\mu_{f}^{2})+\alpha_{s}\ln(\mu_{f}^{2}/m_{q}^{2})$,
and the mass singularity $\alpha_{s}\ln(\mu_{f}^{2}/m_{q}^{2})$ is absorbed in
Eq. (18).

The relation between the high-energy limit of the massive quark contribution
and the $F_{\phi}^{{\rm M0}}$ structure function is nothing but the
factorization theorem Collins (1998) of collinear singularities for the QCD
structure function, which can be utilized to resum the large collinear
logarithms to all orders in $\alpha_{s}$ to make the structure functions
stable at high energy. The extracted collinear logarithm
$\alpha_{s}\ln(\mu_{f}^{2}/m_{q}^{2})$ has a form, that can be resummed to all
orders in $\alpha_{s}$ by means of the standard DGLAP evolution equation,
leading to zero-mass (ZM) scheme structure function $F_{\phi}^{{\rm ZM}}$:

$\displaystyle F_{\phi}^{{\rm ZM}}(x,Q^{2})$ $\displaystyle=$
$\displaystyle\int_{0}^{1}\frac{d\eta}{\eta}\ 2C_{\phi,q}^{{\rm
ZM}}\left(\frac{x}{\eta}\right)\big{[}f_{q/N}(\eta,\mu_{f}^{2})+f_{\bar{q}/N}(\eta,\mu_{f}^{2})\big{]}$
(19) $\displaystyle=$
$\displaystyle\frac{1}{2}\big{[}f_{q/N}(x,\mu_{f}^{2})+f_{\bar{q}/N}(x,\mu_{f}^{2})\big{]}.$

The heavy quark PDFs $f_{q/N}$ and $f_{\bar{q}/N}$ introduced in Eq. (19) are
generated by the gluon splitting into $q\bar{q}$ through the DGLAP evolution
equation, and thus reduces to Eq. (18) at the leading order in $\alpha_{s}$
expansion:

$\displaystyle f_{q/N}(x,\mu_{f}^{2})$ $\displaystyle=$
$\displaystyle\frac{\alpha_{s}}{2\pi}\ln\left(\frac{\mu_{f}^{2}}{m_{q}^{2}}\right)\int_{x}^{1}\frac{d\xi}{\xi}\
{\cal P}_{qg}\left(\frac{x}{\xi}\right)f_{g/N}(\xi,\mu_{f}^{2})+{\cal
O}(\alpha_{s}^{2}).$ (20)

So far we have defined three types of structure function $F_{\phi}^{\rm
M},F_{\phi}^{{\rm M0}}$ and $F_{\phi}^{{\rm ZM}}$. The M-scheme structure
function is reliable in low-$Q^{2}$ region but unreliable in the high-$Q^{2}$
region due to the mass singularity, while the ZM scheme structure function is
reliable in the high-$Q^{2}$ region but unreliable in the low-$Q^{2}$ region.
Therefore these two are complementary to each other. According to these
observations, a scheme for the structure function was constructed, which
includes heavy quark mass effects near $q\bar{q}$ threshold region and also
the large logarithm resummation making the structure function stable even at
high-$Q^{2}$ region. The result is a new structure function which consists of
three terms as

$\displaystyle F_{\phi}^{\rm SACOT}\left(x,Q^{2}\right)$ $\displaystyle=$
$\displaystyle F_{\phi}^{\rm M}\left(x,Q^{2}\right)+F_{\phi}^{{\rm
ZM}}\left(x,Q^{2}\right)-F_{\phi}^{\rm sub}(x,Q^{2}).$ (21)

Here the second and third terms are computed by setting heavy quark masses to
zero, and this is called Simplified ACOT (SACOT) scheme Krämer et al. (2000);
Aivazis et al. (1994b). The first term $F_{\phi}^{\rm M}$ is the contribution
of a heavy quark pair to the structure function where the heavy quarks
$q,\bar{q}$ are massive and produced in the scalar-gluon fusion. The second
term $F_{\phi}^{{\rm ZM}}$ is the contribution of heavy quark excitations, and
plays a role to resum the large-collinear logarithm $\ln(Q^{2}/m_{q}^{2})$ to
all orders in $\alpha_{s}$ by use of the heavy quark and anti-quark PDFs
$\\{f_{q/N},f_{\bar{q}/N}\\}$. By the construction of $F_{\phi}^{{\rm ZM}}$,
there is a double counting of large-logarithm between $F_{\phi}^{\rm M}$ and
$F_{\phi}^{{\rm ZM}}$, and this double counting effect should be subtracted by
the last term:

$\displaystyle F_{\phi}^{\rm sub}(x,Q^{2})$ $\displaystyle=$ $\displaystyle
F_{\phi}^{{\rm M0}}(x,Q^{2}).$ (22)

The physical picture of the SACOT scheme is as follows: the first term
$F_{\phi}^{\rm M}$ contains all the mass effects order by order in the
expansion of powers of $\alpha_{s}$, and the second term $F_{\phi}^{\rm ZM}$
adds the large logarithmic corrections of
$(\alpha_{s}\ln(Q^{2}/m_{q}^{2}))^{n}$ for $n=1,\cdots,\infty$ to improve the
high-$Q^{2}$ behavior avoiding the double counting by the last term
$F_{\phi}^{\rm sub}$.

###  II.5 Improvements for threshold behavior

Although we are working on the leading order formulation for the heavy quark
structure function, there are a number of important effects, which can be
incorporated in the present computation, on the threshold behavior of the
structure functions. We take into account such improvements here.

The first such effect is one by so-called $\chi$-rescaling prescriptionTung et
al. (2002); Kretzer et al. (2004). The $\chi$-rescaling prescription aims to
incorporate threshold kinematics of the heavy quark production into the
massless structure function (ZM) and the massless limit (M0) of the massive
structure function using $\chi(x,Q^{2})$ introduced in Eq.(15) instead of $x$
variable. This defines structure functions in ZM-$\chi$ and M0-$\chi$ schemes:

$\displaystyle F_{\phi}^{{\rm ZM\mathchar 45\relax\chi}}(x,Q^{2})$
$\displaystyle=$
$\displaystyle\frac{1}{2}\big{[}f_{q/N}(\chi(x,Q^{2}),\mu_{f}^{2})+f_{\bar{q}/N}(\chi(x,Q^{2}),\mu_{f}^{2})\big{]},$
(23) $\displaystyle F_{\rm\phi}^{{\rm M0\mathchar 45\relax\chi}}(x,Q^{2})$
$\displaystyle=$
$\displaystyle\frac{\alpha_{s}}{2\pi}\ln\left(\frac{\mu_{f}^{2}}{m_{q}^{2}}\right)\int_{\chi(x,Q^{2})}^{1}\frac{d\xi}{\xi}{\cal
P}_{qg}\left(\frac{\chi(x,Q^{2})}{\xi}\right)f_{g/N}(\xi,\mu_{f}^{2}).$ (24)

The subtraction term with $\chi$-rescaling is similarly defined by
$F_{\phi}^{{\rm sub\mathchar 45\relax\chi}}(x,Q^{2})=F_{\phi}^{{\rm
M0\mathchar 45\relax\chi}}(x,Q^{2})$. With these structure functions
SACOT-$\chi$ scheme is also defined as

$\displaystyle F_{\phi}^{{\rm SACOT\mathchar 45\relax\chi}}(x,Q^{2})$
$\displaystyle\equiv$ $\displaystyle F_{\phi}^{\rm
M}\left(x,Q^{2}\right)+\big{[}F_{\phi}^{{\rm ZM\mathchar
45\relax\chi}}\left(x,Q^{2}\right)-F_{\phi}^{{\rm sub\mathchar
45\relax\chi}}(x,Q^{2})\big{]}.$ (25)

The second improvement is a choice of the factorization scale $\mu_{f}^{2}$.
In the traditional DIS analysis the scale choice $\mu_{f}^{2}=Q^{2}$ is
commonly used assuming $Q^{2}\gg m_{q}^{2}$. However we are interested in not
only high-$Q^{2}$ but also the threshold region of the heavy quark production,
especially the contribution from the gluon fusion of Fig. 3, for which $Q^{2}$
can be the same order with $m_{q}^{2}$ or even smaller than $m_{q}^{2}$. For
such a case, the scale choice $\mu_{f}^{2}=Q^{2}$ is not suitable, and one
needs to take a proper physical scale of the process. To ensure that the
factorization scale does not become too low, we take the scale as
$\mu_{f}^{2}=\mu_{Q}^{2}$ with

$\displaystyle\mu_{Q}^{2}$ $\displaystyle=$ $\displaystyle
Q^{2}\,\big{[}c\,(1-z_{m})^{n}\,\Theta(1-z_{m})+z_{m}\big{]},$ (26)

where $z_{m}=m_{q}^{2}/Q^{2}$, $n=2$ and $c=0.5$ are chosen following
Ref.Aivazis et al. (1994b).

The SACOT-$\chi$ structure function interpolates between massive
$F_{\phi}^{{\rm M}}$ and massless $F_{\phi}^{{\rm ZM\mathchar 45\relax\chi}}$
structure functions. Ideally $F_{\phi}^{{\rm SACOT\mathchar 45\relax\chi}}$ is
supposed to reduce to the massive one in the low-$Q^{2}$ region, while in the
high-$Q^{2}$ region to the massless one. This expectation holds parametrically
at each order in expansion in powers of $\alpha_{s}$, but numerically it can
happen that the $F_{\phi}^{{\rm SACOT\mathchar 45\relax\chi}}$ does not
converge well to $F_{\phi}^{{\rm M}}$ near the heavy quark threshold
$Q^{2}\sim m_{q}^{2}$. If the cancellation between $F_{\phi}^{{\rm ZM\mathchar
45\relax\chi}}$ and $F_{\phi}^{{\rm sub\mathchar 45\relax\chi}}$ in
low-$Q^{2}$ region is not effective, it must be due to unsuppressed higher-
order terms in powers of $\alpha_{s}$ resummed into $F_{\phi}^{{\rm
ZM\mathchar 45\relax\chi}}$. Namely the terms $F_{\phi}^{{\rm ZM\mathchar
45\relax\chi}}-F_{\phi}^{{\rm sub\mathchar 45\relax\chi}}={\cal
O}(\alpha_{s}^{k}L^{k})~{}(k\geq 2)$ are too large in the region where the
massless approximation is not trustable. Easy solution to avoid this trouble
is to suppress the $F_{\phi}^{{\rm ZM\mathchar 45\relax\chi}}-F_{\phi}^{{\rm
sub\mathchar 45\relax\chi}}$ in low-$Q^{2}$ region by hand. Thus we define
improved structure functions for ZM-$\chi$ and M0-$\chi$ schemes:

$\displaystyle F_{\phi,\rm thr}^{{\rm ZM\mathchar
45\relax\chi}}\left(x,Q^{2}\right)$ $\displaystyle=$ $\displaystyle
F_{\phi}^{{\rm ZM\mathchar 45\relax\chi}}\left(x,Q^{2}\right)S_{\rm
thr}(m_{q}^{2}/Q^{2}),$ (27) $\displaystyle F_{\phi,\rm thr}^{{\rm M0\mathchar
45\relax\chi}}\left(x,Q^{2}\right)$ $\displaystyle=$ $\displaystyle
F_{\phi,\rm thr}^{\rm sub\chi}\left(x,Q^{2}\right)=F_{\phi}^{{\rm M0\mathchar
45\relax\chi}}(x,Q^{2})S_{\rm thr}(m_{q}^{2}/Q^{2}),$ (28)

where $S_{\rm thr}(m_{q}^{2}/Q^{2})$ is a function which suppress the
structure functions in ZM-$\chi$ and M0-$\chi$ schemes forcing them to
smoothly match with correct threshold behavior. The functional form of $S_{\rm
thr}(z)$ is somewhat arbitrary but the only requirement is $S_{\rm
thr}(z)\stackrel{{\scriptstyle z\to 0}}{{\to}}1$ for the large-logarithm
resummation for high $Q^{2}$. For simplicity we choose

$\displaystyle S_{\rm thr}(z)=\left(1-z\right)^{2}~{}\Theta\left(1-z\right),$
(29)

in the same form introduced in Ref. Forte et al. (2010). Taking all the
improvements we define the SACOT-$\chi$(thr) structure function by

$\displaystyle F_{\phi,{\rm thr}}^{{\rm SACOT\mathchar
45\relax\chi}}(x,Q^{2})$ $\displaystyle\equiv$ $\displaystyle F_{\phi}^{{\rm
M}}\left(x,Q^{2}\right)+\big{[}F_{\phi,\rm thr}^{{\rm ZM\mathchar
45\relax\chi}}\left(x,Q^{2}\right)-F_{\phi,\rm thr}^{{\rm sub\mathchar
45\relax\chi}}(x,Q^{2})\big{]}.$ (30)

The combination $\big{[}F_{\phi,\rm thr}^{{\rm ZM\mathchar
45\relax\chi}}-F_{\phi,\rm thr}^{{\rm sub\mathchar 45\relax\chi}}\big{]}$
ensures that the SACOT-$\chi$(thr) structure function reduces to massive one
near heavy quark threshold. For the SACOT-$\chi$(thr) scheme, we always adopt
the scale setting $\mu_{f}^{2}=\mu_{Q}^{2}$ and the threshold factor $S_{\rm
thr}(m_{q}^{2}/Q^{2})$.

##  III Numerical analysis: Structure functions

We analyze the structure functions for the scalar interaction, i.e. $\phi=S$.
The pseudoscalar case ($\phi=P$) is much the same as the scalar case, and we
refrain from showing the numerical results for the pseudoscalar case. In the
present article we use CT14 LO PDFs Dulat et al. (2016), and present the
numerics for the proton with $m_{N}=938\,{\rm MeV}$ as a target nucleon. The
scale choice $\mu_{f}^{2}=\mu_{Q}^{2}$ is adopted for all the structure
functions.

###  III.1 Effect of $\chi$-rescaling on the structure functions

Figure 5:  The structure functions $F_{\phi}$ computed in the ZM-$\chi$,
ZM-$x$, M0-$\chi$, and M0-$x$ schemes are plotted as functions of $Q$ for
$x=0.3$ and $x=0.03$ in the left and right columns, respectively. The upper
and lower rows show the contributions of the bottom ($q=b$) and charm ($q=c)$
quarks, respectively.

Let us first discuss the importance of $\chi$-rescaling for the structure
functions in the ZM and M0 schemes in low-$Q^{2}$ region. For the QCD
structure functions effects of $\chi$-rescaling are discussed in Refs. Tung et
al. (2002); Kretzer et al. (2004); Stavreva et al. (2012). To make the
difference explicit between the structure functions using $x$ and $\chi$
variables, we introduce the following notations: $F_{\phi}^{{\rm ZM\mathchar
45\relax}x}(=F_{\phi}^{{\rm ZM}}),F_{\phi}^{{\rm M0\mathchar
45\relax}x}(=F_{\phi}^{{\rm M0}})$. Hereafter we call them $x$-scheme
structure functions when the distinction is necessary. The ZM and M0 structure
functions ($x$-scheme) in Eqs.(18) and (19) do not contain any information on
the heavy quark threshold. Thus there is no reason to trust $F_{\phi}^{{\rm
ZM\mathchar 45\relax}x},F_{\phi}^{{\rm M0\mathchar 45\relax}x}$ near the heavy
quark threshold, but it is still expected to have an improvement on the
threshold behavior near $Q^{2}=m_{q}^{2}$ by the $\chi$-rescaling. In Fig. 5
the ZM and M0 structure functions are plotted as functions of $Q$. The
structure functions for $x=0.3$ and $0.03$ are shown in the left and right
columns, and those for bottom and charm quarks are shown in the upper and
lower rows. The solid lines represent the ZM-$\chi$ and M0-$\chi$ structure
functions, and the dot-dashed lines represent the corresponding $x$-scheme
structure functions. One can see that the effect of $\chi$-rescaling is huge
and plays an essential role for the threshold suppression near $Q\sim m_{q}$
($m_{b}=4.75\,{\rm GeV}$ for bottom quark and $m_{c}=1.3\,{\rm GeV}$ for charm
quark). The $x$-scheme structure functions are unrealistically large for
$Q\sim m_{q}$, and it is remarkable that the $\chi$-rescaling improves the
unphysical behavior of the massless structure functions nicely. The effect of
$\chi$-rescaling is decreasing in high-$Q^{2}$ region, and the difference
between the use of $x$ and $\chi$-rescaling is negligible at $Q=100\,{\rm
GeV}$ ($Q=50\,{\rm GeV}$) for bottom (charm) quark. We conclude that the
$\chi$-rescaling is effective only in the low-$Q^{2}$ region of $Q\leq
100\,{\rm GeV}~{}(Q\leq 50\,{\rm GeV})$ for bottom (charm) quark.

###  III.2 Massive vs. zero-mass schemes, and SACOT scheme

Figure 6:  The structure functions $F_{\phi}$ in the ZM-$\chi$, M0-$\chi$, M,
and SACOT-$\chi$(thr) schemes are plotted as functions of $Q$ for $x=0.3$ and
$x=0.03$ in the left and right columns, respectively. The upper and lower rows
show the contributions of bottom ($q=b$) and charm ($q=c$) quarks,
respectively. Figure 7:  Same as Fig. 6 but for low-$Q^{2}$ region: $x=0.3$
(left column) and $x=0.03$ (right column) for bottom ($q=b)$ and charm ($q=c$)
quarks in the two rows.

Here we will compare the structure functions in M, ZM-$\chi$(thr), and
SACOT-$\chi$(thr) schemes (for the last two schemes $\chi$-rescaling and the
threshold factor are adopted). In Fig. 6 the heavy quark contribution to the
structure functions are plotted as functions of $Q$ (up to $Q=300\,{\rm GeV}$)
for $x=0.3$ and $x=0.03$. For $x=0.3$ it is seen that the curves for
SACOT-$\chi$(thr) and ZM-$\chi$(thr) are broadly similar to each other for
bottom and charm quarks, and their values are much larger than the massive
scheme result. The difference between the curves of the ZM-$\chi$(thr) and
SACOT-$\chi$(thr) schemes is explained by the difference between the M and
M0-$\chi$ schemes because $(F_{\phi,\rm thr}^{{\rm SACOT\mathchar
45\relax\chi}}-F_{\phi,\rm thr}^{{\rm ZM\mathchar
45\relax\chi}})=(F_{\phi}^{\rm M}-F_{\phi,\rm thr}^{{\rm M0\mathchar
45\relax\chi}})$. Remember that $F_{\phi,\rm thr}^{{\rm M0\mathchar
45\relax\chi}}$ is the subtraction term of the SACOT-$\chi$(thr) scheme. For
$x=0.03$ this agreement between the SACOT-$\chi$(thr) and ZM-$\chi$(thr)
schemes becomes tighter, because $(F_{\phi}^{{\rm M}}-F_{\phi,\rm thr}^{{\rm
sub\mathchar 45\relax\chi}})_{x=0.03}\simeq 0$. It should be noted that the
structure functions for $x=0.03$ are more than two orders of magnitude larger
than those of $x=0.3$.

The same plots as Fig. 6, focusing on the range below $Q=30\,{\rm GeV}$, are
shown in Fig. 7. It will be seen later that the contributions of the structure
functions to the CLFV DIS cross section for $E_{e}\leq 1\,{\rm TeV}$ are
dominated by the $Q$ values of this range. Therefore Fig. 7 is more relevant
than Fig. 6 for the fixed target experiments at present and in near future.
For bottom quark, the M scheme structure function is closer to the
SACOT-$\chi$(thr) scheme than ZM-$\chi$(thr) for $x=0.03$. For $x=0.3$ both
the curves of M and ZM-$\chi$(thr) are away from the SACOT-$\chi$(thr) but
their magnitudes are less than percent level compared to those at $x=0.03$.
For charm quark at $x=0.03$, the M scheme structure function is closer to the
SACOT-$\chi$(thr) for very low $Q$. For $Q\geq 8.6\,{\rm GeV}$ the
ZM-$\chi$(thr) scheme becomes closer to SACOT-$\chi$(thr) especially for the
large-$Q^{2}$ region. For charm quark at $x=0.3$, the magnitudes of the
structure functions are less than a percent level compared to those at
$x=0.03$.

###  III.3 Threshold factor

In Figs. 6 and 7 the threshold factor $S_{\rm thr}$ had been taken into
account for the structure functions $F_{\phi}^{{\rm SACOT\mathchar
45\relax\chi}}$, $F_{\phi}^{{\rm ZM\mathchar 45\relax\chi}}$, and
$F_{\phi}^{{\rm M0\mathchar 45\relax\chi}}$. The effect of the threshold
factor is limited in low-$Q^{2}$ region: $S_{\rm
thr}(m_{b}^{2}/Q^{2})=0.60,0.89,0.95$ for $Q=10,20,30\,{\rm GeV}$ for bottom
quark, and $S_{\rm thr}(m_{c}^{2}/Q^{2})=0.87,0.97,0.99$ for $Q=5,10,15\,{\rm
GeV}$ for charm quark. These values give about $40\%,11\%,5\%$ suppressions
for the structure functions $F_{\phi,\rm thr}^{{\rm ZM\mathchar
45\relax\chi}},F_{\phi,\rm thr}^{{\rm M0\mathchar 45\relax\chi}}$ of bottom
quark at $Q=10,20,30\,{\rm GeV}$ compared to those without the threshold
factor, and about $13\%,3\%,1\%$ suppressions for the charm quark case at
$Q=5,10,15\,{\rm GeV}$. For the SACOT-$\chi$(thr) scheme the relative size of
$(F_{\phi,\rm thr}^{{\rm M0\mathchar 45\relax\chi}}-F_{\phi,\rm thr}^{{\rm
ZM\mathchar 45\relax\chi}})=S_{\rm thr}(m_{q}^{2}/Q^{2})(F_{\phi}^{{\rm
M0\mathchar 45\relax\chi}}-F_{\phi}^{{\rm ZM\mathchar 45\relax\chi}})$ to
$F_{\phi}^{{\rm M}}$ determines the effect of $S_{\rm thr}$:

$\displaystyle\frac{F_{\phi,{\rm thr},b}^{{\rm SACOT\mathchar
45\relax\chi}}}{F_{\phi,b}^{{\rm SACOT\mathchar
45\relax\chi}}}=\bigg{\\{}\begin{array}[]{ll}0.94,~{}0.97,~{}0.98&(x=0.3)\\\
0.99,~{}0.99,~{}1.0&(x=0.03)\end{array}~{}~{}~{}\mbox{for}~{}Q=10,20,30\,{\rm
GeV},$ (33)

for bottom quark, and

$\displaystyle\frac{F_{\phi,{\rm thr},c}^{{\rm SACOT\mathchar
45\relax\chi}}}{F_{\phi,c}^{{\rm SACOT\mathchar
45\relax\chi}}}=\bigg{\\{}\begin{array}[]{ll}0.95,~{}0.98,~{}0.99&(x=0.3)\\\
0.99,~{}1.0,~{}1.0&(x=0.03)\end{array}~{}~{}~{}\mbox{for}~{}Q=5,10,15\,{\rm
GeV},$ (36)

for charm quark. It is understood that a major part of the threshold
suppression is already taken care by the $\chi$-rescaling, and the effect of
$S_{\rm thr}$ became minor for the structure function in the SACOT-$\chi$(thr)
scheme.

##  IV Numerical Analysis: Cross sections

In this section, we investigate how effective the SACOT-$\chi$ scheme and the
others are for the description of CLFV process associated with the heavy quark
pair productions. As a continuation of the previous section only the scalar
case ($\phi=S$) will be studied. As is explained in the previous section, the
structure functions are functions of $x$ and $Q^{2}$, and each scheme of the
structure functions has validity regions for a specific $Q^{2}$ range. However
our concerns are the total cross sections $\sigma$ and differential
distributions $d\sigma/d\bm{p}_{\tau}$ of the final $\tau$-momentum. There
arises a question of which scheme is the most relevant, and which scheme is
the most effective for the description of the CLFV DIS in the full kinematical
range of the cross section, which will be answered in this section.

For definiteness we take electron and tau lepton as initial and final leptons,
respectively, namely $i=e$ and $j=\tau$. In the numerical analysis we ignore
the electron mass, and take the following mass values:

$\displaystyle m_{b}=4.75\,{\rm GeV},~{}m_{c}=1.3\,{\rm
GeV},~{}m_{\tau}=1.78\,{\rm GeV}.$ (37)

The CLFV couplings $\rho_{ij}^{\phi}$ and quark-mediator coupling
$\rho_{qq}^{\phi}$ are a priori not known and we set their values as

$\displaystyle|\rho^{\phi}_{qq}|^{2}=|\rho_{ij}^{\phi}|^{2}+|\rho_{ji}^{\phi}|^{2}=1.$
(38)

These coupling constants determine the normalization of the CLFV cross
section, and therefore our numerical results need to be multiplied by mode-
dependent prefactors to match them with experimental values to be measured.
For the choice of the factorization scale $\mu_{f}^{2}$ we adopt
$\mu_{f}^{2}=\mu_{Q}^{2}$ defined in Eq. (26). In our numerical analysis, we
have applied a kinematical cut of $Q\geq 1.3\,{\rm GeV}$ and
$W\equiv\sqrt{(P+q)^{2}}\geq 1.4\,{\rm GeV}$ to ensure that the processes we
are considering are in perturbative and deep-inelastic régime, though it
turned out that the effect of the cut is tiny and numerically negligible for
the CLFV DIS associated with the heavy quark pair productions.

###  IV.1 Zero-mass schemes

The ZM schemes are the most frequently used schemes for DIS involving heavy
quarks as well as the light quarks. This is so even for the CLFV DIS
associated with bottom and charm quark productions because of their
computational simplicities. However, the use of massless approximation cannot
be justified for low $Q^{2}$, and a reliable computational scheme should be
the massive scheme there. Nevertheless, it is worthwhile to know limitations
of the ZM schemes for the CLFV cross sections. Here we investigate the ZM-$x$
and ZM-$\chi$ schemes to clarify their applicability for the cross section in
relatively low collision energies. As example cases, we simulate the cross
section for $E_{e}=200\,{\rm GeV}$ and $E_{e}=1\,{\rm TeV}$. Here we do not
include the threshold factor $S_{\rm thr}$ for the ZM schemes because it cuts
away the low-$Q^{2}$ region and the difference between ZM-$x$ and ZM-$\chi$
are naturally suppressed.

Figure 8:  The $\tau$-momentum distributions Eq. (11) for the bottom quark
production in the ZM-$x$(left column) and ZM-$\chi$ (right column) schemes.
The distributions for the electron beam energies of $E_{e}=200\,{\rm GeV}$
(upper row) and $E_{e}=1\,{\rm TeV}$ (lower row) with scalar mass
$m_{S}=10\,{\rm GeV}$ are shown.

In Fig. 8 we show the $\tau$-momentum distributions for the CLFV DIS
associated with bottom quark production for $E_{e}=200\,{\rm GeV}$ and
$E_{e}=1\,{\rm TeV}$ in the two rows. The scalar mass is set to
$m_{S}=10\,{\rm GeV}$. The left and right columns show the distributions in
the ZM-$x$ and ZM-$\chi$ schemes, respectively. The contour lines are labeled
by percentages ($20\%,40\%,60\%,80\%$) of the cross section of the enclosed
region normalized to their total cross section. The colored region contains
$99\%$ of total events of the process. The value of the scalar mass $m_{S}$,
the beam energy $E_{e}$, and the total cross section $\sigma$ are shown inside
each panel. For $E_{e}=200\,{\rm GeV}$ the effect of $\chi$-rescaling is huge
suppression for the overall normalization $\sigma$, and the physical regions
of $\chi$-scheme distributions are shrunk into a smaller region than those of
$x$-schemes. For $E_{e}=1\,{\rm TeV}$ the effect of $\chi$-rescaling is still
large for the overall normalization but weak compared to the case of
$E_{e}=200\,{\rm GeV}$. The ratio of the total cross section in $x$-scheme to
that in $\chi$-scheme is $\sigma^{{\rm ZM\mathchar
45\relax}x}_{b}/\sigma^{{\rm ZM\mathchar 45\relax\chi}}_{b}\sim 70~{}(3.8)$
for $E_{e}=200\,{\rm GeV}~{}(1\,{\rm TeV})$. The large enhancements of the
total cross sections in $x$-schemes hold even in the case of heavy scalar
mass. For instance, taking $m_{S}=10^{5}\,{\rm GeV}$, the cross section ratio
is $\sigma_{b}^{{\rm ZM\mathchar 45\relax}x}/\sigma_{b}^{{\rm ZM\mathchar
45\relax\chi}}\sim 73~{}(3.0)$ for $E_{e}=200\,{\rm GeV}~{}(1\,{\rm TeV})$ for
the bottom quark production.

Figure 9:  Same as Fig. 8, but heavy quark is charm quark ($q=c$): ZM-$x$
(left column) and ZM-$\chi$ (right column) for $E_{e}=200\,{\rm GeV}$ and
$E_{e}=1\,{\rm GeV}$ in the two rows.

In Fig. 9 we show the $\tau$-momentum distributions for $m_{S}=10\,{\rm GeV}$
in the case of charm quark production. Comparing the $x$ and $\chi$ schemes,
the shapes of the contour lines look quite similar to each other, but the
effect of $\chi$-rescaling is still sizable for the normalization of total
cross sections. The ratio is $\sigma_{c}^{{\rm ZM\mathchar
45\relax}x}/\sigma_{c}^{{\rm ZM\mathchar 45\relax\chi}}\sim 1.7~{}(1.4)$ for
$E_{e}=200\,{\rm GeV}~{}(1\,{\rm TeV})$. This value is not changed much even
for the heavy scalar case. For instance, taking $m_{S}=10^{5}\,{\rm GeV}$, the
ratio is $\sigma_{c}^{{\rm ZM\mathchar 45\relax}x}/\sigma_{c}^{{\rm
ZM\mathchar 45\relax\chi}}\sim 1.6~{}(1.2)$ for $E_{e}=200\,{\rm
GeV}~{}(1\,{\rm TeV})$.

Figure 10:  The cross sections for the CLFV DIS in the M, ZM-$x$, and
ZM-$\chi$ schemes normalized to the SACOT-$\chi$(thr). The scale of the upper
horizontal axis in each panel shows the beam energy in the fixed target
experiments. The results for bottom ($q=b$) and charm ($q=c$) quarks are shown
in the upper and lower rows, and the results for the scalar masses
$m_{S}=10\,{\rm GeV}$ and $m_{S}=10^{5}\,{\rm GeV}$ are shown in the left and
right columns.

In Fig. 10 the total cross sections associated with bottom and charm quark
productions are plotted respectively in the upper and lower rows, where each
cross section is normalized to the value of SACOT-$\chi$(thr). The left and
right columns are two cases of scalar mass, $m_{S}=10\,{\rm GeV}$ and
$10^{5}\,{\rm GeV}$, respectively. Numerical values of the cross sections are
listed in Table 1. In the plots we observe the following:

  * •

For the bottom quark production the curves of M scheme are close to the one of
SACOT-$\chi$(thr), i.e. $\sigma_{b}^{\rm M}/\sigma_{{\rm thr},b}^{{\rm
SACOT\mathchar 45\relax\chi}}\sim 1$. The cross section in the ZM-$\chi$
scheme for the small scalar mass $m_{S}=10\,{\rm GeV}$ is quite off from that
of SACOT-$\chi$(thr) irrespective of the collision energy: $\sigma_{b}^{{\rm
ZM\mathchar 45\relax\chi}}/\sigma_{{\rm thr},b}^{{\rm SACOT\mathchar
45\relax\chi}}\lesssim 0.5.$ For the case of the large scalar mass
$m_{S}=10^{5}\,{\rm GeV}$, the ZM-$\chi$ and ZM-$x$ curves are gradually
approaching to the SACOT-$\chi$ , and around $\sqrt{s}\sim 10^{3}\,{\rm GeV}$
they meet at one point. Notably, the curve of the ZM-$x$ scheme for the bottom
production is far off from that of SACOT-$\chi$(thr) in low collision energy.

  * •

Even for the charm quark production, inadequacy of the ZM-$x$ scheme in low
collision energy is the same as the case of bottom quark, but the behaviors of
the cross sections in the M and ZM-$\chi$ schemes are quite different.
Specifics of the charm cross sections are as follows. For the small scalar
mass $m_{S}=10\,{\rm GeV}$ the M scheme curve grows with $\sqrt{s}$ and
overshoots the SACOT-$\chi$(thr) at $\sqrt{s}\sim 10^{2}\,{\rm GeV}$, while
for the large scalar mass $m_{S}=10^{5}\,{\rm GeV}$ it is almost constant and
small by a sizable amount compared to the SACOT-$\chi$(thr). For the large
scalar mass, the value of the ZM-$\chi$ is close to the one of
SACOT-$\chi$(thr) for arbitrary collision energy, and it is expected that
charm quark can be treated as massless provided that the $\chi$-rescaling is
adopted and the scalar mass is large enough. It will be shown in Fig. 17 that
the scalar mass of $m_{S}\sim 50\,{\rm GeV}$ is large enough to validate the
treatment of massless charm with $\chi$-rescaling adopted for it.

Table 1: The CLFV total cross sections associated with bottom and charm quark productions in ZM-$x$, ZM-$\chi$, and M schemes. The coupling constants are set to one by Eq. (38). $m_{S}~{}[\rm{GeV}]$ | $E_{e}~{}{\rm[GeV]}$ | $\sigma_{b}^{{\rm ZM\mathchar 45\relax}x}~{}{\rm[fb]}$ | $\sigma_{b}^{{\rm ZM\mathchar 45\relax\chi}}~{}{\rm[fb]}$ | $\sigma_{b}^{\rm M}~{}{\rm[fb]}$
---|---|---|---|---
| $10^{2}$ | $6.49\times 10^{2}$ | $1.02\times 10^{-2}$ | $2.08\times 10^{-1}$
| $10^{3}$ | $8.67\times 10^{4}$ | $2.28\times 10^{4}$ | $4.93\times 10^{4}$
$10$ | $10^{4}$ | $7.84\times 10^{5}$ | $3.72\times 10^{5}$ | $7.32\times 10^{5}$
| $10^{5}$ | $3.24\times 10^{6}$ | $1.90\times 10^{6}$ | $3.72\times 10^{6}$
| $10^{6}$ | $9.12\times 10^{6}$ | $5.87\times 10^{6}$ | $1.18\times 10^{7}$
| $10^{7}$ | $2.11\times 10^{7}$ | $1.43\times 10^{7}$ | $2.99\times 10^{7}$
| $10^{2}$ | $1.51\times 10^{-13}$ | $2.02\times 10^{-18}$ | $3.17\times 10^{-17}$
| $10^{3}$ | $5.64\times 10^{-11}$ | $1.89\times 10^{-11}$ | $2.23\times 10^{-11}$
$10^{5}$ | $10^{4}$ | $1.88\times 10^{-9}$ | $1.41\times 10^{-9}$ | $1.27\times 10^{-9}$
| $10^{5}$ | $3.35\times 10^{-9}$ | $3.10\times 10^{-8}$ | $2.57\times 10^{-8}$
| $10^{6}$ | $4.66\times 10^{-7}$ | $4.57\times 10^{-7}$ | $3.66\times 10^{-7}$
| $10^{6}$ | $5.76\times 10^{-6}$ | $5.73\times 10^{-6}$ | $4.50\times 10^{-6}$
$m_{S}~{}[\rm{GeV}]$ | $E_{e}~{}{\rm[GeV]}$ | $\sigma_{c}^{{\rm ZM\mathchar 45\relax}x}~{}{\rm[fb]}$ | $\sigma_{c}^{{\rm ZM\mathchar 45\relax\chi}}~{}{\rm[fb]}$ | $\sigma_{c}^{\rm M}~{}{\rm[fb]}$
| $10^{2}$ | $6.57\times 10^{4}$ | $2.99\times 10^{4}$ | $2.74\times 10^{4}$
| $10^{3}$ | $8.67\times 10^{5}$ | $6.41\times 10^{5}$ | $6.59\times 10^{5}$
$10$ | $10^{4}$ | $4.10\times 10^{6}$ | $3.46\times 10^{6}$ | $4.15\times 10^{6}$
| $10^{5}$ | $1.15\times 10^{7}$ | $1.02\times 10^{7}$ | $1.46\times 10^{7}$
| $10^{6}$ | $2.45\times 10^{7}$ | $2.24\times 10^{7}$ | $3.83\times 10^{7}$
| $10^{7}$ | $4.68\times 10^{7}$ | $4.33\times 10^{7}$ | $8.68\times 10^{7}$
| $10^{2}$ | $1.00\times 10^{-11}$ | $4.75\times 10^{-12}$ | $3.86\times 10^{-12}$
| $10^{3}$ | $3.03\times 10^{-10}$ | $2.55\times 10^{-10}$ | $1.93\times 10^{-10}$
$10^{5}$ | $10^{4}$ | $4.92\times 10^{-9}$ | $4.70\times 10^{-9}$ | $3.51\times 10^{-9}$
| $10^{5}$ | $6.37\times 10^{-8}$ | $6.31\times 10^{-8}$ | $4.66\times 10^{-8}$
| $10^{6}$ | $7.52\times 10^{-7}$ | $7.50\times 10^{-7}$ | $5.48\times 10^{-7}$
| $10^{6}$ | $8.45\times 10^{-6}$ | $8.45\times 10^{-6}$ | $6.09\times 10^{-6}$

###  IV.2 SACOT-$\chi$ and its components

In this subsection we study the cross section and the $\tau$-momentum
distribution in the SACOT-$\chi$(thr) scheme and its components, for which the
$\chi$-rescaling and the threshold factor $S_{\rm thr}$ are adopted. Here and
hereafter the word “components” denotes the three contributions, M,
ZM-$\chi$(thr), and M0-$\chi$(thr), which constitute the SACOT-$\chi$(thr)
scheme.

Figure 11:  The $\tau$-momentum distribution in the CLFV DIS associated with
bottom quark production in the SACOT-$\chi$ (thr) scheme. The threshold factor
$S_{\rm thr}$ is included. The results for the scalar masses $m_{S}=10\,{\rm
GeV}$ and $m_{S}=10^{5}\,{\rm GeV}$ are shown in the left and right columns,
and the results for the initial electron beam energies $E_{e}=200\,{\rm GeV}$
and $E_{e}=1\,{\rm TeV}$ are shown in the two rows.

Figure 12:  The $\tau$-momentum distribution for the components of
SACOT-$\chi$ scheme for the bottom quark production at $E_{e}=1\,{\rm TeV}$.
The results for the M, ZM-$\chi$, and M0-$\chi$ without the threshold factor
are shown respectively in the first, second, and third columns. The scalar
masses are $m_{S}=10\,{\rm GeV}$ and $m_{S}=10^{5}\,{\rm GeV}$ in the upper
and lower rows.

In Fig. 11 we show the $\tau$-momentum distribution in the CLFV DIS associated
with the bottom quark production in the SACOT-$\chi$(thr) scheme, where a
combination of energies $E_{e}=200\,{\rm GeV},~{}1\,{\rm TeV}$ and the scalar
masses $m_{S}=10\,{\rm GeV},~{}10^{5}\,{\rm GeV}$ is taken for each plot. One
can see that the scalar mass does not affect much the shape of distribution
for $E_{e}=200\,{\rm GeV}$, but the overall normalization. For $E_{e}=1\,{\rm
TeV}$ there are sizable differences in the shape of the distribution between
$m_{S}=10\,{\rm GeV}$ and $10^{5}\,{\rm GeV}$. Decomposing the cross section
of SACOT-$\chi$(thr) scheme into the components one obtains the contributions:

$\small\begin{split}\mbox{(i)}~{}~{}~{}~{}&\sigma_{b}^{{\rm SACOT\mathchar
45\relax\chi}}=\big{[}3.37+0.32(0.77)-0.30(0.75)\big{]}\times 10^{\,2}\,{\rm
fb}~{}~{}~{}~{}=3.39(3.38)\times 10^{\,2}\,{\rm fb},\\\
\mbox{(ii)}~{}~{}~{}&\sigma_{b}^{{\rm SACOT\mathchar
45\relax\chi}}=\big{[}4.93+1.53(2.28)-1.35(2.05)\big{]}\times 10^{\,4}\,{\rm
fb}~{}~{}~{}~{}=5.11(5.15)\times 10^{\,4}\,{\rm fb},\\\
\mbox{(iii)}~{}~{}&\sigma_{b}^{{\rm SACOT\mathchar
45\relax\chi}}=\big{[}6.61+1.02(2.24)-0.93(2.12)\big{]}\times 10^{-14}\,{\rm
fb}~{}=6.70(6.73)\times 10^{-14}\,{\rm fb},\\\
\mbox{(iv)}~{}~{}&\sigma_{b}^{{\rm SACOT\mathchar
45\relax\chi}}=\big{[}2.23+1.47(1.89)-1.22(1.60)\big{]}\times 10^{-11}\,{\rm
fb}~{}=2.47(2.52)\times 10^{-11}\,{\rm fb},\end{split}$ (39)

where the first/second/third number in the square parenthesis represents the
M/ZM-$\chi$/M0-$\chi$ component with (without) the threshold factor $S_{\rm
thr}$ for cases (i) $E_{e}=200\,{\rm GeV},~{}m_{S}=10\,{\rm GeV}$, (ii)
$E_{e}=1\,{\rm TeV},~{}m_{S}=10\,{\rm GeV}$, (iii) $E_{e}=200\,{\rm
GeV},~{}m_{S}=10^{5}\,{\rm GeV}$, and (iv) $E_{e}=1\,{\rm
TeV},~{}m_{S}=10^{5}\,{\rm GeV}$. It is observed that the values of the
ZM-$\chi$ and M0-$\chi$ components receive sizable threshold suppressions by
the $S_{\rm thr}$, but the SACOT-$\chi$ cross section is approximately the
same irrespective of inclusion of the threshold factor. It is because the
contributions of the ZM-$\chi$(thr) and M0-$\chi$(thr) are the same size and
cancel each other in the combination of $\big{[}F_{\phi,\rm thr}^{{\rm
ZM\mathchar 45\relax\chi}}-F_{\phi,\rm thr}^{{\rm sub\mathchar
45\relax\chi}}\big{]}$ in the SACOT-$\chi$(thr) scheme.

In Fig. 12 we show the $\tau$-momentum distribution of the components for the
bottom quark production at $E_{e}=1\,{\rm TeV}$, but without the threshold
factor $S_{\rm thr}$. The upper and lower rows are the results for
$m_{S}=10\,{\rm GeV}$ and $m_{S}=10^{5}\,{\rm GeV}$ respectively, and the
first column is the result of M scheme, and the second and third columns are
the results of the ZM-$\chi$ and M0-$\chi$ schemes respectively. It turns out
that the largest contribution is coming from the massive scheme cross section.
These observations lead that the massive scheme cross section is effective and
nearly equal to the SACOT-$\chi$(thr) in the range of collision energy up to
$E_{e}=1{\rm TeV}$ ($\sqrt{s}\simeq 45\,{\rm GeV}$). Effectiveness of massive
scheme cross section in a wider range of collision energy will be discussed
later (see Fig.18).

Figure 13:  Same as Fig. 11, but for the charm quark productions:
$m_{S}=10\,{\rm GeV}$ (left column) and $m_{S}=10^{5}\,{\rm GeV}$ (right
column) for $E_{e}=200\,{\rm GeV}$ and $E_{e}=1\,{\rm TeV}$ in the two rows.

Figure 14:  Same as Fig. 12, but for the charm quark productions. The results
for the M, ZM-$\chi$, and M0-$\chi$ are shown in the three columns, and the
results for $m_{S}=10\,{\rm GeV}$ and $m_{S}=10^{5}\,{\rm GeV}$ are shown in
the two rows.

In Fig. 13 the $\tau$-momentum distribution in CLFV DIS associated with charm
quark production is shown for $E_{e}=200\,{\rm GeV}$ and $E_{e}=1\,{\rm TeV}$.
The distribution is more concentrated in the high-$p_{Z}$ region compared to
one of the bottom quark production because the charm quark ($m_{c}=1.3\,{\rm
GeV}$) is more relativistic than the bottom quark ($m_{b}=4.75\,{\rm GeV}$).
The size of scalar mass affects the normalization and also the shape of the
distribution. The cross sections in terms of components are given by:

$\small\begin{split}\mbox{(i)}~{}~{}\sigma_{{\rm thr},c}^{{\rm SACOT\mathchar
45\relax\chi}}&=\big{[}9.30+8.64(9.98)-6.49(7.60)\big{]}\times 10^{\,4}\,{\rm
fb}~{}~{}~{}~{}~{}=1.15(1.17)\times 10^{\,5}\,{\rm fb},\\\
\mbox{(ii)}~{}~{}\sigma_{{\rm thr},c}^{{\rm SACOT\mathchar
45\relax\chi}}&=\big{[}6.60+5.85(6.41)-4.86(5.38)\big{]}\times 10^{\,5}\,{\rm
fb}~{}~{}~{}~{}~{}=7.59(7.63)\times 10^{\,5}\,{\rm fb},\\\
\mbox{(iii)}~{}~{}\sigma_{{\rm thr},c}^{{\rm SACOT\mathchar
45\relax\chi}}&=\big{[}1.56+1.78(1.99)-1.26(1.43)\big{]}\times 10^{-11}\,{\rm
fb}~{}~{}=2.08(2.12)\times 10^{-11}\,{\rm fb},\\\
\mbox{(iv)}~{}~{}\sigma_{{\rm thr},c}^{{\rm SACOT\mathchar
45\relax\chi}}&=\big{[}1.93+2.43(2.55)-1.75(1.85)\big{]}\times 10^{-10}\,{\rm
fb}~{}~{}=2.61(2.62)\times 10^{-10}\,{\rm fb},\end{split}$ (40)

where the first/second/third number in the square parenthesis represents the
M/ZM-$\chi$/M0-$\chi$ contribution with (without) the threshold factor for (i)
$E_{e}=200\,{\rm GeV},~{}m_{S}=10\,{\rm GeV}$, (ii) $E_{e}=1\,{\rm
TeV},~{}m_{S}=10\,{\rm GeV}$, (iii) $E_{e}=200\,{\rm
GeV},~{}m_{S}=10^{5}\,{\rm GeV}$, and (iv) $E_{e}=1\,{\rm
TeV},~{}m_{S}=10^{5}\,{\rm GeV}$. In the components of SACOT-$\chi$(thr) cross
section, the rates of M and ZM-$\chi$(thr) cross sections are the same size.
For the charm quark production, dominance of only one component does not hold
for $E_{e}=200\,{\rm GeV}$ and $E_{e}=1\,{\rm TeV}$.

It should be remembered that we have fixed the CLFV couplings by Eq. (38),
which control the overall normalization of the cross section. Thus it is of
great importance to find a sensitivity to the scalar mass in the shape of the
distribution, which can be utilized for a detailed study to discriminate the
structure of the CLFV interactions, namely for the separation of effects of
coupling constants and of scalar mass.

###  IV.3 SACOT-$\chi$: $Q$-distribution

Here we rewrite the cross section as

$\displaystyle\frac{d\sigma}{dQ}$ $\displaystyle=$
$\displaystyle\frac{1}{m_{\phi}^{4}}\,W_{\phi}(m_{\phi}^{2},Q^{2})\widetilde{M}_{\phi}(s,Q^{2}),$
(41) $\displaystyle\widetilde{M}_{\phi}(s,Q^{2})$ $\displaystyle=$
$\displaystyle\frac{(Q^{2})^{\frac{3}{2}}(Q^{2}+m_{\tau}^{2})}{8\pi
s^{2}}M_{\phi}(s,Q^{2}),$ (42) $\displaystyle W_{\phi}(m_{\phi}^{2},Q^{2})$
$\displaystyle=$
$\displaystyle\left(\frac{m_{\phi}^{2}}{Q^{2}+m_{\phi}^{2}}\right)^{2},$ (43)

where we have introduced a modified inverse moment $\widetilde{M}$ and a
weighting factor $W_{\phi}$. The mediator mass can be set to $m_{\phi}=m_{S}$.
Note that the product of $W_{\phi}$ and $\widetilde{M}_{\phi}$ gives
$d\sigma/dQ$ not $d\sigma/dQ^{2}$ (up to $m_{\phi}^{4}$). The inverse moment
$\widetilde{M}_{\phi}(s,Q^{2})$ is more adequate than the structure function
to see which region of $Q$ contributes to the cross section.

Figure 15:  The inverse moment $\widetilde{M}_{\phi}(s,Q^{2})$ for each
component of the SACOT-$\chi$(thr), and the weighting factor
$W_{\phi}(m_{S}^{2},Q^{2})$ for $m_{S}=10\,{\rm GeV}$ and $m_{S}=50\,{\rm
GeV}$. For visualization the weighting factor normalized to appropriate values
are plotted. The results for bottom quark ($q=b$) and charm quark ($q=c$) are
shown respectively in the upper and lower rows, and the beam energies are
$E_{e}={\rm 200\,GeV}$ and $E_{e}={\rm 1\,TeV}$ in the left and right columns.

In Fig. 15, the inverse moments $\widetilde{M}_{\phi}(s,Q^{2})$ are plotted as
functions of $Q$ for the bottom and charm quark productions in upper and lower
rows, respectively. The results for $E_{e}=200\,{\rm GeV}$ and $E_{e}=1\,{\rm
TeV}$ are shown in the left and right columns, respectively. In the plots one
can see the following features of the inverse moment:

  * •

The support of the M scheme inverse moment starts at low $Q$: the curve of the
$\widetilde{M}_{\phi}$ in the M scheme rises from 0 at $Q\sim 2\,{\rm GeV}$
($Q\sim 1\,{\rm GeV}$) for the bottom (charm) case. On the other hand, the
curve of the $\widetilde{M}_{\phi}$ in the ZM-$\chi$(thr) scheme rises from 0
at $Q\sim 6\,{\rm GeV}$ ($Q\sim 2\,{\rm GeV}$) for the bottom (charm) case.
Thus the cross section of the M scheme is superior in magnitude to that of the
ZM-$\chi$(thr) in the very low-$Q^{2}$ region.

  * •

The larger the beam energy $E_{e}$ the higher the maximum $Q$ for the support
of the inverse moment in all the schemes. The relative size of ZM-$\chi$(thr)
to that of the M scheme tends to grow with the beam energy.

These features due to the dynamics of the QCD (properties of the structure
functions) together with the $Q$-dependence of the weighting function
$W_{\phi}(m_{\phi}^{2},Q^{2})$ determine the relative importance of the M
scheme contribution to that of ZM-$\chi$(thr) as a function of $m_{\phi}^{2}$
and $s$.

Figure 16:  The CLFV DIS cross sections $m_{S}^{4}\,(d\sigma/dQ)$ are plotted
as functions of $Q$. The results for bottom quark ($q=b$) and charm quark
($q=c$) are shown respectively in the upper and lower rows, and the beam
energies are $E_{e}=200\,{\rm GeV}$ and $E_{e}=1\,{\rm TeV}$ in the left and
right columns. The scalar masses are $m_{S}=10\,{\rm GeV}$(black lines) and
$m_{S}=50\,{\rm GeV}$(blue lines).

In Fig. 16, the cross section $m_{S}^{4}\,d\sigma/dQ$ is plotted as a function
of $Q$ for CLFV DIS associated with the heavy quark production. A prefactor
$m_{S}^{4}$ is multiplied to scale out the leading $m_{S}$-dependence in the
large-$m_{S}$ limit. The results for the bottom ($q=b$) and charm ($q=c$)
quark productions are presented in the upper and lower rows, and the results
for the beam energies of $E_{e}=200\,{\rm GeV}$ and $E_{e}=1\,{\rm TeV}$ are
presented in the left and right columns. Two representative cases
$m_{S}=10\,{\rm GeV}$ (black curves) and $m_{S}=50\,{\rm GeV}$ (blue curves)
for the scalar mass are simulated. Note that $m_{S}=50\,{\rm GeV}$ is
sufficiently large compared to typical momentum transfer $Q$ for
$E_{e}=200\,{\rm GeV}$ and $E_{e}=1\,{\rm TeV}$ (see Figs. 15, 17), thus it
corresponds approximately to a case of the large-$m_{S}$ limit. In the plots
we observe the following:

  * •

For the bottom quark production, the supports of the cross sections of M and
ZM-$\chi$(thr) schemes start respectively at $Q\sim 2\,{\rm GeV}$ and $Q\sim
6\,{\rm GeV}$. The contribution from the region ${\cal R}_{Q}=\\{Q|~{}2\,{\rm
GeV}\leq Q\leq 6\,{\rm GeV}\\}$ to the M scheme cross section is given by
$\sigma^{{\rm M}}({\cal R}_{R})=\int_{{\cal R}_{Q}}(d\sigma^{{\rm M}}/dQ)dQ$
and it is large, while the contribution from the same region to ZM-$\chi$(thr)
is $\sigma^{{\rm ZM\mathchar 45\relax\chi}}({\cal R}_{Q})=0$. This yields
significantly large contribution exclusively to the M scheme cross section,
leading the dominance of the M component in the SACOT-$\chi$(thr) cross
section for $E_{e}=200\,{\rm GeV}$ and $E_{e}=1\,{\rm TeV}$. The difference
$(\sigma_{\phi,{\rm thr}}^{{\rm ZM\mathchar 45\relax\chi}}-\sigma_{\phi,{\rm
thr}}^{{\rm M0\mathchar 45\relax\chi}})$ is approximately 0 except the case of
$m_{S}=50\,{\rm GeV}$ for $E_{e}=1\,{\rm TeV}$. The effect of the $Q^{2}$
evolution of the bottom PDF to the SACOT-$\chi$(thr) cross section is
negligible for $m_{S}=10\,{\rm GeV}$ irrespective of the beam energy, and it
is moderate for $m_{S}=10^{5}\,{\rm GeV}$ with $E_{e}=1\,{\rm TeV}$.

  * •

For the charm quark production, all the components are approximately equal in
magnitude, and the difference $(\sigma_{\phi,{\rm thr}}^{{\rm ZM\mathchar
45\relax\chi}}-\sigma_{\phi,{\rm thr}}^{{\rm M0\mathchar 45\relax\chi}})$ is
large, thus $(\sigma_{\phi,{\rm thr}}^{{\rm SACOT\mathchar
45\relax\chi}}-\sigma_{\phi}^{{\rm M}})$ is also large. This means that the
effect of $Q^{2}$ evolution for the charm PDF is large, therefore the
deviation of M component from the SACOT-$\chi$(thr) becomes sizable. Breakdown
of the M scheme dominance occurs depending on the value of scalar mass and
also on beam energy $E_{e}$. Actually the cross section in ZM-$\chi$(thr)
becomes dominant for large scalar mass $m_{S}=50\,{\rm GeV}$ (see also the
case of $m_{S}=10^{5}\,{\rm GeV}$ in Eq. (40)).

  * •

The M0-$\chi$(thr) curves are close to the ZM-$\chi$(thr) curves in the
low-$Q^{2}$ region, while in the higher $Q$ they approach to the M scheme
curves. This is expected by its construction; the subtraction term
(=M0-$\chi$(thr)) should agree with M scheme contribution for $Q^{2}\gg
m_{q}^{2}$ because the leading mass singularities are the same in both
schemes, and the subtraction term should agree with ZM-$\chi$(thr) for
$Q^{2}\sim m_{q}^{2}$ because the heavy quark PDFs are born in the region
$\mu_{f}^{2}\sim m_{q}^{2}$ and $f_{q/N}(x,\mu_{f}^{2})$ are not evolved much.
Indeed it is seen in all the plots that curves of M0-$\chi$(thr) interpolate
the two schemes from low-$Q^{2}$ to high-$Q^{2}$ region between ZM-$\chi$ and
M schemes.

There is one important difference between the CLFV DIS mediated by the massive
scalar $\phi$ and standard neutral current DIS $eN\to eX$ where the massless
photon is exchanged. The total cross section for the CLFV DIS contains the
contribution from all the $Q$ region, and the relative importance of the
low-$Q^{2}$ region versus large-$Q^{2}$ region is controlled not only by the
collision energy $\sqrt{s}$ but also by the scalar mass $m_{S}$. Here the
scalar mass $m_{S}$ plays a role of cut-off for the momentum transfer $Q$, and
the contribution from the region of $Q\gtrsim m_{S}$ to the CLFV cross section
is suppressed. This is contrasted with the normal DIS where the $Q$ region
which contributes to the cross section is controlled solely by the collision
energy $\sqrt{s}$.

###  IV.4 SACOT-$\chi$: Total cross section

Here we discuss the dependences of the total cross sections on the collision
energy $\sqrt{s}$ and on the scalar mass $m_{S}$ in the SACOT-$\chi$(thr)
scheme. In Fig. 17 the total cross sections $m_{S}^{4}\,\sigma_{q}$ for CLFV
DIS associated with heavy quark productions are plotted as functions of scalar
mass $m_{S}$. The results for bottom and charm quark productions are presented
in the upper and lower rows, respectively, and the beam energies are
$E_{e}=200\,{\rm GeV}$ and $E_{e}=1\,{\rm TeV}$ respectively in the left and
right columns. The following behaviors concerning to the $m_{S}$-dependence
can be read from the plots:

  * •

For the bottom quark production, all the curves for SACOT-$\chi$(thr), M,
ZM-$\chi$(thr), and sub-$\chi$(thr) converge to their asymptotic (constant)
values in large-$m_{S}$ limit. The SACOT-$\chi$(thr) cross section fitted at
the large-$m_{S}$ limit is $\sigma_{{\rm thr},b}^{{\rm SACOT\mathchar
45\relax\chi}}\approx(6.6+1.0-0.9)\cdot 10^{6}\,{\rm fb}\times(m_{S}/1\,{\rm
GeV})^{-4}$, decomposing the components for M, ZM-$\chi$(thr),
sub-$\chi$(thr), for $E_{e}=200\,{\rm GeV}$ and $\sigma_{{\rm thr},b}^{{\rm
SACOT\mathchar 45\relax\chi}}\approx(2.2+1.5-1.2)\cdot 10^{9}\,{\rm
fb}\times(m_{S}/1\,{\rm GeV})^{-4}$ for $E_{e}=1\,{\rm TeV}$, which agree with
the values for $m_{S}=10^{5}\,{\rm GeV}$ in Eq. (39). The SACOT-$\chi$(thr)
cross section evaluated at $m_{S}=50\,{\rm GeV}$ deviates from this asymptotic
form by $4\%$ ($13\%$) for $E_{e}=200\,{\rm GeV}$ ($E_{e}=1\,{\rm TeV}$). This
bears out that the value of $m_{S}=50\,{\rm GeV}$ can be regarded in a good
approximation as the large-$m_{S}$ limit for beam energies $E_{e}\lesssim
1\,{\rm TeV}$. The plots for the bottom cross section also show that the M
scheme cross section is a good approximation to the SACOT-$\chi$(thr),
regardless of the value of $m_{S}$ for $E_{e}\lesssim 1\,{\rm TeV}$.

  * •

For the charm quark production, all the curves for $E_{e}=200\,{\rm GeV}$ and
$E_{e}=1\,{\rm TeV}$ converge to their asymptotic values similarly to the case
of bottom quark production. The SACOT-$\chi$(thr) cross section fitted as the
large-$m_{S}$ limit is $\sigma_{{\rm thr},c}^{{\rm SACOT\mathchar
45\relax\chi}}\approx(1.6+1.8-1.3)\cdot 10^{9}\,{\rm fb}\times(m_{S}/1\,{\rm
GeV})^{-4}$ for $E_{e}=200\,{\rm GeV}$ and $\sigma_{{\rm thr},c}^{{\rm
SACOT\mathchar 45\relax\chi}}\approx(1.9+2.4-1.8)\cdot 10^{10}\,{\rm
fb}\times(m_{S}/1\,{\rm GeV})^{-4}$ for $E_{e}=1\,{\rm TeV}$, which agree with
the values for $m_{S}=10^{5}\,{\rm GeV}$ in Eq. (40). Comparing the asymptotic
large-$m_{S}$ limit with the values at $m_{S}=50\,{\rm GeV}$ bears out that
$m_{S}=50\,{\rm GeV}$ can be regarded in a good approximation as the
asymptotic large $m_{S}$-limit for beam energies $E_{e}\lesssim 1\,{\rm TeV}$.
For the charm case, the ZM-$\chi$(thr) cross section is the dominant component
for $m_{S}\gtrsim 15\,{\rm GeV}$ and $E_{e}\lesssim 1\,{\rm TeV}$, but there
is still sizable corrections from the components M and sub-$\chi$(thr) to
match with the value of the SACOT-$\chi$(thr) scheme cross section.

Figure 17:  The total cross sections $m_{S}^{4}\,\sigma_{q}$ are plotted as
functions of $m_{S}$. The results for bottom quark ($q=b$) and charm quark
($q=c$) are shown respectively in the upper and lower rows, and the beam
energies are $E_{e}=200\,{\rm GeV}$ and $E_{e}=1\,{\rm TeV}$ in the left and
right columns.

Figure 18:  The components M, ZM-$\chi$, and M0-$\chi$ normalized to the
SACOT-$\chi$ as functions of the collision energy $\sqrt{s}$: the ticks of
upper horizontal axis indicate the corresponding beam energy $E_{e}$. The
cross sections for the bottom and charm quark production are shown in the
upper and lower rows. The scalar masses are $m_{S}=10\,{\rm GeV}$ and
$m_{S}=10^{5}\,{\rm GeV}$ respectively in the left and right columns.

For comparison among the components of SACOT-$\chi$(thr) cross section in a
wide range of the collision energy, the total cross sections are plotted in
Fig. 18 as functions of $\sqrt{s}$ for bottom and charm quark productions.
Numerical values for the cross sections in the SACOT-$\chi$(thr) scheme and
its components are listed in Table 2. Concerning to the collision energy
dependences following features can be observed from the plots:

  * •

For the bottom quark production the dominance of the M scheme contribution
holds for the small scalar mass $m_{S}=10\,{\rm GeV}$ in all the range of
$\sqrt{s}$. But for the large scalar mass $m_{S}=10^{5}\,{\rm GeV}$ the
ZM-$\chi$(thr) contribution becomes the dominant component for
$\sqrt{s}\gtrsim 100\,{\rm GeV}$ ($E_{e}\gtrsim 5\,{\rm TeV}$). In large
$\sqrt{s}$ limit for the case of $m_{S}=10^{5}\,{\rm GeV}$ the cross sections
in M and ZM-$\chi$(thr) schemes coincide each other above $\sqrt{s}\approx
1\,{\rm TeV}$. Thus the M and sub-$\chi$(thr) terms cancels each other and the
SACOT-$\chi$(thr) scheme cross section is approximated by the ZM-$\chi$(thr)
cross section for $\sqrt{s}\gtrsim 1\,{\rm TeV}$, i.e. $\sigma_{{\rm
thr},b}^{{\rm SACOT\mathchar 45\relax\chi}}\simeq\sigma_{{\rm thr},b}^{{\rm
ZM\mathchar 45\relax\chi}}$ for $m_{S}=10^{5}\,{\rm GeV}$ and $\sqrt{s}\gtrsim
1\,{\rm TeV}$.

  * •

For the charm quark production for small scalar mass $m_{S}=10\,{\rm GeV}$,
there is no unique component which dominates the cross section in all the
energy range, and all the components M, ZM-$\chi$(thr), and sub-$\chi$(thr)
are equally important for the total cross section $\sigma_{{\rm thr},c}^{{\rm
SACOT\mathchar 45\relax\chi}}$. On the other hand, for the large scalar mass
$m_{S}=10^{5}\,{\rm GeV}$, the ZM-$\chi$ cross section describes the
SACOT-$\chi$(thr) cross section well for all the energy range except
$\sqrt{s}\lesssim 100\,{\rm GeV}$. Starting at $\sqrt{s}\sim 100\,{\rm GeV}$
the M and sub-$\chi$(thr) cross sections coincide so that they cancel each
other in the SACOT-$\chi$(thr) cross section, i.e. $\sigma_{{\rm thr},c}^{{\rm
SACOT\mathchar 45\relax\chi}}\simeq\sigma_{{\rm thr},c}^{{\rm ZM\mathchar
45\relax\chi}}$ for $m_{S}=10^{5}\,{\rm GeV}$.

Table 2: The total cross sections for CLFV DIS associated with bottom quark production in SACOT-$\chi$-thr and its components. For the ZM-$\chi$, M0-$\chi$ and SACOT-$\chi$ schemes, the threshold factor $S_{\rm thr}$ is taken into account. $m_{S}~{}[\rm{GeV}]$ | $E_{e}~{}{\rm[GeV]}$ | $\sigma_{b}^{\rm M}~{}{\rm[fb]}$ | $\sigma_{{\rm thr},b}^{{\rm ZM\mathchar 45\relax\chi}}~{}{\rm[fb]}$ | $\sigma_{{\rm thr},b}^{{\rm M0\mathchar 45\relax\chi}}~{}{\rm[fb]}$ | $\sigma_{{\rm thr},b}^{{\rm SACOT\mathchar 45\relax\chi}}~{}{\rm[fb]}$
---|---|---|---|---|---
| $10^{2}$ |  $2.08\times 10^{-1}$ |  $1.95\times 10^{-3}$ |  $2.32\times 10^{-3}$ |  $2.07\times 10^{-1}$
| $10^{3}$ | $4.93\times 10^{4}$ | $1.53\times 10^{4}$ | $1.35\times 10^{4}$ | $5.11\times 10^{4}$
$10$ | $10^{4}$ | $7.32\times 10^{5}$ | $2.87\times 10^{5}$ | $2.66\times 10^{5}$ | $7.53\times 10^{5}$
| $10^{5}$ | $3.72\times 10^{6}$ | $1.54\times 10^{6}$ | $1.51\times 10^{6}$ | $3.75\times 10^{6}$
| $10^{6}$ | $1.18\times 10^{7}$ | $4.90\times 10^{6}$ | $5.18\times 10^{6}$ | $1.15\times 10^{7}$
| $10^{7}$ | $2.99\times 10^{7}$ | $1.22\times 10^{7}$ | $1.39\times 10^{7}$ | $2.81\times 10^{7}$
| $10^{2}$ | $3.17\times 10^{-17}$ | $4.06\times 10^{-19}$ | $4.74\times 10^{-19}$ | $3.17\times 10^{-17}$
| $10^{3}$ | $2.23\times 10^{-11}$ | $1.47\times 10^{-11}$ | $1.22\times 10^{-11}$ | $2.47\times 10^{-11}$
$10^{5}$ | $10^{4}$ | $1.27\times 10^{-9}$ | $1.32\times 10^{-9}$ | $1.07\times 10^{-9}$ | $1.51\times 10^{-9}$
| $10^{5}$ | $2.57\times 10^{-8}$ | $3.03\times 10^{-8}$ | $2.44\times 10^{-8}$ | $3.16\times 10^{-8}$
| $10^{6}$ | $3.66\times 10^{-7}$ | $4.54\times 10^{-7}$ | $3.61\times 10^{-7}$ | $4.59\times 10^{-7}$
| $10^{7}$ | $4.50\times 10^{-6}$ | $5.72\times 10^{-6}$ | $4.49\times 10^{-6}$ | $5.73\times 10^{-6}$
$m_{S}~{}[\rm{GeV}]$ | $E_{e}~{}{\rm[GeV]}$ | $\sigma_{c}^{\rm M}~{}{\rm[fb]}$ | $\sigma_{{\rm thr},c}^{{\rm ZM\mathchar 45\relax\chi}}~{}{\rm[fb]}$ | $\sigma_{{\rm thr},c}^{{\rm M0\mathchar 45\relax\chi}}~{}{\rm[fb]}$ | $\sigma_{{\rm thr},c}^{{\rm SACOT\mathchar 45\relax\chi}}~{}{\rm[fb]}$
| $10^{2}$ |  $2.74\times 10^{4}$ |  $2.45\times 10^{4}$ |  $1.79\times 10^{4}$ |  $3.40\times 10^{4}$
| $10^{3}$ | $6.59\times 10^{5}$ | $5.85\times 10^{5}$ | $4.86\times 10^{5}$ | $7.58\times 10^{5}$
$10$ | $10^{4}$ | $4.15\times 10^{6}$ | $3.25\times 10^{6}$ | $3.13\times 10^{6}$ | $4.27\times 10^{6}$
| $10^{5}$ | $1.46\times 10^{7}$ | $9.73\times 10^{6}$ | $1.11\times 10^{7}$ | $1.32\times 10^{7}$
| $10^{6}$ | $3.83\times 10^{7}$ | $2.15\times 10^{7}$ | $2.96\times 10^{7}$ | $3.02\times 10^{7}$
| $10^{7}$ | $8.68\times 10^{7}$ | $4.19\times 10^{7}$ | $6.78\times 10^{7}$ | $6.09\times 10^{7}$
| $10^{2}$ | $3.86\times 10^{-12}$ | $3.99\times 10^{-12}$ | $2.83\times 10^{-12}$ | $5.03\times 10^{-12}$
| $10^{3}$ | $1.93\times 10^{-10}$ | $2.43\times 10^{-10}$ | $1.75\times 10^{-10}$ | $2.61\times 10^{-10}$
$10^{5}$ | $10^{4}$ | $3.51\times 10^{-9}$ | $4.64\times 10^{-9}$ | $3.41\times 10^{-9}$ | $4.75\times 10^{-9}$
| $10^{5}$ | $4.66\times 10^{-8}$ | $6.29\times 10^{-8}$ | $4.62\times 10^{-8}$ | $6.33\times 10^{-8}$
| $10^{6}$ | $5.48\times 10^{-7}$ | $7.50\times 10^{-7}$ | $5.47\times 10^{-7}$ | $7.50\times 10^{-7}$
| $10^{7}$ | $6.09\times 10^{-6}$ | $8.45\times 10^{-6}$ | $6.09\times 10^{-6}$ | $8.44\times 10^{-6}$

##  V Summary

In this article, we have considered a scenario where a CLFV (pseudo-)scalar
mediator couples dominantly to heavy quarks and studied the CLFV DIS
associated with heavy quark productions $\ell_{i}N\to\ell_{j}q\bar{q}X$
($q=b,c$). The CLFV DIS cross sections are written in terms of the leptonic
and the hadronic parts. We have computed the heavy quark contributions to the
structure function in the SACOT-$\chi$(thr), M, ZM-$\chi$(thr), and
M0-$\chi$(thr) schemes and presented their results. We have examined three
improvements for the threshold behavior of the structure function: the
$\chi$-rescaling, choice of factorization scale $\mu_{f}$, and inclusion of
the threshold factor $S_{\rm thr}$. To our best knowledge the present article
is the first application of the SACOT-$\chi$(thr) scheme for the CLFV DIS
associated with heavy quark productions. We have shown that only the
SACOT-$\chi$(thr) scheme provides a reliable theory prediction for the CLFV
DIS cross section in the wide kinematical region and in the full parameter
space. Thus the systematic study for the CLFV signal search including the full
DIS kinematical information will be available in the SACOT-$\chi$(thr) scheme.

For the structure function we have made a comparison among the different
computational schemes and observed that the $\chi$-rescaling is effective in
$Q\lesssim 100\,(50)$ GeV for the bottom (charm) quark production. It is
mandatory to incorporate the $\chi$-rescaling to predict the structure
function especially on the threshold behavior of the heavy quark productions.

We also made a detailed analysis on the collision energy dependence and the
mediator mass dependence of the heavy quark production cross sections focusing
on the beam energies up to $E_{e}=1\,{\rm TeV}$. For the bottom quark
production, we showed that the M scheme cross section approximates that of the
SACOT-$\chi$(thr) irrespective of the size of the scalar mass. For the charm
quark production, we showed that the ZM-$\chi$(thr) cross section approximates
that of the SACOT-$\chi$(thr) for $m_{S}\gtrsim 50\,{\rm GeV}$, but for the
small scalar mass $\sim 10\,{\rm GeV}$ the M scheme cross section becomes the
dominant component for SACOT-$\chi$(thr). We found that the ratio of the
contributions in the massive and the zero-mass schemes strongly depends on the
mediator mass. This is because when the mediator mass is small, the
contributions from the low-$Q^{2}$ region are enhanced, for which the M scheme
contribution is superior. We conclude that the SACOT-$\chi$(thr) prescription
is indispensable to obtain the reliable sensitivity to the CLFV interactions
in the next generation experiments of the energy range $E_{e}\lesssim 1\,{\rm
TeV}$.

To utilize the processes for the CLFV signal searches, we propose the
measurements of the total cross sections and the momentum distributions
$d^{2}\sigma/d\bm{p}$ of the final lepton. The normalizations of the total
cross sections depend on the combination of CLFV mediator couplings
$|\rho_{qq}^{\phi}|^{2}(|\rho_{ij}^{\phi}|^{2}+|\rho_{ji}^{\phi}|^{2})$ and
$m_{S}$, while the (normalized) momentum distributions depend on $m_{S}$. We
have shown for the case of $eN\to\tau q\bar{q}X$ that the momentum
distributions can have a sensitivity on the size of scalar mass. To make a
definite conclusion on the feasibility of the simultaneous determination of
the CLFV couplings and the scalar mass, more efforts have to be devoted to
improve the precision of the theory computations. The issues of the scale
dependence, PDF uncertainties, QCD radiative corrections, etc., have to be
addressed together with more detailed simulation study taking the experimental
uncertainties into account, which are beyond the scope of the present article,
but we render them for a future work.

For the present model Lagrangian, it is known that the other types of
subprocesses also contribute to the CLFV DIS via the gluonic operator
($\sim\phi\,G^{a}_{\mu\nu}G^{a\,\mu\nu}$) and the photonic dipole operator
($\sim\bar{\ell}_{i}\sigma_{\mu\nu}\ell_{j}F^{\mu\nu}$) which generate the
processes $\ell_{i}g\to\ell_{j}g$ and $\ell_{i}q_{l}\to\ell_{j}q_{l}$ ($q_{l}$
being the light quarks). The comprehensive analysis of the DIS observables
taking these subprocesses into account is required to disentangle the type of
the CLFV operators and eventually to unravel the properties of CLFV mediators.
We leave these issues for our future works.

## Acknowledgements

This work was partly supported by MEXT Joint Usage/Research Center on
Mathematics and Theoretical Physics JPMXP0619217849 (M.Y.) and JSPS KAKENHI
Grant Numbers JP18K03611, JP16H06492, JP20H00160 (M.T.); JP18H01210,
JP21H00081 (Y.U.); JP20H05852 (M.Y.).

##  Appendix A Physical region of DIS kinematics

###  A.1 Range of $(x,y)$ in generic case

We derive the physical region of kinematical variable $x$ and $y$ for CLFV DIS
process $\ell_{i}(k_{i})p(k_{p})\to\ell_{j}(k_{j})\hat{X}(k_{X})$, where
$\hat{X}$ can be one particle or a multi-particle system. We assume that the
initial QCD parton $p$ is massless with momentum $k_{p}=\xi P$, and other
particles are massive: $k_{i}^{2}=m_{i}^{2},k_{p}^{2}=0,k_{j}^{2}=m_{j}^{2}$,
and $k_{X}^{2}=w^{2}$. Similar analyses can be found in literatures Albright
and Jarlskog (1975); Hagiwara et al. (2003).

First we express the kinematical variables in the center-of-mass frame of
lepton and parton, where lepton energies are given by

$\displaystyle E_{i}~{}=~{}\frac{\hat{s}+m_{i}^{2}}{2\sqrt{\hat{s}}},\hskip
11.38109ptE_{j}~{}=~{}\frac{\hat{s}+m_{j}^{2}-w^{2}}{2\sqrt{\hat{s}}},$ (44)

where $\hat{s}=(k_{p}+k_{i})^{2}$ is related to the lepton-nucleon collision
energy squared $s=(P+k_{i})^{2}$ by $\hat{s}-m_{i}^{2}=\xi\,(s-m_{i}^{2})$.
The momentum transfer $Q^{2}=-q^{2}=-(k_{i}-k_{j})^{2}$ is given by
$Q^{2}=-(m_{i}^{2}+m_{j}^{2})+2E_{i}E_{j}-2|\bm{k}_{i}||\bm{k}_{j}|\cos\theta_{ij}$
in the center-of-mass frame, where the product of spacial momenta in the last
term can be replaced by
$|\bm{k}_{i}||\bm{k}_{j}|=\sqrt{E_{i}^{2}-m_{i}^{2}}\sqrt{E_{j}^{2}-m_{j}^{2}}$.
Now the angle $\theta_{ij}$ can be written as

$\begin{split}\cos\theta_{ij}=\frac{2E_{i}E_{j}-xy(s-m_{i}^{2})-(m_{i}^{2}+m_{j}^{2})}{2\sqrt{E_{i}^{2}-m_{i}^{2}}\sqrt{E_{j}^{2}-m_{j}^{2}}},\end{split}$
(45)

where $Q^{2}=xy(s-m_{i}^{2})$ is used. The constraint $|\cos\theta_{ij}|\leq
1$ leads to an inequality

$\begin{split}\bigl{[}2E_{i}E_{j}-xy\left(s-m_{i}^{2}\right)-\left(m_{i}^{2}+m_{j}^{2}\right)\bigr{]}^{2}\leq
4(E_{i}^{2}-m_{i}^{2})(E_{j}^{2}-m_{j}^{2}).\end{split}$ (46)

Substituting the lepton energies by Eq. (44), we obtain

$\displaystyle
y\,\big{\\{}(1-y)\big{[}x(s-m_{i}^{2})+m_{i}^{2}\big{]}-m_{j}^{2}\big{\\}}\geq
0,$ (47)

where we used $w^{2}=(k_{p}+q)^{2}=y(\hat{s}-m_{i}^{2})(1-x/\xi)$. This
condition determines the upper bound of the inelasticity $y$

$\begin{split}y\leq 1-\frac{r_{j}}{x+r_{i}}.\end{split}$ (48)

Here we introduced dimensionless masses $r_{a}~{}(a=i,j)$

$\displaystyle r_{i}$ $\displaystyle=$
$\displaystyle\frac{m_{i}^{2}}{s-m_{i}^{2}},\hskip
14.22636ptr_{j}~{}=~{}\frac{m_{j}^{2}}{s-m_{i}^{2}}.$ (49)

The lower one is bound by the partonic phase space. We express the
inelasticity parameter $y$ in terms of $x$ and $w^{2}$

$\displaystyle y$ $\displaystyle=$
$\displaystyle\frac{1}{\xi-x}\left(\frac{w^{2}}{s-m_{i}^{2}}\right).$ (50)

The momentum fraction $\xi$ is expressed in terms of Lorentz invariance as
$\xi=x\,\left(Q^{2}+\omega^{2}\right)/Q^{2}$, and its minimum needs to be less
than $x$;

$\begin{split}\xi=x\left(\frac{Q^{2}+w_{\text{min}}^{2}}{Q^{2}}\right)\geq
x.\end{split}$ (51)

Thus the lower bound of $y$ is given by the production threshold of
$w^{2}_{\rm min}$ with maximum momentum fraction $\xi=1$:

$\displaystyle y\geq\frac{1}{1-x}\left(\frac{w^{2}_{\rm
min}}{s-m_{i}^{2}}\right)$ (52)

Thus the physical range of $y$ for fixed $x$ is given by $y_{\rm min}\leq
y\leq y_{\rm max}$ with

$\displaystyle y_{\rm min}(x)~{}=~{}\frac{r_{\rm min}}{1-x},\hskip
28.45274pty_{\rm max}(x)~{}=~{}1-\frac{r_{j}}{x+r_{i}},$ (53)

where $r_{\rm min}\equiv w_{\rm min}^{2}/(s-m_{i}^{2})$.

The physical range of $x$ can be obtained by requiring that the lower bound on
$y$ should be smaller than the upper one, namely $y_{\rm min}(x)\leq y_{\rm
max}(x)$ should hold. This yields a relation

$\begin{split}x^{2}-(1-r_{i}+r_{j}-r_{\rm min})x-(r_{i}-r_{j}-r_{i}\,r_{\rm
min})\leq 0.\end{split}$ (54)

Solving the quadratic inequality for $x$, we find the range $x_{-}\leq x\leq
x_{+}$ with

$\displaystyle x_{\pm}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\bigg{[}\left(1-r_{i}+r_{j}-r_{\rm
min}\right)\pm\sqrt{(1-r_{i}+r_{j}-r_{\rm min})^{2}+4(r_{i}-r_{j}-r_{i}r_{\rm
min})}\bigg{]}.$ (55)

###  A.2 Heavy quark pair production in terms of $(x,y)$

Here we give concrete expression for the physical region by taking electron
and tau leptons as initial and final leptons, and a heavy quark pair of mass
$m_{q}$ in the hadronic part, i.e. $\hat{X}=q\bar{q}$. We ignore the electron
mass, but tau and heavy quark masses are kept. In this case Eq. (53) reduces
to

$\displaystyle y_{\rm min}(x)$ $\displaystyle=$
$\displaystyle\frac{4m_{q}^{2}}{(1-x)s},\hskip 28.45274pty_{\rm
max}(x)~{}=~{}1-\frac{m_{\tau}^{2}}{xs},$ (56)

and Eq. (55) reduces to

$\displaystyle x_{\pm}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\bigg{[}1+\frac{m_{\tau}^{2}}{s}-\frac{4m_{q}^{2}}{s}\pm\sqrt{\left(1+\frac{m_{\tau}^{2}}{s}-\frac{4m_{q}^{2}}{s}\right)^{2}-4\frac{m_{\tau}^{2}}{s}}~{}\bigg{]},$
(57)

where the minimum of $w^{2}$ has been substituted by $4m_{q}^{2}$.

###  A.3 Heavy quark pair production in terms of $(x,Q^{2})$

When one takes $(x,Q^{2})$ as independent variables instead of $(x,y)$, the
relation $xy=Q^{2}/s$ can be used. The physical region for $x$ is given by
$x_{\rm min}(s,Q^{2})\leq x\leq x_{\rm max}(Q^{2})$ with

$\displaystyle x_{\rm min}(s,Q^{2})~{}=~{}\frac{Q^{2}+m_{\tau}^{2}}{s},\hskip
28.45274ptx_{\rm max}(Q^{2})~{}=~{}\frac{Q^{2}}{Q^{2}+4m_{q}^{2}},$ (58)

and for $Q^{2}$ the physical region is $Q_{-}^{2}\leq Q^{2}\leq Q_{+}^{2}$
with

$\displaystyle Q^{2}_{\pm}$ $\displaystyle=$ $\displaystyle
sx_{\pm}-m_{\tau}^{2},$ (59)

where $x_{\pm}$ is defined in Eq. (57).

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