# Sensitivities on the anomalous quartic $\gamma\gamma\gamma\gamma$ and
$\gamma\gamma\gamma Z$ couplings at the CLIC

E<EMAIL_ADDRESS>Department of Physics, Sinop University,
Turkey.

###### Abstract

It is essential to directly investigate the self-couplings of gauge bosons in
the Standard Model (SM) due to its non-Abelian nature, as these couplings play
a significant role in comprehending the gauge structure of the model. The
discrepancies between the Standard Model’s expectations and the measured value
of gauge boson self-couplings serve as strong evidence for new physics
phenomena extending beyond the Standard Model. Such deviations provide
valuable insights into the nature of new physics and potentially lead to a
deeper understanding of fundamental particles and their interactions. This
study examines the sensitivities of anomalous couplings associated with
dimension-8 operators that affect the $\gamma\gamma\gamma\gamma$ and
$Z\gamma\gamma\gamma$ quartic vertices. The study focuses on the process
$e^{-}\gamma\to e^{-}\gamma\gamma$ with the incoming photon under
Weizsäcker–Williams approximation at the stage-3 scenario of Compact Linear
Collider (CLIC) that refers to a CoM energy of 3 TeV. Due to the CLIC options,
we take into account both unpolarized and $\mp 80\%$ polarized electron beam
with the related integrated luminosities of ${\cal L}=5,4,1$ $\rm ab^{-1}$
under the systematic uncertainties of $\delta_{sys}=0,3,5$. Obtained
sensitivities on the anomalous quartic gauge couplings (aQGCs) for the process
$e^{-}\gamma\to e^{-}\gamma\gamma$ at $\sqrt{s}=3$ TeV and various
polarizations are improved by a factor of 2-200 times better for the couplings
$f_{T,j}/\Lambda^{4}$ compared with the experimental results reported by CMS
Collaboration in Ref.JHEP10-2021

—–

###### pacs:

12.60.-i, 14.70.Bh, 14.70.Hp
Keywords: Electroweak interaction, Anomalous couplings, Models beyond the
Standard Model.

##  I Introduction

The non-Abelian nature of the SM implies the gauge boson’s self-interactions
as triple and quartic gauge couplings(TGC and QGC). Also, possible deviations
from triple and quartic gauge interactions as anomalous TGC and QGC (aTGC and
aQGC) have an essential role in checking the validity of SM and predicting the
new physics contribution coming from the Beyond Standard
Model(BSM)arXiv:2106.11082 ; PRD93-2016 ; JHEP10-2017 ; JHEP10-2021 . In the
SM, neutral gauge boson
$ZZ\gamma\gamma,Z\gamma\gamma\gamma,\gamma\gamma\gamma\gamma$ couplings are
excluded. In this manner, the contributions from these vertices are sources of
new physics and a sign beyond the Standard Model.

These effects can be determined with Effective Field Theory (EFT) by adding
high-dimension operators to the SM Lagrangian. It provides a model-independent
framework to parametrize the effects of new physics systematically. It is
discussed in the papers CPC44-2020 ; PRD107-2023  to generate the neutral TGC
(NTGC) $ZV\gamma$ $(V=\gamma,Z)$ by dim-8 operators. Numerous experimental and
phenomenological studies have been conducted to explore high-dimensional gauge
operators, both at present and future colliders, focusing on proton-proton,
electron-proton, and electron-positron collisions Eboli-PRD93-2016 ; Marantis-
JPCS2105-2021 ; Degrande-AP335-2013 ; Hao-PRD104-2021 ; Degrande-JHEP02-2014 ;
Hays-JHEP02-2019 ; Gounaris-PRD65-2002 ; Wrishik-JHEP08-2022 ; Murphy-
JHEP10-2020 ; Ellis-China64-2021 ; Murphy-JHEP04-2021 ; Green-RMP89-2017 ;
PLB760-2016 ; JHEP12-2018 ; PLB540-2002 ; PRD70-2004 ; PRD62-2000 ; PRD88-2013
; PRL113-2014 ; PRL115-2015 ; PRD96-2017 ; EPJC77-2017 ; PRD95-2017 ;
PRD90-2014 ; PLB774-2017 ; PLB770-2017 ; JHEP06-2017 ; PRL120-2018 ;
PLB795-2019 ; PLB798-2019 ; PLB812-2021 ; JHEP06-2020 ; PLB809-2020 ;
PLB811-2020 ; Stirling ; PRD89-2014 ; EPJC13-2000 ; PLB515-2001 ; Eboli ;
PRD75-2007 ; AHEP2016-2016 ; JHEP10-121-2021 ; EPJC81-2021 ; EPJP130-2015 ;
PRD104-2021 ; JHEP06-142-2017 ; JPG26-2000 ; Eboli2 ; PRD81-2010 ; PRD85-2012
; arXiv:0908.2020 ; PRD86-2012 ; Eboli4 ; arXiv:2109.12572 ; Han-PLB1998 ;
Daniele-RNC1997 ; Boos-PRD1998 ; Boos-PRD2000 ; Beyer-EPJC2006 ; Christian-
EPJC2017 ; Stephen-arxiv:9505252 ; Stirling-PLB1999 ; Belanger-PLB1992 ;
Senol-AHEP2017 ; Barger-PRD1995 ; Cuypers-PLB1995 ; Cuypers-IJMPA1996 ; Ji-
arXiv:2204.08195 ; ATLAS1 ; ATLAS2 ; CMS1 ; JPG2022 ; JPG2023 ; Koksal-son .
One of the current phenomenological studies aims to investigate anomalous
$\gamma\gamma\gamma\gamma$ couplings via the
$\mu^{+}\mu^{-}\to\mu^{+}\gamma\gamma\mu^{-}$ process at future muon collider
with the center-of-mass energies $\sqrt{s}=$3, 14 and 100 TeV arXiv:2306.03653
. In this context, performing this work in CLIC will be a complementary study
of future lepton colliders.

The current CLIC scenarios supposed 1.0 $\rm ab^{-1}$, 2.5 $\rm ab^{-1}$, and
5.0 $\rm ab^{-1}$ of luminosities with the center-of-mass energies of
$\sqrt{s}=0.38$ TeV, 1.5 TeV, and 3 TeV, respectively franc . Also, CLIC
enables $\pm 80\%$ polarisation options for the electron beam, but there is no
option for positron polarisation. Here, unpolarized and polarized electron
beam of $P_{e^{-}}=-80\%$, $80\%$ are pertinent with the luminosities of
${\cal L}=5$ ab-1, ${\cal L}=4$ ab-1 and ${\cal L}=1$ ab-1, respectively
CLIC-1812.06018 . Polarization options provide a better signal-background
ratio to reach better sensitivities on aQGC with high luminosities. Besides,
the collisions at the fundamental level in the lepton collider provide a
cleaner environment than the hadron colliders. This gives us an advantage of
minor systematic uncertainties, easy analysis, and precise measurements.

The cross-section of any process consists of the cross-sections related to
electron and positron beam polarizations ($P_{e^{-}}$,$P_{e^{+}}$) given in
the following equation kek2017 .

$\displaystyle\sigma\left(P_{e^{+}},P_{e^{-}}\right)=\frac{1}{4}\\{\left(1+P_{e^{+}}\right)\left(1+P_{e^{-}}\right)\sigma_{RR}+\left(1-P_{e^{+}}\right)\left(1-P_{e^{-}}\right)\sigma_{LL}$
(1)
$\displaystyle+\left(1-P_{e^{+}}\right)\left(1+P_{e^{-}}\right)\sigma_{RL}+\left(1+P_{e^{+}}\right)\left(1-P_{e^{-}}\right)\sigma_{LR}\\}\,,$

Here, $\sigma_{LL}$ and $\sigma_{RR}$ are the cross-sections of both left and
right-handed electron-positron beams. Similarly, $\sigma_{RL}$ and
$\sigma_{LR}$ represent the cross-conditions of the beam polarizations.

In this study, the process $e^{-}\gamma\to e^{-}\gamma\gamma$ with a CoM
energy $\sqrt{s}=3$ TeV at CLIC for both unpolarized and polarized
$P_{e^{-}}=-80\%$, $80\%$ is taken into account to examine the new physics
effects via aQGC under various systematic uncertainties. The paper consists of
the following parts. Section II defines the dim-8 EFT for the anomalous
$Z\gamma\gamma\gamma$ and $\gamma\gamma\gamma\gamma$ couplings. Applied
analyzing techniques on the process and obtained sensitivities on the
$f_{T,j}/\Lambda^{4}$ couplings at $95\%$ Confidence Level (C.L.) are
discussed in Section-III and Section-IV, respectively. Finally, we sum up our
results in Section V.

##  II Dim-8 operators for the anomalous $Z\gamma\gamma\gamma$ and
$\gamma\gamma\gamma\gamma$ couplings

EFT in the SM is a framework that extends the reach of the SM of particle
physics. It provides a systematic way to describe the effects of new
physics(NP). Here, the starting point is to identify the effective operators.
The EFT Lagrangian consists of the SM Lagrangian plus an infinite tower of
higher-dimension operators, which are suppressed by a cut-off scale. The EFT
Lagrangian is given as follows.

${\cal L}_{eff}={\cal L}_{SM}+\sum_{i}\frac{c_{i}^{(6)}}{\Lambda^{2}}{\cal
O}_{i}^{(6)}+\sum_{j}\frac{c_{j}^{(8)}}{\Lambda^{4}}{\cal O}_{j}^{(8)}+...,$
(2)

The processes, including aQGCs, can be produced via tri-boson productions and
the vector boson scattering (VBS) processes. Here, VBS processes refer to the
scattering of two vector bosons with each other that can produce quartic gauge
boson interactions. On the other hand, Tri-boson production refers to the
production of three vector bosons. Compared with the tri-boson processes, the
VBS processes exhibit a higher sensitivity to aQGCs.ATLAS1 ; ATLAS2 ; CMS1 .

In this study, we focused on the dim-8 anomalous $Z\gamma\gamma\gamma$ and
$\gamma\gamma\gamma\gamma$ quartic vertices using Effective Lagrangian
techniques by adding dim-8 operators to the generic SM Lagrangian. Generally,
the formalism for aQGC has been discussed in the literature Eboli1 ; Eboli ;
Degrande ; Eboli3 ; Eboli-PRD101-2020 . The effective Lagrangian, including
the dim-8 operators for quartic couplings, is as follows.

${\cal L}_{eff}={\cal
L}_{SM}+\sum_{k=0}^{2}\frac{f_{S,k}}{\Lambda^{4}}O_{S,k}+\sum_{i=0,1,2,3,4,5,7}^{7}\frac{f_{M,i}}{\Lambda^{4}}O_{M,i}+\sum_{j=0,1,2,5,6,7,8,9}\frac{f_{T,j}}{\Lambda^{4}}O_{T,j},$
(3)

Here, $O_{S,k}$, $O_{M,i}$ and $O_{T,j}$ is the operators of dim-8 and
$\frac{f_{S,k}}{\Lambda^{4}}$, $\frac{f_{M,i}}{\Lambda^{4}}$ and
$\frac{f_{T,j}}{\Lambda^{4}}$ are the related parameters running by the
effective operators. In Eq.3, there are three types of aQGC operators.

The first set of operators ($\frac{f_{S,k}}{\Lambda^{4}}O_{S,k}$) is
contribute with the covariant derivative which induces the $ZZZZ$, $WWZZ$ and
$WWWW$ couplings. On the other hand, $\frac{f_{T,j}}{\Lambda^{4}}$ and
$\frac{f_{M,i}}{\Lambda^{4}}$ operators are contributing with Gauge boson
field strength tensors and both field (Mixed field), respectively. Below, the
aQGC operators corresponding to these three type operators are presented
Degrande .

$\bullet$ Scalar field:

$\displaystyle O_{S,0}$ $\displaystyle=$
$\displaystyle[(D_{\rho}\Phi)^{\dagger}(D_{\sigma}\Phi)]\times[(D^{\rho}\Phi)^{\dagger}(D^{\sigma}\Phi)],$
(4) $\displaystyle O_{S,1}$ $\displaystyle=$
$\displaystyle[(D_{\rho}\Phi)^{\dagger}(D^{\rho}\Phi)]\times[(D_{\sigma}\Phi)^{\dagger}(D^{\sigma}\Phi)],$
(5) $\displaystyle O_{S,2}$ $\displaystyle=$
$\displaystyle[(D_{\rho}\Phi)^{\dagger}(D_{\sigma}\Phi)]\times[(D^{\sigma}\Phi)^{\dagger}(D^{\rho}\Phi)].$
(6)

$\bullet$ Tensor field:

$\displaystyle O_{T,0}$ $\displaystyle=$ $\displaystyle
Tr[\widehat{W}_{\sigma\lambda}\widehat{W}^{\sigma\lambda}]\times
Tr[\widehat{W}_{\alpha\beta}\widehat{W}^{\alpha\beta}],$ (7) $\displaystyle
O_{T,1}$ $\displaystyle=$ $\displaystyle
Tr[\widehat{W}_{\lambda\mu}\widehat{W}^{\nu\beta}]\times
Tr[\widehat{W}_{\nu\beta}\widehat{W}^{\lambda\mu}],$ (8) $\displaystyle
O_{T,2}$ $\displaystyle=$ $\displaystyle
Tr[\widehat{W}_{\lambda\nu}\widehat{W}^{\nu\sigma}]\times
Tr[\widehat{W}_{\sigma\mu}\widehat{W}^{\mu\lambda}],$ (9) $\displaystyle
O_{T,5}$ $\displaystyle=$ $\displaystyle
Tr[\widehat{W}_{\lambda\sigma}\widehat{W}^{\lambda\sigma}]\times
B_{\mu\nu}B^{\mu\nu},$ (10) $\displaystyle O_{T,6}$ $\displaystyle=$
$\displaystyle Tr[\widehat{W}_{\lambda\mu}\widehat{W}^{\nu\sigma}]\times
B_{\nu\sigma}B^{\lambda\mu},$ (11) $\displaystyle O_{T,7}$ $\displaystyle=$
$\displaystyle Tr[\widehat{W}_{\lambda\nu}\widehat{W}^{\nu\sigma}]\times
B_{\sigma\mu}B^{\mu\lambda},$ (12) $\displaystyle O_{T,8}$ $\displaystyle=$
$\displaystyle B_{\sigma\lambda}B^{\sigma\lambda}B_{\mu\nu}B^{\mu\nu},$ (13)
$\displaystyle O_{T,9}$ $\displaystyle=$ $\displaystyle
B_{\lambda\nu}B^{\nu\sigma}B_{\sigma\mu}B^{\mu\lambda}.$ (14)

$\bullet$ Mixed field:

$\displaystyle O_{M,0}$ $\displaystyle=$ $\displaystyle
Tr[\widehat{W}_{\nu\lambda}\widehat{W}^{\nu\lambda}]\times[(D_{\sigma}\Phi)^{\dagger}(D^{\sigma}\Phi)],$
(15) $\displaystyle O_{M,1}$ $\displaystyle=$ $\displaystyle
Tr[\widehat{W}_{\nu\lambda}\widehat{W}^{\lambda\sigma}]\times[(D_{\sigma}\Phi)^{\dagger}(D^{\nu}\Phi)],$
(16) $\displaystyle O_{M,2}$ $\displaystyle=$
$\displaystyle[B_{\nu\lambda}B^{\nu\lambda}]\times[(D_{\sigma}\Phi)^{\dagger}(D^{\sigma}\Phi)],$
(17) $\displaystyle O_{M,3}$ $\displaystyle=$
$\displaystyle[B_{\nu\lambda}B^{\lambda\sigma}]\times[(D_{\sigma}\Phi)^{\dagger}(D^{\nu}\Phi)],$
(18) $\displaystyle O_{M,4}$ $\displaystyle=$
$\displaystyle[(D_{\nu}\Phi)^{\dagger}\widehat{W}_{\sigma\lambda}(D^{\nu}\Phi)]\times
B^{\sigma\lambda},$ (19) $\displaystyle O_{M,5}$ $\displaystyle=$
$\displaystyle[(D_{\nu}\Phi)^{\dagger}\widehat{W}_{\sigma\lambda}(D^{\lambda}\Phi)]\times
B^{\sigma\nu}+h.c.,$ (20) $\displaystyle O_{M,7}$ $\displaystyle=$
$\displaystyle[(D_{\nu}\Phi)^{\dagger}\widehat{W}_{\sigma\lambda}\widehat{W}^{\sigma\nu}(D^{\lambda}\Phi)].$
(21)

In above equations, the subscripts $S$, $M$, $T$ refer to scalar (or
longitudinal), $M$ mixed field and $T$ transversal. In these equations, $\Phi$
is the Higgs doublet,
$D_{\mu}\Phi=(\partial_{\mu}+igW^{j}_{\mu}\frac{\sigma^{j}}{2}+\frac{i}{2}g^{\prime}B_{\mu})\Phi$
is the covariant derivatives of the Higgs field and finally Pauli matrices are
denoted by $\sigma^{j}(j=1,2,3)$. Here, $B_{\mu\nu}$ and
$\widehat{W}_{\mu\nu}\equiv\sum_{i}\frac{1}{2}\sigma^{i}W^{i}$ represent the
gauge field strength tensors.

As mentioned above, we aimed at the dim-8 aQGC operators. A list of quartic
vertices altered with the dim-8 operators is given in Table I. In this study,
the evaluated process is sensitive to T-type operators. Because of this, we
only consider the $f_{T,j}/\Lambda^{4}$ parameters with $j=0,1,2,5,6,7,8,9$.
Obtained sensitivities on $f_{T,8}/\Lambda^{4}$ and $f_{T,9}/\Lambda^{4}$
parameters are important due to the relation on the electroweak neutral bosons
only. On the other hand, the analytical expressions of the anomalous
$Z\gamma\gamma\gamma$ coupling and related anomalous parameters for the
process $e^{-}\gamma\to e^{-}\gamma\gamma$ are given in Eqs. (22)-(25)
PRD104-2021 ; Ji-arXiv:2204.08195 .

$\displaystyle
V_{Z\gamma\gamma\gamma,1}=F^{\mu\nu}F_{\mu\nu}F^{\alpha\beta}Z_{\alpha\beta},$
(22) $\displaystyle
V_{Z\gamma\gamma\gamma,2}=F^{\mu\nu}F_{\nu\alpha}F^{\alpha\beta}Z_{\beta\mu}.$
(23)

Here, $Z^{\mu\nu}=\partial^{\mu}Z^{\nu}-\partial^{\nu}Z^{\mu}$. The related
coefficients are given in the following equations.

$\displaystyle\alpha_{Z\gamma\gamma\gamma,1}=\frac{c_{W}^{3}s_{W}}{\Lambda^{4}}(f_{T,5}+f_{T,6}-4f_{T,8})+\frac{c_{W}s_{W}^{3}}{\Lambda^{4}}(f_{T,0}+f_{T,1}-f_{T,5}-f_{T,6}),$
(24)

$\displaystyle\alpha_{Z\gamma\gamma\gamma,2}=\frac{c_{W}^{3}s_{W}}{\Lambda^{4}}(f_{T,7}-4f_{T,9})+\frac{c_{W}s_{W}^{3}}{\Lambda^{4}}(f_{T,2}-f_{T,7}).$
(25)

H

Table 1: The aQGCs related with dim-8 operators are represented with $\bullet$
Degrande .

| $WWWW$ | $WWZZ$ | $ZZZZ$ | $WW\gamma Z$ | $WW\gamma\gamma$ | $ZZZ\gamma$ | $ZZ\gamma\gamma$ | $Z\gamma\gamma\gamma$ | $\gamma\gamma\gamma\gamma$
---|---|---|---|---|---|---|---|---|---
$O_{S0}$, $O_{S1}$ | $\bullet$ | $\bullet$ | $\bullet$ |  |  |  |  |  |
$O_{M0}$, $O_{M1}$, $O_{M7}$ | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$ |  |
$O_{M2}$, $O_{M3}$, $O_{M4}$, $O_{M5}$ |  | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$ |  |
$O_{T0}$, $O_{T1}$, $O_{T2}$ | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$
$O_{T5}$, $O_{T6}$, $O_{T7}$ |  | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$
$O_{T8}$, $O_{T9}$ |  |  | $\bullet$ |  |  | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$

##  III Analysis for the aQGC via the process $e^{-}\gamma\to
e^{-}\gamma\gamma$

Linear colliders offer a powerful means to explore physics beyond the Standard
Model by utilizing $\gamma\gamma$ and $e^{-}\gamma$ interactions. In these
colliders, emitted photons from incoming electrons undergo minimal scattering
angles with the beam pipe, resulting in almost real, low-virtuality photons.
The Weizsacker-Williams approximation, also known as the Equivalent Photon
Approximation (EPA), plays a pivotal role in phenomenological studies,
allowing researchers to estimate cross sections for processes like
$e^{-}\gamma\to X$ by studying the primary $e^{-}e^{+}\to e^{-}\gamma e^{+}\to
Xe^{+}$ interaction. This ”X” represents particles observed in the final
state, and these interactions benefit from exceptionally clean experimental
conditions due to the collision of elementary particles (not composed of
smaller constituents), free from hadronic activities. With the emission of an
elastic photon, incoming charged particles, be they electrons or protons,
scatter at small angles and evade detection by central detectors. This creates
a distinctive missing energy signature known as a forward large-rapidity gap
in the forward region of the central detector. Using forward detectors with
well-synchronized central detectors is highly effective in mitigating
background. It’s worth noting that the program at CLIC has embraced forward
physics and incorporated extra detectors located at varying distances from the
interaction point, enhancing their capabilitiesCLIC . In the context of this
study, the EPA, commonly referred to as the Weizsacker-Williams Approximation
(WWA)WWA1 ; WWA2 , serves as the primary theoretical tool for estimating
sensitivity in the total cross-section of the process $e^{-}\gamma\to
e^{-}\gamma\gamma$ to analyze the potential of CLIC-based electron-photon
colliders to probe the anomalous $Z\gamma\gamma\gamma$ and
$\gamma\gamma\gamma\gamma$ quartic vertices in the stage-3 scenario with
electron polarizations of $P_{e^{-}}=-80\%$, $0\%$, $80\%$.

Here, the incoming photon from the positron beam is taken under
Weizsäcker–Williams approximation (WWA). The following equation presents the
photon’s spectrum emitted by a positron beam. Budnev ; Chen2 .

$\displaystyle f_{\gamma}(x)=\frac{\alpha}{\pi
E_{e}}\\{[\frac{1-x+x^{2}/2}{x}]log(\frac{Q_{max}^{2}}{Q_{min}^{2}})-\frac{m_{e}^{2}x}{Q_{min}^{2}}$
(26)
$\displaystyle(1-\frac{Q_{min}^{2}}{Q_{max}^{2}})-\frac{1}{x}[1-\frac{x}{2}]^{2}log(\frac{x^{2}E_{e}^{2}+Q_{max}^{2}}{x^{2}E_{e}^{2}+Q_{min}^{2}})\\}$

Here $x=E_{\gamma}/E_{e}$ and $Q_{max}^{2}$ is the photon’s maximum
virtuality. On the other hand, the expression of $Q_{min}^{2}$ is given as:

$\displaystyle Q_{min}^{2}=\frac{m_{e}^{2}x^{2}}{1-x}.$ (28)

Using the methodology, the cross-section of the $e^{-}\gamma\to
e^{-}\gamma\gamma$ process at CLIC can be calculated as follows.

$\displaystyle\sigma=\int f_{\gamma}(x)d\hat{\sigma}dE_{1}.$ (29)

Possible Feynman diagrams of the process $e^{-}\gamma\to e^{-}\gamma\gamma$
are shown in Fig.1. Here, the black dots show the effective vertices. In the
analysis, we aimed at the cross-sections to constrain the anomalous
$\frac{f_{T,j}}{\Lambda^{4}}$ parameters related to $Z\gamma\gamma\gamma$ and
$\gamma\gamma\gamma\gamma$ quartic vertices. The total cross-section of the
process $e^{-}\gamma\to e^{-}\gamma\gamma$, including anomalous interaction,
is composed as the summation of SM, purely new physics contribution, and the
interference part. This is given as follows.

$\displaystyle\sigma_{Tot}\Biggl{(}\sqrt{s},\frac{f_{T,j}}{\Lambda^{4}}\Biggr{)}$
$\displaystyle=$
$\displaystyle\sigma_{SM}(\sqrt{s})+\sigma_{(INT)}\Biggl{(}\sqrt{s},\frac{f_{T,j}}{\Lambda^{4}}\Biggr{)}$
$\displaystyle+$
$\displaystyle\sigma_{NP}\Biggl{(}\sqrt{s},\frac{f^{2}_{T,j}}{\Lambda^{8}}\Biggr{)},$
$\displaystyle\hskip 8.53581ptj=0-2,5-9,$

In the above equation, $\sigma_{SM}$ is the SM cross-section. On the other
hand, $\sigma_{(INT)}$ consists of the contributions from the interference
between SM and the EFT operators. Lastly, $\sigma_{NP}$ is the cross-section
purely contributed by the EFT operators. During the analysis, every
coefficient is taken to zero at a time.

To evaluate the total cross-section
$\sigma_{Tot}\biggl{(}\sqrt{s},\frac{f_{T,j}}{\Lambda^{4}}\biggr{)}$ of the
process $e^{-}\gamma\to e^{-}\gamma\gamma$, we used the
MadGraph5_aMC@NLOMadGraph . The operators described in Equations (7)-(14) were
embedded into MadGraph5_aMC@NLO using the Feynrules packageAAlloul , which
served as a Universal FeynRules Output (UFO) moduleCDegrande .

In the first step, we evaluate the SM background and the signals cross-
sections for each anomalous parameter at a time with the no-cut situation to
determine the optimized kinematic cuts for the next step. In Fig.2, we give
the transverse momentum of the final state photons $p_{T}^{\gamma}$. It must
be mentioned that the given transverse momentum is the scalar sum of all
photons. Here, $p_{T}^{\gamma}$ cut is very decisive, and a chosen value of
350 GeV is ideal for the separation area for the signals and SM background.
The rest of the selected cuts came with the default values of the
MadGraph5_aMC@NLO. $p_{T}^{l}$ and $\eta^{\gamma}$ are the transverse momentum
of the final state leptons and the pseudorapidities of the photons,
respectively. We consider $p_{T}^{l}>10$ GeV and $\eta^{\gamma}<2.5$ with
labeled as Cut-1. Next, we applied an angular separations $(\Delta
R=((\Delta\phi)^{2}+(\Delta\eta)^{2})^{1/2}$ for the final state charged
lepton and photons that are $\Delta R(\gamma,\gamma)>0.4$, $\Delta
R(\gamma,l)>0.4$ which is labeled as Cut-2. The final state photons of the
process $e^{-}\gamma\to e^{-}\gamma\gamma$ are useful to separate the signal
and SM background events because, in the large values of $p_{T}^{\gamma}$,
high dimensional operators can affect the transverse momentum of the photon.
However, $p_{T}^{\gamma}>350$ GeV is applied for the final state photons with
labeled Cut-3.

H  Kinematic cuts Cut-1 $p^{l}_{T}>10$ GeV , $|\eta^{\gamma}|<2.5$ Cut-2
$\Delta R(\gamma,\gamma)>0.4$ , $\Delta R(\gamma,l)>0.4$ Cut-3
$p^{\gamma}_{T}>350$ GeV

Table 2: Selected cuts for the process $e^{-}\gamma\to e^{-}\gamma\gamma$.

In Table III, a cut flow chart is given to see the effect of the selected
kinematic cuts on the events for the signals and the SM background step-by-
step. In table, we give the $f_{T,0}/\Lambda^{4}$ $f_{T,2}/\Lambda^{4}$,
$f_{T,5}/\Lambda^{4}$, $f_{T,7}/\Lambda^{4}$, $f_{T,8}/\Lambda^{4}$ for
signals and the SM background. All coupling values are taken 1 TeV-4 one at a
time. In the table, cut-0 refers to the default kinematic cuts to arrange the
singularities and divergences in the phase space. After applying the selected
cuts, we can see that the SM background events dramatically decrease compared
with the signals. Also, we give the total cross-section as a function of
anomalous parameters $f_{Tj}/\Lambda^{4}$ in Fig.3-5 to compare cross-sections
of signals each other for the different polarizations of electron beams
$P_{e^{-}}=-80\%,0\%,80\%$ after applying cuts given in Table-III.

Another critical topic is the systematic uncertainties while doing the
analysis. We have obtained the sensitivities of anomalous parameters
$f_{Tj}/\Lambda^{4}$ under the systematic uncertainties of $0\%$, $3\%$, and
$5\%$ at CLIC. Possible sources of the systematic uncertainties are the jet-
photon misidentification, integrated luminosities, and photon efficiencies.
These are listed in detail in Table II, given in the reference PLB478-2000 .
Other phenomenological studies also used the same systematic uncertainty
values to obtain the sensitivities PLB478-2000 ; PRD98-2018 ;
PRD98-015017-2018 . In the next section, we will take our analysis one step
further and obtain the sensitivities on anomalous parameters of
$\frac{f_{T,j}}{\Lambda^{4}}$.

H  Unpolarized electron beams Kinematic Cuts $f_{T0}/\Lambda^{4}$ = 1 TeV-4
$f_{T2}/\Lambda^{4}$ $f_{T5}/\Lambda^{4}$ $f_{T7}/\Lambda^{4}$
$f_{T8}/\Lambda^{4}$ Standard Model Cut-0 840 600 8100 2300 293000 530 Cut-1
840 600 8100 2300 293000 530 Cut-2 840 600 8100 2300 293000 530 Cut-3 287 74
6799 1588 263424 12

Table 3: Number of events for the process $e^{-}\gamma\to e^{-}\gamma\gamma$
for various couplings and SM background for every step of selected kinematic
cuts given in Table II for the unpolarized electron beams.

##  IV Expected sensitivity on the anomalous $\frac{f_{T,j}}{\Lambda^{4}}$
parameters at the CLIC

The chi-square method directly quantifies the parameters
$f_{T,j}/\Lambda^{4}$, $j=0-2,5-9$ of new physics that is related to anomalous
$Z\gamma\gamma\gamma$ and $\gamma\gamma\gamma\gamma$ couplings.

$\chi^{2}(f_{T,j}/\Lambda^{4})=\Biggl{(}\frac{\sigma_{SM}(\sqrt{s})-\sigma_{Total}(\sqrt{s},f_{T,j}/\Lambda^{4})}{\sigma_{SM}(\sqrt{s})\sqrt{(\delta_{st})^{2}+(\delta_{sys})^{2}}}\Biggr{)}^{2},$
(31)

Here, $\sigma_{Total}(\sqrt{s}f_{T,j}/\Lambda^{4})$ is the total cross section
which is contributed by the anomalous couplings and the SM part. Besides,
$\sigma_{SM}(\sqrt{s})$ is the SM cross-section. On the other hand,
$\delta_{sys}$ and $\delta_{st}=\frac{1}{\sqrt{N_{SM}}}$ are the systematic
and statistical error, respectively. In addition, $N_{SM}={\cal
L}\times\sigma_{SM}$ is the number of events that ${\cal L}$ is the integrated
luminosity.

H  $P_{e^{-}}$ $0\%$ $-80\%$ $80\%$ Couplings (TeV-4) ${\cal L}=5$ ab-1 ${\cal
L}=4$ ab-1 ${\cal L}=1$ ab-1 $\delta=0\%$ $[-0.56;0.62]\times 10^{-1}$
$[-4.92;4.48]\times 10^{-2}$ $[-1.93;1.90]\times 10^{-1}$ $f_{T0}/\Lambda^{4}$
$\delta=3\%$ $[-0.63;0.69]\times 10^{-1}$ $[-5.35;4.92]\times 10^{-2}$
$[-1.98;1.95]\times 10^{-1}$ $\ \,\delta=5\%$ $[-0.72;0.78]\times 10^{-1}$
$[-5.93;5.50]\times 10^{-2}$ $[-2.06;2.03]\times 10^{-1}$ $\delta=0\%$
$[-0.56;0.62]\times 10^{-1}$ $[-4.92;4.48]\times 10^{-2}$ $[-1.93;1.90]\times
10^{-1}$ $f_{T1}/\Lambda^{4}$ $\delta=3\%$ $[-0.63;0.69]\times 10^{-1}$
$[-5.35;4.92]\times 10^{-2}$ $[-1.98;1.95]\times 10^{-1}$ $\ \,\delta=5\%$
$[-0.72;0.78]\times 10^{-1}$ $[-5.93;5.50]\times 10^{-2}$ $[-2.06;2.03]\times
10^{-1}$ $\delta=0\%$ $[-0.98;1.56]\times 10^{-1}$ $[-0.94;1.03]\times
10^{-1}$ $[-3.99;3.92]\times 10^{-1}$ $f_{T2}/\Lambda^{4}$ $\delta=3\%$
$[-1.11;1.70]\times 10^{-1}$ $[-1.03;1.12]\times 10^{-1}$ $[-4.10;4.02]\times
10^{-1}$ $\ \,\delta=5\%$ $[-1.29;1.88]\times 10^{-1}$ $[-1.15;1.24]\times
10^{-1}$ $[-4.26;4.18]\times 10^{-1}$ $\delta=0\%$ $[-1.47;0.97]\times
10^{-2}$ $[-1.31;0.84]\times 10^{-2}$ $[-2.57;2.33]\times 10^{-2}$
$f_{T5}/\Lambda^{4}$ $\delta=3\%$ $[-1.60;1.10]\times 10^{-2}$
$[-1.19;1.11]\times 10^{-2}$ $[-2.63;2.29]\times 10^{-2}$ $\ \,\delta=5\%$
$[-1.77;1.27]\times 10^{-2}$ $[-1.32;1.24]\times 10^{-2}$ $[-2.74;2.50]\times
10^{-2}$ $\delta=0\%$ $[-1.47;0.97]\times 10^{-2}$ $[-1.31;0.84]\times
10^{-2}$ $[-2.57;2.33]\times 10^{-2}$ $f_{T6}/\Lambda^{4}$ $\delta=3\%$
$[-1.60;1.10]\times 10^{-2}$ $[-1.19;1.11]\times 10^{-2}$ $[-2.63;2.39]\times
10^{-2}$ $\ \,\delta=5\%$ $[-1.77;1.27]\times 10^{-2}$ $[-1.32;1.24]\times
10^{-2}$ $[-2.74;2.50]\times 10^{-2}$ $\delta=0\%$ $[-0.37;0.17]\times
10^{-1}$ $[-0.28;0.16]\times 10^{-1}$ $[-0.55;0.46]\times 10^{-1}$
$f_{T7}/\Lambda^{4}$ $\delta=3\%$ $[-0.40;0.20]\times 10^{-1}$
$[-0.30;0.19]\times 10^{-1}$ $[-0.56;0.48]\times 10^{-1}$ $\ \,\delta=5\%$
$[-0.44;0.23]\times 10^{-1}$ $[-0.33;0.21]\times 10^{-1}$ $[-0.58;0.50]\times
10^{-1}$ $\delta=0\%$ $[-1.79;2.05]\times 10^{-3}$ $[-2.88;2.49]\times
10^{-3}$ $[-2.67;2.15]\times 10^{-3}$ $f_{T8}/\Lambda^{4}$ $\delta=3\%$
$[-2.01;2.27]\times 10^{-3}$ $[-3.13;2.74]\times 10^{-3}$ $[-2.74;2.21]\times
10^{-3}$ $\ \,\delta=5\%$ $[-2.28;2.55]\times 10^{-3}$ $[-3.47;3.08]\times
10^{-3}$ $[-2.84;2.31]\times 10^{-3}$ $\delta=0\%$ $[-0.50;0.32]\times
10^{-2}$ $[-0.59;0.51]\times 10^{-2}$ $[-0.51;0.49]\times 10^{-2}$
$f_{T9}/\Lambda^{4}$ $\delta=3\%$ $[-0.54;0.36]\times 10^{-2}$
$[-0.65;0.56]\times 10^{-2}$ $[-0.53;0.50]\times 10^{-2}$ $\ \,\delta=5\%$
$[-0.60;0.42]\times 10^{-2}$ $[-0.72;0.63]\times 10^{-2}$ $[-0.55;0.52]\times
10^{-2}$

Table 4: Sensitivities at $95\%$ C.L. on the anomalous $Z\gamma\gamma\gamma$
and $\gamma\gamma\gamma\gamma$ couplings via the process $e^{-}\gamma\to
e^{-}\gamma\gamma$ with the electron polarizations of
$P_{e^{-}}=0\%,-80\%,80\%$ under the systematic uncertainties of
$\delta_{sys}=0\%,3\%,5\%$ are represented.

H  Linear Linear+Quadratic Couplings Experimental Bound Systematic Errors Our
Projection Our Projection $\delta=0\%$ $[-1.11;1.11]\times 10^{-1}$
$[-0.56;0.62]\times 10^{-1}$ $f_{T0}/\Lambda^{4}$ $[-5.70;5.46]$ $\delta=3\%$
$[-1.37;1.37]\times 10^{-1}$ $[-0.63;0.69]\times 10^{-1}$ $\delta=5\%$
$[-1.75;1.75]\times 10^{-1}$ $[-0.72;0.78]\times 10^{-1}$ $\delta=0\%$
$[-1.11;1.11]\times 10^{-1}$ $[-0.56;0.62]\times 10^{-1}$ $f_{T1}/\Lambda^{4}$
$[-5.70;5.46]$ $\delta=3\%$ $[-1.37;1.37]\times 10^{-1}$ $[-0.63;0.69]\times
10^{-1}$ $\delta=5\%$ $[-1.75;1.75]\times 10^{-1}$ $[-0.72;0.78]\times
10^{-1}$ $\delta=0\%$ $[-2.58;2.58]\times 10^{-1}$ $[-0.98;1.56]\times
10^{-1}$ $f_{T2}/\Lambda^{4}$ $[-11.40;10.90]$ $\delta=3\%$
$[-3.20;3.20]\times 10^{-1}$ $[-1.11;1.70]\times 10^{-1}$ $\delta=5\%$
$[-4.09;4.09]\times 10^{-1}$ $[-1.29;1.88]\times 10^{-1}$ $\delta=0\%$
$[-2.88;2.88]\times 10^{-2}$ $[-1.47;0.97]\times 10^{-2}$ $f_{T5}/\Lambda^{4}$
$[-2.92;2.92]$ $\delta=3\%$ $[-3.57;3.57]\times 10^{-2}$ $[-1.60;1.10]\times
10^{-2}$ $\delta=5\%$ $[-4.54;4.54]\times 10^{-2}$ $[-1.77;1.27]\times
10^{-2}$ $\delta=0\%$ $[-2.88;2.88]\times 10^{-2}$ $[-1.47;0.97]\times
10^{-2}$ $f_{T6}/\Lambda^{4}$ $[-3.80;3.88]$ $\delta=3\%$ $[-3.57;3.57]\times
10^{-2}$ $[-1.60;1.10]\times 10^{-2}$ $\delta=5\%$ $[-4.54;4.54]\times
10^{-2}$ $[-1.77;1.27]\times 10^{-2}$ $\delta=0\%$ $[-0.70;0.70]\times
10^{-1}$ $[-0.37;0.17]\times 10^{-1}$ $f_{T7}/\Lambda^{4}$ $[-7.88;7.72]$
$\delta=3\%$ $[-0.87;0.87]\times 10^{-1}$ $[-0.40;0.20]\times 10^{-1}$
$\delta=5\%$ $[-1.11;1.11]\times 10^{-1}$ $[-0.44;0.23]\times 10^{-1}$
$\delta=0\%$ $[-1.40;1.40]\times 10^{-2}$ $[-1.79;2.05]\times 10^{-3}$
$f_{T8}/\Lambda^{4}$ $[-1.06;1.10]$ $\delta=3\%$ $[-1.73;1.73]\times 10^{-2}$
$[-2.01;2.27]\times 10^{-3}$ $\delta=5\%$ $[-2.21;2.21]\times 10^{-2}$
$[-2.28;2.55]\times 10^{-3}$ $\delta=0\%$ $[-0.92;0.92]\times 10^{-2}$
$[-0.50;0.32]\times 10^{-2}$ $f_{T9}/\Lambda^{4}$ $[-1.82;1.82]$ $\delta=3\%$
$[-1.13;1.13]\times 10^{-2}$ $[-0.54;0.36]\times 10^{-2}$ $\delta=5\%$
$[-1.44;1.44]\times 10^{-2}$ $[-0.60;0.42]\times 10^{-2}$

Table 5: Sensitivities at $95\%$ C.L. on the anomalous
$\gamma\gamma\gamma\gamma$ and $Z\gamma\gamma\gamma$ couplings via the process
$e^{-}\gamma\to e^{-}\gamma\gamma$ for linear and linear+quadratic terms with
the $P_{e^{-}}=0\%$ under the systematic uncertainties of
$\delta_{sys}=0\%,3\%,$ and $5\%$ are represented.

The anomalous parameters $f_{T,j}/\Lambda^{4}$ are obtained at the $95\%$ C.L.
using the cross-sections of the process $e^{-}\gamma\to e^{-}\gamma\gamma$
after the selected cuts given in Table II for each coupling at a time. The
process are performed in $\sqrt{s}=3$ TeV option with an integrated
luminosities of ${\cal L}=1$ ${\rm ab^{-1}}$ ($P_{e^{-}}=80\%$), ${\cal L}=4$
${\rm ab^{-1}}$ ($P_{e^{-}}=-80\%$) and ${\cal L}=5$ ${\rm ab^{-1}}$
($P_{e^{-}}=0\%$) under the systematic uncertainties of
$\delta_{sys}=0\%,3\%,5\%$ at CLIC collider.

Figs. 3-5 shows the variation of the anomalous couplings to the total cross-
section after applying the selected cuts in Table II for different electron
polarization options of $P_{e^{-}}=-80\%$, $P_{e^{-}}=0\%$ and
$P_{e^{-}}=80\%$, respectively. In those figures, the dim-8 operators have
strong dependencies and highly affect the cross-section to increase with its
value.

Sensitivities on anomalous $Z\gamma\gamma\gamma$ and
$\gamma\gamma\gamma\gamma$ couplings at $95\%$ C.L. with the process
$e^{-}\gamma\to e^{-}\gamma\gamma$ at the CLIC are given in Table IV. The
sensitivities are obtained for different electron polarization options
$P_{e^{-}}=-80\%$, $0\%$, $80\%$ and under various systematic uncertainties of
$\delta_{sys}=0\%,3\%,5\%$. Here can be seen, $f_{T,5}/\Lambda^{4}$,
$f_{T,8}/\Lambda^{4}$, and $f_{T,9}/\Lambda^{4}$ couplings have the
restrictive sensitivities of $[-1.31;0.84]\times 10^{-2}\hskip 2.84526pt{\rm
TeV^{-4}}$ , $[-1.79;2.05]\times 10^{-3}\hskip 2.84526pt{\rm TeV^{-4}}$ and
$[-0.50;0.32]\times 10^{-2}\hskip 2.84526pt{\rm TeV^{-4}}$, respectively.

In Figs. 6-13, we give the comparison of the anomalous $f_{T,j}/\Lambda^{4}$
parameters with the experimental results in Ref. JHEP10-2021  via the process
$pp\to Z\gamma\gamma\to l^{+}l^{-}\gamma\gamma$ at $\sqrt{s}=13$ TeV and
integrated luminosity of $\cal L=$ 137 fb-1 that is reported by the CMS
Collaboration. In those figures, we consider the electron polarizations of
$P_{e^{-}}=80\%$ with the integrated luminosities of ${\cal L}=0.1,0.5$ and
$1$ $\rm ab^{-1}$. Similarly, $P_{e^{-}}=-80\%$ polarizations with the
integrated luminosities of ${\cal L}=0.1,1,4$ $\rm ab^{-1}$ and ${\cal
L}=0.1,1,5$ $\rm ab^{-1}$ for unpolarized electron beam are also taken into
account. Sensitivities on $f_{T,0}/\Lambda^{4}$, $f_{T,1}/\Lambda^{4}$,
$f_{T,2}/\Lambda^{4}$, $f_{T,5}/\Lambda^{4}$, $f_{T,6}/\Lambda^{4}$ and
$f_{T,7}/\Lambda^{4}$ couplings at $P_{e^{-}}=-80\%$ are more stringent than
the obtained for the other options $P_{e^{-}}=0\%$ and $80\%$. On the other
hand, $f_{T,8}/\Lambda^{4}$ and $f_{T,9}/\Lambda^{4}$ couplings have its best
sensitivities at $P_{e^{-}}=0\%$. Also, in this study, we have extended our
results with the sensitivities obtained via linear term (interference of dim8
operators with SM), given in Table V, and compared them with the full effects
(linear+quadratic) and the experimental results. Obtained sensitivities via
linear terms are worse than the full effects as expected. Here, the quadratic
terms dominated the interference term. On the other hand, linear terms
improved the experimental results by up to 200 times, as shown in Table V.

##  V CONCLUSIONS

In this study, a specific class of interactions aQGCs are handled within the
EFT framework, which is described by the non-Abelian gauge structure of the
SM. In this context, the study may lead to an opportunity to test the validity
of SM and give important clues to the presence of new physics. Additionally,
lepton colliders are very suitable due to their compact design and cleaner
background environment without any hadronic activity due to the nature of
lepton collision compared with the LHC. These are the main motivations for
performing this simulation in CLIC. Furthermore, the CLIC program gives high
CoM energy up to a 3 TeV in the stage-3 scenario with high integrated
luminosity. Also, we consider the electron polarization options of CLIC to see
the effect on aQGC couplings.

With these motivations, we evaluate the process $e^{-}\gamma\to
e^{-}\gamma\gamma$ at the CLIC to probe the dim-8 anomalous
$Z\gamma\gamma\gamma$ and $\gamma\gamma\gamma\gamma$ couplings. Obtained
sensitivities on dim-8 parameters $f_{T,j}/\Lambda^{4}$ that are between 2-200
times stronger than the experimental limits given in the Ref.JHEP10-2021 . On
the other hand, anomalous $Z\gamma\gamma\gamma$ and $ZZ\gamma\gamma$ couplings
are studied at CLIC via the process $e^{-}e^{+}\to
Z\gamma\gamma$arXiv:2112.03948 . Obtained results in this study via the
process $e^{-}\gamma\to e^{-}\gamma\gamma$ improved the sensitivities of aQGCs
by a factor between 2-12 times compared with the Ref.arXiv:2112.03948 .
Stringent bounds on the anomalous $f_{T,5,8,9}/\Lambda^{4}$ couplings are
$\frac{f_{T5}}{\Lambda^{4}}=[-1.31;0.84]\times 10^{-2}\hskip 2.84526pt{\rm
TeV^{-4}}$ with ${\cal L}=4\hskip 2.27621pt\rm ab^{-1}$, $P_{e^{-}}=-80\%$ and
$\delta_{sys}=0\%$, $\frac{f_{T8}}{\Lambda^{4}}=[-1.79;2.05]\times
10^{-3}\hskip 2.84526pt{\rm TeV^{-4}}$ and
$\frac{f_{T9}}{\Lambda^{4}}=[-0.50;0.32]\times 10^{-2}\hskip 2.84526pt{\rm
TeV^{-4}}$ with ${\cal L}=5\hskip 2.27621pt\rm ab^{-1}$, $P_{e^{-}}=0\%$ and
$\delta_{sys}=0\%$. Consequently, $O_{T,8}$ and $O_{T,9}$ operators have the
optimal sensitivities related on anomalous $Z\gamma\gamma\gamma$ and
$\gamma\gamma\gamma\gamma$ couplings for the process $e^{-}\gamma\to
e^{-}\gamma\gamma$. Also, we give the obtained sensitivities via linear terms
and compare the full contributions (linear+quadratic) and the experimental
results. Both linear and linear+quadratic contributions improved the
experimental results remarkably.

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Figure 1:  Diagrams for the process $e^{-}\gamma\to e^{-}\gamma\gamma$
involving the anomalous $Z\gamma\gamma\gamma$ and $\gamma\gamma\gamma\gamma$
couplings. New physics contributions are shown by a black circle.

Figure 2:  The number of events as a function of the transverse momentum of
the final state photon $p^{\gamma}_{T}$ for the process $e^{-}\gamma\to
e^{-}\gamma\gamma$ and SM background at $\sqrt{s}=3$ TeV and $P_{e^{-}}=0\%$.

Figure 3:  Cross-sections of the process $e^{-}\gamma\to e^{-}\gamma\gamma$ in
terms of the anomalous parameters $f_{T,j}/\Lambda^{4}$ for the polarized
beams $P_{e^{-}}=-80\%$.

Figure 4:  Same as in Fig. 3, but for $P_{e^{-}}=0\%$.

Figure 5:  Same as in Fig. 3, but for $P_{e^{-}}=80\%$.

Figure 6:  Comparison of experimental limits given in Ref.JHEP10-2021  and
obtained sensitivity on $f_{T,0}/\Lambda^{4}$ for the integrated luminosities
of ${\cal L}=100,500,1000,4000,5000\hskip 2.27621ptfb^{-1}$. We take into
account $P_{e^{-}}=-80\%,0\%,80\%$. terms.

Figure 7:  Same as in Fig. 6, but for $f_{T,1}/\Lambda^{4}$.

Figure 8:  Same as in Fig. 6, but for $f_{T,2}/\Lambda^{4}$.

Figure 9:  Same as in Fig. 6, but for $f_{T,5}/\Lambda^{4}$.

Figure 10:  Same as in Fig. 6, but for $f_{T,6}/\Lambda^{4}$.

Figure 11:  Same as in Fig. 6, but for $f_{T,7}/\Lambda^{4}$.

Figure 12:  Same as in Fig. 6, but for $f_{T,8}/\Lambda^{4}$.

Figure 13:  Same as in Fig. 6, but for $f_{T,9}/\Lambda^{4}$.