# Probe of a Randall-Sundrum-like model from muon pair production at high
energy muon collider

S.C. İnan
Department of Physics, Sivas Cumhuriyet University, 58140, Sivas, Turkey
and
A.V. Kisselev
A.A. Logunov Institute for High Energy Physics, NRC “Kurchatov Institute”,
142281, Protvino, Russian Federation Electronic address:
<EMAIL_ADDRESS>address<EMAIL_ADDRESS>
###### Abstract

We have examined $\mu^{+}\mu^{-}\rightarrow 2(\mu^{+}\mu^{-})$ and
$\mu^{+}\mu^{-}\rightarrow\mu^{+}\mu^{-}$ collisions at future high energy
muon colliders in the framework of the Randall-Sundrum-like model with a small
curvature of space-time. The collision energies of 3 TeV, 14 TeV and, 100 TeV
are addressed. Both differential and total cross sections are calculated, and
excluded bounds on a 5-dimensional gravity scale are obtained depending on
collision energy and integrated luminosity of the muon collider.

##  1 Introduction

The Standard Model (SM) has been proven in a lot number of collider
experiments. Nevertheless, we are still searching for solutions for many
problems that SM cannot give a satisfactory solution. One of such problem is
the so-called hierarchy problem which means the large energy gap between the
electroweak scale and gravity scale. The most elegant answer to this
phenomenon has been given in the framework of the Randall-Sundrum (RS) model
[1] which is based on a 5D theory with one extra dimension compactified in an
orbifold $S_{1}/Z_{2}$. The main parameters of the RS model are the
compactification radius $r_{c}$ and AdS5 curvature parameter $k$ (hereinafter
referred to as the _curvature_ $k$). The model predicts Kaluza-Klein (KK)
gravitons which are heavy resonances with masses around the TeV scale. The
most stringent limits on KK graviton masses come from the LHC searches for
heavy resonances. The experimental limits depend on a ratio $k/M_{P}$, where
$M_{P}$ is the Planck mass. The CMS collaboration have excluded KK graviton
masses below 2.3 to 4.0 TeV for the diphoton final state [2]. For the dilepton
final state the CMS have excluded the RS graviton masses in the region
2.47-4.78 TeV [3]. The best lower limit of the ATLAS collaboration, 4.6 TeV,
has been obtained in searching for the diphoton final state [4].

In papers [5, 6] the Randall-Sundrum-like model with a small curvature of the
5-dimensional space-time (RSSC model) has been proposed. In particular, a
general solution for the warped metric has been obtained [7]. In contrast to
the original RS model, the RSSC model has an almost continuous graviton mass
spectrum which is similar to the spectrum of the ADD model [8]-[10], if $k\ll
M_{P}$. Thus, the above mentioned experimental bounds are not applied to the
RS scenario with a small value of $k$. A probe of the RSSC model at the LHC
can be found in [11, 12]. A detailed comparison of the RSSC model with the RS
model is given in section 2.

In the present paper we intend to examine the RSSC model through the
$\mu^{+}\mu^{-}\rightarrow 2(\mu^{+}\mu^{-})$ and
$\mu^{+}\mu^{-}\rightarrow\mu^{+}\mu^{-}$ processes at a future muon collider.
The idea of the muon collider was proposed by F. Tikhonin and G. Budker in the
late 1960’s [13, 14], and it was also discussed in the early 1980’s [15, 16].
At present, a great physical potential of the muon collider for collisions of
elementary particles at very high energies is being actively examined. Its
advantage lies in the fact that muons can be accelerated in a ring without
limitation from synchrotron radiation compared to linear or circular electron-
positron colliders [17]-[22]. For instance, the muon collider may provide a
determination of the electroweak couplings of the Higgs boson which is
significantly better than what is considered attainable at other future
colliders [23]-[29]. Interest in designing and building a muon collider is
also based on its capability of probing the physics beyond the SM. In a number
of recent papers searches for SUSY particles [30], WIMPs [31]-[33], and dark
matter [34], vector boson fusion [35], leptoquarks [36], lepton flavor
violation [37]-[39], vector-like leptons [40], heavy leptons [41, 42], and
heavy neutrinos [43], top Yukawa couplings [44], multi-boson processes [45],
and physics of the muon $(g-2)$ [46, 47]. In our recent paper we have probed
axion-like particles (ALPs) at high energy muon colliders [48]. In a number of
papers anomalous quartic [49]-[51] and triple [52, 53] gauge couplings at the
muon collider were studied. For more details on a spectacular opportunity of
the muon collider in the direct exploration of the energy frontier, see [54].

In the $\mu^{+}\mu^{-}\rightarrow 2(\mu^{+}\mu^{-})$ scattering one pair of
the outgoing muons with large transverse momenta and large dimuon invariant
mass is detected, while the other two scattered muons escape a detector. Our
goal is to obtain excluded bounds on the 5-dimensional gravity scale $M_{5}$
which can be probed at TeV and multi-TeV muon colliders. The gravity
contribution to this process comes from the subprocess $VV\rightarrow
G\rightarrow\mu^{+}\mu^{-}$, where $V$ is the $\gamma$ or $Z$ boson, $G$ is
the KK graviton, and a summation over all KK gravitons is assumed. We will
also study the $\mu^{+}\mu^{-}\rightarrow\mu^{+}\mu^{-}$ scattering, taking
into account contributions from $s$\- and $t$-channel graviton exchanges, to
derive the bounds on $M_{5}$. Note that the processes we are interested in can
be easily distinguished experimentally from each other, since they have quite
different distributions in invariant mass of the detected dimuon pair.

The paper is organized as follows. In the next section, the detailed
description of the RSSC model is presented. The production of four muons via
vector boson fusion at the muon collider is examined in section 3. The bounds
on 5-dimensional Planck scale $M_{5}$ are obtained. In section 4 we study the
$\mu^{+}\mu^{-}\rightarrow\mu^{+}\mu^{-}$ process and we calculate the values
of $M_{5}$ which can be probed at the muon collider.

##  2 Model of warped extra dimension with small curvature

In this section, we describe the RSSC model in detail and compare it with the
original RS model. The RS scenario with one extra dimension and two branes [1]
was proposed as an alternative to the ADD scenario with large flat extra
dimensions (EDs) [8]-[10]. It has the following background warped metric

$\quad ds^{2}=e^{-2\sigma(y)}\,\eta_{\mu\nu}\,dx^{\mu}\,dx^{\nu}-dy^{2}\;,$
(1)

where $\eta_{\mu\nu}$ is the Minkowski tensor with the signature $(+,-,-,-)$,
$y$ is an extra coordinate, and $\sigma(y)$ is the warp factor. The
periodicity condition $y=y+2\pi r_{c}$ is imposed, and the points
$(x_{\mu},y)$ and $(x_{\mu},-y)$ are identified. Thus, we have a model of
gravity in a slice of the AdS5 space-time compactified to the orbifold
$S^{1}\\!/Z_{2}$. This orbifold has two fixed points, $y=0$, and $y=\pi
r_{c}$. There are two branes located at these points (called Planck and TeV
brane, respectively). The SM fields are confined to the TeV brane.

The classical action of the RS model is given by [1]

$\displaystyle S$ $\displaystyle=\int\\!\\!d^{4}x\\!\\!\int_{-\pi r_{c}}^{\pi
r_{c}}\\!\\!dy\,\sqrt{G}\,(2\bar{M}_{5}^{3}\mathcal{R}-\Lambda)$
$\displaystyle+\int\\!\\!d^{4}x\sqrt{|g^{(1)}|}\,(\mathcal{L}_{1}-\Lambda_{1})+\int\\!\\!d^{4}x\sqrt{|g^{(2)}|}\,(\mathcal{L}_{2}-\Lambda_{2})\;,$
(2)

where $G_{MN}(x,y)$ is the 5-dimensional metric, with $M,N=0,1,2,3,4$,
$\mu=0,1,2,3$. The quantities

$g^{(1)}_{\mu\nu}(x)=G_{\mu\nu}(x,y=0)\;,\quad
g^{(2)}_{\mu\nu}(x)=G_{\mu\nu}(x,y=\pi r_{c})$ (3)

are induced metrics on the branes, $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ are
brane Lagrangians, $G=\det(G_{MN})$, $g^{(i)}=\det(g^{(i)}_{\mu\nu})$. The
parameter $\bar{M}_{5}$ is a _reduced_ 5-dimensional Planck scale related to
the fundamental gravity scale $M_{5}$, namely,
$\bar{M}_{5}=M_{5}/(2\pi)^{1/3}$. The parameter $\Lambda$ is a 5-dimensional
cosmological constant, and $\Lambda_{1,2}$ are brane tensions.

For the first time, the solution for $\sigma(y)$ in (1) has been obtained in
[1],

$\sigma_{\mathrm{RS}}(y)=k|y|\;,$ (4)

where $k$ is a parameter with a dimension of mass. It defines the curvature of
the 5-dimensional space-time. Later on in [7] the generalized solution for
$\sigma(y)$ was derived

$\sigma(y)=\frac{kr_{c}}{2}\left[\left|\mathrm{Arccos}\left(\cos\frac{y}{r_{c}}\right)\right|-\left|\pi-\mathrm{Arccos}\left(\cos\frac{y}{r_{c}}\right)\right|\right]+\frac{\pi\,|k|r_{c}}{2}-C\;,$
(5)

where $C$ is a $y$-independent quantity. Note that the brane tensions obey the
fine-tuning relations

$\Lambda=-24\bar{M}_{5}^{3}k^{2}\;,\quad\Lambda_{1}=-\,\Lambda_{2}=24\bar{M}_{5}^{3}k\;.$
(6)

Here $\mathrm{Arccos(z)}$ is a principal value of the multivalued inverse
trigonometric function $\arccos(z)$. Let us underline that the generalized
solution (5) (i) obeys the orbifold symmetry $y\rightarrow-y$; (ii) makes the
jumps of $\sigma^{\prime}(y)$ on both branes; (iii) has explicit symmetry with
respect to the branes. More details can be found in [7].

By taking $C=0$ in (5), we get the RS model (4), while taking $C=\pi kr_{c}$,
we come to the RS-like scenario with the small curvature of the space-time
(RSSC model, see [5]-[7]).

It is worth to remind the main features of the RSSC model in comparison with
those of the RS model. The interactions of the Kaluza-Klein (KK) gravitons
$h_{\mu\nu}^{(n)}$ with the SM fields on the TeV brane are given by the
effective Lagrangian density

$\mathcal{L}_{\mathrm{int}}=-\frac{1}{\bar{M}_{\mathrm{Pl}}}\,h_{\mu\nu}^{(0)}(x)\,T_{\alpha\beta}(x)\,\eta^{\mu\alpha}\eta^{\nu\beta}-\frac{1}{\Lambda_{\pi}}\sum_{n=1}^{\infty}h_{\mu\nu}^{(n)}(x)\,T_{\alpha\beta}(x)\,\eta^{\mu\alpha}\eta^{\nu\beta}\;,$
(7)

were $\bar{M}_{\mathrm{Pl}}=M_{\mathrm{Pl}}/\sqrt{8\pi}$ is the reduced Planck
mass, $T^{\mu\nu}(x)$ is the energy-momentum tensor of the SM fields. The
coupling constant is equal to

$\Lambda_{\pi}=\bar{M}_{5}\sqrt{\frac{\bar{M}_{5}}{k}}\;.$ (8)

The hierarchy relation looks like

$\bar{M}_{\mathrm{Pl}}^{2}=\frac{\bar{M}_{5}^{3}}{k}\left[e^{2\pi
kr_{c}}-1\right].$ (9)

To compare, in the original RS model the hierarchy relation is different,
$\bar{M}_{\mathrm{Pl}}^{2}=(\bar{M}_{5}^{3}/k)\left[1-e^{-2\pi
kr_{c}}\right]$. It is usually assumed that $k\pi r_{c}\gg 1$.

The masses of the KK gravitons in the RSSC model are proportional to the
curvature $k$ [5, 6],

$m_{n}=x_{n}k\;,\quad n=1,2,\ldots\;,$ (10)

where $x_{n}$ are zeros of the Bessel function $J_{1}(x)$. Should we take
$k\ll\bar{M}_{5}\sim 1$ TeV, the mass splitting $\Delta m$ will be very small,
$\Delta m\simeq\pi k$, and we come to an _almost continuous_ mass spectrum,
similar to the mass spectrum of the ADD model [8]. That is why, the RSSC model
can be considered as a distorted ADD model with one ED [5]. On the contrary,
in the RS model the gravitons are _heavy resonances_ with masses above one-few
TeV. For the first time heavy graviton searches were discussed in [55].

Let us underline once more that graviton mass spectra are quite different in
the RSSC and RS models. It means that all experimental bounds and
phenomenological predictions for parameters of the original RS model can not
be applied to the RS-like scheme examined in the present paper.

As was shown in a number of phenomenological papers on the RSSC model [11,
12], the cross sections weakly depend on the parameter $k$, if
$k\ll\bar{M}_{5}$. That is why, in what follows we will fix this parameter to
be $k=1$ GeV.

##  3 Production of two muon pairs in muon collisions

Let us consider the process
$\mu^{-}\mu^{+}\rightarrow\mu^{-}V_{1}V_{2}\mu^{+}\rightarrow
2(\mu^{+}\mu^{-})$ shown in Fig. 1. Our goal is to estimate a contribution of
the KK gravitons to the gauge boson fusion $VV\rightarrow l^{-}l^{+}$ in the
framework of the RSSC model described in the previous section. The amplitude
of this subprocess is defined as $M=M_{\mathrm{SM}}+M_{\mathrm{KK}}$, where
$M_{\mathrm{SM}}$ is the SM term, and $M_{\mathrm{KK}}$ is given by the sum of
$s$-channel KK gravitons

$M_{KK}=\frac{1}{2\Lambda_{\pi}^{2}}\sum_{n=1}^{\infty}\left[\bar{u}(p_{1})\Gamma_{2}^{\mu\nu}v(p_{2})\,\frac{B_{\mu\nu\alpha\beta}}{s-m_{n}^{2}+i\,m_{n}\Gamma_{n}}\,\Gamma_{1}^{\alpha\beta\rho\sigma}e_{\rho}(k_{1})e_{\sigma}(k_{2})\right].$
(11)

Figure 1: The Feynman diagrams describing contribution of the KK graviton $G$
to the collision of two vector bosons $V_{1},V_{2}=\gamma$ or $Z$, with two
outgoing charged leptons at the muon collider.

Here $k_{1},k_{2}$, $p_{1},p_{2}$ and $e_{\rho}(k_{1})$, $e_{\sigma}(k_{2})$
are, respectively, incoming photon momenta, outgoing lepton momenta and
polarization vectors of photons. $\Gamma_{n}$ is the total width of the
graviton with the mass $m_{n}$. The coherent sum in (11) is over all massive
KK modes. The Feynman rules for the KK graviton were derived in [56, 57] (see
also [58]). In particular, the $KK$–$VV$ vertex looks like

$\Gamma_{1}^{\alpha\beta\rho\sigma}=-\frac{i}{2}\,\\{[m_{V}^{2}+(k_{1}\cdot
k_{2})]\,C^{\alpha\beta\rho\sigma}+D^{\alpha\beta\rho\sigma}\\}\;,$ (12)

where

$\displaystyle C^{\alpha\beta\rho\sigma}$
$\displaystyle=\eta^{\alpha\rho}\eta^{\beta\sigma}+\eta^{\alpha\sigma}\eta^{\beta\rho}-\eta^{\alpha\beta}\eta^{\rho\sigma}\;,$
(13) $\displaystyle D^{\alpha\beta\rho\sigma}$
$\displaystyle=\eta^{\alpha\beta}k^{\sigma}_{1}k^{\rho}_{2}-(\eta^{\alpha\sigma}k^{\beta}_{1}k^{\rho}_{2}+\eta^{\alpha\rho}k^{\sigma}_{1}k^{\beta}_{2}-\eta^{\rho\sigma}k^{\alpha}_{1}k^{\beta}_{2})$
$\displaystyle-(\eta^{\beta\sigma}k^{\alpha}_{1}k^{\rho}_{2}+\eta^{\beta\rho}k^{\sigma}_{1}k^{\alpha}_{2}-\eta^{\rho\sigma}k^{\beta}_{1}k^{\alpha}_{2})\;.$
(14)

The $KK$–$l^{-}l^{+}$ vertex is defined as

$\Gamma_{2}^{\mu\nu}=-\frac{i}{8}\,[\gamma^{\mu}(p^{\nu}_{1}-p^{\nu}_{2})+\gamma^{\nu}(p^{\mu}_{1}-p^{\mu}_{2})]\;.$
(15)

Finally, $B_{\mu\nu\alpha\beta}$ in (11) is a tensor part of the KK graviton
propagator. Its explicit expression was derived in [56, 57]. We can safely
omit terms in $B_{\mu\nu\alpha\beta}$ which give zero contribution to eq.
(11). Then we find

$\displaystyle
B_{\mu\nu\alpha\beta}=\eta_{\mu\alpha}\eta_{\nu\beta}+\eta_{\mu\beta}\eta_{\nu\alpha}-\frac{2}{3}\,\eta_{\mu\nu}\eta_{\alpha\beta}\;.$
(16)

The $s$-channel contribution of the KK gravitons is equal to

$\mathcal{S}(s)=\frac{1}{\Lambda_{\pi}^{2}}\sum_{n=1}^{\infty}\frac{1}{s-m_{n}^{2}+i\,m_{n}\Gamma_{n}}\;.$
(17)

This sum has been calculated in ref. [59],

$\mathcal{S}(s)=-\frac{1}{4\bar{M}_{5}^{3}\sqrt{s}}\;\frac{\sin(2A)+i\sinh(2\varepsilon)}{\cos^{2}\\!A+\sinh^{2}\\!\varepsilon}\;,$
(18)

where

$A=\frac{\sqrt{s}}{k}\;,\quad\varepsilon=0.045\left(\frac{\sqrt{s}}{\bar{M}_{5}}\right)^{\\!\\!3}.$
(19)

Note that contrary to the ADD model with the flat metric in one ED, the series
in (17) converges. It is a consequence of nonzero value of the space-time
curvature $k$.

The squared amplitude of the subprocess $VV\rightarrow l^{-}l^{+}$ is a sum of
three terms,

$|M|^{2}=|M_{\mathrm{SM}}|^{2}+|M_{\mathrm{KK}}|^{2}+|M_{\mathrm{int}}|^{2}\;,$
(20)

where $M_{\mathrm{SM}}$ denotes the SM amplitude, while $M_{\mathrm{KK}}$ and
$M_{\mathrm{int}}$ denote pure KK graviton and interference terms. In [58] the
quantities $|M_{\mathrm{SM}}(\gamma\gamma\rightarrow l^{-}l^{+})|^{2}$,
$|M_{\mathrm{KK}}(\gamma\gamma\rightarrow l^{-}l^{+})|^{2}$, and
$|M_{\mathrm{int}}(\gamma\gamma\rightarrow l^{-}l^{+})|^{2}$ were calculated
for _massless_ leptons. The results of our calculations of the squared
amplitudes $|M(VV\rightarrow l^{-}l^{+})|^{2}$ for _nonzero_ $m_{l}$ and
$m_{V}$ ($V=\gamma,Z$) are presented in Appendix A.

The virtual KK graviton production should lead to deviations from the SM
predictions in a magnitude of the cross section. The cross section of our
process $\mu^{-}\mu^{+}\rightarrow\mu^{-}V_{1}V_{2}\mu^{+}\rightarrow
2(\mu^{+}\mu^{-})$ is defined by the formula

$d\sigma=\int\limits_{\tau_{\min}}^{\tau_{\max}}\\!\\!d\tau\\!\\!\\!\int\limits_{x_{\min}}^{x_{\max}}\\!\\!\frac{dx}{x}\,\sum_{V_{1},V_{2}=\gamma,Z_{T},Z_{L}}\\!\\!f_{V_{1}/\mu^{+}}(x,Q^{2})f_{V_{2}/\mu^{-}}(\tau/x,Q^{2})\,d\hat{\sigma}(V_{1}V_{2}\rightarrow\mu^{+}\mu^{-})\;.$
(21)

Here

$x_{\max}=1-\frac{m_{\mu}}{E_{\mu}}\;,\
\tau_{\max}=\left(1-\frac{m_{\mu}}{E_{\mu}}\right)^{\\!2},\
x_{\min}=\tau/x_{\max}\;,\ \tau_{\min}=\frac{p_{\bot}^{2}}{E_{\mu}^{2}}\;,$
(22)

and $p_{\bot}$ is the transverse momenta of the outgoing photons. The
unpolarized boson distributions inside the in unpolarized muon beam,
$f_{\gamma/{\mu^{\pm}}}(x,Q^{2})$, $f_{Z_{T}/{\mu^{\pm}}}(x,Q^{2})$, and
$f_{Z_{L}/{\mu^{\pm}}}(x,Q^{2})$ are given by [60]

$f_{\gamma/\mu^{\pm}}(x,Q^{2})=\frac{\alpha}{2\pi}\frac{1+(1-x)^{2}}{x}\ln\frac{Q^{2}}{m_{\mu}^{2}}\;,$
(23)

and [61, 62]

$\displaystyle f_{Z_{T}/\mu^{\pm}}(x,Q^{2})$
$\displaystyle=\frac{\alpha_{Z}^{\pm}}{2\pi}\frac{1+(1-x)^{2}}{x}\ln\frac{Q^{2}}{m_{Z}^{2}}\;,$
$\displaystyle f_{Z_{L}/\mu^{\pm}}(x,Q^{2})$
$\displaystyle=\frac{\alpha_{Z}^{\pm}}{\pi}\frac{(1-x)}{x}\;,$ (24)

where

$\alpha_{Z}^{\pm}=\frac{\alpha}{(\cos\theta_{W}\sin\theta_{W})^{2}}\left[(g_{V}^{\pm})^{2}+(g_{A}^{\pm})^{2}\right],$
(25)

$g_{V}^{\pm}=\pm 1/4\mp\sin^{2}\theta_{W}$, $g_{A}^{\pm}=\mp 1/4$, and
$m_{\mu}$ is the muon mass. The variable $x$ in (23), (3) is the ratio of the
boson energy and energy of the incoming muon $E_{\mu}$. Note that the $Z$
boson has different distributions for its transverse ($T$) and longitudinal
($L$) polarizations (3). Note that the mixed $\gamma$–$Z_{T}$ and
$\gamma$–$Z_{L}$ terms are present in (21) along with the $\gamma$–$\gamma$
and $Z$–$Z$ terms.

Let us underline that we examine the _exclusive_ process
$\mu^{+}\mu^{-}\rightarrow\mu^{+}\mu^{-}+\gamma\gamma$. Should one study the
_inclusive_ process $\mu^{+}\mu^{-}\rightarrow\gamma\gamma+X$, where $X$ is an
unspecified remnant, one has to use electroweak (EW) PDFs, see [63, 64] and
references therein. In other words, there is no need to accounting for EW
evolution of vector boson distributions (23)–(25).

The differential cross section of the subprocess
$V_{1}V_{2}\rightarrow\mu^{+}\mu^{-}$, where $V_{1,2}=\gamma$ or $Z$, is a sum
of helicity amplitudes squared

$\frac{d\hat{\sigma}}{d\Omega}=\frac{1}{64\pi^{2}\hat{s}}\sum_{\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4}}\\!\\!|M_{\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4}}^{V_{1}V_{2}}|^{2},$
(26)

where $\sqrt{\hat{s}}$ is a collision energy of this subprocess, and
$\lambda_{1,2}$ ($\lambda_{3,4}$) are boson (muon) helicities. In boson
distributions (23), (3) we put $Q^{2}=\hat{s}$, where
$\sqrt{\hat{s}}=2E_{\mu}\sqrt{\tau}$ is the invariant energy of the subprocess
$V_{1}V_{2}\rightarrow\mu^{+}\mu^{-}$.

As it was mentioned in [65], the scattering angle for high energy initial
muons is peaked near $\theta_{\mu}\approx 0.02^{\circ}-1.2^{\circ}$. These
very forward muons would most likely escape a muon detector away from
colliding beams. Thus, only muons produced in the boson fusion (as those shown
in Fig. 1) will be detected. We apply the cuts to the final muons, $p_{t}>50$
GeV, $|\eta|<2.5$. The main purpose of these cuts is to ensure that only two
muons are detected in the final state. As was already mentioned at the end of
section 2, we take $k=1$ GeV.

The results of our numerical calculations of the differential cross sections
for the $\mu^{+}\mu^{-}\rightarrow 2(\mu^{+}\mu^{-})$ scattering at the future
muon collider are presented in Fig. 2. The predictions for three collision
invariant energies of the muon collider are shown. As one can see, for each
energy the cross sections rise as the invariant mass of the detected muons
grows, while the SM cross sections decrease rapidly with an increase of
$m_{\mu^{+}\mu^{-}}$. We have also calculated the differential cross sections
via transverse momentum of the detected muons, see Fig. 3.

Figure 2: The differential cross sections for the process
$\mu^{+}\mu^{-}\rightarrow 2(\mu^{+}\mu^{-})$ via invariant mass of two
detected muons at the muon collider. The left, middle and right panels
correspond to the colliding energy of 3 TeV, 14 TeV, and 100 TeV. The curves
(from the top down) correspond to $\bar{M}_{5}=1$ TeV, $\bar{M}_{5}=2$ TeV,
and $\bar{M}_{5}=3$ TeV, respectively. The SM cross sections (low curves) are
also shown. Figure 3: The differential cross sections for the process
$\mu^{+}\mu^{-}\rightarrow 2(\mu^{+}\mu^{-})$ via transverse momentum of the
detected muons at the muon collider.

The total cross sections as functions of the minimal invariant mass of two
detected muons at the muon collider $m_{\mu^{+}\mu^{-},\min}$ are shown in
Fig. 4. The cross sections strongly depend on the fundamental gravity scale
$\bar{M}_{5}$. If $\bar{M}_{5}=1$ TeV, the cross section exceeds the SM one
for all three collision energies. For larger values of $\bar{M}_{5}$ the total
cross section strongly dominates over the SM cross section for $\sqrt{s}=14$
TeV and $\sqrt{s}=100$ TeV.

Figure 4: The total cross sections for the process $\mu^{+}\mu^{-}\rightarrow
2(\mu^{+}\mu^{-})$ via minimal invariant mass of two detected muons at the
muon collider $m_{\mu^{+}\mu^{-},\min}$.

All this enables us to derive the excluded bounds on the 5-dimensional reduced
Planck scale $\bar{M}_{5}$. To derive them, we apply the following formula for
the statistical significance $SS$ [66]

$SS=\sqrt{2[(S-B\,\ln(1+S/B)]}\;,$ (27)

where $S$ is the number of signal events and $B$ is the number of background
(SM) events. We define the regions $SS\leqslant 1.645$ as the regions that can
be excluded at the 95% C.L. To reduce the SM background, we used the cuts
$m_{\mu^{-}\mu^{+}}>1$ TeV, $m_{\mu^{-}\mu^{+}}>5$ TeV, and
$m_{\mu^{-}\mu^{+}}>50$ TeV for the colliding energy of 3 TeV, 14 TeV, and 100
TeV, respectively. The results are shown in Fig. 5. Our best limits for
$\sqrt{s}=3$ TeV, $14$ TeV and $100$ TeV are $\bar{M}_{5}=3.8$ TeV, 13.1 TeV,
and 106.4 TeV, respectively.

Figure 5: The excluded bounds on the reduced fundamental gravity scale
$\bar{M}_{5}$ via integrated luminosity of the muon collider for the process
$\mu^{+}\mu^{-}\rightarrow 2(\mu^{+}\mu^{-})$. The left, middle and right
panels correspond to the colliding energy of 3 TeV, 14 TeV, and 100 TeV.

##  4 Production of one muon pair in muon collisions

As was said in the previous section, we expect that in the
$\mu^{+}\mu^{-}\rightarrow 2(\mu^{+}\mu^{-})$ process only two final muons are
detected, while two scattered muons escape the detector. In the process
$\mu^{-}\mu^{+}\rightarrow\mu^{+}\mu^{-}$ two outgoing muons with high
transverse momenta are also detected. However, in such a case the invariant
mass of the dimuon system $m_{\mu^{+}\mu^{-}}$ is fixed and close to the
collision energy $\sqrt{s}$, that does not occur for the
$\mu^{-}\mu^{+}\rightarrow 2(\mu^{+}\mu^{-})$ scattering. Note that the muon
collider has the low level of beamstrahlung and synchrotron radiation compared
to linear or circular electron-positron colliders. As a result, an energy
spread in the collision is significantly reduced, and it enables an improved
energy resolution. That is why one can easily discriminate between two
mentioned above processes experimentally by measuring the invariant mass of
the detected muon pair.

The virtual KK graviton exchanges have to give a contribution to the cross
sections of the $\mu^{-}\mu^{+}\rightarrow\mu^{+}\mu^{-}$ scattering. It is
shown in Fig. 6.

Figure 6: The Feynman diagrams describing contribution of the KK graviton $G$
to the $\mu^{-}\mu^{+}\rightarrow\mu^{+}\mu^{-}$ scattering.

The analytical expressions for amplitudes squared of this collision are given
in Appendix B. Using them we have calculated the differential cross sections
for the $\mu^{+}\mu^{-}\rightarrow\mu^{+}\mu^{-}$ scattering at the muon
collider depending on transverse momentum of the final muons $p_{t}$, taking
into account gravity contribution. Our result are presented in Fig. 7 for
three values of the collision energy $\sqrt{s}$ and different values of the
reduced 5-dimensional Planck scale $\bar{M}_{5}$. As we can see, for
$\sqrt{s}=14$ TeV and $\sqrt{s}=100$ TeV, the cross section significantly
dominates the SM one, especially for large $p_{t}$. It is worth to compare
this figure with the $p_{t}$-distribution in Fig. 3. The oscillations of the
curves in Fig. 3 originate from the function $S(s)$ (18) which describes the
$s$-channel contribution of all KK gravitons, since the invariant energy of
the detected dimuon pair is not a constant. On the other hand, in the process
$\mu^{-}\mu^{+}\rightarrow\mu^{+}\mu^{-}$ the invariant energy of the dimuon
pair is fixed, and we have no oscillations.

Figure 7: The differential cross sections for the process
$\mu^{+}\mu^{-}\rightarrow\mu^{+}\mu^{-}$ via transverse momentum of the
detected muons at the muon collider. The left, middle and right panels
correspond to the colliding energy of 3 TeV, 14 TeV, and 100 TeV. The curves
(from the top down) correspond to $\bar{M}_{5}=1$ TeV, $\bar{M}_{5}=2$ TeV,
and $\bar{M}_{5}=3$ TeV, respectively. The SM cross sections (low curves) are
also shown.

The differential cross sections integrated in $p_{t}$ from the minimal
transverse momentum of the detected muons $p_{t,\min}$ are presented in Fig.
8.

Figure 8: The total cross sections for the process
$\mu^{+}\mu^{-}\rightarrow\mu^{+}\mu^{-}$ via minimal transverse momentum of
the outgoing muons $p_{t,\min}$.

As before, we have calculated the excluded bounds on $\bar{M}_{5}$ which can
be probed in the process $\mu^{+}\mu^{-}\rightarrow\mu^{+}\mu^{-}$ depending
on the integrated luminosity of the future muon collider, see Fig. 9. We have
used eq. (27) for the statistical significance. In doing so, the cuts
$p_{t,\min}=0.5$ TeV, $p_{t,\min}=2.5$ TeV, and $p_{t,\min}=25$ TeV were
applied for the 3 TeV, 14 TeV, and 100 TeV center-of-mass energies,
respectively.

Figure 9: The excluded bounds on the reduced fundamental gravity scale
$\bar{M}_{5}$ via integrated luminosity of the muon collider for the process
$\mu^{+}\mu^{-}\rightarrow\mu^{+}\mu^{-}$. The left, middle and right panels
correspond to the colliding energy of 3 TeV, 14 TeV, and 100 TeV.

We see that in the process $\mu^{+}\mu^{-}\rightarrow\mu^{+}\mu^{-}$ the
scales up to $\bar{M}_{5}=3.85$ TeV, 17.8 TeV and 126.3 TeV can be probed for
$\sqrt{s}=3$ TeV, $14$ TeV, and $100$ TeV, respectively. We conclude that
these limits on $\bar{M}_{5}$ are stronger than the limits obtained in the
previous section for the $\mu^{-}\mu^{+}\rightarrow 2(\mu^{+}\mu^{-})$
scattering.

##  5 Conclusions

We have examined two collisions at future TeV and multi-TeV muon colliders in
the Randall-Sundrum-like model with the small curvature (RSSC model) [5, 6].
It is the model with one ED and warped metric whose 5-dimensional space-time
curvature $k$ is about one GeV. The other main parameter of the RSSC model,
the 5-dimensional Planck scale $\bar{M}_{5}$, is equal to (larger than) one
TeV.

We have studied the $\mu^{-}\mu^{+}\rightarrow 2(\mu^{+}\mu^{-})$ scattering
first. The collision goes via the $VV\rightarrow\mu^{-}\mu^{+}$ scattering,
where $V=\gamma,Z$. For this scattering, the analytical expressions for the
squared amplitudes, including the gravity, SM and, interference terms, are
derived for the first time for _massive_ leptons. They are presented in
Appendix A. Then the differential cross sections depending on the invariant
mass of the detected muons $m_{\mu^{+}\mu^{-}}$ are calculated for three
values of the reduced 5-dimensional Planck scale $\bar{M}_{5}$ for 3 TeV, 14
TeV and 100 TeV muon colliders. The total cross section is calculated as a
function of the minimal value of $m_{\mu^{+}\mu^{-}}$. As a result, the
excluded bounds on the scale $\bar{M}_{5}$ are obtained. They are
$\bar{M}_{5}=3.80$ TeV, $13.1$ TeV and $106.4$ TeV, for the collision energy
of $\sqrt{s}=3$ TeV, $14$ TeV and $100$ TeV, respectively.

The $\mu^{-}\mu^{+}\rightarrow\mu^{+}\mu^{-}$ scattering is also studied. As
in the previous case, we have calculated the gravity, SM and interference
squared amplitudes analytically, see Appendix B. It enabled us to estimate
numerically the differential cross sections as functions of the transverse
momenta of the outgoing muons. The total cross sections are also calculated.
Finally, the excluded bounds on the main parameter of the RSSC model, the
scale $\bar{M}_{5}$, have been obtained. We have shown that the values of
$\bar{M}_{5}=3.85$ TeV, $17.8$ TeV and $126.3$ TeV can be probed at 3 TeV, 14
TeV, and 100 TeV muon colliders.

Let us remember that $\bar{M}_{5}=M_{5}/(2\pi)^{1/3}\approx 0.54M_{5}$, where
$M_{5}$ is the fundamental 5-dimensional gravity scale $M_{5}$ in the RSSC
model. It means that the bounds on the scale $M_{5}$ are approximately twice
stronger.

Let us stress that our bounds on $\bar{M}_{5}$ should not be directly compared
with the current experimental bound on the 5-dimensional Planck scale of the
original RS model [1], since the mass spectra of the KK gravitons and,
correspondingly, experimental signatures are quite different in the RSSC and
RS models [5]-[7].

## Appendix A. Squared amplitudes for $VV\rightarrow l^{-}l^{+}$ scattering

Our calculations give the following analytical expressions for the squared
amplitudes for the $\gamma\gamma\rightarrow l^{-}l^{+}$ collision in eq. (20)

$\displaystyle|M_{\mathrm{SM}}|^{2}$
$\displaystyle=\frac{8e^{4}}{(t-m_{l}^{2})^{2}(s+t-m_{l}^{2})^{2}}[-34m_{l}^{8}+m_{l}^{6}(60s+64t)$
$\displaystyle-m_{l}^{4}(31s^{2}+52st+28t^{2})+m_{l}^{2}s(s^{2}-2st-4t^{2})$
$\displaystyle-t(s+t)(s^{2}+2st+2t^{2})]\;,$ (A.1)
$\displaystyle|M_{\mathrm{KK}}|^{2}$
$\displaystyle=-\frac{1}{8}|S(s)|^{2}[2m_{l}^{8}-8m_{l}^{6}t+m_{l}^{4}(s^{2}+4st+12t^{2})-2m_{l}^{2}t(s+2t)^{2}$
$\displaystyle+t(s+t)(s^{2}+2st+2t^{2})]\;,$ (A.2)
$\displaystyle|M_{\mathrm{int}}|^{2}$
$\displaystyle=-\frac{e^{2}[S(s)+S^{\star}(s)]}{2(t-m_{l}^{2})(s+t-m_{l}^{2})}[-2m_{l}^{8}+m_{l}^{6}(3s+4t)+m_{l}^{4}s(3s-4t)$
$\displaystyle-
m_{l}^{2}(s^{3}+2s^{2}t+3st^{2}+4t^{3})+t(s+t)(s^{2}+2st+2t^{2})]\;,$ (A.3)

where $s$, $t$ are Mandelstam variables, $m_{l}$ is the lepton mass, and
$S(s)$ is defined in the text (17)-(19). If we take $m_{l}=0$ we get known
results obtained in [58].

For the $ZZ\rightarrow l^{-}l^{+}$ collision our calculations result in the
following formulas

$\displaystyle|M_{\mathrm{SM}}|^{2}$
$\displaystyle=\frac{g_{Z}^{4}}{(t-m_{l}^{2})^{2}(s+t-m_{l}^{2}-2m_{Z}^{2})^{2}}\\{-2m_{l}^{8}[-300\cos(2\theta_{w})+184\cos(4\theta_{w})$
$\displaystyle-68\cos(6\theta_{w})+17\cos(8\theta_{w})+195]$
$\displaystyle+4m_{l}^{6}[-4m_{Z}^{2}(-94\cos(2\theta_{w})+59\cos(4\theta_{w})-24\cos(6\theta_{w})$
$\displaystyle+6\cos(8\theta_{w})+56)+163s+196t-8(33s+37t)\cos(2\theta_{w})$
$\displaystyle+18(9s+10t)\cos(4\theta_{w})+(15s+16t)(\cos(8\theta_{w})-4\cos(6\theta_{w}))]$
$\displaystyle+m_{l}^{4}[-2m_{Z}^{4}(-164\cos(2\theta_{w})+128\cos(4\theta_{w})$
$\displaystyle+23(-4\cos(6\theta_{w})+\cos(8\theta_{w})+3))$
$\displaystyle+8m_{Z}^{2}(72s+65t-4(32s+27t)\cos(2\theta_{w})+(84s+68t)\cos(4\theta_{w})$
$\displaystyle+(10s+7t)(\cos(8\theta_{w})-4\cos(6\theta_{w})))-301s^{2}-436t^{2}-668st$
$\displaystyle+4(125s^{2}+260st+156t^{2})\cos(2\theta_{w})-8(39s^{2}+78st+46t^{2})\cos(4\theta_{w})$
$\displaystyle-(31s^{2}+52st+28t^{2})(\cos(8\theta_{w})-4\cos(6\theta_{w}))]$
$\displaystyle+m_{l}^{2}[7s^{3}+50s^{2}t+92st^{2}+s(s^{2}-2st-4t^{2})(\cos(8\theta_{w})-4\cos(6\theta_{w}))$
$\displaystyle+80t^{3}-4(3s^{3}+14s^{2}t+24st^{2}+24t^{3})\cos(2\theta_{w})$
$\displaystyle+8(s+2t)(s^{2}+st+3t^{2})\cos(4\theta_{w})+2m_{Z}^{2}(79s^{2}+252st+112t^{2}$
$\displaystyle-4(31s^{2}+108st+52t^{2})\cos(2\theta_{w})+8(9s^{2}+34st+17t^{2})\cos(4\theta_{w})$
$\displaystyle+(5s^{2}+28st+16t^{2})(\cos(8\theta_{w})-4\cos(6\theta_{w})))$
$\displaystyle+2m_{Z}^{4}(-263s-414t+(428s+696t)\cos(2\theta_{w})-16(16s+27t)\cos(4\theta_{w})$
$\displaystyle-21(s+2t)(\cos(8\theta_{w})-4\cos(6\theta_{w})))$
$\displaystyle+2m_{Z}^{6}(-156\cos(2\theta_{w})+96\cos(4\theta_{w})-36\cos(6\theta_{w})+9\cos(8\theta_{w})+91)]$
$\displaystyle-[-28\cos(2\theta_{w})+16\cos(4\theta_{w})-4\cos(6\theta_{w})+\cos(8\theta_{w})+19]$
$\displaystyle\times[4m_{Z}^{8}-4m_{Z}^{6}(s+3t)+m_{Z}^{4}(s^{2}+6st+14t^{2})-2m_{Z}^{2}t(s+2t)^{2}$
$\displaystyle+t(s+t)(s^{2}+2st+2t^{2})]\\}\;,$ (A.4)
$\displaystyle|M_{\mathrm{KK}}|^{2}$
$\displaystyle=\frac{1}{288}|S(s)|^{2}\\{-72m_{Z}^{8}+6m_{Z}^{6}(-40m_{l}^{2}+9s+48t)$
$\displaystyle-4m_{Z}^{4}[-2m_{l}^{2}(5s+96t)+136m_{l}^{4}+9t(7s+12t)]$
$\displaystyle+3m_{Z}^{2}[-80m_{l}^{6}+m_{l}^{4}(256t-14s)-4m_{l}^{2}(s^{2}+29st+68t^{2})$
$\displaystyle+9s^{3}+42s^{2}t+114st^{2}+96t^{3})]-36[m_{l}^{4}-2m_{l}^{2}t+t(s+t)]$
$\displaystyle\times(2m_{l}^{4}-4m_{l}^{2}t+s^{2}+2st+2t^{2})\\}\;,$ (A.5)
$\displaystyle|M_{\mathrm{int}}|^{2}$
$\displaystyle=-\frac{g_{Z}^{2}[S(s)+S^{\star}(s)]}{96(t-m_{l}^{2})(s+t-m_{l}^{2}-2m_{Z}^{2})}\\{-12m_{l}^{8}[\cos(4\theta_{w})-2\cos(2\theta_{w})]$
$\displaystyle+2m_{l}^{6}[(16m_{Z}^{2}+3(3s+4t)(\cos(4\theta_{w})-2\cos(2\theta_{w}))+6(s-2t)]$
$\displaystyle+2m^{4}_{l}[-m^{2}_{Z}((9s+4t)(\cos(4\theta_{w})-2\cos(2\theta_{w}))$
$\displaystyle+24s+60t)+8m^{4}_{Z}(-2\cos(2\theta_{w})+\cos(4\theta_{w})+4)+6(3s^{2}+st+6t^{2})$
$\displaystyle+3s(3s-4t)(\cos(4\theta_{w})-2\cos(2\theta_{w}))]$
$\displaystyle+m^{2}_{l}[4m_{Z}^{2}((2s^{2}-3st+14t^{2})(\cos(4\theta_{w})-2\cos(2\theta_{w}))$
$\displaystyle+2s^{2}+5st+46t^{2})+4m_{Z}^{4}((s-10t)(\cos(4\theta_{w})-2\cos(2\theta_{w}))-38t)$
$\displaystyle+8m_{Z}^{6}(-2\cos(2\theta_{w})+\cos(4\theta_{w})+5)$
$\displaystyle-3(3s^{3}+10s^{2}t+2(s^{3}+2s^{2}t+3st^{2}+4t^{3})$
$\displaystyle\times(\cos(4\theta_{w})-2\cos(2\theta_{w}))+24st(s+t))]$
$\displaystyle+6[-2\cos(2\theta_{w})+\cos(4\theta_{w})+2][2m_{Z}^{8}+m_{Z}^{6}(s-8t)$
$\displaystyle+m_{Z}^{4}(-s^{2}+2st+12t^{2})-m_{Z}^{2}t(s^{2}+7st+8t^{2})$
$\displaystyle+t(s+t)(s^{2}+2st+2t^{2})]\\}\;,$ (A.6)

where $\theta_{w}$ is the Weinberg angle,
$g_{Z}=e/[\sin(\theta_{w})\cos(\theta_{w})]$ is the weak coupling constant,
and $m_{Z}$ is the mass of the $Z$ boson. Because of conservation of helicity,
in the massless limit $m_{l}=m_{Z}=0$ $s$-channel graviton amplitudes squared
(Appendix A. Squared amplitudes for $VV\rightarrow l^{-}l^{+}$ scattering) and
(Appendix A. Squared amplitudes for $VV\rightarrow l^{-}l^{+}$ scattering) are
proportional to the factor $t(s+t)=s(\sin\theta)^{2}/4$, where $\theta$ is the
scattering angle.

Finally, the SM squared amplitude for the $\gamma Z\rightarrow l^{+}l^{-}$
collision looks like

$\displaystyle|M_{\mathrm{SM}}|^{2}$
$\displaystyle=\frac{4g_{Z}^{4}}{(t-m_{l}^{2})^{2}(s+t-m_{l}^{2}-m_{Z}^{2})^{2}}$
$\displaystyle\times\\{-2m_{l}^{8}[184\cos(4\theta_{w})+17\cos(8\theta_{w})-300\cos(2\theta_{w})$
$\displaystyle-68\cos(6\theta_{w})+195]+4m_{l}^{6}[-2(59\cos(4\theta_{w})+6\cos(8\theta_{w})-94\cos(2\theta_{w})$
$\displaystyle-24\cos(6\theta_{w})+56)m_{Z}^{2}+163s+196t-8(33s+37t)\cos(2\theta_{w})$
$\displaystyle+18(9s+10t)\cos(4\theta_{w})+(15s+16t)(\cos(8\theta_{w})-4\cos(6\theta_{w}))]$
$\displaystyle+m_{l}^{4}[(68\cos(2\theta_{w})+44\cos(6\theta_{w})-56\cos(4\theta_{w})-11\cos(8\theta_{w})-25)m_{Z}^{4}$
$\displaystyle+4[72s+65t-4(32s+27t)\cos(2\theta_{w})+(84s+68t)\cos(4\theta_{w})$
$\displaystyle+(10s+7t)(\cos(8\theta_{w})-4\cos(6\theta_{w}))]m_{Z}^{2}-301s^{2}-436t^{2}-668st$
$\displaystyle+4(125s^{2}+260ts+156t^{2})\cos(2\theta_{w})-8(3s^{2}+78st+46t^{2})\cos(4\theta_{w})$
$\displaystyle-(31s^{2}+52st+28t^{2})(\cos(8\theta_{w})-4\cos(6\theta_{w}))]$
$\displaystyle+m_{l}^{2}[7s^{3}+50s^{2}t+92st^{2}+(s^{2}-2st-4t^{2})(\cos(8\theta_{w})$
$\displaystyle-4\cos(6\theta_{w}))s+80t^{3}+(79s^{2}+252st+112t^{2}$
$\displaystyle+((5(\cos(8\theta_{w})-4\cos(6\theta_{w})+11)+56\cos(4\theta_{w})$
$\displaystyle-92(\cos(2\theta_{w})))m_{Z}^{2}-141s-226t+4(57s+94t)(\cos(2\theta_{w}))$
$\displaystyle-8(17s+29t)(\cos(4\theta_{w}))-11(s+2t)(\cos(8\theta_{w})-4\cos(6\theta_{w})))m_{Z}^{2}$
$\displaystyle-4(31s^{2}+108st+52t^{2})\cos(2\theta_{w})+8(9s^{2}+34st+17t^{2})\cos(4\theta_{w})$
$\displaystyle+(5s^{2}+28st+16t^{2})(\cos(8\theta_{w})-4\cos(6\theta_{w})))m_{Z}^{2}$
$\displaystyle-4(3s^{3}+14s^{2}t+24st^{2}+24t^{3})\cos(2\theta_{w})$
$\displaystyle+8(s+2t)(s^{2}+st+3t^{2})\cos(4\theta_{w})]$
$\displaystyle-t[16\cos(4\theta_{w})+\cos(8\theta_{w})-28\cos(2\theta_{w})-4\cos(6\theta_{w})+19]$
$\displaystyle\times(s+t-m_{Z}^{2})(s^{2}+2t^{2}+2st-2tm_{Z}^{2}+m_{Z}^{4})\\}\;.$
(A.7)

The contribution to the $\gamma Z\rightarrow l^{+}l^{-}$ collision from the
gravitons $G$ is zero, since there is no $\gamma ZG$ vertex. Note that in the
limit $m_{l}=m_{Z}=0$ all squared amplitudes depend on variables $s$ and
$t(s+t)=tu$, where $u$ is the Mandelstam variable.

## Appendix B. Squared amplitudes for $l^{+}l^{-}\rightarrow l^{+}l^{-}$
scattering

Here we present the result of our calculations of the squared amplitudes for
the $l^{+}l^{-}\rightarrow l^{+}l^{-}$ process (both incoming and outgoing
leptons have the same flavor). The SM squared amplitude has both the $Z$ boson
and photon contributions. The latter one is given by the formula

$\displaystyle|M_{\mathrm{SM}}|^{2}$
$\displaystyle=\frac{16e^{4}}{s^{2}t^{2}}[m_{l}^{4}(5s^{2}+11st+5t^{2})+4m_{l}^{2}(s+t)(s^{2}+5st+t^{2})$
$\displaystyle+(s^{2}+st+t^{2})^{2}]\;,$ (B.1)

where $m_{l}$ is the lepton mass. Note that the photon contribution to
$|M_{\mathrm{SM}}|^{2}$ is dominant. That is why, we do not present (rather
complicated) analytical expression for the $Z$ boson contribution to
$|M_{\mathrm{SM}}|^{2}$. The graviton squared amplitude is defined by KK
graviton exchanges in $s$-, and $t$-channels,

$\displaystyle|M_{\mathrm{KK}}|^{2}$
$\displaystyle=\frac{1}{4608}\\{|S(s)|^{2}F_{1}(s,t)+|S(t)|^{2}F_{1}(t,s)$
$\displaystyle+[S(s)S(t)^{\star}+S(s)^{\star}S(t)]F_{2}(s,t)\\}\;.$ (B.2)

Finally, the interference term of $|M|^{2}$ is equal to (neglecting small $Z$
boson contribution)

$\displaystyle|M_{\mathrm{int}}|^{2}$
$\displaystyle=-\frac{e^{2}}{24st}\\{[S(s)+S(t)^{\star}]F_{3}(s,t)+[S(s)^{\star}+S(t)]F_{3}(t,s)\\}\;.$
(B.3)

Here $S(s)$ is defined by eqs. (18), (19), and the following functions are
introduced

$\displaystyle F_{1}(s,t)$
$\displaystyle=6656m_{l}^{8}+m_{l}^{6}(8576s+10752t)+m_{l}^{4}(3440s^{2}+7296t^{2}+10752st)$
$\displaystyle+m_{l}^{2}(360s^{3}+2376s^{2}t+4320st^{2}+2304t^{3})$
$\displaystyle+9s^{4}+90s^{3}t+378s^{2}t^{2}+576st^{3}+288t^{4}\;,$ (B.4)
$\displaystyle F_{2}(s,t)$
$\displaystyle=7552m_{l}^{8}+15168m_{l}^{6}(s+t)+m_{l}^{4}(7248(s^{2}+t^{2})+14968st)$
$\displaystyle+m_{l}^{2}(1032(s^{3}+t^{3})+3690st(s+t))$
$\displaystyle+36(s^{4}+t^{4})+225st(s+t)+378s^{2}t^{2}\;,$ (B.5)
$\displaystyle F_{3}(s,t)$
$\displaystyle=m_{l}^{6}(576s+512t)+m_{l}^{4}(552s^{2}+448t^{2}+1120st)$
$\displaystyle+m_{l}^{2}(86s^{3}+360s^{2}t+432st^{2}+144t^{3})$
$\displaystyle+3s^{4}+21s^{3}t+45s^{2}t^{2}+48st^{3}+24t^{4}\;,$ (B.6)

where $m_{l}$ is the lepton mass. Note that $F_{2}(s,t)=F_{2}(t,s)$.

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# Planar Hall effect and quasi - 2D anisotropic superconductivity in
topological candidate 1$T$-NbSeTe

C. Patra  Department of Physics, Indian Institute of Science Education and
Research Bhopal, Bhopal, 462066, India  T. Agarwal  Department of Physics,
Indian Institute of Science Education and Research Bhopal, Bhopal, 462066,
India  Rajeshwari R. Chowdhury  Department of Physics, Indian Institute of
Science Education and Research Bhopal, Bhopal, 462066, India  R. P. Singh
<EMAIL_ADDRESS>Department of Physics, Indian Institute of Science
Education and Research Bhopal, Bhopal, 462066, India

###### Abstract

Superconducting topological materials have generated considerable interest in
condensed matter research due to their unusual gap structures and topological
properties. In this study, we have investigated the normal and superconducting
characteristics of a potential topological semimetal 1$T$-NbSeTe through
comprehensive transport and magnetization measurements on bulk single
crystals. The results suggest the topological semimetallic nature of NbSeTe,
evidenced by the observation of the planar Hall effect. Moreover, it displays
quasi-2D anisotropic superconductivity, which breaks the Pauli limit. The
coexistence of the topological semimetallic nature and superconductivity in
NbSeTe makes it a potential contender for topological superconductivity.

##  I INTRODUCTION

Superconductors and topological materials are distinct types of quantum
materials, both displaying exceptional physical properties with potential
applications. When combined, they can produce a distinct quantum phase called
a topological superconductor (TSC) topology_computation ;
sc_proximity_majarana_Fermi ; Majarana_fermion , representing a new paradigm
for unconventional superconductivity. The TSC offers a novel platform to
realize non-Abelian quasiparticle excitation, such as Majorana fermions, which
can be used as a topological qubit for quantum computation
majarana_quantization ; review_TSupercon . However, the realization of TSC in
bulk topological materials is limited. Superconducting Dirac and Weyl
semimetals have recently gained significant attention due to their non-trivial
topology, as they provide a new platform to innovate TSCs NiS2_mott_insulator
; TSC_chinese ; Dirac_Cd3As2 ; TaAs_weyl ; theory_TSC_TaAs . Meanwhile, doped
semimetallic topological transition metal dichalcogenides (TMDs) have emerged
as a new platform to realize topological superconductivity Theory_MoTe2 .

Recent studies have revealed that the 1$T$ phases of MTe2 (M = Zr, Hf, Ni, Pt,
Pd, Ir) ZrTe2_dirac_semimetal ; HfTe2_dirac_semimetal ; NiTe2_Re_doped ;
PtTe2_dirac_semi ; Topology_Pd_semi ; Pt_doped_sc_IrTe2 , as well as the
distorted 1$T^{\prime\prime}$ phase of NbTe2 NbTe2_quantum_MR , exhibit
topological semimetallic behavior band_NbTe2_TaTe2 . Further research suggests
that superconductivity can be induced in these materials through strain,
pressure, and doping Pt_doped_sc_IrTe2 , highlighting the potential for 1$T$
phase materials as a platform for topological superconductivity. NbTe2 has
garnered significant interest due to its fascinating properties, such as
magnetism, charge-density waves (CDW), superconductivity, and quantum spin
Hall effect band_NbTe2_TaTe2 ; NbTe2_fermi_surface ; 1T_NbSe2 . Though the
1$T$ phase of NbTe2 and NbSe2 does not occur naturally in bulk crystals, the
stable 1$T^{\prime\prime}$ phase of NbTe2 has been identified as a promising
candidate for topological superconductivity, as topological semimetallic
behavior and superconductivity ($T_{c}$ = 0.7 K) coexist in this material
NbTe2_fermi_surface ; NbTe2_50T_field .

However, NbSeTe is unique in that it crystallizes in the 1$T$ phase and
exhibits enhanced superconductivity due to the suppression of CDW. The
disorder is believed to play an important role in both suppressing and
enhancing superconductivity in NbSeTe. Recent studies have shown that
controlled disorder can enhance the superconducting transition temperature in
monolayer NbSe2 and 2$H$-TaSe2-xSx by suppressing competing electronic orders,
such as CDW NbSe2_disorder_effect ; effect_of_disorder ; tases_2H ; TaSe2S2 .
Similar disorder-induced enhanced superconductivity has also been observed in
bulk 2$H$-TaSe2-xSx. These studies suggest that disorder enhanced the
electron-phonon coupling strength angle_resolve_NbSeTe ;
weak_anti_localization ; NbSeTe_reported ; TaSeTe_structure ; NbTe2_pressur .
Thus, NbSeTe presents a unique opportunity to explore superconductivity in
topological semimetals and the impact of the disorder on superconducting
topological semimetals.

In this paper, we report the synthesis and properties of a single crystal of
NbSeTe, which crystallizes in the structure 1$T$ $P\bar{3}m1$ (164). Our
findings reveal that NbSeTe possesses a topological semimetallic nature, as
demonstrated by the observation of the planar Hall effect. Furthermore, NbSeTe
displays 2D-anisotropic superconductivity with a transition temperature of
3.16 K. The upper critical field surpasses the Pauli limiting field, and its
proximity to the unconventional band in the Uemura plot suggests
unconventional superconductivity.

Figure 1:  (a) The unit cell structure of 1$T$-NbSeTe. (b) Shows the image of
the single crystal. (c) The Laue diffraction pattern. (d) EDS scan that
detects Nb, Se, and Te. (e) The X-ray diffraction pattern of the single
crystal indicates that the crystal is oriented along the [00$l$] plane. (f)
Refinement of powder XRD data for determining lattice parameters.

##  II EXPERIMENTAL METHODS

We prepared a single crystal of 1$T$-NbSeTe using the chemical vapor transport
method. We mixed Nb (4N), Se (5N), and Te (5N) powders in a stoichiometric
ratio and sealed them in a quartz tube with iodine (5 mg cm-3) as the
transport agent. The tube was then placed in a two-zone horizontal furnace
with a temperature gradient of 950°C and 850°C. After 15 days; shiny single
crystals were grown in the cold zone of the tube. The orientation of the
crystal plane was determined by X-ray diffraction (XRD) using a PANalytical
diffractometer equipped with Cu$K_{\alpha}$ radiation ($\lambda$ = 1.54056 Å),
and the Laue diffraction pattern was recorded using a Photonic-Science Laue
camera. We used a scanning electron microscope (SEM) with an energy-dispersive
X-ray (EDS) spectrometer to verify the sample compositions, which confirmed
the presence of Nb, Se, and Te. Magnetization measurements were conducted
using a Quantum Design magnetic measurement system (MPMS3), while resistivity
and specific heat measurements were taken using a physical property
measurement system (PPMS).

##  III RESULTS AND DISCUSSION

###  a Sample characterization

Fig. 1 shows the unit cell, single crystal image, Laue diffraction pattern,
EDS measurements, and XRD patterns of 1$T$-NbSeTe. The unit cell, single-
crystal image, and Laue diffraction pattern are depicted in (a), (b), and (c),
respectively. EDS measurements in various areas of single crystals found the
average elemental concentration to be Nb1.01Se1.03Te0.95, as shown in (d). The
XRD pattern of the single crystal, depicted in (e), confirms the crystal’s
orientation along the (00$l$) direction. To determine the phase, we collected
the XRD pattern of the crushed single-crystal powder, shown in (f). The powder
XRD refinement confirms that the sample crystallizes in the trigonal
$P\bar{3}m1$ (164) space group, with lattice parameters of $a=b=3.56(5)$ Å and
$c=6.52(7)$ Å NbSeTe_reported ; TaSeTe_structure ; NbTe2_pressur . These
lattice parameters are identical to those reported for 1$T$-NbSeTe
NbSeTe_reported .

###  b Planar Hall effect

The planar Hall effect (PHE) is a phenomenon exhibited by topological
semimetals (Dirac/Weyl) due to the chiral anomaly and non-trivial Berry
curvature of these materials. PHE has been observed in several topological
semimetals, including WTe2 topology_hall_wte , Cd3As2 Dirac_Cd3As2 , ZrTe5
ZrTe5_planar , PdTe2 Topology_Pd_semi , PtTe2 PtTe2_dirac_semi , and MoTe2
origin_planar_hall_MoTe2  as well as superconducting topological materials
LaOBiSe2_planar ; TaSe3_planar ; Giant_planar_srbi2se3 ; moirte2 . However,
recent research suggests that PHE can also be observed in topological
materials that lack the chiral anomaly planr_hall_TI . Therefore, PHE is
considered a useful tool for studying topological behavior through magneto-
transport studies.

Figure 2:  (a) The planar Hall response for various applied magnetic fields at
a temperature of 5 K. (b) Planar Hall effect at different temperatures under 9
T. (c) The variation $\Delta\rho_{xy}^{chiral}$ with the magnetic field. (d)
The decrease of $\Delta\rho_{xy}^{chiral}$ with increasing temperature, and an
inset showing the Hall resistivity variation with the magnetic field at a
temperature of 6 K.

The schematic diagram of PHE is represented in the inset of Fig. 2 (c). The
standard four-probe technique measures the planar Hall resistivity
($\rho_{xy}$). The magnetic field is applied along the $ab$ plane or
perpendicular to the $c$ axis and rotated around the $c$ axis at an angle
$\phi$ relative to the current $I$. Experimental Hall resistivity data include
normal and planar Hall contributions, with chiral anomaly-induced Hall
resistivity symmetric under an antisymmetric applied field, and the normal
Hall resistivity being antisymmetric under an opposite applied field moirte2 ;
planar_hall_review ; Cu_doped_PdTe2 ; mote2_basic_paper . To extract the
chiral anomaly contribution, the planar Hall resistivity data is averaged
opposite to the magnetic field as follows:

$\rho_{xy}^{PHE}=\left[\frac{\rho_{xy}(B)+\rho_{xy}(-B)}{2}\right]$ (1)

where $\rho_{xy}(B)$ and $\rho_{xy}(-B)$ are Hall resistivity under applied
positive and negative magnetic field directions, respectively. The angular
dependence of $\rho_{xy}^{PHE}$ is shown in Fig. 2 (a) and (b), with a period
of $\pi$ and maximum values at $\pi/4$ and $3\pi/4$, consistent with the
Planar Hall effect (PHE). In Fig. 2 (a), the variation of planar Hall
resistivity is shown at 5 K under different magnetic fields. The variation of
$\Delta\rho_{xy}^{chiral}$ with magnetic field strength is proportional to
$H^{2}$, as shown in Fig. 2 (c), indicating the possible presence of a
topological phase in the 1$T$-NbSeTe system, similar to NbTe2 NbTe2_50T_field
. To extract the chiral anomaly contribution, the $\rho{xy}^{PHE}$ data is
fitted using Eq. (2).

$\rho_{xy}=-\Delta\rho_{xy}^{chiral}sin{\phi}cos{\phi}$ (2)

where $\Delta\rho_{xy}^{chiral}$ = $\rho_{\perp}-\rho_{\parallel}$ gives the
anisotropy in resistivity due to chiral anamoly. $\rho_{\perp}$ and
$\rho_{\parallel}$ represent the resistivity for the magnetic field applied
perpendicular and parallel to the current direction, respectively. In Fig. 2
(b), the variation of planar Hall resistivity data under 9 T magnetic field at
different temperatures is shown. The temperature variation of
$\Delta\rho_{xy}^{chiral}$ continuously decreases with increasing temperature,
as represented in Fig. 2 (d).

Furthermore, the inset of Fig. 2 (d) displays Hall resistivity data
($\rho_{xy}$ vs $H$) fitted with $\rho_{xy}=R_{H}H$ (where $R_{H}$ is the Hall
coefficient) for 6 K. The obtained carrier concentration is 2.22(3) $\times$
1021 cm-3.

###  c Superconducting properties

Resistivity and magnetization measurements were performed on cleaved crystals
(2 mm $\times$ 2 mm). The electrical resistivity vs temperature measurement
$\rho(T)$ was conducted over the temperature range of 300 to 1.9 K in the
absence of an applied magnetic field, as depicted in Fig. 3 (a). The
resistivity shows metallic behavior in the normal state and exhibits a sudden
drop at $T_{c}^{onset}$ of 3.16(1) K due to the superconducting transition, as
shown in Fig. 3 (a). The residual resistivity ratio (RRR) is 1.23. In the
normal state, the behavior of electrons is influenced by phonons, leading to
temperature-dependent behavior. The Bloch-Grüneisen (BG) model BG_model  can
explain the temperature dependence of $\rho(T)$ well. According to this model,
the resistivity is given by

$\rho(T)=\rho_{0}+\rho_{BG}(T)$ (3)

where $\rho_{BG}(T)$ is defined as

$\rho_{BG}(T)=r\left(\frac{T}{\Theta_{D}}\right)^{5}\int_{0}^{\frac{\Theta_{D}}{T}}\frac{x^{5}}{(e^{x}-1)(1-e^{-x})}dx$
(4)

here, $\rho_{0}$ represents the residual resistivity, $r$ is a material-
dependent constant, and $\Theta_{D}$ is the Debye constant. The fitting
parameters obtain using Eq. (4) were $\rho_{0}$ = 0.20(7) m$\ohm$ cm, $r$ =
0.11(2) m$\Omega$ cm, and $\Theta_{D}$ = 169(4) K, as shown in Fig. 3 (a). The
Debye temperature value is found to be similar to the reported value in
specific heat for NbSeTe NbSeTe_reported .

Figure 3:  (a) Temperature variation of resistivity at zero magnetic field,
indicating superconductivity with a transition temperature of 3.16(1) K. (b)
Magnetization data collected in ZFCW-FCC mode, showing the superconducting
transition temperature of 3.11(2) K at an applied magnetic field of 1 mT.

The absence of a charge density wave (CDW) signal in the normal state
resistivity of 1$T$-NbSeTe distinguishes it from its parent compounds,
1$T^{\prime\prime}$-NbTe2 and 1$T$-NbSe2, which exhibit strong CDW even in the
monolayer limit cdw_nbte2 ; 1T_NbSe2 . The substitution of selenium in the
parent compound may have suppressed the CDW and promoted superconductivity, as
reported transition temperatures of 1.3 K NbSeTe_reported  and 3.0 K
angle_resolve_NbSeTe ; weak_anti_localization  have been observed in
1$T$-NbSeTe without CDW NbSeTe_reported ; angle_resolve_NbSeTe ;
weak_anti_localization . In our crystal, the recorded superconducting
transition temperature is $T_{c}$ of 3.16 K, which is comparable to the value
reported in Ref. weak_anti_localization , revealing that $T_{c}$ is dependent
on the composition.

Figure 4:  (a) Low field variation of magnetization data in the perpendicular
direction. (b) The lower critical field variation with temperature is well-
fitted with GL equations, providing values of 0.63(2) and 0.37(4) mT for the
$H\perp c$ and $H\parallel c$ directions, respectively.

The temperature dependence of the magnetic moment measurement of 1$T$-NbSeTe
with a superconducting transition at 3.11(2) K is presented in Fig. 3 (b). A 1
mT magnetic field was applied in zero-field cooling (ZFCW) and field-cooled
cooling (FCC) modes. The lower critical field $H_{c1}$(0) is calculated from
low-field magnetization measurements for the perpendicular and parallel
directions. The $M$-$H$ curves at various temperatures for $H\perp c$ are
illustrated in Fig. 4 (a). The value of $H_{c1}(T)$ is obtained from the point
where the Meissner lines deviate for a specific temperature. The Ginzburg-
Landau equation Eq. (5) is utilized to fit the values of $H_{c1}(T)$ for both
perpendicular and parallel directions of the magnetic field.

$H_{c1}(T)=H_{c1}(0)\left(1-\left(\frac{T}{T_{c}}\right)^{2}\right)$ (5)

In Fig. 4 (b), the anisotropy of $H_{c1}(T)$ in two directions is presented.
The blue and green dotted lines correspond to the values of $H_{c1}(0)$, which
are 0.63 (2) mT and 0.37(4) mT for the $H\perp c$ and $H\parallel c$
directions, respectively.

The upper critical field, $H_{c2}(0)$, was calculated by measuring the
isofield resistivity ($\rho$ vs $T$) in the field range of 0 to 5 T, in both
the in-plane and out-of-plane directions of the crystal. $H_{c2}(T)$ values
are determined from the $96\%$ drop of the normal state resistivity attributed
to disorder. Fig. 5 (a) displays $\rho$ vs $T$ for the perpendicular and
parallel (inset) directions of the magnetic field. The $H_{c2}(T)$ data is
fitted using the Ginzburg-Landau equation:

$H_{c2}(T)=H_{c2}(0)\left[\frac{1-\left(\frac{T}{T_{c}}\right)^{2}}{1+\left(\frac{T}{T_{c}}\right)^{2}}\right].$
(6)

The resulting fits in both directions yield the values of 7.76 (1) T and 2.29
(2) T for the $H\perp c$ and $H\parallel c$ direction, respectively, as shown
in Fig. 5 (b). The 2D Ginzburg-Landau model is used to explain $H_{c2}(T)$,
with $(1-T/T_{c})^{1/2}$ for $H\perp c$ and $(1-T/T_{c})$ for $H\parallel c$.
However, the $H_{c2}$ data points deviated for $H\perp c$, whereas for
$H\parallel c$ direction, it is consistent with the 3D model, indicating that
quasi-2D superconductivity may be present in the system ausn4 .

In type-II superconductors, Cooper pairs can be broken by the orbital effect
and the Pauli paramagnetic effect. In the case of the orbital limiting effect,
the increasing kinetic energy of the system destroys Cooper pairs. The orbital
limiting field ($H_{c2}^{orbital}$) can be expressed using the Wartherm-
Helfand-Hohenberg (WHH) equation paramagnetic_limit :

$H_{c2}^{orbital}=-\alpha T_{c}\frac{dH_{c2}(T)}{dT}\bigg{|}_{T=T_{c}},$ (7)

where $\alpha$ is taken as 0.693 for dirty limit superconductors. For
1$T$-NbSeTe, the initial slope $-\frac{dH_{c2}(T)}{dT}$ near $T_{c}$ is
calculated to be 2.12(1) T/K using the temperature variation of in-plane upper
critical field data, giving an orbital limiting field of 4.64(4) T. However,
in the Pauli paramagnetic limit, it is energetically favorable for electron
spins to align with the magnetic field, breaking the Cooper pairs. The Pauli
paramagnetic limit is expressed as $H_{c2}^{P}=1.86T_{c}=5.87(7)$ T (shown as
the red dotted line in Fig. 5 (b)). Therefore, the upper critical field is
1.32 times higher than the Pauli limiting field. The upper critical fields of
comparative materials are listed in Table 1. In layered superconductors,
violations of the Pauli limit can occur due to strong SOC or finite-momentum
pairing. Strong SOC leads to Ising-type superconductivity, which has recently
been observed in monolayer 2$H$-NbSe2 monolayer_nbse2 . Finite-momentum
pairing can lead to the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state
ba6nb11se28 . More low-temperature angle-dependent measurements and
theoretical inputs are required to confirm the exact mechanism of Pauli limit
violation.

Figure 5:  (a) show the temperature variation of resistivity for different
magnetic fields for H $\perp$ c and H $\parallel$ c in the inset. (b) The
temperature variation of the upper critical field for two different magnetic
field directions fitted with the Ginzburg-Landau equations. The red dotted
line indicates the Pauli limit, H${}_{c2}^{P}$, which is 5.87 T.

In addition, a detailed study of the anisotropy in the upper critical field
for $H\perp c$ and $H\parallel c$ was conducted by measuring the field-
dependent resistivity at 2 K for different angles. The variation of field-
dependent resistivity with the angle for 1$T$-NbSeTe is shown in Fig. 6 (a),
where $\theta$ represents the angle between the magnetic field and the normal
to the sample plane (as depicted in the inset of Fig. 6 (a)). The
$H_{c2}(\theta,T)$ values are obtained from the $98\%$ resistivity drop from
the normal state value ceir3 ; ausn4 . The anisotropic behavior of the cusp-
like upper critical field variation is described using both the 3D anisotropic
Ginzburg-Landau model (AGL) and the 2D Tinkham model.

$\left(\frac{H_{c2}(\theta,T)\sin{\theta}}{H_{c2}^{\perp}}\right)^{2}+\left(\frac{H_{c2}(\theta,T)\cos{\theta}}{H_{c2}^{\parallel}}\right)^{2}=1$
(8)

$\left(\frac{H_{c2}(\theta,T)sin\theta}{H_{c2}^{\perp}}\right)^{2}+\left\arrowvert\frac{H_{c2}(\theta,T)cos\theta}{H_{c2}^{||}}\right\arrowvert=1$
(9)

In Fig. 6 (b), the field-dependent resistivity is fitted using the 3D
anisotropic Ginzburg-Landau model (AGL) and the 2D Tinkham model, represented
by the dotted blue and solid green lines, respectively. The zoomed part in the
inset of Fig. 6 (b) shows that the 2D Tinkham model provides a better fit,
indicating the possible presence of 2D superconductivity in the bulk single
crystal 1$T$-NbSeTe ausn4 ; nbse2_3d_gl .

Figure 6:  (a) The field variation of resistivity at a temperature of 2 K. (b)
The angle-dependent upper critical field, fitted with both the 3D Ginzburg-
Landau model and the 2D Tinkham model.

The Ginzburg-Landau coherence length ($\xi_{\perp c}(0)$ = 119.9(6) Å and
$\xi_{\parallel c}(0)$ = 35.4(5) Å) are obtained through the equations
$H_{c2}^{\parallel}(0)=\frac{\phi_{0}}{2\pi\xi_{\perp c}^{2}(0)}$ and
$H_{c2}^{\perp}(0)=\frac{\phi_{0}}{2\pi\xi_{\parallel c}(0)\xi_{\perp c}(0)}$
zeta_calculation . Utilizing these coherence lengths and the lower critical
field values ($H_{c1}^{\perp}(0)$ = 0.63(2) mT and $H_{c1}^{\parallel}(0)$ =
0.37(4) mT), the Ginzburg-Landau penetration depth is calculated to be
$\lambda_{\perp c}(0)$ = 1482(9) nm and $\lambda_{\parallel c}(0)$ = 937(7) nm
by applying Eq. (10).

$H_{c1}^{\parallel}(0)=\frac{\phi_{0}}{4\pi\lambda^{2}_{\perp
c}(0)}\left[ln\left(\frac{\lambda_{\perp c}(0)}{\xi_{\perp
c}(0)}\right)+0.12\right]$ (10)

Table 1: The comparative superconducting parameters for 1$T$-NbSeTe with some layered compounds Parameter | 1$T$- | 2$H$- | NbS2 | Ba6Nb11Se28
---|---|---|---|---
| NbSeTe | NbSe2pauli_NbSe2  | pauli_2hNbSe2  | ba6nb11se28
$H_{c2}^{\perp}$(T) | 7.76 | 17.3 |  | 8.84
$H_{c2}^{\parallel}$(T) | 2.29 | 5.3 | 1.6 | 0.57
$H_{c2}^{P}$(T) | 5.87 | 13.54 | 10.4 | 4.27
$\xi_{\perp c}(\text{\AA}$) | 119.9 | 78.8 | 143 | 240.4
$\xi_{\parallel c}(\text{\AA}$) | 35.4 | 24.1 | 9.7 | 15.6
$\gamma=\frac{H_{c2}^{\perp}(0)}{H_{c2}^{\parallel}(0)}$ | 3.39 | 3.3 | 7.94 | 10.53

where $\phi_{0}$ (= 2.07$\times$10-15 Tm2) is the quantum magnetic flux. Using
the two coherence lengths, the Ginzburg-Landau parameter $\kappa_{\perp c}$ is
calculated as 123$>$ $\frac{1}{\sqrt{2}}$ by $\kappa_{\perp
c}=\frac{\lambda_{\perp c}(0)}{\xi_{\perp c}(0)}$. This suggests that the
sample exhibits type II superconducting behavior.

The electron-phonon coupling constant ($\lambda_{e-ph}$) is a dimensionless
parameter that provides insight into the strength of interactions within the
system. Using McMillan’s model coupling_electron_phono , $\lambda_{e-ph}$ can
be calculated from $\Theta_{D}$ and $T_{c}$ using 169 K and 3.16 K
respectively in Eq. (11), where $\mu^{*}$ is the Coulomb repulsion constant
(0.13 in our case).

$\lambda_{e-ph}=\frac{1.04+\mu^{*}\ln(\Theta_{D}/1.45T_{c})}{(1-0.62\mu^{*})\ln(\Theta_{D}/1.45T_{c})-1.04}$
(11)

For our sample, the calculated value of $\lambda_{e-ph}$ is 0.66, which is
higher than the reported value of 0.55 for 1$T$-NbSeTe with $T_{c}$ of 1.3 K.
This enhancement in $\lambda_{e-ph}$ suggests that the weak disorder can
increase the electron-phonon coupling and consequently enhance the
superconducting transition temperature. This has been observed in other
systems, such as weak disorder in 2$H$-TaSe2-xSx TaSe2S2 , Ir-doped MoTe2
moirte2 , and Re-doped NiTe2 NiTe2_Re_doped .

###  d Electronic properties and the Uemura plot

A set of equations are solved to determine the electronic mean free path and
to verify the dirty or clean limit of the 1$T$-NbSeTe superconductor
lamda_zeta_calculation . The Sommerfeld coefficient, which is related to the
quasiparticle number density per volume with mean free path, is given by Eq.
(12)

$\gamma_{n}=\left(\frac{\pi}{3}\right)^{2/3}\frac{k_{B}^{2}m^{*}V_{f.u.}n^{1/3}}{\hbar^{2}N_{A}},$
(12)

where $k_{B}$ is the Bolzman constant, $m^{*}$ is effective mass, $V_{f.u.}$
is number density per volume of the quasiparticle, and $N_{A}$ is Avogadro
number. The relationship between the mean free path $l_{e}$, the residual
resitivity $\rho_{0}$, and the Fermi velocity is

$l_{e}=\frac{3\pi^{2}\hbar^{3}}{e^{2}\rho_{0}m^{*2}v_{F}}.$ (13)

The value of the Sommerfeld coefficient used in this study is 5.7 mJ/mol K-2,
obtained from the previous report on specific heat measurement for 1$T$-NbSeTe
NbSeTe_reported . Using carrier concentration and residual resistivity, the
effective mass, Fermi velocity, and mean free path are calculated to be 6.27
me, 0.74 $\times$ 105 m/s and 36.05 (6) Å, respectively, with a coherence
length of 324(2) Å lamda_calculation . The $\frac{\xi_{0}}{l_{e}}$ ratio is 9,
indicating that the 1$T$-NbSeTe sample is in the dirty limit.

According to Uemura et al., superconducting materials can be classified as
conventional or unconventional based on their $T_{c}/T_{F}$ ratio.
Unconventional superconductors such as Chevrel phases, heavy fermions, Fe-
based superconductors, and high $T_{c}$ superconductors have a $T_{c}/T_{F}$
ratio within the range of 0.01 $\leq$Tc/T${{}_{F}}\leq$0.1 uemra_plot_1 ;
umara_plot_2 . To determine the classification of 1$T$-NbSeTe, the Fermi
temperature $T_{F}$ was evaluated using the equation:

$k_{B}T_{F}=\frac{\hbar^{2}}{2}(3\pi^{2})^{2/3}\frac{n^{2/3}}{m^{*}}$ (14)

where n is the carrier density of 2.22 $\times$ 1027 m-3 and m∗, the effective
mass is 6.27me for the sample. The calculated Fermi temperature for NbSeTe is
1151(4) K (as shown in Fig. 7). The obtained $T_{c}/T_{F}$ ratio of 0.0027 is
in the unconventional range, similar to other unconventional materials, which
may indicate the presence of a topological phase.

Figure 7:  The plot of the superconducting transition temperature vs the Fermi
temperature for different superconducting families. In between, two solid blue
lines show the unconventional band of superconductors. NbSeTe lies close to
the unconventional band.

##  IV Conclusion

In summary, we have successfully synthesized single crystals of NbSeTe in the
trigonal 1$T$ phase with space group 164. The material exhibits exotic chiral
anomaly-induced PHE with $H^{2}$ behavior and has a superconducting transition
temperature of 3.16 K. The in-plane upper critical field exceeds the Pauli
limit, and the angle-dependent upper critical field suggests the presence of
2D superconductivity, indicating unconventional superconductivity in bulk
NbSeTe. Furthermore, the Uemura plot also supports the presence of
unconventional superconductivity in this material. The coexistence of
topological semimetallic nature and superconductivity in NbSeTe makes it a
potential candidate for topological superconductivity. However, further
investigations, including angle-resolved photoemission spectroscopy (ARPES),
detailed electronic structure calculations, and low-temperature and thickness-
dependent measurements, are necessary to fully understand this system’s
superconducting pairing mechanism.

##  V Acknowledgments

R. P. S. acknowledge Science and Engineering Research Board, Government of
India, for the CRG/2019/001028 Core Research Grant.

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