# Experimental observation of a Rindler horizon

Morgan H. Lynch<EMAIL_ADDRESS>Center for Theoretical Physics,
Seoul National University, Seoul 08826, Korea

###### Abstract

In this manuscript we confirm the presence of a Rindler horizon at CERN-NA63
by exploring its thermodynamics induced by the Unruh effect in their high
energy channeling radiation experiments. By linking the entropy of the emitted
radiation to the photon number, we find the measured spectrum to be a simple
manifestation of the second law of Rindler horizon thermodynamics and thus a
direct measurement of the recoil Fulling-Davies-Unruh (FDU) temperature.
Moreover, since the experiment is born out of an ultra-relativistic positron,
and the FDU temperature is defined in the proper frame, we find that
temperature boosts as a length and thus fast objects appear colder. The
spectrum also provides us with a simple setting to measure fundamental
constants, and we employ it to measure the positron mass.

##  I Introduction

The pursuits of quantum field theory in curved spacetime have led to a wide
array of surprising discoveries. Most notably, it completely changed our
understanding of such simple questions as how many particles are present in a
given system. This ambiguity in particle number gave rise to a new class of
kinematic particle production induced by horizons; namely the Parker, Hawking,
and the Fulling-Davies-Unruh effects [1, 2, 3, 4, 5, 6, 7]. Most surprising of
all is the fact that the particles which are produced via these effects are
thermalized at a temperature which is characterized by the acceleration scale
of the system. In all cases, this acceleration characterizes the location of a
horizon, either real or apparent, which is also present.

When considering these horizons, one can also ascribe thermodynamic quantities
to the system. Most notably, via the second law of thermodynamics, the
connection between the area of the horizon, or area change, and the entropy
encoded within it [8, 2, 9]. All of these notions combined paint a rather
surprising picture; that for an accelerated observer, the presence of these
thermalized particles along with the energy/momentum flux of particles through
the associated horizon gives rise to an area change in the horizon, which is
completely determined by general relativity; i.e. the Einstein equation is the
thermodynamic equation of state of these effervescent particles [10]. The
implication being that gravitation is an emergent quantity born out of these
vaccuum fluctuations.

The recent experimental observation of radiation reaction and the Unruh effect
by CERN-NA63 [11, 6] provides us with an unprecendented opportunity to
investigate the various tenets of quantum field theory in curved spacetime. In
particular, we will now examine the data via the thermodynamics of the Rindler
horizon.

##  II Rindler horizon thermodynamics

Thermodynamics offers a surprisingly simple technique to examine the
properties of both black holes as well as Rindler horizons [8, 10, 9]. The
temperatures of these systems can also be explored via radiation reaction.
There we have a FDU temperature which is set by the recoil kinetic energy,
$T_{FDU}=\frac{(\hbar\omega)^{2}}{2mc^{2}k_{B}}$. Systems that are thermalized
at this temperature will then, by necessity, obey the second law of
thermodynamics,

$dQ=k_{B}TdS.$ (1)

As we shall see, this statement of the second law not only describes the
relationship between energy flux and entropy generation, but it also describes
the spectrum of thermalized particles due to the Unruh effect. Ultimately,
this second law, and its application to Rindler horizons, lies at the heart of
our understanding of gravity. As it has been shown [10], this second law along
with the Bekenstein-Hawking area-entropy relation,
$S=\frac{1}{4\ell_{P}^{2}}A$, with $\ell_{P}^{2}$ being the Planck area, is
equivalent to the Einstein equation of general relativity.

###  II.1 Experimental methods

The CERN-NA63 experimental site studies various aspects of strong field QED
[12, 13]. Here we analyse their high energy channeling radiation experiement
which successfully measured radiation reaction [11]. There, an ultra-
relativistic $178.2$ GeV positron traverses a 3.8 mm thick sample of single
crystal silicon along the $\braket{111}$ axis. These “channeled” positrons are
repelled by the atomic lattice and undergo a transverse harmonic oscillation
which causes a photon to build up around the positron as it is pumped up the
ladder of harmonic oscillator states. This photon becomes so energetic that
its energy is comparable to the positron rest mass and upon emission, the
positron experiences an enormous recoil acceleration. This acceleration is
sufficiently strong enough to thermalize the system via the Unruh effect [6].

In order to analyze the NA63 data set using the second law, we must first
transform their power spectrum, $\frac{dE}{d(\hbar\omega)dt}$ into a photon
spectrum, $\frac{dN}{d(\hbar\omega)}$. We shall take the crystal crossing time
to be the time interval, $dt=(3.8$ mm)$/c$. Next, the single particle
spectrum, is given by $\frac{dE_{sp}}{d(\hbar\omega)}\frac{1}{\hbar\omega}$
and the differential bin spectrum is given by
$\frac{dE}{d(\hbar\omega)}\frac{1}{\Delta E_{b}}$, where $\Delta E_{b}=1.007$
GeV is the bin width. It is this differential bin spectrum which we will
analyze. Thus we have,

$\frac{dN}{d(\hbar\omega)}=\frac{dE_{data}}{d(\hbar\omega)dt}\frac{3.8\;mm}{c\Delta
E_{b}}$ (2)

A comparison of the different spectra is presented in Fig. (1). We will now
turn to the theoretical description of this data based on Rindler horizon
thermodynamics.

Figure 1: A comparison of the single particle photon spectrum and the
differential bin spectrum, Eqn. (2), of the CERN-NA63 3.8 mm radiation
reaction data set [11].

###  II.2 The Rindler entropy photon spectrum

Although it is comonly believed that thermal radiation contains no
information, there is, in fact, a persistance of information in the particles
radiated [14, 15]. Although small, each photon emitted will carry away a
portion of the information content present prior to burning. This provides us
with a method to link the spectrum of a thermal system to its information
content via the second law of thermodynamics.

We begin by exploring the information content of the Rindler horizon. In
particular, we shall examine the entropy-information content which is
“embedded” into the the surface of the horizon by the radiation that passes
through it. In consideration of the entropy, $S$, we will then have
$\alpha_{S}$ nats of information per $N$ photons, i.e. $S=\alpha_{S}N$.
Finally taking the energy that crosses the horizon to be that of one photon,
$dQ=d(\hbar\omega)$, our second law can be written as

$\frac{dN}{d(\hbar\omega)}=\frac{1}{\alpha_{S}}\frac{1}{k_{B}T}.$ (3)

The above expression is the photon spectrum which has been mapped from the
entropy of the Rindler horizon to the photons which have passed through it.
These photons will be thermalized at the FDU temperature, $T=T_{FDU}$. We must
note that there is still a slight ambiguity in the recoil temperature due to
the photon emission time. From the relativistic Newtons law, we have a proper
acceleration of $a=\frac{1}{m}\frac{\Delta p}{\Delta t}$. We have $\Delta
p=\hbar\omega/c$, however for radiation reaction we also expect the time scale
of emission to be on the order of a photon period, i.e. $\Delta t\sim
1/\omega$. What the precise proportionality is, remains unknown. In the
radiation analysis of the Unruh effect [6], $\Delta t=\pi/\omega$ was used,
and here we have used $\Delta t=1/(\pi\omega)$ such that the temperature is
set to the recoil kinetic energy. The true form of the recoil acceleration,
and thus temperature remains an open problem however. We simply note, that
there is still a constant of order unity which remains to be fixed. As such,
we will adopt a coefficient $\alpha_{T}$ to rescale our emission time, $\Delta
t=\alpha_{T}/(\pi\omega)$ in the temperature,
$T_{FDU}=\frac{(\hbar\omega)^{2}}{\alpha_{T}2mc^{2}k_{B}}$.

Finally, we note the fact that this is an ultra-relativistic system that is
thermalized and thus gives us insight into how a temperature boosts [16, 17,
18, 19]. We find that temperature boosts like a length, i.e.
$T_{lab}=T_{proper}/\gamma$. As such, in order to match the data set, our
temperature must be boosted into the lab frame via, $T_{lab}=T_{FDU}/\gamma$.
Thus, our spectrum is given by,

$\frac{dN}{d(\hbar\omega)}=\alpha_{TS}\frac{2mc^{2}\gamma}{(\hbar\omega)^{2}}.$
(4)

By taking a best fit of $\alpha_{TS}=\alpha_{T}/\alpha_{S}$, we can gain
insight into the number of bits associated with each photon as well as the
unknown emission timescale. With a threshold of $22$ GeV based on the
thermalization time of the system and using the energy range of $30$ GeV to
$120$ GeV, where the chi-squared statistic lies within the 1 standard
deviation threshold in the radiation analysis [6], we find $\alpha_{TS}=1.61$
with a reduced chi-squared statistic of 1.90. We must comment on the fact that
although this chi-squared is excellent, it is not below the threshold of 1.26.
This is most likely due to the fact there are, in principle, additional terms
in our second law which are not included in this analysis that would
correspond to chemical potentials. These reflect the various terms which
comprise the energy gap of the Unruh-DeWitt detector. We also note the entropy
content per photon of blackbody radiation is given by [14, 15],
$\alpha_{S}=2.7\pm 1.7$. Thus if we take $\alpha_{S}=2.7$ then this would
imply that $\alpha_{T}=4.38$ and thus $\Delta t=1.39/\omega$, i.e.
approximately one fourth of a photon period. Therefore, the ambiguity due to
these two parameters is an overall factor of order unity. Given one, the other
is then fixed. What is important to note, is that the entropy content,
$\alpha_{S}=2.7$, is for a 3-dimensional blackbody and its value depends
crucially on the dimensionality of the system [20]. It is an important concept
to consider, that the dimensionality that governs the thermodynamics of the
Rindler horizon may be 1 or perhaps 2-dimensional [21].

Figure 2: Here we present the measured photon bin spectrum compared to the
theoretical prediction based on the second law of Rindler horizon
thermodynamics. Here, the 2nd law spectrum is with $\alpha_{TS}=1$ and the
$\alpha_{TS}$-2nd law spectrum is the best fit with $\alpha_{TS}=1.61$. The
reduced chi-squared per degree of freedom for this fit is 1.90.

###  II.3 Measurement of the positron mass

Given a succesful description of the photon spectrum based on the second law
of thermodynamics, it is interesting to note that within the temperature, one
is able to isolate any of the physical constants present; namely $c$, $k_{B}$,
$\hbar$, or the positron mass $m_{p}$. This provides an interesting technique
for measuring physical constants. Although many of these constants are used as
experimental inputs prior to the measurement, it seems reasonable that one,
say the positron mass, $m_{p}$, may be left out of any preliminary
experimental inputs and then be measured via the overall coefficient on the
frequency dependent temperature, $T_{FDU}$. Thus, we can solve for the
positron mass to yield,

$m=\frac{1}{\alpha_{TS}}\frac{(\hbar\omega)^{2}}{2c^{2}\gamma}\frac{dN}{d(\hbar\omega)}.$
(5)

The measurement of the positron mass is presented in Fig. (3). As such, we
find, via the recoil FDU temperature, a thermometer capable of measuring mass
and/or other physical constants. From the data set, we average the measured
positron mass over the region where the chi-squared statistic remains below
the 1-standard deviation threshold from the radiation analysis; $30-120$ GeV.
We obtain a value of $m_{p}=1.04\pm.221\times 10^{-30}$ kg or $.583\pm.124$
MeV/$c^{2}$. Here the error comes from the standard deviation of the mean.
Note, the statistical error here is the dominant source of error as the
experimental error bars are smaller than the data points. The accepted value
of the positron mass is $m_{p}=9.11\times 10^{-31}$ kg or $.511$ MeV/$c^{2}$
[22].

Figure 3: Here we see the convergence of the experimental data set to the
positron rest mass. Given a recoil FDU temperature,
$T_{FDU}=\frac{(\hbar\omega)^{2}}{2mc^{2}k_{B}}$, one can use an Unruh-
thermalized spectrum to measure a selection of fundamental constants.

##  III conclusions

In this manuscript, we have analyzed the Unruh-thermalized photon spectrum of
channeled positrons undergoing the extreme accelerations of radiation reaction
measured by CERN-NA63. We employed an analysis based entirely on the 2nd law
of Rindler horizon thermodynamics, at the recoil FDU temperature, and found an
excellent agreement with the data. Moreover, the ultra-relativistic nature of
the system revealed that temperatures boost like a length, i.e.
$T_{lab}=T_{proper}/\gamma$. The technique also provides a novel way to
experimentally measure the positron mass.

## Acknowledgments

This work has been supported by the National Research Foundation of Korea
under Grants No. 2017R1A2A2A05001422 and No. 2020R1A2C2008103.

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11institutetext: Department of Physics, SRM University AP, Amaravati 522240,
Andhra Pradesh, India22institutetext: School of Physics and Astronomy,
University of Southampton, Southampton SO17 1BJ, UK33institutetext: Department
of Physics and Astronomy, Uppsala University, Box 516, SE-751 20 Uppsala,
Sweden

# Leptonic CP-violation in the sneutrino sector of the BLSSM with Inverse
Seesaw

Arindam Basu 1 Amit Chakraborty 2 Yi Liu 2,3 Stefano Moretti 3 and Harri
Waltari<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study CP violation (CPV) in the sneutrino sector within the B-L extension
of the Minimal Supersymmetric Standard Model (BLSSM), wherein an inverse
seesaw mechanism has been implemented. CPV arises from the new superpotential
couplings in the (s)neutrino sector, which can be complex and the mixing of
CP-eigenstates induced by those couplings. CPV leads to asymmetries in so
called T-odd observables, but we argue that such asymmetries also lead to a
wider distribution of those observables. We look at a final state where a
sneutrino decays to a lepton, two jets and missing transverse momentum at the
Future Circular Collider operating in hadron-hadron mode at $100$ TeV and with
a luminosity of 3 ab-1. In order to exclude the CP conserving scenario we need
to improve traditional analysis by introducing boosted decision trees using
both standard kinematic variables and T-odd observables and we need
$Z^{\prime}$ boson not too much above current bounds as a portal to produce
sneutrinos efficiently.

##  1 Introduction

Non-zero neutrino masses have been observed by several experiments, in fact,
such observations represent one of the strongest hints for physics Beyond the
Standard Model (BSM), as in the latter scenario neutrinos are assumed to be
massless, for two reasons. Firstly, there are no Right-Handed (RH) neutrinos
as Dirac fermions to which weak currents can couple to. Secondly, the SM
strictly conserves the Baryon minus Lepton ($B-L$) number. To address this,
though, the SM can be extended by a new $B-L$ symmetry, e.g., with a $U(1)$
gauge structure, allowing for a Majorana nature of neutrinos. The emerging
$U(1)_{B-L}$ extended SM, or BLSM (see Ref. Khalil:2012gs ; Khalil:2013in  for
a review) thus leads to new testable signals, including heavy neutrinos, a
neutral gauge boson $Z^{\prime}$ and a new Higgs state that breaks the
$U(1)_{B-L}$ gauge group Wetterich:1981bx ; Buchmuller:1991ce ; Khalil:2007dr
; Basso:2008iv ; Basso:2009gg ; Basso:2009zz ; Basso:2009hf ;
FileviezPerez:2009hdc ; Basso:2010dq ; Basso:2010si ; Basso:2010jm ;
Basso:2010tv ; Basso:2010jt ; Li:2010rb ; Abdallah:2015uba ; Abdallah:2015hma
; Accomando:2016rpc ; Accomando:2017qcs ; Gutierrez-Rodriguez:2024nny ;
Martinez-Martinez:2023qjt ; Iso:2009nw ; Dev:2021qjj .

A seesaw mechanism provides an elegant explanation for neutrino masses. In a
canonical seesaw mechanism, such as, e.g., Type-I seesaw, RH neutrinos acquire
mass at the $B-L$ symmetry breaking scale (TeV or so). Consequently, the
corresponding Yukawa couplings are constrained to be smaller than
$\mathcal{O}(10^{-6})$ Khalil:2007dr . The inverse seesaw mechanism
Khalil:2006yi ; Khalil:2010iu  can allow larger Yukawa couplings, making this
scenario more feasible to be tested at the Large Hadron Collider (LHC).
Specifically, by introducing two SM gauge singlet fermions, the light neutrino
obtains its mass when one of the two singlets interacts with the RH neutrino.
Furthermore, the other singlet can serve as a Dark Matter (DM) particle, for
which there is no viable candidate in the SM El-Zant:2013nta  (see also Ref.
Basso:2012gz ).

There are some hints of a non-zero CP phase in the neutrino sector T2K:2019bcf
; NOvA:2021nfi , although the results come with large error bars. If Charge
and Parity Violation (CPV) arises from the dynamics that generates neutrino
masses, it may eventually manifest itself in specific kinematical observables,
which appears as a difference in the angular distributions of multi-body decay
processes of a particle and corresponding antiparticle. Triple-Product
Asymmetries (TPAs) are proportional to the scalar triple product of the
generic form:

${\rm TPA}=\vec{v}_{1}\cdot(\vec{v_{2}}\times\vec{v_{3}}),$ (1)

where the vectors $\vec{v}_{i}$ are the three-momenta of final state
particles. TPAs are generally T-odd observables and those that can violate CP
symmetry are known as ‘true’ TPAs. Therefore, TPAs can serve as an indicator
of CPV BSM physics with a relatively clean background from the SM (only onset
by Cabibbo-Kobayashi-Maskawa (CKM) phases), which can lead to a deeper
understanding of the CP properties of such new interactions Gronau:2011cf ;
Bevan:2014nva . In this work we show that besides the non-zero expectation
value of the triple product, also the width of its distribution may serve as
an indicator of CPV.

Another flaw of the SM is the so-called hierarchy problem, which can however
be elegantly solved by Supersymmetry (SUSY) Moretti:2019ulc , so it is
tempting to combine the $U(1)_{B-L}$ symmetry with inverse seesaw and SUSY.
This leads to the $B-L$ Supersymmetric SM with IS (BLSSM-IS), which will serve
at the theoretical backdrop of our analysis of TPAs, as it can well embed CPV
in its neutrino sector. (For phenomenological studies of the BLSSM, in both a
Type-I and IS configuration, see Refs. Elsayed:2011de ; Elsayed:2012ec ;
Abdallah:2014fra ; Hammad:2015eca ; Hammad:2016trm ; Abdallah:2016vcn ;
DelleRose:2017hvy ; DelleRose:2017ukx ; Ahmed:2020lua ; Biswas:2018yus ;
Abdallah:2019svm ; Kanemura:2014rpa .)

The structure of this paper is as follows. The BLSSM-IS model and how CPV
signatures can emerge in its leptonic sector is described in Section 2. This
is followed by a section introducing some Benchmark Points (BPs) over the
parameter spaces of this BSM scenario alongside presenting some details of our
Monte Carlo (MC) simulation. In the following part we perform our collider
analysis using both a cut-based analysis and a Boosted Decision Tree (BDT) and
present the ensuing results, including a discussion of the $Z^{\prime}$ portal
used and of the CP-Conservation (CPC) limit. We then conclude.

##  2 The BLSSM-IS scenario and CPV

The BLSSM-IS extends the MSSM by adding three parts. The first part contains
two SM singlet chiral Higgs superfield $\chi_{1}$ and $\chi_{2}$ with $B-L$
charge equal to $+1$ and $-1$, respectively. When the scalar components of the
superfields $\chi_{1}$ and $\chi_{2}$ get their VEVs, they break the
$U(1)_{B-L}$ symmetry spontaneously. The second part includes three SM singlet
chiral superfields: $N_{i}$ ($i=1,2,3$) with $B-L$ charge equal to $-1$. They
are used to cancel the $U(1)_{B-L}$ anomaly and the fermionic component of
$N_{i}$ is considered as a RH neutrino. The third part consists of two chiral
SM singlet superfield $S_{1}$ and $S_{2}$ with $B-L$ charge equal to $+2$ and
$-2$, respectively, in order to implement the inverse seesaw mechanism
Elsayed:2011de ; Khalil:2010iu . The superpotential of the leptonic sector is

$W=Y_{E}LH_{1}E^{c}+Y_{\nu}LH_{2}N^{c}+Y_{S}N^{c}\chi_{1}S_{2}+\mu
H_{1}H_{2}+\mu^{\prime}\chi_{1}\chi_{2}+\mu_{S}S_{2}S_{2}.$ (2)

When the EW symmetry and $B-L$ symmetry break, the Higgs fields and the
$\chi_{i}$ ($i=1,2$) get a VEV. This leads to neutrino mass terms

$\mathcal{L}=m_{D}\bar{\nu}_{L}N^{c}+M_{N}\bar{N}^{c}S_{2}+\mu_{S}S_{2}S_{2},$
(3)

where $m_{D}=Y_{\nu}v\sin{\beta}$ and $M_{N}=Y_{S}v^{\prime}\sin{\theta}$,
with $\tan\beta=\frac{v_{2}}{v_{1}}=\frac{\langle H_{2}\rangle}{\langle
H_{1}\rangle}$ and
$\tan\theta=\frac{v_{1}^{\prime}}{v_{2}^{\prime}}=\frac{\langle\chi_{1}\rangle}{\langle\chi_{2}\rangle}$.
Specifically, $v^{2}=v_{1}^{2}+v_{2}^{2}$ and
${v^{\prime}}^{2}={v^{\prime}_{1}}^{2}+{v^{\prime}_{2}}^{2}$. The smallness of
Left-Handed (LH) neutrino masses are related to a small lepton number
violating term $\mu_{S}S_{2}S_{2}$, with $\mu_{S}\approx\mathcal{O}(1)$ keV,
which can emerge at the $B-L$ scale from the non-renormalizable term in the
superpotential $\frac{\chi_{1}^{4}S_{2}^{2}}{M_{I}^{3}}$, with $M_{I}$ an
intermediate scale of the order $\mathcal{O}(10^{7})$ GeV Elsayed:2011de . The
$3\times 3$ neutrino mass matrix of one generation has the following form (in
the basis $(\nu_{L},N^{c},S_{2})$):

$M_{\nu}=\left(\begin{array}[]{ccc}0&m_{D}&0\\\ m_{D}&0&M_{N}\\\
0&M_{N}&\mu_{S}\end{array}\right).$ (4)

one finds that the light neutrino masses are

$m_{\nu_{L}}=\frac{m^{2}_{D}\mu_{S}}{M^{2}_{N}+m^{2}_{D}}$ (5)

and the heavy ones being nearly degenerate with a mass

$m=\sqrt{M^{2}_{N}+m^{2}_{D}}\simeq M_{N}.$ (6)

This model is of interest for studying the CPV in the sneutrino sector for two
reasons. First, the additional gauge boson $Z^{\prime}$ couples to sneutrinos
and can act as a portal for producing them. Without such a portal the
production cross section for heavy sneutrinos would be low even at future
colliders. Second, the inverse seesaw mechanism allows for large
superpotential couplings in the neutrino-sneutrino sector, which is necessary
for the CPV phases to create large effects in observables. In BLSSM-IS both
$Y_{\nu}$ and $Y_{S}$ can be large. Their relative importance depends on the
detailed decay chain of the sneutrinos.

We shall next show how CPV could be observed in the sneutrino sector. If any
of the superpotential coefficients in Equation (2) is complex, this can lead
to CPV, provided one cannot absorb the phases by redefining the superfields.
The Yukawa terms arising from a complex superpotential coupling
$y_{ijk}X_{i}X_{j}X_{k}$ are of the form

$\mathcal{L}=y_{ijk}\bar{\Psi}_{i}\left(\frac{1-\gamma^{5}}{2}\right)\Psi_{j}\varphi_{k}+y_{ijk}^{*}\bar{\Psi}_{i}\left(\frac{1+\gamma^{5}}{2}\right)\Psi_{j}\varphi_{k}^{*}+\mathrm{permutations}.$
(7)

As long as the coupling $y_{ijk}$ is neither real nor purely imaginary, there
will be both a scalar and a pseudoscalar coupling between the fermions $i,j$
and the scalar $k$ (+permutations in $ijk$). This leads to direct CPV, since
the interference of the scalar and pseudoscalar parts produces a parity
violating distribution. Since $\bar{\Psi}\Psi$ and $i\bar{\Psi}\gamma^{5}\Psi$
are both even under charge conjugation, conservation of CPT tells us that this
part is also T-odd. Hence, triple products of momenta, of the form
$p_{1}\cdot(p_{2}\times p_{3})$, which are odd both under P and T will be
sensitive to such a form of CPV.

There is also another source for CPV effects that arises from the complex
Yukawa couplings. That comes from the mixing of CP-even and CP-odd sneutrino
states in the sneutrino mass matrix induced by the complex Yukawa couplings.
The mixture of CP-even and CP-odd components in the amplitude will again lead
to terms with varying numbers of $\gamma^{5}$ factors in them. In the general
case the mixing between CP-even and CP-odd states can also be a result of
complex soft SUSY breaking mass matrices. Here we take the soft SUSY breaking
masses to be real.

We consider the process $pp\rightarrow
Z^{\prime}/Z\rightarrow\tilde{\nu}\tilde{\nu}$, when one of the sneutrinos
decays into the lightest chargino and a lepton, while the other sneutrino
decays to a neutrino and a neutralino. The charginos cascade into the lightest
neutralino and $W^{\pm}$ boson, leading to a di-jet final state when the $W$
decays hadronically. As a result, we get a final state signature involving di-
jets, at least one isolated lepton, and significantly large MET. The Feynman
diagram of such a process is shown in figure 1. The CPV phases in the
(s)neutrino sector can arise from both the $Y_{\nu}$ and $Y_{S}$ couplings of
Equation (2) and the mixing of the CP-even and CP-odd states.

From the structure of the Superpotential we may notice that direct CPV can
give a detectable final state only for RH sneutrinos through the term
$Y^{\nu}LH_{2}N^{c}$, while for LH sneutrinos this term would lead to the
invisible neutrino+neutralino final state. CPV arising from left-handed
sneutrinos comes hence from mixing and only the wino-components of the
charginos will contribute. If the wino mass parameter is larger than the
sneutrino mass, this contribution is somewhat suppressed.

The most dominant contribution of an SM background here arises from the
process: $pp\rightarrow W^{\pm}+{\rm jets}$, where the $W^{\pm}$ boson decays
into lepton and neutrino. However, processes like $t\bar{t}+{\rm jets}$ with
semi-leptonic/di-leptonic final states can also contribute significantly. As
we discuss below, we are not going to be looking at the expectation value of
T-odd observables, which would limit us to only processes involving the phase
of the CKM matrix. We expect that especially the $Z^{\prime}$ mediated signal
will contain leptons and jets that are on average harder than what is typical
for SM processes. Hence the contribution from $t\bar{t}$, which tends to lead
to hard jets and lepton in the SM, will be an important background.

In this paper, we focus on the CPV effects through the cascaded decays of
heavy Supersymmetric particles. When an unstable heavy particle decays to four
lighter particles through a cascade, the decay rate includes a term of the
form
$\epsilon_{\mu\nu\alpha\beta}p_{0}^{\mu}p_{1}^{\nu}p_{2}^{\alpha}p_{3}^{\beta}$,
where $p_{i}$ is the four-momentum of the $i$-th particle Langacker:2007ur .
This arises from the trace of an odd number of $\gamma^{5}$ matrices together
with four $\gamma$-matrices. Many such combinations exist in the decay
amplitude, e.g. taking the vector parts of the $W$-boson vertices together
with a $\gamma^{5}$ arising from the pseudoscalar part of the complex Yukawa
coupling. In the rest frame of the decaying particle, this term provides the
CP-odd observable given by the expectation value of the triple product,

$\mathcal{T}=\vec{p_{1}}\cdot(\vec{p_{2}}\times\vec{p_{3}})$ (8)

This T-odd observable (triple product) is sensitive to the CP-odd phases and
provides a good discriminator between the signal and background (having only
complex phases in the CKM matrix as the CP-odd phase). We further formulate
the T-odd observables depending on the final state topology of the
signal/background.

Figure 1: The Feynman diagram showing one of the possible production and decay
modes of the sneutrinos. The $W^{\pm}$ will eventually decay hadronically
giving multiple jets in the final state. Here all the three flavours of SM
neutrinos are collectively denoted as $\nu_{\rm SM}$, while all the six
sterile neutrinos are denoted as $\nu_{\rm BSM}$.

Our three momentum vectors will quite often involve one lepton and two jets.
To properly evaluate the triple product of Equation (8), we should be able to
distinguish a quark jet from an antiquark jet. We shall not try this.
Therefore, since at random we pick equal amounts of quark and antiquark jets
for a given $p_{i}$, the expectation value $\langle\mathcal{T}\rangle$ will
vanish. However, the non-zero expectation value has an influence on the
distribution. We assume that there is an intrinsic variance to the measurement
of $\mathcal{T}$, given by

$V(\mathcal{T})=\langle\mathcal{T}^{2}\rangle-\langle\mathcal{T}\rangle^{2},$
(9)

when the momenta are in some given order and that this does not depend on
$\langle\mathcal{T}\rangle$. Then, we assume that there is some distribution
$f(\mathcal{T})$ that gives the probability density of $\mathcal{T}$, when the
momenta are in some particular order, say, lepton–quark–antiquark. Then
$f(-\mathcal{T})$ gives the probability density for the ordering
lepton–antiquark–quark and the overall distribution is given by
$\frac{1}{2}(f(\mathcal{T})+f(-\mathcal{T}))$. Now,

$\displaystyle\langle\mathcal{T}\rangle$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}f(\mathcal{T})\mathcal{T}\,\mathrm{d}\mathcal{T},$
(10) $\displaystyle\langle\mathcal{T}^{2}\rangle$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}f(\mathcal{T})\mathcal{T}^{2}\,\mathrm{d}\mathcal{T}.$
(11)

The variance for the distribution
$\frac{1}{2}(f(\mathcal{T})+f(-\mathcal{T}))$ can be computed from

$\displaystyle\langle\mathcal{T}_{\rm sym}\rangle$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}\frac{1}{2}(f(\mathcal{T})+f(-\mathcal{T}))\mathcal{T}\,\mathrm{d}\mathcal{T}=0,$
(12) $\displaystyle\langle\mathcal{T}_{\rm sym}^{2}\rangle$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}\frac{1}{2}(f(\mathcal{T})+f(-\mathcal{T}))\mathcal{T}^{2}\,\mathrm{d}\mathcal{T}=\langle\mathcal{T}^{2}\rangle=V(\mathcal{T})+\langle\mathcal{T}\rangle^{2},$
(13)

where a change of variables $\mathcal{T}\rightarrow-\mathcal{T}$ has been used
in the second terms. If we assume that $V(\mathcal{T})$ is independent of
$\langle\mathcal{T}\rangle$, then $\langle\mathcal{T}^{2}\rangle$ is larger if
$\mathcal{T}$ has a non-zero expectation value. Since CPV leads to a non-zero
expecation value, we expect the variance of $\mathcal{T}_{\rm sym}$ to be
larger than for a distribution without an expectation value, i.e., the
distribution to be wider in the presence of CPV. Hence we use this as our
indicator of CPV. If one will be able to distinguish between quark and
antiquark jets, the simpler criterium $\langle\mathcal{T}\rangle\neq 0$ is
sufficient.

##  3 The BPs and simulation details

We wish to study the leptonic CPV in the sneutrino sector. We introduce
explicit CPV in terms of two complex Yukawa matrices $Y_{\nu}$ and $Y_{S}$. Of
these, the matrix $Y_{\nu}$ is relevant for decays of the type
$\tilde{\nu}\rightarrow\ell^{\pm}\tilde{H}^{\mp}$. We prepare our particle
spectrum in such a way that there is a neutralino somewhat lighter than the
higgsino such that one gets relatively energetic jets and a reasonably large
amount of missing momentum from the decay of the higgsino. As discussed above,
for left-handed sneutrinos only the wino-component of the chargino is relevant
and the CPV effects will arise from mixing.

For a detailed analysis we choose the three pairs of BPs given in Tables 1–3.
Within a pair we take a CPV BP, where Yukawa matrices are complex and a CPC
one, where the Yukawa couplings are the real parts of the corresponding CPV
couplings and slight adjustments are made to soft SUSY breaking masses so that
the mass spectrums are as identical as possible. These pairs allow us to
compare the widths of the triple product distributions and to show that the
width of the distribution is different, when sizable CPV is involved.

We choose the model parameters in such a manner that the masses of some of the
lighter sneutrinos as well as of the neutralinos and charginos are below a
TeV, while the rest of the particle spectrum is decoupled to a few TeV (see
Table 3). The masses of the $Z^{\prime}$ (Table 1) are chosen slightly above
current experimental bounds since, as we discuss in section 4.5, the cross
section from the SM $Z$ portal alone is too low. The model files are generated
using Sarah v4.14 Staub:2015kfa , which also provides source codes for SPheno
v4.0.4 Porod:2003um ; Porod:2011nf  through which we obtain the particle
spectrum and different couplings relevant to these BPs.

Parameter | BP1 | BP2 | BP3
---|---|---|---
$\tan\beta$ | 20 | 10 | 20
$\tan\theta$ | 1.05 | 1.08 | 1.06
$M_{Z^{\prime}}$ [TeV] | 5.20 | 5.30 | 5.30
Table 1: The values chosen for some of the free parameters of the BLSSM-
IS.Furthermore, we choose $g_{B-L}=0.46$ for all points.

We show some general model parameters in Table 1 and the CPV Yukawa couplings
in Table 2. For BP1 we choose relatively small phases, BP2 has large phases
and BP3 intermediate phases. Kinematically BP1 and BP3 resemble each other,
while the neutralino/chargino spectrum is different for BP2 leading to harder
leptons and jets. Since our CPV enters only through leptonic interactions, the
contribution to hadronic CPV (which is experimentally constrained) comes at
higher loop orders. We expect that contribution to be so small that the
experimental constraints are satisfied even with large CPV phases in the
leptonic sector.

Parameter | BP1 | BP2 | BP3
---|---|---|---
$Y_{\nu 11}$ | 0.22+0.006i | 0.25+0.14i | 0.34+0.08i
$Y_{\nu 12}$ | 0.12+0.0027i | -0.16+0.11i | -0.23+0.067i
$Y_{\nu 13}$ | -0.067-0.004i | 0.06-0.04i | 0.086-0.016i
$Y_{\nu 21}$ | 0.12+0.013i | 0.15+0.01i | 0.25+0.1i
$Y_{\nu 22}$ | 0.31+0.011i | 0.43+0.26i | 0.53+0.061i
$Y_{\nu 23}$ | 0.17-0.006i | 0.39-0.2i | 0.46-0.1i
$Y_{\nu 31}$ | -0.074+0.012i | -0.074-0.046i | -0.094+0.027i
$Y_{\nu 32}$ | 0.16+0.053i | 0.17+0.14i | 0.22+0.083i
$Y_{\nu 33}$ | 0.34+0.014i | 0.37+0.3i | 0.43+0.12i
Table 2: The complex Yukawa matrix elements are shown for the three CPV BPs. For the first one the phases are small, the two others have large phases. Particle | BP1 | BP2 | BP3
---|---|---|---
$\tilde{\nu}_{1}$ | 871.3 | 902.0 | 884.6
$\tilde{\nu}_{2}$ | 871.3 | 902.0 | 884.6
$\tilde{\nu}_{3}$ | 922.5 | 903.0 | 925.2
$\tilde{\nu}_{4}$ | 922.5 | 903.0 | 925.2
$\tilde{\nu}_{5}$ | 940.5 | 904.4 | 951.0
$\tilde{\nu}_{6}$ | 940.5 | 904.4 | 951.0
$\tilde{\chi}_{1}^{\pm}$ | 583.7 | 659.2 | 583.0
$\tilde{\chi}_{1}^{0}$ | 533.6 | 514.6 | 531.4
$\tilde{\chi}_{2}^{0}$ | 587.1 | 664.2 | 586.5
$\tilde{\chi}_{3}^{0}$ | 602.8 | 664.3 | 602.1
$\nu_{4}$ | 16.26 | 31.05 | 35.25
$\nu_{5}$ | 16.26 | 31.05 | 35.25
$\nu_{6}$ | 56.30 | 64.34 | 76.47
$\nu_{7}$ | 56.30 | 64.34 | 76.47
$\nu_{8}$ | 82.20 | 127.6 | 140.2
$\nu_{9}$ | 82.20 | 127.6 | 140.2
Table 3: The masses (in GeV) of some of the sneutrinos, lighter gauginos, and
the six sterile neutrinos ($\nu_{4}$ to $\nu_{9}$) for all the three BPs.

The particle spectrum for the three BPs is shown in Table 3. In addition to
the masses, the Branching Ratios (BRs) are also modified due to the presence
of the CPV phases. In Table 4, we display the BRs of the six lightest
sneutrinos, namely $\tilde{\nu}_{i}$ ($i$ = 1,..,6), to different possible
final states, including the new decay modes originating from the BSM physics.
The decay modes primarily include charginos and neutralinos along with SM
neutrinos and the sterile neutrinos. For the leptonic decay of the sneutrinos
through charginos, we combine the contributions coming from electrons, muons
and leptonically decaying taus. When a sneutrino decays invisibly, we
separately add the contribution coming from the all three flavours of SM
neutrinos, collectively denoted as $\nu_{\rm SM}$, and contributions coming
from the six sterile neutrinos, denoted as $\nu_{\rm BSM}$. The BRs to other
modes that involve other neutralinos than the lightest one are summed over for
a given sneutrino and expressed as $BR(\tilde{\nu}\to{\rm others})$. It is
evident from the table that the presence of imaginary parts in the couplings
affects the BRs (both visible and invisible ones), especially in the case of
the ‘BP2’ where the imaginary parts are relatively large compared to other two
BPs. The complex couplings and the mixing of CP eigenstates also affect the
angular distributions of the decay products, which in turn will impact the
distribution of T-odd observables.

Decay channels | BP1 | BP2 | BP3
---|---|---|---
BR($\tilde{\nu}_{1}\to\tilde{\chi}_{1}^{\pm}l^{\mp}$) | 0.0251 | 0.4912 | 0.0346
BR($\tilde{\nu}_{1}\to\tilde{\chi}_{1}^{0}\nu_{\rm SM}$) | 0.2068 | 0.0018 | 0.2796
BR($\tilde{\nu}_{1}\to\tilde{\chi}_{1}^{0}\nu_{\rm BSM}$) | 0.2931 | 0.4817 | 0.3570
BR($\tilde{\nu}_{1}\to{\rm others}$) | 0.474 | 0.026 | 0.332
BR($\tilde{\nu}_{2}\to\tilde{\chi}_{1}^{\pm}l^{\mp}$) | 0.0251 | 0.4912 | 0.0346
BR($\tilde{\nu}_{2}\to\tilde{\chi}_{1}^{0}\nu_{\rm SM}$) | 0.2086 | 0.0018 | 0.2796
BR($\tilde{\nu}_{2}\to\tilde{\chi}_{1}^{0}\nu_{\rm BSM}$) | 0.2939 | 0.4808 | 0.3568
BR($\tilde{\nu}_{2}\to{\rm others}$) | 0.474 | 0.026 | 0.332
BR($\tilde{\nu}_{3}\to\tilde{\chi}_{1}^{\pm}l^{\mp}$) | 0.0331 | 0.4890 | 0.0368
BR($\tilde{\nu}_{3}\to\tilde{\chi}_{1}^{0}\nu_{\rm SM}$) | 0.072 | 0.0001 | 0.2921
BR($\tilde{\nu}_{3}\to\tilde{\chi}_{1}^{0}\nu_{\rm BSM}$) | 0.3095 | 0.0044 | 0.3988
BR($\tilde{\nu}_{3}\to{\rm others}$) | 0.584 | 0.496 | 0.263
BR($\tilde{\nu}_{4}\to\tilde{\chi}_{1}^{\pm}l^{\mp}$) | 0.0311 | 0.4890 | 0.0368
BR($\tilde{\nu}_{4}\to\tilde{\chi}_{1}^{0}\nu_{\rm SM}$) | 0.072 | 0.0001 | 0.2922
BR($\tilde{\nu}_{4}\to\tilde{\chi}_{1}^{0}\nu_{\rm BSM}$) | 0.3091 | 0.0044 | 0.3985
BR($\tilde{\nu}_{4}\to{\rm others}$) | 0.584 | 0.496 | 0.263
BR($\tilde{\nu}_{5}\to\tilde{\chi}_{1}^{\pm}l^{\mp}$) | 0.1933 | 0.4993 | 0.318
BR($\tilde{\nu}_{5}\to\tilde{\chi}_{1}^{0}\nu_{\rm SM}$) | 0.0258 | – | 0.0403
BR($\tilde{\nu}_{5}\to\tilde{\chi}_{1}^{0}\nu_{\rm BSM}$) | 0.2559 | 0.026 | 0.4024
BR($\tilde{\nu}_{5}\to{\rm others}$) | 0.521 | 0.472 | 0.231
BR($\tilde{\nu}_{6}\to\tilde{\chi}_{1}^{\pm}l^{\mp}$) | 0.1933 | 0.4993 | 0.318
BR($\tilde{\nu}_{6}\to\tilde{\chi}_{1}^{0}\nu_{\rm SM}$) | 0.0258 | – | 0.0403
BR($\tilde{\nu}_{6}\to\tilde{\chi}_{1}^{0}\nu_{\rm BSM}$) | 0.2559 | 0.026 | 0.4009
BR($\tilde{\nu}_{6}\to{\rm others}$) | 0.521 | 0.472 | 0.231
Table 4: The BR of $\tilde{\nu}_{i}$ ($i$ = 1,..,6) to different possible
final states, including the new decay modes originating from the BSM physics,
corresponding to BP1, BP2 and BP3. See the text for more details.

The SM background processes, e.g., $pp\to W^{\pm}+{\rm jets}$, $t\bar{t}+{\rm
jets}$ etc., will receive the CPV effects through the phase present in the CKM
matrix. However, as discussed in section 2, we are looking at the overall
width of the distribution and will be insensitive to the deviation from zero,
which is the traditional measure of CPV. Hence it is important to include the
SM processes, which lead to largest values of our triple products. It is also
critical that the modeling of the CPC BSM physics is accurate enough as we
need to show a wider distribution than would emerge from the sneutrino
processes without CPV.

Therefore, as already mentioned, we will calculate several observables that
are sensitive to the CPV effects. To do so, we simulate 200,000 signal events
and around 500,000 background events, both at Leading Order (LO) in
$\alpha_{s}$, using Madgraph5 (v2.8.2) Alwall:2011uj  using collision energy
$\sqrt{s}$ = 100 TeV and integrated luminosity $\mathcal{L}=3~{}\rm ab^{-1}$
at FCC-hh collider Benedikt:2022kan . These events are then passed to Pythia
(v8.2) Sjostrand:2014zea  for parton showering and hadronization, plus finally
to Delphes3 deFavereau:2013fsa  to consider detector effects. The Delphes card
in the FCC-hh is redesigned to satisfy higher Pile-Up (PU) and enhance the
tagging efficiencies. To manage the simulated background events efficiently,
we apply a generation level cut on the minimum missing transverse momentum
calculated using the invisible particles present at the parton level setting
at 100 GeV. This has helped to generate events within the desired signal
region of interest, especially regions with large missing transverse energy.
The cross section for the signal and background processes are listed in Table
5. The jets are reconstructed using FastJet fastjet  with jet radius R = 0.4
and anti-$k_{T}$ jet algorithm Cacciari:2008gp . The jets are pre-selected
with the jet transverse momentum $p_{T}^{j}>30~{}\rm GeV$ and the
pseudorapidity $|\eta|<4.5$.

| Process | Cross section ($\sigma$) [pb]
---|---|---
|  $pp\to\tilde{\nu}_{i}\tilde{\nu}_{j}$ (BP1) | $5.2\times 10^{-3}$
Signals |  $pp\to\tilde{\nu}_{i}\tilde{\nu}_{j}$ (BP2) | $6.4\times 10^{-3}$
|  $pp\to\tilde{\nu}_{i}\tilde{\nu}_{j}$ (BP3) | $5.3\times 10^{-3}$
| $W^{\pm}+{\rm jets}$ | $2.93\times 10^{3}$
| $t\bar{t}~{}(1L)$ | $1.49\times 10^{3}$
Backgrounds | $t\bar{t}~{}(2L)$ | $3.42\times 10^{2}$
| $tW$ | 1.37
| $t\bar{t}W$ | 0.518
Table 5: The LO cross sections of the simulated signal and background
processes with $\sqrt{s}=100~{}\rm TeV$ at FCC-hh.

##  4 Collider phenomenology and results

In this section, we discuss the details of the collider analysis, including
the multiVariate analysis, followed by the results obtained to probe the
sensitivity of the proposed 100 TeV collider experiment.

###  4.1 Kinematic variables

As already mentioned, we focus on the final state topology that includes at
least one isolated lepton ($e$ or $\mu$), at least two jets along with large
MET, say ${\mathcal{O}}(100)$ GeV. These leptons must satisfy
$p_{T}^{L}>20~{}\rm GeV$ with pseudorapidity $|\eta|<2.5$. In Figure 2, we
show the multiplicity of leptons (left) and jets (right) present in the signal
and background events111In the subsequent analysis we consider the three
dominant backgrounds only, neglecting the two sub-dominant ones, namely, $tW$
and $t\bar{t}W$, as their contribution to the forthcoming classification step
is highly negligible.. The signal events include significantly more leptons
and jets compared to the background events. We check that these leptons
primarily originated from the decay of the sneutrinos to a sterile neutrino
and a neutralino, where these neutrinos further decay to leptons and jets. The
mass difference between these new particles ensures that some of the
lepton/jets are relatively boosted. This is evident from the $p_{T}$
distribution of the leading (i.e., with highest $p_{T}$) lepton and leading
two jets as shown in Figure 3. The sub-leading jet seems to overlap with the
SM backgrounds and, therefore, will not play a significant role in controlling
the latter.

Figure 2: Distribution of the number of leptons (left) and jets (right) for
the signals and SM background.

Figure 3: The left panel describes the transverse momentum of the leading
lepton. The middle panel shows the distribution of transverse momentum for the
leading jet ($p^{j_{1}}_{T}$). The right panel presents the $p_{T}$ of the
sub-leading jet ($p^{j_{2}}_{T}$).

The presence of additional invisible particles, in addition to the SM
neutrinos, and the contribution of mistagging of the reconstructed objects,
leads to higher values of MET (denoted by $\not{E}_{T}$ in the plots) for the
signal events and, therefore, provides a clear separation from the SM
backgrounds. The left panel of Figure 4 displays the MET distribution. It
seems that a choice of selecting events with $\not{E}_{T}>300~{}\rm GeV$ would
allow most of the signal to be retained while discarding a significant
fraction of the backgrounds. Additionally, we also construct two more
important variables, namely $H_{T}$ and $M_{\rm eff}$, where $H_{T}$ denotes
the scalar $p_{T}$ sum of all jets present in the event while $M_{\rm eff}$
includes all visible (lepton and jet) and invisible objects transverse
momentum. In the middle panel of Figure 4, we show the distribution of $H_{T}$
while in the right panel we display the distribution of the ratio
$\frac{\not{E}_{T}}{\sqrt{H_{T}}}$. Both of these distributions show a clear
separation power of the signal with respect to the backgrounds.

Figure 4: Distribution in MET ($\not{E}_{T}$) for the signal and SM
backgrounds (left). Distribution in the $H_{T}$ variable describing the scalar
$p_{T}$ sum of all the jets present in that event (middle). Distribution in
the ratio $\not{E}_{T}/\sqrt{H_{T}}$ (right).

In the left panel of Figure 5, we show the distribution of the transverse mass
$M_{T}$ constructed using the leading lepton and $\not{E}_{T}$, defined as
$M_{T}^{L\nu}=\sqrt{p_{T}^{L}\not{E}_{T}(1-\cos(\Delta\phi_{L\not{E}_{T}}))}$,
where $\phi$ is the azimuthal angle between the lepton momentum and
$\not{E}_{T}$. For the SM backgrounds, the primary source of this lepton and
MET is a $W$-boson which decays leptonically. Therefore, the $M_{T}$
distribution is expected to have an edge around the $W$-mass (i.e., 80 GeV),
which can be seen from the distribution. The signal events arise from the
decay of a heavier object that decays to more than just a lepton and MET (in
the form of a neutralino), so there will be no edges and the distribution
extends to higher values of $M_{T}$, as seen from the figure. Note, for the
$t\bar{t}(2L)$ background, which implies that both the W-bosons are decaying
leptonically, the same distribution gets some shift towards higher values.
This can be understood from the fact from these events have more than one
lepton to choose from the set and also have somewhat higher MET. The variable
$M_{\rm eff}$ also takes into account the effect of large MET, and so have a
clear separation from the SM backgrounds as shown in the right panel of Figure
5. Along with these observables, we also calculate the invariant mass of the
two leading jets and the same using the leading lepton and the two leading
jets, as shown in Figure 6.

Figure 5: Left: distribution in the transverse mass of the leading lepton and
MET. Right: distribution in $M_{\rm eff}$. We refer to the text for more
details.

Figure 6: Left: distribution in the invariant mass of the two leading jets.
Right: distribution in the invariant mass of the leading lepton and the two
leading jets.

Having discussed some of the kinematic variables involving the lepton and
jets, we also need to assess how the leptons and jets are separated among each
other. We calculate the angular separation $\Delta R$ between the leading
lepton with the two leading jets, where $(\Delta
R)^{2}=(\Delta\phi)^{2}+(\Delta\eta)^{2}$ in the ($\eta,\phi$) plane, where
$\phi$ is the azimuthal angle and $\eta$ is the pseudorapidity of the
individual objects. In Figure 7, we show the $\Delta R$ distributions between
the leading lepton and leading jet (left panel), leading lepton and sub-
leading jet (middle panel), and the two leading jets (right panel).

The leading lepton seems to be more aligned with the direction of the leading
jet, while it can be either very close or widely separated from the sub-
leading jet. The lepton may come from the decay of a sneutrino or a heavy
neutrino, both decays can also contain jets. As the mother particle is boosted
if it results from a $Z^{\prime}$ decay, the lepton and the jet(s) tend to
have a small angular separation. For the background or a signal event from the
SM $Z$ portal such a boost is not typical, so the hardest particles are
scattered back-to-back. To understand these distributions better, we also
study the angular separation between the lepton and jets in the pseudo-
rapidity and azimuthal planes separately, see Figure 8 and Figure 9. In both
of these figures, we observe that the signal events prefer to lie within
$|\Delta\phi|<1.5$ for all the three variables, while background events have
more contribution to higher values of $|\Delta\phi|$. Almost similar behavior
can be seen for $|\Delta\eta|$, where signal events prefer to stay closer
while background events have somewhat wider distribution.

Figure 7: $\Delta R$ distribution between the lepton and leading jet (left),
the lepton and second leading jet (middle) as well as the first and second
leading jet (right).

Figure 8: $\Delta\phi$ distribution between the lepton and leading jet (left),
the lepton and second leading jet (middle) as well as the first and second
leading jet (right).

Figure 9: $\Delta\eta$ distribution between the lepton and leading jet (left),
the lepton and second leading jet (middle) as well as the first and second
leading jet (right).

###  4.2 T-odd observables

CPV can manifest itself as an asymmetry that results into an observable
difference in angular distributions of the decay products of a particle and
its antiparticle. The scalar Triple products (TPs) are designed to probe such
asymmetries. Here we construct three TPs, primarily using the 3-momentum of
the leading lepton and the two leading jets. Furthermore, we also calculate
similar TPs using the two MET components: it is therefore a 2-dimensional
projection of the original 3-dimensional TPs.

The T-odd triple products used in this work are listed below,

  * •

$T_{L_{1}j_{1}j_{2}}=\frac{p_{L_{1}}\cdot(p_{j_{1}}\times
p_{j_{2}})}{s^{3/2}}$,
$T_{L_{1}L_{2}j_{1}}=\frac{p_{L_{1}}\cdot(p_{L_{2}}\times
p_{j_{1}})}{s^{3/2}}$, and
$T_{L_{1}L_{2}j_{2}}=\frac{p_{L_{1}}\cdot(p_{L_{2}}\times
p_{j_{2}})}{s^{3/2}}$

  * •

$T_{\not{E}_{T}L_{1}j_{1}}=\frac{\not{E}_{T}\cdot(p_{L_{1}}~{}\times
p_{j_{1}})}{s^{3/2}}$,
$T_{\not{E}_{T}L_{1}j_{2}}=\frac{\not{E}_{T}\cdot(p_{L_{1}}~{}\times
p_{j_{2}})}{s^{3/2}}$, and
$T_{\not{E}_{T}j_{1}j_{2}}=\frac{\not{E}_{T}\cdot(p_{j_{1}}~{}\times
p_{j_{2}})}{s^{3/2}}$,

where, $p_{L_{1}}$, $p_{j_{1}}$, and $p_{j_{2}}$ denote the 3-momentum of the
lepton, leading jet, and sub-leading jet respectively. To convert these
observables into dimensionless quantities, we normalize them with $s^{3/2}$
where $\sqrt{s}=100~{}\rm TeV$ is the center of mass energy. These quantities
are plotted in figures 10 and 11. We may see that the distribution of the
signal benchmarks is a lot wider than that of the SM processes. This is
partially due to the harder final state particles arising from $Z^{\prime}$
decays to sneutrinos and partially due to the widening of the distributions
caused by CPV. Notice that the distributions in figures 10 and 11 are
normalised to one, i.e. when normalised to luminosity the tails arising from
the signal are not as prominent as we shall discuss below. Nevertheless, the
fact that the bulk of the SM background has low absolute values of the TP
observables gives us some chance of analysing the signal and the CPV within
it.

Figure 10: Distribution of the three T-odd observables that are calculated
without using the MET.

Figure 11: Distribution of the set of T-odd observables that are calculated
using the MET.

In section 2 we conjectured that the width of the distributions of T-odd
observables would be wider if CPV was present compared to a similar CP
conserving case. In figure 12 we show a comparison of the distributions of the
T-odd observables for CP conserving and violating benchmarks. This indeed
seems to confirm our conjecture: CP violation widens the distributions of
T-odd observables and the difference seems to be substantial.

Figure 12: Distributions of T-odd observables for CP violating benchmark
points compared to CP conserving ones. We may notice that the distributions
are wider with CP violation.

To quantify the separation between the SM values and the BSM ones, we
calculate the area covered by a given distribution outside a specific range of
the T-odd observable. Higher the region outside the window of the T-odd
observable, stronger the impact of the CPV phase on the T-odd observable. In
Table 6, we show the area of the tails estimated using the following equation:

$\mathcal{A_{T}}=1-\frac{\rm Area~{}with~{}|T|<0.06}{\rm Total~{}area~{}}.$
(14)

From the data displayed in Table 6, one can find observe the impact of the CP
violation and the harder final state objects for the signal BPs. The SM
background consists of softer objects and the distributions do not get much
widened by the CKM phase as its effects gets suppressed by the small mixing
angles of the CKM matrix. In the next section, we will study the impact of
different kinematic variables on these T-odd observables and will make use of
them in obtaining good signal over background ratios.

T-odd observable | BP1 | BP2 | BP3 |  $W$+jets |  $t\bar{t}$(1L) |  $t\bar{t}$(2L)
---|---|---|---|---|---|---
$T_{L_{1}j_{1}j_{2}}$ | 0.114 | 0.279 | 0.157 | 0.025 | 0.025 | 0.022
$T_{L_{1}L_{2}j_{1}}$ | 0.008 | 0.074 | 0.033 | 0.0 | 0.0 | 0.002
$T_{L_{1}L_{2}j_{2}}$ | 0.002 | 0.037 | 0.016 | 0.0 | 0.0 | $5.8\times 10^{-4}$
$T_{\not{E}_{T}L_{1}j_{1}}$ | 0.149 | 0.310 | 0.176 | 0.007 | 0.005 | 0.01
$T_{\not{E}_{T}L_{1}j_{2}}$ | 0.069 | 0.169 | 0.091 | 0.003 | 0.002 | 0.003
$T_{\not{E}_{T}j_{1}j_{2}}$ | 0.241 | 0.286 | 0.267 | 0.038 | 0.038 | 0.039
Table 6:  The area of the distribution tails $\mathcal{A_{T}}$ calculated
using the preselection on the given T-odd observable, as defined in Equation
(14).

###  4.3 Cut-based analysis

We first have a look at what kind of results can be obtained through a
classical cut-based analysis, using standard kinematical variables and the
distributions of the T-odd observables introduced above. We shall improve this
strategy using boosted decision trees in the following subsection.

As we discussed above, CP violation makes the distributions of the T-odd
observables wider than it would be without it. For the SM backgrounds the
distribution is centered around zero, even more than for the CP conserving
versions of our benchmarks, see figures 10–12. Here we analyze the prospects
for a cut-and-count analysis for BP2 which, following the Table 2 and Table 4,
seems to have the largest impact on the additional CP phases.

The two T-odd observables that show the best sensitivity are
$T_{L_{1}j_{1}j_{2}}$ and $T_{\not{E}_{T}L_{1}j_{1}}$ (see Table 6). However,
to maximize the impact of these observables, we first impose the cuts on the
kinematic variables and then focus on different regions of the T-odd
observables. The observable $T_{L_{1}j_{1}j_{2}}$ depends on the transverse
momentum of the leading and sub-leading jets along with leading lepton.
Following the differences observed in the $p_{T}$ distribution of the jets
(see the middle and right panel of Figure 3 respectively), we select events
with $p_{T}^{j_{1}}$ larger than 300 GeV and $p^{j_{2}}_{T}>150$ GeV. We
further constrain the angular separation of the leading lepton and the sub-
leading jet (middle panel of Figure 7) by selecting events with $\Delta
R_{L_{1}j_{2}}\leq 1.5$. Furthermore, we also use a cut on the angular
separation between the two jets i.e., $\Delta R_{j_{1}j_{2}}<1.5$. After
imposing these cuts, we plot the T-odd observable as shown in fig. 13, which
now clearly shows the minimal contamination of the SM backgrounds. Using this
distribution, we now proceed to calculate the signal significance
$\mathcal{S}=\frac{S}{\sqrt{S+B}}$, where S and B represent the number of
signal and background events at the integrated luminosity
$\mathcal{L}=3~{}{\rm ab^{-1}}$. In Table 7, we show how significance changes
with the different choices of cuts on the T-odd observable. We use four
specific regions of the observable, namely $|T_{L_{1}j_{1}j_{2}}|>$ 0.001,
0.01, 0.05, and 0.1. As the $|T_{L_{1}j_{2}j_{1}}|$ increases, the number of
signal events that pass the cut reduces, and the same for the background
events also reduces significantly, and therefore, signal significance
increases.

Figure 13: The distribution of the T-odd observable $T_{L_{1}j_{1}j_{2}}$ after the imposition of the kinematic cuts $p_{T}^{j_{1}}>300$ GeV, $p_{T}^{j_{2}}>150$ GeV, $\Delta R_{L_{1}j_{2}}<1.5$, and $\Delta R_{j_{1}j_{2}}<1.5$. $|T_{L_{1}j_{1}j_{2}}|$ | Signal |  $W$+jets |  $t\bar{t}$(1L) |  $t\bar{t}$(2L) | Total bgnd | $\frac{S}{\sqrt{S+B}}$
---|---|---|---|---|---|---
$>0.001$ | 9220 | $25.3\times 10^{7}$ | $27.1\times 10^{7}$ | $2.56\times 10^{7}$ | $54.96\times 10^{7}$ | 0.39
$>0.01$ | 7230 | $15.8\times 10^{7}$ | $16.5\times 10^{7}$ | $1.64\times 10^{7}$ | $33.94\times 10^{7}$ | 0.39
$>0.05$ | 4150 | $5.15\times 10^{7}$ | $5.81\times 10^{7}$ | $0.62\times 10^{7}$ | $10.10\times 10^{7}$ | 0.41
$>0.10$ | 2860 | $1.19\times 10^{7}$ | $2.03\times 10^{7}$ | $0.32\times 10^{7}$ | $3.54\times 10^{7}$ | 0.48
Table 7: The signal significances after the cuts as well as a cut on the T-odd
observable $T_{L_{1}j_{1}j_{2}}$. For the signal events we use BP2. The
integrated luminosity is 3 ab-1 and the collision energy $\sqrt{s}=$100 TeV.

The second best T-odd observable, based on Table 6, is
$T_{\not{E}_{T}L_{1}j_{1}}$, which involves the leading jet, leading lepton
and MET. Following the strategy adopted for $T_{L_{1}j_{1}j_{2}}$, we first
select events based on the kinematic cuts and later analyze the cuts on
$T_{\not{E}_{T}L_{1}j_{1}}$. The cuts placed on the relevant variables are as
follows: $\not{E}_{T}>300~{}\rm GeV$, $p^{j_{1}}_{T}>300$ GeV and $\Delta
R_{L_{1}j_{1}}\leq 1.5$. The distribution of $T_{\not{E}_{T}L_{1}j_{1}}$ seems
prominent after the imposition of the above-mentioned cuts, as shown in Figure
14. The calculation of signal significances with the different binned
selection of $T_{\not{E}_{T}L_{1}j_{1}}$ is shown in Table 8.

We may note that while the distribution of the T-odd observables for the
signal is wider, the background has a cross section that is several orders of
magnitude larger. Even though the SM background is strongly centered around
zero, there is a minor tail that is larger than that coming from the signal.
Hence the significance of the deviation at the tails remains at a modest
level.

A cut-based selection can be considered to be a decision tree. The efficiency
of such a selection can be enhanced by using machine learning to decide the
optimal boundary. Furthermore, boosting uses an ensemble of shallow decision
trees trained by reweighting incorrectly identified data points and takes the
average of the decision boundaries of such trees smlbook . This allows for
more complicated decision boundaries and usually leads to a better
discrimination power. Therefore, we perform a Multi-Variate Analysis (MVA),
namely the BDT method implemented in “Toolkit for Multivariate Analysis”
(TMVA) framework TMVA:2007ngy , as discussed in the next section. Unlike cut-
based approaches, a BDT can effectively capture complex correlations and
nonlinear relationships within the data, allowing for more nuanced
discrimination between signal and background events, and thereby elevating the
overall significance.

Figure 14: The distribution of the T-odd observable $T_{\not{E}_{T}L_{1}j_{1}}$ after the application of the kinematic cuts $\not{E}_{T}>300~{}{\rm GeV}$, $p^{j_{1}}_{T}>300~{}{\rm GeV}$ and $\Delta R_{L_{1}j_{1}}<1.5$. $|T_{\not{E}_{T}L_{1}j_{1}}|$ | Signal |  $W$+jets |  $t\bar{t}$(1L) |  $t\bar{t}$(2L) | Total bgnd | $\frac{S}{\sqrt{S+B}}$
---|---|---|---|---|---|---
$>0.001$ | 9130 | $9.68\times 10^{7}$ | $5.36\times 10^{7}$ | $0.85\times 10^{7}$ | $15.89\times 10^{7}$ | 0.72
$>0.01$ | 7030 | $5.28\times 10^{7}$ | $3.11\times 10^{7}$ | $0.47\times 10^{7}$ | $8.86\times 10^{7}$ | 0.74
$>0.05$ | 4040 | $0.88\times 10^{7}$ | $0.87\times 10^{7}$ | $0.14\times 10^{7}$ | $1.89\times 10^{7}$ | 0.93
$>0.10$ | 2640 | $0.30\times 10^{7}$ | $0.25\times 10^{7}$ | $0.07\times 10^{7}$ | $0.65\times 10^{7}$ | 1.04
Table 8: The signal significances after the cuts as well as the cut on the
T-odd observable $T_{\not{E}_{T}L_{1}j_{1}}$. For the signal events we use BP2
as in Table 7.

###  4.4 BDT analysis

The cut-based method focuses on the identification of the zones in the feature
space by drawing rectangular boundaries on the features to increase the signal
efficiency over the background. This approach has its limitations. The use of
a BDT trained with samples of (simulated) signal and background data results
in a more complex decision boundary between the classes and hence generally
improves the performance in classifying the data. To bypass the shortcomings
of the simple cut-based analysis, we use a BDT where we consider all the
features or observables in a single go and obtain a Receiver Operating
Characteristics (ROC) curve giving the probability of correctly rejecting
background events as a function of the probability of correctly identifying
signal events. We use the ROOT Library, “Toolkit for Multivariate Analysis”
TMVA:2007ngy  to perform the MVA. More specifically, we use the ”Adaptive
Boosted Decision Tree” (AdaBoost) algorithm FREUND1997119  for the
classification of signal and background events222 Different SM backgrounds are
weighted according to their cross sections while performing the BDT analysis..
The AdaBoost classifier algorithm constructs various dependency matrices book
based on the available features of the dataset provided and arranges these
features in order to their respective importance for the classification.

Two sets of inputs are provided in the dataset: Set-I includes all the
relevant kinematic variables, while Set-II includes all the T-odd observables
in additional to the kinematic variables of Set-I. In Table 9, we list all
these variables of Set-I and Set-II. In Figure 15, we show the ROC curve for
the two sets of input corresponding to BP2. Any difference in the ROC curves
for the two sets is an improvement of the classification due to the T-odd
observables. The blue line portrays the ROC curve for Set II, which clearly
shows the impact of the T-odd observables in improving the signal
efficiencies. The elbow point corresponds to the coordinates of 60% signal
efficiency and a mistagging rate of 0.1%.

Set I |  $\not{E}_{T}$, $M_{\rm eff}$, $\not{E}_{T}/\sqrt{H_{T}}$, $H_{T}$, $M_{T}^{L\nu}$, $M_{L_{1}j_{1}j_{2}}$ ,$\Delta R_{lj_{1}}$, $\Delta R_{lj_{2}}$, $\Delta R_{j_{1}j_{2}}$, $p_{T}^{L}$ ,$p^{j_{1}}_{T}$, $p^{j_{2}}_{T}$
---|---
Set II |  $\not{E}_{T}$, $M_{\rm eff}$, $\not{E}_{T}/\sqrt{H_{T}}$, $H_{T}$, $M_{T}^{L\nu}$, $M_{L_{1}j_{1}j_{2}}$ ,$\Delta R_{lj_{1}}$, $\Delta R_{lj_{2}}$, $\Delta R_{j_{1}j_{2}}$, $p_{T}^{L}$ ,$p^{j_{1}}_{T}$, $p^{j_{2}}_{T}$ $T_{\not{E}_{T}L_{1}j_{1}}$, $T_{\not{E}_{T}L_{2}j_{2}}$, $T_{\not{E}_{T}j_{1}j_{2}}$, $T_{L_{1}j_{1}j_{2}}$, $T_{L_{1}L_{2}j_{1}}$, $T_{L_{1}L_{2}j_{2}}$
Table 9: The two sets of input variables used in the BDT analysis. Figure 15:
The receiver operating characteristic (ROC) curve obtained using the AdaBoost
algorithm. The two above-mentioned sets were used for the input of BDT
algorithm for classifying signal events (BP2) from the SM processes.

To better represent the effect of the T-odd observables, we perform the BDT
analysis for all BPs using the input variables of Set-II . The performance of
the classifier is displayed in Figure 16 (left). Amidst the three BPs related
to new CPV phases, the BP2 captures the maximum CPV effect leading to wider
distributions of the T-odd observables. This results in the higher signal
efficiency for BP2 as shown by the solid magenta line in Figure 16 (left). The
quantity that describes the impact of these new variables is the statistical
signal significance defined as $\frac{S}{\sqrt{S+B}}$ with S and B being the
signal and background events at a given integrated luminosity. We calculate
signal significance for all the BPs varying the cut on the BDT score,
essentially which means calculating the signal efficiencies at different
values of mistagging rates. The right panel of Figure 16 displays the
variation of the signal significance with the variation of BDT cut. As evident
from the figure, we can achieve around 4$\sigma$ statistical significance at 3
${\rm ab^{-1}}$ of integrated luminosity for BP2, which other two BPs give
lower significances.

Figure 16: Left: The receiver operating characteristic (ROC) curve obtained
using Set II as BDT inputs for BP1, BP2 and BP3. Right: The signal
significance with the variation of BDT-cut for the three BPs.

In using machine learning methods we have to use simulated samples of data
both for the SM and our BSM signal to train the BDT. In our case we of course
know the actual features of our benchmark points and can generate accurate BSM
data. One may ask, how well would this be justified if we were to investigate
real data?

In investigating CPV we are doing characterisation of BSM physics. This means
that we should have discovered it first. In the BLSSM with the $Z^{\prime}$ as
a portal to sneutrino production, we would probably have seen the $Z^{\prime}$
as a dilepton and dijet resonance, which would give us its mass. Similarly it
seems reasonable that we would have an idea of the spectrum of the
superpartners, which determines the decay kinematics. Obviously, there would
be some remaining uncertainty in these, which will reduce the discriminating
power of the BDT. However, we expect the uncertainties in the BSM spectrum to
be so small that the reduction in the significance would be small.

###  4.5 The CPC limit and $Z^{\prime}$ impact

Till now we have studied the impact of the new CPV phases through the T-odd
observables. In the previous subsection we showed that we can distinguish the
signal from the SM. We should also see that we can establish that there is CPV
by comparing the CPV case to a corresponding CPC benchmark. We perform the
previously discussed BDT analysis using all the observables presented as Set-
II, our findings are shown in Figure 17. Here we showcase the comparative
analysis of ROC curves for CPC and CPV scenarios for the selected benchmark
points, BP2 and BP3. As indicated in Table 6, these two benchmark points
exhibit greater asymmetry compared to BP1. The magenta and green dashed lines
correspond to the cases when BP2 and BP3 are considered as the signal setting
all the new phases to zero (keeping the CKM phase to the desired non-zero
value), while the solid lines denote the same BP but with non-zero CPV phases.
Comparing the ROC curves of the CPV and CPC cases, we can see that it is much
easier to distinguish a CPV benchmark from the SM than a CPC one.

Figure 17: Comparison of the ROC curves for BP2 (left panel) and BP3 (right
panel) with CPV (solid line) and CPC (dashed line) cases. In both cases, all
the SM processes are used as the background for the BDT analysis.

To quantify the statement made above while comparing the ROC curves for the
CPV and CPC cases as in Figure 17, we display the statistical significance
curves for BP2 and BP3 in Figure 18. The CPC scenario exhibits significantly
lower significance compared to the CPV scenario. Specifically, the
significance curves for the CPC scenario, for both BP2 and BP3, are below
$2.5\sigma$. In contrast, the significance for the CPV scenario is above
$4\sigma$ for BP2 and above $3\sigma$ for BP3. The difference between the
significances between the CPC and CPV cases for BP2 is more than $2\sigma$, so
in the case of large CPV we can at least reject a CP conserving scenario with
more than $95\%$ confidence. For BP3 with smaller CPV one cannot make such a
claim.

Figure 18: Comparison of the statistical significance curves for BP2 (left
panel) and BP3 (right panel) with CPV (solid line) and CPC (dashed line)
cases.

Another point of worth mentioning is the role played by the mediator particle
$Z^{\prime}$ in our analysis. The pair production of the sneutrinos through
proton-proton collisions involves both the SM Z-boson and the $Z^{\prime}$
boson, a new particle associated with the $B-L$ symmetry. However, the mass of
$Z^{\prime}$ has to be quite large in order to satisfy the existing limits
from the collider experiments. Therefore, higher the mass of $Z^{\prime}$
(around 5 TeV), lower the contribution to the total cross section. However,
after being produced on-shell at higher energies, it can provide significant
boost to the sneutrinos of mass around a TeV, and thereby its decay products.
The T-odd observables are constructed using the 4-momentum of the decay
products of sneutrinos, and therefore the presence (or absence) of
$Z^{\prime}$ will definitely play some role in the overall outcome. We find
that the inclusion of $Z^{\prime}$ in the production process increases the
cross section (LO in $\alpha_{s}$) by a factor of 5 at the 100 TeV collider,
as shown in Table 10.

Signal cross section $\sigma_{s}$ for CPV2 | 0.0064 pb
---|---
Signal cross section $\sigma_{s}$ for CPV2 without $Z^{\prime}$ | 0.0011 pb
Table 10: The sneutrino pair production cross section with and without the
$Z^{\prime}$ at the 100 TeV collider.

To quantitatively analyze the overall significance of the $Z^{\prime}$ in the
process, a comprehensive analysis of T-odd observables is pursued, elucidating
the significance curve to formulate definitive conclusions. As already
mentioned, for those signal events without the $Z^{\prime}$ contribution, the
T-odd overvables will be much more restrained around zero, contrary to the
case when $Z^{\prime}$ is present (see figures 19 and 20 for a visual
illustration).

Figure 19: Distribution of the first set of T-odd observables calculated
without using MET.

Figure 20: Distribution of the second set of T-odd observables calculated
using MET.

As discussed earlier, the efficacy of the BDT analysis of discerning the
signal efficiency over the background rejection relies upon the features
offered to the BDT. In a similar fashion we provide the BDT with the same 18
features for the ($pp\rightarrow Z\rightarrow\tilde{\nu}\tilde{\nu}$) process
(the case of without $Z^{\prime}$) along with the previously discussed
production process ($pp\rightarrow
Z^{\prime}/Z\rightarrow\tilde{\nu}\tilde{\nu}$) for BP2. Despite the CPV
effects originating from the Yukawa matrix elements being the same, the only
elevating difference is due to the absence of the $Z^{\prime}$ in the former
process. Evidently, the blue dashed line of Figure 21 (left) depicting the
signal efficiency over the background rejection rate, is reduced if the signal
events don’t contain the $Z^{\prime}$ but includes only the SM $Z$ boson. At
the elbow point, the signal efficiency is approximately 40% for the process
lacking the $Z^{\prime}$, contrasting with the approximately 60% efficiency
observed when the $Z^{\prime}$ is present in the process. Defining S and B as
the effective number of signal and background events respectively, we show the
signal significance $\frac{S}{\sqrt{S+B}}$ with the variation of the BDT cut
in Figure 21 (right). Due to the diminished cross section, the significance
drops severely for the process without the $Z^{\prime}$. Hence it is necessary
to have an additional gauge portal in the form of the $Z^{\prime}$ boson that
is not too heavy to be able to characterise the CPV in the sneutrino sector at
the FCC-hh.

Figure 21: Left: The ROC curve obtained using the AdaBoost algorithm when the
above-mentioned Set II is used for the input for the BP2 (with and without
$Z^{\prime}$ separately) and combined SM processes as background. Right: The
variation of Signal Significance with the variation of BDT-cut.

##  5 Conclusions

We investigated the possibility of finding CPV in the sneutrino sector at the
FCC-hh. We studied this in the BLSSM with inverse seesaw. This is partially
because we needed an efficient portal to produce sneutrinos, which comes in
the form of the new gauge boson $Z^{\prime}$ related to the U$(1)_{B-L}$
symmetry. The inverse seesaw on the other hand allows large superpotential
couplings in the (s)neutrino sector, which can be complex and lead to CPV
effects. These include direct CPV through the couplings and indirect CPV
through the mixing of CP-even and CP-odd sneutrino states.

In general the effects of CPV can be most clearly seen as a non-vanishing
expectation value of T-odd triple product asymmetries. These are products of
three three-momenta, in our case a sneutrino could decay to a lepton and two
jets. Such an approach would therefore require the ability to distinguish
quark jets from antiquark jets, which we did not try to do. Instead we assumed
— and this assumption was backed by our simulated data — that there is an
intrinsic variance in the distribution of the triple product, which is then
widened by the CPV effects. Hence we could use this widening as an indicator
of CPV.

The limits of the BSM searches from the LHC experiments restrict the cross
sections for processes involving new physics to be small at future colliders,
too. Even though the distributions of our triple product observables were much
wider than for any SM process, the tails of the SM distribution still
overshadowed the BSM contribution using a traditional cut-based approach. The
significance was too low to claim any kind of evidence for CPV.

The cut-based approach can be improved by using BDTs and this allowed to
distinguish our BSM scenario from the SM background. At least in optimistic
cases (large CPV) we could also distinguish between a CPV and CPC BSM scenario
being able to exclude a CP conserving scenario with more than $95\%$
confidence level. This of course highlights the need for a sufficiently
precise description of the BSM physics in the CPC limit so that the prediction
of the distributions of T-odd observables is robust.

## Acknowledgements

SM is financed in part through the NExT Institute and STFC Consolidated Grant
No. ST/L000296/ 1. HW is supported by the Carl Trygger Foundation under grant
No. CTS18:164 and Ruth and Nils-Erik Stenbäck’s Foundation. Some of the
authors acknowledge the use of the IRIDIS High Performance Computing Facility
and associated support services at the University of Southampton in the
completion of this work. The work of AB and AC is funded by the Department of
Science and Technology, Government of India, under Grant No. IFA18-PH 224
(INSPIRE Faculty Award).

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