11institutetext: Department of Physics, School of Engineering and Sciences,
SRM University AP, Amaravati, Mangalagiri 522240, India22institutetext: Centre
for High Energy Physics,
Indian Institute of Science, Bangalore 560012, India33institutetext: Particle
Physics Department, STFC Rutherford Appleton Laboratory,
Chilton, Didcot, Oxon OX11 0QX, UK44institutetext: School of Physics and
Astronomy, University of Southampton,
Highfield, Southampton SO17 1BJ, UK

# Extraction of Neutrino Yukawa Parameters from Displaced Vertices of
Sneutrinos

Amit Chakraborty 3,4 Stefano Moretti 3,4 Claire H. Shepherd-Themistocleous 3,4
and Harri Waltari<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study displaced signatures of sneutrino pairs potentially emerging at the
Large Hadron Collider (LHC) in a Next-to-Minimal Supersymmetric Standard Model
supplemented with right-handed neutrinos triggering a Type-I seesaw mechanism.
We show how such signatures can be established through a heavy Higgs portal,
the sneutrinos then decaying to charged leptons and charginos giving rise to
further leptons or hadrons. We finally illustrate how the Yukawa parameters of
neutrinos can be extracted by measuring the lifetime of the sneutrino from the
displaced vertices, thereby characterising the dynamics of the underlying
mechanism of neutrino mass generation. We show our numerical results for the
case of both the current and High-Luminosity LHC.

##  1 Introduction

The Standard Model (SM) has survived nearly all experimental tests. Besides
the strong evidence for Dark Matter (DM), the only unexplained experimental
phenomenon is neutrino oscillations Athanassopoulos:1997pv ; Fukuda:1998mi ;
Aguilar:2001ty ; Ahn:2002up ; Abe:2011sj ; An:2012eh , which in turn imply
that neutrinos have a tiny, but non-zero, mass. The standard explanation is
that neutrino masses are generated through a seesaw mechanism Minkowski:1977sc
; Konetschny:1977bn ; Mohapatra:1979ia ; Magg:1980ut ; Schechter:1980gr ;
Foot:1988aq , where the effective dimension five operator responsible for
neutrino masses is suppressed by a heavy mass scale.

The range of possible seesaw scales varies from eV-scale sterile neutrinos to
those with masses of the order $10^{14}$ GeV. Taking type-I seesaw as an
example, the neutrino Yukawa couplings are $\mathcal{O}(1)$ at the upper end
while they are tiny at the lower end of the possible spectrum of their
interaction strength. One interesting option is that the seesaw scale is
around the Electro-Weak scale (EW), which requires the neutrino Yukawa
couplings to be somewhat smaller than the electron Yukawa coupling. As the
Right-Handed (RH) neutrinos are singlets under the SM gauge group, the only
interactions they have are the Yukawa couplings, which being small can lead to
Displaced Vertices (DVs) Basso:2008iv ; Helo:2013esa ; Izaguirre:2015pga ;
Accomando:2016rpc ; Liu:2019ayx .

Supersymmetry (SUSY) is a well-motivated framework for Beyond the SM (BSM)
physics. SUSY is the only space-time symmetry that can be added to the
Poincaré algebra Haag:1974qh  and it relates particles with different spins,
specifically, bosons to fermions. This relation leads to the cancellation of
quadratic divergences emerging in the calculation of the Higgs boson mass in
the SM (the so-called hierarchy problem). Furthermore, SUSY may induce the
convergence of the Electro-Magnetic (EM), weak and strong couplings at some
high energy scale, unlike the SM, a precondition for a theory embedding
unification of forces. Finally, if one removes the baryon and lepton number
violating couplings by requiring $R$-parity, one can get as a by-product of
SUSY a DM candidate in the form of the Lightest Supersymmetric Particle (LSP).

Needless to say then, in order to pursue BSM physics that addresses all the
aforementioned SM flaws, SUSY is one of the possible paths to follow, so long
that it embeds a mechanism for neutrino mass generation. In doing so, it is
then necessary to surpass its minimal realisation and consider non-minimal
ones Book , wherein the gauge and/or Higgs structures are enlarged with
respect to the case of the Minimal Supersymmetric Standard Model (MSSM). An
attractive framework in this respect is the Next-to-MSSM (NMSSM), wherein a
singlet Superfield, containing an extra singlet Higgs state and its SUSY
counterpart, is added to the MSSM particle content. This way, the so-called
$\mu$-problem Ellwanger:2009dp  of the MSSM is overcome. If such a construct
is supplemented with RH neutrinos and their SUSY counterparts, a viable model
for neutrino mass generation based on a type-I seesaw is established.

By adopting this theoretical framework, we will show that the heavy Higgs
states belonging to it (both CP-even and -odd) can have significant couplings
to RH sneutrinos, even in the alignment limit, as required by measurements of
the SM-like Higgs boson discovered at the Large Hadron Collider (LHC) in 2012.
Furthermore, given that in this model RH (s)neutrinos and higgsinos get their
masses through the same mechanism, one can expect the SUSY states to be rather
degenerate, yet, suitable soft SUSY-breaking mass terms can render the RH
sneutrinos somewhat heavier than the higgsinos. In such a case a RH sneutrino
can decay visibly through its Yukawa interactions to a charged lepton
($\ell=e,\mu$) and chargino or else a neutrino and neutralino. Therefore, a
typical signal that may emerge at the LHC in this theoretical scenario is
heavy Higgs mediated production of a sneutrino pair eventually yielding a di-
lepton signature together with soft jets and missing transverse energy. As the
seesaw mechanism has a source of lepton number violation, we get both
opposite-sign and same-sign dileptons, the latter giving better discovery
potential due to smaller backgrounds. Remarkably, the aforementioned mass
degeneracy may make the sneutrinos long lived, so that the visible tracks of
this signature may be displaced, which in turn implies a smaller background
with respect to the one affecting similar prompt signatures Moretti:2019yln ;
Moretti:2020zbn . Here we shall prove that this signature with two DVs can be
extracted at the LHC and, moreover, we shall also show how the kinematics of
the displaced (visible) tracks could allow for a measurement of the
(s)neutrino Yukawa couplings, thereby enabling one to probe the underpinning
neutrino mass generation dynamics.

Our paper is organised as follows. In the next section we introduce our
theoretical framework. In the following one we illustrate the properties of
the track displacements and how these can be related to the discussed Yukawa
couplings. Then we perform our MC analysis aimed at extracting both the
relevant signature and its underlying (s)neutrino mass parameters. We then
conclude.

##  2 NMSSM with RH neutrinos

We shall study the NMSSM with RH neutrinos. It is based on the following
Superpotential Kitano:1999qb ; Cerdeno:2008ep

$W=y^{u}_{ij}(Q_{i}\cdot H_{u})U^{c}_{j}-y^{d}_{ij}(Q_{i}\cdot
H_{d})D^{c}_{j}-y^{\ell}_{ij}(L_{i}\cdot
H_{d})E^{c}_{j}+y^{\nu}_{ij}(L_{i}\cdot H_{u})N^{c}_{i}+\lambda S(H_{u}\cdot
H_{d})$ $+\frac{\lambda_{Ni}}{2}SN_{i}^{c}N_{i}^{c}+\frac{\kappa}{3}S^{3}.$
(1)

As mentioned, this model cures some problems of the MSSM, namely the
$\mu$-term will be generated through the Vacuum Expectation Value (VEV) of the
scalar component of the singlet Superfield $S$ and we also have a mechanism
for neutrino mass generation. As the $\mu$-term should not be too far above
the EW scale, the RH neutrino masses are at the EW scale too, hence the
neutrino Yukawa couplings need to be very small, of the order $10^{-7}$.

We shall look at a scenario where we have light higgsinos and RH neutrinos
being roughly degenerate with these. The soft SUSY-breaking masses should then
make the RH sneutrinos heavier than the higgsinos and thus the decays
$\tilde{N}\rightarrow\tilde{\chi}^{0}\nu,\tilde{\chi}^{\pm}\ell^{\mp}$ are
kinematically open. The decay width will be given by the neutrino Yukawa
couplings. As mentioned, they will be tiny, and thus may lead to DVs.

This model allows two important features: EW Symmetry Breaking (EWSB)
generates both a lepton-number violating mass term for the RH sneutrinos and a
coupling between Higgs states and the sneutrinos. The coupling in the
alignment limit is (neglecting doublet-singlet mixing)

$\displaystyle C_{h\tilde{N}\tilde{N}}$ $\displaystyle=$
$\displaystyle\pm\frac{1}{2}\lambda\lambda_{N}v\sin 2\beta,$ (2)
$\displaystyle C_{H\tilde{N}\tilde{N}}$ $\displaystyle=$
$\displaystyle\pm\frac{1}{2}\lambda\lambda_{N}v\cos 2\beta,$ (3)

where the upper (lower) sign is for CP-even (CP-odd) sneutrinos. If
$\tan\beta>1.5$, the heavy Higgs state has a stronger coupling to sneutrinos.
If $\lambda$ and $\lambda_{N}$ are large, RH sneutrinos can be pair produced
through the heavy Higgs portal and they can be detected through lepton-number
violating signatures Moretti:2019yln ; Moretti:2020zbn . The singlet field is
essential in achieving this as in the MSSM with RH neutrinos the Higgs-RH
sneutrino couplings would not exist. As intimated, our aim is to use DVs to
both improve background rejection and allow for a quantitative estimate of the
neutrino Yukawa couplings.

##  3 From displacements to Yukawa couplings

We shall assume that the only kinematically available decay channels for the
sneutrino are
$\tilde{N}\rightarrow\tilde{\chi}^{0}\nu,\tilde{\chi}^{\pm}\ell^{\mp}$ and
that the neutralino and chargino are higgsino-like.

The sneutrino-lepton-chargino vertex factor is

$\lambda_{\tilde{N}\ell^{+}\tilde{\chi}^{-}}=\frac{i}{\sqrt{2}}y^{\nu}_{ab}V_{12}\frac{1+\gamma_{5}}{2},$
(4)

where $a,b$ refer to the flavours of the charged lepton, sneutrino and
$V_{12}$ tells the higgsino component of the lightest chargino. For CP-odd
sneutrinos we only need the replacement
$\frac{i}{\sqrt{2}}\rightarrow\frac{1}{\sqrt{2}}$. This leads to the partial
width (neglecting the lepton mass)

$\Gamma(\tilde{N}_{i}\rightarrow\ell^{\pm}_{j}\tilde{\chi}^{\mp})=\frac{(m_{\tilde{N}}^{2}-m_{\tilde{\chi}^{\pm}}^{2})^{2}}{16\pi
m_{\tilde{N}}^{3}}|y^{\nu}_{ji}|^{2}|V_{12}|^{2}.$ (5)

We shall assume $|V_{12}|=1$ in the following.

If we neglect the mixing between Left-Handed (LH) and RH neutrinos111This
mixing introduces a vertex factor $\lambda_{N}N_{j5}$ times the RH neutrino
component of the light eigenstates. The elements of the neutrino left-right
mixing matrix are of the same order as the elements of $y^{\nu}$. As the
singlino component $N_{j5}$ is not negligible, this correction to the vertex
factor is numerically only an order of magnitude smaller than
$|y^{\nu}N_{j3}|$ so this does introduce an $\mathcal{O}(10\%)$ correction to
the partial width., the sneutrino-neutrino-neutralino vertex factor is

$\lambda_{\tilde{N}\nu\tilde{\chi}^{0}}=-\sum_{a}\frac{i}{\sqrt{2}}\left(N_{j3}^{*}P^{*}_{ab}y^{\nu}_{bc}\frac{1-\gamma_{5}}{2}+N_{j3}P_{ab}y^{\nu*}_{bc}\frac{1+\gamma_{5}}{2}\right),$
(6)

where $P$ is the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix, $b$ and $c$
are the flavours of the neutrino and sneutrino, respectively, while $N_{j3}$
gives the $\tilde{H}_{u}$ component of the neutralino $j$. For CP-odd
sneutrinos we shall again replace
$\frac{i}{\sqrt{2}}\rightarrow\frac{1}{\sqrt{2}}$ in the prefactor. This gives
the partial width

$\Gamma(\tilde{N}_{i}\rightarrow\nu\tilde{\chi}^{0}_{j})=\sum_{k}\frac{(m_{\tilde{N}}^{2}-m_{\tilde{\chi}^{0}}^{2})^{2}}{16\pi
m_{\tilde{N}}^{3}}|y^{\nu}_{ki}|^{2}|N_{j3}|^{2}.$ (7)

Here $j$ can get values $1,2$ and $|N_{j3}|^{2}\simeq 1/2$.

From Equations (5) and (7) we see that the total decay width is proportional
to $\sum_{i}|y^{\nu}_{ij}|^{2}$. Hence the measurement of the lifetime of the
sneutrino will give an estimate of the sum of the Yukawa couplings, while the
ratios $|y_{ik}/y_{jk}|^{2}$ are proportional to
BR$(\tilde{N}_{k}\rightarrow\ell_{i}^{\pm}\tilde{\chi}^{\mp})/{\rm
BR}(\tilde{N}_{k}\rightarrow\ell_{j}^{\pm}\tilde{\chi}^{\mp})$. Measuring the
lifetime and the ratios of the Branching Ratios (BRs) would then give us the
absolute values of the individual neutrino Yukawa couplings.

We shall assume that the chargino and neutralinos would have been observed
already and their masses would be known reasonably well. From the decay mode
$\tilde{N}\rightarrow\ell^{\pm}\tilde{\chi}^{\mp}$ and the subsequent chargino
decay $\tilde{\chi}^{\pm}\rightarrow\tilde{\chi}^{0}+$ hadrons, the invariant
mass of the visible decay products should have an endpoint at
$m_{\tilde{N}}-m_{\tilde{\chi}^{0}}$ from which the sneutrino mass could be
estimated. We also expect that the mass of the heavy Higgs bosons would be
known from one of its fermionic decay modes.

In order to extract the lifetime of the sneutrino in its rest frame, we need
to measure the position of the secondary vertex and the relativistic
$\gamma$-factor of the sneutrino, which can be computed, e.g., through
$\gamma_{\tilde{N}}=E_{\tilde{N}}/m_{\tilde{N}}$. This can be done on an
event-by-event basis as follows.

In the Center-of-Mass (CM) frame energy-momentum conservation gives us

$\displaystyle E_{\tilde{N}}^{*}$ $\displaystyle=$
$\displaystyle\frac{m_{H}}{2},$ (8) $\displaystyle p_{\tilde{N}}^{*}$
$\displaystyle=$ $\displaystyle\sqrt{\frac{m_{H}^{2}}{4}-m_{\tilde{N}}^{2}}.$
(9)

To figure out what are the sneutrino momenta in the laboratory frame, we may
use the following facts.

  * •

The initial transverse momentum of the heavy Higgs boson is nearly zero as
long as there are no hard prompt jets in the event.

  * •

The three-momentum of the sneutrino is directed from the primary vertex to the
secondary one, since the sneutrino is a neutral particle and its trajectory
will not be curved.

Figure 1: The kinematics can be solved as shown. As the sneutrinos are
neutral, their momentum vectors are aligned with the displacements of the
secondary vertices from the primary vertex. We then construct the momenta in
the lab frame by using these direction vectors and the fact that the
transverse components are equal. Then we may boost to the center-of-mass
frame, where we know the total momentum $p^{*}$ drom Eq. (9).

We shall draw vectors from the primary vertex to the secondary vertices as
shown in Figure 1. Next we shall scale them so that
$|p_{T,\tilde{N}_{1}}|=|p_{T,\tilde{N}_{2}}|$ based on the small initial
transverse momentum. Due to the difference of initial state gluon momenta, the
final state will have a net momentum in the $z$-direction, but we may boost to
the CM frame simply by adding equal vectors to both laboratory frame momenta.
Once we are in the CM frame, we know $p_{\tilde{N}}^{*}$ from Equation (9) and
hence may solve for $p_{T,\tilde{N}}$. This then allows us to compute

$p_{\tilde{N}}=p_{T,\tilde{N}}\sqrt{1+\frac{d_{Z}^{2}}{d_{\perp}^{2}}},$ (10)

where $d_{z}$ and $d_{\perp}$ are the longitudinal and transverse
displacements, respectively. In Figure 2, we show a representative image of
the trajectory of a long-lived particle in a 2-dimensional plane (from
Allanach:2016pam ).

Figure 2: Schematic view in the transverse plane of a long-lived particle
decay.

We can then deduce $E_{\tilde{N}}=\sqrt{p_{\tilde{N}}^{2}+m_{\tilde{N}^{2}}}$
and from this $\gamma_{\tilde{N}}$. As the total displacement is

$d=\gamma_{\tilde{N}}v\tau,$ (11)

where $v$ is the velocity and $\tau$ the lifetime of the sneutrino, we may
solve for the lifetime

$\tau=\frac{d}{\gamma_{\tilde{N}}\sqrt{1-\frac{1}{\gamma_{\tilde{N}}^{2}}}}.$
(12)

The lifetimes follow an exponential probability distribution

$P(\tau)\propto e^{-\tau/\tau_{0}},$ (13)

where $\tau_{0}$ is the average lifetime. Plotting the lifetime distribution
with a suitable binning with a logarithmic scale should then produce a
straight line, whose slope gives the inverse of the average lifetime222We
refer to Banerjee:2019ktv ; Liu:2020vur  for elaborated discussions on
determining the lifetime of long-lived particles at the LHC.

When Next-to-Leading Order (NLO) and even higher order corrections are taken
into account, the Higgs bosons will have a finite initial transverse momentum
$p_{T}^{H}$. This will introduce a relative error in the measurement of the
momenta, which is of the order of $p_{T}^{H}/p_{\tilde{N}}$. As we are
interested in events that do not have a large longitudinal boost, i.e., all
decay products are in the barrel region of the detector, we expect typically
the $\gamma$-factor of the sneutrino to be rather small. In such a case the
error on $\gamma_{\tilde{N}}$ will be of the order of
$(p_{T}^{H}/m_{\tilde{N}})^{2}$. If we veto for hard jets a typical
$p_{T}^{H}$ is about $10$ GeV Harlander:2014uea  and the mass of the sneutrino
is $200$ GeV or more so the error from the finite initial transverse momentum
on the lifetime measurement is very small.

##  4 Collider analysis

We shall look at the process $pp\rightarrow H,A\rightarrow\tilde{N}\tilde{N}$,
where both sneutrinos decay visibly giving a charged lepton ($\ell=e,\mu$) and
a charged higgsino. In this model the sneutrinos do not have a well-defined
lepton number and the lepton number violating mass splitting is larger than
the decay width, so there is a $50\%$ chance for each sneutrino to decay to
either charge lepton. As the backgrounds for same-sign dileptons are smaller,
we choose events with two same-sign leptons and veto for a third hard lepton
with $p_{T}>15$ GeV.

If the higgsinos have a relatively smaller mass splitting, say close to $5-10$
GeV, then the decay width for the sneutrino to visible decays is small.
Numerically the minimal width is close to $10^{-14}$ GeV leading to a mean
decay length of a couple of millimeters. In contrast, in the region of phase
space where the production of heavy Higgses is possible with a reasonable
cross section and the decays are kinematically allowed, the decay widths are
typically a few times $10^{-13}$ GeV. This implies that the mean decay lengths
are around hundreds of micrometers, which leads to final states with DVs. Thus
for a large fraction of the signal events we should have two DVs with charged
leptons and soft charged tracks.

In this work, we require both of the leptons ($\ell=e,\mu$) to be displaced.
Such a requirement means that the backgrounds come from processes involving
either $b$-quarks or $c$-quarks. The same-sign requirement further reduces
these as in the case of only two heavy-flavour quarks same-sign leptons are
possible only in the case of flavour oscillations.

Our signal events will have missing transverse momentum in the form of the
LSPs and, if the spectrum is not compressed, it is possible to require large
$\not{p}_{T}$ with a rather good signal acceptance. Heavy flavour events
rarely satisfy this requirement. In order to have neutrinos with significant
transverse momentum, the quarks in the hard process must have large $p_{T}$
and the cross section falls down quickly with increasing $p_{T}$. Also,
$t\overline{t}$ events with both the tops decaying hadronically can
kinematically mimic the signal events.

###  4.1 Event generation procedure

For detailed analysis we choose the three Benchmark Points (BPs) given in
Table 1. While they differ slightly in their mass spectrum, their main
difference is in the sneutrino lifetimes. BP-I represents a “typical”
benchmark for EW scale seesaw with a decay width corresponding to a mean decay
length around $1$ mm. The second benchmark has a shorter lifetime with a mean
decay length less than $0.3$ mm, while the third one has a mean decay length
around $2$ mm, which is about as long as one can get without going to very
compressed spectra. For such cases any leptons would be soft and triggering
the event would be more difficult. For interested readers, we refer
Fukuda:2019kbp ; Bhattacherjee:2020nno  for the recent proposals on these
issues.

Regarding the constraints on the spectrum, the higgsinos need to be heavier
than about $160$ GeV with a mass splitting of $10$ GeV Sirunyan:2018iwl .
Since $m_{H}>2m_{\tilde{N}}>2m_{\tilde{H}}$, the heavy Higgses need to be
beyond $400$ GeV. However, the production cross section falls off rather
quickly beyond $500$ GeV. In this range $\tan\beta$ needs to be low to avoid
the constraints from $H\rightarrow\tau^{+}\tau^{-}$ searches Sirunyan:2018zut
; Aad:2020zxo . For our BPs we have $2<\tan\beta<3$ as this both evades the
experimental constraints and gives a large
BR($H\rightarrow\tilde{N}\tilde{N}$).

We simulate 100,000 signal events and $\mathcal{O}(10^{7})$ events each for
the $t\bar{t}$ and $b\bar{b}$ backgrounds using MadGraph5 v2.6.6 Alwall:2011uj
at LO. Parton showering and hadronisation are modelled through Pythia v8.2
Sjostrand:2014zea  and fast detector simulation is obtained by Delphes v3.3.3
deFavereau:2013fsa  with the ATLAS card. We use a modified version of the
default ATLAS card to implement the impact parameter smearing effects. The
event rates are then corrected to NLO accuracy with $k$-factor of 2 for the
signal Spira:1993bb ; Spira:1995rr ; Muhlleitner:2006wx  and to next-to-next-
to-leading order (NNLO) accuracy with $k$-factor of 1.8 for the two dominant
backgrounds Czakon:2011xx ; Aliev:2010zk ; Catani:2020kkl 333Note that, as
pointed out in Ref.Catani:2020kkl , at high transverse momenta of the bottom
quarks, large logarithmic terms of the form $\ln(\frac{p^{b}_{T}}{m_{b}})$
become important and need to be resummed properly while estimating the NNLO
cross section for the $b\bar{b}$ process. For our study, bottom quarks are
pair produced with a minimum $p^{b}_{T}$ = 200 GeV, so we make a conservative
choice of $k$-factor = 1.8.. Further, in order to generate events efficiently,
we demand that the top quarks are decaying hadronically and the $b$-hadrons,
obtained after parton shower and hadronisation of the $b$-quarks, are decaying
through leptonic final states. We generate two sets of $b\bar{b}$ samples by
varying the $p_{T}$ of the bottom quarks at the generation level. We find that
the one with generation level cut $p^{b}_{T,{\rm min}}$ = 200 GeV has better
sensitivity. In Table 2, we show the details of event generation of individual
signal and background events.

Observable | BP-I | BP-II | BP-III
---|---|---|---
Lightest Higgs mass | 125.0 | 125.2 | 125.6
2nd Higgs mass | 338.2 | 322.1 | 370.4
3rd Higgs mass | 462.2 | 484.0 | 483.6
Lightest Pseudoscalar Higgs mass | 259.0 | 256.8 | 261.7
2nd Pseudoscalar Higgs mass | 446.9 | 470.2 | 468.1
Lightest Sneutrino mass | 219.4 | 219.9 | 228.4
Lightest CP-odd Sneutrino mass | 220.0 | 219.8 | 229.2
Lightest Chargino mass | 185.8 | 177.6 | 205.1
Lightest Neutralino mass | 177.3 | 168.2 | 196.0
Next-to Lightest Neutralino mass | 200.3 | 193.2 | 218.6
BR($h_{3}\to\tilde{\nu_{1}}\tilde{\nu_{1}}$) (in %) | 5.3 | 4.9 | 4.9
BR($A_{3}\to\tilde{\nu_{1}}\tilde{\nu_{1}}^{\prime}$) (in %) | 1.2 | 1.3 | 1.6
BR($\tilde{\nu_{1}}\to\ell\tilde{\chi^{\pm}_{1}}$) (in %) | 48.2 | 48.6 | 45.3
BR($\tilde{\nu_{1}}\to\nu\tilde{\chi^{0}_{1}}$) (in %) | 51.8 | 51.4 | 54.7
$\Gamma(\tilde{\nu_{1}})$ (GeV) | $1.6\times 10^{-13}$ | $8.5\times 10^{-13}$ | $9\times 10^{-14}$
Table 1: Details of the BPs (all the masses are in GeV). The leptonic BRs include electron, muons and taus. Process | Cross section (pb) | Events generated | Event weight factor
---|---|---|---
Signal (BP-I) | 0.0666 | 100,000 | 0.09
Signal (BP-II) | 0.0558 | 100,000 | 0.08
Signal (BP-III) | 0.0508 | 100,000 | 0.07
$t\bar{t}$ (hadronic) | 369.0 | 10,000,000 | 5.1
$b\bar{b}$ ($p^{b}_{T,{\rm min}}$ = 30 GeV) | 1183654.3 | 5,000,000 | 32432.1
$b\bar{b}$ ($p^{b}_{T,{\rm min}}$ = 200 GeV) | 378.0 | 10,000,000 | 5.2
Table 2: Event simulation details at $\sqrt{s}=13~{}{\rm TeV}$ and
$\mathcal{L}=137~{}{\rm fb}^{-1}$.

###  4.2 Definition of displaced objects

We now provide the details of the observables used to probe the displaced
signal events.

  * •

Displaced leptons: The isolated leptons ($\ell=e,\mu$) with $p_{T}(\ell)>10$
GeV and $|\eta(\ell)|<2.5$ must satisfy $|{d_{\perp}}|>0.2~{}{\rm mm}$
CMS:2014hka ; Aad:2019tcc  where $d_{\perp}$ is the transverse impact
parameter relative to the primary vertex. The lepton isolation is achieved by
demanding the angular separation between the lepton and jets, $\Delta
R(\ell,jet)$, should be greater than 0.4. Additionally, we demand that the
leptons carry at least 80% (90%) of the transverse momentum within a cone of
radius R = 0.5 in case of a muon (electron). Note that, we have used a
modified version of the default ATLAS card available within Delphes to
implement the impact parameter smearing effects and obtain the displaced
leptons.

  * •

DVs using displaced tracks:

The tracks used in the DV reconstruction must satisfy the following
requirements: $p_{T}>1$ GeV and $|{d_{\perp}}|>2~{}{\rm mm}$. Further, the
significance of $d_{\perp}$ with respect to the beam axis (i.e., $|d_{\perp}|$
divided by its uncertainty $\sigma_{d_{\perp}}$) should be at least 4
Sirunyan:2018pwn ; Aad:2019kiz ; Aad:2019tcc . The final requirement on track
$\frac{|d_{\perp}|}{\sigma_{d_{\perp}}}$ improves the identification of
displaced tracks associated to the DVs.

We collect the displaced tracks and construct the DV. To combine the tracks,
we use the truth information of the vertices (i.e., vertex position) obtained
from the detector emulator and merge those tracks if $|\Delta
X(t_{i},t_{j})|<0.001$, $|\Delta Y(t_{i},t_{j})|<0.001$ and $|\Delta
Z(t_{i},t_{j})|<0.001$, where $t_{i}$ denotes the $i$-th displaced track. The
invariant mass of the DVs are calculated using the summed 4-momenta of the
associated tracks, i.e., $m^{2}_{\rm DV}={(\sum
E_{i})}^{2}-{(\sum{\vec{p}}_{i})}^{2}$, where the sum runs over the tracks
associated to the DV Aad:2019kiz .

  * •

Jets and displaced jets: The jets are constructed from calorimeter tower
elements using Fastjet v3.3.2 Cacciari:2011ma  and the anti-$k_{T}$ jet
clustering algorithm Cacciari:2008gp  with jet radius $R=0.5$. We demand that
the jets must satisfy $p_{T}>20$ GeV and $|\eta|<3.0$. For signal events,
hadronic decay of the charginos leads to displaced hadronic final states. Note
that long-lived hadrons (e.g., $b$-hadrons, $c$-hadrons) as well as soft
particles coming from the prompt decay of the displaced hard processes are
also present in the events. So, we calculate the angular separation $\Delta R$
between the jet and the displaced tracks then demand that the displaced jet
should have at least 2 or more displaced tracks satisfying $\Delta
R(j,t)<0.4$. The displaced jets are constructed following the Refs.
Nemevsek:2018bbt ; LLPtalk . We check that for signal events, the final state
stable objects are not energetic enough to pass the jet $p_{T}$ threshold and
provide significant separation from the background events, therefore, we do
not consider the displaced jets for further analysis.

###  4.3 Distribution of different observables

Here we show the distribution of several kinematic variables relevant for the
collider analysis. All the histograms are drawn for events which satisfy the
basic selections on the leptons and jets discussed in the previous section.
Distributions are scaled for $\mathcal{L}=137~{}{\rm fb}^{-1}$ of integrated
luminosity at the $\sqrt{s}=13$ TeV run of the LHC.

Figure 3: Lepton multiplicity: all (left) and displaced (right).

Figure 4: Transverse impact parameter ($d_{\perp}$) of leading (left) and sub-
leading (right) lepton.  Figure 5: Leading DV mass.

In the left panel of Figure 3, we show the multiplicity of isolated leptons
present in the signal and background events. The right panel displays the same
distribution but for the isolated displaced leptons with minimum transverse
impact parameter $d_{\perp}>0.2~{}{\rm mm}$. The $d_{\perp}$ distribution of
the leading (in $p_{T}$) two leptons are shown in Figure 4. Significant
overlap is overserved expecially at small values of $d_{\perp}$. As mentioned
in the previous section, we construct the DVs using the displaced tracks and
calculate the mass of the DV. In Figure 5, we plot the mass of the leading (in
mass) DV. It is evident that the presence of high $p_{T}$ displaced tracks,
associated with the displaced leptons originating from the sneutrino decay,
results in DVs with relatively larger masses. Therefore, we can control the
background events by selecting events with mass greater than 5 GeV or so.
Also, the distribution has a kinematical endpoint at
$m_{\tilde{N}}-m_{\tilde{\chi}^{\pm}}$ so this can be used to estimate the
sneutrino mass once the chargino mass is known.

###  4.4 Event selection and signal significance

After looking at the distributions of several kinematic observables, we find
the following selection cuts optimised for our process of interest. The cuts
are as follows.

  * •

C1: Two same-sign same-flavour leptons (electron or muons) satisfying basic
lepton selection criteria.

  * •

C2: The leading (in $p_{T}$) lepton must satisfy $p_{T}(\ell_{1})>25$ GeV.

  * •

C3: The subleading (in $p_{T}$) lepton must satisfy $p_{T}(\ell_{2})>15$ GeV.

  * •

C4: Veto on a third lepton with $p_{T}(\ell_{3})>15$ GeV.

  * •

C5: For opposite sign same flavour leptons, veto di-lepton invariant masses
around the $Z$ mass i.e., $m_{\ell^{\pm}\ell^{\mp}}\neq[80,100]$ GeV.

  * •

C6: Select events with $\cancel{p}_{T}$ $>$ 30 GeV.

  * •

C7: Both leptons have a transverse impact parameter $d_{\perp}>0.2~{}{\rm
mm}$.

Even though our main backgrounds are from heavy-flavour jets, in this study we
did not want to impose a $b$-veto so that we would be free of uncertainties
related to the $b$-tagging when we show the viability of our approach. As the
displacement of a secondary vertex is a key input for $b$-tagging algorithms,
the background displacement distributions would be affected by imposing a
$b$-veto. Also the acceptance of the signal events is uncertain. If
experimental collaborations were to do this type of an analysis, they may be
able to further improve the background rejection with the use of a $b$-tagger.

The complete cutflow is presented in Table 3. The signal significance
($\frac{S}{\sqrt{(S+B)}}$), calculated at $\sqrt{s}=13~{}{\rm TeV}$ and
$\mathcal{L}=137~{}{\rm fb}^{-1}$, is shown in Table 4. We may see that the
displacement requirement rejects almost all of the signal of BP-II. For such a
case the prompt signature can still be visible — this is actually BP3 of
Moretti:2019yln  for which a cut-based analysis gave a $\sim 3\sigma$ excess.
From Table 3 it is interesting to note that even though the cuts C4 and C5 do
not reduce the dominant backgrounds, however, they are important to reduce the
sub-dominant backgrouds like WZ, ZZ, tW, tZ and $t\bar{t}Z$ processes where
both prompt and displaced leptons can be present.

| BP-I | BP-II | BP-III | $t\bar{t}$ |  $b\bar{b}$ (30 GeV) |  $b\bar{b}$ (200 GeV)
---|---|---|---|---|---|---
C1 | 420.4 | 354.9 | 160.6 | 2637.8 | 2.7 $\times 10^{7}$ | 440.2
C2 | 354.7 | 335.8 | 71.9 | 857.0 | 2.4 $\times 10^{6}$ | 362.5
C3 | 315.3 | 309.1 | 57.6 | 384.9 | 1.2 $\times 10^{6}$ | 176.1
C4 | 314.7 | 307.8 | 56.9 | 384.9 | 1.2 $\times 10^{6}$ | 176.1
C5 | 314.7 | 307.3 | 56.9 | 384.9 | 1.2 $\times 10^{6}$ | 176.1
C6 | 265.8 | 270.4 | 49.6 | 123.2 | 32432.1 | 150.2
C7 | 35.9 | 1.7 | 13.5 | 5.1 | 0 | 5.2
Table 3: Cutflow table ($\sqrt{s}=13~{}{\rm TeV}$ and $\mathcal{L}=137~{}{\rm fb}^{-1}$). | Signal (S) | B ($t\bar{t}$) | B ($b\bar{b}$) | B (total) | Significance = $\frac{S}{\sqrt{(S+B)}}$
---|---|---|---|---|---
BP-I | 35.9 | 5.1 | 5.2 | 10.3 | 5.3
BP-II | 1.7 | 5.1 | 5.2 | 10.3 | 0.5
BP-III | 13.5 | 5.1 | 5.2 | 10.3 | 2.8
Table 4: Signal significances estimated at $\sqrt{s}=13~{}{\rm TeV}$ and
$\mathcal{L}=137~{}{\rm fb}^{-1}$.

Before we proceed to extract neutrino Yukawa couplings, we discuss a few
kinematic observables which can be used, in addition to cuts C1-C6, to reduce
the SM backgrounds. For example, we can optimise $d_{\perp}$ for both the
leading two leptons, especially the second lepton which, for signal events,
has larger $p_{T}$ and longer decay lengths (see plots in the upper panel of
Figure 6). Another important quantity is the ratio of the missing transverse
momentum and scalar $H_{T}$, defined as
$\alpha=\frac{\cancel{p_{T}}}{\sqrt{H_{T}}}$. Signal events have large missing
transverse energy and, therefore, relatively larger values of $\alpha$. These
additional observables provide us with an extra handle to minimise the SM
backgrounds and improve the sensitivity to sneutrino events.

Figure 6: Distribution of the transverse impact parameter ($d_{\perp}$) of the
leading (top-left) and sub-leading (top-right) lepton. Bottom: The
distribution of $\alpha=\frac{\cancel{p_{T}}}{\sqrt{H_{T}}}$. All the figures
are drawn for events satisfying cuts C1-C6.

##  5 Extraction of Yukawa couplings

In order to extract the Yukawa couplings, we need to know the spectrum. The
heavy Higgs masses can be measured reasonably accurately from their fermionic
decay channels. The higgsinos would also be discovered from other searches and
those would give an estimate for the masses of the neutralinos and charginos.
The sneutrino mass can be estimated by measuring the invariant mass of the DV
related to the sneutrino decay to a lepton and a chargino. The invariant mass
distribution has an endpoint at $m_{\tilde{N}}-m_{\tilde{\chi}^{\pm}}$ (see
Figure 5), which will give us the sneutrino mass once we know the chargino
mass.

We assume that we are able to get an essentially background free sample (say,
over $95\%$ purity) using the cuts given in the previous section (both those
given in Table 3 and discussed at the end of the section) and the usage of
$b$-tagging. We then do an event-by-event correction to the displacements
described in section 3 to get the actual lifetimes. As both the CP-even and
CP-odd Higgs bosons contribute to the signature, we need to pick which mass we
use in the boost correction. As discussed in Moretti:2019yln , the CP-even
Higgs state usually has the largest BR to sneutrinos, although also the amount
of available phase space has an impact. If the CP properties of the two heavy
Higgses have been measured, the CP-even mass would give the better estimate,
otherwise it would be a reasonable choice to pick the heavier one due to the
larger phase space. This ambiguity leads to a systematic error in the
measurement of the Yukawa couplings, which is of the order of $10\%$ estimated
by studying the impact of choosing the other Higgs mass to the lifetime
distribution.

The background originating from heavy flavour hadrons that survives the cuts
is heavily boosted (with the $p_{T}$ of a typical $b$-jet being $200$ GeV or
more) leading to an average displacement greater than $5$ mm. As the boost
correction assumes the particle to be heavy, the lifetime of heavy flavour
hadrons will be overestimated and will on average correspond to mean decay
lengths around $5$ mm, although the exponential distribution of course gives
events at all displacements. We shall use a binning of $0.1$ mm. As the total
number of background events at the HL-LHC is expected to be around $200$
(scaled from Table 4 to $3000$ fb-1) and these are distributed into some $100$
bins or more, there will not be many background events in a single bin. The
background distribution may be estimated, e.g., by looking at events with
small vertex mass and $\not{p}_{T}/\sqrt{H}_{T}$ and the subtract it from the
distribution.

The signal region includes a cut on the transverse distance of the leptons
requiring $d_{\perp}>0.2$ mm. The distribution of lifetimes close to the cut
will be modified and hence we put a lower bound on the lifetime when fitting.
We fit the exponential to the distribution in the interval $0.5$ mm$<c\tau<5$
mm. This leads to rather robust results if the true lifetime is around
$c\tau\simeq 1$ mm but, if the lifetime is shorter, the number of events in
the fit region becomes so low that the statistical error increases. In such a
case it might be reasonable just to use this method to find an upper limit for
the lifetime and derive a lower limit for the neutrino Yukawa couplings. An
upper limit for the Yukawa couplings can be obtained from the constraint on
the sum of neutrino masses. That can be expressed as $\sum
m_{\nu}=\mathrm{Tr}(m^{\nu})=\sum_{i,j}|y^{\nu}_{ij}|^{2}v^{2}\sin^{2}\beta/2m_{N_{j}}$.

Once we had studied the fitting method with one BP, we tried to analyse other
BPs blindly so that one of the authors generated events and another one then
analysed them, the result was then compared to the unknown input. As long as
the decay lenghts were at the mm scale, a reasonable agreement was achieved.

Figure 7: The fit of an exponential to the lifetime distribution of the
sneutrino for BP-I. The amount of data corresponds to $3000$ fb-1 at
$\sqrt{s}=14$ TeV.

We show in Figure 7 a fit of the sneutrino lifetime for BP-I. We fit a simple
exponential to the lifetime distribution. The fit with a number of events
corresponding to HL-LHC energy ($\sqrt{s}=14$ TeV) and luminosity ($3000$
fb-1) gives us $\sum_{i}|y^{\nu}_{i1}|^{2}=(2.62\pm 0.13\pm 0.26)\times
10^{-13}$, where the first error is statistical and the second one the theory
error (assumend to be $10\%$) from the ambiguity in choosing the CP-even or
CP-odd Higgs as discussed above. On top of these there will be experimental
systematic errors, which are mostly related to measuring the primary and
secondary vertices444As this is a simple lifetime measurement, many typical
sources for systematic errors, like parton distribution functions, do not
matter.. The true value is $2.28\times 10^{-13}$, which slightly more than one
standard deviation off. For BP-II the event sample is so small that the Yukawa
couplings cannot be reliably estimated. The best fit value would have been
$3.57\times 10^{-13}$ but, as the actual average decay length is shorter than
the lower end of our fitting region, this estimate is based only on few events
and individual outliers can significantly change the result. The estimate was
off by a factor of two. With such a short decay length it makes sense only to
give a lower bound on the Yukawa couplings. We give the results for our three
BPs in Table 5. When doing the estimate, we added $10\%$ to the invisible
decay width of Equation (7) due to the mixing effect of LH and RH neutrinos
(see footnote on page 1).

| BP-I | BP-II | BP-III
---|---|---|---
Measured $\sum|y^{\nu}_{1i}|^{2}$ | $2.62\pm 0.13\pm 0.26$ | $>2.94$ | $3.73\pm 0.67\pm 0.37$
Actual $\sum|y^{\nu}_{1i}|^{2}$ | $2.28$ | $7.81$ | $2.40$
Table 5: The sums of neutrino Yukawa couplings from displacements and the
actual input values in units of $10^{-13}$. The first uncertainty is
statistical and the second one is a $10\%$ theoretical uncertainty related to
the ambiguity between the CP-even/odd Higgs states as discussed in the text.
For BP-II the decay length was so short that we only quote a lower limit for
the Yukawa couplings.

From this value we can then determine the absolute values of individual Yukawa
couplings as the number of events with a given lepton flavour, which will be
proportional to $\epsilon_{i}|y^{\nu}_{i1}|^{2}$, where $\epsilon_{i}$ is the
efficiency of identifying the lepton of flavour $i$. Since the error on
$|y^{\nu}|^{2}$ will often be below $20\%$ with high enough statistics, the
error on the Yukawa couplings themselves will be around $10\%$. Hence even
this rather simple fitting method can give a reasonably good estimate of the
Yukawa couplings.

##  6 Conclusions and outlook

We have studied sneutrino pair production via the heavy Higgs portal in the
NMSSM with RH neutrinos. In the model both higgsinos and RH neutrinos get
their masses through the singlet VEV, so we expect them to be at the EW scale.
If RH sneutrinos are heavier than higgsinos, they can decay to both a charged
lepton and a chargino or a neutrino and a neutralino, the former one giving a
visible signature at colliders.

Since the RH sneutrinos are gauge singlets, their decay modes are dictated by
the neutrino Yukawa couplings, which are tiny in an EW scale seesaw model. The
smallness of the decay width together with the lepton number violation arising
from the RH (s)neutrino mass term leads to a signature with same-sign
dileptons emerging from displaced vertices together with missing transverse
momentum. The SM backgrounds for such a signal topology are therefore low.

We have indeed performed an alaysis proving that the study of the emerging
signatures with DVs can lead to the extraction of the neutrino Yukawa
couplings, which salient features are as follows.

  * •

We searched for displaced leptons and jets, with the leptons originating from
sneutrino decays. Several BPs of the aforementioned BSM scenarios were
introduced, with varied displacement lengths, and studied by MC analysis
assuming a simple cut-based analysis providing a good signal-to-backgroud
ratio already at the end of the 13 TeV run of the LHC.

  * •

The kinematic distribution of the displaced vertex mass gives an end point
which can be used to estimate the sneutrino mass once the chargino mass is
known.

  * •

Additional variables can make the regions of phase-space effectively
background-free, thereby implying better handle on the extraction of the
intervening Yukawa couplings.

  * •

The decay width of the RH sneutrinos are dictated by the sum of the squares of
the Yukawa couplings so a measurement of the sneutrino lifetime would allow
one to measure the Yukawa couplings. The individual Yukawa couplings can then
be extracted from the sneutrino BRs to different lepton flavours, once
corrected with identification efficiencies.

  * •

We ultimately showed that, for lifetimes corresponding to mm scale
displacements, lifetimes can be measured with reasonable accuracy, which can
then give the absolute values of the Yukawa couplings with a $10\%$ accuracy
so long that sufficiently large data samples can be accrued for the signal,
like, e.g., at the HL-LHC.

In short, such an approach can lead to the extraction of the underlying
neutrino dynamics parameters from the study of sneutrinos signatures at
hadronic colliders, albeit under a specific SUSY paradigm. The results
obtained here from events with DVs therefore complement those obtained in
Refs. Moretti:2019yln ; Moretti:2020zbn  for the case of prompt signatures.

## Acknowledgments

SM, CHS-T and HW are financed in part through the NExT Institute. SM is also
funded by the STFC consolidated Grant No. ST/L000296/1. HW acknowledges
financial support from the Magnus Ehrnrooth Foundation, the Finnish Academy of
Sciences and Letters and STFC Rutherford International Fellowship (funded
through the MSCA-COFUND-FP Grant No. 665593). The work of AC is funded by the
Department of Science and Technology, Government of India, under Grant No.
IFA18-PH 224 (INSPIRE Faculty Award). Some of us also acknowledge the use of
the IRIDIS High Performance Computing Facility and associated support services
at the University of Southampton. We thank Nishita Desai, Terhi Järvinen,
Emmanuel Olaiya, Ian Tomalin and Jose Zurita for useful discussions.

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