aainstitutetext: Harish-Chandra Research Institute, Allahabad 211019,
Indiabbinstitutetext: Regional Centre for Accelerator-based Particle Physics
Harish-Chandra Research Institute, Allahabad 211019, Indiaccinstitutetext:
Physics and Applied Mathematics Unit, Indian Statistical Institute,
203 B.T. Road, Kolkata 700108, India

# Sbottoms of natural NMSSM at the LHC

Jyotiranjan Beuria c Arindam Chatterjee a AseshKrishna Datta
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Search for the bottom squarks (sbottoms) at the Large Hadron Collider (LHC)
has recently assumed a heightened focus in the hunt for Supersymmetry (SUSY).
The popular framework of the Next-to-Minimal Supersymmetric Standard Model
(NMSSM) could conceive a naturally light sbottom which can easily be
consistent with available constraints from the experiments at the LHC.
Phenomenology of such sbottoms could in principle be as striking as that for a
light top squark (stop) thanks to a rather nontrivial neutralino sector (with
appreciable mixing among the neutral higgsinos and the singlino) that the
scenario gives rise to. Nonetheless, finding such sbottoms would require a
moderately large volume of data ($\sim 300$ fb-1) at the 13 TeV run of the
LHC. A multi-channel analysis establishing a generic depletion of events in
the usual $2b$-$jets+\not{E}_{T}$ final state while registering, in
conjunction, characteristically significant rates in various multi-lepton
final states accompanied by $b$-$jets$ might point not only to the presence of
light sbottom(s) but could also shed crucial light on their compositions and
the (singlino) nature of the lightest SUSY particle (LSP).

###### Keywords:

Hadronic Colliders, Beyond Standard Model, Supersymmetry Phenomenology

††preprint: HRI-P-16-03-001
RECAPP-HRI-2016-004

##  1 Introduction

It is well known that squarks from the third generation, in particular, the
top squarks (stops), are capable of providing a ‘natural’ supersymmetric
(SUSY) solution to the notorious hierarchy problem that cripples the scalar
(Higgs) sector of the Standard Model (SM) if they are relatively light
($\lesssim 1\;\mathrm{TeV}$). Over the years, this has motivated the search
for sub-TeV stops at the collider experiments and recently, at the Large
Hadron Collider (LHC), hunt for them has expectedly taken the centre stage.
The latest analyses of the data from the 7 TeV and 8 TeV runs of the LHC
(collectively termed as the LHC Run-I in this work) Khachatryan:2015wza ;
Aad:2015pfx  have resulted in stringent lower bounds on the stop mass, under
varied assumptions over how it decays. The most conservative of these bounds
says that the lighter stop should be heavier than $\sim 700$ GeV Aad:2015pfx .
A reasonably performing 13 TeV run of the LHC (LHC-13), currently underway,
could quickly and substantially improve on these bounds thus pushing them
around a TeV.

On the other hand, over the past few years, there has been a growing interest
in the phenomenology of the bottom squarks (sbottoms) Li:2010zv ; Datta:2011ef
; Lee:2012sy ; Alvarez:2012wf ; Bi:2012jv ; Ghosh:2012wb ; Chakraborty:2013moa
; Belanger:2013gha ; Batell:2013psa ; Huang:2014oza ; Kobakhidze:2015dra ;
Dutta:2015hra ; Han:2015tua , the other squarks from the third generation.
Also, in recent years, implications of the LHC results for the squarks from
the third generation in a ‘natural’ SUSY setup Barbieri:1987fn  are studied
intensively Cheng:2013fma ; Han:2013kga ; Zheng:2014dea ; Grothaus:2015yha .
This has a solid motivation. Although the sbottom sector is not broadly as
instrumental as the stop sector for rendering SUSY ‘natural’ (due to its
relatively much smaller coupling to the Higgs boson), sbottoms are rather
close cousins of the stops. In particular, the left-handed (L-type) stop and
the sbottom ($\tilde{t}_{L}$ and $\tilde{b}_{L}$) live in the same weak
doublet and enjoy an approximate custodial symmetry Lee:2012sy .
Quantitatively, its breaking is restricted by electroweak precision data via
the well-known Peskin-Takeuchi $STU$ (oblique) parameters Peskin:1990zt . On
the theoretical side, the soft masses for these L-type states are
$SU(2)_{L}$-invariant and hence equal. However, a modest splitting occurs due
to $SU(2)_{L}$-breaking $D$-term (which is a function of
$\tan\beta={v_{u}\over v_{d}}$, the ratio of vacuum expectation values of the
neutral components of the two Higgs doublets) and because of the presence of
an $F$-term which goes as the corresponding quark (top or bottom) mass. This
implies that at least one squark state from each flavor (which are dominantly
L-type) would have comparable masses. Only a rather large mixing in the stop
sector could obscure this phenomenon under certain circumstances. On the other
hand, the masses of the right-handed (R-type) stop and the sbottom
($m_{\tilde{t}_{R}}$ and $m_{\tilde{b}_{R}}$) are neither related by any
symmetry nor do they bear any relation to their corresponding L-type partners.
Thus, they can assume any value allowed by applicable searches at the
colliders. Hence if the lighter stop is accessible to the LHC experiments and
happens to be mostly L-type, the same should be true for a sbottom which is
also dominantly L-type. This substantiates the increased interest in the
search for sbottoms at the LHC. On the other hand, a relatively light R-type
sbottom can be present while the other sbottom and both stops are relatively
heavy111 It may, however, be noted that presence of relatively heavy stops
(and/or a gluino) may not be an immediate threat to naturalness as revealed by
certain recent parameterizations in the MSSM context that refer only to weak
scale values of the parameters as long as $|\mu|$ ($|\mu_{eff}|$, in NMSSM) is
small enough which is essential to the scenario presented in this work
Baer:2012up ; Baer:2013ava ; Mustafayev:2014lqa ; Baer:2015rja . .

It is somewhat interesting to note that various possible decay modes of the
sbottom have recently been explored rigorously for the first time by the
experimental searches at the LHC Chatrchyan:2013fea ; Aad:2014qaa ;
Aad:2014pda ; Aad:2015pfx ; Khachatryan:2015wza . These result in lower bounds
on $m_{\tilde{b}_{1}}$ roughly ranging between 500-750 GeV. Subsequently,
these findings are exploited by the phenomenological works providing further
insight Han:2015tua ; Ghosh:2012wb . However, these studies are heavily based
on the popular framework of the Minimal SUSY extension of the Standard Model
(MSSM). In the present work, we look into the phenomenology of possibly light
sbottom(s) in the framework of the next-to-MSSM (NMSSM) Ellwanger:2009dp , a
scenario which has assumed a renewed relevance in view of the recent Higgs
results from the LHC.

Issues with sbottom search vis-a-vis stop search have already been discussed
in the literature Bai:2012gs . For example, search for stop squarks may suffer
from large background from the SM top quark production when the decay
$\tilde{t}_{1}\to t\chi_{1}^{0}$ is dominant. This is not the case with
sbottom search in one of its preferred decay modes, i.e., $\tilde{b}_{1}\to
b\chi_{1}^{0}$. On the other hand, depending on how they decay, stop and
sbottom could lead to the same final states thus making it somewhat difficult
to extract unambiguous information. Combining analyses for the sbottom and
stop to find an optimal sensitivity to a given region in the parameter space
parameter has recently been advocated in reference Han:2015tua .

The popular framework of the NMSSM offers new possibilities in this context.
First, the scope of having a richer electroweak gaugino sector (a mixture of a
singlino, higgsinos and gauginos) with relatively light states could lead to
interesting phenomenology of the stop and the sbottom through their decays to
these states. Secondly, the NMSSM can easily accommodate relatively light
stops without being in conflict with the observations in the Higgs sector
Beuria:2015mta . If such a light stop happens to be dominantly L-type in
nature, there is going to be an L-type sbottom close-by in mass to the light
stop. These two issues render the NMSSM to be a ‘natural’ setup to study the
sbottom sector. Interesting aspects of such a scenario and their phenomenology
have recently been pointed out mainly in the context of (or in terms of)
relatively light stops accompanied by a singlino-like neutralino LSP while the
lighter chargino and a pair of immediately heavier neutralinos are higgsino-
like Beuria:2015mta ; Kim:2015dpa .

In the present work, we focus on relatively light sbottom(s) with varied
compositions whose phenomenology might turn out to be more tractable than that
of the stops. This is since the extent of mixing that could be present in the
sbottom sector is much smaller than that possible in the stop sector. We adopt
a setup inspired by naturalness and hence with low values of the effective
$\mu$ parameter ($\mu_{\mathrm{eff}}$) Ellwanger:2011mu . This exploits the
original scope in the NMSSM to dynamically generate such a low value of
$\mu_{\mathrm{eff}}$ via a suitable choice of the vacuum expectation value
(vev) $v_{S}$ of the singlet scalar field, a defining feature of such a
scenario. As we shall see, this opens up further possibilities of not only
having a singlino-like lightest SUSY particle (LSP) but also nontrivial
mixings among the singlino and the higgsinos. Impact of the singlino on the
NMSSM phenomenology under specific circumstances have been discussed in the
literature Das:2012rr ; Ellwanger:2013rsa ; Kim:2014noa ; Ellwanger:2014hia ;
Cao:2014efa ; Potter:2015wsa . Here, we investigate how the mixed singlino-
higgsino LSP would alter the decay pattern of the sbottom in characteristic
ways. In particular, for a mostly L-type sbottom, its decay to higgsino-like
lighter chargino is driven by the top Yukawa coupling and hence could
dominate, in particular, at low $\tan\beta$. In contrast, decays of a
dominantly R-type light sbottom to lighter higgsinos-like states (both
chargino and neutralinos) are all governed by the bottom Yukawa coupling and
thus, could become comparable.

Light stops when accompanied by a modest value of $\mu_{\mathrm{eff}}$ could
further ensure a lower fine-tuning in the setup. These together add another
perspective to the current issue. A stop lighter than what is achievable in
the MSSM (being compatible with the observed Higgs mass) can be obtained for
larger values of the $\lambda$ parameter in the NMSSM. Note that $\lambda$
couples the singlet state to the doublet Higgs states in the NMSSM
superpotential. However, given the relationship $\mu_{\mathrm{eff}}=\lambda
v_{S}$, a smaller value of $\mu_{\mathrm{eff}}$ would then require a
relatively small $v_{S}$. On the other hand, avoiding appearance of Landau
pole below the unification (GUT) scale restricts $\kappa$ to smaller values
when $\lambda$ is large. As we will see later, a small $\mu_{\mathrm{eff}}$
combined with smaller $\kappa$ but a large $\lambda$, could induce a
significant mixing among the higgsinos and the singlino before reaching a
point when the LSP becomes singlino-like with $m_{LSP}\approx
m_{\tilde{S}}=2\kappa v_{S}$.

The paper is organized as follows. In section 2 we outline the theoretical
setup for the $Z_{3}$-invariant NMSSM by briefly discussing the sbottom (stop)
sector, the Higgs sector, the sectors involving the charginos and the
neutralinos and the relevant interactions among these states which are germane
to the present work. In section 3 we look into the status of sbottom search at
the end of the LHC Run-I when interpreted in the NMSSM framework. We take a
critical look into the cases with sbottoms(s) having varied compositions and
their phenomenological ramifications. Section 4 is devoted to identifying a
few benchmark scenarios based on the findings of section 3 followed by the
presentation of their detailed simulation at the 13 TeV run of the LHC. In
section 5 we conclude.

##  2 The theoretical setup

The present work relies on a popular NMSSM framework known as the
$Z_{3}$-invariant NMSSM. The superpotential is given by

$\mathcal{W}=\mathcal{W_{MSSM}}|_{\mu=0}+\lambda\hat{S}\hat{H}_{u}.\hat{H}_{d}+{\kappa\over
3}\hat{S}^{3}.$ (1)

Here, $\cal{W_{MSSM}}$ is the MSSM superpotential, $\hat{H}_{u}$ and
$\hat{H}_{d}$ stand for the doublet Higgs superfields while the gauge singlet
superfield, characterizing the NMSSM, is denoted by $\hat{S}$. The
corresponding soft-supersymmetry breaking terms are given by

$-\mathcal{L}_{\rm soft}=-\mathcal{L^{\rm MSSM}_{\rm
soft}}|_{B\mu=0}+m_{S}^{2}|S|^{2}+\lambda
A_{\lambda}SH_{u}.H_{d}+\frac{1}{3}\kappa A_{\kappa}S^{3}+{\rm h.c.}+..\,$ (2)

where $\mathcal{L}^{\rm MSSM}_{\rm soft}$ represents the soft SUSY-breaking
terms in the MSSM, $A_{\lambda}$ and $A_{\kappa}$ are the new trilinear soft
SUSY-breaking terms (with dimensions of mass) that are present in the NMSSM
while $m_{S}^{2}$ is the soft SUSY-breaking mass term for the singlet scalar
$S$ of the NMSSM.

In the rest of this section we briefly discuss the salient features of the
sbottom and the stop sectors, the Higgs sector and the sector involving the
neutralinos and the charginos (the electroweakinos). The stop and the Higgs
sector play crucial roles in motivating and defining the overall scenario that
we work in while, in our ‘simplified’ approach, except for some involved
situations, these are only the sbottoms and the electroweakinos that directly
shape up the interesting phenomenology pertaining to the sbottoms in the
NMSSM.

###  2.1 The sbottom and the stop sectors

At the tree level, the mass-squared matrices for the sbottom and the stop
squarks in the NMSSM are similar to the corresponding ones in the MSSM with
$\mu$ in the off-diagonal terms being replaced by $\mu_{\mathrm{eff}}$
($=\lambda v_{S}$). These are given, at the tree level, by the following
$2\times 2$ matrices in the bases $\\{m_{\tilde{b}_{L}},m_{\tilde{b}_{R}}\\}$
and $\\{m_{\tilde{t}_{L}},m_{\tilde{t}_{R}}\\}$, respectively;

${\cal
M}_{\tilde{b}}=\left(\begin{array}[]{cc}m_{\tilde{Q}_{3}}^{2}+y_{b}^{2}v_{d}^{2}+(v_{u}^{2}-v_{d}^{2})\left(\frac{g_{1}^{2}}{12}+\frac{g_{2}^{2}}{4}\right)&y_{b}(A_{b}v_{d}-\mu_{\mathrm{eff}}v_{u})\\\
y_{b}(A_{b}v_{d}-\mu_{\mathrm{eff}}v_{u})&m_{\tilde{D}_{3}}^{2}+y_{b}^{2}v_{d}^{2}+(v_{u}^{2}-v_{d}^{2})\frac{g_{1}^{2}}{6}\end{array}\right)\;$
(3)

and

${\cal
M}_{\tilde{t}}=\left(\begin{array}[]{cc}m_{\tilde{Q}_{3}}^{2}+y_{t}^{2}v_{u}^{2}+(v_{u}^{2}-v_{d}^{2})\left(\frac{g_{1}^{2}}{12}-\frac{g_{2}^{2}}{4}\right)&y_{t}(A_{t}v_{u}-\mu_{\mathrm{eff}}v_{d})\\\
y_{t}(A_{t}v_{u}-\mu_{\mathrm{eff}}v_{d})&m_{\tilde{U}_{3}}^{2}+y_{t}^{2}v_{u}^{2}-(v_{u}^{2}-v_{d}^{2})\frac{g_{1}^{2}}{3}\end{array}\right).$
(4)

In the above two equations $m_{\tilde{Q}_{3}}$, $m_{\tilde{D}_{3}}$ and
$m_{\tilde{U}_{3}}$ denote the soft SUSY breaking mass terms for the $SU(2)$
doublet (L-type) and the singlet (R-type) sbottom and stop squarks,
respectively. $y_{b,t}$ are the respective Yukawa couplings while $A_{b,t}$-s
are the respective trilinear soft SUSY breaking terms. $v_{d}$ and $v_{u}$
stand for the vev’s of the CP-even down- and up-type neutral Higgs bosons,
$H_{d}^{0}$ and $H_{u}^{0}$, respectively. $g_{1}$ and $g_{2}$ denote the
gauge couplings for the $U(1)_{Y}$ and $SU(2)_{L}$ gauge groups, respectively.
We also define a generic rotation matrix $\mathcal{R}_{\tilde{f}}$ that
diagonalizes the sbottom and stop mass matrices as follows:

${\cal
R}_{\tilde{f}}=\left(\begin{array}[]{rr}\cos\theta_{\tilde{f}}&\>\sin\theta_{\tilde{f}}\\\
-\sin\theta_{\tilde{f}}&\>\cos\theta_{\tilde{f}}\end{array}\right)$ (5)

where $\tilde{f}$ stands for $\tilde{b}$ or $\tilde{t}$, respectively and
$\theta_{\tilde{f}}$, at the tree level, is still given by its standard MSSM
expression

$\sin
2\theta_{\tilde{f}}={2m_{f}X_{f}\over{m_{\tilde{f}_{2}}^{2}-m_{\tilde{f}_{1}}^{2}}}\quad,\qquad
X_{f}=A_{\tilde{f}}-\mu_{\mathrm{eff}}\;r\quad,$ (6)

where $r=\tan\beta\,(\cot\beta)$ for the sbottom (stop) sector, $m_{f}$ is the
corresponding fermion mass while $m_{\tilde{f}_{1,2}}$ stand for the masses of
the lighter and the heavier sbottom/stop eigenstates, respectively. In this
convention, $\theta_{\tilde{f}}=0({\pi\over 2})$ corresponds to the unmixed
L-type (R-type) state to be the lightest mass eigenstate.

Clearly, if $m_{\tilde{Q}_{3}}$ is relatively small, one could expect a light
sbottom along with a light stop close-by in mass, both of them being
dominantly L-type. It could also, in general, be noted that since
$y_{b}<<y_{t}$, mixing between the L- and the R-type sbottom states can at
best be modest (when compared to the stop sector). Thus, the physical sbottoms
are nearly pure ‘chiral’ states and hence their phenomenology is more
tractable. Nonetheless, as we will later see, even such a small mixing could,
under certain circumstances, lead to very interesting phenomenology for the
sbottoms.

###  2.2 The Higgs sector

The Higgs sector of the NMSSM also gets extended nontrivially when compared to
its MSSM counterpart with the inclusion of a singlet scalar $S$ belonging to
the singlet superfield $\hat{S}$. On electroweak symmetry breaking, CP even
components of the three neutral scalar fields, $H_{u}$, $H_{d}$ and $S$, mix
to give rise to three CP-even Higgs bosons of the NMSSM. The mass of the SM-
like Higgs boson is given by Ellwanger:2011sk

$m_{h}^{2}=m_{Z}^{2}\cos^{2}2\beta+\lambda^{2}v^{2}\sin^{2}2\beta+\Delta_{\mathrm{mix}}+\Delta_{\rm
rad.corr.}\,$ (7)

with $v=\sqrt{v_{u}^{2}+v_{d}^{2}}\simeq 174\,{\rm GeV}$. The first term in
the right hand side of 7 gives the tree level squared mass of the SM-like
Higgs boson in the MSSM. The term proportional to $\lambda^{2}$ is the NMSSM
contribution at the tree level. The term $\Delta_{\mathrm{mix}}$ originates in
the so-called singlet-doublet mixing. In the limit of weak mixing, this is
given by

$\Delta_{\mathrm{mix}}=\frac{4\lambda^{2}v_{S}^{2}v^{2}(\lambda-\kappa\sin
2\beta)^{2}}{\tilde{m}_{h}^{2}-m_{ss}^{2}}$ (8)

with $\tilde{m}_{h}^{2}=m_{h}^{2}-\Delta_{\mathrm{mix}}$ and
$m_{ss}^{2}=\kappa v_{S}(A_{\kappa}+4\kappa v_{S})$. The additional NMSSM
contribution at the tree level could enhance the mass of the SM-like Higgs
boson to a level such that to conform to the observed mass, it does not need
to bank much upon the radiative corrections Ellwanger:2011sk ; Beuria:2015mta
. Hence smaller values of stop masses (and/or smaller mixing in the stop
sector) can be easily afforded thus rendering the framework more ‘natural’.

Although the sbottom sector could only play a subdominant role in issues
pertaining to ‘naturalness’, its phenomenology, nonetheless, could
significantly exploit such upshots in the NMSSM stop sector. To conform to the
observed mass of the SM-like Higgs boson, a scenario with light stops in the
NMSSM prefers a genuinely low $\tan\beta\,(\lesssim 5)$ King:2012is ;
King:2012tr ; Beuria:2015mta . As we will discuss in the subsequent sections,
this shapes the phenomenology of the sbottoms in a novel way. It may be
reiterated that a light stop which is mostly L-type is accompanied by an
L-type sbottom having a close-by mass. An sbottom which is mostly R-type,
however, can be light independent of the masses of the stops. As we shall see,
in both cases, light sbottoms could easily escape experimental bounds in a
simplified NMSSM framework that we adopt in this work.

###  2.3 The neutralinos and the charginos

The compositions of the neutralinos and the charginos are crucial to the
phenomenology of the sbottoms (and of the stops, as well). While the structure
of the chargino sector of the NMSSM (at the tree level) is identical to that
of the MSSM, the neutralino sector has some essential difference. The
difference arises from the presence of the fermionic component $\widetilde{S}$
corresponding to the singlet superfield $\hat{S}$ present in the NMSSM
superpotential (see equation 1). The singlino could mix with the higgsinos and
the gauginos. As we shall see, this could lead to an LSP which has a
significant singlino admixture thus affecting crucially the cascade decay
patterns of the heavier SUSY particles. We assume conserved $R$-parity and
hence a stable LSP.

The symmetric $5\times 5$ neutralino mass matrix is given by

${\cal
M}_{0}=\left(\begin{array}[]{ccccc}M_{1}&0&-\dfrac{g_{1}v_{d}}{\sqrt{2}}&\dfrac{g_{1}v_{u}}{\sqrt{2}}&0\\\
&M_{2}&\dfrac{g_{2}v_{d}}{\sqrt{2}}&-\dfrac{g_{2}v_{u}}{\sqrt{2}}&0\\\
&&0&-\mu_{\mathrm{eff}}&-\lambda v_{u}\\\ &&&0&-\lambda v_{d}\\\ &&&&2\kappa
v_{S}\end{array}\right)\;$ (9)

in the basis
$\psi^{0}=\\{\widetilde{B},~{}\widetilde{W}^{0},~{}\widetilde{H}_{d}^{0},~{}\widetilde{H}_{u}^{0},~{}\widetilde{S}\\}$
Ellwanger:2009dp ; Kraml:2005nx . In the above expression, $M_{1}$ and $M_{2}$
stand for the soft SUSY-breaking masses of the $U(1)$ ($\widetilde{B}$) and
the $SU(2)$ ($\widetilde{W}$) gauginos, respectively. The rest of the
variables appearing in equation 9 have already been introduced in the text.
The above mass-matrix can be diagonalized by a matrix $N$ Ellwanger:2009dp ;
Kraml:2005nx :

$N^{*}{\cal
M}_{0}N^{\dagger}=\mathrm{diag}(\chi_{1}^{0},\chi_{2}^{0},\chi_{3}^{0},\chi_{4}^{0},\chi_{5}^{0})$
(10)

such that the five neutralino mass-eigenstates (in order of increasing mass as
‘$i$’ varies from 1 to 5, in our present study) can be written in a compact
form in terms of the five weak eigenstates ($\psi_{j}$, with $j=1,....,5$) as

$\chi_{i}^{0}=N_{ij}\psi_{j}^{0}$ (11)

As can be gleaned from the entries of the mass matrix, the singlino state
mixes with the gaugino states only via the higgsino sector. Hence these
mixings are rather suppressed. In contrast, the singlino-higgsino mixing is
direct and could be appreciable depending upon the relative values of
$\mu_{\mathrm{eff}}$, $2\kappa v_{S}\,(=m_{\widetilde{S}})$ and the value of
$\lambda$. Thus, in a setup where $\mu_{\mathrm{eff}}$ and $2\kappa v_{S}$ are
small but of comparable magnitudes, the neutralino sector becomes rather
involved with the presence of low-lying neutralino states (including the LSP)
having widely varying singlino admixtures. This can alter the phenomenology of
the light squarks in an essential manner, in particular, that of the light
sbottoms which is the subject of the present work.

The chargino mass matrix of the NMSSM scenario under consideration, in the
basis

$\psi^{+}=\left(\begin{array}[]{c}-i\widetilde{W}^{+}\\\
\widetilde{H}_{u}^{+}\end{array}\right)\quad,\quad\psi^{-}=\left(\begin{array}[]{c}-i\widetilde{W}^{-}\\\
\widetilde{H}_{d}^{-}\end{array}\right)\quad,$ (12)

is given by

${\cal M}_{C}=\left(\begin{array}[]{cc}M_{2}&\quad g_{2}v_{u}\\\
g_{2}v_{d}&\quad\mu_{\mathrm{eff}}\end{array}\right).$ (13)

This differs from ${\cal M}_{C}$ of the MSSM in the entry $\mu_{\mathrm{eff}}$
(which is $\mu$ for the MSSM). As in the MSSM, this asymmetric matrix is
diagonalized by two $2\times 2$ unitary matrices $U$ and $V$:

$U^{*}{\cal
M}_{C}V^{\dagger}=\mathrm{diag}(m_{\tilde{\chi}_{1}^{\pm}},m_{\tilde{\chi}_{2}^{\pm}})$
(14)

with $m_{\tilde{\chi}_{1}^{\pm}}<m_{\tilde{\chi}_{2}^{\pm}}$.

Relevant interactions involving the sbottom, the neutralinos, the chargino and
the Higgs boson are discussed in section 2.4. In the present work, we restrict
ourselves in a ‘simplified’ setup where the lighter three neutralinos are
typically light and are mixtures of the singlino and the higgsino states while
the lighter chargino is almost purely higgsino. The heavier states from both
the sectors are made to be heavy enough by choosing $M_{1}$ and $M_{2}$ to be
rather large so that they mostly decouple from the phenomenology.

###  2.4 Relevant interactions

In the context of the present work, the important interactions are those
involving the sbottoms, the neutralinos and the charginos. While we would rely
on the strong production modes of the sbottoms (in pairs), the phenomenology
would crucially depend not only on how the sbottoms decay to neutralinos and
the charginos, but also on how these decay products cascade down to the LSP.
Below we outline these interactions in brief.

Interactions of sbottom with neutralinos are of the generic form

$b\tilde{b}_{i}\tilde{\chi}^{0}_{n}:\quad
g_{2}\,\bar{b}\left(a_{in}^{\tilde{b}}P_{R}+b_{in}^{\tilde{b}}P_{L}\right)\tilde{\chi}_{n}^{0}\tilde{b}_{i}+{\mathrm{h.c.}}$

which is similar to the MSSM case except for the fact that the neutralino
index ‘$n$’ now runs from 1 to 5 (see equation 9) Kraml:2005nx  to include a
singlino-like state. $P_{R,L}$ are the standard projection operators given by
${{1\pm\gamma_{5}}\over 2}$. However, couplings of sbottoms to charginos are
identical to those in the MSSM and are given by Bartl:2003pd

$t\tilde{b}_{i}\tilde{\chi}^{+}_{j}:\quad
g_{2}\,\bar{t}\left(l_{ij}^{\tilde{b}}P_{R}+k_{ij}^{\tilde{b}}P_{L}\right)\tilde{\chi}_{j}^{+}\tilde{b}_{i}+{\mathrm{h.c.}}\hskip
10.0pt.$

In the above expressions, $a_{in}^{\tilde{b}}$, $b_{in}^{\tilde{b}}$,
$l_{ij}^{\tilde{b}}$ and $k_{ij}^{\tilde{b}}$ are all as given in reference
Bartl:2003pd . Nonetheless, for future purposes, we write down the factors
$l_{ij}^{\tilde{b}}$ and $k_{ij}^{\tilde{b}}$ (that appear in the decays of
sbottoms to the chargino states) explicitly as follows:

$l_{ij}^{\tilde{b}}=-{\cal R}_{i1}U_{j1}+y_{b}{\cal R}_{i2}U_{j2},\qquad
k_{ij}^{\tilde{b}}=y_{t}{\cal R}_{i1}V_{j2}^{*}$

and $y_{b}={m_{b}\over{\sqrt{2}m_{W}\cos\beta}}$ and
$y_{t}={m_{t}\over{\sqrt{2}m_{W}\sin\beta}}$.

The decays of neutralinos and the lighter chargino that involve the Higgs
bosons (in particular, the light, neutral ones) and the gauge ($Z$ and $W$)
bosons are governed by the following interactions. The Higgs-neutralino-
neutralino coupling is given by

$\displaystyle H_{a}\chi^{0}_{i}\chi^{0}_{j}$ $\displaystyle:$
$\displaystyle\frac{\lambda}{\sqrt{2}}(S_{a1}\Pi_{ij}^{45}+S_{a2}\Pi_{ij}^{35}+S_{a3}\Pi_{ij}^{34})-\sqrt{2}\kappa
S_{a3}N_{i5}N_{j5}$
$\displaystyle+\frac{g_{1}}{2}(S_{a1}\Pi_{ij}^{13}-S_{a2}\Pi_{ij}^{14})-\frac{g_{2}}{2}(S_{a1}\Pi_{ij}^{23}-S_{a2}\Pi_{ij}^{24})$

where $\Pi_{ij}^{ab}=N_{ia}N_{jb}+N_{ib}N_{ja}$ Ellwanger:2009dp  and $N$ is
given by equations 10 and 11. On the other hand, the $Z$-neutralino-neutralino
coupling is determined by the factor Choi:2004zx

$\displaystyle Z\chi^{0}_{i}\chi^{0}_{j}$ $\displaystyle:$ $\displaystyle
g_{2}\left(N_{i3}N_{j3}-N_{i4}N_{j4}\right)$

while $W^{\pm}$-chargino-neutralino interaction is determined by the two
bilinear charges ${\cal W}_{L}$ and ${\cal W}_{R}$ where Choi:2004zx

$W^{\pm}\chi_{i}^{\mp}\chi^{0}_{j}:\quad\left\\{g_{2}{\cal
W}_{Lij}=g_{2}(U^{*}_{i1}N_{j2}+{1\over\sqrt{2}}U^{*}_{i2}N_{j3}),\quad{g_{2}\cal
W}_{Rij}=g_{2}(V^{*}_{i1}N^{*}_{j2}-{1\over\sqrt{2}}V^{*}_{i2}N^{*}_{j4})\right\\},$

$U$ and $V$ being the unitary matrices given by equation 14. All through,
$g_{1}$ and $g_{2}$ stand for $U(1)$ and $SU(2)$ gauge couplings,
respectively.

##  3 Sbottoms at the LHC Run-I

As has been mentioned earlier, given that only a modest mixing between L- and
R-type sbottom states is possible, broad phenomenological studies involving
sbottoms can safely be carried out presuming them to be pure ‘chiral’ states.
Hence in the subsequent discussions, we would systematically take up the cases
of a light sbottom which is either L- or R-type. However, in the light of the
discussion above, we would also demonstrate the impact of even a minuscule
admixture of L-type sbottom in the lighter sbottom state which is otherwise
dominantly R-type.

###  3.1 The case with a light $\tilde{b}_{1}\equiv\tilde{b}_{L}$

The published bounds on the mass of $\tilde{b}_{1}\equiv\tilde{b}_{L}$ from
the LHC Run-I Khachatryan:2015wza ; Aad:2015pfx  vary depending on how it
decays222While the present work is being finalized, The ATLAS Collaboration
Aad:2016tuk  has come up with some tighter lower bounds on the masses of stop
and sbottom squarks from the 13 TeV run of the LHC under similar sets of
theoretical assumptions. This might have some limited numerical bearing on the
present work. We, however, continue to use the analyses of references
Khachatryan:2015wza ; Aad:2015pfx  which are incorporated and validated in a
popular framework like CheckMATE Drees:2013wra ; Kim:2015wza . The analyses
discussed here and the conclusions drawn thereof are expected to remain
broadly unaltered when subjected to the latest experimental results.. As
indicated in the Introduction, it touches $\sim 650$ GeV when $\tilde{b}_{L}$
decays 100% of the times into the $b$-LSP mode. However, in a scenario with
relatively light higgsino-like neutralinos, $\tilde{b}_{L}$ could decay into
these states. Unlike in the MSSM, where these decays would be suppressed since
these are driven by the weak bottom Yukawa coupling (which cannot compete with
an analogous setup with, say, a bino/wino-like LSP neutralino), in the case of
the NMSSM these would dominate. This is precisely since, in our scenario, the
LSP is more like a singlino and thus its coupling to sbottom would be
suppressed. Thus, in the present case, $\tilde{b}_{L}$ could dominantly decay
to higgsino-like neutralinos followed by the latter decaying to the singlino-
like LSP. The other possibility, under such a circumstance, is that
$\tilde{b}_{L}$ decays to a higgsino-like chargino and a top quark. Note that
once the phase space for this decay is available, this could be the dominant
two-body decay mode for $\tilde{b}_{L}$ since it is driven by the top Yukawa
coupling. Hence, in any case, the assumption of BR[$\tilde{b}_{L}\to b$ LSP]
would not hold for our scenario and the mass-bound on $\tilde{b}_{L}$
mentioned above is easily evaded.

It can now be noted that the published bound on the light sbottom mass touches
$\sim 470$ GeV when the two-body decay branching ratios BR[$\tilde{b}_{1}\to
t\chi_{1}^{-}$] followed by BR[$\chi_{1}^{\mp}\to W^{\mp}\chi_{1}^{0}$] are
both 100% Aad:2014pda ; Aad:2015pfx . Such decays can give rise to same-sign
dilepton (SSDL) final states (which are used to put the above bound) thanks to
two sources of $W$-boson in an sbottom cascade; the top quark and the
chargino. An SSDL pair originates from two $W$-bosons of the same sign that
would come from the cascades of the $\tilde{b}\tilde{b}^{*}$ pair originally
produced in the hard scattering. Being known to be a very clean search
channel, the SSDL final state offers an efficient probe to the sbottom sector.
From the above discussion it is apparent that, in our scenario, this could
only be the case if $\tilde{b}_{1}$ is dominantly L-type thus attracting the
bound mentioned above.

Here, we would like to make a very general observation (valid in the MSSM as
well) which is, to the best of our knowledge, new. In a setup with lighter
higgsino-like neutralinos and chargino (the lightest states in case of the
MSSM), there could be a region of phase space where the two-body decay
$\tilde{b}_{1}\to t\chi_{1}^{-}$ is closed but the three-body decays
$\tilde{b}_{1}\to tW^{-}\chi_{1}^{0}$ (via off-shell chargino) and/or
$\tilde{b}_{1}\to bW^{+}\chi_{1}^{-}$ (via off-shell top), which exploit(s)
the same enhanced coupling $\tilde{b}_{1}$-$t$-$\chi_{1}^{-}$ could compete
and, under favorable circumstances (at low virtuality), beat the two-body
$\tilde{b}_{1}\to b\chi_{2,3}^{0}$ decay rate.

Figure 1:  Variations of branching fractions of $\tilde{b}_{1}$ into various
two- and three-body decay modes as functions of $\Delta
m(\tilde{b}_{1},t+\chi_{1}^{-})$ with $\mu_{\mathrm{eff}}$ being indicated in
the color palette. The scan is performed with $1<\tan\beta<3$,
$0.6<\lambda<0.75$, $0.01<\kappa<0.4$, $100<|\mu_{\mathrm{eff}}|<400$ GeV,
$|A_{\lambda,\kappa}|<1$ TeV, $m_{\tilde{Q}_{3}}=570$ GeV,
$m_{\tilde{U}_{3}}=m_{\tilde{D}_{3}}=1500$ GeV and $A_{t,b}=0$,
$M_{1},M_{2}=1$ TeV and $M_{3}=2$ TeV. The value of $m_{\tilde{b}_{1}}$ is
fixed around 400 GeV. Only those points are chosen for which the singlino
fraction in the LSP is larger than 50%. ‘3B’ stands for 3-body decays. See
text for further details.

The phenomenon is illustrated in figure 1 in our NMSSM scenario with a light
sbottom which is dominantly L-type. It shows the variations of decay branching
fractions of $\tilde{b}_{1}$ in various 2- and 3-body decay modes as functions
of $\Delta m(\tilde{b}_{1},t+\chi_{1}^{-})$, i.e., the difference of mass of
$\tilde{b}_{1}$ and the sum of the masses of the top quark and the lighter
chargino. $\Delta m(\tilde{b}_{1},t+\chi_{1}^{-})$ is varied by varying
$\mu_{\mathrm{eff}}$ which is reflected in the color palette. The scatter plot
on the left of the upper panel shows how the 3-body decay branching ratio (in
red) takes off significantly in advance of the 2-body threshold of
$\tilde{b}_{1}\to t\chi_{1}^{-}$ (in light brown) and how it overtakes the
collective 2-body branching ratio into $b\chi_{1,2,3}^{0}$ final states (in
blue). This then indicates the onset of the regime where bounds from both
$\tilde{b}_{1}\to b$ LSP and $\tilde{b}_{1}\to t\chi_{1}^{-}$ cease to exist.
The right plot of the upper panel reveals that the BR[$\tilde{b}_{1}\to
b\chi_{2,3}^{0}$] has the dominant share in the collective 2-body branching
ratio over most of the parameter space. The 2-body branching ratio for the
decay $\tilde{b}_{1}\to b\chi_{1}^{0}$ (singlino) is mostly small and only
becomes significant when $\Delta m$ is large negative thus implying the decays
$\tilde{b}_{1}\to b\chi_{2,3}^{0}$ suppressed by phase space.

Our goal is to find how light the sbottom could be under such a circumstance.
We thus choose an sbottom mass of 400 GeV for the illustration. This is close
to but smaller than the current bound of around 470 GeV obtained assuming
BR[$\tilde{b}_{1}\to t\chi_{1}^{-}$]=100%. A good check to ensure that such a
relaxation is indeed possible is to subject the parameter point to a CheckMATE
analysis. We find that this is indeed the case. In principle, under such a
circumstance, an even lighter sbottom could have escaped the current
experimental searches.

In any case, it is important to realize that such a light
$\tilde{b}_{1}\equiv\tilde{b}_{L}$ would imply a nearby light stop. The latter
could generically escape the latest LHC bounds obtained assuming
$\tilde{t}_{1}\to t\chi_{1}^{0}$ or $\tilde{t}_{1}\to b\chi_{1}^{-}$ only if
it shares its decay branching fractions among available modes Aad:2015pfx ;
Khachatryan:2015wza . This, in turn, is possible in two ways: first, if there
is some mixing present in the stop sector leading to some admixture of R-type
stop in a predominantly L-type $\tilde{t}_{1}$; secondly, even for an unmixed
L-type $\tilde{t}_{1}$ if it could decay to heavier neutralinos.

A careful look at the left plot of the upper panel of figure 1 reveals a
narrow slit within the red and the blue bands. The plot in the bottom panel
reveals that the slits actually separate two strands which have
$\mu_{\mathrm{eff}}<0$ (in blue) and $\mu_{\mathrm{eff}}>0$ (in orange). A
priori, this may not be unusual given that the sign on $\mu_{\mathrm{eff}}$ is
known to affect phenomenology in a modest way Han:2015tua . In fact, we found
that the 3-body decay widths of $\tilde{b}_{1}$ remain comparable for either
sign of $\mu_{\mathrm{eff}}$ in the regime under consideration. Rather, the
slits have their origin in the palpably different 2-body decay widths of
$\tilde{b}_{1}$ for negative and positive $\mu_{\mathrm{eff}}$, the latter
resulting in a larger width. This effect, in turn, can be traced back to
different couplings of sbottoms to the gaugino components of the light
neutralino states for the two signs on $\mu_{\mathrm{eff}}$. Note that even
though we work in a regime where $\mu_{\mathrm{eff}}<<M_{1}=M_{2}$ (=1.5 TeV),
i.e., in a somewhat ‘deep higgsino region’, a suppressed bottom Yukawa
coupling (as is expected for low values of $\tan\beta$) makes way for even
small gaugino admixtures to play the lead role. Indeed, this effect can be
minimized by increasing the values of $M_{1}$ and $M_{2}$ further. Also note
that, the small values of $\Delta m$ on the left part of the plot indeed
correspond to large $\mu_{\mathrm{eff}}$ (mostly negative) and hence to a
heavier chargino.

The Feynman diagrams of the two possible modes of 3-body decay of sbottom are
shown in figure 2: $\tilde{b}_{1}\to tW^{-}\chi_{1}^{0}$ proceeds through a
virtual $\chi_{1}^{-}$ while $\tilde{b}_{1}\to bW^{+}\chi_{1}^{-}$ proceeds
through a virtual top quark. The former is dominant at low values of $\Delta
m$ for which the chargino is heavier. The latter dominates for low values of
$\mu_{\mathrm{eff}}$ where the 3-body final state competes with the 2-body
one.

Figure 2: Feynman diagrams representing 3-body decays of $\tilde{b}_{1}$.

It turns out, however, that the scattered points in the left part of these
plots with the 3-body decays reaching up to $\sim 40$% cannot survive bounds
from stop searches at the LHC. This is because for these points the LSP is
relatively light ($\sim 50$ GeV) and thus BR[$\tilde{t}_{1}\to
t\;{\mathrm{LSP}}$] could still be 100% even though the LSP is mostly
singlino-dominated (since $\kappa$ is small and $\mu_{\mathrm{eff}}$ is large)
thanks to the large top Yukawa coupling. Also, since
$\mu_{\mathrm{eff}}\approx m_{\tilde{b}_{1}}\approx m_{\tilde{t}_{1}}\sim 400$
GeV (both $\tilde{b}_{1}$ and $\tilde{t}_{1}$ being dominantly L-type) and
thus the chargino is very close to the stop mass, the 2-body decays of stop to
higgsino-like states would be suppressed. Given the nature of the couplings
there could be a competition among various decay modes of stop. An appropriate
LHC study on the stop sector Aad:2015pfx  rules these points out.

In the context of the NMSSM, it is particularly noteworthy that published LHC
results Aad:2015pfx  do not constrain $m_{\tilde{b}_{1}}$ at all when
$m_{\tilde{\chi}_{2}^{0}}\lesssim 280$ GeV for both BR[$\tilde{b}_{1}\to
b\chi_{2}^{0}$] and BR[$\chi_{2}^{0}\to\chi_{1}^{0}h$] being 100%. Presence of
a singlino-like LSP in the spectrum and a pair of, rather degenerate,
higgsino-like neutralinos could naturally fulfil such conditions in the
cascade decay of $\tilde{b}_{1}\equiv\tilde{b}_{L}$ if its decay to the
$t\chi_{1}^{-}$ final state is not only kinematically disallowed but
$m_{\tilde{b}_{1}}$ is also significantly below the $t\chi_{1}^{-}$ threshold
(see section 3.3).

###  3.2 The case with a light $\tilde{b}_{1}\equiv\tilde{b}_{R}$

The important difference between the phenomenology of
$\tilde{b}_{1}\equiv\tilde{b}_{R}$ and that of
$\tilde{b}_{1}\equiv\tilde{b}_{L}$ is that all the relevant decays of
$\tilde{b}_{R}$ to higgsino-like states are governed by the small bottom
Yukawa coupling and thus they all compete and saturate above the top-chargino
threshold. This is shown in figure 3. When compared to figure 1, a sharp
contrast is clearly visible in the vicinity of the bottom-chargino threshold
and beyond. Note that while for $\tilde{b}_{L}$ the decay to the chargino
final state quickly dominates and becomes 100% above the threshold, the same
for $\tilde{b}_{R}$ saturates to a value of around 40%.

As a result, in this regime, bound on a light sbottom, which is R-type,
obtained from the top-chargino mode would be much relaxed when compared to the
case of an L-type sbottom. In practice, the amount of such a relaxation can
only be estimated explicitly using a framework like CheckMATE. On the
theoretical side, under such a situation, the lower bound on the mass of the
$\tilde{b}_{R}$ is restricted by how low one could go down in the chargino
mass. The latter is constrained by LHC analyses involving the chargino-LSP
system when the chargino decays 100% of the times to $W$-LSP, which is the
case in the NMSSM scenario we are discussing. References Aad:2015eda ;
Khachatryan:2014qwa  indicate that under such a situation chargino mass as low
as 200 GeV is allowed333It has recently been reported in Xiang:2016ndq  that
mass-reaches like $m_{\chi_{2,3}^{0},\,\chi_{1}^{\pm}}\sim 320(500)$ GeV are
possible for the current 13 TeV, 30 fb-1 (future 14 TeV, 300 fb-1) run of the
LHC. Thus, masses for these excitations that are used in most of the benchmark
points are within the reach of 13 TeV LHC run. for a LSP mass around 100 GeV.

Figure 3: Same as in the plots in the upper panel of figure 1 but for an
R-type $\tilde{b}_{1}$.

###  3.3 The case with $m_{\tilde{b}_{1}}<m_{t}+m_{\tilde{\chi}_{1}^{-}}$

Issues with sbottom having mass below the top-chargino threshold (large
$\mu_{\mathrm{eff}}$) are more or less similar for $\tilde{b}_{L}$ and
$\tilde{b}_{R}$. The important exception is that for the latter case the
3-body decays are not at all favored below the top-chargino threshold. In both
cases, the two body decays to $b\chi_{i}^{0}$ dominate. In this regime, it is,
however, important to study how the these latter BRs are shared among $b$-LSP
($\chi_{1}^{0}$; singlino-like) and $b\chi_{2,3}^{0}$ (higgsino-like) since
these rates determine which kind of experimental constraints on sbottom mass
would be pertinent. It can be noted here that sbottoms are searched for at the
LHC assuming their three pure modes of decays, viz., $\tilde{b}_{1}\to
b\chi_{1}^{0}$, $\tilde{b}_{1}\to b\chi_{2}^{0}$ followed by $\chi_{2}^{0}\to
h\chi_{1}^{0}$ and $\tilde{b}_{1}\to t\chi_{1}^{-}$ Aad:2014pda ;
Chatrchyan:2013fea . For sbottom mass below the top-chargino threshold the
last decay mode simply does not exist.

In the right plot of the upper panel of figure 1 and in the right plot of
figure 3 we illustrate these shares of BRs for $\tilde{b}_{L}$ and
$\tilde{b}_{R}$, respectively. As before, the outer edges of the green and the
pink regions correspond to the maximum admixture (50%) of higgsino in the LSP
that we allow for in this analysis. The figures show that BR in bottom-LSP
drops sharply as $\mu_{\mathrm{eff}}$ decreases (as we go from left to right)
since this opens up the decays to $b\chi_{2,3}^{0}$. Clearly, thus, the bounds
on sbottom mass that assume BR[$\tilde{b}_{1}\to b\,$ LSP]=100% would not hold
in this regime. The next regime with BR[$\tilde{b}_{1}\to
b\chi_{2,3}^{0}$]=100%, where sbottom searches put bound on its mass, also
gets affected because of such a sharing of branching fractions thus resulting
in a relaxed bound. The situation is a bit worse for $\tilde{b}_{L}$ for which
the 3-body decays also get their shares in this regime.

###  3.4 Effect of mixing in the sbottom sector

The discussions in section 3.1 could well point to how things change
drastically even for a small admixture of $\tilde{b}_{L}$ in an otherwise
$\tilde{b}_{R}$-dominated $\tilde{b}_{1}$. It may be reiterated that, unlike
in the case of the MSSM, one could afford a significantly low $\tan\beta$
($\sim 2$) in the NMSSM when $\lambda$ is large. As a consequence, $y_{t}\over
y_{b}$ could become large. Given that the mixing in the sbottom sector is
naturally restrained, it is all the more interesting to observe how
significantly this affects the decay pattern of $\tilde{b}_{1}$ having only
very small $\tilde{b}_{L}$ admixture.

In the left plot of figure 4 we illustrate the implication of such a small
$\tilde{b}_{L}$-admixture in $\tilde{b}_{1}$ for the decay branching fraction
BR[$\tilde{b}_{1}\to t\chi_{1}^{-}$]. We take close values of
$m_{\tilde{Q}_{3}}$ ($\sim 680$ GeV) and $m_{\tilde{D}_{3}}$ ($\sim 630$ GeV)
which facilitate mixing in the sbottom sector. Also, we fix
$\mu_{\mathrm{eff}}$ at $-300$ GeV, a value which is representative of a
‘natural’ setup. Variations of colour along the individual curves reflect the
varying chiral-mixing (in terms of $\cos\theta_{\tilde{b}}$) which can be
gleaned from the colour palette. Note that $\cos\theta_{\tilde{b}}=0$ implies
a pure R-type $\tilde{b}_{1}$ (see section 2.1).

Variations of BR[$\tilde{b}_{1}\to t\chi_{1}^{-}$] are then shown for three
representative values of $\tan\beta$ ranging over small to intermediate
values. It can be seen from the figure that for all values of $\tan\beta$, the
said branching fraction increases with increasing $|A_{b}|$. This is expected
since increasing $A_{b}$ can indeed induce larger $\tilde{b}_{L}$ admixture in
$\tilde{b}_{1}$ which is predominantly R-type in our present setup. The
positions of the troughs in all these curves are primarily determined by the
values of $A_{b}$ for which the $\tilde{b}_{L}$ admixtures are the smallest.
For small $\tan\beta\,(\approx 2)$, this can be clearly seen in the figure.
The relative locations of the troughs for curves with different values of
$\tan\beta$, however, have complicated dependence on both $A_{b}$ and
$\tan\beta$. Also, one could find that, for large values of $|A_{b}|$,
BR[$\tilde{b}_{1}\to t\chi_{1}^{-}$] could get as large as 95%, 70% and 55%
for $\tan\beta=2,5$ and 10, respectively with a small to moderate mixing.
Furthermore, one can see that the gradients of the variations are steeper for
smaller $\tan\beta$ values. This can be seen as a result of how the ratio
$y_{t}\over y_{b}$ varies as a function of $\tan\beta$. This variation is
shown in the right plot of figure 4.

Such large branchings of $\tilde{b}_{1}$ to top quark and chargino, under the
present setup, are definite outcomes of possible low values of $\tan\beta$
which can be easily accommodated in a ‘natural’ NMSSM scenario. In contrast,
in the MSSM, such a large branching fraction for $\tilde{b}_{1}\to
t\chi_{1}^{-}$ is rather difficult to attain since the observed mass of the
SM-like Higgs boson restricts $\tan\beta$ from the bottom.

Figure 4:  Left: Variations of BR($\tilde{b}_{1}\to t\chi_{1}^{-}$) as a
function of $A_{b}$ for various values of $\tan\beta$. The color palette
indicates the varying magnitude of sbottom mixing along each curve in terms of
$\cos\theta_{\tilde{b}}$. Other parameters are fixed as follows:
$\mu_{\mathrm{eff}}=-300$ GeV, $m_{\tilde{b}_{1}}\approx 700$ GeV,
$m_{\tilde{b}_{2}}\approx 750$ GeV, $m_{\tilde{Q}_{3}}=680$ GeV,
$m_{\tilde{U}_{3}}=1.5$ TeV, $m_{\tilde{D}_{3}}=630$ GeV, $A_{t}=0$, $M_{1}=1$
TeV, $M_{2}=1.5$ TeV and $M_{3}=2$ TeV. Right: Variation of the ratio
$y_{t}\over y_{b}$ as a function of $\tan\beta$. The color palette indicates
the varying magnitude of $y_{b}$ along the curve.

##  4 Benchmark scenarios and the 13 TeV LHC

As has been already pointed out, studying the phenomenology of sbottoms is
important in its own right and could shed crucial light on the physics of the
squarks from the third generation by complementing/supplementing the studies
in the stop sector. The subject gets further involved in nontrivial ways in a
scenario like ‘natural’ NMSSM, in particular, in the light of available
results from the LHC Run-I. A systematic first study into this should be based
on some salient aspects that the sbottom sector possesses in such a setup.
Some such issues are discussed in the previous section. In this section we
first present a few benchmark points which are representative of the
situations described in sections 3.1, 3.2 and 3.4. We then study the collider
aspects of these scenarios at the ongoing LHC-13.

###  4.1 Benchmark scenarios

In table 1 we present six benchmark scenarios/points (BPs). For all of them,
we fix the gaugino soft mass parameters to be $M_{1}=$ 1 TeV, $M_{2}$= 1.5 TeV
and $M_{3}$ = 2 TeV such that the heavier neutralinos and the heavier chargino
are absent in the cascades of a much lighter $\tilde{b}_{1}$ (a simplified
scenario with only singlino- and higgsino-like lighter neutralinos and a
higgsino-like lighter chargino). Large values of $\lambda$ ($\gtrsim 0.6$) are
chosen in order to enhance the tree level NMSSM contribution to the SM-like
Higgs boson mass. As discussed earlier, this requires low values of $\kappa$
to ensure the absence of Landau pole below the GUT scale which, in turn,
results in an LSP which is singlino-like. We thus choose $\kappa\lesssim
0.25$. Except for BP6, all other benchmark points have $A_{t}=0$, as discussed
in section 3.1.

Two common but salient features of these scenarios are that they all have
small $\tan\beta\,(\sim 2)$ and that the LSP is mostly a singlino ($\sim
75\%-90\%$), albeit with a non-negligible higgsino admixture. Such a
composition is achieved by choosing relatively large values of
$\lambda(\gtrsim 0.6)$ while keeping $\mu_{\mathrm{eff}}$ somewhat smaller.
Note that such a setup generically guarantees a low fine-tuning (i.e., more
‘natural’) and singlino-domination in the LSP can thus be seen to be directly
connected to this fact. Palpable mixing among the higgsino and the singlino
states is also unavoidable for large $\lambda$ and since
$\mu_{\mathrm{eff}}\approx\lambda v_{S}$ is a possibility. The benchmark
scenarios presented here are checked for their viability via CheckMATE against
the available analyses of the LHC Run-I data. These points also satisfy all
Higgs-related constraints and other phenomenological bounds that are built-in
in NMSSMTools-v4.8.2 Ellwanger:2004xm ; Ellwanger:2005dv ; Ellwanger:2006rn ;
Das:2011dg ; Muhlleitner:2003vg . However, constraints pertaining to muon
$g-2$ are ignored while in the dark matter sector only the upper bound on the
relic density ($\Omega_{c}h^{2}<0.131$) as incorporated in NMSSMTools is
respected.

We have presented three benchmark scenarios (BP1-BP3) having $\tilde{b}_{1}$
which is purely $\tilde{b}_{R}$ while for the scenarios BP5 and BP6 it is
purely $\tilde{b}_{L}$. In BP4 we allow for a little admixture of
$\tilde{b}_{L}$ in $\tilde{b}_{1}$. Scenarios BP1 and BP2 differ in the masses
of $\tilde{b}_{1}$ and in the fact whether $h_{1}$ or $h_{2}$ is the SM-like
Higgs boson. BP2 and BP3 are rather similar except for the possibility of an
additional decay of $\chi_{2,3}^{0}$ to $h_{1}$ that is available for the
latter. For these three benchmark points we set
$m_{\tilde{Q}_{3}}=m_{\tilde{U}_{3}}=1500$ GeV444Choice of stops and an
sbottom as heavy as 1.5 TeV, whenever possible, is made just to ensure that
they effectively decouple for our purpose while the other sbottom may still
remain light in the scenario under consideration. It may, however, be noted
that stops with mass $\lesssim 1$ TeV would not affect yields in the final
states we consider and, hence our results, in any significant way. As
discussed in the Indroduction, relatively heavier stops may not necessarily be
in conflict with ‘naturalness’ when $|\mu_{eff}|$ is small enough. This has
been checked explicitly by using NMSSTools where the related finetuning
parameter yields values $\sim 10$, which is indicative of a healthy
naturalness, for all the benchmark points with heavy stop(s). and
appropriately small values of $m_{\tilde{D}_{3}}$ are chosen.

In BP4 we present a scenario where a small admixture of $\tilde{b}_{L}$ in
$\tilde{b}_{1}\,(\simeq\tilde{b}_{R})$ which is achieved by making $A_{b}$
large even as $m_{\tilde{Q}_{3}}\sim m_{\tilde{D}_{3}}$. As pointed out
earlier, even a tiny left admixture ($\lesssim 2\%$), when assisted by large
$\frac{y_{t}}{y_{b}}$ (driven by small $\tan\beta$), could result in a very
large branching fraction for $\tilde{b}_{1}\to t\chi_{1}^{+}$ (reaching $\sim
90\%$). This is something not usually expected of
$\tilde{b}_{1}\equiv\tilde{b}_{R}$, in particular, when other two-body decay
modes (to light neutralinos) are kinematically accessible to it.

In BP5 $\tilde{b}_{1}$ is made to resemble $\tilde{b}_{L}$ by lowering the
$m_{\tilde{Q}_{3}}$ (=500 GeV $<<m_{\tilde{U}_{3}}=m_{\tilde{D}_{3}}=1500$
GeV). BP6 is a point representative of the cross-over region for the curves
that illustrate the variations of the 2-body and 3-body branching fractions of
$\tilde{b}_{1}$ in figure 3. Note that, in this case, the lighter stop has a
mass of around 400 GeV. As pointed out in section 3.1, this could escape the
latest bound from the LHC only if the stop sector is attributed with some
mixing. Also, BP6 features sub-TeV $\tilde{t}_{1}$ and $\tilde{t}_{2}$ whose
origins are discussed in section 3.1.

As far as the cascades of the light neutralinos and the lighter chargino are
concerned, it can be seen from table 1 that the former decay predominantly to
$\chi_{1}^{0}Z$ followed by $\chi_{1}^{0}h$ (SM-like) while the latter decays
100% of the time to $\chi_{1}^{0}W^{\pm}$. Scenario BP5 is an exception for
which BR[$\chi_{2}^{0}\to\chi_{1}^{0}h$] dominates over
BR[$\chi_{2}^{0}\to\chi_{1}^{0}Z$]. For all the benchmark points, current
bounds on the chargino-neutralino and the third generation squark sectors are
respected, using the public software CheckMATE wherever dimmed necessary.

Parameters | BP1 | BP2 | BP3 | BP4 | BP5 | BP6
---|---|---|---|---|---|---
$\lambda$ | 0.63 | 0.65 | 0.65 | 0.67 | 0.68 | 0.70
$\kappa$ | 0.15 | 0.20 | 0.17 | 0.20 | 0.19 | 0.25
$A_{\lambda}$ (GeV) | 435 | -600 | -600 | -580 | 660 | -320
$A_{\kappa}$ (GeV) | -50 | 50 | 50 | 50 | -50 | 85
$\mu_{{}_{eff}}$ (GeV) | 200 | -300 | -300 | -300 | 350 | -250
$\tan\beta$ | 2 | 2 | 2 | 2 | 2 | 1.6
$A_{t}$ (GeV) | 0 | 0 | 0 | 0 | 0 | 300
$A_{b}$ (GeV) | -1000 | -1000 | -1000 | -2500 | 0 | 0
$m_{\tilde{Q}_{3}}$ (GeV) | 1500 | 1500 | 1500 | 550 | 500 | 260
$m_{\tilde{U}_{3}}$ (GeV) | 1500 | 1500 | 1500 | 1500 | 1500 | 400
$m_{\tilde{D}_{3}}$ (GeV) | 330 | 440 | 440 | 520 | 1500 | 1500
Observables | BP1 | BP2 | BP3 | BP4 | BP5 | BP6
Singlino | 0.78 | 0.87 | 0.90 | 0.87 | 0.89 | 0.77
fraction in the LSP |  |  |  |  |  |
$\tilde{b}_{R}$ fraction in $\tilde{b}_{1}$ | 1 | 1 | 1 | 0.98 | 0 | 0
$m_{h_{1}}$ (GeV) | 107.1 | 124.4 | 125.7 | 124.6 | 123.2 | 125.9
$m_{h_{2}}$ (GeV) | 125.2 | 190.2 | 163.8 | 184.4 | 202.6 | 173.1
$m_{\tilde{b}_{1}}$ (GeV) | 407.7 | 506.0 | 506.0 | 598.3 | 588.5 | 405.0
$m_{\tilde{b}_{2}}$ (GeV) | 1553.3 | 1552.4 | 1552.4 | 633.9 | 1557.6 | 1549.0
$m_{\tilde{t}_{1}}$ (GeV) | 1552.8 | 1555.1 | 1555.1 | 650.0 | 605.6 | 396.2
$m_{\tilde{t}_{2}}$ (GeV) | 1567.4 | 1564.5 | 1564.5 | 1556.7 | 1557.2 | 533.6
$m_{{}_{\chi_{1}^{0}}}$ (GeV) | 113.7 | 200.1 | 171.7 | 194.3 | 209.5 | 195.1
$m_{{}_{\chi_{2}^{0}}}$ (GeV) | 212.0 | 322.0 | 319.6 | 319.5 | 356.0 | 264.0
$m_{{}_{\chi_{3}^{0}}}$ (GeV) | 238.3 | 328.2 | 329.2 | 327.1 | 376.0 | 274.4
$m_{{}_{\chi_{1}^{-}}}$ (GeV) | 201.0 | 309.2 | 309.2 | 306.4 | 348.8 | 250.0
BR($\tilde{b}_{1}\to t\chi_{1}^{-}$) | 0.29 | 0.30 | 0.30 | 0.93 | 1 | 0
BR($\tilde{b}_{1}\to b\chi_{1}^{0}$) | 0.02 | 0.01 | 0 | 0 | 0 | 0.18
BR($\tilde{b}_{1}\to$3-body) | 0 | 0 | 0 | 0 | 0 | 0.57
BR($\tilde{b}_{2}\to t\chi_{1}^{-}$) | 0.73 | 0.76 | 0.76 | 1 | 0 | 0
BR($\chi_{2}^{0}\to\chi_{1}^{0}Z$) | 1 | 1 | 0.65 | 0.85 | 0.40 | 0.95
BR($\chi_{2}^{0}\to\chi_{1}^{0}h$) | 0 | 0 | 0.35 | 0.15 | 0.60 | 0
BR($\chi_{3}^{0}\to\chi_{1}^{0}Z$) | 1 | 0.99 | 0.79 | 0.96 | 0.80 | 0.98
BR($\chi_{3}^{0}\to\chi_{1}^{0}h$) | 0 | 0.01 | 0.21 | 0.04 | 0.20 | 0
Table 1: Benchmark points studied in the present work. The slepton and the
first two generation squark masses are all set to 1.5 TeV and the $SU(3)$
gaugino mass, $M_{3}$ is set to 2 TeV. The $SU(2)$ gaugino mass $M_{2}$ is set
to 1.5 TeV and $U(1)_{Y}$ gaugino mass $M_{1}$ is set to 1 TeV. Remaining
sbottom branching fraction in each case is attributed to its decays to
$b\chi_{2,3}^{0}$. See text for details.

###  4.2 Sbottoms at the LHC-13

Cascade modes | Intermediate products | Partonic final states
---|---|---
$\tilde{b}_{1}\rightarrow b\chi_{2,3}^{0},\;\;\tilde{b}_{1}\rightarrow b\chi_{2,3}^{0}$ | $2b+2(Z/h)$ | $4\ell+2b+\not{E}_{T}$
$\tilde{b}_{1}\rightarrow t\chi_{1}^{-},\;\;\tilde{b}_{1}\rightarrow b\chi_{2,3}^{0}$ | $2b+W^{+}W^{-}+(Z/h)$ | $(3\ell,4\ell)+2b+\not{E}_{T}$
$\tilde{b}_{1,2}\rightarrow t\chi_{1}^{-},\;\;\tilde{b}_{1,2}\rightarrow t\chi_{1}^{-}$ | $2b+2(W^{+}W^{-})$ | (SSDL, $3\ell,4\ell$) $+2b+\not{E}_{T}$
$\tilde{t}_{1,2}\rightarrow t\chi_{2,3}^{0},\;\;\tilde{t}_{1,2}\rightarrow t\chi_{2,3}^{0}$ | $2b+W^{+}W^{-}+2(Z/h)$ | $(3\ell,4\ell)+4b+\not{E}_{T}$
$\tilde{t}_{1,2}\rightarrow t\chi_{2,3}^{0},\;\;\tilde{t}_{1,2}\rightarrow b\chi_{1}^{+}/t\chi_{1}^{0}$ | $2b+W^{+}W^{-}+Z/h$ | $(3\ell,4\ell)+2b+\not{E}_{T}$
Table 2: Possible decay modes of $\tilde{b}_{1,2}$, $\tilde{t}_{1,2}$. The decays $\chi_{2,3}^{0}\rightarrow\chi_{1}^{0}Z/h$ and $\chi_{1}^{-}\rightarrow W^{-}\chi_{1}^{0}$ have 100% branching fraction. Channel | Search channel | Dominant
---|---|---
ID |  | backgrounds
$\mathrm{SRSSDL}$ | $\mathrm{SSDL}+\geq 2j(2b)+\not{E}_{T}$ |  $t\bar{t}$, $t\bar{t}Z$, $t\bar{t}W$
$\mathrm{SR3\ell 2b}$ | $3\ell+\geq 4j(2b)+\not{E}_{T}$ |  $t\bar{t}$, $t\bar{t}Z$, $t\bar{t}W$
$\mathrm{SR4\ell 1b}$ | $3\ell+\geq 2j(1b)+\not{E}_{T}$ |  $t\bar{t}Z$, $t\bar{t}W$
Table 3:  Classification of signal regions in terms of the actual search
channels undertaken in the present analysis. Leptons have their origins in the
$Z$\- and the $W$-bosons. The last column presents the dominant SM background
processes corresponding to each final state. These are inclusive of two hard
jets except for the $ZW$ and $t\bar{t}$ processes for which three-jet
inclusive samples are used.

In table 2 we present the possible decay modes of the sbottoms and the stops
and indicate the intermediate products in the cascades and the partonic final
states thereof. It may be apparent from the table that a simultaneous analysis
involving various multi-lepton ($n_{\ell}\geq$3) and SSDL final states with
low $b$-$jet$ multiplicity may be a useful probe to such a scenario. The
cascade decays of stops could yield a larger $b$-$jet$ multiplicity. This
might eventually help recognizing the presence of stops. In all these cases
the leptons have their origins in the decays of the SM gauge bosons. The
cascade of higgsino-like neutralino will vary crucially determine the signal
topology. Identifying $b$-$jets$ will certainly be helpful in this specific
scenario. Table 4 describes suitable cuts for the above mentioned signal
topologies.

The SSDL signal arises only from the cascade $\tilde{b}_{1}\to t\chi_{1}^{-}$
followed by $\chi_{1}^{-}\to\chi_{1}^{0}W^{-}$. The pure L-type stop quark
will never give an SSDL signal in this set up. So the observation of the SSDL
signal would definitely point to a $\tilde{b}_{1}$ cascade. But as we can see
from BP4, due to the effect of large $\frac{y_{t}}{y_{b}}$, it may not be easy
to estimate the mixing in the sbottom sector.

###  4.3 Simulation setup and selection of final states

The lowest order (LO) parton-level signal events are generated using
MadGraph_aMc@NLO v2.3.3 Alwall:2014hca . Background events are obtained using
MadGraph_aMc@NLO v2.1.2 Alwall:2014hca . In both cases, parton distribution
function nn23lo1 Ball:2014uwa , default to MadGraph, is used with the
factorization/renormalization scale set at the default MadGraph setting of
transverse mass. For the signal, the next-to-leading-order (NLO) cross
sections are estimated via Prospino v2.1 Beenakker:1996ed . For the
backgrounds, to take into the account higher order effects, we employ a flat
$K$-factor of 1.6 for the inclusive $t\bar{t}$ samples and 1.3 for the rest.

Signal events are showered with Pythia v8.2 Sjostrand:2014zea . Background
events are showered with Pythia v6.426 Sjostrand:2006za  embedded in the
MadGraph setup. For background events, we employ the MLM Mangano:2006rw ;
Alwall:2007fs  scheme for matching jets in order to avoid double counting in
the presence of hard partonic jets and parton shower. Background events are
produced with up to three inclusive jets.

Subsequently, both signal and background events are subjected to detector
simulation via DELPHES v3.2.0 Selvaggi:2014mya  which includes Fastjet v3.1.0
Cacciari:2011ma  for jet reconstruction. For merging of jets, we employ the
anti-$k_{T}$ algorithm with cone size set to 0.4 and require a minimum
$p_{T}^{jet}$ of 20 GeV with pseudorapidity in the range $|\eta_{jet}|<2.5$.
$b$-tagging efficiency is set to 70%. Furthermore, we consider the probability
of a $c$-$jet$ and a light quark jet being tagged as a $b$-$jet$ to be 20% and
1%, respectively.

Leptons (electrons and muons) are reconstructed with minimum $p_{T}^{\ell}$ of
10 GeV and with $|\eta_{\ell}|<2.5$. Leptons having neighbouring
(reconstructed) jets lying within a cone of $\Delta R\leq 0.2$ about them are
rejected. To increase the purity of electrons further, we require the ratio of
total $p_{T}$ of the stray tracks within the cones of their identification to
their own $p_{T}$’s is less than 0.1. For muons the maximum total $p_{T}$ of
other tracks is required to be below 1.8 GeV.

Variables | $\mathrm{SRSSDL}$ | $\mathrm{SR3\ell 2b}$ | $\mathrm{SR4\ell 1b}$
---|---|---|---
$n_{\ell}$ | 2 (SSDL) | 3 | 4
$n_{jet}$ | $\geq 2$ | $>=4$ | $>=2$
$n_{b\text{-}jet}$ | $>=2$ | $>=2$ | $>=1$
$p_{T}^{j_{(n)}}$ (GeV) | $p_{T}^{j_{(1,2,3,\geq 4)}}>(30,30,30,20)$
$p_{T}^{b\text{-}jet(n)}$ (GeV) | $p_{T}^{b\text{-}jet(1,2)}>(40,30)$
$p_{T}^{\ell}$ (GeV) | $p_{T}^{\ell(1,2,3,4)}>(30,20,15,15)$
$\not{E}_{T}$ (GeV) | $>100$
$m_{T2}$ (GeV) |  $>$ 90 | - | -
$H_{T}$ (GeV) |  $>$ 400 |  $>$ 500 |  $>$ 500
Table 4:  Definitions of the signal regions (SR) indicating the final states
they represent and the sets of selection cuts on the physics objects that are
independent of the benchmark scenarios. By leptons only electrons and muons
are referred to. Other notations follow standard conventions.

In table 4 we define three distinct signal regions namely $\mathrm{SRSSDL}$,
$\mathrm{SR3\ell 2b}$ and $\mathrm{SR4\ell 1b}$ and the signal selection cuts
used for them in the framework of MadAnalysis 5v1.1.12 Conte:2012fm ;
Conte:2014zja ; Dumont:2014tja  . Jets and leptons are $p_{T}$-ordered with
the hardest jet being denoted as $j_{1}$and the hardest lepton as $\ell_{1}$.
To have a better handle on some important backgrounds like $t\bar{t}+jets$, we
employ standard kinematic variables like $H_{T}=\sum_{{}_{jets}}|p_{T}^{j}|$
and $m_{T2}$ Cheng:2008hk , where

$\displaystyle
m_{T2}(p_{T}^{\ell_{1}},p_{T}^{\ell_{2}},\not{p}_{T})=\min_{\not{p}_{T,1}+\not{p}_{T,2}=\not{p}_{T}}\\{\max\\{m_{T}(p_{T}^{\ell_{1}},\not{p}_{T,1}),m_{T}(p_{T}^{\ell_{2}},\not{p}_{T,2})\\}\\}$
(15)

with

$\displaystyle
m_{T}(p_{T}^{\ell},\not{p}_{T})=\sqrt{2p_{T}^{\ell}\not{p}_{T}(1-\cos\phi_{\ell,\not{p}_{T}})}$
(16)

Figure 5:  Differential distributions for the kinematic variables
$\not{E}_{T}$, $m_{T2}$ and $H_{T}$ for both signal (in scenario BP6) and the
background at LHC-13 for inclusive final states. See text for details.

###  4.4 Results and discussion

In this subsection we discuss the results of our simulations for the LHC-13 in
different signal regions described in table 4. These signal regions satisfy a
general set of cuts on lepton and jet $p_{T}$’s that are appropriate for the
LHC-13 along with the multiplicities of the same (including $b$-$jets$) that
characterize the concerned final states. In addition, these regions also
feature cuts on kinematic variables like the missing transverse energy
($\not{E}_{T}$), $m_{T2}$ and $H_{T}$ which are chosen by studying the
respective distributions as shown in figure 5. Note that the plots in figure 5
are obtained using the benchmark scenario (BP6) which, as we would find later
(see table 5), shows a good sensitivity to the rather characteristic multi-
lepton ($>2$ leptons) final states. Furthermore, the set of kinematic cuts
extracted from these distributions is also found to be optimal for other
benchmark scenarios that we consider in this work.

We find that the LHC Run-I may not be generically sensitive to scenarios
having somewhat low sbottom masses ($\lesssim$ 400 GeV). The benchmark point
BP1 having an R-type sbottom with mass $\approx$ 400 GeV, a near-degenerate
pair of higgsino-like neutralinos ($\chi_{2}^{0}$, $\chi_{3}^{0}$) with
$m_{\chi_{{}_{2,3}}}^{0}\approx 200$ GeV and with an appreciable
BR[$\tilde{b}_{1}\to b\chi_{2,3}^{0}$] is perfectly allowed by the recent LHC
analyses. This is further confirmed through an explicit check using CheckMATE.
Also, $\tilde{b}_{1}\equiv\tilde{b}_{L}$ with mass $\approx$ 400 GeV (see
scenario BP6) escapes the current bound that assumes BR[$\tilde{b}_{1}\to
b\chi_{1}^{0}$]=100%. This is because in the present case, $\tilde{b}_{1}$ has
shared branching fractions distributed among available two- and three-body
decay modes, in all of which the final states are different from the one
(2$b$-$jets+\not{E}_{T}$) considered by the ATLAS collaboration Aad:2015pfx ;
Khachatryan:2015wza . Note that while some such situations discussed in the
framework of the MSSM are perfectly allowed departures from some simplifying
assumptions, these are just the natural expectations in the NMSSM in the
presence of a light singlino-like LSP.

Benchmark | Number of events and signal significance
---|---
scenario | $\mathrm{SRSSDL}$ | $\mathrm{SR3\ell 2b}$ | $\mathrm{SR4\ell 1b}$
BP1 |  | (88,1200) | (120, 146)
|  $<$ 2 | 2.5 | 9.0
BP2 |  |  | (56, 146)
|  $<$ 2 |  $<$ 2 | 4.4
BP3 |  |  | (37, 146)
|  $<$ 2 |  $<$ 2 | 3.0
BP4 | (58, 47) | (158, 1200) |
| 7.3 | 4.3 |  $<$ 2
BP5 | (36,47) |  |
| 4.7 |  $<$ 2 |  $<$ 2
BP6 |  | (240, 1200) | (80, 146)
|  $<$ 2 | 6.7 | 6.1
Table 5:  Number of signal and background events after cuts (given in the
parentheses, top sub-rows) and the corresponding signal significance (the
bottom sub-row; for an integrated luminosity of 300 fb-1) in different final
states for the six benchmark points at the LHC-13.

Such NMSSM scenarios, however, can be studied at the LHC-13 in its early phase
in various multi-lepton final states. The signal yields are presented in table
5 along with the corresponding expectations for the backgrounds for an
integrated luminosity of 300 fb-1. Also indicated in each case is the value of
the signal-significance calculated using the Poisson distribution given by

$\sigma=\sqrt{2\left[(S+B)\,\mathrm{ln}(1+{S\over B})-S\right]}.$ (17)

To summarize, scenarios with $\tilde{b}_{1}\equiv\tilde{b}_{R}$ (BP1, BP2 and
BP3) are only sensitive to 4-lepton+$b$-$jet$ (SR4$\ell$1b) final state. Final
states requiring three $b$-$jets$ (e.g., SR3$\ell$3b) are generally less
promising irrespective of the scenarios considered. This is since the
corresponding rates get suppressed by the extra $b$-tag efficiency factors. It
should be stressed here that unravelling such a scenario would require
corroborative signatures in multiple final states. Note that even with 300
fb-1of data, one could have not more than two simultaneous final states for
which the signal significances reach $~{}5\sigma$ (SRSSDL and SR3$\ell$2b for
scenario BP4 and SR3$\ell$2b and SR4$\ell$1b for scenario BP6).

In such scenarios with $\tilde{b}_{1}\equiv\tilde{b}_{R}$, the SSDL final
state is not at all sensitive since the decay $\tilde{b}_{1}\to t\chi_{1}^{-}$
is heavily suppressed. For an R-type $\tilde{b}_{1}$, its decay shares
branching among the modes $\tilde{b}_{1}\to b\chi_{2,3}^{0}$ significantly.
For BP1 and BP2, the 4-lepton final states are reasonably sensitive as
$\chi_{2,3}^{0}$ decays 100% of the times to $\chi_{1}^{0}Z$ and leptons come
from the decays of the $Z$-bosons. Still, the sensitivity is smaller for BP2
sheerly because of a heavier $\tilde{b}_{1}$. On the other hand, scenario BP3,
in which decays of $\chi_{2,3}^{0}$ share branchings between $Z\chi_{1}^{0}$
and $h\chi_{1}^{0}$, naturally loses sensitivity to this final state.

Benchmark points BP5 and BP6 present the cases where
$\tilde{b}_{1}\equiv\tilde{b}_{L}$. For BP5, BR($\tilde{b}_{1}\to
t\chi_{1}^{-}$) is 100%. Naturally, as discussed earlier in section 3.1, the
most sensitive final state is the one with SSDL. In BP6, we present a case
where 3-body decays of $\tilde{b}_{1}$ (to $tW^{-}\chi_{1}^{0}$ and
$bW^{+}\chi_{1}^{-}$) compete with its 2-body decays (to $b\chi_{1,2,3}^{0}$).
This is the reason why BP6, albeit comes with an L-type $\tilde{b}_{1}$, has a
poor sensitivity in the SSDL final state. On the other hand, it has much
better sensitivities in the 3-lepton and the 4-lepton final states.

It has been pointed out earlier that a relatively light L-type sbottom will
always be accompanied by a (L-type) stop close-by in mass when the mixing in
the stop sector is not large. In BP5 and BP6 we have such light
$\tilde{t}_{1}$-s which are not excluded by the usual stop-search as they
share branchings among different decay modes like $t\chi_{1,2,3}^{0}$ and
$b\chi_{1}^{+}$ Aad:2015pfx . Final states with high $b$-$jet$ multiplicity
like SR3$\ell$3b and SR4$\ell$3b are arising only from the stop cascade. We
find that they are hardly significant with data worth 300 fb-1.

In scenario BP4, $\tilde{b}_{1}$ and $\tilde{b}_{2}$ are close in mass. As we
find, both of them decay dominantly to $t\chi_{1}^{-}$ with branching
fractions larger than 90%. Note that this is a unique feature in the NMSSM and
is in sharp contrast to the MSSM where, to obtain the mass of the SM-like
Higgs boson in the right ballpark, tan$\beta$ needs to be large and hence
$\frac{y_{t}}{y_{b}}$ cannot become large enough to make both sbottoms decay
dominantly to $t\chi_{1}^{-}$. Naturally, this scenario shows enhanced
sensitivity ($\approx 7\sigma$) to the SSDL final state as both
$\tilde{b}_{1}$ and $\tilde{b}_{2}$ contribute with similar strengths. This is
clearly reflected in the signal corresponding significance as presented in
table 5. In addition, BP4 is also found to be moderately sensitive to the
$3\ell 2b$ final state.

It may be mentioned here that, in general, one may expect contributions to the
final states under consideration from stop pair production as well. However,
we find that except for BP6 these contributions are negligibly small. In BP6,
it is $\tilde{t}_{2}$ that may contribute to 3- and 4-lepton final states but
these do not exceed $\sim 25\%$.

A noteworthy omission in the list of final states is the canonical search mode
involving 2$b$-$jets+\not{E}_{T}$. An important upshot of such an NMSSM setup
with a singlino dominated LSP is that a light sbottom, irrespective of its
composition, would not generically have a healthy decay branching fraction to
such an LSP. This would then deplete the event count in the
2$b$-$jet+\not{E}_{T}$ final state severely while multi-lepton final states
arising from the cascades of sbottom(s) via higgsino-like neutralinos would
tend to dominate. In contrast to an MSSM scenario with small $\mu$
($\sim\mu_{\mathrm{eff}}$ and $<<M_{1},M_{2}$) thus having an LSP which is
nearly mass-degenerate $\chi_{2}^{0}$ (and $\chi_{1}^{\mp}$), such cascades in
a ‘natural’ NMSSM setup of the present kind could lead to leptons which are
hard enough. As can be expected, the contrast becomes rather prominent when
the decay $\tilde{b}_{1}\to t\chi_{1}^{-}$ is kinematically forbidden.

In principle, such a depletion in the event rate in the
2$b$-$jets+\not{E}_{T}$ final state, when contrasted against its expected rate
in the MSSM, could be rather illuminating. This is best demonstrated in the
variations of the effective branching fraction of a pair of sbottoms decaying
to this final state in the NMSSM and in the MSSM. The spectral setup adopted
for the purpose is shown in the left of figure 6 in the form of a level
diagram. Note that, in the MSSM case, we deviate from the scheme with
higgsino-like LSP described in the last paragraph and introduce a bino-like
LSP which can have some mixing with the higgsinos. This choice is rather
conservative in the sense that it mimics the NMSSM spectrum thus allowing us
to have a faithful estimate of the extent by which the signals from an
analogous MSSM setup could masquerade as the NMSSM one. As is just pointed
out, the mass-splitting between $\tilde{b}_{1}$ and the higgsino-like chargino
is restricted to 150 GeV such that the decay $\tilde{b}_{1}\to t\chi_{1}^{-}$
is prohibited and $\tilde{b}_{1}$ always decays to bottom quark and
neutralinos. In the right of figure 6 we present the variations of the
branching fraction as a function of the singlino/bino content of the LSP
($N^{2}_{1\tilde{S}/\widetilde{B}})$ while the palette shows the
simultaneously varying mass-splitting ($\Delta m$) between $\tilde{b}_{1}$ and
the LSP.

Figure 6:  Left: Level diagram representing the spectrum comprising of a
relatively light sbottom and a bino- ($\widetilde{B};$ in the MSSM) or a
singlino-like ($\widetilde{S}$; in the NMSSM) LSP along with higgsino-like
neutralinos and chargino sandwiched between them. Right: Variations of the
effective branching fraction of a pair of sbottoms decaying into
2$b$-$jets+\not{E}_{T}$ final state as a function of bino (singlino) content
of the LSP, $N_{1\tilde{B}(\tilde{S})}^{2}$ in the MSSM (NMSSM). The upper
(lower) pair of curves correspond to MSSM (NMSSM). In both cases
$\mu/\mu_{\mathrm{eff}}>0(<0)$ is represented by the upper (lower) curve. The
color palette indicates the simultaneously changing mass-splitting ($\Delta
m$) between $\tilde{b}_{1}$ and the LSP. The following fixed values of various
parameters are used: $\lambda=0.7$, $\kappa=0.18$ and $\tan\beta=5$.

To conform to our broad scenario, we choose $\lambda=0.7$ and $\kappa=0.18$
with $\tan\beta=5$ while $A_{\lambda,\kappa}$ are adjusted to obtain the mass
of the SM-like Higgs boson in the right ballpark. Variations in both $\Delta
m$ and $N^{2}_{1\tilde{S}/\widetilde{B}}$ are achieved by varying
$|\mu_{\mathrm{eff}}(\mu)|$ over the range 150-500 GeV. Note that in the NMSSM
with $\mu_{\mathrm{eff}}=\lambda v_{S}$ and $m_{\tilde{S}}\simeq 2\kappa
v_{S}$, varying $\mu_{\mathrm{eff}}$ results in a varying $m_{\tilde{S}}$ and
hence in a changing LSP mass. At the same time, such a variation alters the
splitting between the higgsino-like states and the LSP which is roughly given
by $\Delta m_{2}\approx(\lambda-2\kappa)v_{S}$, as indicated in the level
diagram of figure 6.

In the MSSM case we make a conservative choice of $M_{1}=150$ GeV for which
any chargino mass is allowed Aad:2015eda . Further, a minimum difference of 10
GeV between $\mu$ and $M_{1}$ is required with $|\mu|>M_{1}$. With this, the
minimum bino fraction in the LSP that we could achieve in the MSSM is $\sim
50$% (which is manifested in the abrupt termination of the MSSM curves on the
left side of the right plot of figure 6) when $|\mu|\simeq 160$ GeV. Note that
for a similar higgsino content of the LSP the bino-higgsino mass difference in
the MSSM is usually smaller than the singlino-higgsino mass difference in the
NMSSM. Thus, with $m_{\tilde{b}_{1}}=150+\mu_{\mathrm{eff}}(\mu)$, the decay
mode $\tilde{b}_{1}\to b\chi_{1}^{0}$ gets more phase space in the NMSSM when
compared to the MSSM case555Note, however, that with these choices the
lightest sbottom might become as light as $\sim 300$ GeV which may still be
allowed under certain circumstances Aad:2015pfx . In any case, given that the
purpose here is to demonstrate how BR[$\tilde{b}_{1}\to b$ LSP] could get
heavily suppressed for a singlino-like LSP, we choose not to get into the
intricacies involving the possibility of such a light sbottom. Furthermore, in
this discussion on variation of branching fraction, the more important
parameter is the mass-splitting $\Delta m$ (rather than the absolute mass of
the LSP) and as long as it allows for a large enough $\not{E}_{T}$ and render
other final state objects visible, the discussion goes through..

The right plot of figure 6 shows that the effective branching fraction to the
$2b+\not{E}_{T}$ final state (arising from $\tilde{b}_{1}$ pair production)
can get severely suppressed in the NMSSM (with light singlino and higgsinos)
when compared to an analogous neutralino spectrum in the MSSM (with light bino
and higgsinos). This is so even though, as pointed out in the last paragraph,
the available phase space is larger in the case of the NMSSM as compared to
the MSSM. This suppression can clearly be attributed to the absence of any
tree-level coupling of $\tilde{b}_{1}$ to a singlino-like LSP in the NMSSM as
opposed to the presence of a tree-level coupling between the sbottom and the
bino-like LSP in the MSSM. Furthermore, it is clear, as expected, that the
NMSSM rates are decreasing with increasing singlino-content of the LSP. In the
MSSM, because of similar couplings of $\tilde{b}_{1}\equiv\tilde{b}_{L}$ to
bino and the higgsinos (for our choice of $\tan\beta$), the branching ratio to
the LSP is characteristically larger.666 Further, note that this branching
ratio does not initially vary appreciably as the bino fraction in the LSP
increases since it is accompanied by a similar increase in the higgsino
component of the NLSP neutralinos which are also kinematically accessible to
$\tilde{b}_{1}\equiv\tilde{b}_{L}$ to decay to. However, with increasing
$\mu$, in our setup, the mass-splitting between $\tilde{b}_{L}$ and the LSP
grows while the same with respect to the NLSPs remains roughly unaltered.
Thus, the growing phase space explains the quick pick-up in
BR[$\tilde{b}_{1}\to b$ LSP] at the right edge of the plot.

Note that only L-type $\tilde{b}_{1}$ has been considered in figure 6. R-type
$\tilde{b}_{1}$ with a larger hypercharge would always enhance the partial
width for $\tilde{b}_{1}\to b\chi_{1}^{0}$ in the MSSM with a bino-like LSP
thus making the difference with NMSSM more drastic. Similarly, our generic
choice of a smaller $\tan\beta$ would also lead to a smaller $y_{b}$ thus
suppressing the $\tilde{b}_{1}$ decays to higgsino-like states in the MSSM
which would further favor its decay to the bino-like LSP in such a scenario.

##  5 Conclusions

In this work we discuss the characteristic phenomenology that the sbottoms
could derive in a ‘natural’ NMSSM setup even though they do not actively
address the issue of naturalness. We point out the generic possibility of
having light sbottom(s) in such a scenario which could evade current bounds
from the LHC experiments. The NMSSM setup considered in this work is
characterized by relatively small $\mu_{\mathrm{eff}}$ and light stop(s) which
facilitate compliance with the standard notion of naturalness. The latter
requires one to consider moderate to large values of $\lambda$ in order to
have the mass of the SM-like Higgs boson in the experimentally observed range.

It is stressed how the two ingredients that render the setup ‘natural’, i.e.,
small $\mu_{\mathrm{eff}}$ ($|\mu_{\mathrm{eff}}|\lesssim 350$ GeV) and large
$\lambda$ ($\sim 0.6-0.7$), could have an immediate impact on the sector
comprising of lighter neutralinos. While this combination tends to require a
small $v_{S}$, large $\lambda$ by itself implies small $\kappa$ once absence
of Landau pole below a high (unification) scale is demanded. These, in turn,
drive the singlino mass ($=2\kappa v_{S}$) small thus making it comparable or
even smaller than $\mu_{\mathrm{eff}}$. Such a situation, in the presence of a
large $\lambda$, could potentially make the lighter neutralino sector
phenomenologically nontrivial. As far as the LSP is concerned, its composition
could range between that like a nearly-pure singlino to a heavy mixture of
singlino and higgsinos. This is accompanied by two neutralinos (immediately
heavier than the LSP) whose compositions range between pure higgsinos and a
similar mixture of higgsinos and singlino as in the case of the LSP. In the
present context, this could crucially alter the ways in which sbottoms could
decay. In our ‘simplified’ scenario we assume the soft gaugino masses, $M_{1}$
and $M_{2}$ (and $M_{3}$ as well) to be rather heavy. The lighter chargino is
thus higgsino-like and the heavier electroweakinos and the gluino all
decouple.

A further interesting observation in such an NMSSM framework (with large
$\lambda$ and light stops) pertains to the opening up of the rather low
$\tan\beta$ ($\lesssim 2$) regime which is disfavored in the MSSM as it fails
to find the mass of the SM-like Higgs boson in the right ballpark. A
$\tan\beta$ value as low as 2 boosts the ratio $y_{t}\over y_{b}$ thus quickly
enhancing the $y_{t}$ driven decays of $\tilde{b}_{1}$ over the $y_{b}$
induced ones. This is found to be an important issue in the phenomenology of
an L-type sbottom whose decay to higgsino-like chargino and a top quark is
induced by $y_{t}$. Even more interestingly, the effect can be so strong that
a 3-body decay of an L-type sbottom involving a $y_{t}$-driven vertex could
overwhelm its $y_{b}$-induced 2-body decays to $b\chi_{1,2,3}^{0}$ for low to
moderate virtuality of the propagating state ($\chi_{1}^{\mp}$ or the top
quark). On the other hand, an enhanced $y_{t}\over y_{b}$ could also ensure
that a mostly R-type sbottom, with a tiny admixture of L-component, can have a
significant branching fraction to $t\chi_{1}^{-}$.

We choose a few benchmark NMSSM scenarios satisfying the criteria mentioned
above. These have the singlino-content of the LSP varying between 75% and 90%.
The lighter sbottom can be (almost) purely R- or L-type. As pointed out in the
last paragraph, we also discuss the interesting case of R-dominated lighter
sbottom with a tiny admixture of the L-component. The mass of the lighter
sbottom lies between 400 GeV and 600 GeV in these representative scenarios.

On the other hand, the heavier sbottom is arranged to be rather heavy
($\gtrsim 1.5$ TeV), in general. However, we showed (scenario BP4) that
requiring even a small admixture of the L-component in an otherwise
R-dominated light sbottom would result in a heavier sbottom not so different
in mass from its lighter partner. Furthermore, scenarios with an L-type light
sbottom inevitably have a stop squark in the spectrum of similar mass and
hence should be simultaneously accessible at the LHC. In these two cases, a
given final state may receive some limited contributions from more than one
excitation (the heavier sbottom and the stop squarks) thus making its
interpretation potentially involved. In the latter case, however, it is
important to ensure that such a light sbottom indeed escapes the indirect
bound that could be derived from non-observation of an accompanying light
stop. Evading such a bound could be possible when the stop sector has an
appreciable mixing and/or, for an unmixed L-type $\tilde{t}_{1}$, if it could
decay to heavier neutralinos. The former would, in turn, prompt the lighter
stop to decay in multiple ways, with shared branching fractions.

For an L-type lighter sbottom, its $t\chi_{1}^{-}$ decay mode, if
kinematically open, would dominate and would result in a healthy SSDL rate. A
much enhanced SSDL rate might point to both sbottoms contributing to this
final state with one of the sbottoms being L-type, for all practical purposes,
while the mostly R-type one has a slight admixture of $\tilde{b}_{L}$. If
$t\chi_{1}^{-}$ decay mode is not kinematically accessible, an L-type lighter
sbottom would rather show up strongly in several multi-lepton final states.
For a light NMSSM sbottom which is mostly R-type, the only resort to see its
signature, in such a scenario, could be in the four-lepton final state.

The bottom line of the present study is that the signals of sbottoms in a
natural NMSSM framework could be characteristically stubborn in not showing up
promptly at the LHC experiments. This feature is crucially governed by the
nature of the LSP. At the LHC Run-I this might have helped sbottoms escape the
searches. On the other hand, the LHC-13 would require moderately large data
(worth $\sim$300 fb-1) to hint/establish their presence in various multi-
lepton plus $b$-$jets$ final states with $\not{E}_{T}$. There, it would be
further corroborative if one experiences a dearth of events in the
2$b$-$jets+\not{E}_{T}$. Improved techniques proposed recently
Schlaffer:2016axo  could help sharpen the search for the sbottoms (and stops)
in an involved situation. These observations should be helpful in planning
future experimental analyses to uncover possible spectral configurations with
light sbottoms and an LSP with a singlino-admixture in a ‘natural’ NMSSM
setup.

###### Acknowledgements.

JB and AC are partially supported by funding available from the Department of
Atomic Energy, Government of India for the Regional Centre for Accelerator-
based Particle Physics (RECAPP), Harish-Chandra Research Institute. AC
acknowledges financial support from the Department of Science and Technology,
Government of India through the INSPIRE Faculty Award
/2016/DST/INSPIRE/04/2015/000110. The authors acknowledge the use of the
cluster computing setup available at RECAPP and at the High Performance
Computing facility of HRI. JB like to thank B. Fuks for very helpful
discussions. The authors thank U. Chattopadhyay for his kind hospitality at
the Theoretical Physics Group of the Indian Association for the Cultivation of
Science, Kolkata where a part of the work is done. They also thank Amit Khulve
and Ravindra Yadav for technical support.

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