TIFR/TH/22-22

Indian Contributions to LHC Theory

Sreerup Raychaudhuri

Department of Theoretical Physics, Tata Institute of Fundamental Research,

Homi Bhabha Road, Mumbai, 400005, India

Email<EMAIL_ADDRESS>
Published in: Eur. Phys. J. Spec. Top. (March 9, 2023)
https://doi.org/10.1140/epjs/s11734-022-00736-x

> Indian scientists began to work on the theoretical aspects of LHC physics
> from the early 1980s, at the same time when the rest of the world started
> taking interest in this then-futuristic topic. From this point grew a whole
> school of collider phenomenologists, who now form a significant fraction of
> the Indian high-energy physics community. This article briefly reviews the
> growth and contributions of the Indian school, on the way describing some of
> the physics ideas while placing the work in the international context, and
> proceeding thus, brings the story up to date in mid-2022.

Dedicated to the memory of the late D.P. Roy (1941-2017), the father of Indian
collider theory.

##  1 The expectant 80s

The LHC owes its origin [1] to CERN’s far-sighted leaders, Léon van Hove, who
served from 1976-1980 as the Scientific Director-General, and Technical
Director-General Sir John Adams (1976-1981), who resolved that it would be
feasible and scientifically rewarding to build the next-generation storage
ring at CERN. Thus, in 1977, Adams wrote the crucial concept note that the LEP
collider111The Large Electron-Positron collider. tunnel, which would be
excavated in the 1980s, should be made large enough to accommodate the
superconducting magnets required for a future phase of proton acceleration at
higher energies. This 1977 note may be thought of as the genesis of the LHC.
By 1978, it was already well known in the community, even though the formal
approval for the LEP came only in 1981, and the machine itself started running
in 1989 [2]. However, the fact that the 20 TeV LHC would eventually come and
that the physics at such a machine should be studied, was in the air right
from the early 1980s onward. Moreover, the USA was supposed to build a 40 TeV
Superconducting Super Collider (SSC) which, sadly, never materialised222It was
sanctioned in 1989, and eventually cancelled in 1993..

The 1980s were, in fact, a period of great optimism for high-energy physics —
or, as it was then more commonly known, elementary particle physics. One
trigger for this may have lain in the decision of the Nobel Prize committee,
in 1979, to recognise the work of Glashow (1961) [3], of Salam (1966-1968) [4]
and of Weinberg (1967) [5] in developing a working model of electroweak
unification. Although the Higgs boson [6, 7, 8], the cornerstone of the theory
developed by Weinberg and Salam, would be discovered only 3 decades later [9,
10], the Nobel Prize committee made an unusual departure in deviating from its
usual conservative stand and honouring a theory which still lacked five of its
essential particles — the $W^{\pm}$ and $Z^{0}$ bosons, the Higgs boson, the
top quark and the tau neutrino. Their confidence was brilliantly reinforced by
the 1983 discovery, at CERN’s Super Proton Synchrotron (SpS), of the $W^{\pm}$
[11, 12] and $Z^{0}$ [13] bosons, which prompted another Nobel Prize, the very
next year, to Carlo Rubbia and Simon van der Meer, who were the leaders of
this experiment. This marked the beginning of the reign of the Standard Model
(SM), a name which had been coined in 1975 by Treiman and Pais [15]. That the
SM would prove so durable was not, however, quite apparent at the time, and
the literature was simply buzzing with alternative ideas.

The theoretical idea which attracted the most attention in the early 1980s was
definitely that of grand unification, which was inspired by the success of the
electroweak unification achieved by Glashow, Salam and Weinberg. Proposed by
Georgi and Glashow [16] in 1975, building on pioneering work by Pati and Salam
(1974) [17], this sought to unify the strong interactions with the
electroweak, within the framework of a single spontaneously-broken gauge
theory. However, it was recognised that any such theory would immediately lead
to fast decay of protons, unless the scale of the symmetry-breaking lies
around at least $10^{1}6$ GeV333The lower bound on the proton decay then was
$3\times 10^{30}$ years, as determined in Ref. [18]. It has since risen by
about four orders of magnitude. For the latest results, see Ref. [19].. This
immediately brought to the fore a problem with the Higgs boson. The fact that
the mass of an elementary scalar is not stable under quantum corrections, but
tends to rise to the cutoff scale for the theory, had been recognised by
Wilson as early as 1971 [20]. In Grand Unified Theories (GUTs), this cutoff
scale would then be around $10^{16}$ GeV — clearly an intolerable situation
for the Higgs particle whose mass was already known to be restricted by
perturbation theory to lie within a TeV, and is now known, thanks to the LHC,
to be around 125 GeV. This problem acquired the name of the ’gauge hierarchy
problem’ [21], or simply, the hierarchy problem.

It may be noted in passing that the hierarchy problem does not exist if the
Standard Model holds all the way to the Planck scale, the energy scale around
$10^{19}$ GeV at which the effects of spacetime curvature, i.e. gravity,
become as strong as the gauge interactions which describe the electroweak and
the strong interactions. For then there exists a well-known mechanism – the
mass renormalisation technique – to absorb the large quantum corrections into
the bare mass of the Higgs boson, leaving a small quantity which is
measurable. However, if there is an intermediate scale, like the GUT scale,
where there are physical fields with physical masses, then, even if the same
huge quantum corrections are absorbed into the Higgs mass at one energy scale,
they will reappear (somewhat diminished but still intolerably large) when the
physical masses run to another scale. In the 1980s, when most high-energy
physicists believed that a GUT of some form does exist, this was the obvious
situation to be faced, with the Higgs boson mass getting dragged up to the GUT
scale. Today, we are not quite so sure [22].

Two different solutions of the hierarchy problem had emerged by the end of the
1970s. One was a rather well-traversed idea, based on the behaviour of strong
interactions, which bind the quarks into hadrons. These can be split into
other hadrons, but cannot be broken back into quarks — a phenomenon called
quark confinement. The strong interaction (QCD) coupling, in fact, grows
stronger as the energy scale falls and appears to diverge at a value around a
quarter of the nucleon mass, the so-called QCD scale444Technically, we would
call it a Landau pole in the running of the strong coupling constant, and it
really indicates that the interaction has become too strong to be described
correctly by perturbation theory.. It is this nonperturbative interaction that
binds quarks into hadrons, and hence the scale of hadron masses is set by the
scale of this breakdown point [23, 24]. Now hadron fields are known to form
Lorentz scalars, as well as vectors, tensors and fermions. Could the Higgs
boson be just such a composite object, bound by a much stronger (maybe gauge,
like QCD?) interaction, where the scale (or Landau pole) lies around the Higgs
boson mass? This, in a nutshell, is the contention of a class of theories
collectively dubbed technicolor theories [25, 26]. The beauty of such theories
lies in the fact that the Standard Model breaks down at a scale around a TeV,
or a few TeV, and the new gauge interaction comes into play. Quantum
corrections to the Higgs boson mass would, then be completely manageable and
there would be no hierarchy problem. The foundations of this theory had been
laid down as early as 1973, by Jackiw and Johnson [27], and independently by
Cornwall and Norton [28]. However, it was the 1976 work of Weinberg [29],
describing the gauge field theories necessary to drive technicolor, followed
by the 1978 work of Susskind [30], applying this to the electroweak model,
which made technicolor a viable means of theorising beyond the Standard Model.
Obviously, such a model would make naive grand unification redundant, it being
assumed that the electroweak and strong interactions are effective theories in
the same way as Yukawa theory is an effective theory of QCD555In fact for
hadronic physics, this idea has been developed into what is called chiral
perturbation theory [31, 32, 33]..

Compositeness is always an attractive idea, given that atoms, nuclei and
hadrons were all found to be composites as soon as experiments began to probe
inside them. The ideas of primitive technicolor were soon taken ahead by
Harari [34] and, independently, by Shupe [35], who developed the first so-
called ’preon’ models, where all the Standard Model particles are taken to be
composites, not just the Higgs boson. Preon models flourished in the 1980s,
with one of the most dedicated exponents being Pati [36], whose 1974 paper
with Salam [17] contained one crucial sentence which inspired the whole genre
of preon models.

By far the most popular solution of the hierarchy problem, however, was
supersymmetry (SUSY). It is still a popular model, though somewhat the worse
for wear after the second run of the LHC. The inspiration for this came from
string theory, where the basic excitations are all bosonic, and, therefore,
one needs supersymmetry to have fermions [37, 38, 39]. It was already known
from the 1975 work of Haag, Lopuszanski and Sohnius [40, 41] that if we want
to have a super unified theory, where gravity is united with the gauge
interactions, then this will have to be supersymmetric. However, this might
never have become relevant in the LHC context, had it not been for the
discovery that the quadratic divergences in the Higgs boson self-energy cancel
out in SUSY theories, leaving behind only logarithmic divergences, which are
easy to absorb into the measured mass. This ’miracle’ was discovered in the
simplest SUSY model by Wess and Zumino in 1974 [42, 43] and extended to a SUSY
version of QED. Extension to a nonAbelian gauge theory was achieved a year
later by Ferrara and Zumino [44, 45]. In 1976, Pierre Fayet wrote down a SUSY
version of the Standard Model [46], but it was not till the work of Dimopoulos
and Georgi (1981) that it was established that the SUSY must be explicitly
broken to be phenomenologically viable [47]. Soon after, several authors
established the cancellation of quadratic divergences in this so-called
Minimal Supersymmetric SM, or MSSM. Among them were two Indians, viz. Kaul
(then at the IISc, Bengaluru) and Majumdar (then at the TIFR, Mumbai), who
published a series of papers [48, 49, 50], discussing the cancellation of
quadratic divergences in anomaly-free SUSY gauge theories, including the MSSM.
Though not directly connected with the LHC, these may be regarded as one of
the first important contributions from India to physics which was – and still
is – central to the analysis of LHC data.

In the 1980s, therefore, the efforts of theoretical particle physicists
centred around four major themes.

  * •

One was the Standard Model itself and its hitherto undiscovered components —
the top quark, the Higgs boson and the tau neutrino. In this genre, we may
include bottom-up extensions of the Standard Model, such as a fourth
generation of fermions [51, 52], exotic fermions [53, 54, 55, 56, 57, 58],
two-Higgs doublet models [59, 60, 61], extra gauge bosons ($W^{\prime}$ and
$Z^{\prime}$) [62, 63], and left-right symmetry [64, 65, 66, 67, 68, 69]. The
study of neutrinos and neutrino oscillations may also be included in this
category [70, 71, 72, 73].

  * •

The next was GUT model building [74, 75], which sought to understand various
features of the SM masses and mixings as consequences of higher gauge
symmetries and their spontaneous-breaking patterns. This received a major
boost from string theories, which brought into the limelight large symmetry
groups such as $E_{8}\times E_{8}$ [76], $E_{6}$ [77, 78] and $SO(10)$ [79,
80], as well as large representations, such as the 126-plet of Higgs bosons in
$SO(10)$ models. These may otherwise have lain beyond the wildest dreams of
particle phenomenologists.

  * •

The study of compositeness, especially technicolor, was pursued by some small
but prolific groups of particle physicists, the most well-known group being
centred around Harvard University and led by Howard Georgi, Ken Lane, Bill
Bardeen, and others. Technicolor models predicted the discovery of technipions
and techni-rho particles at colliders, which would look like heavy scalars or
vector bosons [81, 82]. Preon models were somewhat peripheral to the
mainstream literature, as they were thought of then – as now – as premature
and somewhat speculative [83].

  * •

The study of supersymmetry, especially the MSSM, and its most restrictive
version, then known as the minimal SuperGravity Model (mSUGRA) [84] and later
as the constrained MSSM, or cMSSM, grew enormously in this period, and may
have reached its climax in the 1990s. The most popular cliché in this context
was to describe the MSSM as the ’standard model’ of beyond-SM physics. It was
during this period that the well-known collider signals [85, 86, 87], like
trileptons, missing energy and like-sign dileptons, were predicted and
studied.

It was widely expected in the 1980s that just as the SM had been vindicated at
the SpS, some signal from these new theories would be found at the next
generation of machines, viz. the LEP at CERN and the Tevatron at FNAL. After
that would come the third generation of colliders, viz. the LHC at CERN and
the SSC in Texas. It was definitely time to start thinking about the physics
of these futuristic machines [88].

In the following discussion, we will describe Indian contributions to these
themes under different subheadings, together with brief notes on the general
developments in those areas. Of course all of these were being pursued at the
same time – with often the same person pursuing more than a single theme – so
that such a description will not be strictly linear or follow a rigid
timeline. Moreover, over and above the above list, new ideas surfaced towards
the turn of the century, such as brane worlds, unparticles and little Higgs
models. The lack of evidence for any of these has created a hostile
environment when researchers take shelter within the bunker of effective field
theories and simplified bottom-up approaches to LHC physics. At the same time,
there have been momentous developments in neutrino physics, and in cosmology
and astroparticle physics, which will not be described in any detail in this
article but will still find some mention in passing, for they have influenced
LHC studies as, indeed, all of high energy physics. At the cost of repeating a
well-worn cliché, it is still worth stating that scientific research is not
conducted in a vacuum. Nevertheless – and this may be treated as a disclaimer
– it has not been possible to include all Indian efforts in the list of
citations in this article. Nor has that been the purpose of the albeit long
list of references, which is for the purpose of illustration and guidance, and
is not intended to form a compendium. Furthermore, this article does not
describe the interesting, even exciting, results being found at the LHCb and
ALICE experiments — partly for lack of space, but also because the author
lacks expertise to comment on these in any insightful manner.

One final remark before we embark on our voyage of discovery. In this article,
the work ’Indian’ has been used in a fairly general sense to include all high-
energy physicists of Indian origin who are working or have worked in India in
any capacity, whether as graduate student, postdoc or faculty member. The
overseas work of such individuals has also been considered. The work of
Indians and scientists of Indian origin who have settled abroad have, by and
large, been omitted, with some notable exceptions.

##  2 Taming the top

It is now generally accepted that the prediction of a third generation of
fermions was made as early as 1973 by Kobayashi and Maskawa [89]. Their work,
even though it eventually won the authors a Nobel Prize, did not make much of
an impact at the time it was proposed. Thus, when the tau lepton ($\tau$) was
discovered at SLAC in 1975 [90], it came as something of a surprise. However,
within 2 years, in 1977, the bottom quark ($b$) was also discovered at
Fermilab [91]. There was, thus, an incomplete generation, where it was assumed
that there was a leptonic and a quark doublet of $SU(2)_{L}$ with missing
partners — dubbed, for obvious reasons, the tau neutrino ($\nu_{\tau}$) and
the top quark ($t$), respectively. The standard argument for the existence of
these then-undiscovered fermions arose from the requirement of anomaly
cancellation in the Standard Model [92, 93]. More specifically, the
requirement is that $\sum_{f}N_{f}Y_{f}=0$ where $Y_{f}$ is the hypercharge of
the fermion and $N_{f}$ is the colour multiplicity (1 for leptons, 3 for
quarks), and the sum runs over one generation. The cancellation must happen
over each generation separately. Taking the Standard Model values for the
third generation, i.e. $Y(\nu_{\tau})=-1$, $Y(\tau_{L})=-1$, $Y(\tau_{R})=-2$,
$Y_{tL}=1/3$, $Y_{bL}=1/3$, $Y_{tR}=4/3$ and $Y_{bR}=-2/3$, and appropriate
colour factors, it is easy to verify that the required cancellation does
occur. Obviously, it will not happen if one or more of the fields are missing.
This was the most compelling reason to believe that the missing top quark and
the missing tau neutrino must exist. However, there did exist speculations
which assigned different quantum numbers to these fermions, e.g. making the
$b_{L}$ a singlet of $SU(2)_{L}$. Most of these ’topless models’ [94] were
ruled out by data from the early runs of the LEP Collider at CERN, which
started running in 1989. However, very similar ideas are now being reworked in
the context of an extra $Z^{\prime}$ in models beyond the SM.

The fact that a high-energy hadron collider would be an ideal place to look
for a top quark, because it can be produced in strong interactions, was
already recognised by 1983, when the first Indian paper on the topic was
published by the trio of Roy, Godbole and Pakvasa [95]. At the time, it was
already known that the mass of the top quark must be more than about 20 GeV,
since the PETRA – an $e^{+}e^{-}$ collider at DESY, operating at a centre-of-
mass energy between 27 and 46 GeV – had failed to see any signal for top-quark
pair production. In their pioneering work, Roy et al. pointed out that the
semileptonic decay of the top quark, viz. $t\to b+W^{+}\to
b+(\ell^{+}\nu_{\ell})$ will lead to a final state with a hard transverse
lepton $\ell=e,\mu,\tau$ which is isolated from the jet which will originate
from the b quark in the decay. In their studies, isolation meant a minimum
angular distance $\Delta R>0.4$, where $\Delta
R^{2}=\Delta\eta^{2}+\Delta\varphi^{2}$, in terms of the pseudorapidity
difference $\Delta\eta=\eta_{b}-\eta_{\ell}$ and the azimuthal angle
$\Delta\varphi=\varphi_{b}-\varphi_{\ell}$ between the lepton $\ell$ and the
thrust axis of the $b$-jet666These quantities remain invariant under
longitudinal boosts and make a better measure than the opening angle.
Nowadays, other invariant measures of distance, such as $k_{T}$ are more
popular.. The point made by Roy et al. – which was also independently arrived
at by Barger et al. [96] around the same time – is that in the similar
semileptonic decays of the bottom or charm quarks, which are much lighter, the
decay products will be much more strongly boosted along the direction of the
parent particle, leading to a non-isolated lepton, i.e. which will appear as
part of the jet. Thus, the heavy mass of the top quark, which renders its
production difficult (well-nigh impossible at the energies available in the
1980s), also leads to a clear signal, once it is produced.

The isolated lepton signal dominated the top-quark searches at Fermilab’s
Tevatron machine, which ran at 1.8 (2.0) TeV from 1987 till 2011. It was also
the channel through which the top quark was finally discovered by the CDF and
D0 Collaborations at the Tevatron in 1995. Meanwhile, in 1984, Roy alone [97],
and also in collaboration with Lindfors [98], had shown that electroweak
production of top quarks at a hadron collider is suppressed by an order of
magnitude. A detailed analysis of some candidate events seen at the UA1
detector at CERN’s Super Proton Sychrotron (SpS) was published by Roy with
Godbole [99], in which they concluded that both semileptonic and fully
hadronic (three-jet) decay modes of a top quark of mass around 35 GeV could
explain the candidate events. However, these events were unlikely to have
originated from top quarks, since the SpS energy was only about 500 GeV, where
we know with hindsight that the probability of producing a pair of 172 GeV top
quarks is negligibly small [100]. In fact, the first results on
$B^{0}$-$\bar{B}^{0}$ mixing – which came in 1987 from the ARGUS Collaboration
at the DORIS II ring at DESY, Hamburg – immediately showed that the top quark
mass must lie above 60 GeV in the Standard Model [101]. Five years later, the
early runs from the Tevatron at Fermilab had pushed the lower bound to 91 GeV
[102]. By 1994, the lower bound rose to 131 GeV [103].

At around the same time, the LEP Collider began to run at CERN [2]. It had
been hoped that it would lead to the rapid discovery of the Higgs boson, but
the elementary scalar proved frustratingly elusive. However, LEP did provide
accurate measurements of the masses of the $Z$ and the $W$ bosons, which are
affected at the loop level by both the Higgs boson and the top quark. Within
2-3 years of running, global fits to the LEP data were able to push the top-
quark mass limit to somewhere between 150 and 190 GeV [104]. These, of course,
assumed that there are no particles other than the Standard Model ones. When
the top quark was finally discovered at the Tevatron – by the CDF
Collaboration [105], followed some months later by the D0 Collaboration [106]
– its mass came out as around 175 GeV, i.e. right in the middle of the
expected range.

The top quark is definitely an odd fish among the fermions in the Standard
Model. As it considerably heavier than its closest rival, the $b$ quark, it is
the only parton which decays before it can hadronise. Another consequence of
this large mass is that in the Standard Model the Yukawa coupling of a top-
quark pair with the Higgs boson come out as almost exactly unity — by far the
largest coupling constant in the Standard Model at the electroweak scale. A
third consequence is the large separation between the measured mass around 172
GeV and its MS mass, which is around 165 GeV. We may recall that like the
other quarks we cannot define a pole mass for the $t$ quark, since quark
states are never asymptotically free. The heavy mass of the top quark also
makes it the principal driver of loop corrections in a variety of processes,
including $B^{0}$-$\bar{B}^{0}$ mixing, the Higgs boson self-energy, and the
RG evolution of the quartic scalar coupling $\lambda$ in the Standard Model.
In models beyond the Standard Model, such as the constrained minimal
Supersymmetric Standard Model, it is such top-quark-mediated loops which
provide an elegant mechanism to drive electroweak symmetry-breaking. As a
result, the top quark and its properties have been a matter of great interest
to high energy physicists every since its discovery.

One reason why it is imperative to know the mass of the top quark precisely is
mentioned above, which is its pivotal role in determining if the $\lambda$
parameter remains positive and the electroweak vacuum remains stable all the
way to the Planck scale [107]. For, if it does not, then there would be a
high-energy scale where all particles would go down to indefinitely large
negative energies and emit huge amounts of radiation, leading to a ’fiery
death’ of the Universe. Conversely, since the Universe began in such a ’fiery’
situation, well above the scale of instability, it could never have cooled
down close to absolute zero — which it has. Therefore, one would need to
postulate the existence of new physics beyond the Standard Model in order to
provide the required vacuum stability. With the present level of accuracy of
top quark mass measurements, however, we cannot yet make a definitive
statement on this issue [107].

The mainstream in top-quark physics at the LHC has been the measurements of
the production cross-section and the couplings of the top quark to the gauge
bosons, as well as the Higgs boson. Theoretical studies of these have centred
around calculating higher order QCD corrections to the production cross-
section of $t\bar{t}$ [108] as well as $t\bar{t}X$ where $X=Z,H$ [109, 110].
The Indian community has, however, focussed on the top quark as a portal to
BSM physics. The pioneer in this has been Rindani, who has worked on
polarisation effects, mostly in the context of $e^{+}e^{-}$ colliders from
1995 onward [111]. In 2010, he also teamed up with Godbole et al. [112] to
discuss the LHC signals for nonstandard couplings which would generate a
nonzero top-quark polarisation, showing that asymmetries in the azimuthal
angle distribution of leptons arising from top semileptonic decays can be a
powerful probe of such effects. Top polarisation studies were then carried
forward by Godbole, with her collaborators [113, 114], mainly analysing the
$t\bar{t}H$ coupling and developing angular variables where BSM effects would
show.

Detailed studies of nonstandard top-quark couplings in one-loop effects have
been carried out by Choudhury and Saha, who explored chromomagnetic and
chromoelectric dipole interactions of the top quark in a model-independent
framework [115] and then the possible effects of anomalous top-quark couplings
in Higgs boson production at the LHC [116]. On a slightly different note,
Bardhan et al., discussed flavour-changing decays of the top quark at one
loop, including the process $t\to c+Z^{0}$, setting up a general framework in
which BSM predictions could be compared with the data [117].

The top quark has also featured in other kinds of BSM studies, where it is
generally a decay product of some heavier nonstandard particle. Indian
contributions in these directions have been described in the following
sections. However, this section would not be complete without a mention of the
new paradigm for top-quark identification at the LHC, namely jet substructure.
It was pointed out by Kaplan et al. in 2008 [118] that hadronically decaying
top quarks could be tagged by a consideration of the jet substructure, since
the final-state jets at the LHC energies would be likely to be highly boosted,
compared to earlier machines. This has since become a standard technique.
However, another obvious decay channel in this context is the semileptonic
decays, originally touted as the ’smoking gun’ signals for top quarks, but
which remained unexplored in the context of boosted top decays. This lacuna
was filled in by Chatterjee et al. [119], who, in 2019, developed a technique
to tag ’electron-rich’ jets as originating from a top-quark decay — or
otherwise.

##  3 Hunting for Higgses

The Higgs boson has been described as the keystone of the Standard Model, and
its discovery [9, 10] in 2012 was one of the greatest triumphs of modern
science, both theoretical and experimental. Nevertheless, this success came
only at the end of a protracted search lasting several decades. Being the only
elementary scalar in the SM, the Higgs boson has been the subject of intense
speculation, from its mass, spin and magnetic moment, to its couplings with
all the other SM fields, as well as its vacuum structure. Something of this
massive level of questioning can be gauged by a perusal of the (somewhat-
dated) monograph The Higgs Hunter’s Guide [59, 60, 61], which provides a
comprehensive introduction to the subject as it was in the 1990s.

While the role of a Higgs doublet in engineering spontaneous symmetry-breaking
in the SM gauge sector had been worked out in the 1960s, it was the discovery
of the $W$ and $Z$ bosons which made the presence of a Higgs boson imperative,
for without it the scattering amplitude for, say, $e^{+}e^{-}\to WW$ becomes
non-unitary in the high-energy limit. In fact, purely from unitarity
considerations one could infer that the Higgs boson would have a mass within
about 1 TeV. At that stage the unitarity problem could have been explained
away by invoking a strongly-interacting electroweak sector [120], but such
hopes faded away when such models became severely constrained by the data from
LEP in the 1990s. Once the top quark mass became known in 1994, the 1998
global fit of all electroweak data by Erler and Langacker [121] also brought
the Higgs boson mass down to below 225 GeV. The direct search data at LEP had
already established 114 GeV as a lower bound. Thus, by the turn of the
century, a discovery within this narrow window by the LHC was more-or-less
guaranteed, so far as the SM was concerned. As a matter of fact, this goal was
achieved rather early in the LHC run, with the 2012 discovery

As with the top quark, Indian contributions to the physics of Higgs bosons in
the LHC context has mostly been in the area of BSM models. A notable
exception, however, has been the crucial work of Ravindran, who, with his
collaborators, has been steadily working out QCD corrections in the SM to the
dominant $g+g\to H^{0}$ partonic process for Higgs production at the LHC —
with increasing accuracy over the years. These corrections were at NLO in 2002
[122], at NNLO in 2003 [123], at full two loops in 2005 [124], with threshold
corrections beyond NNLO in 2007 [125], at NNNLO in 2014 [126], and at NNLO +
NNLL in 2018 [127] and 2021 [128, 129]. Further efforts include similar series
of calculations of the $b+b\to H^{0}$ process to three loops and beyond [130,
131, 132, 133, 134] and calculations of the $g+g\to H^{0}+H^{0}$ process [135,
136, 137], the latter being vital for the direct measurement of the quartic
Higgs couplings $\lambda$ in future runs of the LHC. It may be noted that not
only do these extremely sophisticated calculations make the SM prediction more
accurate, but even in variables (such as Higgs signal strengths) when the
cross-section eventually cancels out, these make the result almost independent
of the scale of renormalisation.

In the field of BSM Higgs studies, there have been a few studies of exotics
such as singlet [138], triplet [139] and composite Higgses [140], but most of
the Indian work has centred around two-Higgs doublet models (2HDM), especially
with regard to the additional charged Higgses and the pseudoscalar Higgs
predicted in these models. Moreover, since SUSY models are 2HDMs with a
severely constrained parameter space, a fair portion of the 2HDM analyses have
been carried out in a SUSY framework. An interesting early study of charged
Higgs bosons by Biswarup Mukhopadhyaya and collaborators identified the
possibility of observing radiation amplitude zeroes in the production and
decay of a charged Higgs boson [141]. Though written in the context of the
defunct SSC, the result is equally applicable to some processes at the LHC
[142].

Nevertheless, the pioneer in detailed collider studies of nonstandard Higgs
bosons, as in the case of the top quark, was undoubtedly Roy, who initially
considered a charged Higgs boson $H^{\pm}$ decaying into a top quark, both at
the partonic level [143] and with QCD corrections [144]. Then it was the
pioneering paper of Bullock et al. [145], in which they pointed out that the
polarisation of $\tau$ leptons arising in $H^{\pm}$ decays is different from
that of those arising from $W^{\pm}$ decays (their principal SM background at
hadron colliders including the LHC) which inspired a series of articles from
Roy and his collaborators, in which this idea was explored with ever-
increasing degrees of sophistication [146, 147, 148, 149, 150, 151, 152, 153].
On a variant note, Roy, with his collaborators, also studied a possible
multi-$b$ quark signal for the $H^{\pm}$ [154, 155, 156] and $\tau$
polarisation for the pseudoscalar in 2HDMs [157], as well as the rarely
studied process $H^{\pm}\to W^{\pm}h^{0}$ [158]. Other studies of charged
Higgs bosons include a study by AseshKrishna Datta and collaborators on H±
produced in cascade decays of SUSY particles [159]. Novel signatures for
$H^{\pm}$ have also been explored in the context of extra Yukawa couplings
[160] and a chromophobic $H^{\pm}$ in a $Z_{2}$-extension of the SM [161].
Similar studies of neutral scalar and pseudoscalar Higgs bosons in the context
of stop decays in SUSY models have been explored in 2000 by Sridhar [162], in
2015 by Mukhopadhyaya [163] and by Ghosh [164], each with their respective
collaborators.

With the 2012 discovery of the obviously-neutral scalar of mass 125 GeV at the
LHC came immediate questions about the spin and parity of this new particle.
In a 2011 study [165], Mukhopadhyaya and collaborators had already showed how
final states in Higgs production via vector boson fusion could provide the
clue to this, as well as indicate the presence of anomalous HWW vertices.
Post-Higgs discovery analyses on similar lines were carried out by Godbole
with her collaborators in the context of the LHC [166, 167] and a possible
LHeC [168]. In all these studies, the azimuthal angle distribution of final
states is the key to identification of anomalous couplings.

In the days preceding the Higgs boson discovery, the absence of Higgs signals
engendered fears that the Higgs may actually have been produced and remained
undetected because it dominantly decays ’invisibly’. An early idea was a Higgs
boson decaying into heavy neutrinos [169, 170], but this possibility receded
with the LEP data. LHC signals for an invisibly decaying Higgs were also
explored fairly early by Roy and collaborators [149, 171]. Post-discovery, the
possibility of a subdominant but measurable ’invisible’ branching fraction of
the observed Higgs was investigated by Ghosh et al. in 2013 [172]. Invisible
decay modes of the Higgs boson have become highly relevant today in the
context of dark matter searches, though all we have, till now, is a fairly
stringent upper bound [173]. As a matter of fact, after the Higgs boson, dark
matter has become the new ’Holy Grail’ of high-energy physics, and the Higgs
boson is often considered as a portal to this unknown world. In this context,
Datta et al. made a detailed study of multi-lepton signatures [183] of the
’inert doublet model’ where a decoupled scalar doublet plays the role of dark
matter [174]. Several other studies on similar lines followed [174, 175, 176,
177, 178, 179, 180, 181, 182].

The discovery of the Higgs boson has also created renewed interest in the
electroweak vacuum and eventual fate of the Universe. Ever since Coleman and
Weinberg [184] pointed out the scale-dependence of the shape of the
electroweak potential, the fate of the electroweak vacuum has been a matter of
intense speculation. However, no definite statements could be made since the
mass of the Higgs boson was not known at any scale. Once this became known in
2012, a clutch of articles [185, 186, 187, 188] appeared, pointing out the
possibility that the electroweak vacuum might destabilise at a scale somewhere
between $10^{11}$ GeV and the Planck scale.

Vacuum stability was soon recognised to be a powerful tool to probe BSM
physics, and several Indian groups took this up. Anindya Datta and Sreerup
Raychaudhuri studied universal extra dimensions [189, 190], while neutrino
mass models were studied by Chakrabortty et al., who considered Majorana
neutrinos [191], Khan et al., who considered a minimal seesaw model with a
scalar singlet [192] and Bhupal Dev et al., who considered a Type-II seesaw
[193]. The SUSY Higgs vacuum was considered by Chowdhury et al. [194, 195], a
similar non-SUSY model was discussed by Chakrabarty et al. [196], and a
singlet scalar dark matter candidate was considered by Khan and Rakshit [176].
Other work of similar nature followed, for vacuum stability has now become a
standard constraint to check for any new physics model involving the Higgs
sector.

##  4 Seeking supersymmetry

The rise and fall of supersymmetry (SUSY) must, when histories of the current
period are written, become the dominant theme of the 3 decades from 1990–2020.
For no theory of elementary particles – not even the SM – has come packed with
so many phenomenological goodies as SUSY. Let us quickly list the main ones.

On the aesthetic side, SUSY not only does away with the ugly distinction
between bosons and fermions which is a legacy of quantum mechanics, but also
provides the only way to unify gravity with the other interactions in a gauge
theoretic framework [40, 41]. On the practical side, we have already discussed
how it eases the Hierarchy Problem in GUTs [48, 49, 50]. Great excitement
arose in 1991 when Amaldi et al. [197] showed that when measurements of the
three SM coupling constants at the $Z$-pole were combined with low-energy
measurements, the running of the couplings indicated that there was no scale
where all three would become identical, as predicted in a GUTs. However, when
a SUSY $SU(5)$ GUT model was taken, with the threshold for sparticles around 1
TeV, proper unification could be achieved at a scale around $10^{16}$ GeV.
Since then, the bounds on proton decay have pushed up the unification scale
roughly an order of magnitude higher [198], but this can be accommodated by a
change in the GUT group [199, 200]. SUSY also permits the running of the mass
parameters in the Higgs sector. A beautiful feature of this is that one of
these mass-squared parameters can go negative, breaking the electroweak
symmetry at precisely the right scale. To achieve this, one requires a Higgs
boson mass less than about 135 GeV — a robust prediction of SUSY models [201,
202, 203, 204, 205, 206]. Indeed the Higgs boson does have a mass of 125 GeV,
which is a tantalising, but in no way compelling, agreement with SUSY. Last,
but not least, SUSY models can incorporate an elegant $Z_{2}$ symmetry called
$R$-parity which makes the lightest supersymmetric particle (LSP) both stable
and invisible — the perfect WIMP candidate for dark matter [207].

Given the runaway popularity of SUSY as the canonical BSM theory, it has come
as a huge shock and surprise to the high-energy physics community that even a
decade after the 2012 Higgs triumph, the LHC has not found a shred of evidence
for supersymmetry [208, 209], or indeed, of anything which can definitely be
described as BSM [210]. This has led to a reaction of sorts, with
supersymmetry sometimes being castigated as ’a nightmare scenario’ [211] with
some even calling for a total rethink of the traditional synergy between
mathematical theory and experiment [212]. However, such pessimism is
premature, as has been pointed out by many experts, including review articles
in the SUSY context by Datta [213] and by Mukhopadhyaya [214]. Nevertheless,
the focus of the high-energy community has definitely drifted away from SUSY
towards areas such as dark matter, effective field theories and minimal
models, which are touched upon in Sect. 6.

Before proceeding further, it is necessary to mention the nomenclature of
different low-energy SUSY scenarios. Since supersymmetry must be broken, one
has to introduce a slew of ’soft’ SUSY-breaking terms, which will give masses
to the superpartners, making them heavier than their SM counterparts. This
leads to an explosion in the number of free parameters in the theory. The MSMM
has 37 $CP$-conserving free parameters—and another 67 if $CP$ is violated. As
a result, a Higgs boson below about 135 GeV [201, 202, 203, 204, 205, 206] is
perhaps the only robust prediction of SUSY models. Theorists, therefore, have
tried to model the SUSY breaking through different mechanisms, each idea
individually leading to a drastic reduction in free parameters — at the cost
of a proliferation of models. The earliest of these was a minimal
supergravity, or mSUGRA framework, which assumed a GUT and a ’hidden sector’
where SUSY is broken, presumably spontaneously [84]. Combining this with a
mechanism for electroweak symmetry-breaking produced the so-called
’constrained’ MSSM, or cMSSM [215, 216, 217]. Some of the GUT constraints were
relaxed to produce the so-called ’non-universal’ models [218]. Alternative
mechanisms for SUSY-breaking led to the ’gauge-mediated’ SUSY breaking (GMSB)
[219] and ’anomaly-mediated’ SUSY breaking (AMSB) models [220, 221]. This was
followed by the idea of ’split SUSY’, with an unabashedly ’unnatural’ mass
spectrum [222, 223, 224, 225]. Null results at the LHC have prompted the
exploration of various ’phenomenological’ MSSM, or pMSSM versions, with
subsets of the 37 parameters being considered independent – typically 10, 15,
or 21 of them – while the rest are either fixed or set to zero [226, 227,
228].

As may be expected, a theory, or, rather, class of models, with so much to
offer rapidly became the mainstream for BSM physics in the 1990s and
continued, as we see, till fairly recently. As with the rest of the world,
contributions proposing detailed or novel signatures for SUSY in all its
avatars form the bulk of high-energy phenomenology contributions from India.
The scope of this article does not permit a comprehensive description of this.
Hence, in the discussion that follows, a general portrayal will be attempted,
highlighting some specific works and perforce omitting many, many others.

The pioneering Indian SUSY studies in the context of hadron colliders were
made, as in other areas, by Roy and his collaborators, starting from the
1980s. In particular, Roy and Reya wrote a series of papers [229, 230, 231,
232, 233] highlighting the fact that the Majorana nature of gluinos would
result, when their cascade decays into final states with leptons are
considered, in like-sign dilepton signals. Roy continued these studies with
Drees et al. [234, 235, 236, 237, 238, 239, 240, 241, 242] till the advent of
the LHC. On another note, he carried his experience of using $\tau$
polarisation in H± decays to SUSY signals, especially with staus [243, 244,
245, 246]. These and similar ideas were carried ahead by Godbole with Bélanger
and her other collaborators [247, 248, 249] in the changing scenario as the
LHC (null) results started coming in. Datta and his collaborators were among
the earliest to point out that ’invisible’ decays of other SUSY particles,
such as the next-to-LSP or NLSP, could enhance the missing-energy signals from
the LSP. With the increasing automatisation of cascade decay calculations,
such pinpointed studies have merged into the general SUSY searches for
invisible decays or, equivalently, SUSY dark matter searches [250, 251, 252,
253, 254, 255, 256, 257, 258, 259, 260]. Datta, with his collaborators also
carried out a series of studies of the ’non-universal’ varieties of SUSY,
under different ansätze for the soft SUSY breaking parameters [261, 262, 263,
264, 265, 266, 267, 268, 269], culminating, as may be expected, in the pMSSM
[270], as well as a series of studies of stability of the electroweak
potential within SUSY models and corresponding constraints on the SUSY mass
spectrum [271, 272, 273].

Many different aspects of SUSY models were studied over the years in the
school set up by Mukhopadhyaya with his students and collaborators. An
exhaustive list would, perhaps, require a separate article, but some of the
LHC-related work is highlighted here to indicate the range and variety of
their oeuvre. One of the earliest of these studies was the consideration of
vector boson fusion to produce charginos and neutralinos, as well as sleptons
at the LHC in the MSSM as well as in AMSB models [274, 275, 276, 277, 278,
279]. Several studies of split SUSY followed [280, 281, 282, 283, 284]. The
impact of neutrino masses and mixings on SUSY models and corresponding
collider signals was explored over the years in a series of papers [161, 285,
286, 287, 288, 289, 290, 291, 292, 293, 294, 295, 296, 297, 298, 299]. Another
topic of interest was ’non-universal’ SUSY models [292, 300, 301, 302, 303,
304, 305], in addition to a collection of more purely phenomenological
investigations [306, 307, 308, 309, 310, 311, 312, 313, 314, 315, 316, 317,
318], as the focus of the subject shifted away from UV-complete model building
to the study of low-energy mass spectra. This inevitably has led to multiple
studies of dark matter in the SUSY context [163, 288, 297, 299, 319, 320, 321,
322, 323, 324, 325, 326, 327]. In addition, many other ramifications of SUSY
models in the LHC context have also been discussed over the years by Choudhury
[328, 329, 330, 331, 332], Sridhar [162, 333, 334, 335], Raychaudhuri [336,
337, 338, 339, 340, 341, 342] and Guchait [343, 344, 345, 346, 347, 348, 349,
350, 351, 352, 353, 354, 355, 356], followed by Ghosh [357, 358, 359, 360,
361, 362, 363, 364, 365, 366], Datta [367, 368] and Datta [369, 370, 371, 372,
373, 374, 375, 376, 377, 378, 379], with their respective students and
collaborators. Younger entrants to this field include Rai [380, 381, 382, 383,
384, 384, 385, 386, 387], Konar [388, 389, 390] and Bhattacherjee [391, 392,
393, 394, 395, 396, 397, 398, 399, 400, 401, 402, 403, 404, 405]. It would
take a full volume even to list the abstracts of these papers, indicating the
enormous effort which has gone into the study of supersymmetry. Further,
noting that the Indian SUSY oeuvre is merely a microcosm of the worldwide
efforts (of which it is a small part) helps to realise that the experimental
non-observation of SUSY has been little short of a débacle.

The Indian effort has not failed to touch two alternative viewpoints of
supersymmetry, viz. the GMSB model [357, 406, 407, 408, 409, 410, 411, 412,
413, 414, 415] and the $CP$-violating version of SUSY [308, 416, 417, 418,
419, 420, 421, 422, 423, 424, 425, 426, 427, 428, 429], though it must be said
that no studies have been undertaken in a coordinated or systematic fashion.
On the other hand, the off-mainstream version of SUSY which seems to have had
particular fascination for the Indian community is the version where R-parity
is violated, i.e. there is no stable LSP, and hence no SUSY candidate for dark
matter. Given that this removes one of the most attractive features of the
theory, it is natural to ask why this less rewarding version has become so
popular. The answer may be sought in the SM itself, where baryon number ($B$)
and lepton number ($L$) are conserved, but purely accidentally, since they do
not arise from the assumed symmetries, but just happen to be a symmetry of the
Lagrangian when the other symmetries and renormalisability are imposed. The
moment one tries to go beyond the SM, one encounters the problem that these
may be violated. In SUSY, the second Higgs doublet superfield (required by
holomorphicity of the superpotential) is forced (by the absence of tree-level
flavour-changing neutral currents) to have the same gauge quantum numbers as
the lepton doublet superfield, which opens up the possibility of mixing
between these representations with different lepton numbers, i.e. violation of
lepton number. To prevent this, which would lead to fast proton decay and rule
out all SUSY models, $R$-parity was introduced [430] as a compound of these
symmetries, imposed on SUSY models. However, it was then pointed out by
Weinberg and others, including Auluck and Mohapatra [431, 432, 433] that
proton decay requires violation of both baryon and lepton number and hence any
one of the two can be violated while conserving the other. This opened up the
possibility for $R$-parity violation. In such models, the LSP is no longer
stable, but decays into multi-lepton final states if $L$ is violated, or, into
multiple jets if $B$ is violated. Either way, one could predict spectacular
signals at a machine like the LHC [434, 435, 436, 437, 438, 439, 440, 441],
and this is presumably where the Indian interest originated.

The Indian HEP community was introduced to the possibilities lying in R-parity
violation by a talk given at the First Workshop on High Energy Physics
Phenomenology (WHEPP-1) in Jan 1989 by Arnowitt [442], which was shortly
followed by the publication of a magisterial analysis of the phenomenological
implications by Barger et al. [443]. As elsewhere, Indian contributions to
this field originated with D.P. Roy, who studied the collider signals [444,
445, 446, 447], and by P. Roy, who considered indirect signals [448, 449, 450,
451, 452] for $R$-parity violation. The mantle was then taken up by others.
The different aspects explored included novel signals at hadron colliders
[282, 453, 454, 455, 456, 457, 458, 459, 460, 481, 462, 463, 464, 465, 466,
467, 468, 469, 470, 471, 472, 473, 474, 475, 476, 477, 478, 479, 480, 481,
482], collider bounds on $R$-parity-violating couplings [484, 485, 486, 487,
488, 489, 490, 491, 492, 493, 494, 495, 496, 497, 498, 499] and $R$-parity
violation as an explanation for some (mostly transient) ’anomalies’ in
collider data [500, 501, 502, 503, 504, 505]. Several studies of bilinear
$R$-parity violation were also made [285, 506, 507, 508, 509, 510, 511, 512].
Interestingly, even with the decline of interest in SUSY, $R$-parity-violating
sparticles continue to be studied independently as the eponymous dileptons,
diquarks and leptoquarks [513, 514, 515, 516, 517, 518, 519, 520, 521, 522,
523, 524, 525, 526, 527, 528, 529, 530].

##  5 Exploring extra dimensions

It has been related above that in the 1990s, it became quite clear that
technicolor models could not be made to work without adding multiple ad hoc
features, and, therefore, SUSY became the only plausible solution of the
hierarchy problem in the SM. It was at this point (1998) that a new
theoretical idea erupted on the high energy physics scene like a volcano which
had long lain dormant. This was the idea of extra dimensions of spacetime.

By itself, the idea of extra dimensions was not new. An extra dimension of
space had been dreamt of by philosophers [531, 532], but it only became
serious physics with the invention of Minkowski space and its subsequent
application by Nordström et al. in separate attempts to unify Maxwellian
electrodynamics with general relativity [533, 534, 535, 536, 537]. These
attempts died a natural death when they failed to correctly predict the $e/m$
ratio of known particles. However, extra dimensions were revived by the advent
of superstring theories in the 1970s [37, 538, 539, 540]. At this point, they
did not seem to have much connection with particle physics, since the extra
dimensions were assumed to be tiny, and the new particles were predicted to
have masses around the Planck scale, i.e. $10^{16}$ TeV. Though there had been
some speculations that there could be extra dimensions which were much bigger
[541, 542, 543], these did not attract much attention at the time, even though
a series of papers by I. Antoniadis and collaborators [544, 545, 546, 547,
548] went on to suggest that one or more of the extra dimensions in a string
theory could be large and have observable effects.

The watershed was a 1998 paper [549] by Arkani-Hamed, Dimopoulos and Dvali,
now collectively shortened to ’ADD’, in which it was suggested that the
presence of large extra dimensions could eliminate the hierarchy problem by
bringing down the Planck scale to a few TeV. To ensure that this remained
compatible with the weak strength of gravity, it was necessary to assume that
our familiar four-dimensional Minkowski world is actually a wafer-thin
topological defect in the higher dimensional ’bulk spacetime’, on which the SM
fields have somehow got trapped. Almost immediately after, the authors teamed
up with Antoniadis to place this in the context of string theories [550]. As a
matter of fact, string theories can provide just the sort of topological
defect with trapped fields as imagined by ADD — these are called $D$-branes
and had been invented a decade earlier by Polchinski and collaborators [551]
and by Horava [552]. From this point, it became common for the fourdimensional
topological defect to be called ’the brane’, even if there was no actual use
of string theory in the problem. However, it is likely that this model would
have remained a curiosity, had not the low-energy theory with multiple-
graviton emission and exchange been explicitly worked out by two different
groups [553, 554], later in the same year. It was these, together with a paper
by Mirabelli et al. [555], that set off the hunt for extra dimensions at
colliders, including the LHC, which was then definitely on the way.

Indian involvement with extra dimensions began early – in 1998 itself – with
the trio of Mathews, Raychaudhuri and Sridhar [556, 557, 558], who
investigated Kaluza-Klein graviton effects in $t\bar{t}$ production, $ep$
scattering and dijet production, in the context of the Tevatron and HERA
colliders. This set off an explosion of studies in the Indian HEP community,
some of which, relating to the LHC, is summarised below. As a matter of fact,
extra dimensional theories soon split into three different paradigms, viz.

  * •

The Large Extra Dimensions (LED) paradigm, which was essentially the model of
ADD, where two or more extra dimensions are of size as large as, perhaps, a
millimetre or a few hundred microns, and the SM fields are confined to a
single brane of thickness less than $(100~{}GeV)^{-1}$, while gravitons move
freely in the toroidal bulk.

  * •

The Warped Extra Dimensions (RS) paradigm, proposed by Randall and Sundrum
[559] in 1999, in which there are two branes at the ends of a single extra
dimension with the topology of a once-folded circle ($S(1)/Z_{2}$)and fine-
tuned vacuum energies in the bulk and on the branes. This configuration
permits a solution of the Einstein equations where the Planck mass can be
small on one brane and exponentially large on the other.

  * •

The Universal Extra Dimensions (UED) paradigm, proposed by Appelquist et al.
[560] in 2001, which envisages a bulk geometry similar to the RS model, but
places all the SM particles in the bulk. The existence of a discrete symmetry
due to the specific geometry acts in a way similar to $R$-parity and is indeed
known as KK-parity. This allows this model to have a candidate for dark matter
in the same way as SUSY does.

If we go by these divisions, work on the LED paradigm was somewhat limited,
since the other, more interesting options were soon invented. Nevertheless,
some studies of possible LED signals at the LHC, and associated phenomena were
made over the next decade or so [561, 562, 563, 564, 565, 566, 567, 568, 569,
570, 571]. Studies of QCD corrections to collider processes in LED models have
mostly been made by Mathews and Ravindran with their students and
collaborators [571, 572, 573, 574, 575, 576, 577, 578, 579, 580, 581, 582,
583, 584, 585, 586, 587, 588].

The minimal RS model, which predicts heavy graviton resonances, has a
phenomenology more akin to more conventional electroweak models [589].
Therefore, there were a few studies from India of the early versions of the RS
model [590, 591, 592, 593, 594, 595, 596, 597, 598], including QCD corrections
to the relevant processes [598, 599, 600, 600, 601, 602, 603, 604]. Later came
the so-called bulk RS models, where all the SM particles can be in the bulk
(as in UED models) and have Kaluza-Klein modes on the brane. These came is
stages, with the gauge bosons, and then the fermions, and finally the Higgs
boson field, being moved into the bulk. These models have been mostly studied
in India by Sridhar and his collaborators [605, 606, 607, 608, 609, 610, 611,
612, 613, 614, 615, 616, 617, 618, 619, 620, 621, 622, 623, 624]. Extensions
of the minimal RS model have been attempted in the gravity sector by Choudhury
and collaborators [625, 626, 627, 628, 629, 630, 631, 632], while effects of a
bulk torsion field have been discussed at length by Mukhopadhyaya and SenGupta
in a series of papers [633, 634, 635, 636, 637, 638, 639, 640, 641, 642, 643,
644, 645, 646, 647, 648, 649]. The latter has also studied the possibility of
Gauss-Bonnet and higher curvature terms in the bulk [650, 651, 652, 653, 654,
655, 656, 657, 658, 659, 660, 661, 662].

It was also realised, soon after the RS model was proposed, that it requires
an extreme fine-tuning of the inter-brane separation to get the right
hierarchy of mass scales. To obtain this more naturally, Goldberger and Wise
added an additional bulk scalar to the model [663, 664], which appears on the
brane as a kind of dilatonic field, dubbed the radion. The phenomenology of a
light radion is quite similar to that of the Higgs boson, and in fact, it can
mix with the Higgs boson [665]. Partly for this reason, there has been
considerable interest in the Indian community on radion physics and signals
for radions at colliders, including the LHC [666, 667, 668, 669, 670, 671,
672, 673, 674, 675].

The minimal UED models (mUED) have also been studied quite intensively in
India [189, 370, 371, 676, 677, 678, 679, 680, 681, 682, 683, 684, 685, 686,
687, 688]. Attempts to extend this to non-minimal versions have been made in
two directions — one by assuming more than one universal extra dimension [689,
690, 691, 692, 693, 694, 695] and one by assuming boundary-localised terms in
the action [696, 697, 698, 699, 700, 701, 702, 703]. In either case, changes
happen in the mass spectrum and the couplings and these reflect in the
collider phenomenology.

Two offshoots of theories with extra dimensions bear mentioning. One was the
rather arcane idea of unparticles [704, 705, 706] — which are field theoretic
configurations in four dimensions with a behaviour similar to higher
dimensional fields and a phenomenology rather similar to that of the ADD model
(2007). This idea inspired quite a few Indian phenomenological studies of
unparticles [707, 708, 709, 710, 711, 712, 713, 714, 715, 716, 717, 718, 719,
720, 721, 722, 723, 724, 725, 726, 727] before the idea quietly faded out of
the limelight. Somewhat more lasting were an ingenious class of models,
collectively known as ’little Higgs models’ [728, 729, 730, 731]. In these
models, developed between 2000 and 2005, a gauge group $[SU(2)\times
U(1)]^{n}$ breaks in stages to the $SU(2)_{L}\times U(1)_{Y}$ of the SM,
creating a hierarchy of heavy gauge bosons, and heavy fermions to match. The
beauty of this model – or class of models – lies in the fact that group-
theoretic factors provide the negative signs required for cancellation of
quadratic divergences in the Higgs boson self-energy, which can, therefore,
happen separately [732] between pairs of bosons and pairs of fermions777Unlike
SUSY, where these happen between boson and fermion pairs.. In some versions of
these models, the extra fields have a 2 symmetry called $T$-parity, as a
result of which the lightest particle with non-zero $T$ becomes stable [733,
734, 735, 736]. This model, therefore, has a solution to the hierarchy problem
together with a dark matter candidate, just as SUSY has. Indian HEP theorists
were somewhat late getting on to this bandwagon, but eventually this model, or
rather class of models, did inspire a whole series of investigations [161,
307, 337, 737, 738, 739, 740, 741, 742, 743, 744, 745, 746, 747, 748, 749,
750, 751, 752, 753, 754, 755, 756, 757], before they, too, faded into the
twilight with the others.

##  6 Continuing with caution

Gradually, as the LHC continued to run beyond 2012, the euphoria accompanying
the Higgs boson discovery began to fade. As we have noted before, there have
been no signs in the intervening decade of any signals for compositeness,
SUSY, extra dimensions, or any kind of BSM physics whatsoever [210]. So
uniformly have the data matched with the predictions of the SM, that the LHC
is already being dubbed ’the Standard Model collider’. Moreover, the entire
paradigm of hypothesis and prediction preceding discovery has been called –
perhaps prematurely – into question [212]. And yet, theorists are not
comfortable with a world where the SM is the ultimate theory of elementary
particles and their interactions. It is useful, at this juncture, to briefly
review the reasons for this profound dissatisfaction.

Apart from discomfort with the large number of experimentally fitted
parameters in the SM, which seem to follow no known order or system, theorists
have seven major issues with the SM as a theory. These can be classified into
three groups, viz.

  * •

Missing components: The biggest puzzle is the nature of (I) dark matter and
(II) dark energy. If these are particulate in nature – as seems to be the
general consensus, for dark matter at least – then there are no candidates in
the SM which fit the bill.

  * •

Inadequate parameters: One of these issues is (III) baryogenesis, or the
matter-antimatter asymmetry of the Universe, which may be tracked to the
measured level of $CP$-violation in the SM being some $10^{-11}$ times too
small to explain the observed baryon asymmetry. On the other hand, we also
have the (IV) strong $CP$ problem, where the $\theta_{\rm QCD}$ coupling has
to be tuned to less than $10^{-10}$ in the SM without apparent justification.
There is also the issue of (V) neutrino masses, which were originally believed
to vanish, but have been proved otherwise. Accommodating them in the SM leads
to multiple issues of naturalness and lepton number non-conservation.

  * •

Inconsistency at quantum level: It is beyond question that the SM is a quantum
field theory, for this has been vindicated by a large body of evidence,
including neutral pion decay, precision measurements of electroweak
parameters, the running of the strong coupling constant and the occurrence of
rare flavour-changing decays. However, as described in Sect. 1, this leads to
a (VI) hierarchy problem, with a fine-tuning of the running Higgs boson self-
energy to some 16 orders of magnitude. Another problem arises with the running
of the Higgs boson self-coupling, which tends to destroy (VII) vacuum
stability at some high-energy scale, something that can only be countered by
invoking BSM physics.

Earlier UV-complete theories like SUSY provided simultaneous answers to
several of these questions at the same time. Unfortunately, the whole paradigm
of UV-complete theories predicting light particles discoverable as resonances
at the LHC – though by no means disproved – has lost popularity as the best
way to probe BSM physics at the LHC. Instead we have about half-a-dozen new
approaches, which eschew the top-down approach, and instead, concentrate on
minimal extensions, effective field theories, simplified models, and so on,
i.e. essentially bottom-up approaches which do not worry about issues of UV
completeness or nonunitarity. The high-energy physics community has certainly
learnt the virtues of caution! In this section, we briefly discuss a few of
these approaches, focussing on the Indian contributions to this field and
restricting ourselves to the LHC context. Since this is an active area of
research, with new ideas being proposed on an almost daily basis, this section
will necessarily be incomplete, even sketchy.

###  6.1 Minimal extensions

The SM has three sectors, viz. the gauge boson sector, the fermion sector and
the scalar sector. Minimal extensions have been contemplated in all of these.
The oldest and perhaps the best-motivated of these is the left-right symmetric
model (and its SUSY version), based on a group $SU(2)_{L}\times
SU(2)_{R}\times U(1)_{B-L}$ which breaks spontaneously to $SU(2)_{L}\times
U(1)_{Y}$, thereby explaining the origin of parity violation [64, 65, 66, 67,
68, 69]. This class of models predicts the existence of heavy $W_{R}$ bosons,
and there is a considerable body of work by Indians discussing collider
signatures for these or closely related phenomena [758, 759, 760, 761, 762,
763, 764, 765, 766, 767, 768, 769, 770, 771, 772, 773, 774, 775, 776, 777,
778, 779, 780]. Some gauge extensions, as well as a few recent ’anomalies’ in
the flavour sector seem to call for the existence of an extra U(1) symmetry
and this predicts a heavy $Z^{\prime}$ boson, which has also been studied
quite intensively by Indian theorists [781, 782, 783, 784, 785, 786, 787].

In the fermion sector, the LEP precision measurements of electroweak
parameters [788] permit the existence of just three kinds of heavy BSM
fermions, viz. sequential fermions with left-handed doublets and right-handed
singlets, mirror fermions with left-handed singlets and right-handed doublets,
and also vectorlike fermions, where both left- and right-handed fermions are
either singlets or doublets. A fourth sequential generation, however, was
ruled out by the LEP data except for a highly fine-tuned version [789, 790,
791, 792, 793], and even that was ruled out by the LHC data soon after the
Higgs boson discovery [794, 795, 796, 797]. Mirror fermions, however, and
vectorlike fermions, survive to this day, and, together with other classes of
exotic fermions, have been investigated by several Indian theorists [776, 798,
799, 800, 801, 802, 803, 804, 805, 806, 807, 808, 809, 810, 811, 812, 813,
814, 815, 816, 817, 818, 819, 820], most notably Gopalakrishna [57, 821, 822,
823, 824, 825, 826] and his collaborators. Another class of exotic fermions
are the bulk fermions in a higher dimensional theory, which have already been
mentioned in the previous section.

Finally, we come to the Higgs sector. Since we know least about this sector,
except that a Higgs boson exists, we have the greatest freedom to propose
modifications in this part of the SM. Many of these have been discussed in
Sect. 3. In addition, a doubly charged Higgs boson, such as arises whenever
there is a scalar triplet or bidoublet in the model, has received considerable
attention [327, 827, 828, 829, 830, 831, 832, 833, 834] in the Indian
community.

###  6.2 Modelling Dark Matter

There is a wide consensus today that the dark matter component of the Universe
is particulate in nature, and that these are BSM particles having
gravitational and weak interactions, but not strong and electromagnetic
interactions. The gravitational interactions have been proved from
astrophysical observations, and weak interactions are necessary for enough
dark matter to be produced to satisfy the relic density inferred from CMB
studies. The absence of electromagnetic interactions makes them ’dark’ and the
absence of strong interactions makes them ’non-baryonic’, both of which are
proved from astrophysical observations [835, 836, 837, 838].

There has been much speculation about these particles. The Indian contribution
to this has mostly been along two lines, viz. ’invisible’ particles and
’simplified models’. In the first paradigm, the existence of weakly
interacting massive particles (WIMPs) has been assumed and the analyses
discuss how to search for and/or constrain such theories [172, 396, 569, 839,
840, 841]. In the second paradigm, the SM is minimally extended to include a
dark matter particle, and one or two ’mediators’ to ensure that the dark
matter has weak interactions [841, 842, 843, 844, 845, 846, 847, 848, 849,
850, 851, 852, 853, 854, 854]. Until there is some experimental evidence from
the LHC or from direct search experiments, the field is wide open for
speculation, somewhat as the whole of BSM physics was in the 1980s.

###  6.3 Compressed spectra

When heavy particles are produced at high-energy colliders, they immediately
decay into lighter particles, which in turn, decay into lighter particles, and
the process continues until there are stable particles in the final state.
Such a cascade decay process creates a ’signature’ of the final particle in
the form of a pattern of stable co-produced final states [877, 878, 879]. This
has always been the classic technique to look for new particles in the SM, and
now beyond it, in models such as SUSY and extra dimensions. However, if the
mass gap between the initial and final states is small, i.e. the mass spectrum
of all the intermediate states are ’compressed’ into a small range, then the
(meta)stable particles in the final state, viz. electrons, photons, muons,
etc. will have very low energies and may fall below the triggering thresholds
of the detectors. As a result, in the case of compressed mass spectra, a BSM
model becomes very difficult to discover. SUSY models, in particular, have
some parts of the allowed parameter space which predict such spectra. It has,
thus, been a part of the international effort, to which Indians have also
contributed [315, 316, 390, 395, 880, 881], to identify novel variables and
signatures which would enable us to find new particles, even in the case of
such compressed spectra.

###  6.4 New variables for BSM studies

Inspiration from the study of compressed spectra has lead researchers to
explore new kinematic variables where signatures for new physics may be
lurking and to devise ways and means of exploiting the existent data to the
maximum using such variables [337, 855, 856, 857, 858, 859, 860, 861, 862,
863, 864, 865, 866, 867, 868, 869, 870, 871]. Inevitably, such searches have
led to the use of machine learning tools to sift and find patterns in the
data. This field has been attracting an increasing amount of interest from
India [327, 872, 873, 874, 875, 876] in recent times and holds out the promise
of being a major tool in the future runs of the LHC.

###  6.5 Effective Field Theories

In this class of models, the SM is assumed to be the low-energy limit of a
deeper, presumably UV-complete, theory at a higher energy scale. The prototype
of this is the Fermi effective theory of weak interactions, which is the low-
energy limit of the gauge theory of Glashow, Salam and Weinberg. Just as the
Fermi current-current operator has mass dimension 6 and the coupling constant
$G_{F}$ has a negative mass dimension, so do these effective field theories,
or EFTs, involve higher dimensional operators with dimensional couplings,
presumably reflecting the scale of the new physics. The total number of such
operators at dimensions 5, 6 and 8 (which have mostly been studied) is very
large [882], but there are gauge-invariant subsets under renormalisation group
evolution, which makes the problem more tractable [883]. Moreover, for a
particular set of processes, the dominant contributions may come from a
limited set of operators, leading to an EFT version of simplified models. Many
of these aspects have been investigated by Indian physicists [884, 885, 886,
887, 888, 889, 890, 891, 892, 893, 894, 895, 896, 897, 898, 899], though much
remains to be done, as this is an active, ongoing line of research.

##  7 Pontifications & predictions

Having reviewed the happenings in this field during the past 40-odd years, one
can essay to make some remarks of a general nature. Perhaps the first and most
important observation to make is that the element of uncertainty in high-
energy physics (HEP), is perhaps, the highest in any current branch of
science, since what is being probed is a length scale never ’seen’ before and,
if our current understanding of a Big Bang origin of the Universe is correct,
a time scale in the early Universe which has never before been ’accessible’.
The history of science in the nineteenth and twentieth centuries is replete
with examples of the big surprises which awaited researchers when they probed
smaller scales. Of these, the discovery of atomic sub-structures and the
composite nature of hadrons were perhaps the most dramatic, but of equal
importance was the discovery that it is continuous fields, and not corpuscular
entities, that form the key to understanding Nature at such small lengths. The
importance of symmetry as a framework to develop fundamental theories has also
been gradually forced upon us, until it has now become the first tool of the
HEP theorist. And yet we do not really know what lies at length scales below
$10^{-17}$ cm, and most of our guesses do not seem to be working. High-energy
physicists are, therefore, once again as they were in the first half of the
twentieth century, like pioneers pushing into an unknown world, unaware of
what exotic landforms or animals may be found there.

During the second half of the twentieth century, and indeed, until a decade
ago, HEP witnessed a very different paradigm compared to the rest of the
scientific world. In HEP, theoretical ideas surged ahead of experimentation,
with new theories making predictions far beyond the capacity of contemporary
experiments to test. Of these, the set of ideas clubbed together as the
Standard Model, have proved successful far beyond the dreams of anyone who
proposed it. Today, in the sheer number of experimental results which match
with the Standard Model, arguably the only competitor would be Newtonian
gravity.

However, this success has been very hard-earned. From the 1970s, as high-
energy physics moved away from cosmic ray and fixed target to colliding-beam
experiments, the complexity and cost of high-energy physics experiments has
shot up by many orders of magnitude. Modern HEP experiments – especially of
the collider type – have become notorious for their size, complexity, long
gestation time — and, therefore, cost. The size and scope of these experiments
requires a small army of research workers, from engineers to data analysts, to
run the machines, extract the data and interpret them. Given that the quantity
of data to process increases rapidly with the energy, this makes for a long
time before we can expect results from a new experiment. The involvement of
more people also increases the possibility of human errors, and thus further
delays are introduced while the results go through multiple checks before
obtaining a stamp of approval from the relevant collaboration. Thus, it took a
full half century as the Standard Model and its different predictions were
verified, particle by particle and process by process, at different
experiments. All the while, theorists were left free to let their minds range
over the smaller length scales which would one day become accessible to
experiment, and make predictions for the same.

###  7.1 Modern Times

In the post-Higgs discovery era – essentially the past decade – we have seen
the runaway success of theoretical ideas, especially those based on new
symmetries, come to a grinding halt. The LHC has been slowly, but surely,
checking the predictions of these exotic theories, and, till date, has come up
with a total blank. However, the LHC has barely scratched the surface of the
new energy regime, and even if it goes deeper with the same kind of results,
all that this would mean is that physics at the new length scale must be
different from what has been imagined – or can be imagined – and it will
require experimental data to create a new understanding of it. That is, and
always has been, the sober scientific view. However, there are always
sociological ripples, including detractors who would argue that mathematical
approaches to physics are fundamentally wrong, or that too much tax-payers’
money is being spent, or that scientists should not dare to speculate beyond a
certain limit. Generally, in science as in any other field of human endeavour,
such voices rise during a period of stagnation and fall silent whenever the
next major advance/breakthrough comes.

In truth, as the saying goes, rumours of the death of BSM physics have been
greatly exaggerated. What has mostly been eliminated is the most ’hopeful’
part of the vast parameter space of these models, where the word ’hopeful’ is
used in the sense of that which would have given new physics in the early runs
of the LHC. However, Nature has not proved to be so generous, and in fact,
rarely is. In the 1970s, the $W$ and $Z$ bosons were expected to lie around 2
GeV and in the 1970s, the Higgs boson and the top quark were believed to lie
around $10-20$ GeV. These were the ’hopeful’ values at the time, but it
required much higher energies and much higher costs to actually find these
particles. Nevertheless, they do exist and the entire human race would have
been the poorer if their search had been abandoned in the 1980s. Thus, all
that we can say is that there is a long road ahead, where new physics of the
expected, deviant or totally unexpected kind may appear at any juncture.

Another point to note is that the style of theoretical work in HEP has been
evolving quite rapidly over the past two decades. Till the end of the
twentieth century, theorists contented themselves with calculating total and
differential cross-sections, and making fairly crude numerical estimates using
some broad kinematic cuts to simulate the experimental constraints. It was
left to the experimentalists to simulate the events, including all peripheral
effects and the limitations of their own detectors. However, mostly after the
turn of the century, there has been a proliferation of ’user-friendly’
software packages, with different levels of sophistication, to be used in the
analyses by theorists. Thus, we have packages like, to name only a few,
MadGraph, Pythia and Herwig to simulate collider processes, SuperISO and EOS
for flavour physics, MicrOMEGAs to calculate the dark energy relic density and
DELPHES to simulate detector effects. This is accompanied by mathematical
calculation tools like Form, FeynRules and, of course, the all-pervading
Mathematica. Recently, as mentioned in Sect. 6.4 above, the use of Machine
Learning tools has also come into vogue, and no doubt the near future will see
these incorporated into some of the HEP packages.

Thus, a young theorist now requires to master a large number of software tools
to remain competitive, but once this is done, the calculational process is
accelerated, leaving time for the researcher to think about deeper issues. It
may thus be hoped that with these powerful tools, the next generation of
theorists will be able to make inroads into problems which have eluded their
predecessors. On the flip side is the caveat that the power and beauty of
these software tools should not intoxicate researchers into creating
unnecessary sophistication where it does not add any insight888In this
context, readers may appreciate Richard Feynman’s observations regarding his
colleague Stan Frenkel, in Ref. [900]..

###  7.2 The Indian context

Turning now to the Indian context, HEP theory and experiment have grown
greatly over the past 4 decades. For the experimental effort, the 1970s and
1980s saw a drastic decline in numbers, as cosmic ray experiments receded into
the background, and accelerator physics grew in proportion. The closure of the
Kolar gold mines in 1991 led to the abandonment of India’s front-line proton
decay experiment. Henceforth, Indian experimental participation in cutting-
edge HEP would be through international collaborations — at CERN, FNAL and
KEK, among others. The first groups to get into this were from TIFR and BARC
and the Universities of Delhi and Punjab, but others soon joined in. Many of
the graduate students in the early years moved abroad, but increasingly many
are coming back to India after postdoctoral stints. The growth in the number
of science-oriented institutions in the country has also helped many of these
young HEP experimentalists to find positions and set up laboratories under the
aegis of one international collaboration, or another. The pool of experience
in India has steadily grown till it now includes seniors who have worked on
the ISR and fixed target experiments at CERN and at the proton decay
experiment at Kolar, and a large number of others who have worked on the
collider experiments at SLAC and FNAL in the USA, the LEP and LHC machines at
CERN, as well as the BELLE experiment at KEK. Sadly, India’s own attempt to
set up a large HEP experiment in the form of an underground neutrino detector
to be called the India-based Neutrino Observatory (INO) has remained bogged
down in socio-political issues till the present juncture.

On the theory side, too, there has been a massive growth in the number of
researchers. An interesting phenomenon, indicative of a mature community, is
that there are now four generations of HEP theorists who are active in the
country. The author notes that this includes his own teachers and mentors, his
own colleagues and contemporaries, as well as his students and their students.
This is a healthy sign, no less than the fact that the present HEP community
is spread over a wide variety of academic institutions across the country and
there exist groups where earlier there used to be a solitary practitioner.
Another sign of maturity is the fact that there are now many close
collaborations between theorists and experimentalists, something which would
have been very rare in the 1980s. There is an increasing trend for close
collaborations between particle physicists and astrophysicists, as well as
cosmologists, as the two subjects show signs of merging, especially in the
area of Dark Matter. On the other hand, HEP theory and string theory have
moved away from the close synergy they had in the past century, partly because
string theory is moving closer to condensed matter physics, but also because
supersymmetry, the main tie between the two areas, is getting pushed into the
background.

The new HEP theorists in India have, in general, embraced the software-
dominated culture of HEP theory very well. Fortunately most of these softwares
lie in the public domain, and those that do not are quite reasonably priced.
As a result, Indian HEP theorists are not financially constrained as Indian
HEP experimentalists are, and can acquire as much skill as their foreign
counterparts in more scientifically advanced countries. This shows up in the
high quality of theory HEP publications from India. The next step would be for
some of the software packages to actually be developed in India. Given the
success of Indians and Indian-origin workers in the IT industry, such a
consummation may be predicted with some level of confidence.

###  7.3 The future

Now, finally, we come to the paramount question — what results can we expect
in HEP in the years to come? One answer is obvious, viz. we shall see the SM
being tested at higher energies and luminosities, i.e. in effect, at a smaller
length scale. It is also easy to predict that most of these tests will be
along the lines of the past, yielding more confirmation of the SM predictions.
The excitement, if any, will lie in anything which deviates from the SM and
raises hopes that new physics has – at last – been discovered.

Over the years, there have been several false alarms and abortive signals for
BSM physics. These have generally disappeared after the collection of more
data, or a re-evaluation of the SM prediction. Among the few which have
persisted are the muon anomalous magnetic moment $(g-2)_{\mu}$ [901, 902] and
a handful of flavour anomalies [903]. Most recently, the complete data from
the now-defunct CDF experiment at Fermilab has been analysed to yield a value
of the $W$ boson mass which shows as much as a $7\sigma$ discrepancy with the
SM prediction [904]. Though it has immediately become the subject of intense
speculation, including by Indians [905, 906, 907, 908, 909, 910, 911, 912,
913, 914], this result will need to be confirmed in the further runs of the
LHC before it can be taken in all seriousness. One can only say that, taken
together, all these anomalies constitute a tantalising set of hints that the
SM may not be the final theory of elementary particles, after all.

The LHC is scheduled to run for another decade and a half [915], and there is
already much speculation about the next generation of high-energy colliders.
Given the pool of talents and expertise in the burgeoning high-energy physics
community in India, we shall no doubt see a great deal more of LHC-oriented
research and beyond coming out of India, and – who knows? – maybe a major
discovery. It is an exciting thought.

Acknowledgements: The author acknowledges support of the Department of Atomic
Energy, Government of India, under Project Identification No. RTI 4002a.

## References

  * [1] C. Llewellyn, Smith. Philos. Trans. R. Soc. A373, 20140037 (2015).
  * [2] R. A$\beta$mann, M. Lamont, S. Myers, Nucl. Phys. Proc. Suppl. B109, 17 (2002)
  * [3] S.L. Glashow, Nucl. Phys. 22, 579 (1961)
  * [4] A. Salam, J.C. Ward, Phys. Lett. 13, 168 (1964)
  * [5] S. Weinberg, Phys. Rev. Lett. 19, 1264 (1967)
  * [6] F. Englert, R. Brout, Phys. Rev. Lett. 13, 321 (1964)
  * [7] P.W. Higgs, Phys. Rev. Lett. 13, 508 (1964)
  * [8] G. Guralnik, C.R. Hagen, T.W.B. Kibble, Phys. Rev. Lett. 13, 585 (1964)
  * [9] ATLAS Collaboration, G. Aad, et al., Phys. Lett. B 716, 1 (2012)
  * [10] C.M.S. Collaboration, S. Chatrchyan et al., Phys. Lett. B 716, 30 (2012)
  * [11] UA1 Collaboration, G. Arnison, et al., Phys. Lett. B 122, 103 (1983)
  * [12] UA2 Collaboration, M. Banner, et al., Phys. Lett. B 122, 476 (1983)
  * [13] UA1 Collaboration, G. Arnison, et al., Phys. Lett. B 126, 410 (1983)
  * [14] UA2 Collaboration, P. Bagnaia, et al., Phys. Lett. B 129, 130 (1983)
  * [15] A. Pais, S.B. Treiman, Phys. Rev. Lett. 35, 1556 (1975)
  * [16] H. Georgi, S.L. Glashow, Phys. Rev. Lett. 32, 438 (1974)
  * [17] J.C. Pati, A. Salam, Phys. Rev. D 10, 275 (1974)
  * [18] F. Reines, M.F. Crouch, Phys. Rev. Lett. 32, 493 (1974)
  * [19] The Super-Kamiokande Collaboration, A. Takenaka, et al., Phys. Rev. D 102, 112011 (2020)
  * [20] K.G. Wilson, Phys. Rev. D 3, 1818 (1971)
  * [21] G. ’tHooft, NATO Adv. Study Inst. Ser. B 59 (Phys.), 135 (1980)
  * [22] A. Echhorn et al., JHEP 64, 022 (2015)
  * [23] K.G. Wilson, Phys. Rev. D 10, 2445 (1974)
  * [24] R. Akolfer, J. Greensite, J. Phys. G34, S3 (2007)
  * [25] K. Lane, Boston University Physics Department Preprint BUHEP-94-2 (1993). arXiv:hep-ph/ 9401324v2
  * [26] C.D. Carone, Phys. Rev. D 86, 055011 (2012)
  * [27] R. Jackiw, K. Johnson, Phys. Rev. D 8, 2386 (1973)
  * [28] J.M. Cornwall, R.E. Norton, Phys. Rev. D 8, 3338 (1973)
  * [29] S. Weinberg, Phys. Rev. D 13, 974 (1976)
  * [30] L. Susskind, Phys. Rev. D 20, 2619 (1979)
  * [31] S. Weinberg, Physica A 96, 327 (1979)
  * [32] V. Bernard, U.G. Mei$\beta$ner, Ann. Rev. Nucl. Part. Sci. 57, 33 (2007)
  * [33] B. Ananthanarayan, Pramana 61, 911 (2003)
  * [34] H. Harari, Phys. Lett. B 86, 83 (1979)
  * [35] M.A. Shupe, Phys. Lett. B 86, 87 (1979)
  * [36] J.C. Pati, A. Salam, J. Strathdee, Phys. Lett. B 59, 265 (1975)
  * [37] P. Ramond, Phys. Rev. D 3, 2415 (1971)
  * [38] A. Neveu, J.H. Schwarz, Phys. Lett. B 34, 517 (1971)
  * [39] C.B. Thorn, Phys. Rev. D 4, 1112 (1971)
  * [40] Yu.A. Golfand, E.P. Likhtman, JETP Lett. 13, 323 (1971)
  * [41] R. Haag, J.T. Lopuszanski, M. Sohnius, Nucl. Phys. B 88, 257 (1975)
  * [42] J. Wess, B. Zumino, Nucl. Phys. B 70, 1 (1974)
  * [43] J. Wess, B. Zumino, Nucl. Phys. B 70, 39 (1974)
  * [44] S. Ferrara, B. Zumino, Nucl. Phys. B 70, 413 (1974)
  * [45] B. de Wit, D.Z. Freedman, Phys. Rev. D 12, 2286 (1975)
  * [46] P. Fayet, Phys. Lett. B 64, 159 (1976)
  * [47] S. Dimopoulos, H. Georgi, Nucl. Phys. B 193, 150 (1981)
  * [48] R.K. Kaul, P. Majumdar, CTS preprint Print-81-0373 (1981)
  * [49] R.K. Kaul, P. Majumdar, Phys. Lett. B 109, 19 (1982)
  * [50] R.K. Kaul, P. Majumdar, Nucl. Phys. B 199, 36 (1982)
  * [51] P.H. Frampton, P.Q. Hung, M. Sher, Phys. Rep. 330, 263 (2000)
  * [52] P.A. Zyla et al., Particle data group. Prog. Theor. Exp. Phys. 2020, 083C01 (2020)
  * [53] P. Langacker, D. London, Phys. Rev. D 38, 886 (1988)
  * [54] E. Nardi, E. Roulet, D. Tommasin, Nucl. Phys. B 386, 239 (1992)
  * [55] T.P. Cheng, L.F. Li, Phys. Rev. D 16, 1429 (1977)
  * [56] F. del Aguila, M.J. Bowick, Phys. Lett. Bl19, 142 (1982)
  * [57] S.A.R. Ellis, R.M. Godbole, S. Gopalakrishna, J.D. Wells, JHEP 09, 130 (2014)
  * [58] F.F. Freitas, J. Goncalves, A.P. Moraia, R. Pasechnik, JHEP 01, 076 (2021)
  * [59] C.T. Hill, Ph.D. thesis, p. 100. https://thesis.library.caltech.edu/4505/
  * [60] J.F. Gunion, H. Haber, G. Kane, S. Dawson, The Higgs Hunter’s Guide (CRC Press, Boca Raton, 1990). (Chap. 4)
  * [61] F. Kling, S. Shu, W. Su, JHEP 06, 163 (2020)
  * [62] M. Cvetic, S. Godfrey, U. Penn. preprint UPR-648-T (1995). arXiv: hep-ph/9504216 [hep-ph]
  * [63] P.A. Zyla et al. Particle data group. Prog. Theor. Exp. Phys. 2020, 083C01 (2020) Sec 87-88
  * [64] J.C. Pati, A. Salam, Phys. Rev. D 10, 275 (1975)
  * [65] R.N. Mohapatra, J.C. Pati, Phys. Rev. D 11, 576 (1975)
  * [66] R.N. Mohapatra, J.C. Pati, Phys. Rev. 2558 (1975)
  * [67] W. Dekens, et al., Los Alamos preprint LA-UR-21-26789 (2021). arXiv:2107.10852 [hep-ph]
  * [68] C.S. Aulakh, K. Benakli, G. Senjanovic, Phys. Rev. Lett. 79, 2188 (1997)
  * [69] C.S. Aulakh, A. Melfo, G. Senjanovic, Phys. Rev. D 57, 4174 (1998)
  * [70] B. Pontecorvo, Z. Eksp, Teor. Fiz. 33, 549 (1957)
  * [71] B. Pontecorvo, JETP 6, 429 (1958)
  * [72] B. Pontecorvo, Z. Eksp, Teor. Fiz. 34, 247 (1957)
  * [73] B. Pontecorvo, JETP 7, 172 (1958)
  * [74] G.G. Ross, Grand Unified Theories (Benjamin/Cummings Pub Co., California, 1984)
  * [75] D. Croon et al., Front. Phys. 7, 76 (2019)
  * [76] D.J. Gross, J.A. Harvey, E. Martinec, R. Rohm, Phys. Rev. Lett. 54, 502 (1985)
  * [77] F. Gursey, P. Ramond, P. Sikivie, Phys. Lett. B 60, 177 (1976)
  * [78] R. Nevzorov, S. Pakvasa, Phys. Lett. B 728, 210 (2014)
  * [79] H. Fritzsch, P. Minkowski, Ann. Phys. 93, 193 (1975)
  * [80] S.F. King, S. Pascoli, J. Turner, Y.-L. Zhou, JHEP 10, 226 (2021)
  * [81] K. Lane, M.V. Ramana, Phys. Rev. D 44, 2678 (1991)
  * [82] A. Belyaev, A. Coupe, N. Evans, M. Scott, Phys. Rev. D 99, 095006 (2019)
  * [83] R. Raitio, Nucl. Phys. B 931, 283 (2018)
  * [84] A.H. Chamseddine, R.L. Arnowitt, Pran Nath. Phys. Rev. Lett. 49, 970 (1982)
  * [85] H. Baer, X. Tata, Weak Scale Supersymmetry (Cambridge University Press, Cambridge, 2010)
  * [86] M. Drees, R.M. Godbole, P. Roy, Theory and Phenomenology of Sparticles (World Scientific, New Jersey, 2005)
  * [87] G. Kane (ed.), Perspectives in Supersymmetry (World Scientific, New Jersey, 1998)
  * [88] G.G. Ross, Nature 352, 21 (1991)
  * [89] M. Kobayashi, T. Maskawa, Prog. Theor. Phys. 49, 652 (1973)
  * [90] M.L. Perl et al., Phys. Rev. Lett. 35, 1489 (1975)
  * [91] S.W. Herb et al., Phys. Rev. Lett. 39, 252 (1977)
  * [92] T.-P. Cheng, L.-F. Li, Gauge Theory of Elementary Particle Physics (Oxford University Press, Oxford, 1984), p.347
  * [93] J.F. Donoghue, E. Golowich, B.R. Holstein, Dynamics of the Standard Model (Cambridge University Press, Cambridge, 1992). (Sec III-3)
  * [94] V.D. Barger, W.-Y. Keung, R.J.N. Phillips, Phys. Rev. D 24, 1328 (1981)
  * [95] R.M. Godbole, S. Pakvasa, D.P. Roy, Phys. Rev. Lett. 50, 1539 (1983)
  * [96] V.D. Barger, A.D. Martin, R.J.N. Phillips, Phys. Lett. B 125, 339 (1983)
  * [97] D.P. Roy, Z. Phys. C 21, 333 (1984)
  * [98] J. Lindfors, D.P. Roy, Z. Phys. C 24, 271 (1984)
  * [99] R.M. Godbole, D.P. Roy, Z. Phys. C 29, 83 (1985)
  * [100] https://twiki.cern.ch/twiki/bin/view/LHCPhysics/LHCTopWGSummaryPlots
  * [101] ARGUS Collaboration, H. Albrecht, et al., Phys. Lett. B 192, 245 (1987)
  * [102] F. Abe et al., Phys. Rev. Lett. 68, 447 (1992)
  * [103] S. Abachi et al., Phys. Rev. Lett. 72, 2138 (1994)
  * [104] A. Blondel, Nuov. Cim. A109, 771 (1996)
  * [105] F. Abe et al., Phys. Rev. Lett. 74, 2626 (1995)
  * [106] S. Abachi et al., Phys. Rev. Lett. 72, 2632 (1994)
  * [107] G. Degrassi et al., JHEP 08, 098 (2012)
  * [108] M. Czakon, D. Heymes, A. Mitov, D. Pagani, I. Tsinikos, JHEP 10, 186 (2017)
  * [109] M. Zaro, PoS in TOP2015, 040 (2016)
  * [110] U. Baur, A. Juste, D. Rainwater, L.H. Orr, Phys. Rev. D 73, 034016 (2006)
  * [111] F. Cuypers, S.D. Rindani, Phys. Lett. B 343, 333 (1995)
  * [112] R.M. Godbole, K. Rao, S.D. Rindani, R.K. Singh, JHEP 1011, 144 (2010)
  * [113] R.M. Godbole, L. Hartgring, I. Niessen, C.D. White, JHEP 01, 011 (2012)
  * [114] F. Boudjema, R.M. Godbole, D. Guadagnoli, K.A. Mohan, Phys. Rev. D 92, 015019 (2015)
  * [115] D. Choudhury, P. Saha, Pramana 77, 1079 (2011)
  * [116] D. Choudhury, P. Saha, JHEP 1208, 144 (2012)
  * [117] D. Bardhan, G. Bhattacharyya, D. Ghosh, M. Patra, S. Raychaudhuri, Phys. Rev. D 94, 015026 (2016)
  * [118] D.E. Kaplan, K. Rehermann, M.D. Schwartz, B. Tweedie, Phys. Rev. Lett. 101, 142001 (2008)
  * [119] S. Chatterjee, R.M. Godbole, T.S. Roy, JHEP 01, 170 (2020)
  * [120] J.F. Donoghue, E. Golowich, B.R. Holstein, Dynamics of the Standard Model (Cambridge University Press, Cambridge, 1992). (Chapter XV)
  * [121] J. Erler, P. Langacker in Proceedings of WEIN 98 . arXiv: 9809352 [hep-ph]
  * [122] V. Ravindran, J. Smith, W.L. van Neerven, Nucl. Phys. B 634, 247 (2002)
  * [123] V. Ravindran, J. Smith, W.L. van Neerven, Nucl. Phys. B 665, 325 (2003)
  * [124] V. Ravindran, J. Smith, W.L. van Neerven, Nucl. Phys. B 704, 332 (2005)
  * [125] V. Ravindran, J. Smith, W.L. van Neerven, Nucl. Phys. B 767, 100 (2007)
  * [126] T. Ahmed, M.K. Mandal, N. Rana, V. Ravindran, Phys. Rev. Lett. 113, 212003 (2014)
  * [127] P. Banerjee, G. Das, P.K. Dhani, V. Ravindran, Phys. Rev. D 97, 054024 (2018)
  * [128] A.H. Ajjath, P. Mukherjee, V. Ravindran, A. Sankar, S. Tiwari, Phys. Rev. D 100, 114016 (2019)
  * [129] A.H. Ajjath, P. Mukherjee, V. Ravindran, A. Sankar, S. Tiwari, Phys. Rev. D 103, 111502 (2021). and arXiv: 2109.12657 [hep-ph]
  * [130] T. Ahmed, M. Mahakhud, P. Mathews, N. Rana, V. Ravindran, JHEP 1408, 075 (2014)
  * [131] T. Ahmed, N. Rana, V. Ravindran, JHEP 10, 139 (2014)
  * [132] T. Ahmed, M.K. Mandal, N. Rana, V. Ravindran, JHEP 02, 131 (2015)
  * [133] A.H. Ajjath, A. Chakraborty, G. Das, P. Mukherjee, V. Ravindran, JHEP 1911, 006 (2019)
  * [134] A.H. Ajjath et al., Phys. Rev. D 100, 114016 (2019)
  * [135] P. Banerjee, S. Borowka, P.K. Dhani, T. Gehrmann, V. Ravindran, JHEP 1811, 130 (2018)
  * [136] A.H. Ajjath et al., Phys. Rev. D 100, 114016 (2019)
  * [137] T. Ahmed, V. Ravindran, A. Sankar, S. Tiwari, arXiv: 2110.11476 [hep-ph]
  * [138] A. Datta, A. Raychaudhuri, S. Raychaudhuri, S. Chakrabarti, Z. Phys. C 72, 449 (1996)
  * [139] K. Cheung, D.K. Ghosh, JHEP 0211, 048 (2002)
  * [140] G. Cacciapaglia, S. Gascon-Shotkin, A. Lesauvage, N. Manglani, K. Sridhar, arXiv: 2108.03005 [hep-ph]
  * [141] B. Mukhopadhyaya, M.A. Samuel, J. Reid, G. Tupper, Phys. Lett. B 247, 607 (1990)
  * [142] P. Bandyopadhyay, S. Dutta, A. Karan, Eur. Phys. J. C 81, 315 (2021)
  * [143] R.M. Godbole, D.P. Roy, Phys. Rev. D 43, 3640 (1991)
  * [144] M. Drees, D.P. Roy, Phys. Lett. B 269, 155 (1991)
  * [145] B.K. Bullock, K. Hagiwara, A.D. Martin, Phys. Rev. Lett. 67, 3055 (1991)
  * [146] D.P. Roy, Phys. Lett. B 277, 183 (1992)
  * [147] D.P. Roy, Phys. Lett. B 283, 403 (1992)
  * [148] V.D. Barger, R.J.N. Phillips, D.P. Roy, Phys. Lett. B 324, 236 (1992)
  * [149] D. Choudhury, D.P. Roy, Phys. Lett. B 322, 368 (1994)
  * [150] S. Raychaudhuri, D.P. Roy, Phys. Rev. D 52, 1556 (1995)
  * [151] S. Raychaudhuri, D.P. Roy, Phys. Rev. D 53, 4902 (1996)
  * [152] D.P. Roy, Phys. Lett. B 459, 607 (1999)
  * [153] M. Guchait, R. Kinnunen, D.P. Roy, Eur. Phys. J. C 52, 665 (2007)
  * [154] E. Ma, D.P. Roy, J. Wudka, Phys. Rev. Lett. 80, 1162 (1998)
  * [155] D.J. Miller, S. Moretti, D.P. Roy, W.J. Stirling, Phys. Rev. D 61, 055011 (2000)
  * [156] S. Moretti, D.P. Roy, Phys. Lett. B 470, 209 (1999)
  * [157] D.K. Ghosh, R.M. Godbole, D.P. Roy, Phys. Lett. B 628, 131 (2005)
  * [158] M. Drees, M. Guchait, D.P. Roy, Phys. Lett. B 471, 39 (1999)
  * [159] A. Datta, A. Djouadi, M. Guchait, Y. Mambrini, Phys. Rev. D 65, 015007 (2002)
  * [160] D.K. Ghosh, W.-S. Hou, T. Modak, Phys. Rev. Lett. 125, 221801 (2020)
  * [161] S. Gabriel, B. Mukhopadhyaya, S. Nandi, S.K. Rai, Phys. Lett. B 669, 180 (2008)
  * [162] G. Bélanger, F. Boudjema, K. Sridhar, Nucl. Phys. B 568, 3 (2000)
  * [163] N. Chakrabarty, D.K. Ghosh, B. Mukhopadhyaya, I. Saha, Phys. Rev. D 92, 015002 (2015)
  * [164] A. Chakraborty, D.K. Ghosh, S. Mondal, S. Poddar, D. Sengupta, Phys. Rev. D 91, 115018 (2015)
  * [165] N. Desai, D.K. Ghosh, B. Mukhopadhyaya, Phys. Rev. D 83, 113004 (2011)
  * [166] A. Djouadi, R.M. Godbole, B. Mellado, K. Mohan, Phys. Lett. B 723, 307 (2013)
  * [167] R.M. Godbole, D.J. Miller, K. Mohan, C.D. White, Phys. Lett. B 730, 275 (2014)
  * [168] S.S. Biswal, R.M. Godbole, B. Mellado, S. Raychaudhuri, Phys. Rev. Lett. 109, 261801 (2012)
  * [169] D. Choudhury, R.M. Godbole, P. Roy, Phys. Lett. B 308, 394 (1993)
  * [170] D. Choudhury, R.M. Godbole, P. Roy, Phys. Lett. B 314, 482 (1993). (erratum)
  * [171] R.M. Godbole, M. Guchait, K. Mazumdar, S. Moretti, D.P. Roy, Phys. Lett. B 571, 184 (2003)
  * [172] D. Ghosh, R.M. Godbole, M. Guchait, K. Mohan, D. Sengupta, Phys. Lett. B 725, 344 (2013)
  * [173] ATLAS Collaboration, M. Aaboud, et al., Phys. Rev. Lett. 122, 23 (2019)
  * [174] N.G. Deshpande, E. Ma, Phys. Rev. D 18, 2574 (1978)
  * [175] M. Gustaffson, in PoS Charged 2010 . pp. 030 (2010), arXiv:1106.1719 [hep-ph]
  * [176] N. Khan, S. Rakshit, Phys. Rev. D 90, 113008 (2014)
  * [177] N. Khan, S. Rakshit, Phys. Rev. D 92, 055006 (2015)
  * [178] S. Jangid, P. Bandyopadhyay, Eur. Phys. J. C 80, 573 (2020)
  * [179] S. Jangid, P. Bandyopadhyay, Eur. Phys. J. C 80, 715 (2020)
  * [180] S. Bhattacharya, P. Ghosh, A.K. Saha, A. Sil, JHEP 03, 090 (2020)
  * [181] A. Ghosh, P. Konar, S. Seth, Phys. Rev. D 105, 115038 (2022)
  * [182] A. Bhardwaj, P. Konar, T. Mondal, S. Sadhukhan, Phys. Rev. D 100, 055040 (2019)
  * [183] A. Datta, N. Ganguly, N. Khan, S. Rakshit, Phys. Rev. D 95, 015017 (2017)
  * [184] S.R. Coleman, E.J. Weinberg, Phys. Rev. D 7, 1888 (1973)
  * [185] J. Elias-Miro et al., Phys. Lett. B 709, 222 (2012)
  * [186] G. Degrassi et al., JHEP 1208, 098 (2012)
  * [187] F. Bezrukov, M.Y. Kalmykov, B.A. Kniehl, M. Shaposhnikov, JHEP 1210, 140 (2012)
  * [188] M. Holthausen, K.S. Lim, M. Lindner, JHEP 1202, 037 (2012)
  * [189] A. Datta, S. Raychaudhuri, Phys. Rev. D 87, 035018 (2013)
  * [190] A. Datta, A. Patra, S. Raychaudhuri, Phys. Rev. D 89(9), 093008 (2014)
  * [191] J. Chakraborty, M. Das, S. Mohanty, Mod. Phys. Lett. A 28, 1350032 (2013)
  * [192] S. Khan, S. Goswami, S. Roy, Phys. Rev. D 89, 073021 (2014)
  * [193] P.S. Bhupal. Dev, D.K. Ghosh, N. Okada, I. Saha, JHEP 1303, 150(2018) (2013)
  * [194] D. Chowdhury, R.M. Godbole, K.A. Mohan, S.K. Vempati, JHEP 02, 110 (2014)
  * [195] D. Chowdhury, R.M. Godbole, K.A. Mohan, S.K. Vempati, JHEP 03, 149 (2018). (erratum)
  * [196] N. Chakrabarti, U.K. Dey, B. Mukhopadhyaya, JHEP 12, 166 (2014)
  * [197] U. Amaldi, W. de Boer, H. Furstenau, Phys. Lett. B 260, 447 (1991)
  * [198] A. Super-Kamiokande Collaboration, Takenaka, et al., Phys. Rev. D 102, 112011 (2020).
  * [199] S.F. King, S. Pascoli, J. Turner, JHEP 10, 225 (2021)
  * [200] J. Ellis, J.L. Evans, N. Nagata, D.V. Nanopoulos, K.A. Olive, Eur. Phys. J. C 81(12), 1109 (2021)
  * [201] S.P. Li, M. Sher, Phys. Lett. B 140, 339 (1984)
  * [202] J.F. Gunion, A. Turski, Phys. Rev. D 39, 2701 (1989)
  * [203] M.S. Berger, Phys. Rev. D 41, 225 (1990)
  * [204] Y. Okada, M. Yamaguchi, T. Yanagida, Prog. Theor. Phys. 85, 1 (1991)
  * [205] Y. Okada, M. Yamaguchi, T. Yanagida, Phys. Lett. B 262, 54 (1991)
  * [206] P.N. Pandita, Phys. Lett. B 318, 338 (1993)
  * [207] J. Ellis, J.S. Hagelin, D.V. Nanopoulos, K. Olive, M. Srednicki, Nucl. Phys. B 238, 453 (1984)
  * [208] ATLAS Collaboration, A. Canepa, Rev. Phys. 4, 100033 (2019)
  * [209] CMS Collaboration, U. Sarkar, arXiv:2105.01629 [hepex]
  * [210] S. Dawson, Online Talk Planned for DIS 2020 (March 2021). http://quark.phy.bnl.gov/˜dawson/dis_2021.pdf
  * [211] N. Wolchover, Quanta Mag. (November 2012)
  * [212] S. Hossenfelder, Lost in Math: How Beauty Leads Physics Astray (Basic Books, New York, 2018)
  * [213] A. Datta, Pramana 89, 54 (2017)
  * [214] B. Mukhopadhyaya, Eur. Phys. J. ST 229, 3143 (2020)
  * [215] L.E. Ibañez, G.G. Ross, Phys. Lett. B 110, 215 (1982)
  * [216] J. Ellis, D.V. Nanopoulos, K. Tamvakis, Phys. Lett. B 121, 123 (1983)
  * [217] L. Alvaréz-Gaumé, J. Polchinski, M. Wise, Nucl. Phys. B 221, 495 (1983)
  * [218] L.J. Hall, V.A. Kostelecky, S. Raby, Nucl. Phys. B 267, 415 (1986)
  * [219] G.F. Giudice, R. Rattazzi, Phys. Rep. 322, 419 (1999) (and references therein).
  * [220] L. Randall, R. Sundrum, Nucl. Phys. B 557, 79 (1999)
  * [221] G.F. Giudice, M.A. Luty, H. Murayama, R. Rattazzi, JHEP 12, 027 (1998)
  * [222] J.D. Wells, in Proc. SUSY 2003 (June, 2003). arXiv: hep-ph/0306127
  * [223] N. Arkani-Hamed, S. Dimopoulos, JHEP 0506, 073 (2005)
  * [224] G.F. Giudice, A. Romanino, Nucl. Phys. B 699, 65 (2004)
  * [225] G.F. Giudice, A. Romanino, Nucl. Phys. B 706, 65 (2005). (erratum)
  * [226] A. Djouadi, J.L. Kneur, G. Moultaka, Comp. Phys. Commun. 176, 426 (2007)
  * [227] C.F. Berger et al., JHEP 02, 023 (2009)
  * [228] M. Carena, J. Lykken, S. Sekmen, N.R. Shah, C.E.M. Wagner, Phys. Rev. D 86, 075025 (2012)
  * [229] E. Reya, D.P. Roy, Phys. Lett. B 141, 442 (1984)
  * [230] E. Reya, D.P. Roy, Phys. Rev. Lett. 53, 881 (1984)
  * [231] E. Reya, D.P. Roy, Phys. Rev. D 32, 645 (1985)
  * [232] E. Reya, D.P. Roy, Phys. Lett. B 166, 223 (1986)
  * [233] E. Reya, D.P. Roy, Z. Phys. C 32, 615 (1986)
  * [234] H.K. Dreiner, M. Guchait, D.P. Roy, Phys. Rev. D 49, 3270 (1994)
  * [235] M. Guchait, D.P. Roy, Phys. Rev. D 52, 133 (1995)
  * [236] M. Drees, R.M. Godbole, M. Guchait, S. Raychaudhuri, D.P. Roy, Phys. Rev. D 54, 5598 (1996)
  * [237] M. Drees, M. Nojiri, D.P. Roy, Y. Yamada, Phys. Rev. D 56, 276 (1997)
  * [238] M. Drees, M. Nojiri, D.P. Roy, Y. Yamada, Phys. Rev. D 64, 039901 (2001). (erratum)
  * [239] U. Chattopadhyay, D. Choudhury, M. Drees, P. Konar, D.P. Roy, Phys. Rev. B 632, 114 (2006)
  * [240] S.F. King, J.P. Roberts, D.P. Roy, JHEP 10, 106 (2007)
  * [241] U. Chattopadhyay, D. Das, D.P. Roy, Phys. Rev. D 79, 095013 (2009)
  * [242] M. Guchait, D.P. Roy, D. Sengupta, Phys. Rev. D 85, 035024 (2012)
  * [243] M. Guchait, D.P. Roy, Phys. Lett. B 535, 243 (2002)
  * [244] S. Moretti, D.P. Roy, Phys. Lett. B B545, 329 (2002)
  * [245] R.M. Godbole, M. Guchait, D.P. Roy, Phys. Lett. B 618, 193 (2005)
  * [246] R.M. Godbole, M. Guchait, D.P. Roy, Phys. Rev. D 79, 095015 (2009)
  * [247] G. Bélanger, F. Boudjema, F. Donato, R.M. Godbole, S. Rosier-Lees, Nucl. Phys. B 581, 3 (2000)
  * [248] G. Bélanger, R.M. Godbole, L. Hartgring, I. Niessen, JHEP 05, 167 (2013)
  * [249] G. Bélanger, R.M Godbole, S. Kraml, S. Kulkarni, arXiv: 1304.2987 [hep-ph]
  * [250] A. Datta, M. Guchait, B. Mukhopadhyaya, Mod. Phys. Lett. A 10, 1011 (1995)
  * [251] A. Datta, S. Chakravarti, M. Guchait, Z. Phys. C 68, 325 (1995)
  * [252] A. Datta, A. Datta, S. Raychaudhuri, Phys. Lett. B 349, 113 (1995)
  * [253] A. Datta, A. Datta, S. Raychaudhuri, Eur. Phys. J. C 1, 375 (1998)
  * [254] A. Datta, M. Guchait, M. Drees, Z. Phys. C 69, 347 (1996)
  * [255] A. Datta, S. Maity, Phys. Rev. D 59, 055019 (1999)
  * [256] N. Bhattacharya, A. Datta, S. Poddar, Phys. Rev. D 82, 035022 (2010)
  * [257] A. Choudhury, A. Datta, JHEP 06, 006 (2012)
  * [258] A. Choudhury, A. Datta, Mod. Phys. Lett. A 27, 1250188 (2012)
  * [259] A. Choudhury, A. Datta, JHEP 09, 119 (2013)
  * [260] A. Chatterjee, A. Choudhury, A. Datta, B. Mukhopadhyaya, JHEP, 154 (2015)
  * [261] A. Datta, M. Guchait, N. Parua, Phys. Lett. B 395, 54 (1997)
  * [262] A. Datta, A. Datta, M.K. Parida, Phys. Lett. B B431, 347 (1998)
  * [263] A. Datta, A. Datta, M. Drees, D.P. Roy, Phys. Rev. D 61, 055003 (2000)
  * [264] A. Datta, S. Maity, A. Datta, J. Phys. G27, 1547 (2001)
  * [265] U. Chattopadhyay, A. Datta, A. Datta, A. Datta, D.P. Roy, Phys. Lett. B 493, 127 (2000)
  * [266] S.P. Das, A. Datta, M. Maity, Phys. Lett. B B596, 293 (2004)
  * [267] U. Chattopadhyay, D. Das, A. Datta, S. Poddar, Phys. Rev. D 76, 055008 (2007)
  * [268] N. Bhattacharya, A. Choudhury, A. Datta, Phys. Rev. D 83, 115025 (2011)
  * [269] N. Bhattacharya, A. Choudhury, A. Datta, Phys. Rev. D 84, 095006 (2011)
  * [270] M. Mitra, U. Chattopadhyay, A. Choudhury, A. Datta, S. Poddar, JHEP 07, 019 (2014)
  * [271] A. Datta, A. Kundu, A. Samanta, Phys. Rev. D 63, 015008 (2001)
  * [272] A. Datta, A. Samanta, Phys. Lett. B 526, 111 (2002)
  * [273] A. Datta, A. Samanta, Phys. Lett. B 607, 144 (2005)
  * [274] A. Datta, P. Konar, B. Mukhopadhyaya, Phys. Rev. D 65, 055008 (2002)
  * [275] A. Datta, P. Konar, B. Mukhopadhyaya, Phys. Rev. Lett. 88, 181802 (2002)
  * [276] D. Choudhury et al., Phys. Rev. Lett. D68, 075007 (2003)
  * [277] P. Konar, B. Mukhopadhyaya, Phys. Rev. Lett. D70, 115011 (2004)
  * [278] T. Han, S. Mukhopadhyaya, B. Mukhopadhyaya, Y. Wu, JHEP 05, 128 (2017)
  * [279] T. Biswas, A. Datta, B. Mukhopadhyaya, arXiv:2107. 05503 [hep-ph]
  * [280] B. Mukhopadhyaya, S. Sengupta, Phys. Rev. D 71, 035004 (2005)
  * [281] B. Mukhopadhyaya, S. Sengupta, Mod. Phys. Lett. A 22, 1391 (2007)
  * [282] S.K. Gupta, P. Konar, B. Mukhopadhyaya, Phys. Lett. B 606, 384 (2005)
  * [283] K.S. Babu, Ts. Enkhbat, B. Mukhopadhyaya, Nucl. Phys. B 720, 47 (2005)
  * [284] S.K. Gupta, B. Mukhopadhyaya, S.K. Rai, Phys. Rev. D 73, 075006 (2006)
  * [285] B. Mukhopadhyaya, S. Roy, F. Vissani, Phys. Lett. B 443, 191 (1998)
  * [286] B. Mukhopadhyaya, P. Roy, R. Srikanth, arXiv: 0510298 [hep-ph]
  * [287] B. Mukhopadhyaya, R. Srikanth, Phys. Rev. D 74, 075001 (2006)
  * [288] B. Mukhopadhyaya, S. Sengupta, R. Srikanth, Phys. Rev. D 76, 075001 (2007)
  * [289] S.K. Gupta, B. Mukhopadhyaya, S.K. Rai, Phys. Rev. D 75, 075007 (2007)
  * [290] T. Han, B. Mukhopadhyaya, Z. Si, K. Wang, Phys. Rev. D 76, 075013 (2007)
  * [291] D. Choudhury, S.K. Gupta, B. Mukhopadhyaya, Phys. Rev. D 78, 015023 (2008)
  * [292] S. Bhatttacharya, A. Datta, B. Mukhopadhyaya, Phys. Rev. D 78, 115018 (2008)
  * [293] S. Biswas, B. Mukhopadhyaya, Phys. Rev. D 79, 11500 (2009)
  * [294] S. Biswas, B. Mukhopadhyaya, Phys. Rev. D 81, 015003 (2010)
  * [295] PS Bhupal. Dev, S. Mondal, B. Mukhopadhyaya, S. Roy, JHEP 09, 110 (2012)
  * [296] S. Banerjee, PS Bhupal. Dev, S. Mondal, B. Mukhopadhyaya, S. Roy, JHEP 10, 221 (2013)
  * [297] A. Chatterjee, N. Chakrabarty, B. Mukhopadhyaya, Phys. Lett. B 754, 14 (2016)
  * [298] S. Banerjee, G. Bélanger, B. Mukhopadhyaya, P.D. Serpico, JHEP 07, 095 (2016)
  * [299] S. Banerjee, G. Bélanger, A. Ghosh, B. Mukhopadhyaya, JHEP 09, 143 (2018)
  * [300] S. Bhatttacharya, A. Datta, B. Mukhopadhyaya, JHEP 10, 080 (2007)
  * [301] S. Bhatttacharya, A. Datta, B. Mukhopadhyaya, Phys. Rev. D 78, 035011 (2008)
  * [302] P. Bandyopadhyay, A. Datta, B. Mukhopadhyaya, Phys. Lett. B 670, 5 (2008)
  * [303] S. Bhatttacharya, U. Chattopadhyay, D. Choudhury, D. Das, B. Mukhopadhyaya, Phys. Rev. D 81, 075009 (2010)
  * [304] S. Mukhopadhyaya, B. Mukhopadhyaya, S. SenGupta, arXiv: 1103.3678 [hep-ph]
  * [305] S. Bhatttacharya, S. Biswas, B. Mukhopadhyaya, M. Nojiri, JHEP 02, 104 (2012)
  * [306] J.A. Aguilar-Saavedra et al., Eur. Phys. J. C 46, 43 (2006)
  * [307] A. Datta, P. Dey, S.K. Gupta, B. Mukhopadhyaya, A. Nyffeler, Phys. Lett. B 659, 308 (2008)
  * [308] P. Bandyopadhyay, A. Datta, A. Datta, B. Mukhopadhyaya, Phys. Rev. D 78, 015017 (2008)
  * [309] N. Desai, B. Mukhopadhyaya, Phys. Rev. D D80, 055019 (2009)
  * [310] S. Biswas, B. Mukhopadhyaya, Phys. Rev. D D79, 115009 (2009)
  * [311] S. Biswas, B. Mukhopadhyaya, Phys. Rev. D D81, 015003 (2010)
  * [312] B. Mukhopadhyaya, S. Mukhopadhyaya, Phys. Rev. D 82, 031501 (2010)
  * [313] K. Ghosh, B. Mukhopadhyaya, S. Mukhopadhyaya, JHEP 10, 096 (2010)
  * [314] N. Desai, B. Mukhopadhyaya, JHEP 05, 057 (2012)
  * [315] J. Dutta, P. Konar, S. Mondal, B. Mukhopadhyaya, S.K. Rai, JHEP 01, 051 (2016)
  * [316] J. Dutta, P. Konar, S. Mondal, B. Mukhopadhyaya, S.K. Rai, JHEP 09, 026 (2017)
  * [317] J. Dutta, B. Mukhopadhyaya, S.K. Rai, Phys. Rev. D 101, 075040 (2020)
  * [318] A. Bhardwaj, J. Dutta, P. Konar, B. Mukhopadhyaya, S.K. Rai, JHEP 10, 083 (2020)
  * [319] K. Ghosh, B. Mukhopadhyaya, U. Sarkar, Phys. Rev. D D84, 015017 (2011)
  * [320] G. Bélanger, S. Biswas, C. Boehm, B. Mukhopadhyaya, JHEP 12, 076 (2012)
  * [321] A. Chaudhuri, N. Khan, B. Mukhopadhyaya, S. Rakshit, Phys. Rev. D D91, 055024 (2015)
  * [322] S. Banerjee, G. Bélanger, B. Mukhopadhyaya, P. Serpico, JHEP 07, 095 (2016)
  * [323] A. Ghosh, T. Mondal, B. Mukhopadhyaya, JHEP 12, 136 (2017)
  * [324] A. Kar, S. Mitra, B. Mukhopadhyaya, T. Roy Choudhury, Phys. Rev. D 99, 021302 (2019)
  * [325] J. Dutta, B. Mukhopadhyaya, S.K. Rai, Phys. Rev. D D101, 075040 (2020)
  * [326] A. Dey, J. Lahiri, B. Mukhopadhyaya, JHEP 09, 004 (2019)
  * [327] A. Dey, J. Lahiri, B. Mukhopadhyaya, JHEP 06, 126 (2020)
  * [328] D. Choudhury, S. Majhi, V. Ravindran, Nucl. Phys. B 660, 343 (2003)
  * [329] D. Choudhury, S. Majhi, V. Ravindran, JHEP 027 (2006)
  * [330] D. Choudhury, A. Datta, K. Huitu, P. Konar, S. Moretti, Phys. Rev. D 68, 075007 (2003)
  * [331] S.S. AbdusSalam, D. Choudhury, Univ. J. Phys. Appl. 2, 155 (2014)
  * [332] D. Choudhury, K. Ghosh, arXiv:1605.00013 [hep-ph]
  * [333] G. Bélanger, F. Boudjema, K. Sridhar, Nucl. Phys. B 576, 660 (2000). (erratum)
  * [334] B.C. Allanach, M. Guchait, K. Sridhar, Phys. Lett. B 586, 373 (2004)
  * [335] S. Chakraborty, S. Niyogi, K. Sridhar, JHEP 07, 105 (2017)
  * [336] M. Bisset, S. Raychaudhuri, P. Roy, arXiv: hep-ph/9602430 [hep-ph]
  * [337] B. Bhattacherjee, A. Kundu, S.K. Rai, S. Raychaudhuri, Phys. Rev. D 81, 035021 (2010)
  * [338] B. Bhattacherjee, A. Dighe, D. Ghosh, S. Raychaudhuri, Phys. Rev. D D83, 094026 (2011)
  * [339] D. Ghosh, M. Guchait, S. Raychaudhuri, D. Sengupta, Phys. Rev. D D86, 055007 (2012)
  * [340] A. Dighe, D. Ghosh, K.M. Patel, S. Raychaudhuri, Int. J. Mod. Phys. A 28 (2013)
  * [341] S. Chakrabarty, A. Chakrabarty, S. Raychaudhuri, Phys. Rev. D 94, 035014 (2016)
  * [342] D. Bardhan, G. Bhattacharyya, D. Ghosh, M. Patra, S. Raychaudhuri, Phys. Rev. D D94, 015026 (2016)
  * [343] M. Guchait, Z. Phys. C 57, 157 (1993)
  * [344] M. Bisset, M. Guchait, S. Moretti, Eur. Phys. J. C 19, 143 (2001)
  * [345] M. Guchait, J. Kalinowski, P. Roy, Eur. Phys. J. C C21, 163 (2001)
  * [346] A. Djouadi, M. Guchait, Y. Mambrini, Phys. Rev. D 64, 095014 (2001)
  * [347] A. Datta, A. Djouadi, M. Guchait, Y. Mambrini, Phys. Rev. D D65, 015007 (2002)
  * [348] A. Datta, A. Djouadi, M. Guchait, F. Moortgat, Nucl. Phys. B 681, 31 (2004)
  * [349] M. Guchait, D. Sengupta, Phys. Rev. D 84, 055010 (2011)
  * [350] D. Ghosh, M. Guchait, D. Sengupta, Eur. Phys. J. C 72, 2141 (2012)
  * [351] R.M. Chatterjee, M. Guchait, D. Sengupta, Phys. Rev. D 86, 075014 (2012)
  * [352] M. Guchait, A.M. Iyer, R. Samanta, Phys. Rev. D D93, 015018 (2016)
  * [353] M. Guchait, J. Kumar, Int. J. Mod. Phys. A 31, 165006 (2016)
  * [354] M. Guchait, J. Kumar, Phys. Rev. D 95, 035036 (2017)
  * [355] M. Guchait, A. Roy, Phys. Rev. D 102, 075023 (2020)
  * [356] M. Guchait, A. Roy, S. Sharma, Phys. Rev. D D104, 055032 (2021)
  * [357] J.L. Diaz-Cruz, D.K. Ghosh, S. Moretti, Phys. Rev. D 68, 014019 (2003)
  * [358] D.K. Ghosh, T. Honkavaara, K. Huitu, S. Roy, Phys. Rev. D D79, 055005 (2009)
  * [359] D.K. Ghosh, T. Honkavaara, K. Huitu, S. Roy, Phys. Rev. D D82, 115006 (2010)
  * [360] D.K. Ghosh, T. Honkavaara, K. Huitu, S. Roy, Phys. Rev. D D84, 075001 (2011)
  * [361] A. Chakrabarty, D.K. Ghosh, D. Ghosh, D. Sengupta, JHEP 10, 122 (2013)
  * [362] B. Bhattacherjee et al., Phys. Rev. D 88, 035011 (2013)
  * [363] M. Frank, D.K. Ghosh, K. Huitu, S.K. Rai, I. Saha, Phys. Rev. D D90, 115021 (2014)
  * [364] A. Chakrabarty, S. Mondal, S. Poddar, D. Sengupta, Phys. Rev. D D91, 115018 (2015)
  * [365] S. Bahrami, M. Frank, D.K. Ghosh, N. Ghosh, I. Saha, Phys. Rev. D D95, 095024 (2017)
  * [366] D.K. Ghosh, K. Huitu, S. Mondal, M. Mitra, Phys. Rev. D D99, 075014 (2019)
  * [367] A. Datta, K. Huitu, Phys. Rev. D 67, 115006 (2003)
  * [368] M. Chaichian, A. Datta, K. Huitu, S. Roy, Z.-H. Yu, Phys. Lett. B 594, 355 (2004)
  * [369] A. Datta, A. Djouadi, J.-L. Kneur, Phys. Lett. B 509, 299 (2001)
  * [370] A. Datta, K.C. Kong, K.T. Matchev, Phys. Rev. D 72, 096006 (2005)
  * [371] A. Datta, K.C. Kong, K.T. Matchev, Phys. Rev. D 72, 119901 (2005). (erratum)
  * [372] A. Datta, G.L. Kane, M. Toharia, arXiv: hep-h/0510204 [hep-ph]
  * [373] A. Datta, S. Niyogi, arXiv: 1111.0200 [hep-ph]
  * [374] J. Beuria, A. Chatterjee, A. Dutta, S.K. Rai, JHEP 09, 073 (2015)
  * [375] S. Chakraborty, A. Dutta, K. Huitu, S. Roy, H.Waltari, Phys. Rev. D 93, 075005 (2016)
  * [376] J. Beuria, A. Chatterjee, A. Dutta, JHEP 08, 004 (2016)
  * [377] U. Chattopadhyay, A. Datta, S. Mukherjee, A.K. Swain, JHEP 08, 113 (2022)
  * [378] W. Abdallah, A. Chatterjee, A. Datta, JHEP 09, 095 (2019)
  * [379] W. Abdallah, A. Datta, S. Roy, JHEP 04, 122 (2021)
  * [380] D.A. Demir, M. Frank, K. Huitu, S.K. Rai, I. Turan, Phys. Rev. D 78, 035013 (2008)
  * [381] M. Arai, K. Huitu, S.K. Rai, K. Rao, JHEP 08, 082 (2010)
  * [382] S. Chakdar, T. Li, S. Nandi, S.K. Rai, Phys. Lett. B 718, 121 (2012)
  * [383] S. Chakdar, T. Li, S. Nandi, S.K. Rai, Phys. Lett. D87, 096002 (2013)
  * [384] M. Frank, B. Fuks, K. Huitu, S.K. Rai, H. Waltari, JHEP 05, 015 (2017)
  * [385] A. Chatterjee, J. Dutta, S.K. Rai, JHEP 06, 042 (2018)
  * [386] A. Chatterjee et al., Phys. Rev. D 99, 035017 (2019)
  * [387] A. Chatterjee et al., Phys. Rev. D 101, 115014 (2020)
  * [388] P. Konar, D. Zeppenfeld, Phys. Lett. B 647, 460 (2007)
  * [389] P. Konar, K.T. Matchev, M. Park, G.K. Sarangi, Phys. Rev. Lett. 105, 22180 (2010)
  * [390] P. Konar, T. Mondal, A.K. Swain, JHEP 04, 024 (2018)
  * [391] P. Bandyopadhyay, B. Bhattacherjee, Phys. Rev. D 84, 035020 (2011)
  * [392] B. Bhattacherjee, B. Feldstein, M. Ibe, S. Matsumoto, T.T. Yanagida, Phys. Rev. D D87, 015028 (2013)
  * [393] B. Bhattacherjee, K. Ghosh, arXiv: 1207.6289 [hep-ph]
  * [394] B. Bhattacherjee, S.K. Mindal, M. Nojiri, JHEP 04, 031 (2013)
  * [395] B. Bhattacherjee, A. Choudhury, K. Ghosh, S. Poddar, Phys. Rev. D 89, 037702 (2014)
  * [396] B. Bhattacherjee, M. Ibe, K. Ichikawa, S. Matsumoto, K. Nishiyama, JHEP 07, 080 (2014)
  * [397] R.K. Barman, B. Bhattacherjee, A. Chakrabarti, A. Choudhury, Phys. Rev. D 94, 075013 (2016)
  * [398] R.K. Barman, B. Bhattacherjee, A. Choudhury, D. Chowdhury, Eur. Phys. J. Plus 134, 150 (2019)
  * [399] B. Bhattacherjee, S. Mukhopadhyay, M. Nojiri, Y. Sakaki, JHEP 01, 044 (2017)
  * [400] R.K. Barman, G. Bélanger, B. Bhattacherjee, R.M. Godbole, G. Mendiretta, Phys. Rev. D 95, 095018 (2017)
  * [401] R.K. Barman, B. Bhattacherjee, A. Chatterjee, A. Choudhury, A. Gupta, JHEP 05, 177 (2019)
  * [402] B. Bhattacherjee, N. Khan, A. Patra, Phys. Rev. D 100, 075010 (2019)
  * [403] A. Adhikary, B. Bhattacherjee, R.M. Godbole, N. Khan, S. Kulkarni, JHEP 04, 284 (2021)
  * [404] R.K. Barman, B. Bhattacherjee, I. Chakraborty, A. Choudhury, N. Khan, Phys. Rev. D 103, 015003 (2021)
  * [405] R.K. Barman, G. Bélanger, B. Bhattacherjee, R.M. Godbole, D. Sengupta, Phys. Rev. D 103, 015029 (2021)
  * [406] G. Bhattacharyya, A. Romanino, Phys. Rev. D 5, 7015 (1997)
  * [407] A. Datta, A. Datta, A. Kundu, B. Mukhopadhyaya, S. Roy, Phys. Lett. B 416, 117 (1998)
  * [408] B. Mukhopadhyaya, S. Roy, Phys. Rev. D 57, 6793 (1998)
  * [409] D.K. Ghosh, Int. J. Mod. Phys. A 15, 2415 (2000)
  * [410] D. Choudhury, D.K. Ghosh, JHEP 05, 016 (2001)
  * [411] P. Byakti, D. Ghosh, Phys. Rev. D 86, 095027 (2012)
  * [412] G. Bhattacharyya, B. Bhattacherjee, T.T. Yanagida, N. Yokozaki, Phys. Lett. B 730, 231 (2014)
  * [413] G. Bhattacharyya, T.T. Yanagida, N. Yokozaki, Phys. Lett. B 749, 82 (2015)
  * [414] G. Bhattacharyya, T.T. Yanagida, N. Yokozaki, Phys. Lett. B 784, 118 (2018)
  * [415] U. Chattopadhyay, D. Das, S. Mukherjee, JHEP 01, 158 (2018)
  * [416] J.M. Gérard, W. Grimus, A. Raychaudhuri, G. Zoupanos, Phys. Lett. B 140, 349 (1984)
  * [417] J.M. Gérard, W. Grimus, A. Masiero, D.V. Nanopoulos, A. Raychaudhuri, Phys. Lett. B 141, 79 (1984)
  * [418] J.M. Gérard, W. Grimus, A. Masiero, D.V. Nanopoulos, A. Raychaudhuri, Nucl. Phys. B 253, 93 (1985)
  * [419] D. Chang, W.F. Chang, W.-Y. Keung, N. Sinha, R. Sinha, Phys. Rev. D 65, 055010 (2002)
  * [420] A. Arhib, D.K. Ghosh, O.C.W. Kong, Phys. Lett. B 537, 217 (2002)
  * [421] T. Gadjosik, R.M. Godbole, S. Kraml, JHEP 09, 051 (2004)
  * [422] D.K. Ghosh, S. Moretti, Eur. Phys. J. C 42, 341 (2005)
  * [423] G. Bélanger, A. Pukhov, R.K. Singh, Contribution to: PhysTeV 2009 (2009)
  * [424] B. Bhattacherjee, A. Chakraborty, D.K. Ghosh, S. Raychaudhuri, Phys. Rev. D 86, 075012 (2012)
  * [425] A. Chakraborty, B. Das, J.L. Diáz-Cruz, D.K. Ghosh, S. Moretti, Phys. Rev. D 90, 055005 (2014)
  * [426] A. Chakraborty, D.K. Ghosh, Pramana 87, 42 (2016)
  * [427] J. Berger, M.W. Cahill-Rowley, D. Ghosh, J.L. Hewett, A. Ismail, Phys. Rev. D 93, 035017 (2016)
  * [428] A. Arbey, J. Ellis, R.M. Godbole, F. Mahmoudi, Eur. Phys. J. C 75, 85 (2015)
  * [429] A. Arbey, J. Ellis, R.M. Godbole, F. Mahmoudi, Nucl. Part. Phys. Proc. 285, 160 (2017)
  * [430] G.R. Farrar, P. Fayet, Phys. Lett. B 76, 575 (1978)
  * [431] S. Weinberg, Phys. Rev. Lett 26, 1303 (1982)
  * [432] N. Sakai, T. Yanagida, Nucl. Phys. B 197, 533 (1982)
  * [433] C.S. Auluck, R. Mohapatra, Phys. Lett. B 119, 136 (1983)
  * [434] L.J. Hall, M. Suzuki, Nucl. Phys. B 231, 419 (1984)
  * [435] J.R. Ellis, C. Jarlskog, G.G. Ross, J.W.F. Vallé, Phys. Lett. B 150, 142 (1985)
  * [436] G.G. Ross, J.W.F. Valle, Phys. Lett. B 151, 375 (1985)
  * [437] S. Dawson, Nucl. Phys. B 261, 297 (1985)
  * [438] R. Barbieri, A. Masiero, Nucl. Phys. B 267, 679 (1986)
  * [439] M. Carena, C.E.M. Wagner, Phys. Lett. B 186, 361 (1987)
  * [440] S. Dimopoulos, L.J. Hall, Phys. Lett. B 207, 210 (1988)
  * [441] S. Dimopoulos, R. Esmailzadeh, L.J. Hall, G. Starkman, Phys. Rev. D 41, 2099 (1990)
  * [442] R.L. Arnowitt, P. Nath, in Proc. WHEPP-1, p.145 Available at https://lib-extopc.kek.jp/preprints/PDF/ 1989/8910/8910405.pdf
  * [443] V.D. Barger, G.F. Giudice, T. Han, Phys. Rev. D 40, 2987 (1989)
  * [444] D.P. Roy, Phys. Lett. B 283, 270 (1992)
  * [445] H. Dreiner, M. Guchait, D.P. Roy, M. Guchait, D.P. Roy, Phys. Rev. D 54, 3276 (1996)
  * [446] H. Dreiner, M. Guchait, D.P. Roy, M. Guchait, D.P. Roy, Phys. Rev. D 57, 4453 (1998)
  * [447] D. Choudhury, D.P. Roy, Phys. Rev. D D54, 6797 (1996)
  * [448] R.M. Godbole, P. Roy, X. Tata, Nucl. Phys. B 401, 67 (1993)
  * [449] B. Brahmachari, P. Roy, Phys. Rev. D 50, R39 (1994)
  * [450] B. Brahmachari, P. Roy, Phys. Rev. D 51, 3974 (1995) (erratum)
  * [451] C.E. Carlson, P. Roy, M. Sher, Phys. Lett. B 357, 99 (1995)
  * [452] D. Choudhury, P. Roy, Phys. Lett. B 378, 153 (1996)
  * [453] G. Bhattacharyya, D. Choudhury, K. Sridhar, Phys. Lett. B 349, 118 (1995)
  * [454] D. Choudhury, Phys. Lett. B 376, 201 (1996)
  * [455] D.K. Ghosh, S. Raychaudhuri, Phys. Lett. B 422, 187 (1998)
  * [456] M. Carena, D. Choudhuri, S. Lola, C. Quigg, Phys. Rev. D 58, 095003 (1998)
  * [457] D. Choudhuri, S. Raychaudhuri, arXiv: hep-ph/9807373 [hep-ph] (1998)
  * [458] G. Bhattacharyya, A. Datta, Phys. Rev. Lett. 83, 2000 (1999)
  * [459] M. Carena, D. Choudhuri, C. Quigg, S. Raychaudhuri, Phys. Rev. D 62, 095010 (2000)
  * [460] D.K. Ghosh, R.M. Godbole, S. Raychaudhuri, arXiv: hep-ph/9904233 [hep-ph] (1999)
  * [461] S. Rakshit, G. Bhattacharyya, A. Raychaudhuri, Phys. Rev. D 59, 091701 (1999)
  * [462] D.K. Ghosh, S. Roy, Phys. Rev. 63, 055005 (2001)
  * [463] R.M. Godbole, S. Pakvasa, S.D. Rindani, X. Tata, Phys. Rev. D 61, 113003 (2000)
  * [464] A. Datta, B. Mukhopadhyaya, F. Vissani, Phys. Lett. B 492, 324 (2000)
  * [465] A. Datta, B. Mukhopadhyaya, Phys. Rev. Lett. 85, 248 (2000)
  * [466] A. Datta, P. Konar, B. Mukhopadhyaya, Phys. Rev. D 63, 095009 (2001)
  * [467] A. Datta, R. Gandhi, B. Mukhopadhyaya, P. Mehta, Phys. Rev. D 64, 015011 (2001)
  * [468] S.P. Das, A. Datta, M. Guchait, Phys. Rev. D 70, 015009 (2004)
  * [469] A. Datta, J.P. Saha, A. Kundu, A. Samanta, Phys. Rev. D 72, 055007 (2005)
  * [470] S.P. Das, A. Datta, S. Poddar, Phys. Rev. D 73, 075014 (2006)
  * [471] A. Datta, S. Poddar, Phys. Rev. D 75, 075013 (2007)
  * [472] A. Datta, S. Poddar, Phys. Rev. D 79, 075021 (2009)
  * [473] S.-L. Chen, D.K. Ghosh, R.N. Mohapatra, Y. Zhang, JHEP 02, 036 (2011)
  * [474] G. Bhattacharyya, P.B. Pal, Phys. Rev. D 82, 055013 (2010)
  * [475] D.K. Ghosh, G. Senjanovi´c, Y. Zhang, Phys. Lett. B 698, 420 (2011)
  * [476] D. Choudhury, M. Datta, M. Maity, JHEP 10, 004 (2011)
  * [477] B. Bhattacherjee, G. Bhattacharyya, S. Raychaudhuri, Phys. Rev. D 84, 075006 (2011)
  * [478] B. Bhattacherjee, J.L. Evans, M. Ibe, S. Matsumoto, T.T. Yanagida, Phys. Rev. D 87, 115002 (2013)
  * [479] B. Bhattacherjee, A. Chakraborty, Phys. Rev. D 89, 115016 (2014)
  * [480] S. Biswas, D. Ghosh, S. Niyogi, JHEP 06, 012 (2014)
  * [481] R. Bose, A. Datta, A. Kundu, S. Poddar, Phys. Rev. D 90, 035007 (2014)
  * [482] P.S. Bhupal Dev, D.K. Ghosh, W. Rodejohann, Phys. Lett. B 762, 116 (2016)
  * [483] D. Bardhan, A. Chakraborty, D. Choudhury, M. Maity, Phys. Rev. D 96, 035024 (2017)
  * [484] G. Bhattacharyya, D. Choudhury, Mod. Phys. Lett. A 10, 1699 (1995)
  * [485] G. Bhattacharyya, D. Choudhury, K. Sridhar, Phys. Lett. B 355, 193 (1995)
  * [486] G. Bhattacharyya, A. Raychaudhuri, Phys. Lett. B 374, 93 (1996)
  * [487] G. Bhattacharyya, A. Raychaudhuri, Phys. Rev. D 57, 3837 (1998)
  * [488] D.K. Ghosh, S. Raychaudhuri, K. Sridhar, Phys. Rev. B 396, 177 (1997)
  * [489] G. Bhattacharyya, P.B. Pal, Phys. Rev. B 439, 81 (1998)
  * [490] G. Bhattacharyya, P.B. Pal, Phys. Rev. D 59, 097701 (1999)
  * [491] G. Bhattacharyya, Phys. Rev. D 57, 3944 (1998)
  * [492] G. Bhattacharyya, A. Datta, Phys. Rev. Lett. 83, 2300 (1999)
  * [493] G. Bhattacharyya, A. Datta, A. Kundu, Phys. Lett. B 514, 47 (2001)
  * [494] G. Bhattacharyya, A. Datta, A. Kundu, J. Phys. G30, 1947 (2004)
  * [495] D. Chakraverty, D. Choudhury, Phys. Rev. D 63, 075009 (2001)
  * [496] D. Chakraverty, D. Choudhury, Phys. Rev. D 63, 112002 (2001)
  * [497] D. Choudhury, R.M. Godbole, G. Polesello, JHEP 08, 004 (2002)
  * [498] D.K. Ghosh, X.-G. He, B.H.J. McKellar, JHEP 07, 067 (2002)
  * [499] R.M. Godbole, S.K. Rai, S.D. Rindani, Phys. Lett. B 678, 395 (2009)
  * [500] P.H. Chankowski, D. Choudhury, S. Pokorski, Phys. Lett. B 389, 677 (1996)
  * [501] D.K. Ghosh, R.M. Godbole, S. Raychaudhuri, Z. Phys. C 75, 357 (1997)
  * [502] D. Choudhury, S. Raychaudhuri, Phys. Lett. B 401, 54 (1997)
  * [503] A. Datta, E. Paschos, L.P. Singh, Phys. Lett. B 548, 146 (2002)
  * [504] D.K. Ghosh, M. Maity, S. Roy, Phys. Rev. D 84, 035022 (2011)
  * [505] D. Das, C. Hati, G. Kumar, N. Mahajan, Phys. Rev. D 96, 095033 (2017)
  * [506] S. Roy, B. Mukhopadhyaya, Phys. Rev. D 55, 7020 (1997)
  * [507] B. Mukhopadhyaya, S. Roy, Phys. Rev. D 60, 115012 (1999)
  * [508] A. Datta, B. Mukhopadhyaya, S. Roy, Phys. Rev. D 61, 055006 (2000)
  * [509] A. Abada, G. Bhattacharyya, Phys. Rev. D 66, 071701 (2002)
  * [510] A.S. Joshipura, R.D. Vaidya, S.K. Vempati, Nucl. Phys. B 639, 290 (2002)
  * [511] R.S. Hundi, Phys. Rev. D 83, 115019 (2011)
  * [512] R.S. Hundi, Nucl. Phys. D 87, 115005 (2013)
  * [513] R. Mohanta, Phys. Rev. D 89, 014020 (2014)
  * [514] S. Sahoo, R. Mohanta, Phys. Rev. D 91, 094019 (2015)
  * [515] T. Mondal, S. Mitra, S. Seth, JHEP 07, 028 (2015)
  * [516] T. Mondal, S. Mitra, S. Seth, Phys. Rev. D 93, 035018 (2016)
  * [517] D. Das, C. Hati, G. Kumar, N. Mahajan, Phys. Rev. D 94, 055034 (2016)
  * [518] U. Dey, S. Mohanty, G. Tomar, arXiv: 1606.07903 [hep-ph]
  * [519] S. Sahoo, R. Mohanta, A.K. Giri, Phys. Rev. D 95, 035027 (2017)
  * [520] B. Chauhan, B. Kindra, A. Narang, Phys. Rev. D 97, 095007 (2018)
  * [521] U.K. Dey, D. Kar, M. Mitra, M. Spannowsky, A.C. Vincent, Phys. Rev. D 98, 035014 (2018)
  * [522] S. Mandal, M. Mitra, N. Sinha, Phys. Rev. D 98, 095004 (2018)
  * [523] K. Chandak, T. Mandal, S. Mitra, Phys. Rev. D 100, 075019 (2019)
  * [524] R. Padhan, S. Mandal, M. Mitra, N. Sinha, Phys. Rev. D 101, 075037 (2020)
  * [525] A. Bhaskar, D. Das, B. De, S. Mitra, Phys. Rev. D 102, 035002 (2020)
  * [526] B. Pascual-Dias, P. Saha, D. London, JHEP 07, 144 (2020)
  * [527] P. Bandyopadhyay, S. Dutta, A. Karan, Eur. Phys. J. C 80, 573 (2020)
  * [528] P. Bandyopadhyay, S. Dutta, M. Jakkapu, A. Karan, Nucl. Phys. B 971, 115524 (2021)
  * [529] P. Bandyopadhyay, A. Karan, R. Mandal, arXiv:2108.06506 [hep-ph]
  * [530] P. Bandyopadhyay, S. Jangid, A. Karan, arXiv:2111.03872 [hep-ph]
  * [531] H. More, On the Immortality of the Soul (1659). Republished by Springer (1987)
  * [532] C.H. Hinton, in Scientific Romances (1884). Republished by Dover Books (1980)
  * [533] H. Minkowski, Phys. Z. 10, 75 (1909)
  * [534] G. Nordström, Phys. Z. 15, 504 (1914)
  * [535] Th. Kaluza (1921), Reprinted in Int. J. Mod. Phys. D 27, 1870001 (2018)
  * [536] O. Klein, Nature 118, 516 (1926)
  * [537] O. Klein, Z. Phys. 37, 895 (1926)
  * [538] A. Neveu, J. Schwartz, Nucl. Phys. B 31, 86 (1971)
  * [539] T. Yoneya, Prog. Theor. Phys. 51, 1907 (1974)
  * [540] J. Scherk, J. Schwarz, Nucl. Phys. B 81, 118 (1974)
  * [541] K. Akama, Lect. Notes Phys. 176, 267 (1982)
  * [542] V.A. Rubakov, M.E. Shaposhnikov, Phys. Lett. B 125, 136 (1983)
  * [543] M. Visser, Phys. Lett. B 159, 22 (1985)
  * [544] I. Antoniadis, Phys. Lett. B 246, 377 (1990)
  * [545] I. Antoniadis, C.Muñoz, M. Quiros, Nucl. Phys. B 397, 515 (1993)
  * [546] I. Antoniadis, K. Benakli, Phys. Lett. B 326, 69 (1994)
  * [547] I. Antoniadis, K. Benakli, M. Quiros, Phys. Lett. B 331, 313 (1994)
  * [548] I. Antoniadis, S. Dimopoulos, Nucl. Phys. B 516, 70 (1998)
  * [549] N. Arkani-Hamed, S. Dimopoulos, G. Dvali, Phys. Lett. B 429, 263 (1998)
  * [550] I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos, G. Dvali, Phys. Lett. B 436, 257 (1998)
  * [551] J. Dai, R.G. Leigh, J. Polchinski, Mod. Phys. Lett. A 04, 2073 (1989)
  * [552] P. Horava, Phys. Lett. B 231, 251 (1989)
  * [553] G.F. Giudice, R. Rattazzi, J.D. Wells, Nucl. Phys. B 544, 3 (1999)
  * [554] T. Han, J.D. Lykken, R.-J. Zhang, Phys. Rev. D 59, 105006 (1999)
  * [555] E.A. Mirabelli, M. Perelstein, M.E. Peskin, Phys. Rev. Lett. 82, 2236 (1999)
  * [556] P. Mathews, S. Raychaudhuri, K. Sridhar, Phys. Lett. B 450, 343 (1999)
  * [557] P. Mathews, S. Raychaudhuri, K. Sridhar, Phys. Lett. B 455, 115 (1999)
  * [558] P. Mathews, S. Raychaudhuri, K. Sridhar, JHEP 07, 008 (2000)
  * [559] L. Randall, R. Sundrum, Phys. Rev. Lett. 83, 3370 (1999)
  * [560] T. Appelquist, H.-C. Cheng, B. Dobrescu, Phys. Rev. D 64, 035002 (2001)
  * [561] S. Lola, P. Mathews, S. Raychaudhuri, K. Sridhar, arXiv: 0010010 [hep-ph] (2000)