Muon and Electron $(g-2)$ Anomalies with Non-Holomorphic Interactions in MSSM

Md. Isha Ali 1, Manimala Chakraborti2, Utpal Chattopadhyay1 and Samadrita
Mukherjee3 111emails<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS>
1School of Physical Sciences, Indian Association for the Cultivation of
Science, 2A & 2B Raja S.C. Mullick Road, Jadavpur, Kolkata 700 032, India

2Astrocent, Nicolaus Copernicus Astronomical Center of the Polish Academy of
Sciences, ul.Rektorska 4, 00-614 Warsaw, Poland

3Department of Theoretical Physics, Tata Institute of Fundamental Research, 1,
Homi Bhabha Road, Colaba, Mumbai 400005, India

###### Abstract

The recent Fermilab muon $g-2$ result and the same for electron due to fine-
structure constant measurement through ${}^{133}{\rm Cs}$ matter-wave
interferometry are probed in relation to MSSM with non-holomorphic (NH)
trilinear soft SUSY breaking terms, referred to as NHSSM. Supersymmetric
contributions to charged lepton $(g-2)_{l}$ can be enhanced via the new
trilinear terms involving a wrong Higgs coupling with left and right-handed
scalars. Bino-slepton loop is used to enhance the SUSY contribution to $g-2$
where wino mass stays at 1.5 TeV and the left and right slepton mass
parameters for the first two generations are considered to be the same. Unlike
many MSSM-based analyses completed before, the model does not require a light
electroweakino, or light sleptons, or unequal left and right slepton masses,
or a very large higgsino mass parameter. In absence of popular UV complete
models, we treat the NH terms at par with MSSM soft terms, in a model
independent framework of Minimal Effective Supersymmetry. The first part of
the analysis involves the study of $(g-2)_{\mu}$ constraint along with the
limits from Higgs mass, B-physics, collider data, direct detection of dark
matter (DM), while focusing on a higgsino DM which is underabundant in nature.
We then impose the constraint from electron $g-2$ where a large Yukawa
threshold correction (an outcome of NHSSM) and opposite signs of trilinear NH
coefficients associated with $\mu$ and $e$ fields are used to satisfy the dual
limits of $\Delta{a_{\mu}}$ and $\Delta{a_{e}}$ (where the latter comes with
negative sign). Varying Yukawa threshold corrections further provide the
necessary flavor-dependent enhancement of $\Delta{a_{e}}/m_{e}^{2}$ compared
to that of $\Delta{a_{\mu}}/m_{\mu}^{2}$. A larger Yukawa threshold correction
through $A^{\prime}_{e}$ for $y_{e}$ also takes away the direct
proportionality of $a_{e}$ with respect to $\tan\beta$. With a finite
intercept, $a_{e}$ becomes only an increasing function of $\tan\beta$. We
identified the available parameter space in the two cases while also
satisfying the ATLAS data from slepton pair production searches in the plane
of slepton mass parameter and the mass of the lightest neutralino.

##  1 Introduction

The discovery of the Higgs boson[1, 2] at the Large Hadron Collider (LHC)
almost a decade ago gave the Standard Model (SM) of particle physics[3] a
strong foundation. However, SM has its limitations both in the theoretical as
well as in the observational sides. The gauge hierarchy problem, matter-
antimatter asymmetry, no candidate for dark matter[4, 5] are to name a few in
this regard. This demands the existence of a Beyond the Standard Model (BSM)
physics. Low energy Supersymmetry (SUSY)[6, 7, 8, 9, 10, 11, 12, 13, 14] is
especially attractive in this context since it can address the gauge hierarchy
problem associated with the Standard Model (SM) and also it is able to provide
with particle dark matter candidates. Additionally, we must not forget that
the Higgs boson is found to have a mass of 125 GeV, which is well below the
predicted upper limit for an SM-like Higgs particle of the Minimal
Supersymmetric Standard Model (MSSM)[9, 10, 11, 12, 13, 14]. Thus, over the
past decades, SUSY, with its strong theoretical appeal and its ability to
influence a variety of observables of phenomenological interest, continues to
remain as the most attractive candidate for a BSM physics.

Undoubtedly, a BSM physics demands nothing less than direct observations of
new particles at the LHC. This will then lead us to the possible new
symmetries and interactions present in nature. However, even after a decade of
running of the LHC, we are yet to see a cherished new particle. We may as well
need to accept the fact that BSM particles could perhaps be staying quite far
from our experimental reach. Keeping hope for a collider discovery, we must at
the same time continue to look for possible indirect signatures of SUSY. This
may come from flavor physics, electroweak physics precision tests and dark
matter. Concerning the above, we remember that the anomalous magnetic moment
of muon, $a_{\mu}=\frac{1}{2}(g-2)_{\mu}$, stands out prominently over the
past two decades showing some degree of disagreement (over 2 to 3$\sigma$) of
the experimental result as obtained in Brookhaven[15] with that of SM
evaluations performed at different times. The hadronic vacuum polarization
part of the SM result has a large uncertainty, particularly the lowest order
part of the same that requires analysis in the non-perturbative regime. The
non-perturbative aspect may require input from effective field theory like
chiral perturbation theory, hadronic models, dispersion relations together
with experimental data like $e^{+}e^{-}\rightarrow{\rm hadrons}$, and Lattice
Quantum Chromodynamics (LQCD). A comprehensive analysis explaining the break-
up of different contributions to the SM result of $a_{\mu}$ may be seen in
Ref.[16]222A chosen set of references among various important past
contributions to $(g-2)_{\mu}$ evaluation may be found in Ref.[17]. While the
recent Fermilab $a_{\mu}$ data[18, 19] is consistent with the same from
Brookhaven, the difference $\Delta a_{\mu}$ has grown larger. The combined
data from Fermilab and Brookhaven show a $4.2\sigma$ level of discrepancy[19]
as given below.

$\Delta a_{\mu}=a_{\mu}^{\rm exp}-a_{\mu}^{\rm SM}=(251\pm 59)\times
10^{-11}.$ (1)

Interestingly, $\Delta a_{\mu}$ can be ascribed to $a_{\mu}^{\rm SUSY}$, the
SUSY contributions to $a_{\mu}$ which in turn will help us to constrain the
SUSY model parameter space.

With further results to come from the Fermilab in the near future and the data
from upcoming experiment JPARC[20], muon $g-2$ can shed light on various BSM
physics models. In this context, we must point out the recent Lattice
result[21] for the hadronic vacuum polarization. This has effectively shifted
$a^{\rm SM}_{\mu}$ to move toward $a^{\rm exp}_{\mu}$ rather closely causing
tension between the dispersive and LQCD modes of evaluations of the hadronic
uncertainty amount within $a^{\rm SM}_{\mu}$. We would also like to point out
that an agreement between $a_{\mu}^{\rm exp}$ and $a_{\mu}^{\rm SM}$ may
invite issues with global electroweak fits to electroweak precision
observables. This is because the existing deviation of the above two $a_{\mu}$
values is related to precision electroweak predictions via the common
dependence on hadronic vacuum polarization effects[22, 23, 24]. In any case,
such important issues will be transparent in future, but at this point we will
use Eq.1 for $a_{\mu}^{\rm SUSY}$.

On the top of $(g-2)_{\mu}$, we would also include the existing deviation for
$(g-2)_{e}$, the anomalous magnetic moment for electron. A smaller but not
insignificant discrepancy exists for the electron $g-2$ anomaly arising out of
the measurement of the fine-structure constant that used ${}^{133}{\rm Cs}$
matter-wave interferometry. An approximately $2.5\sigma$ level of discrepancy
is given below[25],

$\Delta a_{e}=a_{e}^{\rm exp}-a_{e}^{\rm SM}=(-8.8\pm 3.6)\times 10^{-13}.$
(2)

Unlike the above cases of muon and electron anomalies of Eqs.1 and 2 where
they come with opposite signs, a newer measurement of fine-structure constant
based on ${}^{87}{\rm Rb}$[26] shows a 1.6$\sigma$ deviation in the positive
side.

$\Delta{a_{e}}^{Rb}=a_{e}^{\rm exp}-a_{e}^{\rm SM}=(4.8\pm 3.0)\times
10^{-13}.$ (3)

In many new physics models with flavor universality, one finds
$\frac{m_{\mu}^{2}}{m_{e}^{2}}\frac{\Delta a_{e}}{\Delta a_{\mu}}\simeq 1$.
This is also true in SUSY333except in cases involving an appreciably large
Yukawa threshold corrections as we will see.. In contrast to the above, the
measurement values referred to in Eqs.1 and 2 lead to an appreciably larger
negative value for the above quantity, namely:

$R_{e,\mu}=\frac{m_{\mu}^{2}}{m_{e}^{2}}\frac{\Delta a_{e}}{\Delta
a_{\mu}}\simeq-15.$ (4)

Thus, using the two constraints simultaneously leads to a rather difficult
situation. We note that the right hand side of Eq.4 is only the central value.
Appropriate error estimates of the two magnetic moments may be used for
obtaining the combined uncertainty values. On the other hand, use of Eqs.1 and
3 lead to $R_{e,\mu}\simeq 8$. Clearly, the later case of having simultaneous
positive values for the two deviations with also a smaller $R_{e,\mu}$ is
easier to accommodate in SUSY analyses. In the absence of a resolution of the
$\Delta a_{e}$ puzzle, we choose to consider the rather difficult
${}^{133}{\rm Cs}$-based value of Eq.2.

This will analyze ${(g-2)}_{\mu,e}$ in the framework of Non-standard soft SUSY
breaking terms[27, 28] contrast it with other SUSY-based analyses that also
used the ${}^{133}{\rm Cs}$-based result of Eq.2. We will also show the result
of using the ${}^{87}{\rm Rb}$ data briefly just for the sake of completeness.

Besides ${(g-2)}_{\mu,e}$ we will also include dark matter constraints in our
analysis. The plan of our work is given below. In Section 2 we will describe
the Non-holomorphic MSSM (NHSSM) model and its signature on the SUSY spectra.
We will also discuss the constraints arising from avoiding charge and color
breaking (CCB) minima. Apart from the above, we will also mention the
difference of status between the non-holomorphic soft parameters with the ones
of regular MSSM soft terms in the context of ultraviolet (UV) completion. In
Section 3 we will discuss the SUSY contributions to the magnetic moment of
charged leptons. The above will also emphasize the role of Yukawa threshold
corrections in MSSM that may be important for satisfying Eq.2. We will then
discuss the effect of non-holomorphic trilinear interactions on leptonic
magnetic moments ${(g-2)}_{l}$ and how the trilinear NH terms may provide the
necessary threshold effects appropriate for Eq.2. We will particularly outline
the parameter zone that would be consistent with a higgsino dark matter as a
multi-component dark matter element with relic density obeying only the upper
limit from the PLANCK data. Then we will discuss the combined case of obeying
${(g-2)}_{\mu}$ and ${(g-2)}_{e}$ in MSSM and see how NHSSM effects can
generate appropriate threshold corrections to $y_{e}$ and also to some lesser
extent to $y_{\mu}$. We will discuss the essential points of past MSSM-based
analyses in contrast to the features of NHSSM. We will see that there is no
need to assume a flavor unfriendly choice for slepton masses in NHSSM, neither
we do we need to consider any superheavy higgsino state to generate Yukawa
threshold corrections. In Section 4 we will present the results in two
separate parts namely ${(g-2)}_{\mu}$ with DM and inclusion of ${(g-2)}_{e}$
for the combined analysis. We will also use LHC constraints from slepton pair-
production. Finally, we will conclude in Section 6.

##  2 MSSM With Non-Holomorphic Soft Terms

###  2.1 MSSM: Superpotential and Soft terms

The MSSM Superpotential is given by[9],

$\displaystyle\mathcal{W}=\mu
H_{D}.H_{U}-Y_{ij}^{e}H_{D}.L_{i}{\bar{E}}_{j}-Y_{ij}^{d}H_{D}.Q_{i}{\bar{D}}_{j}-Y_{ij}^{u}Q_{i}.H_{U}{\bar{U}}_{j}.$
(5)

Here, for two doublet chiral superfields $A$ and $B$, one has
$A.B=\epsilon_{\alpha\beta}A^{\alpha}B^{\beta}$, where
$\epsilon_{\alpha\beta}$ is an antisymmetric (Levi-Civita) tensor in
2-dimension. $Y_{ij}^{e}$, $Y_{ij}^{u}$ and $Y_{ij}^{d}$ are lepton, up and
down type of Yukawa matrices respectively. $H_{D}$ and $H_{U}$ with
hypercharges -1 and 1 respectively refer to down and up type of doublets of
Higgs chiral superfields that contain both the Higgs scalars and and their
fermionic partners higgsinos. $L_{i},Q_{i}$ and $E_{i},U_{i}$ are left handed
doublet and right handed singlet chiral superfields of applicable fermions and
their scalar superpartners.

The MSSM soft terms including the non-holomorphic scalar mass terms and the
holomorphic trilinear coupling terms are as given below[9].

$\displaystyle-{\mathcal{L}}_{soft}$ $\displaystyle=$
$\displaystyle[{\tilde{q}}_{iL}.h_{u}{(Y^{u}A_{u})}_{ij}{\tilde{u}}^{*}_{jR}+h_{d}.{\tilde{q}}_{iL}{(Y^{d}A_{d})}_{ij}{\tilde{d}}^{*}_{jR}+h_{d}.{\tilde{l}}_{iL}{(Y^{e}A_{e})}_{ij}{\tilde{e}}^{*}_{jR}+h.c.]$
(6) $\displaystyle+$ $\displaystyle\left(B\mu
h_{d}.h_{u}+h.c.\right)+m_{d}^{2}{|h_{d}|}^{2}+m_{u}^{2}{|h_{u}|}^{2}$
$\displaystyle+$
$\displaystyle{\tilde{q}}_{iL}^{*}{({M}_{\tilde{q}}^{2})}_{ij}{\tilde{q}}_{jL}+{\tilde{u}}_{iR}^{*}{({M}_{\tilde{u}}^{2})}_{ij}{\tilde{u}}_{jR}+{\tilde{d}}_{iR}^{*}{({M}_{\tilde{d}}^{2})}_{ij}{\tilde{d}}_{jR}+{\tilde{l}}_{iL}^{*}{({M}_{\tilde{l}}^{2})}_{ij}{\tilde{l}}_{jL}++{\tilde{e}}_{iR}^{*}{({M}_{\tilde{e}}^{2})}_{ij}{\tilde{e}}_{jR}$
$\displaystyle+$ $\displaystyle{\rm gaugino~{}mass~{}terms.}$

Here, $h_{d}$ and $h_{u}$ are doublets of Higgs scalar fields. The other terms
contain mass terms and trilinear terms involving the scalar parts of the
associated matter superfields. Finally, there are Majorana mass terms
involving the gauginos. With $v_{u},v_{d}$ as the vacuum expectation values
(vevs) of the neutral components of Higgs scalar fields $h_{u}$ and $h_{d}$,
one has $\tan\beta=v_{u}/v_{d}$ and
$M_{Z}^{2}=\frac{1}{4}(g_{Y}^{2}+g_{2}^{2})(v_{u}^{2}+v_{d}^{2})$, leading to
$\sqrt{(v_{u}^{2}+v_{d}^{2})}\simeq 246$ GeV. The Yukawa couplings and masses
are related via the vevs as $y_{e}=\frac{m_{e}}{(v_{d}/\sqrt{2})}$,
$y_{u}=\frac{m_{u}}{(v_{u}/\sqrt{2})}$ etc. The above $y_{e}$ relates to
$Y_{ij}^{e}$ the leptonic Yukawa matrix as
$Y_{11}^{e}=y_{e},Y_{22}^{e}=y_{\mu},Y_{33}^{e}=y_{\tau}$444The Yukawa
matrices are assumed to be diagonal and the trilinear soft terms considered
for leptons given at the low scale are also diagonal corresponding to the
Yukawa matrices.. Similar notation holds good for the quark Yukawa matrices.

###  2.2 Non-holomorphic soft terms

Ref.[27] enumerated the MSSM soft terms shown as $-{\mathcal{L}}_{soft}$ in
Eq.6. It also listed a few additional SUSY breaking interactions in a general
sense that would be regarded as hard SUSY breaking terms in presence of a
gauge singlet scalar field[29]. On the other hand, in absence of any such
singlets as in MSSM, such terms grouped within
$-{\mathcal{L}^{\prime}}_{soft}$ shown as below, are no longer of hard SUSY
breaking type[29, 28, 30] and these are labelled as non-holomorphic soft SUSY
breaking terms or the so-called “C-terms” in SUSY texts[10]. In MSSM these are
regarded as soft SUSY breaking terms.

$\displaystyle-{\mathcal{L}^{\prime}}_{soft}=h_{d}^{c}.{\tilde{q}}_{iL}{(Y^{u}A_{u}^{\prime})}_{ij}{\tilde{u}}^{*}_{jR}+{\tilde{q}}_{iL}.h_{u}^{c}{(Y^{d}A_{d}^{\prime})}_{ij}{\tilde{d}}^{*}_{jR}+{\tilde{l}}_{iL}.h_{u}^{c}{({Y^{e}A}_{e}^{\prime})}_{ij}{\tilde{e}}^{*}_{jR}+\mu^{\prime}{\tilde{h}}_{u}.{\tilde{h}}_{d}+h.c.$
(7)

Here, instead of the Higgs scalar doublets one has their conjugates
$h_{d}^{c}$ and $h_{u}^{c}$. With appropriate hypercharges, $h_{d}^{c}$
couples with the up-type of squarks and $h_{u}^{c}$ goes with the down type of
squarks and sleptons. This is why the above is often referred as a scenario
with wrong Higgs coupling.

Ref.[28] analyzed such terms (along with the MSSM soft terms) in a model
independent way. Instead of a model based analysis, in an agnostic point of
view the authors named the framework as “Minimal Effective Supersymmetry”
where the new soft parameters considered were to be treated at par with the
ones of Eq.6 and these were left to be determined from low energy data only.

In the quest of a model, Ref.[30] considered a hidden sector based F-type
supesymmetry breaking scenario including two chiral superfields. The author
obtained terms of $-{\mathcal{L}^{\prime}}_{soft}$ terms to be of the order
$\frac{{|F|}^{2}}{M^{3}}\sim\frac{M_{W}^{2}}{M}$ indicating suppression by the
scale of mediation $M$ of SUSY breaking. Arising from the same analysis, there
is no such supression for the MSSM soft terms of Eq.6. Clearly, according to
the model of Ref.[30] these are highly suppressed terms if the scale of
mediation is close to the grand unification theory (GUT) scale or the Planck
scale. However, Ref.[30] also pointed out that the above does not make the
terms of $-{\mathcal{L}^{\prime}}_{soft}$ irrelevant for issues starting from
a possible incorrectness in using a simplistic form of spontaneous SUSY
breaking to involvement of multiple high scales etc.

It is known that building models for nonstandard soft SUSY breaking terms or
the “C-terms” is difficult[10]. We cite a few available analyses here.
Refs.[31, 32, 33] may be seen for analyses with generalized supersoft SUSY
breaking that have relations to non-holomorphic soft terms. Generating non-
holomorphic terms based on gauge mediated SUSY breaking may be seen in
Ref.[34]. Here, the authors discussed the possibility of finding gauge
invariant supersymmetric direct Yukawa couplings between the Higgs and the
messenger fields. One-loop corrections involving messenger fields in the loop
may lead to the wrong-Higgs gaugino operators[34] that may become important in
the low energy theory below the scale of SUSY breaking555A gauge mediated susy
breaking (GMSB) inspired analysis was studied in Ref.[35], however here the
bilinear higgsino mass soft term was assumed to have an unknown origin..
Ref.[36] relates to non-holomorphic terms arising out of D-brane instantons
stretched between SUSY breaking and visible sectors.

It is clear that the NH terms of $-{\mathcal{L}^{\prime}}_{soft}$ are not
friendly to supergravity[37, 30] types of scenarios or they may be difficult
to be found from a popular UV complete theory. Similar to Ref.[10, 30] we are
also of the opinion that the NH terms are hardly irrelevant in spite of their
difficulty in model building and we may consider the approach of “Minimal
Effective Supersymmetry” of Ref.[28]. Based on the inputs given at a high or a
low scale we classify below the past phenomenological analyses with non-
standard soft terms that considered no explicit model or in other words were
consistent with the “Effective” approach of Ref.[28]. Analyses of Refs.[38,
39, 40, 41, 42, 43, 44, 45, 46, 47] used renormalization group evolutions
within a Constrained MSSM (CMSSM[9]) like setup. Here the NH parameters were
considered to be of unknown origin and were at par with other CMSSM mass and
trilinear parameters. Similarly, there are works with phenomenological MSSM
(pMSSM)[48] like inputs[49, 50, 51, 52] where the NH parameters given at a low
scale were treated at par with the MSSM soft SUSY breaking parameters. Thus as
with previous analyses, we consider the new parameters to be of unknown origin
given at a low scale. The tree-level Higgs potential is unaffected, but this
is not so for the charge and color breaking terms of the scalar potential[50].
The presence of the higgsino mass soft terms with coupling $\mu^{\prime}$ may
cause isolation of a fine-tuning from the Higgsino mass $\mu$ since at tree
level higgsino mass would have components from the superpotential (Eq.5) as
well as from the soft term of (Eq.7)[45, 46, 49]. The mass matrices for the
scalars get modified in the off-diagonal component involving L-R mixing. For
example, a slepton mass matrix may be written as,

$\displaystyle M_{\tilde{e}}^{2}=$
$\displaystyle\left(\begin{matrix}M_{\tilde{l_{L}}}^{2}+M_{Z}^{2}(T_{3L}^{\tilde{e}}-Q_{e}\sin^{2}\theta_{W})\cos
2\beta+m_{e}^{2}&\hskip 2.84526pt-
m_{e}({A}_{e}-(\mu+{A}_{e}^{\prime})\tan\beta)\\\
-m_{e}({A}_{e}-(\mu+{A}_{e}^{\prime})\tan\beta)&M_{\tilde{l_{R}}}^{2}+M_{Z}^{2}Q_{e}\sin^{2}\theta_{W}\cos
2\beta+m_{e}^{2}\end{matrix}\right).\hskip 17.07164pt$ (8)

We note that going from MSSM to NHSSM, $\mu\tan\beta$ gets replaced by
$(\mu+{A}_{l}^{\prime})\tan\beta$ in the off-diagonal entries. In the
electroweakino sector the higgsino mass entries are altered from $\mu$ to
$\mu+\mu^{\prime}$ leading to the following neutralino and chargino mass
matrices in NHSSM.

$\displaystyle M_{\widetilde{\chi^{0}}}=$
$\displaystyle\left(\begin{matrix}M_{1}&0&-M_{Z}\cos\beta\sin\theta_{W}&M_{Z}\sin\beta\sin\theta_{W}\\\
0&M_{2}&M_{Z}\cos\beta\cos\theta_{W}&-M_{Z}\sin\beta\cos\theta_{W}\\\
-M_{Z}\cos\beta\sin\theta_{W}&M_{Z}\cos\beta\cos\theta_{W}&0&-(\mu+\mu^{\prime})\\\
M_{Z}\sin\beta\sin\theta_{W}&-M_{Z}\sin\beta\cos\theta_{W}&-(\mu+\mu^{\prime})&0\end{matrix}\right).\hskip
17.07164pt$ (9) $\displaystyle M_{\widetilde{\chi^{\pm}}}=$
$\displaystyle\left(\begin{matrix}M_{2}&\sqrt{2}M_{W}\sin\beta\\\
\sqrt{2}M_{W}\cos\beta&-(\mu+\mu^{\prime})\\\ \end{matrix}\right).$ (10)

The present analysis will consider vanishing $\mu^{\prime}$.

###  2.3 Charge and Color Breaking:

Avoiding a Charge and Color Breaking (CCB) minima in NHSSM[50] while
considering both the holomorphic and non-holomorphic trilinear couplings
requires a 4-vev scenario, like the vevs for $H_{u}$, $H_{d}$,
${\tilde{f}}_{L}$ and ${\tilde{f}}_{R}$. Here $\tilde{f}$ stands for the
concerned sfermion. With $A_{f}$ and $A_{f}^{\prime}$ both present there is no
possibility of considering a 3-vev scenario unlike in MSSM. A rather
straigtforward computation as shown in Ref.[50] results into the following
inequalities for avoiding a CCB minima. It is seen that unlike MSSM, there is
no D-flat direction so that terms with $g_{1}^{2}+g_{2}^{2}$ come into the
picture arising out of the D-term potential.

$\displaystyle\left[|A_{t}|+|\mu|+|A_{t}^{\prime}|\right]^{2}$
$\displaystyle<$ $\displaystyle
3\left(m_{1}^{2}+m_{2}^{2}+m_{\tilde{t}_{L}}^{2}+m_{\tilde{t}_{R}}^{2}-2B_{\mu}\right),$
$\displaystyle\left[|A_{b}|+|\mu|+|A_{b}^{\prime}|\right]^{2}$
$\displaystyle<$ $\displaystyle
3\left\\{1+\frac{g_{1}^{2}+g_{2}^{2}}{24y_{b}^{2}}\right\\}\left(m_{1}^{2}+m_{2}^{2}+m_{\tilde{b}_{L}}^{2}+m_{\tilde{b}_{R}}^{2}-2B_{\mu}\right),$
$\displaystyle\left[|A_{\tau}|+|\mu|+|A_{\tau}^{\prime}|\right]^{2}$
$\displaystyle<$ $\displaystyle
3\left\\{1+\frac{g_{1}^{2}+g_{2}^{2}}{24y_{\tau}^{2}}\right\\}\left(m_{1}^{2}+m_{2}^{2}+m_{\tilde{\tau}_{L}}^{2}+m_{\tilde{\tau}_{R}}^{2}-2B_{\mu}\right),$
(11)

where $m_{1,2}^{2}=m_{{H_{d}},{H_{u}}}^{2}+\mu^{2}$. One finds that even with
a very large $A_{e}^{\prime}$ as in our analysis, the above constraint is
easily satisfied because of the $g_{1}^{2}+g_{2}^{2}$ term that becomes very
large due to a small $y_{e}$ in the denominator. This is entirely different
from the MSSM case that has a D-flat direction coming out in a 3-vev based
scenario666For MSSM one has the following:
$A_{\tau}^{2}<3(m_{1}^{2}+m_{\tilde{\tau}_{L}}^{2}+m_{\tilde{\tau}_{R}}^{2})$[11,
53, 54, 55]. .

Apart from a global vacuum stability where a lot of MSSM parameter space can
be excluded for a large value of $A_{f}$ corresponding to the first two
generations of leptons ($f\equiv e,\mu$), we must point out that for a long-
lived universe, the CCB conditions corresponding to the light fermion cases
are readily evaded. The above is quite commonly used in SO(10) based analyses
(e.g Ref.[56]) to label a parameter point valid even when the absolute
stability is affected. This is true as long as the CCB inequalities are
satisfied for the large Yukawa coupling cases i.e. the inequalities involving
the third generation trilinear couplings like $A_{t}$. The rate of tunneling
from the Standard Model like false vacuum to a CCB true vacuum is proportional
to $e^{-a/y^{2}}$[55] (follows from Eq. 9 and 11 of Ref.[57]), where $a$ is a
constant and $y$ is the associated Yukawa coupling for the colored/charged
fields. If the dangerous third generation of sfermion constraints are already
avoided, the rate of tunneling corresponding to a small Yukawa case may be
very small. As mentioned in Ref.[57] these are the cases where D-term
contributions to the potential cannot be neglected. NHSSM with large values of
$A_{e}^{\prime}$ also falls into this class and it is thus additionally
consistent with a long-lived universe consideration. Thus for NHSSM, the
inqualities involving $A_{e}$, $A_{e}^{\prime}$ and $A_{\mu}$,
$A_{\mu}^{\prime}$ are either satisfied for absolute stability or they can be
ignored as in MSSM for a long-lived universe leading to cosmological
stability[56, 57].

##  3 Leptonic $(g-2)$ and Yukawa Threshold Corrections in MSSM

In MSSM, as shown in Fig.1, at the one-loop level the leading contributions to
$a_{\mu}^{\rm SUSY}$ come from ${\tilde{\chi}}^{0}-\tilde{\mu}$ and
${\tilde{\chi}}^{\pm}-\tilde{\nu}_{\mu}$ loops[58, 59]. The required chirality
flip may be found from a SUSY Yukawa coupling of a higgsino to a lepton, an
appropriate slepton ($\tilde{\mu}$) or sneutrino ${\tilde{\nu}}_{\mu}$.
Otherwise, the chirality flip may be associated at a slepton $\tilde{\mu}$
line corresponding to the transition
${\tilde{\mu}}_{L}-{\tilde{\mu}}_{R}$[60]. Using Ref.[61] the one-loop
contributions are given as follows.

Figure 1: One-loop contributions to $a_{\mu}^{\rm SUSY}$

$\displaystyle a_{\mu}^{{\tilde{\chi}}^{0}}$ $\displaystyle=$
$\displaystyle\frac{m_{\mu}}{16\pi^{2}}\sum_{i,m}\left\\{-\frac{m_{\mu}}{12m^{2}_{\tilde{\mu}_{m}}}(|n^{L}_{im}|^{2}+|n^{R}_{im}|^{2})F^{N}_{1}(x_{im})+\frac{m_{{\tilde{\chi}}^{0}_{i}}}{3m^{2}_{\tilde{\mu}_{m}}}{\rm
Re}[n^{L}_{im}n^{R}_{im}]F^{N}_{2}(x_{im})\right\\}\phantom{xxxx}$ (12)
$\displaystyle a_{\mu}^{{\tilde{\chi}}^{\pm}}$ $\displaystyle=$
$\displaystyle\frac{m_{\mu}}{16\pi^{2}}\sum_{k}\left\\{\frac{m_{\mu}}{12m^{2}_{\tilde{\nu}_{\mu}}}(|c^{L}_{k}|^{2}+|c^{R}_{k}|^{2})F^{C}_{1}(x_{k})+\frac{2m_{{\tilde{\chi}}^{\pm}_{k}}}{3m^{2}_{\tilde{\nu}_{\mu}}}{\rm
Re}[c^{L}_{k}c^{R}_{k}]F^{C}_{2}(x_{k})\right\\}\phantom{xxxx}$ (13)

where $i$ and $m$ refer to the four neutralino and two chargino states whereas
$k$ indicates the two smuon states. The referred couplings are given by,

$\displaystyle n^{R}_{im}$ $\displaystyle=$
$\displaystyle\sqrt{2}g_{1}N_{i1}X_{m2}+y_{\mu}N_{i3}X_{m1},$ (14)
$\displaystyle n^{L}_{im}$ $\displaystyle=$
$\displaystyle{1\over\sqrt{2}}\left(g_{2}N_{i2}+g_{1}N_{i1}\right)X_{m1}^{*}-y_{\mu}N_{i3}X^{*}_{m2},$
(15) $\displaystyle c^{R}_{k}$ $\displaystyle=$ $\displaystyle y_{\mu}U_{k2},$
(16) $\displaystyle c^{L}_{k}$ $\displaystyle=$ $\displaystyle-g_{2}V_{k1}.$
(17)

The loop functions for the neutralino and chargino loops namely
$F^{N,C}_{1,2}$ may be seen in Ref.[61]. A simplified result follows when the
loops that contribute most are the ones with chargino-sneutrino and bino-smuon
fields[62].

$\displaystyle a_{\mu}^{{\tilde{\chi}}^{\pm}}$ $\displaystyle\simeq$
$\displaystyle\frac{\alpha_{2}\,m^{2}_{\mu}\,\mu\,M_{2}\tan\beta}{4\pi\sin^{2}\theta_{W}\,m_{\tilde{\nu}_{\mu}}^{2}}\left(\frac{f_{\chi^{\pm}}(M_{2}^{2}/m_{\tilde{\nu}_{\mu}}^{2})-f_{\chi^{\pm}}(\mu^{2}/m_{\tilde{\nu}_{\mu}}^{2})}{M_{2}^{2}-\mu^{2}}\right)\,,$
(18) $\displaystyle a_{\mu}^{{\tilde{\chi}}^{0}}$ $\displaystyle\simeq$
$\displaystyle\frac{\alpha_{1}\,m^{2}_{\mu}\,\,M_{1}(\mu\tan\beta-{A}_{\mu})}{4\pi\cos^{2}\theta_{W}\,(m_{\tilde{\mu}_{R}}^{2}-m_{\tilde{\mu}_{L}}^{2})}\left(\frac{f_{\chi^{0}}(M^{2}_{1}/m_{\tilde{\mu}_{R}}^{2})}{m_{\tilde{\mu}_{R}}^{2}}-\frac{f_{\chi^{0}}(M^{2}_{1}/m_{\tilde{\mu}_{L}}^{2})}{m_{\tilde{\mu}_{L}}^{2}}\right)\,,$
(19)

where the loop functions $f$ are as given in Ref. [62].

###  3.1 Yukawa Threshold Corrections

In MSSM, the Yukawa couplings are modified because of soft interactions. We
would now like to discuss Yukawa threshold corrections for fermions that
affects ${(g-2)}_{l}$ as a higher order effect. At one-loop level, the lepton
Yukawa coupling in MSSM can be given as,

$y_{l}=\frac{m_{l}}{(v/\sqrt{2})}\sqrt{1+\tan^{2}\beta}\frac{1}{1+\Delta_{l}}\left(1+O(\cot\beta)\right).$
(20)

Here $\Delta_{l}$ in MSSM is given by [63],

$\displaystyle\Delta_{l}$ $\displaystyle=$ $\displaystyle-\frac{g_{2}^{2}\
M_{2}}{16\pi^{2}}\tan\beta\Big{[}\mu
I(m_{1},m_{2},m_{{{\tilde{\nu}}_{l}}})+\mu\frac{1}{2}I(m_{1},m_{2},m_{\tilde{l}_{L}})\Big{]}$
(21) $\displaystyle-\frac{g_{1}^{2}\ M_{1}}{16\pi^{2}}\tan\beta\Big{[}\mu
I(\mu,M_{1},m_{\tilde{l}_{R}})-\mu\frac{1}{2}I(\mu,M_{1},m_{\tilde{l}_{L}})-\mu
I(M_{1},m_{\tilde{l}_{L}},m_{\tilde{l}_{R}})\Big{]}.$

Here, $m_{1,2}$ are the chargino masses and the loop function $I$ is as given
in Ref. [63]. As we will discuss for NHSSM, the last term $\mu
I(M_{1},m_{\tilde{l}_{L}},m_{\tilde{l}_{R}})$ will be altered via
$\mu\rightarrow(\mu+A_{l}^{\prime})$. Including these corrections, the one-
loop contributions get modified as,

$\displaystyle a_{l}^{\rm SUSY,1L}\longrightarrow a_{l}^{\rm
SUSY,1L}\frac{1}{1+\Delta_{l}}.$ (22)

$\Delta_{l}$, which is proportional to $\tan\beta$, contains the all order re-
summation of the $\tan\beta$ enhanced contributions[64, 65]. We further note
that with the assumption that all the SUSY masses are equal and much larger
than the W-boson mass $M_{W}$, $\Delta_{\mu}=-0.0018\tan\beta~{}{\rm
sign(\mu)}$ [63, 64]. $\Delta_{\mu}$ may be much larger for unequal SUSY
particle mass parameters [63]. A larger $\Delta_{l}$ that is itself
proportional to $\tan\beta$, can even influence $a_{l}$’s
$\tan\beta$-dependence via $y_{l}$. In the limit of radiative mass generations
of fermions, $a_{l}$ can become almost independent of $\tan\beta$[66]. Such
effects of the radiative generation of mass of fermions in analyzing $g-2$
were considered in Refs.[66, 63, 67, 68, 69, 70, 71]. The authors pointed out
the role of non-holomorphic trilinear interactions on lepton Yukawa couplings
$y_{l}$ and $(g-2)_{l}$. A similar control of $(g-2)_{e}$ via enhancement of
Yukawa threshold corrections in MSSM was used in Ref.[72] where the authors
considered very large values of $\mu$ (up to 500 TeV). We will discuss some
details of the work in Sec.3.2.

Following Sec.2, an off-diagonal element of a slepton matrix would look like
$m_{l}(A_{l}-(\mu+A_{l}^{\prime})\tan\beta)$, indicating a generic alteration
of $\mu$ of MSSM going to $\mu+A_{l}^{\prime}$ in NHSSM wherever there is an
L-R mixing. The same will be true in the last term (L-R) of the last line of
Eq. 21. Since for an electron the aforesaid off-diagonal term is multiplied by
$m_{e}$ which is too small, one must have a very large value of
$(\mu+A_{e}^{\prime})$ so as to get a finite L-R mixing effect in relation to
the diagonal terms. With an electroweak fine-tuning-friendly low $\mu$, we see
that $A_{l}^{\prime}$ has to be very large so that $\mu$ may be ignored in
$\mu+A_{l}^{\prime}$. Thus, with an appropriately large ${A}^{\prime}_{l}$,
$\Delta_{l}$ in NHSSM may be approximately proportional to ${A}^{\prime}_{l}$,
and the effect on $y_{l}$ can be much larger than what may be possible in MSSM
due to a potentially large enhancement factor $1/(1+\Delta_{e})$. We will
demonstrate how the radiatively corrected Yukawa coupling $y_{e}$ may become
very important while discussing the case of $a_{e}$ in Sec.4. Additionally,
the sign correlation of $a_{l}$ with respect to $A_{l}^{\prime}$ can be
ascertained from the relevant bino-slepton loop result of Eq.19 when one
changes $\mu$ to $\mu+A_{l}^{\prime}$. Since $\Delta_{e}$ is proportional to
$\tan\beta$ we will see that the threshold corrections to $y_{e}$ will cause
$a_{e}$ to be a slowly increasing function of $\tan\beta$ with some intercept,
but unlike MSSM it would no longer be proportional to $\tan\beta$ (Eq.22).

There have been an appreciable number of SUSY-based analyses on
${(g-2)}_{\mu}$ after the announcement of Fermilab result[73, 74, 75, 76, 77,
78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96,
97, 99, 100, 101, 102, 98]. Some of the above works also involve dark matter
constraints. DM relic density could be satisfied via higgsino or wino type of
LSPs provided one considers underabundant scenarios with the possibility of
multiple candidates for DM. In this case,
${\widetilde{\chi}}^{\pm}-{\tilde{\nu}}_{\mu}$ loop may contribute dominantly
to $a_{\mu}$. One can similarly consider bino-wino mixed LSP in this regard.
One can also consider bino type of LSP which obtains correct relic density by
self-annihilation via s-channel H/A boson, with $a_{\mu}$ constraint being
satisfied via the contribution from the bino-smuon loop with relatively light
smuons. Alternatively, bino-stau coannihilation may be used for DM relic
density generation and $a_{\mu}$ constraint may be similarly addressed. As we
will see in our work with nonstandard trilinear soft terms, we consider
higgsino to be the LSP (in an underabundant choice for DM). Regarding
$a_{\mu}$, we use the contribution from the bino-smuon loop that is enhanced
by larger L-R mixing due to the above soft terms. We will further see that the
large threshold corrections (due to the nonstandard soft terms) to leptonic
Yukawas, particularly $y_{e}$, can be useful to accommodate the $a_{e}$
constraint simultaneously.

###  3.2 Accommodating $a_{e}$ constraint in addition to $a_{\mu}$ limits

Eq.4 summarizes the requirement for a new physics to accommodate both the
constraints. The ratio is required to be not only large but also negative. We
would like to address the essential parts of a few past MSSM analyses in this
regard. Refs.[62, 72, 97] analyzed the $a_{\mu}$ and $a_{e}$ constraints in
the context of MSSM. Ref.[97] used 1-3 flavor violation in the bino-slepton
loop to get the desired outcome for $a_{e}$ while satisfying the
$\tau\rightarrow e\gamma$ bound. There was no flavor violation to use for
$a_{\mu}$. The essential focus of Ref.[62] was to satisfy $(g-2)_{\mu}$ and
$(g-2)_{e}$ either via the lighter chargino-sneutrino loop or via the
neutralino-slepton loop diagrams with appropriate signs of the U(1) and SU(2)
gaugino masses $M_{1}$ and $M_{2}$. The analysis of Ref.[62] that could only
have a light electroweakino spectra, used different mass parameters for the
sleptons of the first two generations (apart from the sneutrinos). The work
considered differing right and left slepton mass parameters, a highly
unfriendly choice for flavor. Clearly, such large mass splittings in the
sleptons of the first two generations are prone to create large flavor-
violating off-diagonal entries in the slepton mass matrices when lepton
matrices become diagonalized. It can easily give rise to lepton flavor
violation, which is severely limited via the $\mu\rightarrow e\gamma$
constraint. Alignment of slepton and lepton matrices were to be invoked in
order to meet the above constraint. Apart from the above, a generally light
SUSY spectrum in satisfying the $(g-2)_{l}$ magnetic moment data while having
strong LHC constraints on sparticle masses forces one to have only a
compressed scenario involving light sleptons and wino-like chargino along with
a bino-like LSP. Ref.[98] also used differing mass parameters between the two
generations of sleptons, whereas an overabundant bino DM was avoided by
considering a superWIMP dark matter scenario 777Going beyond MSSM, i) Ref.[99]
is an analysis with supersymmetric gauged ${U(1)}_{L_{\mu}-L_{\tau}}$ model,
ii) Ref.[100] pointed out the level of phenomenological stringency of the
issue while citing varieties of generic models including MSSM, leptoquarks and
little Higgs inspired models and showed the usefulness of abelian
${L_{\mu}-L_{\tau}}$ to explain the anomalies. Ref.[100] also commented on the
possibility of non-holomorphic trilinear terms to address the issue. iii)
Ref.[101] used additional $U(1)^{\prime}$ SUSY model with family dependent
(non-universal) $U(1)^{\prime}$ charges, iv) Ref.[102] used inverse-seesaw
extended NMSSM to address the anomalies. . As mentioned earlier, Ref.[72]
explored the above magnetic moment constraints also in the MSSM context using
a very large $\mu$ (up to $\sim 500$ TeV) to generate large threshold
corrections to the relevant Yukawa couplings in the analysis. A large degree
of sensitivity of the above corrections to the slepton masses is used to
accommodate the $(g-2)$ constraints. Selectron masses were considered to be
heavy (multi TeV) whereas smuon masses are less than a TeV in order to satisfy
both the $g-2$ constraints. The analysis required a very large
$\tan\beta$($=70$) and light wino ( $500$ GeV) and massive higgsinos. The work
that is associated with a large value for the electroweak fine-tuning complied
with the vacuum stability condition to find the valid parameter space. On the
theoretical side, both the analyses ([72, 62]) discussed the flavor dependence
of the slepton masses by considering a Higgs mediation scenario[103, 104] so
as to have an alignment of the slepton and lepton matrices. Ref.[105]
addressed the magnetic moment constraint pair by considering CP-violating
phases and the constraint from electron dipole moment (EDM).

In contrast to the above analyses that had to manage the flavor issues
carefully and are strongly constrained by $\mu\rightarrow e\gamma$ or may have
to live with light SUSY spectra or very large $\tan\beta$, we focus on
including nonstandard soft terms. We must emphasize here that as mentioned
before, the MSSM soft terms and the nonstandard soft SUSY breaking terms do
not have the same status. While the regular soft terms are highly supported by
popular UV complete models, the NH terms are in general difficult for
modelling. As mentioned earlier, in a model independent standpoint, we analyze
the effect of the nonstandard terms considering an approach of “Minimal
Effective Supersymmetry” of Ref.[28]. An analysis with NH terms would not
demand any large $\mu$, or light spectra and we will use a flavor friendly
scenario of having identical slepton masses for the first two generations with
equal coefficients for the left and right mass parameters. However, we will
use two different non-holomorphic trilinear parameters for the first two
generations of leptons, both given as inputs at the low scale. In general, had
these been given at a much higher scale like the grand unification scale we
might run into flavor issues. Nevertheless, a detailed analysis of the exact
degree of flavor violation, is beyond the scope of this work.

##  4 Results

###  4.1 Muon magnetic Moment in NHSSM

Our NHSSM analysis on lepton $g-2$ in its first part involves studying the
effects of the nonstandard soft terms on the enhancement of the SUSY
contributions to muon g-2. We will explore at the beginning the constraints
from $(g-2)_{\mu}$ data and analyze the parameter space that would be
consistent with dark matter (DM) and other constraints. We will demonstrate
the mechanism that enhances $(g-2)_{\mu}$ in NHSSM. For DM, we will consider
the case where the lightest supersymmetric particle (LSP) is expected to obey
only the upper limit of the relic density bound from PLANCK data[106]. We will
identify the parameter space that would satisfy all the direct detection
limits from XENON1T namely both the spin-independent[107] and spin-
dependent[108] type of data. Once the above analysis delineates the parameter
space, we will explore the effect of imposing the constraint from the
electron’s magnetic moment $(g-2)_{e}$ due to fine-structure constant
measurement. We will see how the Yukawa threshold corrections due to NHSSM
effects can affect leptonic $g-2$, particularly $(g-2)_{e}$. Because of the
above, we will see that it is indeed possible to find NHSSM parameter regions
such that the scaled magnetic moment ratio of interest $R_{e,\mu}$ as
discussed before may be consistent with Eq.4. We use SARAH-4.14.4[109, 110]
and SPheno 4.0.4[111, 112] to implement the model and for computing SUSY
spectra and observables.

In order to generate a sufficient amount of SUSY contributions to leptonic
$g-2$ in NHSSM we take the help of chirality flip via the Left-Right scalar
mixing due to the non-holomorphic trilinear SUSY breaking interaction
parametrized by ${A}^{\prime}_{l}$. We use $T_{l}^{\prime}$ as an input
defined via $T^{\prime}_{l}=y_{l}{A}^{\prime}_{l}$ (with
$y_{l}=\frac{m_{l}}{(v_{d}/\sqrt{2})}$). This follows the input convention
used in the codes SARAH-SPheno (for example see Ref.[45]) for all the
trilinear soft parameters. We should point out that with this parametrization
a value of a few GeV for $T^{\prime}_{l}$ may mean a large value for
${A}^{\prime}_{l}$. The largeness is most prominent for electron, whereas for
top quark, the associated values of $T_{t}$ and $A_{t}$ are not so different
from each other. We must also remember that because of the appearance of the
fermionic mass $m_{e}$ in the off-diagonal entry of Eq.8, any non-negligible
L-R mixing of selectrons would need a significantly large ${A}^{\prime}_{e}$
because of the smallness of the electronic mass. In one-loop vertex correction
radiative diagrams (Figure 1), this is achieved via the NH trilinear
interactions of scalars in the bino-slepton loops888We use vanishing
coefficients $A_{l}$ for the trilinear SUSY breaking parameters throughout our
study.. Thus, both the above trilinear parameters and the mass of bino $M_{1}$
will have significant roles in our analysis apart from the masses of sleptons
for which we consider equal left and right mass parameters $m_{L},m_{R}$ for
the first two generations. We must emphasize that the above NH terms may
potentially enhance $a_{l}$ significantly in comparison with the MSSM
contributions involving chargino or neutralino loops.

In order to probe NHSSM effects clearly we keep the SU(2) gaugino mass to be
sufficiently heavy, much above $\mu$ or $M_{1}$. A large part of parameter
space where $(g-2)_{e}$ can be accommodated in our work involves i) direct
effect of larger $A_{e}^{\prime}$ on the bino-seletron loop of Fig.1 (i.e.
through Eq.19 with $\mu\rightarrow\mu+A_{e}^{\prime}$) and ii) via the effect
of the enhancement factor $\frac{1}{1+\Delta_{e}}$ directly on $a_{e}$
(Eq.22). Of course, $\frac{1}{1+\Delta_{e}}$ influences the Yukawa coupling
for electron ($y_{e}$) directly via $A_{e}^{\prime}$ through the same L-R
mixing. The effect of the said NH trilinear mixing may easily supersede the
contribution from the higgsino mixing superpotential term characterized by
$\mu$. On the other hand, the corrections may become smaller for larger
slepton mass values. Keeping our choice of scalar mass spectra in tact, we
will rather limit the associated NH trilinear SUSY breaking parameters such as
$A^{\prime}_{e}$ so as to limit $y_{e}$. We will label the above as the
“Limited Threshold Corrections of Yukawa Coupling (LTCYC)” zone. To specify
explicitly, by LTCYC zone of $y_{e}$ we mean that the radiatively corrected
value of $y_{e}$ can at most be twice the corresponding MSSM value, or in
other words $\frac{1}{1+\Delta_{e}}<2$. However, with the chosen SUSY mass
parameters we will see that the required agreement with data for ${(g-2)}_{e}$
range is possible even with a factor quite smaller than two. As we will point
out in Section 4.3.2 the chosen SUSY parameter space that satisfies
${(g-2)}_{\mu}$ requires only moderately large $A_{\mu}^{\prime}$ in contrast
to $A_{e}^{\prime}$ for the case of ${(g-2)}_{e}$. Thus, LTCYC for $y_{\mu}$
is automatically satisfied for all the parts of our analysis meant for
${(g-2)}_{\mu}$. We will explore explicitly how far LTCYC for $y_{e}$ as a
condition becomes important when we incorporate $(g-2)_{e}$ as a constraint in
Section 4.3.

Toward the end, we will impose the constraints from ATLAS search for slepton-
pair production[113] in the LSP-slepton mass plane and explore a few benchmark
points that would satisfy all the constraints. While the neutralino-slepton
loop dominated by bino type of neutralino would enhance $(g-2)_{l}$, we will
primarily consider higgsino as the LSP which would be suitable for satisfying
the DM relic density data from PLANCK (only the upper limit, in a multi-
component DM scenario) as well as the spin independent (SI) direct-detection
(DD) scattering cross section data of LSP-nucleon scattering. The bino-
higgsino mixed region would obviously be disfavored or discarded via the SI DD
constraints. Below, we focus on the minimal requirement for enhancing SUSY
contributions to leptonic $g-2$ and the variables that are closely connected
to $g-2$ and dark matter. With the above in mind, we keep the squark masses
decoupled to large values and we do the same for the tau-sleptons too. All the
left and right handed scalars including that of the first two generations of
sleptons will assume equal SUSY breaking mass parameter input values
($m_{L}=m_{R}$). We keep the trilinear coefficient for top-squark namely
$T_{t}$ at a high negative value that would be consistent with Higgs mass
data. The Table 1 refers to the input parameter values/ranges999The relevant
SM parameters are: $m_{t}^{\rm pole}=173.5,~{}m_{b}^{{\overline{\rm
MS}}}=4.18,~{}m_{\tau}=1.776$ all in GeV..

Parameters | Value | Parameters | Value
---|---|---|---
$M_{1}$ | [100 to 1000] | $M_{2}$ | 1500
$M_{3}$ | 2800 | $\mu$ | [150 to 1000]
$m_{A}$ (as tree-level input toward ${B_{\mu}}$ ) | 2500 | $\tan\beta$ | 10 & 40
$M_{\tilde{q}_{33}}/M_{\tilde{u}_{33}}$ | 4000 | $M_{\tilde{d}_{33}}$ | 4000
$M_{\tilde{q}_{11,22}}/M_{\tilde{u}_{11,22}}$ | 2500 | $M_{\tilde{d}_{11,22}}$ | 2500
$m_{L}\equiv M_{\tilde{L}_{11,22}}=M_{\tilde{e}_{11,22}}\equiv m_{R}$ | [200 to 1000] | $M_{\tilde{L}_{33}}=M_{\tilde{e}_{33}}$ | 2000
$T_{t}$ | -3500 | $T_{t}^{\prime}$ | 0
$T_{e,\mu,\tau}$ | 0 |  $T^{\prime}_{e}$,$T^{\prime}_{\mu}$, $T^{\prime}_{\tau}$ | [-25 to 25], [0 to 1000],0
Table 1: SUSY scale input ranges of Soft masses and Trilinear SUSY breaking
parameters. Quantities labelled by $T$ are scaled trilinear parameters, e.g.
$T_{t}=y_{t}{A}_{t}$ or $T^{\prime}_{\mu}=y_{\mu}{A}^{\prime}_{\mu}$. Masses
and trilinear couplings are in GeV. All the trilinear parameters not mentioned
here are understood to assume zero values.

With the above parameter space been defined, some of our constraints are that
from the Higgs data, $Br(b\rightarrow s\gamma)$, relic density upper limit
from PLANCK. This is apart from the $(g-2)_{\mu}$ and $(g-2)_{e}$ constraints
mentioned earlier and the SI and SD direct detection constraints from XENON1T
that depend on the mass of the LSP. With $M_{\rm SUSY}$ being large, we
consider a 3 GeV theoretical uncertainty in SUSY Higgs mass leading to the
following as the acceptable range[114] for the SM-like Higgs in MSSM.

$122<m_{h}<128~{}{\rm GeV}.$ (23)

The flavor limits at 2$\sigma$ are $3.02\times 10^{-4}<Br(b\rightarrow
s\gamma)<3.62\times 10^{-4}$[115] and $2.23\times
10^{-9}<Br(B_{s}\rightarrow\mu^{+}\mu^{-})<3.63\times 10^{-9}$ [116].
Considering the PLANCK result for dark matter namely
$\Omega_{{\tilde{\chi}}^{0}}h^{2}=0.120\pm 0.001$[106], the $2\sigma$ level
limits are ${[\Omega_{{\tilde{\chi}}^{0}}h^{2}]}_{\rm min}=0.118$ and
${[\Omega_{{\tilde{\chi}}^{0}}h^{2}]}_{\rm max}=0.122$. The DM relic density
is expected to satisfy only the upper bound. In our generally underabundant
scenario, for the purpose of direct detection cross-section, we define a scale
factor $\xi$[117] as given below.

$\xi=\Omega_{{\tilde{\chi}}^{0}}h^{2}/{[\Omega_{{\tilde{\chi}}^{0}}h^{2}]}_{\rm
min}~{}{\rm
if}~{}\Omega_{{\tilde{\chi}}^{0}}h^{2}\leq{[\Omega_{{\tilde{\chi}}^{0}}h^{2}]}_{\rm
min},{\rm else}~{}\xi=1.$ (24)

We plot Figure 2 that describes our result of the SUSY contributions to
$a_{\mu}$ (referred hereafter as $a_{\mu}$ itself) in relation to the relevant
SUSY breaking parameter $M_{1}$, the mass of bino. Fig. 2 shows the variation
of $a_{\mu}$ with the mass of bino ($M_{1}$) for $\tan\beta=$ 10 to 50 in
steps of 10. Here, the fixed parameters used are $T^{\prime}_{\mu}=200$ GeV,
$m_{L}(=m_{R})=600$ GeV, $\mu=600$ GeV, and $M_{2}=1.5$ TeV. As we will see
later (Figure 17), the NHSSM contributions to $a_{\mu}$ are much larger than
the corresponding MSSM ones (i.e. with vanishing $T^{\prime}_{\mu}$) for the
chosen regions of values of mass and coupling parameters of Table 1. The black
horizontal lines are the 1$\sigma$ limits of $a_{\mu}$. For each $\tan\beta$,
$a_{\mu}$ increases with $M_{1}$, and then it decreases. The peaks occur at
around a given value of $M_{1}$ and the corresponding locations are almost
independent of $\tan\beta$. Fig.2 shows the variation of $a_{\mu}$ with
$M_{1}$ for specific values of slepton mass parameters $m_{L}~{}(=m_{R})$
corresponding to the first two generations. The fixed parameters chosen are
$T^{\prime}_{\mu}=200$ GeV, $\mu=1$ TeV and $\tan\beta=10$ with $M_{2}$ taking
an identically large value as before. The curves show similar peaks and their
locations depend on the slepton mass parameter $m_{L}$.

Figure 2: Fig.2: Variation of SUSY contributions to $a_{\mu}$ (referred as
$a_{\mu}$ itself) with respect to the mass of bino ($M_{1}$) for a few values
of $\tan\beta$. Fixed parameters are as mentioned in the plots. The black
horizontal lines are the 1$\sigma$ limits of $a_{\mu}$. For each $\tan\beta$,
$a_{\mu}$ has an ascending and a descending part over the range of variation
of $M_{1}$. The location of the peaks depend on $m_{L}(=m_{R})$, and these are
essentially unchanged with respect to $M_{1}$. Fig.2: Same as Fig.2 except for
a given value of $\tan\beta$ and varying $m_{L}$. There are similar peaks in
$a_{\mu}$, but they shift with varying $m_{L}$.

We continue to study the behavior of $a_{\mu}$ concerning the relevant SUSY
breaking parameters in Figure 3. Undoubtedly, $a_{\mu}$ is enhanced most
prominently via the trilinear parameter $A_{\mu}^{\prime}$ via its strong
effect through chirality flipping L-R scalar interaction in the bino-smuon
loop. For $\tan\beta=10$, $m_{L}=\mu=600$ GeV and $M_{2}=1.5$ TeV, Fig.3 shows
the variation of $a_{\mu}$ with respect to $M_{1}$ for a few different values
of $T_{\mu}^{\prime}$, 100 GeV to 500 GeV in steps of 100 GeV. A substantial
amount of enhancement of $a_{\mu}$ occurs due to a change in
$T_{\mu}^{\prime}$. In this analysis, we satisfy all the charge-breaking
related constraints for NHSSM (Eq.2.3)[50] and stay within the LTCYC zone for
the threshold corrections to $y_{l}$ due to the variation over
$T_{l}^{\prime}$. Fig.3 refers to the variation of $a_{\mu}$ over
$T_{\mu}^{\prime}$ for different values of $M_{1}$. $|a_{\mu}|$ generally
decreases with $M_{1}$ for values above 400 GeV. The exceptions are the cases
of $M_{1}=200$ and 400 GeV, which are flipped because they belong to the
ascending and the descending parts of the corresponding curve for $m_{L}=600$
GeV of Fig.2. Furthermore, the sign of $a_{\mu}$ is approximately given by the
sign of $T^{\prime}_{\mu}$. In the NHSSM scenario of interest where non-
vanishing ${A}^{\prime}_{\mu}$ enhances the bino-smuon loop contribution
causing the same loop to dominate over all the other diagrams, at the lowest
order, $\mu\tan\beta$ gets replaced by $(\mu+{A}^{\prime}_{\mu})\tan\beta$ in
Eq. 19. With vanishing ${A}_{\mu}$, and
${A}^{\prime}_{\mu}~{}(=T^{\prime}_{\mu}/y_{\mu})$ large enough to offset
$\mu$, the sign of $a_{\mu}$ is determined by the sign of $A^{\prime}_{\mu}$.
Numerically this is seen to be valid as long as $|T^{\prime}_{\mu}|\ ^{>}\hbox
to0.0pt{\hss${}_{\sim}$}\ 20$ GeV or so. Consistency with the 1$\sigma$ band
demands that $(g-2)_{\mu}$ data can be satisfied with only positive
$T^{\prime}_{\mu}$. Fig.3 displays the variation of $a_{\mu}$ with $M_{1}$ for
$\mu$ satisfying $150<\mu<1000$ GeV. The blue and red regions refer to
$\tan\beta=10$ and 40 respectively. The limited level of thickening of lines
even for a large $\tan\beta$ demonstrates that there is only a mild degree of
dependence of $a_{\mu}$ on $\mu$ over the range mentioned above. This is
consistent with the fact that a higgsino-smuon or a charged higgsino-sneutrino
loop for $a_{\mu}$ are hardly important in our analysis with a dominant effect
due to a NH trilinear term.

Figure 3:  Fig.3: Variation of $a_{\mu}$ with the mass of bino ($M_{1}$) for a
few values of $T_{\mu}^{\prime}$ where $T_{\mu}^{\prime}$ relates to
${A}^{\prime}_{\mu}$ of Eq.7 via $T^{\prime}_{\mu}=y_{\mu}{A}^{\prime}_{\mu}$.
Fig.3: Variation of $a_{\mu}$ over $T_{\mu}^{\prime}$ for different values of
$M_{1}$. $|a_{\mu}|$ generally decreases with $M_{1}$ for values above 400
GeV. The exceptions are the cases of $M_{1}=200$ and 400 GeV, which are
flipped because they belong to the ascending and the descending zones of the
corresponding curve for $m_{L}=600$ GeV of Fig.2. Fig.3: Variation of
$a_{\mu}$ with $M_{1}$ for the scanned range of $\mu$ satisfying
$150<\mu<1000$ GeV. The blue and red regions refer to $\tan\beta=10$ and 40
respectively. The limited level of thickening of lines even for a large
$\tan\beta$ demonstrates only a mild degree of dependence of $a_{\mu}$ on
$\mu$ over the range. The black horizontal lines are the 1$\sigma$ limits of
$a_{\mu}$.

We now use the $(g-2)_{\mu}$ constraint in the most important
$M_{1}-T^{\prime}_{\mu}$ plane in Fig.4. The fixed parameter under the study
are $\tan\beta=10$, $\mu=300$ GeV, $m_{L}=600$ GeV and $M_{2}=1.5$ TeV. We
divide the region into 1$\sigma$ (blue), 2$\sigma$ (green), and 3$\sigma$
(gray) zones with respect to the $(g-2)_{\mu}$ constraint. The 1$\sigma$
region extends from $T^{\prime}_{\mu}=175$ GeV to $450$ GeV while $M_{1}$
spans the entire space chosen in our analysis. The region bents toward the
left and this is indeed consistent with what follows from Fig.3. Fig.4 shows
similarly constrained region with the same color convention as above. A
generic decrease of $a_{\mu}$ with increase in $m_{L}$ is apparent since large
$T^{\prime}_{\mu}$ is necessary to stay in a given fixed colored zone for
large $m_{L}$ values.

Figure 4: Fig.4: Display of 1$\sigma$ (blue), 2$\sigma$ (green), and 3$\sigma$
(gray) regions with respect to the $(g-2)_{\mu}$ constraint in the
$(T^{\prime}_{\mu}-M_{1})$ plane for fixed parameters mentioned in the figure.
Fig.4: Similar display in the $(T^{\prime}_{\mu}-m_{L})$ plane.

Figure 5 shows a few $a_{\mu}$ contours in the plane of $M_{1}-m_{L}$ for
$\tan\beta=10$ and $40$ for $T^{\prime}_{\mu}=400$ and 700 GeV. The cyan-
colored points represent unconstrained parameter values that only satisfy the
basic constraints like Higgs data, $Br(b\rightarrow s\gamma)$, lightest
neutralino to be the LSP, and restriction from charge-breaking minima. The
blue and green shaded regions are $1\sigma$ and $2\sigma$ bands for the
${(g-2)}_{\mu}$ constraint. Three lines are drawn for $a_{\mu}=5\times
10^{-9}$, $a_{\mu}=10\times 10^{-9}$, and $a_{\mu}=15\times 10^{-9}$. Clearly,
$a_{\mu}$ is large for small $m_{L}$ or small smuon mass regions. We must note
that the reach of the smallness of $m_{L}$ to obtain an enhanced $a_{\mu}$
effectively may lead to picking up small $M_{1}$ regions. This is particularly
true for higher $\mu$ cases. The requirement of ${\tilde{\chi}}^{0}_{1}$ to be
the LSP, which in our case is either a bino or a higgsino dominated state, to
a reasonable degree of approximation demands $m_{L}$ to be higher than $M_{1}$
or $\mu$ whichever is the lowest among the last two. There are other reasons
why the bottom white regions of each of the figures are excluded. These may be
due to the requirements of avoidance of tachyonic sleptons or charge-breaking
minima. On the other hand, small $m_{L}$ regions find stringent constraints
from the LHC data which we will discuss later. Comparing the figures, we can
further see that, as expected, larger $T^{\prime}_{\mu}$ values enhance
$a_{\mu}$. For a given $a_{\mu}$, and a fixed $M_{1}$, a larger
$T^{\prime}_{\mu}$ creates a possibility to accommodate a larger $m_{L}$. A
large $\tan\beta$ like $40$ enhances $a_{\mu}$.

Figure 5: Fig. 5: Plot in $M_{1}-m_{L}$ plane for $\tan\beta=10$, $\mu=400$
GeV and $T^{\prime}_{\mu}=400$ GeV with all other fixed parameters same as
mentioned in Table 1. The cyan-colored points represent unconstrained
parameter values that only satisfy the basic constraints like Higgs data,
$Br(b\rightarrow s\gamma)$, lightest neutralino to be the LSP, and restriction
from charge breaking minima. The blue and green shaded regions are $1\sigma$
and $2\sigma$ bands for the ${(g-2)}_{\mu}$ constraint. Three lines are drawn
for $a_{\mu}=5\times 10^{-9}$, $a_{\mu}=10\times 10^{-9}$, and
$a_{\mu}=15\times 10^{-9}$. Clearly, $a_{\mu}$ is large for small $m_{L}$ or
small smuon mass regions. The white region at the bottom refers to a discarded
parameter zone. This refers to not satisfying the requirements like LSP has to
be the lightest neutralino ${\tilde{\chi}}^{0}_{1}$, avoidance of tachyonic
sleptons or any presence of charge-breaking minima. Fig. 5: Same as above
except $T^{\prime}_{\mu}=700$ GeV. A larger $T^{\prime}_{\mu}$ enhances
$a_{\mu}$, leading the possibility to accommodate a larger $m_{L}$ value for a
given $a_{\mu}$ at a given $M_{1}$ in comparison with Fig. 5. There is however
no $a_{\mu}=15\times 10^{-9}$ line because of unavailability of valid
parameter space. Fig. 5 and Fig. 5 are similar figures as above for
$\tan\beta=40$. The lower discarded (white) regions extend with respect to
$\tan\beta=10$.

###  4.2 Constraints from Dark Matter

We now include the constraints from dark matter in this NHSSM $(g-2)_{\mu}$
analysis. Of course, with our choice of an underabundant scenario of a
higgsino dark matter, no NHSSM trilinear parameter would directly affect the
DM analysis. We will study DM for its effects on the combined parameter space.
In our analysis, wino mass $M_{2}$ is chosen to be quite heavy (1.5 TeV),
whereas the higgsino mass has a range of $150<\mu<1000$ GeV, and for bino, we
have $100<M_{1}<1000$ GeV. Our parameter space further consists of heavy tau-
sleptons (2 TeV) along with decoupled squarks, whereas we choose the range of
the first two generations of sleptons to vary within 200 GeV to 1 TeV. With
$m_{A}=2.5$ TeV as an input the neutral CP-even or CP-odd Higgs masses are way
above to encounter the so-called funnel region that is characterized by bino-
dominated LSPs undergoing self-annihilation via s-channel processes involving
$A$ or $H$-bosons. Coming to the higgsinos, the chosen mass range is such that
the upper limit of 1 TeV is what is necessary for higgsino LSP to become a
single component DM but the upper limit of $M_{1}$ is also chosen to be the
same. A chance equality of $M_{1}$ and $\mu$ would produce a large bino-
higgsino mixing, but this would not be friendly with the SI direct detection.
On the other hand, regions with $M_{1}<\mu$ would produce overabundance.
Hence, in most of the available parameter space, the LSP is of higgsino type,
except in a few occasions when there is a possibility of
$\widetilde{\chi}_{1}^{0}$-slepton coannihilations. With higgsino LSPs to have
mass below a TeV the associated relic density is supposed to satisfy only the
upper limit of DM relic density constraint from PLANCK data. One of the
important channels for DM production would be the coannihilation channel
between the higgsno LSP with the lighter chargino state
(${\tilde{\chi}}^{0}_{1}-{\tilde{\chi}}^{\pm}_{1}$) which is also higgsino
dominated in nature. Fig.6 shows the Spin-independent scattering cross-section
of the LSP with a proton ($\sigma^{\rm SI}_{\chi p}$) as a function of the LSP
mass for $\tan\beta=10$ for the parameter space mentioned in Table 1. We used
micrOMEGAs 5.2.13 for dark matter-related computations[118, 119, 120].
Considering the large number of points with underabundance of DM we multiply
$\sigma^{\rm SI}_{\chi p}$ with a scale factor $\xi$ which was defined in
Eq.24. Only those points satisfying the DM relic density upper bound, Higgs
mass data, B-physics related limits and $(g-2)_{\mu}$ at 2$\sigma$ level are
plotted and these are shown in green. The blue line is the constraint from the
spin-independent (SI) direct detection (DD) experiment of XENON1T indicating
discarded regions above the line (at 90%CL). Using interpolation we will
further use the XEONON1T line to judge whether a parameter point with a given
mass of the LSP would survive the SI DD experiment limit. Fig.6 shows the
results for $\tan\beta=40$.

Figure 6: Fig.6: Spin-independent (SI) scattering cross-section of the LSP
with a proton $\sigma^{\rm SI}_{\chi p}$ as a function of the LSP mass for
$\tan\beta=10$. Considering the large number of points with underabundance of
DM we multiply $\sigma^{\rm SI}_{\chi p}$ with a scale factor $\xi$ (Eq.24).
The points satisfying the DM relic density upper bound and $(g-2)_{\mu}$
limits at 2$\sigma$ level are shown in green. The blue line is the constraint
from $\sigma^{\rm SI}_{\chi p}$, the spin-independent (SI) direct detection
(DD) experiment of XENON1T indicating discarded regions above the line (at
90%CL). Fig.6: Similar figure for $\tan\beta=40$.

Fig.7 shows the projection of the analysis of Fig.6 on the $M_{1}-\mu$ plane.
Points shown in green satisfy the DM relic density upper bound, the XENON1T
spin-independent direct detection cross-section limit at 90%CL as well as
$(g-2)_{\mu}$ values within 2$\sigma$ level. The parameter zone with bino-
higgsino mixed type of LSPs ($M_{1}\simeq\mu$) are typically discarded via the
XENON1T SI-DD limits. Generally, LSPs are visibly higgsino-dominated in nature
and satisfy all the constraints within the mass range of 150 (the chosen lower
limit of $\mu$) to 780 GeV. Apart from higgsinos, additionally, there are a
very few parameter points corresponding to bino-like LSPs undergoing
occasional coannihilations with sleptons for $m_{\widetilde{\chi}_{1}^{0}}$
between 250 to 350 GeV. Fig.7, shows a similar plot for $\tan\beta=40$. Here
$m_{\widetilde{\chi}_{1}^{0}}$ ranges from 150 GeV to 760 GeV. Both for
$\tan\beta=10$ and $40$, the upper limit of the mass of
$\widetilde{\chi}_{1}^{0}$ is restricted via the XENON1T SI-DD data.

Figure 7: Fig.7: Scatter plot in $M_{1}-\mu$ plane for the analysis of Fig.6.
Points shown in green satisfy the $(g-2)_{\mu}$ and the DM relic density
limits both at $2\sigma$ level while also having $\sigma^{\rm SI}_{\chi p}$
value below the XENON1T limit at 90% CL. The LSP is generally of higgsino
type. There are a few isolated points in the small $M_{1}$($<\mu$) zone with
$m_{\widetilde{\chi}_{1}^{0}}$ between 250 to 350 GeV that correspond to bino-
dominated LSPs satisfying the DM relic density limits via slepton
coannihilations. An appropriately large choice of $m_{A/H}$ avoids pair-
annihilations of essentially binos via the s-channels Higgs process. Fig.7:
Similar figure for $\tan\beta=40$.

We now include the Spin-dependent direct detection of DM study in our
analysis. Fig.8 shows the Spin-dependent (SD) scattering cross-section of the
LSP with a neutron ($\sigma^{\rm SD}_{\chi n}$) as a function of the LSP mass
for $\tan\beta=10$. As before, we multiply $\sigma^{\rm SD}_{\chi n}$ with a
scale factor $\xi$ (Eq.24). Only those points satisfying the DM relic density
upper bound as well as $(g-2)_{\mu}$ at 2$\sigma$ level are plotted and these
are shown in green. The blue line is the constraint from the SD direct-
detection experiment of XENON1T indicating discarded regions above the line at
90% CL. We further checked that the constraint from $\sigma^{\rm SD}_{\chi p}$
is weaker than $\sigma^{\rm SD}_{\chi n}$. Fig.8 is a similar plot for
$\tan\beta=40$. Compared to the SI DD cross-sections of Fig.6, here we do not
find any new discarded region. This is true for both choices of $\tan\beta$.
Hence, all our conclusion for the SI DD cross-section analysis remain valid
and analyzing $\sigma^{\rm SI}_{\chi p}$ itself is sufficient for a conclusion
regarding the direct detection of DM.

Figure 8:  Fig.8: Spin-dependent (SD) scattering cross-section of the LSP with
a neutron $\sigma^{\rm SD}_{\chi n}$ multiplied by the previously mentioned
scale factor $\xi$, as a function of the LSP mass, for $\tan\beta=10$. The
points satisfying the DM relic density upper bound and $(g-2)_{\mu}$ limits at
2$\sigma$ level are shown in green. The blue line refers to the constraint
from the SD direct-detection experiment of XENON1T at 90% CL. The constraint
from $\sigma^{\rm SD}_{\chi p}$ is weaker than $\sigma^{\rm SD}_{\chi n}$.
Fig.8: Similar figure for $\tan\beta=40$.

Below, we probe how large $a_{\mu}$ can go depending on the variation of basic
MSSM parameters like $M_{1}$ and $m_{L}$ and NHSSM parameter
$T^{\prime}_{\mu}$. Thus, Fig.9 shows a scatter plot of $a_{\mu}$ vs. $M_{1}$
for $\tan\beta=10$. The parameter points shown in green satisfy the dark
matter constraints including the direct detection limits from XENON1T. For
comparison purposes, we draw maroon points located near the bottom of the
$a_{\mu}$ axis. These points refer to the result of a similar scanning in an
MSSM parameter space (by using $T^{\prime}_{\mu}=0$). None of the dark matter
constraints are however applied here in the MSSM case. The smallness of the
MSSM $a_{\mu}$ values shows that the bino-smuon loop contribution with effects
from $T^{\prime}_{\mu}$ supersedes all the MSSM loop contributions. This
indeed helps us in explaining the sign-correlation of $a_{\mu}$ with
$T^{\prime}_{\mu}$ as mentioned earlier. The regions with large $a_{\mu}$
typically arise in small $M_{1}$ zones and this is likely to be case due to
the associated smallness of slepton masses (see Figure 5). The figure shows
that the SI direct detection cross-section is below the XENON1T limit
throughout the domain of variation chosen for $M_{1}$. The results shown in
Fig.9 confirm larger MSSM contributions to $a_{\mu}$ for an increase of
$\tan\beta$ to 40, although the enhancement is much smaller than the NHSSM
effects toward $a_{\mu}$, hence not so visible.

Figure 9: Fig.9: Scatter plot of $a_{\mu}$ for varying $M_{1}$ and other
parameters as mentioned in the text for $\tan\beta=10$. The parameter points
shown in green satisfy the dark matter constraints including the direct
detection limits from XENON1T. In order to probe the extent to which NHSSM can
enhance $a_{\mu}^{\rm SUSY}$, the maroon points refer to the result of a
similar scanning in an MSSM parameter space (using $T^{\prime}_{l}=0$). None
of the dark matter constraints are applied here. Throughout the analyses, we
confirm that the points that satisfy $\sigma^{\rm SI}_{\chi p}$ also respect
both n and p SD DD cross-sections namely, $\sigma^{\rm SD}_{\chi n}$ and
$\sigma^{\rm SD}_{\chi p}$ respectively. The $1\sigma$ allowed band for
$a_{\mu}$ is shown as black horizontal lines. Fig.9: Similar figure for
$\tan\beta=40$.

Figure 10 and Figure 11 are the scatter plots for the dependence of $a_{\mu}$
on $T^{\prime}_{\mu}$ and $m_{L}$ when other parameters are varied. Fig.10
displays a scatter plot of $a_{\mu}$ for varying $T^{\prime}_{\mu}$ for
$\tan\beta=10$. The points shown in green satisfy the dark matter relic
density upper bound and their SI direct detection (DD) cross-section
$\sigma^{\rm SI}_{\chi p}$ values fall below the XENON1T limit. The large
$a_{\mu}$ region of Figure 10 refers to small $m_{L}$ values. This in turn
correspond to small $M_{1}$ zones with enhanced $a_{\mu}$. The blanck (white)
strip near the $T^{\prime}_{\mu}$ axis i.e., near the region with very small
$a_{\mu}$ denotes the absence of valid parameter points, and this is related
to the blanck region for smaller $m_{L}$ values (near the $m_{L}$ axis) of
Fig.11. In combination, the resulting off-diagonal slepton mass values for
large $T^{\prime}_{\mu}$ and small $m_{L}$ are likely to generate excluded
regions because of the appearance of tachyonic slepton states or vacuum
instability. With $T^{\prime}_{\mu}=y_{\mu}{A}_{\mu}$, the effect becomes more
prominent for a larger $\tan\beta$ as may be seen in Fig.10 and Fig.11. We
further note that the large $a_{\mu}$ regions corresponding to small $m_{L}$
values in each of the parts of Figure 11 refer to small $M_{1}$ and large
$T^{\prime}_{\mu}$ zones. The LSP that satisfies the DM constraints is of
higgsino type and associated with underabundance.

Figure 10:  Fig.10: Scatter plot of $a_{\mu}$ vs. $T^{\prime}_{\mu}$ for
$\tan\beta=10$ for varying parameters as mentioned in the text. All the green
points satisfy the same constraints as mentioned in Fig.9. Fig.10: Similar
figure for $\tan\beta=40$.

Figure 11:  Fig.11: Same as Fig.10 for $\tan\beta=10$ except for varying
$m_{L}$ instead of $T^{\prime}_{\mu}$. Fig.11: Similar plot for
$\tan\beta=40$.

###  4.3 Inclusion of Electron Magnetic Moment limits; Yukawa coupling
enhancement in NHSSM

We will now probe the effect of including the ${}^{133}{\rm Cs}$-based fine-
structure constant measurement derived electron $g-2$ data in our analysis on
top of the muon $g-2$ constraint. We will only consider the necessary amount
for $T^{\prime}_{e}$ that would generate the required value for $a_{e}$. We
will also highlight the threshold corrections to the electron Yukawa coupling
$y_{e}$ due to $T^{\prime}_{e}$ and constrain the NHSSM parameter space
accordingly. This will involve both cases namely a free enhancement of $y_{e}$
as well as limiting $y_{e}$ by requiring to stay within the LTCYC zone as
mentioned earlier. Fig.12 shows the effect of $T^{\prime}_{e}$ on the electron
Yukawa coupling $y_{e}$ for $\mu=400$ GeV and $m_{L}(=m_{R})=500$ GeV. The
blue lines are for $\tan\beta=10$, and the red lines are for $\tan\beta=40$.
For each $\tan\beta$, the solid lines refer to $M_{1}=300$ GeV whereas the
dotted lines are drawn for $M_{1}=700$ GeV. No LTCYC condition for $y_{e}$ is
applied here in order to display the extent of the Yukawa threshold correction
of $y_{e}$. A large negative value for $T^{\prime}_{e}$ gives rise to a larger
$|y_{e}|$ compared to the same for an identical positive value of
$T^{\prime}_{e}$. $y_{e}$ is also seen to flip its sign as $T^{\prime}_{e}$
becomes positive. A larger $M_{1}$ also enhances $|y_{e}|$. We point out that
no LTCYC condition is applied here in order to show the extent how $y_{e}$ is
affected due to an increased $T^{\prime}_{e}$. Depending on the value of
$m_{L}$ and $M_{1}$, the above may in turn cause appearance of tachyonic
selectron states. We now try to understand how $a_{e}$ behaves as
$T^{\prime}_{e}$ is varied. Fig.12 displays $a_{e}$ for two different values
of $\tan\beta$, and two different values of $M_{1}$ identical with those of
Fig.12. The other SUSY parameters are $\mu=400$ GeV and $m_{L}(=m_{R})=500$
GeV as used before. Unless $T^{\prime}_{e}$ is tiny, negative value of $a_{e}$
that is required due to the experimental data is correlated with negative
value of $T^{\prime}_{e}$. We note that the solid lines of $a_{e}$ for
$\tan\beta=10$ and $40$ for $M_{1}=300$ GeV are very close to each other. The
same is true for the two dotted lines corresponding to $M_{1}=700$ GeV. The
reason for the approximate independence of $a_{e}$ on $\tan\beta$ would be
clear in Section 4.3. The display of $y_{\mu}$ for a variation of
$T^{\prime}_{\mu}$ as shown in Fig. 12 indicates a larger available zone of
variation of $T^{\prime}_{\mu}$ compared to that of $T^{\prime}_{e}$ of Fig.
12. This arises because of the difference of mass values of electron and muon.
No LTCYC for $y_{\mu}$ is an issue here since the threshold correction of
$y_{\mu}$ is moderate corresponding to the given span of variation of
$T^{\prime}_{\mu}$ and the chosen SUSY mass parameters. A large negative value
for $T^{\prime}_{\mu}$ gives rise to a larger $|y_{\mu}|$ compared to the same
for a similar positive value for $T^{\prime}_{\mu}$. Unlike the case of
$y_{e}$ of Fig.12, here $y_{\mu}$ does not change sign when $T^{\prime}_{\mu}$
becomes positive, albeit within its given range of interest for our study. A
larger $M_{1}$ also enhances $|y_{\mu}|$. Fig. 12 shows the variation of
$a_{\mu}$ over $T^{\prime}_{\mu}$. We note that with a given value
$T^{\prime}_{(e,\mu)}$, varying $M_{1}$ will alter $y_{(e,\mu)}$ which in turn
would affect $a_{e,\mu}$.

Figure 12: Fig.12: Display of the effect of $T^{\prime}_{e}$ on the electron
Yukawa coupling $y_{e}$ for the shown values of the SUSY parameters. For each
$\tan\beta$, the solid lines refer to $M_{1}=300$ GeV whereas the dotted lines
are drawn for $M_{1}=700$ GeV. A large negative value for $T^{\prime}_{e}$
gives rise to a larger $|y_{e}|$ compared to the same for a similar positive
value for $T^{\prime}_{e}$. $y_{e}$ is also seen to flip its sign as
$T^{\prime}_{e}$ becomes positive. A larger $M_{1}$ also enhances $|y_{e}|$.
No LTCYC condition for $y_{e}$ is applied here in order to display the extent
of the Yukawa threshold correction of $y_{e}$. Fig.12: Display of the electron
magnetic moment $a_{e}$ for two different values of $\tan\beta$ as well as
$M_{1}$. The other SUSY parameters are shown within the figure. The two solid
lines refer to $\tan\beta=10,40$ for $M_{1}=300$ GeV, and the two dotted lines
stand for $\tan\beta=10,40$ where $M_{1}=700$ GeV. Unless $T^{\prime}_{e}$ is
tiny, negative values of $a_{e}$ that is required via the experimental limits
follow from negative values of $T^{\prime}_{e}$. Fig.12: A similar plot like
Fig.12 for electron. Fig.12: A similar plot like Fig.12 for muon.

####  4.3.1 Absence of $\tan\beta$ scaling in $a_{l}$ within NHSSM because of
large ${A}^{\prime}_{l}$

In Section4.3, we discussed the effects on the $\tan\beta$ related behavior of
magnetic moments of leptons in scenarios that may induce large threshold
corrections to the associated Yukawa couplings. In the above context, an NHSSM
study of $a_{l}$ becomes quite relevant. With no LTCYC conditions applied
Figure 13 shows the scaling-related behaviors of $a_{e}$,$a_{\mu}$ with
respect to a variation over $\tan\beta$ for the MSSM and the NHSSM cases.
Here, $a_{e}$,$a_{\mu}$ are appropriately multiplied by powers of 10 so that
all of them may be plotted in a single graph. The solid blue line refers to
$a_{e}\times 10^{13}$ for $T^{\prime}_{e}=-5$ GeV, whereas the dashed blue
line corresponds to $a_{e}\times 10^{14}$ for the MSSM case (i.e.
$T^{\prime}_{e}=0$). Clearly, $a_{e}$ satisfies proportional relationship with
$\tan\beta$ in MSSM, but this is no longer true in NHSSM, where $a_{e}$ is
only a slowly increasing function of $\tan\beta$ with a large intercept
101010This is consistent with what was first commented on non-holomorphic
interactions in relation to $a_{l}$ in radiative generation of fermion mass
context in Ref.[66] (see their discussion followed by Eq.38). . Showing
identical behavior, the solid and dashed red lines are similar results for
$a_{\mu}$.

Figure 13: Display of scaling related behavior of $a_{e}$,$a_{\mu}$ with
respect to $\tan\beta$ for the MSSM and the NHSSM cases. No LTCYC conditions
are used here. Here, $a_{e}$,$a_{\mu}$ are appropriately multiplied by
appropriate powers of 10. The solid blue line refers to $a_{e}\times 10^{13}$
for $T^{\prime}_{e}=-5$ GeV, whereas the dashed blue line corresponds to
$a_{e}\times 10^{14}$ for the MSSM case (i.e.$T^{\prime}_{e}=0$). Clearly,
$a_{e}$ satisfies proportional relationship with $\tan\beta$ in MSSM, but
$a_{e}$ is merely a slowly increasing function of $\tan\beta$ with large
intercept. Showing identical behavior, the solid and dashed red lines are
similar results for $a_{\mu}$.

####  4.3.2 Limiting the degree of threshold corrections of Yukawa couplings
for leptons: LTCYC criterion vs the $a_{e}$ constraint.

The non-holomorphic soft terms may be able to cause large Yukawa corrections
for fermions, particularly the leptons so that the radiatively corrected
values may be a few times the MSSM specified value namely $y_{l,(ref)}\equiv
y_{l}=\frac{m_{e}}{(v_{d}/\sqrt{2})}$. This was already discussed for
phenomenological analyses in Refs.[66, 63, 67, 68, 69, 70, 71] where lepton
masses were generated radiatively, obviously requiring a large radiative
corrections. We should also point out that the present Higgs decay to
$e^{+}e^{-}$ from the LHC can only give an upper bound for $y_{e}$ which is
about a few hundred times of its SM value[121, 122, 123]111111 As described in
Ref.[122] Yukawa couplings may not be proportional to fermion mass in presence
of higher dimensional operators (6 or more )that respect SM gauge symmetry.
Thus a very different Higgs-electron-electron coupling may be possible to
arise from new physics effects from these operators. There may be a
significant amount of cancellation between the contributions to the mass of
electron coming from the SM Yukawa coupling and that obtained from higher
dimensional operators. Similarly, the Higgs-electron-electron coupling will
have two components, one from SM and the other from the higher dimensional
operators. See also Ref.[69] for a SUSY context.. In our analysis, a negative
$a_{e}$ may be accommodated by a relatively large radiative corrections to
$y_{e}$. Of course a choice of a very large slepton mass would be able to
suppress the radiative corrections, but here we want to explore the minimal
zone of values of $|T^{\prime}_{e}|$ that with the chosen SUSY mass spectra of
our analysis would be consistent with limits from $a_{e}$. Since $a_{e}$ is
driven by $y_{e}$, in general, both for muon and electron, as mentioned before
we choose to discard any parameter point that leads to ${y_{l}/y_{l(ref)}}>2$
with $l\equiv e,\mu$ and denote it as Limited Threshold Corrections of Yukawa
Coupling (LTCYC) criterion. Fig.14 shows a scatter plot in the
$(y_{e}/y_{e(ref)}-T^{\prime}_{e})$ plane satisfying the LTCYC criterion in
addition to all other constraints like DM relic density upper bound and direct
detection limits, Higgs mass data, B-physics related limits and
$(g-2)_{e,\mu}$ at 2$\sigma$ level for $\tan\beta=10$. The result is a bound
of 14 GeV for $|T^{\prime}_{e}|$ and there are significant amount of parameter
space with much smaller values like a range of 1.5 to 1.8 for
${y_{e}/y_{e(ref)}}$. Fig.14 shows a similar result for $\tan\beta=40$ showing
the same cut-off at 13 GeV. We observe that for both the values of $\tan\beta$
there are appreciable amount of parameter space where the ratio
${y_{e}/y_{e(ref)}}$ stays between 1.5 to 1.8 (much below the maximum of 2).
This corresponds to $-4{\rm~{}GeV}<T^{\prime}_{e}<-2{\rm~{}GeV}$. Fig.14 and
Fig.14 display the required values of the above Yukawa coupling ratio for
correct $a_{e}$. LTCYC issue is important for $a_{e}$ only. For muon, the
experimental limits of the magnetic moment is hardly stringent. Unlike the
case of electron, the chosen region of variation of $T^{\prime}_{\mu}$ does
not produce any appreciably large amount of radiative corrections to
$y_{\mu}$. Hence, LTCYC for $y_{\mu}$ is not an issue for $a_{\mu}$, so that
the condition gets automatically satisfied for the full parameter space under
consideration. Indeed it may be seen that a positive $T^{\prime}_{\mu}$ as
required for positive $a_{\mu}$ leads to values below unity for
$y_{\mu}/y_{\mu(ref)}$.

Figure 14: Limiting the level of leptonic Yukawa radiative corrections in
NHSSM at the weak scale: $y_{l}$ refers to output Yukawa coupling value that
is obtained after radiative corrections whereas $y_{l(ref)}$ is the tree level
MSSM input value out of leptonic mass $m_{l}$. We choose to discard any
parameter point that leads to $y_{l}/y_{l(ref)}>2$ and denote it as Limited
Threshold Corrections of Yukawa Coupling (LTCYC) criterion. Fig.14: Scatter
plot in $(y_{e}/y_{e(ref)}-T^{\prime}_{e})$ plane satisfying the LTCYC
criterion in addition to all other constraints like DM relic density upper
bound and direct detection limits, Higgs mass data, B-physics related limits
and $(g-2)_{e,\mu}$ at 2$\sigma$ level for $\tan\beta=10$. A larger
$|T^{\prime}_{e}|$ above 14 is seen not to satisfy the said limit. Fig.14:
Same as (a) for $\tan\beta=40$. Fig.14: $(a_{e}-y_{e}/y_{e(ref)})$ plot
showing a possibility of finding the desired $a_{e}$ for radiative corrections
above 45%. Fig.14: Same as (c) for $\tan\beta=40$.

The fact that NHSSM parameter $T^{\prime}_{e}$ is able to generate a
significant amount of correction to $y_{e}$, while on the other hand there is
an approximate sign correlation between $a_{e,\mu}$ with $T^{\prime}_{e,\mu}$
gives one a reasonable expectation for the scaled magnetic moment ratio
$R_{e,\mu}=\frac{(a_{e}/m_{e}^{2})}{(a_{\mu}/m_{\mu}^{2})}$ to be within the
desired zone, namely $R_{e,\mu}\simeq-15$ (see Eq.4). Figure 15 is a scatter
plot of the above ratio with parameters in the x-axis, namely $T^{\prime}_{e}$
and $M_{1}$ that are important for the radiative corrections to Yukawa
coupling $y_{e}$. All other parameters including also $T^{\prime}_{\mu}$ are
varied according to the mentioned respective ranges of Table 1 considering
$\tan\beta=10$ while also applying the LTCYC conditions for the two leptons as
mentioned earlier. All the points (shown in green) satisfy the DM relic
density constraint for its upper limit, the XENON1T data for SI direct
detection, $a_{e}$ and $a_{\mu}$ limits showing the desired $R_{e,\mu}$ values
well within its uncertainty limits. Fig.15 for $\tan\beta=10$ shows scatter
points in the $M_{1}-R_{e,\mu}$ plane. The points with large $R_{e,\mu}$
correspond to smaller $M_{1}$ values. The DM satisfied (green) region satisfy
$250\ ^{<}\hbox to0.0pt{\hss${}_{\sim}$}\ M_{1}\ ^{<}\hbox
to0.0pt{\hss${}_{\sim}$}\ 800$ GeV.

Figure 15: Fig.15: Scatter plot of magnetic moment ratio $a_{e}/a_{\mu}$
scaled with ${m_{\mu}}^{2}/{m_{e}}^{2}$ against $T_{e}^{\prime}$ for
$\tan\beta=10$. All the points (shown in green) satisfy the DM relic density
constraint for its upper limit, the XENON1T data for SI direct detection,
$a_{e}$ and $a_{\mu}$ limits. LTCYC conditions are applied in addition.
Fig.15: Similar scatter plot for a variation of $M_{1}$.

We will now investigate how the conclusion of Section 4.2 gets modified when
we further restrict the NHSSM parameter space toward satisfying the
$(g-2)_{e}$ constraint. First, since we demand $y_{e}$ to be within the LTCYC
zone, $|T_{e}^{\prime}|$ should be small and this comes to around 14 GeV or
less. Considering the fact that SUSY contributions to $a_{e}$ needs to be
negative, the limits on $T^{\prime}_{e}$ becomes $-14\ ^{<}\hbox
to0.0pt{\hss${}_{\sim}$}\ T^{\prime}_{e}<0$. Fig. 16 shows the combined
results (green points) of imposing all the constraints like Higgs mass data,
B-physics limits, DM constraints for relic density and SI direct detection in
addition to $a_{\mu}$ and $a_{e}$ limits drawn in the plane of
$M_{1}-T_{e}^{\prime}$. Here and henceforth in all the subsequent figures of
this work LTCYC limits for $y_{e}$ and $y_{\mu}$ are understood to have been
applied by default. The valid region for $M_{1}$ that satisfy all the
constraints comes out to be $230<M_{1}<800$ GeV for $\tan\beta=10$. Fig. 16, a
similar plot for $\tan\beta=40$ has a valid $M_{1}$ zone satisfying
$230<M_{1}<660$ GeV.

Figure 16: Fig.16: Scatter plots in the $M_{1}-T^{\prime}_{e}$ plane for
$\tan\beta=10$. The points shown in green arise from imposing all the
constraints like Higgs mass data, B-physics limits, DM constraints for relic
density and SI direct detection in addition to $a_{\mu}$ and $a_{e}$ limits.
Here and henceforth in all the subsequent figures of this work LTCYC limits
for $y_{e}$ and $y_{\mu}$ are understood to have been applied by default.
Fig.16: Similar plot for $\tan\beta=40$.

Figure 17 shows similar scatter plots in the $M_{1}-T^{\prime}_{\mu}$ plane.
The larger range of variation of $T^{\prime}_{\mu}$ compared to
$T^{\prime}_{e}$ arises from the way $T^{\prime}_{e,\mu}$ are defined, namely
these are scaled with the respective Yukawa couplings. The conclusion for
valid upper limit of $M_{1}$ remains the same as before for both values of
$\tan\beta$.

Figure 17: Fig.17: Scatter plots in the $M_{1}-T^{\prime}_{\mu}$ plane for
$\tan\beta=10$. The color scheme is same as that of Figure16. Fig.17: Similar
plot for $\tan\beta=40$.

We are to explore now the effect of imposing ${(g-2)}_{e}$ constraint on the
$M_{1}-\mu$ plane of Figure 18. We will compare the above with the
corresponding result of Figure 7 drawn for ${(g-2)}_{\mu}$. The availability
of NHSSM parameter space gets reduced significantly because the threshold
corrections to $y_{e}$ should not become too large. We remember that the above
corrections at least involve masses of bino, sleptons, as well as the
trilinear NH coupling $T^{\prime}_{e}$. Hence, a given set of $M_{1}$ and
$m_{L}$ that is available in the muon $(g-2)$ analysis, may not be consistent
when combined with non-vanishing $T^{\prime}_{e}$ to satisfy ${(g-2)}_{e}$.
Thus, when one imposes the ${(g-2)}_{e}$ constraint, that in turn is sensitive
on the corrections to $y_{e}$, the reduction of parameter space is rather
unavoidable. The green points of Fig. 18 satisfy $(g-2)_{e,\mu}$ constraints
at 2$\sigma$ level in addition to obeying the dark matter relic density bound.
The associated $\sigma^{\rm SI}_{\chi p}$ values also fall below the XENON1T
limit. Fig. 18 shows a similar result for $\tan\beta=40$ where a largeness of
${\rm tan}\beta$ further takes away a significant amount of parameter space so
as to have $y_{e}$ within LTCYC zone.

Figure 18:  Fig.18: Scatter plot in $M_{1}-\mu$ plane for $\tan\beta=10$. All
the points shown in green satisfy $(g-2)_{e,\mu}$, dark matter relic density
upper bound and XENON1T provided $\sigma^{\rm SI}_{\chi p}$ bounds. Fig.18:
Similar plot for $\tan\beta=40$.

Fig.19 is a scatter plot in the plane of trilinear NH parameters for electron
and muon for $\tan\beta=10$. The points shown in green satisfy the DM relic
density constraint, the XENON1T $\sigma^{\rm SI}_{\chi p}$ data along with
$(g-2)_{\mu}$ and $(g-2)_{e}$ limits at $2\sigma$ level. The points also have
$y_{e}$ in the LTCYC zone. Clearly, $T^{\prime}_{e}$ should be appreciably
large so as to satisfy $(g-2)_{e}$ data. We find the valid region to have
$-14~{}{\rm GeV}<T^{\prime}_{e}<0$.

Figure 19: Fig.19: Scatter plots in $T^{\prime}_{e}-T^{\prime}_{\mu}$ plane
for $\tan\beta=10$. The color scheme is same as that of Figure16. Fig.19:
Similar plot for $\tan\beta=40$.

###  4.4 LHC Constraints from slepton pair production and constraint from
compressed higgsino LSP scenario

####  4.4.1 Both $(g-2)_{\mu}$ and $(g-2)_{e}$

With no difference in SUSY sparticle content between NHSSM and MSSM, we
directly apply the SUSY constraints from LHC data on our analysis. We intend
to identify the exclusion region of NHSSM parameter space from the ATLAS data
for slepton pair production that considered selectron and smuon in the
analysis[113]. The later gives an exclusion region in the
$(m_{\tilde{\chi}^{0}_{1}}-m_{L})$ plane. We note here that by directly
applying the ATLAS bound on our parameter space we are taking a conservative
approach as the exclusion can potentially get somewhat weaker for the higgsino
LSP scenario. The reason is as follows. The ATLAS limit is derived for a
simplified model assuming $BR(\tilde{l}\rightarrow
l\widetilde{\chi}_{1}^{0})=100\%$, This criterion is not strictly satisfied in
the higgsino LSP region, where the proximity of $m_{\widetilde{\chi}_{1}^{0}}$
and $m_{{\widetilde{\chi}_{1}}^{\pm}}$ allows for a significant branching
ratio of the sleptons to final states involving $\nu_{l}$ and
${\widetilde{\chi}_{1}}^{\pm}$. This may lead to a reduction in the number of
signal leptons and thus to the weakening of the exclusion limit.

We will further analyze the constraint from a compressed scenario with
higgsino as LSP that is associated with closely spaced values for the masses
$m_{\widetilde{\chi}_{1}^{0}}$, $m_{{\widetilde{\chi}_{1}}^{\pm}}$ and
$m_{\widetilde{\chi}_{2}^{0}}$. This is based on the ATLAS result given in
Fig.14a of Ref.[124].

Figure 20: Allowed (cyan) and disallowed (red) regions in the plane of
$m_{\widetilde{\chi}_{2}^{0}}-\Delta_{21}$ with respect to the 1$\sigma$-level
contour (black line) as given by the ATLAS data namely Fig.14a of Ref.[124].
The corresponding lower limit of $m_{\widetilde{\chi}_{1}^{0}}$ satisfying the
above constraint is about 185 GeV in this higgsino dominated
$\widetilde{\chi}_{1}^{0}$-$\widetilde{\chi}_{2}^{0}$-${\widetilde{\chi}_{1}}^{\pm}$
scenario.

Focusing on higgsino type of LSPs, we show Figure 20 drawn in the plane of
$m_{\widetilde{\chi}_{2}^{0}}-\Delta_{21}$ where
$\Delta_{21}=m_{\widetilde{\chi}_{2}^{0}}-m_{\widetilde{\chi}_{1}^{0}}$. The
above-mentioned 1$\sigma$ contour of the ATLAS result is shown in black. The
red colored points appearing to the left of the contour form the discarded
zone, whereas the cyan colored points survive the constraint. Clearly, a
compressed scenario with $\Delta_{21}$ having values up to 10 to 15 GeV are
eliminated for $m_{\widetilde{\chi}_{2}^{0}}\ ^{<}\hbox
to0.0pt{\hss${}_{\sim}$}\ 200$ GeV. Larger values of $\Delta_{21}$ in the same
zone of $m_{\widetilde{\chi}_{2}^{0}}$ are alright. The corresponding lower
limit of $m_{\widetilde{\chi}_{1}^{0}}$ is approximately 185 GeV. Thus, we may
conclude that not all values of LSP masses below 185 GeV are valid. Hence,
with a conservative standpoint we henceforth consider the lowest value for a
higgsino type of LSP to be 185 GeV. 121212On the other hand, regarding a
compressed LSP-slepton scenario there is hardly any constraint from LHC (Fig.
16a of Ref.[113]) in our analysis.

Fig.21 shows a scatter plot in $m_{L}-m_{\tilde{\chi}^{0}_{1}}$ plane for
$\tan\beta=10$. All the points satisfy the perturbativity of $y_{e}$.
Considering the constraint for compressed higgsino states as mentioned above,
$m_{\tilde{\chi}^{0}_{1}}$ is allowed to have values above 185 GeV. The points
shown in green satisfy the DM relic density constraint, the XENON1T
$\sigma^{\rm SI}_{\chi p}$ data along with $(g-2)_{\mu}$ and $(g-2)_{e}$
limits at $2\sigma$ level. The points also have $y_{e}$ in the LTCYC zone. On
the top of the figure we draw the black line that indicates the current
exclusion bounds in the $m_{L}-m_{\tilde{\chi}^{0}_{1}}$ plane at the LHC. The
green points that stay outside the contour are the residual parameter points
that would also satisfy the LHC limit. On the higher LSP mass side,
irrespective of slepton masses, for $\tan\beta=10$ a value of
$m_{{\tilde{\chi}}^{0}_{1}}$ between 400 to 500 GeV would satisfy all the
constraints considered in this analysis. For slepton masses above 700 GeV,
valid LSP mass would lie below 350 GeV. Fig.21 shows a similar result for
$\tan\beta=40$. Here, the LHC contour engulfs a lot of region of the parameter
space with a larger mass of the LSP as shown in green. The region with slepton
masses above 700 GeV allows valid LSP mass up to 275 GeV. Apart from the
above, the LSP mass values for valid parameter points outside and near the
black contour in its lower side are correlated with $m_{L}$ in an
approximately linear fashion. One finds that with $\tan\beta=10$,
$m_{{\tilde{\chi}}^{0}_{1}}$ can be as large as 400 GeV when $m_{L}=550$ GeV
whereas the later numbers are 400 GeV and 500 GeV respectively for
$\tan\beta=40$.

Figure 21: Fig.21 Scatter plot in $m_{\tilde{\chi}^{0}_{1}}-m_{L}$ plane for
$\tan\beta=10$. The points shown in green satisfy the DM relic density
constraint, the XENON1T $\sigma^{\rm SI}_{\chi p}$ data along with
$(g-2)_{\mu}$ and $(g-2)_{e}$ limits at $2\sigma$ level. The points also have
$y_{e}$ in the LTCYC zone. The black line indicates the current exclusion
bounds in the slepton-neutralino plane as obtained from slepton pair
production data from ATLAS, LHC[113]. All the points pass the compressed
higgsino constraint as mentioned in the text. Fig.21: Similar plot for
$\tan\beta=40$.

####  4.4.2 Only $(g-2)_{\mu}$

Since the LHC data is seen to affect the combined analysis of ${(g-2)}_{\mu}$
and ${(g-2)}_{e}$ with dark matter rather strongly, we feel that it is
important to go back and examine the situation of considering only the
${(g-2)}_{\mu}$ constraint with dark matter (as in Section4.2) in relation to
the ATLAS constraints described above. The results are seen in Figure 22 and
22. Figure 22 shows the ATLAS data[113] satisfied appropriate mass values of
sparticles in the $m_{L}-m_{\tilde{\chi}^{0}_{1}}$ plane. As before, we also
impose the constraint from the compresed higgsino scenario. For
$\tan\beta=10$, satisfying all the constraints, valid
$m_{\tilde{\chi}^{0}_{1}}$ ranges from 400 GeV to 750 GeV for any value of
$m_{L}$. On the other hand when $m_{L}>700$ GeV, $m_{\tilde{\chi}^{0}_{1}}$
assumes values up to the same upper limit namely 750 GeV. For $\tan\beta=40$,
as seen in Figure 22 the above numbers for $m_{\tilde{\chi}^{0}_{1}}$ are in
between 400 to 775 GeV, whereas for $m_{L}>700$ GeV, valid
$m_{\tilde{\chi}^{0}_{1}}$ zone extends to 775 GeV. Apart from the above, the
LSP mass values for valid (green) parameter points outside and near the black
contour in its lower side are correlated with $m_{L}$ in an approximately
linear fashion. One finds for both the values of $\tan\beta$,
$m_{{\tilde{\chi}}^{0}_{1}}$ can be as large as 400 GeV when $m_{L}=550$ GeV.

Figure 22: Fig.22 Scatter plot in $m_{\tilde{\chi}^{0}_{1}}-m_{L}$ plane for
$\tan\beta=10$. The points shown in green satisfy the DM relic density
constraint, the XENON1T $\sigma^{\rm SI}_{\chi p}$ data along with
$(g-2)_{\mu}$ and $(g-2)_{e}$ limits at $2\sigma$ level. The points also have
$y_{e}$ in the LTCYC zone. The black line indicates the current exclusion
bounds in the slepton-neutralino plane as obtained from slepton pair
production data from ATLAS, LHC[113]. All the points pass the compressed
higgsino constraint as mentioned in the text. Fig.22: Similar plot for
$\tan\beta=40$.

We now show two representative points of our analysis in Table 2 for two
values of ${\rm tan}\beta$. Referring to Figure 21 that corresponds to
satisfying both $(g-2)_{\mu}$ and $(g-2)_{e}$ constraints, the two points are
chosen based on the ATLAS provided contours[113] for slepton pair production
shown in black. We choose $tan\beta=10$ and 40 for specifying two benchmark
points BP1 and BP2 both of which correspond to moderate values of
$m_{\widetilde{\chi}_{1}^{0}}$ and slepton mass parameter $m_{L}$. The points
are far away from having any effect due to the constraints from compressed
spectra. We remind that all the left and right slepton mass parameters are
taken to be equal. The LSP is almost a higgsino in nature. The relic density
is satisfied via like $\tilde{\chi}_{1}^{0}-\tilde{\chi}_{2}^{0}$ or
$\tilde{\chi}_{1}^{0}-\tilde{\chi}_{1}^{\pm}$ coannihilation mechanisms. Apart
from other constraints like ${(g-2)}_{e,\mu}$ and dark matter, the constraints
from B-physics are easily satisfied because of large squark masses of the
third generation. With a larger stop-mixing via $A_{t}$, the Higgs mass is
satisfied within the limits of Eq.23. Notably, $a_{e}$ value of BP2 for
$\tan\beta=40$ is similar to that of BP1 corresponding to $\tan\beta=10$. This
is because, as discussed earlier, in NHSSM, the SUSY contributions to $a_{e}$
has a milder $\tan\beta$ dependence unlike a proportional relationship as seen
in MSSM. With LSP as a higgsino it is a candidate of a subdominant component
of DM. Hence we use a scale factor $\xi$ (Eq.24) that further reduces the SI
and SD direct detection cross sections. Certainly, the valid parameter space
would be larger for both small and large $\tan\beta$ if we had chosen only
$(g-2)_{\mu}$ among the two magnetic moment constraints.

Table 2: Representative Points in NHSSM corresponding to SUSY scale input of soft parameters that satisfy all the constraints considered in the analysis along with $a_{\mu}$ and $a_{e}$ (using data from ${}^{133}{\rm Cs}$-based measurement). All the dimensionful parameters are in GeV. Here, $T_{e}^{\prime}=y_{e}A_{e}^{\prime}$ and $T_{t}=y_{t}A_{t}$. Parameters | BP1 | BP2
---|---|---
$M_{1}$ | $553$ | $539$
$\mu$ | $411$ | $401$
$B_{\mu}$ | $6.19\times 10^{5}$ | $1.56\times 10^{5}$
$\tan\beta$ | 10 | 40
$T_{t}$ | $-3500$ | $-3500$
$T_{e}^{\prime},T_{\mu}^{\prime}$ |  $-3.7$, $180.8$ |  $-3.1$, $190.1$
$m_{L}\equiv M_{\tilde{L}_{11,22}}=M_{\tilde{e}_{11,22}}\equiv m_{R}$ | 594 | 526
$a_{\mu}^{SUSY}$ | $1.87\times 10^{-9}$ | $3.17\times 10^{-9}$
$a_{e}^{SUSY}$ | $-1.60\times 10^{-13}$ | $-1.63\times 10^{-13}$
$m_{h}$ | 122.8 | 123.9
$m_{H},m_{A},m_{H^{\pm}}$ | 2481,2481,2482 | 2373,2373,2377
$m_{\tilde{t}_{1,2}}$ | 3947, 4111 | 3943, 4102
$m_{\tilde{b}_{1,2}}$ | 4056, 4091 | 4043, 4067
$m_{\tilde{e}_{1,2}}$ | 609, 640 | 543, 577
$m_{\tilde{\mu}_{1,2}}$ | 593, 652 | 522,592
$m_{\tilde{\tau}_{1,2}}$ | 2004, 2016 | 1983, 2008
$m_{\tilde{\chi}_{1,2}^{0}}$ | 410, 424 | 401, 414
$m_{\tilde{\chi}_{1,2}^{\pm}}$ | 421,1534 | 411, 1534
$m_{\tilde{g}}$ | 3026 | 3026
$BR(B\rightarrow X_{s}+\gamma)$ | $3.19\times 10^{-4}$ | $3.13\times 10^{-4}$
$BR(B_{s}\rightarrow\mu^{+}\mu^{-})$ | $3.22\times 10^{-9}$ | $3.18\times 10^{-9}$
$\Omega_{DM}h^{2}$ | $0.0243$ | $0.0230$
$\sigma_{\chi p}^{SI}$ | $1.53\times 10^{-9}$ | $1.29\times 10^{-9}$
$\xi\sigma_{\chi p}^{SI}$ | $3.15\times 10^{-10}$ | $2.50\times 10^{-10}$
$\sigma_{\chi n}^{SD}$ | $1.50\times 10^{-5}$ | $1.72\times 10^{-5}$
$\xi\sigma_{\chi n}^{SD}$ | $3.09\times 10^{-6}$ | $3.35\times 10^{-6}$

##  5 Results for $a_{e}$ constraint from the ${}^{87}{\rm Rb}$-based
experiment

Finally, we briefly discuss the effects of using the $a_{e}$ constraint from
Eq.3, i.e. from the ${}^{87}{\rm Rb}$-based experiment as mentioned earlier.
There is a 1.6$\sigma$ deviation from the SM result and the spread is more
toward the positive side, unlike the case of Eq.2 corresponding to the
experiment using ${}^{133}{\rm Cs}$ matter-wave interferometry. Focusing on
only $\tan\beta=10$, the scatter-plot of Figure 23 in the
$T^{\prime}_{e}-T^{\prime}_{\mu}$ plane shows the requirement of both positive
and negative signs of $T^{\prime}_{e}$ in relation to the results shown in
Figure 12. The points shown in green satisfy the DM relic density constraint,
the XENON1T $\sigma^{\rm SI}_{\chi p}$ data along with $(g-2)_{\mu}$ and
$(g-2)_{e}$ limits at $2\sigma$ level. The points also have $y_{e}$ in the
LTCYC zone. A milder deviation corresponding to Eq.3 makes the result to have
similarity with the case of using only the $a_{\mu}$ constraint. Similarly, we
plot Figure 23 indicating parameter points in the
$m_{\tilde{\chi}^{0}_{1}}-m_{L}$ plane. Further details regarding constraint
from the ATLAS data are same as those of Figure 21. Expectedly, the result is
closer to that of Figure 22 for the only $(g-2)_{\mu}$ case.

Figure 23: Fig.23: Analysis with ${}^{87}{\rm Rb}$ data for $a_{e}$ for
scatter-plots in the $T^{\prime}_{e}-T^{\prime}_{\mu}$ plane $\tan\beta=10$.
The color scheme and constraints imposed are same as that of Figure19. Fig.23:
Analysis with ${}^{87}{\rm Rb}$ data for $a_{e}$ for scatter-plots in the
$m_{\tilde{\chi}^{0}_{1}}-m_{L}$ plane for $\tan\beta=10$. Further details are
same as those of Figure 21.

##  6 Conclusion

The Standard Model of particle physics has achieved a remarkable degree of
success ever since the discovery of the Higgs Boson. Apart from the above
discovery itself, the measurements of Higgs couplings, various results from
the ATLAS and the CMS experiments of the LHC, several results from low energy
precision measurements involving flavor physics also point towards a strong
validity of SM. However, theoretical issues such as the gauge hierarchy
problem, and observational evidences like the dark matter, baryon asymmetry,
non-zero neutrino mass etc. demands the SM to be extended beyond its current
boundary. Hence, a search for a BSM physics is highly relevant in the present
era. In this connection, we note that the recent Fermilab experiment on muon
$g-2$ measurement has confirmed the deviation from the SM value obtained in
the previous measurement at BNL. Assuming that the SM result that depends on
low energy $e^{+}e^{-}$ data will not get drastically altered in the near
future, or in other words, it would take a while for the lattice-based
computations of the hadronic error to reach a definite conclusion, it is
important to pursue the implications of the Fermilab announcement from the BSM
physics perspective.

The combined discrepancy of the Fermilab and the BNL data is at an impressive
level of 4.2$\sigma$. The anomaly amount itself is of the order of electroweak
corrections in SM. Thus, it is very crucial to probe a BSM physics scenario
that may produce a large contribution to the above observable at the same
energy scale. SUSY, undoubtedly one of the most important candidates for new
physics, is well-known to be capable of producing a large correction to
${(g-2)}_{\mu}$. There are regions of parameter space in MSSM with appropriate
wino, bino, higgsino and left and right handed smuon masses that can explain
the Fermilab result, typically for a large $\tan\beta$. There are also
dedicated beyond the MSSM models that can enhance the $(g-2)_{\mu}$
contribution by virtue of the model properties, including new symmetries. In
this analysis, staying within the MSSM sparticle spectra, we augment MSSM with
non-holomorphic trilinear soft breaking interactions. We also pointed out that
unlike MSSM soft terms, the non-holomorphic terms which for MSSM are soft SUSY
breaking in nature do not have the privilege of being supported by popular UV
complete models. The appropriate NH term generates an enhancement of the SUSY
contributions to $(g-2)_{\mu}$ via a coupling effect which helps to easily
accommodate the data. The model neither demands any flavor unfriendly choice
of different right and left slepton masses nor does it demand a large
$\tan\beta$ or a light chargino. One uses a large bino-smuon loop contribution
with stronger mixing of the L-R smuons to produce the desired effect. Using
the Fermilab data to constrain the model, we further impose constraints due to
dark matter related observables. It is found that a higgsino-dominated dark
matter having an underabundant contribution to the DM relic density can be
consistent with direct detection cross-section as well as ${(g-2)}_{\mu}$. For
${(g-2)}_{\mu}$, a scenario with dominant bino-smuon loop contribution can
satisfy the data via a large L-R mixing caused by the wrong Higgs couplings of
the non-holomorphic terms.

We further impose the relevant direct search bounds from the LHC, e.g. the
constraints from the slepton pair production searches from ATLAS as well as
the constraint coming from the compressed higgsino searches. The conclusion of
the combined ${(g-2)}_{\mu}$ analysis with dark matter and collider limits is
the following (Figure 22). For $\tan\beta=10$ and $40$, satisfying all the
constraints, valid $m_{\tilde{\chi}^{0}_{1}}$ ranges from 400 GeV to 750 GeV
for any value of $m_{L}$. On the other hand when $m_{L}>700$ GeV,
$m_{\tilde{\chi}^{0}_{1}}$ stays in between the lower limit of $\sim 185$ GeV
to the same upper limit namely 750 GeV. Apart from the above, the LSP mass
values for the valid parameter points outside and near the ATLAS provided
contour in its lower side are correlated with $m_{L}$ in an approximately
linear fashion. One finds for both the values of $\tan\beta$,
$m_{{\tilde{\chi}}^{0}_{1}}$ at its minimum can be around 400 GeV when
$m_{L}\simeq 600$ GeV.

We then extend the muon $g-2$ analysis in NHSSM to include the electron $g-2$
data as derived from the fine-structure constant measurement based on
${}^{133}{\rm Cs}$ matter-wave interferometry. The resulting ${(g-2)}_{e}$
shows an approximately $2.5\sigma$ level of discrepancy, but interestingly, it
comes with a negative sign. It becomes very challenging to accommodate the
muon and electron g-2 anomalies simultaneously in MSSM. This measurement is in
contrast to the ${}^{87}{\rm Rb}$-based measurement which shows a smaller
positive deviation. Here, we use the more unfriendly ${}^{133}{\rm Cs}$-based
data in our work to compare our results with previous MSSM-based analyses and
for probing the potential of the non-standard soft terms in relation to any
related phenomenology. The present work approaches to accommodate $a_{e}$ via
appropriately enhancing the Yukawa threshold corrections for $y_{e}$ by using
minimum possible $T^{\prime}_{e}$ values for the chosen mass spectra. The
analysis satisfies constraints from vacuum stability and charge and color
breaking minima. The negative sign for the SUSY contributions to $a_{e}$ could
be generated by negative $T^{\prime}_{e}$. An interesting feature of $a_{e}$
in NHSSM is that one cannot enhance it appreciably by increasing $\tan\beta$.
$a_{e}$ at best can be a slowly increasing function of $\tan\beta$, but unlike
MSSM, it is no longer proportional to $\tan\beta$, an effect that arises out
of a large threshold correction to $y_{e}$.

The conclusion of the combined ${(g-2)}_{\mu}$ and ${(g-2)}_{e}$ analysis with
dark matter and collider limits is the following (see Figure 21). For
$\tan\beta=10$, satisfying all the constraints, valid
$m_{\tilde{\chi}^{0}_{1}}$ ranges from 400 GeV to 500 GeV for any value of
$m_{L}$. On the other hand when $m_{L}>700$ GeV, $m_{\tilde{\chi}^{0}_{1}}$
stays in between the lower limit of $\sim$185 GeV to the same upper limit
namely 500 GeV. For $\tan\beta=40$, the ATLAS contour engulfs a lot of
parameter space with larger mass of the LSP leaving a small window of
$m_{\widetilde{\chi}_{1}^{0}}$ between 400 to 425 GeV for any $m_{L}$. For
$m_{L}>700$ GeV, the valid $m_{\tilde{\chi}^{0}_{1}}$ zone is from 185 GeV to
275 GeV. Apart from the above, the LSP mass values for valid parameter points
outside and near the ATLAS provided contour in its lower side are correlated
with $m_{L}$ in an approximately linear fashion. One finds that for
$\tan\beta=10$, $m_{{\tilde{\chi}}^{0}_{1}}$ can be as large as 410 GeV for
$m_{L}$ below 550 GeV. For $\tan\beta=40$, $m_{{\tilde{\chi}}^{0}_{1}}$ can be
as large as 425 GeV when $m_{L}$ is below 440 GeV. Finally, we presented two
representative points that satisfy all the constraints considered in this
analysis. We also discussed briefly the results of considering the limits of
$a_{e}$ as coming from the ${}^{87}{\rm Rb}$-based experiment indicating a
smaller deviation from the SM result.

## Acknowledgements

IA would like to thank the Council of Scientific and Industrial Research,
India for financial support. UC would like to acknowledge the use of the High
Performance Cluster at IACS. The work of MC is supported by the project
AstroCeNT: Particle Astrophysics Science and Technology Centre, carried out
within the International Research Agendas programme of the Foundation for
Polish Science financed by the European Union under the European Regional
Development Fund.

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