Muon g-2, Dark Matter and the Higgs mass in No-Scale Supergravity

Adam K. Forster⋆ 111E-mail<EMAIL_ADDRESS>and Stephen F. King⋆
222E-mail<EMAIL_ADDRESS>
⋆ Department of Physics and Astronomy, University of Southampton,

SO17 1BJ Southampton, United Kingdom

We discuss the phenomenology of no-scale supergravity (SUGRA), in which the
universal scalar mass is zero at the high scale, focussing on the recently
updated muon g-2 measurement, and including dark matter and the correct Higgs
boson mass. Such no-scale supergravity scenarios arise naturally from string
theory and are also inspired by the successful Starobinsky inflation, with a
class of minimal models leading to a strict upper bound on the gravitino mass
$m_{3/2}<10^{3}$ TeV. We perform a Monte Carlo scan over the allowed parameter
space, assuming a mixture of pure gravity mediated and universal gaugino
masses, using the SPheno package linked to FeynHiggs, MicrOmegas and
CheckMate, displaying the results in terms of a Likelihood function. We
present results for zero and non-zero trilinear soft parameters, and for
different signs of gaugino masses, giving a representative set of benchmark
points for each viable region of parameter space. We find that, while no-scale
SUGRA can readily satisfy the dark matter and Higgs boson mass requirements,
consistent with all other phenomenological constraints, the muon g-2
measurement may be accommodated only in certain regions of parameter space,
close to the LHC excluded regions for light sleptons and charginos.

###### Contents

  1. 1 Introduction
  2. 2 The no-scale SUGRA model parameters
  3. 3 Method
  4. 4 No-scale SUGRA with $A_{0}=0$ (Case I)
    1. 4.1 Positive $k$ (Scan 1)
    2. 4.2 Negative $k$ (Scan 2)
  5. 5 No-scale SUGRA with non-zero $A_{0}$ (Case II)
    1. 5.1 Positive $k$ (Scan 3)
    2. 5.2 Negative $k$ (Scan 4)
  6. 6 Conclusion
  7. 7 Acknowledgments

##  1 Introduction

Recent experimental results have propelled the anomalous muon magnetic moment
$g-2$ back into the centre stage of particle physics. Following the original
Brookhaven National Laboratory measurements [1], the Fermilab Muon
collaboration has recently affirmed these findings [2], with the combined
results now showing a $4.2\sigma$ discrepancy with the Standard Model (SM)
calculation [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23],

$a_{\mu}^{EXP}-a_{\mu}^{SM}=(2.51\pm 0.59)\times 10^{-9}$ (1)

where $a_{\mu}=\frac{g-2}{2}$. One of the classic explanations of the
discrepancy is provided by supersymmetry (SUSY), due to loops of light
sleptons, $\widetilde{\mu}$, $\widetilde{\nu}_{\mu}$, and light electroweak
gauginos [24]. These days such an explanation must be made consistent with the
measurement of the Higgs boson mass, and LHC constraints on superpartners,
both of which point towards rather heavy squarks and gluinos, and such studies
have been recently performed in various SUSY models [25, 26, 27, 28, 29, 30,
31, 32].

Here we shall be interested in the phenomenology of no-scale supergravity
(SUGRA) (for a review and early references see [33]), focussing on the
interplay between the muon $g-2$, dark matter and the Higgs boson mass in
particular. No-scale SUGRA is an ultraviolet (UV) completion of the minimal
supersymmetric standard model (MSSM), in which the scalar masses are zero at
the high scale and are subsequently generated by renormalisation group (RG)
running via couplings to the non-zero gaugino masses. Apart from its
minimality, it is motivated by string theory and more recently by the fact
that the hidden sector contains the ingredients for inflation. However, its
phenomenological viability is not straightforward, given the fact the Higgs
mass requires large stop masses, while the muon g-2 requires light slepton
masses, and dark matter is known to be non-trivial to achieve in the general
MSSM, and all this must be achieved in no-scale SUGRA starting from zero
scalar masses at the high scale. This provides the main motivation for a
detailed phenomenological study of no-scale SUGRA. In our approach, there will
be an important constraint coming from inflation, to which we now turn.

The idea of inflation is one of the key concepts in modern cosmology [34, 35,
36, 37, 38, 39]. Not only does it explain the vast size of the universe (the
flatness problem), it also explains the extreme homogeneity and isotropy of
the universe on cosmological scales (the horizon problem), as well as diluting
cosmological relics (the monopole problem). Furthermore, the slow rolling
inflaton provides small quantum fluctuations which eventually lead to large
scale structure [40, 41]. However, despite its successes, the precise
mechanism that causes inflation is unknown. Clues to the theory behind
inflation, may come from current observational data [42], where the spectral
index is measured to be $n_{s}\approx 0.96\pm 0.007$ with a low tensor-to-
scalar ratio $r<0.08$. These data exclude the simplest chaotic models with
inflaton potentials $\phi^{2}$ or $\phi^{4}$ [43]. Amongst the successful
models which are consistent with these data is Starobinsky inflation [36, 44],
which may have a link to SUSY models [45, 46, 47]. Since inflation is
sensitive to ultraviolet (UV) scales, one must also consider SUGRA when
dealing with inflation, and the no-scale SUGRA models [48] in particular are
well suited for maintaining the flatness of the inflaton potential (thereby
solving the $\eta$ problem), although other approaches have also been
discussed [49, 50, 51, 52, 53]. Also the Lyth bound [54] on the tensor-to-
scalar ratio also suggests a scale of inflation below the Planck scale,
leading to testable at collider tests of inflation.

Ellis, Nanopoulos, Olive (ENO) have shown that no-scale SUGRA can behave like
the successful Starobinsky inflationary model [55, 56, 57]. However, in the
ENO approach, a term with constant modular weight is used to break SUSY, and
there is no connection between inflation and SUSY breaking. Subsequently one
of us has considered the above ENO model, but with a linear Polonyi term added
to the superpotential [58]. The purpose of adding this term was to provide an
explicit mechanism for breaking SUSY in order to provide a link between
inflation and SUSY breaking. Indeed we showed that inflation requires a strict
upper bound for the gravitino mass $m_{3/2}<10^{3}$ TeV [58]. It was
subsequently shown [59] how the model in [58] may be generalised to include
the fields in the visible sector of the minimal supersymmetric standard model
(MSSM). In such a framework the soft-SUSY breaking parameters depend on the
modular weights in the superpotential and lead to new phenomenological
possibilities for supersymmetry (SUSY) breaking, based on generalisations of
no-scale SUSY breaking and pure gravity mediated SUSY breaking. The strict
upper limit on the gravitino mass $m_{3/2}<10^{3}$ TeV provides an important
phenomenological constraint, bearing in mind that the gaugino masses are
typically suppressed by a loop factor $1/(16\pi^{2})$ in gravity mediated
scenarios.

In the present paper, motivated by the desire to relate inflation to collider
physics, including dark matter and the muon $g-2$ measurement, we study the
phenomenology of the no-scale SUGRA inflation model in [59]. Although related
phenomenological studies of similar works have been undertaken [55, 56, 57] a
full phenomenological study of the model in [59], which is subject to the
upper bound on the gravitino mass, has not yet been undertaken. Here we shall
perform a phenomenological study of two of the simplest cases suggested in
[59]: the first case consisting of no-scale SUGRA with zero soft scalar masses
$m_{0}=0$ and zero trilinear soft parameter $A_{0}=0$, where the only source
of SUSY breaking in the visible sector is via the gaugino masses $M_{i}$; the
second case switches on a small soft trilinear mass $A_{0}\neq 0$, while
maintaining zero soft scalar masses $m_{0}=0$. In both cases we assume that
the gaugino masses $M_{i}$ (which are not fixed by the Kahler potential and
are therefore independent of the details of inflation) to arise from a hybrid
of anomaly mediated and universal sources [60]. Such gaugino masses $M_{i}$ at
the high scale will act as the seed of all soft squark and slepton masses at
low energy via renormalisation group (RG) running effects. Given the small
number of input parameters, we shall use a Monte Carlo scan over parameter
space, using the SPheno [61, 62] package linked to FeynHiggs [63], MicrOmegas
[64, 65, 66, 67, 68] and CheckMate [69, 70]. For each case above, we find
viable regions of parameter space displaying the results in terms of a
Likelihood function, including the requirement of successful dark matter relic
density. We then consider a set of representative benchmark points from viable
regions of parameter space, and discuss the prospects for discovering the
resulting SUSY spectrum at colliders. In particular we show that the no-scale
SUGRA case allows for dark matter while satisfying all phenomenological
constraints, including the correct Higgs mass and dark matter relic density,
leading to the prospect that SUSY may be discovered at the LHC or FCC.

The layout of the remainder of the paper is as follows. In section 2, we
summarise the basic parameters of the no-scale SUGRA models that we analyse.
In section 1 we describe our calculational approach and numerical tools and
algorithms that we employ in the analysis. In section 4 we present our results
for non-scale SUGRA case I, with all soft parameters equal to zero at the high
scale apart from the gaugino masses. In section 5 we present our results for
non-scale SUGRA case II, where we allow in addition (small) non-zero values of
$A_{0}$, which relaxes the collider constraints somewhat. Section 6 concludes
the paper.

##  2 The no-scale SUGRA model parameters

In the considered model [59], the soft supersymmetry breaking parameters are
well approximated by

$\displaystyle m_{0}$ $\displaystyle=0$ (2)
$\displaystyle\frac{A_{0}}{m_{\frac{3}{2}}}$ $\displaystyle=-6\alpha,$
$\displaystyle\frac{B_{0}}{m_{\frac{3}{2}}}$ $\displaystyle=2(1-\beta)$

where $\alpha,\beta$ are undetermined modular weights in the Kähler potential.

Turning to the gauginos, the model does not constrain the gaugino mass
parameters. We therefore examine a general breaking scenario where the gaugino
mass parameters are partly derived from "anomaly mediated" susy breaking
giving three mass parameters. Each one is determined partly by loop
corrections given below [71] and partly by a universal term parameterised
below by the dimensionless coefficient $k$ [72],

$\displaystyle M_{1}$
$\displaystyle=(\frac{33}{5}\frac{g_{1}^{2}}{16\pi^{2}}+k)m_{\frac{3}{2}},$
(3) $\displaystyle M_{2}$
$\displaystyle=(\frac{g_{2}^{2}}{16\pi^{2}}+k)m_{\frac{3}{2}},$ $\displaystyle
M_{3}$ $\displaystyle=(-3\frac{g_{3}^{2}}{16\pi^{2}}+k)m_{\frac{3}{2}}$

Note that the sign of $k$ plays an important phenomenological role in
determining the gaugino mass spectrum. For small and positive $k$ the
electroweak gaugino masses $M_{1,2}$ are enhanced while the magnitude of the
gluino mass $M_{3}$ is reduced due to the partial cancellation against the
negative anomaly mediated contribution. This yields a spectrum with a
relatively light gluino and heavy winos and binos. On the other hand, for
small and negative $k$, the electroweak gaugino masses $M_{1,2}$ are reduced
due to a partial cancellation, while the magnitude of the gluino mass $M_{3}$
is increased. This yields a spectrum with relatively light winos and binos
which is more suited to explaining the muon $g-2$.

The supersymmetric theory produced by this model gives a very small set of
high scale parameters; the scalar mass scale, $m_{0}$, the bilinear coupling,
$B_{0}$, and the trilinear coupling, $A_{0}$. In turn the two unified coupling
parameters are determined by the gravitino scale $m_{\frac{3}{2}}$ and our
choice of the modular weights $\alpha$ and $\beta$. Furthermore, $m_{0}$ is
determined by our choice of Kähler potential [59]. In addition we choose a
suitable value of $tan(\beta)$ and $\mu$ to minimise the broken higgs
potential and thus satisfy the higgs potential minimisation conditions given
in [73]. We are therefore left with just five input parameters; $\alpha$,
$\beta$, $tan(\beta)$, $m_{\frac{3}{2}}$ and $\mu$, that must be selected in
order to fully characterise the high scale model and its spontaneously broken
characteristics. We shall mainly focus on the simplest cases $\alpha=0$ and
$\beta=1$ which lead to zero soft parameters $A_{0}=0$ and $B_{0}=0$ at the
high scale.

##  3 Method

The aim of this project is to Monte-Carlo scan over the small set of input
parameters for the two cases given above. We do not use the Metropolis-
Hastings algorithm [74]. This is simply because the input parameter set is so
small that a random scan on the IRIDIS computer cluster will be suffice to
cover the parameter landscape. We then calculate various experimental outcomes
including collider phenomenology, mass spectrum, and dark matter relic density
to calculate a likelihood associated with each parameter point.

We hope that this will give us a greater understanding of the model and its
physical viability. We aim to place some lower bound on the gravitino mass
scale thus constraining the model from above, via inflation constraints, and
below, via Collider phenomenolgy. Furthermore, we hope to find some best fit
parameter points that might satisfy the latest $g-2$ results.

In order to understand the parameter space, we start with the simplest
possible case, and subsequently increase its complexity. Therefore, initially
we fix $\alpha=0$ such that $A_{0}=0$. As $B_{0}$ is determined by electroweak
symmetry breaking, we leave $\beta$ and $tan(\beta)$ as free parameters.
SPheno also requires the $sign(\mu)$ to be prescribed. Although previous
analyses suggest positive $\mu$ is more in-keeping with modern results, we
chose to allow for both signs.

Figure 1: Shows the algorithm flow used for this analyses.

Generally speaking, the algorithm works as shown in Fig. 1. Four primary open
source softwares were used to perform the calculations. For calculating the
SUSY spectrum, $g-2$, and a number of other , we chose to use a modified
version of SPheno that printed the value of $B_{0}$. Although for the most
part our use of this software was very standard, we encountered a technical
problem regarding the $\overline{DR}$ renormalisation scheme. Large values of
the gluino mass induce large logs that must be re-summed. To solve this we
calculate the Higgs mass on-shell using FeynHiggs. In order to calculate the
relic density, we use MicrOmegas.

For post data collection analysis we simulate collider effects using CheckMate
[70]. This software combines a number of subsidiary packages to simulate the
events in a collider and exclude any given model to a 95% confidence level. We
chose to use $13TeV$ and $8TeV$ $pp$ collision data with the ATLAS detector in
a signal region that focused on the search for squarks and gluinos in final
states with jets and missing transverse momentum using $36fb$. This parameter
space is commonly used in SUSY searches. However, further information could be
garnered by broadening the analysis.

Instead of predefining $\beta$, as we did with $\alpha$, and therefore fixing
a value of $B_{0}$, we decided to leave $tan(\beta)$ as a free parameter to be
scanned over. This affords us some flexibility in understanding the parameter
space. Furthermore, with the results in hand it is simple to solve Equation 2
and find the relevant $\beta$ parameter.

case | $m_{0}$ | $A_{0}$ | $B_{0}$ | Section
---|---|---|---|---
I | 0 | 0 | $2(1-\beta)m_{\frac{3}{2}}$ | 4
II | 0 | $-6\alpha m_{\frac{3}{2}}$ | $2(1-\beta)m_{\frac{3}{2}}$ | 5
Table 1: Table showing the two cases considered in this work. Gaugino mass
terms (with a value for $k$), $tan(\beta)$, the sign of $\mu$, and the
gravitino mass scale are also generated depending on the model being
considered.

In order to fully explore the parameter space, a number of scans (and
accordant parameter limits) were initialised. Each scan is assigned to a
subsection and summarised in Table 2. Note that each scan is performed twice;
once for $\mu>0$ and again for $\mu<0$.

| $m_{\frac{3}{2}}\ (\rm{TeV})$ | $\alpha$ | $tan(\beta)$ | $k$ | Subsection
---|---|---|---|---|---
Scan 1 | $[1,1000]$ | $0$ | $[1.5,30]$ | $[0,0.1]$ | 4.1
Scan 2 | $[1,400]$ | $0$ | $[1.5,50]$ | $[-0.01,-0.04]$ | 4.2
Scan 3 | $[1,1000]$ | $[-0.166,0.166]$ | $[1.5,50]$ | $[0,0.1]$ | 5.1
Scan 4 | $[1,200]$ | $[-0.005,0.005]$ | $[1.5,50]$ | $[-0.014,-0.035]$ | 5.2
Table 2: Show the parameters for the 4 primary scans conducted in this paper.
Links to the subsections in which each scan is presented are also included. In
general, parameter ranges were chosen by trial an error so as to be
representative of the parameter scape without losing excessive efficiency.

##  4 No-scale SUGRA with $A_{0}=0$ (Case I)

Completely no-scale SUGRA with $A_{0}=0$ represents a fascinating scheme for
this model. Previously thought to be ruled out, such scale-less supersymmetry
has seen something of a resurrection motivated by inflationary model building
[75]. As shown in Table 1, $\alpha=0$, and therefore $A_{0}=0$, precludes
trilinear terms from the model. Although we allow variance in $\beta$ in order
to understand the parameter space, we shall see a strong preference for
$\beta\approx 1$ implying $B_{0}=0$ bilinear coupling too. With $m_{0}=0$ the
scalar masses are all zero at the high scale, however, the gauginos get mass
terms by Equation 3 and generate non-zero low energy scalar masses via the
RGEs. Remarkably, it turns out that such a scheme is phenomenologically
viable, as we shall see.

We begin by analysing the results of Scan 1 in Table 2, which assumes positive
universal gaugino masses $k$, and allows the gravitino mass $m_{\frac{3}{2}}$
to vary up to its upper bound from inflation of $1000$ TeV.

###  4.1 Positive $k$ (Scan 1)

As discussed earlier, positive universal gaugino mass parameter $k$ will tend
to yield a spectrum with a relatively light gluino and relatively heavy winos
and binos, so we do not expect this choice to explain the muon $g-2$, so we
will focus mainly on the Higgs mass and dark matter in this case. Figure 2
shows a scatter plot between the two most influential scan variables, $k$ and
$m_{\frac{3}{2}}$. It should be noted that, although $tan(\beta)$ is a scan
parameter, its influence on the overall likelihood is limited.

(a) $\mu<0$, case I

(b) $\mu>0$, case I

Figure 2: Shows $k$ against $m_{\frac{3}{2}}$ for case I, $\alpha=0$ data from
a Monte-Carlo scan with parameter ranges ranges
$tan(\beta)\in[1.5,30],k\in[0,0.1]$, and
$m_{\frac{3}{2}}\in[10^{3}GeV,10^{6}GeV]$. Colour denotes likelihood, with
hotter colours corresponding with high likelihoods. The likelihood is
dominated by the relic density calculation. The range of $k$ is naturally
restricted to $k\lesssim 0.006$ due to $|\mu|^{2}<0$. Two bands of high
likelihood points; one from above, one from below, converge at
$(1000TeV,0.0044)$ in $(m_{\frac{3}{2}},k)$ space. The 6 benchmarks that are
presented in Table 3 are also marked. The central colour of the benchmark
denotes its likelihood.

As points that failed to produce correct electroweak symmetry breaking or a
suitable dark matter candidate have been excluded from these plots, a
fascinating structure of allowed, disallowed, and high likelihood points
emerges. Firstly, for values of $k\gtrsim 0.006$ electroweak symmetry breaking
cannot be satisfied (except for some anomalous points that, due to numeric
instability, achieve electroweak symmetry breaking). Secondly, for values of
$k\lesssim 0.004$ with $m_{\frac{3}{2}}\gtrsim 400$ TeV the LSP becomes
charged. This can be seen in Figure 3 where "cold" points tend to 0 mass
difference and therefore, a charged LSP for large $m_{\frac{3}{2}}$ values.

In addition to the structure of excluded points, a band of hot points in the
($m_{3/2},k)$ plane from (350 TeV, 0.0025) up to (250 TeV, 0.004) and across
to (1000 TeV, 0.0045) where the relic density likelihood is maximised. An
additional strip of high likelihood points begins at (400 TeV, 0.0055) and
tracks down to (1000 TeV, 0.0046). We speculate that there could be some
symmetry about $k\approx 0.0045$ that is broken above $k\approx 0.006$ due to
electroweak symmetry breaking.

(a) $\mu<0$, case I,

(b) $\mu>0$, case I,

Figure 3: Shows the mass difference between the LSP and the nLSP for case I,
$\alpha=0$. The colour denotes the value of $k$. Small values of k lead to a
negative difference between the $\widetilde{\chi}_{1+}$ and
$\widetilde{\chi}_{10}$ at $400TeV$ implying a charged LSP.

The masses of the first two neutralinos are given by

$\displaystyle m_{\widetilde{\chi}_{10}}=M_{1}-\frac{m_{W}^{2}(M_{1}+\mu
sin(2\beta))}{\mu^{2}-M_{1}^{2}}+...$ (4) $\displaystyle
m_{\widetilde{\chi}_{20}}=M_{2}-\frac{m_{W}^{2}(M_{2}+\mu
sin(2\beta))}{\mu^{2}-M_{2}^{2}}+...$

And the first chargino by,

$\displaystyle m_{\widetilde{\chi}_{1+}}=M_{2}-\frac{m_{W}^{2}(M_{2}+\mu
sin(2\beta))}{\mu^{2}-M_{2}^{2}}+...$ (5)

From Table 5 we can see that as $k$ reduces, the LSP becomes increasingly
"wino-like". Therefore, the mass of the lightest neutralino is dominated by
the mass of $m_{\widetilde{\chi}_{20}}$ (Equation 4). In turn, the mass of
lightest chargino is given, to leading order, by the same expression (Equation
5). Therefore, their masses are exceptionally close. Although this can be
helpful in reducing the relic density by some co-annihilation processes, for
high $m_{\frac{3}{2}}$ the contributing factors can lead to a switch in
hierarchy between the two mass states. This is further confirmed in Figure 3
where the mass difference between the LSP and the nLSP tends to 0 for small
$k$.

In this case $m_{0}=0$ and thus the susy scale is relatively low. Therefore,
it is important to consider the spectrum masses, their potential collider
signature, and the effect new SUSY diagrams could have on certain branching
ratios. We begin by looking at the mass distributions of the SUSY spectrum.

(a) $\mu<0$, case I

(b) $\mu>0$, case I

Figure 4: Shows the squark mass spectrum for case I, $\alpha=0$. Particles are
plotted in the same colour if they are sufficiently mass degenerate as to be
indistinguishable in this plot. Typically, this would entail masses within
15GeV of each other. Increasing $k$ increases the mass of the squarks. Low $k$
points cut-off just below $400TeV$.

From Fig 4, we see the mass distribution of the squarks with respect to
$m_{\frac{3}{2}}$. Although it is not explicitly plotted we can see the effect
of $k$ variance as a part of the mass distribution seems to cut-off just below
$400TeV$. This is a reflection of the point demonstrated in Figure 3, where
low $k$ points tend to having a charged LSP for large $m_{\frac{3}{2}}$. This
same dependance of $k$ can also be seen in Figures 5 and 6.

We see that a reduction in $k$ increases the overall scale of the squarks.
Their mass is given predominantly by $M_{3}$ contributions. Therefore, a
reduction of $k$ will lead to an increase in the absolute value of $M_{3}$ as,
by Equation 3, the anomaly mediated term is negative. This increase in mass
scale leads to an increase in the mass scale of the squarks. We also note that
the left handed $\widetilde{t}$ tends to be the the lightest squark with the
first generation squarks making up the heaviest squarks as is typical in such
models without flavour mixing.

(a) $\mu<0$, case I

(b) $\mu>0$, case I

Figure 5: As in Figure 4 but showing the slepton masses. Increasing $k$
decreases the mass of the sleptons. Low $k$ points cut-off just below
$400TeV$.

Unlike the squarks, a reduction in $k$ decreases the mass scale of the
sleptons. As they are uncharged under $SU(3)$, they receive their mass from
contribution from $M_{1}$ and $M_{2}$. Furthermore, their anomaly mediated
terms are positive, so a reduction in $k$ leads to a reduction in their
absolute scale and thus a reduction in the slepton masses. Analogously to the
squarks, the left handed $\widetilde{\tau}$ states tend to be the lightest
sleptons.

(a) $\mu<0$, case I

(b) $\mu>0$, case I

Figure 6: As in Figure 4 but showing the gaugino masses. Increasing $k$
decreases the mass of the gauginos. Low $k$ points cut-off just below
$400TeV$.

Perhaps predictably, given the scaleless nature of the model, Figure 6 shows
the most interesting structure with regard to the mass spectrum. Firstly, as
$m_{\frac{3}{2}}$ increases, the hierarchy of the heaviest neutralino and the
gluino reverse. Furthermore, as $k$ increases, the gluino mass increases
significantly. This can be understood by a similar argument to that of the
squarks; an increase in $k$ leads to a decrease in the absolute scale of
$M_{3}$ and thus a decrease in the gluino mass.

Furthermore, for large values of $k$ the first neutralino mass, $\chi_{10}$,
does not depend on $m_{\frac{3}{2}}$. The large values of $k$ increase the
proportion of the first neutralino that is "higgsino-like". As this mass
depends predominantly on $\mu$, this stabilises the mass of the neutralino
with respect to $m_{\frac{3}{2}}$. Conversely, small $k$ creates a "wino-like"
neutralino whose mass is proportional to $M_{2}$ and thus to
$m_{\frac{3}{2}}$.

(a) $\mu<0$, case I

(b) $\mu>0$, case I

Figure 7: Shows the values of the modular weight, $\beta$, produced for case
I, $\alpha=0$. There is a natural tendency for values very close to $1$.
According to Eq. 2, this suggests $B_{0}\approx 0$, in-keeping with scale-less
supersymmetry or no-scale SUGRA.

Due to the different natures of the model weights, we decided to allow
$tan(\beta)$, and therefore $B_{0}$ and $\beta$333$\beta$ is used both as the
inverse tangent of the ratio of the higgs vevs, and as the modular weight.
From context it should be clear which is being referred to., to vary as a free
parameter but keep $\alpha$ fixed. Therefore, it is interesting to see the
implications of the scan for the free $\beta$ parameter. Looking at Fig 7 we
find a striking prediction for the model. For both signs of $\mu$ this model
predicts that $\beta\approx 0.998$ (excluding some anomalous points). Recall
that, by Eq. 2, $\beta$ is connected to the bilinear coupling such that if
$\beta=1\implies B_{0}=0$. As can be seen, the model clearly favours a
bilinear coupling very close to 0 making a fully scale-less model.

(a) $\mu<0$, case I

(b) $\mu>0$, case I

Figure 8: Shows a scatter plot of the higgs mass against the anomalous muon
magnetic moment for a scan with ranges $tan(\beta)\in[1.5,30],k\in[0,0.1]$,
and $m_{\frac{3}{2}}\in[10^{3}GeV,10^{6}GeV]$. The $2\sigma$ region of
$a_{\mu}$ is marked with dotted lines while the $2\sigma$ region of $m_{h}$ is
marked with dashed lines. The colour denotes the relic density. No point
satisfies the observed values of the muon $g-2$.

From Figure 8 we see that this model cannot satisfy this condition for either
sign of $\mu$. In the $\mu<0$ case, we see that the contributions are in fact
negative. The contributions generally take the form $\Delta a_{\mu}\propto{\rm
sign}[\mu]\frac{1}{(m_{\widetilde{\mu},\widetilde{\nu}_{\mu}})^{2}}$.
Therefore, a change in the sign of $\mu$ reverses the sign of these
contributions. In general, large $m_{\frac{3}{2}}$ leads to a large mass
spectrum which in turn suppresses the $g-2$ contributions. From Figure 5 we
can see that the masses of the $\widetilde{\mu}$s and $\widetilde{\nu}_{\mu}$s
grow linearly with $m_{\frac{3}{2}}$ leading to quadratic suppression of
$\Delta a_{\mu}$. Although in the positive $\mu$ case, the contributions are
themselves positive, no point can satisfy the measured discrepancy.

To gain further insight into this model, a set of benchmark points is
presented. Table 3 shows the input parameters for each benchmark point
selected. The benchmark points were selected to reflect a variety of viable
regions in the model.

Benchmark points 1,2, and 3 all have their sign of $\mu$ positive while the
reverse is true for points 4,5, and 6. BP1 and BP4 are the highest likelihood
points of the model for their respective signs of $\mu$. BP4 comes from the
second band of high likelihood points at $k>0.0045$. They both have very high
$m_{\frac{3}{2}}$ values existing just below the cut-off set by the CMB in the
context of inflation [59]. BP2 and BP5 have the lowest values of
$m_{\frac{3}{2}}$ whilst staying in a high likelihood range and BP3 and BP6
have the lowest values of $k$ whilst staying in a high likelihood range.

Quantity |  BP1 |  BP2 |  BP3 |  BP4 |  BP5 |  BP6
---|---|---|---|---|---|---
$\alpha$ |  0 |  0 |  0 |  0 |  0 |  0
$\beta$ |  0.996 |  0.997 |  0.997 |  0.998 |  0.998 |  0.998
$m_{\frac{3}{2}}$ [TeV] |  903 |  268 |  359 |  995 |  269 |  358T
$k$ |  0.00443 |  0.00323 |  0.00251 |  0.00457 |  0.00324 |  0.0026
SPheno: |  |  |  |  |  |
$m_{0}$ [GeV] |  0 |  0 |  0 |  0 |  0 |  0
$tan(\beta)$ |  18.1 |  23.9 |  19.6 |  29.3 |  20.0 |  29.6
$sign(\mu)$ |  1 |  1 |  1 |  -1 |  -1 |  -1
$A_{0}$ [GeV] |  0 |  0 |  0 |  0 |  0 |  0
$M_{1}$ [GeV] |  20800 |  5700 |  7600 |  23000 |  5900 |  7600
$M_{2}$ [GeV] |  6800 |  1700 |  2000 |  7600 |  1700 |  2000
$M_{3}$ [GeV] |  -4400 |  -1600 |  -2400 |  -4700 |  -1600 |  -2400
Table 3: Shows six benchmark points representing six different areas of interest in the parameter where case I, $\alpha=0$. We present the model parameters and the resultant SPheno input parameters. BP1 and BP4 show the highest likelihood points for $\mu>0$ and $\mu<0$ respectively. BP2 and BP5 show points where $m_{\frac{3}{2}}$ is minimised whilst still satisfying our main constraints for $\mu>0$ and $\mu<0$ respectively. BP3 and BP6 show points for minimal values of $k$ whilst still satisfying our main constraints for $\mu>0$ and $\mu<0$ respectively. Dimensions of the parameters are given where "-" means dimensionless. $m_{\frac{3}{2}}$ is given in units of TeV, and $m_{0},A_{0},M_{i}$ are given in GeV. Masses |  BP1 |  BP2 |  BP3 |  BP4 |  BP5 |  BP6
---|---|---|---|---|---|---
$\widetilde{e}_{L}$ |  5690 |  1540 |  1900 |  6340 |  1540 |  1910
$\widetilde{e}_{R}$ |  7590 |  2160 |  2790 |  8400 |  2170 |  2790
$\widetilde{\nu}_{eL}$ |  5690 |  1530 |  1890 |  6340 |  1540 |  1910
$\widetilde{\mu}_{L}$ |  5690 |  1540 |  1900 |  6340 |  1540 |  1910
$\widetilde{\mu}_{R}$ |  7590 |  2160 |  2790 |  8400 |  2170 |  2790
$\widetilde{\nu}_{\mu L}$ |  5690 |  1530 |  1890 |  6330 |  1540 |  1910
$\widetilde{\tau}_{1}$ |  5690 |  1490 |  1860 |  6100 |  1510 |  1820
$\widetilde{\tau}_{2}$ |  7460 |  2100 |  2740 |  8040 |  2130 |  2680
$\widetilde{\nu}_{\tau L}$ |  5610 |  1490 |  1860 |  6100 |  1510 |  1820
$\widetilde{d}_{L}$ |  8580 |  3190 |  4500 |  9260 |  3190 |  4450
$\widetilde{d}_{R}$ |  7780 |  3060 |  4390 |  8310 |  3060 |  4330
$\widetilde{u}_{L}$ |  8580 |  3190 |  4500 |  9260 |  3190 |  4450
$\widetilde{u}_{R}$ |  8920 |  3300 |  4680 |  9620 |  3310 |  4630
$\widetilde{s}_{L}$ |  8580 |  3190 |  4500 |  9260 |  3190 |  4450
$\widetilde{s}_{R}$ |  7780 |  3060 |  4390 |  8310 |  3060 |  4330
$\widetilde{c}_{L}$ |  8580 |  3190 |  4500 |  9260 |  3190 |  4450
$\widetilde{c}_{R}$ |  8920 |  3300 |  4680 |  9620 |  3310 |  4630
$\widetilde{b}_{1}$ |  7670 |  2910 |  4160 |  8040 |  2940 |  4090
$\widetilde{b}_{2}$ |  7870 |  2980 |  4320 |  8420 |  3020 |  4230
$\widetilde{t}_{1}$ |  7600 |  2840 |  4050 |  8190 |  2840 |  4010
$\widetilde{t}_{2}$ |  7880 |  2940 |  4170 |  8420 |  2960 |  4110
$\widetilde{g}$ |  8940 |  3510 |  5080 |  9540 |  3510 |  5010
$\widetilde{\chi}_{10}$ |  1110 |  1370 |  1750 |  1110 |  1370 |  1770
$\widetilde{\chi}_{20}$ |  1111 |  1410 |  2170 |  1110 |  1400 |  2110
$\widetilde{\chi}_{30}$ |  5810 |  1500 |  2170 |  6520 |  1500 |  2110
$\widetilde{\chi}_{40}$ |  9650 |  2620 |  3440 |  10800 |  2640 |  3440
$\widetilde{\chi}_{1+}$ |  1110 |  1370 |  1750 |  1110 |  1370 |  1770
$\widetilde{\chi}_{2+}$ |  5810 |  1500 |  2180 |  6520 |  1500 |  2110
${h_{0}}$ |  124.5 |  120 |  122 |  125 |  121 |  123
${H_{0}}$ |  5510 |  1860 |  2690 |  5410 |  1970 |  2520
$A_{0}$ |  5510 |  1860 |  2690 |  5410 |  1970 |  2520
${H_{\pm}}$ |  5520 |  1870 |  2700 |  5410 |  1970 |  2520
$\mu$ |  1100 |  1400 |  2100 |  -1070 |  -1390 |  -2090
$B_{0}$ |  7610 |  1510 |  1970 |  3890 |  1240 |  1700
Table 4: Shows the spectrum of SUSY masses for the benchmark points given in
Table 3. The difference between the mass of $\widetilde{\chi}_{1+}$ and
$\widetilde{\chi}_{10}$ is also given as this pertains to the production of
dark matter. We also include the highscale bilinear coupling value $B_{0}$ for
its relevance to the high scale parameters of the model. All massses are given
in GeV.

Table 4 shows the resultant SUSY mass spectrum for the 6 benchmark points
presented so far. As changing the sign of $\mu$ plays little role, we will
refer to the combination of BP1 and BP4 as BP14, and similarly for the other
benchmark points. Many points do not achieve a the precise value for the higgs
mass. However, there is a relatively large theoretical uncertainty on this
parameter leaving all these points with relatively high likelihoods.

BP14 shows a hugely inflated mass spectrum for the squarks and sleptons.
Furthermore, we see huge masses for the heaviest gauginos. However, although
much of the spectrum is high in mass, the lighter gauginos are only about
1TeV. These could leave a tell tale signature in current or future colliders.
We also see a relatively low $\mu$ value. This causes the high "higgsino-like"
proportions of the LSP. $\widetilde{\nu}_{\tau L}$ is the smallest
supersymmetric scalar (excluding the higgs boson).

BP25 shows a significantly reduced SUSY spectrum due to the reduced scale of
$m_{\frac{3}{2}}$ and therefore the reduction in $M_{1,2,3}$. Although the
majority of the spectrum is still mostly out of range of modern detectors, the
model still produces low mass light gauginos. Although the mass gap between
$\widetilde{\chi}_{10}$ and $\widetilde{\chi}_{1+}$ is still small, the gap
between the first two charged states has increased due to a change in the
mixing of the gauginos. Again, $\widetilde{\nu}_{\tau L}$ is the smallest
supersymmetric scalar.

Finally, BP36 shows a slight increase in the overall SUSY scale in comparison
to the BP25. Although the mass gap between the heavy and light gaugino states
has increased, the overall scale for the states has increased significantly as
the LSP becomes almost exclusively "wino-like". In general, a high wino-like
state can lead to excessive co-annihilation with the first chargino;
especially when, as can be seen in Table 5, the mass gap between said states
is so low. However, for sufficiently high mass neutralinos, the freeze out
temperature is high thus ending these problematic processes early in the
universes cosmological past and thereby preventing the excessive annihilation
of the candidate. Although $\widetilde{\chi}_{1+}$ is still the second lowest
mass particle, the $\widetilde{\nu}_{\tau L}$ is lighter than the
$\widetilde{\chi}_{20}$ unlike in th other benchmark points where a clear
hierarchy existed between the fermionic and scalar states.

Quantity |  BP1 |  BP2 |  BP3 |  BP4 |  BP5 |  BP6
---|---|---|---|---|---|---
$\Omega_{DM}h^{2}$ |  0.119 |  0.124 |  0.115 |  0.119 |  0.119 |  0.124
$\widetilde{\chi}_{10}$ [GeV] |  1109.5 |  1372.7 |  1748.3 |  1110.5 |  1372.1 |  1767.0
$\widetilde{\chi}_{1+}-\widetilde{\chi}_{10}$ [GeV] |  0.48072 |  1.5 |  0.0329 |  0.59656 |  1.699 |  0.03567
$|\alpha_{1}|^{2}$ |  0.000016 |  0.000441 |  0.000004 |  0.000009 |  0.0004 |  0.000004
$|\alpha_{2}|^{2}$ |  0.000144 |  0.312481 |  0.982081 |  0.0001 |  0.228484 |  0.976144
$|\alpha_{3}|^{2}$ |  0.499849 |  0.352836 |  0.011025 |  0.499849 |  0.394384 |  0.014161
$|\alpha_{4}|^{2}$ |  0.499849 |  0.335241 |  0.007396 |  0.499849 |  0.375769 |  0.009604
Table 5: Shows the relic density of the LSP for each benchmark point. The
difference between the LSP and the nLSP is also given. Finally, we give the
probability of finding the given LSP in a particular flavour state. That is to
say; we give $|\alpha_{i}|^{2}$ where
$\widetilde{\chi}_{10}=\alpha_{1}\widetilde{B}+\alpha_{2}\widetilde{W}+\alpha_{3}\widetilde{H}_{1}+\alpha_{4}\widetilde{H}_{2}$
and $\sum|\alpha_{i}|^{2}=1$. Dimensionful parameters are given in GeV.

Table 5 shows information specifically regarding the relic density for the
benchmarks points. In all cases we see a very compressed gaugino mass spectrum
inducing the requisite co-annihilarion processes. BP36 shows an exceptionally
compressed mass gap. As was previously argued, the high mass leads to an early
freeze out temperature preventing these co-annihilartion processes eradicating
the dark matter too efficiently. We also present the proportion of bino, wino,
and higgsino for the given particle. We see a very interesting shift between
the respective benchmark points with BP14 being mostly higgsino, BP25 being
evenly split between higgsino and wino, and BP36 being majority wino.

Tables 6, 7, 8 show phenomenological information regarding potential
constraints and collider physics. BP1 and BP4, BP2 and BP5 , and BP3 and BP6
are paired together as they belong to similar regions of parameter space. We
implemented a checkmate analysis for both 8, 13, and 14 TeV ATLAS and CMS
analyses. However, we chose not to include 14TeV analysis in our tables as
they are not based on LHC runs but rather Monte-Carlo simulations. Instead,
where appropriate, we have simply noted the $r_{max}$ value (defined below)
produced as suggestive of the type of the effects future colliders could have.
We use MadGraph_2.6.7 [76, 77] to generate events with SUSY final states.
Pythia_8.2.45 [78, 79] is then used to shower and hadronise the events.
Finally, Delphes_3.4.2 [80] and some subsidiary tools [81, 82, 83, 84] are
used to perform event and detector analysis. This approach allows us to assess
a given benchmark points viability in comparison with experimental data. We
find that all presented benchmark points cannot be ruled out by the LHC at
$\sqrt{s}\leq 13TeV$. We include a quantity $r_{max}$ defined by

$r_{max}=\frac{S-1.64\cdot\Delta S}{S95}$ (6)

where $S$ is the number of signal events, $\Delta S$ is its uncertainty, and
$S95$ is the experimental upper limit on the number of signal events. This
quantity indicates whether a point is ruled out by the analyses or not. Points
with $r_{max}\geq 1$ are ruled out; those with $r_{max}<1$ are not. We also
include the total proton to proton collision cross section, $\sigma_{LO}$ at
the given centre of mass energy. Finally, the most important signal regions
analysed by CheckMATE are given. We find a variety of different analyses are
important due to the changing mixing matrices and SUSY spectrum.

We include a number important beyond the standard model constraints; $\Delta
a_{\mu}$, BR $(B_{s}\rightarrow\mu^{+}\>\mu^{-})$, the relic density, and BR
$(b\rightarrow s\>\gamma)$. We find that the two branching ratios agree with
experimental data very well. However, the lack of light $\widetilde{\mu}$s and
$\widetilde{\nu}_{\mu}$s lead to insufficient contributions to the $\Delta
a_{\mu}$ loop diagrams as these are suppressed by the propagator mass squared.
Indeed, BP4, 5, and 6 have negative contributions to the anomalous muon
magnetic moment. This is due to the reversal of sign of the $\mu$ parameter,
and the dependancy on $\mu$ in the gaugino mediated diagrams that contribute
to $\Delta a_{\mu}$. As the relic density is our strongest constraint, all
points have been chosen to satisfy modern cosmological constraints on the
production of dark matter.

Finally, we include the decay width and branching ratios of the two lightest
particles (excluding the LSP who is stable). In BP14 and BP25 these are light
gaugino states. However, in BP36, the $\widetilde{\nu}_{\tau}$ is lighter than
the $\chi_{20}$. Only branching ratios that contribute by more than 1% are
included.

Although BP14 shows a long lifetime it is drastically insufficient for the
particle to escape the detector. Therefore, the particles decay products are
very important in assessing the signature of this model. The analysis region
of most significance is one that focuses on mono-jets. This analysis has a
very large luminosity. It should be noted that, unlike BP25 and BP36, 14TeV
analysis yields an $r_{max}$ value of $0$, perhaps due to the lack of
significance at such high energy scales. The branching ratios for both the two
lightest non-LSP particles are dominated by $\widetilde{\chi}_{10}$ decays.
This would suggest large corresponding missing momenta. We find an excellent
fit for the BSM constraints as well as dark matter. However, $\Delta a_{\mu}$
cannot be satisfied.

Quantity |  BP1 |  BP4
---|---|---
$\Gamma\chi_{1+}$ [GeV] |  $8.4\times 10^{-14}$ |  $1.7\times 10^{-13}$
BR $(\chi_{1+}\rightarrow\chi_{10}\>\pi^{+})$ [%] |  92.9 |  89.3
BR $(\chi_{1+}\rightarrow\chi_{10}\>e^{+}\>\nu_{e})$ [%] |  4.00 |  5.71
BR $(\chi_{1+}\rightarrow\chi_{10}\>\mu^{+}\>\nu_{\mu})$ [%] |  3.14 |  4.90
$\Gamma\chi_{20}$ [GeV] |  $5.12\times 10^{-12}$ |  $2.9\times 10^{-12}$
BR $(\chi_{20}\rightarrow\chi_{10}\>\pi^{0})$ [%] |  63.8 |  74
BR $(\chi_{20}\rightarrow\chi_{1+}\>\pi^{-})$ [%] |  6.1 |  2.64
BR $(\chi_{20}\rightarrow\chi_{1-}\>\pi^{+})$ [%] |  6.1 |  2.64
BR $(\chi_{20}\rightarrow\chi_{10}\>\gamma)$ [%] |  4.9 |  5.79
BR $(\chi_{20}\rightarrow\chi_{10}\>e^{-}\>e^{+})$ [%] |  2.0 |  1.81
BR $(\chi_{20}\rightarrow\chi_{10}\>\mu^{-}\>\mu^{+})$ [%] |  2.0 |  1.72
BR $(\chi_{20}\rightarrow\chi_{10}\>\nu_{e}\>\overline{\nu_{e}})$ [%] |  2.0 |  10.8
BR $(b\rightarrow s\>\gamma)$ [%] |  0.032 |  0.032
BR $(B_{s}\rightarrow\mu^{+}\>\mu^{-})$ [%] |  $2.9\times 10^{-7}$ |  $2.9\times 10^{-7}$
$\Delta a_{\mu}$ |  $8.39\times 10^{-12}$ |  $-1.15\times 10^{-11}$
$\Omega_{DM}h^{2}$ |  0.119 |  0.119
$\chi_{10}$ [GeV] |  1110 |  1110
$\sigma_{q\overline{q}\rightarrow\chi_{10}\chi_{10}}$ [pb] |  $0$ |  $0$
$r_{max}$ |  $3.72\times 10^{-4}$ |  $2.03\times 10^{-4}$
$\sqrt{s}$ [TeV] |  13 |  13
Analysis |  atlas_conf_2017_060 |  atlas_conf_2017_060
Signal Region |  EM7 |  IM6
Ref. |  [85] |  [85]
$\sigma_{LO}$ [pb] |  $4.126\times 10^{-4}$ |  $4.036\times 10^{-4}$
Table 6: Shows branching ratios for lightest supersymmetric particles in the spectrum for the benchmarks points with highest likelihood, BP1 and BP4. Only branching ratios greater than $1\%$ are included. We also include some the the beyond the standard model observables BR $(b\rightarrow s\>\gamma)$, BR $(B_{s}\rightarrow\mu^{+}\>\mu^{-})$, $\Delta a_{\mu}$, and $\Omega_{DM}h^{2}$, where $\Delta a_{\mu}$ is a calculation of the SUSY contribution beyond the standard model. The model successfully predicts the b decays discrepancy as well as the relic density. However, the anomalous muon magnetic moment cannot be satisfied. CheckMATE runs using 13TeV and 8TeV analyses cannot rule out these points. The cross section $\sigma_{LO}$ is calculated by pythia as a summation of subprocesses giving the total p+ p+ cross section. Decay widths and masses are given in GeV, branching ratios are given in %, and cross sections are given in pb.  Quantity |  BP2 |  BP5
---|---|---
$\Gamma\chi_{1+}$ [GeV] |  $8.9\times 10^{-12}$ |  $1.4\times 10^{-11}$
BR $(\chi_{1+}\rightarrow\chi_{10}\>\overline{d}\>u)$ [%] |  60.0 |  58.9
BR $(\chi_{1+}\rightarrow\chi_{10}\>\overline{s}\>c)$ [%] |  0.34 |  2.02
BR $(\chi_{1+}\rightarrow\chi_{10}\>e^{+}\nu_{e})$ [%] |  20.1 |  19.7
BR $(\chi_{1+}\rightarrow\chi_{10}\>\mu^{+}\nu_{\mu})$ [%] |  19.6 |  19.4
$\Gamma\chi_{20}$ [GeV] |  $2.45\times 10^{-4}$ |  $6.1\times 10^{-5}$
BR $(\chi_{20}\rightarrow\chi_{10}\>u\>\overline{u})$ [%] |  3.92 |  4.22
BR $(\chi_{20}\rightarrow\chi_{10}\>c\>\overline{c})$ [%] |  3.90 |  4.18
BR $(\chi_{20}\rightarrow\chi_{10}\>d\>\overline{d})$ [%] |  5.09 |  5.49
BR $(\chi_{20}\rightarrow\chi_{10}\>s\>\overline{s})$ [%] |  5.09 |  5.49
BR $(\chi_{20}\rightarrow\chi_{10}\>b\>\overline{b})$ [%] |  4.68 |  4.74
BR $(\chi_{20}\rightarrow\chi_{10}\>e^{-}\>e^{+})$ [%] |  1.17 |  1.26
BR $(\chi_{20}\rightarrow\chi_{10}\>\mu^{-}\>\mu^{+})$ [%] |  1.17 |  1.26
BR $(\chi_{20}\rightarrow\chi_{10}\>\tau^{-}\>\tau^{+})$ [%] |  1.16 |  1.24
BR $(\chi_{20}\rightarrow\chi_{10}\>\nu_{e}\>\overline{\nu_{e}})$ [%] |  7.03 |  7.57
BR $(\chi_{20}\rightarrow\chi_{1+}\>d\>\overline{u})$ [%] |  11.2 |  10.8
BR $(\chi_{20}\rightarrow\chi_{1-}\>\overline{d}\>u)$ [%] |  11.2 |  10.8
BR $(\chi_{20}\rightarrow\chi_{1+}\>s\>\overline{c})$ [%] |  11.1 |  10.7
BR $(\chi_{20}\rightarrow\chi_{1-}\>\overline{s}\>c)$ [%] |  11.1 |  10.7
BR $(\chi_{20}\rightarrow\chi_{1+}\>e^{-}\>\overline{\nu_{e}})$ [%] |  3.72 |  3.60
BR $(\chi_{20}\rightarrow\chi_{1-}\>e^{+}\>\nu_{e})$ [%] |  3.72 |  3.60
BR $(\chi_{20}\rightarrow\chi_{1+}\>\mu^{-}\>\overline{\nu_{\mu}})$ [%] |  3.72 |  3.60
BR $(\chi_{20}\rightarrow\chi_{1-}\>\mu^{+}\>\nu_{\mu})$ [%] |  3.72 |  3.60
BR $(\chi_{20}\rightarrow\chi_{1+}\>\tau^{-}\>\overline{\nu_{\tau}})$ [%] |  3.68 |  3.54
BR $(\chi_{20}\rightarrow\chi_{1-}\>\tau^{+}\>\nu_{\tau})$ [%] |  3.68 |  3.54
BR $(b\rightarrow s\>\gamma)$ [%] |  0.033 |  0.033
BR $(B_{s}\rightarrow\mu^{+}\>\mu^{-})$ [%] |  $2.9\times 10^{-7}$ |  $3.0\times 10^{-7}$
$\Delta a_{\mu}$ |  $1.15\times 10^{-10}$ |  $-9.6\times 10^{-11}$
$\Omega_{DM}h^{2}$ |  0.124 |  0.119
$\chi_{10}$ [GeV] |  1370 |  1370
$\sigma_{q\overline{q}\rightarrow\chi_{10}\chi_{10}}$ [pb] |  $8.629\times 10^{-8}$ |  $2.718\times 10^{-8}$
$r_{max}$ |  $1.12\times 10^{-2}$ |  $8.10\times 10^{-3}$
$\sqrt{s}$ [TeV] |  13TeV |  13TeV
Analysis |  atlas_1712_02332 |  atlas_1712_02332
Signal Region |  2j-3600 |  2j-3600
Ref. |  [86] |  [86]
$\sigma_{LO}$ [pb] |  $1.611\times 10^{-4}$ |  $1.630\times 10^{-4}$
Table 7: As in Table 6 but for BP2 and BP5; points with low $m_{\frac{3}{2}}$.

The most constraining region for BP25 focuses on squarks and gluinos, with 0
leptons, and 2-6 jets at 13Tev. This is due to the relatively low mass of the
strongly coupled particles. We see the highest value of $r_{max}$ of all the
benchmark points due to the abundance of these lighter particles.
$\widetilde{\chi}_{20}$ shows a huge variety in decay channels as the mass gap
is too small for the squark pairs to hadronise. We find that 14TeV analysis
yields very high $r_{max}$ values of 0.276 in the positive $\mu$ case. This
suggests that future colliders could probe regions of interest in this model.
Again, the relic and BSM branching ratios can be fitted well, but $\Delta
a_{\mu}$ cannot.

As previously eluded to, BP36 shows a change in the spectrum hierarchy. From
Table 5, the mass difference between $\widetilde{\chi}_{10}$ and
$\widetilde{\chi}_{1+}$ is the smallest. Therefore, a smaller phase space is
available leading to fewer decay channels. In contrast to
$\widetilde{\nu_{\tau}}$, whose lifetime is very small, the lifetime of
$\chi_{1+}$ is sufficiently long that the particle could escape the detector.
As it is a charged particle, this would appear as a charge track in the
calorimeter. As was the case for BP25, 14TeV analysis gives a whole order of
magnitude increase in $r_{max}$ hinting at the exciting prospects for physics
to come. Again, the BSM constraints and the relic are satisfied; however,
$\Delta a_{\mu}$ is not.

Quantity |  BP3 |  BP6
---|---|---
$\Gamma\chi_{1+}$ [GeV] |  $7.2\times 10^{-20}$ |  $1.0\times 10^{-19}$
BR $(\chi_{1+}\rightarrow\chi_{10}\>\overline{d}\>u)$ [%] |  72 |  72
BR $(\chi_{1+}\rightarrow\chi_{10}\>e^{+}\>\nu_{e})$ [%] |  28 |  28
$\Gamma\widetilde{\nu_{\tau}}$ [GeV] |  $2.9\times 10^{-1}$ |  $7.7\times 10^{-2}$
BR $(\widetilde{\nu_{\tau}}\rightarrow\chi_{10}\>\nu_{\tau})$ [%] |  33.3 |  33.0
BR $(\widetilde{\nu_{\tau}}\rightarrow\chi_{1+}\>\tau^{-})$ [%] |  66.7 |  67.0
BR $(b\rightarrow s\>\gamma)$ [%] |  0.032 |  0.032
BR $(B_{s}\rightarrow\mu^{+}\>\mu^{-})$ [%] |  $2.9\times 10^{-7}$ |  $3.0\times 10^{-7}$
$\Delta a_{\mu}$ |  $5.50\times 10^{-11}$ |  $-8.30\times 10^{-11}$
$\Omega_{DM}h^{2}$ |  0.115 |  0.124
$\chi_{10}$ [GeV] |  1750 |  1750
$\sigma_{q\overline{q}\rightarrow\chi_{10}\chi_{10}}$ [pb] |  $7.805\times 10^{-8}$ |  $6.467\times 10^{-8}$
$r_{max}$ |  $3.458\times 10^{-3}$ |  $3.189\times 10^{-3}$
$\sqrt{s}$ [TeV] |  13TeV |  13TeV
Analysis |  atlas_conf_2017_060 |  atlas_conf_2017_060
Signal Region |  EM10 |  EM10
Ref. |  [85] |  [85]
$\sigma_{LO}$ [pb] |  $1.520\times 10^{-5}$ |  $1.422\times 10^{-5}$
Table 8: As in Table 6 but for BP3 and BP6; points with low $k$.

###  4.2 Negative $k$ (Scan 2)

In this case we expect a negative universal gaugino mass parameter $k$ to tend
to yield a spectrum with a heavy gluino and relatively light winos and binos,
possibly suitable to explain the muon $g-2$, as well as the Higgs mass and
dark matter. Moreover, in order to explain the measured $\Delta a_{\mu}$,
small values of the $\widetilde{\mu}$ mass are required. As seen in Figures 4,
5, and 6, a reduction in $k$ reduces the slepton masses, whilst increasing the
gluon and squark masses. Further reductions in $k$ reduce the
$\widetilde{\mu}$ mass sufficiently but keep the mass of the higgs boson high
as this is dependant on the SU(3) charged squarks and sleptons.

In Scan 2, the $k$ parameter is varied from $0$ to $-0.04$. As noted, negative
values of $k$ will increase the absolute scale of $M_{3}$ but reduce the scale
of $M_{2}$. This will further contribute to the effect described above,
increasing squark masses whilst decreasing slepton masses. As previously
argued, the $\Delta a_{\mu}$ contributions depend on
$m_{\widetilde{\mu}}^{-2}$ and therefore large values of $m_{\frac{3}{2}}$
will lead to a suppression of the contributions. We therefore scan for lower
values of $m_{\frac{3}{2}}$ between 0 and 400 TeV.

Fig 9 shows the distribution of $k$ against $m_{\frac{3}{2}}$ for both signs
of $\mu$. As we are now focussing on $a_{\mu}$ we redefine the likelihood as
$L=L_{a_{\mu}}\times L_{m_{h}}$. However, we do not want relic particles whose
abundance would rule out the model entirely. Therefore, we impose the
condition that $\Omega h^{2}<0.12+2\times 0.0078$ such that no point can be
ruled out by leaving a non-phenomenologically large relic abundance.

From Fig 9, we see two distinct areas of points that give viable results;
$k\approx-0.016$ and $k\approx-0.023$. However we find that $k\approx-0.023$
with $\mu<0$ is phenomenologically preferred. In this region, we also find
$30TeV<m_{\frac{3}{2}}<100TeV$ to be preferred.

(a) $\mu<0$, case I

(b) $\mu>0$, case I

Figure 9: Shows the distribution of $k$ against $m_{\frac{3}{2}}$ with
$a_{\mu}$ and $m_{h}$ log likelihood for $m_{\frac{3}{2}}\in[0TeV,400TeV]$,
$\alpha=0$ and $k\in[-0.035,-0.014]$. A region with $\mu<0$, $k\approx-0.023$
and $m_{\frac{3}{2}}\in[40TeV,60Tev]$ is preferred.

Finally, we present the results of a scan focussing on the good region as seen
in Figure 10 where k varies between -0.022 and -0.027 and the sign of $\mu$ is
fixed as negative. Negative results for $\Delta a_{\mu}$ are now ruled out as
the absolute value of the gaugino mass parameters is large, suppressing some
key SUSY contributions to $(g-2)$ that involve these particles. This leaves us
with one first order loop diagram that contributes to this result whose sign
is fixed by the sign of $\mu$.

(a) $\mu<0$, case I

(b) $\mu<0$, case I, zoomed

Figure 10: Shows a scatter plot of $a_{\mu}$ and $m_{h}$ for a focussed scan
with ranges $m_{\frac{3}{2}}\in[0TeV,160TeV]$, $k\in[-0.022,-0.027]$ and
$tan(\beta)\in[2,12]$. The $2\sigma$ region of $a_{\mu}$ is marked with dotted
lines while the $2\sigma$ region of $m_{h}$ is marked with dashed lines. The
colour denotes the relic density.

Figure 10 shows the distribution of $\Delta a_{\mu}$ against the higgs mass.
We see many points sit within the $2\sigma$ region. Furthermore, we find that
many of these points have excellent dark matter relic densities. Two benchmark
points are marked on Figure 10 who are presented later.

Until now, the relic abundance of dark matter has played a pivotal role in
assessing the veracity of any given point. However, we find large parts of the
parameter space give the right handed stau state as the lightest sparticle. In
all preceding plots and in Figure 10 we present results where the neutralino
is the LSP. However, it is also interesting to consider potential R-parity
violating models in which these light, charged particles decay into standard
model particles. Although a detailed discussion of the viability of such
points is beyond the scope of this paper, we include Figure 11 as these states
can lead to excellent fits for $a_{\mu}$ and $m_{h}$.

(a) $\mu<0$, case I

(b) $\mu<0$, case I, zoomed

Figure 11: As for Fig 10 but where points that have charged LSP states are
shown. Such states are marked in blue to indicate that no relic density could
be calculated.

Continuing our analysis, we took a number of points from the allowed region
and ran them through CheckMATE. Finding that a sample of points were excluded
we wrote a code to systematically check points in the allowed region. We found
that all points were either excluded by CheckMATE or in a region near
exclusion (allowing for uncertainties in the analysis). We include some of the
more promising benchmark points below. It should be noted that we ran
CheckMATE using the default 5000 points. However, if a warning was presented
speculating that more data could exclude the point in question we increased
the number of events by the specified amount.

In Table 9 we present the input parameters for two benchmarks points from Case
I with negative $k$ values. One point was selected for its relatively high
values of $\Delta a_{\mu}$ whilst the other was selected for it’s relatively
high value of $\Omega h^{2}$. Both points are $k=-0.023$ in accordance with
the highest likelihood region. Indeed, both points are very similar in a
variety of ways suggesting a small input deviations can give large variance in
output; particularly for the LSP relic abundance.

Quantity |  BP7 |  BP8
---|---|---
$\alpha$ |  0 |  0
$\beta$ |  1 |  1
$m_{\frac{3}{2}}$ [TeV] |  51 |  57
$k$ |  -0.023 |  -0.023
SPheno: |  |
$m_{0}$ [GeV] |  0 |  0
$tan(\beta)$ |  9.72 |  9.37
$sign(\mu)$ |  -1 |  -1
$A_{0}$ [GeV] |  0 |  0
$M_{1}$ [GeV] |  -263 |  -263
$M_{2}$ [GeV] |  -1058 |  -1152
$M_{3}$ [GeV] |  -1694 |  -1863
Table 9: Shows two benchmark points representing two different areas of
interest in the parameter space. We present the model parameters and the
resultant SPheno input parameters. BP7 shows a point with high $a_{\mu}$ and
BP8 shows a point with high $\Omega$. $m_{\frac{3}{2}}$ is given in TeV and
all other dimensionful parameters are given in GeV.

Table 10 shows the SUSY spectrum for benchmark points 7 and 8. Strikingly, we
see that the lightest stau state is the nLSP behind the usual neutralino LSP.
This ofcourse has implications for the most powerful diagrams for the relic
density. Furthermore, the nnLSP is given by $\widetilde{\mu}_{R}$ as
anticipated by our efforts to generate a high $\Delta a_{\mu}$ value using
diagrams involving this particle. Unlike the sleptons, the squarks are very
high in mass contributing to the phenomenologically viable higgs mass. In both
cases $\widetilde{\tau}_{1}-\widetilde{\chi}_{10}\approx 10GeV$. This is
important for critical co-annihilation diagrams for the relic density. Again
we see that $B_{0}$ is very small in comparison with $m_{\frac{3}{2}}$
suggesting a fully scale-less model.

Masses |  BP7 |  BP8
---|---|---
$\widetilde{e}_{L}$ |  661 |  717
$\widetilde{e}_{R}$ |  133 |  138
$\widetilde{\nu}_{eL}$ |  656 |  712
$\widetilde{\mu}_{L}$ |  661 |  717
$\widetilde{\mu}_{R}$ |  133 |  137
$\widetilde{\nu}_{\mu L}$ |  656 |  712
$\widetilde{\tau}_{1}$ |  104 |  109
$\widetilde{\tau}_{2}$ |  661 |  717
$\widetilde{\nu}_{\tau L}$ |  654 |  710
$\widetilde{d}_{L}$ |  3110 |  3390
$\widetilde{d}_{R}$ |  3050 |  3330
$\widetilde{u}_{L}$ |  3110 |  3390
$\widetilde{u}_{R}$ |  3050 |  3320
$\widetilde{s}_{L}$ |  3110 |  3390
$\widetilde{s}_{R}$ |  3050 |  3330
$\widetilde{c}_{L}$ |  3110 |  3390
$\widetilde{c}_{R}$ |  3050 |  3320
$\widetilde{b}_{1}$ |  2890 |  3150
$\widetilde{b}_{2}$ |  3040 |  3310
$\widetilde{t}_{1}$ |  2580 |  2820
$\widetilde{t}_{2}$ |  2910 |  3170
$\widetilde{g}$ |  3560 |  3890
$\widetilde{\chi}_{10}$ |  100 |  99
$\widetilde{\chi}_{20}$ |  855 |  933
$\widetilde{\chi}_{30}$ |  1890 |  2060
$\widetilde{\chi}_{40}$ |  1890 |  2060
$\widetilde{\chi}_{1+}$ |  855 |  933
$\widetilde{\chi}_{2+}$ |  1890 |  2060
${h_{0}}$ |  121.5 |  122
${H_{0}}$ |  1990 |  2170
${A_{0}}$ |  1990 |  2170
${H_{\pm}}$ |  2000 |  2170
$\mu$ |  -1777 |  -1931
$B_{0}$ |  -12 |  -26.5
$\widetilde{\tau}_{1}-\widetilde{\chi}_{10}$ |  4.2 |  9.6
Table 10: Shows the spectrum of SUSY masses for the benchmark points given in
Table 9. The difference between the mass of $\widetilde{\tau}_{1}$ and
$\widetilde{\chi}_{10}$ is also given as this pertains to the production of
dark matter. We also include the high scale bilinear coupling value $B_{0}$
for its relevance to the high scale parameters of the model. All parameters
are given in GeV.

Both benchmarks produce a completely bino-like LSP state caused by the large
scale difference between $M_{1}$ and $M_{2}$ and have very similar dark matter
physics in general. Although BP7 has a dark matter relic density within the
allowed region, BP8 gives an excellent value. We see a much larger mass gap
between the LSP and the nLSP than for Benchmark points 1 to 6. This is because
the points produce a much smaller mass LSP. Therefore, the freeze out
temperature is low and as such, there is more cosmological time for co-
annihilation to occur. A larger mass gap reduces the strength of these co-
annihilation channels allowing for the phenomenological Higgs boson mass.

Quantity |  BP7 |  BP8
---|---|---
$\Omega_{DM}h^{2}$ |  0.020 |  0.112
$\widetilde{\chi}_{10}$ [GeV] |  100 |  99
$\widetilde{\tau}_{1}-\widetilde{\chi}_{10}$ [GeV] |  4.2 |  9.6
$|\alpha_{1}|^{2}$ |  1 |  1
$|\alpha_{2}|^{2}$ |  0 |  0
$|\alpha_{3}|^{2}$ |  0 |  0
$|\alpha_{4}|^{2}$ |  0 |  0
Table 11: Shows the relic density of the LSP for each benchmark point. The
difference between the LSP and the nLSP is also given. Finally, we give the
probability of finding the given LSP in a particular flavour state. That is to
say; we give $|\alpha_{i}|^{2}$ where
$\widetilde{\chi}_{10}=\alpha_{1}\widetilde{B}+\alpha_{2}\widetilde{W}+\alpha_{3}\widetilde{H}_{1}+\alpha_{4}\widetilde{H}_{2}$
and $\sum|\alpha_{i}|^{2}=1$. Dimensionful parameters are given in GeV.

Table 12 shows some key observables for the given benchmark points. As
previously stated, both points are at the borderline of exclusion by Checkmate
analysis, since, both points still have relatively low $r_{max}\sim 1$ values,
suggesting that both points might be viable within the uncertainties of the
analysis, as we discuss further below.

We see short life times for $\widetilde{\tau}_{1}$ with only one decay channel
therefore potentially leaving a strong collider signature. Both points get
their strongest constraints from the CMS analysis [87] that focuses on direct
electroweak production of charginos and neutralinos leading to final state
leptons, little hadronic activity, and a large missing momentum. As the colour
charged particles are far more massive than the leptons and gauginos, the
latter will represent the dominant production mechanism for the sparticles.
The signal region focuses on a final state with three light leptons where two
of the three are either $e$ or $\mu$ and as we have a light gaugino LSP with
light sleptons for $g-2$ this is likely to be constraining. Furthermore this
region focuses on states where the leptonic pair have invariant mass greater
than $105GeV$ and transverse mass of the third lepton greater than $160GeV$.
$\widetilde{\chi}_{20}$ has a $32\%$ branching ratio into
$\widetilde{\mu}^{\pm}\mu^{\mp}$ / $\widetilde{e}^{\pm}e^{\mp}$ combined with
$\widetilde{\chi}_{1+}$’s tendency to decay into $\widetilde{\tau}\nu_{\tau}$
pairs creates a strong signal in this region placing the parameter point on
the edge of exclusion. Furthermore, the relatively large mass of
$\widetilde{\chi}_{20}$ means that the resultant lepton pair will exceed the
required $105$ GeV invariant mass and the production of the $\tau$ are on the
edge of exclusion of the transverse mass limit. The b-type decays satisfy the
experimental constraints for both points.

Quantity |  BP7 |  BP8
---|---|---
$\Gamma\widetilde{\tau}_{1}$ [GeV] |  $3.30\times 10^{-3}$ |  $1.63\times 10^{-2}$
BR $(\widetilde{\tau}^{-}_{1}\rightarrow\chi_{10}\tau^{-})$ [%] |  100 |  100
$\Gamma\widetilde{\mu}_{R}$ [GeV] |  $1.32\times 10^{-1}$ |  $1.67\times 10^{-1}$
BR $(\widetilde{\mu}^{-}_{R}-\rightarrow\chi_{10}\mu^{-})$ [%] |  100 |  100
BR $(b\rightarrow s\>\gamma)$ [%] |  0.032 |  0.032
BR $(B_{s}\rightarrow\mu^{+}\>\mu^{-})$ [%] |  $2.97\times 10^{-7}$ |  $2.96\times 10^{-7}$
$\Delta\frac{(g-2)_{\mu}}{2}$ |  $1.42\times 10^{-9}$ |  $1.30\times 10^{-9}$
$\Omega_{DM}h^{2}$ |  0.020 |  0.112
$\chi_{10}$ [GeV] |  100 |  99
$\sigma_{q\overline{q}\rightarrow\chi_{10}\chi_{10}}$ [pb] |  $1.796\times 10^{-14}$ |  $3.562\times 10^{-14}$
$r_{max}$ |  $1.22$ |  $1.04$
$\sqrt{s}$ [TeV] |  13 |  13
Analysis |  cms_sus_16_039 |  cms_sus_16_039
Signal Region |  SR_A44 |  SR_A44
Ref. |  [87] |  [87]
$\sigma_{LO}$ [pb] |  $7.127\times 10^{-11}$ |  $5.744\times 10^{-11}$
Table 12: Shows branching ratios for lightest supersymmetric particles in the
spectrum for BP7 and BP8. Only branching ratio greater than $1\%$ are
included. We also include some beyond the standard model observables BR
$(b\rightarrow s\>\gamma)$, BR $(B_{s}\rightarrow\mu^{+}\>\mu^{-})$,
$\Delta\frac{(g-2)_{\mu}}{2}$, and $\Omega_{DM}h^{2}$, where
$\Delta\frac{(g-2)_{\mu}}{2}$ is a calculation of the SUSY contribution beyond
the standard model. The model successfully predicts the b decays discrepancy
and satisfies the anomalous muon magnetic moment to $2\sigma$. The relic
density is too small and is therefore not ruled out phenomenologically.
CheckMATE runs using 13 TeV and 8 TeV analyses rule out these points. Decay
widths and masses are given in GeV, branching ratios are given in %, and cross
sections are given in pb.

##  5 No-scale SUGRA with non-zero $A_{0}$ (Case II)

In order to fully explore the parameter space we can also allow for non-zero
values of $A_{0}$. Indeed, non-zero $A_{0}$ will serve to increase the Higgs
boson mass, encouraging a better fit to the current experimental results. We
therefore present a subsequent scan where we vary $\alpha$ between $-0.016$
and $0.016$. These values were chosen by trial and error such that computer
time would not be wasted by producing many points with large $A_{0}$ values
breaking colour charge symmetry.

Analogously to the previous, we start by analysing the positive $k$ values.
Further to this analogy, we revert to the previous definition of the
likelihood to emphasise the relic density as key constraint in this paradigm
and highlight the fact that $g-2$ will not be satisfied for positive values of
$k$.

###  5.1 Positive $k$ (Scan 3)

(a) $\mu<0$, case II

(b) $\mu>0$, case II

Figure 12: Shows the distribution of $A_{0}$ where case I data from a Monte-
Carlo scan with parameter ranges $tan(\beta)\in[1.5,50],k\in[0,0.1]$,
$\alpha\in[-0.166,0.166]$, and $m_{\frac{3}{2}}\in[10^{3}GeV,10^{6}GeV]$.
Colour denotes likelihood, with hotter colours corresponding with high
likelihoods. As before, the likelihood is dominated by the relic density
calculation.

(a) $\mu<0$, case II

(b) $\mu>0$, case II

Figure 13: Shows $k$ against $m_{\frac{3}{2}}$ where case II data from a
Monte-Carlo scan with parameter ranges ranges
$tan(\beta)\in[1.5,50],k\in[0,0.1]$, $\alpha\in[-0.166,0.166]$, and
$m_{\frac{3}{2}}\in[10^{3}GeV,10^{6}GeV]$. Colour denotes likelihood, with
hotter colours corresponding with high likelihoods. As before, the likelihood
is dominated by the relic density calculation.

Fig 12 shows the distribution of $A_{0}$ against $k$ for both signs of $\mu$.
Interestingly, the shape of the two plots for the two signs of $\mu$ is quite
different, with the $\mu<0$ sign yielding a mushroom shaped distribution of
points, while the $\mu>0$ sign results in a wigwam shaped plot. However, there
are some key features of both that are shared. In neither do we see any
particular likelihood increase for non-0 $A_{0}$. This suggests that moving to
case II will not give significant improvement compared to the previous case.
Furthermore, in both we observe a preferred region of $k\approx 0.003$.

Fig 13 closely mirrors the structure of Fig 7(b); with $A_{0}$ variation
causing some spread in the likelihoods. The structure of LSP make-up as well
as spectrum is generally similar. For this reason we refer you to the previous
for detailed analysis. Furthermore, the density of points is greatly reduced
as variation in $A_{0}$ produces many low-likelihood points. This does support
claims of naturalness for the exclusively scaleless model previously
presented. Although this scan was not truly exhaustive, no gains were made in
terms of the overall likelihood.

As in case I with positive $k$, we see that no point can satisfy the higgs
mass and $g-2$ simultaneously. The same is true of $g-2$ and the relic
density. In order to achieve the correct higgs mass, $m_{\frac{3}{2}}$ must be
at least 200TeV. This has the effect of increasing the slepton masses and thus
decreasing $\Delta a_{\mu}$. We therefore, turn our attention to $k<0$ values.

###  5.2 Negative $k$ (Scan 4)

As was previously argued, the inclusion of negative $k$ values should allow
for larger Higgs boson masses while keeping slepton masses low; thus
incorporating a mechanism for generating the anomalous muon magnetic moment.
As was done previously, we loosen the relic density constraint insisting only
that the relic density is sufficiently small so as not to completely rule out
the given point. Furthermore, we find large parts of the parameter space give
the right handed stau state as the lightest sparticle. In Figures 14 to 16 we
present results where the neutralino is the LSP. However, it is also
interesting to consider potential R-parity violating models in which these
light, charged particles decay into standard model particles. As was
previously stated, a detailed discussion of the viability of such points is
beyond the scope of this paper, but we do include Figures 17 and 18 as these
states can lead to excellent fits for $a_{\mu}$.

Below we present the results of two scans (over positive and negative $\mu$
values). We limit the range of $m_{\frac{3}{2}}$ under the upper limit set by
the starobinsky-like inflation limits previously discussed. This eliminates
particularly massive gaugino states and thus large slepton mass states.

(a) $\mu<0$, case II

(b) $\mu>0$, case II

Figure 14: Shows the distribution of $k$ against $m_{\frac{3}{2}}$ with
$a_{\mu}$ and $m_{h}$ log likelihood for $m_{\frac{3}{2}}\in[0TeV,200TeV]$,
$\alpha\in[-0.005,0.005]$ and $k\in[-0.035,-0.014]$.

In both cases, two values of k are favoured by the scan. Analogously to case
I, $k\approx-0.016$ and $k\approx-0.0023$ with the latter snf $\mu<0$
representing a greater likelihood region of parameter space. This gives an
approximate ratio of the high scale gaugino mass parameters as
$|M_{1}|:|M_{2}|:|M_{3}|\approx 1:4:7$.

(a) $\mu<0$, case II

(b) $\mu>0$, case II

Figure 15: Shows a scatter plot of the higgs mass against the anomalous muon
magnetic moment for a scan with ranges $m_{\frac{3}{2}}\in[0TeV,200TeV]$,
$\alpha\in[-0.005,0.005]$ and $k\in[-0.035,-0.014]$. The $2\sigma$ region of
$a_{\mu}$ is marked with dotted lines while the $2\sigma$ region of $m_{h}$ is
marked with dashed lines. The colour denotes the relic density.

From Fig. 15 and 16 we again see that $\mu<0$ is favoured with a proportion of
points sitting well within the $2\sigma$ range. We also observe that some
points have a remarkably high relic density almost satisfying the $1\sigma$
region. This is a tantalising suggestion that this effectively scaleless model
may be able to satisfy the higgs mass, relic density, and $a_{\mu}$
simultaneously. Having said this, the inclusion of $A_{0}$ does not give any
discernible improvement in the allowed values of the Higgs boson mass due to
the previously discussed natural tendency toward a scale-less model. However,
these results are still included as the inclusion of a small $A_{0}$ parameter
can modify the SUSY parameters sufficiently such that some points pass the
CheckMATE collider constraints as presented in benchmark points 9 and 10
(Table 13).

(a) $\mu<0$, case II

(b) $\mu>0$, case II

Figure 16: As for Fig 15 but zoomed in on the $3\sigma$ region for both
$a_{\mu}$ and $m_{h}$. Two benchmark points (presented below are marked)

From Fig 17 we can see that allowing for these RPV parameter points slightly
increases the $a_{\mu}$ values. Conversely to the previous, $\mu<0$ now
represents the higher likelihood points. As the stau is the lightest particle
in the cases, no relic density can be calculated, and therefore we mark all
points in blue to indicate this issue. Having said this, some parameter points
perfectly attain the higgs mass and $a_{\mu}$. For a discussion of the
phenomenology of such points see [60].

(a) $\mu<0$, case II

(b) $\mu>0$, case II

Figure 17: As for Fig 15 but where points that have charged LSP states are
shown. Such states are marked in blue to indicate that no relic density could
be calculated.

(a) $\mu<0$, case II

(b) $\mu>0$, case II

Figure 18: As for Fig 17 but zoomed in on the $3\sigma$ region for both
$a_{\mu}$ and $m_{h}$.

We now present the two benchmark points plotted on Fig 16. Both were selected
to represent a particularly high value of $a_{\mu}$ as well as interesting
collider results.

Both benchmarks 9,10 have the same value of k as shown in Fig 12 where
$k=-0.23$ had clear phenomenological advantages. Furthermore, the values of
$m_{\frac{3}{2}}$ and $tan(\beta)$ are also approximately similar. They are
both selected from the $\mu<0$ scan due to its more promising phenomenology.
In both cases $\beta\approx 1$ and $\alpha$ is very small. This is consistent
with the no-scale model we are examining. The high scale gaugino mass
parameters are also presented. After running to the low scale the ratio
becomes far more extreme causing a great hierarchical divergence between the
colour charged and the non-colour charged sparticles as can be seen below.

Quantity |  BP9 |  BP10
---|---|---
$\alpha$ |  0.0007 |  0.0003
$\beta$ |  1 |  1
$m_{\frac{3}{2}}$ [TeV] |  56 |  57
$k$ |  -0.023 |  -0.023
SPheno: |  |
$m_{0}$ [GeV] |  0 |  0
$tan(\beta)$ |  10.5 |  10.0
$sign(\mu)$ |  -1 |  -1
$A_{0}$ [GeV] |  -227 |  -109
$M_{1}$ [GeV] |  -275 |  -271
$M_{2}$ [GeV] |  -1144 |  -1163
$M_{3}$ [GeV] |  -1841 |  -1877
Table 13: Shows two benchmark points representing two different areas of
interest in the parameter space. We present the model parameters and the
resultant SPheno input parameters. BP9 shows a point with high $a_{\mu}$ and
BP10 shows a point with high $\Omega$. Dimensions of the parameters are given
where "-" means dimensionless. $m_{\frac{3}{2}}$ is given in units of TeV, and
$m_{0},A_{0},M_{i}$ are given in GeV.

Table 14 shows the mass spectrum for the two benchmark points BP9, BP10. Large
values of $M_{3}$ give large values of squark and gluon masses and contribute
to the relatively large higgs mass. Conversely, the smaller values of the
other two gaugino mass parameters, in combination with the absence of $m_{0}$,
gives the requisite small slepton states. Furthermore, the hierarchy between
$M_{1}$ and $M_{2}$ gives a notable disparity between the right and left
handed states. Due to the small value of $tan(\beta)$ and the negative sign of
$\mu$, $\widetilde{\tau}_{1}$ is predominantly right handed. We note the very
light smuon states that give key contributions to $a_{\mu}$. Finally, the
small value of $M_{1}$ leads to predominantly bino-like, light dark matter
candidate.

Masses |  BP9 |  BP10
---|---|---
$\widetilde{e}_{L}$ |  712 |  724
$\widetilde{e}_{R}$ |  139 |  140
$\widetilde{\nu}_{eL}$ |  708 |  719
$\widetilde{\mu}_{L}$ |  712 |  724
$\widetilde{\mu}_{R}$ |  139 |  140
$\widetilde{\nu}_{\mu L}$ |  708 |  719
$\widetilde{\tau}_{1}$ |  107 |  110
$\widetilde{\tau}_{2}$ |  712 |  724
$\widetilde{\nu}_{\tau L}$ |  706 |  717
$\widetilde{d}_{L}$ |  3360 |  3420
$\widetilde{d}_{R}$ |  3290 |  3350
$\widetilde{u}_{L}$ |  3360 |  3420
$\widetilde{u}_{R}$ |  3290 |  3350
$\widetilde{s}_{L}$ |  3360 |  3420
$\widetilde{s}_{R}$ |  3290 |  3350
$\widetilde{c}_{L}$ |  3360 |  3420
$\widetilde{c}_{R}$ |  3290 |  3350
$\widetilde{b}_{1}$ |  3120 |  3180
$\widetilde{b}_{2}$ |  3280 |  3340
$\widetilde{t}_{1}$ |  2800 |  2840
$\widetilde{t}_{2}$ |  3140 |  3190
$\widetilde{g}$ |  3850 |  3920
$\widetilde{\chi}_{10}$ |  103 |  103
$\widetilde{\chi}_{20}$ |  926 |  942
$\widetilde{\chi}_{30}$ |  2000 |  2050
$\widetilde{\chi}_{40}$ |  2000 |  2060
$\widetilde{\chi}_{1+}$ |  926 |  943
$\widetilde{\chi}_{2+}$ |  2009 |  2060
${h_{0}}$ |  122 |  122
${H_{0}}$ |  2110 |  2160
${A_{0}}$ |  2110 |  2160
${H_{\pm}}$ |  2110 |  2170
$\mu$ |  -1870 |  -1930
$B_{0}$ |  -68 |  -46
$\widetilde{\tau}_{1}-\widetilde{\chi}_{10}$ |  3.87 |  6.77
Table 14: Shows the spectrum of SUSY masses for the benchmark points given in
Table 13. The difference between the mass of $\widetilde{\tau}_{1}$ and
$\widetilde{\chi}_{10}$ is also given as this pertains to the production of
dark matter. We also include the high scale bilinear coupling value $B_{0}$
for its relevance to the high scale parameters of the model. All parameters
are given in GeV.

Table 15 shows the key parameters for the relic density calculation for the
two points BP9, BP10. As the stau is the nLSP, the mass difference between it
and $\widetilde{\chi}_{10}$ is presented. As its mass gap is so small, the
$\widetilde{\tau}$ plays a critical role in the mechanism for dark matter
annihilation. Indeed the dominant decay channels contributing to the relic
density calculation are
$\widetilde{\tau}_{1}\>\widetilde{\tau}_{1}\rightarrow\tau\>\tau$ and
$\widetilde{\tau}_{1}\>\widetilde{\chi}_{10}\rightarrow\tau\>\gamma$. As the
mass gap is especially small in the BP9, dark matter is over-annihilated by
these channels leading to a relic density below the desired value.

Quantity |  BP9 |  BP10
---|---|---
$\Omega_{DM}h^{2}$ |  0.0397 |  0.020
$\widetilde{\chi}_{10}$ [GeV] |  103 |  103
$\widetilde{\tau}_{1}-\widetilde{\chi}_{10}$ [GeV] |  3.87 |  6.77
$|\alpha_{1}|^{2}$ |  1 |  1
$|\alpha_{2}|^{2}$ |  0 |  0
$|\alpha_{3}|^{2}$ |  0 |  0
$|\alpha_{4}|^{2}$ |  0 |  0
Table 15: Shows the relic density of the LSP for each benchmark point. The
difference between the LSP and the nLSP is also given. Finally, we give the
probability of finding the LSP in a particular flavour state. That is to say;
we give $|\alpha_{i}|^{2}$ where
$\widetilde{\chi}_{10}=\alpha_{1}\widetilde{B}+\alpha_{2}\widetilde{W}+\alpha_{3}\widetilde{H}_{1}+\alpha_{4}\widetilde{H}_{2}$
and $\sum|\alpha_{i}|^{2}=1$. Dimensionful parameters are given in GeV.

Table 16 shows the key collider and phenomenological findings resulting from
these parameter points BP9, BP10. Both the nnLSP, $\widetilde{\mu}_{R}$, and
the LSP $\widetilde{\tau}_{1}$, have short lifetimes with only one decay
channel perhaps suggesting a strong collider signature. Indeed, the strongest
signal region for both BP9 and BP10 is one that focuses on dileptonic final
states with missing transverse energy in the context of "electroweakinos".
Both b-type branching ratios are very well fitted in both cases. Furthermore,
$a_{\mu}$ is within the $2\sigma$ region for BP9 and BP10. We see relatively
high values of $r$ calculated by CheckMATE due to the low mass sleptons.

Quantity |  BP9 |  BP10
---|---|---
$\Gamma\widetilde{\tau}_{1}$ [Gev] |  $2.61\times 10^{-3}$ |  $8.18\times 10^{-3}$
BR $(\widetilde{\tau}^{-}_{1}\rightarrow\chi_{10}\tau^{-})$ [%] |  100 |  100
$\Gamma\widetilde{\mu}_{R}$ [Gev] |  $1.47\times 10^{-1}$ |  $1.57\times 10^{-1}$
BR $(\widetilde{\mu}^{-}_{R}-\rightarrow\chi_{10}\mu^{-})$ [%] |  100 |  100
BR $(b\rightarrow s\>\gamma)$ [%] |  0.032 |  0.032
BR $(B_{s}\rightarrow\mu^{+}\>\mu^{-})$ [%] |  $2.96\times 10^{-7}$ |  $2.95\times 10^{-7}$
$\Delta\frac{(g-2)_{\mu}}{2}$ |  $1.39\times 10^{-9}$ |  $1.32\times 10^{-9}$
$\Omega_{DM}h^{2}$ |  0.0397 |  0.020
$\chi_{10}$ [Gev] |  103 |  103
$\sigma_{q\overline{q}\rightarrow\chi_{10}\chi_{10}}$ [pb] |  $3.982\times 10^{-14}$ |  $2.290\times 10^{-14}$
$r_{max}$ |  $0.40$ |  $0.57$
$\sqrt{s}$ [TeV] |  13 |  13
Analysis |  cms_sus_16_039 |  cms_sus_16_039
Signal Region |  SR_A44 |  SR_A44
Ref. |  [87] |  [87]
$\sigma_{LO}$ [pb] |  $6.342\times 10^{-11}$ |  $5.893\times 10^{-11}$
Table 16: Shows branching ratios for lightest supersymmetric particles in the
spectrum for BP7 and BP8. Only branching ratio greater than $1\%$ are
included. We also include some beyond the standard model observables BR
$(b\rightarrow s\>\gamma)$, BR $(B_{s}\rightarrow\mu^{+}\>\mu^{-})$,
$\Delta\frac{(g-2)_{\mu}}{2}$, and $\Omega_{DM}h^{2}$, where
$\Delta\frac{(g-2)_{\mu}}{2}$ is a calculation of the SUSY contribution beyond
the standard model. The model successfully predicts the b decays discrepancy
and satisfies the anomalous muon magnetic moment to $2/3\sigma$. The relic
density is too small and is therefore not ruled out phenomenologically.
CheckMATE runs using 13TeV and 8TeV analyses do not rule out these points.
Decay widths and masses are given in GeV, branching ratios are given in %, and
cross sections are given in pb.

##  6 Conclusion

Inflation represents a very attractive solution to a number of cosmological
problems and, in combination with supersymmetry, a very attractive model for
Beyond the Standard Model physics emerges based on no-scale SUGRA. We have
focussed on the particular case where the Polonyi term in the superpotential
acts as a slow roll inflaton for Starobinsky inflation, leading to an upper
bound on the gravitino mass $m_{3/2}<1000$ TeV.

The recent Fermilab muon $g-2$ result further motivates a no-scale model,
where all the dimensionful parameters the model are zero (except the gaugino
masses which arise via mixed modulus and anomaly mediation), naturally leading
to light slepton masses for certain gaugino masses. For negative universal
gaugino masses, $k<0$, we find relatively light bino/wino masses together with
light sleptons, as suggested by the muon $g-2$ measurement. In general, such a
model is also capable of providing a good dark matter candidate whilst
satisfying constraints from collider physics, as well as yielding the correct
Higgs boson mass, but it turns out to be non-trivial to achieve this while
satisfying the desired muon $g-2$ constraints.

We have conducted a Monte Carlo parameter scan over the given model in two
conditions: case I and case II, corresponding to zero or non-zero trilinear
soft parameter $A_{0}$. We show that case I with $k>0$ can give excellent fits
for the Higgs boson mass and relic density whilst easily satisfying the
collider and flavour constraints. Furthermore, with a reversal in the sign of
$k$ we show that $g-2$ can be satisfied to $2\sigma$, as exemplified by BP7
and BP8, where BP8 also satisfies the desired relic density. Since BP7 and BP8
are on the edge of exclusion, with $r_{max}$ values slightly above unity, this
motivates the study of case II where we allow a non-zero $A_{0}$ trilinear
soft parameter. By including a small $A_{0}$ parameter at the high scale, we
find that muon $g-2$ can be satisfied whilst satisfying the collider
constraints that threaten case I. However we find, while the case II benchmark
points BP9 and BP10 satisfy the collider constraints and the muon $g-2$, they
both predict a relic density which is below the desired value, which does not
exclude these points of course, but is somewhat disappointing from the point
of view of dark matter. Interestingly, we find that, in both cases, a large
part of parameter space is dominated by RPV-style points, where the LSP
becomes the lightest $\widetilde{\tau}$ state. Although such states are beyond
the scope of this paper, a detailed discussion of these kind of points can be
found [60].

In general, we find that the lightest neutralino $\widetilde{\chi}_{10}$
should be a purely bino-like state with a mass of around 100 GeV for parameter
points BP7-BP10 that satisfy the muon $g-2$ constraint. We highlight in
particular the fully no-scale SUGRA point with zero $A_{0}$, $B_{0}$ and
$m_{0}$, namely BP8, which can explain not only the recent Fermilab muon $g-2$
measurement, and has the correct Higgs boson mass, but also yields the desired
dark matter relic density, albeit with $r_{max}=1.04$ on the edge of
exclusion. We remark that current LHC limits are more easily evaded due to
right handed sleptons being almost degenerate with the neutralino LSP, while
the left handed counterparts are much higher in mass, suppressing chargino
decay channels into first and second generation sleptons. However, even for
BP9 and BP10, the $r_{max}$ value is not too far from unity, suggesting that
future LHC runs are capable of discovering such SUSY particles for all the
interesting benchmarks that satisfy the muon $g-2$.

In conclusion, no-scale SUGRA is not only well motivated theoretically from
string theory and provides an elegant framework for accounting for
cosmological Starobinsky inflation, but also has very interesting
phenomenological implications as well. Ignoring the muon $g-2$ to begin with,
and assuming positive universal gaugino mass contributions, in addition to the
anomaly mediated contributions, we show that no-scale SUGRA can readily
satisfy the dark matter and Higgs boson mass requirements, consistently with
all other phenomenological constraints. We then show that the recent Fermilab
measurement of the muon $g-2$ may be accommodated, together with the correct
Higgs boson mass, for no-scale SUGRA with negative universal gaugino mass
contributions in addition to the anomaly mediated contributions. For the fully
no-scale SUGRA case, with all soft parameters equal to zero at the high scale,
apart from gaugino masses, we find that successful points which satisfy the
muon $g-2$, and can sometimes yield the desired relic density, although such
points tend to be near the edge of LHC collider exclusion. Analysing no-scale
SUGRA with a non-zero $A_{0}$, we find that the muon $g-2$ can still be
explained, with the collider constraints somewhat relaxed. However, even in
this case, light sleptons and charginos are still predicted, with good
prospects for discovering these SUSY particles in LHC Run 3.

##  7 Acknowledgments

The authors acknowledge the use of the IRIDIS High Performance Computing
Facility, and associated support services at the University of Southampton, in
the completion of this work. SFK acknowledges the STFC Consolidated Grant
ST/L000296/1 and the European Union’s Horizon 2020 Research and Innovation
programme under Marie Sklodowska-Curie grant agreement HIDDeN European ITN
project (H2020-MSCA-ITN-2019//860881-HIDDeN).

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