#  MS-TP-21-26 Future searches for SUSY at the LHC post Fermilab $(g-2)_{\mu}$

Amin<EMAIL_ADDRESS>, Michael
<EMAIL_ADDRESS>, Pran
<EMAIL_ADDRESS>and Raza M<EMAIL_ADDRESS>

aInstitut für Theoretische Physik, Westfälische Wilhelms-Universität Münster,
Wilhelm-Klemm-Straße 9, 48149 Münster, Germany
bDepartment of Physics, Northeastern University, Boston, MA 02115-5000, USA
cDepartment of Physics, American University of Sharjah, P.O. Box 26666,
Sharjah, UAE

###### Abstract

We assess the future directions for the search for supersymmetry at the Large
Hadron Collider in view of the new precision results on the muon anomaly by
the Fermilab Collaboration. The existence of a deviation of size 4.1$\sigma$
from the Standard Model prediction points to light sleptons and light weakinos
in the mass range of few hundred GeV while the observation of the Higgs boson
mass at $\sim 125$ GeV points to squark masses lying in the few TeV range.
Thus a split sparticle spectrum is indicated. We discuss the possibility of
such a split sparticle spectrum in the supergravity unified model and show
that a splitting of the sfermion spectrum into light sleptons and heavy
squarks naturally arises within radiative breaking of the electroweak symmetry
driven by heavy gluinos ($\tilde{g}$SUGRA). We discuss the possible avenues
for the discovery of supersymmetry at the LHC within this framework under the
further constraint of the recent muon anomaly result from the Fermilab
Collaboration. We show that the most likely candidates for early discovery of
a sparticle at the LHC are the chargino, the stau, the smuon and the
selectron. We present a set of benchmarks and discuss future directions for
further work. Specifically, we point to the most promising channels for SUSY
discovery and estimate the integrated luminosity needed for the discovery of
these benchmarks at the High Luminosity LHC and also at the High Energy LHC.

Submitted to the Proceedings of the US Community Study

on the Future of Particle Physics (Snowmass 2021)

###### Contents

  1. 1 Introduction
  2. 2 Fermilab $g-2$ constraint on SUGRA models
  3. 3 Sparticle mass hierarchy under the $g-2$ constraint
  4. 4 Benchmarks for future SUSY searches at the LHC consistent with $g-2$
  5. 5 A deep neural network analysis for SUSY discovery
  6. 6 Conclusion

##  1 Introduction

The anomalous magnetic moment is one of the most precisely computed quantities
in the Standard Model and is predicted to have the value [2, 3, 4, 5, 6, 7, 8]

$a^{\rm SM}_{\mu}=116591810(43)\times 10^{-11}.$ (1.1)

The recent combined Fermilab [9] and Brookhaven [10, 11] measurement result
gives

$a^{\rm exp}_{\mu}=116592061(41)\times 10^{-11}.$ (1.2)

The difference between the experimental and the Standard Model result is

$\Delta a^{\rm FB}_{\mu}=a_{\mu}^{\rm exp}-a_{\mu}^{\rm SM}=251(59)\times
10^{-11}.$ (1.3)

The above corresponds to a $4.1\sigma$ deviation from the Standard Model. This
deviation strengthens the previous Brookhaven observation of $3.7\sigma$ (see,
however, [12]). The implications of these for supersymmetry in the framework
of supergravity unified models were analyzed in [13]. These works are
motivated by the fact that it has been known since the early eighties [14, 15]
that a correction to the muon anomaly from supersymmetry can arise which is
comparable to the electroweak correction in the Standard Model. Further, it is
known that the muon anomaly is sensitive to soft parameters in supergravity
models [16, 17, 18]. Thus the soft parameters directly affect the sparticle
masses that enter in the supersymmetric correction to the muon anomaly, and
specifically when the smuon and muon-sneutrino as well as the chargino and the
neutralino are light with masses lying in the sub-TeV region, their loop
corrections to the muon $g-2$ anomaly can be sizable. The above implies that
within the framework of high scale models, the $4.1\sigma$ deviation seen by
Fermilab can be explained provided we have light weakinos and light sleptons.
On the other hand, the observation of the Higgs boson mass at $\sim 125$ GeV
[19, 20] indicates that the size of weak scale supersymmetry lies in the
several TeV region in order to generate a large enough loop correction to the
Higgs boson mass to lift its tree value that lies below the $Z$-boson mass to
the experimentally observed value [21, 22]. The supergravity based analysis
discussed here proposes a natural solution to these apparently conflicting
constraints on the model. Thus a possible solution to this disparity in the
scale of the sfermion mass spectrum can come about if the radiative breaking
of the electroweak symmetry in supergravity (SUGRA) grand unified models is
driven by a large gluino mass lying in the several TeV region while the
universal scalar mass is relatively small of order a few hundred GeV. In this
case squarks, which carry color, acquire large masses the size of several TeV
in renormalization group (RG) running while the color neutral sleptons remain
light down to the electroweak scale.

In the numerical analysis we use an artificial neural network to search for
regions of the parameter space consistent with the Fermilab result within the
SUGRA parameter space and the search leads to the $\tilde{g}$SUGRA model
proposed in earlier works [23, 24, 25]. We will also discuss a specific
$SO(10)$ model, where a similar conclusion is reached [26].

##  2 Fermilab $g-2$ constraint on SUGRA models

To discuss the Fermilab constraint on SUGRA models, we utilize an artificial
neural network to identify the region of the parameter space compatible with
the muon anomaly result. Neural networks are efficient in analyses when one
deals with a large parameter space (see, e.g., Refs. [27, 28]) and this is the
case for models we investigate. Specifically we investigate the SUGRA model
[29] with non-universalities [30, 33, 31, 32] where the non-universalities are
defined by the soft parameters consisting of the set
$m_{0},m_{1},m_{2},m_{3},A_{0},\tan\beta$ and sign$(\mu)$. Here $m_{0}$
defines the universal scalar mass for the sfermions and for the Higgs,
$m_{1},m_{2},m_{3}$ are the soft masses for the $U(1)$, $SU(2)$ and $SU(3)$
gauginos, $A_{0}$ is the universal scalar coupling all taken at the grand
unification scale. Further, $\tan\beta=\langle H_{2}\rangle/\langle
H_{1}\rangle$ where $\langle H_{2}\rangle$ gives mass to the up quarks and
$\langle H_{1}\rangle$ gives mass to the down quarks and the leptons.

As noted earlier there are apparent conflicting constraints that arise on the
sparticle spectrum from the Higgs boson mass measurement at $\sim 125$ GeV and
the muon $g-2$ anomaly measurement. The first one points to a large size of
weak scale SUSY lying in the several TeV region while the muon anomaly result
points to a relatively light scale for the sparticles in order to produce the
desired correction to the anomaly in supersymmetry. Thus on the surface these
two constraints appear to be at odds with each other. However, without any a
priori assumption on the sizes of the soft parameters one can investigate
regions of the parameter space which may lead to consistency using a neural
network. An analysis within this framework shows that the preferred region
within the prescribed parameter space is one where the universal scalar mass
at the GUT scale is relatively small lying in the few hundred GeV region but
the gluino mass is large and lying in the few TeV region.

As a consequence of the large gluino mass, in the RG evolution the squark
masses are driven to TeV scale due to their color interactions, while the
color neutral sleptons remain light. This mechanism then leads to a natural
splitting between the slepton and squark masses as shown in Fig. 1 as noted in
the early work of [23]. The large mass of the squarks allows one to satisfy
the Higgs boson mass constraint, while the relative smallness of the slepton
masses allows us to produce a significant correction to the muon $g-2$ anomaly
indicated by the new measurements. In Fig. 2 we give a scatter plot of SUGRA
model points consistent with the Fermilab constraint with plots of
$m_{0},m_{1},m_{2}$ vs. $\Delta a_{\mu}$, and in Fig. 3 we give the
probability densities of $m_{0}$, $A_{0}/m_{0}$, $m_{1}$ and $m_{2}$
consistent with the $g-2$ constraint. We note that in these plots a
significant population of models exist within the $1\sigma$ range of $\Delta
a^{\rm FB}_{\mu}$ given by Eq. (1.3). Thus the satisfaction of the constraint
occurs over a region of the SUGRA parameter space rather than just over few
model points. Examining the distributions of Fig. 3, we notice an apparent
anticorrelation between $m_{0}$ and $m_{2}$ and general broad distributions
for $m_{1}$ and $A_{0}/m_{0}$. The reason behind this bimodal distribution in
$m_{0}$ and $m_{2}$ is the LHC constraints. Limits on sparticle masses from
LHC analyses disfavor small $m_{0}$ while a larger number of points at higher
values of $m_{0}$ remain viable. The opposite is true for $m_{2}$ where larger
$m_{2}$ values are disfavored. In particular, for $m_{2}>m_{1}$, the right
handed slepton becomes lighter than the left handed one which becomes more
constrained by LHC analyses.

Figure 1:  Exhibition of a natural splitting between the slepton and squark
masses in the gluino-driven radiative breaking of the electroweak symmetry.
The running of the gaugino masses is also shown. The SUSY breaking scale,
$Q_{\rm SUSY}$, is defined as the geometric average of the two stop masses.
Based on the analysis of [23].

##  3 Sparticle mass hierarchy under the $g-2$ constraint

As noted above the satisfaction of the Higgs boson mass and the muon anomaly
constraints leads to a mass hierarchy between the color neutral sfermions and
the sfermions with color as seen from Fig. 1. A priori, a large mass gap
between the sleptons and the squarks appears puzzling. This is so since
sleptons and squarks belong to common multiplets in grand unification, e.g., a
single generation of leptons and quarks and their superpartners lie in one
16-plet representation of $SO(10)$, and soft breaking does not distinguish
between colors. Thus the remarkable aspect of $\tilde{g}$SUGRA is that
renormalization group naturally leads to a split spectrum between sleptons and
squarks, because the squarks couple to the gluino at the loop level and the
large mass of the gluino thus drives their masses to large values while
sleptons are largely unaffected. A display of the sparticle mass hierarchy
generated by the RG running is given in Fig. 4. Here the left panel shows the
light sparticle spectrum while the right panel shows the heavy sparticle
masses. Corrections to the muon $g-2$ are governed by the light spectrum of
the left panel and this mass spectrum lies in the mass range of few hundred
GeV while the heavy mass spectrum lies in the mass range of few TeV. Because
of this split mass spectrum, the sparticles most accessible for SUSY searches
at the LHC are the ones in the left panel. For the low-lying spectrum
consistent with $\Delta a_{\mu}^{\rm FB}$, the mean value (shown by the dashed
line) of the chargino mass is $\sim 445$ GeV, the lightest neutralino mass
$\sim 235$ GeV, the left handed slepton mass $\sim 480$ GeV, the right handed
slepton mass $\sim 590$ GeV, the sneutrino mass $\sim 470$ GeV and two stau
mass eigenstates, $\tilde{\tau}_{1}$ and $\tilde{\tau}_{2}$, are $\sim 300$
GeV and $\sim 670$ GeV, respectively. The heavy spectrum includes the second
chargino, the squarks, the gluino, the lightest stop and the CP even Higgs
$H_{0}$. Note that the charged and CP odd Higgses have masses comparable to
$H_{0}$. The dashed lines on either side of the mean correspond to the
quartiles of this distribution.

Figure 2: A scatter plot of $m_{0},m_{1},m_{2}$ vs. $\Delta a_{\mu}$ using an
artificial neural network analysis consistent with the combined Fermilab-
Brookhaven result within $1\sigma$. The model points satisfy the Higgs boson
mass, the dark matter relic density and limits from dark matter direct
detection experiments and the LHC sparticle mass limits.

Figure 3:  Probability density vs. the range of the soft parameters,
$m_{0},A_{0}/m_{0}$ (top panels) and $m_{1},m_{2}$ (bottom panels) that emerge
in the artificial neural network analysis of SUGRA parameter space which
generates the desired correction to the muon anomaly indicated by the Fermilab
experiment.

Light Spectrum Heavy Spectrum

Figure 4:  A display of the split sparticle spectrum consisting of light
weakinos, sleptons, sneutrino and staus (left panel) and the heavy chargino,
squarks, gluino, lightest stop and the CP even Higgs (right panel) that emerge
in the gluino-driven radiative breaking of the electroweak symmetry in
$\tilde{g}$SUGRA grand unified models. Each subplot shows a probability
density distribution in the particles’ masses for two cases: the region
consistent with $\Delta a_{\mu}^{\rm FB}$ (orange) and the region outside of
the muon $g-2$ error bars (blue).

It is also of interest to investigate the effect of the $g-2$ constraint on
specific grand unified models. As a model example, we consider the case of an
$SO(10)$ model where the doublet-triplet mass splitting is obtained in a
natural fashion via the missing partner mechanism. For $SU(5)$ this is
accomplished in [34, 35] while for $SO(10)$ in [36, 37, 25, 26] (for a review
of other grand unified models see [38]). The $SO(10)$ model also allows for a
$b-t-\tau$ unification [39, 40] along with other desirable properties. A
neural network analysis again leads us to conclusions similar to the above,
i.e., that the radiative electroweak symmetry breaking is gluino driven.
Further, we note that in the analyses of the cases discussed above, the light
sparticle spectrum falls in three classes of mass hierarchies which we label
as (A), (B) and (C) as given below.

### (A):

Here $\tilde{\chi}^{0}_{2},\tilde{\chi}^{\pm}_{1}$ are essentially degenerate,
and $\tilde{\tau}_{1}$ is the next lightest supersymmetric particle (NLSP).
This leads to the mass hierarchy

$m_{\tilde{\tau}_{1}}<m_{\tilde{\chi}^{0}_{2}},m_{\tilde{\chi}^{\pm}_{1}}<m_{\tilde{\ell}},$

where $\tilde{\ell}$ stands for selectron or smuon.

### (B):

Here there is a reversal in the hierarchy for the first two inequalities,
i.e., between $\tilde{\tau}_{1}$ and $\tilde{\chi}^{0}_{2}$ or
$\tilde{\chi}^{\pm}_{1}$ which leads to the following possibilities

$\displaystyle
m_{\tilde{\chi}^{0}_{2}},m_{\tilde{\chi}^{\pm}_{1}}<m_{\tilde{\tau}_{1}}<m_{\tilde{\ell}},$
$\displaystyle
m_{\tilde{\chi}^{\pm}_{1}}<m_{\tilde{\chi}^{0}_{2}}<m_{\tilde{\tau}_{1}}<m_{\tilde{\ell}},$
$\displaystyle
m_{\tilde{\chi}^{\pm}_{1}}<m_{\tilde{\tau}_{1}}<m_{\tilde{\chi}^{0}_{2}}<m_{\tilde{\ell}}.$

### (C):

In this case the selectron and the smuon are lighter than the chargino and the
second neutralino while the stau is the NLSP. Thus here we have

$m_{\tilde{\tau}_{1}}<m_{\tilde{\ell}}<m_{\tilde{\chi}^{0}_{2}},m_{\tilde{\chi}^{\pm}_{1}}.$

The Fermilab result also puts constraints on the allowed region of CP phases
arising from the soft parameters specifically the gaugino masses. Thus the
gaugino masses $m_{1}$ and $m_{2}$ can be complex and one may write
$m_{1}=|m_{1}|e^{i\xi_{1}}$ and $m_{2}=|m_{2}|e^{i\xi_{2}}$. It turns out that
the electroweak corrections to the muon $g-2$ are very sensitive to the CP
phases [41, 42, 43]. As a consequence the measured $\Delta{a^{\rm FB}_{\mu}}$
puts rather stringent constraints on the allowed range of the CP phases. This
is shown in Fig. 5 which exhibits the disallowed region (shaded) and the
allowed region (white) in the $\xi_{1}-\xi_{2}$ plane. It is to be noted that
the allowed region is further constrained by the experimental limits on the
EDM of the electron and of the neutron. Thus the allowed region resulting from
imposing both constraints becomes very narrow. We label this region by a star
in Fig. 5 which can be obtained via the cancellation mechanism [44]. The CP
phases can also affect dark matter analyses (see, e.g., [45]) and proton
stability [46] as well as the production cross sections and decays of the
sparticles. These effects are also worth investigation in the future in view
of the current experimental result on the possible deviation $\Delta{a^{\rm
FB}_{\mu}}$ from the Standard Model prediction.

Figure 5:  A display of the region that is excluded (shaded) and the region
that is allowed (white) by the $g_{\mu}-2$ constraint. The allowed region is
further constrained by the limits on the EDM of the electron and the EDM of
the neutron. The star in the white region represents the small region of the
parameter space where both the electron EDM and the muon $g-2$ are satisfied.
Based on analysis of [13].

##  4 Benchmarks for future SUSY searches at the LHC consistent with $g-2$

As noted earlier, the Fermilab $g-2$ measurement has important implications
for the discovery of supersymmetry and future searches at colliders and
elsewhere. Here we present a set of SUGRA benchmarks consistent with the
recent muon $g-2$ anomaly result to test for a potential SUSY discovery at HL-
LHC and HE-LHC. The benchmarks are given in Table 1 and arranged according to
the categories (A), (B) and (C) described earlier. Thus benchmark (a) belongs
to case (A), benchmarks (b), (c), (d) belong to case (B) and benchmarks (e),
(f), (g) belong to case (C). The benchmarks have an $m_{0}$ of
$\mathcal{O}(100)$ GeV while $m_{3}$ lies in the several TeV range consistent
with a gluino-driven radiative electroweak symmetry breaking. It is noted that
benchmark (b) is taken from Ref. [13] (labeled there as (d)) while (a), (c),
(d), (e), (f), (g) are from Ref. [26] (labeled there as (f), (i), (b), (h),
(g), (d), respectively).

Table 1:  SUGRA benchmarks for future SUSY searches. All masses are in GeV. Model | $m_{0}$ | $A_{0}$ | $m_{1}$ | $m_{2}$ | $m_{3}$ | $\tan\beta$
---|---|---|---|---|---|---
(a) | 688 | 1450 | 852 | 634 | 8438 | 16.8
(b) | 389 | 122 | 649 | 377 | 4553 | 8.2
(c) | 452 | 648 | 624 | 346 | 4843 | 13.1
(d) | 673 | 1127 | 939 | 570 | 8833 | 8.2
(e) | 206 | 603 | 842 | 1298 | 7510 | 8.0
(f) | 106 | 22.6 | 523 | 1309 | 5240 | 9.3
(g) | 164 | 197 | 632 | 1539 | 6171 | 12.2

Along with the satisfaction of the recent muon $g-2$ result, the benchmarks
satisfy the Higgs boson mass, the dark matter relic density as well as the LHC
mass limits constraints. We present in Table 2 the low-lying sparticle
spectrum along with the SM-like Higgs boson mass, the relic density and the
muon $g-2$ anomaly calculated at the two-loop level. Here the left handed
slepton is lighter than the right handed one except in benchmarks (e), (f) and
(g) due to $m_{2}$ being significantly greater that $m_{1}$ as seen in Table
1. Furthermore, the benchmarks respect current constraints on the proton-
neutralino spin-independent cross section and all allow for a multicomponent
dark matter scenario.

Table 2:  The SM-like Higgs mass, the light sparticle spectrum and the dark matter relic density $\Omega h^{2}$ for the benchmarks of Table 1. Also shown is the muon $g-2$.  Model | $h^{0}$ | $\tilde{\ell}_{\rm L}$ | $\tilde{\ell}_{\rm R}$ | $\tilde{\nu}_{\rm L}$ | $\tilde{\tau}_{1}$ | $\tilde{\chi}^{0}_{1}$ | $\tilde{\chi}^{\pm}_{1}$ | $\Omega h^{2}$ | $\Delta a_{\mu}(\times 10^{-9})$
---|---|---|---|---|---|---|---|---|---
(a) | 123.0 | 508.1 | 762.0 | 502.3 | 331.9 | 324.2 | 404.3 | 0.004 | 2.11
(b) | 123.4 | 305.0 | 463.0 | 295 | 251.7 | 237.4 | 237.6 | 0.002 | 2.33
(c) | 123.7 | 346.8 | 511.9 | 338.0 | 240.3 | 205.6 | 205.8 | 0.001 | 2.67
(d) | 125.3 | 422.8 | 763.8 | 415.7 | 370.4 | 337.3 | 337.6 | 0.003 | 2.14
(e) | 124.5 | 628.7 | 402.2 | 623.6 | 338.3 | 326.8 | 998.4 | 0.082 | 1.94
(f) | 123.4 | 722.8 | 262.9 | 718.2 | 206.5 | 195.5 | 1038.4 | 0.103 | 2.57
(g) | 123.9 | 856.4 | 327.4 | 852.4 | 243.5 | 240.1 | 1227 | 0.016 | 1.94

We investigate three main SUSY production channels at the LHC: slepton and
sneutrino pair production and slepton associated production with a sneutrino.
The latter has a significant cross section since it proceeds via the charged
current. We calculate the production cross sections at 14 TeV and 27 TeV at
the aNNLO+NNLL accuracy using Resummino-3.0 [47, 48]. The results are
displayed in Table 3. Notice that the contribution from the right handed
sleptons becomes important in benchmarks (e), (f) and (g) where
$m_{\tilde{\ell}_{\rm R}}<m_{\tilde{\ell}_{\rm L}}$.

Table 3:  The aNNLO+NNLL production cross-sections of slepton pair $(\tilde{\ell}=\tilde{e},\tilde{\mu})$ and sneutrino pair as well as slepton associated production with a sneutrino at $\sqrt{s}=14$ TeV and $\sqrt{s}=27$ TeV for the benchmarks of Table 1. The cross section is in fb. Model | $\sigma(pp\rightarrow\tilde{\ell}_{\rm L}\,\tilde{\ell}_{\rm L})$ | $\sigma(pp\rightarrow\tilde{\ell}_{\rm R}\,\tilde{\ell}_{\rm R})$ | $\sigma(pp\rightarrow\tilde{\nu}_{\rm L}\,\tilde{\ell}_{\rm L})$ | $\sigma(pp\rightarrow\tilde{\nu}_{\rm L}\,\tilde{\nu}_{\rm L})$
---|---|---|---|---
| 14 TeV | 27 TeV | 14 TeV | 27 TeV | 14 TeV | 27 TeV | 14 TeV | 27 TeV
(a) | 1.084 | 4.515 | 0.057 | 0.335 |  |  |  |
(b) | 9.810 | 30.80 | 0.656 | 2.54 | 41.78 | 127.80 | 10.52 | 33.10
(c) | 5.805 | 19.31 | 0.417 | 1.72 | 24.40 | 78.97 | 4.94 | 15.10
(d) | 2.486 | 9.188 | 0.056 | 0.332 | 10.44 | 37.47 | 2.10 | 7.21
(e) | 0.388 | 1.915 | 1.22 | 4.32 |  |  |  |
(f) | 0.188 | 1.065 | 6.91 | 20.15 |  |  |  |
(g) | 0.074 | 0.506 | 2.87 | 9.22 |  |  |  |

In LHC analyses with simplified models, unit branching ratios for specific
production channels are often used. However, this simplified assumption does
not hold in high scale models where the branching ratios exhibit a rich
diversity of final states. Thus the exclusion limits on sparticle masses do
not apply directly to high scale models, but rather limits on
$\sigma\times\text{BR}$ become the relevant ones. To illustrate this we
display the different decay channels of the sleptons and sneutrinos in Table
4.

Table 4:  The branching ratios of relevant decay channels of the left and the right handed slepton and the sneutrino for the benchmarks of Table 1.  Model | $\tilde{\ell}_{\rm L}\to\ell\tilde{\chi}^{0}_{1}$ | $\tilde{\ell}_{\rm L}\to\ell\tilde{\chi}^{0}_{2}$ | $\tilde{\ell}_{\rm L}\to\nu_{\ell}\tilde{\chi}^{\pm}_{1}$ | $\tilde{\ell}_{\rm R}\to\ell\tilde{\chi}^{0}_{1}[\tilde{\chi}^{0}_{2}]$ | $\tilde{\nu}_{\rm L}\to\tilde{\chi}^{+}_{1}\ell^{-}$ | $\tilde{\nu}_{\rm L}\to\tilde{\chi}^{0}_{1}\nu_{\ell}$
---|---|---|---|---|---|---
(a) | 22% | 26% | 52% | 100% [-] | 51% | 24%
(b) | 31% | 7% | 62% | \- [100%] | 62% | 30%
(c) | 31% | 6% | 63% | \- [100%] | 62% | 31%
(d) | 31% | 6% | 63% | \- [100%] | 63% | 32%
(e) | 100% | - | - | 100% [-] | - | 100%
(f) | 100% | - | - | 100% [-] | - | 100%
(g) | 100% | - | - | 100% [-] | - | 100%

##  5 A deep neural network analysis for SUSY discovery

As can be seen from Table 4, there are numerous decay channels for the
sleptons and sneutrinos which result in rich final states. Leptonic channels
are the most prevalent and for these we consider final states consisting of
two leptons of the same flavour and opposite sign (SFOS) and missing
transverse energy (MET) due to the neutralinos (and neutrinos). We design two
signal regions, where in one we require exactly one non-b-tagged jet and in
the other we require at least two non-b-tagged jets. One can easily see how
slepton pair production can give us exactly such final states but it may be
less trivial for sneutrino pair production and for slepton-sneutrino
associated production. In Table 3 the cross section for sneutrino pair
production and slepton associated production for benchmarks (b), (c) and (d)
are shown. As mentioned earlier, those benchmarks belong to case (B) where the
chargino is nearly mass degenerate with the LSP. Thus it is not easy to
distinguish between these particles at the LHC and so we assume that the
chargino escapes undetected and adds to the MET in an event. With a 62%
branching ratio for $\tilde{\nu}_{\rm L}\to\tilde{\chi}^{+}_{1}\ell^{-}$ (see
Table 4), sneutrino pair production will result in the same final states with
a significant $\sigma\times\text{BR}$. The same applies for slepton associated
production with a sneutrino which has the largest cross section among the
different production channels considered. The rest of the benchmarks do not
have a degenerate chargino-LSP and so the decay of the chargino in those
benchmarks will result in very different final states.

The signal, which involves the different production channels presented in
Table 3, and the Standard Model background events are simulated at LO with
MadGraph5 [49] and showered with PYTHIA8 [50] with the addition of ISR and FSR
jets. Cross sections are scaled to their NLO values for the background and to
aNNLO+NNLL for the signal. Detector effects are added by the help of
DELPHES-3.4.2 [51].

The analysis presented here shows that a discrimination between the signal and
background events can be done using deep neural network (DNN) as part of the
‘Toolkit for Multivariate Analysis’ (TMVA) [52] framework within ROOT6 [53].
Thus the possibility of detection of light sparticles appears strong at HL-LHC
and with much greater certainty at HE-LHC.

We use a set of kinematic variables to train the DNN on the signal and
background events. The list of variables include

$\displaystyle E^{\rm miss}_{\rm T},~{}~{}~{}p_{\rm T}(j_{1}),~{}~{}~{}p_{\rm
T}(\ell_{1}),~{}~{}~{}p_{\rm T}(\ell_{2}),~{}~{}~{}p_{\rm T}^{\rm
ISR},~{}~{}~{}M_{\rm T2},~{}~{}~{}M^{\rm min}_{\rm T},~{}~{}~{}m_{\ell\ell},$
$\displaystyle\hskip 71.13188pt~{}~{}~{}\Delta\phi(\textbf{p}_{\rm
T}^{\ell},\textbf{p}^{\rm miss}_{\rm T}),~{}~{}~{}\Delta\phi_{\rm
min}(\textbf{p}_{\rm T}(j_{i}),\textbf{p}^{\rm miss}_{\rm T}),$

where $M^{\rm min}_{\rm T}=\text{min}[m_{\rm T}(\textbf{p}_{\rm
T}^{\ell_{1}},\textbf{p}^{\rm miss}_{\rm T}),m_{\rm T}(\textbf{p}_{\rm
T}^{\ell_{2}},\textbf{p}^{\rm miss}_{\rm T})]$, $\Delta\phi(\textbf{p}_{\rm
T}^{\ell},\textbf{p}^{\rm miss}_{\rm T})$ is the opening angle between the MET
system and the dilepton system and $\Delta\phi_{\rm min}(\textbf{p}_{\rm
T}(j_{i}),\textbf{p}^{\rm miss}_{\rm T})_{i=1,2,3}$ is the smallest opening
angle between the first three leading jets in an event and the MET system.
Note that $\textbf{p}_{\rm T}^{\ell}=\textbf{p}_{\rm
T}^{\ell_{1}}+\textbf{p}_{\rm T}^{\ell_{2}}$. After reconstructing the
momentum of the dilepton system, we determine the angle between the dilepton
system and each non-b-tagged jet in the event. We select up to two jets that
are closest to the dilepton system and tag them as possible jets arising from
the decay of the SUSY system while the rest are classified as ISR jets. Here
$p_{\rm T}^{\rm ISR}$ denotes the total transverse momentum of all the ISR
jets in an event.

Table 5: The preselection criteria and the analysis cuts on a set of kinematic
variables at 14 TeV (27 TeV) grouped by the benchmarks of Table 1 in two
signal regions SR-$2\ell 1$j and SR-$2\ell 2$j. Entries with a dash (-) imply
that no requirement on the variable is considered. Cuts are optimized for each
center-of-mass energy.

Observable | (b), (c), (d) | (a) | (e), (f), (g) | (b), (c), (d) | (a) | (e), (f), (g)
---|---|---|---|---|---|---
| Preselection criteria (SR-$2\ell 2$j) | Preselection criteria (SR-$2\ell 1$j)
$N_{\ell}$ (SFOS) | $2$ | $2$
$N_{\rm jets}^{\rm non-b-tagged}$ | $\geq 2$ | $1$
$p_{T}(j_{1})$ [GeV] | $>20$ | $>20$
$p_{T}(\ell_{1})$ (electron, muon) [GeV] |  $>15$, $>10$ |  $>15$, $>10$
$E^{\rm miss}_{T}$ [GeV] | $>100$ | $>100$
| Analysis cuts | Analysis cuts
 $m_{\ell\ell}~{}\text{[GeV]}>$ | 136 (110) | 150 | 150 (110) | 110 | 120 | 150
$E^{\rm miss}_{T}/\textbf{p}^{\ell}_{\rm T}>$ | 1.9 (2.8) | - | - | 1.0 | - | -
$\Delta\phi_{\rm min}(\textbf{p}_{\rm T}(j_{i}),\textbf{p}^{\rm miss}_{\rm T})~{}\text{[rad]}>$ | - | 0.85 (1.5) | - | - | 0.85 (1.5) | -
$p_{T}(\ell_{2})~{}\text{[GeV]}>$ | - | - | 190 (370) | - | - | 190 (300)
$M_{T2}~{}\text{[GeV]}>$ | \- (140) | \- (120) | 200 (300) | 130 (230) | 100 | 200 (300)
DNN response $>$ | 0.95 | 0.95 | 0.95 | 0.95 | 0.95 | 0.95

We present in Table 5 the set of preselection criteria as well as the analysis
cuts used on the signal and background events. The cuts are optimized
depending on the category the benchmarks belong to and the center-of-mass
energy.

After training the DNN on the signal and background events and making sure no
overtraining has occurred, the testing phase is done on a separate set of
signal and background events. The end result is the construction of a new
discriminating variable called the ‘DNN response’ which tends to take values
closer to one for the signal. Signal (S) and background (B) distributions in
the ‘DNN response’ variable are shown in Fig. 6 for benchmarks (a) (top
panels) and (c) (bottom panels) at 14 TeV and 27 TeV. In the bottom pad of
each panel, we show the distribution in the signal significance $Z$ as a
function of the cut on ‘DNN response’ and taking into consideration the
systematics, i.e.,

$\displaystyle Z=\frac{S}{\sqrt{S+B+(\delta_{S}S)^{2}+(\delta_{B}B)^{2}}},$
(5.1)

where $\delta_{S}$ and $\delta_{B}$ are the systematic uncertainties in the
signal and background estimates. The recommendations on systematic
uncertainties (known as ‘YR18’ uncertainties) published in the CERN’s yellow
reports [54, 55] suggest an overall 20% uncertainty in the background and 10%
in the SUSY signal. Notice that a $5\sigma$ value is reached at the integrated
luminosities shown which can be attained at HL-LHC and HE-LHC.

Figure 6: Distributions in the ‘DNN response’ variable for benchmark (a) (top
panels) and benchmark (c) (bottom panels) at 14 TeV (left) and 27 TeV (right).
A distribution in the signal significance as a function of the cut on the ‘DNN
response’ is shown in the bottom pads of each panel.

Of the two signal regions considered, we find that the most optimal one is the
single jet signal region, which has also been shown to be true in previous LHC
searches [56]. This is evident from Table 6 where the integrated luminosity
for discovery of the benchmarks in SR-$2\ell 2$j is much larger than in
SR-$2\ell 1$j after combining all production channels. The presented
integrated luminosities include the effect of systematics.

Table 6:  The estimated integrated luminosities, in fb-1, for discovery of
benchmarks of Table 1 at 14 TeV and 27 TeV after combining all production
channels and including systematics in the signal and background.

Model | SR-$2\ell 1$j | SR-$2\ell 2$j
---|---|---
|  $\mathcal{L}$ at 14 TeV |  $\mathcal{L}$ at 27 TeV |  $\mathcal{L}$ at 14 TeV |  $\mathcal{L}$ at 27 TeV
(a) | 880 | 310 | 1262 | 694
(b) | 200 | 50 | 1860 | 715
(c) | 148 | 75 | 1887 | 1320
(d) | 425 | 252 | $\cdots$ | 2804
(e) | 1040 | 232 | 1738 | 1194
(f) | 730 | 152 | 2074 | 689
(g) | 970 | 202 | $\cdots$ | 1031

As a comparison between HL-LHC and HE-LHC and to see the effect of systematic
uncertainties, we plot the estimated integrated luminosities for discovery of
benchmarks (a)$-$(g) at 14 TeV and 27 TeV in Fig. 7. Benchmarks (b) and (c)
require 200 fb-1 and 148 fb-1 at 14 TeV which should be attained in the coming
run of LHC. The rest require more than 400 fb-1 but are all within the reach
of HL-LHC. The same benchmarks require much smaller integrated luminosities
for discovery at HE-LHC.

Figure 7: The integrated luminosities, $\mathcal{L}$, needed for discovery of
SUSY at HL-LHC and HE-LHC assuming that $\Delta{a^{\rm FB}_{\mu}}$ arises from
SUSY loops. Values of $\mathcal{L}$ are shown before and after including the
‘YR18’ uncertainties on the signal and background.

We note that recently several works, including those within SUSY, have come
out regarding an explanation of the Fermilab muon anomaly [57, 58, 59, 60, 61,
62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80,
81, 82, 83, 90, 89, 88, 87, 86, 85, 84, 91, 92].

##  6 Conclusion

The recent results from the Fermilab Collaboration confirm the earlier result
from the Brookhaven experiment regarding a deviation from the Standard Model
prediction of the muon $g-2$ anomaly. Specifically, the combined Fermilab and
Brookhaven results give a $4.1\sigma$ deviation from the Standard Model
prediction compared to $3.7\sigma$ prediction for Brookhaven which both
supports the observation and further strengthens it.

In this paper we have pointed to the implications of the Fermilab result for
SUSY discovery at the LHC. To this end we have analyzed the effect of the
Fermilab constraint on the SUGRA parameter space. This is done in a
comprehensive way with the help of neural networks where we have used the
constraints of the Higgs boson mass and the relic density along with the muon
$g-2$ to delineate the region of the parameter space consistent with those
constraints. The neural network analysis identifies the allowed region as the
one that is generated by gluino-driven radiative breaking of the electroweak
symmetry. This region of the parameter space naturally leads to a split
sparticle spectrum constituted of light sleptons and light weakinos but a
heavy gluino and heavy squarks which allows one to satisfy muon $g-2$
constraint and the Higgs boson constraint within a high scale model. We
emphasize the existence of a light spectrum and their possible early discovery
at the LHC. We propose a set of benchmarks relevant for Snowmass study related
to the energy frontier. Further, we demonstrate that a deep neural network
with the ‘DNN response’ variable constitutes a powerful discriminant to reject
the Standard Model background which aids in the discovery of new physics
signals. We exhibit the importance of this technique by analyzing the
integrated luminosities needed for the discovery of the benchmarks, and show
that some of the benchmarks can be discovered at HL-LHC with integrated
luminosities as low as 150$-$300 fb-1 while much smaller integrated
luminosities are required at HE-LHC.

We propose that it would be fruitful if the future SUSY analyses at the LHC
are carried out with the inclusion of the Fermilab $g-2$ constraint and
further such analyses need to go beyond generic simplified models to well
motivated high scale models. The basic reason for such analyses is that in
well motivated models the branching ratio to any production channel is almost
never unity as generally assumed in simplified models. Thus dedicated studies
for the production and signal analysis of sleptons and weakinos at the LHC
within Snowmass study are timely.

The research of AA and MK was supported by the BMBF under contract 05H18PMCC1.
The research of PN was supported in part by the NSF Grant PHY-1913328.

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