#  Likelihood Analysis of Supersymmetric SU(5) GUTs

E. Bagnaschi, J.C. Costa, K. Sakurai,, M. Borsato, O. Buchmueller[Imperial],
R. Cavanaugh,, V. Chobanova[USdC], M. Citron[Imperial], A. De Roeck, M.J.
Dolan, J.R. Ellis, H. Flächer, S. Heinemeyer, G. Isidori, M. Lucio[USdC], D.
Martínez Santos[USdC], K.A. Olive, A. Richards[Imperial], K.J. de
Vries[Imperial], G. Weiglein[DESY]  DESY, Notkestraße 85, D–22607 Hamburg,
Germany  High Energy Physics Group, Blackett Laboratory, Imperial College,
Prince Consort Road, London SW7 2AZ, UK  Institute for Particle Physics
Phenomenology, Department of Physics, University of Durham, Science
Laboratories, South Road, Durham, DH1 3LE, UK  Institute of Theoretical
Physics, Faculty of Physics, University of Warsaw, ul. Pasteura 5, PL–02–093
Warsaw, Poland  Universidade de Santiago de Compostela, E-15706 Santiago de
Compostela, Spain  Fermi National Accelerator Laboratory, P.O. Box 500,
Batavia, Illinois 60510, USA  Physics Department, University of Illinois at
Chicago, Chicago, Illinois 60607-7059, USA  Experimental Physics Department,
CERN, CH–1211 Geneva 23, Switzerland;
Antwerp University, B–2610 Wilrijk, Belgium  ARC Centre of Excellence for
Particle Physics at the Terascale, School of Physics, University of Melbourne,
3010, Australia  Theoretical Particle Physics and Cosmology Group, Department
of Physics, King’s College London, London WC2R 2LS, UK;
Theoretical Physics Department, CERN, CH–1211 Geneva 23, Switzerland  H.H.
Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol BS8
1TL, UK  Campus of International Excellence UAM+CSIC, Cantoblanco, E–28049
Madrid, Spain;
Instituto de Física Teórica UAM-CSIC, C/ Nicolas Cabrera 13-15, E–28049
Madrid, Spain;
Instituto de Física de Cantabria (CSIC-UC), Avda. de Los Castros s/n, E–39005
Santander, Spain  Physik-Institut, Universität Zürich, CH-8057 Zürich,
Switzerland  William I. Fine Theoretical Physics Institute, School of Physics
and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455, USA

###### Abstract

We perform a likelihood analysis of the constraints from accelerator
experiments and astrophysical observations on supersymmetric (SUSY) models
with SU(5) boundary conditions on soft SUSY-breaking parameters at the GUT
scale. The parameter space of the models studied has 7 parameters: a universal
gaugino mass $m_{1/2}$, distinct masses for the scalar partners of matter
fermions in five- and ten-dimensional representations of SU(5), $m_{5}$ and
$m_{10}$, and for the $\mathbf{5}$ and $\mathbf{\bar{5}}$ Higgs
representations $m_{H_{u}}$ and $m_{H_{d}}$, a universal trilinear soft SUSY-
breaking parameter $A_{0}$, and the ratio of Higgs vevs $\tan\beta$. In
addition to previous constraints from direct sparticle searches, low-energy
and flavour observables, we incorporate constraints based on preliminary
results from 13 TeV LHC searches for jets + $\,/E_{T}$ events and long-lived
particles, as well as the latest PandaX-II and LUX searches for direct Dark
Matter detection. In addition to previously-identified mechanisms for bringing
the supersymmetric relic density into the range allowed by cosmology, we
identify a novel ${\tilde{u}_{R}}/{\tilde{c}_{R}}-\tilde{\chi}^{0}_{1}$
coannihilation mechanism that appears in the supersymmetric SU(5) GUT model
and discuss the role of ${\tilde{\nu}_{\tau}}$ coannihilation. We find
complementarity between the prospects for direct Dark Matter detection and
SUSY searches at the LHC.

KCL-PH-TH/2016-57, CERN-PH-TH/2016-217, DESY 16-156, IIFT-UAM/CSIC-16-105
FTPI-MINN-16/29, UMN-TH-3609/16, IPPP/16/97

##  1 Introduction

In the absence so far of any experimental indications of supersymmetry (SUSY)
[1, 2, 3, 4, 5], nor any clear theoretical guidance how SUSY may be broken,
the building of models and the exploration of phenomenological constraints on
them [6, 7, 8, 9, 10, 11] have adopted a range of assumptions. One point of
view has been to consider the simple parametrization of soft SUSY breaking in
which the gaugino and scalar masses, as well as the trilinear soft SUSY-
breaking parameters, are all constrained to be universal at the SUSY GUT scale
(the CMSSM [12, 13, 14, 15, 6, 7]). An alternative point of view has been to
discard all universality assumptions, and treat the soft SUSY-breaking
parameters as all independent phenomenological quantities (the pMSSM [16, 9]),
imposing diagonal mass matrices and the minimal flavour violation (MFV)
criterion. Intermediate between these extremes, models with one or two non-
universal soft SUSY-breaking contributions to Higgs masses (the NUHM1 [17, 18,
19, 20, 6, 7] and NUHM2 [21, 22, 18, 19, 20, 8]) have also been considered.

It is interesting to explore also models that are less (over-)simplified than
the CMSSM, but not as agnostic as the pMSSM, in that they incorporate a
limited number of simplifying assumptions. GUTs motivate the assumption that
the gaugino masses are universal, and constraints on flavour-changing neutral
interactions suggest that the soft SUSY-breaking masses for scalars with
identical quantum numbers are also universal. However, there is no compelling
phenomenological reason why the soft SUSY-breaking masses for scalars with
different quantum numbers should be universal.

Specific GUTs may also provide some guidance in this respect. For example, in
an SO(10) GUT the scalar masses of all particles in a given generation
belonging to a single $\mathbf{16}$ representation of SO(10) would be
universal, as would those for the $\mathbf{5}$ and $\mathbf{\bar{5}}$ SU(5)
Higgs representations that belong to a single $\mathbf{10}$ of SO(10) and
break electroweak symmetry, as in the NUHM1. In contrast, the SU(5) framework
is less restrictive, allowing different masses for scalars in
$\mathbf{\bar{5}}$ and $\mathbf{10}$ representations [23], and also for the
$\mathbf{5}$ and $\mathbf{\bar{5}}$ Higgs representations. Thus it is a
1-parameter extension of the NUHM2. In this paper we explore the theoretical,
phenomenological, experimental and cosmological constraints on this
SU(5)-based SUSY GUT model.

This relaxation of universality is relevant for the evaluation of several
different constraints from both the LHC and elsewhere. For example, the most
powerful LHC constraints on the CMSSM, NUHM1 and NUHM2 are those from the
classic $\,/E_{T}$ searches [1, 4]. These constrain principally the right-
handed squarks, whose decays are dominated by the ${\tilde{q}_{R}}\to
q\,\tilde{\chi}^{0}_{1}$ channel that maximizes the $\,/E_{T}$ signature. On
the other hand, the decay chains of left-handed squarks are more complicated,
typically involving the $\tilde{\chi}^{\pm}_{1}$, resulting in a dilution of
the $\,/E_{T}$ signature and more importance for final states including
leptons. In a SUSY SU(5) GUT, the left-handed squarks and the right-handed up-
type squarks appear in $\mathbf{10}$ representations whereas the right-handed
down-type squarks appear in $\mathbf{\bar{5}}$ representations, with
independent soft SUSY-breaking masses. Hence the impacts of the LHC $\,/E_{T}$
and other constraints need to be re-evaluated.

The possible difference between the soft SUSY-breaking contributions to the
masses of the squarks appearing in a $\mathbf{10}$ of SU(5), i.e., up-type
squarks and left-handed down-type squarks, and those appearing in a
$\mathbf{\bar{5}}$ of SU(5), i.e., right-handed down-type squarks, may also be
relevant to the possibility of a compressed stop spectrum. Also, as we shall
see, with $m_{5}\neq m_{10}$ there is the possibility that
$m_{{\tilde{u}_{R}},{\tilde{c}_{R}}}$ are much smaller than the other squark
masses, leading to another type of compressed spectrum.

In principle, the constraints from flavour observables may also act
differently when $m_{5}\neq m_{10}$. For example, the soft SUSY-breaking
masses of the left- and right-handed charge +2/3 quarks are independent, and
flavour observables such as BR($b\to s\gamma$) and ${\rm
BR}(B_{s}\to\mu^{+}\mu^{-})$ depend on both of them, in general.

Another experimental constraint whose interpretation may be affected by the
non-universality of scalar masses is $(g-2)_{\mu}$. A priori, a SUSY
explanation of the discrepancy between the Standard Model (SM) prediction and
the experimental measurement of $(g-2)_{\mu}$ requires relatively light
smuons, either right- and/or left-handed, which are in $\mathbf{10}$ and
$\mathbf{\bar{5}}$ representations, respectively. It is interesting to
investigate to what extent the tension between a SUSY interpretation of
$(g-2)_{\mu}$ and the LHC constraints on squarks that is present in more
constrained SUSY models could be alleviated by the extra degree of freedom
afforded by the $\mathbf{\bar{5}}-\mathbf{10}$ disconnect in SU(5).

Finally, we recall that in large parts of the regions of the CMSSM, NUHM1 and
NUHM2 parameter spaces favoured at the 68% CL the relic $\tilde{\chi}^{0}_{1}$
density is brought into the range allowed by Planck [24] and other data via
coannihilation with the stau and other sleptons [25, 26]. In an SU(5) GUT, the
left- and right-handed sleptons are in different representations,
$\mathbf{\bar{5}}$ and $\mathbf{10}$, respectively. Hence they have different
masses, in general, providing more flexibility in the realization of
coannihilation. Specifically, as mentioned above, the freedom to have
$m_{5}\neq m_{10}$ allows the possibility that the right-handed up- and charm-
flavour squarks, ${\tilde{u}_{R}}$ and ${\tilde{c}_{R}}$, are much lighter
than the other squarks, opening up the novel possibility of
${\tilde{u}_{R}}/{\tilde{c}_{R}}-\tilde{\chi}^{0}_{1}$ coannihilation, as we
discuss below.

Our analysis of the available experimental constraints largely follows those
in our previous studies of other variants of the MSSM [6, 7, 8, 9, 10, 11],
the main new feature being that we incorporate the constraints based on the
preliminary results from LHC searches for jets + $\,/E_{T}$ events with $\sim
13$/fb of data at 13 TeV [5]. For this purpose, we recast available results
for simplified models with the mass hierarchies $m_{\tilde{g}}\gg
m_{\tilde{q}}$ and vice versa. We also include the preliminary constraints
from LHC searches in 13-TeV data for the heavy MSSM Higgs bosons and long-
lived charged particles, and incorporate in combination the recent PandaX [27]
and LUX [28] data.

The SUSY SU(5) GUT model we study is set up in Section 2, and our
implementations of constraints and analysis procedure are summarized in
Section 3. Section 4 describes how we characterize different Dark Matter (DM)
mechanisms, including the novel
${\tilde{u}_{R}}/{\tilde{c}_{R}}-\tilde{\chi}^{0}_{1}$ coannihilation
mechanism, ${\tilde{\nu}_{\tau}}$ coannihilation and various hybrid
possibilities. Section 5 contains our results in several model parameter
planes, and Section 6 describes various one-dimensional likelihood functions
including those for several sparticle masses, $(g-2)_{\mu}$ and various other
observables. Higgs boson branching ratios (BRs) and are presented in Section
7, followed by a comparison of the SU(5) with the NUHM2 results in Section 8.
The possibility of a long-lived $\tilde{\tau}_{1}$ is discussed in Section 9,
and the prospects for direct DM detection are discussed in Section 10.
Finally, Section 11 presents a summary and some conclusions.

##  2 Supersymmetric SU(5) GUT Model

We assume a universal, SU(5)-invariant gaugino mass parameter $m_{1/2}$, which
is input at the GUT scale, as are the other SUSY-breaking parameters listed
below.

We assume the conventional multiplet assignments of matter fields in the
minimal superymmetric GUT:

$(q_{L},u^{c}_{L},e^{c}_{L})_{i}\;\in\;\mathbf{10}_{i},\;\;(\ell_{L},d^{c}_{L})_{i}\;\in\;\mathbf{\bar{5}}_{i}\,,$
(1)

where the subscript $i=1,2,3$ is a generation index. The only relevant Yukawa
couplings are those of the third generation, particularly that of the $t$
quark (and possibly the $b$ quark and the $\tau$ lepton) that may play an
important role in generating electroweak symmetry breaking. In our discussion
of flavour constraints, we assume the MFV scenario in which generation mixing
is described by the Cabibbo-Kobayashi-Maskawa (CKM) model. This is motivated
by phenomenological constraints on low-energy flavour-changing neutral
interactions, as is our assumption that the soft SUSY-breaking scalar masses
for the different $\mathbf{10}_{i}$ and $\mathbf{\bar{5}}_{i}$ representations
are universal in generation space, and are denoted by $m_{10}$ and $m_{5}$,
respectively. In contrast to the CMSSM, NUHM1 and NUHM2, we allow $m_{5}\neq
m_{10}$. We assume a universal soft trilinear SUSY-breaking parameter $A_{0}$.

We assume the existence of two Higgs doublets $H_{u}$ and $H_{d}$ in
$\mathbf{5}$ and $\mathbf{\bar{5}}$ representations that break electroweak
symmetry and give masses to the charge +2/3 and charge -1/3 and -1 matter
fields, respectively. It is well known that this assumption gives a
(reasonably) successful relation between the masses of the $b$ quark and the
$\tau$ lepton [29], but not for the lighter charge -1/3 quarks and charged
leptons. We assume that whatever physics resolves this issue is irrelevant for
our analysis, as would be the case, for instance, if corrections to the naive
SU(5) mass relations were generated by higher-dimensional superpotential terms
[30]. In the absence of any phenomenological constraints, we allow the soft
SUSY-breaking contributions to the $H_{u}$ and $H_{d}$ masses, $m_{H_{u}}$ and
$m_{H_{d}}$, to be different from each other, as in the NUHM2, as well as from
$m_{5}$ and $m_{10}$. As in the CMSSM, NUHM1 and NUHM2, we allow the ratio of
Higgs vacuum expectation values, $\tan\beta$, to be a free parameter.

In addition to these electroweak Higgs representations, we require one or more
Higgs representations to break the SU(5) GUT symmetry. The minimal possibility
is a single $\mathbf{24}$ representation $\Sigma$, but we do not commit
ourselves to this minimal scenario. It is well known that this scenario has
problems with rapid proton decay 111We note that this problem becomes less
severe for supersymmetry-breaking scales beyond a TeV [20]. and GUT threshold
effects on gauge coupling unification. We assume that these issues are
resolved by the appearance of additional fields at or around the GUT scale
that are otherwise irrelevant for TeV-scale phenomenology. The effective low-
energy Higgsino mixing coupling $\mu$ is a combination of an input bilinear
$H_{u}H_{d}$ coupling and possible trilinear and higher-order couplings to
GUT-scale Higgs multiplets such as $H_{u}\Sigma H_{d}$. We assume that these
combine to yield $\mu={\cal O}(1)\,\,\mathrm{TeV}$ and positive, without
entering into the possibility of some dynamical mechanism, and commenting
below only briefly on the case $\mu<0$.

##  3 Implementations of Constraints and Analysis Procedure

Our treatments in this paper of many of the relevant constraints follow very
closely the implementations in our previous analyses of other supersymmetric
models [6, 7, 8, 9, 10]. For the convenience of the reader, we summarise the
constraints in Table 1. In the following subsections we review our
implementations, highlighting new constraints and instances where we implement
constraints differently from our previous work.

###  3.1 Electroweak and Flavour Constraints

We treat as Gaussian constraints all electroweak precision observables, all
$B$-physics and $K$-physics observables except for ${\rm
BR}(B_{s,d}\to\mu^{+}\mu^{-})$. The $\chi^{2}$ contribution from ${\rm
BR}(B_{s,d}\to\mu^{+}\mu^{-})$, combined here in the quantity $R_{\mu\mu}$
[7], is calculated using a combination of the CMS [31] and LHCb [32] results
described in [33] with the more recent result from ATLAS [34]. The
corresponding $\chi^{2}$ contribution is labelled as “2D likelihood” in Table
1. We calculate the elements of the CKM matrix using only experimental
observables that are not included in our set of flavour constraints.

We have updated our implementations of all the flavour constraints, and now
use the current world average value of $m_{t}$ [35]. These and all other
constraints whose implementations have been changed are indicated by arrows
and boldface in Table 1.

Observable | Source | Constraint
---|---|---
| Th./Ex. |
$\to\quad\mathbf{m_{t}}$ [GeV] | [35] |  $\mathbf{173.34\pm 0.76}$
$\Delta\alpha_{\rm had}^{(5)}(M_{Z})$ | [36] |  $0.02771\pm 0.00011$
$M_{Z}$ [GeV] | [37, 38] | $91.1875\pm 0.0021$
$\Gamma_{Z}$ [GeV] |  [39] /[37, 38] | $2.4952\pm 0.0023\pm 0.001_{\rm SUSY}$
$\sigma_{\rm had}^{0}$ [nb] |  [39] /[37, 38] | $41.540\pm 0.037$
$R_{l}$ |  [39] /[37, 38] | $20.767\pm 0.025$
$A_{\rm FB}(\ell)$ |  [39] /[37, 38] | $0.01714\pm 0.00095$
$A_{\ell}(P_{\tau})$ |  [39] /[37, 38] |  0.1465 $\pm$ 0.0032
$R_{\rm b}$ |  [39] /[37, 38] |  0.21629 $\pm$ 0.00066
$R_{\rm c}$ |  [39] /[37, 38] |  0.1721 $\pm$ 0.0030
$A_{\rm FB}({b})$ |  [39] /[37, 38] |  0.0992 $\pm$ 0.0016
$A_{\rm FB}({c})$ |  [39] /[37, 38] |  0.0707 $\pm$ 0.0035
$A_{b}$ |  [39] /[37, 38] |  0.923 $\pm$ 0.020
$A_{c}$ |  [39] /[37, 38] |  0.670 $\pm$ 0.027
${A_{\rm LR}^{e}}$ |  [39] /[37, 38] |  0.1513 $\pm$ 0.0021
$\sin^{2}\theta_{\rm w}^{\ell}(Q_{\rm fb})$ |  [39] /[37, 38] |  0.2324 $\pm$ 0.0012
$M_{W}$ [GeV] |  [39] /[37, 38] | $80.385\pm 0.015\pm 0.010_{\rm SUSY}$
$a_{\mu}^{\rm EXP}-a_{\mu}^{\rm SM}$ |  [40] /[41] | $(30.2\pm 8.8\pm 2.0_{\rm SUSY})\times 10^{-10}$
$\to\quad\mathbf{M_{h}}$ [GeV] |  [42, 43] / [44] |  $\mathbf{125.09\pm 0.24\pm 1.5_{\rm SUSY}}$
$\to\quad$ BR${{}_{{b\to s\gamma}}^{\rm EXP/SM}}$ |  [45]/[46] | $\mathbf{1.021\pm 0.066_{\rm EXP}}$
|  |  1.021$\mathbf{\pm 0.070_{\rm TH,SM}\pm 0.050_{\rm TH,SUSY}}$
$\to\quad\mathbf{R_{\mathbf{\mu\mu}}}$ |  [47]/[33, 34] | 2D likelihood
$\to\quad$ BR${{}_{B\to\tau\nu}^{\rm EXP/SM}}$ | [46, 48] | $\mathbf{1.02\pm 0.19_{\rm EXP}\pm 0.13_{\rm SM}}$
$\to\quad\mathbf{BR}_{B\to X_{s}\ell\ell}^{\rm EXP/SM}$ |  [49]/[46] | $\mathbf{0.99\pm 0.29_{\rm EXP}\pm 0.06_{\rm SM}}$
$\to\quad\mathbf{BR}_{K\to\mu\nu}^{\rm EXP/SM}$ |  [50, 51] /[36] | $\mathbf{0.9998\pm 0.0017_{\rm EXP}\pm 0.0090_{\rm TH}}$
$\to\quad\mathbf{BR}_{K\to\pi\nu\bar{\nu}}^{\rm EXP/SM}$ |  [52]/[53] | $\mathbf{2.2\pm 1.39_{\rm EXP}\pm 0.20_{\rm TH}}$
$\to\quad\mathbf{\Delta M}_{B_{s}}^{\rm EXP/SM}$ |  [54, 50] /[46] | $\mathbf{1.016\pm 0.074_{\rm SM}}$
$\to\quad{\frac{\mathbf{\Delta M}_{B_{s}}^{\rm EXP/SM}}{\mathbf{\Delta M}_{B_{d}}^{\rm EXP/SM}}}$ |  [54, 50] /[46] | $\mathbf{0.84\pm 0.12_{\rm SM}}$
$\to\quad\mathbf{\Delta\epsilon}_{K}^{\rm EXP/SM}$ |  [54, 50] /[36] | $\mathbf{1.14\pm 0.10_{\rm EXP+TH}}$
$\to\quad\mathbf{\Omega_{\mathbf{\rm}CDM}h^{2}}$ |  [55, 56]/[24] | $\mathbf{0.1186\pm 0.0020}_{\rm EXP}\mathbf{\pm 0.0024_{\rm TH}}$
$\to\mathbf{\sigma^{\rm SI}_{p}}$ | [27, 28] |  $\mathbf{(m_{\tilde{\chi}^{0}_{1}},\sigma^{\rm SI}_{p})}$ plane
$\to\quad$ Heavy stable charged particles | [57] |  Fast simulation based on [57, 58]
$\to\quad\mathbf{\tilde{q}\to q\tilde{\chi}^{0}_{1}},\mathbf{\tilde{g}\to f\bar{f}\tilde{\chi}^{0}_{1}}$ | [5] |  $\mathbf{\sigma\cdot{\rm BR}}$ limits in the $\mathbf{(m_{\tilde{q}},m_{\tilde{\chi}_{1}^{0}})},\mathbf{(m_{\tilde{g}},m_{\tilde{\chi}_{1}^{0}})}$ planes
$\to\mathbf{H/A\to\tau^{+}\tau^{-}}$ | [59, 60, 61] | 2D likelihood, $\mathbf{\sigma\cdot{\rm BR}}$ limit
Table 1: List of experimental constraints used in this work, including
experimental and (where applicable) theoretical errors: supersymmetric theory
uncertainties are indicated separately. Instances where our implementations
differ from those in Table 1 in [9] are indicated by arrows and boldface.

###  3.2 Higgs Constraints

We use the combination of ATLAS and CMS measurements of the mass of the Higgs
boson: $M_{h}=125.09\pm 0.24\,\,\mathrm{GeV}$ [44]. We employ the FeynHiggs
2.11.2 code [42, 43] to evaluate the constraint this imposes on the parameter
space, assuming a one-$\sigma$ theoretical uncertainty of
$1.5\,\,\mathrm{GeV}$ 222We use a modified version of FeynHiggs 2.11.2 that
includes two-loop QCD corrections in the evaluation of the
$\smash{\overline{\mathrm{DR}}}$ running top mass and an improved evalution of
the top mass in the $\smash{\overline{\mathrm{DR}}}$-on-shell conversion for
the scalar tops..

The $\chi^{2}$ contributions of 77 Higgs search channels from the LHC and the
Tevatron are evaluated using HiggsSignals, see [62], where a complete list of
references can be found. The $\chi^{2}$ contributions from the limits from
searches for the heavy neutral MSSM Higgs bosons in the
$H/A\to\tau^{+}\tau^{-}$ channels are evaluated using the code HiggsBounds
[63, 60], which incorporates the results of CMS searches [59, 60] with $\sim
25~{}{\rm fb}^{-1}$ of 8 TeV data. The contributions from the two possible
production modes, $gg\to H/A$ and $b\bar{b}\to H/A$, are combined in a
consistent manner, depending on the MSSM parameters. The results from
HiggsBounds have been compared with the published CMS analysis, and are in
very good agreement [60]. The corresponding $\chi^{2}$ contribution is
labelled as “2D likelihood” in Table 1. For the corresponding constraint with
$~{}13$ fb-1 of 13 TeV data, we implement an approximate treatment of the
$\chi^{2}$ contribution using the preliminary result of ATLAS [61], as we
describe in more detail below. Limits from other Higgs boson searches are not
relevant for the investigation in this paper and are therefore not included.

###  3.3 LHC $\,/E_{T}$ constraints at 13 TeV

ATLAS and CMS have recently announced preliminary results from $\,/E_{T}$
searches with $\sim 13$/fb of data at 13 TeV, using simplified models for
gluino and squark pair production [3, 5]. These searches assume
$m_{\tilde{g}}\ll m_{\tilde{q}}$ and $m_{\tilde{q}}\ll m_{\tilde{g}}$,
respectively, and 100% BRs for the decays ${\tilde{g}}\to
f{\bar{f}}\tilde{\chi}^{0}_{1}$ ($f=q,b,t$) and $\tilde{q}\to
q\tilde{\chi}^{0}_{1}$, respectively, which maximize the possible
corresponding $\,/E_{T}$ signatures. Neither of these assumptions is valid in
the SUSY SU(5) GUT model: as we will see in more detail later, the
$m_{\tilde{g}}$ and $m_{\tilde{q}}$ masses are quite similar in much of the
favoured region of parameter space 333An exception is provided by the
${\tilde{u}_{R}}$ and ${\tilde{c}_{R}}$, which may be much lighter than the
gluino and other squarks in some regions of parameter space. We will discuss
this possibility in detail below., and in general other decay modes dilute the
$\,/E_{T}$ signature, although larger-multiplicity final states may compensate
through an increase in transverse energy $H_{T}$ [64]. These other decay modes
populate other search channels including leptons, which we do not consider in
this paper as they were of limited importance in our previous analyses of the
CMSSM, NUHM1 and NUHM2, having impact only for relatively large squark masses
and small $m_{1/2}$.

Fig. 1 displays the ratios of the ${\tilde{g}}{\tilde{g}}$ cross section (left
panel) and the $\tilde{q}\tilde{q}+\tilde{q}\bar{\tilde{q}}$ cross section
(right panel) that we find in ranges of $m_{\tilde{q}}$ and $m_{\tilde{g}}$
that are representative of those favoured in our analysis before implementing
the LHC 13-TeV $\,/E_{T}$ constraint, relative to the cross sections found in
the simplified models with $m_{\tilde{g}}\ll m_{\tilde{q}}$ and
$m_{\tilde{q}}\ll m_{\tilde{g}}$, respectively. We have used NLL-fast-3.1 [65]
to obtain the cross section at NLO + NLL level. In both plots a large area at
higher squark masses is visible, as well as a thin strip at $\sim
500\,\,\mathrm{GeV}$. The latter corresponds to lighter ${\tilde{u}_{R}}$ and
${\tilde{c}_{R}}$ discussed below. We see that the ${\tilde{g}}{\tilde{g}}$
cross section (left panel) is generally smaller than in the corresponding
simplified model by a factor $>2$ due to the destructive interference between
the $s$-channel gluon exchange diagram and the $t$-channel squark exchange
diagram in $q\bar{q}\to{\tilde{g}}{\tilde{g}}$, thus weakening the LHC
constraints as discussed below. On the other hand, the
$\tilde{q}\tilde{q}+\tilde{q}\bar{\tilde{q}}$ cross section (right panel) is
generally a factor $\gtrsim 10$ larger than in the simplified model, except in
the ${\tilde{u}_{R}}/{\tilde{c}_{R}}-\tilde{\chi}^{0}_{1}$ coannihilation
strip at small
$m_{\tilde{u}_{R}},m_{\tilde{c}_{R}},m_{\tilde{\chi}^{0}_{1}}\sim
500\,\,\mathrm{GeV}$ and $m_{1/2}\sim 2500\,\,\mathrm{GeV}$, to which we
return later. The enhancement of the squark cross-section is due to the fact
that in the squark-neutralino simplified model there is no production mode
with total baryon number $B=2/3$; $qq\to\tilde{q}\tilde{q}$, because gluinos
are assumed to be absent. On the other hand, in our model
$m_{\tilde{g}}\sim\min(m_{\tilde{q}})$, and $qq\to\tilde{q}\tilde{q}$ (with
$t$-channel ${\tilde{g}}$ exchange) becomes the dominant squark production
mode in the large $m_{\tilde{q}}$ region, due to the valence quark-parton
dominance in the proton in the large $x$ regime.

Figure 1: Left panel: the ratio of the ${\tilde{g}}{\tilde{g}}$ cross section
that we find in the range of $m_{\tilde{q}}$ and $m_{\tilde{g}}$ favoured in
our analysis before implementing the LHC 13-TeV $\,/E_{T}$ constraint,
relative to the cross section found in the simplified model with
$m_{\tilde{g}}\ll m_{\tilde{q}}$. Right panel: the corresponding ratio of the
$\tilde{q}\tilde{q}+\tilde{q}\bar{\tilde{q}}$ cross section, relative to the
cross section for $\tilde{q}\bar{\tilde{q}}$ found in the simplified model
with $m_{\tilde{q}}\ll m_{\tilde{g}}$.

Fig. 2 displays the CMS 95% confidence limits in the
$(m_{\tilde{g}},m_{\tilde{\chi}^{0}_{1}})$ plane from a hadronic jets plus
$\,/E_{T}$ search [5] within a simplified model assuming that the decay mode
${\tilde{g}}\to q{\bar{q}}\tilde{\chi}^{0}_{1}$ occurs with 100% BR (solid
black lines). These limits are compared with the best-fit points (green stars)
and the regions in the fits that are preferred at $\Delta\chi^{2}=2.30$ and
$\Delta\chi^{2}=5.99$ (red and blue contours, respectively). Here and in the
following analogous parameter planes, we use the $\Delta\chi^{2}=2.30$ and
$\Delta\chi^{2}=5.99$ contours as proxies for the boundaries of the 68% and
95% CL regions in the fit.

Figure 2: The solid lines show the CMS 95% CL exclusion in the
$(m_{\tilde{g}},m_{\tilde{\chi}^{0}_{1}})$ plane [5], assuming a simplified
model with heavy squarks and 100% BR for ${\tilde{g}}\to
q{\bar{q}}\tilde{\chi}^{0}_{1}$. The left (right) panel shows the best-fit
point (green star), 68 and 95% CL contours (red and blue lines, respectively)
for $(m_{\tilde{g}},m_{\tilde{\chi}^{0}_{1}})$ obtained without (with) the CMS
13-TeV constraint. The dominant ($>50$%) ${\tilde{g}}$ decays into first- and
second-generation quarks and squarks $\tilde{q}_{L,R}$ and third-generation
quarks and squarks ${\tilde{t}/\tilde{b}}_{1,2}$ found in the SUSY SU(5) model
are colour-coded as indicated.

In addition, within the 95% CL region in Fig. 2 we have indicated the dominant
($>50$%) ${\tilde{g}}$ decays found in our analysis. We note that many model
points do not have any decay mode with BR $>50$% within the 95% CL region and
that, for those that do, the dominant decays are two-body
${\tilde{g}}\to\tilde{q}{\bar{q}}$ modes that were not considered in [5].
Because of this and the fact that the ${\tilde{g}}{\tilde{g}}$ cross section
is always smaller than in the gluino simplified model by a factor $>2$ (see
the left panel of Fig. 1), the LHC 13-TeV $\,/E_{T}$ constraint from the
gluino simplified model has only negligible impact. Our LHC 13-TeV $\,/E_{T}$
constraint on the gluino mass actually comes indirectly from the squark mass
constraint estimated using the squark simplified model discussed below, since
the the squark and gluino masses are related via renormalization group
evolution in the SU(5) model. The left panel in Fig. 2 was obtained before
implementing the LHC 13-TeV $\,/E_{T}$ 95% confidence limit on gluino and
squark pair-production, while in the right panel this constraint is included.
We note that the simplified model exclusion in this analysis extended to
$m_{\tilde{g}}<1900\,\,\mathrm{GeV}$, below the gluino mass at the pre-LHC 13
TeV best-fit point, and barely reaching the 68% CL contour (solid red line).

Fig. 3 contains an analogous set of planes for CMS $\,/E_{T}$ searches for
squarks, where the CMS limit assuming a simplified model with heavy gluino and
100% BRs for $\tilde{q}\to q\tilde{\chi}^{0}_{1}$ is displayed (black lines):
the solid lines assume that all the squarks of the first two generations are
degenerate, the dashed lines assume two degenerate squarks, and the dotted
lines assume just one squark. The planes in the upper panels display
$m_{\tilde{\chi}^{0}_{1}}$ and the masses of the first- and second-generation
right-handed up-type squarks (here commonly denoted as $\tilde{u}_{R}$), while
the planes in the lower panels are for the down-type squarks (here commonly
denoted as $\tilde{d}_{R}$). The main decay modes of the $\tilde{u}_{R}$
(upper) and the $\tilde{d}_{R}$ (lower) are indicated over much of the
preferred parameter space, and we note that the dominant ($>50$%) decay modes
of both right-handed up- and down-type squarks are indeed into the
corresponding quark flavour + $\tilde{\chi}^{0}_{1}$ for nearly the whole 68%
CL regions, as assumed in the squark simplified-model search. This is,
however, not the case for the left-handed up- and down-type squarks (not
shown), whose dominant decays are into $\tilde{\chi}^{\pm}_{1}$ and
electroweak doublet partner quark flavours. Furthermore, within the displayed
95% CL regions there are also large areas where decays into gluinos, not
considered in the simplified model, are dominant.

Figure 3: The black lines show the CMS 95% CL exclusion in the
$(m_{\tilde{q}},m_{\tilde{\chi}^{0}_{1}})$ plane [5], assuming a simplified
model with heavy gluinos and 100% BR for $\tilde{q}\to q\tilde{\chi}^{0}_{1}$:
the solid lines assume that all the squarks of the first two generations are
degenerate, the dashed lines assume two degenerate squarks, and the dotted
lines assume just one squark. The left panels show the best-fit point (green
star), 68 and 95% CL contours (red and blue lines, respectively) for
$m_{\tilde{\chi}^{0}_{1}}$ and the masses of the first- and second-generation
right-handed up-type squarks $\tilde{u}_{R}$ (upper panel) and the down-type
squarks $\tilde{d}_{R}$ (lower panel). In both cases, the left panels were
obtained without the CMS 13-TeV constraint, and the right panels include it.
The dominant ($>50$%) $\tilde{q}$ decays found in the SUSY SU(5) model are
colour-coded as indicated.

Because the ${\tilde{q}_{R}}\to q\,\tilde{\chi}^{0}_{1}$ decays are important,
and also because the $\tilde{q}\tilde{q}+\tilde{q}\bar{\tilde{q}}$ cross
section in our sample is much larger than that found at large $m_{\tilde{q}}$
for $\tilde{q}\bar{\tilde{q}}$ in the simplified model with $m_{\tilde{q}}\ll
m_{\tilde{g}}$, as seen in the right panel of Fig. 1, we have implemented a
recast of this search in our global analysis 444The
$\tilde{u}_{R}/\tilde{c}_{R}-\tilde{\chi}^{0}_{1}$ coannihilation strip
visible in the left panels of Fig. 3 at $m_{\tilde{u}_{R}}\simeq
m_{\tilde{\chi}^{0}_{1}}\sim 500\,\,\mathrm{GeV}$ is the subject of a later
dedicated discussion., and the comparison between the upper panels (without
this contribution) and the lower panels (with this contribution) in Fig. 3
shows the importance of this constraint.

Our implementation of the LHC 13-TeV $\,/E_{T}$ constraint is based on [5]. In
this analysis, the CMS Collaboration provides a map of the 95% CL cross-
section upper limit as a function of $m_{\tilde{q}}$ and
$m_{\tilde{\chi}_{1}^{0}}$ assuming $pp\to\tilde{q}\bar{\tilde{q}}$ and 100%
BR for $\tilde{q}\to q\tilde{\chi}_{1}^{0}$. This is indeed the dominant
production and decay mode in most parts of the 68% CL regions of the
considered model, as can be seen in Figs. 1 and 3. For each point we compare
our calculation of
$(\sigma_{\tilde{q}{\bar{\tilde{q}}}}+\sigma_{\tilde{q}\tilde{q}})\;{\rm
BR}^{2}_{\tilde{q}\to q\tilde{\chi}_{1}^{0}}$ with the CMS 95% CL upper limit
on the cross section: $\sigma_{\rm UL}(m_{\tilde{q},\tilde{\chi}_{1}^{0}})$.
We model the $\chi^{2}$ penalty as

$\chi^{2}_{\tilde{q}}({\,/E_{T}})=5.99\cdot\Big{[}\frac{(\sigma_{\tilde{q}{\bar{\tilde{q}}}}+\sigma_{\tilde{q}\tilde{q}})\;{\rm
BR}^{2}_{\tilde{q}\to q\tilde{\chi}_{1}^{0}}}{\sigma_{\rm
UL}(m_{\tilde{q},\tilde{\chi}_{1}^{0}})}\Big{]}^{2}\,,$ (2)

so that the CMS 95% CL upper limit corresponds to $\chi^{2}({\,/E_{T}})=5.99$
and $\chi^{2}$ scales as the square of the number of signal events, $N_{\rm
sig}$, which gives the right scaling.

The aforementioned CMS analysis [5] also looks at three simplified gluino
models assuming 100% BR for $\tilde{g}\to f\bar{f}\tilde{\chi}_{1}^{0}$ with
$f=q,b,t$, respectively, and provides corresponding cross-section upper limit
maps as a function of $m_{\tilde{g}}$ and $m_{\tilde{\chi}_{1}^{0}}$. We
implement these constraints by defining $\chi^{2}_{\tilde{g}\to
f\bar{f}\tilde{\chi}_{1}^{0}}({\,/E_{T}})$ by analogy with Eq. (2).

We also consider the $pp\to\tilde{q}\tilde{g}$ process, treating it as
follows. This process is only relevant when $m_{\tilde{q}}\sim m_{\tilde{g}}$.
In this regime, if $m_{\tilde{q}}>m_{\tilde{g}}$
($m_{\tilde{g}}>m_{\tilde{q}}$), $\tilde{q}$ ($\tilde{g}$) tends to decay into
$\tilde{g}$ ($\tilde{q}$), radiating soft jets. If these soft jets are
ignored, we are left with the ${\tilde{g}}{\tilde{g}}$ ($\tilde{q}\tilde{q}$)
system. In this approximation, the impact of $pp\to\tilde{q}\tilde{g}$ can
therefore be estimated by adding an extra contribution
$\sigma_{\tilde{q}\tilde{g}}BR_{\tilde{q}\to q\tilde{g}}$
($\sigma_{\tilde{q}\tilde{g}}BR_{\tilde{g}\to q\tilde{q}}$) to
$\sigma_{{\tilde{g}}{\tilde{g}}}$
($\sigma_{\tilde{q}\tilde{q}}+\sigma_{\tilde{q}\bar{\tilde{q}}}$). In general,
SUSY searches are designed to look for high $p_{T}$ objects, and one loses a
small amount of sensitivity by ignoring soft jets. We therefore believe that
our implementation of the $pp\to\tilde{q}\tilde{g}$ process is conservative.

Finally, we estimate the total $\chi^{2}$ penalty from the LHC 13-TeV
$\,/E_{T}$ constraint to be
$\chi^{2}(\,/E_{T})=\chi^{2}_{\tilde{q}}(\,/E_{T})+\sum_{f=q,b,t}\chi^{2}_{\tilde{g}\to
f\bar{f}\tilde{\chi}_{1}^{0}}(\,/E_{T})$.

###  3.4 Constraints on long-lived charged particles

We also include in our analysis LHC constraints from searches for heavy long-
lived charged particles (HLCP) that are, in general, relevant to
coannihilation regions where the mass difference between the lightest SUSY
particle (LSP) and the next-to-lightest SUSY particle (NLSP) may be small and
the NLSP may therefore be long-lived. As we discuss below, important roles are
played in our analysis by $\tilde{\tau}_{1}$, $\tilde{\chi}^{\pm}_{1}$ and
$\tilde{u}_{R}/\tilde{c}_{R}$ coannihilation, but only in the
$\tilde{\tau}_{1}$ case is the NLSP - LSP mass difference small enough to
offer the possibility of a long-lived charged particle. We implement in our
global analysis the preliminary CMS 13-TeV result [57] using tracking and
time-of-flight measurements, based on the recipe and the efficiency map as a
function of the pseudo-rapidity and velocity of the HLCP given in [58]. We use
Pythia 8 [66] and Atom [67] to generate and analyse the events, and assume
that the efficiencies for detecting slow-moving $\tilde{\tau}_{1}$s are
similar at 8 and 13 TeV. 555A similar recasting method was used in [68]. See
also [69] for another approach using simplified model topologies. The
efficiency contains a lifetime-dependent factor $\propto{\rm exp}(-dm/p\tau)$,
where $d$ is a distance $d\simeq 10$ m that depends on the pseudorapidity, and
$m,p$ and $\tau$ are the mass, momentum and lifetime of the long-lived
particle. This factor drops rapidly for particles with lifetimes $\lesssim 10$
ps, corresponding to $m_{\tilde{\tau}_{1}}-m_{\tilde{\chi}^{0}_{1}}\gtrsim
1.6\,\,\mathrm{GeV}$.

###  3.5 Constraints on heavy neutral Higgs bosons from Run II

Concerning the production of heavy neutral Higgs bosons, in addition to the
$8$ TeV constraints on $H/A\to\tau^{+}\tau^{-}$ provided by HiggsBounds, we
also take into account the preliminary exclusion limits obtained by ATLAS from
searches for generic spin-0 bosons $\phi$ in the $\tau\tau$ final state with
an integrated luminosity of 13.3 fb-1 at 13 TeV that were presented at the
ICHEP 2016 conference and described in [61]. Upper bounds on $\sigma\times{\rm
BR}({\phi}\to\tau\tau)$ are reported for each ${M_{\phi}}$ separately for the
gluon fusion production channel and for production in association with a
$b{\bar{b}}$ pair assuming there is no contamination between the modes,
assuming a single resonance. We compute the cross sections and the BRs in the
MSSM using FeynHiggs, adding the contributions for $\phi=H$ and $\phi=A$,
using the average of the two masses, which are degenerate within the
experimental resolution. This result is compared with the upper limit from the
corresponding channel neglecting contamination. This approach leads to a
conservative limit since we underestimate the signal yield in each channel by
neglecting the contamination (the events from the other production mode). As
in Eq. (2), the $\chi^{2}$ penalties are modelled as

$\chi^{2}({Y_{i}})=4\cdot\Big{(}\frac{\sigma_{X_{i}}\cdot{\rm
BR}_{\tau^{+}\tau^{-}}}{\sigma_{Y_{i}}^{\rm UL}(M_{A})}\Big{)}^{2}\,,$ (3)

where $X_{i}=(gg\to{H/A}$, $pp\to b\bar{b}{H/A})$ is the production mode,
$Y_{i}=(ggF,bb{\phi})$ is the corresponding search channel and $\sigma^{\rm
UL}(M_{A})$ is the 95% CL upper limit evaluated at $M_{A}{(\approx M_{H})}$ by
ATLAS [61]. Finally we take the stronger $\chi^{2}$ rather than combining them
in order to be on the conservative side:
$\chi^{2}({H/A\to\tau^{+}\tau^{-}})=\max(\chi^{2}({ggF}),\chi^{2}({bb{\phi}}))$.

###  3.6 Other constraints

The most important other constraint update is that on spin-independent DM
scattering. We incorporate in our global fit the recent result published by
the PandaX-II experiment [27], which we combine with the new result from the
LUX Collaboration [28], as discussed in more detail in Section 8.

For the electroweak observables we use FeynWZ [39], and for the flavour
constraints we use SuFla [50]. For the Higgs observables, we use FeynHiggs
2.11.2 [42, 43] (including the updates discussed in Sect. 3.2), HiggsBounds
4.3.1 [63] and HiggsSignals 1.4.0 [62]. We calculate the sparticle spectrum
using SoftSusy 3.10.10 [70] and sparticle decays using SDECAY 1.3b [71] and
StauDecay 0.1[26]. The DM density and scattering rate are calculated using
micrOMEGAs 3.2 [55] and SSARD [56], respectively. Finally, we use SLHALib 2.2
[72] to interface the different codes.

###  3.7 Sampling procedure

As discussed in the previous Section, the SUSY SU(5) GUT model we study has 7
parameters: $m_{1/2}$, $m_{5}$, $m_{10}$, $m_{H_{u}}$, $m_{H_{d}}$, $A_{0}$
and $\tan\beta$. The ranges of these parameters that we scan in our analysis
are listed in Table 2. The quoted negative values actually correspond to
negative values of $m^{2}_{5},m^{2}_{10},m^{2}_{H_{u}}$ and $m^{2}_{H_{d}}$:
for convenience, we use the notation ${\rm sign}(m^{2})\times\sqrt{|m^{2}|}\to
m$. The negative values of $m_{5}$ and $m_{10}$ that are included in the scans
may be compatible with early-Universe cosmology [73], and yield acceptable
tachyon-free spectra. In the portions of the scans with negative values of
$m_{H_{u}}$ and $m_{H_{d}}$, although the effect of the top quark Yukawa
coupling in the renormalization group equations is important, it may not be
the dominant mechanism for generating electroweak symmetry breaking.

Parameter | Range | Number of
---|---|---
|  | segments
$m_{1/2}$ | ( 0 , 4) | 2
$m_{5}$ | ( - 2.6 , 8) | 2
$m_{10}$ | ( - 1.3 , 4) | 3
$m_{H_{u}}$ | ( -7 , 7) | 3
$m_{H_{d}}$ | ( -7 , 7) | 3
$A_{0}$ | ( -8 , 8) | 1
$\tan\beta$ | ( 2 , 68) | 1
Total number of boxes |  | 108
Table 2: Ranges of the SUSY SU(5) GUT parameters sampled, together with the
numbers of segments into which each range was divided, and the corresponding
total number of sample boxes. The mass parameters are expressed in TeV units.

We sample this parameter space using MultiNest v2.18 [74], dividing the
7-dimensional parameter space into 108 boxes, as also described in Table 2.
For each box, we choose a prior such that 80% of the sample has a flat
distribution within the nominal range, while 20% of the sample is in normally-
distributed tails outside the box. Our resultant total sample overlaps
smoothly between boxes, avoiding any spurious features at the box boundaries.
The total number of points in our sample is $\sim 125\times 10^{6}$, of which
$\sim 8\times 10^{6}$ have $\Delta\chi^{2}<10$.

##  4 Dark Matter Mechanisms

The relic density of the LSP, assumed here to be the lightest neutralino,
$\tilde{\chi}^{0}_{1}$, which is stable in supersymmetric SU(5) because of
$R$-parity, may be brought into the narrow range allowed by the Planck
satellite and other measurements [24] via a combination of different
mechanisms. It was emphasized previously [10] in studies of the CMSSM, NUHM1
and NUHM2 that simple annihilations of pairs of LSPs into conventional
particles would not have been sufficient to bring the relic
$\tilde{\chi}^{0}_{1}$ density down into the Planck range for values of
$m_{\tilde{\chi}^{0}_{1}}$ compatible with the LHC search limits and other
constraints on these models. Instead, there has to be some extra mechanism for
suppressing the LSP density. Examples include enhanced, rapid annihilation
through direct-channel resonances such as $Z,h,H/A$. Another possibility is
coannihilation with some other, almost-degenerate sparticle species:
candidates for the coannihilating species identified in previous studies
include the
$\tilde{\tau}_{1},{\tilde{\mu}},{\tilde{e}},{\tilde{\nu}},{\tilde{t}_{1}}$ and
$\tilde{\chi}^{\pm}_{1}$.

We introduced in [10] measures on the sparticle mass parameters that quantify
the mass degeneracies relevant to the above-mentioned coannihilation and rapid
annihilation processes, of which the following are relevant to our analysis of
the SUSY SU(5) GUT model 666We note that the focus-point mechanism does not
play a role in the SU(5) model.:

$\displaystyle{\tilde{\tau}_{1}}{\rm~{}coann.~{}(pink):}\hskip
17.07164pt\left(\frac{m_{\tilde{\tau}_{1}}}{m_{\tilde{\chi}^{0}_{1}}}-1\right)$
$\displaystyle\,<\,0.15\,,$
$\displaystyle\tilde{\chi}^{\pm}_{1}{\rm~{}coann.~{}(green):}\hskip
5.69054pt\left(\frac{m_{\tilde{\chi}^{\pm}_{1}}}{m_{\tilde{\chi}^{0}_{1}}}-1\right)$
$\displaystyle\,<\,0.1\,,$ $\displaystyle
A/H{\rm~{}funnel~{}(pale~{}blue):}\hskip
5.69054pt\left|\frac{M_{A}}{m_{\tilde{\chi}^{0}_{1}}}-2\right|$
$\displaystyle\,<\,0.4\,.$ (4)

We also indicate above the colour codes used in subsequent figures to identify
regions where each of these degeneracy conditions applies. We have verified in
a previous study [10] that CMSSM, NUHM1 and NUHM2 points that satisfy the DM
density constraint fulfill one or more of the mass-degeneracy conditions, and
that they identify correctly the mechanisms that yield the largest fractions
of final states, which are usually $\gtrsim 50$% [8, 75].

In much of the region satisfying the $\tilde{\tau}_{1}$ degeneracy criterion
above, the ${\tilde{\nu}_{\tau}}$ has a similar mass, and can contribute to
coannihilation [22]. We highlight the parts of the sample where sneutrino
coannihilation is important by introducing a shading for regions where the
${\tilde{\nu}_{\tau}}$ is the next-to-lightest sparticle (NLSP), and obeys the
degeneracy condition

$\displaystyle{\tilde{\nu}_{\tau}}^{\rm
NLSP}{\rm~{}coann.~{}(orange):}\left(\frac{m_{\tilde{\nu}_{\tau}}}{m_{\tilde{\chi}^{0}_{1}}}-1\right)$
$\displaystyle\,<\,0.1\,.$ (5)

We discuss later the importance of this supplementary DM mechanism.

As we discuss in this paper, a novel possibility in the SU(5) SUSY GUT is
coannihilation with right-handed up-type squarks, ${\tilde{u}_{R}}$ and
${\tilde{c}_{R}}$, which may be much lighter than the other squarks in this
model, as a consequence of the freedom to have $m_{5}\neq m_{10}$. We quantify
the relevant mass degeneracy criterion by

$\displaystyle{\tilde{u}_{R}/\tilde{c}_{R}}{\rm~{}coann.~{}(yellow):}\;\left(\frac{m_{\tilde{u}_{R}/\tilde{c}_{R}}}{m_{\tilde{\chi}^{0}_{1}}}-1\right)$
$\displaystyle\,<\,0.2\,.$ (6)

As we shall see in the subsequent figures, this novel degeneracy condition can
play an important role when $m_{5}\gg m_{10}$. The existence of this new
coannihilation region was verified using SSARD [56].

We also distinguish in this analysis ‘hybrid’ regions where the
$\tilde{\tau}_{1}$ coannihilation and $H/A$ funnel mechanisms may be relevant
simultaneously:

$\displaystyle\tilde{\tau}_{1}~{}{\rm coann.}+H/A~{}{\rm funnel:}$
$\displaystyle{\rm(purple)}\,,$ (7)

also with the indicated colour code.

##  5 Results

###  5.1 Parameter Planes

We display in Fig. 4 features of the global $\chi^{2}$ function for the SUSY
SU(5) GUT model in the $(m_{5},m_{1/2})$ plane (left panel) and the
$(m_{10},m_{1/2})$ plane (right panel), profiled over the other model
parameters. Here and in subsequent parameter planes, the best-fit point is
shown as a green star, the 68% CL regions are surrounded by red contours, and
the 95% CL regions are surrounded by blue contours (as mentioned above, we use
the $\Delta\chi^{2}=2.30$ and $\Delta\chi^{2}=5.99$ contours as proxies for
the boundaries of the 68% and 95% CL regions in the fit). The regions inside
the 95% CL contours are shaded according to the dominant DM mechanisms
discussed in the previous Section, see the criteria (4, 6, 7). In the
(relatively limited) unshaded regions there is no single dominant DM
mechanism.

Figure 4: The $(m_{5},m_{1/2})$ plane (left panel) and the $(m_{10},m_{1/2})$
plane (right panel) in the SUSY SU(5) GUT model. The best-fit point is shown
as a green star, the red contour surrounds the 68% CL region, and the blue
contour surrounds the 95% CL region. The coloured shadings represent the
dominant DM mechanisms, as indicated in the lower panel and described in the
text.

As we see in Fig. 4, the best-fit point is at relatively small values of
$m_{5},m_{10}$ and $m_{1/2}$, close to the lower limit on $m_{1/2}$, whereas
the 68% CL region extends to much larger values of $m_{5},m_{10}$ and
$m_{1/2}$. The values of the model parameters at the best-fit point are listed
in Table 3 777The SLHA files for the best-fit point and other supplementary
material can be found in [11].. The upper row of numbers are the results from
the current fit including the latest LHC 13-TeV and PandaX-II/LUX constraints,
and the numbers in parentheses in the bottom row were obtained using instead
the previous LHC 8-TeV and XENON100 constraints, but the same implementations
of the other constraints. The most significant effect of the new LHC data has
been to increase the best-fit value of $m_{1/2}$ by $\sim
300\,\,\mathrm{GeV}$: the changes in the other fit parameters are not
significant, in view of the uncertainties. As we discuss in more detail later,
the favoured fit regions are driven towards the $\,/E_{T}$ boundary by the
$(g-2)_{\mu}$ constraint. Away from this boundary, the global $\chi^{2}$
function is quite flat.

The best-fit point and much of the 68% CL region lie within the pink shaded
region where $\tilde{\tau}_{1}-\tilde{\chi}^{0}_{1}$ coannihilation is the
dominant DM mechanism. At larger values of $m_{5}$ and $m_{10}$ we encounter a
blue shaded region where rapid annihilation via direct-channel $H/A$ poles is
dominant. We also see darker shaded hybrid regions where $\tilde{\tau}_{1}$
and/or $\tilde{\chi}^{\pm}_{1}-\tilde{\chi}^{0}_{1}$ coannihilation and $H/A$
annihilation are important simultaneously. At larger values of $m_{1/2}\gtrsim
3000\,\,\mathrm{GeV}$, in the green shaded regions, the dominant DM mechanism
is $\tilde{\chi}^{\pm}_{1}-\tilde{\chi}^{0}_{1}$ coannihilation. There is also
a band in the $(m_{10},m_{1/2})$ plane with $m_{10}\gtrsim
1500\,\,\mathrm{GeV}$ and $m_{1/2}\sim 1000\,\,\mathrm{GeV}$, allowed at the
95% CL, where ${\tilde{\nu}_{\tau}}^{\rm NLSP}$ coannihilation is important.

$m_{1/2}$ | $m_{5}$ | $m_{10}$ | $m_{H_{u}}$ | $m_{H_{d}}$ | $A_{0}$ | $\tan\beta$
---|---|---|---|---|---|---
1050 | -220 | 380 | -5210 | -4870 | -5680 | 12
(890) | (-80) | (310) | (-4080) | (-4420) | (5020) | (11)
Table 3: Parameters of the best-fit point in the SUSY SU(5) GUT model, with
mass parameters given in GeV units. The numbers in parentheses in the bottom
row are for a fit that does not include the LHC 13-GeV constraints and the
recent PandaX-II and LUX constraints on DM scattering. Note that we use the
same convention for the sign of $A_{0}$ as in [6, 7, 8, 9, 10], which is
opposite to the convention used in, e.g., SoftSUSY, and that we use the
notation ${\rm sign}(m^{2})\times\sqrt{|m^{2}|}\to m$ for
$m_{5},m_{10},m_{H_{u}}$ and $m_{H_{d}}$.

We also note the appearance within the 95% CL region at $m_{1/2}\sim
1000\,\,\mathrm{GeV}$, and $m_{10}\sim-1000\,\,\mathrm{GeV}$ of the novel
${\tilde{u}_{R}}/{\tilde{c}_{R}}-\tilde{\chi}^{0}_{1}$ coannihilation region
(shaded yellow). To understand the origin of this novelty, consider the one-
loop renormalization-group equations for the states in the $\mathbf{10}$
representations of SU(5), namely $(q_{L},u^{c}_{L},e^{c}_{L})_{i}$, above the
highest MSSM particle mass (all masses are understood to be scalar fermion
masses, and we suppress subscripts L):

$\displaystyle 16\pi^{2}\frac{\partial m^{2}_{q_{i}}}{\partial t}$
$\displaystyle=$
$\displaystyle\delta_{i3}(X_{t}+X_{b})-\frac{32}{3}g_{3}^{2}|M_{3}|^{2}$ (8)
$\displaystyle-6g_{2}^{2}|M_{2}|^{2}-{\frac{2}{15}}g_{1}^{2}|M_{1}|^{2}+\frac{1}{5}g_{1}^{2}S\,,$
$\displaystyle 16\pi^{2}\frac{\partial m^{2}_{u_{i}^{c}}}{\partial t}$
$\displaystyle=$
$\displaystyle{2}\delta_{i3}X_{t}-\frac{32}{3}g_{3}^{2}|M_{3}|^{2}$ (9)
$\displaystyle-\frac{32}{15}g_{1}^{2}|M_{1}|^{2}-\frac{4}{5}g_{1}^{2}S\,,$
$\displaystyle 16\pi^{2}\frac{\partial m^{2}_{e_{i}^{c}}}{\partial t}$
$\displaystyle=$
$\displaystyle{2}\delta_{i3}X_{\tau}-\frac{24}{5}g_{1}^{2}|M_{1}|^{2}+\frac{6}{5}g_{1}^{2}S\,,$
(10)

where $t\equiv\ln(Q/Q_{0})$ with $Q$ the renormalization scale and $Q_{0}$
some reference scale,

$\displaystyle X_{t}$ $\displaystyle\equiv$ $\displaystyle
2|y_{t}|^{2}(m^{2}_{H_{u}}+m^{2}_{q_{3}}+m^{2}_{t^{c}})+2|A_{t}|^{2}\,,$ (11)
$\displaystyle X_{b}$ $\displaystyle\equiv$ $\displaystyle
2|y_{b}|^{2}(m^{2}_{H_{d}}+m^{2}_{q_{3}}+m^{2}_{b^{c}})+2|A_{b}|^{2}\,,$ (12)
$\displaystyle X_{\tau}$ $\displaystyle\equiv$ $\displaystyle
2|y_{\tau}|^{2}(m^{2}_{H_{d}}+m^{2}_{l_{3}}+m^{2}_{\tau^{c}})+2|A_{\tau}|^{2}\,,$
(13)

and

$\displaystyle S$ $\displaystyle\equiv$
$\displaystyle(m^{2}_{H_{u}}-m^{2}_{H_{d}})$ (14) $\displaystyle+$
$\displaystyle{\rm
Tr}\left({m^{2}_{q}-m^{2}_{l}}-2m^{2}_{u^{c}}+m^{2}_{d^{c}}+m^{2}_{e^{c}}\right)\,,$

where the trace in $S$ sums over the generations. The
${\tilde{u}_{R}}/{\tilde{c}_{R}}-\tilde{\chi}^{0}_{1}$ coannihilation
mechanism becomes important in a region of the SUSY SU(5) GUT parameter space
where $m_{5}^{2}$ is very large and positive, $m_{10}^{2}$ is small and
negative, $m_{H_{u}}^{2}$ is very large and negative, and $m_{H_{d}}^{2}$ is
very large and positive. In this region, therefore, $X_{t}$ is very large and
negative, $X_{b}$ and $X_{\tau}$ are suppressed because of small Yukawa
couplings ($\tan\beta$ is not large in this region), and $S$ is also very
large and negative, since $m_{H_{u}}^{2}-m_{H_{d}}^{2}$ is large and negative
and ${\rm Tr}(m^{2}_{q}-m^{2}_{l}-2m^{2}_{u^{c}}+m^{2}_{d^{c}}+m^{2}_{e^{c}})$
vanishes at the GUT scale. Inspection shows that the $X_{t}$ terms in (8) and
(9) drive the stop and sbottom masses upwards, and the $S$ terms in (8) and
(10) drive the left-handed squark and right-handed slectron masses upwards. On
the other hand, the $S$ term in (9) drives the right-handed squark masses
downwards. Since there are no counteracting $X$ terms for the
${\tilde{u}_{R}}$ and ${\tilde{c}_{R}}$, these have lower masses than the
other sfermions, opening the way to a
${\tilde{u}_{R}}/{\tilde{c}_{R}}-\tilde{\chi}^{0}_{1}$ coannihilation region.

Figure 5: The $(m_{5},m_{10})$ plane in the SUSY SU(5) GUT model. The line
colours and shadings are the same as in Fig. 4.

As discussed in more detail later, we used the Atom [67] simulation code for a
dedicated verification that points in this region escape all the relevant LHC
constraints. These points avoid exclusion by the LHC constraints through a
combination of a strong mass degeneracy,
$m_{{\tilde{u}_{R}}/{\tilde{c}_{R}}}-m_{\tilde{\chi}^{0}_{1}}\lesssim
50\,\,\mathrm{GeV}$, leading to strong suppression of the standard $\,/E_{T}$
signature, and the reduction of the production rate compared to the simplified
model that assumes mass degeneracy of all 8 light flavour squarks (see Fig.
1). These effects are clearly visible in Fig. 18 of [2].

Fig. 5 displays the corresponding information in the $(m_{5},m_{10})$ plane of
the SUSY SU(5) GUT model. As already reported in Table 3, here we see directly
that the best-fit point has very small (and slightly negative) $m_{5}$, and
that $m_{10}$ is somewhat larger, exploiting the possibility that $m_{5}\neq
m_{10}$ that is offered in this model. We also see again that the 68% CL
region extends to values of $m_{5}$ and $m_{10}$ beyond the $\tilde{\tau}_{1}$
coannihilation region. The novel
${\tilde{u}_{R}}/{\tilde{c}_{R}}-\tilde{\chi}^{0}_{1}$ coannihilation region
is clearly visible as a band where
$m_{5}\;\raisebox{-3.00003pt}{$\stackrel{{\scriptstyle\displaystyle>}}{{\sim}}$}\;2000\,\,\mathrm{GeV}$
and $m_{10}\lesssim-1000\,\,\mathrm{GeV}$. We also note that in most of the
rest of this plane $\tilde{\chi}^{\pm}_{1}-\tilde{\chi}^{0}_{1}$
coannihilation is dominant, with only scattered regions where rapid $H/A$
annihilation is important, even in combination with $\tilde{\tau}_{1}$
coannihilation.

Projections of our results in the $(\tan\beta,m_{1/2}),(\tan\beta,m_{5})$ and
$(\tan\beta,m_{10})$ planes are shown in Fig. 6. We see that values of
$\tan\beta\gtrsim 4$ are allowed at the 95% CL, that the range
$\tan\beta\in(8,57)$ is favoured at the 68% CL, and that there is no
phenomenological upper limit on $\tan\beta$ at the 95% CL888The RGE evolution
of the Yukawa couplings blows up for
$\tan\beta\;\raisebox{-3.00003pt}{$\stackrel{{\scriptstyle\displaystyle>}}{{\sim}}$}\;60$..
The best-fit point has $\tan\beta=13$, as also reported in Table 3.

Figure 6: The $(\tan\beta,m_{1/2})$ plane (upper panel), the
$(\tan\beta,m_{5})$ plane (lower left panel) and the $(\tan\beta,m_{10})$
plane (lower right panel) in the SUSY SU(5) GUT model. The line colours and
shadings are the same as in Fig. 4.

The pink $\tilde{\tau}_{1}-\tilde{\chi}^{0}_{1}$ coannihilation region is very
prominent in the $(\tan\beta,m_{1/2})$ projection shown in the upper panel of
Fig. 6, as is the blue rapid $H/A$ annihilation region and the purple
$\tilde{\tau}_{1}-\tilde{\chi}^{0}_{1}$ coannihilation + $H/A$ funnel hybrid
region at large $\tan\beta$ and $m_{1/2}$. As previously, the
$\tilde{\chi}^{\pm}_{1}-\tilde{\chi}^{0}_{1}$ coannihilation region appears
when $m_{1/2}\gtrsim 3000\,\,\mathrm{GeV}$, and this may hybridize with
$\tilde{\tau}_{1}-\tilde{\chi}^{0}_{1}$ coannihilation and/or the $H/A$
funnel, as seen in the mid and dark blue regions at $\tan\beta\sim 20$ and
$m_{1/2}\in(1500,2500)\,\,\mathrm{GeV}$. While the $H/A$ funnel appears in the
CMSSM only when
$\tan\beta\;\raisebox{-3.00003pt}{$\stackrel{{\scriptstyle\displaystyle>}}{{\sim}}$}\;45$
for $\mu>0$ [12], in the SU(5) SUSY GUT model, it is found at significantly
lower $\tan\beta$, due to the separation of $m_{H_{u}}$ and $m_{H_{d}}$ from
$m_{5}$ and $m_{10}$, effectively making $m_{A}$ (and $\mu$) free parameters
as in the NUHM2. There is also a region in the $(\tan\beta,m_{1/2})$ plane
with $\tan\beta\lesssim 10$ and $m_{1/2}\sim 1000\,\,\mathrm{GeV}$ where
${\tilde{\nu}_{\tau}}^{\rm NLSP}$ coannihilation is important.

The $\tilde{\tau}_{1}-\tilde{\chi}^{0}_{1}$ coannihilation region and the
purple $\tilde{\tau}_{1}-\tilde{\chi}^{0}_{1}$ coannihilation + $H/A$ funnel
hybrid region are prominent for $|m_{5}|\lesssim 3000\,\,\mathrm{GeV}$ in the
$(\tan\beta,m_{5})$ and $(\tan\beta,m_{10})$ planes shown in the lower part of
Fig. 6, with $\tilde{\chi}^{\pm}_{1}-\tilde{\chi}^{0}_{1}$ coannihilation
dominant at smaller values of $\tan\beta$, in particular. The
${\tilde{u}_{R}}/{\tilde{c}_{R}}-\tilde{\chi}^{0}_{1}$ coannihilation region
is prominent for $\tan\beta\in(8,20)$ and $m_{10}\sim-1200\,\,\mathrm{GeV}$ in
the $(\tan\beta,m_{10})$ plane shown in the lower right panel of Fig. 6.

We display in Fig. 7 projections of our results for $M_{h}$ versus $m_{1/2}$
(upper left), $\tan\beta$ (upper right), $m_{5}$ (lower left) and $m_{10}$
(lower right). The predicted values of $M_{h}$ are well centred within the
expected FeynHiggs uncertainty range around the value measured at the LHC,
$M_{h}=125.09\pm 0.24\,\,\mathrm{GeV}$ [44]. Moreover, most of the Dark Matter
mechanisms do not exhibit any preference for values of $M_{h}$ above or below
the nominal central value. Thus, there is no apparent tension between this LHC
measurement and the other constraints on the SUSY SU(5) GUT model.

Figure 7: The $(M_{h},m_{1/2})$ plane (upper left panel), the
$(\tan\beta,M_{h})$ plane (upper right panel), the $(M_{h},m_{5})$ plane
(lower left panel) and the $(M_{h},m_{10})$ plane (lower right panel) in the
SUSY SU(5) GUT model. The line colours and shadings are the same as in Fig. 4.

As is well known, the calculation of $M_{h}$ in the MSSM is particularly
sensitive to the value of the trilinear soft SUSY-breaking parameter $A_{0}$
as well as the stop squark masses. The latter depend in the SUSY SU(5) GUT
model on $m_{10}$ and $m_{1/2}$, but are insensitive to $m_{5}$.

The $(m_{H_{u}},m_{H_{d}})$ plane is shown in Fig. 8. We see that the best-fit
point lies in the quadrant where both $m_{H_{u}}$ and $m_{H_{d}}$ are
negative, and that the 68% CL region extends also to the quadrant where
$m_{H_{d}}$ is negative and $m_{H_{u}}$ is positive, as does the
$\tilde{\tau}_{1}-\tilde{\chi}^{0}_{1}$ coannihilation region. On the other
hand, the $\tilde{\chi}^{\pm}_{1}-\tilde{\chi}^{0}_{1}$ coannihilation region
lies in the upper quadrants where $m_{H_{d}}>0$. There is also an intermediate
‘hybrid’ region, much of which is also allowed at the 68% CL. There is also a
region with $M_{H_{u}}\sim
4000\,\,\mathrm{GeV},M_{H_{d}}\sim-3000\,\,\mathrm{GeV}$ where
${\tilde{\nu}_{\tau}}^{\rm NLSP}$ coannihilation is important.

Figure 8: The $(m_{H_{u}},m_{H_{d}})$ plane in the SUSY SU(5) GUT model. The
line colours and shadings are the same as in Fig. 4.

Fig. 9 displays the $(M_{A},\tan\beta)$ plane in the supersymmetric SU(5) GUT
model. We see that $M_{A}\gtrsim 800(1000)\,\,\mathrm{GeV}$ at the 95 (68) %
CL, which is largely due to the interplay of the indirect constraints on
$(M_{A},\tan\beta)$ rather than the direct constraints from the LHC heavy MSSM
Higgs searches. Even for large $\tan\beta$, where these constraints impose the
strongest lower limit on $M_{A}$, it is much weaker than our global limit,
which is $M_{A}\gtrsim 2800(>4000)\,\,\mathrm{GeV}$ at the 95 (68) % CL. The
best-fit point in the global fit has
$(M_{A},\tan\beta)\simeq(1600\,\,\mathrm{GeV},13)$: this is considerably
beyond the present and projected LHC reach, though poorly determined.

Figure 9: The $(M_{A},\tan\beta)$ plane in the SUSY SU(5) GUT model. The line
colours and shadings are the same as in Fig. 4.

##  6 One-Dimensional Likelihood Functions

We now discuss the one-dimensional $\Delta\chi^{2}$ functions for various
observable quantities.

Fig. 10 displays those for $m_{\tilde{g}}$ (top left), $m_{\tilde{q}_{L}}$
(top right), $m_{\tilde{d}_{R}}$ (centre left), $m_{\tilde{u}_{R}}$ (centre
right), $m_{\tilde{t}_{1}}$ (bottom left) and $m_{\tilde{\tau}_{1}}$ (bottom
right). The solid blue line in each panel corresponds to the current analysis
of the supersymmetric SU(5) model including LHC Run 2 data at 13 TeV, the
dashed blue line shows the result of an SU(5) fit in which the LHC 13-TeV
results are not included, and the solid grey line corresponds to ‘fake’
NUHM2-like results obtained by selecting a subset of the SU(5) sample with
$m_{5}/m_{10}\in[0.9,1.1]$, which we discuss in more detail later 999The
$\Delta\chi^{2}$ functions for the NUHM2 subsample are calculated relative to
its minimum $\chi^{2}$, which is $\sim 0.4$ higher than the minimum $\chi^{2}$
for the full SU(5) sample.

The current SU(5) fit exhibits minima of $\chi^{2}$ at masses $\lesssim
2.5\,\,\mathrm{TeV}$: $m_{\tilde{g}}\simeq 2600\,\,\mathrm{GeV}$, common
squark mass $m_{\tilde{q}}\simeq 2200\,\,\mathrm{GeV}$, $m_{\tilde{u}_{R}}$,
$m_{\tilde{d}_{R}}$, $m_{\tilde{t}_{1}}\simeq 2200\,\,\mathrm{GeV}$ and
$m_{\tilde{\tau}_{1}}\simeq 540\,\,\mathrm{GeV}$, followed by a rise at higher
mass towards a plateau with
$\Delta\chi^{2}{\;\raisebox{-3.00003pt}{$\stackrel{{\scriptstyle\displaystyle<}}{{\sim}}$}\;}2$.
The minimum is relatively sharp for $m_{\tilde{g}}$, $m_{\tilde{q}}$ and
$m_{\tilde{\tau}_{1}}$, whereas it is broader for $m_{\tilde{t}_{1}}$. The
exact values are listed in Table 4 and depicted in Fig. 11. In this figure we
also indicate decay BRs exceeding 20% by dashed lines. Fig. 12 displays the 68
and 95% CL ranges for the Higgs and sparticle masses in the supersymmetric
SU(5) model as darker and lighter coloured bands, with the best-fit values
shown as blue lines.

Figure 10: The $\chi^{2}$ likelihood functions in the SUSY SU(5) GUT model
(blue lines) for the gluino mass (top left panel), the left-handed squark mass
(top right panel), the right-handed down squark mass (centre left panel), the
right-handed up squark mass (centre right panel), the lighter stop squark mass
(lower left panel) and the lighter stau slepton mass (lower right panel). The
dashed blue lines shows the result of omitting the LHC 13-TeV constraints, and
the grey lines represent ‘fake’ NUHM2 results obtained by selecting a subset
of the SU(5) sample with $m_{5}/m_{10}\in[0.9,1.1]$.

Concerning future $e^{+}e^{-}$ colliders, one can see that many best-fit
masses of electroweak particles are at $\sim 500\,\,\mathrm{GeV}$, offering
the possibility of pair production at a collider with $\sqrt{s}\sim
1\,\,\mathrm{TeV}$, as envisaged for the final stage of the ILC [76, 77].
Going to higher centre-of-mass energies, e.g.,
$\sqrt{s}\;\raisebox{-3.00003pt}{$\stackrel{{\scriptstyle\displaystyle<}}{{\sim}}$}\;3\,\,\mathrm{TeV}$
[78, 77] as anticipated for CLIC, significant fractions of the 68% CL ranges
of electroweak sparticle masses can be covered.

${\tilde{\tau}_{1}}$ | ${\tilde{\tau}_{2}}$ | ${\tilde{e}_{L}}$ | ${\tilde{e}_{R}}$ | ${\tilde{\nu}_{\tau}}$ | ${\tilde{q}_{L}}$ | ${\tilde{t}_{1}}$ | ${\tilde{t}_{2}}$
---|---|---|---|---|---|---|---
470 | 660 | 630 | 678 | 570 | 2130 | 1840 | 2180
${\tilde{b}_{1}}$ | ${\tilde{b}_{2}}$ | ${\tilde{u}_{R}}$ | ${\tilde{d}_{R}}$ | ${\tilde{g}}$ | $M_{H,A}$ | $m_{\tilde{\chi}^{0}_{1}}$ | $m_{\tilde{\chi}^{0}_{2},\tilde{\chi}^{\pm}_{1}}$
1940 | 2090 | 2000 | 1980 | 2310 | 1620 | 460 | 860
Table 4: Particle masses [at the best-fit point in the SUSY SU(5) GUT model
(in GeV units).

Figure 11: The spectrum at the best-fit point in the SUSY SU(5) GUT model.
Decay branching ratios exceeding 20% are denoted by dashed lines.

Figure 12: The 68 and 95% CL ranges of masses we obtain for the current fit in
the supersymmetric SU(5) model, shown in dark and light orange respectively.
The best-fit point is represented by blue lines.

As already noted, a novel feature of the SUSY SU(5) GUT model with ($m_{5}\neq
m_{10}$) is that the ${\tilde{u}_{R}}$ and ${\tilde{c}_{R}}$ may be much
lighter than the other squarks. This leads to the possibility of a
${\tilde{u}_{R}}/{\tilde{c}_{R}}-\tilde{\chi}^{0}_{1}$ coannihilation strip
where $m_{\tilde{u}_{R}}$ and $m_{\tilde{c}_{R}}\sim 500\,\,\mathrm{GeV}$,
which is visible as a second local minimum of $\chi^{2}$ with
$\Delta\chi^{2}<4$ in the centre right panel of Fig. 10.

We have checked specifically whether this strip is allowed by the available
LHC constraints. To this end, we verified using the Atom simulation code that
points along this strip are consistent with the published constraints from the
LHC 8-TeV data. We have also checked that this strip is consistent with the
preliminary simplified model search for
$\tilde{q}\tilde{q}+\tilde{q}\bar{\tilde{q}}$ at 13 TeV reported by CMS. The
left panel of Fig. 13 displays as a solid/dashed blue line the one-dimensional
$\chi^{2}$ function for $m_{\tilde{u}_{R}}-m_{\tilde{\chi}^{0}_{1}}$
including/omitting the 13-TeV data (the corresponding lines for
$m_{\tilde{c}_{R}}-m_{\tilde{\chi}^{0}_{1}}$ are very similar), and the right
panel of Fig. 13 shows the region of the
$(m_{\tilde{u}_{R}},m_{\tilde{\chi}^{0}_{1}})$ plane where
$\Delta\chi^{2}<{5.99}$, i.e., allowed at the 95% CL. We find that
$\sigma(\tilde{q}\tilde{q}+\tilde{q}\bar{\tilde{q}})<0.1$ pb in this region,
whereas the cross section upper limit as given in [5] is $\gtrsim 1$ pb. We
conclude that this simplified model search does not affect the likelihood in
this ${\tilde{u}_{R}}/{\tilde{c}_{R}}-\tilde{\chi}^{0}_{1}$ coannihilation
strip region. However, it will be explored further by future LHC data with
increased luminosity.

Figure 13: Left panel: the $\chi^{2}$ likelihood function in the SUSY SU(5)
GUT model for $m_{\tilde{u}_{R}}-m_{\tilde{\chi}^{0}_{1}}$ in the
${\tilde{u}_{R}}/{\tilde{c}_{R}}-\tilde{\chi}^{0}_{1}$ coannihilation strip
region (the solid/dashed line includes/omits the 13-TeV LHC data). Right
panel: the region of the $(m_{\tilde{u}_{R}},m_{\tilde{\chi}^{0}_{1}})$ plane
where $\Delta\chi^{2}<{5.99}$.

Another novel feature of the SUSY SU(5) GUT model is visible in Table 4 and
Fig. 11. Having $m_{5}\neq m_{10}$ allows the possibility of strong mixing
between the ${\tilde{\tau}_{R}}$ in the $\mathbf{10}$ representation and the
${\tilde{\tau}_{L}}$ in the $\mathbf{\bar{5}}$ representation. For example, at
the best-fit point the ${\tilde{\tau}_{1}}$ is an almost equal mixture of
$\tilde{\tau}_{L}$ and $\tilde{\tau}_{R}$:

$\tilde{\tau}_{1}\;=\;0.70\,\tilde{\tau}_{L}+0.72\,\tilde{\tau}_{R}\,.$ (15)

This large mixing explains the level repulsion $\Delta m\simeq
200\,\,\mathrm{GeV}$ between the ${\tilde{\tau}_{1}}$ and ${\tilde{\tau}_{2}}$
seen in Table 4, which is much larger than the splitting $\Delta m\simeq
50\,\,\mathrm{GeV}$ between the almost unmixed
${\tilde{e}_{1}}\sim{\tilde{e}_{R}}$ and ${\tilde{e}_{2}}\sim{\tilde{e}_{L}}$
that is also seen in Table 4.

We show in Fig. 14 the contribution to the global $\chi^{2}$ function of
$(g-2)_{\mu}$ (in teal), as a function of $m_{5}$ (left panel), $m_{10}$
(middle panel) and $m_{1/2}$ (right panel). In each case, there is a well-
defined minimum that is lower than the plateau at large mass values by
$\Delta\chi^{2}\gtrsim 2$. In contrast, the contributions to the global
$\chi^{2}$ function of the other observables are relatively featureless over
large ranges of $m_{5}$, $m_{10}$ and $m_{1/2}$, with the exception of the
contribution from the LHC 13-TeV data (mainly due to the $\,/E_{T}$
constraint), which rises sharply at low $m_{1/2}$, as shown in red in the
right panel of Fig. 14. Because we profile over the other parameters, this
does not have much impact on the dependence of $\chi^{2}$ on $m_{5}$ and
$m_{10}$, as seen in the left and middle panels. The well-defined minima seen
in the $(g-2)_{\mu}$ contributions in the left and middle panels of Fig. 14
occur at quite small values of $m_{5}$ and $m_{10}$, reflecting the fact that
$(g-2)_{\mu}$ is sensitive to the soft symmetry-breaking contributions to the
masses of both the ${\tilde{\mu}_{L}}$ and the ${\tilde{\mu}_{R}}$. These are
$m_{5}$ and $m_{10}$, respectively, so maximizing the SUSY contribution to
$(g-2)_{\mu}$ and thereby minimizing the $(g-2)_{\mu}$ contribution to
$\chi^{2}$ prefers small values of both $m_{5}$ and $m_{10}$. Similarly, the
SUSY contribution to $(g-2)_{\mu}$ is suppressed for large gaugino masses,
explaining the aversion to large $m_{1/2}$ seen in the right panel of Fig. 14.

Figure 14: The $\chi^{2}$ contributions of $(g-2)_{\mu}$ (teal) and LHC 13-TeV
data (red) in the SUSY SU(5) GUT model, as functions of $m_{5}$ (left panel),
$m_{10}$ (middle panel) and $m_{1/2}$ (right panel).

The principal contributions to the global $\chi^{2}$ function at the best-fit
point for the SUSY SU(5) GUT model are given in Table 5, and the corresponding
pulls at the best-fit point are displayed graphically in Fig. 15. Apart from
$(g-2)_{\mu}$, the other contributions deserving of comment include the
following. The large contribution from HiggsSignals reflects the large number
of channels considered, and has negligible variation for most of the points in
our sample. We note that $A_{\rm FB}(b)$ makes a contribution that is not much
smaller than that of $(g-2)_{\mu}$ at the best-fit point, and that $A_{\rm
LR}^{e}$ and $\sigma^{0}_{\rm had}$ also make relatively large contributions
to the global $\chi^{2}$ function. These observables reflect the residual
tensions in the electroweak precision observables at the $Z$ peak, which are
present in the SM and the SUSY SU(5) GUT model is unable to mitigate.

$A_{\rm LR}^{e}$ | $A_{b}$ | $A_{\rm FB}(\ell)$ | $A_{\rm FB}(b)$ | $A_{\rm FB}(c)$ | $A_{l}(P_{\tau})$
---|---|---|---|---|---
3.40 | 0.35 | 0.78 | 6.79 | 0.82 | 0.08
$R_{b}$ | BR($b\to s\gamma$) | BR($B_{u}\to\tau\nu_{\tau}$) | $\Omega_{\tilde{\chi}^{0}_{1}}h^{2}$ | $\sigma^{\rm SI}_{p}$ | ${\rm BR}(B_{s,d}\to\mu^{+}\mu^{-})$
0.26 | 0.00 | 0.18 | 0.00 | 0.00 | 2.09
$\sin^{2}\theta_{\mathrm{eff}}$ | $M_{W}$ | $R_{l}$ | $R(K\to l\nu)$ | $(g-2)_{\mu}$ | $M_{h}$
0.60 | 0.07 | 1.04 | 0.0 | 8.28 | 0.01

$\sigma_{\rm had}^{0}$ | ${\frac{\Delta M_{B_{s}}}{\Delta M_{B_{d}}}}$ | $\epsilon_{K}$ | $H/A\to\tau^{+}\tau^{-}$ | HiggsSignals | LHC $\,/E_{T}$ | Total
---|---|---|---|---|---|---
2.54 | 1.78 | 1.94 | 0.00 | 67.95 | 0.3 | 100.34

Table 5: The principal $\chi^{2}$ contributions of observables at the best-fit
point in the SUSY SU(5) GUT model, together with the total $\chi^{2}$
function.

Figure 15: The $\chi^{2}$ pulls for different observables at the best-fit
point in the SUSY SU(5) model.

In order to compare the quality of the SU(5) fit to the results of previous
MasterCode analyses of competing models [9], we follow the prescription used
there of subtracting from the total $\chi^{2}$ given in Table 5 and Fig. 15,
namely 100.34, the $\chi^{2}$ contributions originating from HiggsSignals
[62], which dominate the global $\chi^{2}$ function and would bias the
analysis. Fig. 15 lists 36 separate contributions to the total $\chi^{2}$
function. The first 3 ($m_{t},M_{Z}$, and $\Delta\alpha_{\rm
had}^{(5)}(M_{Z})$) are treated as nuisance parameters and the two LHC MET
constraints at 8 and 13 TeV are applied as a single constraint. Omitting the
HiggsSignals constraints in our determination of the number of degrees of
freedom leaves 30 constraints, with 7 parameters for the SU(5) model and hence
23 degrees of freedom.The $\chi^{2}$ contributions from the relevant
constraints sum to $32.39$, corresponding to a $\chi^{2}$ probability of 9%.
This can be compared with the $\chi^{2}$ probability values of $11,12,11$ and
31% found in [9] for the CMSSM, NUHM1, NUHM2 and pMSSM10, respectively, using
LHC Run 1 constraints. However, as in [9], we stress that these $\chi^{2}$
probabilities are only approximate since, for example, they neglect
correlations between the observables. A more complete treatment using toys is
beyond the scope of this work.

There are a couple of important corollaries to this observation, one
concerning $m_{\tilde{t}_{1}}$. It is sensitive to $A_{0}$ as well as the soft
SUSY-breaking contributions to the ${\tilde{t}_{L}}$ and ${\tilde{t}_{R}}$
mass parameters (which are both given by $m_{10}$ in the SUSY SU(5) GUT
model). Since $A_{0}$ is relatively poorly determined, the $\chi^{2}$ minimum
for $m_{\tilde{t}_{1}}$ is relatively shallow, as seen in the lower left panel
of Fig. 10.

The second observation concerns the sign of $\mu$. All our analysis has been
for $\mu>0$, which is the sign capable of mitigating the discrepancy between
the experimental value of $(g-2)_{\mu}$ and the SM prediction. For $\mu<0$,
the large-mass plateau would have a similar height as in Fig. 14, but the
$\chi^{2}$ function would rise monotonically at low values of $m_{5}$,
$m_{10}$ and $m_{1/2}$, instead of featuring a dip. Thus, the $\mu<0$
possibility would be disfavoured by $\Delta\chi^{2}\gtrsim 2$, and the global
minimum would lie at large masses and be ill defined.

The $\chi^{2}$ distributions for some more observables are shown in Fig. 16,
We see that the minima for $m_{\tilde{\chi}^{0}_{1}}$ (upper left panel) and
$m_{\tilde{\chi}^{\pm}_{1}}$ (upper right panel) are quite well defined,
mirroring the structure in the $\chi^{2}$ function for $m_{\tilde{\tau}_{1}}$
shown in the lower right panel of Fig. 10. The preference for a (very) small
${\tilde{\tau}_{1}}-\tilde{\chi}^{0}_{1}$ mass difference is seen in the lower
left panel of Fig. 16, and reflects the fact, commented on in connection with
many previous figures, that the best-fit point and much of the 68% CL region
lies in the ${\tilde{\tau}_{1}}-\tilde{\chi}^{0}_{1}$ coannihilation region.
On the other hand, no preference for a small
$m_{\tilde{t}_{1}}-m_{\tilde{\chi}^{0}_{1}}$ mass difference can be observed,
as seen in the lower right plot of Fig. 16, reflecting the fact that stop
coannihiliation does not play a significant role.

Figure 16: The $\chi^{2}$ likelihood functions in the SUSY SU(5) GUT model for
the $\tilde{\chi}^{0}_{1}$ mass (upper left panel), the
$\tilde{\chi}^{\pm}_{1}$ mass (upper right panel), the
${\tilde{\tau}_{1}}-\tilde{\chi}^{0}_{1}$ mass difference (lower left panel)
and the ${\tilde{t}_{1}}-\tilde{\chi}^{0}_{1}$ mass difference (lower right
panel). The dashed blue lines shows the result of omitting the LHC 13-TeV
constraints, and the grey lines represent ‘fake’ NUHM2 results obtained by
selecting a subset of the SU(5) sample with $m_{5}/m_{10}\in[0.9,1.1]$.

The $\tilde{\chi}^{\pm}_{1}-\tilde{\chi}^{0}_{1}$ coannihilation region is
prominent in the previous figures, and also contains parameter sets that are
preferred at the 68% CL. Hence a small
$\tilde{\chi}^{\pm}_{1}-\tilde{\chi}^{0}_{1}$ mass difference is also allowed
at the $\Delta\chi^{2}\gtrsim 1$ level, as seen in the left panel of Fig. 17,
although the best-fit point has
$m_{\tilde{\chi}^{\pm}_{1}}-m_{\tilde{\chi}^{0}_{1}}\sim 470\,\,\mathrm{GeV}$.
However, values of the $\tilde{\chi}^{\pm}_{1}$ lifetime that are allowed at
the 95% CL are all too short to provide a long-lived particle signal, as seen
in the right panel of Fig. 17. 101010For conditions to have a long-lived
$\tilde{\chi}^{\pm}_{1}$ with a bino-like LSP: see, e.g., [79].

Figure 17: The $\chi^{2}$ likelihood functions in the SUSY SU(5) GUT model for
the $\tilde{\chi}^{\pm}_{1}-\tilde{\chi}^{0}_{1}$ mass (left panel) and the
$\tilde{\chi}^{\pm}_{1}$ lifetime (right panel). The dashed blue lines shows
the result of omitting the LHC 13-TeV constraints, and the grey lines
represent ‘fake’ NUHM2 results obtained by selecting a subset of the SU(5)
sample with $m_{5}/m_{10}\in[0.9,1.1]$.

We now discuss the one-dimensional likelihood functions for electroweak
precision observables and observables in the flavour sector. The upper left
panel of Fig. 18 shows that for $(g-2)_{\mu}$. We see that the global minimum
occurs for $\Delta(g-2)_{\mu}\simeq 0.4\times 10^{-9}$, with
$\Delta\chi^{2}\lesssim-2$ compared to the case $\Delta(g-2)_{\mu}=0$. We see
again that the SUSY SU(5) GUT model is able to mitigate slightly the
discrepancy between the SM and the measurement of $(g-2)_{\mu}$, although it
does not provide a substantial improvement over the SM prediction.

Figure 18: The $\chi^{2}$ likelihood functions in the SUSY SU(5) GUT model for
$(g-2)_{\mu}/2$ (upper left panel), $M_{h}$ (upper right panel), $M_{W}$
(lower left panel), and ${\rm BR}(B_{s,d}\to\mu^{+}\mu^{-})$ (lower right
panel). The dashed blue lines shows the result of omitting the LHC 13-TeV
constraints, and the grey lines represent ‘fake’ NUHM2 results obtained by
selecting a subset of the SU(5) sample with $m_{5}/m_{10}\in[0.9,1.1]$.

As for $M_{h}$, as shown in the upper right panel of Fig. 18 the $\chi^{2}$
function is minimized close to the nominal experimental value, and is quite
symmetric, showing no indication of any tension in the SUSY SU(5) GUT model
fit. Likewise, the best-fit value of $M_{W}$ (lower left panel of Fig. 18) is
highly compatible with the experimental measurement, and that for ${\rm
BR}(B_{s,d}\to\mu^{+}\mu^{-})$ (lower right panel) is very close to the SM
prediction, and hence also compatible with the experimental measurement. We
note that, whereas values of ${\rm BR}(B_{s,d}\to\mu^{+}\mu^{-})$ that are
slightly larger than the SM value are possible, smaller values are strongly
disfavoured in the SUSY SU(5) GUT model.

##  7 Higgs Branching Ratios

We present in Fig. 19 the one-dimensional likelihood functions for the ratios
of supersymmetric SU(5) and SM predictions for the BRs of $h\to\gamma\gamma$
(left panel), $h\to ZZ^{*}$ (middle panel) 111111The likelihood function for
$h\to WW^{*}$ is very similar to that for $h\to ZZ^{*}$, because of custodial
symmetry. and $h\to gg$ decays (right panel). We see that in each case the
preferred region in the fit corresponds to a prediction in the SU(5) model
that deviates from the SM case by at most a few %, whereas the present
experimental uncertainties in the different coupling modifiers (employing some
theory assumptions) are typically ${\cal O}(30)$% [80], and a precision of
${\cal O}(5-10\%)$ (with the same theory assumptions) can be reached by the
end of the LHC programme. On the other hand, future $e^{+}e^{-}$ colliders
such as the ILC, CLIC or FCC-ee anticipate a precision at the percent level
for couplings to fermions and at the permille level for couplings to massive
gauge bosons [77, 81]. This offers the possiblity that deviations from the SM
in the SUSY SU(5) GUT model can be measured in the future.

Figure 19: The $\chi^{2}$ likelihood functions for the ratios of the SUSY
SU(5) and SM predictions for the BRs of $h\to\gamma\gamma$ (left panel), $h\to
ZZ^{*}$ (middle panel) and $h\to gg$ decays (right panel). The dashed blue
lines shows the result of omitting the LHC 13-TeV constraints, and the grey
lines represent ‘fake’ NUHM2 results obtained by selecting a subset of the
SU(5) sample with $m_{5}/m_{10}\in[0.9,1.1]$.

##  8 Comparison with Previous Results

In previous papers we have studied the CMSSM, NUHM1 and NUHM2 using the LHC
8-TeV results and earlier DM scattering constraints. None of these models are
directly comparable to the supersymmetric SU(5) model studied here, which has
4 different soft SUSY-beaking scalar mass parameters, $m_{5},m_{10},m_{H_{u}}$
and $m_{H_{d}}$. The most similar is the NUHM2, which has the 3 parameters
$m_{0}=m_{5}=m_{10},m_{H_{u}}$ and $m_{H_{d}}$. Here we compare the
supersymmetric SU(5) results found in this paper using LHC 13-TeV data with
‘fake’ NUHM2 results obtained by selecting a subset of this SU(5) sample with
$m_{5}/m_{10}\in[0.9,1.1]$ (which were also displayed as grey lines in Fig.
10) and with previous NUHM2 results [8].

Fig. 20 compares the one-dimensional $\chi^{2}$ likelihood functions for
$m_{\tilde{g}}$ (upper left), $m_{\tilde{q}_{R}}$ (upper right),
$m_{\tilde{t}_{1}}$ (lower left) and $m_{\tilde{\tau}_{1}}$ (lower right)
found in the SU(5) model including LHC 13-TeV constraints (solid blue lines)
with the restricted fake NUHM2 sample (solid grey lines) and, for comparison,
results from our previous NUHM2 analysis that used only the LHC 7- and 8-TeV
constraints (dashed grey lines) [8]. We see here and in Fig. 10 that the
restricted ‘fake’ NUHM2 sample exhibits, in general, best-fit masses that are
similar to those found in the full SU(5) sample. The most noticeable
differences are that lower masses are disfavoured in the restricted sample
relative those in the full SU(5) model, indicating that the latter has some
limited ability to relax the NUHM2 lower bounds on sparticle masses, e.g., at
the 95% CL. The previous NUHM2 analysis [8] also yielded similar best-fit
masses but, as could be expected, gave 95% CL lower limits on sparticle masses
that were further relaxed. Similar features can also be observed in Figs. 16
\- 19, where we have also included the ‘fake’ NUHM2 subsample.

Restricting further our SU(5) to mimic the NUHM1, let alone the CMSSM, is not
useful because of the increased sampling uncertainties in such restricted
samples. However, we showed in [8] that our NUHM2 LHC 7- and 8-TeV results for
the exhibited sparticle masses were broadly similar to those for the NUHM1 and
the CMSSM [7], and we expect the impacts of the LHC 13-TeV data on these
models to be comparable to that in the NUHM2.

Finally, we ask whether or not there is a significant improvement in the SU(5)
fit compared to that in the NUHM2 subsample, thanks to the additional
parameter ($m_{5}$ and $m_{10}$ replacing $m_{0}$). The NUHM2 subsample has a
total $\chi^{2}=100.8$, which is reduced to 32.8 when we remove the
contributions from HiggsSignals, as discussed earlier. It should be noted that
the NUHM2 subsample is statistically significantly smaller than that of the
SU(5) sample. The quoted NUHM2 $\chi^{2}$ represents only an upper bound on
the $\chi^{2}$ of the best-fit point that would be found in a more complete
sample of the NUHM2. Since the NUHM2 model has one less parameter than the
SU(5) model, it has 24 degrees of freedom, and its $\chi^{2}$ probability is
11%. According to the Wilks test, the probability that the data are
represented better by the SU(5) model than by the NUHM2 subsample is 50%,
i.e., there is no evidence that the extra parameter of SU(5) provides a
significant improvement.

Figure 20: The one-dimensional $\chi^{2}$ likelihood functions for the full
SU(5) sample (solid blue lines) and in the restriction of the SUSY SU(5) GUT
model sample to $m_{5}/m_{10}\in[0.9,1.1]$ (solid grey lines) compared to
those in our previous NUHM2 analysis [8] (dashed grey lines) for
$m_{\tilde{g}}$ (upper left panel), $m_{\tilde{q}}$ (upper right panel),
$m_{\tilde{t}_{1}}$ (lower left panel), and $m_{\tilde{\tau}_{1}}$ (lower
right panel).

##  9 The Possibility of a Long-Lived $\tilde{\tau}_{1}$

The possibility of a very small ${\tilde{\tau}_{1}}-\tilde{\chi}^{0}_{1}$ mass
difference opens up the possibility that the ${\tilde{\tau}_{1}}$ might have a
long lifetime, as discussed in the contexts of the CMSSM, NUHM1 and NUHM2 in
[10]. This would occur if
$m_{\tilde{\tau}_{1}}-m_{\tilde{\chi}^{0}_{1}}<m_{\tau}$. As seen in the lower
left panel of Fig. 16, the best-fit point has a mass difference $\sim
20\,\,\mathrm{GeV}$, outside this range, but
$m_{\tilde{\tau}_{1}}-m_{\tilde{\chi}^{0}_{1}}<m_{\tau}$ is allowed with
$\Delta\chi^{2}\sim 1$. In Fig. 21 we analyze the lifetime of the
$\tilde{\tau}_{1}$. We see in the upper left panel of Fig. 21 that there is
essentially no $\chi^{2}$ penalty for $10^{-9}$ s
$\lesssim\tau_{\tilde{\tau}_{1}}\lesssim 10^{-2}$ s, with lifetimes $\sim
10^{-10}$ s and $\lesssim 10^{3}$ s allowed with $\Delta\chi^{2}\lesssim 1$.
Distinguishing a separated-vertex signature at the LHC would be challenging
for smaller values of $\tau_{\tilde{\tau}_{1}}$, and there would be
significant disruption of the successful conventional Big Bang nucleosynthesis
calculations for $\tau_{\tilde{\tau}_{1}}\gtrsim 10^{3}$ s [82].

Figure 21: Upper left panel: The global $\chi^{2}$ function in the SUSY SU(5)
GUT model as a function of the $\tilde{\tau}_{1}$ lifetime. Upper right panel:
The $(m_{\tilde{\tau}_{1}},\tau_{\tilde{\tau}_{1}})$ plane, shaded according
to the values of $\tau_{\tilde{\tau}_{1}}$, as indicated. Lower panels: The
$(m_{5},m_{1/2})$ and $(m_{10},m_{1/2})$ planes, coloured according to the
values of $\tau_{\tilde{\tau}_{1}}$. The 68% and 95% CL contours in these
three planes are coloured red and blue, respectively.

The upper right plot of Fig. 21 compares the $\tilde{\tau}_{1}$ lifetime with
its mass. A striking feature is the separation of the 68% CL region into two
bands: a broad one at smaller $m_{\tilde{\tau}_{1}}$ and a narrower band at
$m_{\tilde{\tau}_{1}}\simeq 2000\,\,\mathrm{GeV}$. These bands are separated
by a strip with $1300\,\,\mathrm{GeV}\lesssim m_{\tilde{\chi}^{0}_{1}}\lesssim
2000\,\,\mathrm{GeV}$ where a long-lived ${\tilde{\tau}_{1}}$ is still allowed
at the 95% CL, but at the 68% CL the $\tau_{1}$ lifetime falls outside the
range displayed.

The lower panels of Fig. 21 display the regions of the $(m_{5},m_{1/2})$
(left) and $(m_{10},m_{1/2})$ (right) planes in the SUSY SU(5) GUT model where
the lowest-$\chi^{2}$ points have $10^{-10}\,{\rm
s}<\tau_{\tilde{\tau}_{1}}<10^{3}\,{\rm s}$. The colour-coding indicates the
lifetimes of these points, as indicated in the legends. The contours for
$\Delta\chi^{2}<2.30(5.99)$ relative to the best-fit point in our sample are
shown as solid red and blue lines, respectively. One can see that larger
lifetimes occur all over the displayed parameter space, with a slight
preference for larger $m_{5}$ or $m_{10}$ values.

##  10 Direct Dark Matter Detection

As already mentioned, the PandaX-II experiment [27] has recently published
results from its first 98.7 days of data, which currently provide the most
stringent upper limits on the spin-independent DM scattering cross section on
protons, $\sigma^{\rm SI}_{p}$. In parallel, the LUX Collaboration [28] has
presented preliminary constraints on $\sigma^{\rm SI}_{p}$ from 332 days of
data. We have combined these two constraints on $\sigma^{\rm SI}_{p}$ into a
single experimental likelihood function, which we have then convoluted with an
estimate of the theoretical uncertainty in the calculation of $\sigma^{\rm
SI}_{p}$, as described in [10], to constrain the SUSY SU(5) GUT parameter
space. This constraint has been used in obtaining the global fit whose results
we have presented in the previous Sections. Here we discuss the future
prospects for direct DM detection in light of our global fit.

Fig. 22 displays our results for the SUSY SU(5) GUT model in the
$(m_{\tilde{\chi}^{0}_{1}},\sigma^{\rm SI}_{p})$ plane. The combined PandaX-
II/LUX constraint (black line) establishes a 95% CL that reaches $\sigma^{\rm
SI}_{p}$ $\simeq 2\times 10^{-46}$ cm2 for
$m_{\tilde{\chi}^{0}_{1}}=50\,\,\mathrm{GeV}$ and $\simeq 10^{-45}$ cm2 for
$m_{\tilde{\chi}^{0}_{1}}=500\,\,\mathrm{GeV}$, providing the upper boundary
of the 95% CL region in the $(m_{\tilde{\chi}^{0}_{1}},\sigma^{\rm SI}_{p})$
plane seen in Fig. 22. We see that there are regions favoured at the 68% CL
that lie relatively close to this boundary, whereas the main 68% CL region and
the best-fit point have smaller values of $\sigma^{\rm SI}_{p}$. We also note
that the $H/A$ funnel and $\tilde{\chi}^{\pm}_{1}-\tilde{\chi}^{0}_{1}$ DM
mechanisms and their hybridization favour values of $\sigma^{\rm SI}_{p}$ that
are relatively close to the PandaX-II/LUX boundary, whereas the
${\tilde{\tau}_{1}}-\tilde{\chi}^{0}_{1}$ mechanism and its hybridization with
the $H/A$ funnel favour smaller values of $\sigma^{\rm SI}_{p}$.

Figure 22: The $(m_{\tilde{\chi}^{0}_{1}},\sigma^{\rm SI}_{p})$ plane in the
SUSY SU(5) GUT model. The solid green line is the 95% CL upper limit from the
XENON100 experiment, and the dashed black solid line is the new 95% CL upper
limit from the LUX experiment. The solid black line shows the 95% CL exclusion
contour for our combination of the PandaX-II and LUX experiments, the solid
purple line shows the projected 95% exclusion sensitivity of the LUX-Zepelin
(LZ) experiment, and the dashed orange line shows the astrophysical neutrino
‘floor’, below which astrophysical neutrino backgrounds dominate (yellow
region). The other line colours and shadings within the 68% and 95% CL regions
are the same as in Fig. 4.

We also display in Fig. 22 the projected 95% exclusion sensitivity of the
future LUX-Zepelin (LZ) experiment (solid purple line) [83] and the
astrophysical neutrino ‘floor’ (dashed orange line) [84, 85], below which
astrophysical neutrino backgrounds dominate (yellow region). We see that much
of the ${\tilde{\tau}_{1}}-\tilde{\chi}^{0}_{1}$ coannihilation region and the
region of its hybridization with the $H/A$ funnel lie below the projected
sensitivity of the LZ experiment, and substantial portions of them also lie
below the neutrino ‘floor’. On the bright side, however, we recall that the
${\tilde{\tau}_{1}}-\tilde{\chi}^{0}_{1}$ region, in particular, lies at
relatively small values of $m_{5},m_{10}$ and $m_{1/2}$, offering greater
prospects for detection at the LHC than, e.g., the
$\tilde{\chi}^{\pm}_{1}-\tilde{\chi}^{0}_{1}$ region, so there is
complementarity in the prospects of the LHC and direct DM experiments for
probing the SUSY SU(5) GUT model, as was noted previously for other SUSY
models [10].

##  11 Summary and Conclusions

We have explored in this paper the experimental, phenomenological,
astrophysical and cosmological constraints on the minimal SUSY SU(5) GUT
model. In this scenario the GUT-scale universal soft SUSY-breaking scalar mass
$m_{0}$ is replaced by independent masses for the $\mathbf{10}$ and
$\mathbf{\bar{5}}$ sfermions. This flexibility introduces some features that
are novel compared to the GUT-universal CMSSM, NUHM1 and NUHM2.

In general we observe that many best-fit values of the coloured particles are
within the reach of the HL-LHC, but that the preferred regions clearly extend
beyond the reach of the final stage of the LHC. On the other hand, many best-
fit masses of electroweak particles are at $\sim 500\,\,\mathrm{GeV}$,
offering the possibility of pair production at a collider with $\sqrt{s}\sim
1\,\,\mathrm{TeV}$, as envisaged for the final stage of the ILC. Going to
higher centre-of-mass energies,
$\sqrt{s}\;\raisebox{-3.00003pt}{$\stackrel{{\scriptstyle\displaystyle<}}{{\sim}}$}\;3\,\,\mathrm{TeV}$
as anticipated for CLIC, significant fractions of the 68% CL ranges of
electroweak sparticle masses can be covered.

One novelty is the appearance of a
${\tilde{u}_{R}}/{\tilde{c}_{R}}-\tilde{\chi}^{0}_{1}$ coannihilation region
that appears where $m_{5}^{2}$ is large and positive, $m_{10}^{2}$ is small
and negative, and $m_{H_{u}}^{2}$ and $m_{H_{d}}^{2}$ are large and negative.
On the other hand, we find that ${\tilde{t}_{1}}-\tilde{\chi}^{0}_{1}$
coannihilation is not important in the SUSY SU(5) GUT model, nor are the
focus-point region and rapid $\tilde{\chi}^{0}_{1}\tilde{\chi}^{0}_{1}$
annihilation via direct-channel $h$ and $Z$ poles. We have checked that the
${\tilde{u}_{R}}/{\tilde{c}_{R}}-\tilde{\chi}^{0}_{1}$ coannihilation region
is not yet excluded by searches for $\,/E_{T}$ events at the LHC, because the
production rate is reduced compared to the case where all 8 squarks are mass
degenerate and the small
${\tilde{u}_{R}}/{\tilde{c}_{R}}-\tilde{\chi}^{0}_{1}$ mass difference
suppresses this signature. However, this region may be accessible with future
LHC runs.

We have also highlighted the possibility that a ${\tilde{\nu}_{\tau}}$ NLSP
might have an important coannihilation role. Another novelty is the
composition of the ${\tilde{\tau}_{1}}$ NLSP in a significant region of the
model parameter space. In the GUT-universal CMSSM, NUHM1 and NUHM2 models, the
universality of $m_{0}$ and the greater renormalization for SU(2) doublets
impose a substantial mass difference between the ${\tilde{\tau}_{2}}$ and the
${\tilde{\tau}_{1}}$, with the latter being predominantly a
${\tilde{\tau}_{R}}$. However, in the SUSY SU(5) GUT model with $m_{5}\neq
m_{10}$, the ${\tilde{\tau}_{R}}$ and ${\tilde{\tau}_{L}}$ may have similar
masses, and the off-diagonal entries in the ${\tilde{\tau}}$ mass matrix may
cause large mixing and repulsion between the ${\tilde{\tau}_{1}}$ and
${\tilde{\tau}_{2}}$ masses.

On the other hand, one experimental signature that is shared by the SUSY SU(5)
GUT model and GUT-universal models is the possible appearance of a long-lived
(metastable) ${\tilde{\tau}_{1}}$. This is a feature of a significant fraction
(but not all) of the ${\tilde{\tau}_{1}}-\tilde{\chi}^{0}_{1}$ coannihilation
region.

The prospects for direct DM detection are mixed: they are relatively good in
the $\tilde{\chi}^{\pm}_{1}-\tilde{\chi}^{0}_{1}$ coannihilation region, but
less promising in the rapid $H/A$ annihilation and hybrid regions, though
potentially detectable in the planned LUX-Zepelin experiment. On the other
hand, the ${\tilde{\tau}_{1}}-\tilde{\chi}^{0}_{1}$ coannihilation region
probably lies beyond the reach of this experiment, as does part of the hybrid
region. Indeed, portions of these regions lie below the neutrino ‘floor’. On
the other hand, substantial parts of these regions are accessible to LHC
searches for long-lived particles and $\,/E_{T}$.

## Acknowledgements

The work of R.C. is supported in part by the National Science Foundation under
Grant No. PHY- 1151640 at the University of Illinois Chicago, and in part by
Fermilab, operated by Fermi Research Alliance, LLC under Contract No. De-AC02-
07CH11359 with the United States Department of Energy. This work of M.J.D. is
supported in part by the Australian Research Council. The work of J.E. is
supported in part by STFC (UK) via the research grant ST/L000326/1, and the
work of H.F. is also supported in part by STFC (UK). The work of S.H. is
supported in part by CICYT (grant FPA 2013-40715-P) and also by the Spanish
MICINN Consolider-Ingenio 2010 Program under grant MultiDark CSD2009-00064.
The work of D.M.-S. is supported by the European Research Council via Grant
BSMFLEET 639068. The work of K.A.O. is supported in part by DOE grant de-
sc0011842 at the University of Minnesota. The work of K.S. is partially
supported by the National Science Centre, Poland, under research grants
DEC-2014/15/B/ST2/02157 and DEC-2015/18/M/ST2/00054. The work of G.W. is
supported in part by the Collaborative Research Center SFB676 of the DFG,
“Particles, Strings and the early Universe”, and in part by the European
Commission through the “HiggsTools” Initial Training Network PITN-
GA-2012-316704.

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