#### Likelihood analysis of the sub-GUT MSSM in light of LHC 13-TeV data

Eur. Phys. J. C
Likelihood analysis of the sub-GUT MSSM in light of LHC 13-TeV data
J. C. Costa 8
E. Bagnaschi 7
K. Sakurai 6
M. Borsato 13
O. Buchmueller 8
M. Citron 8
A. De Roeck 11 12
M. J. Dolan 10
J. R. Ellis 9 14 15
H. Flächer 4
S. Heinemeyer 2 3 5
M. Lucio 13
D. Martínez Santos 13
K. A. Olive 1
A. Richards 8
G. Weiglein 7
0 CSIC , Cantoblanco, 28049 Madrid , Spain
1 School of Physics and Astronomy, William I. Fine Theoretical Physics Institute, University of Minnesota , Minneapolis, MN 55455 , USA
2 Instituto de Física Teórica UAM-CSIC , C/ Nicolas Cabrera 13-15, 28049 Madrid , Spain
3 Instituto de Física de Cantabria (CSIC-UC) , Avda. de Los Castros s/n, 39005 Santander , Spain
4 H.H. Wills Physics Laboratory, University of Bristol , Tyndall Avenue, Bristol BS8 1TL , UK
5 Campus of International Excellence UAM
6 Faculty of Physics, Institute of Theoretical Physics, University of Warsaw , ul. Pasteura 5, 02-093 Warsaw , Poland
7 DESY , Notkestraße 85, 22607 Hamburg , Germany
8 High Energy Physics Group, Blackett Laboratory, Imperial College , Prince Consort Road, London SW7 2AZ , UK
9 Theoretical Physics Department , CERN, 1211 Geneva 23 , Switzerland
10 School of Physics, ARC Centre of Excellence for Particle Physics at the Terascale, University of Melbourne , Parkville 3010 , Australia
11 Antwerp University , 2610 Wilrijk , Belgium
12 Experimental Physics Department , CERN, 1211 Geneva 23 , Switzerland
13 Instituto Galego de Física de Altas Enerxías, Universidade de Santiago de Compostela , Santiago de Compostela , Spain
14 National Institute of Chemical Physics and Biophysics , Rävala 10, 10143 Tallinn , Estonia
15 Theoretical Particle Physics and Cosmology Group, Department of Physics, King's College London , London WC2R 2LS , UK
We describe a likelihood analysis using MasterCode of variants of the MSSM in which the soft supersymmetry-breaking parameters are assumed to have universal values at some scale Min below the supersymmetric grand unification scale MGUT, as can occur in mirage mediation and other models. In addition to Min, such 'subGUT' models have the 4 parameters of the CMSSM, namely a common gaugino mass m1/2, a common soft supersymmetrybreaking scalar mass m0, a common trilinear mixing parameter A and the ratio of MSSM Higgs vevs tan β, assuming that the Higgs mixing parameter μ > 0. We take into account constraints on strongly- and electroweakly-interacting sparticles from ∼ 36/fb of LHC data at 13 TeV and the LUX and 2017 PICO, XENON1T and PandaX-II searches for dark matter scattering, in addition to the previous LHC and dark matter constraints as well as full sets of flavour and electroweak constraints. We find a preference for Min ∼ 105 to 109 GeV, with Min ∼ MGUT disfavoured by χ 2 ∼ 3 due to the BR(Bs,d → μ+μ−) constraint. The lower limits on strongly-interacting sparticles are largely determined by LHC searches, and similar to those in the CMSSM. We find a preference for the LSP to be a Bino or Higgsino with m ˜10 ∼ 1 TeV, with annihilation via heavy Higgs bosons χ H/ A and stop coannihilation, or chargino coannihilation, bringing the cold dark matter density into the cosmological range. We find that spin-independent dark matter scattering is likely to be within reach of the planned LUX-Zeplin and XENONnT experiments. We probe the impact of the (g − 2)μ constraint, finding similar results whether or not it is included.
1 Introduction
Models invoking the appearance of supersymmetry (SUSY)
at the TeV scale are being sorely tested by the negative results
of high-sensitivity searches for sparticles at the LHC [
1,2
]
and for the scattering of dark matter particles [
3–6
]. There
have been many global analyses of the implications of these
experiments for specific SUSY models, mainly within the
minimal supersymmetric extension of the Standard Model
(MSSM), in which the lightest supersymmetric particle
(LSP) is stable and a candidate for dark matter (DM). This
may well be the lightest neutralino, χ˜10 [
7,8
], as we assume
here. Some of these studies have assumed universality of the
soft SUSY-breaking parameters at the GUT scale, e.g., in the
constrained MSSM (the CMSSM) [
9–14
] and in models with
non-universal Higgs masses (the NUHM1,2) [
12,15
]. Other
analyses have taken a phenomenological approach, allowing
free variation in the soft SUSY-breaking parameters at the
electroweak scale (the pMSSM) [
16–27
].
A key issue in the understanding of the implications of
the LHC searches for SUSY is the exploration of regions
of parameter space where compressed spectra may reduce
the sensitivity of searches for missing transverse energy, E/T .
These regions also have relevance to cosmology, since
models with sparticles that are nearly degenerate with the LSP
allow for important coannihilation processes that suppress
the relic LSP number density, allowing heavier values of
mχ0 . The accompanying heavier SUSY spectra are also more
˜1
challenging for the LHC E/T searches.
The CMSSM offers limited prospects for coannihilation,
and examples that have been studied in some detail include
coannihilation with the lighter stau slepton, τ˜1 [
28–35
], or
the lighter stop squark, t˜1 [
36–45
]. Other models offer the
possibilities of different coannihilation partners, such as the
lighter chargino, χ˜1± [
25,46–50
], some other slepton [27] or
squark flavour [
51
], or the gluino [
52–65
]. In particular, the
pMSSM allows for all these possibilities, potentially also in
combination [27].
In this paper we study the implications of LHC and
DM searches for an intermediate class of SUSY models, in
which universality of the soft SUSY-breaking parameters is
imposed at some input scale Min below the GUT scale MGUT
but above the electroweak scale [
66–70
], which we term
‘subGUT’ models. Models in this class are well motivated
theoretically, since the soft SUSY-breaking parameters in the
visible sector may be induced by some dynamical
mechanism such as gluino condensation that kicks in below the
GUT scale. Specific examples of sub-GUT models include
warped extra dimensions [
71
] and mirage mediation [
72–83
].
Mirage mediation can occur when two sources of
supersymmetry breaking play off each other, such as moduli
mediation based, e.g., on moduli stabilization as in [
84
] and
anomaly mediation [
85–90
]. The relative contributions of
each source of supersymmetry breaking can be parametrized
by the strength of the moduli mediation, α, and allows one to
interpolate between nearly pure moduli mediation (large α)
and nearly pure anomaly mediation (α → 0). For example,
gaugino masses, Mi , can be written as Mi = Ms (α + bi gi2)
where Ms is related to the gravitino mass in anomaly
mediation (m3/2 = 16π 2 Ms ), and bi , gi are the beta functions
and gauge couplings. This leads to a renormalization scale,
Min = MGU T e−8π2/α at which gaugino masses and soft
scalar masses take unified values, although there is no
physical threshold at Min in this model. We are not concerned here
with the detailed origin of Min , simply postulating that there
is a scale below the GUT scale where the supersymmetry
breaking masses are unified.
Sub-GUT models are of particular phenomenological
interest, since the reduction in the amount of
renormalizationgroup (RG) running below Min, compared to that below
MGUT in the CMSSM and related models, leads naturally to
SUSY spectra that are more compressed [
66–68
]. These may
offer extended possibilities for ‘hiding’ SUSY via suppressed
E/T signatures, as well as offering enhanced possibilities for
different coannihilation processes. Other possible effects of
the reduced RG running include a stronger lower limit on
mχ0 because of the smaller hierarchy with the gluino mass,
˜1
a stronger lower limit on the DM scattering cross section
because of a smaller hierarchy between mχ0 and the squark
˜1
masses, and greater tension between LHC searches and a
possible SUSY explanation of the measurement of (g −2)μ [
91–
100
], because of the smaller hierarchies between the gluino
and squark masses and the smuon and χ˜10 masses.
We use the MasterCode framework [
9–12,15,25,27,
51,101–104
] to study these issues in the sub-GUT
generalization of the CMSSM, which has 5 free parameters,
comprising Min as well as a common gaugino mass m1/2, a
common soft SUSY-breaking scalar mass m0, a common
trilinear mixing parameter A and the ratio of MSSM Higgs vevs
tan β, assuming that the Higgs mixing parameter μ > 0, as
may be suggested by (g − 2)μ.1 Our global analysis takes
into account the relevant CMS searches for strongly-and
electroweakly-interacting sparticles with the full 2016
sample of ∼ 36/fb of data at 13 TeV [
105–107
], and also
considers the available results of searches for long-lived charged
particles [
108,109
].2 We also include a complete set of direct
DM searches published in 2017, including the PICO limit on
the spin-dependent scattering cross section, σ pSD [
4
], as well
as the first XENON1T limit [
5
] and the most recent
PandaXII limit [
6
] on the spin-independent scattering cross section,
σ pSI, as well as the previous LUX search [
3
]. We also include
full sets of relevant electroweak and flavour constraints.
We find in our global sub-GUT analysis a distinct
preference for MW Min MGUT, with values of Min ∼ 105 or
∼ 108 to 109 GeV being preferred by χ 2 ∼ 3 compared to
the CMSSM (where Min = MGUT). This preference is driven
principally by the ability of the sub-GUT MSSM to
accommodate a value of BR(Bs,d → μ+μ−) smaller than in the
Standard Model (SM), as preferred by the current data [
110–
112
]. As discussed later, this effect can be traced to the
different RGE evolution of At in the sub-GUT model, which
enables it have a different sign from that in the CMSSM.
The lower limits on strongly-interacting sparticles are
similar to those in the CMSSM, being largely determined by
LHC searches. The favoured DM scenario is that the LSP
1 We have also made an exploratory study for μ < 0 with a limited
sample, finding quite similar results within the statistical uncertainties.
2 The ATLAS SUSY searches with ∼ 36/fb of data at 13 TeV [
2
] yield
similar constraints.
is a Bino or Higgsino with mχ0 ∼ 1 TeV, with the cold
˜1
DM being brought into the cosmological range by
annihilation via heavy Higgs bosons H/ A and stop coannihilation,
or chargino coannihilation. In contrast to the CMSSM and
pMSSM11, the possibility that mχ0 1 TeV is strongly
dis˜1
favoured in the sub-GUT model, so the LHC constraints have
insignificant impact. The same is true of the LHC searches
for long-lived charged particles.
The likelihood functions for fits with and without the
(g − 2)μ constraint are quite similar, reflecting the
anticipated difficulty in accounting for the (g − 2)μ anomaly in the
sub-GUT MSSM. Encouragingly, we find a preference for a
range of σ pSI just below the current upper limits, and within
the prospective sensitivities of the LUX-Zeplin (LZ) [
113
]
and XENONnT [
114
] experiments.
The outline of this paper is as follows. In Sect. 2 we
summarize the experimental and astrophysical constraints we
apply. Since we follow exactly our treatments in [
27
], we
refer the interested reader there for details. Then, in Sect. 3
we summarize the MasterCode framework and how we
apply it to the sub-GUT models. Our results are presented
in Sect. 4. Finally, Sect. 5 summarizes our conclusions and
discusses future perspectives for the sub-GUT MSSM.
2 Experimental and astrophysical constraints
2.1 Electroweak and flavour constraints
Our treatments of these constraints are identical to those
in [
27
], which were based on Table 1 of [
51
] with the updates
listed in Table 2 of [
27
]. Since we pay particular attention in
this paper to the impact on the sub-GUT parameter space of
the (g − 2)μ constraint [
91,92
], we note that we assume
aμEXP − aμSM = (30.2 ± 8.8 ± 2.0MSSM) × 10−10
(1)
to be the possible discrepancy with SM calculations [
93–
100
] that may be explained by SUSY. As we shall see,
the BR(Bs,d → μ+μ−) measurement [
110–112
] plays an
important role in indicating a preferred region of the
subGUT parameter space.
2.2 Higgs constraints
In the absence of published results on the Higgs boson based
on Run 2 data, we use in this global fit the published results
from Run 1 [
115
], as incorporated in the HiggsSignals
code [
116,117
].
Searches for heavy MSSM Higgs bosons are incorporated
using the HiggsBounds code [
118–121
], which uses the
results from Run 1 of the LHC. We also include the ATLAS
limit from ∼ 36/fb of data from the LHC at 13 TeV [122].
2.3 Dark matter constraints and mechanisms
Cosmological density
Since R-parity is conserved in the MSSM, the LSP is a
candidate to provide the cold DM (CDM). We assume that the LSP
is the lightest neutralino χ˜10 [
7,8
], and that it dominates the
total CDM density. For the latter we assume the Planck 2015
value: CDMh2 = 0.1186 ± 0.0020EXP ± 0.0024TH [123].
Density mechanisms
As in [
27
], we use the following set of measures related to
particle masses to indicate when specific mechanisms are
important for bringing CDMh2 into the Planck 2015 range, which
have been validated by checks using Micromegas [
124
].
• Chargino coannihilation
This may be important if the χ˜10 is not much lighter than
the lighter chargino, χ˜1±, and we introduce the following
coannihilation measure:
mχ±
˜1
m 0 − 1
χ
˜1
chargino coann. :
and shade in blue the parts of the 68 and 95% CL regions of
the two-dimensional plots in Sect. 4 where (3) is satisfied.
• Stau coannihilation
We introduce the following measure for stau
coannihilation:
τ˜ coann. :
mτ˜1
m 0 − 1
χ
˜1
< 0.15,
and shade in pink the corresponding area of the 68 and
95% CL regions of the two-dimensional sub-GUT
parameter planes. We do not find regions where coannihilation with
other charged slepton species, or with sneutrinos, is
important.
• Stop coannihilation
We introduce the following measure for stop
coannihilation:
t˜1 coann. :
mt˜1
m 0 − 1
χ
˜1
< 0.15,
and shade in yellow the corresponding area of the 68 and
95% CL regions of the two-dimensional sub-GUT parameter
(3)
(4)
(5)
planes. We do not find regions where coannihilation with
other squark species, or with gluinos, is important.
• Focus-point region
The sub-GUT parameter space has a focus-point region
where the DM annihilation rate is enhanced because the LSP
˜10 has an enhanced Higgsino component as a result of
nearχ
degeneracy in the neutralino mass matrix. We introduce the
following measure to characterize this possibility:
(6)
focus point:
μ
mχ0
˜1
and shade in cyan the corresponding area of the 68 and
95% CL regions of the two-dimensional sub-GUT
parameter planes.
• Hybrid regions
In addition to regions where one of the above DM
mechanisms is dominant, there are also various ‘hybrid’ regions
where more than one mechanism is important. These are
indicated in the two-dimensional planes below by shadings
in mixtures of the ‘primary’ colours above, which are shown
in the corresponding figure legends. For example, there are
prominent regions where both chargino coannihilation and
direct-channel H/ A poles are important, whose shading is
darker than the blue of regions where H/ A poles are
dominant.
Direct DM searches
We apply the constraints from direct searches for
weaklyinteracting dark matter particles via both spin-independent
and -dependent scattering on nuclei. In addition to the 2016
LUX constraint on σ pSI [
3
], we use the 2017 XENON1T [
5
]
and PandaX-II [
6
] constraints on the spin-independent DM
scattering, which we combine in a joint two-dimensional
likelihood function in the (mχ0 , σ pSI) plane. We estimate the
˜1
spin-independent nuclear scattering matrix element
assuming σ0 = 36 ± 7 MeV and π N = 50 ± 7 MeV as in [
125–
128
],3 and the spin-dependent nuclear scattering matrix
element assuming u = +0.84 ± 0.03, d = −0.43 ± 0.03
and s = −0.09±0.03 [
125–128
]. We implement the recent
PICO [
4
] constraint on the spin-dependent dark matter
scattering cross-section on protons, σ pSD.
Indirect astrophysical searches for DM
As discussed in [
27
], there are considerable uncertainties in
the use of IceCube data [132] to constrain σ pSD and, as we
discuss below, the global fit yields a prediction that lies well
below the current PICO [
4
] constraint on σ pSD and the current
3 We note that a recent analysis using covariant baryon chiral
perturbation theory yields a very similar central value of π N [129]. However,
we emphasize that there are still considerable uncertainties in the
estimates of σ0 and π N and hence the N |s¯s|N matrix element that is
important for σpSI [130,131].
IceCube sensitivity, so we do not include the IceCube data
in our global fit.
2.4 13 TeV LHC constraints
Searches for gluinos and squarks
We implement the CMS simplified model searches with
∼ 36/fb of data at 13 TeV for events with jets and E/T
but no leptons [
105
] and for events with jets, E/T and a
single lepton [
106
], using the Fastlim approach [133].
We use [
105
] to constrain g˜ g˜ → [qq¯ χ˜10]2 and [bb¯χ˜10]2,
and q˜q˜¯ → [qχ˜10][q¯ χ˜10], and use [
106
] to constrain g˜ g˜ →
[t t¯χ˜10]2. Details are given in [
27
].
Stop and sbottom searches
We also implement the CMS simplified model searches with
∼ 36/fb of data at 13 TeV in the jets + 0 [
105
] and 1
[
106
] lepton final states to constrain t˜1t˜¯1 → [t χ˜10][t¯χ˜10],
[cχ˜10][c¯χ˜10] in the compressed-spectrum region, [bW +χ˜1 ]
0
[b¯W −χ˜10] via χ˜1± intermediate states and b˜1b˜¯1→[bχ˜1 ][b¯χ˜10],
0
again using Fastlim as described in detail in [
27
].
Searches for electroweak inos
We also consider the CMS searches for electroweak inos in
multilepton final states with ∼ 36/fb of data at 13 TeV [
107
],
constraining χ˜1±χ˜20 → [W χ˜10][Z χ˜10], 3 ± + 2χ˜10 via ˜±/ν˜
intermediate states, and 3τ ± + 2χ˜10 via τ˜± intermediate
states using Fastlim [133] as described in [
27
]. These
analyses can also be used to constrain the production of
electroweak inos in the decays of coloured sparticles, since
these searches do not impose conditions on the number of
jets. However, as we discuss below, in the sub-GUT model
the above-mentioned searches for strongly-interacting
sparticles impose such strong limits on the mχ0 and mχ± that the
searches for electroweak inos do not have˜1significa˜1nt impact
on the preferred parameter regions.
Searches for long-lived or stable charged particles
We also consider a posteriori the search for long-lived
charged particles published in [
108
], which are sensitive to
lifetimes ns, and the search for massive charged particles
that escape from the detector without decaying [
109
].
However, these also do not have significant impact on the preferred
parameter regions, as we discuss in detail below, and are not
included in our global fit.
3 Analysis framework
3.1 Model parameters
As mentioned above, the five-dimensional sub-GUT MSSM
parameter space we consider in this paper comprises a
gaugino mass parameter m1/2, a soft SUSY-breaking scalar mass
parameter m0 and a trilinear soft SUSY-breaking parameter
Table 1 The ranges of the sub-GUT MSSM parameters sampled,
together with the numbers of segments into which they are divided,
together with the total number of sample boxes shown in the last row.
This sample is for positive values of the Higgs mixing parameter, μ. As
already noted, a smaller sample for μ < 0 gives similar results. Note that
our sign convention for A is opposite to that used in SoftSusy [134]
Parameter
A0 that are assumed to be universal at some input mass scale
Min, and the ratio of MSSM Higgs vevs, tan β. Table 1
displays the ranges of these parameters sampled in our analysis,
as well as their divisions into segments, which define boxes
in the five-dimensional parameter space.
3.2 Sampling procedure
We sample the boxes in the five-dimensional sub-GUT
MSSM parameter space using the MultiNest
package [135–137], choosing for each box a prior such that 80% of
the sample has a flat distribution within the nominal box, and
20% of the sample is in normally-distributed tails extending
outside the box. This eliminates features associated with the
boundaries of the 96 boxes, by providing a smooth overlap
between them. In total, our sample includes ∼ 112 million
points with χ 2 < 100.
3.3 The MasterCode
The MasterCode framework [
9–12,15,25,27,51,101–104
],
interfaces and combines consistently various private and
public codes using the SUSY Les Houches Accord (SLHA) [138,
139]. This analysis uses the following codes: SoftSusy
3.7.2 [134] for the MSSM spectrum, FeynWZ [140,141]
for the electroweak precision observables, SuFla [142,
143] and SuperIso [144–146] for flavour observables,
FeynHiggs 2.12.1-beta [
147–153
] for (g − 2)μ
and calculating Higgs properties, HiggsSignals 1.4.0
[
116,117
] and HiggsBounds 4.3.1 [
118–121
] for
experimental constraints on the Higgs sector, Micromegas
3.2 [124] for the DM relic density, SSARD [128] for the
spin-independent and -dependent elastic scattering
crosssections σ pSI and σ pSD, SDECAY 1.3b [
154
] for sparticle
branching ratios and (as already mentioned) Fastlim [133]
to recast LHC 13 TeV constraints on events with E/T .
4 Results
4.1 Results for Min, m0 and m1/2
The top left panel of Fig. 1 displays the one-dimensional
profile χ 2 likelihood function for Min, as obtained under various
assumptions.4 In this and subsequent one-dimensional plots,
the solid lines represent the results of a fit including results
from ∼ 36/fb of data from the LHC at 13 TeV (LHC13),
whereas the dashed lines omit these results, and the blue
lines include (g − 2)μ, whereas the green lines are obtained
when this constraint is dropped.
We observe in the top left panel of Fig. 1 a preference
for Min 4.2 × 108 GeV when the LHC 13-TeV data
and (g − 2)μ are both included (solid blue line), falling to
5.9×105 GeV when the 13-TeV data are dropped (dashed
blue line). There is little difference between the global χ 2
values at these two minima, but values of Min < 105 GeV are
strongly disfavoured. The rise in χ 2 when Min increases to
∼ 106 GeV and the LHC 13-TeV data are included (solid
lines) is largely due to the contribution of BR(Bs,d →
μ+μ−). At lower Min, the H → τ +τ − constraint allows a
larger value of tan β, which leads (together with an increase
in the magnitude of A) to greater negative interference in
the supersymmetric contribution to BR(Bs,d → μ+μ−), as
preferred by the data.
For both fits including the LHC 13-TeV data (solid lines),
the χ 2 function ∼ 1 for most of the range Min ∈
(105, 1011) GeV, apart from localized dips, whereas χ 2
rises to 2 for Min 1012 GeV. As already mentioned
and discussed in more detail later, the reduction in the global
χ 2 function for Min 1012 GeV arises because for these
values of Min the sub-GUT model can accommodate
better the measurement of BR(Bs,d → μ+μ−), whose central
experimental value is somewhat lower than in the SM.
When the (g − 2)μ constraint is dropped, as shown by the
green lines in top left panel of Fig. 1, there is a minimum
of χ 2 around Min 1.6 × 105 GeV, whether the LHC
13TeV constraint is included, or not. The values of the other
input parameters at the best-fit points with and without these
data are also very similar, as are the values of χ 2. On the
other hand, the values of χ 2 for Min ∈ (105, 108) GeV
are generally smaller when the LHC 13-TeV constraints are
dropped, the principal effect being due to the H/ A → τ +τ −
constraint.
In contrast, when Min 109 GeV the χ 2 function in
the top left panel of Fig. 1 is quite similar whether the LHC
13-TeV and (g − 2)μ constraints are included or not, though
χ 2 0.5 lower when the (g −2)μ constraint is dropped, as
seen by comparing the green and blue lines. This is because
4 This and subsequent figures were made using Matplotlib [
155
],
unless otherwise noted.
Fig. 1 Profile likelihood functions in the sub-GUT MSSM. Top left:
one-dimensional profile likelihood function for Min. Top right:
twodimensional projection of the likelihood function in the (m0, m1/2)
plane. Middle left: two-dimensional projection of the likelihood
function in the (Min, m0) plane. Middle right: two-dimensional
projection of the likelihood function in the (Min, m1/2) plane. Bottom left:
one-dimensional profile likelihood function for m0. Bottom right:
onedimensional profile likelihood function for m1/2. Here and in
subsequent one-dimensional plots, the solid lines include the constraints from
∼ 36/fb of LHC data at 13 TeV and the dashed lines drop them, and
the blue lines include (g − 2)μ, whereas the green lines drop these
constraints. Here and in subsequent two-dimensional plots, the red (blue)
(green) contours are boundaries of the 1-, 2- and 3-σ regions, and the
shadings correspond to the DM mechanisms indicated in the legend
the tension between (g − 2)μ and LHC data is increased
when M3/M1 is reduced, as occurs because of the smaller
RGE running when Min < MGUT. Conversely, lower Min is
relatively more favoured when (g − 2)μ is dropped, leading
to this increase in χ 2 at high Min though the total χ 2 is
reduced.
We list in Table 2 the parameters of the best-fit points when
we drop one or both of the (g − 2)μ and LHC13 constraints,
as well as the values of the global χ 2 function at the best-fit
points. We see that the best-fit points without (g − 2)μ are
very similar with and without the LHC 13-TeV constraint.
On the other hand, the best-fit points with (g − 2)μ have
quite different values of the other input parameters, as well
as larger values of Min, particularly when the LHC 13-TeV
data are included.
The top right panel of Fig. 1 displays the (m0, m1/2) plane
when the (g − 2)μ and LHC13 constraints are applied. Here
and in subsequent planes, the green star indicates the best-fit
point, whose input parameters are listed in Table 2: it lies
in a hybrid stop coannihilation and rapid H / A annihilation
region.
This parameter plane and others in Fig. 1 and subsequent
figures also display the 68% CL (1-σ ), 95% CL (2-σ ) and
99.7% (3-σ ) contours in the fit including both (g − 2)μ and
the LHC13 data as red, blue and green lines, respectively.
We note, here and subsequently, that the green 3-σ contours
are generally close to the blue 2-σ contours, indicating a
relatively rapid increase in χ 2, and that the χ 2 function is
relatively flat for m0, m1/2 1 TeV. The regions inside the
95% CL contours are colour-coded according to the dominant
DM mechanisms, as shown in the legend beneath Fig. 1.5
Similar results for this and other planes are obtained when
either or both of the (g − 2)μ and LHC13 constraints are
dropped.
We see that chargino coannihilation is important in the
upper part of the (m0, m1/2) plane shown in the top right
panel of Fig. 1, but rapid annihilation via the H / A bosons
becomes important for lower m1/2, often hybridized with
other mechanisms including stop and stau coannihilation.
5 In regions left uncoloured none of the DM mechanism dominance
criteria are satisfied.
We also note smaller regions with m1/2 ∼ 1.5 to 3 TeV
where stop coannihilation and focus-point mechanisms are
dominant.
The middle left panel of Fig. 1 shows the corresponding
(Min, m0) plane, where we see a significant positive
correlation between the variables that is particularly noticeable in
the 68% CL region. In most of this and the 95% CL region
with Min 1013 GeV the relic LSP density is controlled by
chargino coannihilation, though with patches where rapid
annihilation via the A/ H bosons is important, partly in
hybrid combinations. In contrast, the (Min, m1/2) plane
shown in the middle right panel of Fig. 1 does not exhibit
a strong correlation between the variables. We see again the
importance of chargino coannihilation, with the A/ H
mechanism becoming more important for lower m1/2 and larger
Min, and for all values of m1/2 for Min 1014 GeV.
Also visible in the middle row of planes are small regions
with Min ∼ 1013 to 1014 GeV where stau coannihilation is
dominant, partly hybridized with stop coannihilation. The
reduction in the global χ 2 function for Min 1012 GeV
visible in the top left panel of Fig. 1 is associated with the
68% CL regions in this range of Min visible in the two middle
planes of Fig. 1.
The one-dimensional profile likelihood functions for m0
and m1/2 are shown in the bottom panels of Fig. 1. We note
once again the similarities between the results with/without
(g − 2)μ (blue/green lines) and the LHC13 constraints
(solid/dashed lines). The flattening of the χ 2 function for
m0 at small values reflects the extension to m0 = 0 of the
95% CL region in the top right panel of Fig. 1. On the other
hand, the χ 2 function for m1/2 rises rapidly at small
values, reflecting the close spacing of the 95 and 99.7% CL
contours for m1/2 ∼ 1 TeV seen in the same plane. The
impact of the LHC13 constraints is visible in the differences
between the solid and dashed curves at small m0, in
particular. The (g − 2)μ constraint has less impact, as shown by the
smaller differences between the green and blue curves. We
see that the χ 2 function for m0 rises by 1 at large mass
values, whereas that for m1/2 falls monotonically at large
values. The χ 2 function for m1/2 exhibits a local maximum at
m1/2 ∼ 3 TeV, which corresponds to the separation between
the two 68% CL regions in the top right plane of Fig. 1. These
are dominated by chargino coannihilation (larger m1/2, green
shading) and by rapid annihilation via A/H bosons (smaller
m1/2, blue shading) and other mechanisms, respectively.
both be considerably lighter than the gluino and the
firstand second-generation squarks, with 95% CL lower limits
mt˜1 ∼ 900 GeV and mb˜1 ∼ 1.5 TeV, respectively.
4.2 Squarks and gluinos
The various panels of Fig. 2 show the limited impact of the
LHC 13-TeV constraints on the possible masses of
stronglyinteracting sparticles in the sub-GUT model, comparing the
solid and dashed curves. The upper left panel shows that the
95% CL lower limit on mg˜ ∼ 1.5 TeV, whether the LHC
13-TeV data and the (g − 2)μ constraint are included or not.
However, the best-fit value of mg˜ increases from ∼ 2 TeV
to a very large value when (g − 2)μ is dropped, although the
χ 2 price for mg˜ ∼ 2 TeV is ∼ 1. The upper right panel
shows similar features in the profile likelihood function for
mq˜R (that for mq˜L is similar), with a 95% CL lower limit of
∼ 2 TeV, which is again quite independent of the inclusion
of (g − 2)μ and the 13-TeV data. The lower panels of Fig. 2
show the corresponding profile likelihood functions for mt˜1
(left panel) and mb˜1 (right panel). We see that these could
4.3 The lightest neutralino and lighter chargino
The top left panel of Fig. 3 shows the profile likelihood
function for mχ0 , and the top right panel shows that for mχ± .
˜1 ˜1
We see that in all the cases considered (with and without the
(g−2)μ and LHC13 constraints), the value of χ 2 calculated
using the LHC constraints on strongly-interacting sparticles
is larger than 4 for mχ˜10 750 GeV and mχ˜1± 800 GeV.
Therefore, the LHC electroweakino searches [
107
] have no
impact on the 95% CL regions in our 2-dimensional
projections of the sub-GUT parameter space, and we do not include
the results of [
107
] in our global fit.
We now examine the profile likelihood functions for the
fractions of Bino, Wino and Higgsino in the χ˜10 composition:
χ˜10 = N11 B˜ + N12W˜ 3 + N13 H˜u + N14 H˜d ,
(7)
τχ± ≥ 10−15 s that are allowed in the fit including the (g − 2)μ and
˜1
LHC 13-TeV constraints at the 68 (95) (99.7)% CL in 2 dimensions, i.e.,
χ 2 < 2.30 (5.99) (11.83), enclosed by the red (blue) (green) contour
which are shown in Fig. 4. As usual, results from an
analysis including the 13-TeV data are shown as solid lines and
without them as dashed lines, with (g − 2)μ as blue lines and
without it as green lines. The top left panel shows that in the
LHC 13-TeV case with (g − 2)μ an almost pure B˜
composition of the χ˜10 is preferred, N11 → 1, though the possibility
that this component is almost absent is only very slightly
disfavoured. Conversely, before the LHC 13-TeV data there
was a very mild preference for N11 → 0, and this is still the
case if (g − 2)μ is dropped. The upper right panel shows that
a small W˜ 3 component in the χ˜10 is strongly preferred in all
cases. Finally, the lower panel confirms that small H˜u,d
components are preferred when the LHC 13-TeV and (g − 2)μ
constraints are applied, but large H˜u,d components are
preferred otherwise.
The χ˜10 compositions favoured at the 1-, 2- and 3-σ levels
(blue, yellow and red) are displayed in Fig. 5 for fits including
LHC 13-TeV data with (without) the (g−2)μ constraint in the
left (right) panel. We see that these regions are quite similar
in the two panels, and correspond to small Wino admixtures.
On the other hand, the Bino fraction N121 and the Higgsino
fraction N123 + N124 are relatively unconstrained at the 95%
CL. The best-fit points are indicated by green stars, and the
left panel shows again that in the fit with (g − 2)μ the LSP is
an almost pure Bino, whereas an almost pure Higsino
composition is favoured in the fit without (g − 2)μ, as also seen
in Table 3. These two extremes have very similar χ 2 values
in each of the fits displayed.
The global χ 2 function is minimized for mχ 0 1.0 TeV,
˜1
which is typical of scenarios with a Higgsino-like LSP whose
density is brought into the Planck 2015 range by
coannihilation with a nearly-degenerate Higgsino-like chargino χ˜1±.
Indeed, we see in the top right panel of Fig. 3 that χ 2 is
minimized when also mχ ± mχ˜10 1.0 TeV. Table 3 displays
the LSP compositio n˜1 of the sub-GUT model at the best-fit
points with and without (g − 2)μ and the LHC 13-TeV data.
We see again that the χ˜10 LSP is mainly a Higgsino with
almost equal H˜u and H˜d components, except in the fit with
both LHC 13-TeV data and (g − 2)μ included, in which case
it is an almost pure Bino.
Looking at the middle left panel of Fig. 3, we see that the
best-fit point has a chargino-LSP mass difference that may
be O(1) GeV or ∼ 200 to 300 GeV, with similar χ 2 in all
the cases considered, namely with and without the (g − 2)μ
and LHC13 constraints. As seen in the middle right panel of
Fig. 3, in the more degenerate case the preferred chargino
lifetime τχ ± ∼ 10−12 s. The current LHC searches for
long˜1
lived charged particles [
108
] therefore do not impact this
chargino coannihilation region, and are also not included in
our global fit.
The top right panel of Fig. 3 displays an almost-degenerate
local minimum of χ 2 with mχ ± ∼ 1.3 TeV, corresponding
˜1
to a second, local minimum of χ 2 where mχ˜1± − m ˜10 ∼ 200
χ
to 300 GeV, as seen in the middle left panel. In this region
the relic density is brought into the Planck 2015 range by
rapid annihilation through A/ H bosons, as can be inferred
from the bottom left panel of Fig. 3, where we see that at
this secondary minimum M A 2 TeV 2mχ 0 . The χ˜1±
˜1
lifetime in this region is too short to appear in the middle
and bottom right panels of Fig. 3, and too short to have a
separated vertex signature at the LHC.
Finally, the bottom right panel of Fig. 3 shows the regions
of the (mχ ± , τχ ± ) plane with τχ˜1± ∈ (10−16, 10−10) s that are
˜1 ˜1
allowed in the fit including the (g − 2)μ and LHC 13-TeV
constraints at the 68 (95) (99.7) % CL in 2 dimensions, i.e.,
χ 2 < 2.30(5.99)(11.83). Since the chargino would decay
into a very soft track and a neutralino, detecting a separated
4.4 Sleptons
The upper left panel of Fig. 6 shows the profile likelihood
function for mμ˜ R (that for me˜R is indistinguishable, the μ˜ L
and e˜L are slightly heavier). We see that in the sub-GUT
model small values of mμ˜ R were already disfavoured by
earlier LHC data (dashed lines), and that this tendency has been
reinforced by the LHC 13-TeV data (compare the solid lines).
The same is true whether the (g − 2)μ constraint is included
or dropped (compare the blue and green curves).
The upper right panel Fig. 6 shows the corresponding
profile likelihood function for mτ˜1 , which shares many similar
features. However, we note that the χ 2 function for mτ˜1 is
generally lower than that for mμ˜ R ∈ (1, 2) TeV, though the
95% lower limits on mτ˜1 and mμ˜ R are quite similar, and both
are 1 TeV when the LHC 13-TeV constraints are included
in the fit.
The lower left panel of Fig. 6 shows that very small values
of mτ˜1 − mχ 0 in the stau coannihilation region are allowed at
˜1
the χ 2 ∼ 1 level in all the fits with the (g − 2)μ constraint,
rising to χ 2 2 for mτ˜1 − mχ 0 20 GeV when the LHC
13-TeV data are included. ˜1
The lower right panel of Fig. 6 shows the (mτ˜1 , ττ˜1 ) plane,
where we see that ττ˜1 ∈ (10−7, 103) s is allowed at the 68%
CL, for 1600 GeV mτ˜1 2000 GeV and at the 95%
CL also for mτ˜1 ∼ 1100 GeV. This region of parameter
space is close to the tip of the stau coannihilation strip. Lower
τ˜1 masses are strongly disfavoured by the LHC constraints,
particularly at 13 TeV, as seen in the upper right panel of
cated in the right-hand legend. The 68 (95) (99.7)% CL regions in 2
dimensions, i.e., χ 2 < 2.30 (5.99) (11.83), are enclosed by the red
(blue) (green) contours
down of the contributions to the global χ 2 as functions of Min. The
shadings correspond to the different classes of observables, as indicated
in the legend
Fig. 9 One-dimensional profile likelihood function for BR(b → sγ ),
showing the experimental constraint as a dotted line
Fig. 10 One-dimensional profile likelihood function for Mh , where
the dotted line shows the χ 2 contribution due to the (g − 2)μ constraint
alone
Fig. 6. The heavier τ˜1 masses with lower χ 2 seen there
do not lie in the stau coannihilation strip, and have larger
mτ˜1 − mχ 0 and hence smaller lifetimes that are not shown
˜1
in the lower right panel of Fig. 6. Because of the lower limit
on mτ˜1 seen in this panel, neither the LHC search for
longlived charged particles [
108
] nor the LHC search for
(meta)stable massive charged particles that exit the detector [
109
]
are relevant for our global fit.
In view of this, and the fact that the search for long-lived
particles [
108
] is also insensitive in the chargino
coannihilation region, as discussed above, the results of [
108, 109
]
are not included in the calculation of the global likelihood
function.
4.5 (g − 2)μ
We see in the left panel of Fig. 7 that only a small
contribution to (g − 2)μ is possible in sub-GUT models, the profile
likelihood functions with and without the LHC 13-TeV data
and (g − 2)μ being all quite similar. This is because in the
sub-GUT model with low Min the LHC searches for
stronglyinteracting sparticles constrain the μ˜ mass more strongly than
in the GUT-scale CMSSM. The dotted line shows the χ 2
contribution due to our implementation of the (g − 2)μ
constraint alone. We see that in all cases it contributes χ 2 9
to the global fit.
Fig. 11 Left panel: two-dimensional profile likelihood function for
the nominal value of σpSI calculated using the SSARD code [128]
in the (mχ˜10 , σpSI) plane, displaying also the upper limits established
by the LUX [3], XENON1T [
5
] and PandaX-II Collaborations [
6
]
shown as solid black, blue and green contours, respectively. The
projected future 90% CL sensitivities of the LUX-Zeplin (LZ) [
156
] and
XENON1T/nT [
157
] experiments are shown as dashed magenta and
blue lines, respectively, and the neutrino background ‘floor’ [
158,159
]is
4.6 The (MA, tan β) plane
The right panel of Fig. 7 shows the (MA, tan β) plane when
the LHC 13-TeV data and the (g−2)μ constraint are included
in the fit. We see that MA 1.3 TeV at the 95% CL and that,
whereas tan β ∼ 5 is allowed at the 95% CL. Larger values
tan β 30 are favoured at the 68% CL, and the best-fit point
has tan β 36. (This increases to tan β ∼ 45 if either the
LHC 13-TeV and/or (g − 2)μ constraint is dropped.) As in
the previous two-dimensional projections of the sub-GUT
parameter space, the 99.7% (3-σ ) CL contour lies close to
that for the 95% CL.
4.7 B decay observables
We see in the left panel of Fig. 8 that values of BR(Bs,d →
μ+μ−) smaller than that in the SM are favoured. The
subGUT models with μ > 0 that we have studied can
accommodate comfortably the preference seen in the data (dotted line)
for such a small value of BR(Bs,d → μ+μ−),6 which is not
the case in models such as the CMSSM that impose
universal boundary conditions on the soft supersymmetry-breaking
parameters at the GUT scale, if μ > 0. The right panel of
6 This is also the case for the smaller sub-GUT sample with μ < 0 that
we have studied.
shown as a dashed orange line with yellow shading below. Right panel:
Two-dimensional profile likelihood function for the nominal value of
σpSD calculated using the SSARD code [128] in the (mχ0 , σpSD) plane,
˜1
showing also the upper limit established by the PICO Collaboration [
4
].
We also show the indirect limits from the Icecube [132] and
SuperKamiokande [
160
] experiments, assuming χ˜1 χ˜10 → τ +τ − dominates,
0
as well as the ‘floor’ for σpSD calculated in [
161
]
Fig. 8 shows how the contributions of the flavour (blue
shading) and other observables to the global likelihood function
depend on Min for values between 104 and 1016 GeV. This
variation in the flavour contribution (which is dominated by
BR(Bs,d → μ+μ−)) is largely responsible for the sub-GUT
preference for Min < MGUT seen in the top left panel of
Fig. 1. Values of Min ∈ (105, 1012) GeV can accommodate
very well the experimental value of BR(Bs,d → μ+μ−).
This preference is made possible by the different RGE
running in the sub-GUT model, which can change the sign of
the product At μ that controls the relative signs of the SM and
SUSY contributions to the Bs,d → μ+μ− decay amplitudes,
permitting negative interference that reduces BR(Bs,d →
μ+μ−). As already discussed, the reduction in BR(Bs,d →
μ+μ−) and the global χ 2 function for 108 GeV Min
1012 GeV is associated with the blue 68% CL regions with
Min 1012 GeV seen in the middle panels of Fig. 1. On
the other hand, we see in Fig. 9 that sub-GUT models favour
values of BR(b → sγ ) that are close to the SM value.
The contributions to the global χ 2 function of other classes
of observables as functions of Min are also exhibited in
the right panel of Fig. 8. In addition to the aforementioned
reduction in the flavour contribution when Min 1012 GeV
(blue shading), there is a coincident (but smaller) increase
in the contribution of the electroweak precision
observFig. 12 Two-dimensional projections of the global likelihood function
for the sub-GUT MSSM in the (mq˜R , mg) plane (upper left panel), the
˜
(mq˜R , mχ0 ) plane (upper right panel), the (mg, mχ0 ) plane (lower left
˜1 ˜ ˜1
panel), and the (mχ˜10 , σpSI) plane (lower right panel). In each panel we
ables (orange shading) related to tension in the electroweak
symmetry-breaking conditions. The other contributions to
the global χ 2 function, namely the nuisance parameters (red
shading), Higgs mass (light green), (g − 2)μ (teal) and DM
(red), vary smoothly for Min ∼ 1012 GeV.
4.8 Higgs mass
We see in Fig. 10 that the profile likelihood function for Mh
lies within the contribution of the direct experimental
constraint convoluted with the uncertainty in the FeynHiggs
calculation of Mh (dotted line). We infer that there is no
tension between the direct experimental measurement of Mh
and the other observables included in our global fit. We have
also calculated (not shown) the branching ratios for Higgs
decays into γ γ , Z Z ∗ and gg (used as a proxy for gg → h
production), finding that they are expected to be very similar
compare the projections of the sub-GUT parameter regions favoured
at the 68% (red lines), 95% (blue lines) and 99.7% CL (green lines) in
global fits with the LHC 13-TeV data and results from LUX, XENON1T,
and PandaX-II [
3,5,6
] (solid lines), and without them (dashed lines)
to their values in the SM, with 2-σ ranges that lie well within
the current experimental uncertainties.
4.9 Searches for dark matter scattering
The left panel of Fig. 11 shows the nominal predictions for the
spin-independent DM scattering cross-section σ pSI obtained
using the SSARD code [128]. We caution that there are
considerable uncertainties in the calculation of σ pSI, which are
taken into account in our global fit. Thus points with
nominal values of σ pSI above the experimental limit may
nevertheless lie within the 95% CL range for the global fit. We
see that sub-GUT models favour a range of σ pSI close to
the present limit from the LUX, XENON1T and PandaX-II
experiments.7 Moreover, at the 95% CL, the nominal
subGUT predictions for σ pSI are within the projected reaches
of the LZ and XENON1/nT experiments. However, they are
subject to the considerable uncertainty in the σ pSI matrix
element, and might even fall below the neutrino ‘floor’ shown
as a dashed orange line in [
158,159
].
We see in the right panel of Fig. 11 that the sub-GUT
predictions for the spin-dependent DM scattering cross-section
σ pSD lie somewhat below the present upper limit from the
PICO direct DM search experiment. Spin-dependent DM
scattering is also probed by indirect searches for neutrinos
produced by the annihilations of neutralinos trapped inside
the Sun after scattering on protons in its interior. If the
neutralinos annihilate into τ +τ −, the IceCube experiment sets
the strongest such indirect limit [132], and we also show the
constraint from Super-Kamiokande [
160
]. These constraints
are currently not sensitive enough to cut into the range of the
7 We also show, for completeness, the CRESST-II [
162
],
CDMSlite [
163
] and CDEX [
164
] constraints on σpSI, which do not impact
range of mχ0 found in our analysis.
˜1
(mχ0 , σ pSD) plane allowed in our global fit. We also show the
˜1
neutrino ‘floor’ for σ pSD, taken from [
161
]: wee that values
of σ pSD below this floor are quite possible in the sub-GUT
model.
5 Impacts of the LHC 13-TeV and new direct detection constraints
We show in Fig. 12 some two-dimensional projections of the
regions of sub-GUT MSSM parameters favoured at the 68%
Fig. 14 The spectra of Higgs bosons and sparticles at the best-fit points
in the sub-GUT model including LHC 13-TeV data, including the (g −
2)μ constraint (upper panel) and dropping it (lower panel), with dashed
lines indicating the decay modes with branching ratios > 5%. These
plots were made using PySLHA [
166
]
(red lines), 95% (blue lines) and 99.7% CL (green lines),
comparing the results of fits including the LHC 13-TeV
data and recent direct searches for spin-independent dark
matter scattering (solid lines) and discarding them (dashed
lines). The upper left panel shows the (mq˜R , mg) plane, the
˜
upper right plane shows the (mq˜R , mχ0 ) plane, the lower left
˜1
plane shows the (mg˜ , mχ0 ) plane, and the lower right panel
˜1
shows the (mχ0 , σ pSI) plane. We see that in the upper
panels that the ne˜w1 data restrict the favoured parameter space
for mq˜R ∼ 2 TeV, the two left panels show a restriction for
mg˜ ∼ 1.3 TeV, and the right and lower panels show that the
new data also restrict the range of mχ0 to 800 GeV.
How˜1
ever, the lower right panel does not show any new restriction
on the range of possible values of σ pSI.
6 Best-fit points, spectra and decay modes
The values of the input parameters at the best-fit points with
and without the (g − 2)μ and LHC 13-TeV constraints have
been shown in Table 2. The best fits have Min between
1.6 × 105 and 4.1 × 108 GeV, and we note that the input
parameters are rather insensitive to the inclusion of the
13TeV data when (g − 2)μ is dropped. Table 4 displays the
mass spectra obtained as outputs at the best-fit point including
the 13-TeV data (quoted to 3 significant figures) and
including (left column) or dropping (right column) the (g − 2)μ
constraint. As could be expected, the sparticle masses are
generally heavier when (g − 2)μ is dropped. However, the
differences are small in the cases of the χ˜10, χ˜20 and χ˜1±,
being generally < 10 GeV. We also give in the
next-tolast line of Table 4 the values of the global χ 2 function at
these best-fit points, dropping the HiggsSignals
contributions, as was done previously [
51,102
] to avoid biasing the
analysis.
The contributions of different observables to the global
likelihood function at the best-fit points including LHC13
data are shown in Fig. 13. We compare the contributions
when (g − 2)μ is included (pink histograms) and without
(g − 2)μ (blue histograms). We note, in particular, that the
contribution of BR(Bs,d → μ+μ−) is very small in both
cases, which is a distinctive feature of sub-GUT models.
The last line of Table 4 shows the p-values for the best
fits with and without (g − 2)μ, which were calculated
as follows. In the case with (without) (g − 2)μ, setting
aside HiggsSignals so as to avoid biasing the
analysis [
51,102
], the number of constraints making non-zero
contributions to the global χ 2 function (not including
nuisance parameters) is 29 (28), and the number of free
parameters is 5 in each case. Hence the numbers of degrees
of freedom are 24 (23) in the two cases. The values of
the total χ 2 function at the best-fit points, dropping the
HiggsSignals contribution, are 28.9 (18.0) and the
corresponding p-values are 23% (76%). The qualities of the global
fits with and without (g − 2)μ are therefore both good. and
the fit including (g − 2)μ is not poor enough to reject this fit
hypothesis.
The spectra for the best fits are displayed graphically in
Fig. 14, including the (g − 2)μ constraint (upper panel) and
dropping it (lower panel). Also shown are the decay modes
with branching ratios > 5%, as dashed lines whose
intensities increase with the branching ratios. The heavy Higgs
bosons decay predominantly to SM final states, hence no
dashed lines are shown. We see that in both cases the squarks
and gluino are probably too heavy to be discovered at the
LHC, and the sleptons are too heavy to be discovered at any
planned e+e− collider. The best prospects for sparticle
discovery may be for χ˜1± and χ˜20 production at CLIC running
at ECM 2 TeV [
165
].
The global likelihood function is quite flat at large
sparticle masses, and very different spectra are consistent with
the data, within the current uncertainties. The 68 and 95%
CL ranges of Higgs and sparticle masses are displayed in
Fig. 15 as orange and yellow bands, respectively, with the
best-fit values indicated by blue lines. The upper panel is for
a fit including the (g − 2)μ constraint, which is dropped in
the lower panel. At the 68% CL there are possibilities for
squark and gluino discovery at the LHC and the τ˜1, μ˜ R and
e˜R become potentially discoverable at CLIC if it operates at
ECM = 3 TeV [
165
].
7 Summary and perspectives
We have performed in this paper a frequentist analysis of
sub-GUT models in which soft supersymmetry-breaking
parameters are assumed to be universal at some input scale
Min < MGUT. The best-fit input parameters with and without
(g − 2)μ and the LHC 13-TeV data are shown in Table 2. The
physical sparticle masses including the LHC data, with and
without (g − 2)μ, are shown in Table 4 and in Fig. 14, where
decay patterns are also indicated. As seen in the bottom line
of Table 4, the p-values for the fits with and without (g − 2)μ
are 23 and 76%, respectively.
Compared to the best fits with Min = MGUT, we have
found that the minimum value of the global χ 2 function may
be reduced by χ 2 ∼ 2 in the sub-GUT model, with the
exact amount depending whether the (g − 2)μ constraint
and/or LHC13 data are included in the fit. Whether these
observables are included, or not, the global χ 2 minimum
occurs for Min ∼ 107 GeV, and is due to the sub-GUT
model’s ability to provide a better fit to the measured value of
BR( Bs,d → μ+μ−) than in the CMSSM. Although
intriguing, this improvement in the fit quality is not very significant,
but it will be interesting to monitor how the experimental
measurement of BR( Bs,d → μ+μ−) evolves.
In all the scenarios studied (with/without (g − 2)μ and/or
LHC13), the profile likelihood function for m g (mq ) varies
˜ ˜
by 1 for m g˜ 1.9 TeV (mq˜ 2.2 TeV). The
corresponding slowly-varying ranges of χ 2 for mt˜1 (mb˜1 ) start at
∼ 1 TeV (∼ 1.6 TeV), respectively. On the other hand, we
find a more marked preference for m ˜10 ∼ 1 TeV, with the
χ
χ˜1± and χ˜20 being slightly heavier and large mass values being
disfavoured at the χ 2 ∼ 3 level. The best-fit point is in a
region where rapid annihilation via H / A poles is hybridized
with stop coannihilation, with chargino coannihilation and
stau coannihilation also playing roles in both the 68 and 95%
CL regions. Within the 95% CL region, the chargino lifetime
may exceed 10−12 s, and the stau lifetime may be as long
as one second, motivating continued searches for long-lived
sparticles at the LHC.
Taking the LHC13 constraints into account, we find that
the spin-independent DM cross-section, σ pSI, may be just
below the present upper limits from the LUX, XENON1T
and PandaX-II experiments, and within the reaches of the
planned XENONnT and LZ experiments. On the other hand,
the spin-dependent DM cross-section, σ pSD, may be between
some 2 and 5 orders of magnitude below the current upper
limit from the PICO experiment.
Within the sub-GUT framework, therefore, we find
interesting perspectives for LHC searches for strongly-interacting
sparticles via the conventional missing-energy signature.
Future E/T searches for electroweakly-interacting
sparticles and for long-lived massive charged particles may also
have interesting prospects. The best-fit region of parameter
space accommodates the observed deviation of BR( Bs,d →
μ+μ−) from its value in the SM, and it will be interesting
to see further improvement in the precision of this
measurement. A future e+e− collider with centre-of-mass energy
above 2 TeV, such as CLIC [
165
], would have interesting
perspectives for discovering and measuring the properties of
electroweakly-interacting sparticles. There are also
interesting perspectives for direct DM searches via spin-independent
scattering.
Acknowledgements The work of E. B. and G. W. is supported in part
by the Collaborative Research Center SFB676 of the DFG, “Particles,
Strings and the early Universe”. The work of M. B. and D. M. S. is
supported by the European Research Council via Grant BSMFLEET
639068. The work of J. C. C. is supported by CNPq (Brazil). The work
of M. J. D. is supported in part by the Australia Research Council.
The work of J. E. is supported in part by STFC (UK) via the research
Grant ST/L000326/1 and in part via the Estonian Research Council via
a Mobilitas Pluss grant, and the work of H. F. is also supported in part
by STFC (UK). The work of S. H. is supported in part by the MEINCOP
Spain under contract FPA2016-78022-P, in part by the Spanish Agencia
Estatal de Investigación (AEI) and the EU Fondo Europeo de
Desarrollo Regional (FEDER) through the project FPA2016-78645-P, in part
by the AEI through the Grant IFT Centro de Excelencia Severo Ochoa
SEV-2016-0597, and by the Spanish MICINN Consolider-Ingenio 2010
Program under Grant MultiDark CSD2009-00064. The work of M. L.
and I. S. F. is supported by XuntaGal. The work of K.A.O. is
supported in part by DOE Grant de-sc0011842 at the University of
Minnesota. K. S. thanks the TU Munich for hospitality during the final
stages of this work and has been partially supported by the DFG cluster
of excellence EXC 153 “Origin and Structure of the Universe, by the
Collaborative Research Center SFB1258. The work of K. S. is also
partially supported by the National Science Centre, Poland, under research
Grants DEC-2014/15/B/ST2/02157, DEC-2015/18/M/ST2/00054 and
DEC-2015/19/D/ST2/03136. The work of G. W. is also supported in part
by the European Commission through the “HiggsTools” Initial Training
Network PITN-GA-2012-316704.
Open Access This article is distributed under the terms of the Creative
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