CNU-HEP-14-04, IPMU14-0344

arXiv:1411.7329v1 [hep-ph] 26 Nov 2014

Dark Matter in Split SUSY with Intermediate Higgses
Kingman Cheunga,b, Ran Huoc, Jae Sik Leed , Yue-Lin Sming Tsaicab

Department of Physics, National Tsing Hua University, Hsinchu 300, Taiwan

Division of Quantum Phases and Devices, School of Physics, Konkuk University, Seoul
143-701, Republic of Korea
c

Kavli IPMU (WPI), The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba
277-8583, Japand

Department of Physics, Chonnam National University, 300 Yongbong-dong, Buk-gu,
Gwangju, 500-757, Republic of Korea(November 27, 2014)
ABSTRACTThe searches for heavy Higgs bosons and supersymmetric (SUSY) particles at the
LHC have left the minimal supersymmetric standard model (MSSM) with an unusual spectrum of SUSY particles, namely, all squarks are beyond a few TeV while the Higgs bosons
other than the one observed at 125 GeV could be relatively light. In light of this, we study
a scenario characterized by two scales: the SUSY breaking scale or the squark-mass scale(MS ) and the heavy Higgs-boson mass scale (MA ). We perform a survey of the MSSM
< 1010 GeV and MA ∼< 104 GeV such that the lightest Higgs
parameter space with MS ∼boson mass is within the range of the observed Higgs boson as well as satisfying a numberof constraints. The set of constraints include the invisible decay width of the Z bosonand that of the Higgs boson, the chargino-mass limit, dark matter relic abundance from
Planck, the spin-independent cross section of direct detection by LUX, and gamma-ray flux
from dwarf spheroidal galaxies and gamma-ray line constraints measured by Fermi LAT.
Survived regions of parameter space feature the dark matter with correct relic abundance,
which is achieved through either coannihilation with charginos, A/H funnels, or both. We
show that future measurements, e.g., XENON1T and LZ, of spin-independent cross sections can further squeeze the parameter space.

1

1

Introduction

Supersymmetry (SUSY) is one of the most elegant solutions, if not the best, to the gauge
hierarchy problem. SUSY provides an efficient mechanism to break the electroweak symmetry dynamically with a large top Yukawa coupling. Another virtue is that the lightest
SUSY particle (LSP) is automatically a dark matter (DM) candidate to satisfy the relic
DM abundance assuming the R-parity conservation. The fine-tuning argument in the gauge
hierarchy problem requires SUSY particles at work at the TeV scale to stabilize the gap between the electroweak scale and the grand unified theory (GUT) scale or the Planck scale.
With this scale the gauge coupling unification is also naturally achieved in renormalizationgroup equation (RGE) running.
Although SUSY has quite a number of merits at least theoretically, the biggest drawback of SUSY is that so far we have not observed any sign of SUSY. Nevertheless, we haveobserved a light standard model (SM) like Higgs boson, which is often a natural predictionof SUSY. The null results for all the searches of SUSY particles have pushed the mass scaleof squarks beyond a few TeV [1]. Such a high-scale SUSY scenario also draws more andmore attention on CP problems [2], cosmological problems [3], and DM search [4]. On theother hand, the searches for the SUSY Higgs bosons provide the less stringent mass limits
and it still seems possible to find them in the range of a few hundred GeV [5]. Consequently, we are left with an unusual spectrum of SUSY particles and Higgs bosons: (i) all
squarks are heavy beyond a few TeV [1], (ii) the gluino is heavier than about 1 TeV [6],(iii) neutralinos and charginos can be of order O(100 − 1000) GeV, (iv) heavy Higgs bosons
can be of order O(100 − 1000) GeV [5], and (v) a light Higgs boson with a mass 125 GeV[7]. The spectrum is somewhat similar to the proposal of split SUSY [8], except that theheavy Higgs bosons need not be as heavy as those of split SUSY. We name the scenario
the “modified split SUSY” framework, with two distinct scales: the SUSY breaking scale
MS and the heavy Higgs-boson mass scale MA . In the following, for simplicity we callthis “modified split SUSY” as scenario A in which MS and MA are independent, while theoriginal split SUSY as scenario B in which MA and MS are set to be equal.
We perform a survey of the parameter space of the minimal supersymmetric standard model (MSSM) characterized by two scales: (i) the SUSY breaking scale MS with< 1010 GeV, and (ii) the heavy Higgs-boson mass scale (MA ) with MA ∼< 104 GeV,MS ∼such that the lightest Higgs boson mass with large radiative corrections from heavy squarks
is within the range of the mass of the observed Higgs boson. We choose MA smaller thanor at most equal to MS . Specifically, we assume the MSSM above the SUSY breaking scale
MS . Then we do the matching at the scale MS while we decouple all the sfermions. We
evolve from MS down to MA with a set of RGEs comprising of two-Higgs doublet model(2HDM), gauge couplings, and gaugino couplings. For this purpose, we derive the RGEs
governing the range between MS and MA and present them in Appendix A. Then we do2the matching at the scale MA while we decouple all the heavy Higgs bosons. We evolvefrom MA down to the electroweak scale with a set of RGE comprising of the SM and thegauginos. The matching is then done at the electroweak scale. Once we obtain all the
relevant parameters at the electroweak scale, we calculate all the observables and compare
to experimental data.
In this work the LSP of the MSSM is the DM candidate, which is the lightest neutralino in the current scenario. Since we are strongly interested in DM, we include a numberof other existing constraints on SUSY particles and DM:
1. the invisible decay width of the Z boson and that of the Higgs boson,
2. the chargino-mass limit,
3. dark matter relic abundance from Planck,
4. the spin-independent cross section of direct detection by LUX, and5. gamma-ray flux from dwarf spheroidal galaxies (dSphs) and gamma-ray line constraints measured by Fermi LAT.
Due to multidimensional model parameters involved in this work, it will be advantageous to adopt a Monte Carlo sampling technique to perform a global scan. In order
to assess the robustness of our Monte Carlo results, we investigate both Bayesian maps
in terms of marginal posterior (MP) and frequentist ones in terms of the profile likelihood(PL) technique. However, the likelihood functions of experimental constraints are the samefor both approaches.
The organization is as follows. In the next section, we describe the theoretical framework of the modified split SUSY, including the matching conditions at the scales of MS
and MA , and the corresponding interactions of the particles involved. In Sec. 3, we list theset of constraints from collider and dark matter experiments that we use in this analysis.
In Sec. 4, we present the results of our analysis using the methods of PL and MP. We
discuss and conclude in Sec. 5.

2Theoretical Framework

In the case under consideration, we have the two characteristic scales: the high SUSY scale
MS and the Higgs mass scale MA . The relevant phenomenology may be described by theeffective Lagrangians depending on scale Q as follows:
MS < Q: L = LMSSM(1)

MA < Q < MS : L = L2HDM + Lχ̃
(2)

Q < MA : L = LSM + Lχ̃
3

(1)

2.1

Interactions for MA < Q < MS

At the scale MS all the sfermions decouple when we assume that they are heavier than orequal to the scale MS . We are left with the spectrum of the Higgs sector of the 2HDM,
gauginos, and higgsinos.
In this work, we take the general 2HDM potential as follows:†
V2HDM = −µ21 (Φ†1 Φ1 ) − µ22 (Φ†2 Φ2 ) − m212 (Φ†1 Φ2 ) − m∗2
12 (Φ2 Φ1 )+λ1 (Φ†1 Φ1 )2 + λ2 (Φ†2 Φ2 )2 + 2λ3 (Φ†1 Φ1 )(Φ†2 Φ2 ) + 2λ4 (Φ†1 Φ2 )(Φ†2 Φ1 )
+λ5 (Φ†1 Φ2 )2 + λ∗5 (Φ†2 Φ1 )2 + 2λ6 (Φ†1 Φ1 )(Φ†1 Φ2 ) + 2λ∗6 (Φ†1 Φ1 )(Φ†2 Φ1 )+2λ7 (Φ†2 Φ2 )(Φ†1 Φ2 ) + 2λ∗7 (Φ†2 Φ2 )(Φ†2 Φ1 )(2)

with the parameterization
Φ1 =−iσ2 Hd∗ =Φ2 = Hu =

0 −1
1 0

!

√1 (vd + H 0 − iA0 )
d
d
2
−Hd−

Hu+
√1 (vu + H 0 + iA0 )
u
u
2

!

!∗

Hd+
√1 (vd + H 0 + iA0 )
d
d
2

=

!

;

;

(3)

and vd = v cos β = vcβ , vu = v sin β = vsβ , and v ≃ 245 GeV. Then we have
MA2 = MH2 ± + λ4 v 2 − ℜe(λ5 )v 2 ,MH2 ± =

(4)i
ℜe(m212 )v2 h
λ4 cβ sβ + cβ sβ ℜe(λ5 ) + c2β ℜe(λ6 ) + s2β ℜe(λ7 ) ,−cβ s βcβ sβwhere A = −sβ A0d + cβ A0u and H + = −sβ Hd+ + cβ Hu+ .
The wino(bino)-Higgsino-Higgs interactions are given by
(1)

Lχ̃



1
f a + g̃ ′ Be Hf= √ Hu† g̃u σ a W
u
u
2


1
f a − g̃ ′ B
e H
f + h.c.
+ √ Hd† g̃d σ a Wdd2where σ a are the Pauli matrices. We note Hd† = −ΦT1 iσ2 =2.2

√1 (vd + H 0 + iA0 ), −H +
d
d
d
2

(5)


.

Matching at MS

The couplings of the interactions when MA < Q < MS are determined by the matchingconditions at MS and the RGE evolution from MS to Q. Assuming that all the sfermions4are degenerate at MS , the quartic couplings at the scale MS are given by ∗
λ1









4
2
A2b
Nc
1 2
4 µ
4 Ab
(1
−
)
−
y
(g + g ′2 ) +y=t
8(4π)2 b MS2
12MS2
12MS4

λ2 =
λ3 =
λ4 =
λ5 =
λ6 =
λ7 =

4
2
1 2
A2t
Nc
4 µ
4 At
(1
−
)
−
y
y
(g + g ′2 ) +b
8(4π)2 t MS2
12MS2
12MS4


2
Nc
µ2
µ2 A2tµ2 A2b1 2
2 2 Atb
′2
4
4 µ
y y
(g − g ) ++ yt (−) + yb (−)8(4π)2 b t 2
4MS2 12MS4
4MS2 12MS4


2
2
1
Nc
µ2 A2tµ2 A2b
2 2 Atb
4 µ
4 µ− g2 +
−
y
y
+
y
(
−
)
+
y
(
−
)
b t
t
b
4
(4π)2
2
4MS2 12MS4
4MS2 12MS4

2 2 
2 2
Nc
4 µ Ab4 µ At+y,y−b(4π)2 t 12MS4
12MS4


3
Nc
1
A2b
4 µ At4 µAb(− +) + yt,y(4π)2 b MS2
2 12MS2
12MS4


3
1
A2t
Nc
4 µ Ab4 µAt(− +) + yb,(6)y(4π)2 t MS2
2 12MS2
12MS4

with
Atb =





1
6µ2 (µ2 − Ab At )2 3(Ab + At )2
.
− 2−
+
6
MS
MS4
MS2

(7)

We note that the quartic couplings at MS consist of its tree level values and the threshold corrections induced by the A and µ terms. We further observe λ5,6,7 vanish without
including the threshold corrections.
On the other hand, for the wino(bino)-Higgsino-Higgs couplings at the scale MS , we
have
(′)

We note the relation g ′ =g̃u(′) = g̃d = g (′) .

q(8)

3/5 g1 .

The threshold corrections to the gauge and Yukawa couplings at MS also vanish in
the framework under consideration or when all the sfermions are degenerate at MS .

2.3

Interactions for Q < MA

When the scale drops below MA , all the heavy Higgs bosons decouple. We are left with
the SM particles, a light Higgs boson, gauginos, and higgsinos.
The SM Higgs potential is given by"v2
VSM = λ (Φ Φ) −2∗†

We neglect the stau contributions.

5

2

#2

(9)

with
Φ=

G+
√1 (v + h + iG0 )
2

!

(10)

where G±,0 denotes the would-be Goldstone bosons and h the physical neutral Higgs state.
We note m2h = 2λv 2. The wino(bino)-Higgsino-Higgs interactions are then given by
(2)

Lχ̃

2.4



1
f a + ĝ ′ Be Hf= √ Φ† ĝu σ a W
u
u
2


1
f a − ĝ ′ B
e H
f + h.c. .
+ √ (−ΦT iσ2 ) ĝd σ a W
d
d
2

(11)

Matching at MA

The couplings of the interactions when Q < MA are determined by the matching conditions
at MA and the RGE evolution from MA to Q.
At the scale MA , the quartic coupling λ of the SM Higgs potential is given by
λ = λ1 cos4 β + λ2 sin4 β + 2λ̃3 cos2 β sin2 β+ 4λ6 cos3 β sin β + 4λ7 cos β sin3 β + δλ(12)where λ̃3 = λ3 +λ4 +λ5 and δλ denotes the threshold correction. We find that the thresholdcorrection to λ is given by
1
δλ =
4π 2




v2
1
v4
λ3A 2 − λ4A 4
MA 3 MA

!

+

1
MH ±
MH
+
+ 2λ2± ln
λ2H ln
2
8π
MA
MA

λ3H

v2
1 4 v4λ−MH2
3 H MH4

!

+2

λ3±

v2
1 4 v4−λMH2 ±3 ± MH4 ±



!

(13)

where MH denotes the mass of the heavier CP-even neutral Higgs boson and the couplings
λA,H,± are defined as follows:
λA = (λ3 + λ4 − λ5 ) + (λ1 + λ2 − 2λ̃3 ) cos2 β sin2 β + (−λ6 + λ7 ) sin 2β cos 2β ,λH = λ̃3 + 3(λ1 + λ2 − 2λ̃3 ) cos2 β sin2 β + 3(−λ6 + λ7 ) sin 2β cos 2β ,λ± = λ3 + (λ1 + λ2 − 2λ̃3 ) cos2 β sin2 β + (−λ6 + λ7 ) sin 2β cos 2β.

(14)

The wino(bino)-Higgsino-Higgs couplings at MA are given by(′)

ĝu(′) = g̃ (′) sin β ;ĝd = g̃ (′) cos β .

(15)

The threshold corrections to the gauge and Yukawa couplings at MA are neglected becauseof the approximated degeneracy among MA , MH , and MH ± .
6

Measurement
ΓZ
inv
Γhinv
mχ±
relic abundance
LUX (2013)
dSphs γ-ray
Monochromatic Z 0 γ and γγ

central value µ
499 MeV
0.1 MeV103.5 GeV
0.1186see Ref. [14]see Ref. [16]95% upper limits

Error: (σ, τ )1.5 MeV, 0.0+0.51−0.41 MeV, 0.0
0.0 GeV, 1%
0.0031, 10%
see Ref. [14]see Ref. [16]95% upper limits

Distribution
Gaussian
Gaussian
half-Gaussianhalf-Gaussian
PoissonPoissonhard-cut

Ref.
[10]
[11]
[12]
[13]
[15]
[17]
[18]

Table 1: The experimental constraints and the likelihoods. Where it is applicable, thecentral value µ, experimental error σ, and theoretical error τ are given.

2.5

Matching at the electroweak scale

Matching at the electroweak scale is exactly the same as in the original split SUSY framework. We closely follow Ref. [9] to include the threshold corrections to the gauge couplingsat the electroweak scale and to calculate the pole masses for the Higgs boson and the top
quark.

3

Experimental Constraints and Likelihoods

In this section, we describe how to construct the likelihood functions involved with experimental constraints which are used in both MP and PL approaches. For the experimental
constraints considered in this work, we assume either half-Gaussian or Gaussian distribution when the central values µ, experimental errors σ, and theoretical errors τ are available.
Otherwise, we take Poisson distributions.
In Table 1, in the second last column, we show the likelihoods of each experimental
constraint. Here “hard cut” means we apply the 95% upper limits instead of constructing
its likelihood. For the details of our statistical treatment, we refer to Appendix B. In
the following subsections, we give more details of the constraint and likelihood of eachmeasurement.

3.1

Colliders

3.1.1

Invisible decay widths

The invisible decay width of the Z boson was accurately measured by taking the difference
between the total width and the visible width, and is well explained by the three light active
neutrino species of the SM. Any additional invisible decays of the Z boson are strongly
constrained by this data. In the current framework, the additional invisible width comes7

from Z → χ01 χ01 . With the invisible width given in the PDG [10], ΓZinv = 499 ± 1.5 MeV,
we can constrain Z → χ01 χ01 .
If the neutralino mass is below mh /2, the Higgs boson can decay into a pair ofneutralinos, thus contributing to an invisible width of the Higgs boson. From a global fit
using the Higgs-boson data at the 7 and 8 TeV runs of the LHC, the invisible width of the
Higgs boson is constrained to be Γhinv < 0.6 MeV [11] at 1-σ level if all other parameters are
fixed at their SM values. If other parameters are allowed to vary, the Γhinv would have a morerelaxed limit, which is about the same as the bound from the direct search on the invisiblemode of the Higgs boson, which has a branching ratio about 50% [19]. Nevertheless, weuse Γhinv < 0.6 MeV in this work, as shown in Table 1.
3.1.2

Chargino mass

The mass limits on charginos come either from direct search or indirectly from the constraint
set by the non-observation of χ02 states on the gaugino and higgsino MSSM parameters M2
and µ. For generic values of the MSSM parameters, limits from high-energy e+ e− collisions√< s/2.
coincide with the highest value of the mass allowed by phase space, namely mχ± ∼√The combination of the results of the four LEP collaborations of LEP2 running at s up
to 209 GeV yields a lower mass limit of mχ± ≥ 103.5 GeV, which is valid for general MSSM
1models. However, it could be weakened in certain regions of the MSSM parameter space
where the detection efficiencies or production cross sections are suppressed, e.g., when themass difference mχ± − mχ01 becomes too small. Regardlessly, we simply employ the mass1
limit of mχ± ≥ 103.5 in this work. We do not use the LHC constraint since it is more
1model dependent and does not give any bounds when mχ01 >∼ 70 GeV [20]. Furthermore,
for mχ01 < 70 GeV region, the H/Z resonance region (see next subsection) is not sensitive
to this search [21]. Note that the χχ± coannihilation is strongly forbidden by this limit< 90 GeV.
especially when mχ01 ∼To deal with the chargino mass limit without detector simulations, we adopt the half
Gaussian distribution when mχ± < 103.5 GeV to describe the tail of the chargino mass
1
likelihood function. For the likelihood, we assume ∼ 1% theoretical uncertainty. Whenmχ± ≥ 103.5 GeV, we always assume the maximum likelihood.
1

3.2

Relic abundance

The half-Gaussian distribution for relic abundance likelihood in Table 1 suits the wellmotivated moduli decay scenario [22, 23, 24, 25, 26, 27, 28]. In this scenario, the relicabundance can be reproduced by moduli decay after the freeze-out, which is different fromthe usual multi-component DM scenario, in which the total relic abundance is shared amonga few DM candidates, such as the axion. In the moduli decay scenario, all the DM is still8

assumed to be the neutralino, and the DM local density need not be rescaled with respect
to the neutralino fraction as implemented in the multi-component DM scenario, so thatthe DM direct and indirect detection constraints will be stronger.
Very often, the neutralino DM in most of the MSSM parameter space over-produces
the relic abundance, because the annihilation in the early Universe is too inefficient. Generally speaking, by opening the W + W − final state the wino-like neutralino can very efficiently
reduce relic abundance for wino mass up to 3 − 4 TeV, e.g. see Ref. [27, 29, 30]. However,it requires some specific mechanisms for bino-like, Higgsino-like, or mixed neutralinos tofulfill correct relic abundance. Sometimes more than one mechanisms are needed. In mostcases the (non-wino) regions both of correct relic abundance and still allowed by the current
LHC direct searches in our modified split SUSY parameter space are:
• The Z/h resonance region, where the neutralinos annihilate through the resonancewith the Z boson at mχ01 ∼ 45 GeV and Higgs boson at mχ01 ∼ 62.5 GeV. In thisregion, neutralinos are governed mainly by the bino fraction but with a small mixing
with the higgsino fraction.
• The chargino-neutralino coannihilation region, where the µ parameter is usually0closed to gaugino parameters M1 or M2 so that the χ01 , χ±
1 , and χ2 are almost degen±erate. If the masses between χ01 and χ1 or χ01 and χ02 are very close to each other, thenumber densities of the next-to-lightest supersymmetric particle(s) (NLSP(s)) have
only slight Boltzmann suppression with respect to the LSP number density. There00fore, all the interactions among the LSP and NLSP(s), such as χ01 − χ±
1 , χ1 − χ2 and±χ1 − χ02 , play important roles to reduce the relic abundance. Note that χ01 in thisregion shall have nonnegligible fractions of wino or higgsino in order to coannihilate0with χ±1 and χ2 .
• The A/H funnel region, where neutralinos annihilate through the resonance of thepseudoscalar Higgs boson A or the heavy scalar Higgs boson H. In the original
split SUSY framework with MA = MS , because of the large mass of A/H as wellas their large decay width, this mechanism becomes irrelevant. On the other hand,in our modified split SUSY scenario with light MA , this A/H funnel can still play asignificant role in reducing the relic abundance. Nevertheless, we shall see later that
the A/H-funnel for mχ01 > 1 TeV is not efficient enough to reduce the relic abundance
because of the larger A/H decay width.
In split SUSY scenario, because of the very heavy sfermion masses, all the f˜ − χcoannihilation channels have been closed. On the other hand, the chargino annihilation isstill allowed but the chargino mass must be above the LEP limit, mχ± > 103.5 GeV. We
found in our viable parameter space the majority of bino-like neutralino and chargino isalways close to each other (χχ± coannihilation on). Besides, χχ annihilation can have a few
9other choices. Lowering MA to less than 1 TeV, the A/H-funnel region can be important,especially for higgsino and mixed neutralino. For mχ01 < 100 GeV, Z- and h- resonances
can also significantly reduce relic abundance. Finally, the wino-like neutralinos can easily
annihilate into the W + W − final state, which can sufficiently reduce relic abundance as
well.

3.3

LUX: spin-independent cross section

At present the most stringent 90% C.L. limit on the spin-independent component of theelastic scattering cross section comes from LUX [15]. However, it did not take into accountthe systematic uncertainties from nuclear physics and astrophysics, otherwise the constraint
becomes much less straightforward. The astrophysical uncertainties mainly come from ourpoor knowledge of the DM local density and velocity distribution. In order to accountfor the uncertainties of all the astrophysical parameters, we adopt the phase-space density
factor and its associated error bars as computed in Ref. [31]. Nuclear physics uncertainties
enter the systematic uncertainties through the nuclear matrix elements, mainly the pionnucleon sigma term ΣπN and the strange quark content of the nucleon fT s , which promote
the spin-independent cross sections from quark level into nucleon level. In Table 2, we treat
the ΣπN and fT s as nuisance parameters and distribute as Gaussian with central values
and error bars obtained by recent lattice QCD calculations. Regarding the reconstructionof the LUX likelihood including the astrophysical and nuclear uncertainties, we refer toRef. [14] for more detailed explanations.

3.4

Fermi LAT gamma ray

3.4.1

Continuous gamma ray from dSphs

The most luminous gamma-ray source is the Galactic Center (GC) in the Milky Way, but it
is also subject to higher astrophysical backgrounds. Better constraints were obtained fromthe diffuse gamma rays from the dSphs of the Milky Way. They are less luminous and dominated by DM, with little presence of gas or stars. Recently, the Fermi LAT Collaboration
improved significantly the previous sensitivities to DM searches from dSphs [17].
Unlike the published limit from the Fermi LAT collaboration, we only include theeight classical dSphs in our analysis, because the DM halo distribution in the classicaldSphs is measured with a higher accuracy from the velocity dispersion of the luminous
matter [32]. We use the 273 weeks Fermi-LAT data and the Pass-7 photon selection criteria,
as implemented in the FermiTools. The energy range of photons is chosen from 200 MeV
to 500 GeV, and the region-of-interest is adopted to be a 14◦ × 14◦ box centered on eachdSphs. The J-factors are taken from Table-I in Ref. [17].
10In the likelihood analysis, the Fermi-LAT data are binned into 11 energy bins logarithmically spaced between 0.2 and 500 GeV, and we calculate the likelihood map of
Fermi-LAT dSphs on the Ebin-flux plane following the method developed in Ref. [16].
3.4.2

Fermi photon line measured from GC

The experimental signature of monochromatic lines over the continuous spectrum is a clean
signal of DM annihilation. In MSSM, the annihilation of χ01 χ01 into photons induced byloop diagrams also provides stringent constraints on parameter space, especially when χ01
is wino-like and the annihilation cross section is enhanced. However, we do not reconstruct
the likelihood for the Fermi-LAT photon line experiment but simply take the publishedlimit at 5 GeV < mχ01 < 300 GeV. In addition, we adopted the Isothermal profile since it
is known to be more conservative than NFW or Einasto profile [18].

4

Numerical Analysis

In this section, after describing the input parameters over which we perform the scan of
the MSSM, we present the results of our numerical study.
To compute the DM observables such as the relic abundance Ωχ h2 , DM-proton elastic
scattering cross section σpSI , annihilation cross section hσvi at the present time, and branching ratios of DM annihilation, we calculate couplings and mass spectra at the neutralino√mass scale Mχ ≡ µ × M2 , where µ and M2 denote the values at the scale Mχ . First, we
solve the RGEs from MS to MA with those given in Appendix A. For the evolution fromMA to Mχ , which is required when Mχ < MA , we employ the split SUSY RGE code † .
Then we generate the SLHA output and feed it into DarkSUSY 5.1.1 [34] to compute theDM observables. Finally, we use the DM annihilation information from DarkSUSY 5.1.1
to compute the likelihoods for direct and indirect detections by following the method developed in Ref. [14].
We perform the MSSM parameter space scan, including nuisance parameters, by useof MultiNest v2.18 [35] taking 15, 000 living points with a stop tolerance factor of 0.01
and an enlargement factor of 0.8.

4.1

Input Parameters

In this subsection, we provide detailed description of our MSSM input parameters and thenuisance parameters. For the SM input parameters we take the PDG values [10].
†We thank Pietro Slavich for providing us the SplitSuSpect code [33].

11

MSSM Parameterbino masswino massµgluino mass
tan β
MA

Range
10 < |M1 |/TeV < 5
9 × 10−2 < |M2 |/TeV < 5
9 × 10−2 < µ/TeV < 5
1 < |M3 |/TeV < 5
2 < tan β < 62
0.2 TeV < MA < min [10 TeV, MS ]
MA = MS

Prior distribution
Log
Log
Log
Log
Flat
Flat (Scenario A)
Fixed (Scenario B)

Nuisance Parametermh ( GeV)ΣπN ( MeV)fT s

Central value and systematic uncertainty
125.1 ± 2.0 [36, 37]
41.0 ± 6.4 [38]0.043 ± 0.011 [39]Prior distribution
Gaussian
Gaussian
Gaussian−2

Table 2: The prior ranges and distributions of the input parameters over which we performthe scan of the MSSM.

In Table 2, the input parameters, their prior ranges and types of prior distributions
are shown. We take |M1,2 | , |µ| < 5 TeV because it is hard to satisfy the relic abundance
constraint with the LSP heavier than 3 − 4 TeV. We apply the same maximum value forthe gluino mass parameter, which does not affect our results much. The smallest values of|M2 | and µ are chosen by taking into account the LEP limit on the chargino mass. We aretaking |M3 | > 1 TeV because of the LHC limit on the gluino mass. We cover the range oftan β up to 62 and fix the trilinear parameter A0 = µ cot β assuming the no-mixing scenario
in the stop sector. The MSSM input parameters M1,2,3 , µ, and A0 are given at the scale
MS while tan β is the value at the scale MA .
Note that, in this work, we are using mh as an input nuisance parameter and, accordingly, the value of the high SUSY scale MS is an output. Numerically, we solve the
RGEs to find the value of MS which gives the input value of mh . The Higgs boson massmeasurements in the diphoton decay channel now give mh = 125.4 ± 0.4 GeV (ATLAS)[40] and mh = 124.70 ± 0.31 (stat) ± 0.15 (syst) GeV (CMS) [41]. On the other hand, thetheoretical error of Higgs mass is estimated to be around 2 − 3 GeV [42] which is muchlarger than the experimental errors of ∼ 0.4 GeV. Therefore, in this work, we are taking
mh = 125.1 GeV with a Gaussian experimental uncertainty of σ = 2 GeV.
Depending on the relative size of MA to MS , we are taking two scenarios:• scenario A: MA ≤ min[10 TeV, MS ],• scenario B: MA = MS (the same as the original split SUSY).
In the scenario A, we are taking the maximum value of 10 TeV for MA , because the A/H12

funnel (MA ∼ 2 mχ01 ) mechanism becomes ineffective for neutralino annihilation when MA
is beyond 10 TeV. Smaller values of MA may help to obtain the correct Higgs-boson mass
when MS is too large to give mh ∼ 125 GeV in the original split SUSY framework. On
the other hand, the choice of MA in scenario B is the same as in the original split SUSY
framework. We note that the scenario B is a part of scenario A if MS < 10 TeV.
We further need inputs for the pion-nucleon sigma term ΣπN and the strange quark
content of the nucleon fT s . To account for the systematic uncertainties involved in theevaluation of the relevant nuclear matrix elements, we also treat them as nuisance parameters, as mentioned before. The central values and errors are obtained by recent lattice
QCD calculations.

4.2

Numerical Results

We are taking both the PL and MP methods and make comparisons where it is informative. We note that, when we present our result based on the MP method, the systematic
uncertainties of the input parameters are automatically included by utilizing a Gaussianprior distribution, see the nuisance parameters in Table 2. On the other hand, when we areusing the PL method, the systematic uncertainties are added to the likelihood function.
In Fig. 1 we show the scatter plot on the (MA , MS ) plane by varying input parameters
as in Table 2, while requiring mh to be in the 2-σ range: 121.1 GeV < mh < 129.1 GeV.
Different colors represent different tan β ranges. We observe that a larger MS is required> 10, MS becomes almost
for small values of tan β and also as MA decreases. When tan β ∼independent of MA and it lies between ∼ 3 TeV and ∼ 15 TeV. When MA = MS is taken as
in the scenario B, the value of MS is smaller in order to achieve mh ∼ 125 GeV. Therefore,
in the split-SUSY framework with the intermediate Higgses lighter than ∼ 10 TeV, MS isgenerally predicted to be higher especially when tan β is small.
In Fig. 2 we present the probability density functions (PDFs) for marginalized posterior and profiled likelihood in the (|M2 |/µ, |M1 |/µ) plane. All the experimental constraints
in Table 1 are applied and we make comparisons of the scenarios A (left) and B (right). We
represent the bino-like, wino-like, higgsino-like and mixed neutralinos in red, blue, green
and gray, respectively. Precisely, we identify the lightest neutralino χ01 as bino-, wino- or
higgsino-like when the corresponding fraction gb > 0.9, gW > 0.9 or gh > 0.9, respectively.
‡
Otherwise we identify it is the mixed lightest neutralino. Comparing the scenarios A andB, we can see that the difference lies in the bino region. This is because the bino-like χ01
‡

2
2
2
2
The parameters gb,W,h are defined as gb = Zbino, gW = Zwino, and gh = ZH
+ ZH
when χ01 isuddecomposed into bino, wino, and higgsinos as follows

χ01 = Zbino B̃ + Zwino W̃ + ZHu H˜u + ZHd H̃d .

13

1010
109

. <mh /

121 1

GeV

< .

MA <

129 1

MS , TeV

min[

10

2
3
5

< β<
< β<
< β<
β>

tan

tan

3

tan

5

tan

]

10

10

MS (GeV)

108
107
106
105
104
10

3

MA >MS103

MA (GeV)104

Figure 1: The scatter plot on the (MA , MS ) plane varying input parameters as in Table 2
while requiring mh to be in the 2-σ range: 121.1 < mh / GeV < 129.1. The color scheme
are: 2 < tan β < 3 (red circle), 3 < tan β < 5 (blue square), 5 < tan β < 10 (green
triangle), and tan β < 10 (gray cross). In the pink region, MA > MS which is out of ourcurrent consideration.
can satisfy the relic abundance constraint only through Z/h-resonance in the scenario B,
where A/H-funnel does not work because MA = MS >∼ 3 TeV. In fact, the mechanism of
Z/h-resonance requires a small fraction of higgsino but it cannot be too large because of
the constraint from the Fermi dSphs gamma ray measurement. In particular, we find thatthe higgsino composition is between 0.06 to 0.1 in the h resonance region which leads to
the ratio |M1 |/µ ∼ 0.4.
Furthermore, we find that the chargino-neutralino coannihilation working in reducing the relic abundance in both scenarios. Being different from the original split SUSY
framework (scenario B), one can obtain the correct relic abundance in scenario A without
resorting to the coannihilation mechanism thanks to the intermediate Higgses A and H. To
14

101

bino, gb >0.9
wino, gW >0.9
higgsino, gh >0.9
mixed

100

|

1|

100

|

1

|

M /µ

101

102

bino, gb >0.9
wino, gW >0.9
higgsino, gh >0.9
mixed

M /µ

102

10-1

10-1

-2

10 -2
10

MA <
10-1

100

M /µ

|

MS , TeV

min[

10

101

MA = MS

]

-2

10 -2
10

102

10-1

|
2

100

M /µ

|

101

102

2|

Figure 2: The marginalized posterior (contours) and the profiled likelihood (scatter points) PDFs
in the (|M2 |/µ, |M1 |/µ) plane for the scenarios A (left) and B (right). All the three parameters
are the values at the scale MS . The inner (outer) contour corresponds to 2σ (3σ) credible region(CR) but the scatter points represent the 2σ profile likelihood region. The regions with gb > 0.9(bino-like), gW > 0.9 (wino-like), and gh > 0.9 (higgsino-like) are colored in red, blue, and green.
The gray region is for the mixed χ01 , see the text.

MA = MS
103±mχ (GeV)103±mχ (GeV)MA <min[10 TeV,MS ]δχ <5.99

δχ <5.99

2

102

102

mχ (GeV)

2

102

103

102

mχ (GeV) 10

3

Figure 3: The points with δχ2 < 5.99 scattered on the (mχ01 , mχ± ) plane for the scenario
A (left) and B (right).
address this point, we show in Fig. 3 the points with δχ2 < 5.99 on the (mχ01 , mχ± ) plane15

for the scenario A (left) and B (right). In addition to the Z/h-resonance regions aroundmχ01 ∼ 50 , 60 GeV and the chargino-neutralino coannihilation region along the mχ01 = mχ±line, we observe there are more points appearing in the scenarios A (left panel) due to the
A/H-funnel. We find that the A/H-funnel region disappears when mχ01 > 1 TeV, because
the decay widths of A and H become too large and the Breit-Wigner resonance effect is> 2 TeV.
not strong enough to reduce the relic abundance when MH,A ∼In Fig. 4, we show the marginalized 2D posterior 2- and 3-σ credible regions (CRs)for the scenario A (left) and B (right) in the (mχ01 , hσvi) plane. We also show the PL 2-σregion (scattered points) for the bino-like (red) and mixed (gray) χ01 in the upper frames
and the wino-like (blue) and higgsino-like (gray) χ01 in the lower frames. Here hσvi denotes
the annihilation cross section at the present time which is relevant to the DM indirect
detections and through which one may easily identify different mechanisms for the relicabundance.

When mχ01 < 100 GeV, via the Z/h resonances, the marginalized posterior CRs arelocated at the bino-like neutralino region with a small amount of higgsino component inboth scenarios (see the upper frames). Although the Z/h-resonance channels have very
good likelihoods, they only fall into the 3σ (99.73%) CR owing to the small prior volume
effect. The similar effect happens for the bino-like χ01 when mχ01 > 100 GeV and the correct
relic abundance is obtained by the χχ± coannihilation. The fact that more parameter
space survives in the scenario A (left) than scenario B (right) is due to the A/H-funnel.
Nevertheless, most of the additional parameter space is a result of the mixture mechanism
between A/H-funnel and coannihilation. In the lower frames, we observe that the 2σ CR
has the wino-like branch (blue) with the higher hσvi than the higgsino-like one (green).
For the wino-like branch, the relic abundance is mainly reduced by the wino-like DM> 3 TeV, the wino DM cannot giveannihilation into W + W − pairs. However when mχ01 ∼the correct relic abundance as is well known. This mass limit can be slightly extended ifcoannihilation is taken into account. Since the wino DM have higher annihilation crosssections, the indirect detection constraint is stringent. Indeed, the lower bound for the
wino-like neutralino mass is about 300 GeV from the Fermi dSphs gamma ray constraints.
Incidentally, the lower bound for the higgsino-like neutralino mass is about 100 GeV, set
by the LEP limit of mχ± > 103.5. We further see there is no particular lower bound for1
the bino-like or mixed neutralino, as seen from the upper frames.
Finally, in Fig. 5 we show the marginalized 2D posterior PDF 2σ and 3σ contours inthe (mχ01 , σpSI ) plane. The red solid line denotes the recent LUX result, the black dashedline the XENON1T projected sensitivity, and the blue dash-dotted line the LZ projected
sensitivity [43]. The orange dashed line represents the approximate line below which theDM signal becomes hardly distinguishable from the signals from the coherent scattering of
the 8 B solar neutrinos, atmospheric neutrinos and diffuse supernova neutrinos with nuclei.
We observe that a part of 2-σ CR is below the LZ projected sensitivity. We can see that,16

10-23MA <MS , TeV

min[

10

10-23

10-25

10-25

1
3

3

1

(cm s− )

today

10-28

10-23

102

MA <

mχ (GeV)MS , TeV

min[

10

102

10-23

MA MS

10-24

10-25

10-25

3

1

(cm s− )

10-24



10-27

10-29

10-26

σv

today

10-26

10-28

mχ (GeV)bino, gb >0.9
mixed
103

=

3

1

10-29

]

(cm s− )
today

10-27
10-28

bino, gb >0.9
mixed
103

σv


10-26

σv

10-27





σv

today

10-26

10-29

-24

(cm s− )

10

10

MA MS
=

]

-24

10-28

wino, gW >0.9
higgsino, gh >0.9102mχ (GeV)

10-27

10-29

103

wino, gW >0.9
higgsino, gh >0.9102mχ (GeV) 10

3

Figure 4: Marginalized posterior PDF (contours) and profiled likelihood PDF (scatter points) inthe (mχ0 , hσvi) plane for the scenario A (left) and B (right). The inner (outer) contours bounded1
the 2(3)-σ CR. All the scatter points superimposing on the contours agree with likelihood in thecriteria δχ2 < 5.99. The red dots, blue squares, green stars, and gray triangle are for the bino-like,wino-like, higgsino-like, and mixed neutralino, respectively.

in the 2-σ CRs, there is no significant difference between the scenarios A and B. The 3-σ
CRs are slightly different in the lower σpSI region. Moreover, in both scenarios, the future
7-tons experiments, LZ, can set a lower limit on the neutralino DM at mχ01 > 100 GeV.

17

10-6

Marginal Posterior
MA <min[MS ,10 TeV]

-7

10-8

SI

σp (pb)(
LUX

2 01 3

10-8

)

10-9

(
LUX

2013

)

10-9

ted)
rojec
1T (p

N
ENO

10-10

X

L

10-11
10-12

10

Marginal Posterior, MA = MS

-7

σpSI (pb)

10

10-6

te
ojec
Z (pr

10-10

d)

mχ (GeV)

X

10-12

103

te
ojec
T (pr

d)

te
rojec
LZ (p

10-11

g
terin
scat
rent
e
h
o
νc

102

N1
ENO

d)

g
terin
scat
rent
e
h
o
νc

102

mχ (GeV) 10

3

Figure 5: The marginal posterior for the 95% and 99.73% CRs in (mχ01 , σpSI ) plane. The left
(right) panel is for the scenario A (B).

5

Discussion

In this work, we have studied a “modified split SUSY” scenario, characterized by two
separate scales – the SUSY-breaking scale MS and the heavy Higgs-boson mass scale MA .
This is different from the split SUSY scenario, in which the scale MA is also set at MS . The
current scenario is motivated by (i) the absence of direct SUSY signals from the searches ofscalar quarks up to a few TeV, (ii) the observed Higgs boson is somewhat on the heavy sidewhich needs a large radiative correction to the tree-level mass from heavy stops, and (iii)
absence of signals from heavy Higgs bosons A/H and H ± which can be as light as a few
hundred GeV. Therefore, the choice of MA need not be as large as MS . We have studied
two scenarios: (i) MA ≤ min(MS , 10 Tev) and (ii) MA = MS (the same as split SUSY).
Because of two distinct scales MS and MA the running of the soft parameters andcouplings are separated in two steps. We start with the set of RGEs given in appendix A
to run from MS down to MA and perform the matching at the scale MA . Then run from√MA down to the electro-weakino scale Mχ ≡ µ × M2 with the set of RGEs of split SUSY.
Because of this two-step RGEs the predictions for DM observables and the Higgs boson
mass are more reliable than just a single-step RGE.
We have scanned the MSSM parameter space characterized by the two scales: MS
and MA subjected to many existing experimental constraints: invisible widths of the Zboson and the Higgs boson, the chargino mass limit, relic abundance of the LSP, spinindependent cross sections from direct detection, and the gamma-ray data from indirect
detection. We found interesting survival regions of parameter space with features of either
18

chargino-neutralino coannihilation, the A/H funnel, or wino-like. These regions survive
because of the large enough annihilation to reduce the relic abundance to the observed
values, as well as give a large enough Higgs boson mass to fit to the observed value.
Finally, the survived parameter space can be further scrutinized by near future direct
detection experiments such as XENON1T and LZ.
We offer a few important comments as follows.
1. We used the Higgs boson mass in the range range 121.1 < mh < 129.1 GeV to searchfor suitable MS . Since mh is on the rather heavy side, it requires a large radiative
correction to the tree-level mass. This can be achieved by a large stop mass and/or
large mixing in the stop sector. Since the radiative correction is proportional to somepowers of tan β, a smaller tan β requires then a larger MS in order to achieve a large> 105−6 GeV for tan β < 3. For large enough tan β theenough mh . Typically, MS ∼values of MS is more or less independent of MA .
2. On the other hand, if we set MA = MS as we do in scenario A, the allowed MS israther short from about 103 − 104 GeV with large tan β (see Fig. 1).
3. An interesting region that satisfies the relic abundance constraint is characterized bynearly degenerate mass among the first two neutralinos and the lightest chargino,indicated by M2 /µ ≈ M1 /µ ≈ 1. The increased effective annihilation cross section
can help reducing the relic abundance.
4. Another interesting region is the Z/h resonance region (mχ01 ∼ 50 − 60 GeV), though
it is relatively fine-tuned region because of the narrow width of the Z boson and the
Higgs boson.
5. Yet, another interesting survival region is the A/H funnel region. If mχ01 falls aroundthe vicinity of mA/H /2 the resonance effect is strong, provided that the width is not< 1 TeV, that is MA/H ∼< 2 TeV. In scenario
too large. This can be achieved for mχ01 ∼B, where MS = MA , large values of MS then cannot be accepted because the A/Hfunnel is not working efficiently. However, in scenario A, where MA < MS , the A/Hfunnel can be very effective in reducing the relic abundance, thus more parameter
space is allowed.
6. Both wino-like and higgsino-like LSPs have large annihilation cross sections. The
allowed mass for mχ01 ranges from about 300 GeV to 3 TeV for wino-like LSP whilefrom about 100 GeV to 2 TeV for higgsino-like LSP.
7. The current allowed parameter space has a large region below the current LUX limit−9
σpSI <∼ 10 pb. Although the future XENON1T can improve the limit by an orderof magnitude, there is still a sizable region below the XENON1T sensitivity. Yet,19

there still exist some allowed regions even with the future 7-tons size direct detection
experiment LZ. Therefore, this modified split SUSY scenario is hard to be excludedin the future.
8. We have used both the methods of profile likelihood and marginal posterior. Thoughthese two statistical approaches have very different methodology, the resulting 2- and3-σ regions are quite consistent, as shown in the figures.

Acknowledgment
R.H. is grateful to Carlos. E.M. Wagner, Stephen P. Martin and Alessandro Strumia for
useful discussions. K.C. was supported by the National Science Council of Taiwan underGrants No. NSC 102-2112-M-007-015-MY3. J.S.L. was supported by the National Research Foundation of Korea (NRF) grant (No. 2013R1A2A2A01015406) and by Chonnam
National University, 2012. R.H. and Y.S.T. were supported by World Premier International
Research Center Initiative (WPI), MEXT, Japan.

Appendix
A

RGEs from MS to MA

Here we present the one-loop RGEs governing the running of couplings from the high SUSY
scale MS to the intermediate Higgs mass scale MA .
We write the RGE for each coupling gi present in the theory, in the MS or DR scheme(the same up to one-loop level), asdgi
β1 (gi )=.
d ln Q
(4π)2

(A.1)

The relevant coupling constants gi include the gauge couplings (gs , g, g ′), the gaugino couplings (g̃d′ , g̃u′ , g̃d , g̃u ), the third-generation Yukawa couplings (yt , yb, yτ ), and the Higgs
quartic (λ1 , λ2 , λ3 , λ4 , λ5 , λ6 , λ7 ).
At one loop the β functions of gauge couplings below the SUSY scale are given by
β1 (gs ) = −5gs3 ,β1 (g) = −1g 3 ,β1 (g ′ ) =

23 ′3
g .
3

(A.2)

The β functions of gauge couplings defined by the fermion-scalar-gaugino interaction below
the SUSY scale are given by
β1 (g̃u ) = g̃u



3
11
1
3
33
− g 2 − g ′2 + g̃u2 + g̃d2 + g̃u′2 + 3yt2
4
4
4
2
4
20



(A.3)



33
3
11
1
3
β1 (g̃d ) = g̃d − g 2 − g ′2 + g̃d2 + g̃u2 + g̃d′2 + 3yb2 + yτ2
4
4
4
2
4


3
9
5
1
9
2
′2
2
′2
′2
2
′
′
β1 (g̃u ) = g̃u − g − g + g̃u + g̃u + g̃d + 3yt
4
4
4
4
2


3
9
5
1
9
β1 (g̃d′ ) = g̃d′ − g 2 − g ′2 + g̃d2 + g̃d′2 + g̃u′2 + 3yb2 + yτ2
4
4
4
4
2



(A.4)
(A.5)
(A.6)

The β functions of 3rd generation Yukawa interactions below the SUSY scale are
given by




9
17
3
1
9 2 1 2
yt + yb − 8gs2 − g 2 − g ′2 + g̃u2 + g̃u′2
β1 (yt ) = yt
2
2
4
12
2
2


9
5
3
1 ′2
9 2 1 2222
′2
2
y + y + yτ − 8gs − g − g + g̃d + g̃dβ1 (yb ) = yb
2 b 2 t
4
12
2
2


9
15
3
1
5 2
y + 3yb2 − g 2 − g ′2 + g̃d2 + g̃d′2
β1 (yτ ) = yτ
2 τ
4
4
2
2

(A.7)
(A.8)
(A.9)

The β functions of Higgs quartic couplings defined by Haber and Hempfling [44] aregiven byβ1 (λ1 ) =

24λ21 + 4λ23 + 4(λ3 + λ4 )2 + 4λ25 + 48λ26

+

 

3 4
1
2g + (g 2 + g ′2 )2 − 2g̃d4 + (g̃d2 + g̃d′2 )2 − 2Nc yb4 − 2yτ482
+4λ1 γd ,β1 (λ2 ) =

24λ22 + 4λ23 + 4(λ3 + λ4 )2 + 4λ25 + 48λ27

+

 

1
3 4
2g + (g 2 + g ′2 )2 − 2g̃u4 + (g̃u2 + g̃u′2 )2 − 2Nc yt482
+4λ2 γu ,β1 (λ3 ) =



 

1
3 4
2g + (g 2 − g ′2 )2 − 2g̃u2 g̃d2 + (g̃u2 − g̃u′2 )(g̃d2 − g̃d′2 ) − 2Nc yb2yt2
82
+λ3 (2γd + 2γu ) ,(A.12)

4λ4 (λ1 + λ2 + 4λ3 + 2λ4 ) + 16λ25 + 20λ26 + 20λ27 + 8λ6 λ73+ g 2 g ′2 + 2g̃d2 g̃u2 − g̃d2 g̃u′2 − g̃d′2g̃u2 + 2Nc yb2yt22+λ4 (2γd + 2γu ) ,β1 (λ5 ) =(A.11)(λ1 + λ2 )(3λ3 + λ4 ) + 8λ23 + 4λ24 + 4λ25 + 8λ26 + 8λ27 + 32λ6 λ7+β1 (λ4 ) =(A.10)



4λ5 (λ1 + λ2 + 4λ3 + 6λ4 ) + 20(λ26 + λ27 ) + 8λ6 λ7
21(A.13)+λ5 (2γd + 2γu ) ,β1 (λ6 ) =(A.14)

4λ6 (6λ1 + 3λ3 + 4λ4 + 5λ5 ) + 4λ7 (3λ3 + 2λ4 + λ5 )
+λ6 (3γd + γu ) ,β1 (λ7 ) =(A.15)

4λ7 (6λ2 + 3λ3 + 4λ4 + 5λ5 ) + 4λ6 (3λ3 + 2λ4 + λ5 )
+λ7 (γd + 3γu ) ,(A.16)where13γd = Nc yb2 + yτ2 − (3g 2 + g ′2 ) + (3g̃d2 + g̃d′2 ),
4
2
1 2
3 2
′2
2
γu = Nc yt − (3g + g ) + (3g̃u + g̃u′2 ).
4
2(A.17)(A.18)The β functions of gaugino mass parameters and the SUSY µ term are given byβ1 (M3 ) = −18gs2 M3
β1 (M2 ) =β1 (M1 ) =β1 (µ) =B(A.19)

2(−12g + g̃u2 + g̃d2 )M2(g̃u′2 + g̃d′2 )M1
(A.20)


9
3
3
3
1
1
− g 2 + g̃u2 + g̃d2 − g ′2 + g̃u′2 + g̃d′2 µ
2
4
4
2
4
4

(A.21)(A.22)The statistical framework

To calculate the probability of MSSM parameter given the experimental data, one can
employ Bayes’s’ theorem to compute the posterior probability density function,p(θ, φ|d) =L(d|θ, φ)π(θ, φ)
.
Z(d)

(B.1)

Here, we denote the MSSM parameters and DM direct detection nuisance parameters as θand φ, respectively. The likelihood L(d|θ, φ) is the probability of obtaining experimental
data for observables given the MSSM parameters. The prior knowledge of MSSM parameter
space is presented as prior distribution π(θ, φ). Our MSSM prior ranges and distributions
are tabulated in Table 2. Finally, the evidence of the model in the denominator can be
merely a normalization factor, because we are not interested in model comparison.
The Bayesian approach allows us to simply get ride of the unwanted parameters byusing marginalization. For example, if there would be n free model parameters, ri=1,...,n,but one is only interesting in the two-dimensional figure (r1 , r2 ), the marginalization canbe written asZnp(r1 , r2 |d) =p(r1 , ..., rn |d)
22

Y

i=3

dri .

(B.2)

An analogous procedure can be performed with the observables. One should keep in mind
that a poor prior knowledge or likelihood function can raise a volume effect. In otherwords, some regions gain more weight from higher prior probability but fine-tuning regions
such as resonance regions for relic abundance likelihood only have lower prior probability.
Although this is the feature of Bayesian statistics, in order to manifest these fine-tuning
regions, we still present both profile likelihood and marginal posterior method at the sametime.
In Bayesian statistics, a credible region (CR) is the smallest region, R, in the best
agreement with experiments bounded with the fraction ̺ of the total probabilities. For
example at MSSM (M1 , M2 ) plane, the ̺ credible region can be written as
R

R p(M1 , M2 |d)dM1 dM2normalization

= ̺,

(B.3)

where the normalization in the denominator is the total probability with R → ∞. In
this paper, we have shown ̺ = 0.95 and ̺ = 0.9973 corresponding to 2σ and 3σ credible
region. As the comparison, we also present the scatter points with selected criteria δχ2 =−2 ln L/Lmax ≤ 5.99. This criteria is 2σ confidence region of Profile Likelihood method in
2 degrees of freedom. We can see from our result that most of 2σ confidence region of PL
method is similar to the 3 σ credible region in MP method. We would like to note thatthe total profile likelihood here takes the likelihoods including the nuisance parameters
distribution, which is the prior distribution in marginal posterior method.

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