††institutetext: 1 Department of Physics and Engineering, ZhengZhou
University, ZhengZhou 450001, China
2 Institute of Theoretical Physics, College of Applied Science, Beijing
University of Technology, Beijing 100124, China
3 State Key Laboratory of Theoretical Physics, Institute of Theoretical
Physics, Academia Sinica, Beijing 100190, China

# Singlet extension of the MSSM as a solution to the small cosmological scale
anomalies

Fei Wang1, Wenyu Wang2, Jin Min Yang3, Sihong Zhou2<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this work we show that the general singlet extension of the MSSM can
naturally provide a self-interacting singlino dark matter to solve the small
cosmological scale anomalies (a large Sommerfeld enhancement factor can also
be obtained). However, we find that the NMSSM (the singlet extension of the
MSSM with $Z_{3}$ symmetry) cannot achieve this due to the restricted
parameter space. In our analysis we introduce the concept of symmetric and
antisymmetric viscosity cross sections to deal with the non-relativistic
Majorana-fermion dark matter scattering.

##  1 Introduction

As the standard model of the Big Bang cosmology, the $\Lambda$CDM model can
account for most observations of the Universe. A crucial ingredient of this
model is the existence of cold dark matter (CDM), which, with a proper
cosmological constant, can successfully predict the large scale structure of
the Universe. However, the predictions on small scale structures seem not so
successful and some anomalies exist: Bringmann:2013vra  1) missing satellites
– There should be many more dwarf-sized subhalos (satellites) than observed in
the DM halo of the Milky Way (MW). And the observed galaxy luminosity and Hi-
mass functions beyond the MW show shallower faint-end slopes than predicted.
Klypin:1999uc  2) cusp vs core – It seems to have cored inner density profiles
in the low surface brightness and dwarf galaxies, this is at odds with CDM
cusps predicted by simulations deNaray:2011hy . 3) too big to fail – In
comparison with the densest and most massive satellites found in simulations,
the observed brightest satellites of the MW attain their maximum circular
velocity at a too large radii. BoylanKolchin:2011de .

There are various ways to solve these small scale problems, such as the
nonthermal production of warm dark matter Lin:2000qq  or the baryon feedback
in the galaxies to make small halos dark Bullock:2010uy . Also, recently the
author of Tulin:2012wi ; Ko:2014nha  proposed another self-interacting Dirac-
fermion DM scenario with a light mediator ($<$$\sim$ 100 MeV) to solve these
small scale anomalies. With a light force carrier, the dark matter scattering
cross section could have a non-trivial velocity dependence. All of the small
scales (the dwarf size, the Milky Way size as well as the galaxy cluster size)
can have appropriate cross sections, thus leaving enough parameter space for
the mass of DM, the force carrier and the coupling strength. Besides, the
authors also showed that the DM self-interactions can be correlated with the
effect of Sommerfeld enhancement in DM annihilation which is being probed
through indirect detection experiments.

This self-interacting DM scenario perfectly explain the anomalies in the
simulations of small scale structures. Therefore, it is necessary to check if
such a scenario can be realized in popular new physics theories like low
energy supersymmetry (SUSY). In SUSY the better known DM candidate is the
Majorana-type neutralino, which is composed of bino, wino and higgsinos.
Apparently, if the neutralino can have self-interactions through a light force
carrier, such a carrier can not have sizable standard model (SM) interaction
due to the stringent constraints from both collider and DM detection
experiments. Thus, this light force carrier should be composed mainly of a
singlet with respect to the SM gauge groups, which cannot be found in the
minimal supersymmetric standard model (MSSM).

Fortunately, there are various singlet extensions of the MSSM, among which the
next-to-minimal supersymmetric standard model (NMSSM) seems most attractive
fayet ; NMSSM . In the NMSSM, all the parameters in the superpotential are
dimensionless and electroweak symmetry breaking is triggered by the TeV-scale
soft SUSY breaking terms. The SUSY preserving $\mu$ term in the superpotential
of the MSSM is generated by the vacuum expectation values (VEV) of a singlet
superfield $S$. It is shown that in the NMSSM a light singlet scalar at
several GeV can survive the DM detection limits and the collider constraints
NMSSM2 . On the other hand, if we do not impose any discrete symmetry (in the
NMSSM it is $Z_{3}$) and allow for all possible interactions of the singlet
field, then we have the general singlet extension of the MSSM (GMSSM), (more
detail can be seen in Kaminska:2014wia ) which was used to explain the PAMELA
anomaly Hooper:2009gm ; Wang:2009rj . Compared with the NMSSM, the GMSSM has a
larger parameter space. In the GMSSM, the singlet can form a dark sector in
case of a very small $\lambda$. The singlino-like dark matter can annihilate
into the light singlet-like scalar, which can give the correct DM relic
density and a proper Sommerfeld enhancement factor. In this model, the singlet
scalar can be even lighter than in the NMSSM due to a larger parameter space.
So it is intriguing to check if such a singlet scalar in the NMSSM or GMSSM
can serve as the light force carrier mediated in the DM self-interactions,
which is the aim of this work.

In this work we focus on the NMSSM and GMSSM to check if the self-interacting
DM scenario can be realized. In our study we will take into account the
constraints from DM relic density, the DM direct detection experiments as well
as the proper non-relativistic scattering cross sections between DM. We
organize the content as follows. In Sec. 2, we will discuss the general DM
interactions. In Sec. 3 and Sec. 4, we will respectively check the NMSSM and
GMSSM to figure out the possibility of realizing the self-interacting DM
scenario to solve the small cosmological scale anomalies. Sec. 5 contains our
conclusions.

##  2 Dark matter interactions

As mentioned in the introduction, in order to explain both the large scale and
small scale structures of the Universe, we can introduce the self-interacting
DM scenario. The interactions between DM and SM particles can be summarized as
(shown in Fig. 1):

  1. 1.

The annihilation to the SM particles (the left diagram of Fig. 1), whose cross
section at high energy determines the relic density of dark matter and whose
cross section at low energy is being probed by the indirect detection
experiments like PAMELA pamela  and AMS02 AMS-2013 .

  2. 2.

The elastic scattering off the SM particles (the middle diagram of Fig. 1),
which is being probed by various direct detection experiments like CDMS,cdms2
XENON XENON100  and LUX lux .

  3. 3.

The non-relativistic self-scattering (the right diagram of Fig. 1), where
$l=0$ in the partial wave expansion gives the Sommerfeld enhancement relative
to the relativistic annihilation while $l\raise
1.29167pt\hbox{$<$\kern-7.5pt\raise-4.73611pt\hbox{$\sim$}}25$ can account for
the anomalies in the small cosmological scales.

Figure 1: Dark matter interactions: annihilation to SM particles in the left
diagram, scattering off quarks in the middle diagram, and self-scattering in
the right diagram.

These interactions have subtle correlations among each other. A complete study
on DM properties needs to combine all these interactions from various
experiments.

We now briefly review the calculation of DM relic density and the DM-nucleon
cross sections. When the early universe was cooling down, the equilibrium
between DM and SM particles in the thermal bath can no longer be maintained.
The DM will annihilate to SM particles until the annihilation rate falls below
the expansion rate of the universe. Thus the key point in the DM relic density
calculation is the annihilation rate of DM. In our following calculation, we
use the standard method susy-dm-review  to calculate the relativistic
annihilation cross section and the degrees of freedom at the freezing out
temperature.

It is sufficient to consider only the spin independent (SI) elastic cross
sections between DM (denoted by $\chi$) and nucleon (denoted by $f_{p}$ for
proton and $f_{n}$ for neutron susy-dm-review  ) because of its high
sensitivity in current DM direct detection experiments. These interactions are
dominantly induced by scalar exchange processes at tree level, as shown in the
middle diagram of Fig.(1). Note that the vector boson exchange interactions
are also possible and we concentrate on the scalar interactions only. For
moderately light scalar bosons, $f_{p}$ is approximated by (similarly for
$f_{n}$)

$\begin{split}f_{p}\simeq\sum_{q=u,d,s}\frac{f_{q}^{\phi}}{m_{q}}m_{p}f_{T_{q}}^{(p)}+\frac{2}{27}f_{T_{G}}\sum_{q=c,b,t}\frac{f_{q}^{\phi}}{m_{q}}m_{p},\end{split}$
(1)

where $f_{Tq}^{(p)}$ denotes the fraction of $m_{p}$ (proton mass) from the
light quark $q$, $f_{T_{G}}=1-\sum_{u,d,s}f_{T_{q}}^{(p)}$ is the heavy quark
contribution through gluon exchange, and $f_{q}^{\phi}$ is the coefficient of
the effective scalar operator given by

$f_{q}^{\phi}=\frac{C_{\phi\chi\chi}C_{\phi qq}}{m_{\phi}^{2}},$ (2)

with $C$ being the corresponding interaction vertex and $m_{\phi}$ the mass of
the exchanging particle. The DM-nucleus scattering rate is then given by

$\sigma^{SI}=\frac{4}{\pi}\left(\frac{m_{\chi}m_{T}}{m_{\chi}+m_{T}}\right)^{2}\times\bigl{(}n_{p}f_{p}+n_{n}f_{n}\bigr{)}^{2},$
(3)

where $m_{\chi}$ is the DM mass, $m_{T}$ is the mass of target nucleus, and
$n_{p}(n_{n})$ is the number of protons (neutrons) in the target nucleus. We
can see the dependence of SI cross section on the mass of exchanging particle
$\sigma^{SI}\propto 1/m_{\phi}^{4}$, which is very important in our following
discussions.

Note that the annihilation cross section in relic density is calculated at
high energy. To explain the DM indirect detection results (such as the PAMELA
result) and the small cosmological scale anomalies, the DM scattering at low
energies are needed. We will discuss their relations in the following.

###  2.1 The Sommerfeld enhancement effect

The Sommerfeld enhancement effect in dark matter annihilation is proposed to
explain some DM indirect detection results, such as the positron excess
observed by PAMELA pamela  or AMS AMS-2013 . The explanation of positron
excess requires a very large DM annihilation rate which on the other hand can
not explain the DM relic abundance if the DM is produced thermally in the
early universe. The Sommerfeld effect can greatly enhance the annihilation
rate when the velocity of DM is much smaller than the velocity at freeze-out
temperature. Note that it is shown that the positron excess can be more
naturally explained by the pulsar wind and, further, the gamma-rays produced
by inverse Compton scattering on interstellar radiation field of electrons and
positrons produced by dark matter basically exclude the dark matter
interpretation Cholis:2013psa .

The Sommerfeld enhancement is a common effect in non-relativistic quantum
mechanics (QM). When DM has self-interactions, it will lead to an effective
potential $V(r)$ in the non-relativistic limit. Thus the Schrödinger equation
of DM particle can be written as

$-\frac{1}{2m_{\chi}}\nabla^{2}\psi_{k}+V(r)\psi_{k}=\frac{k^{2}}{2m_{\chi}}\psi_{k},$
(4)

where $k$ is the relative momentum of the DM particle. A non-relativistic DM
moves and annihilates around the origin, namely near $r=0$. Therefore, the
only effect of the potential $V(r)$ is the change of the modulus for the wave-
function at the origin comparing to that without $V(r)$. Then, the
annihilation cross section is enhanced to

$\sigma=\sigma_{0}S_{k},$ (5)

where the Sommerfeld enhancement factor $S$ is given by

$S_{k}=\frac{|\psi_{k}(0)|^{2}}{|\psi^{(0)}_{k}(0)|^{2}}=|\psi_{k}(0)|^{2}$
(6)

with $\psi_{k}(0)$ and $\psi_{k}^{(0)}(0)$ are respectively the perturbed
(unperturbed) wave function of DM.

We can find a solution for the Schrödinger equation in non-relativistic QM. As
$R_{kl}(r)\sim r^{l}$ as $r\to 0$ and the $l>0$ part will not change the
annihilating wave function at the origin, we need only to calculate the $l=0$
part of the radial wave $R_{kl}(r)$. We can numerically solve the Schrödinger
equation with the boundary condition

$\chi(r)\equiv rR_{kl}(r)\to\sin(kr+\delta)~{}~{}{\rm as}~{}~{}r\to\infty.$
(7)

In case of a Yukawa-type effective potential induced by scalar exchange

$V(r)=-\alpha_{\chi}\frac{e^{-m_{\phi}r}}{r},$ (8)

with $\alpha_{\chi}={|C_{\phi\chi\chi}|^{2}}/{4\pi}$, the Sommerfeld
enhancement factor is given by

$S_{k}=\left|\frac{\frac{d\chi}{dr}(0)}{k}\right|^{2}.$ (9)

Figure 2: The Sommerfeld enhancement factor at different velocities with the
coupling strength $\alpha_{\chi}=0.01$. Here $v=10\rm km/s,200\rm km/s,1000\rm
km/s$ correspond to the characteristic speed of dwarf halos, the Milky Way and
clusters, respectively.

Though there is no analytical solution for the Schrödinger equation with the
Yukawa-type potential, there are three distinguishable regions for the
Sommerfeld enhancement, depending on the value of $m_{\chi}/m_{\phi}$. If the
mediator scalar mass is comparable to the DM mass, $S_{k}$ is negligible at
all scales. However, if the mediator scalar mass is much smaller than the DM
mass, the enhancement factor becomes independent of $m_{\chi}/m_{\phi}$ and
corresponds to the Coulomb limit. In this limit, the Sommerfeld enhancement
factor is essentially given by $S_{k}\sim\pi\alpha_{\chi}/v$. In the resonance
regions where $m_{\chi}/m_{\phi}\simeq\pi^{2}n^{2}/6\alpha_{\chi}$ with
$n=1,2,3,\cdots$, the DM annihilation cross section can be enhanced. The
enhancement factor is approximately given by
$S_{k}\sim\pi^{2}\alpha_{\chi}m_{\phi}/(6m_{\chi}v^{2})$, which is very
sensitive to the DM velocity. In our numerical calculation we reproduced the
results of Tulin:2013teo , as shown in Fig. 2. This figure shows the
Sommerfeld enhancement factor at velocity $10,~{}200,~{}1000$ km/s,
corresponding to the characteristic velocities of the halo of the dwarf, the
Milky Way and the cluster (these are the three small cosmological scales at
which the $\Lambda$CDM model seems not work well).

###  2.2 The transfer and viscosity cross sections of self-interacting dark
matter

As pointed in Tulin:2013teo ; Laha:2013gva , in order to solve the small scale
simulation anomalies, self-interaction between the DM is necessary. In the
non-relativistic limit, the scattering between DM can be described by QM. The
most recent simulations have shown that $\sigma/m_{\chi}\sim 0.1-10\,\,{\rm
cm^{2}/g}$ on dwarf scales (the characteristic velocity is 10 km/s) is
sufficient to solve the core-vs-cusp and too-big-to-fail problems, while the
Milky Way (the characteristic velocity is 200 km/s) and cluster scales (the
characteristic velocity is 1000 km/s) require $\sigma/m_{\chi}\sim
0.1-1\,\,{\rm cm^{2}/g}$. It appears that all the data may be accounted for
with a constant scattering cross section around $\sigma/m_{\chi}\sim
0.5\,\,{\rm cm^{2}/g}$. On the other hand, the self-interacting DM models
generically predict a velocity-dependent scattering cross section over a wide
range of parameter space.

The numerical input for the simulation of small scales is the differential
cross section $d\sigma/d\Omega$ as a function of the DM relative velocity $v$.
The standard cross section $\sigma=\int d\Omega(d\sigma/d\Omega)$ receives a
strong enhancement in the forward-scattering limit ($\cos\theta\to 1$), which
does not change the DM particle trajectories. Thus two additional cross
sections are defined to parameterize transport krstic:1999 , the transfer
cross section $\sigma_{T}$ and the viscosity (or conductivity) cross section
$\sigma_{V}$:

$\sigma_{T}=\int
d\Omega\,(1-\cos\theta)\,\frac{d\sigma}{d\Omega}\,,\qquad\sigma_{V}=\int
d\Omega\,\sin^{2}\theta\,\frac{d\sigma}{d\Omega}\,.$ (10)

The transfer cross section is weighted by $(1-\cos\theta)$ which is the
fractional longitudinal momentum transfer while the viscosity cross section is
weighted by $\sin^{2}\theta$ which is the energy transfer in the transverse
direction. The transfer cross section was used in the DM literature to
regulate the forward-scattering divergence in case of Dirac-fermion DM
candidate, while the viscosity cross section was used in case of Majorana-
fermion DM candidate. This is because both forward and backward scatterings
diverge, corresponding to poles in the $t$\- and $u$-channel diagrams for the
identical DM candidate.

In Tulin:2013teo  the authors considered a Dirac-fermion DM and calculated
$\sigma_{T}$. Within the resonance region, no analytic formula exists for
$\sigma_{T}$ and it must be computed by solving the Schrödinger equation
directly with partial wave expansion method. The scattering amplitude is given
by

$f(\theta)=\frac{1}{k}\sum_{\ell=0}^{\infty}(2\ell+1)e^{i\delta_{\ell}}P_{\ell}(\cos\theta)\sin\delta_{\ell}\,.$
(11)

The differential scattering cross section is given by

$\frac{d\sigma}{d\Omega}=\frac{1}{k^{2}}\Big{|}\sum_{\ell=0}^{\infty}(2\ell+1)e^{i\delta_{\ell}}P_{\ell}(\cos\theta)\sin\delta_{\ell}\Big{|}^{2}\,,$
(12)

where $\delta_{\ell}$ is the phase shift for a partial wave $\ell$. In terms
of the phase shifts, the transfer cross section is given by

$\frac{\sigma_{T}k^{2}}{4\pi}=\sum_{\ell=0}^{\infty}(\ell+1)\sin^{2}(\delta_{\ell+1}-\delta_{\ell})~{}.$
(13)

To obtain $\delta_{\ell}$, one must solve the Schrödinger equation to
calculate the radial wave function $R_{\ell}(r)$ for the reduced two-DM
particle system

$\frac{1}{r^{2}}\frac{d}{dr}\Big{(}r^{2}\frac{dR_{\ell}}{dr}\Big{)}+\Big{(}k^{2}-\frac{\ell(\ell+1)}{r^{2}}-2\mu
V(r)\Big{)}R_{\ell}=0.$ (14)

The Schrödinger equation can be recast into the form

$\left(\frac{d^{2}}{dx^{2}}+a^{2}-\frac{\ell(\ell+1)}{x^{2}}\pm\frac{1}{x}\,e^{-x/b}\right)\chi_{\ell}(x)=0\;$
(15)

with new variables

$\chi_{\ell}\equiv rR_{\ell}\,,\quad x\equiv\alpha_{\chi}m_{\chi}r\,,\quad
a\equiv\frac{v}{2\alpha_{\chi}}\,,\quad
b\equiv\frac{\alpha_{\chi}m_{\chi}}{m_{\phi}}\;.$ (16)

Figure 3: The birds-eye view of transfer and viscosity cross sections
$\sigma_{T}$, $\sigma_{VS}$, $\sigma_{VA}$ in the parameter space $(a,b)$.

When the DM candidate is a Majorana fermion, the amplitude and Schrödinger
equation are the same as in Eq.(11) and Eq.(15). However, the total wave
function of the spin-1/2 fermionic DM must be antisymmetric with respect to
the interchange of two identical particles. Then the spatial wave function
should be symmetric when the total spin is 0 (singlet) while the spatial wave
function should be antisymmetric when the total spin is 1 (triplet). The
viscosity cross section should be defined with two variables:

$\displaystyle\frac{d\sigma_{VS}}{d\Omega}$ $\displaystyle=$
$\displaystyle\left|f(\theta)+f(\pi-\theta)\right|^{2}=\frac{1}{k^{2}}\left|\sum_{\ell(\mbox{\tiny\rm
EVEN\
number})}^{\infty}(2\ell+1)(\exp(2i\delta_{l})-1)P_{\ell}(\cos\theta)\right|^{2}$
(17) $\displaystyle\frac{d\sigma_{VA}}{d\Omega}$ $\displaystyle=$
$\displaystyle\left|f(\theta)-f(\pi-\theta)\right|^{2}=\frac{1}{k^{2}}\left|\sum_{\ell(\mbox{\tiny\rm
ODD\
number})}^{\infty}(2\ell+1)(\exp(2i\delta_{l})-1)P_{\ell}(\cos\theta)\right|^{2}$
(18)

Using the orthogonality relation for the Legendre polynomials, we can obtain

$\displaystyle\frac{\sigma_{VS}k^{2}}{4\pi}$ $\displaystyle=$
$\displaystyle\sum_{\ell(\mbox{\tiny EVEN\
number})}^{\infty}4\sin^{2}(\delta_{\ell+2}-\delta_{\ell})(\ell+1)(\ell+2)/(2\ell+3),$
(19) $\displaystyle\frac{\sigma_{VA}k^{2}}{4\pi}$ $\displaystyle=$
$\displaystyle\sum_{\ell(\mbox{\tiny ODD\
number})}^{\infty}4\sin^{2}(\delta_{\ell+2}-\delta_{\ell})(\ell+1)(\ell+2)/(2\ell+3).$
(20)

From the expressions of the transfer and viscosity cross sections, we can see
that both $\sigma_{T}$ and $\sigma_{V}$ will converge to a static value as the
phase shift $\delta_{\ell}$ approaching to a same value when the partial wave
$\ell$ grows up. We adopt the numerical method in Tulin:2013teo  to calculate
all the cross sections. The results are shown in Fig. 3 in which we have the
same definitions of input variables. The left plot is the reproduced results,
same as in Tulin:2013teo . We can see that $\sigma_{T}$ and $\sigma_{VS}$ are
almost same while $\sigma_{VA}$ shows differences in some region ($ab\ll
0.1$). The $ab>0.5$ and $b\gg 1$ is the classical regime in which all the
cross sections can be very large. There are also the resonance regions
($b\gtrsim 1$ and $ab\lesssim 0.5$) in which all the cross sections exhibit
patterns of resonance, making the cross section much complicated. Self-
interacting DM scenario used in Tulin:2013teo  and in this paper works around
the resonance regions. In the region $ab\ll 0.1$, $\sigma_{VA}$ is much
smaller than the other two cross sections. The reason is that the summation of
the phase shift in $\sigma_{VA}$ begins at $\ell=1$ as shown in Eq. (20), and
in this region the dominant phase shift is $\delta_{0}$ and other phase shifts
are much smaller. That is why we always calculate $\delta_{0}$ for partial
wave expansion in QM. In our following analysis, we assume that the DM
scatters with random orientations, thus the triplet is three times as likely
as the singlet and the average cross section will be

$\displaystyle\sigma_{V}=\frac{1}{4}\sigma_{VS}+\frac{3}{4}\sigma_{VA}\;.$
(21)

This is simple but sufficient to estimate the viscosity cross section for
Majorana DM.

##  3 Can self-interacting dark matter scenario be realized in the NMSSM ?

SUSY can not only give a solution to the hierarchy problem but also provide a
good dark matter candidate and realize gauge coupling unification. Among the
SUSY models, the MSSM has been intensively studied. This model, however, has
the little hierarchy problem since the newly discovered 125 GeV Higgs boson
requires a heavy stop or a large trilinear coupling $A_{t}$. Besides, the MSSM
also suffers from the $\mu$-problem muew . It is remarkable that both the
little hierarchy problem and the $\mu$-problem can be solved in the NMSSM
fayet ; NMSSM , in which an additional gauge singlet $S$ is introduced. In
this model the $\mu$-term is dynamically generated through the coupling
$SH_{u}H_{d}$ after $S$ develops an electroweak scale VEV, while the little
hierarchy problem is solved through an additional tree-level contribution to
the Higgs mass. With the additional singlet, it might be possible for the
NMSSM to give DM a proper non-relativistic cross section by tuning the singlet
mediator. In the following we check this possibility.

###  3.1 Dark matter and Higgs bosons in the NMSSM

In the NMSSM the relevant superpotential containing $\hat{S}$ is given by

$\displaystyle\lambda\hat{S}\hat{H_{u}}\cdot\hat{H_{d}}+\frac{\kappa}{3}\hat{S}^{3}\,,$
(22)

where $\hat{H}_{u}$ and $\hat{H}_{d}$ are the Higgs doublet superfields, and
$\lambda$ and $\kappa$ are dimensionless parameters. Note that there is no
explicit $\mu$-term and an effective $\mu$-parameter is generated when the
scalar component ($S$) of $\hat{S}$ develops a VEV $s$: $\mu_{eff}=\lambda s$.
The corresponding soft SUSY breaking terms are given by

$\displaystyle A_{\lambda}\lambda SH_{u}\cdot H_{d}+\frac{A_{\kappa}}{3}\kappa
S^{3}+h.c.\,.$ (23)

So the scalar Higgs potential is given by

$\displaystyle V_{F}$ $\displaystyle=$ $\displaystyle|\lambda H_{d}\cdot
H_{u}-\kappa S^{2}|^{2}+|\lambda
S|^{2}\left(|H_{d}|^{2}+|H_{u}|^{2}\right)\,,$ (24) $\displaystyle V_{D}$
$\displaystyle=$
$\displaystyle\frac{g_{2}^{2}}{2}\left(|H_{d}|^{2}|H_{u}|^{2}-|H_{d}\cdot
H_{u}|^{2}\right)+\frac{g_{1}^{2}+g_{2}^{2}}{8}\left(|H_{d}|^{2}-|H_{u}|^{2}\right)^{2}\,,$
(25) $\displaystyle V_{\rm soft}$ $\displaystyle=$ $\displaystyle
m_{d}^{2}|H_{d}|^{2}+m_{u}^{2}|H_{u}|^{2}+m_{s}^{2}|S|^{2}-\left(A_{\lambda}\lambda
SH_{d}\cdot H_{u}+\frac{\kappa}{3}A_{\kappa}S^{3}+h.c.\right)\,,$ (26)

where $g_{1}$ and $g_{2}$ are the coupling constant of $U_{Y}(1)$ and
$SU_{L}(2)$, respectively. Assuming

$H_{u}^{0}=h_{u}+\frac{H_{uR}+iH_{uI}}{\sqrt{2}},\quad
H_{d}^{0}=h_{d}+\frac{H_{dR}+iH_{dI}}{\sqrt{2}},\quad
S=s+\frac{S_{R}+iS_{I}}{\sqrt{2}}$ (27)

with $h_{u}$, $h_{d}$, $s$ being the corresponding VEVs and using the
minimization conditions, one can obtain a $3\times 3$ CP-even Higgs matrix
${\cal M}_{h}$, a $3\times 3$ CP-odd Higgs matrix ${\cal M}_{a}$ and a
$2\times 2$ charged Higgs matrix ${\cal M}_{c}$. Note that there are three
Goldstone bosons in ${\cal M}_{a}$ and ${\cal M}_{c}$ which imply that after
diagonalization these two matrices must have 0 eigenvalues.

From the superpotential in Eq. (22), we can see that the interactions between
the singlet and the SM sector are controlled by the parameter $\lambda$. If a
light singlet Higgs exists, the constraints from both collider and DM
detections can be satisfied only if $\lambda$ is small enough. Then the
singlet Higgs will be a dark sector. The spectrum of the NMSSM has been widely
studied in the literature NMSSM , so we only present our conventions and list
two necessary terms for our analysis, concentrating on the dark singlet
sector:

  * •

The CP-even Higgs mass matrix ${\cal M}_{h}$ in the interaction basis
$S^{bare}=(H_{uR},H_{dR},S_{R})$ can be diagonalized by an orthogonal matrix
$S_{ij}$ to obtain 3 CP-even mass eigenstates (ordered in mass)
$h_{i}=S_{ij}S^{bare}_{j}$ with the corresponding masses denoted by
$m_{h_{i}}$. The elements involve the singlet component are

$\displaystyle{\cal M}_{h,33}$ $\displaystyle=$ $\displaystyle\lambda
A_{\lambda}\frac{h_{u}h_{d}}{s}+\kappa s(A_{\kappa}+4\kappa s),$ (28)
$\displaystyle{\cal M}_{h,13}$ $\displaystyle=$ $\displaystyle
2\lambda\mu_{\rm eff}h_{u}-\lambda h_{d}(A_{\lambda}+2\kappa s),$ (29)
$\displaystyle{\cal M}_{h,23}$ $\displaystyle=$ $\displaystyle
2\lambda\mu_{\rm eff}h_{d}-\lambda h_{u}(A_{\lambda}+2\kappa s).$ (30)

  * •

Neutralino matrix ${\cal M}_{0}$ is composed of the $U(1)_{Y}$ gaugino
$\lambda_{1}$, the neutral $SU(2)$ gaugino $\lambda_{2}$, the singlino
$\psi_{s}$ and the neutral higgsinos $\psi_{u,d}^{0}$. In the basis
$\psi^{0}=(-i\lambda_{1},-i\lambda_{2},\psi_{u}^{0},\psi_{d}^{0},\psi_{s})$
one can rewrite

${\cal L}=-\frac{1}{2}\,(\psi^{0})^{T}{\cal M}_{0}(\psi^{0})+\mathrm{h.c.}$
(31)

where

${\cal
M}_{0}=\left(\begin{array}[]{ccccc}M_{1}&0&\frac{g_{1}h_{u}}{\sqrt{2}}&-\frac{g_{1}h_{d}}{\sqrt{2}}&0\\\
0&M_{2}&-\frac{g_{2}h_{u}}{\sqrt{2}}&\frac{g_{2}h_{d}}{\sqrt{2}}&0\\\
\frac{g_{1}h_{u}}{\sqrt{2}}&-\frac{g_{2}h_{u}}{\sqrt{2}}&0&-\mu_{\rm
eff}&-\lambda h_{d}\\\
-\frac{g_{1}h_{d}}{\sqrt{2}}&\frac{g_{2}h_{d}}{\sqrt{2}}&-\mu_{\rm
eff}&0&-\lambda h_{u}\\\ 0&0&-\lambda h_{d}&-\lambda h_{u}&2\kappa
s\end{array}\right).$ (32)

After diagolization, one obtains 5 eigenstates (ordered in mass)
$\chi^{0}_{i}=N_{ij}\psi^{0}_{j}$, among which the lightest one $\chi^{0}_{1}$
is usually assumed to be the lightest SUSY particle and serve as a good DM
candidate.

From the spectrum we can see that if $\lambda$ approaches zero, one can tune
the masses of the singlet-dominant scalars and neutralinos to arbitrary values
by varying the parameters $\kappa,~{}A_{\kappa}$.

DM (the lightest neutralino) in the NMSSM has three-type components: gaugino,
higgsino and singlino. Assuming the gaugino unification relation
$M_{2}/M_{1}\approx 2$, we have three possibilities for the DM:

  * •

Bino-dominant DM. As shown in Cao:2010ph , under current collider and DM relic
density constraints, the SI cross section can exclude a large part of
parameter space, leaving only a bino-dominant DM candidate below TeV.

  * •

Higgsino-dominant DM. As pointed in Wang:2013rba , the higgsino-dominant DM
candidate around 1.1 TeV can satisfy all the constraints, including the relic
density and current DM direct detections.

  * •

Singlet-dominant DM. In order to explain the observation of CoGeNT
Aalseth:2010vx , the analysis in Draper:2010ew  showed that in the Peccei-
Quinn limit there can exist three light singlet-like particles (0.1-10 GeV): a
scalar, a pseudoscalar and a singlino-like DM candidate. For a certain
parameter window, through annihilation into the light pseudoscalar the
singlino DM can give the correct relic density, and through exchanging the
light scalar in scattering off the nucleon a large cross section suggested by
CoGeNT and DAMA/LIBRA Bernabei:2010mq  can be attained.

###  3.2 Allowed parameter space and dark matter self scattering in the NMSSM

We use the package NMSSMTools nmssmtools  to study all the scenarios
numerically. As the package requires the mass of light Higgs to be heavier
than 1 GeV, we modify the package so that the light Higgs mass can be
arbitrarily low and also we include our own codes in the calculation of SI
cross section. Since it is nontrivial to obtain a light singlet-dominant
Higgs, we randomly scan the parameter space under the condition
$m_{h_{1}}<5{\rm GeV}$ and $m_{h_{2}}>120{\rm GeV}$. In our scan we consider
constraints from DM relic density and DM-nucleon scattering cross section.
Since the SM Higgs ($h_{2}$) is much heavier than the singlet-like Higgs
($h_{1}$), we only need to slightly tune the parameter $A_{\kappa}$ to get a
light Higgs with mass 100 MeV. Finally, we use a light scalar with
$m_{h_{1}}=100{\rm MeV}$ for the calculation of DM-nucleon SI cross section
$\sigma^{\rm SI}$. Note that the singlet sector will be dark with respect to
the SM sector when $\lambda$ is sufficiently small, and then the SM sector is
very insensitive to the tuning in the dark sector.

Figure 4: The spin-independent cross section of DM scattering off the proton
versus the DM mass in different scenarios in the NMSSM. $\Box$ denotes
singlino-dominant DM scenario, most of which are excluded by the DM direct
detection limits; $\bigtriangleup$ denotes bino-dominant DM scenario, among
which the red ones are excluded while the green ones survive; $\circ$ (around
1 TeV) denotes the higgsino-dominant DM scenario, among which the pink ones
are excluded while the green ones survive. All the samples satisfy the DM
relic density constraints.

The numerical results of the three possibilities are shown in Fig. 4, in which
the SI cross section of DM-nucleon scattering is shown and current constraints
of XENON and LUX are displayed. We can see that the DM direct detection
constraints exclude most of the samples. However, there are still some
surviving parameter space, especially for bino-dominant and higgsino-dominant
scenarios. Future XENON1T will further cover the surviving parameter space.

Now we present some details related to our scan for different scenarios:

  * •

For the singlino-dominant DM scenario, we define the parameter
$\varepsilon\equiv{\lambda\mu}/{m_{Z}}$,
$\varepsilon^{\prime}\equiv{A_{\lambda}}/{\mu\tan\beta}-1$ and scan the
parameter space as in Draper:2010ew :

$\displaystyle 2\leq\tan\beta\leq 50,~{}0.05\leq\lambda\leq
0.2,~{}0.0005\leq\kappa\leq 0.05,-0.1\leq\varepsilon\leq 0.1,$
$\displaystyle\varepsilon\sim\varepsilon^{\prime},|A_{\kappa}|<500{\rm
GeV},|\mu|<1{\rm TeV}.$ (33)

The sfermion mass is set to be 6 TeV so that we can easily get a 125 GeV SM
Higgs. As noted in Draper:2010ew , the DM annihilates to SM particles mainly
through the resonance of the singlet pseudo-scalar $a_{1}$ (the Feynman
diagram is shown in Fig. 5). The result of the scan is shown in the left panel
of Fig. 6. As a consequence of resonance, the singlet pseudo-scalar mass is
almost twice of the DM mass. This result is consistent with Draper:2010ew  and
also implies that the coupling of $a_{1}f\bar{f}$ should not be completely
neglected and thus the parameter $\lambda$ can not be too small. On the other
hand, as shown in sec.2, the SI cross section has the inverse quartic power
dependence on the mediated scalar mass (here it is the singlet Higgs $h_{1}$).
Though the coupling of $h_{1}f\bar{f}$ is suppressed by a small $\lambda$, the
cross section will be enhanced greatly when the mass of $h_{1}$ is below GeV.
Therefore, this scenario can hardly satisfy the direct detection limits on the
SI cross section.

Figure 5: Feynman Diagram of DM annihilation to SM fermions through a pseudo-
scalar $a$.

Figure 6: The scatter plots of parameter space allowed by DM relic density
shown in the plane of $M_{a_{1}}$ versus the DM mass. The left panel is the
singlino-dominant DM scenario, where most of the samples are excluded points
by direct detection limits of LUX-2013. The right panel is the bino-dominant
DM scenario, where the red ones are excluded points by direct detection limits
of LUX-2013 while the green ones survived.

  * •

For the bino-dominant DM scenario, we scan the parameter space

$\displaystyle 10^{-5}\leq|\lambda|,|\kappa|<1,~{}~{}2\leq\tan\beta\leq
50,~{}~{}|\mu|,|A_{\lambda}|<1\rm TeV,~{}~{}|M_{2}|<500{\rm GeV}~{},$ (34)

and set $A_{\kappa}\sim-4\kappa\mu/\lambda$ to get a light singlet Higgs (see
Eq. (29)). We also keep the gaugino unification assumption
$M_{1}=M_{2}/2,~{}M_{3}=3M_{2}$ and set the soft parameters to 6 TeV.

Since bino does not couple to singlet Higgs directly, the coupling between
bino-dominant DM and singlet Higgs arises from its higgsino and singlino
components. Therefore, the SI cross section of bino-dominant DM can be much
less than the LUX limits, as shown in Fig. 4.

The right panel of Fig. 6 shows the samples that satisfy the DM relic density
in the bino-dominant DM scenario. The samples can be divided into two regions:
one is in the diagonal region($m_{a}\sim 2m_{\chi}$) and the other is a
vertical belt around $m_{\chi}\sim 50\rm GeV$. The first region is excluded by
the direct detection limits of LUX (the reason is similar to the singlino-
dominant DM scenario). In the second region, the DM annihilates to SM fermions
through the $Z$-boson resonance, independent of singlet coupling $\lambda$.
The coupling $\lambda$ is constrained only by the SI cross section limits and
can only take a very small value, namely $\lambda\raise
1.29167pt\hbox{$<$\kern-7.5pt\raise-4.73611pt\hbox{$\sim$}}0.005$. This can be
seen from the left panel of Fig. 7.

Figure 7: Same as Fig.4, but showing the spin-independent cross section versus
$\lambda$ in the NMSSM.

  * •

For the higgsino-dominant DM scenario, we scan the parameter space

$\displaystyle 10^{-5}\leq|\lambda|,|\kappa|<1,~{}~{}2\leq\tan\beta\leq
50,~{}~{}|A_{\lambda}|<1\rm TeV,$ $\displaystyle|\mu|<1.5\rm
TeV,~{}~{}A_{\kappa}\sim-4\kappa\mu/\lambda,~{}~{}|M_{2}|=10{\rm TeV}~{}.$
(35)

As above, the gaugino unification assumption is used and the soft parameters
are set to be 6 TeV. In this scenario, the DM mass need to be around 1100 GeV
to satisfy the relic density, and thus no resonance through $Z$-boson or
pseudo-scalar happens in the DM annihilation.

The singlet Higgs can couple directly to the higgsino through the $\lambda
S\psi_{u}\psi_{d}$ term in the Lagrangian, so the SI cross section limits only
constrain $\lambda$. From the right panel of Fig. 7 we can see that
$\lambda\raise
1.29167pt\hbox{$<$\kern-7.5pt\raise-4.73611pt\hbox{$\sim$}}0.001$ for the
survived samples.

As a brief summary, the singlino-dominant DM scenario is almost excluded
because of the $\lambda$-value correlation between relic density and DM-
nucleon SI cross section (the relic density needs a not-too-small $\lambda$
which is too large for the DM-nucleon SI cross section). On the other hand,
some parameter space still survived the SI cross section limits in the bino-
dominant and higgsino-dominant DM scenarios, in which the $\lambda$-value
correlation between relic density and DM-nucleon SI cross section is relaxed
and $\lambda$ is constrained by the DM-nucleon SI cross section limits to take
a very small value.

In order to figure out if the NMSSM can realize the DM self-interaction
scenario to explain the small cosmological scale anomalies, we now calculate
the DM self scattering cross section. For each allowed sample in Fig. 7, we
use three velocities corresponding to the three small cosmological scales
(dwarf, Milky Way and cluster) to calculate the DM self scattering cross
section. Then for each sample we have three values of DM self scattering cross
section and we keep the maximal one to display it in Fig. 8. We see that due
to a very small $\lambda$ value, $\sigma_{V}/m_{\chi}$ is too small to explain
the small cosmological scale anomalies (its value must be in $0.1-10\,\,{\rm
cm^{2}/g}$ to explain dwarf scale anomaly and in $0.1-1\,\,{\rm cm^{2}/g}$ to
explain MW and cluster anomalies). For such a weak scattering, we can regard
the DM as a collisionless particle. So we conclude that NMSSM can not realize
the self-interacting DM scenario to solve the small cosmological scale
anomalies.

Figure 8: Same as Fig.7, but showing the non-relativistic viscosity cross
section between DM in the NMSSM. For each allowed sample in Fig. 7, we use
three velocities corresponding to the three small cosmological scales (dwarf,
Milky Way and cluster) to calculate the cross section and here we plot the
maximal one.

##  4 Can self-interacting dark matter scenario be realized in the GMSSM ?

The main reason for the failure of the NMSSM in solving the small cosmological
scale anomalies is a too small $\lambda$ (constrained by DM-nucleon SI cross
section limits), which determines the coupling strength of $h_{1}\chi\chi$ in
DM self-interaction (Note that when the DM is singlino-dominant, the coupling
$h_{1}\chi\chi$ can be proportional to $\kappa$. However, a singlino-dominant
DM is obtained under the condition $\kappa\ll\lambda$ Draper:2010ew , which
means a very small $\kappa$). In the NMSSM the $\lambda SH_{u}\cdot H_{d}$
term in the superpotential is the origin of the $\mu$-term. In the GMSSM,
however, the ${\cal Z}_{3}$ discrete symmetry is not imposed and the $\mu$
term can exist in the superpotential, together with the $\lambda SH_{u}\cdot
H_{d}$ term (several other terms of the singlet superfield can also exist in
the superpotential). Then the singlet sector can be a completely dark sector
in case of a nearly vanishing $\lambda$. Unlike the NMSSM, the dark Higgs
sector (including a singlino-dominant DM) can be easily realized in the GMSSM,
which does not need the condition $\kappa\ll\lambda$ Draper:2010ew . This
means that a singlino-dominant DM can be obtained with a sizable $\kappa$ and
in this case the coupling $h_{1}\chi\chi$ in DM self-interaction, which is
proportional to $\kappa$, can be large. So it should be easy to realize the
self-interacting DM scenario in GMSSM.

As shown in Wang:2009rj , in the GMSSM a singlino-dominant dark matter
annihilates to a singlet-dominant Higgs, which can give the correct DM relic
density, and also the DM-nucleon SI cross section limits from the direct
detections can be easily satisfied. In addition, an appropriate Sommerfeld
enhancement can explain the positron excess observed by PAMELA. The non-
relativistic DM scattering and Sommerfeld enhancement are similar to what
discussed in Tulin:2012wi  except that there can mediated an additional
pseudo-scalar in the GMSSM.

###  4.1 Dark matter and Higgs bosons in the GMSSM

The superpotential of the GMSSM is

$\displaystyle
W=\mu\widehat{H}_{u}\cdot\widehat{H}_{d}+\lambda\widehat{S}\widehat{H}_{u}\cdot\widehat{H}_{d}+\eta\widehat{S}+\frac{1}{2}\,\mu_{s}\widehat{S}^{2}+\frac{1}{3}\kappa\widehat{S}^{3}\
,$ (36)

which involves the parameters $\mu,~{}\lambda,~{}\eta,~{}\mu_{s},~{}\kappa$.
We set $\lambda\sim 0$ so that the singlet sector almost decouple from the SM
sector. In the following discussion, we will concentrate on the singlet
sector.

The soft SUSY breaking terms take the form

$\displaystyle-{\cal L}_{\mathrm{soft}}$ $\displaystyle=$ $\displaystyle
m_{s}^{2}|S|^{2}+\left(C_{\eta}\eta
S+\frac{1}{2}\,B_{s}\mu_{s}S^{2}+\frac{1}{3}\,\kappa A_{\kappa}\
S^{3}+\mathrm{h.c.}\right)\,.$ (37)

The main difference between the NMSSM and GMSSM is reflected in their Higgs
sectors which contain different singlet Higgs mass matrices and self-
interactions. The difference mainly comes from the F-term $F_{s}$:

$\displaystyle V_{F_{s}}$ $\displaystyle=$
$\displaystyle|F_{s}|^{2}=|\eta+\mu_{s}S+\kappa S^{2}|^{2}$ (38)
$\displaystyle=$ $\displaystyle|\kappa
S^{2}|^{2}+\eta^{2}+\mu_{s}^{2}|S|^{2}+\left(\eta\mu_{s}S+\kappa\eta
S^{2}+\kappa\mu_{s}S^{2}S^{\ast}+\mathrm{h.c.}\right).$

Since $\eta^{2}$ is a constant, the
$\mu_{s}^{2}|S|^{2},\eta\mu_{s}S,\kappa\eta S^{2}$ terms can be absorbed by
the redefinition of the soft SUSY breaking parameters
$m_{s}^{2}|S|^{2},~{}C_{\eta}\eta S,\frac{1}{2}\,B_{s}\mu_{s}S^{2}$.

The singlet Higgs potential is

$\displaystyle V$ $\displaystyle=$ $\displaystyle V_{F}+V_{\rm soft}$ (39)
$\displaystyle=$ $\displaystyle m_{s}^{2}|S|^{2}+\left(C_{\eta}\eta
S+\frac{1}{2}\,B_{s}\mu_{s}S^{2}+\frac{1}{3}\,\kappa A_{\kappa}\
S^{3}+\kappa\mu_{s}S^{2}S^{\ast}+\mathrm{h.c.}\right).$

The singlet chiral supermultiplet contains a complex scalar and a Majorana
fermion $\chi$. After the scalar component getting a VEV, we can get one CP-
even Higgs $h$ and one CP-odd Higgs $a$. The mass spectrum and the relevant
Feynman rules are

$\displaystyle m_{\chi}$ $\displaystyle=$ $\displaystyle 2\kappa s+\mu_{S},$
(40) $\displaystyle m_{h}^{2}$ $\displaystyle=$ $\displaystyle\kappa s(4\kappa
s+A_{\kappa}+3\mu_{S})-C_{\eta}\eta/v,$ (41) $\displaystyle m_{a}^{2}$
$\displaystyle=$ $\displaystyle-2B_{s}\mu_{s}-\kappa
s(3A_{\kappa}+\mu_{S})-C_{\eta}\eta/s,$ (42) $\displaystyle V_{hhh}$
$\displaystyle=$ $\displaystyle-4ik(6\kappa
s+A_{\kappa}+3\mu_{S})=-4i\kappa(3m_{\chi}+A_{\kappa}),$ (43) $\displaystyle
V_{haa}$ $\displaystyle=$ $\displaystyle-4i\kappa(2\kappa
s-A_{\kappa}+\mu_{S})=-4i\kappa(m_{\chi}-A_{\kappa}),$ (44) $\displaystyle
V_{h\chi\chi}$ $\displaystyle=$ $\displaystyle-4i\kappa,$ (45) $\displaystyle
V_{a\chi\chi}$ $\displaystyle=$ $\displaystyle-4\kappa\gamma^{5}.$ (46)

We do not give the Feynman rules for the four-scalar vertex for they are
irrelevant to our analysis. From the spectrum and Feynman rules above, we can
see that the mass parameters of the three particles ($h,a,\chi$) can be set
freely because the relevant input soft parameters can take arbitrary values.
This makes the following calculation much easier and we can choose the input
parameters easily without fine-tuning.

Then we have only five input parameters in our calculation, namely
$m_{\chi},~{}m_{h},~{}m_{a},~{}\kappa,~{}A_{\kappa}$. It is useful to note
that the three CP-even Higgs vertex $V_{hhh}$ can vanish if we set
$A_{\kappa}=-3m_{\chi}$.

Figure 9: Feynman diagram of singlino DM annihilation to scalars in the GMSSM.
The final state of $\phi~{}\phi$ can be $hh$, $ha$ and $aa$.

The calculation of non-relativistic DM scattering cross section is already
given in Sec. 2 with the coupling strength of DM to mediator
($\alpha_{\chi}=4\kappa^{2}/\pi$). Here we just give some details relevant to
the calculation of DM relic density. The Feynman diagrams of the DM
annihilation are shown in Fig. 9. The general form of the annihilation cross
section is given by Drees:1992am

$\sigma v=\frac{1}{4}\frac{\bar{\beta}_{f}}{8\pi
sS}\left[|A(^{1}S_{0})|^{2}+\frac{1}{3}\left(|A(^{3}P_{0})|^{2}+|A(^{3}P_{1})|^{2}\right)+|A(^{3}P_{2})|^{2}\right],$
(47)

where $S$ is the symmetry factor,
$A(^{1}S_{0}),~{}A(^{3}P_{0}),~{}A(^{3}P_{1}),~{}A(^{3}P_{2})$ are the
contributions from different spin states of DM and $\beta_{f}$ is given by

$\displaystyle\bar{\beta}_{f}=\sqrt{1-2(m_{X}^{2}+m_{Y}^{2})/s+(m_{X}^{2}+m_{Y}^{2})^{2}/s^{2}}~{},$
(48)

with $X,Y$ being the final states. The amplitudes from different final states
are given by

  1. 1.

$\chi\chi\to h~{}h$ :

$\displaystyle A(^{3}P_{0})$ $\displaystyle=$ $\displaystyle
16\sqrt{6}v\kappa^{2}\left[\frac{R(3m_{\chi}+A_{\kappa})}{4-R(m_{h})^{2}+iG_{h}}-2\frac{1+R(m_{\chi})}{P_{\chi}}+\frac{4}{3}\frac{\bar{\beta}_{f}^{2}}{P_{\chi}^{2}}\right],$
(49) $\displaystyle A(^{3}P_{2})$ $\displaystyle=$
$\displaystyle-(128/\sqrt{3})v\kappa^{2}\bar{\beta}_{f}^{2}/P_{\chi}^{2}~{}.$
(50)

  2. 2.

$\chi\chi\to a~{}a$ :

$\displaystyle A(^{3}P_{0})$ $\displaystyle=$ $\displaystyle
16\sqrt{6}v\kappa^{2}\left[\frac{R(m_{\chi}-A_{\kappa})}{4-R(m_{h})^{2}+iG_{h}}-2\frac{1-R(m_{\chi})}{P_{\chi}}+\frac{4}{3}\frac{\bar{\beta}_{f}^{2}}{P_{\chi}^{2}}\right],$
(51) $\displaystyle A(^{3}P_{2})$ $\displaystyle=$
$\displaystyle-(128/\sqrt{3})v\kappa^{2}\bar{\beta}_{f}^{2}/P_{\chi}^{2}~{}.$
(52)

  3. 3.

$\chi\chi\to h~{}a$ :

$\displaystyle A(^{3}S_{0})$ $\displaystyle=$
$\displaystyle-32\sqrt{2}\kappa^{2}\frac{R(m_{\chi}-A_{\kappa})}{4-R(m_{a})^{2}}\left(1+\frac{v^{2}}{8}\right)$
(53)
$\displaystyle+64\sqrt{2}\kappa^{2}\frac{R(m_{\chi})}{P_{\chi}}\left[1+v^{2}\left(\frac{1}{8}-\frac{1}{2P_{\chi}}+\frac{\bar{\beta}_{f}^{2}}{3P_{\chi}^{2}}\right)\right]$
$\displaystyle+16\sqrt{2}\kappa^{2}\left(R(m_{a})^{2}-R(m_{h})^{2}\right)\left[1+v^{2}\left(-\frac{1}{8}-\frac{1}{2P_{\chi}}+\frac{\bar{\beta}_{f}^{2}}{3P_{\chi}^{2}}\right)\right],$
$\displaystyle A(^{3}P_{1})$ $\displaystyle=$ $\displaystyle
64v\kappa^{2}\bar{\beta}_{f}^{2}/P_{\chi}^{2}~{}.$ (54)

In the above formulas

$\displaystyle
R(m_{X})=\frac{m_{X}}{m_{\chi}},~{}P_{\chi}=1+R(m_{\chi})^{2}-\frac{1}{2}(R(m_{X})^{2}+R(m_{Y})^{2}),~{}G_{h}=\frac{\Gamma_{h}m_{h}}{m_{\chi}^{2}}.$
(55)

Given the annihilation amplitudes, the relic density $\Omega h^{2}$ can be
calculated in the standard way. From the expression of the DM relic density
Tulin:2013teo

$\sigma
v=\frac{3}{8}\frac{\pi\alpha_{\chi}^{2}}{m_{\chi}^{2}}v^{2}\sqrt{1-\frac{m_{h}^{2}}{m_{\chi}^{2}}}~{},$
(56)

we can see that the ratio $\alpha_{\chi}/m_{\chi}$ is almost a constant when
$m_{h}\ll m_{\chi}$ in order to give the correct DM relic density.

###  4.2 Allowed parameter space and dark matter self scattering in the GMSSM

We first change our Feynman rules to those in Tulin:2013teo  and check if we
can reproduce the results in Tulin:2013teo . Our results are shown in Fig. 10,
which agree with Tulin:2013teo . Fig. 10 shows the transfer cross section
$\sigma_{T}$ allowed by solving the small cosmological scale anomalies (dwarf,
Milky Way and cluster) with their corresponding characteristic velocities.
Here all the samples satisfy the DM relic density given by PLANCK planck . We
can see that $\sigma_{T}/m_{\chi}$ can satisfy simultaneously the requirements
of dwarf, Milky Way and galaxy cluster scales to solve the small scale
anomalies. On the other hand, as can be seen from the figure, the requirement
to solve such small scale structure anomalies can also give strong constraints
on the parameter space of self-interacting DM scenario. This means that subtle
relations should be satisfied between $m_{h},m_{\chi}$ and $\alpha_{\chi}$.

Figure 10: Scatter plots of the parameter space in the DM self-interaction
model Tulin:2013teo  allowed by DM relic density plus dwarf scale structures
(blue region), Milky Way size structures (red region) or cluster size
structures (green region). We obtained these results by changing our Feynman
rules to those in Tulin:2013teo .

Figure 11: The scatter plots of the parameter space allowed by DM relic
density and all three small scale structures (dwarf, Milky Way, cluster). The
red ones are for the DM self-interaction model Tulin:2013teo  while the green
ones are for the GMSSM.

Then we perform a numerical calculation in the GMSSM. Since in the GMSSM the
singlino DM is of Majorana type, we need to use viscosity cross section
$\sigma_{V}$. In Fig. 11 we show the GMSSM parameter space allowed by DM relic
density plus all three small scale structures. The corresponding parameter
space for the DM self-interaction model Tulin:2013teo  is also displayed for
comparison. We can see that compared with the DM self-interaction model
Tulin:2013teo , the GMSSM has a larger parameter space to solve the anomalies
of all three small scales (the mass of DM can be heavier than 10 GeV and the
coupling strength $\alpha_{\chi}$ can be at order $10^{-3}$). The reason is
that in the DM self-interaction model Tulin:2013teo  DM can only annihilate
into $hh$ via $t$-channel and $u$-channel while in the GMSSM DM can annihilate
into $hh$, $ha$ and $aa$ via $t$-channel, $u$-channel and $s$-channel, as
shown in Fig.9.

Figure 12: Same as Fig.10, but showing the Sommerfeld enhancement factor. The
left frame is for the DM self-interaction model Tulin:2013teo  while the right
frame is for the GMSSM.

Finally, in Fig. 12 we show the Sommerfeld enhancement in the allowed
parameter space. We see that compared with the DM self-interaction model
Tulin:2013teo , the GMSSM allows for a larger $m_{\chi}/m_{h}$ and thus can
give a larger Sommerfeld enhancement factor. The ongoing indirect detections
of DM can probe the DM annihilation rate (shed light on the Sommerfeld
enhancement factor) and thus help to distinguish different DM self-interaction
models.

##  5 Conclusions

In this paper we studed the possibility of the DM self-interaction to solve
the small cosmological scale anomalies in the singlet extensions of the MSSM.
We first checked the NMSSM and found that the correlation between the DM
annihilation rate and DM-nucleon SI cross section strongly constrains this
model so that it cannot realize the DM self-interacting scenario. For the
GMSSM, the parameter space was found to be large enough to realize the DM
self-interacting scenario and at the same time can give a large Sommerfeld
enhancement factor. Also, we found that for Majorana-fermion DM, we must use
viscosity cross sections ($\sigma_{VS}$ and $\sigma_{VA}$) in DM simulations.

## Acknowledgments

This work was supported by the Natural Science Foundation of China under grant
numbers 11105124,11105125, 11275245, 10821504, 11135003, 11005006, 11172008,
11375001 and Ri-Xin Foundation of BJUT.

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