Spindependent constraints on blind spots for thermal singlinohiggsino dark matter with(out) light singlets
Received: May
Spindependent constraints on blind spots for thermal singlinohiggsino dark matter with(out) light singlets
Marcin Badziak 0 1 2 3 4
Marek Olechowski 0 1 2 4
Pawel Szczerbiak 0 1 2 4
0 University of California , Berkeley, CA 94720 , U.S.A
1 and Theoretical Physics Group, Lawrence Berkeley National Laboratory
2 ul. Pasteura 5, PL02093 Warsaw , Poland
3 Berkeley Center for Theoretical Physics, Department of Physics
4 Institute of Theoretical Physics, Faculty of Physics, University of Warsaw
The LUX experiment has recently set very strong constraints on spinindependent interactions of WIMP with nuclei. These null results can be accommodated in NMSSM provided that the e ective spinindependent coupling of the LSP to nucleons is suppressed. We investigate thermal relic abundance of singlinohiggsino LSP in these socalled spinindependent blind spots and derive current constraints and prospects for direct detection of spindependent interactions of the LSP with nuclei providing strong constraints on parameter space. We show that if the Higgs boson is the only light scalar the new LUX constraints set a lower bound on the LSP mass of about 300 GeV except for a small range around the half of Z0 boson masses where resonant annihilation via Z0 exchange dominates. XENON1T will probe entire range of LSP masses except for a tiny Z0resonant region that may be tested by the LZ experiment. These conclusions apply to general singletdoublet dark matter annihilating dominantly to tt. Presence of light singlet (pseudo)scalars generically relaxes the constraints because new LSP (resonant and nonresonant) annihilation channels become important. Even away from resonant regions, the lower limit on the LSP mass from LUX is relaxed to about 250 GeV while XENON1T may not be sensitive to the LSP masses above about 400 GeV.
Supersymmetry Phenomenology

HJEP07(21)5
1 Introduction
2
3
4
5
6
Blind spots without interference e ects and relic density
Blind spots and relic density with light singlets
Z3symmetric NMSSM
5.1
5.2
Heavy singlet scalar
Light singlet scalar
Conclusions
A LSPnucleon cross sections
B The LSP (co)annihilation channels
B.1 Resonance with the Z0 boson (unitary gauge)
B.2 Annihilation into tt via Z0
C Improved formula for
h2 near a resonance
eter space still exist but they reside very close to blind spots for SI scattering crosssection.
In the present article we study implications of the assumption that the SI scattering of
the LSP is so small (below the neutrino background) that probably it will never be detected
in direct detection of its SI interactions with nuclei. We also demand that the LSP has
W = W MSSM + SHuHd + f (S) ;
Lsoft =
Lsoft
MSSM + m2Hu jHuj2 + m2Hd jHdj2 + m2S jSj2
1
3
A
1
2
+
A
HuHdS +
S3 + m32HuHd +
m0S2S2 + SS + h:c: ;
(2.1)
(2.2)
general models f (S)
F S + 0S2=2 + S3=3.
where S is a chiral SMsinglet super eld. In the simplest version, known as the
scaleinvariant or Z3symmetric NMSSM, m23 = m0S2 = S = 0 while f (S)
S3=3. In more
{ 2 {
The mass squared matrix for the neutral CPeven scalar elds, in the basis h^; H^ ; s^
related to the interaction basis by a rotation by angle
(see [29] for details), reads:
+
H^ H^ ;
S +
vs
vs
s^s^ ;
Ms^2s^ =
vh2 sin 2
+ A
vs
1
2
1
2
Mh^2H^ =
(MZ2
Mh^2s^ = vh(2
2vh2) sin 4 +
h^H^ ;
2
A + h@Sf i sin 2 ) +
h^s^;
H^ s^;
vs
h^ih^j are radiative corrections, vs, vh sin
and vh cos
are VEVs of the singlet and the
two doublets, respectively. The mass eigenstates of Ms2 are denoted by hi with hi = h; H; s
(h is the 125 GeV scalar discovered by the LHC experiments). These mass eigenstates are
expressed in terms of the hatted elds with the help of the diagonalization matrix S~:
The mass squared matrix for the neutral pseudoscalars, after rotating away the
Goldstone boson, has the form
where
MA2^A^ =
2
sin (2 )
A +
vs
MA2^a^ = vh A
A^a^ ;
Ma^2a^ =
1
2
vh2 sin 2
3A
vs
vs
2m0S2
S +
vs
a^a^; :
hi = S~hih^h^ + S~
hiH^ H^ + S~his^s^ :
Mp2 =
MA2^A^ MA2^a^ !
MA2^a^
Ma^2a^
;
+
A^A^ ;
{ 3 {
vs
(2.3)
(2.4)
(2.5)
(2.6)
(2.7)
(2.8)
(2.9)
(2.10)
(2.11)
(2.12)
(2.13)
(2.14)
Diagonalizing Mp2 with the matrix P~ one gets the mass eigenvalues aj = a; A:
aj = P~aja^a^ + P~
ajA^A^ :
After decoupling of the gauginos (assumed in this work) the neutralino mass submatrix
describing the three lightest states takes the form:
M 0 = BB
0
vh sin
0
vh cos
Trading the model dependent term h@S2 f i for one of the eigenvalues, m j , of the above
neutralino mass matrix we nd the following (exact at the tree level) relations for the
neutralino diagonalization matrix elements:
cos
2
sin
2
;
;
1
CC :
A
where Nj3, Nj4 and Nj5 denote, respectively, the two higgsino and the singlino components
of the jth neutralino mass eigenstate while j = 1; 2; 3 and jm 1 j
jm 2 j
jm 3 j. Using
the last two equations and neglecting contributions from decoupled gauginos one can
express the composition of the three lighter neutralinos in terms of:
vh, m =
and tan .
Later we will be interested mainly in the LSP corresponding to j = 1, so to simplify the
notation we will use m
m 1 . The physical (positive) LSP mass is given by mLSP
In the present work we are interested mainly in two properties of the LSP
particles: their crosssections on nucleons and their relic abundance. The spindependent
LSPnucleon scattering cross section is dominated by the Z0 boson exchange and equals
S(ND) = C(N) 10 38 cm2 (N123
N124)2 ;
where C(p)
3:1 [42, 43]. The spinindependent crosssection for the LSP
interacting with the nucleus with the atomic number Z0 and the mass number A is given by
where red is the reduced mass of the nucleus and the LSP. When the squarks are heavy,
as assumed in the present work, the e ective couplings f (N) (N = p; n) are dominated by
the tchannel exchange of the CPeven scalars [44]:
SI =
f (N)
4 r2ed
Zf (p) + (A
A2
Z)f (n) 2
;
3
X f (N)
hi
i=1
3
X
i=1
hi
2m2hi
hiNN :
Further details may be found in appendix A. Formulae for the LSP annihilation cross
section and its relic density are much more complicated (some of them are collected in
appendices B and C).
{ 4 {
(2.15)
(2.16)
(2.17)
(2.18)
jm j.
(2.19)
(2.20)
(2.21)
The main goal of the present work is to identify regions of the NMSSM parameter
space for which the singlinohiggsino LSP particles ful ll three conditions: 1) have very
small, below the neutrino background, SI crosssection on nuclei (SI blind spots); 2) have
small SD crosssections to be consistent with present experimental bounds; 3) have relic
density close to the experimentally favored value
0:12 so the LSP can play the role
of a dominant component of DM.1 Of course, such points in the parameter space must be
consistent with other experimental constrains e.g. those derived from the LHC [48] and LEP
h
2
data [49, 50]. In the next sections we discuss solutions ful lling all the above mentioned
conditions, starting with the simplest case of blind spots without interference e ects and
then for blind spots for which such e ects are crucial. We investigate also modi cations
Blind spots without interference e ects and relic density
In this section we consider situation when the SI blind spot (BS) takes place without
interference e ects i.e. when all three contributions to the e ective coupling f (N) in eq. (2.21)
are very small. Two of them are small because the corresponding scalars, s and H, are
very heavy while fh is suppressed due to smallness of
h . In the next subsection we
discuss the simplest case in which the mixing among the scalars may be neglected. Then
the e ects of such mixing will be taken into account.
3.1
Without scalar mixing
When the interference e ects and scalar mixing may be neglected the SI blind spot
condition has the following simple form:
m
sin 2
= 0 ;
(3.1)
which corresponds to vanishing HiggsLSPLSP coupling for ms; mH ! 1. In gure 1 the
dependence of
h2 on mLSP and tan
for such blind spots is shown for some speci c values
of
and
(the latter parameter does not in uence the situation as long as the resonance
with the lightest pseudoscalar is not considered). Some parts of the (mLSP; tan ) plane
are excluded by the upper bounds on
SD from LUX [41] and IceCube [51] experiments
(and also by the LEP data). Particularly important are the new LUX constraints which
exclude large part of the parameter space with
h2 = 0:12
0:02. We note that in the
allowed part of the parameter space presented in gure 1 correct thermal relic abundance
is obtained for singlinodominated LSP. It is not di cult to understand the results shown
in gure 1 using (approximate) analytic formulae.
For given values of m and tan , the blind spot condition (3.1) together with eqs. (2.17)
and (2.18) may be used to obtain the LSP composition. For example, the combination
1In order to take into account theoretical uncertainties in the relic abundance calculations we consider
parameters leading to
h
0:02 when calculated with the help of MicrOMEGAs 4.3.1 [45] and
NMSSMTools 5.0.2 [46, 47] which we use to calculate the NMSSM spectrum.
{ 5 {
.80 ¡ iZ n 0v
L U X
X E N O N 1 T
04
.01
.012
1
.014
05
03
47]) as functions of mLSP and tan
for blind spots with decoupled all Higgs particles except the
SMlike one. Dashed and dotted blue lines correspond to eq. (3.6) and (3.7), respectively, after
substituting (3.2), for
h
2 = 0:12. Red region depicts the points with m
< 103 GeV which are
ruled out by LEP [49]. Yellow area is forbidden because of eq. (3.9), whereas green/cyan one due
to LUX/IceCube (IC) limits on spindependent LSP interaction with nucleons. Vertical red lines
1
correspond to (from left to right) mLSP = mZ0 =2; mh=2; mW ; mt.
which determines the LSP coupling to the Z0 boson is given by (see appendix A)2
N123
The above expressions are valid as long as blind spot condition (3.1) is satis ed. For
the LSP masses for which the annihilation cross section is dominated by the schannel
Z0 exchange this is enough to calculate the LSP relic density to a good accuracy. The
approximate formulae are (see the appendix B for the details):
h
2
0:1
0:3
N123
N124
2There are some corrections caused by not totally decoupled particles. In our numerical scan we took
I c e C u b e ( W + W ¡ )
j
j
N
N
¡
¡
2 14
2 13
jN
0 : 1 4
0 : 1
I c e C u b e ( t „t )
05
06
LSP mass regimes (blue and red dashed lines in the left plot were obtained from eq. (3.6) whereas
dotted lines in the right plot from eq. (3.7)). Yellow area is forbidden because of eq. (3.9). Grey
color in the left plot denotes the regions in which h resonance and W +W
=Z0Z0 channels may
be important and a ect the results. Green/cyan areas correspond to LUX/IceCube limits on
spindependent LSP interaction with nucleons [41], [51]. Dashed (continuous) green lines in both plots
correspond to the precision of the future XENON1T (LZ) experiment of SD direct interaction of
the LSP with neutrons [52].
for mLSP of order mZ0 =2 (and below the W +W
threshold) and
h
2
0:1
0:05
N123
N124
2
s
2
6
6
4
1
above the tt threshold. They reproduce very well the relic density calculated numerically
using MicrOMEGAs, as may be seen in
gure 1.
The composition of the LSP is crucial not only for its relic density but also for some
of the experimental constraints. One gets the following upper bounds on the
combination (3.2):
from the LEP bound for invisible decays of Z0 [18, 50] and
N123
.
(N) exp
SD
C(N) 10 38cm2
(3.8)
(3.9)
from experiments sensitive to the SD interactions of the LSP with protons or neutrons for
which C(p)
3:1, respectively [42, 43]. These upper bounds on N123
functions of the LSP mass are shown in gure 2.
One can nd the allowed range of mLSP in the vicinity of mZ0 =2. The points in the left
plot in gure 2 at which the blue and red dashed lines enter the green region (excluded by
LUX constraints on the SD crosssection) determine the limiting values of mLSP for which
{ 7 {
n
a
t
02
01
51
5
04
0 : 1
.40
.45
.50
.25
.35
.20
.15
(dashed lines in the left plot was obtained from eq. (3.6) after substituting (3.2)
whereas dotted lines in the right plot from eq. (3.7) after substituting (3.2)) for blind spots with
decoupled all Higgs particles except the SMlike one. Left plot: green areas and thick red lines
denote the points excluded by LUX and the LEP chargino searches, respectively; thick blue points
correspond to eq. (3.10) and (3.11). Right plot: green region and thick orange line depict the points
excluded by LUX and points for which the stop masses above 5 TeV are necessary to obtain the
correct Higgs mass (even when the contribution from the stop mixing is maximized), respectively.
The red lines denoted by LP depict regions with a Landau pole below the GUT scale.
the Z0 resonance may give the correct relic density of the LSP. To be more accurate we
used the results for
h2 obtained from MicrOMEGAs and found the limiting LSP masses to
be approximately 41 and 46.5 GeV.3 Substituting the corresponding values of jN123
into eq. (3.2) we nd the following linear dependence:
tan
tan
42
44
for m
for m
41 GeV ;
46:5 GeV :
(3.10)
(3.11)
The values of tan
necessary to obtain good relic abundance of the LSP become larger
when moving to values of mLSP closer to the peak of the resonance (which is slightly below
mZ0 =2). The situation is illustrated in the left plot of gure 3. One can see that in the
region allowed by LUX large tan
is required unless
is small (
0:1 in this case). Of
course the above limits may become stronger (i.e. for a given
larger tan
might be
required) when SD direct detection experiments (especially based on the interactions with
neutrons) gain better precision. For instance, the LZ experiment will be able to explore
the entire region considered here  see the left plot in gure 2.
In the LSP mass range between the W +W
and tt thresholds the annihilation cross
section is dominated by gauge boson (W +W
=Z0Z0) nal states with the chargino/neutralino
exchanged in the t channel. The related couplings are proportional to the higgsino
components of the LSP (the gauginos are decoupled) which for the blind spot (3.1) are related to
the LSPZ0 coupling by eqs. (3.3) and (3.4). The values of N14 (N13 is smaller by factor
3These results were obtained for the case of
h2 = 0:12 but they do not change much when the
uncertainty in the calculation of the relic abundance is taken into account.
{ 8 {
1= tan ) necessary to get h
0:12 lead to too large SD and are excluded by both LUX
and IceCube data (see gure 1). Thus, the LSP masses in the range mW + . mLSP . mt are
excluded. The only way to have correct relic abundance consistent with all experimental
constraints is to go to very small values of
in order to suppress SI crosssection below the
LUX constraint also away from the blind spot (3.1) and increase tan
such that the Higgs
mass constraint is ful lled.4 More exibility in the parameter space may appear if some
additional particles are exchanged and/or appear in the nal state of the LSP annihilation
(such situations will be discussed in the next sections).
For mLSP & 160 GeV annihilation into tt (via schannel Z0 exchange) starts to be
kinematically accessible so smaller higgsino component su ces to have large enough
ansection is predicted. As a result, IceCube [51] constraints are satis ed for mLSP & 175 GeV.
depending on tan , translates to N125
However, the new LUX constraints exclude mLSP up to about 300 GeV for
This lower bound on mLSP may change by about 50 GeV when the uncertainties in the
calculation of the relic abundance are taken into account. It becomes stronger (weaker) for
smaller (bigger) values of
h2. We should also emphasize that the lower bound on the LSP
mass from SD constraints are the same for the whole class of singletdoublet fermion DM
as long as it annihilates dominantly to tt. In particular, similar lower bound of 300 GeV on
the LSP mass was recently set by LUX on the welltempered neutralino in MSSM [31]. One
can also see in gure 2 that the correct relic abundance requires jN123
0:95 (see eq. (3.5)) and such values may be
explored by XENON1T. The right panel of gure 3 shows values of tan
necessary to get
h
2 = 0:12 as a function of mLSP and . Contrary to the Z0 resonance case small values
of tan
are preferred and hence moderate or large
(in order to have big enough Higgs
mass at the tree level). However, too small values of tan
lead, for a given big value of ,
to a Landau pole below the GUT scale. Thus, the assumption of perturbativity up to the
GUT scale and the requirement
h2 = 0:12 give constraints which result in a dependent
upper bound on the mass of the LSP. For example, mLSP . 700 GeV for
= 0:7 (see
gure 3) and mLSP . 800 GeV for
= 0:6. Let us also note that for large LSP masses
coannihilation becomes nonnegligible. This e ect relaxes the upper bound on tan
and
is increasingly important as
decreases, as can be seen in
gure 3 from comparison of full
result by MicrOMEGAs and the approximated one with only tt included.
h
2
Let us comment on two features of the
0:12 curves in gure 1. First: there are
no signs of a resonant annihilation with the h boson exchange in the schannel. This is
simply the consequence of the blind spot condition leading to vanishing (or at least very
small) LSPHiggs coupling. This is characteristic of all blind spots without interference
e ects. Second: tan
decreases with m
for all m
> mZ0 =2 with the exception of
the vicinity of the tt threshold. This is related to the fact that the annihilation cross
section is directly (Z0 in the s channel) or indirectly (the V V
nal states) connected to
4Note that a welltempered binohiggsino LSP in MSSM with the mass between the W and t masses
cannot accommodate all constraints since in that case the mixing in the neutralino sector is controlled by
the gauge coupling constant which is xed by experiment, see e.g. refs. [9, 31].
{ 9 {
the value of (N123
N124) given by eq. (3.2). The r.h.s. of (3.2) is a decreasing function
of m
and decreasing function of tan
(for tan
> 1 +
vh=m ). Thus, in order to keep
it approximately unchanged the increase of m
must be compensated by the decrease of
tan
(other parameters determining the annihilation cross section may change this simple
relation only close to the tt threshold and below the Z0 resonance).
Another comment refers to constraints obtained from the indirect detection
experiments. The IceCube upper bounds on
SD change by orders of magnitude depending on
what channels dominate the LSP annihilation. This can be already seen in the simple case
discussed in this subsection. The lower bound on tan
obtained from the IC data (as a
function of mLSP) visible in gure 1 drops substantially above the tt threshold because the
tt pairs give softer neutrinos as compared to the W +W
pairs [51]. Moreover, the latest
LUX results on SD lead to stronger bounds in almost all cases. Only for quite heavy LSP
the IC limits are marginally stronger, as may be seen in the right panel of gure 2.
To sum up, in this section we identi ed two crucial mechanisms (Z0 resonance and
annihilation into tt) which may give correct relic density and are allowed by the experiments.
However, both of them rely on the Z0 boson exchange in the s channel and therefore are
proportional to the LSPZ0 coupling, which controls also the SD cross section of the LSP
scattering on nucleons. Therefore, the future bounds on such interaction will be crucial in
order to constrain the parameter space. In fact, XENON1T is expected to entirely probe
regions of the parameter space in which annihilation into tt dominates while LZ will be
able to explore the entire region of Z0 resonance. It is also worth noting that the situation
presented in gure 1 may change if we consider light pseudoscalar a with mass ma
2m .
Such resonance for singlinodominated LSP (we require
6
= 0) is controlled mainly by
the mixing in (pseudo)scalar sector and hence may not be so strongly limited by the SD
direct detection experiments. For instance, we checked with MicrOMEGAs that for ma in a
few hundred GeV range we can easily obtain points in parameter space with correct relic
density and SD below the future precision of the LZ experiment. In principle, the e ect
of light pseudoscalar may be also important for 2m
& mh + ma when the LSP starts
to annihilate into ha state which, depending on , may suppress the tt channel and may
weaken the IceCube limits. However, in the case considered in this subsection, the
contribution from the ha channel may be important only for large mixing in the pseudoscalar
sector. This requires quite large values of A
which leads to unacceptably small values of
the Higgs mass. We will come to these points in the next sections where the annihilation
channels involving the singletdominated pseudoscalar may play a more important role.
3.2
With scalar mixing
Next we consider the case when the mixing among scalars is not negligible and a ects the
blind spot condition (3.1), which is now of the form:
where , de ned by
S~hs^
S~hh^
N13N14
N125
N15(N13 sin
+ N14 cos )
;
(3.12)
(3.13)
depends on the LSP composition (some formulae expressing
in terms of the model
parameters may be found in appendix A). In equation (3.12) we introduced also parameter
describing the mixing of the SMlike Higgs with the singlet scalar. This mixing can be
expressed (for ms
mh assumed in this section) in terms of the NMSSM parameters as:
S~hs^
S~hh^
v
2
ms2
sgn
2
A + h@Sf i sin 2
2
p2
j mixjmh
ms
;
(3.14)
where
mix
mh
M^ hh is the shift of the SMlike Higgs mass due to the mixing [53]. For
ms > mh this shift is negative so we prefer it has rather small absolute value.
The scenario of higgsinodominated LSP with
0:12 is very similar to the
analh
2
ogous case in the MSSM model and requires j j
1:2 TeV. Even the present results from
the direct and indirect detection experiments constrain possible singlino admixture in the
higgsinodominated LSP to be at most of order 0.1. So small singlino component leads to
negligible changes of
necessary to get the observed relic density of DM particles.5
Thus, similarly as before, we focus on SI blind spots with
h
2
0:12 for
singlinodominated LSP. In this case and for nonnegligible
j j N125) the blind spot condition may be approximated by:
(more precisely, when jN13N14j
m
sin 2
1
m
2!
:
This condition is a quadratic equation for
and has solutions only when
(3.15)
(3.16)
# 2
:
(3.17)
One can see from (3.15) that for
> 0 we have always m
> 0 and the LSP has
more higgsino fraction than when condition (3.1) holds. In the opposite case i.e.
< 0
we can have either m
dominated LSP with m
> 0 with slightly smaller higgsino fraction or strongly
singlino
< 0. However, for values of j j small enough not to induce large
negative
mix the higgsino component of the LSP with m
< 0 is too small to obtain
h
2
0:12. Therefore, from now on we focus on the case m
> 0. Solving eq. (3.15) for
the ratio m = and substituting the solution into eq. (3.2) one can
nd the di erence of
two higgsino components of the LSP. For small j j it can be approximated as (3.2) with a
small correction:
5In fact, bigger changes of come from not totally decoupled gauginos, e.g. for M1, M2 of order 5 TeV.
m Z 0 = 2
54
‚ = 0 : 6 ; • = 0 : 3 ; j j • 0 : 0 2 5 ; fi h ´ ´ = 0
04
06
07
08
+ As =
< 0 (> 0) refers to the lower (upper) limit of a given area. The shift of the orange
region with respect to the yellow one for LSP masses above about ms is due to the annihilation
channels containing s in the nal state and coannihilation e ects which become more important
for larger LSP masses.
As we already mentioned discussing eq. (3.2), the rst term in the r.h.s. of the last equation
is a decreasing function of tan
(unless tan
is very close to 1). So, in order to keep the
value of N123
N124 necessary to get
h
2
0:12, the contribution from the second line
of (3.17) may be compensated by increasing (decreasing) the value of tan
when
> 0
(< 0).6 This e ect from the sh mixing is illustrated in
gure 4. One can see that indeed,
depending on the sign of
, we can have smaller or larger (in comparison to eq. (3.1))
values of tan
for a given LSP mass, while keeping
h
2
0:12. In particular,
nonnegligible Higgssinglet mixing may relax the upper bound on the LSP mass arising from
the perturbativity up to the GUT scale. It is also important to emphasize that the resonant
annihilation with s scalar exchanged in the s channel is quite generic for this kind of blind
spot (see the right plot in
gure 4, where we have chosen ms = 600 GeV) because it is
easier to have substantial sh mixing when the singletdominated scalar is not very heavy.
Moreover, the presence of resonant annihilation via s exchange can relax the lower limit
on the LSP mass from LUX constraints on the SD crosssection, as seen in the right panel
of gure 4, because in such a case smaller higgsino component is required to obtain correct
relic density.
4
Blind spots and relic density with light singlets
Now we move to the case when the singletdominated scalar is lighter than the SM Higgs.
Neglecting the e ect from the heavy doublet H exchange for the SI cross section (i.e. setting
fH to zero) the blind spot condition may be written in the following form [29]:
6The sign of the second term in the r.h.s. of (3.17) is determined by the sign of product
m but for
considered here case of m
> 0 it is the same as that of
(4.1)
measure (in the large tan
limit) the ratio of the couplings, normalized to the SM values,
of the hi (= s; h) scalar to the b quarks and to the Z0 bosons.
In the rest of this section we will consider singlinolike LSP, because the case of
higgsinodominated LSP does not di er much from the one described in section 3.2. The blind spot
condition (4.1), analogously to a simpler case (3.12), may be approximated by a quadratic
equation for
m
sin 2
1
+ As
As
which has solutions only if
cos2
>
1
2
1
2
+
(4.2)
(4.3)
(4.4)
(4.5)
As
1 + cs
1 + ch
mh
ms
chi
1
tan
1
1+cs
1+ch
m
2!
mh 2
ms
mh 2 :
ms
vh
1
m
vh 1 + 2 1+cs
where
and parameters
The last condition may be interpreted as the upper bound on tan
(lower bound on cos ).
It is nontrivial when its r.h.s. is positive i.e. when the second term under the absolute
value is negative but bigger than
1. The bound is strongest when that term equals
However, usually the absolute value of that term is smaller then 12 because the hs mixing
measured by
is rather small. So, typically the bound on tan
becomes stronger with
increasing LSP mass or increasing j j (with other parameters xed). We focus again on
the more interesting case m
> 0 because for m
< 0 the blind spot condition may be
satis ed only for the LSP strongly dominated by singlino which typically leads to too large
relic density.7 Since in this section we consider ms < mh, jAsj is typically larger than j j
(unless cs and/or ch deviate much from 1, which under some conditions may happen which
we discuss in more detail later in this section) { in such a case the condition (4.5) is always
ful lled for
< 0 (see gure 5), whereas for
> 0 (see comment in footnote 6 ) there
1
2
.
is an upper bound on tan
which gets stronger for larger values of j
j
.
In the following discussion we focus on big values of j j because they lead to a relatively
big positive contribution to the Higgs mass from the Higgssinglet mixing [53]. In our
numerical analysis we take j j = 0:4 which corresponds to
mix
4 GeV. In order to
emphasize new features related to the modi cation of the BS condition we also consider
rather large values of j j
O(0:1). For such choices of the parameters there are no viable
blind spots for
> 0 in accord with the discussion above so we focus on
< 0.
7For m
< 0, h
2
0:12 may be obtained only when resonant annihilation via singlet (pseudo)scalar
exchange is dominant (we describe this phenomenon for m
> 0) or
which is less interesting from
the point of view of the Higgs mass.
HJEP07(21)5
at 4
fl
n
fl
n
01
8
8
3
5
5
7
7
2
2
05
05
at 4
3 L H C
L E P
01
01
I C
.01
.012
.014
510
510
.01
.012
.014
I C
L H C
520
520
.01
.014
I C
L U X
510
510
520
520
L H C
02
02
mix
4 GeV. In brown areas NMSSMTools
reports unphysical global minimum (UM) while in grey ones the LHC constraints on the Higgs
production and decay are violated (LHC). Black regions in the upper right corners denote the
points where the condition (4.5) does not hold.
The fact that our blind spot condition now comes from destructive interference between
fh and fs amplitudes (rather than vanishing HiggsLSP coupling) in uences strongly the
relation between the DM relic density, especially in small LSP mass regime, and other
experimental constraints. Now the HiggsLSP coupling is not negligible so the LSP mass
below mh=2 is forbidden, or at least very strongly disfavored, by the existing bounds on
the invisible Higgs decays [54]. Thus, resonant annihilation with Z0 or s (in this section
we consider s lighter than h) exchanged in the s channel can not be used to obtain small
enough singlinolike LSP relic abundance. As concerning the h resonance: it may be used
but only the \half" of it with mLSP & mh=2 (this e ect is visible in all panels of gure 5).
However, we found that other experimental constraints, such as the ones from the LHC
and/or LUX exclude even this \half" of the h resonance when the mixing parameter j j
is large.
In general NMSSM the masses of singletlike scalar s and pseudoscalar a are
independent from each other so let us
rst consider the situation when a is heavy.
The
case with ma = 1 TeV is presented in the upper left panel of
gure 5. The contours
of
h
2 = 0:10; 0:12; 0:14 above the tt threshold are quite similar to the case with heavy
singlet. The only di erence is that now somewhat larger values of tan
are preferred but
even in this case they cannot exceed about 5.
Let us now check what happens when the lighter pseudoscalar is also singletdominated
(i.e. a1 = a) and relatively light. The existence of such light pseudoscalar is very important
for both the relic abundance of the LSP and the constraints from the IceCube experiment.
Let us now discuss these e ects in turn.
The DM relic density is in uenced by a pseudoscalar in two ways. First is given
by possible resonant annihilation with a exchanged in the s channel. This possibility is
interesting only for ma & mh because for lighter a one still has problems with nonstandard
Higgs decays constrained by the LHC data (see above).8
However, as for any narrow
resonance, the DM relic density may be in agreement with observations only for a quite
small range of the DM mass (for a given a mass). One can see this in three panels (except
the upper left one) of gure 5. Second e ect is related to new annihilation
nal states
including the singletdominated pseudoscalar, namely sa, ha, aa (in addition to similar
HJEP07(21)5
channels involving only scalars: ss, sh, hh). It is best illustrated in the upper right panel
of gure 5 for ma = 260 GeV. In this case the sa threshold roughly coincides with the tt
threshold. Near this threshold the curves of constant
h
2
0:12 go up towards bigger
and leave the region excluded by the LUX data for smaller LSP mass than
in the case with heavy singlets. The reason is quite simple. With increasing contribution
from the annihilation channels mediated by (nonresonant) pseudoscalar exchange smaller
contribution from the channels mediated by Z0 exchange is enough to get the desired value
h
2
0:12. Moreover, smaller LSPZ0 coupling is obtained for bigger values of tan
so
larger values of tan
are preferred than in the case with heavy a. As a result the lower
possible LSP mass consistent with the LUX SD limits is almost 100 GeV smaller when a
is relatively light. The precise values depend on the relic density and for
around 200 GeV instead of around 300 GeV as in the case with heavy singlets.
The behavior of the
h2 = 0:12 0:02 curves close to and slightly above the tt threshold
depends on the parameters. Particularly important is the sign of . We see from the right
panels of
gure 5 that even for the same mass of the pseudoscalar, ma = 260 GeV in
this case, the plots are very di erent for di erent signs of . Most di erences originate
from the fact that
> 0 implies ch > 1 and cs < 1 while for
< 0 the inequalities are
reversed. There are two important implications of these correlations which we describe in
h
2 = 0:12 is
the following.
Firstly, the LHC constraints from the Higgs coupling measurements are stronger for
> 0 because in such a case the Higgs coupling (normalized to the SM) to bottom quarks
is larger than the one to gauge bosons. In consequence, the Higgs branching ratios to gauge
bosons is suppressed as compared to the SM. Moreover, nonzero
results in suppressed
Higgs production crosssection so if j j is large enough the Higgs signal strengths in gauge
boson decay channels is too small to accommodate the LHC Higgs data which agree quite
well with the SM prediction. Moreover, a global t to the current Higgs data shows some
suppression of the Higgs coupling to bottom quarks [48]9 which disfavors ch > 1, hence
also large
> 0. It can be seen from the upper right panel of gure 5 that for
= 0:4 the
8The situation may be di erent if the blind spot condition is of the standard form (3.1)  we will come
to this point at the end of this section.
9The current t also indicates an enhancement of the top Yukawa coupling. In NMSSM suppression of
the bottom Yukawa coupling is correlated with enhancement of the top Yukawa coupling which has been
recently studied in refs. [55, 56].
LHC excludes some of the interesting part of the parameter space which is allowed by LUX
due to the LSP annihilations into sa nal state. The LHC constraints can be satis ed for
small values of
but this comes at a price of smaller
mix, hence somewhat heavier stops.
Secondly, jAsj is larger for
< 0 than in the opposite case (see eq. (4.2)). Moreover,
since deviations of cs and ch from 1 grow with tan , jAsj increases (decreases) with tan
for negative (positive) . For
> 0, this implies that for large enough tan
the r.h.s. of the
blind spot condition (4.4) changes sign and
< 0 is no longer preferred. Equivalently,
the upper bound on tan
in eq. (4.5) gets stronger as tan
grows so it is clear that
at some point condition (4.5) is violated. The appearance of the violation of the blind
spot condition is clearly visible at large tan
in the upper panels of gure 5 (the black
< 0 instead the blind spot condition may be always ful lled for
(by taking e.g. appropriate value of ).10 Moreover, there is interesting phenomenon that
may happen for
< 0 above the tt threshold which is well visible in the lower right panel
of gure 5. The values of tan
corresponding to
h2 = 0:12 grow rapidly just above the
tt threshold and there is a gap in the LSP masses for which there are no solutions with
SI BS and observed value of . Such solutions appear again for substantially bigger mLSP
(above 300 GeV in this case). The reason why
h
2 = 0:12 curve is almost vertical near
mLSP
mt and the gap appears is related to the fact that m =
varies very slowly with
tan , which results in fairly constant jN123
N124j which determines the tt annihilation
crosssection, hence also
h2. The weak dependence of m = on tan
originates from the
fact that for increasing tan
both sides of the blind spot condition (4.4) grow. The l.h.s.
grows because of decreasing sin(2 ) while the r.h.s. due to increasing cs. Of course, the
fact that these two e ects approximately compensate each other relies on speci c choice of
parameters and does not necessarily hold e.g. for di erent values of .
The presence of light a in uences also the IceCube constraints in a way depending
on the LSP mass. For m
& (ma + ms)=2 (assuming ma > ms) the IceCube constraints
are very much relaxed and become practically unimportant for the cases discussed in this
section. This is so because the additional annihilation channels (into sa, ha, ss, sh) at
v = 0 lead to softer neutrinos as compared to otherwise dominant V V channels (or tt
channel for even heavier pseudoscalar). The situation is di erent (and more complicated) for
LSP masses between the W +W
and as thresholds. In this region one can have
destructive/constructive interference between Z0 and amediated amplitudes11 for bb annihilation
at v = 0 which strengthens/reduces the constraints (IceCube limit on SD cross section are
two orders of magnitude stronger for V V than for bb). In our case the e ect depends on
the sign of : for
and vice versa for
< 0 the IceCube limits are strengthen (relaxed) for m
. ma (& ma)
> 0  see the lower panels of gure 5.
In the examples considered in this section and shown in gure 5 we have chosen the
sign of
to be positive (then m is also positive because, as discussed earlier, there are
no interesting solutions with m
< 0). The corresponding solutions with negative , and
also changed signs of other parameters like ,
and A , are qualitatively quite similar.
10The only exception is when ch <
1 but this may happen only for very large tan
and/or very light H.
11Note that these are the only nonnegligible amplitudes which have nonzero a term in
v expansion {
see (B.1).
Of course there are some quantitative changes. Contours in the corresponding plots are
slightly shifted towards smaller or bigger (depending on the signs of other parameters)
values of tan .
As explained in detail in ref. [29], one can also get vanishing spinindependent cross
section when the standard BS condition (3.1) is ful lled. Then, the scalar sector has no
e ect on the blind spot condition. In such a case, we can have as large
mix as is allowed
solely by the LEP and LHC constraints, irrespective of the DM sector. Interestingly,
the standard BS condition appears also, in a nontrivial way, when j j is relatively small
(we still consider singlinodominated LSP) and both terms in the denominator of (3.13)
are comparable and approximately cancel each other. The blind spot condition (3.12)
xed and not very large. From eq. (A.4) we see that in such a case
a small denominator, for a singlinodominated LSP, may be compensated by small factor
(m =
sin 2 ) which means that the simplest BS condition (3.1) is approximately ful lled.
In both cases, the analysis performed in subsection 3.1 holds. However, it should be noted
S3 and hence may decrease the
abovethat small j j weakens the singlet selfinteraction
mentioned e ects from the a exchange.
5
All the analysis presented till this point apply to a general NMSSM. In this section we
focus on the most widely studied version of NMSSM with Z3 symmetry. In this model
there is no dimensionful parameters in the superpotential:
WNMSSM =
SHuHd + S3=3 ;
while the soft SUSY breaking Lagrangian is given by (2.2) with m23 = m0S2 =
S = 0.
This model has ve free parameters less than general NMSSM which implies that some
physical parameters important for dark matter sector are correlated. The main features of
Z3symmetric NMSSM relevant for phenomenology of neutralino dark matter are
summarized below:
(5.1)
(5.2)
(5.3)
sgn(m
) = sgn( )
LSP dominated by singlino implies
j j <
1
2
:
Neither singletlike scalar nor singletlike pseudoscalar can be decoupled due to the
following treelevel relation (for singlino LSP after taking into account the leading
contributions from the mixing with both scalars coming from the weak doublets, h^
and H^ ):
1
3
ms2 +
m2a
m2LSP + 2(ms2
m2h) :
Masses of both singletdominated scalar and pseudoscalar are at most of order mLSP.
0 : 7
0 : 6
0 : 6
03
M S U S Y > 5 T e V
04
> 0 in the Z3invariant NMSSM for several values of
with ms = 200 (left panel) and
500 GeV (right panel). The color code is the same as in the right panel of
gure 3. The lines
denoted by UM depict regions in which NMSSMTools reports unphysical global minimum.
Phenomenologically viable (small) Higgssinglet mixing leads to the following
treelevel relation:
MA
jMA^A^j
2j j
sin(2 )
r
1
sin(2 )
2
2j j ;
sin(2 )
(5.4)
where the last approximation is applicable for large tan
and/or singlinolike LSP and
forbids resonant LSP annihilation via heavy Higgs exchange. Such resonance may be
present only for
1 since only in such a case signi cant deviation from relation (5.4)
is possible. Important constraints on dark matter sector of Z3symmetric NMSSM follow
from relation (5.3) which we discuss in more detail in the following subsections.
5.1
Heavy singlet scalar
Let us rst discuss the case of heavy singlet scalar in which only the Higgs exchange is
relevant in the SI scattering amplitude and the SI blind spot has the standard form (3.1).
In this case j j must be close to zero to avoid large negative correction to the Higgs mass
and eq. (5.3) implies mLSP > ms. This is demonstrated in gure 6 where it is clearly seen
that for mLSP . ms there are no solutions (due to a tachyonic pseudoscalar).12
We also note that eq. (5.3) implies that resonant LSP annihilation via singletlike scalar
or pseudoscalar is typically not possible in this case.13 On the other hand, eq. (5.3) implies
that the LSP annihilation channel into sa via a exchange is almost always open (for small
and ms > mh this channel is kinematically forbidden only in a small region of
the parameter space for which ma
3ms). This allows for smaller annihilation rate into
12In
gure 6 vanishing h^s^ mixing is assumed but for small nonzero mixing
(preferred by the Higgs
mass) the results are similar.
13Due to loop corrections to eq. (5.3) one may nd some small regions of resonant annihilation mediated
by a singlet for ms not far above mh and large
close to the perturbativity bound. We discuss this e ect
in more detail in subsection 5.2 because it is more generic for ms < mh.
tt, hence also for smaller higgsino component of the LSP and larger tan . In consequence,
larger LSP masses consistent with
h
2 = 0:12 and perturbativity up to the GUT scale
are possible than in the case with both singlets decoupled (compare gure 6 to gure 3).
For the same reason large enough LSP masses are beyond the reach of XENON1T, as seen
from
gure 6.
The situation signi cantly changes when singletlike scalar is light, especially if the
Higgssinglet mixing is not small (which enhances the Higgs mass if ms < mh). This is because
the blind spot condition changes to eq. (4.4). Moreover, for light singlet the loop corrections
to condition (5.3) can no longer be neglected which under some circumstances allows for
resonant LSP annihilation via the schannel exchange of a.
In the Z3symmetric NMSSM the singletdominated pseudoscalar a plays quite
important role for the relic density of the singlinodominated LSP. First we check if and when the
schannel exchange of a may dominate the LSP annihilation cross section and lead to the
observed relic density. Of course, this may happen if we are quite close to the resonance,
i.e. when ma
2mLSP. It occurs that it is not so easy to ful ll this requirement in the
Z3symmetric model. This is related to the condition (5.3) which, for ma
2mLSP and
after taking into account the loop corrections in eqs. (2.6) and (2.14), may be rewritten in
the form
1
3
ms2 +
m2LSP + 2 m2h
ms2
1
3
s^s^ +
a^a^ :
The l.h.s. of the above expression is positive so this condition can not be ful lled without
the loop contributions. The last equation may be treated as a condition for the size of
the loop corrections necessary to have resonant annihilation of the LSP mediated by the
pseudoscalar a. In order to understand qualitatively the impact of condition (5.5) on our
analysis it is enough to consider the following simple situation: we assume that the scalar
is negligible and the BS is approximated by (3.1). On the r.h.s. of eq. (5.5) we
take into account only the rst term of the loop correction
s^s^ [57]
(5.5)
(5.6)
(5.7)
s^s^
1
ms2
m2LSP
"
tan
2
2
ln
2MSUSY
mLSP tan
MS2USY
m2LSP
1
3
#
:
(the second term is subdominant because m2LSP
2 < 14 2 due to condition (5.2)). In such approximation and for tan
2 for a singlinodominated LSP and
1 condition (5.5)
simpli es to
For given values of
and ms any change of mLSP must be compensated by appropriate
change of tan . The expression in the square bracket has a maximum as a function of
tan
approximately at 1:2MSUSY=mLSP. Thus, to keep the r.h.s. constant in order to stay
close to the resonance one has to decrease tan
for small mLSP and increase for large mLSP.
In our numerical examples presented in
gure 7 we x MSUSY = 4 TeV so the maximum
of the square bracket corresponds to tan
about 30 (10) for the LSP masses of 150 (500)
01
21
nta 5
9
6
8
4
3
7
2
‚ = 0 : 5 ; = 0 ; m s = 9 5 G e V ; „ > 0 ; 3 ¡ N M S S M ; f s + f h = 0
‚ = 0 : 5 ; = 0 : 1 ; m s = 9 5 G e V ; „ > 0 ; 3 ¡ N M S S M ; f s + f h = 0
01 150 20 250
L U X
41
61
31
51
21
at
nfl 7
9
8
6
4
3
5
I C
I C
01
4
9
6
8
3
7
2
GeV. For small j j the a resonance occurs at the blind spot for tan
of order 10. That is
why tan
typically decreases with mLSP, as can be seen from
gure 7. Local minimum for
tan
is present only in the lower panel of gure 7 because more negative values of
lead in
general to larger tan
(see gure 8 and discussion at the end of this section). Nevertheless,
in every case the
h2 = 0:12 curves corresponding to the a resonance have horizontallike
behavior: do not change very much with the LSP mass (and have values of tan
of order
10 for MSUSY = 4 TeV that we use in our numerical examples). This should be compared
to the general case when such curves are almost vertical (narrow ranges of the LSP mass
but wide ranges of tan )  see gure 5. This di erence comes from the fact that in the
general model there are more parameters and eq. (5.3) is not ful lled.
Figure 7 shows that there are two situations for which BS and correct value of DM
relic density are still compatible with the latest bound on DM SD crosssection. One is
the above discussed case of resonant annihilation with the light pseudoscalar exchanged in
the s channel. The second one occurs for smaller tan
but bigger mLSP and corresponds
to annihilation via nonresonant exchange of particles in the s channel. Usually the main
contribution to the annihilation crosssection in such a case comes from the exchange of
Z0 boson decaying into tt
nal state. This process allows to avoid the LUX bounds on
SD for mLSP above about 300 GeV but is not su cient to push SD below sensitivity of
XENON1T, as discussed in section 3. The situation changes when new
nal state channels,
especially as, open. Then not only the present bounds on SD may be easily ful lled but
some parts of the parameter space are beyond the XEXON1T reach. We see from
gure 7
that for light singlets the lower limit on the LSP mass from LUX may be relaxed to about
250 GeV. The e ect of annihilation into light singlets is even more important for heavier
LSP so XENON1T may not be sensitive to LSP masses above about 400 GeV.
02
12
41
2
61
81
91
01
31
71
51
fln 12
at 1
9
6
8
7
.02
.02
.0
21.0
01.
tan 4.0
.90
.80
.60
.70
.20
.50
.30
.15
‚ = 0 : 5 ; m s = 9 5 G e V ; „ > 0 ; 3 ¡ N M S S M ; f s + f h = 0
‚ = 0 : 5 ; m s = 9 5 G e V ; „ > 0 ; 3 ¡ N M S S M ; f s + f h = 0
m L S P = 1 5 0 G e V
m L S P = 3 0 0 G e V
m L S P = 4 5 0 G e V
.30
.10
.35
.05
.0
NMSSM. Left panel: resonant annihilation via a exchange for mLSP = 150 GeV. Right panel:
nonresonant annihilation for mLSP = 300 and 450 GeV. The green parts of the contours are excluded
by LUX. The parabolalike curves show dependence of
mix (on the right horizontal axes) on .
In both cases discussed above the allowed values of tan
are correlated with the LSP
mass. The exact form of such correlation depends on the s^h^ mixing parameter . Quite
generally values of tan
decrease with . This is illustrated in gure 8 where the bands of
allowed tan
between tan
as functions of
are shown for a few values of the LSP mass. This correlation
and
can be easily understood from eqs. (4.2){(4.4). The rst factor on the
r.h.s of (4.4) grows in the rst approximation like
. This can not be compensated by
decreasing
because in the Z3symmetric NMSSM
is xed by the LSP mass. The BS
condition (4.4) with increasing r.h.s. may be ful lled by decreasing the absolute value of
the negative contribution to its l.h.s. i.e. by increasing tan .
6
Motivated by the recent strong LUX constraints we investigated consequences of the
assumption that the spinindependent crosssection of singlinohiggsino LSP scattering o
nuclei is below the irreducible neutrino background. We determined constraints on the
NMSSM parameter space assuming that the LSP is a thermal relic with the abundance
consistent with Planck observations and studied how present and future constraints on
spindependent scattering crosssection may probe blind spots in spinindependent direct
detection.
In the case when all scalars except for the 125 GeV Higgs boson are heavy the new
LUX constraints exclude the singlinohiggsino masses below about 300 GeV unless the LSP
mass is very close to the half of the Z0 boson mass (between about 41 and 46 GeV). In
the allowed region LSP dominantly annihilates to tt and tan
must be below about 3.5
(assuming perturbative values of
up to the GUT scale) with the upper bound being
stronger for smaller
and heavier LSP. There is also an upper bound of about 700 GeV
assuming perturbativity up to the GUT scale. We found that XENON1T has sensitivity
to exclude the entire region of dark matter annihilating dominantly to tt. This conclusion
apply to general models of singletdoublet dark matter. On the other hand, the LSP
resonantly annihilating via Z0 boson exchange is possible only for large tan
unless
is very small e.g. for
> 0:5, tan
& 20. Only small range of LSP masses around the
resonance of about 2 GeV is beyond the XENON1T reach while LZ is expected to probe Z0
resonance completely. In all of the above cases the LSP is dominated by singlino. Current
and future constraints can be avoided also for very pure higgsino with mass in the vicinity
of 1.1 TeV.
The situation signi cantly changes when singletlike (pseudo)scalars are light. Firstly,
the presence of light CPeven singlet scalar modi es the condition for spinindependent
blind spot when its mixing with other Higgs bosons is nonnegligible. Depending on the
sign of the mixing angle between the singlet and the 125 GeV Higgs preferred values of
tan
may be either increased or decreased, as compared to the case with heavy singlet.
Interestingly, tan is increased when the Higgs coupling to bottom quarks is smaller than that to gauge bosons which is somewhat favored by the LHC Higgs coupling measurements. Secondly, the presence of light singlets opens new annihilation channels for the LSP.
As a result, correct relic abundance requires smaller higgsino component of the LSP which
relaxes spindependent constraints. We found that resonant annihilation via exchange of
singlet pseudoscalar is possible even in the Z3invariant NMSSM. Interestingly, even far
away from the resonant region the lower limit on the mass of LSP annihilating mainly to tt
may be relaxed to 250 GeV. For larger LSP masses sa may become dominant annihilation
channel and the LSP masses above 400 GeV may be beyond the reach of XENON1T.
Acknowledgments
This work has been partially supported by National Science Centre, Poland, under
research grants DEC2014/15/B/ST2/02157, DEC2015/18/M/ST2/00054 and
DEC2012/04/A/ST2/00099, by the O
ce of High Energy Physics of the U.S. Department
of Energy under Contract DEAC0205CH11231, and by the National Science
Foundation under grant PHY1316783.
MB acknowledges support from the Polish Ministry
of Science and Higher Education through its programme Mobility Plus (decision no.
1266/MOB/IV/2015/0). PS acknowledges support from National Science Centre, Poland,
grant DEC2015/19/N/ST2/01697.
A
LSPnucleon cross sections
In this appendix we collect several expressions useful in discussing the SI and SD
crosssections of LSP on nuclei.
The couplings of the ith scalar to the LSP and to the nucleon, appearing in the
formula (2.19) for the SI crosssection, after decoupling the gauginos are approximated,
respectively, by
hi
hiNN
ai
p
2
mN
p2v
p
i 2
S~hih^N15 (N13 sin
+ N14 cos ) + S~
hiH^ N15 (N14 sin
N13 cos )
+ S~his^ N13N14
;
S~hih^ F (N) + Fu(N) + S~hiH^ tan F (N)
d d
1
tan
Fu(N)
The LSP couplings to pseudoscalars, important for the relic abundance calculation, are
approximated by ref. [57]
where P~ij are elements of the matrix diagonalizing the pseudoscalar mass matrix de ned
in eq. (2.15).
Parameter
de ned by eq. (3.13) and convenient for the discussion of SI blind, using
eqs. (2.17) and (2.18), may be written in the form
=
v 2
1 +
1
m
2
m
2
sin 2
2
m
m
sin 2
1
B
The LSP (co)annihilation channels
In this appendix we will use the following expansion of v around v = 0:
v = a + bv2 + O(v4) :
With the help of eqs. (2.17) and (2.18), the combination of the LSP components crucial
for SD, (N123
N124), may be written as:
N123
h
1
(m = )2i (1
1 + (m = )
2
N125) cos 2
2 (m = ) sin 2
We can see immediately that the crosssection disappears in the limit of tan
= 1 or a
pure singlino/higgsino LSP. The ratio of the higgsino to the singlino components of the
LSP may be calculating from eqs. (2.17) and (2.18):
1
N125
N125 =
vh
2 1 + (m = )
2
2(m = ) sin 2
:
h
1
(m = )
2i2
Using this relation we may rewrite formula (A.5) in the form:
Then, the relic density may be written as [58]:
2
9:4
10 12 GeV 2 xf
a + 3b=xf
;
where xf
Resonance with the Z0 boson (unitary gauge)
Let us consider the LSP annihilation into the SM fermions (except the t quark14) via Z0
exchange in s channel. The expansion coe cient a and b in eq. (B.1) are equal to:
a =
b0 =
g
4
g
4
32 (4m2
X cF (2 F2
F
2 F + 1) ;
(B.2)
(B.3)
(B.4)
b0v2
(B.5)
where g
p(g12 + g22)=2, cF = 1 for leptons and 3 for quarks, whereas F = 2jqF j sin W2.
The 0 index in b0 parameter means that we put fermion masses to 0 (which is a very good
approximation for m
the t quark) in (B.4) equal
4m2 =m2Z0 ) which means that b
mZ0 =2; of course a0 = 0). The sum over the SM fermions (except
14:6. It is worth noting that b0
m2 and a
m2F (1
0 (in contrary to naive expectation). Moreover,
the terms proportional to higher powers of v2 in
v (for m
mF ) are suppressed with
respect to bv2 term in geometric way by v2=4. Therefore we can approximate
v
and hence expressed the relic density in the form of eq. (B.2). We will however improve
slightly this approach (see appendix C) and write our formula in the following form:
h
2
0:1
0:3
N123
N124
Z0
m2Z0 5 :
where the term proportional to v2
0:52 stems from the fact that the dark matter particles
posses some thermal energy during the freezeout. Eq. (B.5) reproduces very well the results
obtained from MicrOMEGAs far from the resonance (see e.g.
gure 1), however very close
to the resonance, especially for m
. mZ0 =2, the di erence may be sizable ( gure 3).
B.2
Annihilation into tt via Z0
In this case the dominant contribution also comes from Z0 exchange in s channel but in
contrary to the previous paragraph m
mF (= mt). Therefore the statement that b
a
is now longer true. It becomes clear when we write down the expression for a and b terms
14The e ect from the t quark appears for m
discuss this case separately in the next paragraph.
mt which is quite far from the resonance  we will
One can see that for m
mt both terms are comparable whereas for larger m
we have
a=b
4 and eq. (B.6) su ces (as we would expect, the terms proportional to higher powers
of v2 are suppressed for m
mt; mZ0 as v2=4). Similarly to eq. (B.5) we can nd the
expression for
h2. Combining (B.2) with (B.6) and (B.7) we get:
2
0:1
0:05
N123
N124
2
mm2t2 +
3 1
4 xf
1
mt2
2m2
3 1
r
1
1
7
m2
(B.8)
The above equation works well for m
& 175 GeV (see gure 1), however for m
mt we
have to be more careful because the expansion in v
2 breaks down. One can see that for
mt; mZ0 the square bracket in (B.8) equals roughly 1 and
h2 depends on jN123
N124j
only. Similarly to the case of the resonance with Z0, the crucial experimental bounds comes
from SD direct detection (see right plot in gure 2).
It is worth pointing out that both a and b coe cients in (B.6) and (B.7) come purely
from
into account this term is also crucial for DM annihilation in galactic halos (v2
10 6) for
p p =m2Z0 term in Z0 propagator. It was noticed long time ago [59, 60] that taking
mZ0 =2. This is because the a coe cient in (B.3) vanishes which causes large dip in
the annihilation cross section.
Improved formula for
h2 near a resonance
The method described below may be found e.g. in [58, 61]. Let us consider a general
expression for v for scalar dark matter (with mass m) annihilating via s channel exchange
of a particle with mass M and total decay width :
in the limit m
mZ0 :
b
3g4
32
3g4
32
(N123
(N123
mt2
m2
:
ing dimensionless quantities
approximation s = 4m2=(1
For simplicity we assume we assume
= const which is generally not the case, however
we are mainly focused on the e ect on
h2 coming from the denominator in (C.1).
Us4m2=M 2
1,
=M and considering nonrelativistic
v =
(s
M 2)2 + 2M 2
v2=4)
4m2(1 + v2=4) we get:
v =
=M 4
(B.6)
(B.7)
(C.1)
(C.2)
Let us now de ne Y (x)
ns , where x = T =m, and write
1
Y (1)
1
Y (xd)
= m MPl pg
gs r
45 xd
Z 1 h vi dx :
x2
Parameter xd is de ned as a moment in thermal evolution of DM when the term 1=Y (xd)
starts to be small and can be safely neglected. Dark matter relic abundance can be then
calculated by double integration over v and x:
h2 =
GeV
MPl gs
1
p
2
Z 1
0
dv ( v)v2
Z 1
xd
dx
e v2x=4 # 1
p
x
Note that we changed the usual order of integration. We will now perform the simpler
integral over x, obtaining:
1
p
2
Z 1
0
dv ( v)v2
Z 1
xd
dx
e v2x=4
p
x
Z 1
0
( v)v erfc(v=2pxd) dv :
Substituting here eq. (C.2) and erfc(v=2pxd)
pxd= v + : : : we can easily nd simple
expressions for
h2 for some hierarchical values of
and
e.g.
etc.
In the case of fermionic dark matter our expression (C.1) generalizes to:
(C.3)
(C.4)
(C.5)
(C.6)
(C.7)
(C.8)
(C.9)
(C.10)
v =
(s
Z 1 f (v)v erfc(v=2pxd) dv :
( + v2=4)2 + 2
1
f (s)
For xd = 25 we have v
0:5. The above method e ectively includes the fact that the dark
matter particles posses some thermal energy during their freezeout. Other cases of f (v)
can be also easily analyzed and compared with numerical results.
Now we have to perform the following integral
In order to proceed further we have to specify the formula for f (s). In the case of LSP
annihilation into fermions via Z0 exchange the dominant contribution is f (v)
v2 { see
appendix B.1. Analytical form of the above integral is very complicated even for such
simple expression for f (v). The numerator in eq. (C.8) has a maximum for some speci c
value of v. Therefore we will take the denominator in front of the integral, substituting
v ! v, where v is de ned as a mean value of the numerator. Then we have:
: : :
M 4 ( + v2=4)2 + 2 0
1
v
Z 1 v3 erfc(v=2pxd) dv = : : :
R01 v v3 erfc(v=2pxd) dv
R01 v3 erfc(v=2pxd) dv
64
15p
3=x2d
M 4 ( + v2=4)2 + 2
d
x 1=2 :
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