VOLUME 88, N
UMBER 9 PHYSICAL REVIEW LETTERS 4M
ARCH 2002
Thermally Generated Gauge Singlet Scalars as Self-Interacting Dark Matter
John McDonald*
CERN, Theory Division, 1211 Geneva, 23, Switzerland
(Received 5 July 2001; revised manuscript received 23 July 2001; published 15 February 2002)
We show that a gauge singlet scalar S, with a coupling to the Higgs doublet of the form l
S
S
y
SH
y
H
and with the S mass entirely generated by the Higgs expectation value, has a thermally generated relic
density V
S
艐 0.3 if m
S
艐共2.9
10.5兲共V
S
兾0.3兲
1兾5
共h兾0.7兲
2兾5
MeV. Remarkably, this is very similar to
the range 关m
S
苷 共6.6
15.4兲h
2兾3
MeV兴 required in order for the self-interaction 共h兾4兲共S
y
S兲
2
to account
for self-interacting dark matter when h is not much smaller than 1. The corresponding coupling is l
S
艐
共2.7 3 10
210
3.6 3 10
29
兲共V
S
兾0.3兲
2兾5
共h兾0.7兲
4兾5
, implying that such scalars are very weakly coupled
to the standard model sector.
DOI: 10.1103/PhysRevLett.88.091304 PACS numbers: 95.35. +d, 12.60.Fr, 98.80.Cq
1. Introduction.— It has become apparent that conven-
tional collisionless cold dark matter (CCDM) may have
problems accounting for the observed structure of galax-
ies.
N-body simulations with CCDM indicate that galaxies
should have singular halos [1,2] with large numbers of sub-
halos [3,4]. Observationally, the density profile of galaxies
in the inner few kiloparsecs appears to be much shallower
than predicted by numerical simulations (the central den-
sity of dark matter halos being 50 times smaller than the
CCDM prediction for dwarf galaxies and roughly indepen-
dent of halo mass [2,5]), while the number of dwarf galax-
ies in the local group is an order of magnitude fewer than
predicted [3,4]. In addition, the CCDM predictions for
the Tully-Fisher relation [6,7] and the stability of galac-
tic bars in high surface brightness spiral galaxies [8] are
not in agreement with what is observed, indicating lower
density galaxy cores than predicted by CCDM. Although
there is at present considerable uncertainty regarding the
interpretation of observations and simulations [9,10], it has
nevertheless been argued that all the discrepancies between
observations and simulations may be understood as indi-
cating that dark matter halos in CCDM simulations are
more centrally concentrated than observed [11].
In order to overcome the possible deficiencies of CCDM
halos, one suggestion has been that the cold dark mat-
ter particles have a nondissipative self-interaction [12,13],
and it has been shown that such cold, nondissipative self-
interacting dark matter (SIDM) can be effective in alle-
viating the various problems of CCDM [11]. Scattering
of dark matter particles stops gravitational accretion at the
center of the halo and so allows a smooth core to form.
Simulations with SIDM [11,14] are able to simultaneously
account for the observed density profiles of galactic ha-
los and the number of subhalos [11]. In the future SIDM
may be strongly constrained by gravitational lensing ob-
servations of the shape of cluster halos [15–17] and by the
formation of massive black holes at the centers of galax-
ies, which is enhanced by self-interactions of dark matter
particles [18].
In order to be able to account for the observed properties
of dark matter halos, the requirement on the mass
M and
self-interaction scattering cross section s of the SIDM
particles is that [13]
r
S
苷
s
M
苷 共2.05 3 10
3
2.57 3 10
4
兲 GeV
23
. (1)
The upper bound corresponds to the limit at which galaxy
halos in massive clusters are destroyed by interacting with
hot particles in the cluster halo (evaporation) [13], while
the lower bound corresponds to the limit where the SIDM
particle would not interact within a typical galactic halo
during a Hubble time [12,13].
The canonically simplest dark matter particle is arguably
a gauge singlet scalar S. The possibility that gauge singlet
scalars, interacting with the standard model sector via a
coupling to the Higgs doublet of the form S
y
SH
y
H, could
naturally constitute dark matter has been pointed out by
a number of authors in the past [19,20] as well as more
recently [21]. These calculations consider the case of mas-
sive 共.1 GeV兲 scalars which freeze out of thermal equi-
librium when nonrelativistic [22]. However, the range of
S masses considered is too large to account for SIDM with
perturbative S couplings.
It has recently been noted that gauge singlet scalars have
a natural self-interaction via an S
4
-type coupling and so in
principle could account for SIDM [23–25]. An estimate
of the upper bound on the coupling of S scalars to the
Higgs doublets for S mass of the order of 10
100 MeV
(which is of the greatest interest in the case of perturbative
S self-interactions) was derived in [24] by requiring that
S scalars do not come into thermal equilibrium and so
overpopulate the Universe.
In this Letter we consider thermal generation of SIDM
S scalars which do not achieve equilibrium. We will show
that such nonequilibrium thermal generation can naturally
account for a dark matter density of S scalars with the right
properties to account for SIDM.
The Letter is organized as follows. In Section 2 we dis-
cuss the perturbative upper limit on the S scalar mass. In
Section 3 we consider the thermal generation of a relic
density of S scalars. In Section 4 we consider the case of
zero bare S mass and the resulting consistency of the relic
091304-1 0031-9007兾02兾 88(9)兾091304(4)$20.00 © 2002 The American Physical Society 091304-1
VOLUME 88, N
UMBER 9 PHYSICAL REVIEW LETTERS 4M
ARCH 2002
density, S mass, and Spergel-Steinhardt SIDM cross sec-
tion for natural values of the S self-coupling. In Section 5
we present our conclusions.
2. Limit on m
S
for perturbative SIDM.—We first con-
sider the perturbative upper limit on the S mass if it is to
play the role of SIDM. We will consider the case of com-
plex gauge singlet scalars for consistency with the cross
sections and discussion given in [20], which we will use
here. We expect that the results for real scalars will be
very similar. The model is described by
L 苷 ≠
m
S
y
≠
m
S 2 m
2
S
y
S 2l
S
S
y
SH
y
H 2
h
4
共S
y
S兲
2
.
(2)
The total center-of-mass S scattering cross section is the
sum of SS
y
! SS
y
and SS ! SS,
s ⬅ s
SS
y
!SS
y
1s
SS!SS
苷
3h
2
128pm
2
S
. (3)
Therefore
r
S
苷
s
m
S
苷
3h
2
128pm
3
S
, (4)
which implies that
m
S
苷 35.8a
1兾3
h
µ
2.05 3 10
3
GeV
23
r
S
∂
1兾3
MeV , (5)
where a
h
苷 h
2
兾4p. (Similar expressions have been ob-
tained in [24,25].) Thus if we require that a
h
& 1 in order
to have a perturbative theory, then the condition Eq. (1) re-
quires that m
S
苷 a
1兾3
h
共15.4
35.8兲 MeV & 30 MeV. (We
refer to this as the Spergel-Steinhardt mass range.) We note
that this puts a severe bound on the coupling l
S
, since the
S scalar gains a mass from the Higgs expectation value,
m
2
S
苷 m
2
1
l
S
y
2
2
, (6)
where y 苷 250 GeV. Thus the requirement that m
S
&
30 MeV imposes an upper bound on l
S
,
l
S
, 2.9 3 10
28
µ
m
S
30 MeV
∂
2
. (7)
The Spergel-Steinhardt mass range assumes a perturba-
tive S self-coupling. A nonperturbative self-coupling
may be possible, but it would be difficult to calculate
the properties of such a model, so we must restrict
ourselves to the perturbative case. In addition, the
only known scalar self-coupling, that of the stan-
dard model Higgs doublet, l
H
共H
y
H兲
2
, is given by
l
H
苷 m
2
h
兾4y
2
苷 0.053共m
h
兾115 GeV兲
2
. (The experi-
mental lower bound on the Higgs mass is m
h
. 113 GeV
[26], while an upper bound for the pure standard model
is obtained from radiative corrections to electroweak
observables, m
h
, 165 GeV [27]. The upper bound in
extensions of the standard model can be 1 TeV or larger
[28].) This is typically perturbative but not very much
smaller than 1, suggesting that a natural value for the S
scalar self-couplings is around 0.1.
2. Thermal generation of S scalars.—There are two
processes which can produce a density of S scalars: 2 $ 2
annihilation processes and decay of a thermal equilibrium
density of Higgs scalars to SS
y
pairs, h
o
! SS
y
.We
first consider 2 $ 2 annihilations. The relic density from
scattering processes in a radiation dominated Universe is
found by solving the Boltzmann equation [20,22],
df
dT
苷
具s
ann
y
rel
典
K
共f
2
2 f
2
0
兲; K 苷
∑
4p
3
g共T 兲
45M
2
Pl
∏
1兾2
,
(8)
where f 苷 n
S
兾T
3
, f
0
苷 n
0
兾T
3
, and g共T兲 苷 g
B
1 7g
F
兾
8, where g
B
and g
F
denote the number of relativistic
bosonic and fermionic degrees of freedom, respectively.
n
S
is the number density of S scalars and n
0
is the thermal
equilibrium S number density; for relativistic S scalars
n
0
苷
µ
1.2
p
2
∂
T
3
. (9)
We consider the case where the S scalar density is very
small compared with the equilibrium density and solve the
Boltzmann equation with f 苷 0 on the right-hand side,
df
dT
苷 2
具s
ann
y
rel
典
K
f
2
0
. (10)
We take the Universe to be initially at a high temperature,
T ¿ m
W
, and calculate the resulting relic density of S
scalars as the Universe cools. The annihilation cross sec-
tions of relativistic SS
y
pairs to t quarks, W and Z bosons,
and the h
0
Higgs scalars (lighter quarks and leptons do
not contribute significantly due to their very small Yukawa
couplings) are estimated by using the center-of-mass anni-
hilation cross sections calculated for S scalars with typical
energy E
T
艐 T . We will see that the core results of the
paper are not very sensitive to uncertainties in the calcu-
lation of the annihilation cross section and thermal relic
S density. The relativistic annihilation cross sections s
i
may be obtained from the nonrelativistic 具s
ann
y
rel
典 given
in [20] via the relation s
i
苷 共1兾2兲具s
ann
y
rel
典共m
S
! E
T
兲
(where i ⬅ t, W, Z, h
0
denotes the standard model particle
in question), which may be confirmed by directly calculat-
ing the cross sections. For T , m
i
the contribution of
s
i
to the total cross section is zero, which models Boltz-
mann suppression. Then 具s
i
y
rel
典 苷 2s
i
, where we take
y
rel
苷 2 for relativistic annihilations [29]. F
or relativistic
S scalars, f
0
苷 1.2兾p
2
is a constant so Eq. (10) can be
integrated as
f
i
苷 22f
2
0
Z E
T
f
E
T
0
dE
T
s
i
K
, (11)
where E
T
苷 T, E
T
f
苷 m
i
, and the initial thermal energy
E
T
0
! `. We will take K ~ g共T兲
1兾2
to be constant with
g共T兲 苷 g共T
i
兲, where T
i
苷 m
i
, since most of the integral
comes from E
T
close to m
i
. The total contribution to f
is then
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f
T
苷
X
f
i
苷 1.3 3 10
12
l
2
S
∑
1 1 0.27
µ
115 GeV
m
h
∂
1 0.20
µ
m
h
115 GeV
∂
2
∏
.
(12)
In this we have taken g共T
i
兲 苷 106.75, corresponding to
the standard model degrees of freedom and l
t
苷 0.7 (cor-
responding to m
t
苷 175 GeV). In addition, in order to
obtain an analytical result we have expanded the Higgs
propagators in s
i
共i 苷 W, Z, t兲 assuming that 4E
2
T
is large
compared with m
2
h
, which is generally satisfied if m
2
h
is
small compared with 4m
2
W
. (We refer to this as the small
Higgs mass limit.)
The S number density from the decay of thermal equi-
librium h
0
scalars at temperatures less than the electroweak
phase transition (where T
EW
* 1.5m
h
[30]) is given by
dn
S
dt
1 3Hn
S
苷 具G
h
0
典n
h
0
eq
, (13)
where H is the expansion rate and the thermal equilibrium
density of h
0
, n
h
0
eq
, is given by
n
h
0
eq
苷
1
2p
2
Z
`
m
h
E共E
2
2 m
2
h
兲
1兾2
共e
E兾T
2 1兲
dE , (14)
and the decay rate for h
0
scalars with energy E is
G
h
0
苷
l
2
S
y
2
16pE
. (15)
Thus the thermal average of the decay rate is
具G
h
0
典 苷
1
n
h
0
eq
l
2
S
y
2
T
2
e
2m
h
兾T
h共m
h
兾T兲
32p
3
;
h共a兲 苷
Z
`
0
t
1兾2
共t 1 2a兲
1兾2
共e
t
2 e
2a
兲
dt . (16)
Therefore in terms of f, the S density from h
0
decays is
given by
df
dT
苷 2
具G
h
0
典f
h
0
eq
KT
3
⬅ 2
h共m
h
兾T兲
KT
4
l
2
S
y
2
e
2m
h
兾T
32p
3
. (17)
h共a兲 is a slowly varying function of a, with h共0兲 苷 1.64,
h共1兲 苷 1.87, and h共5兲 苷 3.00. Since most of the contri-
bution to f comes from m
h
兾T ⬃ 1, we take h共m
h
兾T兲 to
be equal to h共1兲, in order to obtain an analytical expres-
sion. Therefore the density of S scalars from h
0
decay,
f
dec
, is given by
f
dec
苷
l
2
S
y
2
h共1兲
16p
3
Km
3
h
⯝ 1.08 3 10
14
l
2
S
µ
115 GeV
m
h
∂
3
. (18)
In this we have assumed that y is given by its T 苷 0 value,
y 苷 250 GeV. Since most of the contribution to f
dec
comes from T & m
h
, T
EW
, this should be a reasonable
approximation. We see that the S density from h
0
decays
is generally much larger than that from 2 $ 2 annihilation
processes in the small Higgs mass limit. For larger values
of the Higgs mass, it is possible that s-channel pole anni-
hilations [29] may result in 2 $ 2 processes dominating
the h
0
decays [31], in which case f
dec
is a lower bound on
the number of S scalars produced thermally.
The resulting density of S plus S
y
scalars is then the
sum of scattering and decay contributions,
V
S
苷
2m
S
r
c
g共T
g
兲T
3
g
X
f
i
g共T
i
兲
, (19)
where r
c
苷 7.5 3 10
247
h
2
GeV
4
is the critical density,
T
g
苷 2.4 3 10
24
eV is the present photon temperature,
T
i
艐 m
i
, and g共T
g
兲 苷 2. In this we have used the fact
that the S number density to entropy is conserved once the
scattering and decay processes are Boltzmann suppressed,
such that g共T兲n
S
兾T
3
is constant. Therefore with f 艐
f
dec
and g共T
i
兲 苷 106.75 for T
i
艐 m
h
, the thermal relic
S density V
S
is related to l
S
by
l
S
苷 2.0 3 10
210
h
h
1兾3
µ
V
S
0.3
∂
1兾2
3
µ
10h
2兾3
MeV
m
S
∂
1兾2
µ
m
h
115 GeV
∂
3兾2
. (20)
Thus, for Higgs masses in the range 115 GeV to 1 TeV
and with expansion rate h 艐 0.7, the upper bound on
l
S
from requiring that V
S
& 0.3 is in the range 共1.4 3
10
210
3.6 3 10
29
兲h
21兾3
. This is in broad agreement
with the upper bound estimated in [24], based on the
weaker condition that the S scalars do not come into ther-
mal equilibrium. More importantly, we see that it is pos-
sible to generate a thermal relic density with V
S
艐 0.3 and
m
S
艐 10 MeV (typical of SIDM scalars) purely within
the minimal gauge singlet scalar extension of the stan-
dard model.
4. Naturally consistent thermal relic SIDM for zero bare
mass.—The value of l
S
from requiring that V
S
艐 0.3
is satisfied is not very much smaller than the upper limit
Eq. (7) from the requirement that the Higgs expectation
value contribution to the S mass is compatible with pertur-
bative SIDM S scalars. This suggests that it is quite likely
that all the S mass might come from its interaction with
the Higgs scalar when its relic density is sufficient to ac-
count for dark matter. If we assume that all the S mass is
due to the Higgs expectation value, then we find that the
S mass is fixed by the thermal relic density
m
S
苷 2.9
µ
V
S
0.3
∂
1兾5
µ
h
0.7
∂
2兾5
µ
m
h
115 GeV
∂
3兾5
MeV . (21)
We refer to this as the thermal relic S mass. Comparing
with the Spergel-Steinhardt range for SIDM,
m
S
苷 共6.6
15.4兲h
2兾3
MeV , (22)
we see that the thermal relic mass for S scalars is within the
range required to account for SIDM when the self-coupling
constant h is equal to about 0.1, a natural value which
is consistent with the Higgs doublet self-coupling in the
standard model. The thermal relic mass is not strongly de-
pendent upon cosmological parameters, nor is it strongly
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dependent upon the Higgs mass. In particular, it is rela-
tively insensitive to uncertainties in the calculation of f,
since a change in f by a factor d produces a change in V
S
by the same factor, and so a change in the thermal relic
mass by d
1兾5
. The coupling corresponding to the thermal
relic mass is
l
S
苷 2.7 3 10
210
µ
V
S
0.3
∂
2兾5
µ
h
0.7
∂
4兾5
µ
m
h
115 GeV
∂
6兾5
.
(23)
This suggests a scenario for dark matter in which stable
gauge singlet scalars couple very weakly to the standard
model sector but self-couple with a relatively strong
coupling of about 0.1, of the order expected from the
example of the standard model Higgs self-coupling.
5. Conclusions.—We have considered the thermal gen-
eration of a relic density of SIDM gauge singlet scalars.
The dominant process for small Higgs mass is the decay
of thermal equilibrium Higgs scalars to gauge singlet scalar
pairs. For SIDM scalars with perturbative self-interactions,
the mass must be no greater than around 30 MeV. For
such light scalars, the requirement of an acceptable relic
density of S scalars requires that the S coupling to the
standard model Higgs satisfies l
S
& 10
2共9
10兲
. This limit
comes from the requirement that S scalars are not ther-
mally overproduced, which is a stronger condition than
requiring that they do not come into thermal equilibrium.
In the case where S scalars account for dark matter and
where the S mass is entirely due to the Higgs expectation
value, we find that the S mass is fixed by the thermal relic
dark matter density to be between about 2.9 and 10.5 MeV
for Higgs masses ranging from 115 GeV to 1 TeV. (The
upper limit on the S mass may be smaller if 2
$ 2 annihi-
lations dominate h
0
decays for large Higgs mass.) This is
very similar to the range of masses 关共6.6
15.4兲h
2兾3
MeV兴
required by self-interacting dark matter with self-coupling
h of the order of the natural value (based on compari-
son with the standard model Higgs doublet self-coupling)
of around 0.1. This result is not strongly sensitive to un-
certainties either in the cosmological parameters or in the
calculation of the thermal relic S density. We find this co-
incidence remarkable and a possible hint that light gauge
singlet scalars with very weak coupling to the standard
model sector may play an important role in cosmology and
particle physics.
The author thanks the CERN theory division for its hos-
pitality and PPARC for a Travel Fund award.
*Present address: Theoretical Physics Division, Department
of Mathematical Sciences, University of Liverpool, Liver-
pool L69 3BX, UK.
Electronic address: mcdonald@sune.amtp.liv.ac.uk
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