Phenomenology of a Light Cold Dark Matter Two-Singlet Model

0.35 MB·English

by  Abdessamad Abada

Phenomenology of a Light Cold Dark Matter Two-Singlet Model
Abdessamad Abadaa,b∗ and Salah Nasrib†
aLaboratoire de Physique des Particules et Physique Statistique,
Ecole Normale Sup´erieure, BP 92 Vieux Kouba, 16050 Alger, Algeria
bPhysics Department, United Arab Emirates University,
POB 17551, Al Ain, United Arab Emirates
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1 (Dated: January 9, 2012)
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Abstract

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We study the implications of phenomenological processes on a two-singlet extension of the Stan-
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dard Model we introduced in a previous work to describe light cold dark matter. We look into

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h the rare decays of Υ and B mesons, most particularly the invisible channels, and study the decay
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- channels of the Higgs particle. Preferred regions of the parameter space are indicated, together
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h with others that are excluded. Comments in relation to recent Higgs searches and finds at the
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LHC are made.
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PACS numbers: 95.35.+d;98.80.-k; 12.15.-y;11.30.Qc.
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Keywords: cold dark matter. light WIMP. extension of Standard Model.
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∗Electronic address: [email protected]
†Electronic address: [email protected]
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I. INTRODUCTION
While still elusive, dark matter is believed to contribute about 23% to the energy budget
of the Universe [1]. We know it should be massive, stable on cosmic time scales and nonrel-
ativistic when it decouples from the thermal bath in order to be consistent with structure
formation. Although its mass and spin are not yet known, masses in the range of 5 10
−
GeV seem to be favored by the direct detection experiments CoGeNT [2], DAMA [3] and
CRESST II [4].
A number of models have been proposed to try to explain the results of these experiments
[5]. As it turns out, having a light dark-matter candidate in supersymmetric theories is quite
challenging. For instance, in mSUGRA, the constraint from WMAP and the bound on the
pseudo-scalar Higgs mass from LEP give m 50GeV [6]. Also, in the MSSM, a lightest
χ0
1 ≥
supersymmetric particle with a mass around 10GeV and an elastic scattering cross-section
off a nuclei as large as 10−41cm2 is needed in order to fit the CoGeNT data, which in turn
requires a very large tanβ and a relatively light CP-odd Higgs. However, such a choice of
parameters leads to a sizable contribution to the branching ratios of some rare decays, which
then disfavors the scenario of light neutralinos in the context of the MSSM [7] (see also [8]).
In a recent work [9], we proposed a two-singlet extension of the Standard Model as a
Z
simple model for light cold dark matter. Both scalar fields were -symmetric, with one
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undergoing spontaneous symmetry breaking while the other remaining unbroken to ensure
stability of the dark-matter candidate. We studied the behavior of the model, in particular
the effects of the dark-matter relic-density constraint and the restrictions from experimental
direct detection. We concluded that the model was capable of bearing a light dark-matter
weakly-interacting massive particle (WIMP) in mass regions where other models may find
difficulties. We should mention in passing that there are scenarios with unstable light Higgs-
like particles that have been previously studied in certain extensions of the Standard Model,
see for example [10]. There is also the possibility of having a light pseudo-scalar in the
NMSSM, see for example [11].
The present work studies the effects and restrictions on the two-singlet model coming
from particle phenomenology. A limited selection of low-energy processes has to be made,
and we choose to look into the rare decays of Υ and B mesons. We limit ourselves to small
dark-matter masses, in the range 0.1 10 GeV. We also study the implications of the model
−
2
on the decay channels of the Higgs particle and make quick comments in relation to recent
finds at the LHC.
The theory starts effectively with eight parameters [9]. The spontaneous breaking of
the electroweak and Z symmetries introduces the two vacuum expectation values v and v
2 1
respectively. The value of v is fixed experimentally to be 246GeV and we take v = 100GeV.
1
Four of the parameters are the three physical masses m (dark-matter singlet S ), m (the
0 0 1
second singlet S ) and m (Higgs h), plus the mixing angle θ between h and S . We let m
1 h 1 1
vary in the interval 0.1 10 GeV and fix the Higgs mass to m = 125GeV [12, 13], except in
h
−
the part about the Higgs decays where we let m vary in the interval 100 200 GeV1. For
h
−
the purpose of our discussions, it is sufficient to let θ vary in the interval 1o 40o. The last
−
(4)
parameters are the two physical mutual coupling constants λ (dark matter – Higgs) and
0
(4) (4)
η (dark matter – S particle). In fact, η is not free as it is the smallest real and positive
01 1 01
solution to the dark-matter relic density constraint [9], which is implemented systematically
(4)
throughout this work. Thus we are left with four parameters, namely, m , m , θ and λ .
0 1 0
(4)
To ensure applicability of perturbation theory, the requirement η < 1 is also imposed
01
(4)
throughout, as well as a choice of rather small values for λ .
0
II. UPSILON DECAYS
We start by looking at the constraints on the parameter space of the model coming from
the decay of the meson Υ in the state nS (n = 1,3) into one photon γ and one particle S .
1
For m . 8GeV, the branching ratio for this process is given by the relation:
1
G m2sin2θ 4α
Br(Υ γ +S ) = F b x 1 sf(x ) Br(µ)Θ(m m ). (2.1)
nS → 1 √2πα n − 3π n ΥnS − 1
(cid:18) (cid:19)
In this expression, x 1 m2/m2 with m = 9.46(10.355)GeV the mass of Υ ,
n ≡ − 1 Υns Υ1(3)S 1(3)S
the branching ratio Br(µ(cid:0)) Br Υ (cid:1) µ+µ− = 2.48(2.18) 10−2 [15], α is the QED
1(3)S
≡ → ×
couplingconstant, αs = 0.184the(cid:0)QCDcouplingc(cid:1)onstant atthescalemΥnS, thequantityGF
is the Fermi coupling constant and m the b quark mass [16]. The function f(x) incorporates
b
the effect of QCD radiative corrections given in [17].
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The exclusionmassrangereportedby the CMSandATLAS Collaborationsappliesto the SMHiggs and
can be weakened or evaded in models where the Higgs production and/or decay channels are suppressed
[14]. We will comment on this possibility within our model in the last section.
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However, a rough estimate of the lifetime of S indicates that this latter is likely to
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decay inside a typical particle detector, which means we ought to take into account its most
dominant decay products. We first have a process by which S decays into a pair of pions,
1
with a decay rate given by:
G m m2 11m2 2
Γ(S ππ) = F 1 sin2θ 1 1+ π
1 → 4√2π "27 (cid:18) 2m21 (cid:19)
1
4m2 2
1 π Θ[(m 2m )(2m m )]
× − m2 1 − π K − 1
(cid:18) 1 (cid:19)
3
4m2 2
+3 M2 +M2 1 π Θ(m 2m ) . (2.2)
u d − m2 1 − K
(cid:18) 1 (cid:19) #
(cid:0) (cid:1)
Here, m is the pion mass and m the kaon mass. Also, chiral perturbation theory is used
π K
below the kaon pair production threshold, and the MIT bag model above, with the dressed
u and d quark masses M = M = 0.05GeV. Note that this rate includes all pions, charged
u d
and neutral. Above the 2m threshold, there is the production of both a pair of kaons and
K
η particles. The decay rate for K production is:
3
9 3G M2m 4m2 2
Γ(S KK) = F s 1 sin2θ 1 K Θ(m 2m ). (2.3)
1 → 13 4√2π − m2 1 − K
(cid:18) 1 (cid:19)
In the above rate, M = 0.45GeV is the s quark bag-mass [18, 19]. For η production, replace
s
m by m and 9 by 4 .
K η 13 13
The particle S decays also into c and b quarks (mainly c). Including the radiative QCD
1
corrections, the corresponding decay rates are given by:
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3G m¯2m 4m¯2 2 α¯
Γ(S qq¯) = F q 1 sin2θ 1 q 1+5.67 s Θ(m 2m¯ ). (2.4)
1 → 4√2π − m2 π 1 − q
(cid:18) h (cid:19)
(cid:16) (cid:17)
The dressed quark mass m¯ m (m ) and the running strong coupling constant α¯
q q 1 s
≡ ≡
α (m ) are defined at the energy scale m [20]. There is also a decay into a pair of gluons,
s 1 1
with the rate:
G m3sin2θ α′ 2 4m2 32 4m2 32
Γ(S gg) = F 1 s 6 2 1 π 1 K Θ(m 2m ).
1 → 12√2π (cid:18) π (cid:19) " − (cid:18) − m21 (cid:19) −(cid:18) − m21 (cid:19) # 1 − π
(2.5)
Here, α′ = 0.47 is the QCD coupling constant at the bag-model scale.
s
We then have the decay of S into leptons, the corresponding rate given by:
1
3
G m2m 4m2 2
Γ S ℓ+ℓ− = F ℓ 1 sin2θ 1 ℓ Θ(m 2m ), (2.6)
1 → 4√2π − m2 1 − ℓ
(cid:18) 1 (cid:19)
(cid:0) (cid:1)
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where m is the lepton mass. Finally, S can decay into a pair of dark matter particles, with
ℓ 1
a decay rate:
2
(3)
η
01 4m2
Γ(S S S ) = 1 0Θ(m 2m ). (2.7)
1 → 0 0 (cid:16)32πm(cid:17)1 s − m21 1 − 0
(3)
The coupling constant η is given in [9]. The branching ratio for Υ decaying via S into
01 nS 1
a photon plus X, where X represents any kinematically allowed final state, will be:
Br(Υ γ +X) = Br(Υ γ +S ) Br(S X). (2.8)
nS nS 1 1
→ → × →
In particular, X S S corresponds to a decay into invisible particles.
0 0
≡
Λ H4L =0.02,m =1.5GeV,Θ=20°
0 0
10-5 X=ΤΤ
X=ΠΠ
10-7
X=KK
L
X
+ ExpΤΤ
Γ 10-9
®
s
1 ExpΠΠ
U
H
r
B 10-11
ExpKK
10-13
1 2 3 4 5 6 7 8
m HGeVL
1
FIG. 1: Typical branching ratios of Υ decaying into τ’s, charged pions and charged kaons as
1S
functions of m . The corresponding experimental upper bounds are shown.
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The best available experimental upper bounds on 1S–state branching ratios are: (i)
Br(Υ γ +ττ) < 5 10−5 for 3.5GeV < m < 9.2GeV [21]; (ii) Br(Υ γ +π+π−) <
1S 1 1S
→ × →
6.3 10−5 for 1GeV < m [22]; (iii) Br(Υ γ +K+K−) < 1.14 10−5 for 2GeV < m <
1 1S 1
× → ×
3GeV [23]. Figure 1 displays the corresponding branching ratios of Υ decays via S as
1S 1
functions of m , together with these upper bounds. Also, the best available experimental
1
upper bounds on Υ branching ratios are: (i) Br(Υ γ +µµ) < 3 10−6 for 1GeV <
3S 3S
→ ×
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m < 10GeV; (ii) Br(Υ γ +Invisible) < 3 10−6 for 1GeV < m < 7.8GeV [24].
1 3S 1
→ ×
Typical corresponding branching ratios are shown in figure 2.
Λ H4L =0.02,m =1.5GeV,Θ=20°
0 0
10-6 X=ΜΜ
X=S S
0 0
10-8
ExpΜΜ
L
X
+
®Γ ExpInv
s 10-10
3
U
H
r
B
10-12
10-14
1 2 3 4 5 6 7 8
m HGeVL
1
FIG. 2: Typical branching ratios of Υ decaying into muons and dark matter as functions of m .
3S 1
The corresponding experimental upper bounds are shown.
A systematic scan of the parameter space indicates that the main effect of the Higgs-
(4)
dark-matter coupling constant λ and the dark-matter mass m is to exclude, via the relic
0 0
density and perturbativity constraints, regionsof applicability of the model. This is shows in
figures 1 and 2 where the region m . 1.4GeV is excluded. Otherwise, these two parameters
1
have little effect on the shapes of the branching ratios themselves. The onset of the S S
0 0
channel for m 2m abates sharply the other channels and this one becomes dominant by
1 0
≥
far. The effect of the mixing angle θ is to enhance all branching ratios as it increases, due to
the factor sin2θ. The dark matter decay channel reaches the invisible upper bound already
for θ 15o, for fairly small m , say 0.5GeV. The other channels find it hard to get to their
0
≃
respective experimental upper bounds, even for large values of θ.
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III. B MESON DECAYS
Next we look at the flavor changing process in which the meson B+ decays into a K+
plus invisibles. The corresponding Standard-Model mode is a decay into K+ and a pair of
neutrinos, withabranchingratioBrSM(B+ K+ +νν¯) 4.7 10−6 [25]. Theexperimental
→ ≃ ×
upper bound is BrExp(B+ K+ +Inv) 14 10−6 [26]. As in the Υ decays, the most
→ ≃ ×
prominent B invisible decay in this model is into S S via S . The process B+ K++S
0 0 1 1
→
has a the branching ratio:
9√2τ G3m4m2m2m2
Br B+ K+ +S = B F t b + − V V∗ 2f2 m2
→ 1 1024π5m3 (m m )2 | tb ts| 0 1
B b − s
(cid:0) (cid:1) (cid:0) (cid:1)
(m2 m2)(m2 m2) sin2θ Θ(m m ). (3.1)
× + − 1 − − 1 − − 1
q
Here m = m m where m is the B+ mass, τ its lifetime, andV and V are flavor
± B K B B tb ts
±
changing CKM coefficients. The function f (s) is given by the relation:
0
0.63s 0.095s2 0.591s3
f (s) = 0.33exp + . (3.2)
0 m2 − m4 m6
(cid:20) B B B (cid:21)
The different S decay modes are given in (2.2) - ( 2.7) above. The branching ratio of B+
1
decaying into K++S S via the production and propagation of an intermediary S will be:
0 0 1
Br(S1) B+ K+ +S S = Br B+ K+ +S Br(S S S ). (3.3)
0 0 1 1 0 0
→ → × →
(cid:0) (cid:1) (cid:0) (cid:1)
Figure 3 displays a typical behavior of Br(S1)(B+ K+ +S S ) as a function of m .
0 0 1
→
The branching ratio is well above the experimental upper bound, and θ as small as 1o will
not help with this, no matter what the values for λ(4) and m are. So, for m . 4.8GeV, this
0 0 1
process excludes the two-singlet model for m < m /2. For m & 4.8GeV or m m /2,
0 1 1 0 1
≥
the decay does not occur, so no constraints on the model from this process.
Another process involving B mesons is the decay of B into predominately a pair of
s
muons. The Standard Model branching ratio for this process is BrSM(B µ+µ−) = (3.2
s
→ ±
0.2) 10−9 [27], and the experimental upper bound is BrExp(B µ+µ−) < 1.08 10−8
s
× → ×
[28]. In the present model, two additional decay diagrams occur, both via intermediary S ,
1
yielding together the branching ratio:
9τ G4f2 m5 1 4m2/m2 3/2
Br(S1)(B µ+µ−) = Bs F Bs Bsm2m4 V V∗ 2 − µ Bs sin4θ. (3.4)
s → 2048π5 µ t | tb ts| m(cid:0)2 m2 2 +m(cid:1) 2Γ2
Bs − 1 1 1
(cid:0) (cid:1)
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Λ H4L =0.01,m =0.2GeV,Θ=10°
0 0
0.1
S S
0 0
ExpInv
0.01 SM
L
0
S
0
S
+
+ 0.001
K
®
+
B
H
r
B 10-4
10-5
0 1 2 3 4 5
m HGeVL
1
FIG. 3: Typical branching ratio of B+ decaying into dark matter via S as a function of m . The
1 1
SM and experimental bounds are shown.
In this relation, τ is the B life-time, m = 5.37GeV its mass, and f a form factor that
Bs s Bs Bs
we take equal to 0.21GeV. The quantity Γ is the total width of the particle S [9].
1 1
A typical behavior of Br(S1)(B µ+µ−) as a function of m is shown in figure 4. The
s 1
→
(4)
peak is at m . All three parameters λ , m and θ combine in the relic density constraint
Bs 0 0
to exclude regions of applicability of the model. For example, for the values of figure 4, the
region m < 2.2 GeV is excluded. However, a systematic scan of the parameter space shows
1
(4)
that outside the relic density constraint, λ has no significant direct effect on the shape of
0
Br(S1)(B µ+µ−). As m increases, it sharpens the peak of the curve while pushing it
s 0
→
up. This works until about 2.7GeV, beyond which m ceases to have any significant direct
0
effect. Increasing θ enhances the values of the branching ratio without affecting the width.
Also, for all the range of m , all of Br(S1) +BrSM stays below BrExp as long as θ < 10o.
1
As θ increases beyond this value, the peak region pushes up increasingly above BrExp and
thus gets excluded. Hoping for a clear signal if any, figure 5 displays a density plot showing
BrSM+Br(S1)(B µ+µ−) inthe plane(m ,θ), squeezed between BrSM+5σ frombelow and
s 1
→
BrExp from above. The behavior we see in this figure is generic across the ranges of m and
0
8
Λ H4L =0.05,m =3GeV,Θ=15°
0 0
Μ+Μ-
Exp
10-8
SM
L
-Μ 10-9
+
Μ
®
s
B
H
Br 10-10
10-11
0 2 4 6 8 10
m HGeVL
1
FIG. 4: Typical behavior of Br(S1)(B µ+µ−) as a function of m , together with the SM and
s 1
→
experimental bounds.
(4)
λ : the V-shape structure in gray developing from m = m is the allowed region. The
0 1 Bs
white region in the middle is due to BrExp, and the white region outside to BrSM+5σ. It can
happenthat some ofthegrayVis eatenupby therelic-density constraint andperturbativity
(4)
requirement for larger values of λ .
0
From this process, there is probably one element to retain if we want the model to
contribute a distinct signal to B µ+µ− for the range of m chosen, and that is to restrict
s 0
→
4GeV . m . 6.5GeV. No additional constraint on m is necessary while keeping λ(4) . 0.1
1 0 0
to avoid systematic exclusion from direct detection is safe.
IV. HIGGS DECAYS
We finally examine the implications of the model on the Higgs different decay modes. In
this part of the work, we allow the Higgs mass m to vary in the interval 100GeV 200GeV.
h
−
First, h can decay into a pair of leptons ℓ, predominantly τ’s. The corresponding decay
rate Γ(h ℓ+ℓ−) is given by the relation (2.6) where we replace m by m . It can also
1 h
→
9
Λ H4L =0.03,m =5GeV
0 0
30
25
20
Θ
15
10
4.5 5.0 5.5 6.0 6.5
m HGeVL
1
FIG. 5: A density plot of BSM +Br(S1) squeezed between BSM +5σ from below and BExp from
above.
decay into a pair of quarks q, mainly into b’s and, to a lesser degree, into c’s. Here too the
decay rate Γ(h qq¯) is given in (2.4) with the replacement m instead of m . Then the
h 1
→
Higgs can decay into a pair of gluons. Including the next-to-next-to-leading QCD radiative
corrections, the corresponding decay rate can write like this:
2
G m3 m2 1 1−x 1 4xy
Γ(h gg) = F h q dx dy −
→ 4√2π (cid:12)(cid:12)(cid:12)Xq m2h Z0 Z0 mm2h2q −xy(cid:12)(cid:12)(cid:12)
α¯ (cid:12)2 215α¯ α¯2 (cid:12) m2
s (cid:12) 1+ s + s 156.8 5.7(cid:12)log t cos2θ, (4.1)
(cid:12) (cid:12)
× π 12 π π2 − m2
(cid:20) (cid:18) h(cid:19)(cid:21)
(cid:16) (cid:17)
where the sum is over all quark flavors q. A systematic study of the double integral above
shows that, with m in the range 100GeV – 200GeV, the t quark dominates in the sum over
h
q, with non-negligible contributions from the c and b quarks.
For m smaller than the W or Z pair-production threshold, the Higgs can decay into a
h
pair of one real and one virtual gauge bosons, with rates given by:
3G2m4 m m2
Γ(h VV∗) = F V h cos2θA R V Θ[(m m )(2m m )]. (4.2)
→ 16π3 V m2 h − V V − h
(cid:18) h(cid:19)
10