Table Of Content130 GeV gamma-ray line and enhancement of h γγ in the Higgs
→
triplet model plus a scalar dark matter
Lei Wang, Xiao-Fang Han
2
1
Department of Physics, Yantai University, Yantai 264005, China
0
2
Abstract
p
e
S With a discrete Z symmetry being imposed, we introduce a real singlet scalar S to the Higgs
2
1
triplet model with the motivation of explaining the tentative evidence for a spectral feature at E
1 γ
] = 130 GeV in the Fermi LAT data. The model can naturally satisfy the experimental constraints
h
p of the dark matter relic density and direct detection data from Xenon100. The doubly charged
-
p
e and one charged scalars can enhance the annihilation cross section of SS γγ via the one-loop
→
h
[ contributions,andgivethenegligiblecontributionstotherelicdensity. < σv > form = 130
SS→γγ S
2
GeV can reach (1) 10−27cm3s−1 for the small charged scalar masses and the coupling constant
v
O ×
6
7 of larger than 1. Besides, this model also predict a second photon peak at 114 GeV from the
3
0 annihilation SS γZ, and the cross section is approximately 0.76 times that of SS γγ, which
→ →
.
9
is below the upper limit reported by Fermi LAT. Finally, the light charged scalars can enhance
0
2
1 LHC diphoton Higgs rate, and make it to be consistent with the experimental data reported by
:
v
ATLAS and CMS.
i
X
r
a PACS numbers: 12.60.Fr,95.35.+d, 95.85.Pw,14.80.Ec
1
I. INTRODUCTION
Recently, several groups [1–3] have reported a line spectral feature at E = 130 GeV
γ
in publicly available data from the Fermi Large Area Telescope (LAT) [4]. Moreover, Ref.
[3, 5] reported the hints of a second line at around 111 GeV with less statistically significant.
The sharp pick of the gamma-ray around 130 GeV can be explained by the 130 GeV dark
matter (DM) annihilation into two photons, whose cross section < σv > is 1.27
SS→γγ
±
0.32+0.18 10−27cm3s−1 (2.27 0.57+0.32 10−27cm3s−1) for Einasto (NFW) DM profile
−0.28 × ± −0.51 ×
employed [1]. Besides, the line at 130 GeV can also be produced by the 142 GeV (155
GeV) DM annihilating to γZ (γh with h being a 125 GeV Higgs boson). The Fermi LAT
collaborationtakes slightly different search regions andmethodology, and sets anupper limit
of < σv > < 1.4 10−27cm3s−1, which is in mild tension with the claimed signal [6].
SS→γγ
×
The cross section of SS γγ (1.27 10−27cm3s−1) required by the claimed 130 GeV
→ ×
gamma-ray line signal is approximately 0.042 in units of the thermal relic density value,
< σv > = 3 10−26cm3s−1 [7]. Since the DM is in general electrically neutral, SS γγ
0
× →
should arise at one-loop through virtual massive charged particles. If the charged particles
at the loop are lighter than the DM particle, the corresponding tree-level cross section for
annihilations to these charged particles will exceed that of the loop-level process to γγ by
many orders of magnitude, which conflicts with the the total annihilation cross section to
generate the observed relic density. In addition, an enormous annihilation cross section
to charged particles is disfavored by the gamma-ray constraints from observations of the
Galactic Center and elsewhere [6, 8]. A variety of DM models have been proposed to
solve this issue [9–11]. Ref. [9] shows a multi-charged and colored scalar X can enhance
< σv > to (1) 10−27cm3s−1 via the interaction of λ SSXX at one-loop, and not
SS→γγ X
O ×
lead to the conflict with the relic density for its mass is larger than that of DM. In addition,
the LHC diphoton Higgs rate is also enhanced by the scalar.
To construct the DM models economically, a real singlet scalar is respectively added
to the standard model [12] and two Higgs doublet model [13] with a discrete Z symmetry
2
being imposed. These models cansatisfy naturally the constraints fromthe DMrelic density
and direct detection data, but hardly accommodate the claimed 130 GeV gamma-ray line
signal [14, 15]. In this paper, we introduce such a scalar S to the Higgs triplet model
(HTM) which contains a complex doublet Higgs field and a complex triplet Higgs field with
2
hypercharge Y = 2 [16]. In the original HTM, several physical Higgs bosons remain after
the spontaneous symmetry breaking, including two CP-even (h and H), one CP-odd (A),
one charged (H±) and one doubly charged Higgs scalars (H±±). The charged scalars H±±
and H± can enhance the cross section of SS γγ at one-loop. Besides, the SM-like Higgs
→
decay into two photoncan beenhanced by these charged scalars, which is favored by thenew
ATLAS and CMS data. The new Higgs data has been discussed in the HTM [17–19], the
minimal supersymmetric standard model (MSSM) [20], the next-to-MSSM [21], and other
extensions of Higgs models [22].
This work is organized as follows. In Sec. II, we introduce a real single scalar DM
to the Higgs triplet model. In Sec. II, we study the constraints of DM relic density and
direct detection data. In Sec. III, we calculate the cross sections of < σv > and
SS→γγ
< σv > . In Sec. IV, we discuss the enhancement of LHC diphoton Higgs rate.
SS→γZ
Finally, we give our conclusion in Sec. V.
II. THE HIGGS TRIPLET MODEL PLUS A SCALAR DM (HTMD)
In the HTM [16], a complex SU(2) triplet scalar field ∆ with Y = 2 is added to the SM
L
Lagrangian in addition to the doublet field Φ. These fields can be written as
δ+/√2 δ++ φ+
∆ = , Φ = . (1)
δ0 δ+/√2 φ0
−
The renormalizable scalar potential can be written as [23]
λ
V = m2Φ†Φ+ (Φ†Φ)2 +M2Tr(∆†∆)+λ (Φ†Φ)Tr(∆†∆) (2)
− Φ 4 ∆ 1
+ λ (Tr∆†∆)2 +λ Tr(∆†∆)2 +λ Φ†∆∆†Φ+[µ(ΦTiτ ∆†Φ)+h.c.].
2 3 4 2
The Higgs doublet and triplet field can acquire vacuum expectation values
1 0 1 0 0
Φ = , ∆ = (3)
h i √2 v h i √2 v 0
d t
with v2 = v2 +4v2 (246 GeV)2.
SM d t ≈
After the spontaneous symmetry breaking, the Lagrangian of Eq. (2) predicts the seven
physical Higgs bosons, including two CP-even (h and H), one CP-odd (A), one charged
(H±) and one doubly charged Higgs scalars (H±±). These mass eigenstates are in general
3
mixturesofthedoubletandtripletfields. Theexperimental valueoftheρparameterrequires
v2/v2 to be much smaller than unity at tree-level, which gives a upper bound of v < 8 GeV.
t d t
[17, 24]. For a very small v , the mixing angle in the CP-even sector α and charged Higgs
t
sector β are approximately,
sinα 2v /v , sinβ √2v /v , (4)
t d t d
≃ ≃
and the mixing of the doublet and triplet fields is nearly absent. For this case, the seven
Higgs masses can be obtained from the Lagrangian of Eq. (2) [17, 18],
λ
m2 v2,
h ≃ 2 d
λ λ
m2 M2 +( 1 + 4)v2 +3(λ +λ )v2,
H ≃ ∆ 2 2 d 2 3 t
λ λ
m2 M2 +( 1 + 4)v2 +(λ +λ )v2,
A ≃ ∆ 2 2 d 2 3 t
λ λ
m2 = M2 +( 1 + 4)v2 +(λ +√2λ )v2,
H± ∆ 2 4 d 2 3 t
λ
m2 = M2 + 1v2 +λ v2. (5)
H±± ∆ 2 d 2 t
In the following discussions, we always assume the value of v is very small. We take h as
t
the 125 GeV SM-like Higgs boson, which is from the Higgs doublet field. H, A, H± and
H±± are heavier than h, which are from the Higgs triplet field. The h field couplings to ff¯,
WW and ZZ equal to those of SM nearly. In addition, the scalar potential terms in Eq. (2)
contain the SM-like Higgs boson coupling to the charged scalars [18],
λ
4
ghH++H−− λ1vd, ghH+H− (λ1 + )vd. (6)
≈ − ≈ − 2
However, the similar couplings for H are suppressed by the factor sinα, v or sinβ. Thus,
t
the H production cross section at the collider is very small, which satisfies the constraints
of the present Higgs data easily.
Now we introduce the renormalizable Lagrangian of the real single scalar S,
1 m2 κ κ
= ∂µS∂ S 0SS 1Φ†ΦSS κ Tr(∆†∆)SS sS4. (7)
S µ 2
L 2 − 2 − 2 − − 4
The linear and cubic terms of the scalar S are forbidden by the Z symmetry S S. S
2
→ −
has a vanishing vacuum expectation value which ensures the DM candidate S stable. κ
s
is the coupling constant of the DM self-interaction, which does not give the contributions
4
to the DM annihilation and Higgs signal. In order to explain the 130 GeV gamma-ray
line, we take DM mass as 130 GeV, which determines the value of m by the relation of
0
m = (m2 + 1κ v2 +κ v2)1/2. The total DM annihilation cross sections mainly depend on
S 0 2 1 d 2 t
the κ , which determines the couplings hSS and hhSS. κ determines the couplings HSS,
1 2
HHSS, AASS, H±H∓SS and H±±H∓∓SS, where the coupling HSS is suppressed by v .
t
The couplings H±H∓SS and H±±H∓∓SS give the important contributions to XX γγ
→
at one-loop.
For v < 10−4 GeV, H±± ℓ±ℓ± is the dominant decay mode of H±±. Assuming
t
→
Br(H±± ℓ±ℓ±) = 1, CMS presents the low bound 383 GeV on mH±± from the searches
→
for H±± ℓ±ℓ± via qq¯ H±±H∓∓ and qq¯ H±±H∓ production processes [25]. However,
→ → →
H±± W±W± and H±± H±W∗ are the dominant modes for v > 10−4 GeV [17, 18, 26],
t
→ →
for which there have been no direct searches. Therefore, the above bound on mH±± can not
be applied to the case of v > 10−4 GeV, and H±± could be much lighter in this scenario.
t
In this paper, we take vt = 0.1 GeV and mH±± to be as low as 140 GeV. LEP searches
for the charged scalar give the constraints on the possible existence of light scalars [27]. A
conservative lower bound on mH± should be larger than 100 GeV due to the absence of
non-SM events at LEP. To simplify the parameter space, we take the triplet scalars to be
degenerate, namely λ = 0. We can neglect the contributions of λ and λ to the triplet
4 2 3
masses which are suppressed by v2.
t
III. DARK MATTER RELIC DENSITY AND DIRECT DETECTION
A. calculation of relic density
The degeneracy masses of the triplet scalars are taken to be larger than 140 GeV. Thus,
the annihilation processes SS HH, AA, H±H∓, H±±H∓∓ are forbidden for m =130
S
→
GeV. When the triplet scalars masses are slightly larger than DM mass, the cross section of
the forbidden annihilation channel is important [28]. Here, we do not consider this scenario.
¯
Since the H field couplings to SS, ff, WW, ZZ are suppressed by v or sinα, the s-channel
t
annihilationprocessesmediatedbyH giveanegligiblecontributionstototalDMannihilation
¯
cross sections. Therefore, the main annihilation processes include SS ff, SS WW,
→ →
SS ZZ which proceed via an s-channel h exchange, and SS hh which proceeds via a
→ →
5
4-point contact interaction, an s-channel h exchange and t- and u-channel S exchange. The
total annihilation cross sections time the relative velocity v for these processes are given as
[29],
σv = σ v +σ v +σ v +σ v, (8)
ff WW ZZ hh
κ2 m2 4m2
σ v = 1 f (1 f)3/2,
ff 4π(s m2)2 − s
f − h
X
κ2 s 4m2 4m2 12m4
σ v = 1 1 W 1 W + W ,
WW 8π(s m2)2 − s − s s2
− h r (cid:18) (cid:19)
κ2 s 4m2 4m2 12m4
σ v = 1 1 Z 1 Z + Z ,
ZZ 16π(s m2)2 − s − s s2
− h r (cid:18) (cid:19)
κ2 4m2 s+2m2 2 8κ v2 s+2m2
σ v = 1 1 h h 1 hF(ξ)
hh 16πs − s s m2 − s 2m2 s m2
r "(cid:18) − h (cid:19) − h − h
8κ2v4 1
+ 1 +F(ξ) . (9)
(s 2m2)2 1 ξ2
− h (cid:18) − (cid:19)(cid:21)
where F(ξ) arctanh(ξ)/ξ with ξ (s 4m2)(s 4m2 )/(s 2m2), and s is the squared
≡ ≡ − h − D − h
center-of-mass energy. p
The thermally averaged annihilation cross section times the relative velocity, < σv >, is
well approximated by a non-relativistic expansion,
T
< σv >= a+b < v2 > + (< v4 >) a+6b (10)
O ≃ m
S
The freeze-out temperature T is defined by solving the following equation [30],
f
0.038gm m < σv >
pl S
x = ln . (11)
f 1/2 1/2
g x
∗ f
Where x = mS and m = 1.22 1019 GeV. g is the total number of effectively relativistic
f Tf pl × ∗
degrees of freedom at the time of freeze-out [31]. g = 1 is the internal degrees of freedom
for the scalar DM S. The present-day abundance of S is approximately [30]
1.07 1019 x 1
Ωh2 × f . (12)
≃ m √g (a+3b/x )
pl ∗ f
The relic density from the WMAP 7-year result [32] is
Ω h2 = 0.1123 0.0035. (13)
DM
±
6
B. Calculation of the spin-independent cross section between S and nucleon
The results of DM-nucleus elastic scattering experiments are presented in the form of
a normalized DM-nucleon scattering cross section in the spin-independent case. In the
HTMD, the elastic scattering of S on a nucleon receives the dominant contributions from
the h exchange diagrams, which is given as [33],
m2
σSI = p(n) fp(n) 2, (14)
Sp(n) 2
4π m +m
S p(n)
(cid:2) (cid:3)
where (cid:0) (cid:1)
m 2 m
fp(n) = fp(n) p(n) + fp(n) p(n), (15)
Tq CSq m 27 Tg CSq m
q q
q=u,d,s q=c,b,t
X X
with = κ1mq [34],
CSq m2h
(p) (p) (p) (p)
f 0.020, f 0.026, f 0.118, f 0.836,
Tu ≈ Td ≈ Ts ≈ Tg ≈
(n) (n) (n) (n)
f 0.014, f 0.036, f 0.118, f 0.832. (16)
Tu ≈ Td ≈ Ts ≈ Tg ≈
In fact, here σSI σSI. The recent data on direct DM search from Xenon100 put the most
Sp ≈ Sn
stringent constraint on the cross section [35].
C. results and discussions
In our calculations, m = 130 GeV and m = 125 GeV are fixed. Thus, both the relic
S h
density and the spin-independent cross section between S and the nucleon are only sensitive
to the parameter κ . In Fig. 1, we plot Ωh2 and σSI versus the κ , respectively. The left
1 Sn 1
panel of Fig. 1 shows that κ should be around 0.04 to get the correct DM relic abundance.
1
For such value of κ , the right panel shows that σSI is around 1.2 10−45cm2, which is below
1 Sn ×
the upper bound presented by Xenon100 data and accessible at the future Xenon1T.
IV. GAMMA-RAY LINES FROM SS γγ AND SS γZ
→ →
A. 130 GeV gamma-ray line from SS γγ
→
The annihilation SS γγ may be radiatively induced by massive charged particles
→
in the loop. The charged scalars H±± and H± can give the dominant contributions to
7
0.2 10
0.18
XENON100
0.16 1
0.14 2)m
c
2h0.12 450( 10-1
1
W
SI· Sn XENON1T
0.1
s
-2
0.08 10
m =130GeV m =130GeV
S S
0.06 m =125GeV m =125GeV
h h
-3
0.04 10
0.02 0.025 0.03 0k.035 0.04 0.045 0.05 0.01 0.02 0.03 0.04 0.0k5 0.06 0.07 0.08 0.09 0.1
1 1
FIG. 1: Left panel: The dark matter relic density versus κ . The horizontal lines show the corre-
1
spondingboundsfrom experimental data of the WMAP 7-year. Right panel: the spin-independent
cross section between S and the nucleon versus κ . The horizontal lines show the upper bound
1
from Xenon100 and the sensitivity of projected Xenon1T. The vertical lines show the range of κ
1
constrained by relic density.
this annihilation process via the couplings H±±H∓∓SS and H±H∓SS, and the relevant
Feynman diagrams are depicted in Fig. 2. Besides, there is another type Feynman diagram
for SS γγ in which s-channel h or H exchange is combined with a charged particle loop.
→
The contributions of the diagram can be sizably enhanced for m (m ) 2m =260 GeV
h H S
∼
and the charged particle with an around 130 GeV mass [10]. For the SM-like Higgs h, its
mass is 125 GeV and the relic density requires κ to be around 0.04, which suppresses the
1
coupling hSS. Although we may take m =260 GeV, the coupling HSS is suppressed by v .
H t
Therefore, the contributions from the type diagram are negligible compared to those of Fig.
2. The annihilation cross section corresponding to the diagrams of Fig. 2 is approximately
given by
2
α2κ2 25α2κ2
< σv >SS→γγ≃ 32π3m22 4E(τH±±)+E(τH±) = 32π3m22 E(τH±±)2 (17)
S(cid:12) (cid:12) S
(cid:12) (cid:12)
(cid:12) (cid:12)
with τH±± = mm2H±2S±, τH± = mm2H2S±, and(cid:12)(cid:12)E(τ) = 1−τ[sin−1((cid:12)(cid:12)1/√τ)]2. For the second equation,
we take mH±± = mH±.
The H± and H±± contributions areconstructive each other. Because H±± has an electric
8
S H±±(H±) γ S H±±(H±) γ
H±±(H±)
S H±±(H±) γ S H±±(H±) γ
(a) (b)
FIG. 2: The Feynman diagrams for SS γγ, which give the dominant contributions to the
→
annihilation process.
8
1.4
7
1.66 1.27
6 2.27 0.88
5
2
k
4
3
2 m =130GeV
S
1
140 160 180 200 220 240
m ++(GeV)
H
FIG. 3: Thecontours for < σv >SS→γγ in the plane of κ2 versus mH±±. The numberson the cures
denote < σv > /1.0 10−27cm3s−1.
SS→γγ
×
charge of 2, the H±± contributions are enhanced by a relative factor 4 in the amplitude.
±
Fig. 3showssomecontoursfor< σv > = 0.88 10−27cm3s−1, 1.27 10−27cm3s−1, 1.4
SS→γγ
× × ×
10−27cm3s−1, 1.66 10−27cm3s−1, and 2.27 10−27cm3s−1 in the plane of κ2 versus mH±±.
× ×
From Fig. 3, we find that, in order to obtain < σv > = 1.27 10−27cm3s−1, the
SS→γγ
×
minimal value of κ2 should be from 1.7 to 4.0 for mH±± in the range of 140 GeV and 180
GeV. As the increasing of mH±±, the corresponding κ2 are required to increase, which will
be constrained by the perturbation of the theory.
9
B. 114 GeV gamma-ray line from SS γZ
→
We can obtain the Feynman diagrams of SS γZ and SS γh by replacing a γ
→ →
with Z and h in the Fig. 2, respectively. The cross section of SS γh is zero due to the
→
charge-conjugation invariance of the interactions involved. The cross section of SS γZ is
→
related to that of SS γγ, which is approximately given by
→
< σv > m2
SS→γZ 2(cot2θ )2(1 Z )1/2 = 0.76 (18)
< σv > ≃ W − 4m2
SS→γγ S
m2
TheenergyofthissinglephotonisgivenbyE = m (1 Z ) = 114GeV. ThecurrentFermi
γ S −4m2S
LAT upper limit on< σv > for E = 110 GeVis 2.6 10−27cm3s−1 (3.6 10−27cm3s−1
SS→γZ γ
× ×
) for Einasto (NFW) DM profile employed [6]. For < σv > = 1.27 10−27cm3s−1, the
SS→γγ
×
prediction value of < σv > is below the upper bound presented by Fermi LAT.
SS→γZ
V. LHC DIPHOTON HIGGS RATE
To some extent, the h γγ decay is related to the SS γγ annihilation, since the
→ →
doubly charged and one charged scalars can contribute to both SS γγ and h γγ. It
→ →
is necessary to restudy the LHC diphoton Higgs rate although it has been studied in detail
[17, 18].
Since the new scalars and DM are heavy than the SM-like Higgs h, h does not have any
new important decay modes compared to that of SM. Except for the decay h γγ, the
→
decay modes and their widths of h are nearly the same both in HTMD and SM. The decay
width of h γγ is expressed as [36]
→
2
α2m3
Γ(h → γγ) = 256π3hv2(cid:12)F1(τW)+ NcfQ2fF1/2(τf)+gH±F0(τH±)+4gH±±F0(τH±±)(cid:12) ,(19)
(cid:12) Xi (cid:12)
(cid:12) (cid:12)
where (cid:12) (cid:12)
(cid:12) (cid:12)
4m2 4m2 4m2 4m2
τW = m2W, τf = m2f, τH± = mH2±, τH±± = mH2±±,
h h h h
v v
gH± = −2m2 ghH+H−, gH±± = −2m2 ghH++H−−. (20)
H± H±±
N , Q are the color factor and the electric charge respectively for fermion f running in
cf f
the loop. The dimensionless loop factors for particles of spin given in the subscript are:
F = 2+3τ +3τ(2 τ)f(τ), F = 2τ[1+(1 τ)f(τ)], F = τ[1 τf(τ)], (21)
1 1/2 0
− − − −
10
