Quantum corrections to the spin-independent cross section of the inert doublet dark matter

3.4 MB·English

by  Tomohiro Abe

KEK-TH-1787
Quantum corrections to the spin-independent cross section
of the inert doublet dark matter
Tomohiro Abe1 and Ryosuke Sato1
5 1Theory Center, Institute of Particle and Nuclear Studies,
1
High Energy Accelerator Research Organization (KEK), Tsukuba, 305-0801, Japan
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The inert Higgs doublet model contains a stable neutral boson as a candidate of dark
n
a
matter. We calculate cross section for spin-independent scattering of the dark matter on
J

7 nucleon. We take into account electroweak and scalar quartic interactions, and evaluate
1
effects of scattering with quarks at one-loop level and with gluon at two-loop level. These

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h contributionsgiveanimportanteffectforthedarkmattermasstobearoundm /2, because
h
p
- a coupling with the standard model Higgs boson which gives the leading order contribution
p
e shouldbesuppressedtoreproducethecorrectamountofthethermalrelicabundanceinthis
h
[ mass region. In particular, we show that the dark matter self coupling changes the value of

1 the spin-independent cross section significantly.
v
1
6
1
4
0
.
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0
5
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:
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i
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r
a
1 Introduction
The discovery of the Higgs boson at the Large hadron collider (LHC) in 2012 [1, 2] is one of the
biggest achievements of the standard model (SM). In spite of its success, the SM does not include
a candidate of the dark matter which has many evidences for existing in our universe [3]. Hence,
we need some extension of the SM to explain the dark matter as an elementary particle.
The inert two-Higgs doublet model [4, 5] is a simple extension of the SM with a dark matter
candidate. It was originally discussed in an analysis of electroweak symmetry breaking in the two
Higgs doublet model by Deshpande and Ma [4], and recently, it draws attention as a model of dark
matter [5]. In this model, an additional SU(2) doublet scalar field with Y = 1/2, which is called
L
inert doublet, and a Z parity are introduced. Under this parity, all of the SM fields are even
2
and the inert doublet is odd. Then the lightest neutral boson with the Z odd charge becomes
2
the dark matter candidate. The Z odd particles have electroweak interaction and scalar quartic
2
interactions with the SM Higgs boson. Thus, they are thermalized in the early universe, and the
amount of the dark matter in the present universe is generated as a thermal relic [6–8].
TheHiggssectorintheinertdoubletmodelsometimesappearsinapartofbeyondthestandard
models, e.g., left-right Twin Higgs model [9–11], a composite Higgs model [12], a radiative seesaw
model [13–15] and models of neutrino flavor with non-Abelian discrete symmetry [16–19]. Also,
the inert doublet model is analyzed in contexts of strong first order electroweak phase transition
[20–24], Coleman-Weinberg mechanism driven by the inert doublet [25], and inflation [26]. In
spite of its simplicity, the inert doublet model has rich phenomenology. In addition to the dark
mattercandidate, themodelhasaheavierneutralscalarandachargedscalarboson. TheseZ odd
2
particlescanbeprobeddirectlyattheLHCRunII[27–31]andtheILC[32,33]. Themeasurements
ofthebranchingfractionoftheHiggsdecaye.g.,diphotonsignalandinvisibledecaywillbeaprobe
of the Z odd sector [33–36]. Also, there is a possibility of the inert doublet dark matter to be
2
probed by indirect search [37–40]. Thus, the inert doublet model is well motivated dark matter
model in both theoretical and phenomenological points of view.
The direct detection experiments give an important constraint on the inert doublet dark matter
[5, 41, 42]. At the leading order, the inert doublet dark matter scatters with the quarks at the
tree level, and with the gluon at the one-loop level by exchanging the SM Higgs boson. These
contributions to the cross section for scattering of the dark matter on nucleon can be calculated in
the same manner as the singlet scalar dark matter model [43–45]. It is proportional to λ2, where
A
2
λ is the effective Higgs-dark matter coupling which is defined in Sec. 2. If λ is not so small, they
A A
give dominant contribution to the cross section. However, if the dark matter mass m is around a
A
half of the SM Higgs boson mass, λ should be suppressed because the SM Higgs boson s-channel
A
exchange diagrams significantly contribute to the annihilation cross section which determines the
relic amount of the dark matter. In this case, contributions which does not depends on λ become
A
important for the spin-independent cross section. For example, as shown in Ref. [46], one-loop
electroweak correction for the scattering with the light quarks gives an important correction.
In this paper, we revisit the radiative correction on the spin-independent cross section in the
inert two-Higgs doublet model for the dark matter mass to be around a half of the Higgs boson
mass. In particular, Ref. [46] does not take into account for the effect of various scalar quartic
couplings. We take into account for the non-zero values of the inert doublet couplings, which are
equivalenttothemassdifferencebetweenthedarkmatterandotherZ oddparticles. Theycannot
2
be neglected in a viable parameter region in the light of the LEP II collider constraint [47, 48]. In
addition to them, there is an interesting coupling, namely the self-coupling of the Z odd particles,
2
λ . This coupling is irrelevant for the phenomenology at the tree level, but we find it also plays
2
the significant role here. Furthermore, we also evaluate contributions from twist-2 quark operators
and two-loop diagrams of dark matter-gluon scattering. These contributions give the same order
corrections as the scattering with quark at the one-loop level.
This paper is organized as follows. We briefly review the inert two-Higgs doublet model in
Sec. 2. In Sec. 3, we review the calculation of the spin-independent cross section at the tree level,
and introduce our strategy to incorporate the loop corrections to it. In Sec. 4, we show our result.
We conclude in Sec. 5. The details of the loop calculations are in the Appendices.
2 Model
In this section, we briefly review the inert doublet model. In addition to the SM Higgs field H,
we introduced a new SU(2) doublet scalar field Φ with Y = 1/2. We impose Z parity, under
L 2
which the scalar fields behave as,
H H, Φ Φ. (2.1)
→ → −
Other quark and lepton fields are also invariant under the Z parity as the SM Higgs field. Hence,
2
Φ cannot have Yukawa interactions with the SM fermions. The generic potential of H and Φ under
3
the Z parity is,
2
V(H,Φ) = m2H†H m2Φ†Φ λ (H†H)2 λ (Φ†Φ)2
1 2 1 2
− − − − −
(cid:18) (cid:19)
λ
λ (Φ†Φ)(H†H) λ (Φ†H)(H†Φ) 5(Φ†H)2+h.c. . (2.2)
3 4
− − − 2
We assume that Φ does not get any vacuum expectation value (VEV), then, the Z parity which
2
we have imposed is unbroken in the vacuum, and m2 is related to the Higgs VEV and the coupling
1
λ as,
1
m2 = 2λ v2, (2.3)
1 1
−
where v is the Higgs VEV, v2 = (√2G )−1 (246 GeV)2. G is the Fermi constant. Compared
F F
(cid:39)
to the SM, we have additional five free parameters, m2, λ , λ , λ and λ . For the stability of this
2 2 3 4 5
potential, the following relations are required [4]:
(cid:112) (cid:112)
λ > 0, λ > 0, λ > 2 λ λ , λ +λ λ > 2 λ λ . (2.4)
1 2 3 1 2 3 4 5 1 2
− −| | −
Wecanalwaystakeλ asarealpositivebyaredefinitionofthephaseofΦfield. Forexample, when
5
argλ = θ = 0, we redefine Φ as eiθ/2Φ. Therefore, the inert doublet Higgs does not contribute to
5
(cid:54)
CP violation. Hereafter we take a basis in which λ is a real positive. In this basis, we parametrize
5
the component fields of H and Φ as follows,
   
iπ+ iH+
H =  − W , Φ = − , (2.5)
v+h√+iπZ S√+iA
2 2
where each component fields correspond to mass eigenstates. We can find mass eigenvalues of each
particles and interaction terms. The mass eigenvalues are,
m2 =2v2λ , (2.6)
h 1
1
m2 =m2+ λ v2, (2.7)
H± 2 2 3
1
m2 =m2+ (λ +λ +λ )v2, (2.8)
S 2 2 3 4 5
1
m2 =m2+ (λ +λ λ )v2. (2.9)
A 2 2 3 4− 5
4
As we mentioned in the above, we take λ > 0 in this paper, hence A is the lightest neutral Z
5 2
odd particle, and it is the dark matter candidate 1.
The three-point interaction terms for the Higgs boson and the Z odd particles are,
2
1 1
(λ +λ λ )vhA2 λ vhH+H− (λ +λ +λ )vhS2. (2.10)
3 4 5 3 3 4 5
L (cid:51)− 2 − − − 2
The Higgs coupling to the dark matter is important to study dark matter phenomenology, and it
is proportional to λ +λ λ . So we denote it as
3 4 5
−
λ λ +λ λ . (2.11)
A 3 4 5
≡ −
We also introduce other short-handed notations,
∆m m m , (2.12)
H± H± A
≡ −
∆m m m . (2.13)
S S A
≡ −
We treat (m , ∆m , ∆m , λ ) as input parameters and determined (m2, λ , λ , λ ) from these
A H± S A 2 3 4 5
input parameters. Note that λ is not related with these input parameters, and irrelevant for the
2
analysis at tree level. However, λ plays an important role at the loop level as we will see later.
2
The loop correction to the dark matter mass is small for the light dark matter mass regime [51],
so we keep using the above tree level relations among the mass and couplings in this paper.
In the following of this paper, we assume almost all of the energy density of the dark matter
is comprised of the inert doublet dark matter which is generated as a thermal relic. The amount
of thermal relic is controlled by the annihilation cross section of the dark matter [6–8]. There
are some comprehensive studies on viable parameter regions [42, 49–53]. Because of its SU(2)
L
charge, AA WW(∗) channel gives a significant contribution to the annihilation cross section
→
for the case of m (cid:38) m [54], and it tends to be too large to obtain the correct abundance
A W
Ω h2 = 0.1196 0.0031 [55]. It is known that there are two parameter regions to obtain the
DM
±
correct relic abundance [52, 53]. One region is the light mass region with m (cid:46) 72 GeV, in which
A
AA WW∗ becomes less significant because it is well below energy threshold of two body WW
→
1 Some references assume S is the lightest Z odd particle. However, this is just a difference of the basis of Φ. For
2
example, if we define Φ(cid:48) ≡iΦ, we can see S(cid:48) =−A and A(cid:48) =S. Hence, there is no physical difference.
5
mode. The other region is the heavy mass region with m (cid:38) 600 GeV, in which the annihilation
A
cross section is suppressed by its mass2.
SincetheinertdoubletdarkmattercoupleswiththeSMHiggsfieldviathecouplingλ ,thedark
A
matter can scatter with nucleus and the direct detection experiment gives an important constraint
on the coupling λ [5, 41, 42]. Especially, this constraint gives a large impact on the light mass
A
region. This is because the amount of the relic abundance is also controlled by the same coupling.
As a result, the region with m (cid:46) 53 GeV is already excluded by the LUX experiment, and viable
A
region in the light mass range is 53 GeV (cid:46) m (cid:46) 72 GeV [52, 53]. In this viable range, although
A
the coupling λ is small, the annihilation cross section is enhanced because of the propagator of
A
the SM Higgs boson in s-channel. However, the scattering of a nucleon and a dark matter does
not hit the SM Higgs pole, and thus the spin-independent cross section is just suppressed by the
coupling λ . Therefore the contributions which is independent of λ , i.e., the radiative corrections
A A
on the spin-independent cross section becomes important in this mass range.
3 Spin-independent cross section
In this section, we formulate how to include radiative corrections to the spin-independent cross
section. To calculate the cross section of elastic scattering of dark matter and nucleon, first, we
construct the effective interaction of the dark matter and quark/gluon. The relevant terms for our
calculation are written as,
=1 (cid:88) ΓqA2(m q¯q) 1αsΓGA2Ga Gaµν
Leff. 2 q − 24π µν
q=u,d,s
+ 1 (cid:88) (cid:2)(∂µA)(∂νA)Γq q A(∂µ∂νA)Γ(cid:48)q q (cid:3), (3.1)
2m2 t2Oµν − t2Oµν
A q=u,d,s,c,b
where q is the quark twist-2 operator which is defined as,
µν
O
(cid:18) (cid:19)
i 1
q q¯ ∂ γ +∂ γ g /∂ q. (3.2)
Oµν ≡ 2 µ ν ν µ− 2 µν
IntheeffectiveLagrangiangiveninEq.(3.1), weneglecthighertwistgluonoperatorsbecausetheir
contributions are suppressed by α compared to the twist-0 gluon operator [57]. The coefficients Γ
s
2 Ref. [56] pointed out another parameter region in which some of diagrams of AA→WW cancel out. However,
this parameter region is severely constrained by the LUX experiment. See, Ref. [52].
6
are determined by matching with UV Lagrangian, which will be explained later. To calculate the
scattering amplitude of nucleon, we also need matrix elements of quark/gluon operators, which are
given as,
N m q¯q N = m f , (3.3)
q N q
(cid:104) | | (cid:105)
9α
s N Ga Gaµν N = m f , (3.4)
− 8π (cid:104) | µν | (cid:105) N g
(cid:18) (cid:19)
1 1
N q N = p p m2 g (q(2)+q¯(2)). (3.5)
(cid:104) |Oµν| (cid:105) m µ ν − 4 N µν
N
f is related to f as,
g q
(cid:88)
f = 1 f . (3.6)
g q
−
q=u,d,s
This relation is derived by using the relation obtained from the trace anomaly [58],
m = N Tµ N = 9αs N Ga Gaµν N + (cid:88) N m q¯q N . (3.7)
N (cid:104) | µ| (cid:105) − 8π (cid:104) | µν | (cid:105) (cid:104) | q | (cid:105)
q=u,d,s
From this discussion, we can see N m q¯q N and (α /4π) N Ga Gaµν N are same order. Thus,
q s µν
(cid:104) | | (cid:105) (cid:104) | | (cid:105)
the calculation at the n-loop order requires the (n+1)-loop order calculation for diagrams with
Ga Gaµν. For q(2) and q¯(2), we can see that they are the second moments of the quark and
µν
anti-quark parton distribution functions by using a discussion of operator product expansion as3,
(cid:90) 1
q(2)+q¯(2) = dx(q(x)+q¯(x)). (3.8)
0
We use the CTEQ parton distribution functions [60] to evaluate them, and use the same value
used in [61].
We have checked that the spin-independent cross section of a dark matter and a proton is the
almost same as of the a dark matter and a neutron. Their difference is smaller than a few percent
in almost all of the parameter region. In the following of this paper, we calculate the scattering
cross section of a dark matter and a neutron. The matrix elements which are used are summarized
in Tab. I. By using the above matrix elements and the coefficients Γ’s in the effective interaction
3 For example, see section 18.5 in Peskin-Schroeder’s textbook [59].
7
u(2) 0.11 u¯(2) 0.036
f 0.0110 d(2) 0.22 d¯(2) 0.034
u
f 0.0273 s(2) 0.026 s¯(2) 0.026
d
f 0.0447 c(2) 0.019 c¯(2) 0.019
s
b(2) 0.012 ¯b(2) 0.012
Table I: Matrix elements for neutron. Left panel shows the matrix elements for quark twist-0 operators,
whicharetakenfromthedefaultvaluesof micrOMEGAs[62]. Rightpanelshowsthesecondmomentsforquark
distribution function, which are evaluated at the scale of µ = m by using the CTEQ parton distribution
Z
functions [60].
A A
A A
h
h
q q Q
(a) (b)
Figure 1: The diagrams which contribute to the spin-independent cross section at the leading order.
given in Eq. (3.1), the scattering amplitude of the nucleon and the dark matter is given as,
(cid:34) (cid:35)
i =im (cid:88)Γqf + 2ΓGf + 3 (cid:88)(Γq +Γ(cid:48)q )(q(2)+q¯(2)) , (3.9)
M N q 9 g 4 t2 t2
q q
µ2
σ = 2, (3.10)
SI 4πm2 |M|
A
where µ is the reduced mass, which is defined as µ m m /(m +m ). Hence, what we have
N A N A
≡
to calculate is the effective coupling Γ’s.
3.1 At the leading order
We start to give a brief review on the calculation at the leading order. We need to calculate
the elastic scattering cross section for the dark matter and nucleon system, σ(DM N DM N),
→
where N stands for the nucleon. As described before, we construct the effective Lagrangian with
the gluon and the light quarks q = u,d,s by integrating out the heavy quarks Q = c,b,t and
the SM Higgs boson. We should take into account the one-loop diagrams for the scattering with
gluon, because their contributions are same order as the tree-level scattering with the light quarks.
8
The dark matter scatters with the SM quarks at the tree level and the gluon at the one-loop level
as shown in Fig. 1(a) and 1(b), respectively. Their amplitudes are proportional to the effective
Higgs-dark matter coupling λ . From these processes, the following relevant operators for the
A
spin-independent cross section are generated,
A2q¯q, A2Ga Gaµν. (3.11)
µν
The coefficients of the effective Lagrangian given at the leading order is determined as,
λ
Γq = ΓG = A, Γq = Γ(cid:48)q = 0. (3.12)
m2 t2 t2
h
Using these coefficients and Eq. (3.10), we can calculate the amplitude of the process and the
spin-independent cross section as,
1 λ2µ2m2 f2
σ = A N N, (3.13)
SI 4π m2m4
A h
where,
2 7 (cid:88)
f + f . (3.14)
N q
≡ 9 9
q
3.2 At the next leading order
We move to calculate the loop corrections to the spin-independent cross section. We need to
consider the loop corrections to the four relevant operators for the spin-independent cross section,
A2q¯q, A2Ga Gaµν, (∂µA)(∂νA) q , A(∂µ∂νA) q . (3.15)
µν µν µν
O O
There are some remarks on this calculation. First, trace anomaly relation Eq. (3.7) is suffered
from QCD correction at the next-leading order. However, we consider λ is not so large, and
A
assume corrections of the order of λ α /4π can be neglected. Also, for the contribution which is
A s
independent of λ , we only take into account the leading order of α . Thus, for the scattering with
A s
the gluon, we can still use Eq. (3.7) even in the loop level calculation. Second, we evaluate the
9
A A
A A
A A
H±,S
h W,Z W,Z
W,Z W,Z
q q q q q q
(a) (b) (c)
A A
A A A A
H±,S
h
W,Z W,Z
W,Z W,Z
Q
(d) (e) (f)
Figure 2: The diagrams we calculate. The shaded region is one-loop correction.
effect of twist-2 operator q at the scale µ = m . Thus, we take into account q = u,d,s,c and b
µν Z
O
and evaluate the matrix element of q by using the parton distribution functions at µ = m .
µν Z
O
The diagrams we need to calculate are shown in Fig. 2. The diagrams with gluons are two-loop
diagrams but contribute to the spin-independent cross section as the one-loop order correction as
we mentioned in Sec. 3.1. There are some diagrams which are the same order but not shown in
Fig. 2. They are proportional to the Higgs coupling to the dark matter, λ . We are interested
A
in the case that this coupling is very small. Thus the diagrams with this coupling give much
smaller contributions than the diagrams shown in Fig. 2, and do not need to be calculated. Here
we parametrized the loop corrections to the λ as δΓ (q2), and denote the correction from the
A h h
box and triangle diagrams as Γq . Here q2 is the momentum squared of the Higgs boson. What
Box h
we need is the scattering amplitude in the non-relativistic limit. In the limit of zero momentum
transfer, the amplitudes of the diagrams given in Fig. 2 are written as,
iδΓ (0)
h
Fig. 2(a) = m u¯u, (3.16)
m2 q
h
(cid:18) (cid:19)
i 1
Fig. 2(b)+Fig. 2(c) = iΓq m u¯u+ (Γq +Γ(cid:48)q )u¯ (pq)p/ p2/q u, (3.17)
Box q m2 t2 t2 − 4
A
(cid:18) (cid:19)
iδΓ (0) 2 9α
Fig. 2(d) = h sGa Gaµν , (3.18)
m2 × 9 − 8π µν
h
(cid:18) (cid:19)
2 9α
Fig. 2(e)+Fig. 2(f) = iΓG sGa Gaµν . (3.19)
Box× 9 − 8π µν
10