#### Scalar dark matter search from the extended νTHDM

Revised: May
Scalar dark matter search from the extended THDM
Seungwon Baek 0 1 2 3
Arindam Das 0 1 3
Takaaki Nomura 0 1 3
0 Seoul 02841 , Korea
1 Seoul 02455 , Korea
2 Department of Physics , Korea University
3 School of Physics , KIAS
We consider a neutrino Two Higgs Doublet Model ( THDM) in which neutrinos obtain naturally small Dirac masses from the soft symmetry breaking of a global U(1)X symmetry. We extended the model so the soft term is generated by the spontaneous breaking of U(1)X by a new scalar eld. The symmetry breaking pattern can also stabilize a scalar dark matter candidate. After constructing the model, we study the phenomenology of the dark matter: relic density, direct and indirect detection.
Beyond Standard Model; Higgs Physics; Neutrino Physics
1 Introduction
2 The Model
3 DM phenomenology
4 Conclusion
1
Introduction
tively there is a simple model, neutrino Two Higgs Doublet Model ( THDM) [8, 9], which
can generate the Dirac mass term for the light neutrinos as well as for the other fermions
in the SM. In this model we have two Higgs doublets; one is the same as the SM-like Higgs
doublet and the other one is having a small VEV (O(
1
)) eV to explain the tiny neutrino
mass correctly. Due to this fact, the neutrino Dirac Yukawa coupling could be order 1.
It has been discussed in [8] that a global softly broken U(
1
)X symmetry can forbid the
Majorana mass terms of the RHNs; a hidden U(
1
) gauge symmetry can be also applied to
realize THDM as in ref. [10]. In this model all the SM fermions obtain Dirac mass terms
via Yukawa interactions with the SM-like Higgs doublet ( 2) whereas only the neutrinos
get Dirac masses through the Yukawa coupling with the other Higgs doublet (
1
). Another
scenario of the generation of Dirac neutrino mass through a dimension
ve operator has
been studied in [11]. The corresponding Yukawa interactions of the Lagrangian can be
written as
LY =
QLY u e2uR
QLY d 2dR
LLY e 2eR
LLY e1 R + H:c:
(1.1)
where ei = i 2 i (i = 1; 2), QL is the SM quark doublet, LL is the SM lepton doublet,
eR is the right handed charged lepton, uR is the right handed up-quark, dR is the right
handed down-quark and R are the RHNs. The
1 and
R are assigned with the global
charge 3 under the U(
1
)X group. The global symmetry forbids the Majorana mass term
between the RHNs. In the original model [8], the global symmetry is softly broken by the
mixed mass term between 1 and 2 (m212 y1 2) such that a small VEV is obtained by
seesaw-like formulas
mM212A2v2 ;
SU(2)L
U(
1
)Y
U(
1
)X
1
2
1
2
3
2
2
1
2
0
S
1
0
3
X
1
0
1
R
1
0
3
where MA is the pseudo-scalar mass in [8]. If MA
100 GeV and m12
scalar S which breaks the U(
1
)X symmetry. The soft term m212 is identi ed with
v1 can be obtained as O(
1
) eV. In the paper [12], the model is extended to include singlet
hSi
where
is the Higgs mixing term,
y1 2S + h:c:. It has been studied in [12] that an SM
singlet fermion being charged under U(
1
)X could be a potential DM candidate.
O(100) keV then
In this paper we extend the model with a natural scalar Dark Matter (DM)
candidate (X). In this model the global U(
1
)X symmetry is spontaneously broken down to Z2
symmetry by VEV of a new singlet scalar S. The remnant of the Z2 symmetry makes
the DM candidate stable. The Z2 symmetry would be broken by quantum gravity e ect
and DM would decay via e ective interaction [13]. This can be avoided if the U(
1
)X is a
remnant of local symmetry at a high energy scale and we assume the Z2 symmetry is not
broken. A CP odd component of S becomes the Goldstone boson and hence we study the
DM annihilation from this model and compare with the current experimental sensitivity.
The papers is organized as follows. In section 2 we describe the model. In section 3
we discuss the DM phenomenology and nally in section 4 we conclude.
2
The Model
We discuss the extended version of the model in [8] with a scalar eld (X). We write the
scalar and the RHN sectors of the particle content in table 1 The gauge singlet Yukawa
interaction between the lepton doublet (LL), the doublet scalars ( 1; 2) and the RHNs
( R) can be written as
L
YiejLLi 2eRj
Yij LLi ~ 1 Rj + H:c:
We assume that the Yukawa coupling constants Yiej and Yij are real. The scalar potential
can be written by
V ( 1; 2; S) =
m121 y1 1
m222 y2 2
m2SSyS + M X2 XyX
(
y1 2S + h:c:)
+ 1( y1 1)2 + 2( y2 2)2 + 3( y1 1)( y2 2) + 4( y1 2)( y2 1)
+ S(SyS)2 + 1S y1 1SyS + 2S y2 2SyS + X (XyX)2 + 1X y1 1XyX
+ 2X y2 2XyX + SX SySXyX
( 3X SyXXX + H:c:):
(2.1)
(2.2)
The Dirac mass terms of the neutrinos are generated by the small VEV of 1. According
to [8, 9] we assume that the VEV of 1 is much smaller than the electroweak scale. The
{ 2 {
vacuum stability analysis of a general scalar potential has been studied in [14]. Additionally,
a remaining Z3 symmetry is also involved when U(
1
)X is broken by non-zero VEV of S.
Here X is the only Z3 charged stable (scalar) particle and as a result X could be considered
as a potential Dark Matter (DM) candidate. The mass term MX of X in eq. (2.2) is positive
de nite which forbids X to get VEV and as a result the Z3 symmetry promotes the stability
of X as a DM candidate. It has already been discussed in [12] that a CP-odd component
in S becomes massless Goldstone boson. Then we write scalar elds as follows
+ vS. We assume X does not develop a VEV while the VEVs of 1
, 2 and
2m121v1 + 2 1v13 + v1( 1SvS2 + 3v22 + 4v2)
2
2m222v2 + 2 2v23 + v2( 2SvS2 + 3v12 + 4v1)
2
2m2SvS + 2 SvS3 + vS( 1Sv12 + 2Sv22)
p
p
p
2 v2vS = 0;
2 v1vS = 0;
2 v1v2 = 0:
We then nd that these conditions can be satis ed with v1 '
fv2; vSg and SM Higgs
VEV is given as v ' v2 ' 246 GeV. From the rst one of the eq. (2.5) we nd that v1 is
proportional to and of the same order with
such that
v1 '
p
2 v2vS
1SvS2 + ( 3 + 4)v22
2m211 :
The small order of v1(
) is required to keep v2 and vS in the electroweak scale.
Considering the neutrino mass scale as m
0:1 eV, the value of =v2 should be small such
as =v2
O(10 12) ensuring Y
as O(
1
) such that me=v2
O(10 6). Hence v1 is
considered to be smaller than the other VEVs. It also interesting to notice that
= 0 restores
the symmetry of the Lagrangian hence a technically natural small value of
is
acceptable [15, 16]. It is also interesting to notice that
= 0 enhances the symmetry of the
Lagrangian in the sense that we can assign arbitrary U(
1
)X charge to
1, which ensures
the radiative generation of the -term is proportional to
itself. Hence a small value of
is technically natural [15, 16]. Now we identify mass spectra in the scalar sector.
Charged scalar: in this case we calculate the mass matrix in the basis ( 1
; 2 ) where
1 is approximately physical charged scalar while 2 is approximately NG boson absorbed
by W
boson. In the following we write physical charged scalar eld as H
charged scalar mass matrix can be written as
The charged Higgs mass can be written as
m2H
'
p
v2( 2 vS
2v1
4v1v2) :
{ 3 {
(2.3)
(2.4)
(2.5)
(2.6)
ical. Hence the mass matrix can be written in the basis of (h1; h2; ) as
0 pv22vv1S
0
0
1Sv1vS
0
p
v2
2
p
2
0
We nd that all the masses of the mass eigenstates, Hi(i = 1; 2; 3), are at the electroweak
scale and the mixings between h1 and other components are negligibly small while the h2
and
can have sizable mixing. The mass eigenvalues and the mixing angle for h2 and
system can be given by
m2H2;H3 =
0 sin
0
cos
Here H2 is the SM-like Higgs, h, and mH2 ' mh where the mixing angle
between H2
and H3 is constrained as sin
0:2 by the LHC Higgs data [17{19] using the numerical
analyses on the Higgs decay followed by [20, 21].
CP-odd neutral scalar: calculating the mass matrix of the pseudo-scalars in a basis
(a1; a2; aS) we get the mass matrix as
MA2 = p
0 v2vS
v1
B vS
vS
v21
v2 v1
0 pv22vv1S 0 01
0
0
0 0C ;
A
nd three mass eigenstates,
vS+p+iaS . In the last step we used the approximation, v1(
)
v2; vS. We
tan 2 =
1
2
from the interaction ae 5e, etc., because it interacts with the SM particles only via
highlysuppressed (
v1=v2;S) mixing with the SM Higgs. Note that, in our analysis below, we
approximate pseudo-scalars as A ' a1, G0
' a2 and a ' aS since we assume v1
v2; vS in
realizing small neutrino mass. Here we also discuss decoupling of the physical Goldstone
boson from thermal bath where we assume it is thermalized via Higgs portal interaction.
e ective interaction among the Goldstone boson a and the SM fermions
HJEP05(218)
where mf is the mass of the SM fermion f , and we used as ' a. The temperature, Ta, at
which a decouples from thermal bath is roughly estimated by [22]
(2.16)
(2.17)
(2.18)
(2.19)
collision rate
expansion rate '
22Smf2 Ta5mP L
m4H2 m4H3
1;
2S
where mP L denotes the Planck mass and mf should be smaller than Ta so that f is in
thermal bath. The decoupling temperature is then calculated by
Ta
2 GeV
mH3
decoupling and does not contribute to the e ective number of active neutrinos1 [23]. Note
that the Goldstone boson should be in thermal bath at temperature below that of freeze-out
of DM when we consider the relic density of DM, X, is explained by the process, XX ! aa,
in our analysis below. Taking minimum DM mass as
100 GeV freeze-out temperature
Tf is larger than
100=xf GeV
4 GeV where xf = mDM=Tf
25. Therefore we can
get Tf > Ta even with small 2S(= 0:01) as long as mH3 is not much heavier than the
electroweak scale.
As the phenomenology of the Higgs sector has been discussed in [8, 12, 24, 25], we
concentrate on the DM phenomenology in the following analysis.
1If mH3
500 MeV and 2S
0:005, then a can make sizable contribution:
Ne = 4=7 [22].
{ 5 {
Dark matter interaction. Firstly masses of dark matter candidates X is given by [27]
1
4vS2
2
1SH3
{ 6 {
m2X = M X2 +
2
1X v12 +
2
2X v22 +
2
SX 2
v
S
where the real and imaginary part of X has the same mass and X is taken as a complex
scalar eld; this is due to remnant Z3 symmetry. The interactions relevant to DM physics
are given by
L
1
vS
+
+
1X
2
4
SSH33 +
1
vS
where we ignored terms proportional to v1 since the value of VEV is tiny, SS
1S
sin
1SvS, 2S
2SvS, and omitted scalar mixing sin (cos ) assuming cos
1. Thus relevant free parameters to describe DM physics are summarized as;
m2H3 =(2vS),
' 1 and
fmX ; mH1 ; mH3 ; mA; mH ; vS; 1X ; 2X ; SX ; 3X ; 1S; 2Sg;
(3.1)
(3.2)
(3.3)
2000
1500
D
V
e
G
VS 1000
500
200
300
400
500
where we choose 1S;2S as free parameter instead of 1S;2S and we use SS = m2H3 =(2vS). In
our analysis, we focus on several speci c scenarios for DM physics by making assumptions
for model parameters to illustrate some particular processes of DM annihilations. These
scenarios are given as follows:
Scenario-I: 100 GeV < vS < 2000 GeV, f 1X ; 2X ; SX ; 3X ; 1S=vg
through contact interaction with coupling 1X as shown
gure 1-(III). Finally
scenarioIV represents semi-annihilation processes XX ! XH3 as shown in
gure 1-(IV). In our
analysis, we assumed
2S
O(
1
) so that we can neglect the case of DM annihilation via
the SM Higgs portal interaction since it is well known and constraints from direct detection
experiments are strong.
Relic density.
Here we estimate the thermal relic density of DM for each scenario given
above. The relic density is calculated numerically with micrOMEGAs 4.3.5 [30] to solve
the Boltzmann equation by implementing relevant interactions. In numerical calculations
we apply randomly produced parameter sets in the following parameter ranges. For all
{ 7 {
HJEP05(218)
1.000
0.500
Scenario-III
200
400
600
800
1000
Scenario-IV
in Scenario-III. Right: that for parameters on mX - 12X in Scenario-IV.
scenarios we apply parameter settings as
mX 2 [50; 500] GeV;
2S = 1 GeV;
MH1 = MA = MH
2 [100; 1000] GeV;
2X
1;
where the setting for 2X is to suppress the SM Higgs portal interactions and small value of
2S is to suppress scalar mixing. Then we set parameter region for each scenarios as follows:
Scenraio I : vS 2 [100; 2000] GeV;
1S 2 [0:001; 0:1] GeV;
SX;1X;3X 2 [10 8; 10 4];
MH3 2 [10; 30] GeV;
(3.4)
(3.5)
{ 8 {
Then we search for the parameter sets which can accommodate with observed relic density.
Here we apply an approximated region [31]
In gure 2, we show parameter points on mX -vS plane which can explain the observed
relic density of DM in Scenario-I. In this scenario, relic density is mostly determined by
the cross section of XX ! aSaS process which depends on mX =vS via second term of the
Lagrangian in eq. (3.2). Thus preferred value of vS becomes larger when DM mass increases
as seen in
gure 2. In left and right panel of gure 3, we respectively show parameter
points on mX - SX and 1S- SX planes satisfying correct relic density in Scenario-II. In
this scenario, the region mX . 100 GeV requires relatively larger SX coupling since scalar
boson modes fH3H3; H1H1; AA; H
H g are forbidden by our assumption for scalar boson
masses. On the other hand the region mX > 100 GeV allow wider range of SX around
0:01 .
SX . 1:0 since DM can annihilate into other scalar bosons if kinematically allowed.
In left (right) panel of gure 4, we show parameter region on mX - 1X ( 3X ) satisfying the
relic density in Scenario-III(IV). In scenario-III, DM mass should be larger than
to annihilate into scalar bosons from
1 and required value of the coupling is 0:2 .
1X .
1:0 for mX
500 GeV. In scenario-IV, the required value of the coupling 3X has similar
behavior as 1X in the scenario-III for mX > 100 GeV but slightly larger value. This
is due to the fact that semi-annihilation process require larger cross section than that of
annihilation process.
Direct detection.
Here we brie y discuss constraints from direct detection experiments
estimating DM-nucleon scattering cross section in our model.
Then we focus on our
scenario-III since DM can have sizable interaction with nucleon via H2 and H3 exchange
and investigate upper limit of mixing sin . The relevant interaction Lagrangian with
mixing e ect is given by
(3.6)
(3.7)
(3.8)
(3.9)
L
2
SX vS X X(c H3
s H2) + X mq qq(s H3 + c H2);
(3.10)
q
v
where q denote the SM quarks with mass mq, and we assumed
X
SX vS as in the
relic density calculation. We thus obtain the following e ective Lagrangian for DM-quark
interaction by integrating out H2 and H3;
mq hN jqqjN i:
X
q=c;b;t
fqN =
X
mN q=c;b;t
hN j
12
s G
a G
a
N i;
where mN is nucleon mass and fN is the e ective coupling constant given by
The heavy quark contribution is replaced by the gluon contributions such that
HJEP05(218)
where mH2 ' mh = 125 GeV is used. The e ective interaction can be rewritten in terms
of nucleon N instead of quarks such that
(3.12)
(3.13)
(3.14)
(3.15)
(3.16)
(3.17)
(3.18)
which is obtained by calculating the triangle diagram for heavy quarks inside a loop. Then
we write the trace of the stress energy tensor as follows by considering the scale anomaly;
Combining eqs. (3.14) and (3.15), we get
which leads
= mN N N =
X mqqq
q
8
7 s a G
G
a :
X
q=c;b;t
fqN =
0
2
X
q=u;d;s
1
f N
q A ;
fN =
+
2
9
7
9
X
q=u;d;s
fqN :
Finally we obtain the spin independent X-N scattering cross section as follows;
SI(XN ! XN ) =
1
8
2
NX f N2 m2N 2SX vS2s2c2
v2m2X
1
m2
h
1
m2H3
!2
;
where
NX = mN mX =(mN + mX ) is the reduced mass of nucleon and DM. Here we
consider DM-neutron scattering cross section for simplicity where that of DM-proton case
gives almost similar result. In this case, we adopt the e ective coupling fn ' 0:287 (with
fun = 0:0110, fdn = 0:0273, fsb = 0:0447) in estimating the cross section. In gure 5, we
show DM-nucleon scattering cross section as a function of sin
we take mX = 300 GeV,
mH3 = 300 GeV, vS = 5000 GeV, and
SX = 0:5(0:01) for red(blue) line as reference
values. We nd that some parameter region is constrained by direct detection when
SX
is relatively large and sin
> 0:01. More parameter region will be tested in future direct
detection experiments.
The Higgs portal interaction can be also tested by collider experiments. The
interaction can be tested via searches for invisible decay of the SM Higgs for 2mX < mh
10-43
2 D 10-45
m
MXR = 300 GeV
MH3 = 300 GeV
reference values. The current bounds from XENON1T [32] and PandaX-II [33].
while collider constraint is less signi cant compared with direct detection constraints for
2mX > mh [34{36]. Furthermore DM can be produced via heavier Higgs boson H3 if
2mX < mH3 and the possible signature will be mono-jet with missing transverse
momentum as pp ! H3j ! XXj. However the production cross section will be small when the
mixing e ect sin is small as we assumed in our analysis. Such a process would be tested in
future LHC with su ciently large integrated luminosity while detailed analysis is beyond
the scope of this paper.
Indirect detection.
Here we discuss possibility of indirect detection in our model by
estimating thermally averaged cross section in current Universe with micrOMEGAs 4.3.5
using allowed parameter sets from relic density calculations. Since aSaS
nal state is
dominant in scenario-I, we focus on the other scenarios in the following.
where left and right panels correspond to Scenario-II and Scenario-III/IV. In
Scenario
II, the cross section is mostly
O(10 26)cm 3=s while some points give smaller(larger)
values corresponding to the region with 2mX & (.)MH3 as a consequence of resonant
e ect. The annihilation processes in the scenario provide the SM
nal state via decay of
H3 and fH1; H ; Ag where H3 decay gives mainly bb via mixing with the SM Higgs and
the scalar bosons from second doublet gives leptons. This cross section would be tested
via -ray observation like Fermi-LAT [37] as well as high energy neutrino search such as
IceCube [38, 39], especially when the cross section is enhanced. In Scenario-III, the cross
section is mostly
O(10 26)cm 3=s and the nal states from DM annihilation include
Scenario-II
10-25
D
3sm10-26
v
Σ
<10-27
10-28
10-25
3sD 10-26
m
Right: that for Scenario-III and IV represented by red and blue points.
components of 1 that are fH1; H ; Ag. Thus DM mainly annihilate into neutrinos via the
decay these scalar bosons while little amount of charged lepton appear from H . Therefore
constraints from indirect detection is weaker in this scenario. In Scenario-IV, the values
of cross section is relatively larger due to the nature of semi-annihilation scenario. In this
case
nal states from DM annihilation give mostly bb via decays of H3 in the
nal state.
Then it would be tested by -ray search and neutrino observation as in the scenario-II.
4
Conclusion
We consider a neutrino Two Higgs Doublet Model ( THDM) in which small Dirac
neutrino masses are explained by small VEV, v1
O(
1
) eV, of Higgs H1 associated with
neutrino Yukawa interaction. A global U(
1
)X symmetry is introduced to forbid seesaw
mechanism. The smallness of v1 proportional to soft U(
1
)X -breaking parameter m212 is
technically natural.
We extend the model to introduce a scalar dark matter candidate X and scalar S
breaking U(
1
)X symmetry down to discrete Z2 symmetry. Both are charged under U(
1
)X .
The lighter state of X is stable since it is the lightest particle with Z2 odd parity. The
soft parameter m212 is replaced by
hSi. The physical Goldstone boson whose dominant
component is pseudoscalar part of S is shown to be phenomenologically viable due to small
ratio (
O(10 9)) of v1 compared to electroweak scale VEVs of the SM Higgs and S.
We study four scenarios depending on dark matter annihilation channels in the early
Universe to simplify the analysis of dark matter phenomenology. In Scenario I, Goldstone
modes are important. Scenario II is H3 portal. In Scenario III, the dark matter makes use
of the portal interaction with
1 which generates Dirac neutrino masses. In Scenario IV
the dominant interaction is 3X SyXXX + h:c: which induces semi-annihilation process of
our dark matter candidate. In Scenario II, the dark matter scattering cross section with
neucleons can be sizable and detected at next generation direct detection experiments. We
calculated indirect detection cross section in Scenarios II, III, and IV, which can be tested
by observing cosmic -ray and/or neutrinos.
Acknowledgments
This work is supported in part by National Research Foundation of Korea (NRF) Research
Grant NRF-2015R1A2A1A05001869 (SB).
Open Access.
This article is distributed under the terms of the Creative Commons
Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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