Fermion dark matter in gaugeHiggs unification
HJE
Fermion dark matter in gaugeHiggs unification
Nobuhito Maru 0 3
Takashi Miyaji 0 3
Nobuchika Okada 0 1
Satomi Okada 0 2
0 Tuscaloosa , Alabama 35487 , U.S.A
1 Department of Physics and Astronomy, University of Alabama , USA
2 Graduate School of Science and Engineering, Yamagata University
3 Department of Mathematics and Physics, Osaka City University
We propose a Majorana fermion dark matter in the context of a simple gaugeHiggs Unification (GHU) scenario based on the gauge group SU(3)×U(1)′ in 5dimensional Minkowski space with a compactification of the 5th dimension on S1/Z2 orbifold. The dark matter particle is identified with the lightest mode in SU(3) triplet fermions additionally introduced in the 5dimensional bulk. We find an allowed parameter region for the dark matter mass around a half of the Standard Model Higgs boson mass, which is consistent with the observed dark matter density and the constraint from the LUX 2016 result for the direct dark matter search. The entire allowed region will be covered by, for example, the LUXZEPLIN dark matter experiment in the near future. We also show that in the presence of the bulk SU(3) triplet fermions the 125 GeV Higgs boson mass is reproduced through the renormalization group evolution of Higgs quartic coupling with the compactification scale of around 108 GeV.
Phenomenology of Field Theories in Higher Dimensions

1 Introduction
2
3
4
5
6
1
Dark matter relic abundance
Direct dark matter detection
Higgs boson mass in effective theory approach
Conclusions and discussions
Introduction
particle. Among various possibilities, the socalled Weakly Interacting Massive Particle
is a prime candidate for the DM particle, which is the thermal relic from the early
Universe and whose relic abundance is calculable independently of the history of the Universe
before the DM has gotten in thermal equilibrium. A variety of experiments aiming for
directly/indirectly detecting DM particles is ongoing and planned, and the discovery of the
dark matter may be around the corner. In this paper we consider a fermion DM in the
context of a simple gaugeHiggs Unification (GHU) scenario in 5dimensions and identify
a modelparameter region which is consistent with the current experimental constraints.
The GHU scenario [1–6] is a unique candidate for new physics beyond the SM, which
offers a solution to the gauge hierarchy problem without invoking supersymmetry. An
essential property of the GHU scenario is that the SM Higgs doublet is identified with
an extra spatial component of the gauge field in higher dimensions. Associated with the
higherdimensional gauge symmetry, the GHU scenario predicts various finite physical
observables, irrespective of the nonrenormalizability of the scenario, such as the effective
Higgs potential [7–12], the effective Higgs coupling with digluon/diphoton [13–15], the
anomalous magnetic moment g − 2 [16, 17], and the electric dipole moment [18].
In the previous paper by some of the present authors [15], the oneloop contributions
of KaluzaKlein (KK) modes to the Higgstodigluon and Higgstodiphoton couplings were
calculated in a 5dimensional GHU model by introducing colorsinglet bulk fermions with
a halfperiodic boundary condition, in addition to the SM fermions. It was shown that
the colorsinglet bulk fermions play a crucial role not only to explain the observed
Higgstodigluon and Higgstodiphoton couplings, but also to achieve the 125 GeV Higgs boson
– 1 –
mass. See also refs. [19, 20] for extended analysis including colortriplet bulk fermions. As
a bonus, it was pointed out that the lightest KK mode of the bulk fermions can be a DM
candidate by choosing their hypercharges appropriately. The main purpose of this paper
is to pursue this possibility and investigate the DM physics in the context of the GHU
scenario. For related works on the DM physics in GHU scenarios, see refs. [21–25].
Towards the completion of the GHU scenario as new physics beyond the SM, we need
to supplement a DM candidate to the scenario. In order to keep the original motivation of
the GHU scenario to solve the gauge hierarchy problem, the DM candidate to be introduced
must be a fermion. Since the GHU scenario is defined in a higher dimensional spacetime
with a gauge group into which the SM gauge group is embedded, it would be the most
section of the DM particle off with nuclei from the current DM direct detection
experiments. An effective field theoretical approach of the GHU scenario will be discussed in
section 5, and the 125 GeV Higgs boson mass is reproduced in the presence of the bulk
SU(3) triplet fermions with certain boundary conditions. The compactification scale is
determined in order to reproduce the Higgs boson mass of 125 GeV. The last section is
devoted to conclusions.
2
Fermion DM in GHU
We consider a GHU model based on the gauge group SU(3) × U(
1
)′ [26] in a 5dimensional
flat spacetime with orbifolding on S1/Z2 with radius R of S1. In our setup of bulk
fermions including the SM fermions, we follow ref. [27]: the uptype quarks except for the
top quark, the downtype quarks and the leptons are embedded, respectively, into 3, 6,
and 10 representations of SU(3). In order to realize the large top Yukawa coupling, the
– 2 –
top quark is embedded into a rank 4 representation of SU(3), namely 15. The extra U(
1
)′
symmetry works to yield the correct weak mixing angle, and the SM U(
1
)Y gauge boson
is realized by a linear combination between the gauge bosons of the U(
1
)′ and the U(
1
)
subgroup in SU(3) [26]. Appropriate U(
1
)′ charges for bulk fermions are assigned to yield
the correct hypercharges for the SM fermions.
The boundary conditions should be suitably assigned to reproduce the SM fields as
the zero modes. While a periodic boundary condition corresponding to S1 is taken for
all of the bulk SM fields, the Z2 parity is assigned for gauge fields and fermions in the
representation R by using the parity matrix P = diag(−, −, +) as
Aμ(−y) = P †Aμ(y)P,
Ay(−y) = −P †Ay(y)P,
ψ(−y) = R(P )γ5ψ(y)
(2.1)
where the subscripts μ (y) denotes the four (the fifth) dimensional component. With this
choice of parities, the SU(3) gauge symmetry is explicitly broken down to SU(2) × U(
1
).
A hypercharge is a linear combination of U(
1
) and U(
1
)′ in this setup. One may think
that the U(
1
)X gauge boson which is orthogonal to the hypercharge U(
1
)Y also has a zero
mode. However, the U(
1
)X symmetry is anomalous in general and broken at the cutoff
scale and hence, the U(
1
)X gauge boson has a mass of order of the cutoff scale [26]. As a
result, zeromode vector bosons in the model are only the SM gauge fields.
Offdiagonal blocks in Ay have zero modes because of the overall sign in eq. (2.1), which
corresponds to an SU(2) doublet. In fact, the SM Higgs doublet (H) is identified with
A(y0) = √
1
2
0 H !
H† 0
brane localized fermions with conjugate SU(2) × U(
1
) charges and an opposite chirality
to the exotic fermions, allowing us to write mass terms on the orbifold fixed points. In
the GHU scenario, the Yukawa interaction is unified with the gauge interaction, so that
the SM fermions obtain the mass of the order of the W boson mass after the electroweak
symmetry breaking. To realize light SM fermion masses, one may introduce Z2parity odd
bulk mass terms for the SM fermions, except for the top quark. Then, zero mode fermion
wave functions with opposite chirality are localized towards the opposite orbifold fixed
points and as a result, their Yukawa couplings are exponentially suppressed by the overlap
integral of the wave functions. In this way, all exotic fermion zero modes can be heavy
and the small Yukawa couplings for the light SM fermions can be realized by adjusting the
bulk mass parameters. In order to realize the top quark Yukawa coupling, we introduce a
rank 4 tensor representation, namely, a symmetric 15 without a bulk mass [27]. This leads
to a group theoretical factor 2 enhancement of the top quark mass as mt = 2mW at the
compactification scale [26]. Note that this mass relation is desirable since the top quark
pole mass receives QCD threshold corrections which push up the mass about 10 GeV.
– 3 –
HJEP07(21)48
Now we discuss the DM sector in our model. In addition to the bulk fermions
corresponding to the SM quarks and leptons, we introduce a pair of extra bulk fermions ψ, ψ˜
which are triplet representations under the bulk SU(3) and have a U(
1
)′ charge 1/3. With
this choice of the U(
1
)′ charge, the triplet bulk fermions include electriccharge neutral
components and a linear combination among the charge neutral components serves as the
DM particle. Associated with S1 we impose the periodic boundary condition in the fifth
dimension, while the Z2 parity assignments are chosen as
ψ(−y) = P γ5ψ(y),
ψ˜ = −P γ5ψ˜(y).
After the electroweak symmetry breaking, the lightest mass eigenstate among the bulk
triplets is identified with the DM particle. As we will discuss in section 5, these bulk
fermions also play a crucial role to reproduce the observed Higgs boson mass of 125 GeV.
The Lagrangian relevant to our DM physics discussion is given by
LDM = ψ iD/ ψ + ψ˜ iD/ ψ˜ − M
ψψ˜ + ψ˜ψ
+ δ(y)
m2 ψ3(0R)cψ3(0R) +
2
m˜ ψ˜3(0L)cψ˜3(0L) + h.c. ,
where the covariant derivative and a pair of the bulk SU(3) triplets are given by
D/ = ΓM ∂M − igAM − ig′A′M ,
ψ = (ψ1, ψ2, ψ3)T ,
ψ˜ =
ψ˜1, ψ˜2, ψ˜3
T
ψ(x, y) = √2πR
ψ(x, y) = √
1
1
πR n=1
ψ(0)(x) + √
X ψ(n)(x) cos
for ψ1L,2L,3R, ψ˜1R,2R,3L ,
n
R
y
∞
X ψ(n)(x) sin
for ψ1R,2R,3L, ψ˜1L,2L,3R ,
With the nontrivial orbifold boundary conditions, the bulk SU(3) triplet fermions are
decomposed into the SM SU(2) doublet and singlet fermions. As we will see later, the
DM particle is provided as a linear combination of the second and third components of the
triplet fermions. In eq. (2.4) we have introduced a bulk mass (M ) to avoid exotic massless
fermions. Here we have also introduced Majorana mass terms on the brane at y = 0 for
the zeromodes of the third components of the triplets (ψ3(0R) and ψ˜3(0L)), which are singlet
under the SM gauge group. The superscript “c” denotes the charge conjugation. With
the Majorana masses on the brane, the DM particle in 4dimensional effective theory is a
Majorana fermion, and hence its spinindependent cross section with nuclei through the
Zboson exchange vanishes in the nonrelativistic limit.
Let us focus on the following terms in eq. (2.4), which are relevant to the mass terms
in 4dimensional effective theory:
Lmass = ψiΓ5 (∂y − ighAyi) ψ + ψ˜iΓ5 (∂y − ighAyi) ψ˜ − M
ψψ˜ + ψ˜ψ
+δ(y)
m2 ψ3(0R)cψ3(0R) +
2
m˜ ψ˜3(0L)cψ3(0L) + h.c. ,
where Γ5 = iγ5. Expanding the bulk fermions in terms of KK modes as
∞
1
πR n=1
n
R
y
– 4 –
(2.3)
(2.4)
(2.5)
(2.6)
(2.7)
(2.8)
(2.9)
and integrating out the fifth coordinate y, we obtain the expression in 4dimensional
effective theory. The zeromode parts for the electriccharge neutral fermions are found
to be
Lmass
zero−mode = imW
ψ2(0L)ψ3(0R) + ψ˜(0) ˜(0)
3L ψ2R
− M
ψ2(0L)ψ˜2(0R) + ψ˜3(0L)ψ3(0R)
+ h.c.
2
m˜ ψ˜3(0L)cψ˜3(0L) + h.c.
m ψ3(0R)cψ(0)
3R −
m˜ ψ˜3(0L)cψ˜3(0L) + h.c.
2
2
→ −mW
ψ2(0L)ψ(0)
3R − ψ˜(0) ˜(0)
3L ψ2R
− M
ψ2(0L)ψ˜2(0R) + ψ˜3(0L)ψ3(0R)
+ h.c.
where mW = gv/2 is the W boson mass, and the arrow means the phase rotations ψ3(0R) →
iψ3(0R) and ψ˜3(0L) → iψ˜3(0L). It is useful to rewrite these mass terms in a Majorana basis
defined as
χ ≡ ψ3(0R) + ψ3(0R)c,
ω ≡ ψ2(0L) + ψ2(0L)c,
χ˜ ≡ ψ˜3(0L) + ψ˜(0)c
3L ,
ω˜ ≡ ψ˜2(0R) + ψ˜(0)c
2R ,
and we then express the mass matrix (MN) as
Lmass
zero−mode = − 2
The zeromodes of the charged fermions, ψ1(0L) and ψ˜(0)
1R , have a Dirac mass of M .
To simplify our analysis, we set m = m˜, and in this case we find a simple expression
for the mass eigenvalues of MN as
m1 =
m2 =
m3 =
m4 =
1
2
1
2
1
2
1
2
m −
m +
m −
m +
q
q
q
q
4m2W + (m − 2M )2 ,
4m2W + (m − 2M )2 ,
4m2W + (m + 2M )2 ,
4m2W + (m + 2M )2 ,
– 5 –
(2.10)
HJEP07(21)48
(2.11)
(2.12)
(2.13)
for the mass eigenstates defined as (χ χ˜ ω ω˜)T = UM (η1 η2 η3 η4)T with a unitary matrix
,
,
,
,
Note that without loss of generality we can take M, m ≥ 0. Considering the current
experimental constraints from the search for an exotic charged fermion, we may take M &
1 TeV ≫ mW [28]. In this case, the lowest mass eigenvalue (dark matter mass mDM) is
given by m1. From the explicit form of the mass matrix MN in eq. (2.12) and M ≫ mW ,
we notice two typical cases for the constituent of the DM particle: (i) the DM particle is
mostly an SM singlet when m = m˜ . M , or (ii) the DM particle is mostly a component in
the SM SU(2) doublets when m = m˜ & M . In the case (i), the DM particle communicates
with the SM particle essentially through the SM Higgs boson. On the other hand, the
DM particle is quite similar to the socalled Higgsinolike neutralino DM in the minimal
supersymmetric SM (MSSM) for the case (ii). Since the Higgsinolike neutralino DM has
been very wellstudied in many literatures,1 we focus on the case (i) in this paper. Note
that the case (i) is a realization of the socalled Higgsportal DM from the GHU scenario.
We emphasize that in our scenario, the Yukawa couplings in the original Lagrangian are
not free parameters, but are the SM SU(2) gauge coupling, thanks to the structure of the
GHU scenario.
Now we describe the coupling between the DM particle and the Higgs boson. In the
original basis, the interaction can be read off from eq. (2.12) by v → v + h as
LHiggs−coupling = − 2
h
1In this case, a pair of the DM particles mainly annihilates into the weak gauge bosons through the
SM SU(2) gauge coupling, and the observed DM relic abundance can be reproduced with the DM mass of
around 1 TeV [29].
– 6 –
where h is the physical Higgs boson, and the explicit form of the matrix Ch is given by
C1 =
2 , C2 =
2 , C3 =
2 , C4 =
2 , C5 =
4u2
c
2
4u3
c
3
4u4
c
4
c1c2
2(u1 + u2) , C6 =
2(u3 + u4) .
c3c4
The interaction Lagrangian relevant to the DM physics is given by
LDM−H = − 2
1 mW
v
C1 h ψDM ψDM − 2
1 mW
v
C5 h (η2 ψDM + h.c.) ,
where we have identified the lightest mass eigenstate η1 as the DM particle (ψDM).
3
Dark matter relic abundance
In this section, we evaluate the DM relic abundance and identify an allowed parameter
region to be consistent with the Planck 2015 measurement of the DM relic density [30] (68
% confidence level):
In our model, the DM physics is controlled by only two free parameters, namely, m and
M . As we discussed in the previous section, we focus on the Higgsportal DM case with
0 ≤ m . M . Using M ≫ mW , we can easily derive approximate formulas for parameters
involved in our DM analysis. For the mass eigenvalues listed in eq. (2.13), we find
m2W
m1 ≃ −M + m − 2M − m
,
m2 ≃ M +
m2W
2M − m
.
By using these formulas, we express u1,2 and c1,2 in eqs. (2.15) and (2.16) as
u1 ≃ −
2M − m
mW
mW
, u2 ≃ 2M − m
, c1 ≃
√
2
2M − m
mW
, c2 ≃
√2,
which lead to
C1 ≃ − 2M − m
2mW
, C5 ≃ 1.
For m . M and a fixed value of M ≫ mW , the DM particle can be light when m ≃ M ,
otherwise mDM ≃ M while m2 ≃ M for any values of m . M . The coupling of a DM
particle pair with the Higgs boson is always suppressed by C1 ≪ 1 while C5 ≃ 1.
According to the interaction Lagrangian in eq. (2.20), we consider two main
annihilation processes of a pair of DM particles. One is through the schannel Higgs boson
exchange, and the other is the process ψDMψDM → hh through the exchange of η2 in the
t/uchannel. Since C1 ≪ 1 and C5 ≃ 1, the t/uchannel processes dominate for the DM
(2.18)
(2.19)
(2.20)
(3.1)
(3.2)
(3.3)
(3.4)
pair annihilations when the DM particle is heavier than the Higgs boson. In evaluating
this process, we may use an effective Lagrangian of the form,
which is obtained by integrating η2 out, and calculate the DM pair annihilation cross
section times relative velocity (vrel) as
eff
LDM−H =
1
2
mW
v
m2
2 C52 h h ψDM ψDM,
σvrel =
1
64π
mW
v
4
It is wellknown that the observed DM relic density is reproduced by σ0 ∼ 1 pb. Since we
find σ0 ∼ 0.02 pb for C5 ≃ 1 and m2 ≃ M = 1 TeV, we conclude that the observed relic
density is not reproduced by the process ψDMψDM → hh.
Next we consider the DM pair annihilation through the schannel Higgs boson exchange
when the DM particle is lighter than the Higgs boson. Since the coupling between the a
pair of DM particles and the Higgs boson is suppressed by C1 ≪ 1, an enhancement of the
DM annihilation cross section through the Higgs boson resonance is necessary to reproduce
the observed relic DM density. We evaluate the DM relic abundance by integrating the
Boltzmann equation
dY
dx
= − H(mDM)
xshσvi
Y 2
− YE2Q ,
where the temperature of the Universe is normalized by the DM mass as x = mDM/T ,
H(mDM) is the Hubble parameter as T = mDM, Y is the yield (the ratio of the DM
number density to the entropy density s) of the DM particle, YEQ is the yield of the DM
in thermal equilibrium, and hσvreli is the thermal average of the DM annihilation cross
section times relative velocity for a pair of the DM particles. Various quantities in the
Boltzmann equation are given as follows:
(3.5)
(3.6)
g∗ mx3D3M ,
H(mDM) =
r π2
90 g∗ MP
m2DM ,
sYEQ =
2π2
gDM m3DM K2(x),
x
where MP = 2.44 × 1018 GeV is the reduced Planck mass, gDM = 2 is the number of degrees
of freedom for the DM particle, g∗ is the effective total number of degrees of freedom for
the particles in thermal equilibrium (in our analysis, we use g∗ = 86.25 corresponding
to mDM ≃ mh/2 with the Higgs boson mass of 125 GeV), and K2 is the modified Bessel
function of the second kind. For mDM ≃ mh/2 = 62.5 GeV, a DM pair annihilates into a
pair of the SM fermions as ψDMψDM → h → f f¯, where f denotes the SM fermions. We
calculate the cross section for the annihilation process as
σ(s) =
2
yDM
16π
3
mb 2
v
+ 3
mc 2
v
+
mτ 2
v
qs s − 4m2DM
s − m2 2 + m2 Γ2
h h h
where yDM = (mW /v)C1 (see eq. (2.20)), and we have only considered pairs of bottom,
charm and tau for the final states, neglecting the other lighter quarks, and used the following
– 8 –
values for the fermion masses at the Zboson mass scale [31]: mb = 2.82 GeV, mc =
685 MeV and mτ = 1.75 GeV. The total Higgs boson decay width Γh is given by Γh =
ΓSM +Γnhew, where ΓShM = 4.07 MeV [32] is the total Higgs boson decay width in the SM and
h
Γnew =
h
annihilation cross section is given by
hσvi = (sYEQ)−2gD2M 64π4x 4mDM
mDM Z ∞
dsσˆ(s)√sK1
√
x s
mDM
,
where σˆ(s) = 2(s − 4m2DM)σ(s) is the reduced cross section with the total annihilation
cross section σ(s), and K1 is the modified Bessel function of the first kind. We solve the
Boltzmann equation numerically and find an asymptotic value of the yield Y (∞) to obtain
the present DM relic density as
Ωh2 =
mDMs0Y (∞) ,
ρc/h2
10−5 GeV/cm3 is the critical density.
where s0 = 2890 cm−3 is the entropy density of the present universe, and ρc/h2 = 1.05 ×
In figure 1 we show the resultant DM relic density as a function of the DM mass for
various values of yDM. The solid lines from top to bottom correspond to yDM = 0.005,
0.00692 and 0.01, respectively, while the dashed line denotes the observed DM density
ΩDMh2 = 0.1198 from the Planck 2015 result. For a fixed yDM value, intersections of the
solid and the dashed lines denote the DM mass to reproduce the observed DM density. We
can see that there is a lower bound on yDM ≥ 0.00692 in order to reproduce the observed
DM density.
In the left panel of figure 2, we show yDM as a function of mDM (solid line) along
which the observed DM density ΩDMh2 = 0.1198 is reproduced. Here, the current
experimental upper bound from the LUX 2016 result [33] and the prospective reach in the
future LUXZEPLIN DM experiment [34] are also shown as the dashed and the
dotted lines, respectively, which will be derived in section 4. In order to satisfy the LUX
2016 constraint, we find the parameter regions such as 58.0 ≤ mDM[GeV] ≤ 62.4 and
(0.00692 ≤) yDM ≤ 0.0164. Recall that the Yukawa coupling between the DM particle and
the Higgs boson, yDM = (mW /v)C1, and the DM mass are determined by the two
parameters, M and m, from eqs. (2.13), (2.15) and (2.16). Using these formulas, we can express M
as a function of mDM along the solid line in the left panel. Our result is shown in the right
panel. Since M ≫ mW , the parameter m as a function of mDM is approximately given by
m ≃ M − mDM . Corresponding to the parameter regions of 58.0 ≤ mDM[GeV] ≤ 62.4 and
(0.00692 ≤) yDM ≤ 0.0164, we find 3.14 ≤ M [TeV] (≤ 7.51).
– 9 –
2
M0.20
D0.15
W
0.10
59
60
61
62
along with the observed DM density ΩDMh2 = 0.1198 (horizontal dashed line) The three solid lines
form top to bottom correspond to yDM = 0.005, 0.00692, and 0.01, respectively.
0.050
0.030
0.020
M0.015
D
y
0.010
56
58
60
62
56
58
60
62
ΩDMh2 = 0.1198 is reproduced. Here, the current experimental upper bound from the LUX 2016
result [33] and the prospective reach in the future LUXZEPLIN DM experiment [34] are also shown
as the dashed and the dotted lines, respectively. Right panel: M as a function of mDM, along the
solid line in the left panel.
4
Direct dark matter detection
A variety of experiments are underway and also planned for directly detecting a dark matter
particle through its elastic scattering off with nuclei. In this section, we calculate the
spinindependent elastic scattering cross section of the DM particle via the Higgs boson exchange
to lead to the constraint on the model parameters from the current experimental results.
7.0
5.0
D
V
e
M
2.0
1.5
The spinindependent elastic scattering cross section with nucleon is given by
where μψDMN = mN mDM/(mN + mDM) is the reduced mass of the DMnucleon system
with the nucleon mass mN = 0.939 GeV, and
σSI =
1 yDM 2 μψDMN
π
v
2
2
fN
fN =
X
q=u,d,s
fTq + 9 fT G mN
m2
h
2
(4.1)
(4.2)
(4.3)
is the nuclear matrix element accounting for the quark and gluon contents of the nucleon. In
evaluating fTq , we use the results from the lattice QCD simulation [35]: fTu + fTd ≃ 0.056
and fTs ≤ 0.08. For conservative analysis, we take fTs = 0 in the following.
the trace anomaly formula, Pq=u,d,s fTq + fT G = 1 [36–40], we obtain f N2 ≃ 0.0706 m2N
Using
and hence
σSI ≃ 4.47 × 10−7 pb × yDM
2
for mDM = mh/2 = 62.5 GeV.
The LUX 2016 result [33] currently provides us with the most severe upper bound on
the spinindependent cross section, from which we read σSI ≤ 1.2 × 10−10 pb for mDM ≃
62.5 GeV. From eq. (4.3), we find yDM ≤ 0.0164, which is depicted as the horizontal dashed
line in the left panel of figure 2. The nextgeneration successor of the LUX experiment, the
LUXZEPLIN experiment [34], plans to achieve an improvement for the upper bound on
the spinindependent cross section by about two orders of magnitude. When we apply a
conservative search reach to the LUXZEPLIN experiment as σSI ≤ 1.2 × 10−11 pb (just an
order of magnitude improvement from the current LUX bound), we obtain yDM ≤ 0.00518.
This prospective upper bound is shown as the dotted line in the left panel of figure 2. We
can see that the present allowed parameter region all covered by the future LUXZEPLIN
experiment.
5
Higgs boson mass in effective theory approach
In this section, we calculate the Higgs boson mass by using a 4dimensional effective theory
approach of the GHU scenario in 5dimensional Minkowski space, which is developed in
refs. [41, 42]. In this paper, it has been shown that an effective Higgs quartic coupling
derived from the 1loop effective Higgs potential after integrating out all KK modes coincides
with a running Higgs quartic coupling at low energies obtained from the renormalization
group (RG) evolution with a vanishing Higgs quartic coupling at the compactification
scale (“gaugeHiggs condition” [41, 42]). This vanishing Higgs quartic coupling indicates
a restoration of the 5dimensional gauge invariance at the compactification scale. With
this approach, we can easily calculate the Higgs quartic coupling at low energies by solving
the RG equations, once the particle contents and the mass spectrum of the model below
the compactification are defined. Assuming that the electroweak symmetry breaking is
correctly achieved, the Higgs boson mass is calculated by the Higgs quartic coupling value
at the electroweak scale.
There are two scales involved in our RG analysis, namely, the bulk mass M ≃ m
and the compactification scale MKK = 1/R. In the following analysis, we ignore the mass
splitting among the bulk fermion zero modes and set all of their masses as M . As we will
show in the following, a hierarchy M ≪ MKK is necessary to reproduce the 125 GeV Higgs
boson mass, and hence this treatment is justified.
For the renormalization scale smaller than the bulk mass μ < M , all bulk fermions are
decoupled and we employ the SM RG equations at two loop level [43–49]. For the three
SM gauge couplings gi (i = 1, 2, 3), we have
Here, among the SM Yukawa couplings, we have taken only the top Yukawa coupling (yt)
into account. The RG equation for the top Yukawa coupling is given by
where the oneloop contribution is
while the twoloop contribution is given by
The RG equation for the Higgs quartic coupling is given by
β(2) = −12yt4 +
t
38903 g12 +
21265 g22 + 36g32 y
2
t
+
d ln μ
=
1
16π2 βλ(
1
) +
(16π2)2 βλ(2),
1
with
β(
1
) = 12λ2 −
λ
59 g12 + 9g22 λ +
(5.1)
(5.2)
(5.3)
(5.4)
(5.5)
(5.6)
(5.7)
and
In solving these RGEs, we use the boundary conditions at the top quark pole mass (Mt)
,
,
λ(Mt) = 2(0.12711 + 0.00206(mh − 125.66) − 0.00004(Mt − 173.10)).
(5.9)
We employ MW = 80.384 GeV, αs = 0.1184, the central value of the combination of
Tevatron and LHC measurements of top quark mass Mt = 173.34 GeV [51], and the central
value of the updated Higgs boson mass measurement, mh = 125.09 GeV from the combined
analysis by the ATLAS and the CMS collaborations [52].
For the renormalization scale μ ≥
M , the SM RG equations are modified in the
presence of the bulk fermions. In this paper, we take only oneloop corrections from the
bulk fermions into account. In the presence of a pair of the SU(3) triplet bulk fermions,
the beta functions of the SU(2) and U(
1
)Y gauge couplings receive new contributions as
Δb1 = Δb2 = .
2
3
The beta functions of the top Yukawa and Higgs quartic couplings are modified as
β(
1
)
t
→ βt(
1
) + 2ytYS2, β(
1
)
λ → βλ(
1
) + 8λYS2 − 8YS4,
where YS is the universal Yukawa coupling of ψ and ψ˜ with the Higgs doublet in eq. (2.7),
which obeys the RG equation,
16π2 dYS = YS 3yt2 −
290 g12 + g
In our RG analysis, we numerically solve the SM RG equations from Mt to M , at
which the solutions connect with the solutions of the RG equations with the bulk triplet
(5.8)
(5.10)
(5.11)
(5.12)
0.25
(solid line), along with the result in the SM (dashed line). The compactification scale is found
to be MKK = 1.9 × 108 GeV, where the gaugeHiggs condition λ(MKK) = 0 and YS(MKK) =
g2(MKK)/√2 are satisfied. Right panel: the relation between the bulk mass (M ) and the
compactification scale (MKK) so as to reproduce the 125 GeV Higgs boson mass.
fermions. For a fixed M values, we arrange an input YS(M ) value so as to find numerical
solutions which satisfy the gaugeHiggs condition and the unification condition between
the gauge and Yukawa couplings at the compactification scale, such that
λ(MKK) = 0, YS(MKK) =
g2(MKK) .
√
2
(5.13)
is lowered from MKK ≃ 1010 GeV to 108 GeV.
In the left panel of figure 3 we show the RG evolution of the Higgs quartic coupling (solid
line) for M = 1 TeV, along with the one in the SM (dashed line). At MKK = 1.9 × 108 GeV,
the boundary condition in eq. (5.13) is satisfied. For a fixed M value, we numerically find
a MKK value. Our result for the relation between M and MKK is shown in the right
panel of figure 3. For MKK ≃ 108 GeV, the 125 GeV Higgs boson mass is reproduced. We
obtained the hierarchy M ≪ MKK mentioned above. Note that in the absence of the bulk
SU(3) triplet fermions, the RG evolution of the Higgs quartic coupling follows the SM one
and the compactification scale, at which the quartic coupling becomes zero, is found to be
MKK ≃ 1010 GeV [43–49]. In the presence of the bulk fermions, the compactification scale
6
Conclusions and discussions
In this paper, we have proposed a Majorana fermion DM scenario in the context of a
5dimensional GHU model based on the gauge group SU(3) × U(
1
)′ with a compactification
of the 5th dimension on S1/Z2 orbifold. A pair of bulk SU(3) triplet fermions is introduced
along with a bulk mass term and a periodic boundary condition. The bulk fermions are
decomposed into a pair of the SU(2) doublets and a pair of the electriccharge neutral
singlets under the SM gauge group of SU(2) × U(
1
)Y . With Majorana mass terms for the
singlets, which are introduced on a brane at an orbifold fixed point in general, the lightest
mass eigenstate among the doublet and singlet components serves as a DM candidate.
We have focused on the case that the DM particle is mostly composed of the SM
singlet fermions, and have investigated the DM physics. In this case, the DM particle
communicates with the SM particles through the Higgs boson. We have found that an
allowed parameter region to reproduce the observed DM density is quite limited and the
DM particle mass is to be a vicinity of a half of the Higgs boson mass. The allowed region
has been found to be further constrained when we take into account the upper limit of the
elastic scattering of the DM particle off with the nuclei by the LUX 2016 result. We have
found that the entire allowed region will be covered by the LUXZEPLIN experiment in
the near future.
Note that even if the parameter region shown in the leftpanel of figure 2 is entirely
excluded in the future, our DM scenario can be still viable for the case where the DM
particle is mostly a component in the SM SU(2) doublets. As mentioned in section 2, the
DM particle property in this case is very similar to the Higgsinolike neutralino DM in the
MSSM and the observed DM relic abundance is reproduced with the DM mass of around
1 TeV [29]. Since the reduced mass is μψDMN ≃ mN for mDM ≫ mN , we apply eq. (4.3)
for the spinindependent elastic scattering cross section also for the present case. However,
the limit on yDM is weaker since the experimental upper bound on σSI for mDM ∼ 1 TeV is
about an order of magnitude higher than the one for mDM ≃ 62.5 TeV [33]. Furthermore,
we can estimate yDM = (mW /v)C1 with eqs. (2.15) and (2.16) as
yDM ≃ 2
mW
v
mW
m
(6.1)
for m & M ≃ mDM ∼ 1 TeV. Therefore, in the decoupling limit of the SM singlet
components, namely m ≫ M , the spinindependent elastic scattering cross section is highly
suppressed, and therefore the DM particle escapes detection. This limit is analogous to
the pure Higgsino dark matter in the MSSM.
Employing the effective theoretical approach with the gaugeHiggs condition, we have
also studied the RG evolution of Higgs quartic coupling and shown that the observed Higgs
mass of 125 GeV is achieved with the compactification scale of around 108 GeV. In the
presence of the bulk DM multiplets, the compactification scale to reproduce the 125 GeV
Higgs boson mass is reduced by about two orders of magnitude from MKK ≃ 1010 GeV.
However, in terms of providing a solution to the gauge hierarchy problem by the GHU
scenario, MKK ≃ 108 GeV is too high for the scenario to be natural. In fact, as has been
shown in refs. [15, 19, 20], when we introduce a pair of bulk fermions in higher dimensional
SU(3) representations such as 10plet and 15plet, the compactification scale can be as
low as O(1 TeV), while reproducing the 125 GeV Higgs boson mass. Hence, toward a
natural GHU scenario with a fermion DM, it is worth extending our present model and
introducing the bulk DM multiplets in such a higher dimensional representation. In this
case, we will see that the DM physics investigated in this paper remains almost the same
while the Higgs boson mass of 125 GeV can be reproduced with the compactification scale
of order 1 TeV [53].
Acknowledgments
S.O. would like to thank the Department of Physics and Astronomy at the University of
Alabama for hospitality during her visit. She would also like to thank FUSUMA Alumni
Association at Yamagata University for travel supports for her visit to the University of
Alabama. The work of N.O. is supported in part by the United States Department of
Energy (Award No. DESC0013680).
Open Access.
This article is distributed under the terms of the Creative Commons
Attribution License (CCBY 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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