G2HDM: Gauged Two Higgs Doublet Model
Revised: March
G2HDM: Gauged Two Higgs Doublet Model
WeiChih Huang 0 1 3 5 7
YueLin Sming Tsai 0 1 3 6
TzuChiang Yuan 0 1 2 3 4
London 0 1 3
Hsinchu 0 1 3
Taiwan 0 1 3
0 Nangang , Taipei 11529 , Taiwan
1 Kashiwa , Chiba 2778583 , Japan
2 Institute of Physics , Academia Sinica
3 44221 Dortmund , Germany
4 Physics Division, National Center for Theoretical Sciences
5 Fakultat fur Physik, Technische Universitat Dortmund
6 Kavli IPMU (WPI), University of Tokyo
7 Department of Physics and Astronomy, University College London
A novel model embedding the two Higgs doublets in the popular two Higgs doublet models into a doublet of a nonabelian gauge group SU(2)H is presented. The Standard Model SU(2)L righthanded fermion singlets are paired up with new heavy fermions to form SU(2)H doublets, while SU(2)L lefthanded fermion doublets are singlets under SU(2)H . Distinctive features of this anomalyfree model are: (1) Electroweak symmetry breaking is induced from spontaneous symmetry breaking of SU(2)H via its triplet vacuum expectation value; (2) One of the Higgs doublet can be inert, with its neutral component being a dark matter candidate as protected by the SU(2)H gauge symmetry instead of a discrete Z2 symmetry in the usual case; (3) Unlike LeftRight Symmetric Models, the complex gauge elds (W10 iW20) (along with other complex scalar elds) associated with the SU(2)H do not carry electric charges, while the third component W30 can mix with the hypercharge U(1)Y gauge eld and the third component of SU(2)L; (4) Absence of tree level avour changing neutral current is guaranteed by gauge symmetry; and etc. In this work, we concentrate on the mass spectra of scalar and gauge bosons in the model. Constraints from previous Z0 data at LEP and the Large Hadron Collider measurements of the Standard Model Higgs mass, its partial widths of
Beyond Standard Model; Higgs Physics

and Z
modes are discussed.
2.1
2.2
2.3
3.1
3.2
3.3
4.1
4.2
4.3
4.4
4.5
4.6
3
Spontaneous symmetry breaking and mass spectra
Spontaneous symmetry breaking
Scalar mass spectrum
SU(2)H
U(1)X gauge boson mass spectrum
4
Phenomenology
Numerical solutions for scalar and gauge boson masses
Z0 constraints
Constraints from the 125 GeV SMlike Higgs
Dark matter stability
Dark matter relic density
EWPT  S, T and
U
5
Conclusions and outlook
A Analytical expression for v, v and v B
Decay width of SM Higgs to and Z
1 Introduction 2 G2HDM set up
persymmetry (SUSY) chiral structure. In the inert two Higgs doublet model (IHDM) [6],
since the second Higgs doublet is odd under a Z2 symmetry its neutral component can be
{ 1 {
for in the SM since the CKM phase in the quark sector is too small to generate su cient
baryon asymmetry [11{13], and the Higgs potential cannot achieve the strong
rstorder
electroweak phase transition unless the SM Higgs boson is lighter than 70 GeV [14, 15]. A
general two Higgs doublet model (2HDM) contains additional CP violation source [16{20]
in the scalar sector and hence it may circumvent the above shortcomings of the SM for the
baryon asymmetry.
The complication associated with a general 2HDM [21] stems from the fact that there
exist many terms in the Higgs potential allowed by the SM gauge symmetry, including
various mixing terms between two Higgs doublets H1 and H2. In the case of both
doublets develop vacuum expectation values (vevs), the observed 125 Higgs boson in general
would be a linear combination of three neutral scalars (or of two CP even neutral scalars
if the Higgs potential preserves CP symmetry), resulting in
avourchanging neutral
current (FCNC) at treelevel which is tightly constrained by experiments. This Higgs mixing
e ects lead to changes on the Higgs decay branching ratios into SM fermions which must
be confronted by the Large Hadron Collider (LHC) data.
One can reduce complexity in the 2HDM Higgs potential by imposing certain
symmetry. The popular choice is a discrete symmetry, such as Z2 on the second Higgs doublet in
IHDM or 2HDM typeI [22, 23], typeII [23, 24], typeX and typeY [25{27] where some
of SM fermions are also odd under Z2 unlike IHDM. The other choice is a continuous
symmetry, such as a local U(1) symmetry discussed in [28{31]. The FCNC constraints can
be avoided by satisfying the alignment condition of the Yukawa couplings [32] to
eliminating dangerous treelevel contributions although it is not radiatively stable [33].
Alternatively, the aforementioned Z2 symmetry can be used to evade FCNC at treelevel since SM
fermions of the same quantum number only couple to one of two Higgs doublets [34, 35].
Moreover, the Higgs decay branching ratios remain intact in the IHDM since H1 is the only
doublet which obtains the vacuum expectation value (vev), or one can simply make the
second Higgs doublet H2 much heavier than the SM one such that H2 essentially decouples
from the theory. All in all, the IHDM has many merits, including accommodating DM,
avoiding stringent collider constraints and having a simpler scalar potential. The required
Z2 symmetry, however, is just imposed by hand without justi cation. The lack of
explanation prompts us to come up with a novel 2HDM, which has the same merits of IHDM
with an simpler twodoublet Higgs potential but naturally achieve that H2 does not obtain
a vev, which is reinforced in IHDM by the arti cial Z2 symmetry.
In this work, we propose a 2HDM with additional SU(2)H
U(1)X gauge symmetry,
where H1 (identi ed as the SM Higgs doublet) and H2 form an SU(2)H doublet such that
the twodoublet potential itself is as simple as the SM Higgs potential with just a quadratic
mass term plus a quartic term. The price to pay is to introduce additional scalars: one
SU(2)H triplet and one SU(2)H doublet (which are all singlets under the SM gauge groups)
with their vevs providing masses to the new gauge bosons. At the same time, the vev of
the triplet induces the SM Higgs vev, breaking SU(2)L
U(1)Y down to U(1)Q, while H2
do not develop any vev and the neutral component of H2 could be a DM candidate, whose
stability is protected by the SU(2)H gauge symmetry and Lorentz invariance. In order
{ 2 {
to write down SU(2)H
U(1)X invariant Yukawa couplings, we introduce heavy SU(2)L
singlet Dirac fermions, the righthanded component of which is paired up with the SM
righthanded fermions to comprise SU(2)H doublets. The masses of the heavy fermions
come from the vev of the SU(2)H doublet. In this setup, the model is anomalyfree with
respect to all gauge groups. In what follows, we will abbreviate our model as G2HDM, the
acronym of gauged 2 Higgs doublet model.
We here concentrate on the scalar and additional gauge mass spectra of G2HDM and
various collider constraints. DM phenomenology will be addressed in a separate
publication. As stated before, the neutral component of H2 or any neutral heavy fermion inside
an SU(2)H doublet can potentially play a role of stable DM due to the SU(2)H gauge
symmetry without resorting to an adhoc Z2 symmetry. It is worthwhile to point out
that the way of embedding H1 and H2 into the SU(2)H doublet is similar to the Higgs
bidoublet in the LeftRight Symmetric Model (LRSM) [36{40] based on the gauge group
U(1)B L, where SU(2)L gauge bosons connect elds within H1 or H2 ,
whereas the heavy SU(2)R gauge bosons transform H1 into H2 . The main di erences
between G2HDM and LRSM are:
The charge assignment on the SU(2)H Higgs doublet is charged under U(1)Y
U(1)X
while the bidoublet in LRSM is neutral under U(1)B L. Thus the corresponding
Higgs potential are much simpler in G2HDM;
All SU(2)H gauge bosons are electrically neutral whereas the WR of SU(2)R carry
electric charge of one unit;
The SM righthanded fermions such as uR and dR do not form a doublet under
SU(2)H unlike in LRSM where they form a SU(2)R doublet, leading to very di erent
phenomenology.
On the other hand, this model is also distinctive from the twin Higgs model [41, 42] where
H1 and H2 are charged under two di erent gauge groups SU(2)A (identi ed as the SM gauge
group SU(2)L) and SU(2)B respectively, and the mirror symmetry on SU(2)A and SU(2)B,
i.e., gA = gB, can be used to cancel quadratic divergence of radiative corrections to the SM
Higgs mass. Solving the little hierarchy problem is the main purpose of twin Higgs model
while G2HDM focuses on getting an inert Higgs doublet as the DM candidate without
imposing an adhoc Z2 symmetry. Finally, embedding two Higgs doublets into a doublet
of a nonabelian gauge symmetry with electrically neutral bosons has been proposed in
refs. [43{45], where the nonabelian gauge symmetry is called SU(2)N instead of SU(2)H .
Due to the E6 origin, those models have, nonetheless, quite di erent particle contents
for both fermions and scalars as well as varying embedding of SM fermions into SU(2)N ,
resulting in distinct phenomenology from our model.
This paper is organized as follows. In section 2, we specify the model. We discuss
the complete Higgs potential, Yukawa couplings for SM fermions as well as new heavy
fermions and anomaly cancellation. In section 3, we study spontaneous symmetry breaking
conditions (section 3.1), and analyze the scalar boson mass spectrum (section 3.2) and
the extra gauge boson mass spectrum (section 3.3).
We discuss some phenomenology
{ 3 {
1/6
2/3
1=3
1=2
0
1
1=3
0
1
2/3
1/2
0
0
1
1
0
1
0
1
0
0
0
0
1
0
1
HJEP04(216)9
QL = (uL dL)
UR = uR uH T
R
DR = dRH dR
LL = ( L eL)
T
T
T
NR =
ER = eRH eR
H T
R
T
R
u
d
e
H = (H1 H2)
0
3=2
p
T
p
p= 2
3=2
1
A
which contains just two terms (1 mass term and 1 quartic term) as compared to 8 terms
(3 mass terms and 5 quartic terms) in general 2HDM [21];
V ( H ) =
V ( H ) =
2 y
H H +
and nally the mixed term
Vmix (H; H ; H ) = + MH
Hy H H
2 2) +
before performing the minimization of it to achieve spontaneous symmetry breaking (see
next section).
2
H
2
T
H
U(1)X is introduced to simplify the Higgs potential V (H; H ; H ) in eq. (2.1). For
example, a term
H H obeying the SU(2)H symmetry would be allowed in the
absence of U(1)X . Note that as far as the scalar potential is concerned, treating
U(1)X as a global symmetry is su cient to kill this and other unwanted terms.
In eq. (2.4), if
2 < 0, SU(2)H is spontaneously broken by the vev h 3i =
v 6= 0
with h p;mi = 0 by applying an SU(2)H rotation.
The quadratic terms for H1 and H2 have the following coe cients
1
2
MH
v
+
1
doublet H2 does not obtain a vev, its lowest mass component can be potentially a
DM candidate whose stability is protected by the gauge group SU(2)H .
2 , H1 can still develop a vev (0 v=p2)T
H
Similarly, the quadratic terms for two elds 1 and
2 have the coe cients
1
2
M
v
+
1
2
v
2 +
1
2 may acquire nontrivial vev and h 1i = 0 with the help of
(2.7)
(2.8)
respectively. The eld
a large second term.
2.2
Yukawa couplings
We start from the quark sector. Setting the quark SU(2)L doublet, QL, to be an SU(2)H
singlet and including additional SU(2)L singlets uRH and dRH which together with the SM
righthanded quarks uR and dR, respectively, to form SU(2)H doublets, i.e., URT = (uR uRH )2=3
and DRT = (dRH dR) 1=3, where the subscript represents hypercharge, we have2
LYuk
ydQL (DR
H) + yuQL UR H
+ H:c:;
t
= ydQL dRH H2
dRH1
yuQL uRH~1 + uRH H~2 + H:c:;
(2.9)
where H
t
(H~2
u and d obtain their masses but uRH and dRH remain massless since H2 does not get a vev.
To give a mass to the additional species, we employ the SU(2)H scalar doublet
( 1 2)T , which is singlet under SU(2)L, and lefthanded SU(2)L;H singlets u and d as
H~1)T with H~1;2 = i 2H1;2. After the EW symmetry breaking hH1i 6= 0,
LYuk
=
yd0 d (DR
H
yd0 d dR 2
H ) + yu0 u UR
~
H
+ H:c:;
dR 1
yu0 u uR 1 + uRH 2 + H:c:;
(2.10)
2A B is de ned as ijAiBj where A and B are two 2dimensional spinor representations of SU(2)H .
{ 6 {
where
has Y = 0, Y ( u) = Y (UR) = 2=3 and Y ( d) = Y (DR) =
1
)T . With h 2i = v =p2, uRH ( u) and dRH ( d) obtain masses yu0v =p2 and yd0v =p2,
1=3 with ~
H = ( 2
respectively. Note that both v
and v contribute the SU(2)H gauge boson masses.
The lepton sector is similar to the quark sector as
paired up with L having Dirac mass MD = y v=p2, while
where ERT = (eRH eR) 1, NRT = ( R
RH )0 in which R and RH are the righthanded neutrino
and its SU(2)H partner respectively, while e and
are SU(2)L;H singlets with Y ( e) =
1
and Y (
) = 0 respectively. Notice that neutrinos are purely Dirac in this setup, i.e., R
RH paired up with
having
H
= y0 v =p2. As a result, the lepton number is conserved, implying
vanishing neutrinoless double beta decay. In order to generate the observed neutrino masses
of order subeV, the Yukawa couplings for L and
R are extremely small (
compared to the electron Yukawa coupling. The smallness can arise from, for example, the
10 11) even
small overlap among wavefunctions along a warped extra dimension [47, 48].
Alternatively, it may be desirable for the neutrinos to have a Majorana mass term which
can be easily incorporated by introducing a SU(2)H scalar triplet
N with X( N ) =
Then a renormalizable term gN N Rc N NR with a large h N i 6= 0 will break lepton number
and provide large Majorana masses MN = gN h N i to
Rs (and also
for the Ls can be realized via the typeI seesaw mechanism which allows one large mass
RH s). SubeV masses
of order MN and one small mass of order (MD)2=MN . For MD
subeV neutrino masses can be achieved provided that y
1:28
y v and v
10 7pMN =GeV.
246 GeV,
We note that only one SU(2)L doublet H1 couples to two SM fermion
elds in the
above Yukawa couplings. The other doublet H2 couples to one SM fermion and one
nonSM fermion, while the SU(2)H doublet
H couples to at least one nonSM fermion. As a
consequence, there is no avour changing decays from the SM Higgs in this model. This is
in contrast with the 2HDM where a discrete Z2 symmetry needed to be imposed to forbid
avour changing Higgs decays at tree level. Thus, as long as H2 does not develop a vev in
the parameter space, it is practically an inert Higgs, protected by a local gauge symmetry
instead of a discrete one!
2.3
Anomaly cancellation
We here demonstrate the aforementioned setup is anomalyfree with respect to both the SM
and additional gauge groups. The anomaly cancellation for the SM gauge groups SU(3)C
U(1)Y is guaranteed since addition heavy particles of the same hypercharge form
Dirac pairs. Therefore, contributions of the lefthanded currents from
cancel those of righthanded ones from uH , dRH , RH and eRH respectively.
R
u, d
and e
Regarding the new gauge group SU(2)H , the only nontrivial anomaly needed to be
checked is [SU(2)H ]2U(1)Y from the doublets UR, DR, NR and ER with the following
{ 7 {
result
2Tr[T afT b; Y g] =2 ab
X Yl
l
X Yr
r
!
=
2 ab X Yr
r
=
2 ab (3 2 Y (UR) + 3 2 Y (DR) + 2 Y (NR) + 2 Y (ER)) (2.12)
where 3 comes from the SU(3)C color factor and 2 from 2 components in an SU(2)H doublet.
With the quantum number assignment for the various elds listed in table 1, one can check
that this anomaly coe cient vanishes for each generation.
In terms of U(1)X , one has to check [SU(3)C ]2U(1)X , [SU(2)H ]2U(1)X , [U(1)X ]3,
rst three terms are zero due to
cancellation between UR and DR and between ER and NR with opposite U(1)X charges. For
[U(1)Y ]2U(1)X and [U(1)X ]2U(1)Y , one has respectively
2
2
3
3
Y (UR)2X(UR) + Y (DR)2X(DR) + Y (ER)2X(ER) ;
X(UR)2Y (UR) + X(DR)2Y (DR) + X(ER)2Y (ER) ;
both of which vanish.
One can also check the perturbative gravitational anomaly [49] associated with the
hypercharge and U(1)X charge current couples to two gravitons is proportional to the
following sum of the hypercharge
3 (2 Y (QL) + Y ( u) + Y ( d)
2 Y (UR)
2 Y (DR))
+ 2 Y (LL) + Y ( ) + Y ( e)
2 Y (NR)
2 Y (ER);
and U(1)X charge
X(UR) + X(DR) + X(ER) + X(NR);
which also vanish for each generation.
Since there are 4 chiral doublets for SU(2)L and also 8 chiral doublets for SU(2)H for
each generation, the model is also free of the global SU(2) anomaly [50] which requires the
total number of chiral doublets for any local SU(2) must be even.
We end this section by pointing out that one can also introduce QLH = (uLH dLH )T to
pair up with QL and LLH = ( LH eLH )T to pair up with LL to form SU(2)H doublets. Such
possibility is also interesting and will be discussed elsewhere.
3
Spontaneous symmetry breaking and mass spectra
After specifying the model content and fermion mass generation, we now switch to the
scalar and gauge boson sector.
We begin by studying the minimization conditions for
spontaneous symmetry breaking, followed by investigating scalar and gauge boson mass
spectra. Special attention is paid to mixing e ects on both the scalars and gauge bosons.
3[SU(2)L]2U(1)X anomaly does not exist since fermions charged under U(1)X are singlets under SU(2)L.
(2.13)
(2.14)
(2.15)
{ 8 {
To facilitate spontaneous symmetry breaking, let us shift the elds as follows
G+
v+2h + iG0
p
!
H1 =
;
H =
p
GH
are the physical elds.
and H2 = (H2+ H20)T . Here v, v
of the potential;
G
and v
are vevs to be determined by minimization
fG+; G3; GpH ; G0H g are Goldstone bosons, to be absorbed by the
fh; H2; 1; 2; 3; pg
Minimization of the potential in eq. (3.2) leads to the following three equations for the vevs
4
v
3
Note that one can solve for the nontrivial solutions for v2 and v2 in terms of v and other
parameters using eqs. (3.3) and (3.4). Substitute these solutions of v2 and v2 into eq. (3.5)
leads to a cubic equation for v
which can be solved analytically (See appendix A).
3.2
Scalar mass spectrum
The scalar boson mass spectrum can be obtained from taking the second derivatives of the
potential with respect to the various elds and evaluate it at the minimum of the potential.
The mass matrix thus obtained contains three diagonal blocks. The rst block is 3
the basis of S = fh; 3; 2g it is given by
0
H vv
v2 (MH
v2 (M
2 H v )
2
v )
M20 = BB v2 (MH
1
2 H v ) 4v
8
v
3 + MH v2 + M
v
2
v2 (M
This matrix can be diagonalized by a similar transformation with orthogonal matrix O,
Oij jmij with i and j referring to the avour and mass
eigenwhich de ned as jf ii
states respectively,
OT
M20 O = Diag(m2h1 ; m2h2 ; m2h3 ) ;
where the three eigenvalues are in ascending order. The lightest eigenvalue mh1 will be
identi ed as the 125 GeV Higgs h1 observed at the LHC and the other two mh2 and mh3
{ 9 {
H vv
2
v
2
2
v )CC :
1
A
(3.6)
(3.7)
0
M
0
v
p
M002 = BB 12 M
v
1
4v
are for the heavier Higgses h2 and h3. The physical Higgs hi is a linear combination of the
three components of S: hi = OjiSj . Thus the 125 GeV scalar boson could be a mixture of
the neutral components of H1 and the SU(2)H doublet
H , as well as the real component
3 of the SU(2)H triplet
H .
The SM Higgs h1 treelevel couplings to f f , W +W , ZZ and H2+H2 pairs, each will
be modi ed by an overall factor of O11, resulting a reduction by jO11j2 on the h1 decay
branching ratios into these channels. On the other hand, as we shall see later, h1 !
and Z
involve extra contributions from the 3 and 2 components, which could lead to
either enhancement or suppression with respect to the SM prediction.
The second block is also 3
p
3. In the basis of G = fGH ; p; H20 g it is given by
D can be a DM candidate in G2HDM. Note that in the parameter space where the quantity
inside the square root of eq. (3.9) is very small, e would be degenerate with D. In this
case, we need to include coannihilation processes for relic density calculation. Moreover,
it is possible in our model to have RH or
decays to SM lepton and Higgs) to be DM candidate as well.
( R either is too light or is not stable since it
The nal block is 4
4 diagonal, giving
It is easy to show that eq. (3.8) has a zero eigenvalue, associated with the physical Goldstone
boson, which is a mixture of GH ,
of two physical elds e and D. They are given by
p and H20 . The other two eigenvalues are the masses
M e ;D =
1 n
8v
h MH
MH v2 + 4 (MH
+ M
) v2 + M
v
2
v2 +4v2 +M
v2 +4v2
2
16MH
M
v
2 v2 + 4v2 + v2 i 2
for the physical charged Higgs H2 , and
m2
H2
= MH v ;
m2G
= m2G0 = m2G0H = 0 ;
for the three Goldstone boson
elds G , G0 and G0H . Note that we have used the
minimization conditions eqs. (3.3), (3.4) and (3.5) to simplify various matrix elements of the
above mass matrices.
Altogether we have 6 Goldstone particles in the scalar mass spectrum, we thus expect
to have two massless gauge particles left over after spontaneous symmetry breaking. One
is naturally identi ed as the photon while the other one could be interpreted as dark
photon D.
(3.8)
(3.9)
1
:
(3.10)
(3.11)
After investigating the spontaneous symmetry breaking conditions, we now study the mass
spectrum of additional gauge bosons. The gauge kinetic terms for the
H ,
and H are
L
Tr h D0
H y D0
H
i
+ D0
y D0
+ D0 H y D0 H ;
with
and
D0
H
igH W 0 ; H
;
D0
=
W 0pT p + W 0mT m
igH W 03T 3
igX X
;
HJEP04(216)9
D0 H =
D
1
W 0pT p + W 0mT m
igH W 03T 3
igX X
H ;
1
2
i pgH
2
i pgH
2
3
a=1
1
2
where D is the SU(2)L covariant derivative, acting individually on H1 and H2, gH (gX )
is the SU(2)H (U(1)X ) gauge coupling constant, and
W 0 =
X W 0aT a = p
W 0pT p + W 0mT m
+ W 03T 3;
in which T a =
(W 01
iW 0 2)=p2, and
a=2 ( a are the Pauli matrices acting on the SU(2)H space), W 0 (p;m) =
T p =
1 + i 2 =
; T m =
1
i 2 =
1
2
obtained its mass entirely from v, so it is given by
same as the SM.
The SU(2)H gauge bosons W 0a and the U(1)X gauge boson X receive masses from
h 3i, hH1i and h 2i. The terms contributed from the doublets are similar with that from
the standard model. Since
H transforms as a triplet under SU(2)H , i.e., in the adjoint
representation, the contribution to the W 0a masses arise from the term
L
gH2 Tr
W 0 ; H y W 0 ; H
:
All in all, the W 0(p;m) receives a mass from h 3i, h 2i and hH1i
m2W 0(p;m) =
14 gH2 v2 + v2 + 4v2 ;
while gauge bosons X and W 03, together with the SM W 3 and U(1)Y gauge boson B,
acquire their masses from h 2i and hH1i only but not from h H i:
1
8
v
2 2gX X
+ gH W 03
gW 3 + g0B
2
+ v2
2gX X
+ gH W 03 2 ;
(3.21)
where g0 is the SM U(1)Y gauge coupling.
Note that the gauge boson W 0(p;m) corresponding to the SU(2)H generators T
do not
carry the SM electric charge and therefore will not mix with the SM W
B and X will become massive, by absorbing the imaginary part of H10 and
and X do mix with the SM W 3 and B bosons via hH1i. In fact, only two of W 3, W 03,
2. To avoid
undesired additional massless gauge bosons, one can introduce extra scalar elds charged
under only SU(2)H
U(1)X but not under the SM gauge group to give a mass to W 03
and X, without perturbing the SM gauge boson mass spectrum. Another possibility is to
involve the Stueckelberg mechanism to give a mass to the U(1)X gauge boson as done in
refs. [51{54]. Alternatively, one can set gX = 0 to decouple X from the theory or simply
treat U(1)X as a global symmetry, after all as mentioned before U(1)X is introduced to
bosons while W 03
simplify the Higgs potential by forbidding terms like
SU(2)H but not U(1)X .
in the basis V 0 =
B; W 3; W 03; X :
T
H
H H , which is allowed under
From eq. (3.21), one can obtain the following mass matrix for the neutral gauge bosons
g0g v2
4
g2v2
4
As anticipated, this mass matrix has two zero eigenvalues corresponding to m
= 0 and
m D = 0 for the photon and dark photon respectively. The other two nonvanishing
eigenvalues are
where
1
8
M 2 =
v2 + v
2
q
v2 + v
A strictly massless dark photon might not be phenomenologically desirable. One could
have a Stueckelberg extension of the above model by including the Stueckelberg mass
term [51, 52]
= g2 + g02 + gH2 + 4gX2 ;
= gH2 + 4gX2 ;
2
= gH
4gX2 :
1
2
+ MY B )2 ;
(3.22)
(3.24)
(3.25)
where MX and MY are the Stueckelberg masses for the gauge elds X and B of U(1)X
and U(1)Y respectively, and a is the axion eld. Thus the neutral gauge boson mass matrix
is modi ed as
M21 = BBB
0
B
B
g02v2 + MY2
4
It is easy to show that this mass matrix has only one zero mode corresponding to the
photon, and three massive modes Z; Z0; Z00. This mass matrix can be diagonalized by
an orthogonal matrix. The cubic equation for the three eigenvalues can be written down
analytically similar to solving the cubic equation for the vev v
given in the appendix A.
However their expressions are not illuminating and will not presented here.
As shown in ref. [52], MY will induce the mixing between U(1)Y and U(1)X and the
resulting massless eigenstate, the photon, will contain a U(1)X component, rendering the
neutron charge, Qn = Qu + 2Qd, nonzero unless u's and d's U(1)X charges are zero or
proportional to their electric charges. In this model, however, none of the two solutions
can be satis ed. Besides, lefthanded SM elds are singlets under U(1)X while righthanded
ones are charged. It implies the lefthanded and righthanded species may have di erent
electric charges if the U(1)X charge plays a role on the electric charge de nition. Here
we will set MY to be zero to maintain the relations Q = I3 + Y and 1=e2 = 1=g02 + 1=g2
same as the SM in order to avoid undesired features. As a result, after making a rotation
in the 1
2 plane by the Weinberg angle w, the mass matrix M12 can transform into a
block diagonal matrix with the vanishing rst column and rst row. The nonzero 3by3
block matrix can be further diagonalized by an orthogonal matrix O, characterized by
three rotation angles ( 12; 23; 13),
0 Z 1
Z00
0 Z 1
Z00
0ZSM
1
X
(3.27)
0ZSM
1
X
where ZSM is the SM Z boson without the presence of the W30 and X bosons. In this model,
the charged current mediated by the W boson and the electric current by the photon
are
exactly the same as in the SM:
f
g
2
L( ) =
X Qf ef
f A ;
L(W ) = p ( L
eL + uL
dL) W + + H:c: ;
where Qf is the corresponding fermion electric charge in units of e. On the other hand,
neutral current interactions, including ones induced by W 0, take the following form (for
illustration, only the lepton sector is shown but it is straightforward to include the quark sector)
LNC = L(Z) + L(Z0) + L(Z00) + L(W 0) ;
(3.28)
(3.29)
and
gH
2
X
f=e;
JZSM =
1
cos w
JW 03 =
J
X =
X
X
f=NR;ER
f=NR;ER
fR
(I3H )fR ;
QfX fR
fR ;
with I3 (I3H ) being the SU(2)L (SU(2)H ) isospin and QfX the U(1)X charge. Detailed
analysis of the implications of these extra gauge bosons is important and will be presented
elsewhere.
4
Phenomenology
In this section, we discuss some phenomenology implications of the model by examining
the mass spectra of scalars and gauge bosons, Z0 constraints from various experiments, and
Higgs properties of this model against the LHC measurements on the partial decay widths
of the SM Higgs boson, which is h1 in the model.
4.1
Numerical solutions for scalar and gauge boson masses
We rst study the SM Higgs mass (mh1 ) dependence on the parameters in the mass matrix
in eq. (3.6). As we shall see later the vev v has to be bigger than 10 TeV (
v = 246 GeV).
In light of LEP measurements on the e+e
! e+e
crosssection [55], the mass matrix
will exhibit blockdiagonal structure with the bottomright 2by2 block much bigger than
the rest and h basically decouple from
2 and 3
.
To demonstrate this behavior, we simplify the model by setting
to zero and then choose
that one can investigate how the Higgs mass varies as a function of
H , H ,
equal
is very small compared to v , the Higgs mass is simply the
(1,1) element of the mass matrix, 2 H v2, and h1 is just h, i.e., O121 ' 1. Nonetheless, when
becomes comparable to v , and the (1,2) element of the mass matrix gives rise to a
sizable but negative contribution to the Higgs mass, requiring a larger value of
H than
(3.30)
(3.31)
1 0 3
e
M
)
V
mass is basically determined by two parameters MH
and
H only. Other parameters are set as
follows:
SMlike: O121 ' 1. Therefore, one has to measure the quartic coupling
double Higgs production to be able to di erentiate this model from the SM.
the SM one so as to have a correct Higgs mass. In this regime, jO11j is, however, still very
H through the
For the analysis above, we neglect the fact all vevs, v , v
and v are actually functions
of the parameters s, M s and
s in eq. (2.1), the total scalar potential. The analytical
solutions of the vevs are collected in appendix A. As a consequence, we now numerically
diagonalized the matrices (3.6) and (3.8) as functions of , M and
, i.e., replacing all
vevs by the input parameters. It is worthwhile to mention that v
has three solutions as
eq. (3.5) is a cubic equation for v . Only one of the solutions corresponds to the global
minimum of the full scalar potential.
We, however, include both the global and local
minimum neglecting the stability issue for the latter since it will demand detailed study
on the nontrivial potential shape which is beyond the scope of this work.
In order to explore the possibility of a nonSM like Higgs having a 125 GeV mass, we
further allow for nonzero mixing couplings. We perform a grid scan with 35 steps of each
dimension in the range
10 2
10 2
10 2
10 1
1:0
1:05
H
H
H
MH = GeV
v =v
5 ;
5 ;
5 ;
5 ;
2
5:0 :
104 ;
In order not to overcomplicate the analysis, from now on we make
= 0:8MH , unless otherwise stated. In table 2, we show 6
representative benchmark points (3 global and 3 local minima) from our grid scan with the
dark matter mass mD and the charged Higgs mH
of order few hundred GeVs, testable
m Z ′
θ Z ′ −
g H × ′
.
Z0 mixing (blue dotted line) and the LEP constraints on the electronpositron
scattering crosssection (black dashed line) in the mZ0
gH plane.
v for M
understood as
in the near future. It is clear that Global scenario can have the SM Higgs composition
2
signi cantly di erent from 1, as O11
0:8 in benchmark point A, but h1 is often just h in
Local case. On the other hand, the other two heavy Higgses are as heavy as TeV because
their mass are basically determined by v
and v .
For the other mass matrix eq. (3.8), we focus on the mass splitting between the dark
matter and the charged Higgs and it turns out the mass splitting mostly depends on v
and v . In gure 2, we present the mass di erence normalized to mD as a function of
and v
= 10 TeV used in table 2. The behavior can be easily
mH2
mD
mD
'
=
v
2
1
v2 + jv
2
8 v2 + 2 v2
v2 =v2
v2 j
4 + v2 =v2
(4.2)
where we have neglected terms involving v since we are interested in the limit of v , v
Note that this result is true as long as M
= 0:8MH
and v
> v
v regardless of the
other parameters.
4.2
Z0 constraints
Since performing a full analysis of the constraints on extra neutral gauge bosons including
all possible mixing e ects from
; Z; Z0; Z00 is necessarily complicated, we will be contented
in this work by a simple scenario discussed below.
The neutral gauge boson mass matrix in eq. (3.22) can be simpli ed a lot by making
U(1)X a global symmetry. In the limit of gX = MX = 0, the U(1)X gauge boson X
decouples from the theory, and the 3by3 mass matrix in the basis of B, W 3 and W 03 can
be rotated into the mass basis of the SM , Z and the SU(2)H Z0 by:
0
0
4
4
where Rij refers to a rotation matrix in the i
j block; e.g., R12 ( w) is a rotation along
the zaxis (W 03) direction with cos w as the (1; 1) and (2; 2) element and
sin w as the
(1; 2) element (sin w for (2; 1)). The mixing angles can be easily obtained as
sin w =
sin ZZ0 =
g0
pg2 + g02
;
p2pg2 + g02 gH v
2
1=4
(g2 + g02)v2 + gH2 v2 + v2 + 1=2 1=2 ;
where
=
have the approximate result
g2 + g02 + gH2 v2 + gH2 v2 2
4gH2 g2 + g02 v2v2 . In the limit of v
sin ZZ0
pg2 + g02 v2
gH v
2
;
mZ
pg2 + g02
; mZ0
v
2
v
gH 2
:
and
A couple of comments are in order here. First, the SM Weinberg angle characterized by w
is unchanged in the presence of SU(2)H . Second, the vev ratio v2=v2 controls the mixing
between the SM Z and SU(2)H Z0. However, the Z
Z0 mixing for TeV Z0 is constrained
to be roughly less than 0:1%, results from Z resonance line shape measurements [56],
electroweak precision test (EWPT) data [57] and collider searches via the W +W
nal
states [58, 59], depending on underlying models.
Direct Z0 searches based on dilepton channels at the LHC [60{62] yield stringent
constraints on the mass of Z0 of this model since righthanded SM fermions which are part of
SU(2)H doublets couple to the Z0 boson and thus Z0 can be produced and decayed into
dilepton at the LHC. To implement LHC Z0 bounds, we take the Z0 constraint [62] on
the Sequential Standard Model (SSM) with SMlike couplings [63], rescaling by a factor of
gH2 (cos2 w=g2): It is because rst SU(2)H does not have the Weinberg angle in the limit
of gX = 0 and second we assume Z0 decays into the SM fermions only and branching
ratios into the heavy fermions are kinematically suppressed, i.e., Z0 decay branching ratios
are similar to those of the SM Z boson. The direct search bound becomes weaker once
(4.3)
(4.4)
(4.5)
v, we
(4.6)
(4.7)
gH2 (cos2
the couplings.
e+e
! `+`
heavy fermion nal states are open. Note also that Z0 couples only to the righthanded
SM
elds unlike Z0 in the SSM couples to both lefthanded and righthanded
elds as in
the SM. The SSM Z0 lefthanded couplings are, however, dominant since the righthanded
ones are suppressed by the Weinberg angle. Hence we simply rescale the SSM result by
w=g2) without taking into account the minor di erence on the chiral structure of
In addition, Z0 also interacts with the righthanded electron and will contribute to
processes. LEP measurements on the crosssection of e+e
! `+`
can
be translated into the constraints on the new physics scale in the context of the e ective
fourfermion interactions [55]
= 0 (1) for f 6= e (f = e) and ij = 1 ( 1) corresponds to constructive (destructive)
interference between the SM and the new physics processes. On the other hand, in our
model for mZ0
TeV the contact interactions read
Le =
(1 + )
g
2
mH2Z0 eR
eRfR
fR :
+
+
L R ! e e
L R with
(4.8)
(4.9)
(4.10)
It turns out the strongest constraint arises from e e
= 8:9 TeV and
=
1 [55], which implies
gH
mZ0
.
0:2
TeV
and v & 10 TeV :
v
4.3
In gure 3, in the plane of gH and mZ0 , we show the three constraints: the Z
Z0 mixing
by the blue dotted line, the heavy narrow dilepton resonance searches from CMS [62] in red
and the LEP bounds on the electronpositron scattering crosssection of e+e
in black, where the region above each of the lines is excluded. The direct searches are
dominant for most of the parameter space of interest, while the LEP constraint becomes
most stringent toward the highmass region. From this
gure, we infer that for 0:1 .
gH . 1, the Z0 mass has to be of order O(TeV). Note that mZ0
gH v =2 and it implies
! e+e [55]
30 TeV for mZ0
1:5 TeV but v can be 10 TeV for mZ0 & 2:75 TeV.
Constraints from the 125 GeV SMlike Higgs
We begin with the SM Higgs boson h1 partial decay widths into SM fermions. Based on
the Yukawa couplings in eqs. (2.9), (2.10) and (2.11), only the avor eigenstate H1 couples
to two SM fermions. Thus, the coupling of the mass eigenstate h1 to the SM fermions is
rescaled by O11, making the decay widths universally reduced by O121 compared to the SM,
which is di erent from generic 2HDMs.
For the treelevel (o shell) h1 ! W +W , since the SM W bosons do not mix with
additional SU(2)H (and U(1)X ) gauge bosons and
2 and 3 are singlets under the SU(2)L
gauge group, the partial decay width is also suppressed by O121.
h1 ! ZZ receives additional contributions from the Z
On the other hand,
Z0 mixing and thus the 3 and
and h ! Z in this model. Due to the fact the jO11j
1 and
H is always positive, R
is often less than the SM prediction while RZ ranges from 0:9 to 1,
given the ATLAS and CMS measurements on R
: : 1:17
0:27 (ATLAS [64]) and 1:13
0:24
(CMS [65]). Only ATLAS result is shown which completely covers the CMS 1 con dence region.
The left (right) panel is for Global (Local) scenario.
2 components charged under SU(2)H will also contribute. The mixing, however, are
constrained to be small (. 10 3) by the EWPT data and LEP measurement on the
electronpositron scattering crosssection as discussed in previous subsection. We will not consider
the mixing e ect here and the decay width is simply reduced by O121 identical to other
treelevel decay channels.
Now, we are in a position to explore the Higgs radiative decay rates into two photons
and one Z boson and one photon, normalized to the SM predictions. For convenience,
we de ne RXX to be the production cross section of an SM Higgs boson decaying to XX
divided by the SM expectation as follows:
(4.11)
(4.12)
Note that rst additional heavy colored fermions do not modify the Higgs boson production
cross section, especially via gg ! h1, because the SU(2)H symmetry forbids the coupling of
h1f f , where f is the heavy fermion. Second, in order to avoid the observed Higgs invisible
decay constraints in the vector boson fusion production mode: Br(h1 !
invisible) <
0:29 [66, 67], we constrain ourself to the region of 2mD > mh so that the Higgs decay
channels are the same as in the SM. As a consequence, eq. (4.11) becomes,
R
=
(h1 !
SM(h1 !
)
)
; RZ =
(h1 !
SM(h1 !
Z)
)
;
similar to the situation of IHDM.
As mentioned before due to the existence of SU(2)H , there are no terms involving one
SM Higgs boson and two heavy fermions, which are not SU(2)H invariant. Therefore, the
RXX
(pp ! h1)
Br(h1 ! XX)
SM(pp ! h1) BrSM(h1 ! XX)
:
only new contributions to h1 !
and h1 ! Z arise from the heavy charged Higgs boson,
H2 . In addition, h1 consists of 3 and 2 apart from H10. With the quartic interactions
2 H H2yH2H1yH1 +
H
H2yH2 32=2 + H H2yH2 2 2, there are in total three contributions
to the H2 loop diagram. The Higgs decay widths into
and Z including new scalars
can be found in refs. [68{71] and they are collected in appendix B for convenience.
The results of h !
(red circle) and h ! Z
(blue square) are presented in gure 4
for both the Global minimum case (left) and additional points which having the correct
Higgs mass only at Local minimum (right). All the scatter points were selected from
our grid scan described in subsection 4.1. It clearly shows that the mass of the heavy
charged Higgs has to be larger than 100 GeV in order to satisfy the LHC measurements
3jH1j2 jH2j2 can be either positive or negative, in this model we
H as a quartic coupling has to be positive to ensure the
potential is bounded from below. It implies that for jO11j2 being very close to 1 like the
benchmark point C in table 2, h1 is mostly h, the neutral component in H1, and R
is
determined by
H . In this case, R , corresponding to the red curve, is always smaller
than 1 in the region of interest where mH2
values below the SM prediction as well.
charged Higgs in the context of IHDM [72]), while RZ denoted by the blue curve also has
> 80 GeV (due to the LEP constraints on the
In contrast, there are some points such as the benchmark point A where the
contributions from
2 and 3 are signi cant, they manifest in gure 4 as scatter points away from
the lines but also feature R , RZ < 1.
4.4
Dark matter stability
In G2HDM, although there is no residual (discrete) symmetry left from SU(2)H symmetry
breaking, the lightest particle among the heavy SU(2)H fermions (uH , dH , eH ,
H ), the
SU(2)H gauge boson W 0 (but not Z0 because of Z
Z0 mixing) and the second heaviest
eigenstate in the mass matrix of eq. (3.8)4 is stable due to the gauge symmetry and the
p
Lorentz invariance for the given particle content. In this work, we focus on the case of the
second heaviest eigenstate in the mass matrix of eq. (3.8), a linear combination of H20 , p
and GH , being DM by assuming the SU(2)H fermions and W 0 are heavier than it.
At tree level, it is clear that all renormalizable interactions always have even powers of
the potential DM candidates in the set D
fuH ; dH ; eH ; H ; W 0; H20 ; p; GpH g. It implies
they always appear in pairs and will not decay solely into SM particles. In other words,
the decay of a particle in D must be accompanied by the production of another particle in
D, rendering the lightest one among D stable.
Beyond the renormalizable level, one may worry that radiative corrections involving
DM will create nonrenormalizable interactions, which can be portrayed by high order
e ective operators, and lead to the DM decay. Assume the DM particle can decay into
solely SM particles via certain higher dimensional operators, which conserve the gauge
symmetry but spontaneously broken by the vevs of H1,
H ,
H . The SM particles refer
4The lightest one being the Goldstone boson absorbed by W 0.
to SM fermions, Higgs boson h and
2 (which mixes with h).5 The operators, involving
one DM particle and the decay products, are required to conserve the gauge symmetry.
We here focus on 4 quantum numbers: SU(2)L and SU(2)H isospin, together with U(1)Y
and U(1)X charge. DM is a linear combination of three avour states: H20 ,
First, we study operators with H20 decaying leptonically into n1 Ls, n2 eLs, n3 Rs, n4
eRs, n5 hs and n6 2s.6 One has, in terms of the quantum numbers (I3H ; I3; Y; X),
p and GH .
p
This implies the number of net fermions (fermions minus antifermions) is n1+n2+n3+n4 =
1, where positive and negative ni correspond to fermion and antifermion respectively. In
other words, if the number of fermions is odd, the number of antifermions must be even;
and vice versa. Clearly, this implies Lorentz violation since the total fermion number
(fermions plus antifermions) is also odd. Therefore, the
avor state H20 can not decay
solely into SM particles. It is straightforward to show the conclusion applies to
GpH as well. As a result, the DM candidate, a superposition of H20 ,
p and GpH , is stable
p and
as long as it is the lightest one among the potential dark matter candidates D.
Before moving to the DM relic density computation, we would like to comment on
e ective operators with an odd number of D, like three Ds or more. It is not possible that
this type of operators, invariant under the gauge symmetry, will induce operators involving
only one DM particle, by connecting an even number of Ds into loop. It is because
those operators linear in D violate either the gauge symmetry or Lorentz invariance and
hence can never be (radiatively) generated from operators obeying the symmetries. After
all, the procedure of reduction on the power of D, i.e., closing loops with proper vev
insertions (which amounts to adding Yukawa terms), also complies with the gauge and
Lorentz symmetry.
4.5
Dark matter relic density
We now show this model can reproduce the correct DM relic density. As mentioned above,
the DM particle will be a linear combination of GH ,
p
p and H20 . Thus (co)annihilation
channels are relevant if their masses are nearly degenerated and the analysis can be quite
involved. In this work, we constrain ourselves in the simple limit where the G2HDM
becomes IHDM, in which case the computation of DM relic density has been implemented
in the software package micrOMEGAs [73, 74]. For a recent detailed analysis of IHDM, see
ref. [10] and references therein. In other words, we are working in the scenario that the
5The SM gauge bosons are not included since they decay into SM fermions, while inclusion of 3 will
not change the conclusion because it carries zero charge in term of the quantum numbers in consideration.
6The conclusion remains the same if quarks are also included.
DM being mostly H20 and the SM Higgs boson h has a minute mixing with 3 and 2. The
IHDM Higgs potential reads,
VIHDM =
21jH1j2 + 22jH2j2 + 1jH1j4 + 2jH2j4 + 3jH1j2jH2j2 + 4jH1yH2j2
+
2
5 n(H1yH2)2 + h:c:o ;
(4.15)
H2 H20 ! W
the fact H20 (DM particle) also receives a tiny contribution from
where 1 = 2 = 2 3 =
H with 4 = 5 = 0 when compared to our model. It implies the
degenerate mass spectrum for H2 and H20. The mass splitting, nonetheless, arises from
p and GpH as well as loop
contributions. In the limit of IHDM the mass splitting is very small, making paramount
(co)annihilations, such as H2+H2 ! W +W , H20H20 ! W +W , (H2+H2 ; H20H20) ! ZZ,
, and thus the DM relic density is mostly determined by these channels.
Besides, from eq. (4.15), H is seemingly xed by the Higgs mass, 1 = m2h=2v2
0:13,
a distinctive feature of IHDM. In G2HDM, however, the value of H can deviate from 0.13
due to the mixing among h, 3 and 2 as shown in gure 1, where the red curve corresponds
to the Higgs boson mass of 125 GeV with
H varying from 1.2 to 1.9 as a function of MH ,
which controls the scalar mixing. To simulate G2HDM but still stay close to the IHDM
limit, we will treat
H as a free parameter in the analysis and independent of the SM
Higgs mass.
We note that micrOMEGAs requires
ve input parameters for IHDM: SM Higgs
mass ( xed to be 125 GeV), H2 mass, H20 mass (including both CPeven and odd
components which are degenerate in our model), 2 (=
H ) and ( 3 + 4
5)=2 (
H ).
It implies that only two of them are independent in G2HDM, which can be chosen to be
mDM and
H . Strictly speaking, mDM is a function of parameters such as M
, MH , v ,
v , etc., since it is one of the eigenvalues in the mass matrix of eq. (3.8). In this analysis,
we stick to the scan range of the parameters displayed in eq. (4.1) with slight modi cation
as follows
0:12
0:8
M
H
=MH
0:2 ;
1:0 ;
(4.16)
see the IHDM limit (OD2
and also demand the mixing (denoted by OD) between H20 and the other scalars to be
less than 1%. In the exact IHDM limit, OD should be 1. Besides, the decay time of H2
into H20 plus an electron and an electron neutrino is required to be much shorter than one
second in order not to spoil the Big Bang nucleosynthesis.
Our result is presented in
gure 5. For a given
H , there exists an upper bound on
the DM mass, above which the DM density surpasses the observed one, as shown in the
left panel of gure 5. The brown band in the plot corresponds to the DM relic density of
0:1 <
h2 < 0:12 while the yellow band refers to
h2 < 0:1. In the right panel of gure 5,
we show how well the IHDM limit can be reached in the parameter space of M
versus v =v , where the red band corresponds to 0:8 < OD2 < 0:9, the green band refers to
0:9 < OD2 < 0:99 (green) and while the light blue area represents 0:99 < OD2. One can easily
=MH
1) can be attained with M
. MH . For M
> MH , we
.08
.072
Te
)
V
(
M
D
.012
.014
.018
.02
0:99 < OD2, respectively. See the text for detail of the analysis.
with the relic density between 0.1 and 0.12 but the lower yellow region is the relic density less than
0.1. Right: accepted DM mass region projected on (v =v , M
=MH ) plane. The red, green
and light blue regions present the DM inert Higgs fraction 0:8 < OD2 < 0:9, 0:9 < OD2 < 0:99 and
have mH2
< mH20 , implying H20 can not be DM anymore. Finally, we would like to point
out that the allowed parameter space may increase signi cantly once the small H20 mixing
constraint is removed and other channels comprising GpH and
It is worthwhile to mention that not only H20 but also
p are considered.
H (righthanded neutrino's
SU(2)H partner) can be the DM candidate in G2HDM, whose stability can be proved
similarly based on the gauge and Lorentz invariance. If this is the case, then there exists
an intriguing connection between DM phenomenology and neutrino physics. We, however,
will leave this possibility for future work.
4.6
In light of the existence of new scalars and fermions,7 one should be concerned about the
EWPT, characterized by the oblique parameters
S,
T and
U [75]. For scalar contri
butions, similar to the situation of DM relic density computation, a complete calculation
in G2HDM can be quite convoluted. The situation, however, becomes much simpler when
one goes to the IHDM limit as before, i.e., H20 is virtually the mass eigenstate and the inert
doublet H2 is the only contribution to
S,
T and
U , since
H and
H are singlets
under the SM gauge group. Analytic formulas for the oblique parameters can be found in
ref. [8]. All of them will vanish if the mass of H2 is equal to that of H20 as guaranteed by
the SU(2)L invariance.
7For additional gauge bosons in the limit of U(1)X being a global symmetry, the constrain considered
above, sin ZZ0 < 10 3, are actually derived from the electroweak precision data combined with collider
bounds. As a consequence, we will not discuss their contributions again.
On the other hand, the mass splitting stems from the H20 mixing with GpH and
p
therefore, in the limit of IHDM, the mass splitting is very small, implying a very small
deviation from the SM predictions of the oblique parameters. We have numerically checked
that all points with the correct DM relic abundance studied in the previous section have
negligible H2 contributions, of order less than 10 3, to
T and
Finally for the heavy fermions, due to the fact they are singlets under SU(2)L, the
corresponding contributions will vanish according to the de nition of oblique parameters [75].
Our model survives the challenge from the EWPT as long as one is able to suppress the
extra scalar contributions, for instance, resorting to the IHDM limit or having cancellation
among di erent contributions.
5
Conclusions and outlook
In this work, we propose a novel framework to embed two Higgs doublets, H1 and H2 into
a doublet under a nonabelian gauge symmetry SU(2)H and the SU(2)H doublet is charged
under an additional abelian group U(1)X . To give masses to additional gauge bosons, we
introduce an SU(2)H scalar triplet and doublet (singlets under the SM gauge group). The
potential of the two Higgs doublets is as simple as the SM Higgs potential at the cost of
additional terms involving the SU(2)H triplet and doublet. The vev of the triplet triggers
the spontaneous symmetry breaking of SU(2)L by generating the vev for the rst SU(2)L
Higgs doublet, identi ed as the SM Higgs doublet, while the second Higgs doublet does
not obtain a vev and the neutral component could be the DM candidate, whose stability is
guaranteed by the SU(2)H symmetry and Lorentz invariance. Instead of investigating DM
phenomenology, we have focused here on Higgs physics and mass spectra of new particles.
To ensure the model is anomalyfree and SM Yukawa couplings preserve the additional
U(1)X symmetry, we choose to place the SM righthanded fermions and new
heavy righthanded fermions into SU(2)H doublets while SM SU(2)L fermion doublets are
singlets under SU(2)H . Moreover, the vev of the SU(2)H scalar doublet can provide a mass
to the new heavy fermions via new Yukawa couplings.
Di erent from the leftright symmetric model with a bidoublet Higgs bosons, in which
WR carries the electric charge, the corresponding W 0 bosons in this model that connects H1
and H2 are electrically neutral since H1 and H2 have the same SM gauge group charges.
On the other hand, the corresponding Z0 actually mixes with the SM Z boson and the
mixing are constrained by the EWPT data as well as collider searches on the W
nal
state. Z0 itself is also confronted by the direct resonance searches based on dilepton or
dijet channels, limiting the vev of the scalar doublet
to be of order O(TeV).
By virtue of the mixing between other neutral scalars and the neutral component of H1,
the SM Higgs boson is a linear combination of three scalars. So the SM Higgs boson
treelevel couplings to the SM fermions and gauge bosons are universally rescaled by the mixing
angle with respect to the SM results. We also check the h !
decay rate normalized to
the SM value and it depends not only on the mass of H2 but also other mixing parameters.
As a result, H2 has to be heavier than 100 GeV while the h ! Z
decay width is very
close to the SM prediction. We also con rm that our model can reproduce the correct
DM relic abundance and stays unscathed from the EWPT data in the limit of IHDM
where DM is purely the second neutral Higgs H20. Detailed and systematic study will be
pursued elsewhere.
As an outlook, we brie y comment on collider signatures of this model, for which
detailed analysis goes beyond the scope of this work and will be pursued in the future.
Due to the SU(2)H symmetry, searches for heavy particles are similar to those of SUSY
partners of the SM particles with Rparity. In the case of H20 being the DM candidate,
one can have, for instance, uRuR !
W 0pW 0m via tchannel exchange of uH , followed
R
by W 0p
! uRuRH ! uRH20uL and its complex conjugate, leading to 4 jets plus missing
transverse energy. Therefore, searches on charginos or gauginos in the context of SUSY
may also apply to this model. Furthermore, this model can also yield monojet or
monophoton signatures: uRuR ! H20H20 plus
or g from the initial state radiation. Finally,
the recent diboson excess observed by the ATLAS Collaboration [76] may be partially
explained by 2 TeV Z0 decays into W +W
via the Z0
Z mixing.
Phenomenology of G2HDM is quite rich. In this work we have only touched upon its
surface. Many topics like constraints from vacuum stability as well as DM and neutrinos
physics, collider implications, etc are worthwhile to be pursued further. We would like to
return to some of these issues in the future.
A
Analytical expression for v, v
and v
From eqs. (3.3) and (3.4), besides the trivial solutions of v2 = v2 = 0 one can deduce the
following nontrivial expressions for v2 and v2 respectively,
v2 =
v
2 =
H
(2 H
H
H
2
H
2
H
2
)v2 + ( H M
MH )v
+ 2(2
2
H
H )v2 + ( H MH
Substituting the above expressions for v2 and v2 into eq. (3.5) leads to the following cubic
equation for v :
v
3 + a2v
2 + a1v
+ a0 = 0 ;
where a2 = C2=C3, a1 = C1=C3 and a0 = C0=C3 with
C0 = 2 ( H M
C1 = 2 2 (2 H
+2 4 H
C2 = 3 [( H
C3 = 4
H
H
2
2
H
2
H
MH ) 2H + 2 ( H MH
) 2H + 2 (2 H
2 +
H M 2
) M
+ ( H
H
H
H MH
+ 2
H
H
M
H ) 2
) MH ] ;
2
H
M H2
:
2 )
;
2 )
H :
(A.1)
(A.2)
(A.3)
(A.4)
(A.5)
(A.6)
(A.7)
HJEP04(216)9
The three roots of cubic equation like eq. (A.3) are wellknown since the middle of
16th century
where
with
v 1 =
v 2 =
v 3 =
3 a2 + (S + T ) ;
1
1
p
D ;
D ;
Q3 + R2 ;
1
2
1
2
(S + T ) + 1 ip3 (S
(S + T )
2
2
1 ip3 (S
T ) ;
T ) ;
Q
R
3a1
9
9a1a2
a
2
2 ;
27a0
54
(h1 !
) =
with
B
Decay width of SM Higgs to
and Z
Below we summarize the results for the decay width of SM Higgs to
and Z [68{71],
including the mixing among h, 3 and
2 characterized by the orthogonal matrix O, i.e.,
(h; 3; 2)
T = O (h1; h2; h3)T . In general one should include the mixing e ects among Z
and Z0 (and perhaps Z00) as well. As shown in section IV, these mixings are constrained
to be quite small and we will ignore them here.
Taking into account H2 contributions, the partial width of h1 !
is
GF
1282pm23h1 O3121
Ch m2
H v2
H2
A0 ( H2 )+A1 ( W )+X NcQf2 A1=2( f ) ;
The form factors for spins 0, 12 and 1 particles are given by
Ch = 1
O21 2 H v + MH
O11
4 H v
+
O31 H v
O11 2 H v
:
A0 ( ) = [
A1=2( ) = 2[ + (
;
A1 ( ) = [2 2 + 3 + 3(2
;
(A.8)
(A.9)
(A.10)
(A.11)
(A.12)
(A.13)
(A.14)
(A.15)
2
(B.1)
(B.2)
(B.3)
HJEP04(216)9
with the function f ( ) de ned by
f ( ) = <
8> arcsin2 p
>
>
>
:
>
>
1
4
log
1
1
1
i
#2
; for
1 ;
; for
> 1 :
The parameters i = m2h1=4mi2 with i = H2 ; f; W
masses of the heavy particles in the loops.
Including the H2 contribution, we have
(h1 ! Z ) =
G2F m2W
m3h1O121
64 4
Ch m2
H v2
H2
1
m2Z !3
m2h1
v
H2
A0Z ( H2 ; H2 ) + A1Z ( W ; W )
+ X Nc cW
f
with the function f ( ) de ned in eq. (B.4) and the function g( ) can be expressed as
g( ) = < p
8 p
>
>
>
>
>
>
:
1
1
2
1 arcsin p
1 "
log
1 + p
1
p
1
1
1
1
; for
1 ;
#
i
; for
< 1 :
The corresponding SM rates SM(h1 !
) and SM(h1 ! Z ) can be obtained by
omitting the H2 contribution and setting O11 = 1 in eqs. (B.1) and (B.5).
with v
H2
The loop functions are
A0Z ( H ; H ) = I1( H ; H ) ;
where I1 and I2 are de ned as
I1( ; ) =
I2( ; ) =
2(
2(
2(
+
2 2
)2 f ( 1
) f (
1
) +
)2 g( 1
) g(
f ( 1
) f (
1) ;
I2( ; )+
1+
5 +
2
I1( ; ) ;
2 s
Acknowledgments
The authors would like to thank A. Arhrib, F. Deppisch, M. Fukugita, J. Harz and T.
Yanagida for useful discussions. WCH is grateful for the hospitality of IOP Academia
Sinica and NCTS in Taiwan and HEP group at Northwestern University where part of
this work was carried out. TCY is grateful for the hospitality of IPMU where this project
was completed. This work is supported in part by the Ministry of Science and Technology
(MoST) of Taiwan under grant numbers 1012112M001005MY3 and
1042112M001001MY3 (TCY), the London Centre for Terauniverse Studies (LCTS) using funding from
the European Research Council via the Advanced Investigator Grant 267352 (WCH), DGF
Grant No. PA 803/101 (WCH), and the World Premier International Research Center
Initiative (WPI), MEXT, Japan (YST).
Open Access.
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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