Radiative corrections to electroweak parameters in the Higgs triplet model and implication with the recent Higgs boson searches

Abstract

We study radiative corrections to the electroweak parameters in the Higgs
model with the Y=1 triplet field, which is introduced in the scenario of
generating neutrino masses based on the so-called type II seesaw mechanism. In
this model, the rho parameter deviates from unity at the tree level.
Consequently, the electroweak sector of the model is described by the four
input parameters such as $\alpha_{\text{em}}$, $G_F$, $m_Z$ and
$\sin^2\theta_W$. We calculate the one loop contribution to the W boson mass as
well as to the rho parameter in order to clarify the possible mass spectrum of
the extra Higgs bosons under the constraint from the electroweak precision
data. We find that the hierarchical mass spectrum among $H^{\pm\pm}$, $H^{\pm}$
and $A$ (or $H$) is favored by the precision data especially for the case of
$m_A$ $(\simeq m_H)>m_{H^+}>m_{H^{++}}$, where $H^{\pm\pm}$, $H^{\pm}$, $A$ and
$H$ are the doubly-charged, singly-charged, CP-odd and CP-even Higgs bosons
mainly originated from the triplet field. We also discuss phenomenological
consequences of such a mass spectrum with relatively large mass splitting. The
decay rate of the Higgs boson decay into two photons is evaluated under the
constraint from the electroweak precision data, regarding the recent Higgs
boson searches at the CERN LHC.

Full text

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arXiv:1201.6287v1 [hep-ph] 30 Jan 2012
UT-HET 064
Radiative corrections to electroweak parameters in the Y = 1 Higgs triplet model and
implication with the recent Higgs boson searches at the CERN LHC
Shinya Kanemura, Kei Yagyu
Department of Physics, University of Toyama,
3190 Gofuku, Toyama 930-8555, Japan
We study radiative corrections to the electroweak parameters in the Higgs model with the Y = 1
triplet field, which is introduced in the scenario of generating neutrino masses based on the so-
called type II seesaw mechanism. In this model, the rho parameter deviates from unity at the tree
level. Consequently, the electroweak sector of the model is described by the four input parameters
such as αem, GF, mZ and sin2θW. We calculate the one loop contribution to the W boson mass
as well as to the rho parameter in order to clarify the possible mass spectrum of the extra Higgs
bosons under the constraint from the electroweak precision data. We find that the hierarchical
mass spectrum among H±±, H±and A (or H) is favored by the precision data especially for the
case of mA (≃ mH) > mH+ > mH++, where H±±, H±, A and H are the doubly-charged, singly-
charged, CP-odd and CP-even Higgs bosons mainly originated from the triplet field. We also discuss
phenomenological consequences of such a mass spectrum with relatively large mass splitting. The
decay rate of the Higgs boson decay into two photons is evaluated under the constraint from the
electroweak precision data, regarding the recent Higgs boson searches at the CERN LHC.
PACS numbers: 12.60.Fr, 12.15.Lk, 14.80.Cp
I. INTRODUCTION
The Higgs boson search is underway at the LHC. The mass of the Higgs boson in the Standard Model (SM) has
already constrained to be between 115 GeV and 127 GeV or to be higher than 600 GeV at the 95% confidence level
by the recent direct search results [1]. By the combination with the electroweak precision measurement at the LEP [2]
and the SLC [3], we may expect that a light Higgs boson exists as long as the Higgs boson interactions are of SM-like
and that it will be discovered in near future.
The SM is expected to be replaced by a new model at higher energies above the TeV scale, which may give
reasonable explanations for tiny neutrino masses [4], the origin of dark matter [5] and the baryon asymmetry of the
Universe [6]. The low energy effective theory of such a more fundamental model often contains a non-minimal Higgs
sector. The structure of a Higgs sector strongly depends on the property of the corresponding new physics model at
the high energy scale, so that experimental reconstruction of the Higgs sector is extremely important to determine
new paradigm for physics beyond the SM.
Once a Higgs boson is found at the LHC, its properties such as the mass, the width and the decay branching ratios
will be thoroughly measured as precisely as possible. The precision measurement for Higgs boson properties can
also be performed at future linear colliders such as the International Linear Collider (ILC) and the Compact Linear
Collider (CLIC). Each extended Higgs model will be tested by preparing the full set of theoretical predictions on the
observables which will be precisely measured at experiments at these colliders.
The electroweak rho parameter has been one of the most important experimental hints to select possible structures
of extended Higgs sectors. Its experimental value (ρ ≃ 1) would indicate that the Higgs sector is composed of one
or more doublet fields (or with singlet fields). Such models naturally predict ρ = 1 at the tree level because of the
custodial symmetry in the kinetic term of their Higgs sectors. Radiative corrections give the deviation from unity
corresponding to the violation of the custodial symmetry in the physics of dynamics in the loop [7–12]. On the other
hand, as another possibility, we may consider with some physics motivations the other class of Higgs models which
contain scalar fields of larger representations under the SU(2)Lgauge symmetry, in which ρ ?= 1 is predicted at the
tree level. In these models, the vacuum expectation values (VEVs) of such fields with large isospin representations
are highly constrained to be much smaller than 246 GeV to satisfy ρ ≃ 1.
The Higgs triplet model (HTM) is one of the examples which predict ρ ?= 1 at the tree level. In the HTM, a Higgs
triplet field with the hypercharge Y = 1 is added to the minimal model with a Higgs doublet field with Y = 1/2.
The HTM can explain the tiny neutrino masses in a simple way via the so-called type II seesaw mechanism [13]. An
interesting feature of the HTM is the existence of doubly-charged Higgs bosons in addition to singly-charged as well
as extra neutral bosons, all of which are components of the triplet field.
In this paper, we study radiative corrections to the electroweak observables in the HTM. In the model with ρ ?= 1
at the tree level like the HTM, apart from the models with only doublet fields and singlets which predict ρ = 1 at the

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tree level, a new input parameter has to be introduced in addition to the usual three input parameters such as (αem,
GF and mZ). In Ref. [14], the on-shell renormalization scheme is constructed in the Higgs model with the Y = 0
triplet field, in which four input electroweak parameters (αem, GF, mZ and sin2θW) are chosen to describe all the
other electroweak observables. The radiative corrections to the electroweak observables have been calculated in the
Y = 0 triplet model [14–16] and in the left-right symmetric model [16].
In our analysis, we first define the on-shell renormalization scheme for the electroweak sector of the HTM by using
the method in Ref. [14]. We then calculate radiative corrections to the electroweak observables such as mW and
ρ as a function of the four input parameters (αem, GF, mZ and sin2θW)1. We examine the preferable values of
the mass spectrum of the triplet-like Higgs bosons and the VEV of the triplet field under the constraint from the
electroweak precision data. We find that the hierarchical mass spectrum with large mass splitting is favored especially
for the case of mA(≃ mH) > mH+ > mH++, where mA, mH, mH+ and mH++ are the masses of the CP-odd (A),
CP-even (H), singly-charged (H±) and doubly-charged (H±±) Higgs bosons, respectively, which mainly come from
the components of the triplet field. On the contrary, the inverted hierarchical case with mH++ > mH+ > mA(≃ mH)
is relatively disfavored, in particular for the light SM-like Higgs boson. We then discuss implication with the Higgs
boson phenomenology at the LHC in the HTM [18–27]. The deviation due to the quantum effect of doubly- and
singly-charged Higgs bosons is evaluated on the decay rate of h → γγ [28], where h is the SM-like Higgs boson. We
here examine the decay rate in the parameter regions which are indicated by the electroweak precision data for the
mass of the SM-like Higgs boson to be 125 GeV taking into account the recent Higgs boson search at the LHC [1].
The deviation from the SM prediction can be significant although it is destructive. We also give some comments on
the direct searches for the triplet-like Higgs bosons in the case with [17, 24–27] and without [18–23] the mass splitting
among these Higgs bosons.
In Sec. II, we give a brief review for the tree level formulae in the HTM towards the one-loop calculation. In Sec. III,
the on-shell renormalization scheme is introduced for the electroweak precision observables, and the predictions for
mW and ρ at the one-loop level are calculated. In Sec. IV, we evaluate the deviation from the SM prediction of
the decay rate of h → γγ, and we discuss the collider phenomenology of the triplet-like Higgs bosons at the LHC.
Discussions and conclusions are given in Sec. V.
II.THE HIGGS TRIPLET MODEL
The scalar sector of the HTM is composed of the isospin doublet field Φ with hypercharge Y = 1/2 and the triplet
field ∆ with Y = 1. The relevant terms in the Lagrangian are given by
LHTM= Lkin+ LY − V (Φ,∆), (1)
where Lkin, LY and V (Φ,∆) are the kinetic term, the Yukawa interaction and the scalar potential, respectively. The
kinetic term of the Higgs fields is given by
Lkin= (DµΦ)†(DµΦ) + Tr[(Dµ∆)†(Dµ∆)],(2)
where the covariant derivatives are defined as
DµΦ =
?
∂µ+ ig
2τaWa
µ+ ig′
2Bµ
?
Φ,Dµ∆ = ∂µ∆ + ig
2[τaWa
µ,∆] + ig′Bµ∆.(3)
The Yukawa interaction for neutrinos is given by
LY = hijLic
Liτ2∆Lj
L+ h.c., (4)
where Li
that the triplet field ∆ carries the lepton number of 2. The most general form of the Higgs potential under the gauge
symmetry is given by
Lis the i-th generation left-handed lepton doublet, hijis the 3×3 complex symmetric Yukawa matrix. Notice
V (Φ,∆) = m2Φ†Φ + M2Tr(∆†∆) +?µΦTiτ2∆†Φ + h.c.?
+ λ1(Φ†Φ)2+ λ2
?Tr(∆†∆)?2+ λ3Tr[(∆†∆)2] + λ4(Φ†Φ)Tr(∆†∆) + λ5Φ†∆∆†Φ, (5)
1In Ref. [17], the constraint from the electroweak precision data has been discussed in the Y = 1 HTM. However, the renormalization
scheme they used seems to be only valid in models with ρ = 1 at the tree level.

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where m and M are the dimension full real parameters, µ is the dimension full complex parameter which violates the
lepton number, and λ1-λ5are the coupling constants which are real. We here take µ to be real. The scalar fields Φ
and ∆ can be parameterized as
Φ =
?
ϕ+
1
√2(ϕ + vΦ+ iχ)
?
, ∆ =
?
∆+
√2
∆0−∆+
∆++
√2
?
with ∆0=
1
√2(δ + v∆+ iξ), (6)
where vΦ and v∆ are the VEVs of the doublet Higgs field and the triplet Higgs field, respectively which satisfy
v2≡ v2
From the stationary condition at the vacuum (vΦ,v∆), we obtain
Φ+ 2v2
∆≃ (246 GeV)2.
m2=1
2
?
∆−1
−2v2
Φλ1− v2
∆(λ4+ λ5) + 2√2µv∆
?
, (7)
M2= M2
2
?2v2
∆(λ2+ λ3) + v2
Φ(λ4+ λ5)?, with M2
∆≡
v2
√2v∆
Φµ
. (8)
The mass matrices for the scalar bosons can be diagonalized by rotating the scalar fields as
?
ϕ±
∆±
?
=
?cβ±−sβ±
sβ±
cβ±
??
w±
H±
?
,
?χ
ξ
?
=
?cβ0−sβ0
sβ0
cβ0
??
z
A
?
,
?ϕ
δ
?
=
?cα −sα
sα
cα
??
h
H
?
, (9)
with the mixing angles
tanβ±=
√2v∆
vΦ
, tanβ0=2v∆
vΦ
, tan2α =v∆
vΦ
2v2
Φλ1− M2
Φ(λ4+ λ5) − 4M2
∆− v2
∆
2v2
∆(λ2+ λ3), (10)
where we used abbreviation such as sθ= sinθ and cθ= cosθ. In addition to the three Nambu-Goldstone bosons w±
and z which are absorbed by the longitudinal components of the W boson and the Z boson, there are seven physical
mass eigenstates H±±, H±, A, H and h. The masses of these physical states are expressed as
m2
H++ = M2
∆− v2
∆−λ5
?
11sin2α + M2
h= M2
∆λ3−λ5
2v2
??
?
22cos2α + M2
22sin2α − M2
Φ, (11)
m2
H+ =
?
M2
4v2
Φ
1 +2v2
∆
v2
Φ
?
, (12)
m2
A= M2
∆
1 +4v2
∆
v2
Φ
, (13)
m2
m2
H= M2
12sin2α,(14)
11cos2α + M2
12sin2α,(15)
where M2
which are given by
11, M2
22and M2
12are the elements of the mass matrix M2
ijfor the CP-even scalar states in the (ϕ,δ) basis
M2
M2
M2
11= 2v2
22= M2
12= −2v∆
Φλ1,
∆+ 2v2
(16)
∆(λ2+ λ3), (17)
vΦ
M2
∆+ vΦv∆(λ4+ λ5).(18)
The masses of the W boson and the Z boson are obtained at the tree level as
m2
W=g2
4(v2
Φ+ 2v2
∆),m2
Z=
g2
4cos2θW(v2
Φ+ 4v2
∆).(19)
The electroweak rho parameter can deviate from unity at the tree level;
ρ ≡
m2
W
m2
Zcos2θW
=
1 +2v2
∆
v2
4v2
v2
Φ
1 +
∆
Φ
. (20)

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As the experimental value of the rho parameter is near unity, v2
the tree level. In this case, the state h behaves mostly as the SM Higgs boson, while the other states are almost
originated from the components of the triplet field. Then, from Eqs. (11), (12), (13) and (14), there are interesting
relations among the masses;
∆/v2
Φis required to be much smaller than unity at
m2
H++ − m2
m2
H+ ≃ m2
?≃ M2
H+ − m2
?.
A
?
≃ −λ5
4v2
Φ≡ ξ
?
, (21)
H≃ m2
A
∆
(22)
These characteristic mass relations would be used as a probe of the Higgs potential in the HTM [25]. If the masses
of the triplet-like Higgs bosons are hierarchical, there are two patterns of the mass hierarchy among the triplet-like
scalar bosons; i.e., when λ5is positive (negative), the mass hierarchy is mφ0 > mH+ > mH++ (mH++ > mH+ > mφ0),
where mφ0 = mAor mH. We here define the mass difference between H±±and H±as
∆m ≡ mH++ − mH+. (23)
In the HTM, tiny Majorana masses of neutrinos are generated by the Yukawa interaction with the VEV of the
triplet field, which is proportional to the lepton number violating coupling constant µ as
(mν)ij=
√2hijv∆= hijµv2
Φ
M2
∆
. (24)
If µ ≪ M∆the smallness of the neutrino masses are explained by the so-called type II seesaw mechanism [13]. In this
paper, we assume that the lepton number violating parameter µ is sufficiently smaller than the electroweak scale so
that the mass scale of the triplet-like field is O(100-1000) GeV.
III. ONE-LOOP CORRECTIONS TO ELECTROWEAK PARAMETERS
In this section, we calculate one-loop corrected electroweak observables in the on-shell scheme which was at first
proposed by Blank and Hollik [14] in the model with a triplet Higgs field with Y = 0. In the SM, and in all the
models with ρ = 1 at the tree level, the kinetic term of the Higgs field contains three parameters g, g′and v. All
the electroweak parameters are determined by giving a set of three input parameters which are well known; i.e., for
example, αem, GF and mZ [29, 30]. On the other hand, in models with ρ ?= 1 at the tree level like the HTM, an
additional input parameter is necessary to describe electroweak parameters. Therefore, in addition to the three input
parameters αem, GF and mZ, we take the weak angle sin2θW as the fourth input parameter in our calculation as in
Ref. [14]. The experimental values of these input parameters are given by [31]
α−1
em(mZ) = 128.903(15),
mZ= 91.1876(21) GeV,
GF = 1.16637(1)× 10−5GeV−2,
ˆ s2
W(mZ) = 0.23146(12),(25)
where ˆ s2
W(mZ) is defined as the ratio of the coefficients of the vector part and the axial vector part in the Z¯ ee vertex;
1 − 4ˆ s2
W(mZ) =Re(ve)
Re(ae), (26)
here veand aeare defined in Eq. (41). Tree level formulae for the other electroweak parameters are given in terms of
the four input parameters:
g2=4παem
ˆ s2
παem
√2GFˆ s2
1
√2GF
Wˆ c2
2παemm2
W
,(27)
m2
W=
W
, (28)
v2=, (29)
v2
∆=ˆ s2
W
Z−
√2
4GF, (30)

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where ˆ s2
The deviation from the relation in Eq. (28) due to radiative corrections can be parameterized as
W= ˆ s2
W(mZ) and ˆ c2
W= 1 − ˆ s2
W
2.
GF=
παem
√2m2
Wˆ s2
W
(1 + ∆r), (32)
where ∆r is
∆r = −δGF
GF
+δαem
αem
−δˆ s2
ˆ s2
W
W
−δm2
m2
W
W
. (33)
The counter terms are obtained by imposing the renormalization conditions given in Ref. [30] as
δGF
GF
= −ΠWW
T
(0)
m2
W
− δV B, (34)
δαem
αem
δm2
m2
=
d
dp2Πγγ
=ΠWW
T(p2)
???
p2=0+2ˆ sW
ˆ cW
ΠγZ
T(0)
m2
Z
, (35)
W
W
T
(m2
W)
m2
W
, (36)
and in Ref. [14] as
δˆ s2
ˆ s2
W
W
=ˆ cW
ˆ sW
ΠγZ
T(m2
m2
Z)
Z
− δ′
V, (37)
where δV B and δ′
vertex, respectively. These are calculated as [14, 32]
Vare the vertex and the box diagram corrections to GF and the radiative corrections to the Z¯ ee
δV B=
αem
4πˆ s2
W
?
6 +10 − 10ˆ s2
W− 3(R/ˆ c2
2(1 − R)
−ΓZ¯ ee
ae
W)(1 − 2ˆ s2
W)
lnR
?
,R ≡m2
W
m2
Z
, (38)
δ′
V=
ve
2ˆ s2
W
?ΓZ¯ ee
V
(m2
ve
Z)
A(m2
Z)
?
, (39)
where ΓZ¯ ee
V,Aare defined through the renormalized Z¯ ee vertex as [14]
ˆΓZ¯ ee
µ
(m2
Z) = i
g
2ˆ cW
?(ve+ aeγ5)γµ+ γµΓZ¯ ee
2+ 2ˆ s2
V
(m2
Z) + γµγ5ΓZ¯ ee
A (m2
Z)?, (40)
ve= −1
W,ae= −1
2.(41)
In Eqs. (34)-(36), ΠXY
the list of all the analytic expressions for the gauge boson self-energies in the HTM with the Y = 1 triplet field in
Appendix. The radiative correction ∆r can be obtained by
T
(p2) (X,Y = W,Z,γ) are the 1PI diagrams for the gauge boson self-energies. We show
∆r =ΠWW
T
(0) − ΠWW
m2
T
(m2
W)
W
+
d
dp2Πγγ
T(p2)
???
p2=0+2ˆ sW
ˆ cW
ΠγZ
T(0)
m2
Z
−ˆ cW
ˆ sW
ΠγZ
T(m2
m2
Z)
Z
+ δV B+ δ′
V.(42)
2In the limit of v∆→ 0, we obtain the following relation
ˆ s2
Wˆ c2
παem
W
m2
Z=
1
√2GF
. (31)
By using this relation, mW can be expressed by m2
as the SM.
W= m2
Zˆ c2
W. This relation can be found in models with ρ = 1 at the tree level such