APS/123-QED

Unitarity Constraints in the Two Higgs Doublet Model

Andres Castillo and Rodolfo Diaz
Departamento de Fsica. Universidad Nacional de Colombia.
(Dated: May 2, 2011)
We present an overview study of the unitarity constraints in the Two Higgs Doublet Model
(2HDM). For this purpose, rstly we discuss the motivations to introduce an extra Higgs doublet
in the electroweak spontaneous broken mechanism. Among these are: the similarity in the Higgs
sector of supersymmetric and Left-Right models, the explanation of the hierarchy of masses in the
quark sector, and induction of phenomenology as the Flavor Changing Neutral Currents (FCNCs)
and explicit CP violation. From here, we have studied dierents ways, essencially theoretical and
experimental, to constrain the free parameters of the 2HDM. In this point, the unitarity emerges as
a fundamental limiting of the theory, and therefore become a relevant mechanism to nd bounds in
the Higgs masses.
Keywords: Unitarity, renormalization, standard model with two higgs doublets
I. INTRODUCTION AND 2HDM-MOTIVATIONS
Despite the Standard Model (SM) of elementary particles
is highly predictive, there are theoretical and experimental
issues in its formulation. For instance the neutrino oscillation
inexplicability (i.e mixing matrix of this leptons), the mass
hierarchy in the third generation of quarks family and the
universal unknowledge of the Higgs sector (responsible for
Spontaneuos Simmetry Broken (SSB) of the SM local gauge
SU(2)
L
U(1)
Y
). Actually, there is not any fundamental
reason to describe the phenomenology of SM from a Higgs
minimal sector (i.e only one Higgs doublet). In addition,
the experimental constraints indicates with denite accuracy
level that SM Higgs sector have to be generated through of
a no-minimal behaviour [1]. In the rst instance for nd a
solution to the SM problems, we can evocate the next ex-
tension gauge invariant compatible to the minimal sector as
a good candidate. This new sector is denominated as Two
Higgs Doublet Model (2HDM), which consist of adding a sec-
ond Higgs doublet with the same quantum numbers as the
rst one [2].
Therefore, another iso-doublet could give a natural reason
for the hierarchy of Yukawa couplings in the third generation
of quarks. The naturality come from the following fact: if
the bottom received its mass from one doublet and the mass
top from another doublet, and if the free parameters of the
theory acquired the appropiate values, we can explain (in the
theoretical sense) the dierence between the corresponding
masses.
There are other motivations for the Higgs sector exten-
sion. One of them lies in that the 2HDM could induce CP-
violation either explicitly or spontaneously in the Higgs po-
tential. Another reason for this formulation comes from the
study of processes called Flavour Changing Neutral Currents
(FCNCs), which does not violate any fundamental law of na-
ture but they are severely suppresed by experimental data
[13] with remarkable exception of neutrino oscillation. In
addition, there are some models where in their low energy
limit the Higgs sector associated is non minimal. This is
the case the supersymetric models where at least two Higgs
doublets are necessary for achieve the SSB. For instance, the
so called 2HDM type II has the same Yukawa couplings as
the Minimal Supersymmetric Standard Model (MSSM). The
MSSM is the next supersymmetric extension to the SM, in
the which the particles contents is doubled for the so called
partners SUSY of Poincare Group (for each fermionic degree
of freedom there are a bosonic degree of freedom). Other
important study that could be reached with the 2HDM is
its behaviour at non zero temperature. In this regimen the
richness of model can become even more maniest. Here, the
eective parameters evolve with the temperature, which can
lead a thermal phase transitions, with important cosmologi-
cal implications [4].
This overview rstly display the phenomenology of the two
Higgs doublet Lagrangian before of the study of theoretical
and experimental constraints of this model. Actually, we
focus our analysis to the constraints provided by the tree
level unitarity.
II. TWO HIGGS DOUBLET MODEL
The Electroweak Symmetry Breaking (EWSB) in the SM
is described usually with the Higgs mechanism. In the min-
imal behaviour (the simplest variant), an initial Higgs eld
(without Vacuum Expectation Value (VEV)) is an isodou-
blet of scalar elds with weak isospin

. The next no-trivial
extension of the Higgs, our 2HDM consist in introducing two
Higgs weak isodoublets of scalar elds, denoted as
1
and

2
, with hypercharge Y = +1 ( The 2HDM in all exten-
sion is considered in the references [2, 3], where there is a
construction of model since the minimal Higgs sector).
The EWSB via the Higgs mechanism is described with the
Lagrangian of structure [5]
L
SSB
= L
gf
+L
H
+L
Y
: L
H
= L
cin
V
H
(1)
Here, L
gf
describes the SM interactions of gauge bosons
and fermions (as well fermions and gauge bosons propgators),
whose form is independent of the Higgs sector structure; the
Higgs scalar Lagrangian L
H
contains the kinetic term T (i.e
Higgs propagators and interactions with gauge bosons) and
the potential V
H
(responsible of achieve the EWSB and the
Higgs self-interactions). The L
Y
describes the Yukawa inter-
actions of fermions with the Higgs scalars.
In the 2HDM, the Higgs potential have the general form
2
V
H
=m
2
11

1
+m
2
22

2
+
1
2
_

1
_

1
_
2
+
2
_

2
_
2
_
+
3
_

1
__

2
_
+
4
_

2
__

1
_
+
_
1
2

5
_

2
_
2
+
6
_

1
__

2
_

7
_

2
__

2
_
m
2
12

2
+h.c.
_
. (2)
Here,
1
,
2
,
3
,
4
, m
2
11
y m
2
22
are real (due to hermiticity
of the potential), while
5
,
6
,
7
y m
2
12
are, in general, com-
plex parameters. Reverting to the standard representation
of sacalr doublets, the two higgs doublets are given as

1
=
_

1
+i
1

1
+i
1
_
,
2
=
_

2
+i
2

2
+i
2
_
(3)
The potential with real coecients describes the theory
without CP violation in the Higgs sector while the complex
values of some coecients allow the CP-violation in the same
sector (CP-violation means that the CP conjugate elds do
not satisfy the same Euler-Lagrange equation [6]).
The crucial role in the 2HDM is played by the discrete
Z
2
-symmetry, i.e, the behaviour under the transformation
(
1

1
,
2

2
) or (
1

1
,
2

2
) (4)
This symmetry forbids the (
1
,
2
) mixing. With this
symmetry, the CP violation in the Higgs sector is forbbiden
and the FCNC are unnatural. In one realistic theory this
Z
2
symmetry is violated [7].
In order to see in detail the EWSB, we consider the fol-
lowing parametrization of the Higgs doublets

1
=
_

+
i
v
i
+
h
i
+iz
i

2
_
i = 1, 2 (5)
Here, the v

i
s are the VEV respectively. After EWSB, the
W and Z gauge boson acquire masses (through the Goldstone
Theorem and the Higgs-Kibble mechanism). The Goldstone
theorem tell us that the Spontaneous Symmetry Breaking
(SSB) of a Lie Group G symmetry to a Lie Group H Sym-
metry leads to dimG/H massles scalar bosons denominated
Goldstone bosons. In the particular case of SM, the SSB
scheme is SU(2)
L
U(1)
Y
U(1)
Em
. Therefore, there are
three Golstone elds. This value is independent of the Higgs
sector realization. This proccedure generates the following
relations:
m
2
W
=
1
2
g
2
v
2
; m
2
z
=
1
2
(g
2
+g

2
)v
2
, (6)
where g y g

are the SU(2)

weak
and U(1)
Y
gauge couplings
and v
2
= v
2
1
+ v
2
2
. The combination v
2
1
+ v
2
2
is thus xed by
the electroweak scale through of (2

2G
F
)
1
. In this point of
view, we are left the 2HDM with 8 free parameters, namely
the (
i
)
i=1,....,7
y tan = v
2
/v
1
. Meanwhile, three of the
eight degrees of freedom of the isodoublets correspond to the
3 Goldstone bosons (G

, G
0
) and the remaining ve become
physical Higgs bosons: H
0
, h
0
(CP-even), A
0
(CP-odd) and
H

. Their masses are obtained as usual by the shift

i

i
+v
i
[8]
If we restrict ourselves to the particular case
6
=
7
= 0
[4, 7], and after generating the scalar masses in term of scalar
parameters
i
, and using straightforward algebra, we can
express all the
i
as function of the physical masses in the
following form [8, 10]:

4
=
g
2
2m
2
W
m
2
H
,
6
=
g
2
2m
2
W
m
2
A
;
3
=
g
2
8m
2
W
sin cos
sin cos
_
m
2
H
m
2
h
_

5
4

1
=
g
2
8 cos
2
m
2
W
_
m
2
H
cos
2
+m
2
h
sin
2

sin cos
tan
(m
2
H
m
2
h
)
_

5
4
_
tan
2
1
_

2
=
g
2
8 sin
2
m
2
W
_
m
2
H
sin
2
+m
2
h
cos
2
sin cos tan (m
2
H
m
2
h
)
_

5
4
_
1
tan
2

1
_
(7)
The angle diagonalizes both the CP-odd and charged
scalar mass matrices, leading to the physical states H

and
A
0
. The angle diagonalizes the CP-even mass matrix lead-
ing to the physical states H
0
y h
0
. Now, we are free to take
as 7 independent parameters (
i
)
i=1,..,6
and tan or equiv-
alently the four scalar masses, tan , and one
i
. Usually
in this basis,
5
is taken as a free parameter.
3
III. CONSTRAINTS ON 2HDM
Due to the number of free parametes (or physical degrees of
freedom in the new Higgs sector), it is neccesary impose a set
of constraints in the model. Actually, we can consider two
phenomenological regimens of these bounds: Experimental
and theoretical constraints. The following part of this review
will discuss about these regions of the theory.
A. Experimental Constraints
These constraints arise of the dierents experiments with
the dierents collaborations set in the electroweak energy
scale (in the order of MeV-TeV). In fact, they are all empty
Higgs searches made with dierent decay channels and the
scattering processes with the concerning energy regimes.
Therefore, since no Higgs boson was observed so far either at
LEP-II or the Tevatron, stringent limits have been derived
on the masses and couplings allowed in the Higgs sector, both
within and beyond the SM [11, 12]. For instance, in the H

bounds, the principal constraints come from the pair produc-

tion e
+
e

H
+
H

, followed by the decays in the H

and H

cs, while for the neutral Higgs the most

studied chanels are the Higgsstrahlung e
+
e

h
0
(H
0
)Z and
the double Higgs production e
+
e

H
0
h
0
(Both searches
carried out by LEP [11]).
For instance we put a schematic representation of the
bounds in the Higgs masses for diferents values of the param-
eters. In this case, this experiment come from of the OPAL
collaboration (CERN). The theoretical inaccesible regimes
will be discused in the next section.
FIG. 1: Behaviour of the bounds Higgs masses with respect a the
other Higgs masses and diferent parameters values. Figure taken
of [13]. We can see at 95% CL of accurancy that the lighter neu-
tral Higgs mass is found in the interval 85 m
h
(GeV ) 125 with
tan > 6 and the other region (most probable in the phenomeno-
logical sense) with the values between 115 m
h
(GeV ) 125
with 0 < tan < 0.2
B. Theoretical Constraints
It is also interesting to know what could be possible theo-
retical limitations for masses of the so far elusive Higgs parti-
cles within such a quasi realistic model (in the phenomeno-
logical sense of this). For this purpose, some rather general
method have been raised, based mostly on the requirements
on internal consistency of the quantum eld theoretical de-
scription of the relevant physical quatities (e.g Higgs masses
and free parameters in the Lagrangian). Between the con-
straints most used, we have:
Vacuum Stability (Positivity): For the vacuum
conguration to be stable the Higgs potential, must
be positive in all eld space directions for asymptoti-
cally large values of the elds. At large eld values, the
potential is dominated by the quartic terms (i.e only
should be an electroweak phase transition in all energy
regimes). Here the bounds obtained are
2
,
1
> 0,

3
>

2
and if
6
=
7
= 0 (where the Z
2
is not
violated)
3
+
4
+
5
>

2
[14].
Oblique parameters: The electroweak oblique pa-
rameters S, T, U [15] constitute a sensitive probe of
new physics coupling to the Electroweak gauge bosons.
Values on these parameters are constrained by the
precision measurements at LEP [1]. In general, the
contribution to the oblique parameters from the ex-
tra SU(2) doublet in the 2HDM is small, since scalar
doublets (or singlets) do not break the custodial sym-
metry [16] which protects the tree-level relation
m
W
/(m
Z
cos
w
) = 1. However, large mass splittings
among the new Higgs states can still induce sizeable
contributions at the loop level [12].
Perturbativity: This method lies in the fact that the
self interacting parameters in the Higgs potential and
couplings in the Yukawa sector remain perturbative up
to a large scale. This property impose that |
i
| < 4.
Unitarity: As is known from Optical Theorem (as we
will see in the next section), the complete all-order scat-
tering (S) matrix has to be unitary. Essentially there
are two dierent ways of achieving this. In a weakly cou-
pled theory higher order contributions to the S-matrix
typically become smaller and smaller. If the theory
on the other hand is strongly coupled, the individual
contributions may be arbitrarily large. It is then only
in the sum that they cancel and unitarity is respected.
For a weakly coupled theory it is natural to require that
the S-matrix is unitary already at tree-level. In what
follows we strongly expand these concepts and identi-
es some of the constraints to the parameters of the
theory.
IV. UNITARITY CONSTRAINTS
Again, it is relevant to note that the use of unitarity is not
restricted only to the Higgs potential sector but is extrapo-
lated to the entire Lagrangian. We start with the implica-
tions of the unitarity in the S-matrix and so in the computes
4
of the associated cross sections of dierents theories like SM-
minimal, SM-2HDM, etc. In order to achieve the unitarity
incorporation to our model, rstly we will study the Optical
theorem for then analyze the method called LQT.
A. Optical theorem
The Optical theorem is an explicit consequence of the use
of unitarity in the denition of the S-matrix, wich depend of
the collection of transitions amplitudes with the form
S

= , t |, t (8)
Actually, the unitarity guarantees the permanence of these
states in the same Hilbert space [17, 18]. Diagrammatically
the optical theorem is represented by the gure 2.
FIG. 2: The Optical theorem as schematic representation.
The essence of unitarity is set here: this applies order by
order in the perturbation theory. Finally, the optical theorem
states that the imaginary part of the scattering amplitude -
forward- is proportional to the total cross section [19].
B. 2HDM-Unitarity
To constrain the scalar potential parameters (i.e nding
the upper bounds on the Higgs boson masses) we will the
well-known LQT (Lee, Quigg and Thacker) method [20].
This proccedure relies on imposing the condition of pertur-
bative (in particular at tree level in virtue to the optical
theorem) unitarity on an appropiate set of physical scatter-
ing processes. Within a renormalizable theory, the scattering
amplitudes are aymptotically at, i.e they do not exhibit
any power-like growth in the high energy limit [3]. How-
ever, the dominant couplings are typically proportional to
the scalar boson masses and one can thus obtain useful tech-
nical constraints on their values [10].
In other words, the latter statement coresponds to the re-
quirement that J = 0 partial waves (a
0
) for scalar-scalar and
gauge boson-scalar scattering satisfy |a
0
| 1/2 in the high
energy limit. At very high energy limit, the equivalence the-
orem states that the amplitude of a scattering process involv-
ing longitudinal gauge bosons W

, Z may be approximated
by the scalar amplitude in wich gauge bosons are replaced by
their corresponding Goldstone boson G

, G
0
. We conclude
that unitarity constraints can be implemented by solely con-
sidering pure scalar scattering [8].
In very high energy collisions, it can be shown that the
dominant contribution to the amplitude ot two body scat-
tering s
1
s
2
s
3
s
4
is the one wich is mediated by the quar-
tic coupling. Those contributions mediated by trilinear cou-
plings are suppresed on dimensional grounds. Therefore the
unitarity constraint |a
0
| 1/2 reduces to following con-
straint on the quartic coupling, |Q(s
1
, s
2
; s
3
s
4
) 8 [9].
As a reference, we put in the table 1 dierents upper
bounds for the Higgs spectrum obtained by the LQT method
in the 2HDM for the case in wich there are not FCNCs (type
I-II)
Bounds from M
H
M
A
M
h
M
H
[8] 691 695 435 638
[9] 872 1233 411 712
[10] 712 712 411 712
Table 1: A comparison between the bounds produced by [8] and
those of [9] and [10]. All masses are in GeV and
5
=
6
=
7
= 0
V. CONCLUSIONS AND OUTLOOK
We have considered a status of the Two Higgs Doublet
Model in the sense of characterize the constraints present due
to the quantum internal consistency of the theory, primarily
focusing on the limits on the Higgs mass provided by the tree
level unitarity. For this reason, we discuss the importance of
the unitarity to arrive to the optical theorem. Moreover, this
concept allowed us to extend a relationship between unitarity
and the scattering processes of the model, and therefore with
the paremeter set in the 2HDM Lagrangian. In addition, we
have named the importance of the experimental constraints
in the phenomenological study of the 2HDM-paremeters be-
haviour.
As prospects, and after the ideal compression of LQT
method, we have proposed to reconsider the essence of
this technique to treat scattering processes which involve
fermionic states too. This formulation will allow to consider
not only the high energy regimes but also regions where the
scales are comparable with the masses of the quark content
of the standard model.
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