Radiative corrections to the Higgs potential in the LH model
Antonio Dobado, Lourdes Tabares-Cheluci
Departamento de Física Teórica I,

Universidad Complutense de Madrid, E-28040 Madrid, Spain

Siannah Peñaranda
Departamento de Física Teórica, Universidad de Zaragoza, E-50009, Zaragoza,
Spain

Javier Rodriguez-Laguna
Departamento de Matemáticas, Universidad Carlos III de Madrid, E-28911,
Madrid, Spain

ABSTRACT

In this work we compute the radiative corrections to the Higgs mass and the
Higgs quartic couplings coming from the Higgs sector itself and the scalar
fields $\phi$ in the Littlest Higgs (LH) model. The restrictions that the new
contributions set on the parameter space of the models are also discussed.
Finally this work, together with our three previous papers, complete our
program addressed to compute the relevant contributions to the Higgs low-
energy effective potential in the LH model and the analysis of their
phenomenological consequences.

##  1 Introduction

The discovery of a Higgs boson and the elucidation of the mechanism
responsible for the electroweak symmetry breaking are some of the major goals
of present and future searches in particle physics. Because of the precise
data obtained for a long time to test the Standard Model (SM) of particle
interactions, and the recent measurements of the $W$ and the top masses at the
Fermilab Tevatron [1], the SM has been confirmed as the right model describing
the electroweak phenomena at the current experimental energy scale. However,
the origin of the electroweak symmetry breaking, for which the Higgs boson is
responsible in the SM, remains elusive. The quadratically divergent
contributions to the Higgs mass and the electroweak precision observables
imply different scales for physics beyond the SM, being the first one below
$1$ TeV and the second one above $10$ TeV. This is the so called little
hierarchy problem. As it is well known the mass of the Higgs boson receives
one-loop corrections that are quadratic in the loop momenta. The largest
contributions come from the top quark loop, with smaller corrections coming
from loops of the electroweak gauge bosons and of the Higgs boson itself.
Cancellations between the top sector and other sectors must occur in order to
have the Higgs mass lighter than $200$ GeV as expected from the electroweak
precision test of the SM, which requires a fine-tuning of one part in 100. As
this situation is quite unnatural various theories and models have been
designed to solve this problem.

An interesting attempt to deal with it is the so called Littlest Higgs model
(LH) [2], inspired in an old suggestion by Georgi and Pais [3], which tries to
solve the little hierarchy problem by adding new particles with masses O(TeV)
and symmetries which protect the Higgs mass from those dangerous quadratically
divergent contributions (see [4] and [5] for reviews). These particles include
the Goldstone bosons (GB) corresponding to a global spontaneous symmetry
breaking (SSB) from the $SU(5)$ to the $SO(5)$ group, a new third generation
vector quark called $T$ and the gauge bosons corresponding to an additional
gauge group which contains at least a $SU(2)_{R}$ and eventually a new
hypercharge $U(1)$. In this case, and contrary to the supersymmetric theories,
cancellation occurs between same-statistics particles. However, LH models
typically leave uncanceled logarithmic divergencies which requires additional
new contributions at some higher scale to preserve a small Higgs boson mass.
Many of such models with different theory space have been constructed [2, 6],
and electroweak precision constraints on various little Higgs models have been
investigated by performing global fits to the precision data [7, 8, 9, 10,
11]. The existence of the different new states in these models could give rise
to a very rich phenomenology, which could be probed at the CERN Large Hadron
Collider (LHC) [12, 13].

Nevertheless, it is clear that any viable model has to fulfill the basic
requirement of reproducing the SM model at low energies. In particular, from
the LH model it is possible in principle to compute the Higgs low-energy
effective potential and then, by comparing with the SM potential, to obtain
their phenomenological consequences including new restrictions on the
parameter space of the LH model itself. For example, one can obtain the one-
loop contribution to the parameters of the standard Higgs potential,

$V=-\mu^{2}HH^{{\dagger}}+\lambda(HH^{{\dagger}})^{2};$ (1.1)

where $\mu^{2}$ and $\lambda$ denote the well known Higgs mass and Higgs self-
couplings parameters. Then it is possible to set restrictions over the LH
parameters space by imposing the condition $\mu^{2}=\lambda v^{2}$, where $v$
is the SM vacuum expectation value ($H=(0,v)/\sqrt{2}$). The $\mu^{2}$ sign
and value are well known [2, 13], and effectively they are the right ones to
produce the electroweak symmetry breaking, giving a Higgs mass
$m_{H}^{2}=2\mu^{2}$. However, the full expression for the radiative
corrections to $\lambda$ has not been analyzed in detail so far. In principle
both $\mu^{2}$ and $\lambda$ receive contributions from fermion, gauge boson
and scalar loops, besides others that could come from the ultraviolet
completion of the LH model. We have previously computed the contributions to
the Higgs effective potential in the LH model coming from the fermion sector
and the gauge boson sector [14, 15]. On the other hand, several relations for
the threshold corrections to the $\lambda$ parameter in the presence of a $10$
TeV cut-off, depending on the UV-completion of the theory, have been reported
(see, for example [17]). Besides, we have computed the effective potential for
the doublet Higgs and the triplet $\phi$ [16], coming from the fermionic and
gauge boson one-loop contributions and from the higher order effective
operators needed for the ultraviolet completion of the model.

In [14] and [15] we computed and analyzed the fermion contributions to the low
energy Higgs effective potential together with the effects of virtual heavy
and electroweak gauge bosons present in the LH model. We have illustrated in
these works the kind of constraints on the possible values of the LH
parameters that can be set by requiring the complete LH effective potential to
reproduce exactly the SM potential. The radiative corrections to $\lambda$, at
the one-loop level, had not been previously computed. The computation of
$\lambda$ is important for several reasons: First, it must be positive, for
the low energy effective action to make sense. In addition, from the effective
potential (1.1), one gets the simple formula $m^{2}_{H}=2\lambda v^{2}$ or,
equivalently, $\mu^{2}=\lambda v^{2}$, where $v$ is set by the experiment (for
instance from the muon lifetime) to be $v\simeq 245$ GeV. In our
phenomenological discussion in [14, 15] we have shown that the one-loop
effective potential of the LH model cannot reproduce the SM potential with a
low enough Higgs mass, $m^{2}_{H}=2\lambda v^{2}=2\mu^{2}$, in agrement with
the present experimental constraints.

In order to solve this problem we computed in [16] the effective potential for
the doublet Higgs and the triplet $\phi$; coming from the fermionic and gauge
boson one-loop contributions and also from the higher order effective
operators, as defined in [12]. The relevant terms of this effective potential
can be read as,

$\displaystyle V_{eff}(H,\phi)$ $\displaystyle=$
$\displaystyle-\mu_{fg}^{2}HH^{{\dagger}}+\lambda_{fg}(HH^{{\dagger}})^{2}$
(1.2)
$\displaystyle+\lambda_{\phi^{2}}f^{2}\mbox{tr}(\phi\phi^{{\dagger}})+i\lambda_{H^{2}\phi}f(H\phi^{{\dagger}}H^{T}-H^{*}\phi
H^{{\dagger}})\,,$

where $\mu_{fg}^{2}>0$ and $\lambda_{fg}>0$.

With this potential we studied the regions of the LH parameter space giving
rise to the SM electroweak symmetry breaking. Although radiative corrections
from fermion and gauge boson loops were discussed in [14, 15], the radiative
contributions to $\lambda_{\phi^{2}}$ and $\lambda_{H^{2}\phi}$ have not been
computed so far. New constraints over the LH parameter space emerge once we
impose the new relation between coefficients of the effective Higgs potential
namely;
$v^{2}={\mu_{fg}^{2}}/{\lambda_{fg}-\lambda_{H^{2}\phi}^{2}/\lambda_{\phi^{2}}}$.
In particular, the lowest value found for the $\mu$ parameter was $390$ GeV
[16], which implied a Higgs boson mass of about $m_{H}\simeq 550$ GeV, still
not compatible with the present experimental constraints.

On the other hand it is well known that the radiative corrections coming from
the Higgs itself and the $\phi$ fields could also provide relevant
contributions to the effective potential. Thus the main goal of the present
work is to check wether these corrections could really reduce the Higgs mass
to solve the above mentioned problem, making the LH model compatible with the
present phenomenology.

This work is organized as follows: In Section 2 we briefly explain the LH
model. A summary on the SSB and the mass eigenstates is presented in Section
3. We set the notation in the two aforementioned sections. Section 4 is
devoted to the computation of the radiative corrections contributions to the
Higgs mass and quartic coupling coming from the scalar sector loops. In
Section 5 we analyze the constraints that our computation establishes on the
LH parameters and, finally, in Section 6 we present the conclusions. The
expressions of the coefficients of the effective potential (1.2) coming from
the radiative corrections and the effective operators are listed in the
Appendix.

##  2 The model

The LH model is based on the assumption that there is a physical system with a
global $SU(5)$ symmetry that is spontaneously broken to a $SO(5)$ symmetry at
a high scale $\Lambda$ through a vacuum expectation value (v.e.v) of order
$f$. Thus, 14 Goldstone bosons (GB) are obtained as a consequence of this
breaking. In this work we will consider two different versions of the LH
model. In the first one the $SU(5)$ subgroup $[SU(2)\times U(1)]^{2}$ is
gauged. We refer to this version as _Model I_. In the second one the gauge
group is $[SU(2)^{2}\times U(1)]$ (_Model II_) [14, 15]. In both cases some of
the GB acquire masses through radiative corrections coming from the gauge
bosons and the $t$, $b$ and $T$ fermions loops.

The starting Lagrangian of the LH model is given by [2, 12, 13]:

$\textit{L}=\textit{L}_{\Sigma}+\textit{L}_{YK}$ (2.3)

where $\textit{L}_{\Sigma}$ is the Non Linear Sigma Model (NLSM) lagrangian:

$\textit{L}_{\Sigma}=\frac{f^{2}}{8}\mbox{tr}[(D_{\mu}\Sigma)(D^{\mu}\Sigma)^{\dagger}]\,;$
(2.4)

and $\textit{L}_{YK}$ the Yukawa couplings for fermions and scalars:

$\textit{L}_{YK}=-\frac{\lambda_{1}}{2}f\overline{u}_{R}\epsilon_{mn}\epsilon_{ijk}\Sigma_{im}\Sigma_{jn}\chi_{Lk}-\lambda_{2}f\overline{U}_{R}U_{L}+\mbox{h.c.}\,.$
(2.5)

In the above Lagrangians $\Sigma$ is the GB matrix given by:

$\Sigma=e^{2i\Pi/f}\Sigma_{0}$ (2.6)

where $\Sigma_{0}$ can be chosen to be:

$\Sigma_{0}=\left(\begin{array}[]{ccc}0&0&\textbf{1}\\\ 0&1&0\\\
\textbf{1}&0&0\\\ \end{array}\right)\,,$ (2.7)

with 1 being the $2\times 2$ unit matrix, and the $\Pi$ matrix can be
parameterized as:

$\displaystyle\Pi$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}0&\frac{-i}{\sqrt{2}}H^{{\dagger}}&\phi^{{\dagger}}\\\
\frac{i}{\sqrt{2}}H&0&\frac{-i}{\sqrt{2}}H^{*}\\\
\phi&\frac{i}{\sqrt{2}}H^{T}&0\\\ \end{array}\right),$ (2.11)

where$H=(H^{0},H^{+})$ is the SM Higgs doublet and $\phi$ is the triplet given
by:

$\phi=\left(\begin{array}[]{cc}\phi^{0}&\frac{1}{\sqrt{2}}\phi^{+}\\\
\frac{1}{\sqrt{2}}\phi^{+}&\phi^{++}\end{array}\right)\,.$ (2.12)

The covariant derivative $D_{\mu}$ is defined as:

Model I $\displaystyle D_{\mu}\Sigma$ $\displaystyle=$
$\displaystyle\partial_{\mu}\Sigma-i\sum_{k=1}^{2}g_{k}W^{a}_{k}(Q_{k}^{a}\Sigma+\Sigma
Q_{k}^{aT})-i\sum_{k=1}^{2}g^{\prime}_{k}B_{k}(Y_{k}\Sigma+\Sigma Y_{k}^{T})$
Model II $\displaystyle D_{\mu}\Sigma$ $\displaystyle=$
$\displaystyle\partial_{\mu}\Sigma-i\sum_{k=1}^{2}g_{k}W^{a}_{k}(Q_{k}^{a}\Sigma+\Sigma
Q_{k}^{aT})-ig^{\prime}B(Y\Sigma+\Sigma Y^{T})\,,$ (2.13)

where $g$ and $g^{\prime}$ are the gauge couplings, $W_{k}^{a}$ $(a=1,2,3)$
and $B_{k}\,,B$ are the $SU(2)$ and $U(1)$ gauge fields respectively,
$Q_{1ij}^{a}=\sigma_{ij}^{a}/2$ for $i,j=1,2$ and zero otherwise,
$Q_{2ij}^{a}=\sigma_{i-3,j-3}^{a*}/2$ for $i,j=4,5$ and zero otherwise,
$Y_{1}=diag(-3,-3,2,2,2)/10$, $Y_{2}=diag(-2,-2,-2,3,3)/10$ and
$Y=diag(-1,-1,0,1,1)/2$. The Yukawa Lagrangian in (2.5) describes the
interactions between GB and fermions, more exactly, the third generations of
quarks plus the extra $T$ quark appearing in the LH model. The indices in
$\textit{L}_{YK}$ are defined such that $m,n=4,5$, $i,j=1,2,3$, and

$\displaystyle\overline{u}_{R}$ $\displaystyle=$ $\displaystyle
c\,\overline{t}_{R}+s\,\overline{T}_{R}\,,$ $\displaystyle\overline{U}_{R}$
$\displaystyle=$ $\displaystyle-s\,\overline{t}_{R}+c\,\overline{T}_{R},$
(2.14)

with:

$\displaystyle c$ $\displaystyle=$
$\displaystyle\cos\theta=\frac{\lambda_{2}}{\sqrt{\lambda_{1}^{2}+\lambda_{2}^{2}}},$
$\displaystyle s$ $\displaystyle=$
$\displaystyle\sin\theta=\frac{\lambda_{1}}{\sqrt{\lambda_{1}^{2}+\lambda_{2}^{2}}}\,,$
(2.15)

and

$\chi_{L}=\left(\begin{array}[]{c}u\\\ b\\\ U\\\
\end{array}\right)_{L}=\left(\begin{array}[]{c}t\\\ b\\\ T\\\
\end{array}\right)_{L}.$ (2.16)

In addition to the above terms it is needed to add to the LH Lagrangian the
Yang-Mills terms corresponding to the various gauge fields, and also the gauge
fixing and Faddeev-Popov terms. Some of the gauge fields get massive at the
tree level through the Higgs mechanism associated to the $SU(5)/SO(5)$
symmetry breaking. By using the Landau gauge, which is the most appropriate
for the kind of computations we are presenting here (see [15] for further
details), the quadratic part of the complete gauge boson Lagrangian can be
written as:

$\textit{L}_{\Omega}=\frac{1}{2}\Omega^{\mu}((\Box+M_{\Omega}^{2})g_{\mu\nu}-\partial_{\mu}\partial_{\nu}+2\tilde{I}\,g_{\mu\nu})\Omega^{\nu}\,,$
(2.17)

where $\Omega$ stands for any of the gauge bosons,

Model I $\displaystyle\Omega^{\mu}=({W^{\prime}}^{\mu a},W^{\mu
a},{B^{\prime}}^{\mu},B^{\mu}),$ Model II
$\displaystyle\Omega^{\mu}=({W^{\prime}}^{\mu a},W^{\mu a},B^{\mu})\,,$ (2.18)

being the mass matrix eigenstates,

Model I $\displaystyle M_{\Omega}=(M_{W^{\prime}}1_{3\times 3},0_{3\times
3},M_{B^{\prime}},0),$ Model II $\displaystyle
M_{\Omega}=(M_{W^{\prime}}1_{3\times 3},0_{3\times 3},0)\,,$ (2.19)

with $M_{W^{\prime}}=f\sqrt{g_{1}^{2}+g_{2}^{2}}/2$ and
$M_{B^{\prime}}=f\sqrt{g_{1}^{{}^{\prime}2}+g_{2}^{{}^{\prime}2}}/\sqrt{20}$.
The gauge boson mass eigenstates are defined such as:

$\displaystyle W^{a}$ $\displaystyle=$ $\displaystyle
c_{\psi}W_{1}^{a}+s_{\psi}W_{2}^{a},$ $\displaystyle W^{{}^{\prime}a}$
$\displaystyle=$ $\displaystyle s_{\psi}W_{1}^{a}-c_{\psi}W_{2}^{a},$ (2.20)

where

$\displaystyle s_{\psi}$ $\displaystyle=$
$\displaystyle\sin\psi=\frac{g_{1}}{\sqrt{g_{1}^{2}+g_{2}^{2}}},$
$\displaystyle c_{\psi}$ $\displaystyle=$
$\displaystyle\cos\psi=\frac{g_{2}}{\sqrt{g_{1}^{2}+g_{2}^{2}}},$ (2.21)

and

$\displaystyle B$ $\displaystyle=$ $\displaystyle
c^{\prime}_{\psi}B_{1}+s^{\prime}_{\psi}B_{2},$ $\displaystyle B^{\prime}$
$\displaystyle=$ $\displaystyle
s^{\prime}_{\psi}B_{1}-c^{\prime}_{\psi}B_{2},$ (2.22)

with

$\displaystyle
s^{\prime}_{\psi}=\sin\psi^{\prime}=\frac{g^{\prime}_{1}}{\sqrt{{g^{\prime}}_{1}^{\,2}+{g^{\prime}}_{2}^{\,2}}},$
$\displaystyle
c^{\prime}_{\psi}=\cos\psi^{\prime}=\frac{{g^{\prime}}_{2}}{\sqrt{{g^{\prime}}_{1}^{\,2}+{g^{\prime}}_{2}^{\,2}}}\,.$
(2.23)

$\tilde{I}$ is the interaction matrix between the gauge bosons and the $H$ and
$\phi$ scalars which can be found in our previous works [15, 16].

By adding the appropriate kinetic terms, the complete Lagrangian for the
quarks becomes:

$\displaystyle\textit{L}_{\chi}=\overline{\chi}_{R}(i{\partial\mkern-9.0mu/}-M+\hat{I})\chi_{L}+\mbox{h.c.}\,,$
(2.24)

where

$\chi_{R}=\left(\begin{array}[]{c}t\\\ b\\\ T\\\ \end{array}\right)_{R}\,,$

$M=$diag$(0,0,m_{T})$ with $m_{T}=f\sqrt{\lambda_{1}^{2}+\lambda_{2}^{2}}$ and
$\hat{I}$ is the scalar-quark interaction matrix. The elements of this matrix
can be found in [14, 16]. For more details about the model, including Feynman
rules and also some phenomenological results see for example [12].

##  3 Effective operators

It is well known that the effective Higgs potential receive also contributions
from additional operators coming from the ultraviolet completion of the LH
model. Obviously these operators must be consistent with the symmetries of the
LH model [2, 12, 18]. At the lowest order they can be parameterized by two
unknown coefficients $a$ and $a^{\prime}$ $\sim O(1)$. The form of these
effective operators is, for the fermion sector [12]:

$\textit{O}_{f}=-a^{\prime}\frac{1}{4}\lambda_{1}^{2}f^{4}\epsilon^{wx}\epsilon_{yz}\epsilon^{ijk}\epsilon_{kmn}\Sigma_{iw}\Sigma_{jx}\Sigma^{*my}\Sigma^{*nz}\,,$
(3.25)

where $i,j,k,m,n$ run over 1,2,3 and $w,x,y,z$ run over 4,5 and for the gauge
sector we have for _Model I_ :

$\displaystyle\textit{O}_{gb}=\frac{1}{2}af^{4}\left\\{g_{j}^{2}\sum_{a=1}^{3}\mbox{Tr}\left[(Q_{j}^{a}\Sigma)(Q_{j}^{a}\Sigma)^{*}\right]+g_{j}^{{}^{\prime}2}\mbox{Tr}\left[(Y_{j}\Sigma)(Y_{j}\Sigma)^{*}\right]\right\\}\,,$
(3.26)

with $j=1,2$ and $Q_{j}^{a}$ and $Y_{j}$ being the generators of the
$SU(2)_{j}$ and $U(1)_{j}$ groups, respectively. In the case of _Model II_ :

$\displaystyle\textit{O}_{gb}=\frac{1}{2}cf^{4}\left\\{g_{j}^{2}\sum_{a=1}^{3}\mbox{Tr}\left[(Q_{j}^{a}\Sigma)(Q_{j}^{a}\Sigma)^{*}\right]+g^{{}^{\prime}2}\mbox{Tr}\left[(Y\Sigma)(Y\Sigma)^{*}\right]\right\\}\,,$
(3.27)

where $j=1,2$ and $Y$ is the generator of the unique $U(1)$ group.

By expanding the GB field matrix $\Sigma$ in these effective operators, we
obtain their different contributions to the coefficients of the effective
potential (1.2). The results are presented in the Appendix.

The complete result for the coefficients of the Higgs potential is given by
the sum of the contributions coming from the effective operators, as given
above, and the radiative contributions coming from all sectors of the model,
as will be discussed in the following.

##  4 SSB and mass eigenstates

In the LH model the electroweak symmetry breaking is triggered, in principle,
by the Higgs potential generated by one-loop radiative corrections, including
both, fermion and gauge boson loops, and the effective operators introduced in
the previous section. Obviously, this potential is invariant under the
electroweak gauge group $SU(2)\times U(1)$ and also should have the correct
form to break this symmetry spontaneously to $U(1)_{em}$. The relevant terms
for this work are given in (1.2). Quartic terms involving $\phi^{4}$ and
$H^{2}\phi^{2}$ are not included since they give subleading contributions to
the Higgs mass. These parameters were computed in our previous works [14, 15,
16] and are given in the Appendix for completeness.

The scalar potential, as given in (1.2), reaches its minimum at:

$\langle HH^{{\dagger}}\rangle={v^{2}}/{2}$ and
$\,\,\langle\phi\phi^{{\dagger}}\rangle={v^{\prime}}^{2}$ with:

$v^{2}=\frac{\mu_{fg}^{2}}{\lambda_{fg}-\lambda_{H^{2}\phi}^{2}/\lambda_{\phi^{2}}},\hskip
56.9055pt{v^{\prime}}^{2}=\frac{\lambda_{H^{2}\phi}}{\sqrt{2}\lambda_{\phi^{2}}}\frac{v^{2}}{f}.$
(4.28)

Note that both, the doublet and triplet scalars, get a v.e.v., $v$ and
$v^{\prime}$ respectively. A standard choice for the components of these
fields at the vacuum is:

$\displaystyle H^{+}=0,\hskip 28.45274ptH_{0}=\frac{v}{\sqrt{2}},\hskip
28.45274pt\phi_{0}=-v^{\prime},\hskip 28.45274pt\phi^{+}=\phi^{++}=0.$ (4.29)

Then $H$ and $\phi$ can be parameterized as:

$\displaystyle H=(w^{+},\frac{1}{\sqrt{2}}(v+h+iw_{0}))\hskip
8.5359pt\mbox{and}\hskip
8.5359pt\phi=\left(\begin{array}[]{cc}-v^{\prime}+\frac{1}{\sqrt{2}}(\xi+i\rho)&\frac{1}{\sqrt{2}}\phi^{+}\\\
\frac{1}{\sqrt{2}}\phi^{+}&\phi^{++}\end{array}\right).$ (4.32)

Obviously the new fields describe fluctuations around the vacuum and the
potential written in terms of them can be split in four sectors, namely, the
scalar, the pseudoscalar, the charged and the doubly charged. For the first
three sectors we find that the new fields are not mass eigenstates. By
diagonalizing the corresponding mass matrices we obtain the mass eigenstates
in each case. I.e., for the scalar sector:

$\displaystyle h$ $\displaystyle=$ $\displaystyle
c_{0}\mathcal{H}+s_{0}\Phi_{0},\hskip 56.9055ptm^{2}_{\mathcal{H}}\equiv
m^{2}_{fg}=2\,\mu_{fg}^{2},$ $\displaystyle\xi$ $\displaystyle=$
$\displaystyle c_{0}\Phi_{0}-s_{0}\mathcal{H},\hskip
56.9055ptm^{2}_{\Phi_{0}}=M_{\phi}^{2}+2\,m^{2},$ (4.33)

the pseudoscalar sector:

$\displaystyle w_{0}$ $\displaystyle=$ $\displaystyle
c_{P}G^{0}+s_{P}\Phi^{P},\hskip 56.9055ptm^{2}_{G^{0}}=0,$ $\displaystyle\rho$
$\displaystyle=$ $\displaystyle c_{P}\Phi^{P}-s_{P}G^{0},\hskip
56.9055ptm^{2}_{\Phi^{P}}=M_{\phi}^{2}+2m^{2},$ (4.34)

and the charged sector:

$\displaystyle w^{+}$ $\displaystyle=$ $\displaystyle
c_{+}G^{+}+s_{+}\Phi^{+},\hskip 42.67912ptm^{2}_{G^{+}}=0,$
$\displaystyle\phi^{+}$ $\displaystyle=$ $\displaystyle
c_{+}\Phi^{+}+s_{+}G^{+},\hskip
42.67912ptm^{2}_{\Phi^{+}}=M_{\phi}^{2}+m^{2},$ (4.35)

with $M_{\phi}^{2}=\lambda_{\phi^{2}}f^{2}$,
$m^{2}=v^{2}\lambda_{H^{2}\phi}^{2}/\lambda_{\phi^{2}}$. The doubly charged
sector remains unchanged with a mass $M_{\phi}$.

Where the notation introduced for the mass eigenstates is the following:
$\mathcal{H}$ and $\Phi_{0}$ are neutral scalars, $\Phi^{P}$ is a neutral
pseudoscalar, $\Phi^{+}$ and $\Phi^{++}$ are the charged and doubly charged
scalars, and $G^{+}$ and $G^{0}$ are the would-be Goldstone bosons
corresponding to the SM $W$ and $Z$.

In terms of the mass eigenstates the leading order in the
$\mathcal{O}(v^{2}/f^{2})$ expansion of the potential is given by:

$\displaystyle V_{eff}$ $\displaystyle=$
$\displaystyle\frac{1}{2}m_{fg}^{2}\mathcal{H}^{2}+\frac{1}{2}m_{\Phi_{0}}^{2}\Phi_{0}^{2}+\frac{1}{2}m_{\Phi^{p}}^{2}{\Phi^{P}}^{2}$
$\displaystyle+$ $\displaystyle
m_{\Phi^{+}}^{2}\Phi^{+}\Phi^{-}+v\lambda_{fg}\mathcal{H}^{3}+v\lambda_{fg}{G^{0}}^{2}\mathcal{H}+2v\lambda_{fg}{G^{+}}{G^{-}}\mathcal{H}$
$\displaystyle+$
$\displaystyle\frac{\lambda_{fg}}{4}\mathcal{H}^{4}+\frac{\lambda_{fg}}{2}\mathcal{H}^{2}{G^{0}}^{2}+\lambda_{fg}\mathcal{H}^{2}{G^{+}}{G^{-}}$
$\displaystyle-$
$\displaystyle\frac{\lambda_{H^{2}\phi}}{\sqrt{2}}f\mathcal{H}^{2}\Phi_{0}-\sqrt{2}\lambda_{H^{2}\phi}f\mathcal{H}{G^{0}}\Phi^{P}-\lambda_{H^{2}\phi}f(\mathcal{H}{G^{-}}\Phi^{+}+\mathcal{H}{G^{+}}\Phi^{-})+...$

##  5 Goldstone boson sector contributions

The objective of this section is the computation of the radiative
contributions to the Higgs mass and the Higgs quartic coupling coming from the
GB sector. The relevant Lagrangian is given by:

$\displaystyle\textit{L}_{GB}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\partial_{\mu}\Pi)(\partial^{\mu}\Pi)+\frac{1}{f^{2}}\left((\partial_{\mu}\Pi)(\partial^{\mu}\Pi)\Pi\Pi+\Pi(\partial_{\mu}\Pi)\Pi(\partial^{\mu}\Pi)\right)-V_{eff}.$

In order to calculate the radiative contributions we write this Lagrangian in
terms of the mass eigenstates and we split the Higgs field as
$\mathcal{H}=\mathcal{\overline{H}}+\mathcal{\tilde{H}}$ where
$\mathcal{\overline{H}}$ is the vacuum field and $\mathcal{\tilde{H}}$
describes the field fluctuations around this point. Then the first two terms
of the Lagrangian above become:

$\displaystyle\textit{L}_{Kin}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(1+2\frac{\mathcal{\overline{H}}^{2}}{f^{2}}\right)(\partial_{\mu}\mathcal{\tilde{H}})(\partial^{\mu}\mathcal{\tilde{H}})+\frac{1}{2}\left(1+\frac{\mathcal{\overline{H}}^{2}}{2f^{2}}\right)(\partial_{\mu}\Phi_{0})(\partial^{\mu}\Phi_{0})$
(5.38) $\displaystyle+$
$\displaystyle\frac{1}{2}\left(1+\frac{\mathcal{\overline{H}}^{2}}{2f^{2}}\right)(\partial_{\mu}{G^{0}})(\partial^{\mu}{G^{0}})+\frac{1}{2}\left(1+\frac{\mathcal{\overline{H}}^{2}}{2f^{2}}\right)(\partial_{\mu}\Phi^{P})(\partial^{\mu}\Phi^{P})$
$\displaystyle+$
$\displaystyle\left(1+\frac{\mathcal{\overline{H}}^{2}}{4f^{2}}\right)(\partial_{\mu}\Phi^{+})(\partial^{\mu}\Phi^{-})+\left(1+\frac{\mathcal{\overline{H}}^{2}}{2f^{2}}\right)(\partial_{\mu}{G^{+}})(\partial^{\mu}{G^{-}})$
$\displaystyle+$
$\displaystyle(\partial_{\mu}\Phi^{++})(\partial^{\mu}\Phi^{--}).$

Obviously, all the kinetic terms in this formula, but the last one, are not
properly normalized. Therefore we write the fields in terms of a new set of
properly normalized fields up to order $1/f^{2}$ as:

$\displaystyle\Upsilon$ $\displaystyle=$
$\displaystyle\left(1-\frac{\mathcal{\overline{H}}^{2}}{4f^{2}}\right)\Upsilon^{\prime}\hskip
28.45274pt\mbox{with}\hskip
28.45274pt\Upsilon^{(^{\prime})}={G^{0(^{\prime})}},G^{\pm(^{\prime})},\Phi_{0}^{(^{\prime})},\Phi^{P(^{\prime})},$
(5.39) $\displaystyle\mathcal{\tilde{H}}$ $\displaystyle=$
$\displaystyle\left(1-\frac{\mathcal{\overline{H}}^{2}}{f^{2}}\right)\mathcal{H}^{\prime},$
(5.40) $\displaystyle\Phi^{\pm}$ $\displaystyle=$
$\displaystyle\left(1-\frac{\mathcal{\overline{H}}^{2}}{8f^{2}}\right)\Phi^{\pm^{\prime}},$
(5.41)

so that the Lagrangian is just:

$\displaystyle\textit{L}_{Kin}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\partial_{\mu}\mathcal{H}^{\prime}\partial^{\mu}\mathcal{H}^{\prime}+\frac{1}{2}\partial_{\mu}\Phi_{0}^{\prime}\partial^{\mu}\Phi_{0}^{\prime}+\frac{1}{2}\partial_{\mu}{G^{0}}^{\prime}\partial^{\mu}{G^{0}}^{\prime}+\frac{1}{2}\partial_{\mu}{\Phi^{P}}^{\prime}\partial^{\mu}{\Phi^{P}}^{\prime}$
(5.42) $\displaystyle+$
$\displaystyle\partial_{\mu}G^{+^{\prime}}\partial^{\mu}G^{-^{\prime}}+\partial_{\mu}\Phi^{+^{\prime}}\partial^{\mu}\Phi^{-^{\prime}}+\partial_{\mu}\Phi^{++}\partial^{\mu}\Phi^{--},$

Then, the effective potential $V_{eff}$ is given by:

$\displaystyle
V_{eff}=\frac{1}{2}m_{fg}^{2}\mathcal{\overline{H}}^{2}+\frac{\lambda_{fg}}{4}\mathcal{\overline{H}}^{4}+V_{eff}^{ss}+V_{eff}^{ps}+V_{eff}^{cs}+...$
(5.43)

where

$\displaystyle V_{eff}^{ss}$ $\displaystyle=$
$\displaystyle\frac{1}{2}m_{\Phi_{0}}^{2}\Phi_{0}^{{}^{\prime}2}+\frac{1}{2}m_{fg}^{2}\mathcal{H}^{{}^{\prime}2}+\frac{3}{2}\lambda_{fg}\overline{\mathcal{H}}^{2}\mathcal{H}^{{}^{\prime}2}-\frac{\lambda_{\phi^{2}}}{4}\overline{\mathcal{H}}^{2}\Phi_{0}^{{}^{\prime}2}$
(5.44) $\displaystyle-$
$\displaystyle\sqrt{2}\lambda_{H^{2}\phi}f\overline{\mathcal{H}}\mathcal{H}^{\prime}\Phi_{0}^{\prime}-\frac{\lambda_{H^{2}\phi}}{\sqrt{2}}f\overline{\mathcal{H}}^{2^{\prime}}\Phi_{0}^{\prime},$
$\displaystyle V_{eff}^{ps}$ $\displaystyle=$
$\displaystyle\frac{1}{2}m_{\Phi^{p}}^{2}{\Phi^{P}}^{{}^{\prime}2}+\frac{\lambda_{fg}}{2}\overline{\mathcal{H}}^{2}{G^{0}}^{{}^{\prime}2}-\frac{\lambda_{\phi^{2}}}{4}\overline{\mathcal{H}}^{2}{\Phi^{P}}^{{}^{\prime}2}$
(5.45) $\displaystyle-$
$\displaystyle\sqrt{2}\lambda_{H^{2}\phi}f\overline{\mathcal{H}}{G^{0}}^{\prime}{\Phi^{P}}^{\prime},$
$\displaystyle V_{eff}^{cs}$ $\displaystyle=$ $\displaystyle
m_{\Phi^{+}}^{2}\Phi^{{}^{\prime}+}\Phi^{{}^{\prime}-}+\lambda_{fg}\overline{\mathcal{H}}^{2}G^{{}^{\prime}+}G^{{}^{\prime}-}-\frac{\lambda_{\phi^{2}}}{4}\overline{\mathcal{H}}^{2}\Phi^{{}^{\prime}+}\Phi^{{}^{\prime}-}$
(5.46) $\displaystyle-$
$\displaystyle\lambda_{H^{2}\phi}f\overline{\mathcal{H}}(G^{{}^{\prime}-}\Phi^{{}^{\prime}+}+G^{{}^{\prime}+}\Phi^{{}^{\prime}-}),$

Observe that the third terms in (5.44), (5.45) and (5.46) describe the new
interactions which come from the new normalization of the fields and the fact
that the triplet boson mass is $\mathcal{O}(f^{2})$. These interactions play a
decisive role to cancel the quadratic divergences that come from the GB loops.

Finally, we can see that the split into different scalar sectors is maintained
after diagonalization and normalization. This fact is important in order to
simplify the computation of the radiative contributions coming from the GB.
Thus we can deal with each scalar sector in an independent way being the
computations in all cases similar. We illustrate this by computing the
($\mathcal{H^{\prime}},\Phi_{0}^{\prime}$) contribution and then we apply the
same method to the other scalars.

###  5.1 Scalar sector contribution

The Lagrangian for the scalar sector
($\mathcal{H^{\prime}},\Phi_{0}^{\prime}$) is given by:

$\displaystyle\mathcal{L}^{ss}(\overline{\mathcal{H}},\mathcal{H}^{\prime},\Phi_{0}^{\prime})$
$\displaystyle=$
$\displaystyle\frac{1}{2}\partial_{\mu}\mathcal{H}^{\prime}\partial^{\mu}\mathcal{H}^{\prime}+\frac{1}{2}\partial_{\mu}\Phi_{0}^{\prime}\partial^{\mu}\Phi_{0}^{\prime}-V_{eff}^{ss}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\partial_{\mu}\mathcal{H}^{\prime}\partial^{\mu}\mathcal{H}^{\prime}+\frac{1}{2}\partial_{\mu}\Phi_{0}^{\prime}\partial^{\mu}\Phi_{0}^{\prime}-\frac{1}{2}m_{fg}^{2}\mathcal{H}^{{}^{\prime}2}-\frac{1}{2}m_{\Phi_{0}}^{2}\Phi_{0}^{{}^{\prime}2}$
$\displaystyle-$
$\displaystyle\frac{3}{2}\lambda_{fg}\overline{\mathcal{H}}^{2}\mathcal{H}^{{}^{\prime}2}+\frac{\lambda_{\phi^{2}}}{4}\overline{\mathcal{H}}^{2}\Phi_{0}^{{}^{\prime}2}+\sqrt{2}f\lambda_{H^{2}\phi}\overline{\mathcal{H}}\mathcal{H}^{\prime}\Phi_{0}^{\prime}+\frac{\lambda_{H^{2}\phi}}{\sqrt{2}}\overline{\mathcal{H}}^{2}\Phi_{0}^{\prime}.$

The effective action for the $\overline{\mathcal{H}}$ is:

$e^{iS_{eff}[\overline{\mathcal{H}}]}=\int[d\mathcal{H}^{\prime}][d\Phi_{0}^{\prime}]e^{i\int
dx\mathcal{L}^{ss}},$ (5.48)

From the (5.1) we observe that the integration can be computed in two steps:
First we concentrate on the $\Phi_{0}^{\prime}$ field and then we integrate
the $\mathcal{H^{\prime}}$ field. After integrating $\Phi_{0}^{\prime}$ we get
the $\mathcal{H}$ effective action:

$\displaystyle S_{eff}^{ss}[\overline{\mathcal{H}},\mathcal{H^{\prime}}]$
$\displaystyle=$
$\displaystyle-\frac{i}{2}\mbox{Tr}\log\left[1+G_{\Phi_{0}}\frac{\lambda_{\phi^{2}}}{2}\overline{\mathcal{H}}^{2}\right]$
(5.49) $\displaystyle-$ $\displaystyle f^{2}\lambda_{H^{2}\phi}^{2}\int
dxdy\overline{\mathcal{H}}^{2}\mathcal{H}^{\prime}_{x}G_{\Phi_{0}xy}\mathcal{H}^{\prime}_{y}-\frac{\lambda_{H^{2}\phi}^{2}}{4}f^{2}\int
dxdyG_{\Phi_{0}xy}\overline{\mathcal{H}}^{4}\delta_{yx}$ $\displaystyle=$
$\displaystyle-\frac{i}{2}\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k}\mbox{Tr}\left(G_{\Phi_{0}}\frac{\lambda_{\phi^{2}}}{2}\overline{\mathcal{H}}^{2}\right)^{k}+\tilde{I}_{2}+\tilde{I}_{4}\,,$

where the $\Phi_{0}^{\prime}$ propagator is given by:

$G_{\Phi_{0}}(x,y)=\int
d\tilde{k}e^{ik(x-y)}\frac{1}{k^{2}-m_{\Phi_{0}}^{2}},$ (5.50)

here $\tilde{k}\equiv d^{4}k/(2\pi)^{4}$, and

$\displaystyle\tilde{I}_{2}$ $\displaystyle=$
$\displaystyle-f^{2}\lambda_{H^{2}\phi}^{2}\int
dxdy\overline{\mathcal{H}}^{2}\mathcal{H}^{\prime}_{x}G_{\Phi_{0}xy}\mathcal{H}^{\prime}_{y}\,,$
(5.51) $\displaystyle\tilde{I}_{4}$ $\displaystyle=$
$\displaystyle-\frac{\lambda_{H^{2}\phi}^{2}}{4}f^{2}\int
dxdy\overline{\mathcal{H}}^{4}\delta_{xy}G_{\Phi_{0}xy}\,.$ (5.52)

Observe that we have obtained three terms. The first and the third ones are
$\mathcal{H}^{\prime}$ independent and they will give the $\Phi_{0}^{\prime}$
radiative contributions to the Higgs mass and the quartic coupling.

Now integrating out $\mathcal{H}^{\prime}$ we find its contribution to the
$\overline{\mathcal{H}}$ effective action:

$\displaystyle S^{ss}[\overline{\mathcal{H}}]$ $\displaystyle=$
$\displaystyle-\frac{i}{2}\mbox{Tr}\log\left[1+G_{\mathcal{H}^{\prime}}(-3\lambda_{fg}\overline{\mathcal{H}}^{2}+2\tilde{I}_{2})\right]$
(5.53) $\displaystyle=$
$\displaystyle\frac{i}{2}\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k}\mbox{Tr}\left(G_{\mathcal{H}}(3\lambda_{fg}\overline{\mathcal{H}}^{2}-2\tilde{I}_{2})\right)+...\,,$

where $G_{\mathcal{H}^{\prime}}$ is the $\mathcal{H}^{\prime}$ propagator,

$G_{\mathcal{H}^{\prime}}(x,y)=\int
d\tilde{k}e^{ik(x-y)}\frac{1}{k^{2}-m_{fg}^{2}}\,.$ (5.54)

Finally, by taking into account (5.49) and (5.53), we obtain the
$\overline{{\mathcal{H}}}$ effective action which reads:

$\displaystyle S^{ss}[\overline{\mathcal{H}}]$ $\displaystyle=$
$\displaystyle-\frac{i}{2}\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k}\mbox{Tr}\left(G_{\Phi_{0}}\frac{\lambda_{\phi^{2}}}{2}\overline{\mathcal{H}}^{2}\right)^{k}$
$\displaystyle+\frac{i}{2}\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k}\mbox{Tr}(G_{\mathcal{H}^{\prime}}(3\lambda_{fg}\overline{\mathcal{H}}^{2}+\tilde{I}_{2}))^{k}+\tilde{I}_{4}.$

In order to obtain the scalar contribution to the Higgs mass we only need to
consider the $k=1$ term in the expansion (5.1). The generic loop diagrams are
shown in Fig. 1. Then, for $k=1$,

$\displaystyle S^{(1)ss}[\mathcal{\overline{H}}]$ $\displaystyle=$
$\displaystyle-\frac{i}{2}\lambda_{\phi^{2}}\int
dxdy(G_{\Phi_{0}xy}\overline{\mathcal{H}}^{2}\delta_{yx})+\frac{i}{2}\int
dxdyG_{\mathcal{H}^{\prime}xy}(3\lambda_{fg}\overline{\mathcal{H}}^{2}\delta_{yx}+\tilde{I}_{2}\delta_{yx})$
(5.56) $\displaystyle=$ $\displaystyle-\frac{i}{4}\lambda_{\phi^{2}}\int
dx\overline{\mathcal{H}}^{2}I_{0}(m_{\Phi_{0}}^{2})+\frac{3}{2}i\lambda_{fg}\int
dx\overline{\mathcal{H}}^{2}I_{0}(m_{fg}^{2})$
$\displaystyle+i\lambda_{H^{2}\phi}^{2}f^{2}\int
dx\overline{\mathcal{H}}^{2}I_{3}(m_{\Phi_{0}}^{2},m_{fg}^{2})\,,$

with

$\displaystyle I_{0}(M^{2})$ $\displaystyle\equiv$ $\displaystyle\int
d\tilde{k}\frac{i}{(k^{2}-M^{2})}=\frac{1}{(4\pi)^{2}}\left[\Lambda^{2}-M^{2}\log\left(1+\frac{\Lambda^{2}}{M^{2}}\right)\right],$
(5.57) $\displaystyle I_{3}(M_{a}^{2},M_{b}^{2})$ $\displaystyle\equiv$
$\displaystyle\int d\tilde{p}\frac{i}{(p^{2}-M_{a}^{2})(p^{2}-M_{b}^{2})}$
$\displaystyle=$
$\displaystyle-\frac{1}{(4\pi)^{2}}\frac{1}{M_{a}^{2}-M_{b}^{2}}\left[M_{a}^{2}\log\left(1+\frac{\Lambda^{2}}{M_{a}^{2}}\right)-M_{b}^{2}\log\left(1+\frac{\Lambda^{2}}{M_{b}^{2}}\right)\right].$

$\Gamma_{L},~{}\Gamma_{H}$$\mathcal{\overline{H}}$$\mathcal{\overline{H}}$$\Gamma_{L}$$\Gamma_{H}$$\mathcal{\overline{H}}$$\mathcal{\overline{H}}$(a)$\mathcal{\overline{H}}$$\mathcal{\overline{H}}$$\mathcal{\overline{H}}$$\mathcal{\overline{H}}$(b)

Figure 1: (a) Scalar sector loops contributing to the Higgs mass.
$\Gamma_{L}=\mathcal{H}^{\prime},{G^{0}}^{\prime}$ or $G^{\pm^{\prime}}$ and
$\Gamma_{H}=\Phi_{0}^{\prime},\Phi^{P}$ or $\Phi^{\pm^{\prime}}$. (b)
Contribution to the Higgs quartic coupling from the $\Phi_{0}^{\prime}$
propagator.

For the quartic coupling Higgs correction coming from $\tilde{I}_{4}$ we have:

$\tilde{I}_{4}=\frac{\lambda_{H^{2}\phi}^{2}}{4\lambda_{\phi^{2}}}\int
dx\overline{\mathcal{H}}^{4}+...$ (5.59)

where we have expanded the $\Phi_{0}^{\prime}$ propagator in powers of
$k^{2}/m_{\Phi_{0}^{\prime}}^{2}$ and kept just the first term.

###  5.2 Pseudoscalar sector and charged sector contributions

The computation of the contributions from the pseudoscalar and charged sectors
is similar to the previous ones with only one difference, i.e.: these sectors
do not give a contribution to the Higgs quartic coupling. They just contribute
to the Higgs mass. Then the results for the pseudoscalar sector are:

$\displaystyle S^{(1)ps}[\mathcal{\overline{H}}]$ $\displaystyle=$
$\displaystyle-\frac{i}{4}\lambda_{\phi^{2}}\int
dx\overline{\mathcal{H}}^{2}I_{0}(m_{\Phi^{p}}^{2})+\frac{1}{2}i\lambda_{fg}\int
dx\overline{\mathcal{H}}^{2}I_{0}(0)$ (5.60) $\displaystyle+$ $\displaystyle
i\lambda_{H^{2}\phi}^{2}f^{2}\int
dx\overline{\mathcal{H}}^{2}I_{3}(m_{\Phi^{p}}^{2},0).$

and for the charged sector:

$\displaystyle S^{(1)cs}[\mathcal{\overline{H}}]$ $\displaystyle=$
$\displaystyle-\frac{i}{4}\lambda_{\phi^{2}}\int
dx\overline{\mathcal{H}}^{2}I_{0}(m_{\Phi^{+}}^{2})+i\lambda_{fg}\int
dx\overline{\mathcal{H}}^{2}I_{0}(0)$ (5.61) $\displaystyle+$ $\displaystyle
i\lambda_{H^{2}\phi}^{2}f^{2}\int
dx\overline{\mathcal{H}}^{2}I_{3}(m_{\Phi^{+}}^{2},0).$

Notice the there is no contribution coming from the doubly charged scalar
sector.

###  5.3 Analytical results

Now by adding (5.56), (5.60), (5.61) we obtain the total radiative corrections
to the Higgs mass from the GB sector up to order $\mathcal{O}(v^{2}/f^{2})$
which reads:

$\displaystyle\Delta m_{GB}^{2}$ $\displaystyle=$
$\displaystyle\frac{3}{(4\pi)^{2}}\left\\{\left(-\frac{\lambda_{\phi^{2}}}{4}+\lambda_{fg}\right)\Lambda^{2}+\left(\frac{\lambda_{\phi^{2}}}{4}+\frac{\lambda_{H^{2}\phi}^{2}}{\lambda_{\phi^{2}}}\right)M_{\phi}^{2}\log\left(1+\frac{\Lambda^{2}}{M_{\phi}^{2}}\right)\right.$
(5.62)
$\displaystyle\left.-\frac{1}{2}\lambda_{fg}m_{fg}^{2}\log\left(1+\frac{\Lambda^{2}}{m_{fg}^{2}}\right)\right\\}\,,$

where, in order to simplify the computations, we have considered the heavy
scalar fields as degenerate since $m^{2}/M_{\phi}^{2}$ is of the order of
$\mathcal{O}(v^{2}/f^{2})$ (see eq. (4)).

The coefficients of the Higgs potential $\lambda_{fg},\lambda_{\phi^{2}}$ and
$\lambda_{H^{2}\phi}^{2}$ appearing in eq. (5.62) receive contributions from
both the radiative corrections and the effective operators (see Appendix).
Since the contributions to $\lambda_{fg}$ and $\lambda_{\phi^{2}}$ contain
terms of the order of $\Lambda^{2}$, divergencies $\mathcal{O}(\Lambda^{4})$
and $\mathcal{O}(\Lambda^{2})$ emerge from the first term in (5.62). However,
these divergencies cancel due to the relationship between $\lambda_{\phi^{2}}$
and $\lambda_{fg}$, namely:

$\displaystyle\lambda_{fg}^{\Lambda^{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\lambda_{\phi^{2}}^{\Lambda^{2}},$
$\displaystyle\lambda_{fg}^{EO}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\lambda_{\phi^{2}}^{EO},$ (5.63)

where the index $\Lambda^{2}$ refers to the quadratically divergent terms and
$EO$ represents the part of these coefficients coming from the effective
operators. This fact occurs in the fermionic and gauge boson sectors, where
the quadratic divergences coming from light and heavy modes of the same
statistics cancel [2]. Then the corrections summarized in $\Delta m_{GB}^{2}$
(eq. 5.62) are at most of the order
$\mathcal{O}(\Lambda^{2}\log(\Lambda^{2}/M^{2}))$. It is important to stress
that the above cancellations occur exactly only in _Model I_ (as you can
easily check from the results given in the Appendix). However, in _Model II_
(where only the $SU(2)\times SU(2)\times U(1)$ is gauged), there are
$\mathcal{O}(\Lambda^{2})$ terms coming from the U(1) sector which do not
cancel. However, such terms appear always with a squared gauge coupling
$g^{\prime}$ factor which is very small ($g^{\prime 2}/g^{2}\sim 0.3$ in the
SM) and then their contribution is not expected to be too large.

Finally, from (5.59), the radiative correction to the quartic coupling is:

$\displaystyle\tilde{I}_{4}=\frac{1}{4}\Delta\lambda_{GB}\int
dx\overline{\mathcal{H}}^{4}\,,$

being

$\displaystyle\Delta\lambda_{GB}=\frac{\lambda_{H^{2}\phi}^{2}}{\lambda_{\phi^{2}}}.$
(5.64)

In summary, taking into account (5.59) and (5.62), the Higgs boson potential
can be written as:

$V=\frac{1}{2}m_{\overline{\mathcal{H}}}^{2}{\overline{\mathcal{H}}}^{2}+\frac{1}{4}\lambda_{\overline{\mathcal{H}}}{\overline{\mathcal{H}}}^{4},$
(5.65)

where the Higgs mass is given by,

$m_{\overline{\mathcal{H}}}^{2}=2(\mu_{fg}^{2}-\Delta m_{GB}^{2}),$ (5.66)

and the quartic Higgs couplings is,

$\lambda_{\overline{\mathcal{H}}}=\lambda_{fg}-\frac{\lambda_{H^{2}\phi}^{2}}{\lambda_{\phi^{2}}}.$
(5.67)

It is important to note that we have obtained the GB contributions after
having broken the SM symmetry through the fermion and gauge boson radiative
corrections. In this fact we differ from other analysis performed in the
literature (see for example [2, 18]), where these scalar contributions are
computed at the tree level from the effective operators only. Moreover, in our
case, the coefficients of the potential (1.2) do not depend only on the two
unknown coefficients $a$ and $a^{\prime}$, but also on the scale $f$ and the
cutoff $\Lambda$, thus setting more restrictions on the space parameter as we
will see in the following.

##  6 Numerical Results and Phenomenological Discussion

In this section we continue our study about the allowed region of the
parameter space of the LH model started in our previous papers [14, 15, 16].
In the present one we complete this phenomenological study, taking into
account also the contributions from the Goldstone boson sector to the Higgs
mass and quartic coupling obtained above. The LH parameters different
relationships and their relevant ranges considered are the following:

First, we impose the minimum condition for the complete effective potential
(1.2):

$v^{2}=\frac{\mu_{fg}^{2}}{\lambda_{fg}-\lambda_{H^{2}\phi}^{2}/\lambda_{\phi^{2}}}\,.$
(6.68)

This condition is crucial in order to reproduce the electroweak symmetry
breaking.

If we want to study the allowed region of the parameter space in these models,
we should also take into account other constraints imposed by requiring the
consistency of the LH models with the electroweak precision data. There exist
several studies of the corrections to electroweak precision observables in the
Little Higgs models, exploring whether there are regions of the parameter
space in which the model is consistent with the available data [12, 13, 7, 8,
4, 5, 9, 10, 11]. In _Model I_ with a gauge group $SU(2)\times SU(2)\times
U(1)\times U(1)$ we have a multiplet of heavy $SU(2)$ gauge bosons and a heavy
$U(1)$ gauge boson. The last one leads to large electroweak corrections and
some problems with the direct observational bounds on the $Z^{\prime}$ boson
from Tevatron [7, 8]. Then, a very strong bound on the symmetry breaking scale
$f$, $f>4$ TeV at $95\%$ C.L, is found [7]. However, it is known that this
bound is lowered to $1-2$ TeV for some region of the parameter space [8] by
gauging only $SU(2)\times SU(2)\times U(1)$ (_Model II_). For this reason, in
the following we will concentrate only on this model.

On the other hand, in order to avoid small values for the $W^{\prime}$ mass
and a very strong coupling constant, we set the range of the $\psi$ mixing
angle (for the $SU(2)$ group) to be $0.1<c{{}_{\psi}}<0.9$ [15]. In addition,
the condition
$\lambda_{T}\raisebox{-1.72218pt}{$\stackrel{{\scriptstyle>}}{{\scriptstyle\sim}}$\,}$
0.5 is established from the top mass [12], setting the bounds on the couplings
$\lambda_{1},\lambda_{2}\geq m_{t}/v$ or $\lambda_{1}\lambda_{2}\geq
2(m_{t}/v)^{2}$. In order to avoid a large fine-tuning in the Higgs potential
[2, 13] we set the condition
$m_{T}\raisebox{-1.72218pt}{$\stackrel{{\scriptstyle<}}{{\scriptstyle\sim}}$\,}$
2.5 TeV. Then, since $m_{T}$ grows linearly with $f$, $f$ should be less than
about one TeV [14]. Following the restrictions on the parameters given in
[15], we take $0.8$ TeV $<f<1$ TeV. Finally the usual condition
$\Lambda\raisebox{-1.72218pt}{$\stackrel{{\scriptstyle<}}{{\scriptstyle\sim}}$\,}4\pi
f$ is also imposed.

By using the constraints on the LH parameters given above, taking into account
also that the Higgs mass is experimentally restricted to the range $114$ GeV
$<m_{\overline{\mathcal{H}}}<200$ GeV, and by imposing the minimum condition
(6.68), we analyze the available regions for the remaining LH parameters. To
do that we include the contributions of both radiative corrections and
effective operators. In fact, in order to see the role played for each of
them, we consider three different cases: having just radiative corrections
(RC), just effective operators (EO) and the most general case including both
of them (RC+EO).

Figure 2: (a) Values of $\lambda_{T}$, $\Lambda$ and $c_{\psi}$ which are
possible solutions for the LH model. Here $f$ vary between $0.8$ and $1$ TeV,
and $a$ and $a^{\prime}$ are $\mathcal{O}(1)$. The three separate surfaces
correspond with the three different cases analyzed in this section. (b) Values
of $a$ and $a^{\prime}$ which are possible solutions for the LH model. The
$\lambda_{T}$, $\Lambda$, $c_{\psi}$ and $f$ ranges are described in the text.

In Fig. 2 we show the allowed regions of the parameter space for the three
different cases analyzed; RC (red region), EO (blue region) and RC+EO (green
region). In Fig. 2.a we show the possible solutions to the LH model in the
$(\Lambda,c_{\psi},\lambda_{T})$ space varying $f$ between $0.8$ TeV and $1$
TeV and by assuming that the $a$ and $a^{\prime}$ parameters are of the order
of $\mathcal{O}(1)$. From these results there are two important issues to
remark. First, when only radiative corrections are included we do not find any
solution for the LH model if $\Lambda>6$ TeV. Unfortunately, precision
electroweak data rule out new strong interactions at scales below about $10$
TeV. On the contrary, in the other two cases, RC+EO and EO, the possible
values for the cut-off are larger. This fact implies also that the mass of the
$\phi$ fields must be about $2$ TeV when the model includes only radiative
corrections unlike in the other two cases where it is about $5$ TeV (see Fig.
3). In Fig. 2.b we show the possible values for the unknown $a$ and
$a^{\prime}$ parameters. Here, the other parameters have been varied in the
ranges set above. The two cases considered are RC+EO and EO only. We find that
the set of possible solutions include in both cases positive values for $a$.
In the RC+EO case we obtain large and negative values for $a^{\prime}$,
whereas in the EO case $a^{\prime}$ takes small and positive values. Notice
also that $a$ is always positive. This is important since it is known that
$a<0$ leads to a large v.e.v for the scalar triplet.

The reason for the differences of the parameter solutions for the three cases
come from the $\Delta m_{GB}^{2}$ cutoff dependence when the radiative
contributions are included. For example, in the case where only the radiative
corrections are taken into account, a cut-off $\Lambda$ bigger than $6$ TeV
produces GB contributions resulting in a negative Higgs mass. However, by
dropping the value of $\Lambda$ we get a LH parameter space where the
condition (6.68) is satisfied and the Higgs mass is well inside the
experimental constraints. In the RC+EO case, the $a^{\prime}$ parameter can
take values which help to compensate the big effect of the GB radiative
contributions (see also Fig. 4) thus allowing larger cutoff values.

Figure 3: (a) $m_{T}$ as a function of $\lambda_{T}$, (b) $M_{W^{\prime}}$ as
a function of $\cos\psi$ and (c) $M_{\phi}$ as a function of $\lambda_{T}$ and
$\cos\psi$, where the $\Lambda$, $f$, $a$ and $a^{\prime}$ parameters vary
between ranges described in the text.

For completeness, Fig. 3 shows the mass values for the heavy particles in the
three different cases analyzed. Each point of the figures is a possible
solution of the LH model. In this way, these regions represent the possible
values for the masses of the heavy particles predicted by the LH model, which
are compatible with electroweak symmetry breaking and precision data. The
region of possible values for the masses coming from EO contributions is
clearly larger than in the case of considering RC alone. Notice that the
theoretical lower bounds in the heavy states masses,
$M_{\phi},M_{W^{\prime}}\raisebox{-1.72218pt}{$\stackrel{{\scriptstyle>}}{{\scriptstyle\sim}}$\,}$
1 TeV, and the condition
$m_{T}\raisebox{-1.72218pt}{$\stackrel{{\scriptstyle<}}{{\scriptstyle\sim}}$\,}$
2.5 TeV are fulfilled.

Figure 4: These figures show the average and standard deviation of both the
fermionic and gauge boson contribution $\mu_{fg}$ and the Goldstone boson
contribution $\Delta m_{GB}$ to the Higgs mass as a function of $\lambda_{T}$.
(a) RC case, (b) RC+EO case and (c) EO case.

To complete our study, we compare the contributions to the Higgs mass coming
from the different sectors i.e. fermionic and gauge bosons ($\mu_{fg}$) and on
the other hand the GB contribution ($\Delta m_{GB}$), as a function of
$\lambda_{T}$. We show the average and standard deviation for each
contribution (Fig. 4). In all physical cases it can be seen that
$\mu_{fg}>\Delta m_{GB}$, thus yielding a real value for the Higgs mass (eq.
5.66). It is also remarkable the higher variability of $\Delta m_{GB}$
compared with $\mu_{fg}$. The reason is that both the parameters appearing in
the radiative corrections, i.e. $f,\Lambda,\lambda_{T},\cos\phi$, and the two
EO parameters $a$ and $a^{\prime}$, play an important role in the final
results of $\Delta m_{GB}$ (see the discussion above).

Finally, as an example, we give in the Table.1 the lowest Higgs mass values
found for the three cases considered in this work.

Parameters | RC | RC+EO | EO
---|---|---|---
$m_{\overline{\mathcal{H}}}$ |  $156.66$ GeV |  $114.69$ GeV |  $116.94$ GeV
$\mu_{fg}$ |  $359.54$ GeV |  $236.87$ GeV |  $288.70$ GeV
$\Delta_{Gb}$ |  $342.04$ GeV |  $222.55$ GeV |  $275.53$ GeV
$\lambda_{\overline{\mathcal{H}}}$ | $0.97$ | $0.90$ | $1.42$
$f$ |  $0.86$ TeV |  $0.96$ TeV |  $0.82$ TeV
$\Lambda$ |  $5$ TeV |  $11.64$ TeV |  $10.01$ TeV
$\lambda_{T}$ | $0.6$ | $0.61$ | $0.53$
$c_{\psi}$ | $0.18$ | $0.16$ | $0.3$
$a$ | $0$ | $0.98$ | $1.06$
$a^{\prime}$ | $0$ | $-1.25$ | $0.5$
Table 1: The lowest values for the Higgs mass found for the three cases: RC,
RC+EO and EO.

##  7 Conclusion

In this work we have completed our program of computing the relevant
contributions to the Higgs low-energy effective potential in the context of
the Littlest Higgs models based on the $SU(5)/SO(5)$ coset. To the radiative
corrections coming from the fermions and the gauge bosons considered so far,
we have added here the effect the scalar loops and also the effective
operators emerging from the ultraviolet completion of the model.

In particular we have computed in detail the main contributions to the Higgs
mass and its quartic coupling. From our previous works, in which only
fermionic and gauge boson radiative corrections were included, it was clear
that the effect of the scalar sector could be decisive in order to have the
appropriate cancellations between the different sectors of the model to give a
Higgs mass within the present experimental limits. We have performed our
analytical computations for two different versions of the model called _Model
I_ and _Model II_ having as gauge groups $[SU(2)\times U(1)]^{2}$ and
$SU(2)^{2}\times U(1)$ respectively.

In order to complete our analysis, we have concentrated on studying those
regions of the parameter space where the model could give rise to an
acceptable phenomenology. In particular we have done a detailed numerical
search for _Model II_ since _Model I_ seems to be incompatible with the
present experimental data [7, 8]. We have analyzed three cases: 1) radiative
corrections only (RC), 2) radiative corrections and effective operators
(RC+EO) and 3) effective operator only (EO). From this analysis we get that
this model is compatible with the expected Higgs mass provided that the
contribution of the effective operators is included. We also conclude that the
Goldstone boson contributions are fundamental to obtain a low enough Higgs
particle mass. For example a Higgs mass $m_{H}\simeq 115GeV$ can be obtained
when radiative and effective operator contributions are both taken into
account.

Summarizing, we have arrived to the conclusion that the $SU(5)/S(5)$ Littlest
Higgs model with gauge group $[SU(2)\times U(1)]^{2}$ is phenomenologically
viable through some tuning in the parameter space, assuming a careful
inclusion of fermions, gauge bosons, scalar loops and effective operators.

In any case it will be the LHC, whose main goal is to disentangle the
mechanism of the electroweak symmetry breaking, which will decide if Littlest
Higgs models are appropriate for describing mechanism or not.

Acknowledgments: This work is supported by DGICYT (Spain) under project number
FPA2008-00592 and by the Universidad Complutense/CAM: UCM-BSCH GR58/08 910309.
The work of S.P. is supported by a Ramón y Cajal contract from MEC (Spain) and
partially by CICYT (grant FPA2006-2315) and DGIID-DGA (grant 2008-E24/2).The
work of J.R.L. is supported by project number FIS2006-04885. We would like to
thank J.R.Espinosa for useful discussions.

Appendix

a. Coefficients coming from loops computation

_Model I_

$\displaystyle\mu_{fg}^{2}$ $\displaystyle=$
$\displaystyle\mu^{2}_{f}+\mu^{2}_{g}$ $\displaystyle=$ $\displaystyle
N_{c}\frac{m_{T}^{2}\lambda_{t}^{2}}{4\pi^{2}}\log\left(1+\frac{\Lambda^{2}}{m_{T}^{2}}\right)$
$\displaystyle-$
$\displaystyle\frac{3}{64\pi^{2}}\left[3g^{2}M_{W^{\prime}}^{2}\log\left(1+\frac{\Lambda^{2}}{M_{W^{\prime}}^{2}}\right)+g^{{}^{\prime}2}M_{B^{\prime}}^{2}\log\left(1+\frac{\Lambda^{2}}{M_{B^{\prime}}^{2}}\right)\right]$

$\displaystyle\lambda_{f}$ $\displaystyle=$
$\displaystyle\frac{N_{c}}{(4\pi)^{2}}\left[2(\lambda_{t}^{2}+\lambda_{T}^{2})\frac{\Lambda^{2}}{f^{2}}\right.$
$\displaystyle-$
$\displaystyle\log\left(1+\frac{\Lambda^{2}}{m_{T}^{2}}\right)\left(-\frac{2m_{T}^{2}}{f^{2}}\left(\frac{5}{3}\lambda_{t}^{2}+\lambda_{T}^{2}\right)+4\lambda_{t}^{4}+4(\lambda_{T}^{2}+\lambda_{t}^{2})^{2}\right)$
$\displaystyle-$
$\displaystyle\left.4\lambda_{T}^{2}\frac{1}{1+\frac{m_{T}^{2}}{\Lambda^{2}}}\left(\frac{m_{T}^{2}}{f^{2}}-2\lambda_{t}^{2}-\lambda_{T}^{2}\right)-4\lambda_{t}^{4}\log\left(\frac{\Lambda^{2}}{m^{2}}\right)\right]$
$\displaystyle-$ $\displaystyle\frac{3}{(16\pi
f)^{2}}\left[-\left(\frac{g^{2}}{c_{\psi}^{2}s_{\psi}^{2}}+\frac{g^{{}^{\prime}2}}{c_{\psi}^{{}^{\prime}2}s_{\psi}^{{}^{\prime}2}}\right)\Lambda^{2}\right.$
$\displaystyle+$
$\displaystyle\left.g^{2}M_{W^{\prime}}^{2}\log\left(1+\frac{\Lambda^{2}}{M_{W^{\prime}}^{2}}\right)\left(4+\frac{1}{c_{\psi}^{2}s_{\psi}^{2}}+2g^{{}^{\prime}2}\frac{(c_{\psi}^{2}s_{\psi}^{{}^{\prime}2}+s_{\psi}^{2}c_{\psi}^{{}^{\prime}2})^{2}}{c_{\psi}^{2}s_{\psi}^{2}c_{\psi}^{{}^{\prime}2}s_{\psi}^{{}^{\prime}2}}\frac{f^{2}}{M_{W^{\prime}}^{2}-M_{B^{\prime}}^{2}}\right)\right.$
$\displaystyle+$
$\displaystyle\left.g^{{}^{\prime}2}M_{B^{\prime}}^{2}\log\left(1+\frac{\Lambda^{2}}{M_{B^{\prime}}^{2}}\right)\left(\frac{4}{3}+\frac{1}{c_{\psi}^{{}^{\prime}2}s_{\psi}^{{}^{\prime}2}}+2g^{2}\frac{(c_{\psi}^{2}s_{\psi}^{{}^{\prime}2}+s_{\psi}^{2}c_{\psi}^{{}^{\prime}2})^{2}}{c_{\psi}^{2}s_{\psi}^{2}c_{\psi}^{{}^{\prime}2}s_{\psi}^{{}^{\prime}2}}\frac{f^{2}}{M_{B^{\prime}}^{2}-M_{W^{\prime}}^{2}}\right)\right.$
$\displaystyle+$
$\displaystyle\left.f^{2}\log\left(1+\frac{\Lambda^{2}}{M_{W^{\prime}}^{2}}\right)\left(3g^{4}+2(3g^{2}+g^{{}^{\prime}2})g^{2}\frac{(s_{\psi}^{2}-c_{\psi}^{2})^{2}}{c_{\psi}^{2}s_{\psi}^{2}}\right)\right.$
$\displaystyle+$
$\displaystyle\left.f^{2}\log\left(1+\frac{\Lambda^{2}}{M_{B^{\prime}}^{2}}\right)\left(g^{{}^{\prime}4}+2(g^{2}+g^{{}^{\prime}2})g^{{}^{\prime}2}\frac{(s_{\psi}^{{}^{\prime}2}-c_{\psi}^{{}^{\prime}2})^{2}}{c_{\psi}^{{}^{\prime}2}s_{\psi}^{{}^{\prime}2}}\right)\right.$
$\displaystyle+$
$\displaystyle\left.f^{2}\log\left(\frac{\Lambda^{2}}{m^{2}}\right)\left(3g^{4}+g^{{}^{\prime}4}+8g^{2}g^{{}^{\prime}2}\right)-3f^{2}\frac{g^{4}}{1-\frac{M_{W^{\prime}}^{2}}{\Lambda^{2}}}-f^{2}\frac{g^{{}^{\prime}4}}{1-\frac{M_{B^{\prime}}^{2}}{\Lambda^{2}}}\right]$

$\displaystyle\lambda_{\phi^{2}f}$ $\displaystyle=$
$\displaystyle\frac{8N_{c}}{(4\pi
f)^{2}}(\lambda_{t}^{2}+\lambda_{T}^{2})\left(\Lambda^{2}-m_{T}^{2}\log\left(\frac{\Lambda^{2}}{m_{T}^{2}}+1\right)\right)$
$\displaystyle+$ $\displaystyle\frac{3}{4(4\pi
f)^{2}}\left[\frac{g^{2}}{c_{\psi}^{2}s_{\psi}^{2}}\Lambda^{2}-g^{2}M_{W^{\prime}}^{2}\log\left(\frac{\Lambda^{2}}{M_{W^{\prime}}^{2}}+1\right)\left(\frac{(s_{\psi}^{2}-c_{\psi}^{2})^{2}}{c_{\psi}^{2}s_{\psi}^{2}}-4\right)\right.$
$\displaystyle+$
$\displaystyle\left.\frac{g^{{}^{\prime}2}}{c_{\psi^{\prime}}^{2}s_{\psi^{\prime}}^{2}}\Lambda^{2}-g^{{}^{\prime}2}M_{B^{\prime}}^{2}\log\left(\frac{\Lambda^{2}}{M_{B^{\prime}}^{2}}+1\right)\frac{(s_{\psi^{\prime}}^{2}-c_{\psi^{\prime}}^{2})^{2}}{c_{\psi^{\prime}}^{2}s_{\psi^{\prime}}^{2}}\right]$

$\displaystyle\lambda_{H^{2}\phi}$ $\displaystyle=$
$\displaystyle-\frac{4N_{c}}{(4\pi
f)^{2}}\left[(\lambda_{t}^{2}+\lambda_{T}^{2})\Lambda^{2}-\lambda_{T}^{2}m_{T}^{2}\log\left(\frac{\Lambda^{2}}{m_{T}^{2}}+1\right)\right]$
$\displaystyle+$ $\displaystyle\frac{3}{8(4\pi
f)^{2}}\left[g^{2}\frac{s_{\psi}^{2}-c_{\psi}^{2}}{c_{\psi}^{2}s_{\psi}^{2}}\left(\Lambda^{2}-M_{W^{\prime}}^{2}\log\left(\frac{\Lambda^{2}}{M_{W^{\prime}}^{2}}+1\right)\right)\right.$
$\displaystyle+$
$\displaystyle\left.g^{{}^{\prime}2}\frac{s_{\psi^{\prime}}^{2}-c_{\psi^{\prime}}^{2}}{c_{\psi^{\prime}}^{2}s_{\psi^{\prime}}^{2}}\left(\Lambda^{2}-M_{B^{\prime}}^{2}\log\left(\frac{\Lambda^{2}}{M_{B^{\prime}}^{2}}+1\right)\right)\right]\,,$

_Model II_

$\displaystyle\mu_{fg}^{2}$ $\displaystyle=$
$\displaystyle\mu^{2}_{f}+\mu^{2}_{g}$ $\displaystyle=$ $\displaystyle
N_{c}\frac{m_{T}^{2}\lambda_{t}^{2}}{4\pi^{2}}\log\left(1+\frac{\Lambda^{2}}{m_{T}^{2}}\right)-\frac{3}{64\pi^{2}}\left(3g^{2}M_{W^{\prime}}^{2}\log\left(1+\frac{\Lambda^{2}}{M_{W^{\prime}}^{2}}\right)+g^{{}^{\prime}2}\Lambda^{2}\right)$

$\displaystyle\lambda_{fg}$ $\displaystyle=$
$\displaystyle\frac{N_{c}}{(4\pi)^{2}}\left[2(\lambda_{t}^{2}+\lambda_{T}^{2})\frac{\Lambda^{2}}{f^{2}}\right.$
$\displaystyle-$
$\displaystyle\log\left(1+\frac{\Lambda^{2}}{m_{T}^{2}}\right)\left(-\frac{2m_{T}^{2}}{f^{2}}\left(\frac{5}{3}\lambda_{t}^{2}+\lambda_{T}^{2}\right)+4\lambda_{t}^{4}+4(\lambda_{T}^{2}+\lambda_{t}^{2})^{2}\right)$
$\displaystyle-$
$\displaystyle\left.4\lambda_{T}^{2}\frac{1}{1+\frac{m_{T}^{2}}{\Lambda^{2}}}\left(\frac{m_{T}^{2}}{f^{2}}-2\lambda_{t}^{2}-\lambda_{T}^{2}\right)-4\lambda_{t}^{4}\log\left(\frac{\Lambda^{2}}{m^{2}}\right)\right]$
$\displaystyle-$ $\displaystyle\frac{3}{(16\pi
f)^{2}}\left[-\frac{g^{2}}{c_{\psi}^{2}s_{\psi}^{2}}\Lambda^{2}+\frac{4}{3}{g^{\prime}}^{2}\Lambda^{2}+g^{2}M_{W^{\prime}}^{2}\log\left(\frac{\Lambda^{2}}{M_{W^{\prime}}^{2}}+1\right)\left(4+\frac{1}{c_{\psi}^{2}s_{\psi}^{2}}\right)\right.$
$\displaystyle+$
$\displaystyle\left.f^{2}\log\left(1+\frac{\Lambda^{2}}{M_{W^{\prime}}^{2}}\right)\left(3g^{4}+2(3g^{2}+{g^{\prime}}^{2})g^{2}\frac{(s_{\psi}^{2}-c_{\psi}^{2})^{2}}{s_{\psi}^{2}c_{\psi}^{2}}\right)\right.$
$\displaystyle+$
$\displaystyle\left.f^{2}\log\left(\frac{\Lambda^{2}}{m^{2}}\right)(3g^{4}+{g^{\prime}}^{4}+8g^{2}{g^{\prime}}^{2})-3f^{2}\frac{g^{4}}{1-\frac{M_{W^{\prime}}^{2}}{\Lambda^{2}}}\right]$

$\displaystyle\lambda_{\phi^{2}}$ $\displaystyle=$
$\displaystyle\frac{8N_{c}}{(4\pi
f)^{2}}(\lambda_{t}^{2}+\lambda_{T}^{2})\left(\Lambda^{2}-m_{T}^{2}\log\left(\frac{\Lambda^{2}}{m_{T}^{2}}+1\right)\right)$
$\displaystyle+$
$\displaystyle\frac{3}{64\pi^{2}f^{2}}\left[\frac{g^{2}}{c_{\psi}^{2}s_{\psi}^{2}}\Lambda^{2}-g^{2}M_{W^{\prime}}^{2}\log\left(\frac{\Lambda^{2}}{M_{W^{\prime}}^{2}}+1\right)\left(\frac{(s_{\psi}^{2}-c_{\phi}^{2})^{2}}{c_{\psi}^{2}s_{\psi}^{2}}-4\right)\right]$
$\displaystyle+$ $\displaystyle\frac{3g^{{}^{\prime}2}}{(4\pi
f)^{2}}\Lambda^{2}$

$\displaystyle\lambda_{H^{2}\phi}$ $\displaystyle=$
$\displaystyle-\frac{4N_{c}}{(4\pi
f)^{2}}\left[(\lambda_{t}^{2}+\lambda_{T}^{2})\Lambda^{2}-\lambda_{T}^{2}m_{T}^{2}\log\left(\frac{\Lambda^{2}}{m_{T}^{2}}+1\right)\right]$
$\displaystyle+$
$\displaystyle\frac{3g^{2}}{8(4f\pi)^{2}}\frac{s_{\psi}^{2}-c_{\psi}^{2}}{c_{\psi}^{2}s_{\psi}^{2}}\left(\Lambda^{2}-M_{W^{\prime}}^{2}\log\left(\frac{\Lambda^{2}}{M_{W^{\prime}}^{2}}+1\right)\right)\,,$

b. Coefficients coming from effective operators

_Modelo I_

$\displaystyle\lambda_{fg}^{\rm EO}$ $\displaystyle=$
$\displaystyle\frac{a}{8}\left(\frac{g^{2}}{s_{\psi}^{2}c_{\psi}^{2}}+\frac{g^{{}^{\prime}2}}{s_{\psi}^{{}^{\prime}2}c_{\psi}^{{}^{\prime}2}}\right)+2a^{\prime}(\lambda_{t}^{2}+\lambda_{T}^{2})$
$\displaystyle{\lambda_{\phi^{2}}}^{\rm EO}$ $\displaystyle=$
$\displaystyle\frac{a}{2}\left(\frac{g^{2}}{s_{\psi}^{2}c_{\psi}^{2}}+\frac{g^{{}^{\prime}2}}{s_{\psi}^{{}^{\prime}2}c_{\psi}^{{}^{\prime}2}}\right)+8a^{\prime}(\lambda_{t}^{2}+\lambda_{T}^{2})$
$\displaystyle{\lambda_{H^{2}\phi}}^{\rm EO}$ $\displaystyle=$
$\displaystyle\frac{a}{4}\left(g^{2}\frac{c_{\psi}^{2}-s_{\psi}^{2}}{s_{\psi}^{2}c_{\psi}^{2}}+g^{{}^{\prime}2}\frac{c_{\psi}^{{}^{\prime}2}-s_{\psi}^{{}^{\prime}2}}{s_{\psi}^{{}^{\prime}2}c_{\psi}^{{}^{\prime}2}}\right)+4a^{\prime}(\lambda_{t}^{2}+\lambda_{T}^{2})$

_Modelo II_

$\displaystyle\lambda_{fg}^{\rm EO}$ $\displaystyle=$
$\displaystyle\frac{a}{8}\left(\frac{g^{2}}{s_{\psi}^{2}c_{\psi}^{2}}\right)-\frac{a}{3}g^{{}^{\prime}2}+2a^{\prime}(\lambda_{t}^{2}+\lambda_{T}^{2})$
$\displaystyle{\lambda_{\phi^{2}}}^{\rm EO}$ $\displaystyle=$
$\displaystyle\frac{a}{2}\left(\frac{g^{2}}{s_{\psi}^{2}c_{\psi}^{2}}\right)+4a{g^{{}^{\prime}2}}+8a^{\prime}(\lambda_{t}^{2}+\lambda_{T}^{2})$
$\displaystyle{\lambda_{H^{2}\phi}}^{\rm EO}$ $\displaystyle=$
$\displaystyle\frac{a}{4}g^{2}\frac{c_{\psi}^{2}-s_{\psi}^{2}}{s_{\psi}^{2}c_{\psi}^{2}}+4a^{\prime}(\lambda_{t}^{2}+\lambda_{T}^{2})$
$\displaystyle\mu^{2\,\rm EO}$ $\displaystyle=$ $\displaystyle
af^{2}g^{{}^{\prime}2}$

## References

  * [1] T. Aaltonen et al. [CDF Collaboration], Phys. Rev. Lett. 99 (2007) 151801 [arXiv:0707.0085 [hep-ex]]; T.T.E. Group et al. [Tevatron Electroweak Working Group and CDF Collaboration and D0 Collab], arXiv:0803.1683 [hep-ex].
  * [2] N. Arkani-Hamed, A.G. Cohen, E.Katz, A.E.Nelson, JHEP 0207:034 (2002), hep-ph/0206021.
  * [3] H. Georgi and A. Pais, Phys. Rev. D10, 539 (1974) H. Georgi and A. Pais, Phys. Rev. D12, 508 (1975).
  * [4] M. Schmaltz, D. Tucker-Smith, Ann. Rev. Nucl. Part. Sci. 55 (2005) 229, hep-ph/0502182.
  * [5] M. Perelstein, Prog. Part. Nucl. Phys. 58 (2007) 247, hep-ph/0512128.
  * [6] M. Schmaltz, JHEP 0408 (2004) 056, hep-ph/0407143; H. C. Cheng and I. Low, JHEP 0408, 061 (2004), hep-ph/0405243; D. E. Kaplan and M. Schmaltz, JHEP 0310 (2003) 039, hep-ph/0302049; H. C. Cheng and I. Low, JHEP 0309, 051 (2003), hep-ph/0308199; I. Low, W. Skiba and D. Tucker-Smith, Phys. Rev. D 66, 072001 (2002), hep-ph/0207243.
  * [7] C. Csaki, J. Hubisz, G. D. Kribs, P. Meade, J. Terning, Phys. Rev. D67 (2003) 115002, hep-ph/0211124.
  * [8] C. Csaki, J. Hubisz, G. D. Kribs, P. Meade, J. Terning, Phys. Rev. D68, 035009 (2003), hep-ph/0303236.
  * [9] Z. Han, W. Skiba, Phys. Rev. D72, 035005 (2005), hep-ph/0506206; H.E. Logan, Phys. Rev. D70, 115003 (2004), hep-ph/0405072; T. Han, H. E. Logan, B. McElrath, L. T. Wang, Phys. Lett. B563, 191 (2003) [Erratum-ibid. B603, 257 (2004)], hep-ph/0302188.
  * [10] J. L. Hewett, F. J. Petriello, T. G. Rizzo, JHEP 0310, 062 (2003), hep-ph/0211218. T. Gregoire, D. R. Smith, J. G. Wacker, Phys. Rev. D69, 115008 (2004), hep-ph/0305275; M. C. Chen and S. Dawson, Phys. Rev. D70, 015003 (2004), hep-ph/0311032; W. Kilian, J. Reuter, Phys. Rev. D70, 015004 (2004), hep-ph/0311095; G. Marandella, C. Schappacher, A. Strumia, Phys. Rev. D72, 035014 (2005), hep-ph/0502096.
  * [11] S. K. Kang, C. S. Kim and J. Park, Phys. Lett. B 666 (2008) 38, arXiv:0805.0179 [hep-ph]; M.C. Chen, Mod. Phys. Lett. A21, 621 (2006), hep-ph/0601126; S.R. Choudhury, A.S. Cornell, N. Gaur, A. Goyal, hep-ph/0604162; J.A. Conley, J. Hewett, M. P. Le, Phys. Rev. D72, 115014 (2005), hep-ph/0507198; C.O. Dib, R. Rosenfeld, A. Zerwekh, AIP Conf. Proc. 815, 296 (2006), hep-ph/0509013; Z. Berezhiani, P.H. Chankowski, A. Falkowski, S. Pokorski, Phys. Rev. Lett. 96, 031801 (2006), hep-ph/0509311.
  * [12] T. Han, Heather E. Logan, B. McElrath, Lian-Tao Wang, Phys. Rev. D67, 095005 (2003), hep-ph/0301040.
  * [13] M. Perelstein, M. E. Peskin, A. Pierce, Phys. Rev. D69, 075002 (2004), hep-ph/0310039.
  * [14] A. Dobado, L. Tabares and S. Peñaranda, Eur. Phys. J. C50 (2007) 647, hep-ph/0606031.
  * [15] A. Dobado, L. Tabares, S. Peñaranda, Phys. Rev. D75 (2007) 083527, hep-ph/0612131.
  * [16] A. Dobado, L. Tabares, S. Peñaranda, Eur. Phys. J. C 58 (2008) 471, arXiv:0711.4471 [hep-ph].
  * [17] F. Bazzocchi, M. Fabbrichesi, M. Piai, Phys. Rev. D72 (2005) 095019, hep-ph/0506175.
  * [18] J. Alberto Casas, Jose Ramón Espinosa, Irene Hidalgo, JHEP 0503:038.
  * [19] A. Dobado, A. Gómez-Nicola, A.L. Maroto and J.R. Peláez, Effective Lagrangians for the Standard Model, Springer-Verlag, Heidelberg, (1997).
  * [20] A. Manohar, H. Georgi, Nucl. Phys. B234, 189 (1984); M. A. Luty, Phys. Rev. D57, 1531 (1998), hep-ph/9706235; A. G. Cohen, D. B. Kaplan, A. E. Nelson, Phys. Lett. B412, 301 (1997).
  * [21] J. R. Espinosa and J. M. No, JHEP 0701 (2007) 006, hep-ph/0610255.