# E- & T-Model Hybrid Inflation

C. Pallis Laboratory of Physics, Faculty of Engineering,
Aristotle University of Thessaloniki,
GR-541 24 Thessaloniki, GREECE
<EMAIL_ADDRESS>
###### Abstract

Abstract

We consider the impact of a kinetic pole of order one or two on the non-
supersymmetric model of hybrid inflation. These poles arise due to logarithmic
Kähler potentials which control the kinetic mixing of the inflaton field and
parameterize hyperbolic manifolds with scalar curvature related to the
coefficient $(-N)<0$ of the logarithm. Inflation is associated with the
breaking of a local $SU(2)\times U(1)$ symmetry, which does not produce any
cosmological defects after it, and remains largely immune from the minimal
possible radiative corrections to the inflationary potential. For $N=1$ and
equal values of the relevant coupling constants, $\lambda$ and $\kappa$, the
achievement of the observationally central value for the scalar spectral
index, $n_{\rm s}$, requires the mass parameter, $m$, and the symmetry
breaking scale, $M$, to be of the order of $10^{12}~{}{\mbox{\rm GeV}}$ and
$10^{17}~{}{\mbox{\rm GeV}}$ respectively. Increasing $N$ above unity the
tensor-to-scalar ratio $r$ increases above $0.002$ and reaches its maximal
allowed value for $N\simeq 10-20$.

##  1 Introduction

It is widely believed that the introduction of _supersymmetry_ (SUSY) and its
local extension – _supergravity_ (SUGRA) – can alleviate the shortcomings of
_Standard Model_ (SM) and provide a safe framework for building a variety of
inflationary models – see e.g. sugrareview. However, we have to accept that
there is no direct experimental confirmation of SUSY until now [2]. On the
contrary, there is a strong observational evidence in favor of the
inflationary paradigm [4, 3]. Consequently, it is worthwhile to build
inflationary models compatible with the observations, even without the
presence of SUSY.

One of those models – for reviews see review – is undoubtedly _Hybrid
Inflation_ (HI) [6] which elegantly combines a period of inflation with a
phase transition at its termination. It is termed “hybrid” because the
inflationary vacuum energy density is provided by a waterfall field, different
from the slowly rolling inflaton field. The model allows for values of the
relevant coupling constants larger than those used in chaotic inflation, and
can be successfully handled with subplanckian values for the inflaton field.
It can be, also, embedded in several _Grand Unified Theory_ (GUT) schemes –
mainly in the SUSY framework [7, 8, 9] – and nicely connects inflationary
cosmology with particle physics. A very intriguing possibility of these
embeddings is the production of topological defects at its end via the Kibble
mechanism [10]. Between them, the metastable cosmic strings are currently of
special interest since they may decay generating a stochastic background of
gravitational waves [11] and interpreting, thereby, recent results [12] – for
other sources of gravitational waves’ production after the end of HI see
spanos, dim.

On the other hand, HI suffers from the problem of the enhanced (scalar)
spectral index $n_{\rm s}$ which turns out to be, mostly, well above the
present data [15, 16, 17] – for another point of view see hi22. Indeed,
$n_{\rm s}$ within tree level HI exceeds unity [6] and only in a minor and
tuned region of parameters [19] a reconciliation with data can be achieved.
Inclusion of _radiative corrections_ (RCs) [20] due to a possible coupling of
the inflaton to fermions [21] or a conveniently selected non-minimal coupling
to gravity [19, 22] can reconcile the model’s predictions with observations –
for another mechanism possibly applicable in non-SUSY HI see suphi. However,
the situation remains problematic since in both aforementioned cases, HI of
hilltop type [24] is obtained and so an unavoidable tuning of the initial
conditions emerges. Nonetheless, it is by now well-known [25, 26, 27, 28] that
the presence of a pole in the kinetic term of the inflaton facilitates the
achievement of observation-friendly chaotic inflation. It would be, therefore,
interesting to investigate if this technique can be also applied for HI
improving, thereby, its compatibility with the observations – for a similar
recent work see hlinde.

We find out that two types of HI can be formulated depending on the order $p$
of the inflaton kinetic pole [30, 31]. For $p=1$ we obtain _E-model HI_ (EHI)
whereas for $p=2$, _T-model HI_ (THI) arises. The terms are coined in analogy
to _E-model_ (chaotic) _inflation_ (EMI) [32] (or $\alpha$-Starobinsky model
[33]) and _T-model inflation_ (TMI) [34]. As in the latter cases [28], EHI and
THI can be relied on Kähler potentials and so the relevant particle models can
be established as non-linear sigma models with specific geometry of the moduli
space. In both cases, the minimal possible RCs [20] to the inflationary
potential are considered and subplanckian values for the initial (non-
canonically normalized) inflaton field are required. Unfortunately, our scheme
is inconsistent with the formation of cosmic strings after HI, since the
relevant scale $M$ of the phase transition turns out to be a little lower than
the reduced Planck scale, $m_{\rm P}=2.44\cdot 10^{18}~{}{\mbox{\rm GeV}}$,
and so the tension of the cosmic strings [11] would be unacceptably large.
Therefore, the version of HI with the waterfall field charged under the
continuous group $U(1)$ with the lowest possible dimensionality is not a
representative working example for our setup. However, if there is a $U(1)$
factor in the initial and the remaining gauge group, with the waterfall field
arranged in a convenient representation, then the production of cosmic strings
(and monopoles) after HI can be eluded [35]. In our analysis we adopt the
simplest possible scenario for reference. Other ways to overcome the obstacle
of topological defects within HI are proposed in hlinde, notd.

Below, in hi, we describe how we can formulate these versions of HI. The
dynamics of the resulting inflationary models is studied in inf and these are
tested against observations in res. Finally, con summarizes our conclusions.
In Appendix A we explore the possibility for generating different moduli
kinetic mixings allowed by generic Kähler potentials. Throughout the text, the
subscript $,\chi$ denotes derivation _with respect to_ (w.r.t) the field
$\chi$, charge conjugation is denoted by a star (∗) and we use units where
$m_{\rm P}=1$ unless otherwise stated.

##  2 Models’ Setup

Since E and T models are introduced by means of a non-minimal kinetic mixing
[28], we find it convenient to establish their combination with HI taking as
reference the non-linear sigma models. In these the kinetic mixing is
controlled by a metric $K_{{\alpha}{\bar{\beta}}}$ defined on the moduli
space. Here we employ two (complex) scalar fields $Z^{\alpha}$ with
${\alpha}=1,2$ – the inflaton $Z^{1}=S$ and the waterfall field $Z^{2}=\Phi$.
Therefore, the relevant lagrangian terms are written as

${\cal
L}=\sqrt{-\mathfrak{g}}\left(K_{{\alpha}{\bar{\beta}}}\left(D_{\mu}Z^{\alpha}\right)^{\dagger}D^{\mu}Z^{{\bar{\beta}}}-V(Z^{\alpha})\right),$
(2.1)

where $\mathfrak{g}$ is the determinant of the background Friedmann-Robertson-
Walker metric $g^{\mu\nu}$ with signature $(+,-,-,-)$. We further assume that
$K_{{\alpha}{\bar{\beta}}}$ originates from a Kähler potential $K$ – as in the
context of SUGRA – according to the definition

$K_{{\alpha}{\bar{\beta}}}={K_{,Z^{\alpha}Z^{*{\bar{\beta}}}}}>0\>\>\>\mbox{with}\>\>\>K^{{\bar{\beta}}{\alpha}}K_{{\alpha}\bar{\gamma}}=\delta^{\bar{\beta}}_{\bar{\gamma}}.$
(2.2)

We below, in hi1 and 2.2, we respectively specify the kinetic terms and the
inflationary potential of our models.

###  2.1 Kinetic Mixing

The adopted $K$’s include two contributions, one for the inflaton, $K_{\rm
I}$, and one for the waterfall field, $K_{\rm W}$. Namely, we set

$K=K_{\rm I}+K_{\rm W}\>\>\mbox{with}\>\>K_{\rm
W}=|\Phi|^{2}\>\>\mbox{and}\>\>K_{\rm
I}=\begin{cases}-2N\ln(1-(S+S^{*})/2)&\mbox{for EHI},\\\
-(N/2)\ln(1-|S|^{2})&\mbox{for THI}\,.\end{cases}$ (2.3)

We see that no mixing between the inflaton and waterfall sectors exists and
the non-zero elements of the relevant metric are calculated to be

$K_{\Phi\Phi^{*}}=1\>\>\mbox{and}\>\>K_{SS^{*}}=\frac{N}{2}\cdot\begin{cases}(1-(S+S^{*})/2)^{-2}&\mbox{for
EHI},\\\ (1-|S|^{2})^{-2}&\mbox{for THI}\,.\end{cases}$ (2.4)

For both models, $K_{\rm W}$ parameterizes flat manifold – contrary to the
setting in hlinde, dim – whereas the geometry induced by $K_{\rm I}$ is
hyperbolic, i.e., the curvature of the moduli space is negative. In
particular, the scalar curvatures associated with $K_{\rm W}$ and $K_{\rm I}$
respectively are

${\cal R}_{\rm W}=0\>\>\mbox{and}\>\>{\cal R}_{\rm
I}=-K^{SS^{*}}\partial_{S}\partial_{S^{*}}\ln
K_{SS^{*}}=-({1}/{N})\begin{cases}1&\mbox{for EHI},\\\ 4&\mbox{for
THI}.\end{cases}$ (2.5)

More specifically, $K_{\rm I}$ parameterizes the coset space $SU(1,1)/U(1)$ in
the case of THI, whereas for EHI the space generated by $K_{\rm I}$ is
invariant under the set of transformations related to the group $U(1,1)$ – see
epole.

In Saction1 we include the covariant derivatives for the scalar fields
$D_{\mu}Z^{\alpha}$ which are given by

$D_{\mu}Z^{\alpha}=\partial_{\mu}Z^{\alpha}+igA^{\rm a}_{\mu}T^{\rm
a}_{{\alpha}{\beta}}Z^{\beta},$ (2.6)

with $A^{\rm a}_{\mu}$ being the vector gauge fields, $g$ a gauge coupling
constant and $T^{\rm a}$ with ${\rm a}=1,...,\mbox{\sf\footnotesize dim}G_{\rm
GUT}$ the generators of the gauge group $G_{\rm GUT}$ with dimensionality
$\mbox{\sf\footnotesize dim}G_{\rm GUT}$. To avoid the production of cosmic
defects after the end of HI we have to select $G_{\rm GUT}$ according to the
guidelines of td. For definiteness, we here assume that $\Phi$ belongs to the
$({\bf 2},1)$ representation of the group $G_{21}=SU(2)\times U(1)$.
Interestingly enough, this group may be identified with the $SU(2)_{\rm
R}\times U(1)_{B-L}$ part of a realistic (minimal) version [36] of the GUT
based on the group $G_{\rm LR}=SU(3)_{\rm C}\times SU(2)_{\rm L}\times
SU(2)_{\rm R}\times U(1)_{B-L}$.

If we use the following parameterizations for the two fields

$S=\sigma e^{i\theta}\>\>\mbox{and}\>\>\Phi=\frac{1}{\sqrt{2}}\mbox{
$\left\lgroup\begin{matrix}\phi_{1}+i\varphi_{1}\cr\phi_{2}+i\varphi_{2}\end{matrix}\right\rgroup$},$
(2.7)

we can introduce he canonically normalized (hatted) fields during HI as
follows

$\langle{K_{{\alpha}{\bar{\beta}}}}\rangle_{\rm
HI}\dot{Z}^{\alpha}\dot{Z}^{*{\bar{\beta}}}\simeq\frac{1}{2}\left(\dot{\widehat{}\sigma}^{2}+\dot{\widehat{}\theta}^{2}+\dot{\phi_{\alpha}}^{2}+\dot{\varphi_{\alpha}}^{2}\right)\>\>\mbox{for}\>\>{\alpha}=1,2$
(2.8)

where $K_{{\alpha}{\bar{\beta}}}$ is given by Ksset and assures the canonical
normalization of $\phi_{\alpha}$ and $\varphi_{\alpha}$. The symbol
“$\langle{Q}\rangle_{\rm HI}$” denotes the value of a quantity $Q$ during HI,
where all the fields besides $\sigma$ are set equal to zero – see below – and
dot stands for derivation w.r.t the cosmic time. From Kc the remaining fields
$\widehat{\sigma}$ and $\widehat{\theta}$ are identified as

$\frac{d\widehat{\sigma}}{d\sigma}=J=\frac{\sqrt{N}}{f_{p}}~{}~{}\Rightarrow~{}~{}\widehat{\sigma}=\begin{cases}-\sqrt{N}\ln(1-\sigma)&\mbox{for}~{}~{}p=1,\\\
\sqrt{N}\rm
arctanh{\sigma}&\mbox{for}~{}~{}p=2\end{cases}\>\>\mbox{and}\>\>\widehat{\theta}\simeq
J\sigma\theta,$ (2.9)

where $f_{p}=1-\sigma^{p}$ with $p=1,2$. Note that $\widehat{\sigma}$ gets
increased above unity for $\sigma_{\rm c}\leq\sigma<1$, facilitating, thereby,
the attainment of HI with sub-Planckian $\sigma$ values.

###  2.2 Inflationary Potential

In Saction1 the inflationary potential for our models is also incorporated.
Its form is

$V(S,\Phi)=\kappa^{2}\left(|\Phi|^{2}-M^{2}\right)^{2}+\frac{1}{2}m^{2}|S|^{2}+\lambda^{2}|\Phi|^{2}|S|^{2}+\frac{1}{8}\widetilde{m}|S-S^{*}|^{2},$
(2.10)

where $M$, $m$ and $\widetilde{m}$ are mass parameters whereas $\kappa$ and
$\lambda$ are dimensionless coupling constants. The last unusual term is
adopted to provide the angular mode of $S$, $\theta$, with mass, as we see
below. The global minima of $V$ lie at the direction

$\langle{\phi_{1}}\rangle=\pm\sqrt{2}M\>\>\mbox{and}\>\>\langle{\sigma}\rangle=\langle{\phi_{2}}\rangle=\langle{\varphi_{\alpha}}\rangle=\langle{\theta}\rangle=0$
(2.11)

and ensure a spontaneous breaking of the local symmetry $G_{21}$ down to a
$U(1)^{\prime}$. The scalar spectrum of the theory is composed by three real
scalars $\sigma$, $\delta\phi_{1}=\phi_{1}-\langle{\phi_{1}}\rangle$ and
$\theta$ with masses correspondingly

$\widehat{m}_{\sigma}=\sqrt{(m^{2}+2\lambda^{2}M^{2})/N},\>\>m_{\delta\phi_{1}}=2\kappa
M\>\>\mbox{and}\>\>\widehat{m}_{\theta}=\widetilde{m}/\sqrt{N}.$ (2.12)

Thanks to the selected representation of $\Phi$ in para neither cosmic strings
nor monopoles are left behind the breakdown of $G_{21}$[35]. The truncated
form of $V$ with all the fields, but $\sigma$ and $\phi_{1}$, set at the
origin reduces to its well-known form [6] within HI, i.e.,

$V\left(\phi_{1},\sigma\right)=\kappa^{2}\left(\frac{\phi_{1}^{2}}{2}-M^{2}\right)^{2}+\frac{m^{2}}{2}\sigma^{2}+\frac{\lambda^{2}}{2}\phi_{1}^{2}\sigma^{2}\,.$
(2.13)

Besides the phase transition above, $V$ in Vhi1 gives rise to HI. This is,
because it possesses an almost $\sigma$-flat direction at

$\langle{\phi_{\alpha}}\rangle_{\rm HI}=\langle{\varphi_{\alpha}}\rangle_{\rm
HI}=\langle{\theta}\rangle_{\rm HI}=0\>\>\mbox{with}\>\>\sigma\geq\sigma_{\rm
c}=\sqrt{2}\kappa M/\lambda\,.$ (2.14)

The last inequality here assures the stability of the inflationary path w.r.t
the fluctuations of $\phi_{\alpha}$ and $\varphi_{\alpha}$ during HI. Indeed,
as shown in tab1 – where the mass squared spectrum of the model along the
trough in vevi is arranged – the positivity of $m^{2}_{\Phi}$ is protected by
the last inequality in vevi. From tab1 we can also verify that $\theta$
acquires mass thanks to the last term of Vhi1 and confirm that
$\widehat{m}^{2}_{\theta}\gg H_{\rm HI}^{2}$ and $\widehat{m}^{2}_{\Phi}\gg
H_{\rm HI}^{2}$ for $\sigma_{\rm c}<\sigma\leq\sigma_{\star}$. Here we define

$\mbox{\footnotesize\sf(a)}\>\>H_{\rm HI}=\sqrt{V_{\rm
HI0}/3}\>\>\>\mbox{with}\>\>\>\mbox{\footnotesize\sf(b)}\>\>V_{\rm
HI0}=\kappa^{2}M^{4}\simeq V(S,0)$ (2.15)

the almost constant potential density during HI. Note that no massive gauge
bosons exist in the particle spectrum since $G_{21}$ is unbroken during HI.
Using the derived spectrum we can compute the one-loop RCs (to the tree-level
potential) $\Delta V_{\rm HI}$ which can be written as [20]

$\Delta V_{\rm HI}={1\over
64\pi^{2}}\left(\widehat{m}_{\theta}^{4}\ln{\widehat{m}_{\theta}^{2}\over\Lambda^{2}}+2{\sf
N}_{\rm
G}\widehat{m}_{\phi}^{4}\ln{\widehat{m}_{\phi}^{2}\over\Lambda^{2}}\right).$
(2.16)

Here, ${\sf N}_{\rm G}=2$ is the dimensionality of the representation of
$\Phi$ and $\Lambda$ is a renormalization-mass scale which, is determined by
requiring $\Delta V_{\rm HI}(\sigma_{\star})=0$ or $\Delta V_{\rm
HI}(\sigma_{\rm c})=0$. This determination of $\Lambda$ renders our results
practically independent of $\Lambda$ in the major parameter space of the
models. Indeed, these can be derived exclusively by using the tree level
$V_{\rm HI}=V(S,0)$ with the various quantities evaluated at $\Lambda$ – the
relevant renormalization-group running is expected to be negligible – see res.
On the other hand, as shown in the same section, $\Delta V_{\rm HI}$ is
capable to ruin the successful inflationary predictions for $\kappa\gtrsim
0.003$. This effect may be avoided if extra contributions with opposite sign
are included into $\Delta V_{\rm HI}$ due to possible coupling of $S$ or
$\Phi$ to fermions (such as right-handed neutrinos) – see hirc.

In conclusion, the inflationary potential for EHI and THI is

$V_{\rm HI}=V_{\rm HI0}+{1\over 2}m^{2}\sigma^{2}+\Delta V_{\rm HI}$ (2.17)

with the canonical normalization of $\sigma$ in je being taken into account
and differentiating the models between each other.

Fields | Eigen- | Masses
---|---|---
| states | Squared
$1$ real scalar | $\widehat{\theta}$ | $\widehat{m}^{2}_{\theta}$ | $\widetilde{m}^{2}f_{p}^{2}/N$
$2{\sf N}_{\rm G}$ real scalars | $\phi_{\alpha},~{}\varphi_{\alpha}$ | $\widehat{m}^{2}_{\Phi}$ | $\lambda^{2}\sigma^{2}-2\kappa^{2}M^{2}$

[]Mass spectrum for EHI $(p=1)$ and THI ($p=2$) along the inflationary path in
vevi.

##  3 Inflation Analysis

A period of slow-roll EHI or THI is controlled by the strength of the slow-
roll parameters which can be derived by applying the standard formulae – see
e.g. review. Note that the derivation can be performed employing $V_{\rm HI}$
in Vhi and $J$ in je, without express explicitly $V_{\rm HI}$ in terms of
$\widehat{\sigma}$ – see e.g. nmi. Namely, we find

$\epsilon={1\over 2}\left(\frac{V_{{\rm HI},\widehat{\sigma}}}{V_{\rm
HI}}\right)^{2}\simeq\frac{m^{4}f_{p}^{2}\sigma^{2}}{2NV_{\rm
HI0}^{2}}\>\>\>\mbox{and}\>\>\>\eta=\frac{V_{{\rm
HI},\widehat{\sigma}\widehat{\sigma}}}{V_{\rm HI}}\simeq
m^{2}f_{p}\frac{f_{p}-p\sigma^{p}}{NV_{\rm HI0}}\cdot$ (3.1a) In the
expressions above we replace $V_{\rm HI}$ from Eq. (2.15b) whereas $V_{{\rm
HI},\widehat{\sigma}}$ and $V_{{\rm HI},\widehat{\sigma}\widehat{\sigma}}$ are
obtained by performing derivations of Vhi. As $m$ increases beyond $5\cdot
10^{-6}$ the approximation above becomes less accurate. It can be verified
numerically that $\epsilon(\sigma_{\rm c})\leq
1\>\>\>\mbox{and}\>\>\>\eta(\sigma_{\rm c})\leq 1$ (3.1b)

and so HI is terminated prematurely – i.e., for $\sigma_{\rm f}=\sigma_{\rm
c}$ – in the major parametric space of our model.

To protect our scenario from the production of extra e-foldings [6, 29, 38]
during the waterfall regime, we take into account the behavior of $\sigma$ and
$\phi_{1}$ after the time $H_{\rm HI}^{-1}$ from the moment when
$\sigma=\sigma_{\rm c}$, following the approach in hybrid,hlinde. The absolute
value of the $\phi_{1}$ effective mass after this time can be estimated as

$\Delta m_{\phi_{1}}^{2}\simeq 2\lambda^{2}\Delta\sigma\sigma_{\rm
c}\>\>\mbox{where}\>\>\Delta\sigma=3\sqrt{2}{m^{2}m_{\rm
P}^{2}}/{\lambda\kappa M^{3}}$ (3.2)

is the reduction of the $\sigma$ value below $\sigma_{\rm c}$ as can be
computed by the inflationary equation of motion. No sizable amount of
inflation occurs during the waterfall regime, if

${\Delta m_{\phi_{1}}^{2}}/{H_{\rm HI}^{2}}\simeq 36{m^{2}m_{\rm
P}^{4}}/{\kappa^{2}M^{6}}>1\>\>\Rightarrow\>\>\kappa\lesssim{6mm_{\rm
P}^{2}}/{M^{3}},$ (3.3)

where $H_{\rm HI}^{2}$ is given by Eq. (2.15a). This condition is roughly
identical with that obtained in more accurate computations [38] and can be
easily met in our scenario – see res.

The number of e-foldings $N_{\star}$ that the pivot scale $k_{\star}=0.05/{\rm
Mpc}$ experiences during EHI or THI can be calculated through the relation

$N_{\star}=\int_{\widehat{\sigma}_{\rm
f}}^{\widehat{\sigma}_{*}}d\widehat{\sigma}\frac{V_{\rm HI}}{V_{\rm
HI,\widehat{\sigma}}}=\frac{NV_{\rm HI0}}{pm^{2}}\left(\ln\frac{f_{p\rm
c}}{f_{p\star}}+p\ln\frac{\sigma_{\star}}{\sigma_{\rm
c}}+\frac{1}{f_{p\star}}-\frac{1}{f_{p\rm c}}\right),$ (3.4)

where $f_{p\star}=f_{p}(\sigma_{\star})$ and $f_{p\rm c}=f_{p}(\sigma_{\rm
c})$ with $\sigma_{\star}~{}[\widehat{\sigma}_{*}]$ being the value of
$\sigma~{}[\widehat{\sigma}]$ when $k_{\star}$ crosses the inflationary
horizon. Taking into account that $1\simeq\sigma_{\star}\gg\sigma_{\rm
f}=\sigma_{\rm c}$, we can find the following approximate version of the exact
formula above which can be analytically solved w.r.t $\sigma_{\star}$ as
follows

$N_{\star}\simeq\frac{NV_{\rm
HI0}}{pm^{2}}\left(\frac{1}{f_{p\star}}-1-p\ln\sigma_{\rm
c}\right)\>\>\Rightarrow\>\>\sigma_{\star}\simeq\left(1-\frac{NV_{\rm
HI0}}{NV_{\rm HI0}(1+p\ln\sigma_{\rm c})+N_{\star}pm^{2}}\right)^{1/p}\,.$
(3.5)

The elimination of the terms $p\ln\sigma_{\star}-\ln f_{p\star}$ can be
justified by the fact that their contribution to $N_{\star}$ in Nhi is
subdominant for the $\sigma$ values in the range $0.5\leq\sigma_{\star}<1$
encountered in our numerics.

The amplitude $A_{\rm s}$ of the power spectrum of the curvature perturbations
generated by $\sigma$ at the scale $k_{\star}$ is estimated as follows

$\sqrt{A_{\rm s}}=\>\frac{1}{2\sqrt{3}\,\pi}\;\frac{V_{\rm
HI}(\widehat{\sigma}_{*})^{3/2}}{|V_{\rm
HI,\widehat{\sigma}}(\widehat{\sigma}_{*})|}\simeq\frac{\sqrt{N}V_{\rm
HI0}^{3/2}}{2\sqrt{3}\pi\sigma_{\star}m^{2}f_{p\star}}\,\cdot$ (3.6)

Inserting sgx into the previous equation and expanding the resulting relation
in powers of $N_{\star}$ we can obtain a rough estimation of $\kappa$ as
follows

$\sqrt{A_{\rm s}}\simeq\>\frac{\sqrt{N}V_{\rm HI0}^{3/2}}{2\sqrt{3}\pi
m^{2}}\left(1+\frac{NV_{\rm
HI0}}{p^{2}m^{2}N_{\star}}\right)\>\>\Rightarrow\>\>\kappa\simeq\left(\frac{2\pi\sqrt{3A_{\rm
s}}m^{2}}{M^{6}N^{1/2}}\right)^{1/3}\cdot$ (3.7)

We, also, calculate the remaining inflationary observables via the relations

$\displaystyle n_{\rm s}=\>1-6\epsilon_{\star}\ +\ 2\eta_{\star}\simeq
1-\frac{2}{N_{\star}}-\frac{3N}{p^{2}N_{\star}^{2}}+\frac{V_{\rm
HI0}}{m^{2}N_{\star}^{2}}\left({2N\over p^{2}}+{4N\over p}+N\ln\sigma_{\rm
c}^{2}\right)\,,$ (3.8a) $\displaystyle\alpha_{\rm s}=\>{2\over
3}\left(4\eta_{\star}^{2}-(n_{\rm
s}-1)^{2}\right)-2\xi_{\star}\simeq-\frac{2}{N_{\star}^{2}}-\frac{6N}{p^{2}N_{\star}^{3}}+\frac{NV_{\rm
HI0}}{m^{2}N_{\star}^{3}}\left({4\over p^{2}}+{10\over p}+\ln\sigma_{\rm
c}^{4}\right),\>\>\>\>\>\>\>\>$ (3.8b) $\displaystyle
r=16\epsilon_{\star}\simeq\frac{8N}{p^{2}N_{\star}^{2}}-\frac{16N^{2}V_{\rm
HI0}}{p^{2}m^{2}N_{\star}^{3}}\left({1\over p^{2}}+{1\over p}+\ln\sigma_{\rm
c}\right),$ (3.8c)

where $\xi={V_{\rm HI,\widehat{}\sigma}V_{\rm
HI,\widehat{}\sigma\widehat{}\sigma\widehat{}\sigma}/V_{\rm HI}^{2}}$, the
variables with subscript $\star$ are evaluated at $\sigma=\sigma_{\star}$ and
the approximate expressions are obtained by expanding the exact result in
powers of $1/N_{\star}$. A clear dependence of the observables on the
parameters of $V_{\rm HI}$ in Vhi arises which is expected to modify
considerably the – dominating at leading order in the expansions above –
predictions of the pure E- and T-model inflation [30, 31].

We should, finally, note that the results above can be deliberated from our
ignorance about the Planck-scale physics, if we impose two additional
theoretical constraints on our models:

$\mbox{\footnotesize\sf(a)}\>\>V_{\rm HI}(\sigma_{\star})^{1/4}\leq
1\>\>\>\mbox{and}\>\>\>\mbox{\footnotesize\sf(b)}\>\>\sigma_{\star}\leq 1.$
(3.9)

As we show below, the first from the inequalities above is easily satisfied in
our set-up for $\kappa$ and $M$ values consistent with lda – see also res. Eq.
(3.9b) is fulfilled by construction since the $K$’s in Ket with the
parameterizations in para induce a kinetic pole for $\sigma=1$ and so HI takes
place for $\sigma<1$. The introduction of a possible parameter multiplying $S$
in Ket can be absorbed by a reparametrization of the free parameters of the
model – see e.g. so.

##  4 Numerical Results

The outputs of the analysis above can be refined numerically and employed in
order to delineate the available parameter space of the models. In particular,
we confront the quantities in Eqs. (3.4) and (3.6) with the observational
requirements [3]

$\mbox{\footnotesize\sf(a)}\>\>N_{\star}\simeq
61.3+\frac{1}{12}\ln\frac{\pi^{2}g_{\rm rh*}T_{\rm rh}^{4}}{30V_{\rm
HI}(\sigma_{\rm
f})}\>\>\>\mbox{and}\>\>\>\mbox{\footnotesize\sf(b)}\>\>\sqrt{A_{\rm s}}\simeq
4.588\cdot 10^{-5}.$ (4.1)

In deriving Eq. (4.1a) we assume that HI is followed in turn by a oscillatory
phase, with mean equation-of-state parameter $w_{\rm rh}\simeq 0$, radiation
and matter domination. We expect that $w_{\rm rh}=0$, corresponding to
quadratic potential, approximates rather well $V_{\rm HI}$ for
$\sigma\ll\sigma_{\rm c}$. Motivated by implementations of non-thermal
leptogenesis [39], which may follow HI, we set $T_{\rm rh}\simeq 4.1\cdot
10^{-10}$ for the reheat temperature. Also $g_{\rm rh*}=106.75$ is the energy-
density effective number of degrees of freedom which corresponds to SM
spectrum respectively.

As regards the remaining observables, we take into account the latest data
from _Planck_ (release 4) [15], baryon acoustic oscillations, _Cosmic
Microwave Background_ (CMB) lensing and BICEP/Keck [16]. Adopting the most
updated fitting in gws we obtain approximately the following allowed margins

$\mbox{\footnotesize\sf(a)}\>\>n_{\rm s}=0.965\pm
0.0074\>\>\>\mbox{and}\>\>\>\mbox{\footnotesize\sf(b)}\>\>r\leq 0.032,$ (4.2)

at 95$\%$ _confidence level_ (c.l.) with $|\alpha_{\rm s}|\ll 0.01$. The
constraint on $|\alpha_{\rm s}|$ is readily satisfied within the whole
parameter space of our models. Enforcing Eq. (4.1a) and (b) we can restrict
$\kappa$ and $\sigma_{\star}$ and compute the models’ predictions via Eqs.
(3.8a), (3.8b) and (3.8c), for any selected $N$, $\lambda/\kappa$, $m$ and $M$
– see Vhi1. Note that $\widetilde{m}$ is totally irrelevant for the
computation and can be fixed at a value (e.g., $\widetilde{m}=10m$) throughout
so that $\widehat{m}_{\theta}\gg H_{\rm HI}$. With this arrangement, $\Delta
V_{\rm HI}$ in Vrc is totally dominated by its second term in the equation
above. Given that $\Lambda$ in Vrc can be determined self-consistently as
mentioned in hi2, $\Delta V_{\rm HI}$ is calculated using as inputs the free
parameters of the tree-level potential in Vhi1. Compared to the original model
of HI [6], we here employ one more free parameter, $N$, related to the non-
minimal kinetic terms adopted in Ksset.

We embark on the presentation of our results by comparatively plotting the
variation of $V_{\rm HI}$ as a function of $\sigma$ in Fig. 4-(a) and
$\widehat{\sigma}$ in Fig. 4-(b) for EHI (solid lines) and THI (dashed lines).
In both cases we set $\lambda=\kappa$ and $M=0.1$ and achieve the central
$n_{\rm s}$ in Eq. (4.2a) by selecting $m=4.12\cdot 10^{-6}$ [$m=2.04\cdot
10^{-6}$] for EHI [THI]. The corresponding values of $\kappa$, $N_{\star}$,
$\sigma_{\star}$ and $r$ are listed in the third, fourth, fifth and sixth
leftmost column of the Table below the graphs. These results are obtained by
our numerical code taking into account exact expressions for $J,V_{\rm HI}$
and the other observables – i.e., Eqs. (2.9), (2.17), (3.4), and the leftmost
equalities in Eqs. (3.8a), (3.8b) and (3.8c). These values can be approximated
by those obtained by employing the formulas of inf – i.e., Eqs. (3.5), (3.7),
and the rightmost equalities in Eqs. (3.8a), (3.8b) and (3.8c). The outputs of
this computation are displayed in the five rightmost columns of the Table in
fig1. Note that in the case of analytic expressions we prefer to compare
$\sigma_{\star}$ derived by sgx with its numerical value and let $m$ as input
parameter. We remark that $N_{\star}$ is analytically underestimated in both
cases. This fact leads to an overestimation of $\sigma_{\star}$ and $\kappa$.
On the other hand, $n_{\rm s}$ and $r$ are closer to their numerical values
for EHI than for THI. The non-consideration of the second term in Vhi in the
denominators of sr seems to aggravates the error in analytic expressions of
THI. Despite their low accuracy, though, the analytical expressions of inf
assist us in understanding the general behavior of the inflationary dynamics.

In the plots of fig1 we verify the pretty stable mechanism [34, 26, 27] which
establishes EMI and TMI: $V_{\rm HI}$ expressed in terms of $\widehat{\sigma}$
develops a plateau for $\widehat{\sigma}\gg 1$ but $\sigma<1$ – since
$\widehat{\sigma}$ increases w.r.t $\sigma$ as inferred from je. Indeed, the
$\sigma_{\star}$ values depicted in Fig. 4-(a) – and arranged in the Table of
fig1 – get enhanced according to je, i.e., $\widehat{\sigma}_{*}=0.99$
[$1.66$] for EHI [THI] – see Fig. 4-(b). Contrary to the standard EMI and TMI,
though, we here observe that (i) the inflationary path terminates at
$\sigma=\sigma_{\rm c}$ due to the instability in vevi – this is the same for
both EHI and THI since $M$ and $\lambda/\kappa$ are kept fixed in both cases –
and (ii) the required $\sigma_{\star}$ does not lie extremely close to the
location of the pole at $\sigma=1$ and so the relevant tuning of the initial
conditions, somehow quantified – cf. so,epole – by the quantity

$\Delta_{1*}=\left(1-\sigma_{\star}\right),$ (4.3)

is less severe here. Namely, we obtain $\Delta_{1*}=19\%$ [$\Delta_{1*}=24\%$]
for EHI [THI], whereas $\Delta_{1*}\leq 10\%$ in all the models of chaotic
inflation studied in so,epole, etpole. Also no proximity is needed between
$\sigma_{\star}$ and $\sigma_{\rm c}$ as in the last paper of su5.

Model | $m/10^{-6}$ | Numerical Values | Analytic Values
---|---|---|---
|  | $\kappa/0.001$ | $N_{\star}$ | $\sigma_{\star}$ | $r/0.001$ | $\kappa/0.001$ | $N_{\star}$ | $\sigma_{\star}$ | $n_{\rm s}$ | $r/0.001$
EHI | $4.12$ | $1.07$ | $50.6$ | $0.81$ | $3.83$ | $2$ | $41.7$ | $0.85$ | $0.965$ | $3.1$
THI | $2.04$ | $0.87$ | $50.2$ | $0.76$ | $2.44$ | $1.3$ | $47.7$ | $0.79$ | $0.95$ | $1.45$

[]Variation of $V_{\rm HI}$, $V_{\rm HI}/V_{\rm HI0}-1$, as a function of (a)
$\sigma$ for $\sigma_{\rm c}\leq\sigma<1$ and (b) $\widehat{\sigma}$ for
$\widehat{\sigma}(\sigma_{\rm c})\leq\widehat{\sigma}<\widehat{\sigma}(1)$
fixing $M=0.1$, $N=1$, $\lambda=\kappa$ and $n_{\rm s}=0.965$. We consider EHI
(solid lines) or THI (dashed lines). Values corresponding to $\sigma_{\star}$
and $\sigma_{\rm c}$ (a) or $\widehat{\sigma}_{*}$ and $\widehat{\sigma}_{\rm
c}$ (b) are also depicted. Some of the parameters of our models – derived by
employing our numerical and analytic formulae – are displayed in the Table .

Employing data from the two representative cases depicted in fig1, we can
analyze further the inflationary mechanism of our models. Namely, (i) the
semiclassical approximation, used in our analysis, is perfectly valid avoiding
possible corrections from quantum gravity, since $V_{\rm
HI}^{1/4}(\sigma_{\star})=3.3\cdot 10^{-3}$ [$V_{\rm
HI}^{1/4}(\sigma_{\star})=2.9\cdot 10^{-3}$] for EHI [THI] is less than the
ultraviolet cut-off scale, $m_{\rm P}=1$, of theory – cf. subP; (ii) any
possible running of the quantities measured at $\Lambda$ is negligible, since
the scale $\Lambda$ in Vrc is found to be $\Lambda=1.6\cdot 10^{-3}$
[$\Lambda=9.5\cdot 10^{-4}$] for EHI [THI], i.e., quite close to the
aforementioned $V_{\rm HI}^{1/4}(\sigma_{\star})$’s; (iii) the contribution of
$\Delta V_{\rm HI}$ to $V_{\rm HI}$ is quite suppressed, since $(\Delta V_{\rm
HI}/V_{\rm HI})(\sigma_{\star})=3.6\cdot 10^{-5}$ [$(\Delta V_{\rm HI}/V_{\rm
HI})(\sigma_{\star})=10^{-5}$] for EHI [THI]; (iv) the criterion of kbou is
comfortably fulfilled, since ${\Delta m_{\phi_{1}}^{2}}/{H_{\rm HI}^{2}}\simeq
536.5$ [${\Delta m_{\phi_{1}}^{2}}/{H_{\rm HI}^{2}}\simeq 199.6$] for EHI
[THI] and so no modification of Eq. (4.1a) is necessitated.

A prominent aim of the marriage between E- and T-models and HI is the
diminishment of $n_{\rm s}$ at the acceptable level of Eq. (4.2a). The
achievement of this goal is readily demonstrated in fig2 where we display the
resulting values of $n_{\rm s}$ versus $m$ for EHI – see Fig. 4-(a) – and THI–
see Fig. 4-(b) – with the constraints of ntot and (4.2b) being fulfilled. We
set $M=0.1$, $N=1$ and $\lambda=\kappa$ (solid line), $\lambda=2\kappa$
(dashed line) and $\lambda=\kappa/2$ (dot-dashed line). The observational
allowed region of Eq. (4.2a) is also limited by thin lines. In both cases $m$
turns out to be of the same order of magnitude (around $10^{-6}$) but the
allowed region for EHI is wider than that in THI. With fixed $M$ and $N$,
which keep $\kappa$ roughly unchanged – cf. lda –, we observe that as $m$
increases (without change order of magnitude) the positive contribution in the
approximate part of ns decreases together with $n_{\rm s}$, which enters the
observationally favored region of Eq. (4.2a). For even large $m$ values,
$\sigma_{\star}$ in sgx approaches unity – since the second term of the
denominator overshadows the first one –, and so ntot ceases to be satisfied
and the various lines terminate.

[]The allowed by Eqs. (4.1) and (4.2b) values of $n_{\rm s}$ versus $m$ for
$M=0.1$, $N=1$, several $\lambda/\kappa$’s (indicated in the graphs) and (a)
EHI or (b) THI. The region of Eq. (4.2a) is also limited by thin lines.

Model | $m/10^{12}~{}{\mbox{\rm GeV}}$ | $M/10^{17}~{}{\mbox{\rm GeV}}$ | $\kappa/10^{-3}$ | $\alpha_{\rm s}/10^{-4}$ | $r/10^{-3}$ | $\Delta_{1*}~{}(\%)$
---|---|---|---|---|---|---
EHI | $9.98-29.7$ | $1.5-16.1$ | $0.00047-3.2$ | $(-6)-(-2)$ | $2.3-4.5$ | $1-32$
THI | $3.3-14.8$ | $1.8-17.5$ | $0.0011-1.7$ | $(-6)-10$ | $0.6-2.96$ | $0.2-26$

[]Allowed curves in the $m-M$ [$\kappa-M$] plane (a1 and b1) [(a2 and b2)] for
$N=1$, $n_{\rm s}=0.965$ and various $\lambda/\kappa$’s indicated in the
graphs. We consider EHI (a1 and a2) or THI (b1 and b2). The allowed ranges of
the various parameters for $\lambda=\kappa$ are listed in the table with units
reinstalled.

Encouraged by the acceptable $n_{\rm s}$ results above, we proceed to the
delineation of the allowed parameter space of our models setting $N=1$ and
$n_{\rm s}$ equal to its central value in Eq. (4.2a) – possible variation of
$n_{\rm s}$ within its allowed margin yields relatively narrow strips which
would be indistinguishable especially in the $\kappa-M$ plane. We present our
findings in fig3 devoting fig3-(a1) and (a2) to EHI and fig3-(b1) and (b2) to
THI. In particular, we display in fig3-(a1) and (b1) the allowed contours in
the $m-M$ plane and in fig3-(a2) and (b2) the allowed curves in the $\kappa-M$
plane. The conventions adopted for the various lines are also shown in the
right-hand side of each graph. In particular, the solid, dashed and dot-dashed
lines correspond to $\lambda=\kappa$, $\lambda=2\kappa$ and
$\lambda=\kappa/2$. The various lines terminate at low $M$ (or large $\kappa$)
values due to the augmentation of the contribution of $\Delta V_{\rm HI}$ in
Vrc to $V_{\rm HI}$ in Vhi. Indeed, when $|\Delta V_{\rm HI}/V_{\rm HI}|$
approaches $8\cdot 10^{-4}$, $\Delta V_{\rm HI}$ starts influence the
predictions of HI. Such a situation is encountered for $\kappa\gtrsim 6\cdot
10^{-3}$ [$\kappa\gtrsim 3\cdot 10^{-3}$] in EHI [THI]. On the other side, the
almost horizontal part of the various lines stop for large $m$ (low $\kappa$)
values since $\sigma_{\star}$ reaches the pole region – see sgx – and the
computation related to ntot becomes unstable. At these points we encounter the
lowest possible $\Delta_{1*}$’s which tend to the ugly amount of $0.2\%$. For
$\lambda=2\kappa$ the upper bounds of the corresponding lines are obtained for
$M=1$. We adopt this conservative bound which assures meaningful
$\langle{\phi_{1}}\rangle$ values and comfortable fulfillment of kbou.

In the Table of fig3, we accumulate indicatively explicit values of the
various parameters (restoring units hereafter) for $\lambda=\kappa$. From the
listed values we notice that the allowed $M$’s approach mainly the string
scale contrary to the SUSY versions of HI where $M$ turns out to be [7, 8]
close to the SUSY GUT scale, $M_{\rm GUT}\simeq 2.86\cdot 10^{16}~{}{\rm
GeV}$. This fact, though, neither violates Eq. (3.9a) nor amplifies
dangerously $\Delta V_{\rm HI}$ in Vhi since the low enough $\kappa$ values,
which are compatible with lda, keep $V_{\rm HI}(\sigma_{\star})$ and $\Delta
V_{\rm HI}$ under control. Comparing the data for EHI and THI, we remark that
larger $m$’s and $\Delta_{1*}$’s and wider ranges of $\kappa$ are allowed in
EHI whereas both models share similar $M$’s. For the same inputs of the Table
fig3, we can estimate the inflaton mass $\widehat{m}_{\sigma}$ from mass, with
results

$0.08\lesssim{\widehat{m}_{\sigma}}/{10^{13}~{}{\mbox{\rm GeV}}}\lesssim
66\>\>\>\mbox{}\mbox{for
EHI;}\>\>\>0.27\lesssim{\widehat{m}_{\sigma}}/{10^{13}~{}{\mbox{\rm
GeV}}}\lesssim 44\>\>\>\mbox{for THI.}$ (4.4)

These ranges let, in principle, open the possibility of non-thermal
leptogenesis [39], if we introduce a suitable coupling between $\Phi$ and the
right-handed neutrinos [21].

[]Allowed by ntot values of $N$ versus $m$ for $M=0.1$, $n_{\rm s}=0.965$
several $\lambda/\kappa$’s, indicated in the graphs, and (a) EHI, or (b) THI.
Shown is also the variation of $r$ in grey along the lines.

Varying $N$ beyond unity, we are able to explore another portion of the
parameter space of our models as in fig4. We depict there the allowed curves
in the $m-N$ plane for $M=0.1$, central $n_{\rm s}$ in Eq. (4.2a) and EHI –
see Fig. 4-(a) – or THI – see Fig. 4-(b). We use the same shape code for the
lines as in fig4 with the change that in Fig. 4-(b) the dashed line
corresponds to $\lambda=3\kappa/2$ and not to $\lambda=2\kappa$ as in all the
previous cases. The reason of this replacement is that RCs invalidate almost
the whole parameter space of THI for $\lambda=2\kappa$ and $N>1$. From our
data we see that $r$ increases with $N$ as in the standard EMI and TMI [34,
28] and in agreement with the analytical findings in rs. The progressive
enhancement of $r$ along each line is shown by gray numbers. Note that a
segment of the solid and dot-dashed lines in the left graph of fig4 coincide
between each other. We display the $r$’s corresponding to $\lambda=\kappa/2$
along the common part of these lines. The upper bound on $r$ in Eq. (4.2b)
provides an upper bound on $N$ which is related to the geometry of the moduli
space via Rs. Namely, we find the following maximal $N$ values

$N^{\rm max}=\begin{cases}\begin{matrix}9.6,&13,&18\\\
13.1,&17.5,&33.8\end{matrix}\end{cases}\mbox{for}\>\>\>\lambda/\kappa=\begin{cases}\begin{matrix}2,&1,&0.5\>\>\>&\mbox{({EHI})}\cr
1.5,&1,&0.5\>\>\>&\mbox{({THI})}\end{matrix}\end{cases}.$ (4.5)

We observe that the largest $N^{\rm max}$ values are obtained for
$\lambda<\kappa$. This can be understood by the observation that
$\ln\sigma_{\rm c}<0$ and dominates the parenthesis in rs. For
$\lambda<\kappa$, $|\ln\sigma_{\rm c}|$ is lower than its value for
$\lambda\geq\kappa$ and so, larger $N$ values are needed so as $r$ to reach
its bound of Eq. (4.2b). In all, from the data of fig4 we find the following
allowed ranges of parameters

$\begin{matrix}&1\lesssim{m/10^{13}~{}{\mbox{\rm GeV}}}\lesssim
10.2,&1\lesssim{\kappa}/{10^{-3}}\lesssim
2.8&\mbox{and}&1.4\lesssim{\Delta_{1*}/0.1}\lesssim
3.6&\mbox{for~{}~{}EHI;}\\\ &3.8\lesssim{m/10^{12}~{}{\mbox{\rm GeV}}}\lesssim
84,&6.8\lesssim{\kappa}/{10^{-4}}\lesssim
29&\mbox{and}&1.8\lesssim{\Delta_{1*}/0.1}\lesssim
3.0&\mbox{for~{}~{}THI.}\end{matrix}$ (4.6)

In the ranges above, we obtain $-6\lesssim\alpha_{\rm s}/10^{-4}\lesssim-1.5$
or $-2.8\lesssim\alpha_{\rm s}/10^{-4}\lesssim 6$ for EHI or THI respectively.
We remark that $\Delta_{1*}$ increases with $N$ (and $r$) in both cases.

##  5 Conclusions

Inspired by the successive data releases on CMB perturbations – which favor
the inflationary paradigm – and the negative LHC results regarding SUSY, we
focused on the well-known model of non-SUSY HI (i.e. hybrid inflation) [6]
attempting to reconcile it with data. Our proposal is based on the non-SUSY
potential in Vhi1 and the non-minimal kinetic terms of the inflaton sector
with a pole of order $p=1$ or $2$ – see Eqs. (2.1) and (2.4). These terms
emerge thanks to the adoption of the logarithmic Kähler potentials in Ket
which parameterize hyperbolic moduli manifolds. Therefore, our models although
non-SUSY inherit the Kählerian type of moduli geometry from the SUSY context.
This assumption restricts considerably the allowed $p$ values as shown in
Appendix A. The resulting models are called EHI (for $p=1$) and THI (for
$p=2$) due to the function which relates the initial and the canonically
normalized inflaton – see je.

Selecting the gauge group broken after the end of HI and the representation of
the waterfall field we succeeded to evade the formation of any topological
defects. We also included the minimal possible one-loop RCs (i.e., radiative
corrections) to the inflationary potential and restricted ourselves to the
portion of the parameter space where the waterfall regime occurs suddenly. We
found ample and natural space of parameters compatible with all the imposed
restrictions. E.g., for $N=1$, $\lambda=\kappa$ and the observationally
central value of $n_{\rm s}$, we find the allowed ranges shown in the Table of
fig3. Increasing $N$ beyond unity, for the same $n_{\rm s}$ and
$\lambda/\kappa$ values, we verify an enhancement on the $r$ values which
reaches its maximal allowed value for $N=13$ or $N=17.5$ within EHI or THI,
respectively. It is gratifying that no extensive proximity between the values
of the inflaton field at the horizon crossing of the pivot scale and the
location of the pole is needed – see the $\Delta_{1*}$ values in the
aforementioned Table which increase with $N$. Therefore, EHI and THI can be
characterized as more natural than the original E- and T-model inflation as
regards this tuning. On the other hand, EHI and THI are less predictive due to
the more parameters which let corridors for larger variation of the
inflationary observables. It is also remarkable that the one-loop RCs in Vrc
do not affect the inflationary solutions for low enough values of the relevant
coupling constants $\kappa$ and $\lambda$ – see fig3.

Comparing our work with that of hlinde, we may remark the following
differences-novelties here: (i) We motivated EHI and THI as non-linear sigma
models; (ii) we overcome the problem of the topological defect through the
consideration of a specific representation of the waterfall field; (iii) we
explored not only THI but also EHI; (iv) we considered flat geometry for the
waterfall field; (v) we included the minimal one-loop RCs to the inflationary
potential; (vi) we delineated the parameter space of the models avoiding any
subcritical production of e-foldings. On the other hand, we did not embed our
models in SUGRA as done in hlinde. The standard mechanism [40] of embedding is
not so effective for HI since the stabilization of the waterfall fields at the
origin does not reproduce the non-SUSY potential and a rather different
situation may emerge. Moreover, the $R$ symmetry preserved by the
superpotential [7] restricts further the possible forms of $K$, if it is
imposed also to it. Modification of the present model in order to generate
gravitational waves compatible with the NANOGrav reported signal [12], via a
late production of cosmic strings [11] or via a subcritical stage of fast-roll
inflation [14] could be another interesting target for future analysis.

#### Acknowledgments:

I would like to thank K. Dimopoulos for useful discussions. This research work
was supported by the Hellenic Foundation for Research and Innovation
(H.F.R.I.) under the “First Call for H.F.R.I. Research Projects to support
Faculty members and Researchers and the procurement of high-cost research
equipment grant” (Project Number: 2251).

## Appendix A: Kinetic Poles and Kähler Potentials

It is clear that the consideration of a kinetic pole is crucial for the
successful implementation of our proposal. For this reason it would be
interesting to check if the pole order could be different than the values,
$p=1$ and $2$, considered in our work. In general grounds, the metric of
moduli space $K_{SS^{*}}$ may be a totally arbitrary real function and so it
may include pole of any $p$. This freedom, though, can be drastically reduced
if we confine ourselves to metrics originated from some Kähler potential, $K$.

To highlight this key issue, we generalize the $K$’s considered in Ket
adopting the following

$K=-N_{K}\ln\left(1-\frac{1}{3}\left(|S|^{2n}+S^{q}S^{*l}+S^{l}S^{*q}\right)\right).$
(A.1)

The argument of $\ln$ includes a real function $|S|^{2n}$ and a complex one,
$S^{q}S^{*l}$ – together with its complex conjugate which result to a real
contribution. We set unity inside $\ln$ to assure a well-behaved expansion for
low $S$ values, as required by the consideration of non-renormalizable
operators. We insist on logarithmic $K$’s since these are usually employed in
string theory and assure fractional metrics with possible poles. The metric
$K_{SS^{*}}$, generated by the $K$ above, along the inflationary path in vevi
is found to be

$\langle{K_{SS^{*}}}\rangle_{\rm
HI}=N_{K}\frac{3n^{2}\sigma^{2n}+(l-q)^{2}\sigma^{2(l+q)}+2\sigma^{l+q}\left(3lq+(l-n)(n-q)\sigma^{2n}\right)}{\sigma^{2}\left(3-\sigma^{2n}-2\sigma^{l+q}\right)^{2}}\,.$
(A.2)

Varying the exponents $n,q$ and $l$ in the domain $(-3,3)$ with unit step we
obtain the metrics

$\langle{K_{SS^{*}}}\rangle_{\rm
HI}=N_{K}\cdot\begin{cases}1/f_{1}^{2}&\mbox{for}\>\>(q,l,n)=(1,0,0)\>\&\>(0,1,0),\\\
1/f_{2}^{2}&\mbox{for}\>\>\>(q,l,n)=(0,0,\pm
1),~{}\pm(1,1,0)\>\&\>\pm(1,1,1),\end{cases}$ (A.3)

which reproduce the – already used in Kc – kinetic terms

$N\dot{}\sigma^{2}/2f_{p}^{2}\>\>\>\mbox{with}\>\>f_{p}=1-\sigma^{p}$ (A.4)

for $p=1$ and $2$. The former is obtained for $N=N_{K}/2$ whereas the latter
for $N=2N_{K}$. Note that $\langle{K_{SS^{*}}}\rangle_{\rm HI}$ employed in
THI can be derived not only for $n=1$ as expected – see e.g. Ket – but also
for $n=-1$. On the other hand, our scanning does not reveal any other kinetic
terms of the type in kin for $p>2$. The simplest metrics including $f_{p}$
with $p=3,4,5$ and $6$ in the denominator are

$\langle{K_{SS^{*}}}\rangle_{\rm
HI}=\frac{9N_{K}\sigma^{4}}{4f_{3}^{2}},\>\frac{4N_{K}\sigma^{2}}{f_{4}^{2}},\>\frac{N_{K}(24+\sigma^{5})\sigma^{3}}{4f_{5}^{2}}\>\>\mbox{and}\>\>\frac{9N_{K}\sigma^{4}}{f_{6}^{2}}\cdot$
(A.5)

It is rather uncertain if the $\langle{K_{SS^{*}}}\rangle_{\rm HI}$’s above
yield kinetic terms which support inflationary solutions. As a consequence, it
is highly nontrivial, if not impossible, to achieve kinetic terms of the form
in kin with $p>2$ starting from a Kähler potential.

## References

  * [1] M. Yamaguchi, Supergravity based inflation models: a review, Class. Quant. Grav. 28, 103001 (2011) [ 1101.2488]; J. Ellis, M.A.G. Garcia, N. Nagata, D.V. Nanopoulos, K.A. Olive and S. Verner, Building Models of Inflation in No-Scale Supergravity, Int. J. Mod. Phys. D 29, 16, 2030011 (2020) [2009.01709].
  * [2] H. Pacey [ATLAS and CMS], EW SUSY production at the LHC, PoS PANIC2021, 133 (2022).
  * [3] Y. Akrami et al. [Planck Collaboration], Planck 2018 results. X. Constraints on inflation, Astron. Astrophys. 641, A10 (2020) [1807.06211].
  * [4] N. Aghanim et al. [Planck Collaboration], _Planck 2018 results. VI. Cosmological parameters_ , _Astron. Astrophys._ 641, A6 (2020) – Erratum: _Astron. Astrophys._ 652, C4 (2021) [1807.06209].
  * [5] J. Martin, C. Ringeval and V. Vennin, Encyclopædia Inflationaris, Phys. Dark Univ. 5, 75 (2014) [ 1303.3787]; J. Martin, C. Ringeval R. Trotta and V. Vennin, The Best Inflationary Models After Planck, J. Cosmology Astropart. Phys03201403914 [1312.3529].
  * [6] A.D. Linde, Hybrid inflation, Phys. Rev. D491994748 [astro-ph/9307002]; E.J. Copeland et al., False Vacuum Inflation with Einstein Gravity, Phys. Rev. D 49, 6410 (1994) [astro-ph/9401011].
  * [7] G.R. Dvali, Q. Shafi and R.K. Schaefer, Large scale structure and supersymmetric inflation without fine tuning, Phys. Rev. Lett.7319941886 [hep-ph/9406319].
  * [8] B. Kyae and Q. Shafi, Flipped SU(5) predicts delta T/T, Phys. Lett. B 635, 247 (2006) [0510105]; S. Khalil, M.U. Rehman, Q. Shafi and E.A. Zaakouk, Inflation in Supersymmetric SU(5), Phys. Rev. D 83, 063522 (2011) [1010.3657]; C. Pallis and Q. Shafi, Update on Minimal Supersymmetric Hybrid Inflation in Light of PLANCK, Phys. Lett. B 725, 327 (2013) [1304.5202].
  * [9] G. Lazarides and C. Panagiotakopoulos, Smooth hybrid inflation, Phys. Rev. D521995R559 [hep-ph/9506325]; R. Jeannerot, S. Khalil, G. Lazarides and Q. Shafi, Inflation and monopoles in supersymmetric SU(4)(C) x SU(2)(L) x SU(2)(R), 102000012 [hep-ph/0002151].
  * [10] T.W.B. Kibble, Topology of Cosmic Domains and Strings, J. Phys. A 9, 387 (1976).
  * [11] W. Buchmüller, V. Domcke, and K. Schmitz, From NANOGrav to LIGO with metastable cosmic strings, Phys. Lett. B 811, 135914 (2020) [2009.10649]; W. Buchmüller, V. Domcke, and K. Schmitz, Stochastic gravitational-wave background from metastable cosmic strings, _J. Cosmol. Astropart. Phys._ 12, no. 12, 006 (2021) [2107.04578].
  * [12] Z. Arzoumanian et al. [NANOGrav], The NANOGrav 12.5 yr Data Set: Search for an Isotropic Stochastic Gravitational-wave Background, Astrophys. J. Lett. 905, no. 2, L34 (2020) [2009.04496]; B. Goncharov et al. On the Evidence for a Common-spectrum Process in the Search for the Nanohertz Gravitational-wave Background with the Parkes Pulsar Timing Array, Astrophys. J. Lett. 917, no. 2, L19 (2021) [2107.12112].
  * [13] V.C. Spanos and I.D. Stamou, Gravitational waves and primordial black holes from supersymmetric hybrid inflation, Phys. Rev. D 104, no. 12, 123537 (2021) [2108.05671].
  * [14] K. Dimopoulos, Waterfall “kination” can generate observable primordial gravitational waves, J. Cosmology Astropart. Phys102022027 [2206\. 02264].
  * [15] Y. Akrami et al. [Planck Collaboration], $Planck$ intermediate results. LVII. Joint Planck LFI and HFI data processing, Astron. Astrophys. 643, A42 (2020) [2007.04997].
  * [16] P.A.R. Ade et al. [BICEP and Keck Collaboration], Improved Constraints on Primordial Gravitational Waves using Planck, WMAP, and BICEP/Keck Observations through the 2018 Observing Season, Phys. Rev. Lett. 127, no. 15, 151301 (2021) [2110.00483].
  * [17] M. Tristram et al., _Improved limits on the tensor-to-scalar ratio using BICEP and Planck_ , _Phys. Rev. Lett._ 127, 151301 (2021) [2112.07961].
  * [18] G. Ye, J.Q. Jiang and Y.S. Piao, Towards hybrid inflation with $n_{\rm s}=1$ in light of Hubble tension and primordial gravitational waves, 2205.02478.
  * [19] C. Pallis, _Non-Minimally Gravity-Coupled Inflationary Models_ , Phys. Lett. B 692, 287 (2010) [1002.4765].
  * [20] S.R. Coleman and E.J. Weinberg, _Radiative Corrections as the Origin of Spontaneous Symmetry Breaking_ , Phys. Rev. D719731888.
  * [21] M. Ur Rehman, Q. Shafi and J.R. Wickman, Hybrid Inflation Revisited in Light of WMAP5, Phys. Rev. D792009103503 [0901.4345].
  * [22] S. Koh and M. Minamitsuji, Non-minimally coupled hybrid inflation, Phys. Rev. D 83, 046009 (2011) [ 1011.4655].
  * [23] G. Lazarides and C. Pallis, Reducing the spectral index in F-term hybrid inflation through a complementary modular inflation, Phys. Lett. B 651, 216-223 (2007) [0702260].
  * [24] L. Boubekeur and D.H. Lyth, Hilltop inflation, J. Cosmology Astropart. Phys072005010 [0502047].
  * [25] B.J. Broy, M. Galante, D. Roest and A. Westphal, _Pole inflation, Shift symmetry and universal corrections_ , 122015149 [1507.02277]; T. Terada, _Generalized Pole Inflation: Hilltop, Natural, and Chaotic Inflationary Attractors_ , 7602016674 [1602.07867].
  * [26] C. Pallis, Pole-Induced Higgs Inflation With Hyperbolic Kaehler Geometries, J. Cosmology Astropart. Phys052021043 [2103.05534]; C. Pallis, $SU(2,1)/(SU(2)\times U(1))$ B-L Higgs Inflation, J. Phys. Conf. Ser. 2105, no. 12, 12 (2021) [2109.06618].
  * [27] C. Pallis, An alternative framework for E-model inflation in supergravity, Eur. Phys. J. C 82, no. 5, 444 (2022) [2204.01047].
  * [28] C. Pallis, Pole Inflation in Supergravity, PoS CORFU 2021, 078 (2021) [2208.11757]; C. Pallis, Formulating E- & T-Model Inflation in Supergravity, 2209.08325.
  * [29] R. Kallosh and A. Linde, Hybrid cosmological attractors, Phys. Rev. D 106, no. 2, 023522 (2022) [2204\. 02425].
  * [30] R. Kallosh and A. Linde, _BICEP/Keck and cosmological attractors_ , _J. Cosmol. Astropart. Phys._ 12, no. 12, 008 (2021) [2110.10902].
  * [31] J. Ellis, M.A. G. Garcia, D.V. Nanopoulos, K.A. Olive and S. Verner, _BICEP/Keck constraints on attractor models of inflation and reheating_ , _Phys. Rev. D_ 105, no. 4, 043504 (2022) [2112.04466].
  * [32] R. Kallosh, A. Linde and D. Roest, _Superconformal Inflationary $a$-Attractors_, 112013198 [1311.0472].
  * [33] J. Ellis, D. Nanopoulos and K. Olive, Starobinsky-like Inflationary Models as Avatars of No-Scale Supergravity, J. Cosmology Astropart. Phys102013009 [1307.3537].
  * [34] R. Kallosh and A. Linde, _Universality Class in Conformal Inflation_ , J. Cosmology Astropart. Phys072013002 [1306.5220].
  * [35] T. Watari and T. Yanagida, GUT phase transition and hybrid inflation, Phys. Lett. B 589, 71 (2004) [ 0402125].
  * [36] B. Brahmachari, E. Ma and U. Sarkar, Left-Right Model of Quark and Lepton Masses without a Scalar Bidoublet, Phys. Rev. Lett. 91, 011801 (2003); F. Siringo, Symmetry breaking of the symmetric left-right model without a scalar bidoublet, Eur. Phys. J. C 32, 555 (2004) [0307320].
  * [37] G. Lazarides and C. Pallis, Shift Symmetry and Higgs Inflation in Supergravity with Observable Gravitational Waves, J. High Energy Phys. 11, 114 (2015) [1508.06682].
  * [38] S. Clesse, Hybrid inflation along waterfall trajectories, Phys. Rev. D 83, 063518 (2011) [1006.4522]; H. Kodama, K. Kohri and K. Nakayama, On the waterfall behavior in hybrid inflation, Prog. Theor. Phys.  126, 331 (2011) [1102.5612].
  * [39] G. Lazarides and Q. Shafi, Origin of matter in the inflationary cosmology, 2581991305; G. Lazarides, R.K. Schaefer and Q. Shafi, Supersymmetric inflation with constraints on superheavy neutrino masses, Phys. Rev. D5619971324 [hep-ph/9608256].
  * [40] R. Kallosh, A. Linde and T. Rube, General inflaton potentials in supergravity, Phys. Rev. D832011043507 [1011.5945].
  * [41]


# Oblique Corrections in the MSSM at One Loop. I. Scalars

Yao Yu and Sibo Zheng
Department of Physics, Chongqing University, Chongqing 401331, P.R. China

Abstract

This paper is devoted to reconsider the one-loop oblique corrections arising
from the scalar superpartners in the MSSM, i.e, the squarks, the sleptons and
the scalars in Higgs sector. We explicitly present the complete one-loop forms
of self-energy corrections to the gauge bosons of SM electroweak gauge groups,
as well as their descendants the $S$, $T$ and $U$ parameters, which can be
directly applied to constrain the parameter space of the MSSM. Our results
about one-loop self energies are found to agree with Ref. [7, 14].
Nevertheless, the $S$, $T$ and $U$ parameters aren’t in agreement with Ref.
[8].

##  1 Introduction

With LHC keeping running on searches of the standard model (SM) Higgs with
mass around 125 GeV [1], we are at the dawn when studies of Higgs physics
become a field of the precise test, and we find out whether low-scale
supersymmetry is going to show up.

Among frameworks in which the implication of the recent LHC results in new
physics are explored, the electroweak precise test always stands as an
important way for studying phenomenology of Higgs physics. The logic is that
SM (naturally including the Higgs particle $h$) and its extensions such as
supersymmetric SM can be analyzed through considering the quantum corrections,
more concretely oblique corrections [3, 4] in these theories to the
electroweak precise test observables. In SM the oblique corrections mainly
depend on the Higgs particle mass $m_{h}$ and top quark mass, it follows that
electroweak precise test provides useful constraint on $m_{h}$ [6]. For
example, it is very successful in analyzing technicolor models [4].

This method can be similarly applied in SUSY models, from which the masses of
supersymmetric particles can be constrained (see, e.g, [14]). But this method
is not very efficient due to some reasons. One is that there are so many mass
parameters in SUSY models. Another is that the bounds on these masses are not
yet sufficient before running of the LHC collider [2]. So even in the minimal
supersymmetric standard model (MSSM) it is impossible to make firm claims. In
order to extract useful information, as discussed in the literature, various
limits such as highly degenerate [5, 6] or super heavy masses [8, 9, 10] of
supersymmetric particles are considered in SUSY models.

However, these approximations are too simple to be suitable when the present
LHC data is incorporated into the MSSM. Also, it can be verified that the
analytic results related to our discussion usually do not consistent in the
literature. Due to these observations, we should ascertain the oblique
corrections in the MSSM at first. In this paper, we will address this issue by
using the two-component formalism of Lagrangian for supersymmetric particles
under the electroweak group. The main outcomes are two-fold. At first, the
results we obtain can be applied to most general cases without taking any
assumptions about the quark- and lepton scalar masses. Moreover, our results
can’t reduce to those of Ref. [8] when we use the same limit as the authors
took. The difference is probably due to the missing or over counting of
Feynman loops involving heavy scalars as the internal lines.

We want to emphasize that while this work is being prepared, the LHC results
[1] indicate that its main task has been transformed from discovery to precise
tests of the SM-like Higgs and discovery of supersymmetry. The oblique
corrections to electroweak observables can still serve as a window to explore
the parameter space of superpartners’ masses by combining the data collected
by the LHC and other colliders. The implications to MSSM and NMSSM along this
line will be addressed elsewhere [19].

The paper is organized as follows. In section 2 , we derive the bosonic
oblique corrections in the MSSM. We divide the task into three parts, i.e, the
squark sector, the slepton sector, and the Higgs sector sector. We will
consider the fermionic contributions in the further [17]. In section 3, we
then make preliminary checks on the results presented in section 2 in terms of
the fact that the radiative corrections to electroweak mixing angle is finite.
In section 4, we present the bosonic results of $S$, $T$ and $U$ parameters in
the MSSM. The property that these parameters are also finite is more stringent
examination on the results than the one in the section 3. In section 5, we
conclude and discuss our main results. We find that our results about one-loop
self energies are found to agree with Ref. [7, 14]. Nevertheless, the $S$, $T$
and $U$ parameters do not math with those of Ref. [8]. In appendix A explicit
expressions for the functionals related to one-loop graphs are presented. We
would like to emphasize that the on-shell renormalization scheme is used
throughout this note.

##  2 One-loop Bosonic Contributions

Since we deal with the analytic calculation of one-loop self energy of SM
electroweak gauge bosons, it is convenient to use the two-component formalism
both for the supersymmetric scalars and fermions.

###  2.1 Lagrangian for Electroweak Scalar Doublets

To derive the Feynman rules, we refer to the Lagrangian for scalar doublets
$\Phi^{T}=(\phi_{1},\phi_{2})$ under the representation of electroweak gauge
group, which can be written as,

$\displaystyle{}\mathcal{L}_{H}\sim(\overrightarrow{D}_{\nu}\Phi)^{{\dagger}}(\overrightarrow{D}^{\nu}\Phi)=\Phi^{{\dagger}}\overleftarrow{D}_{\nu}\overrightarrow{D}^{\nu}\Phi$
(1)

with the definitions,

$\displaystyle{}\overrightarrow{D}_{\nu}\Phi=\left(\begin{array}[]{cc}\overrightarrow{\partial}_{\nu}-i(\frac{e}{2sc}-\frac{es}{c}Q_{1})Z_{\nu}-ieQ_{1}A_{\nu}&-i\frac{e}{\sqrt{2}s}W_{\nu}^{+}\\\
-i\frac{e}{\sqrt{2}s}W_{\nu}^{-}&\overrightarrow{\partial}_{\nu}+i(\frac{e}{2sc}+\frac{es}{c}Q_{2})Z_{\nu}-ieQ_{2}A_{\nu}\end{array}\right)\Phi$
(4)

Here $Q_{1}$ and $Q_{2}$ represent the $Y$-charges of up-type $\phi_{1}$ and
down-type $\phi_{2}$ respectively. And

$\displaystyle{}\Phi^{{\dagger}}\overleftarrow{D}_{\nu}=\Phi^{{\dagger}}\left(\begin{array}[]{cc}\overleftarrow{\partial}_{\nu}+i(\frac{e}{2sc}-\frac{es}{c}Q_{1})Z_{\nu}+ieQ_{1}A_{\nu}&i\frac{e}{\sqrt{2}s}W_{\nu}^{+}\\\
i\frac{e}{\sqrt{2}s}W_{\nu}^{-}&\overleftarrow{\partial}_{\nu}-i(\frac{e}{2sc}+\frac{es}{c}Q_{2})Z_{\nu}+ieQ_{2}A_{\nu}\end{array}\right)$
(7)

Parameters $s=\sin\theta_{W}$ and $c=\cos\theta_{W}$.

For EW singlet scalars $S$ such as right-hand squarks and sleptons, the
Lagrangian is given by,

$\displaystyle{}\mathcal{L}_{H}\sim(D_{\nu}S)^{{\dagger}}(D^{\nu}S)=S^{*}\left(\overleftarrow{\partial}_{\nu}-i\frac{s}{c}eQZ_{\nu}+ieQA_{\nu}\right)\left(\overrightarrow{\partial}^{\nu}+i\frac{s}{c}eQZ^{\nu}-ieQA^{\nu}\right)S$
(8)

Here $Q$ denotes the $Y$-charge of $S$ field.

From (1) to (8), we can derive the Feynman rules for MSSM scalars (see,
e.g,[15]). In what follows, we take the ’ t Hooft-Feynman gauge for non-
abelian gauge fields involved. It turns out that there are three type of
graphs as shown in Fig.1 needed to be considered,

Figure 1: Graphs that contribute to the self-energy of SM gauge bosons due to
scalar superpartners.

###  2.2 Squark Sector

The squark contributions are composed of those coming from three-generation
left-hand squarks $\tilde{q}_{Li}=(\tilde{u}_{i},\tilde{d}_{i})$ and their
right-hand $\tilde{u}_{Ri}$, $\tilde{d}_{Ri}$. Accoring to the Lagrangian (1)
we find that there are two types of Feynman diagrams that contribute to the
one-loop self-energy of vector bosons, as shown in fig 1. $(a)$ and fig. 1
$(b)$111 In the next subsection there will be an extra Feynman diagram needed
to be counted due to the Higgs VEVs.. We find 222We have neglected $1/16$
factor in each $\Pi^{VV^{\prime}}$ in this section, which will be restored in
the section 4.

$\displaystyle{}\Pi^{\gamma\gamma}_{L}(q^{2})$ $\displaystyle=$ $\displaystyle
N_{c}\frac{e^{2}}{9\pi^{2}}\sum_{i=1,2,3}\left[-16A(q^{2};m^{2}_{\tilde{u}_{i}},m^{2}_{\tilde{u}_{i}})-4A(q^{2};m^{2}_{\tilde{d}_{i}},m^{2}_{\tilde{d}_{i}})+8a(m^{2}_{\tilde{u}_{i}})+2a(m^{2}_{\tilde{d}_{i}})\right]$
$\displaystyle\Pi^{ZZ}_{L}(p^{2})$ $\displaystyle=$ $\displaystyle
N_{c}\frac{1}{\pi^{2}}\sum_{i=1,2,3}\left[-4\left(\frac{e}{2sc}-\frac{2es}{3c}\right)^{2}A(q^{2};m^{2}_{\tilde{u}_{i}},m^{2}_{\tilde{u}_{i}})-4\left(\frac{e}{2sc}-\frac{es}{3c}\right)^{2}A(q^{2};m^{2}_{\tilde{d}_{i}},m^{2}_{\tilde{d}_{i}})\right.$
(9) $\displaystyle+$
$\displaystyle\left.2\left(\frac{e}{2sc}-\frac{2es}{3c}\right)^{2}a(m^{2}_{\tilde{u}_{i}})+2\left(\frac{e}{2sc}-\frac{es}{3c}\right)^{2}a(m^{2}_{\tilde{d}_{i}})\right]$

and

$\displaystyle{}\Pi^{\gamma Z}_{L}(q^{2})$ $\displaystyle=$ $\displaystyle
N_{c}\frac{e}{3\pi^{2}}\sum_{i=1,2.3}\left[-8\left(\frac{e}{2sc}-\frac{2es}{3c}\right)A(q^{2};m^{2}_{\tilde{u}_{i}},m^{2}_{\tilde{u}_{i}})-4\left(\frac{e}{2sc}-\frac{es}{3c}\right)A(q^{2};m^{2}_{\tilde{d}_{i}},m^{2}_{\tilde{d}_{i}})\right.$
$\displaystyle+$
$\displaystyle\left.4\left(\frac{e}{2sc}-\frac{2es}{3c}\right)a(m^{2}_{\tilde{u}_{i}})+4\left(\frac{e}{2sc}-\frac{es}{3c}\right)a(m^{2}_{\tilde{d}_{i}})\right]$
$\displaystyle\Pi^{WW}_{L}(q^{2})$ $\displaystyle=$ $\displaystyle
N_{c}\frac{1}{2\pi^{2}}\frac{e^{2}}{s^{2}}\sum_{i=1,2.3}\left[\left(a(m^{2}_{\tilde{u}_{i}})+a(m^{2}_{\tilde{d}_{i}})\right)-4A(q^{2};m^{2}_{\tilde{u}_{i}},m^{2}_{\tilde{d}_{i}})\right]$
(10)

for left-hand squark doublets, and

$\displaystyle{}\Pi^{\gamma\gamma}_{R}(q^{2})$ $\displaystyle=$ $\displaystyle
N_{c}\frac{e^{2}}{9\pi^{2}}\sum_{i=1,2,3}\left[-16A(q^{2};m^{2}_{\tilde{u}_{Ri}},m^{2}_{\tilde{u}_{Ri}})-4A(q^{2};m^{2}_{\tilde{d}_{Ri}},m^{2}_{\tilde{d}_{Ri}})+8a(m^{2}_{\tilde{u}_{Ri}})+2a(m^{2}_{\tilde{d}_{Ri}})\right]$
$\displaystyle\Pi^{ZZ}_{R}(p^{2})$ $\displaystyle=$ $\displaystyle
N_{c}\frac{e^{2}}{9\pi^{2}}\frac{s^{2}}{c^{2}}\sum_{i=1,2,3}\left[-16A(q^{2};m^{2}_{\tilde{u}_{Ri}},m^{2}_{\tilde{u}_{Ri}})-4A(q^{2};m^{2}_{\tilde{d}_{Ri}},m^{2}_{\tilde{d}_{Ri}})+8a(m^{2}_{\tilde{u}_{Ri}})\right.$
(11) $\displaystyle+$ $\displaystyle\left.2a(m^{2}_{\tilde{d}_{Ri}})\right]$

together with

$\displaystyle{}\Pi^{\gamma Z}_{R}(q^{2})$ $\displaystyle=$ $\displaystyle
N_{c}\frac{e^{2}}{9\pi^{2}}\frac{s}{c}\sum_{i=1,2.3}\left[16A(q^{2};m^{2}_{\tilde{u}_{Ri}},m^{2}_{\tilde{u}_{Ri}})+4A(q^{2};m^{2}_{\tilde{d}_{Ri}},m^{2}_{\tilde{d}_{Ri}})-8a(m^{2}_{\tilde{u}_{Ri}})-2a(m^{2}_{\tilde{d}_{Ri}})\right]$
$\displaystyle\Pi^{WW}_{R}(q^{2})$ $\displaystyle=$ $\displaystyle 0$ (12)

for right-hand squarks. Here $A(p^{2};x,y)$ and $a(x)$ are one-loop integral
functions, which are defined in appendix A. Note that the right-hand sfermions
$\tilde{u}_{Ri}$ , $\tilde{d}_{Ri}$ and $\tilde{e}_{Ri}$ have no contributions
to the self-energy of W boson. Also we want to mention that there are four-
point vertexes for W bosons from the Lagrangian (1), however they contribute
at two-loop.

The effects on $\Pi$ due to FCNC can be taken into account through the
reexpression of squark and slepton sectors with their corresponding mass
eigenstates. For example, the mixing in mass matrix of $\tilde{u}_{L}$ and
$\tilde{u}_{R}$ can be diagonalized via a unitary matrix $U(\alpha_{1})$

$\displaystyle{}\left(\begin{array}[]{c}\tilde{u}_{L}\\\
\tilde{u}_{R}\end{array}\right)=\left(\begin{array}[]{cc}\cos\alpha_{1}&-\sin\alpha_{1}\\\
\sin\alpha_{1}&\cos\alpha_{1}\end{array}\right)\left(\begin{array}[]{c}\tilde{u}_{1}\\\
\tilde{u}_{2}\end{array}\right)$ (19)

such that the vertexes for sfermions coupled to vector bosons can be read from
the mass eigenstates $\tilde{u}_{1,2}$.

###  2.3 Slepton Sector

In this sector, the contributions to self-energy of SM vector bosons stem from
the three-generation left-hand sleptons
$\tilde{l}_{Li}=(\tilde{\nu}_{i},\tilde{e}_{i})$ and right-hand
$\tilde{e}_{Ri}$. In terms of the general expression in (1), we derive that,

$\displaystyle{}\Pi^{\gamma\gamma}(q^{2})$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{\pi^{2}}\sum_{i=1,2,3}\left[-4A(q^{2};m^{2}_{\tilde{e}_{i}},m^{2}_{\tilde{e}_{i}})-4A(q^{2};m^{2}_{\tilde{e}_{Ri}},m^{2}_{\tilde{e}_{Ri}})+2a(m^{2}_{\tilde{e}_{i}})+2a(m^{2}_{\tilde{e}_{R}})\right]$
$\displaystyle\Pi^{ZZ}(p^{2})$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{\pi^{2}}\sum_{i=1,2,3}\left[\frac{(1-2s^{2})^{2}}{s^{2}c^{2}}\left(-A(q^{2};m^{2}_{\tilde{e}_{i}},m^{2}_{\tilde{e}_{i}})+\frac{1}{2}a(m^{2}_{\tilde{e}_{i}})\right)+\frac{1}{2s^{2}c^{2}}a(m^{2}_{\tilde{\nu}_{i}})\right.$
$\displaystyle-$
$\displaystyle\left.\frac{s^{2}}{c^{2}}\left(4A(q^{2};m^{2}_{\tilde{e}_{Ri}},m^{2}_{\tilde{e}_{Ri}})-2a(m^{2}_{\tilde{e}_{Ri}})\right)-\frac{s^{2}}{c^{2}}4A(q^{2};m^{2}_{\tilde{{\nu}_{i}}},m^{2}_{\tilde{{\nu}_{i}}})\right]$
$\displaystyle\Pi^{\gamma Z}(q^{2})$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{\pi^{2}}\sum_{i=1,2,3}\left[\frac{1-2s^{2}}{sc}\left(-2A(q^{2};m^{2}_{\tilde{e}_{i}},m^{2}_{\tilde{e}_{i}})+a(m^{2}_{\tilde{e}_{i}})\right)+\frac{s}{c}\left(4A(q^{2};m^{2}_{\tilde{e}_{Ri}},m^{2}_{\tilde{e}_{Ri}})-2a(m^{2}_{\tilde{e}_{Ri}})\right)\right]$
$\displaystyle\Pi^{WW}(q^{2})$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi^{2}}\frac{e^{2}}{s^{2}}\sum_{i=1,2.3}\left[\left(a(m^{2}_{\tilde{\nu}_{i}})+a(m^{2}_{\tilde{e}_{i}})\right)-4A(q^{2};m^{2}_{\tilde{\nu}_{i}},m^{2}_{\tilde{e}_{i}})\right]$
(20)

As we have mentioned in the previous subsection, there is no contribution to
$\Pi^{WW}$ from the right-hand $\tilde{e}_{Ri}$. The discussion about taking
the masses mixings among left- and right-hand sclars is similar to that about
(19) via intorducing a unitary matrix $U(\alpha_{2})$

$\displaystyle\left(\begin{array}[]{c}\tilde{e}_{L}\\\
\tilde{e}_{R}\end{array}\right)=\left(\begin{array}[]{cc}\cos\alpha_{2}&-\sin\alpha_{2}\\\
\sin\alpha_{2}&\cos\alpha_{2}\end{array}\right)\left(\begin{array}[]{c}\tilde{e}_{1}\\\
\tilde{e}_{2}\end{array}\right)$ (27)

So far we have dealt with evaluating the first two graphs in fig. 1, now we
proceed to discuss the Higgs sector in which the last graph has to be
included.

###  2.4 Higgs sector

The calculation of contributions coming from the neutral $A^{0}$, $H^{0}$ as
well as the charged real scalar $H^{\pm}$ is similar to those of sfermion
sector but more involved. One reason is that there is an extra Feynman diagram
fig. 1$(c)$ needed to be considered due to the new couplings with the Higgs
VEVs $v_{\mu}$ and $v_{d}$. The other reason is that the expressions when we
transform from Higgs gauge eigenstates to their mass eigenstates are
complicated,

$\displaystyle{}H_{\mu}=\left(\begin{array}[]{c}\sin\beta~{}G^{+}+\cos\beta~{}H^{+}\\\
\frac{1}{\sqrt{2}}v_{\mu}+\frac{1}{\sqrt{2}}\cos\alpha~{}h^{0}+\frac{1}{\sqrt{2}}\sin\alpha~{}H^{0}+\frac{i}{\sqrt{2}}\sin\beta~{}G^{0}+\frac{i}{\sqrt{2}}\cos\beta~{}A^{0}\end{array}\right)$
(30)

and

$\displaystyle{}H_{d}=\left(\begin{array}[]{c}\frac{1}{\sqrt{2}}v_{d}-\frac{1}{\sqrt{2}}\sin\alpha~{}h^{0}+\frac{1}{\sqrt{2}}\cos\alpha~{}H^{0}-\frac{i}{\sqrt{2}}\cos\beta~{}G^{0}+\frac{i}{\sqrt{2}}\sin\beta~{}A^{0}\\\
-\cos\beta~{}G^{-}+\sin\beta~{}H^{-}\end{array}\right)$ (33)

Here $G^{0}$ and $G^{\pm}$ are Nambu-Goldstone modes under the ’t Hooft-
Feynman gauge. $\tan\beta=v_{\mu}/v_{d}$ and angle $\alpha$ is introduced in
the unitary matrix so as to diagonalize the mass matrix of scalars. The mass
spectra for scalars in the Higgs sector can be explicitly found in [16].

Evaluate the Feynman diagrams gives rise to,

$\displaystyle{}\Pi^{\gamma\gamma}(q^{2})$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{\pi^{2}}\left[-4A(q^{2};m^{2}_{G^{+}},m^{2}_{G^{+}})-4A(q^{2};m^{2}_{H^{+}},m^{2}_{H^{+}})\right.$
$\displaystyle+$
$\displaystyle\left.2a(m^{2}_{G^{+}})+2a(m^{2}_{H^{+}})\right.$
$\displaystyle+$
$\displaystyle\left.\frac{\pi^{2}e^{2}}{2s^{2}}\left(\sin\beta
v_{\mu}+\cos\beta v_{d}\right)^{2}b_{0}(q^{2};m^{2}_{W},m^{2}_{G^{+}})\right]$
$\displaystyle\Pi^{ZZ}(q^{2})$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{4\pi^{2}s^{2}c^{2}}\left\\{a(m^{2}_{h^{0}})+a(m^{2}_{H^{0}})+a(m^{2}_{G^{0}})+a(m^{2}_{A^{0}})\right.$
(34) $\displaystyle+$
$\displaystyle\left.2(1-2s^{2})^{2}\left[a(m^{2}_{G^{+}})+a(m^{2}_{H^{+}})\right]\right\\}$
$\displaystyle+$
$\displaystyle\frac{e^{4}}{4\pi^{2}s^{4}c^{4}}\left[(\cos\alpha~{}v_{\mu}-\sin\alpha~{}v_{d})^{2}b_{0}(q^{2};m^{2}_{Z},m^{2}_{h^{0}})\right.$
$\displaystyle+$
$\displaystyle\left.(\sin\alpha~{}v_{\mu}+\cos\alpha~{}v_{d})^{2}b_{0}(q^{2};m^{2}_{Z},m^{2}_{H^{0}})\right.$
$\displaystyle+$
$\displaystyle\left.2s^{4}c^{2}(\sin\beta~{}v_{\mu}+\cos\beta~{}v_{d})^{2}b_{0}(q^{2};m^{2}_{W},m^{2}_{G^{+}})\right]$
$\displaystyle-$
$\displaystyle\frac{e^{2}}{\pi^{2}s^{2}c^{2}}\left\\{(1-2s^{2})^{2}\left[A(q^{2};m^{2}_{G^{+}},m^{2}_{G^{+}})+A(q^{2};m^{2}_{H^{+}},m^{2}_{H^{+}})\right]\right.$
$\displaystyle+$
$\displaystyle\left.\cos^{2}(\alpha-\beta)\left[A(q^{2};m^{2}_{h^{0}},m^{2}_{A^{0}}+A(q^{2};m^{2}_{H^{0}},m^{2}_{G^{0}}))\right]\right.$
$\displaystyle+$
$\displaystyle\left.\sin^{2}(\alpha-\beta)\left[A(q^{2};m^{2}_{h^{0}},m^{2}_{G^{0}})+A(q^{2};m^{2}_{H^{0}},m^{2}_{A^{0}})\right]\right\\}$

and

$\displaystyle{}\Pi^{\gamma Z}(q^{2})$ $\displaystyle=$
$\displaystyle-\frac{e^{2}(1-2s^{2})}{2\pi^{2}sc}\left[2a(m^{2}_{G^{+}})+2a(m^{2}_{H^{+}})\right.$
$\displaystyle+$
$\displaystyle\left.4A(q^{2};m^{2}_{G^{+}},m^{2}_{G^{+}})-4A(q^{2};m^{2}_{H^{+}},m^{2}_{H^{+}})\right]$
$\displaystyle-$ $\displaystyle\frac{e^{4}}{2\pi^{2}sc}(\sin\beta
v_{\mu}+\cos\beta v_{d})^{2}b_{0}(q^{2};m^{2}_{W},m^{2}_{G^{+}})$
$\displaystyle\Pi^{WW}(q^{2})$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{4\pi^{2}s^{2}}\left[2a(m^{2}_{H^{+}})+2a(m^{2}_{G^{+}})+a(m^{2}_{H^{0}})+a(m^{2}_{A^{0}})+a(m^{2}_{G^{0}})+a(m^{2}_{h^{0}})\right]$
$\displaystyle+$ $\displaystyle\frac{e^{4}}{4\pi^{2}s^{2}c^{2}}(\sin\beta
v_{\mu}+\cos\beta
v_{d})^{2}\left[c^{2}b_{0}(q^{2};0,m^{2}_{G^{+}})+s^{2}b_{0}(q^{2};m^{2}_{Z},m^{2}_{G^{+}})\right]$
$\displaystyle+$ $\displaystyle\frac{e^{4}}{4\pi^{2}s^{4}}\left[(\cos\alpha
v_{\mu}-\sin\alpha v_{d})^{2}b_{0}(q^{2};m^{2}_{W},m^{2}_{h^{0}})\right.$
$\displaystyle+$ $\displaystyle\left.(\sin\alpha v_{\mu}+\cos\alpha
v_{d})^{2}b_{0}(q^{2};m^{2}_{W},m^{2}_{H^{0}})\right]$ $\displaystyle-$
$\displaystyle\frac{e^{2}}{\pi^{2}s^{2}}\left\\{\sin^{2}(\beta-\alpha)\left[A(q^{2};m^{2}_{h_{0}},m^{2}_{G^{+}})+A(q^{2};m^{2}_{H_{0}},m^{2}_{H^{+}})\right]\right.$
$\displaystyle+$
$\displaystyle\left.\cos^{2}(\alpha+\beta)\left[A(q^{2};m^{2}_{h_{0}},m^{2}_{H^{+}})+A(q^{2};m^{2}_{H_{0}},m^{2}_{G^{+}})\right]\right.$
$\displaystyle+$
$\displaystyle\left.A(q^{2};m^{2}_{A_{0}},m^{2}_{H^{+}})+A(q^{2};m^{2}_{G_{0}},m^{2}_{G^{+}})\right\\}$

In terms of the corrections arising from the SM , the superpartners’
contributions in MSSM can be separated from (2.4) and (2.4) as,

$\displaystyle{}\Pi^{\gamma\gamma}(q^{2})$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{\pi^{2}}\left[-4A(q^{2};m^{2}_{H^{+}},m^{2}_{H^{+}})+2a(m^{2}_{H^{+}})\right]$
$\displaystyle\Pi^{ZZ}(q^{2})$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{4\pi^{2}s^{2}c^{2}}\left[a(m^{2}_{H^{0}})+2a(m^{2}_{A^{0}})+2(1-2s^{2})^{2}a(m^{2}_{H^{+}})\right]$
$\displaystyle+$
$\displaystyle\frac{e^{2}}{\pi^{2}s^{2}c^{4}}m^{2}_{W}\cos^{2}(\alpha-\beta)\left[b_{0}(q^{2};m^{2}_{Z},m^{2}_{H^{0}})-b_{0}(q^{2};m^{2}_{Z},m^{2}_{H^{0}}\right]$
$\displaystyle-$
$\displaystyle\frac{e^{2}}{\pi^{2}s^{2}c^{2}}\left\\{(1-2s^{2})^{2}A(q^{2};m^{2}_{H^{+}},m^{2}_{H^{+}})\right.$
$\displaystyle+$
$\displaystyle\left.\cos^{2}(\alpha-\beta)[A(q^{2};m^{2}_{h^{0}},m^{2}_{A^{0}})+A(q^{2};m^{2}_{H^{0}},m^{2}_{G^{0}})]\right.$
$\displaystyle-$
$\displaystyle\left.\cos^{2}(\alpha-\beta)A(q^{2};m^{2}_{h^{0}},m^{2}_{G^{0}})+\sin^{2}(\alpha-\beta)A(q^{2};m^{2}_{H^{0}},m^{2}_{A^{0}})\right\\}$
$\displaystyle\Pi^{\gamma Z}(q^{2})$ $\displaystyle=$
$\displaystyle-\frac{e^{2}(1-2s^{2})}{\pi^{2}sc}\left[-2A(q^{2};m^{2}_{H^{+}},m^{2}_{H^{+}})+a(m^{2}_{H^{+}})\right]$
(36) $\displaystyle\Pi^{WW}(q^{2})$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{4\pi^{2}s^{2}}\left[2a(m^{2}_{H^{+}})+a(m^{2}_{H^{0}})+a(m^{2}_{A^{0}})\right]$
$\displaystyle+$
$\displaystyle\frac{e^{2}}{\pi^{2}s^{2}}\cos^{2}(\alpha-\beta)m^{2}_{W}\left[b_{0}(q^{2};m^{2}_{W},m^{2}_{H^{0}})-b_{0}(q^{2};m^{2}_{W},m^{2}_{h^{0}})\right]$
$\displaystyle+$
$\displaystyle\frac{e^{2}}{\pi^{2}s^{2}}\cos^{2}(\alpha-\beta)\left[A(q^{2};m^{2}_{h_{0}},m^{2}_{G^{+}})-A(q^{2};m^{2}_{h_{0}},m^{2}_{H^{+}})-A(q^{2};m^{2}_{H_{0}},m^{2}_{G^{+}})\right]$
$\displaystyle-$
$\displaystyle\frac{e^{2}}{\pi^{2}s^{2}}\left[\sin^{2}(\alpha-\beta)A(q^{2};m^{2}_{H_{0}},m^{2}_{H^{+}})+A(q^{2};m^{2}_{A_{0}},m^{2}_{H^{+}})\right]$

##  3 Preliminary Checks on The Results

In electroweak theory, there are various obeservables that can be precisely
measured, however, they depend only on three parameters which are composed of
the gauge coupling constants parameters $g$, $g^{\prime}$ and VEV $v$
associated with the scale of spontaneous symmetry breaking. For example, one
can define the weak mixing angle as

$\displaystyle{}s^{2}_{W}\equiv 1-\frac{m_{W}^{2}}{m_{Z}^{2}}$ (37)

Another definition uses,

$\displaystyle{}s^{2}_{*}\equiv\frac{g^{{}^{\prime}2}}{g^{2}+g^{{}^{\prime}2}}$
(38)

Alternatively, we do this via the accurately known weak-interaction
obeservables $\alpha$, $G_{F}$ and $m_{Z}$,

$\displaystyle{}\sin
2\theta_{0}\equiv\left(\frac{4\pi\alpha(m_{Z}^{2})}{\sqrt{2}G_{F}m_{Z}^{2}}\right)^{1/2}=0.2307\pm
0.0005$ (39)

which gives us an accurate standard of reference. The corrections to $\alpha$,
$G_{F}$ and $m_{Z}$ due to quantum effects of new particle states beyond SM
will lead to deviation from the reference point. So the measured value can be
used to constrain the content of these new particles and estimate the bounds
of their masses.

Now we use the weak mixing angle as an example to check our results presented
in the previous section. The rational is as follows. The quantum corrections
to observables such as $m_{Z}$, $m_{W}$, $\alpha$ and $G_{F}$ in low-energy
electroweak theory include corrections to the vector boson propagator, vertex
and box corrections. The last two types are usually proportional to the ratio
of masses of external (light) fermions over masses of heavy particle states.
With the limit that these light masses of SM fermions are set to zero, only
the first type is important practically. This type of correction is known as
oblique correction, as they enter the low-energy weak interactions only
indirectly. The different definitions (37) to (40) all agree at zero order but
receive different radiative corrections. However, the differences, for
instance [13]

$\displaystyle{}s^{2}_{W}-s^{2}_{*}\equiv-\frac{c^{2}}{m_{W}^{2}}\left[\Pi^{WW}(m^{2}_{W})-c^{2}\Pi^{ZZ}(m^{2}_{Z})\right]+\frac{sc^{3}}{m_{W}^{2}}\Pi^{\gamma
Z}(m^{2}_{Z})$ (40)

due to the radiative corrections must be finite, and are free of ultraviolet
divergence.

For the part of slepton sector, substituting (2.3) into (40) gives,

$\displaystyle{}s^{2}_{W}-s^{2}_{*}$ $\displaystyle=$
$\displaystyle\frac{c^{2}e^{2}}{\pi^{2}s^{2}m^{2}_{W}}\left[-(1-2s^{2})A(m^{2}_{Z};m^{2}_{\tilde{e}_{i}},m^{2}_{\tilde{e}_{i}})-A(m^{2}_{Z};m^{2}_{\tilde{\nu}_{i}},m^{2}_{\tilde{\nu}_{i}})\right.$
(41) $\displaystyle+$
$\displaystyle\left.2A(m^{2}_{W};m^{2}_{\tilde{e}_{i}},m^{2}_{\tilde{\nu}_{i}})-s^{2}a(m^{2}_{\tilde{e}_{i}})\right]$

for each generation of left-hand sleptons, which then gives us the part of
divergence is proportional to

$\displaystyle-$
$\displaystyle(1-2s^{2})\left(\frac{1}{2}m^{2}_{\tilde{e}_{i}}-\frac{1}{12}m^{2}_{Z}\right)$
$\displaystyle-$
$\displaystyle\left(\frac{1}{2}m^{2}_{\tilde{\nu}_{i}}-\frac{1}{12}m^{2}_{Z}\right)$
$\displaystyle+$ $\displaystyle
2\left(\frac{1}{4}m^{2}_{\tilde{e}_{i}}+\frac{1}{4}m^{2}_{\tilde{\nu}_{i}}-\frac{1}{12}m^{2}_{W}-s^{2}m^{2}_{\tilde{e}_{i}}\right)=0$

For each generation of right-hand slepton, we find that the contribution to
the deference in (40) vanishes.

Now we proceed to examine the results in the squark sector. Substituting (2.2)
into (40) yields,

$\displaystyle{}s^{2}_{W}-s^{2}_{*}$ $\displaystyle=$
$\displaystyle\frac{e^{2}c^{2}}{\pi^{2}s^{2}m^{2}_{W}}\left[-(1-\frac{3}{4}s^{2})A(m^{2}_{Z};m^{2}_{\tilde{u}_{i}},m^{2}_{\tilde{u}_{i}})-\frac{2}{3}s^{2}a(m^{2}_{\tilde{u}_{i}})-\frac{1}{3}s^{2}a(m^{2}_{\tilde{d}_{i}})\right.$
(42) $\displaystyle-$
$\displaystyle\left.(1-\frac{2}{3}s^{2})A(m^{2}_{Z};m^{2}_{\tilde{d}_{i}},m^{2}_{\tilde{d}_{i}})+2A(m^{2}_{W};m^{2}_{\tilde{u}_{i}},m^{2}_{\tilde{d}_{i}})\right]$

for the left-hand squarks, from which the part of divergence is proportional
to

$\displaystyle-$
$\displaystyle(1-\frac{3}{4}s^{2})(\frac{1}{2}m^{2}_{\tilde{u}_{i}}-\frac{1}{12}m^{2}_{Z})-\frac{2}{3}s^{2}m^{2}_{\tilde{u}_{i}}-\frac{1}{3}s^{2}m^{2}_{\tilde{d}_{i}}$
(43) $\displaystyle-$
$\displaystyle(1-\frac{2}{3}s^{2})(\frac{1}{2}m^{2}_{\tilde{d}_{i}}-\frac{1}{12}m^{2}_{Z})+2(\frac{1}{4}m^{2}_{\tilde{u}_{i}}+\frac{1}{4}m^{2}_{\tilde{d}_{i}}-\frac{1}{12}m^{2}_{Z})=0$

For the part of Higgs sector, substituting (2.4) and (2.4) into (40) gives
that the divergent parts in (40) are composed of those arising from $a(x)$,
$A(x)$ and $b_{0}$ functions. It turns out the contribution from $a(x)$
function is proportional to
$-\frac{e^{2}c^{2}}{\pi^{2}m^{2}_{W}}(m^{2}_{G^{+}}+m^{2}_{H^{+}})$, which
exactly cancels the part from $A$ function. And the divergent parts in $b_{0}$
function cancel themselves.

##  4 $S$, $T$ and $U$ in the MSSM

In this section, we derive a set of parameters which estimate the oblique
corrections to precise electroweak observables. These parameters are known as
$S$, $T$ and $U$ in the literature [3, 4]. As a byproduct we also use the
property that these parameters are finite values to examine the results shown
in the section 2, independently of what we have done in the section 3. The
dependence of $S$, $T$ and $U$ parameters on $\Pi^{IJ}(p^{2})$ can be
perturbatively expanded in powers of the external momentum squared. It turns
out that these corrections are quite simple [3, 4],

$\displaystyle{}S$ $\displaystyle\equiv$
$\displaystyle-\frac{16\pi}{e^{2}}sc\left[sc\Pi^{\gamma\gamma^{\prime}}(0)-sc\Pi^{ZZ^{\prime}}(0)+(c^{2}-s^{2})\Pi^{\gamma
Z^{\prime}}(0)\right]$ $\displaystyle T$ $\displaystyle\equiv$
$\displaystyle\frac{4\pi}{e^{2}}\left[\frac{\Pi^{WW}(0)}{m^{2}_{W}}-\frac{\Pi^{ZZ}(0)}{m^{2}_{Z}}-\frac{2s}{c}\frac{\Pi^{\gamma
Z}(0)}{m^{2}_{Z}}\right]$ (44) $\displaystyle U$ $\displaystyle\equiv$
$\displaystyle\frac{16\pi
s^{2}}{e^{2}}\left[\Pi^{WW^{\prime}}(0)-c^{2}\Pi^{ZZ^{\prime}}(0)-s^{2}\Pi^{\gamma\gamma^{\prime}}(0)-2cs\Pi^{\gamma
Z^{\prime}}(0)\right]$

where

$\displaystyle{}\Pi^{IJ^{\prime}}(0)=d^{2}\Pi^{IJ}/dp^{2}\mid_{p^{2}=0}$ (45)

$\Pi^{IJ}$ is the part with metric as the coefficient in
$\Pi_{\mu\nu}^{IJ}=g_{\mu\nu}\Pi^{IJ}+\cdots$.

In this paper, we consider the calculations of the oblique corrections in
MSSM, which are composed of four parts given a specific vacuum polarization,

$\displaystyle{}\Pi_{SUSY}=\Pi_{\tilde{S}}+\Pi_{\tilde{l}}+\Pi_{H}+\Pi_{NC}$
(46)

We derive the first three parts in (46) in this paper, and will explore last
part for the future [17].

Firstly, the slepton sector gives

$\displaystyle{}S_{\tilde{l}}$ $\displaystyle=$
$\displaystyle\frac{1}{12\pi}\sum_{i=1}^{3}\ln\frac{m^{2}_{\tilde{\nu}_{i}}}{m^{2}_{\tilde{e}_{i}}}$
$\displaystyle T_{\tilde{l}}$ $\displaystyle=$ $\displaystyle\frac{1}{16\pi
s^{2}m^{2}_{W}}\sum_{i=1}^{3}\left[m^{2}_{\tilde{\nu}_{i}}+m^{2}_{\tilde{e}_{i}}-\frac{2m^{2}_{\tilde{e}_{i}}m^{2}_{\tilde{\nu}_{i}}\ln(\frac{m^{2}_{\tilde{\nu}_{i}}}{m^{2}_{\tilde{e}_{i}}})}{m^{2}_{\tilde{\nu}_{i}}-m^{2}_{\tilde{e}_{i}}}\right]$
(47) $\displaystyle U_{\tilde{l}}$ $\displaystyle=$
$\displaystyle\frac{1}{\pi}\sum_{i=1}^{3}\left[\frac{m^{2}_{\tilde{e}_{i}}}{3}b^{\prime}_{0}(0,m^{2}_{\tilde{e}_{i}},m^{2}_{\tilde{e}_{i}})-\frac{m^{2}_{\tilde{e}_{i}}+m^{2}_{\tilde{\nu}_{i}}}{6}b^{\prime}_{0}(0,m^{2}_{\tilde{e}_{i}},m^{2}_{\tilde{\nu}_{i}})\right.$
$\displaystyle+$
$\displaystyle\left.\frac{(m^{2}_{\tilde{e}_{i}}-m^{2}_{\tilde{\nu}_{i}})^{2}}{24}b^{\prime\prime}_{0}(0,m^{2}_{\tilde{e}_{i}},m^{2}_{\tilde{\nu}_{i}})+\frac{1}{12}f(m^{2}_{\tilde{e}_{i}},m^{2}_{\tilde{\nu}_{i}})-\frac{1}{12}f(m^{2}_{\tilde{e}_{i}},m^{2}_{\tilde{e}_{i}})\right]$

for the squark sector, we have

$\displaystyle{}S_{\tilde{S}}$ $\displaystyle=$
$\displaystyle\frac{1}{12\pi}\sum_{i=1}^{3}\ln\frac{m^{2}_{\tilde{d}_{i}}}{m^{2}_{\tilde{u}_{i}}}$
$\displaystyle T_{\tilde{S}}$ $\displaystyle=$ $\displaystyle\frac{3}{16\pi
s^{2}m^{2}_{W}}\sum_{i=1}^{3}\left[m^{2}_{\tilde{u}_{i}}+m^{2}_{\tilde{d}_{i}}-\frac{2m^{2}_{\tilde{u}_{i}}m^{2}_{\tilde{d}_{i}}\ln(\frac{m^{2}_{\tilde{u}_{i}}}{m^{2}_{\tilde{d}_{i}}})}{m^{2}_{\tilde{u}_{i}}-m^{2}_{\tilde{d}_{i}}}\right]$
(48) $\displaystyle U_{\tilde{S}}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{3}\frac{1}{\pi}\left[m^{2}_{\tilde{u}_{i}}b^{\prime}_{0}(0,m^{2}_{\tilde{u}_{i}},m^{2}_{\tilde{u}_{i}})+m^{2}_{\tilde{d}_{i}}b^{\prime}_{0}(0,m^{2}_{\tilde{d}_{i}},m^{2}_{\tilde{d}_{i}})-(m^{2}_{\tilde{u}_{i}}+m^{2}_{\tilde{d}_{i}})b^{\prime}_{0}(0,m^{2}_{\tilde{u}_{i}},m^{2}_{\tilde{d}_{i}})\right.$
$\displaystyle+$
$\displaystyle\left.\frac{1}{2}f(m^{2}_{\tilde{u}_{i}},m^{2}_{\tilde{d}_{i}})-\frac{1}{4}f(m^{2}_{\tilde{u}_{i}},m^{2}_{\tilde{u}_{i}})-\frac{1}{4}f(m^{2}_{\tilde{d}_{i}},m^{2}_{\tilde{d}_{i}})\right]$

with the finite function $f(x,y)$ is defined as,

$\displaystyle{}f(x,y)=\frac{1}{x-y}\left(x\ln x-y\ln y\right)$ (49)

Note that the descents of function $b_{0}$, $b^{\prime}_{0}$ and
$b^{\prime\prime}_{0}$ in (4) to (4) are given in the appendix B, which are
also finite.

Finally, the Higgs sector yields,

$\displaystyle{}\pi S_{H}$ $\displaystyle=$ $\displaystyle
h(m^{2}_{H^{+}},m^{2}_{H^{+}})-\sin^{2}(\alpha-\beta)h(m^{2}_{A_{0}},m^{2}_{H_{0}})$
(50) $\displaystyle+$
$\displaystyle\cos^{2}(\alpha-\beta)\left[h(m^{2}_{h_{0}},m^{2}_{G_{0}})-h(m^{2}_{H_{0}},m^{2}_{G_{0}})-h(m^{2}_{h_{0}},m^{2}_{A_{0}})\right]$
$\displaystyle-$
$\displaystyle\frac{m^{2}_{W}}{c^{2}}\cos^{2}(\alpha-\beta)\left[b^{\prime}_{0}(0;m^{2}_{Z},m^{2}_{h_{0}})-b^{\prime}_{0}(0;m^{2}_{Z},m^{2}_{H_{0}})\right]$

and

$\displaystyle{}\pi T_{H}$ $\displaystyle=$
$\displaystyle\frac{1}{4s^{2}m^{2}_{W}}\left\\{m^{2}_{H^{+}}\ln
m^{2}_{H^{+}}+\cos^{2}(\alpha-\beta)\left[m^{2}_{W}(f(m^{2}_{W},m^{2}_{H_{0}})-f(m^{2}_{W},m^{2}_{h_{0}}))\right.\right.$
(51) $\displaystyle+$
$\displaystyle\left.\left.m^{2}_{Z}(f(m^{2}_{Z},m^{2}_{h_{0}})-f(m^{2}_{Z},m^{2}_{H_{0}}))\right]\right.$
$\displaystyle+$
$\displaystyle\left.\left.\cos^{2}(\alpha-\beta)\left[g(m^{2}_{h_{0}},m^{2}_{G^{+}})+g(m^{2}_{h_{0}},m^{2}_{A^{0}})+g(m^{2}_{H_{0}},m^{2}_{G^{0}})\right.\right.\right.$
$\displaystyle-$
$\displaystyle\left.\left.g(m^{2}_{h_{0}},m^{2}_{H^{+}})-g(m^{2}_{H_{0}},m^{2}_{G^{+}})-g(m^{2}_{h_{0}},m^{2}_{G^{0}})\right]\right.$
$\displaystyle-$
$\displaystyle\left.\sin^{2}(\alpha-\beta)g(m^{2}_{H_{0}},m^{2}_{H^{+}})+\sin^{2}(\alpha-\beta)g(m^{2}_{A_{0}},m^{2}_{H^{0}})-g(m^{2}_{A_{0}},m^{2}_{H^{0}})\right\\}$

together with

$\displaystyle{}\pi U_{H}$ $\displaystyle=$ $\displaystyle
h(m^{2}_{H_{+}},m^{2}_{H^{+}})-h(m^{2}_{A_{0}},m^{2}_{H^{0}})$ (52)
$\displaystyle-$
$\displaystyle\cos^{2}(\alpha-\beta)m^{2}_{W}\left[b^{\prime}_{0}(0;m^{2}_{W},m^{2}_{H_{0}})-b^{\prime}_{0}(0;m^{2}_{W},m^{2}_{h_{0}})\right]$
$\displaystyle+$
$\displaystyle\cos^{2}(\alpha-\beta)m^{2}_{Z}\left[b^{\prime}_{0}(0;m^{2}_{Z},m^{2}_{h_{0}})-b^{\prime}_{0}(0;m^{2}_{Z},m^{2}_{H_{0}})\right]$
$\displaystyle+$
$\displaystyle\cos^{2}(\alpha-\beta)\left[h(m^{2}_{h_{0}},m^{2}_{G^{+}})-h(m^{2}_{h_{0}},m^{2}_{H^{+}})-h(m^{2}_{H_{0}},m^{2}_{G^{+}})\right.$
$\displaystyle+$
$\displaystyle\left.h(m^{2}_{h_{0}},m^{2}_{A^{0}})-h(m^{2}_{h_{0}},m^{2}_{G^{0}})+h(m^{2}_{G_{0}},m^{2}_{H^{0}})\right]$
$\displaystyle-$
$\displaystyle\sin^{2}(\alpha-\beta)\left[h(m^{2}_{H_{0}},m^{2}_{H^{+}})-h(m^{2}_{A_{0}},m^{2}_{H^{0}})\right]$

with the finite function $g(x,y)$ and $h(x,y)$ defined as,

$\displaystyle{}g(x,y)$ $\displaystyle=$ $\displaystyle\frac{1}{12}(x\ln
x+y\ln
y)+\frac{x+y}{6}[-1+f(x,y)]+\frac{1}{24}\left(x+y-\frac{2xy}{x-y}\ln\frac{x}{y}\right)$
$\displaystyle h(x,y)$ $\displaystyle=$
$\displaystyle\frac{1}{6}(x+y)b^{\prime}_{0}(0;x,y)-\frac{1}{24}(x-y)^{2}b^{\prime\prime}_{0}(0;x,y)-\frac{1}{12}f(x,y)$
(53)

Note that the combination $\cos(\alpha-\beta)$ is given by,

$\displaystyle{}\cos^{2}(\alpha-\beta)=\frac{m^{2}_{h}}{m^{2}_{A}}\frac{m^{2}_{Z}-m^{2}_{h}}{m^{2}_{Z}+m^{2}_{A}-2m^{2}_{h}}$
(54)

Here a few comments are in order for the results (4) to (52). At first, it is
obvious that each part in MSSM contributes to the finite $S$, $T$ and $U$
values as required. There are some constants terms in the expression of $S$
,$T$ and $U$ parameters, which naively violate the property that large
superpartner masses (compared with $m_{Z}$) lead to the conclusion that the
theory decouples from the SM. But expanding the relevant functions we have
defined implies that these constants are cancelled and the conclusion is
restored. Second, from (50) to (52) it seems that the SM contributions has not
been separated from the MSSM contributions due to some terms involved with SM
fields. But it is crucial to notice that these terms are multiplied by a
factor $\cos^{2}(\alpha-\beta)$, as given by (54). Under the limit that the
superpartners decouple from SM, which results in $\alpha=\beta$, the results
in (50) to (52) reduce to pure MSSM ones. Finally, unlike in the case of SM,
where the sensitivity of these parameters to the SM Higgs mass $m_{h}$ is
logarithmic, the results in (50) to (4) demonstrate that this is not strictly
true in the MSSM anymore. According to the simulations about $S$, $T$, $U$
shown in Ref. [19], the sensitivity of these parameters to the mass of Higgs
boson can be either stronger or weaker, which depends on choices of parameters
left in the Higgs sector. Overall, the discrepancy is not so significant as
one expects roughly.

Under the assumption that the breaking of $SU(2)$ symmetry is weak, i.e,
$\Delta_{q_{i}}=m^{2}_{\tilde{e}_{i}}-m^{2}_{\tilde{\nu}_{i}}<<m^{2}_{\tilde{\nu}_{i}},m^{2}_{\tilde{e}_{i}}$
and
$\Delta_{S_{i}}=m^{2}_{\tilde{d}_{i}}-m^{2}_{\tilde{u}_{i}}<<m^{2}_{\tilde{u}_{i}},m^{2}_{\tilde{d}_{i}}$,
we find

$\displaystyle{}S_{\tilde{q}}$ $\displaystyle\rightarrow$
$\displaystyle\frac{1}{12\pi}\sum_{i=1}^{3}\frac{\Delta_{S_{i}}}{m^{2}_{\tilde{d}_{i}}}$
$\displaystyle T_{\tilde{q}}$ $\displaystyle\rightarrow$
$\displaystyle-\frac{3}{16\pi
s^{2}}\sum_{i=1}^{3}\frac{\Delta_{S_{i}}}{m^{2}_{W}}$ (55) $\displaystyle
U_{\tilde{q}}$ $\displaystyle\rightarrow$
$\displaystyle-\frac{1}{4\pi}\sum_{i=1}^{3}\frac{\Delta_{S_{i}}}{m^{2}_{\tilde{d}_{i}}}$

and

$\displaystyle{}S_{\tilde{l}}$ $\displaystyle\rightarrow$
$\displaystyle-\frac{1}{12\pi}\sum_{i=1}^{3}\frac{\Delta_{q_{i}}}{m^{2}_{\tilde{e}_{i}}}$
$\displaystyle T_{\tilde{l}}$ $\displaystyle\rightarrow$
$\displaystyle-\frac{1}{16\pi
s^{2}}\sum_{i=1}^{3}\frac{\Delta_{q_{i}}}{m^{2}_{W}}$ (56) $\displaystyle
U_{\tilde{l}}$ $\displaystyle\rightarrow$
$\displaystyle-\frac{1}{24\pi}\sum_{i=1}^{3}\frac{\Delta_{q_{i}}}{m^{2}_{\tilde{e}_{i}}}$

From (4) and (4) one sees that both in the squark and slepton sector the
relative ratio $|S/U|$ is approximatively around the unity, while their values
relative to $T$ depend on the ratios of $m^{2}/m^{2}_{\tilde{d}_{i}}$ and
$m^{2}/m^{2}_{\tilde{e}_{i}}$.

##  5 Discussions and Conclusions

In this paper we have revisited the oblique corrections in the context of the
MSSM. the motivation for exploring these contributions is quite clear since
they are useful to interpret the latest LHC data about the Higgs mass and
bounds on superpartner masses 333Note that there is no need to consider the
gauge invariance of these contributions once again, as this problem in the SM
contributions has been properly treated [18]. . Thus, we reconsider the
theoretic calculations under ’t Hooft-Feynman gauge, with the bosonic part as
the first step towards the complete answer.

The results presented in this paper are checked by two examinations. In one
examination, we directly confront our results to the finite radiative
correction to the deference $s^{2}_{W}-s^{2}_{*}$ in each sector. It shows
that the individual contribution deriving from each sector indeed respects
this property. The other examination is by using the property that the $S$,
$T$ and $U$ parameters are finite. We also verify the expectation.

In summary, our results about one-loop self energies are found to exactly
agree with Ref. [7, 14] 444 The formulae presented in [7] take the SM part
into account, however, do not include the contribution arising from the pure
gauge part. As we missed this point, we made the claims about disagreement
with [7] in the previous version of this manuscript.. Nevertheless, the $S$,
$T$ and $U$ parameters do not match with those of Ref. [8], which can not be
explained by the possible difference of renormalization scheme performed. In
particular, our results do not take the scalar mass mixing among left and
right hand into account, therefore correspond to the simple case with
vanishing mass mixing. Take this difference into account, compare our results
with Ref.[8], one finds that

  * •

The $S$ and $U$ parameter arising from squark and slepton sector do not agree.

  * •

The $T$ parameter in (4) and (4) agree with [8].

We leave the study of oblique corrections arising from the fermions, i.e, the
neutralinos and charginos elsewhere [17].

$\bf{Acknowledgement}$
SZ would like to thank Prof. Ken-ichi Hikasa for correspondence. The work is
supported in part by National Natural Science Foundation of China with grant
No.11247031.

$\bf{Conflict~{}of~{}Interests}$
As the authors of the manuscript, we do not have a direct financial relation
with any commercial identity mentioned in our paper that might lead to a
conflict of interest for any of the authors.

##  Appendix A One-loop Integrals

Two-point funtions $A(p^{2};x,y)$, $a(x)$ and $b_{0}(p^{2};x,y)$ are defined
as the integrals [11, 12],

$\displaystyle{}\mu^{\epsilon}\int\frac{d^{n}k}{(2\pi)^{n}}\frac{i}{k^{2}-m^{2}}$
$\displaystyle=$ $\displaystyle\frac{1}{(4\pi)^{2}}a(m^{2})$
$\displaystyle\mu^{\epsilon}\int\frac{d^{n}k}{(2\pi)^{n}}\frac{ik_{\mu}k_{\nu}}{[k^{2}-m_{1}^{2}][(k+p)^{2}-m^{2}_{2}]}$
$\displaystyle=$
$\displaystyle\frac{1}{(4\pi)^{2}}\left[g_{\mu\nu}A(p^{2};m_{1}^{2},m_{2}^{2})+\cdots\right]$
$\displaystyle\mu^{\epsilon}\int\frac{d^{n}k}{(2\pi)^{n}}\frac{i}{[k^{2}-m_{1}^{2}][(k+p)^{2}-m^{2}_{2}]}$
$\displaystyle=$
$\displaystyle\frac{1}{(4\pi)^{2}}b_{0}(p^{2};m_{1}^{2},m_{2}^{2})$ (57)

respectively, with $p^{2}$ is the external line vector boson momentum squared
. Here we have ignored terms that are irrelevant for discussions in the
definition of $A(p^{2};x,y)$. Explicitly,

$\displaystyle{}a(x)$ $\displaystyle=$ $\displaystyle-\eta x+a_{F}(x)$
$\displaystyle b_{0}(p^{2};x,y)$ $\displaystyle=$
$\displaystyle-\eta+b_{F}(p^{2};x,y)$ (58)

where the divergent factor in these functions are carried by $\eta$,

$\displaystyle{}\eta=\frac{1}{\epsilon}+\ln(4\pi)-\gamma_{E}$ (59)

Here $d=4-2\epsilon$, and $\mu$ is the RG scale. The finite parts $a_{F}$ and
$b_{F}$ in (A) are given by,

$\displaystyle{}a_{F}(x)$ $\displaystyle=$ $\displaystyle
x\left(-1+\ln\frac{x}{\mu^{2}}\right)$ $\displaystyle b_{F}(p^{2};x,y)$
$\displaystyle=$
$\displaystyle\int^{1}_{0}dt\ln\frac{tx+(1-t)y-t(1-t)p^{2}}{\mu^{2}}$ (60)

respectively. And

$\displaystyle{}A(p^{2};x,y)$ $\displaystyle=$
$\displaystyle\left(\frac{p^{2}}{12}-\frac{x+y}{4}\right)\eta+\frac{1}{12}[a_{F}(x)+a_{F}(y)]+\frac{1}{6}\left(x+y-\frac{p^{2}}{2}\right)b_{F}(p^{2};x,y)$
(61) $\displaystyle+$
$\displaystyle\frac{x-y}{12p^{2}}\left[a_{F}(x)-a_{F}(y)-(x-y)b_{F}(p^{2};x,y)\right]-\frac{1}{6}\left(x+y-\frac{p^{2}}{3}\right)$

Note that the $A(p^{2};x,y)$ and $b_{0}(p^{2};x,y)$ functions are symmetric in
$x$ and $y$.

It is also useful to notice some descents of $b_{0}$ functions

$\displaystyle{}b^{{}^{\prime}}_{0}(0;x,y)$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\frac{x+y}{(x-y)^{2}}+\frac{xy}{(x-y)^{3}}\ln\frac{x}{y}$
$\displaystyle b^{{}^{\prime\prime}}_{0}(0;x,y)$ $\displaystyle=$
$\displaystyle\frac{x^{3}+9x^{2}y-9xy^{2}-6xy(x+y)\ln\frac{x}{y}-y^{3}}{3(x-y)^{5}}$
(62)

## References

  * [1] F. Gianotti, the ATLAS Collaboration, talk given at CERN on July 4, 2012;
J. Incandela, the CMS Collaboration, talk given at CERN on July 4, 2012.

  * [2] The ATLAS Collaboration, ATLAS-CONF-2012-001, (2012); arXiv:1109.6572 [hep-ex]; The CMS Collaboration, CMS PAS SUS-11-004.
  * [3] M. E. Peskin and T.Takeuchi, Phys. Rev. D46 (1992) 381.
  * [4] M. E. Peskin and T.Takeuchi, Phys. Rev. Lett. 65 (1990) 964;
  * [5] K. Hagiwara, D. Haidt, and S. Matsumoto, Eur. Phys. J. C2 (1998) 95, arXiv: hep-ph/9706331.
  * [6] K. Hagiwara, S. Matsumoto, D. Haidt, C. S. Kim, Z. Phys. C64 (1994) 559, arXiv: hep-ph/9409380.
  * [7] M. Drees , K. Hagiwara and A. Yamada, Phys. Rev. D45 (1992) 1725.
  * [8] A. Dobado, M. J. Herrero and S. Penaranda, Eur. Phys. J. C7 (1999) 313, [arXiv: hep-ph/9710313].
  * [9] A. Dobado, M. J. Herrero and S. Penaranda, Eur. Phys. J. C12 (2000) 673, arXiv: hep-ph/9903211.
  * [10] A. Dobado, M. J. Herrero and S. Penaranda, Eur. Phys. J. C17 (2000) 487, arXiv: hep-ph/0002134.
  * [11] S. Pokorski, Gauge Field Theory (2ed), Cambridge University Press , (2000) .
  * [12] G. ’t Hooft and M. J. G. Veltman, Nucl. Phys. B153 (1979) 365;
G. Passarino and M. J. G. Veltman, Nucl. Phys. B160 (1979) 151.

  * [13] M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory, Addison-Wesley (1995).
  * [14] D. M. Pierce et al, Nucl. Phys. B 491 (1997) 3.
  * [15] J. F. Gunion, H. E. Haber, G. L. Kane, S. Dawson, The Higgs Hunter’s Guide, Front. Phys.80, (2000) 1.
  * [16] S. P. Martin, arXiv: hep-ph/9709356.
  * [17] Y. Yu and S. Zheng, Mod. Phys. Lett. A28 (2013) 1250238, arXiv:1207.4868[hep-ph].
  * [18] G. Degrassi and A. Sirlin, Nucl. Phys. B383 (1992) 73 ;
G. Degrassi and A. Sirlin Phys. Rev. D46 (1992) 3104 ;
G. Degrassi, B.A. Kniehl and A. Sirlin, Phys. Rev. D48 (1993) 3963.

  * [19] S. Zheng and Y. Yu, JHEP1308, 031 (2013), arXiv:1303.1900 [hep-ph],