# The $(g-2)_{\mu}$ anomaly, Higgs bosons and heavy neutrinos

G.G.Boyarkina, O.M.Boyarkin
Byelorussian State Pedagogical University
Soviet Street 18, Minsk, 220050, Belarus E-mail<EMAIL_ADDRESS>
(*)

###### Abstract

Within the model based on the $SU(2)_{L}\times SU(2)_{R}\times U(1)_{B-L}$
gauge group and having the bidoublet and two triplets of the Higgs fields
(left-right model) the Higgs sector impact on the value of the muon anomalous
magnetic moment (AMM) is considered. The contributions coming from the doubly
charged Higgs bosons, the singly charged Higgs bosons and the lightest neutral
Higgs boson are taken into account. The obtained value of the muon is the
function of the Higgs boson masses and the Higgs boson couplings constants
(CC’s). We express the most of part of the CC’s as the functions of the heavy
neutrino sector parameters. We show that at the particular parameters values
the model under study could explain the BNL’00 result.
PACS number(s): +12.15.Cc, 12.15.Ff,-13.40.Em.

##  1 Introduction

Measurements of the spin magnetic dipole moment of particles have a rich
history as harbingers of impressive progress in the quantum theory. Thus,
registration of the anomalous values of the nucleons magnetic moments was
powerful argument for the benefit of the $\pi$-mesonic theory of the nuclear
forces formulated by Yukawa. Determination of the anomalous magnetic moment
(AMM) of the electron has played an important role in development of modern
quantum electrodynamics and renormalization theory. It appeared to be
reasonable that the ongoing muon $(g-2)_{\mu}$ measurement E821 at Brookhaven
National Laboratory (BNL) on Alternating Gradient Synchrotron would be a
sensitive test for the results of the standard model (SM) electroweak
corrections. Because this experiment is the culmination of a series of
measurements of ever-increasing accuracy over the past several years then its
results has triggered the interest in theoretical calculations of the muon
AMM. The first BNL result based on the data taken through 1997 was [1]

$a_{\mu}^{exp}=(116\ 592\ 500\pm 1500)\times
10^{-11}\mu_{0}\qquad(\mbox{BNL'97}),$ $None$

where $\mu_{0}$ is the muon magnetic moment predicted by the Dirac’s theory.
The 1998 and 1999 runs had the much higher statistics and gave the results
with the increased precession

$a_{\mu}^{exp}=(116\ 591\ 910\pm 590)\times
10^{-11}\mu_{0}\qquad(\mbox{BNL'98}\ [2]).$ $None$ $a_{\mu}^{exp}=(116\ 592\
020\pm 160)\times 10^{-11}\mu_{0}\qquad(\mbox{BNL'99}\ [3]).$ $None$

The BNL’98 and BNL’99 results averaged with older measurements made at CERN
[4] brought to the following value of the muon AMM

$a_{\mu}^{exp}=(116\ 592\ 023\pm 151)\times 10^{-11}\mu_{0},$ $None$

In the SM the expression for the muon AMM can be presented as a sum

$a_{\mu}^{SM}=a_{\mu}^{QED}+a_{\mu}^{EW}+a_{\mu}^{had},$ $None$

in which $a_{\mu}^{QED}=11\ 658\ 470.57(0.29)\times 10^{-10}\mu_{0}$ (see [5]
and references therein) and $a_{\mu}^{EW}=15.2(0.4)\times 10^{-10}$ (see [6]
and references therein).

The term $a_{\mu}^{had}$ arises from virtual hadronic contributions to the
photon propagator in $4^{th}$ $a_{\mu}^{had}(VP1)$ and $6^{th}$ order, where
the latter includes hadronic vacuum polarization $a_{\mu}^{had}(VP2)$ and
light-by-light scattering $a_{\mu}^{had}(LbyL)$. The dominant contribution to
$a_{\mu}^{had}$ as well as one of the largest ambiguities in its value come
from the $a_{\mu}^{had}(VP1)$. The $a_{\mu}^{had}(VP1)$ has been derived in
Ref. [7] from the $e^{+}e^{-}$ hadronic cross section and the hadronic $\tau$
decay data

$a_{\mu}^{had}(VP1)=6\ 924(62)\times 10^{-11}.$ $None$

Evolution of 3-loop hadron vacuum polarization contribution
$a_{\mu}^{had}(VP2)$ has given the result [8]

$a_{\mu}^{had}(VP2)=-100(6)\times 10^{-11}\mu_{0}.$ $None$

It is important to keep in mind that all the estimations of the $LbyL$
scattering contribution $a_{\mu}^{had}(LbyL)$ made so far are the model
dependent. The calculations are based on the chiral perturbation or extended
Nambu-Jona-Lasinio model. Also the vector meson dominance is assumed and the
phenomenological parametrization of the pion form factor
$\pi\gamma^{*}\gamma^{*}$ is introduced in order to regularize the divergence.
The previous average value for $a_{\mu}^{had}(LbL)$ is given by [9,10]

$a_{\mu}^{had}(LbyL)=-85(25)\times 10^{-11}\mu_{0}.$ $None$

With this value of the $LbyL$ hadronic correction the total SM prediction of
$a_{\mu}^{SM}$ was

$a_{\mu}^{SM}=116\ 591\ 597(67)\times 10^{-11}\mu_{0}.$ $None$

Comparing Eq. (9) with the experimental average in Eq. (4) one could find

$\delta a_{\mu}\equiv a_{\mu}^{exp}-a_{\mu}^{SM}=426(165)\times
10^{-11}\mu_{0}.$ $None$

Eq. (10) means that there is the 2.6 $\sigma$ deviation between experiment and
the SM prediction.

Recently the theoretical prediction for the $a_{\mu}^{had}(LbyL)$ has
undergone a significant revision because of the change in sign. The re-
evaluations have given the following values for $a_{\mu}^{had}(LbyL)$

$a_{\mu}^{had}(LbyL)=\left\\{\begin{array}[]{lll}83(12)\times
10^{-11}\mu_{0}\qquad[11],\\\ 89(15)\times 10^{-11}\mu_{0}\qquad[12],\\\
83(32)\times 10^{-11}\mu_{0}\qquad[13].\end{array}\right.$ $None$

Taking the average of these new results one finds

$a_{\mu}^{SM}=116\ 591\ 770(70)\times 10^{-11}\mu_{0}.$ $None$

Using Eqs. (4) and (12) one could obtain

$\delta a_{\mu}=260(160)\times 10^{-11}\mu_{0}.$ $None$

Thus the deviation value has droped from 2.6 $\sigma$ up to 1.6 $\sigma$.

On July 30, 2002 Muon g-2 Collaboration announced the new result based on the
$\mu^{+}$ data collected in the year 2000 [14]

$a_{\mu}^{exp}=116\ 592\ 040(70)(50)\times
10^{-11}\mu_{0}.\qquad(\mbox{BNL'00})$ $None$

An uncertainty of BNL’00 is almost two times smaller than in BNL’99 and only
two times larger than the final aim of the E821 experiment. With this new
result the present world average experimental value is

$a_{\mu}^{exp}=116\ 592\ 030(80)\times 10^{-11}\mu_{0}.$ $None$

The improved calculations of the $a_{\mu}^{had}(VP1)$ have been presented
recently [15,16]. These are data-driven analysis using the most recent data
from the $e^{+}e^{-}$ hadronic cross section observed at CMD-2, BES, SND [17].
Their precision $\sim 58\times 10^{-11}$ are now even smaller than those in
Eq. (6). Further on for the $a_{\mu}^{had}(VP1)$ we shall use the result of
Ref. [15] where the experimental input is based only on the $e^{+}e^{-}$ data

$a_{\mu}^{had}(VP1)=6\ 889(58)\times 10^{-11}\mu_{0}.$ $None$

For the estimation of the $a_{\mu}^{had}(LbyL)$ we invoke the new result
obtained in [18]

$a_{\mu}^{had}(LbyL)=80(40)\times 10^{-11}\mu_{0}.$ $None$

Then with help of Eqs. (7), (16) and (17) the full hadronic contributions is
given by

$a_{\mu}^{had}=6\ 869(71)\times 10^{-11}\mu_{0}.$ $None$

This leads us to the SM prediction

$a_{\mu}^{SM}=116\ 591\ 726.7(70.9)\times 10^{-11}\mu_{0}.$ $None$

So, at present the deviation between experimental data and the SM prediction
reached the value

$\delta a_{\mu}=303.3(106.9)\times 10^{-11}\mu_{0},$ $None$

that is, the deviation is roughly about 3$\sigma$.

Since the E821 data have been thoroughly collected and studied over many
years, it is most unlikely that this discrepancy could be also explained as a
mere statistical fluctuation, as several earlier deviations from the SM turned
out to be. Attention is drawn to the fact of the extremely small variation of
the muon AMM central value in all the BNL results presented up to now. This
circumstance could be the weighty argument in favour of a trustworthiness of
the E821 experiment. While it is often argued that the SM should be augmented
by New Physics at higher energy scales because of some unanswered fundamental
questions, the $(g-2)_{\mu}$ anomaly with such phenomena as the neutrino
oscillations [19], 3$\sigma$ departure of $\sin^{2}\theta_{W}$ from the SM
predictions measured in the deep inelastic neutrino-nucleon scattering [20],
and the observation of the neutrinoless double beta decay [21] may serve as
the New Physics signal already at the weak scale. If the deviation of Eq. (20)
can be attributed to effects of the physics beyond the SM, then at 95% $CL$,
$\delta a_{\mu}/\mu_{0}$ must lie in the range

$93.8\times 10^{-11}\leq{\delta a_{\mu}\over\mu_{0}}\leq 512.8\times
10^{-11}.$ $None$

This contribution is positive, and has the same order as the electroweak
corrections to $a_{\mu}$, namely $\sim G_{F}m_{\mu}^{2}/(4\pi^{2}\sqrt{2}).$

Suggestions already made in literature for explaining $\delta a_{\mu}/\mu_{0}$
include supersymmetry [22], additional gauge bosons [23], anomalous gauge
boson couplings [24], leptoquarks [25], extra dimensions [26], muon
substructure [27], exotic flavour-changing interactions [28], exotic
vectorlike fermions [29], possible nonpertubative effects at the 1 TeV order
[30] and so on.

Amongst explanations of E821-experiment at Brookhaven AGL the approach based
on the possibility of the violation of $CPT$ and Lorentz invariance (see, for
example [31]) should be particularly noted. In E821-experiment $a_{\mu}$ is
determined by measuring the difference $\omega_{a}$ between the spin
precession angular frequency $\omega_{s}$ and the cyclotron angular frequency
$\omega_{c}$ of highly polarized muons in a storage ring with a uniform
magnetic field. The field strength is determined from the nuclear magnetic
resonance (NMR) frequency of protons in water, calibrated relative to the free
proton NMR frequency $\omega_{p}$. The quantity $a_{\mu}$ is determined via

$a_{\mu}={\omega_{a}/\omega_{p}\over\mu_{\mu}/\mu_{p}-\omega_{a}/\omega_{p}},$
$None$

where the ratio $\mu_{\mu}/\mu_{p}$ is taken from the measurement by W.Liu et
al. [32]. According to Ref. [31] $CPT$/Lorentz violating terms in the
Lagrangian induce a shift $\delta\omega_{a}$ to frequency $\omega_{a}$. This
shift is predicted to be different for positive and negative muons and to
oscillate with the Earth’s sidereal frequency. The level of $CPT$/Lorentz
violating effects is characterized by the dimensionless quantity
$r=\omega_{a}/m_{\mu}$ which interpret $\delta\omega_{a}$ as an muon energy
shift in respect to the rest energy $m_{\mu}$. Already with the 1999 data set
$r$ could be probed down to the level of $0.19\times 10^{-22}$. Nowadays the
common believe is that $CPT$ and Lorentz invariance are the immovable laws of
Nature and, therefore, detecting of their violation must lead to the radical
alterations of the contemporary quantum field theory.

Some of explanations of E821-experiment turn out to be excluded by the current
experimental data. To cite some examples. The possibility of muon substructure
can be immediately ruled out since the necessary compositeness scale of muon
should already have been seen in processes involving high energetic muons at
LEP, HERA, and the Tevatron.

For anomalous $W$-boson dipole magnetic moment

$\mu_{W}={e\over 2m_{W}}(1+\kappa_{\gamma})$

the additional one loop contribution to $a_{\mu}$ is given by the expression

$a_{\mu}(\kappa_{\gamma})\approx{G_{F}m_{\mu}^{2}\over
4\sqrt{2}\pi^{2}}\ln\left({\Lambda^{2}\over
m_{W}^{2}}\right)(\kappa_{\gamma}-1),$

where $\Lambda$ is the high momentum cutoff required to give a finite result.
For $\Lambda\approx 1$ TeV, in order to obtain the accord between theory and
observation one should demand

$\delta\kappa_{\gamma}\equiv\kappa_{\gamma}-1\approx 0.4.$

However such a big values of $\delta\kappa_{\gamma}$ is already eliminated by
$e^{+}e^{-}\rightarrow W^{+}W^{-}$ data at LEP II which gives [33]

$\delta\kappa_{\gamma}=0.08\pm 0.17.$

In this manner, at the moment the $(g-2)_{\mu}$ anomaly plays the role of an
Occam’s razor for the existing SM extensions.

The purpose of this work is to investigate the $(g-2)_{\mu}$ anomaly within
the left right model (LRM) based on the gauge group $SU(2)_{R}\times
SU(2)_{L}\times U(1)_{B-L}$. One-loop contributions to $a_{\mu}$ from extra
gauge bosons have been calculated in [34]. However, the contribution coming
from the $Z_{2}$ gauge boson is negative while in order to accommodate the
discrepancy in Eq. (20) the mass value of the $W_{2}$ gauge boson should lies
around 100 GeV, which is clearly ruled out by direct searches and precission
measurements [33]. In the LRM the Higgs bosons may apply to the role of the
following candidates which may give significant contributions to the muon AMM.

In the SM, the Higgs boson contribution to $a_{\mu}$ is negligible because
$\overline{\mu}\mu h$ coupling is extremely small, namely $\sim m_{\mu}/v$,
where $v$ is a vacuum expectation value being equal to $246$ GeV. In the LRM
the Higgs sector is much richer than in the SM. It includes four doubly
charged scalars $\Delta^{(\pm\pm)}_{1,2}$, four singly charged scalars
$h^{(\pm)}$ and $\tilde{\delta}^{(\pm)}$, four neutral scalars $S_{i}$
($i=1,2,3,4$) and two neutral pseudoscalars $P_{1,2}$. The current
experimental data allow some of these Higgs bosons to have masses around the
electrowek scale and couplings of at least electroweak strength. It is well to
bear in mind that amongst the extensions of the SM the LRM is of special
interest because its Higgs sector contain the elements belonging to other most
popular nowaday models. The presence of the bidoublet in the LRM causes the
existence of the same physical Higgs bosons as in the two Higgs doublet
modification of the SM (THDM) [35] and in the MSSM [36]. Owing to the
availability of the triplets the LRM has the Higgs bosons which are present in
the model based on the $SU(3)_{L}\times U(1)_{N}$ gauge group [37].

One more a fascinating property of the LRM resides in the fact that the LRM
belongs among the models in which the Higgs bosons coupling constants (CC’s)
determining the interaction of the Higgs bosons both with leptons and with
gauge bosons are connected to the neutrino oscillation parameters (NOP’s).
Therefore, in such models the obtained bounds on the Higgs sector parameters
could be extended to the bounds on the NOP’s.

The paper is organized as follows. In the next Sect. the one-loop electroweak
corrections to the muon AMM caused by the LRM Higgs bosons are calculated.
There we establish the connection between the CC’s and the NOP’s. Then
comparing the theoretical and the experimental values of $a_{\mu}$ we find the
bounds on the Higgs sector parameters which provide in its turn information on
the heavy neutrino masses and the mixing angles. Sec.4 is devoted to analysis
of the results obtained.

##  2 Higgs bosons corrections to $a_{\mu}$

In the LRM the choice of the Yukawa potential has the influence upon the form
of the Lagrangian describing the Higgs boson interactions both with fermions
and gauge bosons. The most general Yukawa potential ${\cal{L}}^{g}_{Y}$ has
been proposed in [38]. In spite of the fact that ${\cal{L}}^{g}_{Y}$ has the
very complicated form the diagonalization of the charged Higgs bosons mass
matrix presents no special problems. However, for the neutral Higgs bosons
mass matrix $M_{n}$ this procedure could be only realized when some
simplifications in ${\cal{L}}^{g}_{Y}$ have been done [39]. For example, the
matrix $M_{n}$ could be diagonalized at the following conditions (we use the
same notation as in Ref.[38])

$\alpha_{1}={2\alpha_{2}k_{2}\over
k_{1}},\qquad\alpha_{3}={2\alpha_{2}k_{-}^{2}\over
k_{1}k_{2}},\qquad\beta_{1}={2\beta_{3}k_{2}\over k_{1}},$ $None$

where $\alpha_{1,2,3}$ and $\beta_{1,2}$ are the constants entering the Yukawa
potential, $k_{1}$ $k_{2}$ are the vacuum expectation values (VEV’s) of the
neutral components of the Higgs bidoublet and $k_{\pm}=\sqrt{k_{1}^{2}\pm
k_{2}^{2}}$ ($k_{+}=174$ GeV).

Of all the Higgs bosons, $\Delta^{(\pm\pm)}_{1,2}$-, $h^{(\pm)}$ and
$\tilde{\delta}^{(\pm)}$\- and $S_{1}$-bosons have been of our main interest
here because the current data allow their masses to lie on the electroweak
scale (recall that the $S_{1}$-boson is the analog of the SM Higgs boson).
Assuming the conditions (23) to be fulfilled one obtain the squared masses of
these particles

$m_{h}^{2}=\alpha(v_{R}^{2}+k_{0}^{2})+{\beta_{1}^{2}k_{+}^{4}k_{0}^{2}\over{k_{-}^{4}(\alpha+\rho_{1}-\rho_{3}/2)}},$
$None$
$m_{\tilde{\delta}}^{2}=(\rho_{3}/2-\rho_{1})v_{R}^{2}-{\beta_{1}^{2}k_{+}^{4}k_{0}^{2}\over{k_{-}^{4}(\alpha+\rho_{1}-\rho_{3}/2)}}.$
$None$ $m_{\Delta_{1}}^{2}={\alpha_{3}k_{-}^{2}+4\rho_{2}v_{R}^{2}\over
2}+{k_{-}^{4}(\beta_{3}k_{+}^{2}+\beta_{1}k_{1}k_{2})^{2}\over{2k_{1}^{4}(4\rho_{2}+\rho_{3}-2\rho_{1})v_{R}^{2}}},$
$None$
$m_{\Delta_{2}}^{2}={\alpha_{3}k_{-}^{2}-(2\rho_{1}-\rho_{3})v_{R}^{2}\over
2}-{k_{-}^{4}(\beta_{3}k_{+}^{2}+\beta_{1}k_{1}k_{2})^{2}\over{2k_{1}^{4}(4\rho_{2}+\rho_{3}-2\rho_{1})v_{R}^{2}}},$
$None$
$m_{S_{1}}^{2}=2\lambda_{1}k_{+}^{2}+8k_{1}^{2}k_{2}^{2}(2\lambda_{2}+\lambda_{3})/k_{+}^{2}-8\lambda_{4}k_{1}k_{2}+$
$+{4k_{1}k_{2}k_{-}^{4}[2(2\lambda_{2}+\lambda_{3})k_{1}k_{2}/k_{+}^{2}-\lambda_{4}]^{2}\over\alpha_{2}v_{R}^{2}k_{+}^{2}},$
$None$

where

$\alpha={\alpha_{3}k_{+}^{2}\over
2k_{-}^{2}}={\alpha_{3}(1+\tan^{2}\beta)\over
2(\tan^{2}\beta-1)},\qquad\beta_{0}={\beta_{1}k_{+}^{2}\over
k_{-}^{2}}={\beta_{1}(1+\tan^{2}\beta)\over(\tan^{2}\beta-1)},$
$k_{0}={k_{-}^{2}\over\sqrt{2}k_{+}}={k_{+}(\tan^{2}\beta-1)\over\sqrt{2}(1+\tan^{2}\beta)},$

$\rho_{1,3}$ are the constants entering the Yukawa potential,
$\tan\beta=k_{1}/k_{2}$ and $v_{R}$ is the VEV of the neutral component of the
right-handed Higgs triplet, $v_{R}\gg\mbox{max}(k_{1},k_{2})$). From the
relations (25) and (27) it follows that the masses $\tilde{\delta}^{(\pm)}$
and $\Delta^{(\pm\pm)}_{2}$-bosons are very close to each other. For the
$h^{(\pm)}$, $\tilde{\delta}^{(\pm)}$ and $\Delta^{(\pm\pm)}_{1,2}$-boson
masses to be around the electroweak scale the constants $\alpha_{3}$,
$\rho_{2}$ and $(\rho_{3}/2-\rho_{1})$ should have the order of $\sim 10^{-2}$
which follows from the expressions (24) — (27).

The Lagrangians which are required for our purposes are given by the
expressions

${\cal{L}}_{\gamma\Delta\Delta}=2ie[(\partial_{\mu}\Delta^{(--)*}_{1}(x))\Delta^{(--)}_{1}(x)-\Delta^{(--)*}_{1}(x)(\partial_{\mu}\Delta^{(--)}_{1}(x))]+(1\rightarrow
2),$ $None$ ${\cal{L}}^{dc}_{l}=-\sum_{a,b}{f_{ab}\over
2}[\overline{l}^{c}_{a}(x)(1+\gamma_{5})l_{b}(x)c_{\theta_{d}}-\overline{l}^{c}_{a}(x)(1-\gamma_{5})l_{b}(x)s_{\theta_{d}}]\Delta^{(--)*}_{1}(x)+$
$+(1\rightarrow 2,\theta_{d}\rightarrow\theta_{d}-{\pi\over 2})+\mbox{conj}.,$
$None$ ${\cal{L}}_{W_{1}\gamma
h}={g_{R}em_{W_{1}}(1-\tan^{2}\beta)(\alpha-\rho_{3}/2+\rho_{1}+1)s_{\xi}\over
g_{L}(1+\tan^{2}\beta)}h^{(-)*}(x)W_{1\mu}(x)A_{\mu}(x)+\mbox{conj}.,$ $None$
${\cal{L}}_{W_{2}\gamma h}={\cal{L}}_{W_{1}\gamma h}(s_{\xi}\rightarrow
c_{\xi}),$ $None$
${\cal{L}}_{W_{1}\gamma\tilde{\delta}}=g_{R}eg_{L}^{-1}\beta_{1}m_{W_{1}}s_{\xi}\tilde{\delta}^{(-)*}(x)W_{1\mu}(x)A_{\mu}(x)+\mbox{conj}.,$
$None$
${\cal{L}}_{W_{2}\gamma\tilde{\delta}}={\cal{L}}_{W_{1}\gamma\tilde{\delta}}(s_{\xi}\rightarrow
c_{\xi}),$ $None$
${\cal{L}}^{sc}_{l}=\sum_{a,b}\\{[{h_{ab}^{\prime}k_{2}-h_{ab}k_{1}\over
2k_{+}}\overline{\nu}_{a}(x)(1-\gamma_{5})l_{b}(x)-{h_{ab}k_{2}-h_{ab}^{\prime}k_{1}\over
2k_{+}}\overline{N}_{a}(x)(1+\gamma_{5})l_{b}(x)]h^{(-)*}(x)+$
$+{f_{ab}\over{\sqrt{2}}}[\overline{l}^{c}_{a}(x)(1+\gamma_{5})\nu_{b}(x)\left({\beta_{0}k^{2}_{0}\over{(\alpha+\rho_{1}-\rho_{3}/2)v_{R}^{2}}}h^{(-)*}(x)-\tilde{\delta}^{(-)*}(x)\right)+\overline{l}^{c}_{a}(x)(1-$
$-\gamma_{5})N_{b}(x)\left({k_{0}\over
v_{R}}h^{(-)*}(x)+{\beta_{0}k_{0}\over{(\alpha+\rho_{1}-\rho_{3}/2)v_{R}}}\tilde{\delta}^{(-)*}(x)\right)]+\mbox{conj}.\\},$
$None$
${\cal{L}}^{n}=-{1\over\sqrt{2}k_{+}}\\{\sum_{a,b}\overline{l}_{a}(x)l_{b}(x)[(h_{ab}k_{1}+h^{\prime}_{ab}k_{2})s_{\theta_{0}}+(h^{\prime}_{ab}k_{1}-h_{ab}k_{2})c_{\theta_{0}}]S_{1}(x),$
$None$

where the superscript $c$ denotes the charge conjugation operation,
$c_{\theta_{d}}=\cos\theta_{d}$, $s_{\theta_{d}}=\sin\theta_{d}$, $\theta_{d}$
is the mixing angle of the doubly charged Higgs bosons
($\tan\theta_{d}\sim{k_{+}^{2}/v_{R}^{2}}$), $f_{ab}$ is the Yukawa triplet
coupling constant, $g_{R}$ is the gauge coupling of the $SU(2)_{R}$ subgroup
(further we shall speculate that $g_{L}=g_{R}$), $N_{a}(x)$ describes the
heavy neutrino with the flavor $a$, $\xi$ is the mixing angle of the charged
gauge bosons, and the angle $\theta_{0}$ is determined by the Yukawa potential
parameters and the VEV’s

$\tan
2\theta_{0}={{4k_{1}k_{2}k_{-}^{2}[-2(2\lambda_{2}+\lambda_{3})k_{1}k_{2}+\lambda_{4}k_{+}^{2}]}\over{k_{1}k_{2}[(4\lambda_{2}+2\lambda_{3})(k_{-}^{4}-4k_{1}^{2}k_{2}^{2})-k_{+}^{2}(2\lambda_{1}k_{+}^{2}-8\lambda_{4}k_{1}k_{2})]+\alpha_{2}v_{R}^{2}k_{+}^{4}}}.$
$None$

The influence of the Yukawa potential choice on the physical results could be
easily seen by the example of the Lagrangian (36). Really, when in the
condition (23) the change $k_{2}\rightarrow-k_{2}$ is carried out, then
instead of (36) one obtains

${\cal{L}}^{n}=-{1\over\sqrt{2}k_{+}}\\{\sum_{a}m_{a}\overline{l}_{a}(x)l_{a}(x)c_{\theta_{0}}+\sum_{a,b}\overline{l}_{a}(x)l_{b}(x)(h_{ab}k_{1}-h^{\prime}_{ab}k_{2})s_{\theta_{0}}\\}S_{1}(x).$
$None$

Since $\tan\beta_{0}\sim k_{-}^{2}/v_{R}^{2}$ and for the muon
$m_{\mu}/k_{+}\sim 6\times 10^{-3}$, then, as the exapmle, the cross section
of the electron-muon recharge

$e^{-}\mu^{+}\rightarrow e^{+}\mu^{-},$

never could have the resonance peak connected with the $S_{1}$-boson when the
Lagrangian (38) is used [39], while the existence of such a peak could be
quite possible when one works with the Lagrangian (36) [40].

We now proceed to the calculations of the contribution to the muon AMM caused
by the Higgs bosons. The diagrams corresponding the exchange of the doubly
charged Higgs bosons are shown in Fig.1. They give the following corrections
to the AMM value

${\delta a_{\mu}^{\Delta}\over\mu_{0}}={1\over 8\pi^{2}}\left(4f_{\mu
e}^{2}\sum_{i=1}^{2}I^{\Delta_{i}}_{e}+f_{\mu\mu}^{2}\sum_{i=1}^{2}I^{\Delta_{i}}_{\mu}+4f_{\mu\tau}^{2}\sum_{i=1}^{2}I^{\Delta_{i}}_{\tau}\right),$
$None$

where

$I^{\Delta_{i}}_{l_{a}}=\int^{1}_{0}\left({2m_{\mu}^{2}(z^{2}-z^{3})\over
m_{\mu}^{2}(z^{2}-z)+m_{\Delta_{i}}^{2}z+m_{l_{a}}^{2}(1-z)}+{m_{\mu}^{2}(z^{2}-z^{3})\over
m_{\mu}^{2}(z^{2}-z)+m_{\Delta_{i}}^{2}(1-z)+m_{l_{a}}^{2}z}\right)dz,$

and $I^{\Delta_{i}}_{l_{a}}>0.$

The singly charged Higgs bosons also influence the value of the AMM. The
relevant diagrams are depicted in Fig.2. For the diagrams which contain the
loops with the $W_{1}^{\pm}-$ and $h^{(\pm)}$-bosons the following relation
takes place

${M_{W_{1}N_{\mu}h}\over M_{W_{1}\nu_{\mu}h}}=s_{\xi},$

where $M_{W_{1}\nu_{\mu}h}$ ($M_{W_{1}N_{\mu}h}$) are the matrix elements
appropriate the diagrams with the exchange of the light (heavy) neutrino. As
the mixing angle of the charged gauge bosons is very small $\mid\xi\mid\approx
10^{-2}-10^{-5}$ [33], then one could neglect the contributions coming from
the diagrams with the virtual heavy neutrino. Taking into account the
analogous relations

${M_{W_{2}\nu_{\mu}h}\over
M_{W_{2}N_{\mu}h}}=s_{\xi},\qquad{M_{W_{1}N_{\mu}\tilde{\delta}}\over
M_{W_{1}\nu_{\mu}\tilde{\delta}}}=s_{\xi},\qquad{M_{W_{2}\nu_{\mu}\tilde{\delta}}\over
M_{W_{2}N_{\mu}\tilde{\delta}}}=s_{\xi},$

we give the dominant contribution from the diagrams shown in Fig.2 to the muon
AMM

${\delta a_{\mu}^{(hh)}\over\mu_{0}}={1\over
8\pi^{2}}\sum_{a=e,\mu,\tau}\left(\alpha_{\mu
N_{a}h}^{2}I^{hh}_{N_{a}}+\alpha_{\mu\nu_{a}h}^{2}I^{hh}_{\nu_{a}}\right),$
$None$ ${\delta a_{\mu}^{(\tilde{\delta}\tilde{\delta})}\over\mu_{0}}={1\over
8\pi^{2}}\sum_{a=e,\mu,\tau}\left(\alpha_{\mu
N_{a}\tilde{\delta}}^{2}I^{\tilde{\delta}\tilde{\delta}}_{N_{a}}+\alpha_{\mu\nu_{a}\tilde{\delta}}^{2}I^{\tilde{\delta}\tilde{\delta}}_{\nu_{a}}\right),$
$None$ ${\delta
a_{\mu}^{(W_{1}h)}\over\mu_{0}}={(\alpha-\rho_{3}/2+\rho_{1}+1)(1-\tan^{2}\beta)s_{\xi}m_{W_{1}}\over
16\sqrt{2}\pi^{2}(1+\tan^{2}\beta)}\alpha_{\mu\nu_{\mu}h}I^{W_{1}h},$ $None$
${\delta
a_{\mu}^{(W_{2}h)}\over\mu_{0}}={(\alpha-\rho_{3}/2+\rho_{1}+1)(1-\tan^{2}\beta)c_{\xi}m_{W_{1}}\over
16\sqrt{2}\pi^{2}(1+\tan^{2}\beta)}\alpha_{\mu N_{\mu}h}I^{W_{2}h},$ $None$
${\delta
a_{\mu}^{(W_{1}\tilde{\delta})}\over\mu_{0}}={\beta_{1}(1-\tan^{2}\beta)s_{\xi}m_{W_{1}}\alpha_{\mu\nu_{\mu}\tilde{\delta}}\over
16\sqrt{2}\pi^{2}(1+\tan^{2}\beta)}I^{W_{1}\tilde{\delta}},$ $None$ ${\delta
a_{\mu}^{(W_{2}\tilde{\delta})}\over\mu_{0}}={\beta_{1}(1-\tan^{2}\beta)c_{\xi}m_{W_{1}}\alpha_{\mu
N_{\mu}\tilde{\delta}}\over
16\sqrt{2}\pi^{2}(1+\tan^{2}\beta)}I^{W_{2}\tilde{\delta}},$ $None$

where

$\alpha_{l_{a}\nu_{b}h}={h_{ab}^{\prime}k_{2}-h_{ab}k_{1}\over
2k_{+}},\qquad\alpha_{l_{a}N_{b}h}={h_{ab}^{\prime}k_{1}-h_{ab}k_{2}\over
2k_{+}},\qquad\alpha_{l_{a}\nu_{b}\tilde{\delta}}={f_{ab}\over{\sqrt{2}}},$
$\alpha_{l_{a}N_{b}\tilde{\delta}}={f_{ab}\beta_{0}k_{0}\over\sqrt{2}(\alpha+\rho_{1}-\rho_{3}/2)v_{R}}={f_{ab}\beta_{1}k_{+}\over
2(\alpha+\rho_{1}-\rho_{3}/2)v_{R}},$
$I^{hh}_{i}=\int^{1}_{0}{m_{\mu}^{2}(z^{3}-z^{2})dz\over
m_{\mu}^{2}z^{2}+(m_{h}^{2}-m_{i}^{2}-m_{\mu}^{2})z+m_{i}^{2}},\qquad
i=\nu_{a},N_{a},$
$I^{\tilde{\delta}\tilde{\delta}}_{i}=I^{hh}_{i}(m_{h}\rightarrow
m_{\tilde{\delta}}),\qquad I^{hh}_{i}<0$ $I^{W_{1}h}={m_{\mu}\over
m_{W_{1}}^{2}-m_{h}^{2}}\\{\ln\left({m_{W_{1}}^{2}\over
m_{h}^{2}}\right)-\int_{0}^{1}{z^{2}[m_{\mu}^{2}(2z-1)+m_{W_{1}}^{2}-m_{\nu_{\mu}}^{2}]dz\over
m_{\mu}^{2}z^{2}+(m_{W_{1}}^{2}-m_{\nu_{\mu}}^{2}-m_{\mu}^{2})z+m_{\nu_{\mu}}^{2}}+(m_{W_{1}}\rightarrow
m_{h})\\},$ $I^{W_{2}h}=I^{W_{1}h}(m_{W_{1}}\rightarrow
m_{W_{2}},m_{\nu_{\mu}}\rightarrow m_{N_{\mu}}),\qquad
I^{W_{k}\tilde{\delta}}=I^{W_{k}h}(m_{h}\rightarrow m_{\tilde{\delta}}),\qquad
I^{W_{k}h}>0.$

The contribution from the neutral Higgs boson $S_{1}$ to the AMM value are due
to the diagram shown in Fig.3 and is given by

${\delta a_{\mu}^{(S_{1})}\over\mu_{0}}={1\over 8\pi^{2}}\sum_{a}\alpha_{\mu
l_{a}S_{1}}^{2}I^{S_{1}}_{l_{a}},$ $None$

where

$\alpha_{l_{a}l_{b}S_{1}}=-{1\over\sqrt{2}k_{+}}[(h_{ab}k_{1}+h^{\prime}_{ab}k_{2})s_{\theta_{0}}+(h^{\prime}_{ab}k_{1}-h_{ab}k_{2})c_{\theta_{0}}],$
$I^{S_{1}}_{l_{a}}=\int^{1}_{0}{[m_{\mu}^{2}(z^{2}-z^{3})+m_{l_{a}}^{2}z^{2}]dz\over
m_{\mu}^{2}(z^{2}-z)+m_{S_{1}}^{2}(1-z)+m_{l_{a}}^{2}z},\qquad
I^{S_{1}}_{l_{a}}>0.$

The total correction value motivated by the Higs bosons to the muon AMM
$\delta a_{\mu}/\mu_{0}$ is defined by the sum of the expressions (39) — (46).
To perform an exhaustive analysis of the obtained result one should have
information both about the coupling constants $\alpha_{L_{a}L_{b}H_{i}}$
($L_{a}=l_{a},\nu_{a},N_{a}$) and the Higgs boson masses $m_{H_{i}}$. At
present such information follows from looking for the deviations from the SM
predictions. It is usually reported in terms of the upper limits for
quantities of the type $\alpha_{L_{a}L_{b}H_{i}}/m_{H_{i}}$ or, that is more
frequent, for quantities of the type

$\sum_{i}C_{i}\epsilon_{i}^{aba^{\prime}b^{\prime}}=\sum_{i}C_{i}{(\alpha_{L_{a}L_{b}H_{i}}\alpha_{L_{a^{\prime}}L_{b^{\prime}}H_{i}})^{2}\over
m_{H_{i}}^{4}},$

where $C_{i}$ are the constants (see, for review [41]). As a rule the
determination of the upper bound only for one quantity
$\alpha_{L_{a}L_{b}H_{i}}/m_{H_{i}}$ is a very involved task. As a case in
point we consider the decay

$\mu^{-}\rightarrow e^{-}\gamma.$ $None$

In the third order of the perturbation theory the appropriate diagrams follow
from those in Figs.1,2 and 3 when in the final state the muon is replaced by
the electron. The width of the decay (47) includes the whole complex
quantities $\epsilon_{i}^{aba^{\prime}b^{\prime}}$. Besides, the presence of
the diagrams with the singly charged Higgs bosons leads to the destructive
interference of the reaction amplitudes that complicates a lot the task of
extracting information about the individual quantities of
$\epsilon_{i}^{aba^{\prime}b^{\prime}}$.

However, the Higgs bosons coupling constants
$\alpha_{\overline{L}_{a}L_{b}H_{i}}$ could be expressed in terms of the
lepton sector parameters. It is easily done at least in the two-flavor
approximation. For this purpose we need the neutrino mass matrix ${\cal M}$.
Once one chooses the basis
$\Psi^{T}=\left(\nu_{aL}^{T},N_{aR}^{T},\nu_{bL}^{T},N_{bR}^{T}\right)$, the
${\cal M}$ takes the form

${\cal M}=\left(\matrix{f_{aa}v_{L}&m^{a}_{D}&f_{ab}v_{L}&M_{D}\cr
m_{D}^{a}&f_{aa}v_{R}&M_{D}&f_{ab}v_{R}\cr
f_{ab}v_{L}&M_{D}&f_{bb}v_{L}&m^{b}_{D}\cr
M_{D}&f_{ab}v_{R}&m_{D}^{b}&f_{bb}v_{R}\cr}\right),$ $None$

where $v_{L}$ is the VEV of the neutral component of the left-handed Higgs
triplet ($v_{L}\ll(\mbox{max}(k_{1},k_{2})$) and

$m_{D}^{a}=h_{aa}k_{1}+h^{\prime}_{aa}k_{2},$ $None$
$M_{D}=h_{ab}k_{1}+h^{\prime}_{ab}k_{2}.$ $None$

In its turn the elements of the matrix ${\cal M}$ are connected with the
neutrino oscillations parameters [39,42]

$\left.\begin{array}[]{lll}m_{D}^{a}=c_{\varphi_{a}}s_{\varphi_{a}}(-m_{1}c^{2}_{\theta_{\nu}}-m_{3}s^{2}_{\theta_{\nu}}+m_{2}c^{2}_{\theta_{N}}+m_{4}s^{2}_{\theta_{N}}),\\\
\\\
m_{D}^{b}=m_{D}^{a}(\varphi_{a}\rightarrow\varphi_{b},\theta_{\nu,N}\rightarrow\theta_{\nu,N}+{\pi\over
2}),\end{array}\right\\}$ $None$
$M_{D}=c_{\varphi_{a}}s_{\varphi_{b}}c_{\theta_{\nu}}s_{\theta_{\nu}}(m_{1}-m_{3})+s_{\varphi_{a}}c_{\varphi_{b}}c_{\theta_{N}}s_{\theta_{N}}(m_{4}-m_{2}),$
$None$
$f_{ab}v_{R}=s_{\varphi_{a}}s_{\varphi_{b}}c_{\theta_{\nu}}s_{\theta_{\nu}}(m_{3}-m_{1})+c_{\varphi_{a}}c_{\varphi_{b}}c_{\theta_{N}}s_{\theta_{N}}(m_{4}-m_{2}),$
$None$
$\left.\begin{array}[]{lll}f_{aa}v_{R}=(s_{\varphi_{a}}c_{\theta_{\nu}})^{2}m_{1}+(c_{\varphi_{a}}c_{\theta_{N}})^{2}m_{2}+(s_{\varphi_{a}}s_{\theta_{\nu}})^{2}m_{3}+(c_{\varphi_{a}}s_{\theta_{N}})^{2}m_{4},\\\
\\\ f_{bb}v_{R}=f_{aa}v_{R}(\varphi_{a}\rightarrow\varphi_{b}+{\pi\over
2},\theta_{N}\rightarrow\theta_{N}+{\pi\over 2}),\end{array}\right\\}$ $None$
$f_{ll^{\prime}}v_{L}=f_{ll^{\prime}}v_{R}(\varphi_{l,l^{\prime}}\rightarrow\varphi_{l,l^{\prime}}+{\pi\over
2}),\qquad l,l^{\prime}=a,b,$ $None$

where $\varphi_{a}$ is the mixing angle in the $a$ generation between the
light and the heavy neutrino entering into the left-handed and the right-
handed lepton doublet

$\left(\matrix{\nu_{a}\cr l_{a}}\right)_{L},\qquad\left(\matrix{N_{a}\cr
l_{a}}\right)_{R},$

respectively, $\theta_{\nu}(\theta_{N})$ is the mixing angle between the
$\nu_{aL}$ and the $\nu_{bL}$ neutrino ($N_{aR}$ and $N_{bR}$),
$c_{\varphi_{a}}=\cos\varphi_{a},\ s_{\varphi_{a}}=\sin\varphi_{a}$ and so on.
As $m_{\nu}\ll m_{N}$, then with the help of Eqs.(53) and (55) it is possible
to find the relationship for an estimation of the mixing angles between light
and heavy neutrinos. Further on we shall assume that the mixing takes place
between $\mu$ and $\tau$ generations ($a=\mu,b=\tau$) only. Then for the
mixing angles we obtain

$\sin 2\varphi_{\mu}\approx{f_{\mu\mu}\sqrt{v_{R}v_{L}}\over
c_{\theta_{N}}^{2}m_{N_{\mu}}+s_{\theta_{N}}^{2}m_{N_{\tau}}},$ $None$ $\sin
2\varphi_{\tau}\approx{f_{\tau\tau}\sqrt{v_{R}v_{L}}\over
s_{\theta_{N}}^{2}m_{N_{\mu}}+c_{\theta_{N}}^{2}m_{N_{\tau}}}.$ $None$

The estimation of $v_{L}$ could be done with the help of quantity

$\rho={m_{Z}^{2}c_{W}^{2}\over m_{W}^{2}}.$

In LRM the quantity $\rho$ is defined by the relation [43]

$\rho={1+4x\over 1+2x},$ $None$

where

$x=\left({v_{L}\over k_{+}}\right)^{2}.$

As the experiment for today yields

$\rho=1.0107\pm 0.0006,$

that the value $v_{L}$ can reach 13 GeV.

Taking into consideration both the definition of $m^{a}_{D}$ (Eqs. (49), (51)
) and the formulae for the charged lepton masses

$m_{l_{a}}=h_{aa}k_{2}+h_{aa}^{\prime}k_{1}$ $None$

it is not difficult to obtain

$\alpha_{\overline{l}_{a}\nu_{a}h}={h_{aa}^{\prime}k_{2}-h_{aa}k_{1}\over
2k_{+}}={1+\tan^{2}\beta\over
2k_{+}(1-\tan^{2}\beta)}\left({2m_{l_{a}}\tan\beta\over
1+\tan^{2}\beta}+m^{a}_{D}\right)\approx$ $\approx{1+\tan^{2}\beta\over
2k_{+}(1-\tan^{2}\beta)}\left[{2m_{l_{a}}\tan\beta\over
1+\tan^{2}\beta}+c_{\varphi_{a}}s_{\varphi_{a}}(m_{2}c^{2}_{\theta_{N}}+m_{4}s^{2}_{\theta_{N}})\right].$
$None$

The analogous mathematics for $\alpha_{\overline{l}_{a}N_{a}h}$ lead to the
expression

$\alpha_{\overline{l}_{a}N_{a}h}={h_{aa}^{\prime}k_{1}-h_{aa}k_{2}\over
2k_{+}}={1+\tan^{2}\beta\over
2k_{+}(1-\tan^{2}\beta)}\left({2m^{a}_{D}\tan\beta\over
1+\tan^{2}\beta}-m_{l_{a}}\right)\approx$ $\approx{1+\tan^{2}\beta\over
2k_{+}(1-\tan^{2}\beta)}\left[{2c_{\varphi_{a}}s_{\varphi_{a}}(m_{2}c^{2}_{\theta_{N}}+m_{4}s^{2}_{\theta_{N}})\tan\beta\over
1+\tan^{2}\beta}-m_{l_{a}}\right].$ $None$

It is pertinent to note that there is the connection between the coupling
constants $\alpha_{\overline{l}_{a}N_{b}h}$ and
$\alpha_{\overline{l}_{a}l_{b}S_{1}}$

$\alpha_{\overline{l}_{a}N_{b}h}\approx{\alpha_{\overline{l}_{a}l_{b}S_{1}}\over\sqrt{2}}.$
$None$

The next step is the determination of the non-diagonal Higgs bosons coupling
constants. To suppress the mixing in the charged lepton sector (betwen $l_{a}$
and $l_{b}$) it is necessary to demand

$h_{ab}k_{2}+h_{ab}^{\prime}k_{1}=0.$ $None$

Then, with the regard to the definitions of the quantity $M_{D}$ (Eqs.(45) and
(47) ) one obtains

$\alpha_{\overline{l}_{a}\nu_{b}h}=-{M_{D}\over
2k_{+}}\approx-{s_{\varphi_{a}}c_{\varphi_{b}}c_{\theta_{N}}s_{\theta_{N}}(m_{4}-m_{2})\over
2k_{+}},$ $None$ $\alpha_{\overline{l}_{a}N_{b}h}=-{M_{D}\tan\beta\over
k_{+}(1+\tan^{2}\beta)}\approx-{s_{\varphi_{a}}c_{\varphi_{b}}c_{\theta_{N}}s_{\theta_{N}}(m_{4}-m_{2})\tan\beta\over
k_{+}(1+\tan^{2}\beta)}.$ $None$

From the expressions (60) — (62), (64) and (65) it is obvious that the values
of the coupling constants $\alpha_{\overline{L}_{a}L_{b}H_{i}}$ are basically
defined by the oscillation parameters of the heavy neutrinos. Nowadays the
information concerning the heavy neutrino sector is very poor. All we have is
the upper bound for the heavy electron neutrino mass resulting from the
experiments aimed at finding the neutrinoless double $\beta$ decay

$m_{N_{e}}>63\ \mbox{GeV}\left({1.6\ \mbox{TeV}\over m_{W_{2}}}\right)^{4}.$
$None$

Hence, our sole way out in an existing situation is to set any minimal number
of parameters of heavy neutrinos sector, and other parameters to express
through them with the help of the equations (51) — (55). As those we shall
take $m_{N_{\mu}}$, $m_{N_{\tau}}$ and $\theta_{N}$.

Now we shall pass to the discussion of the constraints on the Higgs bosons
masses. The current limit on the singly charged Higgs boson mass has been
obtained within the THDM’s under investigation of the reaction

$e^{+}e^{-}\rightarrow H^{+}H^{-}.$ $None$

The lowest value for the mass of the charged Higgs boson, independent of the
its branching ratio, is currently 78.6 GeV [33]. It is evident that this limit
may be broken down to the singly charged Higgs bosons of the LRM
$\tilde{\delta}^{(\pm)}$ and $h^{(\pm)}$. Really, in the THDM the charged
Higgs boson interacts with the quarks at the tree level while in the LRM such
interaction exists for the $h^{(\pm)}$-boson only. Furthermore, in the both
models the coupling constants of the charged Higgs bosons with the $Z$-boson
are not equal to each other

${(g_{HHZ})_{2HDM}\over(g_{\tilde{\delta}\tilde{\delta}Z})_{LRM}}={\cot
2\theta_{W}\over
g^{\prime}\cos^{-1}\theta_{W}(\alpha+\rho_{1}-\rho_{3}/2)(g^{\prime-1}\sin\theta_{W}\cos\Phi+g_{R}^{-1}\sin\Phi)},$
$None$ ${(g_{HHZ})_{2HDM}\over(g_{hhZ})_{LRM}}=-{\cot
2\theta_{W}\over\cos^{-1}\theta_{W}(\sin^{-1}\theta_{W}\cos
2\theta_{W}\cos\Phi+g_{R}g^{\prime-1}\sin\Phi)/2}.$ $None$

However, since the analysis of the process (67) from the LRM poit of view is
absent up to now, we shall assume that the lower bound on the masses of the
singly charged Higgs bosons of the LRM is 78.6 GeV too.

As regards the doubly charged Higgs boson mass the situation is somewhat more
simple. The $\Delta^{(\pm\pm)}_{1,2}$-bosons are typical representatives of
the LRM. For this reason the experiments aimed at their appearence are
analyzed from the LRM poit of view only. The current lower bounds on their
masses obtained by OPAL Collaboration at 95% $CL$ [44] are 98.5 GeV.

We also should discuss the implementation of the lower bound 115 GeV on the
mass of the lightest neutral Higgs (LNH) in the SM extensions. This bound has
been obtained on LEP II under investigation of the reaction

$e^{+}e^{-}\rightarrow Z^{*}\rightarrow Zh$ $None$

from the SM poit of view [45]. The reaction (70) is analyzed for the four $Zh$
decay channels

$Zh\rightarrow
q\overline{q}q^{\prime}\overline{q}^{\prime},q\overline{q}\nu\overline{\nu},q\overline{q}l_{a}\overline{l}_{a}\
(l_{a}=e,\mu),\tau^{+}\tau^{-}q\overline{q},$

where the final state $h\rightarrow q\overline{q}$ includes both the quark-
antiquark and the gluon-gluon pairs. For the THDM’s, as an example, the
substantial deviations from the SM are present both in the cross section
$\sigma_{e^{+}e^{-}\rightarrow Zh}$ and in the decay widths $\Gamma_{h}$.
Since in these models the coupling constant of the neutral CP-even $h$ boson
(analog of the SM Higgs boson) with the $Z$ boson has the form

$g_{ZZh}={g_{L}m_{Z}\sin(\beta-\alpha)\over\mbox{c}_{\theta_{W}}}$ $None$

where $\sin(\beta-\alpha)\sim 0$, then $\left(\sigma_{e^{+}e^{-}\rightarrow
Zh}\right)_{THDM}$ is much less relatively to the SM value. On the other hand,
as the relation takes place

${\left(g_{f\overline{f}h}\right)_{THDM}\over\left(g_{f\overline{f}h}\right)_{SM}}\approx\tan\beta,\qquad
f=b,\tau.$

the $h$ boson decay widths through quarks and leptons have greater values than
in the SM. It is evident, that the analysis of the reaction (70) leads to the
different mass values for the SM Higgs and the LNH of the THDM. Actually, when
$\mid\sin(\beta-\alpha)\mid\leq 0.06$ the LEP data result in $m_{h}\sim 10$
GeV at 98% CL [46].

The only reason for the LRM cross section $\sigma_{e^{+}e^{-}\rightarrow
ZS_{1}}$ not to coincide with that of the SM may be the coupling constant of
the $S_{1}$ boson with the $Z_{1}$ boson

$g_{Z_{1}Z_{1}S_{1}}={g_{L}\mbox{s}_{W}^{2}m_{Z_{1}}[\mbox{c}_{\theta_{0}}(1-\mbox{tg}^{2}\beta)-2\mbox{s}_{\theta_{0}}\tan\beta)]\over
2\mbox{c}_{W}(\tan^{2}\beta+1)}\left(2g_{R}g^{\prime-1}\mbox{c}_{\Phi}\mbox{s}_{\Phi}\mbox{s}_{W}^{-1}-\right.$
$\left.-\mbox{c}_{\Phi}^{2}\mbox{s}_{W}^{-2}-g_{R}^{2}g^{\prime-2}\mbox{s}_{\Phi}^{2}\right),$
$None$

where $\Phi$ is the mixing angle of $Z_{1}$ and $Z_{2}$ bosons ($\Phi\approx
10^{-2}-10^{-3}$) and $g^{\prime}$ is the gauge constant of subgroup
$U(1)_{B-L}$. When

$\Phi=k_{2}=0$

then $g_{Z_{1}Z_{1}S_{1}}$ converts to the constant describing the interaction
between the Higgs bosons and the $Z$ boson in the SM. Since the symmetric LRM
reproduces the SM under the following values of $g_{R}$ and $g^{\prime}$

$g_{L}=g_{R}=e\mbox{s}_{W}^{-1},\qquad
g^{\prime}=e\sqrt{\mbox{c}_{W}^{2}-\mbox{s}_{W}^{2}},$ $None$

then the quantity $g_{R}g^{\prime-1}$ may moderately differ from unity.
Therefore, the major contributor to the deviation $g_{Z_{1}Z_{1}S_{1}}$ from
their SM values is the factor

$\Delta
g={[\mbox{c}_{\theta_{0}}(1-\tan^{2}\beta)-2\mbox{s}_{\theta_{0}}\tan\beta)]\over(\tan^{2}\beta+1)}.$
$None$

From the expression (37) follows that the angle value $\theta_{0}$ is
basically determined by the parameter $\alpha_{2}$ which enters the Higgs
potential. When $\alpha_{2}\sim 10^{-2}$ then the angle $\theta_{0}$ may reach
the value $\pi/4$. At this condition the $S_{1}$ boson could remain light as
usual but the $S_{2}$ boson ceases to be superheavy

$m_{S_{2}}^{2}={\alpha_{2}v_{R}^{2}k_{+}^{2}\over
k_{1}k_{2}}-{4k_{1}k_{2}k_{-}^{4}[2(2\lambda_{2}+\lambda_{3})k_{1}k_{2}/k_{+}^{2}-\lambda_{4}]^{2}\over\alpha_{2}v_{R}^{2}k_{+}^{2}}.$
$None$

Recall, that the demand

$m_{S_{2}}\geq 10\ \mbox{TeV},$ $None$

is caused by the necessity to supress at the tree level the flavor changing
neutral currents (FCNC) in the Lagrangian

${\cal
L}^{n}_{q}=-{1\over\sqrt{2}k_{+}}\sum_{a,b}\overline{u}_{a}\left\\{\left[m_{u_{a}}\left(\mbox{c}_{\theta_{0}}+{2k_{1}k_{2}\over
k_{-}^{2}}\mbox{s}_{\theta_{0}}\right)S_{1}-m_{u_{a}}(\mbox{s}_{\theta_{0}}-{2k_{1}k_{2}\over
k_{-}^{2}}\mbox{c}_{\theta_{0}})S_{2}-\right.\right.$
$\left.\left.-im_{d_{a}}\gamma_{5}P_{1}\right]\delta_{ab}+{k_{+}^{2}\over
k_{-}^{2}}\left({\cal K}{\cal M}_{d}{\cal
K}^{*}\right)_{ab}\left(S_{1}\mbox{s}_{\theta_{0}}+S_{2}\mbox{c}_{\theta_{0}}\right)\right\\}u_{b}+$
$+(u_{a}\rightarrow d_{a},m_{u_{a}}\leftrightarrow
m_{d_{a}},\gamma_{5}\rightarrow-\gamma_{5}),$ $None$

where ${\cal{K}}$ is the Cabibbo-Kobayashi-Maskawa matrix and ${\cal{M}}_{d}$
is the diagonal mass matrix for the down quarks. The absence of the FCNC in
its turn allows to discribe properly the $\overline{K}^{0}\leftrightarrow
K^{0}$ transitions. However, as it is shown in [39] the successfull LRM
building demands the redefinition of the traditional Yukawa Lagrangian for
quarks. The expression (76) must be changed for

${\cal
L}^{n}_{q}=-{1\over\sqrt{2}k_{+}}\sum_{a}\overline{u}_{a}\left\\{m_{u_{a}}\left[\left(\mbox{c}_{\theta_{0}}-{k_{1}\over
k_{2}}\mbox{s}_{\theta_{0}}\right)S_{1}-\left(\mbox{s}_{\theta_{0}}+{k_{1}\over
k_{2}}\mbox{c}_{\theta_{0}}\right)S_{2}\right]+\right.$
$\left.+{im_{u_{a}}k_{1}\over
k_{2}}\gamma_{5}P_{1}\right\\}u_{a}+(u_{a}\rightarrow
d_{a},\theta_{0}\rightarrow-\theta_{0}).$ $None$

Since the Lagrangian (78) does not induce any FCNC, the inequality (76) breaks
down. Then from Eq.(74) follows that with the increasing of the angle
$\theta_{0}$ the deviation $\Delta g$ from unity could be large enough.

From the form of the Lagrangian (78) it is evident that the decay widths of
the $S_{1}$ boson into quarks and gluons may significantly differ from those
of the SM. Since the coupling constant of the $S_{1}$ boson with the $\tau$
lepton is determined by Eq.(62) then the value
$\Gamma_{S_{1}\rightarrow\tau^{+}\tau^{-}}$ also could not coincide with the
corresponding value in the SM. Thus, it is apparent that the LNH mass lower
bound in the LRM may not agree with that in the SM. However, since up to now
any works containing the analysis of the process (70) from the point of view
of the LRM are absent, we shall also take the value $115$ GeV as the low bound
on the $S_{1}$-boson mass.

Now we are ready to embark on the investigation of the Higgs boson
contributions to the muon AMM. Let us determine some key moments in our
strategy at calculation of the couplings constants of the Higgs bosons. To
evaluate the VEV of the right-handed Higgs triplet $v_{R}$ we invoke the
relation [39]

$v_{R}=\sqrt{{m_{W_{2}}^{2}-m_{W_{1}}^{2}\over
g_{L}^{2}(1+\mbox{tg}^{2}2\xi)}}.$ $None$

The current bounds on the $W_{2}$ gauge boson and the mixing angle $\xi$ are
varied within a broad range in relation to what kind of reactions and what
assumptions have been used at analysis [33]. For example, the lower bound on
$m_{W_{2}}$ being equal to 484 GeV is obtained from the investigation of the
polarized muon decay under assumption $\xi=0$. The analysis of the process
$b\rightarrow s\gamma$ leads to the constraints

$-0.01\leq\xi\leq 0.003.$ $None$

Having specified $m_{W_{2}}=0.8$ TeV and $\xi=10^{-2}$ we evaluate $v_{R}$.
Then setting values $m_{N_{\mu}}$, $m_{N_{\tau}}$, $\theta_{N}$ and $v_{L}$ we
can present the quantities $\varphi_{\mu}$, $\varphi_{\tau}$, $f_{\mu\tau}$,
$m^{\mu}_{D}$ and $M_{D}$ as the functions on $f_{\mu\mu}$.

First we assume that the dominant contribution comes from the
$\Delta^{(--)}_{2}$-boson. A negligible value of the corrections from the
$S_{1}$, $\Delta^{(--)}_{1}$, $\tilde{\delta}^{(-)}$ and $h^{(-)}$-bosons
could be caused both by the large values of their masses and by the small
values of their couplings constants. It is natural to require that
$f_{\mu\mu}$ should be less than 1. Then analysis shows that the interval of
the $\Delta_{2}^{(--)}$-boson mass at which the satisfaction to BNL’00-results
is possible, critically depends on the value of $f_{\mu\tau}$. Let us note,
that the value of $f_{\mu\tau}$, as follows from Eq. (53), very weakly depends
on the angles $\varphi_{\mu}$ and $\varphi_{\tau}$ and is basicly determined
by the difference of the heavy neutrino masses. When one sets $v_{L}$ equal to
1.7 GeV then the $\Delta_{2}^{(--)}$-boson mass would reach the greatest value
$(m_{\Delta_{2}})_{max}$ at $f_{\mu\tau}\approx 0.15$. For this case in the
$m_{\Delta_{2}}$ vs. $f_{\mu\mu}$ parameter space two contour lines marked
$93.8$ and $512.8$ corresponding to $95\%$ CL limits for the contribution of
New Physics to $\delta a_{\mu}/\mu_{0}$ are exhibited in Fig.4. The range of
the Higgs sector parameters allowed by the BNL’00 result lies between contours
$93.8$ and $512.8$.

When $f_{\mu\tau}>0.15$ and $m_{\Delta_{2}}>140$ GeV the value of $f_{\mu\mu}$
becomes more than 1 for the upper bound of $\delta a_{\mu}/\mu_{0}$.
Decreasing of $f_{\mu\tau}$ results in the reduction of
$(m_{\Delta_{2}})_{max}$. At fixed $f_{\mu\tau}$ the reduction of $v_{L}$
practically has no effect on the final result. However, increasing of $v_{L}$
up to 10 GeV gives rise to the growth of $f_{\mu\mu}$. For example, at
$m_{\Delta_{2}}=100$ GeV the value of $f_{\mu\mu}$ lies in the interval
$(0.318,0.743)$.

Inasmuch as the masses of the $\Delta^{(--)}_{1}$ and $\tilde{\delta}^{(-)}$\-
bosons are close to each other then the following possibility should be
considered: the observable value of the muon AMM stems from
$\Delta^{(--)}_{1,2}$ and $\tilde{\delta}^{(-)}$-bosons. Notice, that the
quantities $\delta a_{\mu}^{(W_{1,2}\tilde{\delta})}$ change the sign as one
passes from the region $\tan\beta<1$ to the region $\tan\beta>1$. We shall
restrict our consideration to the specific case, namely, when the quantities
$\beta_{1}\alpha_{\mu\nu_{\mu}\tilde{\delta}}$ and $\beta_{1}\alpha_{\mu
N_{\mu}\tilde{\delta}}$ are positive (recall, that
$I^{(W_{1,2}\tilde{\delta})}>0$).

In numerical calculations we shall use the following parameters values

$\left.\begin{array}[]{ll}m_{N_{\mu}}=110\ \mbox{GeV},\qquad m_{N_{\tau}}=125\
\mbox{GeV},\qquad v_{L}=0.17\ \mbox{GeV},\\\
\tan\beta=0.8,\qquad\alpha-\rho_{3}/2+\rho_{1}=1,\qquad\beta_{1}=1,\qquad\theta_{N}=0.78\end{array}\right\\}$
$None$

and shall assume such hierarchy of the Higgs boson masses

$m_{\Delta_{1}}=1.1\ m_{\Delta_{2}},\qquad m_{\tilde{\delta}}=1.05\
m_{\Delta_{2}}.$

In the $m_{\Delta_{2}}$ vs. $f_{\mu\mu}$ parameter space two contour lines
marked $93.8$ and $512.8$ are shown in Fig.5. At the chosen values of the
heavy neutrino masses the quantity $f_{\mu\tau}$ is approximately equal to
0.01.

With the increase of the heavy neutrino masses, but provided that

$m_{N_{\tau}}-m_{N_{\mu}}=\mbox{const}$

the function $f_{\mu\mu}(m_{\Delta_{2}})$ grows faster. However, this rise
becomes essential only when the heavy neutrino masses are approximately
changed on the order of magnitude. So, for example, choosing $m_{N_{\mu}}=900$
and $m_{N_{\tau}}=915$ GeV, we obtain that at $m_{\Delta_{2}}=200$ GeV the
value of $f_{\mu\mu}$ lies in the interval $(0.092,0.444)$. In this case
$f_{\mu\tau}$ is approximately equal to 0.006. On the other hand, with the
increase of $m_{N_{\tau}}-m_{N_{\mu}}$ the rate of growth of the function
$f_{\mu\mu}(m_{\Delta_{2}})$ goes down. For example, when we set

$m_{N_{\mu}}=900\ \mbox{GeV},\qquad m_{N_{\tau}}=1100\ \mbox{GeV}$

and leave all the remaining parameters without change, the value of
$f_{\mu\mu}$ will lie in the interval $(0.029,0.235)$ at $m_{\Delta_{2}}$
being equal to 100 GeV. The reduction of $\tan\beta$ results in decreasing of
$f_{\mu\mu}$ as a function of $m_{\Delta_{2}}$. For example, in the case
$m_{N_{\mu}}=110$ and $m_{N_{\tau}}=125$ GeV (all the remaining parameter
values are unchanged) when $\tan\beta$ has been set to 0.3 we have

$f_{\mu\mu}\in(0.009,0.049)\qquad\mbox{when}\qquad m_{\Delta_{2}}=100\
\mbox{GeV}$

and

$f_{\mu\mu}\in(0.014,0.079)\qquad\mbox{when}\qquad m_{\Delta_{2}}=200\
\mbox{GeV}.$

The obtained results are practically not changed at increasing of $v_{L}$ up
to its maximum value.

At present we assume that the muon AMM value could be explained by the
$S_{1}$-boson contribution only. To suppress the contributions coming from the
remaining Higgs bosons it is enough to assume that

$\alpha\sim 1,\qquad\rho_{2}\sim 1,\qquad\rho_{3}/2-\rho_{1}\sim 1$ $None$

(this will make $h^{(-)}$-, $\Delta^{(--)}_{1,2}-$ and
$\tilde{\delta}^{(-)}$\- bosons to be superheavy). The contours $93.8$ and
$512.8$ in $m_{S_{1}}$ vs. $\tan\beta$ parameter space are represented in
Fig.6. In numerical calculations the following parameters values have been
used

$f_{\mu\mu}=0.04,\qquad m_{N_{\mu}}=110\ \mbox{GeV},\qquad m_{N_{\tau}}=125\
\mbox{GeV},\qquad v_{L}=1.7\ \mbox{GeV}.$

Increasing (decreasing) of $v_{L}$ results in the reduction (the enhancement)
of the allowed values of $\tan\beta$. For example, when $v_{L}$ is equal to 10
GeV we obtain

$\tan\beta\in(0.804,0.913)\qquad\mbox{at}\qquad m_{S_{1}}=115\ \mbox{GeV}$

and

$\tan\beta\in(0.88,0.947)\qquad\mbox{at}\qquad m_{S_{1}}=200\ \mbox{GeV}.$

Increasing of the heavy neutrino masses and of the values of $f_{\mu\mu}$ does
not cause the appreciable change of the obtained results.

In the case when the muon is stipulated by the contributions from $S_{1}$\-
and $h$-bosons the contours $93.8$ and $512.8$ in $m_{S_{1}}$ vs. $\tan\beta$
parameter space are represented in Fig.7. The choice of the model parameters
is as follows

$m_{N_{\mu}}=110\ \mbox{GeV},\qquad m_{N_{\tau}}=125\ \mbox{GeV},\qquad
v_{L}=1.7\ \mbox{GeV},\qquad f_{\mu\mu}=0.04$ $None$

With the decrease of $v_{L}$ the value $\tan\beta$ comes closer and closer to
1, while the increase of $v_{L}$ results in the removal of the values of
$\tan\beta$ from 1 to 0. The increase of the heavy neutrino masses does not
practically influence the behaviour of the contours shown in Fig.7. At the
increase of $f_{\mu\mu}$ there is the reduction of the allowed values of
$\tan\beta$ with the growth of the $m_{S_{1}}$ values. For example, when
$f_{\mu\mu}=0.1$ (all the remaining parameters are unchanged), then at
$m_{S_{1}}=165$ GeV the values of $\tan\beta$ lie within the interval (0.0015,
0.4542) and at $m_{S_{1}}=200$ GeV those lie within the interval (0.0211,
0.5562).

## 3\. Conclusions

We have considered the Higgs sector of the LRM as a source of the muon value
observed at BNL. The contributions from the interactions of the doubly charged
Higgs bosons ($\Delta^{(--)}_{1,2}$), the singly charged ($h^{(-)}$ and
$\tilde{\delta}^{(-)}$) and the neutral ($S_{1}$) Higgs bosons both with
leptons and gauge bosons were taken into the account. The found value of the
muon represents the function of the Higgs boson masses and the Higgs boson
couplings constants (CC’s). For the majority of the SM extensions the
information about the Higgs boson masses is at the level of knowledge of the
lower borders only. The situation with the CC’s is even more pessimistic. The
experimental data derived up to now do not allow to obtain the constraints on
all the CC’s. We managed to show that the most of part of the CC’s is the
functions of the neutrino oscillation parameters. By this it turned out that
the values of these CC’s are practically not sensitive to the masses and the
mixing angles in the light neutrino sector and are mainly defined by the
values of the heavy neutrino masses and by the mixing angles between the light
and the heavy neutrinos. It should be particularly emphasized that this
property is common for all the models with the ”see-saw” mechanism, i.e. for
the models with the heavy neutrino.

To explain the observed value of the muon AMM by the contributions either from
$S_{1}$ and $h^{(-)}$-bosons or from both of them, it is necessary to assume
that $\tan\beta$ is close to 1. To put this another way, the coincidence with
the BNL’00 result will take place at quasi-degeneracy of the bidoublet VEV’s
($k_{1}\approx k_{2}$), i.e. at the fine tuning of the bidoublet VEV ’s. As in
this case obtained borders on the Higgs boson parameters weakly depends on the
neutrino sector parameters then the recovery of some information concerning
the masses and the mixing angles of the neutrinos will be rather difficult.
However the reverse side of this history is the fine capability for detecting
of $S_{1}$ and $h^{(-)}$-bosons. It is appeared that when $\tan\beta$ is close
to 1 then the values of the CC’s for $h^{(-)}$\- and $S_{1}$-bosons are those
that these bosons can be observed as the resonance peaks in the whole series
of the processes. For example, the $S_{1}$-boson could be observed as the
resonance splashes in the cross sections of the reactions

$\mu^{+}\mu^{-}\rightarrow\mu^{+}\mu^{-},\tau^{+}\tau^{-},$ $None$
$\mu^{+}e^{-}\rightarrow\mu^{-}e^{+},$ $None$

which practically have now background. The reactions (84) and (85) may be
investigated right now, because the energy of the muon beams used in the
current experiments is rather high. The Spin Muon Collaboration at CERN has
been working with muon beams having energy 190 GeV [47] and the FNAL
experiments investigating the muon-proton interaction has been using the muons
with energies of 470 GeV [48]. The reactions (84) and (85) can be also
investigated at the muon colliders (MC’s) which are now under design. For the
detection of the $h^{(-)}$-bosons one could employ the reactions

$e^{-}\nu_{e}\rightarrow W^{-}_{1}Z_{1},$ $None$
$e^{-}\nu_{e}\rightarrow\mu^{-}\nu_{\mu},$ $None$

which have the $s$-channel diagrams with the exchange of the $h^{(-)}$-boson
[39]. The ultra high energy cosmic neutrinos could be used for studying these
two reactions at such neutrino telescopes as BAIKAL NT-200, NESTOR and AMANDA.

However, one important point to remember that the fine tuning of the
parameters always belongs to the extremely rare expedient which is used by
Nature. For this reason the variant with wider range of the parameters
ensuring the agreement between theory and experement is preferable. The
situation with the dominating contribution to the muon AMM from
$\Delta^{(--)}_{1,2}$\- and $\tilde{\delta}^{(-)}$-bosons is just such a case.
However and in this case it is still far to the final definition of the heavy
neutrino parameters. Having established the values of $f_{\mu\mu}$, $v_{L}$
and $v_{R}$ we shall obtain only two equations for the definition of the
quantities $\varphi_{\mu}$, $\theta_{N}$, $m_{N_{\mu}}$ and $m_{N_{\tau}}$,
that is obviously not enough. Certainly it is possible to select the
conventional way too. Out of the five parameters of the heavy neutrino sector
($\varphi_{\mu}$, $\varphi_{\tau}$, $\theta_{N}$, $m_{N_{\mu}}$,
$m_{N_{\tau}}$) we may fix the four parameters and vary the one, say, $m_{N}$.
At such an approach instead of the contours shown in Fig.5 we shall have the
contours constructed in $m_{N_{\mu}}$ vs. $m_{\Delta_{2}}$ parameter space,
that will not introduce anything essentially new to our analysis. The most
important here is something else, namely, when the contribution to the muon is
truly caused by the $\Delta^{(-)}_{1}$-, $\tilde{\delta}^{(-)}$\- and/or
$\Delta^{(-)}_{2}$-bosons then the further way of defining the parameters of
the heavy neutrino without their direct observation is evident. For example,
we could investigate the reactions

$\mu^{-}\mu^{-}\rightarrow\mu^{-}\mu^{-},$ $None$
$\mu^{-}\mu^{-}\rightarrow\mu^{-}\tau^{-},$ $None$
$\mu^{-}\mu-\rightarrow\tau^{-}\tau^{-},$ $None$

which may be observed at the MC’s. All these reactions are going through the
$s$-channels with the exchanges of the $\Delta^{(--)}_{1,2}$-bosons.
Therefore, their cross sections have two resonance peaks related to the Higgs
bosons. Detecting of the reaction (88) will allow to determine $f_{\mu\mu}$,
while the invesigation of reactions (89) and (90) will yield the information
about $f_{\mu\tau}$ and $f_{\tau\tau}$ respectively. Then the use of Eqs. (51)
— (55) will allow to define the regions in which the values of the heavy
neutrino masses and the mixing angles are constrained.

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## Fugures Captions

Fig.1. One-loop diagrams contribute to the muon AMM due to the doubly charged
Higgs bosons $\Delta^{(--)}_{1,2}$. The wavy line represent the
electromagnetic field.
Fig.2. One-loop diagrams contribute to the muon AMM due to the singly charged
Higgs bosons $\tilde{\delta}^{(-)}$ and $h^{(-)}$.
Fig.3. One-loop diagrams contribute to the muon AMM due to the lightest
neutral Higgs boson $S_{1}$.
Fig.4. Contours of the one-loop contribution from the
$\Delta^{(--)}_{2}$-boson to the muon AMM.
Fig.5. Contours of the one-loop contribution from the $\Delta^{(--)}_{1,2}$
and the $\tilde{\delta}^{(-)}$-boson to the muon AMM.
Fig.6. Contours of the one-loop contribution from the $S_{1}$-boson to the
muon AMM.
Fig.7. Contours of the one-loop contribution from the $S_{1}$\- and
$h^{(-)}$-boson to the muon AMM.

$\gamma$$l_{a}$$l_{a}$$\mu^{-}$$\Delta^{(--)}_{1,2}$$\mu^{-}$$\gamma$$\Delta^{(--)}_{1,2}$$\Delta^{(--)}_{1,2}$$\mu^{-}$$l_{a}$$\mu^{-}$Fig.1
$\gamma$$h^{(-)}$,$\tilde{\delta}^{(-)}$$h^{(-)}$,$\tilde{\delta}^{(-)}$$\mu^{-}$$\nu_{a}$$\mu^{-}$$\gamma$$h^{(-)}$,$\tilde{\delta}^{(-)}$$h^{(-)}$,$\tilde{\delta}^{(-)}$$\mu^{-}$$N_{a}$$\mu^{-}$$\gamma$$h^{(-)}$,$\tilde{\delta}^{(-)}$$W^{-}_{1,2}$$\mu^{-}$$\nu_{\mu}$,$N_{\mu}$$\mu^{-}$$\gamma$$W^{-}_{1,2}$$h^{(-)}$,$\tilde{\delta}^{(-)}$$\mu^{-}$$\nu_{\mu}$,$N_{\mu}$$\mu^{-}$Fig.
2 $\gamma$$l_{a}$$l_{a}$$\mu^{-}$$S_{1}$$\mu^{-}$Fig. 3

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 100 105 110 115 120 125 130 135 140
$f_{\mu\mu}$ $m_{\Delta_{2}},\mbox{GeV}$93.8512.8

Fig. 4

0.05 0.1 0.15 0.2 0.25 0.3 100 120 140 160 180 200 $f_{\mu\mu}$
$m_{\Delta_{2}},\mbox{GeV}$93.8512.8

Fig. 5

0.9 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 120 130 140 150 160 170 180
190 200 $\tan\beta$ $m_{S_{1}},\mbox{GeV}$93.8512.8

Fig. 6

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 120 130 140 150 160 170 180 190 200
$\tan\beta$ $m_{S_{1}},\mbox{GeV}$93.8512.8

Fig. 7