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\setcounter{chapter}{5}
\chapter{R Parity Violation }


\renewcommand{\theenumi}{\arabic{enumi}}

\section{Introduction}



Most studies of supersymmetric phenomenology have been made
in the framework of the MSSM  which assumes the conservation of a discrete 
symmetry called R--parity ($R_p$) as has been explained in the previous
Section. Under this symmetry all the standard
model particles are R-even, while their superpartners are R-odd.
$R_p$ is related to the spin (S), total lepton (L), and
baryon (B)  number according to $R_p=(-1)^{(3B+L+2S)}$. 
Therefore the requirement of baryon and lepton number  conservation 
implies the conservation of $R_p$. Under this assumption the
SUSY particles must be pair-produced,
every SUSY particle decays into another SUSY particle and
the lightest of them is absolutely stable. These three
features underlie all the experimental searches for
new supersymmetric states.

However, neither gauge invariance nor SUSY require $R_p$ 
conservation. The most general supersymmetric extension of the
standard model contains explicit $R_p$ violating interactions that
are consistent with both gauge invariance and supersymmetry. 
Detailed analysis of the constraints on these models and 
their possible signals have been made\cite{barger:1989rk}.
In general, there are too many independent couplings and some of these
couplings have to be  set to zero to avoid the proton to decay too fast. 

For these reasons we restrict, in this chapter, 
our attention to the possibility
that $R_p$ can be an exact symmetry of the Lagrangian,
broken spontaneously through the Higgs 
mechanism\cite{aulakh:1982yn,
santamaria:1987uq, santamaria:1988zm, santamaria:1989ic,
  Masiero:1990uj,nogueira:1990wz}. 
This may occur via  nonzero
vacuum expectation values for scalar neutrinos, such as
%
\begin{equation}
v_R = \vev {\tilde{\nu}_{R\tau}}\qquad ;
\qquad v_L = \vev {\tilde{\nu}_{L\tau}}\ .
\end{equation}
%
If spontaneous $R_p$ violation occurs
in absence of any additional gauge symmetry, it leads to the
existence of a physical massless Nambu-Goldstone boson, called
Majoron (J)\cite{aulakh:1982yn,santamaria:1987uq, santamaria:1988zm,
  santamaria:1989ic}. In 
these models there is a 
new decay mode for the $Z^0$ boson, $Z^0 \ra \rho + J $, 
where $\rho$ is a light scalar. This decay mode
would increase the invisible $Z^0$ width by an amount 
equivalent to $1/2$ of a light neutrino family. 
The LEP measurement on the number of such 
neutrinos\cite{Abbaneo:1997yb} is enough to exclude any model where the
Majoron is not mainly an isosinglet\cite{gonzalez-garcia:1989zh,
romao:1990yh}. The simplest
way to avoid this limit is to extend the MSSM,
so that the $R_p$ breaking is driven by isosinglet
VEVs, so that the Majoron is mainly a singlet\cite{Masiero:1990uj}. In this
section we will describe in detail this model for Spontaneously Broken
R--Parity (SBRP) and compare its predictions with the
experimental results.








\section{Model with spontaneously broken R parity
  \label{Model with spontaneous broken R parity}}

The most general superpotential terms involving the Minimal 
Supersymmetric Standard Model (MSSM) superfields
in the presence of the \21 singlet superfields $({\widehat \nu^c}_i,
\widehat{S}_i,{\widehat \Phi})$ 
carrying a conserved lepton number assigned as $(-1, 1,0)$, respectively 
is given as~\cite{romao:1992zx} 
%
\begin{eqnarray} \nonumber
{\cal W} &\hskip-4mm=\hskip-4mm& \varepsilon_{ab}\Big(
h_U^{ij}\widehat Q_i^a\widehat U_j\widehat H_u^b 
+h_D^{ij}\widehat Q_i^b\widehat D_j\widehat H_d^a 
+h_E^{ij}\widehat L_i^b\widehat E_j\widehat H_d^a 
+h_{\nu}^{ij}\widehat L_i^a\widehat \nu^c_j\widehat H_u^b %% \\ \nonumber
\!- {\hat \mu}\widehat H_d^a\widehat H_u^b 
\!- (h_0 \widehat H_d^a\widehat H_u^b +\delta^2)\widehat\Phi \Big)  \\
& &+\hskip 5mm   h^{ij} \widehat\Phi \widehat\nu^c_i\widehat S_j +
M_{R}^{ij}\widehat \nu^c_i\widehat S_j 
+ \frac{1}{2}M_{\Phi} \widehat\Phi^2 +\frac{\lambda}{3!} \widehat\Phi^3
\label{eq:Wsuppot} 
\end{eqnarray}
%
The first three terms together with the $ {\hat \mu}$ term define the
R-parity conserving MSSM, the terms in the last row only involve the
\21 singlet superfields 
$({\widehat \nu^c}_i,\widehat{S}_i,{\widehat \Phi})$~\footnote{The term
  linear in $\Phi$ has been included in the first row as it is
  relevant in electroweak breaking.}, while the remaining terms couple
the singlets to the MSSM fields. We stress the importance of the
Dirac-Yukawa term which connects the right-handed neutrino superfields
to the lepton doublet superfields, thus fixing lepton number.


\subsection{Spontaneous Symmetry Breaking}
\label{sec:spontaneous-symmetry-breaking}

The presence of singlets in the model is essential in order to drive
the spontaneous violation of R parity and electroweak symmetries in a
phenomenologically consistent way.  Like all other Yukawa couplings
$h_U, h_D, h_E$ we assume that $h_{\nu}$ is an arbitrary non-symmetric
complex matrix in generation space. For technical simplicity we take
the simplest case with just one pair of lepton--number--carrying \21
singlet superfields, $\widehat\nu^c$
and $\widehat S$, in order to avoid
inessential complication. This in turn implies, $ h_{ij} 
\to h$ and $ h_{\nu}^{ij} \to h_{\nu}^{i}$. 

The full scalar potential along neutral directions is given by
%
\begin{eqnarray}
V_{total}  &=&
|h \Phi \tilde{S} + h_{\nu}^{i} \tilde{\nu}_i H_u  + M_R \tilde{S}|^2 +
|h_0 \Phi H_u + \hat{\mu} H_u|^2 + |h \Phi \tilde{\nu^c}+ M_R
\tilde{\nu^c}|^2 
\label{scalarpot}
\\\nonumber&&
+|- h_0 \Phi H_d  - \hat{\mu} H_d +
h_{\nu}^{i} \tilde{\nu}_i \tilde{\nu^c} |^2+
|- h_0 H_u H_d + h \tilde{\nu^c} \tilde{S} - \delta^2 +M_{\Phi} \Phi 
+\frac{\lambda}{2} \Phi^2|^2 \\\nonumber&&
+\sum_{i=1}^3|h_{\nu}^{i} \tilde{\nu^c} H_u|^2
+ \Big[  A_h h \Phi \tilde{\nu^c} \tilde{S}
- A_{h_0} h_0 \Phi H_u H_d 
+ A_{h_{\nu }}  h_{\nu}^{i}\tilde{\nu}_i H_u \tilde{\nu^c} 
- B \hat{\mu} H_u H_d  \\\nonumber&&
- C_{\delta} \delta^2 \Phi + B_{M_R} M_R \tilde{\nu^c} \tilde{S} 
+ \frac{1}{2} B_{M_{\Phi}} M_{\Phi} \Phi^2 
+ \frac{1}{3!} A_{\lambda} \lambda \Phi^3 
 + h.c. \Big]\\\nonumber&&
+ \sum_{\alpha} \tilde{m}_{\alpha}^2 |z_{\alpha}|^2
+ \frac{1}{8} (g^2 + {g'}^2) 
\Big( |H_u|^2 - |H_d|^2 - \sum_{i=1}^3 |\tilde{\nu}_i|^2\Big)^2,
\end{eqnarray}
%
where $z_{\alpha}$ denotes any neutral scalar field in the theory.

The pattern of spontaneous symmetry breaking of both electroweak and R
parity symmetries works in a very simple way. The spontaneous breaking
of R parity is driven by nonzero vevs for the scalar neutrinos. The
scale characterizing R parity breaking is set by the isosinglet vevs
\begin{equation}
\vev{\tilde{\nu^c}} = \frac{v_R}{\sqrt{2}},\quad
\langle\tilde{S}\rangle=\frac{v_S}{\sqrt{2}},
\label{eq:rpv}
\end{equation}
and 
\begin{equation}
\vev{\Phi} = \frac{v_{\Phi}}{\sqrt{2}}.
\label{eq:rpphi}
\end{equation}
We also have  very small left-handed sneutrino vacuum expectation
values
\begin{equation}
\vev{\tilde{\nu}_{Li}} = \frac{v_{Li}}{\sqrt{2}}\,.
\label{vl}
\end{equation}
The spontaneous breaking of R--parity also entails the spontaneous
violation of total lepton number. This implies that one of the neutral 
CP--odd scalars, which we call majoron, and which is given by 
the imaginary part of
\begin{equation}
\frac{\sum_i v_{Li}^2}{Vv^2} (v_u H_u - v_d H_d) +
\sum_i \frac{v_{Li}}{V} \tilde{\nu_{i}} 
+\frac{v_S}{V} S
-\frac{v_R}{V} \tilde{\nu^c}
\label{maj}
\end{equation}
remains massless, as it is the Nambu-Goldstone boson associated to the
breaking of lepton number. Note that this majoron is quite different
from the one that emerges in the seesaw majoron model, as it is
characterized by a different lepton number (one unit instead of two)
and by a different scale, determined by the combination $V =
\sqrt{v_R^2 + v_S^2} \sim$~TeV .  Note that Eq.~(\ref{eq:rpv}) is the
origin of lepton number violation in this model and plays a crucial
role in determining the neutrino masses. 

On the other hand, electroweak breaking is driven by the isodoublet vevs 
$\vev{{H_u}} = \frac{v_{u}}{\sqrt{2}}$ and 
$\vev{{H_d}}= \frac{v_{d}}{\sqrt{2}}$, with the combination 
$v^2 = v_u^2 + v_d^2 + \sum_i v_{L i}^2$ fixed by the W mass
\begin{equation}
m_W^2 = \frac{g^2 v^2}{4},
\label{mw}
\end{equation}
while the ratio of isodoublet vevs yields
\begin{equation}
\tan \beta = \frac{v_u}{v_d}.
\label{beta}
\end{equation}
This basically recovers the standard tree level spontaneous breaking
of the electroweak symmetry in the
MSSM~\cite{Barbieri:1982eh}~\footnote{ We have verified explicitly,
  however, that radiative electroweak breaking may also occur.  }.

 \subsection{Fermion Mass Matrices}


 \subsubsection{Chargino Mass Matrix}

 The form of the chargino mass matrix is common to 
 a wide class of  SUSY models with spontaneously broken $R_p$
 and is given by~\cite{nogueira:1990wz,romao:1991up,romao:1992ex}

 \beq
 \begin{array}{c|cccccccc}
 & e^+_j & \tilde{H^+_u} & -i \tilde{W^+}\cr
 \hline
 \vb{20}
 e_i & h_{e ij} v_d & - h_{\nu ij} v_{Rj} &  g v_{Li} \cr
 \vb{20}
 \tilde{H^-_d} & - h_{e ij} v_{Li} & \mu & g v_d \cr
 \vb{20}
 -i \tilde{W^-} & 0 &  g v_u & M_2 
 \end{array}
 \eeq

 Two matrices U and V are needed to diagonalize the $5 \times 5$ 
 (non-symmetric) chargino mass matrix
 %
 \beqa
 {\chi}_i^+ &=& V_{ij} {\psi}_j^+ \qquad ; \qquad
 \psi_j^+ = (e_1^+, e_2^+ , e_3^+ ,\tilde{H^+_u}, -i \tilde{W^+})\cr
 \vb{18}
 {\chi}_i^- &=& U_{ij} {\psi}_j^- \qquad ; \qquad 
 \psi_j^- = (e_1^-, e_2^- , e_3^-, \tilde{H^-_d}, -i\tilde{W^-})
 \label{INO}
 \eeqa
 %
 where the indices $i$ and $j$ run from $1$ to $5$.


 \subsubsection{Neutralino Mass Matrix}


 Under reasonable approximations, we can truncate the neutralino 
 mass matrix so as to obtain an effective $7\times 7$
 matrix~\cite{romao:1991up,romao:1992ex}

 \beq
 \begin{array}{c|cccccccc}
 & {\nu}_i & \tilde{H}_u & \tilde{H}_d & -i \tilde{W}_3 & -i \tilde{B}
 \nonumber \\
 \hline
 {\nu}_i & 0 & h_{\nu ij} v_{Rj} & 0 & \frac{\ds g}{\ds \sqrt{2}}
 v_{Li} & -\frac{\ds g'}{\ds \sqrt{2}} v_{Li}\nonumber \\
 \vb{18}
 \tilde{H}_u & h_{\nu ij} v_{Rj} & 0 & - \mu 
 & -\frac{\ds g}{\ds \sqrt{2}} v_u & \frac{\ds g'}{\ds \sqrt{2}} v_u\nonumber \\
 \vb{18}
 \tilde{H}_d & 0 & - \mu & 0 & \frac{\ds g}{\ds \sqrt{2}} v_d 
 & -\frac{\ds g'}{\ds \sqrt{2}} v_d\nonumber \\
 \vb{18}
 -i \tilde{W}_3 & \frac{\ds g}{\ds \sqrt{2}} v_{Li} 
 & -\frac{\ds g}{\ds \sqrt{2}} v_u & \frac{\ds g}{\ds \sqrt{2}} v_d 
 & M_2 & 0\nonumber \\
 \vb{18}
 -i \tilde{B} & -\frac{\ds g'}{\ds \sqrt{2}} v_{Li} & \frac{\ds g'}{\ds
 \sqrt{2}} v_u & -\frac{\ds g'}{\ds \sqrt{2}} v_d & 0 & M_1 \nonumber
 \end{array}
 \label{nino}
 \eeq

 This matrix is diagonalized by a $7 \times 7$ unitary matrix N,
 %
 \beq
 {\chi}_i^0 = N_{ij} {\psi}_j^0 \qquad \hbox{where} \qquad
 \psi_j^0 = ({\nu}_i,\tilde{H}_u,\tilde{H}_d,-i \tilde{W}_3,-i
 \tilde{B})
 \eeq



 \subsubsection{Charged Current Couplings}

 Using the diagonalization matrices we can write the 
 charged current Lagrangian describing the 
 weak interaction between charged lepton/chargino and
 neutrino/neutralinos  as
 %
 \beq
 \lag^{CC}=\frac{g}{\sqrt2} W_\mu \bar{\chi}_i^- \gamma^\mu 
 (K_{Lik} P_L + K_{Rik} P_R) {\chi}_k^0 + h.c.
 \label{CC}
 \eeq
 %
 where the $5\times 7$ coupling matrices $K_{L,R}$ may
 be written as

 \beqa
 K_{Lik} &=& \eta_i (-\sqrt2 U_{i5} N_{k6} - U_{i4}
 N_{k5} - \sum_{m=1}^{3} U_{im}N_{km})\label{KL}\cr
 \vb{18}
 K_{Rik} &=& \epsilon_k (-\sqrt2 V_{i5} N_{k6} + V_{i4} N_{k4})
 \eeqa


 \subsubsection{Neutral Current Couplings}



 The corresponding neutral current Lagrangian may be written as
 %
 \beq
 \label{NC}
 \lag^{NC}=\frac{g}{\cos\theta_W}\ Z_\mu \left[ \bar{\chi}_i^- \gamma^\mu 
 (O'_{Lik} P_L + O'_{Rik} P_R) \chi_k^- 
 + \frac{1}{2} \
 \bar{\chi}_i^0 \gamma^\mu 
 ( O''_{Lik} P_L + O''_{Rik} P_R) \chi_k^0 
 \right]
 \eeq
 %
 where the $7 \times7$ coupling matrices $O'_{L,R}$ and
 $O''_{L,R}$ are given by
 %
 \beqa
 O'_{Lik}
 &\hskip -3mm=\hskip -3mm&
 \eta_i \eta_k \hskip -2mm
 \left( \frac{1}{2} U_{i4} U_{k4} + U_{i5}
 U_{k5} + \frac{1}{2} \sum_{m=1}^{3} U_{im}U_{km} -\hskip -1mm \delta_{ik}
 \sin^2\theta_W \right)\label{OL}\cr
 \vb{18}
 O'_{Rik} 
 &\hskip -3mm=\hskip -3mm&
 \frac{1}{2} V_{i4} V_{k4} + V_{i5}
 V_{k5} - \delta_{ik} \sin^2\theta_W \label{OR}\cr
 \vb{18}
 O''_{Lik} 
 &\hskip -3mm=\hskip -3mm&
 \frac{1}{2} \epsilon_i \epsilon_k   \left( N_{i4} N_{k4} - N_{i5}
 N_{k5} - \sum_{m=1}^{3} N_{im}N_{km} \right)\hskip -2mm =
 \hskip -2mm - \epsilon_i \epsilon_k
 O''_{Rik} 
 \eeqa
 %
 In writing these  couplings we have assumed CP conservation. 
 Under this assumption the diagonalization matrices can be chosen 
 to be real. The $\eta_i$ and $\epsilon_k$ factors are sign 
 factors, related with the relative CP parities of these 
 fermions, that follow from the diagonalization of their 
 mass matrices. 


\subsection{Neutrino masses}
\label{sec:neutrino-masses}

Since neutrino masses are so much smaller than all other fermion mass
terms in the model, once can find the effective neutrino mass matrix
in a seesaw--type approximation. From the full neutral fermion mass
matrix, see Eq.~(\ref{eq:mass}), one calculates the effective $3\times
3$ neutrino mass matrix $(\mathbf{m_{\nu\nu}^{\rm eff}})$ as
\begin{equation}
\mathbf{m_{\nu\nu}^{\rm eff}} = -\mathbf{M_D^T}\mathbf{M_H^{-1}}\mathbf{M_D},
\label{eq:Seesaw}
\end{equation}
where $\mathbf{M_H}$ is the $7\times 7$ matrix of all other neutral
fermion states, see Eq.~(\ref{eq:mass}), and the $3\times 7$ matrix
$\mathbf{m^T_{\chi^0\nu}}$ is given as
\begin{equation}
\mathbf{M_D^T}=
\left(\begin{array}{llll}
\mathbf{m^T_{\chi^0\nu}} & \mathbf{m_{D}} & 
\mathbf{0} & \mathbf{0} 
\end{array} \right),
\label{eq:massDSim}
\end{equation}
where the matrices $\mathbf{m^T_{\chi^0\nu}}$ and $\mathbf{m_{D}}$ are
given in Eqs.~(\ref{eq:mrpv}) and (\ref{eq:mD}). The inverse of
$\mathbf{M_H}$ is too long to be given explicitly here.

After some algebraic manipulation, the effective neutrino mass 
matrix can be cast into a very simple form 
\begin{equation}
(\mathbf{m_{\nu\nu}^{\rm eff}})_{ij} = a \Lambda_i \Lambda_j + 
     b (\epsilon_i \Lambda_j + \epsilon_j \Lambda_i) +
     c \epsilon_i \epsilon_j.
\label{eq:eff}
\end{equation}
%
where one can define the effective bilinear R--parity violating
parameters $\epsilon_{i}$ and $\Lambda_i$ as
\begin{equation}
  \label{eq:eps}
\epsilon_{i} = h_{\nu}^{i}\, \frac{v_R}{\sqrt{2}}  
\end{equation}
and
\begin{equation}
\Lambda_i = \epsilon_i v_d + \mu v_{L_i}
\label{eq:deflam0}
\end{equation}
%
Here the parameter $\mu$ is
\begin{equation}
\mu = \hat{\mu} + h_0 \frac{v_{\Phi}}{\sqrt{2}}
\label{eq:defmu},
\end{equation}
while the coefficients appearing in Eq.~(\ref{eq:eff}) are given by  
\begin{equation}
a= \frac{1}{4\mu \rm{Det}(\mathbf{M_H})}\Big(m_{\gamma}\widehat{M}_R
(-h^2 v_{R} v_{S}\mu + \widehat{M}_{\Phi}\widehat{M}_R\mu 
+h_0^2\widehat{M}_Rv_dv_u) \Big)
\label{eq:co_a}
\end{equation}
\begin{equation}
b= \frac{1}{8\mu \rm{Det}(\mathbf{M_H})}\Big(h_0 m_{\gamma}\widehat{M}_R
(h_0 \widehat{M}_R + h\mu)v_u(v_u^2-v_d^2)\Big)
\label{eq:co_b}
\end{equation}
\begin{equation}
c= \frac{1}{4\mu \rm{Det}(\mathbf{M_H})}\Big(
(h_0 \widehat{M}_R + h\mu)^2v_u^2 (2 M_1M_2\mu -m_{\gamma}v_dv_u)\Big)
\label{eq:co_c}
\end{equation}
and $\rm{Det}(\mathbf{M_H})$ is given as

\begin{eqnarray}\nonumber
\rm{Det}(\mathbf{M_H})&= & \frac{1}{8}\widehat{M}_R
\Big\{8M_1M_2\mu(\widehat{M}_{\Phi}\widehat{M}_R\mu-
h^2\mu v_Rv_S+h_0^2\widehat{M}_R v_dv_u) \\ 
&-&m_{\gamma}\Big(4\mu v_d(\widehat{M}_{\Phi}\widehat{M}_R-h^2v_Rv_S)
v_u +h_0^2\widehat{M}_R(v_d^2+v_u^2)^2\Big)\Big\}
\label{eq:det}
\end{eqnarray}
Note that $\widehat{M}_R$ and $\widehat{M}_{\Phi}$ above are 
defined as 
\begin{eqnarray}
\widehat{M}_R=M_R + h \frac{v_{\Phi}}{\sqrt{2}},\quad
\widehat{M}_{\Phi}=M_{\Phi} + \lambda \frac{v_{\Phi}}{\sqrt{2}}.
\label{eq:effmr}
\end{eqnarray}
The ``photino'' mass parameter is defined as $m_{\gamma} = g^2M_1
+g'^2 M_2$.

Eq.~(\ref{eq:eff}) resembles very closely the corresponding expression 
for the explicit bilinear R-parity breaking model~\cite{diaz:1998xc,chun:1999bq,abada:2001zh,Diaz:2003as,hirsch:2000ef,romao:1999up},  
once the dominant 1-loop corrections are taken into account. 
Note that the tree-level result of the explicit bilinear model 
can be recovered in the limit 
$\widehat{M}_R,\widehat{M}_{\Phi} \to \infty$. 
In this limit the coefficients $b$ and $c$ go to zero, while 

\begin{equation}
a = \frac{m_{\gamma}}{4{\rm Det}(\mathbf{M_{\chi^0}})}
\label{TreeLevelBil}
\end{equation}
In this limit only one non-zero neutrino mass remains. Whether the
1-loop corrections or the contribution from the singlet fields are
more important in determining the neutrino masses depends essentially
on the relative size of the coefficient $c$ in Eq.~(\ref{eq:eff})
compared to the corresponding 1-loop coefficient. Both extremes can be
realized in our model. We note, however, that as discussed below large
branching ratios of the Higgs into invisible final states require
sizeable values of $h$ and $h_0$ (as well as singlets not being too
heavy). For such choices of parameters we have found that the
``singlino'' contribution to Eq.~(\ref{eq:eff}) is usually much more
important than the 1-loop corrections to the neutrino masses.

Note also that the model does not predict whether the atmospheric 
(solar) mass scale is mainly due to the first (third) term in 
Eq.~(\ref{eq:eff}) or vice versa. We have checked numerically that 
both possibilities can be realized and ``good'' points (in the sense 
of being appropriate for neutrino physics) can be found easily 
in either case.


\section{Higgs spectrum}
\label{sec:higgs-spectrum}

Let us first briefly discuss the spectrum of the scalar and
pseudo-scalar sectors in the model. For detailed definitions we refer
the reader to Ref.~\cite{Hirsch:2004rw}. Since these mass matrices are
too complicated for analytic diagonalization, we will solve the exact
eigensystems numerically. However, before doing that, we discuss
certain limits, where some simplifying approximations are made.  This
allows us to gain some insight into the nature of the spectra.

In the SBRP model there are 8 neutral CP-even states $S^0_i$. In the
neutral CP-odd sector there are six massive states $P^0_i$ ($i=1,\ldots,6$), in addition
to the majoron $J$, with $m_J=0$, and the Goldstone $G^0$.  We
introduce the convention, to be discussed below:
\begin{eqnarray}
\label{eq:defstates}
\left(S^0\right)^T &=&\left(
 S_{h^0},S_{H^0},S_J,S_{J_\perp},S_{\Phi},S_{\tilde\nu_i} \right)
\\ \nonumber
\left(P^0\right)^T &=&\left(
P_{A^0},P_{J_\perp},P_{\Phi},P_{\tilde\nu_i},J, G^0\right).
\end{eqnarray}
Note, that the ordering of these states is not by increasing mass,
as we have defined $P^0_i$ ($i=1,\ldots,6$) as the massive states.

First we note that all entries in the sub-matrices which mix the
left-sneutrinos to the doublet Higgses and the singlet states are
proportional to $h^i_{\nu}$. In the region of parameters where the
model accounts for the observed neutrino masses we must have that
$\epsilon_i = h^i_{\nu}v_R/\sqrt{2}$ is necessarily a small number and
therefore $h^i_{\nu} \ll 1$. Thus, left sneutrinos mix very little
with the other (pseudo-)scalars, unless entries in the sneutrino
sector are, by chance, highly degenerate with the ones in the other
sectors. The real (imaginary) parts of these nearly-sneutrino states
are denoted by $S_{\tilde\nu_i}$ ($P_{\tilde\nu_i}$) in the definition
given above.  Barring fine-tuned situations, we conclude that mixing
between Higgses and left sneutrinos will, in general, be small.

Consider now the pseudoscalar sector,
\begin{eqnarray}
  \label{eq:mp2}
\mathbf{M^{P^2}} = \left[
    \begin{array}{lll}
      \mathbf{M^{P^2}_{HH}} & \mathbf{M^{P^2}_{H\widetilde L}} & \mathbf{M^{P^2}_{HS}}\\[+2mm]
      \mathbf{M^{P^2}_{H\widetilde L}}\!{}^{\rm T} & \mathbf{M^{P^2}_{\widetilde L \widetilde L}} 
      & \mathbf{M^{P^2}_{\widetilde L S}}\\[+2mm]
      \mathbf{M^{P^2}_{HS}}\!{}^{\rm T} &\mathbf{M^{P^2}_{\widetilde L S}}\!{}^{\rm T} & \mathbf{M^{P^2}_{SS}}
    \end{array}
    \right],
\end{eqnarray}
%
where $\mathbf{M^{P^2}_{HH}}$ is a symmetric $2\times 2$ matrix, $\mathbf{M^{P^2}_{\widetilde L
\widetilde L}} $ and $\mathbf{M^{P^2}_{SS}} $ are symmetric $3\times 3 $ matrices,
while   $\mathbf{M^{P^2}_{H\widetilde L}}$ and $\mathbf{M^{P^2}_{HS}}$ are $2\times 3$ matrices
and finally $M^{P^2}_{\widetilde L S}$ is (a non-symmetric) $3\times 3$
matrix. In this notation $\widetilde L$ denotes the sneutrinos and $S$
the singlet fields. 

Neglecting terms proportional to $h^i_{\nu}$, $\mathbf{M^{P^2}_{HH}}$ can be 
written as
\begin{eqnarray}
  \label{eq:mphh}
\mathbf{M^{P^2}_{HH}} = \left[
    \begin{array}{ll}
     \Omega \frac{v_u}{v_d}& \Omega \\
     \Omega & \Omega \frac{v_d}{v_u}
    \end{array}
    \right],
\end{eqnarray}
where~\footnote{We correct a misprint in Ref.~\cite{Hirsch:2004rw}.}
%
\begin{equation}
  \label{eq:omega}
   \Omega= B \hat\mu
   -\delta^2 h_0 + \frac{\lambda}{4} h_0 v_{\Phi}^2+\frac{1}{2} h h_0
   v_R v_S + \frac{\sqrt{2}}{2} A_{h_0} h_0 v_{\Phi} +
  \frac{\sqrt{2}}{2} h_0 M_{\Phi} v_{\Phi}\,.
\end{equation}
%
Note the presence of $h_0$-dependent terms in Eq.~(\ref{eq:omega}). If
there were no mixing between the doublet and singlet Higgses,
Eq.~(\ref{eq:mphh}) would yield the eigenvalues,
\begin{eqnarray}
  \label{eq:mphheigs}
m^2_{1,2} = \Big( 0, \Omega(\frac{v_u}{v_d}+\frac{v_d}{v_u})\Big)\,,
\end{eqnarray}
with the massless state identified as the Goldstone boson, $G^0$, 
and the other state as the pseudo-scalar Higgs $A^0$ of the 
MSSM, with 
\begin{equation}
  \label{eq:massAD}
  m^2_{A^0}=\frac{2 \Omega}{\sin 2\beta}\,.
\end{equation}
The state most closely resembling the MSSM $A^0$, i.e. the state 
remaining in the spectrum when singlets are decoupled is called 
$P_{A^0}$ in Eq.~(\ref{eq:defstates}).
%
The sub-matrix $\mathbf{M^{P^2}_{SS}}$, on the other hand, in the limit 
$h^i_{\nu}=0$, can be written as,
\begin{eqnarray}
  \label{eq:mpss}
\mathbf{M^{P^2}_{SS}} = \left[
    \begin{array}{lll}
     M^{P^2}_{SS_{11}} & M^{P^2}_{SS_{12}} & M^{P^2}_{SS_{13}}\\
     M^{P^2}_{SS_{12}} & - \Gamma \frac{v_R}{v_S}& - \Gamma \\
     M^{P^2}_{SS_{13}} & - \Gamma & - \Gamma \frac{v_S}{v_R}
    \end{array}
    \right],
\end{eqnarray}
where,
\begin{eqnarray}
  \label{eq:mpssdef}
M^{P^2}_{SS_{11}} &=& \delta^2  \left(
    C_{\delta} + M_{\Phi} \right) \frac{\sqrt{2}}{v_{\Phi}}
  -\frac{\sqrt{2}}{2} (v_d^2 + v_u^2) \frac{h_0 \hat\mu}{v_{\Phi}} -
\frac{\sqrt{2}}{4} \lambda \left( 3 A_{\lambda} + M_{\Phi}\right)
    v_{\Phi} \nonumber \\
  & - & 2 B_{M_{\Phi}} M_{\Phi} 
- \frac{\sqrt{2}}{2} h \left(A_h +  M_{\Phi} \right)
    \frac{v_R v_S}{v_{\Phi}} 
+ \frac{\sqrt{2}}{2} h_0 \left(A_{h_0} + M_{\Phi}\right)
\frac{v_u v_d }{v_{\Phi}} \nonumber \\
&+& 2 \delta^2 \lambda + \lambda h_0\, v_u v_d -
\lambda h\, v_R v_S - \frac{\sqrt{2}}{2} h\, M_R
\frac{v_S^2+v_R^2}{v_{\Phi}}\, , 
\\[+2mm]
M^{P^2}_{SS_{12}} &=& -\frac{1}{\sqrt{2}} h(A_h-{\hat M}_{\Phi}) v_R\,, \\
M^{P^2}_{SS_{13}} &=& -\frac{1}{\sqrt{2}} h(A_h-{\hat M}_{\Phi}) v_S\,. 
\end{eqnarray}
Here ${\hat M}_{\Phi}= M_{\Phi} + \lambda v_\Phi/\sqrt2$ and
\begin{equation}
  \label{eq:gamma}
  \Gamma=B_{M_R} M_R -\delta^2 h + \frac{1}{4} h \lambda v_{\Phi}^2 -
  \frac{1}{2} h h_0 v_u v_d + \frac{\sqrt{2}}{2} h \left( A_h + 
  M_{\Phi}\right) v_{\Phi}\, .
\end{equation}
%
Eq.~(\ref{eq:mpss}) has one zero eigenvalue, approximately identified 
with the majoron, $J$, and two non-zero eigenvalues. If 
$M^{P^2}_{SS_{12}},M^{P^2}_{SS_{13}} \ll M^{P^2}_{SS_{11}}+ \Gamma$ 
then the eigenvalues of Eq.~(\ref{eq:mpss}) are approximately given by
\begin{eqnarray}
\label{eq:mpsseigs}\nonumber
m^2_{1,2,3} = \Big( 0, &-& \Gamma(\frac{v_R}{v_S}+\frac{v_S}{v_R})-
\frac{1}{2} \frac{h^2(A_h-{\hat M}_{\Phi})^2 v_R^2v_S^2}
                 {M^{P^2}_{SS_{11}}v_R v_S+\Gamma(v_R^2+v_S^2)} + \cdots, \\
& &M^{P^2}_{SS_{11}}
+\frac{1}{2} \frac{h^2(A_h-{\hat M}_{\Phi})^2v_R v_S(v_R^2+v_S^2)}
                 {M^{P^2}_{SS_{11}}v_R v_S+\Gamma(v_R^2+v_S^2)} + \cdots
\Big)\,,
\end{eqnarray}
where the dots stand for higher order terms. The eigenvalue
proportional to $\Gamma$ is mainly a combination of
$\tilde{S}^I,\tilde{\nu}^{c I}$ fields and we call it $P_{J_\perp}$ in
Eq.~(\ref{eq:defstates}) above, because in the limit where $m_{P_\Phi}
\rightarrow \infty$ and $v_{L_i} \to 0$ this massive state is
orthogonal to the majoron.  As we will discuss below, it is this state
which preferably decays invisibly. The third eigenvalue in Eq.~(\ref{eq:mpsseigs}) 
is an approximation to the state called $P_{\Phi}$
above.  Due to mixing between doublet and singlet states both Eq.~(\ref{eq:mphheigs}) 
and Eq.~(\ref{eq:mpsseigs}), are only very crude estimates.

Consider the scalar sector of the model,
\begin{eqnarray}
  \label{eq:1a}
  \mathbf{M^{S^2}}=\left[
    \begin{array}{lll}
      \mathbf{M^{S^2}_{HH}} & \mathbf{M^{S^2}_{H\widetilde L}} & \mathbf{M^{S^2}_{HS}}\\[+2mm]
      \mathbf{M^{S^2}_{H\widetilde L}}\!{}^{\rm T} & \mathbf{M^{S^2}_{\widetilde L \widetilde L}} 
      & \mathbf{M^{S^2}_{\widetilde L S}}\\[+2mm]
      \mathbf{M^{S^2}_{HS}}\!{}^{\rm T} &\mathbf{M^{S^2}_{\widetilde L S}}\!{}^{\rm T} & \mathbf{M^{S^2}_{SS}}
    \end{array}
    \right],
\end{eqnarray}
%
where the blocks have the same structure as before. $\mathbf{M^{S^2}_{HH}}$ 
contains two eigenvalues which, in the limit of zero mixing, would 
be identified with the MSSM states $h^0$ and $H^0$. \footnote{As 
in the MSSM, there is an upper limit for the mass of the $S_{h^0}$, see 
the discussion below.} These states are the ones called $S_{h^0}$ 
and $S_{H^0}$ in Eq.~(\ref{eq:defstates}) above. 

The sub-matrix $\mathbf{M^{S^2}_{SS}}$ contains, in general, three non-zero 
eigenvalues. One can find an approximate analytic expression for 
them in the limit that the state $S_\Phi$ is much heavier than the 
remaining two eigenstates (called $S_J$ and $S_{J_\perp}$). Again 
in the limit of small mixing, the eigenvalues of the latter are 
approximately given by
\begin{eqnarray}
  \label{eq:mssseigs}
m^2_{1,2} = \Big(2 h^2\frac{v_R^2 v_S^2}{(v_R^2+v_S^2)}+ \cdots,
           -\Gamma(\frac{v_R}{v_S}+\frac{v_S}{v_R})-
2 h^2\frac{v_R^2 v_S^2}{(v_R^2+v_S^2)}+\cdots\Big)
\end{eqnarray}
The first (second) of the eigenvalues in Eq.~(\ref{eq:mssseigs}) is
approximately the state $S_J$ ($S_{J_\perp}$).

Fig.~\ref{fig:masses}, to the left, shows an example of the four
lowest lying eigenvalues in the CP-even sector, as a function of
$\Gamma$ for a random but fixed choice of the remaining parameters.
One of the states, $S_{J_\perp}$, which is mainly singlet, is proportional to
$\Gamma$, as indicated by Eq.~(\ref{eq:mssseigs}). There is another
singlet state, corresponding to $S_J$ of Eq.~(\ref{eq:mssseigs}), and
two mainly doublet states, identified with $S_{h^0}$ and $S_{H^0}$. We
note in passing that $m_{S_{H^0}}$ is proportional to $\Omega$, as in
the MSSM. Mixing between singlet and doublet states will be important
always if the eigenvalues are comparable, as for the example shown in
the figure.  Thus, all the discussion above should be taken as
qualitative only.

The right panel in Fig.~\ref{fig:masses} shows an example of the
two lightest massive CP-odd eigenvalues as a function of $\Gamma$ for a fixed
but random set of other parameters. That one eigenvalue is
proportional to $\Gamma$ is obvious from Eq.~(\ref{eq:mpsseigs}). We note that $\Omega$ and
$\Gamma$ are the main parameters which will determine associated
production and influence the branching ratio into invisible states, as
we will discuss in the following sections.
%
\begin{figure}[htbp]
  \centering
%  \begin{tabular}{cc}
  \includegraphics[clip,width=0.47\linewidth]{mH-Gamma.eps}
  \includegraphics[clip,width=0.47\linewidth]{mA-Gamma.eps}
%  \end{tabular}
  \caption{Typical CP-even (left) and CP-odd (right) Higgs masses as
    function of the parameter $\Gamma$. In this example there are four
    light CP-even states and two light massive CP-odd states (plus two
    massless states, $G^0$ and $J$, not shown). Just as in the MSSM
    there is always one light doublet state, coinciding with $h^0$ in
    the limit of zero mixing. Other states can (but need not) be
    light, depending on the parameters $\Omega$ and $\Gamma$, see
    text.}
  \label{fig:masses}
\end{figure}
%
The model clearly exhibits decoupling, just as the MSSM. In the limit
where $\Omega$ goes to infinity the masses of both states 
$P_{A^0}$ and $S_{H^0}$ go to infinity, just as what happens in the
MSSM when $m_{A^0}$ goes to infinity. The states $S_{J_\perp}$ and
$P_{J_\perp}$ are decoupled in the limit as $\Gamma$ goes to infinity.
If, in addition, we require $h\ll 1$ also $S_{J}$ decouples and the SM
Higgs phenomenology is recovered, as in the MSSM.


\section{Higgs boson production}
\label{sec:higgs-prod}

Supersymmetric Higgs bosons can be produced at an $e^+ e^-$ collider
through their couplings to $Z^0$, via the so--called Bjorken process
($e^+ e^- \rightarrow Z^0 S^0_i$), or via the associated production mechanism 
($e^+ e^- \rightarrow S^0_i P^0_j$).
%
In our SBRP model there are 8 neutral CP-even states $S^0_i$ and 6
massive neutral CP-odd Higgs bosons $P^0_i$, in addition to the majoron $J$ and
the Goldstone $G^0$, see Eq.~(\ref{eq:defstates}).

One must diagonalize the (pseudo-)scalar boson mass matrices in order to find the
couplings of the scalars to the $Z^0$. After doing that we obtain
the Lagrangian terms
%
\begin{equation}
\label{HZZ1}
{\cal L}
\supset\sum_{i=1}^8 (\sqrt 2 G_F)^{1/2} M_Z^2 Z^0_{\mu}Z^{0\mu}\, \eta_{{\rm B}_i} S^0_i
+\sum_{i,j=1}^8(\sqrt{2}G_F)^{1/2}M_Z\, \eta_{{\rm A}_{ij}}
\left(Z^{0\mu}S^0_i\leftrightarrow{\partial_{\mu}} P^0_j\right)
\end{equation}
%
with each $\eta_{{\rm B}_i}$ given as a weighted combination of the five \21
doublet scalars,
%
\begin{equation}
  \label{eq:etaB}
  \eta_{{\rm B}_i}= \frac{v_d}{v} R^{S^0}_{i 1} + \frac{v_u}{v} R^{S^0}_{i 2} + 
\sum_{j=1}^3 \frac{v_{Lj}}{v} R^{S^0}_{i j+2} 
\end{equation}
%
and the $\eta_{{\rm A}_{ij}}$ given by
%
\begin{equation}
  \label{eq:etaA}
\eta_{{\rm A}_{ij}}=R^{S^0}_{i1}R^{P^0}_{j1}-R^{S^0}_{i2}R^{P^0}_{j2}+\sum_{k=1}^3
R^{S^0}_{ik+2}R^{P^0}_{jk+2} 
\end{equation}
%
where the subscripts ${\rm B}$ and ${\rm A}$ refer, respectively, to the Bjorken
process or associated production mechanisms. From these Lagrangian terms we
can easily derive the production cross sections. These are simple
generalizations of the MSSM
results~\cite{Accomando:1997wt,pocsik:1981bg} and for completeness we
give them in Appendix~\ref{sec:prod-cross-sect}.

In the MSSM, there are two sum rule rules, one concerning only the CP
even sector
%
\begin{equation}
  \label{eq:sumrule1}
  \eta_{{\rm B}_{h^0}}^2 +\eta_{{\rm B}_{H^0}}^2 = 1\,,
\end{equation}
%
and another relating the Bjorken and the associated production mechanisms,
%
\begin{equation}
  \label{eq:sumrule2}
  \eta_{{\rm B}_{h^0}}^2 +\eta_{{\rm A}_{h^0A^0}}^2 = 1\,,
\end{equation}
%
with $\eta_{{\rm B}_{h^0}}=\sin(\alpha-\beta)$ and $\eta_{{\rm A}_{h^0A^0}}=
\eta_{{\rm B}_{H^0}}= \cos(\alpha-\beta)$, in an obvious notation.


It is very easy, and instructive, to use our expressions for $\eta_{\rm A}$
and $\eta_{\rm B}$ to recover the MSSM result in the limit that we have only 
the $H_d$ and $H_u$ doublets. In fact, in this case
%
\begin{equation}
  \label{eq:3}
  \frac{v_d}{v}=R^{P^0}_{22}\,,\qquad \frac{v_u}{v}=R^{P^0}_{21}\,,
\end{equation}
%
so we have
%
\begin{equation}
  \label{eq:1}
  \eta_{{\rm B}_{h^0}}=R^{P^0}_{22} R^{S^0}_{11} + R^{P^0}_{21} R^{S^0}_{12}\,, \quad
  \eta_{{\rm B}_{H^0}}=R^{P^0}_{22} R^{S^0}_{21} + R^{P^0}_{21} R^{S^0}_{22}\,, \quad
  \eta_{{\rm A}_{h^0A^0}}=R^{P^0}_{21} R^{S^0}_{11} - R^{P^0}_{22} R^{S^0}_{12}
\end{equation}
%
and we get for the sum rule of Eq.~(\ref{eq:sumrule2})
%
\begin{eqnarray}
  \label{eq:2}
  \eta_{{\rm A}_{h^0A^0}}^2 + \eta_{{\rm B}_{h^0}}^2 &=& 
  \left(R^{P^0}_{22} R^{S^0}_{11} + R^{P^0}_{21} R^{S^0}_{12}\right)^2
  +\left(R^{P^0}_{21} R^{S^0}_{11} - R^{P^0}_{22} R^{S^0}_{12}\right)^2\\
&=& \left( R^{S^0}_{11} R^{S^0}_{11} + R^{S^0}_{12} R^{S^0}_{12} \right)
    \left( R^{P^0}_{21} R^{P^0}_{21} + R^{P^0}_{22} R^{P^0}_{22}\right)\\
&=&1\,,
\end{eqnarray}
%
where we have used the orthogonality of the rotation matrices
%
\begin{equation}
  \label{eq:4}
  \sum_{k=1}^2 R^{S^0}_{ik} R^{S^0}_{jk}=\delta_{ij}\,,\quad
  \sum_{k=1}^2 R^{P^0}_{ik} R^{P^0}_{jk}=\delta_{ij}\,, \quad (i,j=1,2)\,.
\end{equation}
%
For the sum rule of the CP-even sector, Eq.~(\ref{eq:sumrule1}), we
get 
\begin{eqnarray}
  \label{eq:6}
  \eta_{{\rm B}_{h^0}}^2 + \eta_{{\rm B}_{H^0}}^2 &=& 
  \cos^2 \beta \left(R^{S^0}_{11} R^{S^0}_{11} + R^{S^0}_{21}
  R^{S^0}_{21}\right)
 + \sin^2 \beta \left(R^{S^0}_{12} R^{S^0}_{12} + R^{S^0}_{22}
  R^{S^0}_{22}\right)\\
&&+2 \sin\beta \cos\beta \left(
  R^{S^0}_{11} R^{S^0}_{12}+R^{S^0}_{21} R^{S^0}_{22}\right)\\
  &=&1\,,
\end{eqnarray}
%
using the result that in an orthogonal matrix also the vectors
corresponding to the columns are orthonormal, that is
%
\begin{equation}
  \label{eq:7}
    \sum_{k=1}^2 R^{S^0}_{ki} R^{S^0}_{kj}=\delta_{ij}\,,\quad (i,j=1,2)\,.
\end{equation}
%

How this differs in our case? The difference is that, in general, 
%
\begin{equation}
  \label{eq:5}
  \sum_{k=1}^2 R^{S^0,P^0}_{ik} R^{S^0,P^0}_{jk} \not= \delta_{ij} \quad
  \hbox{and}\quad
  \sum_{k=1}^2 R^{S^0,P^0}_{ki} R^{S^0,P^0}_{kj} \not= \delta_{ij}\,, 
\end{equation}
%
due to the fact that we now have more than two (pseudo-)scalars. As it was
stated in the last section and will be discussed in more detail when
we consider the decays, to have a sizeable invisible branching ratio
we need the doublets to be close in mass to the singlet states related
to the majoron and orthogonal combinations. This means that, in the CP-even 
sector, the first four states are $(S_{h^{0}}, S_{H^{0}},
S_{J_{\perp}}, S_{J})$, while in the CP-odd sector we should have 
$(P_{A_{0}},P_{J_{\perp}}, J, G^0)$.  If this situation happens then we can
very easily find a generalization of the sum rule of the CP-even
sector, as
%
\begin{equation}
  \label{eq:8}
  \eta_{{\rm B}_{S_{h^{0}}}}^2 + \eta_{{\rm B}_{S_{H^{0}}}}^2 +
  \eta_{{\rm B}_{S_{J_{\perp}}}}^2+\eta_{{\rm B}_{S_{J}}}^2 =1
\end{equation}
to a good approximation.  This is displayed in Fig.~\ref{fig:sumrule}
where we plot the sum $\eta_{{\rm B}_{S_{H^{0}}}}^2 +
\eta_{{\rm B}_{S_{J_{\perp}}}}^2+\eta_{{\rm B}_{S_{J}}}^2$ against
$\eta_{{\rm B}_{S_{h^{0}}}}^2$.
%
\begin{figure}[ht]
  \centering
  \includegraphics[clip,height=70mm]{cpeven-sumrule.eps} 
  \caption{Sum rule in the CP-even sector, for the case explained in
    the text. The four states, $(S_{h^{0}}, S_{H^{0}}, S_{J_{\perp}},
    S_{J})$. For this example all scalar masses are taken below $200$ GeV.}
  \label{fig:sumrule}
\end{figure}
%
The significance of this sum rule should be clear: if the lightest
Higgs boson has a very small coupling to the $Z^0$ and hence a small
production cross section, there should be another state nearby that
has a large production cross section. 

The other sum rule, relating the CP-even and CP-odd sectors,
Eq.~(\ref{eq:sumrule2}), is more difficult to generalize. In fact the
$P_{A_{0}}$ state will now mix with the $P_{J_{\perp}}$ and the
identification of Eq.~(\ref{eq:3}) will be no longer true. However
qualitatively the sum rule still holds in the sense that if the
parameters are such that the production of the CP-odd states is
reduced one always gets a CP-even state produced.

The above discussion has concentrated on Higgs boson production at an
$e^+e^-$ collider. We now briefly comment on the differences with
regards to Higgs production at the LHC~\cite{Romao:1992dc}. It has
been suggested to search for an invisibly decaying Higgs at the LHC in
$WW$ boson fusion \cite{Eboli:2000ze}, in asociated production with a $Z^0$
boson \cite{Godbole:2003it}, or in the $t{\bar t}$ channel
\cite{Gunion:1993jf}. For the production in $WW$ fusion or in asociated
production with a $Z^0$ boson the above discussion applies
straightforwardly, since the relevant coupling in both cases is
$\eta_{B_i}$ (i.e. $sin(\beta-\alpha)$ in the MSSM limit). For the
$t{\bar t}$ channel in the MSSM production cross section the factor
$\cos\alpha$ has to be replaced by $R^{S^0}_{i2}$ for the SBRP model.


\section{Higgs boson decays}
\label{sec:higgs-decays}

In the following we will discuss the decays of light CP-even and CP-odd 
supersymmetric Higgs bosons. Since the phenomenology of Higgs
bosons within the MSSM is well-known~\cite{gunion:1989we,gunion:1986yn}, we
will concentrate on non-standard final states. Of these, the most
important are the majoron Higgs boson decay modes, which are
characteristic of the SBRP model, without an MSSM counterpart.
%
We will limit ourselves to the discussion of light states, i.e.  Higgs
bosons with masses below the $2W$ threshold. As discussed
below, the decays of heavier CP-odd states will be similar to the
situation encountered in the (N)MSSM.

\subsection{CP-even Higgs Boson Decays}
\label{sec:Hdecays}

In the MSSM light CP-even Higgs bosons decay dominantly to $b{\bar b}$
final states. In our calculation we take into account all fermion
final states, including the leading QCD radiative corrections from
\cite{Djouadi:1995gt}. In the SBRP model new decay modes appear, such
as $S^0_i \to JJ$ and, if kinematically allowed $S^0_i \to P^0_jJ$ and $S^0_i
\to P^0_jP^0_k$.  From the latter usually only $S^0_i \to JJ$ has a large
branching ratio (see appendix \ref{sec:non-mssm-decays}).


We now turn to the lightest Higgs boson decays. Given that other MSSM
decay modes are less important, we are particularly interested here in
the ratio
%
\begin{equation}
  \label{eq:ratio}
  R_{Jb}=\frac{\Gamma(h \to JJ)}{\Gamma(h \to b \bar{b})}
\end{equation}
of the invisible decay to the Standard Model decay into b-jets. For
this we have to look separately at the decay widths,
%
\begin{equation}
  \label{eq:JJ}
  \Gamma(h\to  JJ)=\frac{g_{hJJ}^2}{32\pi m_h}
\end{equation}
%
and
%
\begin{equation}
  \label{eq:bb}
  \Gamma(h\to  b \bar{b})=\frac{3 G_F \sqrt{2}}{8\pi\cos^2\beta}\,
  \left(R^S_{11}\right)^2\, m_h\,  m_b^2 \left[ 1-4
  \left(\frac{m_b}{m_h}\right)^2 \right]^{3/2}
\end{equation}
%
From these expressions we see that $\Gamma(h \to b \bar{b})$ will be
small if the component of the lightest Higgs boson along $H_d^0$ is
small. On the other hand the magnitude of $\Gamma(h \to JJ)$ will
depend on the $g_{hJJ}$ coupling. This is in general given by a
complicated expression, but for the situation that we are considering
here with
%
\begin{equation}
  \label{eq:vi}
  v_{Li} \ll v_d,v_u \ll v_R,v_S
\end{equation}
%
we have to a very good approximation
%
\begin{equation}
  \label{eq:14a}
  J\simeq (0,0,0,0,0,0,\frac{v_S}{V},-\frac{v_R}{V})
\end{equation}
%
where $  V^2=v_S^2+v_R^2$.
%
Under this approximation we can write the coupling $g'_{i}$ for the
vertex $h'_i JJ$ of the Majoron with the \textit{unrotated} Higgs
boson $h'_i$, in the following form
\begin{eqnarray}
  \label{eq:16}
  g'_1&=&h h_0 v_u \frac{v_S v_R}{V^2}\nonumber \\[+1mm]
  g'_2&=&h h_0 v_d \frac{v_S v_R}{V^2} -\frac{2 v_u}{V^2}\,
  \sum_{j=1}^3 \epsilon_j^2
\nonumber \\[+1mm]
  g'_i&=&-\frac{2 \epsilon_{i-2}}{V^2}\sum_{j=1}^3 \epsilon_j v_{Lj}
  \quad (i=3,4,5)
\nonumber \\[+1mm]
  g'_6&=& -\sqrt{2}\, h \left(A_h +   \widehat{M}_{\Phi} \right)\,
\frac{v_S v_R}{V^2} -\sqrt{2}\, h\, \widehat{M}_R \nonumber \\[+1mm]
  g'_7&=&-h^2 \frac{v_S v_R^2}{V^2}\nonumber \\[+1mm]
  g'_8&=&-h^2 \frac{v_S^2 v_R}{V^2}
\end{eqnarray}
%
where $\widehat{M}_R$ and $\widehat{M}_{\Phi}$ have been defined in 
Eq.~(\ref{eq:effmr}).

From these expressions we conclude that $g_{hJJ}$ can be large in two
situations. The first is, of course, if the lightest Higgs boson is
mainly a combination of the $\tilde{\nu^c}$ and $\tilde{S}$ fields.
In this case not only $g_{hJJ}$ will be large, but also
$\Gamma(h\to b{\bar b})$ will be small suppressing $h \to b
\bar{b}$. Unfortunately the production would be suppressed, as
singlets do not couple to the $Z$.
%
The phenomenologically novel and interesting situation is when $h$ and
$h_0$ are large. In this case the Higgs boson behaves as the lightest
MSSM Higgs boson (with moderately reduced production cross section)
but with a large branching to the invisible channel $h \to JJ$.

The sensitivities of LEP experiments to the invisible channel $h \to
JJ$ have been discussed since long
ago~\cite{lopez-fernandez:1993tk,deCampos:1997bg} and the current
status has been presented in Ref.~\cite{Abdallah:2003ry}.  In order to
evaluate the experimental sensitivities to the parameters of the model
we must take into account both the production as well as Higgs decays.

\subsubsection{Numerical results}
\label{sec:numerical-results}

In this section we discuss the numerical results on the invisible
decay of the Higgs boson in our model. We start with a brief
discussion of the SBRP parameters.

Unknown parameters of the spontaneous R-parity breaking model fall
into three different groups. First, there are the MSSM parameters,
mainly the unknown soft SUSY breaking terms. The second group of
parameters are the $\epsilon_i$ and left-handed sneutrino vevs
$v_{L_i}$.  We trade the latter for the parameters $\Lambda_i$ using
Eq.~(\ref{eq:deflam0}).  These six parameters occur also in the
explicit bilinear model.  And, finally, there are the parameters of
the singlet sector, namely singlet vevs $v_R$, $v_S$ and $v_{\Phi}$,
Yukawa couplings $h$, $h_0$ and $\lambda$ and the singlet mass terms
$M_R$, $M_{\Phi}$, $\delta^2$, as well as the corresponding soft
terms.

We have checked by a rather generous scan that the results presented 
below qualitatively do not depend on the choice of MSSM parameters, 
as expected. Thus, for definiteness we will fix the MSSM parameters 
in the following to the SPS1a benchmark point \cite{Allanach:2002nj}, 
defined by 

\begin{equation}
m_0 = 100 \rm{GeV} \hskip5mm m_{1/2} = 250 \rm{GeV} \hskip5mm \tan\beta=10 
\hskip5mm A_0 = -100 \rm{GeV}  \hskip5mm \mu <0
\label{sps1a}
\end{equation}
We have run down this set of parameters to the electro-weak scale using 
the program package SPheno \cite{porod:2003um}. We stress again that different 
choices of MSSM parameters will not lead to qualitatively different 
results.

\subsubsection{General case}
\label{sec:general-case}

We first consider the general model defined by the superpotential in
Eq.~(\ref{eq:Wsuppot}) reduced to one generation of $\nu^c$ and $S$
fields.  For the singlet parameters we choose as a starting point
$v_R= v_S = v_{\Phi} = -150$ GeV and $M_R = - M_{\Phi} = \delta =
10^3$ GeV, as well as $h=0.8$, $h_0 = -0.15$ and $\lambda=0.1$. We
have tried other values of parameters and obtained qualitatively
similar results to the ones discussed below.

The explicit bilinear parameters are then fixed approximately such
that neutrino masses and mixing angles~\cite{Maltoni:2004ei} are in
agreement with experimental
data~\cite{fukuda:1998mi,eguchi:2002dm,ahmad:2002jz}. Slightly
different values of parameters are found, depending on whether the
first or the third term in Eq.~ (\ref{eq:eff}) is responsible for the
atmospheric neutrino mass scale. Both possibilities lead to very
similar results for the invisible decay of the Higgs.  This can be
understood quite easily.  The ratio of the atmospheric and solar
neutrino mass scale is only of the order of ($4-7$) \footnote{In a
  hierarchical model, such as the one discussed here, the square roots
  of the $\Delta m_{ij}^2$ are approximately equal to the larger
  mass.} and the changes in parameters $\vec \Lambda$ and $\vec
\epsilon$ are only of the order of the square root of this number.
Such a small change can always be compensated by a slight adjustment
of other parameters, leading to the same (or very similar) final
result.

After having defined our ``preferred'' choice of parameters in the
following we will vary one unknown parameter at a time. We now turn to
a discussion of the results. In Fig. (\ref{fig:plot54}) we show the
ratio $R_{Jb}$ as a function of $\eta^2$ for different choices of $h$ (left) 
and for different choices of $v_R$ (right) and all other parameters fixed. 
Larger values of $R_{Jb}$ are found for smaller values of $\eta$, as 
expected. However, one sees explicitly that even for values of 
$\eta \simeq 1$, $R_{Jb}$ can be larger than $1$. This means that the 
lightest Higgs can decay mainly invisibly, even when the cross section 
for its production is essentially equal to the usual (MSSM) doublet Higgs 
boson cross section. This is the main result of this work.

\begin{figure}[htbp]
\begin{center}
\vspace{5mm}
\includegraphics[width=75mm,height=5cm]{plot-54-i.eps}
\includegraphics[width=75mm,height=5cm]{plot-59-i.eps}
\end{center}
\vspace{-5mm}
\caption{Ratio $R_{Jb}$, defined in Eq.~(\ref{eq:ratio}), as function 
  of $\eta^2$.  a) to the left, for different values of the
  parameter $h$, from top to bottom: $h=1,\, 0.9,\, 0.7,\, 0.5,\, 0.3,\, 0.1$. 
  b) to the right, for different values of the
  parameter $v_R=v_S$: $-v_R= 150,\, 200,\, 300,\, 400,\, 600,\, 800,\, 1000$ GeV. 
  The plots show explicitly that $R_{Jb}>1$ is possible even for 
  $\eta \simeq 1$. This is the main result of the current paper.}
\label{fig:plot54}
\end{figure}

In Fig. (\ref{fig:plot59}) we show $R_{Jb}$ as function of 
$V = \sqrt{v_R^2+v_S^2}$ (left) and as function of $h$ (to the right).
The figure shows that large values of $R_{Jb}$ are obtained for 
small values of $V$ and for large values of $h$.  The decreasing 
of $R_{Jb}$ with increasing values of $V$ can be easily understood, 
since in the limit $V \to \infty$ the majoron should obviously 
decouple. 

\begin{figure}[htbp]
\vspace{5mm}
\begin{center}
\includegraphics[width=75mm,height=5cm]{plot-54-i-a.eps}
\includegraphics[width=75mm,height=5cm]{plot-59-i-a.eps}
\end{center}
\vspace{-5mm}
\caption{Ratio $R_{Jb}$, defined in Eq.~(\ref{eq:ratio}), as function 
  of $V$ (left) and as function of $h$ (to the right). Small (large) 
  values of $V$ ($h$) lead to large values of $R_{Jb}$.}
\label{fig:plot59}
\end{figure}

Other singlet-sector parameters also can have an important impact on
$R_{Jb}$, as demonstrated in Fig. (\ref{fig:plot60}). As shown in the
left panel of this figure, larger values of $h_0$ lead to larger
values of $R_{Jb}$. For values of $h$ smaller than about $h \simeq
0.75$ (for our specific choice of the other parameters) the order of
the lines is exchanged. This is due to a level-crossing in the
eigenvalues. Below this value, the lightest Higgs is mainly a singlet
and thus even though it decays dominantly invisibly its production
cross section is very much reduced.

On the other hand, the right panel of Figure (\ref{fig:plot60}) 
shows that the value of $v_{\Phi}$ is normally somewhat less important
than the value of $V$ in determining $R_{Jb}$.  Again this can be
qualitatively understood since $V$ is the parameter whose magnitude
determines the breaking of lepton number (indeed, with the help of the
approximate couplings $g'_{i}$ in Eq.~(\ref{eq:16}) one can see that
the parameters $h$, $h_0$, $v_R$ and $v_S$ should be the most
important ones).

 \begin{figure}[htbp]
 \vspace{5mm}
   \begin{center}
   \includegraphics[width=75mm,height=50mm]{plot-60-i-a.eps}
\includegraphics[width=75mm,height=5cm]{plot-62-i-a.eps}
   \end{center}
   \vspace{-5mm}
   \caption{Ratio $R_{Jb}$, defined in Eq.~(\ref{eq:ratio}), as 
     a) left figure: function of 
     $|h|$ for $-h_0=0.3,\, 0.1,\, 0.03,\, 0.01,\, 0.001$ 
     (on the right part of the plot from top to bottom). 
     The right panel b) gives $R_{Jb}$ as function of $|v_{\Phi}|$ for 
     different values of the parameter $v_R=v_S$ for $-v_R=
     150,\,175,\,200,\,300,\,400,\,600,\,800,\,1000$ GeV.}
   \label{fig:plot60}
   \end{figure}

   
   As a summary of this section we conclude that large branching
   ratios of the doublet--like Higgs boson into invisible final states
   are possible in the SBRP model, despite the smallness of the
   neutrino masses indicated by oscillation data. Large values of
   $R_{Jb}$ occur for large values of the Yukawa couplings and
   for small values of $v_R$. The presence of the field $\Phi$ plays a
   crucial role in getting the invisible Higgs boson decays that are
   not suppressed by the small neutrino masses.


\subsubsection{Cubic--only superpotential}
\label{sec:cubic-only-superp}

Before concluding we illustrate the results we have obtained for the
case of a restricted SBRP model described by the superpotential in
Eq.~(\ref{eq:Wsuppot}) containing only cubic
terms~\cite{Romao:1997xf}. The restricted model provides a potential
``solution'' to the $\mu$ problem in the context of spontaneous
R--parity violation. We give results for the same parameter choices as
above, except that no mass parameters are now present in the basic
superpotential.

\begin{figure}[htbp]
\begin{center}
\vspace{5mm}
\includegraphics[clip,width=75mm,height=5cm]{plot-143-i.eps}
\includegraphics[width=75mm,height=5cm]{plot-144-i.eps}
\end{center}
\vspace{-5mm}
\caption{Ratio $R_{Jb}$, defined in Eq.~(\ref{eq:ratio}), as function 
  of $\eta^2$, a) to the left, for different values of the parameter
  $h$ and b) to the right, for different values of the parameter
  $v_R=v_S$. As in the general case (Fig.~\ref{fig:plot54}), large
  values of $R_{Jb}$ can be found even for $\eta\simeq 1$ also in the
  cubic-only case.}
\label{fig:plot143}
\end{figure}

\begin{figure}[htbp]
\begin{center}
\includegraphics[clip,width=75mm,height=5cm]{plot-143-i-a.eps}
\includegraphics[width=75mm,height=5cm]{plot-144-i-a.eps}
\end{center}
\vspace{-5mm}
\caption{Ratio $R_{Jb}$, defined in Eq.~(\ref{eq:ratio}), as function 
  of the parameter $V$ (left) and as function of $h$ (right). The
  qualitative behaviour is similar to the general case, compare to
  figure \ref{fig:plot59}.}
\label{fig:plot144}
\end{figure}

Even though acceptable physical solutions consistent with experiment
(supersymmetric particle searches as well as neutrino oscillation
data) are somewhat harder to find, they exist. Figs.
(\ref{fig:plot143}) and (\ref{fig:plot144}) show $R_{Jb}$ as function
of $\eta^2$ and as function of $h$ and $V$ for the cubic-only case,
compare to Figs.  (\ref{fig:plot54}) and (\ref{fig:plot59}) for the
general case. As can be seen, the qualitative behaviour is very
similar in all cases, although the parameters for which acceptable
solutions are found are usually restricted to narrower ranges in the
cubic-only case. These figures demonstrate that also in the cubic-only
case large production cross section and large invisible branching
ratios for the lightest Higgs decay can occur at the same time.






We now extend that discussion so as
to include also the next-to-lightest CP-even state which plays an 
important role, if the lightest CP-even state is mainly singlet. 




It is well known that, in contrast to the Standard Model, in the MSSM
(and in the NMSSM) the mass of the lightest CP-even supersymmetric
Higgs boson obeys an upper bound that follows from the D-term origin
of the quartic terms 
in the scalar potential, contained in Eq.~(\ref{scalarpot}). This mass
acquires a contribution from the top-stop quark
exchange~\cite{haber:1991aw,ellis:1991zd,Okada:1990vk}, a fact that
modifies the numerical value of this upper
bound~\cite{haber:1991aw,ellis:1991zd,Okada:1990vk}.  
Many other loops contribute, for a recent two-loop level calculation see,
e.g.~Ref.~\cite{heinemeyer:1998np}.
This limit is slightly relaxed in the NMSSM as opposed to the
MSSM~\cite{ellwanger:1999ji}. 

How does this bound emerge in the SBRP model? Since the CP-even sector
contains eight scalars, we cannot diagonalize the corresponding mass
matrices analytically. Therefore we calculate the upper bound on the
Higgs mass numerically, and including the most important radiative
corrections, using formulas from \cite{ellis:1991zd}.  In the SBRP
model it is possible that the lightest CP-even Higgs is mainly a
singlet. However, if this happens, there must exist a light, mainly
doublet Higgs, to which the NMSSM bounds apply. This is shown in Fig.~\ref{fig:eta2-eta1}, 
where we plot (to the left) $\eta_{{\rm B}_2}^2$ as
function of the $\eta_{{\rm B}_1}^2$ and (to the right) the upper limit on
the mass of the second lightest Higgs as function of $\eta_{{\rm B}_2}^2$.
As is seen, if the lightest state is mainly singlet, $\eta_{{\rm B}_1}^2 \simeq 0$, 
therefore $\eta_{{\rm B}_2}^2\simeq 1$, then there is an upper
bound on the second lightest state mass. Vice versa the upper bound applies to the lightest
state if it is mainly doublet.

\begin{figure}[htb]
  \centering
%  \includegraphics[width=55mm,height=45mm]{eta2sq-eta1sq.eps}
%  \includegraphics[width=55mm,height=45mm]{mH2-eta2sq.eps}
  \includegraphics[clip,width=0.47\linewidth]{eta2sq-eta1sq.eps}
  \includegraphics[clip,width=0.47\linewidth]{mH2-eta2sq.eps}
\caption{In the left panel we show the parameter characterizing direct
  production of the second lightest neutral CP-even Higgs boson,
  $\eta_{{\rm B}_2}^2$, as function of the corresponding one for the first 
  lightest neutral CP-even Higgs boson, $\eta_{{\rm B}_1}^2$. To the right:
  Upper limit on the mass of the second lightest CP-even Higgs as a
  function of $\eta_{{\rm B}_2}^2$.}
  \label{fig:eta2-eta1}
\end{figure}

\begin{figure}[htb]
  \centering
%  \includegraphics[width=55mm,height=50mm]{R1-eta1sq-line.eps}\hspace{0.5cm}
%  \includegraphics[width=55mm,height=50mm]{R2-eta2sq-line.eps}
  \includegraphics[clip,width=0.47\linewidth]{R1-eta1sq-line.eps}
  \includegraphics[clip,width=0.47\linewidth]{R2-eta2sq-line.eps}
  \caption{To the left (right): Ratio $R_1$ ($R_2$) as a function of 
the direct production parameter, $\eta_{{\rm B}_1}^2$ ($\eta_{{\rm B}_2}^2$), for 
the first (second) lightest neutral CP-even Higgs boson.}
  \label{fig:R-eta}
\end{figure}

As shown previously \cite{Hirsch:2004rw}, one can have large direct
production cross section for the lightest neutral CP-even Higgs boson
as well as a large branching ratio to the invisible final majoron
states. This is demonstrated in the left panel of Fig.~\ref{fig:R-eta} 
for a random but fixed choice of undisplayed parameters. 
We note that a very similar behaviour is also found for the second
lightest state, as seen from the right panel of Fig.~\ref{fig:R-eta}. 
Thus if the lightest state is mainly singlet there
must be a state nearby which is mainly doublet and decays invisibly.

In summary, we have seen that in the SBRP model there is always at
least one light state, which is mainly doublet, and therefore can be 
produced at future colliders. Irrespectively of whether this state is
the lightest or second-lightest Higgs state, it can decay with very
large branching ratio to an invisible final state.


\subsection{CP-odd Higgs Boson Decays}
\label{sec:Adecays}


Light CP-odd Higgs bosons in the MSSM decay according to $P^0_i \to
f{\bar f}$.  The $WW$ channel becomes dominant as soon as kinematically
allowed~\cite{gunion:1989we,gunion:1986yn}, however we will not include it
as we are mainly interested in the possibility of invisible decays of
the lowest-lying pseudoscalar.
%
The formulas for the CP-even and CP-odd Higgs boson MSSM decay 
branching ratios, apart from the larger number of Higgs bosons, are
totally analogous to those of the MSSM~\cite{Djouadi:1995gt}, except
for the prefactors which are determined by the diagonalizing matrices
of our model. The corresponding matrix elements replace the familiar
$\sin (\beta-\alpha)$ and $\cos (\beta-\alpha)$ factors.

In the SBRP we must take into account in addition the decays $P^0_i \to
JJJ$ and, if kinematically allowed, also $P^0_i \to S^0_j J$, $P^0_i \to S^0_j
P^0_k$, $P^0_i \to P^0_j JJ$, $P^0_i \to P^0_j P^0_kJ$ and $P^0_i \to P^0_j P^0_k P^0_m$.
For the lightest Higgs boson we are interested only in $P^0_i
\to JJJ$ and $P^0_i \to S^0_j J$. 
%
The formulas for the CP-even and CP-odd Higgs boson non-MSSM decay
widths are collected in appendix \ref{sec:non-mssm-decays}.

\subsubsection{$P^0_i \to S^0_j J$}

The decay width of the CP-odd Higgs boson to a CP-even Higgs boson and
a majoron is given in Eq.~(\ref{eq:widthPSJ}). Using the approximation
Eq.~(\ref{eq:majntl}) we can find the coupling $g'_{ij}$ for the
vertex $S'^0_i P'^0_jJ$ of the majoron with the \textit{unrotated}
neutral scalar $S'^0_i$ and pseudoscalar $P'^0_j$ to leading order in the
small parameter $\frac{v_L}{v}$ as
\begin{eqnarray}
  \label{eq:16}
  g'_{11}&=&\frac{g^2+g'^2}{4}\;\frac{v_d^2v_L^2}{Vv^2}\,,\nonumber\\[+1mm]
  g'_{12}&=&\left(\frac{g^2+g'^2}{4}-h_0^2\right)\frac{v_dv_uv_L^2}{Vv^2}\,,
  \nonumber\\[+1mm]
  g'_{21}&=&-\left(\frac{g^2+g'^2}{4}-h_0^2\right)\frac{v_dv_uv_L^2}{Vv^2}\,,
  \nonumber\\[+1mm]
  g'_{22}&=&-\frac{g^2+g'^2}{4}\;\frac{v_u^2v_L^2}{Vv^2}\,,
  \nonumber\\[+1mm]
  g'_{i1}&=&\frac{-\mu\epsilon_{i-2}}{V}\qquad (i=3,\ldots,5)\,,\nonumber \\[+1mm]
  g'_{i2}&=&\frac{-\epsilon_{i-2}}{V}
            \left(A_{h_{\nu}}+\frac{v_S}{v_R}\hat{M}_R\right)\qquad 
            (i=3,\ldots,5)\,,\nonumber \\[+1mm]
  g'_{61}&=&\frac{v_L^2}{v^2V}
            \left(\sqrt{2}h_0\mu v_d
            -\frac{1}{\sqrt{2}}\left(h_0\hat{M}_{\Phi}+A_{h_0}\right)v_u\right)\,,\nonumber  \\[+1mm]
  g'_{62}&=&\frac{v_L^2}{v^2V}
            \left(-\sqrt{2}h_0\mu v_u
            +\frac{1}{\sqrt{2}}\left(h_0\hat{M}_{\Phi}+A_{h_0}\right)v_d\right)\,,\nonumber  \\[+1mm]
  g'_{71}&=&\frac{-hh_0v_uv_R}{2V}\,,\nonumber \\[+1mm]
  g'_{72}&=&\frac{-hh_0v_dv_R}{2V}\,,\nonumber \\[+1mm]
  g'_{81}&=&\frac{hh_0v_uv_S}{2V}\,,\nonumber \\[+1mm]
  g'_{82}&=&\frac{hh_0v_dv_S}{2V}\,.
\end{eqnarray}
Note that the first four of the above are suppressed by the smallness
of sneutrino vevs, needed to reproduce the observed neutrino
oscillation data.  The coupling $g_{S^0_iP^0_jJ}$ then appears through
mixing, and is given as
\begin{equation}
g_{S^0_iP^0_jJ} = g'_{71} R^{S^0}_{i7}R^{P^0}_{j1} + g'_{72} R^{S^0}_{i7}R^{P^0}_{j2} +
              g'_{81} R^{S^0}_{i8}R^{P^0}_{j1} + g'_{82} R^{S^0}_{i8}R^{P^0}_{j2} \,.
\label{def:gspj}
\end{equation}

\subsubsection{$P^0_i \to JJJ$}

The decay width of the CP-odd Higgs boson to three majorons is given
in Eq.~(\ref{eq:widthPJJJ}). Using again the approximate equation
giving the profile of the majoron, Eq.~(\ref{eq:majntl}), the coupling
$g'_{i}$ for the vertex $P'^0_i JJJ$ of the majorons with the
\textit{unrotated} neutral pseudoscalar $P'^0_i$, is given as
\begin{eqnarray}
  \label{eq:16y}\nonumber
  g'_1&=&-\frac{3v_L^2}{v^2V^3}h_0 h v_u v_Rv_S\,, \\[+1mm]\nonumber
  g'_2&=&\frac{3v_L^2}{v^2V^3}h_0 h v_d v_Rv_S\,, \\[+1mm]\nonumber
  g'_3 & \sim & g'_4 \sim g'_5\sim {\cal O}(\frac{v_L^3}{v^3})\,,\\[+1mm]\nonumber
  g'_6  & \sim & {\cal O}(\frac{v_L^3\epsilon}{v^3V})\,, \\[+1mm]\nonumber
  g'_7&=&\frac{-3h^2v_Sv_R^2}{V^3}\,, \\[+1mm]
  g'_8&=&\frac{3h^2v_S^2v_R}{V^3}\,.
\end{eqnarray}
Again, the first six of the above vanish in the limit $v_L \to 0$.
Therefore the coupling $g_{P^0_iJJJ}$ for the vertex of the majorons
with the neutral pseudoscalar $P^0_i$ mass eigenstate is
\begin{equation}
g_{P^0_iJJJ}=g'_7 R^{P^0}_{i7}+g'_8 R^{P^0}_{i8}.
\end{equation}

\subsubsection{Numerical results}

We can see from Eq.~(\ref{eq:16}) that if the CP-odd mass eigenstate
is mainly a Higgs doublet (i.e., its main components are
$P'^0_1=H_d^{0I},\; P'^0_2=H_u^{0I}$ so that its production is not
reduced) then its decays to $S^0_j J$ and $JJJ$ are suppressed as the
corresponding couplings are very small, suppressed by two powers in
$\frac{v_L}{v}$. To find sizeable branching ratios for the decays of
the lightest massive pseudoscalar $P^0_1$, mixing between doublet and singlet
states is therefore required.


\begin{figure}[htbp]
  \centering
  \begin{tabular}{cc}
  \includegraphics[clip,width=0.47\linewidth]{inv-prod-23.eps}
  \end{tabular}
  \caption{Production cross section (red/solid curve) and invisible
    final states decay branching ratio (green/dashed curve) for the
    lightest CP-odd Higgs boson.}
  \label{fig:brinv}
\end{figure}
%
As discussed in section \ref{sec:higgs-spectrum}, in order to have
sizeable mixing between doublet and singlet CP-odd Higgs bosons, we
must require that at least one of the singlet states is light, i.e.
the parameter $\Gamma$ should be very roughly of order $\Gamma \sim
\Omega$.  Fig.~\ref{fig:brinv} shows an example. Here, we plot
$\eta^2_{{\rm A}_{21}}$ and ${\rm BR}(P^0_1 \to \hskip1mm {\rm inv})$ as function of
$\sqrt{\Gamma}$ for one fixed, but arbitrary set of other model
parameters. For small values of $\sqrt{\Gamma}$ the lightest massive CP-odd
state is mainly singlet, therefore ${\rm BR}(P^0_1 \to \hskip1mm {\rm inv})$
is close to 1.  However, the production parameter
$\eta^2_{{\rm A}_{21}}$ is small. Increasing $\sqrt{\Gamma}$ increases the
mass of the lightest CP-odd state.  From a certain point onwards, it
is the doublet state which is lightest, compare to Fig.~\ref{fig:masses}. 
This state can have a sizeable production, but the
branching ratio to invisible final states typically is small. Only in
the intermediate region of sizeable mixing between doublet and singlet
states, i.e.  in the region of $\sqrt{\Gamma} \sim 100-115$ GeV of
Fig.~\ref{fig:brinv} one can have both, sizeable production and
sizeable invisible decay.

In summary, the CP-odd Higgs bosons in the SBRP model usually behave
very similar to the situation discussed in the (N)MSSM. However,
sizeable branching ratios to invisible final states are possible when
there are light CP-odd Higgs bosons from both, the doublet and the
singlet sectors.

\section{Conclusions}
\label{sec:discussion}

We have carefully analyzed the mass spectra, production and decay
properties of the lightest supersymmetric CP-even and CP-odd Higgs
bosons in models with spontaneously broken R-parity.  We have compared
the resulting mass spectra with what is predicted in the Minimal
Supersymmetric Standard Model, stressing the validity of the upper
bound on the lightest CP-even Higgs boson mass.
%
We have seen how the presence of the additional scalar singlet states
affects the Higgs production cross sections, both in the Bjorken and
associated modes.

The main difference with respect to the MSSM case comes from the fact
that the spontaneous breaking of lepton number necessarily implies the
existence of the majoron, and this opens new decay channels for
supersymmetric Higgs bosons into ``invisible'' final states.
%
We have found that the invisible decays of CP-even Higgses can be
dominant, despite the small values of the neutrino masses indicated by
neutrino oscillation data. In contrast, although the decays of the
CP-odd bosons into invisible final states can also be sizeable, this
situation is not generic.
%

Therefore the existence of invisibly decaying Higgs bosons should be
taken seriosly in the planning of future accelerators, like the LHC
and the ILC. These decays may signal the weak-scale violation of
lepton number in a wide class of theories. Within the supersymmetric
context they are a characteristic feature of the SBRP models. These
can account for the observed pattern of neutrino masses and mixings in
a way which allows the neutrino mixing angles to be cross checked at
high energy accelerators like LHC/ILC. For example, in our model there
is a $b{\bar b}$ plus missing momentum signal associated to the invisible decay of
the lightest CP-even Higgs boson produced in association with a
pseudoscalar. Although this is a standard topology,
also present in the Standard Model and the MSSM, its kinematical
properties in our model differ, as the $JJ$ add up to the CP-even Higgs 
boson mass and $b{\bar b}$ to the CP-odd Higgs 
boson mass.  Further studies to elucidate the impact of these
decay modes for future colliders, should be conducted.  While for the
LHC we may encounter difficulties associated to missing energy
measurements and/or b-tagging, these potential limitations do not
affect in the same way the ILC.

Last, but not least, as already explained, we have restricted our
analysis to Higgs bosons below the $WW$ threshold. Extension to relax
this restriction is totally straightforward, though somewhat less
interesting. Due to the validity of the supersymmetric Higgs boson
mass upper limit we must have one light CP-even Higgs boson which, as
we have shown, is likely to have an important decay into invisible
final states.


\newpage

\section{Appendix}
 
%\subsection{Production cross sections}


We collect here various useful formulas.

\subsection{Neutrino-Neutralino-Singlino mass matrix}

In the basis 
\begin{equation}
(-i\lambda',-i\lambda^3,{\tilde H_d},{\tilde H_u},\nu_e,\nu_{\mu},\nu_{\tau},
\nu^c,S,\tilde{\Phi})
\label{eq:defbasis}
\end{equation}
the mass matrix of the neutral fermions following from Eq.~(\ref{eq:Wsuppot}) 
can be written as

\begin{equation}
\mathbf{M_N}=
\left(\begin{array}{lllll}
\mathbf{M_{\chi^0}} & \mathbf{m_{\chi^0\nu}}& \mathbf{m_{\chi^0\nu^c}}& 
\mathbf{0}& \mathbf{m_{\chi^0\Phi} } \\
\\
\mathbf{m^T_{\chi^0\nu}} & \mathbf{0} & \mathbf{m_{D}} & 
\mathbf{0} & \mathbf{0} \\
\\
\mathbf{m^T_{\chi^0\nu^c}}&\mathbf{m^T_{D}} & \mathbf{0} &
\mathbf{M_{\nu^c S}} & \mathbf{M_{\nu^c\Phi}} \\
\\
\mathbf{0} &\mathbf{0} &
\mathbf{M^T_{\nu^c S}} &\mathbf{0} &\mathbf{M_{S\Phi}} \\
\\
\mathbf{m^T_{\chi^0\Phi} } & \mathbf{0} & \mathbf{M^T_{\nu^c\Phi}} & 
\mathbf{M^T_{S\Phi}} & \mathbf{M_{\Phi}}
\end{array} \right).
\label{eq:mass}
\end{equation}
\noindent
where the matrix $\mathbf{M_{\chi^0}}$ is the MSSM neutralino mass
matrix:

\begin{equation}
\mathbf{M_{\chi^0}} =
\left(\begin{array}{llll}
M_1 & 0    & - \frac{1}{2} g' v_d & + \frac{1}{2} g' v_u \\ \vb{12}
0   & M_ 2 & + \frac{1}{2} g v_d &  - \frac{1}{2} g v_u \\ \vb{12}
- \frac{1}{2} g' v_d & + \frac{1}{2} g v_d &    0 & -\mu \\ \vb{12}
+ \frac{1}{2} g' v_u & - \frac{1}{2} g v_u & -\mu & 0
\end{array} \right).
\label{eq:mntrl}
\end{equation}
Here, $\mu ={\hat\mu}+h_0v_{\Phi}/\sqrt{2}$. 
$\mathbf{m_{\chi^0\nu}}$ is the R-parity violating neutrino-neutralino 
mixing part, which also appears in explicit bilinear R-parity breaking models:

\begin{equation}
\mathbf{m^T_{\chi^0\nu}} =
\left(\begin{array}{llll}
-\frac{1}{2}g'v_{L e} &\frac{1}{2}gv_{L e} & 0 & \epsilon_e \\[2mm]
-\frac{1}{2}g'v_{L \mu}& \frac{1}{2}gv_{L \mu}& 0& \epsilon_{\mu}\\[2mm]
-\frac{1}{2}g'v_{L \tau} & \frac{1}{2}gv_{L \tau} & 0& \epsilon_{\tau} 
\end{array} \right),
\label{eq:mrpv}
\end{equation}
where $v_{L i}$ are the vevs of the left-sneutrinos, $\epsilon_i$ 
are defined by $\epsilon_i = \frac{1}{\sqrt{2}}h_{\nu}^{i} v_R$, 
and $v_R$ is the vev of the right-sneutrino.

\noindent
Here $\mathbf{m_{\chi^0\nu^c}}$ is given as

\begin{equation}
\mathbf{m^T_{\chi^0\nu^c}} =
\left(
0, \hskip2mm 0, \hskip2mm  0, \hskip2mm 
\frac{1}{\sqrt{2}}\sum h_{\nu}^{i} v_{L i} 
\right).
\label{eq:mchinuc}
\end{equation}
and $\mathbf{m^T_{\chi^0\Phi} }$ is 

\begin{equation}
\mathbf{m^T_{\chi^0\Phi} }
= ( 0 , 0, - \frac{1}{\sqrt{2}}h_0 v_u , - \frac{1}{\sqrt{2}}h_0 v_d)
\label{eq:mchiphi}
\end{equation}
The ``Dirac'' mass matrix is defined in the usual way:

\begin{equation}
(\mathbf{m_{D}})_{i} = \frac{1}{\sqrt{2}}h_{\nu}^{i}v_u
\label{eq:mD}
\end{equation}
The $\nu^c$ and $S$ states are coupled by

\begin{equation}
(\mathbf{M_{\nu^c S}}) =  M_R + h\frac{v_{\Phi}}{\sqrt{2}}
\label{eq:mnucs}
\end{equation}
$\mathbf{M^T_{\nu^c\Phi}}$ and $\mathbf{M^T_{S\Phi}}$ are

\begin{equation}
\mathbf{M^T_{\nu^c\Phi}} = (\langle v_{S}\rangle)
\label{eq:mnucp}
\end{equation}
\begin{equation}
\mathbf{M^T_{S\Phi}} = (\langle v_{R}\rangle)
\label{eq:sp}
\end{equation}
Here, $\langle v_{R}\rangle = h v_R$ and $\langle v_{S}\rangle = h v_S$. 
Finally $\mathbf{M_{\Phi}}$ is 

\begin{equation}
\mathbf{M_{\Phi}} = M_{\Phi} + 
\lambda\frac{v_{\Phi}}{\sqrt{2}}
\label{eq:mp2}
\end{equation}

We briefly comment on the case of three generations of neutral
fermions in the singlet sector. For three copies of $\nu^c$ and $S$
fields the mass matrix of the neutral fermions can be written in
exactly the same form as given in Eq.~(\ref{eq:mass}) with some rather
straight-forward generalizations of the above definitions. These
changes are: $h$ and $h_{\nu}^i$ become $3 \times 3$ matrices $h^{ij}$
and $h_{\nu}^{ij}$. In Eq.~(\ref{eq:mchinuc}) the matrix becomes a $3
\times 4$ matrix, $M_R$ is a symmetric $3 \times 3$ matrix and Eqs.
(\ref{eq:mnucp}) and (\ref{eq:sp}) have to be replaced by

\begin{equation}
\mathbf{M^T_{\nu^c\Phi}} = (\langle v_{S_1}\rangle,\langle v_{S_2}\rangle ,
\langle v_{S_3}\rangle )
\end{equation}
\begin{equation}
\mathbf{M^T_{s\Phi}} = (\langle v_{R_1}\rangle,\langle v_{R_2}\rangle ,
\langle v_{R_3}\rangle )
\end{equation}
where $\langle v_{R i} \rangle = \sum_j h^{ji} v^R_j$ and 
$\langle v_{S_i} \rangle = \sum_j h^{ij} v^S_j$. 

Notice that even with three generations of $\nu^c$ and $S$ fields one
neutrino mass is zero at the tree-level.

\subsection{The Neutral Scalar Mass Matrix}

The $8\times 8$ scalar mass matrix is a symmetric matrix that in the
basis of the real part of 
$(H_d^0,H_u^0,\tilde\nu_i,\Phi,\tilde{S},\tilde\nu^c)$
can be written in the form,
%
\begin{eqnarray}
  \label{eq:1a}
  M^{S^2}=\left[
    \begin{array}{lll}
      M^{S^2}_{HH} & M^{S^2}_{H\widetilde L} & M^{S^2}_{HS}\\[+2mm]
      M^{S^2}_{H\widetilde L}\!{}^T & M^{S^2}_{\widetilde L \widetilde L} 
      & M^{S^2}_{\widetilde L S}\\[+2mm]
      M^{S^2}_{HS}\!{}^T &M^{S^2}_{\widetilde L S}\!{}^T & M^{S^2}_{SS}
    \end{array}
    \right]
\end{eqnarray}
%
where $M^{S^2}_{HH}$ is a symmetric $2\times 2$ matrix, $M^{S^2}_{\widetilde L
\widetilde L} $ and $M^{S^2}_{SS} $ are symmetric $3\times 3 $ matrices,
while   $M^{S^2}_{H\widetilde L}$ and $M^{S^2}_{HS}$ are $2\times 3$ matrices
and finally $M^{S^2}_{\widetilde L S}$ is (a non-symmetric) $3\times 3$
matrix. In this notation $\widetilde L$ denotes the sneutrinos and $S$
the singlet fields. 

We can write the mass matrix by giving the components of the various
blocks. We get,

{$\bullet M^{S^2}_{HH}$}
%
\begin{eqnarray}
  \label{eq:2}
  M^{S^2}_{HH_{11}}&=&\frac{1}{4}(g^2+g'^2) v_d^2 + \Omega \tan\beta +
   \frac{\sqrt{2}}{2} \mu 
   \frac{v_R}{v_d} \sum_{i=1}^3 h_{\nu}^{i}\, v_{Li}\\[+2mm]
  M^{S^2}_{HH_{12}}&=&-\frac{1}{4}(g^2+g'^2) v_d v_u -\Omega +h_0^2 v_u v_d\\[+2mm]
  M^{S^2}_{HH_{22}}&=&\frac{1}{4}(g^2+g'^2) v_u^2 + \Omega \cot\beta -
   \frac{\sqrt{2}}{2} \frac{v_R}{v_u} \sum_{i=1}^3 
A_{h_{\nu}} h_{\nu}^{i}\, v_{Li}
-\frac{\sqrt{2}}{2}\, \widehat{M}_R \frac{v_S}{v_u} \sum_{i=1}^3 
   h_{\nu}^{i} v_{Li}
\end{eqnarray}
%
where,
%
\begin{eqnarray}
  \label{eq:3}
  \Omega&=& B \hat\mu
   -\delta^2 h_0 + \frac{\lambda}{4} h_0 v_{\Phi}^2+\frac{1}{2} h h_0
   v_R v_S + \frac{\sqrt{2}}{2} A_{h_0} h_0 v_{\Phi} +
  \frac{\sqrt{2}}{2} h_0 M_{\Phi} v_{\Phi}
\end{eqnarray}
and $\mu$, $\widehat{M}_R$ and $\widehat{M}_{\Phi}$ are defined 
in Eqs.~(\ref{eq:defmu}) and (\ref{eq:effmr}).

{$\bullet M^{S^2}_{\widetilde L \widetilde L}$}
%
\begin{eqnarray}
  \label{eq:4}
  M^{S^2}_{\widetilde L \widetilde L_{ij}}&= &
\frac{1}{4}(g^2+g'^2) v_{Li} v_{Lj} + \frac{1}{2} \left(v_R^2+v_u^2\right) 
h_{\nu}^{i}
h_{\nu}^{j} + \delta_{ij} \left(
-\frac{\sqrt{2}}{2} \frac{v_u v_R}{v_{Li}}\, A_{h_{\nu}} h_{\nu}^{i}
\right.\nn\\ 
&\hskip-6mm& \left.
+ \frac{\sqrt{2}}{2} \frac{v_d v_R}{v_{Li}}\, h_{\nu}^{i}\, \mu 
- \frac{1}{2}\, \frac{v_R^2+v_u^2}{v_{Li}}\, h_{\nu}^{i}\, \sum_{k=1}^3
h_{\nu}^{k} v_{Lk} -\frac{\sqrt{2}}{2} \widehat{M}_R \frac{v_S v_u}{v_{Li}}
h_{\nu}^{i} \right)
\end{eqnarray}

{$\bullet M^{S^2}_{\widetilde L S}$}
%
\begin{eqnarray}
  \label{eq:5}
  M^{S^2}_{\widetilde L S_{i1}}&=&
-\frac{1}{2}\, h_0 v_d\, v_R h_{\nu}^{i}
+\frac{1}{2}\, h\, v_u\, v_S h_{\nu}^{i} \\[+2mm]
M^{S^2}_{\widetilde L S_{i2}}&=& \frac{\sqrt{2}}{2} \widehat{M}_R\,
v_u  h_{\nu}^{i} \\[+2mm]
M^{S^2}_{\widetilde L S_{i3}}&=& 
\frac{\sqrt{2}}{2}\, v_u A_{h_{\nu}} h_{\nu}^{i} - 
\frac{\sqrt{2}}{2} h_{\nu}^{i} \mu
v_d +   h_{\nu}^{i} v_R  \sum_{k=1}^3
h_{\nu}^{k} v_{Lk}
\end{eqnarray}


{$\bullet M^{S^2}_{H \widetilde L}$}
%
\begin{eqnarray}
  \label{eq:6}
  M^{S^2}_{H \widetilde L_{1i}}& = &
\frac{1}{4}(g^2+g'^2) v_d v_{Li} - \frac{\sqrt{2}}{2}\, \mu\, v_R\,
h_{\nu}^{i}\\[+2mm]
M^{S^2}_{H \widetilde L_{2i}}& = &
-\frac{1}{4}(g^2+g'^2) v_u v_{Li} + \frac{\sqrt{2}}{2}\, v_R\,
A_{h_{\nu}} h_{\nu}^{i} +\frac{\sqrt{2}}{2} \widehat{M}_R\, v_S h_{\nu}^{i} +
v_u\, h_{\nu}^{i}  \sum_{k=1}^3 h_{\nu}^{k} v_{Lk}
\end{eqnarray}


{$\bullet M^{S^2}_{H S}$}
%
\begin{eqnarray}
  \label{eq:7}
  M^{S^2}_{H S_{11}}&=&\sqrt{2} h_0 \mu v_d  
  -\frac{\sqrt{2}}{2} h_0 \left(A_{h_0} + \widehat{M}_{\Phi}\right) v_u 
    - \frac{1}{2}
  h_0 v_R \sum_{k=1}^3  h_{\nu}^{k} v_{Lk}\\[+2mm]
  M^{S^2}_{H S_{12}}&=&-\frac{1}{2} h h_0\,
  v_R v_u \\[+2mm]
  M^{S^2}_{H S_{13}}&=&-\frac{1}{2} h h_0\,
  v_S v_u - \frac{\sqrt{2}}{2} \mu \sum_{k=1}^3  h_{\nu}^{k} v_{Lk}\\[+2mm]
  M^{S^2}_{H S_{21}}&=&\sqrt{2} h_0 \mu v_u  
  -\frac{\sqrt{2}}{2}h_0 \left(A_{h_0}+  \widehat{M}_{\Phi}\right) v_d
  +\frac{1}{2} h\, v_S \sum_{k=1}^3 h_{\nu}^{k} v_{Lk}
 \\[+2mm]
  M^{S^2}_{H S_{22}}&=&-\frac{1}{2} h h_0\, v_R v_d 
  +\frac{\sqrt{2}}{2} \widehat{M}_R \sum_{k=1}^3 h_{\nu}^{k} v_{Lk}
  \\[+2mm]
  M^{S^2}_{H S_{23}}&=&-\frac{1}{2} h h_0\,
  v_S v_d + v_u v_R \sum_{k=1}^3 h_{\nu}^{k} h_{\nu}^{k} +
\frac{\sqrt{2}}{2} \sum_{k=1}^3  A_{h_{\nu}} h_{\nu}^{k} v_{Lk}
\end{eqnarray}


{$\bullet M^{S^2}_{S S}$}
%
\begin{eqnarray}
  \label{eq:8}
    M^{S^2}_{S S_{11}}&\hskip-3mm=\hskip -3mm& 
\frac{1}{2} \lambda^2 v_{\Phi}^2 + \delta^2  \left(
    C_{\delta} + M_{\Phi} \right) \frac{\sqrt{2}}{v_{\Phi}}
  -\frac{\sqrt{2}}{2} (v_d^2 + v_u^2) \frac{h_0 \hat\mu}{v_{\Phi}} +
\frac{\sqrt{2}}{4} \lambda \left( A_{\lambda} + 3  M_{\Phi}\right)
    v_{\Phi}\nn \\
&&  - \frac{\sqrt{2}}{2} h \left(A_h + M_{\Phi} \right)
    \frac{v_R v_S}{v_{\Phi}}
+ \frac{\sqrt{2}}{2} h_0 \left(A_{h_0} + M_{\Phi}\right)
\frac{v_u v_d }{v_{\Phi}} +
\frac{1}{2} h_0 \frac{v_d v_R}{v_{\Phi}} \sum_{k=1}^3 h_{\nu}^{k}
    v_{Lk}\nn \\
&& -\frac{1}{2} h\, \frac{v_S v_u}{v_{\Phi}}  \sum_{k=1}^3 h_{\nu}^{k} v_{Lk}
    -\frac{\sqrt{2}}{2} h\, M_R \frac{v_S^2 +v_R^2}{v_{\Phi}} 
\\[+2mm]
    M^{S^2}_{S S_{12}}&\hskip-3mm=\hskip -3mm& 
 \frac{\sqrt{2}}{2} h \left( A_h +\widehat{M}_{\Phi}\right) v_R 
 +\sqrt{2} h\, \widehat{M}_R v_S + \frac{1}{2} h\, v_u  \sum_{k=1}^3
 h_{\nu}^{k} v_{Lk} 
\\[+2mm]
    M^{S^2}_{S S_{13}}&\hskip-3mm=\hskip -3mm& 
 \frac{\sqrt{2}}{2} h \left( A_h + \widehat{M}_{\Phi}\right) v_S
-\frac{1}{2} h_0 v_d  \sum_{k=1}^3 h_{\nu}^{k} v_{Lk}
+\sqrt{2}\, h\, \widehat{M}_R v_R
\\[+2mm]
    M^{S^2}_{S S_{22}}&\hskip-3mm=\hskip -3mm& 
-\Gamma \frac{v_R}{v_S} - \frac{\sqrt{2}}{2} \frac{v_u}{v_S} 
\widehat{M}_R \sum_{k=1}^3 h_{\nu}^{k} v_{Lk}
\\[+2mm]
    M^{S^2}_{S S_{23}}&\hskip-3mm=\hskip -3mm& \Gamma + h^2 v_R v_S\\[+2mm]
    M^{S^2}_{S S_{33}}&\hskip-3mm=\hskip -3mm& -\Gamma \frac{v_S}{v_R} +
    \frac{\sqrt{2}}{2} \frac{\mu v_d}{v_R} \sum_{k=1}^3 h_{\nu}^{k}
    v_{Lk}  - \frac{\sqrt{2}}{2} \frac{v_u}{v_R} 
    \sum_{k=1}^3 A_{h_{\nu}} h_{\nu}^{k} v_{Lk}  
\end{eqnarray}
%
where
%
\begin{equation}
  \label{eq:9}
  \Gamma=B_{M_R} M_R -\delta^2 h + \frac{1}{4} h \lambda v_{\Phi}^2 -
  \frac{1}{2} h h_0 v_u v_d + \frac{\sqrt{2}}{2} h \left( A_h + 
  M_{\Phi}\right) v_{\Phi} 
\end{equation}


\subsection{The Neutral Pseudo--Scalar Mass Matrix}

The $8\times 8$ pseudoscalar mass matrix is a symmetric matrix that
can be written in the form,
%
\begin{eqnarray}
  \label{eq:11}
M^{P^2} = \left[
    \begin{array}{lll}
      M^{P^2}_{HH} & M^{P^2}_{H\widetilde L} & M^{P^2}_{HS}\\[+2mm]
      M^{P^2}_{H\widetilde L}\!{}^T & M^{P^2}_{\widetilde L \widetilde L} 
      & M^{P^2}_{\widetilde L S}\\[+2mm]
      M^{P^2}_{HS}\!{}^T &M^{P^2}_{\widetilde L S}\!{}^T & M^{P^2}_{SS}
    \end{array}
    \right]
\end{eqnarray}
%
where the blocks have the same structure as before.
We can write the mass matrix by giving the components of the various
blocks. We get,

{$\bullet M^{P^2}_{HH}$}
%
\begin{eqnarray}
  \label{eq:12}
  M^{P^2}_{HH_{11}}&=&\Omega \tan\beta +
   \frac{\sqrt{2}}{2} \mu 
   \frac{v_R}{v_d} \sum_{i=1}^3 h_{\nu}^{i}\, v_{Li}\\[+2mm]
  M^{P^2}_{HH_{12}}&=&\Omega \\[+2mm]
  M^{P^2}_{HH_{22}}&=&\Omega \cot\beta -
   \frac{\sqrt{2}}{2} \frac{v_R}{v_u} \sum_{i=1}^3 A_{h_{\nu}} 
h_{\nu}^{i}\, v_{Li}
-\frac{\sqrt{2}}{2}\, \widehat{M}_R \frac{v_S}{v_u} \sum_{k=1}^3
   h_{\nu}^{k} v_{Lk} 
\end{eqnarray}
%
where $\Omega$ and $\mu$ are given in Eqs.~(\ref{eq:3}) and (\ref{eq:defmu}).

{$\bullet M^{P^2}_{\widetilde L \widetilde L}$}
%
\begin{eqnarray}
  \label{eq:14}
\hskip -2mm
  M^{P^2}_{\widetilde L \widetilde L_{ij}}&=&
\frac{1}{2} \left(v_R^2+v_u^2\right) h_{\nu}^{i}
h_{\nu}^{j} + \delta_{ij} \left(
-\frac{\sqrt{2}}{2} \frac{v_u v_R}{v_{Li}}\, A_{h_{\nu}} h_{\nu}^{i} + 
\frac{\sqrt{2}}{2} \frac{v_d v_R}{v_{Li}}\, h_{\nu}^{i}\, \mu \right.\nn\\
&&\left.
- \frac{1}{2}\, \frac{v_R^2+v_u^2}{v_{Li}}\, h_{\nu}^{i}\, \sum_{k=1}^3
h_{\nu}^{k} v_{Lk} - \frac{\sqrt{2}}{2}\, \widehat{M}_R
  \frac{v_S v_u}{v_{Li}}h_{\nu}^{i} 
\right)
\end{eqnarray}

$\bullet M^{P^2}_{\widetilde L S}$
%
\begin{eqnarray}
  \label{eq:15}
  M^{P^2}_{\widetilde L S_{i1}}&=&
-\frac{1}{2}\, h_0 v_d\, v_R h_{\nu}^{i} + \frac{1}{2}\, h\, v_u\,
v_S h_{\nu}^{i}\\[+2mm] 
M^{P^2}_{\widetilde L S_{i2}}&=& \frac{\sqrt{2}}{2} \widehat{M}_R\,
v_u\, h_{\nu}^{i}  \\[+2mm]
M^{P^2}_{\widetilde L S_{i3}}&=& 
-\frac{\sqrt{2}}{2}\, v_u A_{h_{\nu}} h_{\nu}^{i} + \frac{\sqrt{2}}{2} 
h_{\nu}^{i} \mu v_d 
\end{eqnarray}


{$\bullet M^{P^2}_{H \widetilde L}$}
%
\begin{equation}
  \label{eq:16b}
  M^{P^2}_{H \widetilde L_{1i}}=- \frac{\sqrt{2}}{2}\, \mu\, v_R\,
h_{\nu}^{i},\hskip 10mm
M^{P^2}_{H \widetilde L_{2i}}=-\frac{\sqrt{2}}{2}\, v_R\,
A_{h_{\nu}} h_{\nu}^{i} - \frac{\sqrt{2}}{2}\, v_S \widehat{M}_R\, h_{\nu}^{i}
\end{equation}


{$\bullet M^{P^2}_{H S}$}
%
\begin{eqnarray}
  \label{eq:17}
  M^{P^2}_{H S_{11}}&=&
\frac{\sqrt{2}}{2} h_0 \left(A_{h_0} -  \widehat{M}_{\Phi}\right) v_u 
    + \frac{1}{2}
  h_0 v_R \sum_{k=1}^3  h_{\nu}^{k} v_{Lk}\\[+2mm]
  M^{P^2}_{H S_{12}}&=&-\frac{1}{2} h h_0\,
  v_R v_u \\[+2mm]
  M^{P^2}_{H S_{13}}&=&-\frac{1}{2} h h_0\,
  v_S v_u - \frac{\sqrt{2}}{2} \mu \sum_{k=1}^3  h_{\nu}^{k} v_{Lk}\\[+2mm]
  M^{P^2}_{H S_{21}}&=&
  \frac{\sqrt{2}}{2}h_0 \left(A_{h_0}-  \widehat{M}_{\Phi}\right) v_d 
  + \frac{1}{2}\, h\, v_S \sum_{k=1}^3  h_{\nu}^{k} v_{Lk}
\\[+2mm]
  M^{P^2}_{H S_{22}}&=&-\frac{1}{2} h h_0\,
  v_R v_d +
  \frac{\sqrt{2}}{2}\widehat{M}_R  \sum_{k=1}^3  h_{\nu}^{k} v_{Lk} \\[+2mm]
  M^{P^2}_{H S_{23}}&=&-\frac{1}{2} h h_0\,
  v_S v_d  -
\frac{\sqrt{2}}{2} \sum_{k=1}^3  A_{h_{\nu}} h_{\nu}^{k} v_{Lk}
\end{eqnarray}


{$\bullet M^{P^2}_{S S}$}
%
\begin{eqnarray}
  \label{eq:18}
    M^{P^2}_{S S_{11}}&\hskip-3mm=\hskip -3mm& 
\delta^2  \left(
    C_{\delta} + M_{\Phi} \right) \frac{\sqrt{2}}{v_{\Phi}}
  -\frac{\sqrt{2}}{2} (v_d^2 + v_u^2) \frac{h_0 \hat\mu}{v_{\Phi}} -
\frac{\sqrt{2}}{4} \lambda \left( 3 A_{\lambda} + M_{\Phi}\right)
    v_{\Phi} -2 B_{M_{\Phi}} M_{\Phi}\nn \\
&&  - \frac{\sqrt{2}}{2} h \left(A_h +  M_{\Phi} \right)
    \frac{v_R v_S}{v_{\Phi}}
+ \frac{\sqrt{2}}{2} h_0 \left(A_{h_0} + M_{\Phi}\right)
\frac{v_u v_d }{v_{\Phi}} +
\frac{1}{2} h_0 \frac{v_d v_R}{v_{\Phi}} \sum_{k=1}^3 h_{\nu}^{k}
    v_{Lk}\nn \\
&&+ 2 \delta^2 \lambda + \lambda h_0\, v_u v_d -
\lambda h\, v_R v_S -\frac{1}{2} h\, \frac{v_u v_S}{v_{\Phi}} 
\sum_{k=1}^3  h_{\nu}^{k} v_{Lk} - \frac{\sqrt{2}}{2} h\, M_R
\frac{v_S^2+v_R^2}{v_{\Phi}}  
\\[+2mm]
    M^{P^2}_{S S_{12}}&\hskip-3mm=\hskip -3mm& 
-\frac{\sqrt{2}}{2} h \left( A_h -  \widehat{M}_{\Phi}\right) v_R 
-\frac{1}{2} h\, v_u \sum_{k=1}^3  h_{\nu}^{k} v_{Lk}
\\[+2mm]
    M^{P^2}_{S S_{13}}&\hskip-3mm=\hskip -3mm& 
- \frac{\sqrt{2}}{2} h \left( A_h -  \widehat{M}_{\Phi}\right) v_S
-\frac{1}{2} h_0 v_d  \sum_{k=1}^3 h_{\nu}^{k}
    v_{Lk}\\[+2mm]
    M^{P^2}_{S S_{22}}&\hskip-3mm=\hskip -3mm& 
    -\Gamma \frac{v_R}{v_S}
    -\frac{\sqrt{2}}{2} \widehat{M}_R \frac{v_u}{v_S} 
    \sum_{k=1}^3  h_{\nu}^{k} v_{Lk}
    \\[+2mm]
    M^{P^2}_{S S_{23}}&\hskip-3mm=\hskip -3mm& -\Gamma \\[+2mm]
    M^{P^2}_{S S_{33}}&\hskip-3mm=\hskip -3mm& -\Gamma \frac{v_S}{v_R} +
    \frac{\sqrt{2}}{2} \frac{\mu v_d}{v_R} \sum_{k=1}^3 h_{\nu}^{k}
    v_{Lk}  - \frac{\sqrt{2}}{2} \frac{v_u}{v_R} 
    \sum_{k=1}^3 A_{h_{\nu}} h_{\nu}^{k} v_{Lk}  
\end{eqnarray}
%
where $\Gamma$ is given in Eq.~(\ref{eq:9}).



\subsection{Production cross sections}
\label{sec:prod-cross-sect}

In this section we give the formulas for the production cross section
of both channels at an $e^+ e^-$ machine.

%\subsubsection{Bjorken process}
\subsubsection{Bjorken process}

The cross section for the Bjorken process is~\cite{Accomando:1997wt}

\begin{equation}
  \label{eq:10}
\sigma(e^+e^- \to Z^0 S^0_i) = \eta_{{\rm B}_{i}}^2\ 
\frac{G_F^2 M_Z^4}{96 \pi s}\,
\left(v_e^2+a_e^2\right)\, \beta \frac{\beta^2
  +12M_Z^2/s}{(1-M_Z^2/s)^2+\left( \Gamma_Z M_Z/s\right)^2}\,,  
\end{equation}
%
where 
%
\begin{equation}
  \label{eq:11}
  v_e=-1+4\sin\theta_W^2, \ a_e=-1, 
\quad \beta=\frac{\lambda(s,M_Z^2,M_{S^0_i}^2)}{s}\ ,
\end{equation}
%
$\lambda$ is the 2-body phase space function,
%
\begin{equation}
  \label{eq:12}
  \lambda(a,b,c)=\sqrt{\left(a+b-c\right)^2 -4 a b}
\end{equation}
%
and the $\eta_{{\rm B}_i}$ are given in Eq.~(\ref{eq:etaB}).

%\subsubsection{Associated production}
\subsubsection{Associated production}

The cross section for the associated production is~\cite{pocsik:1981bg}

\begin{equation}
  \label{eq:13}
\sigma(e^+e^- \to S^0_i P^0_j) = \eta_{{\rm A}_{ij}}^2 \ 
\frac{G_F^2 M_Z^4}{96 \pi s}\,
\left(v_e^2+a_e^2\right)\, 
\frac{\beta^3}{(1-M_Z^2/s)^2+\left( \Gamma_Z M_Z/s\right)^2}  
\end{equation}
%
with
%
\begin{equation}
  \label{eq:14}
  \beta=\frac{\lambda(s,M_{P^0_j}^2,M_{S^0_i}^2)}{s}
\end{equation}
%
and the $\eta_{{\rm A}_{ij}}$ are given in Eq.~(\ref{eq:etaA}).

%\subsection{Non-MSSM decays}
\subsection{Non-MSSM decays}
\label{sec:non-mssm-decays}

The most characteristic decays of this model which do not exist in the
(N)MSSM are those involving a majoron. In the following we collect the
formulas for these decays.
The most important ones are:

\begin{eqnarray}
\label{eq:widthSJJ}  
%\begin{equation}
&&\Gamma(S^0_i\to  JJ)=\frac{g_{S^0_iJJ}^2}{32\pi m_{S^0_i}}\,,\\[+2mm]
%\end{equation}
%
%\begin{equation}
\label{eq:widthPSJ}  
&&\Gamma(P^0_i\to S^0_jJ)=\frac{g_{S^0_jP^0_iJ}^2}{16\pi
  m_{P^0_i}^3}\left(m_{P^0_i}^2-m_{S^0_j}^2\right)\,. %\\[+2mm]
%\end{equation}
\end{eqnarray}

For completeness we consider also:

\begin{eqnarray}
%\begin{equation}
\label{eq:widthSPJ}  
&&\Gamma(S^0_i\to P^0_jJ)=\frac{g_{S^0_iP^0_jJ}^2}{16\pi
  m_{S^0_i}^3}\left(m_{S^0_i}^2-m_{P^0_j}^2\right)\,, \\[+2mm]
%\end{equation}
%
%\begin{equation}
\label{eq:widthPSP}  
&&\Gamma(P^0_i\to S^0_jP^0_k)=\frac{g_{S^0_jP^0_iP^0_k}^2}{16\pi
  m_{P^0_i}^3}\lambda(m_{P^0_i}^2,m_{S^0_j}^2,m_{P^0_k}^2)\,, \\[+2mm]
%\end{equation}
%
%\begin{equation}
\label{eq:widthPJJJ}  
&&\Gamma(P^0_i\to JJJ)=\frac{m_{P^0_i}\, g^2_{P^0_iJJJ}}{3072 \pi^3}\,,\\[+2mm]
%\end{equation}
%
%\begin{equation}
\label{eq:widthPPJJ}  
&&\Gamma(P^0_i\to P^0_jJJ)=\frac{g_{P^0_iP^0_jJJ}^2}{1024\pi^3
  m_{P^0_i}^3}\left(m_{P^0_i}^4-m_{P^0_j}^4\right)\,, \\[+2mm]
%\end{equation}
%
%\begin{multline}
\label{eq:widthPPPJ}  
&&\Gamma(P^0_i\to P^0_jP^0_kJ)=\frac{g_{P^0_iP^0_jP^0_kJ}^2}{512\pi^3
  m_{P^0_i}^3}\lambda(m_{P^0_i}^2,m_{P^0_j}^2,m_{P^0_k}^2)\;\times
\left\{ 
  \begin{array}{l}
    \displaystyle
    \frac{1}{2}\, , \quad j=k\cr
    1\, , \quad j \not= k
  \end{array}
\right\}
\nonumber \\[+2mm] 
&&\hskip 5mm \times \, 
\frac{m_{P^0_i}^2\left(m_{P^0_i}^2-2m_{P^0_j}m_{P^0_k}
  \right)+2m_{P^0_j}m_{P^0_k} \left(m_{P^0_j}+m_{P^0_k}\right)^2
  -\left(m_{P^0_j}+m_{P^0_k}\right)^4} {m_{P^0_i}^2-\left(m_{P^0_j}+m_{P^0_k}\right)^2}\,.
%\\[+2mm]
%\end{multline}
%
\end{eqnarray}
%
The decays $P^0_i \to P^0_jP^0_kP^0_l$ are possible, 
but closed kinematically for the light states of interest,
therefore we do not give here the explicit formulas for the widths.




 
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