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\begin{document}
\vspace*{2cm}
\title{Multiple mechanisms in \betabeta decay}
\author{A. Meroni }
\address{SISSA/INFN, Via Bonomea 265, 34136 Trieste, Italy.
}
\maketitle\abstracts{The \betabeta-decay can be induced by more
than one lepton charge nonconserving mechanism. We analyze some
mechanisms contributing to the $\betabeta$ decay amplitude in the
general case of CP nonconservation: light Majorana neutrino
exchange, heavy left-handed (LH) and right-handed (RH) Majorana
neutrino exchanges, lepton charge non-conserving coupling in SUSY
theories with $R_p$ breaking. We show the analysis for the cases of
two ``non-interfering'' and two ``interfering'' mechanisms. This
method can be generalized to the case of more than two $\betabeta$
decay mechanisms and allows to treat the cases of CP conserving and
CP nonconserving couplings generating the $\betabeta$ decay in a
unique way.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Whether the massive neutrinos are Dirac or Majorana particles is
one of the fundamental open questions in neutrino (and particle)
physics today. The Majorana nature of neutrinos can manifest itself
in the existence of processes in which the total lepton
charge is not conserved. At present
the only feasible experiments that can unveil the Majorana nature of
massive neutrinos are the experiments searching for
neutrinoless double beta decay ($\betabeta$):
$(A,Z) \rightarrow (A,Z+2) + e^- + e^-$. In this processes
the total lepton charge changes by two units, $\Delta L= 2$.
If neutrinos are Majorana particles, at some probability level their
exchange should trigger the $\betabeta$ decay. One can consider the light
Majorana neutrino exchange as the ``standard'' mechanism that
induces the decay. In this case the fundamental lepton number
violating parameter describing this mechanism is the
effective Majorana mass \meff:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\be \meff = \left | \sum_j^{light}(U_{ej}^{PMNS})^2 m_j\right |, \ee
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
$m_j$, $j=1,2,3$, being the three light neutrino masses, $m_j
\lesssim 1 $eV and $U^{PMNS}$ is the Pontecorvo-Maki-Nakagawa-Sakata
neutrino mixing matrix which contains a Dirac and two Majorana
CP-violating phases. The observation of \betabeta
decay and the measurement of $\meff$ would prove not only the
Majorana nature of massive neutrinos, but it could give information
on the type of neutrino mass spectrum, on the absolute neutrino mass
scale, and with additional information from other sources ($^3H$
decay experiments or cosmological and astrophysical data
considerations) one might extract unique information on the Majorana
CP-violation phases. Experimentally the isotopes used in the
searches for $\betabeta$ decay are those for which the single
$\beta$-decay is forbidden: $^{48}Ca$, $^{76}Ge$, $^{82}Se$,
$^{100}Mo$, $^{118}Cd$, $^{130}Te$, $^{136}Xe$, $^{150}Nd$. A large
number of projects, aiming at a sensitivity of $\meff \sim (0.01 -
0.05)$ eV, will test the results claimed in \cite{Klapdor} (with
$T_{1/2}^{0\nu}(^{76}Ge) = 2.23^{+0.44}_{-0.31}\times 10^{25}
\mbox{yr}$, corresponding to $\meff = 0.32 \pm 0.03$~eV) such as
CUORE ($^{130}$Te), GERDA ($^{76}$Ge), EXO ($^{136}$Xe), KamLAND-Zen
($^{136}$Xe).
The \betabeta-decay can be triggered, in principle, not only by the
light Majorana neutrino exchange, but by more than one lepton charge
nonconserving mechanism. These mechanisms are, in general,
CP-nonconserving.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{CP-violating mechanisms}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
If the \betabeta-decay will be observed, the question of which
lepton charge nonconserving mechanisms induce the
decay will inevitably arise.
% It will be of fundamental importance to
% understand which mechanisms are triggering the decay and, if more
% than one, whether they interfere or not.
Each of the various $\betabeta$-decay mechanisms considered
in the literature is characterised by its own lepton number violating
(LNV) parameter, $\eta_{\kappa}^{LNV}$, where the idex $\kappa$
lables the mechanism. The mechanisms we will consider in what follows are,
in general, CP-nonconserving. As a consequence, the corresponding
LNV parameters are complex.
% Indeed, they might induce
% $\betabeta$ decay individually or together and if the lepton current
% structure in the amplitude coincides they will interfere. If, on the
% contrary, these are not-interfering mechanisms, i.e. the leptonic
% current structure is different, then the interference term is
% suppressed by a factor depending on the considered nucleus
% \cite{HalprinPetcov}. One can associate a lepton number violating
% (LNV) parameter, $\eta_{\kappa}^{LNV}$, to the considered mechanism
% $k$ and since these mechanisms are, in general, CP-nonconserving,
% the corresponding parameters can be real or complex.
If several mechanisms are involved in $\betabeta$-decay, the inverse
value of the $\betabeta$-decay half-life for a given isotope $(A,Z)$
can be written as:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
\frac{1}{G^{0\nu}(E_0,Z) T^{0\nu}_{1/2}} &=& |\sum_\kappa \eta_{\kappa}^{LNV} {M'}_{\kappa}^{0\nu}|^2\,,
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
where $G^{0\nu}(E_0,Z)$ and ${M'}_{\kappa}^{0\nu}$ are,
respectively, the known phase-space factor ($E_0$ is the energy
release) and the nuclear matrix element of the decay (we list in
Table \ref{tab:table1} the values for the isotopes we will consider her).
The latter depends on the mechanism generating the decay and on the
nuclear structure of the specific isotopes $(A,Z)$, $(A,Z+1)$ and
$(A,Z+2)$ under study.
Depending on the Lorenz structure of, e.g., the currents
describing the two electrons in the final state of $\betabeta$-decay,
two mechanisms generating the $\betabeta$-decay
can be either ''interfering'' or ''non-interfering''.
In the first case the interference term
in the $\betabeta$-decay rate, originating from the
product of the contributions of each of the two mechanisms
to the $\betabeta$-decay amplitude, is not supptressed,
while in the second case - it is suppressed and often can
be neglected. Such a suppression can occure
if, e.g., the electron currents predicted by
the two mechanisms have different chiral structure
and the level of suppression depends
on the decaying nucleus \cite{HalprinPetcov}.
The \betabeta decay is allowed in a wide
range of models. We will consider in this analysis in addition to
the standard case in which \betabeta decay is triggered by the
exchange of light Majorana neutrino, a finite number of models such
as the Left-Right Symmetry model, in which \betabeta decay is
induced by heavy right handed Majorana neutrinos, and for example
$R_p$-parity nonconserving Supersymmetry (SUSY) theories where
Majorana fermions such as gluinos and neutralinos can induce the
decay. The complete analysis in the general case of CP nonconserving
couplings can be found in \cite{Faessler:2011qw}. Here we briefly
discuss the two main cases: \betabeta decay induced by i) two
``non-interfering'' mechanisms, e.g. LH light and RH heavy ($M_k>$
10 GeV) Majorana neutrino exchange whose LNV parameters are denoted
respectively by $|\eta_{\nu}|$ and $|\eta^{R}_{_N}|$ and ii) two
interfering mechanisms, e.g, light Majorana neutrino,
$|\eta_{\nu}|$, and supersymmetric gluino exchange,
$|\eta_{\lambda'}|$.
One can determine and/or sufficiently constrain the fundamental
parameters $ |\eta_\nu|$, $|\eta^{R}_{_N}|$, etc. associated with
the lepton charge nonconserving couplings exploiting the dependence
of the nuclear matrix elements on the decaying nucleus, and using as
input hypothetical values of the $\betabeta$-decay half-life of
$^{76}$Ge satisfying the existing lower limits and the value claimed
in ref. \cite{HMGe76} \cite{Klapdor} as well as the following
hypothetical ranges for $T^{0\nu}_{1/2}$($^{100}$Mo) and
T$^{0\nu}_{1/2}$($^{130}$Te):
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\be
\begin{split} T^{0\nu}_{1/2}(^{76}Ge)\geq 1.9\times 10^{25} y
,&\quad
T^{0\nu}_{1/2}(^{76}Ge)= 2.23^{+0.44}_{-0.31}\times 10^{25} y\\
5.8\times 10^{23}y\leq T^{0\nu}_{1/2}(^{100}Mo)\leq5.8\times
10^{24}y,&\quad 3.0\times 10^{24}y\leq T^{0\nu}_{1/2}(^{130}Te) \leq
3.0\times 10^{25}y
\end{split}
\label{limit} \ee
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
Let us note that $5.8\times 10^{23}$ y and $3.0\times 10^{24}$ y are
the existing lower bounds on the half-lives of $^{100}Mo$ and
$^{130}Te$ \cite{NEMO3,CUORI}.
%In the analysis which follows we will present numerical
%results first for $g_A=1.25$ and using the NMEs calculated with the
%large size single particle basis (``large basis'') and the Charge
%Dependent Bonn (CD-Bonn) potential. Later results for $g_A=1.0$, as
%well as for NMEs calculated with the Argonne potential, will also be
%reported.
As we will see, in certain cases of
at least one more mechanism being operative in $\betabeta$-decay
beyond the light neutrino exchange, one has to take into account the
upper limit on the absolute scale of neutrino masses set by the
$^3H$ $\beta$-decay experiments \cite{MoscowH3,MainzKATRIN}:
$m(\bar{\nu}_e) < 2.3$ eV. In the case of $\betabeta$-decay, this
limit implies a similar limit on the effective Majorana mass
\footnote{We remind the reader that for $m_{1,2,3}\gtrsim 0.1$ eV
the neutrino mass spectrum is quasi-degenerate (QD), $m_1\cong
m_2\cong m_3 \equiv m$, $m^2_j >> \Delta m^2_{21},|\Delta
m^2_{31}|$. In this case we have $m(\bar{\nu}_e) \cong m$ and $\meff
\lesssim m$.} $\meff < 2.3$ eV.
\begin{table}
\centering
\caption{Phase space factors and values of NMEs.}
\label{tab:table1}
\begin{tabular}{lcccc}
\hline
Transition & $G^{0\nu}_i(E, Z) [y^{-1}]$ & $|{M'}^{0\nu}_\nu|$ &
$|{M'}^{0\nu}_N|$ & $|{M'}^{0\nu}_{\lambda'}|$\\
\hline
$^{76}$Ge $\rightarrow$ $^{76}$Se & 7.98$\times10^{-15}$ & 5.82 & 412 & 596\\
$^{100}$Mo $\rightarrow$ $^{100}$Ru &5.73$\times10^{-14}$ & 5.15 & 404 & 589\\
$^{130}$Te $\rightarrow$ $^{130}$Xe &5.54$\times10^{-14}$ & 4.70 & 385 & 540\\
\hline
\end{tabular}
\end{table}
\subsection{Example of two ``non-interfering'' mechanism}
In the case of two ``non-interfering'' mechanisms, the light Majorana neutrino (denoted by $\eta_\nu$)
and the right-handed heavy Majorana
neutrino exchange (denoted by $\eta^{R}_{_N}$), the inverse of the half-life of an isotope $i$ undergoing
\betabeta decay is given by:
\begin{gather}
(T^{0\nu}_{1/2})^{-1}_i = G^{0\nu}_i (|\eta_\nu|^2
|{M'}^{0\nu}_{i,\nu }|^2 + |\eta^{R}_{_N}|^2|{M'}^{0\nu}_{i,N }|^2
),\quad \mbox{with}
\nonumber\\
\eta_{\nu} =\frac{\meff}{m_e}, \quad \eta^{R}_{_N} ~= ~ \left
(\frac{M_W}{M_{W_R}}\right)^{4}\,\sum^{heavy}_k~ V_{e k}^2
\frac{m_p}{M_k}.
\end{gather}
where $G^{0\nu}_i$ and ${M'}^{0\nu}_{i,\kappa}$, $\kappa=\nu, N$
are respectively the phase space factor and the nuclear matrix
element (NMEs), $m_e$ and $m_p$ are the electron and the proton
masses, $V_{ek}$ is the element of the $\nu$- mixing matrix through
which the heavy neutrino $N_k$ couples to the electron in the
hypothetical $V+A$ charged lepton current, and $M_W\cong 80$ GeV
($M_{W_R}>2.5$ TeV) is the LH (RH) weak charged boson mass.
In this ``non-interfering'' case one can see that, in order to
determine the LNV parameters (the unknowns) we can set a system of
two linear equations using as input hypothetical half-lives of two
isotopes ($T_1$ and $T_2$), and reference values for the NMEs
${M'}^{0\nu}_{i,k}$, and the kinematical factor (see Table
\ref{tab:table1}). One finds that the LNV parameters, solutions of
the system of equations, are given by:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\be
\begin{split}
|\eta_\nu|^2 =\frac{|{M'}^{0\nu}_{2,N }|^2/T_1 G_1-
|{M'}^{0\nu}_{1,N }|^2/T_2 G_2} {|{M'}^{0\nu}_{1,\nu
}|^2|{M'}^{0\nu}_{2,N }|^2-|{M'}^{0\nu}_{1,N }|^2
|{M'}^{0\nu}_{2,\nu }|^2},\quad |\eta^{R}_{_N}|^2=\frac{
|{M'}^{0\nu}_{1,\nu }|^2/T_2 G_2 - |{M'}^{0\nu}_{2,\nu }|^2/T_1 G_1}
{|{M'}^{0\nu}_{1,\nu }|^2|{M'}^{0\nu}_{2,N }|^2- |{M'}^{0\nu}_{1,N
}|^2|{M'}^{0\nu}_{2,\nu }|^2}.
\end{split}
\ee
%%%%%%%%%%%%%%%%%%%%%%%%%%
Negative solutions are not physical, so requiring $|\eta_\nu|^2>0$
$|\eta^{R}_{_N}|^2>0$ and fixing one of the two half-lives, e.g.
$T_1$, we can find a range for $T_2$ of physical solutions
\footnote{This results are valid for $A_1 |{M'}^{0\nu}_{1,N }|^2
/|{M'}^{0\nu}_{2,N }|^2$ (see table \ref{tab:table1}). Using as two
isotopes $^{76}Ge$ and $^{100}Mo$ and fixing $T_1\equiv
T^{0\nu}_{1/2}(^{76}Ge)= 2.23\times 10^{25}$ y~\cite{Klapdor}, one
obtains the results shown in the left panel in Fig. \ref{fig1}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\subsection{Example of two ``interfering'' mechanism}
In this second case, considering as interfering mechanisms the light
Majorana neutrino and the supersymmetric gluino exchange, (denoted
by $\eta_{\lambda'}$), the \betabeta decay inverse half-life of a
given nucleus reads:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\be (T^{0\nu}_{1/2,i})^{-1}=G^{0\nu}_i(|\eta_\nu|^2 ({M'}^{0\nu}_{i,
\nu})^2 + |\eta_{\lambda'}|^2({M'}^{0\nu}_{i,\lambda'})^2 +
2\cos\alpha
{M'}^{0\nu}_{i,\lambda'}{M'}^{0\nu}_{i,\nu}|\eta_\nu||\eta_{\lambda'}|)\,,
\label{hlint} \ee
%%%%%%%%%%%%%%%%%%%%%%%%%%
where the LNV parameters are given in \cite{Faessler:2011qw}. From
Eq. (\ref{hlint}) it is possible to extract the values of
$|\eta_\nu|^2$, $|\eta_{\lambda'}|^2$ and $\cos\alpha$ setting up a
system of three equation with these three unknowns using as input
the ``data'' on the half-lives of three different nuclei. The
solutions are given using the Cramer's rule. As well, we must
require that $|\eta_\nu|^2$ and $|\eta_{\lambda'}|^2$ be
non-negative and that the factor
$2\cos\alpha|\eta_\nu||\eta_{\lambda'}|$ in the interference term
satisfies:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\be -2|\eta_\nu||\eta_{\lambda'}| \leq
2\cos\alpha|\eta_\nu||\eta_{\lambda'}|
\leq2|\eta_\nu||\eta_{\lambda'}|. \label{fase} \ee
%%%%%%%%%%%%%%%%%%%%%
If we fix (i.e. have data on) the half-lives of two of the nuclei
and combine these with the condition in Eq. (\ref{fase}), we can
obtain the interval of values of the half-life of the third
nucleus, which is compatible with the data on the half-lives of the
two other nuclei and the mechanisms considered. The minimal
(maximal) value of this interval of half-lives of the third nucleus
is obtained for $\cos\alpha = +1$ ($\cos\alpha = -1$). An example of
such an analysis is plotted in Fig. \ref{fig1} (right panel). One
can notice that the positivity conditions in this case allow to
constrain the region of positive solutions given by the white area.
For a detailed analysis see \cite{Faessler:2011qw}.
\begin{figure}
\centering \subfigure
{\includegraphics[width=0.35\textwidth]{GeFixedA.eps}}
\vspace{5mm}
\subfigure
{\includegraphics[width=0.45\textwidth]{GeminMomaxfixed.eps}}
\vspace{-0.8cm} \caption{\label{fig1} Rescaled values of i)
$|\eta_\nu|^2$ (solid line) and $|\eta^{R}_{_N}|^2$ (dashed line)
for
$T^{0\nu}_{1/2}(^{76}Ge)=2.23\times10^{25}$y~\protect\cite{Klapdor}
(left panel), and of ii) $|\eta_\nu|^2$ (solid line) and
$|\eta_{\lambda'}|^2$ (dashed lined) for the same value of
$T^{0\nu}_{1/2}(^{76}Ge)$ and
$T^{0\nu}_{1/2}(^{100}Mo)=5.8\times10^{24}$y (right panel). The
experimental lower bound \protect\cite{CUORI}
$T^{0\nu}_{1/2}(^{130}Te)> 3\times10^{24}$y is taken into account.
The physical allowed regions correspond to the areas shown in white;
the areas shown in gray are excluded. The horizontal solid (dashed)
line corresponds to the upper limit \protect\cite{MoscowH3}
\protect\cite{MainzKATRIN} $\meff < 2.3$eV (prospective upper limit
\protect\cite{MainzKATRIN} $\meff \leq 0.2$eV). }
\end{figure}
\section*{Acknowledgments}
A. M. acknowledges the Organizers for the opportunity to present
this work and S. T. Petcov for useful comments.
\section*{References}
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%A.~Meroni, S.~T.~Petcov, F.~Simkovic and J.~Vergados,
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%%CITATION = ARXIV:1103.2434;%%
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%%CITATION = HEP-PH 0102276;%%
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%%CITATION = ARXIV:0807.2336;%%
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\end{thebibliography}
\end{document}