References
-
\label{section}
-
Introduction to Neutrinoless Double Beta
Decay\(\mathbf{(0}\mathbf{\nu\beta\beta)}\)
One fundamental question about neutrinos yet to resolve is “Are
neutrinos Dirac particle or Majorana particle?” Neutrinoless double
beta decay\((0\nu\beta\beta\)), a yet unobserved very rare weak process
where for some even-even nuclei two bound neutrons converts into two
protons emitting two electrons in the final states without any
neutrinos, violating lepton no conservations, is the only process
available to prove that neutrinos are their own anti-particles (Majorana
particles) rather than Dirac particles. This process occurs to those
nuclei where single beta decay is energetically forbidden or there is
large spin difference between the nuclear energy levels. A virtual
Majorana neutrino appears in the intermediate states.
Here I will present a brief description of how neutrinoless double beta
decay\((0\nu\beta\beta)\) will tell us the Majorana nature of neutrinos
and how it will hint us in predicting right absolute mass scale of
neutrinos, yet an unsolved problem. Also I will present a brief review
of challenges of calculations of Nuclear Matrix Elements
(NME)\(\ M_{0\nu}\) which is turned out to be a most important tool to
predict half-life\(\ (t_{0\nu}^{1\backslash 2\ })\) of
\(\ \)neutrinoless double beta decay \(\left(0\nu\beta\beta\right).\)
My research will be focused in calculating Nuclear Matrix Elements
(NME)\(\ M_{0\nu}\) with better precision and to predict proper
half-life\(\ (t_{0\nu}^{1\backslash 2\ })\) of \(\ \)neutrinoless double
beta decay \(\left(0\nu\beta\beta\right)\) and absolute mass scale of
neutrinos.
Long story of neutrino starts from the idea of single beta decay where
inside a nucleus for some radioactive isotopes a bound neutron\((n)\)
converts into a bound proton\((p)\) emitting beta particle\((e^{-})\)
and a third particle called anti electron
neutrino\(({\overset{\overline{}}{\nu}}_{e})\).
\(n\rightarrow p+e^{-}+{\overset{\overline{}}{\nu}}_{e}\)
Initially there was no concept of the third particle anti electron
neutrino\(({\overset{\overline{}}{\nu}}_{e})\) and it was expected that
beta decay spectrum should be discrete, although observed spectrum was
continuous and also there was violation of energy and momentum
conservations law. To explain continuous nature of beta decay spectrum
Pauli in 1930 postulated that there must be a third particle which
carries the extra energy. It was Fermi in 1932 who incorporated the
third particle in his famous 4-point beta decay theory and named that
particle neutrino. Thus idea of neutrino was born. At that time concept
of only one type of neutrino (electron neutrino\(\ \nu_{e}\)) and
corresponding antineutrino
(electron-antineutrino\({\overset{\overline{}}{\text{\ ν}}}_{e}\)) was
there. Later two more type of neutrino and anti-neutrino (muon
type\((\nu_{\mu}\text{\ and\ }{\overset{\overline{}}{\nu}}_{\mu})\) and
tau type (\(\nu_{\tau}\text{\ and}{\overset{\overline{}}{\nu}}_{\tau}\)
) was discovered. In Standard Model of Particle Interactions (SMPI)
neutrinos and antineutrinos belongs to family of fundamental leptons and
antileptons. They are charge less, spin \(\frac{1}{2}\) Dirac Fermions
with lepton no +1 for neutrinos and lepton no -1 for antineutrinos.
Electron type neutrinos belong to first generations, muon type neutrinos
belongs to second generation and tau type neutrinos belongs to third
generation. Neutrinos are kept massless in Standard Model of Particle
Interactions (SMPI) although neutrino oscillations data shows that
neutrinos have finite masses.
In 1935 M.Goppert-Mayer proposed that there is a possibility for certain
isotopes to emit via 2nd order weak process called
double beta decay(\(2\nu\beta\beta\ \)) where two bound neutrons\((n)\)
inside the nucleus decay into two protons\((p)\) emitting two
electrons\((e^{-})\) and two electron type
antineutrinos\((\overset{\overline{}}{\nu_{e}})\). This process is
allowed where single beta decay is energetically forbidden or there is
large spins difference between nuclear energy levels. All the
conservation laws are followed in this process and it was observed first
in 1987. After 1987 it was observed in many experiments and now became a
well understood phenomenon.
In 1928 PAM Dirac gave his famous relativistic equation known as Dirac
equation obeyed by Dirac Fermions
\begin{equation}
\left(\text{iℏ}\gamma^{\mu}\partial_{\mu}-m\right)\psi=0\ \nonumber \\
\end{equation}
Here \(\psi\) is four-component Dirac spinor containing both particles
and antiparticles fields in positive and negative time directions. Dirac
was the first person to introduce the concept of anti-particles.
According to Dirac theory Neutrinos and antineutrinos are different
particles\(\ \nu\neq\overset{\overline{}}{\nu}\).
In 1937 Ettore Majorana pointed out that Dirac theory may be valid for
charged fermions and it cannot be applied to charge less Fermions and
gave his theory for charge less fermions considering \(\psi\) as 2
component spinor and suggesting that neutrinos and anti-neutrinos are
rather same particles \(\nu=\overset{\overline{}}{\nu}.\)
New Physics starts evolving after Majorana pointed out the nature of
neutrinos. So considering the Majorana nature of neutrinos where
neutrinos and antineutrinos are rather same particles Giulio Racah
pointed out that if neutrinos are Majorana particles there may be a
third type of beta decay called Neutrinoless Double Beta Decay
(\(0\nu\beta\beta)\) where inside some even-even nucleus two bound
neutron simultaneously decay into two protons emitting two electrons in
the final states without any neutrinos violating lepton no conservation.
Hypothetical Majorana neutrinos come out in the intermediate states as
virtual particles. Calculations of phase space factor for this process
rather comes out to be more preferable than well understood and observed
Double Beta Decay mode\((2\nu\beta\beta)\).
To allow this new type of neutrinoless double beta decay
(\(0\nu\beta\beta)\) we must have
It is quite interesting that a process may allowed although lepton no
conservation is violated. Is it really possible? For the possibility of
this process we must analyze the lepton no conservation rule from the
point of view of Standard Model of Particle Interactions (SMPI) and
Beyond the Standard Model (BSM) physics. In SMPI neutrinos are kept
massless and lepton no conservation is followed strictly by massless
leptons. But form neutrino oscillations we know that neutrinos have
finite masses, thus lepton no conservation rule is not effective on
neutrinos and we need to go Beyond the Standard Model (BSM) Physics . So
to observe this process neutrinoless double beta decay
(\(0\nu\beta\beta)\) we must have
So, neutrino masses became an important factor of observing neutrinoless
double beta decay\(\ \left(0\nu\beta\beta\right)\) . In next section I
will present a brief description of how neutrino oscillations lead to
masses of neutrinos increasing the possibility of observing neutrinoless
double beta decay (\(0\nu\beta\beta).\)
-
Neutrino Mass
Neutrino masses have been always an elusive area of neutrino physics.
2015 was an exciting year for both neutrino masses and neutrinoless
double beta decay\(\ (0\nu\beta\beta)\). Takaaki kajita at Super
Kamiokande, Japan and Arthur B. McDonald at SNO, Canada experiments
independently shows that neutrinos changes flavours among themselves by
neutrino oscillations that proves that neutrinos have finite masses.
Although neutrinoless double beta decay\(\ (0\nu\beta\beta)\) is a yet
to observe phenomena, after the discovery of existence of neutrinos
mass, it becomes a must allowed process. Here I will try to present how
these neutrinos really oscillate that give rise masses to them,
considering both Dirac and Majorana nature of Neutrinos.
We consider that at initial time neutrinos born with three orthogonal
mass Eigen states \(\nu_{1},\nu_{2},\nu_{3}\) whose mixing by various
angle when they travel through space give rise three Neutrino flavours
states\(\ \nu_{e},\nu_{\mu},\nu_{\tau}\). Mixing is associated with an a
unitary transformation as
Where U\(=\begin{bmatrix}U_{e1}&U_{e2}&U_{e3}\\
U_{\mu 1}&U_{\mu 2}&U_{\mu 3}\\
U_{\tau 1}&U_{\tau 2}&U_{\tau 3}\\
\end{bmatrix}\ \)is a unitary mixing matrix given by
\(U=\begin{pmatrix}c_{12}c_{13}&s_{12}c_{13}&s_{13}e^{-i\delta}\\
-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{\text{iδ}}&c_{12}c_{23}-s_{12}s_{23}s_{13}e^{\text{iδ}}&s_{23}c_{13}\\
s_{12}s_{23}-c_{12}c_{23}s_{13}e^{\text{iδ}}&-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{\text{iδ}}&c_{23}c_{13}\\
\end{pmatrix}\)
called Pontecorvo–Maki–Nakagawa–Sakata (PMNS) matrix.
\(S_{\text{ij}}=\sin\theta_{\text{ij}}\text{and~{}}c_{\text{ij}}=cos\theta_{\text{ij}},\ \ and\ \delta\ is\ CP\ violating\ phase\ factor.\)
So mixing of various mass Eigen states \(\nu_{1},\nu_{2},\nu_{3}\) give
rise to Neutrino flavours states \(\nu_{e},\nu_{\mu},\nu_{\tau}\) by
above PMNS unitary matrix and then we calculate probability of
inter-conversion among various flavor Eigen states which is directly
related to masses of neutrinos. Above PMNS matrix is basically for Dirac
nature of neutrino. If we consider Majorana nature of neutrinos
oscillations nature of neutrinos is not changed but PMNS mixing matrix
is modified as
Modified PMNS Matrix of mixing =\(U*\begin{pmatrix}e^{\frac{i\alpha_{1}}{2}}&0&0\\
0&e^{\frac{i\alpha_{2}}{2}\ }&0\\
0&0&1\\
\end{pmatrix}\) where \(\alpha_{1}\text{\ and\ }\alpha_{2}\) are
Majorana phase factor.
From various neutrino oscillations experiments date we can found the
Unitary Matrix \(U\)
Most recent results show that\(\ U=\begin{bmatrix}0.82\pm 0.01&0.54\pm 0.02&0.15\pm 0.03\\
0.35\pm 0.06&0.70\pm 0.06&0.62\pm 0.06\\
0.44\pm 0.06&0.45\pm 0.06&0.77\pm 0.06\\
\end{bmatrix}\). Using this mixing matrix and probability we can predict
square of mass differences
(\(m_{1}^{2}-m_{2}^{2},\ m_{2}^{2}-m_{3}^{2},\ m_{1}^{2}-m_{3}^{2})\)
of various neutrino flavours. Again one problem here is that we are
predicting square of mass differences
\((m_{1}^{2}-m_{2}^{2},\ m_{2}^{2}-m_{3}^{2},\ m_{1}^{2}-m_{3}^{2})\)
rather than absolute value of neutrino
mass\(\text{\ m}_{1},m_{2}\text{\ and\ }m_{3}\).This problem in physics
is known as “mass hierarchy problem”.
There are three cases of considerations
-
If \(m_{1}^{2}<m_{2}^{2}<m_{3}^{2}\) it is called normal hierarchy
-
If \(m_{3}^{2}<m_{1}^{2}<m_{2}^{2}\) it is called inverted
hierarchy
-
If \(m_{1}^{2}\approx m_{2}^{2}\approx m_{3}^{2}\) it is called
quasi degenerate
Normal Hierarchy Inverted Hierarchy
In next section I will discuss how neutrinoless double beta
decay\(\ (0\nu\beta\beta)\) will try to predict absolute mass scale of
neutrinos, solving “mass hierarchy problem”, using modified PMNS
matrix after inclusion of Majorana phases.
-
Effective Majorana Neutrino Mass
(\(\mathbf{m}_{\mathbf{\text{ββ}}}\mathbf{)}\): Towars the
Absolute Neutrino Mass Scale
In predicting absolute neutrino mass scale neutrinoless double beta
decay\(\ (0\nu\beta\beta\)) will play an important role. One important
parameter of \(0\nu\beta\beta\) is effective Majorana neutrino mass
\begin{equation}
m_{\text{ββ}}=\sum_{j=1,3}{U_{\text{ej}}^{2}e^{\frac{i\alpha_{j}}{2}}m_{j}}\nonumber \\
\end{equation}
This effective Majoran mass predicts absolute mass scale of neutrinos.
To find this we use neutrino oscillations data to get the value of
matrix \(U\) and use the calculated value of Majorana
phases\(\ \alpha_{1}\text{and\ }\alpha_{2}\).
Depending upon Inverted hierarchy consideration we find absolte
effective neutrino mass as
\(m_{\text{ββ}}\approx few\ tens\ of\ meV.\ \)
Considering normal hierarchy region effective Majoran mass comes out to
be\(\ m_{\text{ββ}}\approx too\ small\ around\ few\ meV\).
This will be a great achievement for us if neutrinoless double beta
decay\(\ (0\nu\beta\beta\)) is observed and we can predict absolute
neutrino mass scale.
-
Half-Life
(\(\mathbf{T}_{\mathbf{0}\mathbf{\nu}}^{\frac{\mathbf{1}}{\mathbf{2}}}\))
of Neutrinoless Double Beta
Decay\(\mathbf{\ (}\mathbf{0}\mathbf{\text{νββ}}\))
Most important parameter to get fruitful results of observing
neutrinoless double beta decay\(\ (0\nu\beta\beta\) is to predict
correct limit of half-life(\(T_{0\nu}^{\frac{1}{2}}\)) of neutrinoless
double beta decay\(\ (0\nu\beta\beta\)) which in terms of Nuclear Matrix
elements(NME) \(M_{0\nu}\mathbf{\ }\)and Effective Majoran mass
\(m_{\text{ββ}}\) is written as
\begin{equation}
\ \frac{1}{T_{0\nu}^{\frac{1}{2}}}=G_{0\nu}\left|M_{0\nu}\right|^{2}\left(\frac{m_{\text{ββ}}}{m_{e}}\right)^{2}\nonumber \\
\end{equation}
Phase space factor (PSF) \(G_{0\nu}\) for \(0\nu\beta\beta\) is
calculated with great precisions with less approximations in various
methods. Effective Majoran mass \(m_{\text{ββ}}\) is evaluated from
various neutrino oscillations experimental data and Majorana Phase
factor\(\text{\ α}_{1}\text{\ and\ }\alpha_{2}\ \)can also be
calculated. Only problem is with predicting proper Nuclear Matrix
elements (NME) \(M_{0\nu}.\)
In next section I am going to discuss how calculations of Nuclear Matrix
elements (NME) \(M_{0\nu}\) play an important role in predicting
half-life (\(T_{0\nu}^{\frac{1}{2}}\)) of neutrinoless double beta
decay\(\ (0\nu\beta\beta\)) and what sources of uncertainty are there in
the calculations of NME.
-
Nuclear Matrix Elements(NME) \(\mathbf{M}_{\mathbf{0\nu}}\)
Calculations
For observing experimentally the phenomena neutrinoless double beta
decay\(\ (0\nu\beta\beta)\) we must calculate proper
half-life\({(t}_{0\nu}^{\frac{1}{2}}\)) and to calculate
half-life\({(t}_{0\nu}^{\frac{1}{2}}\)) we must calculate proper Nuclear
Matrix elements (NME)\(M_{0\nu}\). Unfortunately precision in
calculation of Nuclear Matrix elements (NME) \(M_{0\nu}\) is not that
much satisfactory. Uncertainty Nuclear Matrix elements
(NME)\(\text{\ M}_{0\nu}\)calculations comes from various sources of
complex nuclear structure. Basically for calculating Nuclear Matrix
elements (NME) \(M_{0\nu}\) we need to find proper wave functions of
initial and final nuclear energy levels. Then we calculate a transition
operator which contains all the information about interaction energy,
nature of coupling and other additional term inside the complex nuclear
structure. Nuclear Matrix elements (NME) \(M_{0\nu}\)contain both Fermi
and Gamow-Teller matrix elements as both type of transition is possible.
If we expand expression of Nuclear Matrix Elements (NME)\(M_{0\nu}\) we
get
\begin{equation}
M_{0\nu}=M_{0\nu}^{GT}-\frac{g_{V}^{2}}{g_{A}^{2}}M_{0F}^{F}\nonumber \\
\end{equation}
\(M_{0\nu}^{F}=<f\left|\sum_{j,k}{H(r_{\text{jk}},\overset{\overline{}}{E}}\right)\tau_{j}^{+}\tau_{k}^{+}|i>\)
is a matrix element of Fermi like operator .
\(M_{0\nu}^{\text{GT}}=<f\left|\sum_{j,k}{H(r_{\text{jk}},\overset{\overline{}}{E}}\right){\overrightarrow{\sigma}}_{j}.{\overrightarrow{\sigma}}_{k}\tau_{j}^{+}\tau_{k}^{+}|i>\)
is matrix elements of Gamow-Teller like operator. \(g_{V}\) And
\(g_{A}\) are weak vector and axial-vector current coupling
respectively. \(H\) Depends upon distance between two initial
neutrons\((r)\) and average excitation energy
\((\overset{\overline{}}{E})\) in the intermediate nucleus, and
sometimes called “neutrino potential” or transition operator.
\begin{equation}
H\left(r,\overset{\overline{}}{E}\right)=\frac{2R}{\text{~{}πr}}\int_{0}^{\infty}{\frac{\text{qsinqr}}{\omega(\omega+\overset{\overline{}}{E}-\frac{\left[M_{i}-M_{f}\right]}{2})}\text{dq}}\nonumber \\
\end{equation}
\(M_{i}\ \)and \(M_{f}\) are masses of initial and final nuclei. This
transition operator \(H\) must contain all the interactions taking place
inside complex nuclear structure. Challenges in calculating Nuclear
Matrix elements (NME) \(M_{0\nu}\) lie in the complex nuclear structure,
we need to know nature of all the interactions taking place inside the
nucleus and modify accordingly transition operator \(H\) that give rise
changes to Nuclear Matrix Elements (NME)\(M_{0\nu}.\)
Some approximation methods and models based on many-body theory or
mean-field theory are developed that are trying to find better wave
functions and transition operator for finding NME and predicting Proper
half-life.
Followings are various models of Nuclear Matrix elements (NME)
calculations based on mean-field theory:
-
Interacting Shell Model(ISM)
-
Quasiparticle Random Phase Approximation(QRPA)
-
Interacting Boson Model(IBM-2)
-
Projected Hartree-Fock Bogoliubov Method(PHFB)
-
Energy Density Function Method(EDF)
Matrix Elements for various isotopes using various models
Various model approaches with different assumptions with common goal-to
find proper wave functions and transition operator of nuclear levels to
calculate Nuclear Matrix Elements (NME). ISM and QRPA method
have been extensively used as they are more sophisticated than others.
Uncertainty is an inherent property in calculating all these things. I
will discuss how this transition operator gets modified in the presence
of various uncertainty sources that ultimately propagate to proper
Nuclear Matrix elements (NME) calculations in next section.
-
Sources of Uncertainty in Calculating Nuclear Matrix
Elements(NME)
Avoiding uncertainty in calculating NME became a challenging task for
us. For these to avoid we must understand various sources of uncertainty
that are coming from complex nuclear structure. Following are some of
the sources of uncertainty.
-
Deformations of Nucleus
-
Short Range Correlations\((SRC)\)
-
Finite Size of Nucleons\((FNS)\)
-
Average Excitation Energy of Intermediate
States\((\overset{\overline{}}{E})\)
-
Higher Order Current Contributions\((HOC)\)
If there is uncertainty in calculations of Nuclear Matrix Elements
(NME)\(M_{0\nu}\), there will be uncertainty in predicting half
life\((t_{0\nu}^{\frac{1}{2}}\mathbf{)}\). Predicting wrong
half life\((t_{0\nu}^{\frac{1}{2}}\mathbf{)}\) may leads to wrong
experiments and that may harm possibility of observing neutrinoless
double beta decay\(\ (0\nu\beta\beta)\) in the lab. So we must be
cautious about all these sources of uncertainty.
6.1 Deformation Effect (Variation in inter-nucleon Distance)
Transition Probability for \(0\nu\beta\beta\) in term of
Matrix-element\((M_{0\nu})\) is written as
\begin{equation}
\int{\frac{P\left(r\right)\text{dr}}{r}=M_{0\nu}}\nonumber \\
\end{equation}
Due to deformation of nuclei there is a variation in inter-nucleon
distance\((r)\) between two initial neutrons thus there is a variation
of probability function \(P(r)\) and Nuclear Matrix Elements
(NME)\(\mathbf{.}\)
6.2 Short Range Correlations (SRC)
When evaluating the 0νββ nuclear matrix elements (NME) in the closure
approximation it is necessary to consider the matrix element of a
two-body transition operator which connects a state with two neutrons in
a specific mean field single particle states with two protons again in
some specific final states and we must include short range
nucleon-nucleon repulsion correlations.
We do this by including in the transition operator (or equivalently in
the two body wave-function) a phenomenological Jastrow-like function
\begin{equation}
f_{\text{Jastrow}}\left(r\right)=1-ce^{-ar^{2}}(1-br^{2})\nonumber \\
\end{equation}
Choice of values of \(a,b,c\ \) as chosen by Miller-Spencer, Argonne and
CD-Boon give rise to three types of correlation as shown in figure.
Sometimes they are called SRC1, SRC2, and SRC3.
6.3 Finite Nucleon Size (FNS)
Neutrons and Protons are not point like particles, they made up of
quarks and they have finite size. This fact modify the
interaction operator “neutrino
potential\(\ H(r,\overset{\overline{}}{E})\)” by introducing the form
factor function
\begin{equation}
f_{\text{FNS}}=\frac{1}{\left(1+\frac{q^{2}}{M_{A}}\right)^{2}}\nonumber \\
\end{equation}
Where \(M_{A}\) varies between \(1.0-1.2\ \)GeV.
6.4 Average Excitation
Energy\(\mathbf{(}\overset{\overline{}}{\mathbf{E}}\mathbf{)}\)
Dependence
In QRPA methods of calculating \(M_{0\nu}\) using closure approximation
\(\overset{\overline{}}{E}\) varies between \(0-12\ MeV\) and
correspondingly Matrix elements\(\left(M_{0\nu}\right)\) varies as
shown in figure for isotopes
Matrix elements M 0ν for several nuclei evaluated within QRPA in the
closure approximation as a function of the assumed average excitation
energy. Black line corresponds to without approximations.
So choices of \(\overset{\overline{}}{E}\) may affect value of NME
(\(M_{0\nu})\).
6.5 Higher Order Currents
Although in neutrinoless double beta decay\(\ (0\nu\beta\beta)\)
interaction at quark level has \(V-A\) current structure, but at
nucleon (being a complicated structure) level nature of interaction has
higher order currents given by
\begin{equation}
J^{\mu+}=\psi_{\tau}^{+}\left[g_{V}\left(q^{2}\right)\gamma^{\mu}-ig_{M}\left(q^{2}\right)\frac{\sigma^{\text{μν}}}{2m_{p}}q_{\nu}-g_{A}\left(q^{2}\right)\gamma^{\mu}\gamma^{5}+g_{P}\left(q^{2}\right)q^{\mu}\gamma_{5}\right]\psi\nonumber \\
\end{equation}
And it contains weak, weak magnetism, axial vector and induced pseudo
scalar currents. It may significantly increase uncertainty in
\(M_{0\nu}\) calculations.
Pseudo scalar Coupling is given by
\begin{equation}
g_{P}\left(q^{2}\right)=\frac{g_{A}\left(q^{2}\right)*2m_{p}}{q^{2}+m_{\pi}^{2}}\nonumber \\
\end{equation}
Weak-magnetism coupling or form factor of is
\begin{equation}
g_{M}\left(q^{2}\right)=\left(\mu_{p}-\mu_{n}\right)g_{V}(q^{2})\nonumber \\
\end{equation}
Considering all these higher order terms (HOT) total coupling correction
term in GT part of “neutrino potential”
\(H(r,\overset{\overline{}}{E})\)is
\begin{equation}
g_{\text{HOT}}^{\text{GT}}\left(q^{2}\right)=1-\frac{2}{3}\frac{{\overset{\overline{}}{q}}^{2}}{{\overset{\overline{}}{q}}^{2}+m_{\pi}^{2}}+\frac{1}{3}\left(\frac{{\overset{\overline{}}{q}}^{2}}{{\overset{\overline{}}{q}}^{2}+m_{\pi}^{2}}\right)^{2}+\frac{2}{3}\left(\frac{g_{V}\left(q^{2}\right)}{g_{A}\left(q^{2}\right)}\right)^{2}\frac{(\mu_{p}^{2}-\mu_{n}^{2}){\overset{\overline{}}{q}}^{2}}{4m_{p}^{2}}\nonumber \\
\end{equation}
6.6 Variation of Interaction Operator “Neutrino
potential\(\mathbf{\ H(r,}\overset{\overline{}}{\mathbf{E}}\mathbf{)}\mathbf{"}\)
considering all these uncertainty
Using above Discussed Uncertainty source of matrix
elements\((M_{0\nu})\) calculations modified GT part of Hamiltonian is
written as
\begin{equation}
H_{\text{GT}}\left(r,{\overset{\overline{}}{E}}_{0\nu}\right)=\frac{2R}{\pi}\int_{0}^{\infty}{j_{0}\left(\text{qr}\right)}\frac{q}{q+{\overset{\overline{}}{E}}_{0\nu}}f_{\text{FNS}}^{2}\left(q^{2}\right)g_{\text{HOT}}^{\text{GT}}\left(q^{2}\right)\text{dq}\nonumber \\
\end{equation}
So various sources of uncertainty abruptly changes the value of
Gamow-Teller part of matrix elements, although Fermi part of matrix
elements remain same. In the limit of neglecting FNS, Higher order
Currents and \(\overset{\overline{}}{E}=0,\) this potential assumes
the form of coulomb potential.
So we understand that various sources of uncertainty inside complex
nuclear structure lead to uncertainty in interaction potentials and thus
in calculations of Nuclear Matrix Elements (NME) \(M_{0\nu}\) for
\(0\nu\beta\beta.\)
So our aim will be to understand these uncertainty sources and try to
reduce the uncertainty in NME\((M_{0\nu})\)
-
Experimental Status of Neutrinoless Double Beta Decay
\(\mathbf{(}\mathbf{0}\mathbf{\nu\beta\beta)}\)
As I mentioned \(0\nu\beta\beta\) is yet a unobserved penenomena,
existence of neutrino mass from neutrino oscillation enhancing the
interest of observing \(0\nu\beta\beta.\) Above chart is the proposed
experiments around the world for observing and predicting half-life of
\(0\nu\beta\beta\)
-
Conclusions
This lepton no violating process neutrinoless double beta decay
\(0\nu\beta\beta\) becomes now the only mechanism available in which we
can predict that neutrinos are Majorana particles rather than Dirac
particles (means\(\ \nu=\overset{\overline{}}{\nu})\) .Many next
generation experiments are proposed and many sophisticated models are
trying to improve the precision of Nuclear Matrix Elements (NME)
\(M_{0\nu}\ \) calculations reducing certain uncertainty coming from
complex nuclear structure and predict proper half-life. Observation of
neutrinoless double beta decay \(0\nu\beta\beta\) will predict the
absolute mass scales of neutrinos, which is yet a unsolved topic of
physics. Many phenomena in Physics Beyond the standard model will be
verified which are based on Majorana Neutrinos.
-
Research Plan
From the above discussion we came to know that one of the biggest
challenges in observing neutrinoless double beta
decay\(\left(0\nu\beta\beta\right)\) is to calculate proper half
life\((t_{0\nu}^{\frac{1}{2}}\mathbf{)}\) theoretically and it is really
tough to predict whether calculated value is right or wrong as there is
nothing to compare with experimental results as it is an unobserved
phenomena. Uncertainty in calculations of proper half
life\((t_{0\nu}^{\frac{1}{2}}\mathbf{)}\) depends upon Uncertainty in
calculations of Nuclear Matrix Elements (NME)\(M_{0\nu}\) which further
depends on finding proper wave functions of initial and final states of
nuclear levels and transition operator between these levels considering
all the sources of uncertainty inherited from complex Nuclear Stricture.
Basically I will use some of the models (Interacting Shell Model and
Quasi particle Random Phase
Approximation) to calculate Nuclear Matrix Elements (NME)\(M_{0\nu}\)
for Neutrinoless Double Beta Decay\((0\nu\beta\beta)\) with better
precisions finding proper wave functions of initial and final states and
Transition operator\(\ (H)\) of nuclear levels considering all the
sources of uncertainty inherited from complex Nuclear Stricture. Using
the calculated Nuclear Matrix Elements (NME)\(M_{0\nu}\) I will predict
the
h\(\text{alf~{}life~{}}\frac{1}{T_{0\nu}^{\frac{1}{2}}}=G_{0\nu}\left|M_{0\nu}\right|^{2}\left(\frac{m_{\text{ββ}}}{m_{e}}\right)^{2}\)of
\(0\nu\beta\beta\) for some isotopes and also predict the absolute mass
scale of neutrinos by predicting effective Majoran neutrino
mass\(\ m_{\text{ββ}}=\sum_{j=1,3}{U_{\text{ej}}^{2}e^{\frac{i\alpha_{j}}{2}}m_{j}}\).
-
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