Contents
  1. Introduction to Neutrinoless Double Beta Decay\(\mathbf{(0}\mathbf{\nu\beta\beta)}\)
  2. Neutrino Mass
  3. Effective Majorana Neutrino Mass (\(\mathbf{m}_{\mathbf{\text{ββ}}}\mathbf{)}\): Towars the Absolute Neutrino Mass Scale
  4. Half-Life (\(\mathbf{T}_{\mathbf{0}\mathbf{\nu}}^{\frac{\mathbf{1}}{\mathbf{2}}}\)) of Neutrinoless Double Beta Decay\(\mathbf{\ (}\mathbf{0}\mathbf{\text{νββ}}\))
  5. Nuclear Matrix Elements(NME) \(\mathbf{M}_{\mathbf{0}\mathbf{\nu}}\) Calculations
  6. Sources of Uncertainty in Calculating Nuclear Matrix Elements(NME)
    1. Deformation Effect (variation in inter-nucleon Distance)
      6.2 Short Range Correlations (SRC)
      6.3 Finite Nucleon Size (FNS)
      6.4 Average Excitation Energy\((\overset{\overline{}}{E})\) Dependence
      6.5 Higher Order Currents
    1. Variation of Interaction Operator “Neutrino Potential\(\ H(r,\overset{\overline{}}{E})"\) Considering all these Uncertainty
  7. Experimental Status of Neutrinoless Double Beta Decay
    \(\mathbf{(}\mathbf{0}\mathbf{\nu\beta\beta)}\)
  8. Conclusions
  9. Research Plan
  10. References
    1. \label{section}
    1. 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 (\(0\nu\beta\beta).\)
    1. 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
    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.
    1. 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.
    1. 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.
    1. 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:
    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.
    1. 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.
    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\((M_{0\nu})\)
    1. 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 \(0\nu\beta\beta\)
    1. 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.
    1. 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\(\ m_{\text{ββ}}=\sum_{j=1,3}{U_{\text{ej}}^{2}e^{\frac{i\alpha_{j}}{2}}m_{j}}\).
    1. References