Advances in High Energy Physics

Volume 2019, Article ID 4863620, 9 pages

https://doi.org/10.1155/2019/4863620

## Constraining the Effective Mass of Majorana Neutrino with Sterile Neutrino Mass for Inverted Ordering Spectrum

Department of Physics, Lucknow University, Lucknow 226007, India

Correspondence should be addressed to Jaydip Singh; moc.liamg@hgnis.pidyaj

Received 10 February 2019; Revised 11 April 2019; Accepted 21 April 2019; Published 6 May 2019

Academic Editor: Sally Seidel

Copyright © 2019 Jaydip Singh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

Inspired by the experimental anomalies in neutrino physics and recent oscillation data from short baseline and another neutrino experiment, the realization of one extra neutrino flavor seems to be favoring. This extra flavor may change the observable, , of current data taking and next-generation -decay experiments aim to probe and possibly look at the Inverted Ordering region (eV) of parameter space. This observation would allow establishing physics beyond the standard model and phenomena like lepton number violation and Majorana nature of neutrino. The range of this observable () is not very well defined for both the ordering of mass spectrum (Normal Ordering and Inverted Ordering). Several attempts have been made for defining exactly the range for three active neutrino states. For contrasting this range, I have worked with an extra mass state, , and its effect on the observable with various combinations of CP violation Majorana phases by taking into account the updated data on the neutrino oscillation parameters for IO case. Based on the Monte Carlo technique, a parameter region is obtained using the fourth Majorana-Dirac phase of sterile parameters that lead to an effective mass below 0.01 eV or .05 eV for inverted mass ordering case which is planned to be observed in the near future experiment.

#### 1. Introduction

In the near future, the positive observation of the neutrino-less double beta decay process would be clear evidence for the lepton number violation and the confirmation of the Majorana nature of neutrinos [1]. In addition to the phenomena discussed earlier, it can also address some of the yet unresolved issues in the neutrino physics and physics beyond the standard model, such as the origin of the tiny neutrino masses, the absolute neutrino mass scale, and the mass ordering and the dark matter physics; see [2–5] for more detail. In general, three active neutrinos are considered in the standard neutrino oscillation picture with mass-squared differences of orders and [6]. From tritium beta decay experiments and cosmological observations, the absolute mass scale of neutrino is also constrained to be below 1 eV. Considering the three neutrino formalisms we have various successful achievements in atmospheric neutrino physics, solar neutrino physics, accelerator, and reactor neutrino experiments but there are experimental anomalies that cannot be explained within the standard three-neutrino framework. Mainly the issue of LSND [7], MiniBooNE[8], the Gallium neutrino anomaly [9, 10], and the reactor antineutrino anomaly [11] are yet not understood clearly. It is frequently interpreted as a hint towards the existence of one or two sterile neutrino states with masses at the eV scale. Such neutrinos are called “sterile” since they cannot participate in the weak interactions due to the collider constraints and point towards the nonstandard neutrino physics. The presence of sterile neutrinos would significantly change the observables in neutrino experiments, specifically the oscillation probabilities in short-baseline experiments and the effective mass in neutrino-less double beta decay [12–14]. Current data taking reactor and solar and gallium experiments support the sterile neutrino oscillations phenomena and recent analysis and discussion can be found in [15, 16]. Constantly growing current and future experiments for verifying the short-baseline neutrino oscillation anomalies and perhaps revealing sterile neutrinos are STEREO [17], DANSS [18], NEOS [19], PROSPECT [20], Neutrino-4 [21], BEST [22], and SOLID [23].

In the oscillation experiment, total lepton number is conserved while in the neutrinoless double beta decay experiment total lepton number changes by two units. So, some of the fundamental questions like lepton number violation and Majorana nature of neutrino cannot be answered through the neutrino oscillation experiments but can be answered through the neutrino-less double beta decay (0) experiment. Therefore, the most promising process which can be unambiguous Majorana nature of neutrinos is neutrino-less double beta decay and the search for this phenomena has a long history [24]. Current running experiments like KamLAND-Zen, CUORE, CUORE-0, Cuoricino, and GERDA-II are trying to find out the lower bounds on the of this decay for several nucleus samples. Recent lower bound obtained by the KamLAND-Zen collaboration for sample (xenon-136) is yr [25], GERDA-II collaboration obtaining the lower bound for sample (germanium-76) is yr [26], and CUORE, CUORE-0 and Cuoricino experiments collectively obtaining the lower bound for sample (tellurium-130) is yr[26].

The -decay rate is proportional to the effective Majorana mass in the three-Majorana neutrino picture. The range of this effective Majorana mass is not very well understood but, based on the present neutrino oscillation data, it is bounded from below eV in the case of three neutrinos mass spectrum with Inverted Ordering [27]. While for the Normal Ordering configuration of the mass spectrum the lower bound is eV [28] and it can be extremely small depending on the values of the Dirac, Majorana phases, and the smallest neutrino mass. In the article [29] the possible range of is determined by varying the Majorana and CPV phases to show the dependency of on the phases. Then a condition is established using the full range of Majorana and Dirac CPV phases and also for some particular Majorana and Dirac CPV phases with NO (Normal Ordering) spectrum, under which in the 3-neutrino mixing exceeds the milli-electron-volt value. In this article, a similar approach is followed to define the possible range of by considering one extra neutrino states (3+1). Neutrino double beta decay experiments are trying to cover the range of from the top and the current reachable upper limit reported by KamLAND-Zen collaboration is [25]. This upper limit is obtained by the KamLAND-Zen collaboration using the lower limit on the half-life of the Xenon-136 sample and they have considered the uncertainties in the NMEs (Nuclear Matrix Elements) in their analysis of the relevant process.

New-generation experiments plan to look the Inverted Ordering region of parameter space and possibly work for the eV energy range. Various running experiments are considered [30, 31] for upgradation and new experiments are also proposed to achieve this goal. Some of those experiments are MAJORANA, LEGEND, CANDLES, AMoRE, PandaX-III, SuperNEMO and DCBA, ZICOS , MOON , COBRA , SNO+, NEXT, and nEXO. If these planned experiments did not find positive response in this energy range (), then next generation experiment will be very interesting for sterile neutrino which will correspond to eV or more below in the -decay experiment.

The introduction of a sterile neutrino at the eV mass scale can change the prediction for the possible range of values in the neutrinoless double-beta decay [12, 32–36]. In the present article, I have determined a sterile parameter region using Monte Carlo technique under which the effective Majorana mass is pushed below a certain value (.01 eV or .05 eV) in the case of 3+1 neutrino mixing with Inverted Ordering mass spectrum. For completeness Normal Ordering scenario of the spectrum with three active and one sterile neutrino is also discussed with the recent global data. Considering the latest global data and various possible CP violation Majorana phases possible ranges of observables in -decay, tritium beta decay experiments and cosmological observations are also discussed.

#### 2. The Effective Mass in 3 and 3+1 Neutrino States

In this section, I will discuss the general formalism of neutrino parameters and evaluate the contributions to the effective mass relevant for neutrino-less double beta decay. First I outline the formalism of three active neutrinos’ mixing and then mixing in the presence of one extra sterile states (3+1). I have worked with the 3+1 scenario when the sterile neutrino is heavier than the active neutrino with Normal Ordering and Inverted Ordering scheme. Another alternative scheme is 1+3; here new nonstandard massive neutrinos is lighter than the three standard massive neutrinos. The 1+3 schemes are disfavored by the cosmological upper bound on the neutrino masses that is smaller than 1 eV, and by the upper bound on the effective neutrino mass in neutrinoless double- decay if neutrinos are Majorana particles, detail discussion can be found in [37, 38]. Hence I will not discuss this scheme further; other possible schemes (3+2/2+3 or 1+3+1) are also available in the literature but not discussed here; more detailed information can be found in [12].

##### 2.1. Three Active Neutrinos’ Mixing

For three active neutrino states configuration of the effective Majorana mass is defined as

with the partial contribution of the massive Majorana neutrino with mass and U being the leptonic mixing matrix and known as Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix, which exactly defined the mixing of the electron neutrino with the three massive neutrinos. The first row of this mixing matrix is the one relevant for -decay, and in standard parameterization it is defined [28] as

where and , where are the mixing angles, and and are the Dirac and Majorana phases [39], respectively, and these are the complex phases and the possible range of these parameters are . The values of these phases are not known and all possible values of must take into account all the possible range of these phases. The results for the neutrino squared-mass differences are expressed in terms of the solar and atmospheric squared mass differences, which are defined aswhere . Given this assignment of the squared mass differences, it is currently unknown if the ordering of the neutrino masses is normal, i.e., , or inverted, i.e., . I also defined in the NO(IO). In terms of the lightest neutrino mass, CPV phases, neutrino mixing angles, and neutrino mass-squared differences, the effective Majorana mass readswhere we have defined . Effective mass for both cases, and , can be written in the form

It is then clear that the effective Majorana mass is the length of the vector sum of three vectors in the complex plane for three neutrino mixing case. And their relative orientations are determined by the CPV phase factors and that can push the range of significantly up and down with Normal or Inverted ordering of the neutrino mass spectrum. Tables 1 and 2 show the best fit, 1, 2, and 3 ranges of the three neutrino oscillation parameters used for this analysis with NO and IO of the mass spectrum obtained from the recently updated global data [6].