Implications of the Dark LMA solution and Fourth Sterile Neutrino for Neutrino-less Double Beta Decay

We analyze the effect of the Dark-large mixing angle (DLMA) solution on the effective Majorana mass ($m_{\beta\beta}$) governing neutrino-less double beta decay ($0\nu\beta\beta$) in the presence of a sterile neutrino. We consider the 3+1 picture, comprising of one additional sterile neutrino. We have checked that the MSW resonance in the sun can take place in the DLMA parameter space in this scenario. Next we investigate how the values of the solar mixing angle $\theta_{12}$ corresponding to the DLMA region alter the predictions of $m_{\beta\beta}$ including a sterile neutrino in the analysis. We also compare our results with three generation cases for both standard large mixing angle (LMA) and DLMA. Additionally, we evaluate the discovery sensitivity of the future ${}^{136}Xe$ experiments in this context.


I. INTRODUCTION
The standard three flavour neutrino oscillation picture has been corroborated by the data from decades of experimentation on neutrinos. However some exceptions to this scenario have been reported over the years, calling for the necessity of transcending beyond the three neutrino paradigm.
The first among these signatures came from the LSNDν µ →ν e oscillation data [1], which could be explained by invoking additional neutrino states (sterile) that mix with active neutrinos [2][3][4][5][6]. This result was supported by the hints obtained : from the appearance data ofν µ →ν e and ν µ → ν e at MiniBooNE experiment [7][8][9][10][11], from the reactor neutrino anomaly [12,13] where a deficit in theν e reactor flux has been reported by short baseline(SBL) oscillation data and also from the missing neutrino flux at GALLEX [14][15][16] and SAGE [17] source experiments. However accelerator experiments like KARMEN [18], ICARUS [19] , NOMAD [20] have not found a positive signal. There are also disappearance experiments using reactors and accelerators as neutrino sources which have not reported any evidences of sterile neutrino [21]. The allowed region from the global analysis including all these data have been obtained in [22,23]. Several new experiments are planned to test the sterile neutrino hypothesis [24].
The basic question whether the neutrinos are Dirac particles or lepton number violating Majorana particles (for which particles and antiparticles are the same) remains as a major puzzle in neutrino physics. Since oscillation experiments do not help us to determine the nature of the neutrinos, one has to rely on studying the processes in which total lepton number is violated. In this regard, neutrino-less double beta decay (0νββ) process ( X A Z → X A Z+2 + 2e − ) stands as a promising probe to establish the Majorana nature of neutrinos. 0νββ decay has not been observed so far and there are several ongoing and upcoming experiments that search for this signal. The best limit on the half life of 0νββ decay is T 1/2 > 1.07×10 26 years coming from the KamLAND-Zen experiment using 136 Xe [25]. This gives a bound on the effective Majorana mass (m ββ ) as, m ββ ≤ 0.061 − 0.165 eV.
The range corresponds to the uncertainty in nuclear matrix elements (NME). This process is suppressed by the proportionality of the transition amplitude to the effective Majorana mass m ββ , which in turn depends on the lowest neutrino mass, neutrino mass ordering, mixing angles and Majorana phases. However, the predictions for m ββ are known to change substantially in a 3+1 mixing scenario when an additional sterile neutrino is introduced [26][27][28][29][30][31][32][33][34][35].
It is also well known that in the presence of non-standard interactions (NSI), solar neutrino data admits a new solution for θ 12 > 45 • , known as the dark large mixing angle (DLMA) solution [36][37][38]. This is nearly a degenerate solution with ∆m 2 21 7.5 × 10 −5 eV 2 and sin 2 θ 12 0.7. The DLMA parameter space was shown to be severely constrained from neutrino-nucleus scattering data from COHERENT experiment [39]. However the bound depends on the mass of the light mediator [40].
In this context, the effect of the DLMA solution on 0νββ for the standard three generation picture has been studied recently in ref. [41] where it was shown that the prediction for m ββ remains unchanged for the inverted mass scheme whereas for normal hierarchy, it becomes higher for the Dark-LMA parameter space and shifts to the "desert region" between the two. This region can be tested in the next generation experiments.
In this work, we have studied the implications of the Dark-LMA solution to the solar neutrino problem for 0νββ in the presence of a fourth sterile neutrino as introduced to explain the LSND/MiniBooNE results (see references [21,42] for recent reviews on the status of eV scale sterile neutrinos.). In this case, m ββ depends on the third mass-squared difference ∆m 2 LSN D , the mixing angle θ 14 and an additional Majorana phase γ/2, in addition to the the two mass squared differences ∆m 2 21 and ∆m 2 31 , two mixing angles θ 12 (degenerate LMA or DLMA solutions) and θ 13 and the Majorana phases α/2 and β/2. Depending on the values of these parameters, there can be enhancement or cancellation of the 0νββ decay rate.
It has to be noted that the sum of masses of all the neutrino species is highly constrained from cosmology, which does not allow an eV scale sterile neutrino (see [42] for a recent review on the status of light sterile neutrinos and the cosmological bounds). To avoid the cosmological constraints, one can invoke "secret neutrino interactions" which can dynamically suppress the production of sterile neutrinos in the early universe by finite temperature effects [43]. One may also avoid the cosmological constraints by assuming a very low reheating temperature (∼ M eV ) after inflation [44][45][46].
The rest of the paper is organized as follows. In the next section, we discuss the DLMA solution and the MSW resonance condition in the presence of a fourth sterile neutrino. In section-III, we discuss the implications of the sterile neutrino and the DLMA solution for 0νββ process.
The discovery sensitivity of 0νββ process in the the new allowed parameter space is discussed in section-IV in the context of 136 Xe based experiments. Finally, we summarize our results in section-V.
(15) of reference [47]. The Majorana phase matrix comes into play while studying 0νββ process, but they are not relevant for oscillation studies. In Table I, we have given the 3σ ranges of the mixing angles and mass squared differences in the three generation [48] as well as four generation schemes [22]. Similar analysis can also be found in references [49,50] for three generation case and in [23] for the four generation case.
The neutral current Lagrangian for NSIs in matter is given by the effective dimension 6 four fermion operator as [51], where f is the charged fermion, P is the projection operator (left and right) and f P αβ are the parameters which govern the NSIs. The NSI affects the neutrino propagation in matter through vector coupling and we can write f αβ = f L αβ + f R αβ . The total matter potential including standard and non-standard interactions is governed by the  of neutrino oscillation data with three light active neutrinos [48] and one extra sterile neutrino [22]. Hamiltonian, Here, we have neglected non-standard interactions in the sterile sector 1 . Following the same approach as in [39] for three generation, we can now construct the Hamiltonian in an effective (2.4) 1 Studies including non-standard interactions of sterile neutrinos have been discussed in [52].
and we have taken θ 34 = 0. Now the new parameters f D , f N are related to the old parameters f αβ through the following equations : .
(2.6) k 1 and k 2 are defined as, Diagonalizing the above effective Hamiltonian gives the matter mixing angle θ M as, Hence, the resonance occurs when,  The half life for 0νββ in the standard scenario with light neutrino exchange is given by [54,55], where G is the phase space factor, M ν is the nuclear matrix element and m e is the electron mass.
The expression for the effective Majorana mass m ββ is given by, In this work, we denote the standard LMA solution as θ 12 and the DLMA solution as θ D12 . The 3σ ranges of these two parameters are shown in Table-III [39,48].  Another important point to be noted is that for the sterile neutrino scenario, there is no desert region between NH and IH unlike in the standard three generation picture [41]. This is true for both LMA and DLMA solutions. Also, the maximum allowed values of m ββ is higher in the case of the four generation picture and is almost independent of whether one take the standard LMA So the cancellation region corresponds to α ∼ π, γ ∼ π and s 2 14 ∼ 0.013. The cancellation is achieved due to large value of ∆m 2 LSN D . In three generation case, such cancellation is not there because of the absence of large valued term which can counter the first positive large term. In this region, the effective mass parameter is independent of the lightest neutrino mass eigenstate (Eqn. 3.7) and is bounded from above and below by, Here the cancellation occurs for α ∼ β ∼ γ ∼ π and s 2 14 ∼ 0.018. In three neutrino mixing the cancellation is not possible due to the absence of high m 4 . Eqn. 3.11 is maximum for α, β, γ = 0, 2π and is independent of θ 12 . Since the value of s 2 13 is small, the minimum value of m ββ is independent of β. Hence the minimum value of m ββ corresponds to α ∼ π and γ ∼ π. The minimum value of m ββ becomes In this section, we calculate the sensitivity in the future 136 Xe experiments for which we have adopted the method discussed in reference [60]. The value of T 1/2 for which an experiment has a 50% probability of measuring a 3σ signal above the background is defined as the 3σ T 1/2 discovery sensitivity. It is given as, . C 3σ stands for the number of counts for which the cumulative Poisson distribution with mean as B obeys, We use the normalized upper incomplete gamma function to define CDF P oisson as a continuous distribution in C as follows, This avoids the discrete variations that would arise in the discovery sensitivity if C 3σ is restricted to be integer valued. Using the above equations, we have calculated the T 1/2 discovery sensitivities of 0νββ as a function of for various values of β for 136 Xe nucleus and the results are shown in In this plot, the red shaded band corresponds to the new allowed region of m ββ ∼ 0.008 − 0.04 eV for the DLMA solution for the NH case with a sterile neutrino. This band in m ββ which is due to the variation of the parameters in the PMNS matrix, is converted to a band in T 1/2 using Eqn. 3.1, by taking into account the NME uncertainty as given in Table IV. The dotted black line corresponds to the future 3σ sensitivity of nEXO, which is T 1/2 = 5.7 × 10 27 years [59]. The yellow, black, brown and blue lines correspond to four different values of the sensitive background levels of 0, 10 −5 , 10 −4 and 10 −3 cts/(kg iso yr) respectively. From the figure, we can see that a large part of this newly allowed region for NH is in the reach of the nEXO experiment. With lower background levels and/or higher sensitive exposure, the next generation experiments can probe this entire region.
In Table IV [56,57] and the phase space factors [58] used in the calculation are also given.

V. SUMMARY
In this paper, we have studied the implications of the DLMA solution to the solar neutrino problem for 0νββ in the 3+1 scenario, including an additional sterile neutrino. We have verified that even in the presence of sterile neutrino, the MSW resonance can take place in the DLMA region. Next, we have studied how for these values of θ 12 , the predictions for 0νββ in 3+1 picture is changed as compared to the predictions for 3+1 scenario assuming ordinary LMA solution. We also compare with the predictions of m ββ for the three generation picture.
We find that for IH, there is no change in m ββ predictions as compared to the 3+1 case assuming θ 12 to be in the standard LMA region. This is because in this case, the maximum value of m ββ is independent of θ 12 and the minimum value of m ββ is of the order of ∼ 10 −4 eV where the difference is not very evident. In particular, the cancellation region which was reported earlier for 3+1 sterile neutrino picture also continues to be present for the DLMA parameter space due to the contribution from the fourth mass eigenstate. This conclusion is similar to the conclusion obtained for the three generation case, for which also the LMA and DLMA solutions gave same predictions for m ββ in the case of IH.
In the case of NH, cancellation can occur for certain values of m lightest and the values for which this happens is higher for the DLMA solution. Also, the maximum value of m ββ is same for the standard LMA and DLMA solutions in the 3+1 scenario and unlike the three generation case there is no desert region between NH and IH. However, the maximum value is higher than that for the three generation DLMA case.
If future experiments with sensitivity reach of ∼ 0.015 eV observe a positive signal for 0νββ then it could be due to IH (three generation or 3+1 generation) or due to NH (3+1 picture) for both LMA and DLMA solutions. If however, no such signal is found then for three generation picture 0νββ experiments can disfavor IH and one moves to the next frontier of 0.001 eV [61,62].
In this regime a demarcation between LMA and DLMA is possible for three generation picture if a signal is obtained for m ββ 0.004 eV [41]. However, if the sterile neutrino hypothesis is true then distinction between NH and IH is not possible from 0νββ experiments. This also spoils the sensitivity to demarcate between LMA and DLMA solutions. If however, the current indication of NH from accelerator experiments is confirmed by future data then the next generation of 0νββ experiments with sensitivity reach up to 10 −3 eV can distinguish between LMA and DLMA solution in presence of a sterile neutrino for m lightest 0.005 eV.