Non-standard neutrino interactions in U(1)′ model after COHERENT data

We explore the potential to prove light extra gauge Z′ boson inducing nonstandard neutrino interactions (NSIs) in the coherent-elastic neutrino-nucleus scattering (CEνNS) experiments. We intend to examine how the latest COHERENT-CsI and CENNS-10 data can constrain this model. A detailed investigation for the upcoming Ge, LAr-1t, and NaI detectors of COHERENT collaboration has also been made. Depending on numerous other constraints coming from oscillation experiments, muon (g −2), beam-dump experiments, LHCb, and reactor experiment CONUS, we explore the parameter space in Z′ boson mass vs coupling constant plane. Moreover, we study the predictions of two-zero textures that are allowed in the concerned model in light of the latest global-fit data.


Introduction
Oscillations of neutrinos among different flavors are now a well established phenomenon from various experimental searches, which implies that the neutrinos carry non-zero masses and their different flavor are substantially mixed [1]. Currently, we have fairly good understanding of all the neutrino oscillation parameters in three-flavor paradigm, except the Dirac CP violating phase [2][3][4]. This led us into an era of precision measurements in the leptonic sector, where it is possible to observe sub-leading effects originating from physics beyond the Standard Model (SM). Furthermore, this may affect the propagation of neutrinos and eventually it may impact the measurements of three-flavor neutrino oscillation parameters. Among various new physics scenarios beyond the standard three-flavor neutrino oscillations, non-standard neutrino interactions (NSIs) can be induced by the new physics beyond the SM (BSM). In literature, they are traditionally described by the dimension-6 four-fermion operators of the form [5], where f αβ represent NSI parameters and α, β = e, µ, τ , f = e, u, d. The importance of NSIs were discussed well before the establishment of neutrino oscillation phenomena by a number of authors in [5][6][7][8]. For a detailed model-independent review of NSIs and their phenomenological consequences see refs. [9][10][11] and the references therein.

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In past a few years there are many BSM models that have addressed NSIs. Some of the popular models where NSIs can be present are the flavor-sensitive, Z mediated U(1) extended gauge models [12][13][14][15][16]. 1 In these scenarios, an extra gauged U(1) is added to the SM gauge group, where the corresponding symmetry breaking leads to a new gauge boson Z . In literature, numerous studies have been performed based on U(1) model, ranging from flavor models to GUTs scenarios [20][21][22][23]. Dark matter phenomenology based on such symmetry has been addressed in [24][25][26][27]. Furthermore, to provide strong constraints between the mass and gauge coupling of associate gauge boson Z , a large variety of measurements have been performed, such as rare decays, anomalous magnetic moments of the electron or muon, electroweak precision tests, and direct searches at the LHC [28][29][30][31][32][33][34][35][36][37][38][39]. However, our main focus is to examine the importance of non-standard neutrino interactions within the gauge extended framework of the SM.
In this work, we investigate non-standard neutrino interactions arising from a new gauge boson Z associated with an extra U(1) = U(1) B−2Lα−L β symmetry, where α = β = e, µ, τ . Considering the combined effect of different experimental constraints coming from the COHERENT collaboration, oscillation data, beam-dump experiments, and the LHCb dark photon searches, we examine the allowed region in M Z vs the coupling constant g plane that can lead to possible NSIs. In addition, we also explore the potential of reactor based CEνNS experiment like the COherent NeUtrino Scattering experiment (CONUS) [43]. Bounds arising from other processes like anomalous magnetic moment of muon i.e., (g − 2) µ , from astrophysical observations such as Big Bang Nucleosynthesis (BBN) and Cosmic Microwave Background (CMB) have also been shown for the comparison. Furthermore, by introducing two different U(1) breaking scalar fields, it has been observed that the neutrino oscillation parameters are in well agreement with the current 1 Note that some recent studies of NSIs considering heavy charged singlet and/or doublet scalars have been performed in [17][18][19].
Each of these four cases have their own NSI structure. We explore the impact for each model considering current COHERENT [40] data and for the future CEνNS experiments. Other neutrino phenomenology, such as the predictions for the neutrino-less double beta (0νββ) decay and the prediction for the lightest neutrino mass have also been discussed.
The remainder of the paper is outlined as follows. In next section 2, we give a brief description of non-standard interactions (NSIs) and their latest bounds. The theoretical set-up of the model has been discussed in section 3. Section 4 is devoted to CEνNS processes as well as other experimental and numerical details. The principle results of the paper has also been discussed in this section. Later, in section 5 phenomenology of two-zero textures have been addressed. We summarize our findings in section 6. Appendix A has dealt with the anomaly cancellation of the U(1) symmetry and the light neutrino mass under type-I seesaw mechanism has been discussed in appendix B.

Non-standard neutrino interactions
Here we present a general description of the non-standard interactions involving neutrinos. We consider the effect of neutral-current NSI in presence of matter which is describe by the dimension-6 four-fermion operators of the form [5], where f C αβ are NSI parameters, α, β = e, µ, τ , C = L, R, denotes the chirality, f = e, u, d, and G F is the Fermi constant. 3 The Hamiltonian in presence of matter NSI, in the flavor basis, can be written as, where U is the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing matrix [77], ∆m 2 ij = m 2 i −m 2 j (i < j = 1, 2, 3), and A ≡ 2 √ 2G F N e E represents the potential due to the standard matter interactions of neutrinos and αβ can be written as where αβ = | αβ |e iφ αβ for α = β. In general, the elements of αβ are complex for α = β, whereas diagonal elements are real due to the Hermiticity of the Hamiltonian as given by Table 1. Recent constraints for the NSI parameters ε u αβ and ε d αβ , at 2σ C.L., obtained from the combined analysis of the oscillation experiments and COHERENT measurements [78]. eq. (2.2). For the matter NSI, αβ can be defined as, where N f is the number density of fermion f and f C αβ = f L αβ + f R αβ . In case of the Earth matter, one can assume that the number densities of electrons, protons, and neutrons are equal (i.e. N p N n = N e ), in such a case N u N d 3N e and one can write, In table 1, we give recent constraints for NSIs obtained from a combined analysis of oscillation experiments and COHERENT measurements [78] at 2σ C.L. Having introduced general descriptions of NSI and its bounds, in next section we describe our model in great details.

The setup
In this work, we extend the SM gauge group to an anomaly free U(1) = U(1) B−2Lα−L β , 4 where α and β can be e, µ and τ . In our framework, two different lepton flavors are coupled to the new gauge interaction. The relevant charge assignments for the lepton fields as well as the scalar fields that trigger the U(1) gauge symmetry breaking are listed in table 2.
In this prescription, we include two scalar fields φ 1 and φ 2 transforming as 1 and 2 under U(1) , respectively. It is worth to mention that this U(1) interaction is not flavor violating. These scalar fields are responsible for the U(1) breaking and therefore to give mass to the Z gauge boson. The scalar potential for the fields in our framework (see table 2) can be split in three parts, The first part is the SM Higgs potential, the second is the coupling of the Higgs doublet with the singlet fields, 2) and the third part is the potential for the two singlet fields, 3) Now once new scalar fields φ i attain their vev (v i / √ 2), we get mass for the Z gauge boson as 1 2 where we have used charges for the φ i as mentioned in table 2. Now to give an order of estimation about the breaking scale, we take mass of Z gauge boson M Z = 0.1 GeV, whereas coupling strength g is taken as ≈ 2.8×10 −5 . Note that we consider these numerical values in such a way that these can be probed in future COHERENT experiments (for a detail discussion see section 4 and figure 3). Using these numerical values in eq. (3.4), one finds the vevs of φ 1 and φ 2 as v 1 ≈ 3 TeV and v 2 ≈ 1 TeV, respectively. It is worth to mention that the Higgs vacuum stability can be obtained when coupled to singlet scalar field, whose vev is at the TeV scale [79]. The Yukawa Lagrangian that is invariant under SM ⊗ U(1) for charged-leptons and neutrinos can be written as where,H = iτ 2 H † . It is clear from eq. (3.5) that the charged lepton mass matrix as well as the Dirac neutrino mass matrix are diagonal.

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x e x µ x τ Neutrino mass matrix Type NSI parameters Table 3. Neutrino mass matrix textures depending on the choices of the charges x α . Notice that only four of these two zero textures are allowed by the latest neutrino oscillation data [2][3][4].
There are several anomaly-free solutions to the U(1) involving baryon numbers, for scenarios where CEνNS and NSI have been explored, see for instance [14][15][16]70]. In our approach, we choose the anomaly-free solution for the U(1) = U(1) B−2Lα−L β , see appendix A for details. In this case, let's take one of the solutions, namely (x e , x µ , x τ ) = (0, −1, −2) for instance, the right-handed (RH) neutrino Lagrangian is given by The six possible assignments under the U(1) charges for the leptons are given in table 3 and each of the charge assignments give rise to a different model, namely different neutrino masses and mixings as well as different NSI.
Having discussed our theoretical set-up, in the subsequent sections we aim to discuss phenomenological importance of the model. In what follows, we first examine the potential of CEνNS processes to explain NSIs as given in table 3. Later, predictions for neutrino oscillation parameters as well as the effective Majorana neutrino mass have been analyzed for the allowed two-zero textures as mentioned in table 3.

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Argon Germanium  Table 4. Proton rms radius (in fm) of the stable isotopes of sodium, iodine, cesium, argon and germanium [83]. Their percentage relative abundance is provided in parenthesis. For detectors made of argon and germanium, we use the average R p = i X i R i p , where X i and R i p stand for the relative abundance and proton rms radius of the i-th isotope, respectively.

Coherent elastic neutrino-nucleus scattering
Coherent elastic neutrino-nucleus scattering has already been measured by the COHER-ENT experiment [40], using a scintillator detector of made of CsI. The low energy neutrino beam was generated from the Spallation Neutron Source (SNS) at Oak Ridge National Laboratory.
The SM differential cross section for CEνNS process is given by [12,80,81] where T is the nuclear recoil energy, E ν is the incoming neutrino energy, and M N is the nuclear mass. Also, Q 2 w is the weak nuclear charge and is given by where Z(N ) is the proton (neutron) number, Q is the momentum transfer, F Z(N ) (Q 2 ) its nuclear form factor, and g V p = 1/2 − 2 sin 2 θ W , g V n = −1/2 are the SM weak couplings. It is important to notice that the cross section depends highly on the mass of the detector and the type of material, especially on the number of neutrons N , since the dependence on Z is almost negligible due to the smallness of g V p (∼ 0.02). For the SNS energy regime, another important feature of the CEνNS cross section are the nuclear form factors. From now onwards, we will adopt the Helm form factor [82], where equal values of proton and neutron rms radius have been used. In table 4 we present the corresponding values for different isotopes. For the analysis of the CsI detector, we will use the best-fit value of R n = 5.5 from ref. [52].
The differential recoil spectrum can be computed as Here, N A is the Avogadro's number, M det is the detector mass, M m is the molar mass of the material, and φ α (E ν ) is the neutrino flux for each flavor. The SNS neutrino flux consists of monochromatic ν µ coming from π + decays, along with delayed ν e andν µ from JHEP06(2020)045 the subsequent µ + decays. Each of these flux components are given by for neutrino energy E ν ≤ m µ /2 52.8 MeV. The normalization constant is η = rN POT /4πL 2 , where r = 0.08 is the fraction of neutrinos produced for each proton on target, N POT represents the total number of protons on target (∼ 2.1×10 23 POT) per year, and L is the distance from the detector. From eq. (4.3) we can compute the expected number of neutrinos per energy bin: where A(T ) is the acceptance function, taken from the COHERENT data released in [84].
In figure 1 we show the measured number of events from the COHERENT collaboration as a function of the nuclear recoil energy T, for the expected number of events in the SM framework.
In presence of NSI, the cross section for CEνNS is affected through the weak nuclear charge (see eq. (4.2)) in the following way: where α = (e, µ, τ ). Notice that with this new contribution, the differential cross section from eq. (4.1) is now flavor dependent. It is possible to write an effective low-energy Lagrangian for the neutrino-fermion interactions with the Z boson as

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where Q 2 is the transferred momentum. Therefore, by comparing this effective Lagrangian with the NSI Lagrangian in eq. (2.1), we can relate the NSI parameters with the Z interaction parameters as . (4.10) In figure 2, we plotted the number of events versus the nuclear recoil energy, for different detectors (Ge, LAr-1t and NaI) considering the future plans of the COHERENT collaboration. The features of the future detectors that are used in our numerical simulations, along with the current CsI detector are presented in table 5. We show the expected events in the SM framework, and compare with the case of NSI terms in the cross section. For this particular example, we considered the A 2 model with M Z = 0.1 GeV and g = 2 × 10 −5 . We give the details about the values of M Z and g that are considered here in figure 3. As expected, the number of events in presence of NSI increases with respect of those in the SM, but this increase is higher for smaller values of the nuclear recoil energy T .
Given the relation in eq. (4.10), it is now clear how the NSIs can be generated from the interactions of a new vector boson Z . By computing the number of events including NSI contributions, we are now able to compare with the COHERENT measurements in order to set boundaries to the coupling and mass of the Z boson.
As mentioned before, the first part of the analysis consists in comparing with the first measurements of CEνNS, provided by the COHERENT collaboration [40]. A CsI detector of 14.6 kg was used at a distance of 19.3 m from the source. The cross section for this type of detector has to be computed separately for cesium (Cs) and iodine (I) in the following way:  Table 5. Specifications of the current COHERENT-CsI [40] and CENNS-10 [42] detectors, along with the future setups using other type of detectors [85]. For CENNS-10, the efficiency function F (T ) is taken from figure 3 of ref. [42] for the analysis B. Since there is no information about the efficiencies of the future detectors (viz, LAr-1t, Ge, and NaI), we have assumed a conservative flat efficiency of 50%.
We perform a fit of the COHERENT-CsI data by means of a least-squares function i ss is the statistical uncertainty. Also, B i on and B i ss are the beam-on and steady-state backgrounds, respectively. We marginalize over the nuisance parameters α and β, which quantify to the signal and background normalization uncertainties σ α and σ β , respectively. Following the COHERENT-CsI analysis, we choose σ α = 0.28, which includes neutrino flux (10%), signal acceptance (5%), nuclear form factor (5%) and quenching factor (25%) uncertainties, and σ β = 0.25 [40]. Since the fit to the quenching factor was done for the bins from i = 4 to 15, we follow our analysis only for these energy bins.
In order to extract information about the Z boson, we compute the expected number of events N th including NSI effects, according to eq. (4.7) and the weak nuclear charge in eq. (4.8). It must be pointed out that in the NSI scenario, the differential cross section is now flavor dependent.
As we have mention before (see section 3 for details), the proposed model has six possibilities depending on the U(1) charges of the charged leptons. Since only four of these cases are allowed by oscillations data (A 1 , A 2 , B 3 and B 4 ), we will perform the χ 2 analysis only for these cases. Note that we will give a detailed phenomenological consequences of these four two-zero textures within the standard three-flavor neutrino oscillation paradigm in the next section.
Since all the quarks have same U(1) charge, we get ε uV αα = ε dV αα , reducing the number of free parameters. Also, the neutrino source does not produce tau neutrinos, and hence, we can not extract any information about ε τ τ .
It is to be noted that the COHERENT collaboration has reported the first measurement of CEνNS with argon by using the CENNS-10 detector, which corresponds to 13.7 × 10 22 POT. The CENNS-10 detector has an energy threshold of 20 keV, an active mass of 24 kg, and is located at 27.5 m from the SNS target. As described in ref. [42], the JHEP06(2020)045 Figure 3. Exclusion regions at 95% C.L. in the (M Z , g ) plane for the different models. The lightgreen shaded area corresponds to the constraint set by the current COHERENT measurement using a CsI detector [40], while the orange solid line comes from the recent COHERENT results using the CENNS-10 detector [42]. The solid purple line shows the limit from oscillation experiments [78]. The limits set by the future detectors setup from the COHERENT collaboration [85], namely, Ge, NaI, and LAr-1t are shown using the red dash-dotted, yellow dotted, and blue dashed lines, respectively. The limits from the CONUS reactor experiment [43] are shown by the magenta (long dashed) lines. The exclusion regions set by the beam dump experiments [86][87][88][89][90][91][92][93][94][95], BBN and CMB [96], and LHCb dark photon searches [97] are presented using color code yellow, gray and sky-blue regions, respectively. The pink shaded band corresponds to the region where the muon (g − 2) anomaly is explained [98] (see text for more details).

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collaboration has performed two independent analyses, labeled A and B. Analysis B yielded a total of 121 CEνNS events, 222 beam-related and 1112 steady-state background events.
To extract exclusion regions for the Z parameters, we perform a single-bin analysis, using a χ 2 function equivalent to eq. (4.12). Following the analysis B of ref. [42], we take σ α = 0.07 and σ β = 0.107. For the calculation of the number of events, we use the efficiency function provided in figure 3 from ref. [42].
In figure 3, we show the exclusion regions at 95% C.L. in the (M Z , g ) plane. Each panel corresponds to one of the four possible models, where the resulting light neutrino mass matrix is of type A 1 , A 2 , B 3 and B 4 . The constraints coming from the COHERENT data, using the CsI and CENNS-10 detectors, has been presented using the light-green shaded region and the orange solid line, respectively. In order to have a more complete study, we also include exclusion regions arising from the future upgrades of the COHERENT collaboration: Ge, NaI, and LAr-1t detectors, considering a 10% SM signal as background and an exposure of four years. For this analysis, we consider a decrease in the quenching factor uncertainty by a factor of two with respect to the CsI detector case (12.5%). This improvement leads to a signal nuisance parameter of σ α = 0.175, while the background parameter remains the same σ β = 0.25. We show the exclusion regions using the red dashdotted, yellow dotted, and blue dashed lines, respectively. We can see how these future setups can improve the current COHERENT limits for the coupling g by almost one order of magnitude.
Notice that the propagation of neutrinos in matter are affected by coherent forward scattering where one have zero momentum transfer. Hence, the effective Lagrangian from eq. (4.9) that is relevant for NSI can be written as irrespective of the Z mass. In this limit eq. (4.10) becomes . (4.14) In figure 3, we also include limits coming from oscillation experiments (see purple solid line) using the relation given in eq. (4.14). For models A 1 and A 2 , we take the smallest value of ε µµ from the first column of table 1, when setting ε ee = ε τ τ = 0. Then we use eq. (4.14) to get a limit for g as a function of M Z . For B 3 , we extract a value for ε ee by taking ε µµ = 0. 5 The limit from BBN +CMB [96] is also presented using gray band in figure 3. For the cases where ε ee = 0 (i.e., for B 3 and B 4 ), we have also included boundaries for a light Z boson, obtained by different electron beam dump experiments as shown by the yellow region. We have used the Darkcast [99] code to translate the beam dump limits to our specific model. In the cases where ε µµ is present, we also consider limits set by dark photon searches for LHCb limits [97] shown using the sky-blue region. We also use the

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Darkcast [99] code to translate these limits to the different cases of our model, which has been shown using the sky-blue regions in figure 3. The interaction of the Z boson with muons leads to an additional contribution to the anomalous magnetic moment: where (4.16) Since the existence of new light vector bosons can explain the inconsistency in the anomalous magnetic moment of the muon, (g − 2) µ , [22,100], we have incorporated boundaries arising from this process in figure 3. The region of the (M Z , g ) plane where our model can explain the discrepancy ∆a µ = (29 ± 9) × 10 −10 [98] is the pink region. Notice that only in the B 3 this region is absent, since there is no interaction between muons and the Z boson (ε µµ = 0).
Furthermore, there are several proposals aiming to measure CEνNS using nuclear reactors, such as CONNIE [101], CONUS [43], MINER [102], RED100 [103], TEXONO [104], etc. For example, the CONUS experiment will consist of a 4 kg Germanium detector with an energy threshold of 300 eV, located at 17 m from the nuclear power plant at Brokdorf, Germany [43]. They expect ∼ 10 5 events over a 5 year run, assuming the SM signal.
We also present limits for the Z boson considering the CONUS experiment. For the calculation of the number of CEνNS events, we have taken into account an antineutrino energy spectrum coming from the fission products 235 U, 238 U, 239 Pu and 241 Pu [105]. For energies below 2 MeV, we use the theoretical results obtained in ref. [106]. Since reactor antineutrinos are produced with energies of a few MeV, the nuclear form factors play no role in the detection of CEνNS events, therefore we safely take them to be equal to one.
For this analysis, we assume a flat detector efficiency of 50%, and the same χ 2 function given by eq. (4.12) with a background equal to 10% of the SM signal, where uncertainties σ α = 0.1 and σ β = 0.25 have been used. Since a nuclear reactor produces only electron antineutrinos, we give an exclusion regions only for the cases where ε ee = 0 (i.e. for B 3 and B 4 ). These regions are shown in the lower panels of figure 3, denoted with the magenta dashed line.
The first panel of figure 3, i.e., A 1 (U(1) B−Lµ−2Lτ ) has µµ and τ τ with τ τ > µµ . In this scenario, it can be seen that the future COHERENT experiment with LAr-1t detector will explore a parameter space for masses between 7 MeV to 3 GeV and couplings as small as g ∼ 10 −5 . For masses between 200 MeV and 4 GeV the future COHERENT bounds will be competitive with the current LHCb exclusion limits. However, we notice that above 3 GeV bounds coming from the LHCb drak-photon searches will give the strongest constraints, where g can be ∼ 10 −3 (see sky-blue region). Bounds arising from the calculation of ∆N eff of BBN will rule out M Z < 7 MeV as shown by the gray band. We now proceed to discuss our results for A 2 (U(1) B−2Lµ−Lτ ) as shown by the second panel of the first row of figure 3. It has µµ and τ τ as in A 1 but in this case τ τ < µµ . Here, we have found that the future JHEP06(2020)045 COHERENT experiment will explore a parameter space for masses between 7 MeV to 0.55 GeV and couplings up to g ∼ 10 −5 . For masses between 200 and 500 MeV the future COHERENT bounds will be comparable as exclusion coming from LHCb. Unlike A 1 , LHCb can explore more parameter space for this scenario, i.e., M Z ≥ 0.55 GeV, compared to COHERENT-LAr-1t bounds. ∆N eff also shows similar bounds as A 1 .
Unlike the scenarios A 1 and A 2 , we also have contributions coming from the beam dump experiments and reactor experiment CONUS that is because of non-zero ee for B 3 and B 4 , which we show at the second row of figure 3, respectively. The model B 3 (U(1) B−Le−2Lτ ) predicts NSI parameters like ee and τ τ (see table 3 for details). It has been observed that CONUS shows the most stringent constraint, compared to the future COHERENT-LAr-1t bounds, for the masses greater than 25 MeV with the coupling constant g ∼ 5 × 10 −6 , as shown by the magenta dashed line. Moreover, the region M Z < 25 MeV and g < 5 × 10 −6 is ruled out by the beam dump bounds (see light-yellow region). In our final scenario, i.e., B 4 (U(1) B−Le−2Lµ ), the contribution from the LHCb is also observed because of nonzero µµ together with ee . We notice that for masses greater 25 MeV up to ∼ 500 MeV and couplings g in the range (5 × 10 −6 -0.5 × 10 −4 ), CONUS will show the strongest exclusion region, whereas masses ≥ 500 MeV will be explored by LHCb. On the other hand, predictions below M Z < 25 MeV remains same as B 3 .
It is worth to mention that the exclusion region coming from the recent results of the CENNS-10 detector is weaker than the future upgrade LAr-1t detector for two main reasons: the greater mass of the latter (∼ 25 times bigger) and the total exposure that has been considered in this work (4 years).
Finally, by investigating all the four scenarios, it has been seen that the bounds arising from (g − 2) µ (see the pink band) is ruled out by the current COHERENT-CsI data, while limits from oscillation experiments (as shown by the solid purple line) will be ruled out by the future COHERENT data. Finally, we present a set of benchmark values that can be explored by different experiments in the table 6.
So far we have discussed the importance of CEνNS processes to investigate NSIs for all the possible allowed cases for the given U(1) charges as given by table 3. Our next section is devoted to the predictions for the standard three flavor neutrino oscillation parameters as well as for the effective Majorana neutrino mass within the formalism of two-zero textures that are appeared in this gauge extended model (see table 3 for allowed possibilities).

Two-zero textures
Here we revisit the phenomenology of the two-zero textures that are allowed in this model, as given in table 3, viz A 1 , A 2 , B 3 , and B 4 in light of the latest global-fit data. The two-zero textures that were classified in [107] are phenomenologically very appealing in the sense that they guarantee the calculability of the neutrino mass matrix M ν from which both the neutrino mass spectrum and the flavor mixing pattern can be determined [108][109][110][111]. In what follows, we first parameterize M ν in terms of the three neutrino mass eigenvalues (m 1 , m 2 , m 3 ) and the three neutrino mixing angles (θ 12 , θ 23 , θ 13 ) together with the three CP violating phases (δ, α, β). Note that here δ is the Dirac type CP-phase, whereas α, and JHEP06(2020)045  β are the Majorana type CP-phases. Therefore, the mass matrix M ν can be diagonalized by a complex unitary matrix U as In the standard PDG formalism, the neutrino mixing matrix U , also known as the PMNS matrix is given by where s ij = sin θ ij and c ij = cos θ ij . Given the parameterization of U , it is now straight forward to write down the elements of neutrino mass matrix M ν with the help of eq. (5.1).

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The two-zero textures of the neutrino mass matrix M ν (see eq. (5.1)) satisfies two complex equations as m ab = 0, m pq = 0 , (5.3) where a, b, p and q can take values e, µ and τ . Above equations can also be written as where V has been defined in eq. (5.2). We notice that these two equations involve nine physical parameters m 1 , m 2 , m 3 , θ 12 , θ 23 , θ 13 and CP-violating phases α, β, and δ. The three mixing angles (θ 13 , θ 12 , θ 23 ) and two mass-squared differences (∆m 2 12 , ∆m 2 23 ) are known from the neutrino oscillation data. Note here that from the latest global-fit results, we have some predictions about the CP-violating phase δ, however at 3σ, full range i.e., 0 • -360 • is still allowed. Therefore, in this study we kept δ as a free parameter. The masses m 2 and m 3 can be calculated from the known mass-squared differences ∆m 2 12 and ∆m 2 23 using the relations m 2 = m 2 1 + ∆m 2 12 , and m 3 = m 2 2 + ∆m 2 23 . Thus, we have two complex equations relating four unknown parameters viz. m 1 , α, β and δ. Therefore, one can have the predictability of all these four parameters within the formalism of two-zero textures.
We numerically solve eq. (5.4) for the concerned types of two-zero textures, see table 3. It has been known from the latest global analysis of neutrino oscillation results [2][3][4] that the least unknown parameter among the three mixing angles is the atmospheric mixing angle θ 23 . Therefore, considering some benchmark values of θ 23 , we calculate remaining unknown parameters, which we present in table 7 and 8. We take the latest best-fit value of θ 23 from [4] as one of our benchmark value, whereas maximal value of θ 23 i.e., θ 23 = 45 • is taken as the second benchmark value. Notice that the seed point θ 23 = 45 • has the great importance in perspective of flavor symmetries as well as flavor models building. Among numerous theoretical frameworks, µ − τ symmetry that explains θ 23 = 45 • has received great attention in the neutrino community, for the latest review see ref. [112]. From table 7, we notice that the textures A 1 , A 2 can explain both the latest best-fit as well as the maximal value of θ 23 . Further, given these benchmark values we calculate unknown parameters m 1 , α, β and δ. It is to be noted from the fourth column that the predicted values of δ for all the cases lies within 1σ of the latest best-fit value [4], which is 237.6 +37.8 • −27.0 • . We also calculate m 2 , m 3 (see second column of the table 7) to find m ν . From the third column, one can find that the measured values of m ν for all the cases are well within the latest value provided by Planck collaboration [113] which gives m ν < 0.12 eV (95%, Planck TT, TE, EE + lowE + lensing + BAO). Notice that recently, the T2K collaboration [114] has published their latest results, which gives the best-fit values of the atmospheric mixing angle sin 2 θ 23 = 0.53 +0.03 −0.04 and the Dirac CP-violating phase δ = −1.89 +0.70 −0.58 (or 252 +39.6 −32.4 in degree) for the normal neutrino mass hierarchy. We find for the textures A 1 and A 2 (see table 7) are in well agreement with the latest T2K measurements within the 1σ confidence level [114].   Currently, number of experiments that are dedicated to look for the signature of 0νββdecay are namely, GERDA Phase II [115], CUORE [116], SuperNEMO [117], KamLAND-Zen [118] and EXO [119]. It is to be noted here that, this process violate lepton number by two-units and the half-life of such decay process can be read as [120,121],

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where G 0ν is the two-body phase-space factor, and M 0ν represents the nuclear matrix element (NME). |m ee | is the effective Majorana neutrino mass and is given by, where U stands for PMNS mixing matrix as mentioned in eq. (5.2). We present our predictions for the effective Majorana neutrino mass |m ee | for both the textures in figure 4. The 3σ allowed parameter space of |m ee | considering the latest global-fit data [4] for the normal neutrino mass hierarchy is shown by the light-orange band. 6 The magenta band shows the latest bounds on |m ee |, arises from the KamLAND-Zen 400 experiment [118] which is read as |m ee | < (61-165) meV at 90% C.L. by taking into account the uncertainty in the estimation of the nuclear matrix elements. We also show the first results of KamLAND-Zen 800 collaboration using the lighter-green band, which was presented in the latest meeting TAUP 2019 [122]. Besides this, the predictions for |m ee | for the textures B 3 and B 4 are shown by the blue (cyan) patch at 3σ (1σ) significance level. We notice from both the panel of figure 4 that the calculated values of |m ee | lie in 6 Note that the present oscillation data tends to favor normal mass hierarchy (i.e., ∆m 2 31 > 0) over inverted mass hierarchy (i.e., ∆m 2 31 < 0) at more than 3σ [2][3][4], therefor, we focus only on the first scenario.
|m e e |

[eV]
KamLAND-Zen 400 KamLAND-Zen 800 Normal B 4 Figure 4. Predictions for the effective Majorana neutrino mass |m ee | vs the lightest neutrino mass m 1 . The 3σ allowed parameter space of |m ee | using the latest global-fit data is shown by the light-orange band [4]. The bound on |m ee | from the KamLAND-Zen 400 [118] collaboration has been shown by the light-magenta horizontal band, whereas the first results of the KamLAND-Zen 800 [122] collaboration is outlined by the lighter-green band. Predictions for |m ee | for B 3 , and B 4 are shown by the blue (cyan) patch at 3σ (1σ).   It is to be noted here that the latest bound on the sum of neutrino masses m ν come from Planck collaboration [113] which gives m ν < 0.12 eV (95%, Planck TT, TE, EE + lowE + lensing + BAO). Now, given the constrained bound on m ν , if one converts them for the lightest neutrino mass m 1 , then it can be seen that the textures B 3 is almost rule out. On the other hand, the textures B 4 is consistent with the latest data. We further examine that none of these textures are able to explain the latest best-fit value of θ 23 . However, both these types are consistent with the maximal value of the mixing angle θ 23 . Considering θ max 23 as a seed point, we calculate remaining unknown in table 8. From the fifth column, one can notice that these textures predict maximal value for the Dirac type CP-phase δ, which is in well agreement with the latest best-fit value within 1σ range [4]. Also, CP-conserving values are predicted for the Majorana type CP-phases α, β. We show the predictions for the sum of neutrino masses m ν and the effective Majorana neutrino mass |m ee | for texture types B 3 , and B 4 in third and fourth column, respectively.

JHEP06(2020)045 6 Conclusion
Physics beyond the Standard Model (BSM), incorporating neutrino masses, are testable in the next generation superbeam neutrino oscillations as well as CEνNS experiments. This work is dedicated to investigating non-standard neutrino interactions (NSIs), a possible sub-leading effects originating from the physics beyond the SM, and eventually can interfere in the measurements of neutrino oscillation parameters. There exists numbers of BSM scenarios give rise to NSIs that can be tested in the oscillation experiments. However, such models undergo numerous constrained arising from the different particle physics experiments. In this work, we focus on an anomaly free U(1) gauge symmetry where a new gauge boson, Z , exchanged has been occurred. Depending on U(1) charge assignments, we find four different scenarios compatible with the current neutrino oscillation data, namely, U(1) B−Lµ−2Lτ , U(1) B−2Lµ−Lτ , U(1) B−Le−2Lτ , and U(1) B−Le−2Lµ . It has been further realized that these four scenarios correspond to four different two-zero textures for the neutrino mass matrix, namely, A 1 , A 2 , B 3 and B 4 . We notice that the NSI parameter ee is obtained under B 3 and B 4 textures, A 1 , A 2 , and B 4 lead to µµ , whereas one finds τ τ from A 1 , A 2 , and B 3 . We summarize our results for possible NSIs considering various experimental limits in figure 3, whereas other neutrino phenomenology are given in figure 4 and in table 7, 8, respectively. Depending on our analysis, we make our final remarks as follows: • Texture A 1 : in this case, we notice that the future COHERENT experiments with NaI or LAr-1t detectors will explore a parameter space for masses 7 MeV ≤ M Z ≤ 3 GeV within the coupling limits 0.8 × 10 −5 ≤ g ≤ 10 −3 . Also, the parameter space below 5.3 MeV can be ruled out using the measurement of ∆N eff coming from the observation of Big Bang nucleosynthesis. Notice here that this observation holds true for remaining cases. Furthermore, it can be seen that above 3 GeV the LHCb can put the strongest bound. Also, in this scenario, the effective mass parameter m ee of the 0νββ-decay is zero.
• Texture A 2 : findings of A 2 is similar as A 1 . However, we notice that the future COHERENT experiments will show the tightest constraint upto the mass limit ∼ 550 MeV and above this the LHCb will give the stringent bound. It is to be noted here that the LHCb can exclude more parameter space for A 2 compared to A 1 , which is simply because µ− field carry 2-units of U(1) charge than of A 1 (in case of A 1 , U(1) charge of µ− field is 1).
• Texture B 3 : outputs of B 3 is very different compared to A 1 and A 2 . Here we notice the CEνNS experiment CONUS can explore the most of the parameter space for the masses of M Z above ∼ 25 MeV and coupling constant g ≥ 5 × 10 −6 . On the other hand, below 25 MeV, the parameter space has been ruled out by the beam dump experiments.
Moreover, one also have predictions for 0νββ-decay which can be explored by the KamLAND-Zen collaboration (see left panel of figure 4).

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Above this mass limit and coupling strength the LHCb can put the tightest constraint. Moreover, the beam dump experiments can exclude the parameter space below 25 MeV. We also have predictions for the 0νββ-decay and the parameter space are marginally consistent with the present limit of both the KamLAND-Zen and the Planck bound as given in the right panel of figure 4.
Finally, we like to emphasize that the U(1) charges that lead to the scenarios A 1 and A 2 , as given in table 3, the LHCb provides the tightest constraint than the CEνNS experiments above 0.55, 3 GeV, respectively. Moreover, it is noteworthy to notice that the predictions of Dirac CP phase δ for A 1 and A 2 (see table 7) are in well agreement with the latest T2K result within the 1σ confidence level [114]. On the other hand, the CEνNS experiment CONUS puts the most stringent limit on

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As it can be seen, conditions in eq. (A.1c) and (A.1e) are already equal to zero. By imposing all the other conditions equal to zero, the U(1) charges of the right-handed neutrinos have to fulfill the following relations By looking at table 2, we can notice that these relations hold, since the charges of the righthanded neutrinos are the same as for the charged leptons, and − x = 0 + 1 + 2 = 3.

B Neutrino mass matrix
In this section we will show an example of how to compute the light neutrino mass matrix, for a specific choice of U(1) charges. Within the type-I seesaw scenario [124], the low energy neutrino mass matrix is given by where M D and M R are the Dirac and Majorana neutrino mass matrices, respectively. In our prescription, the Yukawa Lagrangian invariant under SM ⊗U(1) for the chargedleptons and neutrinos is given by − L Y ⊃ y e L e e H + y µ L µ µ H + y τ L τ τ H + y ν 1 L eH N 1 + y ν 2 L µH N 2 + y ν 3 L τH N 3 . For the Majorana neutrino mass matrix, we need to specify the U(1) fermion charges. For example, with the choice (x e , x µ , x τ ) = (0, −1, −2), the RH neutrino Lagrangian is which corresponds to the type A 1 neutrino mass matrix. One can follow the same procedure for the other charge assignments to get the different light neutrino mass matrices (see table 3 for details).

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Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.