Impact of lepton flavour universality violation on CP violation sensitivity of long baseline neutrino oscillation experiments

The observation of neutrino oscillation as well as the recent experimental result on lepton flavor universality (LFU) violation in $B$ meson decays are indications of new physics beyond the Standard Model. Many theoretical models, which are introduced in the literature as an extension of SM to explain these observed deviations in LFU, lead to new kind of interactions so-called non-standard interaction (NSI) between the elementary particles. In this paper, we consider a model with an additional $Z'$ boson (which is quite successful in explaining the observed LFU anomalies) and analyze its effect in the lepton flavour violating (LFV) $B_d\to \tau^\pm e^\mp$ decay modes. From the present upper bound of the $B_d\to \tau^\pm e^\mp$ branching ratio, we obtain the constraints on the new physics parameters, which are related to the corresponding NSI parameters in the neutrino sector by $SU(2)_L$ symmetry. These new parameters are expected to have potential implications in the neutrino oscillation studies and in this work we investigate the possibility of observing the effects of these interactions in the currently running and upcoming long-baseline experiments, i.e., NOvA and DUNE respectively.


I. INTRODUCTION
The Standard Model of particle physics, which seems to provide a complete picture of interaction and dynamics of elementary particles with the discovery of Higgs boson at LHC [1], predicts the equality of electroweak couplings of electron and muons so-called Lepton Flavor Universality (LFU). However, the observation of neutrino oscillation, which allows mixing between different lepton families of neutrinos, implies that family lepton number is from their corresponding SM values R(D * )| SM = 0.252 ± 0.003 [18] and R(D)| SM = 0.300 ± 0.008 [4]. Since these decays are mediated at tree level in the SM, relatively large new physics contributions are necessary to explain these deviations.
• Observation of 2.6σ deviation of µ/e universality in the dilepton invariant mass bin 1 GeV 2 q 2 6 GeV 2 in b → s transitions [5]: from the SM prediction R SM K = 1.0003 ± 0.0001. • CMS recently also searched for the decay h → τ µ and found a non-zero result of Br(h → τ µ) = 0.84 +0. 39 −0.37 [6] which disagrees by about 2.4σ from 0, i.e. from the SM value. These deviations from the SM have triggered a series of theoretical speculations about possible existence of NP beyond the SM. Some of the prominent NP models which can explain these deviations from the SM are: models with an extra Z boson [7] and/or additional Higgs doublets [8], models with leptoquarks [9] etc. The observation of lepton flavour nonuniversality effects also provide the possibility of the observation of lepton flavour violating (LFV) decays [10]. Although so far, there is no concrete evidence of LFV decays but there exist strict upper bounds in many LFV decays such as µ → eγ µ → eee, etc [11]. Various dedicated experiments are already planned to search for LFV decays. In this paper, we would like to see the implications of the LFV interactions in various long-baseline neutrino oscillation experiments. In other words, we would like to explore whether it is possible to observe these effects in the long-baseline neutrino oscillation experiments or not. In particular, we will focus on the NP contributions which could affect only to the τ sector. This is particularly interesting as the tauonic B decays provide an excellent probe of new physics because of the involvement of heavy τ lepton. There are a few deviations observed in the leptonic/semileptonic B decays with a τ in the final state. We consider the model with an additional Z boson, which can mediate flavour changing neutral current (FCNC) transitions at tree level. Z gauge bosons, which are associated with as extra U (1) gauge symmetry, are predicted theoretically in many extensions of the SM [12], such as grand unified theories (GUTs), left-right symmetric models, E 6 model, supersymmetric models, superstring theories etc. Although the U (1) charges are in general family-universal but it is not mandatory to be so, and the family non-universal Z has been introduced in some models, such as in E 6 model [13]. On the experiment side also there are many efforts undergoing to search for the Z directly at the LEP, Tevatron, and LHC. With the assumption that the coupling of Z to the SM fermions are similar to those of the SM Z boson, the direct searches for the Z can be performed in the dilepton events. At this stage, the lower mass limit has been set as 2.9 TeV at the 95% C.L. with 8 TeV data set by using e + e − and µ + µ − [14] events and this value becomes 1.9 TeV using the τ + τ − events [15]. However, such constraints from the LHC would not be valid if the Z boson couples very weakly with the leptons, and thus one has to rely on the hadronic channels.
The paper is organized as follows. In section II, we discuss the possible hints of new physics from B meson decays and extract the constraints on the lepton flavor violating new NP parameters in the charged lepton sector from the from the decay mode B d → τ ± e ∓ . These parameters are in general related to the corresponding NP parameters in the neutrino sector by the SU (2) L gauge symmetry. The basic formalism of neutrino oscillation including NSI effects are briefly discussed in section III. In section IV, we study the effect of NSI parameters on ν e appearance oscillation probability and the search for the new CP violating signals at long-baseline experiments is presented in section V. Section VI contains the summary and conclusions.

II. NEW PHYSICS EFFECTS FROM B MESON DECAYS
In this section, we would like to see the possible interplay of new physics in the τ -lepton sector considering the decay channels of B meson. For this purpose, we first consider the leptonic decay channel B − → τ −ν . During the last few years, there has been a systematic disagreement between the experimental and SM predicted value for the branching ratio of B → τ ν mode. The branching ratio for B − → τ ν τ is given as This mode is very clean and the only non-perturbative quantity involved in the expression for branching ratio (3) is the decay constant of B meson. However, there is still a tension between the exclusive and inclusive value of V ub at the level of 3σ. This mode has been precisely measured [11] with a value The latest result from Belle Collaboration [16] Br also in the line of the previous measurements. Since there is an uncertainty between the |V ub | values extracted from exclusive and inclusive modes, we use the SM fitted value of its branching ratio from UTfit collaboration [17] Br(B − → τ −ν τ ) = (0.84 ± 0.07) × 10 −4 .
This value agrees well with the experimental value (4). However, the central values of these two results differ significantly. One can eliminate the V ub dependence completely by introducing the LFU probing ratio which has around 2.6σ deviation from its SM prediction of R π,SM τ /l = 0.31(6) [18]. Thus, these deviations may be considered as the smoking gun signal of new physics associated with the tauonic sector. We then proceed to obtain the bound on the lepton flavor violating new physics parameter associated with the τ lepton from the decay mode B d → τ ± e ∓ . A.
Extraction of the NP parameter from the lepton flavour violating decay pro- The violation of lepton flavour universality in principle can induce lepton flavour violation.
In this section, we will consider the lepton flavour violating decay process B d → τ ± e ∓ , which is induced by flavour changing neutral current interactions. As an example, here we will consider a simple and well-motivated model, which would induce lepton flavour violating interactions at the tree level, is the model with an additional Z boson. Many SM extensions often involve the presence of an extra U (1) gauge symmetry and the corresponding gauge boson is generally known as the Z boson. Here we consider the model which can induce the lepton flavour violating decays both in the down quark sector and the charged lepton sector [7,19] at the tree level. Thus, in this model the coupling of Z boson to down type quarks and charged leptons can be written generically as where g is the new U (1) gauge coupling constant, η L/R db are the vector/axial vector FCNC couplings ofdb quark-antiquark pair to the Z boson and η L,R eτ are the LFV parameters. The constraint on the LFV coupling η eτ can be obtained from the lepton flavour violating In the SM this decay mode is loop-suppressed with tiny neutrino mass in the loop. However, in the Z model it can occur at tree level, described by the quark level transition b → dτ ± e ∓ and is expected to have significantly large branching ratio.
The Feynman diagram for this process in the Z model is shown in Fig. 1, where the blobs represent the tree level FCNC coupling of Z boson. The present upper limit on its branching ratio is 2.8 × 10 −5 . The effective Hamiltonian describing this process in the Z model can be given as where M Z is the mass of Z boson. In order to evaluate the transition amplitude we use the following matrix element where f B is the decay constant of B meson and p B its momentum. Thus, with eqns. (9) and (10), one can obtain the transition amplitude for the process and the corresponding branching ratio is given as where τ B is the lifetime of B meson. In order to find out the bound on the LFV couplings η L,R eτ , we need to know the value of the parameter η db , which can be obtained from the decay process B d → µ + µ − . The branching ratio for this decay mode has been recently measured by the LHCb [20] and CMS [21] collaborations and the present world average value [22] is given as The corresponding SM value has been precisely calculated including the corrections of O(α) and O(α 2 s ) with value [23] Br Although the SM predicted value is in agreement with the experimental result but it does not exclude the possible existence of new physics as the central values of these two results differ significantly. The effective Hamiltonian describing this process is given as where C 10 is the Wilson coefficient and its value at the m b scale is given as C 10 = −4.245.
The corresponding Hamiltonian in the Z model is given as where C µ V and C µ A are the vector and axial-vector couplings of the Z boson to µ − µ + pair. Including the contribution arising from the Z exchange to the SM amplitude, one can write the amplitude for B d → µµ process as Thus, from Eq. (17), one can obtain the branching ratio as Assuming the axial-vector coupling of Z to muon pair, i.e., C µ A has the same form as the the corresponding SM Z boson coupling to fermion-antifermion pair with value C µ A = −1/2. Now with Eqn. (18) and considering 1-σ range of experimental and SM predicted branching ratios from (14) and (13), the constraint on the parameter η R db is found to be for M Z =1 TeV, where we have used the particle masses and CKM elements from [11].
Using this allowed range of |η R db |, the bounds on the LFV couplings η L,R eτ can be obtained by comparing (12) with the corresponding branching ratio Br(B d → τ e) < 2.8 × 10 −5 [11] as where we have considered η L eτ = η R eτ . These couplings can be redefined in terms of another set of new couplings as ε eτ = (g 2 M 2 Z /g 2 M 2 Z )η eτ , which can give the relative NP strength in comparison to SM ones as for g g and a TeV scale Z boson, i.e., M Z 1 TeV. Since these parameters are related to the corresponding NSI parameters of the neutrino sector by the SU (2) L symmetry, we now proceed to see their implications in various long baseline neutrino oscillation experiments.
Analogously, one can obtain the bounds on the NSI couplings ε eµ from B d → eµ decay, which are expected to be of the same order as ε eτ .
Moreover, the three flavor framework of neutrino oscillation is very successful in explaining NSIs and their effect on neutrino oscillation. The Lagrangian corresponds to NSIs during the propagation of neutrino is given by [36], where G F is the Fermi coupling constant, ε f C αβ are the new coupling constants known as NSI parameters, f is fermion and P C = (1 ± γ 5 )/2 are the right (C = R) and left (C = L) chiral projection operators. The NSI contributions which are relevant while neutrino propagate through the earth are those coming from the interaction of neutrinos with matter (e, u and d), since the earth matter is made up of these fermions only. Therefore, the effective NSI parameter is given by where ε f αβ = ε f L αβ + ε f R αβ , n f is the number density of the fermion f and n e the number density of electrons in earth. For earth matter, we can assume that the number densities of electrons, protons and neutrons are equal, i.e, n n ≈ n p = n e . Therefore, one can write ε αβ as [37] Thus, with Eqns. (21) and (24), the bound on the NSI parameter ε eτ is found to be where we have assumed that either left-handed or right-handed couplings would be present at a given time.
NSIs and their consequences can be studied in both model-dependent and -independent approaches by which one can obtain the model-dependent and -independent bounds on the NSI parameters. Recently, considering the model independent approach, we have studied the effect of lepton flavor violating NSIs on physics potential of long-baseline experiments [38].
Moreover, the recent works on the effect of NSI on the measurements of various neutrino oscillation experiments can be seen in [39][40][41][42][43][44][45][46][47]. Since, we focus on model-dependent approach in this paper, we consider the LVF decays of B meson in Z model to get the bound on NSI parameter as discussed in Section II A. There are many works in the literature, which are dealt with extensive study of model-dependent NSI parameters and their effect on neutrino oscillation experiments [48,49]. However, in this work we focus on the lepton flavor violating

A. Basic formalism with NSIs
The effective Hamiltonian describing the propagation of neutrinos through matter in the standard three flavor framework is given by where H 0 is the Hamiltonian in vacuum, ∆m 2 ji = m 2 j −m 2 i is neutrino mass squared difference, H M is the Hamiltonian responsible for matter effect, V CC = √ 2G F n e is the matter potential and U is the PMNS mixing matrix which is described by three mixing angles (θ 12 , θ 13 , θ 23 ) and one CP violating phase (δ CP ) is given by where c ij = cos(θ ij ) and s ij = sin(θ ij ). The NSI Hamiltonian, which describes the new interactions between the matter particles as neutrinos propagate through matter is given by where ε αβ = |ε αβ |e iδ αβ are the complex NSI parameters. Then the neutrino oscillation probability in presence of NSI is given by In this paper, we focus on lepton flavor violating NSIs, i.e., the effects of the off-diagonal elements of the matrix (28). Moreover, constraints from terrestrial experiments show that the muon sector is strongly constrained [50], so that one can set ε eµ and ε µτ to zero. Therefore, in our analysis we consider only the contributions from the NSI parameter ε eτ and use a conservative value for ε eτ as ε eτ ≈ 0.3, consistent with the bound obtained from lepton flavour violating B meson decays, as shown in Eqn. (25).

IV. NUMERICAL ANALYSIS
A. Effect of NSI on oscillation probability and event spectra In this section, we discuss the effect of NSI parameter on the neutrino oscillation probability as well as on the event spectra of long baseline experiments like NOνA and DUNE. We use GLoBES package [51,52] for our analysis. We also use snu plugin [53,54]  Baseline length(km) 810 1300 Running time (yrs) 6 (3ν+3ν) 10 (5ν+5ν) we consider in this paper are given in the Table I and the true value of oscillation parameters that we use in our calculations are given in Table II  on oscillation probability, we obtain ∆P = |P N SI − P SI | (where P N SI(SI) denotes the probability with Non-standard (Standard) interactions) for different baseline length and energy using the neutrino oscillation parameters as given in Table II  can see that NSI can significantly affect CPV sensitivity of both experiments. Though there is significant enhancement in the CPV sensitivity in presence of NSI for NOνA, it should be noted that the δ CP coverage for CPV sensitivity above 1σ is reduced in presence of NSI while comparing with that of SO. Whereas for DUNE, the CPV sensitivity is enhanced in the presence of NSI and it is above 5σ for more than 50% allowed values of δ CP in the case of both NH and IH. corresponding NSI parameters in the neutrino sector by SU (2) L symmetry, we have studied the possible implications of these new physics interactions in the long-baseline neutrino oscillation experiments. In our analysis considering a conservative representative value for ε eτ as ε eτ = 0.3 and we have investigated its implications in the CP-violation sensitivity of long-baseline experiments. We found that the NSI parameters in the eτ sector remarkably affect the ν e appearance oscillation probability. Moreover, we found that the presence of NSIs lead to misinterpretation of oscillation data. The δ CP coverage of NOνA for CPV sensitivity above 1σ is reduced in presence NSIs. However, the CPV sensitivity is enhanced in the presence of NSI and it is above 5σ for more than 50% allowed values of δ CP in the case of both NH and IH for DUNE.