# Impact of lepton flavor universality violation on CP-violation sensitivity of long-baseline neutrino oscillation experiments

## Abstract

The observation of neutrino oscillation as well as the recent experimental results 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 a new kind of interactions, the 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 flavor violating (LFV) \(B_d\rightarrow \tau ^\pm e^\mp \) decay modes. From the present upper bound of the \(B_d\rightarrow \tau ^\pm e^\mp \) 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., NO\(\nu \)A and DUNE, respectively.

## 1 Introduction

*B*decays, both in the case of \(b \rightarrow c\) charged-current as well as in the case of \(b \rightarrow s\) neutral current transitions, also point toward physics beyond the SM. These results can be summarized as follows:

- About \(4.0\sigma \) deviation of \(\tau /l\) universality (\(l = \mu , e)\) in \(b \rightarrow c\) transitions [3], i.e.,from their corresponding SM values \(R(D^*)|_\mathrm{SM}=0.252 \pm 0.003\) [4] and \(R(D)|_\mathrm{SM}=0.300 \pm 0.008\) [5]. Since these decays are mediated at tree level in the SM, relatively large new physics contributions are necessary to explain these deviations.$$\begin{aligned} R ({D^*})= & {} \frac{\mathrm{Br} (B \rightarrow D^* \tau \nu _\tau )}{\mathrm{Br} (B \rightarrow D^* l \nu _l)}=0.316 \pm 0.016 \pm 0.010, \nonumber \\ R({D})= & {} \frac{\mathrm{Br} (B \rightarrow D \tau \nu _\tau )}{\mathrm{Br} (B \rightarrow D l \nu _l)}=0.397 \pm 0.040 \pm 0.028,\nonumber \\ \end{aligned}$$(1)
- Observation of \(2.6\sigma \) deviation of \(\mu /e\) universality in the dilepton invariant mass bin \(1~\mathrm{GeV}^2 \leqslant q^2 \leqslant 6~ \mathrm{GeV}^2\) in \(b \rightarrow s\) transitions [6]:from the SM prediction \(R_K^\mathrm{SM}=1.0003 \pm 0.0001\).$$\begin{aligned} R_K = \frac{\mathrm{Br}(B \rightarrow K \mu ^+ \mu ^-)}{\mathrm{Br}(B \rightarrow K e^+ e^-)} = 0.745_{-0.074}^{+0.090} \pm 0.036, \end{aligned}$$(2)
CMS recently also searched for the decay \(h \rightarrow \tau \mu \) and found a non-zero result of \(\mathrm{Br} (h \rightarrow \tau \mu ) = 0.84^{+0.39}_{-0.37}\) [7], which disagrees by about \(2.4 \sigma \) from 0, i.e. from the SM value.

*B*decays provide an excellent probe of new physics, because of the involvement of heavy \(\tau \) lepton. There are a few deviations observed in the leptonic/semileptonic

*B*decays with a \(\tau \) in the final state. We consider the model with an additional \(Z'\) boson, which can mediate flavor changing neutral current (FCNC) transitions at tree level. \(Z'\) gauge bosons, which are associated with an extra \(U(1)'\) gauge symmetry, are predicted theoretically in many extensions of the SM [17, 18, 19, 20, 21, 22, 23, 24], 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, it is not mandatory to be so, and the family non-universal \(Z'\) has been introduced in some models, such as in the \(E_6\) model [25, 26, 27, 28, 29]. On the experimental side also there are many efforts being made to search for the \(Z'\) directly at the LEP, Tevatron, and LHC. With the assumption that the couplings 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 \(\mu ^+ \mu ^-\) [30] events and this value becomes 1.9 TeV using the \(\tau ^+ \tau ^-\) events [31]. 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 Sect. 2, 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 decay mode \(B_d \rightarrow \tau ^\pm e^\mp \). 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 Sect. 3. In Sect. 4, we study the effect of NSI parameters on the \(\nu _{e}\) oscillation probability and the search for the new CP-violating signals at long-baseline experiments is presented in Sect. 4. Section 5 contains the summary and conclusions.

## 2 New physics effects from *B* meson decays

*B*meson. For this purpose, we first consider the leptonic decay channel \( B^- \rightarrow \tau ^- \bar{\nu }\). During the last few years, there has been a systematic disagreement between the experimental and SM predicted value for the branching ratio of \(B \rightarrow \tau \nu \) mode. The branching ratio for \( B ^- \rightarrow \tau \nu _\tau \) is given as

*B*meson. However, there is still a tension between the exclusive and inclusive value of \(V_{ub}\) at the level of \(3 \sigma \). This mode has been precisely measured [16] with a value

### 2.1 Extraction of the NP parameter from the lepton flavor violating decay process \(B_d \rightarrow \tau ^\pm e^\mp \)

*B*decay mode \(B_d \rightarrow \tau ^\pm e^\mp \). 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 \(b \rightarrow d \tau ^\pm e^\mp \) 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 \(2.8 \times 10^{-5}\). The effective Hamiltonian describing this process in the \(Z'\) model can be given as

*B*meson and \(p_B\) its momentum. Thus, with Eqs. (9) and (10), one can obtain the transition amplitude for the process \(B_d \rightarrow \tau ^- e^+\) as

*B*meson. In order to find the bound on the LFV couplings \(\eta _{e \tau }^{L,R}\), we need to know the value of the parameter \(\eta _{db}\), which can be obtained from the decay process \(B_d \rightarrow \mu ^+ \mu ^-\). The branching ratio for this decay mode has been recently measured by the LHCb [36] and CMS [37] collaborations and the present world average value [38] is given as

*Z*boson coupling to fermion–antifermion pair with value \(C_A^\mu =-1/2\). Now with Eq. (18) and considering the 1-\(\sigma \) range of experimental and SM predicted branching ratios from (14) and (13), the constraint on the parameter \(\eta _{db }^R\) is found to be

## 3 Neutrino oscillation in the presence of NSIs

*f*is a fermion and \(P_C=(1\pm \gamma _5)/2\) are the right (\(C = R\)) and left (\(C = L\)) chiral projection operators. The NSI contributions which are relevant while neutrinos 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

*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., \(n_n \approx n_p =n_e\). Therefore, one can write \(\varepsilon _{\alpha \beta }\) as [53]

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 [54]. Moreover, the recent works on the effect of NSI on the measurements of various neutrino oscillation experiments can be found in [55, 56, 57, 58, 59, 60, 61, 62, 63]. 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 Sect. 2.1. There are many works in the literature, dealing with an extensive study of model-dependent NSI parameters and their effect on neutrino oscillation experiments [64, 65]. However, in this work we focus on the lepton flavor violating NSI parameter, where the bound is obtained from the LFV decays of *B* meson in a \(Z'\) model and check its effect on the measurements of CP-violation at the long-baseline experiments like NO\(\nu \)A and DUNE. This would provide an indirect signal for the existence of \(Z'\) boson coming from the long-baseline neutrino experiment results.

### 3.1 Basic formalism with NSIs

*U*is the PMNS mixing matrix which is described by three mixing angles (\(\theta _{12},\theta _{13},\theta _{23}\)) and one CP-violating phase (\(\delta _{CP}\)); it is given by

*B*meson decays, as shown in Eq. (25).

## 4 Numerical analysis

### 4.1 Effect of NSI on oscillation probability and event spectra

The true values of oscillation parameters considered in the simulations

Oscillation parameter | True value |
---|---|

\(\sin ^2\theta _{12}\) | 0.32 |

\(\sin ^2 2\theta _{13}\) | 0.1 |

\(\sin ^2 \theta _{23}\) | 0.5, 0.41 (LO), 0.59 (HO) |

\(\Delta m_{atm}^2\) | \(2.4 \times 10^{-3} ~\mathrm{eV}^2\) for NH |

\(-2.4 \times 10^{-3} ~\mathrm{eV}^2\) for IH | |

\(\Delta m_{21}^2\) | \(7.6 \times 10^{-5}~ \mathrm{eV}^2\) |

\(\delta _{CP} \) | \(-90^\circ \) |

To show the effect of NSI parameter \(\varepsilon _{e\tau }\) on oscillation probability, we obtain \(\Delta P = |P_{NSI}-P_{SI}|\) (where \(P_{NSI(SI)}\) denotes the probability with non-standard (standard) interactions) for different baseline length and energy using the neutrino oscillation parameters as given in Table 2. The contour plots for \(\Delta P\) as a function of neutrino energy and baseline length are given in Fig. 2. The different shades in the figure correspond to different ranges of \(\Delta P\). From the figure, we can see that \(\Delta P \in \) (0.02,0.03) and (0.04,0.05) for NO\(\nu \)A (\(L= 810\) km and \(E = 2\) GeV) and DUNE (\(L= 1300\) km and \(E = 2.5\) GeV), respectively, for NH, whereas for IH, \(\Delta P \in \) (0.02,0.03) for both NO\(\nu \)A and DUNE. This implies that the non-standard interactions can affect the measurement of oscillation parameters at NO\(\nu \)A and DUNE experiments significantly.

Next, we show the oscillation probabilities as a function of the CP-violating phase for NO\(\nu \)A (DUNE) in the left (right) panel of Fig. 3. The dark solid (dashed) curve in the figure corresponds to oscillation probability for NH (IH) in the presence of NSI, whereas the light solid (dashed) curve corresponds to oscillation probability for NH (IH) in the standard oscillation. From the figure, we can see that there is an enhancement (diminution) in the probability for CP- violating phase in the range \(0^{\circ } \leqslant \delta _{CP} \leqslant 180^{\circ }\) (\( -180^{\circ } \leqslant \delta _{CP} \leqslant 0^{\circ }\)) for both mass hierarchies, if the NSI phase \(\delta _{e\tau }\) is zero. Further, the \(\nu _e\) event spectra for NO\(\nu \)A and DUNE are shown in Figs. 4 and 5, respectively. From these figures, we can see that the event rate in the presence of NSI is larger than that in SO for \(\delta _{CP}=0\) or \(90^\circ \). Meanwhile, for \(\delta _{CP}=-90^\circ \), the event rates in the presence of NSI are lesser than that in SO for \(\delta _{e\tau } =0\).

### 4.2 Effect of NSI parameter on \(\delta _{CP}\) sensitivity

## 5 Summary and conclusions

Conservation of lepton flavor universality is one of the unique features of the SM. However, recently there were a series of experimental results in *B* physics pointing toward possible violations of LFU, both in the charged and neutral current mediated semileptonic decays. Such lepton flavor universality violation could in principle also induce lepton flavor violating interactions. Considering the lepton flavor violating decays of *B* meson, i.e., \(B_d \rightarrow \tau ^\pm e^\mp \) decay, we constrain the lepton flavor violating couplings in the \(Z'\) model using the upper limits of the corresponding branching ratios. We obtained the bound \(|\varepsilon _{e \tau }| < 0.7\) from the decay rate. Assuming these NSI parameters in the charged lepton sectors to be related to the 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 we considered a conservative representative value for \(\varepsilon _{e \tau }\) as \(\varepsilon _{e \tau }=0.3\) and we have investigated its implications in the CP-violation sensitivity of long-baseline experiments. We found that the NSI parameters in the \(e \tau \) sector remarkably affect the \(\nu _e\) oscillation probability. Moreover, we found that the presence of NSIs leads to a misinterpretation of the oscillation data. The \(\delta _{CP}\) coverage of NO\(\nu \)A for CPV sensitivity above 1\(\sigma \) is reduced in the presence of NSIs. However, the CPV sensitivity is enhanced in the presence of NSI and it is above 5\(\sigma \) for more than 50% allowed values of \(\delta _{CP}\) in the case of both NH and IH for DUNE.

## Notes

### Acknowledgements

We would like to thank the Science and Engineering Research Board (SERB), Government of India, for financial support through Grant No. SB/S2/HEP-017/2013. SC would like to thank Dr. Sushant K Raut, Dr. Arnab Dasgupta, Dr. Monojit Gosh, and Mr. Mehedi Masud for many useful discussions regarding GLoBES.

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