Lepton Flavor Violating Non-Standard Interactions via Light Mediators

Non-Standard neutral current Interactions (NSIs) of neutrinos with matter can alter the pattern of neutrino oscillation due to the coherent forward scattering of neutrinos on the medium. This effect makes long-baseline neutrino experiments such as NO$\nu$A and DUNE a sensitive probe of beyond standard model (BSM) physics. We construct light mediator models that can give rise to both lepton flavor conserving as well as Lepton Flavor Violating (LFV) neutral current NSI. We outline the present phenomenological viability of these models and future prospects to test them. We predict a lower bound on Br$(H\to \mu \tau)$ in terms of the parameters that can be measured by DUNE and NO$\nu$A, and show that the hint for $H\to \mu \tau$ in current LHC data can be accommodated in our model. A large part of the parameter space of the model is already constrained by the bounds on Br$(\tau \to Z^\prime \mu)$ and rare meson decays and can in principle fully tested by improving these bounds.


Introduction
All observations and experimental evidence so far show that neutrinos only have weak interactions and can be well described within the Standard Model (SM) of particle physics.It is however intriguing to ask whether these rather mysterious particles can have any new interactions that have not been so far detected.In particular, if there is a new neutral current interaction with matter fields (i.e., with the electron or first generation quarks), it can affect neutrino propagation in matter [1].Thus physics beyond the Standard Model can be probed via neutrino oscillations in matter.The Non-Standard neutral current Interaction (NSI) of neutrinos can be effectively described by the following four-Fermi interaction: where f is the matter field (u, d or e), P is the chirality projection matrix and f P αβ is a dimensionless matrix describing the deviation from the SM.Surprisingly the upper bounds on αβ from neutrino oscillation effects are rather weak [2,3].The 90 % C.L. current bounds from neutrino oscillation observations are [4]: and Notice that these are model-independent constraints using only neutrino oscillation data.More model-dependent bounds arise from Tevatron and LHC data which can be stronger than those in Eq. (2) for mediator masses heavier than O(100 GeV) for some flavor combinations [5,6].
In the near-term future, long baseline experiments such as NOνA and the upcoming stateof-the-art DUNE experiment [7,8] will bring about unprecedented opportunities to probe these couplings further.Indeed, the recent establishment of the DUNE collaboration has created renewed interest in NSI [3,[9][10][11].Furthermore, improvements on NSI limits can be made by better measurements of the electron neutrino survival probability near the MSW "upturn" around a few MeV [12,13].In fact, dark matter direct detection experiments may be able to offer improved determination of the behavior of solar electron neutrinos, and hence NSI [14].
The underlying UV-complete model giving rise to Eq. ( 1) has to respect the electroweak symmetry so it will in general also give rise to sizeable new interactions of charged leptons [15], which tend to be much more strongly constrained [16].Further constraints on the underlying model are imposed by non-detection of the new mediator particle [16], which we denote as X, whose exchange gives rise to new effective interactions [16].Let us denote the coupling and mass of this new particle by g X and m X .The parameters describing the deviation from the SM can be estimated as ∼ (g 2 X /M 2 X )G −1 F so to obtain ∼ 1, the ratio g X /m X should be fixed to ∼ G 1/2 F .Non-detection of the new particle can be explained in two limits: (i) m X m Z ; (ii) g X 1.In the former case, which has been the focus of most model builders, perturbativity of g X (i.e., g X ≤ 1) implies that 1.That is we would not expect any sizeable nonstandard effect on neutrino propagation.Ref [17] suggests to invoke the second option with g X ∼ 5 × 10 −5 and m X ∼ 10 MeV to obtain ∼ 1.Notice that since for neutrino propagation only forward scattering of matter is relevant (i.e., t-channel diagrams in which X is exchanged with t = 0), we can use the effective coupling in Eq. ( 1) even for neutrino energies much higher than m X .Ref. [17] introduces a consistent model for the so-called LMA-Dark solution [19] with qP µµ = qP τ τ ∼ 1, qP ee = 0 and αβ | qP α =β = 0.For scattering experiments such as NuTeV with energy exchange of q much higher than m X , the amplitude of new effects will be suppressed by a factor of (m X /q) 2 so the corresponding bounds can be easily satisfied.
The off-diagonal elements of the qP matrix which violate lepton flavor can induce a significant effect on neutrino oscillation in matter.The DUNE and current NOνA experiments will be able to probe the values of these elements well below the present bound.In this paper, we explore possibilities of obtaining nonzero off-diagonal qP elements within U (1) gauge models in which left-handed leptons sit in a non-trivial two-component representation.Similarly to [17], we will assume the new gauge boson is relatively light.As shown in [16], the bound on τ → Z µ significantly constrains the new gauge coupling to leptons.To maintain sizeable non-standard effective couplings between leptons and quarks, we take U (1) charges of the quarks to be much larger than those of leptons.This in turn implies that the contributions of quarks and leptons to U (1) gauge anomalies should be canceled separately.The contribution to anomalies from leptons automatically cancel out.To cancel the anomaly from quarks, we introduce new generation(s) of leptons with appropriate U (1) charges.The lightest new lepton can play the role of the dark matter so we find a dark matter candidate as a bonus.
The paper is organized as follows.In section 2, we explore the possibilities to build a gauge interaction model with off-diagonal couplings to the leptons: Z µ να γ µ ν β with α = β.In section 3, we will specialize to the case of α = µ and β = τ and discuss various observational bounds and constraints.In section 4, we summarize our conclusions.

The Model
In this section we introduce a U (1) gauge model that gives rise to an effective coupling of form in Eq. (1).NSI involving the electron will affect electron-neutrino cross section in solar neutrino experiments such as Borexino and Super-Kamiokande.To avoid any deviation, following [4] we set f = q = {u, d}.To obtain an interaction between neutrinos and quarks, both leptons and quarks need to have nonzero U (1) charges.If NSI with quarks is non-chiral (i.e., if qL αβ = qR αβ ), the rate of deuteron dissociation (ν + Deuteron → ν + n + p) used by the Sudbury Neutrino Observatory (SNO) to derive total solar neutrino flux will be affected.The agreement between the prediction of standard solar model and the total flux measured by SNO sets bounds on the qL αβ − qR αβ .To avoid such a constraint, we assume the coupling of Z to quarks to be non-chiral qL αβ = qR αβ .In fact, the combination relevant for neutrino propagation in matter is the vectorial combination of qL αβ and qR αβ q αβ ≡ qL αβ + qR αβ .As emphasized in the introduction the aim of the present paper is to build a model giving rise to Lepton Flavor Violating (LFV) NSI of neutrinos with matter that can be probed at long baseline experiments.However, for simplicity we take the interaction to be flavor diagonal in the quark sector: That is we assume quarks are in the singlet representation of U (1) and only obtain a phase under U (1) transformation.Putting these together, we conclude that the U (1) charges of quarks of each generation are the same.In other words, the U (1) charges of quarks are given by where B i is the Baryon number of the i generation.With this definition, the couplings of quark to Z can be described as If η i are not equal, in the quark mass basis, the Z couplings can have nonzero off-diagonal elements.From the D − D mixing and Kaon physics, there are strong bounds on the 1-2 components of flavor changing neutral current.Similarly to [20] to avoid these bounds, we set η 1 = η 2 .In case η 3 = η 1 = η 2 , in the mass basis, flavor changing couplings of form g (η 3 − η 1 )V tb V * ts Z µ bγ µ s and g (η 3 − η 1 )V tb V * td Z µ bγ µ d appear which can give rise to b → Z s and b → Z d.As is well-known because of the longitudinal components of Z , the rates of b → Z s and b → Z d will be proportional to m 3 b /m 2 Z .Since we want m Z m b , this causes a huge enhancement.To avoid problems, we can set η 1 = η 2 = η 3 ; i.e., we gauge Baryon number.
Let us now discuss the transformation of leptons under U (1) symmetry.To obtain offdiagonal components, we assume that two generations of left-handed doublets, Lα = (ν α l− α ) and Lβ = (ν β l− β ), form a doublet of U (1) : −→ e iζg σ 1 α L, where σ 1 is the first Pauli matrix and ζ/η i gives the relative strength of coupling of leptons to that of quarks.In this basis the coupling of the Z boson to left-handed leptons will take the following form Notice that the transformation as Eq. ( 4 ( We have used tilded symbols to emphasize that | l− α and | l− β are not necessarily mass eigenstates.Denoting the charged leptons of definite mass by l − α and l − β and corresponding neutrinos with ν α and ν β , we can in general write: In the mass basis, the U (1) gauge interaction will be of form For θ L = 0, π/2, we shall have flavor conserving interactions, too.There will be flavor violating gauge interactions for all values of θ L within physical range except for θ L = ±π/4.Let us assume that the lepton generation denoted by γ is singlet under U (1) .In the next section, we will identify γ with first generation on which there are strong bounds.Notice that the contribution of L (or equivalently L) to the U (1) − SU (2) − SU (2) and U (1) − U (1) − U (1) anomalies automatically cancel out because T r(σ 1 ) = 0; however, there will be a contribution to the U (1) − U (1) − U (1) anomaly from L. The contribution of quarks to the U (1) − SU (3) − SU (3) anomaly is given by + Notice that ν R does not help to cancel the U (1) − U (1) − U (1) anomaly since ν R does not carry any hypercharge.To cancel the remaining U (1) − U (1) − U (1) anomaly from lepton side, we should assign an appropriate transformation to l − Rα and l − Rβ .In the following, we suggest two solutions: Rα and l − Rβ form a doublet of U (1) : Let us define R as a doublet under U (1) which is formed from the right-handed charged leptons and transform under U (1) as follows −→ e iζg σ 1 α R.
In other words, in the basis that the left-handed leptons have opposite U (1) charges (see Eq. 5), two right-handed leptons ( l− 2 have the "same opposite" charges.It is straightforward to show that the contribution from R to U (1) − U (1) − U (1) cancels that from L. We can write the following Yukawa couplings invariant under electroweak as well as U (1) symmetry where H is the SM Higgs and b 0 and b 1 are Yukawa couplings.After electroweak symmetry breaking these Yukawa interactions will induce the following mass matrix: Notice that the 11 and 22 components of the mass matrix B are equal which implies the mixing angle defined in Eq. ( 6) will be equal to π/4.This in turn implies that the coupling of Z conserves flavor (see Eq. ( 7)).To solve this problem, we introduce a 2 × 2 matrix Φ whose components are scalar fields, doublet under electroweak with the same hypercharge as that of the SM Higgs.Under U (1) , Φ transforms as We can write a Yukawa coupling of form The components of Φ can be heavy enough to avoid bounds but we can take c 0 1 to obtain small contribution to the lepton masses.Taking m 2 β m 2 α , we can write which can be in general nonzero, leading to flavor violating Z coupling.Another issue is that the mass structure of charged leptons has to be hierarchical.If we want the main contribution to heavier lepton to come from the vacuum expectation value of the SM Higgs (i.e., if c 0 Φ b i v), b 0 and b 1 should be approximately equal to reconstruct the hierarchical mass pattern in the SM (i.e., m β m α ).Such an equality can be explained by discrete symmetry R † → R † σ 1 and Φ → σ 1 Φ.We can then write b 0 b 1 m β /2v.We can moreover write the couplings of the Higgs to the charged leptons as where θ R is the mixing angle relating R to the right-handed charged lepton mass eigenvectors.This can lead to LFV Higgs decay: and Notice that for general complex Φ with Re( Φ 21 ) = Re( Φ 12 ), the values of θ R and θ L are not the same.
With this charge assignment, the U (1) − U (1) − U (1) anomaly automatically cancels out.The value of q should be assigned such that the contribution from the right-handed leptons to the U (1) − U (1) − U (1) anomaly cancels that from the left-handed leptons.That is −2q 2 (−1) + 2(−1/2)T r(σ 1 σ 1 ) = 0 which implies q = 1.To couple both lRα and lRβ to L, we need to introduce two scalar U (1) doublets whose components are Higgs-like doublets under electroweak symmetry with hyper-charge equal to +1/2: −→ e iζg α e iζg σ 1 α Φ 1 and Φ With this field content, we can write Yukawa couplings of the following forms After Φ 1 and Φ 2 develop vacuum expectation values, these terms give masses to charged leptons.In general, θ L = π/4 so the flavor violating U (1) gauge couplings are obtained.
The SM Higgs can mix with the neutral components of Φ i so it can decay through this mixing to charged lepton pairs.However the rate of the Higgs decay into charged leptons will deviate from the SM prediction and will not be given by (m f /v) 2 .The decay rate of the SM Higgs into τ pair is now measured and found to be consistent with the SM prediction.This solution is not therefore suitable for the case that α or β is identified with τ .
In both of two above cases, going to the lepton flavor basis arranged as (γ , α , β), the qP matrix will be equal to Without loss of generality, we can set η 1 = 1.
Notice that in neither of the above two solutions, right-handed neutrinos are required for anomaly cancelation.Although it is not the main subject of the present paper, let us provide an example to show how type I seesaw mechanism can be implemented within this model.Let us take the right-handed neutrinos singlet under U (1) , too.Majorana mass matrix of right-handed neutrino as well as the Dirac mass term for L γ can be written as in the standard type I seesaw mechanism.To write the Dirac mass terms νRi L α and νRi L β , we however need a scalar doublet of U (1) shown by H N whose components are doublets of electroweak and their hypercharge is equal to that of L. We then obtain the desired Dirac mass terms via νRi H T N cL and νRi H T N cσ 1 L. Vacuum expectation values of H N can be taken to be much smaller than that of the SM Higgs to avoid changing Yukawa coupling of H to fermions m f / H √ 2m f /v where v = 246 GeV).As shown in [17], components of H N can be made heavy despite small H 0 N .As discussed before, the contribution of leptons to anomalies cancel.Taking η 1 = η 2 = η 3 in Eq. ( 3), the quark sector induces a contribution to the U (1) − U (1) − U (1) and U (1) − SU (2) − SU (2) anomalies but the U (1) − SU (3) − SU (3) anomaly automatically cancels out.The anomaly can be canceled by adding new chiral degrees of freedom.On example is to introduce two generations of new leptons (with the same charges under SM gauge symmetry as those of leptons) with U (1) charge equal to −9/2η 1 .1With such field content anomalies will be canceled.These chiral fermions, like quarks, can obtain mass by coupling to the SM Higgs.Perturbativity of their Yukawa coupling then implies an approximate upper bound of O(600 GeV) on their mass.These particles can be produced at colliders via their electroweak interactions.The present lower bound on the mass of such new charged charged leptons is about O(100 GeV) [46].The U (1) symmetry prevent mixing between these new leptons and the SM leptons.Thus, the lightest new lepton, which can correspond to the new neutrino, will be stable and can play the role of dark matter.Notice that this aspect of the scenario is only peripheral to the purpose of the present paper.We will not therefore elaborate on it further.
In the above discussion, we have introduced new scalars that have charges of ζ under U (1) and transform as standard model Higgs under electroweak and develop Vacuum Expectation Value (VEV).Their VEV will both induce a mass for Z and mixings between Z with Z (but not with γ).These mass parameters are of order of g ζ times the VEV of these new scalars.Since the VEV of these new scalars are taken to be smaller than H , the mass terms created by their VEV will be smaller than g ζ H .As we shall see in the next section, g ζ is constrained to be smaller than 3 × 10 −9 (m Z /10 MeV).These mass terms are therefore much smaller than m Z and can be safely neglected.To explain the mass of Z , we can either invoke the Stückelberg mechanism or introduce a scalar (S) singlet under SM gauge symmetry and a U (1) charge of ζ .The mass of Z will be given by g ζ S ∼ few × 10 MeV.Notice that since g ζ can be arbitrarily small, S can be made large.The new scalar can be much heavier than Z .

Phenomenological Implications
In this section, we discuss the observational effects of the model presented in the previous section and discuss the bounds on its parameters from various observations and experiments.The observational imprint of the model depends on the decay modes of Z .Since we take m Z < 2m µ , it cannot decay into muon and tau lepton pairs but Z can decay into ν α να , ν β νβ as well as ν α νβ and ν β να .If α is identified with e, Z can decay into e − e + , too.Moreover if α = e, β = µ and m Z > m µ , we can have Z → µ − e + and Z → e − µ + .Although in our model quarks also couple to Z (and as we shall see, with a coupling much larger than those of leptons), as far as m Z < m π , Z will not have hadronic decay modes because the lightest hadrons (i.e., pions) are heavier than Z .This is the famous mass gap problem which appears in the confined regime of strongly interacting theories.At first sight, it seems that via quark loops Z and photon mix which can give rise to Z → e − e + even for the α = µ and β = τ case with no tree level coupling between the electron and Z .However for energy scale below the QCD scale (∼ few × 100 MeV) instead of quarks, hadrons should propagate in the loops.This is well-known in the calculation of hadronic loop for vacuum polarization of the photon which is needed for precise calculation of (g − 2) µ (see, for example, [39].)On the other hand, mesons (having zero baryon number) are neutral under U (1) so they cannot mix Z and the photon.The lightest hadron charged under both U (1) em and U (1) is the proton which is much heavier than the scale that we are interested in and therefore is decoupled from low energy physics.As a result, for α = µ and β = τ , the only available Z decay mode is into neutrino pairs ν µ νµ , ν τ ντ , ν µ ντ and ν τ νµ .
If Z decays into e − e + , it can be traced in the beam dump fixed target experiments [40][41][42], for a given m Z , these experiments rule out values of coupling between an upper bound and a lower bound.The Z production in these experiments is through their couplings to the quarks.If the coupling to e − e + is higher than the upper bound, the Z decay takes place in the dump and the produced e − e + will not be registered.On the other hand, if the couplings to quarks and leptons are too small, the rates of the Z production and of the subsequent decay into e − e + will be too low to have an observable effect.The beam dump bounds are derived and shown in [40][41][42] for the models that the U (1) charges of the electron and the quarks are equal.The upper limit of the excluded region, determined by Γ(Z → e − e + ), can be readily interpreted as the upper limit of the excluded region of ζg sin 2θ L in our model in case that α is identified with the electron.This bound can be further improved by the SHiP experiment [43].The lower limit of the excluded region in our case should be however stronger than [40][41][42] because the Z production is determined by quark charge which is fixed to +1 (cf.lepton charges are suppressed by ζ).If α, β = e, Z does not decay to e − e + so the beam dump bounds do not apply.
The bounds on the couplings of a new U (1) gauge boson to quarks are mostly derived by looking for the e − e + pair from the decay of Z produced in various density frontier experiments.Most importantly Ref [44] sets a bound g < 10 −3 for m Z ∼ 20 MeV from π 0 → γZ and subsequently Z → e − e + .As we show below, we can find a comparable bound for the case that Z decays into ν ν from Br(π 0 → ν νγ) < 6 × 10 −4 [46].We expect A similar formula can be found in [45] for the case of a U (1) gauge boson mixed with the photon.The factor of 2 reflects the fact that unlike the case of π 0 → γγ, the final particles in π 0 → γZ are distinguishable.Taking Br(π 0 → γZ ) ≤ Br(π 0 → γν ν) < 6 × 10 −6 , we find For m Z > 135 MeV, π 0 cannot decay into Z γ so no bound can be set from pion decay on g .To our best knowledge, all the bounds set on g for 135 MeV < m Z < 200 MeV are based on searching for leptons from the Z decay [47][48][49][50] which do not apply to our case.For m Z > 200 MeV when the Z decay into µ − µ + becomes possible, there are stronger bounds from BaBar [51] and KLOE-2 [52].Notice that for m π < m Z < 200 MeV, the range of the U (1) interaction between nucleons is comparable to that of strong interactions so as long as (g ) 2 /4π < α, its effects will be too small to be discerned in the presence of strong interactions.Throughout this section, we assume that g saturates this bound so that we can obtain a sizeable qP αβ .In this mass range the Z can decay into π 0 γ but since decay takes place inside the dump, the beam dump experiments cannot identify it.
With such large g , the Z particles can be produced at supernova core via N + N → N +N +Z .The Z particles will thermalize in the supernova core via interactions with nucleons with a mean free path smaller than 1 cm in the supernova core and eventually decay into standard model particles.The Z production and decay can take place outside neutrinosphere, too.It will be interesting to study its possible effects on supernova evolution and shock revival but such an analysis is beyond the scope of the present paper.
Case I, β = µ and α = e: If we identify β = µ and α = e (and therefore γ = τ ) we will have µ → eZ and subsequent decay of Z → eē will produce a signal of µ → eee on which there is a strong bound Br(µ → eee) < 10 −12 [46].For m Z ∼ 10 MeV, this bound translates into g ζ((cos 2 2θ L + cos 2 2θ R )/2) 1/2 < 10 −13 which is so strong that kills any hope for sizeable µe .However for m Z > m µ , µ → Z e is not possible but µ → eee can take place via a tree level diagram in which virtual Z is exchanged with Γ(µ → eee) ∼ (ζg ) 4 cos 2 2θ L sin 2 2θ L m 5 µ /(100π 3 m 4 Z ).From the upper bound on Br(µ → eee), we find g ζ √ sin 2θ L cos 2θ L < 6 × 10 −7 m Z /(150 MeV).For m Z ∼ 150 MeV and Z → e + e − , the KLOE experiment finds g < 8 × 10 −4 by studying φ → Z η and subsequently Z → e − e + .Putting these two bounds together we find qP eµ < 5 × 10 −4 .We will not investigate this case further.Case II, β = τ and α = e: Let us now consider the case of α = e and β = τ .The bound Br(τ → eZ ) < 2.7 × 10 −3 [46] sets the bound which along with g < 2 × 10 −3 gives qP τ e < 1.5 × 10 −3 .The beam dump experiment E137 [40] rules out Z with m Z ∼ 10 MeV with a coupling larger than 3×10 −8 to quarks and the electron.In our case, since the coupling to quarks (determining the production of Z in this experiment) is much larger than the coupling to the leptons, the bound from E137 should be reconsidered.Performing this analysis is beyond the scope of the present paper.
Case III, β = τ and α = µ As shown in [16], the present bound on the branching ratio of this mode, Br(τ → Z µ) < 5 × 10 −3 [46], sets a severe bound on The reason why the bound is so strong is the fact that Γ(τ → Z µ) is enhanced by (m τ /m Z ) 2 which is the famous factor due to the production of longitudinally polarized vector boson.At m Z < m π , the bound on the coupling of quarks to Z is also rather strong (see Eq. ( 19)).
As discussed above, we can obtain sizeable qP αβ only for the case of qP µτ .We therefore focus in this case.Using the terminology of the last section, we take γ = e, α = µ and β = τ .We studied observational effects and various bounds from Borexino on neutrino-electron interactions [23], NuTeV neutrino-nucleus scattering [24], cosmic neutrino absorption at IceCube [22,[25][26][27][28][29][30][31][32] and neutrino trident production from CCFR [33,34].We have found that lepton couplings suppressed by ζ ∼ 10 −5 , the effects on these observations and experiments will be negligible.The relevant bounds (also displayed in 4/3 N eff , where ρ rad is the total radiation energy density and ρ γ is the photon energy density.Minimally we will need m Z > 0.1 MeV in order not to contribute all of its entropy directly to N eff (which would result in ∆N eff 1.71).This would considerably exceed the BBN constraint, of ∆N eff = 1.13 (68% C.L.) [35].However, even if the Z is not directly contributing to N eff as radiation itself, the entropy it transfers to ν µ and ν τ can raise their temperatures compared to the standard model case.We use the conservation of entropy to compute the temperatures of the µ and τ type neutrinos after the ν e neutrinos undergo electroweak decoupling at T SM,ν 1 MeV.We find that this rules out m Z < 5 MeV in good agreement with the results of [22].Notice that for m Z > 5 MeV and g ζ > 4 × 10 −11 , the lifetime of Z will be shorter than 1 sec.
2. Neutrino oscillation data: current limits and prospect of DUNE and NOνA: The long baseline of the DUNE and NoνA experiments make them a natural setup to look for NSI [3,[9][10][11].The combinations that are relevant for neutrino oscillation in matter are αβ where n f is the number density of fermion f .In the crust of the Earth, n u /n e = n d /n e = 3 so within our model, we can write αβ 6 u αβ = 6 d αβ .Moreover by αα → αα − τ τ I 3×3 , the neutrino oscillation pattern will remain unaltered.Following [10], let us redefine the diagonal components as ˜ ee ≡ ee − τ τ and ˜ µµ ≡ µµ − τ τ .In our model these parameters are correlated and can be written as ˜ µµ = 2A sin(2θ L ) and µτ = A cos(2θ L ) where Bounds on µτ and ˜ µµ can be translated into a bound on A = 2 µτ + ˜ 2 µµ /4.DUNE (NOνA) sensitivity has been estimated to be µτ ∼ 0.021 (0.031) and ˜ µµ ∼ 0.1 (0.15) [10] which is equivalent to sensitivity to A down to 0.054 (0.081).This can be translated into a bound on √ ζg /m Z .Notice that the bound on √ ζg /m Z is given by the root of A.  2) (which is equivalent to A < 0.108) taking ζ = 3 × 10 −6 .This bound can be written as As we showed in Eq. ( 21), for a given g /m Z , the bound on Br(τ → Z µ) can be interpreted as an upper bound on ζ cos 2θ L .Setting ζ cos 2θ L equal to this bound and asking µτ = 6 u µτ to be larger than certain value we find the following lower bound on g : where µτ = 6 u µτ .In Fig. 1, we show this lower bound for µτ = 6 u µτ = 0.06 (present bound) and µτ = 6 u µτ = 0.021 (DUNE reach).As seen from the figure for 5 MeV < m Z < 9 MeV and for m Z > 130 MeV, we can obtain values of µτ observable at DUNE.  24).The dotted and dot-dashed black lines show the bound from only µτ for µτ = 6 u µτ = 0.021 and µτ = 6 u µτ = 0.06, respectively (see Eq. 25).The red curves shows the upper bound from π 0 → γZ (see Eq. 18).Modifications to the effective number of relativistic degrees of freedom constrain the mass of the mediator to be 5 MeV [22].For additional details see the main body of the text.
Moreover, for 5 MeV < m Z < 20 MeV and m Z > 110 MeV, we can have µµ − τ τ large enough to be discerned at DUNE.
For 200 MeV m Z 10 GeV, B-factories impose relatively strong bounds on g particularly from Υ → γZ [36] and from Υ → invisible [37].In this mass range, the Z can decay into µμ, hadrons, and ν ν pairs.For m Z > 500 MeV, this bound starts to become serious as it pushes to values lower than the present bounds in Eq. ( 2).Throughout our discussion, we have assumed a gauge boson mass, m Z < 200 MeV.In this range, Υ → γZ and Υ → Z Z are negligible.This can be understood as a consequence of the fact that in the limit of m Z → 0, according to the Landau-Yang theorem, the spin one Υ particle cannot decay into two massless or very light vector bosons.
In the following, we will discuss consequences of having lepton flavor violating gauge couplings within the present model.Most of these bounds come from processes of type τ → µ + f 1 + f2 where f 1 and f2 are final fermions and can be any of the pairs uū, d d, ν µ νµ , ν τ ντ , ν τ νµ and ν µ ντ .The process takes place via the exchange of a virtual Z and the amplitude can be written as where, for f 1 = f 2 = u, d, a V = g and a A = 0, for and a A = ζg (sin 2θ L − sin 2θ R )/2, for f 1 = ν µ and f 2 = ν τ as well as for f 1 = ν τ and Using Dirac equation for f 1 and f 2 we can write As a result, when f 1 and f 2 correspond to quarks (for which a A = 0) or to neutrinos (for which m f 1,2 → 0), we can drop the terms proportional to 1/m 2 Z in the propagator of Z which comes from longitudinal degrees of freedom.The amplitude for τ → µq q will be given by ζ(g m τ ) 2 /(q 2 − m 2 Z ) multiplied by a function of sines and cosines of θ L and θ R .Similarly, for neutrinos it will be given by (g ζm τ ) 2 /(q 2 − m Z ) 2 again multiplied by a function of sines and cosines of θ L and θ R .For τ → µµµ longitudinal components of the propagator gives rise to an amplitude given by ) again multiplied by a function of sines and cosines of θ L and θ R .When 4m 2 µ ∼ q 2 m 2 τ , the amplitude will be enhanced.As a result, the integration of |M | 2 over the phase space of the three final particles will yield a log(m 2 τ /4m 2 µ ) ∼ 4 enhancement which does not change the order of magnitude of the decay rate.

Summary and outlook
We have proposed a model for neutrino NSI with matter based on a new U (1) gauge symmetry with a light gauge boson, Z .The model by construction gives LF conserving as well as LFV terms.We have put two generations of left-handed leptons in the doublet representation of the U (1) symmetry and have assumed that the third generation of the leptons are invariant under U (1) .In other words, in a certain basis which does not correspond to the mass basis, two generations of leptons have opposite U (1) charges and the third state which can be a certain flavor (mass) eigenvector has zero U (1) charge (see Eqs. ( 4) and ( 5) for clarification).The U (1) charges of quarks are taken equal to +1 so we do not predict new flavor violating effects in the quark sector.The U (1) − U (1) − U (1) and U (1) − SU (2) − SU (2) anomalies have to be canceled by new heavy degrees of freedom.As an example, we suggest existence of new chiral fermions with masses of order of 400 − 500 GeV with the same electroweak quantum numbers as those of leptons.The existence of such new particles can be tested at accelerators such as the LHC.Moreover, they can provide a dark matter candidate as a bonus.
We have proposed two possibilities for the right-handed leptons.In the first case, the right-handed leptons also transform as a doublet of U (1) (see Eq. ( 8)).Thus, the SM Higgs can couple to these fields and give them mass.In the second case, the right-handed leptons transform as Eq. ( 14) and two new scalar doublets are introduced to give mass to charged leptons.As a result, in the second case the branching ratios of the Higgs to charged leptons will significantly deviate from the SM prediction.Given that the measured Br(H → τ τ ) is in reasonable agreement with the SM prediction, the former solution is more suitable for models in which the τ lepton has nonzero U (1) charge.This case is made all the more intriguing given 2.4σ hint of nonzero Br(H → τ µ) [53], which the model can explain.We have also shown how a seesaw mechanism for neutrino mass production can be added to this model.
If the Z coupling to the electron is nonzero, Z can decay into e − e + which makes the detection of Z in low energy luminosity frontier experiments simpler.Null results for φ → ηZ , Z → e − e + and for π 0 → γZ , Z → e − e + imply an upper bound of ∼ 10 −3 on the Z coupling to quarks.If the electron and electron neutrino are singlets under U (1) , Z cannot decay into electron positron pair so these bounds do not apply.However we have shown that the bound on Br(π 0 → γν ν) again puts a strong bound on the Z coupling to quarks for m Z < m π .However for m π < m Z < 2m µ , the bound on the coupling of quarks to Z is dramatically relaxed.
In the presence of the relevant LFV gauge coupling, charged lepton l − β can decay into lighter charged lepton l − α and Z .The longitudinal component of Z leads to a strong enhancement of (m l β /m Z ) 2 in the rate of this process which puts a strong bound on the LFV gauge coupling.For m Z > m µ although µ → Z e will not be possible but the LFV gauge coupling leads to µ → eee at tree level and again very sever bounds on the µe component of the gauge coupling are obtained.
We have estimated the maximum qP αβ that can be obtained within the present model still satisfying the bounds both from rare meson decays and LFV charged lepton decays.We have found qP µe < 5×10 −4 and qP τ e < 1.5×10 −3 .However, qP µτ and qP µµ − qP τ τ for m π < m Z < 2m µ can be as large as respectively 5 × 10 −3 and 0.05 which can be discerned by upcoming long baseline experiments.For this reason we mainly focus on the case that the first generation of leptons are invariant under the U (1) and the second and third generation of leptons are in the doublet representation of the U (1) .The flavor structure of NSI in our model is shown in Eq (17).In the parameter range in which we are interested, all the bounds on a new gauge boson are avoided.For example, since the coupling of Z to nucleons is relatively large, the mean free path of the produced Z inside supernova will be smaller than 1 cm so the Z production will not directly contribute to supernova cooling.We also studied the phenomenological effects of LFV gauge coupling on τ → µν ν, τ → µ + hadrons, τ → µµµ and τ → µ + γ.The effects appear to be well below the sensitivity limit.
The long baseline NOνA and DUNE experiments as well as high statistics atmospheric neutrino oscillation experiments such as IceCube Deepcore [54] can determine effects of neutral current NSI on neutrino oscillation pattern.If these neutrino experiments find that µτ and τ τ − µµ are nonzero, we shall obtain a significant hint in favor of this model.The case will become stronger if the signal for H → τ µ is confirmed.In fact, we predict a lower bound on Br(H → τ µ) in terms of µτ /( τ τ − µµ ) (see Eq. 29).The model can be tested by improving the bound on τ → Z µ.Another possible test is searching for the Z production in meson decay and its subsequent decay to neutrinos (missing energy) or for m Z > m π to πγ.
would like to thank both the Pennsylvania State University and the Institute for Gravitation and the Cosmos for support.

Fig 1 )
are the following:1.Relativistic Degrees of Freedom: New light degrees of freedom can impact cosmology by changing the relativistic energy density and thus the expansion rate.The cosmological impact of light degrees of freedom is parameterized by the N eff parameter, defined as ρ rad = ρ γ 1 + As a result, the projected sensitivities of NOνA and DUNE to √ ζg /m Z are very close to each other.That is why we are showing them collectively in Fig 1 with a single line.To draw this line we have set ζ = 4 × 10 −5 .Long baseline experiments such as NOνA or DUNE can in principle disentangle µτ and ˜ µµ .That means they can determine not only the value of A but also that of θ L .Fig 1 also shows the constraint from the current oscillation bounds summarized in Eq. (

Figure 1 :
Figure 1: Here we summarize the constraints on the model.The dashed blue and gray lines show the upper bounds from respectively present oscillation data and DUNE experiment incorporating information both on µµ − τ τ and µτ (see Eq. 24).The dotted and dot-dashed black lines show the bound from only µτ for µτ = 6 u µτ = 0.021 and µτ = 6 u µτ = 0.06, respectively (see Eq.25).The red curves shows the upper bound from π 0 → γZ (see Eq. 18).Modifications to the effective number of relativistic degrees of freedom constrain the mass of the mediator to be 5 MeV[22].For additional details see the main body of the text.