On the Role of the $\nu_\tau$ Appearance in DUNE in Constraining Standard Neutrino Physics and Beyond

We consider the $\nu_\mu \to \nu_{\tau}$ appearance channel in the future Deep Underground Neutrino Experiment (DUNE) which offers a good statistics of the $\nu_{\tau}$ sample. In order to measure its impact on constraining the oscillation parameters, we consider several assumptions on the efficiency for $\nu_\tau$ charged-current signal events (with subsequent $\tau \to e$ decay) and the related backgrounds and study the effects of various systematic uncertainties. Two different neutrino fluxes have been considered, namely a CP-violation optimized flux and a $\nu_\tau$ optimized flux. Our results show that the addition of the $\nu_\mu \to \nu_{\tau}$ appearance channel does not reduce the current uncertainties on the standard 3-$\nu$ oscillation parameters while it can improve in a significant way the sensitivity to the Non-Standard Interaction parameter $|\epsilon_{\mu\tau}|$ and to the new mixing angle $\theta_{34}$ of a sterile neutrino model of the $3+1$ type.


Introduction
Neutrino experiments over the last 20 years have established the phenomenon of neutrino oscillations and now we are in the era of precision measurements in the leptonic sector. Although experiments detecting neutrinos from many sources were able to constrain with a very good precision the oscillation parameters, there are still some open questions. In particular, future experiments will be focused on measuring the CP violation in the lepton sector and on determining the sign of the atmospheric mass splitting. Another ambiguity is present in the value of the mixing angle θ 23 , since we still do not know in which octant this angle lies.
Beside checking for the Standard Physics, neutrinos can also be used to test Beyond Standard Model (BSM) physics. Among several scenarios, Non-Standard neutrino Interactions (NSI) with matter [1,2] and the existence of a fourth sterile neutrino have recently attracted a lot of interest, especially in connection to the ability of long baseline neutrino experiments (LBL) to probe for them [3]- [25].
One of the most powerful neutrino experiment that will be built in the near future is the DEEP Underground Neutrino Experiment (DUNE) [26,27]. This experiment consists of a baseline of 1300 km, planned across two sites in North America; the near site, situated at the Fermi National Accelerator Laboratory (FNAL), Batavia in Illinois, will hosts the Long Baseline Neutrino Facility (LBNF) and the Near Detector (ND). LBNF [28] will provide a GeV-scale ν µ beam (with contamination of ν e ) at 1.2 MW, later upgradeable to 2.4 MW. At the opposing end of the baseline, the far site in Sanford Underground Research Facility (SURF) in South Dakota will house four 10 kt Liquid Argon Time Projection Chambers (LArTPC) as the Far Detector (FD).
The DUNE neutrino beam will be able to operate in the Forward Horn Current (FHC, ν mode ) and Reverse Horn Current (RHC,ν mode) modes, in order to look for oscillations of both neutrinos and anti-neutrinos.
DUNE has been designed in order to answer all the questions mentioned before. The proposed neutrino flux (to which we will refer throughout the rest of the paper as the the standard flux [29]) will be optimized for the CP violation measurement and for this reason will provide a relatively large sample of ν e coming from ν µ → ν e oscillations. However this experiment will also be able to collect a huge ν τ sample, even if most of the neutrinos will not reach the threshold energy of 3.4 GeV for the τ production.
Motivated by the interest on τ neutrinos recently triggered by the observation of 8 ν τ interactions by the OPERA experiment 1 [30][31][32], in this paper we consider in detail the effect of adding the ν µ → ν τ appearance channel to the more widely used ν e appearance and ν µ disappearance modes 2 in the study of the sensitivity of the DUNE experiment to the oscillation parameters of the standard 3-ν framework as well as in the investigation of the parameter space of the NSI and of the sterile 3 + 1 neutrino models. Differently from [34], which focused on the τ hadronic decays, we consider the τ → e leptonic decay. In our numerical simulations, performed with the help of the GLoBES software [35,36], we take into account various detection efficiencies, signal to background ratios (S/B) and systematic uncertainties and explore the performances of the DUNE far detector not only for the standard flux but also for a ν τ -optimized flux as described in [37] and [38]. In both cases we have considered 3.5 years of data taking in the neutrino mode and 3.5 years in the anti-neutrino modes for a total of 7 years.
The paper is organized as follows: in section 2 we describe the neutrino fluxes and the efficiencies, systematics and backgrounds of the ν τ appearance channel; in section 3 we focus on the sensitivity reaches of the DUNE detector on the standard oscillation parameters allowed by the ν τ channel alone and for both standard and optimized fluxes. section 4 is devoted to investigate the impact of this additional channel on NSI parameter sensitivities while in section 5 we study in detail the sterile neutrino case. We draw our conclusion in section 6.
2 On the simulation of the ν τ appearance in DUNE In this section we specify the differences between the two fluxes used in our numerical simulations and comment on the values for efficiencies, systematics and backgrounds of the 1 OPERA observed 10 ντ candidates, with an expected background of 2 events. 2 An introductory study about the ντ appearance in DUNE can be found in [33]. ν τ appearance channel taken into account while evaluating the DUNE performance on the mixing parameters measurements, computed for 3.5 + 3.5 years of running time.

Fluxes
The two ν µ fluxes discussed in this paper have been displayed in figure 1; in both panels (neutrinos on the left, anti-neutrino on the right panel), the blue-solid line refers to the DUNE standard flux [29], which is said to be optimized to maximize sensitivity for CP violation measurements [27], while the red-dashed case refers to the optimized ν τ scenario of [37] and [38]. The standard flux consists of LBNF beam delivering 1.47 × 10 21 protons on target (POT) per year with 80 GeV energy running with 1.07 MW beam power and having 1.5 m NuMI (Neutrinos at the Main Injector) style target. The τ optimized flux is as per proposed by the DUNE collaboration [37] and it consists of 1.1 × 10 21 protons on target (POT) per year with 120 GeV energy running with 1.2 MW beam power and having 1 m NuMI style target. The expected un-oscillated charged current (CC) event rates are reported in table 1; for any of the flux options, we computed the ν e and ν µ events (together with their CP conjugate modes) when the LBNF beam is working in the FHC and RHC modes. Notice that for the normalization of the tau optimized flux we referred to the number of ν τ events as reported in Ref. [38].
We observe that the optimized flux in each of the ν andν modes gives a larger number of events. This is mainly due to the more energetic protons involved than in the standard flux which produce higher energy neutrinos and thus implies larger neutrino-nucleus cross sections.  Table 1. Expected ν e and ν µ un-oscillated CC event rates at the DUNE 40-kt far detector for a runtime of total 7 years (3.5 years in neutrino mode and 3.5 years in anti-neutrino mode). Numbers refer to the two flux options analyzed in this paper.

Efficiencies, systematics and backgrounds
In this section we quote the relevant efficiencies, systematics and backgrounds of the transition channels included in our numerical simulations. For the standard flux and the ν e appearance and ν µ disappearance channels, we strictly follow Ref. [27]: the appearance modes in the proposed DUNE experiment have independent systematic uncertainties of 2% each, while the ν µ andν µ disappearance modes have independent systematic uncertainties of 5%. The systematic uncertainties for the backgrounds are 5% for ν e and ν µ CC events, 10% for NC events and 20% for ν τ events. For the standard flux all DUNE collaboration post smearing matrices generated using the DUNE Fast Monte Carlo (MC) [27] have been used for the ν e appearance and ν µ disappearance channels to set the detection efficiencies.
For the optimized flux studies, since to our knowledge there are no official DUNE MC simulations available, background and signal efficiencies for ν e appearance and ν µ disappearance have been set to constant values, obtained from averaging the efficiency factors in various energy bins quoted in [27]. For both fluxes, the energy resolution of the detector is described by resolution functions given by the collaboration in Ref. [27].
As for the ν τ appearance detection, for a zeroth order study we envisage the DUNE detector to be similar to the ICARUS one. The ν τ appearance sample is composed of ν τ CC interactions resulting from ν µ → ν τ oscillations. Backgrounds to this channel come from the ν e CC, ν µ CC and NC interactions. Focusing on the leptonic decay of the tau generated by ν τ CC interactions, we consider the electron channel only (18% branching fraction [39]) since muonic tau decays are affected by large ν µ CC background. From the ICARUS proposal [40] it is clear that choosing the right kinematic cuts, the electron channel background can be reduced to only ν e CC events coming from two main components, which are the intrinsic ν e beam and ν µ → ν e oscillation 3 . Due to the spatial resolution of the DUNE LArTPC detector, very short tau tracks will not be recognized, and for this reason ν e CC background cannot be avoided. In this paper, for both standard and optimized neutrino fluxes, the overall ν τ andν τ appearance signal efficiency has been set to two distinct values (including the branching fraction), that are 6% [41] and, in order to show the full potential of the electron channel, also the maximum reachable efficiency of 18%. The former case has to be considered as a pessimistic case, being related to the expected selection efficiency of an old-generation detector like ICARUS; in fact, we expect DUNE to have better performances in reconstructing electrons tracks and in distinguishing electrons coming from tau decays. On the other hand, the latter efficiency of 18% has to be considered as an optimistic assumption, since it corresponds to DUNE able to reconstruct and recognize all produced electrons.
The number of the ν e CC background has been set to a constant value as to avoid the parameter dependence of the rates that, with such a comparatively small number of ν τ events, can overshadow the effect of the signal in the simulations; in particular, such a background has been chosen in order to reproduce, at the best fit parameter values, two S/B ratios discussed for ICARUS: 18.6 [40] and 2.45 [41]. In order to show the impact of ν τ systematic uncertainties on the results, we study the situation where such an uncertainty for ν τ signal events is fixed to the same value used by the DUNE collaboration when ν τ CC are considered as a background to the ν e appearance and ν µ disappearance channels, that is 20%, and also the situation where an optimistic 10% is taken into account.
Expected total rates for ν τ ,ν τ and ν e ,ν e CC events for both fluxes are shown in tables 3 and 4.
In DUNE the mean neutrino energy in the standard flux has been chosen in order to maximize the atmospheric term; since the minimum ν τ energy needed to be converted in a τ lepton is around 3.4 GeV, the number of ν e and ν µ events will be much bigger than the number of ν τ CC. For this reason, we expect that constraints on θ 23 and ∆m 2 31 will be mainly set by ν µ → ν e and ν µ → ν µ channels. Notice also that next terms in the ∆m 2 21 and θ 13 of eq.(3.1) would exhibit a sin δ CP dependence, so we expect this channel to be also partially sensitive to CP violation searches. However, due to the very large leading term, the changes in probability due to the CP violation phase will be comparatively very small and definitely less important than the corresponding CP violating terms in P µe .
In summary, considering the ν µ → ν τ oscillation probability and the lack of statistics, the ν τ appearance channel is expected to have a negligible impact on standard physics studies.

Expected rates for signal and backgrounds
In this section we estimate the event rates for the ν τ appearance in DUNE. We have used mixing parameters with their error bars from [43] which we summarize in Table 2. The

Parameter
Central Value Relative Uncertainty Table 2. Central values and relative uncertainty of neutrino oscillation parameters from a global fit to neutrino oscillation data [43]. As in [26], for non-Gaussian parameter θ 23 the relative uncertainty is computed using 1/6 of the 3σ allowed range. Normal mass hierarchy (NH) is assumed. Throughout the analysis presented in this paper, we assumed true values of δ CP to be 215 • as per Ref. [43]. We have used these values as central values for our simulation unless otherwise stated explicitly in the text.   Table 3. Expected total number of events after oscillation at the 40-kt far detector for Signals and Backgrounds (Bkg) obtained using no selection efficiencies hypothesis in the case of the standard flux and for Normal Hierarchy (NH). δ CP = 215 • is assumed [43]. The events correspond to DUNE running for a total of 7 years (3.5 years in neutrino mode and 3.5 years in anti-neutrino mode). These numbers must be compared with a total of 2043 (2369) ν µ → ν e ⊕ν µ →ν e CC signal events for the standard (optimized) flux and with a total of 14206 (67143) ν µ → ν µ ⊕ν µ →ν µ CC signal events.
We clearly observe that the DUNE experiment is able by itself to provide a τ sample around 300 events in FHC mode and 150 in RHC mode because of the generous ν µ flux components above the tau production threshold. On top of that, as per the plan for the optimized flux, there is a huge gain in statistics by roughly a factor of 10 with respect to standard taus, thereby justifying the possibility to explore scenarios of new physics with taus.

Details on the χ 2 definition
The confidence regions involving the sensitivity of the measurement of the oscillation parameters are determined based on the standard pull method [44][45][46] as implemented in GLoBES. The χ 2 is calculated by the minimizing over the nuisance parameters ξ. For every transition channel c with energy bin i (in the case of DUNE the number of energy bins suggested by the collaboration is 71 [29]), a Poissonian χ 2 distribution is used of the form: For the i-th energy bin for a given channel c, and for a set of oscillation parameters θ and nuisance parameters ξ, f c,i ( θ, ξ) is the predicted number of events and O c,i , is the observed event rate i.e., the event rate considering assumed true values of the oscillation parameters. Both f c,i and O c,i receive contributions from different sources s, that usually involve signal and background rates given by R c,s,i ( θ), such that The auxiliary parameters a c,s have the form a c,s ≡ k w c,s,k ξ k , in which the coefficients w c,s,k assume the values 1 or 0 corresponding respectively to a particular nuisance parameter ξ k affecting or not affecting the contribution from the source s to channel c. Therefore, the χ 2 is given by in total by: where the overall signal normalization is represented respectively by the last term known as the pull term. The different values used for σ N have been quoted in section 2.2 (for both signal and background).

Numerical results
Standard Flux In this section we exclusively use the standard flux configurations with an exposure of 3.5 + 3.5 years for investigating the sensitivity and correlation among the standard physics parameters obtained from the ν τ appearance channel.
It is clear that when the ν τ detection efficiency is increased from 6% to 18%, the number of signal events is increased and this results in smaller allowed regions in the correlation plots. The allowed range for θ 13 can be reduced by up to 18%, as we can see in the correlation plot in the (θ 13 , ∆m 2 31 ) plane. On the other hand, the θ 23 range can be reduced approximately by 15%, as shown in the (θ 23 , ∆m 2 31 ) plane. For ∆m 2 31 , instead, an improvement of approximately 30% can be reached passing from the worst case (S/B = 2.45, with ν τ detection efficiency at 6%) to the best one (S/B = 18.6, with ν τ detection efficiency at 18%). Notice that the OPERA experiment measured ∆m 2 31 = (2.7±0.7)×10 −3 eV 2 [32,48] while the DUNE setup here discussed with an 18% τ detection efficiency reaches a much better relative uncertainty of about 8%.
Also a larger S/B gives a better sensitivity to the mixing parameters: in particular, for S/B = 18.6 the correlation plots show a reduction of the mixing angles allowed ranges of approximately 5% if compared to the case S/B = 2.45. Roughly the same reduction can be noticed in the relative uncertainty of ∆m 2 31 . Thus we conclude that an increase in efficiency allows a better performance of the DUNE detector than a larger S/B.
In figure 3 we depict the same plots as in figure 2 but with 20% systematic uncertainty on the signal. We see that doubling the systematic uncertainty from 10% to 20% results in a decrease in sensitivity of approximately 8% for all mixing parameters.
Optimized Flux In this section, we exclusively use the tau optimized flux configurations with an exposure of 3.5 + 3.5 years for investigating the sensitivity and correlation among the standard physics parameters as obtained from the ν τ appearance channel only.
From figure 4 we see that an increase in the tau detection efficiency or even in the S/B ratio impacts the parameters sensitivities less than in the case of the standard flux, the reason being that the flux is already optimized for tau as to produce a larger statistics in all cases. Analizing the correlation plots, we observe that the smallest allowed ranges found in the case of the standard flux can be further reduced up to 10% for θ 23 and 15% for θ 13 if the optimized flux is considered. Notice also that with such a flux, DUNE can reach a relative uncertainty of 4.5% in the measurement of the atmospheric mass difference if 10% systematics on the signal is assumed. Finally, in figure 5 we present the results obtained for a 20% signal uncertainty. As before, the improvement in the sensitivity is smaller than in the case of the standard flux. The parameter which is affected the most by the systematics is the atmospheric mass splitting whose relative uncertainty is roughly 10%.

The case of Non-Standard Interaction
Several neutrino experiments have led to the opening for several new physics scenarios among which NSIs are quite popular. If the SM is regarded as a low-energy effective theory of some higher theory in the UV, then BSM would enter as higher-dimensional operators, suppressed by the new physics scale. In neutrino physics these are often written as fourfermion interactions, described by an effective four fermion Lagrangian [49]: where G F is the Fermi constant, ε f P αβ is the parameter which describes the strength of the NSI, f is a first generation SM fermion (e, u or d), P denotes the chiral projector {L, R = (1 ± γ 5 )/2}, and α and β denote the neutrino flavors e, µ or τ .
What is relevant for neutrino propagation in matter is the vector part V of interaction ε f V αβ = ε f L αβ + ε f R αβ since the neutrino propagation in a medium is sensitive to the combination ε αβ = ε eV αβ + N u /N e ε uV αβ + N d /N e ε dV αβ . Following what is usually done in the literature, we discuss the bounds from DUNE in terms of ε αβ . Overall, this Lagrangian describes neutral current (NC) interactions; it basically modifies the matter Hamiltonian and consequently the transition probability of the neutrinos in matter. The strength of the new interaction is parameterized in terms of the complex ε αβ = | αβ |e iφ αβ couplings. Thus the state evolution equations are given by: where ∆ ij = ∆m 2 ij /2E, U P M N S is the neutrino mixing matrix, A ≡ 2 √ 2G F n e with n e being the electron density in the Earth crust. All in all, beside the standard oscillation parameters, the parameter space is enriched by six more moduli | αβ | and three more phases φ αβ .
Direct constraints on NSI can be derived from scattering processes [50] and from neutrino oscillation data [51]. Latest constrains on NSI parameters from global fits were quoted in [52], from which we extracted the limits reported in table 5, used in our numerical analysis. We want to outline that, in order to compute the oscillation probabilities in presence NSI parameters Limits Table 5. Current constraints on the NSI parameters at 90% CL obtained from a global fit to neutrino oscillation data [52]. No bounds on the phases φ αβ are available so far.
of NSI, we may subtract from the diagonal entries any one of the diagonal elements αα as the oscillation phenomenon is insensitive to overall factors. Therefore one may consider, as done in table 5, the shifted parameters ee − µµ and τ τ − µµ instead of ee and τ τ respectively and set µµ to 0, which is also motivated by the strong external bounds as well [53].

The importance of ν µ → ν τ channel
General NSI considerations were already investigated in Refs. [5,6] and the effect of systematics were studied in [54] in the standard DUNE scenario consisting of the ν e appearance and ν µ disappearance channels. The ν µ → ν e oscillation probability is affected by the eµ and eτ parameters, as well as by ee . However, statistics in DUNE is dominated by the disappearance channel ν µ → ν µ which is mainly affected by the presence of τ τ and µτ . The dependence of probabilities on these parameters have been studied, among others, in [55][56][57].
For the appearance probability ν µ → ν τ , the leading analytic behavior in terms of the small parameters is given by [56]: where we neglected the small solar mass squared difference ∆m 2 21 ; P SM µτ is the oscillation probability in absence of NSI already discussed in section 3. Additional terms, which depend on both the real and imaginary parts of µτ , are at second order in the perturbative expansion, and are expected to provide only small sensitivity in the regions with φ µτ ∼ ±π/2. Since the term containing τ τ is very small for an almost maximal atmospheric angle, the probability P µτ is sensitive to large values of τ τ only, unlike for µτ to which we expect a maximal sensitivity.

Effect of Systematics, Experimental Reach and ν τ Detection Efficiency
In figures 6 and 7 we report the ∆χ 2 as a function of | µτ | for 10% (left panel) and 20% (right panel) systematic uncertainties for the ν τ signal, for the standard and optimized fluxes, respectively. Plots have been obtained marginalizing over all standard and non standard parameters according to the constraints reported in tables 2 and 5, except for the solar parameters θ 12 and ∆m 2 21 and the NSI parameter ε µτ , which have been fixed to their best fit values. All the NSI phases have been left free with no bounds.
The common feature of the above figures is that there is a significant improvement in the bound for | µτ | by increasing the efficiency from 6% upto 18%, and the S/B from 2.45 upto 18.6; one can envisage an overall improvement at 90% CL of approximately 10% in the parameter relative uncertainty in the case of the standard flux and 18% using the optimized flux.
Sensitivity limits at 90% CL reached with all the three appearance and disappearance channels in DUNE using the two fluxes are presented in the Tables 6 and 7. The worst case scenario using the standard flux gives us a limit on the | µτ | 0.25 while the most stringent limit can be set using the optimized flux with the best efficiency (18%), the best S/B (18.6) and 10% systematic uncertainty, | µτ | 0.20. This limit is approximately 35% smaller than the one that can be set by DUNE using only ν e appearance and ν µ disappearance channels with standard flux, | µτ | < 0.32, as estimated in Ref. [54]. We  observe that a fit to the OPERA ν τ events [32] did in Ref. [48] predicted the bound | µτ | 0.41 (marginalising over all parameters including NSI parameters) which is almost a factor of two larger than the worst limit DUNE can set. We checked that the other NSI parameters do not benefit so much from the ν τ appearance channel and the sensitivity reach remains roughly the same as in the standard DUNE scenario with ν e appearance and ν µ disappearance only.

The case of 3+1
The 3+1 sterile neutrino scenario is a possible solution to the short-baseline anomalies, which are about an excess of oscillations in short baseline experiments. For this reason, this new physics model is one of the most studied so far, and it is expected to be tested more in future experiments. The 3+1 sterile neutrino model is based on the addition of a fourth mass-eigenstate, m 4 , to the other standard three. Furthermore, the new interaction eigenstate is assumed to be sterile, that is not involved in electro-weak interactions with matter.
Adding a new neutrino to the standard model introduces new parameters (three more mixing angles, θ 14 , θ 24 and θ 34 , two more phases δ 1 and δ 3 and a new independent masssquared splitting ∆m 2 41 ) whose sensitivity will be studied in the following, with particular emphasis to the constraints attainable from the ν τ appearance channel.
The U P M N S matrix in this case is a unitary 4 × 4 matrix. The parameterization used here is as follows [58][59][60]: where the phase δ 3 reduces to the standard δ CP when the new mixing angles are set to zero. In our numerical simulations no external priors on the θ i4 's have been considered.

The importance of ν µ → ν τ channel
The oscillation probabilities ν µ → ν τ in the case of the sterile neutrino model are more complicated than the standard ones. However in the vacuum it is possible to make some useful approximations (matter effects have been taken into account in the numerical simulations). If the new mass-squared splitting satisfies |∆m 2 41 | |∆m 2 32 |, it is possible to average all the trigonometric functions which include ∆m 2 41 . Using the additional approximation of vanishing ∆m 2 21 , the following ν µ → ν τ oscillation formula can be obtained: . produces the following result: Both eqns.(5.2) and (5.3) show that the ν µ → ν τ probability is driven by the combination U * µ4 U τ 4 of new mixing angles that, as we have verified, does not appear in any of the other oscillation probabilities at the leading order. In the parameterization used in this paper, such a combination can be written in terms of mixing angles as: This relation shows that the relevant changes in P µτ with respect to the three neutrino scenario are mainly due to θ 34 and θ 24 . Since the combinations U * µ4 U e4 and |U µ4 | 2 that appear in the other oscillation probabilities contain θ 24 as well [61][62][63][64], we expect θ 34 to be the parameter whose sensitivity will be mostly affected by the addition of the ν τ appearance channel.

Constraints on Sterile Neutrino Parameters
In figures 8 (standard flux case) and 9 (optimized flux case) we show ∆χ 2 as a function of θ 34 computed considering all available oscillation channels in DUNE, including the ν µ → ν τ transition. In the left panels we considered the case where a 10% of systematic uncertainty is assumed for the signal while the 20% case is illustrated in the right panel.
As in the previous sections, we present the four different cases corresponding to 6% of τ detection efficiency and S/B = 2. 45  S/B = 2.45 (Brown, Dashed), 6% efficiency and S/B = 18.6 (Blue, Dotted) and 18% efficiency and S/B = 18.6 (Black, Solid). In the numerical simulations, the new physics parameters true values have been set to zero except for ∆m 2 41 that has been fixed to 1 eV 2 .  For the standard parameters we considered the central values and uncertainties reported in table 2. The mixing angles θ 14 and θ 24 , together with the two phases δ 1,2 have been left free in the fit. From figures 8 and 9 we extracted the bounds at 90% CL on θ 34 that we summarized in tables 8 and 9. As expected, the most stringent limit around 22 • can be set using the optimized flux with the best efficiency (18%), the best S/B (18.6) and 10% systematic uncertainty, a result in line with the recent NC event analysis carried out in [65]. The importance of ν τ appearance relies on the fact that such a limit would be roughly 20% smaller if only ν e appearance and ν µ disappearance channels with standard flux are considered, i.e 27.5 • . An important outcome of our analysis is that while the results with 10% systematic uncertainty and the optimized flux are always better than those obtained with the standard flux, for the 20% systematic this is not always the case; this is because the performances of the other two channels with optimized flux are worse, due to the increased number of ν τ CC background events. Thus, even though the ν τ appearance can help more in putting strong bounds for the optimized flux case, a larger systematic uncertainty brings to a regime where sensitivity limits are mainly set by the other two channels.
We complete our anaysis showing the correlation plots at 90% CL in several interesting planes. In all of them, the blue solid lines show the results obtained including the ν τ appearance channel in the most conservative case, that is with a 6% τ detection efficiency, S/B = 2.45 and signal systematic uncertainty of 20%. The red dashed lines show the situation with no contribution of the τ channel but only ν e appearance + ν µ disappearance.
In figure 10 we focused on the (θ 34 , ∆m 2 41 ) plane; we clearly see that for both flux options the gain in sensitivity using the τ channel is not negligible, amounting to about 6% and 20% for the standard and optimized flux cases, respectively.
In figure 11 we show the correlation plots in the planes (θ 34 , θ 14 ) (upper panels) and (θ 34 , θ 24 ) (lower panels). The new mass difference ∆m 2 41 is set to 10 −5 eV 2 (left panels) and 1 eV 2 (right panels) and the standard flux is assumed. First of all, we confirm that the τ channel does not significantly improve the bounds on θ 14 and θ 24 , which are around 20 • and between 10 • and 20 • , respectively. Instead, the upper limit on θ 34 can be improved by roughly 7% for both values of ∆m 2 41 analyzed here. As for the absolute upper bound on θ 34 , we observe that when the mass splitting is smaller, θ 34 is confined to larger values because, as seen from the expression in eq.(5.3), in this mass regime the non standard matrix elements are always sub leading.
The situation where the optimized flux is taken into account is presented in figure 12; consistently with figure 10, the final sensitivity to θ 34 when adding to the simulations all three transition channels is very similar to that obtained for the standard case. The real improvement carried by the optimized flux are on the (bad) limits set by ν e appearance + ν µ disappearance only, since they suffer by a large τ CC background. Finally, we comment on the fact that the sensitivities to the other sterile neutrino parameters are not affected at all by the addition of the ν τ appearance channel.

Discussion and Conclusions
The DUNE experiment is being proposed as a high precision next-generation neutrino experiment to be built in the USA. The baseline of DUNE is suitable for observing the neutrino mass hierarchy and measuring the CP phase δ CP but only ν e appearance and ν µ disappearance have been considered in most of the analysis. We studied the impact of the inclusion of ν µ → ν τ andν µ →ν τ channels in the DUNE experiment, considering the electronic τ decay mode. This oscillation channel is interesting because, on the one hand, it can confirm the three neutrino paradigm thanks to the redundancy produced by unitarity of the U P M N S and, on the other hand, it can probe several scenarios of new physics thanks to the almost unique dependence of the transition probability on certain non-standard parameters.
Our simulations have taken into account the impact of various systematic uncertainties, ν τ detection efficiencies, two different S/B ratios and two distinct neutrino fluxes on the sensitivities of the oscillation parameters. options have been considered. Based on the simulations performed with the GLoBES software presented in sections 3, 4 and 5, we conclude the following: • For the standard physics, the addition of ν τ appearance channel does not improve the sensitivities of any of the neutrino oscillation parameter set by the other two channels already being considered in DUNE.
• The performances of the tau optimized flux in the ν e appearance and ν µ disappearance channels result generally in a worsening of the sensitivities, thus overshadowing the advantage one may get from the increase in the ν τ statistics. This is mainly due to the increased background events in both the ν e and ν µ channels.
• In the new physics cases, the NSI parameter sensitivities are little affected by the addition of the new channel, except for the coupling | µτ | for which improved limits (about 15% better than the no-τ case) can be found.
• For the sterile neutrino 3+1 case, the only parameter that shows an increase in sensitivity is the mixing angle θ 34 whose improvement is about 20% compared to the case where τ signal events are not considered.
We remark that it may be possible to improve the above-mentioned sensitivities if we add to this analysis the tau hadronic decay channel as well [34], which has a better ν τ detection efficiency and consequently a bigger statistics.