3-flavor and 4-flavor implications of the latest T2K and NO$\nu$A electron (anti-)neutrino appearance results

The two long-baseline experiments T2K and NO$\nu$A have recently presented new findings. T2K has shown the first $\bar \nu_e$ appearance data while NO$\nu$A has released the first $\nu_e$ appearance results. These data are of particular importance because they allow us to probe for the first time in a direct (or manifest) way the leptonic CP-violation. In fact, it is the first time that a hint of CP-violation arises from the comparison of the observations of neutrinos and antineutrinos. We consider the implications of such new results both for the standard 3-flavor framework and for the non-standard 3+1 scheme involving one sterile neutrino species. The 3-flavor analysis shows a consolidation of the previous trends, namely a slight preference for $\sin \delta<0$, disfavoring CP conservation ($\delta =0, \pi$) with a statistical significance close to $90\%$ C.L., and a mild preference (at more than 68\% C.L.) for the normal hierarchy. In a 3+1 framework, the data constrain two CP-phases ($\delta_{13} \equiv \delta$ and $\delta_{14}$), which exhibit a slight preference for the common value $\delta_{13} \simeq \delta_{14} \simeq -\pi/2 $. Interestingly, in the enlarged four neutrino scheme the preference for the normal hierarchy found within the 3-flavor framework completely disappears. This indicates that light sterile neutrinos may constitute a potential source of fragility in the capability of the two LBL experiments of discriminating the neutrino mass hierarchy.


Introduction
Neutrino physics is entering a new era. After the discovery of a relatively large value of θ 13 , the determination of leptonic CP violation (CPV) and the identification of the neutrino mass hierarchy (NMH) have become realistic targets.
In this respect, the long-baseline (LBL) experiments represent the first setups able to probe these properties. One of them (T2K), already operative since 2011, has provided the first observation of the conversion of muon neutrinos into electron neutrinos [1]. This result, in combination with the precise determination of the third mixing angle θ 13 achieved with the dedicated reactor experiments [2,3,4], has provided the first indirect information on CPV. In fact, all the recent global [5,6,7] and partial [8,9] (including only reactor and T2K data) fits indicate a slight preference for CPV, with a best fit value of the fundamental CP-phase δ close to the value −π/2.
A few weeks ago the T2K experiment has presented the first appearance results in the antineutrino channel [10,11]. In the same days the experiment NOνA has released the first appearance data in the neutrino channel [12].
Although both measurements have a limited statistical power it is nonetheless interesting to consider their impact on the preexisting scenario. With this purpose, we here present both a 3-flavor and a 4-flavor analysis of the results of the two long-baseline experiments, updating our previous work [8].
We underline that the latest T2K and NOvA data considered in the present work are of particular importance.
Arguably, they represent the most important piece of information in the current neutrino landscape. In fact, these data allow us to probe for the first time in a direct (or manifest) way the leptonic CPV. It is the first time that a hint of CPV arises from the comparison of the observations of neutrinos and antineutrinos. In contrast, in all the previous analyses the indication on the CP-phase δ was indirect, since it arose from the combination of long-baseline (T2K) data and reactor data (Daya-Bay, RENO and Double-Chooz). Our analysis shows that the role of reactor data, albeit still significant, is now less pronounced in determining the estimate of the CP-phase δ. This trend is likely to be reinforced in the future since the current LBL appearance data represent only a small fraction (∼ 10%) of their total planned exposure.
We recall that the LBL experiments are sensitive to CPV because they can probe the interference of two distinct oscillating frequencies. In the 3-flavor case, the two frequencies in question are related to the atmospheric squaredmass splitting and to the solar one. In the 4-flavor case, the interference occurs between the fast (averaged) oscillations induced by the new O(eV 2 ) splitting and the atmospheric one. As first shown in [8], the new interference term has an amplitude comparable to that of the standard 3-flavor one. Therefore, it is possible to extract from LBL experiments 2 some information on the enlarged CPV sector involved in the 3+1 scheme. 3 In particular, it was shown that the data preferred the common value δ 13 ≃ δ 14 ∼ −π/2 (where δ 13 ≡ δ). Such a behavior was essentially driven by the slight mismatch existing between the two determinations of θ 13 achieved respectively with reactor and T2K data.
We also recall that the LBL experiments are sensitive to the NMH because of the MSW effect [25,26], which induce (tiny) modifications of the mass-mixing parameters in matter that depend on the sign of the atmospheric splitting. In a sense, also this phenomenon may be regarded as a kind of interference, occurring between the MSW potential (whose sign is known) and the atmospheric squared-mass splitting (whose sign is unknown and to be determined). The same phenomenon is at the basis of the planned measurements of the NMH with atmospheric neutri-nos [27,28,29,30], where there is the advantage that the CPV interference effects are less important and there is no degeneracy among the determination of CPV and NMH, which instead notoriously plagues the LBL measurements. It is worth mentioning that the interference of the MSW matter potential with the atmospheric frequency is also present in the propagation of solar neutrinos (see for example [31]), but unfortunately its amplitude is too small to extract any useful information from it (at least with the present data).
Our 3-flavor fit evidences a consolidation of the previous hints, namely a slight preference for sin δ < 0 and a mild preference (at more than 68% C.L.) for the normal hierarchy. In a 3+1 framework, the data are able to constrain two (of the three) CP-phases (δ 13 ≡ δ and δ 14 ), which exhibit a slight preference for the common value δ 13 ≃ δ 14 ≃ −π/2.
Interestingly, in the enlarged four neutrino scheme the preference for the normal hierarchy found within the 3-flavor framework completely disappears. This occurs because the mismatch between the two estimates of θ 13 provided by reactor and LBL experiments (more pronounced in the IH case) can be "cured" with an appropriated choice of the new CP-phase δ 14 . This circumstance indicates that light sterile neutrinos may constitute a potential source of fragility in the capability of the two LBL experiments of discriminating the neutrino mass hierarchy.
The rest of the paper is organized as follows. In Sec. II we introduce the theoretical framework needed to discuss the analytical behavior of the LBL ν µ → ν e andν µ →ν e transition probabilities in the 3-flavor and 4-flavor frameworks. In Sec. III we present the results of the numerical analysis. Finally, in Sec. IV we draw our conclusions.

Theoretical framework
In the presence of a fourth sterile neutrino ν s , the flavor and the mass eigenstates are connected through a 4 × 4 mixing matrix. A convenient parameterization of the mixing matrix is where R i j (R i j ) represents a real (complex) 4 × 4 rotation in the (i, j) plane containing the 2 × 2 submatrix in the (i, j) sub-block, with The parameterization in Eq. (1) enjoys the following properties: i) For vanishing mixing with the fourth state (θ 14 = θ 24 = θ 34 = 0) it returns the 3-flavor matrix in its usual parameterization. ii) The leftmost positioning of the matrixR 34 makes the vacuum ν µ → ν e conversion probability independent of θ 34 (and the related CP-phase δ 34 ). iii) For small values of θ 13 and of the mixing angles involving the fourth mass eigenstate, one has |U e3 | 2 ≃ s 2 13 , |U e4 | 2 = s 2 14 (exact), |U µ4 | 2 ≃ s 2 24 and |U τ4 | 2 ≃ s 2 34 , with a clear physical interpretation of the new mixing angles. Let us now come to the transition probability relevant for T2K and NOνA. From the discussion made in [8], it emerges that the conversion probability can be approximated as the sum of three terms The first term represents the positive definite atmospheric transition probability, the second term is related to the standard solar-atmospheric interference, while the third term is driven by the atmospheric-sterile interference. The probability depends on the three mixing angles s 13 , s 14 , s 24 ≃ 0.15, which can be all assumed to be of the same order ǫ and on the ratio of the solar and atmospheric squared-mass splittings α ≡ ∆m 2 12 /∆m 2 13 ≃ ±0.03, which is of order ǫ 2 . Keeping terms up to the third order, in vacuum, one finds where ∆ ≡ ∆m 2 13 L/4E is the atmospheric oscillating frequency, which depends on the baseline L and the neutrino energy E.
The matter effects slightly modify the transition probability, introducing a dependency on the ratio where is the (constant) matter potential along the neutrino trajectory. Both in T2K and in NOνA the value of v is relatively small, being v ∼ 0.05 in T2K, and v ∼ 0.17 in NOνA, where we have taken as a benchmark the peak energy of the two neutrino beams (E = 0.6 GeV in T2K, E = 2 GeV in NOνA). Therefore, it is natural to treat v as a small parameter 4 of order ǫ. The probability in matter can be simply obtained (see the appendix in [8] and the works [32,33,34]) by making in the vacuum expression the following substitutions where s m 13 and ∆ m represent the mixing angle s 13 and the atmospheric oscillation frequency in matter, which for small values of v, take the form It can be shown that at the peak energy, where |∆| ≃ π/2, the substitutions in Eqs. (10) -(12) produce third order corrections only in the atmospheric term, which reads while the same substitutions in the two interference terms produce only corrections of the fourth order.
The swap of the NMH is parametrized by the replacements One can observe that, while in general, the two interference terms are not invariant under a swap of the NMH, around the oscillation maximum |∆| = π/2, such an invariance is approximately valid. Therefore, in practice the effect of the NMH swap is captured by the modification of the leading term in Eq. (13). Due to the change of sign of v, the transition probability (which acquires the dominant contribution from the atmospheric term) increases (decreases) with respect to the vacuum case in the NH (IH) case. NOνA is expected to be more sensitive than T2K to the NMH because of the larger value of the ratio v.
Finally, we recall that the transition probability for antineutrinos is obtained from that of neutrinos by changing the sign of the MSW potential V and of all the CP-phases. This, for a given choice of the NMH, corresponds to the substitutions 5 In the NH case v > 0 for neutrinos and v < 0 for antineutrinos. According to Eq. (13), in the NH case the leading contribution to the transition probability will increase (decrease) for neutrinos (antineutrinos). In the IH case the opposite conclusion holds. When passing from neutrinos to antineutrinos, the first interference term (around the oscillation maximum) changes sign, since it is essentially proportional to − sin δ 13 . In contrast, the second interference term remains approximately invariant, since it depends on cos(δ 13 − δ 14 ).
Remarkably, for typical values of the mixing angles preferred by the current global 3+1 fits, the amplitude of the (atmospheric-sterile) interference term is almost identical to that of the standard (solar-atmospheric) interference term. As a consequence, a substantial impact on the regions reconstructed by the experiments T2K and NOνA in the pane of the two parameters [θ 13 , δ 13 ] is expected. In addition, a similar sensitivity to the two CP-phases δ 13 and δ 14 is expected in the combination of the two LBL experiments with the reactor data.

Numerical Analysis
In our numerical analysis we include the reactor experiments Daya-Bay and RENO and the two LBL experiments T2K and NOνA. The analysis of LBL data is slightly different for the two cases of three and four flavors, since in this last case there are appreciable oscillation effects not only at the far detector but also at the near detector, which have been taken into account as described in detail in [8].
The analysis of the reactor experiments is performed using the total rate information and following the approach described in [35]. For both experiments we have used the latest data [2,3] based, respectively, on 621 live days (Daya Bay) and 800 live days (RENO). Since the electron neutrino survival probability probed by these experiments is independent of the CP-violating phases (standard and non-standard), their estimate of θ 13 is independent of them.
We recall that such an estimate is extracted using the ratio of the event rates measured at the far and at the near sites.
Since the fast oscillations induced by ∆m 2 14 are averaged out at both detector sites, Daya Bay and RENO are not sensitive to 4-flavor effects. As a result, their estimate of θ 13 is independent of the mixing angle θ 14 as long as it is allowed to vary in the range we are exploring.
Concerning the LBL experiments, we use the T2K results of the ν µ → ν e appearance searches [1], which reported 28 events with an estimated background of 4.92 events and the preliminary results of theν µ →ν e searches presented at the EPS HEP conference [10,11], which report 3 events with an estimated background of 1.68 events. For NOνA we use the preliminary results presented at Fermilab [12]. It must be noticed that the NOνA collaboration, prior to unbinding the data, decided to show the results obtained with two different event selection procedures (LID and LEM). The first is considered the primary method by the collaboration so we conservatively use the results obtained with such a method, which selects 6 events over an estimated background of 0.94 events. For completeness, we will comment on the results that are obtained by using the secondary LEM selector, which identifies a considerably larger number (11) of events (the background being very similar to the LID case) in comparison to the primary LID method.
In order to calculate the theoretical expectation for the total number of T2K ν e events and their binned spectrum in the reconstructed neutrino energy, we convolve the product of the ν µ flux [36] with horns operating in the +250 kA mode (tables provided on the T2K home page [37]), the charged current quasi elastic (CCQE) cross-section (estimated from [1]), and the ν µ → ν e transition probability, with an appropriate energy resolution function. A similar procedure is followed for the antineutrino channel, where we have used theν µ fluxes with horns operating in the -250 kA mode plotted in [11]. Concerning NOνA we used directly the product of the neutrino fluxes with the cross section (extrapolated at the far detector) provided in [12]. We have checked that our prediction for the binned spectra of events are in good agreement with those shown by the collaborations. We have performed the 3-flavor analysis both using the total rate information and the full spectrum, observing very small differences between the two methods. This is due to the limited statistics currently available, and to the effect of the smearing induced by the energy resolution. As explained in [8], in the 4-flavor case, from outside the collaborations, it is possible to consistently perform only a total rate analysis. Therefore, for homogeneity, also in the 3-flavor case, we report the results obtained with the total rate information.

Results of the 3-flavor analysis
In the 3-flavor analysis, the two mixing angles (θ 13 , θ 23 ) and the CP-phase δ 13 are treated as free parameters, taking into account the external prior sin 2 θ 23 = 0.51 ± 0.05 provided by the ν µ → ν µ disappearance measurement [38] performed by T2K. For the atmospheric mass splitting we use the best fit value |∆m 2 13 | = 2.4 × 10 −3 eV 2 obtained in the same analysis. The solar mass-mixing parameters are fixed at the best fit value obtained in the global analysis [5]. Figure 1 shows the results of the analysis for the two cases of NH (upper panels) and IH (lower panels) in the plane spanned by the two variables [sin 2 2θ 13 , δ 13 ], the atmospheric mixing angle θ 23 having been marginalized away. The left panels report the regions allowed by the combination of T2K (ν e andν e ) and NOνA (ν e ) for the confidence levels The narrow vertical band displayed in both panels represents the range allowed at 68% C.L. for θ 13 by the reactor experiments. It is interesting to note that at the 68% C.L., the contours determined by the combination of T2K and NOνA are closed regions around the best fit value δ 13 ≃ −π/2. This means, that thanks to the information coming from the two channels (ν µ → ν e andν µ →ν e ) the LBL experiments are starting to probe the CP symmetry in a direct way. This situation is qualitatively different from the preexisting one, where the extraction of the information on δ 13 was indirect, i.e. not based upon a manifest observation of CPV. In fact, the previous hints in favor of δ 13 ≃ −π/2 derived from the combination of T2K (ν e ) and reactor data. Fig. 1 shows appreciable differences between the two cases of NH and IH, which can be traced to the presence of the matter effects. As discussed in Sec. II, these tend to increase (decrease) the theoretically expected ν e rate in the case of NH (IH). The opposite is true forν e 's but their weight in the analysis is small, so the neutrino datasets dominate.
In addition, as discussed in Sec. II, the NOνA ν e data are more sensitive than the T2K ν e data to the matter effects. Therefore, the neutrino datasets are far more relevant for what concern the sensitivity to NMH. More specifically, the following differences among the two hierarchies emerge, which will persist also in the 4-flavor analysis. The regions obtained for the case of IH: i) are shifted towards larger values of θ 13 . ii) are slightly wider in the variable θ 13 with respect to those obtained in the NH case, and iii) tend to prefer (reject) values of sin δ 13 < 0 (sin δ 13 > 0) in a more pronounced way.
The two right panels of Fig. 1 show that the combination of the reactor experiments with LBLs tends to further reinforce the preference for values of δ ∼ −π/2, disfavoring the case of no CPV (δ 13 = 0, π) at roughly the 90% C.L. In addition, we note that the weak preference for the case of normal hierarchy tends to consolidate, being to be compared with the result χ 2 NH − χ 2 IH = −0.8 obtained in our previous work [8]. However, the statistical significance of the indication is still small and below the 90% C.L.
A final remark is in order concerning the two events selection methods used by NOνA. As already stressed above, we have conservatively adopted the results obtained with the primary selector, based on a likelihood identification (LID) method. By adopting the results obtained with the alternative method, based on a library event matching (LEM) selector, which identifies a relatively larger number of ν e events (11 vs 6), we find that both the indication on the CP-phase δ 13 and that on the NMH are slightly enhanced. More precisely, the rejection of the CP-conservation cases (δ 13 = 0, π) and the rejection of the inverted hierarchy both rise, respectively, at about the 2σ level and 1.5σ level.
A combination of the results obtained with the two events selection methods is not trivial because they are strongly correlated. However, one can guess that such a combined analysis would provide results which are intermediate between those obtained using the two methods. As a benchmark we also report the range allowed for θ 13 by reactors, which is identical to the standard case. A quick comparison of the four panels of Fig. 2 with the 3-flavor case (left upper panel of Fig. 1) shows the noticeable impact of the 4-flavor effects on the allowed regions. The behavior of the curves can be qualitatively understood taking into account that the dominant contribution to the total rate comes from a region of the energies close to the first oscillation maximum, where ∆ ∼ π/2, and that the neutrino datasets dominate over the antineutrino one. Inspection of Eq. (6) shows that the standard interference term is proportional to − sin δ 13 . From Eq. (7) we see that for δ 14 = π/2, the new interference term is proportional to sin δ 13 . Therefore, in this case the two terms are in opposition of phase and having similar amplitude tend to cancel out, making the wiggles structure almost to disappear (right upper panel). Vice versa, for δ 14 = −π/2 (right lower panel) the two inference terms have the same phase and the horizontal excursion of the "wiggles" is increased. In the two cases δ 14 = 0, π (left panels) the new interference term is proportional to ± cos δ 13 = ± sin(π/2 − δ 13 ) and thus it has a ±π/2 difference of phase with respect to the standard one. As a result, in such two cases, the behavior of the allowed regions is intermediate between the two cases δ 14 = (−π/2, π/2). It is interesting to note how, in the presence of 4ν effects, a better agreement among the two estimates of θ 13 derived from reactors and LBLs can be obtained. In particular, this occurs for δ 14 ≃ −π/2. As we have discussed for the 3-flavor case, the mismatch of the θ 13 estimates from LBLs and reactors tends to disfavor the inverted hierarchy.

Results of the 4-flavor analysis
The same is no more true in the 4-flavor scheme, since the two estimates can be brought in agreement for an appro-  projections having as one of the two variables the mixing angle θ 13 . As expected, there is a similar sensitivity to both CP-phases with a preference for values of δ 13 ∼ δ 14 ∼ −π/2. The case δ 13 = 0 is disfavored at a slightly lower confidence level in comparison with the 3-flavor case. This is imputable to the degeneracy among the two CP-phases.

Conclusions and Outlook
We have investigated the impact of the latest data released by the two long-baseline experiments T2K and NOνA.
Specifically, we have included in our analysis the firstν e appearance data from T2K and the first ν e appearance results from NOνA. We have considered the implications of such new results both for the standard 3-flavor framework and for the non-standard 3+1 scheme involving one sterile neutrino species, which is the most popular scenario invoked to explain the short-baseline anomalies (see [39,40,41]).
The 3-flavor analysis shows a consolidation of the preexisting trends, namely a slight preference for sin δ 13 < 0 disfavoring the cases of CP conservation (δ = 0, π) at a confidence level close to 90% C.L., and a mild preference (at more than 68% C.L.) for the normal hierarchy. These hints are slightly reinforced if one adopts the NOνA results obtained with the secondary event selection method (LEM). In a 3+1 framework, two CP-phases (δ 13 ≡ δ and δ 14 ) are constrained, which exhibit a preference for the common value δ 13 ≃ δ 14 ≃ −π/2. Interestingly, in the enlarged four neutrino scheme the preference for the NH found within the 3-flavor framework completely disappears because the mismatch between the two estimates of θ 13 from LBL's and reactors can be lifted (an intriguing circumstance) if the new CP-phase δ 14 lies around its best value −π/2. This fact indicates that light sterile neutrinos may constitute a potential source of fragility in the capability of the two LBL experiments of discriminating between the two neutrino mass hierarchies.
Our results show once again that LBL experiments can give an important contribution in the context of sterile neutrino searches, which should not considered only a short-baseline "affair". In particular, no information on the new CP-phase δ 14 can be extracted from SBL setups, where the interference between the new splitting and the atmospheric one cannot develop (or is completely negligible). Therefore, the two classes of experiments (SBL and LBL) are complementary and synergic in constraining the 4-flavor parameter space. While a discovery of a sterile neutrino can come only from the observation of the oscillating pattern (in energy or/and space) at the new SBL experiments, the full exploration of the 4ν model will become possible only with the contribution of LBL setups, which are sensible interferometers able to probe the new enlarged CPV sector. Also the (present and future) atmospheric neutrino data may help to shed light on this issue, since these neutrinos traverse very long baselines and may present some sensitivity to the new CPV phenomena.
Finally, we would like to underline the other side of the coin, namely the fact that the presence of sterile neutrinos tend to decrease the robustness of the conclusions reached in the simple 3-flavor framework. The present data already show that this is the case for the current hint in favor of NH, which disappears in the 3+1 scheme. In this respect, it would be very interesting to study how the NMH discrimination capabilities of future data expected to come from T2K and NOνA, and from the planned LBL experiments (DUNE, LBNO and T2HK) will be affected by the inclusion of the sterile neutrino effects.