Global Analyses of Neutrino Oscillation Experiments

We summarize the determination of some neutrino properties from the global analysis of solar, atmospheric, reactor, and accelerator neutrino data in the framework of three-neutrino mixing as well as in some extended scenarios such as the mixing with eV-scale sterile neutrinos invoked for the interpretation of the short baseline anomalies, and the presence of non-standard neutrino interactions.


Introduction: the New Minimal Standard Model
Thanks to remarkable discoveries by a number of neutrino oscillation experiments it is now an established fact that neutrinos have mass and leptonic flavors are not symmetries of Nature [1,2]. Historically neutrino oscillations were first observed in the disappearance of solar ν e 's and atmospheric ν µ 's which could be interpreted as flavor oscillations with two very different wavelengths. Over the last 15 years, these effects were confirmed also with terrestrial experiments using man made beams from accelerators and nuclear reactors (see Ref. [3] for an overview). In brief, at present we have observed neutrino oscillation effects in: • atmospheric neutrinos, in particular in the high-statistics results of Super-Kamiokande [4]; • event rates of solar neutrino radiochemical experiments Chlorine [5], Gallex/GNO [6] and SAGE [7], as well as time and energy dependent rates from the four phases in Super-Kamiokande [8][9][10][11], the three phases of SNO [12], and Borexino [13,14]; • disappearance results from accelerator long baseline (LBL) experiments in the form of the energy distribution of ν µ andν µ events in MINOS [15] and T2K [16], and ν µ events in NOνA [17]; • LBL ν e appearance results for both neutrino and antineutrino events in MINOS [18], and ν e appearance in NOνA [17] and T2K [19]; • reactorν e disappearance at medium baselines in the form of the energy distribution of events in Double Chooz [20], Daya Bay [21] and RENO [22]; • the energy spectrum of reactorν e disappearance at LBL in KamLAND [23].
These results imply that neutrinos are massive and there is physics beyond the Standard Model (SM). The fundamental question arises, what is the underlying theory for neutrino masses. In this article, however, we will focus on the more mundane but difficult approach of the detailed determination of the simplest low energy parametrization(s) required to describe the bulk of data. The SM is a gauge theory based on the gauge symmetry SU(3) C × SU(2) L × U(1) Y spontaneously broken to SU(3) C × U(1) EM by the the vacuum expectation value of a Higgs doublet field φ. The SM contains three fermion generations which reside in chiral representations of the gauge group. Right-handed fields are included for charged fermions as they are needed to build the electromagnetic and strong currents. However, no right-handed neutrinos are included in the model since neutrinos are neutral and colourless and therefore the right-handed neutrinos are singlets of the SM group.
In the SM, fermion masses arise from the Yukawa interactions which couple the right-handed fermion singlets to the left-handed fermion doublets and the Higgs doublet. After spontaneous electroweak symmetry breaking these interactions lead to charged fermion masses but leave the neutrinos massless. No Yukawa interaction can be written that would give a tree level mass to the neutrino because no right-handed neutrino field exists in the model. Furthermore, within the SM G global SM = U(1) B × U(1) e × U(1) µ × U(1) τ is an accidental global symmetry. Here U(1) B is the baryon number symmetry, and U(1) e,µ,τ are the three lepton flavor symmetries. Any neutrino mass term which could be built with the particle content of the SM would violate the U(1) L subgroup of G global SM and therefore cannot be induced by loop corrections. Also, it cannot be induced by non-perturbative corrections because the U(1) B−L subgroup of G global SM is non-anomalous. It follows then that the SM predicts that neutrinos are strictly massless. Consequently, there is neither mixing nor CP violation in the leptonic sector. Clearly this is in contradiction with the neutrino data summarized above. So the Standard Model has to be extended at least to include neutrino masses. This minimal extension is what we call the New Minimal Standard Model (NMSM).
The two minimal extensions to give neutrino mass and explain the data are: • to introduce ν R and impose total lepton number (L) conservation. After spontaneous electroweak symmetry breaking we have: In this case mass eigenstate neutrinos are Dirac fermions, i.e., ν c ν; • to construct a mass term only with the SM left-handed neutrinos by allowing L violation: In this case the mass eigenstates are Majorana fermions, ν c = ν. Note that the Majorana mass term above breaks the electroweak gauge invariance, and therefore spoils the renormalizability of the model. In this respect L M can only be understood as a low energy limit of a complete theory, whereas L D is formally self-consistent.
Either way, in the NMSM flavour is mixed in the CC interactions of the leptons, and a leptonic mixing matrix appears analogous to the CKM matrix for the quarks. However the discussion of leptonic mixing is complicated by two factors. First the number massive neutrinos (n) is unknown, since there are no constraints on the number of right-handed (SM-singlet) neutrinos. Second, since neutrinos carry neither color nor electromagnetic charge, they could be Majorana fermions. As a consequence the number of new parameters in the model depends on the number of massive neutrino states and on whether they are Dirac or Majorana particles.
In general, if we denote the neutrino mass eigenstates by ν i , i = 1, 2, . . . , n, and the charged lepton mass eigenstates by l i = (e, µ, τ), in the mass basis, leptonic CC interactions are given by Here U is a 3 × n matrix which verifies UU † = I 3×3 but in general U † U I n×n . This is the case, for example, when considering mixing with non-doublet states, such as discussed in Sec. 4.
In what follows we will review the status of the analysis of the oscillation neutrino data in different frameworks. In Sec. 2 we present the results for the case of three-neutrino mixing, and in Sec. 3 we discuss the implications of such results for observables sensitive to the absolute neutrino mass scale. In Sec. 4 we focus on extended scenarios involving mixing with eV-scale sterile neutrinos, as invoked for the interpretation of the short baseline anomalies. In Sec. 5 we derive limits on the presence of non-standard neutrino-matter interactions.

Analysis in the framework of three-neutrino mixing
The wealth of data listed in the introduction can be consistently described by assuming mixing among the three known neutrinos (ν e , ν µ , ν τ ), which can be expressed as quantum superpositions of three massive states ν i (i = 1, 2, 3) with masses m i . As explained in the previous section this implies the presence of a leptonic mixing matrix in the weak charged current interactions which can be parametrized as [24]: where c i j ≡ cos θ i j and s i j ≡ sin θ i j . In addition to the Dirac-type phase δ CP , analogous to that of the quark sector, there are two extra phases α 1 , α 2 associated to a possible Majorana character of neutrinos. Such phases, however, are not relevant for neutrino oscillations. In this convention, disappearance of solar ν e 's and long baseline reactorν e 's proceeds dominantly via oscillations with wavelength ∝ E/∆m 2 21 (∆m 2 i j ≡ m 2 i − m 2 j and ∆m 2 21 ≥ 0 by convention) and amplitudes controlled by θ 12 , while disappearance of atmospheric and LBL accelerator ν µ 's proceeds dominantly via oscillations with wavelength ∝ E/|∆m 2 31 | E/∆m 2 21 and amplitudes controlled by θ 23 . The angle θ 13 controls the amplitude of oscillations involving ν e flavor with E/|∆m 2 31 | wavelengths. Given the observed hierarchy between the solar and atmospheric wavelengths there are two possible non-equivalent orderings for the mass eigenvalues, which are conventionally chosen as As it is customary we refer to the first option, Eq. (5), as Normal Ordering (NO), and to the second one, Eq. (6), as Inverted Ordering (IO); in this form they correspond to the two possible choices of the sign of ∆m 2 31 . In this convention the angles θ i j can be taken without loss of generality to lie in the first quadrant, θ i j ∈ [0, π/2], and the CP phase δ CP ∈ [0, 2π]. In the following we adopt the (arbitrary) convention of reporting results for ∆m 2 31 for NO and ∆m 2 32 for IO, i.e., we always use the one which has the larger absolute value. Sometimes we will generically denote such quantity as ∆m 2 3 , with = 1 for NO and = 2 for IO. In summary, the 3ν oscillation analysis of the existing data involves six parameters: 2 mass differences (one of which can be positive or negative), 3 mixing angles, and the CP phase δ CP . For the sake of clarity we summarize in Table 1 which experiment contribute dominantly to the present determination of the different parameters.
The consistent determination of these leptonic parameters requires a global analysis of the data described above. Such global fits are presently performed by a few phenomenological groups [25][26][27]; here we summarize the results from Ref. [27,28]. We show in Fig. 1  global analysis as a function of each of the six parameters. The corresponding best fit values and the derived ranges for the six parameters at the 1σ (3σ) level are given in Tab. 2. For each parameter the curves and ranges are obtained after marginalizing with respect to the other five parameters. The results in the table are shown for three scenarios. In the first and second columns we assume that the ordering of the neutrino mass states is known "a priori" to be Normal or Inverted, respectively, so the ranges of all parameters are defined with respect to the minimum in the given scenario. In the third column we make no assumptions on the ordering, so in this case the parameter ranges are defined with respect to the global minimum (which corresponds to Inverted Ordering) and are obtained marginalizing also over the ordering. For this third case we only give the 3σ intervals. Of course in this case the range of ∆m 2 3 is composed of two disconnected intervals, one one containing the absolute minimum (IO) and the other the secondary local minimum (NO).
As already mentioned, all the data described above can be consistently interpreted as oscillations of the three known active neutrinos. In addition to these data, however, several anomalies at short baselines (SBL) have been observed which cannot be explained as 3ν oscillations, but could be interpreted as oscillations involving an O(eV) mass sterile state. They will be discussed in detail in Sec. 4. For what concerns the analysis presented here the only SBL effect which has to be taken into account is the so called reactor anomaly. It turns out that the most recent reactor flux calculations [29,30] fall short at describing the results from reactor experiments at baselines 100 m, such as Bugey4 [31], ROVNO4 [32], Bugey3 [33], Krasnoyarsk [34,35], ILL [36], Gösgen [37], SRP [38], and ROVNO88 [39]. Such reactor short-baseline experiments (RSBL) do not contribute to oscillation physics in the 3ν framework, but they play an important role in constraining the unoscillated reactor neutrino flux if they are used instead of the theoretically calculated reactor fluxes. Thus to account for the possible effect of the reactor anomaly in the determined ranges of neutrino parameters we show the results in Fig. 1 for two extreme choices. The first option (labeled "Free+RSBL") is to leave the normalization of reactor fluxes free and include the RSBL data. The second option (labeled "Huber") is not to include short-baseline reactor data but assume reactor fluxes and uncertainties as predicted in [30].
From the results in the figure and table we conclude that: 1. if we define the 3σ relative precision of a parameter by 2(x up − x low )/(x up + x low ), where x up (x low ) is the upper (lower) bound on a parameter x at the 3σ level, from the numbers in the table we find 3σ relative precision of 14% (θ 12 ), 32% (θ 23 ), 15% (θ 13 ), 14% (∆m 2 21 ) and 11% (|∆m 2 3 |) for the various oscillation parameters; 2. for either choice of the reactor fluxes the global best fit corresponds to IO with sin 2 θ 23 > 0.5, while the second local minima is for NO with sin 2 θ 23 < 0.5; 3. the statistical significance of the preference for Inverted versus Normal ordering is quite small, ∆χ 2 1; 4. the present global analysis disfavors θ 13 = 0 with a ∆χ 2 ≈ 500. Such impressive result is mostly driven by the reactor data from Daya Bay, with secondary contributions from RENO and Double Chooz; 5. the uncertainty on θ 13 associated with the choice of reactor fluxes is at the level of 0.5σ in the global analysis. This is so because the most precise results from Daya Bay and RENO are independent of the reactor flux normalization; 6. a non-maximal value of the θ 23 mixing is slightly favored, at the level of ∼ 1.4σ for Inverted Ordering at of ∼ 1.0σ for Normal Ordering; 7. the statistical significance of the preference of the fit for the second (first) octant of θ 23 is ≤ 1.4σ (≤ 1.0σ) for IO (NO); 8. the best fit for δ CP for all analyses and orderings occurs for δ CP 3π/2, and values around π/2 are disfavored with ∆χ 2 6. Assigning a confidence level to this ∆χ 2 is non-trivial, due to the non-Gaussian behavior of the involved χ 2 function, see Refs. [27,40] for discussions and a Monte Carlo studies.
These results are robust with respect to changes in the statistical interpretation. The Bayesian analysis performed in [41] leads to quantitatively very similar results.
From this global analysis one can also derive the 3σ ranges on the magnitude of the elements of the leptonic mixing matrix to be: The present status of the determination of leptonic CP violation is further illustrated in Fig. 2. On the left panel we show the dependence of ∆χ 2 of the global analysis on the Jarlskog invariant which gives a convention-independent measure of CP violation [42], defined as:  Table 2. Three-flavor oscillation parameters from our fit to global data after the NOW 2014 conference [27]. The results are presented for the "Free Fluxes + RSBL" in which reactor fluxes have been left free in the fit and short baseline reactor data (RSBL) with L 100 m are included. The numbers in the 1st (2nd) column are obtained assuming NO (IO), i.e., relative to the respective local minimum, whereas in the 3rd column we minimize also with respect to the ordering. Note that ∆m 2 3 ≡ ∆m 2 31 > 0 for NO and ∆m 2 3 ≡ ∆m 2 32 < 0 for IO. at 1σ (3σ) for both orderings. The preference of the present data for non-zero δ CP implies a best fit J best CP = −0.032, which is favored over CP conservation at the ∼ 1.2σ level. These numbers can be compared with the size of the Jarlskog invariant in the quark sector, which is determined to be J quarks CP = (3.06 +0. 21 −0.20 ) × 10 −5 [24]. In the right panel of Fig. 2 we recast the allowed regions for the leptonic mixing matrix in terms of one leptonic unitarity triangle. Since in the analysis U is unitary by construction, any given pair of rows or columns can be used to define a triangle in the complex plane. In the figure we show the triangle corresponding to the unitarity conditions on the first and third columns which is the leptonic analogous to the one usually showed for the quark sector. In this figure the absence of CP violation implies a flat triangle, i.e., Im(z) = 0. As can be seen, the horizontal axis marginally crosses the 1σ allowed region, which for 2 dof corresponds to ∆χ 2 2.3. This is consistent with the present preference for CP violation, χ 2 (J CP = 0) − χ 2 (J CP free) = 1.5. A detailed discussion of the status of the CP phase from present data can be found in Ref. [40].

Absolute neutrino mass measurements
Oscillation experiments provide information on the mass-squared splittings ∆m 2 i j and on the leptonic mixing angles U i j , but they are insensitive to the absolute mass scale for the neutrinos. Of course, the results of an oscillation experiment do provide a lower bound on the heavier mass in ∆m 2 i j , |m i | ≥ ∆m 2 i j for ∆m 2 i j > 0, but there is no upper bound on this mass. In particular, the corresponding neutrinos could be approximately degenerate at a mass scale that is much higher than ∆m 2 i j . Moreover, there is neither an upper nor a lower bound on the lighter mass m j .
Information on the neutrino masses, rather than mass differences, can be extracted from kinematic studies of reactions in which a neutrino or an anti-neutrino is involved. In the presence of mixing the most relevant constraint comes from the study of the end point (E ∼ E 0 ) of the electron spectrum in Tritium beta decay 3 H → 3 He + e − +ν e . This spectrum can be effectively described by a single parameter, m ν e , if for all   neutrino states E 0 − E m i . In this case: where R(E) contains all the m ν -independent factors, and where the second equality holds if unitarity is assumed. At present we only have an upper bound, m ν e ≤ 2.2 eV at 95% CL [43], which is expected to be superseded soon by KATRIN [44] with about one order of magnitude improvement in sensitivity. Direct information on neutrino masses can also be obtained from neutrinoless double beta decay (A, Z) → (A, Z + 2) + e − + e − . This process violates lepton number by two units, hence in order to induce the 0νββ decay ν's must Majorana particles. In particular, for the case in which the only effective lepton number violation at low energies is induced by the Majorana mass term for the neutrinos, the rate of 0νββ decay is proportional to the effective Majorana mass of ν e : m ee = i m i U 2 ei = m 1 c 2 13 c 2 12 e i2α 1 + m 2 c 2 13 s 2 12 e i2α 2 + m 3 s 2 13 e −i2δ CP (12) which, unlike Eq. (11), depends also on the three CP violating phases. Recent searches carried out with 76 Ge (GERDA experiment [45]) and 136 Xe (KamLAND-Zen [46] and EXO-200 [47] experiments) have established the lifetime of this decay to be longer than 10 25 yr, corresponding to a limit on the neutrino mass of m ee ≤ 0.2 − 0.4 eV at 90% CL. A series of new experiments is planned with sensitivity of up to m ee ∼ 0.01 eV [48]. Neutrino masses have also interesting cosmological effects. In general, cosmological data mostly give information on the sum of the neutrino masses, i m i , while they have very little to say on their mixing structure and on the ordering of the mass states.
Correlated information on these three probes of the neutrino mass scale can be obtained by mapping the results from the global analysis of oscillations presented previously. We show in Fig. 3 the present status of this exercise. The relatively large width of the regions in the right panel are due to the unknown Majorana phases. Thus from a positive determination of two of these probes information can be obtained on the value of the Majorana phases and/or the mass ordering.

Sterile neutrinos at the eV scale
Besides the huge success of three-flavour oscillations described in Sec. 2 there are some anomalies which cannot be explained within the 3ν framework and which might point towards the existence of additional neutrino flavors (so-called sterile neutrinos) with masses at the eV scale: • the LSND experiment [49] reports evidence forν µ →ν e transitions with E/L ∼ 1 eV 2 , where E and L are the neutrino energy and the distance between source and detector, respectively; • this effect is also searched for by the MiniBooNE experiment [50], which reports a yet unexplained event excess in the low-energy region of the electron neutrino and anti-neutrino event spectra. No significant excess is found at higher neutrino energies. Interpreting the data in terms of oscillations, parameter values consistent with the ones from LSND are obtained; • radioactive source experiments at the Gallium solar neutrino experiments SAGE and GALLEX have obtained an event rate which is somewhat lower than expected. This effect can be explained by the hypothesis of ν e disappearance due to oscillations with ∆m 2 1 eV 2 ("Gallium anomaly") [51,52]; • state-of-the-art calculations of the neutrino flux emitted by nuclear reactors [29,30] predict a neutrino rate which is a few percent higher than observed in short-baseline (L 100 m) reactor experiments. A decreased rate at those distances can be explained by assumingν e disappearance due to oscillations with ∆m 2 ∼ 1 eV 2 ("reactor anomaly") [53].
Here we report the results of a global analysis from Ref. [54] of those data under the hypothesis of additional neutrino species at the eV scale (see [55,56] for similar analyses). We introduce a neutrino state, ν 4 , with a mass-squared difference ∆m 2 41 much larger than |∆m 2 31 |. This situation is called "3+1 mass scheme". In this case the oscillation probabilities for experiments exploring the range E/L ∼ 1 eV 2 are rather simple: where ∆ ≡ ∆m 2 41 L/4E and the effective mixing angles are defined as with α = e, µ and U α4 are the elements of the lepton mixing matrix describing the mixing of the 4th neutrino mass state with the electron and muon flavour. There is no CP violation in 3+1 SBL oscillations and those relations apply for neutrinos as well as antineutrinos. Neglecting quadratic terms in the mixing matrix elements one has the following relation between the effective amplitudes relevant for appearance and disappearance probabilities: 4 sin 2 2θ µe ≈ sin 2 2θ ee sin 2 2θ µµ . . Allowed regions at 95% CL (2 dof) for 3+1 oscillations. We show SBL reactor data [31][32][33][34][35][36][37][38][39] (blue shaded), Gallium radioactive source data [6,[57][58][59] (orange shaded), ν e disappearance constraints from ν e -12 C scattering data from LSND and KARMEN [60,61] (dark red dotted), long-baseline reactor data from CHOOZ, Palo Verde, DoubleChooz, Daya Bay and RENO (blue short-dashed) and solar+KamLAND data (black long-dashed). The red shaded region is the combined region from all these ν e andν e disappearance data sets. See Ref. [54] for details.
Dividing the relevant data into ν e disappearance, ν µ disappearance, and ν µ → ν e appearance searches, this relation implies that the system is over-constrained. Indeed, as will be discussed below, there is significant tension in the global data and Eq. (15) makes it difficult to obtain a good fi to all available data. We consider first the global data including SBL anomalies related toν e and ν e disappearance (reactor and Gallium anomalies) but ignoring for the time being the ν µ → ν e andν µ →ν e appearance anomalies (LSND and MiniBooNE). In this case the relevant SBL phenomenology is determined by the two parameters ∆m 2 41 and |U e4 |. The allowed regions for them is shown in Fig. 4. We find that a consistent region emerges (shown in red), not in conflict with any other data. The best fit point occurs at sin 2 2θ ee = 0.09 and ∆m 2 41 = 1.78 eV 2 , and the no-oscillation hypothesis for the eV-scale is excluded at 3.1σ (∆χ 2 = 12.9/2 dof), driven by the reactor and Gallium anomalies. The θ 13 determination is rather stable with respect to the presence of sterile neutrinos, up to an ambiguity at the level of less than 1σ (see also discussion in section 2). We note, however, that its interpretation becomes slightly more complicated. For instance, using a particular parametrization [54] for the 3+1 scheme, the relation between mixing matrix elements and mixing angles is |U e3 | = cos θ 14 sin θ 13 and |U e4 | = sin θ 14 . Hence, the one-to-one correspondence between |U e3 | and θ 13 as in the three-flavor case is spoiled.
We now address the question whether the hints for ν e disappearance can be reconciled with the appearance hints from LSND and MiniBooNE. As mentioned above, Eq. (15) links those appearance signals to disappearance in the ν e as well as ν µ channels. Despite the possible signal in ν e disappearance, so-far no positive signal has been observed in ν µ disappearance and several experiments set bounds on the relevant mixing parameter |U µ4 |, see Fig. 5 (left). Hence, the combined limits on ν µ and ν e mixing with the eV-scale mass state lead to a tension between appearance signals and disappearance data in the 3+1 scheme. Such tension is illustrated for global data in the right panel of Fig. 5, where we show the allowed region for all appearance experiments, compared to the limit from disappearance experiments in the plane of sin 2 2θ µe and ∆m 2 41 . The preferred values of ∆m 2 41 for disappearance data come from the reactor and Gallium anomalies. The regions for disappearance data, however, are not closed in this projection in the parameter space and include sin 2 2θ µe = 4|U e4 U µ4 | 2 = 0, which can always be achieved by letting U µ4 → 0 due to the non-observation of any positive signal in SBL ν µ disappearance. The upper bound on sin 2 2θ µe from disappearance emerges essentially as the product of the upper bounds on |U e4 | and |U µ4 | from ν e and ν µ disappearance according to Eq. (15). We observe from the plot the clear tension between those data sets, with only marginal overlap  [62], atmospheric neutrinos [63], MiniBooNE disappearance [64], MINOS CC and NC data [65,66], and the combination of them. In red we show the region preferred by LSND and MiniBooNE appearance data combined with reactor and Gallium data, where for fixed |U µ4 | 2 we minimize with respect to |U e4 | 2 . Right: Comparison of the parameter region preferred by appearance data (LSND [49], MiniBooNE appearance analysis [50], NOMAD [67], KARMEN [68], ICARUS [69], E776 [70]) to the exclusion limit from disappearance data (atmospheric, solar, reactors, Gallium, CDHS, MINOS, MiniBooNE disappearance, KARMEN and LSND ν e − 12 C scattering). See Ref. [54] for details. regions at above 99% CL around ∆m 2 41 ≈ 0.9 eV 2 and at 3σ around ∆m 2 41 ≈ 6 eV 2 . We find that the global 3+1 fit leads to χ 2 min /dof = 712/680 with a p-value 19%, whereas the so-called parameter goodness of fit (PG) test [71] indicates that appearance and disappearance data are consistent with each other only with a p-value of about 10 −4 .
A valid question to ask is whether the situation improves if more neutrino states at the eV scale are introduced. Consider the hypothesis of 2 states with eV scale mass splittings, ν 4 and ν 5 , which can be arranged either such that ∆m 2 41 and ∆m 2 51 are both positive ("3+2") and where one of them is negative ("1+3+1"). The new qualitative feature in those 5-neutrino schemes is CP violation at the E/L ∼ eV 2 scale [72,73], which introduces some freedom in fitting neutrino versus anti-neutrino data from LSND and MiniBooNE together. However, the main prediction from the 4-neutrino case remains valid also for 5-neutrinos: a non-zero ν µ → ν e appearance at SBL necessarly predicts SBL disappearance for ν e as well as ν µ . Indeed, the tension between appearance and disappearance data remains severe, and a PG analysis gives a consistency below 10 −4 for 3+2, whereas for 1+3+1 consistency at the 2 permil level can be achieved [54].
In summary, several anomalies at the level of 3σ do not fit into the three-flavour picture and might indicate additional neutrino states at the eV scale. While a consistent fit can be obtained for data on ν e disappearance (reactor and Gallium anomalies) the global data suffers from severe tension between appearance and disappearance data, mostly due to the non-observation of ν µ disappearance at the eV 2 scale. Finally we mention that additional neutrino states with eV-like masses and sizeable mixings (as necessary to explain the oscillation anomalies) have severe implications for cosmology [74,75] and may lead to observable effects in IceCube [76][77][78].

Matter potential: non-standard interactions
Neutrino oscillation experiments can also provide important information on other neutrino properties beyond the SM. As an example we briefly summarize here the results of the most up-to-date determination of new physics in the matter effects in neutrino propagation from the global analysis of neutrino oscillation experiments from Ref. [79], to which we refer the reader for details and related references.
In the three-flavor oscillation picture described above the neutrino evolution equation along trajectory parametrized by coordinate x reads ( ν = (ν e , ν µ , ν τ ) T ):  6. Two-dimensional projections of the 90%, 95%, 99% and 3σ CL (2 dof) allowed regions of the oscillation parameters for f = u after marginalizing over the matter potential parameters and the undisplayed oscillation parameters. The full regions and the star correspond to the global analysis including NSI, while the black-contour void regions and the triangle correspond to the analysis with the usual SM potential. The green and red dotted areas show the 90% and 3σ CL allowed regions from partial analyses where the effects of the non-standard matter potential have been neglected either in the solar+KamLAND (green) or in the atmospheric+LBL (red) sectors. while for antineutrinos the hamiltonian is H¯ν = (H vac − H mat ) * . In the Standard Model H mat is fully determined both in its strength and flavor structure to be H SM mat = √ 2G F N e (r) diag(1, 0, 0) for ordinary matter [80,81]. Generically ordinary matter is composed by electrons (e), up-quarks (u) and down-quark (d), thus in the most general case a non-standard matter potential can be parametrized as: Since this matter term can be determined by oscillation experiments only up to an overall multiple of the identity, without loss of generality one can assume ε f µµ = 0. With this, we have 8 additional parameters (for each f ) since ε f ee and ε f ττ must be real whereas ε f eµ , ε f eτ and ε f µτ can be complex. The theoretical framework for this parametrization of the matter potential is provided by non-standard interactions (NSI) of neutrinos with the matter particles. They can be described by effective four-fermion operators of the form where f is a charged fermion, P = (L, R) and ε f P αβ are dimensionless parameters encoding the deviation from standard interactions. NSI enter in neutrino propagation only through the vector couplings so the induced matter Hamiltonian takes the form Eq. (17) with ε f αβ = ε f L αβ + ε f R αβ . We show in Figs. 6 and 7 some projections of the large parameter space in oscillation parameters and on the NSI parameters (after marginalizing over all oscillation and NSI undisplayed parameters) from a global analysis of oscillation data in terms of 3ν oscillations with general real matter potential (with ∆m 2 21 effects neglected in the analysis of ATM and LBL experiments). From the figures we read the following: • the determination of most the oscillation parameters discussed in the previous section is robust under the presence of NSI as large as allowed by the oscillation data itself with the exception of the octant of θ 12 ; • a solution with θ 12 > 45 • (the "so-called" LMA-D solution [82]) still provides a good fit to the data, as can be seen in the lower-left panel in Fig. 6. Such solution requires large NSI, which nevertheless are fully compatible with the bounds from atmospheric and LBL oscillation data; • the analysis of solar and KamLAND data favours non-vanishing NSI to better fit the fact that neither the SNO nor SK4 low energy threshold analysis nor the 8 B measurement in Borexino seem to show evidence of the low energy turn-up of the spectrum predicted in the standard LMA MSW; • comparing the results in Fig. 7 with the bounds on NSI derived in Refs. [83,84] from non-oscillation data we find that, with the possible exception of ε u,d eµ , the global oscillation analysis presented here yields the most restrictive bounds on the vector NSI parameters, in particular those involving τ flavour.
It is important to notice that in writing the phenomenological Lagrangian in Eq. (18) one assumes that the new physics which induces the NSI operators does not introduce new charge lepton physics at tree level or that charge lepton effects are very suppressed compared to those of neutrinos. This constraints the new physics realizations of this scenario [85] and, generically, the size of the NSI couplings which can be generated.

Conclusions and outlook
Thanks to the remarkable discoveries by neutrino oscillation experiments, such us those awarded by 2015 Nobel prize, it is now an established fact that neutrinos have mass and leptonic flavors are not symmetries of Nature. These results represent, to this date, the only laboratory evidence of physics beyond the Standard Model.
In this contribution we have summarized some results on the present characterization of the low energy parametrization of the neutrino properties as obtained from direct comparison with the data. The relevance of the work lies on the fact that the determination of the flavour structure of the leptons at low energies, is, at this point, our only source of information to understand the underlying new dynamics and it is fundamental to ultimately establish the New Standard Model.