1 Introduction
###### Abstract

The matter effects for solar neutrino oscillations are studied in a general scheme with an arbitrary number of sterile neutrinos, without any constraint on the mixing, assuming only a realistic hierarchy of neutrino squared-mass differences in which the smallest squared-mass difference is effective in solar neutrino oscillations. The validity of the analytic results are illustrated with a numerical solution of the evolution equation in three examples of the possible mixing matrix in the simplest case of four-neutrino mixing.

 2 December 2009

Matter Effects in Active-Sterile Solar Neutrino Oscillations

[0.5cm] C. Giunti, Y.F. Li

[0.5cm] INFN, Sezione di Torino, Via P. Giuria 1, I–10125 Torino, Italy Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China

## 1 Introduction

The possible existence of sterile neutrinos [1] is one of the most interesting open questions in neutrino physics (see Refs. [2, 3, 4, 5, 6, 7, 8]), which may give important information on the physics beyond the Standard Model (see Refs. [9, 10]). Sterile neutrinos are neutral fermions which do not interact weakly, as the three ordinary active neutrinos , , do. Moreover, sterile neutrinos can mix with active neutrinos, leading to active-sterile neutrino oscillations which are observable either through the disappearance of active neutrinos or through flavor transitions generated by a squared-mass difference which is larger than the solar (SOL) and atmospheric (ATM) mass differences (see Refs. [2, 3, 4, 5, 6, 7, 8])

 Δm2SOL≃ 8×10−5eV2, (1) Δm2ATM≃ 2.5×10−3eV2. (2)

In fact, the oscillation data show that , , are mainly mixed with three light neutrinos , , with masses such that

 Δm2SOL=Δm221, (3) Δm2ATM=|Δm231|≃|Δm232|, (4)

with . The order of magnitude of the absolute values of the three neutrino masses , , is bound to be smaller than about 1 eV by -decay, neutrinoless double- decay and cosmological data (see Refs. [11, 6, 7, 8]).

The possible existence of sterile neutrinos has been discussed by many authors in connection with the indication of oscillations with found in the LSND experiment [12]. Since this squared-mass difference is much larger than , the LSND oscillations can only be explained with mixing of more than three neutrinos, leading to the unavoidable introduction of additional sterile neutrinos.

The LSND signal has not been seen in other experiments and is currently disfavored by the negative results of the KARMEN [13] and MiniBooNE [14] experiments [15, 16, 17]. On the other hand, indications in favor of active-sterile neutrino oscillations [18, 19, 20, 21] come from the anomalous ratio of measured and predicted observed in the Gallium radioactive source experiments GALLEX [22, 23] and SAGE [24, 25, 26, 27] and from the MiniBooNE [14] low-energy anomaly. Moreover, sterile neutrinos may constitute the cosmological dark matter and active-sterile oscillations may explain the pulsar velocities [28, 29, 30, 31, 32, 33, 34, 35]. Since the standard cosmological scenario constrains the masses, mixing, and number of neutrinos [36, 37, 38, 39, 40, 41, 42], sterile neutrinos may be heralds of alternative cosmological models [43, 44, 45].

Bearing in mind that the existence of sterile neutrinos which are not related to the LSND anomaly remains possible, the quest for sterile neutrinos is one of the best tools for the exploration of physics beyond the Standard Model. A convenient way to reveal the existence of sterile neutrinos is the search for disappearance of active neutrinos due to active-sterile oscillations [46, 47]. The explanation of the Gallium radioactive source experiments anomaly and the MiniBooNE low-energy anomaly through active-sterile neutrino oscillations requires the disappearance of electron neutrinos into sterile states [18, 19, 20, 21]. If these transitions exist, they contribute also to the disappearance of solar electron neutrinos. This effect, which has not been studied in detail so far, is the object of study of this paper.

The matter effects for solar neutrino oscillations in a four-neutrino scheme was studied in Ref. [48], where it was assumed that the electron neutrino has non-negligible mixing with only two massive neutrinos, and , which generate the solar squared-mass difference . In this approximation, electron neutrinos cannot oscillate into sterile states in the short distances required for the explanation of the Gallium radioactive source experiments anomaly and the MiniBooNE low-energy anomaly. Hence, a combined analysis of the Gallium radioactive source experiments anomaly, the MiniBooNE low-energy anomaly and solar neutrino data in terms of active-sterile oscillations [49] requires an extension of the study in Ref. [48] to the general case in which there is no constraint on the mixing of the electron neutrino.

In this paper we study the matter effects for solar neutrino oscillations in a general scheme with an arbitrary number of sterile neutrinos, without any constraint on the mixing. Hence, we consider the general case of mixing of three active neutrino fields (, , ) and sterile neutrino fields:

 ναL=N∑k=1UαkνkL(α=e,μ,τ,s1,…,sNs), (5)

where , is the field of a massive neutrino with mass , and the subscript denotes the left-handed chiral component. The numbering of the massive neutrinos is chosen in order to satisfy Eqs. (3) and (4) with the first three neutrino masses and we assume that the additional neutrino masses are much larger than , , , in order to generate short-baseline oscillations which could explain the Gallium radioactive source experiments anomaly and/or the MiniBooNE low-energy anomaly [18, 19, 20, 21] and could be observed in future laboratory experiments [50, 51, 52, 53, 15, 54, 55, 46, 47] and through astrophysical measurements [56, 57, 58, 59, 60]. Hence, we have the following hierarchy of squared-mass differences:111 The different case of mixing of two active neutrinos and a sterile neutrino with two small squared-mass differences, both of which contribute to solar neutrino oscillations, has been studied in Ref. [61].

 Δm221≪|Δm231|≪Δm2k1fork≥3. (6)

The unitary mixing matrix can be parameterized in terms of mixing angles (see Ref. [7]). Three mixing angles mix the three active neutrinos among themselves and the other mixing angles mix the three active neutrinos with the sterile neutrinos. There are also Dirac phases which affect neutrino oscillations and Majorana phases which do not affect neutrino oscillations [62, 63, 64]. However, in this paper we do not need an explicit parameterization of . We derive the flavor transition probabilities of solar neutrinos in terms of the values of the elements of the mixing matrix.

The plan of the paper is as follows. Section 2 is the main one, where we derive the flavor transition probabilities. In Section 3 we discuss the peculiarities of the flavor transition probabilities in the extreme non-adiabatic limit. In Section 4 we illustrate the validity of our results in the simplest case of four-neutrino mixing, considering some examples in which we compare the values of the analytic approximation of the flavor transition probabilities derived in Section 2 with those obtained with a numerical solution of the flavor evolution equation. Conclusions are drawn in Section 5.

## 2 Evolution of Neutrino Flavors

Solar neutrinos are described by the state

 |ν(x)⟩=∑α=e,μ,τ,s1,…,sNsψα(x)|να⟩, (7)

where is the distance from the production point in the core of the sun and

 ψα(0)=δαe, (8)

with the obvious normalization .

From the mixing relation (5) between flavor and massive neutrino fields and the structure of the weak charged-current Lagrangian,

 LCC=−g√2∑α=e,μ,τN∑k=1¯¯¯¯¯¯¯¯ℓαLγρUαkνkLW†ρ+H.c., (9)

it follows that in vacuum the flavor states which describe neutrinos created in charged-current weak interactions are unitary superpositions of massive states which are the quanta of the corresponding massive neutrino fields (see Ref. [7]):

 |να⟩=N∑k=1U∗αk|νVk⟩(α=e,μ,τ,s1,…,sNs). (10)

The evolution of the flavor transition amplitudes is given by the Mikheev-Smirnov-Wolfenstein (MSW) equation222 We omit a diagonal term which generates an irrelevant common phase. [65, 66] (see Ref. [7])

 iddxΨ=12E(UM2U†+A)Ψ, (11)

where is the neutrino energy and

 Ψ= (ψe,ψμ,ψτ,ψs1,…,ψsNs)T, (12) M2= diag(0,Δm221,Δm231,Δm241,…,Δm2N1), (13) A= diag(ACC+ANC,ANC% ,ANC,0,…), (14)

with and

 ACC=2EVCC,ANC=2EVNC. (15)

The charge-current and neutral-current matter potentials are given by

 VCC=√2GFNe≃7.63×10−14NeNAcm−3eV,VNC=−12√2GFNn, (16)

where is the Fermi constant, is the electron number density, is the neutron number density, and is Avogadro’s number. It is convenient to use the standard definition of the electron fraction

 Ye=NeNe+Nn. (17)

In a neutral medium we have

 Ne=ρgNAYe,Nn=ρgNA(1−Ye). (18)

where is the density. From Eqs. (16) and (18), we obtain

 ANC=RNCACC,withRNC=−1−Ye2Ye. (19)

For solar densities we have

 ACC∼|ANC|∼Δm221≪|Δm2k1|fork≥3. (20)

In order to decouple the flavor transitions generated by from those generated by the larger squared-mass differences, it is useful to work in the vacuum mass basis

 ΨV=(ψV1,…,ψVN)T=U†Ψ, (21)

which satisfies the evolution equation

 iddxΨV=12E(M2+U†AU)ΨV, (22)

In this basis, the solar neutrino state in Eq. (7) is given by

 |ν(x)⟩=N∑k=1ψVk(x)|νVk⟩, (23)

In the vacuum basis the inequality in Eq. (20) implies that the evolution of each of the amplitudes is decoupled from the others, since the coupling due to the matter effects is negligible. Therefore, we have

 ψVk(x)≃ψVk(0)exp(−iΔm2k1x2E),fork≥3, (24)

with

 ψVk(0)=U∗ek, (25)

from Eqs. (8) and (21).

On the other hand, the evolution of and is coupled by the matter effect. In order to solve it, we consider the truncated 1-2 sector of Eq. (22):

 (26)

where the sums over the flavor index cover only the active neutrinos, which have a matter potential in Eq. (14). It is convenient to subtract the diagonal term

 14E(Δm221+∑α(|Uα1|2+|Uα2|2)Aα), (27)

which generates an irrelevant common phase. The resulting effective evolution equation can be written as

 (28)

with

 Acos2ξ= ∑α(|Uα1|2−|Uα2|2)Aα=ACCX, (29) Asin2ξeiφ= 2∑αU∗α1Uα2Aα=ACCYeiφ, (30)

where

 X= |Ue1|2−|Ue2|2+RNC∑α=e,μ,τ(|Uα1|2−|Uα2|2) = |Ue1|2−|Ue2|2−RNCNs∑i=1(|Usi1|2−|Usi2|2), (31) Y= 2∣∣ ∣∣U∗e1Ue2+RNC∑α=e,μ,τU∗α1Uα2∣∣ ∣∣ = 2∣∣ ∣∣U∗e1Ue2−RNCNs∑i=1U∗si1Usi2∣∣ ∣∣. (32)

In these expressions we used the unitarity of the mixing matrix in order to express the neutral-current contribution either through the active neutrino elements of the mixing matrix or through the sterile neutrino elements.

One can immediately notice that, apart from the complex phase , the effective evolution equation (28) has the same structure as the effective evolution equation in the vacuum mass basis for - or - two-neutrino mixing with the two-neutrino mixing angle replaced by the effective mixing angle and replaced by , with

 tan2ξ=YX,A=ACC√X2+Y2, (33)

In fact, one can easily check that in the limit of - or - two-neutrino mixing, coincides with the two-neutrino mixing angle and becomes . Therefore, the effective mixing angle plays the same role as the usual two-neutrino mixing angle in the solution of the flavor evolution equation. However, one must keep in mind that in general is not a pure mixing angle given by a specific parameterization of the mixing matrix, because it depends on the neutral-current to charged-current ratio .

For simplicity, we consider a medium with constant electron fraction , which implies a constant neutral-current to charged-current ratio . In this case, the effective mixing angle and the phase are constant. Moreover, we can use the phase freedom in the parameterization of the mixing matrix in order to eliminate . In fact, the flavor transition probabilities obtained from the MSW evolution equation (11) are invariant under the phase transformations

 Uαk→eiφαUαkeiφk. (34)

which gives, from the definition in Eq. (30),

 Asin2ξeiφ→Asin2ξei(φ+φ2−φ1). (35)

Therefore, choosing , the evolution equation (28) is simplified to

 iddx(ψV1ψV2)=14E(−Δm221+Acos2ξAsin2ξAsin2ξΔm221−Acos2ξ)(ψV1ψV2), (36)

which has the same structure as the effective evolution equation in the vacuum mass basis for - or - two-neutrino mixing.

In practice, and are not constant in the Sun, as shown in Fig. 1. The electron fraction is almost constant in the radiative () and convective () zones where the Hydrogen/Helium mass fraction is practically equal to the primordial value. In the core () the electron fraction increases towards the center because of the accumulated thermonuclear production of Helium during the 4.5 Gy of the life of the Sun. However, it is plausible that the effects on flavor transitions of the variation of in the core is negligible if the transitions occur mainly in a resonance which is located in the radiative or in the convective zone. Hence we solve the evolution equation of neutrino flavors in the approximation of a constant . In Section 4 we will show with numerical examples that the effects of the variation of in the Sun are indeed negligible.

The effective Hamiltonian in the evolution equation (36) is diagonalized by the rotation

 (ψV1ψV2)=(cosωsinω−sinωcosω)(ψM1ψM2), (37)

with

 tan2ω=Asin2ξΔm221−Acos2ξ. (38)

The angle is the mixing angle between the vacuum mass basis and the effective mass basis in matter. Its expression is similar to the one in - or - two-neutrino mixing. It is therefore useful to note that the evolution equation (36) corresponds to the evolution of the effective interaction amplitudes

 (ψI1ψI2)=(cosξsinξ−sinξcosξ)(ψV1ψV2), (39)

for which the matter potential is diagonal:

 (40)

This basis of effective interaction amplitudes is physically important for the study of the flavor evolution, because it is the basis in which the mixing generates transitions through the off-diagonal term in the evolution equation. Moreover, in this basis one can see that there is a resonance when the diagonal terms are equal, i.e. for , with

 AR=Δm221cos2ξ. (41)

The relations between the effective mass and interaction bases in matter and the flavor basis are discussed in Appendix A.

The connection between the effective interaction amplitudes and and the amplitudes and of the effective energy eigenstates in matter is similar to that in two-neutrino mixing:

 (ψI1ψI2)=(cosξMsinξM−sinξMcosξM)(ψ% M1ψM2), (42)

where is the effective mixing angle in matter given by

 ξM=ξ+ω. (43)

From Eq. (38) we obtain

 tan2ξM=tan2ξ1−(A/Δm221cos2ξ). (44)

The effective mixing is maximal () in the resonance . Notice that Eq. (44) and the resonance condition (41) are similar to the corresponding equations in - or - two-neutrino mixing (see Ref. [7]), with the two-neutrino mixing angle replaced by the effective mixing angle and replaced by .

The interesting limiting cases for the effective mixing angle in matter are:

 ξM≃ξ, ω≃0, forA≪AR, (45) ξM≃π2, ω≃π2−ξ, forA≫AR. (46)

The evolution of the amplitudes and of the effective energy eigenstates in matter is given by

 iddx(ψM1ψM2)=14E(−Δm2M−4iEdω/dx4iEdω/dxΔm2M)(ψM1ψM2), (47)

with the effective squared-mass difference

 Δm2M=√(Δm221cos2ξ−A)2+(Δm221sin2ξ)2. (48)

The non-adiabatic off-diagonal transitions depend on

 dωdx=Δm221Asin2ξ2(Δm2M)2dlnNedx. (49)

It is convenient to define the adiabaticity parameter

 γ=Δm2M4E|dω/dx|=(Δm2M)32EΔm221Asin2ξ|dlnNe/dx|. (50)

The evolution is adiabatic if all along the neutrino path.

The solution of the evolution equation (47) leads to the averaged probabilities

 ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯|ψM1(x)|2= |ψM1(0)|2(1−P12)+|ψM2(0)|2P12, (51) ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯|ψM2(x)|2= |ψM1(0)|2P12+|ψM2(0)|2(1−P12), (52)

where is the probability of non-adiabatic off-diagonal transitions in Eq. (47). The averaging process washes out the interference between and , which is not measurable because of the energy resolution of the detector and the uncertainty of the production region. The initial values and are given by Eqs. (25) and (37) with the effective mixing angle in the production region:

 ψM1(0)= cosω0ψV1(0)−sinω0ψV2(0)=cosω0U∗e1−sinω0U∗e2, (53) ψM2(0)= sinω0ψV1(0)+cosω0ψV2(0)=sinω0U∗e1+cosω0U∗e2. (54)

We can use the phase freedom in Eq. (34) in order to have real and , keeping the reality of in Eq. (30), since there are three arbitrary phases available: , , and . With this choice, and can be written as

 Ue1=cosϑecosχe,Ue2=sinϑecosχe,withsin2χe=N∑k=3|Uek|2. (55)

Then, and are given by

 ψM1(0)=cosϑ0ecosχe,ψM% 2(0)=sinϑ0ecosχe, (56)

with the effective mixing angle in the production region

 ϑ0e=ϑe+ω0. (57)

From Eq. (97), the electron neutrino state is given by

 |νe⟩= [cos(ϑe+ω)|νM1⟩+sin(ϑe+ω)|νM2⟩]cosχe+N∑k=3U∗αk|νVk⟩ = [cos(ϑe−ξ)|νI1⟩+sin(ϑe−ξ)|νI2⟩]cosχe+N∑k=3U∗αk|νVk⟩. (58)

Hence, an electron neutrino contains both and if . In general, this is the case, since the effective mixing angle in the effective evolution equation (28) of the truncated 1-2 sector is different from the electron neutrino mixing angle because of the neutral-current contribution to the matter potential (see the discussion after Eq. (70)). In fact, as one can see from Eqs. (31)–(33), the effective mixing angle does not depend only on the mixing of , but also on the mixings of and which affect the neutral-current contribution to the matter potential.

Since the matter effects in the detector are negligible, the averaged flavor transition probabilities are given by

 ¯¯¯¯Pνe→νβ(x)= ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯|ψβ(x)|2=¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯∣∣ ∣∣N∑k=1UβkψVk(x)∣∣ ∣∣2 = ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯∣∣ ∣∣2∑k=1UβkψMk(x)+N∑k=3UβkψVk(x)∣∣ ∣∣2 = 2∑k=1|Uβk|2¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯|ψMk(x)|2+N∑k=3|Uβk|2|ψVk(x)|2 = 2∑k=1|Uβk|2¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯|ψMk(x)|2+N∑k=3|Uβk|2|Uek|2, (59)

where we have used Eqs. (24) and (25). Writing and as

 |Uβ1|2=cos2ϑβcos2χβ,|Uβ2|2=sin2ϑβcos2χβ,% withsin2χβ=N∑k=3|Uβk|2, (60)

and using Eqs. (51), (52) and (56), we obtain

 ¯¯¯¯Pνe→νβ=[12+(12−P12)cos2ϑβcos2ϑ0e]cos2χβcos2χe+N∑k=3|Uβk|2|Uek|2. (61)

It is clear that in the limit of two-neutrino mixing, in which , the transition probability reduces to the well-known Parke formula [68] (see Ref. [7]).

For neutrinos produced above the resonance, where , from Eqs. (46) and (57) we have

 ϑ0e≃π2+ϑe−ξ⟹cos2ϑ0e≃−cos2(ϑe−ξ). (62)

In this case,

 ¯¯¯¯Pνe→νβ≃[12−(12−P12)cos2ϑβcos2(ϑe−ξ)]cos2χβcos2χe+N∑k=3|Uβk|2|Uek|2. (63)

The crossing probability can be inferred by using the similarity of the evolution equation (47) with that in two-neutrino mixing [69, 70, 71, 72] (see Ref. [7]):

 P12=exp(−π2γRF)−exp(−π2γRFsin2ξ)1−exp(−π2γRFsin2ξ)θ(A0−AR), (64)

where is the adiabaticity parameter at the resonance (41), which corresponds to the minimum of :

 (Δm2M)R=Δm221sin2ξ. (65)

Then, from Eq. (50), is given by

 γR=Δm2sin22ξ2Ecos2ξ∣∣dlnNe/dx∣∣R. (66)

The value of the parameter depends on the electron density profile. For an exponential density profile, which is a good approximation for the solar neutrinos, [69, 70, 71, 73, 74, 75, 76, 72, 77]

 F=1−tan2ξ. (67)

In Eq. (64) there is a -factor which reduces to zero when the neutrino is created at a density which is smaller than the resonance density ( is the value of at production).

Let us finally notice that for solar neutrinos it is sufficient to calculate the survival probability and the sum of the transition probabilities into sterile neutrinos , from which is trivially obtained using the conservation of probability. The separate transition probabilities into and are not needed because and have the same interactions in the detector. In fact, and can be distinguished only through charged-current interactions with production of the associated charged lepton, which is forbidden by the low energy of solar neutrinos (). Hence, and are detected only through neutral-current interactions which are flavor-blind.

It is interesting to note that in the extreme non-adiabatic limit the transition probability is different from the averaged transition probability in vacuum (VAC)

 ¯¯¯¯PVACνe→νβ= N∑k=1|Uβk|2|Uek|2 = [12+12cos2ϑβcos2ϑe]cos2χβcos2χe+N∑k=3|Uβk|2|Uek|2. (68)

In fact, since in the extreme non-adiabatic limit

 P(γR≪1)12≃cos2ξ, (69)

from Eq. (63) we obtain

 ¯¯¯¯P(γR≪1)νe→νβ≃[12+12cos2ϑβcos2(ϑe−ξ)cos2ξ]cos2χβcos2χe+N∑k=3|Uβk|2|Uek|2. (70)

This expression coincides with that in Eq. (68) only if . From Eqs.(31)–(33) one can see that if there is no neutral-current contribution in Eqs.(31) and (32) or if the neutral-current contributions cancel in the ratio . The first possibility is obviously realized in a neutron-free medium (). It is also realized if for , which implies that and . This is obviously the case in three-neutrino mixing (no sterile neutrinos). The second possibility is realized in four-neutrino mixing with [48]: in that case

 Ue1=cosϑe,Ue2=sinϑe, (71)

and

 Us1=−sinϑecosχs,Us2=cosϑecosχs. (72)

Using the unitarity of the mixing matrix, we have

 ∑α=e,μ,τ(|Uα1|2−|Uα2|2)=|Us2|2−|Us1|2=cos2ϑecos2χs, (73) 2∑α=e,μ,τU∗α1Uα2=−U∗s1Us2=sin2ϑecos2χs, (74)

 X= cos2ϑe(1+RNCcos2χs), (75) Y= sin2ϑe(1+RNCcos2χs), (76)

and .

In the case of two-neutrino mixing the averaged flavor transition probability in the extreme non-adiabatic case coincides with the averaged flavor transition probability in vacuum because the flavor states are conserved in the resonance. A created above the resonance is practically equal to a , which travels undisturbed to the resonance, where it is still a (it only develops a harmless phase). If the crossing of the resonance is extremely non-adiabatic, the emerges unchanged from the resonance (apart from an irrelevant phase) and the flavor transition measured on Earth is only due to the vacuum oscillations from the Sun to the Earth. On the other hand, in the case of mixing with sterile neutrinos, we have seen in Eq. (58) that, if , a contains both and , which are conserved if the crossing of the resonance is extremely non-adiabatic. The different phases developed by and before and during resonance crossing make the averaged flavor transition probability different from that in vacuum. We can show that explicitly following the evolution of the solar neutrino state . If the neutrino is created as a above the resonance, the initial state is, from Eqs. (46), (55), and (97)

 |ν(0)⟩=|νe⟩=[−sin(ϑe−ξ)|νM1⟩+cos(ϑe−ξ)|νM2⟩]cosχe+N∑k=3U∗ek|νVk⟩. (77)

Since from the production to the resonance the effective massive states develop different phases, just before the resonance the solar neutrino state is

 |ν(x−R)⟩= [−sin(ϑe−ξ)eiϕ1(x−R)|νM1⟩+cos(ϑe−ξ)eiϕ2(x−R)|νM2⟩]cosχe +N∑k=3U∗ekeiϕk(x−R)|νVk⟩. (78)

Since above the resonance and , we have

 |ν(x−R)⟩= [sin(ϑe−ξ)eiϕ1(x−R)|νI2⟩+cos(ϑe−ξ)eiϕ2(x−R)|νI1⟩]cosχe +N∑k=3U∗ekeiϕk(x−R)|νVk⟩. (79)

If the crossing of the resonance is extremely non-adiabatic, there are no transitions among the effective interaction states and , which develop only different phases. Hence, just after the resonance we have

 |ν(x+R)⟩= [sin(ϑe−ξ)eiϕ1(x+R)|νI2⟩+cos(ϑe−ξ)eiϕ2(x+R)|νI1⟩]cosχe +N∑k=3U∗ekeiϕk(x+R)|νVk⟩. (80)

Since after the resonance the neutrino propagates practically in vacuum, it is convenient to express its state in terms of the propagating massive states:

 |ν(x>xR)⟩= [−sin(ϑe−ξ)sinξeiϕ1(x)+cos(ϑe−ξ)cosξeiϕ2(x)]cosχe|ν% V1⟩ +[sin(ϑ