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Neutrino spin-flip effects in collapsing stars

We study the spin-flavor transitions of neutrinos in the magnetic fields of a collapsing star. For the neutrino mass squared difference (10^{-10} - 10) eV^2 the transitions take place in almost isotopically neutral region of the star, where the effective matter density is suppressed up to 3 - 4 orders of magnitude. This suppression is shown to increase the sensitivity of the neutrino bursts studies to the magnetic moment of neutrino by 1.5 - 2 orders of magnitude, and for realistic magnetic field the observable effects may exist for (2 - 3)10^{-14} Bohr magneton. In the isotopically neutral region the jumps of the effective potential exist which influence the probabilities of transitions. The experimental signatures of the spin-flavor transitions are discussed. In particular, in the case of direct mass hierarchy, the spin-flip effects result in a variety of modifications of the electron antineutrino spectrum. Taking this into account, we estimated the upper bounds on the magnetic moment from the SN1987A data. In the isotopically neutral region the effects of possible twist of the magnetic field on the way of neutrinos can be important, inducing distortion of the neutrino energy spectra and further increasing the sensitivity. However, if the total rotation angle is restricted the absolute change of probabilities is small.


Introduction
At least 100 neutrino bursts from the gravitational collapses of stars in our Galaxy are already on the way to the Earth 1 . A registration of even one of those by the existing detectors, Kamiokande [2], Baksan [4], LSD [5], and especially, by the new installations: LVD [6] (which is already working), SNO [7] and Superkamiokande [8] (starting to operate in 1996), ICARUS [9] (which is at the prototype stage) will give unique and extremely rich information about features of the gravitational collapse, supernova phenomena and neutrinos themselves. Already SN1987A has given a lot. In general, one will be able to get information on the energy spectra (as a function of time) of ν e ,ν e , as well as of neutrinos of the non-electron type: ν µνµ , ν τ ,ν τ . Certain properties of these spectra do not depend on the model of the star thus opening the possibility to study the characteristics of neutrinos. New experiments are discussed to detect the neutrino bursts from collapses in other nearby galaxies [10].
In this paper we consider a possibility to study effects of the neutrino magnetic moments using the neutrino bursts. In the magnetic fields the neutrinos with magnetic moments undergo spin-precession [11,12,13], spin-flavor precession [14] or/and resonant spin-flavor conversion, e.g.
The spin-flip effects are determined by the product, µB, of the neutrino magnetic moment, µ, and the strength of the magnetic field, B. The discovery of the effects corresponding to values of µ near the present upper bound, µ < (1 − 3) · 10 −12 µ B [21] (µ B is the Bohr magneton) will imply rich physics beyond the standard model. From this point of view the studies of the neutrino bursts are of special interest since the magnetic fields in supernova can be as strong as (10 12 − 10 14 ) Gauss.
It is reasonable to address two questions, keeping in mind that the magnetic field profiles of supernovae are essentially unknown, and vary from the star to star. What are the experimental signatures of the spin-flip effects? And what could be the sensitivity of the neutrino burst studies to the magnetic moments of neutrinos, or more precisely to µB?
The spin-flip probability may be of the order 1, if where the strength of the magnetic field is integrated along neutrino trajectory. Suggesting the field profile with B 0 ∼ (10 12 − 10 14 ) Gauss, k = 2 − 3, and r 0 ∼ 10 km, where r is the distance from the center of star, one gets from (1), and µ > ∼ (10 −15 − 10 −17 )µ B . (For a constant field a similar bound is µB ≥ (∆r B ) −1 , where ∆r B is the size of region with the magnetic field). These numbers look encouraging, being 3 -5 orders of magnitude below the present bound.
However, the estimation (1) corresponds to neutrino propagation in vacuum (no matter), where the precession takes place with maximal depth, A P = 1. In this case the "optimal conditions" for the spin-flip are realized: the value of µB needed to change the spin-flip probability by a given amount ∆P is minimal [22]. The presence of dense matter strongly reduces a sensitivity to µ.
is the total potential, G F is the Fermi constant, n is the nucleon number density, ρ is the matter density and m N is the nucleon mass. The level splitting suppresses the depth of precession [13]: and to have A P ≥ 1 2 , one needs µB ≥ 1 2 V SF .
The suppression of the matter density (i.e., approaching the vacuum condition) results in the increase of the sensitivity to µ. This is realized in the case of the resonant spin-flavor conversion [16]. If the left and right handed neutrino components connected by the magnetic moment have different masses, m 1 , m 2 , being also of different flavors, the matter effect can be compensated by mass splitting: (resonance condition), where ∆m 2 ≡ m 2 2 − m 2 1 , and E is the neutrino energy. Now the expression for A P becomes and in the resonance (7) one gets A P = 1. However, since V SF changes with distance, the equality A P = 1 holds only in the resonance point, r R . Strong compensation of the matter effects occurs in some layer around r R whose size depends on gradient of the potential:V SF ≡ dV SF /dr. Indeed, from (8) one finds that A P ≥ 1 2 , if ∆H = ∆V SF ≤ ∆V R , where is the half-width of the resonance layer. The spatial size of the region with A P ≥ 1/2 equals 2∆r R = 2(V SF ) −1 ∆V R = 2(V SF ) −1 2µB. Then the strong spin-flip effect implies that 2∆r R is larger than half of the precession length, l p : 2∆r R > ∼ l p /2 = π/2µB. Substituting the expression for ∆r R in this inequality one gets This is the adiabaticity condition in the resonance [15,16], and κ R is the resonance adiabaticity parameter. (Note that the adiabaticity parameter γ used in [16] is related to that in (10) as κ R = 2γ/π). For large densities the condition (10) is much less restrictive, than (6), allowing for a strong spin-flip effect for smaller values of µB. For example, at B = 10 14 Gauss, ρ = 10 12 g/cm 3 , and V SF /V SF ∼ r 0 ∼ 10 km, one gets µ > 3 · 10 −12 µ B , instead of 10 −7 µ B .
Here we will consider two new phenomena which suppress the matter effect and thus, increase the sensitivity of neutrino burst studies to the neutrino magnetic moment. The potential V SF can be diminished for special nuclear composition of matter. We point out that this is realized in the external region of supernova, where the medium is almost isotopically neutral. Another phenomenon is the magnetic field twist -the change of the direction of the magnetic strength lines on the way of neutrinos. The field twist leads to an additional contribution to the level splitting ∆H which can compensate the matter effect.
The experimental signatures of the spin-flavor transitions of supernova neutrinos have been discussed previously in [17]. The influence of the spin-flip on the relation between directional (induced by ν e e-scattering) and isotropic (ν e p -interaction) signals was studied. Without consideration of the dynamics of the propagation it was suggested that the probabilities of different transitions are either one or zero. The estimated number of events shows that four cases (1) no transitions, (2) only flavor transitions, (3) only spin-flavor transitions, (4) flavor plus spin-flavor transitions can be disentangled by new installations (SNO, Superkamiokande). We will consider a more general situation concentrating on the energy dependence of the transitions.
In this paper we study the dynamics of propagation to estimate the values of µB needed for different effects. We consider signatures of the spin-flip transitions related to specific density distribution in the isotopically neutral region and to possible presence of the magnetic field twist.
The paper is organized as follows. In Sect. 2, we describe the effective potential in supernova. In Sect. 3 the level crossing schemes for different values of neutrino masses are found and the dynamics of neutrino propagation is considered. In Sect. 4, we study the dependence of the transition probabilities on the magnetic field profile and on the neutrino parameters. Sect. 5 is devoted to the interplay of the spin-flavor and flavor transitions. In Sect. 6 we consider some special effects of neutrino propagation and discuss the possible implications of the results. In particular, we describe the distortion of energy spectra of neutrinos and estimate the sensitivity of the studies of ν-bursts to µB. Also, the upper bound on (µB) will be obtained from SN1987A data. In Sect. 7, the effects of the magnetic field twist are considered.
For the spin-flavor transitions, e.g., ν eL →ν µR ,ν eR → ν µL , the matter effect is described by the potential [15,16] where Y e is the number of electrons per nucleon. The value of Y e depends on the nuclear composition of the matter. In isotopically neutral medium (# protons = # neutrons), one has Y e = 1 2 , and according to (11) V SF = 0. Consequently, the matter effect is determined by the deviation from the isotopical neutrality.
For the flavor conversion the potential is proportional to the electron density: Evidently, there is no suppression of V F in the isotopically neutral medium; V F is suppressed in the central strongly neutronized region of the star.

Isotopically neutral region
The progenitor of the type II supernova has the "onion" structure. Below the hydrogen envelope the layers follow which consist mainly of the isotopically neutral nuclei: 4 He, 12 C, 16 O, 28 Si, 32 S. Thus the region between the hydrogen envelope and the core, where elements of iron peak dominate, is almost isotopically neutral. The deviation from the neutrality is stipulated by small abundances, ξ i , of the elements with small excess of neutrons, i = 22 Ne, 23 Na, 25 Mg, 56 Fe [23,24].
It can be written as where Z i and A i are the electric charge and the atomic number of nuclei "i", correspondingly.
Typically, one gets i ξ i ≤ 10 −2 and (1− 2Z A ) ≤ 10 −1 , and therefore, according to (13): (1−2Y e ) ≤ 10 −3 . The value of (2Y e − 1) equals Consequently, in the isotopically neutral region the potential V SF is suppressed by more than 3 orders of magnitude with respect to the total potential V 0 ( fig.1). We calculated the potentials using the model of the progenitor with mass 15M ⊙ (where M ⊙ is the solar mass) [23]. Two remarks are in order. For r < R ⊙ the spatial distributions of the total density, are very similar (up to factor 2) for stars in a wide range of masses: M = (12−35)M ⊙ . The profiles may differ appreciably in the external low density regions which are unessential for our consideration. The collapse of the core and the propagation of the shock wave change both the density profile and the nuclear composition of medium. However, during the neutrino burst (0 -10 seconds after the core bounce) the shock wave can reach the distance ∼ several 1000 km at most. Therefore most part of the isotopically neutral region turns out to be undisturbed and one can use the profile of the progenitor.
Strong convection effects in the inner and intermediate parts of the star, if exist, may result in an injection of the elements in the iron peak into outer layers, thus diminishing the degree of the isotopical neutrality.
This covers the range of ∆m 2 interesting from the point of view of the solar and the atmospheric neutrino problems. For the cosmologically interesting values, ∆m 2 ∼ (4 − 50) eV 2 , the resonance is at the inner part of the isotopically neutral region, where the potential is still suppressed by factor 1/30 -1/100.
Let us stress that the suppression of the effective density takes place for the spin-flavor conversion only, i.e. when both neutrino components are active. In the case of the spin-flip into sterile neutrino ν e →ν s , the effective potential equals V s = V 0 2 (3Y e − 1), and in the isotopically neutral region one has V s = V 0 4 . The potential V s is suppressed in strongly neutronized central part of star, where Y e ≈ 1 3 [18,19].
3 The dynamics of the neutrino transitions

Level crossing scheme
We will consider the system of three massive neutrinos with transition magnetic moments and vacuum mixing (see [30] for details). For definiteness we assume the direct mass hierarchy: m 1 ≪ m 2 ≪ m 3 and the smallness of flavor mixing. The energies of the flavor levels (the diagonal elements of the effective Hamiltonian), H α , (α = ν e ,ν e , ν µ ,ν µ , ν τ ,ν τ ), can be written as Here θ is the e-µ-mixing angle in vacuum. The ν τ energy levels, H τ , Hτ , can be obtained from H µ , Hμ by substituting θ → θ τ and ∆m 2 → ∆m 2 31 . The angle φ(t) (φ ≡ dφ/dr) defines the direction of the magnetic field in the transverse plane. In (15) for later convenience, we have subtracted from all elements the energy H e to make the first diagonal element of the Hamiltonian (i.e., H e ) equal to zero. (This is equivalent to the renormalisation of all wave functions by the same factor which does not change the probabilities). The qualitative dependence of the energy levels H α on distance r forφ = 0 and for different values of ∆m 2 /2E is shown in Fig. 2. The resonance (level crossing) conditions read as H α = H β (α = β).
The peculiar behavior of the effective potential V SF in the isotopically neutral region stipulates a number of features in the level crossing scheme. With increase of ∆m 2 one gets the following changes. (The levels for ν e andν e are fixed, whereas ν µ -andν µ -levels go up).
For ∆m 2 < ∼ 10 −10 eV 2 (E ∼ 20 MeV) the mass splitting can be neglected. Two crossings of the levels which correspond to the spin-flavor conversions ν e →ν µ andν e → ν µ occur practically at the same point, where (2Y e − 1) ≃ 0 (r ≃ 0.8R ⊙ ), i.e. on the border between the H-envelope and 4 He-layer. The crossings are stipulated by change of nuclear composition of the star (Fig. 2a).
The ν τ -level can cross ν e -andν e -levels in the isotopically neutral region. Moreover, for cosmologically interesting values of ∆m 2 13 the ν τ -resonances are in inner edge of this region. They can also be in the central part of the star.

Adiabaticity conditions
In the isotopically neutral region the effective potentials for the flavor conversion and the spin-flavor conversion differ by 3 − 4 orders of magnitude. Therefore, the flavor and spin-flavor resonances are strongly separated in space, in contrast with the case of usual medium. Consequently, one can consider these crossings independently, reducing the task to two neutrino tasks.
If θ is small the flavor transition is essentially a local resonance phenomenon. It takes place at r R determined from (15): H µ (r R ) = 0. Therefore, the effect depends on the local properties of the density distribution in resonance: (This corresponds to the situation when at least half of the oscillation length, is obtained in the resonance layer, in the direct analogy with the definition of the adiabaticity parameter for the spinflip conversion (10)). Using the potential V F ∼ V 0 ( fig. 1), we find that for neutrinos with E ∼ 20 MeV and ∆m 2 = 10 −6 , 10 −5 , 10 −4 and 10 −2 eV 2 , the resonance is situated at r = 0.7R ⊙ , 0.5R ⊙ , 0.17R ⊙ , 0.03R ⊙ , correspondingly, and the condition (16) is satisfied for sin 2 2θ > 0.5, 0.06, 0.02, In contrast, the spin-flip effects can be nonlocal even in case of variable density. Indeed, the helicity mixing angle, θ B , which determines the mixing between the left and the right components (In medium with constant parameters θ B determines the depth of precession, when the initial state has a definite chirality: A P = sin 2 2θ B ). Since in general the field strength depends on distance, the angle θ B can be large enough in a wide spatial region provided µB ∼ V SF ≫ ∆m 2 /2E. For large B the spin-flip effect may not be localized in the resonance layer, and moreover, the resonance itself may not be local [30].
The rapidity of the θ B change on the way of neutrino,θ B ≡ dθ B /dr, determines the adiabaticity condition for spin-flip: The adiabaticity parameter can be written as (18) reduces to (10). Beyond the resonance κ depends on the magnetic field change. In the extreme cases one has where κ R is the resonance adiabatic parameter for a given point (10). According to (19) the adiabaticity condition is strongly relaxed, if V SF ≫ µB above the resonance, and if ∆m 2 /2E ≫ µB below the resonance. For the density and the field profiles under consideration these inequalities are fulfilled, so that the adiabaticity is the most crucial in the resonance.
If the initial and final mixings are small: θ i B ≈ π/2 and θ f B ≈ 0 and the level crossing is local, one can estimate the survival probability using the Landau-Zener formula: where B 2 A ≡ πV /8µ 2 (see (10)). At the adiabatic condition, B = B A , one gets P LZ = 0.085. The probability increases quickly with decrease of the field: P L Z = 0.3, 0.54, 0.8, when B/B A = 0.7, 0.5, 0.3 correspondingly (see Fig. 3). An appreciable effect could be for P LZ = 0.8, when the magnetic field is about 3 times smaller than the adiabatic value.

The precession and the adiabaticity bounds
The propagation of neutrinos with magnetic moment in the magnetic field and matter is an interplay of two processes: the precession and the resonance spin conversion. A relation between them is determined largely by density distribution. Let us define the precession bound for the product (µB) P as At (µB) = (µB) P , the precession may have the depth A P = 1/2 (see (6)); for (µB) ≪ (µB) P the precession effects can be neglected. Let us also introduce the adiabaticity bound (µB) A using the adiabaticity condition (10): where level crossing is adiabatic. Comparing (21) and (22) one finds that for fixed H SF the precession bound increases with V SF faster than the adiabaticity bound. Consequently, for large V SF , the inequality (µB) P ≫ (µB) A holds and the conversion dominates over precession. The suppression of V SF makes the precession effect more profound. If the potential changes as V SF ∝ r −3 , one gets Clearly, (µB) P < (µB) A in the hydrogen envelope, and (µB) P > (µB) A in the center of star. The adiabatic conversion dominates in the inner parts, whereas the precession may dominate in the external layers of star ( Fig. 3). In the isotopically neutral region one has (µB) P ∼ (µB) A , i.e., both processes may be essential (see Sect. 4). In fact, the condition (21) is not sufficient for a strong spin-precession effect. The spatial size of the region where (21) is fulfilled, ∆r, should be large enough, so that the inequality (1) or ∆r > ∼ 1/µB is satisfied. For r > 0.3R ⊙ the latter bound is even stronger than (21). Instead of µB, in further discussion we will give the estimations of the magnetic fields at µ = 10 −12 µ B . Rescaling to other values of µ is obvious.
The adiabaticity and the precession bounds determined by the density distribution should be compared with the magnetic field profile of star. In fig. 3, we depict the profile (2) with k = 2 and B 0 = 1.5·10 13 Gauss. The strength of this field is practically everywhere below both the adiabatic, B A , and the precession, B P , bounds (µ = 10 −12 µ B ) which means that spin-flip effects are rather weak. The effects of the global field (2) may be strong, when B 0 ≥ 10 14 Gauss. However, for k = 3 even extremely large central field, B 0 ∼ 10 17 Gauss, corresponds to very weak magnetic field in the isotopically neutral region. Local mechanisms of the magnetic fields generation due to possible convection and differential rotation may produce the local field being much larger than the global field, as in the case of the Sun.

Effects of jumps of the potential
The electron density number n(2Y e − 1), and consequently, the effective potential V SF change quickly in the ignition layers, so that for reasonable values of µB the adiabaticity is broken. This feature is reflected by peaks in the adiabaticity bound ( fig. 3). To a good approximation one can consider these changes as sudden jumps of the potential, ∆V j ≡ V max − V min , at fixed points r j . in [25]). If ∆V j ≤ ∆V R , the jump leads to the peak of the survival probability with width, ∆E, determined by the width of the resonance layer: ∆E/E ∼ 2µB/V SF (r j ) [25]. The height of the peak is given by the ratio: For 2µB ≫ ∆V j the jump effect is smoothed: since the size of the resonance region is large, the width of peak is also large, but its height turns out to be suppressed.
If ∆V j ≥ 2∆V R ≈ µB, the height of the peak is of the order 1, and the width is fixed by the size of jump: For ∆V SF ≫ ∆V R , the jump suppresses the transition in the interval with minimal and maximal energies determined by V max and V min .
The most profound effect appears, when without jump the adiabaticity is fulfilled and the transition is practically complete. If the width of the resonance is small (2µB < V SF ) and if ∆V j ∼ ∆V R , the jump induces a thin peak of the survival probability. Such conditions can be fulfilled in the inner part of the isotopically neutral region.
Note that in the considered model of the star there are two big jumps of the potential (∆V j /V SF ∼ 10) at r = 7 · 10 −2 R ⊙ and r = 0.3R ⊙ ( fig.1). Also small jumps (∆V j /V SF < ∼ 1) exist.
The elements of the dynamics presented here allow to understand all features of the evolution of the neutrino state.

Neutrino propagation in isotopically neutral region
Let us consider for definiteness the spin-flavor transitionsν e − ν µ and ν e −ν µ . If ∆m 2 ≤ 0.1 eV 2 the level crossing takes place in the isotopically neutral region. For realistic magnetic fields (profile (2) with B 0 ≪ 10 17 Gauss) the effect ofν e − ν µ mixing can be neglected in the inner part of the star. Therefore, the spin-flavor evolution of the neutrino state starts from the inner part of the isotopically neutral region r i ∼ 10 −3 R ⊙ . At r i the states coincide practically with pure helicity/flavor states. Flavor conversion, if efficient, can be taken into account as further independent transformation (sect. 5). Also the transitions which involve ν τ can be considered separately.

Spin-flip effects in global magnetic field
We use the profile (2) with k = 2 and take different values of B 0 . For B 0 < ∼ 2 · 10 13 Gauss the magnetic field strength is below the precession bound (µ = 10 −12 µ B ) in the isotopically neutral region. Consequently, the depth of the precession is suppressed, and the transition has a local resonance character ( fig. 4). The transition occurs only in the resonance region whose width is determined by ∆V R /V SF = 2µB/V SF . The efficiency of the transition depends on κ R . The strongest effect takes place for the energies (over ∆m 2 ), E/∆m 2 , which correspond to the weakest violation of the resonance adiabaticity (B(r) is closer to the adiabaticity bound ( fig. 3, 4)). With the increase of B 0 the E/∆m 2 range of strong conversion expands ( fig. 5). 13 Gauss, the field profile is above both the precession and the adiabatic bounds, and the conversion becomes essentially nonlocal ( fig. 4). For E/∆m 2 > 10 3 MeV/eV 2 initial mixing is still very small: θ B ≈ π/2, sin 2 2θ B ≈ 0. Therefore, the neutrino state coincides with one of the eigenstates in medium. Since the adiabaticity is fulfilled, the survival probability is uniquely determined by the helicity mixing angle: P (ν e → ν e ) ≈ cos 2 θ B . The precession effects are very small ( fig. 4). Before the resonance the angle is determined from tan 2θ B ≈ µB(r)/V SF (r). The mixing increases, when the neutrino enters the isotopically neutral region, and if µB ≫ V SF the probability approaches P ∼ 1/2 ( fig. 4). After the resonance layer the angle diminishes quickly: tan 2θ B ∼ 2µB/(∆m 2 /2E), since the magnetic field decreases with distance, whereas the splitting approaches the asymptotic value ∆m 2 /E. Correspondingly, the probability goes to zero and one has almost complete spin-flip with P (ν e → ν e ) ≈ 0 ( fig. 4).
For large E/∆m 2 the resonance is in the external layers of the star, where the adiabaticity is broken. The final survival probability increases with E/∆m 2 , approaching the value of probability before the resonance. As the result one gets P > ∼ 1/2. At very large E/∆m 2 mass splitting becomes unessential and the dependence of probability on E/∆m 2 disappears.

Spin-flip in local magnetic field
Let us suppose that a strong magnetic field exists in some spatial region ∆r between r 1 and r 2 , so that the transitions take place in ∆r only. For simplicity we consider a constant magnetic field B c . If µB is below the precession bound, the conversion occurs in the resonance layer, and the transition probability is determined by κ R . The effect is appreciable in the energy interval determined via the resonance condition by the maximal and minimal values of the potential in the layer with magnetic field (fig. 6). When the adiabaticity is broken, the maximal effect corresponds to the resonance in the central part of the region with the magnetic field. In this case there is no averaging over the precession phase and smooth dependence of the probability on energy can be modulated. The energy dependence can be distorted also by jumps of the potential.
If the field is strong enough and the adiabaticity condition is fulfilled, the average probability follows the change of helicity mixing angle θ B (r) (17). The survival probability, as a function of E/∆m 2 is determined by the adiabatic formula: where Φ is the precession phase: Here θ i B and θ f B are the values of the mixing angle (17) at r 1 and r 2 . For µB ≫ V SF , the influence of matter is small and the survival probability is large for E/∆m 2 > 1/(µB). If E/∆m 2 ≫ 1/(µB) the precession has maximal depth ( fig. 6).

On the energy dependence of probability
The survival probability as function of the neutrino energy has some common features for different magnetic field profiles. There are two characteristic energies: (E/∆m 2 ) min and (E/∆m 2 ) max . The former, (E/∆m 2 ) min , is determined by the adiabaticity violation in the inner part of the isotopically neutral region, or in the inner part of the region with magnetic field (in the case of the local field).
The latter, (E/∆m 2 ) max , is fixed in such a way that for E > E max the effect of mass splitting can be neglected. For the local field (E/∆m 2 ) max is determined by the potential at the outer edge of the region with magnetic field. In the case of global field one has (E/∆m 2 ) max ≈ 10 −10 eV 2 which follows from the adiabaticity violation due to a fast decrease of µB or V SF at the border of the isotopically neutral region (fig. 3).
The energies (E/∆m 2 ) max and (E/∆m 2 ) min divide the whole energy interval into three parts. 1). Region of matter/vacuum suppression of the spin-flip: E/∆m 2 < (E/∆m 2 ) min . For neutrinos with such energies before the resonance layer the precession amplitude is strongly suppressed by matter effect. There is no appreciable conversion in the resonance region, since the adiabaticity is strongly broken; after the resonance the amplitude is suppressed by vacuum splitting: A P ∼ (µB)/(∆m 2 /E). As a result, the probability of spin-flip is small.
2). Resonance region: (E/∆m 2 ) max < E/∆m 2 < (E/∆m 2 ) min . The resonance is in the isotopically neutral region. If the adiabaticity is unbroken or weakly broken, the survival probability is small. Smooth energy dependence of P due to conversion can be modulated by the effect of density jumps, as well as by the precession effect, if e.g. the initial mixing angle is not small.
3). Precession (asymptotic) region: E/∆m 2 > (E/∆m 2 ) max . Here the mass splitting (∆m 2 /E) can be neglected, and therefore the spin-flip probability does not depend on energy. The spin-flip is due to precession and possible adiabatic (non-resonance) conversion.
If the layers with magnetic field are in the inner part of the isotopically neutral region, the resonance effects can be more profound. Here the adiabaticity condition is fulfilled, even when the width of the resonance is small, 2µB/V SF < 1. For the external layers with small density the adiabaticity holds for 2µB/V SF > 1 only, i.e. when the width of resonance is large. Three neutrino states, (ν e ,ν e , ν µ ), are involved in the transitions. Correspondingly, one can introduce 3×3 matrix of probabilities S which relates the original, F 0 = F 0 (ν e ), F 0 (ν e ), F 0 (ν µ ) , and the final, F = (F (ν e ), F (ν e ), F (ν µ )), fluxes: Let r ′ be some point between the two resonances: r s ≪ r ′ ≪ r f . Then in the region r < r ′ the matrix S depends only on the spin-flavor transition probability, P s (ν e − ν µ ): S = S s (P s ).
(Evidently, (1 − P s ) coincides with the survival probability calculated in the sect. 4). In the region r > r ′ the matrix S depends on the flavor transition probability, P f (ν e − ν µ ) transition: or/and if there is a strong enough magnetic field in the inner part of star: r < 0.3R ⊙ (Fig. 7).
In the overlapping energy region the probabilities P s and P f differ from zero and the electron neutrinos undergo a double transition:ν e → ν µ → ν e (fig. 7).

Effects of tau neutrino
Inclusion of ν τ adds two new level crossings: the flavor, (ν e − ν τ ), and spin-flavor, (ν e − ν τ ) ones ( fig. 2). Other crossings involving ν µ and ν τ do not give observable effects. Due to the assumed mass hierarchy the transitions with ν τ (the inner region of the star) can be considered independently from the transitions involving ν µ (outer part). Consequently, the total transition matrix is the product of the transition matrices in the inner and the outer regions.
Three states, ν e ,ν e , ν τ , are transformed in the inner part ( fig. 2). For monotonously decreasing density, the order of resonances is the same as before: first -spin-flavor and then flavor one. If the ν τ resonances are enough separated, the transition matrix can be found from that in (27) by substitution P f → P ′ f and P s → P ′ s , where P ′ f and P ′ s are the 2ν-probabilities of flavor and spin-flavor transitions in the inner region. (In fact one should consider 4 × 4 transition matrices in the (ν e ,ν e , ν µ , ν τ )-basis in the inner and the outer parts and find their product.) The effects due to the ν τ -resonances can be taken into account as the initial conditions,F i , for the fluxes at the border with the outer region. Using the matrix (27) with substitution mentioned above we get Further on we will concentrate on the cosmologically interesting values m 3 ∼ (2 − 7) eV. This corresponds to ∆m 2 = (4− 50) eV 2 , so that for E ∼ 20 MeV the resonances are in the inner part of the isotopically neutral region: r ∼ (1 − 2) · 10 −3 R ⊙ . Here the total density is (0.3 − 3) · 10 8 g/cm 3 and (1−2Y e ) ∼ (2−4)·10 −2 . According to fig. 3, the precession bound equals (10 13 −3·10 15 ) Gauss, and the adiabaticity bound is (0.5−2)·10 11 Gauss. Thus an appreciable spin-flip transition implies very large magnetic field: B > 10 10 Gauss. In contrast, the flavor transitions can be efficient even for very small mixing angles: sin 2 2θ > 10 −6 . For ∆m 2 under consideration the spin-flip effects may be more probable below the isotopically neutral region.

Transitions in the central region of the star
The collapse and the shock wave propagation lead to strong time dependence of the density profile in the central region of the star. Moreover, the profile may be non-monotonous. With approaching the center of star the total matter density may first diminish from 10 8 g/cm 3 to (10 6 − 10 7 ) g/cm 3 at r ∼ 100 km, and then rapidly increase at the surface of the protoneutron star, r < (20 − 30) km. Moreover, in the central part the neutrino-neutrino scattering should be taken into account [34]. Due to the non-monotonous behavior of V SF , for ∆m 2 = (4 − 50) eV 2 the additional level Gauss. This is only slightly higher than that in the inner edge of the isotopically neutral region.
However, the existence of such a field is more probable in the inner part; for the profile (2) with k = 2 and B 0 = 10 13 Gauss the field is only 2 -3 times smaller than the adiabaticity bound. Thus the magnetic moment µ ∼ 10 −12 µ B may give strong spin-flip effect here.
If in the inner part all four transitions are efficient, there is no observable effect: the final state will coincide with the initial one.

Implications
In principle, future experiments will give information on the energy spectra of different neutrino species. Confronting these spectra with each other, one can get essentially model independent information about possible neutrino transitions. It is convenient to introduce three types of original spectra: soft, F s (E), middle, F m (E), and hard, F h (E), which coincide with original ν e -,ν e -, and ν µ -spectra, respectively. The muon and tau neutrinos and antineutrinos are indistinguishable in the standard electroweak model at low energies, and will be detected by neutral currents. Therefore we will consider the total flux of the "non electron neutrinos", F (ν ne ). If there is no neutrino transitions, then at the exit: Let us find the signatures of the spin-flip transitions, as well as the sensitivity of the ν-burst studies to the neutrino magnetic moments. We estimate the values of B (at µ = 10 −12 µ B ) needed for different effects to be observable.

Sensitivity to magnetic moments and magnetic fields
The suppression of the effective potential by 3 − 4 orders of magnitude diminishes the strength of the magnetic field (magnetic moment) needed for a strong spin-flip effect. The adiabatic and the precession bounds decrease by 1.5 − 2 and 3 − 4 orders of magnitude respectively. We take as the criterion of the sensitivity of the ν-burst studies to the spin-flip effects the magnitude of the magnetic field strength B s (µ = 10 −12 µ B ) at which the probability of transition is ∼ 1/2 at least for some neutrino energy interval. According to Fig. 2, in the most part of the isotopically neutral region (r ≤ 0.3R ⊙ ) the adiabaticity bound is below the precession bound. Only in the external part of star the precession could be more preferable. Consequently, for r ≤ 0.3R ⊙ the sensitivity limit corresponds to inequality 2µB < V SF . The latter means that mixing is small everywhere apart from the resonance layer and the transition is due to the level crossing with not so strong adiabaticity violation. The nonaveraged over the precession phase probability P ∼ 1/2, can correspond to the value of the Landau-Zener probability P LZ ∼ 0.75. Therefore we will define the sensitivity limit B s (r) in a given point as the strength of the magnetic field, for which P LZ (r) ∼ 0.75. According to the estimations in section 3.2, the sensitivity bound can be about 3 times smaller than the adiabaticity bound: fig. 3).
For the effect to be appreciable, the size of the region with magnetic field, ∆r B , should be comparable with the size of the resonance region. If V SF ∝ r −3 , we get ∆r B = (dV SF /dr) −1 2∆V R ≈ 4µBr/3V SF , and in particular, for 2µB/V SF < 0.3: ∆r B < 0.2r. In fact, B s should be considered as the average field in the region ∆r B .
In the external part of the isotopically neutral region, r = (0.3 − 0.7) R ⊙ , the precession bound is essentially below the adiabaticity bound. However, here for 2µB ∼ V SF the precession length is already comparable with the distance to the center of star and the condition (3)  Conversely, if the magnetic field is ∼ 3 · 10 5 Gauss, the ν-burst will be sensitive to µ ∼ 3 · 10 −14 µ B .
Note that in the isotopically neutral region one needs for appreciable ν eL −ν µR conversion 1.5 − 2 orders of magnitude smaller field than for conversion into a sterile state: ν eL → ν sR .
Comparing the above results with those of sect. 5.3 we find that the sensitivity to µ can be even higher in the isotopically neutral region than possible sensitivity to µ in the central region of star. (Of course, the latter corresponds to large values of ∆m 2 ).

Bounds on µB from SN1987A
The spin-flip ν µ →ν e leads to an appearance of the high energy tail in theν e -spectrum at the Earth: where P s is the transition probability. At small mixing angles the flavor conversion does not change this result (see fig. 3). Note that in (30) P s can be close to 1, in contrast with averaged vacuum oscillation effect which gives P ≤ 1/2.
The absence of the distortion of theν e -energy spectrum, and in particular, the absence of the high energy tail, gives the bound on µB as function of ∆m 2 . For a class of supernovae models the data from SN1987A allow to get the restriction P s < 0.35 under the assumption that P S does not depend on energy [32]. In fact, at the border of sensitivity the suppression pit is rather thin (figs. 5, 6) and the energy dependence can not be completely neglected. In this case the strongest observable effect takes place, when a position of the pit coincides with the high energy part of the ν µ -spectrum. The sensitivity limit obtained in sect. 6

Transitions of the degenerate or massless neutrinos
The mass differences ∆m 2< ∼ 10 −10 eV 2 can be neglected for the neutrino energies E ∼ 5 − 50 MeV. This corresponds to the asymptotic region in P -dependence on E/∆m 2 ( fig. 4, 5). As we have noticed in sect. 2, for ∆m 2 ≈ 0 there are two level crossings ν e −ν µ andν e − ν µ at the same point, r ∼ R ⊙ . The crossings are induced by change of the nuclear composition on the border between the H-envelope and the 4 He-layer. However, in this region the effective density changes very quickly, and the adiabaticity condition implies very strong magnetic field B((0.7 − 1)R ⊙ ) > 3 · 10 5 Gauss which exceeds the precession bound. The effect is mainly due to the precession, and the field should be as strong as B ∼ (2 − 5) · 10 4 Gauss. The probabilities of transitions ν e ↔ν µ andν e ↔ ν µ are equal and do not depend on energy. The final spectra of ν e andν e equal: F (ν e ) = (1−P s )F s +P s F h , F (ν e ) = (1 − P s )F m + P s F h . In the case of complete transformation, P s ≈ 1, theν e -and ν e -energy spectra coincide with the initial ν µ (ν µ )-spectrum: For ∆m 2 > (10 −9 − 10 −8 ) eV 2 (which is interesting for the spin-flip of solar neutrinos [33]) the effect of the mass splitting becomes important. The probabilities ofν e → ν µ and ν e →ν µ transitions are different, and depend on energy.

Distortion of neutrino energy spectra
Using the matrix of the transition probabilities for the isotopically neutral region (27) and the initial conditions (28) we get for the final neutrino spectra: The important conclusions can be drawn from (32,33) immediately. Theν e -spectrum is a mixture Thus the distortion of theν e -spectrum, and in particular, the appearance of the hard component can be considered as the signature of the spin-flavor conversion.
According to fig. 8, the spin-flip may result in a variety of distortions of theν e -spectrum. In particular, when the transition probability P s is constant, the permutation of theν e -and ν µspectra can be symmetric, so that F (ν e ) ≃ (1 − P s )F m + P s F h , Such an effect is realized if the spectra are in the asymptotic region of the suppression pit or in the region of strong (complete) transformation. The transition can be asymmetric, so that inν e -spectrum the suppression of the F m -component is weaker than the appearance of F h -component and vice versa. This is realized, when the spectra are at the edges of the suppression pit or in the modulated resonance region.
Note that the typical energy scale of the modulations of the probability can be characterized by factor 2 -3. Therefore the spin-flavor transition results in a smooth distortion of each component.
The fine structure of the energy spectrum (∆E ∼ 1 − 2 MeV) can be due to the jumps of density or/and the field twist in the inner parts of the star (see sect. 7).
Two remarks are in order. The change of theν e -spectrum is expected also if flavor mixing is large. In this case, however, one gets an energy independent interchange of the spectra with the transition probability P < 0.5. If the neutrino mass hierarchy is inverse, the properties of the transitions in the neutrino and antineutrino channels should be interchanged. In particular, ν e -spectrum will acquire the soft component.
According to (33), the final ν e -spectrum is, in general, an energy dependent combination of all three original spectra. However,   The modifications of the spectra are especially simple, when the resonant transitions are either completely efficient or completely inefficient: P i = 0 or 1 [17]. In this case a complete permutation of the original spectra occur. Moreover, as follows from (33,34), there are only four possible types of final spectra (Table I).
According to the Table I Interesting effect can be related to the existence of a region in which the density increases with distance. As we have noticed, here the order of resonances is changed: neutrinos first cross the flavor resonance and then the spin-flavor resonance. In this case ν e can be transformed intoν e ; the finalν e -spectrum will contain the soft component, and moreover the neutronization peak will consist ofν e . Such a modification of spectra can be realized, if two inner resonances s and f , are inefficient. They are at the surface of the protoneutron star, where the density is changed very quickly, and therefore the adiabaticity could be strongly broken.
7 Effects of magnetic field twist 7.1 Field twist. Scale of twist If the direction of the magnetic strength lines changes with distance, the propagating neutrinos feel the rotation (twist) of the magnetic field. The field twist leads to the splitting of the levels with different helicities by the valueφ (see (15)) [30]. The effect can be considered as the modification of the effective potential: Thus the field twist changes the level crossing picture (the positions of resonances). However, since the flavor and spin-flavor resonances are strongly separated due to the isotopical neutrality of the medium, the shift (34) hardly induces the spatial permutation of resonances [30].
The field twist can be characterized by the scale of twist, r φ : so that on the way, r φ , the total rotation angle (in case of uniform rotation) equals ∆φ = π.
It is natural to suggest that the total rotation angle is restricted by For example, the twist appears, when the neutrinos cross the toroidal magnetic field with strength lines winding around the torus. In this case the maximal rotation angle is π, i.e. the bound (36) is fulfilled. Let ∆r B be the size of the region with the magnetic field, then ∆φ ∼φ∆r B , and the condition (36) implies That is the scale of the rotation is comparable or larger than the region with the magnetic field.
For the global field one has ∆r B ∼ r, and therefore As we will see the bound on the total rotation angle (36) restricts strongly the effects of the field twist.

Effect of density suppression
Let us consider possible increase of sensitivity to (µB) due to the field twist. According to (34) aṫ the matter effect is completely compensated at the point r. Let us define the critical rotation scale, r c φ , at which the condition (39) is fulfilled: (Note that r c φ coincides up to the factor 2 with the refraction length). At r φ ∼ r c φ the effect of field twist can be essential, and for r φ < r c φ , it even dominates over the density effect. Comparing r c φ with the distance from the center, r, we find ( fig. 9) that r c φ /r ≥ 1 for r ≥ 5 · 10 −2 R ⊙ , and r c φ ≪ r for r < 5 · 10 −3 R ⊙ . In the inner parts of the star the critical scale is much smaller than the distance from center, e.g., at r = 10 −3 R ⊙ one gets r φ ≃ 20 cm. The condition (38) is fulfilled for r ≃ 5 · 10 −3 R ⊙ . Let us stress that the isotopical neutrality essentially enlarges the region, where (38) is satisfied and therefore the field twist can be important.
Suppose the equality (39) is fulfilled in some layer of size ∆r (evidently, the field twist should be nonuniform). In ∆r the spin-flip effect has a character of precession with maximal depth, and the sensitivity to µB is maximal. Let us estimate µB taking into account the restriction (36). The total rotation angle ∆φ equals and if ∆r ≪ r: The spin-flip probability is of the order one, when ∆r ≃ π(2µB) −1 . Substituting this ∆r in (42), we get the relation: Finally, (43) and the bound on the total angle of the twist (36) give µB > V SF /2 which precisely coincides with the precession bound (6). Thus, the field twist can relax the precession bound by factor 1/2 at most, and to the further increase of the sensitivity one should admit the field rotation on the total angle which exceeds π.
In the external layers (r ≥ 0.1R ⊙ ), ∆r is larger than r, and the compensation (39) implies fine tuned profiles ofφ(r) and V (r) in a wide spatial region which seems rather unnatural. On the other hand, if the compensation takes place in the inner part (r < 0.1R ⊙ ) then ∆r ≪ r. However, here the precession bound is more stringent than the adiabaticity bound, and the compensation does not allow to gain in the diminishing of the field strength. In the inner part the effect of the field twist can be due to influence on the adiabaticity.

Influence of field twist on the adiabaticity
Nonuniform field rotation (φ = 0) modifies the adiabaticity condition. The adiabaticity parameter equals: Atφ there is a strong improvement of the adiabaticity. The field twist results in flattening of the potential, so that dV φ /dr ≃ 0 or V φ ≃ const., in some region ∆r. Let us estimate the minimal value of the total rotation angle in this region. For this we suggest thatφ = 0 at the one of the edges of ∆r. Then from (45), it follows that ∆φ ≈V SF (∆r) 2 /2. Inserting ∆r ≈ π(2µB) −1 (the condition for the strong precession effect) in the last expression, we get i.e. the total rotation angle is the inverse value of the adiabaticity parameter without field twist.
Therefore, if the adiabaticity is strongly broken, κ R ≪ 1, one needs ∆φ ≫ π to get an appreciable spin-flip effect. For restricted values of total rotation angle (36) only weakly broken adiabaticity can be restored. The relation (46) means that if the transition without field twist is weak (κ ≪ 1), then field twist will induce a weak effect for ∆φ < π. On the other hand, if the transition without twist is strong (κ R ∼ 1), then the field twist can make it even stronger but the absolute change of probability turns out to be always small ( fig. 11).
The effect of the adiabaticity restoration due to field twist can be considered as the effect of precession in the region ∆r with flat potential (V φ = const.). For constant B (which we suggest for simplicity) the transition probability equals One can find from (47) that field twist leads to the distortion of the energy dependence of the probability in some energy interval ∆E located at E/∆m 2 ≃ (2V φ ) −1 . Let us estimate the size of ∆E. According to (47) ∆E is determined by the resonance width ∆E/E ∼ 4µB/V SF . In turn, 2µB can be estimated from (46) and the condition ∆φ < π: 2µB > ∼ (πV SF )/2 . So that finally, we get If V SF ∝ r −3 and H SF ∝ r, the relation (48) gives ∆E E ∝ r. The closer the region with field twist to the center of the star the thinner the peak. For r ∼ 10 −2 R ⊙ , one gets ∆E E > ∼ 0.3. In external region of star (r > 0.1R ⊙ ), the field twist results in smooth change of the probability in a wide energy region, and it is impossible to distinguish it from other effects ( fig. 11). In any case the absolute value of ∆P is small, unless ∆φ ≫ π.
As follows from (44) the field twist can also destroy the adiabaticity, when |φ| ≫ |V |. Randomly twisting field inhibit the conversion in such a way. This can result in bumps of the survival probability that may be similar to those due to density jumps.

Conclusion
1. For neutrino mass squared differences ∆m 2 < 10 eV 2 (which are interesting for the cosmology, as well as for the physics of solar and atmospheric neutrinos) the resonant spin-flavor transitions (ν e →ν µ etc.) take place in almost isotopically neutral region of collapsing star. In this region which extends from ∼ 10 −3 R ⊙ to ∼ R ⊙ the deviation from the isotopical neutrality, 2Y e − 1, can be as small as 10 −3 − 10 −4 . Correspondingly, the matter potential for the spin-flavor transitions, being proportional to (2Y e − 1), turns out to be suppressed by 3 -4 orders of magnitude. Moreover, the potential changes the sign at the inner edge of the hydrogen envelope.
2. The suppression of the effective potential in the isotopically neutral region diminishes the values of (µB) needed to induce an appreciable spin-flip effects by 1.5 -2 orders of magnitude.
Thus, the sensitivity of the neutrino burst studies to the transition magnetic moments of neutrinos increases, being of the order 10 −13 µ B for ∆m 2 = (10 −8 − 10 −1 ) eV 2 and for a reasonable strength of the magnetic field. In particular, for ∆m 2 = (10 −8 − 10 −9 ) eV 2 the desired values of (µB) turn out to be 1.5 − 2 orders of magnitude smaller than those for a strong conversion of the solar neutrinos.
3. In the isotopically neutral region the potential changes very quickly in the layers with local ignition (jumps of the potential). The jumps result in distortion of the energy dependence of probabilities. In particular, one can expect the appearance of thin peaks in the survival probability if the jump is situated in the inner part of the star. 4. Depending on the values of the neutrino parameters as well as on the magnetic field profile one expects a variety of the modifications of the neutrino spectra. In the case of a direct mass hierarchy and a small flavor mixing the main signature of the spin-flip effect is the distortion of thē ν e -energy spectrum, and especially the appearance of the high energy tail. In general, the finalν espectrum is the energy dependent combination of the originalν e -spectrum and the hard spectrum resulting again in hard and equal ν e -andν e -spectra. 6. The absence of the considered effects allows one to get the bound on (µB(r)) as a function of ∆m 2 . We estimate such a bound for SN 1987A.
7. In the isotopically neutral region, where the matter potential is strongly suppressed, the effects of (even) small field twist may be important. Field twist can further suppress the potential, thus increasing the sensitivity to the magnetic moment. However, if the total rotation angle is restricted (< π), a possible diminishing of µB can be by factor 1/2 at most. The twist of the field may induce a distortion of the neutrino energy spectra. In particular, improving adiabaticity, in the inner part of the isotopically neutral region the field twist may lead to peaks of the transition probabilities.          (2) with k = 2 and B 0 = 5 · 10 12 Gauss was used. The solid line shows the probability without field twist. The field twist is located in the region r 0 − (r 0 + δr), and the profile of twist is described byφ(r) = (2π/δr)(1 − (r − r 0 )/δr); the total rotation angle equals π.