Resonance refraction and neutrino oscillations

The refraction index and matter potential depend on neutrino energy and this dependence has a resonance character associated to the production of the mediator in the $s-$channel. For light mediators and light particles of medium (background) the resonance can be realized at energies accessible to laboratory experiments. We study properties of the energy dependence of the potential for different C-asymmetries of background. Interplay of the background potential and the vacuum term leads to (i) bump in the oscillation probability in the resonance region, (ii) dip related to the MSW resonance in the background, (iii) substantial deviation of the effective $\Delta m^2$ above the resonance from the low energy value, etc. We considered generation of mixing in the background. Interactions with background shifts the energy of usual MSW resonance and produces new MSW resonances. Searches of the background effects allow us to put bounds on new interactions of neutrinos and properties of the background. We show that explanation of the MiniBooNE excess, as the bump due to resonance refraction, is excluded.


Introduction
The Wolfenstein potential 1 , which describes the matter effect on neutrino oscillations, do not depend on the neutrino energy [1][2][3][4]. This is the consequence of (i) large mass of mediators of interactions, M med , or low energies of neutrinos, so that the total energy in the CMS: √ s M med . Recall that originally the potentials were derived using the 4 fermion point-like interactions.
(ii) the C-(CP-) asymmetry of background. In the C-symmetric medium in the lowest order the potentials are zero.
In general (independently of the C-asymmetry) substantial dependence of the potentials on energy should show up at energies √ s M med . Furthermore, exchange of mediator in the schannel leads to the resonance character of this dependence [5]. We will call this phenomenon the resonance refraction.
In the Standard Model the mediators of neutrino interactions are W , Z 0 as well as H 0 . Z 0 leads to the resonance refraction in theνν− annihilation. In resonance the potential is exactly zero and changes the sign with energy change. Above the resonance energy the potential has 1/E dependence similar to the usual kinetic term related to mass squared difference [5]. In principle, this refraction can be realized in scattering on the ultra high energy cosmic neutrinos on relic neutrino background (E ≥ 10 21 eV in the present epoch) [5]. The W −boson exchange produces the resonance refraction in theν e e− scattering, i.e., in the Glashow resonance. For electrons at rest this requires the neutrino energy ∼ 6.4 PeV. We comment on possibility of observational effects in sect. 3.8.
For light mediators and light scatterers (their existence implies physics beyond the SM) the resonance refraction can be realized at low energies accessible to existing experiments. The resonance refraction leads to increase of the oscillation phase which can dominate over the vacuum phase in the energy range around the resonance. This produces an enhancement of the oscillation effect which would be negligible without resonance refraction. Such an enhancement was used in [6] to explain the low energy excess of the MiniBooNE events [7]. In this explanation the medium was composed of the overdense relic neutrinos.
In this paper we focus on phenomenon of resonance refraction itself presenting results in a model independent way. We study in detail dependence of the resonant potentials on energy for different values of the C− asymmetry of background. We consider interplay of resonance potentials with usual vacuum (kinetic) term as well with usual matter potential. New interesting features are realized, such as shift of the usual MSW resonances, increase or decrease of the effective mass squared difference with energy, etc. We identify signatures of the resonance refraction and outline possible observable effects. As an illustration, we apply our results to the MiniBooNE excess and show that explanation [6] is excluded. In general, applications can include explanations of some energy localized anomalies. In the absence of anomalies the bounds can be established on background parameters (densities, characteristics of scatterers) and neutrino couplings.
The paper is organized as follows. In sect. 2 we introduce interactions of neutrinos with new light sector. We compute potentials due these interactions and study resonances in these potentials.
In sect. 3 we discuss effects of interplay of the background potential with vacuum (kinetic) term and usual matter potential. We consider possible observational effects and applications of the results, in particular, to an explanation of the MiniBooNE excess in sect. 4. Conclusions follow in sect. 5.

Potentials and resonances 2.1 Neutrino interactions with new light sector
In this paper we focus on phenomenon of resonance refraction itself, and present our results in general and universal form valid for different mediators and particles of background. We consider the simplest (minimal) light sector composed of new scalar φ (which can be real or complex) with mass m φ and fermion χ with mass m χ . We comment on some extensions of this sector later.
Interactions of the SM neutrino mass states ν iL (i = 1, 2, 3) with these new particles are described by φ may acquire VEV, thus contributing to neutrino mass. Then for single χ only one neutrino (combination of ν i ) will acquire mass by VEV of φ. We assume that some other sources of χ and neutrino masses exist, e.g. the see-saw mechanism, so that χ and all ν i acquire different masses and in general these masses are not related to g i . The couplings (1) were considered in various contexts before [16, 19-21, 23, 24, 27, 29]. For light new particles m φ , m χ 1 GeV, a number of generic bounds were obtained. The bounds are based on possible transitions ν → χ + φ.
Notice that refraction is induced by the elastic forward scattering being proportional to g 2 /M 2 med . Therefore it does not disappear in the limit g → 0, provided that M med decreases in the same way as g. This allows us to avoid most of the bound based the inelastic processes for which σ ∝ (g 2 /q 2 ) 2 , and the transfer momentum squared q 2 is restricted from below by condition of observability.
The laboratory bounds on g are rather weak: g φ 10 −3 for masses m φ < m K (K− meson mass). They follow, in particular, from additional contribution to the decay K → µχφ. Much stronger bounds follow from Cosmology (BBN, CMB data, structure formation) and astrophysics (star cooling, supernova dynamics and SN87A neutrino observations). They give the bound Elastic forward scattering due to the interactions (1) produces the effective potentials V i for neutrino mass states in medium. There are two possibilities even for simplest case of (1) (i) φ plays the role of mediator while χ form a background, and vice versa: (ii) χ is the mediator while φ form a background.

Potentials in the fermionic background
We consider first the case of strong hierarchy of couplings: g 3 g 2 , g 1 , so that ν 3 couples with background, while interactions of others can be neglected. In this case the interactions, and consequently the potentials, are diagonal in the mass basis. We will discuss couplings of all three neutrinos later in sect. 3.7. Also we comment on the case of three χ j .
We consider background composed of fermions χ and antifermionsχ with number densities n χ andn χ correspondingly. The C-asymmetry of the background can be defined as The mediator is a scalar φ and the diagrams of the neutrino scattering on χ (left) andχ (right) are shown in Figure 1. For m φ > m ν , m χ the right diagram with the s-channel exchange produces resonance.
To obtain the potential, we integrate the matrix element of the process over the momentum of particle χ with distribution function, F χ (k). The latter is normalized as and similarly forχ. The left (u-channel) and right (s-channel) diagrams in Fig. 1 give correspondingly the potentials Here in the propagator we added the term with total width of φ. In vacuum (φ → ν χ), where g = g 3 and we take g 1 = g 2 = 0. In medium the term with Γ is modified (see below).
We assume first that the background particles are at rest which is valid for cold gases like dark matter (DM) or relic neutrinos from the cosmological neutrino background (CνB). Then F χ (k) = nδ( k) and the integrals in (5) and (6) give the total potential This expression differs from expression for potential in [6], but coincides with that in [11].
We obtain similar result for moving χ with the only substitution m χ → E χ , if the angular distribution is isotropic. This is important for the degenerate gas with large overdensity when the The second term in (8) has a resonance dependence on energy (pole of propagator) with the resonance energy At E R the contribution V s is exactly zero and it changes the sign with energy change. The amplitude of scattering becomes purely imaginary, which corresponds to production of the on shell φ. In terms of the resonance energy the potential (8) can be rewritten as where and in vacuum Let us introduce a dimensionless parameter In terms of y the expression for the potential (10) becomes where In this way we can disentangle dependencies of the potential on relevant physical quantities: V 0 depends on parameters of mediator, g and m φ , and on total density of scatterers in a background.
It has a form of the standard matter potential at low energies with G φ = g 2 /2m 2 φ . The parameter ξ is proportional to the coupling constant squared, while the mass of χ enters via E R . V 0 is introduced in such a way that for y → 0 we have V B → V 0 , and consequently, for = ±1: V B = ±V 0 , thus reproducing the standard Wolfenstein potential.

Potentials in the bosonic background
For the scalar particle background and fermionic mediator the potential is similar to the one computed before. In the lowest order in g 2 , up to factor of 2 the potential has the same expression as in (8) with the following substitutions Thus, The resonance is realized if m χ > m ν + m φ , and the resonance energy equals In terms of resonance energy the potential can be written in exactly the same form as in (14) with The difference from the fermionic background case may appear in higher orders in g 2 due to fermionic nature of mediator χ. Now the amplitude of scattering is proportional to / q = / p + / k: The first term gives contribution to renormalization of the wave function of neutrino: ν = (1+Σ ν /2)ν L , while the second one generates the potential: for the background at rest γ 0 m φ Σ χ = γ 0 V . Renormalization leads to change of the potential: V = (1 + Σ * ν /2)V (1 + Σ ν /2) = V (1 + Σ ν ) (as well as usual kinetic term) [11]. The correction is of the order g 2 . In this order one should take into account also loop corrections to external neutrino lines All these corrections have the same nature and can be described by tree level diagrams with multiple scattering on a background: ν + φ * → χ → ν + φ * , ν + φ * → χ.... Alternatively it can be treated as resummation of self-energy loop diagrams. The high order corrections will not change general properties (energy dependence) of potentials. In the lowest order properties of the resonances in the scalar and fermion backgrounds are the same. The difference appears in applications and implications for theory.

Resonance, energy smearing, coherence
In resonance, y = 1, the s−component of the potential (14) is zero for any asymmetry, V s = 0, and only non-resonance component contributes. The potential has extrema at y = 1 ± ξ: So, in resonance the enhancement is given by inverse coupling constant squared. The energy interval between two extrema equals 2ξE R . In these points the ratio of the resonant to non- Zero of the total potential is shifted with respect to y = 1 due to the non-resonant contribution as The width of the peak at the half of height, V s (y 1/2 ) = 0.5V max s , equals For values of couplings (2), ξ ∼ ξ 0 < 10 −15 , the characteristics of resonance in (20) - (22) (width and enhancement in the peak) have no physical sense. One should take into account (i) smearing of the peaks due to integration with distribution of the background χ over momenta, which differ from δ function, (ii) averaging over uncertainty in neutrino energy, (iii) effect of density correction to the width of φ, (iv) dumping due to resonance absorption.
Let σ y be the scale of smearing in variable y. The smearing leads to decrease of heights of the peaks and their widening. If σ y ξ, we can neglect ξ 2 in (14). Then the height of the peak after averaging can be estimated as So that the enhancement factor is given by 1/σ y . The maxima shift to y ≈ σ y /2. Let us consider possible origings of σ y .
The quantity σ y can be the width of F χ (k) distribution. Recall that deriving the potential (10) we assumed that the background particles are at rest, k χ = 0. (This can be still a possibility for condensate of scalar DM). For fermions F (k) is not the δ−function, but distribution with finite width. In Eq. (8) one should use (even for isotropic background) E χ = m 2 χ + k 2 χ . Near the resonance and for non-relativistic background E χ ∼ m χ + k 2 /2m χ , so that ∆E χ ≈ ∆(k 2 )/2m χ . For thermal background with temperature T we can take ∆k 2 = (3T ) 2 , and therefore If T = 1.945 K and m χ = 0.05 eV, we obtain the value of enhancement 1/σ y ∼ 10 4 .
Further smearing of the dependence of potential on energy is due to neutrino energy uncertainty σ E in the oscillation setup. In this case For very narrow resonance one needs to take into account the medium corrections to the φ−propagator. The main correction is given by the loop diagram φ → ν + χ * → φ with the χ propagator in a finite density medium. This medium correction corresponds to scattering of φ on particles of medium via neutrino as mediator: φ + χ → ν → φ + χ. So, whole the process consists These transitions can be treated as the induced decay of φ in medium. The polarization operator equals which should be compared with m φ Γ 0 φ = g 2 m 2 φ /8π. Therefore the width can be written as Ratio of the polarization operator and m 2 φ (the denominator outside the resonance): can be considered as the expansion parameter of the perturbation theory.
Refraction implies coherence: zero transfer momentum by neutrinos, and consequently, the unchanged state of medium |M : M |M ≈ 1. In the resonance region (in the s-channel) ν interacting with χ in some point x produces nearly on-shell φ which propagates for some distance and then decays back into ν and χ. So, the particle of medium reappears in different space-time point x . Then the coherence condition requires χ(x )|χ(x) ≈ 1. That is, the corresponding wave functions of χ before and after scattering should nearly coincide.
The time of propagation of φ between the production and annihilation is determined by the decay rate τ φ = 1/Γ. Taking into account the Lorentz factor γ = E φ /m φ we find the distance of propagation of φ in the rest frame of background: For light background particles the total energy of mediator is d should be smaller than the uncertainty in the position (localization) of the background particle ∆x ≈ 1/∆p χ . This gives the coherence condition which imposes the upper bound on the uncertainty However, for a given neutrino energy most of the particles of a background are not in resonance exactly and produced φ will be out of mass shell. The virtuality can be estimated as χ . So, the condition for coherence can be written as

Properties of resonance and total potential
Outside the resonance, |y − 1| ξ, neglecting ξ we obtain from (14) For y = 0: so that for symmetric background V B = 0. Above the resonance, y 1, independently of Thus, at E E R the potential takes the form of the standard vacuum contribution with 1/E dependence. Therefore, in principle, the standard neutrino oscillations can be reproduced (even for massless neutrinos) provided that (See recent discussion in [32], [13]).
For particular values of we have the following dependence on y (see Fig. 2).
With increase of the low energy part of the potential (y < 1) shifts up.
• = 0 corresponds to symmetric background. The potential equals V = 0 at y = 0, and then V B (y, 0) decreases linearly below the resonance: • > 0: according to (32) for y > , V B has positive values, it vanishes at y = and then becomes negative.
• = 1 corresponds to pure χ background and resonance is absent: describes the asymptotic curve with V B /V 0 = 1 at y = 0. V B (y)/V 0 decreases monotonously from 1 to 0 at y → ∞ and at y = 1 the ratio equals 0.5. Shown also the vacuum kinetic energy V vac /V 0 as function of y.
For < 1 the dependence of potential on y has two branches. In the low energy branch, y < 1, smearing, see eq. (20). In the high energy branch, y > 1, we have V B /V 0 > 0, and it decreases from V B /V 0 ∼ (1 − )/4ξ at y = 1 + ξ down to zero at y → ∞ (without smearing). The two branches are connected in the range y = 1 ± ξ. The largest effect of a background is for = −1.
With increase of both branches approach the non-resonance curve (36) everywhere apart from the region around 1: 3 Resonance refraction and oscillations

Background versus vacuum contributions
Let us consider an interplay of the background V B with kinetic term ("vacuum potential"): We can neglect the usual matter effect if the refraction resonance energy is much smaller than In general, V vac can be positive or negative depending on the mass ordering (sign of ∆m 2 ).
The sign is relevant since now we have two contribution to the phase. In the model where y i correlate with masses, the potentials V vac and V 0 correlate too, having the same sign. If both V 0 and V vac are positive, the potential V vac (y) crosses V B (y) at y > 1 provided that V vac To compare the two contributions we consider the ratio where according to (9), (15) , The parameter r determines the relative strength of the background effect. Notice that r depends on the mass of particles of the background, but does not depend on the mass of mediator. More importantly, r determines the ratio of potentials for y → ∞.
Two contributions to the phase are equal (for r = 1) at This equation gives y eq = 1/(1 − r) for = −1, and y eq = 1/(1 − r) for = 0. With decrease of r, as well as increase of the value of y eq approaches 1. For the non-resonance case ( = 1) y eq = 1/(r − 1) and the equality is realized when r > 2.
For the low energy branch, y < 1, an interesting feature is cancellation of two contributions which corresponds to the MSW resonance on the background. If r = −1 this happens at so that y c = 1/(1 + r) for = −1, and y c = 1/(1 + r) for = 0. With decrease of r and → 1 the cancellation point approaches 1. Also for → 1 we find that y c → 1.
The sum of two contributions in the units of V 0 as function of y, is shown in Fig. 3. It has the following features. In the high energy branch V sum increases from [V vac (1 + r)] at y → ∞ to V 0 (1 − )/4ξ at y = 1 + ξ (in absence of smearing). The two contributions become equal at y eq (40). In the low energy branch V sum /V 0 decreases from V vac /V 0 (1 + ) at y → 0, down to −(1 − )/4ξ at y = 1 − . It crosses zero at y = y c .
Correspondingly, the modulus |V sum | increases with y at y > y c up to V 0 /ξ.
Thus, the background contribution distorts substantially the potential (and consequently, the vacuum phase) dependence on y in the resonance region y ∼ 1: y c ÷ y eq . This region shrinks with increase of r and . Maximal distortion effect is at = −1.

Effective mass splitting
Effect of the background can be treated as modification of the mass squared difference which depends on neutrino energy: so, that V sum = ∆m 2 eff (y)/2E. The ratio of the effective splitting in a background, ∆m 2 eff , and in vacuum, ∆m 2 equals R ∆ (y) = 1 + r y(y − ) For y → 0 the correction disappears R ∆ (y) = 1 + ry.
For high energies with increase of y the ratio converges to constant value independently of . Thus, the key consequence of interaction with background is that ∆m 2 extracted from data above the refraction resonance differs from ∆m 2 extracted from low energy data.
In Fig. 4 we show dependence of the ratio (45) on y for different values of r. Here the important point is y s in which corrections to the modulus of effective mass squared difference changes the sign. It is determined by or according to (43) by V B (y)/V vac (y) = −2. Solution of the corresponding equation gives y s = 1 2(2 + r) r + 2 r 2 + 8r(2 + r) .
For = 0, we find y s = 2/(2 + r). Consequently, for r = 1.5 it equals y s = 0.87, and for = −1: y s = 0.75. In the interval y = 0 ÷ y s the background diminishes splitting: ∆m 2 eff < ∆m 2 , and consequently, the oscillation phase. For y > y s : ∆m 2 eff > ∆m 2 and the phase increases. With decrease of r the correction decreases and the benchmark energies y c , y s and y eq approach 1.

Negative κ
In the previous consideration we assumed that ∆m 2 is positive, or more precisely, ∆m 2 and 1 are positive simultaneously. That is, the potentials follow the hierarchy of masses, which is automatically satisfied if both differences are given by g 2 2 − g 2 1 . As a consequence, κ ≥ 0 and r ≥ 0.
If, however, neutrinos have other sources of masses apart from VEV of φ, the signs and values of ∆m 2 and V B are independent. In this connection let us consider the case of negative κ and r.
Above the resonance the quantities V B and V vac have opposite signs. Therefore Figure 4: The effective mass splitting ∆m 2 eff /∆m 2 as function of y for different values of r. We take = 0. Shown are also lines π/2LV vac (y) which correspond to two different values of baseline π/2LV vac R (numbers at the lines). Crossings of these lines with ∆m 2 eff /∆m 2 give the points where the total phase Φ = π/2 (see text for explanations).
1. The cancellation point y c (the MSW resonance on background) is above the refraction resonance: y c > 1.

Phases and probabilities
In the case of diagonal matrix of potentials in the neutrino mass basis the background potential modifies neutrino oscillations via an extra contributions to the oscillation phase: while the mixing is unchanged. Thus, for two neutrino mixing the ν α − ν β transition probability equals P να− →ν β (L, E) = sin 2 2θ sin 2 0.5Φ.
We assume here constant density of background particles.
Since the phase Φ enters in the observables (probability) as cos Φ or sin 2 Φ/2, the change of sign of V in the resonance does not lead to suppression due to integration over energy. (Notice that this is valid for 2ν case and without matter effect. In the 3ν− case we have interference of different channels with different frequencies and those terms are not even with respect to V .) Observational effects of the background depend on the baseline of experiment. In Fig. 4 we For fixed L the lines correspond to the inverse of the vacuum oscillation phase as function of y.
With increase of L the slope decreases. In Fig. 4 the left (right) line corresponds to the short (long) baseline.
There are four crossings: low energy y l , and y − , y + with left and right branches of the resonance peak as well at the resonance y ≈ 1. The equation for crossing (53) can be written as For parts of the lines R ∆ (y), which are above the crossings the phase is big Φ > π/2, while for the parts below the crossings the phase is small: Φ < π/2.
In Fig. 5 we show the oscillatory factor sin 2 0.5Φ as function of y for three different values of baseline. We performed smearing over energy.
The crossings determine four intervals of y with different observational features.
• y < y l : the oscillatory curve with increasing period when y → 0. At y → 0 oscillations in background nearly coincide with the vacuum oscillations.
With decreases of L: y l → 0, while y − , y + → 1. Thus, the dip widens, whereas the resonance region becomes narrower.

Bump: number of events
The characteristic relevant for observations is not the width of the peak, but the energy range where the background effect is bigger than the standard oscillation effect. It is determined by the tails of resonance where |V B | |V B max |. According to (32) We take r = 1.6 and = 0. The dotted lines correspond to the oscillatory factors for pure vacuum oscillations (r = 0).
As a criteria for strong effect, we can use sin 2 Φ B /2 = 1/2, which gives according to (55) Therefore the region of strong effect has width This region decreases with increase of .
where y max and y min are determined by conditions the phase Φ B (y max ) = Φ B (y min ) = π/2, so that sin 2 0.5Φ B = 1/2. We can approximate the oscillatory factor by its average value: sin 2 Φ/2 ≈ 0.5. Then according to (55), and this result is valid for Φ 0 /π 1.
More precise computation of the integral (59) for any interval of y can be done in the following way. Let us introduce δ y (which depends on Φ 0 ) such that in the range |y − 1|< δ y the phase is very big: Φ 0 /2δ y 1, and consequently, the sine has very fast oscillations (δ y ∆y). Then the integral I can be split in three parts: with integration over y in the intervals [1 − δ y ÷ 1 + δ y ], In the first (central) interval the integrand can be approximated by 1/2, and consequently, The tail integrals can be computed numerically as follows.
In the central (resonance) region, −2Φ 0 /π < (y − 1) < 2Φ 0 /π, we can substitute the integrand sin 2 Φ/2 by 1/2. Outside the resonance region, 0 < y < 1−2Φ 0 /π (lower region) and y > 1+2Φ 0 /π (upper region), the sine squared can be approximated by normalized in such a way that at the borders it equals 1/2. Then for small Φ 0 /π the high and the low energy tails give and the sum equals The ratio of the tails to the resonance (60) contributions equals and it depends on the phase weakly: with increase of Φ 0 the ratio decreases. The contribution from resonance width (22) is negligible.

Adding usual matter effect
The matter potential V e = √ 2G F n e does not depend on energy in the range we are considering.
The equality V e ≈ V vac determines the MSW resonance energy E MSW . Since in this setup the mixing does not change by the background, the MSW resonance condition has usual form: There are three possibilities depending on relative values of V e and V vac R .
I. V e < V vac R : In this case the refraction resonance is below the MSW resonance E B R < E MSW (see Fig. 3). There are three crossing of V vac (y) with V e in the neutrino channel: (i) Standard MSW resonance. It is shifted to higher energies due to background contribution.
The resonance energy with background correction can be found from eq.(64). The expression is simplified in the case y MSW 1, so that we can take the asymptotic value ∆m 2 eff (E) ≈ ∆m 2 (1+r). As a result, where E MSW,0 is the standard resonance energy without background: The shift of MSW resonance can be used to search for the background effect.
(iii) New crossing with the low energy branch of V tot .
In theν− channel there are two crossings: (i) near the refraction resonance; (ii) with low energy branch of V sum .
In the crossing points the mixing in medium (matter plus background) becomes maximal.
If V e V vac R , E B R E MSW , at low energies and short baseline experiments the effects of four new crossings become unobservable because in these crossings Φ 1.
II. V e > V vac R : In this case the refraction resonance is above the MSW resonance: E B R > E MSW . Depending on the shift of the MSW resonance can be to higher or low energies.
As before there are two new crossings in the ν−channel and two new crossings in theν−channel.
In the ν−channel one crossing is near the refraction resonance, and another one is in the high energy branch. The energy of the latter can be substantially larger than y = 1. In theν−channel the two crossings are near the refraction resonance being in the low energy branch.
III. The case of V e ≈ V vac R is of special interest: the standard MSW resonance coincides with the refraction resonance, while two new resonances (at y > 1 and at y < 1) can be far from the refraction resonance y = 1.

Generation of mixing in the background
In the previous consideration the matrix of potentials had only one entry and so it was diagonal in the mass eigenstate basis. If couplings of other mass states with background are not neglected the transition ν 1χ → ν 2χ generates a non-diagonal element of the matrix of potentials which is proportional g 1 g * 2 . In the 2ν case the total Hamiltonian becomes where is the background potential discussed in the previous sections. κ is defined in (38).
Notice that the resonance energies are different for different neutrino mass states ν i : but this difference is still much smaller than the scale of smearing due to motion of scatterers.
Therefore we can neglect dependence of potentials on the neutrino masses and the only relevant dependence on type of neutrino is in the coupling constants.
Diagonalization of the matrix (66) gives the difference of eigenvalues and the mixing angle of mass states The flavor mixing angle becomes Let us consider different limits and benchmark points.
2. y → y c : the cancellation point (V B = −V vac ) becomes the energy of MSW resonance on the background. Here the mixing is maximal sin 2 2θ B = 1 and splitting is non-zero: The transition probability equals P ≈ sin 2 (αΦ vac ).
For small α in comparison to no-mixing case modifications of P are small. The most significant change is in the cancellation region.
Finally, let us comment on the case of three different fermions χ j -each per generation. If VEV of φ is the only source of neutrino mass then the couplings are diagonal in the mass basis.
Furthermore, transition ν iχi → ν jχj will not form potential, since finalχ j differs from initialχ i being orthogonal each other. In this case the matrix of potentials is diagonal and the difference of i |−|g 2 j |, will enter expressions considered above.

Resonance refraction and Glashow resonance
In the standard model, the resonance refraction is realized in the Glashow resonance: that is, in the ν e −e scattering with W boson as the mediator. The resonance energy equals E R = m 2 W /2m e ≈ 6.4 PeV. Dependence of the potential on neutrino energy is described by the Eq. (14) with = −1, At low energies the potential coincides with the Wolfenstein potential. The difference from what we have discussed before is that the coupling is large g W ∼ O(1). Therefore the width of the resonance is not negligible, enhancement is not extremely strong and smearing effect is weaker.
The maxima In the resonance region the vacuum contribution, ∆m 2 /2E is negligible: r = 10 −6 . Vacuum mixing is strongly suppressed. Furthermore, dumping due to absorption can be substantial.
The refraction length in resonance can be reduced by factor 20 in comparison to the Wolfenstein length, being of the order 300 km. However, existence of observable effects at the Earth is questionable.
1. Oscillation effects with usual ∆m 2 and θ are negligible. Refraction index is still very close to 1, so that bending and refraction effects are negligible too.

2.
One can explore possible effect in astrophysical objects -sources of high energy neutrinos.
3. Mixing of active neutrinos with sterile neutrinos of mass 10 2 eV can be considered. In this case ∆m 2 /2E R ∼ V e and the mixing can be enhanced in matter.

Applications to specific experiments 4.1 Signatures and implications
Recall that the oscillatory pattern in terms of universal variables, R ∆ (y) and y depends on (i) rrelative strength of interactions with background (39), (ii) -charge asymmetry of the background, (iii) baseline L. Thus, observing the oscillatory pattern at given L one can determine and r (which is the combination of the fundamental parameters and density of a background (39)) or put bounds on these parameters.
Observable effects of the background vanish completely if r → 0, however, they do not disappear when → 1. At = 1 the resonance is absent, the cancellation point is at y c = 1 and R ∆ (y) = 1 + r y y + 1 .
For large energies the background effects are determined by r and dependence on is weak.
To some extend the effects of and r on the oscillatory pattern correlate, and there is certain degeneracy. However, variations of the pattern with r can be much more substantial than that with . Effect of is restricted by its minimal value −1.
The presence of the resonance bump testifies for = 1. Value of determines the benchmark energies. With → −1 the region of distortions in the resonance interval becomes wider. Measuring the oscillatory pattern in different energy ranges allows to disentangle effects of r and . Let us summarize signatures of interactions with background. For κ > 0 they include: • deviation of the oscillatory pattern from sin 2 (A/y) in the low energy interval; • oscillation dip at y < 1, with zero at y c ; • increase of the probability at y → 1; • bump at y ∼ 1; • tail at y > 1.2, which corresponds to larger ∆m 2 eff than at low energies.
In the presence of usual matter we have in addition For κ < 0 the dip is at higher energies. In asymptotics the effective ∆m 2 eff is smaller than that at low energies.
Thus, to search for effects for fixed L one can consider different energy intervals. For a given neutrino beam one can use different L, e.g., results from near and far detectors.

MiniBooNE excess and resonance refraction
The low energy excess of events reported by the MiniBooNE collaboration [7] could be a manifestation of the resonance refraction [6]. The background is composed of the overdense relic neutrinos.
In this case m χ = 0.05 eV and ≈ 0.
The best fit of the MiniBooNE data is obtained for values of parameters Then Notice that the mediator and background particles are light enough and therefore the astrophysical bounds on g are applicable (see sect. 2.1).
From (71) we obtain and correspondingly, r = 4Y ∆m 2 = 1.59, y c = 0.62. Thus, in the resonance region and above it the background potential dominates. The usual matter potential is very small: V e = 2 · 10 −13 eV.
The MiniBooNE baseline L MB = 541 m corresponds to which means that the phase is very small, Φ 1, everywhere except for narrow region close to y = 1. The resonance peak is smeared by the energy resolution.
Let us show that this solution is excluded because of strong dependence of the effective ∆m 2 eff on energy (y). In Fig. 6 we show ∆m 2 eff as function of energy for E R = 320 MeV, = 0 and different values of r. At low energies y 1, ∆m 2 eff ≈ ∆m 2 (as in vacuum) while above the resonance According to this equation for y = 2 and y = 3, which correspond to E = 680 MeV and 1020 MeV, the enhancement of ∆m 2 eff is given by factors 3.12 and 2.79 respectively. In asymptotics, y → ∞, it converges to 2.59. Fig. 6 shows also results of measurements of the "atmospheric" ∆m 2 ≈ m 2 3 at different energies. At the lowest energies, E = (2 − 5) MeV (y ∼ 10 −2 ) the data on ∆m 2 ee ≈ ∆m 2 31 are provided by the reactor experiments [33][34][35]. Here the background effect can be neglected. The T2K experiment [36,37] measures ∆m 2 32 at (0.3−1.3) GeV which is slightly above the resonance. At higher energies (essentially in asymptotics) the data are given by NOvA [38] and then MINOS and MINOS+ [39].
The main conclusion is that within the experimental error bars ∆m 2 eff does not depend on energy over 4 orders of magnitude. This puts strong bound on strength of interaction with background: which certainly excludes r > 1.6 required by MiniBooNE explanation.
Similar result can be obtained for negative r. Now above the resonance the predicted values of ∆m 2 eff are below the experimental points.
The same consideration with the same conclusion is applied for the bosonic background and fermionic mediator. In particular, the Fig. 6 will be unchanged. The only difference is that the potential is 2 times larger which can be accounted by renormalization g → √ 2g. The latter could have some implications to particle physics model but not to the exclusion.

Bounds on the background effects
We have obtained the upper bound on strength of the background effects r (75) for E R ∼ 320 MeV. According to Fig. 6 similar bound can be established in the interval of E R from 10 MeV to 10 GeV. For E R < 1 MeV -no distortion is expected in the region of observations (i.e. at E > MeV), while for E R > 10 GeV the effect of background in the observable region becomes much smaller than vacuum effect and it decreases with energy decrease.
The strength r (39) can be written as This means that for given E R and r the potential is restricted by The largest value of E R , for which a given bound on r exists, gives the most strong restriction on V 0 . Therefore according to (77) Thus, consideration at higher energies allows to strengthen the bound on r.
For the background particles at rest the strength factor can be written as Is the bound on r we obtained from resonance refraction substantial, or there are other more strong bounds? One such a bound on the system comes from contribution of χ to the dark matter in the Universe: For a given value of m χ this gives the number density of χ which compose ρ χ /ρ DM fraction of the local dark matter: Inserting this expression into (79) and taking for the local energy density of DM ρ DM = 0.4 GeV/cm 3 , we obtain the strength factor r = 2.6 · 10 −7 g 2 10 −3 For g satisfying the bound (2), ρ χ = ρ DM and m χ = 0.05 eV Eq. (82) gives r = 2.6 · 10 −14 which is much below the refraction bound. For these values of parameters n χ = 8 · 10 9 cm −3 . r can be enhanced if we take smaller mass of χ and g = 10 −3 , which satisfies the laboratory bounds but requires more complicated cosmological evolution that allows to avoid BBN and CMB bounds.
This consideration is valid for bosonic background with changing subscripts χ ↔ φ in Eq. (81 -82). For the fermionic background additional restrictions follow from Pauli principle. Indeed, the density indicated above gives the Fermi momentum of the degenerate gas p F = (6π 2 n χ ) 1/3 = 1.3 eV. That is, E χ ≈ p F m χ , and therefore we deal here with strongly degenerate fermion gas.
Thus, r is determined by the coupling constant and fraction of the DM in χ and does not depend on m χ . The value r ≤ 4.7 · 10 −8 , which is much smaller than sensitivity range to the resonance refraction effects of experiments at the laboratory energies. bump in the resonance region, (iv) additional contribution to V vac (y) above refraction resonance which does not disappear in asymptotics.

Conclusions
6. Effects of background can be considered as modification of the effective ∆m 2 eff (y) with peculiar dependence on energy. 7. As an example we applied our results to the MiniBooNE excess interpreted as bump produced by the refraction resonance. We show that this interpretation is excluded because of strong difference of ∆m 2 eff expected at high energies (T2K, NOvA, MINOS, MINOS+, IceCube, ANTARES) and low energies (reactor experiments) in contrast to observations. We obtain the bound on the relative strength of neutrino interactions with background r < (0.001 − 0.01).