Neutrino Oscillation in Dense Matter

As the increasing of neutrino energy or matter density, the neutrino oscillation in matter may undergo"vacuum-dominated","resonance"and"matter-dominated"three different stages successively. Neutrinos endure very different matter effects, and therefore present very different oscillation behaviors in these three different cases. In this paper, we focus on the less discussed matter-dominated case (i.e., $|A^{}_{\rm CC}| \gg |\Delta m^{2}_{31}|$), study the effective neutrino mass and mixing parameters as well as neutrino oscillation probabilities in dense matter using the perturbation theory. We find that as the matter parameter $|A^{}_{\rm CC}|$ growing larger, the effective mixing matrix in matter $\tilde{V}$ evolves approaching a fixed $3 \times 3$ constant real matrix which is free of CP violation and can be described using only one simple mixing angle $\tilde{\theta}$ which is independent of $A^{}_{\rm CC}$. As for the neutrino oscillation behavior, $\nu^{}_{e}$ decoupled in the matter-dominated case due to its intense charged-current interaction with electrons while a two-flavor oscillation are still presented between $\nu^{}_{\mu}$ and $\nu^{}_{\tau}$. Numerical analysis are carried on to help understanding the salient features of neutrino oscillation in matter as well as testing the validity of those concise approximate formulas we obtained. At the end of this paper, we make a very bold comparison of the oscillation behaviors between neutrinos passing through the Earth and passing through a typical white dwarf to give some embryo thoughts on under what circumstances these studies will be applied and put forward the interesting idea of possible"neutrino lensing"effect.


I. INTRODUCTION
When neutrinos pass through a medium, the interactions with the particles in the background give rise to modifications of the properties of neutrinos as well as the oscillation behaviors. This is well known as the matter effect which have been playing important roles in understanding various neutrino oscillation data. In the standard three neutrinos framework, the effective Hamiltoniañ H in the flavor basis responsible for the propagation of neutrinos in matter, differs from the Hamiltonian in vacuum H, where H describes the forward coherent scattering of neutrinos with the constituents of the medium (i.e., electrons, protons and neutrons) via the weak charged-current (CC) and neutralcurrent (NC) interactions [1][2][3][4]. Here A CC = 2EV CC , A NC = 2EV NC (with V CC = √ 2G F N e and V NC = − √ 2 2 G F N n being the effective matter potentials) are parameters of the same unit as the mass-squared difference ∆m 2 ji that measure the strength of the matter effect, and V is just the 3 × 3 unitary Pontecorvo-Maki-Nakagawa-Sakata (PMNS) leptonic mixing matrix [5,6] which is conventionally parametrized in terms of three mixing angles θ 12 , θ 13 , θ 23 where c ij ≡ cos θ ij and s ij ≡ sin θ ij (for ij = 12, 13, 23) have been introduced. Throughout this paper we do not consider the possible Majorana phases, simply because they are irrelevant to neutrino oscillations in both vacuum and matter. For anti-neutrino oscillation in matter, one may simply replace V by V * and A CC by −A CC in the effective Hamiltonian (i.e., A CC is negative in the case of anti-neutrino oscillation).
The intriguing matter effect is a result of the interplay between the vacuum Hamiltonian H and the matter term H . Note that, the diagonal term 1 2E m 2 1 + A NC · 1 in Eq. (1) develops just a common phase for all three flavors, and does not affect the neutrino oscillation behaviors. Therefore it is the interplay among the two mass-squared differences ∆m 2 21 , ∆m 2 31 , the mixing parameters in V (which are all parameters in the vacuum Hamiltonian H and have been well determined from varieties of neutrino oscillation experiments [7,8]) and the matter term A CC (which will vary with the matter density and the energy of neutrino) that give rise to varied neutrino oscillation behaviors.
According to the relative magnitude of ∆m 2 21 , |∆m 2 31 | and |A CC |, the various possible values of A CC can be laid in three main different regions: the vacuum-dominated region (i.e., |A CC | ∆m 2 21 , |∆m 2 31 |), the resonance region (i.e., A CC ∼ ∆m 2 21 , ∆m 2 31 ), and the matterdominated region (i.e., |A CC | ∆m 2 21 , |∆m 2 31 |). Among various studies of the matter effect, neutrino oscillation behaviors in the resonance region attracted the most attention (see e.g., [9][10][11][12][13][14]). The oscillation probabilities may get dramatic corrections owing to the resonances at around A CC ∼ ∆m 2 21 (solar resonance) or A CC ∼ ∆m 2 31 (atmospheric resonance) 1 which are crucially important for the studies of atmospheric neutrinos, accelerator neutrino beams passing through the Earth or the spectrum of solar neutrinos. Also there have been discussions concerning the vacuum-dominated case [16][17][18][19][20], which could be helpful for various long-or medium-baseline neutrino oscillation experiments with neutrino beam energy E below the solar resonance. In this region, the neutrino oscillation probabilities as well as the leptonic CP violation receive predictable small corrections from the matter effect.
Recently interests have been shown in exploring the less discussed matter-dominated case [21][22][23][24], where the matter term H dominates over the vacuum Hamiltonian H, or more specifically, |A CC | |∆m 2 31 |. Such studies are applicable in the case of neutrinos having extremely high energy or going through extremely dense object. Further to these works, we explore in this paper the effective neutrino mass and mixing parameters as well as the neutrino oscillation probabilities in dense matter using the perturbation theory. We find that as the matter parameter |A CC | growing larger, the effective mixing matrix in matterṼ evolves approaching a fixed 3 × 3 constant real matrix which is free of CP violation and can be described using simply one mixing angleθ which is independent of the matter parameter A CC . As for the neutrino oscillation behavior, ν e decoupled in the matter-dominated case due to its intense charged-current interaction with electrons in the medium while a two-flavor oscillation can still present between ν µ and ν τ . As far as the six neutrino oscillation parameters in vacuum are well determined and the condition |A CC | |∆m 2 31 | is satisfied, the neutrino oscillation probabilities in dense matter can be well predicted regardless if the matter density varies along the path.
We plan to organize the remaining parts of this paper as following. In section II we aim to reveal the features of the effective neutrino masses and mixing matrix in matter under the condition |A CC /∆m 2 31 | → ∞ with the help of the perturbation theory. Base on the results of the series expansions, a set of pretty concise approximate formulas of neutrino oscillation probabilities in the matter-dominated region are derived in section III. Numerical analysis are carried on in both sections to help understanding the salient features of neutrino oscillation in matter as |A CC | changes from zero to infinity as well as testing the validity of those concise formulas. Finally, in section IV we make a very bold comparison of the oscillation behaviors between neutrinos passing through the Earth and passing through a typical white dwarf so as to answer the question under what circumstances these studies will be applied and put forward the interesting idea of possible "neutrino lensing" effect.

TERS IN THE MATTER-DOMINATED CASE
As already mentioned above, in the standard three neutrinos framework, the effective Hamil-tonianH in the flavor basis responsible for the propagation of neutrinos in matter can be written where the effective neutrino massesm i (for i = 1, 2, 3) and flavor mixing matrixṼ in mater have been defined. Given a constant matter profile, the exact analytical relations between {Ṽ ,m i } and {V , m i } have been established in many works using different approaches [9,16,[25][26][27][28][29][30]. And the neutrino oscillation probabilities in matter can be written in the same way as those in vacuum by simply replacing V αi and ∆m 2 ji with the corresponding effective parametersṼ αi and ∆m 2 ji . As for any realistic profile of the matter density, it is also possible to numerically calculate the neutrino oscillation probabilities by solving the evolution equations of neutrino flavor states.
However, in the matter-dominated region we are concerning, some useful and more transparent analytical approximations could be obtained by regarding both ∆m 2 21 /|A CC | and |∆m 2 31 /A CC | as small parameters and performing the diagonalization ofH using the perturbation theory. In comparison with the method adopted in previous works [21][22][23][24] that applying further simplification on the exact formulas, the series expansion method can automatically achieve approximate formulas with any required accuracy. Moreover, the values those effective parameters in matter would approach in the limit |A CC | → ∞ are straightforwardly given in the zeroth order expansion. Also note that, different from previous works on series expansions [10-15, 20, 31-33] which usually regard known constant such as α ≡ ∆m 2 21 /∆m 2 31 or sin θ 13 as small expansion parameters, the two expansion parameters ∆m 2 21 /|A CC | and |∆m 2 31 /A CC | we employed in this paper vary with the matter parameter A CC , i.e., vary with neutrino energy E as well as the matter density ρ. As a result, this kind of series expansion relates only to the matter-dominated case, and the accuracies of those approximate formulas given in this paper depend also on the magnitude of A CC . We will have a detailed discussion on this problem later at the end of Sec. III. The details of the diagonalization of the effective HamiltonianH are given in Appendix A, where the approximate expressions of three eigenvalues ofH, the effective mixing matrix and the neutrino oscillation probabilities in matter up to the first order of both ∆m 2 21 /|A CC | and |∆m 2 31 /A CC | are also presented. As the increase of |A CC |, terms proportional to 1/A CC are all approaching zero, and as one can clearly seen from Eqs. (A12) and (A13), three eigenvalues ofH are approaching a set of fixed where the Hermitian matrix Ω is defined as which has the two-flavor-mixing structure and can be parametrized using just one mixing angleθ defined by Considering the strong hierarchy of ∆m 2 21 |∆m 2 31 | and the smallness of s 13 , one can immediately obtain from above equation that the mixing angleθ ≈ θ 23 2 . One may also find that the mixing angleθ defined in Eq. (6) is actually an indicator of the µ-τ symmetry breaking in the Dirac neutrino mass matrix 3 . If the neutrino mass matrix in vacuum M ≡ V diag{m 2 1 , m 2 2 , m 2 3 }V † possess the exact µ-τ symmetry, we then haveθ = π/4. The fixed points in the limit |A CC | → ∞ has been noticed in Refs. [21,23,24,32], in which the evolution behaviors of not only nine elements of the effective mixing matrix |Ṽ αi | but also those mass and mixing parameters are illustrated. It's worth to go a step further drawing a full picture of the evolution behaviors of three effective neutrino masses and the effective mixing matrix in the matter-dominated case. In the limit |A CC /∆m 2 31 | → ∞, one of the eigenstates ofλ 1 is decoupled due to the large potential of A CC and A NC while the other two eigenvalues are nearly degenerate (λ 2 λ 3 ) for they are both dominated by the large neutral-current potential term A NC .

Normal Mass Ordering
FIG. 1: The evolution of three squared effective neutrino massesm 2 i (for i = 1, 2, 3) in matter with respect to the dimensionless ratio A CC /|∆m 2 31 | in the normal mass ordering case for both neutrinos (with A CC > 0, red curves in the right half panel) and anti-neutrinos (with A CC < 0, blue curves in the left half panel) , where the best-fit values of the mass-squared differences and the mixing parameters in Table. I have been input. Note that, the common terms m 2 1 + A NC are omitted from all threem 2 i for the sake of simplicity, while the relation ∆m 2 ji =m 2 j −m 2 i still holds. Both the input values ofm 2 i in vacuum and the fixed points in the limit |A CC | |∆m 2 31 | are given on the plots.
Correspondingly, the 3 × 3 effective mixing matrixṼ in matter presents a nearly two-flavor-mixing structure. It meansṼ asymptotically conserves intrinsic CP and can be well described by just one mixing anglesθ, which can be approximately expressed asθ ≈ θ 23 .
To see the features of fixed points as well as the evolution ofm i andṼ more transparently,

Inverted Mass Ordering
FIG. 2: The evolution of three squared effective neutrino massesm 2 i (for i = 1, 2, 3) in matter with respect to the dimensionless ratio A CC /|∆m 2 31 | in the inverted mass ordering case for both neutrinos (with A CC > 0, red curves in the right half panel) and anti-neutrinos (with A CC < 0, blue curves in the left half panel) , where the best-fit values of the mass-squared differences and the mixing parameters in Table. I have been input. Note that, the common terms m 2 1 + A NC are omitted from all threem 2 i for the sake of simplicity, while the relation ∆m 2 ji =m 2 j −m 2 i still holds. Both the input values ofm 2 i in vacuum and the fixed points in the limit |A CC | |∆m 2 31 | are given on the plots.
γ,k αβγ ijk (for α, β, γ = e, µ, τ and i, j, k = 1, 2, 3) 4 [35,36] with the increasing of the dimensionless ratio A CC /|∆m 2 31 | in both the normal and the inverted mass ordering cases. The best-fit values of the neutrino oscillation parameters from Ref. [8] as summarized in Table I has been adopted as the inputs in vacuum (A CC = 0) in our numerical calculations. One can clearly see that the evolution behaviors of these effective parameters in matter in the region |A CC /∆m 2 31 | 1 are all in good agreement with the predictions of Eqs. be calculated using Eq. (7). Note that the three eigenvaluesλ i are ordered in such a way that in all four scenarios the same correct order {λ 1 ,λ 2 ,λ 3 } = {m 2 1 , m 2 2 , m 2 3 }/2E can be obtained in the limit A CC = 0 through continuous evolution as |A CC | decreasing as one can see in Figs None solar Note that, instead of ordering the eigenvalues according to their magnitude, we choose the order ofλ i in such a way that in the limit |A CC | → 0, the correct mass-squared differences in vacuum are obtained and the ith column ofṼ are corresponding eigenvectors ofλ i . It's well known that in the standard three neutrinos framework there are two possible resonance regions (i.e., the solar resonance at around A CC ∼ ∆m 2 21 and the atmospheric resonance at around A CC ∼ ∆m 2 31 ) when studying the neutrino oscillation in matter. However, because the sign of A CC are different for neutrino or anti-neutrino oscillation and the sign of ∆m 2 31 are different in the normal or inverted mass ordering case, above two resonance conditions are not always satisfied even if the magnitude of A CC could be carefully chosen. When passing through the resonance region, the related two eigenvalues ( as well as the corresponding two eigenvectors) "exchange" their evolution behaviors.
That explains the different patterns of the fixed points in different scenarios. Such a difference originates mainly from the fact that the resonances they experienced are different. To be specific, Normal Mass Ordering The evolution of the absolute value of nine elements of the effective mixing matrix in matter |Ṽ αi | (for α = e, µ, τ and i = 1, 2, 3) with respect to the dimensionless ratio A CC /|∆m 2 31 | in the normal mass ordering case for both neutrinos (with A CC > 0, red curves in each right half panel) and anti-neutrinos (with A CC < 0, blue curves in each left half panel) , where the best-fit values of the mass-squared differences and the mixing parameters in Table. I have been input. Both the input values in vacuum and the fixed points in the limit |A CC | |∆m 2 31 | are given on the plots.
we list in Table II the different resonances neutrinos or anti-neutrinos with different mass orderings may experience together with the resulting pattern of the eigenvaluesλ i and the corresponding effective mixing matrixṼ in the limit |A CC | → ∞ in different scenarios. Anyway, neither the ordering of the eigenvalues nor the omitted common terms would change the neutrino oscillation behaviors in matter which we will discuss in the next section.
One may clearly find from Figs. 1-5 that the evolutions of three effective neutrino masses Inverted Mass Ordering being the effective neutrino masssquared difference in matter. Here the Greek letters α, β are the flavor indices run over e, µ, τ , while the Latin letters i, j are the indices of mass eigenstates run over 1, 2, 3. And E is the energy of the neutrino/anti-neutrino beam.
Here ∆m 2 32 has the same sign as ∆m 2 31 . Again, taking into account the strong hierarchy of ∆m 2 21 |∆m 2 31 | and the smallness of s 13 , we can then obtain that the effective mass-squared difference ∆m 2 32 ≈ ∆m 2 32 (or ∆m 2 31 ) 6 together withθ ≈ θ 23 . These analytical approximations give us a clear picture of neutrino oscillation in the matterdominated region: ν e are decoupled (due to its intense charged-current interaction with electrons in the medium), while oscillation can still happened between ν µ and ν τ 7 . This two-flavor oscillation can be described by one effective mixing angleθ and the effective mass-squared difference ∆m 2 32 whose expressions are given in Eqs. (7) and (10)  = 0, which tells us that 6 In our numerical analysis, we have ∆m 2 32 = 2.349 × 10 −3 eV 2 together with ∆m 2 32 = 2.3772 × 10 −3 eV 2 in the normal mass ordering case, and ∆m 2 32 = −2.501 × 10 −3 eV 2 together with ∆m 2 32 = −2.5859 × 10 −3 eV 2 in the inverted mass ordering case. 7 This is in agreement with the near degeneracy ofλ 2 andλ 3 in the limit |A CC | → ∞.  Table. I have been input. The fixed points of these probabilities in the limit |A CC | |∆m 2 31 | for different L/E are given on the plots. ν e /ν e decouples from the other flavors in the matter-dominated case. Although the oscillation probabilities between ν µ and ν τ are approximately independent of A CC in the matter-dominated region,P µµ andP τ τ (P µτ andP τ µ ) change periodically between 1 and 1 − sin 2θ (0 and sin 2θ) as the variation of L/E. Note that, sin 2θ is actually close to 1, which means if L and E are properly  Table. I have been input. The fixed points of these probabilities in the limit |A CC | |∆m 2 31 | for different L/E are given on the plots.
chosen, a simple but significant two-flavor oscillation between ν µ and ν τ can be observed in the matter-dominated case.
Before ending this section, we would like to test the accuracy of the formulas given in Eq. (9) and discuss the valid region of these formulas. Figures 8 and 9 show the absolute errors |P αβ −P f ixed αβ | of   neutrino/anti-neutrino oscillation probabilities in both the normal and the inverted mass ordering cases, whereP f ixed αβ is calculated using Eq. (9) andP αβ is numerically calculated without any approximation. In previous discussion, we have employed |A CC /∆m 2 31 | 10 as the criterion of the matter-dominated condition, i.e., the matter term H is at least an order larger than the vacuum Hamiltonian H. As we can see from Figs. 8 and 9, under this criterion, the differences ofP ee ,P eµ , P eτ ,P µe andP τ e with respect to their fixed points (1 or 0) are all smaller than 10 −4 . If a more strict criterion |A CC /∆m 2 31 | 100 is adopted, the absolute error of these oscillation probabilities related to electron flavor would be smaller than 10 −7 . And as one can infer from Eq. (A15),   the absolute errors would fall quadratically with the increase of |A CC |. We can then safely make the conclusion that both ν e andν e are decoupled in the matter-dominated case. On the other hand, in addition to the dependence on the matter parameter A CC , the accuracy of the oscillation probabilitiesP µµ ,P τ τ ,P µτ andP τ µ which describe the remaining oscillation between ν µ and ν τ in dense matter depend also crucially on the ratio L/E. If both the conditions |A CC /∆m 2 31 | 10 and L/E [km/GeV] 10|A CC /∆m 2 31 | are satisfied, the absolute error of these four probabilities are all smaller than 10 −3 . And if the more strict constraint L/E [km/GeV] |A CC /∆m 2 31 | together with |A CC /∆m 2 31 | 10 are imposed on, the accuracy of the order 10 −5 or better can be obtained.
The reason for this additional criterion is that the first order correction to the effective masssquared difference ∆m 2 32 is proportional to |∆m 2 31 /A CC | as on can see in Eq. (A16). In the case L/E [km/GeV] 10|A CC /∆m 2 31 | this correction to the oscillation frequency is significant enough and should not be ignored. In this case one may calculate ∆m 2 32 using Eq. (A16) instead of Eq. (10) to further improve the accuracy of Eq. (9). Also note that, when a realistic experiment is discussed especially for those with large ∆m 2 32 L/4E, the energy resolution must be taken into consideration.

IV. OUTLOOK
As the ending section of this manuscript, it is interesting to ask under what circumstances these studies of neutrino oscillation in dense matter will be applied. Here we bring our embryo thoughts by making a very bold comparison of the oscillation behaviors between neutrinos passing through the Earth and passing through a typical white dwarf. Note that, instead of the more accurate PREM model of the Earth [45], we adopted here a simpler two layer mantle-core model [46,47]  that, since the neutrino-nucleon cross section of neutrinos increase with increasing energy [48,49], at such high energies the Earth becomes opaque to neutrinos and the neutrino flux gets attenuated (for more details, see discussions in e.g., [50][51][52][53]). In the case of neutral-current interaction neutrinos P ( However, if we can do the same measurements on a white dwarf whose volume is comparable to P ( FIG. 11: The comparison of the neutrino (anti-neutrino) oscillation probabilities with or without the matter effect as a neutrino (anti-neutrino) beam of energy E go through the Earth along the diameter, where the inverted neutrino mass ordering is assumed and the best-fit values of the mass-squared differences and the mixing parameters in Table. I have been input. The fixed points of the probabilities in the limit |A CC | |∆m 2 31 | given by Eq. (9) (dashed lines) are also plotted in this figure for comparison. Note that all the probabilities are averaged over a Gaussian energy resolution of 5%. that of the Earth but mass is comparable to that of the Sun, things could have been very different. Figures 12 and 13 show the variations of oscillation probabilities as functions of neutrino/antineutrino energy E, when the neutrino/anti-neutrino beam passing through a typical white dwarf [54,55] along its diameter. The corresponding oscillation probabilities in vacuum are also presented in these plots using dotted lines for comparison. Again, all the probabilities are averaged over a Gaussian energy resolution of 5%. The white dwarf is an excellent choice for this thought experiment. On one hand a white dwarf is very dense can give rise to significant matter effect, and on the other hand the material in a white dwarf no longer undergoes fusion reactions which means it does not radiate large amount of neutrinos on its own. In our analysis, the mass M ∼ 0.7M (with M being the mass of the Sun), the radius R ∼ 10 4 km, an uniform density ρ ∼ 2×10 6 g/cm 3 or equivalently an uniform electron number density n e ∼ 6 × 10 29 cm −3 ∼ 10 6 N A cm −3 (with N A being the Avogadro's number) are assumed as the properties of this white dwarf.
Due to the extremely high density, neutrino oscillation experiences the resonances and then enter the matter-dominated region at very low energies (below MeV). One may clearly see from Figs. 12 and 13 that at around E ∼ 0.4 keV (the solar resonance where |A CC | ∆m 2 21 ) the oscillation probabilities start to markedly differ from the vacuum oscillation probabilities and change towards their fixed points. For neutrino oscillation in the normal mass ordering case or anti-neutrino oscillation in the inverted mass ordering case, there is a significant resonance hump at around E ∼ 20 keV (the atmospheric resonance where |A CC | |∆m 2 31 |). After that, at around E ∼ 0.2 MeV (where |A CC |/|∆m 2 31 | 10), it enters the matter-dominated region. In our analysis, the neutrino/anti-neutrino oscillation probabilities in this region are all in perfect agreement with the predictions of Eq. (9) if the same energy resolution is taken into account. In the energy range shown in these two figures, L/E is extremely large, the oscillatory frequencies are all extremely high, therefore only the average oscillatory magnitude can be observed, which is a constant and is markedly different from the vacuum probabilities in the matter-dominated case.
It's worth mentioning that, in the low energy region, the oscillatory frequency ∆m 2 ji L/4E could be high. In this case neutrinos undergo very quick oscillations which can not actually be observed due to the finite energy resolution of the detectors. In our numerical analysis presented in Figs. 10-13, all the probabilities are averaged over a Gaussian energy resolution of 5% (which can be achieved by the upcoming neutrino experiments, such as JUNO [56], at the MeV energy range) to mimic the working of the detector on one hand and uncover features hidden in these fast oscillations on the other. Our numerical analysis also show that even if we choose a worse energy resolution of 15%, the intriguing features discussed above can still be well recognized, since we are looking for the resonance hump and the deviation of the average oscillatory magnitude after neutrinos passing through the white dwarf instead of looking for the oscillation behavior itself. However if we want to trace the remaining oscillation between ν µ and ν τ in this dense matter at a much higher energy range, a good energy resolution could be crucially important.  where the normal neutrino mass ordering is assumed and the best-fit values of the mass-squared differences and the mixing parameters in Table. I have been input. The white dwarf is assumed to have an approximately constant density of ρ 2 × 10 6 g · cm −3 (or equivalently a electron number density of n e 10 6 N A cm −3 with N A being the Avogadros number) and a radius of R 10 4 km. The fixed points of the probabilities in the limit |A CC | |∆m 2 31 | given by Eq. (9) (dashed lines) are also plotted in this figure for comparison. Note that all the probabilities are averaged over a Gaussian energy resolution of 5%.
Note that all the interesting features of the probabilities we discussed above will finally be embodied in the neutrino/anti-neutrino spectrum we observed. The finding of a change of the  where the inverted neutrino mass ordering is assumed and the best-fit values of the mass-squared differences and the mixing parameters in Table. I have been input. The white dwarf is assumed to have an approximately constant density of ρ 2 × 10 6 g · cm −3 (or equivalently a electron number density of n e 10 6 N A cm −3 with N A being the Avogadros number) and a radius of R 10 4 km. The fixed points of the probabilities in the limit |A CC | |∆m 2 31 | given by Eq. (9) (dashed lines) are also plotted in this figure for comparison. Note that all the probabilities are averaged over a Gaussian energy resolution of 5%. slope (around the solar resonance) and a subsequent hump (around the atmospheric resonance) could help to ping down the corresponding resonance energy which can then be turned into the electron density of the compact object. What's more, if both the neutrino and anti-neutrino spectrum can be measured, the present or the absent of the atmospheric resonance hump would be a novel judgement of the neutrino mass ordering. If at a much higher energy range in the matterdominated region, the oscillatory behavior between ν µ and ν τ can be observed, the corresponding oscillation frequency, if well determined, may also reveal some information relating to the size of this compact object.
Since white dwarf is a high-density object, there is a concern about the absorption of neutrinos/anti-neutrinos inside the white dwarf. We give a quick estimate of neutrino's mean free path in a typical white dwarf here to preliminarily discuss the significance of this effect for neutrinos with different initial energies. The absorption of neutrinos inside the white dwarf is dominated by the charged-current interaction between neutrinos and the nucleons in the medium. Without loss of generality, we simply use the ν-n (orν-p) cross section to evaluate this interaction rate which in the low energy region can be approximately calculated by σ νn,νp CC 9.3 × 10 −44 (E/MeV) −2 cm 2 (see e.g., [57,58]). Then the corresponding mean free path of neutrinos/anti-neutrinos can be written as = (σρ/m p ) −1 ∼ 0.9 × 10 13 (E/MeV) −2 cm, where the typical density of the white dwarf ρ ∼ 2 × 10 6 g/cm 3 have been taken into account. We can then infer from this result, for neutrinos with energy E 10 MeV, the mean free path 9 × 10 5 km can be obtained, which is much larger than the length 2R ∼ 2 × 10 4 km the neutrinos transport in the white dwarf. Or in other words, for the neutrino energy of interest to us (E 10 MeV), the white dwarf can be approximately regarded as transparent. Of course if neutrinos with energy higher than 10 MeV are considered, the attenuation of neutrinos/anti-neutrinos due to both the absorption and scattering need to be carefully studied.
Truly, we cannot actually conduct a long baseline neutrino oscillation experiment on a white dwarf. However, we are now observing neutrinos with a broad range of energies from distant objects using varieties of neutrino detectors, many of which cover the MeV range. If there happen to be a compact object sitting in between the source and the observer, this compact object can not only bend the light and produce the gravitational lensing effect, but also "lens" the neutrinos from the source by distorting its spectrum. But different from the gravitational lensing effect which is capable of uncovering the mass distribution in our universe, this "neutrino lensing" effect could be sensitive to the distribution of electrons (or positrons) in the space.
Of course, the discussion so far is just an immature and inaccurate thought. For illustrative purposes, the examples we introduced in this manuscript are very simplified and idealized. Lots of details such as the spectrum and the flavor composition of the neutrino source, the properties of the compact objects and their distribution in the space, the capability of the detector have to be carefully studied before we can finally draw the conclusion if this kind of "neutrino lensing" effect can be actually observed. In our opinion, it is worthwhile to concentrate more efforts on this topic, for it may open a new window to the universe via the weak interaction of neutrino with the compact objects. We believe that with the improving of the detector capabilities and the data analysis techniques, it is possible to site experiments some day to located the hidden compact objects in the space via this "neutrino lensing" effect.
In the case of neutrinos having extremely high energy or going through extremely dense object, we could have |A CC | |∆m 2 31 | which indicates that the matter potential terms dominate over the vacuum terms. In this matter-dominated region, we may regard both |∆m 2 31 /A CC | and ∆m 2 21 /|A CC | as small parameters and perform the diagonalization ofH using the perturbation theory. We can then write down the series expansion of the effective HamiltonianH as The eigenvalues and eigenvectors can also be written asλ i =λ i + ... (for i = 1, 2, 3) correspondingly. One may immediately find that the zeroth order HamiltonianH (0) is diagonal by itself in the flavor basis, which means Note that two eigenvalues ofH (0) (λ 3 ) are identical (degenerate). In this case the corresponding zeroth order mixing matrixṼ (0) should be written as By carefully repeating the derivation, we find that, in the case ofH (0) possessing two degenerate eigenvalues (e.g.,λ conditions are obviously satisfied for any n ≥ 2, since we haveH (n) = 0 (for n ≥ 2) as one can find from Eq. (A2). And further more, from <H > (1) 23 =<H > (1) 32 = 0, it's quite straightforward to haveθ andφ solved as Here the Hermitian matrix Ω is defined as Given that the zeroth order solutions are well determined, the first order corrections to the eigenvalues and eigenvectors can be expressed as One can clearly see that the lowest order corrections toṼ If the matter density can be regarded as a constant along the path neutrinos propagate, we can then write down the neutrino oscillation probabilities in matter simply by replacing the neutrino mass-squared differences and the mixing matrix in neutrino oscillation probabilities in vacuum with the corresponding effective neutrino mass and mixing parameters in matter.
In the matter-dominated region, as the increase of |A CC |, terms proportional to 1/A CC are all approaching zero fast, and therefore as one can clearly seen from Eqs. (A12) which has the two-flavor-mixing structure and can be expressed using just one mixing angleθ as defined in Eq. (A7).