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Spin Flavor Spectral Splits of Supernova Neutrino Flavor Conversions

The supernova neutrino flavor evolution in the presence of the non-trivial neutrino magnetic moment and strong magnetic field is numerically derived using the two-flavor and single-angle approximation. The novel properties of collective neutrino oscillations are studied and distinct patterns of flavor and spin-flavor spectral splits are presented. Finally we also discuss how the neutrino magnetic moment affects the observable supernova neutrino energy spectra.


Introduction
The first observation of neutrinos from the Supernova (SN) 1987A explosion [1][2][3] represents an important milestone for both neutrino physics and neutrino astronomy, which has triggered intensive interest for high-statistics detection of SN neutrinos for the next Galactic SN explosion using large neutrino detectors [4][5][6][7]. The observation of SN neutrinos can not only help us to establish the SN explosion mechanisms [8], but also provide an excellent laboratory to study the neutrino properties in the extreme environment [9].
After the establishment [10] of solar neutrino flavor conversions with the Mikheyev-Smirnov-Wolfenstein (MSW) matter effects [11][12][13], it has been realized that the matter effects may have significant impact on the flavor conversion of SN neutrinos [9]. The ordinary MSW flavor conversions may occur in the outer layers of the SN mantle or envelope, whose main features depend on the SN density profile and the neutrino mass ordering [14,15]. Moreover, the neutrino coherent for-ward scattering on the dense neutrino background may be important in the SN environment [16][17][18], and nonlinear collective neutrino oscillations, including the synchronized and bipolar ones, would arise in the presence of the neutrino self-interaction potential [19], in which the neutrinos may have almost complete flavor conversions.
An interesting observable phenomenon for collective neutrino oscillations is the existence of the spectral splits in the SN neutrino energy spectra of different neutrino flavors [20][21][22][23].
In this work we shall provide a formalism on the neutrino flavor evolution equation in the presence of the nontrivial neutrino magnetic moment and strong magnetic field, and study novel properties of the collective neutrino oscillations. By using the two-flavor and singleangle approximation, we are going to make a numerical calculation of the neutrino evolution equation inside the dense SN medium. We shall also study the patterns of neutrino flavor and spin-flavor spectral splits for both normal ordering (NO) and inverted ordering (IO) of the neutrino mass spectrum and discuss how the neutrino magnetic moment affects the observable neutrino energy spectra.
The remaining part of this work is organized as follows. In Sec. 2, we present our calculation framework for the neutrino flavor evolution equation in the presence of the neutrino magnetic moment and strong magnetic field. Then, results of the numerical calculations for the evolution equation are given in Sec. 3, and different patterns of the collective neutrino oscillations and energy spectral splits will be provided. Finally, we summarize the main results and conclude in Sec. 4.

Calculation Framework
First of all, in this section we shall introduce the calculation framework of this study. We start with the bulb model of supernova neutrino emission, in which the proto-neutron star emits neutrinos and antineutrinos uniformly and isotropically outward in all directions [20].
The emitted neutrino is characterized by its momentum q (|q| = E), with the incident polar angle θ q , azimuthal angle φ q and radial position r in the SN medium.
The differential neutrino number density d n να (q) is then given by where R να = 50 km is the radius of the neutrino sphere, L να = 10 51 erg/s is the ν α luminosity. E να is the average energy of the ν α flux, which is taken as for ν e , ν e , ν x and ν x , respectively, where ν x and ν x denote all the non-electron flavors ν µ , ν τ , and ν µ , ν τ respectively. According to the geometry of the neutrino bulb model and by assuming single angle approximation where the neutrino evolution history is trajectory independent [20,21], the neutrino flux at the radial position r can be obtained using an additional geometric factor D(r) The normalized neutrino spectra φ να (E) can be calculated as where ζ ≈ 1.202, and β νe = 0.315 The flavor evolution of the neutrino states can be calculated using the density matrix formulation where the Hamiltonian H includes four different components: and the density matrix ρ in the flavor basis of both neutrinos and antineutrinos (ν e , ν x , ν e , ν x ) can be written as: The first component H vac (E) in Eq. 5 is the effective vacuum Hamiltonian: where ω = ∆m 2 /4E is the vacuum oscillation frequency with ∆m 2 being chosen as 2.0×10 −3 eV 2 , sin 2 θ = 10 −4 is the mixing angle between two different flavors.
where n e (r) and n n (r) are the number densities of electrons and neutrons at the radial position r, respectively. In our numerical calculation, we employ the time dependent SN density profile in Ref. [21], and the rela-tion of n e = 0.4n b is used [20].
The third Hamiltonian component is from contributions of the neutrino self-interaction H self [24], where ρ c is a transformation of ρ by replacing the where B T is the magnetic field component transverse to the neutrino momentum q, and µ ν is the neutrino magnetic moment. We assume a Standard-Model (SM) size magnetic moment as [26,27] and a typical distribution of the SN magnetic field [28] B T (r) 10 12 50 km r 2 2 gauss .
Therefore the nominal value of µ ν B T is taken as and other choices will be given in the unit of this nominal value.
Note that in order to fulfill the probability conservation, the density matrix satisfies the following relation and the initial condition can be written as where

Spin Flavor Spectral Splits
In this section, we present the results of numerically To understand the property of the collective oscillations induced by the neutrino magnetic moment better, we start with a comparison of two simplified cases with µ ν = 0 or θ = 0. The first case corresponds to the studies in Ref. [21], in which the 4 × 4 neutrino flavor evolution equation will be reduced into two subsystems of (ν e , ν x ) and (ν e , ν x ) with two 2 × 2 evolution equations (see Eqs. (34) and (35) in Ref. [21]). In the left panel of Fig. 1, we re-illustrate the spectral split feature between different flavor neutrinos for the inverted mass ordering, which illustrates almost complete spectral swaps where B eff = (sin 2θ eff , 0, − cos 2θ eff ), tan 2θ eff = µ ν B T /ω, and the new polarisation vectors P + and P − are defined and The These two kinds of neutrino energy spectral splits fall into the category of the so-called single split type [21], in which the detailed analysis has been discussed. For the left panel in Fig. 1, the critical energies can be estimated by requiring the lepton number conversion: where E 1 and E 2 depend on both the mass-mixing parameters and the neutrino energy spectra, and according to the current inputs, one can obtain that E 1 7 MeV and E 2 2 MeV. As for the right panel with the non-zero neutrino magnetic moment, an effective lepton number conservation of each sub-system can be defined as where E 1 and E 2 can be similarly calculated by using the neutrino energy spectra and the effective mixing θ eff . In summary, we conclude that the neutrino magnetic moment is playing the role of effective mixing in the subsystems of (ν e , ν x ) and (ν e , ν x ), which provides an ini-tial perturbation for the transition between different spin states (i.e., neutrino or antineutrino), and when the neu- Meanwhile the single split between the different spinflavor states in the right panel of Fig. 1 for the normal ordering case is also kept in the right panel of Fig. 2 when we consider the flavor mixing, indicating the fact that one effect is dominating over the other for the neutrino energies below 10 MeV. Next let us take a closer look at the spectral splits above 10 MeV. Interestingly, spinflavor spectral split appears for the inverted ordering, but flavor spectral split happens for the normal ordering. However, the split behavior is completely different, which belongs to the type of the multiple spectral split discussed in Ref. [23]. Taking the (ν e , ν x ) pair in the left panel of Fig. 2 as an example, they first swap their spectra at the location of E 1 12 MeV, and finally switch back at around E 3 35 MeV. In between the spectra naturally cross each other at E 2 19 MeV. The location of the multiple spectral splits can also be inferred from the spirit of an effective lepton number conservation: Before finishing this section, we would like to discuss how the size of the neutrino magnetic moment changes the patterns of spectral splits. Taking the normal (inverted) mass ordering and enlarging the magnitude of the magnetic moment to µ ν = 10 −5 µ ν (SM) (µ ν = 10 2 µ ν (SM)) and µ ν = µ ν (SM) (µ ν = 10 4 µ ν (SM)), we illustrate in MeV. Comparing the situations in the normal and inverted mass orderings, one can learn that the neutrino magnetic moment is much more sensitive in the former case, and even that much smaller than the SM prediction, could have significant effects on the neutrino spectra, and may be observable in the future SN neutrino observation.

Conclusion
In this work, we have performed a numerical calculation of the neutrino flavor evolution inside the SN medium in the presence of the neutrino magnetic moment and in the environment of the strong magnetic field. By using the two-flavor and single-angle approximation, we first demonstrate that the neutrino magnetic moment can serve as an effective mixing between differ-ent neutrino spin-flavor states. We observe both flavor and spin-flavor neutrino collective oscillations inside the dense SN medium. Fruitful neutrino spectral splits have been identified. In the low energy range below around 10 MeV, single spectral split between different flavor states occurs for the case of inverted mass ordering, while the single spectral split between different spin-flavor states takes place for the case of normal mass ordering. For the energy range above 10 MeV, we observe multiple spectral splits within sub-systems of two different neutrinos, either pure flavor conversions (normal ordering) or spin-flavor conversions (inverted ordering). We also find that the pattern of the flavor and spin-flavor spectral splits strongly depends on the magnitude of the neutrino magnetic moment, and even if the magnetic moment is smaller than the SM prediction, there will be significant effects on the SN neutrino spectra.
The results in this study are calculated within the two-flavor and single-angle approximation. Including the effects of the three-flavor mixing and multi-angle calculations would make the calculation more complicated and rather time-consuming. According to the experience of the pure flavor neutrino collective oscillations, the general features will preserve, but some new features may emerge [31][32][33]. Therefore, we anticipate it will be similar cases for the spin-flavor collective oscillations, which will be reported in a future separated work.