New Resonances of Supernova Neutrinos in Twisting Magnetic Fields

We investigate the effect of resonant spin conversion of the neutrinos induced by the geometrical phase in a twisting magnetic field. We find that the geometrical phase originating from the rotation of the transverse magnetic field along the neutrino trajectory can trigger a new resonant spin conversion of Dirac neutrinos inside the supernova, even if there were no such transitions in the fixed-direction field case. We have shown that even though resonant spin conversion is too weak to affect solar neutrinos, it could have a remarkable consequence on supernova neutronization bursts where very intense magnetic fields are quite likely. We demonstrate how the flavor composition at Earth can be used as a probe to establish the presence of non-negligible magnetic moments, potentially down to $10^{-15}~\mu_B$ in upcoming neutrino experiments like the Deep Underground Neutrino Experiment (DUNE), and the Hyper-Kamiokande (HK). Possible implications are analyzed.

Introduction.-WolfgangPauli's 1930 letter [1] not only postulates the neutrino's existence as an explanation for the apparent non-conservation of energy in radioactive decay but also suggests that these elusive particles possess a mass, along with non-zero magnetic moments.Later, in 1954, Cowan et al. set the first limits on neutrino magnetic moments [2], even before neutrinos were discovered and Bernstein et al. did a survey [3] of the experimental information on neutrino electromagnetic properties.In the late 80s and early 90s, the study of neutrino magnetic moments became a popular topic in addressing the solar neutrino problem [4][5][6][7].In recent decades, several experiments have discovered neutrino oscillations, which conclusively demonstrated that neutrinos have masses and mixing, indicating the need for physics beyond the standard model (BSM).In such BSM theories, neutrinos establish interactions with photons via quantum loop correction, even though neutrinos are immune to electromagnetic interaction in the Standard Model (SM) 1 .There are now hundreds of potential neutrino mass models.Yet, not all models qualify to conceive large magnetic moments without upsetting neutrino masses (see Ref. [9] and references therein).Neutrinos with large magnetic moments can significantly impact searches at neutrino scattering experiments [10][11][12] and dark matter direct detection experiments [13][14][15], astrophysical neutrino signals [16][17][18][19][20][21], stellar cooling [22][23][24], cosmological imprints [25][26][27] and charged lepton's magnetic moment [28] (for a review, see Ref. [29]).The presence of large transverse magnetic fields within the sun, supernovae, neutron stars, or other astrophysical objects can result in efficient spin-precession [4,5] or resonant spin-flavor precession [6,7] of neutrinos.
The majority of the literature (see Ref. [16][17][18][19][20][21] and references therein), however, assumes that the direction of the transverse magnetic field is fixed.Nevertheless, this is not always the case.In such scenarios, neutrinos traveling from the core of the supernova outwards and crossing such field configurations would encounter a transverse magnetic field whose direction changes continuously throughout their trajectory [30].It will introduce a new geometrical phase2 governed by the magnetic field rotation angle ϕ in addition to the usual dynamical phase, determined by the energy splitting of the neutrino eigenstates.In the early 1990s, although Vidal et al. and Aneziris et al. discussed the effect of such phases in the context of solar neutrino problem [32,33]; Smirnov was the first to correctly recognize [34] the resonant structure of neutrino spin-precession due to the solar magnetic field's geometrical phase, which was later explored by other authors [34][35][36][37].However, we find that the impact of the geometrical phase on neutrino precession in the Sun is negligible, given a magnetic moment less than ∼ 10 −11 µ B , which has been ruled out by current laboratory-based experiments [12,14,15].One needs a perhaps unrealistically large magnetic field and magnetic moment combination for an emphatic effect.Here, we have shown that it could have a remarkable consequence on supernova neutronization bursts where very intense magnetic fields are quite likely.We analyze the neutrino spectra from the neutronization burst phase and demonstrate how its time-variation can be used as a probe to establish the presence of non-negligible magnetic moments, potentially down to 10 −15 µ B in forthcoming neutrino experiments like DUNE [38], and HK [39].Considering a realistic setup, it will be extremely difficult to probe µ ν ≲ 10 −12 µ B in laboratory-based experiments based on neutrino-electron scattering.Moreover, it has been argued that Dirac neutrino magnetic moments over 10 −15 µ B would not be natural [40] because they would produce unacceptable neutrino masses at larger loops.Thus, the results presented here are in the borderline of the conceivable region for µ ν .
Evolution of neutrino system in twisting magnetic field.-Letus consider a system of left-handed neutrinos ν L = (ν eL , ν µL , ν τ L ), and right-handed counterparts ν R = (ν eR , ν µR , ν τ R ), with magnetic moment µ evolving in matter and a transverse magnetic field.If the magnetic field rotates along the neutrino path in the transverse plane, denoted by B = B x + iB y = Be iϕ , where ϕ(r) is the angle of rotation, the resulting evolution equation can be expressed as: where r is the radial coordinate, I is the identity matrix and µ = diag{µ ν , µ ν , µ ν } is the matrix of Dirac magnetic moments.Eq. ( 1) is expressed in a frame [35] rotating with the magnetic field; see Fig. 1.H L represents the Hamiltonian for ν L propagating in matter, given by where n e , n p , n n are the number densities of electron, proton and neutrons respectively and Y e = n e /(n p + n n ) is the electron-fraction.H R is the Hamiltonian for ν R , which does not experience matter interactions, and For antineutrinos, νL are the ones that do not interact with matter.Furthermore, matter potentials for antineutrinos have the opposite sign: Ve = −V e and Vµ,τ = −V µ,τ .Now consider a neutrino system propagating in a background of the non-uniform matter and a rotating transverse magnetic field.As V changes, strong resonant spinflip conversion, ν L ↔ ν R , can occur.In the two-state approximation, the resonance condition for the ν αL ↔ ν αR conversion can be expressed as [35] V α + φ = 0. ( For antineutrinos, ναL ↔ ναR resonance condition is Vα − φ = 0 and occurs at the same location.The dynamics of spin-flip transition in the resonance region is governed by the adiabaticity coefficient γ α .Under twostate approximation, it can be expressed as [35] For our analysis, we study the three-flavor evolution (six states), described in Eq. ( 1).In such a scenario, coupled resonances (in which one resonance interferes with others owing to closeness) will exist between ν αL and all ν R states since the right-handed (RH) states are linked among themselves as a result of mixing.The simplified neutrino energy levels in the resonance zone are depicted in Fig 1 .At high densities (to the right), ν eL is heavier than all RH states since V e + φ(> 0) is very large, but the converse occurs at low densities (to the left) where V e + φ < 0. The primed states in Fig. 1 are the eigenstates of H R and, to a decent approximation, of the whole system.Supernova environment.-Wefocus on neutrinos released during the neutronization-burst phase, which occurs right after the core bounces and lasts for a few-tens of milliseconds.During this phase, the ν e -flux is dominant over other flavors in most of the energy-spectrum [16,41].Moreover, collective neutrino oscillations, a significant source of complication to the flavor evolution, are expected to be suppressed during this stage [42,43].The predicted neutrino-fluxes during this phase have only about O(10%) uncertainty [44][45][46][47].As a result, we anticipate that the estimates of the SN neutrino flavor content during this period are more robust.
In this work, we explore the discovery potential of the Dirac magnetic moments of neutrinos coming from SNe with magnetic field strength 10 10 − 10 12 G in the ironcore.Such magnetic fields are commonly associated with the formation of magnetar-like field structures in SN remnants [30].We assume the radius of the iron-core, r 0 , to lie somewhere in the range of 103 − 10 4 Km (for practical calculations r 0 ≈ 2000 Km [30]) and model the magnetic field as, B(r) = B 0 for r < r 0 , and B(r) = B 0 (r 0 /r) 3 for r > r 0 [30].The matter potentials for V e , V µ , and V τ are determined by the matter density ρ and electron number fraction Y e , which we obtain from a simulation of an 18 M ⊙ progenitor [48] at t = 4.37 ms [49].See Fig. 2 for further details.
The spin-flip transition in the resonance region is dictated by the adiabaticity coefficient γ α , which is heavily influenced by µ ν B 0 .When taking into account the electron-fraction and density of matter in the SN [cf.Fig. 2], Vα (r) varies as a function of r only, while φ(r) can be arbitrary.For simplicity, we assume | φ| = 0.This assumption is valid as long as the fluctuations around the average velocity φ are smaller in scale than the spin-precession scale [50], which is approximately π/µ ν B 0 ∼ O(10) kilometers at the resonance layer for µ ν B 0 = 10 −14 µ B 10 12 Gauss = 10 −2 µ B G. We find the surface of the iron-core to be the most promising region 3 as γ α ≳ 1 for values as small as µ ν B 0 = 10 −2 µ B G, the smallest over the whole profile.This is due to the fact that ρ decreases with r −3 and ρ decreases with −r −4 , so | Vα | reduces as the distance (r) from the SN centre increases.At r = r 0 , the smallest | Vα | occurs when the highest possible field strength, B(r 0 ) = B 0 , is present.At the surface of the iron-core, Y e ≈ 0.5 and n e ≈ n n , see Fig. 2. Thus, at this point, V e ≈ 0.5 √ 2G F n e = −V x , where x = µ, τ .V e is a monotonically decreasing function in the range of r ∼ 10 3 − 10 4 Km, such that 2 m −1 ≳ V e ≳ 0.01 m −1 .Therefore, the rate of rotation of B in the range −0.01 m −1 ≳ φ ≳ −2 m −1 could produce a resonance in the ν e and νe channels, see Eq. ( 2), while the rotation in the opposite direction 2 m −1 ≳ φ ≳ 0.01 m −1 produces a resonance in the ν µ,τ and νµ,τ channels.Note that the magnitude of | φ| ∼ 0.01 m −1 implies that for resonant conversion to occur, it is sufficient for B to undergo approximately one revolution within a width of about one kilometer (which corresponds to the resonance layer).This observation aligns with the simulation pre-sented in [30], where the magnetic field B reverses its direction within a few kilometers around r 0 .
Analysis and results.-Beforeanalyzing the effect of non-zero neutrino magnetic moments, we briefly describe the signal for the standard scenario (µ ν = 0).SN neutrinos are produced at the core where V e ≫ V x and the electron-neutrino (ν e ) becomes the heaviest of the three matter-eigenstates, either as ν 3m for normal mass ordering (NO) or as ν 2m for inverted mass ordering (IO); while the muon-neutrino (ν µ ) and tau-neutrino (ν τ ) are a combination of the two remaining eigenstates.The opposite is true for antineutrinos, where the electron antineutrino (ν e ) becomes the lightest of the three eigenstates in matter, either as ν1m in NO or as ν3m in IO, because Ve ≪ Vx .
) where we used the unitarity of the PMNS matrix to write For antineutrinos, at the production point, νe ≈ ν1m in NO and νe ≈ ν3m in IO.Then, we have: Now we analyze the effect of the new resonance due to twisting magnetic fields at the surface of the SN ironcore.This resonance can happen at r ∼ r 0 , before the L and H resonances take place, and can have profound consequences to the fluxes, see Eqs. (4-7), coming out of the collapsing star during the neutronization-burst phase.The reason is that a sizable fraction of active neutrinos could be converted to their right-handed counterparts, effectively decreasing the initial fluxes before they reach the L and H resonances, i.e.,  for the specific flavor α which has the resonance condition in Eq. ( 2) satisfied.The Landau-Zener factor e − π 2 γα is the "flip" probability that a transition happens between the states ν αL ↔ ν ′ R at the resonance point.In what follows, we analyze the limiting case of total adiabaticity, where γ α > 1.Indeed, Fig. 2 shows that for µ ν B 0 = 10 −14 µ B 10 12 G = 10 −2 µ B G, γ α ≈ 2 and e − π 2 γα ≈ 0.04 in specific locations resulting in dramatic modifications to the expected neutronization-fluxes.
These numbers indicate that twisting magnetic fields can impact the time-variation of the neutronization-burst signal.In particular, Case-1 optimizes the impact of the new resonance for Inverted Ordering (IO), resulting in up to an 81% reduction in the ν e neutronization-peak, which occurs before 20 ms and is significant in the standard case for IO, see the difference between the green and red lines in the panel for DUNE, IO in Fig. 3.However, Case-2 has the highest potential for demonstrating the effects of twisting magnetic fields since it can cause a strong suppression (greater than 70%) of neutrino-fluxes in almost every detection channel and time-window.Fig. 3 depicts cases 1 ( φ = −0.9m −1 ) and 2 ( φ = +0.9m −1 ) in future experiments like DUNE [38], a 40 Kt of liquid argon time-projection chamber, which mainly detects ν e via ν e + 40 Ar → 40 K * + e − , and HK [39], a 374 Kt water-Cherenkov detector which detects mainly νe via inverse-beta decay, νe + p → e + + n.The number of detected events of ν α per energy is where N t denotes number of target particles in the detector, R = 10 Kpc is the distance between Earth and the galactic-SN, Φ α and σ α represent ν α -flux and interaction cross-section, and W is energy-resolution function.For technical details, see Ref. [16].To study the sensitivity of DUNE and HK to a non-zero µ ν in presence of resonant spin-precession, we use the χ 2 -estimator with F i and D i the number of events in the i-th time-bin for finite and null values of µ ν , respectively.We perform a shape-only analysis where the normalization parameter ξ varies in the range [−1, 1000] and adjust the normalization of the test hypothesis to the true hypothesis.In this way, the analysis concentrates on the time-variation of the signal.Discovery reaches for DUNE and HK, as well as the combined analysis of both experiments, are summarized in Table I.The sensitivity of the combination of DUNE and HK can reach µ ν of order 4 − 7.5 × 10 −15 µ B at 90% C.L. for B 0 = 10 12 G4 .
Interstellar magnetic fields only influence the overall flux-normalization [18] and not its temporal variation.Hence, if supernova-flux is measured in the future and the flux-deficit is seen to be different in various timebins, this will occur owing to the above-mentioned resonances caused by the magnetic field configuration's twisting structure.While the intriguing effect could appear in the observed (anti)neutrino spectra at HK and DUNE experiments, the reality is expected to be significantly more intricate, necessitating future research.Even with assumed knowledge of neutrinos as Dirac fermions and known mass ordering, uncontrollable factors, including other BSM effects, may influence the magnetic moment effect.
Final remarks.-Wehave shown that if the magnetic field inside the supernova has a twisting structure, then the rotation of the magnetic field along the neutrino trajectory can induce a resonant spin-conversion, which will affect predictions for the event rates when detecting supernova neutrinos in future neutrino experiments such as DUNE and HK.If neutrinos are Dirac particles possessing large magnetic moments, this resonance effect will present the optimal avenue towards unravelling the scenario at hand.

FIG. 1 .
FIG. 1. Left: schematic representation of the rotating frame where z-axis denotes the direction of the neutrino momentum, φ is the velocity of B field-rotation.The spins of νL,R are shown by thick red arrows.Right: neutrino energy levels in the resonance region for Normal Ordering (NO) and φ < 0. See text for details.
During the subsequent evolution, neutrinos and antineutrinos might cross adiabatically the low (L) and high (H) Mikheyev-Smirnov-Wolfenstein (MSW) resonances [52-54].These happen when |V e − V x | = |∆m 2 n1 | cos θ 1n /2E, where n = 2(n = 3) for L(H) resonance.In NO, L− and H−resonances occur exclusively in the neutrino channel, but in IO, L−resonance occurs for neutrinos and H−resonance for antineutrinos [55].Considering the best-fit values for the oscillation parameters [56], H−resonance occurs at about 3 × 10 4 Km, whereas L−resonance occurs at approximately 2 × 10 5 Km from the centre of the SN.In NO, at the production region ν e ≈ ν 3m , so Φ i3 = Φ ie .At vacuum, however, only a small component of ν 3 , |U e3 | ≈ 0.02, remains as electron flavor.Therefore, only 0.02Φ ie contributes to the final ν e -flux.Assuming the initial ν µ -flux and ν τ -flux to be equal, Φ iµ = Φ iτ = Φ ix , the initial fluxes of the remaining matter-eigenstates ν 1m and ν 2m , which are mixtures of ν µ and ν τ , are also equal to Φ ix .Thus, their respective contributions to the final ν e -flux are |U e1 | 2 Φ ix and |U e2 | 2 Φ ix .The total final ν e -flux in NO is then:

0 FIG. 3 .
FIG.3.Expected number of νe-events at HK (upper) and νe-events at DUNE (lower) for NO (left) and IO (right) in the time-window between −5 ms and 40 ms corresponding to the SN neutronization-burst stage in which 0 ms is the time of core-bounce.Results are shown for the standard scenario (µν = 0) and for µν = 10 −14 µB and B0 = 10 12 G for both cases 1 ( φ < 0) and 2 ( φ > 0).

TABLE I .
Experimental sensitivities on µν for different benchmark scenarios and B0 = 10 12 G.