Maxim Dvornikov
IZMIRAN, Russia

Plan of the talk
•Neutrino spin oscillations in ascattering off a black hole (BH)
•Neutrino flavor oscillations instochastic gravitational waves(GWs)•SummaryPublications• M. Dvornikov, Interaction of supernova neutrinos with stochastic gravitational waves, Phys.

Rev. D 104, 043018 (2021), arXiv:2103.15464.
• M. Dvornikov, Neutrino scattering off a black hole surrounded by a magnetized accretion disk,

JCAP 04 (2021) 005, arXiv:2102.00806.
• M. Dvornikov, Flavor ratios of astrophysical neutrinos interacting with stochastic gravitational

waves having arbitrary spectra, JCAP 12 (2020) 022, arXiv:2009.02195.
• M. Dvornikov, Spin oscillations of neutrinos scattered off a rotating black hole, Eur. Phys. J. C80, 474 (2020), arXiv:2006.01636.
• M. Dvornikov, Spin effects in neutrino gravitational scattering, Phys. Rev. D 101, 056018(2020), arXiv:1911.08317.
• M. Dvornikov, Neutrino flavor oscillations in stochastic gravitational waves, Phys. Rev. D 100,096014 (2019), arXiv:1906.06167.

Introduction and motivation• Neutrinos are left-handed in the standard model, i.e. their spin is opposite to

the neutrino momentum• If the neutrino spin precesses in an external field, i.e. changes its direction withrespect to the neutrino momentum, particles become right-handed• Right-handed neutrinos are sterile in the standard model• We will observe the effective reduction of the initial neutrino flux
• This process is called neutrino spin oscillations• External fields, including gravity, can change polarization of fermions• Recently, the supermassive BH shadow in M87 was observed. What happens if

we look at this SMBH in a neutrino telescope?

Formulation of the problem• Uniform flux of left-polarized neutrinos• Gravitational scattering off BH rotating BH: Kerr metric• Electroweak interaction of neutrinos with dense matter of anaccretion disk, which, as a rule surrounds BH• Accretion disk with realistic radial density distribution• Neutrino has nonzero magnetic moment• Accretion disk is permeated by the magnetic field• Neutrino motion in the equatorial plane only

Schematic illustration ofneutrino scatteringParameters of an accretion disk and a neutrino• Wald (1974) found an electromagnetic field in the vicinity of a Kerr BH which asymptoticallyapproaches to a constant and uniform magnetic field. It acquires an electric component.
However, such magnetic field is unphysical since B should disappear at the edge of a disk

• Beskin (2010) reviewed numerous models of magnetic fields in a disk. Both poloidal andtoroidal components are present. We take that only poloidal exists. Toroidal component
does not contribute significantly to neutrino spin-flip• Blandford & Payne (1982) assumed the equipartition of the energy between the magnetic

field and accreting plasma. If 𝐵 ∝ 𝐵0 𝑟 −𝜅 and 𝑛 ∝ 𝑛0 𝑟 −𝛽 , 𝜅 = (𝛽 + 1)/2.

• Narayan & Yi (1994) found that 𝛽 = 3/2 in an advection dominated accretion disk. Thus 𝜅 =5/4.

• Jiang et al. (2019) obtained that 𝑛0 = 1018 𝑐𝑚−3 for 𝑀 = 108 𝑀⊙• Magnetic field near BH is constrained by the Eddington limit 𝐵𝐸𝑑𝑑 . We take that 𝐵0 =10−2 𝐵𝐸𝑑𝑑 . One gets that 𝐵0 = 3.2 × 102 𝐺. Daly (2019) reports that such magnetic fields are
not excluded by observations• Bell et al. (2005) suggests a model independent constraint on the Dirac neutrino magnetic

moment: 𝜇 = 10−14 𝜇𝐵

Direct and retrograde scatteringsNeutrino spin evolution in the locally
Minkowskian frame• 3D neutrino spin in curved spacetime (Pomeranskii & Khriplovich, 1998;

Dvornikov 2006, 2013):𝑑 𝜁ԦԦ= 2(Ω × 𝜁)𝑑𝑡
• Neutrino velocity in the locally Minkowskian frame changes its direction in
gravitational scattering• Transition and survival probabilities are

1
𝑃𝐿𝑅,𝐿𝐿 = (1 ± cos 𝛼+∞ )2• If a neutrino interacts only with gravity and a magnetic field, 𝛼+∞ can be
expressed in quadratures

Transition probability for ultrarelativistic
neutrinos in Kerr metric• If ultrarelativistic neutrinos interact only with gravity, 𝛼+∞ = −𝜋. Thus

𝑃𝐿𝑅 = 0. Thus result is in agreement with Lambiase et al. (2005), whostudies the neutrino scattering off a Kerr BH in the weak field limit• If a neutrino has nonzero magnetic moment, 𝑃𝐿𝑅 ≠ 0 even for

ultrarelativistic particles• Transition/survival probabilities versus the impact parameter are notmeasurable because of multiple revolutions of a neutrino around BH• The flux of particles versus the scattering angle is measured• Only left-polarized neutrinos interact with a detector. The measured

flux ∝ 𝑃𝐿𝐿 𝐹0• The interaction with magnetic field and/or accretion disk changes theflux compared to “scalar” neutrinos and when only magnetic scattering
is accounted forElectroweak interaction with accretion
disk• Neutrino forward scattering in plasma

𝐿𝑚𝑎𝑡𝑡 = −𝐺𝐹𝜈𝛾ҧ 𝜇 (1 − 𝛾 5 )𝜈 ∙ 𝐺𝜇2• Modification of the precession equationΩ = Ω𝑔 + Ω𝐵 + Ω𝑚𝑎𝑡𝑡

Ω𝑚𝑎𝑡𝑡 ∝ 𝐺𝐹 𝑛𝑒𝑓𝑓

• Rotating accretion disk (Bardeen et al., 1972)

𝑈𝑓𝑡 =2𝑥 3/2 + 𝜆𝑧,

1𝜙
𝑈𝑓 =𝑟𝑔𝜆

2𝑥 3 − 3𝑥 2 + 2𝜆 2𝑧𝑥 3/2
2𝑥 3 − 3𝑥 2 + 2𝜆 2𝑧𝑥 3/2• Corotating disk: 𝜆 = +1; counter-rotating disk: 𝜆 = −1
• The problem cannot be solved in quadratures. The effective Schrodinger equation

should be integrated numericallyFluxes of scattered neutrinos accounting
for spin oscillations: corotating disk(a) Direct scattering(b) Retrograde scattering(d) Retrograde scattering(c) Direct scattering• Maximally rotating SMBH with 𝑎 = 𝑀• 𝜆 = +1
• The maximal effect is for the backwardscattering 𝜒 = 𝜋. The glory flux ofneutrinos experience maximal (several %)reduction

Fluxes of scattered neutrinos accounting forspin oscillations: counter-rotating disk(a) Direct scattering(b) Retrograde scattering(d) Retrograde scattering(c) Direct scattering• Maximally rotating SMBH with 𝑎 = 𝑀• 𝜆 = −1
• There is an upward spike in the total flux inthe direct scattering• 𝑃𝐿𝐿 is greatFluxes of scattered neutrinos accounting forspin oscillations: counter-rotating disk(a) Direct scattering(b) Retrograde scattering(d) Retrograde scattering(c) Direct scattering• SMBH with 𝑎 = 0.2𝑀
• 𝜆 = −1
• There is a downward spike in the total fluxin the retrograde scattering• 𝑃𝐿𝑅 grows

Discussion• There is spin-flip of ultrarelativistic neutrinos in their scattering off a rotating BH

only if they have a magnetic moment and there is a magnetic field in the disk
• Solely gravitational scattering does not make a spin-flip of ultrarelativistic

neutrinos• The form and the size of a BH shadow is unchanged since spin oscillations do notcontribute to the flux at 𝜒 = 0• The major modification of the flux is for backwardly scattered neutrinos 𝜒 = 𝜋• The electroweak interaction with counter-rotating disks result in the up- or

downward spikes in the total fluxes in direct or retrograde scatterings• Possible interpretation: neutrinos interact with matter of a disk which moves

with relativistic velocities. Moving matter is known to affect spin oscillations
including causing the spin-flip

Introduction and motivation• Neutrinos interact with other leptons (e,μ,τ) as flavor eigenstates:ν = (νe, νµ, ντ)• Flavor eigenstates do not have definite masses• We introduce mass eigenstates ψ = (ψ1, ψ2, ψ3)• These bases are related by the unitary matrix transformation

𝜈 = 𝑈𝜓
• These neutrino properties result in neutrino flavor oscillation, i.e. the change of a flavor content of

the neutrino beam, which can happen even in vacuum• External fields, including gravity, can influence neutrino flavor oscillations• It is interesting to check if nonstationary gravitational field, like GWs, which were directly detected by

LIGO-Virgo, can contribute to neutrino flavor oscillations• NANOGrav (2020) reported about a strong evidence of stochastic GWs

Evolution of a mass eigenstate in
GW
Ahluwalia & Burgard (1996); Fornengo et al (1997)
established the evolution of neutrino mass
eigenstates in a gravitational field

𝜓𝑎 𝑥, 𝑡 ~ exp[−𝑖𝑆𝑎 𝑥, 𝑡 ]𝜕𝑆𝑎 𝜕𝑆𝑎𝑔𝜇𝜈= 𝑚𝑎2𝜕𝑥𝜇 𝜕𝑥𝜈

𝑑𝑠 2 = 𝑔𝜇𝜈 𝑑𝑥 𝜇 𝑑𝑥 𝜈 = 𝑑𝑡 2 − 1 − ℎ+ cos 𝜙 𝑑𝑥 2 − 1 + ℎ+ cos 𝜙 𝑑𝑦 2 + 2𝑑𝑥𝑑𝑦ℎ× sin 𝜙 − 𝑑𝑧 2
Dvornikov (2021) found the perturbative solution of the Hamilton-Jacobi equation in a plane GW

Schrodinger equation and the effective
Hamiltonian for neutrino flavor eigenstates𝑖 𝜈ሶ = 𝐻𝑓 𝜈

𝐻𝑓 = 𝑈𝐻𝑚 𝑈 +GW does not contribute to neutrino(𝑔)
𝐻𝑚=−sin2 𝜗 cos(2𝜑 − 𝜙𝑎 ) oscillations if neutrino beam𝑎𝑎
2 𝑝2 + 𝑚𝑎2
propagates along GW (𝜗 = 0)
𝑝2 ℎStochastic GWs• Neutrino interacts with randomly emitted GWs• Density matrix (Loreti & Balantekin, 1994)• Averaging over angles• Gaussian distribution of strain with arbitrary correlator:

𝑓+,× (|𝑡1 − 𝑡2 |)ℎ+,× (𝑡1 )ℎ+,× (𝑡2 ) =• We can find the correction to the probabilities of vacuum oscillations caused byGWs

Δ𝑃𝜆 𝑥 = 2 ෍ 𝑃𝜎 (0) ෍𝜎∗∗+ Im 𝑈𝜆𝑎 𝑈𝜆𝑏
𝑈𝜎𝑎𝑈𝜎𝑏 sin𝑎>𝑏∗∗
𝑈𝜎𝑎𝑈𝜎𝑏 cos
ቊRe 𝑈𝜆𝑎 𝑈𝜆𝑏𝑥
2𝜋𝐿𝑎𝑏

ൠ 1 − exp𝑥2𝜋𝐿𝑎𝑏4𝜋2 𝑥− 2 ‫׬‬0 𝑔෤ 𝑡 𝑑𝑡𝐿𝑎𝑏Initial condition
We study neutrinos emitted in 𝜈𝑒 -burst in core-collapsing SN
SN is almost point-like source
The size of neutrinosphere is ~ 100 km
The contribution of solar oscillations channel is not smeared
Fluxes at a source are (𝐹𝑒 , 𝐹𝜇 , 𝐹𝜏 )𝑆 = (1: 0: 0)
Initial condition 𝜌11 0 = 1, 𝜌22 0 = 0, 𝜌33 0 = 0

GW emitted by randomly coalescencing
supermassive BHs (SMBH)• Spectral function for GW fromdifferent types of merging BHs
is calculated by Rosado (2011)• Ω is the energy density ofstochastic GWs per logarithmicfrequency interval with respect
to the closure density of theuniverseΩ ∝ 𝑓𝛼• We will study the case of SMBH

since they produce stochasticGWs with the major effect onneutrino oscillations

Fluxes of flavor neutrinos

E = 10 MeV𝜏 = 𝑥/𝐿
L = 10 kpc

Neutrino fluxes after 𝜈𝑒 -burst

•
•
•
•

Significant number of neutrinos of different flavors is emitted after 𝜈𝑒 -burst
We study oscillations of such neutrinos in stochastic GWs
0 < ∆𝑡 < 0.1 𝑠 is the time after 𝜈𝑒 -burst. We approximate the fluxes at a source (SN) by exponents
The contributions to the fluxes from stochastic GWs in a detector are vanishing at ∆𝑡~0.1 𝑠

Discussion• We have the analytic expression for the probabilities for all neutrino

flavors interacting with stochastic GWs• Two independent polarizations of GWs are accounted for• The correlators of amplitudes are arbitrary• The results are applied for oscillations of SN neutrinos• The major effect is for neutrinos emitted in 𝜈𝑒 -burst. At subsequent

moments of time, the contribution of stochastic GWs is vanishing• The interaction with stochastic GWs can results in the change of the SN

neutrinos fuxes by ±350 events, in case of the Super-Kamiokande, andby ±3750 events, for the Hyper-KamiokandeSummary• We have studied the influence of spin oscillations on the observed fluxes of neutrinos

scattered off a rotating BH• There is a spin-flip of ultrarelativistic neutrinos in scattering in the Kerr metric caused by

the interaction of the neutrino magnetic moment with the magnetic field in an accretion
disk disk• There is a reduction of fluxes mainly for the backward scattering• The interaction of neutrinos with an accretion disk also modifies the fluxes. There arespikes in the total fluxes, which can result from the neutrino interaction with moving
matter of a disk

• We have examined the relaxation of the fluxes of flavor neutrinos owing to theirinteraction with stochastic GWs• The major contribution is from GWs emitted by merging SMBHs• This effect can be potentially observed for SN neutrinos with E = 10 MeV in our Galaxy

with the propagation length L = 10 kpc