Collective neutrino oscillations accounting for neutrino quantum decoherence

Abstract

In our previous studies (see [1] and references therein) we developed a new theoretical framework that enabled one to consider a new mechanism of neutrino quantum decoherence engendered by the neutrino radiative decay. In parallel, another framework was developed (see [2] and references therein) for the description of the neutrino quantum decoherence due to the non-forward neutrino scattering processes. Both mechanisms are described by the master equations in the Lindblad form. We study the influence of the neutrino quantum decoherence on collective neutrino oscillations. In the present studies we are are not interested in a specific mechanism of neutrino quantum decoherence. Therefore, we use the general Lindblad master equation for the description of the neutrino quantum decoherence and do not fix an analytical expressions for the decoherence and relaxation parameters.

Full text

aa r X i v : . [ h e p - ph ] F e b Collective neutrino oscillations accounting for neutrinoquantum decoherence

Konstantin Stankevich 𝑎, ∗ and Alexander Studenikin 𝑎,𝑏 𝑎 Faculty of Physics, Lomonosov Moscow State University,Moscow 119991, Russia 𝑏 Joint Institute for Nuclear Research,Dubna 141980, Moscow Region, Russia

E-mail: kl.stankevich@physics.msu.ru, studenik@srd.sinp.msu.ru

In our previous studies (see [1] and references therein) we developed a new theoretical frameworkthat enabled one to consider a new mechanism of neutrino quantum decoherence engendered bythe neutrino radiative decay. In parallel, another framework was developed (see [2] and referencestherein) for the description of the neutrino quantum decoherence due to the non-forward neutrinoscattering processes. Both mechanisms are described by the master equations in the Lindbladform.In the present studies we are are not interested in a specific mechanism of neutrino quantumdecoherence. Therefore, we use the general Lindblad master equation for the description of theneutrino quantum decoherence and do not fix an analytical expressions for the decoherence andrelaxation parameters.We study the influence of the neutrino quantum decoherence on collective neutrino oscillations.Collective neutrino oscillations is a phenomenon engendered by neutrino-neutrino interaction. Itis significant in different astrophysical environments where the neutrino density is extremely high.Examples of such environments are the early universe, supernovae explosions, binary neutron stars,accretion discs of black holes. The effect of collective neutrino oscillations attracts the growinginterest in sight of appearance of multi-messenger astronomy and constructing of new large-volumeneutrino detectors that will be highly efficient for observing neutrino fluxes from supernovaeexplosions. Previously, it was shown that neutrino quantum decoherence can significantly modifythe neutrino fluxes from reactors and the sun. Here below we study the influence of the neutrinoquantum decoherence on supernovae neutrino fluxes. The peculiarity of the supernovae fluxes isthat one of the main modes of neutrino oscillations in supernovae is engendered by the collectiveeffects. Note, in the previous studies dedicated to the collective neutrino oscillations only thekinematical decoherence (see [3] and references therein) was accounted for. ∗ Speaker © Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/ ollective neutrino oscillations accounting for neutrino quantum decoherence

Konstantin StankevichConsider the two-flavor neutrino mixing scenarios, i.e. the mixing between 𝜈 𝑒 and 𝜈 𝑥 stateswhere 𝜈 𝑥 stands for 𝜈 𝜇 or 𝜈 𝜏 . Here below we focus on the derivation of the neutrino oscillationprobability and highlight the interplay between collective neutrino oscillations and neutrino quantumdecoherence. We use the simplified model of supernova neutrinos that was considered in [4, 5]. Insuch a model neutrinos are produced and emitted with a single energy and a single emission angle.The neutrino evolution in supernovae environment that accounts for neutrino quantum deco-herence is determined by the following equations 𝑖 𝑑𝜌 𝑓 𝑑𝑡 = [ 𝐻, 𝜌 𝑓 ] + 𝐷 [ 𝜌 𝑓 ] , 𝑖 𝑑 ¯ 𝜌 𝑓 𝑑𝑡 = [ ¯ 𝐻, ¯ 𝜌 𝑓 ] + 𝐷 [ ¯ 𝜌 𝑓 ] , (1)where 𝜌 𝑓 ( ¯ 𝜌 𝑓 ) is the density matrix for neutrino (antineutrino) in the flavour basis and 𝐻 ( ¯ 𝐻 )are total neutrino (antineutrino) Hamiltonian. Neutrino quantum decoherence is described by thedissipation term 𝐷 [ 𝜌 ] that we define in the next section.Hamiltonian 𝐻 contains the three terms 𝐻 = 𝐻 𝑣𝑎𝑐 + 𝐻 𝑀 + 𝐻 𝜈𝜈 , where 𝐻 𝑣𝑎𝑐 is the vacuumHamiltonian, 𝐻 𝑀 and 𝐻 𝜈𝜈 are Hamiltonians that describe matter potential and neutrino-neutrinointeraction correspondingly. The exact expressions one can find in [4, 5]. The dissipation term 𝐷 [ 𝜌 ] we write in the Lindblad form 𝐷 [ 𝜌 ˜ 𝑚 ( 𝑡 )] = Õ 𝑘 = h 𝑉 𝑘 , 𝜌 ˜ 𝑚 𝑉 † 𝑘 i + h 𝑉 𝑘 𝜌 ˜ 𝑚 , 𝑉 † 𝑘 i , (2)where 𝑉 𝑘 are the dissipative operators that arise from interaction between the neutrino system andthe external environment, 𝜌 ˜ 𝑚 is the neutrino density matrix in the effective mass basis. Here below,we omit index “ ˜ 𝑚 ” in order not overload formulas. The operators 𝑉 𝑘 , 𝜌 𝑓 and 𝐻 can be expandedover the Pauli matrices 𝑂 = 𝑎 𝜇 𝜎 𝜇 , where 𝜎 𝜇 are composed by an identity matrix and the Paulimatrices. In this case eq. (1) can be written in the following form 𝜕𝑃 𝑘 ( 𝑡 ) 𝜕𝑡 𝜎 𝑘 = 𝜖 𝑖 𝑗𝑘 𝐻 𝑖 𝑃 𝑗 ( 𝑡 ) 𝜎 𝑘 + 𝐷 𝑘𝑙 𝑃 𝑙 ( 𝑡 ) 𝜎 𝑘 , (3)where the matrix 𝐷 𝑙𝑙 = − 𝑑𝑖𝑎𝑔 { Γ , Γ , Γ } and Γ , Γ are the parameters that describe two dissipativeeffects: 1) the decoherence effect and 2) the relaxation effect, correspondingly. In the case of theenergy conservation in the neutrino system there is an additional requirement on a dissipativeoperators [6] [ 𝐻 𝑆 , 𝑉 𝑘 ] =

0. In this case the relaxation parameter is equal to zero Γ =

0. Herebelow, we consider only the case of the energy conservation, i.e. Γ = 𝛿𝑃 𝑘 around the initial configuration 𝑃 𝑘 and a corresponding variation of the density dependentHamiltonian 𝛿𝐻 𝑘 around the initial Hamiltonian 𝐻 𝑘 : 𝑃 𝑘 = 𝑃 𝑘 + 𝛿𝑃 𝑘 where 𝛿𝑃 𝑘 = 𝑃 ′ 𝑘 𝑒 − 𝑖𝜔𝑡 + 𝐻.𝑐. and 𝐻 𝑘 = 𝐻 𝑘 + 𝛿𝐻 𝑘 where 𝛿𝐻 𝑘 = 𝐻 ′ 𝑘 𝑒 − 𝑖𝜔𝑡 + 𝐻.𝑐. . Where one can write 𝐻 ′ 𝑘 = 𝜕𝐻 𝑘 𝜕𝑃 𝑘 𝑃 ′ 𝑘 + 𝜕𝐻 𝑘 𝜕 ¯ 𝑃 𝑘 ¯ 𝑃 ′ 𝑘 .In the case of high electron density the in-medium eigenstates initially coincide with the flavorstates. Therefore, the initial conditions are given by 𝐻 𝑘 = (cid:0) , , 𝐻 (cid:1) 𝑇 and 𝑃 𝑘 = (cid:0) , , 𝑃 (cid:1) 𝑇 . Putting2 ollective neutrino oscillations accounting for neutrino quantum decoherence Konstantin Stankevicheverything together and considering only the non-diagonal elements ( 𝜌 = 𝑃 𝑥 + 𝑖𝑃 𝑦 ) one obtainsthe following equation for eigenvalues (we neglect the higher-order corrections) ( 𝜔 − 𝑖 Γ ) 𝜌 ′ ¯ 𝜌 ′ ! = 𝐴 𝐵 ¯ 𝐴 ¯ 𝐵 ! 𝜌 ′ ¯ 𝜌 ′ ! , (4)where on the right-hand side of equation is the stability matrix that coincides with one from [4, 7].In case of a single energy and single emission angle it is expressed as 𝐴 = ( 𝐻 − 𝐻 ) − 𝜕𝐻 𝜕 𝜌 ( 𝜌 − 𝜌 ) , 𝐵 = 𝜕𝐻 𝜕 ¯ 𝜌 ( 𝜌 − 𝜌 ) , ¯ 𝐴 = ( ¯ 𝐻 − ¯ 𝐻 ) − 𝜕 ¯ 𝐻 𝜕 ¯ 𝜌 ( ¯ 𝜌 − ¯ 𝜌 ) , ¯ 𝐵 = 𝜕 ¯ 𝐻 𝜕 𝜌 ( ¯ 𝜌 − ¯ 𝜌 ) . (5)The eigenvalues are given by 𝜔 = 𝑖 Γ + (cid:16) 𝐴 + ¯ 𝐴 ± p ( 𝐴 − ¯ 𝐴 ) + 𝐵 ¯ 𝐵 (cid:17) . One cansee that if the eigenvalues have an imaginary part, the non-diagonal elements of the neutrino densitymatrix can grow exponentially and thus the system become unstable, that is, if ( ( 𝐴 − ¯ 𝐴 ) + 𝐵 ¯ 𝐵 < , I 𝑚 (cid:0) ( 𝐴 − ¯ 𝐴 ) + 𝐵 ¯ 𝐵 (cid:1) > Γ . (6)The first condition is the same as was derived in [4, 7]. The second term is a new one that wasnot considered before. From eq. (6) one can see, that neutrino quantum decoherence prevents asystem from an exponential growth of non-diagonal elements, i.e. neutrino quantum decoherenceleads to the damping of the neutrino collective oscillations.In this paper we considered for the first time the effect of the neutrino quantum decoherencein supernovae fluxes. We derive new conditions of the collective neutrino oscillations accountingfor neutrino quantum decoherence which can appear as a result of the physics beyond the StandardModel. This work was supported by the Russian Foundation for Basic Research under Grant No.20-52-53022-GFEN-a. The work of KS was also supported by the Russian Foundation for BasicResearch under Grant No. 20-32-90107. References [1] K. Stankevich, A. Studenikin,

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