Stationary and non-stationary solutions of the evolution equation for neutrino in matter

We study solutions of the equation which describes the evolution of a neutrino propagating in dense homogeneous medium in the framework of the quantum field theory. In the two-flavor model the explicit form of Green function is obtained, and as a consequence the dispersion law for a neutrino in matter is derived. Both the solutions describing the stationary states and the spin-flavor coherent states of the neutrino are found. It is shown that the stationary states of the neutrino are different from the mass states, and the wave function of a state with a definite flavor should be constructed as a linear combination of the wave functions of the stationary states with coefficients, which depend on the mixing angle in matter. In the ultra-relativistic limit the wave functions of the spin-flavor coherent states coincide with the solutions of the quasi-classical evolution equation. Quasi-classical approximation of the wave functions of spin-flavor coherent states is used to calculate the probabilities of transitions between neutrino states with definite flavor and helicity.


Introduction
Lobanov A. E., Theoretical and Mathematical Physics, 2017, 192:1, 1000-1015 The fermions are combined in SU(3)-multiplets 1-patricle wave functions are elements of the representation space of the direct product of Poincarre group and SU (3) We may quantize the model The Fock space for the superposition of mass states can be constructed We obtain the pertubation series in the interaction representation The formulas for neutrino oscillations are in good agreement with those obtained in the phenomenological theory An effective equation describing neutrino oscillations and its spin rotation A. E. Lobanov, Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, 59, No. 11, 141, (2016). [Russ. Phys. J., 59, No. 11, 1891, (2016]. Here, I is a 3 × 3 unit matrix , M Hermitian mass matrix of the neutrino multiplet, which can be written as follows M = 3 l=1 m l P (l) .
(2.2) m l are the eigenvalues of the mass matrix, which have the meaning of the masses of the multiplet components, and the matrices P (l) are orthogonal projectors on the subspaces with these masses. Matrix P (e) is the projector on the state of the neutrino with electron flavor.
The effective potential due to the interaction via charged currents f µ(e) = √ 2G F j µ(e) − λ µ(e) (2. 3) The effective potential due to the neutral current interaction are the 4-vectors of the current, and are the 4-vectors of the polarizarion of the components of the medium. In the 3-flavor model Ψ is a 12-component object. It is convenient to introduce block structure, namely to define this object as three Dirac spinors In the 2-flavor model the wave function Ψ can be presented as a pair of Dirac spinors. We will call the representation of the matrices M, P (e) , which diagonalizes the mass matrix, the mass representation. We also introduce the flavor representation of these matrices as a representation, where the projectors P (e) are diagonal matrices. These representations are connected by the unitary mixing matrix U PMNS .

11)
Mass states and states with definite flavor The mass states are the states, which are described by the wave functions Ψ i (i = 1..3), which can be written in the mass representation as follows where ψ i , i = 1..3 are the Dirac spinors. We say that the neutrinos are in states with definite flavor in a given moment of time if their wave functionsΨ i (i = 1..3) in the flavor representation take the form We will use the notationp = p µ γ µ . In the momentum representation the Greens function may be written as follows If we introduce the notations (1 + γ 5 )(aI + P (e) ) ∓ M , (4.2) the Greens function take the form It can be shown that Here, operator S describes the projection of neutrino spin on its canonical momentum in the matter rest frame. In the reference frame, where the medium is moving, it takes the form We will search for the stationary solutions in the form where the spinor Ψ 0 is an arbitrary 8-component spinor, which does not depend on the coordinates of the event space. Here the component of the 4-momentum p 0 is the neutrino energy, and p is the canonical momentum of the neutrino multiplet. It can be shown that for the moving medium the solutions may be found as the eigenfunctions of the operator which describes the polarization state of the neutrino For the matter at rest f µ = {f 0 , 0, 0, 0} the neutrino helicity is conserved, and the 8-component spinor (5.2) can be chosen as the eigenfunction of the helicity operator.
We will use the standard representation of the γ-matrices. 8-component spinors, which describe the neutrino with definite helicity, take the form The matrix equation on the coefficients A ζ n , B ζ n (n = 1, 2) follows from the neutrino evolution equation.
If the matter is unpolarized, then the dispersion law takes the form where ξ is either 1 or −1.
In the limit f 0 → 0 in the mass representation the solutions of the evolution equation are the wave functions of the mass states In the flavor representation the stationary solutions with the momentum p for ξ = 1 and ξ = −1 take the form (5.16) where θ m = θ + φ is the effective mixing angle in matter, which is defined as follows Here ψ 0 is a constant bispinor, e j is an arbitrary unit vector in the three dimensional vector space over the field of complex numbers, s µ 0 being the 4-vector of neutrino polarization, which satisfies the condition (us 0 ) = 0.
May, 28, 2018 In the flavor representation the solution of this type takes the form Here, the following notations are used (6.4) q µ is the kinetic momentum of the neutrino, q 2 = m 2 .
A. V. Chukhnova, A. E. Lobanov (MSU) May, 28, 2018 In the ultrarelativistic limit the 4-velocity of the neutrino u µ = {u 0 , u}, which is connected to the kinetic momentum as follows u µ = q µ /m, can be related to the coordinates of the particle as follows x µ = u µ τ. Let us consider the spin-flavor transitions between the states with definite flavors. The projectors on these states are given by the following matrices Let's assume that in each these states the neutrino has definite helicity, i. e.
The probability in this case is given by the following expression (7.8) If we assume u 0 ≈ |u|, then for the medium at rest (7.9) where θ eff is the effective mixing angle in matter.
C ±1 = cos τ Z ±1 /2 , S ±1 = sin τ Z ±1 /2 , ω = R(1/2 + a). Generally, these spin-flavor transition probabilities are characterized by six frequences. If u 0 ≈ |u| then is the flavor oscillation length in vacuum, and the parameter is the flavor oscillation length in matter. Four combinational oscillation lengths (7.13) arise due to correlations between flavor transitions and the spin rotation.
The number of such combinational lengths under certain conditions can be equal to two.
The spin-flip probability W is the sum of W 2 and W 4 W = 1 2 A (A 1 (1−cos ω 1 τ )+A 2 (1−cos ω 2 τ )+A 3 (1−cos ω 3 τ )+A 4 (1−cos ω 4 τ )), does not depend on the medium density. It depends on the 4-velocities of the medium v µ and the neutrino u µ . Here, ϑ is the angle between the vectors of the velocities of the medium and the neutrino. If u > v, then the maximum value of this amplitude is obtained when It is equal to If u < v, the maximum value of this amplitude is obtained when

Conclusion
We find the Greens function of neutrino in dense medium.
We show that the stationary states in medium differ from the mass states.
The wave function, describing the state with a definite flavor in matter can be constructed as a linear combination of the stationary states. The coefficients in this linear combination depend on the mixing angle in matter.
We obtain the wave functions of the spin-flavor coherent states, which in quasi-classical limit become similar to the solutions of the quasi-classical evolution equation. These solutions describe the neutrino with definite velocity.
A. V. Chukhnova, A. E. Lobanov (MSU) May, 28, 2018 We obtain the spin-flip probability for ultrarelativistic neutrinos. The pattern of such transitions depends on the initial flavor state of the neutrino, as well as on the density of the medium and the velocities of neutrino and the medium.
There is no spin rotation for the medium at rest.
The spin-flip probability is non-zero only if the directions of neutrino propagation and the movement of the medium are close (but not equal) to each other.