Three Neutrino Oscillations in Matter

Following similar approaches in the past, the Schrodinger equation for three neutrino propagation in matter of constant density is solved analytically by two successive diagonalizations of 2x2 matrices. The final result for the oscillation probabilities is obtained directly in the conventional parametric form as in the vacuum but with explicit simple modification of two mixing angles ($\theta_{12}$ and $\theta_{13}$) and mass eigenvalues.

Following similar approaches in the past, the Schrodinger equation for three neutrino propagation in matter of constant density is solved analytically by two successive diagonalizations of 2x2 matrices. The final result for the oscillation probabilities is obtained directly in the conventional parametric form as in the vacuum but with explicit simple modification of two mixing angles (θ12 and θ13) and mass eigenvalues.

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The MSW effect [1] for the neutrino propagation in matter attracts a lot of experimental and theoretical attention. Most recently, the discussion is focused on the DUNE experiment [2].
On the theoretical side, a large number of numerical simulations of the MSW effect in matter with a constant or varying density has been performed. Although, in principle, sufficient for comparing the theory predictions with experimental data, they do not provide a transparent physical interpretation of the experimental results. Therefore, several authors have also published analytical or semi-analytical solutions to the Schroedinger equation for three neutrino propagation in matter of constant density, in various perturbative expansions [3][4][5]. The complexity of the calculation, the transparency of the final result and the range of its applicability depend on the chosen expansion parameter.
In this short note we solve the Schroedinger equation in matter with constant density, using the approximate seesaw structure of the full Hamiltonian in the electroweak basis. This way one can diagonalize the 3x3 matrix by two successive diagonalizations of 2x2 matrices (similar approaches have been used in the past, in particular in ref. [4] and [5]). We specifically have in mind the parameters of the DUNE experiment but our method is applicable for their much wider range. The final result for the oscillation probabilities is obtained directly in the conventional parametric form as in the vacuum but with modified two mixing angles and mass eigenvalues [11], similarly to the well known results for the two-neutrino propagation in matter. The three neutrino oscillation probabilities in matter have been presented in the same form as here in the recent ref. [6], where the earlier results obtained in ref. [5] are rewritten in this form. The form of our final results can also be obtained after some simplifications from ref. [4]. Our approach can be easily generalized to non-constant matter density.
The starting point is the Schroedinger equation where H is the Hamiltonian in matter. In the electroweak basis it reads The matrix U is the neutrino mixing matrix in the vacuum. The mass squared differences are defined as ∆m 2 ⊙ ≡ m 2 2 − m 2 1 (≈ 7.5 10 −5 eV 2 ) and ∆m 2 a ≡ m 2 3 − m 2 1 (≈ ±2.5 10 −3 eV 2 , positive sign is for normal mass ordering and negative sign for inverted one). Here V (x) is the neutrino weak interaction potential energy V = √ 2G F N e (N e is electron number density) and we take it in this section to be x-independent. The neutrino oscillation probabilities are determined by the S-matrix elements For a constant V and in order to obtain our results in the same form as for the oscillation probabilities in the vacuum, it is convenient to rewrite the S-matrix elements as follows: The matrix H m is the Hamiltonian in matter in the mass eigenstate basis: and the U m is the neutrino mixing matrix in matter. Defining φ 21 = (H 2 − H 1 )L and φ 31 = (H 3 − H 1 )L, we can write The remaining task is to find the eigenvalues of H and the mixing matrix U m : It is convenient to do it in two steps, first calculating the hamiltonian in a certain auxiliary basis. This way, to an excellent approximation, we can diagonalize the 3x3 matrix by two successive diagonalizations of the 2x2 matrices. The auxiliary basis [7,8] is defined by the following equation and the rotations O ij are defined by the decomposition of the mixing matrix U in the vacuum (see eq. 2) as follows: where (c 12 ≡ cos θ 12 , s 12 ≡ sin θ 12 etc). The matrices O ij are orthogonal matrices. It is more convenient to rewrite the matrix U in another form Using eqs. (2,8) we obtain (The term s 2 12 ∆m 2 ⊙ 2E has been subtracted from the diagonal elements; it gives an overal phase in the S-matrix elements.) [12] This matrix has a see-saw structure, with the (13), (31) elements much smaller than the (33) element and can be put in an almost diagonal form by two rotations After the first rotation we have and We can safely neglect the (23), (32) elements which are generated after the first rotation (see Appendix A) and diagonalize the remaining 2x2 sub-matrix with the second rotation sin 2θ m 12 = cos θ ′ 13 sin 2θ 12 (cos 2θ 12 − ǫ ⊙ ) 2 + cos 2 θ ′ 13 sin 2 2θ 12 The eigenvalues of H are Finally, for the mixing matrix U m in matter we obtain cos θ ′ 13 sin 2θ 12 (cos 2θ 12 − ǫ ⊙ ) 2 + cos 2 θ ′ 13 sin 2 2θ 12 , cos 2θ m 12 = cos 2θ 12 − ǫ ⊙ (cos 2θ 12 − ǫ ⊙ ) 2 + cos 2 θ ′ 13 sin 2 2θ 12

(27)
In summary the mixing matrix in matter, U m , is given by the following change of the parameters from the vacuum solution: The mass eigenvalues are given by eqs.21,22,23. The ocillation probabilities P να→ν β (α, β = e, µ, τ ) have the same forms as for the vacuum oscillations with mass eigenstates as above and with replacements θ 12 → θ m 12 and θ 13 → θ m 13 . This approximate solution is valid for all energies. Numerically our result is identical to the approximation of two angles rotation in [4] and the 0th order result of [5]. For anti-neutrino oscillations Pν α →ν β , V→ -V and δ → −δ. For normal mass hierarchy ∆m 2 a is positive and for inverted mass hierarchy it is negative.
For normal hierarchy and for neutrinos sin θ ′ 13 sin 2θ12 2 ∆m 2 ⊙ L 2E φ31 sin θ m 12 reached its maximal value 0.4 % at second resonance. at E=6.5 GeV it is about 0.1%. Other term proportional to cos θ m 12 is much smaller for all energies. Appendix B For easy reference we collect here the formulae for oscillation probabilities in vacuum