Analytical solutions to renormalization-group equations of effective neutrino masses and mixing parameters in matter

Abstract

Neutrino oscillations in matter can be fully described by six effective parameters, namely, three neutrino mixing angles { θ 12 , θ 13 , θ 23 }, one Dirac-type CP-violating phase δ, and two neutrino mass-squared differences ∆ 21 ≡ m 2 2 − m 2 1 and ∆ 31 ≡ m 2 3 − m 2 1 . Recently, a complete set of differential equations for these effective parameters have been derived to characterize their evolution with respect to the ordinary matter term a ≡ 2 √ 2G F N e E, in analogy with the renormalization-group equations (RGEs) for running parameters. Via series expansion in terms of the small ratio α c ≡ ∆ 21 /∆ c with ∆ c ≡ ∆ 31 cos 2 θ 12 +∆ 32 sin 2 θ 12 , we obtain approximate analytical solutions to the RGEs of the effective neutrino parameters and make several interesting observations. First, at the leading order, θ 12 and θ 13 are given by the simple formulas in the two-flavor mixing limit, while θ 23 ≈ θ 23 and δ ≈ δ are not changed by matter effects. Second, the ratio of the matter-corrected Jarlskog invariant J to its counterpart in vacuum J approximates to J /J ≈ 1/( C 12 C 13 ), where C 12 ≡ 1 − 2A * cos 2θ 12 + A 2 * with A * ≡ a/∆ 21 and C 13 ≡ 1 − 2A c cos 2θ 13 + A 2 c with A c ≡ a/∆ c have been defined. Finally, after taking higher-order corrections into account, we find compact and simple expressions of all the effective parameters, which turn out to be in perfect agreement with the exact numerical results. *

On the properties of the effective Jarlskog invariant for three-flavor neutrino oscillations in matter

“…(1)- (7), following the approach suggested in Ref. [15], and demonstrate that the improvement can be simply achieved by relaxing the approximation of cos 2 θ 13 ≈ 1. Moreover, we apply the improved formulas to the study of the basic properties of the matter-corrected Jarlskog invariant J .…”

“…Let us first explain how to improve the analytical solutions to the RGEs in Ref. [15]. For clarity, we focus only on three-flavor neutrino oscillations in matter in the case of normal neutrino mass ordering (NO) with m 1 < m 2 < m 3 (or ∆ 31 > 0).…”

“…It has been shown in Ref. [15] that the series expansions of ∆ ij will be much simpler if the perturbation parameter is chosen as α c ≡ ∆ 21 /∆ c , which is on the same order of α and thus can be equally good for perturbation calculations. More explicitly, to the first order of α c , we have…”

“…In our previous work [15], we have presented for the first time the analytical solutions to those RGEs with some reasonable approximations and obtained compact and simple expressions of all the effective oscillation parameters. For three-flavor neutrino oscillations in matter, the leptonic CP violation can be characterized by the effective Jarlskog invariant J ≡ ε αβγ ε ijk Im V αi V * αj V * βi V βj , where ε αβγ and ε ijk are the totally-antisymmetric tensors with (α, β, γ) and (i, j, k) being the cyclic permutations of (e, µ, τ ) and (1,2,3), respectively.…”

“…For three-flavor neutrino oscillations in matter, the leptonic CP violation can be characterized by the effective Jarlskog invariant J ≡ ε αβγ ε ijk Im V αi V * αj V * βi V βj , where ε αβγ and ε ijk are the totally-antisymmetric tensors with (α, β, γ) and (i, j, k) being the cyclic permutations of (e, µ, τ ) and (1,2,3), respectively. It has been briefly mentioned that the ratio of the matter-corrected Jarlskog invariant J to the Jarlskog invariant J ≡ ε αβγ ε ijk Im U αi U * αj U * βi U βj in vacuum [16,17], where U stands for the leptonic flavor mixing matrix in vacuum, can be approximately expressed as [15]…”