Renormalization-group equations of neutrino masses and flavor mixing parameters in matter

Abstract

Abstract:We borrow the general idea of renormalization-group equations (RGEs) to understand how neutrino masses and flavor mixing parameters evolve when neutrinos propagate in a medium, highlighting a meaningful possibility that the genuine flavor quantities in vacuum can be extrapolated from their matter-corrected counterparts to be measured in some realistic neutrino oscillation experiments. Taking the matter parameter a ≡ 2 √ 2 G F N e E to be an arbitrary scale-like variable with N e being the net electron number density and E being the neutrino beam energy, we derive a complete set of differential equations for the effective neutrino mixing matrix V and the effective neutrino masses m i (for i = 1, 2, 3). Given the standard parametrization of V , the RGEs for { θ 12 , θ 13 , θ 23 , δ} in matter are formulated for the first time. We demonstrate some useful differential invariants which retain the same form from vacuum to matter, including the well-known Naumov and Toshev relations. The RGEs of the partial µ-τ asymmetries, the off-diagonal asymmetries and the sides of unitarity triangles of V are also obtained as a by-product.

Sum rules and asymptotic behaviors of neutrino mixing in dense matter

“…[8,9,10,11,12,13,14]), or in describing matter effects on neutrino mixing and CP violation with the help of a language similar to the renormalization-group equations (see, e.g., Refs. [15,16,17,18]).…”

“…This is the first time that a full and analytical understanding of these matter-corrected quantities in the A → ∞ limit has been achieved purely in terms of the fundamental quantities |U αi | 2 and ∆ ji ≡ m 2 j − m 2 i , although their asymptotic behaviors were partly observed in some previous numerical calculations (see, e.g., Refs. [17,22]).…”

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

“…Recently, a complete set of differential equations of the effective neutrino masses m i and the effective neutrino mixing matrix elements V αi (for i = 1, 2, 3 and α = e, µ, τ ) in ordinary matter with respect to matter parameter a ≡ 2 √ 2G F N e E, where E is the neutrino beam energy, G F is the Fermi constant and N e is the net electron number density, have been derived in Refs. [5,6] to describe the connection between the fundamental neutrino oscillation parameters in vacuum and those effective ones in matter. In particular, a close analogy of these differential equations with the renormalization-group equations (RGEs) has been made in Ref.…”

“…In particular, a close analogy of these differential equations with the renormalization-group equations (RGEs) has been made in Ref. [6]. In the standard parametrization of the effective mixing matrix V in matter [7], the RGEs for three effective mixing angles { θ 12 , θ 13 , θ 23 } and the effective CPviolating phase δ are found to be [6] d θ 12 da = 1 2 sin 2 θ 12 ∆ −1 21 cos 2 θ 13 − ∆ 21 ∆ −1 31 ∆ −1 32 sin 2 θ 13 ,…”

“…d δ da = − ∆ 21 ∆ −1 31 ∆ −1 32 sin 2 θ 12 sin θ 13 sin δ cot 2 θ 23 ; (4) and the RGEs for the effective neutrino mass-squared differences ∆ ij ≡ m 2 i − m 2 j for ij = 21, 31, 32 are given by [6] d ∆ 21 da = − cos 2 θ 13 cos 2 θ 12 ,…”