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 aequiv 2sqrt{2}kern0.5em {G}_{mathrm{F}}{N}_eE to be an arbitrary scale-like variable with Ne 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 {tilde{m}}_i (for i = 1, 2, 3). Given the standard parametrization of V , the RGEs for left{{tilde{theta}}_{12},kern0.5em {tilde{theta}}_{13},kern0.5em {tilde{theta}}_{23},kern0.5em tilde{delta}right} 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.
Highlights
JHEP05(2018)015 crucially important when the neutrino beam propagates in the Earth matter for a long distance
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
In this paper we emphasize that the dependence of the effective mixing parameters Vαi and m2i on the matter term a can perfectly be described by a complete set of differential equations, which are analogous to the renormalizationgroup equations (RGEs) associated with the dependence of fundamental parameters on the renormalization energy scale or distance in quantum field theories [18, 19], solid-state physics [20, 21] and other fields of modern physics [22]
Summary
The essential idea of ours is to study the dependence of the flavor mixing parameters on the scale-like matter term a by following the normal RGE approach. Note that eq (2.11) can be regarded as the formal (integral) solutions to the RGEs of |Vei|2 in eq (2.9) with the mixing matrix elements |Uei|2 and neutrino masses m2i in vacuum as initial conditions. The central results for the RGEs of the leptonic flavor mixing matrix V and neutrino masses mi in matter are given in eqs. It should be stressed that the evolution of |Vμi|2 is qualitatively identical to that of |Vτi|2 for i = 1, 2, 3, comparing the plots in the second row and those in the third row of figure 1 This behavior can be well understood by noticing that the muon and tau flavors are indistinguishable, since muon and tau neutrinos (antineutrinos) experience only the universal neutral-current interactions in ordinary matter. The numerical results in the IO case or those for antineutrinos in both cases can be understood by studying the appearance of the MSW resonances
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