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Register a new userContinuous and Discrete Symmetries of Renormalization Group Equations for Neutrino Oscillations in Matter
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Shun Zhou
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Three-flavor neutrino oscillations in matter can be described by three effective neutrino masses $widetilde{m}^{}_i$ (for $i = 1, 2, 3$) and the effective mixing matrix $V^{}_{alpha i}$ (for $alpha = e, mu, tau$ and $i = 1, 2, 3$). When the matter parameter $a equiv 2sqrt{2} G^{}_{rm F} N^{}_e E$ is taken as an independent variable, a complete set of first-order ordinary differential equations for $widetilde{m}^2_i$ and $|V^{}_{alpha i}|^2$ have been derived in the previous works. In the present paper, we point out that such a system of differential equations possesses both the continuous symmetries characterized by one-parameter Lie groups and the discrete symmetry associated with the permutations of three neutrino mass eigenstates. The implications of these symmetries for solving the differential equations and looking for differential invariants are discussed.
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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 $a equiv 2sqrt{2} G^{}_{rm F} N^{}_e E$ 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 $widetilde{m}^{}_i$ (for $i = 1, 2, 3$). Given the standard parametrization of $V$, the RGEs for ${widetilde{theta}^{}_{12}, widetilde{theta}^{}_{13}, widetilde{theta}^{}_{23}, widetilde{delta}}$ 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 $mu$-$tau$ asymmetries, the off-diagonal asymmetries and the sides of unitarity triangles of $V$ are also obtained as a by-product.
This paper aims at presenting the first steps towards a formulation of the Exact Renormalization Group Equation in the Hopf algebra setting of Connes and Kreimer. It mostly deals with some algebraic preliminaries allowing to formulate perturbative renormalization within the theory of differential equations. The relation between renormalization, formulated as a change of boundary condition for a differential equation, and an algebraic Birkhoff decomposition for rooted trees is explicited.
Expressions for neutrino oscillations contain a high degree of symmetry, but typical forms for the oscillation probabilities mask these symmetries. We elucidate the $2^7=128$ symmetries of the vacuum parameters and draw connections to the choice of definitions of the parameters as well as interesting degeneracies. We also show that in the presence of matter an additional set of $2^7=128$ symmetries exist of the matter parameters for a total of $2^{14}=16,384$ symmetries of the vacuum and/or matter parameters in the oscillation probabilities in matter. Due to the complexity of the exact expressions for neutrino oscillations in matter, we show that under certain assumptions, approximate expressions have at most $2^6=64$ additional symmetries of the matter parameters for a total of $2^{13}=8,192$ symmetries. We investigate which of these symmetries apply to numerous approximate expressions in the literature and show that a more careful consideration of symmetries improves the precision of approximations.
We consider a class of models for the relativistic covariant wave packets which can be used as asymptotically free in and out states in the quantum field theoretical formalisms for description of the neutrino flavor oscillation phenomenon. We demonstrate that the new asymmetric wave packet (AWP) is an appropriate alternative for the more convenient symmetric wave packets, like the so-called relativistic Gaussian packet (RGP) widely used in the QFT-based approaches to neutrino oscillations. We show that RGP is not a particular case of AWP, although many properties of these models are almost identical in the quasistable regime. We discuss some features of AWP distinguishing it from RGP.
We analyze status of ${bf C}$, ${bf P}$ and ${bf T}$ discrete symmetries in application to neutron-antineutron transitions breaking conservation of baryon charge ${cal B}$ by two units. At the level of free particles all these symmetries are preserved. This includes ${bf P}$ reflection in spite of the opposite internal parities usually ascribed to neutron and antineutron. Explanation, which goes back to the 1937 papers by E. Majorana and by G. Racah, is based on a definition of parity satisfying ${bf P}^{2}=-1$, instead of ${bf P}^{2}=1$, and ascribing $ {bf P}=i$ to both, neutron and antineutron. We apply this to ${bf C}$, ${bf P}$ and ${bf T}$ classification of six-quark operators with $|Delta {cal B} |=2$. It allows to specify operators contributing to neutron-antineutron oscillations. Remaining operators contribute to other $|Delta {cal B} |=2$ processes and, in particular, to nuclei instability. We also show that presence of external magnetic field does not induce any new operator mixing the neutron and antineutron provided that rotational invariance is not broken.
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