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In this paper we review different expansions for neutrino oscillation probabilities in matter in the context of long-baseline neutrino experiments. We examine the accuracy and computational efficiency of different exact and approximate expressions. We find that many of the expressions used in the literature are not precise enough for the next generation of long-baseline experiments, but several of them are while maintaining comparable simplicity. The results of this paper can be used as guidance to both phenomenologists and experimentalists when implementing the various oscillation expressions into their analysis tools.
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Motivated by tremendous progress in neutrino oscillation experiments, we derive a new set of simple and compact formulas for three-flavor neutrino oscillation probabilities in matter of a constant density. A useful definition of the $eta$-gauge neutrino mass-squared difference $Delta^{}_* equiv eta Delta^{}_{31} + (1-eta) Delta^{}_{32}$ is introduced, where $Delta^{}_{ji} equiv m^2_j - m^2_i$ for $ji = 21, 31, 32$ are the ordinary neutrino mass-squared differences and $0 leq eta leq 1$ is a real and positive parameter. Expanding neutrino oscillation probabilities in terms of $alpha equiv Delta^{}_{21}/Delta^{}_*$, we demonstrate that the analytical formulas can be remarkably simplified for $eta = cos^2 theta^{}_{12}$, with $theta_{12}^{}$ being the solar mixing angle. As a by-product, the mapping from neutrino oscillation parameters in vacuum to their counterparts in matter is obtained at the order of ${cal O}(alpha^2)$. Finally, we show that our approximate formulas are not only valid for an arbitrary neutrino energy and any baseline length, but also still maintaining a high level of accuracy.
We perform the flavour $SU(3)$ analysis of the recently discovered $Omega(2012)$ hyperon. We find that well known (four star) $Delta(1700)$ resonance with quantum numbers of $J^P=3/2^-$ is a good candidate for the decuplet partner of $Omega(2012)$ if the branching for the three-body decays of the latter is not too large $le 70$%. That implies that the quantum numbers of $Omega(2012)$ are $I(J^P)=0(3/2^-)$. The predictions for the properties of still missing $Sigma$ and $Xi$ decuplet members are made. We also discuss the implications of the ${ overline{ K} Xi(1530)}$ molecular picture of $Omega(2012)$. Crucial experimental tests to distinguish various pictures of $Omega(2012)$ are suggested.
We construct a new perturbative framework to describe neutrino oscillation in matter with the unique expansion parameter epsilon, which is defined as Delta m^2_{21} / Delta m^2_{ren} with the renormalized atmospheric Delta m^2_{ren} equiv Delta m^2_{31} - s^2_{12} Delta m^2_{21}. It allows us to derive the maximally compact expressions of the oscillation probabilities in matter to order epsilon in the form akin to those in vacuum. This feature allows immediate physical interpretation of the formulas, and facilitates understanding of physics of neutrino oscillations in matter. Moreover, quite recently, we have shown that our three-flavor oscillation probabilities P( u_alpha rightarrow u_beta) in all channels can be expressed in the form of universal functions of L/E. The u_e disappearance oscillation probability P( u_e rightarrow u_e) has a special property that it can be written as the two-flavor form which depends on the single frequency. This talk is based on the collaborating work with Stephen Parke [1].
In the work of Mukhin and Varchenko from 2002 there was introduced a Wronskian map from the variety of full flags in a finite dimensional vector space into a product of projective spaces. We establish a precise relationship between this map and the Plucker map. This allows us to recover the result of Varchenko and Wright saying that the polynomials appearing in the image of the Wronsky map are the initial values of the tau-functions for the Kadomtsev-Petviashvili hierarchy.
One can hardly believe that there is still something to be said about cubic equations. To dodge this doubt, we will instead try and say something about Sylvester. He doubtless found a way to solve cubic equations. As mentioned by Rota, it was the only method in this vein that he could remember. We realize that Sylvesters magnificent approach for reduced cubic equations boils down to an easy identity.
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