document: The expression for Pc is also easily found in the same basis, where it becomes apparent that the dynamics of conversion in matter depends only on the relative orientation of the eigenstates of the vacuum and matter Hamiltonians. This allows to directly apply the known analytical solutions for Pc, and, upon rotating back, obtain a generalization of these results to the NSI case. For example, the answer for the infinite exponential profile [18,19] A∝exp(−r/r0) becomes Pc=exp[γ(1−cos2θrel)/2]−1exp(γ)−1, where γ≡4πr0Δ=πr0Δm2/Eν. We further observe that since γ⪢1 the adiabaticity violation occurs only when |θ−α|⪡1 and φ≃π/2, which is the analogue of the small-angle MSW [10,20] effect in the rotated basis. The “resonant” region in the Sun where level jumping can take place is narrow, defined by A≃Δ [21]. A neutrino produced at a lower density evolves adiabatically, while a neutrino produced at a higher density may undergo level crossing. The probability Pc in the latter case is given to a very good accuracy by the formula for the linear profile, with an appropriate gradient taken along the neutrino trajectory, (12)Pc≃Θ(A−Δ)e−γ(cos2θrel+1)/2, where Θ(x) is the step function, Θ(x)=1 for x>0 and Θ(x)=0 otherwise. We emphasize that our results differ from the similar ones given in [5,22] in three important respects: (i) they are valid for all, not just small values of α (which is essential for our application), (ii) they include the angle φ, and (iii) the argument of the Θ function does not contain cos2θ, as follows from [21]. We stress that for large values of α and φ≃π/2 adiabaticity is violated for large values of θ.
extractive_keyphrases: adiabaticity violation analytical solutions for pc conversion in matter generalization of these results to the nsi case matter hamiltonians neutrino pc relative orientation of the eigenstates vacuum
abstractive_keyphrases: nsi