# Effect of extended neutrino production region on collective oscillations in supernovae

## TL;DR

This paper investigates how the finite size of the neutrino emission region in supernovae affects collective oscillations, revealing a significant suppression and delay of flavor transitions due to averaging effects.

## Contribution

It introduces a new model incorporating emission region size into collective oscillation analysis, quantifies the suppression factor, and derives evolution equations with both numerical and analytical solutions.

## Key findings

- Averaging over the emission region suppresses off-diagonal density matrix elements by ~10^{-10}.
- The delay of flavor transition development depends logarithmically on the suppression factor.
- The emission region size modifies the exponential growth of collective oscillations.

## Abstract

In supernovae neutrinos are emitted from a region with a width $r_{\rm eff}$ of a few kilometers (rather than from a surface of infinitesimal width). We study the effect of integration (averaging) over such an extended emission region on collective oscillations. The averaging leads to additional suppression of the correlation (off-diagonal element of the density matrix) by a factor $ \sim 1/r_{\rm eff} V_e \sim 10^{-10}$, where $V_e$ is the matter potential. This factor enters the initial condition for further collective oscillations and, consequently, leads to a delay of the strong flavour transitions. We justify and quantify this picture using a simple example of collective effects in two intersecting fluxes. We have derived the evolution equation for the density matrix elements integrated over the emission region and solved it both numerically and analytically. For the analytic solution we have used linearized equations. We show that the delay of the development of the instability and the collective oscillations depends on the suppression factor due to the averaging (integration) logarithmically. If the instability develops inside the production region, the integration leads not only to a delay but also to a modification of the exponential grow.