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Interactions of Neutrino Wave Packets

Michael J. Cervia

TL;DR

The work investigates neutrino–neutrino interactions within the low-energy Standard Model EFT, formulating the problem as a lattice in momentum space and then examining its continuum limit in the center-of-momentum frame. It demonstrates that the plane-wave continuum limit is effectively trivial unless finite wave-packet widths are incorporated, and shows how wave-packet size controls the balance between forward and non-forward scattering, with a cross section scaling Sigma_nunu ~ G_F^2 sigma_p^4 / E^2 that connects to the standard weak cross section when sigma_p ~ E. A renormalized momentum-space volume, phi, emerges as a key quantity determining interaction time scales and coherence, with phi ~ |p_tot|^3 / (24 pi^2). The results bridge lattice and continuum treatments, clarify when neutrino self-interactions are nonzero, and provide guidance for modeling collective oscillations in dense environments, while outlining future work on many-body effects, antineutrinos, and richer flavor dynamics.

Abstract

The low energy effective field theory of interacting neutrinos derived from the Standard Model may be framed as a pointlike interaction and thereby modeled on a lattice of neutrino momenta. We identify a path to take a continuum limit of this lattice problem in the Center of Momentum frame. In this limit, the weak interaction is found to become trivial between incoming plane waves describing ultrarelativistic particles, unless finite neutrino wave packet sizes are taken into consideration. We follow up with an analytic treatment of interacting neutrino wave packets, demonstrating the importance of the wave packet size for characterizing neutrino-neutrino scattering in dense environments.

Interactions of Neutrino Wave Packets

TL;DR

The work investigates neutrino–neutrino interactions within the low-energy Standard Model EFT, formulating the problem as a lattice in momentum space and then examining its continuum limit in the center-of-momentum frame. It demonstrates that the plane-wave continuum limit is effectively trivial unless finite wave-packet widths are incorporated, and shows how wave-packet size controls the balance between forward and non-forward scattering, with a cross section scaling Sigma_nunu ~ G_F^2 sigma_p^4 / E^2 that connects to the standard weak cross section when sigma_p ~ E. A renormalized momentum-space volume, phi, emerges as a key quantity determining interaction time scales and coherence, with phi ~ |p_tot|^3 / (24 pi^2). The results bridge lattice and continuum treatments, clarify when neutrino self-interactions are nonzero, and provide guidance for modeling collective oscillations in dense environments, while outlining future work on many-body effects, antineutrinos, and richer flavor dynamics.

Abstract

The low energy effective field theory of interacting neutrinos derived from the Standard Model may be framed as a pointlike interaction and thereby modeled on a lattice of neutrino momenta. We identify a path to take a continuum limit of this lattice problem in the Center of Momentum frame. In this limit, the weak interaction is found to become trivial between incoming plane waves describing ultrarelativistic particles, unless finite neutrino wave packet sizes are taken into consideration. We follow up with an analytic treatment of interacting neutrino wave packets, demonstrating the importance of the wave packet size for characterizing neutrino-neutrino scattering in dense environments.
Paper Structure (8 sections, 41 equations, 3 figures)

This paper contains 8 sections, 41 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Effective Field Theory of the Weak Interaction Between Ultrarelativistic Particles
  3. Neutrino Interaction as a Lattice Problem
  4. Plane Wave Neutrinos Interacting in the Continuum
  5. Interacting Neutrino Wave Packets
  6. Ultrarelativistic Particles and Weak Interactions
  7. Conclusions
  8. Volume of Outgoing Momentum Space

Figures (3)

  • Figure 1: The space of allowed momenta resulting from the incoming state $\ket{p_0,-p_0}$ discretized to $M=8$ evenly spaced pairs.
  • Figure 2: Oscillations over (unitless) $G't$ time according to Eqs. \ref{['eq:ham-com-lat']} and \ref{['eq:time-evo-lat']} in the total occupancy numbers per momentum in the lattice of the CoM frame. The incoming momentum state $\ket{p_0,-p_0}$ produces all possible outgoing states with equal likelihood, so we display just $p_1\neq p_0$ for brevity. Shown here are results from allowing for $M=4$ (top) and $M=8$ (bottom) evenly spaced pairs of outgoing momenta in simulations. In all simulations, a dimensionless time step value of $G'\delta=0.002$ is used.
  • Figure 3: In both plots, we identify the trough in oscillations for the occupancy of the initial momentum $p_0$, $n(p_0)$, by the closest time value in numerical simulations. In all simulations, a dimensionless time step value of $G'\delta=0.002$ is used. This method produces uncertainties in measured time values and consequently in survival probabilities of simulations for each plot. Notably, numerical simulations per a program adapted from that of Ref. PhysRevD.110.123028 are in excellent agreement with formulae derived form Eq. \ref{['eq:time-evo-lat']} (depicted as solid trend lines in each plot.) Top: We illustrate a trend in the survival probability of an incoming momentum state $p_0$ as a function of the number $M$ of allowed outgoing momentum as a result of the neutrino-neutrino interaction. In fact, as we refine the discretization of our lattice in the CoM, we find this survival probability to eventually become constant (at 1) in time for $M\to\infty$. Bottom: We illustrate a trend in the time required to reach the first trough in the occupancy of the incoming momentum state $p_0$ as a function of the number $M$ of allowed outgoing momentum as a result of the neutrino-neutrino interaction. We find that the oscillation frequency grows linearly with $M$, and correspondingly this required time value $t_{\min}$ trends to zero.