Interactions of Neutrino Wave Packets

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.

💡 Research Summary

The paper investigates neutrino‑neutrino interactions at low energies using the effective field theory (EFT) derived from the Standard Model. Starting from the neutral‑current Z‑boson exchange Lagrangian, the authors integrate out the heavy Z boson to obtain a four‑fermion point‑like interaction with coupling G_F≈10⁻¹¹ MeV⁻². They write down the full momentum‑space Hamiltonian (Eq. 3) that includes both flavor‑exchange and momentum‑transfer structure factors. The spinor form factor f(p,q) reduces in the ultra‑relativistic limit to |f|²=1−b_p·b_q, where b_p is the unit vector of the neutrino momentum.

To make many‑body calculations tractable, the momentum space is discretized on a lattice with spacing A_p. This lattice spacing simultaneously serves as a regularization of the momentum basis and as the inverse of the spatial wave‑packet width σ_x≈1/σ_p. The total spatial volume V=A_p^{-d} thus plays the role of a wave‑packet normalization volume.

The authors focus on a pair of neutrinos in their center‑of‑mass (CoM) frame, initially in the state |p₀,−p₀⟩. Momentum conservation restricts all possible final states to antipodal pairs |p′,−p′⟩ lying on a circle of radius |p₀| in momentum space. In this frame the Hamiltonian factorizes into a flavor‑exchange part H_flav and a momentum‑transfer part H_mom. H_mom becomes an isotropic “all‑to‑all” matrix: every allowed momentum pair couples to every other with the same amplitude G′=√2 G_F/V. Consequently, the time evolution reduces to a simple unitary rotation in the M‑dimensional subspace spanned by the M lattice points on the circle.

Numerical simulations are performed for increasing numbers of lattice points M (4, 8, 16, …). Two clear trends emerge: (i) the minimum occupation probability of the initial momentum state |p₀⟩ approaches unity as M grows, meaning that the probability of actually transferring momentum to another mode diminishes; (ii) the oscillation frequency of the occupations scales linearly with M. Analytically, the eigenvalues of H_mom are proportional to M, while each transition matrix element scales as 1/M, explaining the observed behavior. In the continuum limit (A_p→0, V→∞) the momentum‑transfer Hamiltonian effectively becomes the projector onto the uniform superposition of all momentum states, and the probability for any specific non‑forward transition vanishes. Hence, for true plane waves (infinite spatial extent) the weak interaction between two ultra‑relativistic neutrinos is trivial.

Recognizing that realistic neutrinos are never perfect plane waves, the authors introduce finite wave‑packet sizes explicitly using the Kiers‑Nussinov‑Weiss formalism. They treat Gaussian wave packets characterized by a momentum width σ_p (or equivalently a spatial width σ_x). Analytic evaluation shows two limiting regimes: (a) σ_p ≪ |p| (broad packets) reproduces the standard weak cross‑section σ∼G_F²E² because the overlap factor |f|² suppresses non‑forward contributions; (b) σ_p ≫ |p| (narrow packets) makes the overlap factor essentially unity, allowing sizable non‑forward scattering. Thus, the cross‑section depends sensitively on the packet size: narrow packets enhance non‑forward scattering, while broad packets reduce it to the familiar forward‑scattering dominated picture.

The paper also discusses scale separation. The weak potential falls off over a distance r∼G_F E⁻², which is tiny compared to the de Broglie wavelength of MeV‑scale neutrinos. Therefore, only when the spatial packet is comparable to or smaller than this interaction range does the potential affect the dynamics. In dense astrophysical environments (core‑collapse supernovae, neutron‑star mergers) where neutrino densities are enormous, the wave‑packet size can be limited by production and scattering processes, potentially placing the system in the narrow‑packet regime where non‑forward neutrino‑neutrino scattering becomes non‑negligible.

In the concluding sections the authors summarize their findings: (1) a lattice formulation of neutrino‑neutrino interactions clarifies how the continuum limit suppresses momentum transfer for plane waves; (2) finite wave‑packet effects restore non‑trivial scattering, with the magnitude controlled by σ_x; (3) numerical simulations corroborate the analytic continuum limit; (4) the results suggest that existing mean‑field treatments, which assume infinite‑extent plane waves and focus on forward scattering, may miss important physics in environments where wave packets are short. They propose future work to incorporate wave‑packet dynamics into many‑body quantum simulations and to explore the interplay with possible non‑standard neutrino self‑interactions.