[Paper Review] Interpolating wave packets and composite wave functions in QFT and neutrino oscillation problem

S. É. Korenblit, D. Taychenachev+1 more|arXiv (Cornell University)|Dec 18, 2017

Neutrino Physics Research15 references4 citations

TL;DR

This paper presents a relativistically covariant wave packet formalism for neutrino oscillations, introducing an 'interpolating' wave packet that smoothly bridges momentum- and coordinate-space states via a Lorentz-invariant width. It defines a composite wave function for intermediate neutrinos, resolving causality and covariance issues in macroscopic Feynman diagrams by providing a one-packet integral representation that reduces to on-shell behavior asymptotically, with explicit overlap function calculations for pion decay vertices matching known limits.

ABSTRACT

A consistent constructive covariant description of neutrino flavour transition amplitude in vacuum is presented. To this end a special generalized relativistic wave packet is constructed with correct extension onto the higher spins. This packet is uniquely defined as an `interpolating' wave packet, which by means of relativistically invariant `width' accurately interpolates between the states localized in momentum space and in coordinate space. The wave packet is unambiguously determined by analytical properties of Wightman functions in complex coordinate space naturally connected with its minimization properties. The packet gives natural relativistic generalization of non relativistic Gaussian wave packet but it contains covariant states of particle (antiparticle) only with positive (negative) energy sign and propagates without their mixing and without changing of its relativistically invariant width. For the diagrammatic treatment of oscillation with the use of these wave packets for external particles, the notion of covariant composite wave function for intermediate neutrino is introduced. It strictly and naturally connects both oscillation pictures, giving an effective language for detailed description of this process, and resolves the problems with causality and with covariant equal time prescription for the intermediate neutrino picture. It is closely related to overlap function of neutrino creation/detection vertices, elucidating a covariant meaning of the `pole integration' procedure. Their space-time asymptotic behaviour in narrow-packets approximation naturally conforms with such approximation of one-packet state and with the asymptotic behaviour of oscillation amplitude. The respective overlap function is explicitly calculated for two-packet example of pion decay vertex. Its correspondence and difference with previous approximate calculations is analyzed.

Motivation & Objective

• To resolve the long-standing problem of Lorentz covariance and causality in neutrino oscillation descriptions using wave packets.
• To construct a relativistic generalization of the non-relativistic Gaussian wave packet that maintains positive-energy particle states without mixing.
• To introduce a covariant composite wave function for intermediate neutrinos that unifies the vacuum and on-shell pictures.
• To provide a rigorous, covariant framework for the overlap function in neutrino creation/detection vertices, replacing ad hoc 'pole integration' procedures.
• To demonstrate consistency with asymptotic behavior and known approximations in the narrow-packet and on-shell limits.

Proposed method

• Constructs a generalized relativistic wave packet via interpolation between momentum- and coordinate-space states using analytical properties of Wightman functions.
• Defines the wave packet as an 'interpolating' state with a relativistically invariant width parameter, ensuring unambiguous construction from minimal physical principles.
• Introduces a covariant composite wave function for intermediate neutrinos as a superposition of on-shell wave packets with varying centers and widths.
• Derives a one-packet integral representation for the on-shell composite wave function using Bessel function identities and Lorentz-invariant parametrization.
• Applies the formalism to pion decay vertices, calculating the overlap function explicitly and comparing it with previous approximate treatments.
• Demonstrates that the asymptotic space-time behavior of the composite wave function matches both the narrow-packet approximation and the standard oscillation amplitude.

Experimental results

Research questions

• RQ1 How can a relativistically covariant wave packet be consistently defined to interpolate between momentum- and coordinate-space states without mixing particle and antiparticle states?
• RQ2 What is the covariant meaning of the 'pole integration' procedure in neutrino oscillation amplitudes, and how can it be replaced by a physically meaningful wave packet construction?
• RQ3 How does the composite wave function for intermediate neutrinos unify the vacuum and on-shell pictures in a way that preserves causality and Lorentz invariance?
• RQ4 What is the exact form of the overlap function between neutrino creation and detection vertices in the context of this wave packet formalism?
• RQ5 How do the asymptotic space-time behaviors of the composite wave function and the standard oscillation amplitude compare in the narrow-packet and on-shell limits?

Key findings

• The interpolating wave packet is uniquely defined by the analytical structure of Wightman functions and possesses a relativistically invariant width, ensuring consistent Lorentz transformation properties.
• The composite wave function for intermediate neutrinos is expressed as a superposition of on-shell wave packets with different centers and widths, providing a natural covariant generalization of the intermediate state.
• The on-shell limit of the composite wave function is exactly reproduced in the asymptotic regime, confirming consistency with standard oscillation theory.
• The overlap function for a two-packet system (e.g., pion decay) is explicitly calculated and shown to reduce to known results in the narrow-packet and on-shell limits.
• The formalism resolves the causality problem in macroscopic Feynman diagrams by replacing adiabatic switching with a physically grounded wave packet structure.
• In the massless limit (mj → 0), the composite wave function reduces to a finite integral involving modified Bessel functions, preserving normalization and yielding a finite, well-defined expression.