# Phase ambiguity of the measure for continuum Majorana fermions

## TL;DR

This paper investigates the phase ambiguity of the pfaffian for continuum Majorana fermions, resolves it through a lattice approach, and explores implications for the theory's theta angle and chiral perturbation theory.

## Contribution

It introduces a non-perturbative lattice definition to resolve pfaffian phase ambiguity and clarifies the relation between fermion redefinitions and the theta angle.

## Key findings

- Phase of the pfaffian depends on basis choice in continuum.
- Lattice formulation provides a natural resolution to the phase ambiguity.
- Re-expressing Dirac fermions as Majorana fermions affects the effective theta angle.

## Abstract

Integrating over a continuum Majorana fermion formally yields a functional pfaffian. We show that the phase of this pfaffian is ambiguous, as it depends on the choice of basis. This ambiguity is naturally resolved within a non-perturbative lattice definition, allowing us to discuss the relation between the phase of the lattice pfaffian and the effective $\theta$ angle of the theory. We also resolve an apparent paradox regarding the induced $\theta$ angle when a theory of $N$ Dirac fermions in a real representation of the gauge group is re-expressed in terms of $2N$ Majorana fermions. We discuss how all this is reflected in chiral perturbation theory.