# The role of the Euclidean signature in lattice calculations of quasi-distributions and other non-local matrix elements

## TL;DR

This paper demonstrates that spatially non-local matrix elements calculated in Euclidean lattice QCD are equivalent to those in Minkowski space, clarifying their relation for nucleon structure studies.

## Contribution

It provides a rigorous proof of the equivalence between Euclidean and Minkowski matrix elements for non-local operators in lattice QCD, including perturbative and all-order arguments.

## Key findings

- Euclidean and Minkowski matrix elements are identical for non-local operators.
- Perturbative calculations confirm the equivalence.
- Proof established for all orders in perturbation theory.

## Abstract

Lattice quantum chromodynamics (QCD) provides the only known systematic, nonperturbative method for first-principles calculations of nucleon structure. However, for quantities such as lightfront parton distribution functions (PDFs) and generalized parton distributions (GPDs), the restriction to Euclidean time prevents direct calculation of the desired observable. Recently, progress has been made in relating these quantities to matrix elements of spatially nonlocal, zero-time operators, referred to as quasidistributions. Even for these time-independent matrix elements, potential subtleties have been identified in the role of the Euclidean signature. In this work, we investigate the analytic behavior of spatially non-local correlation functions and demonstrate that the matrix elements obtained from Euclidean lattice QCD are identical to those obtained using the LSZ reduction formula in Minkowski space. After arguing the equivalence on general grounds, we also show that it holds in a perturbative calculation, where special care is needed to identify the lattice prediction. Finally we present a proof of the uniqueness of the matrix elements obtained from Minkowski and Euclidean correlation functions to all order in perturbation theory.