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

TL;DR

The paper tackles the problem of extracting light-front PDFs from lattice QCD, where Euclidean time hinders direct access to lightlike observables. It develops a general framework showing that matrix elements of time-local operators are signature-agnostic and identical when computed from Euclidean correlators or Minkowski LSZ reductions, supported by a perturbative toy model and an all-orders perturbation-theory proof. A key result is that properly defined Euclidean-time correlators (including mixed time-momentum representations) yield the Minkowski matrix elements for quasi-distributions, resolving previously reported tensions. The findings provide a rigorous justification for using Euclidean lattice calculations to obtain PDFs and GPDs, with implications for renormalization and continuum extrapolation.

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.

Figures (5)

• Figure 1: The analytic structure for the finite- and infinite-volume correlation functions, left- and right-hand diagrams, respectively, carrying the quantum numbers of a nucleon at rest in a Euclidean spacetime. In both cases we show the first threshold, where a nucleon-pion pair with nucleon quantum numbers can go on shell. In the infinite-volume limit this introduces a cut, as is shown in the right-hand diagram.
• Figure 2: Perturbative contributions to the momentum-space correlation function involving a single insertion of a space-dislocated operator, depicted by the crossed circles, in (a) perturbative QCD and (b) the scalar toy model considered in the text.
• Figure 3: Analytic structure of the integrand in the (a) Minkowski and (b) Euclidean coordinate space, together with contours for the $I_M$, $I_E$, and $\Delta I$ integrals discussed in the text.
• Figure 4: Analytic continuation of the momentum-space Euclidean correlator from real $P_4$ (left) to the on-shell point (right).
• Figure 5: An example of the analytic continuation of the fourth component of momentum from Euclidean to Minkowski, discussed in the text. The left-hand panel shows the structure of the integrand for some negative, real value of $P_4$. The Wick rotation then is effected by changing the coordinate system while keeping the contour along the real $\widetilde{k}_4^{[\theta]}$ axis. The smaller middle panel shows the original $k_4$ axes midway through the rotation. As indicated by the arrows attached to the square poles in the middle panel, $P_4$ is continued to imaginary values simultaneously as the contour rotates in a way that ensures that poles never cross the contour. The final result, shown in the right-hand panel, is a Minkowski signature integral with pole locations satisfying the standard $i \epsilon$ prescription.