# Role of the Euclidean signature in lattice calculations of quasidistributions and other nonlocal matrix elements

## Abstract

Lattice quantum chromodynamics (QCD) provides the only known systematic, nonperturbative method for first-principles calculations of nucleon structure. However, for quantities such as light-front 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. Still, 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 nonlocal correlation functions and demonstrate that the matrix elements obtained from Euclidean lattice QCD are identical to those obtained using the Lehmann-Symanzik-Zimmermann 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. Lastly, we present a proof of the uniqueness of the matrix elements obtained from Minkowski and Euclidean correlation functions to all order in perturbation theory.

- Authors:

- Thomas Jefferson National Accelerator Facility (TJNAF), Newport News, VA (United States)
- Johannes Gutenberg-Univ. Mainz, Mainz (Germany)
- The State Univ. of New Jersey, Piscataway, NJ (United States)

- Publication Date:

- Research Org.:
- Thomas Jefferson National Accelerator Facility (TJNAF), Newport News, VA (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), Nuclear Physics (NP) (SC-26)

- OSTI Identifier:
- 1371541

- Alternate Identifier(s):
- OSTI ID: 1369328

- Report Number(s):
- JLAB-THY-17-2434; DOE/OR/23177-4188; arXiv:1703.06072

Journal ID: ISSN 2470-0010; PRVDAQ

- Grant/Contract Number:
- AC05-06OR23177

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Physical Review D

- Additional Journal Information:
- Journal Volume: 96; Journal Issue: 1; Journal ID: ISSN 2470-0010

- Publisher:
- American Physical Society (APS)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS

### Citation Formats

```
Briceno, Raul A., Hansen, Maxwell T., and Monahan, Christopher J.
```*Role of the Euclidean signature in lattice calculations of quasidistributions and other nonlocal matrix elements*. United States: N. p., 2017.
Web. doi:10.1103/PhysRevD.96.014502.

```
Briceno, Raul A., Hansen, Maxwell T., & Monahan, Christopher J.
```*Role of the Euclidean signature in lattice calculations of quasidistributions and other nonlocal matrix elements*. United States. doi:10.1103/PhysRevD.96.014502.

```
Briceno, Raul A., Hansen, Maxwell T., and Monahan, Christopher J. Tue .
"Role of the Euclidean signature in lattice calculations of quasidistributions and other nonlocal matrix elements". United States.
doi:10.1103/PhysRevD.96.014502. https://www.osti.gov/servlets/purl/1371541.
```

```
@article{osti_1371541,
```

title = {Role of the Euclidean signature in lattice calculations of quasidistributions and other nonlocal matrix elements},

author = {Briceno, Raul A. and Hansen, Maxwell T. and Monahan, Christopher J.},

abstractNote = {Lattice quantum chromodynamics (QCD) provides the only known systematic, nonperturbative method for first-principles calculations of nucleon structure. However, for quantities such as light-front 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. Still, 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 nonlocal correlation functions and demonstrate that the matrix elements obtained from Euclidean lattice QCD are identical to those obtained using the Lehmann-Symanzik-Zimmermann 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. Lastly, we present a proof of the uniqueness of the matrix elements obtained from Minkowski and Euclidean correlation functions to all order in perturbation theory.},

doi = {10.1103/PhysRevD.96.014502},

journal = {Physical Review D},

number = 1,

volume = 96,

place = {United States},

year = {Tue Jul 11 00:00:00 EDT 2017},

month = {Tue Jul 11 00:00:00 EDT 2017}

}

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