# Finite-volume effects due to spatially nonlocal operators

## Abstract

Spatially non-local matrix elements are useful lattice-QCD observables in a variety of contexts, for example in determining hadron structure. In order to quote credible estimates of the systematic uncertainties in these calculations, one must understand, among other things, the size of the finite-volume effects when such matrix elements are extracted from numerical lattice calculations. In this work we estimate these effects by considering a simple toy model in which the non-local operator is composed of two currents displaced in a spatial direction by a distance $$\xi$$. We find that the finite-volume corrections depend on the details of the matrix element. If the external state is the lightest degree of freedom in the theory, e.g.~the pion in QCD, then the volume corrections scale as $$ e^{-m_\pi (L- \xi)} $$, where $m$ is the mass of the light state. For heavier external states the usual $$e^{- m_\pi L}$$ form is recovered, but with a polynomial prefactor of the form $$L^a/|L - \xi|^b$$ that can lead to enhanced volume effects. As a result, these observations {might be relevant} for a wide variety of observables being studied using lattice QCD, including parton distribution functions, double-beta decay and Compton scattering matrix elements, and long-range weak matrix elements.

- Authors:

- 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:
- 1461723

- Alternate Identifier(s):
- OSTI ID: 1462022

- Report Number(s):
- [JLAB-THY-18-2697; DOE/OR/23177-4429; arXiv:1805.01034; CERN-TH-2018-109; INT-PUB-18-019]

[Journal ID: ISSN 2470-0010; PRVDAQ; 014511]

- Grant/Contract Number:
- [AC05-06OR23177; SC0008791; FG02-00ER41132]

- Resource Type:
- Published Article

- Journal Name:
- Physical Review D

- Additional Journal Information:
- [Journal Name: Physical Review D Journal Volume: 98 Journal Issue: 1]; Journal ID: ISSN 2470-0010

- Publisher:
- American Physical Society

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS

### Citation Formats

```
Briceño, Raúl A., Guerrero, Juan V., Hansen, Maxwell T., and Monahan, Christopher J. Finite-volume effects due to spatially nonlocal operators. United States: N. p., 2018.
Web. doi:10.1103/PhysRevD.98.014511.
```

```
Briceño, Raúl A., Guerrero, Juan V., Hansen, Maxwell T., & Monahan, Christopher J. Finite-volume effects due to spatially nonlocal operators. United States. doi:10.1103/PhysRevD.98.014511.
```

```
Briceño, Raúl A., Guerrero, Juan V., Hansen, Maxwell T., and Monahan, Christopher J. Thu .
"Finite-volume effects due to spatially nonlocal operators". United States. doi:10.1103/PhysRevD.98.014511.
```

```
@article{osti_1461723,
```

title = {Finite-volume effects due to spatially nonlocal operators},

author = {Briceño, Raúl A. and Guerrero, Juan V. and Hansen, Maxwell T. and Monahan, Christopher J.},

abstractNote = {Spatially non-local matrix elements are useful lattice-QCD observables in a variety of contexts, for example in determining hadron structure. In order to quote credible estimates of the systematic uncertainties in these calculations, one must understand, among other things, the size of the finite-volume effects when such matrix elements are extracted from numerical lattice calculations. In this work we estimate these effects by considering a simple toy model in which the non-local operator is composed of two currents displaced in a spatial direction by a distance $\xi$. We find that the finite-volume corrections depend on the details of the matrix element. If the external state is the lightest degree of freedom in the theory, e.g.~the pion in QCD, then the volume corrections scale as $ e^{-m_\pi (L- \xi)} $, where $m$ is the mass of the light state. For heavier external states the usual $e^{- m_\pi L}$ form is recovered, but with a polynomial prefactor of the form $L^a/|L - \xi|^b$ that can lead to enhanced volume effects. As a result, these observations {might be relevant} for a wide variety of observables being studied using lattice QCD, including parton distribution functions, double-beta decay and Compton scattering matrix elements, and long-range weak matrix elements.},

doi = {10.1103/PhysRevD.98.014511},

journal = {Physical Review D},

number = [1],

volume = [98],

place = {United States},

year = {2018},

month = {7}

}

DOI: 10.1103/PhysRevD.98.014511

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