# Theoretical aspects of quantum electrodynamics in a finite volume with periodic boundary conditions

## Abstract

First-principles studies of strongly interacting hadronic systems using lattice quantum chromodynamics (QCD) have been complemented in recent years with the inclusion of quantum electrodynamics (QED). The aim is to confront experimental results with more precise theoretical determinations, e.g. for the anomalous magnetic moment of the muon and the C P -violating parameters in the decay of mesons. Quantifying the effects arising from enclosing QED in a finite volume remains a primary target of investigations. To this end, finite-volume corrections to hadron masses in the presence of QED have been carefully studied in recent years. This paper extends such studies to the self-energy of moving charged hadrons, both on and away from their mass shell. In particular, we present analytical results for leading finite-volume corrections to the self-energy of spin-0 and spin-$$1\over{2}$$ particles in the presence of QED on a periodic hypercubic lattice, once the spatial zero mode of the photon is removed, a framework that is called QED_{L}. By altering modes beyond the zero mode, an improvement scheme is introduced to eliminate the leading finite-volume corrections to masses, with potential applications to other hadronic quantities. Our analytical results are verified by a dedicated numerical study of a lattice scalar field theory coupled to QED_{L}. Further, this paper offers new perspectives on the subtleties involved in applying low-energy effective field theories in the presence of QED_{L}, a theory that is rendered nonlocal with the exclusion of the spatial zero mode of the photon, clarifying recent discussions on this matter.

- Authors:

- Publication Date:

- Research Org.:
- Univ. of Washington, Seattle, WA (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC)

- OSTI Identifier:
- 1497247

- Alternate Identifier(s):
- OSTI ID: 1609321

- Grant/Contract Number:
- FG02-00ER41132

- Resource Type:
- Published Article

- Journal Name:
- Physical Review D

- Additional Journal Information:
- Journal Name: Physical Review D Journal Volume: 99 Journal Issue: 3; Journal ID: ISSN 2470-0010

- Publisher:
- American Physical Society

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; Astronomy & Astrophysics; Physics

### Citation Formats

```
Davoudi, Z., Harrison, J., Jüttner, A., Portelli, A., and Savage, M. J. Theoretical aspects of quantum electrodynamics in a finite volume with periodic boundary conditions. United States: N. p., 2019.
Web. doi:10.1103/PhysRevD.99.034510.
```

```
Davoudi, Z., Harrison, J., Jüttner, A., Portelli, A., & Savage, M. J. Theoretical aspects of quantum electrodynamics in a finite volume with periodic boundary conditions. United States. doi:10.1103/PhysRevD.99.034510.
```

```
Davoudi, Z., Harrison, J., Jüttner, A., Portelli, A., and Savage, M. J. Thu .
"Theoretical aspects of quantum electrodynamics in a finite volume with periodic boundary conditions". United States. doi:10.1103/PhysRevD.99.034510.
```

```
@article{osti_1497247,
```

title = {Theoretical aspects of quantum electrodynamics in a finite volume with periodic boundary conditions},

author = {Davoudi, Z. and Harrison, J. and Jüttner, A. and Portelli, A. and Savage, M. J.},

abstractNote = {First-principles studies of strongly interacting hadronic systems using lattice quantum chromodynamics (QCD) have been complemented in recent years with the inclusion of quantum electrodynamics (QED). The aim is to confront experimental results with more precise theoretical determinations, e.g. for the anomalous magnetic moment of the muon and the C P -violating parameters in the decay of mesons. Quantifying the effects arising from enclosing QED in a finite volume remains a primary target of investigations. To this end, finite-volume corrections to hadron masses in the presence of QED have been carefully studied in recent years. This paper extends such studies to the self-energy of moving charged hadrons, both on and away from their mass shell. In particular, we present analytical results for leading finite-volume corrections to the self-energy of spin-0 and spin-$1\over{2}$ particles in the presence of QED on a periodic hypercubic lattice, once the spatial zero mode of the photon is removed, a framework that is called QEDL. By altering modes beyond the zero mode, an improvement scheme is introduced to eliminate the leading finite-volume corrections to masses, with potential applications to other hadronic quantities. Our analytical results are verified by a dedicated numerical study of a lattice scalar field theory coupled to QEDL. Further, this paper offers new perspectives on the subtleties involved in applying low-energy effective field theories in the presence of QEDL, a theory that is rendered nonlocal with the exclusion of the spatial zero mode of the photon, clarifying recent discussions on this matter.},

doi = {10.1103/PhysRevD.99.034510},

journal = {Physical Review D},

number = 3,

volume = 99,

place = {United States},

year = {2019},

month = {2}

}

DOI: 10.1103/PhysRevD.99.034510

*Citation information provided by*

Web of Science

Web of Science

Works referenced in this record:

##
Finite volume corrections to the electromagnetic mass of composite particles

journal, February 2016

- Lee, Jong-Wan; Tiburzi, Brian C.
- Physical Review D, Vol. 93, Issue 3

##
Heavy-quark bound states in lattice QCD

journal, January 1991

- Thacker, B. A.; Lepage, G. Peter
- Physical Review D, Vol. 43, Issue 1

##
Uncertainty quantification in lattice QCD calculations for nuclear physics

journal, February 2015

- Beane, Silas R.; Detmold, William; Orginos, Kostas
- Journal of Physics G: Nuclear and Particle Physics, Vol. 42, Issue 3

##
Calculation of the Hadronic Vacuum Polarization Contribution to the Muon Anomalous Magnetic Moment

journal, July 2018

- Blum, T.; Boyle, P. A.; Gülpers, V.
- Physical Review Letters, Vol. 121, Issue 2

##
Electromagnetic Splittings and Light Quark Masses in Lattice QCD

journal, May 1996

- Duncan, A.; Eichten, E.; Thacker, H.
- Physical Review Letters, Vol. 76, Issue 21

##
Baryon chiral perturbation theory using a heavy fermion lagrangian

journal, February 1991

- Jenkins, Elizabeth; Manohar, Aneesh V.
- Physics Letters B, Vol. 255, Issue 4

##
Strange and charm HVP contributions to the muon (g − 2) including QED corrections with twisted-mass fermions

journal, October 2017

- Giusti, D.; Lubicz, V.; Martinelli, G.
- Journal of High Energy Physics, Vol. 2017, Issue 10

##
Locally smeared operator product expansions in scalar field theory

journal, April 2015

- Monahan, Christopher; Orginos, Kostas
- Physical Review D, Vol. 91, Issue 7

##
Improving the volume dependence of two-body binding energies calculated with lattice QCD

journal, December 2011

- Davoudi, Zohreh; Savage, Martin J.
- Physical Review D, Vol. 84, Issue 11

##
SU(N) gauge theories with C-periodic boundary conditions (I). Topological structure

journal, July 1991

- Kronfeld, A. S.; Wiese, U. -J.
- Nuclear Physics B, Vol. 357, Issue 2-3

##
Finite-volume electromagnetic corrections to the masses of mesons, baryons, and nuclei

journal, September 2014

- Davoudi, Zohreh; Savage, Martin J.
- Physical Review D, Vol. 90, Issue 5

##
Electromagnetic structure of light baryons in lattice QCD

journal, September 1997

- Duncan, A.; Eichten, E.; Thacker, H.
- Physics Letters B, Vol. 409, Issue 1-4

##
Double exponential formulas for numerical integration

journal, January 1973

- Takahasi, Hidetosi; Mori, Masatake
- Publications of the Research Institute for Mathematical Sciences, Vol. 9, Issue 3

##
Three-boson problem at low energy and implications for dilute Bose-Einstein condensates

journal, July 2008

- Tan, Shina
- Physical Review A, Vol. 78, Issue 1

##
Isospin breaking corrections to meson masses and the hadronic vacuum polarization: a comparative study

journal, September 2017

- Boyle, P.; Gülpers, V.; Harrison, J.
- Journal of High Energy Physics, Vol. 2017, Issue 9

##
Finite-volume QED corrections to decay amplitudes in lattice QCD

journal, February 2017

- Lubicz, V.; Martinelli, G.; Sachrajda, C. T.
- Physical Review D, Vol. 95, Issue 3

##
Leading isospin breaking effects on the lattice

journal, June 2013

- de Divitiis, G. M.; Frezzotti, R.; Lubicz, V.
- Physical Review D, Vol. 87, Issue 11

##
Two-particle elastic scattering in a finite volume including QED

journal, October 2014

- Beane, Silas R.; Savage, Martin J.
- Physical Review D, Vol. 90, Issue 7

##
An effective field theory for heavy quarks at low energies

journal, April 1990

- Georgi, Howard
- Physics Letters B, Vol. 240, Issue 3-4

##
Full $\mathrm{QED}\mathbf{+}\mathrm{QCD}$ Low-Energy Constants through Reweighting

journal, August 2012

- Ishikawa, Tomomi; Blum, Thomas; Hayakawa, Masashi
- Physical Review Letters, Vol. 109, Issue 7

##
Gauge invariant determination of charged hadron masses

journal, May 2018

- Hansen, M.; Lucini, B.; Patella, A.
- Journal of High Energy Physics, Vol. 2018, Issue 5

##
Finite-volume corrections to electromagnetic masses for larger-than-physical electric charges

journal, May 2017

- Matzelle, Matthew E.; Tiburzi, Brian C.
- Physical Review D, Vol. 95, Issue 9

##
Two-baryon systems with twisted boundary conditions

journal, April 2014

- Briceño, Raúl A.; Davoudi, Zohreh; Luu, Thomas C.
- Physical Review D, Vol. 89, Issue 7

##
$1\mathbf{+}1\mathbf{+}1$ flavor $\mathrm{QCD}\mathbf{+}\mathrm{QED}$ simulation at the physical point

journal, August 2012

- Aoki, S.; Ishikawa, K. -I.; Ishizuka, N.
- Physical Review D, Vol. 86, Issue 3

##
Ab initio calculation of the neutron-proton mass difference

journal, March 2015

- Borsanyi, S.; Durr, S.; Fodor, Z.
- Science, Vol. 347, Issue 6229

##
Volume dependence of the energy spectrum in massive quantum field theories: II. Scattering states

journal, June 1986

- Lüscher, M.
- Communications in Mathematical Physics, Vol. 105, Issue 2

##
Quantum electrodynamics in finite volume and nonrelativistic effective field theories

journal, April 2016

- Fodor, Z.; Hoelbling, C.; Katz, S. D.
- Physics Letters B, Vol. 755

##
openQ*D simulation code for QCD+QED

journal, January 2018

- Campos, Isabel; Fritzsch, Patrick; Hansen, Martin
- EPJ Web of Conferences, Vol. 175

##
The strategy for computing the hadronic mass spectrum

journal, January 1984

- Parisi, Giorgio
- Physics Reports, Vol. 103, Issue 1-4

##
Effective Field Theories of the Standard Model: Proceedings of the Workshop

conference, November 2014

- MeißNer, Ulf-G.
- Workshop on Effective Field Theories of the Standard Model

##
Using infinite-volume, continuum QED and lattice QCD for the hadronic light-by-light contribution to the muon anomalous magnetic moment

journal, August 2017

- Blum, Thomas; Christ, Norman; Hayakawa, Masashi
- Physical Review D, Vol. 96, Issue 3

##
On the calculation of lattice sums

journal, January 1957

- Nijboer, B. R. A.; De Wette, F. W.
- Physica, Vol. 23, Issue 1-5

##
Simulations of QCD and QED with C ^{*} boundary conditions

journal, January 2018

- Hansen, Martin; Lucini, Biagio; Patella, Agostino
- EPJ Web of Conferences, Vol. 175

##
Heavy quark effective theory and nonrelativistic QCD Lagrangian to order ${\alpha}_{s}{/m}^{3}$

journal, July 1997

- Manohar, Aneesh V.
- Physical Review D, Vol. 56, Issue 1

##
C- and G-periodic QCD at finite temperature

journal, May 1992

- Wiese, U. -J.
- Nuclear Physics B, Vol. 375, Issue 1

##
NRQED Lagrangian at order $1/{M}^{4}$

journal, March 2013

- Hill, Richard J.; Lee, Gabriel; Paz, Gil
- Physical Review D, Vol. 87, Issue 5

##
Charged hadrons in local finite-volume QED+QCD with C⋆ boundary conditions

journal, February 2016

- Lucini, B.; Patella, A.; Ramos, A.
- Journal of High Energy Physics, Vol. 2016, Issue 2

##
High statistics analysis using anisotropic clover lattices. II. Three-baryon systems

journal, October 2009

- Beane, Silas R.; Detmold, William; Luu, Thomas C.
- Physical Review D, Vol. 80, Issue 7

##
Feynman diagrams for the Yang-Mills field

journal, July 1967

- Faddeev, L. D.; Popov, V. N.
- Physics Letters B, Vol. 25, Issue 1

##
Deep-inelastic scattering and the operator product expansion in lattice QCD

journal, January 2006

- Detmold, William; Lin, C. -J. David
- Physical Review D, Vol. 73, Issue 1

##
Determination of light quark masses from the electromagnetic splitting of pseudoscalar meson masses computed with two flavors of domain wall fermions

journal, December 2007

- Blum, Thomas; Doi, Takumi; Hayakawa, Masashi
- Physical Review D, Vol. 76, Issue 11

##
Power corrections at short distances

journal, January 1980

- David Politzer, H.
- Nuclear Physics B, Vol. 172

##
Restoration of rotational symmetry in the continuum limit of lattice field theories

journal, September 2012

- Davoudi, Zohreh; Savage, Martin J.
- Physical Review D, Vol. 86, Issue 5

##
QED in Finite Volume and Finite Size Scaling Effect on Electromagnetic Properties of Hadrons

journal, September 2008

- Uno, S.; Hayakawa, M.
- Progress of Theoretical Physics, Vol. 120, Issue 3

##
Review of lattice results concerning low-energy particle physics: Flavour Lattice Averaging Group (FLAG)

journal, February 2017

- Aoki, S.; Aoki, Y.; Bečirević, D.
- The European Physical Journal C, Vol. 77, Issue 2

##
SU suN) gauge theories with C-periodic boundary conditions (II). Small-volume dynamics

journal, July 1993

- Kronfeld, A. S.; Wiese, U. -J.
- Nuclear Physics B, Vol. 401, Issue 1-2

##
Effective field theory approach to processes involving both light and heavy fields

journal, July 1988

- David Politzer, H.; Wise, Mark B.
- Physics Letters B, Vol. 208, Issue 3-4

##
Boundaries forSU(3) c xU(1)el lattice gauge theory with a chemical potential

journal, March 1993

- Polley, L.
- Zeitschrift f�r Physik C: Particles and Fields, Vol. 59, Issue 1

##
Up and Down Quark Masses and Corrections to Dashen’s Theorem from Lattice QCD and Quenched QED

journal, August 2016

- Fodor, Z.; Hoelbling, C.; Krieg, S.
- Physical Review Letters, Vol. 117, Issue 8

##
Electromagnetic mass splittings of the low lying hadrons and quark masses from $2+1$ flavor lattice $\mathrm{QCD}+\mathrm{QED}$

journal, November 2010

- Blum, Tom; Zhou, Ran; Doi, Takumi
- Physical Review D, Vol. 82, Issue 9

##
High statistics analysis using anisotropic clover lattices: Single hadron correlation functions

journal, June 2009

- Beane, Silas R.; Detmold, William; Luu, Thomas C.
- Physical Review D, Vol. 79, Issue 11

##
Isospin splittings of meson and baryon masses from three-flavor lattice QCD + QED

journal, September 2016

- Horsley, R.; Nakamura, Y.; Perlt, H.
- Journal of Physics G: Nuclear and Particle Physics, Vol. 43, Issue 10

##
Leading isospin-breaking corrections to pion, kaon, and charmed-meson masses with twisted-mass fermions

journal, June 2017

- Giusti, D.; Lubicz, V.; Tarantino, C.
- Physical Review D, Vol. 95, Issue 11

##
Effective lagrangians for simulating of heavy quark systems

journal, April 1988

- Lepage, G. P.; Thacker, B. A.
- Nuclear Physics B - Proceedings Supplements, Vol. 4

##
Electromagnetic corrections to the hadronic vacuum polarization
of the photon within QED _{L} and QED _{M}

journal, January 2018

- Bussone, Andrea; Della Morte, Michele; Janowski, Tadeusz
- EPJ Web of Conferences, Vol. 175

##
Two-nucleon systems in a finite volume. II. ${}^{3}S_{1}\mathrm{\text{\u2212}}{}^{3}D_{1}$ coupled channels and the deuteron

journal, December 2013

- Briceño, Raúl A.; Davoudi, Zohreh; Luu, Thomas C.
- Physical Review D, Vol. 88, Issue 11

##
Massive Photons: An Infrared Regularization Scheme for Lattice $\mathrm{QCD}+\mathrm{QED}$

journal, August 2016

- Endres, Michael G.; Shindler, Andrea; Tiburzi, Brian C.
- Physical Review Letters, Vol. 117, Issue 7

##
Isospin Splittings in the Light-Baryon Octet from Lattice QCD and QED

journal, December 2013

- Borsanyi, Sz.; Dürr, S.; Fodor, Z.
- Physical Review Letters, Vol. 111, Issue 25

##
First Lattice Calculation of the QED Corrections to Leptonic Decay Rates

journal, February 2018

- Giusti, D.; Lubicz, V.; Tarantino, C.
- Physical Review Letters, Vol. 120, Issue 7

##
QED corrections to hadronic processes in lattice QCD

journal, April 2015

- Carrasco, N.; Lubicz, V.; Martinelli, G.
- Physical Review D, Vol. 91, Issue 7

##
QED effects in the pseudoscalar meson sector

journal, April 2016

- Horsley, R.; Nakamura, Y.; Perlt, H.
- Journal of High Energy Physics, Vol. 2016, Issue 4

##
Including electromagnetism in *K* → ππ decay calculations

journal, January 2018

- Christ, Norman; Feng, Xu
- EPJ Web of Conferences, Vol. 175