# Lattice simulations with $G$ -parity boundary conditions

## Abstract

We discuss $$\textit{G}$$-parity lattice boundary conditions as a means to impose momentum on the pion ground state without breaking isospin symmetry. This technique is expected to be critical for the precision measurement of $$\textit{K} → (ππ)_{I=0}$$ matrix elements where physical kinematics demands moving pions in the final state and the statistical noise caused by disconnected contributions will make it difficult to use multiexponential fits to isolate this as an excited state. We present a formalism for computing hadronic Green’s functions with $$\textit{G}$$-parity boundary conditions, derive the discretized action and its symmetries, discuss how the strange quark can be introduced and detail techniques for the numerical implementation of these boundary conditions. We demonstrate and test these methods using several 16^{3} × 32 dynamical domain wall ensembles with a 420 MeV pion mass and $$\textit{G}$$-parity boundary conditions in one and two spatial directions.

- Authors:

- Publication Date:

- Research Org.:
- Columbia Univ., New York, NY (United States)

- Sponsoring Org.:
- Intel Corporation; USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR). Scientific Discovery through Advanced Computing (SciDAC)

- Contributing Org.:
- RBC and UKQCD Collaborations

- OSTI Identifier:
- 1592805

- Alternate Identifier(s):
- OSTI ID: 1802296

- Grant/Contract Number:
- SC0011941

- Resource Type:
- Published Article

- Journal Name:
- Physical Review D

- Additional Journal Information:
- Journal Name: Physical Review D Journal Volume: 101 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; Astronomy & Astrophysics; Physics

## Citation Formats

```
Christ, N. H., Kelly, C., Zhang, D., and RBC and UKQCD Collaborations. Lattice simulations with G -parity boundary conditions. United States: N. p., 2020.
Web. doi:10.1103/PhysRevD.101.014506.
```

```
Christ, N. H., Kelly, C., Zhang, D., & RBC and UKQCD Collaborations. Lattice simulations with G -parity boundary conditions. United States. https://doi.org/10.1103/PhysRevD.101.014506
```

```
Christ, N. H., Kelly, C., Zhang, D., and RBC and UKQCD Collaborations. Tue .
"Lattice simulations with G -parity boundary conditions". United States. https://doi.org/10.1103/PhysRevD.101.014506.
```

```
@article{osti_1592805,
```

title = {Lattice simulations with G -parity boundary conditions},

author = {Christ, N. H. and Kelly, C. and Zhang, D. and RBC and UKQCD Collaborations},

abstractNote = {We discuss $\textit{G}$-parity lattice boundary conditions as a means to impose momentum on the pion ground state without breaking isospin symmetry. This technique is expected to be critical for the precision measurement of $\textit{K} → (ππ)_{I=0}$ matrix elements where physical kinematics demands moving pions in the final state and the statistical noise caused by disconnected contributions will make it difficult to use multiexponential fits to isolate this as an excited state. We present a formalism for computing hadronic Green’s functions with $\textit{G}$-parity boundary conditions, derive the discretized action and its symmetries, discuss how the strange quark can be introduced and detail techniques for the numerical implementation of these boundary conditions. We demonstrate and test these methods using several 163 × 32 dynamical domain wall ensembles with a 420 MeV pion mass and $\textit{G}$-parity boundary conditions in one and two spatial directions.},

doi = {10.1103/PhysRevD.101.014506},

journal = {Physical Review D},

number = 1,

volume = 101,

place = {United States},

year = {2020},

month = {1}

}

https://doi.org/10.1103/PhysRevD.101.014506

*Citation information provided by*

Web of Science

Web of Science

Works referenced in this record:

##
$2+1$ flavor domain wall QCD on a $(2\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{fm}{)}^{3}$ lattice: Light meson spectroscopy with ${L}_{s}=16$

journal, July 2007

- Allton, C.; Antonio, D. J.; Blum, T.
- Physical Review D, Vol. 76, Issue 1

##
Volume dependence of the energy spectrum in massive quantum field theories: I. Stable particle states

journal, June 1986

- Lüscher, M.
- Communications in Mathematical Physics, Vol. 104, Issue 2

##
Domain wall fermion QCD with the exact one flavor algorithm

journal, March 2018

- Jung, C.; Kelly, C.; Mawhinney, R. D.
- Physical Review D, Vol. 97, Issue 5

##
Continuum limit physics from $2+1$ flavor domain wall QCD

journal, April 2011

- Aoki, Y.; Arthur, R.; Blum, T.
- Physical Review D, Vol. 83, Issue 7

##
Continuum limit of ${B}_{K}$ from $2+1$ flavor domain wall QCD

journal, July 2011

- Aoki, Y.; Arthur, R.; Blum, T.
- Physical Review D, Vol. 84, Issue 1

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

journal, May 1992

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

##
Axial symmetries in lattice QCD with Kaplan fermions

journal, April 1995

- Furman, Vadim; Shamir, Yigal
- Nuclear Physics B, Vol. 439, Issue 1-2

##
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

##
Lattice determination of the $K\mathbf{\to}(\pi \pi {)}_{I=2}$ decay amplitude ${A}_{2}$

journal, October 2012

- Blum, T.; Boyle, P. A.; Christ, N. H.
- Physical Review D, Vol. 86, Issue 7

##
The BlueGene/Q supercomputer

conference, December 2012

- Boyle, Peter A.
- Proceedings of The 30th International Symposium on Lattice Field Theory — PoS(Lattice 2012)

##
Domain wall QCD with near-physical pions

journal, May 2013

- Arthur, R.; Blum, T.; Boyle, P. A.
- Physical Review D, Vol. 87, Issue 9

##
Physical results from $2+1$ flavor domain wall QCD and SU(2) chiral perturbation theory

journal, December 2008

- Allton, C.; Antonio, D. J.; Aoki, Y.
- Physical Review D, Vol. 78, Issue 11

##
Quenched lattice QCD with domain wall fermions and the chiral limit

journal, April 2004

- Blum, T.; Chen, P.; Christ, N.
- Physical Review D, Vol. 69, Issue 7

##
Weak Transition Matrix Elements¶from Finite-Volume Correlation Functions

journal, May 2001

- Lellouch, Laurent; Lüscher, Martin
- Communications in Mathematical Physics, Vol. 219, Issue 1

##
First results from $2+1$ -flavor domain wall QCD: Mass spectrum, topology change, and chiral symmetry with ${L}_{s}=8$

journal, June 2007

- Antonio, D. J.; Blum, T.; Bowler, K. C.
- Physical Review D, Vol. 75, Issue 11

##
The BAGEL assembler generation library

journal, December 2009

- Boyle, Peter A.
- Computer Physics Communications, Vol. 180, Issue 12

##
Exact pseudofermion action for Monte Carlo simulation of domain-wall fermion

journal, November 2014

- Chen, Yu-Chih; Chiu, Ting-Wai
- Physics Letters B, Vol. 738

##
Two-particle states on a torus and their relation to the scattering matrix

journal, May 1991

- Lüscher, Martin
- Nuclear Physics B, Vol. 354, Issue 2-3

##
K→ππ decays in a finite volume

journal, December 2001

- Lin, C. -J. D.; Martinelli, G.; Sachrajda, C. T.
- Nuclear Physics B, Vol. 619, Issue 1-3

##
$\mathrm{K}\to \pi \pi $ $\mathrm{\Delta}\mathrm{I}=3/2$ decay amplitude in the continuum limit

journal, April 2015

- Blum, T.; Boyle, P. A.; Christ, N. H.
- Physical Review D, Vol. 91, Issue 7

##
I = 2 ππ scattering using G-parity boundary condition

journal, March 2004

- Kim, Changhoan
- Nuclear Physics B - Proceedings Supplements, Vol. 129-130

##
$K\to (\pi \pi {)}_{I=2}$ Decay Amplitude from Lattice QCD

journal, April 2012

- Blum, T.; Boyle, P. A.; Christ, N. H.
- Physical Review Letters, Vol. 108, Issue 14

##
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

##
Standard Model Prediction for Direct $CP$ Violation in $K\to \pi \pi $ Decay

journal, November 2015

- Bai, Z.; Blum, T.; Boyle, P. A.
- Physical Review Letters, Vol. 115, Issue 21

##
Practical all-to-all propagators for lattice QCD

journal, November 2005

- Foley, Justin; Juge, K. Jimmy; Ó Cais, Alan
- Computer Physics Communications, Vol. 172, Issue 3