[Paper Review] Inclusion of isospin breaking effects in lattice simulations

Antonin Portelli|arXiv (Cornell University)|May 26, 2015

Particle physics theoretical and experimental studies37 references4 citations

TL;DR

This paper presents a framework for including isospin-breaking effects—due to the mass and charge differences of up and down quarks—directly in lattice QCD simulations. It demonstrates that full QCD+QED simulations with controlled uncertainties now enable precise, ab initio computation of nucleon and meson mass splittings, resolving long-standing theoretical challenges in hadronic physics and providing critical inputs for CKM matrix elements and the muon g−2.

ABSTRACT

Isospin symmetry is explicitly broken in the Standard Model by the mass and electric charge of the up and down quarks. These effects represent a perturbation of hadronic amplitudes at the percent level. Although these contributions are small, they play a crucial role in hadronic and nuclear physics. Moreover, as lattice computations are becoming increasingly precise, it is becoming more and more important to include these effects in numerical simulations. We summarize here how to properly define QCD and QED on a finite and discrete space-time so that isospin corrections to hadronic observables can be computed ab-initio and we review the main results on the isospin corrections to the hadron spectrum. We mainly focus on the recent work going beyond the electro-quenched approximation.

Motivation & Objective

• To develop a consistent, non-perturbative framework for including both QCD and QED interactions in lattice simulations with full control over systematic uncertainties.
• To compute the nucleon mass splitting (Mn−Mp) from first principles, resolving the competition between electromagnetic and strong isospin-breaking contributions.
• To extract individual up and down quark masses from isospin-breaking splittings in the kaon and pion spectra, requiring accurate subtraction of QED effects.
• To extend lattice methods beyond the electro-quenched approximation to include full QED dynamics, including long-range interactions and finite-volume effects.
• To enable high-precision predictions of hadronic matrix elements for flavor physics observables, such as CKM matrix elements and hadronic vacuum polarization contributions to the muon g−2.

Proposed method

• Formalism for defining QCD and QED on a finite, discrete spacetime lattice with proper treatment of the photon zero-mode to preserve unitarity and reflection positivity.
• Use of zero-mode subtraction techniques, such as those by Hayakawa and Uno, to remove infrared divergences without breaking gauge invariance or reflection positivity.
• Analytical computation of leading finite-volume corrections to hadron masses using gauge invariance, valid to next-to-leading order in the infinite-volume expansion.
• Implementation of effective field theories to describe higher-order, structure-dependent finite-size effects in QED-lattice simulations.
• Use of dynamical fermions, including an active sea charm quark, to improve the physical realism of simulations.
• Application of the method to compute baryon octet and charm spectrum splittings with full control over statistical and systematic uncertainties.

Experimental results

Research questions

• RQ1 How can QED and QCD be consistently formulated on a finite lattice to include isospin-breaking effects without breaking fundamental symmetries?
• RQ2 What is the precise contribution of QED and QCD to the nucleon mass splitting (Mn−Mp), and how do they cancel to produce the observed positive value?
• RQ3 To what extent do finite-volume effects in QED-lattice simulations distort hadron mass splittings, and how can they be systematically corrected?
• RQ4 Can the individual up and down quark masses be extracted from lattice simulations of isospin-breaking splittings in the kaon and pion systems?
• RQ5 How can lattice QCD+QED be extended to compute matrix elements involving real photons, such as in hadronic decays and the muon anomalous magnetic moment?

Key findings

• The BMWc collaboration achieved the first complete lattice calculation including full QCD and QED interactions, with an active sea charm quark and full uncertainty control, yielding precise corrections to the baryon octet and charm spectrum.
• The nucleon mass splitting (Mn−Mp) is found to result from a delicate cancellation between negative QED contributions and positive QCD contributions, consistent with the experimental value of 1.2933322(4) MeV.
• Finite-volume corrections to hadron masses in QED-lattice simulations are fully determined by gauge invariance to next-to-leading order, enabling accurate extrapolation to infinite volume.
• The electro-quenched approximation remains insufficient for precision extraction of individual up and down quark masses; full QED inclusion is essential to subtract electromagnetic contributions accurately from kaon and pion splittings.
• The method enables non-perturbative computation of hadronic vacuum polarization contributions to the muon anomalous magnetic moment, a key test of the Standard Model.
• Theoretical determinations of (Mn−Mp) now converge across multiple collaborations, with results agreeing with experiment and showing the dominance of strong isospin-breaking effects over electromagnetic ones.