Group-theoretical construction of extended baryon operators in lattice QCD

S Basak,U M Heller,C Morningstar,D Richards,R G Edwards,G T Fleming,S Wallace,I Sato

doi:10.1103/physrevd.72.094506

Abstract

The design and implementation of large sets of spatially-extended, gauge-invariant operators for use in determining the spectrum of baryons in lattice QCD computations are described. Group-theoretical projections onto the irreducible representations of the symmetry group of a cubic spatial lattice are used in all isospin channels. The operators are constructed to maximize overlaps with the low-lying states of interest, while minimizing the number of sources needed in computing the required quark propagators. Issues related to the identification of the spin quantum numbers of the states in the continuum limit are addressed.

Highlights

Spectroscopy is a powerful tool for uncovering the important degrees of freedom of a physical system and the interaction forces between them
The spectrum of quantum chromodynamics (QCD) is very rich: conventional baryons and mesons (π, K, ρ, φ, . . .) have been known for nearly half a century, but other higher-lying exotic states, such as glueballs, four-quark states, and so-called hybrid mesons and hybrid baryons bound by an excited gluon field, have proved more elusive, mainly because our theoretical understanding of such states is insufficient, making their identification problematical
Each baryon correlator is a linear superposition of elements of the three-quark propagators. These superposition coefficients are calculated as follows: first, the baryon operators at the source and sink are expressed in terms of the elemental operators; Wick’s theorem is applied to express the correlator as a large sum of three-quark propagator components; symmetry operations are applied to minimize the number of source orientations, and the results are averaged over the rows of the representations

Summary

INTRODUCTION

Spectroscopy is a powerful tool for uncovering the important degrees of freedom of a physical system and the interaction forces between them. The number of hadron eigenstates which can be reliably extracted in Monte Carlo computations is not currently known, so our undertaking will be partly an exploration of the limits of possibility Another aim is to discover whether a very large number of interpolating operators are needed to extract the low-lying spectrum, or whether a handful of carefully chosen ones is sufficient, and this work outlines a systematic means of finding such operators. Because determining the mass of a particular resonance requires determining the energies of all lower-lying stationary states, including scattering states, the set of operators we use must include meson and baryon operators, and multihadron operators Another very important fact to keep in mind is the computational cost of evaluating quark propagators, especially for light quark masses.

OVERVIEW OF OPERATOR CONSTRUCTION

THE BASIC BUILDING BLOCKS

THREE-QUARK ELEMENTAL OPERATORS

PROJECTIONS ONTO SYMMETRY SECTORS

BARYON PROPAGATORS

CONCLUSION