[Paper Review] Spectrum of excited states using the stochastic LapH method

John Bulava, Brendan Fahy+6 more|arXiv (Cornell University)|Oct 29, 2013

Quantum Chromodynamics and Particle Interactions3 citations

TL;DR

This paper presents a novel application of the stochastic LapH method to compute the spectrum of excited baryons and mesons in lattice QCD, using a 56-operator correlation matrix combining 12 spatially extended meson operators and 44 two-meson operators with varying relative momenta. The method efficiently computes Wick contractions via Laplacian Heaviside quark-field smearing, enabling robust extraction of finite-volume energy levels and successful identification of resonance precursors, including the $\rho(770)$, $\rho(1450)$, and $\rho_3(1690)$, in the $I=1$, $S=0$, $T_{1u}^{+}$ channel.

ABSTRACT

Progress in computing the spectrum of excited baryons and mesons in lattice QCD is described. Our first results in the zero-momentum bosonic I=1, S=0, T1u+ symmetry sector of QCD using a correlation matrix of 56 operators are presented. In addition to a dozen spatially-extended meson operators, 44 two-meson operators are used, involving a wide variety of light isovector, isoscalar, and strange meson operators of varying relative momenta. All needed Wick contractions are efficiently evaluated using a stochastic method of treating the low-lying modes of quark propagation that exploits Laplacian Heaviside quark-field smearing. Level identification is discussed.

Motivation & Objective

• To compute the finite-volume stationary-state energies of QCD in the $I=1$, $S=0$, $T_{1u}^{+}$ symmetry sector using lattice QCD simulations.
• To extend the reach of excited-state spectroscopy by including a large number of two-meson operators with diverse quantum numbers and relative momenta.
• To address the challenge of level identification in large correlation matrices by combining optimized single-hadron operators with full matrix analysis.
• To validate the method by comparing extracted energy levels with known experimental resonances, particularly in the $\rho$-meson sector.

Proposed method

• A 56-operator correlation matrix is constructed, combining 12 spatially extended single-meson operators and 44 two-meson operators with varying isospin, strangeness, and relative momenta.
• The stochastic LapH method is employed to efficiently compute all required Wick contractions by leveraging Laplacian Heaviside quark-field smearing to treat low-lying quark propagator modes.
• Irreducible representations of the cubic lattice symmetry group are used to construct gauge-covariant, momentum-projected hadron states with definite quantum numbers.
• Optimization rotations are applied to single-hadron operators to produce 'optimized' operators with improved overlap to individual energy eigenstates, enhancing plateau stability in effective mass plots.
• Level identification is performed by analyzing overlaps of optimized operators with eigenstates of the full correlation matrix, supported by classical operator expansions to suppress unwanted spin components.
• The method is applied to the $(24^3|390)$ ensemble with a pion mass of approximately 390 MeV, enabling comparison with experimental resonance masses.

Experimental results

Research questions

• RQ1 Can the stochastic LapH method enable efficient and accurate computation of a large correlation matrix with 56 operators in a non-trivial QCD symmetry sector?
• RQ2 How well can the method resolve and identify individual energy levels in the presence of strong mixing between single-hadron and two-meson states?
• RQ3 To what extent can optimized single-hadron operators improve the signal-to-noise ratio and enable reliable level identification in finite-volume spectra?
• RQ4 Can the extracted finite-volume energy levels be consistently identified with known experimental resonances such as the $\rho(770)$, $\rho(1450)$, and $\rho_3(1690)$?
• RQ5 What is the impact of unaccounted-for three- and four-meson states on the reliability of higher-lying level extractions?

Key findings

• The lowest-lying energy level (level 0) has a mass of $0.2218(14)\,a_t^{-1}$ and is identified as the precursor of the $\rho(770)$ resonance, with dominant overlap to the first optimized single-hadron operator.
• Level 8, with energy $0.3092(33)\,a_t^{-1}$, is identified as the dominant precursor of the $\rho(1450)$ resonance, showing significant overlap to a single optimized operator.
• Level 31, at $0.3851(35)\,a_t^{-1}$, is identified as the precursor of the spin-3 $\rho_3(1690)$ resonance, supported by the use of operators with suppressed spin-1 components.
• Levels 40 and 46 are also identified as dominant precursors of higher-lying resonances, with energies $0.4154(37)\,a_t^{-1}$ and $0.4442(38)\,a_t^{-1}$, respectively.
• The method successfully extracts all two-meson states below the five lowest-lying single-hadron states, though higher-lying states (hollow boxes) are less certain due to unaccounted-for two-meson states.
• The general pattern of the spectrum, including the ordering and spacing of states, is in qualitative agreement with experimental expectations despite the large pion mass of 390 MeV.