Excited-state uncertainties in lattice-QCD calculations of multi-hadron systems

TL;DR

This paper introduces tighter energy gap bounds for lattice QCD multi-hadron spectroscopy, improving uncertainty estimates and analyzing their effectiveness through theoretical models and high-statistics nucleon-nucleon scattering data.

Contribution

It develops and tests new energy gap bounds that provide more precise constraints on excited-state effects in lattice QCD calculations of multi-hadron systems.

Findings

Gap bounds yield tighter energy constraints than residual bounds.

Different analysis methods show inconsistencies, but gap bounds are consistent across operators.

Assumptions about energy levels are necessary to interpret scattering amplitudes.

Abstract

Excited-state effects lead to hard-to-quantify systematic uncertainties in lattice quantum chromodynamics (LQCD) spectroscopy calculations when computationally accessible imaginary times are smaller than inverse excitation gaps, as often arises for multi-hadron systems with signal-to-noise problems. Lanczos residual bounds address this by providing two-sided constraints on energies that do not require assumptions beyond Hermiticity, but often give very conservative systematic uncertainty estimates. Here, a more-constraining set of gap bounds is introduced for hadron spectroscopy. These bounds provide tighter constraints whose validity requires an explicit assumption about an energy gap. Exactly solvable lattice field theory correlators are used to test the utility of residual and gap bounds at finite and infinite statistics. Two-sided bounds and other analysis methods are then applied to a high-statistics LQCD calculation of nucleon-nucleon scattering at $m_\pi \sim 800$ MeV. Generalized eigenvalue problem (GEVP) and Lanczos energy estimators are compatible when applied to the same correlator data, but analyses including different interpolating operators show statistically significant inconsistencies. However, two-sided bounds from all operators are consistent. Under the assumption that the number of energy levels below $N\Delta$ and $\Delta\Delta$ thresholds is the same as for non-interacting nucleons, gap bounds are sufficient to constrain nucleon-nucleon scattering amplitudes at phenomenologically relevant precision. Lanczos methods further reveal that energy-eigenstate estimates from previously studied asymmetric correlators have not converged over accessible imaginary times. Nevertheless, data-driven examples demonstrate why assumptions are required to draw conclusions about the natures of two-nucleon ground states at these masses.