Excited-state uncertainties in lattice-QCD calculations of hadron masses and scattering phase shifts

Abstract

Lattice QCD has historically produced energy results interpretable as either estimates relying on implicit assumptions about asymptotic behavior or one-sided upper bounds. New Lanczos methods providing two-sided bounds with less-restrictive assumptions are introduced and quantified in a high-statistics calculation with unphysical quark masses. Two-sided bounds without spectral assumptions provide sub-percent constraints on the nucleon mass. Other bounds, which assume all states in a given energy window are resolved, provide meaningful two-sided constraints on nucleon-nucleon scattering phase shifts.

Figures (5)

• Figure 1: Effective masses of GEVP principal correlators compared with highest-weight fit (orange band, left, and first point, right) and Bayesian model averaged Jay:2020jkz (BMA) multi-state fit results (second orange point, right). Median averages of large-$m$ Lanczos energy estimators, residual bounds, and combined gap and variational bounds are shown as blue points, right. Arrows denote one-sided variational bounds. Horizontal lines denote bootstrap medians of two-sided bounds; triangles (chevrons) denote 68% confidence intervals for residual (gap plus variational) bounds.
• Figure 2: Scattering phase shift determinations from the $S$-wave-truncated Lüscher quantization condition Luscher:1986pfLuscher:1990ux for $I=1$ (top) and $I=0$ (bottom) channels. Orange bands denote statistical uncertainties. Chevron-bounded blue bands denote gap plus variational bounds at 68% confidence.
• Figure 3: Results for the $\beta=6.3$ ensemble with $m_{\pi} \approx 180$ MeV and $L=48$, left, and the same parameters with $L=64$, right. Upper: Nucleon effective masses and multi-state fit results (orange) along with Lanczos energy estimators and two-sided residual and gap bounds from a median of large-$m$ results (blue); details are as in \ref{['fig:nn_summary']}. Lower: Iteration dependence of Lanczos energy and residual-norm-square estimators. A vertical dashed lines is placed after the median $m$ where spurious eigenvalues first appear; the previous iteration (at the inner bootstrap level) is used for SLRVL state identification; see Ref. long. Correlator results with $t \in \{0,1\}$ are omitted to avoid complications arising from contact terms.
• Figure 4: Results for the $\beta=6.5$ ensemble with $m_{\pi} \approx 135$ MeV and $L=96$, left, and the $\beta=6.5$ ensemble with $m_{\pi} \approx 180$ MeV and $L=72$, right. Details are as in \ref{['fig:nuc-EMP-64-48']}.
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