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

TL;DR

This work tackles the challenge of excited-state uncertainties in lattice QCD for multi-hadron systems by introducing two-sided bounds derived from Lanczos methods: residual bounds ( Hermiticity-based, largely assumption-free) and gap bounds (require explicit spectral-gap assumptions). It systematically tests these bounds on solvable models and then applies them to a high-statistics two-nucleon calculation at $m_\pi\sim800$ MeV, comparing with GEVP and asymmetric correlator analyses. The results show that gap bounds can provide tighter, more reliable constraints on spectral energies and phase shifts than traditional energy estimators, particularly under no-missing-states assumptions; however, asymmetries and stagnation in finite statistics complicate interpretation. The study emphasizes that robust, physics-guided assumptions are essential to extracting meaningful scattering information from current LQCD data, and it outlines concrete paths for improving uncertainty quantification in multi-hadron spectroscopy.

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 $m_π\sim 800$ 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 $NΔ$ and $ΔΔ$ 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.

Figures (45)

• Figure 1: Example Davis-Kahan and Haas-Nakatsukasa gap parameter estimates $\hat{g}_0^{(m)}$ and $\hat{G}_0^{(m)}$ for $\lambda_0^{(m)}$ (circled point) with estimated $T$ eigenvalues shown as dashed lines and Ritz values shown as points. Since $\hat{G}_1^{(m)}> \hat{g}_1^{(m)}$, Haas-Nakatsukasa gap bounds will be tighter than the Davis-Kahan gap bounds in this case. Note that if the estimated $\lambda$ spectrum depicted was in fact incomplete and additional unexpected eigenvalues were present in the true spectrum, then gap bounds computed using either of these approaches could give incorrect bounds. In the case of Davis-Kahan gap bounds, this will occur if a second true eigenvalue is closer to $\lambda_0^{(m)}$ than $\hat{\lambda}_1$, while Haas-Nakatsukasa gap bounds will additionally fail if a third true eigenvalue is closer to $\lambda_0^{(m)}$ than $\hat{\lambda}_2$.
• Figure 2: Finite-statistics ($N_{\rm cfgs}\in\{10^4,10^5,10^6\}$) and exact results for Lanczos energy estimators, with $E_0$ shown as a dashed red line for comparison. The dotted line is placed before the median values of the smallest $m$ where spurious eigenvalues are identified for $N_{\rm cfgs}=10^6$ by the Hermitian subspace and ZCW tests with $F_{ZCW} = 10$; the previous iteration (at the inner bootstrap level) is used for SLRVL state identification, as described in the main text. Infinite-statistics points are filtered using the ZCW test with $\varepsilon_{\rm ZCW} = 10\, e^{-E_0 L_t}$ in order to remove thermal states; the largest-magnitude non-spurious Ritz value is then used to define $E_0^{(m)}$. For $t=11$ at infinite statistics, this ZCW cut leaves two non-spurious energy estimators $E_0^{(m)}$ and $E_1^{(m)}$ that are close to $E_0$; both are shown.
• Figure 3: Finite-statistics ($N_{\rm cfgs}\in\{10^4,10^5,10^6\}$) and exact results for Lanczos residual-norm-square estimators.
• Figure 4: Comparisons of finite-$m$ truncation errors with the widths of residual bounds (top) and gap bounds (bottom), for both exact and finite-statistics results. The values at $m=1$ are not shown because $\lambda_0^{(1)}$ is closer to $\lambda_1$ than $\lambda_0$ and bounds therefore involve the first excited state rather than the ground state. Non-monotonicity of finite-$m$ truncation errors and the non-monotonicity in $m$ of bound widths both arise from thermal effects discussed in the main text.
• Figure 5: Infinite-statistics results for Lanczos energy-estimators as a function of iteration count are compared to the exact spectrum (gray dashed lines), which, including thermal states, is given by $(n+1)E_0$ for all $n \in \mathbb{Z}$. Energies for states passing the ZCW test with $\varepsilon_{\rm ZCW} = e^{-E_0 L_t}$ are shown as colored squares and diamonds; those for states with overlaps below this ZCW cut are shown as gray circles. Non-monotonic features visible for $m \in \{6,9,15\}$ arise from thermal effects.