Bayesian Analysis and Analytic Continuation of Scattering Amplitudes from Lattice QCD

Abstract

We present a novel procedure for analyzing the lattice-QCD spectrum via the finite-volume formalism to obtain constraints on multi-hadron scattering amplitudes at both real and complex energies. This approach combines a Bayesian reconstruction of the scattering amplitude on the real axis with Nevanlinna interpolation for analytic continuation to complex-valued energies. The method is non-parametric, inherently accounting for parametrization dependence within the uncertainty. We demonstrate the applicability of this approach using both toy data and real lattice-QCD data in resonant systems from the HadSpec and BaSc collaborations.

Figures (15)

• Figure 1: Locations of singularities in the scattering amplitude $\mathcal{M}_\ell$ in the complex momentum and energy planes, inspired by Fig. 1 in Ref. Matuschek:2020gqe.
• Figure 2: The Schwarz--Christoffel mapping, \ref{['eq:schwarz_christoffel_rectangle']}, of a rectangle (left panel) to the unit disk (right panel). A rectangle of width $W$ and height $H$ is centered at point $C$ below the real line and contains a subset of the real line. Black circles denote the schematic locations of input energies on the real line for the analytic continuation problem. The colored triangles fix the relative orientation of arbitrary points in the rectangle under the mapping to the unit disk, , the upward-pointing blue triangle on the left is mapped to the upward-pointing blue triangle on the right.
• Figure 3: Sketch of the uncertainty analysis procedure described in \ref{['sec:pole_locations_Nevanlinna_uncertainty']}. For a fixed sample from the posterior distribution, the function $\mathcal{K}^{-1}$ is defined by a reference point with coordinates $(r, \theta)$ within the Wertevorrat of the Nevanlinna interpolation of $(\bm E, \bm u)$. The coordinates $(r, \theta)$ are chosen from a uniform distribution over the unit disk. They are held fixed independent of $E$, while the location $c_N$ and size $r_N$ of the Wertevorrat obviously depend on $E$.
• Figure 4: Numerical example based on the toy resonance presented in \ref{['sec:toy_example']}. The blue points in the background show the noisy mock data generated as detailed in the text. Each set of triangles represents an independent sample drawn from the (Gaussian approximation of the) posterior distribution. For each sample, the pole position is added as a reference.
• Figure 5: Phase shift for the model in \ref{['eq:phaseshiftBW']} (orange line). The blue points show the noisy mock data generated as detailed in the text, while the purple line and band depict the prior. The green line and band indicate the posterior central value according to \ref{['eq:expectedu']} and the Bayes uncertainty according to \ref{['eq:erroru']}. The Monte-Carlo integration error as well as the size of the Wertevorrat are negligibly small. Note that the effect of $\ell_c$ is not visible in this graphical representation of the prior (purple), although it does affect the results of the Bayesian analysis, the posterior (green). The lower panel displays the absolute difference between the posterior and the truth in units of the Bayes error.