Constrained Curve Fitting

TL;DR

Constrained curve fitting addresses the underconstrained nature of lattice QCD analyses by injecting Bayesian priors into the fit of correlators modeled as $G_{ m th}(t)=\sum_{n=1}^{\infty} A_n e^{-E_n t}$. The method augments the traditional likelihood with a prior term to form ${\chi^2_{\rm aug}}$, allowing many higher-energy states to be included without destabilizing the leading parameter estimates and removing the need to optimally choose a $t_{\min}$. A full Bayesian treatment is presented, with Gaussian priors justified by maximum entropy and practical error estimation via quadratic approximations or bootstrap methods; the framework also supports empirical Bayes and maximum-entropy nonparametric approaches. Demonstrations across meson and heavy-quark correlators show precise extraction of low-lying energies while higher states are controlled by priors, enabling more accurate determinations of lattice spacings, quark masses, and broader extrapolations, with broad applicability to correlator analyses in lattice QCD and related fields.

Abstract

We survey techniques for constrained curve fitting, based upon Bayesian statistics, that offer significant advantages over conventional techniques used by lattice field theorists.

Figures (5)

• Figure 1: Fit values for the lowest two energies from a 2-term fit to a local-local $\Upsilon$ correlator using different ${t_{\rm min}}$'s. The correct values, from other analyses, are indicated by the dotted lines.
• Figure 2: Fit values for the two lowest energies from unconstrained fits with different numbers of terms in ${G_{\rm th}}$. The correlator is a local-local $\Upsilon$ correlator and is fit for all $t$'s.
• Figure 3: Fit values for the two lowest energies from constrained fits with different numbers of terms in ${G_{\rm th}}$. The correlator is a local-local $\Upsilon$ correlator and is fit for all $t$'s.
• Figure 4: The constrained 5-term fit to the local-local $\Upsilon$ correlator. The statistical errors in the data points are too small to be resolved in the plot.
• Figure 5: A constrained fit to a local-local $B$ meson correlator made with NRQCD and staggered quark propagators. The energies of the three lowest-energy states are listed in lattice units. The statistical errors in the data points are too small to be resolved in the plot.