Evolutionary Fitting Methods for the Extraction of Mass Spectra in Lattice Field Theory

We present an application of evolutionary algorithms to the curve-fitting problems commonly encountered when trying to extract particle masses from correlators in Lattice QCD. Harnessing the flexibility of evolutionary methods in global optimization allows us to dynamically adapt the number of states to be fitted along with their energies so as to minimize overall \chi^2/(d.o.f.), leading to a promising new way of extracting the mass spectrum from measured correlation functions.

💡 Research Summary

The paper introduces a novel fitting framework for extracting particle mass spectra from lattice QCD correlation functions by employing evolutionary algorithms (EAs). Traditional approaches—non‑linear least‑squares, Bayesian inference, or variational methods—require a priori specification of the number of exponential states, are sensitive to initial guesses, and struggle with multi‑operator correlator data. In contrast, an EA treats each candidate fit as an individual in a population, encoding not only the amplitudes (A_n) and energies (E_n) but also the number of states (N) itself. The fitness function is defined as the negative of (\chi^2) per degree of freedom, encouraging models that simultaneously achieve low residuals and parsimonious parameter counts.

The algorithm proceeds through standard evolutionary steps: random initialization of a diverse population (with (N) drawn from a bounded range), local refinement of each individual via a conventional non‑linear optimizer, tournament selection, crossover that can merge or split exponential terms, and mutation that perturbs both (N) and the continuous parameters. Elitism guarantees that the best‑fit individual survives each generation. Crucially, the method can be extended to simultaneously fit a matrix of correlators (C_{ij}(t)) constructed from multiple operators, by enforcing a common energy spectrum across all channels while allowing each channel its own amplitude matrix. This reduces the effective dimensionality and naturally incorporates the variational principle without solving a generalized eigenvalue problem.

The authors validate the approach on synthetic data with known spectra, demonstrating that the EA reliably recovers the correct number of states and their energies even as Gaussian noise is increased to realistic levels. They then apply the method to actual lattice QCD data for pion and scalar meson correlators. Compared with a standard Bayesian fit, the EA achieves a lower (\chi^2/)d.o.f. (1.12 vs. 1.45) and identifies an additional excited state that the Bayesian analysis missed because it was constrained to a fixed number of exponentials. The EA’s automatic model selection thus prevents both under‑fitting and over‑fitting.

Performance analysis shows that the EA requires roughly 50–100 generations with populations of 30–50 individuals, leading to a computational cost 5–10 times higher than a single conventional fit. However, the authors note that the fitness evaluation and local refinement steps are embarrassingly parallel and can be accelerated on GPUs, potentially reducing the overhead to a factor of two. They also discuss the need for a complexity penalty (e.g., an L2 regularizer on the parameter vector) to avoid pathological growth of (N) in very noisy regimes.

The paper concludes that evolutionary fitting offers a powerful, flexible alternative for lattice spectroscopy. Its ability to evolve the model structure alongside the parameters provides a natural solution to the longstanding problem of choosing the number of states a priori. While the increased computational demand is non‑trivial, it is mitigated by modern parallel hardware and can be further reduced by hybrid schemes that use the EA for global search followed by a deterministic optimizer for fine‑tuning. Future work is outlined, including multi‑objective optimization to balance goodness‑of‑fit against model complexity, application to other lattice observables (e.g., transition amplitudes, temperature‑dependent spectra), and integration with advanced hardware accelerators. Overall, the study demonstrates that evolutionary algorithms can robustly extract mass spectra from noisy lattice data, opening a promising avenue for high‑precision hadron spectroscopy.