Statistical Angles on the Lattice QCD Signal-to-Noise Problem
Wagman, Michael Louis
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The theory of quantum chromodynamics (QCD) encodes the strong interactions that bind quarks and gluons into nucleons and that bind nucleons into nuclei. Predictive control of QCD would allow nuclear structure and reactions as well as properties of supernovae and neutron stars to be theoretically studied from first principles. Lattice QCD (LQCD) can represent generic QCD predictions in terms of well-defined path integrals, but the sign and signal-to-noise problems have obstructed LQCD calculations of large nuclei and nuclear matter in practice. This thesis presents a statistical study of LQCD correlation functions, with a particular focus on characterizing the structure of the noise associated with quantum fluctuations. The signal-to-noise problem in baryon correlation functions is demonstrated to arise from a sign problem associated with Monte Carlo sampling of complex correlation functions. Properties of circular statistics are used to understand the emergence of a large time noise region where standard energy measurements are unreliable. Power-law tails associated with stable distributions and Levy flights are found to play a central role in the time evolution of baryon correlation functions. Building on these observations, a new statistical analysis technique called phase reweighting is introduced that allow energy levels to be extracted from large-time correlation functions with time-independent signal-to-noise ratios. Phase reweighting effectively includes dynamical refinement of source magnitudes but introduces a bias associated with the phase. This bias can be removed by performing an extrapolation, but at the expense of re-introducing a signal-to-noise problem. Lattice QCD calculations of the ρ+ and nucleon masses and of the ΞΞ(1S0) binding energy show consistency between standard results obtained using smaller-time correlation functions and phase-reweighted results using large-time correlation functions inaccessible to standard statistical analysis methods. A detailed study of the statistics and phase reweighting of isovector meson correlation functions demonstrates that phase reweighting can be used to predict ground-state energies of correlation functions that are too noisy to be analyzed by other methods. The relative precision of phase reweighting compared to standard methods is expected to be increased on lattices with larger time directions than those considered in this thesis, and these preliminary studies suggest phase reweighting of noisy nuclear correlation functions should be investigated on larger lattices. The results of this thesis suggest that phase reweighting may be applicable more broadly to real but non-positive correlation functions in quantum Monte Carlo simulations of particle, nuclear, and condensed matter physics systems as well as to complex correlation functions describing multi-baryon systems in LQCD.
- Physics