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Register a new userStatistical Angles on the Lattice QCD Signal-to-Noise Problem
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Michael L. Wagman
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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 can represent generic QCD predictions in terms of well-defined path integrals, but the sign and signal-to-noise problems have obstructed lattice QCD calculations of large nuclei and nuclear matter in practice. This thesis presents a statistical study of lattice QCD 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. The phases of complex correlation functions are analyzed in the framework of circular statistics, and the time evolution of the phase is shown to resemble a heavy-tailed random walk on the unit circle. Building on these observations, a new technique called phase reweighting is investigated that involves calculations of phase differences over fixed-length time intervals. Phase reweighting allows results for hadronic observables to be extracted from large-time correlation functions with constant signal-to-noise ratios. The signal-to-noise problem re-emerges as the length of the phase-difference interval is increased. Applications of phase reweighting to meson, baryon, and two-baryon systems are discussed.
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Lattice QCD simulations of multi-baryon correlation functions can predict the structure and reactions of nuclei without encountering the baryon chemical potential sign problem. However, they suffer from a signal-to-noise problem where Monte Carlo estimates of observables have quantum fluctuations that are exponentially larger than their average values. Recent lattice QCD results demonstrate that the complex phase of baryon correlations functions relates the baryon signal-to-noise problem to a sign problem and exhibits unexpected statistical behavior resembling a heavy-tailed random walk on the unit circle. Estimators based on differences of correlation function phases evaluated at different Euclidean times are discussed that avoid the usual signal-to-noise problem, instead facing a signal-to-noise problem as the time interval associated with the phase difference is increased, and allow hadronic observables to be determined from arbitrarily large-time correlation functions.
Path integrals describing quantum many-body systems can be calculated with Monte Carlo sampling techniques, but average quantities are often subject to signal-to-noise ratios that degrade exponentially with time. A phase-reweighting technique inspired by recent observations of random walk statistics in correlation functions is proposed that allows energy levels to be extracted from late-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 Quantum Chromodynamics calculations of the $rho$ and nucleon masses and of the $XiXi$ binding energy show consistency between standard results obtained using earlier-time correlation functions and phase-reweighted results using late-time correlation functions inaccessible to standard statistical analysis methods.
Ideas and recent results for light-front Hamiltonian quantisation of lattice gauge theories.
The long standing problem is solved why the number and the location of monopoles observed in Lattice configurations depend on the choice of the gauge used to detect them, in contrast to the obvious requirement that monopoles, as physical objects, must have a gauge-invariant status. It is proved, by use of non-abelian Bianchi identities, that monopoles are indeed gauge-invariant: the technique used to detect them has instead an efficiency which depends on the choice of the abelian projection, in a known and controllable way.
We calculate the electric dipole moment of the nucleon induced by the QCD theta term. We use the gradient flow to define the topological charge and use $N_f = 2+1$ flavors of dynamical quarks corresponding to pion masses of $700$, $570$, and $410$ MeV, and perform an extrapolation to the physical point based on chiral perturbation theory. We perform calculations at $3$ different lattice spacings in the range of $0.07~{rm fm} < a < 0.11$ fm at a single value of the pion mass, to enable control on discretization effects. We also investigate finite size effects using $2$ different volumes. A novel technique is applied to improve the signal-to-noise ratio in the form factor calculations. The very mild discretization effects observed suggest a continuum-like behavior of the nucleon EDM towards the chiral limit. Under this assumption our results read $d_{n}=-0.00152(71) bartheta e~text{fm}$ and $d_{p}=0.0011(10) bartheta e~text{fm}$. Assuming the theta term is the only source of CP violation, the experimental bound on the neutron electric dipole moment limits $left|barthetaright| < 1.98times 10^{-10}$ ($90%$ CL). A first attempt at calculating the nucleon Schiff moment in the continuum resulted in $S_{p} = 0.50(59)times 10^{-4} bartheta e~text{fm}^3$ and $S_{n} = -0.10(43)times 10^{-4} bartheta e~text{fm}^3$.
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