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Statistical Angles on the Lattice QCD Signal-To-Noise Problem

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Statistical Angles on the Lattice QCD Signal-To-Noise Problem c Copyright 2017 Michael L. Wagman arXiv:1711.00062v1 [hep-lat] 31 Oct 2017 Statistical Angles on the Lattice QCD Signal-to-Noise Problem Michael L. Wagman A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy University of Washington 2017 Reading Committee: Martin J. Savage, Chair Silas R. Beane David B. Kaplan Program Authorized to Offer Degree: Physics University of Washington Abstract Statistical Angles on the Lattice QCD Signal-to-Noise Problem Michael L. Wagman Chair of the Supervisory Committee: Prof. Martin J. Savage Physics 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 in the distribution of baryon correlation functions, associated with stable distributions and L´evy flights, are found to play a central role in their time evolution. Building on these observations, a new statistical analysis technique called phase reweight- ing 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 dynam- ical 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 1 of the ΞΞ( S0) 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 applies to real but non-positive correlation functions and 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 corre- lation 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. TABLE OF CONTENTS Page List of Figures . vi Chapter 1: Introduction . .1 1.1 Quarks and Gluons . .6 1.2 Lattice Field Theory . 16 1.3 Meson and Baryon Correlation Functions . 27 Chapter 2: Statistical Distributions of Baryon Correlation Functions . 42 2.1 A Magnitude-Phase Decomposition . 49 2.2 An Improved Estimator . 85 Chapter 3: Phase Reweighting . 100 Chapter 4: Meson Statistics and Real-Valued Sign Problems . 111 4.1 A Magnitude-Sign Decomposition . 113 4.2 Non-Positive Correlation Functions Moments . 119 4.3 Phase Reweighting Non-Positive Correlation Functions . 121 Chapter 5: Conclusion and Outlook . 128 Bibliography . 132 i ACKNOWLEDGMENTS An old proverb says that it takes a village to raise a child. In my experience, raising a graduate student all the way through twenty-first grade requires not just one but several villages. First and foremost, I am grateful to my advisor Prof. Martin Savage for teaching by example how to always stay curious about the physical world, turn fuzzy questions into crisp ideas, and push through whatever unexpected obstacles arise to finish a worthwhile project in however long it takes. Martin's focus, commitment, and sense of justice have been inspirational to me, and his training has given me a much deeper understanding of both how to calculate anything and how to assess what is actually worth calculating in order to learn more about nature. My earliest teachers were my parents Kathy and Ed and my sister Nicole. Their encouragement and willingness to keep answering \why?" built the foundation of my education, and the worlds of spaceships and superheros that I first explored with Nicole taught me to think of science as exciting and anything as possible. A number of other teachers have played essential roles in my education. I am for- ever grateful to my middle school teacher Mrs. Graham, who opened my eyes to the wonders of mathematics and taught me to think critically and independently about every subject from math to literature. I am likewise grateful to the high school teach- ers who nurtured my passion for learning, especially Ms. Hawthorn, Mr. Young, Mr. Davis, Mr. Goldsborough, and my AP physics teacher Mr. Rowe for confusing and de-confusing me about what an electric field is and sharing an infectious enthusiasm for physics. I would also like to wholeheartedly thank my classmates, especially my ii friends from Mrs. Graham's class, my collaborators in TSA, and my band mates, for making learning a fun collaborative project and pondering life's big philosophical questions with me as if we could solve them all. My time at Brown University was full of lessons from great teachers. I am partic- ularly grateful to Prof. James Valles for his encouragement and mentoring and thank him, Ilyong Jung, and Shawna Hollen for introducing me to the world of research. I am also grateful to Prof. Jan Hesthaven and Scott Field for nurturing my interests in theoretical and computational physics research. I learned many lessons about life and about physics from my friends and roommates at Brown, and I cannot thank them enough for their love and support as well as countless stimulating discussions. In particular I thank Matt Dodelson for convincing me to take any math course that looked interesting regardless of the prerequisites and thank him, Adam Coogan, and Tom Iadecola for helping me survive the ensuing all-nighters. During graduate school, I benefited greatly from the teachings of Profs. Silas Beane, David Kaplan, Andreas Karch, Gerald Miller, Ann Nelson, Sanjay Reddy, Martin Savage, Steve Sharpe, and Larry Yaffe. Much of my learning was done along- side my physics-life-buddy Dorota Grabowska, with whom I look forward to many more dinner table conversations about the foundations of quantum field theory, and many dear friends from my graduate-school cohort and fellow residents of Physics House. I am deeply grateful to the other members of the NPLQCD collaboration: Silas Beane, Emmanuel Chang, Zohreh Davoudi, Will Detmold, Kostas Orginos, As- sumpta Parre~no,Martin Savage, Phiala Shanahan, Brian Tiburzi, and Frank Winter for welcoming me into an inspirational family of collaborators. I thank Michael Bu- choff for the countless hours we spent following half of \shut up and calculate" and for guiding me through the early stages of entering the physics community. I also thank Ra´ulBrice~no,Aleksey Cherman, Max Hansen, Natalie Klco, and Alessandro Roggero iii for their guidance and collaboration and for the positive energy that they bring to the world. I am also grateful for my opportunities to meet friends and collaborators outside UW, in particular Sri Sen and my TASI cohort. The research presented in this thesis has also benefited from conversations with Tanmony Bhattacharya, Joe Carl- son, Tom DeGrand, Leonardo Giusti, Rajan Gupta, Dean Lee, Alessandro Lovato, Steve Peiper, Daniel Trewartha, and Bob Wiringa, and would not have been possible without inspiration from David Kaplan to dive into learning about the signal-to-noise problem. Last but far from least, I thank my partner Kelly Buckley for her support during the highs and the lows of research, for helping me find big picture explanations, and for joining me on our next adventure. iv DEDICATION To teachers who can explain quantum field theory and to students who find it beautiful. v LIST OF FIGURES Figure Number Page 1.1 Perturbative fluctuations of the QED vacuum, left, and QCD vacuum, right, about a source of charge in the vacuum. Fluctuations of electron-positron pairs in QED are more likely to be oriented with the opposite charge closer to the test charge at center, and the QED vacuum accordingly acts as a dielec- tric where charge is slightly screened at large distances compared to classical expectations. In QCD, quark-antiquark pairs act similarly, but fluctuations of gluon fields also contribute. A gluon source, right, induces vacuum fluctu- ations from gluons in orthogonal color orientations that behave like charged vector particle loops. Both the QED and QCD configurations shown act as paramagnets as well as dielectrics if the spins of the fluctuations are aligned. Pauli blocking raises the energy of each spin-aligned fermion, but no such ef- fect occurs for bosons. Explicit calculation shows that paramagnetic effects lead to an overall decrease in the vacuum energy if color charge is effectively enhanced, rather than screened, at large distances [19, 256]. 12 1.2 A visualization of the topological charge density of a generic LQCD vacuum gluon configuration, courtesy Daniel Trewartha.
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