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Exploration of New Methods for Lattice QCD Andres Rios Tascon JUN 2 2 2017 Exploration of New Methods for Lattice QCD by Andres Rios Tascon Submitted to the Department of Physics in partial fulfillment of the requirements for the degree of Bachelor of Science in Physics at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2017 @ Massachusetts Institute of Technology 2017. All rights reserved. Author .................................. Signature redacted Department of Physics May 12, 2017 redacted C ertified by ...............................Signature ............. William Detmold Assistant Professor of Physics Thesis Supervisor A ccepted byAcceped ............................. bySignature . .. .. .. -redacted-- - Nergis Mavalvala Senior Thesis Coordinator, Department of Physics MASSACHUSET INSTITUTE OF TECHNOLOGY JUN 2 2 2017 LIBRARIES ARCHIVES 77 Massachusetts Avenue Cambridge, MA 02139 MITLibraries http://Iibraries.mit.edu/ask DISCLAIMER NOTICE Due to the condition of the original material, there are unavoidable flaws in this reproduction. We have made every effort possible to provide you with the best copy available. Thank you. The images contained in this document are of the best quality available. 2 Exploration of New Methods for Lattice QCD by Andres Rios Tascon Submitted to the Department of Physics on May 12, 2017, in partial fulfillment of the requirements for the degree of Bachelor of Science in Physics Abstract We explore two methods aimed at alleviating two difficulties in Lattice QCD: statis- tical noise and data storage. The first method intends to improve the signal-to-noise (S/N) ratio in three-point correlators, by extending previous work by Detmold and Endres. We test the method in the measurement of two observables: the nucleonic axial charge, and a matrix element computation related to the electromagnetic form factor of the rho meson. Only in the case of the rho we see a very slight improvement. We conclude that, in general, a case-by-case study would be needed to determine the effectiveness of the S/N optimization. The second method that we study aims to improve data compression of gauge fields. It consists in generating a set of matrices distributed roughly uniformly along the group manifold, and constructing fine lattices around each of these matrices. We show that this compression can indeed provide better performance for SU(2) and SU(3) than the Lie algebra compression, but the improvement is not very significant. Nevertheless, we show that it is fairly close to the best compression one could hope to achieve with this type of method. We find that the compression procedure is very costly, which makes it currently impractical for machine precision- compression. We conclude that studies must be done to determine if it can be improved by using additional information from the gauge fields. Thesis Supervisor: William Detmold Title: Assistant Professor of Physics 3 4 Acknowledgments I am deeply grateful to Professor William Detmold, my research supervisor. I learned an incredible number of things working with him over the past year. I am also grate- ful to my academic advisor, Professor Hong Liu, and my previous research advisor, Professor Markus Klute. I would also like to thank my parents and brother for their constant support throughout my undergraduate studies, and my friends at MIT and elsewhere. 5 6 Contents 1 Introduction 15 1.1 Lattice Field Theory .... ....... ...... ...... .... 15 1.2 The Discretized Path Integral ....... ............... 16 1.3 Monte Carlo Integral ... ...... ....... ...... ..... 17 1.4 Lattice Q CD ............. ............ ...... 18 1.5 Correlation Functions ...... ........ ........ ..... 19 1.6 Variational Method ..... ........ ......... ...... 22 1.7 Lattice QCD Software .. ...... ....... ...... ..... 24 2 Signal-to-Noise Optimization 25 2.1 S/N Optimization of Two-Point Correlators ... ........ ... 25 2.1.1 S/N Optimization with a Fixed Source ..... .. 28 2.1.2 S/N Optimization with Unconstrained Source and Sink ... 29 2.1.3 Two-State Toy Model ........ ............ ... 29 2.2 S/N Optimization of Three-Point Correlators ........... ... 32 2.2.1 Two-State Toy Model .... ........... ........ 33 2.3 Applications: Hadronic Three-Point Functions .. ........... 35 2.3.1 Nucleon Axial Charge .................. .... 36 2.3.2 Rho Meson Form Factor ...... ............ ... 38 2.4 Discussion and Conclusions ... ..... ..... ..... ..... 40 3 Gauge Field Data Compression 43 3.1 Compression Method Description .... ..... ..... ..... 45 7 3.2 Theoretical performance . .... .... ..... .... .... ... 47 3.3 Implementation . ..... ..... ...... ..... .... .... 48 3.4 Application: SU(2) Field .. .... .... .... .... .... ... 50 3.5 Application: SU(3) Field .. .... .... .... .... .... ... 53 3.6 Discussion and Conclusions .. ...... ..... .... ..... 54 4 Conclusions 57 8 List of Figures 1-1 An example of an effective mass plot. At early times there is excited state contamination, while at late times the noise increases exponentially. 21 1-2 An example of the effectiveness of the variational method. Excited state contamination at early times is reduced, resulting in a larger plateau. ... ........ ........ ........ ....... 24 2-1 Comparison of the purely variational, variational plus S/N optimized, and purely S/N optimized correlators ..... .... ..... .... 29 2-2 S/N landscape of two-point correlators of the two-state system with a = 0.04 and b = c = 0.01, using the parametrization from (2.21). The lower right plot shows a cross-sectional view of the other three plots at 6 = 0. ...... ........ ......... .. ...... 31 2-3 S/N landscape of three-point correlators of the two-state system with a = 0.04 and b = c = 0.01, using the parametrization from (2.21). The lower right plot shows a cross-sectional view of the other three plots at 6 = 0. ..... .... ..... ..... .... ..... ..... 35 2-4 Comparison of the location of the S/N maximum for two- and three- point functions of the two-state system with a = 0.04 and b = c = 0.01, using the parametrization from (2.21). For large times (small A) the locations of the maxima are similar. ... .... .... .... .... 35 9 2-5 Comparison of the variational method, variational plus S/N optimiza- tion, and S/N optimization for the nucleon axial charge for a sink separation of t = 15. The S/N optimization offers no improvement over the variational method in this case. .. .... ..... ..... 37 2-6 Comparison of the results obtained from the fits for the nucleon axial charge for different time separations. The S/N optimization offers no improvement over the variational method in this case. .. ....... 37 2-7 Same as Fig. 2-6, but without enforcing (anti)hermiticity. The S/N optimization for time separations t = 15 and t = 18 offers a slightly more precise result, but it is likely less accurate. .. ....... ... 38 2-8 Comparison of the variational method, variational plus S/N optimiza- tion, and S/N optimization for the rho form factor for a sink separation of t = 15. The S/N optimization offers a very slight improvement. .. 39 2-9 Comparison of the results obtained from the fits for the rho form factor for different sink separation times. The S/N optimization offers only a very slight improvement for t = 15. ...... ...... ...... 39 2-10 Same as Fig. 2-8, but without enforcing hermiticity. The S/N opti- mization significantly reduces the noise of individual data points, but the plateaus become less pronounced. ..... ...... ...... 40 2-11 Same as Fig. 2-9, but without enforcing hermiticity. The S/N op- timization offers practically no improvement overall, since the more subtle plateaus inhibit the improvement in the statistical noise. .. 40 3-1 Distribution of distances between random SU(2) matrices and the clos- est element in E = G120. ....... .................. 49 3-2 Distribution of distances between products of Ri (as in Eq. (3.5)) and the identity, in the case of SU(2). Shown are E = 0.06,0.09,0.12,0.15, w ith 16 bits. ................................ 49 10 3-3 Distance from random matrices to the closest matrix in the lattice. Increasing the number of bits reduces the average distance, as well as improving the behavior of the tail ........ ........... 51 3-4 Average and maximum distance between random SU(2) matrices and the closest element in the lattice as a function of bits used for the fine lattice. The lines correspond to the best exponential fits. ....... 52 11 12 List of Tables 3.1 Results from using 5 sets of matrices Ri and 1000 random SU(2) ma- trices. The increase in kurtosis and decrease in elapsed time at 22 bits is due to the switch to a different partitioning. ............. 51 3.2 Comparison of the estimated number of bits required to describe a ma- trix with this method and other methods, as well as the best theoretical com pression. ................... ............. 53 3.3 Same as Table 3.2, but for SU(3). .......... .......... 54 13 14 Chapter 1 Introduction 1.1 Lattice Field Theory Quantum Field Theories (QFTs) comprise the current theoretical framework for de- scribing the elementary particles and interactions of nature, and also serve to describe composite particles and quasiparticles in condensed matter systems. Many observ- ables, such as lepton-lepton scattering cross sections, can be found using perturbative techniques. However, many other observables can only be studied with nonpertur- bative calculations, which generally present a more formidable challenge. Significant progress in the nonperturbative regime started to be made with the introduction of lattice gauge theory by Wilson in 1974 [1]. The idea behind it was to discretize
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