GENERALIZED POLYLOGARITHMS IN PERTURBATIVE QUANTUM FIELD THEORY A DISSERTATION SUBMITTED TO THE DEPARTMENT OF PHYSICS AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY Jeffrey S. Pennington August 2013 © 2013 by Jeffrey Starr Pennington. All Rights Reserved. Re-distributed by Stanford University under license with the author. This work is licensed under a Creative Commons Attribution- Noncommercial 3.0 United States License. http://creativecommons.org/licenses/by-nc/3.0/us/ This dissertation is online at: http://purl.stanford.edu/zq746zs7783 Includes supplemental files: 1. Ancillary files for Chapter 1 (Chapter1_MRK.tar.gz) 2. Ancillary files for Chapter 4 (Chapter4_R63.tar.gz) 3. Ancillary files for Chapter 5 (Chapter5_R64.tar.gz) ii I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Lance Dixon, Primary Adviser I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Renata Kallosh I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Michael Peskin Approved for the Stanford University Committee on Graduate Studies. Patricia J. Gumport, Vice Provost Graduate Education This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file in University Archives. iii Acknowledgements My experience in graduate school has been truly rewarding andfulfilling,thanksin no small part to the many talented students and supportive mentors I have met along the way. I am particularly grateful to Lance Dixon, whose support, guidance, and expertise have been indispensable over the years. Our collaborations together have taught me a tremendous amount and have helped shape my growth as a research scientist. Iwouldalsoliketoacknowledgemymanyothercollaborators,fromwhomIhave learned a great deal and without whom much of this work would have been impossible. In particular, I would like to thank James Drummond and ClaudeDuhrformany fruitful discussions and for teaching me many tricks of the trade. IamgratefultothetheorygroupatSLACforprovidingsucharich and stimulating research environment. I am particularly indebted to MichaelPeskinforhisdedication to student education and for his time spent overseeing numerous study groups. It was a pleasure to know and work with many amazing students. Thank you to Kassahun Betre, Camille Boucher-Veronneau, Randy Cotta,MartinJankowiak, Andrew Larkoski, Tomas Rube, and many others for countless discussions and plenty of good times. Thank you to all of my friends, especially Jared Schwede and David Firestone, who kindly received more than their fair share of complaints when times were tough, and were always there to help me unwind. IcannotexpressenoughthankstoLimor,whohasbeenaconstant source of sup- port and encouragement, even through the countless working weekends, late nights, and (occasional) foul moods that would result. iv Finally, I thank my Mom, Dad, and brother for being there for meateverystep of this long journey. You gave me the confidence to pursue my intellectual interests to the fullest possible extent, and without your enduring support for my education I never would have made it this far. v Contents Acknowledgements iv Introduction 1 I Functions of two variables 10 1Single-valuedharmonicpolylogarithmsandthemulti-Regge limit 11 1.1 Introduction . 11 1.2 The six-point remainder function in the multi-Regge limit . 20 1.3 Harmonic polylogarithms and their single-valued analogues . 26 1.3.1 Review of harmonic polylogarithms . 26 1.3.2 Single-valued harmonic polylogarithms . .. 29 1.3.3 Explicit construction . 33 1.4 The six-point remainder function in LLA and NLLA . .37 1.5 Thesix-pointNMHVamplitudeinMRK . 48 1.6 Single-valued HPLs and Fourier-Mellin transforms . ...... 54 1.6.1 The multi-Regge limit in (ν, n)space.............. 54 1.6.2 Symmetries in (ν, n)space.................... 56 1.6.3 General construction . 57 1.6.4 Examples . 59 1.7 Applications in (ν, n)space:theBFKLeigenvaluesandimpactfactor 63 1.7.1 TheimpactfactoratNNLLA . 63 vi 1.7.2 The four-loop remainder function in the multi-Regge limit . 66 1.7.3 Analytic results for the NNLL correction to the BFKL eigen- value and the N3LL correction to the impact factor . 71 1.8 Conclusions and Outlook . 80 2Thesix-pointremainderfunctiontoallloopordersinthemulti- Regge limit 83 2.1 Introduction . 83 2.2 The six-point remainder function in multi-Reggekinematics. 87 2.3 Review of single-valued harmonic polylogarithms .............................. 93 2.4 Six-point remainder function in the leading-logarithmic approximation of MRK . .96 2.4.1 The all-orders formula . 96 2.4.2 ConsistencyoftheMHVandNMHVformulas . 100 2.5 Collinear limit . 102 2.5.1 MHV................................ 103 2.5.2 NMHV............................... 107 2.5.3 The real part of the MHV remainder function in NLLA . 108 2.6 Conclusions . 109 3Leadingsingularitiesandoff-shellconformalintegrals 112 3.1 Introduction . 112 3.2 Conformal four-point integrals and single-valued polylogarithms . 121 3.2.1 The symbol . 122 3.2.2 Single-Valued Harmonic Polylogarithms (SVHPLs) . 124 3.2.3 Thex ¯ 0limitofSVHPLs ................... 127 → 3.3 The short-distance limit . 129 3.4 The Easy integral . 134 3.4.1 Residues of the Easy integral . 134 vii 3.4.2 The symbol of E(x, x¯)......................137 3.4.3 The analytic result for E(x, x¯): uplifting from the symbol . 138 3.4.4 The analytic result for E(x, x¯): the direct approach . 139 3.4.5 Numerical consistency tests for E ................ 141 3.5 TheHardintegral............................. 142 3.5.1 ResiduesoftheHardintegral . 142 3.5.2 The symbols of H(a)(x, x¯)andH(b)(x, x¯)............145 3.5.3 The analytic results for H(a)(x, x¯)andH(b)(x, x¯). 146 3.5.4 Numerical consistency checks for H ............... 152 3.6 The analytic result for the three-loop correlator . ....... 153 3.7 A four-loop example . 154 3.7.1 Asymptoticexpansions . 155 3.7.2 A differential equation . 158 3.7.3 Anintegralsolution. 161 3.7.4 Expression in terms of multiple polylogarithms . .... 167 3.7.5 Numerical consistency tests for I(4) ............... 171 3.8 Conclusions . 172 II Functionsofthreevariables 175 4Hexagonfunctionsandthethree-loopremainderfunction 176 4.1 Introduction . 176 (3) 4.2 Extra-pure functions and the symbol of R6 .............. 185 4.3 Hexagon functions as multiple polylogarithms . .... 194 4.3.1 Symbols . 194 4.3.2 Multiple polylogarithms . 196 4.3.3 The coproduct bootstrap . 200 4.3.4 Constructing the hexagon functions . 206 4.4 Integral Representations . 216 4.4.1 General setup . 216 4.4.2 Constructing the hexagon functions . 221 viii 4.4.3 Constructing the three-loop remainder function . .... 230 4.5 Collinear limits . 233 4.5.1 Expanding in the near-collinear limit . .233 4.5.2 Examples . 237 4.5.3 Fixing most of the parameters . 241 4.5.4 Comparison to flux tube OPE results . 242 4.6 Multi-Regge limits . 248 4.6.1 Method for taking the MRK limit . 252 4.6.2 Examples . 255 ′′ 4.6.3 Fixing d1, d2,andγ ....................... 259 (3) 4.7 Final formula for R6 and its quantitative behavior . 262 4.7.1 The line (u, u, 1) . 265 4.7.2 The line (1, 1,w).........................269 4.7.3 The line (u, u, u).........................271 4.7.4 Planes of constant w ....................... 277 4.7.5 The plane u + v w =1..................... 278 − 4.7.6 The plane u + v + w =1..................... 280 4.7.7 The plane u = v .......................... 282 4.7.8 The plane u + v =1 ....................... 284 4.8 Conclusions . 285 5Thefour-loopremainderfunction 288 5.1 Introduction . 288 5.2 Multi-Regge limit . 295 5.3 Quantitative behavior . 300 5.3.1 The line (u, u, 1) . 300 5.3.2 The line (u, 1, 1) . 304 5.3.3 The line (u, u, u).........................308 5.4 Conclusions . 314 ASingle-valuedharmonicpolylogarithmsandthemulti-Regge limit 316 A.1 Single-valuedharmonicpolylogarithms . ... 316 ix A.1.1 Expression of the L± functions in terms of ordinary HPLs . 316 A.1.2 Lyndonwordsofweight1 . 317 A.1.3 Lyndonwordsofweight2 . 317 A.1.4 Lyndonwordsofweight3 . 317 A.1.5 Lyndonwordsofweight4 . 318 A.1.6 Lyndonwordsofweight5 . 318 A.1.7 Expression of Brown’s SVHPLs in terms of the L± functions . 322 A.2 Analyticcontinuationofharmonicsums . 323 A.2.1 The basis in (ν, n) space in terms of single-valued HPLs . 328 BLeadingsingularitiesandoff-shellconformalintegrals 335 B.1 Asymptotic expansions of the Easy and Hard integrals . ... 335 B.2 AnintegralformulafortheHardintegral . .337 B.2.1 Limits . 339 B.2.2 First non-trivial example (weight three) . ... 339 B.2.3 Weight five example . 340 B.2.4 The function H(a) fromtheHardintegral. 341 B.3 A symbol-level solution of the four-loop differential equation . 342 CHexagonfunctionsandthethree-loopremainderfunction 347 C.1 Multiple polylogarithms and the coproduct . .... 347 C.1.1 Multiple polylogarithms . 347 C.1.2 The Hopf algebra of multiple polylogarithms . .350 C.2 Complete basis of hexagon functions through weight five . ...... 354 C.2.1 Φ˜ 6 ................................. 355 C.2.2 Ω(2) ................................ 356 C.2.3 F1 ................................. 357 C.2.4 G .................................. 358 C.2.5 H1 ................................. 358 C.2.6 J1 ................................. 359 C.2.7 K1 ................................. 360 C.2.8 M1 ................................. 361 x C.2.9 N ................................. 362 C.2.10 O .................................. 363 C.2.11 Qep ................................. 364 C.2.12 Relation involving M1 and Qep ................. 366 C.3 Coproduct of Rep ............................. 368 Bibliography 371 xi List of Tables 1.1 All Lyndon words Lyndon(x0,x1)throughweightfive. 28 1.2 Decomposition of functions in (ν, n)spaceintoeigenfunctionsofthe Z Z action.