Analytic tools for Feynman integrals

Cover......Page 1
Analytic Tools for Feynman Integrals......Page 4
Preface......Page 6
Contents......Page 8
14 Appendix E: A Brief Review of Some  Other Methods......Page 11
References......Page 20
2.1 Feynman Rules and Feynman Integrals......Page 21
2.2 Divergences......Page 24
2.3 Alpha Representation......Page 29
2.4 Regularization......Page 31
2.5 Properties of Dimensionally Regularized  Feynman Integrals......Page 36
References......Page 40
3.1 Simple One- and Two-Loop Formulae......Page 42
3.2.1 Recursively One-Loop Feynman Integrals......Page 46
3.2.3 Dealing with Numerators......Page 47
3.3 One-Loop Examples......Page 50
3.4 Feynman Parameters......Page 54
3.5 Two-Loop Examples......Page 57
References......Page 67
4.1 Hepp and Speer Sectors......Page 70
4.2 Recursive Sector Decompositions......Page 76
4.3 Geometrical Sector Decompositions......Page 80
4.4 Analysis of Convergence......Page 83
References......Page 88
5.1 A One-Loop Example......Page 91
5.2 Evaluating Multiple MB Integrals......Page 96
5.3 How to Derive and Check MB Representations......Page 97
5.4 Two Basic Strategies of Resolving Singularities in ε......Page 107
5.5 Two-Loop Examples......Page 109
5.6 Examples with the Old Strategy A......Page 113
5.7 Applying MB Representation to Expand Feynman  Integrals in Momenta and Masses......Page 126
5.8 Combining MB Representation with Sector Decompositions......Page 128
5.9 Conclusion......Page 130
References......Page 131
6.1 Solving IBP Relations by Hand......Page 135
6.2 General Setup for IBP Reduction......Page 141
6.3 Laporta's Algorithm......Page 143
6.4 Algebraic Structure of IBP Relations......Page 145
6.5 Baikov's Method......Page 147
6.6 Shifting Dimension and Gröbner Bases......Page 155
6.7 Obtaining Additional Relations......Page 158
6.8 Conclusion......Page 160
References......Page 161
7.1 One-Loop Examples......Page 163
7.2 Two-Loop Example......Page 169
7.3 Three-Loop Example......Page 173
7.4 Conclusion......Page 175
References......Page 177
8.1 Dimensional Recurrence Relations......Page 179
8.2 General Prescriptions......Page 181
8.3 Multiloop Examples......Page 185
8.4 Evaluating Multicomponent Master Integrals......Page 199
8.5 What Numbers Can Appear in Epsilon Expansions?......Page 206
References......Page 208
9.1 Expansion by Subgraphs......Page 209
9.2 Expansion by Regions......Page 214
9.3 Expansion by Regions in Parametric Representations......Page 220
9.4 How to Reveal Relevant Regions......Page 222
9.5 Revealing Potential Regions......Page 225
9.6 Revealing Glauber Regions......Page 227
9.7 Mathematical Status of Expansion by Regions......Page 231
References......Page 232
10.1 Table of Integrals......Page 233
10.2 Some Useful Formulae......Page 240
References......Page 247
12.1 Some Number Series......Page 249
12.2 Power Series of Weights 3 and 4 in Terms  of Polylogarithms......Page 254
12.3 Inverse Binomial Power Series up to Weight 4......Page 255
12.4 Power Series of Weights 5 and 6 in Terms of HPLs......Page 257
References......Page 262
13.1 MB Integrals with Four Gamma Functions......Page 263
13.2 MB Integrals with Six Gamma Functions......Page 271
13.3 The Gauss Hypergeometric Function  and MB Integrals......Page 276
14.1 Dispersion Integrals......Page 278
14.2 Gegenbauer Polynomial x-Space Technique......Page 280
14.3 Gluing......Page 281
14.4 Star-Triangle Relations......Page 282
14.5 Duality for Planar Graphs......Page 283
14.6 IR Rearrangement and R*......Page 285
14.7 Difference Equations......Page 287
14.8 Experimental Mathematics and PSLQ......Page 288
14.10 Symbols......Page 290
References......Page 293
List of Symbols......Page 296
Index......Page 298