Empirical determinations of Feynman integrals using integer relation algorithms

Abstract

Integer relation algorithms can convert numerical results for Feynman integrals to exact evaluations, when one has reason to suspect the existence of reductions to linear combinations of a basis, with rational or algebraic coefficients. Once a tentative reduction is obtained, confidence in its validity is greatly increased by computing more decimal digits of the terms and verifying the stability of the result. Here we give examples of how the PSLQ and LLL algorithms have yielded remarkable reductions of Feynman integrals to multiple polylogarithms and to the periods and quasi-periods of modular forms. Moreover, these algorithms have revealed quadratic relations between Feynman integrals. A recent application concerning black holes involves quadratic relations between combinations of Feynman integrals with algebraic coefficients.

Analytic integration methods in quantum field theory: an Introduction

“…In many complex applications it is difficult to proof the result analytically. Advanced applications of these and similar methods are discussed in [10].…”

“…with q k (x) = 0 for 1 ≤ k ≤ m. Further, γ m+1 = 0 if ḡ(x) = 0 in (10), and γ m+1 = 1 and q m+1 (x) being a mild variation of ḡ(x) if ḡ(x) = 0. One obtains d'Alembertian solutions [134] since the master integrals appearing in quantum field theories obey differential equations with rational coefficients, the letters h i , which constitute the iterative integrals, have to be hyperexponential and the solution can be computed using the package HarmonicSums.…”

“…The topics of the workshop included both techniques to reduce the number of Feynman diagrams by physical relations, such as the integration by parts relations [5][6][7] as well as the mathematical methods to compute these integrals analytically. The latter include the method of generalized hypergeometric functions [8] and the general theory of contiguous relations [9], the methods of integer relations [10], guessing methods of one-dimensional quantities, hyperlogarithms [11], the solution of master-integrals using difference and differential equations [12][13][14][15], Risch algorithms on nested integrals and rationalization algorithms [16], holonomic integration [17], the multivalued Almkvist-Zeilberger algorithm [18], expansion by regions [19], elliptic integrals and related topics [20,21], cutting techniques [22], and special multi-leg applications [23][24][25]. In different precision calculations these technologies are applied.…”