15 Jan 2024 | Xuhang Jiang, Jiahao Liu, Xiaofeng Xu, Li Lin Yang
This paper presents a novel method for constructing symbol letters for Feynman integrals, which are crucial for analytically calculating these integrals in terms of iterated integrals. The method does not require prior knowledge of the canonical differential equations (CDEs) and is implemented in a Mathematica package. The authors demonstrate the effectiveness of their method by successfully bootstrapping the CDEs for a two-loop five-point family with two external masses and a three-loop four-point family with two external masses, which were previously unknown. The method relies on the recursive structure of Baikov representations and the d-log-integrand construction, identifying rational and irrational symbol letters based on specific criteria. The symbol letters obtained can be used to write the solutions as iterated integrals, which can further be converted to multiple polylogarithms or evaluated using series expansion. The authors also discuss the relationship between their method and the Schubert approach, showing how the cross ratios obtained through Schubert analysis are related to the symbol letters written in terms of Gram determinants. The method has potential applications in a wide range of cutting-edge calculations.This paper presents a novel method for constructing symbol letters for Feynman integrals, which are crucial for analytically calculating these integrals in terms of iterated integrals. The method does not require prior knowledge of the canonical differential equations (CDEs) and is implemented in a Mathematica package. The authors demonstrate the effectiveness of their method by successfully bootstrapping the CDEs for a two-loop five-point family with two external masses and a three-loop four-point family with two external masses, which were previously unknown. The method relies on the recursive structure of Baikov representations and the d-log-integrand construction, identifying rational and irrational symbol letters based on specific criteria. The symbol letters obtained can be used to write the solutions as iterated integrals, which can further be converted to multiple polylogarithms or evaluated using series expansion. The authors also discuss the relationship between their method and the Schubert approach, showing how the cross ratios obtained through Schubert analysis are related to the symbol letters written in terms of Gram determinants. The method has potential applications in a wide range of cutting-edge calculations.