Feynman integrals and critical modular $L$-values

Abstract

Abstract. Broadhurst [12] conjectured that the Feynman integral associated to the polynomial corresponding to t = 1 in the one-parameter family (1 +, where f is a cusp form of weight 3 and level 15. Bloch, Kerr and Vanhove [8] have recently proved that the conjecture holds up to a rational factor. In this paper, we prove that Broadhurst's conjecture is true. Similar identities involving Feynman integrals associated to other polynomials in the same family are also established.

Wrońskian factorizations and Broadhurst–Mellit determinant formulae

“…In 2013, Broadhurst wrote that "we know nothing about the number theory of V 5 " [15, §8.6], which stood in stark contrast with other physically relevant Bessel moments IKM(a, b; 2k + 1) involving a + b = 5 Bessel factors, where k is a non-negative integer. In particular, conjectures on the closed-form expressions of IKM(1, 4; 2k + 1) and IKM(2, 3; 2k + 1) for k ∈ Z ≥0 have been supported by numerical experiments [5] and confirmed by theoretical analyses [5,7,31,36].…”

“…When all the external legs and all the internal lines bear the same parameters (say, M = m 1 = · · · = m n = 1 in the diagram above), we are left with the single-scale Bessel moments [23,5,16,21] IKM(a, b; n) := ∞ 0 [I 0 (t)] a [K 0 (t)] b t n d t (1.2) for certain non-negative integers a, b, n ∈ Z ≥0 . In addition to their important rôles in the computation of anomalous magnetic dipole moment [25,24,27] in quantum electrodynamics, these single-scale Bessel moments are also intimately related to motivic integrations in algebraic geometry [7] and modular forms in number theory [31], thus having stimulated intensive mathematical research. For example, various linear relations among Bessel moments, such as π 2 IKM(3, 5; 1) = IKM(1, 7; 1) [conjectured in 16, (148)] and 9π 2 IKM(4, 4; 1) = 14 IKM(2, 6; 1) [conjectured in 16, (147)] had been discovered by numerical experiments, before their formal proofs [35,36] were constructed by algebraic and analytic methods.…”

Some Algebraic and Arithmetic Properties of Feynman Diagrams

“…On one hand, Feynman diagrams provide us with many physically meaningful multiple integrals over rational functions, which are special cases of motivic integrals [44,3], playing prominent rôles in the arena for algebraic geometers. On the other hand, certain Feynman diagrams are (conjecturally or provably) related to arithmetically interesting objects [42,11,50], such as special values of modular L-functions inside their critical strips, inviting pilgrims to the shrine for number theorists.…”

“…The studies of the Bessel moments IKM(1, 4; 1) and IKM(2, 3; 1) had been initiated by Bailey-Borwein-Broadhurst-Glasser [1, §5]. Back in 2008, it was analytically confirmed that [1, (95)], but was not resolved until Bloch-Kerr-Vanhove carried out a tour de force in motivic cohomology during 2015 [3], and Samart elucidated the computations of special gamma values in 2016 [42]. We have recently simplified […”

On Laporta’s 4-loop sunrise formulae

“…1 It is arguable whether L( f 4,6 ,2) should count as a closed-form evaluation in its own right. As one may recall, Bloch-Kerr-Vanhove [7] and Samart [30] have expressed the 3-loop sunrise diagram 2 3 ∞ 0 I 0 (t)[K 0 (t)] 4 t d t as [28], such a special L-value can be reduced to a product of gamma values at rational arguments, thus leaving us a formula 2 3 ∞ 0 I 0 (t)[K 0 (t)] 4…”