Integrating three-loop modular graph functions and transcendentality of string amplitudes

Abstract

Modular graph functions (MGFs) are SL(2, ℤ)-invariant functions on the Poincaré upper half-plane associated with Feynman graphs of a conformal scalar field on a torus. The low-energy expansion of genus-one superstring amplitudes involves suitably regularized integrals of MGFs over the fundamental domain for SL(2, ℤ). In earlier work, these integrals were evaluated for all MGFs up to two loops and for higher loops up to weight six. These results led to the conjectured uniform transcendentality of the genus-one four-graviton amplitude in Type II superstring theory. In this paper, we explicitly evaluate the integrals of several infinite families of three-loop MGFs and investigate their transcendental structure. Up to weight seven, the structure of the integral of each individual MGF is consistent with the uniform transcendentality of string amplitudes. Starting at weight eight, the transcendental weights obtained for the integrals of individual MGFs are no longer consistent with the uniform transcendentality of string amplitudes. However, in all the cases we examine, the violations of uniform transcendentality take on a special form given by the integrals of triple products of non-holomorphic Eisenstein series. If Type II superstring amplitudes do exhibit uniform transcendentality, then the special combinations of MGFs which enter the amplitudes must be such that these integrals of triple products of Eisenstein series precisely cancel one another. Whether this indeed is the case poses a novel challenge to the conjectured uniform transcendentality of genus-one string amplitudes.

Properties of infinite product amplitudes: Veneziano, Virasoro, and Coon

“…The standard weight assignments of ζ( ) and ζ( , 1) are and + 1, respectively, which justifies assigning weight one to the function H 1 (k). This assignment of non-zero transcendental weight to finite harmonic sums is familiar to the low-energy expansion of one-loop superstring amplitudes [24,25] and to loop amplitudes in N = 4 supersymmetric quantum field theory [26,27].…”

Properties of infinite product amplitudes: Veneziano, Virasoro, and Coon

“…If we assign weight k to the zeta-value ζ(k) (the standard number theoretic assignment) and weight −1 to the Mandelstam variables, then each term in (2.10) has transcendental weight two. Uniform transcendentality is in fact a general property of tree-level superstring amplitudes [23], and the transcendental structure of one-loop superstring amplitudes is under active study [24,25]. In comparison, non-trivial transcendental structure in field theory only arises from loop integrals [26][27][28][29].…”

Snowmass White Paper: String Perturbation Theory

“…Uniform transcendentality, familiar from dimensionally regularized Feynman integrals [345][346][347], enjoys a Type I and Type II superstring amplitude counterpart in their α ′ -expansions at tree-level [161,348] and at one loop [177,338,349]. However, uniform transcendentality of Type II superstrings may get challenged at higher α ′ -orders of one-loop amplitudes [350] and may be violated in two-loop amplitudes [351]. Moreover, tree-level amplitudes of Heterotic and bosonic strings [108] violate uniform transcendentality.…”