Heterotic-string amplitudes at one loop: modular graph forms and relations to open strings

Abstract

We investigate one-loop four-point scattering of non-abelian gauge bosons in heterotic string theory and identify new connections with the corresponding open-string amplitude. In the low-energy expansion of the heterotic-string amplitude, the integrals over torus punctures are systematically evaluated in terms of modular graph forms, certain nonholomorphic modular forms. For a specific torus integral, the modular graph forms in the low-energy expansion are related to the elliptic multiple zeta values from the analogous openstring integrations over cylinder boundaries. The detailed correspondence between these modular graph forms and elliptic multiple zeta values supports a recent proposal for an elliptic generalization of the single-valued map at genus zero. (2,0) 12|34 66 D.2.2 Rewriting the integral I (4,0) 12|34 66 D.2.3 Towards a uniform-transcendentality basis 67 -ii -D.3 Efficiency of the new representations for higher-order expansions 67 E The elliptic single-valued map at the third order in α 68 E.1 Modular transformation of iterated Eisenstein integrals 68 E.2 The third α -order of I (2,0) 1234 and iterated Eisenstein integrals 69

All-order differential equations for one-loop closed-string integrals and modular graph forms

“…Our method in particular never generates modular graph forms with negative entries and does not necessitate holomorphic subgraph reduction to expose holomorphic Eisenstein series. 21 See Section 5.3 of [98] for a recent account on the key challenges in setting up loop-level KLT relations among string amplitudes from the viewpoint of intersection theory [99]. A field-theory version of one-loop KLT relations among loop integrands in gauge theories and supergravity has been derived from ambitwistor strings [100,101].…”

“…seen in earlier literature [9,13,17,21]. Non-holomorphic Eisenstein series give rise to the closed formula for two-column modular graph forms…”

“…, a n are permuted with the permutation ρ in the second slot of the argument of W τ (A|B) and vice versa. This is to ensure consistency with the notation (2.41) of open-string integrals and also to have W τ (A|B) carry modular weight (|A|, |B|), where Here and in the rest of this work, we use the abbreviating notation Component integrals of the type in (2.43) arise from the conformal-field-theory correlators underlying one-loop amplitudes of closed bosonic strings, heterotic strings and type-II superstrings [21]. More specifically, the f (a) ij were found to appear naturally from the spin sums of the RNS formalism [36] and the current algebra of heterotic strings [55].…”

“…There are many non-trivial identities relating modular graph forms and these are crucial in the simplification of α -expansions of Koba-Nielsen integrals [2,4,9,13,14,21,23]. In this appendix, we will review the most important techniques by which these identities can be obtained.…”

“…More specifically, the right-hand side of our definition (2.53) does not include the factor of e∈E Γ ( Im τ π ) 1 2 (ae+be) from[9,13,20,22] and the factor of e∈E Γ ( Im τ π ) be from[21].…”