Two-loop scattering amplitudes from ambitwistor strings: from genus two to the nodal Riemann sphere

Abstract

We derive from ambitwistor strings new formulae for two-loop scattering amplitudes in supergravity and super-Yang-Mills theory, with any number of particles. We start by constructing a formula for the type II ambitwistor string amplitudes on a genus-two Riemann surface, and then study the localisation of the moduli space integration on a degenerate limit, where the genus-two surface turns into a Riemann sphere with two nodes. This leads to scattering amplitudes in supergravity, expressed in the formalism of the two-loop scattering equations. For super-Yang-Mills theory, we import 'half' of the supergravity result, and determine the colour dependence by considering a current algebra on the nodal Riemann sphere, thereby completely specifying the two-loop analogue of the Parke-Taylor factor, including non-planar contributions. We also present in appendices explicit expressions for the Szegő kernels and the partition functions for even spin structures, up to the relevant orders in the degeneration parameters, which may be useful for related investigations in conventional superstring theory. arXiv:1805.05344v3 [hep-th] 17 Dec 2018 6 The amplitude on the bi-nodal Riemann sphere 51 6.1 The measure on the bi-nodal Riemann sphere 52 6.2 The scattering equations on the bi-nodal Riemann sphere 53 6.3 The chiral integrand on the bi-nodal Riemann sphere 55 6.4 Absence of unphysical poles 58 7 Super-Yang-Mills amplitudes 62 7.1 Parke-Taylor factor on the bi-nodal Riemann sphere 62 7.2 Colour trace decomposition 64 8 Discussion 65 -i -A Useful identities 67 A.1 Identities involving the chiral partition function 67 A.2 Useful identities for the PCO gauge slice 68 B Modular transformations and M g,n 68C The separating degeneration q 12 = 0 and uniqueness of f (q 12 )

Two-loop scattering amplitudes: double-forward limit and colour-kinematics duality

“…The expression (3.12) does not yet coincide with the one presented in [19], denoted as I (2,+) SU(Nc) below. We now proceed to the naive-guess formulae for gauge theory and gravity, based on (3.1).…”

“…Before proceeding, let us relate the present discussion, which employs two sets of scattering equations, to previous work at two loops [16,19], which employed a single set of scattering equations (then chosen to be α = +1). The distinction is that previous work considered only supersymmetric theories.…”

“…In the ambitwistor string -being a worldsheet model -two-loop amplitudes arise from correlators on genus-two Riemann surfaces, integrated over the moduli space. As explored in [19], however, a residue theorem on the moduli space can be used to localise the calculation on the bi-nodal sphere; see fig. 12.…”

“…11 Of course, the splitting into two parts as in (3.1) can be considered, but these should be equivalent up to 2 → − 2 . Diagrams of types T1b and T2 do not arise from the particular form of (4.1) constructed in [19], although we will not attempt a general proof here. This would be consistent with the known diagrammatic representations of loop integrands in theories with sufficient amount of supersymmetry.…”

“…Finally, the set of diagrams Γ (2) n+4 excludes those diagrams which are divergent in the double-forward limit. Notice that we do not sketch here versions of these formulae where the two-loop integrands are expressed as integrals over the moduli space M 0,n+4 , analogous to the (maximally) supersymmetric formulae presented in [19]. It turns out that the picture of the bi-nodal sphere is not realised straightforwardly for generic theories.…”