Two-Loop Scattering Amplitudes from Ambitwistor Strings: from Genus Two to the Nodal Riemann Sphere

TL;DR

<3-5 sentence high-level summary> The paper develops two-loop scattering amplitudes in ambitwistor-string theory, deriving genus-two genus-two (type II) amplitudes and localising them on a bi-nodal Riemann sphere to obtain a manifestly rational, CHY-type integrand. It provides explicit gravity and super-Yang-Mills formulae, including a two-loop Parke-Taylor factor for colour, and proves modular invariance and absence of unphysical poles through careful degeneration analyses and spin-structure sums. The work extends the CHY/ambitwistor framework to higher genus, showing how modular properties and Deligne–Mumford degeneration underpin a sphere-based representation, with potential implications for colour-kinematics duality and higher-loop generalisations. It also clarifies the role of partition functions, Szegő kernels, and Pfaffians in constructing consistent two-loop amplitudes across NS-NS and Ramond sectors, and discusses future directions toward non-supersymmetric theories and gluing-operator approaches.

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 Szego 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.

Figures (14)

• Figure 1: The bi-nodal Riemann sphere, with nodes parametrised by $\sigma_{1^\pm}$ and $\sigma_{2^\pm}$ representing the two loops of field theory.
• Figure 2: The residue theorem on the fundamental domain.
• Figure 3: The nodal Riemann sphere, including the labels of the node.
• Figure 4: Homology basis of cycles at genus two. The orientation of the cycles ensures that the intersection form is canonical.
• Figure 5: Separating and non-separating boundary divisors. The pinched cycles are indicated in red.