The classical double copy relates exact solutions in biadjoint scalar, gauge and gravity theories. Recently, nonperturbative solutions have been found in biadjoint theory, which have been speculated to be related to the Wu-Yang monopole in gauge theory. We show that this seems not to be the case, by considering monopole solutions in the infinitely boosted (shockwave) limit. Furthermore, we show that the Wu-Yang monopole is instead related to the Taub-NUT solution, whose previously noted single copy is that of an abelian-like (Dirac) monopole. Our results demonstrate how abelian and non-abelian gauge theory objects can be associated with the same gravity object, and clarify a number of open questions concerning the scope of the classical double copy. 1 n.bahjat-abbas@qmul.ac.uk

Biadjoint scalar field theory has been the subject of much recent study, due to a number of applications in field and string theory. The catalogue of exact non-linear solutions of this theory is relatively unexplored, despite having a role to play in extending known relationships between gauge and gravity theories, such as the double copy. In this paper, we present new solutions of biadjoint scalar theory, corresponding to singular line configurations in four spacetime dimensions, with a power-law dependence on the cylindrical radius. For a certain choice of common gauge group (SU(2)), a family of infinitely degenerate solutions is found, whose existence can be traced to the global symmetry of the theory. We also present extended solutions, in which the pure power-law divergence is partially screened by a form factor. 1 n.bahjat-abbas@qmul.ac.uk

We propose new formulae for the two-loop n-point D-dimensional integrands of scattering amplitudes in Yang-Mills theory and gravity. The loop integrands are written as a double-forward limit of tree-level trivalent diagrams, and are inferred from the formalism of the two-loop scattering equations. We discuss the relationship between the formulae for non-supersymmetric theories and the Neveu-Schwarz sector of the formulae for maximally supersymmetric theories, which can be derived from ambitwistor strings. An important property of the loop integrands is that they are expressed in a representation that includes linear-type propagators. This representation exhibits a loop-level version of the colour-kinematics duality, which follows directly from tree level via the double-forward limit. arXiv:1908.05221v1 [hep-th] Contents 6 Conclusion 32 A Two-loop partition functions and propagators on the Riemann sphere 33 A.1 Two-loop partition functions 33 A.2 Two-loop propagators 341 IntroductionWorldsheet techniques inspired by string theory offer an alternative to the Feynman diagram approach for calculating scattering amplitudes in quantum field theory, in particular for theories of massless particles. This broad programme has seen remarkable advances in recent years. Our aims are to extend the lessons learned at tree level and one loop to two-loop amplitudes, and to interpret previous two-loop results for maximally supersymmetric theories in a more general context, allowing for reduced or no supersymmetry. Even though we will motivate our proposal for two-loop amplitudes from the insights of this 'stringy' approach, the proposal itself will not be written in a worldsheet language. It will instead be written in a (non-Feynman) diagrammatic language, whereby the two-loop integrands are suitably defined double-forward limits of tree-level amplitudes. The worldsheet techniques that inspire our work originated in Witten's twistor string [1] describing four-dimensional super-Yang-Mills theory, and in the corresponding 'connected prescription' to compute scattering amplitudes [2]. In this approach, tree-level scattering amplitudes for n massless particles are computed as integrals over the moduli space of punctured Riemann spheres, M 0,n . The -1 -modern version of these advances, applicable to theories of massless particles in any number of dimensions, was developed by Cachazo, He and Yuan (CHY) [3][4][5], who discovered the general type of formulae, and by Mason and Skinner [6], who constructed the associated type of worldsheet model; the ambitwistor string. For a variety of interesting theories in this framework, see e.g. [7,8]. The ambitwistor string models reproducing Yang-Mills and gravity amplitudes are supersymmetric, but the extraction of amplitudes in non-supersymmetric theories is possible even at loop level, as we shall discuss. For a comparison of the bosonic and supersymmetric ambitwistor strings, see [9,10]. For recent work on moduli-space formulae tuned to a specific number of spacetime dimensions using the spinor...

We propose a procedure to determine the moduli-space integrands of loop-level superstring amplitudes for massless external states in terms of the field theory limit. We focus on the type II superstring. The procedure is to: (i) take a supergravity loop integrand written in a BCJ doublecopy representation, (ii) use the loop-level scattering equations to translate that integrand into the ambitwistor string moduli-space integrand, localised on the nodal Riemann sphere, and (iii) uplift that formula to one on the higher-genus surface valid for the superstring, guided by modular invariance. We show how this works for the four-point amplitude at two loops, where we reproduce the known answer, and at three loops, where we present a conjecture that is consistent with a previous proposal for the chiral measure. Useful supergravity results are currently known up to five loops.

We investigate how loop-level propagators arise from tree level via a forward-limit procedure in two modern approaches to scattering amplitudes, namely the BCFW recursion relations and the scattering equations formalism. In the first part of the paper, we revisit the BCFW construction of one-loop integrands in momentum space, using a convenient parametrisation of the D-dimensional loop momentum. We work out explicit examples with and without supersymmetry, and discuss the non-planar case in both gauge theory and gravity. In the second part of the paper, we study an alternative approach to one-loop integrands, where these are written as worldsheet formulas based on new one-loop scattering equations. These equations, which are inspired by BCFW, lead to standard Feynman-type propagators, instead of the ‘linear’-type loop-level propagators that first arose from the formalism of ambitwistor strings. We exploit the analogies between the two approaches, and present a proof of an all-multiplicity worldsheet formula using the BCFW recursion.