One-loop amplitudes on the Riemann sphere (1511.06315v3)

Abstract: The scattering equations provide a powerful framework for the study of scattering amplitudes in a variety of theories. Their derivation from ambitwistor string theory led to proposals for formulae at one loop on a torus for 10 dimensional supergravity, and we recently showed how these can be reduced to the Riemann sphere and checked in simple cases. We also proposed analogous formulae for other theories including maximal super-Yang-Mills theory and supergravity in other dimensions at one loop. We give further details of these results and extend them in two directions. Firstly, we propose new formulae for the one-loop integrands of Yang-Mills theory and gravity in the absence of supersymmetry. These follow from the identification of the states running in the loop as expressed in the ambitwistor-string correlator. Secondly, we give a systematic proof of the non-supersymmetric formulae using the worldsheet factorisation properties of the nodal Riemann sphere underlying the scattering equations at one loop. Our formulae have the same decomposition under the recently introduced Q-cuts as one-loop integrands and hence give the correct amplitudes.

Overview of "One-loop amplitudes on the Riemann sphere"

The paper "One-loop amplitudes on the Riemann sphere" by Geyer et al. investigates the structure of scattering amplitudes at one-loop level employing the novel framework of ambitwistor string theory. This framework, which provides a refined approach to scattering equations, is utilized to derive one-loop integrands on the Riemann sphere for various field theories, including non-supersymmetric Yang-Mills and gravity theories. The authors successfully extend the ambitwistor string theory from the ambitious yet challenging ambit of ambitwistor torus geometry to the more tractable Riemann sphere, streamlining the computational complexity traditionally involved in such calculations.

The heart of the paper lies in two main contributions: the proposal of alternative formulae for one-loop integrands on the Riemann sphere and a systematic proof verifying these formulae. The proposal extends existing formulae by incorporating one-loop integrands for theories lacking supersymmetry. This extension emerges from the detailed identification of state configurations in the ambitwistor-string correlator, which allows the derivation of non-supersymmetric integrands through worldsheet factorization properties.

Key Numerical Results and Methodological Insights

A significant assertion in the work is the integration of one-loop scattering equations that lack apparent double periodicity into numerically tractable forms on the Riemann sphere. Particularly, the proposed formulae show robust concordance with the Q-cuts formalism recently introduced to decompose loop amplitudes into tree-level amplitudes. Through this alignment, the paper validates the conjecture that the scattering equations intrinsically encode the requisite structure for loop-level amplitudes, matching known results in gauge theories and gravity.

The authors utilize dimensional regularization to manage degenerate solutions of the scattering equations, showing that these do not contribute to Q-cut presentations, enhancing the robustness of their approach. This careful handling facilitates an efficient computation, streamlining the complexity introduced by degenerate solutions in non-supersymmetric theories.

Implications for Theory and Experiment

The theoretical implications of this research are profound, offering a refined perspective on the structure of loop amplitudes. The transformation to the Riemann sphere not only simplifies calculations but also potentially broadens the class of theories that can be analyzed with this method. Future work could explore applications in diverse theoretical contexts, such as higher dimensional theories or theories with lower degrees of supersymmetry.

Practically, this innovation opens the door to simulating higher-loop corrections that are pivotal in accurately predicting scattering processes in high-energy physics experiments. The results provide tools that could anticipate experimental discrepancies arising from loop-level interactions, guiding more precise experimental approaches.

Speculation on Future Developments

The paper's insights may serve as a stepping stone toward a more generalized formalism for loop amplitudes in ambitwistor string theory. Pursuing higher-order loops using the methods refined here could systematically advance the understanding of amplitudes, driving new computational techniques in quantum field theory and beyond. Additionally, the demonstrated compatibility with Q-cut formulations suggests future insights into the unifying principles bridging amplitude representation and physical scattering.

In conclusion, Geyer et al.'s work refines the landscape of amplitude computation in quantum field theory, providing clarity and tractability to the one-loop amplitude problem through manipulation on the Riemann sphere. This approach holds the promise of advancing both theoretical and experimental pursuits in high-energy particle physics.