Loop Integrands for Scattering Amplitudes from the Riemann Sphere (1507.00321v3)

Abstract: The scattering equations on the Riemann sphere give rise to remarkable formulae for tree-level gauge theory and gravity amplitudes. Adamo, Casali and Skinner conjectured a one-loop formula for supergravity amplitudes based on scattering equations on a torus. We use a residue theorem to transform this into a formula on the Riemann sphere. What emerges is a framework for loop integrands on the Riemann sphere that promises to have wide application, based on off-shell scattering equations that depend on the loop momentum. We present new formulae, checked explicitly at low points, for supergravity and super-Yang-Mills amplitudes and for n-gon integrands at one loop. Finally, we show that the off-shell scattering equations naturally extend to arbitrary loop order, and we give a proposal for the all-loop integrands for supergravity and planar super-Yang-Mills theory.

• The paper introduces a residue theorem approach that converts loop integrals from complex elliptic curves to a nodal Riemann sphere for simpler amplitude evaluations.
• The method reformulates loop integrands as rational functions, validated by low-point checks that align with established n-gon results in supergravity and gauge theories.
• The approach sets the stage for extending these techniques to multi-loop amplitudes, potentially advancing our understanding of symmetries and integrability in field theories.

Analysis of "Loop Integrands for Scattering Amplitudes from the Riemann Sphere"

The paper by Geyer, Mason, Monteiro, and Tourkine presents an innovative approach to the evaluation of scattering amplitudes in gauge theories and gravity using loop integrands on the Riemann sphere. This paper builds upon the scattering equations formalism on the Riemann sphere, extending it beyond tree-level to develop formulae for one-loop and multi-loop amplitudes.

Summary and Contributions

The authors start with the foundational idea that scattering amplitudes, particularly at tree-level, benefit from the worldsheet formalism due to its reduction of the complexity implicit in Feynman diagrams. Traditionally, this framework encounters significant mathematical challenges at higher orders due to the involvement of higher-genus worldsheets. Previous work by Adamo, Casali and Skinner proposed a one-loop supergravity formula using scattering equations on a torus, but transforming these into a simpler form on the Riemann sphere was nontrivial.

The key contribution of this paper is the transformation of loop integrals from an elliptic curve setting to the Riemann sphere, utilizing a residue theorem. The novel framework eliminates the need for theta functions by focusing on off-shell scattering equations sensitively dependent on loop momentum. The integrands for supergravity and super-Yang-Mills amplitudes, validated for low-point cases, bring simplicity and potential generalization.

Technical Insights

The procedure presented leverages the residue theorem in the modular $\tau$-plane, simplifying the one-loop proposal initially on the elliptic curve by localization on a Riemann sphere with nodes. This results in loop integrands expressed as rational functions, a significant simplification from the original Jacobi theta function dependencies.

For supergravity amplitudes, the derivation successfully aligns with known results at low points and corroborates the $n$-gon conjecture, extending its speculative reach to all-loop orders. The paper introduces a canonical choice for loop momentum, which is essential for translating standard Feynman integrals into the proposed new language of integrands, albeit requiring careful shifts for consistency.

Numerical Analysis and Formula Validation

Explicit checks conducted at four and five points reinforce the consistency of the Riemann sphere approach with established $n$-gon results and known supergravity calculations. By employing shifts in loop momentum and leveraging permutation symmetry, the novel expressions match the expected permutations of box integrals within the gauge theories. This robust validation makes the proposal well-grounded for further exploration.

Implications and Future Directions

This framework's capacity to simplify loop integrands on nodal Riemann spheres offers a fresh perspective on the field theory amplitude computations. If extended successfully to multi-loop contexts, the result could redefine approaches to symmetries and integrability within both gauge and gravitational theories. The authors propose analogous extensions for planar super-Yang-Mills theories and biadjoint scalar amplitudes at arbitrary loop orders, enhancing the practical utility of these methods.

Future research could focus on elucidating these extensions and optimizing the applicability to various physical theories, potentially bridging conventional formulations with more complex string theory constructs. Additionally, the explicit construction of correlators on multi-loop nodal spheres remains an open challenge, integral for application in realistic scenarios.

Conclusion

The detailed formulation and route from higher-genus surfaces to the manageable Riemann sphere solutions mark a significant advancement in tackling loop amplitudes, particularly for supergravity and planar gauge theories. This work not only enhances theoretical comprehension but also provides a streamlined pathway for future investigation into amplitude computations in fundamental physics.