Propagators, BCFW Recursion and New Scattering Equations at One Loop

TL;DR

This work unifies two modern frameworks for one-loop amplitudes—BCFW recursion in momentum space and worldsheet formulations based on one-loop scattering equations—by showing how loop propagators arise from forward limits of tree-level data. The authors develop a momentum-space recursion for planar one-loop integrands, analyze spurious and boundary terms, and prove that MHV integrands in planar ${ m N}=4$ SYM admit a worldsheet representation with standard quadratic propagators via a new ell^2-deformed scattering equation. They extend the construction to non-planar YM and ${ m N}=8$ supergravity using color-kinematics duality and discuss toy models, toy-nonplanar cases, and extensions beyond MHV. A complementary worldsheet approach with ell^2-deformed equations yields a practical, quadratic-propagator planar MHV formula, which the authors prove using BCFW recursion and detail for higher-point cases. The paper closes with a discussion of non-planar generalizations, lower-supersymmetry challenges, and potential links to broader geometric frameworks, outlining clear directions for future research.

Abstract

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.

Figures (12)

• Figure 1: The above four-particle diagrams illustrate the placement of the loop momentum $\ell$ between $k_1$ and $k_{n=4}$ for planar diagrams with ordering $(1234)$. Note that when $1$ and $n$ occur in a 'massive corner', there is no propagator of the form $1/\ell^{2}$.
• Figure 2: Examples of diagrams for four and five particles that do not contribute to the single cut. This is due to the BCFW shift \ref{['eq:shift']}, which leaves all loop propagators in these diagrams invariant, e.g. $\hat{\ell}+\hat{k}_1=\ell+k_1$.
• Figure 3: Recursion for MHV amplitudes using on-shell diagrams, which is equivalent to \ref{['eq:MHVrecursion']}. We will say more about on-shell diagrams in section \ref{['momtwistrecursion']}. The first term in the recursion is an $L$-loop $(n-1)$-particle MHV amplitude with a soft factor, the second term corresponds to an $(n+2)$-particle $(L-1)$-loop NMHV amplitude in the forward limit. In the figure, the black and white vertices represent 3-particle MHV and $\overline{\text{MHV}}$ amplitudes, and the edges signify integrals over on-shell states.
• Figure 4: On-shell diagram recursion relation for one-loop MHV amplitudes, where the first term corresponds to tree-level factorisation, and the second term corresponds to a forward limit.
• Figure 5: Five-particle amplitude $\mathcal{A}_{5}^{(1)}=K(2,3)+K(2,4)+K(3,4)$ from the momentum twistor recursion. The first term comes from tree-level factorisation, and the other terms come from the forward limit.

Theorems & Definitions (2)

• Lemma 8.1
• Lemma E.1