One-Loop Yang-Mills Integrands from Scattering Equations

TL;DR

The paper develops a CHY-based framework to generate one-loop Yang-Mills integrands with quadratic propagators by applying a double forward limit to linear-propagator CHY integrands. It demonstrates that the resulting integrand matches four-dimensional unitarity cuts and decomposes into the standard box, triangle, and bubble basis. A master color-kinematics numerator and a relation between linear and quadratic numerators underpin the construction, and the approach is validated at four points and argued to generalize. The work suggests pathways to extensions to higher multiplicities, supersymmetric theories, gravity, and rational terms with connections to ambitwistor strings.

Abstract

We investigate in the context of the scattering equations, how one-loop linear propagator integrands in gauge theories can be linked to integrands with quadratic propagators using a double forward limit. We illustrate our procedure through examples and demonstrate how the different parts of the derived quadratic integrand are consistent with cut-integrands derived from four-dimensional generalized unitarity. We also comment on applications and discuss possible further generalizations.

Figures (5)

• Figure 1: Half-ladder tree diagram associated with the one-loop color-kinematic numerator, $n[{\ell^+, \rho_n,\ell^-}]$ for off-shell momenta $\ell^+$ and $\ell^-$.
• Figure 2: Half-ladder tree diagram associated with the one-loop color-kinematic numerator: $n[{\ell^+_1, \ell^+_2, \rho_n,\ell^-_1, \ell^-_2]}$ for on-shell $\ell_1^+$, $\ell_1^-$, $\ell_2^+$ and $\ell_2^-$.
• Figure 3: Double cut discontinuities: $\ell^2=0$ and $(\ell+k_1+k_2)^2=0$.
• Figure 4: Quadruple cut given by the conditions, $\ell^2=(\ell_1)^2=(\ell_{2})^2=(\ell_3)^2=0$. We define, $\ell_1\equiv\ell+k_1, \, \ell_2\equiv \ell+k_{12}$ and $\ell_3\equiv\ell-k_4$.
• Figure 5: Triple cut given by, $\ell^2=(\ell_{2})^2=(\ell_3)^2=0$, where, $\ell_2\equiv \ell+k_{12}$ and $\ell_3\equiv\ell-k_4$.