The Q-cut Representation of One-loop Integrands and Unitarity Cut Method

TL;DR

This paper introduces the Q-cut representation as a complete, on-shell–based algorithm for constructing one-loop integrands in massless theories. It uses a dimensional deformation to convert quadratic propagators into linear ones and sums over all Q-cuts and helicities, with a two-step deformation that removes forward-limit singularities. Through detailed scalar and Yang-Mills examples, it demonstrates that the Q-cut integrand reproduces the same master integral coefficients as the standard unitarity-cut method, providing robust cross-checks and clarifying how to handle non-scalar internal states. The work lays a foundation for extensions to multi-loop and massive theories and highlights links to CHY/ambitwistor formulations and potential advances in integrand reduction and IBP techniques.

Abstract

Recently, a new construction for complete loop integrands of massless field theories has been proposed, with on-shell tree-level amplitudes delicately incorporated into its algorithm. This new approach reinterprets integrands in a novel form, namely the Q-cut representation. In this paper, by deriving one-loop integrands as examples, we elaborate in details the technique of this new representation, e.g., the summation over all possible Q-cuts as well as helicity states for the non-scalar internal particle in the loop. Moreover, we show that the integrand in the Q-cut representation naturally reduces to the integrand in the traditional unitarity cut method for each given cut channel, providing a cross-check for the new approach.

Figures (5)

• Figure 1: (a) Graphic presentation of ${\cal Q}$-cut: the tree amplitudes are evaluated with the rescaled $D$-dimensional loop momenta $\widehat{\ell}_L$ and $\widehat{\ell}_R$, multiplied by two novel propagators $1/\ell^2$ and $1/(2\ell\cdot P_L+P_L^2)$. (b) Graphic presentation of unitarity cut: the tree amplitudes are evaluated with the on-shell loop momenta $\widetilde{\ell}_L$ and $\widetilde{\ell}_R$, with two propagators $1/\widetilde{\ell}_L^2\widetilde{\ell}_R^2$ replaced by $\delta^{+}(\widetilde{\ell}_L^2)\delta^{+}(\widetilde{\ell}_R^2)$.
• Figure 2: Convention of external momenta for scalar integrands of box, triangle and bubble topologies. $p_i$ denotes massless momentum and $P_i$ denotes the sum of several massless momenta.
• Figure 3: The contributing Feynman diagrams of color-ordered 4-point one-loop amplitude in $\phi^3$ theory.
• Figure 4: The contributing Feynman diagrams of non-color-ordered 4-point one-loop amplitude in $\phi^3$ theory.
• Figure 5: All possible internal states for the helicity sum of a given $\mathcal{Q}$-cut term, while the internal particles are $D$-dimensional gluons of $+,-$ and $S_A$$(A=1,\ldots, \dim[\mu])$ physical polarizations.