From loops to trees by-passing Feynman's theorem

TL;DR

The paper establishes a duality between one-loop integrals and single-cut phase-space integrals by introducing a Lorentz-covariant dual i0 prescription, circumventing the need for multiple cuts inherent in the Feynman Tree Theorem. It shows that a d-dimensional one-loop integral can be written as a negative sum of single-cut dual phase-space integrals with dual propagators, with the auxiliary vector lat{} ensuring Lorentz invariance and canceling unphysical singularities when summed. The authors extend the formalism to one-loop Green's functions and amplitudes, incorporating massive and complex-mass propagators, gauge-pole considerations, and forward-scattering regularization, and discuss implications for NLO cross sections and numerical methods. They illustrate the duality with a detailed scalar two-point function example and clarify the relation to the Feynman Tree Theorem, highlighting potential computational advantages for analytic and numerical one-loop calculations.

Abstract

We derive a duality relation between one-loop integrals and phase-space integrals emerging from them through single cuts. The duality relation is realized by a modification of the customary +i0 prescription of the Feynman propagators. The new prescription regularizing the propagators, which we write in a Lorentz covariant form, compensates for the absence of multiple-cut contributions that appear in the Feynman Tree Theorem. The duality relation can be applied to generic one-loop quantities in any relativistic, local and unitary field theories. %It is suitable for applications to the analytical calculation of %one-loop scattering amplitudes, and to the numerical evaluation of %cross-sections at next-to-leading order. We discuss in detail the duality that relates one-loop and tree-level Green's functions. We comment on applications to the analytical calculation of one-loop scattering amplitudes, and to the numerical evaluation of cross-sections at next-to-leading order.

Figures (12)

• Figure 1: Momentum configuration of the one-loop $N$-point scalar integral.
• Figure 2: Location of the particle poles of the Feynman (left) and advanced (right) propagators, $G(q)$ and $G_A(q)$, in the complex plane of the variable $q_0$ or $q_{\pm}$.
• Figure 3: Location of poles and integration contour $C_L$ in the complex $q_0$-plane for the advanced (left) and Feynman (right) one-loop integrals, $L_A^{(N)}$ and $L^{(N)}$.
• Figure 4: The single-cut contribution of the Feynman Tree Theorem to the one-loop $N$-point scalar integral. Graphical representation as a sum of $N$ basic single-cut phase-space integrals.
• Figure 5: The duality relation for the one-loop $N$-point scalar integral. Graphical representation as a sum of $N$ basic dual integrals.