From multiple unitarity cuts to the coproduct of Feynman integrals

TL;DR

<3-5 sentence high-level summary> The paper develops a framework linking multiple unitarity cuts, branch-discontinuities, and the coproduct structure of multiple polylogarithms to analyze and reconstruct Feynman integrals with massless propagators. By extending Cutkosky rules to sequences of cuts in different channels and employing real kinematics, the authors establish precise relations between Cut, Disc, and δ within the Hopf-algebra language, and validate them on one- and two-loop examples such as the three-mass triangle and the two-loop ladder. They show that, in many cases, a full integral or its symbol/coproduct can be recovered from a small set of cuts via dispersion relations or coproduct reconstruction, illustrating a powerful, algebraic approach to the analytic structure of Feynman integrals. The results offer a promising route for computing multiloop integrals and understanding their analytic properties through a unitarity-cut–discontinuity–coproduct dictionary.

Abstract

We develop techniques for computing and analyzing multiple unitarity cuts of Feynman integrals, and reconstructing the integral from these cuts. We study the relations among unitarity cuts of a Feynman integral computed via diagrammatic cutting rules, the discontinuity across the corresponding branch cut, and the coproduct of the integral. For single unitarity cuts, these relations are familiar. Here we show that they can be generalized to sequences of unitarity cuts in different channels. Using concrete one- and two-loop scalar integral examples we demonstrate that it is possible to reconstruct a Feynman integral from either single or double unitarity cuts. Our results offer insight into the analytic structure of Feynman integrals as well as a new approach to computing them.

Figures (12)

• Figure 1: Sequential cuts of a triangle diagram, whose vertices $v$ are labelled by all possible color sequences $(c_1(v),c_2(v))$ encoding the cuts. Energy flows from $0$ to $1$ for each cut, giving the restrictions listed below each diagram.
• Figure 2: An example of crossed cuts, which we do not allow.
• Figure 3: The triangle integral, with loop momentum defined as in the text; and with cuts in the $p_2^2$ and $p_3^2$ channels.
• Figure 4: The four-mass box integral, with pairs of unitarity cuts.
• Figure 5: Cut integral diagrams for sequential discontinuities of the two-mass-hard box, where legs 1 and 2 have null momenta. Here, we do not need the detailed information of physical cut channels or conjugated Feynman rules, since it makes no difference to the results. (a) Channel pairs $(s, p_3^2), (s, p_4^2)$, or $(p_3^2, p_4^2)$. (b) Channel pair $(t, p_3^2)$. (c) Channel pair $(t, p_4^2)$. (d) Channel pair $(s, t)$.