String-inspired BCJ numerators for one-loop MHV amplitudes

TL;DR

This work derives simple, all-multiplicity one-loop MHV BCJ numerators for ${ cal N}=4$ SYM and its gravity counterpart by unifying string-theory based field-theory limits with a kinematic algebra of structure constants $X_{i,j}$. The key innovation is a prescription $oldsymbol{ extscr X}$ that maps self-dual (all-plus) numerators to MHV numerators, reducing loop momentum power and ensuring BCJ duality; gravity amplitudes follow by the double-copy construction. The authors provide explicit N=4,5,6 examples, generalize to arbitrary multiplicity, and connect to the dimension-shifting formula of Bern and collaborators, offering a converse perspective. A string-theory derivation via the open-string one-loop correlator clarifies the irreducible and reducible pieces and yields a worldline representation consistent with Feynman integrals, including a discussion of the BRST anomaly at higher points. The results yield a dimension-agnostic, BCJ-satisfying framework that naturally extends to supergravity and potentially to higher loops and helicities, with implications for ultraviolet behavior and integrability connections.

Abstract

We find simple expressions for the kinematic numerators of one-loop MHV amplitudes in maximally supersymmetric Yang-Mills theory and supergravity, for any multiplicity. The gauge theory numerators satisfy the Bern-Carrasco-Johansson (BCJ) duality between color and kinematics, so that the gravity numerators are simply the square of the gauge theory ones. The duality holds because the numerators can be written in terms of structure constants of a kinematic algebra, which is familiar from the BCJ organization of self-dual gauge theory and gravity. The close connection that we find between one-loop amplitudes in the self-dual case and in the maximally supersymmetric case is the converse to the dimension-shifting formula. The starting point for arriving at our expressions is the dimensional reduction of ten-dimensional amplitudes obtained in the field-theory limit of open superstrings.

Figures (7)

• Figure 1: A diagrammatic representation of color and kinematic Jacobi relations.
• Figure 2: The master topology is the half-ladder diagram at tree level (left) and the $N$-gon at one loop (right).
• Figure 3: An example for our notation for one-loop numerators.
• Figure 4: Our prescription $\mathscr{X}$ versus the dimension-shifting formula of ref. Bern:1996ja.
• Figure 5: Given the impact of worldsheet Green functions on the field-theory limit, antisymmetrized $p$-gon numerators agree with $(p-1)$-gon numerators as long as leg 1 adjacent to the loop momentum is not involved in the antisymmetrization.