One-loop monodromy relations on single cuts

TL;DR

This work develops a comprehensive framework for one-loop monodromy relations via single cuts, linking forward-tree amplitudes to the full loop integrand. By analyzing the forward limit, momentum kernels, and the string-theory origin of monodromies, it shows that a co-rank of $(n-1)!+(n-2)!+2(n-3)!$ governs independent forward amplitudes and identifies three independent forward-tree bases, with irreducible numerators constrained by Jacobi/BCJ structures. It then demonstrates that string-theory monodromies yield exact integrand relations in the field-theory limit, with boundary terms accounting for loop-momentum shifts and enabling a refined planar/non-planar decomposition that remains BCJ-compatible at four and five points. The results provide a principled, integrand-level mechanism to organize loop amplitudes according to colour-kinematic duality and highlight how forward-limit regularisation preserves the essential relations. The approach offers a path toward systematic higher-loop extensions and potential applications to integration-by-parts identities and ultraviolet behavior in gauge and gravity theories.

Abstract

The discovery of colour-kinematic duality has led to significant progress in the computation of scattering amplitudes in quantum field theories. At tree level, the origin of the duality can be traced back to the monodromies of open-string amplitudes. This construction has recently been extended to all loop orders. In the present paper, we dissect some consequences of these new monodromy relations at one loop. We use single cuts in order to relate them to the tree-level relations. We show that there are new classes of kinematically independent single-cut amplitudes. Then we turn to the Feynman diagrammatics of the string-theory monodromy relations. We revisit the string-theoretic derivation and argue that some terms, that vanish upon integration in string and field theory, provide a characterisation of momentum-shifting ambiguities in these representations. We observe that colour-dual representations are compatible with this analysis.

Figures (7)

• Figure 1: Wave-function renormalisation and tadpole graphs divergent in the forward limit.
• Figure 2: Mapping of the annulus to the plane with $x=e^{2i\pi \nu}$ in the $t\to \infty$ limit. In the reduction of the blue contour onto the real axis, a singularity at $x=0$ is encountered.
• Figure 3: The right-hand side integration contour of Figure \ref{['fig:map']} is equivalent to the one on the left-hand side here, which is equivalently represented in a disk picture on the right.
• Figure 4: Left-hand side: boundary terms along contours $A$ and $B$ as defined in figure \ref{['fig:map']}. Right-hand side: conjectured corresponding types of one-loop graphs. Notice the loop-momentum shift.
• Figure 5: Triangles obtained by bringing $\nu_2$ close to $\nu_3$, or to $\nu_1$. The latter is not possible for the upper vertex configuration, in which $\nu_1$ is on the other boundary of the annulus.