Uniqueness of two-loop master contours

TL;DR

The paper develops a systematic two-loop unitarity framework by classifying maximal cuts of the double-box topology, revealing that the associated leading singularities organize into Riemann-surface structures whose topology tracks the number of three-point vertices. It proves the uniqueness of master contours at four points, linking them to a minimal, infrared-finite chiral-basis of master integrals, and demonstrates compact analytic results for these chiral doubles boxes. The work connects geometric (momentum-twistor) and analytic (symbol-based) methods to show how maximal cuts reflect the analytic structure of two-loop amplitudes, including an elliptic-case where polylogarithms are insufficient. These insights point to streamlined strategies for two-loop computations across gauge theories and suggest promising extensions to higher-point and non-planar topologies using finite integral bases.

Abstract

Generalized-unitarity calculations of two-loop amplitudes are performed by expanding the amplitude in a basis of master integrals and then determining the coefficients by taking a number of generalized cuts. In this paper, we present a complete classification of the solutions to the maximal cut of integrals with the double-box topology. The ideas presented here are expected to be relevant for all two-loop topologies as well. We find that these maximal-cut solutions are naturally associated with Riemann surfaces whose topology is determined by the number of states at the vertices of the double-box graph. In the case of four massless external momenta we find that, once the geometry of these Riemann surfaces is properly understood, there are uniquely defined master contours producing the coefficients of the double-box integrals in the basis decomposition of the two-loop amplitude. This is in perfect analogy with the situation in one-loop generalized unitarity. In addition, we point out that the chiral integrals recently introduced by Arkani-Hamed et al. can be used as master integrals for the double-box contributions to the two-loop amplitudes in any gauge theory. The infrared finiteness of these integrals allow for their coefficients as well as their integrated expressions to be evaluated in strictly four dimensions, providing significant technical simplification. We evaluate these integrals at four points and obtain remarkably compact results.

Figures (10)

• Figure 1: The general double-box integral. The $\cdots$ dots at each vertex represent the presence of an arbitrary number of massless legs. Each of the vertices, shown as gray blobs, is given a label $i=1,\ldots,6$ which equals the index of the associated external momentum $K_i$.
• Figure 2: A generic integral belonging to case 1: all three vertical lines are part of some three-point vertex. The shown chirality assignment is forbidden, as explained in the main text.
• Figure 3: The six different classes of kinematical solutions to the heptacut constraints (\ref{['eq:on-shell_constraint_1']})-(\ref{['eq:on-shell_constraint_7']}), illustrated here as Riemann spheres (intended to represent the complex degree of freedom $z$ left unfrozen by the heptacut constraints), in case 1. The kinematical solutions are characterized by the distribution of chiralities at the vertices of the double-box graph (see figure \ref{['fig:general_double_box']}), shown next to each sphere. Each Riemann sphere coincides with the two adjacent spheres in the chain at a single point, illustrated as a black dot. These points are precisely the poles of the heptacut Jacobian. The Riemann spheres contain additional singularities, denoted by $\infty_L$ and $\infty_R$, associated with respectively the left or right loop momentum becoming infinite (respectively occuring as $z=\infty$ or $z=P_2^{(\bullet)}$ in eqs. (\ref{['eq:Jac_poles_spinor_ratios']})-(\ref{['eq:parametrization_of_six_solutions']})). Parity-conjugate kinematical solutions appear antipodally in the chain.
• Figure 4: The four different classes of kinematical solutions to the heptacut constraints in case 2: exactly two vertical lines of the double-box graph are part of some three-point vertex. Figure (a) illustrates the (m, M, m) subcase, whereas figure (b) illustrates the (M, m, m) subcase. The only difference between these two subcases is the number of points contained in each sphere at which one of the loop momenta becomes infinite. We observe that in both subcases the number of independent residues one can take is 8.
• Figure 5: The two different classes of kinematical solutions to the heptacut constraints in subcase (M, M, m) of case 3. The subcases (M, m, M) and (m, M, M) are similar. Again the number of independent residues one can take is 8.