The Soft-Collinear Bootstrap: N=4 Yang-Mills Amplitudes at Six and Seven Loops

TL;DR

The paper develops a soft-collinear bootstrap for the four-point planar ${\cal N}=4$ SYM integrand, showing that the integrand of the logarithm has milder infrared behavior in the soft-collinear limit and can be used, together with dual conformal invariance and dihedral symmetry, to uniquely determine the four-point integrand at any loop order. By constructing a basis of dual conformally invariant (DCI) integrands and enforcing the ${\cal O}(1/\epsilon)$ pole constraint on the log, the authors obtain explicit four-point integrands through seven loops and verify consistency with known results up to five loops. The work includes a complete classification of four-point DCI diagrams through seven loops and provides a scalable, automatable bootstrap procedure that leverages lower-loop data to constrain higher-loop contributions. The results yield detailed counts of contributing DCI diagrams (e.g., 229 at six loops and 1873 at seven loops) and reveal coefficient patterns (including instances of $+2$) that align with two-particle cut expectations, offering a powerful framework for high-precision amplitude construction in planar ${\cal N}=4$ SYM with potential extensions beyond planarity.

Abstract

Infrared divergences in scattering amplitudes arise when a loop momentum $\ell$ becomes collinear with a massless external momentum $p$. In gauge theories, it is known that the L-loop logarithm of a planar amplitude has much softer infrared singularities than the L-loop amplitude itself. We argue that planar amplitudes in N=4 super-Yang-Mills theory enjoy softer than expected behavior as $\ell \parallel p$ already at the level of the integrand. Moreover, we conjecture that the four-point integrand can be uniquely determined, to any loop-order, by imposing the correct soft-behavior of the logarithm together with dual conformal invariance and dihedral symmetry. We use these simple criteria to determine explicit formulae for the four-point integrand through seven-loops, finding perfect agreement with previously known results through five-loops. As an input to this calculation we enumerate all four-point dual conformally invariant (DCI) integrands through seven-loops, an analysis which is aided by several graph-theoretic theorems we prove about general DCI integrands at arbitrary loop-order. The six- and seven-loop amplitudes receive non-zero contributions from 229 and 1873 individual DCI diagrams respectively.

Figures (8)

• Figure 1: Dual conformally invariant diagrams through three-loops. We use the standard notation where a dotted line connecting two faces $x_a,x_b$ corresponds to a numerator factor $x_{ab}$, but for clarity we omit from each diagram an overall factor of $x_{13} x_{24}$. These four diagrams correspond to the rational functions given in eqns. (\ref{['eq:oneloop']}), (\ref{['eq:twoloop']}), (\ref{['eq:threeladder']}) and (\ref{['eq:tennis']}) respectively.
• Figure 2: Two different plane embeddings of the same planar graph (they are related to each other by rotating the bottom edge of the square by 180 degrees out of the plane of the paper while leaving the other three edges fixed). The graph on the left can be made pseudo-conformal with a suitable numerator factor while the graph on the right cannot.
• Figure 3: The only six-loop integrand contributing with coefficient $+2$.
• Figure 4: The 7 seven-loop contributions with coefficient $+2$.
• Figure 5: A degenerate graph is one in which two external legs are attached to the same vertex. Such graphs can be pseudo-conformal (although this example is not), but they cannot be DCI.

Theorems & Definitions (3)

• Lemma 1
• proof
• Corollary 1