The Cusp Limit of Correlators and A New Graphical Bootstrap for Correlators/Amplitudes to Eleven Loops

We consider the universal behavior of half-BPS correlators in $\mathcal{N}=4$ super-Yang-Mills in the cusp limit where two consecutive separations $x_{12}^2,x_{23}^2$ become lightlike. Through the Lagrangian insertion procedure, the Sudakov double-logarithmic divergence of the $n$-point correlator is related to the $(n+1)$-point correlator where the inserted Lagrangian "pinches" to the soft-collinear region of the cusp. We formulate this constraint as a new graphical rulefor the $f$-graphs of the four-point correlator, which turns out to be the most constraining rule known so far. By exploiting this single graphical rule, we bootstrap the planar integrand of the four-point correlator up to ten loops ($n=14$) and fix all 22024902 but one coefficient at eleven loops ($n=15$); the remaining coefficient is then fixed using the triangle rule. We verify the "Catalan conjecture" for the coefficients of the family of $f$-graphs known as "anti-prisms" where the coefficient of the twelve-loop ($n=16$) anti-prism is found to be $-42$ by a local analysis of the bootstrap equations. We also comment on the implication of our graphical rule for the non-planar contributions.