The cusp limit of correlators and a new graphical bootstrap for correlators/amplitudes to eleven loops

A bstract We consider the universal behavior of half-BPS correlators in $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 4 super-Yang-Mills in the cusp limit where two consecutive separations $$ {x}_{12}^2 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>x</mml:mi> <mml:mn>12</mml:mn> <mml:mn>2</mml:mn> </mml:msubsup> </mml:math> , $$ {x}_{23}^2 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>x</mml:mi> <mml:mn>23</mml:mn> <mml:mn>2</mml:mn> </mml:msubsup> </mml:math> 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 rule for 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.