We define and study infinite families of all-loop planar, dual conformal invariant (DCI) integrals, which contribute to four-point Coulomb-branch amplitudes and correlators in \(\mathcal{N}=4\) supersymmetric Yang-Mills theory, by solving “boxing” differential equations via HyperlogProcedures [https://www.math.fau.de/person/oliver-schnetz/]; the resulting single-valued harmonic polylogarithmic functions (SVHPL) are nicely labeled by “binary” strings of 0 and 1 without consecutive 1’s. These functions are special cases of the so-called generalized ladders studied in [JHEP 02 (2013) 092], where extended Steinmann relations (no consecutive 1’s) are imposed due to planarity. Our results can be viewed as “two-dimensional” extensions of the well-known ladder integrals to many more infinite families of DCI integrals: the ladders have strings with a single 1 followed by all 0’s, and the other extreme, which nicely evaluate to the “zigzag” SVHPL functions with alternating 1’s and 0’s, are nothing but the four-point DCI integrals from the very special family of anti-prism f-graphs (while all other binary DCI integrals lie in between these two extreme cases). We also study periods of these integrals: while their periods are in general complicated single-valued multiple zeta values (SVMZV), the “zigzag” DCI integrals from anti-prism gives exactly the famous “zigzag” periods proportional to ζ2L+1, and empirically it provides a numerical lower-bound for L-loop periods of any binary string, with the upper-bound given by that of the ladder (also proportional to ζ2L+1). Based on f-graphs as a tool for studying these periods, we discuss several interesting facts and observations about these (motivic) SVMZV and relations among them to all loops, and enumerate a basis for them up to L = 10.
