On Solving Dual Conformal Integrals in Coulomb-branch Amplitudes and Their Periods

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 \texttt{HyperlogProcedures}~\cite{hyperlogprocedures}; 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~\cite{Drummond:2012bg}, 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 ${\zeta }_{2L+1}$, 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 ${\zeta }_{2L+1}$). 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 $L=10$.