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

Song He, Xuhang Jiang

TL;DR

The paper defines infinite families of all-loop planar dual conformal invariant (DCI) integrals contributing to four-point Coulomb-branch amplitudes in ${\cal N}=4$ SYM and solves boxing differential equations to obtain single-valued harmonic polylogarithms (SVHPL) labeled by binary strings with no consecutive '1's. It introduces the binary Steinmann SVHPL space via inverse boxing, organizes these functions with canonical series tied to antiprism $f$-graphs, and analyzes their periods—often single-valued MZVs—through the graphical-function HyperlogProcedures framework. Key results include the identification of ladder and zigzag extremes within the binary DCI class, explicit period structures up to ten loops, and identities linking periods across different $f$-graphs, revealing a rich interplay between combinatorics of $f$-graphs, SVHPLs, and motivic MZVs. The work lays groundwork for potential all-loop resummations, ternary Steinmann extensions, and systematic bootstrap approaches for (elliptic) MPLs in more general correlator/amplitude contexts in ${\cal N}=4$ SYM.

Abstract

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 ${\cal N}=4$ 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 $ζ_{2L{+}1}$, 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 $ζ_{2L{+}1}$). 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 $L=10$.

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

TL;DR

The paper defines infinite families of all-loop planar dual conformal invariant (DCI) integrals contributing to four-point Coulomb-branch amplitudes in SYM and solves boxing differential equations to obtain single-valued harmonic polylogarithms (SVHPL) labeled by binary strings with no consecutive '1's. It introduces the binary Steinmann SVHPL space via inverse boxing, organizes these functions with canonical series tied to antiprism -graphs, and analyzes their periods—often single-valued MZVs—through the graphical-function HyperlogProcedures framework. Key results include the identification of ladder and zigzag extremes within the binary DCI class, explicit period structures up to ten loops, and identities linking periods across different -graphs, revealing a rich interplay between combinatorics of -graphs, SVHPLs, and motivic MZVs. The work lays groundwork for potential all-loop resummations, ternary Steinmann extensions, and systematic bootstrap approaches for (elliptic) MPLs in more general correlator/amplitude contexts in SYM.

Abstract

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 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 and without consecutive 's. These functions are special cases of the so-called generalized ladders studied in~\cite{Drummond:2012bg}, where extended Steinmann relations (no consecutive '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 followed by all 's, and the other extreme, which nicely evaluate to the ``zigzag'' SVHPL functions with alternating 's and 's, are nothing but the four-point DCI integrals from the very special family of anti-prism -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 , and empirically it provides a numerical lower-bound for -loop periods of any binary string, with the upper-bound given by that of the ladder (also proportional to ). Based on -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 .

Paper Structure

This paper contains 17 sections, 2 theorems, 95 equations, 4 figures, 7 tables.

Table of Contents

  1. Introduction
  2. An invitation: from ladders to zigzag DCI integrals
  3. The special family of "zigzag" DCI integrals from the antiprism f-graphs
  4. Solving boxing differential equations
  5. The periods for antiprism f-graphs: zigzag periods
  6. Inverse boxing: binary DCI integrals as Steinmann-satisfying SVHPL
  7. Binary Steinmann SVHPL function space
  8. A canonical series of binary DCI integrals from inverse boxing
  9. Periods, binary DCI integrals and f-graphs
  10. Periods of binary Steinmann SVHPLs
  11. Identities of periods between different DCI integrals from f-graphs
  12. Conclusions
  13. Relations between integrated antiprism and zigzag periods
  14. Examples of f-graphs and periods not related to binary DCI integrals
  15. Weight-preserving f-graphs and weight-dropping f-graphs
  16. ...and 2 more sections

Key Result

Proposition 1

$P_{n_1,n_2,\ldots,n_{r-1},n_{r}}=P_{n_{r},n_{r-1},\ldots,n_{2},n_{1}}$ holds for arbitrary integers $n_1,\ldots,n_{r}\ge 2$.

Figures (4)

  • Figure 1: box integral which evaluates to Bloch-Wigner function with leading singularity $1/(z-\bar{z})$.
  • Figure 2: Above is one typical type (which corresponds to the canonical type of ladder integrals in Sec. \ref{['sec:canonical']}) of $f$-graph that corresponds to four-loop ladder. The hexagon in the middle plane can be generalized to $n$-gon, with the two dashed line also being generalized to $n-4$, which corresponds to $(n-2)$-loop ladder integral. The four points colored red are taken as external points here.
  • Figure 3: The comparison plot of magnitudes of periods of binary Steinmann SVHPLs. The magnitudes has been rescaled by a monotonic function $1125\log x-4625$. We can see that the periods of different loops are separated. In each loop, the largest value corresponds to the ladder and the smallest value corresponds to the zigzag.
  • Figure 4: We mark the "squall lines" of the third graph in two contrary boxing sequence \ref{['eq:I5f5path']} and \ref{['eq:I5f5pathrev']} out with thick blue lines. They coincide with each other and there is a correspondence between $x_{1},x_{2},x_{3},x_{4}$ and $x_{7},x_{8},x_{6},x_{5}$.

Theorems & Definitions (4)

  • Conjecture 1
  • Proposition 1
  • Conjecture 2
  • Proposition 2