Reciprocity of gauge operators in N=4 SYM
Matteo Beccaria, Valentina Forini
TL;DR
The work probes generalized Gribov-Lipatov reciprocity in ${ m N}=4$ SYM for a class of gluonic twist-3 operators by computing the spin-dependent anomalous dimension at four loops via the asymptotic Bethe Ansatz. A reciprocity-respecting kernel ${ m P}$ is constructed and proven to satisfy reciprocity, using a Mellin-space framework and a reduction algorithm that rewrites results in parity-preserving combinations of nested harmonic sums. The four-loop result comprises no-dressing and dressing (with ${eta=\zeta_3}$) pieces, both individually reciprocity respecting, and matches MVV-type large-$N$ constraints, including a consistent large-$N$ expansion that reproduces the cusp anomalous dimension. Wrapping corrections are discussed as a potential caveat, but reciprocity appears to be a robust structural feature that can help constrain finite-length effects in the planar ${ m N}=4$ theory and hints at deeper algebraic underpinnings of the Bethe Ansatz. Overall, the paper strengthens reciprocity as a guiding principle in high-loop gauge theory as well as its potential role in guiding wrapping analyses and AdS/CFT correspondence checks.
Abstract
A recently discovered generalized Gribov-Lipatov reciprocity holds for the anomalous dimensions of various twist operators in N=4 SYM. Here, we consider a class of scaling psu(2,2|4) operators that reduce at one-loop to twist-3 maximal helicity gluonic operators. We extract from the asymptotic long-range Bethe Ansatz a closed expression for the spin dependent anomalous dimension at four loop order and provide a complete proof of reciprocity. We comment about the interplay with possible, yet unknown, wrapping corrections.