The Curious Case of Leading Transcendentality: Three Point Form Factors

TL;DR

This paper computes and analyzes three-point form factors in ${ m extbf{N}=4}$ super Yang-Mills theory, revealing insights into transcendentality, operator mixing, and violations of the principle of maximal transcendentality at higher loops.

Contribution

It provides the first detailed three-loop analysis of form factors for specific operators in ${ m extbf{N}=4}$ SYM, highlighting transcendentality properties and PMT violations.

Findings

Highest transcendental weight terms match half-BPS for ${ m extbf{O}}^1$

Form factors of ${ m extbf{O}}^3$ violate the principle of maximal transcendentality

Operator mixing and quantum corrections to dilatation operators are computed

Abstract

Form factors are important ingredients to investigate the principle of maximal transcendentality (PMT) and to extract anomalous dimensions of local gauge invariant operators. In this article, we compute several two- and three-point FFs to three- and two-loops, respectively, for three different choices of local gauge invariant operators in ${\cal N}=4$ super Yang-Mills (SYM) theory. The operators ${\cal O}^1$ and ${\cal O}^2$ are flavour and helicity blind configurations of supersymmetric descendant of the half-BPS and Konishi primary, respectively, and ${\cal O}^3$ is the energy-momentum tensor. The operators ${\cal O}^1$ and ${\cal O}^2$ are composed of non-protected dimension-three (classical) fermionic and scalar components belonging to SU($2|3$) closed sub-sector of ${\cal N}=4$ SYM. We analyse the mixing among the non-protected fermionic and scalar components of these operators up to three-loops in perturbation theory and consequently, compute the quantum corrections to the corresponding dilatation operators. The highest transcendental (HT) weight terms of the FFs of ${\cal O}^1$ are found to be independent of the external on-shell states and, moreover, those are equal to that of half-BPS, however, this does not hold true for the FFs of ${\cal O}^2$. FFs of the ${\cal O}^3$ exhibit identical behaviours to that of half-BPS, in concordance with the classical expectations. However, the three-point FFs of ${\cal O}^3$ violate the PMT while comparing with the corresponding quantity in the standard model, observed for the first time at the level of FF.