Four-point functions in N=4 SYM

TL;DR

The paper presents a compact, superconformal formulation for four-point functions of charge $Q$ chiral primary multiplets in $N=4$ SYM using analytic superspace and Schur polynomials of $GL(2|2)$. It shows that the correlator factorizes into a universal spacetime prefactor times a sum over Schur polynomials in a single adjoint-invariant matrix $Z$, with analyticity constraining the allowed representations to short and long sectors. The long-sector contribution yields a function of two conformal invariants, while short-sector exchanges generate functions of a single variable that are non-renormalised by the OPE, leading to a partial non-renormalisation theorem. Crossing symmetry then reduces the independent function basis to a finite set that depends on $Q$, with explicit counts given for $N=2$ and $N=4$, and the free theory serving as the non-renormalised template for the single-variable pieces. The framework extends to $N=0$ and offers a path to generalisations to other BPS sectors and higher dimensions via harmonic/analytic superspace techniques, highlighting the deep role of protected operators in constraining four-point dynamics.

Abstract

A new derivation is given of four-point functions of charge $Q$ chiral primary multiplets in N=4 supersymmetric Yang-Mills theory. A compact formula, valid for arbitrary $Q$, is given which is manifestly superconformal and analytic in the internal bosonic coordinates of analytic superspace. This formula allows one to determine the spacetime four-point function of any four component fields in the multiplets in terms of the four-point function of the leading chiral primary fields. The leading term is expressed in terms of $1/2 Q(Q-1)$ functions of two conformal invariants and a number of single variable functions. Crossing symmetry reduces the number of independent functions, while the OPE implies that the single-variable functions arise from protected operators and should therefore take their free form. This is the partial non-renormalisation property of such four-point functions which can be viewed as a consequence of the OPE and the non-renormalisation of three-point functions of protected operators.