Correlation Functions of Conserved Currents in Four Dimensional Conformal Field Theory

TL;DR

This work tackles the problem of characterizing 3-point functions of higher-spin conserved currents in $D=4$ CFT by deriving generating functions for $G(r_1,r_2,r_3)$ in parity-even and parity-odd sectors, revealing a factorization into a common $G(r,r,r)$ factor that points to realization by free massless fields with spin $s=r/2$. The authors also analyze a rational 4-point function of four abelian $U(1)$ currents of dimension 3, finding two free-field realizations and a unique interacting candidate whose positivity is not yet established but is suggestive of an interacting fermion–gauge theory. The generating functions are built from conformal covariants $R_{ab}$, $L_a$, and $O_{abc}$ and exhibit a structure controlled by a set of differential operators acting on auxiliary variables, with conservation enforcing precise coefficient relations. Together, the results imply a universality of free-field structures for 3-point functions in 4D CFT with higher-spin symmetry, while highlighting a constrained, potentially interacting sector at the four-point level and outlining open problems for positivity and higher-point extensions.

Abstract

We derive a generating function for all the 3-point functions of higher spin conserved currents in four dimensional conformal field theory. The resulting expressions have a rather surprising factorized form which suggest that they can all be realized by currents built from free massless fields of arbitrary (half-)integer spin s. This property is however not necessarily true also for the higher-point functions. As an illustration we analyze the general 4-point function of conserved abelian U(1) currents of scale dimension equal to three and find that apart from the two free field realizations there is a unique possible function which may correspond to an interacting theory. Although this function passes several non-trivial consistency tests, it remains an open challenging problem whether it can be actually realized in an interacting CFT.