Abstract
We show that the four-point functions in conformal field theory are defined as distributions on the boundary of the region of convergence of the conformal block expansion. The conformal block expansion converges in the sense of distributions on this boundary, i.e. it can be integrated term by term against appropriate test functions. This can be interpreted as a giving a new class of functionals that satisfy the swapping property when applied to the crossing equation, and we comment on the relation of our construction to other types of functionals. Our language is useful in all considerations involving the boundary of the region of convergence, e.g. for deriving the dispersion relations. We establish our results by elementary methods, relying only on crossing symmetry and the standard convergence properties of the conformal block expansion. This is the first in a series of papers on distributional properties of correlation functions in conformal field theory.
Highlights
Of many experimentally measurable quantities, such as the critical exponents of the 3d Ising [5,6,7,8,9], O(N ) [9,10,11,12] and other critical points
We show that the four-point functions in conformal field theory are defined as distributions on the boundary of the region of convergence of the conformal block expansion
In this work we studied the properties of the conformal block expansion on the boundary of its region of convergence
Summary
We consider a correlation function of four not necessarily identical scalar operators φi with scaling dimensions ∆i, φ1(x1)φ2(x2)φ3(x3)φ4(x4). We would like to show that partial sums of the conformal block expansion (4.2) satisfy a uniform powerlaw bound. We will prove this by relating g1234(ρ, ρ) to the four-point function where operators are inserted symmetrically with respect to the origin [30]. The key idea is to use OPE in the cross channel to infer the leading singularity of the correlator and to argue that a similar bound holds throughout the range |ρ|, |ρ| < 1 This does not work directly for g1234, but only for 4pt functions with non-negative ρ,ρ expansion coefficients, such as g1221 and g4334. We run the argument for those, and recover the general case by Cauchy-Schwarz
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
