Distributions in CFT. Part II. Minkowski space
Petr Kravchuk, Jiaxin Qiao, Slava Rychkov
Abstract
A bstract CFTs in Euclidean signature satisfy well-accepted rules, such as the convergent Euclidean OPE. It is nowadays common to assume that CFT correlators exist and have various properties also in Lorentzian signature. Some of these properties may represent extra assumptions, and it is an open question if they hold for familiar statistical-physics CFTs such as the critical 3d Ising model. Here we consider Wightman 4-point functions of scalar primaries in Lorentzian signature. We derive a minimal set of their properties solely from the Euclidean unitary CFT axioms, without using extra assumptions. We establish all Wightman axioms (temperedness, spectral property, local commutativity, clustering), Lorentzian conformal invariance, and distributional convergence of the s-channel Lorentzian OPE. This is done constructively, by analytically continuing the 4-point functions using the s-channel OPE expansion in the radial cross-ratios ρ , $$ \overline{\rho} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>ρ</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> . We prove a key fact that | ρ |, $$ \left|\overline{\rho}\right| $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfenced> <mml:mover> <mml:mi>ρ</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:mfenced> </mml:math> < 1 inside the forward tube, and set bounds on how fast | ρ |, $$ \left|\overline{\rho}\right| $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfenced> <mml:mover> <mml:mi>ρ</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:mfenced> </mml:math> may tend to 1 when approaching the Minkowski space. We also provide a guide to the axiomatic QFT literature for the modern CFT audience. We review the Wightman and Osterwalder-Schrader (OS) axioms for Lorentzian and Euclidean QFTs, and the celebrated OS theorem connecting them. We also review a classic result of Mack about the distributional OPE convergence. Some of the classic arguments turn out useful in our setup. Others fall short of our needs due to Lorentzian assumptions (Mack) or unverifiable Euclidean assumptions (OS theorem).