Effective dynamics of open 2D CFTs

We analyze the momentum-space representations of causal response functions of scalar primary operators in an arbitrary 2D CFT, which can be exploited to characterize the effective dynamics of open quantum CFTs. We derive our results by analytically continuing Euclidean correlators to Lorentzian signature. While the two- and three-point functions can be continued straightforwardly, the case of the four-point function is more challenging. In principle, the analytic continuation of the four-point function can be performed in full generality using either the Virasoro conformal block expansion or by mapping the conformal blocks to radial coordinates. However, these frameworks are mathematically intractable for explicit momentum-space calculations. We argue that the global block expansion offers a valuable alternative, as it allows us to recover the time-ordered correlators while preserving a relatively simple analytic structure. From these causal configurations, the four-point response function is systematically constructed and its momentum-space representation is evaluated.