Minimal Distances Between SCFTs

We study lower bounds on the minimal distance in theory space between four-dimensional superconformal field theories (SCFTs) connected via broad classes of renormalization group (RG) flows preserving various amounts of supersymmetry (SUSY). For N=1 RG flows, the ultraviolet (UV) and infrared (IR) endpoints of the flow can be parametrically close. On the other hand, for RG flows emanating from a maximally supersymmetric SCFT, the distance to the IR theory cannot be arbitrarily small regardless of the amount of (non-trivial) SUSY preserved along the flow. The case of RG flows from N=2 UV SCFTs is more subtle. We argue that for RG flows preserving the full N=2 SUSY, there are various obstructions to finding examples with parametrically close UV and IR endpoints. Under reasonable assumptions, these obstructions include: unitarity, known bounds on the c central charge derived from associativity of the operator product expansion, and the central charge bounds of Hofman and Maldacena. On the other hand, for RG flows that break N=2 to N=1, it is possible to find IR fixed points that are parametrically close to the UV ones. In this case, we argue that if the UV SCFT possesses a single stress tensor, then such RG flows excite of order all the degrees of freedom of the UV theory. Furthermore, if the UV theory has some flavor symmetry, we argue that the UV central charges should not be too large relative to certain parameters in the theory.