We revisit two longstanding puzzles in supersymmetric gauge theories. The first concerns the question of the holomorphy of the coupling, and related to this the possible definition of an exact (NSVZ) beta function. The second concerns instantons in p
ure gluodynamics, which appear to give sensible, exact results for certain correlation functions, which nonetheless differ from those obtained using systematic weak coupling expansions. For the first question, we extend an earlier proposal of Arkani-Hamed and Murayama, showing that if their regulated action is written suitably, the holomorphy of the couplings is manifest, and it is easy to determine the renormalization scheme for which the NSVZ formula holds. This scheme, however, is seen to be one of an infinite class of schemes, each leading to an exact beta function; the NSVZ scheme, while simple, is not selected by any compelling physical consideration. For the second question, we explain why the instanton computation in the pure supersymmetric gauge theory is not reliable, even at short distances. The semiclassical expansion about the instanton is purely formal; if infrared divergences appear, they spoil arguments based on holomorphy. We demonstrate that infrared divergences do not occur in the perturbation expansion about the instanton, but explain that there is no reason to think this captures all contributions from the sector with unit topological charge. That one expects additional contributions is illustrated by dilute gas corrections. These are infrared divergent, and so difficult to define, but if non-zero give order one, holomorphic, corrections to the leading result. Exploiting an earlier analysis of Davies et al, we demonstrate that in the theory compactified on a circle of radius beta, due to infrared effects, finite contributions indeed arise which are not visible in the formal limit that beta goes to infinity.
We derive a quantization formula of Bohr-Sommerfeld type for computing quasinormal frequencies for scalar perturbations in an AdS black hole in the limit of large scalar mass or spatial momentum. We then apply the formula to find poles in retarded Gr
een functions of boundary CFTs on $R^{1,d-1}$ and $RxS^{d-1}$. We find that when the boundary theory is perturbed by an operator of dimension $Delta>> 1$, the relaxation time back to equilibrium is given at zero momentum by ${1 over Delta pi T} << {1 over pi T}$. Turning on a large spatial momentum can significantly increase it. For a generic scalar operator in a CFT on $R^{1,d-1}$, there exists a sequence of poles near the lightcone whose imaginary part scales with momentum as $p^{-{d-2 over d+2}}$ in the large momentum limit. For a CFT on a sphere $S^{d-1}$ we show that the theory possesses a large number of long-lived quasiparticles whose imaginary part is exponentially small in momentum.
Recently there has been much progress in building models of gauge mediation, often with predictions different than those of minimal gauge mediation. Meade, Seiberg, and Shih have characterized the most general spectrum which can arise in gauge mediat
ed models. We discuss some of the challenges of building models of General Gauge Mediation, especially the problem of messenger parity and issues connected with R symmetry breaking and CP violation. We build a variety of viable, weakly coupled models which exhibit some or all of the possible low energy parameters.
