Conformal Field Theories in the Epsilon and 1/N Expansions

Abstract

In this thesis, we study various conformal field theories in two different approximation schemes - the epsilon-expansion in dimensional continuation, and the large N expansion. We first propose a cubic theory in d=6-epsilon as the UV completion of the quartic scalar O(N) theory in d>4. We study this theory to three-loop order and show that various operator dimensions are consistent with large-N results. This theory possesses an IR stable fixed point at real couplings for N>1038, suggesting the existence of a perturbatively unitary interacting O(N) symmetric CFT in d=5. Extending this model to Sp(N) symmetric theories, we find an interacting non-unitary CFT in d=5. For the special case of Sp(2), the IR fixed point possesses an enhanced symmetry given by the supergroup OSp(1|2). We also observe that various operator dimensions of the Sp(2) theory match those from the 0-state Potts model. We provide a graph theoretic proof showing that the zero, two, and three-point functions in the Sp(2) model and the 0-state Potts model indeed match to all orders in perturbation theory, strongly suggesting their equivalence. We then study two fermionic theories in d=2+epsilon - the Gross-Neveu model and the Nambu-Jona-Lasinio model, together with their UV completions in d=4-epsilon given by the Gross-Neveu-Yukawa and the Nambu-Jona-Lasinio-Yukawa theories. We compute their sphere free energy and certain operator dimensions, passing all checks against large-N results. We use two sided Pad\'e approximations with our epsilon expansion results to obtain estimates of various quantities in the physical dimension d=3. Finally, we provide evidence that the N=1 Gross-Neveu-Yukawa model which contains a 2-component Majorana fermion, and the N=2 Nambu-Jona-Lasinion-Yukawa model which contains a 2-component Dirac fermion, both have emergent supersymmetry.