Spin chains in $ \mathcal{N} = {2} $ superconformal theories from the $ {\mathbb{Z}_2} $ quiver to superconformal QCD

Abstract

In this paper we find preliminary evidence that $ \mathcal{N} = {2} $ superconformal QCD, the SU(N c ) SYM theory with N f = 2N c fundamental hypermultiplets, might be integrable in the large N Veneziano limit. We evaluate the one-loop dilation operator in the scalar sector of the $ \mathcal{N} = {2} $ superconformal quiver with SU(N c ) × SU(N c ) gauge group, for N c ≡ N c . Both gauge couplings g and ǧ are exactly marginal. This theory interpolates between the $ {\mathbb{Z}_2} $ orbifold of $ \mathcal{N} = {4} $ SYM, which corresponds to ǧ = g, and $ \mathcal{N} = {2} $ superconformal QCD, which is obtained for ǧ → 0. The planar one-loop dilation operator takes the form of a nearest-neighbor spin-chain Hamiltonian. For superconformal QCD the spin chain is of novel form: besides the color-adjoint fields $ \phi_b^a, $ which occupy individual sites of the chain, there are “dimers” $ Q_i^a\overline Q_b^i $ of flavor-contracted fundamental fields, which occupy two neighboring sites. We solve the two-body scattering problem of magnon excitations and study the spectrum of bound states, for general ǧ/g. The dimeric excitations of superconformal QCD are seen to arise smoothly for ǧ → 0 as the limit of bound wavefunctions of the interpolating theory. Finally we check the Yang-Baxter equation for the two-magnon S-matrix. It holds as expected at the orbifold point ǧ = g. While violated for general ǧ ≠ g, it holds again in the limit ǧ → 0, hinting at one-loop integrability of planar $ \mathcal{N} = {2} $ superconformal QCD.