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Spin Chains in N=2 Superconformal Theories: from the Z_2 Quiver to Superconformal QCD

Abhijit Gadde, Elli Pomoni, Leonardo Rastelli

TL;DR

This work analyzes the one-loop dilatation operator in the scalar sector of N=2 SCQCD and an interpolating N=2 SCFT with two marginal couplings, recasting planar dynamics as a nearest-neighbor spin chain with both adjoint and dimer (Qar Q) impurities. It computes the magnon spectrum, maps the interpolating theory between the N=4 Z2 orbifold and N=2 SCQCD via the parameter κ=ĝ/g, and solves two-body scattering to reveal a rich spectrum of bound and antibound states whose existence and dispersion depend on κ. A key finding is that Yang-Baxter integrability holds at the orbifold point (κ=1) and in the SCQCD limit (κ→0) but generally fails for intermediate κ, although the κ→0 limit shows signs of potential one-loop integrability in planar N=2 SCQCD. The results support a bottom-up field-theory perspective on the AdS dual of N=2 SCQCD and highlight the nuanced role of dimers and flavor structure in the emergent spin-chain dynamics.

Abstract

In this paper we find preliminary evidence that N=2 superconformal QCD, the SU(N_c) SYM theory with N_f= 2 N_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 N=2 superconformal quiver with SU(N_c) X SU(N_{\check c}) gauge group, for N_c = N_{\check c}. Both gauge couplings g and \check g are exactly marginal. This theory interpolates between the Z_2 orbifold of N=4 SYM, which corresponds to \check g=g, and N=2 superconformal QCD, which is obtained for \check g -> 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 φ^a_{b}, which occupy individual sites of the chain, there are "dimers" Q^a_{i} \bar Q^i_{b} 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 \check g/g. The dimeric excitations of superconformal QCD are seen to arise smoothly for \check g -> 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 \check g = g. While violated for general \check g \neq g, it holds again in the limit \check g -> 0, hinting at one-loop integrability of planar N=2 superconformal QCD.

Spin Chains in N=2 Superconformal Theories: from the Z_2 Quiver to Superconformal QCD

TL;DR

This work analyzes the one-loop dilatation operator in the scalar sector of N=2 SCQCD and an interpolating N=2 SCFT with two marginal couplings, recasting planar dynamics as a nearest-neighbor spin chain with both adjoint and dimer (Qar Q) impurities. It computes the magnon spectrum, maps the interpolating theory between the N=4 Z2 orbifold and N=2 SCQCD via the parameter κ=ĝ/g, and solves two-body scattering to reveal a rich spectrum of bound and antibound states whose existence and dispersion depend on κ. A key finding is that Yang-Baxter integrability holds at the orbifold point (κ=1) and in the SCQCD limit (κ→0) but generally fails for intermediate κ, although the κ→0 limit shows signs of potential one-loop integrability in planar N=2 SCQCD. The results support a bottom-up field-theory perspective on the AdS dual of N=2 SCQCD and highlight the nuanced role of dimers and flavor structure in the emergent spin-chain dynamics.

Abstract

In this paper we find preliminary evidence that N=2 superconformal QCD, the SU(N_c) SYM theory with N_f= 2 N_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 N=2 superconformal quiver with SU(N_c) X SU(N_{\check c}) gauge group, for N_c = N_{\check c}. Both gauge couplings g and \check g are exactly marginal. This theory interpolates between the Z_2 orbifold of N=4 SYM, which corresponds to \check g=g, and N=2 superconformal QCD, which is obtained for \check g -> 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 φ^a_{b}, which occupy individual sites of the chain, there are "dimers" Q^a_{i} \bar Q^i_{b} 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 \check g/g. The dimeric excitations of superconformal QCD are seen to arise smoothly for \check g -> 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 \check g = g. While violated for general \check g \neq g, it holds again in the limit \check g -> 0, hinting at one-loop integrability of planar N=2 superconformal QCD.

Paper Structure

This paper contains 18 sections, 69 equations, 1 figure, 6 tables.

Table of Contents

  1. Introduction
  2. Lagrangian and Symmetries
  3. ${\cal N} =2$ SCQCD
  4. $\mathbb{Z}_2$ orbifold of ${\cal N} = 4$ and interpolating family of SCFTs
  5. One-loop Dilation Operator in the Scalar Sector
  6. Hamiltonian for ${\cal N}=2$ super QCD
  7. Magnons in the SCQCD spin chain
  8. Hamiltonian for the interpolating SCFT
  9. Magnons in the interpolating spin chain
  10. Protected Spectrum
  11. Protected spectrum in ${\cal N} = 2$ SCQCD
  12. Protected spectrum in the orbifold theory
  13. Away from the orbifold point: matching with ${\cal N} = 2$ SCQCD
  14. $\mathbf{SU(2)_L}$ singlets
  15. $\mathbf{SU(2)_L}$ non-singlets
  16. ...and 3 more sections

Figures (1)

  • Figure 1: Various types of Feynman diagrams that contribute, at one loop, to anomalous dimension. The first diagram is the self-energy contribution. The second diagram represents the gluon exchange contribution whereas the third one stands for the quartic interaction between the fields. The first and the second diagrams are proportional to the identity in the R symmetry space while the third one carries a nontrivial R symmetry index structure.