QCD in AdS

TL;DR

The paper analyzes QCD with fundamental matter on AdS, using broken conformal Ward identities and Witten diagrams to compute one-loop anomalous dimensions of boundary displacement operators for both fermions and scalars. It provides evidence that Dirichlet bc disappear via merger and annihilation in the confining phase, while Neumann bc persist toward flat space, and shows that in the conformal window one boundary operator becomes the IR displacement operator in the Banks–Zaks regime. The work also connects boundary data to the end of the conformal window and explores how chiral symmetry breaking manifests at the AdS boundary, with implications for how AdS data encodes nonperturbative flat-space physics. Overall, it extends YM results to include matter, revealing robust patterns of boundary operator mixing, confinement signatures, and clues for the location and nature of conformal windows.

Abstract

We study QCD on AdS space with scalars or fermions in the fundamental representation, extending earlier results on pure Yang-Mills theory. In the latter, the Dirichlet boundary condition is conjectured to disappear via merger and annihilation, as signaled by the lightest scalar singlet operator approaching marginality as the coupling increases. With matter, there are two candidate operators for this mechanism. We compute their one-loop anomalous dimensions via broken conformal Ward identities and Witten diagrams. In the confining phase, with Dirichlet (Neumann) boundary condition, their anomalous dimensions are negative (positive), consistent with the disappearance (persistence) of the associated boundary CFT in the flat-space limit. In the conformal window, one of these operators becomes the displacement operator of the IR CFT, as signaled by the vanishing of its one-loop anomalous dimension in the perturbative Banks-Zaks regime. Possible scenarios for the lower edge of the conformal window are discussed. Finally, we consider general boundary conditions on fermions and discuss their relation to chiral symmetry breaking in flat space.

Figures (6)

• Figure 1: Witten diagrams that contribute to the two point function of $\mathcal{D}_{\rm F}$ (a, b) and to the mixing two point function between $\mathcal{D}_{\rm YM}$ and $\mathcal{D}_{\rm F}$ (c) at the next-to-leading order.
• Figure 2: One-loop corrections to the fermion two-point function. The counterterm contribution (ii) does not give rise to logarithmic terms and hence does not contribute to the anomalous dimension.
• Figure 3: Witten diagrams that contribute to the two point function of $\mathcal{D}_{\rm SC}$ (a, b) and to the mixing two point function between $\mathcal{D}_{\rm YM}$ and $\mathcal{D}_{\rm SC}$ (c, d) at the next-to-leading order.
• Figure 4: One-loop corrections to the scalar two-point function. The quartic diagram (ii) vanishes in the FY gauge, while the counterterm (iii) contribution does not give rise to logarithmic terms.
• Figure 5: Schematic representation of the evolution of the scaling dimensions of the operators $\overline{\cal D}_{1,2}$ for both Dirichlet and Neumann bc, as a function of the AdS length $L$, in the confining phase. The red dashed line indicates the operator in the boundary conformal theory associated to the Dirichlet$^*$ bc, which becomes marginal at $L=L_\text{crit}$, where both Dirichlet and Dirichlet$^*$ bc merge and disappear. The Dirichlet$^*$ bc must exist for $\Lambda L \lesssim (\Lambda L)_{\text{crit}}$, but it is not known how much it extends towards weak coupling.