Baryons, Skyrmions and $θ$-periodicity anomaly in chiral and vector-like gauge theories

In this paper, we study the baryons and solitons of chiral and vector-like $SU(N)$ gauge theories with matter in mixed one and two-index representations. Focusing on the Color-flavor locked (CFL) phase, we compute the topology of the coset of their low-energy EFT. We find that in the chiral models under consideration, Skyrmions are always absent. We also show, however, that some of these models admit heavy baryons that are expected to be stable, because their decay into the lighter degrees of freedom of the EFT is forbidden by the unbroken symmetry group. This mismatch suggests that some deeper dynamical mechanism must be responsible with either the instability of the seemingly stable heavy baryons or the unreliability of the Skyrme model in the low-energy EFT. In the vector-like models all the expected baryons are mirrored by Skyrmions. Then we turn to the study of domain walls. We determine some aspects of their dynamics by matching the $θ$-periodicity anomaly. We find that, for complete CFL, the $θ$-periodicity anomaly is always matched without introducing new dynamical degrees of freedom in the low-energy EFT. If part of the color group is unbroken, new dynamical degrees of freedom must be added to the low-energy EFT in the domain-wall background with few exceptions.

💡 Research Summary

This paper investigates the spectrum of baryons and topological solitons in a class of strongly‑coupled SU(N) gauge theories that contain fermions in both fundamental (one‑index) and two‑index representations. The authors focus on phases where a color‑flavor locking (CFL) mechanism operates, a situation familiar from high‑density QCD but here applied to a broader set of chiral and vector‑like models.

The first part of the work establishes a general framework for CFL. Starting from a product symmetry G = G_c × G_f (possibly modded by a discrete subgroup), the authors define complete CFL (all color generators are Higgsed) and partial CFL (a subgroup H_c ⊂ G_c remains unbroken). Using the CCWZ construction they derive the low‑energy effective field theory (EFT): the physical Nambu‑Goldstone bosons (NGBs) live on the coset G_f/(H_f × H_{cf}), while the broken color generators give massive gauge bosons. When the CFL scale is well separated from the strong‑coupling scale, the EFT is reliable; otherwise only qualitative statements survive.

Next, the paper examines several concrete chiral gauge theories—ψη, χη, Bars‑Yankielowicz (BY), and Georgi‑Glashow (GG) models—each containing mixed one‑ and two‑index fermions. For each model the authors list the classical global symmetries, the ’t Hooft anomaly structure, and the possible gauge‑invariant baryon operators. Two families of baryons appear: “light” baryons built without the totally antisymmetric ε‑tensor (O(N⁰) fermion fields) and “heavy” baryons that involve the ε‑tensor and therefore carry O(N) units of a conserved U(1) charge. In the UV, heavy baryons are protected from decay by these charges and are expected to be stable.

The central question is whether the low‑energy EFT contains Skyrmions—topological solitons associated with π₃ of the NGB target space—that can be identified with the heavy baryons. The authors compute π₃ for each model’s coset. In all chiral examples π₃ is trivial, so no Skyrmion exists. Nevertheless, the heavy baryons remain protected, creating a mismatch: the usual Skyrme‑baryon correspondence fails. The authors discuss two possible resolutions. First, the low‑energy description may be incomplete; additional non‑perturbative objects such as domain‑wall “pancake” solitons, whose world‑volume hosts a topological quantum field theory (TQFT), could provide the missing dynamics and render the heavy baryons unstable or reinterpret them. Second, the Skyrme model itself may be inapplicable in these chiral settings, perhaps because the required higher‑derivative terms are suppressed or because the EFT lacks the necessary Wess‑Zumino‑Witten (WZW) term.

In contrast, for the vector‑like theories the homotopy group π₃ is non‑trivial (π₃ ≅ ℤ). The EFT admits a WZW term, and Skyrmions exist with the correct quantum numbers to match the heavy baryons. Thus the traditional Skyrmion‑baryon correspondence holds, confirming that in these theories the low‑energy EFT faithfully captures the baryonic sector.

The second major theme of the paper is the analysis of domain walls and the so‑called θ‑periodicity anomaly. A θ‑periodicity anomaly occurs when, in the presence of background gauge fields with fractional topological charge, shifting the θ angle by 2π changes the Euclidean partition function by a non‑trivial phase. Matching this anomaly between the UV theory and the IR EFT provides a stringent consistency check. The authors show that in a completely locked CFL phase the anomaly disappears because all color degrees of freedom are Higgsed; the low‑energy theory automatically reproduces the trivial periodicity. However, in a partially locked phase where some color gauge bosons remain massless, the anomaly survives. To cancel it, the domain‑wall world‑volume must support a Chern‑Simons term (or a more general TQFT) whose variation under θ‑shifts reproduces the required phase. The paper identifies the precise Chern‑Simons level required for several models and notes a few exceptional cases where the anomaly can be matched without extra degrees of freedom.

Overall, the work delivers a comprehensive picture: (i) the Skyrmion‑baryon correspondence is not universal; it fails in chiral gauge theories with mixed representations despite the presence of stable heavy baryons, indicating that additional IR dynamics (perhaps domain‑wall TQFTs) are needed; (ii) in vector‑like theories the correspondence works as expected; (iii) the θ‑periodicity anomaly provides a powerful tool to diagnose whether the low‑energy EFT needs extra topological sectors, especially in partially locked CFL phases. The authors conclude that any realistic description of such gauge theories must go beyond the naive natural anomaly‑matching paradigm and incorporate either new solitonic objects or modified effective actions to resolve the observed mismatches.