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arxiv: 2510.09866 · v3 · submitted 2025-10-10 · ✦ hep-th

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

Stefano Bolognesi , Andrea Luzio , Giacomo Santoni This is my paper

Pith reviewed 2026-05-18 07:17 UTC · model grok-4.3

classification ✦ hep-th
keywords SkyrmionsBaryonsColor-flavor lockingChiral gauge theoriesVector-like gauge theoriesTheta periodicityDomain wallsEffective field theory
0
0 comments X

The pith

In chiral SU(N) gauge theories with mixed representations, the color-flavor locked phase has no Skyrmions yet permits stable heavy baryons protected by unbroken symmetries.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper computes the topology of the coset space in the low-energy effective theory for these models after color-flavor locking. In the chiral cases this topology shows that Skyrmions are absent, but the same symmetry group forbids the decay of certain heavy baryons into the lighter degrees of freedom, leaving them stable. Vector-like models show the expected matching between baryons and Skyrmions. The theta-periodicity anomaly on domain walls is matched by the existing fields when color-flavor locking is complete, but requires extra dynamical degrees of freedom when part of the color group remains unbroken.

Core claim

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. In the vector-like models all the expected baryons are mirrored by Skyrmions. For complete CFL the theta-periodicity anomaly is always matched without introducing new dynamical degrees of freedom in the low-energy EFT.

What carries the argument

The topology of the coset space of the low-energy effective field theory in the color-flavor locked phase, which encodes both the possible solitons and the selection rules from the unbroken global symmetry group.

If this is right

  • Skyrmions are absent throughout the chiral models examined.
  • Certain heavy baryons remain stable because symmetry forbids their decay into lighter states.
  • Vector-like models exhibit a one-to-one correspondence between baryons and Skyrmions.
  • The theta-periodicity anomaly on domain walls is matched without extra fields when color-flavor locking is complete.
  • Incomplete color-flavor locking generally requires new dynamical degrees of freedom on domain walls to match the anomaly.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The mismatch between missing Skyrmions and protected heavy baryons may indicate that the Skyrme model approximation breaks down in these theories.
  • Similar topology-based arguments could be applied to other phases or matter representations to predict stable states.
  • The requirement for extra degrees of freedom on domain walls when color is partially unbroken suggests a general pattern for anomaly matching in gapped phases.
  • These results constrain possible ultraviolet completions that realize the same low-energy coset.

Load-bearing premise

The low-energy effective field theory obtained after color-flavor locking faithfully encodes the relevant topology of the coset space and the selection rules from the unbroken global symmetry group.

What would settle it

Detection of a Skyrmion soliton in one of the chiral models, or observation of a decay channel for a heavy baryon that the unbroken symmetry group is supposed to forbid, would show that the effective theory misses essential non-perturbative effects.

read the original abstract

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 $\theta$-periodicity anomaly. We find that, for complete CFL, the $\theta$-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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper studies baryons and solitons in chiral and vector-like SU(N) gauge theories with mixed one- and two-index matter representations, focusing on the color-flavor locked (CFL) phase. It computes the topology of the coset space in the low-energy EFT, finding that Skyrmions (classified by π3) are absent in the chiral models considered, while some of these models nevertheless admit heavy baryons whose stability is protected by the unbroken global symmetry group forbidding decay into lighter Goldstone modes. This mismatch is presented as evidence for a deeper dynamical mechanism. In vector-like models the expected baryons are mirrored by Skyrmions. The paper further examines domain walls, determining aspects of their dynamics via θ-periodicity anomaly matching; for complete CFL the anomaly is matched without new dynamical degrees of freedom, whereas unbroken color subgroups generally require additional light modes on the wall (with few exceptions).

Significance. If the central claims are substantiated, the work would identify a concrete limitation in the applicability of the Skyrme model to certain chiral gauge theories and illustrate how anomaly inflow constrains domain-wall dynamics in CFL phases. The explicit coset homotopy calculations and the identification of symmetry-protected heavy states provide falsifiable predictions that could guide lattice or holographic studies of baryon spectra in QCD-like theories.

major comments (2)
  1. [§3] §3 (coset topology after CFL): the claim that π3(G/H) is trivial for the chiral models, implying absence of Skyrmions, rests on the assumption that the low-energy EFT coset faithfully captures all relevant topology without UV non-perturbative corrections (e.g., instanton-induced operators that could generate effective π3 configurations or violate the apparent selection rules). The manuscript flags the resulting mismatch with stable heavy baryons but supplies no independent argument or suppression estimate showing such effects are absent or irrelevant at low energies; this assumption is load-bearing for the central mismatch claim.
  2. [§4] §4 (heavy baryon stability): the argument that certain heavy baryonic states are stable because their quantum numbers under the residual global symmetry forbid decay into the lighter EFT degrees of freedom does not include a quantitative discussion of possible higher-dimension operators or mixing induced by the UV completion that could lift the protection.
minor comments (2)
  1. [§2] The notation for the mixed one- and two-index representations and the precise definition of the unbroken subgroup H after CFL could be clarified with an explicit table or diagram in §2.
  2. [§5] In the domain-wall anomaly-matching discussion, a few steps in the inflow calculation would benefit from expanded intermediate equations for readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for the careful and constructive review of our manuscript on baryons and solitons in chiral and vector-like gauge theories. The major comments point to areas where additional justification would enhance the clarity of our arguments regarding the coset topology and baryon stability. We respond to each comment below and outline the planned revisions.

read point-by-point responses

  1. Referee: §3 (coset topology after CFL): the claim that π3(G/H) is trivial for the chiral models, implying absence of Skyrmions, rests on the assumption that the low-energy EFT coset faithfully captures all relevant topology without UV non-perturbative corrections (e.g., instanton-induced operators that could generate effective π3 configurations or violate the apparent selection rules). The manuscript flags the resulting mismatch with stable heavy baryons but supplies no independent argument or suppression estimate showing such effects are absent or irrelevant at low energies; this assumption is load-bearing for the central mismatch claim.

    Authors: We appreciate this observation. Our calculation of the homotopy groups is based on the standard low-energy EFT for the CFL phase, where the coset is determined by the pattern of symmetry breaking. In these specific chiral models, instanton effects are highly suppressed because the representations involved do not permit the formation of instantons that induce relevant operators at low energies, as can be seen from the 't Hooft anomaly matching which is satisfied without additional contributions. To address the referee's concern explicitly, we will add a discussion in the revised version of §3 providing an estimate of the suppression factor and explaining why UV corrections do not alter the triviality of π3 at the scales of interest. This will support the central claim of the mismatch with the heavy baryons. revision: yes

  2. Referee: §4 (heavy baryon stability): the argument that certain heavy baryonic states are stable because their quantum numbers under the residual global symmetry forbid decay into the lighter EFT degrees of freedom does not include a quantitative discussion of possible higher-dimension operators or mixing induced by the UV completion that could lift the protection.

    Authors: The protection mechanism relies on the unbroken global symmetries that assign distinct quantum numbers to the heavy baryons, preventing their decay into lighter modes while conserving those charges. Higher-dimension operators from the UV completion are suppressed by the cutoff scale and must preserve the global symmetries; thus, they cannot induce mixing that violates the selection rules. We will include in the revision of §4 a brief quantitative analysis, estimating the order of magnitude of possible mixing amplitudes and showing that they remain negligible compared to the mass gap of the heavy states. This will provide the requested discussion on the robustness of the stability. revision: yes

Circularity Check

0 steps flagged

No significant circularity; topological and anomaly arguments are self-contained.

full rationale

The paper's central results follow from direct computation of the homotopy groups of the coset space G/H in the CFL phase of the low-energy EFT, combined with unbroken global symmetry selection rules and standard anomaly inflow matching for domain walls. These are independent mathematical facts (e.g., vanishing of pi_3 implying absent Skyrmions, and quantum number mismatch forbidding decays) that do not reduce to self-definitional loops, fitted parameters renamed as predictions, or load-bearing self-citations. The noted mismatch between absent Skyrmions and stable heavy baryons is presented explicitly as an open puzzle requiring deeper UV dynamics, rather than being resolved tautologically within the EFT. No step equates a claimed derivation to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The analysis rests on standard assumptions of effective field theory in gauge theories, topological classification of cosets after symmetry breaking, and anomaly matching; no free parameters, ad-hoc entities, or invented particles are introduced in the abstract.

axioms (2)
  • domain assumption The low-energy EFT after color-flavor locking is described by a nonlinear sigma model whose target space is the coset of the broken global symmetries.
    Invoked when computing the topology that determines presence or absence of Skyrmions.
  • standard math Anomaly matching must be satisfied by the light degrees of freedom in the domain-wall background.
    Used to decide whether new dynamical objects are required when part of the color group remains unbroken.

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Reference graph

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