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

This paper investigates baryons, Skyrmions, and $\theta$-periodicity anomalies in chiral and vector-like $SU(N)$ gauge theories with mixed-representation matter, revealing that Skyrmions are absent in chiral models (where heavy stable baryons suggest a mismatch requiring deeper dynamical mechanisms) but present in vector-like models, while also determining that domain wall dynamics in the Color-flavor locked phase match the anomaly without new degrees of freedom only when the color group is fully broken.

The Gist

Imagine the universe is built from a giant, invisible fabric made of tiny, vibrating strings called "quarks." In some theories, these strings are tied together in specific patterns to form heavier objects called "baryons" (like protons and neutrons). Physicists usually try to understand these heavy objects by looking at the "low-energy" version of the theory, which is like looking at a blurry, simplified map of the territory.

This paper is a detective story where the authors try to match the "heavy objects" they expect to find in the real world (the UV theory) with the "simplified maps" (the low-energy Effective Field Theory) they draw to describe them. They are looking for a specific type of map feature called a Skyrmion.

Here is the breakdown of their investigation using simple analogies:

1. The Skyrmion: The "Swirl" in the Fabric

Think of the low-energy theory as a sheet of fabric. Sometimes, you can twist this fabric into a stable, knot-like swirl that doesn't unravel. In physics, these stable swirls are called Skyrmions.

• The Expectation: Usually, if you have a heavy particle (a baryon) in the real world, the simplified map should show a Skyrmion swirl that represents it. The Skyrmion is the "shadow" of the heavy particle.
• The Twist: The authors studied several complex theories (chiral and vector-like gauge theories) and found a strange mismatch.

2. The Mismatch: Heavy Guests with No Seats

In the Chiral Theories (one type of theory they studied):

• The Reality: They found that some heavy particles (heavy baryons) should be stable. Imagine a heavy guest at a party who is forbidden by the rules from leaving or breaking into smaller pieces. They are stuck there.
• The Map: However, when they looked at the "fabric" of their low-energy map, they found no knots or swirls (Skyrmions) to represent these guests. The fabric is too smooth to hold a knot.
• The Conclusion: This is a problem. If the heavy guest is stuck, the map should show a knot holding them. Since it doesn't, the authors suggest either:
  1. The heavy guest isn't actually stuck (they decay in a way the map doesn't show).
  2. The map itself is unreliable for these specific theories.

In the Vector-Like Theories (the other type they studied):

• The Match: Here, everything works perfectly. The heavy guests are stable, and the map has the exact right number of knots (Skyrmions) to hold them. The "heavy" particles and the "swirls" are perfect mirrors of each other.

3. The Domain Wall: The "Fault Line"

The authors then looked at Domain Walls. Imagine the fabric of the universe has a "fault line" or a seam where the rules of the fabric change slightly from one side to the other.

• The Anomaly: They checked a specific rule called the $\theta$-periodicity anomaly. Think of this as a "tension" in the fabric. If you twist the fabric by a full circle (2$\pi$), does it snap back perfectly, or does it leave a weird residue?
• Complete Locking (CFL): In theories where the color and flavor are "completely locked" together (like a zipper fully closed), the tension is zero. The fabric snaps back perfectly. No extra ingredients are needed to fix the map.
• Partial Locking: In theories where the zipper is only half-zipped (partial locking), the tension remains. The fabric doesn't snap back perfectly on its own.
  • The Fix: To fix this tension on the "fault line" (the domain wall), the authors found you must add new, invisible ingredients to the map. These ingredients behave like a special kind of "topological glue" (mathematically described as Chern-Simons theories) that lives only on the wall itself. Without this glue, the map is broken.

4. The "Pancake" Idea

The paper mentions a fascinating possibility: Pancake Solitons.

• Imagine a heavy particle isn't just a point, but a flat, pancake-shaped object made of a "metastable" (unstable but long-lasting) domain wall.
• In some theories (like $N_f=1$ QCD), these pancakes are known to be stable and act like baryons.
• The authors suggest that in the theories where they found the "Heavy Guest with No Seat" problem, these pancake-like objects might be the real solution. They might be the heavy particles that the smooth fabric map failed to capture as knots. However, the authors admit they don't have enough control over the math yet to prove these pancakes are stable.

Summary

• Chiral Theories: Heavy particles exist, but the low-energy map has no knots (Skyrmions) to represent them. This suggests the map is incomplete or the particles decay in a hidden way.
• Vector-Like Theories: Heavy particles exist, and the map has the perfect knots to match them.
• Domain Walls: If the theory is only partially "locked," the "fault lines" in the universe need special "topological glue" (new degrees of freedom) to fix a mathematical tension (anomaly). If it's fully locked, no glue is needed.

The paper essentially highlights a gap between what we expect to see (stable heavy particles) and what our simplified maps show (no knots), suggesting that our current understanding of how these particles form might need a new, more complex mechanism—perhaps involving these "pancake" structures.

Technical Summary

Problem Statement
This paper investigates the solitonic sector (baryons and domain walls) of chiral and vector-like $SU(N)$ gauge theories containing matter in mixed one-index (fundamental) and two-index (symmetric or antisymmetric) representations. The central problem addresses the consistency between the ultraviolet (UV) spectrum of baryonic operators and the infrared (IR) topological solitons (Skyrmions) predicted by the low-energy effective field theory (EFT). Specifically, the authors examine the Color-Flavor Locked (CFL) phase, where a quark bilinear condensate breaks the global symmetry $G$ down to a subgroup $H$.

A key tension arises in chiral models: while certain "heavy" baryons (constructed using the completely antisymmetric $\epsilon$ tensor of $SU(N)$) appear stable due to selection rules forbidding their decay into lighter degrees of freedom, the topology of the low-energy coset space $G_f / (H_f \times H_{cf})$ may be trivial, implying the absence of Skyrmions. This mismatch challenges the standard Skyrme model correspondence, where stable baryons are expected to be mirrored by topological solitons. Additionally, the paper explores the $\theta$-periodicity anomaly to determine if new dynamical degrees of freedom are required in the IR EFT, particularly in the context of domain walls.

Methodology
The authors employ a combination of symmetry analysis, homotopy theory, and anomaly matching techniques:

1. Symmetry and Baryon Classification: The authors classify baryonic operators in specific chiral models ($\psi\eta$, $\chi\eta$, Bars-Yankielowicz (BY), and Georgi-Glashow (GG)) and vector-like models ($\psi\tilde{\psi}\eta\tilde{\eta}$ and $\chi\tilde{\chi}\eta\tilde{\eta}$). They distinguish between "light" baryons (no $\epsilon$ tensor) and "heavy" baryons (containing $\epsilon$ tensors) and analyze their stability under the unbroken symmetry groups in the CFL phase.
2. Topological Analysis (Skyrmions): Using the Callan-Coleman-Wess-Zumino (CCWZ) construction, they identify the coset space of the Nambu-Goldstone bosons (NGBs). They compute the homotopy groups $\pi_3$, $\pi_4$, and $\pi_5$ of these cosets. A non-trivial $\pi_3$ is a necessary condition for Skyrmions, while specific conditions on $\pi_4$ and $\pi_5$ are required for non-trivial Wess-Zumino-Witten (WZW) terms.
3. Anomaly Matching ($\theta$-periodicity): The authors analyze the $\theta$-periodicity anomaly. They compare the UV partition function (restricted to specific topological sectors where CFL occurs) with the IR partition function. They determine if the anomaly can be matched solely by background gauge fields or if it necessitates new dynamical degrees of freedom on the worldvolume of domain walls.
4. Domain Wall Dynamics: For cases where the anomaly cannot be matched by background fields alone, they construct effective actions on the domain wall worldvolume, proposing Chern-Simons theories as the necessary topological field theories (TQFT) to stabilize "pancake-like" solitons.

Key Contributions and Results

• Absence of Skyrmions in Chiral Models: In the chiral models studied ($\psi\eta$, $\chi\eta$, BY, and GG), the authors find that the homotopy group $\pi_3(G_f / (H_f \times H_{cf}))$ is trivial. Consequently, the low-energy EFT does not admit Skyrmion solutions.
  • In the $\psi\eta$ and BY models, this absence is consistent with numerical checks suggesting that heavy baryons are unstable and can decay into lighter states.
  • In the $\chi\eta$ and GG models, however, the unbroken symmetry group forbids the decay of certain heavy baryons into lighter particles. The authors highlight a "mismatch": stable heavy baryons exist in the UV spectrum, yet the IR EFT lacks the topological solitons (Skyrmions) typically associated with them. They suggest this implies either an instability mechanism not captured by selection rules or a limitation of the Skyrme model in these specific low-energy EFTs.
• Skyrmions in Vector-Like Models: In contrast, the vector-like models ($\psi\tilde{\psi}\eta\tilde{\eta}$ and $\chi\tilde{\chi}\eta\tilde{\eta}$) possess two independent baryon numbers. The coset topology yields $\pi_3 = \mathbb{Z} \times \mathbb{Z}$, allowing for Skyrmions. The authors successfully match the quantum numbers and large-$N$ scaling of these Skyrmions with the heavy baryons of the UV theory via WZW terms.
• $\theta$-Periodicity Anomaly Matching:
  • Complete CFL: For theories with complete CFL (where the entire color group is broken, e.g., $\psi\eta$ and BY models), the $\theta$-periodicity anomaly is always matched by the background gauge fields alone. No new dynamical degrees of freedom are required in the IR EFT.
  • Partial CFL: For theories with partial CFL (where a subgroup of the color group remains unbroken, e.g., $\chi\eta$ and vector-like models), the anomaly matching fails if only background fields are considered. Specifically, for the $\chi\eta$ model with even $N$, and vector-like models where $N$ is a multiple of specific GCDs, the anomaly is non-trivial.
• Domain Wall Dynamics: In cases of partial CFL where the anomaly is non-trivial, the authors demonstrate that the domain wall worldvolume must support a topological field theory. They explicitly construct Chern-Simons actions (e.g., $U(1)_{-4}$ or $U(4)_{-1}$) on the domain wall that reproduce the required anomaly. This suggests the existence of "pancake-like" solitons (metastable domain walls bounded by strings), which could potentially correspond to the heavy baryons missing from the Skyrmion analysis.

Significance
The paper provides a rigorous consistency check for the IR phases of strongly coupled gauge theories using modern anomaly matching and topological tools. Its primary significance lies in identifying a specific class of chiral gauge theories where the standard correspondence between stable heavy baryons and Skyrmions breaks down. The authors argue that this discrepancy necessitates a deeper dynamical explanation, potentially involving the instability of heavy baryons or the emergence of non-perturbative states (like pancake solitons) on domain walls. Furthermore, the work clarifies the conditions under which the $\theta$-periodicity anomaly forces the introduction of new degrees of freedom in the low-energy EFT, specifically linking partial CFL phases to the necessity of Chern-Simons theories on domain walls. This refines the understanding of how topological solitons and anomalies constrain the spectrum of baryons in theories with mixed representations.