Infrared spectra of some strongly--coupled chiral gauge theories

This paper investigates several simple asymptotically-free chiral gauge theories to reveal surprisingly rich infrared effective structures, RG flows, and light spectra by applying recent developments in generalized symmetries and anomaly matching alongside established knowledge of vectorlike theories.

The Gist

Imagine the universe as a giant, complex machine built from invisible threads of force. Physicists usually understand the "easy" parts of this machine, like how magnets stick together or how light behaves. But there is a mysterious, chaotic side to this machine where the forces get incredibly strong and tangled. This is the world of "chiral gauge theories."

Think of these theories as a set of rules for how different types of invisible particles (fermions) interact with invisible forces (gauge groups). The authors of this paper are like explorers trying to map out what happens when these forces get so strong that they crush the particles together, forming new, unexpected structures. They aren't building a new car or a new phone; they are trying to understand the fundamental "engine" of reality.

Here is a breakdown of their journey, using simple analogies:

The Main Idea: A Race of Strength

The researchers set up several different "race tracks" (theoretical models). In each race, there are different teams of forces (like $SU(N)$ or $Sp(6)$) and different runners (particles).

The only thing that changes between races is who gets strong first.

• Imagine two runners, Alice and Bob.
• Scenario A: Alice gets tired (strong) first. She grabs Bob, and they merge into a new team.
• Scenario B: Bob gets tired first. He grabs Alice, and they merge differently.

The paper asks: What does the finish line look like in each scenario? Does the race end with a single winner, a team of friends, or does everyone just stop moving?

The Models They Studied

1. The "Handshake" Model (The $SU(N) - SU(N+4)$ Model)
Imagine two groups of people holding hands in a circle.

• If the first group gets strong: They pull the second group into a tight hug. This "hug" (called a condensate) breaks the circle and leaves behind a smaller, weaker group of people who are still holding hands, plus some loose particles that float away.
• If the second group gets strong first: They pull the first group in a different way. The result is a different kind of leftover group.
• The Surprise: Even though they started with the same ingredients, the order in which they got strong changed the final "family" of particles left over. Sometimes they end up with a "supersymmetric" team (a very special, balanced group), and sometimes they end up with a mix of heavy and light particles.

2. The "Chain Reaction" Model (The Quiver Model)
Imagine a line of people holding hands: Person 1 holds Person 2, who holds Person 3, who holds Person 4, and so on.

• If the first person (Person 1) gets super strong, they pull Person 2 into a tight knot.
• Because Person 2 is now tied up, they can't hold hands with Person 3 in the same way. The chain breaks and reforms.
• The authors found that this chain reaction keeps happening. If you have a long chain, the strong force eats away at the ends, two by two, until you are left with just a few people in the middle.
• The Result: In some cases, the chain shrinks down until you are left with a single, lonely person who doesn't interact with anyone else anymore. In other cases, you end up with a very specific, balanced team that behaves like a "supersymmetric" machine.

3. The "Tug-of-War" Model (The $SU(N) - Sp(6) - Sp(6)$ Model)
Imagine a tug-of-war where one team is huge ($SU(N)$) and two smaller teams ($Sp(6)$) are pulling on the sides.

• If the big team wins first: They pull the rope so hard that the two smaller teams are forced to merge into one diagonal team. The "rope" (the force) gets heavy, and the smaller teams get stuck together, forming heavy balls of matter.
• If one of the small teams wins first: They pull the big team into a different shape. The big team shrinks down, and the other small team is left alone.
• The Result: Depending on who wins the tug-of-war first, you either get a world full of heavy, stuck-together particles, or a world where the forces split apart and leave behind a few light, free-floating particles.

4. The "Solo Act" Model (The $SU(10)$ Model)
This is the strangest race. There is only one runner and one force.

• The force gets so strong that the runner tries to grab themselves.
• Because of the rules of the universe (quantum mechanics), they can't just grab themselves and disappear. Instead, they split into two different "versions" of themselves.
• One version gets heavy and disappears. The other version is left alone, but now it's part of a smaller, weaker force.
• The Result: Eventually, the system breaks down into two separate, invisible "photons" (like beams of light) that don't talk to anything else. It's a world of pure, empty light.

The Big Picture

The authors discovered that the "infrared" (the deep, low-energy end of the universe) is full of surprises.

• Sometimes, the chaos settles down into a single, free particle that just floats around.
• Sometimes, it settles into two free beams of light.
• Sometimes, it creates a gapped world where everything is heavy and nothing moves.

They didn't find a way to build a new engine for a car or a cure for a disease. Instead, they mapped out the possible "landscapes" of the universe's hidden rules. They showed that even with simple starting ingredients, the universe can end up in many different, intricate states depending on the order of events.

In short: They played with the rules of the universe's strongest forces to see what kind of "final state" the universe could end up in. They found that the universe is more flexible and creative than we thought, capable of turning complex tangles of forces into simple, free-floating particles or empty light.

Technical Summary

Technical Summary: Infrared Spectra of Strongly-Coupled Chiral Gauge Theories

Problem Statement
The dynamics of strongly-coupled gauge theories in four dimensions remain poorly understood, particularly for chiral gauge theories which lack the non-trivial global symmetries ("family" symmetries) often found in vectorlike theories like QCD or $\mathcal{N}=2$ supersymmetric theories. While vectorlike theories have been extensively studied via lattice simulations and exact analytical methods, quantitative control over chiral gauge theories—potentially relevant for physics beyond the Standard Model—is limited. The authors aim to explore the infrared (IR) effective theories, renormalization group (RG) flows, and light spectra of simple, asymptotically-free (AF) chiral gauge theories. The models considered possess no non-trivial non-abelian global symmetries; their only free parameters are the choice of gauge groups, matter Weyl fermion representations, and the relative magnitudes of the renormalization-group-invariant scales ($\Lambda_i$) associated with each gauge group.

Methodology
The authors employ a combination of recent theoretical developments regarding generalized symmetries and new types of 't Hooft anomaly-matching considerations, alongside established knowledge from vectorlike theories. The analysis focuses on determining the IR phase of specific models based on the hierarchy of their coupling strengths. The primary mechanism involves identifying which gauge interaction becomes strong first along the RG flow, leading to confinement or dynamical Higgs phases (via bifermion condensates). The study relies on:

• Anomaly Matching: Utilizing mixed anomalies involving discrete symmetries (e.g., $Z_2$ fermion parity) and 1-form center symmetries to rule out certain confining phases and favor dynamical Higgs phases.
• Condensate Formation: Assuming the formation of color-flavor locked bifermion condensates ($\langle \psi \eta \rangle$, $\langle \chi \chi \rangle$, etc.) when a gauge group becomes strongly coupled.
• Symmetry Breaking Patterns: Analyzing how condensates break global and gauge symmetries, determining the resulting massless spectra (Goldstone bosons, massless fermions) and massive sectors.
• RG Flow Analysis: Tracking the evolution of coupling constants ($g_i$) from the ultraviolet (UV) to the IR, identifying sequential strong-coupling scales ($\Lambda_1, \Lambda_2, \dots$) and the resulting effective theories at each stage.

Key Contributions and Results

The paper investigates four distinct classes of models, yielding rich and varied IR structures:

1. The $SU(N) - SU(N+4)$ Model:

   • Setup: A combination of Bars-Yankielowicz and Georgi-Glashow models with fermions $\psi, \eta, \chi$.
   • Scenario A ($SU(N)$ strong first): The system enters a dynamical Higgs phase via a $\langle \psi \eta \rangle$ condensate, breaking $SU(N+4)$ to $SU(4)$ and locking $SU(N)$ to a diagonal subgroup. The low-energy theory contains massless baryons and pseudo-Nambu-Goldstone (NG) bosons. Depending on $N$, the remaining $SU(N)'$ or $SU(4)$ subsequently confines, leading to either a pure $SU(4)$ theory with a self-adjoint fermion or an $SU(N)$ tensor QCD.
   • Scenario B ($SU(N+4)$ strong first): The system follows the Georgi-Glashow dynamics with a $\langle \chi \eta \rangle$ condensate, breaking $SU(N+4)$ to $SU(N) \times SU(4)$. The IR spectrum consists of a one-flavor symmetric tensor QCD and a decoupled $SU(4)$ sector that eventually confines.

2. Quiver Models with Multiple $SU(N)$'s:

   • Setup: $SU(N)_1 \times SU(N)_2 \times \dots \times SU(N)_n$ with bifundamental fermions.
   • Dynamics: Sequential confinement occurs. When one factor becomes strong, it forms a condensate that breaks the adjacent axial symmetry, leaving a diagonal unbroken gauge group.
   • Result: For odd $n$, the chain reduces to an $SU(N)_3$ quiver, eventually flowing to a pure $\mathcal{N}=1$ Supersymmetric Yang-Mills (SYM) theory plus a decoupled free massless fermion. For even $n$, the system reduces to an $SU(N)_1 - SU(N)_2$ quiver, where all fermions eventually acquire dynamical masses, leaving a pure gauge theory or a gapped spectrum depending on the specific scale hierarchy.

3. The $SU(N) - Sp(6) - Sp(6)$ Model:

   • Setup: A generalization of the Pati-Salam model with $SU(N)$ and two $Sp(6)$ factors.
   • Scenario A ($SU(N)$ strong first): The system behaves like $SU(N)$ QCD with $N_f=6$. The condensate $\langle \psi \tilde{\psi} \rangle$ breaks the global symmetry and the weakly gauged $Sp(6)_L \times Sp(6)_R$ to the diagonal $Sp(6)_{diag}$. The IR contains massive mesons/baryons, massive gauge bosons (from broken $Sp(6)_A$), and light pseudo-NG bosons in the $\mathbf{14}$ of $Sp(6)_{diag}$.
   • Scenario B ($Sp(6)$ strong first): A condensate $\langle \psi \psi \rangle$ forms, breaking $SU(N) \to Sp(N)$. The remaining fermions $\tilde{\psi}$ then form a condensate at a lower scale, breaking $Sp(N)$ and leaving a pure $Sp(6)_R$ theory which eventually confines.

4. The $AF-IF$ $SU(10)$ Model:

   • Setup: An $SU(10)$ theory with a single Weyl fermion in the fifth-rank antisymmetric (self-adjoint) representation.
   • Dynamics: Generalized symmetry arguments (mixed anomalies between $Z_{70}$ and 1-form center symmetry) suggest a condensate $\langle \psi \psi \rangle$ forms, breaking $SU(10) \to SU(8) \times SU(2) \times U(1)$.
   • Result: The $SU(8)$ sector is asymptotically free, while $SU(2)$ is infrared free (IF). The $SU(8)$ interaction becomes strong at a lower scale $\Lambda_2$, forming a condensate that breaks the remaining $SU(2)$ to $U(1)$. The final IR state is a free theory of two decoupled photons (one from the original $U(1)$, one from the broken $SU(2)$) with no light matter.

Significance and Claims
The authors claim that these models demonstrate "surprisingly rich and intriguing" structures in their infrared effective theories, RG flows, and light spectra. The work highlights that even simple chiral gauge theories without non-abelian global symmetries can exhibit complex dynamical behaviors, including sequential symmetry breaking, mixed asymptotic freedom/infrared freedom, and the emergence of free massless fermions or photons in the deep IR.

The paper modestly frames these results as "modest steps" toward a deeper understanding of chiral gauge dynamics. The authors explicitly state that while the results are of interest, they do not yet constitute a realistic model of fundamental interactions. They acknowledge that a more elaborated UV model is required to reproduce the Standard Model ($SU(3) \times SU(2) \times U(1)$) with three families, realistic fermion masses, CKM mixing, and a natural Higgs sector. The current work serves as a theoretical exploration of the possible IR outcomes of strongly-coupled chiral systems.