Confinement without symmetry breaking in chiral gauge theories

This paper utilizes the functional renormalisation group to demonstrate that the Bars-Yankielowicz class of chiral gauge theories exhibits two distinct infrared phases, including a novel large-colour regime characterized by confinement without symmetry breaking.

The Gist

Imagine the universe is built from tiny, invisible Lego bricks called particles. Some of these bricks are "fermions" (like electrons and quarks), and they are held together by invisible "glue" called gauge fields (like the strong force that holds atomic nuclei together).

For decades, physicists have understood how these bricks behave when they are "mirror images" of each other (like a left hand and a right hand). But there's a tricky class of theories where the bricks are chiral—meaning the "left-handed" bricks and "right-handed" bricks are completely different species with different rules. It's like trying to build a house where the left-handed bricks only fit with other left-handed bricks, but the right-handed ones have their own separate, incompatible set of rules.

This paper, written by Hao-Lin Li and colleagues, tackles a big mystery: What happens to these chiral theories when the energy gets very low (the "infrared" limit)? Do they lock up into solid structures (confinement), or do they break apart and change their nature (symmetry breaking)?

Here is the story of their discovery, explained simply:

1. The Two Possible Outcomes

The researchers used a powerful mathematical tool called the Functional Renormalization Group (fRG). Think of this tool as a "zoom lens" that lets physicists look at the universe at different scales, from the tiniest sub-atomic level all the way up to larger scales, watching how the rules change as they zoom out.

They studied a specific family of theories (called the Bars–Yankielowicz class) and found that the outcome depends entirely on the number of "colors" (a type of charge, not like paint, but a quantum property) in the theory. Let's call this number $N_c$.

• Scenario A: Few Colors (Small $N_c$)
  Imagine a small, tight-knit community. When the number of colors is small (like 3), the "glue" gets so strong that it forces the particles to stick together. But in doing so, it also forces them to change their identity. They break their original symmetry.

  • The Analogy: It's like a group of dancers who are forced to hold hands so tightly that they can no longer dance individually; they form a rigid, locked chain. The system "breaks" its freedom to move independently. This is called Confinement + Symmetry Breaking.

• Scenario B: Many Colors (Large $N_c$)
  Now, imagine a massive, sprawling city with thousands of people. When the number of colors is large (like 6 or more), the "glue" still works. The particles are still locked together and cannot escape (they are confined).

  • The Twist: However, in this large crowd, the particles do not change their identity. They stay exactly as they were, just stuck together.
  • The Analogy: It's like a massive crowd of people holding hands in a giant circle. They are locked in a circle (confined), but everyone is still free to be themselves; no one has been forced to change who they are. This is Confinement without Symmetry Breaking.

2. Why This Discovery is a Big Deal

For a long time, physicists thought that if particles were confined (locked up), they must have broken their symmetry (changed their nature) to do it. It was like assuming that if a group of people is locked in a room, they must have all agreed to wear the same uniform.

This paper proves that this assumption is wrong.

They found a "Goldilocks zone" (a critical point around $N_c \approx 3.8$) where the rules flip.

• Below this point: The particles lock up and change.
• Above this point: The particles lock up but stay the same.

3. The "Exotic" Possibilities

The second scenario (Confinement without Symmetry Breaking) is a brand-new regime. It suggests the existence of a "spectrum" of particles that are massless (weightless) and behave in ways we haven't seen before.

• The Analogy: Imagine a world where you can build a solid wall out of bricks, but the bricks inside the wall are still "ghosts"—they have no weight and don't interact with the world outside, yet the wall itself is solid. This opens the door to "exotic" physics, potentially explaining things like Symmetric Mass Generation, where particles gain mass without breaking any of the universe's fundamental symmetries.

4. How They Did It

The authors didn't just guess; they used a sophisticated mathematical framework (the fRG) that acts like a high-speed simulation.

• They tracked the "strength" of the glue (gauge coupling).
• They tracked the "tendency" of particles to pair up (four-fermion interactions).
• They watched how these two forces fought against each other as the number of colors changed.

They found that for large numbers of colors, the "glue" gets strong enough to lock the particles up, but the "tendency to pair up" gets too weak to force a change in identity. The balance tips, and a new phase of matter emerges.

The Bottom Line

This paper is a roadmap. It shows us that the universe of particle physics has a hidden landscape. We used to think there was only one way for particles to get stuck together (by breaking their rules). Now we know there is a second, stranger way: they can get stuck together without breaking their rules.

This discovery doesn't just solve a math puzzle; it opens the door to understanding new types of matter that could exist in the early universe or in theories beyond our current Standard Model. It's like discovering a new color that we didn't know existed in the rainbow.

Technical Summary

1. Problem Statement

The infrared (IR) dynamics of four-dimensional chiral gauge theories remain a significant open challenge in Quantum Field Theory (QFT). Unlike vector-like theories (e.g., QCD), where left- and right-handed fermions transform identically, chiral theories feature fermions transforming under distinct representations. This distinction renders standard non-perturbative tools ineffective:

• Lattice Simulations: Currently cannot realize genuinely chiral gauge theories without introducing fermion doublers.
• Anomaly Matching: While 't Hooft anomaly matching provides constraints, it is insufficient to uniquely determine the IR phase structure (e.g., whether the theory confines with symmetry breaking or confines while preserving symmetry).
• Theoretical Gap: There is no systematic, first-principles framework to determine if chiral gauge theories undergo dynamical symmetry breaking (dSB) or realize "symmetric mass generation" (confinement without dSB).

The paper focuses on the Bars–Yankielowicz (BY) class of theories: $SU(N_c)$ gauge theories with two species of left-handed Weyl fermions:

• $\chi$: in the two-index symmetric representation.
• $\psi$: in the anti-fundamental representation (multiplicity $N_c + 4$).
  These theories are asymptotically free and possess a global flavor symmetry $SU(N_c+4) \times U(1)$. Previous conjectures suggested that for large $N_c$, these theories might confine without breaking symmetry, resulting in a spectrum of massless baryons ($\langle \chi \psi \psi \rangle$) and no Goldstone bosons.

2. Methodology

The authors employ the Functional Renormalization Group (fRG) to investigate the non-perturbative IR limit. This approach allows for a direct treatment of chiral fermions and provides well-defined criteria for both confinement and dSB.

Key Technical Components:

• Effective Action Truncation: The authors use a minimal truncation of the effective action $\Gamma_k$ that includes:
  • All classical gauge interactions (gauge-fixing and ghost terms).
  • Gauge-fermion interactions.
  • A Fierz-complete basis of four-fermion interactions (dimension-6 operators). This is crucial for capturing the onset of dSB.
• Confinement Diagnosis (Kugo-Ojima Criterion):
  • Confinement is diagnosed via the Kugo-Ojima criterion in the Landau gauge.
  • This requires specific IR scaling of the gluon ($Z_A \propto (p^2)^{-2\kappa}$) and ghost ($Z_c \propto (p^2)^\kappa$) propagators.
  • The authors utilize a bootstrap approach where a dynamically generated mass gap ($\bar{m}^2_{gap}$) is introduced in the gauge sector. The flow equations are solved imposing this scaling solution in the IR.
• Symmetry Breaking Diagnosis:
  • dSB is signaled by the divergence (singularity) of four-fermion couplings ($\bar{\lambda}_i$) as the RG flow proceeds to the IR.
  • The flow of these couplings is governed by a quadratic structure involving gauge-mediated box diagrams and pure four-fermion interactions.
  • A critical gauge coupling ($\alpha^{\text{crit}}_{SB}$) is defined as the minimum strength required to trigger a fixed-point merger in the four-fermion sector, leading to divergence.
• Regulator and Gauge: The study uses the Landau gauge ($\xi=0$) and the Litim (optimized) regulator to allow for analytical integration of loop momenta. Modified Slavnov-Taylor identities (mSTIs) are used to handle the regulator-induced breaking of gauge invariance at finite cutoff $k$.

3. Key Contributions

1. First Systematic fRG Study of BY Theories: This work provides the first quantitative, first-principles analysis of the IR dynamics of the Bars–Yankielowicz class, explicitly accounting for the interplay between color confinement and dynamical symmetry breaking.
2. Identification of a Novel Phase: The authors demonstrate the existence of a distinct phase where confinement occurs without dynamical symmetry breaking.
3. Critical Color Number ($N_c^{\text{crit}}$): The study determines a critical number of colors separating two distinct IR behaviors.
4. Mechanism of Symmetry Preservation: The paper elucidates how the interplay between the growth of gauge couplings (driving confinement) and the specific symmetry structure of the four-fermion sector (determining the critical coupling for dSB) leads to the suppression of symmetry breaking in the large-$N_c$ limit.

4. Results

The analysis of the coupled RG flow equations for the gauge couplings and four-fermion dressings reveals two distinct phases depending on the number of colors $N_c$:

• Small $N_c$ Phase ($N_c \lesssim 3.8$):

  • The gauge coupling grows strong enough to trigger the fixed-point merger in the four-fermion sector.
  • Result: The theory exhibits both confinement and dynamical symmetry breaking (dSB).
  • Condensate: A diquark-like condensate $\langle \chi \chi \rangle$ forms, breaking the global flavor symmetry.
  • Spectrum: Massive baryons and Goldstone bosons are expected.

• Large $N_c$ Phase ($N_c \gtrsim 3.8$):

  • The gauge coupling grows and induces confinement (signaled by the peak of the gauge propagator and the emergence of a mass gap).
  • However, the gauge strength at the confinement scale is insufficient to reach the critical coupling $\alpha^{\text{crit}}_{SB}$ required to trigger the four-fermion divergence.
  • Result: The theory exhibits confinement without symmetry breaking.
  • Spectrum: Fermions remain ungapped at all scales. The IR spectrum is expected to consist solely of massless baryons ($\langle \chi \psi \psi \rangle$) without accompanying Goldstone bosons. This realizes the "symmetric mass generation" scenario.

• Quantitative Transition:

  • The phase transition occurs at a critical color number:
    $$N_c^{\text{crit}} = 3.81^{+0.23}_{-0.31}$$
  • This transition is driven by the fact that while the gauge dynamics scale with $N_c$, the critical coupling $\alpha^{\text{crit}}_{SB}$ increases with $N_c$ due to the specific symmetry structure of the four-fermion sector (dominated by the $\chi$ fields).

5. Significance

• Validation of Exotic Scenarios: The results provide strong dynamical evidence for the existence of "confinement without symmetry breaking," a scenario previously only conjectured in preon models and symmetric mass generation literature.
• Beyond QCD Paradigms: It challenges the standard QCD intuition that confinement is invariably accompanied by chiral symmetry breaking. It shows that in chiral gauge theories, these two phenomena can be decoupled.
• Phenomenological Implications: These findings are relevant for Beyond the Standard Model (BSM) physics, particularly in models involving composite quarks/leptons (preons) and theories attempting to generate fermion masses without breaking global symmetries.
• Methodological Advancement: The work establishes a robust fRG framework for studying chiral gauge theories, offering a path to systematically explore other chiral models and their exotic dynamical regimes (e.g., topological phases, higher-form symmetries).

In conclusion, the paper demonstrates that for $N_c \gtrsim 4$, the Bars–Yankielowicz theories realize a novel regime of confinement without symmetry breaking, opening a new landscape for studying exotic spectra and phenomena in quantum field theory.