$CP$ phase structure of QCD from functional renormalization group

Using the functional renormalization group, this study demonstrates that in QCD-like theories, a $CP$-violating four-fermion interaction becomes relevant in the chirally broken phase while the running of the $\theta$-parameter is strongly suppressed toward the infrared in the presence of finite quark mass, thereby clarifying how strong-$CP$ effects generated at high energies are transferred to low-energy physics.

The Gist

Imagine the universe is built on a set of fundamental rules, like a giant, complex recipe for making matter. One of the most important ingredients in this recipe is a force called the "Strong Force" (Quantum Chromodynamics, or QCD), which holds the building blocks of atoms (protons and neutrons) together.

For a long time, physicists have noticed a strange mystery: The recipe could include a "twist" that breaks the symmetry between left and right (parity) and between matter and antimatter (CP violation). If this twist were large, it would make particles like the neutron act like tiny magnets in a very specific way. However, experiments show that the neutron is almost perfectly neutral in this regard. The "twist" in the recipe must be incredibly tiny—so tiny it's almost zero. This is known as the "Strong CP Problem."

This paper asks a simple but deep question: If we start with a tiny twist at the very beginning of the universe (the "UV" or high-energy scale), how does that twist behave as the universe cools down and the rules change (the "IR" or low-energy scale)?

Here is the breakdown of their findings using everyday analogies:

1. The Setup: A Recipe with a Secret Ingredient

The authors are studying a simplified version of the Strong Force recipe. They add a specific "forbidden" ingredient: a four-fermion operator.

• The Analogy: Imagine you are baking a cake. The standard recipe uses flour and sugar (normal interactions). But you also add a pinch of a mysterious spice that makes the cake taste slightly different if you flip it upside down (a CP-violating ingredient).
• The Goal: They want to see how this spice interacts with the heat of the oven (the energy scale) and the other ingredients as the cake bakes.

2. The Method: Watching the Recipe Evolve

They use a tool called the Functional Renormalization Group (fRG).

• The Analogy: Think of this as a time-lapse camera watching the cake bake. As the temperature changes (energy scale goes down), the ingredients mix differently. Some ingredients might become dominant, while others fade away. The fRG allows them to mathematically track how the "strength" of every ingredient changes as the universe cools from the hot Big Bang down to the cold world we see today.

3. The Discovery: The "Forbidden" Spice Gains Power

The most surprising finding is about how the "forbidden" spice (the CP-violating interaction) behaves when you include the "glue" of the Strong Force (gluons).

• The Old View: Previously, scientists thought that if you started with a tiny amount of this spice, it would stay tiny or become irrelevant as the universe cooled. It was like adding a drop of food coloring to a giant ocean; it would just disappear.
• The New Finding: The authors found that when you let the "glue" (the gauge coupling) change and flow along with the cooling, the "forbidden" spice does not disappear. Instead, it actually becomes more important (relevant) in the phase where the cake sets (the chirally broken phase).
• The Metaphor: It's like adding a drop of yeast to dough. At first, it seems insignificant. But as the dough rises (the system enters the broken phase), that tiny drop causes the whole structure to expand and change shape. The "forbidden" interaction becomes a driving force in the low-energy world, not just a leftover trace.

4. The Twist Parameter (θ): A Quiet Observer

The paper also looked at the "θ-parameter," which is the mathematical number representing the size of the initial twist.

• The Finding: As the universe cools, the value of this θ-parameter itself doesn't change much. It stays relatively stable.
• The Catch: Even though the number itself doesn't change much, its influence does. The authors found that this stable number acts like a "director" that decides which way the "forbidden" spice will push the dough. It determines whether the final cake leans more toward a "scalar" shape or a "pseudoscalar" shape.
• The Metaphor: Imagine a compass needle (θ) that doesn't move much, but it points the way a strong wind (the four-fermion interaction) blows. The wind does the heavy lifting, but the compass decides the direction.

5. The Conclusion: A New Path for Understanding

The paper concludes that we cannot ignore these CP-violating interactions when trying to understand the low-energy physics of the Strong Force.

• The Takeaway: If you start with a CP-violating effect at the high-energy level, the laws of physics (specifically the running of the gauge coupling) naturally amplify this effect as you go down to lower energies. It doesn't just fade away; it gets woven into the fabric of how matter behaves.

In summary: The authors used a mathematical "time-lapse" to show that a tiny, symmetry-breaking ingredient in the universe's recipe doesn't vanish as the universe cools. Instead, thanks to the dynamic nature of the Strong Force, this ingredient grows in importance and actively shapes the behavior of matter at low energies, acting as a crucial link between the high-energy origins of the universe and the physics we observe today.

Technical Summary

Technical Summary: CP Phase Structure of QCD from Functional Renormalization Group

Problem Statement
The paper addresses the "strong CP problem" in Quantum Chromodynamics (QCD), specifically investigating how CP-violating effects generated at the ultraviolet (UV) scale are transferred to the infrared (IR) physics of QCD-like theories. While the physical $\bar{\theta}$-parameter is constrained experimentally to be extremely small ($\bar{\theta} < 10^{-11}$), its theoretical origin involves the interplay between the topological $\theta$-term and the phase of the quark mass matrix. Previous studies within the Standard Model effective field theory suggested that perturbative RG running induces only small shifts in $\bar{\theta}$ down to the QCD scale. However, this perturbative expectation may fail in the non-perturbative regime below 1 GeV, where higher-dimensional operators, such as four-fermion interactions, become relevant and drive dynamical chiral symmetry breaking. The authors aim to clarify the non-trivial transfer of strong-CP effects to the IR by studying a low-energy effective theory containing a P-odd, U(1) axial-breaking four-fermion operator, $(\bar{\psi}\psi)(\bar{\psi}i\gamma_5\psi)$, coupled to non-Abelian gauge fields.

Methodology
The authors employ the Functional Renormalization Group (fRG) method to analyze the low-energy effective theory.

• Model Construction: They construct a minimal low-energy truncation involving a single quark flavor ($N_f=1$) coupled to an $SU(N_c)$ gauge field. The effective action includes the gluon field, quark fields with a complex mass term ($me^{i\gamma_5\theta/2}$), and four-fermion interactions: scalar ($G_S$), pseudoscalar ($G_P$), and the CP-violating mixed operator ($C_4$).
• Framework: The study utilizes the Wetterich equation for the scale-dependent effective average action $\Gamma_k$. The analysis is performed within the Local Potential Approximation (LPA), setting wavefunction renormalization constants to unity ($Z_A = Z_\psi = 1$).
• Truncation and Approximation: The effective action is truncated to include operators up to four-fermion couplings. The authors adopt the large-$N$ leading approximation for four-fermion induced self-energy terms while explicitly including "nonladder" contributions from gauge-field exchanges. They use Litim-type regulator functions to suppress low-momentum modes.
• Flow Equations: A coupled system of ordinary differential equations (beta functions) is derived for the dimensionless couplings: $\tilde{m}$, $\theta$, $\tilde{G}_S$, $\tilde{G}_P$, $\tilde{C}_4$, and the gauge coupling $g_s$. The gauge coupling evolution is treated in two scenarios: a fixed coupling case and a running coupling case governed by the one-loop QCD beta function (asymptotic freedom).

Key Contributions and Results

1. Relevance of the P-Odd Coupling ($C_4$):

   • In the fixed gauge coupling scenario, the phase structure in the $(\tilde{G}_S, \tilde{C}_4)$ plane exhibits a sequence of qualitative changes as $g_s$ increases. Critical values of the gauge coupling ($g_{c,(1)}$ and $g_{c,(2)}$) are identified where fixed points merge or split, altering the existence of chiral symmetric phases.
   • Crucially, in the running gauge coupling (QCD) scenario, the chiral symmetric phase disappears entirely. The RG flow drives the system into a chirally broken regime regardless of initial conditions.
   • The authors find that the P-odd four-fermion coupling $\tilde{C}_4$ becomes relevant in the chirally broken phase once the gauge coupling exceeds a critical value ($g^*_s = \sqrt{2}\pi$). This implies that even an arbitrarily small initial $\tilde{C}_4$ is amplified along the RG flow toward the IR.

2. RG Evolution of the $\theta$-Parameter:

   • The flow equation for the $\theta$-parameter itself is found to be numerically weak; $\theta$ hardly runs along the RG trajectory.
   • However, the value of $\theta$ acts as a selector for the dominant interaction channel. It controls the orientation of the vacuum alignment (scalar vs. pseudoscalar dominance).
   • For generic initial values of $\theta$ (e.g., $\theta \neq 0, \pi$), the RG flow generates a non-zero $\tilde{C}_4$ dynamically through operator mixing, even if $\tilde{C}_4$ is zero at the UV scale. This indicates that the P-even subspace is generically unstable in QCD-type flows when $\theta$ is non-trivial.

3. Operator Mixing and Symmetry:

   • The study reveals a symmetry under the reflection $\tilde{G}_S \leftrightarrow \tilde{G}_P$ and $\theta \leftrightarrow \pi - \theta$.
   • At the special point $\theta = \pi/2$, the system exhibits a self-dual property where the scalar and pseudoscalar diagonal couplings coincide ($G_S = G_P$), and the interaction becomes diagonal in a rotated basis ($S_\pm$).

4. Bosonization:

   • The paper briefly discusses the bosonization of the model, introducing auxiliary fields $\sigma$ and $\eta$. It notes that Yukawa interactions in the bosonized description can regenerate four-fermion terms, complicating the IR analysis and suggesting the need for dynamical bosonization techniques for a complete treatment.

Significance and Claims
The paper claims to provide a further step in clarifying how strong-CP effects generated at the UV are transferred to the IR properties of QCD-like theories. Its primary significance lies in demonstrating that:

• The inclusion of running gauge interactions fundamentally alters the phase structure compared to fixed-coupling NJL models, eliminating the chiral symmetric phase.
• The P-violating four-fermion coupling $C_4$ is not merely a perturbation but a dynamically relevant operator in the IR, generated radiatively and enhanced by the gauge coupling.
• While the $\theta$-parameter itself does not run significantly, its presence ensures that CP-violating four-fermion interactions are unavoidable in the low-energy effective description of massive QCD.

The authors conclude that these CP-violating four-fermion interactions play a non-trivial role in the IR dynamics and provide a concrete framework for systematically incorporating strong-CP effects into low-energy effective models. They explicitly contrast their findings with approaches focusing on the topological charge in pure Yang-Mills theory, noting that their work focuses on the RG flow projection onto the phase of the fermionic mass. The paper acknowledges that a more complete treatment matching topological and fermionic descriptions in the deep IR is required for future work.