Completely asymptotically free chiral theories with scalars

This paper establishes the specific conditions on gauge group colors and fermion family multiplicities required for generalized chiral gauge theories with fundamental or adjoint scalars to achieve complete asymptotic freedom across all gauge, Yukawa, and quartic couplings.

The Gist

Imagine the universe as a giant, complex machine built from tiny, invisible Lego bricks. Physicists call these bricks "particles," and the rules that dictate how they snap together are called "forces." For decades, our best blueprint for this machine is the Standard Model. It works incredibly well, but it has a major flaw: if you zoom in too far (to extremely high energies, like those right after the Big Bang), the blueprint starts to fall apart. Some of the rules become infinite or nonsensical, suggesting our current understanding is just a temporary patch, not the final, perfect design.

The goal of this paper is to find a "perfect" blueprint—a theory where the rules stay stable and make sense no matter how much you zoom in. The authors call this "Complete Asymptotic Freedom."

Here is a simple breakdown of what they did and what they found:

The Problem: The "Leaky Bucket"

Think of the forces in our universe as water flowing through a bucket. In our current Standard Model, if you pour water in at the top (high energy), some of it leaks out or overflows at the bottom (low energy). Specifically, the "Higgs" force (which gives particles mass) and the "Hypercharge" force (related to electricity) behave badly at high energies. They hit a "Landau pole," which is like a mathematical wall where the theory breaks down.

The authors wanted to see if they could build a new bucket where no water ever leaks, no matter how high you pour it. They focused on two specific, classic designs for these buckets (called the Georgi-Glashow and Bars-Yankielowicz models) and added some new ingredients to see if they could fix the leaks.

The Ingredients: Fermions, Scalars, and "Vector-Like" Twins

To fix the bucket, the authors played with three main ingredients:

1. Chiral Fermions: These are the "left-handed" particles (like our electrons and quarks). They are the main workers in the machine.
2. Scalars: These are like the "glue" or "scaffolding" that holds things together. The Standard Model has one famous scalar (the Higgs). The authors added either a Fundamental Scalar (like a single Lego brick) or an Adjoint Scalar (like a complex, multi-brick structure).
3. Vector-Like Families: These are "twins" of the main particles. They come in pairs (one left-handed, one right-handed) and act as stabilizers. The authors asked: How many of these twin pairs do we need to add to stop the leaks?

The Experiment: Balancing the Scales

The authors ran a massive mathematical simulation. They treated the forces like weights on a scale.

• If you add too many particles, the "gauge force" (the main glue) becomes too heavy and stops working (it loses "asymptotic freedom").
• If you add too few, the "Yukawa" and "Scalar" forces (the glue and scaffolding) become too wild and blow up (they hit a Landau pole).

They looked for the "Goldilocks Zone"—a specific number of colors (types of particles) and a specific number of twin families where all the forces balance perfectly and vanish smoothly as you zoom in to the highest energies.

The Results: Finding the Sweet Spots

The paper is essentially a map showing where these "perfect" theories exist. Here are the key takeaways:

1. The "Fundamental" Scalar (The Single Brick):

• They found that if you add a scalar like the Higgs, you can create a perfect theory, but only if you add a specific number of "twin" particle families.
• The Catch: The number of twins needed depends on how many "generations" of particles you have.
• The Big Discovery: For a model that looks like our universe (with 3 generations of particles), they found a perfect solution!
  • If the forces move in perfect sync (called "fixed flow"), you need 4 twin families.
  • If they move at different speeds ("off fixed flow"), you need 18 twin families.
• This suggests that a Grand Unified Theory (a theory combining all forces) could be mathematically perfect and stable all the way to the beginning of the universe, provided we have these extra "twin" particles.

2. The "Adjoint" Scalar (The Complex Structure):

• This is a more complex type of glue. The rules here are much stricter.
• The Result: You cannot make a perfect theory with just 3 generations of particles and this type of scalar. The math only works if you have at least 5 or 7 generations of particles and a much larger number of twin families.
• Essentially, this specific type of "perfect" machine is much harder to build and requires a much more complex universe than the one we currently observe.

The Conclusion

The authors didn't just say "it's possible." They provided a detailed recipe book. They showed exactly how many particle types and how many "twin" families are needed to build a universe where the laws of physics never break, no matter how high the energy gets.

• Good News: There are mathematically perfect versions of Grand Unified Theories.
• The Catch: To make them work, the universe might need to be populated with extra "twin" particles that we haven't discovered yet.
• The Takeaway: This paper proves that a "perfect" universe is mathematically possible, but it requires a specific, delicate balance of ingredients that is different from our current, imperfect Standard Model. It's like finding a recipe for a cake that never burns, but realizing you need a very specific, rare type of flour to make it work.

Technical Summary

Technical Summary: Completely Asymptotically Free Chiral Theories with Scalars

Problem Statement
The Standard Model (SM) is a chiral gauge-Yukawa field theory that functions effectively as an effective field theory rather than a fundamental one valid at arbitrarily high energies. This is evidenced by the Higgs quartic coupling running negative (suggesting a metastable vacuum) and the hypercharge coupling hitting a Landau pole in the ultraviolet (UV). While Grand Unified Theories (GUTs) often achieve asymptotic freedom for gauge couplings, the associated Yukawa and scalar couplings typically remain uncontrolled at high energies. The central problem addressed is identifying the conditions under which chiral gauge theories, specifically those motivated by GUTs, can be "completely asymptotically free" (CAF). A CAF model is defined as one where all couplings (gauge, Yukawa, and scalar quartic) asymptotically vanish towards the UV, ensuring the theory's validity extends to arbitrarily high energy scales.

Methodology
The authors analyze two specific classes of chiral gauge theories: the Generalised Georgi–Glashow (GG) model and the Bars–Yankielowicz (BY) model. These models are extended to include:

1. Scalar Fields: Either a single scalar in the fundamental representation or a single scalar in the adjoint representation of the $SU(N)$ gauge group.
2. Fermion Content: An arbitrary multiplicity of chiral families ($N_g$) and vector-like fermion pairs in the fundamental representation (parameterized by multiplicity $p$).

The analysis relies on one-loop renormalization group (RG) beta functions. The authors systematically derive the conditions for CAF by examining the interplay between gauge, Yukawa, and quartic scalar couplings. The methodology distinguishes between two regimes:

• Fixed-Flow Regime: All couplings vanish at the same rate as the gauge coupling (i.e., the ratio of couplings approaches a constant).
• Off-Fixed-Flow Regime: Yukawa couplings vanish faster than the gauge coupling.

For models with multiple quartic couplings (specifically the adjoint scalar case), the authors employ a fixed-flow framework that converts coupled differential equations into algebraic equations. This involves analyzing the roots of fourth-degree polynomials to determine the existence of real, positive solutions for the rescaled couplings, utilizing Descartes' rule of signs and discriminant analysis.

Key Contributions and Results

• General Framework: The paper provides a generalized derivation of CAF conditions for gauge-Yukawa theories with scalars, extending previous work on gauge-fermion theories. It explicitly addresses the sign changes in beta function coefficients that occur in the fixed-flow regime, which can render CAF conditions trivially satisfied for certain parameter ranges.

• Fundamental Scalar Models:

  • GG Model: CAF regions exist for a wide range of parameters. For a single chiral family ($N_g=1$), CAF is achievable. When extending to multiple families, the CAF region shrinks as $N_g$ increases. For the phenomenologically relevant case of $SU(5)$ with three chiral families ($N_g=3$), the model is CAF on fixed flow if at least 4 vector-like families ($p \ge 4$) are added. In the off-fixed-flow regime, this requirement increases to $p \ge 18$.
  • BY Model: Similar CAF regions are found, but they are more constrained. CAF is possible for $N_g \le 4$. For $N_g \ge 5$, the gauge coupling loses asymptotic freedom, and no CAF region exists.

• Adjoint Scalar Models:

  • The inclusion of an adjoint scalar introduces two independent quartic couplings, necessitating a more complex root analysis.
  • Fixed-Flow Regime: Viable CAF solutions are rare. For the GG model, solutions only exist for $N_g \in \{5, \dots, 12\}$ and require higher color numbers (e.g., $N \ge 13$ for $N_g=5$). The minimal $SU(5)$ GUT with three families does not admit a CAF solution in this regime.
  • Off-Fixed-Flow Regime: CAF solutions exist for lower $N_g$ values. For the GG model with $N_g=3$, a CAF region exists only for $N \ge 7$ and high vector-like multiplicity ($p \ge 24$). For the BY model, CAF regions exist for $N_g \in \{1, \dots, 4\}$ with $N \ge 7$.
  • Notably, the minimal $SU(5)$ GUT model with three chiral families and an adjoint scalar does not support CAF dynamics in either regime.

Significance and Claims
The paper claims to provide the first systematic classification of CAF parameter spaces for these specific chiral models extended with scalars. The authors state that their results offer a "first glimpse" of the short-distance behavior of these theories, identifying specific regions in the $(N, p, N_g)$ parameter space where the models are truly fundamental (valid at arbitrarily high energies).

The significance is framed within the context of Grand Unified Theories. The fundamental scalar models are noted as potentially containing the SM Higgs doublet, while adjoint scalars are commonly used to break GUT groups down to the SM gauge group. The authors modestly conclude that while their results identify CAF candidates, a realistic model would likely require both scalar types with a complex coupling structure. Thus, this work serves as a foundational step toward finding completely asymptotically free GUTs. Additionally, the adjoint scalar case is highlighted as potentially relevant for UV completions based on non-supersymmetric gauge dualities.