SU(2) gauge theory with one and two adjoint fermions towards the continuum limit

This paper presents an extended lattice study of SU(2) gauge theories with one and two adjoint fermion flavors, utilizing multiple methods to demonstrate that both theories reside in the conformal window with chiral symmetry unbroken, yielding continuum-limit anomalous dimensions of approximately 0.170 and 0.291, respectively.

The Gist

Imagine the universe is built from a giant, invisible LEGO set. The "bricks" are fundamental particles, and the "glue" holding them together is a force called the Strong Force. Physicists have a standard rulebook (the Standard Model) for how these bricks work, but they suspect there might be a hidden, more complex layer of LEGO bricks and glue that explains things like why the Higgs boson has the mass it does, or why the universe has the matter it does.

This paper is a massive, high-tech investigation into two specific, hypothetical versions of this LEGO set. The researchers are trying to figure out if these specific setups behave like a rigid, solid structure (confinement) or like a fluid that never quite settles (conformal).

Here is the breakdown of their adventure, translated into everyday language:

The Two Hypothetical Worlds

The scientists are studying two specific "universes" based on a mathematical group called SU(2). Think of SU(2) as the shape of the LEGO bricks.

1. World A (Nf = 1): A universe with one type of special "glue-fermion" particle.
2. World B (Nf = 2): A universe with two types of these particles.

In these worlds, the particles interact in a way that is supposed to be "scale-invariant." Imagine a fractal pattern (like a fern leaf) where if you zoom in or out, it looks exactly the same. The researchers want to know: Do these worlds actually act like fractals, or do they eventually break down into a solid, rigid structure?

The Problem: The "Zoom" Effect

To study these worlds, the scientists use supercomputers to simulate them on a grid (a lattice). This is like looking at a painting through a magnifying glass.

• The Issue: When you look at the painting through a coarse magnifying glass (a "coarse" lattice), the image looks fuzzy and distorted. You might think the painting is a solid block of color.
• The Goal: They need to keep zooming in (making the lattice finer and finer) until they reach the "continuum limit"—the point where the grid disappears, and they see the true, smooth picture of the universe.

The Key Mystery: The "Anomalous Dimension"

The most important thing they are measuring is something called the anomalous dimension (let's call it the "Stretch Factor").

• What is it? Imagine you have a rubber band. If you pull it, how much does it stretch? In these quantum worlds, the "Stretch Factor" tells us how much the particles change their behavior as you zoom in.
• Why does it matter? If the Stretch Factor is huge, it could explain why the Higgs boson is heavy (a theory called "Walking Technicolor"). If it's small or zero, the theory behaves differently.

The Journey of Discovery

1. The "One Particle" World (Nf = 1)

In this world, previous studies suggested the Stretch Factor was huge (around 1.0), which would be a huge win for new physics.

• What they found: As the researchers zoomed in closer and closer (approaching the continuum limit), the Stretch Factor shrank dramatically.
• The Result: It dropped from a huge number down to about 0.17.
• The Analogy: Imagine you thought a rubber band was super stretchy. But as you looked at it under a microscope, you realized it was actually quite stiff. The "magic" of the huge stretch factor seems to vanish when you look at the true, smooth reality.
• Conclusion: This world likely doesn't have the "super-stretchy" properties needed for the specific new physics theories they were hoping for. It might actually be breaking symmetry (getting rigid) rather than staying fluid.

2. The "Two Particle" World (Nf = 2)

This world has been studied for over a decade. The consensus was that it is a "conformal" (fractal-like) world, but the exact Stretch Factor was hard to pin down.

• What they found: As they zoomed in, the Stretch Factor settled down quickly and consistently to a value of about 0.29.
• The Result: This world does seem to be a "fractal" universe that stays fluid and scale-invariant.
• Conclusion: This confirms that the "Two Particle" world is likely inside the "Conformal Window" (the zone where the universe acts like a fractal), but the Stretch Factor is smaller than some hoped.

The Tools They Used

To get these results, they didn't just guess; they used three different "rulers" to measure the universe:

1. The Mass Spectrum: They weighed the "particles" in the simulation (like weighing different LEGO structures) to see how they scale.
2. The Dirac Mode Number: They counted the "vibrations" of the particles (like counting the notes on a guitar string) to see how the frequency changes with the grid size.
3. The R-Ratio: They compared the weight of a heavy "spin-2" particle to a light "spin-0" particle. This ratio acts like a universal fingerprint that tells you if the theory is conformal or not.

All three rulers agreed on the results, giving them high confidence.

The Big Picture: Why Should We Care?

For years, physicists have been hunting for a "New Physics" mechanism to explain the universe without needing the Standard Model's Higgs boson to do all the heavy lifting. They hoped these "Adjoint Fermion" worlds would be the key.

• The Bad News: The "One Particle" world (Nf = 1) seems to have lost its "super-stretchy" magic as they got closer to the truth. It's likely not the solution they were looking for.
• The Good News: The "Two Particle" world (Nf = 2) is confirmed to be a stable, conformal universe, but with a modest Stretch Factor.

The Final Takeaway:
The researchers pushed their supercomputers to the absolute limit (using millions of hours of GPU time) to get the clearest picture possible. They found that lattice artifacts (the "fuzziness" of the simulation) were tricking previous studies. Once they cleaned up the picture, the "One Particle" world looked very different than expected.

It's a reminder that in physics, what you see depends on how close you look. The universe is subtle, and sometimes the "magic" you see from a distance disappears when you get right up close. The hunt for the perfect "Walking Technicolor" theory continues, but this paper has cleared away some false leads.

Technical Summary

1. Problem Statement

The paper investigates the infrared (IR) dynamics of $SU(2)$ gauge theories coupled to fermions in the adjoint representation. Specifically, it focuses on two cases:

• $N_f = 1$: One Dirac fermion flavor.
• $N_f = 2$: Two Dirac fermion flavors.

These theories are of significant interest in Beyond the Standard Model (BSM) physics, particularly for Walking Technicolor and Composite Higgs models. The central question is whether these theories lie within the conformal window (exhibiting IR conformality with a non-zero anomalous dimension $\gamma_*$) or if they exhibit chiral symmetry breaking (confinement).

Previous lattice studies yielded conflicting results:

• For $N_f=2$, the consensus suggested IR conformality with a small anomalous dimension, but precise determination was lacking due to tensions in data.
• For $N_f=1$, earlier studies suggested a large anomalous dimension ($\gamma_* \sim 1$), but recent overlap fermion studies suggested chiral symmetry breaking. A major issue in previous work was the strong dependence of results on the lattice spacing ($a$), preventing a reliable continuum limit extrapolation.

The goal of this work is to perform an extended lattice study with larger volumes, smaller fermion masses, and finer lattice spacings (higher $\beta$) to resolve these discrepancies and determine the continuum limit value of the mass anomalous dimension $\gamma_*$.

2. Methodology

The authors employed Lattice Quantum Chromodynamics (LQCD) simulations using the Wilson gauge action and Wilson fermions.

• Ensembles: They generated dynamical gauge configurations using the Rational Hybrid Monte Carlo (RHMC) algorithm.
  • $N_f=1$: Simulations were performed at $\beta$ values ranging from 2.05 to 2.4, covering a wide range of lattice spacings and fermion masses.
  • $N_f=2$: Simulations were performed at $\beta = 2.25, 2.3, 2.35$.
• Software: Used HiRep (for heavier ensembles) and Grid (for lighter ensembles, leveraging GPU acceleration) to handle the computational load.
• Key Observables:
  1. Mass Spectrum: Extraction of masses for glueballs, mesons, baryons, and hybrid fermions (gluino-glue).
  2. Finite-Size Hyperscaling (FSHS): Analyzing the scaling of particle masses $M$ with the lattice volume $L$ and fermion mass $m$ according to $L M = f(L m^{1/(1+\gamma_*)})$.
  3. Dirac Mode Number: Analyzing the spectral density of the Dirac operator. The mode number $\nu(\Omega)$ scales as $\nu(\Omega) \sim \Omega^{2/(1+\gamma_*)}$. The authors improved upon previous methods by using Bayesian model averaging with Akaike Information Criteria (AIC) to select fitting windows and constrain parameters, reducing systematic errors.
  4. $R$ Ratio: The ratio of the lightest spin-2 mass to the lightest spin-0 mass ($R = M_{2^{++}}/M_{0^{++}}$), which is a universal observable in IR conformal theories.
  5. Chiral Perturbation Theory (ChPT): Testing if the data fits a ChPT ansatz (indicating chiral symmetry breaking) versus a conformal scaling ansatz.
  6. Topological Susceptibility: Checked for topological freezing and compared against pure Yang-Mills expectations.

3. Key Contributions

• Continuum Limit Extrapolation: This is the first study to push the $N_f=1$ and $N_f=2$ $SU(2)$ adjoint theories significantly closer to the continuum limit ($\beta \to \infty$) with sufficient statistics to perform a robust extrapolation.
• Improved Statistical Framework: The authors introduced a rigorous Bayesian approach for extracting $\gamma_*$ from the Dirac mode number. By using AIC weights to average over different fitting windows, they mitigated the arbitrariness of window selection that plagued previous studies.
• Resolution of $N_f=1$ Discrepancy: They demonstrated that the previously observed large anomalous dimension was an artifact of coarse lattice spacings. As the continuum limit is approached, $\gamma_*$ decreases significantly.
• Universal $R$ Ratio Analysis: They utilized the $R$ ratio to cross-check the conformal nature of the theories, showing how the ratio evolves from the pure gauge limit towards conformal predictions as the mass decreases.

4. Key Results

For $N_f = 1$ (One Adjoint Flavor)

• Anomalous Dimension ($\gamma_*$): The value of $\gamma_*$ is strongly dependent on the lattice spacing. As $\beta$ increases (approaching the continuum), $\gamma_*$ decreases significantly.
  • At coarse lattices ($\beta \approx 2.05$), $\gamma_* \approx 0.9$.
  • In the continuum limit, the extrapolated value is $\gamma_* = 0.170(6)$.
• Conformal vs. Chiral Breaking:
  • The hyperscaling analysis and mode number analysis are consistent with a small anomalous dimension.
  • However, the value is very small ($\sim 0.17$), and the data shows deviations from pure hyperscaling at the finest lattices.
  • Chiral Perturbation Theory analysis shows the data is incompatible with a standard chiral symmetry breaking scenario (where $\gamma_*=0$ and masses scale linearly with mass).
  • Conclusion: The theory is likely in a phase very close to the conformal window (near-conformal) or exhibits a very small breaking of chiral symmetry, but it does not show the large anomalous dimension ($\gamma_* \sim 1$) required for phenomenological Walking Technicolor.

For $N_f = 2$ (Two Adjoint Flavors)

• Anomalous Dimension ($\gamma_*$): The dependence on lattice spacing is much milder compared to $N_f=1$.
  • The results converge rapidly to a stable value.
  • The continuum limit estimate is $\gamma_* = 0.291(9)$ (or $0.308(5)$ depending on the specific extrapolation method used, with the paper favoring the plateau value around 0.29).
• Conformal Nature:
  • The data is consistent with the theory being inside the conformal window.
  • The $R$ ratio and mass spectrum support the IR conformal scenario.
  • ChPT analysis shows deviations at light masses, consistent with conformal scaling rather than chiral symmetry breaking.
• Conclusion: The $N_f=2$ theory is confirmed to be in the conformal window with a small, but non-zero, anomalous dimension.

Other Observables

• Topological Freezing: No significant topological freezing was observed in the $N_f=1$ simulations, ensuring ergodicity. For $N_f=2$ at $\beta=2.35$, some slowing was noted but did not invalidate the results.
• Hybrid Fermions: The spin-1/2 hybrid state (gluino-glue) was found to be lighter than the scalar baryon in both theories, a feature consistent with near-conformal dynamics.
• String Tension: The string tension remains non-zero, but the spectrum scaling suggests the theory is deformed away from a pure conformal fixed point by the fermion mass.

5. Significance

• Phenomenological Impact: The results challenge the viability of the $N_f=1$ $SU(2)$ adjoint theory as a direct candidate for Walking Technicolor, as the required large anomalous dimension ($\gamma_* \sim 1$) is not found in the continuum limit. The $N_f=2$ case remains a candidate but with a small anomalous dimension, which may limit its ability to generate large fermion masses for BSM models.
• Methodological Advancement: The paper sets a new standard for extracting anomalous dimensions in lattice gauge theories by combining multiple observables (hyperscaling, mode number, $R$ ratio) and employing advanced Bayesian statistical techniques to handle systematic errors.
• Future Directions: The authors note that the current Wilson action formulation faces limitations in reaching the chiral limit at very fine lattice spacings. They propose future work using Möbius domain wall fermions or Pauli-Villars bosons to better control lattice artifacts and push closer to the true chiral limit.

In summary, the paper provides a definitive lattice study showing that while $N_f=2$ $SU(2)$ adjoint QCD is likely conformal with a small $\gamma_*$, the $N_f=1$ case does not exhibit the large anomalous dimension previously suggested at coarse lattice spacings, instead converging to a small value consistent with near-conformal dynamics or weak chiral symmetry breaking.