Lattice Study of the Conformal Window in QCD-like Theories (0712.0609v2)

Abstract: Using lattice simulations, we study the extent of the conformal window for an $\text{SU}(3)$ gauge theory with $N_f$ Dirac fermions in the fundamental representation. We present evidence that the infrared behavior is conformal for $12 \leq N_f \leq 16$, governed by an infrared fixed point, while confinement and chiral symmetry breaking are present for $N_f \leq 8$.

• The paper demonstrates lattice evidence that Nf=12 exhibits an IR fixed point, defining the conformal window in QCD-like theories.
• It employs a gauge-invariant running coupling via the Schrödinger functional method to extract continuum behavior across scales.
• The study clearly differentiates confinement observed at Nf=8 from conformal behavior at Nf=12, refining the lower boundary of the conformal window.

Insights into the Conformal Window in QCD-like Theories: A Lattice Approach

The paper "Lattice Study of the Conformal Window in QCD-like Theories" by Appelquist, Fleming, and Neil explores the dynamics of an $\text{SU}(3)$ gauge theory with $N_f$ Dirac fermions in the fundamental representation. This investigation focuses on mapping the conformal window, where the infrared (IR) behavior is governed by a fixed point, distinguishing regions of confinement and chiral symmetry breaking from regions of conformal behavior.

Key Findings

The authors present lattice evidence indicating that the conformal window for this theory lies within $12 \leq N_f \leq 16$. Specifically, they provide evidence that for $N_f = 8$, the theory exhibits confinement and chiral symmetry breaking, while for $N_f = 12$, the infrared dynamics are consistent with conformal behavior, governed by an infrared fixed point (IRFP).

Methodology

The researchers employed a gauge-invariant definition of the running coupling derived from the Schrödinger functional (SF) method. This approach is valid at any coupling strength and aligns with the perturbative running coupling at short distances. The computations involved staggered fermions and employed a systematic step-scaling method to extract the continuum limits of the running coupling, $\overline{g}^{2}(L)$, from lattice simulations. By interpolating lattice data using a Laurent series and performing continuum extrapolations, they assessed the presence of an IRFP.

Results and Implications

For $N_f = 12$, an IRFP was observed, consistent with three-loop perturbation theory results, suggesting that this value of $N_f$ lies within the conformal window. In contrast, for $N_f = 8$, the running coupling exhibited no signs of an IRFP, instead showing behavior indicative of confinement and chiral symmetry breaking. These findings suggest that $N_f^\text{c}$, the lower end of the conformal window, lies between 9 and 12. Furthermore, these results challenge earlier studies suggesting lower bounds for $N_f^\text{c}$.

The implications of understanding the conformal window are significant, particularly regarding theories beyond the Standard Model, where non-perturbative behaviors like dynamical symmetry breaking could play pivotal roles in phenomenological models. Understanding the properties of theories near the conformal window could provide insights into such frameworks.

Future Directions

The paper opens several avenues for further research:

1. Increased Lattice Precision: Continued simulations with greater precision for $N_f = 8$ and $12$ could provide more definitive insights into the running coupling and IR behavior at these fermion counts.
2. Investigation of Other Physical Quantities: Additional calculations focusing on physical observables, such as the static quark-antiquark potential and the chiral condensate at zero temperature, could confirm the conclusions regarding confinement and chiral symmetry breaking for $N_f = 8$.
3. Extension to Other Gauge Groups: Generalizing the paper to other gauge groups and fermion representations could further illuminate the dynamics of gauge theories and refine the boundaries of the conformal window.

This paper provides a crucial lattice-based understanding of the conformal window in QCD-like theories, enhancing our comprehension of non-Abelian gauge theories and their applications in particle physics.