[Paper Review] Conformal dynamics in gauge theories via non-perturbative renormalization group

Haruhiko Terao, Akito Tsuchiya|arXiv (Cornell University)|Apr 27, 2007

Quantum Chromodynamics and Particle Interactions1 references3 citations

TL;DR

This paper investigates conformal dynamics in non-supersymmetric SU(Nc) gauge theories with massless Dirac fermions using the non-perturbative renormalization group (NPRG). It identifies a stable infrared (IR) fixed point with non-trivial Yukawa couplings, demonstrates that large anomalous dimensions suppress scalar mass terms, and shows that hierarchical mass scales can emerge without fine-tuning due to strong-coupling effects.

ABSTRACT

The dynamics at the IR fixed point realized in the $SU(N_c)$ gauge theories with massless Dirac fermions is studied by means of the non-perturbative renormalization group. The analysis includes the IR fixed points with non-trivial Yukawa couplings. The renormalization properties of the scalar field are also discussed and it is shown that hierarchical mass scale may be allowed without intense fine-tuning due to a large anomalous dimension.

Motivation & Objective

• To study the infrared (IR) fixed point dynamics in SU(Nc) gauge theories with massless Dirac fermions beyond perturbation theory.
• To investigate the stability and structure of conformal fixed points when Yukawa interactions are included.
• To assess whether large mass hierarchies in scalar sectors can be naturally realized without fine-tuning via strong-coupling effects.
• To extend the non-perturbative renormalization group (NPRG) framework to include gauge-Yukawa models with a singlet scalar field.
• To clarify the phase diagram and fixed point structure in the presence of non-trivial Yukawa couplings and gauge interactions.

Proposed method

• Employing the non-perturbative renormalization group (NPRG) to analyze RG flows and fixed points in SU(Nc) gauge theories with fermions and a gauge-singlet scalar field.
• Deriving one-loop RG equations for four-fermi couplings via operator expansion and integration of shell modes in momentum space.
• Using the Landau gauge to ensure gauge independence of the RG flow equations, particularly for non-abelian gauge theories.
• Applying Fierz identities to map one-loop diagrams involving gauge bosons into effective four-fermi interactions.
• Solving the NPRG equations in the large Nc and Nf leading order to extract the beta functions for gauge, Yukawa, and four-fermi couplings.
• Analyzing the stability of fixed points by examining the eigenvalues of the linearized flow equations around critical points.

Experimental results

Research questions

• RQ1 Does a stable IR fixed point with non-trivial Yukawa coupling exist in SU(Nc) gauge theories with massless Dirac fermions?
• RQ2 How does the inclusion of a gauge-singlet scalar field affect the stability of the Banks-Zaks fixed point?
• RQ3 Can large anomalous dimensions at the IR fixed point suppress scalar mass terms and reduce the need for fine-tuning?
• RQ4 What is the structure of the phase diagram for SU(Nc) gauge-Yukawa models with varying numbers of flavors?
• RQ5 How does the NPRG framework improve upon the ladder approximation in Dyson-Schwinger and perturbative analyses?

Key findings

• A stable IR fixed point with non-trivial Yukawa coupling emerges when a gauge-singlet scalar field is introduced, replacing the unstable Banks-Zaks fixed point.
• The scalar field acquires a large anomalous dimension at the new IR fixed point, which suppresses the renormalization scale dependence of the mass parameter to nearly logarithmic behavior.
• The large anomalous dimension reduces the need for fine-tuning in the scalar mass term, allowing for hierarchical mass scales without strong sensitivity to UV parameters.
• The NPRG equations reproduce known results from Dyson-Schwinger and perturbative analyses in the large Nc and Nf limit, validating the framework.
• The RG flow equations for four-fermi couplings are derived explicitly using one-loop diagrams and Fierz identities, with gauge-dependent corrections addressed via Landau gauge.
• The fixed point structure is found to be stable under the inclusion of Yukawa and four-fermi interactions, indicating a viable conformal phase in the strong-coupling regime.