[Paper Review] Traces of a fixed point: Unravelling the phase diagram at large Nf

Albert Deuzeman, Maria Paola Lombardo+1 more|arXiv (Cornell University)|Nov 11, 2009

Quantum Chromodynamics and Particle Interactions9 references4 citations

TL;DR

This study investigates the phase structure of SU(3) Yang-Mills theory with twelve fundamental fermion flavours using lattice gauge theory. By analyzing the chiral condensate, meson spectrum, and scaling of transitions, it provides evidence for a bulk transition to a chirally symmetric phase and identifies a Coulomb phase with a positive beta function, supporting the existence of a conformal window at large $N_f$. The results imply the theory exhibits an infrared fixed point, consistent with the Appelquist-Miransky-Yamawaki scenario.

ABSTRACT

With a sufficiently high number of fundamental fermionic flavours present, Yang-Mills theory develops an infrared fixed point and becomes (quasi-)conformal in nature. The range of flavour numbers for which this occurs defines the conformal window, the lower limit of which has yet to be determined. We studied the phase diagram of SU(3) Yang-Mills theory with twelve flavours of staggered fermions. Here we present evidence for the occurrence of a bulk transition towards a chirally broken phase and the existence of a Coulomb phase on the symmetric side of this transition, using results from the measurements of the chiral condensate and spectrum, leading to the determination of a positive sign of the beta function. Assuming the validity of the Appelquist-Miransky-Yamawaki scenario, this implies the existence of a conformal window that comprises the theory under investigation.

Motivation & Objective

• To determine whether SU(3) Yang-Mills theory with twelve flavours lies within the conformal window or exhibits spontaneous chiral symmetry breaking.
• To distinguish between a true thermal phase transition and a lattice-induced bulk transition in the chiral phase structure.
• To probe the nature of the symmetric phase after a bulk transition by measuring the sign of the beta function via scaling of meson masses.
• To assess the validity of the Appelquist-Miransky-Yamawaki scenario for large $N_f$ by identifying the presence of an infrared fixed point.

Proposed method

• Lattice simulations of SU(3) gauge theory with twelve staggered fermion flavours at varying coupling $\beta$ and bare quark masses.
• Measurement of the chiral condensate and its extrapolation to the chiral limit to detect spontaneous chiral symmetry breaking.
• Analysis of the ratio of pseudoscalar to vector meson masses ($m_{\mathrm{ps}}/m_{\mathrm{vt}}$) across different $\beta$ and quark masses to identify fixed-point scaling.
• Use of renormalization group invariance to test whether dimensionless ratios remain constant across scales, indicating conformal behaviour.
• Application of a non-perturbative scheme to infer the sign of the beta function from the scaling of physical observables.
• Comparison of scaling behaviour with known cases (e.g., $N_f=16$) to validate the robustness of the observed phase structure.

Experimental results

Research questions

• RQ1 Does SU(3) Yang-Mills theory with twelve flavours exhibit a bulk transition to a chirally symmetric phase, and is this transition a lattice artifact or physical?
• RQ2 Is the symmetric phase after the bulk transition consistent with a Coulomb phase, as indicated by the scaling of meson masses?
• RQ3 What is the sign of the non-perturbative beta function in the symmetric phase, and does it support the existence of an infrared fixed point?
• RQ4 Does the observed scaling of the $m_{\mathrm{ps}}/m_{\mathrm{vt}}$ ratio across different couplings and masses indicate conformal invariance and a non-zero anomalous dimension?
• RQ5 Can the chiral condensate extrapolation to the chiral limit rule out spontaneous chiral symmetry breaking, supporting a conformal phase?

Key findings

• A bulk transition was observed at $N_f = 12$, with $\lim_{N_\tau \to \infty} \beta_T = \beta_c < \infty$, indicating a lattice artifact rather than a physical thermal transition.
• The chiral condensate extrapolation to the chiral limit showed a slight upward deviation from linearity at higher masses near the transition, consistent with remnants of the transition.
• The data preferred a chirally symmetric scenario with a fitted anomalous dimension exponent of 0.965(1), supporting the absence of spontaneous chiral symmetry breaking.
• The ratio $m_{\mathrm{vt}}/m_{\mathrm{ps}}$ showed a smooth trend across couplings and masses, indicating continuity and scaling consistent with a conformal phase.
• The sign of the beta function was determined to be positive in the symmetric phase, implying the existence of an infrared fixed point and supporting the conformal window scenario.
• The phase after the bulk transition was identified as a Coulomb phase, with scaling behaviour consistent with a non-perturbative fixed point and a negative beta function in the perturbative limit.