Analysis of regulator and cutoff artifacts in the phase diagram of the quark-meson model

TL;DR

This work analyzes regulator and cutoff artifacts in the finite-$T$ and finite-$\mu$ phase structure of a two-flavor quark–meson model within the functional renormalization group using a local potential approximation. It introduces RG-consistency with a regulator- and cutoff-tuned parameter-fixing procedure and studies three-dimensional regulators (Litim, Exp2, SL) across UV cutoffs $\Lambda$ in the range $0.6$–$1.2$ GeV. The authors find weak dependence on regulator and cutoff at low $T$ and $\mu$, but substantial cutoff artifacts at high $T$ and low $\mu$, while regulator effects dominate in the low-$T$, high-$\mu$ regime, notably affecting the first-order transition and back-bending. The results highlight the importance of consistent parameter fixing for meaningful regulator comparisons and point to future work investigating larger cutoffs and higher-order truncations to assess and mitigate regulator artifacts.

Abstract

We study regulator and cutoff artifacts in the quark-meson model at finite temperature and quark chemical potential within the functional renormalization-group approach using the local potential approximation. To this end, we discuss the concept of renormalization-group consistency in effective models, which necessitates a nontrivial parameter-fixing procedure to enable a meaningful comparison of results obtained with different regulators and cutoffs. We employ a standard range of cutoff values used in phenomenological studies and regulators that differ significantly in their analytic properties as well as in their classification according to the principle of strongest singularity. We find that regulator and cutoff dependences are small at low temperatures and quark chemical potentials. At high temperatures and low quark chemical potentials, significant cutoff artifacts arise, whereas the properties of the regulator affect the dynamics in the regime governed by a chiral phase transition of first order at low temperatures and high quark chemical potentials.

Figures (8)

• Figure 1: The value of $m^2$ as a function of $\lambda$ for each regulator and cutoff combination considered in this work. All $(\lambda,m^2)$ values on these curves lead to $\bar{\sigma}_\mathrm{gs}^\mathrm{vac} = 0.093 \,\mathrm{GeV}$.
• Figure 2: Vacuum curvature mass of the sigma meson $m^{\text{vac}}_{\text{c}}=m^{\text{vac}}_{\text{c}}(\lambda,m^2)$ as a function of $m^2$ along the lines in the $(\lambda,m^2)$ plane given in \ref{['fig:lam_vs_m2']}.
• Figure 3: Left: The scale-dependent effective potential for the Litim regulator with $\Lambda=0.6 \,\mathrm{GeV}$ is displayed for various scales at $\mu=0$ and $T=0.0001\,\mathrm{GeV}$. Right: The derivative of the effective potential $\partial_{\bar{\sigma}} U_{k=k_\mathrm{IR}}$ at $\mu=0$ and $T=0.0001\,\mathrm{GeV}$, but for various regulator and -cutoff combinations. The inset in the right panel illustrates the global behavior of $\partial_{\bar{\sigma}} U_{k=k_\mathrm{IR}}$. In both panels, the vertical line indicates the position of the value of $\bar{\sigma}_\mathrm{gs}^\mathrm{vac}$, i.e., the minimum of $U=U_{k_\mathrm{IR}}$ or equivalently the zero of $\partial_{\bar{\sigma}} U$.
• Figure 4: Derivative of the effective potential at $\mu=0$ and $T=0.1\,\mathrm{GeV}$ (left panel) and $T=0.17\,\mathrm{GeV}$ (right panel) for different regulator and -cutoff combinations.
• Figure 5: The consistency measure $\alpha$ as defined in \ref{['eq:RG-cons-cutoff']} evaluated for the minimum $O=\bar{\sigma}_{\text{gs}}$ as a function of the cutoff for the three-dimensional Litim regulator at $\mu=0$ and temperatures $T=0.1\,\mathrm{GeV}$ and $T=0.17\,\mathrm{GeV}$.