Mapping the critical region along the second-order chiral phase boundary

Abstract

We investigate the extent of the critical scaling region of the chiral phase transition at finite chemical potential within the quark-meson (QM) model using the functional renormalization group (fRG) approach. By analyzing the scaling behavior of the chiral order parameter and correlation length with respect to temperature and pion mass near the second-order phase transition, we extract critical exponents from the data and quantify the range over which the scaling relations remain valid. We find that both the leading order and the next-to-leading-order scaling regions systematically shrink as the chemical potential increases. This behavior is observed in both the local potential approximation (LPA) and its extension including anomalous dimensions (LPA'), with qualitatively consistent results, while the scaling region in LPA' is slightly smaller than that in LPA.

Figures (10)

• Figure 1: Dimensionless $O(4)$ invariant meson field as function of RG-scale k. The red and magenta dashed lines indicate the scale k at which the LPA and $\mathrm{LPA}^\prime$ results deviate from the plateau.
• Figure 2: The leading order (LO) fit of the order parameters within LPA and $\mathrm{LPA}^\prime$ under different values of pion mass. The results are obtained at $\mu_B=0,\,300,\,400,\,500,\,550,\,575,\,600$ MeV for LPA and $\mu_B=0,\,300,\,400,\,450,\,\,500,\,530$ MeV for $\mathrm{LPA}^\prime$. The right boundary of the red band is the $k_{\mathrm{IR}}$ we choose in the computation. The gray dashed line is the value of the value of the exponent, $\delta\simeq5.0$ for LPA and $\delta\simeq4.79$ for $\mathrm{LPA}^\prime$. The gray band highlights the parameter region where the deviation of the exponent reaches $1\%$.
• Figure 3: The leading order (LO) fit of the order parameters within LPA and $\mathrm{LPA}^\prime$ under different values of temperature. The results are obtained at $\mu_B=0,\,300,\,400,\,500,\,550,\,575,\,600$ MeV for LPA and $\mu_B=0,\,300,\,400,\,450,\,500,\,530$ MeV for $\mathrm{LPA}^\prime$. The gray dashed line is the value of the exponent, $\beta\simeq0.400$ for LPA and $\beta\simeq0.397$ for $\mathrm{LPA}^\prime$. The gray band highlights the parameter region where the deviation of the exponent reaches $1\%$.
• Figure 4: The leading order (LO) fit of the order parameters within LPA and $\mathrm{LPA}^\prime$ under different values of temperature. The results are obtained at $\mu_B=0,\,300,\,400,\,500,\,550,\,575,\,600$ MeV for LPA and $\mu_B=0,\,300,\,400,\,450,\,500,\,530$ MeV for $\mathrm{LPA}^\prime$. The gray dashed line is the value of the exponent, $\nu\simeq0.804$ for LPA and $\nu\simeq0.759$ for $\mathrm{LPA}^\prime$. The gray band highlights the parameter region where the deviation of the exponent reaches $1\%$.
• Figure 5: Contour lines of the fitted LPA data slope on the phase diagram. In the left panel, the contour lines above the phase boundary (black solid line) indicate where the slope of the correlation length as a function of reduced temperature deviates from the critical exponent $\nu$ by $1\%$, $2\%$, and $3\%$. The lines below the phase boundary corresponding to the values 0.36, 0.37 and 0.38 of the slope of the order parameter $\sigma_0$ as a function of reduced temperature. In the right panel, the lines corresponding to $1\%$, $2\%$, and $3\%$ deviations of the slope of the order parameter as a function of pion mass from the critical exponent $\delta$. The gray area denotes regions where no data are available.