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Crossover or First-Order: Impacts of Regularization

Xin-Peng Li, Hao-Ran Zhang, Zhu-Fang Cui, Thomas Klähn

TL;DR

The work addresses how to determine the phase structure and equation of state of hot, dense QCD matter using a symmetry-preserving vector-vector contact interaction within a Dyson–Schwinger framework. It systematically compares two regularization schemes—noncovariant three-momentum cutoff and covariant proper-time regularization—to assess their impact on in-medium thermodynamics, order parameters ($R_B$, $R_C$) and the effective chemical potential $\mu' = \mu - R_C$. A key finding is that the order of the chiral transition is highly regulator-dependent: the three-momentum cutoff can yield a first-order transition for $\chi = \alpha_{ir} \Lambda_{3M}^2 / m_G^2 \gtrsim 1.7$, while the proper-time scheme tends to produce a crossover; the vector mean field $R_C$ acts repulsively, reducing the density-driven changes and weakening first-order behavior. The results highlight that, in nonrenormalizable models, the regulator effectively becomes part of the physics and underscore the need for symmetry-preserving truncations (e.g., Ball–Chiu vertex) to obtain robust predictions relevant to dense QCD phenomena in astrophysical and heavy-ion contexts.

Abstract

The equation of state of hot, dense nuclear matter plays a fundamental role in many areas. However, owing to the nonperturbative nature of strong interactions, a reliable treatment is still under debate. We use a symmetry-preserving treatment of a vector\,$\otimes$\,vector contact interaction to study related issues at nonzero temperature or quark chemical potential, and carefully compare a noncovariant and a covariant regularization scheme. The numerical results show that the character of the phase transition depends sensitively on the regularization scheme and parameter choices.

Crossover or First-Order: Impacts of Regularization

TL;DR

The work addresses how to determine the phase structure and equation of state of hot, dense QCD matter using a symmetry-preserving vector-vector contact interaction within a Dyson–Schwinger framework. It systematically compares two regularization schemes—noncovariant three-momentum cutoff and covariant proper-time regularization—to assess their impact on in-medium thermodynamics, order parameters (, ) and the effective chemical potential . A key finding is that the order of the chiral transition is highly regulator-dependent: the three-momentum cutoff can yield a first-order transition for , while the proper-time scheme tends to produce a crossover; the vector mean field acts repulsively, reducing the density-driven changes and weakening first-order behavior. The results highlight that, in nonrenormalizable models, the regulator effectively becomes part of the physics and underscore the need for symmetry-preserving truncations (e.g., Ball–Chiu vertex) to obtain robust predictions relevant to dense QCD phenomena in astrophysical and heavy-ion contexts.

Abstract

The equation of state of hot, dense nuclear matter plays a fundamental role in many areas. However, owing to the nonperturbative nature of strong interactions, a reliable treatment is still under debate. We use a symmetry-preserving treatment of a vector\,\,vector contact interaction to study related issues at nonzero temperature or quark chemical potential, and carefully compare a noncovariant and a covariant regularization scheme. The numerical results show that the character of the phase transition depends sensitively on the regularization scheme and parameter choices.

Paper Structure

This paper contains 12 sections, 29 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Framework
  3. Quark DSE and order parameters
  4. Regularization schemes
  5. Three-momentum cutoff regularization
  6. Proper time regularization
  7. In Medium Analyses
  8. Zero chemical potential and nonzero temperature case
  9. Zero temperature and nonzero chemical potential case
  10. Results
  11. Discussion
  12. Conclusions

Figures (5)

  • Figure 1: Solutions of the gap equation at $\mu=0$ GeV using PT (solid blue/dashed red curve for $m=0.007/0$ GeV) and 3M (dot-dashed purple/dotted black curve for $m=0.005/0$ GeV) regularizations.
  • Figure 2: Solutions of the gap equation with different regularization schemes at $T\rightarrow0$ GeV. Panel A. 3M regularization; Panel B. PT regularization. Both: the dot-dashed purple and dashed red curves show $R_B$ and $R_C$ beyond the chiral limit, respectively, while the dotted black and solid blue curves represent the corresponding results in the chiral limit $m=0$ GeV.
  • Figure 3: $R_C$ effect on the phase transition in the 3M regularization scheme. Legend: solid blue curve - ignore the effects of $R_C$ using parameters from Table \ref{['parameters']} and set $\alpha_{ir}=0.73\pi$; dotdashed purple curve - the result of $R_B$ restoring the effects of $R_C$; dashed red curve - the result of $R_C$.
  • Figure 4: Comparison of solutions for different parameter sets. Panel A. Using 6 different parameters listed in the upper panel of Table \ref{['group sets']}. Panel B. Using 4 different parameters listed in the lower panel of Table \ref{['group sets']}.
  • Figure 5: The size of the dimensionless parameter $\chi$, Eq. \ref{['Xdef']}, controls the order of the transition at $m=0.005$ GeV. Legend: blue squares – first-order for both $R_C=0$ GeV and $R_C\neq0$ GeV; black circles – first-order only at $R_C=0$ GeV and crossover at $R_C\neq0$ GeV; red triangles – crossover in both cases.