Confinement and Chiral Phase Transitions: The Role of Polyakov Loop Kinetics Terms

TL;DR

The paper addresses how a nontrivial Polyakov loop kinetic term affects gravitational waves from QCD type phase transitions by deriving a field dependent renormalization factor Z(l) from first principles in finite temperature SU(3) Yang Mills and applying it to Haar measure, polynomial and quasi particle Polyakov loop models. The confinement transition shows a strong sensitivity to the kinetic term, leading to 1–2 order changes in the GW spectrum, while the chiral transition, analyzed within the PNJL framework, is largely insensitive to the Polyakov loop kinetics and is dominated by fermion condensation. The work provides a complete framework for computing bubble nucleation rates with noncanonical kinetic terms, quantifies the impact on S_3/T and β, and demonstrates that the PL sector must be treated with its kinetic term for accurate confinement GW predictions, though quantum corrections and multi-field dynamics remain avenues for future refinement. The results have implications for the interpretation of potential stochastic GW backgrounds from early universe QCD-like transitions and guide the modeling of both confinement and chiral dynamics in beyond the standard model contexts, including dark sectors.

Abstract

We studied a crucial but often oversimplified ingredient in predicting gravitational-wave signals from QCD-type phase transitions: the kinetic term of the Polyakov loop. For the first time, we derive this term from first principles in finite-temperature pure SU(3) Yang-Mills theory, incorporating a field-dependent renormalization factor--a calculation we also extend to theories with more colors. Employing this derived kinetic term alongside three commonly-used effective potentials (the Haar-measure, polynomial, and quasi-particle models), we demonstrate that it substantially modifies the predicted GW energy spectrum from confinement transitions by 1-2 orders of magnitude. Based on this, we provide the first complete analysis of the chiral transition within the Polyakov-Nambu-Jona-Lasinio (PNJL) framework, described by the quark condensate. Our results reveal a clear dichotomy: while the Polyakov-loop kinetic term critically shapes GWs from confinement transitions, it has a negligible impact on the dynamics of the chiral transition, which is dominated by fermion condensation effects.

Figures (7)

• Figure 1: Gravitational wave spectra from different models for $N_c=3$, The left panel corresponds to $T_0 = 0.27,\mathrm{GeV}$, while the right panel corresponds to $T_0 = 10,\mathrm{GeV}$. Here, HM, PL, and QP denote the Haar measure, polynomial, and quasi-particle models, respectively. Solid lines represent the trivial kinetic term, and dotted lines the nontrivial one.
• Figure 2: The left panel shows gravitational wave spectra from polynomial potential model, and the right panel shows that from quasi-particle model, both for $N_c=3,4,6,8$. Solid lines represent the trivial kinetic term, and dotted lines the nontrivial one.
• Figure 3: Temperature dependence of $S_3/T$. The blue curve shows the case with non-trivial kinetic terms ($Z\neq1$), while the orange curve shows the case with only the trivial kinetic term ($Z=1$). The dashed line indicates $S_3/T=140$. The left panel corresponds to the Polynomial Potential, and the right panel corresponds to the Quasi-Particle model.
• Figure 4: The blue solid line is the effective potential, in which shooting without renomalization occurs. The brown dashed line is a new potential transformed by $Z_\sigma$. The two lines have the same positions of extrema.
• Figure 5: The left panel shows the path obtained by solving the equations of motion, along with the path in the gradient direction. Here, $\phi$ is defined by introducing the temperature scale as $\phi=lT$. The right panel shows the corresponding $S_3/T$, including the cases where the kinetic term of $l$ is included and where it is neglected for the solved path. This is computed under the Haar-measure model for the PL effective potential, with the parameters taken as $G_S = 2.2$ and $G_D = -282$.