Universal critical dynamics near the chiral phase transition and the QCD critical point

TL;DR

The paper develops a real-time FRG formulation for dynamic universality classes Model G and Model H to capture the critical dynamics near the chiral transition and the QCD critical point in $2<d<4$. Using a $\,\phi^4$ truncation of the LGW free energy and a regulator that preserves the extended temporal symmetry of the reversible mode couplings, it computes dynamic exponents and scaling behavior, revealing a strong-scaling fixed point in Model G and a weak-scaling fixed point in Model H. In $d=3$, it finds $x_{sigma}\approx0.949$ and $x_{eta}\approx0.051$, yielding $z_{\phi}=d+x_{eta}\approx3.051$, while in $d=2$ the shear viscosity exponent vanishes. For Model G, momentum-dependent mobility leads to a diffusion scaling form $D_n(\vec{p},\tau)=D_n^+\tau^{- u x_{gamma}}\mathcal{L}(p\xi(\tau))$, with $x_{gamma}=2-z_n=2-d/2$ at the strong-scaling fixed point and a universal scaling function $\mathcal{L}(x)$, indicating non-perturbative transport behavior near criticality.

Abstract

We use a novel real-time formulation of the functional renormalization group (FRG) for dynamical systems with reversible mode couplings to study Model G and H, which are the conjectured dynamic universality classes of the two-flavor chiral phase transition and the QCD critical point, respectively. We compute the dynamic critical exponent in both models in spatial dimensions $2<d<4$. We discuss qualitative commonalities and differences in the non-perturbative RG flow, such as weak scaling relations which hold in either case versus the characteristic strong scaling of Model G which is absent in Model H. For Model G, we also extract a novel dynamic scaling function that describes the universal momentum and temperature dependence of the diffusion coefficient of iso-(axial-)vector charges in the symmetric phase.

Figures (2)

• Figure 1: FRG flow diagrams of $3d$ Model G and H. Model G admits one attractive strong-scaling fixed point (blue) at $(w^*,f^*) \approx (4.998,0.527)$, and two unstable weak-scaling fixed points (red and green) at $w^*=0,\infty$. Model H admits one attractive weak-scaling fixed point (blue) at $w^* = 0$. Critical exponents of the kinetic coefficients are plotted as a function of spatial dimension $2<d<4$ in matching colors. Here, solid lines indicate our FRG results with $\phi^4$-truncation, and dashed lines results from the perturbative $\epsilon$-expansion. Gray bands show FRG results with the extended local-potential approximation (LPA') of the free energy, with the upper and lower bounds corresponding to the field expansion points $\phi=0$ and the scale-dependent minimum of the effective potential. Figure taken from Ref. Roth:2024hcu.
• Figure 2: By tuning the temperature close to the critical temperature from above, and rescaling the resulting diffusion coefficient $D_{n}(\vec{p})$ at $k=0$ according to \ref{['eq:Dn']}, we find a scaling function $\mathcal{L}(x)$ which only depends on the scaling variable $x\equiv p\xi(\tau)$. Figure taken from Ref. Roth:2024rbi.