Chiral Phase Transition in QCD

Updated 27 July 2025

Chiral Phase Transition in QCD is the process where chiral symmetry is restored at high temperatures/densities, marked by the reduction of the chiral condensate.
Lattice simulations and holographic models reveal that the transition order, whether crossover, second or first order, depends on quark mass and flavor content.
Quantitative analyses using scaling laws and critical temperatures offer a comprehensive mapping of the phase structure, guiding both theoretical and experimental research.

The chiral phase transition in Quantum Chromodynamics (QCD) refers to the restoration of chiral symmetry at finite temperature (and/or density), a fundamental transformation of the QCD vacuum that has profound implications for the order of the thermal transition, the universality classes it belongs to, and the overall phase structure of QCD matter. In the limit of vanishing light quark masses, this symmetry is exact and its spontaneous breaking is directly tied to the appearance of a nonzero chiral condensate serving as the order parameter. A large body of theoretical, lattice, and holographic studies, as well as scaling analyses and universality arguments, have investigated the critical properties of this transition for various numbers of quark flavors and chemical potentials. This entry synthesizes the main findings, methodologies, and implications regarding the order, critical behavior, and physical mapping of the chiral phase transition in QCD across parameter space.

1. Chiral Symmetry and Its Restoration

Chiral symmetry in QCD with $N_f$ massless quark flavors is described by the global symmetry group $SU(N_f)_L \times SU(N_f)_R \times U_A(1)$ , where the $U_A(1)$ component is broken by the axial anomaly. In the hadronic vacuum at low temperature, spontaneous breaking of the $SU(N_f)_L \times SU(N_f)_R$ symmetry leads to the formation of a chiral condensate $\langle \bar{\psi} \psi \rangle \neq 0$ and the appearance of pseudo-Goldstone bosons (pions for $N_f=2$ ). As temperature increases, chiral symmetry is eventually restored, and the condensate drops towards zero.

The nature of this restoration—whether through a first-order, second-order, or crossover transition—depends on both the quark content and their masses. In the chiral limit, the restoration occurs via a non-analytic phase transition. Depending on the number of light quark flavors and the explicit breaking by quark masses, the system falls into different universality classes. For two massless flavors, the transition is expected to be second order, typically within the $O(4)$ or $U(2) \times U(2) / U(2)$ universality class if the axial anomaly is effectively broken (Bonati et al., 2013, Ding et al., 2015). For three massless flavors, a first-order transition is expected (Li et al., 2016).

2. Order of the Transition: Continuum and Lattice Perspectives

The precise order of the chiral phase transition has been a source of significant interest and debate. Lattice QCD simulations incorporating both staggered and chiral (domain wall, Möbius) fermion formulations have shown that:

For physical quark masses, the transition is a continuous crossover, not first order (Bhattacharya et al., 2014, Ding et al., 2015, Ding et al., 2019).
As the light quark masses are decreased, the transition sharpens, but in the $(2+1)$ -flavor theory (two light and one strange) no first-order behavior is observed for pion masses down to $\sim$ 55–60 MeV (Ding et al., 2015, Ding et al., 2019, Ding et al., 2019).
In three-flavor QCD, the critical pion mass for the onset of first-order behavior is estimated to be $\lesssim50$ MeV, implying that the chiral transition above this mass remains a crossover (Ding et al., 2015, Dini et al., 2021).

The universality classes are probed using scaling analyses of the chiral condensate and its susceptibility, employing the magnetic equation of state: $M(t,h) = h^{1/\delta} f_G(z), \quad \chi_M(t,h) = h^{1/\delta - 1} f_\chi(z)$ with $z = t/h^{1/(\beta\delta)}$ , $t$ the reduced temperature, $h$ the dimensionless symmetry breaking field, and $f_G(z)$ , $f_\chi(z)$ universal scaling functions (Ding et al., 2015). Fits to $O(2)$ and $O(4)$ scaling functions consistently describe the lattice data for $(2+1)$ and three-flavor QCD.

First-order regions observed in simulations with coarse lattices (small $N_\tau$ ) shrink as the lattice spacing is reduced, and vanish in the continuum limit according to tricritical scaling (D'Ambrosio et al., 2022). Tricritical scaling relations, such as

$m_{\text{ud}}^{2/5} = C \left[(\mu/T)^2 - (\mu/T)^2_{\text{tric}}\right]$

characterize the boundary of the first-order region in the quark mass–imaginary chemical potential plane (Bonati et al., 2014).

3. Scaling, Universality, and Critical Temperatures

Scaling analyses of lattice data, including finite-size and continuum extrapolations, provide critical benchmarks for the chiral transition. Universal behavior is confirmed through the collapse of lattice data for several chiral observables—condensates, susceptibilities, and even cumulants—onto predicted scaling curves for $O(2)/O(4)$ universality (Ding et al., 2015, Ding et al., 2019, Ding et al., 2 Jan 2024). The chiral transition temperature in $(2+1)$ -flavor QCD in the chiral limit is now quantitatively extracted: $T_c^0 = 132^{+3}_{-6}\ \mathrm{MeV}$ after thermodynamic, continuum, and chiral extrapolations (Ding et al., 2019, Ding et al., 2019, Kaczmarek et al., 2020). In three-flavor QCD, $T_c$ is lower, with continuum extrapolations indicating $T_c < 90$ MeV (Dini et al., 2021).

Novel estimators, such as $T_{60}$ and $T_\delta$ , minimize sensitivity to sub-leading corrections and enable robust extraction of $T_c^0$ via the ratios: $H \chi_M(T_\delta,H) / M(T_\delta, H, L) = 1/\delta,\qquad \chi_M(T_{60},H) = 0.6\,\chi_M^{\max}$ where $H = m_l/m_s$ (Ding et al., 2019, Ding et al., 2019).

Banks–Casher-type relations link the universal scaling of chiral condensate cumulants to the microscopic infrared Dirac eigenvalue correlations, showing that the critical behavior is encoded in the deep infrared spectrum (Ding et al., 2 Jan 2024).

4. Quark Mass, Flavor Content, and Critical Lines—The Columbia Plot

Dependence on quark masses and the number of light flavors is encapsulated in the Columbia plot, which maps the order of the thermal transition as a function of two light and the strange quark masses (Ding et al., 2015, Li et al., 2016, D'Ambrosio et al., 2022). Key findings include:

For $N_f=2+1$ at the physical point, the transition is a crossover, with the tri-critical point (separating first order from crossover regions) located at a strange quark mass lower than physical ( $m_s^{\text{tri}} < m_s^{\text{phy}}$ ) (Ding et al., 2015).
For $N_f=3$ (degenerate quarks), the transition is second order at low masses and becomes a crossover as $m_q$ increases. Scaling analyses reveal the critical temperature to be as low as 98 MeV at finite lattice spacing, descending to $\lesssim90$ MeV in the continuum (Dini et al., 2021).
Extension to finite (imaginary) chemical potential increases the complexity, as the critical surface in the mass–chemical potential space exhibits tricritical scaling and recedes toward the origin as the continuum limit is approached (D'Ambrosio et al., 2022, Bonati et al., 2014).

5. Imaginary Chemical Potential, Roberge–Weiss Transition, and Scaling Behavior

Analyses at non-zero imaginary chemical potential circumvent the sign problem and enable highly controlled studies of critical behavior (Bonati et al., 2013, Bonati et al., 2014, Cuteri et al., 2022). At imaginary chemical potential fixed to a Roberge–Weiss value, center sector transitions are of first order, terminating at a second order (Z(2)) endpoint (Cuteri et al., 2022). Finite-size scaling analyses show:

The endpoint at the Roberge–Weiss value is universally described by the $3d$ Z(2) universality class for all quark masses $m_l \geq m_s/320$ .
The chiral condensate behaves as an energy-like operator for the RW transition. This implies that for any $m_l\neq0$ , its temperature derivative diverges at $T_{RW}$ , and the disconnected susceptibility diverges with exponent $\alpha$ as expected for specific heat in Z(2), in the infinite volume limit (Cuteri et al., 2022).
The critical temperatures for the chiral and RW transitions coincide, $T_{RW} = T_\chi$ , yielding $T_\chi = T_{RW} = 195(1)$ MeV for $N_\tau=4$ in the chiral limit.

6. Holographic and Effective Field Theory Approaches

Bottom-up and top-down holographic models have been used to investigate the chiral phase transition and its dynamical properties:

Modified soft-wall AdS/QCD models reproduce spontaneous chiral symmetry breaking and the Columbia plot, including the correct location and type (first order, second order, crossover) of transitions for given quark masses (Li et al., 2016, Bartz et al., 2017, Fang et al., 2018).
Inclusion of a finite chemical potential reveals, depending on model specifics, either no QCD critical point (flat critical surface) or a critical end point with a phase boundary separating crossover from first order, and a critical temperature decreasing with increasing chemical potential (Bartz et al., 2017, Fang et al., 2018, Li et al., 2020).
Analytic techniques within the Einstein-Maxwell-Dilaton holographic framework reveal a "bypass mechanism" in which black hole phase transitions govern the chiral phase transition at the boundary (Li et al., 2020).
Real-time critical dynamics are captured using the Schwinger–Keldysh formalism leading to a stochastic effective field theory for the order parameter and hydrodynamic charge densities. The resulting stochastic equations of motion near $T_c$ are of "model F" type in the Hohenberg–Halperin classification, confirmed via holographic construction (Bu et al., 12 Dec 2024).

7. Anomalies, Scale Violation, and Flavor Effects

Axial $U_A(1)$ anomaly and scale anomaly significantly influence the chiral transition:

Restoration of $SU(2)_L \times SU(2)_R$ symmetry occurs above $T_c$ , while anomalous $U(1)_A$ symmetry breaking persists to higher temperatures ( $\sim30$ MeV above $T_c$ ) (Bhattacharya et al., 2014).
Analysis of the interplay between chiral and axial/topological susceptibilities demonstrates that, at physical quark masses, the restoration of chiral symmetry is intimately connected to a profound imbalance ("QCD trilemma") between the chiral, axial, and topological sectors (Cui et al., 2021).
Electromagnetically induced scale anomalies can introduce a new critical endpoint in the chiral phase transition, observable under weak magnetic fields and varying strange quark mass (Kawaguchi et al., 2021). In this scenario, the anomaly-induced tadpole term in the Ginzburg–Landau potential facilitates a crossover–second-order–first-order sequence as one moves from two-flavor to three-flavor massless QCD.

The chiral phase transition in QCD exhibits a rich interplay between global symmetries, quark content, critical scaling, lattice artifacts, and anomaly dynamics. Advances in lattice QCD, effective field theory, and holography together provide a comprehensive and consistent picture of its critical properties, universality, and phase diagram structure. The order of the transition, its universality class, and its critical temperature are now precisely constrained for various flavor sectors and masses, informing both theoretical understanding and experimental searches for critical behavior in QCD matter.