Dynamical Quasiparticle Model (DQPM)

• DQPM is an effective description of deconfined QCD matter that represents the quark-gluon plasma as massive, off-shell quarks, antiquarks, and gluons with broad spectral functions.
• It matches quasiparticle properties and effective couplings to lattice-QCD thermodynamics, enabling real-time transport and dynamical hadronization within the PHSD framework.
• The model provides practical insights into transport coefficients and collective flow by linking microscopic quasiparticle behavior to macroscopic observables like viscosity and electromagnetic emissivity.

The Dynamical Quasiparticle Model (DQPM) is an effective description of deconfined QCD matter in which the strongly interacting quark–gluon plasma is represented by massive, off-shell quarks, antiquarks, and gluons with complex self-energies and broad spectral functions, rather than by a weakly interacting gas of massless on-shell partons. Its central construction is to match quasiparticle properties and effective couplings to lattice-QCD thermodynamics in equilibrium, while retaining a propagator-based formulation suitable for real-time transport, electromagnetic emissivities, and hadronization dynamics in the Parton-Hadron-String Dynamics (PHSD) framework [(Linnyk et al., 2010); (Moreau et al., 2021)].

1. Conceptual basis and scope

The DQPM describes QCD matter in terms of single-particle Green’s functions “in the sense of a two-particle irreducible approach,” and interprets the deconfined phase as a medium of effective strongly interacting partonic quasiparticles with finite masses and widths. In this formulation, the real parts of the self-energies generate quasiparticle masses, whereas the imaginary parts encode damping, interaction rates, and finite lifetimes; the corresponding spectral functions are therefore broad rather than delta-like [(Linnyk, 2010); (Soloveva et al., 2019)].

This construction differs structurally from purely on-shell quasiparticle equations of state and from weak-coupling kinetic descriptions. In the DQPM, only the time-like sector is associated with propagating quasiparticles, while the space-like sector is interpreted as interaction or potential energy. That separation enables a dynamical picture in which quasiparticle propagation, mean fields, and effective interactions are treated within a common framework and can be exported to off-shell transport theory (0704.1410).

The model was developed as an equilibrium description of the strongly interacting QGP near and above the deconfinement temperature, but it has also been used as the microscopic partonic input of PHSD, where it governs off-shell transport, partonic reaction rates, hadronization, and partonic electromagnetic radiation [(0808.0022); (Cassing et al., 2010)].

2. Microscopic quasiparticle structure

A standard DQPM propagator is written in retarded form as

$$G^{R}_j(\omega,\mathbf p)=\frac{1}{\omega^2-\mathbf p^2-m_j^2+2i\gamma_j\omega},$$

with quasiparticle self-energy

$\Pi_j=m_j^2-2i\gamma_j\omega,$

for $j=q,\bar q,g$. The associated spectral function is Lorentzian,

$$\rho_j(\omega,\mathbf p)=\frac{4\omega\gamma_j}{\left(\omega^2-\mathbf p^2-m_j^2\right)^2+4\gamma_j^2\omega^2},$$

or equivalently

$$\rho_j(\omega,\mathbf p)=\frac{\gamma_j}{\tilde E_j} \left[ \frac{1}{(\omega-\tilde E_j)^2+\gamma_j^2} - \frac{1}{(\omega+\tilde E_j)^2+\gamma_j^2} \right], \qquad \tilde E_j^2=\mathbf p^2+m_j^2-\gamma_j^2.$$

The normalization condition is

$$\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\,\omega\,\rho_j(\omega,\mathbf p)=1$$

(Moreau et al., 2021, Soloveva et al., 2023).

The physical interpretation is fixed by the self-energy decomposition. The real part shifts the pole and defines an effective thermal mass; the imaginary part sets the width and thus the damping time. This is the defining “dynamical” ingredient of the DQPM: the QGP constituents are not narrow quasiparticles in a weakly perturbed vacuum, but broadened excitations whose widths remain relevant for thermodynamics, transport, and reaction kinematics (Soloveva et al., 2019, Berrehrah et al., 2016).

In the original DQPM and its PHSD applications, the time-like and space-like sectors are separated explicitly. The time-like density $N^+$ or scalar density $N_s$ characterizes propagating quasiparticles, while the space-like contribution to the energy density is interpreted as potential energy $V$. Mean fields and effective interactions are then obtained from density derivatives,

$$U_s(\rho_s)=\frac{dV}{d\rho_s},\qquad v_{gg}(\rho_g)=\frac{d^2V}{d\rho_g^2},$$

which provides a link between equilibrium quasiparticle physics and dynamical transport forces (0704.1410).

3. Thermodynamic construction and lattice-QCD matching

The DQPM is constrained by lattice-QCD thermodynamics above $\Pi_j=m_j^2-2i\gamma_j\omega,$0. In its standard formulation, the effective coupling is fixed from the entropy density, and the quasiparticle masses and widths are then constructed in HTL-inspired form. A representative $\Pi_j=m_j^2-2i\gamma_j\omega,$1 coupling parameterization is

$\Pi_j=m_j^2-2i\gamma_j\omega,$2

with

$\Pi_j=m_j^2-2i\gamma_j\omega,$3

while the corresponding gluon and light-quark pole masses are written as

$\Pi_j=m_j^2-2i\gamma_j\omega,$4

and the widths as

$\Pi_j=m_j^2-2i\gamma_j\omega,$5

(Soloveva et al., 2023).

Thermodynamics is evaluated in a $\Pi_j=m_j^2-2i\gamma_j\omega,$6-derivable or 2PI quasiparticle framework. The entropy density is expressed in terms of dressed propagators and self-energies, and pressure and energy density are then obtained thermodynamically. In PHSD-oriented DQPM formulations, the potential energy density is identified with the space-like pieces of the energy-momentum tensor,

$\Pi_j=m_j^2-2i\gamma_j\omega,$7

and the scalar mean field is

$\Pi_j=m_j^2-2i\gamma_j\omega,$8

The corresponding force on a partonic quasiparticle is described as

$\Pi_j=m_j^2-2i\gamma_j\omega,$9

(Moreau et al., 2021).

At finite baryon density, later DQPM implementations introduce explicit $j=q,\bar q,g$0 dependence in the coupling, masses, and widths. One operational construction uses

$j=q,\bar q,g$1

with $j=q,\bar q,g$2 and $j=q,\bar q,g$3, and evaluates the effective coupling at the scaled temperature

$j=q,\bar q,g$4

This supplies a local map from thermodynamic variables to quasiparticle properties in finite-density transport calculations (Ingrosso et al., 11 Jun 2026).

4. Transport coefficients and kinetic interpretation

The DQPM provides both spectral-function-based and kinetic-theory-based routes to transport coefficients. In the Kubo formalism, the shear viscosity is written as

$j=q,\bar q,g$5

whereas in the quasiparticle pole approximation one obtains the relaxation-time form

$j=q,\bar q,g$6

with the identification

$j=q,\bar q,g$7

in the quasiparticle limit (Soloveva et al., 2019).

A central result of the transport-coefficient program is that the Kubo and RTA extractions are numerically close, which supports the statement that the quasiparticle limit $j=q,\bar q,g$8 holds sufficiently well in the time-like sector for transport applications. In that regime, $j=q,\bar q,g$9 is low near $$\rho_j(\omega,\mathbf p)=\frac{4\omega\gamma_j}{\left(\omega^2-\mathbf p^2-m_j^2\right)^2+4\gamma_j^2\omega^2},$$0 and rises with temperature, while its dependence on $$\rho_j(\omega,\mathbf p)=\frac{4\omega\gamma_j}{\left(\omega^2-\mathbf p^2-m_j^2\right)^2+4\gamma_j^2\omega^2},$$1 remains modest when plotted versus $$\rho_j(\omega,\mathbf p)=\frac{4\omega\gamma_j}{\left(\omega^2-\mathbf p^2-m_j^2\right)^2+4\gamma_j^2\omega^2},$$2 up to $$\rho_j(\omega,\mathbf p)=\frac{4\omega\gamma_j}{\left(\omega^2-\mathbf p^2-m_j^2\right)^2+4\gamma_j^2\omega^2},$$3 MeV (Soloveva et al., 2019).

Bulk viscosity and electric conductivity are likewise computed from DQPM quasiparticles. In the RTA formulation used for hot and dense matter,

$$\rho_j(\omega,\mathbf p)=\frac{4\omega\gamma_j}{\left(\omega^2-\mathbf p^2-m_j^2\right)^2+4\gamma_j^2\omega^2},$$4

and

$$\rho_j(\omega,\mathbf p)=\frac{4\omega\gamma_j}{\left(\omega^2-\mathbf p^2-m_j^2\right)^2+4\gamma_j^2\omega^2},$$5

(Ingrosso et al., 11 Jun 2026).

Recent work has extended the microscopic scattering sector beyond elastic $$\rho_j(\omega,\mathbf p)=\frac{4\omega\gamma_j}{\left(\omega^2-\mathbf p^2-m_j^2\right)^2+4\gamma_j^2\omega^2},$$6 channels by adding radiative $$\rho_j(\omega,\mathbf p)=\frac{4\omega\gamma_j}{\left(\omega^2-\mathbf p^2-m_j^2\right)^2+4\gamma_j^2\omega^2},$$7 processes with massive DQPM quasiparticles and effective propagators. In that extension, radiative channels reduce $$\rho_j(\omega,\mathbf p)=\frac{4\omega\gamma_j}{\left(\omega^2-\mathbf p^2-m_j^2\right)^2+4\gamma_j^2\omega^2},$$8, $$\rho_j(\omega,\mathbf p)=\frac{4\omega\gamma_j}{\left(\omega^2-\mathbf p^2-m_j^2\right)^2+4\gamma_j^2\omega^2},$$9, $$\rho_j(\omega,\mathbf p)=\frac{\gamma_j}{\tilde E_j} \left[ \frac{1}{(\omega-\tilde E_j)^2+\gamma_j^2} - \frac{1}{(\omega+\tilde E_j)^2+\gamma_j^2} \right], \qquad \tilde E_j^2=\mathbf p^2+m_j^2-\gamma_j^2.$$0, and the baryon diffusion coefficient relative to elastic-only calculations, but the reduction remains moderate because the inelastic rates stay below the elastic ones over the explored $$\rho_j(\omega,\mathbf p)=\frac{\gamma_j}{\tilde E_j} \left[ \frac{1}{(\omega-\tilde E_j)^2+\gamma_j^2} - \frac{1}{(\omega+\tilde E_j)^2+\gamma_j^2} \right], \qquad \tilde E_j^2=\mathbf p^2+m_j^2-\gamma_j^2.$$1 domain (Ingrosso et al., 11 Jun 2026).

5. Embedding in PHSD and nonequilibrium dynamics

The DQPM constitutes the partonic sector of PHSD, an off-shell transport approach based on Kadanoff–Baym equations in first-order gradient expansion. In PHSD, the local medium is treated as partonic when the energy density exceeds approximately

$$\rho_j(\omega,\mathbf p)=\frac{\gamma_j}{\tilde E_j} \left[ \frac{1}{(\omega-\tilde E_j)^2+\gamma_j^2} - \frac{1}{(\omega+\tilde E_j)^2+\gamma_j^2} \right], \qquad \tilde E_j^2=\mathbf p^2+m_j^2-\gamma_j^2.$$2

and the propagated quarks and gluons are DQPM quasiparticles with finite masses and widths (Linnyk et al., 2010).

A defining aspect of PHSD is that DQPM quasiparticles are hadronized dynamically through covariant transition rates for

$$\rho_j(\omega,\mathbf p)=\frac{\gamma_j}{\tilde E_j} \left[ \frac{1}{(\omega-\tilde E_j)^2+\gamma_j^2} - \frac{1}{(\omega+\tilde E_j)^2+\gamma_j^2} \right], \qquad \tilde E_j^2=\mathbf p^2+m_j^2-\gamma_j^2.$$3

with corresponding antiquark channels, while respecting flavor conservation, color neutrality, and energy-momentum conservation (Cassing et al., 2010). Because dynamical quarks and antiquarks become very massive close to the phase transition, the formed color-neutral prehadronic states acquire high invariant masses and decay sequentially into lower hadrons and strings. In model fireball studies, this mechanism increases the total entropy by about $$\rho_j(\omega,\mathbf p)=\frac{\gamma_j}{\tilde E_j} \left[ \frac{1}{(\omega-\tilde E_j)^2+\gamma_j^2} - \frac{1}{(\omega+\tilde E_j)^2+\gamma_j^2} \right], \qquad \tilde E_j^2=\mathbf p^2+m_j^2-\gamma_j^2.$$4, which addresses the entropy problem of naive coalescence pictures (0808.0022).

The same DQPM input also generates repulsive mean fields that drive collective flow in the partonic phase. In expanding fireball calculations, the elliptic flow $$\rho_j(\omega,\mathbf p)=\frac{\gamma_j}{\tilde E_j} \left[ \frac{1}{(\omega-\tilde E_j)^2+\gamma_j^2} - \frac{1}{(\omega+\tilde E_j)^2+\gamma_j^2} \right], \qquad \tilde E_j^2=\mathbf p^2+m_j^2-\gamma_j^2.$$5 shows an approximately linear correlation with the initial spatial eccentricity, with $$\rho_j(\omega,\mathbf p)=\frac{\gamma_j}{\tilde E_j} \left[ \frac{1}{(\omega-\tilde E_j)^2+\gamma_j^2} - \frac{1}{(\omega+\tilde E_j)^2+\gamma_j^2} \right], \qquad \tilde E_j^2=\mathbf p^2+m_j^2-\gamma_j^2.$$6, and the effective quasiparticle dynamics yields $$\rho_j(\omega,\mathbf p)=\frac{\gamma_j}{\tilde E_j} \left[ \frac{1}{(\omega-\tilde E_j)^2+\gamma_j^2} - \frac{1}{(\omega+\tilde E_j)^2+\gamma_j^2} \right], \qquad \tilde E_j^2=\mathbf p^2+m_j^2-\gamma_j^2.$$7, close to ideal-fluid behavior in that setup (0808.0022).

Microscopic equilibration studies in a box with periodic boundary conditions have tested the DQPM against full transport dynamics. In equilibrium, the quark and antiquark spectra and spectral functions are well reproduced by the DQPM ansatz, while the gluon spectral function shows a somewhat different shape because of explicit inelastic partonic interactions. The same calculations confirm that PHSD equilibrates to an equation of state well matched by the DQPM and lattice QCD (Ozvenchuk et al., 2012).

6. Phenomenology, extensions, and open issues

A major phenomenological application of the DQPM has been dilepton production from the nonperturbative QGP. Off-shell cross sections have been derived for the channels

$$\rho_j(\omega,\mathbf p)=\frac{\gamma_j}{\tilde E_j} \left[ \frac{1}{(\omega-\tilde E_j)^2+\gamma_j^2} - \frac{1}{(\omega+\tilde E_j)^2+\gamma_j^2} \right], \qquad \tilde E_j^2=\mathbf p^2+m_j^2-\gamma_j^2.$$8

using DQPM propagators for quarks and gluons. Implemented in PHSD, these channels show that low-mass dileptons are described by hadronic sources with collisional broadening of vector mesons, whereas the intermediate-mass region $$\rho_j(\omega,\mathbf p)=\frac{\gamma_j}{\tilde E_j} \left[ \frac{1}{(\omega-\tilde E_j)^2+\gamma_j^2} - \frac{1}{(\omega+\tilde E_j)^2+\gamma_j^2} \right], \qquad \tilde E_j^2=\mathbf p^2+m_j^2-\gamma_j^2.$$9 is dominated by off-shell quark–antiquark annihilation, gluon Compton scattering, and quark Bremsstrahlung in the nonperturbative QGP; the observed softening of the $$\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\,\omega\,\rho_j(\omega,\mathbf p)=1$$0 spectra is approximately reproduced [(Linnyk et al., 2010); (Linnyk et al., 2011)].

Heavy-flavor transport provides a contrasting use of the DQPM. In elastic $$\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\,\omega\,\rho_j(\omega,\mathbf p)=1$$1 and $$\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\,\omega\,\rho_j(\omega,\mathbf p)=1$$2 scattering, replacing massless perturbative partons by massive DQPM quasiparticles changes thresholds, infrared behavior, and relaxation times. However, when comparing on-shell DQPM-based scattering to its off-shell extension using DQPM spectral functions, the finite widths have little influence on the heavy-quark cross sections except close to thresholds; the dominant nonperturbative effects arise from the effective masses and coupling (1311.0736).

The model has been generalized in several directions. The momentum-dependent DQPM$$\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\,\omega\,\rho_j(\omega,\mathbf p)=1$$3 introduces explicit $$\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\,\omega\,\rho_j(\omega,\mathbf p)=1$$4-dependence in partonic self-energies and can reproduce simultaneously the lattice-QCD equation of state, quark number density, quark susceptibility, shear viscosity, electric conductivity, and the charm spatial diffusion coefficient more successfully than the momentum-independent version, especially because the thermal average of effective quark masses is reduced (Berrehrah et al., 2015, Berrehrah et al., 2016). At large baryon density, DQPM-CP incorporates a critical end-point at

$$\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\,\omega\,\rho_j(\omega,\mathbf p)=1$$5

and a first-order phase transition; in that extension the equation of state remains close to PNJL in the phase diagram, while transport coefficients differ substantially, illustrating that phase-boundary information alone does not determine the dynamical evolution of QGP matter (Soloveva et al., 2021).

Recent machine-learning analyses have also challenged the rigidity of the standard one-coupling DQPM ansatz. A deep-neural-network reconstruction constrained by $$\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\,\omega\,\rho_j(\omega,\mathbf p)=1$$6, $$\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\,\omega\,\rho_j(\omega,\mathbf p)=1$$7, and $$\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\,\omega\,\rho_j(\omega,\mathbf p)=1$$8 indicates that the standard entropy-driven scaling is too restrictive to describe these observables simultaneously. The preferred generalized DQPM-like pattern involves heavier gluons, lighter quarks, gluon widths close to the standard DQPM, larger quark widths, and possibly a temperature-dependent strange–light mass splitting (Soloveva et al., 2023).

The principal limitations follow directly from this status as an effective model. Explicit formulas for masses, widths, and couplings are often imported from earlier DQPM parametrizations rather than rederived in application papers; some transport implementations remain restricted to leading tree-level or elastic-dominant scattering sectors; and the phenomenological imprint of finite-$$\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\,\omega\,\rho_j(\omega,\mathbf p)=1$$9 modifications on bulk hadronic observables can be small because either $N^+$0 is small when the QGP dominates or the QGP fraction is small when $N^+$1 is large (Soloveva et al., 2020). These points do not invalidate the DQPM framework; they delimit its present domain as a lattice-constrained, propagator-based, nonperturbative quasiparticle model that is especially useful where a unified treatment of thermodynamics, off-shell transport, and medium-modified reaction rates is required.