PHSD: Parton-Hadron-String Dynamics

• PHSD is a unified off-shell transport framework that models the dynamic transition between partonic and hadronic matter in heavy-ion collisions.
• It employs the Dynamical QuasiParticle Model to simulate partonic properties with lattice QCD constraints, ensuring realistic mass and width distributions.
• The framework integrates dynamical hadronization and string fragmentation processes to reproduce key experimental observables across different energy regimes.

Searching arXiv for PHSD reviews and applications to ground the article in the relevant literature. arXivSearch query: "PHSD Parton-Hadron-String Dynamics review heavy-ion collisions" Parton–Hadron–String Dynamics (PHSD) is a microscopic off-shell transport approach for the description of the space–time evolution of strongly interacting matter in relativistic heavy-ion collisions, from the initial nucleon–nucleon scatterings and string formation through a strongly interacting partonic phase, dynamical hadronization, and final hadronic rescattering. In the PHSD formulation, the transport problem is derived from the Kadanoff–Baym equations for Green’s functions in phase-space representation in first-order gradient expansion beyond the quasi-particle approximation, while the partonic phase is modeled by the Dynamical QuasiParticle Model (DQPM) matched to lattice QCD thermodynamics (Bratkovskaya et al., 2013, Bratkovskaya et al., 2019).

1. Conceptual scope and historical placement

PHSD was developed as a unified covariant framework in which partons, hadrons, and strings are propagated within a single non-equilibrium transport description, with applications ranging from lower SIS and SPS energies to RHIC and LHC energies, and to $p+p$, $p+A$, and $A+A$ systems (Bratkovskaya et al., 2013, Bratkovskaya et al., 2019). In this sense, its defining feature is not merely the inclusion of explicit partonic degrees of freedom, but the dynamical treatment of the transition between partonic and hadronic sectors.

The approach is built on two central pillars: the DQPM for the partonic phase and a covariant dynamical hadronization scheme (Bratkovskaya et al., 2013). Below the critical regime, PHSD merges smoothly into the Hadron–String Dynamics (HSD) approach, so the hadronic sector is not an external afterburner but part of the same dynamical construction (Bratkovskaya et al., 2013). This continuity is central to PHSD’s use in the beam-energy domain where coexistence of partonic and non-partonic matter is substantial.

PHSD also serves as the baseline for later extensions. In particular, PHQMD extends PHSD by replacing the mean field by density dependent two body interactions in a similar way as in the Quantum Molecular Dynamics models, while off-shell propagation of partons and string dynamics are carried over unchanged from PHSD (Bratkovskaya et al., 2019). This relation clarifies that PHSD is the reference microscopic transport backbone for a broader family of transport calculations involving clusters, hypernuclei, and finite-density hadronic correlations.

2. Off-shell transport structure and the DQPM

The microscopic transport equations of PHSD follow from the off-shell Kadanoff–Baym equations under first-order gradient expansion, often written in Botermans–Malfliet form (Bratkovskaya et al., 2013). In compact notation, the phase-space dynamics of each species is governed by

$$\Bigl\{\,p^2-\Re\Pi_j(x,p)\,,\,G_j^{<}(x,p)\,\Bigr\} \;=\; i\,\bigl[\Pi_j^{>}\,G_j^{<}\;-\;\Pi_j^{<}\,G_j^{>}\bigr],$$

with $G_j^< \sim f_j \rho_j$, so that mean-field propagation and collisional broadening are treated simultaneously (Bratkovskaya et al., 2013). Equivalent transport representations used in the PHSD literature emphasize the drift term, a Poisson-bracket Vlasov term generated by $\Re \Sigma^R$, and a collision integral of BUU type (Bratkovskaya et al., 2019).

In the DQPM, the quark–gluon plasma is represented as a gas of off-shell quasiparticles with broad spectral functions. For $j=q,\bar q,g$,

$$\rho_j(\omega,\mathbf{p}) = \frac{\gamma_j}{E_j} \Bigl[ \frac{1}{(\omega-E_j)^2+\gamma_j^2} - \frac{1}{(\omega+E_j)^2+\gamma_j^2} \Bigr],$$

where

$E_j^2(\mathbf{p})=\mathbf{p}^2+M_j^2-\gamma_j^2.$

The retarded self-energy is written as

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

so that the real part fixes the quasiparticle mass $p+A$0 and the imaginary part $p+A$1 encodes the finite lifetime and interaction rate (Bratkovskaya et al., 2013).

The temperature dependence of $p+A$2 and $p+A$3 is adjusted to reproduce lattice QCD thermodynamics, including pressure and entropy density, down to $p+A$4 MeV (Bratkovskaya et al., 2013). Above $p+A$5, the effective coupling $p+A$6 rises in the infrared, generating large quasiparticle masses and broad widths (Bratkovskaya et al., 2013). This is the basis for PHSD’s description of a strongly coupled rather than weakly interacting partonic medium.

From the DQPM one extracts a scalar potential energy density $p+A$7 as a function of the scalar parton density $p+A$8. The scalar mean field

$p+A$9

acts on each quasiparticle and generates a self-consistent Vlasov force $A+A$0, while the widths determine partonic scattering cross sections via detailed balance (Bratkovskaya et al., 2013). A recurring implication in PHSD phenomenology is that these repulsive partonic mean fields are a major source of transverse acceleration and azimuthal anisotropy.

3. Dynamical hadronization, strings, and entropy production

When the local energy density falls below a critical value $A+A$1 GeV/fm$A+A$2, PHSD converts partons to hadrons through covariant transition rates,

$A+A$3

with exact conservation of four-momentum, flavor currents, and color neutrality (Bratkovskaya et al., 2013). The transition rates form high-invariant-mass pre-hadronic color dipoles or tripoles, which then decay microcanonically to the groundstate hadrons, thus avoiding entropy loss (Bratkovskaya et al., 2013).

This entropy issue is foundational. PHSD was explicitly constructed to avoid the entropy deficit of simple coalescence pictures by fusing massive quasiparticles into high-mass resonant pre-hadrons or strings that subsequently decay, increasing the total entropy (Cassing et al., 2010). Because the dynamical quarks and antiquarks become very massive close to the phase transition, the formed pre-hadronic states are of high invariant mass and sequentially decay to the groundstate meson and baryon octets (Cassing et al., 2010).

The string sector remains active throughout the description. High-mass hadronic states, $A+A$4 GeV for mesons and $A+A$5 GeV for baryons, are treated as strings and fragmented via the JETSET algorithm (Bratkovskaya et al., 2013). At high invariant mass in hadron–hadron collisions, PHSD assumes the creation of color flux tubes, with string fragmentation described by the Lund fragmentation function

$A+A$6

and with produced hadrons first appearing as pre-hadrons that interact with reduced cross sections during a formation time $A+A$7–$A+A$8 fm/$A+A$9 (Bratkovskaya et al., 2019).

The hadronization mechanism is therefore not a late-time projection rule but a dynamical conversion process coupled to off-shell transport. This suggests that PHSD treats the quark–hadron transition as a local, covariant, many-body kinetic process rather than as a global freeze-out hypersurface.

4. Numerical realization and dynamical regimes

In heavy-ion applications, PHSD initializes the incoming nuclei by nucleon test particles at a given $$\Bigl\{\,p^2-\Re\Pi_j(x,p)\,,\,G_j^{<}(x,p)\,\Bigr\} \;=\; i\,\bigl[\Pi_j^{>}\,G_j^{<}\;-\;\Pi_j^{<}\,G_j^{>}\bigr],$$0. At early times, binary nucleon–nucleon encounters excite color strings, which melt into partons if the local energy density exceeds $$\Bigl\{\,p^2-\Re\Pi_j(x,p)\,,\,G_j^{<}(x,p)\,\Bigr\} \;=\; i\,\bigl[\Pi_j^{>}\,G_j^{<}\;-\;\Pi_j^{<}\,G_j^{>}\bigr],$$1 (Bratkovskaya et al., 2013). In practice, the transport equations are solved with a test-particle ansatz for the phase-space distributions and a grid for mean fields (Bratkovskaya et al., 2013).

Time evolution proceeds in small steps $$\Bigl\{\,p^2-\Re\Pi_j(x,p)\,,\,G_j^{<}(x,p)\,\Bigr\} \;=\; i\,\bigl[\Pi_j^{>}\,G_j^{<}\;-\;\Pi_j^{<}\,G_j^{>}\bigr],$$2, with partonic and hadronic propagators propagated through Hamilton’s equations augmented by Vlasov forces from $$\Bigl\{\,p^2-\Re\Pi_j(x,p)\,,\,G_j^{<}(x,p)\,\Bigr\} \;=\; i\,\bigl[\Pi_j^{>}\,G_j^{<}\;-\;\Pi_j^{<}\,G_j^{>}\bigr],$$3 and stochastic scatterings determined by local widths $$\Bigl\{\,p^2-\Re\Pi_j(x,p)\,,\,G_j^{<}(x,p)\,\Bigr\} \;=\; i\,\bigl[\Pi_j^{>}\,G_j^{<}\;-\;\Pi_j^{<}\,G_j^{>}\bigr],$$4 (Bratkovskaya et al., 2013). Dynamical hadronization and string fragmentation occur on the fly wherever $$\Bigl\{\,p^2-\Re\Pi_j(x,p)\,,\,G_j^{<}(x,p)\,\Bigr\} \;=\; i\,\bigl[\Pi_j^{>}\,G_j^{<}\;-\;\Pi_j^{<}\,G_j^{>}\bigr],$$5 (Bratkovskaya et al., 2013). In equilibrium studies, PHSD is run in a cubic box with periodic boundary conditions, which permits direct numerical extraction of transport coefficients after kinetic and chemical equilibration (1212.5393).

The relative weight of the partonic phase is strongly energy dependent. At low SPS energies only a modest partonic volume fraction appears, while at RHIC and LHC energies the fireball is dominated by partonic matter for several fm/$$\Bigl\{\,p^2-\Re\Pi_j(x,p)\,,\,G_j^{<}(x,p)\,\Bigr\} \;=\; i\,\bigl[\Pi_j^{>}\,G_j^{<}\;-\;\Pi_j^{<}\,G_j^{>}\bigr],$$6 (Bratkovskaya et al., 2013). A particularly important SPS result is that, in Pb+Pb reactions from $$\Bigl\{\,p^2-\Re\Pi_j(x,p)\,,\,G_j^{<}(x,p)\,\Bigr\} \;=\; i\,\bigl[\Pi_j^{>}\,G_j^{<}\;-\;\Pi_j^{<}\,G_j^{>}\bigr],$$7 to $$\Bigl\{\,p^2-\Re\Pi_j(x,p)\,,\,G_j^{<}(x,p)\,\Bigr\} \;=\; i\,\bigl[\Pi_j^{>}\,G_j^{<}\;-\;\Pi_j^{<}\,G_j^{>}\bigr],$$8 A GeV, at most $$\Bigl\{\,p^2-\Re\Pi_j(x,p)\,,\,G_j^{<}(x,p)\,\Bigr\} \;=\; i\,\bigl[\Pi_j^{>}\,G_j^{<}\;-\;\Pi_j^{<}\,G_j^{>}\bigr],$$9 of the collision energy is stored in the dynamics of the partons, implying a large fraction of non-partonic, hadronic or string-like matter that can be viewed as a hadronic corona (Linnyk et al., 2010). At RHIC energies, by contrast, the partonic phase carries up to $G_j^< \sim f_j \rho_j$0 of the energy for $G_j^< \sim f_j \rho_j$1–$G_j^< \sim f_j \rho_j$2 fm/$G_j^< \sim f_j \rho_j$3 (Bratkovskaya et al., 2012).

That coexistence pattern has interpretive consequences. At top SPS energies, neither purely hadronic nor purely partonic models can be employed to extract physical conclusions in comparing model results with data, because the relevant space–time history contains both a substantial partonic component and a substantial hadronic corona (Linnyk et al., 2010).

5. Phenomenology from soft hadrons to heavy flavor

PHSD has been validated against a broad set of observables. For soft hadrons, it reproduces pion and kaon transverse-mass spectra from SPS and RHIC energies; in particular, the harder slopes of $G_j^< \sim f_j \rho_j$4 and $G_j^< \sim f_j \rho_j$5 in PHSD relative to HSD without partons match NA49, PHENIX, STAR, and BRAHMS data, demonstrating the role of partonic mean fields and rescattering (Bratkovskaya et al., 2013). In the strangeness sector, PHSD enhances multi-strange antibaryon production relative to HSD through slightly enhanced $G_j^< \sim f_j \rho_j$6 production in the partonic phase from massive time-like gluon decay and more abundant strange antibaryon formation in the hadronization process (Bratkovskaya et al., 2011).

For anisotropic flow, the excitation function $G_j^< \sim f_j \rho_j$7 for midrapidity charged particles rises from SPS to RHIC in line with STAR and PHENIX data, whereas hadronic models such as HSD and UrQMD predict an essentially flat $G_j^< \sim f_j \rho_j$8 (Bratkovskaya et al., 2013). PHSD also reproduces the experimentally observed increase of integrated $G_j^< \sim f_j \rho_j$9 with $\Re \Sigma^R$0 in Au+Au from $\Re \Sigma^R$1 GeV up to $\Re \Sigma^R$2 GeV, and its analysis of higher harmonics shows $\Re \Sigma^R$3 and an enhanced $\Re \Sigma^R$4 relative to purely hadronic transport (Konchakovski et al., 2012). The ratio

$\Re \Sigma^R$5

is obtained in PHSD, significantly above the ideal-hydrodynamic value $\Re \Sigma^R$6 and in better accord with RHIC data (Konchakovski et al., 2012). At top RHIC energy, PHSD also shows approximate constituent quark scaling of elliptic flow when plotted as $\Re \Sigma^R$7 versus $\Re \Sigma^R$8 (Bratkovskaya et al., 2011).

Small-system applications expose a different aspect of the model. In $\Re \Sigma^R$9–Pb collisions at $j=q,\bar q,g$0 TeV, PHSD provides correlations between charged-particle multiplicity at midrapidity and the number of participant nucleons close to results from CGC-based rcBK calculations and substantially different from independent Glauber initial conditions (Konchakovski et al., 2014). However, PHSD predicts that

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

drops below unity for $j=q,\bar q,g$2, whereas CGC-based calculations predict $j=q,\bar q,g$3 at forward rapidity (Konchakovski et al., 2014). Accordingly, multiplicity scaling alone is not a clean discriminator of coherent versus incoherent initial conditions once detailed energy–momentum conservation is enforced, while the rapidity dependence of $j=q,\bar q,g$4 is a robust observable for this purpose (Konchakovski et al., 2014).

Electromagnetic probes are treated microscopically in both hadronic and partonic channels. In In+In collisions at $j=q,\bar q,g$5 A GeV, PHSD describes the NA60 dilepton data by combining collisional broadening of vector mesons with partonic radiation from the strongly coupled QGP; in the intermediate mass range $j=q,\bar q,g$6 GeV $j=q,\bar q,g$7 GeV, the spectra are dominated by quark–antiquark annihilation in the nonperturbative QGP (Linnyk et al., 2011). For direct photons in Au+Au at $j=q,\bar q,g$8 GeV, PHSD predicts that the thermal photon yield scales roughly with $j=q,\bar q,g$9 with $$\rho_j(\omega,\mathbf{p}) = \frac{\gamma_j}{E_j} \Bigl[ \frac{1}{(\omega-E_j)^2+\gamma_j^2} - \frac{1}{(\omega+E_j)^2+\gamma_j^2} \Bigr],$$0, whereas the partonic contribution scales with $$\rho_j(\omega,\mathbf{p}) = \frac{\gamma_j}{E_j} \Bigl[ \frac{1}{(\omega-E_j)^2+\gamma_j^2} - \frac{1}{(\omega+E_j)^2+\gamma_j^2} \Bigr],$$1; the direct-photon $$\rho_j(\omega,\mathbf{p}) = \frac{\gamma_j}{E_j} \Bigl[ \frac{1}{(\omega-E_j)^2+\gamma_j^2} - \frac{1}{(\omega+E_j)^2+\gamma_j^2} \Bigr],$$2 is larger in peripheral collisions because photons from hot deconfined matter carry a much smaller elliptic flow than those from final hadronic interactions (Cassing et al., 2014).

Heavy-flavor implementations extend the same off-shell logic to charmonium, bottomonium, and open heavy flavor. In PHSD, heavy quarks propagate off-shell and interact elastically with DQPM partons; the spatial diffusion coefficient $$\rho_j(\omega,\mathbf{p}) = \frac{\gamma_j}{E_j} \Bigl[ \frac{1}{(\omega-E_j)^2+\gamma_j^2} - \frac{1}{(\omega+E_j)^2+\gamma_j^2} \Bigr],$$3 extracted from the drag coefficient is found to be consistent with lattice-QCD extractions (Song et al., 2023). In Pb+Pb at $$\rho_j(\omega,\mathbf{p}) = \frac{\gamma_j}{E_j} \Bigl[ \frac{1}{(\omega-E_j)^2+\gamma_j^2} - \frac{1}{(\omega+E_j)^2+\gamma_j^2} \Bigr],$$4 TeV, PHSD reproduces the $$\rho_j(\omega,\mathbf{p}) = \frac{\gamma_j}{E_j} \Bigl[ \frac{1}{(\omega-E_j)^2+\gamma_j^2} - \frac{1}{(\omega+E_j)^2+\gamma_j^2} \Bigr],$$5-meson nuclear modification factor $$\rho_j(\omega,\mathbf{p}) = \frac{\gamma_j}{E_j} \Bigl[ \frac{1}{(\omega-E_j)^2+\gamma_j^2} - \frac{1}{(\omega+E_j)^2+\gamma_j^2} \Bigr],$$6 and elliptic flow $$\rho_j(\omega,\mathbf{p}) = \frac{\gamma_j}{E_j} \Bigl[ \frac{1}{(\omega-E_j)^2+\gamma_j^2} - \frac{1}{(\omega+E_j)^2+\gamma_j^2} \Bigr],$$7 only when EPS09 nuclear shadowing is included and both collisional energy loss and coalescence hadronization are active (Song et al., 2023).

6. Equilibrium properties, interpretive issues, and ongoing extensions

A distinctive feature of PHSD is that the same off-shell transport framework used for collisions can be applied to infinite matter in a box with periodic boundary conditions, permitting equilibrium studies of transport coefficients (1212.5393). In the relaxation-time approximation, the shear viscosity is written as

$$\rho_j(\omega,\mathbf{p}) = \frac{\gamma_j}{E_j} \Bigl[ \frac{1}{(\omega-E_j)^2+\gamma_j^2} - \frac{1}{(\omega+E_j)^2+\gamma_j^2} \Bigr],$$8

with $$\rho_j(\omega,\mathbf{p}) = \frac{\gamma_j}{E_j} \Bigl[ \frac{1}{(\omega-E_j)^2+\gamma_j^2} - \frac{1}{(\omega+E_j)^2+\gamma_j^2} \Bigr],$$9, and similar expressions hold for $E_j^2(\mathbf{p})=\mathbf{p}^2+M_j^2-\gamma_j^2.$0 and $E_j^2(\mathbf{p})=\mathbf{p}^2+M_j^2-\gamma_j^2.$1 (Bratkovskaya et al., 2013). PHSD yields a pronounced minimum $E_j^2(\mathbf{p})=\mathbf{p}^2+M_j^2-\gamma_j^2.$2 near $E_j^2(\mathbf{p})=\mathbf{p}^2+M_j^2-\gamma_j^2.$3, with rising $E_j^2(\mathbf{p})=\mathbf{p}^2+M_j^2-\gamma_j^2.$4 both in the hadronic phase and toward the perturbative QCD limit at high $E_j^2(\mathbf{p})=\mathbf{p}^2+M_j^2-\gamma_j^2.$5 (Bratkovskaya et al., 2013). For bulk viscosity, the crucial result is that $E_j^2(\mathbf{p})=\mathbf{p}^2+M_j^2-\gamma_j^2.$6 exhibits a peak around $E_j^2(\mathbf{p})=\mathbf{p}^2+M_j^2-\gamma_j^2.$7 due to infrared enhancement of $E_j^2(\mathbf{p})=\mathbf{p}^2+M_j^2-\gamma_j^2.$8 and mean-field effects, and $E_j^2(\mathbf{p})=\mathbf{p}^2+M_j^2-\gamma_j^2.$9 rises approximately linearly above $\Pi_j = M_j^2 - 2 i \gamma_j \omega,$0 up to $\Pi_j = M_j^2 - 2 i \gamma_j \omega,$1 before saturating (Bratkovskaya et al., 2013). Within statistics, the Kubo formalism and relaxation-time approximation give practically the same $\Pi_j = M_j^2 - 2 i \gamma_j \omega,$2 in PHSD (1212.5393).

These equilibrium results reinforce the interpretation of PHSD matter as a strongly interacting medium, but they also delimit what can be extracted from bulk observables. In the finite-$\Pi_j = M_j^2 - 2 i \gamma_j \omega,$3 extension, PHSD5.0 explicitly computes total and differential partonic cross sections as functions of $\Pi_j = M_j^2 - 2 i \gamma_j \omega,$4 and $\Pi_j = M_j^2 - 2 i \gamma_j \omega,$5, yet finds only a very modest change of $\Pi_j = M_j^2 - 2 i \gamma_j \omega,$6 with baryon chemical potential and only small traces of $\Pi_j = M_j^2 - 2 i \gamma_j \omega,$7-dependence in heavy-ion observables (Moreau et al., 2019). The stated implication is that one needs a sizable partonic density and large space–time QGP volume to explore the dynamics in the partonic phase, but such conditions are fulfilled at high bombarding energies where $\Pi_j = M_j^2 - 2 i \gamma_j \omega,$8 is rather low; when the beam energy is decreased and $\Pi_j = M_j^2 - 2 i \gamma_j \omega,$9 rises, the hadronic phase becomes dominant (Moreau et al., 2019).

PHSD has also been used to assess the character of the QCD transition at Beam Energy Scan and FAIR/NICA energies. For directed flow, the semi-qualitative agreement between data and PHSD/HSD results supports the idea of a crossover type of quark-hadron transition which softens the nuclear equation of state but shows no indication of a first-order phase transition (Cassing et al., 2014). At the same time, kaon and antikaon directed flow at $p+A$00 GeV is highly sensitive to hadronic potentials, emphasizing that the low-energy regime is hadron-dominated even within a transport scheme that includes explicit partons (Cassing et al., 2014).

Several limitations and development lines are explicitly identified in the PHSD literature. The current DQPM commonly uses momentum-independent quasiparticle masses and widths, and the validity of taking equilibrium self-energies unchanged out of equilibrium remains an approximation (Bratkovskaya et al., 2011, Bratkovskaya et al., 2011). Ongoing and future developments include the explicit inclusion of chiral dynamics and critical fluctuations near the conjectured QCD critical point, improved initial conditions based on CGC or glasma at LHC energies, and further study of electromagnetic probes and heavy-flavor dynamics with dynamical off-shell propagators (Bratkovskaya et al., 2013). In the hadronic sector, the incorporation of chiral symmetry restoration into string decay has been used within PHSD to provide a microscopic explanation of the $p+A$01 maximum at about $p+A$02 A GeV, showing that the horn structure appears only when both chiral symmetry restoration in string decay and the deconfinement transition are incorporated in the reaction dynamics (Moreau et al., 2016).

Taken together, these results position PHSD as a transport framework in which lattice-QCD-constrained quasiparticle dynamics, dynamical hadronization, and hadronic rescattering are treated within a common off-shell kinetic formalism. Its central claim is not that heavy-ion matter is everywhere partonic, but that the observable dynamics over a wide energy range is governed by a changing mixture of partonic, string-like, and hadronic sectors whose relative weights must be followed microscopically in space and time.