---
title: 'PHQMD: Parton-Hadron Quantum Molecular Dynamics'
url: https://www.emergentmind.com/topics/parton-hadron-quantum-molecular-dynamics-phqmd
type: topic
---

# PHQMD: Parton-Hadron Quantum Molecular Dynamics

Searching arXiv for PHQMD papers and related recent work.
Parton-Hadron-Quantum-Molecular Dynamics (PHQMD) is a microscopic \(n\)-body transport approach for relativistic heavy-ion collisions that was developed to describe, within a single framework, general hadron production together with the dynamical formation of nuclear clusters and hypernuclei [1907.03860, 1911.09496]. Its defining construction is the combination of the collision integral and partonic/hadronic dynamics of Parton-Hadron-String Dynamics (PHSD) with Quantum Molecular Dynamics (QMD)-type propagation of the baryonic sector, so that many-body correlations and fluctuations needed for fragment formation are preserved during the reaction [1907.03860, 2106.14839]. In this formulation, cluster formation is not introduced solely as a freeze-out afterburner; rather, correlated baryonic aggregates emerge dynamically from interactions and can then be identified by algorithms such as Minimum Spanning Tree (MST), the Simulated Annealing Clusterization Algorithm (SACA), or later advanced MST variants used for deuterons [1911.09496, 2208.11802, 2303.02279].

## 1. Origin, scope, and conceptual motivation

PHQMD was introduced to address a specific deficiency of standard heavy-ion transport descriptions: bulk hadron spectra and collective observables can often be described, but light nuclei and hypernuclei require the retention of explicit \(n\)-body baryonic correlations during the reaction [1907.03860, 2106.14839]. The model is intended for energies from SIS and AGS through SPS, RHIC Beam Energy Scan, and RHIC energies, while retaining the PHSD treatment of hadronic reactions, strings, deconfinement, partonic transport, and hadronization [1907.03860, 2106.14839].

In the authors’ formulation, PHQMD “unites the collision integrals of the Parton-Hadron-String Dynamics (PHSD) approach with 2-body potential interactions between baryons similar as in the Quantum Molecular Dynamics (QMD) approach where baryons are described by Gaussian wave functions” [1911.09496]. This architecture is central because PHSD already contains explicit partonic degrees of freedom, quarks and gluons, a lattice-QCD-based equation of state for the deconfined phase, dynamical hadronization, and hadronic elastic and inelastic rescattering, whereas PHQMD replaces baryon mean-field propagation by QMD-like \(n\)-body dynamics in the baryonic sector [1911.09496, 2106.14839].

The motivating physics domain is the baryon-rich regime relevant for NICA-MPD, BM@N, and CBM, framed in one overview as \(\sqrt{s_{NN}} \lesssim 11\) GeV, where compression and decompression of baryonic matter coexist with substantial hadronic chemistry and threshold strangeness production [1911.09496]. This suggests that PHQMD was designed not only for bulk hadrochemistry but also for observables that depend on the space-time history of baryonic correlations, including cluster production, hyperon capture, hypernuclear channels, and the interpretation of weakly bound objects in a hot medium [1911.09496, 2208.11802].

## 2. Microscopic structure and dynamical degrees of freedom

PHQMD is a hybrid construction. Baryons are propagated by QMD, while mesonic and partonic degrees of freedom continue to follow PHSD dynamics [2106.14839]. Accordingly, the relevant propagated degrees of freedom depend on the reaction stage: strings and partons at early times, hadrons during and after hadronization, and correlated baryons in the hadronic stage from which clusters and hypernuclei can be identified [1911.09496].

In the QMD sector, a baryon is represented by a Gaussian single-particle Wigner density. One PHQMD formulation writes
\[
f ({\bf r}_i, {\bf p}_i,{\bf r}_{i0},{\bf p}_{i0},t) = \frac{1}{\pi^3 \hbar^3 } {\rm e}^{-\frac{2}{L} ({\bf r}_i - {\bf r}_{i0} (t) )^2 } {\rm e}^{-\frac{L}{2\hbar^2} ({\bf p}_i - {\bf p}_{i0} (t) )^2},
\]
with \(L=8.66\ {\rm fm}^2\) in the relativistic cluster-production study [2106.14839], whereas the original 2019 formal development used \(L=2.16\ \mathrm{fm}^2\) [1907.03860]. The centroid variables \(({\bf r}_{i0}(t),{\bf p}_{i0}(t))\) are propagated by a variational QMD dynamics derived from the Dirac–Frenkel–McLachlan variational principle,
\[
\delta \int_{t_1}^{t_2} dt \, \langle\psi(t)|i\frac{d}{dt}-H|\psi(t)\rangle = 0,
\]
which yields Hamilton-like equations for the centroids [1907.03860, 2106.14839].

The expectation-value Hamiltonian contains kinetic energy and effective interaction terms. In the Skyrme-based implementations, the baryon-baryon interaction is written as a sum of local Skyrme and Coulomb terms,
\[
\begin{aligned}
V_{i,j}&= V({\bf r}_i, {\bf r}_j,{\bf r}_{i0},{\bf r}_{j0},t)  = V_{\rm Skyrme}+ V_{\rm Coul} \\
&=\frac{1}{2} t_1 \delta ({\bf r}_i - {\bf r}_j)  +  \frac{1}{\gamma+1}t_2 \delta ({\bf r}_i - {\bf r}_j)  \, \rho^{\gamma-1}(\cdots) +\frac{1}{2}  \frac{Z_i Z_j e^2}{|{\bf r}_i-{\bf r}_j|},
\end{aligned}
\]
with the associated density-functional form
\[
\langle V_{Skyrme}({\bf r}_{i0},t) \rangle \,=\, \alpha \left(\frac{\rho_{int} ({\bf r}_{i0},t)}{\rho_0}\right) + \beta \left(\frac{\rho_{int} ({\bf r}_{i0},t)}{\rho_0}\right)^{\gamma}.
\]
Here \(\rho_{int}\) is the interaction density generated from Gaussian overlaps of surrounding baryons, and \(\alpha,\beta,\gamma\) encode the nuclear equation of state (EoS) [1907.03860, 1911.09496, 2106.14839].

For relativistic beam energies, PHQMD uses a pragmatic Lorentz-contracted modification of the baryon Wigner density along the beam axis rather than a fully covariant molecular dynamics [1907.03860, 2106.14839]. The PHSD inheritance remains substantial: string excitation and fragmentation are treated via FRITIOF 7.02 and PYTHIA 6.4, hadronic collision channels are those of PHSD/HSD, and when the local energy density exceeds \(\varepsilon_C \simeq 0.5\) GeV/fm\(^3\), the system can enter the partonic sector described by the Dynamical Quasi-Particle Model (DQPM) [1907.03860, 2106.14839].

## 3. Equation of state and effective interactions

The EoS enters PHQMD through the density dependence of the effective baryonic interaction. In the original Skyrme implementations, the model employed static soft and hard EoS parameterizations. One detailed formulation gives the soft set as
\[
\alpha=-390\ \mathrm{MeV},\quad \beta=320\ \mathrm{MeV},\quad \gamma=1.14,\quad K=200\ \mathrm{MeV},
\]
and the hard set as
\[
\alpha=-130\ \mathrm{MeV},\quad \beta=59\ \mathrm{MeV},\quad \gamma=2.09,\quad K=380\ \mathrm{MeV}
\]
[1907.03860]. A later synopsis also associates the soft set with \(\alpha=-390\) MeV and \(\gamma=1.14\), and the hard set with \(\alpha=-130\) MeV and \(\gamma=2.09\) [1911.09496]. In this context, a low compression modulus \(K\) corresponds to a soft EoS and a high \(K\) to a hard EoS [1911.09496].

A major later extension introduced a momentum-dependent baryonic potential and a soft momentum-dependent EoS, calibrated to proton–nucleus elastic scattering through the Schrödinger-equivalent optical potential [2411.04969]. In that work, the optical-potential-based two-body interaction is parameterized as
\[
V(\Delta p)=   V({\bf p}_{01},{\bf p}_{02})  =(a \Delta p + b\Delta p^2)\  \exp [-c \sqrt{\Delta p}  ],
\]
with \(\Delta p = \sqrt{({\bf p}_{01}-{\bf p}_{02})^2}\), and is promoted to a local interaction by multiplication with \(\delta({\bf r}_1-{\bf r}_2)\) [2411.04969]. The three EoS realizations used there are soft (S), hard (H), and soft momentum-dependent (SM), with parameter sets:

| EoS | Parameters |
|---|---|
| S | \(\alpha=-383.5\) MeV, \(\beta=329.5\) MeV, \(\gamma=1.15\), \(K=200\) MeV |
| H | \(\alpha=-125.3\) MeV, \(\beta=71.0\) MeV, \(\gamma=2.0\), \(K=380\) MeV |
| SM | \(\alpha=-478.87\) MeV, \(\beta=413.76\) MeV, \(\gamma=1.1\), \(K=200\) MeV |

The momentum-dependent parameters are \(a=236.326\), \(b=-20.73\), \(c=0.901\) in the units stated in that work [2411.04969].

The EoS sensitivity is phenomenologically significant across several PHQMD studies. In Au+Au at \(\sqrt{s_{NN}}=11.5\) GeV, a hard EoS increases the slope of hadron spectra at large \(p_T\) compared with a soft EoS, while the proton slope remains slightly underestimated at large \(p_T\) [1911.09496]. At SIS energies, flow observables show that a soft momentum-dependent EoS can behave more like a hard static EoS than like a static soft EoS in \(v_1\) and \(v_2\), despite sharing \(K=200\) MeV with the soft case [2411.04969]. This suggests that compressibility alone does not exhaust the EoS dependence of PHQMD predictions.

## 4. Cluster-recognition strategies and dynamical cluster formation

A central feature of PHQMD is the distinction between dynamical formation and algorithmic recognition. The interactions among baryons generate correlated aggregates during the transport evolution; MST, SACA, or later advanced MST procedures identify those aggregates in the event record rather than producing them as an external afterburner prescription [1911.09496, 2106.14839, 2411.04969].

MST is the simplest recognition method. In the midrapidity-cluster studies, clusters are identified in coordinate space such that a proton-neutron pair belongs to a cluster when their distance in their rest frame is smaller than
\[
r_0 = 4~\mathrm{fm},
\]
and additional momentum-space cuts were reported not to change the cluster distributions significantly [2208.11802, 2106.14839]. Earlier PHQMD work on light clusters quoted an MST coordinate-space condition
\[
r_0 = 2.5\ \mathrm{fm},
\]
with the interpretation that MST is most reliable late in the reaction when fragments are already spatially separated [1907.03860]. A plausible implication is that the concrete MST implementation evolved across PHQMD applications.

SACA is the more elaborate recognition method emphasized in the original cluster and hypernucleus overviews. At a given time \(t\), an initial pre-clustering is first obtained with MST, after which clusters and single nucleons are recombined into candidate partitions. Among these, the most bound configuration is sought stochastically by a Metropolis simulated-annealing procedure, and repeated iteration automatically leads to the most bound configuration [1911.09496]. SACA is based on the idea of Dorso and Randrup that the most bound configuration at early times evolves toward the final fragment distribution, so it can identify clusters before geometric separation is complete [1911.09496, 1911.11892, 1907.03860].

Later deuteron studies refined MST into an advanced Minimum Spanning Tree (aMST) procedure. There the coordinate-space condition remains
\[
| \mathbf{r}_i^* - \mathbf{r}_j^* | < r_{clus}, \qquad r_{clus}=4\ \mathrm{fm},
\]
and the cluster must have negative binding energy,
\[
E_B<0,
\]
with the cluster binding energy in its rest frame written as
\[
E_{B} = \sum_i^A E_i-\sum_i^A {M_N}_i +\sum_{i\neq j} V_{ij}.
\]
The aMST stabilization logic freezes internal cluster degrees of freedom once the cluster is no longer in contact, by collision or potential interaction, with hadrons outside the cluster, and thereby removes the need for an ad hoc cluster-recognition time in deuteron analyses [2303.02279]. A 2024 conference contribution describes a related “advanced Minimum Spanning Tree (aMST)” algorithm as identifying bound clusters with negative binding energies by correlations of baryons in coordinate space [2411.04296].

The contrast with coalescence models is explicit in the PHQMD literature. Coalescence assembles nuclei at freeze-out by cuts in relative coordinate and momentum space, whereas PHQMD aims to preserve the interactions and correlations that make clusters emerge dynamically during the evolution [1907.03860, 2208.11802]. At the same time, PHQMD has also been used as a host environment for embedded coalescence, enabling same-event comparisons between dynamical clustering and freeze-out coalescence prescriptions [2208.11802, 2411.04969].

## 5. Hypernuclei, deuteron mechanisms, and the “ice in the fire” problem

Hypernuclei in PHQMD arise by extending the clusterization logic into the strange sector. Hyperons are produced through PHSD string, hadronic, or hadronization channels, propagate in the hadronic medium, and can become bound to nucleonic clusters; in the model’s own wording, “the capture of the produced hyperons by clusters of nucleons leads to the hypernuclei formation” [1911.09496]. In one relativistic cluster study, the hyperon–nucleon interaction is approximated by
\[
V_{N\Lambda}= \frac{2}{3} V_{NN},
\]
which is explicitly described as a rough approximation and a limitation of the present hypernuclear treatment [2106.14839].

The deuteron sector became a major area of PHQMD development. A 2023 study formulated two microscopic mechanisms for deuteron production at midrapidity: “kinetic” deuterons produced by hadronic reactions and “potential” deuterons produced by the attractive interaction between nucleons and recognized by aMST [2303.02279]. The explicit kinetic channels include
\[
\pi NN \leftrightarrow \pi d, \qquad NNN \leftrightarrow Nd,
\]
with subdominant \(NN \leftrightarrow d\pi\), plus elastic \(d\pi \to d\pi\) and \(dN \to dN\) channels in the full transport implementation [2303.02279]. A 2024 proceeding likewise states that PHQMD includes catalytic hadronic reactions
\[
\pi NN \leftrightarrow \pi d, \qquad NNN \leftrightarrow N d
\]
for deuterons, while anti-clusters are produced by potential interactions of anti-baryons and recognized by aMST in an analogous way [2411.04296].

A key refinement in the kinetic deuteron treatment is the inclusion of the quantum nature of the deuteron in both coordinate and momentum space. PHQMD imposes a finite-size excluded-volume condition around the candidate deuteron and projects the relative momentum of the interacting nucleon pair onto the deuteron wave function in momentum space [2303.02279]. In that study, the excluded radius is
\[
R_d = 1.803~\mathrm{fm},
\]
while the text also mentions a deuteron rms radius \(\sqrt{\langle r_d^2 \rangle} \simeq 2.1~\mathrm{fm}\) [2303.02279]. The combined effect is strong suppression of kinetic deuteron production in a dense medium: each of the excluded-volume and momentum-projection effects suppresses the yield at midrapidity by about a factor of \(3\), both together give roughly another factor of \(2\), the total suppression is about \(6\) at \(\sqrt{s_{NN}}=7.7\) GeV, and at top RHIC the suppression can reach an order of magnitude [2303.02279].

These time-resolved and geometry-resolved analyses underpin the PHQMD explanation of the “ice in the fire” puzzle. PHQMD studies report that deuterons remain at smaller transverse radii than free nucleons and “follow behind the front of the expanding baryonic fireball,” becoming spatially separated from the hotter and more collision-active front [2208.11802]. More generally, the baryons that end up in clusters are found, on average, closer to the reaction center and effectively behind the front of the fast expanding hadrons [2106.14839]. This suggests that the survival of weakly bound nuclei is attributed in PHQMD not to emission from a homogeneous thermal source but to late formation in special spatial regions where collisions have largely ceased.

## 6. Phenomenology, comparisons to experiment, and limitations

PHQMD has been benchmarked against both elementary and nucleus–nucleus data. In elementary inelastic \(p+p\) collisions, PHQMD with the PHSD-tuned string sector reproduces the energy dependence of \(\pi^+\) and \(K^+\) multiplicities reasonably well and provides predictions for \(p+n\) and \(n+n\) [1911.09496]. In Au+Au at \(\sqrt{s_{NN}}=11.5\) GeV, PHQMD reproduces the transverse-momentum spectra of \(\pi^+\), \(\pi^-\), \(K^+\), \(K^-\), \(p\), and \(\bar p\) across centrality classes reasonably well, while the proton slope remains slightly underestimated at high \(p_T\) [1911.09496].

For clusters and hypernuclei, the phenomenology spans AGS, SPS, RHIC-BES, RHIC fixed-target, and SIS energies. At AGS energy \(E_{\rm kin}=10.6\ A\)GeV, PHQMD describes E864 deuteron, triton, and \({}^3\)He rapidity and \(p_T\) spectra rather well and reproduces coalescence-like observables \(B_2\), \(B_3\), and the penalty factor up to \(A=4\) [2106.14839]. At SPS energy \(\sqrt{s_{NN}}=8.8\) GeV, it reproduces NA49 deuteron and \({}^3\)He rapidity and transverse-momentum spectra and the increase of cluster-production probability with transverse momentum [2106.14839]. Across RHIC-BES energies up to top RHIC, PHQMD reproduces the excitation functions of proton, antiproton, and deuteron midrapidity yields reasonably well, together with deuteron \(p_T\) spectra and \(B_2\) systematics, although it overpredicts deuterons at \(\sqrt{s_{NN}}=200\) GeV and overpredicts \(\bar d/\bar p\) by about a factor of two if antinuclei are read out at the same time as nuclei [2106.14839].

Conference-overview studies of midrapidity cluster formation reported that proton and antiproton midrapidity yields and deuteron \(dN/dy\) are well reproduced from the lowest SPS energies to the highest RHIC energies, that \(d/p\) is reproduced rather well, and that \(\bar d/\bar p\) has the correct shape as a function of \(\sqrt{s_{NN}}\) but is overestimated by about a factor of two because of overpredicted \(\bar d\) [2208.11802]. At \(\sqrt{s_{NN}}=3\) GeV, PHQMD describes the \(p_T\) spectra of \(p,d,t,{}^3\mathrm{He},{}^4\mathrm{He}\) well, underestimates the rapidity distribution of light clusters, reproduces the shape of the \({}^3_\Lambda\mathrm{H}\) and \({}^4_\Lambda\mathrm{H}\) \(p_T\) spectra, overestimates the differential yield of \({}^3_\Lambda\mathrm{H}\), and reproduces the yield of \({}^4_\Lambda\mathrm{H}\) [2208.11802].

At SIS energies, the focus shifted to flow. The momentum-dependent EoS study finds that the best description of HADES and FOPI data on the directed and elliptic flow coefficients of protons and light clusters is obtained with the momentum-dependent EoS, and it also reports a scaling behavior of \(v_2\) versus \(p_T\) with atomic number \(A\) [2411.04969]. The same work argues that flow observables can help identify cluster production mechanisms because deuterons from kinetic plus MST production differ from deuterons formed by coalescence when analyzed on the same underlying events [2411.04969].

PHQMD’s limitations are stated repeatedly. The approach is semiclassical, so clusters are not exact quantum ground states and can “spontaneously” lose nucleons; cluster multiplicities therefore depend somewhat on observation time unless stabilization procedures are applied [2208.11802, 2303.02279]. Hypernuclear yields remain sensitive to the poorly constrained \(\Lambda N\) interaction, often represented only by \(V_{N\Lambda}=\frac{2}{3}V_{NN}\) [2106.14839]. Earlier PHQMD versions used static momentum-independent Skyrme interactions and approximate relativistic extensions of QMD [1907.03860, 2303.02279]. Computational cost is also substantial: one study notes that extending cluster calculations from \(A=4\) to \(A=6\) would require roughly three orders of magnitude more CPU time [2106.14839].

Taken together, the PHQMD literature presents the model as a transport framework in which heavy-ion collisions, partonic and hadronic reaction dynamics, baryonic mean-field and correlation effects, and the emergence of nuclear clusters and hypernuclei are treated within a common microscopic scheme [1907.03860, 1911.09496]. Its enduring significance lies in the attempt to make cluster and hypernucleus formation a consequence of the transport dynamics itself rather than a purely external prescription, while retaining contact with bulk hadron phenomenology from SIS and AGS energies through SPS and RHIC [2106.14839, 2208.11802].

Source: https://www.emergentmind.com/topics/parton-hadron-quantum-molecular-dynamics-phqmd