Dynamically Constrained Phase-Space Coalescence (DCPC)

• DCPC is a phase-space coalescence model that employs dynamic constraints on spatial proximity, momentum, and invariant mass to reconstruct composite nuclei, hypernuclei, resonances, and exotic hadronic candidates.
• It integrates outputs from the PACIAE event generator using tailored kinematic and geometric cuts, replacing integrals with event-by-event summations to filter viable constituent combinations.
• DCPC has been applied to light nuclei, hypernuclei, and exotic states such as pentaquarks and glueballs, providing insights into yields, momentum spectra, and structure-sensitive observables in various collision systems.

Dynamically Constrained Phase-Space Coalescence (DCPC) is a phase-space coalescence model used to reconstruct composite nuclei, hypernuclei, resonances, and exotic hadronic candidates from microscopic event records. In the PACIAE+DCPC framework that dominates the published applications, PACIAE generates a final partonic state or final hadronic state, and DCPC then counts only those constituent combinations whose species content, spatial proximity, and invariant mass are compatible with a target bound state or resonance. The method is therefore an event-by-event constrained coalescence prescription rather than a purely global coalescence law (Ragab et al., 2019, Xu et al., 19 May 2026).

1. Conceptual basis and scope

DCPC is motivated by quantum statistical mechanics and by the uncertainty-principle statement that a particle occupies a finite cell in phase space. In the light-nucleus literature, this is written as

$\Delta\vec q\Delta\vec p\geq h^3,$

which underlies the interpretation of cluster production as counting admissible many-body phase-space configurations rather than imposing a single global proportionality between nucleus and constituent yields. This is why the method is presented as a more realistic version of naive coalescence: it acts on the actual event-by-event freeze-out configuration and imposes dynamical constraints on candidate clusters (She et al., 2015, Li et al., 2016, Ragab et al., 2019).

Across the available arXiv literature, the method first appears in applications to light (anti-)nuclei and (anti-)hypertriton production in heavy-ion and $pp$ collisions, and is then extended to a wide spectrum of exotic-hadron problems. These later applications include $\Omega_c^0$ states interpreted as $\Xi_c^+K^-$ bound states, $P_c$ states from $J/\psi p(\bar p)$, $\chi_{c1}(3872)$ from $J/\psi\pi^+\pi^-$, $X(2370)$ as glueball-, tetraquark-, or molecular-state candidates, $T_{cs0}^{*}(2870)^0$ from $pp$0, sexaquarks from three diquarks, and comparative $pp$1 versus $pp$2 production in compact and molecular scenarios. This breadth suggests that DCPC has become a general post-transport cluster-construction scheme whose essential input is a microscopic phase-space distribution rather than a specific particle species (Xu et al., 2019, Chen et al., 2021, Wu et al., 2023, She et al., 2024, Cao et al., 2024, She et al., 24 Feb 2025, Wang et al., 25 Jun 2025, Xu et al., 19 May 2026).

2. Formal structure of the coalescence prescription

In the surveyed papers, the basic yield formula is written as a phase-space integral. For a single particle,

$pp$3

while for an $pp$4-particle cluster,

$pp$5

A related notation used in exotic-state applications replaces the condition $pp$6 by an energy interval $pp$7, but the operational meaning is the same: only a restricted region of many-body phase space contributes to the cluster yield. Because PACIAE outputs discrete particle lists rather than continuous distributions, these integrals are replaced in practice by sums over candidate combinations in simulated events (Dong et al., 2018, Ragab et al., 2019, Wu et al., 2023).

The “dynamic constraint” is implemented through species selection together with kinematic and geometric cuts. In light-nucleus applications the standard form is

$pp$8

or, in some papers,

$pp$9

The invariant mass is evaluated as

$\Omega_c^0$0

Exotic-hadron studies often replace pairwise distances by constituent distances to the cluster center of mass, $\Omega_c^0$1, or by a maximum pairwise distance, $\Omega_c^0$2. For identical constituents, a combinatorial factor can appear explicitly; the two-gluon $\Omega_c^0$3 glueball construction uses $\Omega_c^0$4. Several implementations also specify iterative bookkeeping rules in which a used parton or hadron is removed from the list before the next search step, preventing reuse of the same constituent in multiple clusters (Xu et al., 2019, Chen et al., 2021, She et al., 2024, Cao et al., 2024, She et al., 24 Feb 2025).

3. Coupling to PACIAE and the event-by-event workflow

Most DCPC studies are performed on top of PACIAE, described in the literature as a parton-and-hadron cascade model based on PYTHIA 6.4 in PACIAE 3.0 and on PYTHIA 8.3 in PACIAE 4.0. The common workflow is: parton initiation, parton rescattering, hadronization, and hadron rescattering until freeze-out. In hadronic-cluster studies, DCPC is applied to the final hadronic state. In compact exotic-state studies, DCPC can instead be applied to the final partonic state. This distinction is central in later literature: compact tetraquark or glueball candidates are built at the partonic level, whereas molecular candidates are typically built from hadrons in the hadronic final state (Xu et al., 2019, She et al., 2024, Wang et al., 25 Jun 2025, Xu et al., 19 May 2026).

A recurring methodological feature is baseline tuning. Before coalescence is applied, PACIAE parameters are adjusted to reproduce hadron yields or spectra such as $\Omega_c^0$5, $\Omega_c^0$6, $\Omega_c^0$7, $\Omega_c^0$8, $\Omega_c^0$9, $\Xi_c^+K^-$0, and $\Xi_c^+K^-$1, depending on the study. Only after this transport baseline is validated are DCPC yields, spectra, rapidity distributions, ratios, or structure-sensitive observables reported. This architecture makes DCPC a post-processing filter whose predictive power is directly tied to the realism of the PACIAE phase-space output (She et al., 2019, Xu et al., 2019, Chen et al., 2021, Wu et al., 2023, Wang et al., 25 Jun 2025).

4. Light nuclei and hypernuclei applications

The canonical DCPC applications concern light nuclei and hypernuclei. In $\Xi_c^+K^-$2 collisions at $\Xi_c^+K^-$3 and $\Xi_c^+K^-$4 TeV, PACIAE+DCPC reproduces the ALICE yields, ratios, and transverse-momentum distributions of $\Xi_c^+K^-$5 and $\Xi_c^+K^-$6, and predicts $\Xi_c^+K^-$7, $\Xi_c^+K^-$8, $\Xi_c^+K^-$9, and $P_c$0. At $P_c$1 TeV the reported integrated yields are

$P_c$2

$P_c$3

$P_c$4

The same study emphasizes that yields decrease sharply with increasing mass number $P_c$5, spanning about three orders of magnitude from $P_c$6 to $P_c$7, and reports the strangeness population factor $P_c$8 and $P_c$9 to be about $J/\psi p(\bar p)$0 in these $J/\psi p(\bar p)$1 systems (Ragab et al., 2019).

In heavy-ion systems, DCPC is used to study centrality, beam-energy, and system-size dependences. In Pb–Pb at $J/\psi p(\bar p)$2 TeV, yields of $J/\psi p(\bar p)$3, $J/\psi p(\bar p)$4, and $J/\psi p(\bar p)$5 decrease rapidly toward peripheral collisions, whereas anti-particle to particle ratios are nearly independent of centrality; the same body of work reports that $J/\psi p(\bar p)$6 is approximately constant from central to peripheral collisions and that $J/\psi p(\bar p)$7 for hypertriton is smaller than for $J/\psi p(\bar p)$8He, indicating a strangeness penalty factor (She et al., 2015). In Au–Au collisions from $J/\psi p(\bar p)$9 to $\chi_{c1}(3872)$0 GeV, PACIAE+DCPC finds a nonmonotonic energy dependence of $\chi_{c1}(3872)$1 with a minimum near $\chi_{c1}(3872)$2 GeV, and $\chi_{c1}(3872)$3 rise below $\chi_{c1}(3872)$4 GeV and then saturate around $\chi_{c1}(3872)$5 at higher energies; the authors identify a transition point near $\chi_{c1}(3872)$6 GeV and discuss $\chi_{c1}(3872)$7 and $\chi_{c1}(3872)$8 as possible probes of QCD critical phenomena (Dong et al., 2018).

The same framework has also been used to test whether cluster observables are sensitive to the chiral magnetic effect in isobaric $\chi_{c1}(3872)$9 and $J/\psi\pi^+\pi^-$0 collisions at $J/\psi\pi^+\pi^-$1 GeV. The reported conclusion is that yields, antiparticle-to-particle ratios, $J/\psi\pi^+\pi^-$2, and strangeness population factors are very similar for Ru+Ru and Zr+Zr, implying insensitivity to the modeled CME input at the level studied. Representative values given are $J/\psi\pi^+\pi^-$3 and $J/\psi\pi^+\pi^-$4 in Ru+Ru, and $J/\psi\pi^+\pi^-$5 and $J/\psi\pi^+\pi^-$6 in Zr+Zr (She et al., 2020). A distinct but related application to Pb–Pb nuclear modification factors shows that DCPC-generated $J/\psi\pi^+\pi^-$7 inherit medium-modified constituent spectra, leading to $J/\psi\pi^+\pi^-$8 patterns similar in gross structure to those of $J/\psi\pi^+\pi^-$9 and $X(2370)$0, with stronger suppression in central than in peripheral collisions (She et al., 2019).

5. Extension to resonances and exotic hadrons

Later work generalizes DCPC far beyond nuclear coalescence. In $X(2370)$1 collisions at $X(2370)$2 and $X(2370)$3 TeV, the five narrow $X(2370)$4 states are modeled as $X(2370)$5 bound states satisfying $X(2370)$6 with $X(2370)$7 fm and a mass window $X(2370)$8; the predicted yields place $X(2370)$9 as the most copiously produced state. Using the same logic, $T_{cs0}^{*}(2870)^0$0, $T_{cs0}^{*}(2870)^0$1, and $T_{cs0}^{*}(2870)^0$2 are built from $T_{cs0}^{*}(2870)^0$3 pairs, with yields of order $T_{cs0}^{*}(2870)^0$4 and a structure hierarchy in which nucleus-like configurations give the largest rates, molecular configurations intermediate rates, and pentaquark configurations the smallest rates (Xu et al., 2019, Chen et al., 2021).

A second cluster of studies uses DCPC as a structure discriminator. For non-prompt $T_{cs0}^{*}(2870)^0$5 from beauty-hadron decays, the same hadronic constituents $T_{cs0}^{*}(2870)^0$6 are retained while the structural hypothesis is encoded in $T_{cs0}^{*}(2870)^0$7: $T_{cs0}^{*}(2870)^0$8 fm for a compact tetraquark, $T_{cs0}^{*}(2870)^0$9 fm for a nuclear-like state, and $pp$00 fm for a molecular state, all with $pp$01 MeV. The reported result is that the compact tetraquark scenario gives the best agreement with LHCb and ATLAS measurements of the non-prompt $pp$02 ratio (Wu et al., 2023). For $pp$03, molecular candidates are reconstructed from $pp$04, $pp$05, and $pp$06, and the qualitative ordering

$pp$07

is found across rapidity, $pp$08, and angular observables (Wang et al., 25 Jun 2025).

A third line of work applies DCPC directly to the final partonic state. $pp$09 is treated either as a two-gluon glueball candidate in the partonic final state or as a multi-meson molecular candidate in the hadronic final state; the reported yields and kinematic distributions differ significantly between these scenarios, and in a later study the comparison is extended to glueball-, tetraquark-, baryon-antibaryon-molecular, and three-meson-molecular hypotheses in both $pp$10 and $pp$11 collisions (She et al., 2024, Cao et al., 2024). The sexaquark $pp$12 is built in two steps, first forming diquarks $pp$13, $pp$14, and $pp$15 in the final partonic state and then coalescing them with DCPC; the compact sexaquark yield is found to be about two orders of magnitude smaller than that of the hadronic molecule $pp$16 under the chosen parameter sets (She et al., 24 Feb 2025). In a comparative study of $pp$17 and $pp$18, compact tetraquark states are formed at the partonic level and loose molecular states at the hadronic level, with tetraquark/molecular yield ratios of $pp$19 for $pp$20, $pp$21 for $pp$22, and $pp$23 for $pp$24 (Xu et al., 19 May 2026).

6. Interpretation, observables, and methodological limitations

The observables extracted from DCPC are broader than integrated yields alone. Depending on the application, the literature reports transverse-momentum spectra, rapidity distributions, antiparticle-to-particle ratios, mixed hypernucleus-to-nucleus ratios, nuclear modification factors, coalescence parameters $pp$25, strangeness population factors $pp$26 and $pp$27, charged-state asymmetries, and angular distributions such as

$pp$28

A consistent pattern is that DCPC tends to preserve the gross kinematic shape of the underlying constituent distributions while filtering their normalization and relative channel populations through phase-space compatibility. This is stated explicitly in the $pp$29 study, where DCPC is described as mainly filtering pre-existing $pp$30 pairs rather than introducing a new momentum scale (Xu et al., 2019, Wang et al., 25 Jun 2025, Xu et al., 19 May 2026).

At the same time, the published literature treats DCPC as a phenomenological rather than first-principles bound-state calculation. Several papers explicitly describe $pp$31, $pp$32, and $pp$33 as free or tunable parameters, often fixed from data or scanned over broad intervals; one study states that no systematic uncertainty analysis is presented, and another shows that yield estimates can depend substantially on whether orbital angular momentum is assigned with $pp$34 or $pp$35 (Wang et al., 25 Jun 2025, Cao et al., 15 Sep 2025). A further limitation is that DCPC, at least in the formulations displayed for $pp$36, does not perform a full quantum-number projection; constituent choice, spatial cuts, and invariant-mass windows are used as the operative filters (Cao et al., 2024). The resulting picture is that DCPC is best understood as a structure-sensitive post-transport coalescence algorithm: more restrictive than naive coalescence, strongly informative when coupled to a realistic event generator, but dependent on phenomenological geometric and mass-window inputs rather than derived from an ab initio few-body dynamics.