# Collisional Broadening within a Hadronic Transport Approach

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## Abstract

**:**

## 1. Introduction

## 2. SMASH Transport Approach

## 3. Results

#### 3.1. Hadron Gas in Equilibrium

#### 3.2. Heavy-Ion Collisions

#### 3.3. Collisional Broadening under Different Vacuum Assumptions

- Particles that decay cannot be absorbed, so a larger vacuum width suppresses collisional broadening. At low masses, the vacuum decay width tapers down to 0 in the mass-dependent assumption. This makes the particles more prone to be absorbed by the medium in comparison with the mass-independent case.
- Inelastic cross-section ${\sigma}_{ab}$ affects the broadening of both a and b, since it determines how much one absorbs the other. It has peaks around the pole mass (${M}_{R}^{0}$) of possible resonances $ab\to R$ [10]. The masses of the incoming particles control the off-shell mass of the outgoing resonance (${m}_{R}=\sqrt{{s}_{ab}}$), so such peaks lead to structures in the collisional width of a and b, as exemplified by Figure 6; the contribution of the process $\rho N\to N\left(1520\right)$ is higher and close3 to ${M}_{N\left(1520\right)}^{0}-{m}_{N}=0.57\mathrm{GeV}$, and heavier resonances lead to peaks in larger ${m}_{\rho}$. This effect is not relevant for very small masses, when ${\mathcal{A}}_{R}^{\mathrm{vac}}\to 0$.
- Absorption cross-section ${\sigma}_{ab\to R}$ is also proportional to ${\Gamma}_{R}^{\mathrm{vac}}$, so that different mass assumptions give different weights to the resonance peaks.
- At high enough masses, the absorption cross-section decreases so much that particles stop undergoing collisional broadening, as detailed in Appendix A, such that the vacuum assumption has no effect.

## 4. Conclusions and Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Interaction Cross-Sections

**Figure A1.**Inelastic cross-sections of (

**a**) $\omega +p$ and (

**b**) $\omega +\pi $ scatterings in a thermal gas for different off-shell masses ${m}_{\omega}$.

**Figure A2.**Cross-sections of $p+p$ scatterings we use in a nuclear collision, including string fragmentation (“2-diff”, “1-diff”, and “non-diff”). Below is an enlargement of smaller contributions.

## Notes

1 | We consider stable the hadrons with ${\Gamma}_{0}\le 10\mathrm{keV}$. |

2 | In collisions with energies higher than $\sqrt{{s}_{NN}}\approx 3.5\mathrm{GeV}$, strange hadrons also come from string fragmentation (see Appendix A). |

3 | The difference from the actual peak is due to the kinetic energy given to the created resonance. |

## References

- Larionov, A.B.; Effenberger, M.; Leupold, S.; Mosel, U. Resonance lifetime in BUU: Observable consequences. Phys. Rev. C
**2002**, 66, 054604. [Google Scholar] [CrossRef] - Rapp, R.; Wambach, J. Low mass dileptons at the CERN SPS: Evidence for chiral restoration? Eur. Phys. J. A
**1999**, 6, 415–420. [Google Scholar] [CrossRef] - van Hees, H.; Rapp, R. Dilepton Radiation at the CERN Super Proton Synchrotron. Nucl. Phys. A
**2008**, 806, 339–387. [Google Scholar] [CrossRef] - Buss, O.; Gaitanos, T.; Gallmeister, K.; van Hees, H.; Kaskulov, M.; Lalakulich, O.; Larionov, A.B.; Leitner, T.; Weil, J.; Mosel, U. Transport-theoretical Description of Nuclear Reactions. Phys. Rept.
**2012**, 512, 1–124. [Google Scholar] [CrossRef] - Linnyk, O.; Bratkovskaya, E.L.; Ozvenchuk, V.; Cassing, W.; Ko, C.M. Dilepton production in nucleus-nucleus collisions at top SPS energy within the Parton-Hadron-String Dynamics (PHSD) transport approach. Phys. Rev. C
**2011**, 84, 054917. [Google Scholar] [CrossRef] - Endres, S.; van Hees, H.; Weil, J.; Bleicher, M. Coarse-graining approach for dilepton production at energies available at the CERN Super Proton Synchrotron. Phys. Rev. C
**2015**, 91, 054911. [Google Scholar] [CrossRef] - Endres, S.; van Hees, H.; Weil, J.; Bleicher, M. Dilepton production and reaction dynamics in heavy-ion collisions at SIS energies from coarse-grained transport simulations. Phys. Rev. C
**2015**, 92, 014911. [Google Scholar] [CrossRef] - Staudenmaier, J.; Weil, J.; Steinberg, V.; Endres, S.; Petersen, H. Dilepton production and resonance properties within a new hadronic transport approach in the context of the GSI-HADES experimental data. Phys. Rev. C
**2018**, 98, 054908. [Google Scholar] [CrossRef] - Hirayama, R.; Staudenmaier, J.; Elfner, H. Effective spectral function of vector mesons via lifetime analysis. Phys. Rev. C
**2023**, 107, 025208. [Google Scholar] [CrossRef] - Weil, J.; Steinberg, V.; Staudenmaier, J.; Pang, L.G.; Oliinychenko, D.; Mohs, J.; Kretz, M.; Kehrenberg, T.; Goldschmidt, A.; Bauchle, B.; et al. Particle production and equilibrium properties within a new hadron transport approach for heavy-ion collisions. Phys. Rev. C
**2016**, 94, 054905. [Google Scholar] [CrossRef] - Olive, K.A.; et al.; [Particle Data Group] Review of Particle Physics. Chin. Phys. C
**2014**, 38, 090001. [Google Scholar] [CrossRef] - Manley, D.M.; Saleski, E.M. Multichannel resonance parametrization of pi N scattering amplitudes. Phys. Rev. D
**1992**, 45, 4002–4033. [Google Scholar] [CrossRef] [PubMed] - Danielewicz, P.; Pratt, S. Delays associated with elementary processes in nuclear reaction simulations. Phys. Rev. C
**1996**, 53, 249–266. [Google Scholar] [CrossRef] - Bass, S.A.; Belkacem, M.; Bleicher, M.; Brandstetter, M.; Bravina, L.; Ernst, C.; Gerland, L.; Hofmann, M.; Hofmann, S.; Konopka, J.; et al. Microscopic models for ultrarelativistic heavy ion collisions. Prog. Part. Nucl. Phys.
**1998**, 41, 255–369. [Google Scholar] [CrossRef] - Li, Q.; Bleicher, M. A Model comparison of resonance lifetime modifications, a soft equation of state and non-Gaussian effects on pi-pi correlations at FAIR/AGS energies. J. Phys. G
**2009**, 36, 015111. [Google Scholar] [CrossRef] - Schumacher, D.; Vogel, S.; Bleicher, M. Theoretical analysis of dilepton spectra in heavy ion collisions at GSI-FAIR energies. Acta Phys. Hung. A
**2006**, 27, 451–458. [Google Scholar] [CrossRef] - Reichert, T.; Bleicher, M. Kinetic mass shifts of ρ(770) and K
^{*}(892) in Au+Au reactions at E_{b}eam = 1.23 AGeV. Nucl. Phys. A**2022**, 1028, 122544. [Google Scholar] [CrossRef] - Sjöstrand, T.; Ask, S.; Christiansen, J.R.; Corke, R.; Desai, N.; Ilten, P.; Mrenna, S.; Prestel, S.; Rasmussen, C.O.; Skands, P.Z. An introduction to PYTHIA 8.2. Comput. Phys. Commun.
**2015**, 191, 159–177. [Google Scholar] [CrossRef] - Agakishiev, G.; et al.; [HADES Collaboration] Study of dielectron production in C+C collisions at 1-A-GeV. Phys. Lett. B
**2008**, 663, 43–48. [Google Scholar] [CrossRef] - Adamczewski-Musch, J.; et al.; [HADES Collaboration] Centrality determination of Au + Au collisions at 1.23A GeV with HADES. Eur. Phys. J. A
**2018**, 54, 85. [Google Scholar] [CrossRef] - Vogel, S.; Bleicher, M. Reconstructing ρ 0 and ω mesons from nonleptonic decays in C+ C collisions at 2 GeV/nucleon in transport model calculations. Phys. Rev. C
**2006**, 74, 014902. [Google Scholar] [CrossRef] - Rapp, R.; Gale, C. ρ properties in a hot meson gas. Phys. Rev. C
**1999**, 60, 024903. [Google Scholar] [CrossRef]

**Figure 1.**Proper lifetime of the $\rho $ meson for a gas in equilibrium at different temperatures and baryochemical potential ${\mu}_{B}=400\mathrm{MeV}$.

**Figure 2.**Effective width of (

**a**) $\omega $ mesons and (

**b**) $\Delta \left(1232\right)$ baryons in thermal equilibrium.

**Figure 3.**Effective width of (

**a**) ${K}^{*}\left(892\right)$ and (

**b**) ${a}_{1}\left(1260\right)$ mesons in thermal equilibrium.

**Figure 4.**Effective width of the (

**a**) $\omega \left(782\right)$ meson and (

**b**) $\Delta \left(1232\right)$ baryon for different nuclear collision systems.

**Figure 5.**Effective width of the (

**a**) ${K}^{*}\left(892\right)$ and (

**b**) $\rho $ mesons for different nuclear collision systems.

**Figure 6.**Contributions of the 5 most significant absorption channels to the collisional width of $\rho $ in central Au+Au collisions at 1.23 GeV.

**Figure 7.**Collisional widths of (

**a**) $\rho $ and (

**b**) $\omega $ in thermal equilibrium under different vacuum decay assumptions.

**Figure 8.**Dynamical spectral function of the (

**a**) $\rho $ and (

**b**) $\omega $ mesons at different temperatures and baryochemical potential ${\mu}_{B}=400\mathrm{MeV}$ in thermal equilibrium under different vacuum decay assumptions.

${\mathbf{\Gamma}}^{\mathbf{eff}}\left[\mathbf{GeV}\right]$ | |||
---|---|---|---|

$\mathit{T}\left[\mathrm{MeV}\right]$ | 120 | 140 | 160 |

N | 0.063 (1) | 0.1082 (8) | 0.1789 (7) |

$\pi $ | 0.0802 (4) | 0.2033 (5) | 0.4376 (6) |

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**MDPI and ACS Style**

Balinovic, B.; Hirayama, R.; Elfner, H.
Collisional Broadening within a Hadronic Transport Approach. *Universe* **2023**, *9*, 414.
https://doi.org/10.3390/universe9090414

**AMA Style**

Balinovic B, Hirayama R, Elfner H.
Collisional Broadening within a Hadronic Transport Approach. *Universe*. 2023; 9(9):414.
https://doi.org/10.3390/universe9090414

**Chicago/Turabian Style**

Balinovic, Branislav, Renan Hirayama, and Hannah Elfner.
2023. "Collisional Broadening within a Hadronic Transport Approach" *Universe* 9, no. 9: 414.
https://doi.org/10.3390/universe9090414