# Fast resonance decays in nuclear collisions

## Abstract

In the context of ultra-relativistic nuclear collisions, we present a fast method for calculating the final particle spectra after the direct decay of resonances from a Cooper–Frye integral over the freeze-out surface. The method is based on identifying components of the final particle spectrum that transform in an irreducible way under rotations in the fluid-restframe. Corresponding distribution functions can be pre-computed including all resonance decays. Just a few of easily tabulated scalar functions then determine the Lorentz invariant decay spectrum from each space-time point, and simple integrals of these scalar functions over the freeze-out surface determine the final decay products. This by-passes numerically costly event-by-event calculations of the intermediate resonances. The method is of considerable practical use for making realistic data to model comparisons of the identified particle yields and flow harmonics, and for studying the viscous corrections to the freeze-out distribution function.

## 1 Introduction

Ultra-relativistic heavy ion collisions create the deconfined state of matter called the Quark–Gluon Plasma (QGP), which has been under intensive experimental and theoretical research in the last two decades [1, 2]. Remarkably, the expanding QGP has been very successfully described as a relativistic fluid, where the system dynamics is completely determined by a few macroscopic fields like fluid velocity \(u^\mu (x)\) or temperature *T*(*x*) [3, 4, 5, 6, 7]. As the fluid expands and cools down below the cross-over temperature \(T_c\approx 155\,\text {MeV}\), quarks and gluons are re-confined in hadronic degrees of freedom. Therefore a systematic comparison between the hydrodynamic models of the QGP and experimental data necessitates the conversion of hydrodynamic fields into hadronic degrees of freedom.

Various techniques of treating the hadronic phase have been developed over the years. Resonances are sampled at the freeze-out surface using the Cooper–Frye formula [8] and then passed to hadronic transport models, which describe both the decays and possible rescatterings of resonances [9, 10]. However direct resonance decays (without rescatterings) are often used in phenomenological studies [11, 12, 13, 14]. The decay processes of resonances are simulated by Monte-Carlo generators [15, 16, 17], or by semi-analytic treatments of decay integrals [18, 19]. From \(\sim 300\) species of hadronic resonances produced in high energy nuclear collisions, only a handful long-lived hadrons (e.g. pion, kaons and protons) reach the particle detectors and are directly observed [20, 21]. In this paper we show how to by-pass the numerically costly procedure of calculating the intermediate resonance decays and to relate directly the hydrodynamic fields on the freeze-out surface to the final decay particle spectra.

Let us remark here that semi-analytic description of resonance decays was studied previously [18, 19, 22, 23]. While these works constitute the basis of our approach, our framework is applicable to arbitrary freeze-out surfaces and more general particle distribution functions.

In Sect. 2 we describe a particle decay process as a linear Lorentz invariant map, which transforms the spectrum of initial (or primary) particles to the spectrum of final decay products. In Sect. 3 we argue that the decay map can be applied directly to the particle distribution function *before* performing the Cooper–Frye integral and we define a distribution function for the decay products. Using group theoretical arguments we find the Lorentz invariant decomposition of the decay particle distribution functions as a sum of frame-independent *weight functions*, which we calculate. In Sect. 4 we show that the same procedure also applies to viscous and linear perturbations of particle distribution function on the freeze-out hypersurface. Then in Sect. 5 we discuss the implementation of fast resonance decay procedure for general freeze-out surfaces and phenomenologically convenient setups of blast-wave freeze-out and mode-by-mode hydrodynamics. We end with discussion of further extensions and applications in Sect. 6. Finally Appendix A gives the derivation of the irreducible decay spectrum components.

## 2 Lorentz invariant decay map

Relativistic particle decays is a well established subject [18, 19, 21, 24, 25, 26] and here we briefly discuss some of the key formulas. In kinetic theory, decays can be treated as \(1\leftrightarrow n\) particle scatterings. The probability for such an event is given by a Lorentz invariant integral over the scattering matrix squared \(|\mathcal M|^2\), the available momentum phase space (constrained by 4-momentum conservation) and the phase space densities of initial and final particles, i.e. the gain and loss terms. In chemical and thermal equilibrium both the decay and the reverse process are equally likely, however, if the system becomes dilute and falls out of the detailed balance, multi-particle scatterings become very improbable and the \(1\rightarrow n\) decays, which are *linear* in the initial particle densities, dominates the subsequent phase space evolution of particles. This is exactly what happens for hadron resonances in heavy ion collisions below the freeze-out temperature. At sufficiently late times, all allowed decays will have taken place and the Lorentz invariant spectrum of the final particle species *b* will be proportional to the primary populations of resonance species *a*, which decay (directly or through intermediate resonances) to particles of type *b*.^{1}

^{2}

*a*with momentum \(\mathbf{q}\) to decay to a particle

*b*with corresponding momentum \(\mathbf{p}\). Summing over all species of primary resonances

*a*then gives the

*total decayed particle spectrum*of particle species

*b*. We note in passing that the decay map \(D_b^a(\mathbf{p},\mathbf{q})\) fulfils certain sum rules as a consequence of conservation laws for energy, momentum, net baryon number, electric charge, etc.

*B*. For the simple case of

*isotropic*two-body decay \(a\rightarrow b+c\) the phase space integral of the decay partner

*c*can be done analytically and the map \(D^a_{b|c}\) is reduced to a Lorentz invariant delta function of the product of initial and final particle momenta \(p^\mu q_\mu \) [19, 21, 24]

*B*is the branching ratio for this process. In the rest-frame of particle

*a*, Eq. (2) is simply a uniform probability distribution on a sphere with radius \(|\mathbf{p}|=p^{a}_{b|c}\) fixed by energy conservation

*a*. However treating the two partner particles

*c*and

*d*as a fictitious particle \(\tilde{c}\) with an effective mass \(m_{\tilde{c}}^2=-(p_c+p_d)^2\), the three body decay map \(D_{b|cd}^a\) can be written as an integral of the 2-body decay map for the allowed values of \(m_{\tilde{c}}\) [19, 21, 24]

*b*in the rest-frame of

*a*and \(p^{\tilde{c}}_{c|d}\) is the momentum of particle

*c*or

*d*in their common rest-frame [21].

*w*.

^{3}

The decay chain map \(D^a_{b}(q^\mu p_\mu )\) is independent of initial particle spectrum and only depends on particle properties and branching ratios listed in the particle data book [21]. This means that the main computational cost is in the evaluation of the decay map, which only needs to be done once, and then the final decay spectrum can be computed from an arbitrary initial particle spectrum according to Eq. (1). In particular, more finer details of resonance decay processes could be thus efficiently treated. The primary example is a finite width of resonances, which can be included in the decay map as an additional integral over resonance mass with, for example, Breit-Wigner distribution [15]. The formalism may also be generalized to anisotropic as well as spin-dependent decays. However, even ignoring these additional complications, the sheer number of primary resonances in heavy ion collisions makes the numerical evaluation of Eq. (1) a burden. Therefore we will now specialize to the decays of initial resonance spectrum specified by a common freeze-out procedure and will leave the inclusions of resonance widths and other improvements of the decay map for future work.

## 3 Cooper–Frye for the final decay spectrum

^{4}

*T*(

*x*), flow velocity \(u^\mu (x)\), and chemical potential \(\mu (x)\). We will discuss more general initial particle distributions arising in dissipative hydrodynamics in Sect. 4.

*vector distribution function*\(g^\mu \), which for the primary resonances is \(g^\mu _a = f_a p^\mu \), while for the decay products it is given by

^{5}and some Lorentz scalars, e.g. temperature

*T*or chemical potentials \(\mu \), then by Lorentz invariance of the decay process, the vector distribution function before and after the decay integral in Eq. (9) can be uniquely written as a sum of two scalar functions

*T*, and decay parameters. In the fluid-restframe, 4-vectors \(\bar{E}_\mathbf{p}u^\mu =(\bar{E}_\mathbf{p}, \mathbf {0})\) and \(p^\mu -\bar{E}_\mathbf{p}u^\mu =(0, \bar{\mathbf{p}})\) are two irreducible SO(3) representations transforming under rotations as a scalar and a vector, respectively. The decay operator in Eq. (9) is a linear map and therefore guarantees that \(f^\text {eq}_1\) and \(f^\text {eq}_2\) components do not mix during (isotropic) decays. The initial hadrons on the freeze-out surface are initialized by \(g^\mu _a = f_a p^\mu \) and for the equilibrium distribution function both components \(f_1^\text {eq}\) and \(f_2^\text {eq}\) are initialized to be either Bose-Einstein or Fermi-Dirac distributions

*b*in the fluid-restframe. The isotropic three body decays \(a\rightarrow b+c+d\) can be easily incorporated by integrating the 2-body transformation rules Eq. (12) over the effective decay partner mass \(m_{\tilde{c}}\) as in Eq. (4). Such one-dimensional integrals can be easily done by standard numerical integration routines [30].

^{6}We see that the \(f^\text {eq}_2\) component, which gives the sole contribution to the particle spectra for time-like (fixed time) freeze-out surface, is larger than thermal pion distribution due to feed-down from resonance decays, while the space-like component \(f^\text {eq}_2\) remains of the same size. In an arbitrary reference frame the decay pion spectrum can be straightforwardly calculated using frame independent formulas Eq. (10) and Eq. (8). We will discuss this further in Sect. 5.

In Fig. 3 we plot the final pion spectra \(\pi ^+\) for a simple freeze-out surface with a constant Bjorken time, freeze-out temperature and radial velocity.^{7} In addition to the total pion spectrum (which includes all decay chains producing \(\pi ^+\)), we also show the pion spectrum from the dominant decay channels \(\rho ^{+,-}\rightarrow \pi ^++\pi ^{0,-}\) and \(\omega ^0\rightarrow \pi ^++\pi ^-+\pi ^0\) (where \(\rho ^{+,0}\) and \(\omega ^0\) spectra themselves include decay contributions from yet heavier resonances). We compare our results with the decay pion spectrum generated by a Monte-Carlo resonance decay generator THERMINATOR 2 [17]. All spectra are in excellent agreement, however we would like to stress that the decay pion spectrum in Fig. 3 is obtained immediately from a simple Cooper–Frye freeze-out procedure Eq. (8). The vector distribution function components \(f_i^\text {eq}\) shown in Fig. 2 only need to calculated once for a particular freeze-out temperature \(T_\text {fo}\) and then can applied to any shape of the freeze-out surface or the fluid velocity field \(u^\mu (x)\), without the need of costly calculations of intermediate resonances.

## 4 Viscous and linear corrections to particle spectrum

*m*– mass of the primary resonance, and \(\tau _\Pi /\zeta \) is the ratio of bulk relaxation time and bulk viscosity.

^{8}Different lines in Fig. 4 correspond to different contributions stemming from components \(f_i\) in Eq. (23) and Eq. (24). The labels next to the lines indicate the required factors for the Cooper–Frye freeze-out integral in the fluid-restframe. Note that we factored in \(|\bar{\mathbf{p}}|^2\) for the shear perturbations. We also factored out the terms proportional to the transport coefficients, so any (small) viscous perturbation will produce the same correction to the particle spectrum (up to the magnitude) in the local fluid-restframe. However the presence of such viscous corrections in the hydrodynamic evolution modifies the fluid velocity and temperature fields, and therefore the freeze-out surface will itself be different. Then evaluating the generalized Cooper–Frye freeze-out integral Eq. (8) will yield different particle spectrum.

^{9}

## 5 Application to blast-wave and mode-by-mode freeze-out

*y*.

*y*and angle \(\varphi \) disappears after the integration. The integral expressions for viscous kernels are given in Appendix B. For the simple constant time freeze-out surface used in Fig. 3 only the temporal part of the freeze-out surface contributes and the decay pion spectrum is proportional to \(K_1^\text {eq}(p_T,\bar{\chi }=\arctan v_T)\).

*y*via the combination \(e^{im\varphi +iky}\) with the same azimuthal wavenumber

*m*and rapidity wavenumber

*k*. This is a direct consequence of \(\text {U}(1)\times \mathbb {R}^1\) symmetry, which prevents different representations labelled by

*m*and

*k*from mixing under linear operations. This way arbitrary linear perturbations in fluid fields can be mapped to the modes of the final particle spectra, which can be straightforwardly incorporated in the formalism of mode-by-mode hydrodynamics [34, 35].

## 6 Conclusions

We presented a method to calculate the final decay spectrum of direct resonance decays *directly* from hydrodynamic fields on a freeze-out surface. By applying the decay map, Eq. (1), to the distribution function of primary particles *before* the Cooper–Frye integration, we found the (vector) distribution function of decay products, Eq. (9). By decomposing this distribution into components transforming differently under SO(3) rotations in the fluid-restframe, we expressed the final decay particle spectrum as a sum of a few Lorentz invariant weight functions and known Lorentz vectors. The explicit procedure to determine the irreducible weight functions for an arbitrary decay chain of isotropic 2-body and 3-body decays was derived and a numerical implementation was made public [30]. We considered primary hadron resonances generated by the equilibrium distribution function, viscous shear and bulk perturbations, and linearised temperature and velocity perturbations. Modifications to the particle spectrum due to variations in the chemical potential and the diffusive part of particle current can be also straightforwardly included in this framework.

The final 1-body particle spectrum of decay products is then calculated from a general Cooper–Frye-type freeze-out integral, Eq. (32). The most important aspect of our method is that intermediated particle decays do not need to be calculated event-by-event. The irreducible components of the decay particle distribution function Eq. (9) are computed only once, and the spectrum of a few relevant hadron species, which includes feed-down of all direct decays, can be computed for an arbitrary freeze-out surface. This significantly reduces the computational costs of direct resonance decays.

Although our method of calculating direct resonance decays is already competitive with other treatments available on the market, the computational efficiency of our approach makes it practical to include finer details of resonances decays. For example, new hadron resonance states can be easily added to improve the agreement between the lattice QCD and hadronic equation of state [11]. Finite widths of the resonances can be incorporated in the decay map [15, 40, 41]. This has recently been shown to reduce the discrepancy between measured and predicted proton yield in the statistical hadronization models [42].

In this work we neglected hadronic rescatterings after the chemical freeze-out, which may change the final particle spectra, but the effect is subleading in comparison with the decay feed-down [43]. Elastic scatterings in the hadronic phase can be modeled by a hydrodynamic evolution of hadron fluid in partial chemical equilibrium [44, 45]. In this scenario, the particle ratios are fixed at the chemical freeze-out temperature \(T_\text {chem}\) for each species *i* of long lived hadrons by introducing an appropriate chemical potentials \(\mu _i(T)\) for \(T<T_\text {chem}\). Subsequently, the kinetic freeze-out may take place at some lower temperature \(T_\text {kin}\). Because the primary resonance spectra are still described only by temperature \(T_\text {kin}\) and the chemical potentials \( \mu _i(T_\text {kin})\), the direct decays can be calculated using the techniques proposed in this work.

Another interesting generalization of the framework is to keep track of the particle spin in the decays. This could be particularly useful for the studies of vorticity polarization in heavy ion collisions [46, 47]. However, in this case one might need to go beyond isotropic *s*-wave decays and consider more general momentum dependent decay patterns, which to our knowledge were not included in phenomenological works so far. Finally, we note that the 1-particle distribution function does not have the information of the connected two-particle function, namely the non-flow correlations of particles produced by the same resonance decay. However, the decay map Eq. (1) can be generalized to two-particle spectrum.

In summary, we believe that the computationally efficient way of computing direct resonance decays, which was presented in this paper, will be of great practical utility for phenomenological studies of heavy ion collisions and make realistic particle yield calculations much more affordable.

## Footnotes

- 1.
- 2.
We also note here that the two-particle distribution \(E_\mathbf{p}E_\mathbf{q}dN_{bc}/(d^3\mathbf{p}d^3\mathbf{q})\) inherits correlations from a decay \(a\rightarrow b+c\) as a consequence of energy and momentum conservation. However, in the present paper we only concentrate on one-particle distributions.

- 3.
In the rest-frame of the particle

*b*the variable*w*has the physical interpretation as the fraction of particle’s*a*momentum in the direction of the fluid velocity. The limit of the massless final state particle \(m_b=0\) can be treated by a simple change of variables \(u=(1-w)\frac{m_e^2}{m_b^2}\). - 4.
The hypersurface element is \(d\sigma _\mu =d^3x\sqrt{h}n_\mu \), where

*h*is the determinant of the induced metric on the freeze-out surface, \(d^3x\sqrt{h}\) is the invariant volume element and \(n^\mu \) is a normal vector on the surface, which we take to be pointing inwards. In this work we use mostly positive metric convention. - 5.
In principle \(u^\mu \) could be any time-like vector, not necessarily associated with a fluid flow.

- 6.
For simplicity of comparison, we used the default list of decay chains included in THERMINATOR 2 package [17], which includes strong and weak decays of hadron resonances with mass \(<2.5\text {GeV}\).

- 7.
We used the following THERMINATOR 2 options for the freeze-out surface: \(\tau _\text {fo}=8.17\,\text {fm}\), \(T_\text {fo}=145\,\text {MeV}\), radius of the surface \(R=8.21\,\text {fm}\) and a constant radial velocity \(v_T=0.341\).

- 8.
- 9.
Note that a constant temperature freeze-out surface depends on \(\delta T\). However, one can also consider a freeze-out at a constant

*background*temperature \(\bar{T}\), which is then independent of the perturbation.

## Notes

### Acknowledgements

The authors would like to thank Gabriel Denicol, Ulrich Heinz, Jacquelyn Noronha-Hostler, Jean-François Paquet, and Iurii Karpenko for useful discussions and comments. This work is part of and supported by the DFG Collaborative Research Centre SFB 1225 (ISOQUANT) (S.F., E.G., A.M.). This work was supported in part by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics under Award Numbers DE–FG02–88ER40388 (D.T.).

## References

- 1.W. Busza, K. Rajagopal, W. van der Schee, Ann. Rev. Nucl. Part. Sci.
**68**, 339 (2018). https://doi.org/10.1146/annurev-nucl-101917-020852. arXiv:1802.04801 [hep-ph]ADSCrossRefGoogle Scholar - 2.U.W. Heinz, Nucl. Phys. A
**685**, 414 (2001). https://doi.org/10.1016/S0375-9474(01)00558-9. arXiv:hep-ph/0009170 ADSCrossRefGoogle Scholar - 3.U. Heinz, R. Snellings, Ann. Rev. Nucl. Part. Sci.
**63**, 123 (2013). https://doi.org/10.1146/annurev-nucl-102212-170540. arXiv:1301.2826 [nucl-th]ADSCrossRefGoogle Scholar - 4.D.A. Teaney. https://doi.org/10.1142/9789814293297_0004. arXiv:0905.2433 [nucl-th]CrossRefGoogle Scholar
- 5.M. Luzum, H. Petersen, J. Phys. G
**41**, 063102 (2014). https://doi.org/10.1088/0954-3899/41/6/063102. arXiv:1312.5503 [nucl-th]ADSCrossRefGoogle Scholar - 6.C. Gale, S. Jeon, B. Schenke, Int. J. Mod. Phys. A
**28**, 1340011 (2013). https://doi.org/10.1142/S0217751X13400113. arXiv:1301.5893 [nucl-th]ADSCrossRefGoogle Scholar - 7.R. Derradi de Souza, T. Koide, T. Kodama, Prog. Part. Nucl. Phys.
**86**, 35 (2016). https://doi.org/10.1016/j.ppnp.2015.09.002. arXiv:1506.03863 [nucl-th]ADSCrossRefGoogle Scholar - 8.F. Cooper, G. Frye, Phys. Rev. D
**10**, 186 (1974). https://doi.org/10.1103/PhysRevD.10.186 ADSCrossRefGoogle Scholar - 9.S.A. Bass et al., Prog. Part. Nucl. Phys.
**41**, 255 (1998). (Prog. Part. Nucl. Phys.**41**(1998) 225). https://doi.org/10.1016/S0146-6410(98)00058-1. arXiv:nucl-th/9803035 ADSCrossRefGoogle Scholar - 10.H. Petersen, D. Oliinychenko, M. Mayer, J. Staudenmaier, S. Ryu, Nucl. Phys. A
**982**, 399 (2019). https://doi.org/10.1016/j.nuclphysa.2018.08.008. arXiv:1808.06832 [nucl-th]ADSCrossRefGoogle Scholar - 11.P. Alba, V. Mantovani Sarti, J. Noronha, J. Noronha-Hostler, P. Parotto, I. Portillo Vazquez, C. Ratti, Phys. Rev. C
**98**(3), 034909 (2018). https://doi.org/10.1103/PhysRevC.98.034909. arXiv:1711.05207 [nucl-th]ADSCrossRefGoogle Scholar - 12.K.J. Eskola, H. Niemi, R. Paatelainen, K. Tuominen, Phys. Rev. C
**97**(3), 034911 (2018). https://doi.org/10.1103/PhysRevC.97.034911. arXiv:1711.09803 [hep-ph]ADSCrossRefGoogle Scholar - 13.H. Niemi, K.J. Eskola, R. Paatelainen, Phys. Rev. C
**93**(2), 024907 (2016). https://doi.org/10.1103/PhysRevC.93.024907. arXiv:1505.02677 [hep-ph]ADSCrossRefGoogle Scholar - 14.P. Bozek, W. Broniowski, G. Torrieri, Phys. Rev. Lett.
**111**, 172303 (2013). https://doi.org/10.1103/PhysRevLett.111.172303. arXiv:1307.5060 [nucl-th]ADSCrossRefGoogle Scholar - 15.G. Torrieri, S. Steinke, W. Broniowski, W. Florkowski, J. Letessier, J. Rafelski, Comput. Phys. Commun.
**167**, 229 (2005). https://doi.org/10.1016/j.cpc.2005.01.004. arXiv:nucl-th/0404083 ADSCrossRefGoogle Scholar - 16.N.S. Amelin, R. Lednicky, T.A. Pocheptsov, I.P. Lokhtin, L.V. Malinina, A.M. Snigirev, I.A. Karpenko, Y.M. Sinyukov, Phys. Rev. C
**74**, 064901 (2006). https://doi.org/10.1103/PhysRevC.74.064901. arXiv:nucl-th/0608057 ADSCrossRefGoogle Scholar - 17.M. Chojnacki, A. Kisiel, W. Florkowski, W. Broniowski, Comput. Phys. Commun.
**183**, 746 (2012). https://doi.org/10.1016/j.cpc.2011.11.018. arXiv:1102.0273 [nucl-th]ADSCrossRefGoogle Scholar - 18.J. Sollfrank, P. Koch, U.W. Heinz, Z. Phys, C
**52**, 593 (1991). https://doi.org/10.1007/BF01562334 CrossRefGoogle Scholar - 19.J. Sollfrank, P. Koch, U.W. Heinz, Phys. Lett. B
**252**, 256 (1990). https://doi.org/10.1016/0370-2693(90)90870-C ADSCrossRefGoogle Scholar - 20.B. Abelev et al. [ALICE Collaboration]. Phys. Rev. Lett.
**109**, 252301 (2012). https://doi.org/10.1103/PhysRevLett. arXiv:1208.1974 [hep-ex] - 21.M. Tanabashi et al., [Particle Data Group], Phys. Rev. D
**98**(3), 030001 (2018). https://doi.org/10.1103/PhysRevD.98.030001 - 22.F. Becattini, G. Passaleva, Eur. Phys. J. C
**23**, 551 (2002). https://doi.org/10.1007/s100520100869. arXiv:hep-ph/0110312 ADSCrossRefGoogle Scholar - 23.W. Broniowski, W. Florkowski, Phys. Rev. C
**65**, 064905 (2002). https://doi.org/10.1103/PhysRevC.65.064905. arXiv:nucl-th/0112043 ADSCrossRefGoogle Scholar - 24.E. Byckling, K. Kajantie, Particle kinematics (University of Jyvaskyla, Jyvaskyla, Finland, 1971)Google Scholar
- 25.S.R. De Groot, W.A. Van Leeuwen, C.G. Van Weert (Elsevier, North-holland, 1980), p. 417Google Scholar
- 26.L.D. Landau, E.M. Lifschits, The classical theory of fields, course of theoretical physics, vol. 2 (Pergamon Press, Oxford, 1975)Google Scholar
- 27.K. Aamodt et al., [ALICE Collaboration]. Eur. Phys. J. C
**71**, 1655 (2011). https://doi.org/10.1140/epjc/s10052-011-1655-9. arXiv:1101.4110 [hep-ex] - 28.A. Kalweit [ALICE Collaboration], J. Phys. G
**38**, 124073, (2011). https://doi.org/10.1088/0954-3899/38/12/124073. arXiv:1107.1514 [hep-ex]ADSCrossRefGoogle Scholar - 29.V.V. Anisovich, M.N. Kobrinsky, YuM Shabelski, J. Nyiri,
*Quark Model and High Energy Collisions*(World Scientic, New Jersey, 2004)CrossRefGoogle Scholar - 30.A. Mazeliauskas, S. Floerchinger, E. Grossi, D. Derek, FastReso, GitHub repository (2018) https://github.com/amazeliauskas/FastReso. Accessed 1 Oct 2018
- 31.D. Teaney, Phys. Rev. C
**68**, 034913 (2003). https://doi.org/10.1103/PhysRevC.68.034913. arXiv:nucl-th/0301099 ADSCrossRefGoogle Scholar - 32.J.F. Paquet, C. Shen, G.S. Denicol, M. Luzum, B. Schenke, S. Jeon, C. Gale, Phys. Rev. C
**93**(4), 044906 (2016). https://doi.org/10.1103/PhysRevC.93.044906. arXiv:1509.06738 [hep-ph]ADSCrossRefGoogle Scholar - 33.S. Borsanyi et al., Nature
**539**(7627), 69 (2016). https://doi.org/10.1038/nature20115. arXiv:1606.07494 [hep-lat]ADSCrossRefGoogle Scholar - 34.S. Floerchinger, U.A. Wiedemann, Phys. Lett. B
**728**, 407 (2014). https://doi.org/10.1016/j.physletb.2013.12.025. arXiv:1307.3453 [hep-ph]ADSCrossRefGoogle Scholar - 35.S. Floerchinger, U.A. Wiedemann, Phys. Rev. C
**89**(3), 034914 (2014). https://doi.org/10.1103/PhysRevC.89.034914. arXiv:1311.7613 [hep-ph]ADSCrossRefGoogle Scholar - 36.S. Floerchinger, M. Martinez, Phys. Rev. C
**92**(6), 064906 (2015). https://doi.org/10.1103/PhysRevC.92.064906. arXiv:1507.05569 [nucl-th]ADSCrossRefGoogle Scholar - 37.G.S. Denicol, C. Gale, S. Jeon, A. Monnai, B. Schenke, C. Shen, Phys. Rev. C
**98**(3), 034916 (2018). https://doi.org/10.1103/PhysRevC.98.034916. arXiv:1804.10557 [nucl-th]ADSCrossRefGoogle Scholar - 38.E. Schnedermann, J. Sollfrank, U.W. Heinz, Phys. Rev. C
**48**, 2462 (1993). https://doi.org/10.1103/PhysRevC.48.2462. arXiv:nucl-th/9307020 ADSCrossRefGoogle Scholar - 39.S. Floerchinger, E. Grossi, JHEP
**1808**, 186 (2018). https://doi.org/10.1007/JHEP08(2018)186. arXiv:1711.06687 [nucl-th]ADSCrossRefGoogle Scholar - 40.P.M. Lo, Phys. Rev. C
**97**(3), 035210 (2018). https://doi.org/10.1103/PhysRevC.97.035210. arXiv:1705.01514 [hep-ph]ADSCrossRefGoogle Scholar - 41.P. Huovinen, P.M. Lo, M. Marczenko, K. Morita, K. Redlich, C. Sasaki, Phys. Lett. B
**769**, 509 (2017). https://doi.org/10.1016/j.physletb.2017.03.060. arXiv:1608.06817 [hep-ph]ADSCrossRefGoogle Scholar - 42.A. Andronic, P. Braun-Munzinger, B. Friman, P.M. Lo, K. Redlich, J. Stachel. arXiv:1808.03102 [hep-ph]
- 43.S. Ryu, J.F. Paquet, C. Shen, G. Denicol, B. Schenke, S. Jeon, C. Gale, Phys. Rev. C
**97**(3), 034910 (2018). https://doi.org/10.1103/PhysRevC.97.034910. arXiv:1704.04216 [nucl-th]ADSCrossRefGoogle Scholar - 44.H. Bebie, P. Gerber, J.L. Goity, H. Leutwyler, Nucl. Phys. B
**378**, 95 (1992). https://doi.org/10.1016/0550-3213(92)90005-V ADSCrossRefGoogle Scholar - 45.T. Hirano, K. Tsuda, Phys. Rev. C
**66**, 054905 (2002). https://doi.org/10.1103/PhysRevC.66.054905. arXiv:nucl-th/0205043 ADSCrossRefGoogle Scholar - 46.F. Becattini, V. Chandra, L. Del Zanna, E. Grossi, Annals Phys.
**338**, 32 (2013). https://doi.org/10.1016/j.aop.2013.07.004. arXiv:1303.3431 [nucl-th]ADSCrossRefGoogle Scholar - 47.F. Becattini, I. Karpenko, M. Lisa, I. Upsal, S. Voloshin, Phys. Rev. C
**95**(5), 054902 (2017). https://doi.org/10.1103/PhysRevC.95.054902. arXiv:1610.02506 [nucl-th]ADSCrossRefGoogle Scholar

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