#
Hard-Core Radius of Nucleons within the Induced Surface Tension Approach^{ †}

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## Abstract

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## 1. Introduction

^{−3}at zero pressure and zero temperature, its binding energy per nucleon ${W}_{0}=-16$ MeV, and its incompressibility factor ${K}_{0}$ are of great importance for various phenomenological approaches, since these characteristics are widely used for determination of the model parameters. However, there exists a significant uncertainty in the ${K}_{0}$ value, since earlier estimates provide ${K}_{0}\simeq 220-260$ MeV [6], while the more recent ones give us ${K}_{0}\simeq 250-315$ MeV [7]. Furthermore, such a parameter of the nuclear matter as the hard-core radius (HCR) of nucleons ${R}_{N}$ plays an important role not only in nuclear physics [1,3], but also in nuclear astrophysics [2,4] and in the physics of heavy ion collisions (HIC) [8,9,10,11,12,13,14,15,16,17,18]. However, in the literature one can find any value of ${R}_{N}$ in the range 0.3–0.7 fm. The problem is partly related to the fact that almost all equations of state (EoS) with the hard-core repulsion employ the Van der Waals (VdW) approximation, which is applicable only at low particle number densities.

## 2. Multicomponent Formulation of HRGM with Hard-Core Repulsion

^{±}-mesons (${R}_{K}\simeq 0.395\pm 0.03$ fm), for nucleons (${R}_{p}\simeq 0.365\pm 0.03$ fm), and for the lightest (anti)$\Lambda $-hyperons (${R}_{\Lambda}\simeq 0.085\pm 0.015$ fm) [20,21]. Nevertheless, there is a confidence that in few years from now the new data of high quality which will be measured at RHIC BNL (Brookhaven) [44], NICA JINR (Dubna) [45], and FAIR GSI (Darmstadt) [46], will help us to find out the HCR of other measured hadrons with unprecedentedly high accuracy. However, one should remember that the traditional multicomponent HRGM based on the VdW approximation is not suited for such a purpose, since for $N\sim 100$ different HCR, where N corresponds to the various hadronic species produced in a collision, one has to find a solution of N transcendental equations. Therefore, an increase of the number of HCR to $N\sim 100$ will lead to hard computational problems for the traditional HRGM with multicomponent hard-core repulsion. To resolve this principal problem, the new HRGM based on the IST concept [23] was recently developed in Refs. [20,21,22].

## 3. Nuclear Matter IST EoS and Proton Flow Constraint

^{−3}, which is the maximal density of the flow constraint [55], the repulsion is suppressed, since at these particle number densities the mean nucleon separation is larger than ${r}_{\mathrm{min}}={\left(\frac{3}{4\pi {n}_{\mathrm{max}}}\right)}^{1/3}\simeq 0.7$ fm. But at such distances the microscopic nucleon-nucleon potential is attractive [56], whereas the remaining repulsive interaction can be safely accounted by the particle hard-core repulsion.

^{−3}and the value of its binding energy per nucleon ${W}_{0}=\frac{{\u03f5}_{N}}{{n}_{N}}-m=-16$ MeV (where ${\u03f5}_{N}$ is the energy density). Hence, the baryonic chemical potential of nucleons is $\mu =923$ MeV. The QIST EoS with the attraction term (16) was normalized to these properties of nuclear matter ground state and, simultaneously, it was fitted [54] to obey the proton flow constraint [55]. The region in the pressure-density plane deduced from flow observables has become an easy-to-use standard to constrain the behavior of phenomenological $T=0$ EoS at supersaturation densities. See, e.g., Ref. [58] for its usage in the context of compact star astrophysics. Although at a given value of particle number density the allowed range of pressure values of symmetric nuclear matter is rather wide, it is not easy to obey it and to simultaneously achieve the available range range of the incompressibility constant ${K}_{0}$, since they anti-correlate with each other.

_{⊙}and thus would not fulfill the constraint from the observed mass of $2.01\pm 0.04$ M

_{⊙}for pulsar PSR J0348+432 [59]. In Ref. [58] it was also shown that an equation of state which should fulfill the maximum mass constraint should follow the upper bound of the flow constraint. As a guideline may serve the ab-initio EoS DBHF from Ref. [58], which is soft enough to explain kaon production data in heavy-ion collisions at SIS energies but at the same time even exceeds the upper limit of the flow constraint at higher densities $n>0.5$ fm

^{−3}and yields a maximum mass larger than 2.3 M

_{⊙}, thus being even stiffer than required by the observed mass of pulsar PSR J0348+432 [59]. It has been shown recently in Ref. [60] that the IST EoS in the parametrization optimized for explaining particle yields from heavy-ion collisions is in accordance with the phenomenology of neutron stars, i.e., at $T=0$.

^{−3}of the proton flow constraint, then such a condition can be written as

^{−3}the fourth and higher virial coefficients are not important and, hence, we can require that up to this nucleon density the coefficient ${a}_{2N}^{\mathrm{eff}}({n}_{N})$ obeys the constraint (18). This leads to the follows range of ${R}_{N}$ values: ${R}_{N}\in [0.275;0.36]$ fm. In other words, for such a range of values of the nucleonic HCR not only the second, but also the third virial coefficient of nucleons will provide the fulfillment of the constraint (18).

^{3}and it is a decreasing function of T. A simple analysis shows that for nucleons ${a}_{2}^{(0)}(T=32\phantom{\rule{0.166667em}{0ex}}\mathrm{MeV})\simeq 1$ fm

^{3}, while for $T=50$ MeV one finds ${a}_{2}^{(0)}(T=50\phantom{\rule{0.166667em}{0ex}}\mathrm{MeV})\simeq 0.5$ fm

^{3}and this coefficient decreases fast with temperature.

^{−3}(see Figure 3 in [19]). Hence, in this range of CFO temperatures the hard-core corrections are negligible. On the other hand, for CFO temperatures above 100 MeV and below 170 MeV the quantum effects are small not only for kaons and heavier mesons as we discussed above, but for all mesons [9,10]. Thus, at CFO the classical formulation of HRGM is rather accurate.

## 4. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Deviations of theoretically predicted hadronic yield ratios from experimental values in units of experimental error $\sigma $ are shown for the center of mass collision energies $\sqrt{{s}_{NN}}=8.8$ GeV and $\sqrt{{s}_{NN}}=130$ GeV. Dashed lines correspond to the induced surface tension (IST) equations of state (EoS) fit, while the solid lines correspond to the original hadron resonance gas model (HRGM) fit [12]. For a comparison the results obtained by the HRGM1 with a single hard-core radius ${R}_{\mathrm{all}}=0.3$ fm for all hadrons are also shown (for more details see text).

**Figure 2.**The fit results obtained by the IST EoS: $\sqrt{{s}_{NN}}$ dependence of ${K}^{+}/{\pi}^{+}$ (left panel) and $\Lambda /{\pi}^{-}$ (right panel) ratios. For more than a decade these ratios were the most problematic one to reproduce by the HRGM.

**Figure 3.**The results obtained by the IST EOS on fitting the ALICE data with the new hard-core radius (HCR) found in [20] from fitting the AGS, SPS, and RHIC data. The found chemical freeze-out (CFO) temperature is ${T}_{CFO}\simeq 148\pm 7$ MeV. The fit quality is ${\chi}^{2}/\mathrm{dof}\simeq 8.92/10\simeq 0.89$. The upper panel shows the fit of the ratios, while the lower panel shows the deviation between data and theory in units of estimated error.

**Table 1.**Several sets of parameters which simultaneously reproduce the properties of normal nuclear matter ($p=0$ and $n={n}_{0}$ = 0.16 fm

^{−3}at $\mu =923$ MeV, see text for details) and obey the proton flow constraint on the nuclear matter equations of state (EoS) along with incompressibility factor ${K}_{0}$ and parameters of critical endpoint (CEP). ${R}_{N},{C}_{d}^{2},{U}_{0}$, and $\kappa $ are the adjustable parameters of quantum induced surface tension (QIST) EoS.

$\mathit{\kappa}$ = 0.1 | $\mathit{\kappa}$ = 0.15 | $\mathit{\kappa}$ = 0.2 | $\mathit{\kappa}$ = 0.25 | |||||
---|---|---|---|---|---|---|---|---|

${R}_{N}$ [fm] | 0.28 | 0.42 | 0.35 | 0.48 | 0.41 | 0.50 | 0.47 | 0.52 |

${C}_{d}^{2}$ [MeV· fm^{3κ}] | 284.98 | 325.06 | 206.05 | 229.57 | 168.15 | 179.67 | 146.97 | 152.00 |

${U}_{0}$ [MeV] | 567.32 | 501.65 | 343.93 | 312.83 | 231.42 | 217.76 | 162.03 | 157.41 |

${K}_{0}$ [MeV] | 306.09 | 465.13 | 272.55 | 405.97 | 242.56 | 322.80 | 217.16 | 256.44 |

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**MDPI and ACS Style**

Bugaev, K.A.; Ivanytskyi, A.I.; Sagun, V.V.; Grinyuk, B.E.; Savchenko, D.O.; Zinovjev, G.M.; Nikonov, E.G.; Bravina, L.V.; Zabrodin, E.E.; Blaschke, D.B.;
et al. Hard-Core Radius of Nucleons within the Induced Surface Tension Approach. *Universe* **2019**, *5*, 63.
https://doi.org/10.3390/universe5020063

**AMA Style**

Bugaev KA, Ivanytskyi AI, Sagun VV, Grinyuk BE, Savchenko DO, Zinovjev GM, Nikonov EG, Bravina LV, Zabrodin EE, Blaschke DB,
et al. Hard-Core Radius of Nucleons within the Induced Surface Tension Approach. *Universe*. 2019; 5(2):63.
https://doi.org/10.3390/universe5020063

**Chicago/Turabian Style**

Bugaev, Kyrill A., Aleksei I. Ivanytskyi, Violetta V. Sagun, Boris E. Grinyuk, Denis O. Savchenko, Gennady M. Zinovjev, Edward G. Nikonov, Larissa V. Bravina, Evgeny E. Zabrodin, David B. Blaschke,
and et al. 2019. "Hard-Core Radius of Nucleons within the Induced Surface Tension Approach" *Universe* 5, no. 2: 63.
https://doi.org/10.3390/universe5020063