# Thermal model description of *p*–*Pb* collisions at \(\sqrt{s_{NN}} = 5.02\) TeV

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## Abstract

The ALICE data on light flavor hadron production obtained in *p*–*Pb* collisions at \(\sqrt{s_{NN}} = 5.02\) TeV are studied in the thermal model using the canonical approach with exact strangeness conservation. The chemical freeze-out temperature is independent of centrality except for the lowest multiplicity bin, with values close to 160 MeV but consistent with those obtained in *Pb*–*Pb* collisions at \(\sqrt{s_{NN}} = 2.76\) TeV. The value of the strangeness non-equilibrium factor \(\gamma _s\) is slowly increasing with multiplicity from 0.9 to 0.96, i.e. it is always very close to full chemical equilibrium.

## 1 Introduction

Over the last three decades, the hadron resonance gas (thermal model for short) in its grand-canonical (GC) and canonical formulations has been very successful in describing the abundances of light flavored hadron obtained in heavy-ion collisions [1, 2, 3] up to the highest beam energies.

The thermal model has a long history going back to papers by Heisenberg [4], Fermi [5] and Hagedorn [6]. The model uses a minimal number of parameters, mainly, the chemical freeze-out temperature \(T_{ch}\), the baryon chemical potential \(\mu _B\) and the volume *V*. The physical description emerging from this is a very consistent one, the freeze-out temperature rises rapidly at low beam energies to reach a plateau at around 160 MeV while the baryon-chemical potential decreases steadily from a very high value to become compatible with zero at the highest energies. The implication is that all hadrons are produced at one temperature and within one volume. It is therefore of interest to test the region of applicability of the thermal model and this is done in the present paper for *p*–*Pb* collisions at the LHC.

In the present paper we investigate in detail the particle abundances in *p*–*Pb* collisions [7, 8, 9, 10] in the hope that this contributes to the understanding and the status of the model description.

At LHC energies, the particle abundances in central heavy-ion collisions have been well described [3, 11, 12] over nine orders of magnitude with only two parameters \(T_{ch}\) and *V*, with \(\mu _B\) being restricted to zero because of the particle-antiparticle symmetry at the LHC.

Our results show that there is no clear dependence on the centrality, as measured by the values of \(dN_{ch}/d\eta \), for the chemical freeze-out temperature. This confirms results obtained earlier on the dependence of thermal parameters on the size of the system obtained [13].

In Sect. 2 we briefly review the main features of the model. Section 3 presents our results for *p*–*Pb* collisions. Section 4 contains a discussion of results and Sect. 5 presents our conclusions.

## 2 The model

*V*of the system. It has been shown that this description is also the correct one [15, 16, 17] for a scaling expansion as first discussed by Bjorken [18]. After integration over \(p_T\) these authors have shown that:

In general, if the number of particles carrying quantum numbers related to a conservation law is small, then the grand-canonical description no longer holds. In such a case the conservation of quantum numbers has to be implemented exactly in the canonical ensemble [19, 20]. Here, we refer only to strangeness conservation and consider charge and baryon number conservation to be fulfilled on the average in the grand canonical ensemble because the number of charged particles and baryons is much larger than that of strange particles [21].

*Grand-Canonical Ensemble*(

*GCE*), the volume

*V*, temperature \(T_{ch}\) and the chemical potentials \(\vec \mu \) determine the partition function \(Z(T,V,\vec \mu )\). In the hadronic fireball of non-interacting hadrons, \(\ln Z\) is the sum of the contributions of all

*i*-particle species

*i*. Introducing the particle’s specific chemical potential \(\mu _i\), one gets

*i*contributes to the yields eventually measured. Therefore, the contributions from all heavier hadrons that decay to hadron

*i*are included.

In the GCE the particle yields are determined by the volume of the fireball, its temperature and the chemical potentials.

*Strangeness Canonical Ensemble*(

*SCE*) in which strangeness conservation is considered and charge and baryon number are conserved on the average. The density of strange particle

*i*carrying strangeness

*s*can be obtained from [21],

*k*carrying strangeness

*s*.

*s*is well approximated by [21]

*V*[22, 23]. In our analysis we will not entertain this possibility and work instead with an overall parameter \(\gamma _s\) to describe the deviation of strange particles from chemical equilibrium [24]. This amounts to replacing each particle density by

*s*| is the sum of the number of strange and anti-strange quarks. Note that the \(\phi \) meson picks up a factor \(\gamma _s^2\) since it contains a strange and an anti-strange quark. This makes the \(\phi \) meson behave in a similar way to the \(\Xi \) baryon which contains two strange quarks, a behavior which is supported by the data. The changes for the other strange mesons and baryons are as follows:

## 3 Scaling of yields with strange quark content

The experimental results were taken from [7, 8, 9], these are obtained for central rapidity in an interval \(\Delta y\) = 1. One of the basic features of the SC approach is that the particle ratios exhibit an approximate scaling with the difference of the strangeness quantum numbers. The support for this scaling is shown in Fig. 1. The left pane shows the ratios of various strange particle to pion yields i as a function of multiplicity. Following references [7, 8, 9] we use \(K/\pi \) as an abbreviation for the ratio \((K^++K^-)/(\pi ^++\pi ^-)\) and similarly for all other ratios, except that \(\phi /\pi \) is an abbreviation for \(\phi /(\pi ^++\pi ^-)/2\). Scaling these ratios with the power of the corresponding number of strange quarks one observes a common trend (right pane of Fig. 1). As can be seen from Fig. 1 scaling with the number of strange quarks is very well supported by the experimental data.

This scaling behavior with strange quark content is motivated by the SCE description including the factor \(\gamma _s\). In particular, the yield of \(\phi \) mesons closely follows the one of e.g. \(\Xi \) baryons. This argues against the use of a different correlation radius for the strangeness suppression as this would not affect the \(\phi \) meson since it has strangeness zero.

## 4 Fits using the thermal model

*p*–

*Pb*collisions at \(\sqrt{s_{NN}} = 5.02\) TeV in seven multiplicity bins such as 0–5, 10–20, 20–40, 40–60, 60–80 and 80–100%.

We have used THERMUS [25] to perform these fits in the strangeness canonical ensemble (SCE) with \(\mu _B\) and \(\mu _Q\) fixed to 0 and using only one radius *R* for the system. The fits to the bins with the most central collisions are shown in Fig. 2. In the following figures \(\pi \) refers to \((\pi ^++\pi ^-)/2\), *K* refers to \(K^++K^-)/2\) and similarly *p* refers to \((p+\bar{p})/2\), \(\Lambda \) refers to \((\Lambda +\bar{\Lambda })/2\), \(\Xi \) refers to \((\Xi ^-+\bar{\Xi })/2\) and finally \(\Omega \) refers to \((\Omega + \bar{\Omega })/2\). All fits were performed including the \(\phi \) meson.

In Fig. 2 we present our results for the two most central multiplicity bins, i.e. 0–5 and 5–10%.

The pattern observed for the bins with the highest multiplicity reproduces itself for the next two bins i.e. 10–20 and 20–40% as shown in Fig. 3. The deviations from the thermal model are very similar to the previous ones. In particular, the yield of \(\phi \) mesons is again reasonably well reproduced. This confirms the use of the strangeness non-equilibrium factor \(\gamma _s\). The use of a different correlation radius for the strangeness suppression would not affect the \(\phi \) meson since it has net zero strangeness.

All yields are compatible with the thermal model description within two standard deviations.The next multiplicity bins are shown in Fig. 4. The left-hand side is consistent with the results shown in Figs. 2 and 3. The right-hand pane is a peripheral one corresponding to 60–80% centrality and shows the discrepancies with the thermal model fits. Especially there are clear deviations for the \(\Xi \) and \(\Omega \) baryons. The \(\phi \) is reasonably well described.

To summarize the results obtained above we show the chemical freeze-out temperature \(T_{ch}\) and the corresponding radius *R* as a function of charged particle multiplicity in Fig. 6. The values of \(T_{ch}\) are remarkably independent of multiplicity except for the most peripheral bin. The radius determined from the yields fixes the normalization of the yields and shows a steady increase with the centrality of the collision. The more particles there are in the final state, the larger the radius and, correspondingly, the volume of the system. It should be noted that the volume increases linearly with the multiplicity. This indicates that the density of the fireball remains constant.

The values of the strangeness non-equilibrium factor \(\gamma _s\) are shown on Fig. 7. It can be seen that \(\gamma _s\) is very slowly increasing with multiplicity, starting from a value of about 0.9 for the most peripheral collision reaching about 0.96 for the bin with the highest multiplicity. This shows that the system is very close to chemical equilibrium, the largest deviation being about 10%. Again, being so close to chemical equilibrium is a remarkable property of the fireball produced.

Chemical freeze-out temperature (\(T_{ch}\)), radius (R), \(\gamma _s\) and \(\chi ^{2}/ndf\) of the fits for various multiplicity classes

Centrality (%) | \(T_{ch}\) (MeV) |
| \(\gamma _s\) | \(\chi ^{2}/ndf\) |
---|---|---|---|---|

0–5 | 161.10 ± 3.10 | 3.06 ± 0.16 | 0.96 ± 0.031 | 23.38/4 |

5–10 | 162.93 ± 3.30 | 2.77 ± 0.15 | 0.94 ± 0.030 | 26.52/4 |

10–20 | 162.53 ± 3.23 | 2.64 ± 0.14 | 0.93 ± 0.029 | 25.44/4 |

20–40 | 162.75 ± 3.03 | 2.42 ± 0.13 | 0.93 ± 0.037 | 27.52/4 |

40–60 | 160.48 ± 3.37 | 2.22 ± 0.12 | 0.91 ± 0.029 | 20.07/4 |

60–80 | 165.00 ± 3.78 | 1.77 ± 0.11 | 0.91 ± 0.029 | 55.40/4 |

80–100 | 152.72 ± 2.96 | 1.69 ± 0.09 | 0.90 ± 0.032 | 74.35/4 |

## 5 Conclusions

The fireball being produced in *p*-*Pb* collisions has simple properties, it is close to full chemical equilibrium, the temperature is independent of the multiplicity in the final state and the radius increases smoothly which is to be expected for a system that increases with the multiplicity in the final state.

It is remarkable that the system is always close to full chemical equilibrium with \(\gamma _s\) never below 0.9 even for the most peripheral bins. The chemical freeze-out temperature is also remarkably constant, except (again) for the most peripheral bin. The radius decreases when going to more peripheral collisions, a feature which is to be expected.

Fits to *pp* collisions have recently been made [26] as a function of the multiplicity of charged particles in the event. It was shown that for the highest multiplicities the thermal model works very well and that a grand canonical description becomes valid.

For such a simple model based on only three parameters it is not surprising that a perfect agreement is not always reached, this can be clearly seen in the underestimation of the pion yield and the overestimate of the proton one. This indicates that the thermal model as presented here is, in our opinion, a very good first order description of the particle composition but that it is open to further improvements.

In conclusion, in this paper we have performed an analysis of the particle composition in *p*–*Pb* collisions at 5.02 TeV using the thermal model. This model provides a good description of the hadronic yields. The largest deviations occurring for the most peripheral collisions. The chemical freeze-out temperature is independent of centrality \(T_{ch} = 162 \pm 3\) MeV, except for the lowest multiplicity bin, this value is consistent with values obtained in *Pb*–*Pb* collisions. The value of the strangeness non-equilibrium factor \(\gamma _s\) is slowly increasing with multiplicity from 0.9 to 0.96, i.e. it is always very close to full chemical equilibrium.

## Notes

### Acknowledgements

The authors acknowledge numerous useful discussions with J. Schukraft, B. Hippolyte and K. Redlich. One of us (J.C.) gratefully thanks the National Research Foundation of South Africa for financial support. N.S. acknowledges the support of SERB Ramanujan Fellowship (D.O. No. SB/S2/RJN- 269084/2015) of the Department of Science and Technology of India. L.K. acknowledges the support of the SERB grant No. ECR/2016/000109 of the Department of Science and Technology of India.

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