Characteristics of Positive Pions Produced in p + C Collisions at 4.2 GeV/c

Abstract

Experimental data from JINR and UrQMD (latest code 3.3p2) model simulations have been used to estimate temperature and other properties of positive pions in collisions of protons with carbon nuclei at incident momentum of 4.2 GeV/c. Transverse mass and transverse momentum spectra have been used to get temperature of said particles, with the help of some fitting equations. These equations are referred as Hagedorn thermodynamic and Boltzmann distribution functions. Such functions or equations are used to describe particles spectra. Temperature of positive pions has been found to be equal to 97 ± 7 and 98 ± 12 MeV in experimental and model, respectively, using Hagedorn function. Results from both experimental and model calculations have also been compared with each other, and thus, most reliable fitting function has been suggested. It is found that Hagedorn thermodynamic function is the best reliable function to get pions’ temperature in said collision system at given momentum. Similarly, temperature obtained in this research has been compared with results from other experiments and worthy conclusions have been reached and reported.

Appendix: Rolf Hagedorn’s Model

Temperature of nuclear medium created in collisions of high energy particles or nuclei, at which hadrons are formed in that medium, is obtained via the inverse slopes T of exponentially rising mass/transverse mass or transverse momentum spectra of hadrons with the help of Hagedorn function (Hagedorn 1965, 1967, 1968).

Hagedorn’s work is based on two themes:

• 1. Hagedorn proposed a Statistical Bootstrap Model (SBM), which is a self-similar scheme for the composition and decay of hadrons and their resonances. Hagedorn called these the “fireballs”. SBM is a connection between hadronic particle momentum distribution and properties of hadronic interactions dominated by resonant scattering and exponentially rising mass spectrum. It is a theoretical framework for study of the properties of the equations of state of dense and hot baryonic matter (nuclear matter at finite temperature).

• 2. The application of the resulting resonance spectrum in an ideal gas containing all possible hadrons and hadron resonances and to the construction of hadron production models based on such a thermal input.

1.1 Exponential Hadron Mass Spectrum

Rolf Hagedorn discovered that the exponential growth of the hadronic mass spectrum could lead to an understanding of the limiting hadron temperature T_H. He observed that T_H could be approximately equal to 160 MeV. The experimental mass spectrum was fitted with the following Hagedorn function, and he got the limiting hadron temperature (Hagedorn 1965, 1967, 1968).

$$ \rho (m) \approx c(m_{a}^{2} + m^{2} )^{a/2} \exp (m/T_{H} ) $$

Here a  = −3, m_a = 0.66 GeV and T_H = 159 ± 9 MeV.

He noticed that exponential or inverse slope was the same for all particles, T_H = 160 MeV, and it did not change as the collision energy varied. He thought this phenomenon just as evaporation of particles from boiling elementary matter. In a boiling process a maximum temperature cannot be exceeded even if we provide more and more heating (Hagedorn 1965, 1967, 1968).

1.2 Resonance Gas and Heavy Ion Collisions

In a strongly interacting medium, one includes the conservation of electric charge, baryon number and strangeness. In this case, the partition function of Hagedorn’s thermal model depends not only on temperature but also on chemical potential μ, which guarantees that charges are conserved on an average. For a non-vanishing μ, the partition function is given by Rafelski et al. (2016):

$$ \ln \;Z(T,V,\mu ) = \sum\limits_{i} {Z_{i}^{1} } (T,V,\mu ) $$

(7)

with $\mu = (\mu_{B} ,\mu_{S} ,\mu_{Q} )$, where μ_i are the chemical potentials related to the baryon number, strangeness and electric charge conservation, respectively. For particle i carrying strangeness S_i, the baryon number B_i, the electric charge Q_i and the spin–isospin degeneracy factor g_i, the one particle partition function, reads

$$ Z_{i}^{1} (T,V,\mu ) = \frac{{Vg_{i} Tm_{i}^{2} }}{{2\pi^{2} }}K_{2} (m_{i} /T)\exp \left( {\frac{{B_{i} \mu_{B} + S_{i} \mu_{S} + Q_{i} \mu_{Q} }}{T}} \right) $$

(8)

The Hagedorn model, formulated in Eq. (7), describes bulk thermodynamic properties and particle composition of a thermal fireball at finite temperature and at non-vanishing charge densities. If such a fireball is created in high energy heavy ion collisions, then yields of different hadron species are fully quantified by thermal parameters. However, following Hagedorn’s idea, the contribution of resonances decaying into lighter particles must be included (Hagedorn 1965, 1971).

In Hagedorn’s thermal model, the average number < N_i> of particles i in volume V and at temperature T that carries strangeness S_i, the baryon number B_i and the electric charge Q_i, is obtained from Eq. (7), in Refs. Hagedorn (1965), Rafelski et al. (2016) and Hagedorn (1971).

$$ \left\langle {N_{i} } \right\rangle \;(T,\mu ) = \left\langle {N_{i} } \right\rangle^{th} (T,\mu ) + \sum\nolimits_{j} {\varGamma_{j \to i} \left\langle {N_{j} } \right\rangle^{th.R} (T,\mu )} $$

(9)

The first term in the above equation describes the thermal average number of particles of species i from Eq. (8), and the second term describes overall contribution from resonances. This term is taken as a sum of all resonances that decay into particle i. The $\varGamma_{j \to i}$ is the corresponding decay branching ratio of $j \to i$. The $\left\langle {N_{j} } \right\rangle^{th.R}$ denotes multiplicities of resonances.

The particle yields in Hagedorn’s model Eq. (9) depend on five parameters. However, in high energy heavy ion collisions, only three parameters are independent. In the initial state the isospin asymmetry fixes the charge chemical potential, and the strangeness neutrality condition eliminates the strange chemical potential. Thus, on the level of particle multiplicity, we are left with temperature T and the baryon chemical potential μ_B as independent parameters, as well as, with fireball volume as an overall normalization factor (Rafelski et al. 2016).

Hagedorn’s thermal model introduced in Eq. (9) was successfully applied to describe particle yields measured in heavy ion collisions. The model was compared with available experimental data obtained in a broad energy range, for many particles, from AGS up to LHC. The systematic studies of particle production extended over more than two decades, using experimental results at different beam energies, have revealed a clear justification that in central heavy ion collisions particle yields are indeed consistent with the expectation of the Hagedorn thermal model. The temperature is increasing with $\sqrt s$, and at the SPS energy essentially saturates at the value, which corresponds to the transition temperature from a hadronic phase to a QGP. The chemical potential, on the other hand, is gradually decreasing with $\sqrt s$ and almost vanishes at the LHC.

At high energy collisions particle yields are quantified entirely by the temperature and the fireball volume. Thus there is transparent prediction of Hagedorn’s model Eq. (9) that yields of heavier particles < N_i > , normalized to their spin degeneracy factor g_i = (2 J + 1) should be quantified by Eq. (10), which gives eq. (1) after some calculations (Rafelski et al. 2016).

$$ \frac{{\left\langle {N_{i} } \right\rangle }}{2J + 1} \approx VT^{3} \left( {\frac{{m_{i} }}{2\pi T}} \right)^{3/2} \exp ( - m_{i} /T) $$

(10)