Advances in High Energy Physics

Volume 2017 (2017), Article ID 2979743, 11 pages

https://doi.org/10.1155/2017/2979743

## Particle Ratios from Strongly Interacting Hadronic Matter

Department of Physics, Jamia Millia Islamia (A Central University), New Delhi, India

Correspondence should be addressed to Waseem Bashir

Received 28 December 2016; Accepted 10 April 2017; Published 4 June 2017

Academic Editor: Juan José Sanz-Cillero

Copyright © 2017 Waseem Bashir et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

We calculate the particle ratios , , and for a strongly interacting hadronic matter using nonlinear Walecka model (NLWM) in relativistic mean field (RMF) approximation. It is found that interactions among hadrons modify and particle ratios, while is found to be insensitive to these interactions.

#### 1. Introduction

Since the discovery of asymptotic freedom [1] in case of nonabelian gauge field theories, it was postulated that a phase transition from nuclear state of matter to quark matter is possible. It was further argued that this phase transition can take place at sufficiently high temperature and/or densities and can result in the transformations of hadrons into a new state of matter dubbed as quark-gluon plasma (QGP). Since then, a considerable effort has been put forward to create and understand the properties of this new state of matter (QGP) and the corresponding phase transition. In order to study the dynamics of any phase transition in general, a complete description of a given state of matter on the basis of some underlying theory is required. To understand the dynamics of quark-hadron phase transition, the equation of state for both QGP phase and the hadronic phase is required. The QGP phase so far has been fairly described using Lattice Gauge theory in case of vanishing or low baryon chemical potential. However, the description of strongly interacting hadronic phase in terms of fundamental theory of strong interactions has proven to be far from being trivial. This is primarily due to strong coupling among hadrons, due to which the conventional methods of quantum field theory, for example, perturbative analysis, do not remain valid for the description of such strongly interacting hadronic phase. Therefore, one has to rely on alternate methods to describe the properties of hadronic phase, for example, hadron resonance gas models, chiral models, and quasi-particle models.

However, one can use another approach to determine the dynamics of strongly interacting hadronic phase and consequently of quark-hadron phase transition. By studying the spectra of hadrons, one can in principle comment on some of the properties of the strongly interacting hadronic matter. For example, by studying ratio, it has been argued that transparency effects in case of high energy heavy-ion collisions may become operative [2]. Recently, it has been found that particle ratios of some of the hadrons, for example, , , and , show a sudden rise for a specified range of center of mass energy in case of heavy-ion collisions [3–6]. Taking into account the dependence of baryon chemical potential and temperature on the variable , one can infer that the behaviour of these particle ratios may be sensitive to the critical region of quark-hadron phase transition. In this article, we therefore evaluate the particle ratios , , and for a strongly interacting hadronic matter and analyse their behaviour near first-order quark-hadron phase transition. For hadronic phase, we use nonlinear Walecka model within relativistic mean field (RMF) approximation. RMF theory has been widely and successfully used to describe the properties of the nuclear matter and finite nuclei. Further RMF theory has been also used to describe the equation of state for strongly interacting dense hadronic matter for the application in supernova and neutron stars [7–15]. In RMF theory, hadrons interact via the exchange of scalar and vector mesons and the interaction strength or coupling among hadrons is determined by different methods. For example, the nucleon-meson coupling constants are determined by reproducing the ground state properties of the finite nuclei or by using nuclear matter properties, which is discussed in Section 2. To describe the quark-gluon plasma (QGP) phase, we use a Bag model equation of state.

#### 2. Model

##### 2.1. Hadronic Phase: Baryons

The equation of state for asymmetric baryonic matter is presented in this section. To describe baryonic matter, we use relativistic nonlinear Walecka model (NLWM). In this model, the interaction between baryons is governed by the exchange of various mesons. We include in this model baryons along with their antiparticles. The interaction between baryons is carried out by the exchange of neutral , isoscalar-vector , isovector-vector , and two additional hidden strangeness mesons and . In this model, the Lagrangian density for baryons readswhere is the fermionic field corresponding to baryon . The interaction between baryons is carried out by the exchange of neutral , isoscalar-vector , isovector-vector , and two additional hidden strangeness mesons and . is the scalar self-interaction term for field. Also , , , and is the Bodmer correction or self-interaction term for the vector field and are the coupling constants that characterise the strength of interaction between mesons and baryons . Here, is in-medium mass of baryon, where is the bare mass of baryon. Also is the mass of exchange mesons and is the isospin operator. Using relativistic mean field (RMF) approximation under which the field variables are replaced by their space-time independent classical expectation values, that is, , , , and , the thermodynamic potential per unit volume corresponding to Lagrangian density (1) can be written aswhere effective baryon energy is and effective baryon chemical potential is . Also parameter is , where is the temperature.

##### 2.2. Hadronic Phase: Bosons (Pions + Kaons)

To incorporate bosons (pions + kaons) in our model, we use an approach similar to the one used to model baryonic phase; that is, we use a meson-exchange type of Lagrangian for bosons as well. The Lagrangian density in a minimal-coupling scheme is [18, 19] where is the bosonic field with summation carried over bosons . Here covariant derivative iswith the four-vector defined asand is the effective mass of bosons. Also are the coupling constants that characterise the strength of interaction between exchange mesons and bosons (pions + kaons). Here, is the isospin operator with its third component defined as

It has to be mentioned that one can use even chiral perturbation theory [20] to describe bosons in the hadronic matter. In an earlier work [21], kaons were incorporated using chiral perturbation theory, whereas baryons were incorporated using Walecka model. However, in [22], it was put forward that this approach of modelling baryonic phase with Walecka model and bosonic phase with chiral Lagrangian has some inconsistency that may influence the final results. In our approach, baryons and bosons are incorporated using similar methodology, that is, using meson-exchange type Lagrangian, and therefore this approach is expected to be more consistent. In RMF approximation, the thermodynamic potential for Lagrangian density (3) can be written as where , with effective energy defined as , and is the temporal component of four-vector . Also is the boson chemical potential and is the spin-isospin degeneracy factor of boson .

##### 2.3. Hadronic Phase: Field Equations

The thermodynamic potential per unit volume for hadronic medium can be therefore written as where and are as defined in (2) and (7), respectively.

The different thermodynamic observables of the hadronic system, for example, entropy, pressure, and number density, can be evaluated as follows: provided the expectation values of the exchange mesons field variables are known.

To evaluate the expectation value of exchange meson field variables, one can solve the following set of coupled equations of motion for different field variables that are obtained after minimising the action integral with respect to different exchange meson field variables; that is,where distribution functions for baryons and antibaryons are given by and net-baryon density is

Similarly, the distribution function of bosons and their antiparticles is where is the effective chemical potential of boson . Also the boson density is

##### 2.4. Hadronic Phase: The Coupling Constants

To fix baryon-meson coupling constants, we use two very successful parameter sets of RMF model, namely, parameter sets TM1 and NL3. These parameter sets are listed in Table 1. These parameters have been obtained by evaluating the ground state properties of finite nuclei [16, 23]. For meson-hyperon coupling constants, we use quark model values of vector couplings. These are given by