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Advances in High Energy Physics
Volume 2017, Article ID 2979743, 11 pages
https://doi.org/10.1155/2017/2979743
Research Article

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; moc.oohay@meesaw_rihsab

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 SCOAP3.

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 [36]. 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 [715]. 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

Table 1: TMI parameter set used in our calculation.

The potential depth for hyperons in baryonic matter is fixed as follows. Representing the potential depth of hyperon in baryonic matter as , we use  MeV,  MeV, and  MeV to determine the value of scalar coupling constants , , and respectively [2426]. The hyperon couplings with strange mesons are restricted by the relation obtained in [27]. For the hyperon-hyperon interactions, we use the square well potential with depth  MeV [28]. In Table 2, we list the values of the coupling constants determined from these hyperon potentials. Next, in Tables 3 and 4, we give kaon-meson and pion-meson coupling constants that are used in our calculation.

Table 2: Meson-hyperon coupling constants.
Table 3: Kaon-meson coupling constants.
Table 4: Pion-meson coupling constants [17].

Regarding antibaryon-meson couplings [29], there is no reliable information particularly for high-density matter. Therefore, we will use in our work the values of the antibaryon-meson couplings that are derived using G-parity transformation. The G-parity transformation is analogous to ordinary parity transformation in configurational space, which inverts the direction of three vectors. The G-parity transformation is defined as the combination of charge conjugation and rotation. The degree of rotation is done around the second axis of isospin space.

It is already known that exchange mesons and have positive G-parity and and have negative G-parity. Therefore, by applying G-parity transformation to nucleon potentials, one can obtain the corresponding potential for antinucleons. The result of G-parity transformation can be written as where and denote baryons and antibaryons, respectively. It is worthwhile to mention here that kaon-meson couplings can be fixed for two different kaonic potentials, namely, strongly attractive potential UK[S] and weakly attractive potential UK[W]. These are given in Table 3. Finally, the pion-meson couplings are given in Table 4. Since the pion is nonstrange particle, its coupling with strange mesons and is essentially zero.

3. Results and Discussions

In this article, our main aim is to evaluate the properties of strongly interacting hadronic matter at finite temperature. Therefore, we make an attempt to analyse the properties of particle ratios for this matter. Using the values of baryon-meson and boson-meson coupling constants as defined in the previous section, one can solve the coupled integral equations for the field variables and consequently one can obtain the thermodynamic observables of the hadronic system for given values of temperature and baryon chemical potential . In the discussion to follow, we use following parameter sets for fixing baryon-meson and boson-meson coupling constants used in this model. For baryon-meson couplings, we use parameter set TMI as listed in Tables 1 and 2. For kaon-meson coupling constants, we use parameter sets TM2 and GL85 that correspond to strongly attractive and weakly attractive kaonic potentials, respectively, and are listed in Table 3. The pion-meson coupling constants are listed in Table 4. In the following analysis, we will impose the strangeness conservation criteria = 0, where and are total strangeness and antistrangeness of the system under consideration.

In Figure 1, we plot the variation of with temperature for fixed values of baryon chemical potential. For  MeV, we show for hadronic matter with coupled baryons and bosons for two parameter sets TM1 and NL3 with kaonic potentials UK[S] and UK[W], respectively. The result obtained for noninteracting hadrons is calculated using hadron resonance gas (HRG) model. The effect of chiral symmetry is analysed by decoupling Nambu-Goldstone modes (pions and kaons) from baryons and result is denoted by RMF[D]. This is obtained by setting . It is worthwhile to mention here that, with this choice of boson-meson couplings, baryons can still interact strongly via the exchange of mesons (), while bosons (pions + kaons) get decoupled from baryons and hence remain in the system as free particle with no interaction. One can see that, for  MeV, the effect of interaction is negligible on ratio even up to relatively high temperature of about  MeV. For higher baryon chemical potential, that is,  MeV, one can see that the interactions modify ratio. For hadronic phase with coupled baryons and bosons, the effect of interaction is to increase ratio. Further, one can see that the rise of ratio is significant only in the critical region (CR) of first-order quark-hadron phase transition, which corresponds to large value of Bag constant . Here, we have calculated critical region with equation of state for quark-gluon plasma which is consistent with Lattice QCD (see Appendix). The Bag value was fixed in the range = 165–200 MeV. However, on decoupling baryons from bosons, one can see that the ratio drops and in fact becomes slightly lower than HRG model’s result.

Figure 1: ratio as a function of temperature for fixed value of baryon chemical potential  MeV and  MeV for strongly interacting hadronic matter with coupled baryons and bosons with two parameter sets TM1 and NL3 for kaonic potentials UK[S] and UK[W], respectively, for strongly interacting hadronic matter with decoupled baryons and bosons RMF[D], and for hadron resonance gas (HRG) model. CR is the critical region of first-order quark-hadron phase transition calculated with Bag value = 165–200 MeV. is the coupling strength for quarks.

In Figure 2, we plot the variation of with baryon chemical potential at fixed value of temperature  MeV and  MeV. At lower temperature, ratio for NL3 parameter set is the same as that for HRG model, while, for TM1 parameter set, ratio is less compared to that of HRG model. However, at higher temperature, the particle ratio for both parameter sets, that is, TM1 and NL3, is less compared to HRG model.

Figure 2: as a function of baryon chemical potential at fixed values of temperature  MeV and  MeV for strongly interacting hadronic matter with coupled baryons and bosons with two parameter sets TM1 and NL3 for kaonic potentials UK[S] and UK[W], respectively, and for noninteracting hadrons using hadron resonance gas (HRG) model.

In Figure 3, we next plot the variation of particle ratio with temperature for fixed values of baryon chemical potential . At lower baryon chemical potential, that is,  MeV, the effect of interaction among hadrons is negligible on particle ratio even for relatively high temperature of  MeV. However, for higher baryon chemical potential, that is,  MeV, the effect of interaction among hadrons increases the ratio . One can again see that the rise of is somewhat significant in the critical region of phase transition which corresponds to a large value of Bag constant . Here, CR denotes the critical region of first-order quark-hadron phase transition which is calculated with Bag value = 165–200 MeV and lattice motivated equation of state (see Appendix). Interestingly, one can see that the effect of interaction on vanishes if bosons decouple from baryons. ratio for a system of strongly interacting hadrons with baryons decoupled from bosons is represented by RMF[D].

Figure 3: as a function of temperature for fixed values of baryon chemical potential  MeV and  MeV for strongly interacting hadronic matter with coupled baryons and bosons with two parameter sets TM1 and NL3 for kaonic potentials UK[S] and UK[W], respectively, for strongly interacting hadronic matter with decoupled baryons and bosons RMF[D], and for hadron resonance gas (HRG) model. CR is the critical region of first-order quark-hadron phase transition calculated with Bag value = 165–200 MeV. is the coupling constant for quarks.

In Figure 4, we next plot the variation of particle ratio with baryon chemical potential at fixed values of temperature  MeV and  MeV. At lower temperature, that is,  MeV, the particle ratio behaves differently under parameter sets TM1 and NL3. However, at large temperature, that is,  MeV, the particle ratio for parameter sets TM1 and NL3 coincides with noninteracting systems even up to very large values of baryon chemical potential.

Figure 4: as a function of baryon chemical potential at fixed values of temperature  MeV and  MeV for strongly interacting hadronic matter with coupled baryons and bosons with two parameter sets TM1 and NL3 for kaonic potentials UK[S] and UK[W], respectively, and for system of noninteracting hadrons modelled using hadron resonance gas (HRG) model.

In Figure 5, we next plot the variation of particle ratio as a function of temperature for fixed baryon chemical potential, that is,  MeV and  MeV. For lower value of baryon chemical potential, that is,  MeV, the effect of interactions among hadrons with coupled baryons and bosons is negligible on the particle ratio . Interestingly, even for higher baryon chemical potential, that is,  MeV, the effect of interaction is still negligible on ratio. However, on decoupling bosons from baryons, the ratio becomes large at higher values of temperature. For a reference, we have again shown the critical region of first-order quark-hadron phase transition CR. Here, we have used equation of state for quark-gluon plasma phase with Bag value = 165–200 MeV and strong coupling constant .

Figure 5: Particle ratio as a function of temperature for different values of baryon chemical potential, that is,  MeV and  MeV, for strongly interacting hadronic matter with coupled baryons and bosons with two model parameter sets TM1 and NL3 for kaonic potentials UK[S] and UK[W], respectively, for strongly interacting hadronic matter with decoupled baryons and bosons RMF[D], and for a system of noninteracting hadrons modelled using hadron resonance gas (HRG) model. CR is the critical region of first-order quark-hadron phase transition calculated with Bag value = 165–200 MeV. is the strong coupling constant for quarks.

Finally, in Figure 6, we show the variation of particle ratio with baryon chemical potential for fixed values of temperature  MeV and  MeV. For the case of strongly interacting hadronic matter with coupled baryons and bosons, the particle ratio is the same as that of a system of noninteracting hadrons even up to very large baryon chemical potential values.

Figure 6: Particle ratio as a function of baryon chemical potential for fixed values of temperature  MeV and  MeV for strongly interacting hadronic matter with coupled baryons and bosons with two parameter sets TM1 and NL3 for kaonic potentials UK[S] and UK[W]s, respectively, and for system of noninteracting hadrons modelled using hadron resonance gas (HRG) model.

In Figure 7, we plot the variation of effective chemical potential and effective mass to bare mass ratio of hadrons as a function of temperature for fixed values of baryon chemical potential  MeV and  MeV. Next, in Figure 8, we have shown the variation of these observables, that is, and , as a function of baryon chemical potential at fixed values of temperature  MeV and  MeV. These observables have been calculated for a system of strongly interacting hadrons with coupled baryons and bosons with parameter set TM1. For kaon-meson coupling, we use parameter set TM2 which corresponds to strongly attractive kaonic potential UK[S]. To complete our discussion, we next plot in Figure 9 the variation of strange chemical potential as functions of baryon chemical potential and temperature at fixed values of temperature and baryon chemical potential, respectively. The strangeness chemical potential is fixed by imposing the constraint of strangeness conservation in the hadronic medium. The values of strangeness chemical potential show a decrease with increasing baryon chemical potential with increasing temperature. Further, we also observe that the strangeness chemical potential decreases with the increase of temperature for increasing values of baryon chemical potential. This is consistent with the hadron resonance gas model [30].

Figure 7: Effective chemical potential and mass ratio of hadrons as a function of temperature for fixed values of baryon chemical potential  MeV and  MeV in case of strongly interacting hadronic matter with coupled baryons and bosons. Here, parameter set TM1 is used.
Figure 8: Effective chemical potential and mass ratio of hadrons as a function of baryon chemical potential for fixed values of temperature  MeV and  MeV in case of strongly interacting hadronic matter with coupled baryons and bosons. Here, parameter set TM1 is used.
Figure 9: Strange chemical potential as a function of baryon chemical potential and temperature for fixed values of temperature and baryon chemical potential, respectively, in case of a strongly interacting hadronic system with coupled baryons and bosons RMF[C]. Here, parameter set TM1 is used.

4. Summary

In this article, we have calculated particle ratios for a strongly interacting hadronic matter using nonlinear Walecka model in relativistic mean field (RMF) approximation. In the hadronic medium, we incorporate baryons and bosons (pions + kaons). To describe baryons and bosons, we use a meson-exchange type of Lagrangian and evaluate thermodynamic observables of hadronic matter in RMF approximation. It is found that the interaction among hadrons which in the present model is mediated by the exchange of and mesons can result in the modification of and ratios, while the particle ratio is found to be independent of the interaction among hadrons.

Appendix

Quark-Gluon Plasma Phase

In this section, we present equation of state for quark-gluon plasma (QGP) phase. We consider three quark flavours, up (u), down (d), and strange (s), and gluons. We use a Bag model equation of state with perturbative corrections of the order of which is consistent with Lattice data [31, 32]. The pressure and energy density are given bywhere and are the effective numbers of gluons and fermions, respectively. The quark chemical potential is and is the coupling constant. For the determination of coupling constant , see, for example, [33]. Here, and are the Fermi-Dirac distribution functions for quarks and antiquarks, respectively, and is the Bag constant. To see the possible effect of Bag constant on various observables, see, for example, [34]. In the above calculation, up (u) and down (d) quarks are considered massless, while strange quark (s) is of finite mass  MeV. In the present discussion, heavy quark flavours have not been considered.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

Waseem Bashir is grateful to Council of Scientific and Industrial Research, New Delhi, for awarding Research Associateship.

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