# Chiral phase transition scenarios from the vector meson extended Polyakov quark meson model

## Abstract

Chiral phase transition is investigated in an symmetric vector meson extended linear sigma model with additional constituent quarks and Polyakov loops (extended Polyakov quark meson model). The parameterization of the Lagrangian is done at zero temperature in a hybrid approach, where the mesons are treated at tree-level, while the constituent quarks at 1-loop level. The temperature and baryochemical potential dependence of the two assumed scalar condensates are calculated from the hybrid 1-loop level equations of states. The order of the phase transition along the and axes are determined for various parameterization scenarios. We find that in order to have a first order phase transition at as a function of a light isoscalar particle is needed.

12.39.Fe, 12.40.Yx, 14.40.Be, 14.40.Df, 14.65.Bt, 25.75.Nq

## 1 Introduction

In [1] it was shown through a zero temperature analysis that scalar states, such as the , , and the two ’s are preferred to have masses above GeV. Similar results was obtained with states in [2], while in [2, 3, 4] it was shown by using tetraquarks instead of states that the (tetraquark) scalar masses are in the range GeV. These results suggests that the physical states , , , and (or ) are predominantly states, while , , , and are predominantly tetraquark states. However, states with the same quantum numbers do mix, thus the physical scalar particles are mixtures of and tetraquarks states. In the current case of the extended linear model (ELM) we have only one scalar nonet, thus we can describe one , one and two (which will be denoted by ) particles. Consequently, one of the most interesting question is that which physical states our fields predominantly are if we investigate the finite temperature/density behavior of our model additionally to the zero temperature properties.

The lightest isoscalar () state, (also called ) is strongly related to the non-strange condensate. Since the larger is the sigma mass compared to the mass of its chiral partner (the pion) the larger is the temperature at which approaches in the chiral symmetry restoration, one would expect that a large mass results in a large pseudocritical temperature () at zero baryochemical potentials. On the other hand it is a common expectation that the chiral phase transition is of first order as a function the baryochemical potential () at , and since with increasing mass the transition weakens, at some point it is possible that the transition becomes crossover [5, 6]. This suggest that for a good thermodynamic description a small mass is needed, and indeed as it turns out our approach supports this requirement.

The paper is organized as follows. In the next subsection we introduce the model by giving the Lagrangian. In Sec. 3 the determination of the model parameters is shown, while the description of the approximation used to calculate the grand potential together with the field equations are presented in Sec. 4.

## 2 The Model

The Lagrangian we shall use has the following form:

(1) | ||||

The covariant derivatives above are given by

(2) |

where with () denoting the group generators given in terms of the Gell-Mann matrices . Here stands for the scalar – pseudoscalar fields, for the left and right handed vector fields (which contain the nonets of vector () and axial vector () meson fields), is the electromagnetic field, while are the gluon fields. The field strength tensors are ()

(3) |

while the external fields related to the scalar and vector fields are , (For more details on the model see [1]).

## 3 Setting the Lagrange parameters

There are unknown parameters in Eq. (2), which will be present in the field equations at finite temperature and/or density, namely, , , , , , , , , , , , , , , . From this set can be melted into thus leaving unknowns. These parameters are determined similarly as in [1], that is we calculate values of various masses and decay widths at tree-level and compare them with the corresponding experimental value taken from the PDG [7] through the minimalization process of Ref. [8]. It is important to note that we artificially increased the errors of the PDG to a minimum level of , since we do not expect that our model to be more precise. Another important points are that now we use a different anomaly term in Eq. (2) (the term proportional to ) as was presented in [1], we fit the total width in case of instead of the amplitudes, we include the masses and decay width into the global fit, we take into account of the effects of the fermion vacuum fluctuations, case of which the expression of the (pseudo)scalar masses are modified. Moreover, since now we also included the constituent quarks in the isospin symmetric limit, we use two additional equations to their tree-level masses with the values MeV, and MeV.

The scalar meson sector below GeV contains more physical particles than we can place into one nonet (consisting of , , , ), since in nature two , two and five particles exist in that energy range. These particles are the and (denoted by ), the and , (denoted by ), the (or ), , , and , (denoted by ). Accordingly there are particle assignment possibilities to pair the physical particles to the members of the nonet in the model. We performed a fit for all the assignments and ordered them according to their values. The results of the best solution along with the particle assignments in two cases (with and without the fermionic vacuum fluctuation) are shown in Table 1.

particle assignment | particle assignment | ||||
---|---|---|---|---|---|

45.0 | 3.0 | 46.2 | 3.1 | ||

51.7 | 3.4 | 52.7 | 3.5 | ||

51.7 | 3.4 | 54.1 | 3.6 | ||

60.4 | 4.0 | 60.7 | 4.0 | ||

61.3 | 4.1 | 61.7 | 4.1 |

By a similar fitting procedure in [1] we argued that the best assignment without fitting the is the , while we reasoned that should correspond to . Now it seems that the situation changes since Table 1 one can see that the two best assignment in case with and without using the fermion vacuum fluctuation are and , respectively. This suggest that using only zero temperature quantities in this model is not enough to point out uniquely a single particle assignment. Thus we investigated the properties of the different particle assignments at finite temperature/densities as well.

## 4 Finite temperature field equations

In our model there are four order parameters, which are the two chiral condensates (non-strange) and (strange) and the two Polyakov loop variables and (For the introduction of the Polyakov loop variables and their potential see [9]). The field equations, which determine the dependence on and of the order parameters are given by the minimalization of the grand canonical potential (see eg. [11]),

(4) |

which will result in four coupled equations. In our approach we only consider vacuum and thermal fluctuations for the constituent quarks and not for the mesons. The explicit expression for the field equations can be found in [10] (Eq. (2)-(5)).

## 5 Results and Conclusion

By solving the field equations Eq. (4) we can investigate the behavior of the
, order parameters as a function of at and as a function
at . It was calculated on the lattice [12]
that at the value of the pseudocritical temperature should
be MeV, while it is a common belief that the order of the
transition in at should be of first order. On
Fig. 1 the and values for all the 40
assignments are shown^{3}

It can be seen that in order to be consistent with the lattice the mass should be below GeV, which can be correspond either to or to . However if we would like to have first order phase transition on the axis, the mass should be even smaller ( MeV). Consequently we investigated the phase boundary for parameterizations with relatively small masses.

On the left panel of Fig. 2 the phase boundary together with the position of the critical endpoint (CEP) is shown, while on the right panel the CEP variation with the mass is presented.

If we increase the mass above MeV the CEP ceases to exist.

In conclusion we can say that in order to have a good thermodynamic description within the framework of the current model we must have the particle in the spectrum.

## Acknowledgments

The authors were supported by the Hungarian OTKA fund K109462 and by the HIC for FAIR Guest Funds of the Goethe University Frankfurt.

### Footnotes

- thanks: Presented at Excited QCD 2015 (8-14 March 2015, Tatranska Lomnica, Slovakia)
- in collaboration with Zsolt Szép
- The lines are only shown to guide the eye.

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