The strange critical endpoint and isentropic trajectories in an extended PNJL model with eight Quark interactions

Abstract

In this work, we explore the possible existence of several critical endpoints in the phase diagram of strongly interacting matter using an extended PNJL model with ’t Hooft determinant and eight quark interactions in the up, down and strange sectors. Besides, we also study the isentropic trajectories crossing both (light and strange) chiral phase transitions and around the critical endpoint in both the crossover and first-order transition regions.

Appendix A:The Mean Field approximation and Meson Masses

We introduce the auxiliary scalar, $s_a$, and pseudoscalar field variables, $p_a$, written in terms of quark bilinear operators, $s_a={\overline{q}} \lambda _a q$ and $p_a={\overline{q}} i \gamma ^5 \lambda _a q$, with indices $a=0,1,2,\ldots ,8$. Writing the Lagrangian density in terms of these new variables, yields:

(A.1)

Here, $f_{abc}$ and $d_{abc}$ are the totally antisymmetric and symmetric structure constants of the special unitary group SU(3), respectively. The constants $A_{abc}$ are defined as:

$$\begin{aligned} A_{abc}&= \frac{2}{3} d_{abc} + \sqrt{ \frac{2}{3} } ( \delta _{a0} \delta _{b0} \delta _{c0} - \delta _{a0} \delta _{bc} \nonumber \\&\quad ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - \delta _{b0} \delta _{ca} - \delta _{c0} \delta _{ab} ) . \end{aligned}$$

(A.2)

In order to derive the thermodynamical potential of the model we consider the mean field approximation. In this approximation, all quark interactions are transformed into quadratic interactions by introducing auxiliary fields whose quantum fluctuations are neglected and only the classical configuration contributes to the path integral i.e., the functional integration is dominated by the stationary point. A quark bilinear operator, $\hat{\mathcal {O}}$, can be written as its mean field value plus a small perturbation, . To linearize the product of $N-$operators, terms superior to $(\delta \hat{\mathcal {O}})^2$ must be neglected. Conveniently, the linear product between $N=n+1$ operators can be written using the following formula⁴:

(A.3)

The Lagrangian density can then be trivially linearized, the quadratic fermion term can be exactly integrated out and the grand canonical potential of the model can be derived to yield Eq. (4).

The meson masses can be calculated by writing an effective Lagrangian, built by expanding the Lagrangian in Eq. (A.1) up to second order in the auxiliary fields, [75]. Following the linear expansion of the Lagrangian, to build the quadratic expansion, terms superior to $(\delta \hat{\mathcal {O}})^3$ must be neglected. More easily, the quadratic product between $N=n+2$ operators, with $n \ge 1$, can be written using the following formula⁴

(A.4)

Having the quadratic expansion of the Lagrangian, the pseudoscalar and scalar inverse propagators are defined as the coefficient of the second order terms in the auxiliary fields. The pseudoscalar and scalar meson propagators are then given by:

(A.5)

(A.6)

Here, the indices $a,b=0,1,2,\ldots ,8$.

The pseudoscalar and scalar meson projectors, $P_{ab}$ and $S_{ab}$, with four, six and eight quark interactions, neglecting pseudoscalar condensates (), can be calculated to yield:

(A.7)

(A.8)

Using the diagonal matrices of SU$(3)_f$ and the identity, we can write the mean field values of the bilinear operators in the $0-3-8$ basis. One can switch to the quark flavour basis, $u-d-s$, doing a rotation as follows:

(A.9)

Here, the elements of the matrix $T_{ai}$ are given by:

(A.10)

The polarization functions can be rotated between basis using,

$$\begin{aligned} \varPi _{ab} = T_{ai} T_{bj} \varPi _{ij}. \end{aligned}$$

(A.11)

The pseudoscalar and scalar polarization functions for two quarks with flavours i and j, are given by [75]:

Here,

(A.12)

(A.13)

Where $\mathrm {p.v.}$ stands for the Cauchy principal value of the integral.

The mass of a given meson, $M_M$, and its decay width, $\varGamma _M$, can then be calculated by searching for the complex pole of its inverse propagator in the rest frame, i.e,

(A.14)

The correspondence between the auxiliary pseudoscalar fields and the physical pseudoscalar mesons can be performed using:

(A.15)

Where the pseudoscalar nonet was represented in the usual way. For the auxiliary scalar fields and the physical scalar fields, we use:

(A.16)

Using these correspondences the inverse propagator of a physical meson can be calculated using Eq. (A.14).

For the neutral mesons one must perform, as usual, a diagonalization of the quadratic contributions coming from the 0–3–8 channels. In the isotopic limit, one therefore obtains the straightforward extension of the results from [75] to include the eight quark contributions.