phase transition diagram is analysed using the lattice calculations [51, 52]. The models of clustered quarks are demonstrated to smoothen the order of phase transition (for instance see [53]). Furthermore, the Hagedorn states might play a significant role in the formation of new class of compact stars at high densities. For instance, it has been proposed that the the hypothetical quark stars are made of very massive quarks rather than ordinary baryons [54]. 

For only the sake of simplification, we do not include the Van der Waals (excluded) volume correction in the present work. Nonetheless, the extension to include the repulsive excluded volume correction is straightforward though it requires more tedious numerical calculation. This kind of calculation has been considered in Ref. [1, 2] and the references therein where the order and the shape of the deconfinement phase transition to quark- gluon plasma have been studied extensively. It has been shown that the order and shape of the phase transition depends on the internal structures of the quark-gluon bags. Different color-flavor configurations lead to different exponents α for the continuous high-lying mass spectrum (Hagedorn states). The bag’s internal structure depends on several constraints which are imposed to restrict the color and flavor degrees of freedom. Since the flavor degree of freedom in the hot nuclear matter tends to be invariance even for bags with more complicated color-flavor coupling, the color confinement is maintained basically by projecting the color-singlet state. Hence, the bound state that exists through the intermediate processes is a colorless state as far the deconfinement phase transition is not reached yet. 

In the present work, we study the color-singlet state canonical ensemble for the quark- gluon bag in the context of various internal symmetry groups such as unimodular-like unitary U ( N$_{c}$ ), orthogonal O$_{(}$$_{S}$$_{)}$ ( N$_{c}$ ) and symplectic Sp ( N$_{c}$ ) symmetry groups as well as U (1) $^{N$_{c}$}$. The intermediate transmutation from the hadronic matter that is dominated by the colorless unitary fireballs to another hadronic matter that is populated by the colorless orthogonal states or the colorless symplectic states is introduced in order to demonstrate the multi- nuclear phases in the phase transition diagram and the existence of the tri-critical point. We discuss possible scenarios for the deconfinement phase transition from the colorless states of unitary, orthogonal and symplectic hadronic matter to the colored SU ( N$_{c}$ ) quark-gluon plasma. These scenarios are relevant to the possible intermediate hadronic phases those take place below the deconfinement phase transition and the emergence of the tri-critical point. 

The outline of the present paper is as follows: In Sec. II, we derive the quark-gluon grand 

