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of freedom form an effective color-flavor symmetry group. In other words, the colorless unitary (or orthogonal) state couples with flavor degree of freedom SU$_{V}$ ( N$_{f}$ ) (or SO ( N$_{f}$ )) and forms an effective colorless symplectic Sp ( N ) state. This new symmetry maintains the maximal flavor invariance (i.e. conserved U (1) N$_{f}$ − $^{1}$) and guarantees the colorless state (i.e. color-singlet) in a nontrivial way. Hence, the effective symplectic symmetry is basically a coupled color-flavor symmetry. Furthermore, the color symmetry remains colorless when the symplectic Sp ( N ) symmetry is projected to the color-singlet state while the flavor symmetry remains to maintain its maximal invariance in the following possible scenario 

<img src='content_image/369724.jpg'>

Hence, the extremely dense nuclear matter seems to favor the phase transition from the unitary Hagedorn states to the symplectic Hagedorn states. 

The mass spectral exponents for the Hagedorn states with different internal symmetries are displayed in Table (I). 

Fig. (1) depicts the phase transition diagram in the µ$_{B}$ − T plane. It is shown that at low temperature the dilute nuclear matter which is dominated by the discrete low-lying hadronic mass spectrum states, such as pions, kaons etc and nucleons, Lambdas etc and so on, forms a gas of free discrete low-lying hadrons. Under certain conditions, those hadrons interact with each others and form nuclear matter (usually in the simplest approximation, the hadrons are treated as ideal gas). When the nuclear matter is heated up and/or compressed, it undergoes higher order Gross-Witten Hagedorn phase transition from the nuclear matter that is dominated by the discrete low-lying mass spectrum hadrons to another nuclear matter that is dominated by the continuous high-lying mass spectrum hadronic (Hagedorn) states. The Gross-Witten line appears as the dotted line (red online) in the lower-left of the phase transition diagram. It is shown that the nuclear matter which is dominated by the discrete low-lying hadrons passes higher order Gross-Witten Hagedorn transition to another nuclear matter that is dominated by the continuous high-lying mass spectrum of the colorless unitary SU ( N$_{c}$ ) quark-gluon bag. The order of the phase transition is likely to be a third order one. The intermediate Hagedorn matter is shown above the Gross-Witten line which appears as the dotted line (left) and below the deconfinement phase transition line which appears as the upper thick solid line. It is evident that the continuous high-lying mass spectrum just above 

