with temperature and density, it is naturally to surmise multiple phase transitions such as SU ( N$_{c}$ ) → SU ( N$_{c}$ − 1) × U (1) → · · · → U (1) N$_{c}$ − 1 (see for instance Ref. [70]). More recent discussions of the Higgs mechanism in the semi-quark-gluon plasma can be found in Ref. [71] and the references therein. Instead of breaking the symmetry SU ( N$_{c}$ ) → SU ( N$_{c}$ − 1) × U (1), the symmetry transmutation undergoes multiple symmetry breaking transitions to SU ( N$_{c}$ ) → O ( N$_{c}$ ) → SU ( N$_{c}$ − 1) × U (1) in order to maintain the quark confinement. The intermediate symmetry transmutation O ( N$_{c}$ ) smoothes the symmetry breaking from SU ( N$_{c}$ ) to U (1) N$_{c}$ − 1 in order to soften the Hagedorn’s equation of state. On the other hand, the microscopic mechanism in the multi-processes for the formation of symplectic symmetry Sp ( N$_{c}$ ) is rather more complicated because of the correlation between the flavor and color symmetries. The symmetry reduction undergoes multiple processes, namely, SU$_{V}$ ( N$_{f}$ ) × SU ( N$_{c}$ ) → O (2 N$_{c}$ ) × · · · → Sp ( N$_{c}$ ). In the process of forming the symmetry Sp ( N$_{c}$ ), the color degrees of freedom favor to couple the flavor degrees of freedom through anti- symmetric channels to form diquark condensates and energy gaps. The adjoint mean-fields acquire degenerate/non-degenerate effective gluon masses depending of the type of the color superconductivity that is formed in the medium. The symmetry breaking leads to massless Nambu-Goldstone bosons which come from flavor sector and massive gluons which stem from the color sector. In principle, the energy gaps of the color-superconductivity can be included explicitly in the calculation of the canonical ensemble. The multiple symmetry breaking processes from the matter that is dominated by SU ( N$_{c}$ ) Hagedorn states to another one that is dominated by U (1) N$_{c}$ − 1 and the corresponding Higgs mechanism is sketched in Fig.(5). In order to simplify the present work, we assume that the liberated gluons are approximated to an ideal gas of massless particles. In this approximation, the gluon effective masses which are acquired through the symmetry breaking process are neglected and this does not change the conclusion of the possible multiple phase transitions. 

## VII. CONCLUSION 

The present work is intended to determine the physics behind the intermediate processes toward the quark-gluon plasma in the µ$_{B}$ − T phase transition diagram and the variation of the mass spectral exponent for the Hagedorn mass spectral density in the medium. The µ$_{B}$ − T phase transition diagram from the Hadronic phase to the quark-gluon plasma is 

