flavor symmetry transmutation. In this case the saturate unitary Hagedorn matter transfers to another nuclear matter that is dominated by the symplectic Hagedorn states. The mass spectral exponent for the symplectic matter is α$_{3}$ = 15 / 2. The physics of the symplectic Hagedorn matter is very rich. In the context of the present model, the highly dense neutron star turns to be dominated by the symplectic hadronic matter rather than the conventional quark matter in order to soften the equation of state. Furthermore, the existence of the color super-conductivity phases such as the color-flavor locking phase can be interpreted in terms of the the Hagedorn complex internal structure and the color-flavor coupling channel. In order to understand the microscopic mechanism of the multiple phase transitions from symmetry group to another, the higher order corrections of the hard thermal loops become crucial. Although these corrections are calculated by perturbation, the higher order correc- tions beyond the one-loop correction are highly non-trivial and also the non-perturbative effect may turn to be essential. The simplest way to understand the symmetry breaking from one group to another is to study the symmetry breaking from SU ( N$_{c}$ ) to U (1) N$_{c}$ − $^{1}$. The program to investigate the local and global symmetry behavior at finite temperature and density such as the symmetry breaking or restoration is a straightforward one. In fact this program relies basically on the dependence of effective mass on the temperature [67, 68]. The symmetry is broken when the effective masses turn to be finite and non-degenerate ones and their corresponding adjoint mean-fields become finite ones when the effect of the higher order terms that are beyond the quadratic part in the effective potential turn to be non-negligible. The gluon effective mass in the present case is defined by the gluon’s Debye mass. In order to simplify the discussion to the minimum, the effective potential for the adjoint fields with perturbative corrections beyond the one loop correction will not be con- sidered. Fortunately, the one loop correction sheds some information about what is needed to understand how the symmetry can be broken under extreme conditions. The effective potential for the adjoint fields can be written roughly in the following way where Π µν a $^{′}$a . = 〈 Π µν a $^{$^{′}$a}$( p$_{0}$, ⃗p ) 〉 is the gluon polarization tensor and λ µµ $^{′}$νν ′ 4 aa $^{′}$bb′ is the quartic gluon coupling tensor. The quartic gluon coupling tensor can be calculated from the pertur- bation but also the non-perturbation effect may become signification. The quartic part in V$_{eff}$ ( ˆ A a µ ) . = 1 2 Π µν a $^{′}$a ˆ A a ′ µ ˆ A a ν + 1 3! λ µνκ 3 abc ˆ A a µ ˆ A b ν ˆ A c κ + 1 4! λ µµ $^{′}$νν ′ 4 aa $^{′}$bb′ ˆ A a µ ˆ A a ′ µ ′ ˆ A b ν ˆ A b ′ ν ′ + · · · , . = 1 2 Π µν a $^{′}$a ˆ A a ′ µ ˆ A a ν + 1 4! λ µµ $^{′}$νν ′ 4 aa $^{′}$bb′ ˆ A a µ ˆ A a ′ µ ′ ˆ A b ν ˆ A b ′ ν ′ + · · · , (163) 54