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process smoothen the order of the phase-transition remarkably at µ$_{B}$ ≈ 0. This process is depicted as the symbol ∗ in Fig. (2). Nonetheless, it is possible for the colorless orthogonal (or unitary) states to pass the phase transition to metastable colored bags via the following reaction: 

<img src='content_image/351255.jpg'>

The overall color charge of each bag remains a color-neutral one (i.e. neutral color charge for N$_{fun}$ fundamental quarks and N$_{adj}$ adjoint gluons) before the bags expand and become colored ones and, eventually, form colored quark-gluon plasma. This kind of phase transition is likely supposed to be of higher order and takes place on the left hand side of the tri-critical point which appears in the phase transition diagram. When µ$_{B}$ increases, the flavor struc- ture becomes more rich and complex. The flavor symmetry invariance turns to be broken and involved. Therefore, the complexity of the color-flavor structure increases as the system is compressed and cooled down to lower temperature. The color-flavor internal symmetry is modified in an optimal way minimizing the flavor dependence and maximizing the preserva- tion of the colorless state. This mechanism explain why the quarks avoid to propagate freely in the medium. When µ$_{B}$ becomes significantly large, the complexity of the color-flavor sym- metry increases and the SU ( N$_{c}$ ) symmetry couples with the SU ( N$_{f}$ ) × SU ( N$_{f}$ ) symmetry. The internal symmetry of the quark and gluon bag turns to be Sp ( N$_{c}$ ) color-flavor symme- try. At high temperature, both the flavor invariance and the color-singlet constraint (i.e. colorless state) modify the quark-gluon bag’s underlying internal symmetry from SU ( N$_{c}$ ) to O ( N$_{c}$ ). The quarkyonic matter is conjectured to be dominated by the continuous high-lying hadronic states (Hagedorns) with varied symmetries such as the orthogonal, unitary and symplectic ones. 

It is shown in Fig. (3) that the mass spectral exponent α is modified with respect to µ$_{B}$ and T below the brink of the deconfinement phase transition borderline to the quark-gluon plasma. The Hagedorn’s mass spectral exponent α is small for the dilute and hot hadronic matter. The exponent α is modified as the hadronic matter is compressed and/or heated. The exponent varies and takes the values α$_{1}$ ≤ α$_{2}$ ≤ α$_{3}$ . The exponents are reduced to α$_{1}$ = 3, α$_{2}$ = 9 / 2 and α$_{3}$ = 15 / 2 for the colorless orthogonal, unitary and symplectic Hagedorn states, respectively, when N$_{c}$ = 3. The mass spectral exponent for the colorless U (1) N$_{c}$ composite states (Hagedorns) is reduced to α = 3 / 2 for N$_{c}$ = 3. Furthermore, the 

