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states. The color-singlet projection with unimodular-like constraint is imposed in the all preceding Hagedorn states. It is reduced to α = 1 2 for color non-singlet SU (3) quark-gluon bags. The values of the mass spectral exponent α with various symmetries are displayed in Table (I). The variation of the chiral Hagedorn’s grand potential density with respect to σ is reduced to 

<img src='content_image/354509.jpg'>

The scalar mean field, namely, σ = < σ > is found by extremizing Ω$_{σ}$ ( T, µ, V, σ ) which is presented in Eq.(149). The value of mean field σ = σ$_{0}$ can be approximated to 

<img src='content_image/354508.jpg'>

The consequence of Eq.(160) when the temperature reaches the deconfinement one is im- portant. The integral term on the right hand side of Eq.(160) diverges for the mass spectral exponent α ≤ 5 2 + 1. It implies that the only possible solution is that ∂ b ( σ ) ∂σ = 0 and σ = 0. This means that the chiral field is certainly restored before the deconfinement phase transition point is reached. Hence, the chiral symmetry is restored far away below the de- confinement point for nuclear matter that is dominated by the orthogonal Hagedorn states (or the Hagedorn states with mass spectral exponent α ≤ 5 2 + 1). On the other hand, it is possible to have a finite value solution for σ for Hagedorn states with mass spectral exponent α > 5 2 + 1. Furthermore, when the exponent exceeds the limit α > 7 2 + 1, then it is pos- sible to have the first order chiral phase transition simultaneously with the deconfinement phase transition point. This support the existence of the tri-critical point at finite µ$_{B}$ and T where the nuclear matter at moderate baryonic densities is supposed to be dominated by the unitary Hagedorn states prior the color deconfinement phase transition. 

## VI. DISCUSSIONS 

In order to study the deconfinement phase transition diagram in QCD, the hadronic density of states must be known for the entire energy domain below the borderline of the 

