The contribution of the liberated gluons due to the modification of the Hagedorn state’s internal symmetry is given as follows ln Z$_{lg}$ ( T ) = (2 J + 1) V T $^{3}$N$_{lg}$ ( π 2 90 ) , ( where J = 1 2 ) . (140) ln Z$_{H}$$_{−}$$_{matter}$ ( T, V, Λ) = ln Z$_{(}$$_{II}$$_{)}$ ( T, V, Λ) + ln Z$_{lg}$ ( T ) . (141) where ϵ ( ⃗ p, σ ) = √ ⃗ p 2 + ( m ∗ $_{q}$( σ )) $^{2}$, m ∗ $_{q}$( σ ) = m$_{q}$ + g$_{σ}$ σ and σ = < σ > is the chiral mean field condensate. Hence in the background of the chiral field, the quark-gluon partition function, ln Z$_{qqσ}$ ( β, V, σ ) = (2 J + 1) V N$_{f}$ ∑ q ∫ d $^{3}$⃗ p (2 π ) 3 ∑ i [ ln ( 1 + e − β ( ϵ$_{q}$ ( ⃗ p,σ ) − µ$_{q}$ − i θ i β ) ) + ln ( 1 + e − β ( ϵ$_{q}$ ( ⃗ p,σ )+ µ$_{q}$ + i θ i $_{β}$) )] , (142) 39 The role of chiral phase transition can be considered explicitly in the context of the present model [1]. The chiral restoration is expected to take place at the Gross-Witten point or at its neighbourhood but below the point of the color deconfinement. The chiral phase transition is likely to be a first order one if it takes place at the Gross-Witten point [65]. We shall demonstrate in the present section that the chiral phase transition takes place below the deconfinement phase transition point for the orthogonal Hagedorn states. If this is the case, then the standard treatment given in the preceding sections is sufficient. However, if the chiral restoration takes place far away above the Gross-Witten point in the matter that is dominated by the Hagedorn states then the effect of the chiral field becomes important. In this case, the chiral Hagedorn states have to be considered explicitly in the calculation. The contribution of the discrete low-lying hadron mass spectrum is negligible since the nuclear matter at extreme conditions turns to be dominated by the Hagedorn states above the Gross-Witten point. The partition function for the quark and antiquark given by Eq.(21) in the absent of the chiral background is modified when the chiral field is considered in the calculation. In the mean field approximation, Eq.(21) is modified to become V. CHIRAL PHASE TRANSITION The liberated gluons due to symmetry reduction is given by N$_{lg}$ = 0, 1 $_{2}$N$_{c}$ ( N$_{c}$ + 1) and N$_{c}$ ( N$_{c}$ − 1) for unitary, orthogonal and U (1) N$_{c}$ Hagedorn states, respectively. The total partition function for the Hagedorn matter becomes