The mixed-grand canonical ensemble for the matter of chiral Hagedorn states reads The total grand potential density for the chiral Hagedorn matter in the presence of the chiral field is given by The chiral Hagedorn’s grand potential density ( i.e. the first term on the right hand side of Eq.(149) ) is given as follows, The chiral potential U ( σ ) ( i.e. the second term on the right hand side of Eq.(149) ) can be introduced by a phenomenological one in the following way where the cubic and quartic terms represent the scalar self-interactions and they were pro- posed by Boguta and Bodmer [66]. In the simplest approximation, the cubic and quartic terms are neglected by taking c$_{2}$ = 0 and c$_{3}$ = 0 and this corresponds to the Walecka-like models. The value of the mean field σ is determined by calculating the extremum (i.e. minimum) of the total grand potential density, namely, In order to calculate the chiral Hagedorn’s grand potential density that is given in Eq.(150), the high-lying mass spectral density for chiral Hagedorn states is rewritten in the following way 41 ρ$_{H}$ ( m, σ ) = cm − $^{α}$eb ( σ ) $^{m}$. (153) 1 V ∂ ∂σ Ω$_{σ}$ ( T, µ, V, σ ) = 0 . (152) U ( σ ) = 1 2 m 2 σ σ 2 + 1 3 c$_{2}$ σ 3 + 1 4 c$_{3}$ σ $^{4}$, (151) 1 V Ω$_{H σ}$ ( T, µ, V, σ ) = − T ∂ ∂V ln Z$_{(}$$_{II}$$_{)}$ ( T, µ, V, σ ) , = − 2(2 J$_{H}$ + 1) cosh ( µ$_{H}$/T ) × T ∫ ∞ m$_{0}$ dmρ$_{H}$ ( m, σ ) ( mT 2 π ) 3 / 2 e − $^{m/T}$. (150) 1 V Ω$_{σ}$ ( T, µ, V, σ ) = − T ∂ ∂V ln Z$_{(}$$_{II}$$_{)}$ ( T, µ, V, σ ) + U ( σ ) , = 1 V Ω$_{H σ}$ ( T, µ, V, σ ) + U ( σ ) . (149) ln Z$_{(}$$_{II}$$_{)}$ ( T, µ, V, σ ) = 2(2 J$_{H}$ + 1) cosh ( µ$_{H}$/T ) × V ∫ ∞ m$_{0}$ dm ρ$_{H}$ ( m, σ ) ( mT 2 π ) 3 / 2 e − $^{m/T}$. (148)