and The color-singlet-state micro-canonical ensemble for the unimodular-like U (1) N$_{c}$ symmetry reads where Furthermore, using the standard MIT bag model approximation, the mass spectral density is reduced to The high-lying mass spectral density for the color-singlet-state (unimodular-like) U (1) N$_{c}$ symmetry quark-bag reads, where α = α$_{qb}$ = N$_{c}$/ 2 and Generally speaking, it is possible to write the high lying mass spectral density as a linear combination of the Hagedorn states with different internal symmetries in the following way, 32 ρ$_{(}$$_{II}$$_{)}$ ( m ) = symmetries ∑ i a$_{(}$$_{i}$$_{)}$ρ ( i ) ( II $_{)}$( m ) , (118) c$_{qb}$ = 4 3 C$_{qb}$   3 4 ( ˜ u (0) qqg B ) 1 4  $^{−}$$^{N}$$^{c}$ 2 +1 , b$_{qb}$ = ( ˜ u (0) qqg B ) 1 4 . (117) ρ$_{(}$$_{II}$$_{)}$ ( m ) = ρ$_{qbag}$ ( m ) , = c$_{qb}$ m − α$_{qb}$ e b$_{qb}$ $^{m}$, ≡ c m − $^{α}$e$^{b m}$, (116) ρ$_{(}$$_{II}$$_{)}$ = ρ$_{qbag}$ ( m ) , = 4 3 C$_{qb}$   3 4 ( ˜ u (0) qqg B ) 1 4  $^{−}$$^{N}$$^{c}$ 2 +1 m − N c 2 exp   ( ˜ u (0) qqg B ) 1 4 m   . (115) C$_{qb}$ = 1 $_{2}$√$_{N}$$_{c}$ (2 π ) − N$_{c}$/ 2 ( 3˜ u (0) qqg ˜ u (2) qqg ) N c 2 − 1 2 . (114) Z$_{qbag}$ ( W, V ) = C$_{qb}$ 1 W [ ( 3˜ u (0) $_{qqg}$V W 3 ) 1 / 4 ] − ( N c 2 − 1) exp [ 4 3 ( 3˜ u (0) $_{qqg}$V W 3 ) 1 4 ] , (113) ˜ u (2) qqg = (2 J + 1) N$_{f}$ ∑ f =1 1 6 [ 1 + 3 π 2 ( µ$_{f}$ T ) 2 ] + (2 J + 1) ( 1 6 ) . (112)