function is reduced to where The partition function for quark and anti-quark in the Fock space becomes The Vandermonde determinant, which appears in the invariance Haar measure, contributes to the action as an additional effective potential term. The Vandermonde effective potential term becomes soft when the color eigenvalues of the stationary condition are distributed uniformly over the entire circle | θ$_{i}$ | ≤ π . The integral of the resultant ensemble can be evaluated trivially under such circumstances. However, this will not be the case under the extreme hot and dense conditions in particular when the color eigenvalues turn to be distributed in a narrow interval | θ$_{i}$ | << π . When the saddle points congregate around the origin rather than are distributed uniformly over the entire color circle range | θ$_{i}$ | ≤ π , the Vandermonde effective potential develops a virtual singularity. Subsequently, the action must be regulated in a proper way in order to remove the Vandermonde determinant divergence. This regulation procedure accommodates the extreme hot and dense conditions. It is usually associated with smooth phase transition. The regulation procedure corresponds the Gross-Witten third order Hagedorn phase transition in the context of U ( N$_{c}$ ) in the limit N$_{c}$ → ∞ . The same procedure can be extended to other symmetry groups such as the orthogonal and symplectic ones. The change in the analytic solution corresponds higher order phase transition from the dilute and relatively cold hadronic matter (i.e. discrete low- lying hadronic mass spectrum phase) to the highly thermally excited hadronic matter (i.e. continuous high-lying hadronic mass spectrum phase). This kind of nuclear phase transition would not mean that the deconfinement phase transition is reached. The highly thermally excited hadronic matter is interpreted as an exotic hadronic phase that is dominated by the Hagedorn states. In the lattice theory, this is implying that the weak and strong coupling is not described by the same analytic function. This strong- and weak-coupling transition 12 Z$_{qq}$ ( β, V ) = exp [ (2 J + 1) V ∫ d $^{3}$⃗ p (2 π ) 3 ( 1 2 ∑ i ln [ a 2 q ¯ q ( θ$_{i}$ ) + b 2 q ¯ q ( θ$_{i}$ ) ] )] . (24) a$_{q}$$_{¯}$ q ( θ$_{i}$ ) = ( 1 + cos( θ$_{i}$ ) ∑ N$_{f}$ q [ e − β ( ϵ$_{q}$ ( ⃗ p ) − µ$_{q}$ ) + e − β ( ϵ$_{q}$ ( ⃗ p )+ µ$_{q}$ ) ] + e − 2 β ϵ$_{q}$ ( ⃗ p ) ) , b$_{q}$$_{¯}$ q ( θ$_{i}$ ) = sin( θ$_{i}$ ) ∑ N$_{f}$ q [ e − β ( ϵ$_{q}$ ( ⃗ p ) − µ$_{q}$ ) − e − β ( ϵ$_{q}$ ( ⃗ p )+ µ$_{q}$ ) ] . (23) ln Z$_{qq}$ ( β, V ) = ℜ e ( (2 J + 1) V ∫ d $^{3}$⃗ p (2 π ) 3 [ ∑ i ln ( a$_{q}$$_{¯}$ q ( θ$_{i}$ ) + ib$_{q}$$_{¯}$ q ( θ$_{i}$ )) ]) , (22)