content_image/341758.jpg content_image/341759.jpg content_image/341760.jpg content_image/341761.jpg
The crucial phenomenon of the model is that the quarks and anti-quarks tends to be confined in the U (1) N$_{c}$ Hagedorn states (i.e. colorless states). These states can be interpreted as quark liquid bubbles. The construction of the color-singlet-state for (unimodular-like) U (1) N$_{c}$ quark-bag is different from those for the color-singlet-state bags with the unitary, orthogonal and symplectic internal symmetry groups with the unimodular-like constraint. The U (1) N$_{c}$ symmetry group maintains the minimal gluonic content in the quark-bag (i.e. the minimal interaction among the quarks and antiquarks since N$_{fun}$ = N$_{c}$ and N$_{adj}$ = N$_{c}$ ). The U (1) N$_{c}$ color-singlet-state canonical ensemble for quark and antiquark bag is constructed in the following way 

<img src='content_image/341758.jpg'>

Instead of considering the U (1) N$_{c}$ symmetry group, the U (1) N$_{c}$ symmetry group with the unimodular-like constraint ( ∑ N$_{c}$ i θ$_{i}$ = 0 ) is adopted in order to be consistent with unitary, orthogonal and symplectic symmetry groups those are restricted to the unimodular-like condition and have the same number of the fundamental color charges. Hence, the canonical ensemble for the U (1) N$_{c}$ color-singlet state of the quark-bag is constructed as follows 

<img src='content_image/341759.jpg'>

where it is restricted to the unimodular-like constraint. At extreme condition, it is reduced to 

<img src='content_image/341760.jpg'>

where 

<img src='content_image/341761.jpg'>

