Strongly coupled fermions in odd dimensions and the running cut-off $\Lambda_d$

I study the fermionic $U(N)$ Gross-Neveu model at imaginary chemical potential and finite temperature for odd $d$ dimensions, in the strong coupling regime, by using the gap (saddle point) equation for the fermion condensate of the model. This equation describes the phase transitions from weak to strong coupling regime. I point out that the higher odd dimensional gap equations are linear combinations of the lower dimensional equations in a way that as the dimension of the model increases the lower dimensions are weaker coupled but still in the strong coupling regime. Interestingly, at a specific value of the chemical potential, exactly in the middle of the thermal windows that separate the fermionic from the bosonic (condensed) state of the fermions, I find the mass of the fermion condensate for $d=3,5,7,9$. An anomaly occurs at the $5$ dimensional theory where it is stronger coupled against other theories in higher dimensions and lower energy. The main idea of this work is that the cut-off $\Lambda$ regulator for the UV divergent parts of the fermion mass saddle point equation, plays the role of a physical parameter. This idea is based on the identity of the asymptotic freedom of the Gross-Neveu model as a toy model for QCD.