$$A_0$$-condensation in quark-gluon plasma with finite baryon density

In the present paper, we return to the problem of spontaneous generation of the $$A_0$$ A 0 -background field in QCD at finite temperature and a quark chemical potential, $$\mu $$ μ . On the lattice, this problem was studied by different approaches where an analytic continuation to the imaginary potential $$i \mu $$ i μ has been used. Here we consider both, real and imaginary chemical potential, analytically within the two-loop gauge-fixing independent effective potential $$W_{eff.}$$ W e f f . . We realize the gauge independence in to ways: (1) on the base of Nielsen’s identity and (2) expressing the potential in terms of Polyakov’s loop. Firstly we reproduce the known expressions in terms of Bernoulli’s polynomials for the gluons and quarks. Then, we calculate the $$\mu $$ μ -dependence, either for small $$\mu $$ μ as expansion or numerically for finite $$\mu $$ μ , real and imaginary. One result is that the chemical potential only weakly changes the values of the condensate fields, but quite strongly deepens the minima of the effective potential. We investigate the dependence of Polyakov’s loop in the minimum of the effective potential, thermodynamic pressure and Debye’s mass on the chemical potential. Comparisons with other results are given.