The chemical potential $\mu$, is introduced in the action as the lagrange multiplier $$ \tag 1 S[Q_{0}] \to S[\mu] = S[Q_{0}]-\int dt \mu Q_{0}(t), $$ where $$ Q_{0}(t) = \int d^{3}\mathbf r J_{0}(\mathbf r, t) $$ is the conserved charge carried by 4-current $J$, associated with $\mu$.
I've heard an idea that it is equal to the homogeneous $\mathbf q = 0$ and zero frequency $\omega \to 0$ limit of zeroth component of the fictious gauge field $V_{\mu}$, coupled to the current $J_{\mu}$: $$ \tag 2 \text{lim}_{\omega \to 0, \mathbf q = 0}V_{\mu}(\mathbf q , \omega) \equiv \delta (\omega)\delta (\mathbf q)(\mu , 0,0,0) $$ It seems that this is formally correct description (although I don't understand why do we need it). However, there are some issues.
Suppose for concreteness the chemical potential associated with the vector charge of fermion matter $\psi$ (for simplicity the one specie). In order to deal with it, let's introduce vector field $B_{\mu}$, so that $$ L(\psi, A,B) = \bar{\psi}\gamma^{\mu}(i\partial_{\mu} +A_{\mu}+B_{\mu})\psi $$ As is well-known, this theory (if we want to treat is as $A$ field gauge invariant) contains the chiral anomaly - non-conservation of the axial current $J^{\mu}_{5}$: $$ \tag 3 \partial_{\mu}J^{\mu}_{5} = c(F_{A}\tilde{F}_{A} + F_{B}\tilde{F}_{B} + 2F_{A}\tilde{F}_{B}), $$ where $F$ is the gauge field strength, and $\tilde{F}$ is the dual gauge field strength.
From $(3)$ we see, that the current $J^{\mu}_{5}$ can be redefined as $$ J^{\mu}_{5} \to \tilde{J}^{\mu}_{5} = J^{\mu}_{5} - 4c\epsilon^{\mu\nu\alpha\beta}B_{\nu}F_{\alpha\beta}^{A} - 2c \epsilon^{\mu\nu\alpha\beta}B_{\nu}F^{B}_{\alpha\beta}, $$ so that $$ \partial_{\mu}\tilde{J}^{\mu}_{5} = cF_{A}\tilde{F}_{A} $$ In the limit $(2)$ the axial current acquires additional contribution $$ \tag 3 \Delta J^{\mu}_{5} \equiv \tilde{J}^{\mu}_{5} - J^{\mu}_{5} = 4\delta^{\mu}_{i}c\mu B_{i} $$ which seems to be absent in the case of properly defined chemical potential $(1)$.
My question is: is the interpretation $(2)$ of the chemical potential valid (in the light of $(3)$)?
