After integration by parts, the quadratic Proca action with an external source isSince $M>0$, this kernel is invertible with the Feynman i-epsilon prescription; the massless gauge redundancy is absent in this unitary gauge quadratic action. Complete the square by $A\mapsto A+K_F^{-1}J$ in the Gaussian functional integral. Its source-independent determinant cancels in the normalized generating functional:In Fourier space, the kernel isFor $k^2\ne 0$, use the transverse projector of a vector field $P_T^{\mu \nu }=g^{\mu \nu }-k^{\mu }{k}^{\nu }/k^2$ and the longitudinal projector of a vector field $P_L^{\mu \nu }=k^{\mu }{k}^{\nu }/k^2$. Their mixed-index products obey $P_T{P}_L=0$, soThe last expression also extends to $k^2=0$; that point only obstructs the intermediate projector decomposition. Two source functional derivatives yield the gauge-boson propagator $D_F=i{K}_F^{-1}$:The pole residue is the physical massive-vector massive vector polarization sum. The longitudinal term is essential for inverting the quadratic kernel, and makes the naive $M\to 0$ limit of this unitary gauge Proca propagator singular.

For the one-loop renormalization-group beta function, write $t=\log \mu$. Separating variables givesThis expression resums the leading logarithms of the one-loop running coupling, while omitting higher-order terms in the renormalization-group beta function. Assume a positive initial coupling.

For $b_0<0$, increasing the renormalization scale decreases the running coupling; the zero-coupling renormalization-group fixed point is attractive in the ultraviolet. This is asymptotic freedom. Toward lower scales the one-loop expression has a formal pole at ${\mu }_i\exp [1/(b_0{\alpha }_i)]<{\mu }_i$. The strong-coupling neighborhood of that pole lies outside perturbative control. QCD provides the main example; the minimal Standard Model $SU(2)$ coupling is also asymptotically free at one loop.

For $b_0>0$, the running coupling grows with the renormalization scale and formally diverges at the ultraviolet Landau pole ${\mu }_i\exp [1/(b_0{\alpha }_i)]>{\mu }_i$. Toward the infrared it decreases to zero within this approximation. The Standard Model $U(1)$ weak hypercharge coupling has this sign, as does QED in a fixed charged-particle regime. A perturbative Landau pole marks failure of the approximation, rather than proving that the exact theory has that divergence.

For $b_0=0$, the running coupling is constant at this order; higher-loop terms decide whether it runs. Vanishing of the leading coefficient alone does not establish exact scale invariance.

For a concrete common convention, define ${\alpha }_i=g_i^2/(4\pi )$ with $g_Y$ in its usual, non-unified weak hypercharge normalization. In the minimal Standard Model with three generations and one Higgs doublet, above its particle thresholds,Thus the question's $b_0$ is $b_i/(2\pi )$, with the sign regimes just described. Crossing a mass threshold changes the active field content and requires the appropriate piecewise renormalization-group beta function.