Reviewing the anomalies of the standard model, one knows that the Baryon number is not conserved because of an anomaly associated to the global $U(1)$ symmetry that quarks have. That is the current $$J_B^\mu = \sum_\alpha \bar{\psi}_\alpha\gamma^\mu\psi_\alpha,$$ where $\alpha$ runs over flavors and colors, is not conserved and its divergence is proportional to products of the field strength tensor times its dual. Specifically we have a symmetry transformation which is $$\psi\rightarrow e^{i\theta}\psi\qquad \text{and}\qquad \bar{\psi}\rightarrow e^{-i\theta}\psi\tag{1}\label{eq:symmetry}$$
In a path integral derivation of the anomaly through Fujikawa's method, one gauges the global symmetry and imposes that the action has a zero functional variation with respect to this "auxiliary" symmetry. Then one studies the Jacobian coming out of the field redefinition involved in this process and usually realizes that it is not trivial. However trying to do this for this case, the Jacobian obtained from the transformations in Eq. \eqref{eq:symmetry} is schematically: $$\mathcal{J} = \left[\det(e^{i\theta(x)})\right]^{-1}\left[\det(e^{-i\theta(x)})\right]^{-1}$$
which doesn't show the anomaly. Am I missing something or is this anomaly "invisible" to Fujikawa's method?
Reference (more details on what I am saying)
- John F. Donoghue, Dynamics of the Standard Model, 2014, Cambridge University Press, Ch. III