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For massless fermions the classical axial rotation has Noether currentPromote $\beta$ to a spacetime-dependent parameter. The classical variation, after integration by parts, is $-\int d^{4}x\beta {\partial }_{\mu }{J}_A^{\mu }$. The chiral anomaly is equivalently encoded by the stated shift $\theta \mapsto \theta +2N_f\beta$, which changes the theta action byThe anomalous Ward identity is thereforeso $\alpha =N_f{g}^2/(8{\pi }^2)$ in the trace convention of the question.

Taking the divergence of the Noether current,because the two gradient-product terms cancel. The field equation and its conjugate are $\square \varphi =V^{\prime }\varphi$ and $\square \bar{\varphi }=V^{\prime }\bar{\varphi }$, with real $V^{\prime }(|\varphi {|}^2)$. Hence

The supercurrent is the spinor-valued Noether current of rigid supersymmetry. Its spatial integral is a supercharge. Localizing the supersymmetry parameter produces ${\partial }_{\mu }\overline{ϵ}{S}^{\mu }$ in the variation of the action; a gravitino coupling cancels this term and starts the Noether gauging procedure.

A virial current $V^{\mu }$ represents the trace of a conserved stress-energy tensor as a divergence. Then the dilation Noether current $D^{\mu }=x_{\nu }{T}^{\mu \nu }-V^{\mu }$ is conserved, since its divergence is $T^{\mu }{}_{\mu }-{\partial }_{\mu }{V}^{\mu }$. This establishes scale invariance; a traceless stress-energy tensor improvement is the extra condition needed for conformal currents.

For the left-parameter convention and ghost kinetic term $(\partial b)Dc$, localization of the BRST symmetry yields this conserved Noether current. Subtract the total-derivative contribution $K^{\mu }=h^a(D^{\mu }c{)}^a$ in the Lagrangian variation. The BRST charge is the spatial integral of $j^0$. The antighost term has the displayed sign because the parameter and antighost field derivative anticommute.