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Say we have some quantum field theory which includes a gauge field, and some matter, and a topological $\theta$ term so that the Lagrangian reads $$L=(stuff)+\frac{\theta}{64\pi^2}\varepsilon^{\mu\nu\rho\sigma}F^a_{\mu\nu}F^a_{\rho\sigma}$$

We know that the Lagrangian behaves as follows under the transformation $\theta\to\theta +a$ $$L\to L+\frac{a}{64\pi^2}\varepsilon^{\mu\nu\rho\sigma}F^a_{\mu\nu}F^a_{\rho\sigma}$$

If we now consider the Wilsonian Effective Lagrangian $L_{\Lambda}$, what can we say about how it transforms under a shift like $\theta\to\theta+a$? Can we say that it transforms the same way?

I ask this because this is what Weinberg assumes in section 29.3 of his Quantum Theory of Fields book, that it does indeed transform the same way. Specifically around equation 29.3.7.

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  • $\begingroup$ The $\theta$ term is topological, whatever stuff you integrate out it still remains there, do you agree? $\endgroup$ – MannyC Apr 13 at 4:57
  • $\begingroup$ I could imagine there being high energy configurations with non-trivial winding number contributing to $L_{\Lambda}$. $\endgroup$ – LucashWindowWasher Apr 13 at 5:13
  • $\begingroup$ Related question by OP: physics.stackexchange.com/q/472393/2451 $\endgroup$ – Qmechanic Apr 13 at 7:31
  • $\begingroup$ Do you have any suggestions for how I can make this question more clear? $\endgroup$ – LucashWindowWasher Apr 13 at 16:38
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Weinberg does not actually assume that the Wilsonian Effective Lagrangian transforms in the same way!

The statement he makes is that the transformation $\theta\to\theta+a$ with $a$ a specific number will cancel the transformation made by the anomaly. This is because these two transformations make equal but opposite contributions to the action before integrating out high energy modes.

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