# How Does A $\theta$ Angle Shift Affect the Wilsonian Effective Lagrangrian?

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.

• The $\theta$ term is topological, whatever stuff you integrate out it still remains there, do you agree? – MannyC Apr 13 '19 at 4:57
• I could imagine there being high energy configurations with non-trivial winding number contributing to $L_{\Lambda}$. – LucashWindowWasher Apr 13 '19 at 5:13
• Related question by OP: physics.stackexchange.com/q/472393/2451 – Qmechanic Apr 13 '19 at 7:31
• Do you have any suggestions for how I can make this question more clear? – LucashWindowWasher Apr 13 '19 at 16:38

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.