Local and non-local counterterms in the ultraviolet and infrared limits of a quantum field theory These is a question about the renormalization of a quantum field theory. See below equation (19.3) in page 176 of Thomas Hartman's notes on Quantum Gravity and Black Holes.

When we renormalize a local quantum field theory, why are we only allowed to add local counterterms to the Lagrangian?
Why do non-local terms come from the infrared physics of a quantum field theory?
 A: In renormalized perturbation theory, one is simply rewriting the Lagrangian; therefore a local bare Lagrangian will yield a renormalized Lagrangian with only local terms.  Take for example $$\mathcal{L} = \frac{1}{2}\partial_\mu\phi_0\partial^\mu\phi_0-\frac{1}{2}m_0^2-\frac{\lambda_0}{4!}\phi_0^4.\hspace{1in}[1]$$
Define $$\phi_r\equiv Z_\phi^{-1/2}\phi_0$$ $$m_r^2\equiv Z_m^{-1}Z_\phi m_0^2$$ $$\lambda_r\equiv Z_\lambda^{-1}Z_\phi^2\lambda_0\mu^{-\epsilon},$$
where $\mu$ is some dimensionful scale that absorbs the change in dimensions of $\lambda$ due to dimensional regularization.  Then $$\mathcal{L} = \frac{1}{2}Z_\phi\partial_\mu\phi_r\partial^\mu\phi_r-\frac{1}{2}Z_m m_r^2\phi_r^2-Z_\lambda\frac{\lambda_r}{4!}\mu^\epsilon\phi_r^4.\hspace{1in}[2]$$
The point you should see is that [2] is exactly equal to [1]; there's only been a redefinition of quantities.  So then when we define counter terms
$$\delta_\phi\equiv Z_\phi-1$$ $$\delta_m\equiv m_r^2Z_m-1$$ $$\delta_\lambda\equiv\lambda_r Z_\lambda-1,$$ then our renormalized Lagrangian with counter terms has only local operators:
$$\mathcal{L} = \frac{1}{2}\partial_\mu\phi_r\partial^\mu\phi_r-\frac{1}{2}m_r^2\phi_r^2-\frac{\lambda_r}{4!}\mu^\epsilon\phi_r^4 \\+\frac{1}{2}\delta_\phi\partial_\mu\phi_r\partial^\mu\phi_r-\frac{1}{2}\delta_m\phi_r^2-\frac{\delta_\lambda}{4!}\mu^\epsilon\phi_r^4.$$
I'm not quite sure what you're referring to in terms of non-local terms and infrared physics; could you please clarify?  There are certainly IR divergences for many QFT processes, but these IR divergences can be absorbed systematically to yield IR finite results.
