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Suppose we have a one dimensional field theory for the field $\phi(r)\;r\in[0,\infty]$ and that the solution for the background (Euler Lagrange equations) give a function $\phi_0$ that goes to a constant at infinity.

Now we want to consider the quantum corrections $\delta\phi$ to this background solution for large values of $r$.

Can we guess a behavior for these solutions? Maybe we can say that the quantum corrections must be lower of the mean background or satisfy some other bound so that for instance if the field goes to a constant like $1/r^n$ than also the derivatives of the quantum corrections must be suppressed by $r^n$, or equivalently is $\phi_0\sim1/r^n$ then $\delta\phi\sim1/r^n$.

For now I would say that if we consider operators obtained by replacing powers of the background $\phi_0$ with $\delta\phi$ in the monomi of the Lagrangian, then if we replace just one $\phi_0'$ with one $\delta\phi'$ we can integrate by parts and find that (since the rest of the monomial depends on the background only) it must be $\delta\phi'/\delta\phi\sim\phi_0'/\phi_0$ at $r\rightarrow\infty$.

Can that be made precise? Are there other bounds one can find? Is there some kind of physical reasoning (like naturalness of the observables perhaps?) to bound the quantum corrections of a field?

Thanks for any help!

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