After regularising a transition amplitude, we end up with an expression which depends on regularisation scale. This means that our physical observables like cross section will be a function of regularisation scale. But I am not sure which value we should choose for such regulator.

Let’s give a simple example of scattering amplitude for $\phi \phi \to \phi \phi$ in $\phi^4$ theory at one loop order:

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Using, say, Pauli Villars Regularisation, four-point function reads: $$ \Gamma_4 = -i \lambda \left( 1-\frac{3\lambda}{16\pi^2} \ln \frac{\Lambda}{m}-\frac{1}{2} \right)$$ Where $m$ is mass, and $\Lambda$ is regularisation scale.

In this case, which value should I use for $\Lambda$ to compute the correct cross-section?

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    $\begingroup$ Measurable quantities (such as cross-sections) should be independent of the regularisation parameters (cut-offs, dimReg scales, etc.). If they are not, you did something wrong. $\endgroup$ – AccidentalFourierTransform Oct 4 '18 at 16:15
  • $\begingroup$ @AccidentalFourierTransform, I double-checked the computation, and the expression for 4-point function written in my question is definitely correct. I did a wick rotation, and replaced infinity with cut-off $\Lambda$, and solved the integral. I might be missing a final step at the end that removes the regularisation scale. $\endgroup$ – Ramtin Oct 5 '18 at 0:50
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    $\begingroup$ you haven't introduced the appropriate counterterm yet. You need to do so to make $\Gamma_4$ finite in the limit $\Lambda \to \infty$ $\endgroup$ – tbt Oct 5 '18 at 11:02

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