# Pauli-Villars regularization for Vacuum Polarization

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Summary:
In most of the textbooks corresponding integral is computed in ##k^2 <4m^2## assumption. How to extend?
Hello!

I am currently reading Itzykson Zuber QFT book and on Chapter 7 where for the first time loops are considered. Particular method of dealing with divergences namely Pauli-Villars regularization is considered in section 7-1-1 considering vacuum polarization diagram. I do understand physics here but lost in math. The integral is computed in the textbook but in the assumption ##k^2<4m^2## and the result is the following

$$\bar{\omega}(k^2,m,\Lambda)=-\frac{\alpha}{3\pi}\Bigg\{-\log \frac{\Lambda^2}{m^2}+\frac{1}{3}+2\Bigg(1+\frac{2m^2}{k^2}\Bigg[\Bigg(\frac{4m^2}{k^2}-1\Bigg)^{\frac{1}{2}} \times\\ \mathrm{arccot}\Bigg(\frac{4m^2}{k^2}-1\Bigg)^{\frac{1}{2}}-1\Bigg]\Bigg\}$$

I am completely fine with this calculation however the following phrase from the textbook is not clear
The calculation has been carried out under the assumptio ##k^2<4m^2##. The corresponding function may be continued in the complex ##k^2## plane. The values for ##k^2>4m^2## are obtained by taking a limiting value form above the cut, starting at the point ##k^2=4m^2##. The discontinuity across the cut is $$\bar{\omega}(k^2+i\varepsilon)-\bar{\omega}(k^2-i\varepsilon)=2i\,\mathrm{Im}\, \bar{\omega}(k^2+i\varepsilon)$$
Please explain why to obtain values for ##k^2>4m^2## one should take this limit. And why discontinuity across the cut does matter here at all. Completely lost...

Please, don’t get me wrong I do understand how analytic continuation works. And all functions in the expression are analytic. The question is why “above the cut”? Analytic continuation below the cut is also available but why it is considered to be unphysical?

vanhees71