- #1

- 48

- 1

- TL;DR 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

Many thanks in advance.

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

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...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 $$\bar{\omega}(k^2+i\varepsilon)-\bar{\omega}(k^2-i\varepsilon)=2i\,\mathrm{Im}\, \bar{\omega}(k^2+i\varepsilon)$$

Many thanks in advance.