Pauli-Villars regularization for Vacuum Polarization

In summary, the integral in the textbook is computed but the assumption of ##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{ar
  • #1
Korybut
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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
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...

Many thanks in advance.
 
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  • #2
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?
 
  • #3
This is indeed a subtle point. When calculating the vacuum polarization from the Feynman diagrams involved you already see that the corresponding "self energy" ##\Pi(k^2)## must be a meromorphic function with an exceptional singularity at ##k^2=4m^2##, where ##m## is the electron mass (assuming you talk about standard QED with electrons, positrons and photons).

To define your function completely you must define a branch cut and the Riemann sheet you want to define the meromorphic function on. In standard vacuum QFT you want to calculate the time-ordered Green's functions or, as here, building blocks to calculate them, which leads to the conclusion that you need to calculate the truncated one-particle irreducible diagrams (aka the vertex functions).

The time-ordering prescription tells you that for the vacuum polarization you want to define the cut along the real ##k^2## axis, i.e., along ##k^2>4 m^2##, and for ##k^2<4m^2## the function should be real. Along the cut the values of the function when taking the limits above and below the cut then yield complex conjugate values. Except along the cut (and maybe some poles on the real axis describing bound states, but these you get only if doing corresponding resummations of "ladder diagrams" to find them) the vacuum polarization is a analytic function and thus you can calculate the function everywhere from knowing the discontinuity along the cut, i.e., from the imaginary part. This leads to the Kallen-Lehmann spectral representation.

You can also renormalize self-energy diagrams in this way: There are Feynman rules to calculate the discontinuity along the cut (the socalled Cutkosky or "cutting" rules), which is also called the spectral function, ##\rho=-2 \mathrm{Im} \Pi(k^2+\mathrm{i} \epsilon)##, ##k^2 \in \mathbb{R}##, and this is usually finite. Now in standard QED the spektral function goes logarithmically with ##k^2##, i.e., the dispersion integral from the Kallen-Lehman representation diverges logarithmically, but you can subtract it according to the physical renormalization description ##\Pi(0)=0##, and then the resulting dispersion integral is finite, leading directly to the renormalized vacuum polarization without the need of a regularization.
 
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Related to Pauli-Villars regularization for Vacuum Polarization

1. What is Pauli-Villars regularization for Vacuum Polarization?

Pauli-Villars regularization is a mathematical technique used in quantum field theory to remove divergences in calculations involving vacuum polarization. It involves adding a set of fictitious particles, called Pauli-Villars regulators, to the theory in order to cancel out the divergences.

2. Why is Pauli-Villars regularization necessary for Vacuum Polarization?

Vacuum polarization is a phenomenon in quantum field theory where the presence of virtual particles in the vacuum affects the behavior of real particles. However, this leads to divergent integrals in calculations. Pauli-Villars regularization is necessary to remove these divergences and make the calculations meaningful.

3. How does Pauli-Villars regularization work?

Pauli-Villars regularization works by introducing a set of fictitious particles with opposite properties to the real particles in the theory. These regulators are then used to cancel out the divergences in the calculations, resulting in finite and meaningful results.

4. What are the advantages of using Pauli-Villars regularization for Vacuum Polarization?

One advantage of Pauli-Villars regularization is that it allows for the removal of divergences without breaking the underlying symmetries of the theory. It also provides a consistent and systematic way to regulate divergent integrals in quantum field theory calculations.

5. Are there any limitations to using Pauli-Villars regularization for Vacuum Polarization?

One limitation of Pauli-Villars regularization is that it is not always applicable to all types of divergences in quantum field theory calculations. In some cases, other regularization methods may need to be used. Additionally, the introduction of fictitious particles can complicate the interpretation of physical results.

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