Pauli-Villars regularization

[zetafunction]

how does this regularization work ?, suppose we have three kinds of divergencies

$$\Lambda$$ ,..$$log \Lambda$$ and $$\Lambda^{2}$$

then according to Pauli-Villars regularization should we add 3 different and ficticious 'Fields' with Masses A,B,C tending to infinity ??

 

[alalaika]

it actually means that you set a cutoff. when the term you subtract is big enough, you render the integral over the functions finite. in your case I'd say that you have sth like -1/(lambda-A), while A is a very big mass. if lambda is small enough the term won't have any effect on your function but if lambda is bigger than A, the term will have an effect and render your function finite. It is important that A is big but not tending to infinity...

 

[Entropia]

Pauli-Villars regularization is a method used in quantum field theory to remove divergences from calculations. It works by introducing additional fields, known as "Pauli-Villars fields", with large masses that cancel out the divergences in the original calculation.

In the case of three types of divergences (Λ, log Λ, and Λ^2), the regularization would involve adding three different Pauli-Villars fields with masses A, B, and C respectively. These masses are chosen to be much larger than the energy scale of the original calculation, effectively suppressing the contributions of the Pauli-Villars fields. As the masses of these fields tend to infinity, their contributions to the calculation become negligible, and the divergences are effectively removed.

The idea behind this approach is that the divergences in the original calculation are cancelled out by the opposite contributions from the Pauli-Villars fields. This allows for a more well-defined and finite result.

However, it should be noted that Pauli-Villars regularization is not the only method used to deal with divergences in quantum field theory. Other methods, such as dimensional regularization, also exist and may be more appropriate in certain situations.

Overall, Pauli-Villars regularization is a useful tool for dealing with divergences in quantum field theory calculations, but it is important to carefully consider which method is most suitable for a particular problem.