Cut-off Regularization of Chiral Perturbation Theory

[quantatanu0]

I was trying to learn renormalization in the context of ChPT using momentum-space cut-off regularization procedure at one-loop order using order of p^2 Lagrangian. So,

1. There are counter terms in ChPT of order of $$p^4$$ when calculating in one-loop order using Lagrangian of order $$p^2$$.

2. Divergences are of polynomial kind and logarithmic kind.

3. The counter terms always take care of polynomial divergences (and $$1/\epsilon$$ kind of div. in dimensional method)

4. The logarythmic divergence gets absorbed during coupling constant renormalization.During my calculation using cut-off method I obtained a result where I have only logarithmic divergence and no other divergent term, then I need to understand what is the use of counter-terms in this case.

In any case, we have to consider the counter-terms in ChPT but here we are not doing dimensional regularization so no $$1/\epsilon$$ to get killed by the counter terms, and in my calculation involving cut-off regularization, I have no polynomial divergence either ! Only logarithmic divergence, then what is the use of the counter-terms here ?

 

[quantatanu0]

OK, I am happy to tell you guys that I have figured it out. Here's what I do:

Let's say the counter terms introduce coupling constants (low energy constants) $$L_i , i=1,2,... n$$ and the divergent term coming from the loop calculations is $$D$$, and this can be any kind of divergence, log, polynomial, and/or any other kind (separately or together). Then:

$$L_i = L^r_i + c_i \frac{D}{n}$$ where $$c_i$$ are constants that one chooses in a way that the divergence gets canceled by the counter terms. And this is all.