Surface terms in loop integrals (2D)

[nidnus]

Hi everyone,

Say I have a 2D one loop integral of the form

$$ I^s_n(\Delta)=\int d^2 l \frac{(l^2)^s}{(l^2-\Delta)^n} $$

Using that $1 = \frac{1}{2} \partial_\mu l^\mu$, I can relate say

$$I^1_2(\Delta)=I^0_1(\Delta)$$
+ total derivative term.

In dimensional regularization one usually throw away surface terms meaning that

$$I^1_2(\Delta)=I^0_1(\Delta)$$

However, looking the appendix of, say, Peskin & Schröder they state the actual value of the different $I^s_n(\Delta)$ integrals in terms of $\Gamma$ functions for general D. Expanding $I^1_2$ and $I^0_1$ show that they agree up to a linear $i \pi$ term.

Thus this term has to correspond to a surface term and one should be able to throw it away on physical grounds. However, all this seem very basic and seem rather an important concept every time you want to make an actual calculation. How come I have never seen this? I don't think it's even mentioned in P&S.

What is more, what happens in a hard cutoff $\Lambda$? If one can use the same arguments as in dim reg, then it follows that $\Lambda^2$ terms from 1-loop tadpoles correspond to surface terms (i.e they don't exist)! (this can be seen by relating $2I^1_1=I^2_2$+ total derivative).

Can someone who knows these kind of things in some detail clarify this for me?

Thanks alot,

Nid

 

[AndyW]

hiHi Nidhi,

Great question! The concept of surface terms and their role in dimensional regularization is definitely an important one in calculations involving Feynman diagrams.

First, let's address your question about why this concept may not have been explicitly mentioned in Peskin & Schröder. This could be due to a few reasons:

1. The authors may have assumed that the reader is already familiar with the concept of surface terms and their role in dimensional regularization.

2. The authors may have chosen to focus on the main principles and techniques of dimensional regularization, rather than delving into specific details like surface terms.

3. The appendix you mentioned may have been added in a later edition, and the authors may not have felt the need to mention it in the main text.

Now, let's address your question about the role of surface terms in a hard cutoff $\Lambda$. In dimensional regularization, we can throw away surface terms because we are working in a space with a non-integer number of dimensions, which allows us to use the properties of the Gamma function to simplify our integrals. However, in a hard cutoff, we are working in a space with a finite number of dimensions, so we cannot use the same arguments to throw away surface terms. Instead, we have to carefully consider the contributions of these surface terms to our calculations.

In fact, the linear $i \pi$ term that you mentioned is known as a "logarithmically divergent" term, and it can arise in hard cutoff calculations. This term is usually associated with a surface term, and it can be canceled out by adding counterterms to the Lagrangian. These counterterms essentially account for the effects of the surface terms, and allow us to obtain finite results in our calculations.

So, in summary, surface terms play a crucial role in calculations involving Feynman diagrams, and they are closely related to the regularization method being used. In dimensional regularization, we can throw them away because we are working in a space with a non-integer number of dimensions, but in a hard cutoff, we have to carefully account for them in our calculations.

I hope this helps clarify things for you. Let me know if you have any further questions.