What is the physical meaning of this simplification to calculate the effective coupling constants for a Gaussian model with quartic interactions?

Question

To calculate the effective coupling constants $u'_2(q)$ and $u'_4(q)$ of the effective Hamiltinian eq (4.9) of this paper

$$ H' = -\frac{1}{2}\int\limits_q u'_2(q)\sigma'_q\sigma'_{-q} - \int\limits_{q_1}\int\limits_{q_2}\int\limits_{q_3}u'_4(q_1,q_2,q_3,-q_1-q_2-q_3) \sigma'_{q_1}\sigma'_{q_2}\sigma'_{q_3}\sigma'_{q_-q_1-q_2-q_3}$$

The following simplifications are introduced into eq (4.20) and (4.21) to calculate $u'_2(q)$ and $u'_4(q)$ respectively

1. $u'_2(q)$ is only evaluated to order u, which means only tree level diagrams are considered

2. Higher order than quartic interactions are neglected

3. $\int\limits_{\frac{1}{2} < ¦p¦ < 1} \frac{1}{p^2+r} \rightarrow \frac{1}{1+r}\int\limits_{\frac{1}{2} < ¦p¦ < 1} 1 $

4. $\int\limits_{\frac{1}{2} < ¦p¦ < 1} \frac{1}{p^2+r}\frac{1}{[(\frac{1}{2}q_1+\frac{1}{2}q_2-p)^2 +r]} \rightarrow \frac{1}{(1+r)^2}\int\limits_{\frac{1}{2} < ¦p¦ < 1} 1 $

What is the physical meaning of 3. and 4. ? Is there an "intuitive explanation for the physical meaning of these two simplifications?

PS: here is an alternative link to the paper that maybe works better.

Answer 1

1. $\int\limits_{\frac{1}{2} < ¦p¦ < 1} \frac{1}{p^2+r} \rightarrow \frac{1}{1+r}\int\limits_{\frac{1}{2} < ¦p¦ < 1} 1 $

2. $\int\limits_{\frac{1}{2} < ¦p¦ < 1} \frac{1}{p^2+r}\frac{1}{[(\frac{1}{2}q_1+\frac{1}{2}q_2-p)^2 +r]} \rightarrow \frac{1}{(1+r)^2}\int\limits_{\frac{1}{2} < ¦p¦ < 1} 1 $

Since the referenced document is behind a paywall, it is a little difficult to know for sure of the context, but when I see these two equations, it would appear they are making a simplification about the amplitudes associated with propagators (internal lines of Feynman diagrams). It appears to be saying the upper half of the range of values associate with momentum follow the rules of geometric progression or more appropriately a geometric series and the region of integration has a constant slope (e.g. there is a uniform accumulation in the identified region of integration) Represented as. $$ \int\limits_{\frac{1}{2} < ¦p¦ < 1} 1 $$

I would understand it as a statement about the behavior of the propagators at high momentum, e.g. they are well behaved, and the contribution of more complicated diagrams diminishes at higher orders.

Since the paper is Wilson's paper on renormalization, this interpretation would at least be consistent.

If anyone would like to further clarify, please do.