Estimating size of loop integrals

In summary, there are ways to measure the magnitude of a numerical value from loop integrals. This involves using techniques such as Wick rotation, regularization, and renormalization. By removing the divergent parts through renormalization and estimating the remaining finite part analytically, the magnitude of the numerical value can be controlled. More advanced techniques, such as those described in Vincent Rivasseau's book "From perturbative to constructive Renormalization", can also be used for this purpose.
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Is there a way to measure how large a numerical value a loop integral will give?

For example, take this integral over loop momenta k:

[tex]\frac{1}{k^2+m^2} \frac{1}{(k+p)^2+m^2} [/tex]

How does it compare to setting p=0:

[tex]\frac{1}{k^2+m^2} \frac{1}{(k)^2+m^2} [/tex]

or to a double integral over loop momenta k and q:

[tex]\frac{1}{k^2+m^2} \frac{1}{q^2+m^2}\frac{1}{(k+q)^2+m^2} [/tex]

All I know how to do is to Wick rotate, regulate, renormalize, and I also know that [tex]\frac{1}{k^2+m^2}[/tex] should really be: [tex]\frac{1}{k^2+m^2-i\epsilon}[/tex] so that the integral over the energy component will not blow up - any blow up will be in the 3-momentum component.

But I'm not sure what's really going on with all these integrals. Some of them are infinite and you have to renormalize, and you're left with a finite part, but how to estimate the magnitude of the finite part?
 
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  • #2
Well you can remove the divergent part via renormalization and then control the remaining finite part with analytic estimates.

State of the art in this regard gets a good (though you will need to be comfortable with Combinatorics, Graph Theory and Analysis) in Vincent Rivasseau's "From perturbative to constructive Renormalization"
 

What is the purpose of estimating the size of loop integrals?

The purpose of estimating the size of loop integrals is to understand the behavior of a physical system at very small scales. Loop integrals are used in quantum field theory to calculate the probability of particle interactions, and estimating their size helps to predict the likelihood of these interactions occurring.

How are loop integrals typically estimated?

Loop integrals are typically estimated using a technique called dimensional regularization. This involves extending the number of dimensions in the calculation to make the integral easier to solve, and then taking the limit as the number of dimensions approaches the physical dimensionality of space.

What are some challenges in estimating loop integrals?

One of the main challenges in estimating loop integrals is dealing with divergences. These occur when the integral approaches infinity, and must be accounted for in order to obtain meaningful results. Another challenge is the complexity of the calculations, which can involve a large number of terms and require advanced mathematical techniques.

How do loop integrals relate to renormalization?

Loop integrals are crucial in the process of renormalization, which is used to remove the infinities that arise in quantum field theory calculations. By estimating the size of loop integrals, physicists can determine the parameters needed to renormalize the theory and make accurate predictions about physical phenomena.

Are there any real-world applications of estimating loop integrals?

Yes, estimating loop integrals has many real-world applications in fields such as particle physics, condensed matter physics, and cosmology. For example, in particle accelerators such as the Large Hadron Collider, loop integrals are used to calculate the probability of particle interactions. In cosmology, they are used to study the behavior of the early universe and the formation of structures such as galaxies.

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