Diverging Integral: 4 Powers of q in Numerator, 2 in Denominator

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In summary, the conversation is discussing the divergence of a mathematical expression with four powers of q in the numerator and two in the denominator. The speaker is concerned about the quadratic and infrared divergences at large and small q values, respectively. They are unsure about the legitimacy of the result and ask for help understanding its implications. The other person suggests analyzing the degree of divergence by looking at the radial coordinate in 4-dimensional spherical coordinates. They also mention that this method can be applied to other types of integrand divergences.
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Elwin.Martin
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Quick question:
What does it mean that this has "four powers of q in the numerator and two in the denominator"? Apparently, this diverges quadratically at large q and has an infrared divergence as q→0 (I'm not concerned about the second one all that much though).
I mean, simply looking at a comparison of powers, since the integration is over all of q, it's feels like they get this result by just saying we have 4/2=2 on top. . .but that's not really legitimate.

[itex] g\int \frac{d^4 q}{\left(2\pi\right)^4}\frac{1}{q^2-m^2}[/itex]

Thanks for any help! I don't doubt the divergence, but I'm just not sure what is meant by it ^^; in one dimension, I know that it blows up at q=±m, I'm just not sure what implications this has on the divergence in d4q
 
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The integrand depends only on |q|. Therefere, imagine the integration domain to be a 4-ball of radius Q. You can use 4d spherical coordinates and perform all angular integrals. You are left with an integral over the radial coordinate |q|. It will have some factors of |q| in the numerator and denominator of its integrand. Then analyse the degree of divergence by checking how the result depends on Q approaches infinity. Similarly can be done e.g for integral divergence caused by integrand divergencies such as what happens at q^2 = m^2
 

FAQ: Diverging Integral: 4 Powers of q in Numerator, 2 in Denominator

What is a diverging integral?

A diverging integral is an integral that does not have a finite value, meaning it either approaches infinity or negative infinity as the limits of integration are taken to infinity.

What does it mean when there are powers of q in the numerator and denominator?

This means that the integrand (the function within the integral) contains terms with q raised to certain powers in both the top and bottom of the fraction. This can make the integral more complicated and may affect its convergence or divergence.

Why are there four powers of q in the numerator and only two in the denominator?

This could be due to the specific function being integrated, but in general, having more powers of q in the numerator than in the denominator can lead to a diverging integral. This is because as q approaches infinity, the terms in the numerator will dominate and cause the integral to approach infinity as well.

How do you determine if a diverging integral is convergent or divergent?

To determine the convergence or divergence of a diverging integral, you can use techniques such as the comparison test, limit comparison test, or integral test. These methods involve comparing the given integral to a known integral with a known convergence or divergence.

How can I solve a diverging integral with powers of q?

Solving a diverging integral with powers of q may require using techniques such as partial fractions or substitution to simplify the integrand. If the integral is still diverging after simplification, you can then use one of the methods mentioned in the previous question to determine its convergence or divergence.

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