Shift of momenta cures IR divergence?

[CAF123]

Consider the following integral $$\int \frac{d^4k}{k^2}$$ It is UV divergent but is it IR finite or IR divergent? The integrand is singular as ##k \rightarrow 0## so this suggest an IR divergence but this is no longer the case if I make a shift of the loop momenta by say ##p_1## and write the same integral as $$\int \frac{d^4k}{(k+p_1)^2}$$

Usually we say an IR divergence can be cured by addition of a mass in the denominator (as then the integrand won't be singular as k goes to 0) but isn't the IR divergence also cured by simply making a lorentz transformation on the momenta (assuming ##p_1^2 \neq 0##) ? I don't understand this result so where is the failure in the reasoning?

 

[mfb]

Don't you get a divergence at k=-p₁ now?

 

[Orodruin]

Don't you get a divergence at k=-p₁ now?

Yes.

 

[CAF123]

What if ##p_1## is large numerically? Then the divergence at ##k=-p_1## is not in the IR?

With a mass M, e.g in the integral $$\int \frac{d^4k}{k^2-M^2},$$ we have a divergence at ##k=\pm M##. The mass is said to cure the IR divergence (because integrand no longer singular as ##k\rightarrow 0##) but there is still this ##k=\pm M## divergence in place that is cured by the Feynman prescription.

So if ##k=\pm M## is not an IR divergence then why is ##k=-p_1## one?

 

[mfb]

What if ##p_1## is large numerically? Then the divergence at ##k=-p_1## is not in the IR?

It is still there, and renaming it won't help.
Substituting k -> k-p₁ doesn't change the integral at all.

With a mass M, e.g in the integral $$\int \frac{d^4k}{k^2-M^2},$$ we have a divergence at ##k=\pm M##. The mass is said to cure the IR divergence (because integrand no longer singular as ##k\rightarrow 0##) but there is still this ##k=\pm M## divergence in place that is cured by the Feynman prescription.

That is a different type of divergence (note the difference of squares instead of a squared difference), and there are ways to deal with it.

 

[CAF123]

It is still there, and renaming it won't help.

Ok,so if I took the IR limit (k->0) in the integral ##\int d^4k/(k+p_1)^2## then I lose all k dependence from the integrand and so the result is like ##\int d^4k## which is infinite. Is it in this sense that the IR divergence is still clear?

That is a different type of divergence (note the difference of squares instead of a squared difference), and there are ways to deal with it.

Are these 'ways' you mention the wick rotation to Euclidean 4 momentum etc?

 

[vanhees71]

This integral is IR save since ##\mathrm{d}^4 k \propto |k|^3## (taking it as a Euclidean/Wick rotated QFT integral).