Schwartz QFT book, Problem 14.3

[wphysics]

I am working on Schwartz QFT book problem 14.3, particularly part (c).
Basically, it asks us to evaluate the following integration.
$$\int \cfrac{d^3 p}{2\pi^3} \omega_p e^{i \vec{p} \cdot (\vec{x}-\vec{y})}$$ where
$$\omega_p = \sqrt{p^2 + m^2}$$
I could perform the angular integration, and the result is
$$\cfrac{2}{(2\pi)^2 |x-y|} \int_0^{\infty} dp \,\,p \sqrt{p^2+m^2} \sin{(p|x-y|)} = \cfrac{2}{(2\pi)^2 |x-y|} \textrm{Im}\left( \int_0^{\infty} dp \,\,p \sqrt{p^2+m^2} e^{i p |x-y|} \right)$$
The textbook says it should be expressed as some sort of Hankel functions, but I am not sure how I get to that point. Do you guys have any suggestion? Thank you.

 

[mfb]

I don't know if it works, but a substitution p²+m² = x could give interesting results. It makes the exponential more complicated but simplifies everything else.

 

[wphysics]

I don't know if it works, but a substitution p²+m² = x could give interesting results. It makes the exponential more complicated but simplifies everything else.

Could you tell me in more detail? In fact, I could not find any integral representations of Hankel functions that matches with my integration.

 

[mfb]

Did you try it? What was the result?

 

[HBChen]

The result is the derivative of the modified Bessel function of the second kind, whose integral representation is

http://mathworld.wolfram.com/ModifiedBesselFunctionoftheSecondKind.html

And the solution for the massive case is given in Weinberg's QFT I page 387, which can be derived from the above integral representation. For the massless case, it seems to me that the result is just the limit $m\rightarrow 0$ of the massive case, is this right? In taking the limit that $m\rightarrow 0$, I didn't encounter any singular behavior.