Evaluating Integral with Spherical Coordinates Using 4-Vectors

[parton]

I want to evaluate the following integral:

$$I(p_{1}, p_{2}, p_{3}) = \int \mathrm{d}^{4} q \mathrm{d}^{4}p \, \dfrac{1}{\left[ p_{2} + q \right]^{2} - i0} \dfrac{1}{\left[ p_{1} - q - p \right]^{2} + i0} \Theta(q^{0}) \delta(q^{2}) \Theta(-p_{2}^{0} -p_{3}^{0} - q^{0} -p^{0}) \delta(\left[p_{2} + p_{3} + q + p \right]^{2}) \Theta(p^{0}) \delta(p^{2})$$.

$$p_{1}, p_{2}, p_{3}$$ are time-like four-vectors, so e.g. $$p_{1}^{2} > 0$$

After some work like exploiting the step- and delta-functions $$\Theta(q^{0}) \delta(q^{2}) \Theta(p^{0}) \delta(p^{2})$$ and by choosing a special frame with
$$p_{2} + p_{3} = (p_{2}^{0} + p_{3}^{0}, \vec{0})$$ I arrived at:

$$I(p_{1}, p_{2}, p_{3}) = \int \dfrac{\mathrm{d}^{3}q \, \mathrm{d}^{3}p}{4 \vert \vec{q} \vert \cdot \vert \vec{p} \vert} \, \dfrac{1}{p_{2}^{2} + p_{2}^{0} \vert \vec{q} \vert + \vert \vec{p}_{2} \vert \cdot \vert \vec{q} \vert \cos \theta_{1} - i0} \, \dfrac{1}{p_{1}^{2} - 2p_{1}^{0} \vert \vec{q} \vert + 2 \vert \vec{p}_{1} \vert \cdot \vert \vec{q} \vert \cos \theta_{2} - 2 p_{1}^{0} \vert \vec{p} \vert + 2 \vert \vec{p}_{1} \vert \cdot \vert \vec{p} \vert \cos \theta_{3} + 2 \vert \vec{q} \vert \cdot \vert \vec{p} \vert - 2 \vert \vec{q} \vert \cdot \vert \vec{p} \vert \cos \eta +i0}$$

$$\times \delta((p_{2}^{0}+p_{3}^{0})^{2} + 2 \vert \vec{q} \vert \cdot \vert \vec{p} \vert - 2 \vert \vec{q} \vert \cdot \vert \vec{p} \vert \cos \eta + 2 (p_{2}^{0}+p_{3}^{0})) \end{split}$$.

So I used spherical coordinates, but I don't know how to integrate that thing. I just know that if $$\eta$$ is the angle between the vectors $$\vec{p}$$ and $$\vec{q}$$ than we must have: $$\theta_{3} = \eta - \theta_{2}$$.

But how do I continue? I think I must somehow specify the elevation and azimuth angles in a special way, but I don't know how to do that.

Could anyone help me please?

 

[diazona]

ewww...

I have to wonder, how did you get that integral? (Obviously from a Feynman diagram, but which one?) Something about it looks off to me.

 

[parton]

The integral is just the Fourier transform of $$G_{F}^{*} G_{F} G G$$

(of course with some arguments), where $$*$$ is complex conjugation, $$G_{F}$$ is the (massless) Feynman-Propagator in p-space and $$G(p) = \dfrac{i}{2 \pi} \Theta(p^{0}) \delta(p^{2})$$.

The first expression of the integral above is correct, but I just don't know how to compute it explicitly. And now there is the problem with spherical coordinates, but I don't know how to continue. Any ideas?

 

[diazona]

Not really... I tried fiddling with it a bit but I couldn't get it much simpler than you did. Though I'm not sure that picking a specific reference frame is the way to go about it... whenever I've done these propagator integrals, there hasn't been any need to specialize to a particular reference frame. But on the other hand, I typically had expressions like $\delta^{(4)}(p_1^\mu + p_2^\mu)$ instead of your $G(p)$. A delta function of a momentum four-vector eliminates four degrees of freedom from the integral, but your $\delta(p^2)$ only eliminates one, which means I'd expect your integral to be rather messier than anything I'm used to dealing with.