# Does this integral converge or diverge?

I have the following integral:

$$I= \int_{-\infty}^\infty d\tau_3 \int_{-\infty}^\infty d\tau_4\ I_{13} Y_{134}, \tag{1}$$

with:

$$I_{12} := \frac{1}{(2\pi)^2 x_{12}^2} \qquad \qquad Y_{134} := \int_{\mathbb{R}^4} d^4 x_5 I_{15} I_{35} I_{45} \tag{2}$$

and with $$x_{ij}:=x_i-x_j$$, $$x_1=(1,0,0,0)$$, $$x_3=(0,0,0,\tau_3)$$, $$x_4=(0,0,0,\tau_4)$$.

The integral $$Y_{134}$$ is known and has an analytical form that can be found here for example (eq. (A.5) - it's somewhat complicated so I'd rather not type it here). I would like to know if $$(1)$$ is divergent or not. At first glance, it seems divergent, since $$Y_{134}$$ diverges logarithmically at $$\tau_3 = \tau_4$$. But we still have to integrate over $$\tau_3$$ and $$\tau_4$$, and sometimes integrals of divergent functions are finite (e.g. $$\int_0^1 dx \log x = -1$$). This is what seems to happen here when I do the integral numerically, as it was discussed in this other post of mine.

So is this integral finite or not, and how can I know in general?

## 1 Answer

I solved the problem. The integral converges, just as the numerical data in the post linked above shows. My confusion about the divergence came from the fact that similar expressions but with a term

$$\partial_{\tau_3} Y_{134}$$

usually diverge.