Is this box integral divergent or finite when "pinched" at one point?

Question

Let us define the following conformal integral:

$$X_{1234} = \int \frac{d^4 x_5}{(2\pi)^8} \frac{1}{x_{15}^2 x_{25}^2 x_{35}^2 x_{45}^2}\tag{1}$$

This is the box integral in position space, and it is finite when the $x_i$'s aren't equal.

I am interested in knowing if the integral $X_{1123}$ is divergent or not. In this paper, they say in Appendix A.$2$ (p.$25$) that $X_{1234}$ diverges logarithmically in the limit $x_1 \rightarrow x_2$. However, since I am only interested in the divergence occurring at $x_5 \sim x_1$, I can set:

$$\left. X_{1123} \right|_\text{div} \sim \frac{1}{(2\pi)^4 x_{12}^2 x_{13}^2} \int \frac{d^4 x_5}{(2\pi)^4} \frac{1}{x_{15}^4}, \tag{2}$$

and the remaining integral is known to be $0$, as shown in several sources such as the QFT book by Schwartz (eq. (B$.48$), p. 829), since the IR and UV divergences cancel each other.

So did the authors of the paper forget that this integral vanishes? Or is $X_{1123}$ indeed divergent, and if yes how can I extract the corresponding divergent term?

Answer 1

I suspect the integral is in Minkowski signature which requires more care, but let me take the simpler Euclidean case. If the points $x_1,x_2,x_3,x_4$ in $\mathbb{R}^4$ are distinct, then the integral $$ \int_{\mathbb{R}^4\backslash\{x_1,x_2,x_3,x_4\}}\frac{d^4 x_5}{(2\pi)^8}\ \frac{1}{(x_1-x_5)^2(x_2-x_5)^2(x_3-x_5)^2(x_4-x_5)^2} $$ converges rigorously in the sense of Lebesgue theory of integration. In particular there is no divergence at $x_5\sim x_1$ (divergence in the sense that the integral is not well defined). The above integral defines a function $X_{1234}(x_1,x_2,x_3,x_4)$ of the distinct points $x_1,x_2,x_3,x_4$.

Now if you take $x_2,x_3,x_4$ fixed and distinct, and then consider the limit $$ \lim_{x_1\rightarrow x_2} X_{1234}(x_1,x_2,x_3,x_4) $$ you will find infinity or more precisely a behavior of the form $$ X_{1234}(x_1,x_2,x_3,x_4)\sim C\log|x_1-x_2| $$ for some constant $C=C(x_2,x_3,x_4)$ when $x_1\rightarrow x_2$.