How can I relate this integral to dimensional regularization?

In the paper "Scattering into the Fifth Dimension of $$\mathcal{N}=4$$ Super Yang-Mills", the authors give the following result for an integral:

\begin{align} I^{(1)}(x_{13}^2,x_{24}^2,m) =& \left( x_{13}^2 +m_{13}^2 \right) \left( x_{24}^2 + m_{24}^2 \right)\cr & \times\int d^4 x_5 \frac{1}{(x_{15}^2+m^2) (x_{25}^2+m^2) (x_{35}^2+m^2) (x_{45}^2+m^2)} \notag \\ =& 2 \ln \left( \frac{m^2}{x_{13}^2} \right) \ln \left( \frac{m^2}{x_{24}^2} \right) - \pi^2 + \mathcal{O}(m^2) \tag{1} \end{align}

where $$x_{ij} := x_i - x_j$$, $$m_{ij} := m_i - m_j$$. I would like to take the limit $$m \to 0$$ of this integral and compare it to a computation done with dimensional regularization. The following relation between cutoff and dim reg is given in the paper "Cutoff Regularization Method in Gauge Theories" :

$$\ln \Lambda^2 = \frac{1}{\epsilon} - \gamma + \ln(4\pi^2 \mu^2) + 1 \tag{2}$$

Can I use this relation for translating the "mass cutoff" of eq.(1)?

A few things:

$$\bullet$$ Your definition of the $$I^{(1)}$$ integral, and then the expression involving $$m$$ that follows, is missing information. The paper you quote says that the latter expression holds only in the case where all the masses are the same ($$m_i = m$$), and also only in the limit where $$m$$ is small compared to $$\hat{x}_{13}^2$$ and $$\hat{x}^2_{24}$$

$$\bullet$$ As I think you are aware, it is in this sense that the latter expression already has the limit $$m \to 0$$ taken. That is what the $$\mathcal{O}(m^2)$$ means.

$$\bullet$$ In general, I don't think there is a consistent mapping between different regularization schemes: ie. Although it is often true that a $$1/\epsilon$$ in dim reg corresponds to a $$\log(\Lambda)$$ in a cutoff scheme, I think this is only true for simple loop integrals and I don't think you can always trust this.

$$\bullet$$ In the second paper, your relation between cutoff $$\Lambda$$ and dim-reg $$\epsilon$$ seems to be for an ultraviolet cutoff $$\Lambda$$ (which is supposed to be large compared to the scales in your integral). The mass $$m$$ in your loop integral parametrizes an infrared divergence (in that $$m$$ is small compared to the other scales in your integral). This further undermines using your last expression. I would definitely be wary of using it.

• Mm okay, yeah that makes sense. I will try to think of another way to proceed. Thanks for the clarification!
– Pxx
Nov 2, 2019 at 0:31