a Richardson extrapolation [@Richardson]. This will be discussed in more detail in the following paragraphs.

#### Box 39

 \

![\[fg:box39\] *Explicit definitions of the virtual momenta in box 39.*](./plots/dia_box39.pdf){width="115.00000%"}

Using the definition of real and virtual momenta as in Fig. \[fg:box39\], the contribution to the tensor $A^{\mu\nu}$ \[see Eq. (\[eq:lomat\])\] of the virtual two-loop corrections is given by $$\begin{aligned}
A_{39}^{\mu\nu} & = & \frac{3}{16}\,
\frac{\alpha_s}{\pi}\, (4\pi)^4\, B_{39}^{\mu\nu} \, , \nonumber \\
B_{39}^{\mu\nu} \!\! & \!\!\!\! = \!\!\!\! & \!\! \int \frac{d^n k d^n
q}{(2\pi)^{2n}} \frac{Tr\Big\{ (\,{\!\!\not\! {k}}+{\!\!\not\! {q}}-{\!\!\not\! {p}}_1+m_t)
(\,{\!\!\not\! {k}}+{\!\!\not\! {q}}+m_t) \gamma^\sigma (\,{\!\!\not\! {k}}+m_t)
({\!\!\not\! {k}}+{\!\!\not\! {p}}_2+m_t) \gamma^\nu
(\,{\!\!\not\! {k}}+{\!\!\not\! {q}}_1-{\!\!\not\! {p}}_1+m_t) \gamma^\rho \Big\}}
{[(k+q)^2-m_t^2] [(k+q-p_1)^2-m_t^2] [(k+p_2)^2-m_t^2]
[(k+q_1-p_1)^2-m_t^2]} \nonumber \\[0.3cm]
& & \qquad \times~\frac{g_{\rho\sigma} (2q-q_1)^\mu - g^\mu_\rho
(q-2q_1)_\sigma - g^\mu_\sigma (q+q_1)_\rho}{(k^2-m_t^2)(q-q_1)^2 q^2}
\, ,
\label{eq:mat39}\end{aligned}$$ where $k,q$ are the loop momenta that are integrated over. The Feynman parametrization is first performed for the integration over $k$. We provide Feynman parameters $x_1,\ldots,x_4$ for the first four propagators in the denominator and $1-\sum_i x_i$ for the last one ($k^2-m_t^2$). Performing the substitutions $$x_1 = (1-x)(1-y)\, , \quad x_2 = (1-x)y\, , \quad x_3 = xzr \, ,
\quad x_4 = xz(1-r) \, ,$$ we arrive at a four-dimensional integral over $x,y,z,r$ 