这是一份uwa西澳大学PHYS2002的成功案例

The behavior of this equation is regular in the limit $B \rightarrow 0:$ the flow of $g_{k}$ is effectively negligible for $B \leqslant D^{-2}$. Hence we can replace by the following simpler equation:
$$
\frac{\mathrm{d} g_{k}}{\mathrm{~d} B}=-\frac{\rho N g_{k}^{2}}{2 B} \mathrm{e}^{-2 B e_{k}^{2}}
$$
with the initial condition now posed at $B=D^{-2}$ :
$$
g_{k}\left(B=D^{-2}\right)=\frac{g}{N}
$$
This equation can be integrated easily
$$
\frac{1}{N}\left(\frac{1}{g_{k}(B)}-\frac{1}{g_{k}\left(D^{-2}\right)}\right)=\frac{\rho}{2} \int_{D^{-2}}^{B} \mathrm{~d} B^{\prime} \frac{\exp \left(-2 B^{\prime} \epsilon_{k}^{2}\right)}{B^{\prime}}
$$
yielding
$$
g_{k}(B=\infty)=\frac{1}{N} \frac{g}{1+\rho g \operatorname{Ei}\left(1,2 \epsilon_{k}^{2} / D^{2}\right) / 2}
$$

MATH2002 COURSE NOTES ：

The approximation that is closest to the conventional scaling approach is to focus only on the infrared limit in the flow. We use the following parametrization
$$
g_{k^{\prime} k}(B)=\frac{g_{\mathrm{IR}}(B)}{N} \mathrm{e}^{-B\left(e_{k^{\prime}}-e_{k}\right)^{2}},
$$
and derive the scaling equation for $g_{\mathrm{IR}}(B)$ by putting it into the flow equation (2.103) for $k^{\prime}=k=0$. This gives
$$
\frac{\mathrm{d} g_{\mathrm{IR}}}{\mathrm{d} B}=-2 g_{\mathrm{IR}}^{2} \rho \int_{0}^{D} \mathrm{~d} \epsilon \epsilon \mathrm{e}^{-2 B \epsilon^{2}}
$$
leading to
$$
\frac{\mathrm{d} g_{\mathrm{IR}}}{\mathrm{d} B}=-\frac{\rho g_{\mathrm{IR}}^{2}}{2 B}
$$
with the initial condition posed at $B=D^{-2}$ :
$$
g_{\mathrm{IR}}\left(B=D^{-2}\right)=\frac{g}{N}
$$
This is exactly the scaling equation derived from a conventional scaling with the identification $B=D_{\text {eff }}^{-2}$. It therefore agrees with the exact solution.