of the large factor $N_v$ or $N_{\rm cor}$ is the vertex correction to the impurity line (the semicircles in Fig.1). If, as an estimation of $N_{\rm cor}$, identifying it with the correlation area resulting from the collective pinning theory \[33\], we find $N_{\rm cor} \simeq (\Delta_{\rm eff})^{-1}$. Further, since the mean field glass transition line $T_{\rm G}^{\rm mf}(B)$ in clean limit will lie just below the freezing transition, or crossover, line $T_m^{(2d)}(B)$ in 2D LLL, $\mu^2$ in eq.(2.13) may be replaced by its value at $T_m^{(2d)}(B)$ $10^{-2} h \, b \, k_{\rm B} T (2 \pi \xi_0^2 d)^{-1}$. Then, since the $\beta_A$-value is insensitive to material parameters, the resulting $I_{\rm irr}$ is almost independent of material and physical parameters. At least, it does [*not*]{} vanish in clean limit ($\Delta_{\rm eff} \to 0$), implying that the disorder-free theory cannot be used even in clean limit below the expected $T_m^{(2d)}(B)$.

Next, let us extend the above analysis to 3D case in which the glass transition will occur more easily. For a while, the case of [*point*]{} disorder will be considered in which the random potentials satisfy ${\overline {u_j({\bf r}) u_l({\bf r}')}}=d^{-1} \Delta^{(p)}
\delta^{(2)}({\bf r} - {\bf r}') \, \delta_{j, l}$ and ${\overline 