complex coupling $\tau$ given in . The coupling $\tau$ is only valid for large $|\varphi|$. For small $|\varphi|$ higher instanton corrections can not be neglected. For this reason, we will construct a solution to  which is valid for large $|\varphi|$. Towards this end, split $\tau_{2}$ into two pieces as follows

$$\tau_{2}=V_{1}-V_{2},\quad V_{1}={4\pi^{2}\over g_{cl}^{2}}
   \equiv\tau_{2cl},\quad V_{2}=
   {3\tau_{2cl}\over 2\pi}m_{1}m_{2}m_{3}m_{4}\Big[
   {q\over a^4}+{\bar{q}\over \bar{a}^4}\Big].$$

We can now solve  perturbatively by writing $f=f_{0}+f_{1}+...$ where

To find the leading corrections to the four derivative terms, it is sufficient to focus attention on $f_{0}$ and $f_{1}$. The solution for $f_{0}$ is

$$f_{0}={4N\pi\over\big[y^{2}+\tau_{2cl}|a|^{2}\big]^{2}}.$$

The function $f_{1}$ satisfies

$$\Big[\tau_{2cl}{\partial^{2}\over\partial y^{2}}+
    {\partial^{2}\over\partial a\partial \bar{a}}\Big]f_{1}=
    {3m_{1}m_{2}m_{3}m_{4}\tau_{2cl}\over 2\pi}
    \Big({q\over a^{4}}+{\bar{q}\over\bar{a}^{4}}\Big)
    \Big(-{16N\pi\over \big[y^{2}+\tau_{2cl}a\bar{a}\big]^{3}}
    +{96N\pi y^{2}\over \big[y^{2}+\tau_{2cl}a\bar{a}\big]^{4}}
    \Big).$$

We will look for solutions to this equation that preserve rotational symmetry in the $y_{i}$ variables. To do this it is useful to move into radial coordinates. Denoting the angular variable in the $a,\bar{a}$ plane by $\theta$ and the radial coordinate in the the $a,\bar{a}$ plane by $r$ and in the $y_{i}$ plane by $\rho$, we find

$${\eqalign{\Big[\tau_{2cl}&
   {\partial^{2}\over\partial \rho^{2}}+\tau_{2cl}
   {3\over\rho}{\partial\over\partial\rho}+
   {\partial^{2}\over\partial r^{2}}+
   {1\over r}{\partial\over\partial r}+{1\over r^{2}}
   {\partial^{2}\over\partial\theta^{2}}\Big]f_{1}\cr
   &\qquad=
   {3m_{1}m_{2}m_{3}m_{4}\tau_{2cl}\over \pi}
   {cos(4\theta)\over r^{4}}
   \Big(-{16N\pi\over \big[\rho^{2}+\tau_{2cl}r^{2}\big]^{3}}
   +{96N\pi \rho^{2}\over \big[\rho^{2}+\tau_{2cl}r^{2}\big]^{4}.}
   \Big).}}$$

By inspection, it is clear that the 