by those contributions of the contours in complex $a$-space (i.e., in complex $t$ space) which have the end points $a_f$ and initial point $a(0)=0$. Looking at (21), we find $$P^{0}_{a}={{\partial S_0}\over {\partial a}}=-Ma_0(2a^{2}_{0}{V_0\over M}-1)^{1/2}$$ which have turning points at $a_0=0$ and $a_0=\sqrt{{M}\over {2V_0}}=a_T$. Thus, $$S_0 (Complex\;\;paths\,)=S_0(a_f,\,0)+S(a_0=0\rightarrow a_T \rightarrow a_0
\rightarrow a_f).$$ Here the first term is the contribution from the path that goes from $a=0$ to $a=a_f$ and the second term when evaluated corresponds to a loop path between $a=0$ and $a_f=\sqrt{\frac {2V_0} {M}}$ times the first term in (29). When the repeated loops contribute we get a geometric series. Correspondingly the wavefunction is given by $$\Psi(a_f,\,\phi_f)={{e^{iS_0(a_T,0)}}\over
{1-e^{2iS_0(a_T,0)}}}\,\Psi_0,$$ considering repeated paths, to and fro, between $a=0$ and $a=a_T$. For the derivation of this result the reader is referred to [@bis:prd]. The contribution $S_0(a_T,0)$ gives the second term in (21) and we observe that the imaginary part of the action determines the amplitude and the real part determines the phase of the wavefunction. The CWKB approach gives the extra term i.e., the denominator in (30). This term correctly takes into account the one loop quantum correction. Applying this technique to particle production in uniform electric field, we obtained [@bis:cqg] the Schwinger’s 