from (\[alambda\]) we get $\bar T$, the irreducible positive frequency amplitude

$$\begin{aligned}
\bar T\left( \lambda \right) &=&-ig^2\tau _c\bar u\left( {\mbox{\boldmath $p
$}}^{\prime }\right) \left[ -\frac{\not k}{2E_d\left( p_d^0+E_d\right) }-%
\frac{\not k}{2E_x\left( p_x^0+E_x\right) }\right.  \label{Tbar}
\nonumber\\
&&+\left. \left( \frac 1{2E_d}-\frac 1{2E_x}\right) \gamma ^0+\frac \lambda {%
gf_\pi }\right] \gamma _5u\left( {\mbox{\boldmath $p $}}\right) .\end{aligned}$$

This allows the proper kernel for the process $\pi N_jN_k$ as shown in fig 4b, to be written as

$$\begin{aligned}
K\left( \lambda \right) &=&-ig^3\left\{ \tau _c\bar u\left( {\mbox{\boldmath
$p $}}^{\prime }\right) \left[ -\frac{\not k}{2E_d\left( p_d^0+E_d\right) }-%
\frac{\not k}{2E_x\left( p_x^0+E_x\right) }\right. \right.  \nonumber \\
&&+\left. \left. \left( \frac 1{2E_d}-\frac 1{2E_x}\right) \gamma ^0+\frac 
\lambda {gf_\pi }\right] \gamma _5u\left( {\mbox{\boldmath $p $}}\right)
\right\} ^{(j)}  \nonumber \\
&&\times \frac 1{q^2-m_s^2}\left[ \bar u\left( {\mbox{\boldmath $p $}}%
^{\prime }\right) u\left( {\mbox{\boldmath $p $}}\right) \right] ^{(k)},
\label{TvwpiNN}\end{aligned}$$

where $q=p_2^{\prime }-p_2$.

Three body force
================

In this section we derive the three-nucleon force due to the simultaneous exchanges of one pion and one effective scalar meson and the basic diagram is shown in figure 5. It produces the following three-nucleon amplitude

$$\begin{aligned}
T^{ijk}(\lambda ) &=&g^4{\mbox{\boldmath $\tau $}}^{(i)}\cdot {%
\mbox{\boldmath $\tau $}}^{(j)}\left[ \bar u\left( {\mbox{\boldmath $p $}}%
^{\prime }\right) \gamma _5u\left( {\mbox{\boldmath $p $}}\right) \right]
^{(i)}\frac 1{k^2-\mu ^2}  