$r $}}_2{\mbox{\boldmath $r $}}%
_3\right\rangle =\delta \left( {\mbox{\boldmath $r $}}_1^{\prime }-{%
\mbox{\boldmath $r $}}_1\right) \delta \left( {\mbox{\boldmath $r $}}%
_2^{\prime }-{\mbox{\boldmath $r $}}_2\right) \delta \left( {\mbox{\boldmath
$r $}}_3^{\prime }-{\mbox{\boldmath $r $}}_3\right) W.$$

and the component $W^{ijk}$ of the local potential $W$ is given by

$$\begin{aligned}
W^{ijk} &=&\frac{g^4}{\left( 4\pi \right) ^2\left( 2m\right) ^2}\frac{\mu m_s%
}m{\mbox{\boldmath $\tau $}}^{\left( i\right) }\cdot {\mbox{\boldmath $\tau
$}}^{\left( j\right) }
\nonumber\\
&&\times \left[ \left( 1-\frac{\lambda m}{gf_\pi }\right) U\left(
m_sr_{kj}\right) {\mbox{\boldmath $\sigma $} }^{\left( i\right) }\cdot {%
\mbox{\boldmath $\nabla $ }}_{ji}{\mbox{\boldmath $\sigma $} }^{\left(
j\right) }\cdot {\mbox{\boldmath$\nabla $ 
}}_{ji}U\left( \mu r_{ji}\right) \right.  \nonumber \\
&& +\left. \frac{\lambda m}{gf_\pi }{\mbox{\boldmath $\sigma $} }^{(i)}\cdot 
{\mbox 
{\boldmath$\nabla$ }}_{ji}U\left( \mu r_{ji}\right) {\mbox{\boldmath$\sigma$
}}^{(j)} \cdot {\mbox{\boldmath $\nabla $}}_{kj}U\left( m_sr_{kj}\right)
\right] , \end{aligned}$$

where $U(\mu r)=\frac{\exp (-\mu r)}{\mu r}$ and $r_{kj}=r_k - r_j$.

We assume the approximate identity $\frac m{gf_\pi }\simeq 1$ and, similarly to the procedure used in the two pion exchange three-nucleon force [@Coe+], we regularize the mesonic exchanges with the following dipole form factor $$G\left( k^2\right) =\left( \frac{\Lambda ^2-\mu ^2}{\Lambda ^2-k^2}\right)
^2,$$ where $\Lambda $ is a cut off parameter.

The regularization changes the Yukawa-type function into $$U\left( \mu r,\Lambda \right) =\frac{e^{-\mu r}}{\mu r}-\frac \Lambda \mu 
\frac{e^{-\Lambda r}}{\Lambda r}-\frac 12\frac \mu 