Speaker
Description
A general three-body interaction could be generated by the two-body linear and nonlinear interactions which could appear in the very short and in the long range regions, although the three-body Faddeev equation is written in terms of a two-body linear interaction.
However, in the very short range, many meson or multi quark/gluon exchanges may take place which are taken into account by a ``three-body short range force: $V_{\rm 3BSF}$''. In the long range region, the linear three-body Faddeev equation can not be exactly described by a two-body long range potential.
In this context, we could employ a general three-body interaction $V_{\rm 3BF}$ by using the $\it{general ~particle ~transfer}$ (GPT) potential $V_{\alpha}({\bf r}_{\beta\gamma};n)$ [1], i.e.,
$\Big[V_{{\rm 3BF}}\Big]_{\alpha\beta}
\equiv \Big[ b_{\alpha\beta}+\big\{1/V_{\alpha}({\bf r}_{\beta\gamma};n)+1/V_{\beta}({\bf r}_{\gamma\alpha};n)\big\} \Big ]^{-1} $
$=V_{\alpha}({\bf r}_{\beta\gamma};n)V_{\beta}({\bf r}_{\gamma\alpha};n)/{\cal V}$
and
${\cal V}=b_{\alpha\beta}V_{\alpha}({\bf
r}_{\beta\gamma};n)V_{\beta}({\bf
r}_{\gamma\alpha};n)
+V_{\alpha}({\bf
r}_{\beta\gamma};n)
+V_{\beta}({\bf
r}_{\gamma\alpha};n)$$\equiv (E-H_0)$,
with the three-body kinetic energy $H_0$ and the total energy $E$, respectively.
$b_{\alpha\beta}$ denotes a parameter which represents a border between the linear and the nonlinear interactions.
We obtain
$\Big[V_{{\rm 3BF}}
\Big]_{\alpha\beta}$
$=V_{\alpha}({\bf r}_{\beta\gamma};n)V_{\beta}({\bf r}_{\gamma\alpha};n)/(E-H_0+i\epsilon).$
This formula is a generalized Alt-Grassberger-Sandhas (AGS) Born term which includes both the three-body short range force (3BSF), and the three-body long range force (3BLF).
[1] Shinsho Oryu, J. Phys. Commun. {\bf 6} 015009 (2022).
