25th European Conference on Few-Body Problems in Physics

A General Three-Body Interaction with the GPT-Potential

Not scheduled

20m

AudiMax (Alte Mensa)

Poster Presentation Poster Session

Speaker

Shinsho Oryu (Tokyo University of Science)

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: $V_{3\mathrm{B}\mathrm{S}\mathrm{F}}$''. 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 $V_{3\mathrm{B}\mathrm{F}}$ by using the $\mathit{g}\mathit{e}\mathit{n}\mathit{e}\mathit{r}\mathit{a}\mathit{l}\mathit{p}\mathit{a}\mathit{r}\mathit{t}\mathit{i}\mathit{c}\mathit{l}\mathit{e}\mathit{t}\mathit{r}\mathit{a}\mathit{n}\mathit{s}\mathit{f}\mathit{e}\mathit{r}$ (GPT) potential $V_{\alpha }({\mathbf{r}}_{\beta \gamma };n)$ [1], i.e.,
$[V_{3\mathrm{B}\mathrm{F}}{]}_{\alpha \beta }\equiv [b_{\alpha \beta }+\{1/V_{\alpha }({\mathbf{r}}_{\beta \gamma };n)+1/V_{\beta }({\mathbf{r}}_{\gamma \alpha };n)\}{]}^{-1}$
$=V_{\alpha }({\mathbf{r}}_{\beta \gamma };n)V_{\beta }({\mathbf{r}}_{\gamma \alpha };n)/\mathcal{V}$
and
$\mathcal{V}=b_{\alpha \beta }{V}_{\alpha }({\mathbf{r}}_{\beta \gamma };n)V_{\beta }({\mathbf{r}}_{\gamma \alpha };n)+V_{\alpha }({\mathbf{r}}_{\beta \gamma };n)+V_{\beta }({\mathbf{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
$[V_{3\mathrm{B}\mathrm{F}}{]}_{\alpha \beta }$
$=V_{\alpha }({\mathbf{r}}_{\beta \gamma };n)V_{\beta }({\mathbf{r}}_{\gamma \alpha };n)/(E-H_0+iϵ).$
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).