Speaker
Description
Recently, higher chiral-order nucleon-nucleon potentials have been developed with the chiral effective fields theory [1]. The three-body Faddeev equation had been extended by involving three-body forces [2]. The four-body Yakubovsky equations have also been extended as well [3]. In order to increase the accuracy of not only its two-body forces but also three-body forces, it is indispensable to study not only three-body systems but also four-nucleon systems using ab initio calculation.
Moreover, it is not ignorable that the effect of relativity in high energy region. We have been studying that in the proton-deuteron scattering the effect reveals at the backward of the scattering angle for the elastic process and three-body breakup [4]. It is, of course, expected that such a relativistic effect also appears in case of four-nucleon system.
I would like briefly to present my oral that I explain the Faddeev-Yakubovsky four-body equations including the three-body force [3]. Furthermore, these equations are extended in the framework of relativity. As the result we have the following coupled equations with three-body force $W$,
$\alpha = -G_0{T}PP_{34} \alpha + G_0 { T} P \beta +(G_0+G_0{ T})(G_0+G_0 t^\alpha)W(-P_{34}P +\tilde P)( \alpha -P_{34} \alpha +\beta),$
$\beta = G_0 \tilde T \tilde P G_0 (1-P_{34}) \alpha, $
where $\alpha$ and $\beta$ are Yakubovsky components for 1+3 and 2+2 partitions, respectively, $G_0$ is Green's function, $T$, $\tilde T$ are transition matrices for 1+3 and 2+2 partitions, respectively. $P (\equiv P_{12}P_{23}+P_{13}P_{23})$, $\tilde P(\equiv P_{13}P_{24})$ and $P_{34}$ are permutation operators. Detail is written in [3]. In particular, these transition matrices are the solutions of the following equations,
${ T}=\tau + \tau G_0 { T}$,
$\tau \equiv t^\alpha P+ (1+t^\alpha G_0)W(1+P),$
$\tilde T=t^\beta + \tilde T \tilde P G_0 t^\beta,$
where $t^\alpha$ and $t^\beta$ are 2-body transition matrix which are relativistically boosted depending on the partition sub-systems in the four-body system.
[1] P. Reinert, H. Krebs, E. Epelbaum, Eur. Phys. J. A 54, 86 (2018).
[2] D. Hueber, H. Kamada, H. Witala, W. Gloeckle, Acta Phisica Polonica B28 1677 (1997).
[3] H. Kamada, to be appeared in Few-Body Syst. (2019). (DOI :10.1007/s00601-019-1501-4)
[4] H. Witala, J. Golak, R. Skibinski, W. Gloeckle, H. Kamada, W.N. Polyzou, Phys. Rev. C 83, 044001 (2011).
