Four-Body Scattering Equations Including a Three-Body Force in the Faddeev–Yakubovsky Theory

Abstract

Faddeev–Yakubovsky four-nucleon scattering equations are derived including a three-body force. From the asymptotic form of the Yakubovsky-components we obtain scattering amplitudes of all outgoing channels in a direct manner. We present these equations in the momentum space representation.

Appendices

Appendix A

Here we derive the scattering operator $U^{[3+1]}$ of Eq. (53). First, Eq. (9) is rewritten as

$$\begin{aligned} G_0 \tau= & {} (G_0+G_0 t G_0) (VP+W(1+P)) \nonumber \\= & {} (G_0+G_0tG_0)(G_0^{-1} - \{\mathcal{G}^{[3+1]}\} ^{-1} -V) \nonumber \\= & {} (1+G_0 t) - (G_0+G_0tG_0) \{\mathcal{G}^{[3+1]}\} ^{-1} - G_0 t \nonumber \\= & {} 1- (G_0+G_0tG_0) \{\mathcal{G}^{[3+1]}\} ^{-1}, \end{aligned}$$

(76)

then we have

$$\begin{aligned} (1-G_0\tau ) ^{-1} = \mathcal{G}^{[3+1]} (G_0 + G_0 t G_0) ^{-1}. \end{aligned}$$

(77)

and

$$\begin{aligned} (1-G_0\tau ) ^{-1} G_0 \tau= & {} \mathcal{G}^{[3+1]} (G_0 + G_0 t G_0) ^{-1} G_0 (tP + (1 + t G_0) W (1+P) ) \nonumber \\= & {} \mathcal{G}^{[3+1]} (VP +W(1+P) ). \end{aligned}$$

(78)

Equation (77) is multiplied with Eq. (41) from the left hand side we have

$$\begin{aligned} \psi _{1}= & {} \psi _t^F \delta _{[3+1]}+ \mathcal{G}^{[3+1]} (VP +W(1+P) ) (-P_{34}\psi _{1}+\psi _{2}) \nonumber \\&+\, \mathcal{G}^{[3+1]} W_{123}^{(3)}(-P_{34}P+\tilde{P})(\psi _{1}-P_{34}\psi _{1}+\psi _{2}),~~~ \end{aligned}$$

(79)

Comparing it with Eq. (51) we obtain Eq. (53).

Appendix B

Here we derive the scattering operator $U^{[2+2]}$ in Eq.(57). To this end we start with $G_0 t \tilde{P}$

$$\begin{aligned} G_0 t \tilde{P}= & {} (G_0+G_0 t G_0) V \tilde{P} \nonumber \\= & {} (G_0+G_0tG_0)(G_0^{-1} - \{\mathcal{G}^{[2+2]}\} ^{-1} - V) \nonumber \\= & {} (1+G_0 t) - (G_0+G_0tG_0) \{\mathcal{G}^{[2+2]}\} ^{-1} - G_0 t \nonumber \\= & {} 1- (G_0+G_0tG_0) \{\mathcal{G}^{[2+2]}\} ^{-1} \end{aligned}$$

(80)

to obtain

$$\begin{aligned} (1-G_0t \tilde{P} ) ^{-1} = \mathcal{G}^{[2+2]} (G_0 + G_0 t G_0) ^{-1} \end{aligned}$$

(81)

and

$$\begin{aligned} (1-G_0t \tilde{P}) ^{-1} G_0 t \tilde{P}= & {} \mathcal{G}^{[2+2]} (G_0 + G_0 t G_0) ^{-1} G_0 t \tilde{P} \nonumber \\= & {} \mathcal{G}^{[2+2]} V \tilde{P}. \end{aligned}$$

(82)

When Eq. (81) is multiplied by Eq. (45) from the left, we get

$$\begin{aligned} \psi _{2}= \phi _{d:12} \phi _{d:34} \delta _{[2+2]}+ \mathcal{G}^{[2+2]} V \tilde{P}(1-P_{34})\psi _{1} \end{aligned}$$

(83)

Comparing it with Eq. (56) we obtain Eq. (57).