Partial-wave expansion of $\mathrm{\Lambda }NN$ three-baryon interactions in chiral effective field theory

An expression of partial-wave expansion of three-baryon interactions in chiral effective field theory is presented. The derivation follows the method by Hebeler et al. [Phys. Rev. C 91, 044001 (2015)], but the final expression is more general. That is, a systematic treatment of the higher-rank spin-momentum structure of the interaction becomes possible. Using the derived formula, a $\mathrm{\Lambda }$-deuteron folding potential is evaluated. This information is valuable for inferring the possible contribution of the $\mathrm{\Lambda }NN$ three-baryon forces to the hypertriton as the basis of further studies by sophisticated Faddeev calculations. A microscopic understanding of $\mathrm{\Lambda }NN$ three-baryon forces together with two-body $\mathrm{\Lambda }N$ interactions is essential for the description of hypernuclei and neutron-star matter.

Figure 1

Diagram of two-pion exchange $\mathrm{\Lambda }NN$ 3BF. The small filled circle denotes the $NN\pi$ vertex with the coupling constant $g_A/f_0^2$, and the large filled circle denotes the $NN\pi \pi$ vertex specified by the coupling constants $3b_0+b_D$ and $2b_2+3b_4$ in Eq. (1).

Figure 2

Three types of the Jacobi momenta. The length of the vectors ${\mathit{p}}_i$ and ${\mathit{q}}_i$ does not correspond to the distance in the figure.

Figure 3

Deuteron $s$- and $d$-wave functions in momentum space described by various $NN$ interactions: ChEFT [14], AV18 [15], CD-Bonn [16], and Paris [17]. The sign of the d-wave function is reversed. The normalization of these wave functions is ${\int }_0^{\infty }dk[{\varphi }_0^2\left(k\right)+{\varphi }_2^2\left(k\right)]=1$.

Figure 4

$\mathrm{\Lambda }$-deuteron folding potential with ${\ell }_{\mathrm{\Lambda }}=0$ from two-pion exchange $\mathrm{\Lambda }NN$ interactions. The upper panel shows the contribution of the deuteron $s$-state pair. The lower panel shows the contribution from the remaining pairs: $sd$, $ds$, and $dd$.