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

M. Kohno  Research Center for Nuclear Physics, Osaka University, Ibaraki
567-0047, Japan  H. Kamada  Department of Physics, Faculty of Engineering,
Kyushu Institute of Technology, Kitakyushu 804-8550, Japan  K. Miyagawa
Research Center for Nuclear Physics, Osaka University, Ibaraki 567-0047, Japan

###### Abstract

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. C91, 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
$\Lambda$-deuteron folding potential is evaluated. This information is
valuable for inferring the possible contribution of the $\Lambda NN$ three-
baryon forces to the hypertriton as the basis of further studies by
sophisticated Faddeev calculations. A microscopic understanding of $\Lambda
NN$ three-baryon forces together with two-body $\Lambda N$ interactions is
essential for the description of hypernuclei and neutron-star matter.

##  I Introduction

Any description of two-body baryon-baryon interactions in which various
degrees of freedom are eliminated or frozen is effective. When the
interactions are applied in many-body systems, the appearance of three-body
interactions is inevitable as induced interactions. The important role of
three-body forces (3BFs) in nuclear physics has been observed in scattering
and binding properties of few-nucleon systems WGH98 ; SS02 ; PPW01  and also
in heavier nuclei and nuclear matter, in particular in connection with
saturation properties PPW01 ; APR98 ; BB12 . The recent development of the
construction of baryon-baryon interactions in chiral effective field theory
(ChEFT) EHM09 ; ME11  provides a systematic way to introduce three-body (and
more-than-three-body) forces in a power-counting scheme and therefore
quantifies the role of 3BFs as opposed to simple phenomenological adjustment.

The inclusions of 3BFs in a microscopic description of nuclei often need
partial-wave expansion in two Jacobi momenta. An efficient method was
developed by Hebeler et al. Heb15  for the local 3BFs. Here the local means
that the interaction is a function of the momentum transfer of each Jacobi
momentum except for the cutoff regularization function that does not depend on
angle variables. In their method, the original eight-dimensional angular
integration, though five dimensional because of the rotational invariance, was
reduced essentially to two dimensional.

In this article, following the derivation in Ref. Heb15 , a different
expression for the partial-wave expansion of 3BFs is presented, which is more
systematic for treating higher-rank coupling of spin and momentum vectors.

Before discussing the partial-wave expansion, the basic structure of leading-
order 3BFs in ChEFT is summarized in Sec. II. The expression of partial wave
decomposition of 3BFs in momentum space concerning the Jacobi momenta is
presented in Sec. III. As an application of the derived expression, a possible
role of the $\Lambda$NN 3BFs in the hypertriton is studied by calculating a
$\Lambda$-deuteron folding potential from $\Lambda NN$ 3BFs. A summary follows
in Sec. IV.

##  II Structure of $\Lambda NN$ 3BFs in ChEFT

Two-pion exchange $\Lambda NN$ 3BFs are considered as a concrete example,
which is relevant for studying hypertriton. The structure of the $\Lambda NN$
force in the lowest-order, namely next-to-next-to-leading order (NNLO), is
particularly simple because the $\pi\Lambda\Lambda$ vertex is not present. The
contribution is only from the diagram shown in Fig. 1. The coordinate 1 is
assigned to the $\Lambda$ hyperon.

Figure 1: Diagram of two-pion exchange $\Lambda NN$ 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).

Following the expression by Petschauer et al. Pet16 , the Born amplitude of
this diagram is written as

$\displaystyle V_{TPE}^{\Lambda
NN}=\frac{g_{A}^{2}}{3f_{0}^{4}}(\mbox{\boldmath$\tau$}_{2}\cdot\mbox{\boldmath$\tau$}_{3})\frac{(\mbox{\boldmath$\sigma$}_{3}\cdot\mbox{\boldmath$q$}_{3d})(\mbox{\boldmath$\sigma$}_{2}\cdot\mbox{\boldmath$q$}_{2d})}{(\mbox{\boldmath$q$}_{3d}^{2}+m_{\pi}^{2})(\mbox{\boldmath$q$}_{2d}^{2}+m_{\pi}^{2})}$
$\displaystyle\times\\{-(3b_{0}+b_{D})m_{\pi}^{2}+(2b_{2}+3b_{4})\mbox{\boldmath$q$}_{3d}\cdot\mbox{\boldmath$q$}_{2d}\\},$
(1)

where $\mbox{\boldmath$q$}_{2d}$ ($\mbox{\boldmath$q$}_{3d}$) is the
difference between the final and initial momenta at the nucleon line 2 (line
3):
$\mbox{\boldmath$q$}_{2d}=\mbox{\boldmath$k$}_{2}^{\prime}-\mbox{\boldmath$k$}_{2}$
and
$\mbox{\boldmath$q$}_{3d}=\mbox{\boldmath$k$}_{3}^{\prime}-\mbox{\boldmath$k$}_{3}$.
$g_{A}$ is the axial coupling constant, $f_{0}$ is the pion decay constant,
$m_{\pi}$ is the pion mass, and $\mbox{\boldmath$\sigma$}_{i}$ and
$\mbox{\boldmath$\tau$}_{i}$ stand for the spin and isospin operators,
respectively, of nucleon $i$ (with $i=2,3$). The coupling constants $b_{0}$,
$b_{D}$, $b_{2}$, and $b_{4}$ inherit those in the underlying Lagrangian.
These coupling constants are to be determined in parametrizing $\Lambda N$
interactions in the next-to-next-to-leading order. However, such an attempt is
not possible at present. In this paper, we use the estimation by Petshauer et
al. Pet16 .

Figure 2: Three types of the Jacobi momenta. The length of the vectors
$\mbox{\boldmath$p$}_{i}$ and $\mbox{\boldmath$q$}_{i}$ does not correspond to
the distance in the figure.

In the following, particle 1 is assigned to the $\Lambda$ hyperon, and the
case of the Jacobi momenta $(\mbox{\boldmath$p$}_{1},\mbox{\boldmath$q$}_{1})$
that is depicted by the leftmost diagram of Fig. 2 is considered. In the
center-of-mass frame, $\mbox{\boldmath$k$}_{1}=\mbox{\boldmath$q$}_{1}$,
$\mbox{\boldmath$k$}_{2}=\mbox{\boldmath$p$}_{1}-r_{NN}\mbox{\boldmath$q$}_{1}$,
an
$\mbox{\boldmath$k$}_{3}=-\mbox{\boldmath$p$}_{1}-r_{NN}\mbox{\boldmath$q$}_{1}$
with $r_{NN}\equiv 1/2$. Then,
$\mbox{\boldmath$q$}_{1d}=\mbox{\boldmath$k$}_{1}^{\prime}-\mbox{\boldmath$k$}_{1}=\mbox{\boldmath$q$}_{1}^{\prime}-\mbox{\boldmath$q$}_{1}$,
$\mbox{\boldmath$q$}_{2d}=\mbox{\boldmath$k$}_{2}^{\prime}-\mbox{\boldmath$k$}_{2}=\mbox{\boldmath$p$}_{1}^{\prime}-\mbox{\boldmath$p$}_{1}-r_{NN}(\mbox{\boldmath$q$}_{1}^{\prime}-\mbox{\boldmath$q$}_{1})$,
and
$\mbox{\boldmath$q$}_{3d}=\mbox{\boldmath$k$}_{3}^{\prime}-\mbox{\boldmath$k$}_{3}=-(\mbox{\boldmath$p$}_{1}^{\prime}-\mbox{\boldmath$p$}_{1})-r_{NN}(\mbox{\boldmath$q$}_{1}^{\prime}-\mbox{\boldmath$q$}_{1})$.
$V_{TPE}^{\Lambda NN}$ is a function of
$\mbox{\boldmath$p$}\equiv\mbox{\boldmath$p$}_{1}^{\prime}-\mbox{\boldmath$p$}_{1}$
and
$\mbox{\boldmath$q$}\equiv\mbox{\boldmath$q$}_{1}^{\prime}-\mbox{\boldmath$q$}_{1}$
and can be organized in the following tensor-product representation.

$\displaystyle V_{TPE}^{\Lambda NN}(\mbox{\boldmath$p$},\mbox{\boldmath$q$})$
$\displaystyle=4\pi(\mbox{\boldmath$\tau$}_{2}\cdot\mbox{\boldmath$\tau$}_{3})\sum_{K=0,1,2}\sum_{\ell_{a},\ell_{b}}V_{TPE}^{(K,\ell_{a},\ell_{b})}(p,q)$
$\displaystyle\times[[Y_{\ell_{a}}(\hat{\mbox{\boldmath$p$}})\times
Y_{\ell_{b}}(\hat{\mbox{\boldmath$q$}})]^{K}\times[\mbox{\boldmath$\sigma$}_{2}\times\mbox{\boldmath$\sigma$}_{3}]^{K}]_{0}^{0},$
(2)

where the standard notation is employed for the tensor product:

$\displaystyle[Y_{\ell_{a}}(\hat{\mbox{\boldmath$a$}})\times
Y_{\ell_{b}}(\hat{\mbox{\boldmath$b$}})]_{m_{c}}^{\ell_{c}}=\sum_{m_{a}m_{b}}(\ell_{a}m_{a}\ell_{b}m_{b}|\ell_{c}m_{c})$
$\displaystyle\times
Y_{\ell_{a}m_{a}}(\hat{\mbox{\boldmath$a$}})Y_{\ell_{b}m_{b}}(\hat{\mbox{\boldmath$b$}}).$
(3)

The explicit expressions of $V_{TPE}^{(K,\ell_{a},\ell_{b})}(p,q)$ for the
Jacobi momenta $(\mbox{\boldmath$p$}_{1},\mbox{\boldmath$q$}_{1})$ are given
in Appendix A. It is straightforward to obtain a similar representation for
the other two sets of the Jacobi momenta,
$(\mbox{\boldmath$p$}_{2},\mbox{\boldmath$q$}_{2})$ and
$(\mbox{\boldmath$p$}_{3},\mbox{\boldmath$q$}_{3})$ in Fig. 2, for the 3BFs of
Eq. (1). Because the mass of the $\Lambda$ hyperon differs from that of the
nucleon, $\mbox{\boldmath$k$}_{i}$ and $\mbox{\boldmath$k$}_{i}^{\prime}$
($i=1,2,3$) are not treated cyclically and the functional form of
$V_{TPE}^{K,\ell_{a},\ell_{b}}(p,q)$ is different from those given in Appendix
A. It is noted that the third sigma operator $\mbox{\boldmath$\sigma$}_{3}$
may appear in a general 3BF. In that case, the rank of $K=3$ can appear.

As for the cutoff, the following regulator function is introduced for the
initial and final Jacobi momenta,
$(\mbox{\boldmath$a$},\mbox{\boldmath$b$})=(\mbox{\boldmath$p$}_{1},\mbox{\boldmath$q$}_{1})$
or $(\mbox{\boldmath$p$}_{1}^{\prime},\mbox{\boldmath$q$}_{1}^{\prime})$, with
the scale of $\Lambda=550$ MeV in present calculations:

$f_{\Lambda}(\mbox{\boldmath$a$},\mbox{\boldmath$b$})=\exp\\{-(\mbox{\boldmath$a$}^{2}+\frac{3}{4}\mbox{\boldmath$b$}^{2})^{2}/\Lambda^{4}\\}.$
(4)

Because this function does not depend on the angles, it does not affect the
calculation of the partial-wave expansion, which is discussed in the next
section.

##  III Partial-wave expansion

A 3BF is in general a function of the initial and final Jacobi momenta,
$V_{3BF}(\mbox{\boldmath$p$}_{1}^{\prime},\mbox{\boldmath$q$}_{1}^{\prime},\mbox{\boldmath$p$}_{1},\mbox{\boldmath$q$}_{1})$
in the case of the leftmost diagram of Fig. 2 with supressing the spin and
isospin indices. The partial-wave expansion requires integrals of the product
of four spherical harmonics and
$V_{3BF}(\mbox{\boldmath$p$}_{1}^{\prime},\mbox{\boldmath$q$}_{1}^{\prime},\mbox{\boldmath$p$}_{1},\mbox{\boldmath$q$}_{1})$
over the solid angles related to Jacobi momenta
$\mbox{\boldmath$p$}_{1}^{\prime}$, $\mbox{\boldmath$q$}_{1}^{\prime}$,
$\mbox{\boldmath$p$}_{1}$, and $\mbox{\boldmath$q$}_{1}$ for
$V_{3BF}(\mbox{\boldmath$p$}_{1}^{\prime},\mbox{\boldmath$q$}_{1}^{\prime},\mbox{\boldmath$p$}_{1},\mbox{\boldmath$q$}_{1})$:

$\displaystyle\frac{1}{\\{(2\pi)^{3/2}\\}^{4}}\int\cdots\int
d\hat{\mbox{\boldmath$p$}_{1}^{\prime}}d\hat{\mbox{\boldmath$q$}_{1}^{\prime}}d\hat{\mbox{\boldmath$p$}_{1}}d\hat{\mbox{\boldmath$q$}_{1}}Y_{\ell_{p}^{\prime}m_{p}^{\prime}}^{*}(\hat{\mbox{\boldmath$p$}_{1}^{\prime}})$
$\displaystyle\times
Y_{\ell_{q}^{\prime}m_{q}^{\prime}}^{*}(\hat{\mbox{\boldmath$q$}_{1}^{\prime}})Y_{\ell_{p}m_{p}}(\hat{\mbox{\boldmath$p$}_{1}})Y_{\ell_{q}m_{q}}(\hat{\mbox{\boldmath$q$}_{1}})V_{3BF}(\mbox{\boldmath$p$}_{1}^{\prime},\mbox{\boldmath$q$}_{1}^{\prime},\mbox{\boldmath$p$}_{1},\mbox{\boldmath$q$}_{1}).$
(5)

To make the expression compact, the following notation is used in the
subsequent derivation, in which the angular momenta $\ell_{p}$ and $\ell_{q}$
are coupled to $L$, and $\ell_{p}^{\prime}$ and $\ell_{q}^{\prime}$ to
$L^{\prime}$, respectively. That is, supplementing the summation

$\sum_{m_{\ell_{p}^{\prime}}m_{\ell_{q}^{\prime}}}\sum_{m_{\ell_{p}}m_{\ell_{q}}}(\ell_{p}^{\prime}m_{\ell_{p}^{\prime}}\ell_{q}^{\prime}m_{\ell_{q}^{\prime}}|L^{\prime}M_{L}^{\prime})(\ell_{p}m_{\ell_{p}}\ell_{q}m_{\ell_{q}}|LM_{L}),$
(6)

the angular integration on a 3BF $V_{3BF}$ is expressed as

$\displaystyle\frac{1}{(2\pi)^{6}}\langle[Y_{\ell_{p}^{\prime}}(\hat{\mbox{\boldmath$p$}_{1}^{\prime}})\times
Y_{\ell_{q}^{\prime}}(\hat{\mbox{\boldmath$q$}_{1}^{\prime}})]_{M_{L}^{\prime}}^{L^{\prime}}|V_{3BF}$
$\displaystyle\times|[Y_{\ell_{p}}(\hat{\mbox{\boldmath$p$}_{1}})\times
Y_{\ell_{q}}(\hat{\mbox{\boldmath$q$}_{1}})]_{M_{L}}^{L}\rangle,$ (7)

where the left- and right-angle brackets represent
$d\hat{\mbox{\boldmath$p$}_{1}^{\prime}}d\hat{\mbox{\boldmath$q$}_{1}^{\prime}}d\hat{\mbox{\boldmath$p$}_{1}}d\hat{\mbox{\boldmath$q$}_{1}}$
integration.

The angular momentum projection in momentum space postulates a complete plane-
wave basis GL96 ,

$\langle\mbox{\boldmath$p$}^{\prime}|p\ell_{p}m_{p}\rangle=\frac{\delta(p^{\prime}-p)}{p^{\prime}p}Y_{\ell_{p}m_{p}}(\hat{\mbox{\boldmath$p$}^{\prime}}).$
(8)

The three-body basis states in jj coupling for the total three-body angular
momentum $J$ are constructed as

$\displaystyle|pq\alpha\rangle\equiv$
$\displaystyle|pq(\ell_{p}s_{p})j_{p}(\ell_{q}1/2)j_{q}(j_{p}j_{q})JM\rangle$
$\displaystyle=$
$\displaystyle\sum_{m_{j_{p}}m_{j_{q}}}\sum_{m_{p}m_{s_{p}}}\sum_{m_{q}m_{s_{q}}}(j_{p}m_{j_{p}}j_{q}m_{j_{q}}|JM)$
$\displaystyle\times(\ell_{p}m_{p}s_{p}m_{s_{p}}|j_{p}m_{j_{p}})(\ell_{q}m_{q}s_{q}m_{s_{q}}|j_{p}m_{j_{p}})$
$\displaystyle\times|p\ell_{p}m_{p}\rangle\chi_{p}^{s_{p},m_{s_{p}}}|q\ell_{q}m_{q}\rangle\chi_{q}^{s_{q},m_{s_{q}}},$
(9)

where $\chi_{p}^{s_{p},m_{s_{p}}}$ and $\chi_{q}^{s_{q},m_{s_{q}}}$ represent
spin states of the $p$ and $q$ degrees of freedom, rspectively. The isospin
state can be treated separately. The basis states satisfy the orthonormality
condition:

$\langle
p^{\prime}q^{\prime}\alpha^{\prime}|pq\alpha\rangle=\frac{\delta(p^{\prime}-p)}{p^{\prime}p}\frac{\delta(q^{\prime}-q)}{q^{\prime}q}\delta_{\alpha^{\prime}\alpha}.$
(10)

For the case of a local 3BF
$V_{3BF}(\mbox{\boldmath$p$}_{1}^{\prime},\mbox{\boldmath$q$}_{1}^{\prime},\mbox{\boldmath$p$}_{1},\mbox{\boldmath$q$}_{1})=V_{3BF}(\mbox{\boldmath$p$}_{1}^{\prime}-\mbox{\boldmath$p$}_{1},\mbox{\boldmath$q$}_{1}^{\prime}-\mbox{\boldmath$q$}_{1})$,
the subtle manipulation Heb15  of adding a radial part to the angle
integration while keeping the absolute value by a delta function is helpful,

$\displaystyle d\hat{\mbox{\boldmath$p$}_{1}^{\prime}}\rightarrow
d\mbox{\boldmath$p$}_{1}^{\prime\prime}\frac{\delta(p_{1}^{\prime\prime}-p_{1}^{\prime})}{{p_{1}^{\prime}}^{2}}\hskip
5.0pt\mbox{and}\hskip 5.0ptd\hat{\mbox{\boldmath$q$}_{1}^{\prime}}\rightarrow
d\mbox{\boldmath$q$}_{1}^{\prime\prime}\frac{\delta(q_{1}^{\prime\prime}-q_{1}^{\prime})}{{q_{1}^{\prime}}^{2}}.$
(11)

Changing the variables $\mbox{\boldmath$p$}_{1}^{\prime\prime}$ and
$\mbox{\boldmath$q$}_{1}^{\prime\prime}$ to $p$ and $q$ by the shifts of
$\mbox{\boldmath$p$}_{1}^{\prime\prime}=\mbox{\boldmath$p$}+\mbox{\boldmath$p$}_{1}$
and
$\mbox{\boldmath$q$}_{1}^{\prime\prime}=\mbox{\boldmath$q$}+\mbox{\boldmath$q$}_{1}$,
Eq. (7) is modified as

$\displaystyle\frac{1}{(2\pi)^{6}}\langle[Y_{\ell_{p}^{\prime}}(\hat{\mbox{\boldmath$p$}_{1}^{\prime}})\times
Y_{\ell_{q}^{\prime}}(\hat{\mbox{\boldmath$q$}_{1}^{\prime}})]_{M_{L}^{\prime}}^{L^{\prime}}|V_{3BF}$
$\displaystyle\times|[Y_{\ell_{p}}(\hat{\mbox{\boldmath$p$}_{1}})\times
Y_{\ell_{q}}(\hat{\mbox{\boldmath$q$}_{1}})]_{M_{L}}^{L}\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{({2\pi})^{6}}\int_{0}^{\infty}p^{2}dp\int_{0}^{\infty}q^{2}dq$
$\displaystyle\hskip
20.00003pt\times\langle[Y_{\ell_{p}^{\prime}}(\widehat{\mbox{\boldmath$p$}_{1}+\mbox{\boldmath$p$}})\times
Y_{\ell_{q}^{\prime}}(\widehat{\mbox{\boldmath$q$}_{1}+\mbox{\boldmath$q$}})]_{M_{L}^{\prime}}^{L^{\prime}}|$
$\displaystyle\hskip
20.00003pt\times\frac{\delta(|\mbox{\boldmath$p$}-\mbox{\boldmath$p$}_{1}|-p_{1}^{\prime})}{{p_{1}^{\prime}}^{2}}\frac{\delta(|\mbox{\boldmath$q$}-\mbox{\boldmath$q$}_{1}|-q_{1}^{\prime})}{{q_{1}^{\prime}}^{2}}V_{3BF}$
$\displaystyle\hskip
20.00003pt\times|[Y_{\ell_{p}}(\hat{\mbox{\boldmath$p$}_{1}})\times
Y_{\ell_{q}}(\hat{\mbox{\boldmath$q$}_{1}})]_{M_{L}}^{L}\rangle.$ (12)

The delta function can be written as follows by using a Legendre polynomial of
the first kind $P_{k}$:

$\displaystyle\delta(|\mbox{\boldmath$p$}-\mbox{\boldmath$p$}_{1}|-p_{1}^{\prime})=\frac{p_{1}^{\prime}}{pp_{1}}\delta\left(\cos\widehat{\mbox{\boldmath$p$}_{1}\mbox{\boldmath$p$}}-\frac{p_{1}^{\prime
2}-p_{1}^{2}-p^{2}}{2p_{1}p}\right)$ $\displaystyle=$ $\displaystyle
2\pi\frac{p_{1}^{\prime}}{pp_{1}}\sum_{k^{\prime}}P_{k^{\prime}}(c_{p})(-1)^{k}\sqrt{\hat{k^{\prime}}}[Y_{k^{\prime}}(\hat{\mbox{\boldmath$p$}_{1}})\times
Y_{k^{\prime}}(\hat{\mbox{\boldmath$p$}})]_{0}^{0},$ (13)
$\displaystyle\delta(|\mbox{\boldmath$q$}-\mbox{\boldmath$q$}_{1}|-q_{1}^{\prime})=\frac{q_{1}^{\prime}}{qq_{1}}\delta\left(\cos\widehat{\mbox{\boldmath$q$}_{1}\mbox{\boldmath$q$}}-\frac{q_{1}^{\prime
2}-q_{1}^{2}-q^{2}}{2q_{1}q}\right)$ $\displaystyle=$ $\displaystyle
2\pi\frac{q_{1}^{\prime}}{qq_{1}}\sum_{k}P_{k}(c_{q})(-1)^{k}\sqrt{\hat{k}}[Y_{k}(\hat{\mbox{\boldmath$q$}_{1}})\times
Y_{k}(\hat{\mbox{\boldmath$q$}})]_{0}^{0},$ (14)

where $\hat{k}\equiv 2k+1$, $c_{p}\equiv\frac{p_{1}^{\prime
2}-p_{1}^{2}-p^{2}}{2p_{1}p}$, and $c_{q}\equiv\frac{q_{1}^{\prime
2}-q_{1}^{2}-q^{2}}{2q_{1}q}$. These $\delta$ functions restrict the $p$ and
$q$ integrations as

$\displaystyle p_{min}\equiv|p_{1}^{\prime}-p_{1}|\leq p\leq p_{max}\equiv
p_{1}^{\prime}+p_{1},$ (15) $\displaystyle
q_{min}\equiv|q_{1}^{\prime}-q_{1}|\leq q\leq q_{max}\equiv
q_{1}^{\prime}+q_{1}.$ (16)

The spherical-harmonic functions
$Y_{\ell_{p}^{\prime}}(\widehat{\mbox{\boldmath$p$}_{1}+\mbox{\boldmath$p$}})$
and
$Y_{\ell_{q}^{\prime}}(\widehat{\mbox{\boldmath$q$}_{1}+\mbox{\boldmath$q$}})$
are also expanded as follows:

$\displaystyle
Y_{\ell_{p}^{\prime}m_{p}^{\prime}}(\widehat{\mbox{\boldmath$p$}_{1}+\mbox{\boldmath$p$}})=$
$\displaystyle\sum_{j_{p}+j_{p}^{\prime}=\ell_{p}^{\prime}}\sqrt{\frac{4\pi(\hat{\ell_{p}^{\prime}})!}{(\hat{j_{p}})!(\hat{j_{p}^{\prime}})!}}\frac{p_{1}^{j_{p}}p^{j_{p}^{\prime}}}{{p_{1}^{\prime}}^{\ell_{p}^{\prime}}}$
$\displaystyle\times[Y_{j_{p}}(\hat{\mbox{\boldmath$p$}_{1}})\times
Y_{j_{p}^{\prime}}(\hat{\mbox{\boldmath$p$}})]_{m_{p}^{\prime}}^{\ell_{p}^{\prime}},$
(17) $\displaystyle
Y_{\ell_{q}^{\prime}m_{q}^{\prime}}(\widehat{\mbox{\boldmath$q$}_{1}+\mbox{\boldmath$q$}})=$
$\displaystyle\sum_{j_{q}+j_{q}^{\prime}=\ell_{q}^{\prime}}\sqrt{\frac{4\pi(\hat{\ell_{q}^{\prime}})!}{(\hat{j_{q}})!(\hat{j_{q}^{\prime}})!}}\frac{q_{1}^{j_{q}}q^{j_{q}^{\prime}}}{{q_{1}^{\prime}}^{\ell_{q}^{\prime}}}$
$\displaystyle\times[Y_{j_{q}}(\hat{\mbox{\boldmath$q$}_{1}})\times
Y_{j_{q}^{\prime}}(\hat{\mbox{\boldmath$q$}})]_{m_{q}^{\prime}}^{\ell_{q}^{\prime}}.$
(18)

A straightforward evaluation of the recoupling of these decoupled spherical
harmonics finally gives

$\displaystyle\frac{1}{(2\pi)^{6}}\langle[Y_{\ell_{p}^{\prime}}(\hat{\mbox{\boldmath$p$}_{1}^{\prime}})\times
Y_{\ell_{q}^{\prime}}(\hat{\mbox{\boldmath$q$}_{1}^{\prime}})]_{M_{L}^{\prime}}^{L^{\prime}}|V_{TPE}^{\Lambda
NN}|[Y_{\ell_{p}}(\hat{\mbox{\boldmath$p$}_{1}})\times
Y_{\ell_{q}}(\hat{\mbox{\boldmath$q$}_{1}})]_{M_{L}}^{L}\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{({2\pi})^{6}}(-1)^{\ell_{p}^{\prime}+\ell_{q}^{\prime}-L^{\prime}}\sum_{JM}(-1)^{M_{L}^{\prime}}(L^{\prime}-M_{L}^{\prime}LM_{L}|JM)\frac{1}{p_{1}^{\prime}q_{1}^{\prime}{p_{1}}{q_{1}}}\int_{p_{min}}^{p_{max}}pdp\int_{q_{min}}^{q_{max}}qdq$
$\displaystyle\times(2\pi)^{2}\sum_{k^{\prime}k}\hat{k^{\prime}}\hat{k}P_{k^{\prime}}(c_{p})P_{k}(c_{q})\sum_{j_{p}+j_{p}^{\prime}=\ell_{p}^{\prime}}\sqrt{\frac{(\hat{\ell_{p}^{\prime}})!}{(\hat{j_{p}})!(\hat{j_{p}^{\prime}})!}}\frac{p_{1}^{j_{p}}p^{j_{p}^{\prime}}}{{p_{1}^{\prime}}^{\ell_{p}^{\prime}}}\sum_{j_{q}+j_{q}^{\prime}=\ell_{q}^{\prime}}(-1)^{j_{p}+j_{q}}\sqrt{\frac{(\hat{\ell_{q}^{\prime}})!}{(\hat{j_{q}})!(\hat{j_{q}^{\prime}})!}}\frac{q_{1}^{j_{q}}q^{j_{q}^{\prime}}}{{q_{1}^{\prime}}^{\ell_{q}^{\prime}}}\sqrt{\hat{\ell_{p}^{\prime}}\hat{\ell_{q}^{\prime}}\hat{L}\hat{L^{\prime}}}$
$\displaystyle\times\frac{1}{4\pi}\sum_{L_{p}L_{q}}\sqrt{\hat{j_{p}^{\prime}}\hat{j_{q}^{\prime}}\hat{j_{p}}\hat{j_{q}}}(k^{\prime}0j_{p}^{\prime}0|L_{p}0)(k0j_{q}^{\prime}0|L_{q}0)(k^{\prime}0j_{p}0|\ell_{p}0)(k0j_{q}0|\ell_{q}0)$
$\displaystyle\times\begin{Bmatrix}\ell_{q}&\ell_{q}^{\prime}&L_{q}\\\
j_{q}^{\prime}&k&j_{q}\end{Bmatrix}\begin{Bmatrix}\ell_{p}&\ell_{p}^{\prime}&L_{p}\\\
j_{p}^{\prime}&k^{\prime}&j_{p}\end{Bmatrix}\begin{Bmatrix}\ell_{p}^{\prime}&\ell_{q}^{\prime}&L^{\prime}\\\
\ell_{p}&\ell_{q}&L\\\ L_{p}&L_{q}&J\end{Bmatrix}\int
d\hat{\mbox{\boldmath$p$}}d\hat{\mbox{\boldmath$q$}}\;[Y_{L_{p}}(\hat{\mbox{\boldmath$p$}})\times
Y_{L_{q}}(\hat{\mbox{\boldmath$q$}})]_{M}^{J}V_{TPE}^{\Lambda
NN}(\mbox{\boldmath$p$},\mbox{\boldmath$q$})$ (19)

Observing
$P_{\ell}(\cos\widehat{\mbox{\boldmath$p$}\mbox{\boldmath$q$}})=(-1)^{\ell}\frac{4\pi}{\sqrt{\hat{\ell}}}[Y_{\ell}(\hat{\mbox{\boldmath$p$}})\times
Y_{\ell}(\hat{\mbox{\boldmath$q$}})]_{0}^{0}$, the above expression with $J=0$
corresponds to Eq. (6) of Ref. Heb15 . It is verified numerically that Eq.
(19) with $J=0$ delivers the same results as Eq. (6) of Ref. Heb15 . The
evaluation of $\hat{I}_{L^{\prime}L}V_{3BF}$ with $J>0$ is straightforward and
transparent.

The angle integration in Eq. (19) for 3BFs in the form of Eq. (2) needs

$\displaystyle\int
d\hat{\mbox{\boldmath$p$}}d\hat{\mbox{\boldmath$q$}}\;[Y_{L_{p}}(\hat{\mbox{\boldmath$p$}})\times
Y_{L_{q}}(\hat{\mbox{\boldmath$q$}})]_{M}^{J}[Y_{\ell_{a}}(\hat{\mbox{\boldmath$p$}})\times
Y_{\ell_{b}}(\hat{\mbox{\boldmath$q$}})]_{\mu}^{K}$
$\displaystyle=(-1)^{K+\mu}\delta_{JK}\delta_{M,-\mu}\delta_{L_{p}\ell_{a}}\delta_{L_{q}\ell_{b}}(-1)^{\ell_{a}+\ell_{b}}.$
(20)

This means that the angular momentum $J$ in Eq. (19) corresponds to the rank
$K$ of the tensor-product structure of the angular-momentum coupling in 3BFs.

For actual calculations of the matrix element of 3BFs in various situations,
it is convenient to define a reduced matrix element in a form similar to the
Wigner-Eckart theorem,

$\displaystyle\frac{1}{(2\pi)^{6}}\langle[Y_{\ell_{p}^{\prime}}(\hat{\mbox{\boldmath$p$}_{1}^{\prime}})\times
Y_{\ell_{q}^{\prime}}(\hat{\mbox{\boldmath$q$}_{1}^{\prime}})]_{M_{L}^{\prime}}^{L^{\prime}}|V_{TPE}^{(K,\ell_{a},\ell_{b})}(p,q)[Y_{\ell_{a}}(\hat{\mbox{\boldmath$p$}})\times
Y_{\ell_{b}}(\hat{\mbox{\boldmath$q$}})]_{\mu}^{K}|[Y_{\ell_{p}}(\hat{\mbox{\boldmath$p$}_{1}})\times
Y_{\ell_{q}}(\hat{\mbox{\boldmath$q$}_{1}})]_{M_{L}}^{L}\rangle$
$\displaystyle=$
$\displaystyle(LM_{L}K\mu|L^{\prime}M_{L}^{\prime})\langle[Y_{\ell_{p}^{\prime}}(\hat{\mbox{\boldmath$p$}_{1}^{\prime}})\times
Y_{\ell_{q}^{\prime}}(\hat{\mbox{\boldmath$q$}_{1}^{\prime}})]^{L^{\prime}}||V_{TPE}^{(K,\ell_{a},\ell_{b})}(p,q)[Y_{\ell_{a}}(\hat{\mbox{\boldmath$p$}})\times
Y_{\ell_{b}}(\hat{\mbox{\boldmath$q$}})]^{K}||[Y_{\ell_{p}}(\hat{\mbox{\boldmath$p$}_{1}})\times
Y_{\ell_{q}}(\hat{\mbox{\boldmath$q$}_{1}})]^{L}\rangle_{pwe}.$ (21)

From Eq. (19), the reduced matrix element is found as

$\displaystyle\langle[Y_{\ell_{p}^{\prime}}(\hat{\mbox{\boldmath$p$}_{1}^{\prime}})\times
Y_{\ell_{q}^{\prime}}(\hat{\mbox{\boldmath$q$}_{1}^{\prime}})]^{L^{\prime}}||V_{TPE}^{(K,\ell_{a},\ell_{b})}(p,q)[Y_{\ell_{a}}(\hat{\mbox{\boldmath$p$}})\times
Y_{\ell_{b}}(\hat{\mbox{\boldmath$q$}})]^{K}||[Y_{\ell_{p}}(\hat{\mbox{\boldmath$p$}_{1}})\times
Y_{\ell_{q}}(\hat{\mbox{\boldmath$q$}_{1}})]^{L}\rangle_{pwe}$
$\displaystyle=$
$\displaystyle\frac{1}{4\pi}\frac{1}{(2\pi)^{4}}\sqrt{\frac{\hat{K}}{\hat{L^{\prime}}}}(-1)^{\ell_{p}+\ell_{q}}\frac{1}{p_{1}^{\prime}q_{1}^{\prime}{p_{1}}{q_{1}}}\sqrt{\hat{\ell_{p}^{\prime}}\hat{\ell_{q}^{\prime}}\hat{\ell_{p}}\hat{\ell_{q}}\hat{L}\hat{L^{\prime}}\hat{\ell_{a}}\hat{\ell_{b}}}\int_{p_{min}}^{p_{max}}pdp\int_{q_{min}}^{q_{max}}qdq\sum_{k^{\prime}k}\hat{k^{\prime}}\hat{k}P_{k^{\prime}}(c_{p})P_{k}(c_{q})$
$\displaystyle\times\sum_{j_{p}+j_{p}^{\prime}=\ell_{p}^{\prime}}\sqrt{\frac{(\hat{\ell_{p}^{\prime}})!}{(\hat{j_{p}})!(\hat{j_{p}^{\prime}})!}}\frac{p_{1}^{j_{p}}p^{j_{p}^{\prime}}}{{p_{1}^{\prime}}^{\ell_{p}^{\prime}}}\sum_{j_{q}+j_{q}^{\prime}=\ell_{q}^{\prime}}(-1)^{j_{p}+j_{q}}\sqrt{\frac{(\hat{\ell_{q}^{\prime}})!}{(\hat{j_{q}})!(\hat{j_{q}^{\prime}})!}}\frac{q_{1}^{j_{q}}q^{j_{q}^{\prime}}}{{q_{1}^{\prime}}^{\ell_{q}^{\prime}}}(k^{\prime}0\ell_{a}0|j_{p}^{\prime}0)(k0\ell_{b}0|j_{q}^{\prime}0)$
$\displaystyle\times(k^{\prime}0\ell_{p}0|j_{p}0)(k0\ell_{q}0|j_{q}0)\begin{Bmatrix}\ell_{q}&\ell_{q}^{\prime}&\ell_{b}\\\
j_{q}^{\prime}&k&j_{q}\end{Bmatrix}\begin{Bmatrix}\ell_{p}&\ell_{p}^{\prime}&\ell_{a}\\\
j_{p}^{\prime}&k^{\prime}&j_{p}\end{Bmatrix}\begin{Bmatrix}\ell_{p}^{\prime}&\ell_{q}^{\prime}&L^{\prime}\\\
\ell_{p}&\ell_{q}&L\\\
\ell_{a}&\ell_{b}&K\end{Bmatrix}V_{TPE}^{(K,\ell_{a},\ell_{b})}(p,q).$ (22)

It should be remembered that this expression is valid for a 3BF depending only
on
$\mbox{\boldmath$p$}=\mbox{\boldmath$p$}_{1}^{\prime}-\mbox{\boldmath$p$}_{1}$
and $\mbox{\boldmath$q$}_{1}^{\prime}-\mbox{\boldmath$q$}_{1}$. Introduction
of the regularization functions given by Eq. (4) does not affect the angular
integrations.

##  IV $\Lambda$-deuteron folding potential

The construction of hyperon-nucleon interactions in the strangeness $S=-1$
sector has a difficulty in lacking sufficient scattering data. The fact that
there is no two-body $\Lambda N$ bound state enhances the difficulty. The
hypertriton is, therefore, an important hyper-nuclear system MK95 ; HMN20  for
adjusting the interaction, where the tuning of $\Lambda N$ interactions in the
spin-singlet and -triplet states together with the strength of $\Lambda
N$-$\Sigma N$ coupling can be carried out. However, the possible role of the
$\Lambda NN$ 3BFs has been investigated little. Expecting the settlement of
the current controversy over the shallow binding energy STAR00 , it is
important to describe the hypertriton including $\Lambda NN$ 3BFs. Before
considering Faddeev calculations for the hypertriton including $\Lambda NN$
3BFs, it is instructive to calculate the $\Lambda$-deuteron folding potential
due to the $\Lambda NN$ interactions, applying the expression derived in the
previous section, to obtain some idea about the contribution of the 3BFs.

The folding potential is evaluated by the following integration:

$\displaystyle U_{\Lambda-d}^{J_{t}}(q_{1}^{\prime},q_{1})=$
$\displaystyle\iint p_{1}^{\prime
2}dp_{1}^{\prime}p_{1}^{2}dp_{1}\langle[\Psi_{d}(\mbox{\boldmath$p$}_{1}^{\prime}),(\ell_{\Lambda}^{\prime}1/2)j_{\Lambda}]J_{t}|$
$\displaystyle\times V_{TPE}^{\Lambda
NN}|[\Psi_{d}(\mbox{\boldmath$p$}_{1}),(\ell_{\Lambda}1/2)j_{\Lambda}]J_{t}\rangle,$
(23) $\displaystyle\Psi_{d}(\mbox{\boldmath$p$}_{1})=$
$\displaystyle\sum_{\ell_{d}=0,2}\frac{1}{p_{1}}\phi_{\ell_{d}}(p_{1})[Y_{\ell_{d}}(\hat{\mbox{\boldmath$p$}_{1}})\times\chi_{d}^{1}]_{m}^{1}.$
(24)

The above expression is in an abbreviated notation. A more detailed
calculational procedure is given in Appendix B.

Figure 3: Deuteron $s$\- and $d$-wave functions in momentum space described by
various $NN$ interactions: ChEFT Epe05 , AV18 AV18 , CD-Bonn CDB , and Paris
PARIS . The sign of the $d$-wave function is reversed. The normalization of
these wave functions is
$\int_{0}^{\infty}dk\>(\phi_{0}^{2}(k)+\phi_{2}^{2}(k))=1$.

Deuteron wave functions, the $s$ and $d$ components, are those of the N3LO
ChEFT interactions with the cutoff of 550 MeV Epe05 . This scale of the wave
functions and the 3BFs may not be soft enough to permit a perturbative
treatment. Nevertheless, without strong short-range singularities, the
resulting folding potential helps intuitively infer the $\Lambda$-deuteron
interaction and therefore the possible role of the $\Lambda NN$ 3BFs to the
hypertriton. It is noted that it is not appropriate to employ deuteron wave
functions of other $NN$ interactions having strong short-range singularities
together with the ChEFT 3BFs. For comparison, deuteron wave functions in
momentum space are shown in Fig. 3 in which those of other $NN$ interactions,
i.e., AV18 AV18 , CD-Bonn CDB , and Paris PARIS , are compared.

Figure 4: $\Lambda$-deuteron folding potential with $\ell_{\Lambda}=0$ from
two-pion exchange $\Lambda NN$ interactions. The upper panel shows the
contribution of the deuteron $s$-state pair. The lower panel shows the
contribution of from the remaining pairs: $sd$, $ds$, and $dd$.

Calculated $s$-wave [$\ell_{\Lambda}^{\prime}=\ell_{\Lambda}=0$ in Eq. (23)]
$\Lambda$-deuteron folding potential with $J_{t}=1/2$ from the leading-order
$\Lambda NN$ interactions is presented in Fig. 4. The upper panel shows the
contribution from the $s$-wave component of the deuteron wave function, in
which $V_{TPE}^{(K=0,\ell_{a},\ell_{b})}$ participates. The lower panel
presents the sum of the remaining contributions from the $s$-$d$, $d$-$s$, and
$d$-$d$ pairs of the deuteron wave function.
$V_{TPE}^{(K=2,\ell_{a},\ell_{b})}$ contributes in the $s$-$d$ and $d$-$s$
pairs. Both $V_{TPE}^{(K=0,\ell_{a},\ell_{b})}$ and
$V_{TPE}^{(K=2,\ell_{a},\ell_{b})}$ contribute in the case of the $d$-$d$
pair. The potential with $J_{t}=3/2$ is identical to that with $J_{t}=1/2$ .

The employed parameters are taken from the estimation by Petshauer et al.
Pet16 ; that is, $(3b_{0}+b_{D})=0$ and $(2b_{2}+3b_{4})=3.0\times 10^{-3}$
MeV-1. In principle, parameters of the two-pion exchange 3BFs should be
determined in fitting the two-body $\Lambda N$ interactions. However, the
present experimental situation of the strangeness $S=-1$ sector does not allow
such an investigation.

The calculated potential is weakly attractive, with a depth below about 200
keV. The experimental separation energy of the hypertriton is very small,
$130\pm 50$ keV, though the actual value is still controversial. Therefore,
its wave function extends and the $\Lambda NN$ 3BF contribution can be
considered hindered. Still, the similar magnitude of the $\Lambda NN$ 3BF
effect shown in Fig. 4 as the separation energy indicates that the effect may
not be negligible in the hypertriton. Thus, it suggests that further study of
the hypertriton in Faddeev formalism with incorporation of the 3BFs is
necessary. The repulsive bump structure seen beyond $q\simeq 2.5$ fm implies
that the scale of 550 MeV employed for the ChEFT description still has
remnants of shorter-range singularities, which should be treated in a Faddeev
framework.

##  V Summary

We have presented an expression of partial-wave decomposition of 3BFs
concerning the relevant Jacobi momenta. The derivation essentially follows
that of Hebeler et al. Heb15 , but the final formula differs in that it can
systematically treat the higher-rank spin and angular-momentum tensor-product
structure of 3BFs. Although the consideration is intended specifically for
$\Lambda NN$ 3BFs and one set of the Jacobi momenta, the formula is general,
as far as 3BFs are a function of the momentum transfer in each Jacobi
momentum. Even if a regularization function is angle-dependent, the 3BFs can
be expressed in the form of Eq. (2) and then the result of Eq. (22) is
applicable.

As an application of the derived expression, the $\Lambda$-deuteron folding
potential from NNLO $\Lambda NN$ 3BFs is evaluated. At the present stage, the
construction of baryon-baryon interactions in the strangeness $S=-1$ sector in
ChEFT is practiced up to the NLO level NLO13 ; HMN20 . Even at this low order,
it is difficult to unambiguously determine coupling constants because of the
lack of sufficient experimental data, and therefore a plausible assumption of
the SU(3) symmetry has to be called for. In addition, there is no conclusive
way to fix the parameters of the two-pion exchange $\Lambda NN$ 3BFs that are
basically of the NNLO. This means that experimental and theoretical
investigations are needed in the future. It is essentially important to
quantitatively establish the contribution of $\Lambda NN$ 3BFs in hypernuclei,
which is also relevant for the understanding of the appearance or absence of
$\Lambda$ hyperons in neutron star matter MK18 ; GKW20 .

In particular, the investigation of the hypertriton is of fundamental
importance. Before doing full Faddeev calculations for the hypertriton
including $\Lambda NN$ 3BFs, it is worthwhile to estimate the possible role of
the 3BFs for the hypertriton. The present folding potential calculation
indicates the necessity of considering $\Lambda NN$ 3BFs in the theoretical
study of the hypertriton because the quantitative estimation of their effects
will influence the parametrization of $\Lambda N$ two-body interactions.

Acknowledgements. This work is supported by JSPS KAKENHI Grants No. JP19K03849
and No. JP22K03597.

##  Appendix A Tensor-product decomposition of $\Lambda NN$ three-body
interactions

$V_{TPE}^{(K,\ell_{a},\ell_{b})}(p,q)$ in Eq. (2) for $K=0$, 1 and 2 in the
case of the leftmost diagram of Fig. 1 are as follows:

$\displaystyle V_{TPE}^{K=0,\ell_{a},\ell_{b}}(p,q)=$
$\displaystyle(-1)^{\ell_{a}}\delta_{\ell_{a}\ell_{b}}\delta_{\ell_{a},even}\sqrt{\frac{\hat{\ell_{a}}}{3}}\frac{\\{C_{0}m_{\pi}^{2}+C_{1}(-p^{2}+r_{NN}^{2}q^{2})\\}(p^{2}-r_{NN}^{2}q^{2})}{2pr_{NN}q(p^{2}+r_{NN}^{2}q^{2}+m_{\pi}^{2})}Q_{\ell_{a}}(z_{pq}),$
(25) $\displaystyle V_{TPE}^{K=1,\ell_{a},\ell_{b}}(p,q)=$
$\displaystyle\delta_{\ell_{a}\ell_{b}}\delta_{\ell_{a},odd}\frac{\\{C_{0}m_{\pi}^{2}+C_{1}(-p^{2}+r_{NN}^{2}q^{2})\\}}{(p^{2}+r_{NN}^{2}q^{2}+m_{\pi}^{2})}\frac{1}{\hat{\ell_{a}}}\sqrt{\frac{\ell_{a}(\ell_{a}+1)}{6\hat{\ell_{a}}}}\\{Q_{\ell_{a}+1}(z_{pq})-Q_{\ell_{a}-1}(z_{pq})\\},$
(26) $\displaystyle V_{TPE}^{K=2,\ell_{a},\ell_{b}}(p,q)=$
$\displaystyle-\sqrt{\frac{2}{15}}\frac{\\{C_{0}m_{\pi}^{2}+C_{1}(-p^{2}+r_{NN}^{2}q^{2})\\}}{2pr_{NN}q(p^{2}+r_{NN}^{2}q^{2}+m_{\pi}^{2})}\sqrt{\hat{\ell_{a}}\hat{\ell_{b}}}(\ell_{a}0\ell_{b}0|20)\\{p^{2}Q_{\ell_{b}}(z_{pq})-r_{NN}^{2}q^{2}Q_{\ell_{a}}(z_{pq})\\},$
(27)

where $C_{0}=-\frac{g_{A}^{2}}{3f_{0}^{4}}(3b_{0}+b_{D})$,
$C_{1}=\frac{g_{A}^{2}}{3f_{0}^{4}}(2b_{2}+3b_{4})$, and
$z_{pq}=\frac{p^{2}+r_{NN}^{2}q^{2}+m_{\pi}^{2}}{2pr_{NN}q}$. $Q_{\ell}$ is a
Legendre polynomial of the second kind.

##  Appendix B Evaluation of $\Lambda$-deuteron folding potential from
$\Lambda NN$ 3BFs

An explicit calculational procedure of the $\Lambda$-deuteron folding
potential, given by Eq. (23), is provided. The abbreviated notation of Eq.
(23) means

$\displaystyle U_{\Lambda-d}^{J_{t}}(q_{1}^{\prime},q_{1})=$
$\displaystyle\iint p_{1}^{\prime
2}dp_{1}^{\prime}p_{1}^{2}dp_{1}\langle[\Psi_{d}(\mbox{\boldmath$p$}_{1}^{\prime}),(\ell_{\Lambda}^{\prime}1/2)j_{\Lambda}]J_{t}|V_{TPE}^{\Lambda
NN}|[\Psi_{d}(\mbox{\boldmath$p$}_{1}),(\ell_{\Lambda}1/2)j_{\Lambda}]J_{t}\rangle$
$\displaystyle=$
$\displaystyle\sum_{\ell_{d}^{\prime}=0,2}\sum_{\ell_{d}=0,2}\iint
p_{1}^{\prime
2}dp_{1}^{\prime}p_{1}^{2}dp_{1}\frac{1}{p_{1}^{\prime}}\phi_{\ell_{d}}(p_{1}^{\prime})\frac{1}{p_{1}}\phi_{\ell_{d}}(p_{1})\langle[[Y_{\ell_{d}^{\prime}}(\hat{\mbox{\boldmath$p$}_{1}^{\prime}})\times\chi_{d}^{1}]^{1}\times[Y_{\ell_{\Lambda}^{\prime}}(\hat{\mbox{\boldmath$q$}_{1}^{\prime}})\times\chi_{\Lambda}^{1/2}]^{j_{\Lambda}}]_{M_{t}}^{J_{t}}|$
$\displaystyle\times V_{TPE}^{\Lambda
NN}|[[Y_{\ell_{d}}(\hat{\mbox{\boldmath$p$}_{1}})\times\chi_{d}^{1}]^{1}\times[Y_{\ell_{\Lambda}}(\hat{\mbox{\boldmath$q$}_{1}})\times\chi_{\Lambda}^{1/2}]^{j_{\Lambda}}]_{M_{t}}^{J_{t}}\rangle,$
(28)

where $\chi_{d}^{1}$ and $\chi_{\Lambda}^{1/2}$ denote spin functions for the
deuteron and the $\Lambda$ hyperon, respectively. The isospin degrees of
freedom are not explicitly shown. Because the isospin of the $\Lambda$ hyperon
is 0 and that of the deuteron is 0, the matrix element of the isospin operator
$(\mbox{\boldmath$\tau$}_{2}\cdot\mbox{\boldmath$\tau$}_{3})$ in Eq. (2)
becomes $-3$. Substituting $V_{TPE}^{\Lambda NN}$ of Eq. (2), the following
angular-momentum recoupling is carried out:

$\displaystyle\langle[[Y_{\ell_{d}^{\prime}}(\hat{\mbox{\boldmath$p$}_{1}^{\prime}})\times\chi_{d}^{1}]^{1}\times[Y_{\ell_{\Lambda}^{\prime}}(\hat{\mbox{\boldmath$q$}_{1}^{\prime}})\times\chi_{\Lambda}^{1/2}]^{j_{\Lambda}}]_{M_{t}}^{J_{t}}|V_{TPE}^{(K,\ell_{a},\ell_{b})}(p,q)[[Y_{\ell_{a}}(\hat{\mbox{\boldmath$p$}})\times
Y_{\ell_{b}}(\hat{\mbox{\boldmath$q$}})]^{K}\times[\mbox{\boldmath$\sigma$}_{2}\times\mbox{\boldmath$\sigma$}_{3}]^{K}]_{0}^{0}$
$\displaystyle\times|[[Y_{\ell_{d}}(\hat{\mbox{\boldmath$p$}_{1}})\times\chi_{d}^{1}]^{1}\times[Y_{\ell_{\Lambda}}(\hat{\mbox{\boldmath$q$}_{1}})\times\chi_{\Lambda}^{1/2}]^{j_{\Lambda}}]_{M_{t}}^{J_{t}}\rangle$
$\displaystyle=$
$\displaystyle\sum_{L^{\prime}L}\sum_{S^{\prime}S}3\hat{j_{\Lambda}}\sqrt{\hat{L^{\prime}}\hat{S^{\prime}}\hat{L}\hat{S}}\begin{Bmatrix}\ell_{d}^{\prime}&\ell_{\Lambda^{\prime}}&L^{\prime}\\\
1&1/2&S^{\prime}\\\
1&j_{\Lambda^{\prime}}&J_{t}\end{Bmatrix}\begin{Bmatrix}\ell_{d}&\ell_{\Lambda}&L\\\
1&1/2&S\\\
1&j_{\Lambda^{\prime}}&J_{t}\end{Bmatrix}\langle[[Y_{\ell_{d}^{\prime}}(\hat{\mbox{\boldmath$p$}_{1}^{\prime}})\times
Y_{\ell_{\Lambda}^{\prime}}(\hat{\mbox{\boldmath$q$}_{1}^{\prime}})]^{L^{\prime}}\times[\chi_{d}^{1}\times\chi_{\Lambda}^{1/2}]^{S^{\prime}}]_{M_{t}}^{J_{t}}|$
$\displaystyle\times
V_{TPE}^{(K,\ell_{a},\ell_{b})}(p,q)[[Y_{\ell_{a}}(\hat{\mbox{\boldmath$p$}})\times
Y_{\ell_{b}}(\hat{\mbox{\boldmath$q$}})]^{K}\times[\mbox{\boldmath$\sigma$}_{2}\times\mbox{\boldmath$\sigma$}_{3}]^{K}]_{0}^{0}|[[Y_{\ell_{d}}(\hat{\mbox{\boldmath$p$}_{1}})\times
Y_{\ell_{\Lambda}}(\hat{\mbox{\boldmath$q$}_{1}})]^{L}\times[\chi_{d}^{1}\times\chi_{\Lambda}^{1/2}]^{S}]_{M_{t}}^{J_{t}}\rangle$
$\displaystyle=$
$\displaystyle\sum_{L^{\prime}L}\sum_{S^{\prime}S}3\hat{j_{\Lambda}}\sqrt{\hat{L^{\prime}}\hat{S^{\prime}}\hat{L}\hat{S}}\begin{Bmatrix}\ell_{d}^{\prime}&\ell_{\Lambda^{\prime}}&L^{\prime}\\\
1&1/2&S^{\prime}\\\
1&j_{\Lambda^{\prime}}&J_{t}\end{Bmatrix}\begin{Bmatrix}\ell_{d}&\ell_{\Lambda}&L\\\
1&1/2&S\\\
1&j_{\Lambda^{\prime}}&J_{t}\end{Bmatrix}\sqrt{\hat{J_{t}}\hat{L^{\prime}}\hat{S^{\prime}}}\begin{Bmatrix}J_{t}&J_{t}&0\\\
L^{\prime}&L&K\\\ S^{\prime}&S&K\end{Bmatrix}$
$\displaystyle\times\langle[Y_{\ell_{d}^{\prime}}(\hat{\mbox{\boldmath$p$}_{1}^{\prime}})\times
Y_{\ell_{\Lambda}^{\prime}}(\hat{\mbox{\boldmath$q$}_{1}^{\prime}})]^{L^{\prime}}||V_{TPE}^{(K,\ell_{a},\ell_{b})}(p,q)[Y_{\ell_{a}}(\hat{\mbox{\boldmath$p$}})\times
Y_{\ell_{b}}(\hat{\mbox{\boldmath$q$}})]^{K}||[Y_{\ell_{d}}(\hat{\mbox{\boldmath$p$}_{1}})\times
Y_{\ell_{\Lambda}}(\hat{\mbox{\boldmath$q$}_{1}})]^{L}\rangle_{pwe}$
$\displaystyle\times
18\sqrt{\hat{S}\hat{K}}(-1)^{K+3/2+S}\begin{Bmatrix}S^{\prime}&S&K\\\
1&1&1/2\end{Bmatrix}\begin{Bmatrix}1&1&K\\\ 1/2&1/2&1\\\
1/2&1/2&1\end{Bmatrix},$ (29)

where
$\mbox{\boldmath$p$}=\mbox{\boldmath$p$}_{1}^{\prime}-\mbox{\boldmath$p$}_{1}$
and
$\mbox{\boldmath$q$}=\mbox{\boldmath$q$}_{1}^{\prime}-\mbox{\boldmath$q$}_{1}$.
Then, Eq. (22) is applied in this expression. The result does not depend on
$M_{t}$. Numerical results of the case of $\ell_{d}^{\prime}=\ell_{d}=0$ are
presented in Sec. IV.

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