Contributions of $2\pi$-exchange, $1\pi$-exchange, and contact three-body forces in NNLO ChEFT to ${}_{\Lambda}^{3}$H

The hypertriton (${}_{\Lambda}^{3}$H) is an isospin $T=0$ $\Lambda$NN bound state with small separation energy to $\Lambda$ + deuteron. Because a $\Lambda$N two-body bound system is unlikely, ${}_{\Lambda}^{3}$H is valuable to investigate $\Lambda$-nucleon interaction. For example, its binding energy helps resolve the spin singlet and triplet strengths of the $\Lambda$N interaction to compensate for the lack of scattering data. Although the separation energy has not been accurately established, the situation will change shortly due to ongoing new experiments. On the theory side, the studies of the $\Lambda$NN system bear some ambiguities from the effects of possible three-body forces (3BFs). In ordinary light nuclei, an attractive contribution of three-nucleon forces is necessary to explain their binding energies accurately. Likely, we may expect a non-negligible effect of the 3BFs in ${}_{\Lambda}^{3}$H. It is needed to study whether this contribution is attractive or repulsive and the order of magnitude.

We have studied the effect of the 3BF in chiral effective field theory (ChEFT) for the hypertriton in Ref. KKM23 . In that article, the $2\pi$-exchange $\Lambda$NN interaction derived by Petschauer et al. PET16 at the next-to-next-to-leading- order (NNLO) was addressed as an initial attempt. Faddeev calculations were carried out for the first time, incorporating the 3BF matrix-elements, to show that the contribution is repulsive of the order of 20 keV. This size is small but not negligible compared with the small separation energy of the hypertriton, the present world average of which is $148\pm 40$ keV MA22 .

The remaining NNLO $\Lambda$NN 3BFs, a $1\pi$-exchange 3BF and a contact 3BF, are considered in this article. Because the expressions of these 3BFs in momentum space presented in Ref. PET16 are not readily applicable in evaluating partial-wave matrix elements, we rewrite them in a form appropriate for use. As for low-energy coupling constants (LECs), we rely on the estimation by Petschauer et al. PET17 in a decouplet-dominant model. Although there are uncertainties, it is worth to evaluate the effect of these 3BFs by using those LECs as reference values for further studies.

Expressions of the $1\pi$-exchange $\Lambda$NN 3BF and the contact $\Lambda$NN term are rewritten in Sec. 2 and Sec. 3, respectively, to be suitable for evaluating partial-wave matrix elements. Before explicit Faddeev calculations are carried out for ${}_{\Lambda}^{3}$H, it is worth obtaining a preliminary idea about the contributions of the 3BFs by evaluating a $\Lambda$-deuteron folding potential as was done for the $2\pi$-exchange 3BF in Ref. KKM23 . Those $\Lambda$-deuteron folding potentials in momentum space are discussed in Sec. 4. Results of Faddeev calculations of L3H, which include these 3BFs along with the 2pi-exchange 3BF, are presented in Sec 5. Summary follows in Sec 6. Summary follows in Sec. 6.

The $1\pi$-exchange $\Lambda$NN interaction in momentum space in the NNLO was shown by Petschauer et al. in Ref. PET16 as follows:

	$\displaystyle V_{1\pi}^{\Lambda NN}=-\frac{g_{A}}{2f_{0}^{2}}(\mbox{\boldmath$\tau$}_{2}\cdot\mbox{\boldmath$\tau$}_{3})$
	$\displaystyle\times\left\{\frac{\mbox{\boldmath$\sigma$}_{2}\cdot\mbox{\boldmath$q$}_{2^{\prime}2}}{\mbox{\boldmath$q$}_{2^{\prime}2}^{2}+m_{\pi}^{2}}(D_{1}^{\prime}\mbox{\boldmath$\sigma$}_{1}+D_{2}^{\prime}\mbox{\boldmath$\sigma$}_{3})\cdot\mbox{\boldmath$q$}_{2^{\prime}2}\right.$
	$\displaystyle\hskip 8.53581pt+\frac{\mbox{\boldmath$\sigma$}_{3}\cdot\mbox{\boldmath$q$}_{3^{\prime}3}}{\mbox{\boldmath$q$}_{3^{\prime}3}^{2}+m_{\pi}^{2}}(D_{1}^{\prime}\mbox{\boldmath$\sigma$}_{1}+D_{2}^{\prime}\mbox{\boldmath$\sigma$}_{2})\cdot\mbox{\boldmath$q$}_{3^{\prime}3}$
	$\displaystyle+P_{23}^{(\sigma)}P_{23}^{(\tau)}P_{13}^{(\sigma)}\frac{\mbox{\boldmath$\sigma$}_{2}\cdot\mbox{\boldmath$q$}_{3^{\prime}2}}{\mbox{\boldmath$q$}_{3^{\prime}2}^{2}+m_{\pi}^{2}}\left[-\frac{D_{1}^{\prime}+D_{2}^{\prime}}{2}(\mbox{\boldmath$\sigma$}_{1}+\mbox{\boldmath$\sigma$}_{3})\cdot\mbox{\boldmath$q$}_{3^{\prime}2}\right.$
	$\displaystyle\left.\hskip 56.9055pt+\frac{D_{1}^{\prime}-D_{2}^{\prime}}{2}i(\mbox{\boldmath$\sigma$}_{3}\times\mbox{\boldmath$\sigma$}_{1})\cdot\mbox{\boldmath$q$}_{3^{\prime}2}\right]$
	$\displaystyle+P_{23}^{(\sigma)}P_{23}^{(\tau)}P_{12}^{(\sigma)}\frac{\mbox{\boldmath$\sigma$}_{2}\cdot\mbox{\boldmath$q$}_{2^{\prime}3}}{\mbox{\boldmath$q$}_{2^{\prime}3}^{2}+m_{\pi}^{2}}\left[-\frac{D_{1}^{\prime}+D_{2}^{\prime}}{2}(\mbox{\boldmath$\sigma$}_{1}+\mbox{\boldmath$\sigma$}_{2})\cdot\mbox{\boldmath$q$}_{2^{\prime}3}\right.$
	$\displaystyle\left.\left.\hskip 56.9055pt-\frac{D_{1}^{\prime}-D_{2}^{\prime}}{2}i(\mbox{\boldmath$\sigma$}_{1}\times\mbox{\boldmath$\sigma$}_{2})\cdot\mbox{\boldmath$q$}_{2^{\prime}3}\right]\right\},$		(1)

in which a label of 1 assigned to the $\Lambda$ hyperon. $g_{A}$ is the axial-vector coupling constant, $f_{0}$ is the pion decay constant, $\mbox{\boldmath$q$}_{i^{\prime}j}$ is a momentum transfer at each $\pi$NN vertex, and $P_{ij}^{(\sigma)}$ ($P_{ij}^{(\tau)}$) is an exchange operator in spin (isospin) space. Explicit numbers of the low-energy constants (LECs), $D_{1}^{\prime}$ and $D_{2}^{\prime}$, are set in Se. IV. Noting $P_{ij}^{\sigma}=\frac{1}{2}(1+\mbox{\boldmath$\sigma$}_{i}\cdot\mbox{\boldmath$\sigma$}_{j})$ and $\sigma_{k}\sigma_{\ell}=\delta_{k\ell}+i\epsilon_{k\ell m}\sigma_{m}$, the above expression is rewritten as

	$\displaystyle V_{1\pi}^{\Lambda NN}=$	$\displaystyle-\frac{g_{A}}{2f_{0}^{2}}\left(1-P_{23}^{(\sigma)}P_{23}^{(\tau)}P_{23}^{(p)}\right)(\mbox{\boldmath$\tau$}_{2}\cdot\mbox{\boldmath$\tau$}_{3})$
		$\displaystyle\times\left\{\frac{\mbox{\boldmath$\sigma$}_{2}\cdot\mbox{\boldmath$q$}_{2^{\prime}2}}{\mbox{\boldmath$q$}_{2^{\prime}2}^{2}+m_{\pi}^{2}}(D_{1}^{\prime}\mbox{\boldmath$\sigma$}_{1}+D_{2}^{\prime}\mbox{\boldmath$\sigma$}_{3})\cdot\mbox{\boldmath$q$}_{2^{\prime}2}\right.$
		$\displaystyle\left.\hskip 10.00002pt+\frac{\mbox{\boldmath$\sigma$}_{3}\cdot\mbox{\boldmath$q$}_{3^{\prime}3}}{\mbox{\boldmath$q$}_{3^{\prime}3}^{2}+m_{\pi}^{2}}(D_{1}^{\prime}\mbox{\boldmath$\sigma$}_{1}+D_{2}^{\prime}\mbox{\boldmath$\sigma$}_{2})\cdot\mbox{\boldmath$q$}_{3^{\prime}3}\right\},$		(2)

where $P_{23}^{(p)}$ stands for the exchange operator for the momenta of the nucleon pair, and $P_{23}^{(\sigma)}P_{23}^{(\tau)}P_{23}^{(p)}$ means an exchange operator for the two nucleons. The factor $1-P_{23}^{(\sigma)}P_{23}^{(\tau)}P_{23}^{(p)}$ can be taken care of by the anti-symmetrization of the pair of two-nucleon wave functions. In that case, $P_{23}^{(\sigma)}P_{23}^{(\tau)}P_{23}^{(p)}$ is unnecessary, and the $1\pi$-exchange $\Lambda$NN interaction should be considered as follows.

	$\displaystyle V_{1\pi}^{\Lambda NN}=$	$\displaystyle-\frac{g_{A}}{2f_{0}^{2}}(\mbox{\boldmath$\tau$}_{2}\cdot\mbox{\boldmath$\tau$}_{3})$
		$\displaystyle\times\left\{\frac{\mbox{\boldmath$\sigma$}_{2}\cdot\mbox{\boldmath$q$}_{2^{\prime}2}}{\mbox{\boldmath$q$}_{2^{\prime}2}^{2}+m_{\pi}^{2}}(D_{1}^{\prime}\mbox{\boldmath$\sigma$}_{1}+D_{2}^{\prime}\mbox{\boldmath$\sigma$}_{3})\cdot\mbox{\boldmath$q$}_{2^{\prime}2}\right.$
		$\displaystyle\left.\hskip 10.00002pt+\frac{\mbox{\boldmath$\sigma$}_{3}\cdot\mbox{\boldmath$q$}_{3^{\prime}3}}{\mbox{\boldmath$q$}_{3^{\prime}3}^{2}+m_{\pi}^{2}}(D_{1}^{\prime}\mbox{\boldmath$\sigma$}_{1}+D_{2}^{\prime}\mbox{\boldmath$\sigma$}_{2})\cdot\mbox{\boldmath$q$}_{3^{\prime}3}\right\}.$		(3)

The first (second) term in the curly brackets of the above equation corresponds to the left (right) diagram in Fig. 1.

Now, this expression is transformed to the partial-wave expansion form applicable in evaluating matrix elements using the expression given in Ref. KKM22 .

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

where an abbreviated notation for a tensor product with Clebsch-Gordan coefficients is used:

$\displaystyle[T_{j_{1}}\times T_{j_{2}}]_{M}^{J}=\sum_{m_{1}m_{2}}(j_{1}m_{1}j_{2}m_{2}|JM)T_{j_{1}m_{1}}T_{j_{2}m_{2}}.$

(5)

$p$ and $q$ are differences of the final and initial Jacobi momenta. That is, denoting each momentum of the $i$-th initial particle by $\mbox{\boldmath$k$}_{i}$, Jacobi momenta are defined as $\mbox{\boldmath$p$}_{1}=\mbox{\boldmath$k$}_{2}-\mbox{\boldmath$k$}_{3}$ and $\mbox{\boldmath$q$}_{1}=\mbox{\boldmath$k$}_{1}$ in the center-of-mass frame. Jacobi momenta for the final configuration are represented with a prime. Then, $\mbox{\boldmath$p$}\equiv\mbox{\boldmath$p$}_{1}^{\prime}-\mbox{\boldmath$p$}_{1}$ and $\mbox{\boldmath$q$}^{\prime}\equiv\mbox{\boldmath$q$}_{1}^{\prime}-\mbox{\boldmath$q$}_{1}$. The explicit expression of $V_{1\pi,(i,j)}^{K,\ell_{a},\ell_{b}}(p,q)$ is given in Appendix A.

The contact $\Lambda$NN term in the NNLO was shown by Petschauer et al. in Ref. PET16 as follows:

	$\displaystyle V_{ct}^{\Lambda NN}=$	$\displaystyle C_{1}^{\prime}(1-\mbox{\boldmath$\sigma$}_{2}\cdot\mbox{\boldmath$\sigma$}_{3})(3+\mbox{\boldmath$\tau$}_{2}\cdot\mbox{\boldmath$\tau$}_{3})$
		$\displaystyle+C_{2}^{\prime}(\mbox{\boldmath$\sigma$}_{\Lambda}\cdot(\mbox{\boldmath$\sigma$}_{2}+\mbox{\boldmath$\sigma$}_{3}))(1-\mbox{\boldmath$\tau$}_{2}\cdot\mbox{\boldmath$\tau$}_{3})$
		$\displaystyle+C_{3}^{\prime}(3+\mbox{\boldmath$\sigma$}_{2}\cdot\mbox{\boldmath$\sigma$}_{3})(1-\mbox{\boldmath$\tau$}_{2}\cdot\mbox{\boldmath$\tau$}_{3}),$		(6)

in which a label of 1 is assigned to the $\Lambda$ hyperon. The exchange of the nucleon pair is explicitly taken care of, and the expression is antisymmetric under the exchange of two nucleons; that is, $P_{23}^{(\sigma)}P_{23}^{(\tau)}P_{23}^{(p)}V_{ct}^{\Lambda NN}=-V_{ct}^{\Lambda NN}$. Therefore, $V_{ct}^{\Lambda NN}$ is written as $\frac{1}{2}(1-P_{23}^{(\sigma)}P_{23}^{(\tau)}P_{23}^{(p)})V_{ct}^{\Lambda NN}$. It means that when the interaction is applied to the $\Lambda$NN wave function in which two nucleons are antisymmetrized, factor $\frac{1}{2}$ is necessary.

The above expression is rewritten in the following partial-wave expansion form.

	$\displaystyle V_{ct}^{\Lambda NN}=4\pi\sum_{\tau=0,1}\sum_{k,m,n}V_{ct}^{(k,m,n,\tau)}(\mbox{\boldmath$\tau$}_{2}\cdot\mbox{\boldmath$\tau$}_{3})^{\tau}$
	$\displaystyle\times[[\mbox{\boldmath$\sigma$}_{\Lambda}^{(k)}\times[\mbox{\boldmath$\sigma$}_{2}^{(m)}\times\mbox{\boldmath$\sigma$}_{3}^{(n)}]^{k}]^{0}\times[Y_{0}(\hat{\mbox{\boldmath$p$}})\times Y_{0}(\hat{\mbox{\boldmath$q$}})]^{0}]^{0}.$		(7)

The coefficients $V_{ct}^{(k,m,n,\tau)}$ are given in Table I.

It is instructive to evaluate the $\Lambda$-deuteron folding potential provided by the $1\pi$-exchange and the contact 3BFs to demonstrate the contribution of these 3BFs in the hypertriton. The folding potential is calculated 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$}_{1}^{\prime}),(\ell_{\Lambda}^{\prime}1/2)j_{\Lambda}]J_{t}|$
		$\displaystyle\times V_{1\pi(ct)}^{\Lambda NN}|[\Psi_{d}(\mbox{\boldmath$p$}_{1}),(\ell_{\Lambda}1/2)j_{\Lambda}]J_{t}\rangle,$		(8)
	$\displaystyle\Psi_{d}(\mbox{\boldmath$p$}_{1})=$	$\displaystyle\sum_{\ell_{d}=0,2}\frac{1}{p_{1}}\phi_{\ell_{d}}(p_{1})[Y_{\ell_{d}}(\hat{\mbox{\boldmath$p$}_{1}})\times\chi_{d}^{1}]_{m}^{1},$		(9)

where $\Psi_{d}(\mbox{\boldmath$p$})$ represents a deuteron wave function. A detailed calculational procedure for the 3BF in the form of Eq. 3 or 7 is given in Appendix B in Ref. KKM22 .

Figure 2 shows the result of the $1\pi$-exchange $\Lambda$NN 3BF given in Eq. (3) with the total angular momentum $J_{t}=1/2$. The LECs are taken from the estimation by Petschauer et al. PET17 : $D_{1}^{\prime}=0$ and $D_{2}^{\prime}=\frac{2CH^{\prime}}{9\Delta}$ with $C\approx\frac{3}{4}g_{A}$, $H^{\prime}\approx 1/f_{0}^{2}$, and the decuplet-octet baryon mass splitting $\Delta$. The numerical value of $D_{2}^{\prime}$ is set as $D_{2}^{\prime}=3.268\times 10^{-3}$ fm²MeV^-1 using $g_{A}=1.29$, $f_{0}=92.4$ MeV, and $\Delta=300$ MeV. As discussed in Ref. PET17 , the sign of $D_{2}^{\prime}$ could be the opposite. The upper panel of Fig. 2 depicts the contribution of the $s$-wave pair of the bra and ket deuteron wave functions. The repulsive magnitude is smaller than the corresponding strength of the $2\pi$-exchange $\Lambda$NN reported in Ref. KKM23 . The result of the $s$-$d$ pair of the bra and ket deuteron wave functions is shown in the lower panel of Fig. 2, which is attractive, and the magnitude is unexpectedly large despite the small $d$-wave component of the deuteron wave function. The $d$-$s$ pair provided the same contribution. The contribution of the $d$-wave pair of the bra and ket deuteron wave functions is negligible. The net $\Lambda$-deuteron folding potential, including the contribution of the $2\pi$-exchange reported in Ref. KKM23 , is found to be attractive and to have a value of about $-0.35$ MeV at the origin.

Figure 3 shows the result of the contact term of the $\Lambda$NN 3BF given in Eq. (7). The low-energy constants are set as $C_{1}^{\prime}=C_{3}^{\prime}=\frac{1}{f_{0}^{4}\Delta}$ and $C_{2}^{\prime}=0$ with $f_{0}=92.4$ MeV and $\Delta=300$ MeV, following the estimation by Petschauer et al. PET17 . Other contributions from the $s$-$d$, $d$-$s$, and $d$-$d$ pairs of the bra and ket deuteron wave functions are negligibly small.

Therefore, the sum of the contributions of the $1\pi$-exchange and contact $\Lambda$NN 3BFs is attractive. Even if the repulsive contribution of the $2\pi$-exchange reported in Ref. KKM23 is included, the net contribution of the three types of the NLO $\Lambda$NN 3BFs with the LECs estimated by Petschauer et al. PET17 is expected to be attractive at around $-300$ keV.

In this section, we present the results of the explicit Faddeev calculations of the separation energy of ${}_{\Lambda}^{3}$H that include all NNLO $\Lambda$NN interactions in chiral effective field theory. The method of calculation is explained in Ref. KKM23 . Although the LECs employed are speculative, the calculated results serve as a reference number for possible 3BF contributions in ${}_{\Lambda}^{3}$H.

Uncertainties in the prediction of the ${}_{\Lambda}^{3}$H separation energy due to the NN interaction employed were discussed in Ref. KKM23 , which is qualitatively similar to the finding by Gazda et al. GHF22 . In this article, we show only the results using N⁴LO⁺(550) and N⁴LO⁺(400) for the NN interaction, where the number in parentheses is a cutoff scale in MeV. As for the YN interactions, two versions of the chiral NLO interactions, NLO13 NLO13 and NLO19 NLO19 with a cutoff scale of 550 MeV parametrized by the Jülich-Bonn group, are employed. The Nijmegen NSC89 potential NSC89 is also applied, though it may not be appropriate to use with the regularized chiral interactions at around 500 MeV.

Calculated results for each YN interaction are depicted in Fig. 4. The leftmost entry is the separation energy without 3BFs. The second entry from the left is the result including the $2\pi$-exchange $\Lambda$NN, which was reported already in Ref. KKM23 . The second entry from the right is the result with the $1\pi$-exchange 3BF and the $2\pi$-exchange $\Lambda$NN. The $1\pi$-exchange 3BF acts attractively due to the matrix element of the $s$-$d$ pair of the deuteron wave function, as shown by the folding potential in Sec. 4. The rightmost entry shows the result in which all the 3BFs are included. The net effect after including NNLO 3BFs turns out to be sensitive to the NN interaction. In the case of N⁴LO⁺(550) in which the separation is narrower, the 3BFs bring about the attraction in the order of 20 keV for ${}_{\Lambda}^{3}$H. On the other hand, the 3BFs tend to work repulsively in the case of N⁴LO⁺(400). It seems that the 3BFs work attractively in the case that the ${}_{\Lambda}^{3}$H wave function is wide-spreading. With the wave function shrinking, the net attraction begins to diminish. The trend of the 3BFs contribution is opposite in the NSC89 YN potential, for which the interpretation is difficult because the NN interactions are defined in a low momentum with the cutoff scale of around 500 MeV and the NSC potential in an entire space.

In Ref. KKM23 , we investigated the effect of the $2\pi$-exchange $\Lambda$NN 3BF in ChEFT in the ${}_{\Lambda}^{3}$H hypernuclei by carrying out Faddeev calculations that include the 3BF for the first time. In the present article, the remaining $1\pi$-exchange and contact term $\Lambda$NN 3BFs derived by Petschauer et al. PET17 at the NNLO level are addressed. Because the expressions of these 3BFs in Ref. PET16 are not readily applicable in calculating matrix elements of $\Lambda$ hypernuclei, we rewrite them to be applied in the formula given in Ref. KKM22 . Then, Faddeev calculations for ${}_{\Lambda}^{3}$H are carried out, including NNLO 3BFs.

The net effect of these 3BFs is found to be small because of the cancellation between the attractive contribution of the $1\pi$-exchange 3BF and the repulsive ones of the other 3BFs under the tentative assignment for the sign of the coupling constant $D_{2}^{\prime}$. It is interesting to see that the difference in the separation energies with N⁴LO⁺(550) and N⁴LO⁺(400) becomes smaller when the 3BFs are included for the chiral YN interactions. The attraction from the $1\pi$-exchange 3BF is responsible for the matrix element between the $s$- and $d$-wave components of the NN wave function due to the tensor part of the $1\pi$-exchange 3BF. The coupling effect is sizable despite the small $d$-wave component, as is demonstrated by the folding potential discussed in Sec. 3. Naturally, the quantitative results serve only as a reference because of the speculative nature of the LECs employed. However, the result provides basic information about the contribution of the 3BFs because it enables us to infer what changes are induced by modifying each coupling constant. It is necessary to do similar ab-initio calculations in heavier $\Lambda$-hypernulcei, although the task is computationally demanding.

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