1. Introduction
The Λc baryon as the lightest charm baryon has attracted a lot of attention, which may exist in finite nuclei to form Λc hypernuclei. The HAL QCD Collaboration performed the first lattice QCD simulations of the ΛcN and ΣcN interactions for unphysical light quark masses (mπ = 410, 570, 700 MeV) [1], which provided vital information on the interaction between a nucleon and a charmed baryon Λc or Σc. Employing these lattice QCD results, extrapolations to the physical point have been performed using either the non-relativistic chiral effective field theory (ChEFT) at next-to-leading order (NLO) [2] or the covariant ChEFT at leading order (LO) [3]. In the covariant ChEFT, Lorentz covariance is maintained by employing the covariant chiral Lagrangians, the full form of Dirac spinors, and the relativistic scattering equation (the Kadyshevsky equation). It has been shown that the covariant ChEFT approach can provide reasonable descriptions of octet baryon-octet baryon interactions already at LO, including all the systems from strangeness S = 0 to S = −4, at least in the low energy region [4-14]. 8(8 The next-to-next-to-leading order relativistic chiral nucleon-nucleon interaction is shown to be able to describe the neutron-proton scattering phaseshifts up to Tlab. = 200 MeV as well as the next-to-next-to-next-to-leading order non-relativistic chiral nucleon-nucleon interactions [15].) A recent study [11] showed that one could reproduce both the physical 1S0 and 3S1-3D1 and the lattice QCD nucleon-nucleon partial wave phase shifts fairly well. In particular, for the physical nucleon-nucleon phase shifts and lattice QCD data at mπ = 469 MeV, if one only fits to the 3S1 phase shifts, the 3D1 phase shifts and inelasticity η1 can be predicted and vice versa, as shown in [16]. It implies that indeed the correlations induced by the imposed constraint of covariance in the covariant chiral potentials is reasonable.
In our previous study of the ΛcN interaction in the covariant ChEFT [3], the low energy constants (LECs) were determined by fitting to the lattice QCD data from the HAL QCD Collaboration, where the S-wave phase shifts up to Ec.m. = 30 MeV for mπ = 410 and 570 MeV were considered. The results showed that the covariant ChEFT can describe the lattice QCD data fairly well at low energies. In addition, the phase shifts of the ΛcN 3D1 partial wave and the inelasticity η1, as well as their physical counterparts were predicted.
In a recent study [17], it was shown that the predicted 3D1 phase shifts by the NLO non-relativistic ChEFT are in agreement with the lattice QCD data of [18] at higher energies, but not those of [3]. A closer examination of the lattice QCD data revealed, however, that although at higher energies, the predictions of [3] do not agree with the lattice QCD data, but at low energies close to the threshold, they do agree, both for the 3D1 phase shifts and the inelasticity, at least qualitatively. On the other hand, the predictions of the NLO non-relativistic ChEFT [2] do not agree with the lattice QCD data at low energies.
In this work, we revisit the fits to the lattice QCD data and the corresponding extrapolations to the physical point. We study in detail the differences between the non-relativistic ChEFT and covariant ChEFT in the description of the ΛcN 3D1 phase shifts and inelasticity η1, including the effects of baryon masses and SD coupling in the contact terms, and the retardation effects in the one meson exchange term. 9(9 We found that the retardation effects are quite small and therefore refrain from explicit discussions about these effects from now on, but they are always included in our study.) In addition, we study extrapolations to the physical point employing different fitting strategies to the lattice QCD data. These results are important to better understand the ΛcN interaction and might be helpful to guide future hypernuclei experiments.
The paper is organized as follows. In section 2 , we briefly introduce the non-relativistic and the covariant chiral EFT. In section 3 we perform fits to the lattice QCD data of [18], focusing on the low energy region, where ChEFT is expected to work. We summarize in section 4 .
2. Theoretical framework
In this section, we briefly introduce the non-relativistic ChEFT and covariant ChEFT for the YcN interactions, where Yc = Λc, Σc, and highlight the differences relevant for the present study.
In the non-relativistic ChEFT, the next-to-leading order potentials consist of non-derivative four-baryon contact terms (CT) and one-meson-exchanges (OME). The CT potentials for the 1S0 and 3S1-3D1 partial waves are [19]
$\begin{eqnarray}\begin{array}{rcl}{V}_{\mathrm{CT},1{\rm{S}}0}^{{Y}_{c}N} & = & {\tilde{C}}_{{}^{1}{S}_{0}}+{\tilde{D}}_{{}^{1}{S}_{0}}{m}_{\pi }^{2}\\ & & +({C}_{{}^{1}{S}_{0}}+{D}_{{}^{1}{S}_{0}}{m}_{\pi }^{2})({p}^{2}+p{{\prime} }^{2}),\\ {V}_{\mathrm{CT},3{\rm{S}}1}^{{Y}_{c}N} & = & {\tilde{C}}_{{}^{3}{S}_{1}}+{\tilde{D}}_{{}^{3}{S}_{1}}{m}_{\pi }^{2}\\ & & +({C}_{{}^{3}{S}_{1}}+{D}_{{}^{3}{S}_{1}}{m}_{\pi }^{2})({p}^{2}+p{{\prime} }^{2}),\\ {V}_{\mathrm{CT},3{\rm{D}}1-3{\rm{S}}1}^{{Y}_{c}N} & = & {C}_{{\varepsilon }_{1}}p{{\prime} }^{2},\\ {V}_{\mathrm{CT},3{\rm{S}}1-3{\rm{D}}1}^{{Y}_{c}N} & = & {C}_{{\varepsilon }_{1}}{p}^{2},\end{array}\end{eqnarray}$
where p = ∣p∣ and $p^{\prime} =| {\boldsymbol{p}}^{\prime} | $ are the initial and final center-of-mass (c.m.) momenta of the YcN system, respectively. ${\tilde{C}}_{i}$, ${\tilde{D}}_{i}$, Ci, Di(i = 1S0, 3S1, and ϵ1) are LECs that need to be fixed by fitting to either experimental or lattice QCD data. The OME potential reads, $\begin{eqnarray}\begin{array}{rcl}{V}_{\mathrm{OME}}^{{Y}_{c}N\to {Y}_{c}^{\prime} N} & = & -\displaystyle \frac{{g}_{A}^{{Y}_{c}{Y}_{c}^{\prime} }{g}_{A}^{{NN}}}{4{f}_{\pi }^{2}}\displaystyle \frac{\left({\sigma }_{1}\cdot {\boldsymbol{q}}\right)\left({\sigma }_{2}\cdot {\boldsymbol{q}}\right)}{{{\boldsymbol{q}}}^{2}+{m}_{\pi }^{2}}\\ & & \times {{ \mathcal I }}_{{Y}_{c}N\to {Y}_{c}^{\prime} N},\end{array}\end{eqnarray}$
where ${\boldsymbol{q}}={{\boldsymbol{p}}}^{{\prime} }-{\boldsymbol{p}}$ is the transferred momentum. The coupling constants ${g}_{A}^{{Y}_{c}{Y}_{c}^{\prime} }$ and gNNA and the isospin factor ${ \mathcal I }$ can be found in, e.g. [2, 20]. The scattering amplitudes are then obtained by solving the coupled-channel Lippmann-Schwinger equation $\begin{eqnarray}\begin{array}{l}{T}_{\rho ^{\prime} \rho }^{\nu ^{\prime} \nu ,J}(p^{\prime} ,p;\sqrt{s})={V}_{\rho ^{\prime} \rho }^{\nu ^{\prime} \nu ,J}(p^{\prime} ,p)\\ \quad +\sum _{\rho ^{\prime\prime} ,\nu ^{\prime\prime} }{\displaystyle \int }_{0}^{\infty }\displaystyle \frac{{\rm{d}}{p}^{\prime\prime} {p}^{{\prime\prime} 2}}{{\left(2\pi \right)}^{3}}{V}_{\rho ^{\prime} \rho ^{\prime\prime} }^{\nu ^{\prime} \nu ^{\prime\prime} ,J}(p^{\prime} ,p^{\prime\prime} )\\ \quad \times \displaystyle \frac{2{\mu }_{\rho ^{\prime\prime} }}{{p}_{\rho }^{2}-{p}^{{\prime\prime} 2}+i\eta }{T}_{\rho ^{\prime\prime} \rho }^{\nu ^{\prime\prime} \nu ,J}(p^{\prime\prime} ,p;\sqrt{s}),\end{array}\end{eqnarray}$
where the labels $\nu ,\nu ^{\prime} ,\nu ^{\prime\prime} $ denote the particle channels, $\rho ,\rho ^{\prime} ,\rho ^{\prime\prime} $ denote the partial waves, and μρ is the pertinent reduced mass. The on-shell momentum in the intermediate state, pρ, is defined by $\sqrt{s}=\sqrt{{M}_{{B}_{1,\rho }}^{2}+{p}_{\rho }^{2}}+\sqrt{{M}_{{B}_{2,\rho }}^{2}+{p}_{\rho }^{2}}$. The potentials are regularized with an exponential form factor $\begin{eqnarray}{f}_{{{\rm{\Lambda }}}_{F}}(p,p^{\prime} )=\exp \left[-{\left(\displaystyle \frac{p}{{{\rm{\Lambda }}}_{F}}\right)}^{4}-{\left(\displaystyle \frac{p^{\prime} }{{{\rm{\Lambda }}}_{F}}\right)}^{4}\right],\end{eqnarray}$
where ΛF is the cutoff whose value is in the range of 500-600 MeV. The partial wave S matrix is related to the on-shell T matrix by $\begin{eqnarray}{S}_{\rho ^{\prime} \rho }^{\nu ^{\prime} \nu }={\delta }_{\rho ^{\prime} \rho }{\delta }^{\nu ^{\prime} \nu }-2{{\rm{i}}{aT}}_{\rho ^{\prime} \rho }^{\nu ^{\prime} \nu },\quad a=\displaystyle \frac{\sqrt{{{p}}_{\mathrm{cm}}^{\nu ^{\prime} }{{p}}_{\mathrm{cm}}^{\nu }}{\mu }^{\nu ^{\prime} \nu }}{16{\pi }^{2}},\end{eqnarray}$
where pcm is the C.M. three-momentum of the ΛcN system. The phase space factor a is determined by the elastic unitarity of the scattering equation. For single channels, the phase shifts δ can be obtained from the on-shell S matrix $\begin{eqnarray}S=\exp (2{\rm{i}}\delta ).\end{eqnarray}$
In order to calculate the phase shifts in coupled channels (J > 0), we use the ‘Stapp'- or ‘bar'- phase shifts parametrisation [21] of the S matrix, which can be written as
$\begin{eqnarray}\begin{array}{rcl}S & = & \left(\begin{array}{cc}{S}_{--} & {S}_{-+}\\ {S}_{+-} & {S}_{++}\end{array}\right)\,=\,\left(\begin{array}{cc}\exp ({\rm{i}}{\delta }_{-}) & 0\\ 0 & \exp ({\rm{i}}{\delta }_{+})\end{array}\right)\\ & & \times \left(\begin{array}{cc}\cos (2\epsilon ) & {\rm{i}}\sin (2\epsilon )\\ {\rm{i}}\sin (2\epsilon ) & \cos (2\epsilon )\end{array}\right)\left(\begin{array}{cc}\exp ({\rm{i}}{\delta }_{-}) & 0\\ 0 & \exp ({\rm{i}}{\delta }_{+})\end{array}\right),\end{array}\end{eqnarray}$
where the subscript ‘+' is J + 1, ‘−' for J − 1. The resulting phase shifts and mixing angles are $\begin{eqnarray}\begin{array}{l}\tan (2{\delta }_{\pm })=\displaystyle \frac{\mathrm{Im}({S}_{\pm \pm }/\cos (2{\epsilon }_{J}))}{\mathrm{Re}({S}_{\pm \pm }/\cos (2{\epsilon }_{J}))},\\ \quad \tan (2{\epsilon }_{J})=\displaystyle \frac{-{{\rm{i}}{S}}_{+-}}{\sqrt{{S}_{++}{S}_{--}}}.\end{array}\end{eqnarray}$
In the covariant ChEFT, as discussed in [3], the 1S0 and 3S1-3D1 CT potentials for the YcN system read,
$\begin{eqnarray*}\begin{array}{l}{V}_{\mathrm{CT},1{\rm{S}}0}^{{Y}_{c}N}={\xi }_{{Y}_{c}N}\left[{C}_{1S0}\left({R}_{p^{\prime} }^{N}{R}_{p^{\prime} }^{{Y}_{c}}+{R}_{p}^{N}{R}_{p}^{{Y}_{c}}\right)\right.\\ \quad \left.+\,C^{{\prime} }_{1S0}\left({R}_{p^{\prime} }^{N}{R}_{p}^{N}{R}_{p^{\prime} }^{{Y}_{c}}{R}_{p}^{{Y}_{c}}+1\right)\right],\\ {V}_{\mathrm{CT},3{\rm{S}}1}^{{Y}_{c}N}=\displaystyle \frac{1}{9}{\xi }_{{Y}_{c}N}\left\{2\left({C}_{1S0}-C^{{\prime} }_{1S0}\right)\right.\\ \quad \times \,\left({R}_{p^{\prime} }^{{Y}_{c}}{R}_{p}^{{Y}_{c}}-{R}_{p^{\prime} }^{N}{R}_{p}^{N}\right)\\ \quad +\,{C}_{3S1}\left(-6{R}_{p^{\prime} }^{N}{R}_{p}^{N}+9{R}_{p^{\prime} }^{N}{R}_{p^{\prime} }^{{Y}_{c}}\right.\\ \quad \left.+\,9{R}_{p}^{N}{R}_{p}^{{Y}_{c}}+6{R}_{p^{\prime} }^{{Y}_{c}}{R}_{p}^{{Y}_{c}}\right)\\ \quad \left.+\,9C^{{\prime} }_{3S1}\left[{R}_{p^{\prime} }^{{Y}_{c}}{R}_{p}^{{Y}_{c}}\left({R}_{p^{\prime} }^{N}{R}_{p}^{N}-2\right)+2{R}_{p^{\prime} }^{N}{R}_{p}^{N}\,9\right]\right\},\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}{V}_{\mathrm{CT},3{\rm{D}}1-3{\rm{S}}1}^{{Y}_{c}N}=\displaystyle \frac{{\xi }_{{Y}_{c}N}}{9\sqrt{2}}\left\{\left({C}_{1S0}-C{{\prime} }_{1S0}\right)\right.\\ \quad \left[{R}_{p}^{N}\left({R}_{p^{\prime} }^{N}+3{R}_{p^{\prime} }^{{Y}_{c}}\right)-{R}_{p}^{{Y}_{c}}\left(3{R}_{p^{\prime} }^{N}+{R}_{p^{\prime} }^{{Y}_{c}}\right)\right]\\ \quad +\,{C}_{3S1}\left[9{R}_{p}^{{Y}_{c}}\left({R}_{p^{\prime} }^{N}+4{R}_{p}^{N}\right)\right.\\ \quad \left.+\,3{R}_{p^{\prime} }^{N}{R}_{p}^{N}-3{R}_{p^{\prime} }^{{Y}_{c}}\left(3{R}_{p}^{N}+{R}_{p}^{{Y}_{c}}\right)\right]\\ \quad +\,9C{{\prime} }_{3S1}\left\{{R}_{p^{\prime} }^{{Y}_{c}}\left[{R}_{p}^{N}\left(4{R}_{p^{\prime} }^{N}{R}_{p}^{{Y}_{c}}+3\right)+{R}_{p}^{{Y}_{c}}\right]\right.\\ \quad \left.\left.-\,{R}_{p^{\prime} }^{N}\left({R}_{p}^{N}+3{R}_{p}^{{Y}_{c}}\right)\right\}\right\},\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}{V}_{\mathrm{CT},3{\rm{S}}1-3{\rm{D}}1}^{{Y}_{c}N}=\displaystyle \frac{{\xi }_{{Y}_{c}N}}{9\sqrt{2}}\left\{\left({C}_{1S0}-C{{\prime} }_{1S0}\right)\left[{R}_{p^{\prime} }^{N}\left({R}_{p}^{N}+3{R}_{p}^{{Y}_{c}}\right)\right.\right.\\ \quad \left.-\,{R}_{p^{\prime} }^{{Y}_{c}}\left(3{R}_{p}^{N}+{R}_{p}^{{Y}_{c}}\right)\right]\\ \quad +\,{C}_{3S1}\left[9{R}_{p^{\prime} }^{{Y}_{c}}\left({R}_{p}^{N}+4{R}_{p^{\prime} }^{N}\right)+3{R}_{p}^{N}{R}_{p^{\prime} }^{N}\right.\\ \quad \left.-\,3{R}_{p}^{{Y}_{c}}\left(3{R}_{p^{\prime} }^{N}+{R}_{p^{\prime} }^{{Y}_{c}}\right)\right]\\ \quad +\,9C{{\prime} }_{3S1}\left\{{R}_{p}^{{Y}_{c}}\left[{R}_{p^{\prime} }^{N}\left(4{R}_{p}^{N}{R}_{p^{\prime} }^{{Y}_{c}}+3\right)+{R}_{p^{\prime} }^{{Y}_{c}}\right]\right.\\ \quad \left.\left.-\,{R}_{p}^{N}\left({R}_{p^{\prime} }^{N}+3{R}_{p^{\prime} }^{{Y}_{c}}\right)\right\}\right\},\end{array}\end{eqnarray*}$
$\begin{eqnarray}\begin{array}{l}{V}_{\mathrm{CT},3{\rm{D}}1}^{{Y}_{c}N}=\displaystyle \frac{2}{9}{\xi }_{{Y}_{c}N}\left\{\left({C}_{1S0}-C{{\prime} }_{1S0}+3{C}_{3S1}\right)\right.\\ \quad \times \,\left({R}_{p^{\prime} }^{N}{R}_{p}^{N}-{R}_{p^{\prime} }^{{Y}_{c}}{R}_{p}^{{Y}_{c}}\right)\\ \quad \left.+\,9C{{\prime} }_{3S1}\left[{R}_{p^{\prime} }^{N}{R}_{p}^{N}\left(4{R}_{p^{\prime} }^{{Y}_{c}}{R}_{p}^{{Y}_{c}}-1\right)+{R}_{p^{\prime} }^{{Y}_{c}}{R}_{p}^{{Y}_{c}}\right]\right\},\end{array}\end{eqnarray}$
where $\begin{eqnarray*}\begin{array}{l}{\xi }_{{Y}_{c}N}=4\pi \displaystyle \frac{\sqrt{\left({E}_{p^{\prime} }^{{Y}_{c}}+{M}_{{Y}_{c}}\right)\left({E}_{p}^{{Y}_{c}}+{M}_{{Y}_{c}}\right)\left({E}_{p^{\prime} }^{N}+{M}_{N}\right)\left({E}_{p}^{N}+{M}_{N}\right)}}{4{M}_{N}{M}_{{Y}_{c}}}\,\mathrm{and}\\ \quad {R}_{p(p^{\prime} )}^{{Y}_{c},N}=\displaystyle \frac{p(p^{\prime} )}{{E}_{p(p^{\prime} )}^{{Y}_{c},N}+{M}_{{Y}_{c},N}}.\end{array}\end{eqnarray*}$
It should be noted that there exist three differences between the covariant and the non-relativistic ChEFT potentials as presented above: (1) the covariant chiral potentials explicitly contain the baryon masses ${M}_{{Y}_{c}}$ and MN, where ${M}_{{Y}_{c}}=({M}_{{{\rm{\Lambda }}}_{c}}+{M}_{{{\rm{\Sigma }}}_{c}})/2$, whose values are the same as those given in table 1 of [3]; (2) because of the fact ${M}_{{Y}_{c}}\ne {M}_{N}$, the LECs from the 1S0 partial wave also contribute to those of the 3S1-3D1 partial waves; (3) the LECs responsible for the SD coupling are correlated with those of the 1S0 and 3S1 potentials. It should be noted that the contributions from the ΣcN intermediate state in the CT potentials were set zero in both the non relativistic ChEFT [2] and covariant ChEFT [3], since the limited lattice QCD data could not fix these contributions.Table 1. Seven fitting strategies studied in this work, where ✓ indicates that the SD coupling in the CT potential is turned on, while × denotes that the SD coupling is turned off in the covariant ChEFT approach. |
| Strategy | Lattice QCD data fitted | Approach | ${M}_{{Y}_{c}}[\,\,\,\,]{M}_{N}$ | SD coupling |
|---|---|---|---|---|
| 1 | 3S1 | cov. ChEFT | ≠ | ✓ |
| 2 | ≠ | × | ||
| 3 | Ec.m. ≤ 30 MeV | = | ✓ | |
| 4 | = | × | ||
| 5 | 3S1, 3D1, η1 | cov. ChEFT | ≠ | ✓ |
| 6 | Ec.m. ≤ 5 MeV | = | ✓ | |
| 7 | non-rel. ChEFT |
The leading-order OME potential reads4 ). The relation between the phase shifts and T-matrix is the same as explained above except for the phase space factor a, which appears in the Kadyshevsky equation as $a=\frac{1}{8 \pi^{2}} \frac{\sqrt{p_{\mathrm{cm}}^{\nu} \ p_{\mathrm{cm}}^{\nu^{\prime}} \ M_{B_{1}, \nu^{\prime}} \ M_{B_{2}, \nu^{\prime}} \ M_{B_{1}, \nu} \ M_{B_{2}, \nu}}}{\left(E_{1, \nu^{\prime}} \ +E_{2, \nu^{\prime}}\right)\left(E_{1, \nu} \ +E_{2, \nu}\right)}$. More details about the covariant ChEFT approach can be found in [4-15].
$\begin{eqnarray}\begin{array}{l}{V}_{\mathrm{OME}}^{{Y}_{c}N\to {Y}_{c}^{\prime} N}=-{{\rm{i}}{g}}_{A}^{{Y}_{c}{Y}_{c}^{\prime} }{g}_{A}^{{NN}}{\bar{u}}_{Y{{\prime} }_{c}}\left(p^{\prime} \right)\\ \quad \times \left(\displaystyle \frac{{\gamma }^{\mu }{\gamma }_{5}{q}_{\mu }}{2{f}_{\pi }}\right){u}_{{Y}_{c}}(p)\displaystyle \frac{{\rm{i}}}{{\rm{\Delta }}{E}^{2}-{q}^{2}-{m}^{2}+{\rm{i}}\epsilon }\\ \quad \times {\bar{u}}_{N}\left(-p^{\prime} \right)\left(\displaystyle \frac{{\gamma }^{\nu }{\gamma }_{5}{{\boldsymbol{q}}}_{\nu }}{2{f}_{\pi }}\right){u}_{N}(-p)\times {{ \mathcal I }}_{{Y}_{c}N\to {Y}_{c}^{\prime} N},\end{array}\end{eqnarray}$
where ${\rm{\Delta }}E={E}_{p^{\prime} }-{E}_{p}$ is the transferred kinetic energy, i.e. the retardation effect, and we adopt the complete form of the Dirac spinor for the baryons involved $\begin{eqnarray*}{u}_{B}({\boldsymbol{p}},s)=\left(\begin{array}{c}1\\ \displaystyle \frac{{\boldsymbol{\sigma }}\cdot {\boldsymbol{p}}}{{E}_{p}+{M}_{B}}\end{array}\right){\chi }_{s}.\end{eqnarray*}$
The coupled-channel Kadyshevsky equation [32] is solved to obtain the scattering amplitudes $\begin{eqnarray}\begin{array}{l}{T}_{\rho \rho ^{\prime} }^{\nu \nu ^{\prime} ,J}(p^{\prime} ,p;\sqrt{s})={V}_{\rho \rho ^{\prime} }^{\nu \nu ^{\prime} ,J}(p^{\prime} ,p)\\ \quad +\,\sum _{\rho ^{\prime\prime} ,\nu ^{\prime\prime} }{\displaystyle \int }_{0}^{\infty }\displaystyle \frac{{\rm{d}}{p}^{\prime\prime} {p}^{{\prime\prime} 2}}{{\left(2\pi \right)}^{3}}\\ \quad \times \,\displaystyle \frac{{M}_{{B}_{1,\nu ^{\prime\prime} }}{M}_{{B}_{2,\nu ^{\prime\prime} }}\,{V}_{\rho \rho ^{\prime\prime} }^{\nu \nu ^{\prime\prime} ,J}(p^{\prime} ,p^{\prime\prime} )\,{T}_{\rho ^{\prime\prime} \rho ^{\prime} }^{\nu ^{\prime\prime} \nu ^{\prime} ,J}(p^{\prime\prime} ,p;\sqrt{s})}{{E}_{1,\nu ^{\prime\prime} }{E}_{2,\nu ^{\prime\prime} }\left(\sqrt{s}-{E}_{1,\nu ^{\prime\prime} }-{E}_{2,\nu ^{\prime\prime} }+{\rm{i}}\epsilon \right)},\end{array}\end{eqnarray}$
where $\sqrt{s}$ is the total energy of the two-baryon system in the center-of-mass frame and ${E}_{n,\nu ^{\prime\prime} }=\sqrt{{p}^{{\prime\prime} 2}+{M}_{{B}_{n,\nu ^{\prime\prime} }}^{2}}$, (n = 1, 2). In the numerical study, the potentials are regularized with the same exponential form factor as that of equation (3. Fitting procedure
In [33], the HAL QCD Collaboration presented the 1S0 and 3S1 phase shifts of the ΛcN interaction obtained from lattice QCD simulations with mπ = 410, 570, and 700 MeV. In addition, the corresponding 3D1 partial wave phase shifts and inelasticity η1 can be found in the PhD thesis of Takaya Miyamoto [18]. The results show that the ΛcN 3D1 phase shifts for Mπ = 410, 570, and 700 MeV are slightly repulsive for the center of mass energy no larger than 15, 30, and 40 MeV, respectively and become attractive as Ec.m. increases, and the inelasticity η1 is close to unity in the whole energy region. Fittings to the S partial waves of mπ = 410 and 570 MeV (with Ec.m. ≤ 30 MeV), and extrapolations to the physical point were performed in both the non-relativistic ChEFT [2] and covariant ChEFT [3]. The predictions for the 3D1 phase shifts and inelasticity η1 turn out to be dramatically different. The 3D1 interaction in the former approach is attractive, while that in the latter is repulsive. In addition, both approaches predict a SD coupling stronger than that shown by the lattice QCD data.
In this study, we first investigate where such differences in the predicted ΛcN 3D1 phase shifts between the two approaches originate. In particular, we focus on the masses of Yc and N and the SD coupling in the CT potential. We note that there are no baryon mass terms in the CT potential of the non-relativistic ChEFT, while MB (${M}_{{Y}_{c}}$, MN) appears in the baryon spinors of the covariant ChEFT. As ${M}_{{Y}_{c}}\geqslant {M}_{N}$, we used the ‘physical' masses for Yc and N in our previous study, which has the consequence that C1S0 ($C{{\prime} }_{1S0}$) also contributes to the 3S1-3D1 partial waves [3]. In addition, the SD coupling in the covariant ChEFT is correlated to the 3S1 potential, while a free LEC appears in the non-relativistic ChEFT. These two differences lead to in total 22 = 4 combinations that will be examined. In addition to our previous study [3], we perform three more fits to the same lattice QCD data, and make a systematic comparison of the results, to better understand how the results depend on the baryon masses and SD coupling in the CT potential.
Moreover, since both approaches fail to precisely reproduce the lattice QCD 3D1 phase shifts of ΛcN at low energies, we adopt a new fitting strategy where the phase shifts of ΛcN 3S1, 3D1 partial waves and inelasticity η1 with Ec.m. ≤ 5 MeV are simultaneously fitted. The new strategy can provide a closer look at the two approaches in the descriptions of low energy lattice QCD data. Note that we only consider the lattice QCD data with mπ = 410 and 570 MeV in all the aforementioned fittings. Details of the fitting strategies in this work are shown in table 1.
4. Results and discussions
4.1. Origin of the difference in predicting the ΛcN 3D1 phase shifts
The fitted results of strategies 1-4 as described in the previous section are summarized qualitatively in table 2 and quantitatively in figure 1. It is noted that the treatment of the potentials in strategy 1 is that adopted in [3] 10(10 The χ2 shown in table 2 is larger than that in [3] because of different fitting strategies. The χ2 in [3] is obtained by fitting to the 3S1 phase shits, while the χ2 in table 2 includes the 3D1 and mixing angle data as well.), and strategy 4 is approximately the same as that of the non-relativistic ChEFT. The following conclusions can be obtained from the table: first, the baryon masses affect the 3D1 phase shifts for the large pion mass (mπ = 570 MeV), where negative phase shifts are obtained in strategies 1, 2 and they become positive if ${M}_{{Y}_{c}}$ is taken to be the same as MN (strategies 3, 4). Second, only when ${M}_{{Y}_{c}}={M}_{N}$ and the SD coupling in the CT potential is turned off, the 3D1 interaction becomes attractive in the unphysical region (strategy 4). Third, the SD coupling in the covariant ChEFT reduces the attraction in the 3S1 partial wave in the physical region, compared with the non-relativistic case, as shown in strategies 1 and 3.
Figure 1. ΛcN 3S1, 3D1 phase shifts and inelasticity η1 for different pion masses. The results are obtained by fitting to the lattice QCD ΛcN S—wave phase shifts for Ec.m. ≤ 30 MeV. The bands are generated from the variation of ΛF from 600 to 700 MeV. Different labels denote the ΛcN phase shifts of strategies 1-4: ‘w/.' is the abbreviation for ‘with', and ‘w/o.' is the abbreviation for ‘without'. |
Table 2. Dependence of the ΛcN 3S1 and 3D1 phase shifts on the baryon masses and SD coupling for different pion masses (in units of MeV). The ‘+' and ‘−' indicate the sign of the ΛcN 3S1 and 3D1 phase shifts within the fitting region Ec.m. ≤ 30 MeV, where ‘+' and ‘−' denote attractive and repulsive potentials, respectively. The values of the χ2/d. o. f. (in units of 10−2) are obtained with ΛF = 600/700 MeV. |
| Strategy | ${M}_{{Y}_{c}}[\,\,\,\,]{M}_{N}$ | SD coupling | mπ | ${\delta }_{{}^{3}{S}_{1}}$ | ${\delta }_{{}^{3}{D}_{1}}$ | χ2/d. o. f. b |
|---|---|---|---|---|---|---|
| 1 | ≠ | ✓ | 138 | + − a | − | |
| 410 | + | − | 1.30/1.32 | |||
| 570 | + | − | 25.9/30.9 | |||
| 2 | ≠ | × | 138 | + | − | |
| 410 | + | − | 1.30/1.08 | |||
| 570 | + | − | 0.16/0.08 | |||
| 3 | = | ✓ | 138 | − | − | |
| 410 | + | − | 3.41/18.8 | |||
| 570 | + | + | 7.31/11.0 | |||
| 4 | = | × | 138 | + | + | |
| 410 | + | + | 2.41/2.07 | |||
| 570 | + | + | 0.17/0.14 |
aIndicate that the ΛcN 3S1 interaction for mπ = 138 MeV is weakly attractive only at the very low energy region (about Ec.m. = 3 MeV) and then becomes repulsive as the kinetic energy increases. | |
bThe small χ2's compared with those of table 3 imply that it is easy to reproduce the lattice QCD 3S1 phase shifts than the coupled channel results. |
4.2. Simultaneous fits to the ΛcN 3S1-3D1 partial waves
In this subsection, we simultaneously fit to the phase shifts of ΛcN 3S1, 3D1 and inelasticity η1 of the lattice QCD data for mπ = 410 and 570 MeV with a smaller energy range from threshold up to Ec.m. = 5 MeV in order to achieve a χ2/d. o. f. ≈ 1. With this new strategy, we aim to check whether the covariant ChEFT approach or the non-relativistic ChEFT approach can precisely describe the lattice QCD data at low energies, where they are believed to work the best. In the covariant ChEFT, we only consider two strategies: either ${M}_{{Y}_{c}}\ne {M}_{N}$ or ${M}_{{Y}_{c}}={M}_{N}$. The SD coupling appears naturally in the CT potentials, therefore we did not manually turn it off. The non-relativistic ChEFT approach is also applied to perform the fits for comparison. The fitted results of strategies 5 − 7, as described in table 1, are qualitatively shown in table 3 and quantitatively shown in figures 2 and 4.
Figure 2. Same as figure 1, but the results are obtained by fitting to the lattice QCD phase shifts of 3S1, 3D1 partial waves and inelasticity η1 simultaneously for Ec.m. ≤ 5 MeV. |
Table 3. Phase shifts of ΛcN 3S1 and 3D1 partial waves for different pion masses based on the covariant ChEFT approach and non-relativistic ChEFT approach. The former depends on the baryon masses used, either physical or the lattice QCD ${M}_{{Y}_{c}}$ and MN or their average. These results are obtained by fitting to the phase shifts of 3S1, 3D1 and inelasticity η1 simultaneously for Ec.m. ≤ 5 MeV from lattice QCD simulations, and mπ is in units of MeV. The ‘+' and ‘−' indicate the sign of ΛcN 3S1 and 3D1 partial waves phase shifts within the fitting region, where ‘+' and ‘−' stand for attractive and repulsive potentials, respectively. The values of the χ2/d. o. f. are obtained with ΛF = 600/700 MeV in the covariant ChEFT and ΛF = 500/600 MeV in the non-relativistic ChEFT. |
| Strategy | Approach | ${M}_{{Y}_{c}}[\,\,\,\,]{M}_{N}$ | mπ | ${\delta }_{{}^{3}{S}_{1}}$ | ${\delta }_{{}^{3}{D}_{1}}$ | χ2/d. o. f. |
|---|---|---|---|---|---|---|
| 5 | cov. ChEFT | ≠ | 138 | + | − | |
| 410 | + | − | 2.65/0.92 | |||
| 570 | + | − | 3.32/3.25 | |||
| 6 | = | 138 | + | + | ||
| 410 | + | + | 3.29/3.30 | |||
| 570 | + | + | 5.33/5.35 | |||
| 7 | non-rel. ChEFT | 138 | − | + | ||
| 410 | + | + | 2.22/2.25 | |||
| 570 | + | + | 5.37/5.45 |
4.2.1. Covariant ChEFT
First, we study how the use of ‘physical' baryon masses affects the description of the ΛcN interactions in the covariant ChEFT. The relevant fitting details and the corresponding values of the χ2/d. o. f. are summarized in table 3. For strategy 5, with lattice QCD ${M}_{{Y}_{c}},{M}_{N}$ in the ΛcN CT potentials within the fitting region Ec.m. ≤ 5 MeV, we presented the phase shifts of ΛcN 3S1 and 3D1 partial waves and inelasticity in figure 2. One can see that the ΛcN 3S1 and 3D1 phase shifts agree quantitatively with the lattice QCD data within uncertainties, and the asymptotic behaviors of inelasticity are in good agreement with the lattice QCD data. Comparing these results with those of strategy 6 where ${M}_{{Y}_{c}}={M}_{N}$, shown in figure 2, one can see that the ΛcN 3D1 interactions are attractive, contrary to the repulsive potential obtained in strategy 5.
In both cases, the extrapolation of the relativistic ΛcN 3S1 and 3D1 partial waves phase shifts and inelasticity to the physical point shows that the ΛcN interaction is attractive in the 3S1 partial wave within the fitting region. Comparing the above results with strategy 1 (our previous study), where the ΛcN 3S1 potential is repulsive, we conclude that the extrapolated phase shifts of ΛcN 3S1 are not very stable.
To investigate whether the energy region fitted can affect the extrapolations, we also fitted the lattice QCD data up to Ec.m. ≤ 20 MeV in the covariant ChEFT approach. The results are shown in figure 3 in comparison with the results obtained by fitting only up to Ec.m. ≤ 5 MeV. The two fits are qualitatively consistent with each other . Only η1 is closer to unity in the new fit. In addition, the extrapolated δ3S1 and η1 show some visible differences. At mπ = 138 MeV, δ3S1 becomes smaller, and η1 becomes more dependent on the cutoff.
Figure 3. ΛcN 3S1, 3D1 phase shifts and inelasticity η1 as functions of Ec.m.. The results are obtained by fitting to the lattice QCD δ3S1, δ3D1 and η1 simultaneously up to Ec.m. ≤ 5 MeV (blue bands) and Ec.m. ≤ 20 MeV (magenta bands) in covariant ChEFT. |
4.2.2. Non-relativistic ChEFT
Focusing on the lattice QCD data with Ec.m. ≤ 5 MeV, we show in figure 4 (the blue bands) the ΛcN 3S1,3D1 partial wave phase shifts and inelasticity obtained from strategy 7 in the non-relativistic ChEFT. The corresponding χ2/d. o. f. are listed in table 3. Here, we find that the non-relativistic phase shifts of ΛcN 3S1 partial wave and inelasticity are in qualitative agreement with the lattice QCD data in the region fitted. On the other hand, the ΛcN 3D1 phase shifts turn out to be positive, while the lattice QCD data are negative, though quite small. This is very different from the covariant case as shown in figure 3, where the 3D1 phase shifts are negative for the energy region studied. According to the previous experience in the NN sector [11, 16], the two EFTs should behave similarly in the low-energy regime, while the covariant EFT usually agrees better with the lattice QCD data than the non-relativistic EFT in the relatively high-energy regime. The present results are in conflict with such expectations to some extent. A better understanding can only be achieved once more precise lattice data with realistic uncertainties become available.
Figure 4. Same as figure 3, but for the non-relativistic ChEFT. |
When we extrapolate the non-relativistic results to the physical point, we find that the ΛcN interaction is repulsive in the 3S1 partial wave within the fitting region. Comparing these results with those of the covariant ChEFT and the left panel in Fig.4 of [2], we again conclude that the extrapolated ΛcN 3S1 interactions are not very stable, i.e. sensitive to the adopted fitting strategies.
The HB ΛcN 3S1, 3D1 phase shifts and inelasticity with LECs obtained by fitting to the lattice QCD data up to Ec.m. ≤ 20 MeV (Fit 2) are compared to those obtained by fitting only up to Ec.m. ≤ 5 MeV (Fit 1) in figure 4. Compared to Fit 1, the descriptions of δ3D1 remain almost unchanged in Fit 2, but the 3S1 phase shifts are very different. In Fit 1, δ3S1 increases with Ec.m. for the case of mπ = 570 MeV, while in Fit 2, it increases with Ec.m. for Ec.m. ≤ 5 MeV and then decreases with Ec.m., which are in better agreement with the lattice QCD simulations at least for the energy region shown in this figure. Moreover, the difference for the case of mπ = 570 MeV eventually contributes to the completely different prediction of the physical δ3S1, where the phase shifts become positive in Fit 2. As for the case of mπ = 410 MeV, the two results show no qualitative difference. For the inelasticity, the results obtained in Fit 2 are closer to unity and larger than the lattice QCD simulations for the energy region fitted and become more independent on Ec.m. as the pion mass increases.
5. Conclusion
The ΛcN 3S1-3D1 interactions were studied in leading order covariant ChEFT and next-to-leading order non-relativistic ChEFT. The low-energy constants were determined in two different strategies by fitting to the HAL QCD lattice data, i.e. (a) by only fitting to the ΛcN 3S1 partial wave phase shifts, (b) by a combined fit to the phase shifts of 3S1, 3D1 and inelasticty η1. It was shown that for the first strategy, the predicted ΛcN 3D1 phase shifts from the covariant ChEFT were consistent with the low-energy lattice QCD data by using lattice QCD ${M}_{{Y}_{c}}$, MN and retaining the SD coupling in the contact potentials, while for the second strategy, one obtained results similar to those of the first strategy in the covariant ChEFT. However, the non-relativistic ChEFT predicts an attractive ΛcN 3D1 interaction in both cases, which is inconsistent with the low-energy lattice QCD data. In addition, we found that the extrapolated ΛcN 3S1 phase shifts in the physical region were very sensitive to the fitting strategies and the theoretical approaches used. The covariant ChEFT predicts a repulsive/attractive 3S1 interaction depending on the fitting strategy (a)/(b), while the non-relativistic ChEFT predicts the opposite, which also depends on the energy region fitted. These results indicate that more refined lattice QCD data are needed to reach a firm conclusion about the ΛcN 3S1-3D1 interactions.
It is necessary to point out that there are ongoing discussions on the validity of the HAL QCD method [34-36]. In our present work, we have of course assumed that the method is valid and the 3D1 phase shifts and particularly the inelasticity are correctly extracted with the precision claimed in [18]. Hopefully, the present study can motivate a closer look at the ΛcN interaction in the 3S1-3D1 coupled channel.