Λ N and Ξ N Interactions Studied with Lattice QCD

We present our recent studies of Lambda-Nucleon ( ΛN) as well as Cascade-Nucleon ( ΞN) interactions by using lattice QCD. The equal-time Bethe-Salpeter (BS) amplitude of the lowest energy scattering state of baryon numberB = 2 system (protonΛ and protonΞ0) is calculated from lattice QCD. For the calculation of the ΛN potential, two di fferent types of gauge configurations are employed: (i) 2 +1 flavor full QCD configurations generated by the PACS-CS collaboration at β = 1.9 (a = 0.0907(13) fm) on a 32 3 × 64 lattice, whose spatial volume is (2.90 fm) 3. (ii) Quenched QCD configurations at β = 5.7 (a = 0.1416(9) fm) on a 32 3 × 48 lattice, whose spatial volume is (4.5 fm) . The spin-singlet central potential is calculated from the BS wave function for the spin J = 0 state, whereas the spin-triplet central potential as well as the tensor potential are deduced simultaneously from the BS wave function for the spin J = 1 state by dividing it into theS -wave and theD-wave components. For the calculation of theΞN potential, we employ quenched QCD configurations, at β = 5.7 (a = 0.1416(9) fm) on a 323 × 32 lattice, whose spatial volume is (4.5 fm) 3. The effective central potential in the spin triplet channel as well as the central potential in the spin singlet channel are calculated for the ΞN. The scattering lengths are obtained from the asymptotic behavior of the BS wave function by using the Lüscher’s formula.


Introduction
Study of hyperon-nucleon (YN) and hyperon-hyperon (YY) interactions is one of the keys to explore strange nuclear systems such as hypernuclei and also hyperonic matter inside neutron stars. Hyperons (or strange quarks) would play a characteristic role in normal nuclear systems as "impurities" [1]. A number of spectroscopic studies of the Λ and Σ hypernuclei have been performed experimentally and theoretically; They lead a qualitative conclusion that the Σ-nucleus interaction is repulsive [2,3] whereas the Λnucleus interaction is attractive and the strength of Λ potential in nuclear systems is about 2/3 of the strength of the normal nuclear potential. Moreover, recent systematic study (e.g., Ref. [4]) for light (s-shell) Λ hypernuclei ( 3 Λ H, 4 Λ H, 4 Λ He and 5 Λ He) suggests that the ΛN interaction in the 1 S 0 channel is more attractive than that in the 3 S 1 channel. These results is useful to study the composition of hyperonic matter inside the neutron stars [5]: the Λ particle instead of Σ − would be the first strange baryon to appear in the core of the neutron stars. The ΞN interaction is also interesting and important, in order to explore the existence of Ξ hypernuclei and the dense hyperonic matter in neutron stars. Despite their importance, however, YN and YY intera e-mail: nemura@nucl.phys.tohoku.ac.jp actions have still large uncertainties because direct YN and YY scattering experiments are either difficult or impossible due to the short life-time of hyperons. Consequently, phenomenological YN and YY interaction models are not well constrained from experimental data even under some theoretical guides [6][7][8][9][10][11].
Under these circumstances, it should be desirable theoretically to understand the YN and YY interaction (or, in more general, baryon-baryon interaction) based on the dynamics of quarks and gluons as fundamental degrees of freedom. If one can perform such an appropriate deduction along the theory of quantum chromodynamics (QCD), they should have a reliable prediction regarding the YN and YY potentials.
The lattice QCD would be a valuable theoretical tool to perform a first-principle calculation of baryon-baryon interactions. Previously, scattering parameters based on the Lüscher's formula have been reported for the NN system [12,13] and for the YN system [14,15]. Recently, a new approach to the NN interaction from the lattice QCD has been proposed [16]. In this approach, the NN potential can be directly obtained from lattice QCD through the Bethe-Salpeter (BS) amplitude and the observables such as the phase shift and the binding energy can be calculated by using the resultant potential. The purpose of this report is EPJ Web of Conferences to show the recent results for the YN potentials calculated from lattice QCD.

Formulation
The basic formulation has already been given in Refs. [16,17,20,21]. (See also Refs. [27,28].) and a recent comprehensive accounts for the lattice NN potential is found in Ref. [18]. We start from an effective Schrödinger equation for the equal-time BS wave function of two-baryon system (B 1 B 2 ): Here are the square of asymptotic momentum in the center-of-mass frame and the reduced mass of the (B 1 B 2 ) system, respectively. The total energy is given by We consider the low-energy scattering state so that the nonlocal potential can be rewritten by derivative expansion [29], The general expression of the YN potentials (V NΛ , V NΞ ) are known to be [30] Here S 12 = 3(σ N · n)(σ Y · n) − σ N · σ Y is the tensor operator with n = r/|r|, S ± = (σ N ± σ Y )/2 are symmetric (+) and antisymmetric (−) spin operators, L = −ir × ∇ is the orbital angular momentum operator, and τ N (τ Ξ ) is isospin operator for N = (p, n) T (Ξ = (Ξ 0 , Ξ − ) T ). Note that τ Λ = 0 due to the isospin-singlet nature of the Λ. The antisymmetric spin-orbit forces appear when the two baryons are not identical. V 0,σ,τ,στ,T,T τ are the leading order (LO) potentials while V LS ,LS τ,ALS ,ALS τ are the next-to-leading-order (NLO) potentials in the velocity expansion. For the present work of the ΞN interaction, since we focus on the isospin triplet (I = 1) channel a particular combination of the potential terms with (τ N · τ Ξ ) = 1 is taken into account, For the sake of simplicity we rewrite them in what follows for the ΞN (I = 1) potential The LO potentials have been calculated along the line as follows. For the spin singlet state, we consider the central potential, For the spin triplet state, on the other hand, the wave function φ(r; J = 1) comprises the S -and the D-wave components because of the tensor force. These partial waves can be extracted from the φ αβ (r; J = 1) such that Therefore, the effective Schrödinger equation with the LO potentials becomes: with The r dependence of the central and the tensor potentials, V C (r; J = 0), V C (r; J = 1) and V T (r), are determined once we obtain the wave function, the asymptotic momentum and the reduced mass in lattice QCD. In order to obtain the φ αβ (r; J), we firstly calculate the four-point correlator from the lattice QCD where the summation over X is to select the state with zero total momentum. The B 1,α (x) and B 2,β (y) denote the interpolating fields of the baryons such as is the source to create a B 1 B 2 state with performing the spin projection for the total angular momentum J, M. The baryon sources B 1,α , B 2,β can be given by, The wall source ( f (x) = 1) is employed to calculate the lowest scattering state of the system.
In the Monte Carlo calculations, noise reductions are made for the four-point correlator (12) obtained from lattice QCD in order to restore (i) the rotational (cubic group) symmetry, (ii) the spatial reflection symmetry, (iii) the charge conjugation and time-reversal symmetry.
The desirable wave function φ αβ (r; J) is obtained from the projected four-point correlator at large t − t 0 : Here E n (|E n ) is the eigen-energy (eigen-state) of the sixquark system with the particular quantum number (i.e., J π , M, strangeness S and isospin I), and  Table 1. To calculate the BS wave function, the wall source is placed at the timeslice t 0 with the Coulomb gauge fixing, and the Dirichlet boundary condition is imposed in the temporal direction at the time-slice t − t 0 = 32. In order to improve the statistics, multiple sources at t 0 = 8n with n = 0, 1, 2, · · · , 8 are employed on each gauge configuration.

Quenched calculation
In quenched QCD calculation we employ the plaquette gauge action and the Wilson quark action at β = 5.7 on a 32 3 × 48 (32 3 × 32) lattice for the recent study of ΛN [22] (for the relatively earlier study of pΞ 0 [20]). The periodic boundary condition is imposed for quarks in the spatial direction. The source is placed at t 0 with the Coulomb gauge fixing and the Dirichlet boundary condition is imposed in the temporal direction. The lattice spacing at the physical point is determined as a = 0.1416(9) fm (1/a = 1.393(9) GeV) from m ρ = 770 MeV. The spatial lattice volume is about (4.5fm) 3 . The hopping parameter for the strange quark mass is given by κ s = 0.16432 (6)

ΛN interaction
We first show the ΛN potential. Figures 1 and 2 taken from [24] show the ΛN potentials obtained from 2+1 flavor QCD calculation as a function of r. The central (V C (J = 1)) and the tensor (V T ) potentials in the 3 S 1 − 3 D 1 channels are given in Fig. 1 while the central potential in the 1 S 0 channel (V C (J = 0)) is given in Fig. 2. We also show the central potential multiplied by volume factor (4πr 2 V C (r)) in the left panel in addition to the normal V(r) given in the right panel, in order to compare the strength of the repulsive force between two quark masses. These figures contain results with (m π , m K ) ≈ (699, 787) and (414, 637) MeV, which are obtained at t−t 0 = 13 and 10, respectively. These time-slices are chosen so that the ground state saturation is achieved.
As can be seen in both figures, the attractive well of the central potential moves to outer region as the u, d quark mass decreases while the depth of these attractive pockets do not change so much. The present results show that  the tensor force is weaker than the NN case [21], and the quark mass dependence of the tensor force seems to be small. Both of the repulsive and attractive parts increase in magnitude as the u, d quark mass decreases.
For m π ≈ 699 MeV, the central potentials reach V C → 0 at the radial distance r ∼ 1.3 fm, which is smaller than the half of the physical lattice length (aL/2 ≈ 1.45 fm). Therefore the Lüscher's formula can be applied to extract the scattering phase shift, which will be discussed in the latter section. For m π ≈ 400 MeV, on the other hand, the interaction range of the V C , which is about 1.4 fm, almost reaches to the half of the lattice. Therefore we must be very careful to extract the scattering phase shift at this or lighter quark masses from the Lüscher's formula, though no sign of the violation against the Lüscher's condition was observed within errors for the effective central potential even at m π ≈ 300 MeV. (See Fig. 1 in an earlier report [22].) Calculation on larger spatial volume will be needed to correctly extract the scattering phase shift at m π ≈ 300 MeV.

Quenched calculation
The ΛN potential is also calculated by quenched QCD. quenched QCD calculation.The central (V C (J = 1)) and the tensor (V T ) potentials for J = 1 channel are given in the Fig. 3 while the central potential for J = 0 channel (V C (J = 0)) is given in the Fig. 4. Comparing the strength of the repulsive cores between two quark masses, the central potential multiplied by volume factor (4πr 2 V C (r)) is also shown in the left panel as well as the normal V(r) in the right panel for each Figure. These figures contain results with (m π , m K ) ≈ (512, 606) and (407, 565) MeV, which are obtained at t − t 0 = 7. As can be seen in both figures, qualitative behavior of the potentials is similar to those of the full QCD potentials. Namely, the attractive well of the central potential moves to outer region as the u, d quark mass decreases while the depth of these attractive pockets do not change so much. The tensor force is weaker than the NN case, and the quark mass dependence of the tensor force seems to be small. Both of the repulsive and attractive parts increase in magnitude as the u, d quark mass decreases.

ΞN interaction
For ΞN interaction, we focus on the isovector (I = 1) channel, which has no strong decay mode into other B ′ 1 B ′ 2 systems. (Note that, on the other hand, NΞ in the isoscalar (I = 0) channel is above the ΛΛ threshold.) As is also seen in Table 1, the baryon masses calculated from the lattice QCD are consistent with the experimentally observed ordering of the two-baryon threshold in the strangeness S = −2 sector; E th (ΛΛ) < E th (NΞ) < E th (ΛΣ) < E th (ΣΣ). This guarantees that the ΞN in the I = 1 channel presented here is actually the lowest energy scattering state. Figure 5 taken from Ref. [20] compares the (effective) central pΞ 0 potential at m π ≃ 368 MeV with that at m π ≃ 511 MeV in the 3 S 1 channel (left) and in the 1 S 0 channel (right). At m π ≃ 368 (511) MeV, the potentials are evaluated at t − t 0 = 6 (7). The height of the repulsive core increases as the ud quark mass decreases, while the significant difference is not seen in the medium to long distances within the error bars. The solid lines in Fig.5 are the one pion exchange potential (OPEP),  [6]. Also we define g πNN ≡ f πNN m π 2m N . Unlike the NN potential in the S -wave, the OPEP in the present case has opposite sign between the spinsinglet channel and the spin-triplet channel. Also, the absolute magnitude of OPEP is weak due to the factor 1 − 2α. From Fig.5, clear signature of OPEP at long distance (r > 1.2 fm) is hardly found within statistical errors. On the other hand, there is a clear departure from OPEP at medium distance (0.6fm < r < 1.2fm) in both 1 S 0 and 3 S 1 channels. These observations may indicate a mechanism of state-independent attraction such as the correlated two pion exchange.

Scattering length
The scattering parameters can be obtained from the asymptotic momentum k of the BS wave function according to Lüscher's formula [27,28], which is given by with Z 00 (s; where Z 00 (1; q 2 ) is obtained by the analytic continuation in s. The asymptotic momentum k on the finite lattice volume 07009-p. 6 19 th International IUPAP Conference on Few-Body Problems in Physics is determined by fitting the asymptotic region of the BS wave function in terms of the Green's function which is the solution to the Helmholtz equation with δ L (r) being the periodic delta function [27,28]. Figure 6 (Figure 7) shows the scattering lengths for ΛN (ΞN (I = 1)) as a function of m 2 π . Note that the sign of the S -wave scattering length a 0 defined in Eq. (23) becomes positive when the interaction is weakly attractive (i.e., there is no bound state.). What the Figures show at present is that both of ΛN and pΞ 0 interactions are attractive on the whole. For the present results, the scattering lengths are almost constant for larger u, d quark mass region (corresponding to m π > ∼ 500 MeV). On the other hand, for lighter u, d quark mass (corresponding to 400 MeV < ∼ m π < ∼ 500 MeV), the present results seems to show that the scattering lengths increase as the u, d quark mass decreases. However, the actual quantities in the Fig. 6 are much smaller than the empirical values, a ΛN 0 ( 1 S 0 ) ∼ 1.5 − 2.5 fm, a ΛN 0 ( 3 S 1 ) ∼ 1.5 − 2.5 fm, estimated from the measurement of the ΛN total cross section and the theoretical studies of Λ-hypernuclei. And also diverse predictions on the pΞ 0 scattering lengths from other approaches are summarized as follows: The chiral effective field theory [11] predicts a 0 ( 1 S 0 ) ∼ −0.2 fm and a 0 ( 3 S 1 ) ∼ −0.02 fm. The phenomenological boson exchange model (e.g., SC97f) [6] gives a 0 ( 1 S 0 ) = −0.4 fm and a 0 ( 3 S 1 ) = 0.030 fm. The quark cluster model (fss2) [8] gives a 0 ( 1 S 0 ) = −0.3 fm and a 0 ( 3 S 1 ) = 0.2 fm, while QCD sum rules [10] gives a 0 ( 1 S 0 ) = 3.4 ± 1.4 fm and a 0 ( 3 S 1 ) = 6.0 ± 1.4 fm. More extensive and systematic analysis with smaller lattice spacing, larger spatial volume and/or improved lattice action, etc., towards the physical quark mass point should be needed.

Summary
Recent studies of ΛN and ΞN interactions based on lattice QCD have been presented. The central and the tensor potentials are calculated from the BS wave function of the lowest energy scattering state for the ΛN system. The lightquark mass dependence of the central potentials shows that the interaction range becomes larger whereas the depth of the attractive well hardly changes as the u, d quark mass decreases. On the other hand, the tensor force of the ΛN interaction has relatively a weak light-quark mass dependence. It is also interesting to compare the present results with the NN potential found in Refs. [18,21]; The tensor force of the NN potential is significantly enhanced as the u, d quark mass decreases. For the ΞN system, our lattice calculation shows that the pΞ 0 interaction is attractive for both 1 S 0 and 3 S 1 states, whereas some phenomenological models predicts that the ΞN interaction for I = 1 channel is very weak or repulsive. A slight tendency that the 3 S 1 interaction is relatively more attractive than that in the 1 S 0 channels is found.
Finally, there are a few remarks before closing. Several related studies have been already carried out and/or started; (i) The study of the NN interaction is considerably important. See, for example, Refs. [18,21] for recent developments. (ii) In order to see how well the LO part in derivative expansion of the non-local potential U(r, r ′ ) describes the interaction, the LO NN potential obtained at E c.m. ≈ 45 MeV have been compared with that at E c.m. ≈ 0 MeV [19,25]. The potentials are almost identical between these energies. It seems that the energy dependence of the NN potential is weak between these energies. (iii) Baryonbaryon potentials at the flavor S U(3) limit have been calculated [23]. The flavor symmetry would be a good guide to understand the role of strangeness degree of freedom. The lattice QCD potential is the best tool to see the flavor symmetric world because it is a first-principle approach. (iv) K + N (strangeness S = +1) channel potential is one of the interesting targets of this approach [26]. It is directly connected to searching for Θ + . (v) Three-nucleon force is an important puzzle to be clarified to make a nucleus comprising more than two nucleons from QCD. The calculation for the NNN system will be performed in the near future.