Physics Letters B 673 (2009) 136–141 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb Hyperon–nucleon force from lattice QCD ∗ Hidekatsu Nemura a, ,1, Noriyoshi Ishii b, Sinya Aoki c,d, Tetsuo Hatsuda e a Advanced Meson Science Laboratory, Nishina Center for Accelerator-Based Science, RIKEN, Wako 351-0198, Japan b Center for Computational Sciences, University of Tsukuba, Tsukuba 305-8571, Japan c Graduate School of Pure and Applied Sciences, University of Tsukuba, Tsukuba 305-8571, Japan d Riken BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973, USA e Department of Physics, The University of Tokyo, Tokyo 113-0033, Japan article info abstract Article history: We calculate potentials between a proton and a Ξ 0 (hyperon with strangeness −2) through the equal- Received 6 June 2008 time Bethe–Salpeter wave function, employing quenched lattice QCD simulations with the plaquette Received in revised form 2 February 2009 gauge action and the Wilson quark action on (4.5 fm)4 lattice at the lattice spacing a 0.14 fm. The Accepted 4 February 2009 ud quark mass in our study corresponds to mπ 0.37 and 0.51 GeV, while the s quark mass corresponds Available online 6 February 2009 to the physical value of m . The central pΞ 0 potential has a strong (weak) repulsive core in the 1 S Editor: J.-P. Blaizot K 0 3 ( S1) channel for r 0.6 fm, while the potential has attractive well at medium and long distances 0 PACS: (0.6fm r 1.2 fm) in both channels. The sign of the pΞ scattering length and its quark mass 12.38.Gc dependence indicate a net attraction in both channels at low energies. 13.75.Ev © 2009 Published by Elsevier B.V. 21.30.-x 21.80.+a Keywords: Lattice QCD Hyperon–nucleon interaction Nuclear forces Hypernuclei 1. Introduction in Ref. [11], where the NN scattering lengths were extracted from the quenched simulations by using the Lüscher’s finite volume method [12]. The same method was applied later to the (2 + 1)- Modern nucleon–nucleon (NN)potentials[1] give successful flavor simulations with the mixed action [13]. The scattering length description of the NN scattering data at low energies and have of ΛN system was first examined in [14] in quenched simula- been used for precision calculations in light nuclei [2]. Similarly, tions on a small lattice box. Subsequently, ΛN and Σ N scattering hyperon–nucleon (YN) and hyperon–hyperon (YY)potentialsin lengths were studied in (2 + 1)-flavor simulations with the mixed the coordinate space are known to be quite useful and impor- action [15]. Such low-energy scattering parameters calculated in tant for studying the properties of hypernuclei [3] as well as the lattice QCD would be valuable inputs to construct phenomenologi- equation of state for hyperonic matter in neutron stars [4].How- cal YN and YY potentials to be used for studying hypernuclei and ever, there are still large experimental uncertainties in YN and YY hyperonic matter. interactions at present, because the short life-time of the hyper- Recently, an alternative but closely related approach to [12] has ons makes the scattering experiments difficult. Accordingly, phe- been proposed to define the NN potential from lattice QCD [16– nomenological YN and YY potentials are not well constrained 18]. Since the potential is not a direct physical observable, one from data even under some theoretical guides [5–10]. cannot make quantitative comparison of this lattice NN poten- In such a situation, it would be desirable to analyse hyperon tial with the phenomenological NN potentials. However, they are interactions on the basis of the first principle lattice QCD simula- both designed to reproduce the correct scattering phase shifts, and tions. Studies along this line for nucleon interactions were initiated their spatial structures are found to have common features [16]; the attraction at long and intermediate distances and the strong repulsion at short distance in the S-wave channel [19]. * Corresponding author. The purpose of the present Letter is to report our first attempt E-mail address: [email protected] (H. Nemura). to apply the above approach to the hyperon–nucleon systems. For 1 Present address: Strangeness Nuclear Physics Laboratory, Nishina Center for Accelerator-Based Science, RIKEN, Wako 351-0198, Japan. the NN potential, only two channels (isovector and isoscalar) exist 0370-2693/$ – see front matter © 2009 Published by Elsevier B.V. doi:10.1016/j.physletb.2009.02.003 H. Nemura et al. / Physics Letters B 673 (2009) 136–141 137 in the flavor SU(2) space, i.e., 2 ⊗ 2 = 3 ⊕ 1. Including the strange + V LS(r)(L · S+) + V LSτ (r)(L · S+)(τN · τΞ ) ∗ quark extends this to 8 ⊗ 8 = 27 ⊕ 10 ⊕ 1 ⊕ 8 ⊕ 10 ⊕ 8 in the flavor + V (r)(L · S−) + V (r)(L · S−)(τ · τ ) SU(3) space. The isovector (isoscalar) channel in the NN sector is ALS ALSτ N Ξ ∗ assigned to be a subset in the 27-plet (10 -plet) representation. + O ∇2 . (3) The potentials in newly arising channels are hardly determined = · · − · from experiments so far. Here S12 3(σ1 n)(σ2 n) σ1 σ2 isthetensoroperatorwith = || = ± + In this Letter, we focus on the NΞ potential in the isovector n r/ r , S± (σ1 σ2)/2 are symmetric ( ) and antisymmetric − =− × ∇ (I = 1) channel as a first step. (A preliminary account of this sys- ( ) spin operators, L ir is the orbital angular momentum = T = tem has been reported in [20].) There are two main reasons for operator, and τN (τΞ ) is isospin operator for N (p,n) (Ξ 0 − T picking up this channel: (i) Theoretically, it is the simplest gener- (Ξ ,Ξ ) ). We note that the antisymmetric spin–orbit forces alization of the NN system; pΞ 0 is obtained from pn by replacing (V ALS and V ALSτ ) do not arise in the NN case because of the iden- the d-quarks in the neutron by the s-quarks. Also it is a channel tical nature of the nucleon within the isospin symmetry. which does not have strong decay into other YN systems.2 (ii) Ex- According to the above expansion, the wave function should be = perimentally, not much information has been available on the NΞ classified by the total isospin I, the total angular momentum J + interaction except for a few studies; a recent report gives the upper L S+ and the parity. A particular spin (isospin) projection is made · · limit of elastic and inelastic cross sections [21], and earlier pub- in terms of σN σΞ (τN τΞ ), e.g., for the isospin projection we have (I=0) = − · (I=1) = + · lications suggest weakly attractive Ξ –nucleus potential [22].The P (1 τN τΞ )/4andP (3 τN τΞ )/4. = = Ξ –nucleus interaction will be soon studied as one of the day-one The equal-time BS wave function for (I, Iz) (1, 1) and L 0 − + experiments at J-PARC [23] via (K , K ) reaction with nuclear tar- (S-wave) on the lattice is obtained by get. 1 1 0 0 This Letter is organized as follows. In Section 2,wedescribethe φ(r) = P σ 0|p R[r]+x Ξ (x) pΞ ;k , (4) 3 αβ α β 0 24 L basic formulation to derive the pΞ potential through the Bethe– R∈O x Salpeter amplitude measured in the lattice QCD simulations. Our pα(x) = εabc ua(x)Cγ5db(x) ucα(x), (5) lattice setup is explained in Section 3. In Section 4,weshownu- 0 = merical results of the potentials in the spin-singlet and spin-triplet Ξβ (y) εabc ua(y)Cγ5sb(y) scβ (y), (6) channels and their quark mass dependence. We also show the es- where α and β denote the Dirac indices, a, b and c the color in- timate of the scattering lengths in these channels. Section 5 is dices, and C = γ γ the charge conjugation matrix. The summation devoted to summary and concluding remarks. 4 2 over R ∈ O is taken for cubic transformation group to project out the S-wave state.3 The summation over x is to select the state 2. Basic formulation with zero total momentum. Here we take local field operator pα(x) and Ξ 0(y) for the proton and Ξ 0. The wave function and the Our methodology to obtain the baryon potentials is along the β potential (or equivalently the off-shell behavior of the scattering lines of [16–18]. (See also [12,24] for the seminal attempts to intro- amplitude) depend on the choice of interpolating operators. This is duce similar notion of the potential.) We consider the low-energy the situation common to any field theories. In this Letter, we focus N–Ξ scattering and start with an effective Schrödinger equation exclusively on the local operators as introduced above and leave for the equal-time Bethe–Salpeter (BS) wave function φ(r) ob- further discussions on the operator dependence to [25].Wetake tained from the Lippmann–Schwinger equation [18,25]: the upper components of the Dirac indices to construct the spin = = 1 (σ 0) = (σ 1) = − ∇2φ(r) + U (r,r )φ(r ) d3r = Eφ(r). (1) singlet (triplet) channel by Pαβ (σ2)αβ (Pαβ (σ1)αβ ). The 2μ BS wave function φ(r) is understood as a probability amplitude to 2 find “nucleon-like” three-quarks located at point x +r and “Ξ -like” Here μ = mNmΞ /(mN + mΞ ) and E ≡ k /(2μ) are the reduced three-quarks located at point x. Our BS wave function has infor- mass of the NΞ system and the non-relativistic energy in the mation not only of the elastic amplitude N → N but also of center-of-mass frame, respectively.