Lattice QCD studies of strangeness S = -2 baryon-baryon interactions

Transcription

1 Lattice QCD studies of strangeness S = -2 baryon-baryon interactions Kenji Sasaki (University of Tsukuba) for HAL QCD collaboration HAL (Hadrons to Atomic nuclei from Lattice) QCD Collaboration S. Aoki (Univ. of Tsukuba) T. Doi (Univ. of Tokyo) T. Hatsuda (RIKEN) Y. Ikeda (Tokyo Inst. Tech.) T. Inoue (Nihon Univ.) N. Ishii (Univ. of Tsukuba) K. Murano (RIKEN) H. Nemura (Tohoku Univ.)

2 Introduction Strangeness in nuclei opened the new frontier of nuclear physics. Experimental side Exploration of the multi-strangeness hadronic systems is planned at J-PARC Generalized BB interaction Hypernuclear structure Search of exotic hadrons and so on Theoretical side J-PARC Study the hyperon-nucleon (YN) and hyperon-hyperon (YY) interactions One of the most important subject in the (hyper-) nuclear physics The phenomenological description of them has large uncertainties due to the shortage of experimental data. Lattice QCD simulation can produce BB potential directly from QCD complementary to an experiment.

3 Introduction This work : Baryon-baryon interactions in strangeness S = -2 system The first step towards the multi-strangeness world. Structures of double-λ hypernuclei and Ξ-hypernuclei The SU(3) breaking effects in the BB interaction. H-dibaryon at physical point. Information of ΛΛ interaction and H-dibaryon from experiment Conclusions of the NAGARA Event (The double- hypernuclear event) Lower limit of H mass : m H 2m Λ - 6.9MeV. The Λ Λ interaction is weakly attractive. K.Nakazawa and KEK-E76 & E373 collaborators NAGARA event

4 SU(3) Classification of B-B states Within S-wave total anti-symmetric states are constructed by combination of spin and flavor. 8 x 8 = + 8 S A + + * Flavor symmetric Spin singlet Flavor anti-symmetric Spin triplet Hyper-charge (Strangeness) Y=(S=-) Y=2(S=) pσ + +Σ + p : flavor symmetric pσ + -Σ + p : flavor anti-symmetric Y=(S=-2) Isospin (I z ) There are flavor combinations ΛΛ, pξ, nξ, Ξ p, Ξ n, Σ + Σ, Σ Σ, Σ Σ +, ΛΣ, Σ Λ

5 Classification of B-B states with S=-2 Flavor-Symmetric : spin singlet I=2 state 27 I= states S=-2 BB states 8s I= states Flavor-Anti-symmetric : spin triplet S=-2 BB states * 8a I= states I= state

6 Classification of B-B states with S=-2 Flavor-Symmetric : spin singlet 27 8s SU(3) breaking SU(3) IR Flavor-Anti-symmetric : spin triplet ΣΣ ΝΞ ΛΛ ΛΣ ΝΞ ΣΣ I= states 8s, 27 mixing I= states, 8s, 27 mixing * SU(3) breaking ΣΣ ΛΣ ΝΞ I= states 8a,, * mixing 8a ΝΞ

7 Channel coupling Energy levels of baryon-baryon system in the real world 24 ΣΣ(Ι=) ΣΣ(Ι=) ΣΣ(Ι=2) 23 NΞ(Ι=) ΛΣ(Ι=) small energy difference 2 N (I=) NΣ(Ι=/2) NΣ(Ι=3/2) ΛΛ(Ι=) NΞ(Ι=) NΛ(Ι=/2) The effects of the neighboring states can not be ignored. 9 NN(I=) NN(I=) 8 S= S=- S=-2 We have to extend our method to the coupled channel formalism.

8 HAL QCD strategy Calculate Bethe-Salpeter (BS) wave function on any gauge configuration. t t, x = y Define the non-relativistic Schrödinger equation (general form) Performing the derivative expansion for the interaction kernel U x y V x x y V x, x y The potential is given as B t, x y B t, x BB t 2 E 2 x = U x y y d 3 y V x = E 2 2 x x This technique is widely applicable for hadronic systems Extention to the YN and YY systems

9 Coupled channel Schrödinger equation Using four-point correlator W with an optimized source such as, The coupled channel Schrödinger equation can be rewritten as ( α) p 2 α H 2μ W α( x, E)=V αα ( x)w α ( x, E)+V αβ ( x)w β ( x, E)+V α γ ( x)w γ ( x, E) α Define R α ( x, E) W α( x, E) C α (t ) t R α ( x, E) = p α R α ( x, E) 2 μ α W α ( x, E )= A Ψ α ( x, E )e Et Thus the potential matrix can be obtained as (V Λ Λ V N Ξ V Σ Σ Taking time derivative of R, 2 exp ( (E M α )t ) exp( p t) 2 α 2μ α Λ Λ Λ Λ( x, E ) W N Ξ( x, E ) W Σ Σ( x, E ) Λ Λ W Λ Λ ( x, E ) W N Ξ ( x, E ) W Σ Σ ( x, E ) Λ ( x))=(w Λ W Λ Λ ( x, E 2 ) W N Ξ ( x, E 2 ) W Σ Σ ( x, E 2 )) See details in S.Aoki et al arxiv:6.228 [hep-lat] Product of single baryon correlators ( C Λ Λ RΛ Λ( x, E ) H α W Λ Λ ( x, E ) )) C Λ Λ R Λ Λ ( x, E ) H α W Λ Λ ( x, E ) C Λ Λ R Λ Λ ( x, E 2 ) H α W Λ Λ ( x, E 2

10 Numerical setup 2+ flavor gauge configurations by CP-PACS/JLQCD collaboration. RG improved gauge action & O(a) improved clover quark action β =.83, a - =.632 [GeV], a =.29 [fm] 6 3 x32 lattice, L =.934 [fm]. κ ud =.3825, κ s =.37 was chosen (named Set3). 8 / 8 configurations are used. Flat wall source is considered to produce S-wave B-B state. 6 shifted sources every 2 time-slices are considered to enhance the S/N ratio. The USQCD computer resources are used. Cluster at FNAL We acknowledge the USQCD for providing of computer resources. π K m π /m Κ N Λ Σ Ξ Set3 66± 768± ±3 557±3 576±3 64±3 In unit of MeV

11 Isospin combinations of BB operator ΛΛ, pξ, nξ, Ξ p, Ξ n, Σ + Σ, Σ Σ, Σ Σ +, ΛΣ, Σ Λ I= operators Flavor symmetric N = 2 p 2 n = I= operators N = 2 p 2 n = 2 2 Flavor anti-symmetric I=2 operators =

12 Lists of channels I= states Spin BB channels SU(3) representation S ΛΛ NΞ ΣΣ 8s 27 3 S -- NΞ -- 8a Strong attraction (H-dibaryon) I= states Attraction Strong repulsion Spin BB channels SU(3) representation S NΞ -- ΛΣ -- 8s 27 3 S NΞ ΣΣ ΛΣ 8a * Similar to The NN potential I=2 states Repulsion Spin BB channels SU(3) representation S ΣΣ S

13 Baryon-baryon potential in the flavor SU(3) limit. T. Inoue Accessible by non-strange sector Strong flavor dependence turns out with irreducible rep. Various interaction are seen by extending to SU(3).

14 ΣΣ (I=2) S channel Set 3 : mπ= 66 V Σ Σ Σ Σ Direct correspondence to the 27plet in SU(3) irreducible representation Similar behavior to the NN potential Short range repulsion and mid-range attraction NΞ (I=) 3 S channel Direct correspondence to 8 a plet. V Ν Ξ Ν Ξ Repulsive core is not so high More attractive than 27 plet potential.5.5

15 NΞ, ΛΣ (I=) S channel Set 3 : mπ= 66 V Ν Ξ Ν Ξ V Λ Σ Λ Σ s-27 mixing channel. - Origin of strong repulsion - - V Ν Ξ Λ Σ No attractive pocket in diagonal elements

16 NΞ, ΣΣ, ΛΣ (I=) 3 S channel Set 3 : mπ= 66 V Ν Ξ Ν Ξ V Σ Σ Σ Σ V Σ ΛΣ ΛΣ Σ Σ Σ Relatively small transition potential V Σ Σ Λ Σ V Ν Ξ Σ Σ V Ν Ξ Λ Σ Coupling of ΛΣ state to the other states are quite small.

17 ΛΛ, ΝΞ, ΣΣ (I=) S channel Set 3 : mπ= V Λ Λ Λ Λ Relatively small V Λ Λ Ν Ξ.5.5 V Ν Ξ Ν Ξ Most attractive V Λ Λ Σ Σ V Σ Σ Σ Σ Totally repulsive V Ν Ξ Σ Σ In this channel, our group found the H-dibaryon in the SU(3) limit. T. Inoue [HAL QCD coll.]prl6(2)62.

18 3 S Lists of channels I= operators Spin dependence of potential Spin BB channels SU(3) representation S ΛΛ NΞ ΣΣ 8s 27 3 S -- NΞ -- 8a I= operators Isospin dependence of potential Spin BB channels SU(3) representation S NΞ -- ΛΣ -- 8s 27 3 S NΞ ΣΣ ΛΣ 8a * I=2 operators Spin BB channels SU(3) representation S ΣΣ

19 Spin dependence of NΞ, ΛΣ potentials Set 3 : mπ= Ν Ξ Ν Ξ (I=) S 3 (I=) S Ν Ξ Ν Ξ (I=) S 3 (I=) S Λ Σ Λ Σ (I=) S 3 (I=) S.5.5 Spin triplet potentials are more attractive than the spin singlet potentials The tensor potential is not separated yet In spin triplet channel..5.5

20 Isospin dependence of NΞ, ΣΣ potentials Set 3 : mπ= Ν Ξ Ν Ξ (I=) S (I=) S Ν Ξ Ν Ξ 3 (I=) S 3 (I=) S Σ Σ Σ Σ (I=) S (I=2) S.5.5 In ΝΞ potentials the I= potentials are more attractive than the I= potentials. The short range behavior of potentials are strongly depend on the choice of state..5.5

21 Summaries and outlooks We have investigated the S=-2 BB interactions from lattice QCD. They are complement to experiments. In order to deal with a variety of interactions, we extend our method to the coupled channel formalism. Asymptotic momentum can be determined from time derivative of R-correlator. The source optimization is not necessary in our formalism by employing the time derivative treatment of the energy part. We have found the strong state dependence of ΝΞ potential especially at the short range region. Realistic. potentials with lighter quark masses and large volume Toward the physical point! Look for the physical H-dibaryon. Coupled channel technique is powerful and widely applicable We will tackle to reveal all baryon-baryon interactions with S=, -, -6 below the pion production threshold. By the world's fastest K computer