H-dibaryon & Hyperon interaction

Transcription

1 H-dibaryon & Hyperon interaction from recent Lattice QCD Simulations Takashi Inoue, Nihon Univ. for HAL QCD Collaboration S. Aoki T. Doi T. Hatsuda Y. Ikeda N. Ishii K. Murano H. Nemura K. Sasaki Univ. Tsukuba CNS, Univ. Tokyo U.Tokyo, RIKEN Tokyo Inst. Tech. Univ. Tsukuba RIKEN Univ. Tsukuba Univ. Tsukuba JAEA, Sep 9, 0

2 Introduction H-dibaryon: predicted compact 6-quark state R. L. Jaffe, Phys. Rev. Lett. 8 (977) [MIT Bag model] One of most famous candidates of exotic-hadron. No Pauli exclusion, Flavor singlet, Attraction from OGE Does H exists in nature? BH > 7 MeV is ruled out by ΛΛHe. Possibility of a shallow bound state or a resonance remain. Hyperon interaction (YN, YY int.) are important for neutron-star, super-nova phys. and so on. however, are not well known due to lack of experimental data. Our purpose and goal. We reveal BB int., including existence of the H-dibaryon, directly from QCD by using lattice simulation.. We get deeper(or intuitive) understanding of BB int.. People can apply them to many physics (super-nova etc).

3 Plan of this talk Introduction Formulas & Setup NBS w.f. and Potential Lattice, Action and Facility, Five ensembles Results Brief background, Our purpose and goal Lattice QCD Simulation Hadron system by LQCD Our approach BB int. at the SU()F limit H-dibaryon Hyperon interactions Summary & Outlook

4 Lattice QCD Simulation a μν μ a a L = Gμ ν G a + q γ ( i μ g t A μ ) q m q q 4 quarks q on the sites gluons U = e i a Aμ on the links Vacuum expectation value O( q, q,u ) path integral = du d q d q e S (q, q, U ) O( q, q,u ) 4-dim Euclid Lattice L = du det D(U )e S = lim N N U (U ) O(D (U )) N i= O( D (U i )) quark propagator { Ui } generated w/ probability det D(U) e SU(U) a Well defined (reguralized) Manifest gauge invariance Fully non-perturvative Highly predictable 4

5 Hadron system by LQCD Conventional use energy eigenstate (eigenvalue) Lüscher's finite volume method for phase-shift Infinite volume extrapolation to get bound state energy HAL use the potential V(r) +... from the NBS w.f. Solve effective theory which reproduce T matrix from QCD Advantages No need to separate E eigenstate. Just need to measure NBS w.f. Then extract the potential. Demand a minimal lattice volume. No need to extrapolate to V=. Can output many observables. 5

6 Hadron system by LQCD Conventional use energy eigenstate (eigenvalue) Lüscher's finite volume method for phase-shift Infinite volume extrapolation to get bound state energy HAL use the potential V(r) +... from the NBS w.f. Solve effective theory which reproduce T matrix from QCD Advantages No need to separate E eigenstate. Just need to measure NBS w.f. Then extract the potential. Demand a minimal lattice volume. No need to extrapolate to V=. Can output many observables. saturation NG 6

7 Hadron system by LQCD Conventional use energy eigenstate (eigenvalue) Lüscher's finite volume method for phase-shift Infinite volume extrapolation to get bound state energy HAL use the potential V(r) +... from the NBS w.f. Solve effective theory which reproduce T matrix from QCD Advantages No need to separate E eigenstate. Just need to measure NBS w.f. Then extract the potential. Demand a minimal lattice volume. No need to extrapolate to V=. Can output many observables. 7

8 Hadron system by LQCD Conventional use energy eigenstate (eigenvalue) Lüscher's finite volume method for phase-shift Infinite volume extrapolation to get bound state energy HAL use the potential V(r) +... from the NBS w.f. Solve effective theory which reproduce T matrix from QCD Advantages No need to separate E eigenstate. Just need to measure NBS w.f. Then extract the potential. Demand a minimal lattice volume. No need to extrapolate to V=. Can output many observables. 8

9 Hadron system by LQCD Conventional use energy eigenstate (eigenvalue) Lüscher's finite volume method for phase-shift Infinite volume extrapolation to get bound state energy HAL use the potential V(r) +... from the NBS w.f. Solve effective theory which reproduce T matrix from QCD Advantages No need to separate E eigenstate. Just need to measure NBS w.f. Then extract the potential. Demand a minimal lattice volume. No need to extrapolate to V=. Can output many observables. Need to check validity of the leading term V(r) 9

10 Hadron system by LQCD Conventional use energy eigenstate (eigenvalue) Lüscher's finite volume method for phase-shift Infinite volume extrapolation to get bound state energy HAL use the potential V(r) +... from the NBS w.f. Solve effective theory which reproduce T matrix from QCD Advantages No need to separate E eigenstate. Just need to measure NBS w.f. Then extract the potential. Demand a minimal lattice volume. No need to extrapolate to V=. Can output many observables. Need to check validity of the leading term V(r) We can probe the H in this approach! (as far as H has a baryonic component) 0

11 Formulas and Setup

12 Nambu-Bethe-Salpeter w.f. NBS wave function def (a) the same 0 Bi ( x + r, t) B j ( x, t) B=, a-plet ψ ( r, t) = x QCD eigenstate G ( x + r, x, t) (a) x (a) x, y, t t 0) = 0 Bi ( x,t ) B j ( y,t ) BB 4-point function G ( sink (t 0 ) 0 source Point type octet baryon field operator at sink p α (x ) = ϵ c c c (C γ5 )β β δβ α u(ξ )d (ξ )u(ξ ) Λα ( x) = ϵ c c c (C γ5 )β β δβ α (a) with ξ i={c i, βi, x } [ d (ξ) s(ξ)u(ξ)+s(ξ)u(ξ )d(ξ ) u(ξ)d (ξ) s(ξ)] 6 Quark wall type BB source in the flavor irreducible rep. e.g for flavor-singlet BB () = 4 ΛΛ + ΣΣ + ΝΞ 8 8 8

13 Potential S. Aoki, T. Hatsuda, N. Ishii, Prog. Theo. Phys. 89(00) N. Ishii etal. [HAL QCD coll.] in preparation EGr t NBS wave function ψ( r,t) = ϕgr ( r ) e Est t + ϕst ( r ) e Euclidian space-time Schrödinger eq. of energy-eigen-compnent [ [ ] ] [ E t E MB ϕgr ( r )e + d r ' U ( r, r ') ϕgr ( r ' )e μ Gr E MB ϕst ( r )e μ By adding equations st t + d r ' U ( r, r ') ϕst ( r ' )e t = E Gr ϕgr ( r )e Est t = E st ϕst ( r ) e Gr EGr t E st t Non-local but energy independent ] MB ψ( r, t) + d r ' U ( r, r ' ) ψ( r ', t) = ψ( r,t) μ t expansion U ( r, r ') = δ( r r ')V ( r, ) = δ( r r ')[V ( r ) ] & truncation ψ( r,t ) Therefor, in ψ( r, t ) t the leading V ( r ) = μ ψ( r,t) ψ( r,t ) M B must be t-indep. (non-trivial) t-derivative at each r different from previous method

14 Lattice, Action and Facility β.8 a [fm] 0.() Lattice x L [fm].87 Renormalization group improved Iwasaki gauge and Non-perturbatively O(a) improved Wilson quark We thank K.-I. Ishikawa and the PACS-CS group for providing their DDHMC/PHMC code to generate gauge configuration, and the Columbia Physics System for their lattice QCD simulation code. We enhance S/N of data by averaging on 4x4=6 source, and forward/backward propagation in time. All numerical computation carried at the supercomputer system TK-Tsukuba. 4

15 Five ensembles T.I. et.al. Phys. Rev. Lett. 06, 600(0) K_uds N_cfg M_P.S. [MeV] M_Vec [MeV] M_Bar [MeV] (7) 50.4(0.9) 74() (6) 60.6(.) 0() (5) 88.9(0.9) 749() (6) 07.6(.0) 484() (8) 80.6(.5) 6() relative to New add cfg New We've made five ensembles with different hopping parameter, (=quark mass) corresponding to MPS =.7 [GeV] to 470 [MeV]. With lightest quark (bottom row of the table), p.s. meson is a little lighter than the physical kaon. baryon is a little lighter than the physical sigma baryon. Now, the simulated hadron world is not so far from the real world, although the SU()F breaking is not taken into account. 5

16 Results 6

17 BB int. at SU()F limit In the limit, convenient basis exist to describe the int. 8 8 = 7 + 8s * a flavor irreducible rep. Symmetric In S-wave, no off-diagonal interaction exists. S 0 : V (7) (r), V (8s) (r), V () (r) Fermi statistics leads to Anti-symmetric S : V (0 *) (r), V (0) (r), V (8a) (r) Six V(a)(r) contain essential flavor-spin structure of BB int. We can reconstruct all baryon-base interaction (eg. ΛN) by using these V(a)(r) with SU() C.G. coefficients. The V(a) is useful to pin down physical origin of particular feature, since effective models assume flavor symmetry. 7

18 BB int. at SU()F limit S0 S D u+d u+d+s At the SU()F limit corresponding to Mπ = MK = 87 [MeV]. QM is true at small r. Especially, no repulsion in F channel. This indicate possibility of a bound H-dibaryon in the limit. 8

19 H-dibaryon 9

20 NBS w.f. & potential Left: Measured NBS w.f. of the F channel The finite value at large distance is excited states contribution as well as a finite volume effect. (demonstrated in later) Right: Extracted potential of the F channel V()(r) become more attractive as quark mass decrease. 0

21 Observables Preliminary Left: scattering phase shift v.s. Ecm shows existence of one discrete state below threshold. Right: obtained ground state which is 0-50 MeV below from free BB ie. q-q. This means that there is a 6-quark bound state in the F channel. A stable(bound) H-dibaryon exists in these SU()F limit world!

22 Size of H-dibaryon Left: Wave function of the lowest two states of V(). the measured NBS w.f. is a superposition of red and green with the finite volume effect. Right: Comparison between the H and the physical deuteron. One can get feeling of the H-dibaryon: It is compact.

23 Hyperon interaction

24 Potential in baryon-base In flavor SU() broken world, e.g. the physical one, the baryon-basis are used instead of flavor-basis. In the SU() limit, the baryon-base potential Vij(r) can be obtained by a unitary rotation of the potential V(a)(r). e.g. S=, I=0 sector Λ Λ ( ) Σ Σ = Ξ N ( ) ( 7 U 8, U coupled channel V (7) ) ( = U () t V (8) V V Λ Λ V ΛΣΛΣ V ΛΞΛN V Σ Σ V ΣΞΣN ΞN V ) I show you potentials Vij(r) at the lightest quark mass (Kuds = 0.840, Mps=469 MeV) obtained with the V(a) in an analytic function fitted to data. 4

25 Uncoupled (exclusive) channels Some BB channels belong to a flavor irr-rep. exclusively. Potential of such BB channel is nothing but V(a)(r). S=0 and S= 4 BB channels are completely exclusive. Large uncertainty 5

26 S=, I=/ sector NΛ - NΣ(I=/) coupled. S0 VNΛ has attractive well. VNΣ is strongly repulsive. JP = + Both have an attractive well. NΛ - NΣ strongly coupled. Off-diagonal VT is stronger. 6

27 YN scattering in S=, I=/ Phase-shift and mixing-angle as a function of energy flavor SU() limit small SU() breaking from V(r) at (Kud, Ks)=(0.760, 0.760) and MB at (0.760, 0.70) where MN = 79., MΛ = 87.0, MΣ = 8.8, MΞ = [MeV] Effective central potential is used for S channel. Essential features must remain at the physical point. Future experiment will confirm them. Exciting! 7

28 S=, I=0 sector ΛΛ ΝΞ ΣΣ coupled. Left: diagonal. Right: Off-diagonal. It is flavor symmetric(spin singlet), and involve flavor singlet ch. Channel coupling int. are comparable to diagonal ones, except for the ΛΛ - ΝΞ transition (small sign change in VΛΛ-ΝΞ must be artifact). Interaction is most attractive in NΞ channel, although it has not much real meaning because channel coupling is strong. 8

29 S=, I= sector ΝΞ ΣΛ ( ΣΣ) coupled. S0 VNΞ VΣΛ, moderate coupling JP = + Need to solve 6-channel-coupled Schrödinger eq. for observables. 9

30 S=, I=/ sector ΛΞ ΣΞ coupled. S0 VΛΞ = VΛN, VΣΞ = VΣN JP=+ strong channel coupling by VC. hyperon-changing VT vanishs. 0

31 Summary We've introduced motivation and purpose. We want to explore for the H-dibaryon in QCD by using Lattice. We start with flavor SU() limit to avoid complication. We've explained our approach. We'll extract observables exclusively through the interaction potential. By using both time and spacial derivative of the NBS w.f. at each point, we can extract the potential even without the ground-state-saturation. We've tested and confirmed our new technique. (t-indep. and L-indep.) We've done full QCD lattice simulation to probe the H. x lattice, L=.87 [fm], Iwasaki gauge, clover quark We've found a bound=stable H-dibaryon in flavor SU() symmetric world at MPS = 470 [MeV].7 [GeV], its binding energy is 0 50 [MeV] depending on the quark mass, and size is compact compared to usual B= system.

32 Summary & Outlook Plot of H binding energy from recent full QCD simulations. SR. Beane etal [NPLQCD colla.] Phys. Rev. Lett. 06, 600 (0) Obtained binding energy from the two groups looks consistent. But, all data points are still away from the physical point. We'll continue this study and put more data on this plot. We'll take flavor SU() breaking into account soon. We'll get information of H-dibaryon in the physical world in near future.

33 Thank You!

34 Backup slides 4

35 Binding energy of H Choice of time slice t = 9, 0, [a] All results agree within statistical error. Error from choice of fit function is less than few % ie. negligible. Final results for biding energy of H-dibaryon: m P.S. = 05 MeV : B H = 7. (.7) (.4) MeV m P.S. = 87 MeV : B H = 7.8 (.) (4.) MeV mp.s. = 67 MeV : B H =.6 (4.8) (.5) MeV 5

36 FAQ. Does the potential depend on the choice of source?. Does the potential depend on choice of field operator at sink?. Does the potential U(x,y) or V(r) depend on energy? 6

37 FAQ. Does the potential depend on the choice of source? No. Some sources may enhance excited states in NBS w.f. However, the new method has no problem with excited states at all.. Does the potential depend on choice of field operator at sink? Yes. It can be regarded as the scheme to define potential. Note that the potential itself is not physical observable. We'll obtain unique result for observables irrespective to the choice, as long as the potential U(x,y) is deduced exactly.. Does the potential U(x,y) or V(r) depend on energy? By construction, U(x,y) is non-local but energy independent. While, its leading order truncation V(r) obtained here, is local but velocity dependent. However, we know that the dependence in NN case is very small and negligible at least at E = 0 45 MeV. If we get dependence, we''ll determine the next leading terms from it. 7

38 FAQ 4. Do you think energy dependence of V()(r) is also small? 5. Is the H a compact six-quark object or a tight BB bound state? 8

39 FAQ 4. Do you think energy dependence of V()(r) is also small? Yes. Because a large energy dependence means that [ [ ] ] E M B + V Gr ( r ) ϕgr ( r )e μ E MB + V st ( r ) ϕst ( r )e μ then Gr t = E Gr ϕgr ( r )e E Gr t st t = E st ϕst ( r )e E st t ψ( r, t) t ψ( r, t ) V ( r ) MB μ ψ( r, t) ψ( r, t ) would have a large t-dep. 5. Is the H a compact six-quark object or a tight BB bound state? Both. There is no distinct separation between two, because baryon is nothing but a -quark in QCD. Imagine a compact 6-quark object in (0S)6 configuration. This configuration can be rewritten in a form of (0S) (0S) Exp( a r) with relative coordinate r. This shows that a compact six-quark object can have a baryonic component, which we measure in the NBS w.f. We've established existence of a stable QCD eigenstate which couples to BB state. We do NOT insist that another H doesn't exist which cannot couple to BB. 9

40 FAQ 6. Do you insist such a deeply bound H exists in the real world? 7. What is the meaning of r of H? 40

41 FAQ 6. Do you insist such a deeply bound H exists in the real world? No. With SU()F breaking, three BB thresholds in S=,I=0 sector Th split as E ΛThΛ < E Th < E Σ Σ. Therefore, we expect that the binding NΞ Th energy of H measured from E Λ Λ is much smaller than the present value, or even H is above E ΛThΛ in the real world. 7. What is the meaning of r of H? It is a measure of spacial distribution of baryonic matter in H. It corresponds to the point matter root mean square distance of deuteron (= x.9 [fm]). 4

42 Problem Free -body energy spectrum in finite volume w/ periodic B.C. nx, y, z π px, y, z = L then π K nx, ny, nz = (n x +n y +nz ) L μ ( ) e.g. in L = [fm], μ = M = 750 [MeV] case, 0 [MeV] therefor EGr = M + 0, E st = M + 0, E nd = M Even with interaction, spectrum is essentially the same. State realized in lattice at time t t = Gr + st EGr t NBS w.f. ψ( r, t ) = ϕgr ( r )e Est t + ϕst ( r )e Depending on Δ E E st E Gr, if t is large enough EGr t ψ( r, t) ϕgr ( r )e Ground State Saturation Exponential tail On L = [fm] lattice, G.S.S. is realized at t 0 [a]. 4

43 SU() Octet Baryon operator p = c c c C 5 d d d u d u n = c c c C 5 d d d u d d = c c c C 5 d d d u s u 0 = c c c C 5 d d d = c c c C 5 d d d = c c c C 5 d d d [ d s u u s d ] d s d d s u s u d u d s ] [ 6 0 = c c c C 5 d d d s u s = c c c C 5 d d d s d s With corrected phase = ϵ = (ds sd)=sd ds 4

44 Irreducible BB source operator BB 7 = 7 N BB 8s = N BB = 4 N = BB 0* = pn n p BB 0 = p + + p BB 8a = pn n p with N = or p p n n p p n 0 n

45 Various Theoretical Approaches to Nuclei PWA, meson-ex. etc. Exp. Data Potential Shell Model, Variational etc. Lagrangian of N Exp. Data Fix LEC (QCD) Fix LEC χpt Lagrangian of N Lagrangian of N Potential Lattice BS eq. Nuclei (n 4) Lattice QCD Lattice Nuclei and Nuclear Matter Potential Shell Model Variational etc Nuclei and N.M.