J/ψ-Φ interaction and Y(4140) on the lattice QCD

J/ψ-Φ interaction and Y(4140) on the lattice QCD Sho Ozaki (Univ. of Tokyo) in collaboration with Shoichi Sasaki (Univ. of Tokyo) Tetsuo Hatsuda (Univ. of Tokyo & Riken)

Contents Introduction Charmonium(J/ψ)-Φ scattering and Y(4140) Bottomonium-hadron scattering (Preliminary) Summary

Introduction Recently many charmonium(cc) like particles XYZ are observed in several big facilities in the world. Among them, some Y resonances have interesting features. 1) Although these resonances are heavy, these are very stable. Widths are quite narrow as compared to typical hadron resonances. 2) Open charm channel decays seem to be suppressed. Y D D

Introduction Recently many charmonium(cc) like particles XYZ are observed in several big facilities in the world. Among them, some Y resonances have interesting features. 1) Although these resonances are heavy, these are very stable. Widths are quite narrow as compared to typical hadron resonances. 2) Open charm channel decays seem to be suppressed. Y D D

Example 1) Initial State Radiation(ISR)-produced 1^-- Y families (including Y(4260)): e + e γ γ Y J/ψ ππ many Y states e + e γ γ D D No such Y states

Example 2) Y(4140). T. Aaltonen et al, PRL 102, 242002 (2009) B J/ψφK Y (4140) M Y = 4143.0 ± 2.9 ± 1.2MeV Γ Y = 11.6 +8.3 5.0 ± 3.7MeV quite narrow width 23MeV 60MeV 200MeV Y (4140) J/ψφ D + s D s D + s D s It seems that some Y states do not couple to open charm channels. Is there a specific selection rule? These features should be related to the structure of Y states, and charmonium(j/ψ)-hadron interactions.

Example 2) Y(4140). T. Aaltonen et al, PRL 102, 242002 (2009) B J/ψφK Y (4140) M Y = 4143.0 ± 2.9 ± 1.2MeV Γ Y = 11.6 +8.3 5.0 ± 3.7MeV quite narrow width 23MeV 60MeV 200MeV Y (4140) J/ψφ D + s D s D + s D s It seems that some Y states do not couple to open charm channels. Is there a specific selection rule? These features should be related to the structure of Y states, and charmonium(j/ψ)-hadron interactions.

Charmonium-hadron interactions J/ψ vacuum quantum number π ρ, exchanges are forbidden! hadron

Charmonium-hadron interactions J/ψ gluon exchange dominance hadron So if there exist a (Y) resonance in charmonium-hadron system, gluons would play very interesting role! Non-perturbative method such as lattice QCD is really needed to study this system.

J/ψ-Φ scattering and Y(4140) E 20 MeV J/ψ J/ψ low energy scattering φ Y (4140) φ with J P = (0 ±, 1 ±, 2 ± ) channels on QCD Today, we focus on s-wave: J P = (0 +, 1 +, 2 + )

Finite size method M. Luscher, NPB354 (1991) 531-578 Finite size formula tan δ 0 (k) = π3/2 q Z 00 (1,q 2 ) q = Lk 2π Generalized zeta-function Z 00 (s, q 2 )= 1 4π n Z 3 (n 2 q 2 ) s Finite size formula is the relation which connects energy eigenvalue in a finite volume with scattering phase shift in an infinite volume. This method successfully describe ρ meson from π-π scattering. S. Aoki et al (CP-PACS) PRD 76, 094506 (07)

Finite size method M. Luscher, NPB354 (1991) 531-578 Finite size formula tan δ 0 (k) = π3/2 q Z 00 (1,q 2 ) q = Lk 2π Generalized zeta-function Z 00 (s, q 2 )= 1 4π n Z 3 (n 2 q 2 ) s Finite size formula is the relation which connects energy eigenvalue in a finite volume with scattering phase shift in an infinite volume. We would like to successfully describe Y(4140) from J/ψ-Φ scattering in terms of the finite size method.

k = 2µ E E = E J/ψ φ (M J/ψ + M φ ) E nv = 2π 2 L n + MV 2 =[E J/ψ φ (E nj/ψ + E nφ )] + [E nj/ψ M J/ψ ]+[E nφ M φ ] E = δe n + nj/ψ + nφ Interaction strength Energy of free 2 particles n =2 δe 2 2J/ψ + 2φ n =1 δe 1 1J/ψ + 1φ n =0 δe 0 0: J/ψ φ threshold

Measurement of δe n Two-point function G φ (t, t src )=Ôφ(t)Ô φ (t src) with G J/ψ (t, t src )=ÔJ/ψ(t)Ô J/ψ (t src) Four-point function G J/ψ φ (t, t src )=Ôφ(t)ÔJ/ψ(t)[Ôφ(t src )ÔJ/ψ(t src )] Ô φ (t) = s(t)γ i s(t) Ô J/ψ (t) = c(t)γ i c(t) G J/ψ φ (t, t src ) G J/ψ (t, t src )G φ (t, t src ) e δe nt J/ψ δe n J/ψ φ ( ) φ

Twisted Boundary Condition Periodic Boundary Condition P.F. Bedaque, PLB593 (2004) 84 φ(x + L i )=φ(x), 2π k = L n E 1 = k 2 1/2µ = 115 MeV i = x, y, z Bad resolution Twisted Boundary Condition φ(x + L i )=e iθ i φ(x) k = 2π L (n + d), d =( θ x 2π, θ y 2π, θ z 2π )

Lattice set up PACS-CS 2+1 flavor dynamical gauge configurations at m π = 156 MeV 32^3 x 64 lattice a = 0.0907(13) fm La ~ 2.9 fm S. Aoki et al, PRD79, 034503, 2009 198 configs κ s =0.13640 Wall source Relativistic Heavy Quark (RHQ) action for charm Y. Namekawa et al, PRD84:074505, 2011 Tsukuba type RHQ action (5 parameters) κ charm ν r s c B c E

Result of 0 δe J =0 δe δe J=0-0.001-0.002 0 10 20 30 40 50 60 ΔE [MeV] E = E (M J/ψ + M φ ) J/ψ-Φ interaction is attractive. The strength of the interaction is E-independent.

Scattering phase shift J =0 0.25 δ0 [rad] δ 0 J=0 (k) 0.2 0.15 0.1 0.05 0 0 10 20 30 40 50 60 ΔE [MeV] E = E (M J/ψ + M φ )

J =1 J =2 0.25 0.25 δ0 [rad] 0.2 δ 0 J=1 (k) 0.15 0.1 0.2 δ 0 J=2 (k) 0.15 0.1 0.05 0.05 0 0 10 20 30 40 50 60 0 0 10 20 30 40 50 60 ΔE [MeV] ΔE [MeV] E = E (M J/ψ + M φ ) These seem typical s-wave behaviors: δ l ( E) l+ 1 2 No structure in s-wave J/ψ-Φ system.

J =1 J =2 0.25 0.25 δ0 [rad] 0.2 δ 0 J=1 (k) 0.15 0.1 0.2 δ 0 J=2 (k) 0.15 0.1 0.05 0.05 0 0 10 20 30 40 50 60 0 0 10 20 30 40 50 60 ΔE [MeV] ΔE [MeV] E = E (M J/ψ + M φ ) These seem typical s-wave behaviors: δ l ( E) l+ 1 2 No structure in s-wave J/ψ-Φ system. We will study P-wave J/ψ-Φ scattering as a next step. [ Our homework ]

Scattering lengths a J=0 J/ψ φ = 0.151(20) fm Our definition a J=1 J/ψ φ a J=2 J/ψ φ = 0.130(18) fm = 0.109(18) fm tanδ 0 k k 0 = a J/ψ φ a J/ψ φ < 0: attractive Quenched results a J=0 J/ψ φ a J=1 J/ψ φ a J=2 J/ψ φ = 0.178(21) fm = 0.152(23) fm = 0.123(16) fm Quenched results and full QCD results are quite close within 1σ. This reflects that J/ψ-Φ system is indeed governed by gluon-dynamics.

Charmonium c c Φ meson s s

Charmonium c c Φ meson s s Bottomonium b b

Charmonium c c Φ meson s s Bottomonium b b

Charmonium c c Φ meson s s Bottomonium b b Results are preliminary.

Result of δe J =0 0 φ Υ 0 J/ψ Υ δe -0.0002-0.0004-0.0006 δe J=0-0.0008 0 10 20 30 40 50 0 5 10 15 20 ΔE [MeV] -0.0001-0.0002 E

δ0 [rad] 0.25 0.15 0.2 0.1 0.05 0 0 10 20 30 40 50 0.25 0.15 0.2 0.1 0.05 0 0 10 20 30 40 50 0.25 0.2 0.15 0.1 0.05 φ Υ δ0 [rad] 0.2 δ 0 J=0 (k) δ 0 J=1 (k) 0.15 0.1 0.05 0 0.15 0.05 0 bottom-charm 0 5 10 15 20 ΔE [MeV] bottom-charm 0 5 10 15 20 ΔE [MeV] 0 0 10 20 30 40 50 0 0 5 10 15 20 ΔE [MeV] 0.2 0.1 E 0.2 δ 0 J-=2 (k) 0.15 0.1 0.05 J/ψ Υ bottom-charm

Summary We investigate the low energy s-wave J/ψ-Φ scattering with PACS-CS 2+1 dynamical gauge configurations (mπ=156 MeV). Their interactions are attractive, but no E-dependence. In terms of the finite size method, we calculate scattering phase shifts, but there is no resonance in s-wave systems. We also calculate scattering lengths, and compare with quenched results. gluon-dynamics seem important in this system. Bottomonium scatterings are now ongoing.

Prospects As a next step we will perform P-wave calculations ( J^P = ( 0^-, 1^-, 2^-) ) and search Y(4140) resonance. If Y(4140) resonance exist on the lattice... Determine parity and spin of the resonance. Construct J/ψ-Φ potential, and investigate the structure of Y(4140) resonance in term of its BS wave function.