J/ψ-Φ interaction and Y(4140)
onthe 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 areobserved 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.
YDD̄Introduction
Recently many charmonium(cc) like particles XYZ areobserved 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.
YDD̄Example 1) Initial State Radiation(ISR)-produced 1^-- Yfamilies (including Y(4260)):eγ+γe−e+e−YJ/ψmany Y statesππγγDD̄

No such Y states

Example 2) Y(4140).

T. Aaltonen et al, PRL 102, 242002 (2009)

B → J/ψφK

MY = 4143.0 ± 2.9 ± 1.2MeV

23MeV ∼Y (4140)
J/ψφY (4140)

ΓY = 11.6+8.3−5.0 ± 3.7MeV quite narrow width

60MeV ∼200MeV ∼Ds+ Ds∗−
Ds+ Ds−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

MY = 4143.0 ± 2.9 ± 1.2MeV

23MeV ∼Y (4140)
J/ψφY (4140)

ΓY = 11.6+8.3−5.0 ± 3.7MeV quite narrow width

60MeV ∼
200MeV ∼?

Ds+ Ds∗−Ds+ Ds−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 areforbidden!

hadronCharmonium-hadron interactions

J/ψgluon exchange
dominance

hadronSo 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)

J/ψ
E ∼ 20 MeV

low energy
scatteringJ/ψ
Y (4140)φφ

with

J P = (0± , 1± , 2± ) channels

onQCD
Today, we focus on s-wave: J P = (0+ , 1+ , 2+ )Finite size method

M. Luscher, NPB354 (1991) 531-578

Finite size formula

π 3/2 qtan δ0 (k) =Z00 (1, q 2 )Lkwhere q =2πGeneralized zeta-function
�1Z00 (s, q 2 ) = √(�n2 − q 2 )−s4π �n∈Z 3
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 formulaπ 3/2 qtan δ0 (k) =Z00 (1, q 2 )Lkwhere q =2πGeneralized zeta-function�
1Z00 (s, q 2 ) = √(�n2 − q 2 )−s4π �n∈Z 3Finite 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-Φscattering in terms of the finite size method.

�k = 2µ∆E

EnV =∆E = EJ/ψ−φ − (MJ/ψ + Mφ )��2πnL�2+ MV2= [EJ/ψ−φ − (EnJ/ψ + Enφ )] + [EnJ/ψ − MJ/ψ ] + [Enφ − Mφ ]∆E = δEn + �nJ/ψ + �nφInteraction strengthn=2n=1n=0

δE2

δE1

δE0

Energy of free 2 particles

�2J/ψ + �2φ

�1J/ψ + �1φ0 : J/ψ − φ threshold

Measurement of δEn
Two-point function
Gφ (t, tsrc ) = �Ôφ (t)Ôφ† (tsrc )�

†GJ/ψ (t, tsrc ) = �ÔJ/ψ (t)ÔJ/ψ
(tsrc )�withÔφ (t) = s̄(t)γi s̄(t)ÔJ/ψ (t) = c̄(t)γi c̄(t)Four-point function

GJ/ψ−φ (t, tsrc ) = �Ôφ (t)ÔJ/ψ (t)[Ôφ (tsrc )ÔJ/ψ (tsrc )]† �GJ/ψ−φ (t, tsrc )−δEn t∼eGJ/ψ (t, tsrc )Gφ (t, tsrc )δEn =J/ψ

φ-(J/ψ+φ)Twisted Boundary Condition

P.F. Bedaque, PLB593 (2004) 84

Periodic Boundary Condition

φ(�x + L��i ) = φ(�x) ,�k = 2π �nLi = x, y, z

E1 = k12 /2µ = 115 MeV

Bad resolution

Twisted Boundary Condition
φ(�x + L��i ) = eiθi φ(�x)�k = 2π (�n + d)
�,Lθx θy θz�d=( ,,)2π 2π 2π

Lattice set up
PACS-CS 2+1 flavor dynamical gauge configurations
at mπ = 156 MeV
S. Aoki et al, PRD79, 034503, 200932^3 x 64 latticea = 0.0907(13) fm
La ~ 2.9 fm
198 configs

κs = 0.13640
Wall source
Relativistic Heavy Quark (RHQ) action for charm
Y. Namekawa et al, PRD84:074505, 2011

Tsukuba type RHQ action (5 parameters)κcharm

ν

rscB

cE

0.10821.2153

1.21312.02681.7911

Result of δE
J =0-0.001-0.001
δEJ=0δE0 0Y(4140) energy-0.002-0.00200 010202030
ΔE [MeV]

4040506060∆E = E − (MJ/ψ + Mφ ) MeV

J/ψ-Φ interaction is attractive.
The strength of the interaction is E-independent.

Scattering phase shift
0.3Y(4140) energy

J =0(k)0.20.2
J=00.15δ0δ0 [rad]0.25

0.10.10.05

00 0010202030
ΔE [MeV]

404050∆E = E − (MJ/ψ + Mφ ) MeV6060

0.25

0.2
0.2

0.20.2(k)J=20.15

J =20.15

δ0J=10.25(k)

0.3

δ0

δ0 [rad]0.3

J =10.10.1

0.1
0.10.10.05

0.0500 0010202030
ΔE [MeV]

404050606000 0010202030
ΔE [MeV]

404050∆E = E − (MJ/ψ + Mφ ) MeV

These seem typical s-wave behaviors: δl ∼ (∆E)
No structure in s-wave J/ψ-Φ system.

l+ 126060

0.25

0.2
0.2

0.20.2(k)J=20.15

J =20.15

δ0J=10.25(k)0.3δ0δ0 [rad]0.3

J =10.1
0.1

0.1
0.10.10.05

0.0500 0010202030
ΔE [MeV]

404050606000 0010202030ΔE [MeV]

40405060∆E = E − (MJ/ψ + Mφ ) MeV

These seem typical s-wave behaviors: δl ∼ (∆E)
No structure in s-wave J/ψ-Φ system.

60l+ 12

We will study P-wave J/ψ-Φ scattering
as a next step. [ Our homework ]Scattering lengths
aJ=0J/ψ−φ = −0.151(20) fm

aJ=1J/ψ−φ = −0.130(18) fm
aJ=2J/ψ−φ = −0.109(18) fm

Our definitiontanδ0−|k→0 = aJ/ψ−φkaJ/ψ−φ < 0 : attractive

Quenched results
aJ=0J/ψ−φ = −0.178(21) fm
aJ=1
J/ψ−φ = −0.152(23) fm

aJ=2J/ψ−φ = −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.

Charmoniumcc̄Φ mesonss̄Charmoniumcc̄Φ mesonBottomonium

ss̄

bb̄

Charmoniumcc̄
Φ mesonBottomonium

ss̄

bb̄

Charmoniumcc̄
Φ mesonBottomonium

ss̄

bb̄

Results are preliminary.

Result of δE

J =0
φ−Υx10^-400J/ψ − Υx10^-400-0.0002-1-4δEJ=0δE-0.0001-0.0004yranim-2ile-0.0002

Pr-0.0006-80-0.0008

01010202030304040-350
0500∆E MeV

5510
ΔE [MeV]

1015152020

φ−Υ

0.3J/ψ − Υbottom-charm

0.2
0.2

0.25
0.15

0.2

0.2(k)

0.1
0.1

δ0J=00.15

0.1

0.1
0.05

0.05

01020

30400.3
0.25

0.2

0.2

Pr0.1

0.1

0.05

000510ΔE [MeV]152015201520

bottom-charm

0.2

0.2yr
ani
0.15

01020304050(k)0.1
miel0.1J=10.15

0050

δ00

δ0 [rad]0.05

000510ΔE [MeV]bottom-charm

0.3

0.2

0.2

0.25

0.15

0.2

0.2

J-=2(k)

0.1

0.15

0.1

δ0

δ0 [rad]

00.1

0.1

0.05

0.05

00

00

1020

30254050

500000∆E MeV

510ΔE [MeV]1020

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 phaseshifts, but there is no resonance in s-wave systems.
We also calculate scattering lengths, and compare withquenched 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 ofY(4140) resonance in term of its BS wave function.