Low energy D*+D10 scattering and
charmonium radiative decay from
lattice QCD
Institute of Theoretical Physics
Peking University
Chuan Liu
Outline

Low-energy D*D1 scattering and the
nature of Z(4430)(PRD80:034503, 2009)
 J/psi
radiative decay to scalar
glueball (still going…)
 Other works…

Summary
Basic status of the
collaboration

People







Funding


Y. Chen (IHEP, CAS)
C. Liu (Peking University)
Y.B. Liu (Nankai University)
J.P. Ma (ITP, CAS)
J.B. Zhang (Zhejiang University)
Postdocs(Yi-bo Yang)+students
Funded by NSFC, etc.
Computing facilities: Shanghai, Tianjin, Beijing…
D*D1 scattering and Z(4430)
Signal for Z(4430)
S.K. Choi et al., PRL 100,142001 (2008)

Resonance
structure Z(4430)

Close to threshold D* D10
JP=0-,1-,2 M=4433 MeV
 G=45 MeV
 Q=+1,
Theoretical investigations
 Phenomenology
 Shallow
bound state (S.L. Zhu et al, PRD77,034003)
 Tetra-quark resonance above threshold (X.-H Liu et al, PRD77,
094005)
 Threshold enhancement (J.L. Rosner, PRD76,114002)
 Need
to study scattering near threshold
 Scattering
length a0 and effective range r0
 Lattice QCD study (quenched)
Utilizing Lüscher’s formalism
 Consider
two hadrons in a finite box
 interaction：shift of two-particle energy
 interaction： scattering phase shifts
 Lüscher’s formula:
 ( E12 )  E12 ( L)
Basic formulae

A box of size L, periodic in all three spatial directions:
k  (2 / L) n , n  Z
 Two interacting hadrons
3
E12 (k )  m  k  m  k , q  k L /(2 )
2
1
2
2
2
2

3/ 2
2
q
tan  (q ) 
2
Z 00 (1; q )
2 2
2
Taking advantage of asymmetric box
Use asymmetric volumes: L×(h2 L)×(h3 L)
 Take: h2=1, h3>1
 The symmetry for the box is D4


qh2h3
tan  (q) 
2
Z 00 (1; q ;h2 ,h3 )
3/ 2
Single hadron operators

Single particle operators
D*+

Angular momentum decomposition
D10
Double hadron operators
Correlation matrix

Only the correlation matrix in A1 channel shows signal

We have computed 5 lowest non-zero momentum
modes together with the zero momentum mode.
Simulation setup

Tadpole improved clover action on anisotropic lattices
Checking single particle states
Extract two-hadron energy

Appropriate
ratios are taken:
R (t ) 
i (t )
C D* (t )C D1 (t )
 Eieff  E12  mD*  mD1
Obtaining a0 & r0

Scattering length a0 &
effective range r0
extrapolations
After all the
extrapolations

discussions

Possible bound state?
 For
a0  
shallow bound state,
 But our scattering lengths are all positive
 On the verge of developing a bound state
 Using the square well potential model to estimate,
the potential well is V0=70(10) MeV, R=0.7fm.

Our results is not in favor of a shallow bound
state
Glueballs in J/psi radiative decays
I. Introduction
• QCD predicts the existence of
glueballs
• Quenched LQCD predicts
glueball spectrum in the range
1~3GeV
• candidates: f0(1370), f0(1500),
f0(1710) etc.
•J/psi radiative decay can be the
best hunting ground.
• BESIII is producing 1010 J/psi
events
Y. Chen et al,
Phys. Rev. D 73, 014516 (2006)
Hadronic matrix element


The computation of J/psi radiative decay into
glueballs breaks up into perturbative part (QED
part) and non-perturbative part
The non-perturbative part requires the
computation of matrix element
( e.m.)
G | j
Glueball state
( x) | J / 
EM current
J/ state
Three-point function
Scalar glueball
operator
Electromagnetic
current operator
Charmonium
operator
After the intermediate state insertion, the three-point function
can be written as
Two-point function of vector meson
II. Numerical details
1. Lattice and parameters
Anisotropic lattice:
Strong coupling:
2. Actions
L3 T  83  96
  2.4
  as / at  5
as  0.222(2) fm
3. Configurations and quark propagators
In order to get fair signals of the three point functions,
a large enough statistics is required.
• 5000 gauge configurations, separated by 100 HB sweeps
• Charm quark mass is set by the physical mass of J/psi
• On each configuration, 96 charm quark propagators are
calculated with point sources on all the 96 time slices.
The periodic boundary conditions are used both for the
spatial and temporal directions.
T 1
 




1
( 3) i
iq  y
G ( p f , q; t f , t )   e
OG ( p f , t f   ) j  ( y, t   )OJi ,/ ( )
T  0 y
4. The glueball operators
Building prototypes
(various Wilson loops)
Smearing: Single link scheme (APE) and double link scheme (fuzzying)
The essence of the VM is to
find a set of combinational
coefficients
v ,  1,2,...24
such that the operator
couples mostly to a specific
state.

4
a E ( p )  at2 m 2  2
2
t
2
th

3
 sin 2
i 1
pi
,
2
 2
p
(n1 , n2 , n3 )
L
The energies of 27 momentum
modes of scalar glueballs are
calculated. Plotted are the
plateaus using the optimized
operators
The horizontal line is the
theoretical prediction
with   5
.
It is seen that the deviations
are less than 5%
5. Three point functions
G

J /

G
Temporarily, we only analyze the following cases:
 
p  q  (000), (100), , (222)

pi  (0,0,0)
  i  1,2,3
2
Q  ( p f  pi )  p f  (M i  E f )2
2
2
J /
The plots for the ratios
G (3) (t ,40,0) / C2 (40)
Exponential
fit by taking the
glueball energies
as known
parameters

G ( p f ; t, t )


ii
2
ii
2


(
M
,
p
,
M
)
E
(
Q
,
t
)


(
M
,
p
,
M
)
C
'
(
Q
, t)
f
f
i
1
f
f
i
1
ii 
C2 ( pi  0; t )
( 3),ii
These are known functions
6. The form factor and the decay width
Polynomial fit:
E1 (Q 2 )  E1 (0)  aQ 2  bQ 4
E1 (0)  0.0145(13)GeV
Preliminary !
The branch ratio is
4
| p|
2
G( J /  G0 )   2 E1 (0)  0.030(5)keV
27 M J /
G
 0.030(5) / 93.2  3.2(5) 10  4
Gtot
6. Renormalization constant
• In this study, naïve local current operator
is used which needs a multiplicative
renormalization ZV
6. Renormalization constant (cont.)
Summary

Quenched lattice QCD study on J P = 0 – channel scattering
yields scattering length a0 and effective range r0. Two meson
interaction is attractive, but unlikely to form a bound state.


Unquenched study desired
Preliminary quenched study on J/psi radiative decay to
scalar glueball is performed with the result: G=0.030(5)keV.

More to be checked (current renormalization, extrapolations, other
channels etc.)

Larger lattices (12^3*144, working…)