The hyperfine splitting of charmonium
on the lattice

26 January 2004 Tokyo-Univ. (Hongo campus)

Takashi Umeda ( YITP, Kyoto-Univ. )

for the QCD-TARO Collaboration

Seminar in Tokyo-univ. / 26 Jan. 2004

created by

Preliminary

2

Seminar in Tokyo-univ. / 26 Jan. 2004

Verification of

Quantum Chromodynamics (QCD)

Lattice regularization

Hadron spectroscopy

Light hadron spectrum ( with/without dynamical quarks )

some projects in progress

by CP-PACS, JLQCD, MILC, UKQCD, ∙∙∙

Charmonium spectrum

Charmonia properties are well established

by experiments

Contents

3

Charmonium hyperfine splitting (HFS) on the lattice

Seminar in Tokyo-univ. / 26 Jan. 2004

1) Introduction

problems of the charmonium HFS

2) Quenched charmonium spectrum

systematic study for the charmonium HFS

3) Discussion

remaining uncertainties of HFS

4) Contribution of a disconnected diagram

5) Conclusion

Status of experiments

4

Seminar in Tokyo-univ. / 26 Jan. 2004



c



1

S



J

PC

=

0

-

+

mass = 2979.7 ± 1.5 MeV

width = 16.0 +3.6 – 3.2 MeV

J

/





1

S



mass = 3096.87 ± 0.04 MeV

width = 87 ±5 KeV

J

PC

=

1

-

-

m



J

/





−

m





c



=

117.2

±

1.5

MeV

Seminar in Tokyo-univ. / 26 Jan. 2004

43

Part-I Quenched charmonium spectrum

Problems for heavy quarks on a lattice

5

Charmonium spectrum on lattice QCD

lattice cutoff is necessary

large computational cost !

1

/

a

≫

m

charm

Effective theories

Relativistic framework

NRQCD, etc.

e.g. Trottier, Phys. Rev. D55 (1997) 6844

Fermilab action, anisotropic lattice

e.g. CP-PACS, Phys. Rev. D65 (2002) 094508

Seminar in Tokyo-univ. / 26 Jan. 2004

Status of lattice results

6

Good agreement with the experimental values

except for the hyperfine splitting

M.Okamoto et al. (CP-PACS), Phys. Rev. D65 (2002)094508

quenched anisotropic lattice

tree-level clover quark

continuum extrapolation

with = 1.0 ~ 2.9 GeV

Seminar in Tokyo-univ. / 26 Jan. 2004

HFS in NRQCD

7

X is not so heavy

X no continuum limit

H. Trottier, Phys. Rev. D55 (1997) 6844

quenched simulation

next-to-leading order correction

lattice cutoff in the range

of 1/a = 0.5 ~ 1.2 GeV

m

charm

Seminar in Tokyo-univ. / 26 Jan. 2004

HFS in relativistic frameworks

8

1

/

a

s

≫

m

charm

quenched calculations

Fermilab action

A.El-Khadra et al., [Lattice'91]

Anisotropic lattice action

CP-PACS, PRD65('02)094508

P.Chen, PRD64('01)034509

anisotropic lattice also needs

for reliable continuum extrapolations

large dependence on Dirac operators

Seminar in Tokyo-univ. / 26 Jan. 2004

A study by QCD-TARO Collaboration

9

QCD-TARO Collabration, JHEP08 (2003) 022

Quenched QCD (without dynamical quarks)

Isotropic lattices with large cutoff

Nonperturbatively improved clover quark

M.Lüscher et al., Nucl.Phys.B491 (1997) 323

Continuum extrapolation

1

/

a

≫

m

charm

Seminar in Tokyo-univ. / 26 Jan. 2004

Charmonium correlation functions

43

Seminar in Tokyo-univ. / 26 Jan. 2004

Name mass (GeV)

2.980(2)

3.097

3.526

3.415

3.511



2

s



1



L

J

J

PC





c

J

/



h

c



c0



c1

1

S

0

3

S

1

1

P

1

3

P

0

3

P

1

1

+

+

0

+

+

1

+

-

1

-

-

0

-

+



5



i



i



5

1



ij

point source & point sink correlators

zero momentum projection

†

Lattice setup

43

Gauge field : plaquette gauge

Quark field : nonperturbatively improved clover quark

( tree level clover quark, standard Wilson )

a(fm) La(fm) Csw #conf

6.0 0.0931 1.68 1.769 190

6.2 0.0677 1.62 1.614 90

6.4 0.0513 1.64 1.526 60

6.6 0.0397 1.27 1.467 130

scale set by

r

0

Seminar in Tokyo-univ. / 26 Jan. 2004



Charmonium spectrum

12

nonperturbative clover

continuum limit

quark mass set by m( )

scale set by

J

/



r

0

The results are consistent with the previous works

hyperfine splitting is smaller than the Exp. value

Seminar in Tokyo-univ. / 26 Jan. 2004

Contained uncertainties

13

statistical error

quark mass determination ( )

continuum extrapolation

finite volume effects

scale determination (1/a)

choice of Dirac operators

dynamical quark effects

disconnected diagram contributions

Seminar in Tokyo-univ. / 26 Jan. 2004

Effective mass plot of HFS

14

Seminar in Tokyo-univ. / 26 Jan. 2004

R



t



≡

〈

C



t



〉

〈

C



t



1



〉

=

cosh

[



T

/

2

−

t



M



t



]

cosh

[



T

/

2

−

t

−

1



M



t



]

1

:



M



t



≡

M

B



t



−

M

A



t



2

:

R

A



t



R

B



t



~

e

−



M



t



1: and 2: are consistent

each other within error

2: is less noisy than 1:

we use the definition 2:

Scale determination

43

Lattice cutoff

Sommer scale [R.Sommer,NPB411('94)839]

: Static quark potential

using the nucleon mass amounts to

~ 10 % uncertainty ( due to dynamical quark effects )

Hadron mass ( e.g. m(1P)−m(1S) )

for example CP-PACS results ~ 16 % uncertainty

This discrepancy caused by

scaling violation & dynamical quark effects

r

0

≃

0.5

fm

r

0

Seminar in Tokyo-univ. / 26 Jan. 2004

=

0

r

0

dV



r



dr

r

r

=

1.65

V



r



r

0

≃

0.55

fm

Charm quark mass determination

43

Seminar in Tokyo-univ. / 26 Jan. 2004

matching with a physical charmonium mass

m( ) = 3.097 GeV

m( ) = 2.980 GeV

J

/





c

}

~ 6% uncertainty

only mass splittings of chamonia can be predicted

alternative possibilities

meson mass or decay constant

free of OZI ambiguities

D

,

D

s

Finite volume effect

43

Seminar in Tokyo-univ. / 26 Jan. 2004

L La (fm) (MeV) (MeV) (MeV)

8 0.75 2958(10) 3019(12) 61.4(4.4)

10 0.93 2953(5) 3023(6) 70.6(2.5)

12 1.12 2957(4) 3032(5) 75.4(2.7)

14 1.30 2947(3) 3020(4) 72.6(1.9)

16 1.49 2952(3) 3025(4) 74.9(2.1)

18 1.68 2949(2) 3021(3) 72.5(1.5)

1

S

0

3

S

1

3

S

1

−

1

S

0

Lattice spacing is fixed to 0.093fm ( =6.0)

nonperturbatively improved clover Dirac op. with =0.11925

averaged over 100conf. (190 for L=18)





finite volume effect is negligible at La > 1.1fm

this effect is discussed again using wave functions

Continuum extrapolation

43

Seminar in Tokyo-univ. / 26 Jan. 2004

Nonperturbative Clover

lattice artifact ~

O



a

2



Continuum extrapolation

linearly in



M

=

77



2



MeV

different extrapolations including or not = 6.0

systematic uncertainty of extrapolation ~ 3 MeV



a

2

at the continuum limit

Choice of Dirac operators : I

43

nonperturbative clover

linearly in

for tree-level clover

including and terms

for Wilson

including and terms

O



g

2

a



,

O



a

2



O



a



Seminar in Tokyo-univ. / 26 Jan. 2004

a

2

a

2

a

(1) using the all four



scale violations to be:

for the Wilson Dirac operator

for the tree-level clover Dirac operator

a

a

2

Choice of Dirac operators : II

43

nonperturbative clover

linearly in

for tree-level clover

linearly in

for Wilson

linearly in

Seminar in Tokyo-univ. / 26 Jan. 2004

a

2

a

2

a

(1) using the three largest



Wilson & tree-level Clover need very large cutoff

for a reliable continuum extrapolation

Dirac operator dependence is well controlled

Contained uncertainties

21

statistical error ~ 2 MeV

quark mass determination ( ) ~ 5 MeV

continuum extrapolation ~ 3 MeV

finite volume effects ~ negligible

scale determination (1/a) ~ dynamical quark effects

choice of Dirac operators ~ well controlled

dynamical quark effects

disconnected diagram contributions

Seminar in Tokyo-univ. / 26 Jan. 2004

m( ) ‒ m( ) = 77(2)(6) MeV (117MeV in Experiment)

J

/





c

Wave function

Seminar in Tokyo-univ. / 26 Jan. 2004

43

HFS in non-relativistic approximation

solving the Schredinger eq. with non-rela. Coulomb pot.

0-th order : HFS = 0

degeneracy is removed by spin-spin interaction

1-st order : HFS =

where non-rela. wave function

32





s



m

q



9

m

q

2

∣



NR



0



2

∣





c



0



=



1





NP





1

2

−





8



s

2







9





NR



0





J

/





0



=



1





NP

−



1

6







8



s

2







9





NR



0





NR



r



=

1



8





3

e

−

r

2



,



=

3

4



s

m

q



NP

: non-perturb. correction

: strong coupling at scale

S.Titard et al, Phys. Rev. D51 (1995) 6348.



≈

7.241

×

10

−

2



s









Gauge invariant wave function

Seminar in Tokyo-univ. / 26 Jan. 2004

43

∣







r



2

∣

=

〈

c



c

∣





c



c





0









c





c







r



∣

c



c

〉

=

lim

t



∞

N

∑



x

〈

Tr

[

S





0

,

0

;



x

,

t



S





0

,

2

t

;



x

,

t







5

S





0

,

2

t

;



x





r

,

t



S





0

,

0

;



x





r

,

t





5



]

〉

C.Alexandrou et al., Phys. Rev. D66 (2002) 094503

Charmonium wavefunctions

Seminar in Tokyo-univ. / 26 Jan. 2004

43

scaling violations are very small

's agree with phenomenological expectations qualitatively

In a heavy quark model :





c





0







J

/







0



Finite volume effect : 2

Seminar in Tokyo-univ. / 26 Jan. 2004

43

Discussion

26

dynamical quark effects

disconnected diagram contributions

2-nd part of this talk

Seminar in Tokyo-univ. / 26 Jan. 2004

Charmonium hyperfine splitting is 30~‒40% smaller than Exp.

Possible uncertainties of HFS

Dynamical quark effects

Improved NRQCD on Nf=2 staggered full QCD config.

C.Stewart and R.Koniuk, Phys. Rev. D63 (2001) 054503.

NRQCD on Nf=2 Wilson full QCD config.

CP-PACS, Phys. Rev. D62 (2000) 114508.

Fermilab action on Nf=2+1 staggered full QCD config.

M.di Pierro et al., Nucl. Phys. B(PS)119 (2003) 586.

Seminar in Tokyo-univ. / 26 Jan. 2004

43

There is no systematic study including continuum extrapolation

Dynamical quark effects ~ at most 10%

Dynamical quark effects

M.di Pierro et al., hep-lat/0310042

valence quark :

Fermilab action

tadpole improved tree-level

sea quark :

Improved staggered quarks

Nf = 2+1 ( = 0.05 )

fixed lattice cutoffs 1/a=1.55GeV

Seminar in Tokyo-univ. / 26 Jan. 2004

43

am

s

theory/exp.=0.82(2) ( =0.6 quenched result at 1/a=1.5GeV )

clover coefficient : (tadpole improved) tree-level

Seminar in Tokyo-univ. / 26 Jan. 2004

43

Part-II Contribution of disconnected diagram

OZI forbidden “disconnected” diagram

43

Disconnected diagrams are neglected

because high cost & very small contribution

however, it may contribute to HFS ~ O(10) MeV ?

Seminar in Tokyo-univ. / 26 Jan. 2004

Charmonium correlators

43

C

con



t



=

∑



x

〈

Tr

[



D

−

1



0,



0

;

t

,



x





D

−

1



t

,



x

;

0,



0



]

〉

C

dis



t



=

∑



x

〈

Tr

[



D

−

1



0,



0

;

0,



0



]

Tr

[



D

−

1



t

,



x

;

t

,



x



]

〉

D

−

1



t

,



x

;

t

'

,



x

'



: quark propagator

( Pseudoscalar, Vector )

source & sink operators are extended with

disconnected diagrams are evaluated with

the Z2-noise method







x



∝

exp



a

∣



x

∣

p





=



5,





Seminar in Tokyo-univ. / 26 Jan. 2004

Z2 noise method

43

Seminar in Tokyo-univ. / 26 Jan. 2004

Stochastic estimation of

Tr

[

D

−

1



t

,



x

;

t

,



x





]

using noise vectors

1

N

NV

∑

i

=

1

N

NV

R

i



x





0

,

1

N

NV

∑

i

=

1

N

NV



R

i



x



R

i



y









x

,

y

R

i



x



we use complex Z2 noise

1

N

NV

∑

i

=

1

N

NV

R

i



x



D

−

1



t

,



x

;

t

,



y





R

i



y





Tr

[

D

−

1



t

,



x

;

t

,



x





]

†

†

This method can treat smeared operators

Error of the method can be controlled by

N

NV

Lattice setup

43

Sea quark : Nf=2 KS quark : = 0.1

plaquette gauge : = 5.50

lattice size :

lattice spacing : a=0.16fm (1/a=1.2GeV) set by

16,000 traj. ( measurement at every 5 traj. )

cutoff is not so sufficient for ,this is an exploratory study

Valence quark : Fermilab action

Csw : tadpole improved tree-level ( in Landau gauge)

quark mass set by m( )

Z2-noise method

J

/



u

0

r

0

a

m

q

m

charm

Seminar in Tokyo-univ. / 26 Jan. 2004

Parameter tuning

43

Seminar in Tokyo-univ. / 26 Jan. 2004

On shell matching with

M

rest

=

M

kinetic

is tuned by the dispersion relation of mesons



F

≡



s

/



t

Fermilab action become

standard clover action

in light quark region



F



1

at

m

q



0

*

Z2-noise estimation

43

Seminar in Tokyo-univ. / 26 Jan. 2004

0.11294 0.867 100

0.11013 1.094 200

0.10732 1.323 200

0.10476 1.542 300

0.09342 2.578 600



m

V

N

NV

number of noise vector dependence

N

NV

Smeared operators

43

Effective mass plot

of connected diagram C(t)

Smearing functions are

determined from wavefunction

Ground states dominate at t~2

a( m( ) – m( ) ) = 0.0676 ( ~ 81 MeV )

J

/



Seminar in Tokyo-univ. / 26 Jan. 2004

Effective masses of full correlator (1/5)

43

Seminar in Tokyo-univ. / 26 Jan. 2004

= 0.11294

1920 conf.

= 100

smeared operators



N

NV

m

q

m

charm

~ 0.34

Effective masses of full correlator (2/5)

43

Seminar in Tokyo-univ. / 26 Jan. 2004

= 0.11013

1920 conf.

= 200

smeared operators



N

NV

m

q

m

charm

~ 0.42

Effective masses of full correlator (3/5)

43

Seminar in Tokyo-univ. / 26 Jan. 2004

= 0.10732

1920 conf.

= 200

smeared operators



N

NV

m

q

m

charm

~ 0.51

Effective masses of full correlator (4/5)

43

Seminar in Tokyo-univ. / 26 Jan. 2004

= 0.10476

1920 conf.

= 300

smeared operators



N

NV

m

q

m

charm

~ 0.60

Effective masses of full correlator (5/5)

43

Seminar in Tokyo-univ. / 26 Jan. 2004

= 0.09342

3200 conf.

= 600

smeared operators



N

NV

m

q

m

charm

~ 1.00

Contribution of disconnected diagram

43

Seminar in Tokyo-univ. / 26 Jan. 2004

very small contribution

for vector channel

contribution of

pseudoscalar channel

m

full



t

=

2



−

m

con



t

=

2



The contribution of disconnected diagram is

quickly suppressed as quark mass increase

Summary & Outlook

43

We study the problem of charmonium HFS

and consider a possibility of disconnected diagram contributions

Isotropic lattice + Nonperturbative improved clover quark

is good choice for the charmonium spectrum

In quenched QCD, m( ) − m( ) is

30 − 40% smaller than the experimental value

Disconnected diagram contributions are very small

or hidden by large error

calculations with small sea quark mass & large lattice cutoff

Full QCD simulations including the continuum limit

J

/



Seminar in Tokyo-univ. / 26 Jan. 2004

Takashi Umeda

2003-09-04T17:47:02

Takashi Umeda