The hyperfine splitting of charmonium
in lattice QCD

Workshop on quarkonium

20 - 22 September 2003 Fermilab, IL, USA

Takashi Umeda ( Kyoto-Univ. )

for the QCD-TARO Collaboration

1

Workshop on Quarkonium 2003 Sep. @Fermilab

Contents

Introduction

Hyperfine splitting in quenched lattice QCD

Numerical results

Neglected effects

Contributions of disconnected diagrams

Numerical results

Summary & Outlook

1

Charmonium hyperfine splitting on the lattice

Workshop on Quarkonium 2003 Sep. @Fermilab

Introduction (1)

1

Charmonium spectrum on lattice QCD

lattice cutoff is necessary

large computational cost !

NRQCD

HQET Recent progress will be presented

Fermilab in the next talk !

Anisotropic lattice next page

where the condition may be relaxed

to ?

}

Workshop on Quarkonium 2003 Sep. @Fermilab

1

/

a

s

~

m

charm

,

1

/

a

t

≫

m

charm

Introduction (2)

1

Good agreement with the experimental value

except for the hyperfine splitting

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

reliable extrapolation

needs

large dep. on Dirac op.

quenched anisotropic lattice

tree-level clover quark

continuum extrapolation

with = 1.0 ~ 2.9 GeV

Workshop on Quarkonium 2003 Sep. @Fermilab

1

/

a

s

≫

m

charm

A study by QCD-TARO Collaboration

1

QCD-TARO Collabration, JHEP08 (2003) 022

Quenched QCD (without dynamical quarks)

Isotropic lattices with large cutoff

Nonperturbative improved clover quark

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

Continuum extrapolation

Workshop on Quarkonium 2003 Sep. @Fermilab

1

/

a

≫

m

charm

Lattice setup

1

Gauge field : plaquette gauge

Quark field : nonperturbative improved clover quark

( tree level clover quark, standard Wilson )

beta 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 , quark mass set by m( )

Workshop on Quarkonium 2003 Sep. @Fermilab

finite volume effect is negligible at La > 1.1 fm

Scale determination

1

Lattice cutoff

Sommer scale

~ 10 % uncertainty ( due to dynamical quark effects )

Hadron mass ( e.g. m(1P)−m(1S) )

for example CP-PACS results ~ 16 % uncertainty

discrepancy caused by

scaling violation & dynamical quark effects

Quark mass

matching with a physical charmonium mass

m( ) = 3.097 GeV

m( ) = 2.980 GeV

Workshop on Quarkonium 2003 Sep. @Fermilab

J

/



~ 6 % uncertainty

r

0

≃

0.5

fm

Charmonium spectrum

1

Workshop on Quarkonium 2003 Sep. @Fermilab

nonperturbative clover

continuum limit

quark mass set by m( )

scale set by

The results are consistent with the previous works

hyperfine splitting is smaller than the Exp. value

Hyperfine splitting in the a=0 limit

1

m( ) – m( ) = 77 MeV in the continuum limit

30−40% smaller than the experimental value = 117 MeV

Workshop on Quarkonium 2003 Sep. @Fermilab

nonperturbative clover

tree-level clover

standard Wilson

O



g

2

a



,

O



a

2



O



a



nonperturbative clover shows

small scale violation

even for m( ) a ~ 1.4

other Dirac op. consistent with

each other in the a=0 limit

for m( ) a < 1

J

/



J

/



Contained uncertainties

1

statistical error ~ 2 MeV

quark mass determination ( ) ~ 5 MeV

continuum extrapolation ~ 3 MeV

finite volume effects ~ checked & negligible

scale determination (1/a) ~ dynamical quark effects

choice of Dirac operators ~ well controlled

dynamical quark effects

disconnected diagram contributions

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

Workshop on Quarkonium 2003 Sep. @Fermilab

J

/





c

Dynamical quark effects

1

Nucl. Phys. B(PS)119(2003)586

M.di Pierro et al.

valence quark : Fermilab action

sea quark : KS quark

Nf = 0, 3, 2+1

no continuum limit a = 0.13 fm

Other full QCD studies Dynamical quark effects ~ 10%

T.Manke et al., Phys. Rev. D62 (2000) 114508.

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

Workshop on Quarkonium 2003 Sep. @Fermilab

OZI forbidden “disconnected” diagram

1

Disconnected diagrams are neglected

because high cost & very small contribution

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

Workshop on Quarkonium 2003 Sep. @Fermilab

Charmonium correlators

1

: quark propagator

( Pseudoscalar, Vector )

source & sink operators are extended with

disconnected diagrams are evaluated with

the Z2-noise method

Workshop on Quarkonium 2003 Sep. @Fermilab

Lattice setup

1

Workshop on Quarkonium 2003 Sep. @Fermilab

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

=600

m

charm

Smeared operators

1

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 )

Workshop on Quarkonium 2003 Sep. @Fermilab

Disconnected diagram contributions

1

very preliminary

When ground state dominates,

We have

Too noisy at large t

Contributions of the disconnected diagram may be very small ?

Workshop on Quarkonium 2003 Sep. @Fermilab

Summary & Outlook

1

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

Same calculations with smaller valence quark mass

extrapolation from light quark mass region

Workshop on Quarkonium 2003 Sep. @Fermilab

Appendix: Wave function

1

Workshop on Quarkonium 2003 Sep. @Fermilab

scaling violations are very small

's agree with phenomenological expectations qualitatively

In a heavy quark model :





c





0







J

/







0



Appendix: lighter valence quarks

1

Workshop on Quarkonium 2003 Sep. @Fermilab

Appendix: Z2-noise method

1

Workshop on Quarkonium 2003 Sep. @Fermilab

Takashi Umeda

2003-09-04T17:47:02

Takashi Umeda