Charm quark physics from lattice QCD
arXiv:1104.4600,1111.0142

Yusuke Namekawa(Univ. of Tsukuba)
for the PACS-CS collaboration
S.Aoki, K-I.Ishikawa, N.Ishizuka, T.Izubuchi, K.Kanaya, Y.Kuramashi, Y.Namekawa, M.Okawa,
Y.Taniguchi, A.Ukawa, N.Ukita, T.Yamazaki, T.Yoshie– 1 / 17 –Contents
1 Introduction

32 Simulation setup

73 Results
3.1 Charmonium spectrum . . . . . . . . . . . . . . . . . . .
3.2 Charm-strange spectrum . . . . . . . . . . . . . . . . . .
3.3 Charm quark mass . . . . . . . . . . . . . . . . . . . . .
3.4 Decay constants and CKM matrix elements . . . . . . .
3.5 Charmed baryon – Preliminary – . . . . . . . . . . . . .

9910
12
13
144 Summary

16– 2 / 17 –

1Introduction[Charm quark system]
Charm quark system is charming for physicists.
• Charm quark mass and Cabibbo-Kobayashi-Maskawa(CKM) matrix elements
are important for the standard model of elementary particles, because they
are fundamental parameters which are difficult to be determined precisely
due to mcharm ∼ ΛQCD . Lattice QCD is needed.
← In addition, mcharm and VCKM are also needed as inputs not only forthe standard model but also for a new theory beyond the standard model.
• Exotic hadrons such as Z + (4430), made of udcc̄ !?, have been observed.

Kobayashi and Maskawa got
the Nobel prize in 2008.

– 3 / 17 –[Model and lattice QCD]
So far, many model studies have been performed.
• Correctness of a model must be always checked, because the resultis model-dependent.
← In addition to experiments, lattice QCD can judge a model.
• Since a lattice QCD simulation is expensive, a first trial must be
performed by simple models. Then, we should improve models
by experiments and lattice QCD.
← Judgment on models by lattice QCD will be given later.
ResultInput
αs , mquark
Heavy quark
Cost

Model
Model-dependent
Many parameters
Artificial
1/M expansion
Low– 4 / 17 –LatticeModel-independent
αs , mquark (or hadron masses)
QCD running
Full orderHigh

[Recent progress in lattice QCD]
Simulations become realistic, thanks to the development of computers
and algorithms.
• Nf = 2 + 1 full QCD simulation is performed, which includes
dynamical effects of up-down and strange quarks.
• Dynamical up-down and strange quark masses can be set to their
physical values by use of reweighting (i.e. mπ = 135 MeV).
← So far, up-down quark masses are higher than their physicalvalue, because of the computational cost.

– 5 / 17 –

[Recent progress in lattice QCD(continued)]Light hadron spectrum has been reproduced within 5% accuracy.
→ As a next step, we move on to the heavy quark system.

(mh/mΩ)lat / (mh/mΩ)exp−1

0.20

0.10

0.00
originaltarget-0.10
π Kηss ρ K* φNΛΣΞ∆Σ∗ Ξ∗ Ω0Nf = 2 + 1, a−1 = 2.2 GeV
PACS-CS,2010– 6 / 17 –2Simulation setup

We perform a Nf = 2 + 1 full QCD simulation of the charm quark
system on the physical point.
• Action : Iwasaki gauge + O(a) improved Wilson fermion for light sea quarks+ relativistic heavy fermion for valence charm quark• Lattice size : 323 × 64 (L = 3 fm, a−1 = 2.2 GeV (β = 1.90))• Sea and valence quark masses : on the physical point (i.e. mπ = 135 MeV)
• Inputs : mπ ,mK ,mΩ for mud ,ms ,a; m(1S) ≡ 14 (mηc + 3mJ/ψ ) for mcharmmMS
ud (µ = 2GeV)[MeV]
3mMSs (µ = 2GeV)[MeV]
93– 7 / 17 –Nconf (MD time)80 (2000)

[Operators]• We employ the following two quark operators for mesons.

•MΓf g (x)=q̄f (x)Γqg (x),Γ=I, γ5 , γµ , iγµ γ5 , i[γµ , γν ]/2,f, g:labels for quark flavors.

Smearing function : Ψ(r) = A exp(−Br) at r 6= 0, Ψ(0) = 1.
♦

A = 1.2, B = 0.07 for ud quark.

♦A = 1.2, B = 0.18 for strange quark.

♦A = 1.2, B = 0.55 for charm quark.

– 8 / 17 –

3Results

3.1

Charmonium spectrum• Since m(1S) ≡ 14 (mηc + 3mJ/ψ ) is used as an input for mcharm , differencesfrom m(1S) are predictions.
• Our results agree with experiments except for the hyperfine splitting.
•The hyperfine splitting is underestimated by 3σ from the experimental value of BES III(2011).
→ We have not evaluated the following systematic errors: scaling violations, dynamical charm
quark effects, disconnected loop contributions.

0.60

m-m(1S)[GeV]

0.40

0.20

PDG(2010)
BaBar(2008)
Belle(2008)
BES III(2011)
This work(Nf=2+1,a−1=2.2 GeV)
CP-PACS(Nf=2,a=0)
CP-PACS(Nf=0,a−1=2.0 GeV)

0.00

ExperimentThis work

-0.20

-0.40

ηcJ/ψ

χ0χ1hc0.06– 9 / 17 –0.07

0.08

0.09

0.10

mJ/ψ − mηc [GeV]0.11

0.120.13

3.2

Charm-strange spectrum• Our calculation reproduces the charm-strange spectrum in 2σ level.
•from DK scattering states can be considerably large, which
Contaminations to mD ∗ , mD
s1s0
have not been included yet.

•∗ ,D(Ds0s1 decays are prohibited in our Nf = 2 + 1 lattice QCD.)

Mass(our result) / Mass(experiment)

1.04
1.03
1.02
1.01
1.00
0.99
0.98
0.97

Ds

*

Ds

*

Ds0

0.96

– 10 / 17 –Ds1

[Judgment on models by lattice QCD]
∗ , many models are not good.
• For Ds0
← The standard potential model by Godfrey et al, 1983 fails to reproduce∗ masses.
Ds0• A model by Matsuki et al, 1997;2006 is good.
•∗ (2317) have been confirmed experimentally by BaBar, Belle, CLEO.)(Ds0Experiment
Lattice(This work)
Model(Godfrey et al, 1983)
Model(Ebert et al, 1998)
Model(Di Pierro et al, 2001)
Model(Matsuki et al, 1997;2006)
2.20

2.30

2.40

2.50

*
Ds0 [GeV]– 11 / 17 –

2.60

2.70

2.80

3.3

Charm quark mass

• Charm quark mass is determined from the axial Ward-Takahashi identity.
• Our result is consistent with other lattice and continuum calculations.
•Our systematic error is still large. The main source of our error is the non-perturbative
renormalization factors.
(The renormalization factor is calculated non-perturbatively at the massless point. The massdependent part is calculated perturbatively.)•(Charm quark mass is renormalized at µ = 1/a, and evolved to µ = mMS
charm using Nf = 4
four-loop beta function.)

This work(Nf=2+1, a−1=2.2 GeV)HPQCD(Nf=2+1, a=0, 2010)
ETMC(Nf=2, a=0, 2010)
HQE(BABAR,2010)
Sum rule(Kuhn et al,2007)
Sum rule(Bodenstein et al,2010)
Sum rule(Dehnadi et al,2010)
Sum rule(Narison,2010)
1.10

1.15

1.20
1.25
MSmcharm (µ = mMScharm)– 12 / 17 –

1.30

1.35

3.4

Decay constants and CKM matrix elements• Our fDs agrees with the experimental value and other lattice QCD results.
• CKM matrix elements are extracted from our mass and pseudoscalar decay
constant of charmed-strange meson combined with experimental values forthe leptonic decay width of charmed-strange mesons.
222GFml 
222 Γ(Ds → lν) =m l m D f D 1 − |Vcs |ss
8πm2DsExperiment(2008)
Experiment(2010)
This work(Nf=2+1,a−1=2.2 GeV)HPQCD+UKQCD(Nf=2+1,a=0,2010)
Fermilab+MILC(Nf=2+1,a=0,2010)
ETMC(Nf=2,a=0,2011)
0.22

0.23

0.24

0.25

0.26

0.27

0.28

0.29

PDG(2010)
This work(Nf=2+1,a =2.2 GeV)HPQCD+UKQCD(Nf=2+1,a=0,2010)−10.30

0.90

0.95

1.00

|Vcs|fDs– 13 / 17 –1.05

1.103.5

Charmed baryon – Preliminary –• Our results agree with experiments in 2σ level, except for Ξcc .
♦ Only SELEX(2002,2005) found Ξcc = 3519 [MeV].
♦ BABAR, BELLE and FOCUS found no evidence for Ξcc .
→ Ξcc has been omitted from PDG 2011.
(Σc decay is prohibited on our lattice.)
1.10

Mass(our result) / Mass(experiment)

•

1.05

1.000.95

ΛcΣc0.90– 14 / 17 –Ωc(Ξcc)[Judgment on models by lattice QCD] – Preliminary –• For Ξcc , many models are not bad.
← A model by Martynenko, 2008 seems to be wrong.
• SELEX experiment seems to be wrong.

Experiment(2005)
Lattice(This work)
Model(Ebert et al,2002)
Model(Roberts et al,2008)
Model(Martynenko,2008)
HQET(Korner et al,1994)
3.40

3.50

3.60

3.70

Ξcc [GeV]– 15 / 17 –

3.80

3.90

4.00

4

Summary

We performed a Nf = 2 + 1 full QCD simulation of the charm quark system on thephysical point at a−1 = 2.2 GeV.
• Our calculation reproduces meson mass spectrums of the ground states
except for hyperfine splittings.
♦Our data of the charmonium hyperfine splitting is 3σ smaller than experiments.
← Possible origins of the discrepancy are O(a) effects in our relativistic heavy quark
action, dynamical charm quark effects, and disconnected loop contributions.

• Our results for charm quark mass and CKM matrix elements are presented.
• Our calculation reproduces baryon mass spectrums of the ground states
except for Ξcc .
♦Our preliminary data of Ξcc shows a significant deviation from the experimental valueof SELEX group.
← SELEX value seems to be wrong.

– 16 / 17 –[Future work]• We are going to a finer lattice (a−1 = 3 GeV) to take a continuum limit.
• Excited states of charmonium separating D D̄ contamination (calculations forX, Y, Z are hard, in practice ).
[New computer]• K-computer will be used for lattice QCD, which is the fastest computer inthe world at present(Nov. 2011).
• Many lattice QCD theorists in Japan(including me) are working for this Kcomputer project.

– 17 / 17 –Appendix– 18 / 17 –