Charmonium spectrum including higher spin and exotic states · charm quark mass parameter set by tuning 1 4 mηc + 3 4 mJ/Ψ runs performed on 16 node partitions on the local QCDOC

Charmonium spectrum including higherspin and exotic states

Christian Ehmann

Universitat Regensburg

In collaboration with

Gunnar Bali

19 October 2007

Christian Ehmann Universit at Regensburg

Charmonium spectrum including higher spin and exotic state s

Charmonia on the lattice Implementation Results Outlook

Outline

1 Charmonia on the lattice

2 Implementation

3 Results

4 Outlook




Outline


2 Implementation

3 Results

4 Outlook




Challenges

calculation of the spectra with high precision

hyperfine splitting

configuration mixing

disconnected contributions




Lattice results from JLab

3000

3250

3500

3750

4000

4250

4500

4750

5000

5250

5500 q.m. F-wavesq.m. P-waves

this workexpt. (PDG)

0++

1++

2++

3++

4++

Dudek et al.

arXiv:0707.4162




Fermion action

crucial question: Which fermion action to choose?

chiral symmetry (mostly) plays minor role in charmsystems: chiral actions like overlap or domain wall areoverkilleffective actions like NRQCD:

no true continuum limitlarge radiative and relativistic corrections (〈v2〉 ≅ 0.5)

happy medium: (Clover-)Wilson action with fine lattices




Outline


2 Implementation

3 Results

4 Outlook




Simulation details

valence & sea quark action: Clover-Wilson

gluon action: plaquette

QCDSF configurations in use:

β κ volume mπ[GeV] a[fm] L[fm]

5.20 0.13420 163 × 32 1.007(2) 0.1145 1.85.25 0.13460 163 × 32 0.987(2) 0.0986 1.65.29 0.13500 163 × 32 0.929(2) 0.0893 1.45.40 0.13500 243 × 48 1.037(1) 0.0767 1.7

A. Ali Khan et al., Phys.Lett.B564:235-240

charm quark mass parameter set by tuning 14 mηc + 3

4 mJ/Ψ

runs performed on 16 node partitions on the local QCDOC using theChroma software library (see arXiv:hep-lat/0409003)




Variational method

Choose basis of operators and construct a cross correlator matrix:

Cij(t) = 〈Oi (t)O†j (0)〉 =

X

n

vni vn∗

j e−t En

Solve symmetric generalized eigenvalue problem:

C(t0)−1/2 C(t) C(t0)−1/2 ~ψα = λα(t, t0) ~ψα

Eigenvalues behave like:

λα(t, t0) ∝ e−(t−t0) Eα [1 + O(e−(t−t0) ∆Eα)]

see C. Michael, NPB259, 58; M. Luscher and U. Wolff, NPB339, 222




Operators

name Oh repr. JPC state operator

a0 A1 0++ χc0 1π A1 0−+ ηc γ5ρ T1 1−− J/ψ γia1 T1 1++ χc1 γ5γib1 T1 1+− hc γiγj

π ×∇ T1 1+− hc γ5∇i(ρ ×∇)T1 T1 1++ χc1 ǫijkγj∇k(a1 ×∇)T2 T2 2−− γ5sijkγj∇k(b1 ×∇)T1 T1 1−+ exotic γ4γ5ǫijkγj∇k(π × D)T2 T2 2−+ γ4γ5Di(ρ× D)A2 A2 3−− γi Di(a1 × D)A2 A2 3++ γ5γi Di(b1 × D)A2 A2 3+− γ4γ5γi Di(a1 × B)T2 T2 2+− exotic γ5sijkγj Bk

A1 → J = 0, 4

A2 → J = 3

T1 → J = 1, 3, 4

T2 → J = 2, 3, 4

Di = sijk∇j∇k

Bi = ǫijk∇j∇k

see X. Liao and T.Mankehep-lat/0210030




APE Link Smearing

measure for effect of APE Smearing: 〈Ps〉

0 5 10 15 20 25 30n

iter

0,95

0,955

0,96

0,965

0,97

0,975

0,98

0,985

0,99

0,995

1

<P s>

α = 2.0 α = 2.5 α = 3.0

APE smearing quiteexpensive⇒ we run at α = 2.5and niter = 15




Jacobi Smearing

optimize each operator separately

0 2 4 6 8 10 12t/a

1,2

1,4

1,6

1,8

2

2,2

2,4

amef

f

pointnarrowwide




APE + Jacobi Smearing

What APE smearing does to the trial wavefunctions: Plot of |ψ|2κ = 0.3, nJacobi = 100; α = 2.5, nAPE = 15

ordinary gauge fieldwithout APE smearing

ordinary gauge fieldwith APE smearing




All to all propagators

create N random noise vectors

χiα,a,x =

1√2(v + iw) v ,w ∈ {±1}, i = 1, . . . ,N

define the random contraction

1

N

X

i

χiα,a,xχ

i ∗β,b,y = δx,yδa,bδα,β + O(

1√N

)

by inverting the Dirac Operator M on these sources we obtain N solution vectors

ηi = M−1χi , i = 1, . . . ,N.

naive estimate for A2AP

X

i

ηiχi† =X

i

M−1χiχi† = M−1„

1 + O(1√N

)

«




Improvements

dilution (time, spin, color, even-odd, ...)

hybrid method

maximal variance reduction / domain decomposition

truncation

hopping parameter acceleration

M−1W = (1 − κD)−1 = 1 + κD + . . .+ (κD)n−1 +

∞X

i=n

(κD)i

M−1W = 1 + κD + . . .+ (κD)n−1 + (κD)nM−1

W

⇒ (κD)nM−1W = M−1

W − (1 + κD + . . .+ (κD)n−1)




Outline


2 Implementation

3 Results

4 Outlook




Effective masses

0,5

1

1,5

2

2,5

3

3,5

amef

f

0 2 4 6 8t/a

0,5

1

1,5

2

2,5

3

3,5

amef

f

0 2 4 6 8t/a

0 2 4 6 8t/a

0 2 4 6 8t/a

π (0−+) ρ (1--) a

0 (0

++) a

1 (1

++)

(ρx∇) Τ2 (2++) b

1 (1

+-)

(b1x∇) Τ1 (1

−+) (a1x∇) Τ2 (2

−−)

t0=1 t

0=1 t

0=2 t

0=2

t0=1 t

0=2

t0=1

t0=2

β = 5.20

a−1 ≈ 1730MeV

#conf = 100




Spectrum

0-+

1-- 0

++ 1

++ 2

++ 1

+- 1

-+ 2

-- 2

-+ 2

+- 3

-- 3

+- 3

++

2,8

3,2

3,6

4

4,4

4,8

5,2

M[G

eV]

latticelattice(exotic)experiment?

ηc

J/ψ

χc0

χc1 h

cχ

c2

DDX(3943)

Y(4350)Y(4260)

Y(3940)X(3872)

Z(3930)

∆m1S = 73(2)MeV

∆m2S = 47(6)MeV




”Wavefunctions”

fix to coulomb gauge

reconstruct the wavefunction from the eigenvector components:Ψα(x) =

P

j ψαj Sj (x)

rms(1S) ≈ 0.39fm

1S2S 3S




HPA for the Wilson Operator

Tr(γ5M−1) = Tr γ5 + κTr(γ5D) + κ2Tr(γ5D2) + . . .

0 1 2 3 4 5 6 7 8n

-80

-60

-40

-20

0

20

40

60

80

Tr[γ5(κD)

nM

-1]

L=50




Disconnected contribution to ηc

0 1 2 3 4 5t/a

-0,025

0

0,025

0,05

0,075

0,1

0,125

0,15

D(t

)

Clover action:Tr(γ5D2

cl) ∝ FF

⇒ n = 2

L = 100

σstochσgauge

≈ 0.85




Outline


2 Implementation

3 Results

4 Outlook




Outlook

continuum limit

four quark/molecule operators

wavefunctions optimized for excited states in CCM

further a2a propagator improvements

heavy-light system: spectrum, matrix elements, ...



Documents

Charmonium spectrum including higher spin and exotic states · charm quark mass parameter set by tuning 1 4 mηc + 3 4 mJ/Ψ runs performed on 16 node partitions on the local QCDOC