αs, mc, and mb fromQuarkonium Correlators

G. C. Donald, C. T. H. Davies, R. J. Dowdall, E. Follana, K.Hornbostel, J. Koponen, G. P. Lepage, C. McNeile 1

For the HPQCD Collaboration

1Speaker mcneile@uni-wuppertal.deFor the HPQCD collaboration Short Title 1/24

Introduction

• Since 2003 the HPQCD collaboration has been computingthe properties of mesons containing charm and “bottom”quarks using improved staggered fermions.

• I will report progress on calculations with HISQ quarks forboth the heavy and light quarks.

The talk will largely be based on results from

• Precision tests of the J/ψ from full lattice QCD: mass,leptonic width and radiative decay rate to ηc . Donald et al.(1208.2855).

• High-Precision c and b Masses, and QCD Coupling fromCurrent-Current Correlators in Lattice and Continuum QCD(1004.4285)

For the HPQCD collaboration Short Title 2/24

Basic idea of lattice QCD

What needs to be doneThe equations of lattice QCD are numerically solved inEuclidean space via a Monte Carlo process.

Supercomputer Generate gauge configurations (“snapshots ofthe QCD vacuum”)

Supercomputer Compute quark propagators.Supercomputer Combine quark propagators into meson

correlators and average over spatial volume c(t).PC Fit the meson correlators c(t) ∼ Ae−mt to get

masses m and amplitude A (from which decayconstants).

PC Repeat calculations for different quark massesand lattice spacings and take the continuum limit.

For the HPQCD collaboration Short Title 3/24

The HISQ action

• There are many different choices for the lattice version ofthe Dirac operatror.

• The HISQ (Highly Improved Staggered Action) action wasdesigned to reduced lattice spacing dependence overprevious improved staggered actions.

• The leading lattice spacing corrections to continuum for theHISQ action are O(αsa2).

• The next “best” fermion action has leading lattice spacingcorrections to continuum of O(a2).

• For example at a = 0.09 fm, amc = 0.4, the discretizationerrors in the decay constant of the ηc are O(2%)(1203.3862).

For the HPQCD collaboration Short Title 4/24

Parameters of lattice QCD calculation• In 2001 MILC collaboration started generating gauge configs

with ASQTAD staggered sea quarks (0903.3598).

• In 2008 MILC collaboration started generating gaugeconfigurations with HISQ staggered sea quarks.

0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20a (lattice spacing) fm

100

200

300

400

500

600

Mπ M

eV

Ongoing runs

Valence HISQ, ASQTAD sea nf = 2+1Valence HISQ, HISQ sea nf = 2+1+1

For the HPQCD collaboration Short Title 5/24

Moments methodj5 =ψhγ5ψh:

G(t) = a6∑

x

(am0h)2〈0|j5(x, t)j5(0,0)|0〉

where m0h is the heavy quark’s bare mass.The moments of G(t) are simple to analyze:

Gn ≡∑

t

(t/a)nG(t)

Gn =gn(αMS(µ), µ/mh)

(amh(µ))n−4 +O((amh)m)

for small n ≥ 4,• mh(µ) is the heavy quark’s MS mass at scale µ.• The dimensionless factor gn is computed using continuum

perturbation theory known through α3s for low moments

(lattice perturbation theory is hard).• Lattice spacing from r1 tuned from fπ, Υ.

For the HPQCD collaboration Short Title 6/24

More moments

• First paper with unquenched QCD (0805.2999) HPQCD +Chetyrkin, Kuhn, Steinhauser,Sturm.

• In the second paper (1004.4285) HPQCD used heavierquarks amQ < 0.86 to approach the mass of the bottomquark.

• Ratio further normalised by tree level lattice results.

For the HPQCD collaboration Short Title 7/24

Summary of mc masses (1301.7202)

1.22 1.24 1.26 1.28 1.3 1.32 1.34mc(mc, nf=4) (GeV)

HPQCD HISQ 1004.4285

ETMC 1010.3659nf=2

Chetyrkin et al0907.2110

Dehnadi et al1102.2264

u, d, s sea

u, d sea

contnm

For the HPQCD collaboration Short Title 8/24

Cross-check on mb/mcTo reduce lattice spacing errors slightly extrapolate

(mbmηc )

(mcmηb )

to the ηb mass. Mηb = 9.395(5) Gev and Mηc = 2.985(3) GeV(Mηb/Mηc = 3.15).

mb/mc → 4.49(4)

For the HPQCD collaboration Short Title 9/24

αs from moments (1004.4285 )4th moment is insensitive to charm mass, hence good for αs.

For the HPQCD collaboration Short Title 10/24

Summary of αs from lattice QCD

0.08 0.09 0.10 0.11 0.12 0.13 0.14

nf=2+1+1ETMC ghost-gluonnf=2+1HPQCD (moments)Maltman et al. (Wilson loops)HPQCD (Wilson loops)JLQCD vac polPACS-CS (Schrod)Bazavov et al. (Pot)

PDG 2012

cont

chira

l

0.11 0.12 0.13α (Μ )s Ζ

LatticeDIS e+e- annihilation

τ-decays

Z pole fits

(a) From 2012 PDG

For the HPQCD collaboration Short Title 11/24

αs and lattice QCDA value “from PDG” that summarizes these differentresults is

αs = 0.1185± 0.0007.

With all due respect to lattice people I think this smallerror is totally umplausible 2.

The QCD Running Coupling and its Measurement GuidoAltarelli (1303.6065).

• The PDG summary for αs is stable on removing the latticenumbers.

• Altarelli’s comments illustrates that it is important tocontinue to test lattice QCD against experiment (CLEO-C→ BESIII...).

2There is no explanation for this statementFor the HPQCD collaboration Short Title 12/24

Lattice vs expt for moments(1208.2855)

• Compare the lattice results against expt. moments fromKuhn et al. hep-ph/0702103

• R(s) = σ(e+e−→hadrons)σpth

andMn =∫ ds

sn+1 R(s)

0.0 0.1 0.2 0.3 0.4 0.5(amc)

2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4(n

thm

omen

t)1/

(n−

2)(G

eV−

1 )

n = 4

n = 6

n = 8

n = 10

For the HPQCD collaboration Short Title 13/24

Charmonium hyperfine splitting.

0.0 0.1 0.2 0.3 0.4 0.5(amc)

2

0.110

0.115

0.120

MJ/

ψ−

Mη c

(GeV

)

Donald et al., 1208.2855 connected diagrams only

MJ/ψ −Mηc = 116.5(3.2)MeV

compared to the 2012 PDG summary value of 115.9(1.2) MeV.

For the HPQCD collaboration Short Title 14/24

Decay constant of J/ψ

Γ(vh → e+e−) =4π3α2

QEDe2h

f 2v

mv

where eh is the electric charge of the heavy quark in units of e(2/3 for c).

〈0|ψγ iψ|v〉 = fv mvεi

fvZv

= a0

√2

M0,

c(t) ∼ a0e−M0t

• Renormalization Zv computed using a moments method.

For the HPQCD collaboration Short Title 15/24

J/ψ decay constant

Donald et al., 1208.2855

0.0 0.1 0.2 0.3 0.4 0.5(amc)

2

0.38

0.40

0.42

0.44

0.46

0.48

0.50

f J/ψ

(GeV

)

380 390 400 410 420 430 440J/ decay constant / MeV

HISQ this paper

Twisted mass1206.1445

Particle Data Groupaverage

u, d, s sea

u, d sea

experiment

For the HPQCD collaboration Short Title 16/24

Decay constants (1207.0994)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8D

ECA

Y C

ON

STA

NT

(GEV

)

UNFLAVORED FLAVORED

+

J/c

’

b

’

’’Bc

BsB+

Ds

D+

K+

Lattice QCD predictionsLattice QCD postdictions

experiment

For the HPQCD collaboration Short Title 17/24

Radiative tranistion J/ψ → ηcγDonald et al., 1208.2855

〈ηc(p′)|cγµc|J/ψ(p)〉 =2V (q2)

(MJ/ψ + Mηc )εµαβγp′αpβεJ/ψ,γ

Γ(J/ψ → ηcγ) = αQED64|~q|3

27(Mηc + MJ/ψ)2 |V (0)|2,

ηc J/ψ

T

t

01

23 V

Monday, 13 August 2012

For the HPQCD collaboration Short Title 18/24

Radiative tranistion J/ψ → ηcγ

Donald et al., 1208.2855 Signal seen for radiative decays:hc → ηc and χ0 → J/ψ.

1.2

1.4

1.6

1.8

2

2.2

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014

V(0

)

a2 / fm2

HISQtwisted mass

expt

(b) Continuum extrap. 1.2 1.4 1.6 1.8 2 2.2

Vector form factor V(0)

HISQ this paper

Twisted mass1206.1445

CLEO0805.0252

u, d, s sea

u, d sea

experiment

For the HPQCD collaboration Short Title 19/24

Preliminary results from HPQCDmc(mc) from 2+1+1

1.1

1.15

1.2

1.25

1.3

1.35

1.4

0 5 10 15 20 25 30

mc(

mc)

/ G

eV

a4 (10-5 fm4)

2+12+1 result

2+1+1

For the HPQCD collaboration Short Title 20/24

Decay constants from 2+1+1Preliminary results (1210.8431) from the MILC and Fermilablattice collaboration.

For the HPQCD collaboration Short Title 21/24

Conclusions

• The HPQCD collaboration has computed mc , αs and mb.• To further validate the calculation HPQCD has computed

MJ/ψ - Mηc , fJ/ψ and Γ(J/ψ → ηcγ).

The future• The first preliminary results with 2+1+1 sea quarks are

consistent with those with 2+1 sea quarks.• The new 2+1+1 lattice QCD calculations include “physical

pion masses”.• Updated results for mc and αs require finer lattice

spacings. These runs are ongoing.

For the HPQCD collaboration Short Title 22/24

Backup

Backup

For the HPQCD collaboration Short Title 23/24

Summary of lattice methods tocompute αs

Method scale range GeV pert. non-perturb.Wilson loops Υ fπ 2.1 - 14.7 α3

s a4〈αsG2/π〉Charm moments Υ fπ 3 α3

s a4〈αsG2/π〉Light vaccum pol r0 fπ Ω 1.8 α3

s ψψ , 〈ss〉〈uu〉 ,a

4〈αsG2/π〉Static energy r1 fπ 0.8 - 2.9 N3LL renormalons ?

Schrodinger funct. Ω 16 α3s -

Glue/ghost fπ 1.7 - 6.8 4 loop dp6

Table: Summary of lattice methods to compute αs

For the HPQCD collaboration Short Title 24/24