Charm physics on the lattice with highly improved staggered quarks - - PowerPoint PPT Presentation

Charm physics on the lattice with highly improved staggered quarks Eduardo Follana

The Ohio State University

(Fermilab, April 2008)

HPQCD collaboration Work in collaboration with: C.T.H. Davies (University of Glasgow)

• K. Hornbostel (Dallas Southern Methodist University)

G.P. Lepage (Cornell University)

• Q. Mason (Cambridge University)
• J. Shigemitsu (The Ohio State University)
• H. Trottier (Simon Fraser University)
• K. Wong (University of Glasgow)

Thanks: The MILC collaboration for making their configurations publicly available.

Outline

◮ Motivation. ◮ Staggered quarks.

◮ HISQ (Highly improved staggered quarks.)

◮ Heavy quarks. ◮ Charmed systems: masses and decay constants. ◮ Outlook.

Phys.Rev.D75:054502,2007, Phys.Rev.Lett.100:062002,2008.

Motivation

◮ Low-energy QCD is a strongly-coupled QFT. We need

non-perturbative tools to deal with it.

◮ Other strongly-coupled sectors BSM?

◮ Lattice QCD provides a non-perturbative definition of QCD. It

also provides a quantitative calculational tool. And lately it is also becoming a precise tool.

Goals

◮ To make precise calculations in QCD.

◮ To test lattice field theory as a tool for studying strongly

coupled field theories. (CLEO-c).

◮ fD, fDs ◮ To calculate theoretical quantities needed in the analysis of

experimental data, for example, in the determination of elements of the CKM matrix.

◮ To further test QCD as the theory of strong interactions.

◮ To deepen our understanding of the physics of QCD, for

example, confinement.

LQCD: Quenched vs Unquenched

◮ Fermions are numerically very hard to include. ◮ Ignore fermion pair production ⇒ quenched QCD.

0.9 1 1.1 Quenched 0.9 1 1.1 nf=2+1 Υ(3S-1S) Υ(1P-1S) Υ(2P-1S) Υ(1D-1S) 2mBs,av-mΥ ψ(1P-1S) 2mDs-mηc mΩ 3mΞ - mN fK fπ

Plus the successful prediction of mBc (I. Allison et al).

(Some) systematic errors

◮ Finite volume: m−1 π

≪ L. In practice, L ≈ 2.5, 3fm

◮ Finite lattice spacing: we need simulations at different values

• f a, to extrapolate to the continuum limit a → 0.

◮ To simulate at small values of a, while keeping the physical L

constant is very expensive.

◮ Typically, error ∝ a, a2 ◮ Improved actions (and operators) decrease the error, making

the extrapolation from a given set of lattice spacings more precise.

◮ Chiral extrapolation: In practice, we are not able to simulate

at physical values of the light quark masses mu,d.

◮ Lattice spacing determination: Error in the determination of

the lattice spacing in physical units (r1).

Improved Staggered Quarks

◮ The staggered action describes 4 tastes (in 4D). The spectrum

• n the lattice has a multiplicity of states corresponding to the

same continuum state. There are unphysical taste-changing interactions that lift the degeneracy between such states.

◮ These effects are lattice artifacts, of order a2, and vanish in

the continuum limit a → 0. They involve at leading order the exchange of a gluon of momentum q ≈ π/a.

◮ Such interactions are perturbative for typical values of the

lattice spacing, and can be corrected systematically a la Symanzik.

p=0 p=0 p=π/a p=-π/a

Smear the gauge field to remove the coupling between quarks and gluons with momentum π/a.

◮ In an unquenched simulation,

4

√

• det. → ”Rooting trick”.

Improved Staggered Actions

◮ FAT7(TAD)

+ + + = c1 (Fat link) c5 c7 c3

Improved Staggered Actions

◮ ASQ(TAD)

+ + + + = = (Naik) c5’ c1 (Fat link) c3’ c5 c7 c3

(S. Naik, the MILC collaboration, P. Lepage.)

◮ Discretization errors ≈ O(αsa2, a4).

Improved Staggered Actions

◮ HISQ

Two levels of smearing: first a FAT7 smearing on the original links, followed by a projection onto SU(3), then a modified ASQ on these links. FAT7SU(3) ⊗ ASQ’ (E.F., Q. Mason, C. Davies, K. Hornbostel, P. Lepage, H. Trottier.)

◮ Discretization errors ≈ O(αsa2, a4). ◮ Substantially reduced taste-changing with respect to

ASQTAD.

Heavy Quarks

◮ The discretization errors grow with the quark mass as powers

• f am.

◮ For a direct simulation, we need:

amh ≪ 1 (heavy quarks) La ≫ m−1

π

(light quarks)

◮ Two scales. Difficult to do directly. ◮ Instead take advantage of the fact that mh is large: ⇒

effective field theory (NRQCD, HQET). Very successful for b quarks.

Charm Quarks

◮ The charm quark is in between the light and heavy mass

regime.

◮ Quite light for an easy application of NRQCD. ◮ Quite large for the usual relativistic quark actions, amc <

∼ 1.

◮ However, if we use a very accurate action (HISQ) and fine

enough lattices (MILC), it is possible to get accurate results.

◮ Errors for HISQ: O((am)4, αs(am)2). ◮ Non-relativistic system: can be tuned for further suppression

by factors of (v/c).

◮ Can reduce the errors to the few percent level. ◮ Simple: use the same action in the heavy and the light sector.

◮ We will use this action both for heavy-heavy and heavy-light

systems ⇒ consistency check.

Fixing the parameters

The free parameters in the lattice formulation are fixed by setting a set of calculated quantities to their measured physical values.

◮ Scale: lattice spacing a: Fixed through the upsilon (b¯

b) spectrum, mΥ(2S) − mΥ(1S).

◮ Quark masses: mu,d, ms, mc. Fixed by mπ, mK, mηc. ◮ In the HISQ charm quark formulation: improvement parameter

ǫ. Fixed by requiring relativistic dispersion relation, c2 = 1.

Configurations

MILC ensembles: 2 + 1 ASQTAD sea quarks: (ml, ml, ms)

◮ Very coarse: a ≈ 0.16 fm, 163x 48

◮ ml = ms/2.5, ms/5 ◮ Valence HISQ: amc = .85

◮ Coarse: a ≈ 0.12 fm

◮ ml = ms/2, ms/4

203x 64

◮ ml = ms/8,

243x 64

◮ Valence HISQ: amc = .66

◮ Fine: a ≈ 0.09 fm,

283x 96.

◮ ml = ms/2.5, ms/5 ◮ Valence HISQ: amc = .43.

c2

◮ We adjust the coefficient of the Naik term to have c2 = 1.

This further reduces the discretization errors by factors of v

c .

0.9 0.95 1 1.05 1.1 0.6 0.8 1 1.2 1.4 1.6 1.8 2 c2

• coeff. Naik term

c2, fine MILC, asqtad ma=0.38

Masses

◮ We use the mass of the ηc to fix the mass of the charm quark. 2.8 3 3.2 3.4 3.6 3.8 4 mass /GeV charmonium masses, HISQ on fine MILC ηc(1S) ψ(1S) ηc(2S) ψ(2S) χc0 χc1

Decay constants

◮ Meson decay constants:

Γ(P → lνl(γ)) = G 2

F|Vab|2

8π fP2m2

l mP

• 1 − m2

l

m2

P

2 0|Aµ|P(p) = fPpµ PCAC: fPmP2 = (ma + mb) < 0|¯ aγ5b|P >

◮ We do a simultaneous bayesian fit of the masses and decay

constants to the chiral and continuum limits.

◮ Essentially the same calculation for fπ, fK, fD, fDs.

Masses and decay constants

mDs = 1.963(5) (exp. 1.968) GeV. mD = 1.869(6) (exp. 1.869) GeV.

(2mDs −mηc ) (2mD−mηc ) = 1.249 (14)

(exp. 1.260(2)) GeV

Mass differences

◮ We plot mDs(ml) − mD(ml) and mBs(ml) − mB(ml) as a

function of the sea light quark mass, ml.

20 40 60 80 100 120 140 0.2 0.4 0.6 0.8 1 (mDs/Bs - mD/B ) / MeV sea mu/d/ms HPQCD B + D meson masses PRELIM FEB07 D coarse expt D B coarse expt B

Decay constants

1 1.05 1.1 1.15 1.2 1.25 1.3 0.2 0.4 0.6 0.8 1 fK/fπ mu,d / ms very coarse coarse fine extrapoln

fπ = 132(2) (Exp 130.5(4)) MeV fK = 157(2) (Exp 156.0(8))MeV fK fπ = 1.189(7) (Exp 1.196(6)) Using experimental leptonic branching fractions (KLOE) Vus = 0.2262(13)(4) This gives the unitarity relation 1 − V 2

ud − V 2 us − V 2 ub = 0.0006(8)

Decay constants

1 1.05 1.1 1.15 1.2 1.25 1.3 0.2 0.4 0.6 0.8 1 fDs/fD mu,d / ms very coarse coarse fine extrapoln

fDs = 241(3) MeV fD = 208(4) (Exp 223(17)) MeV Using experimental values from CLEO-c for µ decay: Vcs = 1.07(1)(7) (PDG : 0.96(9)) r1 stat a2 ml ms evol vol isospin, QED tot % error fDs 1.0 0.6 0.5 0.3 0.3 0.1 0.0 1.3

Decay constants

fDs fD = 1.162(9)

Using experimental values from CLEO-c for µ decay: Vcs Vcd = 4.42(4)(41) Double ratios: fDs/fD fK/fπ = 0.977(10) fBs/fB fDs/fD = 1.03(3)

fDs

220 240 260 280 300 320 340 PDG tau mu average lattice

(Bogdan A. Dobrescu, Andreas S. Kronfeld, arXiv:0803.0512)

HISQ2 and hyperfine splitting

PRELIMINARY

• 5

5 10 15 20 25 30 0.05 0.1 0.15 0.2 0.25 0.3 Taste-split / MeV (a/r1)2 Taste-splittings vs a2 fηc HISQ locng - gold fηc DHISQ locng - gold 0.1 0.12 0.14 0.16 0.18 0.2 0.05 0.1 0.15 0.2 0.25 0.3 hyperfine splitting / GeV (a/r1)2 Hyperfine splittings 1-link ψ local ψ 1-link Ds

*

local Ds

*

DHISQ local ψ expt corrected expt ψ

Conclusions

◮ The use of a highly improved quark action and fine enough

lattices provides a very good way of studying systems with charm quarks from first principles.

◮ We can calculate accurately a number of interesting

• quantities. At present all but fDs agree with experiment.

Outlook

◮ Direct determination of mc from the lattice. Needs

perturbative calculation (underway.) Accurate mc/ms.

◮ New method for the calculation of mc (in collaboration with

• K. Chetyrkin et al, Karlsruhe.) combining continuum

perturbation results for the moments of the ηc correlator with lattice data. Preliminary, work in progress.

◮ Leptonic decay width ψ → e+e−. Known accurately from

experiment (∼ 2%).

◮ Semileptonic form factors: D → πlν, D → Klν