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). ◮ f D , f D s ◮ 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. f π f K 3m Ξ - m N m Ω 2m D s -m η c ψ (1P-1S) 2m B s,av -m Υ Υ (3S-1S) Υ (2P-1S) Υ (1P-1S) Υ (1D-1S) 0.9 1 1.1 0.9 1 1.1 n f =2+1 Quenched Plus the successful prediction of m B c (I. Allison et al).
(Some) systematic errors ◮ Finite volume: m − 1 ≪ L . In practice, L ≈ 2 . 5 , 3 fm π ◮ Finite lattice spacing: we need simulations at different values of 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 , a 2 ◮ 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 m u , d . ◮ Lattice spacing determination: Error in the determination of the lattice spacing in physical units ( r 1 ).
Improved Staggered Quarks ◮ The staggered action describes 4 tastes (in 4D). The spectrum on 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 a 2 , 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. Smear the gauge field p=0 p= π /a to remove the coupling between quarks and gluons with momentum p=0 π/ a . p=- π /a √ ◮ In an unquenched simulation, 4 det. → ”Rooting trick”.
Improved Staggered Actions ◮ FAT7(TAD) + = + + (Fat link) c3 c5 c7 c1
Improved Staggered Actions ◮ ASQ(TAD) + + = + + (Fat link) c5’ c1 c3 c5 c7 = (Naik) c3’ (S. Naik, the MILC collaboration, P. Lepage.) ◮ Discretization errors ≈ O ( α s a 2 , a 4 ).
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. FAT7 � SU (3) ⊗ ASQ’ (E.F., Q. Mason, C. Davies, K. Hornbostel, P. Lepage, H. Trottier.) ◮ Discretization errors ≈ O ( α s a 2 , a 4 ). ◮ Substantially reduced taste-changing with respect to ASQTAD.
Heavy Quarks ◮ The discretization errors grow with the quark mass as powers of am . ◮ For a direct simulation, we need: am h ≪ 1 (heavy quarks) La ≫ m − 1 (light quarks) π ◮ Two scales. Difficult to do directly. ◮ Instead take advantage of the fact that m h 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, am c < ∼ 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 , α s ( 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, m Υ(2 S ) − m Υ(1 S ) . ◮ Quark masses: m u , d , m s , m c . Fixed by m π , m K , m η c . ◮ In the HISQ charm quark formulation: improvement parameter ǫ . Fixed by requiring relativistic dispersion relation, c 2 = 1.
Configurations MILC ensembles: 2 + 1 ASQTAD sea quarks: ( m l , m l , m s ) ◮ Very coarse: a ≈ 0 . 16 fm , 16 3 x 48 ◮ m l = m s / 2 . 5 , m s / 5 ◮ Valence HISQ: am c = . 85 ◮ Coarse: a ≈ 0 . 12 fm ◮ m l = m s / 2 , m s / 4 20 3 x 64 ◮ m l = m s / 8 , 24 3 x 64 ◮ Valence HISQ: am c = . 66 ◮ Fine: a ≈ 0 . 09 fm , 28 3 x 96. ◮ m l = m s / 2 . 5 , m s / 5 ◮ Valence HISQ: am c = . 43.
c 2 ◮ We adjust the coefficient of the Naik term to have c 2 = 1. This further reduces the discretization errors by factors of v c . c 2 , fine MILC, asqtad ma=0.38 1.1 1.05 1 c 2 0.95 0.9 0.6 0.8 1 1.2 1.4 1.6 1.8 2 coeff. Naik term
Masses ◮ We use the mass of the η c to fix the mass of the charm quark. charmonium masses, HISQ on fine MILC 4 3.8 ψ (2S) 3.6 η c (2S) χ c1 mass /GeV 3.4 χ c0 3.2 ψ (1S) 3 η c (1S) 2.8
Decay constants ◮ Meson decay constants: � 2 G 2 F | V ab | 2 1 − m 2 � f P 2 m 2 l Γ( P → l ν l ( γ )) = l m P m 2 8 π P � 0 | A µ | P ( p ) � = f P p µ PCAC: f P m P 2 = ( m a + m b ) < 0 | ¯ a γ 5 b | P > ◮ We do a simultaneous bayesian fit of the masses and decay constants to the chiral and continuum limits. ◮ Essentially the same calculation for f π , f K , f D , f D s .
Masses and decay constants m D s = 1 . 963(5) (exp. 1.968) GeV. m D = 1 . 869(6) (exp. 1.869) GeV. (2 m Ds − m η c ) (2 m D − m η c ) = 1 . 249 (14) (exp. 1.260(2)) GeV
Mass differences ◮ We plot m D s ( m l ) − m D ( m l ) and m B s ( m l ) − m B ( m l ) as a function of the sea light quark mass, m l . HPQCD B + D meson masses PRELIM FEB07 D coarse 140 expt D B coarse 120 expt B (m D s /B s - m D/B ) / MeV 100 80 60 40 20 0 0 0.2 0.4 0.6 0.8 1 sea m u/d /m s
Decay constants f π = 132(2) ( Exp 130 . 5(4)) MeV = 157(2) ( Exp 156 . 0(8)) MeV f K 1.3 very coarse coarse fine f K 1.25 extrapoln = 1 . 189(7) ( Exp 1 . 196(6)) f π 1.2 Using experimental leptonic f K /f π branching fractions (KLOE) 1.15 1.1 V us = 0 . 2262(13)(4) 1.05 This gives the unitarity relation 1 0 0.2 0.4 0.6 0.8 1 m u,d / m s 1 − V 2 ud − V 2 us − V 2 ub = 0 . 0006(8)
Decay constants 1.3 very coarse coarse f D s = 241(3) MeV fine 1.25 extrapoln f D = 208(4) ( Exp 223(17)) MeV 1.2 Using experimental values from CLEO-c for µ decay: f D s /f D 1.15 V cs = 1 . 07(1)(7) ( PDG : 0 . 96(9)) 1.1 1.05 1 0 0.2 0.4 0.6 0.8 1 m u,d / m s a 2 r 1 stat m l m s evol vol isospin, QED tot % error f D s 1.0 0.6 0.5 0.3 0.3 0.1 0.0 1.3
Decay constants f Ds f D = 1 . 162(9) Using experimental values from CLEO-c for µ decay: V cs = 4 . 42(4)(41) V cd Double ratios: f D s / f D = 0 . 977(10) f K / f π f B s / f B = 1 . 03(3) f D s / f D
f D s PDG tau mu average lattice 220 240 260 280 300 320 340 (Bogdan A. Dobrescu, Andreas S. Kronfeld, arXiv:0803.0512)
HISQ2 and hyperfine splitting PRELIMINARY Taste-splittings vs a 2 Hyperfine splittings 30 0.2 f η c HISQ locng - gold 1-link ψ f η c DHISQ locng - gold local ψ * 25 1-link D s * 0.18 local D s DHISQ local ψ hyperfine splitting / GeV 20 expt Taste-split / MeV corrected expt ψ 0.16 15 10 0.14 5 0.12 0 -5 0.1 0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.05 0.1 0.15 0.2 0.25 0.3 (a/r 1 ) 2 (a/r 1 ) 2
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