Charm decay constants and semileptonic form factors from lattice QCD Ruth Van de Water Fermilab CKM 2006 December 14, 2006 B K
Decays of charmed mesons Leptonic Decays Semileptonic Decays ν } q 2 l W D,Ds π , K D B.R.( D → �ν ) = (known factor) × f 2 D | V cd | 2 f D,Ds are the D-meson decay constants: } if P ci p µ = � 0 | cγ µ γ 5 q i | P ci � B.R.( D s → �ν ) = (known factor) × f 2 D s | V cs | 2 � q 2 ( q 2 ) 2 × (known factor) max B.R.( D → π�ν ) = | V cd | 2 dq 2 f D → π f + (q 2 ) are the D-meson } + 0 form factors: � q 2 max ( q 2 ) 2 × (known factor) B.R.( D → K�ν ) = | V cs | 2 dq 2 f D → K + 0 In both cases, experiments measure a hadronic M.E. times a CKM element R. Van de Water Charm decay constants and semileptonic form factors from lattice QCD 2 /19
Why calculate f D,Ds and D → π ,K on the lattice? (1) Can combine experimental measurements of branching fractions with lattice calculations of decay constants & form factors to extract |V cd |, |V cs | (2) Can combine experimental measurements of branching fractions with values of |V cd |,|V cs |from elsewhere to experimentally determine decay constants or form factors, then compare with lattice QCD calculations Approach #2 provides a test of lattice QCD methods, e.g. : Dynamical (sea) quark effects Light quark formalism Heavy quark formalism Chiral extrapolations Correct lattice QCD results for D-mesons give confidence in similar lattice calculations with B-mesons R. Van de Water Charm decay constants and semileptonic form factors from lattice QCD 3 /19
Current lattice measurements of D decays CAVEAT : This talk will be restricted to three-flavor unquenched lattice calculations Currently two groups calculating heavy-light meson quantities with three dynamical quark flavors: Fermilab/MILC & HPQCD Both use the publicly available “2+1 flavor” MILC configurations [Phys.Rev.D70:114501,2004] which have three flavors of improved staggered quarks: Two degenerate light quarks and one heavy quark ( ≈ m s ) Light quark mass ranges from m s / 10 ≤ m l ≤ m s Groups use different heavy quark discretizations : Fermilab/MILC uses Fermilab quarks HPQCD uses nonrelativistic (NRQCD) heavy quarks R. Van de Water Charm decay constants and semileptonic form factors from lattice QCD 4 /19
Systematics in lattice calculations Lattice calculations typically quote the following sources of error: 1. Monte carlo statistics & fitting 2. Tuning lattice spacing, , and quark masses a 3. Matching lattice gauge theory to continuum QCD (Sometimes split up into relativistic errors, discretization errors, perturbation theory, ...) 4. Extrapolation to continuum 5. Chiral extrapolation to physical up, down quark masses Errors #3 and #5 are dominant sources of uncertainty in current heavy-light lattice calculations -- will discuss them in turn R. Van de Water Charm decay constants and semileptonic form factors from lattice QCD 5 /19
Heavy quarks on the lattice PROBLEM: Generic lattice quark action will have discretization errors ∝ ( am Q ) n SOLUTION: Use knowledge of the heavy quark/nonrelativistic quark limits of QCD to systematically eliminate HQ discretization errors order-by-order FERMILAB METHOD LATTICE NRQCD [Phys.Rev.D55:3933-3957,1997] [Phys.Rev.D46:4052-4067,1992] Continuum QCD Continuum QCD (using Nonrelativistic QCD HQET) Lattice gauge theory Lattice gauge theory Both methods require tuning parameters of lattice action For heavy-light decays, must also match lattice currents to continuum Typically calculate matching coefficients in lattice perturbation theory [Phys.Rev.D48:2250-2264,1993] R. Van de Water Charm decay constants and semileptonic form factors from lattice QCD 6 /19
Matching errors In principle, can remove errors of any order in heavy quark mass, but, in practice, becomes increasingly difficult at each higher order ⇒ Must estimate size of errors due to inexact matching FERMILAB METHOD LATTICE NRQCD QCD QCD “relativistic errors” , “heavy quark e.g. O( α S Λ QCD / m Q ) & O( Λ QCD 2 / m Q 2 ) discretization effects” NRQCD LGT “ perturbation theory errors” , e.g. O( α S 2 ) Combine all errors associated LGT with discretizing action Estimate errors using knowledge of Estimate errors using power-counting short-distance coefficients and power-counting R. Van de Water Charm decay constants and semileptonic form factors from lattice QCD 7 /19
Chiral extrapolation of lattice data Must extrapolate lattice results to physical values of up, down quark mass For MILC 2+1 flavor lattices, must use staggered chiral perturbation theory [Lee & Sharpe, Aubin & Bernard, Sharpe & RV] Accounts for next-to-leading order light quark mass dependence Also accounts for light quark discretization effects through O( α S 2 a 2 Λ QCD 2 ) Extremely successful for light-light meson quantities such as f π Comment: Staggered results agree with experimental values after chiral extrapolation in large part because the simulated quark masses are light and the lattice results are already close to the correct answer R. Van de Water Charm decay constants and semileptonic form factors from lattice QCD 8 /19
Lattice results for f D,Ds B K
HPQCD calculation of f Ds f Ds = 290 ± 20 ± 29 ± 29 ± 6 MeV generic perturbation relativistic statistics & fitting discretization theory corrections effects Agrees with experiment: f Ds = 279 ± 17 ± 20 MeV [BaBar] Statistical error dominated by f Q m Q1/2 (GeV 3/2 ) extrapolation of m Q to charm quark mass Perturbation theory error from 1-loop lattice-to-continuum B s D s operator matching [Phys.Rev.Lett.92:162001,2004] 1/m Q (GeV -1 ) R. Van de Water Charm decay constants and semileptonic form factors from lattice QCD 10 /19
Fermilab/MILC calculation of f D,Ds chiral lattice spacing & m c tuning extrapolation 320 hep-ph/9711426 [Fermilab] 300 hep-lat/0206016 [MILC] 280 hep-lat/0506030 [Fermilab + MILC] f D+ = 201 ± 3 ± 6 ± 9 ± 13 MeV BaBar (Moriond 2006) f Ds (MeV) 260 240 f Ds = 249 ± 3 ± 7 ± 11 ± 10 MeV 220 200 180 statistics heavy quark 160 0 1 2 3 n f (a) discretization effects [Phys.Rev.Lett.95:122002,2005] 320 hep-ph/9711426 [Fermilab] 300 hep-lat/0206016 [MILC] Simulate directly at charm quark mass 280 hep-lat/0506030 [Fermilab + MILC] hep-ex/0508057 [CLEO-c] f D+ (MeV) 260 Current matching partly nonperturbative 240 220 f D +, f Ds calculations preceded Cleo-c 200 measurements ⇒ lattice predictions 180 160 0 1 2 3 n f Results finalized since CKM 2005 (b) R. Van de Water Charm decay constants and semileptonic form factors from lattice QCD 11 /19
Potential sources of improvement For HPQCD: 2-loop perturbative (or nonperturbative) matching Highly-improved staggered quark (HISQ) action to simulate directly at charm ( in progress -- hep-lat/0610092) For Fermilab/MILC: 2-loop matching of heavy-light current ρ -factor Nonperturbative determination of clover coefficient in heavy-quark action ( e.g. see Lin & Christ) Improved heavy-quark action ( in progress -- Kronfeld & Oktay) In general: Lighter quark masses and finer lattice spacings Heavy-light calculations with different light quark action , e.g domain-wall (RBC) or overlap fermions (JLQCD) R. Van de Water Charm decay constants and semileptonic form factors from lattice QCD 12 /19
Extension to f B Successful predictions of f D , f Ds lend confidence in lattice methods The ratio of decay constants, in which R d/slat. = 0.786 ± 0.043 several lattice uncertainties cancel, is particularly compelling: R d/sexp. = 0.779 ± 0.093 [lat: Phys.Rev.Lett.95:122002,2005; exp: Cleo-c/BaBar] HPQCD f B better than f D because can simulate directly at b quark mass f B = 216(9)(19)(4 )(6 )MeV HPQCD: Phys.Rev.Lett.95:212001,2005 f Bs /f B = 1.20(3)(1) Fermilab/MILC f B comparable to f D , and heavy quark discretization errors somewhat smaller Fermilab/MILC: f Bs /f Ds = 0.99(2)(6) Simone, Lattice ’06 f B /f D = 0.95(3)(6) (Preliminary) R. Van de Water Charm decay constants and semileptonic form factors from lattice QCD 13 /19
Lattice results for D → π ,K B K
Fermilab/MILC calculation of D → π sys. stat. "+, -./0%1234 ? @ (567-%89:;< f +D → π (0) = 0.64(3)(6) " |V cd | = 0.239(10)(24)(20) ? ! = ! > ! !+, ! " # stat. sys. exp. # %&'() # * $ (Statistical errors only) [Phys.Rev.Lett.94:011601,2005] Given |V cd |, result for f(0) consistent with experiment Conversely, 14% measurement of |V cd | -- error dominated by discretization effects: 5% from lattice momenta 7% from heavy quark discretization R. Van de Water Charm decay constants and semileptonic form factors from lattice QCD 15 /19
Fermilab/MILC calculation of D → K 2 2 q max / m Ds * Form factor shape and normalization 2.5 consistent with experiment D → Kl ν 2 Calculations preceded Focus, Belle, 1.5 BaBar measurements 2 ) f + ( q ⇒ lattice prediction 1 0.5 experiment [Belle, hep-ex/0510003] lattice QCD [Fermilab/MILC, hep-ph/0408306] sys. stat. 0 0 0 0.1 0.2 0.3 0.4 0.05 0.15 0.25 0.35 0.45 2 / m Ds 2 q (b) * 11% measurement of |V cs | -- f +D → K (0) = 0.73(3)(7) error dominated by |V cs | = 0.969(39)(94)(24) discretization effects: 5% from lattice momenta stat. sys. exp. 7% from heavy quark discretization [Phys.Rev.Lett.94:011601,2005] R. Van de Water Charm decay constants and semileptonic form factors from lattice QCD 16 /19
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