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Weak Charm Decays with Lattice Q C D

Weak Charm Decays with Lattice Q C D. Aida X. El-Khadra University of Illinois. Charm 2007 workshop, Aug. 5-8, 2007. Outline. 1. Introduction 2. Light Quark Methods 3. Heavy Quark Methods 4. Semileptonic Decays 5. Leptonic Decay Constants 6. Conclusions and Outlook.

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Weak Charm Decays with Lattice Q C D

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  1. Weak Charm Decays with Lattice QCD Aida X. El-Khadra University of Illinois Charm 2007 workshop, Aug. 5-8, 2007

  2. Outline • 1. Introduction • 2. Light Quark Methods • 3. Heavy Quark Methods • 4. Semileptonic Decays • 5. Leptonic Decay Constants • 6. Conclusions and Outlook A. El-Khadra, Charm 2007, Aug 5-8, 2007

  3. Collaborations Fermilab Lattice collaboration: Kronfeld, Mackenzie, Simone, Di Pierro, Gottlieb, AXK, Freeland, Gamiz, Laiho, Van de Water, Evans, Jain Computations at FNAL Lattice QCD clusters MILC: Bernard, DeTar, Gottlieb, Heller, Hetrick, Sugar, Toussaint Levkova, Renner HPQCD: Davies, Hornbostel, Lepage,Shigemitsu, Trottier Follana, Wong A. El-Khadra, Charm 2007, Aug 5-8, 2007

  4. ne e+ D d c p _ u  Introduction Example:Semileptonic D meson decay parameterize the matrix element in terms of form factors A. El-Khadra, Charm 2007, Aug 5-8, 2007

  5. “Gold-Plated” Quantities or What are the “easy” lattice calculations ? For stable (or almost stable) hadrons, masses and amplitudes with no more than one initial (final) state hadron, for example: • p, K, D, Ds, B, Bsmesons • masses, decay constants, weak matrix elements for mixing, • semileptonic and rare decays • charmonium and bottomonium (hc, J/y, hc, …, hb, U(1S), U(2S), ..) • states below open D/B threshold • masses, leptonic widths, electromagnetic matrix elements This list includes most of the important quantities for CKM physics. Excluded arermesons and other resonances. A. El-Khadra, Charm 2007, Aug 5-8, 2007

  6. Vud Vus Vub Vcd Vcb Vcs Vtd Vts Vtb mixing Lattice QCD program relevant to many CKM elements K  p ln B  p ln p mn , K  mn D  K ln Ds ln D p ln D ln B  D, D* ln A. El-Khadra, Charm 2007, Aug 5-8, 2007

  7. L x a … discretize the QCD action (Wilson) e.g.discrete derivative in QCD Lagrangian wherec(a) = c(a;as,m)depends on the QCD parameters calculable in pert. theory Introduction to Lattice QCD fermion fieldlives on sites:y(x) A. El-Khadra, Charm 2007, Aug 5-8, 2007

  8. Introduction to Lattice QCD, cont’d in general:n  1 errors scale with the typical momenta of the particles, e.g.(LQCDa)nfor gluons and light quarks. keep1/a LQCD LQCD~ 200 – 300 MeV typical lattice spacinga 0.1 fm  1/a ~ 2 GeV in practice: need to consider a range of a’s. Improvement: add more terms to the action to make n large A. El-Khadra, Charm 2007, Aug 5-8, 2007

  9. finite lattice spacing,a: take continuum limit: computational effort grows like ~(L/a)6 L a (fm) • perturbation theory errors, errors, errors, … • statistical errors: from monte carlo integration • finite volume • mldependence: chiral extrapolation • nfdependence: sea quark effects: results with nf = 2+1 exist A. El-Khadra, Charm 2007, Aug 5-8, 2007

  10. In numerical simulations, ml > mu,dbecause of the computational cost for smallm. • use chiral perturbation theory to extrapolate tomu,d need ml < ms/2and several different values forml (easier with staggered than Wilson-type actions) f ml ms/2 Decay constants, form factors: chiral logs contribute ~ systematic errors, cont’d • chiral extrapolation, mldependence: Staggered chiral perturbation theory:(Bernard, Sharpe, Aubin,…) remove leadingO(a2)errors in fits A. El-Khadra, Charm 2007, Aug 5-8, 2007

  11. Light Quark Methods • Asqtad (improved staggered):(Kogut+Susskind, Lepage, MILC) • errors: ~O(asa2), O(a4), but large due to taste-changing interactions • has chiral symmetry; uses square root of the determinant in sea • computationally efficient • tested against experiment at the few percent level A. El-Khadra, Charm 2007, Aug 5-8, 2007

  12. Testing the rooted Asqtad action HPQCD+MILC+FNAL, C. Davies, et al,Phys. Rev. Lett. 92:022001,2004 lattice QCD/experiment before 2004 works quite well! A. El-Khadra, Charm 2007, Aug 5-8, 2007

  13. Light Quark Methods • Asqtad (improved staggered):(Kogut+Susskind, Lepage, MILC) • errors: ~O(asa2), O(a4), but large due to taste-changing interactions • has chiral symmetry; uses square root of the determinant in sea • computationally efficient • HISQ (Highly Improved Staggered Action):(Follana, Hart, Davies, Follana et. al) • errors: ~O(asa2), O(a4),×1/3smaller than Asqtad • comp. cost: efficicient,×2 Asqtad • improvedWilson (Clover, …):(Wilson, Sheikholeslami + Wohlert, etc …) • errors: ~O(asa), O(a2)if tree-level (tadpole) imp.;O(a2)if nonpert. imp. • Wilson term breaks chiral symmetry • comp. cost: ×4Asqtad formlight ~ mstrange , but inefficient at small quark masses • Domain Wall Fermions:(Kaplan) • errors: ~O(a2), O(mresa) • almost exact chiral symmetry; breaking ~mres ~ 3 ×10-3 • comp. cost: ×L5Asqtad,L5~ 16- 20 • Overlap Fermions:(Neuberger) • errors: ~O(a2) • exact chiral symmetry • comp. cost: ×5-10DWF A. El-Khadra, Charm 2007, Aug 5-8, 2007

  14. Simulation parameters Ensembles generated by MILC using the Asqtad action. Each point has 400 – 800 configurations. A. El-Khadra, Charm 2007, Aug 5-8, 2007

  15. lattice NRQCD (Lepage, et al., Caswell+Lepage): • discretize NRQCD lagrangian: valid whenamQ > 1 • errors: (ap)n , (p/mQ)n • good for b quarks, but not charm Fermilab(Kronfeld, Mackenzie, AXK): • rel. Wilson action has the same heavy quark limit as QCD • add improvement: preserve HQ limit • smoothly connects light and heavy mass limits, valid for allamQ • errors: as(aL), (aL)2orasL/mQ, ,(L/mQ)n • good for for charm and beauty HISQ(Follana, Hart, Davies, Follana et. al): • errors: ~as(amc)2, (amc)4 • good for charm, but not beauty Heavy Quark Methods mQ LQCD and amQ1: A. El-Khadra, Charm 2007, Aug 5-8, 2007

  16. pp (q2) dependence: • pp 1GeV improved actions help (keepnlarge) • finite volume (L): • fora = 0.1 fm, L = 20, pmin =620 MeV Semileptonic Decays: D p (K) ln • Lattice Result with nf = 2+1 from Fermilab Lattice and MILC • collaboration (C. Aubin et al, PRL 2005). • using MILC coarse ensembles with msea = 1/8 ms, …., ¾ ms • Asqtad action for light valence quarks • Fermilab action for charm quarks • staggered chiral pert. thy (mvalence, msea, a2) • but:q2 dependence parameterized using BK model • only one lattice spacing (a 0.12 fm) A. El-Khadra, Charm 2007, Aug 5-8, 2007

  17. Semileptonic Decays cont’d From V. Pavlunin @ FPCP 2007: The shape is determined more accurately than the normalisation in the Lattice QCD calculation A. El-Khadra, Charm 2007, Aug 5-8, 2007

  18. Semileptonic Decays cont’d Kronfeld (Fermilab Lattice and MILC, 2005): A. El-Khadra, Charm 2007, Aug 5-8, 2007

  19. Semileptonic Decays cont’d Kronfeld (Fermilab Lattice and MILC, 2005): A. El-Khadra, Charm 2007, Aug 5-8, 2007

  20. Semileptonic Decays: Improvements • add more lattice spacings to analysis:  reduce discretisation errors • q2dependence: z-expansion(Arnesen et al, Becher+Hill, Flynn+Nieves, …) based on unitarity and analyticity  model independent analysis of the shape (fig) A. El-Khadra, Charm 2007, Aug 5-8, 2007

  21. PRELIMINARY Semileptonic Decays: Improvements Van de Water + Mackenzie (Fermilab Lattice and MILC, 2006/7): q2dependence fit using z-expansion A. El-Khadra, Charm 2007, Aug 5-8, 2007

  22. Semileptonic Decays: Improvements • add more lattice spacings to analysis:  reduce discretisation errors • q2dependence: z-expansion(Arnesen et al, Becher+Hill, Flynn+Nieves, …) based on unitarity and analyticity  model independent analysis of the shape (fig) • finite volume: twisted boundary conditions(Tantalo, Bedaque, Sachrajda,….)  arbitrarily small momentapi < 2p/L • further technical improvements: • random wall sources(MILC): • improve statistics • double ratios to extract form factors(FNAL, RBC/UKQCD, Becirevic, Haas, improve statistics Mescia) • reduce systematic errors • ….. • repeat calculation with other valence quarks, e.g. HISQ (HPQCD) A. El-Khadra, Charm 2007, Aug 5-8, 2007

  23. Leptonic D and Ds Meson Decay Constants • important test of LQCD methods • results from two groups (FNAL/MILC & HPQCD) using MILC • ensembles at a = 0.09 fm, 0.12 fm, 0.15 fm A. El-Khadra, Charm 2007, Aug 5-8, 2007

  24. Comparison of parameters FNAL/MILC Fermilab action for charm Asqtad light valence quarks a = 0.09 fm: msea = 1/10 ms, 1/5 ms, 2/5 ms a = 0.12 fm: msea = 1/8 ms, 1/4 ms, 1/2 ms, ¾ ms a = 0.15 fm: msea = 1/10 ms, 1/5 ms, 2/5 ms, 3/5 ms + 8-12 valence quark masses/ensemble partial nonpert. renormalisation (~1.5% error) staggered chiral PT fits to all valence and sea quark masses and lattice spacings together blind analysis for Lattice 2007 HPQCD HISQ action for charm and light valence quarks a = 0.09 fm: msea = 1/5 ms, 2/5 ms a = 0.12 fm: msea = 1/8 ms, 1/4 ms, 1/2 ms a = 0.15 fm: msea = 1/5 ms, 2/5 ms + mvalence = msea nonpert. renormalization from PCAC (no error) cont. chiral PT +O(a2)terms, fit to all lattice spacings and masses together A. El-Khadra, Charm 2007, Aug 5-8, 2007

  25. Chiral fits Example: FNAL/MILC fits withmvalence = msea ata = 0.09 fm PREMLIMINARY A. El-Khadra, Charm 2007, Aug 5-8, 2007

  26. Chiral fits FNAL/MILC: staggered chiral PT fit to all data with extrapolation to physical masses and removal of O(a2) errors (left most point) compared to chiral fits at each lattice spacing. PREMLIMINARY A. El-Khadra, Charm 2007, Aug 5-8, 2007

  27. FNAL/MILC Error Budget PRELIMINARY PRL 2005 Lattice 2007 (PRELIMINARY) fDfDs/fDfD fDs/fD HQ disc. 4.2% 0.5% 2.7% 0.2% Light quark + Chiral fits 4-6% 5% 1.3% 1.3% stat + fits 1.5% 0.5% ~1% Inputs (a, mc, ms) 2.8% 0.6% 2.4% 1.0% + finite volume, PT matching ≤1.5% each Total 8.5% 5.0% 4.3% 1.7% A. El-Khadra, Charm 2007, Aug 5-8, 2007

  28. fD+ in comparison HPQCD PRELIMINARY FNAL/MILC (Lattice 2007) FNAL/MILC (PRL 2005) Exp. Av. (Pavlunin @FPCP 2007) A. El-Khadra, Charm 2007, Aug 5-8, 2007

  29. fDs in comparison HPQCD PRELIMINARY FNAL/MILC (Lattice 2007) FNAL/MILC (PRL 2005) Exp. Av. (Pavlunin @FPCP 2007) A. El-Khadra, Charm 2007, Aug 5-8, 2007

  30. fDs / fD+ in comparison HPQCD PRELIMINARY FNAL/MILC (Lattice 2007) FNAL/MILC (PRL 2005) Exp. Av. (Pavlunin @FPCP 2007) A. El-Khadra, Charm 2007, Aug 5-8, 2007

  31. Conclusions and Outlook • Charm physics important test bed for LQCD methods • leptonic decay constants: HPQCD: ~1-2 % error FNAL/MILC: ~4% error (PRELIMINARY) • semileptonic decay form factors: existing result from FNAL/MILC can be improved to allow a quantitative comparison of the shapes with experiments: first fit LQCD and exp. separately to test LQCD then fit together for best determination of CKM elements HPQCD plans to use HISQ on MILC ensembles • Outlook: nf= 2+1ensembles using other sea quark actions are currently being generated. Important test. A. El-Khadra, Charm 2007, Aug 5-8, 2007

  32. nf= 2+1 ensembles used/available today A. El-Khadra, Charm 2007, Aug 5-8, 2007

  33. nf= 2+1 ensembles used/available today …. and soon available … A. El-Khadra, Charm 2007, Aug 5-8, 2007

  34. Back-up slides A. El-Khadra, Charm 2007, Aug 5-8, 2007

  35. Kogut-Susskind: keep the naïve action, and “stagger” 4 tastes on a hypercube and combine into one quark flavor • still leaves 4 degenerate “tastes” • O(a2)errors are large due to taste changing interactions •  remove perturbatively: improved staggered action • (Lepage, MILC) • chiral symmetry is preserved •  light quarks computationally efficient Introduction to Lattice QCD, cont’d the fermion doubling problem naïve lattice action in momentum space:m  sin pm zeroes atp = (0,0,0,0), (p/a,0,0,0), …  16 degenerate particles (“tastes”) A. El-Khadra, Charm 2007, Aug 5-8, 2007

  36. Wilson: add a dim 5 term to the action: • breaks the doubling degeneracy • introduces O(a) error  remove with improvement (Sheikholeslami+Wohlert) • breaks chiral symmetry •  light quarks computationally inefficient Light Quark Methods the fermion doubling problem naïve lattice action in momentum space:m  sin pm zeroes atp = (0,0,0,0), (p/a,0,0,0), …  16 degenerate particles (“tastes”) A. El-Khadra, Charm 2007, Aug 5-8, 2007

  37. L a (fm) systematic errors • finite lattice spacing,a: take continuum limit: • by brute force: • computational • effort grows • like ~(L/a)6 • by improving the action: • computational effort grows much more slowly improved actions are much better … A. El-Khadra, Charm 2007, Aug 5-8, 2007

  38. mixing: b _ d B0 u,c,t W W u,c,t _ d _ b b m W fB: B- _ _ u nm Introduction, cont’d B0 A. El-Khadra, Charm 2007, Aug 5-8, 2007

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