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

2. A. El-Khadra, Charm 2007, Aug 5-8, 20072 Outline 1. Introduction 2. Light Quark Methods 3. Heavy Quark Methods 4. Semileptonic Decays 5. Leptonic Decay Constants 6. Conclusions and Outlook

3. A. El-Khadra, Charm 2007, Aug 5-8, 20073 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 HPQCD: Davies, Hornbostel, Lepage, Shigemitsu, Trottier Follana, Wong MILC: Bernard, DeTar, Gottlieb, Heller, Hetrick, Sugar, Toussaint Levkova, Renner

4. A. El-Khadra, Charm 2007, Aug 5-8, 20074 Introduction parameterize the matrix element in terms of form factors  c d u _ D  ee e Example: Semileptonic D meson decay

5. A. El-Khadra, Charm 2007, Aug 5-8, 20075 “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: , K, D, D s, B, B s mesons masses, decay constants, weak matrix elements for mixing, semileptonic and rare decays charmonium and bottomonium (  c, J/ , h c, …,  b,  (1S),  (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 are  mesons and other resonances.

6. A. El-Khadra, Charm 2007, Aug 5-8, 20076 B   l K   l D   l D  l D  K l D s  l B  D, D * l mixing V ud V us V cd V td V ub V cs V cb V ts V tb Lattice QCD program relevant to many CKM elements    K  

7. A. El-Khadra, Charm 2007, Aug 5-8, 20077 Introduction to Lattice QCD fermion field lives on sites:  (x) L a x … discretize the QCD action (Wilson) e.g. discrete derivative in QCD Lagrangian where c(a) = c(a;  s,m) depends on the QCD parameters calculable in pert. theory

8. A. El-Khadra, Charm 2007, Aug 5-8, 20078 Introduction to Lattice QCD, cont’d in general : n  1 errors scale with the typical momenta of the particles, e.g. (  QCD  a) n for gluons and light quarks.  keep 1  a   QCD  QCD ~ 200 – 300 MeV typical lattice spacing a  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

9. A. El-Khadra, Charm 2007, Aug 5-8, 20079 errors, errors, errors, … finite lattice spacing, a: a (fm) L take continuum limit: computational effort grows like ~ (L/a) 6  m l dependence: chiral extrapolation n f dependence: sea quark effects: results with n f = 2+1 exist statistical errors: from monte carlo integration finite volume  perturbation theory

10. A. El-Khadra, Charm 2007, Aug 5-8, 200710 systematic errors, cont’d chiral extrapolation, m l dependence: In numerical simulations, m l > m u,d because of the computational cost for small m.  use chiral perturbation theory to extrapolate to m u,d need m l < m s /2 and several different values for m l (easier with staggered than Wilson-type actions) mlml f m s /2 Decay constants, form factors: chiral logs contribute ~ Staggered chiral perturbation theory: (Bernard, Sharpe, Aubin,…) remove leading O(a 2 ) errors in fits

11. A. El-Khadra, Charm 2007, Aug 5-8, 200711 Light Quark Methods Asqtad (improved staggered): (Kogut+Susskind, Lepage, MILC) errors: ~ O(  s a 2 ), O ( a 4 ), 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

12. A. El-Khadra, Charm 2007, Aug 5-8, 200712 lattice QCD/experiment Testing the rooted Asqtad action works quite well! before2004 HPQCD+MILC+FNAL, C. Davies, et al, Phys. Rev. Lett. 92:022001,2004

13. A. El-Khadra, Charm 2007, Aug 5-8, 200713 Light Quark Methods Asqtad (improved staggered): (Kogut+Susskind, Lepage, MILC) errors: ~ O(  s a 2 ), O ( a 4 ), 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(  s a 2 ), O(a 4 ), × 1/3 smaller than Asqtad comp. cost: efficicient, × 2 Asqtad improved Wilson (Clover, …): (Wilson, Sheikholeslami + Wohlert, etc …) errors: ~ O(  s a), O(a 2 ) if tree-level (tadpole) imp.; O(a 2 ) if nonpert. imp. Wilson term breaks chiral symmetry comp. cost: × 4 Asqtad for m light ~ m strange, but inefficient at small quark masses Domain Wall Fermions: (Kaplan) errors: ~ O(a 2 ), O(m res a) almost exact chiral symmetry; breaking ~ m res ~ 3 ×10 -3 comp. cost: × L 5 Asqtad, L 5 ~ 16 - 20 Overlap Fermions: (Neuberger) errors: ~ O(a 2 ) exact chiral symmetry comp. cost: × 5-10 DWF

14. A. El-Khadra, Charm 2007, Aug 5-8, 200714 Simulation parameters Ensembles generated by MILC using the Asqtad action. Each point has 400 – 800 configurations.

15. A. El-Khadra, Charm 2007, Aug 5-8, 200715 m Q   QCD and am Q  1 : Heavy Quark Methods 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 all am Q errors:   s (a , (a  ) 2 or   s  /m Q,, (  /m Q ) n good for for charm and beauty Fermilab (Kronfeld, Mackenzie, AXK) : lattice NRQCD (Lepage, et al., Caswell+Lepage) : discretize NRQCD lagrangian: valid when am Q > 1 errors:  (ap) n, (p/m Q ) n good for b quarks, but not charm HISQ (Follana, Hart, Davies, Follana et. al): errors: ~  s (am c ) 2, (am c ) 4 good for charm, but not beauty

16. A. El-Khadra, Charm 2007, Aug 5-8, 200716 Semileptonic Decays: D   l p  (q 2 ) dependence:  p   1GeV improved actions help (keep n large) finite volume ( L ): for a = 0.1 fm, L = 20, p min = 620 MeV Lattice Result with n f = 2+1 from Fermilab Lattice and MILC collaboration (C. Aubin et al, PRL 2005). using MILC coarse ensembles with m sea = 1/8 m s, …., ¾ m s Asqtad action for light valence quarks Fermilab action for charm quarks staggered chiral pert. thy (m valence, m sea, a 2 ) but: q 2 dependence parameterized using BK model only one lattice spacing ( a  0.12 fm)

17. A. El-Khadra, Charm 2007, Aug 5-8, 200717 From V. Pavlunin @ FPCP 2007: Semileptonic Decays cont’d The shape is determined more accurately than the normalisation in the Lattice QCD calculation

18. A. El-Khadra, Charm 2007, Aug 5-8, 200718 Kronfeld (Fermilab Lattice and MILC, 2005): Semileptonic Decays cont’d

19. A. El-Khadra, Charm 2007, Aug 5-8, 200719 Kronfeld (Fermilab Lattice and MILC, 2005): Semileptonic Decays cont’d

20. A. El-Khadra, Charm 2007, Aug 5-8, 200720 Semileptonic Decays: Improvements  add more lattice spacings to analysis:  reduce discretisation errors  q 2 dependence: z-expansion (Arnesen et al, Becher+Hill, Flynn+Nieves, …) based on unitarity and analyticity  model independent analysis of the shape (fig)

21. A. El-Khadra, Charm 2007, Aug 5-8, 200721 Semileptonic Decays: Improvements Van de Water + Mackenzie (Fermilab Lattice and MILC, 2006/7): PRELIMINARY q 2 dependence fit using z-expansion

22. A. El-Khadra, Charm 2007, Aug 5-8, 200722 Semileptonic Decays: Improvements  add more lattice spacings to analysis:  reduce discretisation errors  q 2 dependence: 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 momenta p i < 2  /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)

23. A. El-Khadra, Charm 2007, Aug 5-8, 200723 Leptonic D and D s 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

24. A. El-Khadra, Charm 2007, Aug 5-8, 200724 FNAL/MILC Fermilab action for charm Asqtad light valence quarks a = 0.09 fm : m sea = 1/10 m s, 1/5 m s, 2/5 m s a = 0.12 fm: m sea = 1/8 m s, 1/4 m s, 1/2 m s, ¾ m s a = 0.15 fm: m sea = 1/10 m s, 1/5 m s, 2/5 m s, 3/5 m s + 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: m sea = 1/5 m s, 2/5 m s a = 0.12 fm: m sea = 1/8 m s, 1/4 m s, 1/2 m s a = 0.15 fm: m sea = 1/5 m s, 2/5 m s + m valence = m sea nonpert. renormalization from PCAC (no error) cont. chiral PT + O(a 2 ) terms, fit to all lattice spacings and masses together Comparison of parameters

25. A. El-Khadra, Charm 2007, Aug 5-8, 200725 Chiral fits Example: FNAL/MILC fits with m valence = m sea at a = 0.09 fm PREMLIMINARY

26. A. El-Khadra, Charm 2007, Aug 5-8, 200726 Chiral fits FNAL/MILC: staggered chiral PT fit to all data with extrapolation to physical masses and removal of O(a 2 ) errors (left most point) compared to chiral fits at each lattice spacing. PREMLIMINARY

27. A. El-Khadra, Charm 2007, Aug 5-8, 200727 FNAL/MILC Error Budget PRL 2005 Lattice 2007 (PRELIMINARY) f D f Ds /f D f D f Ds /f D 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, m c, m s ) 2.8% 0.6% 2.4% 1.0% + finite volume, PT matching ≤1.5% each Total 8.5% 5.0% 4.3% 1.7% PRELIMINARY

28. A. El-Khadra, Charm 2007, Aug 5-8, 200728 HPQCD FNAL/MILC (Lattice 2007) FNAL/MILC (PRL 2005) Exp. Av. (Pavlunin @FPCP 2007) PRELIMINARY f D+ in comparison

29. A. El-Khadra, Charm 2007, Aug 5-8, 200729 HPQCD FNAL/MILC (Lattice 2007) FNAL/MILC (PRL 2005) Exp. Av. (Pavlunin @FPCP 2007) PRELIMINARY f Ds in comparison

30. A. El-Khadra, Charm 2007, Aug 5-8, 200730 HPQCD FNAL/MILC (Lattice 2007) FNAL/MILC (PRL 2005) Exp. Av. (Pavlunin @FPCP 2007) PRELIMINARY f Ds / f D+ in comparison

31. A. El-Khadra, Charm 2007, Aug 5-8, 200731 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: n f = 2+1 ensembles using other sea quark actions are currently being generated. Important test.

32. A. El-Khadra, Charm 2007, Aug 5-8, 200732 n f = 2+1 ensembles used/available today

33. A. El-Khadra, Charm 2007, Aug 5-8, 200733 n f = 2+1 ensembles used/available today …. and soon available …

34. A. El-Khadra, Charm 2007, Aug 5-8, 200734 Back-up slides

35. A. El-Khadra, Charm 2007, Aug 5-8, 200735 Introduction to Lattice QCD, cont’d the fermion doubling problem naïve lattice action in momentum space:    sin p  zeroes at p = (0,0,0,0), (  /a,0,0,0), …  16 degenerate particles (“tastes”) 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(a 2 ) errors are large due to taste changing interactions  remove perturbatively: improved staggered action (Lepage, MILC) chiral symmetry is preserved  light quarks computationally efficient

36. A. El-Khadra, Charm 2007, Aug 5-8, 200736 Light Quark Methods the fermion doubling problem naïve lattice action in momentum space:    sin p  zeroes at p = (0,0,0,0), (  /a,0,0,0), …  16 degenerate particles (“tastes”) 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

37. A. El-Khadra, Charm 2007, Aug 5-8, 200737 systematic errors finite lattice spacing, a: a (fm) L take continuum limit: improved actions are much better … by brute force: computational effort grows like ~ (L/a) 6 by improving the action: computational effort grows much more slowly

38. A. El-Khadra, Charm 2007, Aug 5-8, 200738 mixing: b d d _ b _ W W u,c,t B0B0 _ B0B0 b u _ W B-B-   _ fB:fB: Introduction, cont’d