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B Physics with Lattice Q C D :

B Physics with Lattice Q C D :. Status and Prospects. Aida X. El-Khadra. (UIUC and Fermilab). SLAC workshop on V xb and V tx , Dec 4-6, 2002. Outline: Motivation Introduction to lattice QCD f B and B B Semileptonic B meson decays B  D,D * lv B  p lv

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B Physics with Lattice Q C D :

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  1. B Physics with Lattice QCD: Status and Prospects Aida X. El-Khadra (UIUC and Fermilab) SLAC workshop on Vxb and Vtx, Dec 4-6, 2002

  2. Outline: • Motivation • Introduction to lattice QCD • fB and BB • Semileptonic B meson decays • B  D,D* lv • B  p lv • some recent developments • Prospects for the near future • Conclusions A. El-Khadra, SLAC workshop, Dec. 4-6, 2002

  3. present (2001) status: errors are dominated by theory (CLEO-c 2001, 95% cl) After B factories: assuming only incremental progress for theory errors  1/2 Motivation A. El-Khadra, SLAC workshop, Dec. 4-6, 2002

  4. The problem: for example need the hadronic matrix elements from lattice QCD to determine theCKM matrix elements goal: 2-3% theory errors from lattice QCD Motivation cont’d A. El-Khadra, SLAC workshop, Dec. 4-6, 2002

  5. L x a … discretize the QCD action (Wilson) e.g.discrete derivative in QCD Lagrangian where c(a) = c(a;as,m) depends on the QCD parameters calculable in pert. theory Introduction discretize space-time … fermion fieldlives on site:y(x) gauge field lives on link:Um(x) = exp[iagAm(x)] A. El-Khadra, SLAC workshop, Dec. 4-6, 2002

  6. Lattice Lagrangian • in general:Llat =Lcont + O(a n) n  1 • errors scale with the typical momenta of the particles, • e.g.(LQCDa)nfor gluons and light quarks. keep1/a LQCD • typical lattice spacinga 0.1 fm. • Improvement: add more terms to the action to makenlarge • gluons: • Wilson:a2errors (n = 2) • Lüscher + Weisz: as2a2errors • light quarks (am 1): • Wilson:aerrors (n = 1) • Clover (SW):as2aerrors,a2errors(Sheikholeslami+Wohlert) • staggered:a2errors • improved staggered (Asqtad):asa2errors(Lepage, MILC) A. El-Khadra, SLAC workshop, Dec. 4-6, 2002

  7. Lattice Lagrangian • heavy quarks (mQ LQCD and amQ1): SW + HQ expansion(APE, UKQCD): start with light quark action, keepamQ < 1mQ  mcharm then use HQ expansion to reachmQ = mb corresponds to expandingciinmQ:ci = ci(0) + amQci(1) + … errors: (ap)n , (p/mQ)n,(amQ)n  lattice NRQCD (Lepage, et al., Caswell+Lepage): discretize NRQCD lagrangian: valid when amQ > 1 corresponds toexpanding ciin1/mQ: ci = ci(0) + 1/(amQ) ci(1) + … errors:  (ap)n , (p/mQ)n  Fermilab(Kronfeld, Mackenzie, AXK): start with rel. Wilson action (+ improvement) keep full mass dependence ofci, add time-space asymmetry smoothly matches heavy and light mass limits:valid for allamQ errors: (ap)n , (p/mQ)n  A. El-Khadra, SLAC workshop, Dec. 4-6, 2002

  8. 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, SLAC workshop, Dec. 4-6, 2002

  9. MILC 1999: compare different light quark actions example: rmeson mass vs. a2 a = 0.14 fm A. El-Khadra, SLAC workshop, Dec. 4-6, 2002

  10. In numerical simulations, ml > mu,d because of the computational cost for small m. • use chiral perturbation theory to extrapolate to mu,d need ml < ms/2and several different values forml (easier with staggered than Wilson-type actions) f ml ms/2 systematic errors, cont’d • chiral extrapolation, mldependence: • finite Volume: • L ~ 2 fm okay for B’s. A. El-Khadra, SLAC workshop, Dec. 4-6, 2002

  11. nf= 0: quenched approximation introduces systematic error  10 – 30 % nf 0: computationally difficult need improved actions keep a large, a  0.1 fm nf= 2 with staggered and SW fermions (a2 errors) so far: systematic errors, cont’d • nf dependence new MILC (2002): nf= 3withms mlightandmlight = ms/8, ms/4 … , ms/2,…, ms using an improved staggered action (as a2 errors) A. El-Khadra, SLAC workshop, Dec. 4-6, 2002

  12. Need 2-loop lattice perturbation theory for ~ few% errors difficult, especially with improved actions! use: • automated perturbation theory(Lüscher+Weisz, Lepage, et al) • computational methods(di Renzo, et al) • numerical methods(Lepage, Mackenzie, Trottier, …) • nonperturbative methods(Alpha, APE, …) example: static self energy to 3-loops (Trottier, et al, di Renzo+Scorzato) as to 2-loops (in progress) systematic errors, cont’d • perturbation theory: • for example Renormalization: fromp ~ p/a calculable in perturbation theory if ais small enough. For a ~ 0.1 fm, as ~ 0.25: with 1-loop pert. thy, errors ~ O(as2) ~ 5% A. El-Khadra, SLAC workshop, Dec. 4-6, 2002

  13. 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, D*, Ds, Ds*, B, B*, Bs, Bs*mesons • 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 are r mesons and other resonances. A. El-Khadra, SLAC workshop, Dec. 4-6, 2002

  14. “easy” quantities for most CKM elements … Vud Vus Vub K  p ln B  p ln Vcd Vcb Vcs B  D, D* ln D p ln D ln D  K ln Ds ln Vtd Vts Vtb mixing A. El-Khadra, SLAC workshop, Dec. 4-6, 2002

  15. fB and BB • nf= 0: fB= 173 (23) MeV(Yamada average at Lattice 2002) • has been stable in the last four years • dominant error: nf dependence • nf 0: most results have nf= 2 • “heavy” (valence) light quarks withml  ms/2 • fB (nf 0) / fB (nf= 0) = 1.1 – 1.2 • fBs / fBd = 1.16 (5) agrees with nf= 0 • new in 2002: • includechiral logs in chiral extrapolation • increases fBs / fBd  1.3 • increases the systematic error due to ml • dependence • 1st result with nf= 3 (MILC, Lattice 2002) • but also with “heavy” valence light quarks A. El-Khadra, SLAC workshop, Dec. 4-6, 2002

  16. Yamada review at Lattice 2002 fB quenched fB = 173 (23) MeV fBs= 200 (20) MeV fD = 203 (14) MeV fDs= 230 (14) MeV A. El-Khadra, SLAC workshop, Dec. 4-6, 2002

  17. Yamada Lattice 2002 review:  (nf= 0) fBs / fBd = 1.15 (3) BBs / BBd = 1.00 (3)  = 1.15 (5) A. El-Khadra, SLAC workshop, Dec. 4-6, 2002

  18. forfB (Grinstein, et al, Booth, Sharpe, Zhang): whereg~ gB*Bp is theB*Bpcoupling, which is poorly known. from theD*width (CLEO)g2 ~ 0.35. ml dependence: chiral logarithms • Whenml > mu,d, use ChPT to extrapolatefp(ml)to the physical • u,dquark masses(Gasser+Leutwyler, Sharpe, Bernard, Golterman, Shoreish): wherexl= 2 B0ml/ (4p f)2  1, andbis known. chiral log becomes important for smallml needml < ms/2 • the problem: • (valence)ml > ms/2 in most simulations to date A. El-Khadra, SLAC workshop, Dec. 4-6, 2002

  19. Hashimoto (Lattice 2002) fBs / fBd = 1.24 – 1.38 agrees with Kronfeld + Ryan (2002) analysis of chiral logs. ms chiral logarithms cont’d • Kronfeld+Ryan: chiral logs are small forBB • Becirevic, et al: use double ratios:(fBs / fBd )  (fK / fp ) • solution: simulations withml< ms/2 A. El-Khadra, SLAC workshop, Dec. 4-6, 2002

  20. dG(B  D* ln) dw = (known) (w2 – 1)1/2  |Vcb|2 |FBD*(w) |2 • calculateF(1)in lattice QCD from double ratios, • e.g. Semileptonic B meson decays B D, D* ln: e.g. • whenmb = mc: R 1. • systematic errors scale withR – 1, notR. • with O(a) O(1/m) improved action R is correct through 1/m2 • (Kronfeld) A. El-Khadra, SLAC workshop, Dec. 4-6, 2002

  21. a ml nf pert. thy stat. B D, D* lncont’d FNAL (2001): • lattice result is obtained • at nf = 0 • nf=3will be donesoon • chiral extrapolation error • will be reduced by having • ml < ms/2 A. El-Khadra, SLAC workshop, Dec. 4-6, 2002

  22. measuredG/dppfor0  pp < mB/2 • experiment: B p ln • pp (q2) dependence: • pp 1GeV improved actions help (keepnlarge) • old solution: extrapolate topp < mB/2 (q2 = 0) by assumingshape (pole dominance) —introduces model dependence! • better solution: limit recoil momentum range, e.g. 400 MeV < pp < 800 MeV Note: okay for D decays A. El-Khadra, SLAC workshop, Dec. 4-6, 2002

  23. FNAL (2000): partial differential decay rate A. El-Khadra, SLAC workshop, Dec. 4-6, 2002

  24. B p lncont’d • even better solution: moving NRQCD (Foley, Lepage, 2002): give the B meson momentumpB write the b quark momentum as pbm= mb um+ km removemb um from the dynamics reduces to regular NRQCD forbquark at rest. no numerical simulations with this formalism yet This idea can also be adapted to the Fermilab approach using HQET. A. El-Khadra, SLAC workshop, Dec. 4-6, 2002

  25. B p lncont’d to date: nf = 0 only errors are large, e.g. FNAL: stat sys also: new calculation on anisotropic lattices(Shigemitsu, et al) A. El-Khadra, SLAC workshop, Dec. 4-6, 2002

  26. B p lncont’d FNAL (2000): chiral extrapolation dominant systematic error ~ 15 – 20% A. El-Khadra, SLAC workshop, Dec. 4-6, 2002

  27. Some Recent Developments • highly improved actions: • light quarks: • improved staggered action: correct through ~ O(a2) • computationally affordable  nf= 3feasible! • MILC (2002): nf= 3configurations at a = 0.12 fm • with ms mlightandmlight = ms/8, ms/4 … , ms/2,…, ms • heavy quarks: • NRQCD: correct through ~ O(a2), O(v4) • O(v6) in progress • Fermilab: correct through ~O(asa), O(p/m) • O(a2), O(p2/m2) in progress • automated perturbation theory • get pert. results for new actions quickly • 2-loop calculations in progress A. El-Khadra, SLAC workshop, Dec. 4-6, 2002

  28. Recent Developments cont’d • the new MILC configurations include realistic sea quark • effects. • strategy: • the only free parameters in lattice QCD lagrangian: • quark masses andas • tune the lattice QCD parameters using experiment: • mu,d , ms , mc , mbusingp, K, J/y, Umeson masses • asusing1P-1Ssplitting inUsystem • all other quantities should agree with experiment … • try this for some easy quantities … A. El-Khadra, SLAC workshop, Dec. 4-6, 2002

  29. Recent Developments cont’d HPQCD + MILC (preliminary) lattice QCD/experiment works quite well! A. El-Khadra, SLAC workshop, Dec. 4-6, 2002

  30. Prospects for the near future work currently in progress using the MILC configurations within the next year we can expect first results for … • UandJ/ysystems using NRQCD and Fermilab actions • test the new heavy quark actions •  as, mb, mc • p, Kmeson systems using improved staggered light quarks withml < ms/2 masses, decay constants, mixing, SL form factors A. El-Khadra, SLAC workshop, Dec. 4-6, 2002

  31. expect initial accuracy of < 10% errors with an ultimate goal of 2-3% errors. Prospects for the near future cont’d … and for … • D,Ds, B, Bsmeson systems • using improved staggered light quarks withml < ms/2 • masses (splittings), decay constants, mixing, SL form • factors •  comparison with CLEO-c essential to test lattice • results A. El-Khadra, SLAC workshop, Dec. 4-6, 2002

  32. Conclusions • lattice QCD calculations are an important component of the physics program of the B factories. • current status: fB, BB, fBp(E) to 10-30% accuracy FBD*(1) to few % accuracy • lattice results with realistic sea quark effects are here! expect to see a growing number of results within the next year • made possible with improved staggered action • improved heavy quark actions NRQCD/Fermilab • 2-3% accuracy requires 2-loop pert. matching need to redo pert. calculations for the new actions automated pert. theory methods help • CLEO-c experiment is important for testing lattice QCD A. El-Khadra, SLAC workshop, Dec. 4-6, 2002

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