1. Radiative and rare decays Shoji Hashimoto (KEK) @ BaBar-Lattice workshop, SLAC, Sep 16, 2006

2. Sep. 16, 2006Shoji Hashimoto (KEK) Decay modes of interest Radiative: B  K * ,  only q 2 =0 is relevant SU(3) breaking ratio  /K *  is also interesting for |V td /V ts |. Rare: B  K (*) ll, K (*) may contain interesting new physics effects K (*) ll suffer from long distance contribution

3. Sep. 16, 2006Shoji Hashimoto (KEK) Form factors Main operators for B  K*ll also appear in B  l Main operator for B  K*  Heavy quark scaling: V~M 1/2, A 1 ~M -1/2, A 2 ~M 1/2 Heavy quark symmetry: V(q 2 )=2T 1 (q 2 ), A 1 (q 2 )=2iT 2 (q 2 ) from O 9, O 10 O7O7

4. Sep. 16, 2006Shoji Hashimoto (KEK) Previous lattice calculations Bernard, Hsieh, Soni, PRL72 (1994) 1402. UKQCD (Burford et al.), NPB447, 425 (1995). APE (Abada et al.), PLB365, 275 (1996). B  K* , quenched UKQCD (Flynn et al.), NPB461, 327 (1996). B  l, also K * , quenched most extensive study, so far. UKQCD (Del Debbio et al.), PLB416, 392 (1998). Reanalysis of the above data; testing parametrizations. SPQcdR (Abada et al.), Lattice 2002, hep-lat/0209116. finer lattice than UKQCD; still quenched.

5. Sep. 16, 2006Shoji Hashimoto (KEK) Some results All quenched. Lattice cutoff ~ typically 2.7 GeV. Use (O(a)-improved) Wilson fermion for heavy quark; need heavy quark extrapolation. UKQCD (1996) confirmed the HQS relations Heavy quark symmetry: V(q 2 )=2T 1 (q 2 ), A 1 (q 2 )=2iT 2 (q 2 )

6. Sep. 16, 2006Shoji Hashimoto (KEK) Some more results Form factor shape constant, pole, or dipole? 1/M extrapolation Linear or quadratic? again, from UKQCD (1996)

7. Sep. 16, 2006Shoji Hashimoto (KEK) Not much progress since then 1.More complicated than B  P. May want to do B  first. 2.Statistical noise is much larger. Even worse for lighter light quarks and heavier heavy quarks.  extrapolations are harder. 3.No solid theoretical guide for the chiral extrapolation of vector particles. 4.Decay width of K * is significant (50 MeV): not sure how to treat on the lattice. 5.Only q 2 =0 is relevant for B  K * . Must introduce model dependence when extrapolate. WHY?

8. Sep. 16, 2006Shoji Hashimoto (KEK) Then, what we have been doing … 1996-2006: Dynamical fermion simulations become a standard. No more quenching. Chiral extrapolation is more delicate than we naively thought. We don ’ t care much about it in the quenched theory (a wrong theory, anyway), but full theory is a different story. To perform reliable chiral extrapolation, one must come closer to the chiral limit.

9. Sep. 16, 2006Shoji Hashimoto (KEK) An example Pion decay constant Early Wilson fermion simulations are limited in a heavy pion mass region. Staggered fermion groups went ahead to reduce sea quark mass, but a complicated ChPT analysis is necessary. Exact chiral symmetry simulation is now available. Can get to small enough sea quark masses. JLQCD (2002) JLQCD (2006)

10. Sep. 16, 2006Shoji Hashimoto (KEK) Step by step Instead of going to very complicated quantities like B  K*ll, one should (I want to) make sure that the chiral extrapolation goes right using easier quantities. 1.Pion decay constant, PCAC relation: the simplest quantities. 2.Pion EM form factor, kaon form factors: simplest non-trivial form factors. Detailed comparison with ChPT is possible. 3.Heavy-light decay constants: need the B*B  coupling to control the chiral extrapolation. 4.Heavy-to-heavy form factors: more symmetry than heavy-to-light; easier to get precise results. 5.Heavy-to-light form factors, but B  P. 6.Heavy-to-light form factors, then B  V.

11. Sep. 16, 2006Shoji Hashimoto (KEK) Possibilities in the future Besides the steady progress in the standard direction, I am interested in … K * ll at low recoil. Long distance effect? Higher recoil region by the moving frame technique.

12. Sep. 16, 2006Shoji Hashimoto (KEK) K * ll at low recoil Lattice QCD is most suitable for high q 2 region. But, long distance contrib arises when q 2 is around the charmonium resonances. b s c c l l O 1, O 2 Morozumi, Lim, Sanda (1989) LEET or SCET resonances Lattice calculation (?)

13. Sep. 16, 2006Shoji Hashimoto (KEK) OPE technique Not possible to insert time- like momentum to the charm loop. Grinstein and Pirjol, PRD70, 114005 (2004). Proposal to expand the charm loop by operator products:  /m b, m c /m b. Then, it becomes the usual semileptonic topology; easier to calculate on the lattice. Impact of the higher order terms on the A FB analysis, for example? b s c c l l b s l l

14. Sep. 16, 2006Shoji Hashimoto (KEK) Moving frame? A big limitation of the lattice calculation = spatial momentum of initial and final hadrons must be small (< 1 GeV/c) to avoid large discretization errors. B  l K (*) ll form factor restricted in the large q 2 region. B  K * ,  are beyond the reach. A possible solution is to discretize after boosting.

15. Sep. 16, 2006Shoji Hashimoto (KEK) Moving NRQCD Moving HQET: Mandula- Ogilvie (1992) Moving NRQCD: SH, Matusfuru (1996), Sloan (1998), Foley, Davies, Dougall, Lepage (2004~) Work ongoing by Wong, Davies, Lepage (at Lattice 2006) How much recoil needed? Any limitations? Can we boost the B meson to the light-cone? Isgur-Wise function (SH, Matsufuru, 1996)

16. Sep. 16, 2006Shoji Hashimoto (KEK) Moving HQET Mandula, Ogilvie (1992) Generalization of the usual lattice HQET Write the b quark momentum as p b =m b u+k and discretize the residual momentum k. Notation HQET: The original field is recovered by The lattice version: Extension to NRQCD is straightforward.

17. Sep. 16, 2006Shoji Hashimoto (KEK) How much boost possible? Intuitively, light degrees of freedom should also move fast in the boosted frame: momentum (  QCD /m B )p B =  QCD  v Lorentz contraction means a wider distribution in the momentum space. How is it realized in the lattice calculation? A toy example: E tot = E v (k Q ) + E q (k q ) k Q + k q = 0 boost v = 0 v = 0.5 v = 0.8  QCD  QCD  (v  1)

18. Sep. 16, 2006Shoji Hashimoto (KEK) How large velocity needed? To avoid large discretization effect, is necessary. Namely, v < 0.7~0.8. Need finer lattice to boost further. B  l at maximum recoil (q 2 =0) B  K *  pp v 2.64 GeV0 1.0 GeV0.75 0.5 GeV0.93 0.0 GeV0.999 p K* v 2.57 GeV0 1.0 GeV0.67 0.5 GeV0.85 0.0 GeV0.94 Seems feasible. On-going work by Wong et al for (non- )perturbative matching of the parameters.

19. Questions, and partial answers

20. Sep. 16, 2006Shoji Hashimoto (KEK) SU(3) breaking 1. Can lattice QCD calculate the ratio of (and also the individual) form factors T 1 (B  K*)/T 1 (B  ) and if so, what precision can be achieved. Related to that, is there a new program to calculate the ratio of (and also the individual) longitudinal to transverse decay constants for K* and  ? Yes, we can, but currently with model uncertainty in the q 2 extrapolation. Precision??? Decay constants of K* and , to input in the LCSR calc, are much easier to calculate. A related work: Becirevic, Lubicz, Mescia, Tarantino, JHEP05 (2003) 007. I don ’ t know about new programs.

21. Sep. 16, 2006Shoji Hashimoto (KEK) Annihilation diagram 2. Can lattice QCD reliably estimate the u quark contribution to the B  decay (annihilation diagram and u quark loop). I don ’ t have any good idea. Need double insertion of operators. The best one can do is to measure B  l . from Beyer, Melikov, Niktin, Stech, PRD64, 094006 (2001).

22. Sep. 16, 2006Shoji Hashimoto (KEK) Inclusive 3. In the light of Super-B factories, is the inclusive ratio of b  d  / b  s  easier to understand theoretically? Probably. I ’ m not sure how much uncertainty cancels in the ratio, but some of them should. Don ’ t give up the exclusive modes.

23. Sep. 16, 2006Shoji Hashimoto (KEK) Asymmetry 4. Since we are entering precision measurement of asymmetris (CP, isospin) in the B  K*  decay, can lattice QCD say anything about these quantities? Basically, all we can provide is the form factors. Since the single form factor dominates the B  K*  rate, the form factor is irrelevant in the CP asymmetry.

24. Sep. 16, 2006Shoji Hashimoto (KEK) Asymmetry (cont) Grinstein, Grossman, Ligeti, Pirjol, PRD71, 011504R (2005), argued that the contribution of wrong helicity photon could be ~10%. Namely, suppressed only by  QCD /m b rather than m s /m b. comes from charm loop.

25. Sep. 16, 2006Shoji Hashimoto (KEK) Another ratio 5. How about the ratio of form factors for Bs  K*  /Bs  ? Could this be calculated in principle better than B  /B  K*  ? Yes. I think so. Lattice calculation would be easier, because the chiral extrapolation for the spectator quark is not necessary.

26. Sep. 16, 2006Shoji Hashimoto (KEK) Vts/Vub 6. Is it possible to have a theoretically precise estimator of Vts/Vub by measuring ratio of branching fractions B  Kll/B  l at high q 2 ? Yes, if we can control the long distance contribution using the OPE method.

27. Sep. 16, 2006Shoji Hashimoto (KEK) 7. Are SU(3) breaking in the form factors more or less managable than for the vector meson case? Yes. A lot easier than the decaying particles.