B Dlν and B πlν on the Lattice

Transcription

1 B Dlν and B πlν on the Lattice Paul Mackenzie Fermilab mackenzie@fnal.gov Thanks, Ruth van de Water, Richard Hill, Thomas Becher BaBar/Lattice QCD Workshop SLAC Sept. 16, 006 1

2 Lattice calculations Quarks are defined on the sites of the lattice, and gluons are SU3 matrices on the links, U=exp(igA). Calculations are done at finite lattice spacing a, finite volume V, too high a light quark mass m, etc. Theory is used to derive expected functional form for extrapolations in a (OPE), m (chiral PT), V (finite volume chiral PT)... Systematic errors.

3 Progress for simple quantities Quantities that used to agree decently, ~10%, in the quenched approximation agree to a few % in recent unquenched calculations. Gold-plated quantities. Staggered fermions, the least CPU-intensive. Paul Mackenzie Fermilab Wine and Cheese, July 14,

4 Three families of lattice fermions Staggered (Kogut-Susskind)/naive Good chiral behavior (can get to light quark masses). Fermion doubling introduces nonlocal effects which must be theoretically understood. Cheap. Wilson/clover No fermion doubling but horrible chiral behavior. Overlap/domain wall Nice chiral behavior at the expense of adding a fifth space-time dimension. Expensive. Staggered fermion unquenched calculations are the cheapest and currently most advanced phenomenologically, (probably a temporary situation). Paul Mackenzie Fermilab Wine and Cheese, July 14,

5 Gold-plated quantities of lattice QCD Quantities that are easiest for theory and experiment to both get right. Stable particle, one-hadron processes. Especially mesons. More complicated methods are required for multihadron processes: - unstable particles are messy to interpret, - multihadron final states are different in Euclidean and Minkowski space. Paul Mackenzie Fermilab Wine and Cheese, July 14,

6 Many of the most important quantities for lattice QCD are golden quantities. E.g., measurements determining the fundamental parameters of the Standard Model. Paul Mackenzie Fermilab Wine and Cheese, July 14,

7 B Dlν dγ dω (B D( ) ) V cb F B D ( )(ω) Form factors are well described by the Isgur-Wise function. Governed by two parameters to good approximation: normalization and slope. Slope parameter is well measured by experiment. = To obtain Vcb from data, theory must supply only normalization, which can be obtained from B V 0 D at zero recoil. 7 7

8 B Dlν Z B V 0 D Ratio method: determine from a ratio that goes to 1 with vanishing errors in the symmetry limit. Hashimoto et al. (99), (Works for K πlν, too, Becirevic et al.) C DV 0 B (t)c BV 0 D (t) C DV 0 D (t)c BV 0 B (t) D V 0 B B V 0 D D V 0 D B V 0 B Uncertainties cancel in ratio in the symmetry limit. Used in renormalization of the vector current. 1.1 F B D (1) = (18) sta (15) sys F(1) B!>D N f =+1 (FNAL/MILC) N f =0 (FNAL 99) Using HFAG 04 avg for V cb F (1), V cb Lat05 =3.91(09) lat (34) exp m l F ( ) = ( ) ( ) Fermilab/MILC 05. Okamoto, Lattice

9 s f - e- is e F + - y I o B πlν 10 BABAR 5 I also review the form factors in other processes. Some of the recent work on the lattice QCD calculations of the B πlν B ρlν form factors in relativistic formalism are presented. Very precise calculations of semileptonic form factors for B D ( ) lν at zero recoil and the calculations of q Data has nontrivial shape. the slope of the Isgur Wise function are presented. B πlν Theory and experimental uncertainties are q dependent, severely so on the lattice. Harder and more important to understand shape. The exclusive semileptonic decay B πlν determines the CKM matrix element V ub through the following formula, e e- o f e dγ dq = G F 4π 3 k π V ub f + (q ), (1) where the form factor f + is defined as 9 9

10 B πlν, quenched approximation f 0,+ (q ) UKQCD APE Fermilab JLQCD NRQCD LCSR LCSR Onogi, CKM q (GeV ) 10 10

11 B πlν, unquenched.5 N f =+1 (HPQCD) N f =+1 (FNAL/MILC) 1.5 f + f 0 Onogi, Lattice 006. Results agree well with quenched results. Probably not significant; not true for all quantities B!>πlν q [GeV ] 11 11

12 B πlν, finite range of q Proposals to address:.5 N f =+1 (HPQCD) N f =+1 (FNAL/MILC) f + f 0 *) Moving NRQCD (Davies, Lepage, et al.) B!>πlν *) Calculate in charm region, extrapolate to bottom (Abada et al.) F q [GeV ] 10 BABAR B πlν q Lattice data extend over only a fraction of the q range on the physical B πlν decay. With standard methods, discretization errors go like O(ap), signal goes like exp(-eπt). *) Gibbons: global simultaneous fit of all experimental and lattice data. *) Unitarity and analyticity (Lellouch, Fukunaga-Onogi, Arnesen et al., Becher-Hill,...) 1 1

13 B πlν, unitarity fits 3 10 BABAR F + 5 P! F q P (t) φ(t, t 0 ) f(t) = z a k (t 0 )z(t, t 0 ) k k=0 Vanishes at subthreshold (e.g. B*) poles Arbitrary analytic function -- choice only affects particular values of coefficients (a s) Pronounced q dependence in form factor is due to calculable effects. When those are factored out, two parameters suffice to describe the current experimental data. (Just like B Dlν, K πlν?!!) 13 13

14 B πlν, unitarity fits B->" form factor data normalized by P(t) x!(t,t 0 ) vs. z(t) P(t)!(t,t 0 )f(t) P(t)!(t,t 0 )f 0 (t) P(t)!(t,t 0 )f + (t) Coefficients in z expansion are compatible with experiment z(t) q max a0: a1: a:

15 B πlν, unitarity fits f 0 (q ) and f + (q ) B->" semileptonic form factors vs. q 3 param. fit constrained such that f + (0)=f 0 (0) --! /d.o.f. = 0.35 f + (q ) from constrained fit unconstrained 3 parameter fit--! /d.o.f. = 0.35 f + (q ) from unconstrained fit q (GeV ) Combined fits of f+ and f0 may give surprisingly good prediction for form factors well beyond the range of lattice data. - Raw lattice data, - Not extrapolated in m or a, - Momentum dependent discretization errors not yet included. How can the results of such fits best be compared with experiment? 15 15

16 Not covered, but interesting B ρlν, B ωlν, etc. Honest methods for treating unstable particles on the lattice exist (Lüscher,...) but they are much more demanding. B Klν, non-standard Model effects Lattice calculation are no more difficult as long as effective operators are local

17 Summary and to-do list For lattice theorists: how well do lattice methods agree? staggered vs. clover vs. overlap, etc. Will Moving NRQCD allow calculation of the form factors for B πlν in the whole decay region? For theorists and experimentalists: how should lattice data be reported; how should lattice and experiment be compared? Raw lattice data in large global fit (Gibbons) Normalization and slope in the z expansion Form factor and slope at several fiducial points (Becher and Hill) All of the above 17 17