

 14 Sep 1995

LATTICE RESULTS FOR HEAVY QUARK PHYSICS

H. WITTIG (UKQCD COLLABORATION) Physics Department, The University, Southampton SO17 1BJ, UK

The status of lattice calculations for heavy quark systems is reviewed, focussing on weak matrix elements for leptonic and semi-leptonic decays of heavy mesons. After an assessment of the main systematic errors, results for the decay constants fD and fB, the B parameter describing B \Gamma _B mixing, the Isgur-Wise function and the spectroscopy of heavy baryons are discussed.

1 Introduction The physics of hadrons containing heavy quarks plays an important r^ole in the study of some of the CKM matrix elements, and therefore serves to test the consistency of the Standard Model. Lattice simulations of QCD provide a non-perturbative treatment of hadronic processes, and are thus capable of dealing with large strong interaction effects in weak decay amplitudes.

Lattice QCD replaces space-time by a four-dimensional hypercubic lattice of size L3 \Delta T . The sites are separated by the lattice spacing a, which acts as an UV cut-off. One problem encountered in current simulations is that typical values of a\Gamma 1 lie in the range 2 \Gamma 3:5 GeV. Therefore one expects that discretisation errors ("lattice artefacts") will distort the results already for charm physics. Also, b quarks cannot be studied directly, since their mass is above the UV cut-off. One way of addressing this problem is to subtract the leading discretisation error by employing an O(a)-improved lattice action1. Quantities can then be computed safely around mcharm and extrapolated to the b quark mass. Alternatively, one can use the "static approximation" and perform the simulation at infinite heavy quark mass, using the leading term of an expansion of the heavy quark propagator in 1=mQ. Furthermore, b quarks can be formulated using a non-relativistic expression for the QCD action. These three methods provide complementary information for b quark physics on the lattice.

The other main systematic errors which affect our results include using the Quenched Approximation, in which the effects of quark loops are neglected. Furthermore, matrix elements computed on the lattice are related to their continuum counterparts via finite renormalisation constants, due to explicit chiral symmetry breaking induced by the fermionic lattice action. The numerical values of these matching factors are subject to large uncertainties. Finally, lattice estimates of dimensionful quantities are affected by uncertainties in the lattice scale, which arise from the fact that different quantities used to estimate

1

Table 1: Lattice estimates for the pseudoscalar decay constants from different collaborations

Collab. fB [MeV] fD [MeV] fBs=fB fDs=fD

MILC3 148(5)(14)(19) 180(4)(18)(16) 1.13(2)(9) 1.09(1)(4) UKQCD4 160 + 6\Gamma 6 +53\Gamma 19 185 + 4\Gamma 3 +42\Gamma 7 1.22 + 4\Gamma 3 1.18 + 2\Gamma 2

PCW5 180(50) 170(30) 1.09(2)(5) 1.09(2)(5)

BLS6 187(10)(34)(15) 208(9)(35)(12) 1.11(6) 1.11(6) ELC7 205(40) 210(15) 1.08(6) 1.10(4)

a\Gamma 1 [GeV] give different results.

Most of the results discussed here come from a simulation by the UKQCD Collaboration, using a lattice of size 243 \Delta 48 at fi = 6=g2 = 6:2 for which a\Gamma 1 = 2:9(2) GeV. 2 For propagating quarks an O(a)-improved fermion action was used. Results for b physics were obtained using propagating heavy quarks and also the static appoximation.

2 Leptonic B decays and B0 \Gamma B0 mixing The decay constant of a heavy-light pseudoscalar meson, fP , can be extracted from the matrix elements of the lattice axial current via

h0jAlatt4 (0)jP i , MP fP =ZA (1) where MP is the pseudoscalar mass, and ZA = O(1) is the matching factor relating the lattice matrix element to its continuum counterpart. In Fig. 1 we show an example for the scaling behaviour of fP pMP with increasing heavy quark mass. According to HQET, this quantity behaves like a constant, but, as Fig. 1 shows, there are large deviations from this scaling law4. The estimates for fB; fD and the ratios fBs=fB, fDs=fD from various collaborations are shown in Tab. 1. A weighted average of the results in the table yields

fB = 168 \Sigma 30 MeV; fD = 196 \Sigma 20 MeV (90% C:L:) (2) The phenomenologically interesting quantity for the study of CKM matrix elements is the combination fBpBB, which requires knowledge of the B parameter BB, defined via BB = ffs(_)\Gamma 2=fi0 hB0 j OL(_) j B0i= 83 f 2B M 2B; where OL is the \Delta B = 2 four-fermion operator. In a recent study, UKQCD presented results for BB and fB using the static approximation2. The results for the B parameter are

BBd = 0:99 + 5\Gamma 6 + 3\Gamma 2; BBs = 1:01 + 4\Gamma 5 + 2\Gamma 1; (3)

2

Figure 1: UKQCD's data for the quantity fP pMP ffs(MP )2=fi0 plotted versus 1=MP in lattice units. The solid line denotes the extrapolation of the four points using propagating

quarks. The point at infinite meson mass is obtained using the static approximation.Error: /undefinedresult in --currentpoint--
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Execution stack:
%interp_exit .runexec2 --nostringval-- --nostringval-- --nostringval-- 2 %stopped_push --nostringval-- --nostringval-- --nostringval-- false 1 %stopped_push 2 3 %oparray_pop 2 3 %oparray_pop 2 3 %oparray_pop 2 3 %oparray_pop .runexec2 --nostringval-- --nostringval-- --nostringval-- 2 %stopped_push --nostringval-- --nostringval-- --nostringval-- --nostringval--
Dictionary stack:
--dict:1100/1123(ro)(G)-- --dict:0/20(G)-- --dict:74/200(L)-- --dict:128/250(L)-- --dict:42/200(L)-- --dict:42/50(L)--
Current allocation mode is local

