B-meson properties from lattice QCD

Volume 256, number 1 PHYSICS LETTERS B 28 February 1991

B-meson properties from lattice QCD

C. A l e x a n d r o u , F. J e g e f l e h n e r PSI, CH-5232 Villigen, Switzerland

S. Gf i sken , K. Sch i l l ing a n d R. S o m m e r Physics Department, University of Wuppertal, W-5600 Wuppertal 1, FRG

Received 17 September 1990; revised manuscript received 30 November 1990

A high statistics computation of the decay constant fB is presented to study scaling and finite size effects. Smooth wave functions for the B-meson allow to suppress excited state contributions effectively. We observe the scaling violation in the static approximation to be 15% between fl= 5.7 and 6.0. The conventional light quark methods appear to remain applicable, at fl= 6.0 even beyond the D-meson mass. The value obtained in the static approximation, fB = 366 ± 22 (stat.) + 55 (syst.) MeV, cannot be reached by smooth extrapolation from the D-meson range, where we find fo = 198 _+ 17 MeV.

I. Introduction

One of the pending problems in the explorat ion o f the s tandard model of electroweak and strong inter- act ions is the theoret ical and exper imenta l investi- gat ion of the B-meson system. This system, due to the presence o f a heavy quark (b -quark) and a light quark (u, d or s) , confronts us with major theoret ical difficulties concerning the evaluat ion of t ransi t ion ampli tudes . Latt ice QCD offers, in principle, the possibi l i ty of comput ing proper t ies of B-mesons in a s traightforward manner [ 1,2 ]. The most ambi t ious a t tempt in this direct ion was made some t ime ago by Bernard et al. [ 2 ], who s tudied B-physics on 30 lattices o f size 123×33 at f l=6.1. They observed non- asymptot ic mass behav iour of fp on the bulk of their da ta - with an indica t ion of a t ransi t ion into a scaling regime at the upper edge of their mass range.

It is of great interest to substant iate these results, which were obta ined in the s i tuat ion where Mp is larger than the inverse latt ice spacing. Today ' s com- puting power l imits quenched calculat ions to latt ice sizes of about 204 space - t ime points. On the other hand, we know from experience with hadron spec-

Work supported in part by Deutsche Forschungsgemeinschaft grant Schi 257/1-2.

t roscopy that a spatial volume L 3 with L = 1.5 fm is

sufficiently large to keep finite size effects small Un- fortunately, for this value of L the available inverse

lat t ice spacing a - ' is thus less than 3 GeV and the B- meson correlat ion length ( in units of a ) still remains

smaller than one! In this si tuation, two al ternat ive

approaches can be followed: ( l ) Eichten suggested [ 3 ] to remove the b-quark

mass scale analytical ly before performing the latt ice calculation. This is done by expanding the b-quark

propagator in terms of the inverse quark mass 1 ~rob

in the presence of an external gauge field [4,5 ]. The approx imat ion implies a renormal isa t ion of the re-

sulting effective operator , which is carr ied out per-

turbat ively [6,7 ]. (2) We propose here to study the finite size effects

of B-meson propert ies and estimate the minimal value

of L needed to keep them small. In case this value - what we hope for - turns out to be significantly smaller than 1.5 fm, a direct calculat ion with a correlat ion length larger than one would be within the

reach. In this letter we study the appl icabi l i ty of both ap-

proaches. In doing so, we concentrate our investiga- t ion on the phenomenological ly interest ing decay

constant fB. There have been four previous a t tempts

60 0370-2693/91/$ 03.50 © 1991 - Elsevier Science Publishers B.V. (North-Holland)


to calculate this quantity in the static approximation ( f ~at), which is the lowest order in the 1 ~rob expan- sion [ 3-10 ]. Boucaud et al. [ 8 ] pointed out that the "local" B)meson correlation function has large contaminations from excited states. For this reason the first calculation did not really produce the ground state matrix element, f~"L In ref. [9] a two-link smearing has been attempted in order to reduce the contributions from excited states. The smearing technique has been improved subsequently by Allton et al. [ 10 ], who used the APE idea [ 11 ] o f smearing over three-dimensional cubes in Coulomb gauge. They were able to demonstrate that the excited states contaminations can be sufficiently overcome.

Our aim here is to clarify the scaling behaviour o f fB in a systematic fashion. Using smooth wave functions [ 12,13 ] for the B-meson, we are able to present not only f ~at with a precision of a few percent, but also the pseudoscalar-scalar mass splitting, A. This is described in the third section, where we also discuss scaling and finite size effects. We consider four lattices that correspond to two different physical sizes and three values of the lattice spacing a, as will be explained in the second section. In the last section we will check the reliability of the static approximation by comparing f ~at with the fp values obtained from a conventional lattice treatment of heavy quarks with varying heavy quark mass mh.

2. Calibration of the lattices

We work with the standard Wilson action [ 14 ]. In order to separate scaling from finite size effects we choose four lattices with the parameters given in table 1. We will refer to them as lattices 1-4 in the order they appear in table 1. The quark and standard hadron propagators are calculated at six different values o f the hopping parameter x (bare quark mass) for

Table 1 The parameters used for the runs.

fl # config's. L/a Lt/a a~ -1/GeV L/fm

5.74 100 8 24 1.533 1.04 5.82 100 6 28 1.725 0.70 6.00 100 12 36 2.300 1.04 6.00 100 8 36 2.300 0.70

L

each lattice. The statistical errors are estimated with the jacknife method and are checked for possible au- tocorrelation effects by blocking into groups of five subsequent configurations. Lattices 1 and 3 and lattices 2 and 4 are tuned (by help o f previous APE data [ 15 ] ) to have identical physical sizes. This can be seen in fig. la, where M v L is plotted versus M 2 L 2 with Mv the vector mass and M p the pseudoscalar meson (P) mass. Results from the same size lattices fall on universal curves. In order to obtain the physical scale o f lattices 3 and 4, we use the value of the lattice spacing ap at f l= 6.0 from the Mpap data of ref. [ 15 ]. The values of ap for the remaining lattices l and 2 follow simply from the geometrical ratios o f the lattice sizes. They are compiled in table 1.

From lowest order chiral perturbation theory we know that M~, (ml, m2) ~ (ml + m2) for small quark masses m~, me [ 16]. Therefore the quark masses are determined by giving the value of M~, for equal quark masses ml and m2. We have checked that finite size corrections to this relation are only significant for our lowest quark mass in the L = 0 . 7 fm lattices #1. In the

#J With the quenched approximation chiral logarithms and therefore also finite size effects are suppressed [7 ].

16 -~-

,.,, (b) + + + + +

lO

j

(a) **

-.,,k- ~ - 0 - -'1- 4

..~.X.~---9 -'4>" '~' L=8 , f l = 5 . 7 4

~ L = 1 2 , f l ~ 6 . 0 0

2 • L = 6 , ,8=~.82

o L = 8 , .8=6 .00

0 I r I I I I I I 4 8 12 16 20 24 28 52 35

(M, L) 2

Fig. 1. Finite size scaling plot (a) of MvL and (b) of f ~"~/~p( fig. )L 3/2.

61


following we will not refer to quark masses directly but rather give the value o f M 2.

3. The static approximation: scaling and finite size effects

The pseudoscalar decay constant is calculated from the correlation functions

C ~ ( t ) = Y~ (O~o(X, t)[O~,o(O,O)]*) (1) x of composite axial-vector operators

OJAo(X, t)

= ~ K(X, t)~o~sFJ(x,y; ~¢(t))l(y, t) . (2) Y

Here we have introduced trial wave functions F j for the light quark l. In this section the heavy quark h is treated in the static approximation. Its propagator reads [4]

&(x; y) =,L,~

)< { O( xO_ yO) ~l(x; x o, yO) exp[ - m ° ( x ° - y °) ]y+

+ O ( y ° - x °) o?/*(x; 9 , x °) exp[m°(x°-Y °) ]Y }, (3)

with o//(x; x °, yO) a product of path-ordered link matrices along the straight path from (x ,x °) to ( x , y ° -a ) and 7 + = ½ ( 1 + ~ o ) . The correlators C~ (t) must be renormalised with a factor

[Z,ta,(mh, a)]2 = ( a~(C/a) _)~2/z5 k,c%(exp ( - 2 / 3 ) m h )

[6 ]. The value of ln C z = 13.55 can be retrieved from the numbers quoted in ref. [7]. z~'a'(mh, a) is ap- proximately 0.8 for heavy quark masses mh of the order of the b-quark mass and for our values of the lattice spacing a ~2. The trivial exp ( - m o t) dependence on the bare quark mass in eq. (3) has been removed in the following.

The existence of a positive hermitian transfer matrix [ 19 ] is also guaranteed for fermions in the static

~2 Although the discussion between the authors of ref. [6] and ref. [7] about the correct perturbative renormalisation of C~,(t) has not been completely settled, there is a general agreement that Z st~'(mb,a) -~ 0.8 for a - ~ = 1.5...3.00 GeV [ 18 ].

approximation. Therefore, neglecting effects due to the finite time extent of the lattice, the correlation function ofeq. ( 1 ) can be written as:

C ~ ( t ) = Z (OlO~Ao(O)ln)(nlO~Ao(O)*lO) n )<exp( - M~,tat t) (4)

where In) are pseudoscalar eigenstates of the transfer matrix with momentum zero.

Denoting the lowest pseudoscalar state I 1 ) by [P) , the decay constantfp (we use a normalisation where f~= 132 MeV) is given by

< 0 1 ' . . . . tat 1 OAo(O) I e ) - f p a~/~Mp([;[fQa,

(F'°C(x,y; ~l/(t) ) =6x,) • (5)

The combinat ion

f~at f ~ a t ~ (~0,: ((;Xx; ((-2/3)MP)2/3)MB)j~6/25

is considered in the following because it approaches a finite value in the M p - ~ limit, where we can ne- glect the difference between the mass of the heavy quark and the mass o f the meson, f ~atN~ P can be extracted from C~ °c'~°c (t) provided the states I n ) for n > 1 do not contribute significantly in eq. (4). The range of t-values where this holds can be found by searching for a plateau in the local masses

M~(t) = l n ( C ~ ( t - 1 )/C~(t) ) (6)

as a function of t. A calculation with local operators has shown [8] that such a plateau does not exist up to distances t~ 8a at f l= 6.0. We checked the situation on an 83)<24 lattice at f l=5.65 (see fig. 2) and on the lattices listed in table 1 and did not find any plateau either. This indicates that P mesons contain- ing one infinitely heavy quark are accompanied by low lying excited states.

In previous work, some of us have successfully ap- plied smooth hadron wave functions to eliminate the contributions of excited states [ 13 ]. In the present case, the static quark propagator is available without matrix inversion. Therefore it is convenient to ex- ploit translational invariance in (1), (2) to replace the smearing of the light quark source by the smearing of the static quark source. Several wave functions can be considered:

62


1.1

0 . 9

0 . 8

0 . 7

• M ~ t ( t ) , K = 0 . 1 8 6 o

• M t ' t ( t ) , K = 0 . 1 8 6

o M~-~"( t ) O o M ~ - a ( t ) , n = 1 0 a = 4 .

~x Me'.a(t), n = 1 0 , a = 4 . +

, , , , ! / , , 5. 1 5.

[ "10.

t / o

Fig. 2. Local masses [eq. (6) ] for the B-meson correlation in the static approximation on an 83 × 24 lattice at fl= 5.65 and x= 0.166.

FG(x,y; ~ ( t ) ) = ( l + a H ) " ( x , y ; '~'(t)), (7)

FE(x,y; ~'(t) )=(Kc~y-H(x , y ; ; / / ( t ) ) ) - ' , (8)

FCG(x,y; ~ ( t ) )=FG(x,y; ~°tl(t) ) . (9)

Here

H(x,y; ql(t) )

3

= ~ [U~(x,t)&~,~_;+U~,(x-{,t)O~+,] (10) i= l

is the hopping matrix and cgo~(t) is the time slice of link matrices after an adiabatic cooling process [20] that locally minimizes the space part of the action. Note that (contrary to the cooling procedures ap- plied conventionally) the cooling operator ~ acts only onto the links in the wave function and therefore leaves the transfer matrix unaffected.

The labels G and E, used as superscripts on F, have been selected since F G and F z show respectively gaussian and exponential fall-off when all link vari- ables are unit matrices. The parameters in the wave functions were adjusted such that the local masses reach an early plateau. In fig. 2, we have plotted the local masses M ~,~(t) and M ~°~,~(t) for i = loc, G and E for the optimal parameters. We make two observa- tions: both wave functions F G and F E yield a sub- stantial improvement over the local operator and, while a plateau is reached for both F G and F E in the

case o f M ''z, there is an early plateau in M ~°c,i only for the exponential wave function F E.

The second observation is of extreme importance, since both correlators C i,i and C j°c'i are needed to extract f staL It is therefore essential and a non-trivial task to find a wave function which fulfills the require- ment to produce such an early plateau for both Mi"(t) and Ml°C.i(t).

For the lattices listed in table 1, we have used the wave function F E and tuned the parameter K to achieve early ground-state dominance. This always coincided with a (gauge invariantly defined) RMS radius [ 13] of the wave function of CrRMS--~0.3 fm. For the f l= 6.0, L/a = 8 lattice there were large fluc- tuations of the critical K, which made it practically impossible to invert the scalar matrix at the values of K required to obtain aRMS ~-0.3 fro. We therefore re- sorted to FcG (cf. table 2) which indeed gave the required RMS radius ~3. It seems that gaussian wave functions work sufficiently well for fl>~ 6.0.

Our final analysis to obtain f star proceeds as fol- lows. The mass M] t"t [eq. (4) ] is obtained by fitting C" (t >/2a ), assuming all excited states f n ) with n > 1 can be neglected. Then a constrained fit to Ci'l°c(t~tmin) is carried out, based on this value of M ~ TM. The stability and the X 2 per degree of freedom of the constrained fit are controlled by varying tmin- Table 2 gives the fit parameters obtained in this way. In addition this table contains the pseudoscalar-scalar splitting A, for which the divergent renormalisation contributions cancel.

We note that, also in the case of the scalar meson, our results show a clear plateau which we attribute to the use o f smooth wave functions. The observable A does however involve both large scaling violations and significant finite size effects.

We have included the values f o r f ~ a t t 3/2 in fig. lb. The quality of the data allows to test scaling on the level of a few percent. Indeed we observe a scaling violation of 15% between fl= 5.74 and fl=6.0. For a lattice of given physical size, the "scaling combina- tion", f statt 3/2, decreases with increasing ft. Since one observes for example that the rotational symmetry of the SU (3) potential sets in just between fl= 5.4 and f l=5.8 [21 ], one would expect sizeable scaling vio-

.3 It remains to be checked whether cooling is actually necessary for the success of this wave function.

63


Table 2 Decay constant, mass and pseudoscalar-scalar mass splitting in lattice units, using the static approximation. An asterisk (*) indicates that the Z 2 per degree of freedom is larger than 0.5 and the fit is not acceptable.

lattice fl x K f star a 3/2 aM] tat aA

size t m i n = 2 a 3a 4 a 5a 6 a

123X36 6 .00 0.1525 0.1854 * 0.361(05)* 0.332(11)* 0.315(15) 0.312(20) 0.687(12) 0.212(22) 0.1540 0.1854 * 0.345(05)* 0.316(11)* 0.297(16) 0.295(21) 0.666(13) 0.216(26) 0.1550 0.1854 * 0.333(06)* 0.305(11)* 0.287(16) 0.287(22) 0.653(14) 0.229(43)

83X24 5 .74 0.1560 0.1866 0.699(13) 0.707(21) 0.703(23) 0.707(47) 1.26(1) 0.423(12) 0.16Q0 0.1866 0.672(14) 0.669(21) 0.658(23) 0.663(51) 1.24(1) 0.448(8) 0.1620 0.1866 0.655(20) 0.646(28) 0.635(35) 0.642(62) 1.25(1) 0.474(9) 0.1635 0.1866 0.644(26) 0.632(34) 0.618(45) 0.624(76) 1.27(2) 0.512(22)

63X28 5 .82 0.1557 0.1856 * 0.508(58) 0.494(55) 0.472(65) 0.764(13) 0.282(30) 0.1574 0.1856 * 0.494(50) 0.477(57) 0.457(65) 0.749(15) 0.261(60) 0.1587 0.1856 * 0,487(58) 0.465(65) 0.446(78) 0.746(18) 0.240(60)

K Ot n

83X36 6 .00 0.1500 4.025 0.266(13)* 0.279(18)* 0.294(25) 0.301(40) 0.704(9) 0.1525 4.025 0.281(14) 0.285(19) 0.291(30) 0.669(10) 0.154(13)

lations around f l= 5.7. In view of this fact, the amount of fl dependence is rather moderate. We would thus

predict scaling violations beyond fl= 6.0 to be no more than our small statistical errors. Therefore it makes sense to convert our results for ffstat into physical

units, using the value of the lattice spacing given in table 1. This is done in fig. 3, which exhibits a small yet significant dependence on the mass of the light quark. Furthermore a finite size effect of order 10% is seen when changing the spacelike lattice extent from 0.7 fm to 1.05 fm. We have included the results ofref. [ 10 ] in fig. 3, using our value of the lattice spacing.

e. 7 >

e.,_ ~

1.1

1.

0.9

0.8

0.7

0.6

0.5 O.

++%+ + ++ zx L =1 .04 fm

o L = .70 fm

• L = . 8 8 / 1 . 7 5 f r o ,

I T I I I 0.2 Q.4 0.6 0.8 1.

rer. 10 I I I

1.2 1.4 1.6 1.8

(MP2( I I ) / G e V ) 2

Fig. 3. f ~at in physical units. The different symbols correspond to the same parameters as in fig. 1. The squares correspond to a 10ZX20 spatial lattice [ 10].

They are in nice agreement with our data, which is a confirmation of the fact, that the results are indepen- dent of the wave function used. Taking the combined data at face value, one observes a smooth volume dependence i n f ~tat and our L = 1.05 fm data appear to

be close to the infinite volume result. Assuming a linear dependence on M~ (££) and tak-

ing as physical scale the average of the lattice spac- ings determined from the p mass and the kaon decay constant, we arrive at the estimates ~4.

f ~tsat = 4 0 2 ( 2 2 ) MeV,

f~t~t = 3 6 6 ( 2 2 ) MeV. ( 11 )

The systematical error which enters by the choice of the lattice constant is estimated to be 8%. On top of that, we expect the systematic uncertainty due to the perturbative renormalisat ion to be = 10%.

~4 In the experimental analysis of ref. [22] the large value of 250+44~ MeV was obtained for fBo.~ ~/02 where .~ is the B-parameter. We have calculated ~ and found .~ -~ 1 with a small statistical error, but, since the renormalisation constant relat- ing our lattice result to the continuum is not known we will not discuss this point further in this work.

64


4. The mass dependence offl,

Before we can interpret the lattice results given in eq. ( 11 ) as predictions for B-meson physics, we have to account for the effects of a finite value of 1/mb. TO this end we c o m p a r e ,?stat to the decay constants as obtained by standard lattice methods ~5. For future reference this data is compiled in table 3.

The comparison is done for two light quark masses corresponding to M 2 = 0 . 5 0 GeV 2 and M 2 = l . 1 0 GeV 2 in fig. 4. Interpolating lines have been drawn

~s The values forfe were determined in the same way as in ref. [24 ] except that here we are using the perturbative renormalisation ZA of the axial vector current [23 ].

through the data points to help relate the various en- tries. As a function of the heavy quark mass, we took data up to a value slightly beyond the charm quark mass. There is a clear gap by a factor five between the static result, ( f~,t)2, and thef2p values computed in the charm quark region.

In this situation, it is important to clarify to what extent one can trust the standard method. As seen in the inlet of fig. 4, throughout the range spanned by ourfp data, scaling violations are no larger than 15%. This is rather small in view of the fact that we are dealing here with a quantity of dimension three. As a result, we find fr" rather than fr, to be mass indepen- dent in the range 0.9 G e V < M p < 3 . 0 GeV. This is clear evidence for violation of scaling in this regime.

Table 3 Pseudoscalar masses and decay constants from the conventional approach (same convention as in ref. [24], eq. (30), except that the factor x/2 is now included in the definition offp. )

Lattice fl Kj K2 Mpa Z A 2 f ~ Mp a 3

size

123×36 6.00 0.1550 0.1150 1.256(7) 0.0203(9) 0.1550 0.1250 1.047(11) 0.0163(8) 0.1550 0.1350 0.824(5) 0.0136(8) 0.1550 0.1500 0.454(10) 0.00628(55) 0.1550 0.1525 0.378(9) 0.00459(47) 0.1525 0.1150 1.294(2) 0.0244(12) 0.1525 0.1250 1.083(7) 0.0198(6) 0.1525 0.1350 0.872(7) 0.0164(8) 0.1525 0.1400 0.761(5) 0.0136(8) 0.1525 0.1450 0.641(7) 0.0105(6) 0.1525 0.1500 0.516(7) 0.00813(60) 0.1525 0.1525 0.449(9) 0.00641(60)

83x 36 6.00 0.1550 0.1250 1.065(21) 0.0202(44) 0.1550 0.1350 0.850(20) 0.0161(36) 0.1550 0.1500 0.497(20) 0.0067(16) 0.1550 0.1525 0.430(21) 0.0042(11)

83x24 5.74 0.1560 0.0900 2.017(4) 0.0991(22) 0.1560 0.1100 1.634(3) 0.0798(19) 0.1560 0.1250 1.345(4) 0.0705(23) 0.1560 0.1400 1.060(7) 0.0542(38) 0.1600 0.0900 1.980(4) 0.0822(25) 0.1600 0.1100 1.578(3) 0.0655(19) 0.1600 0.1250 1.282(6) 0.0575(30) 0.1600 0.1400 0.992(7) 0.0442(31) 0.1620 0.0900 1.925(6) 0.0734(31) 0.1620 0.1100 1.532(6) 0.0578(31) 0.1620 0.1250 1.252(7) 0.0538(31) 0.1620 0.1400 0.958(7) 0.0397(28) 0.1620 0.1560 0.614(8) 0.0209(17)

65


%

d

<,.,~

1.2-

0.8

0.6

0,4

0.2

D 0-1=1.53, GeV, L=1.05 fm 0.24

-~ • o'1=1.55 GeV. L=|.05 frn

~ [ ~ • o-~=2.30 GeV, L=l.05 fm 0.2 o o-~=2.30 GeV, L=1.05 fm

0.{6 I ~ ! . " 0 " -2 .30 GeV. L-0.TO fm

0.08

0,04

0.3, 0.5 0.7 0.9 1.1

O. t 1 t .1 J . - - . . . ~ o. o.2 o . , 0 .6 0 .8 ~. ~

(Ivtp / GeV) -I

Fig. 4. f2=-f2Mr,([i~)[as(exp(-2/3)Mp)/as(exp(- 2/3)MB) ] 12/25 as a function of Mbt (//~). Here A~g = 100 MeV

has been chosen. The My i ( ~ ) = 0 values originate from the static approximation. Full symbols correspond to a light quark mass of roughly the strange quark mass [ M 2 ( ~ ) = 0 . 5 0 GeV2; x~ (fl= 6.0) = 0.1525 and an interpolation between K~ = 0.1600 and K~ = 0.1620 for fl= 5.74 ]. Open symbols correspond to twice the strange quark mass [M~,(~£)= 1.10 GeV 2, x~(fl=6.0) =0.1500 and K~ (fl= 5.74)= 0.1560]. Lines have been drawn through the data points in order to facilitate the identification of the different parameters. The inlet contains the same data as well as our results for a -~ = 1.533 GeV (squares) on an extended scale.

Note that the data points with a - 1 = 2.3 GeV and a - 1 = 1.5 GeV at 1/Mp -~ 0.4 G e V - ~ agree on the 15% level. This is a justification that we can indeed move up in the quark mass to a point where the (heavy- light) meson correlation length is as low as 1.5 GeV × 0.4 G e V - 1 = 0.6 - a surprisingly small value ~6 Furthermore, fig. 4 shows that finite size effects are rather moderate ( < 10% in fp). Our L - - 1.05 fm lattice may therefore be large enough to allow predictions for D- and B-mesons.

#6 The authors of ref. [2] argue that their apparent scaling behaviour in mass presents sufficient evidence for the use of such large quark masses.

It is plausible that a reliable calculation offB can be done on an L/a,,~20 lattice at f l_6 .2-6 .3 . This is within reach of todays resources.

Again assuming a linear dependence on M 2 ( ~ ) we extract the values for the charm meson decay constant f rom our data:

afcs =0 .100 (2), af~u = 0 . 0 9 4 ( 4 ) , (12)

fcs = 2 0 9 ( 1 8 ) MeV, fcu = 198(17) MeV. (13)

Here we have used the lattice scale as obtained from the kaon decay constant a - ~ = 2.10 ( 18 ) G e V - 1, since this, although increasing the statistical error, should essentially eliminate the systematic error due to the perturbative renormalisation.

We conclude this section by putting the lattice re- suits into the perspective of the QCD sum rule sce- nario: Narison [25 ] has pointed out that the scaling law of the static approximation is found in QCD sum rules only if the cont inuum threshold is close to the B-meson mass. This condition can be related to the excitation spectrum in the P sector: if the lowest excitation is close to the ground state, the asymptotic scaling law according to him will be valid. This pre- dicts the contamination from low lying excited states in the static approximation. On the other hand, within the standard approach and in the D region the first excited state appears to be well separated and we find that it is not in the range of validity of the asymptotic scaling law. In this respect the lattice results are con- sistent with the picture derived from QCD sum rules.

5. Summary and conclusions

We have found that the pseudoscalar decay constant in the static approximation, f stat, shows only 15% scaling violations in the range f l=5.7-6.0 . Changing the spatial extent of the lattice from 0.7 fm to 1.05 fm, we find only a slight volume dependence.

A scaling investigation of other standard fp values appears to justify the use o f surprisingly large quark masses in lattice units: fp scales on the 8% level down to heavy-light meson correlation lengths of 0.6. This is encouraging for a conventional lattice approach to treat the beauty sector.

Furthermore we have seen a clear violation of (mass-)scaling for masses up to the D-meson mass,

66


es tabl i sh ing p rev ious results [ 2 ]. T h e au thors o f ref.

[2] see an ind ica t ion o f a f la t tening b e y o n d this mass.

I f this was the onse t o f scaling it wou ld be in conf l ic t

to the v a l u e r ~at.

Clear ly, the next i m p o r t a n t step in the s tudy o f B-

m e s o n proper t i es on the la t t ice will be the b r idg ing o f

the gap shown in fig. 4. So far 1 / m cor rec t ions have

only been ca lcu la ted in the c o n t i n u u m [26] . It ap-

pears unl ikely that the knowledge o f these correc-

t ions on the la t t ice (which co r r e spond to a s t raight

l ine in fig. 4) will by i tse l f a l low for a t rus twor thy

pred ic t ion o f the B-meson propert ies . Ra the r both the

next to lead ing o rde r t e rms in the 1 / m h expans ion

and a s t anda rd ca lcu la t ion at fl--- 6.3 should be con-

s idered. W o r k a long this l ine is in progress [27] .

Acknowledgement

We wou ld like to t hank G. Sch ie rho lz for his up-

da t ing p rog ram tha t was the basis for the over re laxa-

t ion code used in this work. We en joyed i n f o r m a t i v e

d iscuss ions wi th G. Mar t ine l l i and O. Pene. The nu-

mer ica l ca lcu la t ions were car r ied out on the C R A Y

X - M P o f the E T H - Z f i r i c h and the C R A Y Y - M P o f

the H L R Z in Ji i l ich. We thank the s taf f o f these insti-

tu t ions , in pa r t i cu la r S. K n e c h t f r o m H L R Z , for the i r

k ind suppor t .

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[ 27 ] Cern-Orsay-Rome-Southampton-Wuppertal Collaboration and PSI-Wuppertal Collaboration, work in progress.

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B-meson properties from lattice QCD

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