Meson spectroscopy and decay constants with Wilson fermions at β = 6.4

NUCLEAR

PH Y S I CS B

Nuclear Physics B 376 (1992) 172—198 North-Holland

Meson spectroscopy and decay constants with Wilson fermions at /3 6.4 =

A. Abada a, C.R. Aliton h Ph. Boucaud a D.B. Carpenter M. Crisafulli (1 J. Galand a S. Güsken e G. Martinelli d 0. Pène C.T. Sachrajda b R. Sarno K. Schilling e and R. Sommer ~ C

“,

d

LPTHE, Orsay, France Physics, The Unit’ersity, Southampton S09 5NH, UK C Department of Electronics and Computer Science, The UniLersity, Southampton S09 5NH, UK d Dipartimento di Fisica, Unir’ersitâ di Roma ‘La Sapienza ‘~ 1-00185 Rome, and INFN, b Department of

C

Sezione di Roma, Italy Physics Department, Unicersity of Wuppertal, D-5600 Wuppertal 1, Germany CERN Theory Ditision, CH-1211 Genei’a 23, Switzerland Received 6 November 1991 Accepted for publication 2 December 1991

We present an extensive lattice study of the physical properties of mesons, composed of a heavy, H, and a light, q, quark, at /3 = 6.4 on a 24~x60 lattice, using the Wilson action in the quenched approximation. We have studied the mass spectrum and the decay constants of vector and pseudoscalar mesons. We find significant violations of the mass scaling law f/~~= const. (‘-‘ 50% for D-mesons and 20% for B-mesons). The results using quenched but propagating quarks are remarkably consistent with the static results when the scale is taken from the pion decay constant. Combining the results obtained by several calculations of the pseudoscalar decay constants as a function of the meson mass, at different values of the lattice spacing, we obtain by 2O5±4O)MeV and BB= 1.16±0.07),where BB extrapolation fB%/~=(220±40)MeV (fB=( is the renormalization group invariant B-parameter relevant for Bd—Bd mixing. We also find f~BB/f~dBB = 1.19±0.10.This last result is relevant in experimental studies of B~—B~ mixing. The vector—pseudoscalar mass splittings do not follow the predicted behaviour, M~— const., which is expected (and found experimentally) in the limit of large heavy quark masses (i.e. when m 0>> A0~).

1. Introduction In this paper we present the results of a lattice calculation of meson masses and decay constants at /3 6.4 on a 24~x 60 lattice using the standard Wilson action [1], in the quenched approximation. We focus our attention on the spectroscopy, decay constants and scaling behaviour of mesons composed of a heavy and a light quark. The dependence on the heavy quark mass of physical quantities such as the vector—pseudoscalar meson mass splitting, the pseudoscalar and vector decay constants and the ~1F = 2 B-parameters has been studied in detail. From the =

0550-3213/92/$05.00 © 1992

—

Elsevier Science Publishers B.V. All rights reserved

A. Abada eta!.

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Meson spectroscopy

173

theoretical point of view the problem of the scaling dependence on the heavy quark mass is important because many interesting properties of matrix elements can be derived on rather general assumptions using the spin symmetry present in the limit in which the mass of the heavy quark m~ cc[2—6]. Moreover extrapolation of these results, which were obtained for quark masses in the range 1—2 GeV, to the b quark would provide relevant information for B°—B° mixing and CP-violation in B K~J/~i decays as discussed for example in refs. [7,8]. The scaling laws can be shown to be valid in the lattice regularization and thus provide direct test of the approach to the asymptotic behaviour when the mass of the quark becomes large. We have also combined the results of this work, with the results of similar calculations performed at different values of /3, /3 = 6.0 and /3 6.2 [9—11],and with results obtained by using the static approximation (at lowest order in 1/mH) [11,12]. The main results of this study are the following: (i) The vector—pseudoscalar mass splitting for charmed mesons (MD* MD, MJ/~ M~, etc.) do not agree with the experimental values and are typically smaller by a factor of two. Moreover the expected scaling behaviour, M~ M~ const. is not well satisfied in the region m11 1—2 GeV. —~

—*

=

—

—

—

—‘

(ii) By combining results obtained using propagating heavy quarks with masses in the range m~ 1—2 GeV, with those computed in the static approximation we obtain fB~ (205 ±40) MeV, BBd 1.16 ±0.07, fB/fB. 1.08 ±0.06 and f~BB/f~BB 1.19 ±0.10. (iii) A remarkable consistency of the results obtained for the pseudoscalar decay constant, f~,,in the full, but quenched, theory and in the static approximation is found if the lattice spacing is calibrated using f5., thus giving confidence in the validity of the lattice calculation. The study of the scaling behaviour of f~,as a function of the pseudoscalar meson mass, M~,demonstrates that there are large violations to the asymptotic scaling relation f~/M constant for charmed mesons (—~ 40—100%) and sizeable ones 15—30%) for B-mesons, in agreement with ref. [10] but contrary to previous claims [131.Extrapolation of decay constants and form factors from charmed to bottom quarks, using the scaling laws of refs. [61, is therefore doubtful. The paper is organized as follows. In sect. 2 we give the relevant information on the parameters of the numerical simulation and on the correlation functions which have been used in order to compute the physical quantities of interest. In sect. 3 we describe the analysis of the meson spectrum and report the results for light—light and heavy—light mesons. In sect. 4 we discuss pseudoscalar and vector meson decay constants, whose scaling properties are analyzed in sect. 5. Finally, in sect. 6 we present the results for the D°—D°and B°—B°B-parameters. =

=

=

(‘-S

=

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Meson spectroscopy

2. Generalities on the Monte Carlo simulation The results of this paper have been obtained from a Monte Carlo study performed at /3 6.4 using the standard Wilson action for the gauge fields and the quark propagators [11, in the quenched approximation. We have generated 15 independent gauge field configurations on a 24~x 30 lattice, separated by 500—1600 sweeps, using the overrelaxed algorithm [14]. The 15 configurations were produced in groups of 5, with three independent initial conditions. In each of the three cases the first of the useful configurations was obtained after an initial thermalization of at least 3000 sweeps (500 with the Metropolis algorithm and 2500 with the overrelaxed algorithm). On each configuration we have computed the quark propagators for 7 different values of the Wilson hopping parameter K~, corresponding to “heavy” quarks, KH = 0.1275, 0.1325, 0.1375, 0.1425, and “light” quarks, KL = 0.1485, 0.1490 and 0.1495. Periodic boundary conditions on a 24 x3 x 60 lattice have been imposed in the calculation of the quark propagators, by using appropriate combinations of periodic and antiperiodic (in the time direction) quark propagators calculated on a 24~x 30 lattice [151. To study the meson masses and decay constants we have computed the following correlation functions: =

G

55(t)

=

E(P5(x, t)P(O, 0)),

G05(t)

=

E(A0(x, t)P;(o, 0)),

G50(t)

=

~(P5(x, t)A~(O,0)),

G00(t)

=

~(A0(x,

t)A~(O,0)),

(1)

for pseudoscalar mesons and ~

Gkk(t)=

~(Vk(x,t)V~(O,0)),

k=1,3

(2)

x

for vector mesons. P5, A0 and Vk are the local pseudoscalar density, axial vector and vector currents respectively P5(x,

t) =it~i1(x,t)y5~/i2(x,t),

A0(x,

t)

=

i/11(x, t)y0y5~fJ2(x,t),

Vk(x,

t)

=

~1(x, t)yk~/J2(x,t),

(3)

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where

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Meson spectroscopy

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is the spinor corresponding to the quark with hopping parameter K12. We have also computed the three-point correlation functions corresponding to the ~F = 2 neutral meson weak transitions 1/11,2

2(O,0)P G3(t1,

t2)

=

E(P~(x,t1)O~

5(y,

t2)),

(4)

where ~ is the combination of local lattice four fermion operators which corresponds to the continuum ~iF = 2 operator [16,17] Q~1F2 = (~~y~(1 Y5)c~2)(~lY~(1 Ys)~2). (5) —

—

The coefficients of the lattice four fermion operators were taken from perturbation theory [16,17]. We have only studied the heavy—light matrix elements of ~ which are relevant for D°—D°and B°—B°transitions. The two-point correlation functions have been computed in the degenerate quark case (~qor HH) for all the values of Kw and for two different quarks in all the possible combinations of a heavy and a light quark (~Tq). All the statistical errors quoted below have been obtained with the jacknife method, by decimating 3 configurations at a time. We have verified that, by decimating one configuration at a time, one obtains essentially the same results.

3. Vector and pseudoscalar meson masses In order to extract the meson masses and decay constants we have fitted the two point correlation functions in eq. (1) and (2) to the expressions G1(t)

=

(Z1/M1) exp( —M,T/2) cosh(M,(T/2

G,(t)

=

(Z1/M1) exp( —M,T/2) sinh(M1(T/2

—

—

t)),

i = 55, 00, kk,

t)),

i

=

05, 50,

(6)

in the time interval t = 14—24. In eqs. (6) T represents the lattice size in time, i.e. T = 60. To improve the stability, the correlation functions have been symmetrized (anti-symmetrized) around t = T/2. G05 and G50 have been combined together, and all the results below will simply be referred with the label 05, e.g. X05, where X is any physical result. For all the possible choices of K1 and K2, the Wilson parameters of the two quarks in the meson, the interval in time over which the correlation functions have been fitted ensures that only the lightest state propagating in that channel has survived. This is verified by checking that effective mass tn(t) = log(G(t)/G(t + 1)) is constant in the I-interval considered and agrees well with the result of the fit. The good agreement among the values of the mass found by fitting correlation

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Meson spectroscopy

TABLE 1 Pseudoscalar and vector meson masses for mesons composed of two degenerate light quarks (dimensionless units). The subscripts 55, 05, 00 and kk denote the two-point correlations from which the masses have been calculated, eqs. (1) and (2)

K

1

=

K2

0.1485 0.1490 0.1495 KCr

aM5~

aM05

aM00

aMkk

0.269(6) 0.234(8) 0.196(11) 0

0.267(8) 0.233(10) 0.197(14)

0.265(8) 0.228(10) 0.182(13)

0.321(7) 0.294(8) 0.269(9) 0.211(7)

functions of different interpolating operators, e.g. G55, G05, etc. (cf. tables 1 and 2) provides further support that the lightest pseudoscalar particle has been isolated. The results and errors for the meson masses M and source couplings Z are reported in tables 1—4. The various labels, 55, 00, 05, kk, identify the corresponding correlation functions. From the results given in table 1, we can extract Kcr, the critical value of Kw, corresponding to the point at which the pseudoscalar meson becomes the massless Goldstone boson of QCD. From M55 we find Kcr=0.1506(2),

(7)

TABLE 2 Pseudoscalar and vector meson masses for mesons composed of heavy—heavy and heavy—light quarks (dimensionless units). The subscripts 55, 05, 00 and kk denote the two-point correlations from which the masses have been calculated, eqs. (1) and (2)

K1

K2

aM55

aM1)5

aM50

aMkk

0.1275 0.1275 0.1275 0.1275 0.1275

0.1275 0.1485 0.1490 0.1495

1.267(6) 0.812(5) 0.802(5) 0.793(6) 0.772(7)

1.266(6) 0.808(6) 0.798(6) 0.788(6)

1.264(7) 0.805(6) 0.793(6) 0.782(7)

1.271(7) 0.819(6) 0.809(6) 0.800(6) 0.779(6)

0.1325 0.1325 0.1325 0.1325 0.1325

0.1325 0.1485 0.1490 0.1495

1.056(6) 0.698(5) 0.687(5) 0.678(6) 0.655(7)

1.054(6) 0.695(5) 0.683(5) 0.673(6)

1.053(7) 0.692(5) 0.679(5) 0.667(6)

1.061(6) 0.709(6) 0.699(6) 0.689(6) 0.667(6)

0.1375 0.1375 0.1375 0.1375 0.1375

0.1375 0.1485 0.1490 0.1495

0.835(5) 0.578(4) 0.567(5) 0.557(5) 0.531(6)

0.833(6) 0.575(4) 0.564(6) 0.552(6)

0.832(6) 0.573(5) 0.562(7) 0.547(7)

0.844(6) 0.595(5) 0.585(6) 0.574(6) 0.551(6)

0.1425 0.1425 0.1425 0.1425 0.1425

0.1425 0.1485 0.1490 0.1495

0.601(4) 0.449(4) 0.437(4) 0.425(5) 0.396(6)

0.599(5) 0.447(4) 0.434(5) 0.422(6)

0.599(5) 0.445(5) 0.432(6) 0.418(8)

0.617(6) 0.475(5) 0.464(6) 0.453(6) 0.427(6)

KCr

KCr

KCr

KCr

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TABLE 3 Source coupling, cf. eq. (6), for correlation functions of mesons composed of two degenerate light quarks (dimensionless units)

K 1=K2 0.1485 0.1490 0.1495

4Z 3 a 3.9(6) 55X10 3.4(6) 3.0(7)

a4Z

3

a4Z

05X10 1.1(2) 0.9(2) 0.6(2)

3

a4ZkkXlO3

00X10 0.33(7) 0.21(6) 0.10(4)

4.0(5) 3.2(4) 2.5(4)

Another method to derive Kcr is to use the matrix element of the divergence of the axial current [18], (0 3~A~~-) ~ This matrix element vanishes when the pseudoscalar meson, conventionally denoted as ir, becomes massless. Using this method we find Kcr

=

0.1505(1),

(8)

which is compatible with the previous determination. The value of the inverse lattice spacing a~1can be evaluated from any physical quantity with the dimensions of a mass. In the quenched approximation the most popular choices are the value of the vector meson mass (the p-meson), the derivative of the vector meson mass with respect to the squared pseudoscalar meson mass [19] and the pseudoscalar decay constant, f~.Usually different determinations of this quantity give values which are not compatible within the

TABLE 4 Source couplings, cf. eq. (6), for correlation functions of mesons composed of two heavy and a heavy and a light quark (dimensionless units)

K 1 0.1275 0.1275 0.1275 0.1275

K2 0.1275 0.1485 0.1490 0.1495

4Z 2 a 3.7(3) 55X10 1.10) 1.0(1) 0.90)

0.1325 0.1325 0.1325 0.1325

0.1325 0.1485 0.1490 0.1495

0.1375 0.1375 (1.1375 0.1375 0.1425 0.1425 0.1425 0.1425

a4Z

2

a4Z~

2

a4ZkkXlO2

05X10 2.6(2) 0.62(6) 0.56(6) 0.51(6)

5Xl0 1.80) 0.35(4) 0.31(3) 0.28(3)

8.1(7) 1.9(2) 1.8(2) 1.6(2)

2.8(2) 1.00) 0.90) 0.90)

1.90) 0.53(5) 0.48(5) 0.43(5)

1.20) 0.28(3) 0.25(3) 0.22(3)

5.9(5) 1.7(2) 1.5(2) 1.40)

0.1375 0.1485 0.1490 0.1495

2.0(2) 0.86(9) 0.80(8) 0.74(9)

1.2(1) 0.42(4) 0.38(4) 0.34(4)

0.73(6) 0.21(2) 0.19(2) 0.16(2)

3.8(3) 1.30) 1.20) 1.10)

0.1425 0.1485 0.1490 0.1495

1.2(1) 0.69(7) 0.64(7) 0.60(8)

0.63(5) 0.29(3) 0.26(3) 0.24(4)

0.33(3) 0.13(2) 0.11(2) 0.10(2)

2.0(2) 0.94(9) 0.85(8) 0.76(8)

178

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/ Meson spectroscopy

quoted statistical errors [19]. The discrepancy can be attributed to systematic effects (the quenched approximation, effects of 0(a) or perturbative renormalization of the axial current, etc.) and show the systematic uncertainty present in the determination of a’. In the case of light mesons, from the results reported in table 1 (M55 and Mkk), we have fitted the vector meson mass according to

(9)

2+Bk. Mkka=Ak(M55a)

We find Ak 1.53(13) and Bk determination of a =

-

a’(GeV)

a’(GeV)=

0.211(12). From Ak and Bk we get the following

=

1:

=

M~(GeV)/Bk

=

0.77 GeV/Bk

=

(3.7 ±0.2) GeV,

(10)

(M~-M2) ,~ —

~ (GeV)Ak=1.87GeVAk=(2.9±0.2) GeV. (11) M~)

Finally, from the pseudoscalar decay constant, see sect. 4, we find =

(3.3 ±0.6) GeV.

(12)

For meson spectroscopy, we will use the scale derived from the p-mass, eq. (10), (this includes the determination of the bare lattice quark masses, i.e. the K~ corresponding to the up, strange and charm quarks), which we call calibration “b”. For the determination of the pseudoscalar decay constants we prefer calibration “a” based on eq. (12), since this method reduces the systematic uncertainties, cf. sect. 4. However, for a better evaluation of the error on f B, involved in the extrapolation to the B-meson, we also study the scaling behaviour of the decay constants by taking the scale from the p-mass, see sect. 5. We have not used the determination of the scale from eq. (11), because in this case it is not possible to reproduce the spectrum of the light mesons [19]. We have also fitted the mass-squared of light pseudoscalar mesons (table 1) assuming that its dependence on the quark masses is well described by [19,20] A 2— 25

M~=a

K~ 1

K1

—+—————

2

2

.

(13)

Kcr

We estimate A5 0.76(4). By taking a~ from eq. (10), we find that the value of K~for the strange quark is =

K5=0.1495(2)

(14)

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Meson spectroscopy

179

corresponding to a “bare” lattice quark mass 1 m5(a)

1

x

=

a

1

-~-

(88 ±5) MeV.

=

— S

cr

(15)

From the value of m5(a) it is possible to estimate the renormalized mass, in the MS scheme [21,22]: m~(ji

=

2GeV) =Zm(iia)ms(a)

=

(100 ±6) MeV

(16)

where we have taken the perturbative value of Zm(~a)Is~...2Gev 1.13 from ref. [22]. The value in eq. (16) can be compared with the value reported in refs. [23,24] m~(p. 2 GeV) (170 ±30) MeV. As usual the strange quark mass from the lattice is smaller than the estimates in the continuum theory. It has been argued that this is due to the use of the quenched approximation [251. Using K5 from eq. (14), we get the value of K and ‘i masses =

=

=

*

(870 ±40) MeV

(17)

M~=(980±30)MeV

(18)

MK*

=

in fair agreement with the physical values MK* 890 MeV and M~ 1020 MeV [24]. For any given value of the heavy quark hopping parameter, KH, corresponding to K~ 0.1275—0.1425, we have fitted the heavy—light meson masses (table 2) linearly as a function of l/KL, the Wilson parameter of the light quark. From this fit we have extracted the values of the pseudoscalar and vector particle masses at K~r,i.e. in the limit in which the light quark is massless. The extrapolated results are reported in table 2. In the following we will denote the values of the heavy meson masses in the limit KL Kcr as MD and MD*. If instead we take KL K5, the “strange” quark hopping parameter, we will call the corresponding heavy meson masses MD and M0~. We have fitted the dependence of MD on KH to the expressions =

=

=

=

=

MD=at(AamH+B),

(19)

C A’amH+B’+— (20) am 2)(l/KH 1/Kcr). Given a’, the above equations provide a where mHa of (l/ the charm quark mass. With a determination 3.7 GeV, using the physical value for D meson mass, we obtain Kcharm 0.1379(6) from eq. (19) and Kcharm MD=at

,

—

=

—‘

=

=

=

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Meson spectroscopy

TABLE 5 Comparison of charmed meson masses from this calculation and their experimental values. MD is used as an input to fix the charm quark mass

M

0

Lattice Experim.

input 1.87 GeV

MD,

MD*

1.96(2) GeV 1.97 GeV

1.95(1) GeV 2.01 0eV

M0~ 2.03(2) GeV 2.11 GeV

M~

MJ/~

2.93(2) GeV 2.98 GeV

2.96(2) GeV 3.10 GeV

0.1383(6) from eq. (20), corresponding to m~rm(~r 2 GeV) 1.27(4) GeV and m~~~(jx 2 GeV) 1.22(4) GeV respectively (cf. eq. (16)). The charm quark mass fixes the whole spectroscopy of the charmonium states and of the D-mesons. In table 5 we report the results obtained by using eq. (20) for the meson masses and taking m~rm(!L 2 GeV) 1.22 GeV. A better method to compute mass differences, ~iM, when they are small with respect to the masses themselves (cf. table 5) is to take the ratio of two correlation functions, e.g. =

=

=

=

=

R(t)

=

=

(21)

Gkk(t)/Gss(t),

and fit R(t) to exp(—~Mt). In this way it is possible to obtain directly MDa M~ etc. with smaller statistical errors. The results obtained with this —

MD, MJ/4

—

TABLE 6 Pseudoscalar and vector meson mass differences in dimensionless units

3 K1 0.1275 0.1275 0.1275 0.1275

K, 0.1275 0.1485 0.1490 0.1495

a(Mv 3.6±0.7 M~)xi0

0.1325 0.1325 0.1325 0.1325

0.1325 0.1485 0.1490 0.1495

5.5±0.9 10.9±1.4 11.2±1.0 11.3±0.8

0.1375 0.1375 0.1375 0.1375

0.1375 0.1485 0.1490 0.1495

8.8±1.3 16.5±1.8 17.5±2.5 17.4±0.8

0.1425 0.1425 0.1425 0.1425

0.1425 0.1485 0.1490 0.1495

16.1±2.2 26.1 ±2.6 26.8±2.3 27.3±1.6

0.1485 0.1490 0.1495

0.1485 0.1490 0.1495

54.1 ±4.5 63.5±4.9 80.1 ±4.9

6.9±1.2 7.0±1.0 6.9±2.2

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TABLE 7 Comparison between charmed vector—pseudoscalar mass differences from this calculation and their experimental values

_____

MD~.—MD,

_________

Lattice Experim.

72±9MeV 141±1MeV

Mi/k—MO,____

_____

68±7MeV 142±3MeV

35±6MeV 117±2MeV

method are reported in table 6 and 7. The mass differences in table 7 are our best estimate of the charm spectroscopy. The vector—pseudoscalar mass splittings are significantly smaller than the experimental values, as was also found at /3 6.2 [26]. It should be noted that the masses of the heavy quarks used in this simulation range from about 0.2 to about 0.6 in lattice units and therefore 0(a) effects may be significant at the largest quark masses. This may also have consequences for the dependences of the vector—pseudoscalar splitting on the heavy quark mass, which we now discuss. Asymptotically, as m~ the vector—pseudoscalar meson mass splitting satisfies the scaling relation M~ M~ constant [5,6](the relation is satisfied experimentally to a surprising accuracy. The squared mass differences for Hq mesons only, obtained by taking ratios of the corresponding two point correlation functions, eq. (21), are reported in fig. 1 as a function of l/(Mv + Me), where M~and M~denote the vector pseudoscalar masses respectively. Notice that, contrary to the experimental findings, M~ M~ decreases as the meson mass increases, a =

—~ ~,

—

=

—

ff~

0.025

-

p3=6.4

0.020

—

—

0.015

—

—

0.010

—

—

-

c’1

I 1.25

I

1.5

1.75

I

I

2

I

I

I

I

2.25

2/(aMp+aMy) Fig. 1. The vector—pseudoscalar squared mass difference in dimensionless units is reported as a function of inverse average mass 2/a(Mv + Me). The points correspond to heavy—light mesons only.

182

A. Abada et al.

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Meson spectroscopy

TABLE 8 Mass difference between strange and non-strange heavy flavor mesons. H refers to the infinite-mass limit obtained via a linear extrapolation in 1/M 5. The scale is fixed from the p-mass; the K’s corresponding to strange and charm quarks are fixed from the K and D meson masses

M01— Mod Lattice Experim.

MB,

Ma,— MH,

62 (7) MeV

47 (8) MeV

MB,—

86 (5) MeV 99.5 (0.7) MeV

—

—

behaviour which was also seen at /3 6.0 and 6.2 [26,27]. A comparison of our results with those of refs. [26,27] does not show any significant improvement with increasing /3. A decrease of the mass splitting with the meson mass has been also observed for light quarks [28]. The lattice results for the K* and P vector mesons are nevertheless in fairly good agreement with the experimental values, cf. eqs. (17) and (18). It is important to repeat the calculation of the mass splitting with an “improved” action, to verify whether the problem persists in that case. Preliminary results at /3 6.0 [29]seem to show that indeed the dependence of M~, M~ agrees better with the experimental results, when one uses an “improved” action. Finally we have considered the mass difference between a pseudoscalar meson composed of a heavy and a strange quark and a pseudoscalar meson composed of a heavy and a massless quark. The results are reported in table 8 and compared to the experimental number in the case of the D5—D mass difference. These results are consistent, although somewhat smaller than those obtained at /3 6.0 using the static approximation [26]. =

=

—

=

4. Meson decay constants To compute the pseudoscalar decay constant f~,defined as (0

~IYEY51/’2

IF) =f~p~,

(22)

we have studied the ratio G05(t) Rf(t) =ZA G55(t)

(23)

where ZA is the axial current renormalization constant, which is needed to relate the lattice current to the continuum one, A~0nt ZAAE [30—32].We have taken ZA(13 6.4) 0.88 from first-order perturbation theory. =

=

=

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Meson spectroscopy

183

TABLE 9 Vector and pseudoscalar decay constants for light mesons

0.1485 0.1490 0.1495 Kcr

af~

1/f~

0.062(5) 0.058(5) 0.05 1(6) 0.040(8)

0.299(6) 0.316(7) 0.33500) 0.375(22)

At large time separations, the ratio in eq. (23) goes as

Rf

0(t)

—

Z4 ~tanh(M~(T/2

—

t))

=

ZA (0

~iYoY5~2

P) tanh(Mp(T/2

—

55

(24) where M~is the meson mass. The last factor on the r.h.s. of eq. (24) is essentially constant and equal to ±1 for the heavy mesons, while it varies significantly in time for the lightest ones (i.e. K~ 0.1485—0.1495). The results for the decay constants have been obtained using eq. (24) as follows. We define, cf. eq. (22) =

~

(25)

where

f~(t)

=

T -~-_coth(Mss(~ _t))Rf~(t).

(26)

In eq. (25), the sum is extended over the same time interval which was used to fit the masses (cf. sect. 3) and N is the number of terms in the sum Rf(t) is computed from the two point correlation functions of eqs. (1) and we have used the value of Z55 (0 j P> j and M55 obtained from the fit of G55 (table 1—4). The results for f~,are given in tables 9 and 10. For light quarks, the values of ft,, extrapolated linearly in l/K~to the chiral limit, give ~.

=

2

fp(Kcr)a

from which, by fixing *

=f,Ta

=

0.040(8),

f,. to its physical value, f,.

=

(27)

132 MeV, we obtain the result

We have not done an average in time weighting the terms in the sum of eq. (25) by their errors, because the errors, in the range of

t

used in eq. (25), are comparable.

184

A. Abada et a!.

/

Meson spectroscopy

TABLE 10 Vector and pseudoscalar decay constants for heavy—heavy and heavy—light mesons

K

1

K2

af~

l/f~

U

0.1275 0.1275 0.1275 0.1275 0.1275

0.1275 0.1485 0.1490 0.1495 K~,

0.097(3) 0.069(3) 0.067(4) 0.065(4) 0.061(5)

0.086(3) 0.101(4) 0.099(4) 0.097(4) 0.092(4)

0.85(3)

0.054(4)

0.1325 0.1325 0.1325 0.1325 01325

0.1325 0.1485 0.1490 0.1495 K,~

0.096(3) 0.071(4) 0.069(4) 0.067(4) 0.063(5)

0.104(4) 0.124(5) 0.122(5) 0.120(5) 0.115(4)

0.83(4)

0.051(4)

0.1375 0.1375 0.1375 0.1375 0.1375

0.1375 0.1485 0.1490 0.1495 Kcr

0.093(3) 0.073(4) 0.070(4) 0.069(5) 0.064(5)

0.132(5) 0.158(5) 0.157(5) 0.154(5) 0.150(5)

0.79(3)

0.047(4)

0.1425 0.1425 0.1425 0.1425 0.1425

0.1425 0.1485 0.1490 0.1495 Kcr

0.086(3) 0.072(4) 0.070(4) 0.068(5) 0.064(5)

0.179(6) 0.208(6) 0.208(6) 0.207(6) 0.206(7)

0.75(4)

0.040(4)

in eq. (12)

*~

f,~.With a’

Alternatively, we can take another determination of a 3.7 GeV (cf. eq. (10)) we obtain

— ~,

and predict

=

f,.=(145±30)MeV

(28)

A natural method to compute the decay constants of mesons other than the pion is from the ratio f~/f,1. lattice

(29)

where f, is the experimental pion decay constant. One expects that in this ratio many systematic errors, for example due to the choice of ZA, are reduced. This is equivalent to taking a’ from eq. (12). For fK, since its value is so close to f,., it is *

We can compute the decay constants also by using the quantity f~= ZAVGOU(t)/Gss(t) and modifying accordingly eq. (26). The statistical errors on f~,are in general slightly worse if we use instead of G115, with central values always compatible within errors. For example, from G50 we get afr = 0.043(9), cf. eq. (27).

A. Abada et a!.

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Meson spectroscopy

185

TABLE 11 Charmed vector and pseudoscalar decay constants

f0

Lattice Experim.

f~, 230(50) MeV

210(40) MeV <290 MeV

1/f0 0.37(2) 0.28(1)

—

more significant to report fK/f,. respect to the light quark mass

—

fK/f~

1/fk

1/f~*

1/ft/k

0.34(1) 0.232(5)

0.148(1)

0.130(5) 0.124(5)

—

1 which is related to the derivative of

—

1

=

0.16 ±0.07,

f~with

(30)

to be compared with the experimental value 0.22(1). From chiral perturbation theory, one can argue that the quenched value of fK/f,~. 1 has to be about one half of its value in the full theory [33]. We report in table 11 the predictions from our calculation for fr,, under the assumption a~ 3.3 GeV, K 5 0.1495 and Kcharm 0.1383. The results for fDD are in good agreement with previous lattice determinations [9—11,34]and with QCD sum rules [35]. To evaluate the effect of the mass of the strange quark, it is more instructive to compute directly the ratio fp(KL KS)/fp(KL Kcr). The results reported in fig. 2 as a function of the inverse pseudoscalar meson mass M~,are compatible with a —

=

=

=

=

1.150

1.125

1111111

,

1

I

II~

L

This work

+

Alexandrou

—

—

I ‘4—I

=

et al.

1.100

—

—

1.075

—

—

1.050

—

—

1.025

—

—

p.4 ‘4—4

1.000

I

I 0

11111111

~I

I

I

I

I

I

I

1/ (Mp+Mp5) (GeV~’

Fig. 2. The ratio of the pseudoscalar decay constant for a meson composed of a heavy and a strange quark, ft,, to the pseudoscalar decay constant for a meson composed by a heavy and a massless quark, fp, is reported as a function of the inverse meson mass, 1/(M5+ M0).

186

/ Meson spectroscopy

A. Abada et a!.

constant. In fig. 2 we have also plotted the results obtained in ref. [11] at /3 6.0. The latter agree with those at /3 6.4. Extrapolating to the B-meson, we obtain =

=

fB,/fB,

=

1.06 ±0.04.

(31)

The same quantity has also been computed at lowest order in the static quark limit [4], at /3 6.0 on a 20 x 102 x 40 lattice [12], and on a 12~>< 36 lattice [11], obtaining =

fB,/fB,

I static

=

1.09 ±0.04,

(32)

fB~/fB,

static

=

1.08 ±0.04,

(33)

and

respectively, cf. fig. 2. From the above results we quote fB,/fB.

=

1.08 ±0.06.

(34)

This result, together with the calculation of the relevant B-parameter, is important for the phenomenology of B—B mixing. The analysis of the vector meson decay constant, ~ proceeds in a very similar way. f~is defined by M~3

(0~fr~V)=~,—~-—, iv

(35)

where is the vector meson polarization and M~ its mass. The matrix element of the vector current is extracted from a fit to Gkk(t) i iv

2 M kk

(36)

where Z~ is the renormalization constant of the local lattice vector current, Z~(f3 6.4) 0.84 from first order perturbation theory, and Zkk and Mkk as given in tables 1—4. Our results for ~ f,. 4 etc., using the same values for K5 charm as for the pseudoscalar decay constants, are reported in table 11. It is known that different methods of calculating Z~lead to results which differ by 40—50% at lower values of /3 [18]. For example, the non-perturbative evaluation of Z~at /3 6.0 is 0.57 to be compared to 0.83 from perturbation theory. Consequently, in an attempt to reduce the systematic error, we present also f,~and f~, using =

=

=

=

(f~/fV)alh~f~t,

(37)

A. Abada et a!.

/

Meson spectroscopy

187

where f,~is the experimental value, 0.28 ±0.01, in analogy to what has been done for the pseudoscalar decay constant. We obtain

f~=0.250±0.012,

(38)

0.095 ±0.006.

(39)

fi)ct

=

5. Scaling properties of heavy meson decay constants In this section we study the scaling properties of the decay constants of the heavy—light mesons. This allows us to verify whether, at values around the charm quark mass, the decay constants follow the predicted asymptotic behaviour. It can be shown that, in the limit m~—p the vector and pseudoscalar decay constants scale with the mass of the heavy quark, m~,as [2—4] ~,

M

f~,

C2/~o(M),

(40)

~~a

where M M~ M~ m~. We construct then the quantity =

=

=

(41)

U(M)=f~f~/M,

which should be equal to one in the asymptotic limit, eq. (40) We report U(M) and its error in table 10. A linear fit in 1/M, with M (M~+ M~)/2, gives U(M ~) 0.96 ±0.04, U(M (MB + MB*)/2) 0.90 ±0.04 and U(M (MD + MD*)/2) 0.80 ±0.04. We now discuss the dependence of cJ5(M~)=f~/~i~ on 1/Me. Our new results are reported in table 10 and fig. 3. To increase the information, besides the numbers reported in table 10, we have also used the results from refs. [9—11], where the decay constants were computed with propagating heavy quarks, and from refs. [11,12] where f~,was computed at lowest order in the Eichten expansion, see table 12. Since lattice artefacts are expected to become important for large masses, we have used only the points of ref. [11] corresponding to am~ 0.7. As mentioned before, in the study of the pseudoscalar decay constant, we prefer to fix the scale from f,. (case “a”). However, in the literature, the results have been presented using both calibrations “a”, eq. (12) and “b”, eq. (10). In order to compare our results with previous studies, we have then normalized all the data *•

=

=

=

=

=

=

=

©

*

Up to systematic errors due to the fact that we have used the perturbative values of the renormalization constants of the axial vector and vector currents, Z 4~.

188

A. Abada et a!. ~csi

0.6

,I~

/ Meson spectroscopy

1111

1111

III

1111

II

122

p=6.06.2,6.4 05

Alexandrou et al. Gavela et al. It Ailton et at. X this work

—

0

—

+

Co

-

:I

~

III

II

~I

I

I

I~

III

I

1/Mp

I

I~

II

I ~I

I

(GeV~1

Fig. 3. The quantity fp~/~is reported as a function of the inverse pseudoscalar mass. The results from several calculations at /3 = 6.0, 6.2 and 6.4, with fully propagating quarks only, are shown. The lattice results have been converted in physical units by fixing the scale with method “a”, i.e. using f,,..

TABLE 12 1/Me in dimensionless units and with the two different calibrations, “a” and “b”, of the lattice spacing. The static results, obtained at /3 = 6, have been multiplied by a factor

f~/A~ and

(aS(MB)/aS(l /a))— 6/33

x

a3”2

l/M~

fp~/~i~

Xa~

GeV3~2 “a”

GeV’ “a”

0eV3”2 “b”

1/Me GeV1 “b”

1/Me

fps/~

/3 = 6.0 ref. [11]

0.256(28) 0.092(8) 0.084(7)

0.0 1.00 1.28

0.56(9) 0.20(3) 0.18(3)

0.0 0.60 0.76

0.86(10) 0.31(3) 0.28(3)

0.0 0.44 0.57

ref. [12]

0.251(20)

0.0

0.55(6)

0.0

0.85(8)

0.0

ref. [9]

0.105(10)

1.23

0.23(3)

0.73

0.35(4)

0.55

/3 = 6.2

0.066(2) 0.064(2) 0.061(2) 0.055(2) 0.047(4)

1.10 1.25 1.46 1.78 2.33

0.33(3) 0.32(3) 0.30(3) 0.27(3) 0.23(3)

0.38 0.43 0.50 0.61 0.80

0.28(2) 0.27(2) 0.26(2) 0.23(2) 0.20(2)

0.42 0.48 0.56 0.68 0.90

ref. [9]

0.073(4)

1.51

0.29(4)

0.61

0.37(4)

0.51

/3 = 6.4 this work

0.054(4) 0.051(4) 0.047(4) 0.040(4)

1.30 1.53 1.88 2.53

0.33(6) 0.31(5) 0.28(5) 0.24(4)

0.39 0.46 0.57 0.76

0.38(3) 0.36(3) 0.33(3) 0.28(3)

0.35 0.42 0.51 0.69

ref. [10]

A. Abada et a!.

/

Meson spectroscopy

189

consistently either with “a” or “b”. Notice that since i a3”~it is very sensitive to the choice of scale. In order to combine the results of the static theory with those obtained with fully propagating heavy quarks, we have to take into account the anomalous dimension of the decay constant in the infinite-mass limit, cf. eq. (40). The anomalous dimension implies that cI’(M~) a 2~’3°as M~ 5(MpY with fully propagating quarks. On the other hand, in the static limit, the axial current operator is logarithmic divergent in the ultraviolet cutoff and its matrix elements diverge as a 2~’/3.For this reason in refs. [11,12], the bare axial 5(1/aY current has been multiplied by the factor (aS(MBY2/~°/aS(1/aY2”3°), i.e. it has been renormalized at the scale j.t MB. To obtain the same normalization in the full and static case, we have then multiplied all the results in the full case by the factor (a 6”33 5(]t’Ip)/a5(M~)) This defines ‘~

‘~

=

*.

‘J~(M~) (a

(42)

6”33~(MP),

=

5( Mp)/aS(MB)) which is finite in the infinite-mass limit. The correction to the value of P(M~), due to the anomalous dimension, amounts to few %, in the range of M~considered, and do not alter significantly the dependence of cb(M~)on the meson mass. The results with propagating quarks alone suggest a linear dependence in 1/Me. By fitting all the points in figs. 4 and 5, including those from the static theory, to ~15(M~)=I~+

(43)

~,

M~ one finds (case “a”) =

=

(0.47 ±0.03) GeV3”2,

(44)

(—0.38 ±0.05) GeV5~2,

(45)

with a x2 per degree of freedom, ~2/d.o.f.0.9. To obtain fD we have then multiplied the result, i.e. ‘D,,. + D’/MD, by (aS(MD)/aS(MB)Y6733. The values in eqs. (44) and (45) correspond to fB (175 ± 10) MeV and (212 ±4) MeV. The errors on and fD have been evaluated by allowing a variation of one on x2. The correction due to the 1/Me term is 14% for the B-meson and 40% for the D-meson. =

=

fD

-,

*

—

=

fB

‘~

—

We have used the leading-logarithmic expression for

with A 0c0 = 200 MeV. 6/33 is the

appropriate anomalous dimension in the quenched approximation.

190

A. Abada et a!.

/ Meson spectroscopy

If instead we take the points of fig. 5 (case “b”) we find: (0.48 ±0.04) GeV3”2,

(46)

(—0.37 ±0.05) GeV5”2,

(47)

=

‘ii’

=

with a ,y2/d.o.f. 5.0. The quality of the fit is much worse than in the previous case. Of course a ~2/d.o.f.5 is worrying per se and moreover we would expect a ~2/d.o.f. even ~ 1, given the fact that some of the points are computed on the same sets of gauge field configurations. The values in eqs. (46), (47) correspond to fB 176i~MeV and fD 214~~ MeV. The errors on fB and fD have been evaluated by allowing a variation of 5 on x2. The correction due to the 1/Me term is 15% for the B-meson and 40% for the D-meson. Consistency of lattice QCD requires that the points from below join those obtained in the 1/Me expansion at large values of M~.The static results for P, although not incompatible with the numbers obtained at lower masses, appear rather high, in the case “b” (cf. fig. 5) The authors of refs. [8,11] have observed a decrease of 27 with /3 which they attribute to rather strong 0(a) corrections. To explore the stability of the results for fB’ we have also performed a quadratic fit in 1/Me assuming that, since linear terms are quite large around the D-mass, quadratic corrections (at least) should be included. The result of a quadratic fit =

=

=

—

—

*~

tp’

cP(M~)=cP

+—+-——

M~

~“

(48)

M~

gives (case “a”) (0.55 ±0.02) GeV3”2,

(49)

(—0.76 ±0.03) GeV5”2,

(50)

(0.39 ±0.07) GeV7”2,

(51)

=

CP’

=

=

with a ~2/d.o.f. 0.7. The values in eqs. (49)—(51) correspond to fB (183 ±7) MeV and fD (200 ±20) MeV. The errors on fB and fD have been evaluated by allowing a variation of 1 on x2. The correction due to the 1/Me term is 25% for the B-meson (—~ 70% for the D-meson); the quadratic correction is +2% (+20%). =

=

=

—

—

The difference may be due to several systematic effects (0(a) corrections or perturbative matching, for example) which are present in the different approaches.

A. Abada et a!.

I

0.6

I

I

I

I

/

Meson spectroscopy

191

I

scale from

—

—

~=6.0.6.26.?

11)

0 Alexandrou et a1. + Gavela et al.

0.5

—

a Ailton et a!. x this work

01

i~I

I

I

I

I

I

—0.25

0

I

I

—

I~~II~IIIIIIII

0.25

Ii

0.5 1/Mp

0.75

1

(GeV~1

Fig. 4. The quantity f~~/7/?~ is reported as a function of the inverse pseudoscalar mass. The results from several calculations at /3 = 6.0, 6.2 and 6.4, with fully propagating quarks and in the static limit, are shown. The scale has been fixed using method “a”, i.e. using f,~. The curves refer to the linear and quadratic fit described in the text. The points in the full case have been multiplied by the factor (cr 6”33. The two points computed in the static limit have been multiplied by the factor (a 6”33. The vertical line identify the point corresponding to the B-meson. 5(Mp)/a(M5)) 5(1/a)/a/M~))

In order to monitor the systematic uncertainties, we have repeated the analysis above using calibration “b”. In this case we find =

=

=

(0.75 ±0.03) GeV3”2,

(52)

(—1.37 ±0.04) GeV5”2,

(53)

(0.84 ±0.08) GeV7”2.

(54)

Again the fit is rather poor with a X2/d.o.f. 2.5. The values in eqs. (52)—(54) correspond to fB (230 ±10) MeV and fD (200 ±25) MeV. The errors on fB and fD have been evaluated by allowing a variation of 3 on ~2. The correction due to the 1/Me term is —35% for the B-meson —95% for the D-meson); the quadratic correction is + 5% (+ 30%). The linear and quadratic fits are given in figs. 4 and 5 as continuous curves. The inverse mass corresponding to the physical value for the B-meson is also explicitly indicated in the figures. The authors of ref. [8] have tried a simple ansatz for the dependence of the on M~and the lattice spacing. They conclude that in this way it is possible to reconcile in the static and full case, when using calibration “b”. =

=

=

‘~

(~*

192

A. Abada et aL Co ~

1.0

____________ II III!

/

Meson spectroscopy

___________________________ 1111 1111

scale from

M

11)

p 0

/36.0,6.2,6.4

—

0

Co

IS + P-I

1111

0.6

+

—

Alexandrou et a!. A1!ton et at. Gavela et a!. this work

—

—

0.0II1~I11IhIIIIII

1/Mp (GeV)~ Fig. 5. Same as in fig. 4, but now the scale is taken with method ‘b”, i.e. using M

0.

At the B-mass, taking the values between two curves, we estimate 165 MeV ©fB ~ 190 MeV (“a”) or 165 MeV ~f6 ~ 240 MeV (“b”), from which we quote (cf. sect. 1) fB

=

(205 ±40) MeV.

(55)

For fD we only take the results of the linear fits, which are dominated by the points close to MD, corresponding to fully propagating quarks. We quote 2l0±lS)MeV. (56) Our conclusions about the fD( scaling properties of P~,are the following: (i) Matching between the asymptotic expansion and the propagating quark case is remarkably good, if the scale is taken using the lattice value of f,., eq. (29). (ii) There are large violations to the asymptotic scaling behaviour for m 11 mcharm. This result is supported by calculations done at three different values of /3, /3 6.0, 6.2 and 6.4. (iii) A linear dependence of 1 in 1/me is well satisfied with propagating heavy quarks of mass m~in the range m11a 0.7. (iv) If the p-mass is used to set the scale, then the quality of the fits is significantly reduced. Nevertheless, a quadratic fit results in a value of P,, which is consistent with the value obtained using the static approximation. (v) It will be very important to repeat the above calculations using an 0(a) improved action. This will help to check whether the heavy quark masses are too large and lead to significant systematic errors. =

©

A. Abada et of

/ Meson spectroscopy 13

TABLE

Determinations of the lattice spacing from

193

f,,., “a”, and

M

5, “b”, at several values of /3

a~(GeV)

/3 “a”

“b”

ref.

6.0

1.67±0.13 1.70±0.20

2.25±0.16 2.30±0.25

[19] [11]

6.2

2.48±0.25 2.90±0.20

2.96±0.17 2.60±0.10

[9] [101

3.50±0.14

[8]

6.26 6.4

3.32±0.60

3.66±0.14

this work

Before closing this section we want to add some infonnation which may be useful to the reader. Since we are combining results which have been obtained at different values of 13, and thus may different 0(a) we 13. haveIn also t have method “a” and “b”,effects, cf. table the studied the scaling behaviour of a quenched approximation, as a 0, asymptotic freedom predicts —~

ln(at) =const.+S/3

+

O(ln(/3)),

(57)

where S 4~.2/33 1.20. A linear fit of ln(a _!) as a function of /3, using the data in column “b” of table 13 gives S 1.17 ±0.25 in good agreement with the expected slope in /3, fig. 6. If we take the scale from f,, we find S 2.15 ±0.70, fig. 7. Although the large error does not allow us to draw any firm conclusion, it is likely that the larger slope found in case “a” is related to the observation that the value of a~1from M~is systematically larger than the value obtained using f,. and that the difference decreases as /3 is increased (or using the improved action [29]). Thus for example M~/f,. 4.5 at /3 6.0 whilst we find M~/f,. 5.3 at /3 6.4, to be compared to the experimental value 5.8. =

~,

=

=

—‘

=

=

6. Heavy—light meson B-parameters We have computed the B-parameter of the heavy-light 1~1F 2 four-quark operator in eq. (5), B~,by taking the ratio of the three-point correlation function in eq. (4) to G50(t1) X G50(t2) [19]: =

3G3(t1, R3(t1,

t2)

=

Q ‘72 °~A

*

1

12)

\f~’

I

\

—s B~,

G50k 11) ‘-‘50~2)

We have also used the results of the APE group, which give a

— 1(/3

= 6.0) = 2.30 ±0.06 [28].

(58)

194

A. Abada et aL

/ Meson spectroscopy

5

scale from M~

Fig. 6. logO/a), with a in 0eV~, as determined from M,, is reported as a function of /3 together with the linear fit given in eq. (57). 9.0 8.0

111111h1111h11h1h111

1111

0.8~1jj

Fig. 7. logO/a), with a in GeV

~,

as determined from f,r is reported as a function of /3 together with the linear fit given in eq. (57).

as the time separations t12 become large (12 ~‘+ 0 <
©

*

t

<0 corresponds to

t > T/2.

=

A. Abada et a!.

/ Meson spectroscopy

195

TABLE 14 B-parameters of mesons composed of a heavy and a light quark

K

1

K2

1/(aM0)

B0

0.1275 0.1275 0.1275

0.1485 0.1490 0.1495

1.23 ±0.01 1.25 ±0.01 1.26±0.01

0.85 ±0.03 0.85 ±0.03 0.84±0.03

0.1325 0.1325 0.1325

0.1485 0.1490 0.1495

1.43±0.01 1.46 ±0.01 1.47±0.01

0.84±0.04 0.84 ±0.04 0.83±0.04

0.1375 0.1375 0.1375

0.1485 0.1490 0.1495

1.73 ±0.01 1.76 ±0.02 1.80±0.02

0.82 ±0.04 0.80 ±0.04 0.81 ±0.05

0.1425 0.1425 0.1425

0.1485 0.1490 0.1495

2.23±0.02 2.29±0.02 2.35±0.03

0.78±0.05 0.77±0.05 0.76±0.05

which B~has been computed. In fig. 8 we report B~as a function of the inverse heavy meson mass M~.The points in the figure have been obtained by extrapolating B~and M~to the zero-mass limit for the light quark in the meson. We have fitted the points in fig. 8 as a function of the heavy meson mass according to BP(MP) =B~+B,/(M~a).

(59)

III

I,IIIIIIIIIIIII

1/aMp Fig. 8. The B—B B-parameter, B(1~= 3.7 GeV), is shown together with a linear fit in 1/aM0. The points refer to heavy—light mesons only.

196

A. Abada et at

/

Meson spectroscopy

With a from eq. (10) and fixing the meson masses to the physical values appropriate for D°and B°mesons we found —

BDO(~t=

3.7 GeV)

=

0.78 ±0.06,

(60)

=

3.7 GeV)

=

0.86 ±0.05,

(61)

0.90 ±0.05.

(62)

BBO(~L

B~(~t 3.7 GeV) =

=

The physical predictions for D°—D°and B°—B°mixings are obtained from the renormalization group invariant B-parameters, BD and BB. These can be computed from eqs. (60), (61) by multiplying BDo(~r) and B8o(p,) by the factor 6”33 [36]. By taking AQCD 200 MeV, corresponding to F GeVY F a5(~.t 1.35, we3.7 obtain =

= =

=

BDO=

1.05 ±0.08,

(63)

BB1

1.16 ±0.07.

(64)

=

These results confirm that the B-parameter is very close to one for heavy—light mesons. Combining the result in eq. (55) with eq. (64) we predict fB,fA~

=

(220 ±40) MeV.

(65)

From the study of the dependence of the B-parameter on the light quark mass we have also found (66)

B~,/BP, 1.02 ±0.02, =

at all values of the heavy quark mass. From the results in eq. (66) and eq. (33) we find RSd =f~,B B/f ~,BB, 1.19 ±0.10.

(67)

=

This ratio can be used to predict the B 5—B5 mixing parameter:

x(B.) ~

I”tb~”~2 =

~b~dI

Xx(Bd) XR.d S

2(1+p2—2p 1.19 cos A

x(Bd),

(68)

where x(B) LIMB/FB and A, p and 6 are the parameters of the CKM matrix in the Wolfenstein parametrization [37] By taking x(Bd) 0.7, A 0.221, p 0.5 =

*•

*

=

=

The relation between the notation used here and in ref. [37] is p exp(— iö) = Pw subscript W refers to the parameters introduced by Wolfenstein.

=

—

~‘1w,where the

A. Abada et a!.

and a positive cos 6, cos 6 obtain

=

/

Meson spectroscopy

0.7, as suggested by a large value of x(B5)

‘~

30.

197 fB

[7,8], we

(69)

which is practically unmeasurable experimentally. We warmly acknowledge V. Lubicz for an early participation to this work and for many useful discussions. We thank L. Maiani for discussions. C.R.A. and C.T.S. acknowledge the support of the SERC and D.B.C. that of Esprit project 2701 (Puma); G.M. acknowledges the partial support of the MURST, Italy and INFN. We also acknowledge the following computing centers: CCVR (Palaiseau, France), CINECA (Bologna, Italy) and HLRZ (Jülich), where these calculations were performed and thank their staff for their precious help.

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