B meson decay constants from NRQCD

14 May 1998

Physics Letters B 427 Ž1998. 132–140

B meson decay constants from NRQCD A. Ali Khan a , T. Bhattacharya b, S. Collins c , C.T.H. Davies c , R. Gupta b, C. Morningstar d , J. Shigemitsu a , J. Sloan e a

c d

Physics Department, The Ohio State UniÕersity, Columbus, OH 43210, USA b T-8, Los Alamos National Laboratory, Los Alamos, NM 87545, USA Department of Physics & Astronomy, UniÕersity of Glasgow, Glasgow G12 8QQ, UK Physics Department, UniÕersity of California at San Diego, La Jolla, CA 92093, USA e Physics Department, UniÕersity of Kentucky, Lexington, KY 40506, USA Received 11 February 1998 Editor: M. Dine

Abstract We present quenched results for B meson decay constants using NRQCD b quarks and O Ž a. tadpole improved clover light quarks. For the first time, one-loop matching factors between lattice and continuum currents are incorporated through O Ž arM . taking operator mixing fully into account. This includes an important O Ž a a. discretization correction to the q8 .Ž .Ž . q4 . heavy-light axial vector current. We find f B s 147Ž11.Žy12 9 6 MeV and f B srf B s 1.20Ž4.Žy0 . q 1998 Published by Elsevier Science B.V. All rights reserved. PACS: 12.38.Gc; 12.39.Hg; 13.20.He; 14.40.Ndi

1. Introduction The B meson decay constant f B and the bag parameter BB play crucial roles in analyses of B0 y B0 mixing phenomena and the extraction of CKM matrix elements. Experimental determination of f B is severely constrained by the very small leptonic branching fraction, while BB is not a direct outcome of experiments. Their calculation has proven difficult due to the possibility of large QCD corrections to the basic weak process. Lattice QCD provides a non-perturbative method for calculating the relevant hadronic matrix elements from first principles w1x. This paper presents a quenched lattice calculation of f B and f B s using the non-relativistic QCD ŽNRQCD. approach to b-quarks w2x. Much of our discussion of the

systematic improvement of f B , however, applies to other lattice calculations of this quantity. In the rest frame of the B meson, the momentum of the b-quark is of O Ž LQ C D .. The b-quark velocity is hence Õ f LQC D rM f 0.1, where M is the heavy quark mass, and the nonrelativistic approach is sensible. The heavy quark action that we employ includes relativistic corrections through O ŽŽ L QC D rM . 2 .. For the light quarks we use the tree-level tadpole improved clover action which reduces O Ž a LQ C D . errors to O Ž a Ž a LQ C D .. w3x. In the heavy-light axial current we investigate both 1rM and 1rM 2 corrections. The one-loop matching factors relating the lattice and full Žrelativistic. continuum QCD currents have been calculated to O Ž arM . w4x and incorporated in this analysis. Mixing among heavy-light

0370-2693r98r$19.00 q 1998 Published by Elsevier Science B.V. All rights reserved. PII S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 0 3 2 3 - 2

A. Ali Khan et al.r Physics Letters B 427 (1998) 132–140

current operators and contributions from an O Ž a Ž a LQ C D .. discretization correction to the current are taken into account. In the nonrelativistic approach the 1rM expansion and O Ž a. improvements go hand in hand. Previous calculations of f B using NRQCD b-quarks have included only subsets of the above ingredients w5–7x, as have calculations using the relativistic formalism for the b-quark w8– 12x. Here we include all of them and carry out a comprehensive study with improvements through O ŽŽ LQ C D rM . 2 . and O Ž a L QC D . in the action and th ro u g h O ŽŽ L Q C D r M . 2 . , O Ž a r Ž a M .. , O Ž a Ž LQ C D rM .. and O Ž a Ž a L QC D .. in the currents. 2. Details of the simulation The NRQCD heavy quark action density used in our lattice simulations is a variant of the one proposed in w2x, L s ct c t

ž

yct 1 y

ž

ad H

/ž / ž

2

aH0

1y

t

n

aH0 2n

ad H

n

/ /

U4† t

1y c ty1 , Ž 1. 2 n ty1 2 ty1 where c denotes a two-component Pauli spinor, H0 is the nonrelativistic kinetic energy operator, = 1y

D Ž2. H0 s y

2 M0

,

Ž 2.

and d H includes relativistic and finite-lattice-spacing corrections, g ig dHsy sPBq =PEyEP= . 2 Ž 2 M0 8 Ž M0 . g y sP Ž==EyE== . 2 8 Ž M0 . y

Ž DŽ2. .

2

8 Ž M0 .

3

a2D Ž4. q

y 24 M0

a Ž D Ž2. .

2

16 n Ž M0 .

2

.

Ž 3.

In addition to all 1rM 2 terms we have also included the leading 1rM 3 relativistic correction as well as the discretization corrections appearing at the same order in the momentum expansion. = and D Ž2. are the symmetric gauge-covariant lattice derivative and laplacian, while D Ž4. is a discretized version of the

133

continuum operator Ý Di4 . The parameter n is introduced to remove instabilities in the heavy quark propagator due to the highest momentum modes w2x. For the heavy quark action, we use two-leaf E fields w13x rather than the four-leaf fields of previous NRQCD simulations. The tree-level coefficients are tadpole-improved by rescaling the gauge fields by the fourth root of the plaquette, u 0 s 0.87779 at b s 6.0 w14x. For the light quarks we use the tree-level .. tadpole improved clover action Ž c SW s uy3 0 Simulations were carried out on 102 quenched configurations on 16 3 = 48 lattices at b s 6.0. Both forward and time reversed quark propagators were calculated on each configuration in order to improve statistics. Error estimates are obtained using a bootstrap procedure. Meson correlators are calculated for 30 combinations of quark masses; five values for k and six values for the bare heavy quark mass. Two of the k values are selected to simulate the charm quark, and the remaining three are around the strange quark mass. Heavy quark masses range from below the bottom quark mass to about five times the bottom quark mass. Both the heavy and the light quark propagators were smeared at the sink andror the source. Further details are given in Ref. w6x. There are four input parameters in the simulations, the light quark mass Ž k l ' k u, d ., the strange quark mass Ž k s ., the scale ay1 , and the heavy quark mass Ž M0 .. These need to be fixed via experimental input. To set the scale one can use a variety of quantities like Mr , fp , string tension Ž s ., or quarkonium S–P splittings. In quenched calculations, however, different physical quantities can lead to different ay1 ’s and one must make a choice. In principle one would like to choose an observable whose characteristic momenta are similar to those of the quantities being calculated and which has comparable quenching and discretization errors. The first choice would be to fix ay1 from the heavy-light spectrum itself but experimental and numerical errors will have to be reduced before this becomes practical. Based on the ‘‘brown muck’’ picture of HQET for heavy-light systems w15x, however, we believe that the scales relevant for energy splittings in heavy-light spectroscopy and for the decay constant are determined by the light quark dynamics. For f B and spectroscopy we shall therefore consider observables in the light quark and

A. Ali Khan et al.r Physics Letters B 427 (1998) 132–140

134

gluonic sector to set the scale. For clover fermions at b s 6.0, the values for ay1 from Mr , MK , MN , MD , s , and fp range between 1.8 GeV and 2.0 GeV. We Ž . use ay1 m r s 1.92 7 GeV as it is close to the mean, and can be extracted most reliably from our data. We also calculate the final values using ay1 s 1.8 and 2.0 GeV and use the spread as the uncertainty due to the determination of the scale. We determine k l s 0.13917Ž9. by linearly extrapolating the pion mass, calculated on the same set of configurations, to the physical value. To set k s we use the K, the K ) and the f , which yields the values 0.13755Ž13., 0.13719Ž25., and 0.13717Ž25., respectively. We quote central values for our Bs meson decay constant using the K, and the difference between the K and f as a systematic error. The bare heavy quark mass is fixed by linearly interpolating between the meson masses with the three lightest M0 values to the experimental B meson mass. Since the NRQCD action omits the heavy quark rest mass, heavy-light pseudoscalar meson masses are not direct outputs of the simulations. Instead, correlation functions fall off with an energy Esi m that is related to the full meson mass MP S and the bare b-quark mass M0 as, MP S s D q Esi m ' w Zm M0 y E0 x q Esi m , Ž 4. where Zm is the mass renormalization constant and E0 an energy shift. Zm and E0 have been calculated perturbatively and Esi m is obtained nonperturbatively from the simulations. An alternate way to obtain the meson mass is to calculate energy splittings between correlators with and without spatial momentum,

(

Esi m Ž p . y Esi m Ž 0 . s p 2 q Mk2i n y Mk i n .

Ž 5.

In Table 1 we list results for MP S of Eq. Ž4. using perturbation theory to calculate D Ždenoted ‘‘M p er t ’’

Fig. 1. Heavy-light spectrum with b-quarks. Horizontal dashed lines bracket experimental values taken from the Particle Data Book. Horizontal dotted lines indicate first experimental observaX tions of these states Ž w17x Ž2S & 1q r B2) . and w18x Ž S b ... Errors on simulation results include statistical plus, where appropriate, errors from extrapolation in the light quark mass to the physical pion. The two results for the strange B mesons correspond to fixing the strange quark mass from either the K Žfilled circles. or the f Žopen circles.. For the J s1 P states only one level is shown, since we are currently unable to reliably separate the X physical 1q and 1q states.

in the table. and for Mk i n of Eq. Ž5., both measured in units of ay1 . D can also be obtained nonperturbatively from simulations of quarkonium QQ systems, whose correlations fall off with an energy EsiQQm related to the full QQ meson mass as, QQ MQQ s MQQ k i n s 2 D q Esi m .

Ž 6.

QQ Both MkQQ i n and Esi m can be extracted from the simulations, leading to a nonperturbative determination of D. The resulting masses for the heavy-light mesons are also listed in Table 1 under aM X . One finds consistency among the three determinations of

Table 1 Simulation energies and pseudoscalar meson masses in lattice units for light and strange clover quark masses. Meson masses are determined X from Eq. Ž4. with D coming from perturbation theory Ž M p e r t . or heavy-heavy spectroscopy Ž M ., and from the dispersion relation Eq. Ž5. Ž Mk i n .

kl

aM0 1.6 2.0 2.7 4.0

ks X

aEsi m

aMk i n

aM p e r t

aM

0.422Ž6. 0.438Ž7. 0.452Ž8. 0.461Ž9.

2.17Ž24. 2.57Ž33. 3.34Ž58. 5.2Ž1.5.

2.08Ž09. 2.46Ž10. 3.12Ž11. 4.32Ž10.

2.11Ž5. 2.52Ž6. 3.26Ž8. 4.59Ž9.

X

aEs i m

aMk i n

aM p e r t

aM

0.483Ž8. 0.499Ž8. 0.513Ž8. 0.522Ž8.

2.21Ž13. 2.63Ž18. 3.37Ž29. 4.86Ž67.

2.14Ž09. 2.52Ž10. 3.18Ž11. 4.38Ž10.

2.17Ž5. 2.58Ž6. 3.32Ž8. 4.65Ž9.

A. Ali Khan et al.r Physics Letters B 427 (1998) 132–140

the meson mass, aMk i n , aM p er t and aM X , in most cases, to within 1 s . The errors for aMk i n are much larger than for aM X since finite momentum correlators have much larger fluctuations for heavy-light mesons than for heavy-heavy mesons. The errors on aM p er t were estimated by squaring the one-loop corrections and by varying q ) , the scale in the coupling a . We will use the most accurately determined aM X to fix aM0 . With ay1 s 1.92Ž7. GeV and using linear interpolation between the three lightest heavy quark masses, one finds that aM0 s 2.22Ž11. gives the correct B meson mass. Had one used aMk i n rather than aM X , the number changes to aM0 s 2.28Ž42.. The effect of this difference on f B is invisible. The results for the heavy-light spectrum, including radially excited states, P-states and heavy baryons, based on the same data set, are summarized in Fig. 1. We find that the combination, NRQCD heavy and clover light quarks, reproduces the gross features of the B mesons and the heavy baryons. This is a prerequisite for a reliable calculation of f B . Details of the spectrum calculations will be given in a separate publication w16x.

3. Heavy-light axial currents in NRQCD and matching to continuum full QCD The goal is to calculate the pseudoscalar decay constant f P S , defined in Euclidean space through the relation, ² 0 < Am < PS : s pm f P S .

Ž 7.

Here Am is the heavy-light axial vector current in full continuum QCD. One needs to relate this current to the current operators JLŽ i. in lattice NRQCD. Heavy-light currents in full continuum QCD have the form q G h. For A 0 , we use G s g 5g 0 . The four-component Dirac spinor for the heavy quark, h, is related to the two-component NRQCD heavy quark Žheavy anti-quark. fields, c Ž c˜ ., via an inverse Foldy-Wouthuysen transformation: y1 h s UFy1 WCF W s UF W

c

ž/ c˜

.

Ž 8.

135

Through O Ž1rM 2 . this gives, 1 qG h s qG Q y q G Ž gPD. Qq 2M 1 q G Ž D 2 q g S P B y 2 ig a P E . Q, 8M2

Ž 9. 1 2

where Q s Ž1 q g 0 .CF W and a ' g 0 g, S s diag Ž s,s .. In our simulations we have included all tree-level contributions to the current through O Ž1rM 2 . using the terms in Eq. Ž9.. We use perturbation theory to match between the lattice currents in our simulations and those of continuum QCD and employ on-shell matching conditions. One starts by considering the process in which a heavy quark of momentum p is scattered by the heavy-light current into a light quark of momentum pX . Matching takes place by requiring that the amplitude for this process calculated in full QCD agrees with that in the effective theory through the order in a and 1rM that one is working in. Here we work through O Ž arM .. The amplitude in full QCD can be expanded in terms of prM, pXrM etc. The next step is to identify current operators in the effective theory that would reproduce the same terms in the amplitude. Finally, a one-loop mixing matrix calculation must be done within the effective theory. Details of such a matching calculation are given in w4x. Here we summarize the main features. The full QCD calculation uses naive dimensional regularization ŽNDR. in the MS scheme. A gluon mass l is introduced at intermediate stages of the calculation to regulate IR divergences. This is permissible here since no non-Abelian vertices appear in this one-loop calculation. A one-loop calculation in full QCD finds, upon expanding through O Ž1rM ., that the following three current operators are required in the lattice effective theory: JLŽ0. s qg 5g 0 Q, 1 JLŽ1. s y qg g Ž g P = . Q, 2M 5 0 § 1 JLŽ2. s q g P = g 5g 0 Q. Ž 10 . 2M JLŽ0. and JLŽ1. correspond to discretized versions of the first two terms in Eq. Ž9., JLŽ2. appears only at one loop. After carrying out a one-loop mixing

ž

/

A. Ali Khan et al.r Physics Letters B 427 (1998) 132–140

136

matrix calculation in lattice NRQCD, one ends up with the final matching relation between ² A 0 : in full QCD and matrix elements evaluated in the lattice simulations, ² A 0 :Q C D s Ý C j ² JLŽ j. : .

Once O Ž1rM 2 . corrections to the current are included at tree level, Eq. Ž11. is modified to, ² A 0 :Q C D s

Ý js0,1,2

C j ² JLŽ j. : q

Ý

² JLŽ j. : ,

js3,4,5

Ž 11 .

Ž 14 .

C0 , C1 and C2 are the three matching coefficients. Relation Ž11. is independent of the external states up to lattice artefacts. An interesting outcome of the one-loop calculation is that the mixing between JLŽ0. and JLŽ2. does not vanish in the limit M ™ `. This is because of the generation of a term, at orderŽ a a ., which is a lattice artefact. This term can be absorbed into an O Ž a Ž a LQ C D .. discretization correction to JLŽ0. :

with JLŽ j. , j s 3,4,5, corresponding to the three 1rM 2 current corrections in Eq. Ž9.. The matching coefficients C j , j s 0,1,2, do not include one-loop feedback from these latter 1rM 2 current operators. Such contributions come in at O Ž arM 2 .. Eq. Ž14. is the NRQCDrClover heavy-light axial current we use to extract f P S . It includes terms through O ŽŽ LQ C D rM . 2 ., O Ž a LQ C D rM ., O Ž arŽ aM .. and O Ž a Ž a LQ C D ... We note that powers of both 1rŽ aM . and LQ C D rM are contained in our expressions. These ratios are both considered small in the NRQCD formalism Žin practice 1rŽ aM . is sometimes allowed to become as large as ; 1.. In particular, one does not take the limit aM ™ 0. The cutoff and therefore momenta allowed cannot become larger than ; M in a non-relativistic theory. We end this section with two brief comments. The first concerns non-perturbative effects that arise because we have power-law contributions to the matching coefficients that go as arŽ aM . n. These terms are not separated out explicitly in our calculations but they are evident if coefficients are plotted as a function of aM. They are there to cancel unphysical ultra-violet pieces in the NRQCD current matrix elements, which do not match full QCD. It is possible for the arŽ aM . n terms in, say, C0 to multiply non-perturbative lattice errors, A Ž a LQ C D . n , in the matrix element ² JLŽ0. : to give contributions that look like physical Ž LQ C D rM . n corrections to f B . This limits our ability to extract the physical M dependence of f B . The source of these non-perturbative errors is a mis-match between the infra-red physics of lattice NRQCD and that of full QCD. After taking the improvement of Eq. Ž12. into account our ² JLŽ0. : is designed to match full QCD through O Ž a a.. So these errors appear at O Ž a 2 Ž a L QC D .. or O ŽŽ a LQC D . 2 .. The leading contribution to f B from these non-perturbative lattice artefacts is then arŽ aM . = a 2 Ž a LQ C D . s a 3 = LQC D rM or arŽ aM . = Ž a LQ C D . 2 s a Ž a LQC D . = LQC D rM giving an error in the slope of O Ž a 3 . or

j

JLŽ0.

™

JLŽ0. q CA JLŽ d i sc. ,

Ž 12 .

with §

JLŽ d i sc. s a q g P = g 5g 0 Q.

ž

/

Ž 13 .

The coefficient CA is also calculated in w4x. JLŽ d i sc. is the analogue in heavy-light physics of the a Em P discretization correction to the light quark axial current Žhere P is the pseudoscalar density. of Ref. w19x. The matching coefficients C j are functions of the coupling a and of the heavy quark mass and one needs to specify the precise definitions of these parameters. We use a V Ž q ) . of Ref. w14x as our perturbative expansion parameter. The scale q ) is not yet known for this matching calculation, however, a value for the M ™ ` limit has been calculated in Ref. w20x and gives q ) s 2.18ra. We shall extract f B for q ) s 1ra and q ) s pra and use the difference between the two as an estimate of the a 2 uncertainty. It is also important to use a common definition for the heavy quark mass in the continuum and lattice calculations. This can be ensured in perturbation theory by employing on-shell quark mass renormalization and using the pole mass in both theories. After writing everything in terms of the pole mass, we convert to the bare lattice mass M0 , which is a well defined quantity even beyond perturbation theory and the mass that appears in our lattice simulations. We use one-loop perturbation theory for this conversion.

A. Ali Khan et al.r Physics Letters B 427 (1998) 132–140

137

O Ž arŽ aM . 2 .. There is no indication from the behavior of our matrix elements that such contributions are particularly large. Any uncertainty arising from them will be covered by the systematic errors that we assign in the next section to higher order perturbative and relativistic corrections.

O Ž a Ž a LQ C D ... Both these terms are of the same order as higher order corrections not yet included in our matching calculations; they do not lead to any fundamental difficulties. Better precision can be attained in principle by going to more highly improved actions and current operators. Conversely we note that the situation is considerably worse for Wilson light quarks where discretization errors in the action mean that the currents cannot be improved to the level we have achieved here. Then non-perturbative lattice artefacts could induce an a r Ž aM . = Ž a LQC D . s a Ž LQ C D rM . error in f B . The second comment concerns cancellation of the power-law contributions referred to above. For instance, our matrix elements will contain unphysical contributions that behave as arŽ aM . and which are cancelled by the perturbative matching procedure. This might be a delicate operation if the terms behaving as arŽ aM . in the matching coefficients and in the matrix elements were much larger than the physical terms of O Ž LQC D rM .. In the region in which we work none of the perturbative coefficients of the arŽ aM . terms are unduly large. No delicate cancellation is necessary. Furthermore the current matrix elements are clearly dominated by their physical components, in particular the matrix elements of the higher dimension operators, JLŽ1. and JLŽ2., show no sign of significant arŽ aM . n terms. This will be discussed in more detail elsewhere w21x. In our calculations power-law terms have been subtracted perturbatively through O Ž arŽ aM ... Uncancelled contributions would come in at O Ž a 2 rŽ aM .. and

4. Results Details of our data analysis for decay constants are described in w5,6x and will not be repeated here. We start with the discussion of the size of the various terms in the lattice current defined in Eq. Ž14.. Table 2 summarizes our results for f Ž j. MP S s

(

1

(M

²0 < JLŽ j. < PS :

Ž j s 0 . . . 5. ,

PS

Ž 15 . evaluated at zero 3-momentum. The light quark has been extrapolatedrinterpolated linearly to k l or k s . The matrix elements of JLŽ2. are not shown as they are identical to JLŽ1. at zero 3-momentum. The data show that around the physical B Ž aM0 s 2.22Ž11.. the matrix elements of JLŽ1. are ; 12% of JLŽ0.. The analogous ratio for the sum of the three 1rM 2 currents Žwhich partially cancel. is ; 3%. It is therefore reasonable to assume that the neglected 1rM 3 and higher terms are F 1%. The results for a3r2 f P S MP S with the one-loop matching coefficients are summarized in Table 3.

(

Table 2 Tree-level matrix elements, in lattice units. f Ž j. is defined in Eq. Ž15.

kl

ks

aM0

a 3r 2 f Ž0.'M

a 3r2 f Ž1.'M

a 3r2 f Ž3.'M

a3r2 f Ž4.'M

a3r2 f Ž5.'M

1.6 2.0 2.7 4.0 7.0 10.0 1.6 2.0 2.7 4.0 7.0 10.0

0.164Ž7. 0.167Ž9. 0.170Ž11. 0.174Ž11. 0.191Ž8. 0.198Ž9. 0.194Ž5. 0.198Ž6. 0.205Ž7. 0.212Ž8. 0.230Ž6. 0.237Ž6.

y0.0245Ž12. y0.0206Ž10. y0.0162Ž10. y0.0113Ž8. y0.0075Ž4. y0.0055Ž2. y0.0278Ž8. y0.0236Ž7. y0.0187Ž6. y0.0134Ž5. y0.0088Ž2. y0.0065Ž17.

y0.0053Ž4. y0.0036Ž2. y0.00220Ž14. y0.00111Ž8. y0.00049Ž2. y0.000272Ž12. y0.00635Ž17. y0.00446Ž15. y0.00276Ž10. y0.00144Ž6. y0.000620Ž17. y0.000344Ž9.

0.00363Ž15. 0.00216Ž10. 0.00109Ž6. 0.00045Ž3. 0.000144Ž8. 0.000072Ž5. 0.00420Ž11. 0.00253Ž7. 0.00129Ž4. 0.00054Ž2. 0.000171Ž5. 0.000083Ž3.

y0.0050Ž3. y0.00325Ž19. y0.00181Ž13. y0.00082Ž5. y0.000287Ž15. y0.000145Ž8. y0.00576Ž17. y0.00375Ž12. y0.00210Ž8. y0.00098Ž3. y0.000341Ž10. y0.000171Ž5.

A. Ali Khan et al.r Physics Letters B 427 (1998) 132–140

138

Table 3 Decay matrix elements f'M in lattice units. The first column gives tree-level results, the second includes renormalization constants using a V at q ) s pra, and the third using a V at q ) s 1ra

kl

aM0 1.6 2.0 2.7 4.0 7.0 10.0

)

ks )

tree-level

q s pra

q s 1ra

tree-level

q s pra

q ) s 1ra

0.133Ž7. 0.142Ž8. 0.151Ž10. 0.162Ž10. 0.183Ž8. 0.193Ž9.

0.121Ž6. 0.128Ž7. 0.135Ž9. 0.144Ž10. 0.162Ž7. 0.170Ž8.

0.114Ž6. 0.121Ž7. 0.127Ž9. 0.135Ž9. 0.150Ž7. 0.158Ž8.

0.158Ž5. 0.169Ž5. 0.182Ž7. 0.196Ž7. 0.220Ž6. 0.231Ž6.

0.144Ž4. 0.153Ž5. 0.164Ž6. 0.176Ž6. 0.195Ž5. 0.204Ž6.

0.136Ž4. 0.144Ž5. 0.153Ž6. 0.164Ž6. 0.181Ž5. 0.189Ž5.

We show results for three values of a : 0, a V Ž q ) s 1ra. and a V Ž q ) s pra.. Two of the matching coefficients, C0 and C1 , include a term a lnŽ aM .rp . In Table 3 we have used the dimensionless bare heavy quark mass, aM0 , for the argument of these logarithms. There are no problems with large logarithms around the physical B meson, however, if one were to work at large aM0 , resumming the logarithms would be more appropriate. At the B meson, the one-loop correction decreases the decay constant by ; 9 y 14% depending on q ) of which ; 3 y 5% are due to the one-loop contributions from JLŽ1., JLŽ2. and JLŽ d i sc.. The ; 5% uncertainty coming from the variation in q ) will be taken as a measure of the uncertainty coming from higher order perturbative corrections. This is a conservative assignment of ; 50% of the one-loop correction. The perturbative matching coefficients employed in this article used the full 1rM 2 action of Eq. Ž1. with d H given by Eq. Ž3. Žwith, however, four-leaf E fields.. Matching coefficients from an O Ž1rM . NRQCD action Žwith d H s y 2 Mg 0 s P B . are also available w4x and lead to a difference in f B of about ; 1%. This comparison gives us some idea of the effect from O Ž arM 2 . contributions to matching coefficients on f B . We will assign a ; 4% error to the full O Ž arM 2 . corrections from the 1rM 2 current operators. This is the size of arŽ aM0 . 2 and is more conservative than using the tree level value of the 1rM 2 corrections. Adding this in quadrature to the ; 5% O Ž a 2 . and ; 1% O Ž1rM 3 . errors gives a ; 6% systematic error from higher order perturbative and relativistic corrections. The dominant discretization errors in the present decay constant calculation are of O ŽŽ a LQC D . 2 ., giving an additional systematic error of ; 4%.

)

Table 4 shows results for f P S at the physical b-quark mass for several choices of the scale. For each scale, aMb0 denotes the value of the dimensionless bare heavy quark mass which gives the correct Ž . B meson mass. Using the scale ay1 m r s 1.92 7 GeV and averaging over the two q ) ’s, our final estimate in the quenched approximation is then, q8

f B s 147 Ž 11 . Ž y12 . Ž 9 . Ž 6 . MeV, q7

f B s s 175 Ž 08 .

ž / Ž 11. Ž 7. Ž y1 0

q7 y0

. MeV.

Ž 16 .

The first error comes from statistics plus extrapolationrinterpolations in k and aM0 . Scale uncertainties are reflected in the second error with the upper and lower limits coming from ay1 s 2.0 GeV or ay1 s 1.8 GeV respectively. The third error is due to higher order relativistic and perturbative corrections and the fourth due to discretization corrections. The central value of f B s has been calculated fixing the strange quark mass from the K meson; the fifth error bar comes from the difference in k s from the f and the K. Another quantity of interest is the ratio f B srf B . It can be determined to a greater accuracy than f B and

Table 4 Decay constants f B and f B s in MeV. Statistical errors only are shown. aMb0 is the dimensionless bare heavy quark mass which leads to the correct B meson mass ay1 aMb0 wGeVx 1.8 1.92 2.0

f B wMeVx f B s wMeVx q ) sp r a q ) s1r a q ) sp r a q ) s1r a

2.39Ž7. 140Ž9. 2.22Ž6. 151Ž9. 2.11Ž6. 159Ž9.

131Ž8. 142Ž9. 150Ž9.

171Ž5. 180Ž6. 187Ž6.

160Ž5. 169Ž6. 176Ž6.

A. Ali Khan et al.r Physics Letters B 427 (1998) 132–140

f B s separately, since the direct dependence on the lattice scale drops out, and the statistical errors partly cancel. We obtain this ratio for each value of the heavy mass using, fBs fB

s

Ž f'M . ks k Ž f'M . ks k

s

(M

l

(M

B

,

Ž 17 .

Bs

where MB and MB s are the experimental meson masses. Table 5 shows our results. One sees that one-loop renormalization has negligible effect on f B srf B . This also indicates that the errors from higher orders in perturbation theory are negligible. In the range studied, the data do not show any significant dependence on aM0 . Linearly interpolating to aMb0 , our best estimate of the ratio is fBs fB

q4

s 1.20 Ž 4 . Ž y0 . .

Ž 18 .

The first error is statistical and the second error comes from the uncertainties in fixing k s . Eqs. Ž16. and Ž18. give the main results of this article and represent the most complete results to date for f B and f B srf B using NRQCD b-quarks. It is interesting to compare them with other calculations that have used a relativistic formalism for the heavy b-quark. Table 6 gives such a comparison with the APE w8x, Fermilab w9x, JLQCD w10x, MILC w11x and UKQCD w12x collaborations. Details of the simulation parameters and analyses are different among these groups Žfor recent reviews see w1x.. For example MILC uses the Wilson quark action, UKQCD and APE results are for the clover action with c SW s 1, Fermilab uses tree-level tadpole improved clover and JLQCD uses one-loop tadpole modified clover. Only JLQCD, MILC and Fermilab extrapolate to

Table 5 f B s r f B , without renormalization constants Žleft., q ) sp r a Žmiddle., and q ) s1r a aM0

tree-level

q ) sp r a

q ) s1r a

1.6 2.0 2.7 4.0 7.0 10.0

1.18Ž4.Ž5. 1.19Ž4.Ž4. 1.20Ž5.Ž5. 1.21Ž5.Ž5. 1.19Ž4.Ž5. 1.19Ž4.Ž4.

1.18Ž4.Ž5. 1.19Ž4.Ž4. 1.20Ž5.Ž5. 1.21Ž5.Ž5. 1.20Ž4.Ž4. 1.19Ž4.Ž5.

1.19Ž4.Ž4. 1.19Ž4.Ž5. 1.21Ž5.Ž5. 1.21Ž5.Ž5. 1.20Ž4.Ž4. 1.19Ž4.Ž5.

139

Table 6 Comparison with quenched results for f B , f B s and f B s r f B from recent lattice calculations using relativistic quarks. Errors include statistical and systematic errors, added in quadrature Collaboration

f B wMeVx

f B s wMeVx

fBs r fB

This work

7. 147Žq1 y2 0

. 175Žq18 y18

. 1.20Žq6 y4

APE w8x Fermilab w9x JLQCD w10x MILC w11x UKQCD w12x

180Ž32. 6. 164Žq1 y1 4 163Ž16. 0. 153Žq4 y1 6 q53 . 160Žy2 0

205Ž35. . 185Žq16 y11 175Ž18. . 164Žq50 y16 q62 . 194Žy10

1.14Ž8. . 1.13Žq5 y4 1.10Ž6. 1.22Ž4.

a s 0, and JLQCD and Fermilab include M-dependent renormalization constants at O Ž a .. They do not take into account one-loop mixing with 1rM current operators or the discretization correction that arises at O Ž a Ž a LQ C D ... We expect that at comparable b and M these two corrections should be of similar size in their formalism as we find here Ž; 3 y 5%.. In spite of the range of approaches and approximations used, it is, nevertheless, encouraging to see that very different formulations of heavy quarks give results for f B and f B s that are consistent within errors. The ratio f B srf B , in which many of the errors cancel, shows a larger spread among the groups. The NRQCD number lies 1 ; 2 s higher than the more recent relativistic heavy quark results in Table 6 by APE, Fermilab and MILC. Our B meson decay constant results are still within the quenched approximation and at a fixed value of the lattice spacing. Calculations are underway to address these sources of uncertainty. The first results from n f s 2 dynamical configurations Žwithout the 1rM 2 tree-level corrections. show that, within current statistical and systematic errors, unquenching effects are not yet visible w22x. Higher statistics and smaller dynamical quark masses are required to resolve this issue. Since we are using improved actions and currents we expect scaling violations to be small in quenched calculations at b s 6.0. Further quenched simulations at other lattice spacings Ž b s 5.7 and b s 6.2. have been initiated to check this. Preliminary results at b s 5.7 on UKQCD configurations are encouraging w23x, however, to make any definitive statement requires a full analysis at all three b values.

140

A. Ali Khan et al.r Physics Letters B 427 (1998) 132–140

5. Summary This article describes a quenched calculation of B meson decay constants, f B and f B s, using NRQCD b-quarks and clover light quarks. Tree-level corrections to the heavy quark action and the heavy-light currents have been included through O Ž1rM 2 .. The full O Ž arM . matching between continuum QCD and lattice NRQCD currents has been incorporated together with an O Ž a a . discretization correction to the heavy-light axial current. This is the first time both mass-dependent matching factors and the O Ž a a . current correction have been included in a lattice determination of f B . Our final results for f B , f B s and f B srf B are given in Eqs. Ž16. and Ž18.. The next step in our program is to study unquenching, scaling and finite volume effects, in addition to improving statistics. Work is already in progress on these fronts.

Acknowledgements This work has been supported by grants from the US Department of Energy, ŽDE-FG02-91ER40690, DE-FG03-90ER40546, DE-FG05-84ER40154, DEFC02-91ER75661, DE-LANL-ERWE161., NATO ŽCRG 941259., PPARC and the Royal Society of Edinburgh. A. Ali Khan is grateful to the Graduate School of the Ohio State University for a University Postdoctoral Fellowship. C. Davies thanks the Institute for Theoretical Physics, Santa Barbara, for hospitality and the Leverhulme Trust and Fulbright Commission for funding while this work was being completed. J. Sloan would like to thank the Center for Computational Sciences, University of Kentucky, for support. We thank Joachim Hein, Peter Lepage and Tetsuya Onogi for useful discussions. The simulations reported here were done on CM5’s at the Advanced Computing Laboratory at Los Alamos under a DOE Grand Challenges award, and at NCSA under a Metacenter allocation.

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