---

Properties of Low-Lying Heavy-Light Mesons∗

arXiv:hep-ph/9211263v1 17 Nov 19921Properties of Low-Lying Heavy-Light Mesons∗Anthony Duncan,a Estia Eichten,b Aida X. El-Khadra, b Jonathan M. Flynn,c Brian R. Hill,d andHank Thacker.eaDept. of Physics, Univ.

of Pittsburgh,Pittsburgh, PA 15260 USAbFermilab, MS 106, PO Box 500, Batavia, IL 60510 USAcPhysics Dept., Univ. of Southampton, Southampton SO9 5NH UKdDept.

of Physics, Univ. of California, Los Angeles, CA 90025 USAeDept.

of Physics, Univ. of Virginia, Charlottesville, VA 22901 USAWe present preliminary results for fB and masses of low-lying heavy-light mesons in the static limit.

Cal-culations were performed in the quenched approximation using multistate smearing functions generated from aHamiltonian for a spinless relativistic quark. The 2S–1S and 1P–1S mass splittings are measured.

Using the1P–1S charmonium splitting to set the overall scale, the ground state decay constant fB, is 319 ± 11(stat) MeV.1. INTRODUCTIONLattice gauge calculations of heavy-light me-son structure are of both theoretical and phe-nomenological interest.

[1] One immediate goal ofthese calculations is to obtain precise quantita-tive results for masses, decay constants, and formfactors in the static approximation, where theheavy quark propagator is replaced by a time-like Wilson line.One difficulty which plaguedearly, exploratory calculations of the pseudoscalardecay constant fB was the problem of isolatingthe ground state contribution to the propagatorof the local weak current. Because of the prox-imity of excited states and their sizeable over-lap with the local current, a large separation intime was required, with an accompanying lossof statistics.

Recent attempts to overcome thisproblem have employed nonlocal ¯Qq operators (ina fixed gauge[2]) smeared over a cube[3] or wallsource[4]. By measuring the asymptotic behaviorof both the smeared-smeared (SS) and smeared-local (SL) propagators, one can reduce the sys-tematic error associated with excited state contri-butions.

However, it is likely that such smearingfunctions are too crude to obtain accurate val-∗Based on talks presented by E. Eichten and B. Hillues of the parameters of the low-lying heavy-lightstates. [5] This is illustrated in Table 1.Table 1The overlap amplitudes between various cubesmearing functions and the approximate wave-functions for the ground state |0⟩and first threeexcited states (|1⟩, |2⟩, and |3⟩) for β = 5.9 andκ = 0.158 on a 163 lattice.cube|0⟩|1⟩|2⟩|3⟩point0.2080.2340.2850.34430.5680.4800.3620.12650.7640.3040.010-0.22070.800-0.024-0.296-0.19790.746-0.330-0.301-0.122110.663-0.546-0.099-0.207wall0.459-0.6870.4770.002It is important, therefore, to develop new tech-niques which allow the extraction of the prop-erties of heavy-light states from relatively shorttimes.Here we report preliminary results ob-tained using the multistate smearing method dis-cussed elsewhere [6,7].

The hallmark of a pure,isolated ground state meson is an effective mass

2plot which is constant in time. In Figs.

1 and 2,we show our results for both the SS and SL localeffective mass plots at β = 6.1, κ = 0.151, on aset of (48) 243 × 48 lattices. On the horizontalaxis is the time T in lattice units.

On the verticalaxis is ln(CSS(T −1)/CSS(T )).The SS local effective mass reaches its asymp-totic value around T = 2, while the SL propaga-tor is nearly asymptotic after T = 3.The results exhibit a single consistent plateauat ma = 0.619 ± 0.006 over a large range of T forboth SS and SL propagators. The determinationof the fitted value of the effective mass will bediscussed in Section 4.We note here that theeffect of the off-diagonal entries in the covariancematrix of the data for the two correlators has beenincluded.

[8]These mass plots convincingly demonstrate theeffectiveness of our smearing method in isolatingthe ground state.135791113150.20.40.60.81Figure 1. κ = 0.151 SS effective mass plot.2. WAVEFUNCTIONSThe basics of the multistate smearing methodand the choice of the Hamiltonian to gener-ate the smearing wavefunctions are reported byThacker [7].

The agreement between the spinlessrelativistic quark model (SRQM) wavefunctionsand the lattice QCD wavefunctions is discussedin detail there.135791113150.40.50.60.7Figure 2. κ = 0.151 SL effective mass plot.In the analysis presented here a two statesmearing matrix was used. After a few iterationsof the multistate smearing method a rough valueof the mass parameter µ (in the SRQM) was de-termined.

The output approximate ground statewavefunction at each κ value |0⟩was extractedand used with the first excited state |1⟩gener-ated from the SRQM. These were the two statesused as smearing functionsTwo points need to be made about this varia-tion of the multistate analysis:(1) The states are not exactly orthogonal.

Ifx = ⟨0|1⟩, then an orthonormal pair of states canbe obtained by replacing |1⟩by |1′⟩given by:|1′⟩= (|1⟩−x|0⟩)/p1 −x2(1)This effect is included here but was not includedin previously presented results. The effect on fBis approximately 12 percent.

(2) The state |0⟩varies with light quark hop-ping parameter κ and this variation has some sta-tistical fluctuation, which is reflected in the vari-ation with κ of our results.3. RENORMALIZATIONIn order to extract continuum results from ourlattice calculation, the matrix element calculatedin the lattice effective theory must be relatedto the corresponding quantity in the continuum.This matching is done in perturbation theory intwo steps.

The first step is to relate the operator(in this case the axial current) in the continuum

3effective theory to its counterpart in the full the-ory [9,10]:Zeff= 1 −g212π2 (32 ln m2bµ2 −2). (2)We use the 1992 particle data book average [11]for αS ≡g2/4π at the scale µ = a−1 and massmb = 5 GeV.

In the second step the lattice cur-rent is compared to the corresponding current inthe continuum effective theory [9,10]:Zlat = 1 +g212π2 20.37. (3)The renormalization of the axial current is thenZ−1A= ZeffZlat.

Table 2 lists the renormalizationconstants for all the lattices analyzed here. Weuse the results for the 1P–1S splitting in charmo-nium to obtain a−1 [12].Table 2The current renormalizations for different latticeslatticeβZeffZlatZA123 × 245.70.901.330.83163 × 325.90.961.310.79243 × 486.11.001.300.774.

ANALYSISWe now discuss the statistical analysis leadingto our results for fB. In the first and second sub-sections below, we will explain the methodologyleading to the central values and statistical errorsquoted in our abstract and conclusions.

In thethird subsection, we will look at variations in theanalysis procedure and their impact on these re-sults.Variations are partly attributable to theremaining systematic errors in the computation,allowing an estimate of the magnitude of theseerrors.We present results from three different latticesas listed in table 3.The gauge configurationsare separated by 500 (123 × 24, β = 5.7), 2000(163 × 32, β = 5.9), and 4000 (243 × 48, β = 6.1)pseudo-heat-bath sweeps respectively. They werefixed to machine accuracy in Coulomb gauge us-ing Fourier acceleration.

We use the Wilson ac-tion for the light quarks.We present resultsfrom a single κ value at β = 5.7 and β = 6.1.At β = 5.9, the range κ = 0.154 −0.159 in-cludes 0.154, 0.156, 0.157, 0.158, and 0.159.Table 3The lattices and parameterslatticeβconfs.κlight123 × 245.7480.165163 × 325.9480.154–0.159243 × 486.1480.1514.1. Effective Masses and Selection of FitRangeThe effective mass plots for the 243×48 latticesshown in Figs.

1 and 2 exhibit the asymptoticground state signal starting at quite small times.However, at small times, even with carefully cho-sen wave functions, there are significant contribu-tions to smeared source–smeared sink (SS) andsmeared source–local sink (SL) correlators fromhigher energy states with the same quantum num-bers as the B meson. This is apparent below time3 in the β = 5.9 κ = 0.158 SS effective mass plot,depicted in Figure 3.1357911130.20.40.60.81Figure 3. κ = 0.158 SS effective mass plot.From time slice 3 on, the local effective mass

4is consistent with a constant function of time.This is plausible from inspection of the plot itself,which has the rms (not jackknife) errors plottedin addition to the means.To confirm this, wefit the logarithm of the SS correlator to a linearfunction over the range of times 2–9, and foundχ2=8.9 for 6 degrees of freedom.If the lowerlimit of the fit range is reduced to 1, χ2 increasesto 12.1 for 7 degrees of freedom. These values ofχ2 are obtained from a single fit using 48 gaugefield configurations.

To obtain the effective massand its statistical error, this fit is jackknifed us-ing 12 subensembles, each with 4 of the 48 lat-tices removed. Using the fit range 2–9, we findma=0.664 ± 0.011 (statistical).We now turn to the corresponding SL effectivemass plot, depicted in Figure 4.

This correlator1357911130.50.60.70.8Figure 4. κ = 0.158 SL effective mass plot.has much smaller fluctuations (note the change inscale for the ordinate), and is capable of reveal-ing statistically significant variations in the localeffective mass over the range 2–9.Such varia-tions are apparent in effective mass plots at lowervalues of κ (not depicted), and can be ascribedto an admixture of higher energy states with thesame quantum numbers as the B meson. Calcu-lations of χ2 for various fits at κ = 0.154 suggestthat the lower limit of the fit range should beincreased to 5.

Using the fit range 5–9, for theκ=0.158 SL correlator we find that ma = 0.667±0.009 (jackknife).The κ = 0.158 effective masses from the twocorrelators appear to be consistent to within onestandard deviation. However, monitoring χ2 for asimultaneous fit that requires the two correlatorsto have a common effective mass gives a more pre-cise indication, since the statistical fluctuations ofthe SS and SL correlators may be correlated.

Wetherefore performed fits on all 163 × 32 data (fiveκ values) using a large variety of fit intervals (de-manding a common slope for the correspondingSS and SL correlators). We looked for fit inter-vals of at least three units of time which had χ2per degree of freedom near one.

We selected thelowest value of the lower limit of the fit range thatmet these conditions. For the 163×32 data, the fitrange so selected was 5–9.

For the single κ valueat 123 × 24, we selected 3–9, and for 243 × 48,we selected 4–12. Intra-kappa correlations wereincluded[8].

We will discuss the effect of varia-tions of the fit range and inter-kappa correlationsin Subsection 4.3.4.2. Results for fBIn Figures 3 and 4 the horizontal line superim-posed on the local effective masses is the commonmass obtained from a simultaneous jackknife fitto the SS and SL correlators using the fit inter-val 5–9.

The κ=0.158 results are representativeof the results for each of the five κ values. OnceTable 4Fitted parameters as a function of κ at β = 5.9κ0.1590.1580.1570.1560.154fB3243163563443641110111010ma0.6590.6620.6870.6870.7100.0140.0100.0090.0080.007m′a0.9400.9350.9500.9450.9540.0150.0130.0130.0120.012the common mass and two intercepts have beenobtained, the value of fB at each κ value is deter-mined from the intercepts of the SS and SL fits.These results are presented in Table 4, which alsocontains the effective mass of the meson in latticeunits as a function of κ.

Below each quantity is

5its jackknife uncertainty. m′ will be discussed inSection 5.We then extrapolate fB to κc the critical valueof κ.

We measured κc = 0.1597 ± 0.0001. Sincethe dependence on κ is weak, the uncertainty inκc has a negligible (of order 1 MeV) effect onthe results.

The results for the extrapolation as154 155 156 157 158 159 160280300320340360380Figure 5. fB as a function of κ.well as the results in Table 4 are plotted in Fig-ure 5.At κc, we find for the decay constantfB = 319 ± 11 MeV (jackknife). The slope withrespect to κ−1 is 188 ± 32 MeV.

The extrapolatedvalue of fB and slope with respect to κ−1 are su-perimposed on the Monte Carlo data. The valueplotted at κ=0.1597 is the extrapolated value offB with its errors.

The nonlinearity from using κas the horizontal axis rather than κ−1 is imper-ceptible in the superimposed fit, and were we tohave fit linearly in κ rather than κ−1 we wouldhave changed the extrapolated value of fB andthe (transformed) slope negligibly relative to thestatistical errors. We note that the χ2 was 51 for44 degrees of freedom.We conclude this section by noting our resultsfor fB in the static limit from β = 5.7 κ = 0.165and β = 6.1 κ = 0.151.

In the former case us-ing the fit range 3–9, we find fB=351 ± 13 MeV(a−1= 1.15 GeV) and in the latter case using thefit range 4–12, we find fB=359 ± 7 MeV (a−1=2.43 GeV). Using the pion mass to determine thecorresponding value of κ for β = 5.9 we obtainfrom our fit to the β = 5.9 data the correspond-ing values of fB.

They are fB = 330 MeV forβ = 5.7 and fB = 367 MeV for β = 6.1.4.3. Dependence on Analysis ProcedureIn this subsection, we investigate the depen-dence of our results on (1) the fit interval, and(2) the inclusion of inter-kappa entries in the co-variance matrix.

Perhaps the most interesting ofthe variations in analysis is changing the admix-ture of the excited state. This will be discussedin the Section 5.The dependence of fB on fit interval has beeninvestigated by rerunning the analysis over theother fit ranges, in addition to the primaryrange 5–9.We have reproduced the analog ofFigure 5 in Figure 6.

The results from the addi-154 155 156 157 158 159 160280300320340360380400Figure 6. fB as a function of κ.tional fit ranges are displaced slightly from theirtrue κ values. From left to right, they are 6–10,5–9, 4–8, and 3–7.

In general, the values are con-sistent with those of the primary fit range, 5–9,except at small values of κ where it has alreadybeen noted that the χ2 per degree of freedom in-dicates that excited states are contributing whenthe lower limit of the fit range is less than 5.We now consider the impact of inter-kappa cor-relations.It is apparent from comparing theκ = 0.158 effective mass plots with those of otherκ values (not depicted), that there are strong cor-relations between corresponding quantities at dif-ferent κ values.The reason these were not in-

6cluded in our analysis is that with 50 data pointsper configuration (5 κ’s and 5 T’s for SS and SL)it is impossible to compute a nonsingular covari-ance matrix. To investigate the effect on the ex-trapolated value of fB, we therefore reduced thenumber of κ values to 2, selecting κ=0.158 andκ=0.156.We then performed the fit with andwithout the inter-kappa entries of the covariancematrix, and found that a decrease in the fittedand extrapolated values of fB resulted from in-cluding inter-kappa correlations.

However theseeffects are in all cases less than the one sigmalevel. Larger statistics would allow us to studythe impact of inter-kappa correlation with moreκ values in the simultaneous fit.5.

EXCITED STATESIn this section, we will examine evidence forthe first radially excited state of the B meson,and study the impact of the admixture of thiswave function.Preliminary results for the or-bitally excited states are presented in the secondsubsection. Finally, finite volume systematics arediscussed in the last subsection.5.1.

Radial ExcitationsWe begin this subsection by looking at the ef-fect on the fitted values of fB of the admixtureof the states created by the first radially excitedstate smearing function. The smearing functionfor the analyses in the preceding section was thelinear combination of smearing functions whichdiagonalized a two-by-two matrix of SS correla-tors which was averaged over the same range asthe primary fit range.

For the 163 × 32 data, theadmixture of the trial first excited state increasedfrom 5% to 7% with increasing κ.To see the impact of this 5 to 7% admixtureon the final results, in Figure 7 we have replottedthe data shown in Figure 5 along with the valuesof fB obtained from the undiagonalized smear-ing function.The additional data is displacedslightly from its correct κ value for visibility. Theextrapolated value of fB using the undiagonal-ized smearing function is 333 ± 12 MeV, a 4%increase.We now examine the linear combination of the154 155 156 157 158 159 160280300320340360380400Figure 7. fB as a function of κ.trial ground and first excited states which is or-thogonal to that used for the ground state.

Sincewe do not have correlators involving states withhigher radial excitations, we expect that thisorthogonal state is missing contributions fromhigher radial excitations of the same order as themixing of the trial ground and first excited state.It is neverthelesss interesting to study the effec-135790.50.70.91.11.3Figure 8.κ = 0.158 SS excited state effectivemass.tive mass plots obtained from this orthogonal lin-ear combination, which is mostly the trial firstexcited state. In Figure 8 we plot the SS localeffective mass for this correlator at κ = 0.158,and in Figure 9 we plot the corresponding SLcorrelator.The same criteria to determine thefit interval as in the ground state were used, and

7135790.50.70.91.11.3Figure 9.κ = 0.158 SL excited state effectivemass.gave a fit interval from 3–5. The excited stateeffective masses in lattice units, m′a, using thisfit range are presented in Table 4.

As usual, thisis a common effective mass fitted to both the SSand SL correlator. The 2S–1S splitting in latticeunits, m′a−ma, decreases from 0.281 to 0.244 asκ goes from 0.159 to 0.154.The decay constants of the excited state for thefive κ values are approximately 600 MeV with avariation in κ and jackknife errors both less than±15 MeV.

A slight trend is toward larger decayconstants with increasing κ.The value of thedecay constant, f ′B, extrapolated to κc is 618 ±11 MeV (stat). The quality of the plateau is muchless convincing than for the β = 5.9 ground state.The two-state approximation may induce largersystematic errors here than for the ground state.In addition, finite volume effects may be largerfor the excited states.

One indication supportingthese preliminary results is that the values of theeffective mass and decay constant are increasedby about one sigma only if the fit range is changedto 4–6.5.2. Orbital ExcitationsThe lowest P wave heavy-light mesons havelight quark total angular momenta jl = 1/2and 3/2.

Each of these states is degenerate withthe J = 0, 1 for the jl = 1/2 state and J = 1, 2for the jl = 3/2 state.Smearing functions forthese P wave heavy-light mesons can be gener-ated from the SRQM with the mass parameterµ which gives the best fit for the S wave groundstate.To date only a preliminary analysis has beenperformed on the P waves. The results for themass splittings are given in Table 5.

ConsiderableTable 5The mass splittings (in MeV) for the lowestheavy-light P states at β = 5.9 for various lightquark κ values.κ1P1/2 −1S1P3/2 −1P1/20.158386 ± 9043 ± 1020.156395 ± 8659 ± 950.154416 ± 8162 ± 90refinement will be required before the splittingbetween the jl = 1/2 and 3/2 states can be ob-served clearly.5.3. Finite Volume CorrectionsThe systematic effects of finite volume are un-der study.It would be expected that these ef-fects are more pronounced for the excited statesthan for the ground state because the RMS ra-dius for excited states is larger than for theground state.

The excellent agreement betweenthe SRQM wavefunctions and the measured wave-functions of the 1S, 1P, and 2S heavy-light statesreported by Thacker [7]. allows us to estimate theeffects of finite volume with our periodic bound-ary conditions using SRQM results.We calcu-lated the static energies using 48 Coulomb gaugefixed configurations at β = 5.9 for lattices of spa-tial size 123, 163, and 203.

Using the mass param-eter µ needed in the SRQM (obtained from ourstudy of the heavy-light mesons on the 163 × 32lattices) we estimated the effects on various massand wavefunction parameters at the other twovolumes. The results are shown in Table 6.

Thetypical variation between results at 123 and 163are 10% or more, while the variation between 163and 203 have dropped to a few percent.Thevalidity of these model results is presently be-ing checked by a complete analysis of heavy-lightmesons on the 123 × 24 and 203 × 40 lattices.

8Table 6Finite volume effects in the SRQM. Variation ofeigenvalues and properties of the eigenstates forthe ground and radially excited states.

Resultsare at β = 5.9. ǫn denotes the eigenvalue for the1S(n = 0) and 2S(n = 1) states.Measure123163203ǫ00.9000.9120.911ǫ1 −ǫ00.2510.2780.277|Ψ0(0)|0.1930.2090.206|Ψ1(0)|0.2390.2360.236< r20 >1/23.3203.013.03< r21 >1/25.095.505.536. CONCLUSIONSBy using good approximate wave functions ina multistate smearing calculation, it is possible tocontrol the systematic errors associated with ex-tracting the decay constant from nonasymptotictime.

Our results are very encouraging, not onlyfor the present calculations, but for the determi-nation of other B-meson decay parameters andform factors.All such calculations require theexternal meson to be in a pure eigenstate (i.e.on shell). By using the optimized smearing func-tions discussed here, this requirement can be metwith a minimum separation between the B-mesonsource and the electroweak vertex, greatly im-proving the precision of the results.We find that at β = 5.9 and κ = κcriticalfB = 319±11(stat)× ZA0.79×a−11.75GeV3/2MeV (4)The efficacy of our smearing method is evengreater at β = 6.1 as can be seen by comparingFigs.

1 and 2 with Figs. 3 and 4 with the samephysical volume.In addition to ground state properties, thismethod allows extraction of the properties of thelowest radially and orbitally excited states.

Ourresults for these quantities are preliminary, andwe expect more precise estimates to be obtainedby further analysis.Further numerical studies are required to checkthe finite volume effects, determine the effect ofusing an improved action for the light quarks, andinclude additional light quark mass values at β =6.1 which will allow a better determination of thescaling behaviour of these properties of low-lyingheavy-light states.ACKNOWLEDGEMENTSWe thank George Hockney, Andreas Kron-feld, and Paul Mackenzie for joint lattice ef-forts without which this analysis would not havebeen possible.JMF thanks the Nuffield Foun-dation for support under the scheme of Awardsfor Newly Appointed Science Lecturers. The nu-merical calculations were performed on the Fer-milab ACPMAPS computer system developed bythe CR&D department in collaboration with thetheory group.This work is supported in partby the Department of Energy under ContractNos.

DE–AT03–88ER 40383 Mod A006–Task Cand DE-AS05-89ER40518, and the National Sci-ence Foundation under Grant No. PHY-90-24764.REFERENCES1.See E. Eichten, Nucl.

Phys. B(Proc.Suppl.

)20 (1991) 475 and the references therein.2.See C. Alexandrou, et al, Phys. Lett B 256(1991) 60 for gauge invariant smearing.3.C.

R. Allton et al, Nucl. Phys.

B376 (1992)172.4.C. Bernard, C. M. Heard, J. Labrenz, andA.

Soni, Nucl. Phys.

B (Proc. Suppl.) 26(1992) 384.5.S.

Hashimoto and Y. Saeki, Mod. Phys.

Lett.A7 (1992) 509.6.A. Duncan, E. Eichten, and H. Thacker,Nucl.

Phys. B(Proc.Suppl.) 26 (1992) 391; 26(1992) 394.7.A.

Duncan, E. Eichten, and H. Thacker (theseproceedings).8.D. Toussaint, in From Actions to Answers–Proceedings of the 1989 Theoretical Ad-vanced Study Institute in Elementary ParticlePhysics, T. DeGrand and D. Toussaint, eds.,(World, 1990).9.E.

Eichten, B. Hill, Phys. Lett.

B 234 (1990)253, and Phys. Lett.

B 240 (1990) 193.10. Ph.

Boucaud, C. L. Lin, and O. Pene, Phys.Rev. D 40 (1989) 1529, and Phys.

Rev. D 41(1990) 3541(E).

911. The Particle Data Group, Phys.

Rev. D 45(1992).12.

A. X. El-Khadra, G. M. Hockney, A. S. Kro-nfeld, P. B. Mackenzie, Phys. Rev.

Lett. 69(1992) 729.

---

출처: arXiv:9211.263 • 원문 보기