Heavy-Quark Mass and Heavy-Meson Decay Constants from QCD Sum Rules
aa r X i v : . [ h e p - ph ] D ec Heavy-Quark Mass and Heavy-Meson Decay Constants fromQCD Sum Rules
Wolfgang LUCHA ∗ , Dmitri MELIKHOV ∗ ,† and Silvano SIMULA ∗∗ ∗ Institute for High Energy Physics, Austrian Academy of Sciences, Nikolsdorfergasse 18, A-1050 Vienna, Austria † Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria ∗∗ INFN, Sezione di Roma III, Via della Vasca Navale 84, I-00146 Roma, Italy
Abstract.
We present a sum-rule extraction of decay constants of heavy mesons from the two-point correlator of heavy-lightpseudoscalar currents. Our primary concern is to control the uncertainties of the decay constants, induced by both input QCDparameters and limited accuracy of the sum-rule method. Gaining this control is possible by applying our novel procedure forthe extraction of hadron observables utilizing Borel-parameter-depending dual thresholds. For the charmed mesons, we obtain f D = ( . ± . ( OPE ) ± . ( syst ) ) MeV and f D s = ( . ± . ( OPE ) ± . ( syst ) ) MeV . In the case of the beauty mesons, thedecay constants prove to be extremely sensitive to the exact value of the b -quark MS mass m b ( m b ) . By matching our sum-ruleprediction for f B to the lattice outcomes, the very accurate b -mass value m b ( m b ) = ( . ± . ) GeV is found, which yields f B = ( . ± . ( OPE ) ± . ( syst ) ) MeV and f B s = ( . ± . ( OPE ) ± . ( syst ) ) MeV . Keywords: nonperturbative QCD, QCD sum rules, quark–hadron duality, continuum threshold, charmed or beauty meson, heavy-quark mass
PACS:
QUARK–HADRON DUALITY
The calculation of the decay constants f P of ground-stateheavy pseudoscalar mesons P by QCD sum rules [1, 2] isa complicated problem: First, a reliable operator productexpansion (OPE) for the “Borelized” correlation functionof two pseudoscalar heavy-light currents has to be found.Second, even if all the parameters of this OPE are knownprecisely, the knowledge of only a truncated OPE for thecorrelator allows to extract bound-state observables withonly finite accuracy, reflecting an inherent uncertainty ofthe QCD sum-rule approach. Controlling this uncertaintyconstitutes a delicate problem for actual applications [3].Recall one essential feature of the sum-rule extractionsof decay constants: the quark–hadron duality assumptionentails a (merely approximate) relation between hadronicground-state contribution and OPE with the “QCD-level”correlator cut at some effective continuum threshold s eff : f Q M Q exp ( − M Q t ) = P dual ( t , s eff ) (1) ≡ s eff Z ( m Q + m ) d s exp ( − s t ) r pert ( s ) + P power ( t ) . Here, the perturbative spectral density r pert ( s ) is obtainedas a series expansion in powers of the strong coupling a s : r pert ( s ) = r ( ) ( s ) + a s p r ( ) ( s ) + a p r ( ) ( s ) + · · · . Obviously, in order to extract a decay constant f Q one hasto find a way to fix the effective continuum threshold s eff . A crucial albeit very trivial observation is that s eff mustbe a function of t , otherwise the l.h.s. and the r.h.s. of (1)would exhibit a different t -behaviour. The exact effectivecontinuum threshold — corresponding to the true valuesof hadron mass and decay constant on the l.h.s. of (1) —is, of course, not known. Therefore, our idea of extractinghadron parameters from sum rules consists in attempting(i) to find a reliable approximation to the exact threshold s eff and (ii) to control the accuracy of this approximation.In a recent series of publications [4], we have constructedall the associated procedures, techniques and algorithms.We define a dual invariant mass M dual and a dual decayconstant f dual ( M Q still denoting the true hadron mass) by M ( t ) ≡ − dd t log P dual ( t , s eff ( t )) , (2) f ( t ) ≡ M − Q exp ( M Q t ) P dual ( t , s eff ( t )) . (3)In case the ground-state mass M Q is known, the deviationof the dual ground-state mass M dual from its actual value M Q yields an indication of the excited-state contributionspicked up by our dual correlator. Assuming some specificfunctional shape for our effective threshold and requiringleast deviation of the dual mass (2) from its actual valuein the Borel window leads to a variational solution for theeffective threshold. With s eff ( t ) at our disposal we get thedecay constant from (3). The standard assumption for theeffective threshold is that it is a ( t -independent) constant.In addition to such crude approximation we also considerpolynomials in t . In fact, t -dependent thresholds greatlyfacilitate reproducing the actual mass value. This impliesthat a dual correlator with t -dependent threshold isolateshe ground state much better and is less contaminated byexcited states than a dual correlator with the conventional t -independent threshold. As consequence, the accuraciesof extracted hadron observables are drastically improved.Recent experience from potential models reveals that theband of values obtained from linear, quadratic, and cubicAnsätze for the effective threshold encompasses the truevalue of the decay constant [4]. Moreover, we could showthat the extraction procedures in quantum mechanics andin QCD are even quantitatively very similar [5]. Here, wereport our results [1, 6] for heavy-meson decay constants. OPE AND HEAVY-QUARK MASSES
For heavy-light correlators and emerging decay constantsthe choice of the precise scheme adopted for defining theheavy-quark mass has a great impact. We utilize the OPEfor this correlator to three-loop accuracy [7], obtained interms of the pole mass of the heavy quark. The pole-massscheme is standard and has been used for a long time [8].An alternative is to reorganize the perturbative expansionin terms of the running MS mass [9]. Since the correlatoris known up to O ( a ) , also the relation between pole andMS mass is applied to such accuracy. Figure 1 depicts theresulting B -meson decay constant f B for these two cases.In each case, a constant effective continuum threshold is Τ- H Τ ,s L (cid:144) O H L O H Α L O H Α L power total0.1 0.12 0.14 0.16 0.18 Τ- H Τ ,s (cid:143)(cid:143)(cid:143) L (cid:144) O H L O H Α L O H Α L power total FIGURE 1.
Decay constants f B extracted from the correlatorgiven in terms of the b -quark pole (top) and MS (bottom) mass. fixed by requiring maximum stability of the found decayconstant. Thus, the constant thresholds differ for pole ( s )and MS ( s ) mass schemes. Several lessons can be learnt:(a) In the pole-mass scheme, the perturbative series forthe decay constant shows no sign of convergence: each ofthe LO, NLO, NNLO terms contributes with similar size.Consequently, the pole-mass-scheme result for the decayconstant may significantly underestimate the exact value.(b) Reorganizing the perturbative series in terms of theMS mass of the heavy quark yields a distinct hierarchy ofthe perturbative contributions [9]. Moreover, the absolutevalue of the decay constant extracted in this scheme turnsout to be some 40% larger than in the pole-mass case (a).(c) Note that, in both cases, the decay constant exhibitsperfect stability in a wide range of the Borel parameter t .Thus, mere “Borel stability” is not sufficient to guaranteethe reliability of some sum-rule extraction of bound-stateparameters. We have pointed out this observation alreadyseveral times [3]. Nevertheless, some authors still regardBorel stability as a proof of the reliability of their results.In the light of our above findings, we adopt in the nextsections the OPE formulated in terms of the MS mass [9]. DECAY CONSTANTS OF D AND D s The application of our extraction procedures leads to thefollowing values of the charmed-meson decay constants: f D = ( . ± . ( OPE ) ± . ( syst ) ) MeV , (4) f D s = ( . ± . ( OPE ) ± . ( syst ) ) MeV . (5) f D ( M e V ) QCD-SR LATTICE c o n s t a n t ! - d e p e n d e n t N f = 2 N f = 3m c = 1.279(13) GeV PDG
FIGURE 2.
Comparison of our results for the decay constant f D with lattice findings. For a detailed list of references, cf. [1]. he OPE-related errors in (4,5) are obtained by bootstrapstudies allowing for the variation of all QCD parameters,that is, quark masses, a s , and condensates, in the relevantranges. We observe perfect agreement of our predictionswith the corresponding lattice results (Fig. 2). It has to beemphasized that our t -dependent effective threshold is acrucial ingredient for a successful extraction of the decayconstant from the sum rule (1). Obviously, the (standard) t -independent approximation entails a much lower valuefor the D -meson decay constant f D that resides rather farfrom both the experimental data and the lattice outcome. DECAY CONSTANTS OF B AND B s Our QCD sum-rule findings for the beauty-meson decayconstants turn out to be extremely sensitive to the chosenvalue of the b -quark MS mass m b ( m b ) . For instance, therange m b ( m b ) = ( . ± . ) GeV [10] entails resultsthat are barely compatible with recent lattice calculationsof these decay constants (Fig. 3). Requiring our sum-rule f B result to match the average of the lattice computationsprovides the rather precise value of the b -quark MS mass m b ( m b ) = ( . ± . ) GeV . For this value of the b -quark mass, our sum-rule estimatefor the B - and B s -meson decay constants f B and f B s reads f B = ( . ± . ( OPE ) ± . ( syst ) ) MeV , f B s = ( . ± . ( OPE ) ± . ( syst ) ) MeV . f B ( M e V ) QCD-SR LATTICE N f = 2 N f = 3m b =4.163(16) GeVm b =4.245(25) GeV FIGURE 3.
Comparison of our results for the decay constant f B with lattice findings. For a detailed list of references, cf. [1]. SUMMARY AND CONCLUSIONS
Applying (in an attempt to improve the sum-rule method)our above modifications, we realize several serendipities:1. The t -dependence of the effective thresholds emergesnaturally when one attempts to render the duality relationexact: the dependence is evident from (1). We emphasizetwo facts: (a) In principle, such dependence on t is not inconflict with any properties of quantum field theories. (b)Our analysis of D mesons shows that it indeed improves decisively the quality of the related sum-rule predictions.2. Our study of charmed mesons clearly demonstratesthat using Borel-parameter-dependent thresholds leads tolots of essential improvements: (i) The accuracy of decayconstants predicted by sum rules is drastically improved.(ii) It has become possible to obtain a realistic systematicerror and to diminish it to the level of, say, a few percent.(iii) Our prescription brings QCD sum-rule findings intoperfect agreement with both lattice QCD and experiment.3. The beauty-meson decay constants f B ( s ) are extremelysensitive to the choice of the b -quark mass: Matching ourQCD sum-rule f B outcome to the corresponding averageof lattice computations provides a truly accurate estimateof m b ( m b ) , in good agreement with several lattice resultsbut, interestingly, not at all overlapping with a recent veryaccurate determination [10] (for details, consult Ref. [1]). Acknowledgments.
D. M. is grateful for support by theAustrian Science Fund (FWF) under Project No. P20573.
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