Decay constants of the charmed vector mesons $D^*$ and $D^*_s$ from QCD sum rules

We present a sum-rule calculation of the decay constants of the charmed vector mesons $D^*$ and $D^*_s$ from the two-point correlator of vector currents. First, we show that the perturbative expansion in terms of the pole mass exhibits no sign of convergence whereas the reorganization of this expansion in terms of the $\overline{\rm MS}$ mass leads to a distinct hierarchy. Second, making use of the operator product expansion in terms of the $\overline{\rm MS}$ mass, we determine the decay constants of the $D^*$ and $D^*_s$ mesons with an emphasis on the uncertainties in these theoretically predicted quantities related both to the input QCD parameters and to the limited accuracy of the method of sum rules. Our results are $f_{D^*}=(252.2 \pm 22.3_{\rm OPE}\pm 4_{\rm syst})\; {\rm MeV}$ and $f_{D_s^*}=(305.5 \pm 26.8_{\rm OPE}\pm 5_{\rm syst})\; {\rm MeV}$. For the ratios of the vector-to-pseudoscalar decay constants we report $f_{D^*}/f_D= 1.221\pm 0.080_{\rm OPE}\pm 0.008_{\rm syst}$ and $f_{D^*_s}/f_{D_s}= 1.241\pm 0.057_{\rm OPE}\pm 0.007_{\rm syst}$.


Introduction
The extraction of the decay constants of ground-state vector mesons within the method of QCD sum rules [1,2] is based on the analysis of the two-point correlation function of the vector heavy-light currents for a heavy quark Q of mass m Q and a light quark q of mass m j μ (x) =q(x)γ μ Q (x), (1.2) or, more precisely, on the Borel transform Π(p 2 ) → Π(τ ) of its transverse structure. Equating Π(τ ) as calculated within QCD and the expression obtained by inserting a complete set of hadron states yields the sum rule   In perturbation theory, the correlation function is obtained as expansion in powers of the strong coupling constant α s (μ). The best known three-loop perturbative spectral density has been calculated in [3] in terms of the pole mass of the heavy quark Q  For two massive quarks, the two-loop spectral density in terms of their pole masses was obtained in [4]. However, already for the case of the pseudoscalar correlator it was found that the perturbative expansion in terms of the heavyquark pole mass does not exhibit any sign of convergence; this problem was cured by rearranging the perturbative expansion in terms of the corresponding running MS mass [5]. We show that precisely the same happens in the case of the vector correlator (1.1).
Another subtlety -related to the truncation of the perturbative expansion -is the unphysical dependence of the obtained ground-state parameters on the renormalization scale μ: of course, the full correlator (1.1) does not depend on μ; however, both the perturbative expansion truncated at fixed order in α s and the truncated power corrections Π power (τ , μ) depend on μ. For the pseudoscalar-meson decay constants, this dependence was found to be rather mild [6]. Unfortunately, as we shall demonstrate in this analysis, for the vector-meson decay constants the μ dependence is rather pronounced; this leads to a larger corresponding error in the decay constants of vector mesons obtained from QCD sum rules.
Furthermore, the truncated operator product expansion (OPE) does not allow one to calculate the correlator for sufficiently large τ , such that the continuum states give a sizable contribution to Π(τ ) in the corresponding τ -range. In order to get rid of the continuum contribution, the concept of duality is invoked: Perturbative-QCD spectral density ρ pert (s) and hadron spectral density ρ hadr (s) resemble each other at large values of s; thus, for sufficiently large values of the parameters, (far) above the resonance region, one arrives at the duality relation  where s eff (τ ) is different from the physical threshold s phys . Obviously, for the same reason which causes s eff (τ ) = s phys , s eff (τ ) must be a function of the parameter τ [7,8]. By virtue of (1.7), we may rewrite the sum rule (1.3) as Obviously, the exact effective threshold is unknown. Thus, the extraction of the decay constant requires, in addition to ρ pert (s, μ) and Π power (τ , μ), as further input, a criterion for obtaining an approximation to the exact effective threshold. In [8] we developed the algorithm for fixing s eff (τ ) which allows one to reliably extract the ground-state parameters on the basis of (i) an accurate OPE for the Green functions and (ii) the known value of the ground-state mass.
We shall demonstrate that QCD sum rules armed with this algorithm allow a very satisfactory extraction of the vector-meson decay constants, with an accuracy that is certainly competitive to that found using lattice QCD.

Operator product expansion and choice of scheme for heavy-quark masses
We start with the OPE for the correlation function (1.1). We may use the perturbative spectral density ρ pert (s, M) of [3] in terms of the pole mass of the heavy quark. An alternative option is to reorganize the perturbative expansion in terms of the running MS mass; the relevant analytic expressions are given in [9], see also the discussion in Appendix A. Fig. 1 illustrates the sum-rule estimates for f D * arising from (1.8) for these two choices of the c-quark mass: the pole mass M c and the running MS mass m c (μ). The numerical OPE-parameter values entering this game read [5,6,10,11] The pole mass, recomputed from the O (α 2 s ) relation between m c and M c [12], reads M c = 1.699 GeV. The sum-rule estimates shown in Fig. 1 are obtained for a τ -independent effective threshold s eff . Its values, different for pole-mass OPE and MS-mass OPE, are found by requiring maximal stability of the extracted decay constant in the chosen Borel window (as detailed in Section 3). Let us emphasize that, for the moment, a constant effective threshold and the stability criterion for determining its numerical value are adopted only for illustration: As we have demonstrated in many examples [7], using a constant effective threshold provides rather inaccurate estimates for the decay constant and does not allow one to probe the systematic error of this extraction.
Nevertheless, the results of Fig. 1 illustrate some of the essential features of the extraction procedures. First, using the pole-mass OPE, one observes no hierarchy of the perturbative contributions have the same size. Obviously, there is no reason to expect the unknown higher-order perturbative corrections to be small; the pole-mass OPE truncated at order O (α 2 s ) and the corresponding ground-state parameters suffer from large uncertainties. On the other hand, reorganizing the perturbative expansion in terms of the MS mass of the heavy quark leads to a clear hierarchy and allows a reliable extraction of the ground-state parameters. This is precisely the same feature that has been observed for the pseudoscalar correlator. Second, there is a huge numerical difference between the decay constants obtained using the pole-mass OPE and the running-mass OPE if one compares calculations obtained for the values of m c (m c ) and its pole-mass O (α 2 s ) counterpartner given above. However, comparing the results of the truncated pole-mass and runningmass OPE requires some caution, as the perturbative expansion of the pole mass in terms of the running mass displays its asymptotic nature already at lowest orders [12]: ing the uncertainty of the pole-mass value that corresponds to a specific running-mass value as, e.g., the size of the last included term in the perturbative relation, in our case of the O (α 2 s ) term, amounts to a 15% uncertainty in M c . Due to a large sensitivity of the extracted decay constant to the precise value of the charmquark mass, the uncertainty of 15% in M c leads to a 100% uncertainty in the dual pole-mass correlator. With such an uncertainty, the results obtained from the pole-mass and the running-mass OPE in Fig. 1 are compatible with each other, but suggest that the accuracy of the O (α 2 s )-truncated pole-mass OPE is rather bad.
We therefore make use of the OPE in terms of the running MS mass for the analysis of f V . Accordingly, henceforth the quark masses m Q and m, and the strong coupling α s denote the MS running quantities.

Extraction of the decay constants
In order to extract the decay constants from our QCD sum rule, we first have to fix the working τ -window where the OPE provides a sufficiently accurate description of the exact correlator (i.e., all higher-order radiative and power corrections are under control) and the ground state gives a "sizable" contribution to the correlator. We shall adopt the window fixed in our previous analysis of the decay constants of the D and D s mesons [6].
Next, we must fix the effective continuum threshold s eff (τ ). The corresponding algorithm was developed and verified in quantummechanical potential models [8,13] and proven to work successfully for the decay constants of the heavy pseudoscalar mesons [14].
We define the dual invariant mass M dual and the dual decay constant f dual by For a properly constructed Π dual (τ , s eff (τ )), the dual mass coincides with the actual ground-state mass M V . Therefore, any deviation of the dual mass from M V is an indication of the contamination of the dual correlator by excited states.
For any trial functional form of the effective threshold, one obtains a variational solution by minimizing the difference between the dual mass (3.1) and the actual (experimental) mass in the Borel window. This variational solution provides the decay constant then via (3.1). We consider a set of τ -dependent Ansätze for the effective continuum threshold, viz., (3.2) and fix the parameters on the right-hand side of (3.2) by minimizing A detailed analysis of quantum-mechanical models for different potentials indicated that it is sufficient to consider polynomials up to third order: In this case, the band delimited by the results obtained for linear, quadratic, and cubic Ansätze for s eff (τ ) contains the true value of the decay constant. Even the good knowledge of the truncated OPE does not allow to determine the decay constant precisely, but it allows us to provide the range of values containing the true value of this decay constant. The width of this range may be then treated as a systematic error related to a principally limited accuracy of the method. Presently, we do not see other possibilities to obtain a more reliable estimate for the systematic error. Noteworthy, considering a merely τ -independent threshold does not allow one to probe the accuracy of the obtained estimate for f V .
On top of the systematic error comes the OPE-related error of the decay constant: the OPE parameters are known with some errors, inducing a corresponding error of f V . This OPE-related (or statistical) error is determined by averaging the results for the decay constant assuming for the OPE parameters Gaussian distributions with the central values and standard deviations quoted in (2.1) and a flat distribution over the scale μ in the range 1 < μ (GeV) < 3.

Decay constant of the D * meson
Following [6], we choose for the τ -window for the charmed mesons the interval τ = (0.1-0.5) GeV −2 . Fig. 2 shows the application of our procedure for fixing the effective continuum threshold  Here, μ * is the average scale defined in the standard way: assuming a flat probability distribution for μ in the range 1 < μ (GeV) < 3. The corresponding standard deviation of f D * is 18.7 MeV. For comparison, we also provide the μ dependence and the average scale μ * for f D from [6]: f dual D (μ) = 208.3 MeV 1 + 0.06 log μ/μ * − 0.11 log 2 μ/μ * + 0.08 log 3 μ/μ * , μ * = 1.62 GeV. (3.7) Obviously, the μ dependence of the pseudoscalar correlator is much weaker. This effect has the following origin: both the truncated perturbative dual correlator Π dual pert (s eff , τ , μ) and the truncated Π power (τ , μ) exhibit a rather pronounced μ dependence. For the pseudoscalar correlator, these μ dependences to a large extent cancel each other, whereas for the vector correlator the cancellation does not occur.  (3.8) The OPE uncertainty is composed as follows: 18.7 MeV are due to the variation of the scale μ, 10 MeV arise from the error in m c ≡ m c (m c ), 2 MeV from α s (M Z ), 6 MeV from the quark condensate, and 3 MeV from the gluon condensate. Higher condensates contribute less than 1 MeV to this error. Combining our above results with those for f D from our earlier analysis [6], we obtain f D * / f D = 1.221 ± 0.080 (OPE) ± 0.008 (syst) . (3.9) The OPE uncertainty of this ratio is fully dominated by the impact of the μ dependence.

Decay constant of the D * s meson
For the D * s , we take the same Borel-parameter window as for   [15].
Making use of our result for f D s from [6], we obtain, for the ratio of the vector and the pseudoscalar decay constants, (syst) . (3.14) The OPE uncertainty in this ratio is dominated by the errors arising from the μ dependence (0.043) and the gluon condensate (0.026).
Finally, for the ratio of the D * s and D * decay constants, we get (syst) . (3.15) The error here arises mainly from the errors in the strange-quark mass and the condensates ratio ss / qq = 0.8 ± 0.3. The value (3.15) is slightly larger than but not in disagreement with the lattice result f D * s / f D * = 1.16 ± 0.02 ± 0.06 [15]. 1

Summary and conclusions
Exploiting the tools offered by QCD sum rules, we analyzed in great detail the decay constants of charmed vector mesons, paying special attention to the involved uncertainties of the predicted decay-constant values: the OPE error (related to the precision with which the QCD parameters are known) and the systematic error intrinsic to the sum-rule approach as a whole (reflecting the limited accuracy of the extraction procedure). We thus gained important insights: (i) As was already noted in the case of heavy pseudoscalar mesons [6], also for the vector correlator the perturbative expansion in terms of the heavy-quark pole mass does not seem to converge whereas reorganizing it in terms of the corresponding running mass leads to a clear hierarchy of the perturbative contributions. (ii) The dependence of the vector correlator, known at three-loop accuracy, on the renormalization scale μ turns out to be sizeably stronger compared to the pseudoscalar correlator. Respectively, the error related to the remaining scale dependence of the vector-meson decay constant proves to be twice as large as that for the pseudoscalar-meson decay constant. (iii) We allowed for a Borel-parameter-dependent effective threshold for the decay-constant extractions. Obviously, such a τ -dependent effective threshold visibly improves the stability of the dual mass in the Borel window. This means that the dual correlator is much less contaminated by excited states than the one inferred upon confining oneself to τ -independent effective thresholds. We thus get, as our estimates for the vector-meson decay constants, and, for the various ratios of decay constants, f D * / f D = 1.221 ± 0.080 (OPE) ± 0.008 (syst) , (4.4) f D * s / f D s = 1.241 ± 0.057 (OPE) ± 0.007 (syst) . (4.5) The OPE uncertainties in the decay constants of D * and D * s and in the above ratios are, to a large extent, due to the remaining dependence on the renormalization scale μ.
Our results are in agreement with the recent estimates presented in Ref. [9], which also make use of our idea of a τ -dependent effective threshold. However, in our opinion, the estimates of [9] are not fully trustworthy: first, the OPE used in [9] contained errors which we correct (see (A.3) and (A.4)); second, the authors of [9] do not take properly into account the τ -dependence of the effective threshold when calculating the dual mass.
We stress that our algorithm for fixing τ -dependent effective thresholds allows us to provide, in addition to the OPE errors, also the systematic errors intrinsic to the QCD sum-rule technique. Although not entirely rigorous in the mathematical sense, our algorithm for obtaining the systematic errors has been verified in several examples within quantum mechanics, and proved to work well for decay constants of pseudoscalar mesons. The good news is that the systematic uncertainty turns out to be small and to be under control. These equations replace the corresponding equations from [9].