c,b quark masses and f_D_(s), f_B_(s) decay constants from pseudoscalar sum rules in full QCD to order alpha_s^2

The pseudoscalar sum rules of the heavy-light quark systems are used for extracting simultaneously the c,b quark masses and the decay constants f_D_(s),f_B_(s) of the D_(s),B_(s) mesons. To order alpha_s^2, one obtains the running quark masses: m_c(m_c)=(1.10 \pm 0.04) GeV, m_b(m_b)=(4.05\pm 0.06) GeV, the perturbative pole masses: M_c=(1.46\pm 0.04) GeV, M_b=(4.69\pm 0.06) GeV, and the decay constants: f_D=(205\pm 20) MeV, f_B=(203\pm 23) MeV and f_D_s=(235\pm 24) MeV, f_B_s=(236\pm 30) MeV, in the normalization where f_\pi= 130.56 MeV. The fitted values of the pole and running masses satisfy quite well their three-loop perturbative relation. The value f_D = f_B confirms earlier findings from the sum rule that the 1/M_P^1/2 heavy quark symmetry scaling law is affected by large 1/M_P corrections.


Introduction
One of the most important parameters of the standard model is the quark masses. However, contrary to the leptons, where the physical mass can be identified with the pole of the propagator, the quark masses are difficult to define because of confinement. Some attempts have been made in order to define the heavy quark pole mass within perturbation theory, where it has been shown to be IR-finite [1] and independent of the choice of the regularization and renormalization schemes used [2]. More recently, it has been noticed, in the limit of a large number of flavours, that the resummation of perturbative series can induce a non-perturbative term, which can affect the truncated perturbative result, and can, then, limit the accuracy of the pole mass determination (for reviews see e.g. [3,4]). One may bypass the previous problems, by working, at a given order of perturbative QCD, with the running quark masses, which are treated like coupling constants of the QCD Lagrangian (see e.g. [5]), and where non-perturbative-like effect is expected to be absent. It is also known that the decay constants f D,B of the pseudoscalar D, B mesons play an essential rôle in the neutralD-D andB-B oscillations, and in D, B decays, and can directly be measured in the D − , B − → lν l leptonic decays. In addition, it is important for your knowledge of heavy quark symmetry, how the value of these decay constants deviate from the naïve 1/ √ M P scaling law, expected to occur when the pseudoscalar meson mass is infinitely large. A lot of efforts has been furnished in the literature [6] for extracting directly from the data the running masses of the light and heavy quarks and the heavy quark "perturbative" pole masses using the SVZ QCD spectral sum rules (QSSR) [7] (for a complete review, see e.g.: [8]), whilst f D,B come from different forms of the pseudoscalar sum rules [8]- [15] since the pioneering work of [16]. In this note, I shall consider a direct extraction of the running charm and bottom quark masses from pseudoscalar two-point function sum rules where the α 2 s correction has been recently obtained in [17] and where we shall also use the observed values of the meson masses: M D − = 1.869 GeV and M B − = 5.279 GeV. In the same time, we shall simultaneously estimate the decay constants f D (s) , f B (s) of the D (s) and B (s) mesons. The simultaneous extraction of the quark mass and f P together with the extraction of f P using the running quark mass has been initiated in our previous work [14,13] to order α s , and will be improved here to order α 2 s . All previous works [8]- [13] with the exception of the recent work in [15] have used, as input, the pole mass value for extracting f P , where, as we have mentioned previously, the definition of the pole mass might be affected by some non-perturbative contributions [3,4]. Moreover, the extraction of the quark mass value from the pseudoscalar sum rule itself which is an improvement of our previous works [14,13] is not done in all previous works.

The QCD spectral sum rules
We shall work with the pseudoscalar two-point correlator: built from the heavy-light quark current: J d (x) = (m Q + m d )Q(iγ 5 )d, and which has the quantum numbers of the D and B mesons. m Q is the heavy quark mass, and we shall neglect the d quark mass here. The corresponding Laplace transform sum rules are: where t ≤ is the hadronic threshold. The latter sum rule, or its slight modification, is useful, as it is equal to the resonance mass squared, in the simple duality ansatz parametrization of the spectral function: where the "QCD continuum comes from the discontinuity of the QCD diagrams, which is expected to give a good smearing of the different radial excitations 1 . The decay constant f D is analogous to f π = 92.32 MeV. However, in order to avoid some confusion, and for a more direct comparison with the lattice results, we shall abandon our favorite normalization, and adopt in the rest of the paper, the one: a normalization where f π = 130.56 MeV, and which the different experimental groups have also adopted; t c is the QCD continuum threshold, which is, like the sum rule variable τ , an (a priori) arbitrary parameter.
In this paper, we shall impose the t c and τ stability criteria for extracting our optimal results 2 . The QCD expression of the correlator is well-known to two-loop accuracy (see e.g. [8] and the explicit expressions given in [9]), in terms of the perturbative pole mass M Q , and including the non-perturbative condensates of dimensions less than or equal to six 3 . The sum rule reads: where R2s is the new α 2 s -term obtained semi-analytically in [17] and is available as a Mathematica package program Rvs.m. The other terms are: The previous sum rules can be expressed in terms of the running massm Q (ν) 4 , through the perturbative three-loop relation [1,2,19]: , where, in the RHS, M pole ≡ M is the pole mass and: For 0 ≤ r ≤ 1, ∆(r) can be approximated, within an accuracy of 1% by: Throughout this paper we shall use the values of the parameters [8,20] given in Table 1. Table 1: Different sources of errors in the estimates of the decay constants (in MeV) and quark masses (in GeV). We have exagerately enlarged the error bars of different input in order to have conservative errors.
Sources We have used for the mixed condensate the parametrization: and we deduce the value of the QCD scale Λ from the value of α s (M Z ) = (0.1184 ± 0.031) given in [21,6].

The D-meson channel
We study in Fig. 1, the prediction of f D from the Laplace sum rules L and the one of M D from ratio of moments R for given value of the running charm quark massm c (m c ). The influences of the choice of the continuum threshold t c and of the sum rule scale τ are shown in details. Our optimal results correspond to the case where both stability in τ and in t c are reached. However, for a more conservative estimate of the errors we allow deviations from the stability points, and we take: and where the lowest value of t c corresponds to the beginning of the τ -stability region. One can inspect that the dominant non-perturbative contribution is due to the dimension-four M c d d light quark condensate, and test that the OPE is not broken by high-dimension condensates at the optimization scale. However, the perturbative radiative corrections converge slowly, as the value of f D increases by 12% after the inclusion of the α s correction and the sum of the lowest order plus α s -correction increases by 21 % after the inclusion of the α 2 s term, indicating that the total amount of corrections of 21% is still a reasonnable correction despite the slow convergence of the perturbative series. However, as the radiative corrections are both positive, we expect that this slow convergence will not affect in a sensible way the final estimate. In order to improve the perturbative contributions, we have estimated the possible contribution of the α 3 s -term by assuming that its coefficient is the geometric sum of the α s and α 2 s contributions. This effect is shown in Table 1, which is has a quite reasonnable value. A more precise answer on the higher order perturbative contribution needs an evaluation of the α 3 s term which we hope to be available in the near future. In doing the analysis, one can also notice that the relative size of the perturbative corrections is smaller in the physical observable f D (perturbative+non-perturbative) than in the perturbative graph alone. This is due to fact that the rôle of the ψ ψ condensate is important at the optimization scale, which then decreases the relative weight of the perturbative radiative corrections in the OPE. The estimate of M D from the ratio of moments R is less affected by radiative corrections, which tend to cancel each others due to the form of the sum rule.
where as mentioned previously, we have used the normalization f π = 130.56 MeV. These optimal values correspond to t c = 6.5 GeV 2 and τ = 1.2 GeV −2 . The value of t c roughly corresponds to a radial excitation with a mass-splitting relative to the ground state mass of about M ρ which is phenomenologically acceptable. A similar analysis shown in Figs. 1e to 1h is done for the pole mass. The discussions presented previously apply also here, including the one of the radiative corrections. We quote the final result: where the error is slightly smaller here due to the absence of the subtraction scale uncertainties. For f D , we consider as a final estimate the mean value of the two predictions and taking the largest errors: Using our previous estimate of f Ds /f D = 1.15 ± 0.04 [13], we can also deduce: One can immediately compare the present predictions with the one obtained to order α s using the same procedure [14]:m c (m c ) = (1.08 ± 0.11) GeV , where one can notice a good agreement between the α s and α 2 s results, though the error in [14] is smaller as the effect of the subtraction point ν has not been taken into account. However, the agreement seems paradoxal in view of the fact that radiative corrections tend to increase the value of f D compared to the lower orders result. The different truncations of the expression of α s and of the relation between the pole and running mass also affect the absolute value of f D , which tend to compensate the increase due to the radiative corrections of the correlator. Therefore, a naïve comparison becomes misleading. As one can see in Table 1, the main source of errors is due to the variations of the quark mass and to a lesser extent to the ones of ν, Λ and ψ ψ . The effect of t c on the result is relatively small from the value t c larger than 6 GeV 2 , where one starts to have a τ stability. This result for the mass is also in agreement with the one from M J/ψ [22,6]:m but lower than the one from [23] using non-relativistic Balmer formula for thecc bound state. One can cross-check that the two values ofm c (m c ) and M c give the ratio: which satisfies quite well the three-loop perturbative relation M c /m c (m c ) = 1.33 obtained from the previous Eq. (8). This could be a non-trivial result if one has in mind that the quark pole mass definition can be affected by non-perturbative corrections not present in the standard SVZ-OPE. In particular, it may signal that 1/q 2 correction of the type discussed in [3,24], if present, will only affect weakly the standard SVZphenomenology as observed explicitly in the light quark, gluonia and hybrid channels [24]. Recent results to order α 2 s [12], using the analogous sum rule and using as input the pole mass value, gives f D = (195 ± 20) MeV. The result is slightly lower than the result given here, though in agreement within the errors. We understand this slight difference as due to the lower value of the QCD continuum threshold used there, which corresponds to t c ≈ 5.6 GeV 2 if one uses our pole mass value M c = 1.47 GeV. As shown in Fig. 1e, this t c value is on the boarder of the τ stability region. It also indicates that the error introduced by the choice of t c could have been underestimated in this result. Fig. 1g also indicates that f D is quite sensitive in full QCD to the change of the M c value contrary to the remark given in [12]. Quenched and unquenched lattice results for f D are compiled in [25]. The quenched results range from (192 ± 18) MeV to (221 ± 17) MeV, in fair agreement with our results within the errors. The two available unquenched results [26,27] lead to the average: which is again in agreement within the errors with our previous estimate. Finally, one can also compare the value of f Ds with the experimental value [?]: which agrees within 1σ with our prediction. Improvements of our predictions for f Ds need an estimate of the ratio f Ds /f D to order α 2 s .

The B-meson channel
We extend the previous analysis to the case of the B-meson, which again is an update of our previous work in [14,8,9]. The analysis is still similar to the one done in the D-channel, and is summarized in Fig 2 and in Table 1. Using the running b-quark mass, as a free parameter, we obtain at the optimization scale τ = 0.375 GeV −2 and t c = 38 GeV 2 : while using the pole mass as a free parameter, we get: from which we deduce the average: where we have taken the largest errors. One can again cross-check that the two values ofm b (m b ) and M b lead to to be compared with 1.15 from the three-loop perturbative relation in Eq. (8), and might indirectly indicate the smallness of the 1/q 2 correction if any. Our result ofm b can be compared with our previous estimate from Υ-systems [22]:m b (m b ) = (4.23 ± 0.05) GeV , and with similar values from recent estimates [6,30]. Our value of the perturbative pole mass is in agreement within the errors with the one in [6,22,30] but is again lower than the one in [23]. Our value of f B is in fair agreement with the recent results f B = (206 ± 20) MeV obtained in [12] using HQET sum rules and the one f B = (197 ± 23) MeV obtained in [15] using the Laplace sum rule like in this work. The slight difference is due to the different appreciations of the continuum threshold t c and τ stability regions in each papers. More specifically, errors related to the choice of t c at their choice of lower τ -values appear to be underestimated in these works. At such a choice of low τ -values, the ground state contribution to the sum rule is smaller than in the present analysis. Unquenched lattice results [27,28] give the mean value: where the largest error has been taken. These values agree with our previous determination in Eq. (24) and with our earlier estimate [13]: which has been confirmed from the recent analysis of [15]. Combining this SU (3) breaking ratio with our estimate of f B , one obtains: The extension of the previous analysis to the D * and B * channels is in progress.

Summary
We have updated our previous estimate of the quark masses and decay constants in [14,9,8] using the recent expression [17] of α 2 s corrections for the heavy-light pseudoscalar correlators. Our results for the masses in Eqs. (13,14,22,23) and for the decay constants in Eqs. (15,16,24,29) confirm previous estimates to two-loops. The results for the running masses are: Eqs (13,22) .