Updating m_c,b(m_c,b) from SVZ-Moments and their Ratios

Using recent values of \alpha_s, the gluon condensates<\alpha_s G^2>andand the new data on the \psi/\Upsilon-families, we update our determinations of the MS-bar running quark masses m_c,b(m_c,b) from the SVZ-Moments M_n(Q^2) and their ratios by including higher order perturbative (PT) corrections, non-perturbative (NPT) terms up to dimension d=8 and using the degree n-stability criteria of the (ratios of) moments. Optimal results from different (ratios of) moments converge to the accurate mean values: m_c(m_c)=1264(6) MeV} and m_b(m_b)=4188(8) MeV in Table 4, which improve and confirm our previous findings [1,2] and the recent ones from Laplace sum rules [3]. Comments on some other determinations of m_c(m_c) and<\alpha_s G^2>from the SVZ-(ratios of) moments in the vector channel are given in Section 5.


Introduction and SVZ-moments
In Refs. [1,2], we have used different M n (Q 2 ) moments and their ratios r n /r n+ j introduced by SVZ [4,5] 1 for extracting the values of the charm and bottom running quark masses m c,b (m c,b ) and the dimension 4: α s G 2 and 6: g 3 f abc G 3 gluon condensates. Using the recent values of the gluon condensates from Laplace sum rules [3,15] and new data on the ψ/ϒ-families masses and leptonic widths [16], we shall improve in this paper our previous results for the quark masses. Here, we shall be concerned with the two-point correlator: associated to the J μ =¯ γ μ ( ≡ c, b) heavy quark neutral vector current. The corresponding moments are 2 : E-mail address: snarison@yahoo.fr. 1 For reviews, see e.g. [6][7][8][9][10][11][12][13][14]. 2 We shall use the same normalization as [17] and some of the expressions given there.
Their ratios read: where the experimental sides are more precise than that of the moments M n (Q 2 ). It has been noticed by [18,19] that the OPE of M n (0) breaks down for higher values of n, while it has also been mentioned in [1,2] that low moments n ≤ 3 are sensitive to the way for parametrizing the high-energy part of the spectral function (hereafter called QCD continuum) making the results obtained from low moments model-dependent. Therefore, one should look for compromise values of n (stability in n) where both problems are avoided. Another way out is to work with the Q 2 = 0 moments [11] where the OPE converges faster while the QCD continuum contributions are strongly suppressed.

Expressions of the SVZ-moments M n ( Q 2 )
The QCD expressions of the moments can be derived from the ones of R. The on-shell expression of the spectral function is transformed into the M S-scheme by using the known relation between   Table 2 Masses and electronic widths of the J /ψ family from PDG 16 [16].
where we have used the recent determinations from a recent global fit of the (axial-)vector and (pseudo)scalar charmonium and bottomium systems using Laplace sum rules [3]: The low-energy part of the spectral function is well described by the sum of different resonances contributions within a narrow width approximation (NWA). For the c-quark channel, it reads: where N c = 3; M ψ and ψ→e + e − are the mass and leptonic width of the J /ψ mesons; Q c = 2/3 is the charm electric charge in units of e; α = 1/133.6 is the running electromagnetic coupling evaluated at M 2 ψ . We shall use the experimental values of the J /ψ parameters compiled in Table 2. We shall parametrize the contributions from √ t c ≥ (4.5 ± 0.1) GeV using either: -Model 1: The approximate PT QCD expression of the spectral function to order α 2 s up to order (m 2 c /t) 6 given in [24] and the α 3 s contribution from non-singlet contribution up to order (m 2 c /t) 2 given in [25].
-Model 2: The asymptotic PT expression of the spectral function known to order α 3 s where the quark mass corrections are neglected. 3 -Model 3: Fits of different data above the ψ(2S) mass: we shall take e.g. the results in [25] where a comparison of results from different fitting procedures can be found in this paper (see e.g. [26]). 3 Original papers are given in Refs. 317 to 321 of the book in Ref. [7].  Table 1 and the three models given previously for the QCD continuum parametrization.

Fig. 2.
Values of m c (m c ) from the ratios of moments r n/n+1 (0) and r n/n+2 (0) for different values of n using the QCD input parameters in Table 1 and Model 1 given previously for the QCD continuum parametrization. In the n axis:

Running m c (m c ) charm quark mass from M n (0)
-Using the previous models for parametrizing the QCD continuum, we show in Fig -We do a similar analysis for the ratios of moments r n/n+1 (0) and r n/n+2 (0). The results versus the degree of moments are shown in Fig. 2. We deduce, at the stability point n 4, the value (in units of MeV): where one can notice that the experimental error is reduced compared to the moment results while the ones induced by the QCD parameters have increased.

Fig. 4.
Values of m c (m c ) from the moments M n (8m 2 c ) and their ratios r n/n+1 (8m 2 c ) and r n/n+2 (8m 2 c ) for different values of n using the QCD input parameters in Table 1 and Model 1 given previously for the QCD continuum parametrization. In the n axis:

Running m c (m c ) charm quark mass from M n ( Q 2 = 0)
Previous analysis can be extended to the case of Q 2 = 0 moments where a better convergence of the OPE is expected [11] and where the QCD continuum contribution to the moments is smaller as we shall work with higher moments at which the n-stability The previous results are collected in Table 4.

Comments on m c (m c ) and α s G 2 from M n ( Q 2 )
-One can notice in Fig. 1 that the values of m c (m c ) from the moments M n≤2 (0) are strongly affected by the QCD continuum parametrization though agree within the errors with the ones in [25][26][27][28]. For the case of n = 1 moment used by previous authors to extract their final results, one can deduce from Fig. 1: -Instead, in the n−stability region, the QCD continuum-modeldependence of the result disappears (see Fig. 1) and leads to the optimal and more accurate value given in Eq. (7): The error due to the parametrization of the spectral function is even reduced when working with the ratio of moments (see Fig. 2) leading to the result in Eq. (8): Indeed apart the Wilson coefficient of α s G 2 known to NLO [30], the ones of the high-dimension condensates are only known to LO. Refs. [26][27][28] choose to work with the pole mass in the OPE which, as emphasized in [25] is ambiguous due to the IR renormalon contribution. Then, the use of the running mass in the OPE can be better justified which is also consistent with the use of the has not yet been carefully studied. In order to circumvent a such large enhancement, which does not arise when working with the Laplace sum rule [3] where an optimal value of μ has been derived, we limit here to the (usual and natural) choice μ = m c and do not try to move it arbitrarily around this value.
-Coulombic corrections have been roughly estimated in Ref. [1]. However, it has been also argued in Ref. [17] that this contribution, which is not under a good control, can be safely neglected in the relativistic sum rules. Therefore, we shall not consider such corrections in this paper.
This feature has been also observed from the analysis of the same vector charmonium using Laplace sum rules [3] where constraints from some other charmonium channels are needed for reaching more accurate results.
-To the value of α s G 2 given in Table 1 which is consistent within the errors with our previous results in Table 4.
-The authors deduce their favorite result in Eq. (13) from a common solution of the moments and of their ratios, where one can notice, from our Figs. 3 and 4, that, at a fixed value of α s G 2 , the value of m c (m c ) from the ratios of moments meets the moments outside the n-stability of M n (Q 2 ), while the ratios increase rapidly with n. This fact indicates that a such requirement may not be reliable.
-Beyond the OPE, we can also have some contributions due to the so-called Duality Violation, which is model-dependent. It can be parametrized (within our normalization) as [31,32]: where the coefficients are free parameters and come from a fitting procedure. For an approximate estimate of this additional effect, we compare its contribution with the QCD continuum one parametrized by the asymptotic expression of PT spectral function  ) and their ratios r n/n+1 (8m 2 b ) and r n/n+2 (4m 2 b ) for different values of n using the QCD input parameters in Table 1 and Model 1 given previously for the QCD continuum parametrization. In the n axis: 10 ≡ r 8/10 , 11 ≡ r 9/10 , 12 ≡ r 9/11 , 13 ≡ r 10/11 , 14 ≡ r 10/12 15 ≡ r 11/12 , 16 ≡ r 12/13 , 16 ≡ r 15/17 , 17 ≡ r 12/14 , 18 ≡ r 13/14 .

Table 3
Masses and electronic widths of the ϒ family from PDG 16 [16].
fixed from τ -decay data by assuming that they can be applied here. We found that, in the example n = 1 and Q 2 = 0, this effect is completely negligible even allowing a low value of √ t c = 1.65 GeV at which the fit of the coefficients has been performed.

Running m b (m b ) bottom quark mass from M n ( Q 2 )
The previous analysis is extended to the b-quark mass. We shall use the data input in Table 3. Behaviours of the (ratios of) moments versus the degree of the moments are given in Figs. 5 to 7. We deduce as optimal values the overlapping regions of the one from the moments and the ratios of moments. We obtain to order These results are quoted in Table 4.

Conclusions
We have updated our previous results in Refs. [1,2] from SVZ-(ratios of) moments. These results are confirmed and improved by the new ones summarized in Table 4. The simultaneous Table 4 Charm and bottom running masses m c,b (m c,b ) from (ratios of) moments.

Observables
Mass 4188(6.7) Mean 4188 (8) use of the higher moments and their ratios reduce notably the errors in the mass determinations. Though it is difficult to estimate the systematic errors of the approach, we can expect that they are at most equal to the ones quoted in this paper. These new results are also in perfect agreement with the ones quoted in Eq. (5) from a recent global fit of the (axial-)vector and (pseudo)scalar charmonium and bottomium systems using Laplace sum rules [3]. Some comments on the existing estimates of the quark masses and gluon condensates from SVZ-(ratios of) moments are given in Section 5. Our results are comparable with recent results from non-relativistic approaches [33] but more accurate.