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Register a new userModern status of heavy quark sum rules in QCD
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We briefly report the modern status of heavy quark sum rules (HQSR) based on stability criteria by emphasizing the recent progresses for determining the QCD parameters (alpha_s, m_{c,b} and gluon condensates)where their correlations have been taken into account. The results: alpha_s(M_Z)=0.1181(16)(3), m_c(m_c)=1286(16) MeV, m_b(m_b)=4202(7) MeV,<alpha_s G^2> = (6.49+-0.35)10^-2 GeV^4, < g^3 G^3 >= (8.2+-1.0) GeV^2 <alpha_s G^2> and the ones from recent light quark sum rules are summarized in Table 2. One can notice that the SVZ value of <alpha_s G^2> has been underestimated by a factor 1.6, <g^3 G^3> is much bigger than the instanton model estimate, while the four-quark condensate which mixes under renormalization is incompatible with the vacuum saturation which is phenomenologically violated by a factor (2~4). The uses of HQSR for molecules and tetraquarks states are commented.
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We report results of our recent works [1,2] where we where the correlations between the c,b-quark running masses{m}_{c,b}, the gluon condensate<alpha_s G^2> and the QCD coupling alpha_s in the MS-scheme from an analysis of the charmonium and bottomium spectra and the B_c-meson mass. We use optimized ratios of relativistic Laplace sum rules (LSR) evaluated at the mu-subtraction stability point where higher orders PT and D< 6-8-dimensions non-perturbative condensates corrections are included. We obtain [1] alpha_s(2.85)=0.262(9) and alpha_s(9.50)=0.180(8) from the (pseudo)scalar M_{chi_{0c(0b)}}-M_{eta_{c(b)}} mass-splittings at mu=2.85(9.50) GeV. The most precise result from the charm channel leads to alpha_s(M_tau)=0.318(15) and alpha_s(M_Z)=0.1183(19)(3) in excellent agreement with the world average: alpha_s(M_Z)=0.1181(11)[3,4]. Updated results from a global fit of the (axial-)vector and (pseudo)scalar channels using Laplace and Moments sum rules @ N2LO [1] combined with the one from M_{B_c} [2] lead to the new tentative QCD spectral sum rules (QSSR) average : m_c(m_c)|_average= 1266(6) MeV and m_b(m_b)|_average=4196(8) MeV. The values of the gluon condensate <alpha_s G^2> from the (axial)-vector charmonium channels combined with previous determinations in Table 1, leads to the new QSSR average [1]: <alpha_s G^2>_average=(6.35pm 0.35)x 10^{-2} GeV^4. Our results clarify the (apparent) discrepancies between different estimates of <alpha_s G^2> from J/psi sum rule but also shows the sensitivity of the sum rules on the choice of the mu-subtraction scale. As a biproduct, we deduce the B_c-decay constants f_{B_c}=371(17) MeV and f_{B_c}(2S)< 139(6) MeV.
These talks review and summarize our results in [1,2] on $XYZ$-like spectra obtained from QCD Laplace Sum Rules in the chiral limit at next-to-next-leading order (N2LO) of perturbation theory (PT) and including leading order (LO) contributions of dimensions $dleq 6-8$ non-perturbative condensates. We conclude that the observed $XZ$ states are good candidates for $1^{+}$ and $0^+$ molecules or / and four-quark states while the predictions for $1^-$ and $0^-$ states are about 1.5 GeV above the $Y_{c,b}$ experimental candidates and hadronic thresholds. We (numerically) find that these exotic molecules couple weakly to the corresponding interpolating currents than ordinary $D,B$ heavy-light mesons while we observe that these couplings decrease faster [$1/m_b^{3/2}$ (resp. $1/m_b$) for the $1^+,0^+$ (resp. $1^-,0^-)$ states] than $1/m_b^{1/2}$. Our results do not also confirm the existence of the $X(5568)$ state in agreement with LHCb findings.
In these lectures, I present several important applications of QCD sum rules to the decay processes involving heavy-flavour hadrons. The first lecture is introductory. As a study case, the sum rules for decay constants of the heavy-light mesons are considered. They are relevant for the leptonic decays of $B$-mesons. In the second lecture I describe the method of QCD light-cone sum rules used to calculate the heavy-to-light form factors at large hadronic recoil, such as the $Bto pi ell u_ell$ form factors. In the third lecture, the nonlocal hadronic amplitudes in the flavour-changing neutral current decays $Bto K^{(*)}ellell$ are discussed. Light-cone sum rules provide important nonfactorizable contributions to these amplitudes.
We review our results in Refs.[1,2] for the masses and couplings of heavy-light DD(BB)-like molecules and (Qq)(Qq)-like four-quark states from relativistic QCD Laplace sum rules (LSR) where next-to-next-to-leading order (N2LO) PT corrections in the chiral limit, next-to-leading order (NLO) SU3 PT corrections and non-perturbative contributions up to dimension d=6-8 are included. The factorization properties of molecule and four-quark currents have been used for the estimate of the higher order PT corrections. New integrated compact expressions of the spectral functions at leading order (LO) of perturbative QCD and up to dimensions d< (6 - 8) non-perturbative condensates are presented. The results are summarized in Tables 5 to 10, from which we conclude, within the errors, that the observed XZ states are good candidates for being 1^{++} and 0^{++} molecules or/and four-quark states, contrary to the observed Y states which are too light compared to the predicted 1^{-pm} and 0^{-pm} states. We find that the SU3 breakings are relatively small for the masses (< 10(resp. 3)%) for the charm (resp. bottom) channels while they are large (< 20%) for the couplings which decrease faster (1/m_{b}^{3/2}) than 1/m_{b}^{1/2} of HQET. QCD spectral sum rules (QSSR) approach cannot clearly separate (within the errors) molecules from four-quark states having the same quantum numbers. Results for the BK (DK)-like molecules and (Qq)(us)-like four-quark states from [3] are also reviewed which do not favour the molecule or/and four-quark interpretation of the X(5568). We suggest to scan the charm (2327 ~ 2444) MeV and bottom (5173 ~ 5226) MeV regions for detecting the (unmixed)(cu)ds and (bu)ds states. We expect that future experimental data and lattice results will check our predictions.
We present a global analysis of the observed Z_c, Z_cs and future Z_css-like spectra using the inverse Laplace transform (LSR) version of QCD spectral sum rules (QSSR) within stability criteria. Integrated compact QCD expressions of the LO spectral functions up to dimension-six condensates are given. Next-to-Leading Order (NLO) factorized perturbative contributions are included. We re-emphasize the importance to include PT radiative corrections (though numerically small) for heavy quark sum rules in order to justify the (ad hoc) definition and value of the heavy quark mass used frequently at LO in the literature. We also demonstrate that, contrary to a naive qualitative 1/N_c counting, the two-meson scattering contributions to the four-quark spectral functions are numerically negligible confirming the reliability of the LSR predictions. Our results are summarized in Tables III to VI. The Z_c(3900) and Z_cs(3983) spectra are well reproduced by the T_c(3900) and T_cs(3973) tetramoles (superposition of quasi-degenerated molecules and tetraquark states having the same quantum numbers and with almost equal couplings to the currents). The Z_c(4025) or Z_c(4040) state can be fitted with the D*_0D_1 molecule having a mass 4023(130) MeV while the Z_cs bump around 4.1 GeV can be likely due to the (D^*_s0D_1+ D^*_0D_s1) molecules. The Z_c(4430) can be a radial excitation of the Z_c(3900) weakly coupled to the current, while all strongly coupled ones are in the region (5634-6527) MeV. The double strange tetramole state T_css which one may identify with the future Z_css is predicted to be at 4064(46) MeV. It is remarkable to notice the regular mass-spliitings of the tetramoles due to SU(3) breakings M_{T_cs}-M_{T_c}= M_{T_css}-M_{T_cs= (73- 91) MeV.
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