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Register a new userApplications of QCD Sum Rules to Heavy Quark Physics
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Alexander Khodjamirian
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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.
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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.
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.
It is argued that it is valid to use QCD sum rules to determine the scalar and pseudoscalar two-point functions at zero momentum, which in turn determine the ratio of the strange to non-strange quark condensates $R_{su} = frac{<bar{s} s>}{<bar{q} q>}$ with ($q=u,d$). This is done in the framework of a new set of QCD Finite Energy Sum Rules (FESR) that involve as integration kernel a second degree polynomial, tuned to reduce considerably the systematic uncertainties in the hadronic spectral functions. As a result, the parameters limiting the precision of this determination are $Lambda_{QCD}$, and to a major extent the strange quark mass. From the positivity of $R_{su}$ there follows an upper bound on the latter: $bar{m_{s}} (2 {GeV}) leq 121 (105) {MeV}$, for $Lambda_{QCD} = 330 (420) {MeV} .$
We test the validity of the QCD sum rules applied to the light scalar mesons, the charmed mesons $D_{s0}(2317)$ and $D_{s1} (2460)$, and the X(3872) axial meson, considered as tetraquark states. We find that, with the studied currents, it is possible to find an acceptable Borel window only for the X(3872) meson. In such a Borel window we have simultaneouly a good OPE convergence and a pole contribution which is bigger than the continuum contribution. We interpret these results as a strong argument against the assignment of a tetraquark structure for the light scalars and the $D_{s0}(2317)$ and $D_{s1} (2460)$ mesons.
The QCD up- and down-quark masses are determined from an optimized QCD Finite Energy Sum Rule (FESR) involving the correlator of axial-vector current divergences. In the QCD sector this correlator is known to five loop order in perturbative QCD (PQCD), together with non-perturbative corrections from the quark and gluon condensates. This FESR is designed to reduce considerably the systematic uncertainties arising from the hadronic spectral function. The determination is done in the framework of both fixed order and contour improved perturbation theory. Results from the latter, involving far less systematic uncertainties, are: $bar{m}_u (2, mbox{GeV}) = (2.6 , pm , 0.4) , {mbox{MeV}}$, $bar{m}_d (2, mbox{GeV}) = (5.3 , pm , 0.4) , {mbox{MeV}}$, and the sum $bar{m}_{ud} equiv (bar{m}_u , + , bar{m}_d)/2$, is $bar{m}_{ud}({ 2 ,mbox{GeV}}) =( 3.9 , pm , 0.3 ,) {mbox{MeV}}$.
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