Di-Gluonium Sum Rules, Conformal Charge and I = 0 Scalar Mesons

We revisit, improve and confirm our previous results [1-3] from the scalar digluonium sum rules within the standard SVZ-expansion at N2LO {it without instantons} and {it beyond the minimal duality ansatz} : one resonance $oplus$ QCD continuum parametrization of the spectral function. We select different unsubtracted sum rules (USR) moments of degree $leq$ 4 for extracting the two lowest gluonia masses and couplings. We obtain in units of GeV: $(M_{G},f_G)=[1.04(12),0.53(17)]$ and $[1.52(12),0.57(16)]$. We attempt to predict the masses of their first radial excitations to be $M_{sigma} simeq 1.28(9)$ GeV and $M_{G_2}simeq 2.32(18)$ GeV. Using a combination of the USR with the subtracted sum rule (SSR), we estimate the conformal charge (subtraction constant $psi_G(0)$ of the scalar gluonium two-point correlator at zero momentum) which agrees completely with the Low Energy Theorem (LET) estimate. Combined with some low-energy vertex sum rules (LEV-SR), we confront our predictions for the widths with the observed $I=0$ scalar mesons spectra. We confirm that the $sigma$ and $f_0(980)$ meson can emerge from a maximal (destructive) ($bar uu+bar dd$) meson - $(sigma_B$) gluonium mixing [10]. The $f_0(1.37)$ and $f_0(1.5)$ indicate that they are (almost) pure gluonia states (copious decay into $4pi$) through $sigmasigma$, decays into $etaeta$ and $etaeta$ from the vertex $U(1)_A$ anomaly with a ratio $div$ to the square of the pseudoscalar mixing angle sin$^2theta_P$.

Modern status of heavy quark sum rules in QCD

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.

Z_c -like spectra from QCD Laplace sum rules at NLO

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.

$X_{0,1}$(2900) and $(D^-K^+)$ invariant mass from QCD Laplace sum rules at NLO

We revisit, improve and complete some recent estimates of the $0^{+}$ and $1^-$ open charm $(bar c bar d)(us)$ tetraquarks and the corresponding molecules masses and decay constants from QCD spectral sum rules (QSSR) by using QCD Laplace sum rule (LSR) within stability criteria where the factorised perturbative NLO corrections and the contributions of quark and gluon condensates up to dimension-6 in the OPE are included. We confront our results with the $D^-K^+$ invariant mass recently reported by LHCb from $B^+to D^+(D^-K^+)$ decays. We expect that the bump near the $D^-K^+$ threshold can be originated from the $0^{++}(D^-K^+)$ molecule and/or $D^-K^+$ scattering. The prominent $X_{0}$(2900) scalar peak and the bump $X_J(3150)$ (if $J=0$) can emerge from a {it minimal mixing model}, with a tiny mixing angle $theta_0simeq (5.2pm 1.9)^0$, between a scalar {it Tetramole} (${cal T_M}_0$) (superposition of nearly degenerated hypothetical molecules and compact tetraquarks states with the same quantum numbers) having a mass $M_{{cal T_M}_0}$=2743(18) MeV and the first radial excitation of the $D^-K^+$ molecule with mass $M_{(DK)_1}=3678(310)$ MeV. In an analogous way, the $X_1$(2900) and the $X_J(3350)$ (if $J=1$) could be a mixture between the vector {it Tetramole} $({cal T_M}_1)$ with a mass $M_{{cal T_M}_1}=2656(20)$ MeV and its first radial excitation having a mass $M_{({cal T_M}_1)_1}=4592(141)$ MeV with an angle $theta_1simeq (9.1pm 0.6)^0$. A (non)-confirmation of the previous {it minimal mixing models} requires an experimental identification of the quantum numbers of the bumps at 3150 and 3350 MeV.

The First Months of COVID-19 in Madagascar

Using the official data and aware of the uncertain source and insufficient number of samples, we present a first and (for the moment) unique attempt to study the first two months spread of COVID-19 in Madagascar. The approach has been tested by predicting the number of contaminated persons for the next week after fitting the inputs data collected within 7 or 15 days using standard least $chi^2$-fit method. Encouraged by this first test, we study systematically during 67 days , 1-2 weeks new data and predict the contaminated persons for the coming week. We find that the first month data are well described by a linear or quadratic polynomial with an increase of about (4-5) infected persons per day. Pursuing the analysis, one note that data until 46 days favour a cubic polynomial behaviour which signals an eventual near future stronger growth as confirmed by the new data on the 48th day. We complete the analysis until 67 days and find that the data until 77 days confirm the cubic polynomial behaviour which is a remarkable feature of the pandemic spread in Madagascar. We expect that these results will be useful for some new model buildings. A comparison with some other SI-like models predictions is done.These results may also be interpreted as the lowest values of the real case due to the insufficient number of samples (12907 for 27 million habitants on 05/06/20). The data analysis of the absolute number of cured persons until 67 days shows an approximate linear behaviour with about 3 cured persons per day. However, the number of percentage number of cured persons decreases above 42-46 days indicating the limits of the hospital equipment and care to face the 2nd phase of the pandemic for the 67th first days. Some comments on the social, economical and political impacts of COVID-19 and confinement for Madagascar and, in general, for Worldwide are shortly discussed.

Doubly-hidden scalar heavy molecules and tetraquarks states from QCD at NLO

Alerted by the recent LHCb discovery of exotic hadrons in the range (6.2 -- 6.9) GeV, we present new results for the doubly-hidden scalar heavy $(bar QQ) (Qbar Q)$ charm and beauty molecules using the inverse Laplace transform sum rule (LSR) within stability criteria and including the Next-to-Leading Order (NLO) factorized perturbative and $langle G^3rangle$ gluon condensate corrections. We also critically revisit and improve existing Lowest Order (LO) QCD spectral sum rules (QSSR) estimates of the $({ bar Q bar Q})(QQ)$ tetraquarks analogous states. In the example of the anti-scalar-scalar molecule, we separate explicitly the contributions of the factorized and non-factorized contributions to LO of perturbative QCD and to the $langlealpha_sG^2rangle$ gluon condensate contributions in order to disprove some criticisms on the (mis)uses of the sum rules for four-quark currents. We also re-emphasize the importance to include PT radiative corrections 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. Our LSR results for tetraquark masses summarized in Table II are compared with the ones from ratio of moments (MOM) at NLO and results from LSR and ratios of MOM at LO (Table IV). The LHCb broad structure around (6.2 --6.7) GeV can be described by the $overline{eta}_{c}{eta}_{c}$, $overline{J/psi}{J/psi}$ and $overline{chi}_{c1}{chi}_{c1}$ molecules or/and their analogue tetraquark scalar-scalar, axial-axial and vector-vector lowest mass ground states. The peak at (6.8--6.9) GeV can be likely due to a $overline{chi}_{c0}{chi}_{c0}$ molecule or/and a pseudoscalar-pseudoscalar tetraquark state. Similar analysis is done for the scalar beauty states whose masses are found to be above the $overlineeta_beta_b$ and $overlineUpsilon(1S)Upsilon(1S)$ thresholds.

Spectra and Decay Constants of $B_c$-like and $B^*_0$ Mesons in QCD

Using the existing state of art of the QCD expressions of the two-point correlators into the Inverse Laplace sum rules (LSR) within stability criteria, we present a first analysis of the spectra and decay constants of B_c-like scalar (0^{++}) and axial-vector (1^{++}) mesons and revisit the ones of the B^*_c(1^{--}) vector meson. Improved predictions are obtained by combining these LSR results with the some mass-splittings from Heavy Quark Symmetry (HQS). We complete the analysis by revisiting the B^*_{0}(0^{++}) mass which might be likely identified with the B^*_J(5732) experimental candidate. The results for the spectra collected in Table 2 are compared with some recent lattice and potential models ones. New estimates of the decay constants are given in Table 3.

QCD parameters and f_{B_c} from heavy quark sum rules

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.

m_c and m_b from M_B_c and improved estimate of f_B_c and f_B_c (2S)

We extract (for the first time) the correlated values of the running masses m_c and m_b from M_Bc using QCD Laplace sum rules (LSR) within stability criteria where pertubative (PT) expressions at N2LO and non-perturbative (NP) gluon condensates at LO are included. Allowing the values of m_{c,b}(m_{c,b}) to move inside the enlarged range of recent estimates from charmonium and bottomium sum rules (Table 1) obtained using similar stability criteria, wee deduce : m_c(m_c) = 1286(16) MeV and m_b(m_b) = 4208(8) MeV. Combined with previous estimates (Table 2), we deduce a tentative QCD Spectral Sum Rules (QSSR) average m_c(m_c) = 1266(6) MeV and m_b(m_b) = 4197(8) MeV, where the errors come from the precise determinations from J/psi and Upsilon sum rules. As a result, we present an improved prediction of f_B_c =371(17)MeV and the tentative upper bound f_B_c(2S)<139(6) MeV, which are useful for a further analysis of B_c-decays.

$alpha_s(mu)$ from $M_{chi_{0c(0b)}}-M_{eta_{c(b)}}$@N2LO

This note complements and clarifies the results obtained in the original paper {it QCD Parameters Correlations from Heavy Quarkonia} [1] where, here, we present a more detailed discussion of the alpha_s-results obtained @ N2LO at two different subtraction scales mu=2.85 and 9.50 GeV from the (pseudo)scalar heavy quarkonia mass-spliitings M_{chi_{0c(0b)}}-M_{eta_{c(b)}}. We obtain from the M_{chi_{0c}}-M_{eta_{c}} sum rule: alpha_s(2.85)=0.262(9) --> alpha_s(M_tau)=0.318(15) --> alpha_s(M_Z)=0.1183(19)(3) and from the M_{chi_{0b}}-M_{eta_{b}} one: alpha_s(9.50)=0.180(8) --> alpha_s(M_tau)=0.312(27) --> alpha_s(M_Z)=0.1175(32)(3), in complete agreement with the world average: alpha_s(M_Z)=0.1181(11).