Abstract
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 $(\bar QQ) (Q\bar Q)$ 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 $\langle G^3\rangle$ 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 $\langle\alpha_sG^2\rangle$ 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 $\overline\eta_b\eta_b$ and $\overline\Upsilon(1S)\Upsilon(1S)$ thresholds.
Highlights
In a previous series of papers [18,19,20,21,22,23], we have used the inverse Laplace transform (LSR) [24,25,26,27] of QCD spectral sum rules (QSSR) to predict the couplings and masses of different heavy-light molecules and tetraquarks states by including next-to- nonleading order (N2LO) factorized perturbative (PT) corrections where we have emphasized the importance of these corrections for giving a meaning of the input heavy quark mass which plays an important role in the analysis though these corrections are small in the MS-scheme
Using the Shifman-Vainshtein-Zakharov [1,2] operator product expansion (OPE), we give below the QCD expression of the two-point correlators associated to the χ0χ0 molecule to lowest order (LO) of PT QCD and up to dimension-four condensates which can be extracted from the Feynman diagrams in Figs. 1–3: II
Gluon condensate hαsG2i We use the recent estimate obtained from a correlation with the values of the heavy quark masses and αs which can be compared with the QSSR average from different channels [55]
Summary
QCD spectral sum rules (QSSR) ala Shifman-VainshteinZakharov [1,2,3] have been applied since 41 years to study. In a previous series of papers [18,19,20,21,22,23], we have used the inverse Laplace transform (LSR) [24,25,26,27] of QSSR to predict the couplings and masses of different heavy-light molecules and tetraquarks states by including next-to- nonleading order (N2LO) factorized perturbative (PT) corrections where we have emphasized the importance of these corrections for giving a meaning of the input heavy quark mass which plays an important role in the analysis though these corrections are small in the MS-scheme This feature (a posteriori) can justify the uses of the MS running masses at LO in some channels [28] if the αns -corrections are small, especially in the ratios of moments used to extract. We include the triple gluon condensate hG3i contributions in the operator product expansion (OPE) We use these QCD results using the LSR sum rules within different stability criteria used successfully in some other channels to extract the masses and couplings of the previous molecules and tetraquarks states assumed to be resonances. ; ð4Þ where mQ is the heavy quark mass, τ is the LSR variable, n 1⁄4 0, 1 is the degree of moments, tc is the threshold of the “QCD continuum” which parametrizes, from the discontinuity of the Feynman diagrams, the spectral function Im ΠHMðt; m2Q; μ2Þ where ΠHMðt; m2Q; μ2Þ is the scalar correlator defined as: ΠHMðq2Þ 1⁄4 d4x e−iqxh0jT OHMðxÞðOHMð0ÞÞ†j0i: ð5Þ
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