We present new compact integrated expressions of QCD spectral functions of heavylight molecules and four-quark XY Z-like states at lowest order (LO) of perturbative (PT) QCD and up to d = 8 condensates of the Operator Product Expansion (OPE). Then, by including up to next-to-next leading order (N2LO) PT QCD corrections, which we have estimated by assuming the factorization of the four-quark spectral functions, we improve previous LO results from QCD spectral sum rules (QSSR), on the XY Z-like masses and decay constants which suffer from the ill-defined heavy quark mass. PT N3LO corrections are estimated using a geometric growth of the PT series and are included in the systematic errors. Our optimal results based on stability criteria are summarized in Tables 11 to 14 and compared, in Section 10, with experimental candidates and some LO QSSR results. We conclude that the masses of the XZ observed states are compatible with (almost) pure J P C = 1 +± , 0 ++ molecule or/and four-quark states. The ones of the 1 −± , 0 −± molecule / four-quark states are about 1.5 GeV above the Y c,b mesons experimental candidates and hadronic thresholds. We also find that the couplings of these exotics to the associated interpolating currents are weaker than that of ordinary D, B mesons (f DD ≈ 10 −3 f D ) and may behave numerically as 1/m 3/2 b (resp. 1/m b ) for the 1 + , 0 + (resp. 1 − , 0 − ) states which can stimulate further theoretical studies of these decay constants.

We use QCD sum rules to test the nature of the recently observed mesons Y (4260), Y (4350) and Y (4660), assumed to be exotic four-quark (ccqq) or (ccss) states with J P C = 1 −− . We work at leading order in αs, consider the contributions of higher dimension condensates and keep terms which are linear in the strange quark mass ms. We find for the (ccss) state a mass mY = (4.65±0.10) GeV which is compatible with the experimental candidate Y (4660), while for the (ccqq) state we find a mass mY = (4.49 ± 0.11) GeV, which is higger than the mass of the experimental candidate Y (4350). With the tetraquark structure we are working we can not explain the Y (4260) as a tetraquark state. We also consider molecular Ds0D * s and D0D * states. For the Ds0D * s molecular state we get m D s0D * s = (4.42 ± 0.10) GeV which is consistent, considering the errors, with the mass of the meson Y (4350) and for the D0D * molecular state we get m D 0D * = (4.27 ± 0.10) GeV in excelent agreement with the mass of the meson Y (4260).

In the past decade, due to the experimental observation of many charmonium-like states, there has been a revival of hadron spectroscopy. In particular, the experimental observation of charged charmonium-like, Z c states, and bottomonium-like, Z b states, represents a challenge since they can not be accommodated within the naive quark model. These charged states are good candidates of either tetraquark or molecular states and their observation motivated a vigorous theoretical activity. This is a rapidly evolving field with enormous amount of new experimental information. In this work, we review the current experimental progress and investigate various theoretical interpretations of these candidates of the multiquark states. The present review is written from the perspective of the QCD sum rules approach, where we present the main steps of concrete calculations and compare the results with other approaches and with experimental data.

We scrutinize recent QCD spectral sum rules (QSSR) results to lowest order (LO) predicting the masses of the BK molecule and (su)(bd) four-quark states. We improve these results by adding NLO and N2LO corrections to the PT contributions giving a more precise meaning on the b-quark mass definition used in the analysis. We extract our optimal predictions using Laplace sum rule (LSR) within the standard stability criteria versus the changes of the external free parameters (τ -sum rule variable, tc continuum threshold and subtraction constant µ). The smallness of the higher order PT corrections justifies (a posteriori) the LO order results ⊕ the uses of the ambiguous heavy quark mass to that order. However, our predicted spectra in the range (5173 ∼ 5226) MeV, summarized in Table 7, for exotic hadrons built with four different flavours (buds), do not support some previous interpretations of the D0 candidate , 1 X(5568), as a pure molecule or a four-quark state. If experimentally confirmed, it could result from their mixing with an angle: sin2θ ≈ 0.15. One can also scan the region (2327 ∼ 2444) MeV (where the D * s0 (2317) might be a good candidate) and the one (5173 ∼ 5226) MeV for detecting these (cuds) and (buds) unmixed exotic hadrons (if any) via, eventually, their radiative or π + hadrons decays.

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 ( QQ)(Q 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 G 3 gluon condensate corrections. We also critically revisit and improve existing Lowest Order (LO) QCD spectral sum rules (QSSR) estimates of the ( Q 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 αsG 2 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 η c ηc, J/ψJ/ψ and χ c1 χc1 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 χ c0 χc0 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 η b η b and Υ(1S)Υ(1S) thresholds.