Laplace Sum Rules Research Articles

Light scalar quarkonia from Laplace sum rule at NLO

We review our results on light scalar quarkonia (q¯q and four-quark states) from (inverse) QCD Laplace sum rules (LSR) and their ratios R within stability criteria and including higher order perturbative (PT) corrections up to the (estimated) O(αs5). As the Operator Product Expansion (OPE) usually converges for D⩽6−8, we evaluated the QCD spectral functions at Lowest Order (LO) of PT QCD and up to the D=6 dimension vaccum condensates. We request that the optimal results obey the constraint: Pole (Resonance) contribution to the spectral integral is larger than the QCD continuum one which excludes an on-shell mass around (500−600)MeV obtained for values of the QCD continuum threshold tc⩽(1∼1.5)GeV2. Our results for the different assignments of the scalar mesons are compiled in Tables 1 to 3.

QCD parameters and SM-high precisions from e+e−→ Hadrons: Summary

In this talk, I summarize the results obtained recently in Ref. [1] using the PDG 22 compilation of the e+e−→ Hadrons ⊕ the recent CMD3 data for the pion form factor. Using the gluon condensate 〈αsG2〉=(6.49±0.35)×10−2 GeV4 from heavy quark sum rules, the extracted QCD four-quark and dimension eight condensate condensates values are: ραs〈ψ¯ψ〉2=(5.98±0.64)×10−4 GeV6 and d8=(4.3±3.0)×10−2 GeV8 from the ratio R10 of Laplace sum rules to order αs4. Inversely using these estimated values of the condensates, we obtain from R10: 〈αsG2〉=(6.12±0.61)×10−2 GeV4 which leads to the average (6.40±0.30)×10−2 GeV4. Using the lowest τ-like decay moment, the mean result of Fixed Order (FO) and Contour Improved (CI) PT series within the standard OPE is: αs(Mτ)=0.3385(50)(136)syst [resp. 0.3262(37)(78)syst] to order αs4 [resp. αs5] leading to αs(MZ)=0.1207(17)(3) [resp. 0.1193(11)(3)], while the sum of the non-perturbative contribution at Mτ is: δNPV(Mτ)=(2.3±0.2)×10−2. Using the same data, one also obtains the LO hadronic vacuum polarization to the muon and τ anomalous magnetic moments: aμ|l.ohvp=(7036.5±38.9)×10−11,aτ|l.ohvp=(3494.8±24.7)×10−9 which leads to: Δaμ≡aμexp−aμth=(143±42th±22exp)×10−11 indicating about 3σ discrepancy between the SM predictions and experiment. One also finds: α(5)(MZ)|had=(2766.3±4.5)×10−5.

2++ Tensor di-gluonium from Laplace sum rules at NLO

We evaluate the next-to-leading (NLO) corrections to the perturbative (PT) and 〈αsG2〉 condensate and the LO constant term of the 〈G3〉 contributions to the 2++ tensor di-gluonium two-point correlator. Using these results into the inverse Laplace transform sum rules (LSR) moments and their ratio, we estimate the mass and coupling of the lowest ground state. We obtain: MT=3028(429) MeV and the renormalization group invariant (RGI) coupling fˆT=167(40) MeV within a vacuum saturation estimate of the D=8 dimension gluon condensates (kG=1). We study the effect of kG on the result and find: MT=3188(405) MeV and fˆT=164(28) MeV for kG=(3±2). Absolute upper bounds for the mass and coupling from the positivity of the spectral function are also derived. Our result does not favour the pure gluonia/glueball nature of the observed f2(2010,2300,2340) states.

Formula omitted] from QCD Laplace sum rule

We present a review of our recent works on heavy-light exotic states, with new compact integrated expressions of the spectral functions at lowest order (LO) of perturbative (PT) QCD and up to d=6 condensates of the Operator Product Expansion (OPE). Including higher order (HO) PT corrections, which we have estimated by assuming the factorization of the molecule/four-quark spectral functions, we evaluate masses and couplings with the Laplace sum rules (LSR) and finite energy (FESR) QCD spectral sum rules (QSSR) approaches. Our optimal results based on stability criteria are summarized and compared with experimental candidates and some LO QSSR results. We conclude that, our predicted spectra, do not confirm that the X(5568) is a pure molecule or a four-quark state. The masses of the XZ observed states are compatible with (almost) pure JPC=1+±,0++ molecule or/and four-quark states. The SU3 corrections on the meson masses are tiny: ≤10% (respectively ≤3%) for the c (respectively b)-quark channel but can be large for the couplings (≤20%). We expect that the resonance near the D−K+ threshold can be originated from the 0+(D−K+) molecule and/or D−K+ scattering. The X0(2900), XJ(3150) (if J = 0), X1(2900) and the XJ(3350) (if J = 1) can emerge from a minimal mixing model between tetramoles and first radial excitation. The Zc(3900,4025,4040,4430) and Zcs(3983) spectra are well reproduced by the molecules and tetramoles candidates, or radial excitation.

Formula omitted] and TQQ within QCD Laplace sum rules

We present a review of our results for the masses and couplings of the scalar fully heavy four-quarks and TQQq¯q¯′(JP=0±,1±) tetraquarks states from (inverse) QCD Laplace sum rule (LSR), their ratios R and double ratio of sum rules (DRSR) within stability criteria and including Factorized Next-to-Leading Order (FNLO) Perturbative (PT) corrections. As the Operator Product Expansion (OPE) usually converges for d⩽6−8, we evaluated the QCD spectral functions at Lowest Order (LO) of PT QCD and up to 〈G3〉. Our results for the 0++ fully heavy four-quark states may explain the LHCb broad structure around (6.2−6.7)GeV which can be due to ηc‾ηc,χc1‾χc1 and J/ψ‾J/ψ molecules or/and their analogue ScSc,AcAc and VcVc tetraquarks. The peak at (6.8−6.9)GeV can be identified to the χc0‾χc0 molecule or/and the PcPc tetraquark state. Then, combining R and DRSR we focus on the analysis of the four-quark nature of Tccq¯q¯′1± and 0± states. We show that combining R and DRSR can provide more precise results: MTcc1+=3886(6)MeV and MTcc0+=3883(3)MeV. In a similar way, one obtain the improved mass predictions: MTccs¯u¯1+=3931(7)MeV, MTccs¯u¯0+=3983(7)MeV and MTccs¯s¯0+=3993(11)MeV. From our estimates of the masses of the Pseudoscalar and Vector Tccq¯q¯′ states, we observe that the interpolating currents lead to two classes: Class H (Heavy) states with masses around 6GeV and Class L (Light) states around (3.8−4.4)GeV where the pseudoscalar (resp. all vector states) are below the D‾D0,D‾sDs0 (resp. D‾D1,D‾sDs1) open charm thresholds. Finally, we extend the whole study to the bottom sector and confront our results with the ones from different LSR predictions and some other approaches (lattice, potential models, ⋯) in the literature.

QCD parameters and SM-high precisions from e+e−→ Hadrons

Using the PDG 22 compilation of the e+e−→ Hadrons ⊕ the recent CMD3 data for the pion form factor and the value of gluon condensate 〈αsG2〉 from heavy quarkonia, we extract the value of the four-quark condensate: ραs〈ψ¯ψ〉2=(5.98±0.64)×10−4 GeV6 and the dimension eight condensate: d8=(4.3±3.0)×10−2 GeV8 from the ratio R10 of Laplace sum rules to order αs4. We show the inconsistency in using at the same time the standard SVZ value of the gluon and the vacuum saturation of the four-quark condensates. Using the previous values of the four-quark and d8 condensates, we re-extract 〈αsG2〉 from R10 to be: (6.12±0.61)×10−2 GeV4 in perfect agreement with the one from heavy quarkonia. We also use the lowest τ-like decay moment Rτee to extract the value of the QCD coupling. We obtain to O(αs4) the mean of FO (fixed order) and CI (contour improved) PT series within the standard OPE: αs(Mτ)=0.3385(50)(136)syst where the last error is an added conservative systematic from the distance of the mean to the FO and CI central values. Assuming that the αs-coefficients of the PT series grow geometrically as observed in the calculated case, we obtain to O(αs5): αs(Mτ)=0.3262(37)(78)syst. These results lead to: αs(MZ)=0.1207(17)(3) [resp. 0.1193(11)(3)] to order αs4 [resp. αs5]. The corresponding value of the sum of the non-perturbative contribution at Mτ is: δNPV(Mτ)=(2.3±0.2)×10−2. Reciprocally, using αs(Mτ), 〈αsG2〉 and d8 as inputs, we test the stability of the value of the four-quark condensate obtained from the lowest τ-like moment. We complete our analysis by updating our previous determinations of the lowest order hadronic vacuum polarization contributions to the lepton anomalies and to α(MZ2). We obtain in Table 3: aμ|l.ohvp=(7036.5±38.9)×10−11, aτ|l.ohvp=(3494.8±24.7)×10−9 and α(5)(MZ)|had=(2766.3±4.5)×10−5. This new value of aμ leads to: Δaμ≡aμexp−aμth=(142±42th±41exp)×10−11 which reduces the tension between the SM prediction and experiment.

Scrutinizing the light scalar quarkonia from LSR at higher orders

We scrutinize, improve some determinations of the masses and couplings of light scalar quarkonia (q¯q and four-quark states) and present new results for the π+π−,K+K−,… molecules using QCD Laplace Sum Rule (LSR) truncated at the D=6 dimension vacuum condensates. We pay a special attention on the higher order perturbative (PT) corrections up to the (estimated) O(αs5) which improve the quality of the analysis. We request that the optimal results obey the rigorous constraint: Resonance ≥ QCD continuum contributions (RP/C≥1) in the LSR which excludes a Breit-Wigner/on-shell (not to be confused with a complex pole) scalar meson mass around (500-600) MeV obtained for values [tc≤(1∼1.5) GeV2] of the QCD continuum threshold. Mass-splittings due to SU(3) breakings are small. We discuss the different assignments of the observed scalar mesons below 2 GeV in the Conclusions where the I=0 states are compared with the scalar gluonia. The results are compiled in Tables 3 to 6. We hope that the systematic analysis done in this paper helps to clarify the complex spectra of the light scalar mesons.

Borel–Laplace sum rules with τ decay data, using OPE with improved anomalous dimensions

We perform numerical analysis of double-pinched Borel–Laplace QCD sum rules for the strangeless semihadronic τ-decay data. The D = 0 contribution to the theoretical contour integral in the sum rules is evaluated by the (truncated) Fixed Order perturbation theory method (FO) and by the Principal Value (PV) of the Borel integration. We use for the full Adler function the operator product expansion (OPE) with the terms ∼〈O D 〉 of dimension D = 2n where 2 ≤ n ≤ 5 for the (V+A)-channel, and 2 ≤ n ≤ 7 for the V-channel data. In our previous works Ayala, Cvetič and Teca (2021, Eur. Phys. J. C 81 930), Ayala, Cvetič and Teca (2022, Eur. Phys. J. C 82 362), only the (V+A)-channel data was analysed. In this work, the analysis of a new set of V-channel data is performed as well. Further, a renormalon-motivated construction of the D = 0 part of the Adler function is improved in the u = 3 infrared renormalon sector, by involving the recently known information on the two principal noninteger values of the effective leading-order anomalous dimensions. Additionally, the OPE of the Adler function has now the D = 6 contribution with the principal anomalous dimension (), and terms of higher dimension (with zero anomalous dimension). Cross-checks of the obtained extracted values of α s and of the condensates were performed by reproduction of the (central) experimental values of several double-pinched momenta a (2,n). The averaged final extracted values of the () coupling are: , corresponding to .

Pseudoscalar and vector [formula omitted] spectra and couplings from LSR at NLO

We present systematic and improved estimates of the masses and couplings of the Pseudoscalar (0−) and Vector (1−)TQQq¯q¯′ states (Q≡c,b;q,q′≡u,d,s) using (inverse) QCD Laplace sum rules (LSR) and their ratios R within τ and tc-stabilities criteria complemented by the (rigorous) condition: RP/C≡ Pole (Resonance) over the QCD continuum contributions ≥1. Next-to-leading order (NLO) factorized perturbative (PT) QCD corrections are included for giving a meaning on the choice of the used running MS‾ heavy quark mass, while the operator product expansion (OPE) is truncated at the under-controlled d=6 dimension condensates. Our results are compiled in Tables and compared with some LO existing ones. We observe that the interpolating currents lead to two classes: Class H (Heavy) states with masses around 6 (resp. 13) GeV for charm (resp. bottom) channels. Class L (Light) states Tccq¯q¯′(3.8∼4.4) where the pseudoscalar (resp. all vector states) are below the open charm thresholds and Tbbq¯q¯′(≃10.4) where all of them are below the open beauty thresholds. Mass-splittings due to SU3 breakings are tiny (≤ 50 MeV). Though more accessible experimentally, Class L states have weaker couplings to the currents than the Class H ones and may be difficult to observe. The mass-splittings between the 1st radial excitation and the ground state of TQQq¯d¯ (q≡,u,d) are about 2 GeV which are (almost) heavy flavor and current-type independent while their couplings are large signaling new dynamics of these exotic states. Quark masses behaviors of the masses and couplings based on empirical observation are discussed. The eventual findings of the Tccu¯d¯(6.3) pseudoscalar ground state which may not be obscured by the Class L 1st radial excitations can be an alternative way to test the vacuum saturation violation of the four-quark d=6 condensates.

1+ XTZ States within QCD Sum Rules

We present improved estimates of the couplings, masses and mass ratios of the XQ, ZQ and TQQq¯q¯′ states (Q≡c,b; q,q′≡u,d,s) using (inverse) QCD Laplace sum rules (LSR), their ratios R and double ratios (DRSR), within stability criteria. We conclude that the observed Xc(3872) and Zc(3900) are tetramoles states (superposition of quasi-degenerated molecule and tetraquark states having similar couplings to the currents) with the predicted masses: MTXc=3876(44)MeV and MTZc=3900(42)MeV. We also do an extensive analysis of the four-quark nature of different TQQq¯q¯′ axial-vector states. Then, combining R and DRSR, we reanalyze the observed state Xc(3872) and we obtain a precise prediction of MTcc1+=3886(6)MeV. Extending to the beauty sector, we find the results: MTZb=10579(99)MeV and MXb=10545(131)MeV. Finally, we confront our combined LSR ⊕ DRSR results with the ones from some other approaches (lattices and quark models).

0+ XTZ states from QCD spectral sum rules

We review our results in [1] for the masses and couplings of Tccqq′ (JP=0+) states from (inverse) QCD Laplace sum rule (LSR), their ratios R and double ratio of sum rules (DRSR) within stability criteria and including Factorized Next-to-Leading Order (FNLO) Perturbative (PT) corrections and Lowest Order (LO) QCD condensates up to 〈G3〉. We show that combining R and DRSR can provide more precise results. Calibrated to the observed Xc(3872) and Tcc1+(3875), R combined with DRSR lead to a more precise prediction of MTcc0+=3883(3)MeV. In a similar way, calibrated to the new prediction of Tcc0+R⊕DRSR lead to the improved mass predictions: MTccs¯u¯0+=3927(6)MeV and MTccs¯s¯0+=3993(11)MeV. We extend our analysis to the bottom sector and compare our results with the ones from different LSR predictions and some other determinations (lattice, quark and potential models, …) in the literature.

Heavy-light hybrid mesons with different spin-parities

The spectroscopic parameters of the heavy-light hybrid mesons with different spin-parities and different quark contents are investigated in the framework of the QCD sum rule method. The mass and current coupling of these states are calculated by taking into account the quark, gluon and mixed vacuum condensates up to dimension 10. The obtained results are compared with the existing QCD Laplace sum rule predictions. The results of mass and current coupling for all the considered channels are obtained to be stable and reliable. Our results can be useful for future experimental searches as well as theoretical studies of different parameters related to the hybrid mesons and their decays and interactions with other particles.

DK and BK-like spectra from Laplace sum rule at NLO

Encouraged by the agreement, with the recent LHCb data on the D−K+ invariant mass from B→D+D−K+ decay, of our results for the masses of the 0+ and 1− open charm (c¯d¯)(us) tetraquarks and molecules states from QCD spectral sum rules within stability criteria, which we review here, we extend our analysis to the b-quark channel. We find, in the 0+ case the lowest mass MBK=5195(15)MeV with fBK=8.3(2.4)keV and three (almost) degenerate states having respectively the masses MSS=5702(60)MeV, MAA=5661(75)MeV and MB⁎K⁎=5720(71)MeV and couplings fSS=22.2(2.3)keV, fAA=30.1(3.1)keV and fB⁎K⁎=26.5(2.8)keV, from which we can associate a scalar tetramole with MTM0=5694(69)MeV and fTM0=26.5(2.7)keV. In the spin 1 case, we find four (almost) degenerate states associated with a tetramole having MTM1=5700(81)MeV and fTM1=16.2(2.6)keV. For the first radial excitation of the BK molecule, we have M(BK)1=6265(146)MeV and f(BK)1=22.8(3.2)keV. For the remaining states, we associate a scalar and vector tetramoles having respectively M(TM0)1=7439(314)MeV, f(TM0)1=74.7(8.4)keV and M(TM1)1=7544(345)MeV, f(TM1)1=33.0(6.7)keV.

Zc,b-like states from QCD Laplace sum rules at NLO

We review our results on Zc-like states [1] which we complete with the ones on Zb-like states by using relativistic QCD Laplace Sum Rules (LSR) within stability criteria and including Factorized Next-to-Leading Order (FNLO) Perturbative (PT) corrections and Lowest Order (LO) QCD condensates up to 〈G3〉. We emphasize the importance of PT radiative corrections for heavy quark sum rules in order to justify the use of the running heavy quark mass in the analysis. Our estimates are compiled in Tables 3 and 4. From our results, the observed Zcs(3983) state are good candidate for being Tcs tetramole (superposition of nearly degenerated molecules and tetraquark states having the same quatum numbers JPC and with almost the same couplings to the currents). The Zcs bump around 4100 MeV can be interpreted as a combination of D0⁎Ds1 and Ds0⁎D1 molecules. The physical states Zcs(4000) and Zcs(4220) found by LHCb are too low to be considered as the first radial excitations of Zcs(3983). For the future Zb, Zbs and Zbss, we suggest to scan the region around (10.3 ∼ 10.9) GeV while the 1st radial excitations are about 2.4 GeV above the ground states.

Next-to-leading order (NLO) perturbative effects in QCD sum-rule analyses of light tetraquark systems: A case study in the scalar-isoscalar channel

QCD sum-rule mass predictions for tetraquark states provide insights on the interpretations and internal structure of experimentally-observed exotic mesons. However, the overwhelming majority of tetraquark QCD sum-rule analyses have been performed at leading order (LO), which raises questions about the underlying theoretical uncertainties from higher-loop corrections. The impact of next-to-leading order (NLO) perturbative effects are systematically examined in scalar (JPC=0++) isoscalar light-quark tetraquark systems where comprehensive LO sum-rule analyses have been performed and NLO perturbative corrections to the correlators have previously been calculated. Using the scalar-isoscalar state as a detailed case study to illustrate the differences between LO and NLO analyses, it is shown that NLO effects in individual Laplace sum-rules are numerically significant and have an important role in improving the reliability of the sum-rule analyses by widening the Borel window. However, ratios of sum-rules are found to be less sensitive to NLO effects with the additional advantage of cancelling the anomalous dimension that emerges from the NLO corrections. NLO mass predictions based on these sum-rule ratios are thus remarkably robust despite the slow perturbative convergence of the underlying correlator. The mass predictions 0.52GeV<mσ<0.77GeV for the lightest scalar-isoscalar σ state are in good agreement with the four-quark interpretation of the f0(500), and the relative coupling strengths of the f0(980) and f0(500) to the tetraquark current agree with the pattern found in chiral Lagrangian analyses. Effects of the σ resonance width are studied for different models, including resonance shapes inspired by chiral Lagrangians.

Tests of the Zc -like Laplace sum rule results using finite energy sum rule at NLO

In this note, we use local duality Finite Energy Sum Rule (FESR) to test the validity of the Laplace sum rules (LSR) results truncated at the dimension-six condensates for the estimates of the masses and couplings of the Z_c-like ground states in Ref.1 by taking the example of the D^*D molecule configuration. We confirm the existence of an eventual (D^*D)_1 radial excitation with a mass around 5700 MeV and coupling of 197(25) keV to the current which may mask the eventual Z_c(4430) radial excitation candidate (named (D^*D)_0 in Ref.1) having a relatively small coupling f_{(D^*D)_0}=46(56) keV. We add more explanations on the estimates in Ref.1 from LSR and comment the results in Ref.2.

Improved XTZ masses and mass ratios from Laplace sum rules at NLO

We present improved estimates of the couplings, masses and mass ratios of the ZQ,XQ and TQQq¯q¯′ states (Q≡c,b;q,q′≡u,d,s) using (inverse) QCD Laplace sum rules (LSR), their ratios R and double ratios DRSR within stability criteria, where the NLO factorized PT QCD corrections are included which is important for giving a meaning on the running MS‾ heavy quark mass used in the analysis. We show that combined R and DRSR can provide more precise results. In the 1st part of the paper, we conclude that the observed Xc(3872) and Zc(3900) are tetramoles states (superposition of quasi-degenerated molecule and a tetraquark states having (almost) the same coupling to the currents) with the predicted masses: MTXc=3876(44)MeV and MTZc=3900(42)MeV. In the 2nd part, we focus on the analysis of the four-quark nature of different TQQq¯q¯′1+ and 0+ states within the 3¯c3c interpolating currents. The final results from R and R⊕ DRSR are summarized in Table 7. Combined R and DRSR calibrated to the observed Xc(3872) lead to a precise prediction of e.g. MTcc1+=3886(6)MeV. In a similar way, the DRSR for the MTcc0+/MTcc1+ calibrated to MTcc1+ gives MTcc0+=3883(3)MeV. The SU3 breaking ratios MTccs¯s¯0+/MTcc0+ lead to the improved mass predictions: MTccs¯s¯0+=3988(12)MeV. In the 3rd part, the analysis is extended to the beauty mesons, where we find the tetramole masses: MTZb=10579(99)MeV and MXb=10545(131)MeV. We also observe that the Tbbq¯q¯′1+,0+ (q,q′≡u,d,s) states are (almost) stable (within the errors) against strong interactions. In the 4th part, we (critically) review and correct some recent LSR estimates of the TQQq¯q¯′1+,0+ masses. Our combined LSR ⊕ DRSR results are confronted with the ones from some other approaches (lattices and quark models) in Fig. 26.

Using improved operator product expansion in Borel–Laplace sum rules with ALEPH tau decay data, and determination of pQCD coupling

We use improved truncated operator product expansion (OPE) for the Adler function, involving two types of terms with dimension D=6, in the double-pinched Borel–Laplace sum rules and finite energy sum rules for the V + A channel strangeless semihadronic tau decays. The generation of the higher order perturbative QCD terms of the D=0 part of the Adler function is carried out using a renormalon-motivated ansatz incorporating the leading UV renormalon and the first two leading IR renormalons. The trunacted D=0 part of the sum rules is evaluated by two variants of the fixed-order perturbation theory (FO), by principal value of the Borel resummation (PV), and by contour-improved perturbation theory (CI). For the experimental V + A channel spectral function we use the ALEPH tau -decay data. We point out that the truncated FO and PV evaluation methods account correctly for the renormalon structure of the sum rules, while this is not the case for the truncated CI evaluation. We extract the value of the {overline{mathrm{MS}}} coupling alpha _s(m_{tau }^2) = 0.3235^{+0.0138}_{-0.0126} [alpha _s(M_Z^2)=0.1191 pm 0.0016] for the average of the two FO methods and the PV method, which we consider as our main result. If we included in the average also CI extraction, the value would be alpha _s(m_{tau }^2) = 0.3299^{+0.0232}_{-0.0225} [alpha _s(M_Z^2)=0.1199^{+0.0026}_{-0.0028}]. This work is an extension and improvement of our previous work (Ayala et al. in Eur Phys J. C 81(10): 930, 2021, arXiv:2105.00356 [hep-ph]) where we used for the truncated OPE a more naive (and widely used) form and where the extracted values for alpha _s(M_Z^2) were somewhat lower.

The New Charm-Strange Resonances in the D−K+ Channel

We evaluate the masses and decay constants of the 0+ and 1− open-charm (c¯d¯)(us) tetraquarks and molecular states from QCD spectral sum rules (QSSR) by using QCD Laplace sum rule (LSR). This method takes into account the 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+ → D+(D−K+) decays. We expect that the resonance near the D−K+ threshold can be originated from the 0++(D−K+) molecule and/or D−K+ scattering. The X0(2900) scalar state and the resonance XJ(3150) (if J = 0) can emerge from a minimal mixing model, with a tiny mixing angle θ0 ≃ (5.2 ± 1.9)0, between a scalar Tetramole (TM0) (superposition of nearly degenerated hypothetical molecules and compact tetraquarks states with the same quantum numbers), having a mass MTM0=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 X1(2900) and the XJ(3350) (if J = 1) could be a mixture between the vector Tetramole (TM1), with a mass MTM1=2656(20)MeV, and its first radial excitation having a mass MTM1=4592(141)MeV with an angle θ0 ≃ (9.1 ± 0.6)0. A (non)-confirmation of these statements requires experimental findings of the quantum numbers of the resonances at 3150 and 3350 MeV.

Doubly hidden 0++ molecules and tetraquarks states from QCD at NLO

Motivated by the LHCb-group discovery of exotic hadrons in the range (6.2 ∼ 6.9) GeV, we present new results for the masses and couplings of 0++ fully heavy (Q‾Q)(QQ‾) molecules and (QQ)(QQ‾) tetraquaks states from relativistic QCD Laplace Sum Rule (LSR) within stability criteria where Next-to-Leading Order (NLO) Factorized (F) Perturbative (PT) corrections is included. As the Operator Product Expansion (OPE) usually converges for d ⩽ 6–8, we evaluated the QCD spectral functions at Lowest Order (LO) of PT QCD and up to 〈G3〉. We also emphasize the importance of PT radiative corrections for heavy quark sum rules in order to justify the use of the running heavy quark mass value in the analysis. We compare our predictions in Table 3 with the ones from ratio of Moments (MOM). The broad structure arround (6.2 ∼ 6.9) GeV can be described by the η‾cηc, J/ψ‾J/ψ and χc1‾χc1 molecules or/and Sc‾Sc, Ac‾Ac and Vc‾Vc tetraquarks lowest mass ground states. The narrow structure at (6.8 ∼ 6.9) GeV if it is a 0++ state can be a χc0‾χc0 molecules or/and its analogue Pc‾Pc tetraquark. The χc1‾χc1 predicted mass is found to be below the χc1χc1 threshold while for the beauty states, all of the estimated masses are above the ηbηb and ϒ(1S)ϒ(1S) threshold.