1-- and 0++ heavy four-quark and molecule states in QCD

We estimate the masses of the 1^{--} heavy four-quark and molecule states by combining exponential Laplace (LSR) and finite energy (FESR) sum rules known perturbatively to lowest order (LO) in alpha_s but including non-perturbative terms up to the complete dimension-six condensate contributions. This approach allows to fix more precisely the value of the QCD continuum threshold (often taken ad hoc) at which the optimal result is extracted. We use double ratio of sum rules (DRSR) for determining the SU(3) breakings terms. We also study the effects of the heavy quark mass definitions on these LO results. The SU(3) mass-splittings of about (50 - 110) MeV and the ones of about (250 - 300) MeV between the lowest ground states and their 1st radial excitations are (almost) heavy-flavour independent. The mass predictions summarized in Table 4 are compared with the ones in the literature (when available) and with the three Y_c(4260,~4360,~4660) and Y_b(10890) 1^{--} experimental candidates. We conclude (to this order approximation) that the lowest observed state cannot be a pure 1^{--} four-quark nor a pure molecule but may result from their mixings. We extend the above analyzes to the 0^{++} four-quark and molecule states which are about (0.5-1) GeV heavier than the corresponding 1^{--} states, while the splittings between the 0^{++} lowest ground state and the 1st radial excitation is about (300-500) MeV. We complete the analysis by estimating the decay constants of the 1^{--} and 0^{++} four-quark states which are tiny and which exhibit a 1/M_Q behaviour. Our predictions can be further tested using some alternative non-perturbative approaches or/and at LHCb and some other hadron factories.


Introduction and a short review on the 1 ++ channel
A large amount of exotic hadrons which differ from the "standard" cc chamonium and bb bottomium radial excitation states have been recently discovered in B-factories through J/ψπ + π − and Υπ + π − processes and have stimulated different theoretical interpretations.Most of them have been assigned as four-quarks and/or molecule states [3].In previous papers [1,2], some of us have studied, using exponential QCD spectral sum rules (QSSR) [4] 2 and the double ratio of sum rules (DRSR) [7] 3 , the nature of the X(3872) 1 ++ states found by Belle [11] and confirmed by Babar [12], CDF [13] and D0 [14].If it is a (cq)(cq) four-quark or D − D * molecule state, one finds for m c = 1.23 GeV [1]  4 : corresponding to a t c -value common solution of the exponential Laplace (LSR) and Finite Energy (FESR) sum rules: while in the b-meson channel, using m b = 4.26 GeV, one finds [1]: MeV with √ t c = (10.4±0.02)GeV, (3) where a similar result has been found in [15] using another choice of interpolating current.However, in the case of the X c (3872), the previous two configurations are not favoured by its narrow hadronic width (≤ 2.3 MeV), which has lead some of us to propose that it could be, instead, a λ − J/ψ-type molecule [2] described by the current: where λ a is the colour matrix , while g and Λ are coupling and scale associated to an effective Van Der Vaals force.In this case, the narrow width of the X c is mainly due to the extra-gluon exchange which gives a suppression of the order α 2 s compared to the two former configurations, if one evaluates this width using vertex sum rules.The corresponding mass is slightly lower than the one in Eq. (1) [2]: which is much lower than the ones of ordinary charmonium and bottomium states: (7) and suggests a completely different dynamics for these exotic states.Comparing the previous results with the observed Z b (10610) and Z b (10650) states whose quantum numbers have been assigned to be 1 ++ , one can conclude that these observed states are heavier than the 1st radial excitation of the X b (10.14) expected from QSSR to lowest order in α s [1].

QCD Analysis of the 1 −− and 0 ++ channels
In the following, we extend the previous analysis to the case of the 1 −− and 0 ++ channels and improve some existing analysis from QCD (spectral) sum rules in the 1 −− channel [18,19].The results will be compared with the experimental 1 −− candidate states: seen by Babar [16] and Belle [17,23] and which decay into J/ψπ + π − and Υπ + π − around the Υ(5S ) mass.These states cannot be identified with standard cc charmonium and bb bottomium radial excitations and have been assigned in the literature to be four-quark or molecule states or some threshold effects.

• QCD input parameters
The QCD parameters which shall appear in the following analysis will be the charm and bottom quark masses m c,b , the light quark masses m d,s , the light quark condensates qq and ss , the gluon condensates g 2 G 2 ≡ g 2 G a µν G µν a and g 3 G 3 ≡ g 3 f abc G a µν G b νρ G c ρµ , the mixed condensate qgσGq ≡ qgσ µν (λ a /2)G a µν q and the four-quark condensate ρ qq 2 , where ρ indicates the violation of the four-quark vacuum saturation.Their values are given in Table 1 and we shall work with the running light quark parameters: where β 1 = −(1/2)(11 − 2n/3) is the first coefficient of the β function for n flavours; ms and μq are renormalization group invariant light quark mass and condensate [25,26].

• Interpolating currents
We assume that the Y state is described either by the lowest dimension (without derivative terms) four-quark and molecule Ds D * s vector currents J µ given in Tables 2 and 3. Unlike the case of baryons where both positive and parity states can couple to the same operator [24], the situation is simpler here as the vector and axial-vector currents have a well-defined quantum numbers to which are associated the 1 −− (resp. 1 ++ ) states Table 1: QCD input parameters.For the heavy quark masses, we use the range spanned by the running MS mass m Q (M Q ) and the on-shell mass from QCD (spectral) sum rules compiled in pages 602 and 603 of the book in [5] and recently obtained in Ref. [28].The values of Λ and μq have been obtained from α s (M τ ) = 0.325 (8) [29] and from the running masses: (m u + m d )(2) = 7.9(3) MeV [31].The original errors have been multiplied by 2 for a conservative estimate of the errors.[29,35,37] for the transverse part and the 0 ++ (resp.0 −− ) states for the longitudinal part.In the case of four-quark currents, we can have two-types of lowest derivative vector operators which can mix through the mixing parameter b5 .Another possible mixing can occur through the renormalization of operators [26,27] though this type of mixing will only induce an overall effect due to the anomalous dimension which will be relevant at higher order in α s but will disappear in the ratio of sum rules used in this paper.

Parameters
For the molecule current, we choose the product of local bilinear current which has the quantum number of the corresponding meson state.In this sense, we have only an unique interpolating current.Observed states can be a mixing of different states associated to each choice of operators and their selection can only be done through the analysis of their decays [2] but this is beyond the scope of this paper.

• The two-point function in QCD
The two-point functions of the Y Q (Q ≡ c, b) (assumed to be a 1 −− vector meson) is defined as: where J µ are the interpolating vector currents given Tables 2 and 3. We assume that the Y state is described either by the lowest dimension (without derivative terms) four-quark and molecule Ds D * s currents given in Tables 2 and 3.The two invariants, Π (1) and Π (0) , appearing in Eq. ( 10) are independent and have respectively the quantum numbers of the spin 1 and 0 mesons.We can extract Π (1)  Q and Π (0) (q 2 ) or the corresponding spectral functions from the complete expression of Π µν Q (q) by applying respectively to it the projectors: Due to its analyticity, the correlation function, Π (1,0) (q 2 ) in Eq. ( 10), obeys the dispersion relation: where Im Π (1,0) (s) are the spectral functions.The QCD expressions of these spectral functions are given in Tables 2 and 3. 1/q 2 terms discussed in [44,45],which are dual to higher order terms of the QCD series will not be included here as we work to leading order.

1 −− four-quark state mass Y Qq from QSSR
In the following, we shall estimate the mass of the 1 −− fourquark state (Qq)(Qq) (Q ≡ c, b and q ≡ u, d quarks), hereafter denoted by Y Qd .In so doing, we shall use the ratios of the Laplace (exponential) sum rule: and of FESR: t< dt t n−1 1 π ImΠ (1) (t) where t < is the hadronic (quark) threshold.Within the usual duality ansatz "one resonance" + θ(t − t c ) × QCD continuum parametrization of the spectral function, the previous ratios of sum rules give: For a discussion more closed to the existing literature which we shall test the reliability in the following, we start to work with the current corresponding to b = 0. We shall discuss the more general choice of current when b is a free parameter at the end of this section.
• The Y cd mass from LSR and FESR for the case b=0 Using the QCD inputs in Table 1, we show the τ-behaviour of M Y cd from R LS R cd in Fig. 1a for m c = 1.26 GeV and for different values of t c .One can notice from Fig. 1a that the τ-stability is obtained from √ t c ≥ 5.1 GeV, while the t c -stability is reached for √ t c = 7 GeV.The most conservative prediction from the LSR is obtained in this range of t c -values for m c = 1.26 GeV and gives in units of GeV: We compare in Fig. 1b), the t c -behaviour of the LSR results obtained at the τ-stability points with the ones from R FES R cd for the charm quark mass m c =1.23 GeV (running) and 1.47 GeV (on-shell).One can deduce the common solution in units of GeV: In order to fix the values of M Y cd obtained at this lowest order PT calculations, we can also refer to the predictions of the J/ψ mass using the LSR at the same lowest order PT calculations and including the condensate contributions up to dimensionsix.We observe that the on-shell c-quark mass value tends to overestimate M J/ψ [2,28].The same feature happens for the evaluation of the X(1 ++ ) four-quark state mass [1].Though this observation may not be rigorous as the strength of the radiative corrections is channel dependent, we are tempted to take as a final result in this paper the prediction obtained by using the running mass m c (m c ) = 1262 (17) MeV within which it is known, from different examples in the literature, that the PT series converge faster [28] 6 .Including different sources of errors, we deduce in MeV7 : Using the fact that the 1st FESR moment gives a correlation between the mass of the lowest ground state and the onset of continuum threshold t c , where its value coincide approximately with the value of the 1st radial excitation mass (see e.g.ref. [39] and some other examples in [5]), we shall approximately identify its value with the one of the radial excitation.In order to take into account the systematics of the approach and some eventual small local duality violation advocated by [46] which can only be detectable in a high-precision analysis like the extraction of α s from τ-decay [29,47], we have allowed t c to move around this intersection point.Assuming that the mass of the radial excitation is approximately √ t c , one can deduce the mass-splitting: which is similar to the one obtained for the X(1 ++ ) four-quark state [1].This splitting is much lower than the one intuitively used in the current literature: for fixing the arbitrary value of t c entering in different Borel (exponential) sum rules of the four-quark and molecule states.This difference may signal some new dynamics for the exotic states compared with the usual cc charmonium states and need to be tested from some other approaches such as potential models, heavy quark symmetry, AdS/QCD and lattice calculations.where the lower (resp.higher) values of t c correspond to the beginning of τ (resp.t c )-stability.We compare in Fig. 2b), the t c -behaviour of the LSR results obtained at the τ-stability points with the ones from R FES R bd for the b quark mass m b =4.17 GeV (running) and 4.70 GeV (on-shell).One can deduce the common solution in units of GeV: .57 (7) and m b = 4.17 From the previous result, one can deduce the approximate value of the mass-splitting between the 1st radial excitation and the lowest mass ground state: which are (almost) heavy-flavour independent and also smaller than the one of the bottomium splitting: • Effect of the current mixing b on the mass In the following, we shall let the current mixing parameter b defined in Table 2 free and study its effect on the results obtained in Eqs.(18) and (22).In so doing, we fix the values of τ around the τ-stability point and t c around the intersection point of the LSR and FESR.The results of the analysis are shown in Fig. 3.We notice that the results are optimal at the value b = 0 which a posteriori justifies the results obtained previously for b = 0. the same as a) but for M Y bd and for m b = 4.17 GeV.

• Effect of the current mixing b on the decay constant f Y Qd
For completing the analysis of the effect of b, we also study the decay constant f Y Q d defined as: We show the analysis in Fig. 4 giving M Y Qd and the corresponding t c obtained above.One can deduce the optimal values at b = 0: which are much smaller than MeV [48].On can also note that the decay constant decreases like 1/M Q which can be tested in HQET or/and lattice QCD.
We show the τ-behaviour of r c sd and r b sd respectively in Figs.5a and 5b for m c = 1.26 GeV and m b = 4.17 GeV for different values of t c .We show, in Fig. 5c and Fig. 5d, the t c -behaviour of the stabilities or inflexion points for two different values (running and on-shell) of the quark masses.One can see in these figures that the DRSR is very stable versus the t c variations in the case of the running heavy quark masses.We deduce the corresponding DRSR:  Y Qd in Eqs. ( 18) and ( 22) and the values of the S U(3) breaking ratio in Eq. ( 28), we can deduce the mass of the Y Qs state in MeV: leading to the S U(3) mass-splitting: which is also (almost) heavy-flavour independent.
We   = 11302 (30) , Using the previous results in Eq. ( 32), one obtains in MeV : corresponding to a S U(3) mass-splitting: These results for M DD * are in the upper part of the range given in [18] due both to the smaller values of m c = 1.23 GeV and √ t c =5.5 GeV used in that paper.Though the DD * molecule mass is above the DD * threshold which is similar to the e.g. the case of the ππ continuum and ρ-meson resonance in e + e − to the I=1 hadrons channel, one expects that at the τ-stability point or inside the sum rule window, where the QCD continuum contribution is minimum while the OPE is still convergent, the lowest ground state dominates the sum rule.
• The J/ψS 2 and ΥS 2 molecules Combining LSR and FESR, we consider the mass of the J/ψS   ūu + dd is a scalar meson 9 .In so doing, we work with the LO QCD expression obtained in [19].We show the results versus the LSR variable τ in Fig. 8a,b.The t c -behaviour of different τ-extremas is given in Figs.8c,d from which we can deduce for the running quark masses for √ t c = 5.30(2) and 10.23(3) GeV in units of MeV: The splitting (in units of MeV) with the first radial excitation approximately given by √ t c is: In the same way, we show in Figs. 9 the τ and t c behaviours of the S U(3) breaking ratios, from which, we can deduce: 9 The low-mass π + π − invariant mass due to the σ meson is expected to result mainly from its gluon rather than from its quark component [49,50] such that an eventual quark-gluon hybrid meson nature of the Y c is also possible.
where S 3 ≡ ss is a scalar meson.Then, we obtain in MeV: corresponding to the S U(3) mass-splittings: The mass-splittings in Eq. ( 39) are comparable with the ones obtained previously.Doing the same exercise for the octet current, we deduce the results in Table 4 where the molecule associated to the octet current is 100 (resp.250) MeV above the one of the singlet current for J/ψ (resp Υ) contrary to the 1 ++ case discussed in [2].The ratio of S U(3) breakings are respectively 1.022(5) and 1.010(2) in the c and b channels which are comparable with the ones in Eq. 37. When comparing our results with the ones in Ref. [19], we notice that the low central value of M J/ψS 2 obtained there (which we reproduce) corresponds to a smaller value of m c = 1.

0 ++ four-quark and molecule masses from QSSR
In the following, we extend the previous analysis to the case of the 0 ++ mesons.
• Y 0 Qd mass and decay constant from LSR and FESR We do the analysis of the Y 0 cd and Y 0 bd masses using LSR and FESR.We show the results in Figs 10 for the current mixing parameter b = 0 from which we deduce in MeV, for the running quark masses, and respectively for √ t c = 6.5 and 13.0 GeV where LSR and FESR match: One can notice that the splittings between the lowest ground state and the 1st radial excitation approximately given by which are comparable with the ones of the spin 1 case in Eq. ( 26).
• S U(3) breaking for M 0 Y Qs from DRSR We show in Figs. 13 the τ and t c behaviours of the S U(3) breaking ratios for the current mixing parameter b = 0: from which we deduce: leading (in units of MeV) to: and the S U(3) mass-splittings: One can notice that the splittings between the lowest ground state and the 1st radial excitation approximately given by √ t c is in MeV: ) which, like in the case of the 1 −− states are smaller than the ones of the J/ψ and Υ, and almost heavy-flavour independent.extremas.Therefore, we deduce:

Using the previous values of M D d D d and M B d B d , we deduce in
MeV: which corresponds to a S U(3) splitting:

Summary and conclusions
We have studied the spectra of the 1 −− and 0 ++ four-quarks and molecules states by combining Laplace (LSR) and finite energy (FESR) sum rules.The S U(3) mass-splittings have been obtained using double ratios of sum rules (DRSR).We consider the present results as improvement of the existing ones in the literature extracted only from LSR where the criterion for   fixing the value of the continuum thresholds are often ad hoc or based on the ones of the standard charmonium/bottomium systems mass-splittings which are not confirmed by the present analysis.Our results are summarized in Table 4.We find that : • The three Y c (4260, 4360, 4660) 1 −− experimental candidates are too low for being pure four-quark or/and molecule DD * and J/ψS 2 states but can result from their mixings.The Y b (10890) is lower than the predicted values of the four-quark and BB * molecule masses but heavier than the predicted ΥS 2 and ΥS 3 molecule states.Our results may indicate that some other natures (hybrids, threshold effects,...) of these states are not excluded.On can notice that our predictions for the masses are above the corresponding meson-meson thresholds indicating that these exotic states can be weakly bounded.
• For the 1 −− , there is a regularity of about (250-300) MeV for the value of the mass-splittings between the lowest ground state and the 1st radial excitation roughly approximated by the value of the continuum threshold √ t c at which the LSR and FESR match.These mass-splittings are (almost) flavour-independent and are much smaller than the ones of 500 MeV of ordinary charmonium and bottomium states and do not support some ad hoc choice used in the literature for fixing the t c -values when extracting the optimal results from the LSR.
• There is also a regularity of about 50-90 MeV for the S U( 3  mass-splittings of the different states which are also (almost) flavour-independent.
• The spin 0 states are much more heavier (≥ 400 MeV) than the spin 1 states, like in the case of hybrid states [5].
• The decay constants of the 1 −− and 0 ++ four-quark states obtained in Eqs (26)and (42) are much smaller than f π , f ρ and f D,B .Unlike f B expected to behave as 1/ M Q , the four-quark states decay constants exhibit a 1/M Q behaviour which can be tested using HQET or/and lattice QCD.It is likely that some other non-perturbative approaches such as potential models, HQET, AdS/QCD and lattice calculations check the previous new features and values on mass-splittings, mass and decay constants derived in this paper.We also expect that present and future experiments (LHCb, Belle, Babar,...) can test our predictions.

Acknowledgment
This work has been partly supported by the CNRS-IN2P3 within the project Non-Perturbative QCD and Hadron Physics, by the CNRS-FAPESP program and by CNPq-Brazil.S.N. has been partly supported by the Abdus Salam ICTP-Trieste (Italy) as an ICTP consultant for Madagascar.We thank the referee for his comments which lead to the improvements of the original manuscript.

Current
Table 3: QCD expression of the Molecule Spectral Functions to lowest order in α s and up to dimension-six condensates: Q ≡ c, b is the heavy quark field, while g ′ and Λ ′ are coupling and scale associated to an effective Van Der Vaals force.

26 Figure 1 :
Figure 1: a) τ-behaviour of M Y cd (1 −− ) from R LS R cd for the current mixing parameter b = 0, for different values of t c and for m c = 1.26 GeV; b) The same as a) but for m c = 1.47 GeV; c) t c -behaviour of the LSR results obtained at the τ-stability points and comparison with the ones from R FES R cd for m c = 1.26 and 1.47 GeV.

Figure 2 :
Figure 2: a) τ-behaviour of M Y bd (1 −− ) from R LS R bd for the current mixing parameter b = 0, for different values of t c and for m b = 4.17 GeV.b) t c -behaviour of the LSR results obtained at the τ-stability points and comparison with the ones from R FES R bd for m b = 4.17 and 4.70 GeV.

=
12.09 for √ t c = 12.2 and m b = 4.70.(21)One can notice, like in the case of the charm quark that the value of the on-shell quark mass tends to give a higher value of M Y bd within this lowest order PT calculations.Considering, like in the case of charm, as a final estimate the one from the running bquark mass m b (m b ) = 4177(11) MeV[28], we deduce in MeV:

2 Figure 3 :
Figure 3: a) b-behaviour of M Y cd for given values of τ and t c and for m c = 1.26 GeV; b)

Figure 4 :
Figure 4: a) τ-behaviour of f Y cd for given values of b = 0 and t c and for m c = 1.26 GeV; b) the same as a) but for f Y bd and for m b = 4.17 GeV; c) b-behaviour of f Y cd for given values of τ at the stability and t c d) the same as c) but for f Y bd respectively for √ t c = 5.1 and 11.6 GeV.Using the results for a) 0.05 0.10 0.15 0.20 0.25 0

Figure 5 :
Figure 5: a) τ-behaviour of r c sd for the current mixing parameter b = 0, for different values of t c and for m c = 1.26 GeV; b) τ-behaviour of r b sd for different values of t c and for m b = 4.17 GeV; c) t c -behaviour of the inflexion points (or minimas) of r c sd from Fig 5a; d) The same for the b quark using r b sd from Fig 5b.

4. 1
−− molecule masses from QSSR • The D * d(s) D d(s) and B * d(s) B d(s) molecules 8 Like in the previous case, we use LSR and FESR for studying a)

Figure 6 :
Figure 6: a) τ-behaviour of M D * d D d for different values of t c and for m c = 1.26 GeV; b) τ-behaviour of M B * d B d for different values of t c and for m b = 4.17 GeV; c) t c -behaviour of the extremas in τ of M D * d D d and for m c = 1.26 − 1.47 GeV; d) the same as c) but for M B * d B d show their τ-behaviour for different values of t c and for m c = 1.26 GeV and m b = 4.17 GeV respectively in Figs.6a,b and 7a,b.The t c -behaviour of the τ-minimas is shown in Fig. 6c,d for the masses and in Fig. 7c,d for the S U(3) breaking ratios.Using the sets (m c = 1.26 GeV, √ t c = 5.58 GeV) and (m b = 4.17 GeV, √ t c = 11.64(3)GeV) common solutions of LSR and FESR, one can deduce in MeV: Hereafter, for simplifying notations, D and B denote the scalar D * 0 and B *

Figure 7 :
Figure 7: a) τ-behaviour of r D sd for different values of t c and for m c = 1.26 GeV; b) τbehaviour of r B sd for different values of t c and for m b = 4.17 GeV; c) t c -behaviour of the inflexion points (or minimas) of r D sd from Fig 7a; d) the same for the b quark using r B sd from Fig 7b.
2 and ΥS 2 molecules in a colour singlet combination, where S 2 ≡

Figure 8 :
Figure 8: a) τ-behaviour of M J/ψS 2 for different values of t c and for m c = 1.26 GeV; b) τ-behaviour of M ΥS 2 for different values of t c and for m b = 4.17 GeV; c) t c -behaviour of the extremas in τ of M J/ψS 2 for m c = 1.26 − 1.47 GeV; d) the same as c) but for M ΥS 2 for m b = 4.17 − 4.70 GeV.

Figure 9 :
Figure 9: a) τ-behaviour of r ψ sd for different values of t c and for m c = 1.26 GeV; b) τ-behaviour of r Υ sd for different values of t c and for m b = 4.17 GeV; c) t c -behaviour of the extremas in τ of r ψ sd for m c = 1.26 − 1.47 GeV; d) the same as c) but for r Υ sd for m b = 4.17 − 4.70 GeV.
23 GeV and mainly to a low value of √ t c = 5.1 GeV which does not coïncide with the common solution √ t c = 5.3 GeV from LSR and FESR.On the opposite, the large value of M ΥS 2 = 10.74 (resp.11.09) GeV obtained there corresponds to a too high value √ t c = 11.3 (resp.11.7) GeV compared with the LSR and FESR solution √ t c = 10.23 (resp.10.48) GeV for the singlet (resp.octet) current.

Figure 10 :
Figure 10: a) τ-behaviour of Y 0 cd for the current mixing parameter b = 0, for different values of t c and for m c = 1.26 GeV ; b) τ-behaviour of Y 0 bd for different values of t c and for m b = 4.17 GeV; c) t c -behaviour of the extremas in τ of Y 0 cd for m c = 1.26 − 1.47 GeV; d) the same as c) but for Y 0 bd for m b = 4.17 − 4.70 GeV.

Figure 11 :
Figure 11: a) b-behaviour of M Y 0 cd for given values of τ and t c and for m c = 1.26 GeV; which is larger than the ones of the 1 −− states, comparable with the ones of the J/ψ and Υ, and are (almost) heavy-flavour independent.We show in Fig11the effect of the choice of b operator mixing parameter on the mass predictions, indicating an optimal value at b = 0.For completeness, we show in Fig.12the τ and b behaviours of the decay constants from which we deduce:

Figure 12 :
Figure 12: a) τ-behaviour of f Y 0 cd for given values of b = 0 and t c and for m c = 1.26 D s and M Bs B s from DRSR We show in Fig. 15 the τ-behaviour of the S U(3) mass ratios for different values of t c and the t c behaviour of their τ-

Figure 13 :
Figure 13: a) τ-behaviour of r 0c sd for the current mixing parameter b = 0, for different values of t c and for m c = 1.26 GeV; b) τ-behaviour of r 0b sd for different values of t c and for m b = 4.17 GeV; c) t c -behaviour of the extremas in τ of r 0c sd for m c = 1.26 − 1.47 GeV; d) the same as c) but for r 0b sd for m b = 4.17 − 4.70 GeV.

Figure 14 :
Figure 14: a) τ-behaviour of M D d D d for different values of t c and for m c = 1.26 GeV; b) τ-behaviour of M B d B d for different values of t c and for m b = 4.17 GeV; c) t c -behaviour of the extremas in τ of M D d D d and for m c = 1.26 − 1.47 GeV; d) the same as c) but for M B d B d and for m b = 4.17 − 4.70 GeV.

Figure 15 :
Figure 15: a) τ-behaviour of r D sd for different values of t c and for m c = 1.26 GeV; b) τ-behaviour of r B sd for different values of t c and for m b = 4.17 GeV; c) t c -behaviour of the inflexion points (or minimas) of r D sd from Fig 7a; d)the same for the b quark using r B sd from Fig 7b.

Table 4 :
Masses of the four-quark and molecule states from the present analysis combining Laplace (LSR) and Finite Energy (FESR).We have used double ratios (DRSR) of sum rules for extracting the S U(3) mass-splittings.The results correspond to the value of the running heavy quark masses but the S U(3) mass-splittings are less affected by such definitions.As already mentioned in the text for simplifying notations, D and B denote the scalar D * 0 and B * 0 mesons.The errors do not take into account the unknown ones from PT corrections.

Table 2 :
QCD expression of the Four-Quark Spectral Functions to lowest order in α s and up to dimension-six condensates: Q ≡ c, b is the heavy quark field.