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.

In this paper, we extend the analysis of B c done in Ref. [27][28][29] for studying the spectra and decay constants of the B c -like 1 −− , 0 ++ and 1 ++ mesons. In so doing, we shall work with the LSR. The analysis is very similar to the one in Ref. [29] which can be consulted. We shall improve the LSR results for the masses by using the flavour independence properties of some mass-splittings from Heavy Quark Symmetry (HQS) [31,32].

The QCD Inverse Laplace sum rules (LSR)
• The QCD interpolating currents We shall be concerned with the following QCD interpolating current: where: J S (x) is the local heavy-light scalar current; J µ H (x) [H ≡ V(A)] is the (axial)-vector currents; µ is the (axial) vector polarization; m c,b are renormalized masses of the QCD Lagrangian; f S , f H are the decay constants related to the leptonic width Γ[S , H → l + ν l ] and normalised as f π = 132 MeV.
Email address: snarison@yahoo.fr (Stephan Narison) • Form of the sum rules We shall work with the Finite Energy version of the QCD Inverse Laplace sum rules (LSR) and their ratios : where τ is the LSR variable, n = 0, 1 is the degree of moments, t c is the threshold of the "QCD continuum" which parametrizes, from the discontinuity of the Feynman diagrams, the spectral function Im[ψ S , Π T H (t, m 2 Q , µ 2 ) where ψ 5 (t, m 2 Q , µ 2 ) is the scalar and (axial) vector correlators defined as : ψ S (q 2 ) = i d 4 x e −iqx 0|T J 5 (x) (J 5 (0)) † |0 , with Π (1,0) H corresponds to the spin 1, 0 meson contributions.

The QCD two-point function within the SVZ-expansion
Using the SVZ [1] Operator Product Expansion (OPE), the Inverse Laplace tranform of the two-point correlator can be written in the form: Im [ψ S ; Π (1) H ](t, µ)| PT is the perturbative part of the spectral function; C G 2 S ,H and C ψ S ,H are (perturbatively) calculable Wilson coefficients; α s G 2 and cc are the non-pertubative gluon and quark condensates where G 2 ≡ G a µν G µν a . We have not retained higher dimension condensates as these contributions are negligible in the working regions. The b b condensate contribution where the b is considered as a heavy quark here is already included in C G 2 as explicitly shown in Ref. [23] through the relation [1,38,39] The charm quark is considered as a light quark where an expansion in m 2 c /Q 2 and m c /m b has been done for the nonperturbative contributions. The corresponding condensate is estimated from the analogue previous relation with the gluon condensate.
4. The q 2 = 0 behaviour of the two-point function To NLO, the perturbative part of ψ S (0) reads [4,5,20,40] with : where i ≡ c, b; µ is the QCD subtraction constant and a s ≡ α s /π is the QCD coupling. This PT contribution which is present here has to be added to the well-known non-perturbative contribution: for absorbing log n (−m 2 i /q 2 ) mass singularities appearing during the evaluation of the PT two-point function, a technical point not often carefully discussed in some papers. Working with ψ S (q 2 ) and Π (0) H defined previously is safe as ψ S (0), Π (1) H (0), which disappear after successive derivatives, do not affect the sum rule. This is not the case of the longitudinal part of the vector two-point function Π (0) V (q 2 ) built from the vector current which is related to ψ S (q 2 ) through the Ward identity [4,5,8,20]: and which is also often (uncorrectly) used in literature.

The Two-Point Function at large q 2
We have given in details the QCD expression of the pseudoscalar spectral function in Ref. [29]. Some (not lengthy) expressions of the other spectral functions are given below.
• Perturbative contributions -The complete expressions of the PT spectral function in terms of the on-shell quark masses has been obtained to LO in [41]: with the phase space factor : -The lengthy expressions at NLO are given in Refs [11,23,40]. The ones for the states of opposite parities can be obtained by a careful change of the sign of one of the quark mass (chirality transformation due to the (non)-presence of the γ 5 Dirac matrix).
-We shall use the N2LO contributions obtained in the limit where one of the quark mass is zero [42,43] which we expect to be a good approximation as the N2LO correction is relatively small.This expression is available as a Mathematica program Rvs.m.
-We estimate the error due to the truncation of the PT series from the N3LO contribution from a geometric growth of the PT series which is expected to mimic the phenomenological 1/q 2 dimension-two contribution [44] parametrizing the uncalculated large order terms of PT series [45][46][47][48].
• Non-perturbative contributions The complete non-perturbative contributions due to the gluon condensate have been obtained by [11,23] at LO. These expressions are also lengthy and will not be reported here. However, as these contributions are relatively small in the analysis, it is a good approximation to work with the approximate expressions where linear and quadratic corrections in term of m c are retained. However, one should be careful in using the expressions given by [40,49,51,52] for the (axial)vector currents as the decomposition of the correlator is slightly different of the one in Eq. 10 (H ≡ V, A): The relevant component associated to the spin 1 meson used in [27] and we shall use in the following, is the combination : To avoid singularities at q 2 = 0 (see e.g. [40,51]) and some (non) perturbative effects due to Π (1,0) (0), we shall work with the Inverse Laplace transform of the rescaled function : -The cc quark condensate contribution to ψ S (q 2 ) is given to LO by [40,49,50] and to NLO by [52,53]: where : z ≡ m 2 b τ, l µb ≡ log (µ/m b ) and Γ(n, z) is the n-th incomplete Γ-function.
-The m c -corrections have been derived from the expression given by [40]. The functions f n (z) and g n (x) are respectively the Inverse Laplace transform of the functions Y n (x) log(x) and The epxressions for the correlators associated to the pseudoscalar and axial-vector currents can be deduced from the former by the chiral transformation : m c → −m c .

• From the On-shell to the MS -scheme
We transform the pole masses m Q to the running masses m Q (µ) using the known relation in the MS -scheme to order α 2 s [55][56][57][58][59][60][61][62][63]: for n l = 3 : u, d, s light flavours. In the following, we shall use n f = 5 total number of flavours for the numerical value of α s .

QCD input parameters
The QCD parameters which shall appear in the following analysis will be the QCD coupling α s the charm and bottom quark masses m c,b , the gluon condensates α s G 2 . Their values are given in Table 1.  Parameters Values Sources Ref.
Average [34] • QCD coupling α s We shall use from the M χ 0c −M η c mass-splitting sum rule [34]: which is more precise than the one from These lead to the mean value quoted in Table 1, which is in complete agreement with the world average [54]: but with a larger error.

• c and b quark masses
For the c and b quarks, we shall use the recent determinations [33] of the running masses and the corresponding value of α s evaluated at the scale µ obtained using the same sum rule approach from charmonium and bottomium systems.
• Gluon condensate α s G 2 We use the recent estimate obtained from a correlation with the values of the heavy quark masses and α s which can be compared with the QSSR average from different channels [34].

Parametrisation of the spectral function
-In the present case, where no complete data on the spectral function are available, we use the duality ansatz: for parametrizing the spectral function. M H and f H are the lowest ground state mass and coupling analogue to f π . p = 0 for H ≡ V, A and p = 2 for H ≡ S . The "Continuum" or "QCD continuum" is the imaginary part of the QCD correlator from the threshold t c . Within a such parametrization, one obtains: indicating that the ratio of moments appears to be a useful tool for extracting the mass of the hadron ground state [4][5][6][7]9]. -This simple model has been tested in different channels where complete data are available (charmonium, bottomium and e + e − → I = 1 hadrons) [4,5,13]. It was shown that, within the model, the sum rule reproduces well the one using the complete data, while the masses of the lowest ground state mesons (J/ψ, Υ and ρ) have been predicted with a good accuracy. In the extreme case of the Goldstone pion, the sum rule using the spectral function parametrized by this simple model [4,5] and the more complete one by ChPT [64] lead to similar values of the sum of light quark masses (m u +m d ) indicating the efficiency of this simple parametrization.
-An eventual violation of the quark-hadron duality (DV) [65,66] has been frequently tested in the accurate determination of α s (τ) from hadronic τ-decay data [66][67][68], where its quantitative effect in the spectral function was found to be less than 1%. Typically, the DV behaves as: where κ, α, β are model-dependent fitted parameters but not based from first principles. Within this model, where the contribution is doubly exponential suppressed in the Laplace sum rule analysis, we expect that in the stability regions where the QCD continuum contribution to the sum rule is minimal and where the optimal results in this paper will be extracted, such duality violations can be safely neglected.
-Therefore, we (a priori) expect that one can extract with a good accuracy the masses and decay constants of the B c -like mesons within the approach. An eventual improvement of the results can be done after a more complete measurement of the B c -like spectral function which is not an easy experimental task.
-In the following, in order to minimize the effects of unkown higher radial excitations smeared by the QCD continuum and some eventual quark-duality violations, we shall work with the lowest ratio of moments R c 0 for extracting the meson masses and with the lowest moment L c 0 for estimating the decay constant f H . Moment with negative n will not be considered due to their sensitivity on the non-perturbative contributions such as ψ S (0).

Optimization Criteria
For extracting the optimal results from the analysis, we have used in previous works the optimization criteria (minimum sensitivity) of the observables versus the variation of the external variables namely the τ sum rule parameter, the QCD continuum threshold t c and the subtraction point µ.
One should notice in the previous works that these criteria have lead to more solid theoretical basis and noticeable improvement of the sum rule results. The quoted errors in the results are conservative as the range covered by t c from the beginning of τ-stability to the one of t c -stability is quite large. However, such large errors induces less accurate prediction compared with some other approaches (potentiel models. lattice calculations) especially for the masses of the mesons. This is due to the fact that, in most cases, there are no available data for heavy-light radial excitations which can used to restrict the range of t c -values. However, the value of t c used in the "QCD continuum" model does not necessarily coïncide with the 1st radial excitation mass as the "QCD continuum" is expected to smear all higher states contributions to the spectral function. This feature has been explicitly verified by [80] in the ρ-meson channel. In the case of the B c meson, we have seen [29] that the optimal result has been obtained for √ t c (7.8 − 8.4) GeV which is about 1 GeV above the recent B c (2S )-mass found at 6872(1.5) MeV by CMS [81].
In order to slightly restrict the large range of variations of t c and to minimize the dependence on the form of the "QCD continuum" model, we shall require that its contribution to the spectral function does not exceed (20-25)% of the lowest resonance one.

0 ++ Scalar channel
The analysis here and in the follwing sections is very similar to the case of pseusoscalar channel studied in details in [29]. The results are summarized in different figures.

• τ-stability
In a first step, fixing the value of µ = 7.5 GeV which we shall justify later and which is the central value obtained in [27,29], we show in Fig. 1 Table 1 have been used. We see that f B c * 0 but not M B c * 0 presents inflexion points at τ (0.11 − 0.12) GeV −2 which appear for t c ≥ 55 GeV 2 . We shall use these inflexion points to fix the values of M B c * 0 .

• t c -stability
We study the t c -behaviour of of f B c * 0 and M B c * 0 in Fig. 2 where we see that f B c * 0 starts to stabilize from t c = 70 GeV 2 .

• µ-stability
Fixing t c = 70 GeV 2 and τ = 0.11 GeV −2 , we show in Fig. 7 the µ behaviour of f B c * 0 , where we note an inflexion point at : in agreement with the one quoted in [27,29] using different ways and/or from different channels. The µ-behaviour of M B c * 0 is not shown as it is almost constant in this range of t c -values.   Table 1.

• QCD continuum versus lowest resonance contribution
To have more insights on the QCD continuum contribution, we show in Fig. 4 the ratio of the continuum over the lowest ground state contribution as predicted by QCD : The curve started from t c = 56.5 GeV 2 where the QCD continuum contribution to the spectral function is half of the resonance contribution where one can also note from Fig. 1 that the τ-stability is reached from this value.
• Predictions for M B c * 0 and f B c * 0 -From the previous analysis and taking the large range of t c from 56.5 GeV 2 where the τ-stability starts and where the QCD continuum is less than 50% of the resonance one to t c =75 GeV 2 where the t c -stability is (almost) reached at which the lowest resonance dominates the spectral function (Meson Dominance Model), we obtain, for τ 0.  of the resonance one. In this way, the t c -values is restricted to be (70 ± 5) GeV 2 . In this way, we deduce the improved predictions for τ (0.10 − 0.12) GeV −2 in units of MeV : -We test the accuracy of the approximate expression expanded up to order m 2 c by taking the example of the pseudoscalar channel where the complete expression of the nonperturbative contribution is used. We notice that the approximate result overestimate the mass prediction by about 1.01%. We take into account this effect by adding to the prediction in Eq. 28 a systematic error of about 1% to the mass prediction  and dividing by 1.01 the estimate from the analysis. 2 -Examining the analytical form of the (pseudo)scalar sum rules, one can deduce the approximate LO relation: We have evaluated the running mass at µ = 7.5 GeV. The result is comparable to the more involved estimate f B c = 371(17) MeV in [29],

1 −− Vector channel
We do a similar analysis for the vector channel B * c which is summarized in the different figures shown below. We think that it is important to present the figures in each channel to give a better understanding of the results as they have not the same behaviours in (τ, t c , µ).
• τ-stability We show in Fig. 5 the τ-behaviour of f B * c and M B * c for different values of t c . We see that f B * c presents inflexion points and M B * c τ-minimas for t c ≥ 52 GeV 2 .
• t c -stability We study the t c -behaviour of f B * c and M B * c in Fig. 6 where we see that both quantities start to stabilize in t c for t c 65 GeV 2 .

• µ-stability
Fixing t c = 60 GeV 2 , we show in Fig. 7 the µ-behaviour of M B * c at the τ-minimas where we note a net inflexion point at : in agreement with the ones obtained in the previous section and quoted in [27][28][29] indicating the self-consistency of the whole approach. The µ-behaviour of M B * c is not shown as it is almost constant in this range of t c -values.

• QCD continuum versus lowest resonance contribution
We show in Fig. 4 the ratio of the continuum over the lowest ground state contribution as predicted by QCD :  From the previous analysis we take t c (52 − 65) GeV 2 for extracting our optimal results. The lowest value of t c corresponds to the beginning of τ-stability and also here to the "QCD continuum" contribution which is less than 20% of the resonance one. The highest value corresponds to the beginning of t c -stability. We obtain in units of MeV:

1 ++ Axial-Vector channel
We do a similar analysis for the B c1 axial-vector meson. The expression of the two-point function can be deduced from the vector one by changing m c to −mc. The anaysis is summarized through the different figures shown below.  We show in Fig. 5 the τ-behaviour of f B c1 and M B c1 for different values of t c . We see that both quantities present inflexion points for τ (0.09 − 0.10) GeV −2 which appear for t c ≥ 50 GeV 2 . Imposing that the "QCD continuum" contribution is less than 20-25% of the resonance one leads to t c ≥ 65 GeV 2 .

• t c -stability
We study the t c -behaviour of f B c1 and M B c1 in Figs. 10 and 11 where both quantities start to stabilize for t c 65 GeV 2 . The beginning of t c -stability is reached for t c ≈ 75 GeV 2 . For definiteness, we shall work in the range of t c (65 − 75) GeV 2 .       a decreasing function of µ while f B c1 presents a net inflexion point at : This value agrees with the ones obtained in the previous sections and in [27,29] showing again the self-consistency of the whole approach.
• QCD continuum versus lowest resonance contribution We show in Fig. 4 the ratio of the continuum over the lowest ground state contribution as predicted by QCD:

• Predictions for M B c1 and f B c1
From the previous analysis we take t c (65 − 75) GeV 2 for extracting our optimal results. The lowest value of t c corresponds to the case where the QCD contribution is less than 25% of the resonance one. The highest value corresponds to the beginning of t c -stability. We obtain in units of MeV :

Comments on the results
We notice from previous analysis that : • The results from different channels stabilize at a common value of µ around 7.5 GeV which is consistent with previous analysis in [27,29] indicating the self-consistency of the whole approach.
• The value of t c (60 ± 5) GeV 2 where the B * c parameters are optimally extracted are about the same as the one of B c but lower than the ones t c (70 ± 5) GeV 2 where the B * 0c and B * c1 sum rules are optimized. This feature is dual to the low masses of (B c , B * c ) compared to the ones of (B * 0c , B * c1 ). • The errors due to the QCD parameters are relatively small.The ones from the sum rule external parameters (t c , τ) are dominant. In addition to these, the ones on the decay constants are strongly affected by the error on the mass determination.
• As mentioned earlier, we have added the systematic error of 1% on the mass determination and divided the prediction by 1.01 for quantifying the approximate expression expanded in terms of m c for the non-perturbative contributions.

Mass-splittings from Heavy Quark Symmetry (HQS)
To improve the predictions on the meson masses, we shall use the properties of Heavy Quark Symmetry (HQS) [31,32]. In so doing, we confront the observed values of the D (s) , B (s)like mass-splittings to the expectations of HQS to LO in the (1/M Q ) expansion where : Q ≡ c, b and extrapolate this result for predicting the B c -like meson masses.

• Hyperfine splittings
From spin symmetry, one expects that, to LO, the hyperfine splittings are independent of the flavour of the "brown muck" [31,32] which is realized experimentally [54] : • Heavy Flavour independence of excitation energies -One also expects from HQS that the excitation energies for states with different quantum numbers of the light degrees of freedom are heavy flavour independent [31,32], which is approximately reproduced by the data [54]: -For estimating the scalar meson mass M B * 0,c , we assume the flavour independence (within the errors) of the mass-splitting of chiral multiplets as given by the sum rule results [4,22,82]: and the data [54]: Using the previous experimental value, we deduce : The value of M B * 0 suggests that the experimental candidate B * J (5732) can be fairly identified with a 0 ++ B-like meson. The decay constant of B * 0 has been already estimated in [28]. It is quoted in Table 3 and agrees with [83].

Summary and Comparison with some Other Estimates
-We collect the results obtained in the previous sections in Table 2 and 3 which we compare with recent Lattice calculations and Potential Models (PM) results. We notice that, within the errors, the results from different approaches agree each other except the one for the 2 ++ meson. Note that the orginal numbers from [85] are quoted without any errors but we expect that the PM results are known within 20 MeV error as estimated in [23].  Masses (17)  -One should mention that the errors from LSR are relatively large which can only be improved after a complete measurement of the spectral functions which is out of reach at present.
-As the predictions on the decay constants are largely affected by the value of the masses, we have also reported the predictions using the relatively precise predictions on the masses from HQS. The difference of the LSR and HQS results for f H is due to the LO factor Exp[M 2 H τ/2]/M 2 H entering in the LSR. 9