In the four-quark scenario, scalar mesons can be regarded as S-wave bound states of diquark–antidiquark pairs, where the diquark was assumed to be a spin-zero color anti-triplet. However, we can expect that working with a particular choice of current given above will give a general feature of the four-quark model predictions for X(3872), provided we can work with quantities less affected by radiative corrections and where OPE converges quite well 1 As pointed out in [1], isospin forbidden decays are possible if X is not a pure isospin state. In this work, we want to test under which conditions the results of the sum rules are compatible with the above predictions.
1 In the well-known case of baryon sum rules, a simplest choice of operator [33] and a more general choice [34] are given in the literature. Although technically apparently different, mainly for the region of convergence of the OPE, the two choices of interpolating currents have given the same predictions for the proton mass and the mixed condensate, but differ only for values of the higher-dimensional four-quark condensate. 3 quark level, the complex structure of the QCD vacuum leads us to use the Wilson operator product expansion (OPE).
On the OPE side, we work to leading order in αs and consider condensate contributions up to dimension eight. We keep the term which is linear in the light-quark mass mq, in order to evaluate the mass change in Eq. The light-quark part of the correlation function is calculated in coordinate space, and then Fourier transformed in momentum space in D dimensions.
2 We have not included the effects of a dimension 2 term caused by the UV renormalization, which we expect to be numerically negligible as in other channels [37], although this result should be checked.
CAP´ITULO 2. REGRAS DE SOMA DA QCD (QCDSR) 17
Resumo das Contribui¸c˜ oes da OPE
Integral I nmkl (Q 2 )
N˜ao necessos nos preocupar com ela pois nas QCDSR considers apenas a parte imagin´aria destas integrais. The points indicate higher contributions of the axial vector resonance that will be parameterized, as usual, through the introduction of a continuity threshold parameter s0. Keeping the charm-quark mass bounded, we use the momentum-space expression for the charm-quark propagator.
The resulting light-quark part is combined with the charm-quark part before being dimensionally adjusted to D = 4. After doing an inverse-Laplace (or Borel) transform of both sides and transferring the continuum contribution to the OPE side, Rule of the sum for the X-axis vector meson up to eight-dimensional condensates can be written as 2. We start with the perturbative contribution (plus a very small mq contribution) and each subsequent row represents the addition of an additional dimension of the expanding condensate.
To derive the mass MX without worrying about the value of the decay constant fX, we take the derivative of Eq. 3 We must emphasize that a full assessment of these contributions requires a more involved analysis, including a non-trivial choice of the basis of the factorization assumption [38].
LSR PREDICTIONS OF M X
- Transformada de Borel
5 This quantity has the advantage that it is less sensitive to the disruptive radiation corrections than the individual moments. This analysis allows us to determine the lower bound for M2 in the sum rules window. This figure also shows that although there is a change of sign between the contributions of condensates of dimension six and dimension eight, the contribution of the latter is smaller, where, as we have assumed, in Fig.
The relatively small contribution of the dimension eight condensates can justify the validity of our approach, unlike the case of the 5-quark current correlator, as noted in [42]. However, the partial compensation of these two terms indicates the sensitivity of the central value of the mass prediction to the way the OPE is truncated. We will use here the values of the quark masses obtained within the same QCD methods for spectral sum rules compiled in [19].
They are defined in the M S scheme and are obtained within the same abbreviation of the QCD series from different channels and by different authors.
Pr eli mi na ry
The systematic error includes contributions from the same sources as the uncertainty in the branching fraction of the ¯B0 → ψ!K−π+ decay and the amplitude model dependence of the K∗(892) fit fraction. We express our gratitude to our colleagues in the CERN accelerator departments for the excellent performance of the LHC. We acknowledge support from CERN and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC (China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG (Germany); SFI (Ireland);.
We also acknowledge the support received from EPLANET, Marie Curie Actions and the ERC under FP7. We are also grateful for the computing resources and the access to software R&D tools provided by Yandex LLC (Russia).
2 based on the fact that the mass difference correspond-
4 for X (3872) (which is the same for Z c (3900)) shows good
13) where
14) Therefore we obtain
On the OPE side we consider CC diagrams of the same type of diagram in Fig. A good Borel window is defined when the parameter to be extracted from the sum rule is as independent as possible of the Borel measure. 3 we show, through circles, the right-hand side (RHS) of equation (9), as a function of the Borel measure.
Using the definition of A in equation (10), the value obtained for the coupling constant is gZcψπ = 3.89 GeV, which is in excellent agreement with the estimate from [17] based on dimensional arguments.
CONCLUSIONS
To extract the coupling constant, we again fit the QCDSR results using the exponential form in Eq. This value for this link is again in excellent agreement with the estimate presented in [17]. In the case of the Zc+(3900)J/ψπ+ vertex we used the sum rule at the pawn pole and the coupling was taken directly from the sum rule.
In the three cases, we only considered the color-connected diagrams, since we expect the Zc(3900) to be a genuine tetraquark state with a non-trivial color structure.
