and heavy-light exotic mesons at N2LO in the chiral limit
We use QCD spectral sum rules (QSSR) and the factorization properties of molecule and four-quark currents to estimate the masses and couplings of the and molecules and four-quark at N2LO of PT QCD. We include in the OPE the contributions of non-perturbative condensates up to dimension-six. Within the Laplace sum rules approach (LSR) and in the -scheme, we summarize our results in Table 2, which agree within the errors with some of the observed XZ-like molecules or/and four-quark. Couplings of these states to the currents are also extracted. Our results are improvements of the LO ones in the existing literature.
keywords:QCD Spectral Sum Rules, molecule and four-quark states, heavy quarkonia.
The recent discovery of the (3900) by Belle BELLE () and BESIII BES () has motivated different theoretical analysis REVMOL (). However, all of the previous analysis like e.g. in RAPHAEL (); HEP1 (); HEP2 () from QCD Spectral Sum Rules (QSSR) SVZ (); SNB1 () have been done at LO of PT QCD. In this paper, we are going to use QSSR to evaluate the masses and couplings of some and molecules at N2LO in the PT series and compare the results with those obtained at lowest order and with experiments. This work is a part of the original papers in ARTICLE () and also in HEP3 (); ARTICLE2 ().
2 QCD analysis of the ones in molecule states
Currents and two-point fonctions
The currents used for these molecules states are given by:
- For :
- For :
where , represent respectively light and heavy quarks. The associated two-point correlation function is:
where and are associated to the spin 1 and 0 molecule states. Parametrizing the spectral function by one resonance plus a QCD continuum, the lowest resonance mass and coupling normalized as:
where is the heavy quark on-shell mass, the LSR parameter, the continuum threshold and is the QCD expression of the molecule spectral function.
The QCD two-point function at N2LO
To derive the results at N2LO, we assume factorization and then use the fact that the two-point function of a molecule state can be written as a convolution of the spectral functions associated to quark bilinear currents. We have CONV (); SNPIVO ():
For spin 0:
For spin 1:
with the phase space factor:
Im and Im are respectively the spectral functions associated to the (axial)vector and to the (pseudo)scalar bilinear currents. The QCD expression of the spectral functions for bilinear currents are already known up to order and including non-perturbative condensates up to dimension 6. It can be found in SNFB12 (); SNFBST14 (); GENERALIS (); CHET () for the on-shell mass . We shall use the relation between the on-shell and the running mass to transform the spectral function into the -scheme SPEC1 (); SPEC2 ():
where is the number of light flavours and at the scale .
The PT QCD parameters which appear in this analysis are , the charm and bottom quark masses (the light quark masses have been neglected). We also consider non-perturbative condensates which are the quark condensate , the two-gluon condensate , the mixed condensate , the four-quark condensate , the three-gluon condensate , and the two-quark multiply two-gluon condensate where indicates the deviation from the four-quark vacuum saturation. Their values are given in Table 1 and more recently in SN18 ().
|SNTAU (); BNPa (); BNPb (); BETHKE ()|
|GeV||SNB1 (); SNTAU (); SNmassa (); SNmassb (); SNmass98a (); SNmass98b (); SNLIGHT ()|
|MeV||average SNmass02 (); SNH10a (); SNH10b (); SNH10c (); PDG (); IOFFEa (); IOFFEb ()|
|MeV||average SNmass02 (); SNH10a (); SNH10b (); SNH10c (); PDG ()|
|MeV||SNB1 (); SNmassa (); SNmassb (); SNmass98a (); SNmass98b (); SNLIGHT ()|
|GeV||JAMI2a (); JAMI2b (); JAMI2c (); HEIDb (); HEIDc (); SNhl ()|
|GeV||SNTAU (); LNT (); SNIa (); SNIb (); YNDU (); SNH10a (); SNH10b (); SNH10c (); SNG2 (); SNGH ()|
|GeV||SNH10a (); SNH10b (); SNH10c ()|
|GeV||SNTAU (); LNT (); JAMI2a (); JAMI2b (); JAMI2c ()|
3 Mass of the molecule state
We study the behavior of the mass in term of LSR variable at different values of as shown in Fig.1. We consider as a final and conservative result the one corresponding to the beginning of the stability for =23 GeV and 0.25 GeV until the one where stability is reached for 32 GeV and 0.35 GeV.
Convergence of the PT series
According to these analysis, we can notice that the -stability begins at GeV and the -stability is reached from GeV. Using these two extremal values of , we study in Fig. 2 the convergence of the PT series for a given value of GeV. We observe that from LO to NLO the mass increases by about +1 while from NLO to N2LO, it only increases by +0.1. This result indicates a good convergence of PT series which validates the LO result obtained in the literature when the running quark mass is used RAPHAEL ().
We improve our previous results by using different values of (Fig. 3). Using the fact that the final result must be independent of the arbitrary parameter , we consider as an optimal result the one at the inflexion point for GeV:
where the second error comes from the localisation of the inflexion point, QCD condensates and higher dimension contributions.
4 Coupling of molecule state
We can do the same analysis to derive the decay constant defined in Eq. (4). Noting that the bilinear pseudoscalar heavy-light current acquires an anomalous dimension, then the decay constant runs as:
where is a scale invariant coupling; is the first coefficient of the QCD -function for flavors and . The QCD corrections numerically read:
. Taking the Laplace transform of the correlator, this definition will lead us to the expression of the running coupling in Eq. (6). We show in Fig. 4 the -behaviour of the running coupling for two extremal values of where and stabilities are reached. These values are the same as in the mass determination.
One can see in this figure that the corrections to the LO term of PT series are still small though bigger than in the case of the mass determination from the ratio of sum rules. It is about +5 from LO to NLO and +2 from NLO to N2LO.
In the Fig. 5, we show the behaviour of the invariant coupling . Taking the optimal result at the minimum for GeV, we obtain in units of MeV:
5 and four-quark states
We can do the same analysis for the case of four-quark states. The interpolating currents used are:
where (respectively ) in the charm (resp. bottom) channel and . is the mixing of the two operators. We use k=0 as shown in RAPHAEL (); RAPHAEL1 ().
The behavior of the curves of masses and couplings are very similar to the molecules ones. Considering all the possible currents and channels configurations, we have in Table 2 the results for and molecule and four-quark states.
We have presented improved predictions of QSSR for the masses and couplings of the and molecule and four-quark states at N2LO of PT series and including up to dimension six non-perturbative condensates. We can see a good convergence of the PT series after including higher correction. This good convergence confirms the veracity of our results. The results are improvements of all the precedent works about the masses of exotic hadrons obtained at LO. Our analysis has been done within stability criteria with respect to the LSR variable , the QCD continuum threshold and the subtraction constant which have provided successful predictions in different hadronic channels. The optimal values of the masses and couplings have been extracted at the same value of these parameters where the stability appears as an extremum and/or inflection points. The ill-defined heavy quark mass definition used at LO is not enough to have results. The effects are often large for the coupling. The masses of and are also around the state. We do not include higher dimensioon condensates contributions in our estimate but only consider them as a source of the errors. One can conclude that can be well described with an almost pure molecule. One can notice that the masses of the states are most of them below the corresponding , -like thresholds and are compatible with some of the observed XYZ masses suggesting that these states can be interpreted as almost pure molecules or/and four-quark states.
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