]>PLB34364S0370-2693(19)30007-310.1016/j.physletb.2019.01.001The AuthorPhenomenologyFig. 1The various dimension OPE contribution as a function of M2 in sum rule (4) for s0=2.8GeV for the scalar–scalar case with ρ = 1.Fig. 1Fig. 2The phenomenological contribution in sum rule (4) for s0=2.8GeV for the scalar–scalar case with ρ = 1. The solid line is the relative pole contribution (the pole contribution divided by the total, pole plus continuum contribution) as a function of M2 and the dashed line is the relative continuum contribution.Fig. 2Fig. 3The mass of 0+ tetraquark state with the scalar–scalar configuration as a function of M2 from sum rule (5) with ρ = 1. The continuum thresholds are taken as s0=2.7∼2.9GeV. The ranges of M2 are 0.8 ∼ 1.5 GeV2 for s0=2.7GeV, 0.8 ∼ 1.6 GeV2 for s0=2.8GeV, and 0.8 ∼ 1.7 GeV2 for s0=2.9GeV.Fig. 3Fig. 4The various dimension OPE contribution as a function of M2 in sum rule (4) for s0=2.8GeV for the axial–axial case with ρ = 1.Fig. 4Fig. 5The phenomenological contribution in sum rule (4) for s0=2.8GeV for the axial–axial case with ρ = 1. The solid line is the relative pole contribution (the pole contribution divided by the total, pole plus continuum contribution) as a function of M2 and the dashed line is the relative continuum contribution.Fig. 5Fig. 6The mass of 0+ tetraquark state with the axial–axial configuration as a function of M2 from sum rule (5) with ρ = 1. The continuum thresholds are taken as s0=2.7∼2.9GeV. The ranges of M2 are 0.9 ∼ 1.5 GeV2 for s0=2.7GeV, 0.9 ∼ 1.6 GeV2 for s0=2.8GeV, and 0.9 ∼ 1.7 GeV2 for s0=2.9GeV.Fig. 6Fig. 7The various dimension OPE contribution as a function of M2 in sum rule (4) for s0=2.8GeV for the pseudoscalar–pseudoscalar case with ρ = 1.Fig. 7Fig. 8The various dimension OPE contribution as a function of M2 in sum rule (4) for s0=2.8GeV for the vector–vector case with ρ = 1.Fig. 8Fig. 9The mass of 0+ tetraquark state with the scalar–scalar configuration as a function of M2 from sum rule (5) with ρ = 3. The continuum thresholds are taken as s0=2.7∼2.9GeV. The ranges of M2 are 0.9 ∼ 1.9 GeV2 for s0=2.7GeV, 0.9 ∼ 2.0 GeV2 for s0=2.8GeV, and 0.9 ∼ 2.1 GeV2 for s0=2.9GeV.Fig. 9Fig. 10The mass of 0+ tetraquark state with the axial–axial configuration as a function of M2 from sum rule (5) with ρ = 3. The continuum thresholds are taken as s0=2.7∼2.9GeV. The ranges of M2 are 0.9 ∼ 1.8 GeV2 for s0=2.7GeV, 0.9 ∼ 1.9 GeV2 for s0=2.8GeV, and 0.9 ∼ 2.0 GeV2 for s0=2.9GeV.Fig. 10Revisiting Ds0⁎(2317) as a 0+ tetraquark state from QCD sum rulesJian-RongZhangDepartment of Physics, College of Liberal Arts and Sciences, National University of Defense Technology, Changsha 410073, Hunan, People's Republic of ChinaDepartment of PhysicsCollege of Liberal Arts and SciencesNational University of Defense TechnologyChangshaHunan410073People's Republic of ChinaEditor: J.-P. BlaizotAbstractStimulated by the renewed observation of Ds0⁎(2317) signal and its updated mass value 2318.3±1.2±1.2MeV/c2 in the process e+e−→Ds⁎+Ds0⁎(2317)−+c.c. by BESIII Collaboration, we devote to reinvestigate Ds0⁎(2317) as a 0+ tetraquark state from QCD sum rules. Technically, four different possible currents are adopted and high condensates up to dimension 12 are included in the operator product expansion (OPE) to ensure the quality of QCD sum rule analysis. In the end, we obtain the mass value 2.37−0.36+0.50GeV with the factorization parameter ρ=1 (or 2.23−0.24+0.78GeV with ρ=3) for the scalar–scalar current, which agrees well with the experimental data of Ds0⁎(2317) and could support its explanation as a 0+ scalar–scalar tetraquark state. The final result for the axial–axial configuration is calculated to be 2.51−0.43+0.61GeV with ρ=1 (or 2.52+0.76−0.52GeV with ρ=3), which is still consistent with the mass of Ds0⁎(2317) considering the uncertainty, and then the possibility of Ds0⁎(2317) as a axial–axial tetraquark state can not be excluded. For the pseudoscalar–pseudoscalar and the vector–vector cases, their unsatisfactory OPE convergence makes that it is of difficulty to find rational work windows to further acquire hadronic masses.1IntroductionVery recently, BESIII Collaboration announced the observation of the process e+e−→Ds⁎+Ds0⁎(2317)−+c.c. for the first time with the data sample of 567pb−1 at a center-of-mass energy s=4.6GeV [1]. For the Ds0⁎(2317) signal, the statistical significance is reported to be 5.8σ and its mass is measured to be 2318.3±1.2±1.2MeV/c2. Historically, Ds0⁎(2317) was first observed by BABAR Collaboration in the Ds+π0 invariant mass distribution [2,3], which was confirmed by CLEO Collaboration [4] and by Belle Collaboration [5]. In theory, Ds0⁎(2317) could be proposed as a conventional c¯s meson with JP=0+ (e.g. see [6]). However, one has to confront an approximate 150MeV/c2 difference between the measured mass and the theoretical results from potential model [7] and lattice QCD [8] calculations. In addition, the absolute branching fraction 1.00−0.14+0.00±0.14 for Ds0⁎(2317)−→π0Ds− newly measured by BESIII [1] shows that Ds0⁎(2317)− tends to have a significantly larger branching fraction to π0Ds− than to γDs⁎−, which differs from the expectation of the conventional c¯s state [9]. As a feasible scenario resolving the above discrepancy, one can suppose Ds0⁎(2317) to be some multiquark system, such as a DK molecule candidate [10], a c¯sqq¯ tetraquark state [11], or a mixture of a c¯s meson and a tetraquark state [12]. In a word, it is still undetermined and even unclear for the nature of Ds0⁎(2317).Especially inspired by the BESIII's new experimental result on Ds0⁎(2317) [1], we devote to study it in the tetraquark picture, which is also helpful to deepen one's understanding on nonperturbative QCD. One reliable way for evaluating the nonperturbative effects is the QCD sum rule method [13], which is an analytic formalism firmly entrenched in QCD and has been fruitfully applied to many hadrons [14–18]. Concerning Ds0⁎(2317), there have appeared several QCD sum rule works to compute its mass basing on a c¯s meson picture [19–26], or taking a point of tetraquark view from QCD sum rules in the heavy quark limit [27] as well as from full QCD sum rules involving condensates up to dimension 6 or 8 [28–30]. It is known that one key point of the QCD sum rule analysis is that both the OPE convergence and the pole dominance should be carefully inspected. It has already been noted that some high dimension condensates may play an important role in some cases [31–34]. To say the least, even if high condensates may not radically influence the OPE's character, they are still beneficial to stabilize Borel curves. Therefore, in order to further reveal the internal structure of Ds0⁎(2317), we endeavor to perform the study of Ds0⁎(2317) as a 0+ tetraquark state in QCD sum rules adopting four different possible currents and including condensates up to dimension 12.The rest of the paper is organized as follows. In Sec. 2, Ds0⁎(2317) is studied as a tetraquark state in the QCD sum rule approach. The last part is a brief summary.2QCD sum rule study of Ds0⁎(2317) as a 0+ tetraquark state2.10+ tetraquark state currentsAs one basic point of QCD sum rules, hadrons are represented by their interpolating currents. For a tetraquark state, its current ordinarily can be constructed as a diquark–antidiquark configuration. Thus, one can present following forms of 0+ tetraquark currents:j(I)=ϵabcϵdec(qaTCγ5sb)(q¯dγ5CQ¯eT) for the scalar–scalar case,j(II)=ϵabcϵdec(qaTCsb)(q¯dCQ¯eT) for the pseudoscalar–pseudoscalar case,j(III)=ϵabcϵdec(qaTCγμsb)(q¯dγμCQ¯eT) for the axial vector–axial vector (shortened to axial–axial) case, andj(IV)=ϵabcϵdec(qaTCγ5γμsb)(q¯dγμγ5CQ¯eT) for the vector–vector case. Here q denotes the light u or d quark, Q is the heavy flavor charm quark, and the subscripts a, b, c, d, and e indicate color indices.2.2Tetraquark state QCD sum rulesThe two-point correlator(1)Πi(q2)=i∫d4xeiq.x〈0|T[j(i)(x)j(i)†(0)]|0〉,(i=I,II,III,orIV) can be used to derive QCD sum rules.Phenomenologically, the correlator can be written as(2)Πi(q2)=λH2MH2−q2+1π∫s0∞Im[Πiphen(s)]s−q2ds+..., where s0 is the continuum threshold, MH denotes the hadron's mass, and λH shows the coupling of the current to the hadron 〈0|j|H〉=λH.Theoretically, the correlator can be expressed as(3)Πi(q2)=∫(mc+ms)2∞ρi(s)s−q2ds+Πicond(q2), where mc is the mass of charm quark, ms is the mass of strange quark, and the spectral density ρi(s)=1πIm[Πi(s)].After matching Eqs. (2) and (3), assuming quark–hadron duality, and making a Borel transform Bˆ, the sum rule can be(4)λH2e−MH2/M2=∫(mc+ms)2s0ρi(s)e−s/M2ds+BˆΠicond, with M2 the Borel parameter.Taking the derivative of Eq. (4) with respect to −1M2 and then dividing by Eq. (4) itself, one can arrive at the hadron's mass sum rule(5)MH={∫(mc+ms)2s0ρi(s)se−s/M2ds+d(BˆΠicond)d(−1M2)}/{∫(mc+ms)2s0ρi(s)e−s/M2ds+BˆΠicond}.In detail, the spectral densityρi(s)=ρipert(s)+ρi〈q¯q〉(s)+ρi〈g2G2〉(s)+ρi〈gq¯σ⋅Gq〉(s)+ρi〈q¯q〉2(s)+ρi〈g3G3〉(s)+ρi〈q¯q〉〈g2G2〉(s) and the termBˆΠicond=BˆΠi〈q¯q〉〈g2G2〉+BˆΠi〈q¯q〉〈gq¯σ⋅Gq〉+BˆΠi〈q¯q〉3+BˆΠi〈q¯q〉〈g3G3〉+BˆΠi〈g2G2〉〈gq¯σ⋅Gq〉+BˆΠi〈gq¯σ⋅Gq〉2+BˆΠi〈q¯q〉2〈g2G2〉+BˆΠi〈q¯q〉2〈gq¯σ⋅Gq〉+BˆΠi〈q¯q〉〈g2G2〉2+BˆΠi〈g3G3〉〈gq¯σ⋅Gq〉+BˆΠi〈q¯q〉2〈g3G3〉+BˆΠi〈q¯q〉〈g2G2〉〈gq¯σ⋅Gq〉 including condensates up to dimension 12 can be derived with the similar techniques as Refs. e.g. [18,35]. In reality, their concrete expressions for ρi(s) and BˆΠicond are the same as our previous work [36] other than that mQ should be replaced by the charm quark mass mc, which are not intended to list here for conciseness. Note that in Ref. [36] we have already applied the factorization hypothesis 〈q¯qq¯q〉=ρ〈q¯q〉2 [15,18] and taken the factorization parameter ρ=1.2.3Numerical analysis with ρ=1In the first instance, we set ρ=1 for the 〈q¯qq¯q〉=ρ〈q¯q〉2 factorization. To extract the numerical value of MH, we perform the analysis of sum rule (5) and take mc as the running charm quark mass 1.27±0.03GeV along with other input parameters as ms=96−4+8MeV, 〈q¯q〉=−(0.24±0.01)3GeV3, 〈s¯s〉=m02〈q¯q〉, 〈gq¯σ⋅Gq〉=m02〈q¯q〉, m02=0.8±0.1GeV2, 〈g2G2〉=0.88±0.25GeV4, and 〈g3G3〉=0.58±0.18GeV6 [13,15,37]. As a standard procedure, both the OPE convergence and the pole dominance should be considered to find proper work windows for the threshold s0 and the Borel parameter M2: the lower bound for M2 is obtained by analyzing the OPE convergence; the upper bound is gained by the consideration that the pole contribution should be larger than QCD continuum contribution. Moreover, s0 characterizes the beginning of continuum states and can not be taken at will. It is correlated to the energy of the next excited state and approximately taken as 400∼600MeV above the extracted mass value MH, which is consistent with the existing QCD sum rule works on the same tetraquark state (such as Refs. [27,28,30]).Taking the scalar–scalar case as an example, its different dimension OPE contributions are compared as a function of M2 in Fig. 1. Graphically, one can see that there are three main condensate contributions, i.e. the dimension 3 two-quark condensate, the dimension 5 mixed condensate, and the dimension 6 four-quark condensate. These condensates could play an important role on the OPE side. The direct consequence is that it is of difficulty to choose a so-called “conventional Borel window” namely strictly satisfying that the low dimension condensate should be bigger than the high dimension contribution. Coming to think of it, these main condensates could cancel each other out to some extent. Meanwhile, most of other high dimension condensates involved are very small, for which can not radically influence the character of OPE convergence. All of these factors make that the perturbative term could play an important role on the total OPE contribution and the convergence of OPE is still under control at the relatively low value of M2, and the lower bound of M2 is taken as 0.8GeV2 for this case.In the phenomenological side, a comparison between pole contribution and continuum contribution of sum rule (4) for the threshold s0=2.8GeV is shown in Fig. 2, which manifests that the relative pole contribution is about 50% at M2=1.6GeV2 and decreases with M2. In a similar way, the upper bounds of Borel parameters are M2=1.5GeV2 for s0=2.7GeV and M2=1.7GeV2 for s0=2.9GeV. Thereby, Borel windows for the scalar–scalar case are taken as 0.8∼1.5GeV2 for s0=2.7GeV, 0.8∼1.6GeV2 for s0=2.8GeV, and 0.8∼1.7GeV2 for s0=2.9GeV. In Fig. 3, the mass value MH as a function of M2 from sum rule (5) for the scalar–scalar case is shown and one can visually see that there are indeed stable Borel plateaus. In the chosen work windows, MH is calculated to be 2.37±0.33GeV. Furthermore, in view of the uncertainty due to the variation of quark masses and condensates, we have 2.37±0.33−0.03+0.17GeV (the first error is resulted from the variation of s0 and M2, and the second error reflects the uncertainty rooting in the variation of QCD parameters) or briefly 2.37−0.36+0.50GeV for the scalar–scalar tetraquark state.For the axial–axial case, its OPE contribution in sum rule (4) for s0=2.8GeV is shown in Fig. 4 by comparing various dimension contributions. Similarly, the dimension 3, 5, and 6 condensates could cancel each other out to some extent and most of other dimension condensates are very small. On the other hand, the phenomenological contribution in sum rule (4) for s0=2.8GeV is pictured in Fig. 5. Eventually, work windows for the axial–axial case are chosen as 0.9∼1.5GeV2 for s0=2.7GeV, 0.9∼1.6GeV2 for s0=2.8GeV, and 0.9∼1.7GeV2 for s0=2.9GeV. The corresponding Borel curves for the axial–axial case are displayed in Fig. 6 and its mass is evaluated to be 2.51±0.41GeV in the chosen work windows. With an eye to the uncertainty from the variation of quark masses and condensates, for the axial–axial tetraquark state we achieve 2.51±0.41−0.02+0.20GeV (the first error reflects the uncertainty from the variation of s0 and M2, and the second error roots in the variation of QCD parameters) or shortly 2.51−0.43+0.61GeV.For the pseudoscalar–pseudoscalar case, its various dimension OPE contribution in sum rule (4) for s0=2.8GeV is shown in Fig. 7. One may see that there are also three main condensates, i.e. the dimension 3, 5, and 6 condensates. However, what apparently distinct from the foregoing two cases is that two main condensates (i.e. the dimension 3 and 6 condensates) have a different sign comparing to the perturbative term, which leads that the perturbative part and the total OPE even have different signs at length. The dissatisfactory OPE property causes that related Borel curves are rather unstable visually, and it is difficult to find reasonable work windows for this case. Accordingly, it is not advisable to continue extracting a numerical result.For the vector–vector case, its different dimension OPE contribution in sum rule (4) for s0=2.8GeV is shown in Fig. 8. There appears the analogous problem as the pseudoscalar–pseudoscalar case, and the most direct consequence is that corresponding Borel curves are quite unstable. Hence it is hard to find appropriate work windows to grasp an authentic mass value for the vector–vector case.2.4Numerical analysis with ρ=3 and other discussionsFrom the analysis in part C, one could see that high-dimension condensates have been included in the OPE to improve the M2-stability of the sum rules. It is needed to point out that the included condensates are a part of more general condensate contributions at a given dimension. There is another source of uncertainty in the numerical results. Namely for the d=6 four-quark condensate, a general factorization 〈q¯qq¯q〉=ρ〈q¯q〉2 has been hotly discussed in the past [38,39], where ρ is a constant, which may be equal to 1, to 2, or be even smaller than 1. (In particular, in Ref. [40] it is argued that this factorization may not happen at all.) Furthermore, there may be a poor quantitative control of the four-quark condensate due to the violation of factorization parameter which could be about ρ=3∼4 [41]. This feature indicates that the error quoted in the final result which does not take into account such a violation may be underestimated. Therefore, it is very necessary to investigate the effect of the factorization breaking.Compromisingly, in this part we set ρ=3 for the 〈q¯qq¯q〉=ρ〈q¯q〉2 factorization. Thus, there will be a factor 3 for the four-quark condensate 〈q¯q〉2 and for the related classes of these high-dimension condensates (i.e. 〈q¯q〉3, 〈q¯q〉2〈gq¯σ⋅Gq〉, 〈q¯q〉2〈g2G2〉, and 〈q¯q〉2〈g3G3〉) in the OPE side. From the similar analysis process as above, the relevant working windows for the scalar–scalar case are taken as: M2=0.9∼1.9GeV2 for s0=2.7GeV, M2=0.9∼2.0GeV2 for s0=2.8GeV, and M2=0.9∼2.1GeV2 for s0=2.9GeV. The Borel curves for this case are shown in Fig. 9 and in the chosen windows its mass is computed to be 2.23±0.18GeV. Considering uncertainty due to the variation of quark masses and condensates, one can achieve the final result 2.23−0.24+0.78GeV.Similarly, with ρ=3 the working windows for the axial–axial case are taken as: M2=0.9∼1.8GeV2 for s0=2.7GeV, M2=0.9∼1.9GeV2 for s0=2.8GeV, and M2=0.9∼2.0GeV2 for s0=2.9GeV. The corresponding Borel curves are shown in Fig. 10 and its mass is evaluated to be 2.52±0.47GeV in the work windows. With a view to uncertainty due to the variation of quark masses and condensates, we have the eventual result 2.52−0.52+0.76GeV.By this time, note that the mc value is taken as the running charm quark mass 1.27±0.03GeV, which is often used in the existing literature. Without any evaluation of the perturbation theory radiative corrections, one can equally use the pole mass Mc≐1.4∼1.5GeV [41]. One should take into account a such ambiguity of the charm quark mass definition to clarify the effects on numerical results for the choice of mass (running or pole). After setting ρ=3, replacing the charm running mass by the pole mass, and carrying on the same analysis process as above, one can obtain the mass ranges 2.11∼3.16GeV for the scalar–scalar configuration and 2.11∼4.31GeV for the axial–axial case.Additionally, the way to construct a scalar out of two axial–vector currents or two vector currents could be not unique. In general, when one combines two spin 1 currents one may obtain states with spin=0,1 and 2. To be sure that one is dealing with a scalar, it is needed to project the combination of the currents into the spin 0 channel, which can be done with the help of the projection operators [42]. Since the direct contraction used here contains the overlap with the rigorous projection, the results found in this work can be close enough to the projection disposal. Certainly, one should note that on the final results there may exist the source of uncertainty from different treatments of currents.3SummaryTriggered by the new observation of Ds0⁎(2317) by BESIII Collaboration, we investigate that whether Ds0⁎(2317) could be a 0+ tetraquark state employing QCD sum rules. In order to insure the quality of sum rule analysis, contributions of condensates up to dimension 12 have been computed to test the OPE convergence. We find that some condensates, i.e. the two-quark condensate, the mixed condensate, and the four-quark condensate are of importance to the OPE side. Not bad for the scalar–scalar and the axial–axial cases, their main condensates could cancel each other out to some extent. Most of other condensates calculated are very small, which means that they could not radically influence the character of OPE convergence. All these factors bring that the OPE convergence for the scalar–scalar and the axial–axial cases is still controllable.To the end, we gain the following results: firstly, the final result for the scalar–scalar case is 2.37−0.36+0.50GeV with the factorization parameter ρ=1 (or 2.23−0.24+0.78GeV with ρ=3), which is in good agreement with the experimental value of Ds0⁎(2317). This result supports that Ds0⁎(2317) could be deciphered as a 0+ tetraquark state with the scalar–scalar configuration. Secondly, the eventual result for the axial–axial case is 2.51−0.43+0.61GeV with ρ=1 (or 2.52−0.52+0.76GeV with ρ=3), which is still coincident with the data of Ds0⁎(2317) considering the uncertainty although its central value is somewhat higher. In this way, one could not preclude the possibility of Ds0⁎(2317) as an axial–axial configuration tetraquark state. Meanwhile, one should note the weakness of convergence in the OPE side while presenting these results. Thirdly, the obtained mass ranges are 2.11∼3.16GeV for the scalar–scalar configuration and 2.11∼4.31GeV for the axial–axial case while setting ρ=3 and taking the charm pole mass, which are both in accord with the experimental value of Ds0⁎(2317). 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