\{Q\bar{s}\}\{\bar{Q}^{(^{\prime})}s\} molecular states in QCD sum rules Supported by the National Natural Science Foundation of China (10675167 and 10975184).

# $\{Q\bar{s}\}\{\bar{Q}^{(')}s\}$ molecular states in QCD sum rules

## Abstract

We systematically investigate the mass spectra of molecular states in the framework of QCD sum rules. The interpolating currents representing the molecular states are proposed. Technically, contributions of the operators up to dimension six are included in operator product expansion (OPE). The masses for molecular states with various configurations are presented. The result for the molecular state is consistent with the mass of the newly observed , which could support interpreted as a molecular state.

###### pacs:
11.55.Hx, 12.38.Lg, 12.39.Mk

## I Introduction

The field of heavy hadron spectroscopy is experiencing a rapid advancement mainly propelled by the continuous observations of hadronic resonances, for example, (1), (2), (3), (4), (5), (6), (7), (7), (8) etc. (for experimental reviews, e.g., see (9); (10)), some of which are not easy to accommodate within the quark model picture and may not be conventional charmonium states. Masses for some of these hadrons are very close to the meson-meson thresholds, for which are interpreted as possible molecular candidates in (11); (12); (13); (14); (15). For instance, the charmonium-like state has been interpreted as a molecular state (14); (15). Considering the symmetry of the light flavor quarks, there may also exist and have a rich spectroscopy for molecular states acting as the corresponding partners of molecular states. In fact, some authors have deciphered the newly observed as a molecular state (15); (16), just as the molecular partner of . Furthermore, QCD itself does not exclude the existence of molecular states besides conventional mesons and baryons, so studies of them may deepen one’s understanding of the strong interaction. On all accounts, it is interesting to study mass spectra for the molecular states. However, it is far from clear how to generate hadron masses from first principles in QCD since it is highly noperturbative in the low energy region where futile to attempt perturbative calculations, and then one has to treat a genuinely strong field in nonperturbative methods. Under such a circumstance, one could resort to QCD sum rule (17) (for reviews see (18); (19); (20); (21) and references therein), which is a comprehensive and reliable way for evaluating the nonperturbative effects. Up to now, there have been some works testing the state from QCD sum rules (22); (23); (24). Presently, we extend the work on molecular states (24) to various molecular states.

The paper is organized as follows. In Sec. II, QCD sum rules for the molecular states are introduced, and both the phenomenological representation and QCD side are derived, followed by the numerical analysis to extract the hadronic masses in Sec. III, and a brief summary in Sec. IV.

## Ii Molecular state QCD sum rules

### ii.1 interpolating currents

Following the standard scheme (10), the mesons with are named , , , and for charmed mesons, with , , , and for bottom mesons, respectively. In this work, the corresponding configurations for these mesons are represented as , , , and . In full theory, the interpolating currents for these mesons can be found in Refs. (25); (26). Presently, one constructs the molecular state current from meson-meson type of fields, while constructs the tetraquark state current from diquark-antidiquark configuration of fields. The currents constructed from meson-meson type of fields can be related to those composed of diquark-antidiquark type of fields by Fiertz rearrangements. However, the relations are suppressed by a typical color and Dirac factors so that one could obtain a reliable sum rule only if one has chosen the appropriate current to have a maximum overlap with the physical state, which is expected to be particularly true for multiquark configuration with special molecular or diquark structures. Concretely, it will have a maximum overlap for the molecular state using the meson-meson current and the sum rule can reproduce the physical mass well, whereas the overlap for the molecular state employing a diquark-antidiquark type of current will be small and the sum rule will not be able to reproduce the mass well. Likewise, the opposite is also true (there are some more concrete calculations and discussions in the XII. Appendix in Ref. (27)). Consequently, the interpolating currents for the related molecular states are constructed. For one type of hadrons, with

 j(Q¯s)(¯Q′s)=(¯saiγ5Qa)(¯Q′biγ5sb),

coupling to , , , or molecular state,

 j(Q¯s)∗(¯Q′s)∗=(¯saγμQa)(¯Q′bγμsb),

for , , , or state,

 j(Q¯s)∗0(¯Q′s)∗0=(¯saQa)(¯Q′bsb),

for , , , or state,

 j(Q¯s)1(¯Q′s)1=(¯saγμγ5Qa)(¯Q′bγμγ5sb),

for , , , or state,

 j(Q¯s)(¯Q′s)∗0=(¯saiγ5Qa)(¯Q′bsb),

for , , , or state, and

 j(Q¯s)∗(¯Q′s)1=(¯saγμQa)(¯Q′bγμγ5sb),

for , , , or state, where and denote heavy quarks ( or ), with and are color indices. For another type, with

 jμ(Q¯s)∗(¯Q′s)=(¯saγμQa)(¯Q′biγ5sb),

for , , , or state,

 jμ(Q¯s)1(¯Q′s)=(¯saγμγ5Qa)(¯Q′biγ5sb),

for , , , or state,

 jμ(Q¯s)∗(¯Q′s)∗0=(¯saγμQa)(¯Q′bsb),

for , , , or state, and

 jμ(Q¯s)1(¯Q′s)∗0=(¯saγμγ5Qa)(¯Q′bsb),

for , , , or state.

### ii.2 the molecular state QCD sum rule

For the former case, the starting point is the two-point correlator

 Π(q2)=i∫d4xeiq.x⟨0|T[j(x)j+(0)]|0⟩. (1)

The correlator can be phenomenologically expressed as

 Π(q2)=λ2HM2H−q2+1π∫∞s0dsImΠphen(s)s−q2+% subtractions, (2)

where denotes the mass of the hadronic resonance, and gives the coupling of the current to the hadron . In the OPE side, the correlator can be written as

 Π(q2)=∫∞(mQ+mQ′+2ms)2dsρOPE(s)s−q2  (mQ=mQ′ or mQ≠mQ′), (3)

where the spectral density is given by . After equating the two sides, assuming quark-hadron duality, and making a Borel transform, the sum rule can be written as

 λ2He−M2H/M2 = ∫s0(mQ+mQ′+2ms)2dsρOPE% (s)e−s/M2, (4)

where indicates Borel parameter. To eliminate the hadronic coupling constant , one reckons the ratio of derivative of the sum rule and itself, and then yields

 M2H = ∫s0(mQ+mQ′+2ms)2dsρOPEse−s/M2∫s0(mQ+mQ′+2ms)2dsρOPEe−s/M2. (5)

For the latter case, one starts from

 Πμν(q2)=i∫d4xeiq.x⟨0|T[jμ(x)jν+(0)]|0⟩. (6)

Lorentz covariance implies that the correlator (6) can be generally parameterized as

 Πμν(q2)=(qμqνq2−gμν)Π(1)(q2)+qμqνq2Π(0)(q2). (7)

The of the correlator proportional to will be chosen to extract the mass sum rule here. Similarly, one can finally yield

 M2H=∫s0(mQ+mQ′+2ms)2dsρOPEse−s/M2∫s0(mQ+mQ′+2ms)2dsρOPEe−s/M2, (8)

with .

### ii.3 spectral densities

Calculating the OPE side, one works at leading order in and considers condensates up to dimension six, utilizing the similar techniques in Refs. (28); (29). The quark is dealt as a light one. To keep the heavy-quark mass finite, one uses the momentum-space expression for the heavy-quark propagator. One calculates the light-quark part of the correlation function in the coordinate space, which is then Fourier-transformed to the momentum space in dimension. The resulting light-quark part is combined with the heavy-quark part before it is dimensionally regularized at . For the heavy-quark propagator with two and three gluons attached, the momentum-space expressions given in Ref. (25) are used. After some tedious calculations, the concrete forms of spectral densities can be derived, which are collected in the Appendix. In detail, some different currents lead to the similar OPE, for example, the terms for and are similar, the ones for and are similar and so on. Although the terms for them are similar respectively, the corresponding signs may be different, such as a term for may be “plus” sign while the related one for may be “minus”, which caused by the differences of -matrices in the interpolating currents and the differences of the trace results. Numerically, the two quark condensate is the most important condensate correction, the absolute value of which is bigger than the absolute values of the four quark condensate as well as the mixed condensate . Meanwhile, the two gluon condensate and the three gluon condensate are very small and almost negligible.

## Iii Numerical analysis

In this part, the sum rules (5) and (8) will be numerically analyzed. The input values are taken as , , and (10), with , , , , , and (20); (28). Complying with the standard procedure of sum rule analysis, the threshold and Borel parameter are varied to find the optimal stability window. Namely, we try to consider the Borel curve stability’s dependence on the Borel plateaus (the threshold and Borel parameter ), and find the Borel windows where the perturbative contribution should be larger than the condensate contributions in the OPE side while the pole contribution should be larger than continuum contribution in the phenomenological side. Thus, the regions of thresholds are taken as values presented in the related figure captions, with for , , , , , , and , for , , , , , , , , , , , , , , , and , and for , , , , , , and , respectively. Tables 1-2 collect all the numerical results. Note that uncertainties are owing to the sum rule windows (variation of the threshold and Borel parameter ), not involving the ones from the variation of quark masses and QCD parameters for which are appreciably smaller in comparison with the ones from the sum rule windows here. The numerical result for agrees well with the mass for (24), which supports the molecular configuration for . After the completion of the calculations here, evidence for a new resonance (named as ) has been observed by the Belle Collaboration (30). Note that the predicted value for the molecular state here is consistent with the mass of the newly observed structure.

## Iv Summary

In summary, QCD sum rules have been employed to compute the masses of molecular states, including the contributions of operators up to dimension six in OPE. Ultimately, we have arrived at mass spectra for molecular states with various configurations. The numerical result for agrees well with the mass for (24), which supports the interpretation of as a molecular state. The predicted value for the molecular state is consistent with the mass of the newly observed , which could support interpreted as a molecular state. More experimental evidence on molecular states besides and may appear if they do exist, and the data on molecular states are expecting further experimental identification, which may be searched for experimentally at facilities such as Super-B factories in the mass spectrum in the future.

## Appendix

It is defined that . With

 ρpert(s) = 3211π6∫αmaxαmindα∫1−αβmindβ(1−α−β)[1α3β3r(mQ,mQ′)2 −22mQ′msα3β2r(mQ,mQ′)−22mQmsα2β3r(mQ,mQ′)+3⋅22mQmQ′m2sα2β2] ×r(mQ,mQ′)2, ρ⟨¯ss⟩(s) = 3⟨¯ss⟩27π4∫αmaxαmindα{∫1−αβmindβ[−mQ′α2βr(mQ,mQ′)−mQαβ2r(mQ,mQ′) +22mQmQ′msαβ]r(mQ,mQ′)+{msα(1−α)[αm2Q+(1−α)m2Q′−α(1−α)s] −mQm2s1−α−mQ′m2sα}[αm2Q+(1−α)m2Q′−α(1−α)s]}, ρ⟨¯ss⟩2(s) = ⟨¯ss⟩225π2{[2mQmQ′−(mQ+mQ′)ms]√(s−m2Q+m2Q′)2−4m2Q′s/s +3m2s∫αmaxαmindαα(1−α)}, ρ⟨g¯sσ⋅Gs⟩(s) = 3⟨g¯sσ⋅Gs⟩28π4{∫αmaxαmindα{(−mQ′α−mQ1−α)[αm2Q+(1−α)m2Q′−α(1−α)s] +2ms3[2αm2Q+2(1−α)m2Q′−3α(1−α)s]} +[2mQmQ′ms−(mQ+mQ′)m2s3]√(s−m2Q+m2Q′)2−4m2Q′s/s}, ρ⟨g2G2⟩(s) = ⟨g2G2⟩211π6∫αmaxαmindα∫1−αβmindβ(1−α−β)[m2Q′α3r(mQ,mQ′)+m2Qβ3r(mQ,mQ′) −3mQ′msα3r(mQ,mQ′)−m3Q′msβα3−3mQmsβ3r(mQ,mQ′)−m3Qmsαβ3 −mQm2Q′msα2−m2QmQ′msβ2+3mQmQ′m2sα2+3mQmQ′m2sβ2], ρ⟨g3G3⟩(s) = ⟨g3G3⟩213π6∫αmaxαmindα∫1−αβmindβ(1−α−β)[1α3r(mQ,mQ′)+2m2Q′βα3 +1β3r(mQ,mQ′)+2m2Qαβ3−6mQ′msβα3−6mQmsαβ3−mQmsα2−mQ′msβ2],

for ,

 ρpert(s) = 3212π6∫αmaxαmindα∫1−αβmindβ(1−α−β)[1α3β3(1+α+β)r(mQ,mQ′)2 −22mQ′msα3β2(1+α+β)r(mQ,mQ′)−23mQmsα2β3r(mQ,mQ′) +3⋅23mQmQ′m2sα2β2]r(mQ,mQ′)2, ρ⟨¯ss⟩(s) = 3⟨¯ss⟩27π4∫αmaxαmindα{∫1−αβmindβ[−mQ′α2β(α+β)r(mQ,mQ′)−mQαβ2r(mQ,mQ′) −msαβr(mQ,mQ′)+22mQmQ′msαβ+mQ′m2sα]r(mQ,mQ′) +{msα(1−α)[αm2Q+(1−α)m2Q′−α(1−α)s]−mQm2s1−α−mQ′m2sα}[αm2Q +(1−α)m2Q′−α(1−α)s]}, ρ⟨¯ss⟩2(s) = ⟨¯ss⟩226π2[(22mQmQ′−2mQms)√(s−m2Q+m2Q′)2−4m2Q′s/s +∫αmaxαmindα(−2mQ′ms+3m2sα)(1−α)], ρ⟨g¯sσ⋅Gs⟩(s) = 3⟨g¯sσ⋅Gs⟩28π4{∫αmaxαmindα{∫1−αβmindβmQ′αr(mQ,mQ′)−(mQ′α+mQ1−α)[αm2Q +(1−α)m2Q′−α(1−α)s]+2ms3[αm2Q+(1−α)m2Q′−2α(1−α)s]} +(2mQmQ′ms−mQm2s3)√(s−m2Q+m2Q′)2−4m2Q′s/s −mQ′m2s3∫αmaxαmindα(1−α)}, ρ⟨g2G2⟩(s) = ⟨g2G2⟩212π6∫αmaxαmindα∫1−αβmindβ(1−α−β)[m2Q′α3(1+α+β)r(mQ,mQ′) +m2Qβ3(1+α+β)r(mQ,mQ′)−3mQ′msα3(1+α+β)r(mQ,mQ′) −m3Q′msβα3(1+α+β)−6mQmsβ3r(mQ,mQ′)−