PHYSICAL REVIEW D 101, 034502 (2020)
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Tetraquark interpolating fields in a lattice QCD investigation ð Þ of the Ds0 2317 meson Constantia Alexandrou,1,2 Joshua Berlin,3 Jacob Finkenrath,4 Theodoros Leontiou ,5 and Marc Wagner 3 1Department of Physics, University of Cyprus, P.O. Box 20537, 1678 Nicosia, Cyprus 2Computation-based Science and Technology Research Center, The Cyprus Institute, 20 Kavafi Street, 2121 Nicosia, Cyprus 3Goethe-Universität Frankfurt am Main, Institut für Theoretische Physik, Max-von-Laue-Straße 1, D-60438 Frankfurt am Main, Germany 4Computation-based Science and Technology Research Center, The Cyprus Institute, 20 Kavafi Street, 2121 Nicosia, Cyprus 5Department of Mechanical Engineering, Frederick University, 1036 Nicosia, Cyprus
(Received 4 December 2019; accepted 15 January 2020; published 4 February 2020)
We investigate the Ds0ð2317Þ meson using lattice QCD and considering correlation functions of several cs¯ two-quark and cs¯ ðuu¯ þ dd¯ Þ four-quark interpolating fields. These interpolating fields generate different structures in color, spin and position space including quark-antiquark pairs, tetraquarks and two-meson scattering states. For our computation we use an ensemble simulated with pion mass mπ ≈ 0.296 GeV and spatial volume of extent 2.90 fm. We find in addition to the expected spectrum of two-meson scattering states another state around 60 MeV below the DK threshold, which we interpret as the Ds0ð2317Þ meson. This state couples predominantly to a quark-antiquark interpolating field and only weakly to a DK two- meson interpolating field. The coupling to the tetraquark interpolating fields is essentially 0, rendering a tetraquark interpretation of the Ds0ð2317Þ meson rather unlikely. Moreover, we perform a scattering analysis using Lüscher’s method and the effective range approximation to determine the Ds0ð2317Þ mass for infinite spatial volume. We find this mass 51 MeV below the DK threshold, rather close to both our finite volume result and the experimentally observed value.
DOI: 10.1103/PhysRevD.101.034502
I. INTRODUCTION as e.g., suggested by Refs. [12,13]. This picture is also supported by recent papers [14–17], where DK molecular The D ð2317Þ meson with quantum numbers IðJPÞ¼ s0 components from around 60% to 75% are found. Other 0ð0þÞ, strangeness S ¼ 1 and charm C ¼ S has mass interesting approaches, which are able to explain the mD ¼ 2.3178ð5Þ GeV, around 45 MeV below the DK s0 surprisingly low mass of the Ds0ð2317Þ meson, are e.g., threshold [1–4]. This experimental result is in contrast to presented in Ref. [18], where the Ds0ð2317Þ meson is a theoretical predictions from quark models (see e.g., standard cs¯ configuration with the coupling to the nearby – D ð2317Þ Refs. [5 7]), where the s0 meson is treated as a DK threshold taken into account, and in Refs. [19–21], cs¯ quark-antiquark pair, which leads to a significantly which are based on an SU(3) chiral Lagrangian. For a larger mass in the range of 100 to 200 MeV above the more detailed discussion of the properties of the Ds0ð2317Þ experimental value. Because of that discrepancy, there is meson and existing literature, we refer to the review an ongoing debate about the quark composition of the articles [22,23]. D ð2317Þ s0 meson. Besides a standard quark-antiquark Early quenched lattice QCD studies [24–30] of the structure it could also have a four-quark structure. For D ð2317Þ meson found masses significantly larger than – s0 example Refs. [8 10] propose a tetraquark structure, while the experimental result, similar to quark model predictions. Ref. [11] provides arguments against such a scenario. There are also more recent lattice QCD studies [31–39], DK Another possibility is a mesonic molecule structure where only quark-antiquark interpolating fields of flavor structure cs¯ were taken into account. The majority of these studies also find masses for the Ds0ð2317Þ meson, which Published by the American Physical Society under the terms of are larger than the experimental value, in particular, if the Creative Commons Attribution 4.0 International license. extrapolations to physical quark masses and to the con- Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, tinuum were performed (see e.g., Ref. [37]). If, however, and DOI. Funded by SCOAP3. in addition to quark-antiquark interpolating fields also
2470-0010=2020=101(3)=034502(14) 034502-1 Published by the American Physical Society CONSTANTIA ALEXANDROU et al. PHYS. REV. D 101, 034502 (2020) two-meson DK interpolating fields are included, as is done the correlation function with the particular advantage of in the recent precision lattice QCD computations presented exploiting also data at small temporal separations, where in Refs. [40–42], Ds0ð2317Þ masses below the DK thresh- statistical errors are typically small. In addition to AMIAS old and close or consistent with the experimental result are we also use the standard generalized eigenvalue problem found. In these investigations almost physical u and d (GEVP) method; i.e., we solve generalized eigenvalue quark masses were used, corresponding to pion masses problems and extract the spectrum from effective energy mπ ≈ 0.156 GeV and mπ ≈ 0.150 GeV, respectively, and plateaus (cf. e.g., [48] and references therein). Note that Lüscher’s method was employed, to obtain the meson mass both the GEVP and AMIAS provide information on the at infinite spatial volume. In this context it is also relative importance of the considered interpolating fields. interesting to mention two closely related lattice QCD Combining both methods allows one to check the robust- investigations. In Ref. [43] scattering of charmed and light ness of our results. pseudoscalar mesons was studied, including DK¯ scattering, This paper is organized as follows: In Sec. II we describe and by using SU(3) flavor symmetry the Ds0ð2317Þ mass the lattice setup and techniques with particular focus on the was obtained in agreement with experiment and support for implemented interpolating fields. In Sec. III we discuss the the interpretation as a DK molecule was found. In Ref. [44] spectral decomposition of the corresponding two-point scattering in the D0 sector was studied, however, with correlation functions. A short description of our two rather heavy quark masses, somewhere between the physi- analysis methods, the GEVP and AMIAS, is provided in cal light and strange quark, corresponding to a pion mass Sec. IV. Section V is the main section of this work, where mπ ≈ 0.391 GeV, which led to some insights on the our numerical results are presented. First, in Sec. VA,we qualitative difference between the D0ð2400Þ meson and show several finite volume spectra for the sector with D ð2317Þ the Ds0ð2317Þ meson. s0 quantum numbers corresponding to different In this work we also study the Ds0ð2317Þ meson using sets of interpolating fields. Based on these results the lattice QCD with particular focus on tetraquark interpolat- importance of each interpolating field is discussed. Then, in ’ ing fields. As in the aforementioned lattice QCD inves- Sec. VB, we perform a scattering analysis using Lüscher s tigations [40–42] we consider both quark-antiquark and method and the effective range expansion to determine the D ð2317Þ two-meson interpolating fields. In addition, we include for s0 mass at infinite volume. In Sec. VI we sum- the first time also tetraquark interpolating fields, where marize our findings and give our conclusions. the four quarks are centered at the same point in space. We implemented color and spin contractions, where two II. INTERPOLATING FIELDS standard meson interpolating fields of quark-antiquark type AND LATTICE SETUP are put on top of each other (resembling DK and Dsη To investigate the D ð2317Þ meson, we consider a 7 × 7 mesonic molecules), as well as contractions, which have a s0 correlation matrix, diquark-antidiquark structure. Including such tetraquark interpolating fields might be essential, as it has recently j k† CjkðtÞ¼hO ðt2ÞO ðt1Þi;t¼ t2 − t1: ð1Þ been reported in Ref. [45] for the positive parity mesons a0ð980Þ and K ð700Þ. In both cases a low-lying energy 0 The interpolating fields Oj, j ¼ 1; …; 7 have either a two- level is missed, if they are not taken into account. quark cs¯ orpffiffiffi a four-quark cs¯ qq¯ structure, where qq¯ ¼ Moreover, we compute the couplings of the low-lying ðuu¯ þ dd¯ Þ= 2 states to different types of two-quark and four-quark . In detail we consider the interpolating interpolating fields and compare the spectra obtained from fields X different subsets of interpolating fields. This might shed 1 qq;¯ 1 O ¼ O ¼ N1 ðc¯ðxÞsðxÞÞ; ð2Þ additional light on the question whether the Ds0ð2317Þ x meson is predominantly a cs¯ state or rather has a large X tetraquark component. 2 qq;¯ γ O ¼ O 0 ¼ N2 ðc¯ðxÞγ0sðxÞÞ; ð3Þ We perform our computations in a single spatial volume x of extent 2.90 fm and at unphysically heavy u and d quark mass corresponding to mπ ≈ 296 MeV. For the analysis O3 ¼ ODK;point; of correlation functions we apply the Athens Model X ¼ N ðc¯ðxÞγ qðxÞÞðq¯ðxÞγ sðxÞÞ; ð Þ Independent Analysis Scheme (AMIAS), an analysis 3 5 5 4 method based on statistical concepts for extracting excited x states from correlation functions. AMIAS is a novel O4 ¼ ODsη;point; analysis method, which has previously been used in a X study of the nucleon spectrum and the a0ð980Þ meson ¼ N4 ðc¯ðxÞγ5sðxÞÞðq¯ðxÞγ5qðxÞÞ; ð5Þ [46,47]. AMIAS utilizes all the information encoded in x
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¯ a ð980Þ O5 ¼ OQQ;γ5 ; 0 quantum numbers. In this work, we also compute X and compare the overlaps of the corresponding trial states T j ¼ N5 ϵabcðc¯bðxÞðCγ5Þq¯c ðxÞÞ O jΩi (jΩi denotes the vacuum) to the lowest energy x eigenstate, to obtain certain information about the quark T D ð2317Þ ϵadeðqd ðxÞðCγ5ÞseðxÞÞ; ð6Þ composition of the s0 meson. This might contrib- ute to the ongoing debate whether the Ds0ð2317Þ meson is O6 ¼ ODK;2part; predominantly a quark-antiquark pair or a tetraquark (see X the discussion in Sec. I). ¼ N ðc¯ðxÞγ qðxÞÞðq¯ðyÞγ sðyÞÞ; ð Þ 6 5 5 7 Note that the interpolating fields O3 to O7 do not x;y generate orthogonal trial states. For example the terms x ¼ y O7 ¼ ODsη;2part; with in Eqs. (7) and (8) also appear in Eqs. (4) X and (5). Similarly, one can relate two-meson combinations ¼ N7 ðc¯ðxÞγ5sðxÞÞðq¯ðyÞγ5qðyÞÞ: ð8Þ to diquark-antidiquark combinations via a Fierz identity; x;y i.e., some of the terms present in Eqs. (4) and (5) are also part of Eq. (6) and vice versa. Even though the seven C denotes the charge conjugation matrix and the normali- interpolating fields do not generate orthogonal trial states, zation factors Nj are chosen such that Cjjðt ¼ aÞ¼1 (no they are not linearly dependent either, because each of them sum over j; a is the lattice spacing), i.e., in a way that the contains terms not present in any of the other six. Their interpolating fields generate trial states with similar norm. nonorthogonality does not cause any particular problems All interpolating fields couple to the Ds0ð2317Þ meson and during our analyses, because the two methods we use, the to other states with the same quantum numbers. As in GEVP and AMIAS, are both able to deal with correlation previous lattice QCD computations [40–42] we consider matrices based on nonorthogonal trial states. We remark quark-antiquark interpolating fields, Oqq;¯ 1 and Oqq;¯ γ0 , that on a technical level this work is similar to our lattice DK;2part as well as two-meson interpolating fields, O and QCD investigation of the a0ð980Þ meson [47], because to a ODsη;2part. In Refs. [40–42] it was shown that the latter large extent the same interpolating fields are used, just with interpolating fields are essential to determine the energy different quark flavors. of the ground state and the first excitation reliably. In To increase the coupling of the interpolating fields to the addition we implemented the tetraquark interpolating fields low-lying energy eigenstates, quark fields in Eqs. (2)–(8) ¯ ODK;point, ODsη;point, and OQQ;γ5 with the four quark are Gaussian smeared with APE smeared spatial gauge operators located at the same point in space. links (cf. Refs. [50,51]). The smearing parameters are DK;2 D η;2 κ ¼ 0 5 N ¼ 50 α ¼ 0 45 N ¼ 20 The interpolating fields O part and O s part mostly Gauss . , Gauss , APE . and APE , generate DK and Dsη scattering states, which are where detailed equations are given in [52]. expected to have energies somewhat above the mass of To compute the correlation functions, we use an ensem- D ð2317Þ m þ m − m ≈ 45 ble of around 500 gauge link configurations generated with the s0 meson ( D K Ds0 MeV and DK;2part Nf ¼ 2 þ 1 dynamical Wilson clover quarks and the mD þ mη − mD ≈ 200 MeV [4]). In contrast to O s s0 Iwasaki gauge action by the PACS-CS Collaboration ODsη;2part and , where both mesons have zero momentum, [53]. The lattice size is 64 × 323 with lattice spacing a ¼ ODK;point ODsη;point the interpolating fields and represent 0.0907ð14Þ fm, i.e., the spatial lattice extent L is around two mesons centered at the same point in space and, 2.90 fm. The u and d quark mass and the s quark mass thus, resemble mesonic molecules. Similarly, due to the m ≈ 0 296 ¯ correspond to the pion mass π . GeV and the kaon OQQ;γ5 different color structure, resembles a diquark- mass mK ≈ 0.597 GeV; i.e., they are both heavier than in antidiquark pair. the real world, while the c quark mass corresponds to the D ODK;point ODsη;point Tetraquark interpolating fields like , meson mass mD ≈ 1.845 GeV, i.e., is slightly lighter (see ¯ and OQQ;γ5 were not considered in previous lattice QCD the detailed discussion in Sec. VB). Note that the c quark studies of the Ds0ð2317Þ meson. Thus, the main goal of this only appears as a valence quark. work is to explore whether the inclusion of these tetraquark In a recent publication [54], we implemented and interpolating fields has an effect on the lattice QCD compared various combinations of techniques for the determination of the low-lying spectrum. Similar recent computation of propagators and correlation functions investigations of systems not including the Ds0ð2317Þ including point-to-all propagators, stochastic timeslice-to- meson have led to different findings regarding the impor- all propagators, the one-end trick and sequential propa- tance of tetraquark interpolating fields. While in gators. For each diagram of a similar 6 × 6 correlation Refs. [47,49] only marginal differences in the resulting matrix, which we used to study the a0ð980Þ meson [47],we spectra of the I ¼ 1 hidden-charm and doubly charmed determined the most efficient combination of techniques. sectors and the a0ð980Þ sector were found, Ref. [45] We have applied the same combinations of techniques in observed additional energy levels both with K0ð700Þ and this work to compute the 7 × 7 correlation matrix (1) with
034502-3 CONSTANTIA ALEXANDROU et al. PHYS. REV. D 101, 034502 (2020) P truncated the interpolating fields (2) to (8). Finding efficient methods Em ¼ Em − EΩ and m denotes the sum over a finite is particularly important for diagrams, where quarks propa- number of low-lying energy eigenstates in the sector with gate within a timeslice, e.g., diagrams containing closed Ds0ð2317Þ quantum numbers, which is probed by the quark loops. These diagrams are significantly more noisy interpolating fields (2) to (8) (in the following we assume than their counterparts, where quarks do not propagate the ordering E0 ≤ E1 ≤ E2 ≤ …). within a time slice. Their noise-to-signal ratio grows For temporal separations t around T=2 the correlation exponentially with increasing temporal separation as dis- functions CjkðtÞ have a more complicated expansion than cussed in Ref. [54]. Eq. (12), if there are low-lying multihadron states with the same quantum numbers. In our case, i.e., for Ds0ð2317Þ III. CORRELATION FUNCTIONS FOR PERIODIC quantum numbers, the lowest multihadron state is a DK TEMPORAL DIRECTION scattering state, which has an energy only slightly above the mass of the Ds0ð2317Þ meson. Clearly, the interpolating A correlation function computed on a lattice with DK T fields (2) to (8) not only excite such a scattering state, periodic temporal direction of extension can be expanded jΩi according to when applied to the vacuum , but also yield non- vanishing matrix elements hDjOjjKi and hKjOjjDi, i.e., j k† D CjkðtÞ¼hO ðtÞO ð0Þi annihilate a kaon and create a meson and vice versa. For X example a significant contribution to CjjðtÞ is 1 j ¼ e−EmðT−tÞc e−Entðck Þ ð Þ Z m;n m;n 9 m;n 2 j −ðEDþEK ÞT=2 2 e ðcD;KÞ coshððED − EKÞðt − T=2ÞÞ with energy eigenstates jmi, corresponding energy Z j j j ≈ 2e−ðmDþmK ÞT=2ðc Þ2 ððm − m Þðt − T=2ÞÞ eigenvaluesP Em, possibly complex cm;n ¼hmjO jni and D;K cosh D K Z ¼ e−EmT m . ð13Þ Using the QCD symmetries charge conjugation and time reversal, one can show that all elements of the correlation as can be seen from Eq. (10). Assuming coefficients matrix (1) with interpolating fields (2) to (8) are real. j j Moreover, one can rewrite Eq. (9) in more convenient form, jcD;Kj ≈ jcDK;Ωj, where DK denotes a low-lying DK t ≈ T=2 X scattering state, one can see that in the region of 1 j C ðtÞ¼ e−ðEmþEnÞT=2c ck the corresponding terms in Eq. (10) are comparable in jk Z m;n m;n m;n magnitude. Therefore, terms as in Eq. (13) have to be taken into account, when extracting energy levels from correla- HjkððEm − EnÞðt − T=2ÞÞ; ð10Þ tion functions at large temporal separations t ≈ T=2. j For smaller temporal separations t such contributions with real cm;n and may be neglected, since they are exponentially suppressed ∝ e−2mK ðt−T=2Þ t −sinhðxÞ for j ¼ 2;k≠ 2 and j ≠ 2;k¼ 2 with decreasing . Analytical estimates as H ðxÞ¼ : jk þ ðxÞ well as numerical experiments have shown that within our cosh otherwise setup this is the case for t ≲ 15a, which is an upper bound ð11Þ for all t fitting ranges used in the following.
Since the elements of the correlation matrix are either IV. ANALYSIS METHODS symmetric with respect to the reversal of time, CjkðtÞ¼ þCjkðT − tÞ for HjkðxÞ¼þcoshðxÞ, or antisymmetric, To analyze the 7 × 7 correlation matrix discussed in CjkðtÞ¼−CjkðT − tÞ for HjkðxÞ¼− sinhðxÞ, it is suffi- Sec. III and various submatrices, we use both the GEVP cient to restrict the following discussion to temporal method and the AMIAS method. While the GEVP separations 0 ≤ t ≤ T=2. method is quite common and very well known, the For sufficiently large T, where Z ≈ e−EΩT (Ω denotes the AMIAS method has proven to be particularly suited to vacuum), and for sufficiently large t, Eq. (10) reduces to study excited states [46] and was successfully used in our related previous lattice QCD study of the a0ð980Þ truncatedX j meson [47].InSec.V we show that both methods yield −EmT=2 k CjkðtÞ¼ 4e cm;Ωcm;ΩHjkðEmðt − T=2ÞÞ; consistent results, which we consider to be an important m cross-check, in particular, due to the fact that the signal- ð12Þ to-noise ratios of the elements of the correlation matrix grow rapidly with increasing temporal separations. In the if the correlation function is not contaminated by effects following we summarize both methods and discuss the related to multihadron states as discussed below. details of our analyses.
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t ≥ t A. GEVP method for min and that its statistical errors are still reasonably t ¼ t E A commonly used method to extract several energy small at max. The energy level m is then determined Ef ðtÞ levels from an N × N correlation matrix is to solve the by averaging eff;m over the fitting region, generalized eigenvalue problem, Xtmax 1 f CðtÞv ðt; t Þ¼λ ðt; t ÞCðt Þv ðt; t ÞðÞ E ¼ E ðtÞðÞ m 0 m 0 0 m 0 14 m ðt − t Þ=a þ 1 eff;m 21 max min t¼tmin (see e.g., Ref. [48] and references therein), where CðtÞ is the correlation matrix with entries CjkðtÞ (j; k ¼ 1; …;N), (see also Ref. [55], where a similar procedure was used). v ðt; t Þ vmðt; t0Þ the eigenvector corresponding to the eigenvalue The components of the eigenvectors m 0 obtained λmðt; t0Þ (m ¼ 0; …;N− 1) and t0 ≥ a an input parameter. by solving the GEVP (14) provide information about the We use t0 ¼ a, which is a typical choice. structure of the corresponding energy eigenstates, A number of N effective energies E ;mðtÞ can be X eff j j† obtained by solving jmi ≈ vmðt; t0ÞO jΩi; ð22Þ j λ ðt; t Þ ðE ðtÞðt − T=2ÞÞ m 0 cosh eff;m ¼ ð15Þ for sufficiently large t, where the ≈ sign denotes the λmðt − a; t0Þ coshðE ;mðtÞðt − a − T=2ÞÞ eff expansion of the energy eigenstate jmi within the subspace j† for each eigenvalue λmðt; t0Þ. At sufficiently large, but not spanned by the trial states O jΩi. We found that for t ≥ t too large temporal separations, i.e., in a t-region, where min the eigenvector components are constant within Eq. (12) is a valid parametrization of the correlation matrix, statistical errors. Thus, we average the eigenvector com- E ðtÞ j the effective energies eff;m exhibit plateaus. The values ponents vmðt; t0Þ according to of these plateaus correspond to the N lowest energy levels t E Xmax in the sector probed by the interpolating fields, i.e., to m. j 1 j vm ¼ vmðt; t0Þð23Þ We determine each energy level Em by first fitting ðt − t Þ=a þ 1 max min t¼tmin fðtÞ¼A ðE ðt − T=2ÞÞ þ A ðE ðt − T=2ÞÞ 0 cosh 0 1 cosh 1 j j and normalize via vm → vm=jvmj. ð16Þ λ ðt; t Þ t ≤ t ≤ t to the eigenvalue m 0 in the region min max, B. AMIAS method A A E
034502-5 CONSTANTIA ALEXANDROU et al. PHYS. REV. D 101, 034502 (2020) Z ¯ truncatedX truncatedX Ar ¼ dArArΠðArÞ; ð25Þ j† j† j O jΩi ≈ jmihmjO jΩi¼ cΩ;mjmi: ð30Þ m m Z 1=2 ¯ 2 ΔAr ¼ dArðAr − ArÞ ΠðArÞ : ð26Þ More interesting, however, is inverting Eq. (30) and writing the extracted energy eigenstates in terms of the trial states, X AMIAS is able to handle a rather large number of j j† jmi ≈ v˜ mO jΩi: ð31Þ parameters using Monte Carlo techniques; i.e., it is suited j to study several energy eigenstates, if the lattice QCD j results for correlation functions are sufficiently precise. One can show that the matrix formed by the coefficients v˜ m The PDF for the complete set of fit parameters is j is the inverse of the matrix formed by the coefficients cΩ;m defined by up to exponentially small corrections, i.e., X 1 −χ2=2 j j PðA1; A2; …Þ¼ e ð27Þ v˜ mcΩ;n ≈ δm;n: ð32Þ N j with appropriate normalization N and j Note that the coefficients v˜ m are equivalent to the eigen- j t 2 vm X Xmax ðC ðtÞ − CfitðtÞÞ vector components obtained by solving a GEVP [see χ2 ¼ jk jk ; ð Þ 2 28 Eq. (23)] and, thus, the results from the two methods can be ðΔCjkðtÞÞ j;k t¼tmin compared in a meaningful way, after choosing the same 2 normalization ðv˜mÞ ¼ 1. which is the well-known χ2 used in uncorrelated χ2 minimizing fits. CjkðtÞ denotes the correlation functions V. ANALYSIS OF THE CORRELATION computed using lattice QCD with corresponding statistical MATRIX AND NUMERICAL RESULTS ΔC ðtÞ Cfit ðtÞ ð Þ errors jk , while j;k is given by Eq. (24).In FOR THE Ds0 2317 MESON 2 principle, one can also use a correlated χ . Then, however, A. Extraction of energy levels and amplitudes one has to estimate a covariance matrix, which requires in a finite volume rather precise data and computations on a large number of Our goal in this section is to determine the two lowest gauge link configurations (cf. e.g., Ref. [58] for a detailed energy levels in the sector with D ð2317Þ quantum discussion). s0 numbers in the finite spatial volume L3 of the lattice. To obtain the PDF ΠðArÞ for a specific fit parameter Ar, – one has to integrate Eq. (27) over all other parameters. In From previous results [40 42] we expect that one of the corresponding energy eigenstates is the lowest DK scatter- particular, the probability for the parameter Ar to be inside a; b ing state, while the other has a somewhat smaller energy the interval [ ]is and represents the Ds0ð2317Þ meson. A precise determi- Z R R Q 2 b b þ∞ −χ =2 nation of these two energy levels is necessary to study the a dAr −∞ s≠rdAse dArΠðArÞ¼ R Q : ð29Þ infinite volume limit using Lüscher’s finite volume method þ∞ dA e−χ2=2 a −∞ s s in Sec. VB. Moreover, in this section we also investigate the quark This multidimensional integral can be computed with content and arrangement of the Ds0ð2317Þ state by study- standard Monte Carlo methods. We use a parallel temper- j j ing the eigenvector components vm and the coefficients v˜ m ing scheme combined with the Metropolis algorithm as introduced in Sec. IV. described in detail in Ref. [46]. The parallel tempering scheme prevents the algorithm from getting stuck in a 1. D and K meson masses and DK threshold region around a local minimum of χ2 and guarantees ergodicity of the algorithm. As a preparatory step we computed the masses of the t ¼ 15a pseudoscalar mesons D and K within our lattice setup. It is While we use max in Eq. (28), we vary in our t rather straightforward to obtain precise values for mK and analyses both min and the number of terms in the truncated sum in Eq. (24), until we find a stable region with no mD from correlation functions of standard interpolating observable change in the PDFs for the low-lying energy fields eigenstates of interest. For a detailed example see Ref. [46]. X OD ¼ c¯ðxÞγ uðxÞ; ð Þ cj ¼hΩjOjjmi¼hmjOj†jΩi 5 33 The coefficients Ω;m in the x fit function (24) are the coefficients of the expansions of X j† K the trial states O jΩi in terms of the energy eigenstates O ¼ u¯ðxÞγ5sðxÞ: ð34Þ jmi, i.e., x
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¯ Dsη;point We find u¯γ5u þ dγ5d, which is part of O as well as of ODsη;2part (note that a lattice QCD study of the η mD ¼ 1.8445ð9Þ GeV; ð35Þ meson is quite challenging by itself, partly because of strong statistical fluctuations; see e.g., mK ¼ 0.5965ð4Þ GeV; ð36Þ Refs. [60,61] for a detailed discussion and a recent computation). Moreover, the mass of the Ds0ð2317Þ using the AMIAS analysis method (extracting the masses meson is close to the DK threshold, while the from the corresponding effective energies, as explained in Dsη threshold is around 155 MeV above [4]. the context of the GEVP method in Sec. IVA, leads to This suggests that including the interpolating fields D η; D η;2 compatible results, however, with somewhat larger statis- O s point and O s part is not essential to resolve the tical errors) [59]. Consequently, two lowest energy eigenstates. This is supported by Refs. [14–17], where the molecular components D ð2317Þ mD þ mK ¼ 2.4411ð10Þ GeV; ð37Þ for the s0 were found to be from around 60% to 75% for DK and below 15% for Dsη. Thus, ODsη;point ODsη;2part which is the lowest two-meson threshold in the sector with we do not use and in any of the following analyses. Ds0ð2317Þ quantum numbers and, thus, plays an important Oqq;¯ 1 ODK;point role in the interpretation of further results. Also of interest is The remaining four interpolating fields , , QQ;¯ γ DK;2 the energy of a noninteracting DK pair with one quantum O 5 and O part [Eqs. (2), (4), (6) and (7)] are used in of relative momentum p ¼ 2π=L, the following to determine the two lowest energy levels in min the sector with Ds0ð2317Þ quantum numbers. ðm2 þ p2 Þ1=2 þðm2 þ p2 Þ1=2 ¼ 2 6271ð9Þ : D min K min . GeV 3. GEVP analysis ð38Þ The results of a GEVP analysis of the 4 × 4 correlation matrix containing the four interpolating fields identified in 2. Preselection of interpolating fields the previous subsection are collected in Fig. 1. The upper To reduce the seven interpolating fields (2) to (8) to a plot shows effective energies as functions of the temporal somewhat smaller set of interpolating fields, which are separation. The four plots below contain the squared j 2 most important to resolve the lowest energy eigenstates eigenvector components ðvmÞ . with Ds0ð2317Þ quantum numbers, we performed GEVP as There are two convincing effective energy plateaus with well as AMIAS analyses using individual correlation small statistical errors close to the DK threshold. One is functions or 2 × 2 correlation matrices. From these analy- around 60 MeV below, while the other is somewhat above, ses it became clear that three of the interpolating fields (2) but significantly closer to the DK threshold than to the to (8) are less relevant. energy of a noninteracting DK pair with one quantum of (i) Oqq;¯ γ0 [Eq. (3)]: One can determine the energy of a relative momentum as indicated by the horizontal gray low-lying state, which we identify below as the lines [see also Eqs. (37) and (38)]. Thus, there is an Ds0ð2317Þ meson, using the correlation function additional low-lying state compared to the noninteracting of one of the two quark-antiquark interpolating DK spectrum. The eigenvector components clearly indicate fields, i.e., either of Oqq;¯ 1 or of Oqq;¯ γ0 . For the that the lowest state resembles a quark-antiquark pair qq;¯ 1 2 latter, however, the effective energy plateau is [ðv0 Þ ≳ 0.90; black bar chart], while the first excitation reached at larger temporal separation and statistical is a DK scattering state similar to a noninteracting two- DK;2part 2 errors are larger as well. An analysis of the corre- meson state with both mesons at rest [ðv1 Þ ≈ 0.85; sponding 2 × 2 correlation matrices gives the same red bar chart]. These eigenvector components suggest low-lying state and a second rather noisy effective identifying the lowest state as the Ds0ð2317Þ meson. energy significantly above, which most likely re- Energy levels Em are determined from effective energies ceives contributions from several excited states. The as discussed in detail in Sec. IVA.Form ¼ 0, 1 they are j eigenvector components vm indicate a strong domi- listed in Table I. qq;¯ 1 E ðtÞ nance of the interpolating field O for the ground The third effective energy eff;2 has large statistical state. In view of these findings we consider Oqq;¯ 1 errors and is somewhat above the estimated energy of a superior to Oqq;¯ γ0 and do not use the latter inter- noninteracting DK pair with one quantum of relative polating field in any of the following analyses. momentum. It seems likely that it corresponds to a linear (ii) ODsη;point and ODsη;2part [Eqs. (5) and (8)]: Correla- superposition of several DK scattering states with non- tion functions containing either ODsη;point or vanishing relative momenta. This interpretation is supported ODsη;2part exhibit large statistical errors. This seems by the eigenvector components, which indicate a dominance DK;point DK;point 2 to be a consequence of the “η interpolator” of the O interpolating field [ðv2 Þ ≈ 0.90; green
034502-7 CONSTANTIA ALEXANDROU et al. PHYS. REV. D 101, 034502 (2020)
¯ QQ;γ5 2 From ðvm Þ ≲ 0.05 for m ¼ 0, 1, 2 one can conclude ¯ that the diquark-antidiquark interpolating field OQQ;γ5 is not important to resolve any of the three lowest energy eigen- states. In particular, the ground state seems to be predomi- qq;¯ 1 2 nantly a standard quark-antiquark pair [ðv0 Þ ≳ 0.90] with DK;2part 2 only a small DK component [ðv0 Þ ≲ 0.10]. There is no significant contribution from the tetraquark interpolating ¯ DK;point 2 QQ;γ5 2 fields; i.e., both ðv0 Þ and ðv0 Þ are almost vanishing. We interpret this as indication that the Ds0ð2317Þ meson has no sizable tetraquark component.
4. AMIAS analysis The results of an AMIAS analysis of the 4 × 4 corre- lation matrix are collected in Fig. 2. The upper plot shows the PDFs generated with the fit function given in Eq. (24) and six terms in the truncated sum [62]. The four plots j 2 below show the squared coefficients ðv˜ mÞ . When comparing the PDFs to the effective energies in Fig. 1 one can see that the energy levels obtained with AMIAS are consistent with those from the GEVP analysis. Statistical errors for the AMIAS results are somewhat smaller than for the GEVP results (see Table I). The j 2 coefficients ðv˜ mÞ are also in reasonable agreement with j 2 the GEVP eigenvector components ðvmÞ from Fig. 1, supporting that the ground state is mostly a quark anti- quark pair. FIG. 1. GEVP analysis of the 4 × 4 correlation matrix with To cross-check the obtained results, in particular, to ¯ interpolating fields Oqq;¯ 1, ODK;point, OQQ;γ5 and ODK;2part. (Top) confirm our findings regarding the quark composition and E t Effective energies eff;m as functions of the temporal separation interpretation of the low-lying energy eigenstates, it is together with the DK threshold [see Eq. (37)] and the energy of a useful to compare the above 4 × 4 AMIAS analysis to noninteracting DK pair with one quantum of relative momentum analogous analyses using the four possible 3 × 3 subma- ðvj Þ2 [see Eq. (38)]. (bottom) Squared eigenvector components m . trices as input [for the latter five terms in the truncated sum in Eq. (24) are sufficient]. The corresponding PDFs are 4 4 bar chart], which by construction excites DK states with shown in Fig. 3, with the × PDFs in the background colored in light gray. many different relative momenta. The fourth effective ¯ (A) 3 × 3 correlation matrix without OQQ;γ5 : There is energy E ;2ðtÞ has even larger statistical errors and is eff essentially no difference between the 3 × 3 and around 1 GeV above the DK threshold. Most likely it 4 × 4 PDFs for the three lowest energy levels. This represents a superposition of a larger number of highly confirms that the diquark-antidiquark interpolating excited states. ¯ field OQQ;γ5 is not important to resolve the low-lying energy eigenstates. This in turn supports our con- D ð2317Þ TABLE I. The lowest two energy levels E0 and E1 in the sector clusion from Sec. VA3 that the s0 meson 3 with Ds0ð2317Þ quantum numbers in the finite volume L of the does not have a sizable tetraquark component. lattice obtained by various analyses. (A)–(D) refer to the 3 × 3 (B) 3 × 3 correlation matrix without ODK;point: The AMIAS analyses discussed in Sec. VA 4. lowest two energy levels are consistent with the 4 4 E = E = × result within statistical errors. The energy level Analysis 0 GeV 1 GeV of the second excitation is, however, significantly GEVP, 4 × 4 2.3803(78) 2.4780(50) larger. This indicates that the interpolating field AMIAS, 4 × 4 2.3790(28) 2.4854(44) ODK;point is useful to resolve higher momentum 3 3 AMIAS, × , (A) 2.3765(33) 2.4837(49) excitations, while it is not essential for a determi- AMIAS, 3 × 3, (B) 2.3794(35) 2.4946(36) 3 3 naton of the lowest two energy levels. AMIAS, × , (C) 2.3857(94) 2.5028(135) 3 3 Oqq;¯ 1 AMIAS, 3 × 3, (D) 2.3953(69) 2.7840(456) (C) × correlation matrix without : The lowest two energy levels are slightly larger compared to the
034502-8 TETRAQUARK INTERPOLATING FIELDS IN A LATTICE QCD … PHYS. REV. D 101, 034502 (2020)
ε0 ε1
ε2
2.2 2.4 2.6 2.8 3 3.2 3.4 εj (GeV)
ε0 ε1
ε2
2.2 2.4 2.6 2.8 3 3.2 3.4 εj (GeV)
ε0 ε1 ε2
2.2 2.4 2.6 2.8 3 3.2 3.4 εj (GeV)
ε0
FIG. 2. AMIAS analysis of the 4 × 4 correlation matrix with ε ¯ 1 Oqq;¯ 1 ODK;point OQQ;γ5 ODK;2part interpolating fields , , and . (Top) ε2 PDFs for the energy levels together with the DK threshold [see 2.2 2.4 2.6 2.8 3 3.2 3.4 Eq. (37)] and the energy of a noninteracting DK pair with one ε quantum of relative momentum [see Eq. (38)]. (Bottom) Squared j (GeV) j 2 coefficients ðv˜ mÞ for the four lowest energy levels. FIG. 3. PDFs for the energy levels from 3 × 3 AMIAS analyses. (A)–(D) refer to the 3 × 3 correlation matrices dis- 4 4 cussed in Sec. VA4. The light gray PDFs in the background × result, but they are still compatible, because of 4 4 their drastically larger statistical errors (see Table I). correspond to the × AMIAS analysis and are shown to facilitate comparison. Thus, it is possible to excite the Ds0ð2317Þ meson with only four-quark interpolating fields; i.e., it DK seems to have a nonvanishing, but small 5. Summary of finite volume results and conclusions component, which is in agreement with our findings from Sec. VA3. A summary plot of the obtained energy levels with DK;2 4 4 4 4 3 3 (D) 3 × 3 correlation matrix without O part: The the × GEVP as well as the × and × AMIAS lowest energy level is consistent with the 4 × 4 analyses is shown in Fig. 4. Again it can be seen that the result, even though it has a much larger statistical most important interpolating fields to determine the two qq;¯ 1 DK;2part error. The energy level of the first excitation, lowest energy levels are O and O . Analyses however, cannot be determined reliably anymore. using these two interpolating fields [4 × 4 GEVP, 4 × 4 The PDF has a large width and its peak is localized at AMIAS, 3 × 3 AMIAS (A) and (B)] yield consistent energies significantly above the DK threshold. This energy levels with small statistical errors. j confirms that the interpolating field ODK;2part is of From the GEVP eigenvector components vm and the j central importance for a determination the energy of AMIAS coefficients v˜ m we conclude that the lowest energy the lowest DK scattering state. level mostly corresponds to a quark-antiquark bound state,
034502-9 CONSTANTIA ALEXANDROU et al. PHYS. REV. D 101, 034502 (2020)
3.3 Here Z00 denotes the generalized zeta function and L ≈ 2 90 3.2 . fm the spatial lattice extent (see Sec. II). k ðδ ðkÞÞ k2 3.1 cot 0 can be written as a Taylor series in , 3 1 r k ðδ ðkÞÞ ¼ þ 0 k2 þ Oðk4Þ; ð Þ 2.9 cot 0 41 a0 2 2.8 (GeV) j a S r S ε 2.7 where 0 is the wave scattering length and 0 the wave 2 2.6 effective range. For sufficiently small k one can neglect k4 k ðδ ðkÞÞ 2.5 terms of order and parametrize cot 0 by the first two terms on the right-hand side of Eq. (41). a0 and r0 2.4 are then fixed by the two data points cotðδ0ðk0ÞÞ and 2.3 GEVP AMIAS AMIAS AMIAS AMIAS AMIAS cotðδ0ðk1ÞÞ obtained via Eq. (40). This parametrization is 4x4 4x4 3x3 (A) 3x3 (B) 3x3 (C) 3x3 (D) called effective range expansion. D ð2317Þ FIG. 4. Comparison plot of finite volume energy levels ob- A stable s0 meson manifests itself as a pole in tained from 3 × 3 and 4 × 4 correlation matrices using the GEVP the scattering amplitude, method and the AMIAS method. (A)–(D) refer to the 3 × 3 AMIAS analyses discussed in Sec. VA 4. 1 f0k ¼ ; ð42Þ cotðδ0ðkÞÞ − i possibly similar to the D ð2317Þ meson in the infinite s0 i.e., corresponds to cotðδ0ðkD ÞÞ ¼ i, where kD denotes volume. There seems to be a small DK component, but no s0 s0 the position of the pole. Combining this condition with the sign of any sizable tetraquark component. Moreover, the j j parametrization (41) leads to components vm and the coefficients v˜ m clearly indicate that DK the first excitation is a scattering state. These two 1 r0 2 ikD ¼ þ k ; ð43Þ energy levels are used for the finite volume analysis in s0 D a0 2 s0 Sec. VB. 2 which can easily be solved with respect to k , Ds0 B. Scattering analysis and infinite volume limit 1 1 2 1=2 2 The energy levels collected in Table I were computed at k2 ¼ − þ ð Þ D 2 44 3 s0 r r a r finite spatial volume L with periodic boundary conditions. 0 0 0 0 One can determine the mass of the Ds0ð2317Þ meson, which is the infinite volume limit of the ground state [note that for our data one of the two solutions has to be discarded, because it is far outside the region of validity of energy, from the lowest two energy levels E0 and E1 at finite Oðk4Þ volume by performing a scattering analysis continued to the effective range expansion (41), where terms D ð2317Þ imaginary momenta, i.e., using Lüscher’s finite volume cannot be neglected]. The mass of the s0 meson is 2 method [63]. This approach has been used in a lattice QCD given by the right-hand side of Eq. (39) with kn replaced 2 D ð2317Þ by k , i.e., by study of the s0 meson for the first time in Ds0 Refs. [40,41] and later also in Ref. [42]. It was also used 2 2 1=2 2 2 1=2 to study other systems (see e.g., Refs. [44,64]). For a recent mD ¼ðmD þ k Þ þðmK þ k Þ : ð45Þ s0 Ds0 Ds0 review on scattering in lattice QCD see Ref. [65]. The first step is to determine the squared scattering In Table II we show the results obtained for the lowest 2 2 momenta k0 and k1 via two energy levels E0 and E1, for the squared scattering 2 2 momenta k0 and k1, for the phase shifts, for the S wave 2 2 1=2 2 2 1=2 En ¼ðmD þ knÞ þðmK þ knÞ ; ð39Þ scattering length a0 and effective range r0, as well as for the position of the pole. To verify that the effective range where E0 and E1 can be taken from Table I and mD and mK expansion (41) is a reasonable approximation, we also D K 2 2 2 2 2 are the meson and meson masses obtained within the provide sðk0Þ, sðk1Þ and sðk Þ, where sðk Þ¼ja0r0k =2j Ds0 same lattice setup [see Eqs. (35) and (36)]. With Lüschers corresponds to the ratio of the Oðk2Þ term and the Oðk0Þ finite volume method one can then compute the corre- ≪ 1 δ ðk Þ δ ðk Þ term in Eq. (41). We find values for the two scattering sponding two phase shifts 0 0 and 0 1 , momenta as well as for the position of the pole, which gives 2 certain indication that higher order terms are suppressed, 2Z00ð1; ðknL=2πÞ Þ Oðk4Þ kn cotðδ0ðknÞÞ ¼ pffiffiffi : ð40Þ i.e., that terms in Eq. (41) are indeed negligible. In πL m D Table II we also list Ds0 , the resulting mass of the s0
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TABLE II. Results of the scattering analysis.
2 2 2 2 E0=GeV E1=GeV k0=GeV k1=GeV k0 cotðδ0ðk0ÞÞ=GeV k1 cotðδ0ðk1ÞÞ=GeV GEVP, 4 × 4 2.3803(78) 2.4780(50) −0.0531ð66Þþ0.0340ð46Þ −0.2101ð195Þ −0.2350ð272Þ AMIAS, 4 × 4 2.3790(28) 2.4854(44) −0.0542ð25Þþ0.0408ð38Þ −0.2133ð72Þ −0.1984ð187Þ
2 2 2 2 2 a0=fm r0=fm kD =GeV sðk0Þ sðk1Þ sðkD Þ mD =GeV ðmD þ mK − mD Þ=MeV s0 s0 s0 s0 GEVP, 4 × 4 −0.876ð76Þ −0.113ð152Þ −0.0451ð99Þ 0.07 0.04 0.06 2.3897(116) 51.3(11.7) AMIAS, 4 × 4 −0.964ð34Þþ0.062ð84Þ −0.0449ð27Þ 0.04 0.03 0.03 2.3900(64) 51.1(6.5)
m þ m − m k ðδ ðkÞÞ meson, and D K Ds0 , the binding energy with In Fig. 5 we show the parametrization of cot 0 respect to the DK threshold. All results are provided both with the effective range expansion (41) together with the for the 4 × 4 GEVP and the 4 × 4 AMIAS determination two data points k0 cotðδ0ðk0ÞÞ and k1 cotðδ0ðk1ÞÞ. Note of the lowest two energy levels E0 and E1 discussed in that the effective range expansion is equivalent to the right- Secs. VA3 and VA4. hand side of Eq. (43).p Weffiffiffiffiffiffiffiffi also show the left-hand side of that equation, ik ¼ − −k2. The intersection of the two 2 curves corresponds to k , the binding momentum of the Ds0 Ds0ð2317Þ meson. In Fig. 6 we illustrate the pole in the scattering amplitude by plotting
−1 1 r0k jf0kj¼ þ − i ð46Þ a0k 2
FIG. 5. The parametrization of k cotðδ0ðkÞÞ with the effective FIG. 6. jf0kj according to Eq. (46) in the complex k plane. The range expansion [right-hand side of both Eq. (41) and Eq. (43); pole corresponds to the Ds0ð2317Þ meson. The color of the 2 2 green curve] together with the two data points k0 cotðδ0ðk0ÞÞ and plotted surface is related to the value of sðk Þ [green, sðk Þ < 0.1; 2 2 k1 cotðδ0ðk1ÞÞ (in magenta).pffiffiffiffiffiffiffiffi The intersection with the left-hand yellow, 0.1 ≤ sðk Þ < 0.2; red, 0.2 ≤ sðk Þ]. Thus, it reflects the 2 2 side of Eq. (43), ik ¼ − −k (blue curve), corresponds to kD quality of the effective range expansion (41) and indicates that the s0 4 4 4 k Oðk Þ (indicated by the orange error band). (Top) × GEVP analysis. pole at Ds0 is in a region, where terms should be (Bottom) 4 × 4 AMIAS analysis. negligible.
034502-11 CONSTANTIA ALEXANDROU et al. PHYS. REV. D 101, 034502 (2020) in the complex k plane, i.e., Eq. (42) with the para- In the finite spatial volume of our lattice with extent metrization (41) inserted. The color reflects the quality L ≈ 2.90 fm we find two low-lying energy eigenstates, one of the effective range expansion (41) and indicates that the around 60 MeV below the DK threshold, the other slightly pole is in a region, where Oðk4Þ terms should be negligible. above the DK threshold. The GEVP eigenvector compo- It is important to note that a direct comparison of our nents and the AMIAS coefficients and PDFs clearly result for mD to the corresponding experimental result indicate that the state below threshold, which corresponds s0 m ¼ 2317 8ð5Þ to the Ds0ð2317Þ meson, is mostly of quark-antiquark type Ds0; . MeV [4] is not meaningful, because exp with only a small DK component, while the state above the quark masses in our simulation differ from their DK experimental counterparts. threshold is a scattering state. The tetraquark inter- (i) The light u and d quark mass is unphysically heavy, polating fields explored in this work turned out to be essentially irrelevant, when extracting the corresponding reflected by the pion mass mπ ≈ 0.296 GeV. (ii) The s quark mass is unphysically heavy, as indicated two energy levels; i.e., the couplings of the state below 2m2 − m2 ≈ 0 62 2 threshold to these interpolating fields are close to 0. We by K π . GeV (which is approxi- D ð2317Þ mately proportional to the s quark mass) compared interpret this as indication that the s0 meson is 2 2 2 mainly a quark-antiquark state and not a tetraquark, as to 2m − mπ; ≈ 0.47 GeV . K;exp exp discussed or proposed by various existing papers. (iii) The c quark is unphysically light, because mD ≈ 1 845 m ≈ 1 867 It is important to keep in mind that our computation was . GeV, i.e., below D;exp . GeV. D ð2317Þ carried out for a single spatial volume and at quark masses Also for the binding energy of the s0 meson with different from those in the real world, in particular, a u respect to the DK threshold, mD þ mK − mD ,itisnot s0 and d quark mass corresponding to a heavier pion, clear a priori whether quark masses, which differ from their mπ ≈ 0.296 GeV. We performed a scattering analysis using physical values, result in a value similar to the correspond- ’ D ð2317Þ Lüscher s method to determine the mass of the s0 ing experimental value mD; þmK; −m ≈45 MeV. exp exp Ds0;exp in the infinite volume limit. We find this mass 51 MeV One reason for this is that the threshold mD þ mK clearly below the DK threshold, rather close to our finite volume m depends on the light quark mass, while Ds0 , which result as well as to the experimental value 45 MeV. Thus we according to Sec. VA is mostly a cs¯ state, should be expect that our findings concerning the importance of almost independent of the light quark mass (see also various interpolating fields as well as the quark composi- the discussion in Ref. [41]). Note, however, that we find tion of the Ds0ð2317Þ meson also apply to infinite volume m þ m − m ≈ 51 and physical quark masses at least on a qualitative level. Of D K Ds0 MeV rather close to the experimen- tally observed 45 MeV, which indicates that with respect to course, it would be worthwhile and interesting to perform the Ds0 meson we might be in a similar situation as in real similar computations at physical quark masses and for world QCD, even though we are not precisely at physical several volumes in the future, in particular, to check the quark masses. Because of this and since mD is close to the approximate independence of the GEVP eigenvector com- s0 ponents or the AMIAS coefficients from the spatial lowest energy level E0 obtained at finite lattice volume (around 10 MeV difference as can be seen from Table II), volume. we expect that our findings and statements from Sec. V about the importance of the two-quark and the four-quark ACKNOWLEDGMENTS interpolating fields also apply for physical quark masses J. F. acknowledges financial support by the PRACE Fifth and the infinite volume limit. This is further supported by and Sixth Implementation Phase (Grants No. PRACE-5IP the qualitative agreement of our results for a0 and r0 and corresponding results obtained in lattice QCD computa- and No. PRACE-6IP) program of the European tions at almost physical quark masses [41,42]. Commission under Grants No. 730913 and No. 823767. M. W. acknowledges funding by the Heisenberg Programme of the Deutsche Forschungsgemeinschaft VI. SUMMARY AND CONCLUSIONS (DFG, German Research Foundation), Grant We studied the Ds0ð2317Þ meson with lattice QCD using No. 399217702. This work was supported in part by the interpolating fields of different structure. In addition to Helmholtz International Center for FAIR within the frame- quark-antiquark interpolating fields and two-meson inter- work of the LOEWE program launched by the State of polating fields, which were already considered in previous Hesse. Calculations on the LOEWE-CSC and on the lattice QCD studies, we implemented and explored the FUCHS-CSC high-performance computer of the importance of tetraquark interpolating fields. For these Frankfurt University were conducted for this research. tetraquark interpolating fields the four quark operators are We thank HPC-Hessen, funded by the State Ministry of centered at the same point in space and their color and spin Higher Education, Research and the Arts, for programming structure corresponds to either a meson-meson pair or a advice. Computations have been performed using the diquark-antidiquark pair. Chroma software library [66] with a multigrid solver [67].
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