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Tetraquark Interpolating Fields in a Lattice QCD Investigation of The PHYSICAL REVIEW D 101, 034502 (2020) Editors' Suggestion Tetraquark interpolating fields in a lattice QCD investigation à 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) à 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 à 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 à tetraquark interpretation of the Ds0ð2317Þ meson rather unlikely. Moreover, we perform a scattering à 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 Dà ð2317Þ meson with quantum numbers IðJPÞ¼ s0 components from around 60% to 75% are found. Other 0ð0þÞ, strangeness S ¼1 and charm C ¼ S has mass interesting approaches, which are able to explain the mDà ¼ 2.3178ð5Þ GeV, around 45 MeV below the DK à s0 surprisingly low mass of the Ds0ð2317Þ meson, are e.g., threshold [1–4]. This experimental result is in contrast to à 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 – Dà 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 à 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]. Dà 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 Dà 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 à 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 à 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 à 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 à 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 à qualitative difference between the D0ð2400Þ meson and show several finite volume spectra for the sector with à Dà ð2317Þ the Ds0ð2317Þ meson. s0 quantum numbers corresponding to different à 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 Dà 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 Dà 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 à 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 à 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 034502-2 TETRAQUARK INTERPOLATING FIELDS IN A LATTICE QCD … PHYS. REV. D 101, 034502 (2020) ¯ a ð980Þ O5 ¼ OQQ;γ5 ; 0 quantum numbers.
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