We study the light scalar mesons a_0(980) and kappa using N_f = 2+1+1 flavor lattice QCD. In order to probe the internal structure of these scalar mesons, and in particular to identify, whether a sizeable tetraquark component is present, we use a large set of operators, including diquark-antidiquark, mesonic molecule and two-meson operators. The inclusion of disconnected diagrams, which are technically rather challenging, but which would allow us to extend our work to e.g. the f_0(980) meson, is introduced and discussed

Alexandrou, Constantia

Daldrop, Jan Oliver

Dalla Brida, Mattia

Gravina, Mario

Scorzato, Luigi

Urbach, Carsten

Wagner, Marc

Wiese, Christian

English

arXiv

Hochschulschriftenserver - Universität Frankfurt am Main

arXiv:1212.1648v1  [hep-lat]  7 Dec 2012Scalar mesons and tetraquarks by means of latticeQCDEuropean Twisted Mass Collaboration (ETMC):Marc Wagner∗Goethe-Universität Frankfurt am Main, Institut für Theoretische Physik,Max-von-Laue-Straße1, D-60438 Frankfurt am Main, GermanyConstantia Alexandrou, Mario GravinaDepartment of Physics, University of Cyprus, P.O. Box 20537, 1678 Nicosia, CyprusComputation-based Science and Technology Research Center, Cyprus Institute, 20 Kavaﬁ Street,Nicosia 2121, CyprusJan Oliver Daldrop, Carsten UrbachHelmholtz-Institut für Strahlen- und Kernphysik (Theorie) and Bethe Center for TheoreticalPhysics, Universität Bonn, D-53115 Bonn, GermanyMattia Dalla BridaSchool of Mathematics, Trinity College Dublin, Dublin 2, IrelandLuigi ScorzatoECT⋆, Strada delle Tabarelle, 286, I-38123, Trento, ItalyChristian WieseNIC, DESY Zeuthen, Platanenallee 6, D-15738 Zeuthen, GermanyHumboldt-Universität zu Berlin, Institut für Physik, Newtonstraße 15, D-12489 Berlin, GermanyWe study the light scalar mesons a0(980) and k using Nf = 2+1+1 ﬂavor lattice QCD. In or-der to probe the internal structure of these scalar mesons, and in particular to identify, whethera sizeable tetraquark component is present, we use a large set of operators, including diquark-antidiquark, mesonic molecule and two-meson operators. The inclusion of disconnected dia-grams, which are technically rather challenging, but which would allow us to extend our work toe.g. the f0(980) meson, is introduced and discussed.Xth Quark Conﬁnement and the Hadron Spectrum8-12 October 2012TUM Campus Garching, Munich, Germany∗Speaker.c ￿ Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/Scalar mesons and tetraquarks by means of lattice QCD Marc Wagner1. IntroductionThe nonet of light scalar mesons formed by s ≡ f0(500), k ≡ K∗0(800), a0(980) and f0(980)is poorly understood. Compared to expectation all nine states are rather light and their ordering isinverted. For example in a standard quark antiquark picture the a0(980) states, which have I = 1,must necessarily be composed of two light quarks, e.g. a0(980) ≡ ¯ du, while the k states withI = 1/2 are made from a strange and a light quark, e.g. k ≡ ¯ su. Consequently, k should be heavierthan a0(980), since ms >mu,d. Inexperiments, however, theopposite isobserved, i.e. m(k)=682±29MeV, while m(a0(980)) = 980±20MeV [1]. On the other hand in a four-quark or tetraquarkpicture the quark content could be a0(980) ≡ ¯ du¯ ss and k ≡ ¯ su(¯ uu+ ¯ dd) naturally explaining theobserved ordering. Moreover, certain decay channels, e.g. a0(980) → K+ ¯ K, indicate that besidesthe two light quarks also an s¯ s pair is present and, therefore, also support a tetraquark interpretation.A detailed discussion of light scalar mesons can be found e.g. in [2, 3].In addition to the light scalar mesons there are also various tetraquark candidates among theheavy mesons, e.g. D∗s0(2317) and Ds1(2460) (cf. e.g. [4, 5]) or the charmonium states X(3872),Z(4430), Z(4050), Z(4250).Here we report about the status of an ongoing long-term project with the aim to study possibletetraquark candidates from ﬁrst principles using lattice QCD. Parts of this work have already beenpublished [6, 7].2. Lattice setupWe use gauge link conﬁgurations with Nf = 2+1+1 dynamical quark ﬂavors generated bythe European Twisted Mass Collaboration (ETMC).We have studied several ensembles with the same rather ﬁne lattice spacing a≈ 0.086fm. Theensembles differ in the volume (L/a)3×(T/a)=203×48,...,323×64and the unphysically heavylight quark mass corresponding to mp ≈ 280MeV...460MeV. Details regarding these gauge linkconﬁgurations can be found in [8, 9, 10, 11, 12, 13].Currently we ignore disconnected diagrams, which are technically rather challenging (cf. theoutlook in section 5). An important physical consequence is that the quark number and the anti-quark number are separately conserved for each ﬂavor. Therefore, there is no mixing between ¯ uu,¯ dd and ¯ ss resulting in an hs meson with ﬂavor structure ¯ ss instead of h and h′.Further lattice details and technicalities can be found in [7].3. Four-quark creation operatorsIn the following we focus on the a0(980) sector, which has quantum numbers I(JP) = 1(0+).As usual in lattice QCD we extract the low lying spectrum in that sector by studying the asymptoticexponential behavior of Euclidean correlation functionsCjk(t) =D(Oj(t))†Ok(0)E. (3.1)Oj and Ok denote suitable creation operators, i.e. operators generating the a0(980) quantum num-bers, when applied to the vacuum state.2Scalar mesons and tetraquarks by means of lattice QCD Marc WagnerAssuming that the experimentally measured a0(980) with mass 980 ± 20MeV is a ratherstrongly bound four quark state, suitable creation operators to excite such a state areOK ¯ K moleculea0(980) = åx￿¯ s(x)g5u(x)￿￿¯ d(x)g5s(x)￿(3.2)Odiquarka0(980) = åx￿eabc¯ sb(x)Cg5 ¯ dc,T(x)￿￿eadeud,T(x)Cg5se(x)￿. (3.3)The ﬁrst operator has the spin/color structure of a K ¯ K molecule (¯ s(x)g5u(x) and ¯ d(x)g5s(x) corre-spond to akaon K and an antikaon ¯ K at the sameposition x). Thesecond resembles abound diquarkantidiquark pair, where spin coupling viaCg5 corresponds to the lightest diquarks/antidiquarks (cf.e.g. [2, 14, 15]).Further low lying states in this sector are the two particle states K + ¯ K and hs+p. Suitablecreation operators to resolve these states areOK+ ¯ K two-particlea0(980) =￿åx¯ s(x)g5u(x)￿￿åy¯ d(y)g5s(y)￿(3.4)Ohs+p two-particlea0(980) =￿åx¯ s(x)g5s(x)￿￿åy¯ d(y)g5u(y)￿. (3.5)4. Numerical results an their interpretationWeﬁrstdiscuss numerical results for theensemble withthesmallest volume, (L/a)3×(T/a)=203 ×48, which corresponds to a spatial extension of L ≈ 1.72fm. This ensemble is particularlysuited to distinguish two-particle states with relative momentum from states with two particles atrest and from possibly existing a0(980) tetraquark states (two-particle states with relative momen-tum have a rather large energy because one quantum of momentumpmin = 2p/L ≈ 720MeV).Figure 1a shows effective mass plots from a 2×2 correlation matrix with a K ¯ K moleculeoperator (3.2) and a diquark-antidiquark operator (3.3). The corresponding two plateaus are around1100MeV and, therefore, consistent both with the expectation for possibly existing a0(980) tetra-quark states and with two-particle K + ¯ K and hs+p states, where both particles are at rest(m(K + ¯ K) ≈ 2m(K) ≈ 1198MeV; m(hs+p) ≈ m(hs)+m(p) ≈ 1115MeV in our lattice setup).Increasing this correlation matrix to 4×4 by adding two-particle K + ¯ K and hs+p operators(eqs. (3.4) and (3.5)) yields the effective mass results shown in Figure 1b. Two additional statesare observed, whose plateaus are around 1500MeV...2000MeV. From this 4×4 analysis weconclude the following:• We do not observe a third low-lying state around 1100MeV, even though we provide opera-tors, which are of tetraquark type as well as of two-particle type. This suggests that the twolow-lying states are the expected two-particle K + ¯ K and hs +p states, while an additionalstable a0(980) tetraquark state does not exist.• The effective masses of the two low-lying states are of much better quality in Figure 1bthan in Figure 1a. We attribute this to the two-particle K + ¯ K and hs +p operators, which3Scalar mesons and tetraquarks by means of lattice QCD Marc Wagner 0 500 1000 1500 2000 2500 3000 3500 0  2  4  6  8  10  12  14meffective in MeVt/aa)  KK molecule  ,  diquark  (2´2 matrix) 0 500 1000 1500 2000 2500 3000 3500 0  2  4  6  8  10  12  14meffective in MeVt/ab)  KK molecule  ,  diquark  ,  K+K two-particle  ,  hs+p  (4´4 matrix)two-particle states with relative momentum 0 0.2 0.4 0.6 0.8 1 0  2  4  6  8  10  12  14|vj(0)|2t/ac)  operator content, state 0j = KK moleculej = diquarkj = K+K two-particlej = hs+p two-particle 0 0.2 0.4 0.6 0.8 1 0  2  4  6  8  10  12  14|vj(1)|2t/ad)  operator content, state 1j = KK moleculej = diquarkj = K+K two-particlej = hs+p two-particleFigure 1: a0(980) sector, (L/a)3 ×(T/a) = 203 ×48. a) Effective masses as functions of the temporalseparation, 2×2 correlation matrix (operators: K ¯ K molecule, diquark-antidiquark, eqs. (3.2) and (3.3)).b) 4×4 correlation matrix (operators: K ¯ K molecule, diquark-antidiquark, two-particle K + ¯ K, two-particlehs +p, eqs. (3.2) to (3.5)). c), d) Squared eigenvector components of the two low-lying states from b) asfunctions of the temporal separation.presumably create larger overlap to those states than the tetraquark operators. This in turnconﬁrms the interpretation of the two observed low-lying states as two-particle states.• To investigate the overlap in a more quantitative way, we show the squared eigenvector com-ponents of the two low-lying states in Figure 1c and Figure 1d (cf. [13] for a more detaileddiscussion of such eigenvector components). Clearly, the lowest state is of hs +p type,whereas the second lowest state is of K + ¯ K type. On the other hand, the two tetraquarkoperators are essentially irrelevant for resolving those states, i.e. they do not seem to con-tribute any important structure, which is not already present in the two-particle operators.These eigenvector plots give additional strong support of the above interpretation of the twoobserved low lying states as two-particle states.• The energy of two-particle excitations with one relative quantum of momentum can be esti-mated according tom(1+2,p = pmin) ≈qm(1)2+ p2min+qm(2)2+ p2min , pmin =2pL. (4.1)Inserting the meson masses corresponding to our lattice setup, m(K) ≈ 599MeV,m(hs) ≈ 774MeV and m(p) ≈ 341MeV, yields m(K+ ¯ K,p = pmin) ≈ 1873MeV and4Scalar mesons and tetraquarks by means of lattice QCD Marc Wagnerm(hs +p,p = pmin) ≈ 1853MeV. These numbers are consistent with the effective massplateaus of the second and third excitation in Figure 1b. Consequently, we also interpretthem as two-particle states.We obtained qualitatively identical results, when varying the light quark mass and the space-time volume, as discussed in section 2. Corresponding plots are shown in [7].Using exactly the same techniques, i.e. four-quark operators of tetraquark and of two-particletype, we also studied the k-sector (for details cf. [7]). Again we ﬁnd no sign of any four-quarkbound state besides the expected two-particle spectrum (in this case K +p states). Note that thisresult is in contradiction to a very similar recent lattice study of the k meson [16], where an addi-tional low lying four-quark bound state has been observed.5. Inclusion of singly disconnected diagramsAs mentioned in section 2 disconnected diagrams have been ignored for the results presentedso far. In this section we brieﬂy discuss our strategy for computing such diagrams for speciﬁcfour-quark operators.For correlation functions of four-quark operators with ﬂavor structure ¯ q1q1 ¯ q2q3 a so-calledsingly-disconnected diagram has to be computed. Tetraquark candidates with this ﬂavor structureare e.g. the previously discussed a0(980) ≡ ¯ ss ¯ du (cf. Figure 2) orD∗s0(2317),Ds1(2460) ≡ (¯ uu+ ¯ dd)¯ cs.Whilefor connected four-quark diagrams (i.e.all fourquark propagators connect the timeslicesat time 0 and time t) standard point-to-all propagators can be used, applying exclusively suchpropagators is not possible in practice for singly disconnected diagrams. The reason is that one hasto include a sum over space, åx, at least on one of the two timeslices (wlog. at time t in Figure 2),to project to zero momentum. This in turn requires a all-to-all propagator of quark ﬂavor q1 onthat timeslice, due to åx ¯ q1(x,t)q1(x,t)... (the s quark propagator at timeslice t represented by thesolid red line in Figure 2).Since all-to-all propagators are prohibitively expensive to compute, they are typically esti-mated stochastically. While using a single stochastic propagator for a speciﬁc diagram typicallyresults in a favorable or at least acceptable signal-to-noise ratio, using a larger number of suchpropagators drastically increases statistical errors. Therefore, we decided for the following strat-egy: three quark propagators (the q1-loop at timeslice 0 and the q2 and q3 propagators connectingthe timeslices 0 and t) are realized by exact point-to-all propagators, while the remaining propaga-tor (the q1-loop at timeslice t) is estimated stochastically, using random Z2×Z2 timeslice sources.First numerical tests of this strategy performed for both a0(980) and D∗s0(2317) have beenpromising inthe sense that thestatistical errors ofthesingly disconnected diagrams for themoleculetype operator (3.2) are of the same order of magnitude as the statistical errors of the correspondingconnected diagrams, when investing a comparable amount of HPC resources.6. Conclusions and future plansWe have studied the a0(980) and the k channel by means of Nf = 2+1+1 ﬂavor latticeQCD using four-quark operators of molecule, diquark and two-particle type. Besides the expected5Scalar mesons and tetraquarks by means of lattice QCD Marc Wagnertime 0time tåx can beomitted, because oftranslational invariance¯ su ¯ ds¯ su ¯ ds ¯ su ¯ dsåx needed,to have p = 0Figure 2: The singly disconnected diagram of the a0(980) ≡ ¯ ss ¯ du correlator.two-particle spectrum (two essentially non-interacting pseudoscalar mesons) no indication of anyadditional low lying state, in particular no sign of a four-quark bound state could be observed. Thissuggests that both the a0(980) and k meson have either no sizeable tetraquark component or theyare rather weakly bound unstable states.Toinvestigate the latter one needs tostudy the volume dependence ofthe two-particle spectrumin the corresponding sectors (“Lüscher’s method”, cf. e.g. [17, 18,19]). Such computations are verychallenging using lattice QCD, but ﬁrst results have recently been published (cf. [20, 21]). We planto perform similar computations with our setup in the near future.Moreover, certain possibly present systematic errors need to be studied, quantiﬁed and re-moved: (1) disconnected diagrams have to be computed and included (cf. section 5); (2) latticediscretization errors and the continuum limit has to be studied; (3) computations at even lighterand, therefore, more realistic u/d quark masses would be desirable.AcknowledgmentsM.W.acknowledges support by the EmmyNoether Programme of the DFG (German ResearchFoundation), grant WA 3000/1-1. M.G. acknowledges support by the Marie-Curie European train-ing network ITN STRONGnet grant PITN-GA-2009-238353. M.D.B. is currently funded by theIrish Research Council, acknowledges support by STRONGnet and the AuroraScience project, andis grateful for the hospitality at ECT* and the University of Cyprus, where part of this work wascarried out. L.S. acknowledges support from the AuroraScience project funded by the Province ofTrento and INFN.This work was supported in part by the Helmholtz International Center for FAIR within theframework of the LOEWE program launched by the State of Hesse and by the DFG and the NSFCthrough funds provided to the Sino-German CRC 110 “Symmetries and the Emergence of Structurein QCD”.Part of the computations presented here were performed on the Aurora system in Trento.6Scalar mesons and tetraquarks by means of lattice QCD Marc WagnerReferences[1] J. Beringer et al. [Particle Data Group], “Review of particle physics,” Phys. Rev. D86, 010001 (2012).[2] R. L. Jaffe, “Exotica,” Phys. Rept. 409, 1 (2005) [Nucl. Phys. Proc. Suppl. 142, 343 (2005)][arXiv:hep-ph/0409065].[3] J. R. Pelaez, “Review of light scalars: from hints to precision,” talk given at “Xth Quark Conﬁnementand the Hadron Spectrum,” October 8-12, 2012, Munich, Germany.[4] G. Moir and M. 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Scalar mesons and tetraquarks by means of lattice QCD

Scalar mesons and tetraquarks by means of lattice QCD

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