The potential between static-light mesons forming a meson-meson or a
meson-antimeson system is calculated in quenched and unquenched SU(3) gauge
theory. We use the Sheikholeslami-Wohlert action and statistical estimators of
light quark propagators with maximal variance reduction. The dependence of the
potentials on the light quark spin and isospin and the effect of meson exchange
is investigated. Our main motivation is exploration of bound states of two
mesons and string breaking. The latter also involves the two-quark potential
and the correlation between two-quark and two-meson states.Comment: Contribution to LATTICE99 (QCD spectrum). 3 pages, 4 eps figure

Green, AM

Michael, C

Pennanen, P

UKQCD Collaboration

English

arXiv

The potential between static-light mesons forming a meson-meson or a meson-antimeson system is calculated in quenched and unquenched SU(3) gauge theory. We use the Sheikholeslami-Wohlert action and statistical estimators of light quark propagators with maximal variance reduction. The dependence of the potentials on the light quark spin and isospin and the effect of meson exchange is investigated. Our main motivation is exploration of bound states of two mesons and string breaking. The latter also involves the two-quark potential and the correlation between two-quark and two-meson states

University of Liverpool Repository

arXiv:hep-lat/9908032v1  22 Aug 19991NORDITA-1999/50 HEInteractions of heavy-light mesons∗UKQCD Collaboration: P. Pennanena, C. Michaelb† and A.M. Greenc†aNordita, Blegdamsvej 17, 2100 Copenhagen Ø, DenmarkbTheoretical Physics Division, Dept. of Math. Sciences, University of Liverpool, Liverpool, UKcDept. of Phys. and Helsinki Inst. of Phys., P.O. Box 9, FIN-00014 University of Helsinki, FinlandThe potential between static-light mesons forming a meson-meson or a meson-antimeson system is calculatedin quenched and unquenched SU(3) gauge theory. We use the Sheikholeslami-Wohlert action and statisticalestimators of light quark propagators with maximal variance reduction. The dependence of the potentials on thelight quark spin and isospin and the effect of meson exchange is investigated. Our main motivation is explorationof bound states of two mesons and string breaking. The latter also involves the two-quark potential and thecorrelation between two-quark and two-meson states.We have been working on lattice calculations ofmultiquark systems in order to understand theirproperties from first principles. Here “multi”means that the system can be decomposed intomore than one colour singlet, the simplest casebeing four quarks. Previously, we have studiedfour quarks in the static approximation in SU(2),getting energies for a general set of geometrieswhich have been reproduced by a model based ontwo-body potentials and multiquark interactionterms [1]. We have also looked at the flux distri-bution corresponding to the binding energy [2].These results for the static case are discussed inlast year’s proceedings.In the current work only two of the quarks arestatic and SU(3) is used, also unquenched. Thebinding of this system of two heavy-light mesonsis studied as a function of the heavy quark sep-aration. Different cases of spin and isospin ofthe light quarks are measured and the effect frommeson exchange is extracted from the potentials.The model for energies of static quarks is ex-tended for this more dynamic system as discussedin Ref. [3] and by A.M. Green in these proceed-ings. The meson-antimeson case with light quarkisospin Iq = 0 is relevant to string breaking.In the future we plan to explore breaking of anexcited string, which is interesting from the pointof view of hybrid meson phenomenlogy.∗Presented by P. Pennanen, petrus@hip.fi†e-mails: cmi@liv.ac.uk, anthony.green@helsinki.fi1. TWO HEAVY-LIGHT MESONSExperimental candidates for bound states offour quarks (of which two are antiquarks) lie closeto meson-antimeson thresholds. In our case e.g.a heavy Υ particle could be a B∗B¯∗ system. Sys-tems with heavy quarks should be more easilybound as the repulsive kinetic energy is smallerwith the attractive potential being flavour inde-pendent. The binding of four-quark systems hasbeen studied e.g. in bag, string-flip and deusonmodels, states with two b quarks being stable inmost models.In the lattice calculation we measure twodiagrams for both meson-meson and meson-antimeson cases – see Fig. 1. One of them is theunconnected one without light quark interchangeand the other connected, where the light quarkshop from one meson to another. The light quarkmass we use is approximately that of s. Becauseof the static approximation for the heavy quarkstheir spin and isospin decouples making the pseu-doscalar B and the vector B∗ degenerate, whilephysically they have a 46 MeV (1%) separation.We call this degenerate set B. For the two-mesonsystem combinations of B and B∗ do have differ-ent energies in our case. We measure wavefunc-tions symmetric under interchange of the mesonswith the light quark spin and isospin being sin-glet or triplet; these then couple to B,B∗ combi-nations [4].We obtain estimates of light quark propaga-2Figure 1. Diagrams for BB and BB¯ systems.tors using maximally variance reduced pseudo-fermionic ensembles [6], 24 of which are generatedfor each gauge configuration. A second nestedMonte Carlo calculation is then performed for thepseudofermions of each gauge configuration. Thevariance reduction requires “halving” of the lat-tice into two separate regions with the propaga-tors going from one region to the other. Thusthe connected diagram for BB¯ and the cross cor-relator between QQ¯ and BB¯ need to have oneof the normally spatial axes as the temporalaxis, introducing considerable technical compli-cations. The parameters of our calculation areβ = 5.2, CSW = 1.76, a ≈ 0.14 fm, MPS/MV =0.72 for unquenched [5] and β = 5.7, CSW =1.57, a ≈ 0.17 fm, MPS/MV = 0.65 for thequenched case with a 163 × 24 lattice being usedfor both. We have 20 and 54 gauge configura-tions for quenched and unquenched respectively;with pseudofermionic fields these take some 60GB of diskspace. A variational basis of local andfuzzed mesonic operators is used, diagonalizationof which maximizes overlap with the ground stateof the system.We measure the different spin and isospin com-ponents as discussed in Ref. [4]. For the BB¯case Iq = 1 has only the unconnected diagram,whereas Iq = 0 has the connected one subtracted.2. RESULTSThe raw correlators show that for the BB sys-tem the unconnected diagram is much noisier anddoes not contribute to the binding at R > 1. Theconnected diagram, on the other hand, gives asmall binding for larger R and also contributes tothe observables where the spin of the light quarkschanges.At R = 0 the heavy quarks are at the samepoint and the BB case looks like a baryon withan antitriplet string. We can compare to previ-ously measured [6] energies of the Λb, Σb baryonsfor Iq, Sq = (0, 0) and Iq, Sq = (1, 1) respectively,finding excellent agreement [4]. States with a sex-tet string (0, 1; 1, 0) lie higher. The BB¯ singlet atR = 0 looks like a pion, and we find agreementwith the energy of a pion with non-zero momen-tum.The meson-meson potentials are shown in Fig.2. The Iq, Sq = (1, 1) case is similar to (0, 0)but less bound; the level ordering observed atR = 0 is retained and the attraction disappearsearlier. For the (1, 0) a remnant of the sextetstring makes the small-R potential repulsive. The(0, 1) at R = 0 is attractive for unquenched andrepulsive for quenched; this is the only qualitative(≈ 2.5σ) difference between the quenched and un-quenched results visible in our data. Both (0, 1)and (1, 0) seem to have attraction at r ≈ 0.3 fm,which is a meson exchange effect as opposed tothe small distance behaviour governed by gluoniceffects. From a crude two-body Schro¨dinger ap-proach using these potentials we expect bindingfor all of these cases except perhaps (1, 0).The meson-antimeson potential for Iq, Sq =(1, 0) is shown in Fig. 3, the (1, 1) case being sim-ilar. For both of these a QQ¯+ qq¯ state with thesame quantum number is lighter for small R. For(1, 0); (1, 1) the relevant energies are V (R)+pi andV (R) + ρ respectively, where V (R) is the staticQQ¯ potential. An estimate of V (R) + pi is in-cluded in Fig. 3.The contribution from meson exchange can beexamined e.g. by looking at the crossed dia-gram measured in the quenched approximationat fixed T as a function of R. The BB → BBcase should have a contribution from ρ exchange,and we indeed find agreement by using the pre-viously measured mρ and normalizing by hand.For BB∗ → B∗B we should have a contributionfrom pi exchange. In this case we can use a re-cent determination of the BB∗pi coupling [7], theexperimental decay constant and our mpi. In theone-pi exchange formula everything is thus known,and we find excellent quantitative agreement forR ≥ 0.5 fm. This is strong support for deusonmodels [8] in this distance range.30 1 2 3R/R00.00.20.4 dynamical, 54 confs.quenched, 20 confs.−0.20.00.20.4−0.8−0.40.0−1.2−0.8−0.40.0BB potentialDF: B=5.2, K=0.1395    Q: B=5.7, K=0.14077Iq,Sq=(0,0)Iq,Sq=(1,1)Iq,Sq=(0,1)Iq,Sq=(1,0)R0EBindingFigure 2. BB potentials. Here 2MBR0 =4.98(1), 5.83(3) for quenched, unquenched re-spectively and aR0 = 0.49 fm.3. STRING BREAKINGIn our quenched and unquenched calculationsthe ground state BB¯ and QQ¯ potentials cross atr ≈ 1.2 fm. We are investigating the break-ing of the QQ¯ string by using a variational ap-proach similar to that used in Higgs models byseveral groups. The cross correlator between two-meson and two-quark states allows us to studytheir mixing also in the quenched theory – in theunquenched case additional fermion bubbles in-duce corrections. The quenched mixing matrixelement can then be used to estimate the splittingof energy levels at the string breaking point, eventhough no actual splitting occurs with quenching.With an unquenched calculation the energy split-ting can be studied directly using the full varia-tional approach.One might think that an excited string wouldbreak at a smaller distance than the ground state.This is not necessarily the case, as e.g. the firstexcited state has Jz = 1 with quark separationalong z and only breaks into mesons BLB¯L′ withL + L′ > 0. In general it is an open question if0 1 2 3R/R0−2.5−2.0−1.5−1.0−0.50.00.5R0EBindingBBbar potential Iq=1, Sq=0DF: B=5.2,K=0.1395  Q: B=5.7, K=0.14077dynamical, 54 confs.quenched, 20 confs.dynamical, V(R)+mPSquenched, V(R)+mPSFigure 3. BB¯ potentials. For small R a state witha QQ¯ and a pion should be lighter.a state with particular quantum numbers has thelowest energy at a particular heavy quark separa-tion as a) a hybrid QQ¯ meson with excited glue,b) a ground state QQ¯ meson and a qq¯ meson or c)two heavy-light mesons. These energy levels andtheir mixing can be studied on the lattice withour techniques.REFERENCES1. A. M. Green and P. Pennanen, Phys. Rev.C57, 3384 (1998), hep-lat/9804003.2. P. Pennanen, A. M. Green and C. Michael,Phys. Rev. D59, 14504 (1999),hep-lat/9804004.3. A. M. Green, J. Koponen and P. Pennanen,hep-ph/9902249.4. UKQCD Collaboration, C. Michael andP. Pennanen, Phys. Rev. D60, 54012 (1999),hep-lat/9901007.5. UKQCD Collaboration, C. R. Allton et al.,Phys. Rev. D60, 34507 (1999),hep-lat/9808016.6. UKQCD Collaboration, C. Michael andJ. Peisa, Phys. Rev. D58, 34506 (1998),hep-lat/9802015.7. UKQCD Collaboration, G. M. de Divitiiset. al., JHEP 10, 010 (1998),hep-lat/9807032.8. N. A. To¨rnqvist, Z. Phys. C61, 525–537(1994), hep-ph/9310247.

Interactions of heavy-light mesons

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