The isospin, spin and parity dependent potential of a pair of static-light
mesons is computed using Wilson twisted mass lattice QCD with two flavors of
degenerate dynamical quarks. From the results a simple rule can be deduced
stating, which isospin, spin and parity combinations correspond to attractive
and which to repulsive forces.Comment: 7 pages, 2 figures, talk given at the XXVIII International Symposium
  on Lattice Field Theory (Lattice 2010), June 14-19 2010, Villasimius, Ital

Wagner, Marc

English

arXiv

The isospin, spin and parity dependent potential of a pair of static-light mesons is computed using Wilson twisted mass lattice QCD with two flavors of degenerate dynamical quarks. From the results a simple rule can be deduced stating, which isospin, spin and parity combinations correspond to attractive and which to repulsive forces

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PoS(Lattice 2010)162Forces between static-light mesonsMarc Wagner∗Humboldt-Universität zu Berlin, Institut für Physik, Newtonstraße 15, D-12489 Berlin, GermanyE-mail: mcwagner@physik.hu-berlin.deThe isospin, spin and parity dependent potential of a pair of static-light mesons is computedusing Wilson twisted mass lattice QCD with two ﬂavors of degenerate dynamical quarks. Fromthe results a simple rule can be deduced stating, which isospin, spin and parity combinationscorrespond to attractive and which to repulsive forces.The XXVIII International Symposium on Lattice Field Theory, Lattice2010June 14-19, 2010Villasimius, Italy∗Speaker.c ￿ Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/PoS(Lattice 2010)162Forces between static-light mesons Marc Wagner1. IntroductionLattice computations of the potential of a pair of static-light mesons (in the following alsoreferred to as B mesons) are of interest, because they constitute ﬁrst principles determinations of ahadronic force. Until now interactions between static-light mesons have exclusively been studiedin the quenched approximation [1, 2]. Here I report on the status of an investigation with twoﬂavors of dynamical Wilson twisted mass quarks. Forces are not only studied between the lighteststatic-light mesons (denoted by S), but also ﬁrst excitations are taken into account (denoted by P−).Note that there is another ongoing study of static-light meson interactions with dynamical quarks,which has also been reported during this conference [3].2. Trial states and quantum numbers2.1 Static-light mesonsHere I consider static-light mesons, which are made from a static antiquark ¯ Q and a light quarky ∈ {u, d}. Consequently, isospin I = 1/2 and Iz ∈ {−1/2, +1/2}. Since there are no interac-tions involving the static quark spin, it is appropriate to classify static-light mesons by the angularmomentum of their light degrees of freedom j. I do not consider non-trivial gluonic excitations,hence j = 1/2 and jz = {−1/2, +1/2}, which is the spin of the light u/d quark. Parity is also aquantum number, P ∈ {+, −}.The lightest static-light meson has quantum numbers jP = (1/2)− (denoted by S). The ﬁrstexcitation, which is ≈ 400MeV heavier, has quantum numbers jP = (1/2)+ (denoted by P−).Examples of corresponding static-light meson trial states are ¯ Qg5y|W  and ¯ Qgjy|W  for S mesonsand ¯ Qy|W  and ¯ Qgjg5y|W  for P− mesons respectively.For a more detailed discussion of static-light mesons I refer to [4, 5].2.2 BB systemsThe aim of this work is to determine the potential of a pair of B mesons as a function of theirseparation R (without loss of generality I choose the axis of separation to be the z axis). To this endone has to compute the energy of eigenstates of the Hamiltonian containing two static antiquarks¯ Q(r1) and ¯ Q(r2), r1 = (0,0,−R/2) and r2 = (0,0,+R/2), which deﬁne the positions of the two Bmesons, and which will be surrounded by light quarks and gluons.These BB states are characterized by several quantum numbers. Since there are two lightu/d valence quarks, isospin I ∈ {0, 1} and Iz ∈ {−1, 0, +1}. Due to the separation of the staticantiquarks along the z axis, rotational symmetry is restricted to rotations around this axis. Con-sequently, states can be classiﬁed by the z component of total angular momentum. However, asalready mentioned in section 2.1 there are no interactions involving the static quark spin. There-fore, it is appropriate to label BB states by the z component of the angular momentum of the lightdegrees of freedom jz ∈ {−1, 0, +1}. Parity is also a symmetry and, therefore, a quantum num-ber, P ∈ {+, −}. For states with jz = 0 there is an additional symmetry, reﬂection along an axisperpendicular to the axis of separation (without loss of generality I choose the x axis). The cor-responding quantum number is Px ∈ {+, −}. When using |jz| instead of jz, Px is a quantum2PoS(Lattice 2010)162Forces between static-light mesons Marc Wagnernumber for all states. To summarize, BB states can be characterized by the following ﬁve quantumnumbers: (I,Iz,|jz|,P,Px).I use BB trial states(CG)AB￿¯ QC(r1)y(1)A (r1)￿￿¯ QC(r2)y(2)B (r2)￿|W , (2.1)where the lower indices A, B and C denote spinor indices, C = g0g2 is the charge conjugationmatrix and G is a combination of g matrices. Note that it is essential to couple the light degrees offreedom of both mesons in spinor space, because these degrees of freedom determine the quantumnumber |jz|. Proceeding in a naive way by coupling light and static degrees of freedom in both Bmesons separately will not result in a well deﬁned angular momentum |jz| and, therefore, will mixdifferent sectors. To obtain I = 0, the ﬂavors of the light quarks have to be chosen according toy(1)y(2) = ud −du, while for I = 1 three possibilities exist, y(1)y(2) ∈ {uu, dd, ud +du}. BBtrial states are collected in Table 1 together with their quantum numbers.y(1)y(2) = ud−du y(1)y(2) = ud+du y(1)y(2) ∈ {uu, dd}G |jz| P, Px result P, Px result P, Px resultg5 0 −, + A, SS +, + R, SS +, + R, SSg0g5 0 −, + A, SS +, + R, SS +, + R, SS1 0 +, − A, SP −, − R, SP −, − R, SPg0 0 −, − R, SP +, − A, SP +, − A, SPg3 0 +, − R, SS −, − A, SS −, − A, SSg0g3 0 +, − R, SS −, − A, SS −, − A, SSg3g5 0 +, + A, SP −, + R, SP −, + R, SPg0g3g5 0 −, + R, SP +, + A, SP +, + A, SPg1/2 1 +, ± R, SS −, ± A, SS −, ± A, SSg0g1/2 1 +, ± R, SS −, ± A, SS −, ± A, SSg1/2g5 1 +, ∓ A, SP −, ∓ R, SP −, ∓ R, SPg0g1/2g5 1 −, ∓ R, SP +, ∓ A, SP +, ∓ A, SPTable 1: quantum numbers of BB trial states; due to explicit isospin breaking, (I = 1,Iz = 0) and(I = 1,Iz = ±1) states are not degenerate in twisted mass lattice QCD (cf. section 3) and, therefore, listedseparately; “result” characterizes the shapes of the numerically computed BB potentials (A: attractive poten-tial; R: repulsive potential; SS: lower asymptotic value 2m(S); SP: higher asymptotic value m(S)+m(P−);cf. section 4).3. Lattice setupI use 243×48 gauge ﬁeld conﬁgurations generated by the European Twisted Mass Collabora-tion (ETMC). The fermion action is Nf = 2 Wilson twisted mass,SF[c, ¯ c,U] = a4åx¯ c(x)￿DW+imqg5t3￿c(x) (3.1)3PoS(Lattice 2010)162Forces between static-light mesons Marc Wagner[6, 7], where DW is the standard Wilson Dirac operator and c = (c(u),c(d)) is the light quarkdoublet in the so-called twisted basis. In the continuum the twisted basis is related to the physicalbasis by the twist rotation y = eig5t3w/2c, where w is the twist angle. w has been tuned to maximaltwist, i.e. w = p/2, where static-light mass differences are automatically O(a) improved. Thegauge action is tree-level Symanzik improved [8]. I use b = 3.9 and mq = 0.0040 corresponding toa lattice spacing a = 0.079(3)fm and a pion mass mPS = 340(13)MeV [9]. For details regardingthese gauge ﬁeld conﬁgurations I refer to [10, 11].In twisted mass lattice QCDat ﬁnite lattice spacing SU(2) isospin is explicitely broken to U(1),i.e. Iz is still a quantum number, but I is not. Moreover, parity P has to be replaced by twisted massparity P(tm), which is parity combined with light ﬂavor exchange. The consequence is that twistedmass BB sectors are either labeled by (Iz,|jz|,P(tm)P(tm)x ) for Iz =±1 or by (Iz,|jz|,P(tm),P(tm)x )for Iz = 0. A comparison with the set of quantum numbers discussed in section 2.2 shows that inthe twisted mass formalism there are only half as many BB sectors as in QCD, i.e. QCD BB sectorsare pairwise combined. Nevertheless, it is possible to unambiguously interpret states obtainedfrom twisted mass correlation functions in terms of QCD quantum numbers. The method hassuccessfully been applied in the context of static-light mesons [12] and is explained in detail forkaons and D mesons in [13]. For a detailed discussion of twisted mass symmetries in the contextof BB systems I refer to an upcoming publication [14].When computing correlation functions, I use several techniques to improve the signal qualityincluding operator optimization by means of APE and Gaussian smearing and stochastic propaga-tors combined with timeslice dilution. These techniques are very similar to those used in a recentstudy of the static-light meson spectrum [4, 5] and will also be explained in detail in [14].In contrast to spectrum calculations for static-light mesons [4, 5] and static-light baryons [15],where we have always used the HYP2 static action, I perform computations both with the HYP2static action and with unsmeared links representing the world lines of the static antiquarks. In par-ticular for small ¯ Q ¯ Q separations R<∼2a ultraviolet ﬂuctuations are important, which are, however,ﬁltered out, when using HYP smeared links. The effect of HYP smearing is shown in Figure 1.For all results presented in the following potential values corresponding to R ≤ 2a have been com-puted by means of unsmeared links, while for larger separations HYP smearing has been appliedto improve the signal-to-noise ratio.2m(S)−800MeV2m(S)−400MeV2m(S)0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8BBpotentialinMeVB meson separation R in fmHYP2 static actionno smearingFigure 1: the BB potential correspondingto y(1)y(2) =uu, G=g3 computedwith unsmearedlinks and withthe HYP2 static action.4PoS(Lattice 2010)162Forces between static-light mesons Marc Wagner4. Numerical resultsThe BB potentials presented and discussed in the following have been obtained by ﬁtting con-stants to effective mass plateaus obtained from temporal correlation functions of trial states (2.1).In twisted mass lattice QCD there are 24 independent Iz = 0 trial states (i.e. trial states not relatedby symmetries) and 12 independent Iz = ±1 trial states, i.e. 36 resulting potentials, which are notrelated by symmetries (cf. Table 1). Some of these potentials are quite similar, while others arenot. In total there are four signiﬁcantly different types of potentials: two of them are attractive, theother two are repulsive; two have have asymptotic values for large separations R, which are largerby around 400MeV compared to the other two (cf. the “result” columns of Table 1). For each ofthe four types an example is plotted in Figure 2.2m(S)−800MeV2m(S)−400MeV2m(S)m(S)+m(P−)2m(S)+800MeV0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8BBpotentialinMeVB meson separation R in fmy(1)y(2) = uu, G = 1y(1)y(2) = uu, G = g0y(1)y(2) = uu, G = g5y(1)y(2) = uu, G = g3Figure 2: examples of BB potentials as functions of the separation R.To understand the asymptotic behavior, it is convenient to express the BB creation operatorsappearing in (2.1) in terms of static-light meson creation operators. For the potentials shown inFigure 2 one ﬁnds after some linear algebra(C1)AB￿¯ QC(r1)uA(r1)￿￿¯ QC(r2)uB(r2)￿== −S↑(r1)P−↓(r2)+S↓(r1)P−↑(r2)−P−↑(r1)S↓(r2)+P−↓(r1)S↑(r2) (4.1)(Cg0)AB￿¯ QC(r1)uA(r1)￿￿¯ QC(r2)uB(r2)￿== −S↑(r1)P−↓(r2)+S↓(r1)P−↑(r2)+P−↑(r1)S↓(r2)−P−↓(r1)S↑(r2) (4.2)(Cg5)AB￿¯ QC(r1)uA(r1)￿￿¯ QC(r2)uB(r2)￿== −S↑(r1)S↓(r2)+S↓(r1)S↑(r2)−P−↑(r1)P−↓(r2)+P−↓(r1)P−↑(r2) (4.3)(Cg3)AB￿¯ QC(r1)uA(r1)￿￿¯ QC(r2)uB(r2)￿== −iS↑(r1)S↓(r2)−iS↓(r1)S↑(r2)+iP−↑(r1)P−↓(r2)+iP−↓(r1)P−↑(r2). (4.4)5PoS(Lattice 2010)162Forces between static-light mesons Marc WagnerAt large separations R the BB potentials are expected to approach the sum of the masses of thetwo individual B mesons. When considering (4.1) to (4.4) and Figure 2, one can see that the twopotentials with the lower asymptotic value (y(1)y(2) = uu, G = g5 and y(1)y(2) = uu, G = g3)contain SS combinations. These are signiﬁcantly lighter than the also present P−P− combinationsand should, therefore, dominate the correlation functions and effective masses at large temporalseparations. The asymptotic value of the corresponding potentials should be around 2m(S), whichis the case. In contrast to that the other two potentials with the higher asymptotic value(y(1)y(2) = uu, G = 1 and y(1)y(2) = uu, G = g0) exclusively contain SP− combinations. Theirasymptotic value is expected at around m(S)+m(P−), which is also reﬂected by Figure 2.This expansion of BB creation operators in terms of static-light meson creation operators alsoprovides an explanation, why potentials computed with different operators, but which have iden-tical quantum numbers, are of different type. An example is given by y(1)y(2) = uu, G = g3 andy(1)y(2) = uu, G = 1, both having quantum numbers (I = 1,Iz = +1,|jz| = 0,P = −,Px = −).The G = g3 potential is attractive with an asymptotic value at around 2m(S), while the G = 1potential is repulsive with an asymptotic value at around m(S)+m(P−). From (4.1) and (4.4)one can read off that the static-light meson content is essentially “orthogonal”: the G = g3 oper-ator contains SS and P−P− combinations, whereas the G = 1 operator is exclusively made fromSP− combinations. While the corresponding G = g3 correlator yields the ground state in the(I = 1,Iz = +1,|jz| = 0,P = −,Px = −) sector, which closely resembles a pair of S mesons,the G = 1 operator mainly excites the ﬁrst excitation, which is similar to an SP− combination. Thegenerated ground state overlap is, therefore, rather small and, consequently, very large temporalseparations would be needed to extract the ground state potential. Presumably, the potential corre-sponding to the G = 1 operator has a small ground state contribution, which contaminates the ﬁrstexcited state potential. This is supported by the observation that the asymptotic value of the G = 1potential is slightly lower than m(S)+m(P−). For a clean extraction of this ﬁrst excited state ananalysis of a 2×2 correlation matrix is needed.From the 36 independent potentials one can also deduce a rule stating, whether a BB potentialis attractive or repulsive. The rule is quite simple.A BB potential is attractive, if the trial state is symmetric under meson exchange, repulsive,if the trial state is antisymmetric under meson exchange.Here meson exchange means exchange of ﬂavor, spin and parity. One can easily verify this rule forthe examples discussed above: the operators (4.2) and (4.4) are symmetric under meson exchangeand give rise to attractive potentials, while the operators (4.1) and (4.3) are antisymmetric undermeson exchange and yield repulsive potentials. This more general rule is in agreement to what hasbeen observed in quenched BB computations for SS potentials [1, 2].5. ConclusionsI have presented results of an ongoing computation of BB potentials. Various channels char-acterized by the quantum numbers (I,Iz,|jz|,P,Px) have been investigated. The computationshave been performed with dynamical, rather light quark masses (mPS ≈ 340MeV). The resultshave been interpreted in terms of individual S and P− mesons. A simple rule has been establishedstating, whether a BB potential is attractive or repulsive.6PoS(Lattice 2010)162Forces between static-light mesons Marc WagnerThe statistical accuracy of the correlation functions needs to be improved. BB systems arerather heavy and, hence, effective masses are quickly lost in noise. At the present level of statisticsslight contamination from excited states cannot be excluded. To this end contractions are ongoing.Future plans include studying the light quark mass dependence, the continuum limit and ﬁnitevolume effects. Moreover, also BBs and BsBs potentials could be computed. To treat the s quark asa fully dynamical quark, such computations should be performed on Nf = 2+1+1 ﬂavor gaugeﬁeld conﬁgurations currently produced by ETMC [16]. It would also be interesting to supplementthe lattice computation by a perturbative calculation of BB potentials at small separations R<∼2.Finally, one could use the obtained BB potentials as input for phenomenological considerations toanswer e.g. the question, whether two B mesons are able to form a bound state.AcknowledgmentsI acknowledge useful discussions with Pedro Bicudo, William Detmold, Rudolf Faustov,Roberto Frezzotti, Vladimir Galkin, Chris Michael and Attila Nagy. This work has been supportedin part bythe DFGSonderforschungsbereich TR9Computergestützte Theoretische Teilchenphysik.References[1] C. Michael and P. Pennanen [UKQCD Collaboration], Phys. Rev. D 60, 054012 (1999)[arXiv:hep-lat/9901007].[2] W. Detmold, K. Orginos and M. J. Savage, Phys. Rev. D 76, 114503 (2007) [arXiv:hep-lat/0703009].[3] G. Bali and M. Hetzenegger, talk by M. Hetzenegger at “XXVIIIth International Symposium onLattice Field Theory”, Villasimius, Sardinia (2010).[4] K. Jansen, C. Michael, A. Shindler and M. Wagner [ETM Collaboration], JHEP 0812, 058 (2008)[arXiv:0810.1843[hep-lat]].[5] C. Michael, A. Shindler and M. Wagner [ETM Collaboration], arXiv:1004.4235[hep-lat].[6] R. Frezzotti, P. A. Grassi, S. Sint and P. Weisz [Alpha collaboration], JHEP 0108, 058 (2001)[arXiv:hep-lat/0101001].[7] R. Frezzotti and G. C. Rossi, JHEP 0408, 007 (2004) [arXiv:hep-lat/0306014].[8] P. Weisz, Nucl. Phys. B 212, 1 (1983).[9] R. Baron et al. [ETM Collaboration], arXiv:0911.5061[hep-lat].[10] Ph. Boucaud et al. [ETM Collaboration], Phys. Lett. B 650, 304 (2007) [arXiv:hep-lat/0701012].[11] P. Boucaud et al. [ETM collaboration], Comput. Phys. Commun. 179, 695 (2008) [arXiv:0803.0224[hep-lat]].[12] B. Blossier, M. Wagner and O. Pene [European Twisted Mass Collaboration], JHEP 0906, 022 (2009)[arXiv:0903.2298[hep-lat]].[13] R. Baron et al. [ETM Collaboration], arXiv:1005.2042[hep-lat].[14] M. Wagner [ETM Collaboration], to be published.[15] M. Wagner and C. Wiese [ETM Collaboration], arXiv:1008.0653[hep-lat].[16] R. Baron et al. [ETM Collaboration], JHEP 1006, 111 (2010) [arXiv:1004.5284[hep-lat]].7

Forces between static-light mesons

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