We present results for the spectrum of static-light mesons from Nf=2 lattice
QCD. These results were obtained using all-to-all light quark propagators on an
anisotropic lattice, yielding an improved signal resolution when compared to
more conventional lattice techniques. In particular, we consider the inversion
of orbitally-excited multiplets with respect to the `standard ordering', which
has been predicted by some quark models.Comment: 3 pages with 3 figures. Talk by JF at "Quarks and Nuclear Physics",
  Madrid 5th-10th June 200

Cais, Alan O

Foley, Justin

Peardon, Mike

Ryan, Sinead M

English

arXiv

Submitted to Cornell University’s online archive www.arXiv.org in 2006 by Justin Foley.Post-print sourced from www.arxiv.orgWe present results for the spectrum of static-light mesons from Nf=2 lattice QCD. These results were obtained using all-to-all light quark propagators on an anisotropic lattice, yielding an improved signal resolution when compared to more conventional lattice techniques. In particular, we consider the inversion of orbitally-excited multiplets with respect to the `standard ordering', which has been predicted by some quark models.Justin Foley, Alan O Cais, Mike Peardon, Sinead M Ryanhttp://www.springer.com/physics/elementary/book/978-3-540-72515-

Cais, Alan O.

Peardon, Michael J.

Ryan, Sinead M.

Adelaide Research &amp; Scholarship

arXiv:hep-lat/0610062v1  9 Oct 2006EPJ manuscript No.(will be inserted by the editor)Static-light mesons on a dynamical anisotropic latticeJustin Foley1, Alan o´ Cais2, Mike Peardon2, Sine´ad M. Ryan21 Department of Physics, Swansea University, Singleton Park, Swansea SA2 8PP, UK2 School of Mathematics, Trinity College, Dublin 2, IrelandReceived: date / Revised version: dateAbstract. We present results for the spectrum of static-light mesons from Nf = 2 lattice QCD. These resultswere obtained using all-to-all light quark propagators on an anisotropic lattice, yielding an improved signalresolution when compared to more conventional lattice techniques. In particular, we consider the inversionof orbitally-excited multiplets with respect to the ‘standard ordering’, which has been predicted by somequark models.PACS. PACS-key 11.15.Ha – PACS-key 12.38.Gc1 IntroductionIt has long been known that hadrons containing a singlecharm or bottom quark exhibit approximate heavy quarkspin and flavour symmetries and can be described to areasonable approximation by an idealised system in whichthe heavy quark is taken to be infinitely massive [1]. Inthis limit the heavy quark symmetries become exact andthe system consists of light degrees of freedom bound to astatic colour point source. The static approximation canbe systematically improved using a power-counting rulein ΛQCD/mQ, which forms the basis for heavy quark ef-fective theory (HQET). For hadrons containing a bottomquark this approach is particularly suitable. In this caseone expects leading-order corrections to be already quitesmall, at the level of ten percent.On the lattice, mass-dependent errors which arise fromthe discretisation of the Dirac operator can limit the ac-curacy of simulations of heavy-light systems. In principle,one can avoid such errors by using the HQET lagrangianto describe the heavy quark dynamics. An alternative ap-proach is to perform a number of simulations both atunphysically light heavy quark masses using the latticeDirac operator and in the static limit and to interpolatethe results to the physical heavy quark mass guided bythe predictions of HQET. Therefore, precise calculationsin the static limit are of considerable interest to the lat-tice community. In addition, little is known experimen-tally of the excited state spectrum of heavy-light hadronsand accurate simulations of their simpler static-light coun-terparts will have considerable phenomenological impact.However, traditionally, simulations of static-light systemshave been extremely noisy. Ultimately, this stems from thefact that the use of conventional point-to-all propagatorsfor the light quarks restricts interpolating operators forthe static-light hadron to a single spatial lattice site.This problem could be overcome if it were possible tocompute all elements of the light quark propagator. Thiswould allow source and sink operators for the static-lightcorrelators to be placed at every spatial lattice site, yield-ing a dramatic increase in statistics. Indeed, simulationsof static-light mesons are the natural testing ground forall-to-all techniques since they require a single light quarkpropagator per configuration while the static quark prop-agator is easy to evaluate. It is therefore not surprisingthat a number of groups have attempted to tackle thisparticular problem [2,3].In this contribution, we describe a preliminary study ofthe excited state spectrum of static-light mesons using theall-to-all propagator technique introduced in Ref. [4]. Inthis approach, the important contribution of the low-lyingeigenmodes of the Dirac operator to the fermion propaga-tor is computed exactly. A noisy estimate is then used tocorrect for the contribution of the remaining eigenmodes.This estimate is obtained by inverting the Dirac operatoron a set of Z4 noise sources which have been partitionedin some subset of quarkfield indices. This partitioning or‘dilution’ yields a substantial reduction in the variance ofthe stochastic estimate. This study has been performedon a 3+1 anisotropic lattice, where the temporal latticespacing at is much finer than the spatial lattice spacingas. The fine temporal lattice spacing proves particularlyuseful when determining a plateau in the effective massesof the higher excited states. The correlators have beencomputed on Nf = 2 background configurations.2 Interpolating operatorsDue to heavy quark spin symmetry, states which lie in thesame hyperfine multiplet are degenerate and in the con-tinuum it is conventional to label these multiplets by the2 Justin Foley, Alan o´ Cais, Mike Peardon, Sine´ad M. Ryan: Static-light mesons on a dynamical anisotropic latticeLattice irrep Dimension JG1 212, 72...G2 252, 72...H 4 32, 52, 72..Table 1. Irreducible representations of the group of 24 properspatial lattice rotations with low-lying angular momentum con-tent.angular momentum and parity of the light degrees of free-dom [5]. For example, in the static limit the pseudoscalarand vector correspond to JℓPℓ = 12−.At finite lattice spacing, it makes sense to classify statesaccording to the irreducible representations (irreps) of theoctahedral point group Oh [6]. This 48 element groupis the direct product of the 24 proper spatial rotationsallowed by the lattice with the group consisting of theidentity and the space inversion operator. The relation-ship between these representations and continuum quan-tum numbers is not difficult to determine. Restricting thecontinuum irreps to the elements of the lattice symmetrygroup generates representations for Oh which are in gen-eral reducible and identification of the constituent irreps isa straight-forward exercise in group theory. Here, we takeadvantage of the degeneracies which arise from the heavyquark spin symmetry and construct interpolating opera-tors specifically for the light degrees of freedom. Therefore,our operators transform according to the double-valued orspinorial representations of Oh. There are 6 such represen-tationsG1u,G1g,G2u,G2g,Hu,Hg. (1)where the subscripts u and g denote representations whichare odd and even under spatial inversion respectively.Table 1 lists the irreps of the subgroup of proper rota-tions and the angular momenta of their lower-lying con-stituent states. From this table we see that states withthe same angular momentum but with different Jz maybe scattered across the lattice irreps. For example, stateswhich transform according to the rows of the 6-dimensionalspin 52representation are divided between the G2 and Hirreps. In a numerical study we should therefore expect toobserve energy levels in these irreps which are degenerate,up to lattice artifacts, corresponding to the Jℓ =52states.In this study, we were particularly interested in orbital ex-citations of the static-light meson. There is a single S-wavestate with JPℓℓ quantum numbers12−, the P-wave doubletis 12+, 32+and the D-wave states are labelled 32−, 52−. Ofthese, only the S-wave and the 12+P-wave which lie in theG1u and G1g irreps can be accessed using local operators,while all other irreps require the use of extended interpo-lating operators. It is worth noting that when one usesall-to-all light quark propagators the construction of ex-tended interpolating fields requires no additional fermionmatrix inversions and the extra computational cost in-curred is minimal. In order to determine accurately theexcited state spectrum it is important to choose opera-tors which couple strongly to the states of interest andwe tested a number of operators across the lattice repre-sentations. Further details of our choice of interpolatingoperators will be presented in Ref. [7].3 Simulation detailsThe lattice Dirac and gauge actions used in this study havepreviously been described in detail in Refs. [8,9]. Bothare specifically formulated for 3+1 anisotropic lattices.The Wilson-like fermion action is accurate to O(at, a3s)and employs stout-smeared spatial links to reduce radia-tive corrections. The gauge action incorporates tree-levelSymanzik and tadpole improvement and is accurate toO(a2t , a4s, αsa2s).The anisotropic lattice breaks hypercubic symmetryand introduces two new parameters: the bare quark andgauge anisotropies, denoted ξ0q and ξ0g , which must betuned such that the measured anisotropy, as/at, assumesa consistent value. In dynamical QCD the tuning proce-dure is quite complicated and is detailed in Ref. [10]. Inthis study we use bare anisotropy values taken from thatpaper of ξ0q = 7.52 and ξ0g = 8.06 which yields a measuredanisotropy of 6 with a three percent error.We employ the standard Eichten-Hill action [11] forthe static quark and the corresponding static quark prop-agator is a product of unsmeared temporal links.Measurements have been performed on 249 backgroundconfigurations on an 83 × 80 lattice with a spatial latticespacing of about 0.2fm. The mass of the sea quarks andthe light valence quark was found to be approximately thestrange quark mass. For the light quark propagator, wecomputed the contribution of 100 low-lying eigenvectorsof the Dirac operator. Two independent Z4 noise sourceswhich were diluted in time and colour were used in thestochastic estimate.To extract excited state energies we applied the stan-dard variational analysis to correlation matrices whichwere obtained by applying 5 or 6 levels of Jacobi smearingto the light quark fields. This was facilitated by the useof all-to-all propagators which meant that no additionalinversions were required to smear the quark fields.4 ResultsFig. 1 shows effective masses for the lowest-lying statesacross a number of the lattice irreps. In this case the lowesteffective mass corresponds to the S-wave 12−state and thehigher-lying states are orbital excitations. The highest-lying states lie in D-wave channels. Clear plateaux areevident in each channel.In addition, we obtain strong signals for a numberstates in each of the lattice irreps. Fig. 2 shows the groundstate and the first two excited states in the G1u (S-wave)irrep. These excited states shown here are almost certainlyradial excitations in the S-wave channel.Since we are working in the static limit the energiesdetermined by fitting to the effective masses contain anJustin Foley, Alan o´ Cais, Mike Peardon, Sine´ad M. Ryan: Static-light mesons on a dynamical anisotropic lattice 30 5 10 15 20 25 30 35 40 45 50 55t/at0.10.150.20.250.30.350.4a tMeffG1uG1gHgHuG2uFig. 1. Effective masses for the lowest-lying states in 5 latticeirreps. These include P-wave and D-wave excitations.0 5 10 15 20 25 30 35 40 45 50 55 60 65 70t/at00.10.20.30.40.5a tMeffground statefirst excited statesecond excited stateFig. 2. Effective masses for the lowest-lying states in the G1u(S-wave) irrep.unphysical shift. This shift cancels in energy differences,which are therefore physically meaningful. Fig. 3 plotsthe energy differences between the lowest-lying state inthe G1u representation (i.e. the S-wave multiplet) and anumber of excited states. We have not presented thesepreliminary results in physical units because we have per-formed our simulation on quite a small spatial volumewhich might distort the energies of the orbitally and ra-dially excited states. The main point here is the precisionto which we can compute the spectrum. Nevertheless, it isinteresting to note the position of the lowest-lying statesin the Hu and G2u irreps. Naively, one expects the light-est state in the Hu representation to correspond to the32−D-wave while the 52−ought to be the lightest state inthe G2u irrep. However, there appears to be no significantsplitting between these states which may indicate that the52−multiplet is lighter than the 32−. Such a scenario hasbeen predicted by quark models [12,13].G1u G1g Hg Hu G2u00.050.10.150.2a tδMground statesfirst excited statessecond excited statesPreliminaryFig. 3. Mass differences between the ground state S-wave mul-tiplet and higher lying states.5 Conclusions and outlookWe have presented preliminary results for the spectrum ofstatic-light mesons in Nf = 2 QCD. We have been able tocompute energy differences between the S-wave multipletand a number of orbital and radial excitations. The use ofall-to-all propagators allows us to minimise the statisticaluncertainty in our measurements and facilitates the use ofspatially-extended interpolating operators and the varia-tional approach. The results presented here have allowedus to make some tentative remarks about the nature ofthe spectrum; however, we have not yet made a serious at-tempt to identify the continuum quantum numbers of theobserved states. More recently we have performed similarsimulations using larger operator bases on two differentspatial volumes. These runs should allow us to assess fi-nite volume effects and identify multi-particle states. Weare currently analysing the results of these simulations andwe intend to present our findings in the near future [7].References1. N. Isgur and M.B. Wise, Phys. Lett. B232, (1989) 113.2. A. M. Green et al., Phys. Rev. D69, (2004) 094505.3. T. Burch and C. Hagen, hep-lat/0607029.4. J. Foley et al., Comput. Phys. Commun. 172, (2005) 145-162.5. N. Isgur and M. B. Wise, Phys. Rev. Lett. 66, (1991) 1130-1133.6. S. Basak et al, Phys. Rev. D72, (2005) 094506; Phys. Rev.D72, (2005) 074501.7. J. Foley et al., in preparation.8. J. Foley, A. O´ Cais, M. Peardon and S. M. Ryan, Phys. Rev.D72, (2005) 074503.9. C. Morningstar and M. J. Peardon, Nucl. Phys. Proc. Suppl.83, (2000) 887.10. R. Morrin, A. O´ Cais, M. Peardon, S. M. Ryan and J.I.Skullerud, Phys. Rev. D74, (2006) 014505.11. E. Eichten and B. Hill, Phys. Lett. B240, (1990) 193.12. H. J. Schnitzer, Phys. Lett. B226 (1989) 171.13. N. Isgur, Phys. Rev. D57, (1998) 4041-4053.

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