Static-light meson masses from twisted mass lattice QCD
ETM Collaboration, Karl Jansen, Chris Michael, Andrea Shindler, Marc Wagner
aa r X i v : . [ h e p - l a t ] A ug SFB/CPP-08-60, DESY 08-115, LTH 801, HU-EP-08/28
Static-light meson masses from twisted mass latticeQCD
Karl Jansen
DESY, Platanenallee 6, D-15738 Zeuthen, GermanyE-mail: [email protected]
Chris Michael, Andrea Shindler
Theoretical Physics Division, Department of Mathematical Sciences, University of Liverpool,Liverpool L69 3BX, UKE-mail: [email protected]
E-mail: [email protected]
Marc Wagner ∗ † Humboldt-Universität zu Berlin, Institut für Physik, Newtonstraße 15, D-12489 Berlin, GermanyE-mail: [email protected]
We compute the static-light meson spectrum using two-flavor Wilson twisted mass latticeQCD. We have considered five different values for the light quark mass corresponding to300 MeV < ∼ m PS < ∼
600 MeV. We have extrapolated our results, to make predictions regarding thespectrum of B and B s mesons. The XXVI International Symposium on Lattice Field TheoryJuly 14 - 19, 2008Williamsburg, Virginia, USA ∗ Speaker. † On behalf of the ETM Collaboration. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ tatic-light meson masses from twisted mass lattice QCD
Marc Wagner
1. Introduction
A systematic way to study B and B s mesons from first principles is lattice QCD. Since am b > b quark a formalism such as Heavy Quark Effective Theory (HQET). In this paper we considerthe leading order of HQET, which is just the static limit. In this limit a “ B meson” will be the“hydrogen atom” of QCD. States are either labeled by J P = ( L ± / ) P , where L denotes orbitalangular momentum, ± / P parity, or by S ≡ ( / ) − , P − ≡ ( / ) + , P + ≡ ( / ) + , D − ≡ ( / ) − , ... In the static limit these states will be doubly de-generate, since there is no interaction with the heavy quark spin, i.e. the total spin for given J P is F P = J P ± / N f = N f = u / d quark mass allowing more reliable extrapolations.
2. Static-light meson creation operators
In the continuum an operator creating a static-light meson with well defined quantum numbers J P is given by O ( J P ) ( x ) = ¯ Q ( x ) Z d ˆ n G ( J P ) ( ˆ n ) U ( x ; x + r ˆ n ) y ( x + r ˆ n ) . (2.1)¯ Q creates an infinitely heavy antiquark, R d ˆ n denotes an integration over the unit sphere, U is aspatial parallel transporter, and y creates a light quark separated by a distance r from the antiquark. G ( J P ) is an appropriate combination of spherical harmonics and g -matrices yielding a well definedtotal angular momentum J and parity P . The meson creation operators we use in this paper arelisted in Table 1. J P F P G ( J P ) ( x ) O h lattice J P ( / ) − [ S ] − , − g , g g j x j A ( / ) − , ( / ) − , . . . ( / ) + [ P − ] + , + , g j x j A ( / ) + , ( / ) + , . . . ( / ) + [ P + ] + , + g x − g x (and cyclic) E ( / ) + , ( / ) + , . . . ( / ) − [ D − ] − , − g ( g x − g x ) (and cyclic) E ( / ) − , ( / ) − , . . . ( / ) − [ D + ] − , − g x x + g x x + g x x A ( / ) − , ( / ) − , . . . ( / ) + [ F − ] + , + g ( g x x + g x x + g x x ) A ( / ) + , ( / ) + , . . . Table 1: static-light meson creation operators.
When constructing lattice versions of the operators (2.1), one has to replace the integrationover the unit sphere by a discrete sum over lattice sites with fixed distance r from the static anti-2 tatic-light meson masses from twisted mass lattice QCD Marc Wagner quark. For J = / J = / O ( J P ) ( x ) = ¯ Q ( x ) (cid:229) n = ± ˆ e , ± ˆ e , ± ˆ e G ( J P ) ( ˆ n ) U ( x ; x + r ˆ n ) y ( x + r ˆ n ) , (2.2)and for J = / O ( J P ) ( x ) = ¯ Q ( x ) (cid:229) n = ± ˆ e ± ˆ e ± ˆ e G ( J P ) ( ˆ n ) U ( x ; x + r ˆ n ) y ( x + r ˆ n ) . (2.3)The spatial parallel transporters U in (2.2) are straight paths of links, while in (2.3) we use “diagonallinks”, which are averages over the six possible paths around a cube projected back to SU(3). Thestates created by these lattice meson creation operators do not form irreducible representations ofthe rotation group SO ( ) , but representations of its cubic subgroup O h . Therefore, these stateshave no well defined total angular momentum J . They are linear superpositions of an infinitenumber of total angular momentum eigenstates. The common notation of the corresponding O h representations together with their angular momentum content are also listed in Table 1. Note thatwe do not consider O h representations T and T , because these representations yield correlationfunctions, which are numerically identical to those obtained from the operators listed.
3. Simulation setup
We use 24 ×
48 gauge configurations produced by the European Twisted Mass Collaboration(ETMC). The fermion action is N f = b = . a = . ( ) fm.We consider five different values of the twisted mass m q (cf. Table 2). Tuning to maximal twist hasbeen performed at the lightest m q value yielding k cr = . < ∼ m PS < ∼
600 MeV. For details regarding this setup we refer to [11, 12, 13]. m q m PS in MeV number of gauge configurations0 . ( ) . ( ) . ( ) . ( ) . ( ) Table 2: m q values and corresponding pion masses m PS .
4. The static-light meson spectrum
We determine the static-light meson spectrum from 6 × h representation. Due to parity breaking discretization errors of the Wilson twisted mass formulationwe use states with P = + and states with P = − in the same correlation matrix.3 tatic-light meson masses from twisted mass lattice QCD Marc Wagner
To reduce statistical noise and to simplify quark smearing, we use stochastic propagators ob-tained by inverting four random spin diluted timeslice sources per gauge configuration [13]. Weuse the HYP2 static action [14, 15] and apply Gaussian smearing [16, 17] to the dynamical quarkoperators with APE smeared spatial links [18]. For details regarding the construction and the com-putation of these correlation matrices we refer to [19].Static-light meson masses are determined both by solving a generalized eigenvalue problemand computing effective masses, and by fitting a suitable ansatz of exponentials to the correla-tion matrices. Results obtained from both approaches are consistent. Note that static-light mesonmasses diverge in the continuum limit due to the self energy of the static quark. Therefore, wealways consider mass differences of static light mesons, where this self energy exactly cancels, andwhich are physically meaningful in the continuum limit. Mass differences between various statesand the lightest static-light meson ( J P = ( / ) − ground state) are shown in Figure 1 as functionsof ( m PS ) . D m = m - m ( S ) i n M e V (m PS ) in GeV linear extrapolation to physical quark massesu/d- and s-quark extrapolationsF - (J P = (5/2) + )D + (J P = (5/2) - )D - (J P = (3/2) - )P + (J P = (3/2) + )P - (J P = (1/2) + )S* (J P = (1/2) -, *) Figure 1: static-light mass differences linearly extrapolated to the u / d quark mass and the s quark mass. We extrapolate linearly in ( m PS ) to the physical u / d quark mass ( m PS =
135 MeV) as wellas to the physical s quark mass (taken here as m PS =
700 MeV). Note that we consider the unitarysector in both cases, where valence quarks and sea quarks are of the same mass. This implies forthe s quark extrapolation a sea of two degenerate s instead of a sea of u and d . We plan to improvethese computations by using N f = + + c / dofindicating that a straight line is a suitable ansatz.Performing a similar u / d extrapolation for the P wave mass difference yields m ( P + ) − m ( P − ) = ( ) MeV, i.e. the ( / ) + state is lighter than the ( / ) + state as is usuallyexpected. We see evidence for a reversal of this level ordering as the light-quark mass increases ob-taining a difference m ( P − ) − m ( P + ) = ( ) MeV at the strange quark mass. It will be interestingto check this in the continuum limit. 4 tatic-light meson masses from twisted mass lattice QCD
Marc Wagner u / d quark extrapolation: s quark extrapolation: J P m ( J P ) − m ( S ) in MeV m ( J P ) − m ( S ) in MeV c / dof ( / ) − , ∗ [ S ∗ ] ( ) ( ) . ( / ) + [ P − ] ( ) ( ) . ( / ) + [ P + ] ( ) ( ) . ( / ) − [ D − ] ( ) ( ) . ( / ) − [ D + ] ( ) ( ) . ( / ) + [ F − ] ( ) ( ) . Table 3: static-light mass differences linearly extrapolated to the u / d quark mass and the s quark mass.
5. Predictions for B and B s meson masses To make predictions regarding the spectrum of B and B s mesons, we interpolate between ourstatic-light lattice results and experimental data for charmed mesons [21]. To this end we assumea linear dependence in 1 / m Q , where m Q is the mass of the heavy quark. Results for P wave B and B s states are shown in Figure 2 and Table 4. Note that the lines labeled by “ S ( J P = ( / ) − ) ” inFigure 2 together with the experimental values for B ∗ and B ∗ s (the triangles intersected by theselines) indicate that straight lines are suited for interpolation. D m = m - m ( F P = - ) i n M e V m c / m Q a) prediction for excited B statesP + (J P = (3/2) + )P - (J P = (1/2) + )S (J P = (1/2) - )experiment (PDG, J P unknown)experiment (CDF and C ˘ , J P = (3/2) + ) 0 100 200 300 400 500 600 0 0.2 0.4 0.6 0.8 1 D m = m - m ( F P = - ) i n M e V m c / m Q b) prediction for excited B s statesP + (J P = (3/2) + )P - (J P = (1/2) + )S (J P = (1/2) - )experiment (PDG, J P unknown)experiment (CDF and C ˘ , J P = (3/2) + )BK and B * K thresholds
Figure 2: static-light mass differences linearly interpolated to a heavy b quark. a) B mesons. b) B s mesons. tatic-light meson masses from twisted mass lattice QCD Marc Wagner
In Table 4 we compare our lattice results to experimental data available for P wave B and B s mesons [21, 22, 23, 24, 25]. The lattice results for B states are larger by around 15% thantheir experimental counterparts, while there is significantly better agreement for B s mesons. For aconclusive comparison it will be necessary to investigate the continuum limit, which amounts toconsidering other values for the lattice spacing. m − m ( F P = − ) in MeV m − m ( F P = − ) in MeVstate lattice PDG ∗ CDF CØ state lattice PDG ∗ CDF CØ B ∗ ( ) - - B ∗ s ( ) - - B ∗ ( ) ↑ - - B ∗ s ( ) ↑ - - B ( ) ( ) ( ) ( ) B s ( ) ( ) ( ) - B ∗ ( ) ↓ ( ) ( ) B ∗ s ( ) ↓ ( ) ( ) Table 4: lattice and experimental results for P wave B and B s states ( ∗ : J P unknown). In Figure 2b we also plot the BK and B ∗ K threshold (406 MeV and 452 MeV respectively). Our P wave B s results indicate that the corresponding decays are energetically allowed. Consequently,one can expect that these states have a rather large width compared to e.g. certain excited D s mesons.
6. Conclusions
We have studied the static-light meson spectrum by means of N f = < ∼ m PS < ∼
600 MeV. We have performed an extrapolation in the light quark mass to thephysical u / d mass and s mass respectively, as well as an interpolation in the heavy quark mass tothe physical b mass. Our results agree within < ∼
15% with currently available experimental resultsfor P wave B and B s states.Future plans regarding this project include an investigation of the continuum limit, whichamounts to considering other values for the lattice spacing. Moreover, we plan to perform similarcomputations on N f = + + u and d in B s computations. We also intend todetermine the static-light decay constants f B and f B s . Acknowledgments
MW would like to thank Carsten Urbach for help in retrieving and handling ETMC gaugeconfigurations. Moreover, we acknowledge useful discussions with Benoit Blossier and CarstenUrbach. This work has been supported in part by the DFG Sonderforschungsbereich/TransregioSFB/TR9-03.
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