The spectrum of static-light baryons in twisted mass lattice QCD
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The spectrum of static-light baryons in twistedmass lattice QCD
Marc Wagner, Christian Wiese ∗ Humboldt-Universität zu Berlin, Institut für Physik, Newtonstraße 15, D-12489 Berlin, GermanyE-mail: [email protected]
E-mail: [email protected]
We compute the static-light baryon spectrum with N f = < ∼ m PS < ∼
525 MeV, as wellas partially quenched quarks, which have the mass of the physical s quark. We extract massesof states with isospin I = , / ,
1, with strangeness S = , − , −
2, with angular momentumof the light degrees of freedom j = , P = + , − . We present a preliminaryextrapolation in the light u / d and an interpolation in the heavy b quark mass to the physical pointand compare with available experimental results. The XXVIII International Symposium on Lattice Field Theory, Lattice2010June 14-19, 2010Villasimius, Italy ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ he spectrum of static-light baryons in twisted mass lattice QCD
Christian Wiese
1. Introduction
A systematic way to study bottom baryons 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 the static limit. The spectrum of static-light baryons hasbeen studied by lattice methods in the quenched approximation [1, 2, 3] and recently also withdynamical sea quarks [4, 5, 6]. Here we use N f = L b , S b / S ∗ b , X − b and W − b we also predict a number of not yet measuredstates, mainly of negative parity.
2. Simulation setup
We use 24 ×
48 gauge field configurations generated by the European Twisted Mass Collab-oration (ETMC). The fermion action is N f = O ( a ) improved [9]. The gauge action is tree-level Symanzik improved [10] with b = . a = . ( ) fm[11]. At the moment we have considered three different values of the twisted mass m q with corre-sponding pion masses listed in Table 1. For details regarding these gauge field configurations werefer to [12, 13].In the valence sector we use light quarks, which have the same mass as the sea quarks (corre-sponding to u / d quarks), as well as partially quenched quarks with a mass m q , val = . s quark [14, 15]. m q m PS in MeV number of gauge field configurations0 . ( ) . ( ) . ( ) Table 1: twisted masses m q , corresponding pion masses m PS and number of gauge field configurationsconsidered.
3. Static-light baryon creation operators
To create static-light baryons, we use operators O twisted G = e abc Q a (cid:16) ( c b , ( ) ) T C G twisted c c , ( ) (cid:17) , (3.1)where Q is a static quark operator and c ( n ) are light quark operators in the so-called twisted basis.The upper indices a , b and c are color indices, C = g g is the charge conjugation matrix and G twisted is an appropriately chosen combination of g matrices.2 he spectrum of static-light baryons in twisted mass lattice QCD Christian Wiese
When discussing quantum numbers of static-light baryons, it is more convenient to first trans-form the operators (3.1) into the physical basis, in which they have the same structure, i.e. O physical G = e abc Q a (cid:16) ( y b , ( ) ) T C G physical y c , ( ) (cid:17) . (3.2)In the continuum the relation between the physical and the twisted basis is given by the twistrotation y = exp ( i g t w / ) c , where the twist angle w = p / O ( a ) . Nevertheless,it is possible to unambiguously interpret states obtained from correlation functions of twisted basisoperators in terms of QCD quantum numbers. The method has successfully been applied in thecontext of static-light mesons [16] and is explained in detail for kaons and D mesons in [17]. Fordetails regarding its application to static-light baryons we refer to an upcoming publication [18].Since there are no interactions involving the static quark spin, it is appropriate to label static-light baryons by angular momentum j and parity P of the light degrees of freedom, which aredetermined by G physical . Consequently, j = J = / J = / y ( ) y ( ) = ud − du (corresponding to I = S = y ( ) y ( ) ∈ { uu , dd , ud + du } (correspond-ing to I = S = y ( ) y ( ) ∈ { us , ds } (corresponding to I = / S = −
1) and y ( ) y ( ) = ss (corresponding to I = S = − G physical j P J I S name
I S name
I S name g + / L b / − X − b X X X g g + / L b / − X − b X X X1 0 − / − / − − X X X g − / − / − − − − g j + /
2, 3 / S b , S ∗ b / − − − W − b g g j + /
2, 3 / S b , S ∗ b / − − − W − b g j g − /
2, 3 / − / − − X X X g g j g − /
2, 3 / − / − − − − Table 2: static-light baryon creation operators and their quantum numbers ( j P : angular momentum andparity of the light degrees of freedom; J : total angular momentum; I : isospin; S : strangeness); operatorsmarked with “X” are identically zero, i.e. do not exist.
4. Numerical results
We compute correlation matrices C G j , G k ( t ) = h W | O twisted G j ( t ) (cid:16) O twisted G k ( ) (cid:17) † | W i . (4.1)3 he spectrum of static-light baryons in twisted mass lattice QCD Christian Wiese
We use several techniques to improve the signal quality, including operator optimization by meansof APE and Gaussian smearing, the HYP2 static action and stochastic propagators combined withtimeslice dilution. These techniques are very similar to those used in a recent study of the static-light meson spectrum [9, 19] and will be explained in detail in [18].From the correlation matrices (4.1) we extract effective masses by solving a generalized eigen-value problem (cf. e.g. [20] and references therein). Mass values are obtained by fitting constantsto effective mass plateaus at sufficiently large temporal separations. To exemplify the quality ofour results, we show effective masses and corresponding mass fits for the L b and the W − b baryon inFigure 1.00 . . . . . . . . . e ff ec ti v e m a ss temporal separation m ( L b ) = . ± . . . . . . . . . . e ff ec ti v e m a ss temporal separation m ( W − b ) = . ± . Figure 1: effective masses and corresponding mass fits; left: L b at m q = . × W − b at m q = . × Static-light baryon masses diverge in the continuum limit, because of the infinite self energy ofthe static quark. Therefore, we consider mass differences to another static-light system, the lighteststatic-light meson (quantum numbers j P = ( / ) − , corresponding to the B and the B ∗ meson,which are degenerate in the static limit). In such mass differences the self energies of the staticquarks exactly cancel.We use our results at three different light quark masses (cf. Table 1) by extrapolating massdifferences linearly in ( m PS ) to the physical u / d quark mass ( m PS =
135 MeV). Figure 2 (left)shows this extrapolation for the L b , the S b / S ∗ b and the W − b .Our results for the static-light baryon spectrum are collected in Table 3. In Wilson twistedmass lattice QCD isospin is explicitely broken by O ( a ) . Therefore, states with I = I z will have slightly different masses differing by O ( a ) . Because of this, in Table 3 two values arelisted for such states. Note, however, that within statistical errors there is no difference betweenany two such values, which is in agreement with the expectation that isospin breaking effects are O ( a ) and, hence, should be rather small.
5. Interpolation in the heavy quark mass and comparison to experimental data
To make predictions regarding the spectrum of b baryons and to perform a meaningful compar-4 he spectrum of static-light baryons in twisted mass lattice QCD Christian Wiese m ( b a r yon ) − m ( B ) i n M e V ( m PS ) in MeV W − b S b / S ∗ b L b m ( b a r yon ) − m ( B ) i n M e V m c / m Q W − b S b / S ∗ b L b experimental results Figure 2: left: linear extrapolation of static-light mass differences in ( m PS ) to the physical u / d quark masscorresponding to m PS =
135 MeV; right: linear interpolation between lattice static-light and correspondingexperimental charm-light mass differences in m c / m Q to the physical b quark mass corresponding to m c / m b = . j P I S name m ( baryon ) − m ( B ) in MeV0 + L b ( ) + S b / S ∗ b ( ) / ( ) − − ( ) − − ( ) / ( ) + / − X − b ( ) / ( ) + / − − ( ) / ( ) − / − − ( ) / ( ) − / − − ( ) / ( ) + − W − b ( ) − − − ( ) Table 3: mass differences of static-light baryons and the lightest static-light meson extrapolated to thephysical u / d quark mass. ison with experimental data, we interpolate linearly m c / m Q , where m Q is the heavy quark mass, tothe physical point m c / m b ≈ m ( D ) / m ( B ) = .
35. We do this by using our static-light lattice results(cf. Table 3) corresponding to m c / m Q = m c / m Q =
1. In Figure 2 (right) we show the interpolationfor L b , S b / S ∗ b and W − b .In Table 4 we compare results of these interpolations with experimentally available data for b baryons. Our lattice results are around 15% larger than the corresponding experimental results, atendency, which has already been observed in our recent study of the static-light meson spectrum5 he spectrum of static-light baryons in twisted mass lattice QCD Christian Wiese [9, 19]. In view of this discrepancy it is interesting to compare with results obtained by other latticegroups. When comparing mass differences in units of r , i.e. dimensionless quantities, e.g. with [4],we find agreement within statistical errors. Note, however, that the scale setting is rather different: r = .
49 fm in [4], while the corresponding ETMC value is r = .
42 fm.name m ( baryon ) − m ( B ) in MeV (lattice) m ( baryon ) − m ( B ) in MeV (experiment) L b ( ) ( ) (from [21]) S b ( ) ( ) (from [21]) S ∗ b ( ) ( ) (from [21]) X − b ( ) ( ) (from [21]) W − b ( ) ( ) (from [22]) Table 4: lattice predictions versus experimental results for mass differences between various b baryons andthe B meson. For the negative parity states listed in Table 3 experimental results for b baryons do not exist.There are also no lattice predictions for these states we are aware of. However, one can comparewith phenomenological models, e.g. with [23], for which one finds qualitative agreement.
6. Conclusions
We have studied the static-light baryon spectrum by means of N f = < ∼ m PS < ∼
525 MeV. We have performed an extrapolation in the light quark mass to thephysical u / d mass as well as an interpolation in the heavy quark mass to the physical b mass. Ourresults agree within around 15% with currently available experimental results for b baryons.Future plans regarding this project include increasing the statistical accuracy of our correlationmatrices to a level, where also excited states can reliably be extracted. Moreover, we intend toinvestigate the continuum limit, which amounts to considering other finer values of the latticespacing. Finally we plan to perform similar computations on N f = + + Acknowledgments
We acknowledge useful discussions with Jaume Carbonell, William Detmold, Vladimir Galkin,Karl Jansen, Andreas Kronfeld, Chris Michael and Michael Müller-Preussker. This work hasbeen supported in part by the DFG Sonderforschungsbereich TR9 Computergestützte TheoretischeTeilchenphysik.
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