Untitled ar X iv :h ep -l at /9 40 70 27 v1 2 9 Ju l 19 94 PSI-PR-94-22 UCY-PHY-94/3 CERN-TH.7382/94 Beautiful Baryons from Lattice QCD 1 C. Alexandroua, A. Borrellib, S. Güskenc, F. Jegerlehnerb, K. Schillingd,c, G. Siegertc, R. Sommerd a Department of Natural Sciences, University of Cyprus, Nicosia, Cyprus b Paul Scherrer Institute, CH-5232 Villigen PSI, Switzerland c Physics Department, University of Wuppertal D-42097 Wuppertal, Ger- many d CERN, Theory Division, CH-1211 Geneva-23, Switzerland Abstract We perform a lattice study of heavy baryons, containing one (Λb) or two b-quarks (Ξb). Using the quenched approximation we obtain for the mass of Λb MΛb = 5.728 ± 0.144 ± 0.018GeV. The mass splitting between the Λb and the B-meson is found to increase by about 20% if the light quark mass is varied from the chiral limit to the strange quark mass. 1work supported in part by DFG grant Schi 257/3-2. 1 http://arxiv.org/abs/hep-lat/9407027v1 1 Introduction Considerable progress has been achieved in the computation of low lying hadronic states from lattice QCD (LQCD). It turned out that quenched LQCD reproduces the hadron spectrum in the light quark sector surpris- ingly well, once finite volume corrections are carefully taken into account [1]. This holds in particular for the ratio mN/mρ, which has been a notorious problem for LQCD over quite some years. Beauty physics is attracting much attention because the origin of CP violation in the B-system is still an open question. The investigation of such a system is a great challenge to LQCD. Considerable progress has been made in the lattice studies of heavy-light mesons like D- and B-mesons [2]. Heavy-light systems from the baryonic sector so far have been studied with lattice methods exploratively in the limit of infinite heavy quark mass [3, 4]. Throughout the current study the mass of the heavy quark has been kept finite. We will present results for the mass of Λb, a baryon composed of a b- quark and two light quarks, as well as for the mass of Ξb, a baryon composed of two b-quarks and one light quark. As in the case of heavy-light mesons, the most dangerous source of sys- tematic error stems from the fact that one is forced — in order to enter the region of heavy quark masses — to actually push the heavy mass close to the very limit of current lattice resolutions. We will attempt, however, to keep control on these effects of a finite lattice spacing a by a variety of precautions: 1. We avoid to compute the masses of the Λb and the Ξb directly, but rather calculate the mass splittings ∆Λ = MΛb − MB and ∆Ξ = MΞb − 2MB, with respect to the B-meson mass MB. These splittings do not depend on the heavy quark mass in the infinite mass limit and are therefore less prone to contamination by finite a effects in the b and c quark mass regions. 2. We monitor the dependence of the splittings on the lattice spacing for our three β values, β = 5.74, 6.0, 6.26. This enables us (a) to check the assumption of weak a dependence and (b) to perform an a → 0 extrapolation. 3. We will not calculate directly the mass splittings too close to the b quark mass. Instead we stop the calculation at approximately twice the charm quark mass and then extrapolate our data to the b mass. Moreover, we investigate finite size effects at β = 5.74, on three lattices of spatial volumes 83, 103 and 123, in lattice units. Our lattice parameters 2 are detailed in table 1. To obtain a good signal for the ground state, we use smeared gauge invariant interpolating quark fields [5], defined for the standard Wilson action. NS NT no. configs. β a mρ 8 24 175 5.74 0.542 ± 0.014 10 24 213 5.74 12 24 113 5.74 12 36 204 6.00 0.355 ± 0.016 18 48 67 6.26 0.260 ± 0.014 Table 1: Lattice parameters: space and time extents NS and NT , number of configurations, β and the ρ-mass in lattice units. 2 Smearing Techniques and Volume Effects For the reasons given above, we will compute the baryonic masses with refer- ence to the mass of the B-meson, by proper combinations that would elimi- nate the b-quark mass in the heavy quark limit. So we consider the splittings ∆Λ = MΛb −MB and ∆Ξ = MΞb −2MB, respectively, in the single and double beauty sectors. In the first case, we need to compute the correlators, for the heavy-light pseudoscalar meson, CP (t) = ∑ ~x 〈( Q̄(x)γ5qJ(x) ) ( q̄J(0)γ5Q(0) )〉 , (1) and for the Λ baryon [6] CΛ(t) = ∑ ~x 〈( ǫabcQa(x) ( qJb (x)Cγ5q J c (x) )) ( ǫabcQa(0) ( qJb (0)Cγ5q J c (0) ))† 〉 , (2) where Q(x) = Q(~x, t) is the heavy quark field, and qJ(x) = qJ(~x, t) is the light one, to which smearing of type J [5] has been applied2. C is the charge 2The correlator for the Ξ is obtained from the one for the Λ, by replacing the light quark qJ b by a heavy one Qb. 3 conjugation operator given by C = iγ4γ2. Given the lattice results for these correlators, we perform a direct fit to their ratio RΛ(t) = CΛ(t) CP (t) → Ae−∆Λt (3) in the large t limit. It is by now well known that smearing [3, 7, 8] is crucial to obtain a decent overlap of the operators with the ground state. In this work, we make use of the experience acquired previously while computing properties of the heavy- light pseudoscalar states [3], where we found gauge-invariant ‘Gaussian type’ wave functions (of r.m.s radius 0.3 fm) to provide sufficient overlap. The smearing was applied to the light quark source in the mesonic case. In this study, we are using precisely this procedure for the heavy-light baryons as well, without any further optimization. In order to establish ground state dominance we look for a plateau in the local mass of the ratio RΛ. In fig. 1a we show as an example the local mass of RΛ, for the two largest lattices: the solid line shows the fitted value for the plateau. Fig. 1b shows the corresponding local mass for the ratio RΞ(t) = CΞ(t)/C 2 P (t) used to determine ∆Ξ. These figures show the quality of the plateaus for representative κ values. Worse quality is found only in a few cases, and it resulted in larger statistical errors; for instance the two largest κ values at β = 5.74 given in table 3. The pseudoscalar mass was extensively studied in ref. [9] and the values have been taken from there. For checking the finite volume effects, we computed ∆Λ on three lattices with NT = 24, β = 5.74 and sizes NS =8, 10, 12, for κl = 0.156, and κh = 0.125, 0.140, 0.150. We compare the results for the splitting in table 2. The values exhibit deviations of at most 4% . In fig. 2 we plot the dimensionless ratio ∆Λ/mρ as a function of L, the lattice size in units of mρ; as can be seen, for these values of κh, the finite size effects of this ratio are smaller than our statistical errors. In the following, we will fix the volume to about 1 fm (which corresponds to NS ≃8, 12 and 18 for β = 5.74, 6.0 and 6.26 respectively) and carry out a detailed study of the extrapolation to the continuum limit. As a tribute to possible finite size effects we will add the above maximal variation of 4% as an uncertainty to all results. 4 κl κh ∆Λ, 8 3 × 24 ∆Λ, 10 3 × 24 ∆Λ, 12 3 × 24 0.156 0.125 0.564 ± 0.008 0.554 ± 0.005 0.569 ± 0.005 0.140 0.592 ± 0.007 0.575 ± 0.005 0.588 ± 0.004 0.150 0.621 ± 0.007 0.601 ± 0.005 0.612 ± 0.006 Table 2: Results for three different spatial lattice extents, NS =8, 10, and 12, NT = 24, β = 5.74. 3 Continuum Limit Λ Splitting. The results for ∆Λ at the three β values 5.74, 6.00 and 6.26 at fixed volume are listed in table 3, all in lattice units. The extrapolations to u and s-type light quarks, — for given heavy quark κh — are performed as described in [3]. Since we are evaluating masses, it is natural to use mρ to set the lattice scale. The values of mρ for the various lattices have been listed in table 1; the experimental value used is: mρ = 768 MeV. In fig. 3a we plot ∆Λ in GeV extrapolated to the chiral limit as a function of 1/MP in GeV −1. Within the statistical precision achieved in this compu- tation, the points show no dependence on a, although a is varied by about a factor two (cf. Table 1). In the continuum, the 1/MP expansion for ∆Λ gives ∆cont.Λ (MP ) = c0 + c1 MP + O(MP −2) . (4) Assuming no dependence on the lattice spacing, this form can be fitted directly to the points in fig. 3a yielding ∆Λb = 431(28) MeV at the mass of the B-meson. This result is included in the figure as the inverted triangle. The error bar of this point does not account for the fact that the simulation results exclude an a-dependence only within their precision. A realistic error that includes the uncertainty of extrapolating the lattice data to the continuum is obtained by allowing for the leading a-dependence at each value of MP [9]: We start out from a selected value of MP (in physical units) and interpolate the lattice results from each (fixed) β-value to the value of MP . This enables us to compare ∆Λ at different values of a. A subsequent 5 83 × 24, β = 5.74 κh a∆Λ a∆Λ(ms) a∆Ξ a∆Ξ(ms) ∆Λ(ms)/∆Λ(mu) 0.06 0.25 ± 0.13 0.37 ± 0.08 −0.24 ± 0.02 −0.31 ± 0.02 1.46 ± 0.40 0.09 0.34 ± 0.10 0.42 ± 0.06 −0.17 ± 0.02 −0.23 ± 0.02 1.23 ± 0.18 0.125 0.38 ± 0.04 0.46 ± 0.02 −0.12 ± 0.02 −0.17 ± 0.01 1.20 ± 0.07 0.14 0.42 ± 0.02 0.50 ± 0.01 −0.01 ± 0.01 −0.009 ± 0.009 1.18 ± 0.04 0.15 0.47 ± 0.04 0.54 ± 0.02 0.09 ± 0.01 −0.080 ± 0.008 1.13 ± 0.05 123 × 36, β = 6.00 κh a∆Λ a∆Λ(ms) a∆Ξ a∆Ξ(ms) ∆Λ(ms)/∆Λ(mu) 0.1 0.22 ± 0.02 0.28 ± 0.01 −0.14 ± 0.02 −0.18 ± 0.01 1.23 ± 0.05 0.115 0.21 ± 0.02 0.28 ± 0.01 −0.10 ± 0.01 −0.147 ± 0.008 1.29 ± 0.08 0.125 0.25 ± 0.01 0.300 ± 0.008 −0.09 ± 0.01 −0.130 ± 0.007 1.19 ± 0.03 0.135 0.27 ± 0.01 0.310 ± 0.007 −0.06 ± 0.01 −0.101 ± 0.006 1.17 ± 0.03 0.145 0.31 ± 0.02 0.348 ± 0.007 0.01 ± 0.01 −0.049 ± 0.009 1.11 ± 0.04 183 × 48, β = 6.26 κh a∆Λ a∆Λ(ms) a∆Ξ a∆Ξ(ms) ∆Λ(ms)/∆Λ(mu) 0.09 0.15 ± 0.01 0.188 ± 0.009 −0.07 ± 0.01 −0.107 ± 0.008 1.25 ± 0.07 0.10 0.15 ± 0.02 0.19 ± 0.01 −0.073 ± 0.008 −0.110 ± 0.006 1.29 ± 0.07 0.12 0.16 ± 0.01 0.201 ± 0.007 −0.062 ± 0.06 −0.010 ± 0.004 1.25 ± 0.04 0.135 0.18 ± 0.01 0.216 ± 0.008 −0.032 ± 0.005 −0.071 ± 0.004 1.21 ± 0.04 0.145 0.21 ± 0.01 0.243 ± 0.009 0.022 ± 0.006 −0.024 ± 0.005 1.14 ± 0.04 Table 3: The mass splittings ∆Λ and ∆Ξ at the chiral limit and at the strange quark mass in lattice units for β = 5.74, 6.00, 6.26. The last column gives the ratio ∆Λ(ms)/∆Λ(mu) for the same β values. 6 linear extrapolation in a will then yield the continuum estimate for ∆Λ, at the chosen physical value of MP . Reiteration of the procedure on a set of masses MP determines the nu- merical dependence of ∆cont.Λ on MP . In order to remain in compliance with the assumption of linearity in a, we used only a subset of the data listed in table 3, by excluding the data for the two heaviest masses at any value of β. A final fit of the continuum limit values ∆cont.Λ to eq.(4) yields ∆Λb = 458 ± 144 ± 18 MeV . (5) Here the first error is purely statistical and stems from a full jacknife analysis of our data. The second error represents the 4% uncertainty due to finite vol- ume effects. Together with the known experimental value MB = 5.27 GeV, we obtain an estimate for the mass of Λb : MΛb = 5.728 ± 0.144 ± 0.018 GeV. (6) We note in passing, that the parameter c0 is in agreement with the estimates obtained directly in the static approximation [3], albeit within the larger uncertainties of the latter. In the same way we obtain the mass splitting ∆Λb(ms) where the light quark mass is extrapolated to the strange quark mass. The data fitted are shown in fig. 3b, and the result of the extrapolations described above is ∆Λb(ms) = 597 ± 91 ± 24 MeV , (7) with the same meaning of errors. Instead of looking directly at the splitting at the strange quark mass, one may consider the ratio r = ∆Λ(ms) ∆Λ(mu) , (8) which is expected to have an even weaker a-dependence. Indeed the ratio for the different β values shown in fig. 3c follows a “universal curve” with quite small statistical errors. A linear fit to the (weak) mass dependence yields r = 1.25 ± 0.03 ± 0.05. (9) This ratio represents the change in the Λb mass when replacing the u quark by an s quark. 7 Ξ Splitting. The mass splitting for the Ξ baryon is investigated using the same techniques as for the Λ. Table 3 displays the results after extrapolation of the light quark to the d and s mass respectively. We then convert all data to physical units, and plot ∆Ξ vs. 1/MP at the chiral limit in fig. 4. In this case the data show a statistically significant a-dependence, especially for heavy masses. We extrapolate to the continuum limit as above for pseudoscalar masses MP below the charm mass. We use the same ansatz as in the case of ∆Λ to fit the 1/MP dependence of ∆Ξ and obtain: ∆Ξb = −90 ± 170 MeV , (10) This mass shift together with the experimental value for MB determines MΞb = 10.45 ± 0.17 GeV , (11) for the physical Ξb–mass. Another possibility to compute the Ξb mass, is to consider the ratio R ′Ξ(t) = CΞ(t)/C h̄h P (t), which should yield the splitting between the Ξb mass and the Υ-meson mass. We evaluated this quantity but the resulting plateaus turned out to be of worse quality than those for the other chan- nel studied; this is due to the fact that smearing was applied only to light quarks, and so the heavy-heavy channels suffer from relevant contamination by higher-mass states. 4 Discussion The mass splitting technique is a viable method to compute the Λb mass on the lattice. Both the lattice spacing dependence of the mass splitting and its dependence on the heavy quark mass are weak. Thus an extrapolation to the continuum and to the b–quark mass is possible. Our actual value of MΛb = 5728 ± 144 ± 18 MeV can be compared to the value 5630 MeV suggested by Martin et al. within the naive quark model approach [4]. Experimentally, the determination of the Λb mass has a somewhat con- troversial history ever since 1981 [12]. Recently, the UA1 collaboration has measured the mass from 16 events in the decay channel J/ΨΛ3. Their value 3It is an open question, why this decay channel has not been observed in the CDF experiment. 8 is [13] MΛb = 5640 ± 50 ± 30MeV. The DELPHI collaboration is presently quoting [14] a preliminary mass value, MΛb = 5635 +38 −29 ± 4MeV, which is based on one candidate event in the Λ+c π − and in the D0pπ− decay modes. These numbers are in agreement with our result. For MΛc we obtain 2433±88±18 MeV, which is in rough accord with the experimental value of 2285 MeV [12]. Looking at the light quark dependence we found a 20% increase as we lift the quark mass from the chiral limit to the strange quark mass. Finally, an estimate for the Ξb mass is given in eq.(11). Our errors do include – as the dominant part – the uncertainty induced by an extrapolation to the continuum limit. Nevertheless, it is desirable to further check these extrapolations through simulations with higher lattice resolutions and/or different lattice actions. Acknowledgements We thank the Personal at the Computer Centers CSCS in Manno Switzerland and Jülich, Germany for their support. KS thanks A. Martin for a valuable discussion. References [1] F. Butler, H. Chen, J. Sexton, A. Vaccarino and D. Weingarten, IBM- Report 1994 (HEP-LAT-9405003) [2] see e.g. C. Bernard in Lattice 93 Nucl. Phys. B (Proc. Suppl.) 34 (1994) 47; R. Sommer,Beauty Physics in Lattice Gauge Theory, preprint DESY 94-011, to be published in Physics Reports C. [3] C. Alexandrou, S. Güsken, F. Jegerlehner, K. Schilling, and R. Sommer, Nucl. Phys. B414 (1994) 815. [4] A. Martin and J.M. Richard, Phys.Lett. B185 (1987) 426. [5] S.Güsken, in Lattice 89, Nucl. Phys. B (Proc. Suppl.) 17 (1990) 361 [6] M. Bochicchio, G. Martinelli, C. R. Allton, C. T. Sachrajda and D. B. Carpenter, Nucl. Phys. B372 (1992) 403 9 [7] A. Duncan, E. Eichten and H. Thacker, Nucl. Phys. B (proc. Supp.) 30 (1993) 441. [8] UKQCD Collaboration, S. Collins et al., Nucl. Phys. B (Proc. Supp.) 30 (1993) 393. [9] C. Alexandrou, S. Güsken, F. Jegerlehner, K. Schilling, G. Siegert and R. Sommer, DESY preprint 93-179, Z. Phys. C in press. [10] S. Aoki et al., Nucl. Phys. B (Proc. Suppl.) 34 (1994) 363. [11] C. Alexandrou, S. Güsken, F. Jegerlehner, K. Schilling, and R. Sommer, Nucl. Phys. B374 (1992) 263. [12] Particle Data Group, K. Hikasa et al., Phys. Rev. D45 (1992) 4365. [13] C. Albajar et al., UA1 collaboration, Phys. Lett. B273 (1991) 540. [14] M. Battaglia et al., DELPHI collaboration, DELPHI note DELPHI 94- 30 PHYS 363, March 1994. Figure captions Figure 1a The ∆Λ local masses, given by µ loc Λ (t) = log[ RΛ(t−a) RΛ(t) ] for κl = 0.1525, κh = 0.125 at β = 6.0, for a 12 3 × 36 lattice, and for κl = 0.1492, κh = 0.12 at β = 6.26, for a 18 3 × 48 lattice. Figure 1b The same as figure 1a, but for µlocΞ (t) = log[ RΞ(t−a) RΞ(t) ]. Figure 2 ∆Λ/mρ is shown vs L for three heavy quark masses at β = 5.74. The light quark mass was fixed to about twice the strange quark mass. Figure 3a The Λ mass splitting is shown vs 1/MP in GeV −1 at the chiral limit. The solid line is a global fit assuming no a effects. It gives the value 10 shown with the inverted triangle at the B-meson mass. The value ob- tained after extrapolation to the continuum limit and to the B-meson mass, is shown by the open circle. This extrapolation is discussed in the text. Figure 3b As for figure 3a but at the strange quark mass. Figure 3c The ratio ∆Λ(ms)/∆Λ(mu) is shown vs 1/MP . The notation is the same as for figure 3a. Figure 4 The Ξ mass splitting, ∆Ξ = MΞ − 2MP is shown vs 1/MP in GeV −1 at the chiral limit. The notation is the same as for figure 3a. 11 This figure "fig1-1.png" is available in "png" format from: http://arxiv.org/ps/hep-lat/9407027v1 http://arxiv.org/ps/hep-lat/9407027v1 This figure "fig1-2.png" is available in "png" format from: http://arxiv.org/ps/hep-lat/9407027v1 http://arxiv.org/ps/hep-lat/9407027v1 This figure "fig1-3.png" is available in "png" format from: http://arxiv.org/ps/hep-lat/9407027v1 http://arxiv.org/ps/hep-lat/9407027v1 This figure "fig1-4.png" is available in "png" format from: http://arxiv.org/ps/hep-lat/9407027v1 http://arxiv.org/ps/hep-lat/9407027v1 This figure "fig1-5.png" is available in "png" format from: http://arxiv.org/ps/hep-lat/9407027v1 http://arxiv.org/ps/hep-lat/9407027v1 This figure "fig1-6.png" is available in "png" format from: http://arxiv.org/ps/hep-lat/9407027v1 http://arxiv.org/ps/hep-lat/9407027v1 This figure "fig1-7.png" is available in "png" format from: http://arxiv.org/ps/hep-lat/9407027v1 http://arxiv.org/ps/hep-lat/9407027v1