Heavy-light spectrum and decay constant from NRQCD with two ?avors of dynamical quarks?
arXiv:hep-lat/9911039v1 29 Nov 1999
CP-PACS Collaboration : A. Ali Khana , S. Aokib , R. Burkhaltera,b, S. Ejirib , M. Fukugitac , S. Hashimotod , N. Ishizukab , Y. Iwasakia,b, K. Kanayaa,b, T. Kanekoa, Y. Kuramashid, T. Mankea , K. Nagaia , M. Okawad, H.P. Shanahane , A. Ukawaa,b , and T. Yoshi?a,b e
a b c
Center for Computational Physics, University of Tsukuba, Tsukuba, Ibaraki 305-8577, Japan Institute of Physics, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan
Institute for Cosmic Ray Research, University of Tokyo, Tanashi, Tokyo 188-8502, Japan High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki 305-0801, Japan
DAMTP, 21 Silver St., University of Cambridge, Cambridge, CB3 9EW, England, U.K.
We report on a study of B mesons on Nf = 2 full QCD con?gurations using an RG-improved gauge action, NRQCD heavy quark action and tadpole-improved clover light quark action. Results on the heavy-light spectrum and the decay constants from 163 × 32 lattices at a?1 ≈ 1.5 GeV are presented, and compared with quenched results obtained with the same action combination at matching lattice spacings.
1. Introduction The decay constant fB is being studied extensively on the lattice because of its importance for the determination of CKM matrix elements. The spectrum of excited B mesons and b baryons is being measured in present experiments, whereas there exist only few lattice results on this subject. In this article we report on our study of B mesons in two-?avor full QCD employing the NRQCD action for heavy quark and a tadpoleimproved clover action for light quark. The dynamical con?gurations have been generated using the same light quark action and an RG-improved gauge action with a plaquette and a rectangular term. Details on our full QCD con?gurations can be found in Refs. . A parallel study of B mesons using the clover action for heavy quark is presented in Ref. . 2. Simulation Details We present results for two sets of dynamical lattices corresponding to the heaviest and the light? talk
Table 1 Parameters of lattices. The statistics for the dynamical lattices has been increased since Lat√ tice’99. The scale is ?xed by σ = 427 MeV (for each sea quark for dynamical con?gurations) κsea mP S /mV a?1 [GeV] σ #conf. 0.1375 0.8048(9) 0.937(6) 648 0.1410 0.586(3) 1.127(10) 490 ∞ – 0.919(7) 195
presented by A. Ali Khan
est sea quark in our con?guration set at β = 1.95. The results are compared to those from quenched lattices generated with the same RG-improved gauge action at β = 2.187, the lattice spacing from the string tension matched to the dynamical lattice with κsea = 0.1375. Some details on these runs are given in Table 1. We take 5 κ values for the light valence quark corresponding to mPS /mV ≈ 0.8 ? 0.5. The strange quark mass ms is ?xed using the K and the φ meson. Our results for the Bs meson are obtained with ms from the K, and the φ is used to estimate the systematic error.
2 Table 2 Results for decay constants. Errors given in this table are statistical (including the statistical uncertainty in Mb ), and, where applicable, the uncertainty in ?xing the strange quark mass. Other systematic errors are discussed in the text. κsea ∞ 0.1375 0.1410 fB [MeV] 193(4) 216(4) 215(6) fBs [MeV] 221(4)(+7) 250(4)(+8) 251(6)(+6) fBs /fB 1.147(10)(35) 1.157(9)(+35) 1.166(14)(+31)
For the heavy quarks, we use NRQCD at O(1/M ) with a symmetric evolution equation as de?ned in . We employ 5 bare heavy quark masses, covering a range of roughly 2.5?4.5 GeV. The heavy-light meson mass M is determined from the di?erence of the meson energy at ?nite momentum and at rest, assuming the dispersion relation, E(p) ? E(0) = p2 + M 2 ? M . As a consistency check, we use both the Bd and the Bs meson to determine the b quark mass. In our calculation of decay constants, the heavy-light current is corrected through O(α/M ). The mixing coe?cients between the lattice operators  contributing at this order to the time component of the axial vector current J4 , and the matching factor to the continuum current has been calculated  in one-loop perturbation theory, J4 = (1 + αρ0 )J4,lat + (1 + αρ1 )J4,lat + αρ2 J4,lat .
(2) (0) (1)
√ Figure 1. Φ ≡ (αs (M )/αs (MB ))(2/β0 ) f M √ (top), and one-loop corrections to f M (bottom) as a function of the inverse pseudoscalar meson mass. In the upper plot, squares stand for κsea = 0.1375, diamonds for κsea = 0.1410 and fancy squares for quenched. In the lower plot, (0) (1) circles denote αρ0 J4,lat /J4 , squares, αρ1 J4,lat /J4 , and diamonds, αρ1 J4,lat /J4 .
(1) corrections to be ? 2% ; we estimate our error from the truncation of the 1/M expansion to be ? 4%. The leading discretization e?ects from the light quarks and gluons of O(αaΛQCD ) and O(a2 Λ2 QCD ) are 5%. Added in quadrature, these estimates give 7%. Our two-?avor results for fB and fBs given in Table 2 show a 10% increase compared to the quenched values (see also Fig. 1). We do not resolve any sea quark mass dependence. The dependence on the value of αs is weaker than for the plaquette gauge action, and the di?erence between renormalized and bare decay constants is only about 5%. In Fig. 1 we show the one-loop corrections to
For the RG-improved gluon action, αV has not been calculated, and we use a tadpoleimproved one-loop expression for the M S coupling, αT I (1/a). MS 3. Decay Constants Our preliminary results for fB , fBs and fBs /fB are given in Table 2, along with the statistical error and, where applicable, the uncertainty in the determination of ms . Additional systematic errors are estimated as follows: O(α2 ) corrections, taken to be α2 × O(1), are 5%. A previous NRQCD calculation using the plaquette gluon action at a?1 ? 1 GeV ?nds the tree level O(1/M 2 )
3 the current J4 as a function of the heavy-light meson mass. In the B region, 1/M ? 0.2, we ?nd the (1) correction to J4,lat to be very small and the two other terms to contribute about the same amount. (2) The J4,lat contribution also contains a discretization correction to the current ?rst pointed out in . We note that this discretization correction is considerably smaller for the RG gauge action than for the plaquette gauge action . For fBs /fB , we cannot resolve a di?erence between the three lattices. In a parallel study of B mesons using clover heavy quarks , we have obtained fB and fBs taking the chiral limit for sea quark at β = 1.8, 1.95 and 2.1. The results from that study at β = 1.95 agree within the estimated errors with the present results from NRQCD. 4. Spectrum In Fig. 2, we give our results for several B splittings from the lattices with nf = 0 and nf = 2, κsea = 0.1375. The top part of the ?gure shows the B ? ? B splitting. At present, we cannot resolve any unquenching e?ects. For quarkonia, on the same lattices, the hyper?ne splitting is found to increase from the quenched value only by a few MeV . We ?nd the B ? ? B splitting to be ? 30% smaller than the experimental value. Possible sources of systematic error are the ?niteness of the sea quark mass, the O(α) correction to the coe?cient of the σ · B operator, and higher order relativistic corrections. In the middle part of Figure 2, we show re? sults for the B2 ? B splitting, and in the lower part, the spin-averaged Λb ? B splitting. We do not ?nd signi?cant unquenching e?ects. However, for de?nite conclusions, we need to study several sea quark masses and lattice spacings, which is in progress. This work is supported in part by the Grantsin-Aid of Ministry of Education, Science and Culture (Nos. 09304029, 10640246, 10640248, 10740107, 11640250, 11640294, 11740162). SE and KN are JSPS Research Fellows. AAK and TM are supported by the Research for the Future Program of JSPS.
Figure 2. Meson and baryon splittings in MeV. Circles denote results from CP-PACS at √ a?1 ( σ) ? 0.9 GeV. Diamonds stand for results from  (quenched) and  (nf = 2). Only statistical errors are shown. The solid line denotes the experimental value, the dashed lines, its error. REFERENCES 1. R. Burkhalter, Nucl Phys. B (Proc. Suppl.) 73 (1999) 3. 2. C. J. Morningstar and J. Shigemitsu, Phys.Rev. D57 (1998) 6741. 3. K.-I. Ishikawa, these proceedings. 4. J. Hein et al., Nucl. Phys. B (Proc. Suppl.) 73 (1999) 366. 5. H.P. Shanahan et al., CP-PACS collaboration, these proceedings. 6. T. Manke et al., CP-PACS collaboration, these proceedings. 7. A. Ali Khan et al., hep-lat/9809140. 8. S. Collins et al., hep-lat/9901001.