- Email: [email protected]
Lattice QCD results on B physics
2Q& __ __ @
PROCEEDINGS SUPPLEMENTS Nuclear
ELSEVIER
Physics
B (Proc.
Suppl.)
111 (2002)
75-8 1
Lattice QCD Results on B physics T. Onogi” “Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan I review the status of Lattice QCD results on B physics.
1. Introduction Determination of ]I&], IVubl, lVcbl from B factory experiment requires a model independent calculation of hadron matrix elements. One of the most promising approaches is lattice &CD, which is a formalism for nonperturbative calculation of matrix elements (O(O]Hi), (Hz]I?]Hi) from first principles of QCD with controlled errors. In general, the matrix elements can be decomposed into two parts as (OMS(p))
=
-%4(OV4)>
(I)
where Z(p,a) is the short distance correction to match the scheme difference in lattice and continuum theory. The matching factors are obtained usually perturbatively, but in some cases nonperturbatively. (Olat (a)) is the long distance part which is computed by the numerical path integral
nf 2-lJ n
det(D + mi) exp(-S,)O.
Dd
(2)
such as masses and decay constants unquenched calculation have started for the last few years. Numerous efforts have been devoted in lattice studies for decay constants of B mesons and it has been shown that different approaches to B meson give consistent results within the systematic errors, so that the lattice methods have been proved to work. At present it is time for us to aim at further precisions or look for new phenomenological applications. In this report, I would like to cover two topics. The first topic is the bottom quark mass determination and the decay constants and bag parameters of the B meson, which are the basic quantities. For these quantities, results in quenched QCD calculation have been established. I report the recent developments in unquenched calculation and in matching at two-loop level or nonperturbative level. The second topic is the progress in other new physical quantities such as form factors in exclusive semileptonic decays or HQET parameters in quenched calculations.
i
Since the fermion determinant is highly nonlocal, the numerical cost for the path integral is huge. Thus we often employ the “quenched approximation” with which sea-quark effect is neglected, i.e. the determinant is replaced by unity. Although this drastic approximation which introduces uncontrollable systematic errors, the calculation in quenched approximation is useful since it serves as a test ground for precise calculation in an exploratory stage, since one can apply the same numerical technique to unquenched calculation to obtain the physical answer just by replacing the gauge configuration from the quenched one to the unquenched one. In fact, for simple quantities 0920-5632/02/$ - see front matter 0 2002 Published PI1 SO920-5632(02)01687-O
2. Lattice formalism for heavy quarks There are three approaches to the heavy quark on the lattice with different systematic errors. The first is NRQCD [l], which exploits an nonrelativistic effective action with certain orders in The problem in this ap0(1/m) corrections. proach is that the theory is nonrenormalizable so that no continuum limit can be taken. The matching of full relativistic theory and the effective theory is performed by perturbation theory practically through O((Y,) so that the higher order perturbative error is another problem. The second is the Fermilab approach [2]. This is a reinterpretation of O(a)-improved Wilson ac-
by Elsevier Science B.V.
T Onogi/Nuclear
Heavy Static [15] Static & O(a)-W O(a)-W [16] O(a)-W 1181 1 NRQCD ‘[19]
[16]
Physics
B (Proc. Suppl.)
BBd (mb ) 0.81(5)(3)(2) 0.87(4)(i) 0.93(8) (:) 0.92f4) 1 0.852(45)
BB,
111 (2002)
7541
Bs(mb)
IBB,
1.01(l) 0.99(2) 0.98(5) 0.98(3) 1 1.015(6)(4)
0.96(8)(5)(5) 0.80(l)(4) 0.86(2)(i) [ 0.87(l)(9)
Table 1 Recent results for the bag parameters.
tion. The theory is smoothly connected to relativistic theory so that it is passable to take the continuum limit. However when the quark mass is heavier than the lattice cutoff, the lattice spacing dependence of the discretization errors are complicated so that it is nontrivial how to extrapolate the simulation results. Since the matching is performed with one loop perturbation theory, the perturbative error is another problem as in NRQCD. The third is the relativistic approach, where Symanzik O(a)one uses nonperturbative improvement of Wilson quark action and compute physical quantities around charm quark mass region and extrapolate to bottom, which is exploited by UKQCD and APE collaborations. In this approach, O((am)2) error and extrapolation error are the problems. 3. B meson decay constants rameters
and bag pa-
The decay constant and the bag parameters of the B meson are the key quantities for extracting the CKM elements l&l and IVt,l from the mass differences of the neutral B mesons using the following relations
Am,
=
I&&b12f&BBq,
. AmB, = _Ivtsl”fgaBBs Am&
I%i12~&BB~
9 =
d,s
(3) (4)
The bag parameter of the B, meson is also useful in predicting the width difference in the B, - ??, system as follows
FW@sl~~lBsj
- Gs(z)(BsIOsIBs)
+&,,dcq.
(5)
The decay constants in quenched calculation have been studied extensively. The recent calculations using Fermilab formalism [3-71, NRQCD [8-lo] and extrapolation method [ll-131 show nice agreement within errors of lo%, which can be summarized as fB
f&/f& fD
=
173* 23 MeV
=
1.15 f 0.03
=
203 f 14 MeV.
The bag parameters in quenched calculation have been studied in several different approaches: the static limit, extrapolation method and NRQCD, and recent results are listed in Table 1. The O(a)-improved Wilson Quenched value for the Bag parameter has an error of 10% either from perturbative error (Static, NRQCD) or extrapolation (Relativistic), which are consistent with each other. However, BB, / BB~ has small errors because there is a drastic cancellation in the renormalization factors(Static, NRQCD) and because extrapolation in l/M becomes more stable in the ratio (Relativistic). In the last few years, unquenched calculation for fB has started by several groups using various methods. The unquenched calculation for the bag parameter has been first studied with the heavy quark in the static limit, but the error is still large compared to the quenched calculation. Recently JLQCD has started two-flavor QCD project to compute the decay constant and the bag parameters in which they perform a realistic simulation with small lattice spacing a (p=5.2, a-l N 2 GeV) and modest lattice size (203 x 48 N, 2.0 fm) in search for sea quark effect [19]. The action for the gauge field is the standard plaque-
T Onogi/Nuclear Physics B (Proc. Suppl.) Ill
(2002) 7541
Mp- 6 GeV (-B meson) 0.8
A -
_
I
unquench lin
’
I
I
I
I
’
’
8
CCollins99
’
’
I
’
I
I
-
OMILCOO
qua
WP-PACSOO oCP-PACS(NR) oJLQCDO1 I
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I
0.2
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0.4
I
I
>
0.6
I
0.8
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1
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1
200
150
100
I Figure 2. nf = 2 unquenched results for
2
i0
fs.
mZ2[GeV2] Figure 1. The chiral behavior for the decay constant with nf = 2 unquenched simulation by JLQCD.
tte action and nonperturbatively improved Wilson fermion is used for the light valence and sea quarks with ~=0.1340-0.1355, which corresponds to mps/mv=0.8-0.6 . The heavy quark is described by NRQCD heavy quark action with l/m correction. Figure 3 shows the sea quark mass dependence of the decay constants in unquenched calculation, where 9f, denotes &f~ times the renormalization group factor. They employ three fitting functions for the chiral extrapolations of ‘Pf,, which are the linear function a0 x (1 + a;X), the quadratic function bo x (1 + blX + b2X2), and the linear plus log function ~0 x (1 + clX + c2XZnX) with X = rim2 The bag p&meters Bg are also studied by JLQCD. They tried four different operator matchings which differ either at O(cwz) or O(a,h/mb). The mass dependences are found to be small, the chiral extrapolation error are smaller than that for f~. The preliminary results from JLQCD are =
190 f 14 f 7 f 19 MeV,
fB,/fBd
=
1.184 f 0.026 f 0.020 f 0.015,
BBd(mb)
=
0.872 f 0.039 f 0.004 f 0.073,
fBd
=
0.999 f 0.012 f 0.004,
fBsB;: f&B;:
=
1.183 f 0.027 f 0.020 f 0.015,
Bs, (ma)
=
0.858 f 0.033 f 0.007 f 0.072,
AFS 1’S
=
0.107 f 0.026 f 0.014 f 0.017.
BB,/BB,
<
q
In Figure 3, the nf = 2 unquenched results [14, 6,10,19,7] of fB is shown. They are consistent within errors and collecting all the results, I arrive the following world averages
fB = f&/f& =
198f
30MeV
1.15 f 0.05
for the unquenched (n, = 2) results. 4.
Bottom quark mass
Another major progress in lattice QCD is the precise determination of the bottom quark mass mp(mb). The bottom quark mass is determined from the binding energy E of the B meson using the following relation (6) where 2, is the multiplicative mass renormalization factor and bm is the additive energy shift. An important observation by Martinelli and Sachrajda [20] is that although these renormalization factors have renormalon ambiguities, they cancel each other when combined so that the
l? Onogi/Nuclear
78
Physics
B (Proc. Suppl.)
short distance relation between the lattice binding energy and the MS mass is renormalon free. Although the factor 2, is known to three loop from calculation in the continuum, the energy shift which requires lattice perturbative calculation has been known only to one-loop order for a long time. Recently, lattice part of the perturbative coefficients have been computed in static limit analytically to two-loop order by Martinelli and Sachrajda [23] and to three-loop order using numerical technique by DiRenzo et al. [21] and Lepage et al. [22]. APE [23] and GLOK [24] performed a lattice calculation of B meson in the static limit to obtain quenched mp(mb) with two-loop and three-loop precision respectively. APE also performed a nf = 2 unquenched calculation to obtain mp(m,) curacy, which gives consistent results [25,26].
with two-loop acresults by NRQCD
More recently, quenched mp(mb) from B meson in the static limit with nonperturbative accuracy has been obtained using the Schroedinger functional technique by Heitger and Sommer [27], where they match the relativistic quark with RG invariant mass M and static quark in a small volume. It is found that the mF(mb) with twoloop, three-loop, and nonperturbative give consistent results within quoted errors. Comparison of these quenched calculations and APE’s nf = 2 unquenched results the current estimate of the bottom quark mass is mp(rm,)
=
5. B -+ D(*)lv
4.30f
0.10 GeV.
(7)
form factors
B + D(*)Zv form factor is crucial for extracting ]Vcb] from the following relation dlm ~v,b~2~~13+iX*&“)~2, (8) ZJ where the form factors FB+.D(*) (w) are given as
FB+D*(~)
=
VA
f-
2:;;m,
- &
1’ (9)
111 (2002)
7.541
Hashimoto et al. (Fermilab collaboration) [30] obtained the coefficients in l/m2 corrections performed a quenched lattice calculation of the following double ratio of matrix elements
-Pl~“~l~H~lbyo4D)
lh+(‘)12 =
(Dlq”clD)(BlbyoblB)
(11)
(D*I~“bl~*)(~*IbyOclD*) Ih(
=
JhA1(1)J2
(D*IE~~c~D*)(~?* IbyOblB*) ’ (12) (D*I~~yiy5blB)(B*Ibyiy5clD) (D*I~~yiy5clD)(B*Ibyiy5blB)
=
‘(13)
The idea is that from the symmetry it is guaranteed that the deviation from unity in the double ratio only start from l/m2 including systematic errors. There are power divergences, but the double ratio cancels the O(1) and O(l/m) contribution. So that the coefficients can be obtained with reasonable accuracy which enables to obtain the form factor of l-2% accuracy as follows
where the first and second term denotes the statistical and systematic errors within quenched approximation. In addition to the above errors, this calculation contains unknown quenching errors. However, this method is expected to provide the best estimate without model uncertainties once unquenched calculation is performed. 6. B -+ dv
form factors
The study of B -+ dv form factor f+(q2) has been one of the main issues in lattice &CD, since it is necessary for extracting Iv,&,] from exclusive semileptonic decay from the following formula dr @
G2 - ~(lc,121Vub121f+(q2))2, - 24rr3
where the form factor is defined
f+(q2) [(Pp +h)‘” m&-m;
+f0(q2)
q2
~ Q >
(14) as
mk;““qp] (15)
T Onogi/Nuclear
19
Physics B (Proc. Suppl.) 11 I (2002) 7541
Burdman et al. which manifests the heavy quark symmetry and chiral symmetry. Fitting the lattice data as a function of the light quark mass, inverse heavy quark mass and the pion recoil energies they obtain
(1.18 f 0.37 f 0.08 f 0.31) psec-’ for the partially integrated differential decay rate. 0.0
”
”
”
15
”
”
20
”
”
”
25
’
”
30
q2 (Gd) Figure 3. q2 dependence of B + dv form factors. Upward triangle, downward triangle, square, circle symbols denotes the data from UKQCD, APE, Fermilab and JLQCD collaborations respectively. Open and filled symbols corresponds to f”, f+.
with pB and k, the momenta of the initial and final pseudo-scalar mesons and q = PB - k,. Recently four groups have carried out quenched calculation for semileptonic form factors. UKQCD [33] and APE [34] used relativistic quark action for heavy quark and extrapolated the results around charm quark mass region to the physical b quark mass for a fine lattice with P = 6.2. Fermilab [35] used Fermilab action for the heavy quark and simulated with three different values of /I (5.7, 5.9, 6.1). JLQCD [31] used NRQCD action for the heavy quark and performed a high statistics calculation with @ = 5.9. Figure 3 shows the results from the four groups they give consistent values for f+ although some discrepancies is observed in f”. The difference originates from the differences in the heavy quark mass and light quark mass dependence of the data from each group as well as the extrapolation method. Further understanding of the systematic errors in mass extrapolation is necessary. Since all the groups give consistent values for f+, I present the results from JLQCD collaboration. They used chiral extrapolation method based on the parameterizations of form factor by
7. HQET parameters There are also pioneering studies on HQET parameters which are useful for the theoretical prediction of inclusive decays. The lifetime ratio of Ab and B is one of the problem in understanding the beauty hadron system. If the large deviation from unity observed in current experimental data remains true with further precise data, it casts an important question on the heavy quark symmetry Assuming the quark hadron duality, the life time ratio is
-T(Ab) = +‘I
1
+
xl (Ab) A1(B”) 2mi -
+3cBA2(Ab)
-
X2(B”)
2mt (16)
+O(Ilm% where
(17) A2W)
=
(Hp7.Bbp) 2M
.
(18)
H
JLQCD collaboration [38] performed a quenched calculation on a p = 6.0, 203 x 48 lattice, with NRQCD heavy quark and obtained the following results A(&,) - ii(B)
=
393(31)MeV
- X1(B)
=
-0.21(21)GeV2,
Xi(&)
which are small as expected from the quark model or sum rule estimations.
80
Group Gimenez Kronfeld
1: Onogi/Nuclear
et al [36] Simone [37]
Table 2 Lattice calculation
A (GeV) 0.18t;:g 0.68?~:$
of HQET
Physics B (Proc. Suppl.) II I (2002) 75-81
Xi (GeV2) -(0.09 f 0.14) -(0.45 f 0.12)
parameters.
The semileptonic inclusive decay B + XJu is another place which requires nonperturbative determination of HQET parameters A, Xi, X2. which denotes the binding energy, the kinetic energy and the energy from the chromomagnetic interaction so that the relation ma=mFP-iT+
xl
+ CGA2
2m
+ O( 1/m3)
(19)
holds. Two groups has performed lattice calculations in quenched approximation. Gimenez et al. [36] computed the matrix element in the static limit, where Kronfeld and Simone [37] computed the B meson masses as a function of bare mass to obtain A and Xi from the mass formula. Table 2 gives the results from the two groups. The current result shows a disagreement. Further studies on the perturbative error and the subtraction of power divergence are necessary. 8. Summary In summary, quenched calculation in basic quantities fr,, BE, Bs are under control with 10% accuracy. For these basic quantities, improvements such as unquenched calculations or computations of two-loop or nonperturbative renormalization factors has started giving reliable high precision results with more realistic simulations. Calculations of new quantities in are being developed in quenched approximation. The double ratio method is proved to be a promising tool for a precise calculation of l/m2 corrections of B + D*Zu form factors. Application to unquenched calculation is strongly desired. The current efforts of B + du form factor at 20% acFurther studies in reducing errors and curacy. understanding the difference between different groups is required. The determinations of various HQET parameters in quenched approxima-
tion are in progress. As future problems, nonperturbative renormalization with heavy quark is an important problem for precise calculation. Computing new quantities such as B + plv, K*(p)l+l-, the light-cone wavefucntion, the shape function would be very useful. Unquenched calculations with better control of chiral extrapolation using lighter sea quark is necessary. For this purpose developments in both the computer resources and the simulation algorithm are awaited.
Acknowledgments I would like to thank A. Kronfeld, S. Ryan, C.T. Sachrajda, L. Lellouch, C.-J.D. Lin, R. Sommer, H. Matsufuru, S. Hashimoto, and N. Yamada for discussions and private communications.
REFERENCES 1.
W. E. Caswell and G. P. Lepage, Phys. Lett. B167 (1986) 437; B.A.Thacker and G. P. Lepage, Phys. Rev. D43 (1991) 196; G. P. Lepage, et al., Phys. Rev. D46 (1992) 4052. 2. A. El-Khadra et al., Phys. Rev. D55 (1997) 3933. et al., Phys. Rev. D58 (1998) 3. A. El-Khadra 014506. 4. S. Aoki et al., Nucl. Phys. B (Proc. Suppl.) 63 (1998) 356. 5. C. Bernard et al., Phys. Rev. Lett. 81 (1998) 4812. 6. A. Ali Khan et al., Phys. Rev. D64 (2001) 034505. 7. S. Datta et al., Nucl. Phys. B (Proc. Suppl.) 94 (2001) 346. 8. A. Ali Khan et al.Phys. Lett. B427 (1998) 132. 9. K-I. Ishikawa et al., Phys. Rev. D61 (2000) 074501. 10. A. Ali Khan et al., Phys. Rev. D64 (2001) 054504. 11. D. Becirevic et al., Phys. Rev. D60 (1999) 074501. 12. K. Bowler et al., Nucl. Phys. B619 (2001) 507.
E Onogi/Nuclear Physics B (Proc. Suppl.) 111 (2002) 7541
13. L. Lellouch and C.-J. D. Lin, Phys. Rev. D64 (2001) 094501. 14. S. Collins et al., Phys. Rev. D60 (1999) 074504. 15. V. Gimenez and J. Reyes, Nucl. Phys. B545 (1999) 576. 16. D. Becirevic et al., hep-lat/0110091. 17. D. Becirevic et al., Nucl. Phys. B618 (2001) 241. 18. L. Lelouch adn C.-J. D. Lin, Phys. Rev. D64 (2001) 0011086. 19. N. Yamada et al., Nucl. Phys. B (Proc. Suppl.) 106 (2002) 397. 20. G. Martinelli and C. T. Sachrajda, Nucl. Phys. B478 (1996) 660. 21. DiRenzo et al., Nucl. Phys. B578 (2000) 383, hep-lat/0010064. 22. P. Lepage et al., Nucl. Phys. B (Proc. Suppl.) 83-84 (2000) 866. 23. G. Martinelli and C. Sachrajda, Nucl. Phys. B559 (1999) 429. 24. C. Davies, et al, Phys. Lett. B345 (1995) 42. 25. K. Hornbostel et al., Nucl. Phys. B (Proc. Suppl.) 73 (1999) 339. 26. S. Collins, hep-lat/0009040. 27. J. Heitger and R. Sommer, Nucl. Phys. B (Proc. Suppl.) 106 (2002) 358. 28. V. Gimenez et al., Nucl. Phys. B (Proc. Suppl.) 83 (2000) 286; JHEP 0003 (2000) 018. 29. M. Kurth and R. Sommer, Nucl. Phys. B597 (2001) 488. 30. S. Hashimoto et al., Phys. Rev. D61 (2000) 014502; hep-ph/0110253. 31. S. Aoki et al., Phys. Rev. D64 (2001) 114505. 32. G. Burdman et al., Phys. Rev. D49 (1994) 2331. 33. K. C. Bowler et al., Phys. Lett. B486 (2000) 111. 34. A. Abada et al., Nucl. Phys. B619 (2001) 565. 35. A. X. El-Khadra et al., Phys. Rev. D64 (2001) 014502. 36. V. Gimenez et al., Nucl. Phys. B486 (1997) 227; Phys. Lett. B393 (1997) 124. 37. A .S. Kronfeld and J. N. Simone Phys. Lett. B490 (2000) 228. 38. N. Tsutsui et al., Nucl. Phys. B (Proc. Suppl.) 106 (2002) 376
PROCEEDINGS SUPPLEMENTS Nuclear
ELSEVIER
Physics
B (Proc.
Suppl.)
111 (2002)
75-8 1
Lattice QCD Results on B physics T. Onogi” “Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan I review the status of Lattice QCD results on B physics.
1. Introduction Determination of ]I&], IVubl, lVcbl from B factory experiment requires a model independent calculation of hadron matrix elements. One of the most promising approaches is lattice &CD, which is a formalism for nonperturbative calculation of matrix elements (O(O]Hi), (Hz]I?]Hi) from first principles of QCD with controlled errors. In general, the matrix elements can be decomposed into two parts as (OMS(p))
=
-%4(OV4)>
(I)
where Z(p,a) is the short distance correction to match the scheme difference in lattice and continuum theory. The matching factors are obtained usually perturbatively, but in some cases nonperturbatively. (Olat (a)) is the long distance part which is computed by the numerical path integral
nf 2-lJ n
det(D + mi) exp(-S,)O.
Dd
(2)
such as masses and decay constants unquenched calculation have started for the last few years. Numerous efforts have been devoted in lattice studies for decay constants of B mesons and it has been shown that different approaches to B meson give consistent results within the systematic errors, so that the lattice methods have been proved to work. At present it is time for us to aim at further precisions or look for new phenomenological applications. In this report, I would like to cover two topics. The first topic is the bottom quark mass determination and the decay constants and bag parameters of the B meson, which are the basic quantities. For these quantities, results in quenched QCD calculation have been established. I report the recent developments in unquenched calculation and in matching at two-loop level or nonperturbative level. The second topic is the progress in other new physical quantities such as form factors in exclusive semileptonic decays or HQET parameters in quenched calculations.
i
Since the fermion determinant is highly nonlocal, the numerical cost for the path integral is huge. Thus we often employ the “quenched approximation” with which sea-quark effect is neglected, i.e. the determinant is replaced by unity. Although this drastic approximation which introduces uncontrollable systematic errors, the calculation in quenched approximation is useful since it serves as a test ground for precise calculation in an exploratory stage, since one can apply the same numerical technique to unquenched calculation to obtain the physical answer just by replacing the gauge configuration from the quenched one to the unquenched one. In fact, for simple quantities 0920-5632/02/$ - see front matter 0 2002 Published PI1 SO920-5632(02)01687-O
2. Lattice formalism for heavy quarks There are three approaches to the heavy quark on the lattice with different systematic errors. The first is NRQCD [l], which exploits an nonrelativistic effective action with certain orders in The problem in this ap0(1/m) corrections. proach is that the theory is nonrenormalizable so that no continuum limit can be taken. The matching of full relativistic theory and the effective theory is performed by perturbation theory practically through O((Y,) so that the higher order perturbative error is another problem. The second is the Fermilab approach [2]. This is a reinterpretation of O(a)-improved Wilson ac-
by Elsevier Science B.V.
T Onogi/Nuclear
Heavy Static [15] Static & O(a)-W O(a)-W [16] O(a)-W 1181 1 NRQCD ‘[19]
[16]
Physics
B (Proc. Suppl.)
BBd (mb ) 0.81(5)(3)(2) 0.87(4)(i) 0.93(8) (:) 0.92f4) 1 0.852(45)
BB,
111 (2002)
7541
Bs(mb)
IBB,
1.01(l) 0.99(2) 0.98(5) 0.98(3) 1 1.015(6)(4)
0.96(8)(5)(5) 0.80(l)(4) 0.86(2)(i) [ 0.87(l)(9)
Table 1 Recent results for the bag parameters.
tion. The theory is smoothly connected to relativistic theory so that it is passable to take the continuum limit. However when the quark mass is heavier than the lattice cutoff, the lattice spacing dependence of the discretization errors are complicated so that it is nontrivial how to extrapolate the simulation results. Since the matching is performed with one loop perturbation theory, the perturbative error is another problem as in NRQCD. The third is the relativistic approach, where Symanzik O(a)one uses nonperturbative improvement of Wilson quark action and compute physical quantities around charm quark mass region and extrapolate to bottom, which is exploited by UKQCD and APE collaborations. In this approach, O((am)2) error and extrapolation error are the problems. 3. B meson decay constants rameters
and bag pa-
The decay constant and the bag parameters of the B meson are the key quantities for extracting the CKM elements l&l and IVt,l from the mass differences of the neutral B mesons using the following relations
Am,
=
I&&b12f&BBq,
. AmB, = _Ivtsl”fgaBBs Am&
I%i12~&BB~
9 =
d,s
(3) (4)
The bag parameter of the B, meson is also useful in predicting the width difference in the B, - ??, system as follows
FW@sl~~lBsj
- Gs(z)(BsIOsIBs)
+&,,dcq.
(5)
The decay constants in quenched calculation have been studied extensively. The recent calculations using Fermilab formalism [3-71, NRQCD [8-lo] and extrapolation method [ll-131 show nice agreement within errors of lo%, which can be summarized as fB
f&/f& fD
=
173* 23 MeV
=
1.15 f 0.03
=
203 f 14 MeV.
The bag parameters in quenched calculation have been studied in several different approaches: the static limit, extrapolation method and NRQCD, and recent results are listed in Table 1. The O(a)-improved Wilson Quenched value for the Bag parameter has an error of 10% either from perturbative error (Static, NRQCD) or extrapolation (Relativistic), which are consistent with each other. However, BB, / BB~ has small errors because there is a drastic cancellation in the renormalization factors(Static, NRQCD) and because extrapolation in l/M becomes more stable in the ratio (Relativistic). In the last few years, unquenched calculation for fB has started by several groups using various methods. The unquenched calculation for the bag parameter has been first studied with the heavy quark in the static limit, but the error is still large compared to the quenched calculation. Recently JLQCD has started two-flavor QCD project to compute the decay constant and the bag parameters in which they perform a realistic simulation with small lattice spacing a (p=5.2, a-l N 2 GeV) and modest lattice size (203 x 48 N, 2.0 fm) in search for sea quark effect [19]. The action for the gauge field is the standard plaque-
T Onogi/Nuclear Physics B (Proc. Suppl.) Ill
(2002) 7541
Mp- 6 GeV (-B meson) 0.8
A -
_
I
unquench lin
’
I
I
I
I
’
’
8
CCollins99
’
’
I
’
I
I
-
OMILCOO
qua
WP-PACSOO oCP-PACS(NR) oJLQCDO1 I
I
o’20
I
0.2
I
0.4
I
I
>
0.6
I
0.8
I
I
I
1
I
1
200
150
100
I Figure 2. nf = 2 unquenched results for
2
i0
fs.
mZ2[GeV2] Figure 1. The chiral behavior for the decay constant with nf = 2 unquenched simulation by JLQCD.
tte action and nonperturbatively improved Wilson fermion is used for the light valence and sea quarks with ~=0.1340-0.1355, which corresponds to mps/mv=0.8-0.6 . The heavy quark is described by NRQCD heavy quark action with l/m correction. Figure 3 shows the sea quark mass dependence of the decay constants in unquenched calculation, where 9f, denotes &f~ times the renormalization group factor. They employ three fitting functions for the chiral extrapolations of ‘Pf,, which are the linear function a0 x (1 + a;X), the quadratic function bo x (1 + blX + b2X2), and the linear plus log function ~0 x (1 + clX + c2XZnX) with X = rim2 The bag p&meters Bg are also studied by JLQCD. They tried four different operator matchings which differ either at O(cwz) or O(a,h/mb). The mass dependences are found to be small, the chiral extrapolation error are smaller than that for f~. The preliminary results from JLQCD are =
190 f 14 f 7 f 19 MeV,
fB,/fBd
=
1.184 f 0.026 f 0.020 f 0.015,
BBd(mb)
=
0.872 f 0.039 f 0.004 f 0.073,
fBd
=
0.999 f 0.012 f 0.004,
fBsB;: f&B;:
=
1.183 f 0.027 f 0.020 f 0.015,
Bs, (ma)
=
0.858 f 0.033 f 0.007 f 0.072,
AFS 1’S
=
0.107 f 0.026 f 0.014 f 0.017.
BB,/BB,
<
q
In Figure 3, the nf = 2 unquenched results [14, 6,10,19,7] of fB is shown. They are consistent within errors and collecting all the results, I arrive the following world averages
fB = f&/f& =
198f
30MeV
1.15 f 0.05
for the unquenched (n, = 2) results. 4.
Bottom quark mass
Another major progress in lattice QCD is the precise determination of the bottom quark mass mp(mb). The bottom quark mass is determined from the binding energy E of the B meson using the following relation (6) where 2, is the multiplicative mass renormalization factor and bm is the additive energy shift. An important observation by Martinelli and Sachrajda [20] is that although these renormalization factors have renormalon ambiguities, they cancel each other when combined so that the
l? Onogi/Nuclear
78
Physics
B (Proc. Suppl.)
short distance relation between the lattice binding energy and the MS mass is renormalon free. Although the factor 2, is known to three loop from calculation in the continuum, the energy shift which requires lattice perturbative calculation has been known only to one-loop order for a long time. Recently, lattice part of the perturbative coefficients have been computed in static limit analytically to two-loop order by Martinelli and Sachrajda [23] and to three-loop order using numerical technique by DiRenzo et al. [21] and Lepage et al. [22]. APE [23] and GLOK [24] performed a lattice calculation of B meson in the static limit to obtain quenched mp(mb) with two-loop and three-loop precision respectively. APE also performed a nf = 2 unquenched calculation to obtain mp(m,) curacy, which gives consistent results [25,26].
with two-loop acresults by NRQCD
More recently, quenched mp(mb) from B meson in the static limit with nonperturbative accuracy has been obtained using the Schroedinger functional technique by Heitger and Sommer [27], where they match the relativistic quark with RG invariant mass M and static quark in a small volume. It is found that the mF(mb) with twoloop, three-loop, and nonperturbative give consistent results within quoted errors. Comparison of these quenched calculations and APE’s nf = 2 unquenched results the current estimate of the bottom quark mass is mp(rm,)
=
5. B -+ D(*)lv
4.30f
0.10 GeV.
(7)
form factors
B + D(*)Zv form factor is crucial for extracting ]Vcb] from the following relation dlm ~v,b~2~~13+iX*&“)~2, (8) ZJ where the form factors FB+.D(*) (w) are given as
FB+D*(~)
=
VA
f-
2:;;m,
- &
1’ (9)
111 (2002)
7.541
Hashimoto et al. (Fermilab collaboration) [30] obtained the coefficients in l/m2 corrections performed a quenched lattice calculation of the following double ratio of matrix elements
-Pl~“~l~H~lbyo4D)
lh+(‘)12 =
(Dlq”clD)(BlbyoblB)
(11)
(D*I~“bl~*)(~*IbyOclD*) Ih(
=
JhA1(1)J2
(D*IE~~c~D*)(~?* IbyOblB*) ’ (12) (D*I~~yiy5blB)(B*Ibyiy5clD) (D*I~~yiy5clD)(B*Ibyiy5blB)
=
‘(13)
The idea is that from the symmetry it is guaranteed that the deviation from unity in the double ratio only start from l/m2 including systematic errors. There are power divergences, but the double ratio cancels the O(1) and O(l/m) contribution. So that the coefficients can be obtained with reasonable accuracy which enables to obtain the form factor of l-2% accuracy as follows
where the first and second term denotes the statistical and systematic errors within quenched approximation. In addition to the above errors, this calculation contains unknown quenching errors. However, this method is expected to provide the best estimate without model uncertainties once unquenched calculation is performed. 6. B -+ dv
form factors
The study of B -+ dv form factor f+(q2) has been one of the main issues in lattice &CD, since it is necessary for extracting Iv,&,] from exclusive semileptonic decay from the following formula dr @
G2 - ~(lc,121Vub121f+(q2))2, - 24rr3
where the form factor is defined
f+(q2) [(Pp +h)‘” m&-m;
+f0(q2)
q2
~ Q >
(14) as
mk;““qp] (15)
T Onogi/Nuclear
19
Physics B (Proc. Suppl.) 11 I (2002) 7541
Burdman et al. which manifests the heavy quark symmetry and chiral symmetry. Fitting the lattice data as a function of the light quark mass, inverse heavy quark mass and the pion recoil energies they obtain
(1.18 f 0.37 f 0.08 f 0.31) psec-’ for the partially integrated differential decay rate. 0.0
”
”
”
15
”
”
20
”
”
”
25
’
”
30
q2 (Gd) Figure 3. q2 dependence of B + dv form factors. Upward triangle, downward triangle, square, circle symbols denotes the data from UKQCD, APE, Fermilab and JLQCD collaborations respectively. Open and filled symbols corresponds to f”, f+.
with pB and k, the momenta of the initial and final pseudo-scalar mesons and q = PB - k,. Recently four groups have carried out quenched calculation for semileptonic form factors. UKQCD [33] and APE [34] used relativistic quark action for heavy quark and extrapolated the results around charm quark mass region to the physical b quark mass for a fine lattice with P = 6.2. Fermilab [35] used Fermilab action for the heavy quark and simulated with three different values of /I (5.7, 5.9, 6.1). JLQCD [31] used NRQCD action for the heavy quark and performed a high statistics calculation with @ = 5.9. Figure 3 shows the results from the four groups they give consistent values for f+ although some discrepancies is observed in f”. The difference originates from the differences in the heavy quark mass and light quark mass dependence of the data from each group as well as the extrapolation method. Further understanding of the systematic errors in mass extrapolation is necessary. Since all the groups give consistent values for f+, I present the results from JLQCD collaboration. They used chiral extrapolation method based on the parameterizations of form factor by
7. HQET parameters There are also pioneering studies on HQET parameters which are useful for the theoretical prediction of inclusive decays. The lifetime ratio of Ab and B is one of the problem in understanding the beauty hadron system. If the large deviation from unity observed in current experimental data remains true with further precise data, it casts an important question on the heavy quark symmetry Assuming the quark hadron duality, the life time ratio is
-T(Ab) = +‘I
1
+
xl (Ab) A1(B”) 2mi -
+3cBA2(Ab)
-
X2(B”)
2mt (16)
+O(Ilm% where
(17) A2W)
=
(Hp7.Bbp) 2M
.
(18)
H
JLQCD collaboration [38] performed a quenched calculation on a p = 6.0, 203 x 48 lattice, with NRQCD heavy quark and obtained the following results A(&,) - ii(B)
=
393(31)MeV
- X1(B)
=
-0.21(21)GeV2,
Xi(&)
which are small as expected from the quark model or sum rule estimations.
80
Group Gimenez Kronfeld
1: Onogi/Nuclear
et al [36] Simone [37]
Table 2 Lattice calculation
A (GeV) 0.18t;:g 0.68?~:$
of HQET
Physics B (Proc. Suppl.) II I (2002) 75-81
Xi (GeV2) -(0.09 f 0.14) -(0.45 f 0.12)
parameters.
The semileptonic inclusive decay B + XJu is another place which requires nonperturbative determination of HQET parameters A, Xi, X2. which denotes the binding energy, the kinetic energy and the energy from the chromomagnetic interaction so that the relation ma=mFP-iT+
xl
+ CGA2
2m
+ O( 1/m3)
(19)
holds. Two groups has performed lattice calculations in quenched approximation. Gimenez et al. [36] computed the matrix element in the static limit, where Kronfeld and Simone [37] computed the B meson masses as a function of bare mass to obtain A and Xi from the mass formula. Table 2 gives the results from the two groups. The current result shows a disagreement. Further studies on the perturbative error and the subtraction of power divergence are necessary. 8. Summary In summary, quenched calculation in basic quantities fr,, BE, Bs are under control with 10% accuracy. For these basic quantities, improvements such as unquenched calculations or computations of two-loop or nonperturbative renormalization factors has started giving reliable high precision results with more realistic simulations. Calculations of new quantities in are being developed in quenched approximation. The double ratio method is proved to be a promising tool for a precise calculation of l/m2 corrections of B + D*Zu form factors. Application to unquenched calculation is strongly desired. The current efforts of B + du form factor at 20% acFurther studies in reducing errors and curacy. understanding the difference between different groups is required. The determinations of various HQET parameters in quenched approxima-
tion are in progress. As future problems, nonperturbative renormalization with heavy quark is an important problem for precise calculation. Computing new quantities such as B + plv, K*(p)l+l-, the light-cone wavefucntion, the shape function would be very useful. Unquenched calculations with better control of chiral extrapolation using lighter sea quark is necessary. For this purpose developments in both the computer resources and the simulation algorithm are awaited.
Acknowledgments I would like to thank A. Kronfeld, S. Ryan, C.T. Sachrajda, L. Lellouch, C.-J.D. Lin, R. Sommer, H. Matsufuru, S. Hashimoto, and N. Yamada for discussions and private communications.
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