- Email: [email protected]
B → πlν at three lattice spacings
UCLEAR PHYSIC c
ELSEVIER
PROCEEDINGS SUPPLEMENTS
Nuclear Physics B (Proe. Suppl.) 73 (1999) 390--392
) 7rlP at t h r e e l a t t i c e s p a c i n g s S. Ryan a ' , A. EI-Khadra b, S. Hashimoto a, A. Kronfeld% P. Mackenzie a and J. Simone ~ a Theoretical Physics Department, Fermilab, P.O. Box 500, Batavia, I1 60510, U.S.A. b Loomis Laboratory of Physics, 1110 W. Green Street, Urbana, I1 61801-3080, U.S.A. The increasing accuracy of experimental results for the exclusive, semileptonic decay /} ----r rrh9 requires a similarly accurate calculation of the hadronic matrix elements, to determine [V~bl. Vv'epresent preliminary results for the form factors of the B to light meson decay mode. Using results from three lattices in the range 5.7 < 3 < 6.1 we study the dependence on the lattice spacing.
.,,3 Vol. # cfgs. C~w
1. I n t r o d u c t i o n In this report we present preliminary results from our study of semileptonic/~ -4 a-lt) decays. This exclusive decay mode can be used to extract the C K M matrix element ]Vubl which is currently known to only ,~ 20% accuracy. T h e experimental error will be greatly reduced at B-factories, requiring a sinfilar reduction in the theoretical error to place meaningful constraints on the unitarity triangle. For tile first tiIne tile lattice spacing dependence of the matrix elements for such decays is studied. A similar analysis of D mesons is described in Ref. [1]. Many of the simulation details are the same as those described in Ref. [2], but a brief s u m m a r y is given in Table 1. t Our strategy is to study the a-dependence and perform the contixmum extrapolations with a light quark (active and spectator) at strange. At 3 = 5.7 results are shown after the chiral extrapolation and an estimate of the remaining lattice spacing dependence is m a d e based on our study with strange light quarks. In a forthcoming p a p e r we will also perform a chiral extrapolation at ;3 = 5.9, allowing us to verify this adependence. For b o t h light aJld heavy quarks the action is the tadpole-improved, SheikholeslamiWohlert (SW) action with the plaquette value of uo. For heavy quarks it is interpreted in the Fer*talk presented by S. Ryan. tNote that we have increased our statistics at :3 =- 6.1 to 200 configurations.
.-1 (6eV)
6.1 243 x 4 8 200 1.46
5.9 163 x 3 2 350 1.50
5.7 123 x 2 4 300 1.57
2.62 +_ 98
1.80 +
1.16 +
0.093 0.1385
0.089 0.1405 0.1410 0.1415 0.1419
nh( = ~b) ~t
0.099 0.1373
3
Table 1. Lattice details; a = alP-18. milab formalism [3]. Ttle matrix elements and currents are calculated at tadpole-improved tree level as described in [3]. The calculation of the current nornmlisation at one loop is underway [4] and will be incorporated in our final results. 2.
The
Calculation
At each of the lattice spacings and for each mom e n t u m the required matrix element is extracted froln tile three-point correlation function
C3pt( t,y, U)
-'+
2E~ (/7~) 2E~(/Yt~)
x
T h e energies and amplitudes of the two-point functions were determined as described in Ref. [2]. In this sinmlation tile B meson is at rest and the light meson has m o m e n t u m in lattice units of (0,0,0), (1,0,0), (1,1,0), (1,1,1) and (2,0,0).
0920-5632/99/$ - see front matter © 1999 Elsevier Science B.V. All rights reserved. PII S0920-5632(98)00608-2
S. Ryan et at~Nuclear Physics B (Proc. Suppl.) 73 (1999) 390--392
391
1.4
0.9
• V 4 (temporal component) • V i (spatial component)
1.2
Li
1.0
1
K
¢-
03
E 0.8 (1)
E
(1) X
"~ 0.5 X .¢-
"= 0.6
S E
E
t
0.4
p = 700MeV
0.2 0.0
• < (ss) (p) I V 4 I B, (0)> * < (ss) (p) I Vii B, (0) >
0.1 0.0
0.2
0.4
0.6
0.8
1.0
0.0
Figure 1..Matrix elements at ,.3 = 5.7 interpolated to p E {0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.85, 1.0} GeV.
0.2
0.4
0.6
0.8
1.0
a ( GeV-1 )
p(,,)2 ( G e V 2)
Figure 2. Spatial and temporal components of the heavy-light matrix element at/Y,~ = 700 MeV extrapolated to the continuum limit.
3. R e s u l t s Tile spatial and temporal coinponents of tile matrix eleinents are interpolated in lattice moInenta to fixed physical values, which, being matched at different lattice spacings, can be ext r a p o l a t e d to a = 0. This introduces a dependence on the quantity used to set the scale, although it is found to be mild. Results are quoted using the 1P-1S splitting in charmonium to determine a -1. T h e linearly interpolated points are shown in Figure 1; tile final result depends only mildly on using a different interpolating function. Typical linear continuum extrapolations of the m a t r i x elements are in Figure 2. T h e y show very little a-dependence however, tile lattice spacing dependence does increase with/7~, as expected. At 3 = 5.7 tile matrix eleinents are extrapolated quadratically to the chiral limit. For mom e n t a between 400 and 850 MeV the d a t a fit a quadratic form very well and have good chisquared. At /~r = 0 the results hecoine less reliable because tile B" pole causes tile extrapolation to rise rapidly with decreasing light quark mass. It would be useful to have still-lighter quarks to control this fit further, however exceptional configurations would have to be cured as in eg. Ref. [5]. At higher m o m e n t a tile fits once more become less reliable due to large cutoff effects.
Form factors are defined through
(rr(p),V~[B(p,)) : f+(q2) [p, + p
+/O(q2)
m 2 - m2q]
¢,,
with q = p ' - p . One could then proceed to determine ttle form factors at q2 = 0. However, we choose not to do so. Figure 3 illustrates firstly that our results for f0(q2) and f+(q2) agree with previous calculations using different approaches [6,7] and secondly the huge range in q2 over which lattice results must be extrapolated to reach q2 = 0, thereby increasing tile statistical error and introducing an unknown degree of model dependence in the final result. Instead [8] we focus oil a quantity which can be determined without recourse to a q2 extrapolation and which can be used with experimental d a t a to determine IVub]. The differential decay rate is such a quantity, given by d r _ 2mBG~lVusl 2 Iff.Ia ly+(q2)12 dlff~l 247r3 E~r "
(X)
This is shown in Figure 4 where the dotted lines define tile raalge in pion m o m e n t u m for which we
S. Ryan et al./Nuclear Physics B (Proc. Suppl.) 73 (1999) 390-392
392 2.4
1.5
2.0
• B - > n Iv
• f.(qZ) [this w o r k ] • fn(qZ) [this w o r k ]
1.6
!
! >
o Light Cone S u m Rules U K Q C D , 1995
d I I
1.0
Q)
~- 1.2 .E +"-
4.
I
0.8
or
0.5
I
0.4
[
]; 0.0
*
0.0 0
4
8
12
16
20
24
* 0.0
.
i i
0.2
0.4
*
q2 ( G e V )
Figure 3. The q2-dependence of fo and f + (in the continuum limit) and a comparison with other calculations. Note that the lattice data only include statistical errors.
believe our lattice calculation is most reliable, and tbr which there are experimental results. Note that the p . -- 0 region is experimentally inaccessible since the event rate is zero here. 4. S y s t e m a t i c E r r o r s Comparing data as in Figure 4 with the experimental measurements of exclusive branching ratios of B mesons to pions, eg. in the range 400 _< p~ _ 850 MeV, it is possible to determine IVubl . Estimates of the contributions to the theoretical error in IVubl are tabulated below. V(e expect to improve upon these in our finai results. Statistics Pert". th. chiral extrap.
= 8(~ [[ a-dependence = 5(~ [[ Excited states = 7(~ mQ tuning
0.6
0.8
1.0
p. (GeV)
= 5~ = 2(~ = I(~
5. C o n c l u s i o n s Our preliminary results indicate that lattice spacing errors are under control and continuum extrapolations are possible. The form factors we extract agree with previous calculations for tile range of q2 where a lattice simulation is possible. VVe present an alternative way to use lattice data to determine IVubl, namely by calculating the pax'-
Figure 4. The differential decay rate (less tile momentum independent pre-factors) as a function of /7~ at :3 = 5.7, chirally extrapolated.
tial widths, which should reduce statistical and systematic theoretical uncertainties [8]. Acknowledgements
Fernfilab is operated by Universities R.esearch Association, Inc. for the U.S. Department of Energy. REFERENCES
1. J. Simone et al., these proceedings. 2. A. E1-Khadra, A. Kronfeld, P. Mackenzie, S. Ilyan, J. Simone, Phys. Rev. D 5 8 (1998). 3. A. EI-Khadra, A. Kronfcld, P. Mackenzie, Phys. Rev. D 5 5 (1997), 3933. 4. A. Kronfeld and S. Hashimoto, these proceedings. 5. W. Bardeen, et al., Phys. tlev. D 5 7 (1998) 1633. 6. UKQCD Collaboration, Nucl. Phys. B 4 4 7 (1995) 425. 7. P. Ball, hep-ph/9802394. 8. J. Simone, Xucl. Phys. B Proc. Suppl. 47 (1996) 17.
ELSEVIER
PROCEEDINGS SUPPLEMENTS
Nuclear Physics B (Proe. Suppl.) 73 (1999) 390--392
) 7rlP at t h r e e l a t t i c e s p a c i n g s S. Ryan a ' , A. EI-Khadra b, S. Hashimoto a, A. Kronfeld% P. Mackenzie a and J. Simone ~ a Theoretical Physics Department, Fermilab, P.O. Box 500, Batavia, I1 60510, U.S.A. b Loomis Laboratory of Physics, 1110 W. Green Street, Urbana, I1 61801-3080, U.S.A. The increasing accuracy of experimental results for the exclusive, semileptonic decay /} ----r rrh9 requires a similarly accurate calculation of the hadronic matrix elements, to determine [V~bl. Vv'epresent preliminary results for the form factors of the B to light meson decay mode. Using results from three lattices in the range 5.7 < 3 < 6.1 we study the dependence on the lattice spacing.
.,,3 Vol. # cfgs. C~w
1. I n t r o d u c t i o n In this report we present preliminary results from our study of semileptonic/~ -4 a-lt) decays. This exclusive decay mode can be used to extract the C K M matrix element ]Vubl which is currently known to only ,~ 20% accuracy. T h e experimental error will be greatly reduced at B-factories, requiring a sinfilar reduction in the theoretical error to place meaningful constraints on the unitarity triangle. For tile first tiIne tile lattice spacing dependence of the matrix elements for such decays is studied. A similar analysis of D mesons is described in Ref. [1]. Many of the simulation details are the same as those described in Ref. [2], but a brief s u m m a r y is given in Table 1. t Our strategy is to study the a-dependence and perform the contixmum extrapolations with a light quark (active and spectator) at strange. At 3 = 5.7 results are shown after the chiral extrapolation and an estimate of the remaining lattice spacing dependence is m a d e based on our study with strange light quarks. In a forthcoming p a p e r we will also perform a chiral extrapolation at ;3 = 5.9, allowing us to verify this adependence. For b o t h light aJld heavy quarks the action is the tadpole-improved, SheikholeslamiWohlert (SW) action with the plaquette value of uo. For heavy quarks it is interpreted in the Fer*talk presented by S. Ryan. tNote that we have increased our statistics at :3 =- 6.1 to 200 configurations.
.-1 (6eV)
6.1 243 x 4 8 200 1.46
5.9 163 x 3 2 350 1.50
5.7 123 x 2 4 300 1.57
2.62 +_ 98
1.80 +
1.16 +
0.093 0.1385
0.089 0.1405 0.1410 0.1415 0.1419
nh( = ~b) ~t
0.099 0.1373
3
Table 1. Lattice details; a = alP-18. milab formalism [3]. Ttle matrix elements and currents are calculated at tadpole-improved tree level as described in [3]. The calculation of the current nornmlisation at one loop is underway [4] and will be incorporated in our final results. 2.
The
Calculation
At each of the lattice spacings and for each mom e n t u m the required matrix element is extracted froln tile three-point correlation function
C3pt( t,y, U)
-'+
2E~ (/7~) 2E~(/Yt~)
x
T h e energies and amplitudes of the two-point functions were determined as described in Ref. [2]. In this sinmlation tile B meson is at rest and the light meson has m o m e n t u m in lattice units of (0,0,0), (1,0,0), (1,1,0), (1,1,1) and (2,0,0).
0920-5632/99/$ - see front matter © 1999 Elsevier Science B.V. All rights reserved. PII S0920-5632(98)00608-2
S. Ryan et at~Nuclear Physics B (Proc. Suppl.) 73 (1999) 390--392
391
1.4
0.9
• V 4 (temporal component) • V i (spatial component)
1.2
Li
1.0
1
K
¢-
03
E 0.8 (1)
E
(1) X
"~ 0.5 X .¢-
"= 0.6
S E
E
t
0.4
p = 700MeV
0.2 0.0
• < (ss) (p) I V 4 I B, (0)> * < (ss) (p) I Vii B, (0) >
0.1 0.0
0.2
0.4
0.6
0.8
1.0
0.0
Figure 1..Matrix elements at ,.3 = 5.7 interpolated to p E {0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.85, 1.0} GeV.
0.2
0.4
0.6
0.8
1.0
a ( GeV-1 )
p(,,)2 ( G e V 2)
Figure 2. Spatial and temporal components of the heavy-light matrix element at/Y,~ = 700 MeV extrapolated to the continuum limit.
3. R e s u l t s Tile spatial and temporal coinponents of tile matrix eleinents are interpolated in lattice moInenta to fixed physical values, which, being matched at different lattice spacings, can be ext r a p o l a t e d to a = 0. This introduces a dependence on the quantity used to set the scale, although it is found to be mild. Results are quoted using the 1P-1S splitting in charmonium to determine a -1. T h e linearly interpolated points are shown in Figure 1; tile final result depends only mildly on using a different interpolating function. Typical linear continuum extrapolations of the m a t r i x elements are in Figure 2. T h e y show very little a-dependence however, tile lattice spacing dependence does increase with/7~, as expected. At 3 = 5.7 tile matrix eleinents are extrapolated quadratically to the chiral limit. For mom e n t a between 400 and 850 MeV the d a t a fit a quadratic form very well and have good chisquared. At /~r = 0 the results hecoine less reliable because tile B" pole causes tile extrapolation to rise rapidly with decreasing light quark mass. It would be useful to have still-lighter quarks to control this fit further, however exceptional configurations would have to be cured as in eg. Ref. [5]. At higher m o m e n t a tile fits once more become less reliable due to large cutoff effects.
Form factors are defined through
(rr(p),V~[B(p,)) : f+(q2) [p, + p
+/O(q2)
m 2 - m2q]
¢,,
with q = p ' - p . One could then proceed to determine ttle form factors at q2 = 0. However, we choose not to do so. Figure 3 illustrates firstly that our results for f0(q2) and f+(q2) agree with previous calculations using different approaches [6,7] and secondly the huge range in q2 over which lattice results must be extrapolated to reach q2 = 0, thereby increasing tile statistical error and introducing an unknown degree of model dependence in the final result. Instead [8] we focus oil a quantity which can be determined without recourse to a q2 extrapolation and which can be used with experimental d a t a to determine IVub]. The differential decay rate is such a quantity, given by d r _ 2mBG~lVusl 2 Iff.Ia ly+(q2)12 dlff~l 247r3 E~r "
(X)
This is shown in Figure 4 where the dotted lines define tile raalge in pion m o m e n t u m for which we
S. Ryan et al./Nuclear Physics B (Proc. Suppl.) 73 (1999) 390-392
392 2.4
1.5
2.0
• B - > n Iv
• f.(qZ) [this w o r k ] • fn(qZ) [this w o r k ]
1.6
!
! >
o Light Cone S u m Rules U K Q C D , 1995
d I I
1.0
Q)
~- 1.2 .E +"-
4.
I
0.8
or
0.5
I
0.4
[
]; 0.0
*
0.0 0
4
8
12
16
20
24
* 0.0
.
i i
0.2
0.4
*
q2 ( G e V )
Figure 3. The q2-dependence of fo and f + (in the continuum limit) and a comparison with other calculations. Note that the lattice data only include statistical errors.
believe our lattice calculation is most reliable, and tbr which there are experimental results. Note that the p . -- 0 region is experimentally inaccessible since the event rate is zero here. 4. S y s t e m a t i c E r r o r s Comparing data as in Figure 4 with the experimental measurements of exclusive branching ratios of B mesons to pions, eg. in the range 400 _< p~ _ 850 MeV, it is possible to determine IVubl . Estimates of the contributions to the theoretical error in IVubl are tabulated below. V(e expect to improve upon these in our finai results. Statistics Pert". th. chiral extrap.
= 8(~ [[ a-dependence = 5(~ [[ Excited states = 7(~ mQ tuning
0.6
0.8
1.0
p. (GeV)
= 5~ = 2(~ = I(~
5. C o n c l u s i o n s Our preliminary results indicate that lattice spacing errors are under control and continuum extrapolations are possible. The form factors we extract agree with previous calculations for tile range of q2 where a lattice simulation is possible. VVe present an alternative way to use lattice data to determine IVubl, namely by calculating the pax'-
Figure 4. The differential decay rate (less tile momentum independent pre-factors) as a function of /7~ at :3 = 5.7, chirally extrapolated.
tial widths, which should reduce statistical and systematic theoretical uncertainties [8]. Acknowledgements
Fernfilab is operated by Universities R.esearch Association, Inc. for the U.S. Department of Energy. REFERENCES
1. J. Simone et al., these proceedings. 2. A. E1-Khadra, A. Kronfeld, P. Mackenzie, S. Ilyan, J. Simone, Phys. Rev. D 5 8 (1998). 3. A. EI-Khadra, A. Kronfcld, P. Mackenzie, Phys. Rev. D 5 5 (1997), 3933. 4. A. Kronfeld and S. Hashimoto, these proceedings. 5. W. Bardeen, et al., Phys. tlev. D 5 7 (1998) 1633. 6. UKQCD Collaboration, Nucl. Phys. B 4 4 7 (1995) 425. 7. P. Ball, hep-ph/9802394. 8. J. Simone, Xucl. Phys. B Proc. Suppl. 47 (1996) 17.