- Email: [email protected]
Neutral B meson mixing and heavy-light decay constants from quenched lattice QCD
H
U~.L E,",R Pt tYSIC~
m
PROCEEDINGS SUPPLEMENTS ELSEVIER
Nuclear Physics B (Proc. Suppl.) 73 (1999) 357-359
Neutral B Meson Mixing and Heavy-Light Decay Constants from Quenched
Lattice
QCD
*
Laurent Lellouch a t a n d C.-J. David Lin b ~ (UKQCD Collaboration) aTheory Division,CERN,CH-1211 Geneva 23, Switzerland b D e p a r t m e n t of Physics & Astronomy, The University of Edinburgh, Edinburgh EH9 3JZ, Scotland We present high-statistics results for neutral B-meson mixing and heavy-light-meson leptonic decays in the quenched approximation from tadpole-improved clover actions at fl = 6.0 and/3 = 6.2. We consider quantities such as BBa(,), fDa(,), fBa(~) and the full AB = 2 matrix elements as well as the corresponding SU(3)-breaking ratios. These quantities are important for determining the CKM matrix element IV~d[.
1. I N T R O D U C T I O N
The matrix elements
The study of B d0 - B-0d oscillations allows a clean extraction of the poorly known C K M matrix element IVtdl. However, the accuracy of this determination is currently limited by the theoretical uncertainy in the calculation of the matrix element,
as
.A~bd
:
-0 0 (Sd[O dA B = 2 [Bd)
=
(S l z,(1 -
2
Amd = GZF M 2 IVtdVt*bl2S( ~ ) C ( # ) . M b g ( I z ) 87r2
w
w
,
2
where ]B~(,) is the decay constant, MBa(,) the mass, and BBa(,) the B - p a r a m e t e r of B d0( s ) mesons. In this work, we obtain the ratio rsd from the direct calculation of ~.A4bd as well as from the Cal-
2. S I M U L A T I O N
m2
I
IrM
*CPT-98/PE.3689; CERN-TH/98-289. Support from EPSRC and PPARC under grants GR/K41663 and GR/L29927 is acknowledged. tOn leave from: Centre de Physique Th6orique, Case 907, CNRS Luminy, F-13288 Marseille Cedex 9, France. SPresenter.
DETAILS
"
-zgocsw~
where GF is the Fermi constant, M w the W S
"
We use the tadpole-improved SheikholeslamiWohlert (SW) quark action,
=
boson mass, mt the top-quark mass, ( ~ w ) the relevant Inami-Lim function, # the renormalisation scale, and C(#) the Wilson coefficient. An alternative approach, in which m a n y theoretical uncertainties cancel, is to look at the ratio
be parametrised
J~bd(s) "= -3f B,i(,,) MBa(,) B Ba(o)
culations of ~.fB a and BBB, Ba
- -r )dlB °) ,
which is related to the mass difference of the two mass eigenstates of the B ° - / ~ 0 system, 2
8 2
J~bd(s) C a n
E ~l(x)P~,,(x)a,~,q(x) :gJZ,V
to perform simulations on a 24 a × 48 lattice at /~ = 6.2 and a 163 × 48 lattice at ~ = 6.0. Here S W is the standard Wilson action, go the bare gauge coupling, c s w the clover coefficient, ~ the hopping parameter, and P ~ a lattice definition of the gauge-field strength tensor. Table 1 gives the simulation parameters. We use K L M normalisation for the quark fields. 3. O P E R A T O R
MATCHING
Matching onto the MS scheme is performed at one-loop in perturbation theory using the coupling a~--~(#) defined from the plaquette [1].
0920-5632/99/$ - see front matter © 1999 ElsevierScience B.V. All rights reserved. PII S0920-5632(98)00597-0
358
L. Lellouch, C.-J. DavidLin/NuclearPhysicsB (Proc. Suppl.) 73 (1999)357-359
/~ 6.0 6.2
Table I: Simulation parameters, aQ and aq are the heavy- and light-quark hopping parameters. ://= configs, csw I~q aQ 498 1.48 0.13700 0.13810 0.13856 0.114 0.118 0.122 0.126 0.130 188 1.44 0.13640 0.13710 0.13745 0.120 0.123 0.126 0.129 0.132
Since the clover-leaf interaction term is proportional to go, we can use the perturbative results obtained from a tree-level clover action [2] with modifications appropriate for tadpoleimprovement and KLM normalisation. For the matching of four-fermion operators, we use the basis O~at
=
% x %, + 7 # % x % , % ,
012at
=
7. x7~--7.75 x7~75,
O~a~
=
I x I + Ts x % ,
O~~
=
015at :
I x l-%
Table 2: Critical and strange hopping parameters and inverse lattice spacings. /~ 6.0 6.2 ~c 0.13924(1) 0.13793(1) ~s 0.13757(8) 0.13670(9) a -1 (Mp)(GeV) 1.96(5) 2.57(8) a-l(f,r) (GeV) 1.92(4) 2.58(9)
1.0, . . . .
....
x %,
:,20 O:12S
G/iv X G#v.
We set the coupling and matching scales to # = 1 and, for consistency with the literature, run divergent operators to 5 GeV, using 2-1oop continuum RG in the MS scheme with the appropriate number of flavours. To estimate the systematic error associated with the one-loop matching, we vary the scale # in a range from 1/a to fla. Decay constants are not affected since they are normalized b y / ~ and B-parameters change by about 3% ( / . varies by approximately 3%). Since we are mainly interested in SU(3)-breaking ratios for which these effects are even smaller, we neglect these small variations in what follows. 4. A N A L Y S I S A N D R E S U L T S We determine ~c and s8 from pseudoscalar meson masses. We set the scale with Mp for spectral quantities and f,~ for decay constants. In fact, these two quantities yield remarkably similar scales. (See Table 2.) We then linearly extrapolate and interpolate heavy-light decay constants, B-parameters and AB = 2 matrix elements to ~c and ~8, keeping ~ , ~ , aMp and af~ in the bootstrap loop. Fig. 1 shows examples of these extrapolations.
'¢='
I ....
I '
p = 6.0
'812' ....
L 'J[: ~
A
°"l , 0 .-1 1=4XL ~h . 0 . 1 1 8 ~----.TL-'dt-..~ ±
~ o.81- 0.122 Jlc-.~-4~m::::m~
0.7
0.132 ~
o.128~ ~ .
=
T I
°'vl _ 0 . 1 3 0
~-T-'~T
O:data
O:data x:~ c 0:~, , , , I .... I .... 7.~ 7.3o
0.8=
v.~
v.t
. . . .
~I"
x:~ o I
D:~,
. . . .
1
v 'l=l / ~
.
v.s
Figure 1. Light-quark-mass dependence of the heavy-light B-parameter, Bp, and extrapolation (interpolation) to ~l = ac (~8) at /~ = 6.0 and 6.2.
For heavy-quark (HQ) extrapolations, we define
afp M/-~p! as(Mp) } ~ ~A V ~ [ as(MB)
Cf(Mp)
}
a4MM. a,(Mp) CAF=2(Mp)
--
Mp
[. a, (MB) -
"
Then for X(Mp)=~/(Mp),~AF=2(Mp), B(Mp) and SU(3)-breaking ratios, HQET predicts
X(Mp) = Ax { I + Bx(-~p ) + Cx( M" )2 + } Mp .... Fig. 2 shows examples of the HQ extrapolations.
L. Lellouch, C-J. David Lin/Nuclear Physics B (Proc. Suppl.) 73 (1999) 357-359
0.~i
0,20
....
~
I .... [ .... I .... HQ Scaling
~
''''1
-meson
.... I .... HQ Scaling
I
i.o ~ m l s o n
,$. 0.15
~ =' 6.2 - f l
O. IO.o 0
~eso:
I
I
0.2
O.4
O.O
I 0.e
0.0
'
0.0
Mp/Mp(m~ = ra.)
'
I O.2
. O.4
MJMp(m]
=
0.6
m,)
Figure 2. HQ scaling of ~I and B-parameter.
. . . .
0.1~
i
l
. . . .
i
. . . .
l
. . . .
Chiral Extrapolation
0.04
r3:t¢,
0.1~
.... I .... I .... I HQ Scaling of ~ . ~
359
Table 3: Summary of results. Errors are statistical only. /3 6.0 6.2 fDo (MeV) 239(6) 221(9) fD(MeV) 213(6) 193(10) lB, (MeV) 221(7) 190(12) fB(MeV) 191(10) 161(16) 1.12(1) 1.15(4) 1.15(4) 1.18(8) Ye B~I°(5GeV) 0.86(2) 0.85(2) B~f°(5GeV) 0.83(4) 0.85(3) Be, 1.03(3) 0.99(3) Be (, M__~s ~ ~2Bso 1.38(7) 1.37(13) Me Be ~h~ 1.52(19) 1.70(28)
0.03
0.0~
~ O.03
7.15
1
1
4
x:~. ~ = 6.0 ,.,I .... ~.... I.,, 7.20
7.~ ~1-1
7.SO
0,01
.... 7.05
0.0
I .... O.2
I .... O.4
I O.O
MJM.(m, = m.)
Figure 3. Chiral and HQ extrapolation of the matrix element.
For SU(3)-breaking ratios, we find that taking the ratio before or after the HQ extrapolation leads to nearly indistinguishable results. We use the former for our final results since SU(3)-breaking ratios have milder HQ-mass dependences. Our main results are summarised in Table 3. We obtain rsd from the direct calculation of ~ " .tvIbd as well as from ~f B d and ~B e d • Our results for the direct calculation are consistent with those of [3], obtained with propagating Wilson quarks, and, at/~ = 6.0, with the static result of [4]. However, as Fig. 3 suggests, it is more difficult to control the chiral and HQ extrapolations of the matrix elements in the direct calculation because these extrapolations are more pronounced. Because we have results at only two values of the lattice spacing, we cannot extrapolate to the continuum limit. We therefore consider the )3 ---- 6.2 results to be our best, noting that de-
cay constants may still suffer from relatively large discretisation errors (roughly a 2a effect between 6.0 and 6.2) while SU(3)-breaking ratios and Bparameters are consistent within errors at the two /3 values. Although formally one need not include the aO~P correction to the axial current when using a mean-field improved, tree-level clover action, it would be interesting to investigate its effect on our results in view of understanding how nonperturbatively, O(a)-improved decay constants may behave. We plan to do so in the future. For a comparison of our results with other recent results, we refer the reader to [5]. REFERENCES
1. G. P. Lepage, P. B. Mackenzie, Phys. Rev. D48 (1993) 2250. 2. E. Gabrielli et al., Nucl. Phys. B362 (1991) 475; R. Frezzotti et aL, Nucl. Phys. B373 (1992) 781; and R. Gupta et aL, Phys. Rev. D55 (1997) 4036, for DRED to NDR. 3. C. Bernard, T. Blum and A. Soni, Phys. Rev. D 5 8 (1998) 014501. 4. V. Gim~nez and G. Martinelli, Phys. Lett. B398 (1997) 135. 5. T. Draper, in these proceedings.
U~.L E,",R Pt tYSIC~
m
PROCEEDINGS SUPPLEMENTS ELSEVIER
Nuclear Physics B (Proc. Suppl.) 73 (1999) 357-359
Neutral B Meson Mixing and Heavy-Light Decay Constants from Quenched
Lattice
QCD
*
Laurent Lellouch a t a n d C.-J. David Lin b ~ (UKQCD Collaboration) aTheory Division,CERN,CH-1211 Geneva 23, Switzerland b D e p a r t m e n t of Physics & Astronomy, The University of Edinburgh, Edinburgh EH9 3JZ, Scotland We present high-statistics results for neutral B-meson mixing and heavy-light-meson leptonic decays in the quenched approximation from tadpole-improved clover actions at fl = 6.0 and/3 = 6.2. We consider quantities such as BBa(,), fDa(,), fBa(~) and the full AB = 2 matrix elements as well as the corresponding SU(3)-breaking ratios. These quantities are important for determining the CKM matrix element IV~d[.
1. I N T R O D U C T I O N
The matrix elements
The study of B d0 - B-0d oscillations allows a clean extraction of the poorly known C K M matrix element IVtdl. However, the accuracy of this determination is currently limited by the theoretical uncertainy in the calculation of the matrix element,
as
.A~bd
:
-0 0 (Sd[O dA B = 2 [Bd)
=
(S l z,(1 -
2
Amd = GZF M 2 IVtdVt*bl2S( ~ ) C ( # ) . M b g ( I z ) 87r2
w
w
,
2
where ]B~(,) is the decay constant, MBa(,) the mass, and BBa(,) the B - p a r a m e t e r of B d0( s ) mesons. In this work, we obtain the ratio rsd from the direct calculation of ~.A4bd as well as from the Cal-
2. S I M U L A T I O N
m2
I
IrM
*CPT-98/PE.3689; CERN-TH/98-289. Support from EPSRC and PPARC under grants GR/K41663 and GR/L29927 is acknowledged. tOn leave from: Centre de Physique Th6orique, Case 907, CNRS Luminy, F-13288 Marseille Cedex 9, France. SPresenter.
DETAILS
"
-zgocsw~
where GF is the Fermi constant, M w the W S
"
We use the tadpole-improved SheikholeslamiWohlert (SW) quark action,
=
boson mass, mt the top-quark mass, ( ~ w ) the relevant Inami-Lim function, # the renormalisation scale, and C(#) the Wilson coefficient. An alternative approach, in which m a n y theoretical uncertainties cancel, is to look at the ratio
be parametrised
J~bd(s) "= -3f B,i(,,) MBa(,) B Ba(o)
culations of ~.fB a and BBB, Ba
- -r )dlB °) ,
which is related to the mass difference of the two mass eigenstates of the B ° - / ~ 0 system, 2
8 2
J~bd(s) C a n
E ~l(x)P~,,(x)a,~,q(x) :gJZ,V
to perform simulations on a 24 a × 48 lattice at /~ = 6.2 and a 163 × 48 lattice at ~ = 6.0. Here S W is the standard Wilson action, go the bare gauge coupling, c s w the clover coefficient, ~ the hopping parameter, and P ~ a lattice definition of the gauge-field strength tensor. Table 1 gives the simulation parameters. We use K L M normalisation for the quark fields. 3. O P E R A T O R
MATCHING
Matching onto the MS scheme is performed at one-loop in perturbation theory using the coupling a~--~(#) defined from the plaquette [1].
0920-5632/99/$ - see front matter © 1999 ElsevierScience B.V. All rights reserved. PII S0920-5632(98)00597-0
358
L. Lellouch, C.-J. DavidLin/NuclearPhysicsB (Proc. Suppl.) 73 (1999)357-359
/~ 6.0 6.2
Table I: Simulation parameters, aQ and aq are the heavy- and light-quark hopping parameters. ://= configs, csw I~q aQ 498 1.48 0.13700 0.13810 0.13856 0.114 0.118 0.122 0.126 0.130 188 1.44 0.13640 0.13710 0.13745 0.120 0.123 0.126 0.129 0.132
Since the clover-leaf interaction term is proportional to go, we can use the perturbative results obtained from a tree-level clover action [2] with modifications appropriate for tadpoleimprovement and KLM normalisation. For the matching of four-fermion operators, we use the basis O~at
=
% x %, + 7 # % x % , % ,
012at
=
7. x7~--7.75 x7~75,
O~a~
=
I x I + Ts x % ,
O~~
=
015at :
I x l-%
Table 2: Critical and strange hopping parameters and inverse lattice spacings. /~ 6.0 6.2 ~c 0.13924(1) 0.13793(1) ~s 0.13757(8) 0.13670(9) a -1 (Mp)(GeV) 1.96(5) 2.57(8) a-l(f,r) (GeV) 1.92(4) 2.58(9)
1.0, . . . .
....
x %,
:,20 O:12S
G/iv X G#v.
We set the coupling and matching scales to # = 1 and, for consistency with the literature, run divergent operators to 5 GeV, using 2-1oop continuum RG in the MS scheme with the appropriate number of flavours. To estimate the systematic error associated with the one-loop matching, we vary the scale # in a range from 1/a to fla. Decay constants are not affected since they are normalized b y / ~ and B-parameters change by about 3% ( / . varies by approximately 3%). Since we are mainly interested in SU(3)-breaking ratios for which these effects are even smaller, we neglect these small variations in what follows. 4. A N A L Y S I S A N D R E S U L T S We determine ~c and s8 from pseudoscalar meson masses. We set the scale with Mp for spectral quantities and f,~ for decay constants. In fact, these two quantities yield remarkably similar scales. (See Table 2.) We then linearly extrapolate and interpolate heavy-light decay constants, B-parameters and AB = 2 matrix elements to ~c and ~8, keeping ~ , ~ , aMp and af~ in the bootstrap loop. Fig. 1 shows examples of these extrapolations.
'¢='
I ....
I '
p = 6.0
'812' ....
L 'J[: ~
A
°"l , 0 .-1 1=4XL ~h . 0 . 1 1 8 ~----.TL-'dt-..~ ±
~ o.81- 0.122 Jlc-.~-4~m::::m~
0.7
0.132 ~
o.128~ ~ .
=
T I
°'vl _ 0 . 1 3 0
~-T-'~T
O:data
O:data x:~ c 0:~, , , , I .... I .... 7.~ 7.3o
0.8=
v.~
v.t
. . . .
~I"
x:~ o I
D:~,
. . . .
1
v 'l=l / ~
.
v.s
Figure 1. Light-quark-mass dependence of the heavy-light B-parameter, Bp, and extrapolation (interpolation) to ~l = ac (~8) at /~ = 6.0 and 6.2.
For heavy-quark (HQ) extrapolations, we define
afp M/-~p! as(Mp) } ~ ~A V ~ [ as(MB)
Cf(Mp)
}
a4MM. a,(Mp) CAF=2(Mp)
--
Mp
[. a, (MB) -
"
Then for X(Mp)=~/(Mp),~AF=2(Mp), B(Mp) and SU(3)-breaking ratios, HQET predicts
X(Mp) = Ax { I + Bx(-~p ) + Cx( M" )2 + } Mp .... Fig. 2 shows examples of the HQ extrapolations.
L. Lellouch, C-J. David Lin/Nuclear Physics B (Proc. Suppl.) 73 (1999) 357-359
0.~i
0,20
....
~
I .... [ .... I .... HQ Scaling
~
''''1
-meson
.... I .... HQ Scaling
I
i.o ~ m l s o n
,$. 0.15
~ =' 6.2 - f l
O. IO.o 0
~eso:
I
I
0.2
O.4
O.O
I 0.e
0.0
'
0.0
Mp/Mp(m~ = ra.)
'
I O.2
. O.4
MJMp(m]
=
0.6
m,)
Figure 2. HQ scaling of ~I and B-parameter.
. . . .
0.1~
i
l
. . . .
i
. . . .
l
. . . .
Chiral Extrapolation
0.04
r3:t¢,
0.1~
.... I .... I .... I HQ Scaling of ~ . ~
359
Table 3: Summary of results. Errors are statistical only. /3 6.0 6.2 fDo (MeV) 239(6) 221(9) fD(MeV) 213(6) 193(10) lB, (MeV) 221(7) 190(12) fB(MeV) 191(10) 161(16) 1.12(1) 1.15(4) 1.15(4) 1.18(8) Ye B~I°(5GeV) 0.86(2) 0.85(2) B~f°(5GeV) 0.83(4) 0.85(3) Be, 1.03(3) 0.99(3) Be (, M__~s ~ ~2Bso 1.38(7) 1.37(13) Me Be ~h~ 1.52(19) 1.70(28)
0.03
0.0~
~ O.03
7.15
1
1
4
x:~. ~ = 6.0 ,.,I .... ~.... I.,, 7.20
7.~ ~1-1
7.SO
0,01
.... 7.05
0.0
I .... O.2
I .... O.4
I O.O
MJM.(m, = m.)
Figure 3. Chiral and HQ extrapolation of the matrix element.
For SU(3)-breaking ratios, we find that taking the ratio before or after the HQ extrapolation leads to nearly indistinguishable results. We use the former for our final results since SU(3)-breaking ratios have milder HQ-mass dependences. Our main results are summarised in Table 3. We obtain rsd from the direct calculation of ~ " .tvIbd as well as from ~f B d and ~B e d • Our results for the direct calculation are consistent with those of [3], obtained with propagating Wilson quarks, and, at/~ = 6.0, with the static result of [4]. However, as Fig. 3 suggests, it is more difficult to control the chiral and HQ extrapolations of the matrix elements in the direct calculation because these extrapolations are more pronounced. Because we have results at only two values of the lattice spacing, we cannot extrapolate to the continuum limit. We therefore consider the )3 ---- 6.2 results to be our best, noting that de-
cay constants may still suffer from relatively large discretisation errors (roughly a 2a effect between 6.0 and 6.2) while SU(3)-breaking ratios and Bparameters are consistent within errors at the two /3 values. Although formally one need not include the aO~P correction to the axial current when using a mean-field improved, tree-level clover action, it would be interesting to investigate its effect on our results in view of understanding how nonperturbatively, O(a)-improved decay constants may behave. We plan to do so in the future. For a comparison of our results with other recent results, we refer the reader to [5]. REFERENCES
1. G. P. Lepage, P. B. Mackenzie, Phys. Rev. D48 (1993) 2250. 2. E. Gabrielli et al., Nucl. Phys. B362 (1991) 475; R. Frezzotti et aL, Nucl. Phys. B373 (1992) 781; and R. Gupta et aL, Phys. Rev. D55 (1997) 4036, for DRED to NDR. 3. C. Bernard, T. Blum and A. Soni, Phys. Rev. D 5 8 (1998) 014501. 4. V. Gim~nez and G. Martinelli, Phys. Lett. B398 (1997) 135. 5. T. Draper, in these proceedings.