Post on 18-Sep-2018
Paul MackenzieFermilab
mackenzie@fnal.gov
B→Dlν and B→πlν on the Lattice
BaBar/Lattice QCD WorkshopSLAC
Sept. 16, 2006Thanks, Ruth van de Water, Richard Hill, Thomas Becher
1
Paul Mackenzie BaBar/Lattice QCD Workshop, Sept. 16, 2006
Lattice calculations
2
Quarks are defined on the sitesof the lattice, and gluons are SU3 matrices on the links, U=exp(igA).
Calculations are done at finite lattice spacing a, finite volume V, too high a light quark mass m, etc.Theory is used to derive expected functional form for extrapolations in a (OPE), m (chiral PT), V (finite volume chiral PT) ...➱ Systematic errors.
2
Paul Mackenzie Fermilab Wine and Cheese, July 14, 2006 3
Quantities that used to agree decently, ~10%, in the quenched approximation...
Gold-plated quantities.
Staggered fermions,the least CPU-intensive.
... agree to a few % in recent unquenched calculations.
Progress for simple quantities
3
Paul Mackenzie Fermilab Wine and Cheese, July 14, 2006
Three families of lattice fermions• Staggered (Kogut-Susskind)/naive
• Good chiral behavior (can get to light quark masses). Fermion doubling introduces nonlocal effects which must be theoretically understood. Cheap.
• Wilson/clover
• No fermion doubling but horrible chiral behavior.
• Overlap/domain wall
• Nice chiral behavior at the expense of adding a fifth space-time dimension. Expensive.
4
Staggered fermion unquenched calculations are the cheapest and currently most advanced phenomenologically, (probably a temporary situation).
4
Paul Mackenzie Fermilab Wine and Cheese, July 14, 2006
“Gold-plated quantities” of lattice QCD
5
Quantities that are easiest for theory and experiment to both get right.
Stable particle, one-hadron processes. Especially mesons.
More complicated methods are required for multihadron processes: - unstable particles are messy to interpret, - multihadron final states are different in Euclidean and Minkowski space.
5
Paul Mackenzie Fermilab Wine and Cheese, July 14, 2006 6
Many of the most important quantities for lattice QCD aregolden quantities.
E.g., measurements determining the fundamental parameters of the Standard Model.
6
Paul Mackenzie BaBar/Lattice QCD Workshop, Sept. 16, 2006
To obtain Vcb from data, theory must supply only normalization, which can be obtained from at zero recoil.
B➙Dlν
7
Constraint on Isgur–Wise function for B ! D semileptonic decays 2025
Figure 1. The function F(!) for " = 0 (full line), 0.02 (short broken) and 0.08 (long broken).
These values of " result from setting g = 0.27 (see main text) and E = 0, 0.5 and 1 GeV,
respectively.
For #$ we then obtain the expression†,
#$ (%, !) = 1
2
!d4p$
(2$)3&(p2$ )&(% " v# · p$ )Tr('+(v)($'+(v#)($ )
= 3
8$2g2
f 20%) 2(!)
"! " 1$
!2 " 1log
#! +
$!2 " 1
%&. (11)
Thus, inequality (7) can be explicitly written as,
)(!) ! F(!) %"1+ !
2+ "
'! " 1$
!2 " 1log
#! +
$!2 " 1
%(&" 12
(12)
" = 3
16$2g2E2
f 20. (13)
The value of E must meet the requirement that contributions to F(!) from higher order
corrections in the perturbative chiral expansion must be small. If, as we expect, the
expansion parameter is gE/(4$f0), a value of E = 0.5 GeV should be appropriate. In
fact, given the relative smallness of g, such choice may be somewhat conservative.
Setting E = 0.5 GeV yields " = 0.275g2. There is a recent determination of g from
D& ! D$ decay data [12], g = 0.27+0.04+0.05"0.02"0.02, which leads to " = 0.020. (Note, however,
that larger values of g are not completely ruled out by current data [12].) The function
F(!) is plotted in figure 1 for several values of ".
The derivative of F(!) at zero recoil is given by
F #(! = 1) = "14
'1+ 8"
3
(. (14)
† #$ vanishes at the zero-recoil point ! = 1 due to the factor in square brackets in (11), a result which holds
true also in the case of many pion emission [13].
Form factors are well described by the Isgur-Wise function.Governed by two parameters to good approximation: normalization and slope.Slope parameter is well measured by experiment.
B! D(!)l! decay
B(B! D(!)l!) " |Vcb|2|FB"D(!)(1)|2Zdw f (!)(w)
where w= vB · vD. Use double ratio (FNAL’99): CDV0B(t)CBV0D(t)
CDV0D(t)CBV0B(t)
! !D|V0|B"!B|V0|D"!D|V0|D"!B|V0|B"
B! Dl!
n f =2+1, FNAL/MILC
0 0.01 0.02 0.03
ml
1
1.1
F(1
)
Nf=2+1 (FNAL/MILC)
Nf=0 (FNAL’99)B!>D
FB"D(1) = 1.074 (18)sta(15)sys
Using HFAG’04 avg for |Vcb|F (1),|Vcb|Lat05=3.91(09)lat(34)exp"10#2
B! D!l!
S#PT calc. completed (Laiho’s talk)
0 0.025 0.05 0.075 0.1 0.125 0.15m_pi^2
0.89
0.9
0.91
0.92
0.93
h_A1
Cusp (in #PT) disappears in S#PT
n f = 2+1 calc. underway (FNAL)=# more precise |Vcb|
4 Workshop on the CKM Unitarity Triangle, IPPP Durham, April 2003
2010-10-20
4
3
2
1
2010-10-20
4
3
2
1
Figure 4. Dispersive bound for f 0 and f + with UKQCD lattice
data. Model-independent QCD bounds with 90%, 70%, 50% and
30% confidence levels are given by the pair of curves. Figure
taken from Ref. [ 7].
in two di!erent methods and equate the results. The two
computational methods are : (1) light cone expansion
which is expressed by the pion light-cone wavefunction
φπ(u) and (2) the dispersion relation which takes the fol-lowing sum of the physical poles
CFv !m2BfB
mb
f +(q2)1
m2B" p2
B
+ higher poles, (5)
where higher poles are suppressed by Borel transformation
with M and approximated by the light-cone expansion re-
sults above a threshold s20.
The theoretical input parameters are the parameters ai’s
of light-cone wavefunctions in the Gegenbauer polynomial
expansion,
φπ = 6u(1 " u)[1 + a2C3/22 (2u " 1) + · · ·], (6)
the B meson decay constant fB, the b quark mass mb, the
threshold s20, and the parameter M for the Borel transfor-
mation.
We here give the new results of Ref. [ 12] as an example.
It is found that the radiative correction is about 10% and
the correction from higher twists ( twist 3) is ! 30%. Theresults for q2 < 14 GeV2 are well fitted by
f +(q2) =F(0)
1 " aq2/m2B+ b(q2/m2
B)2. (7)
Light-cone QCD sum rule results for q2 < 14 GeV2 canalso be fitted by the pole dominance ansatz.
f +(q2) =c
1 " q2/m2B#
(8)
where c $ fB#gBB#π/(2mB#) = 0.414+0.016"0.018 plus systematic
errors.
Fig. 2.1 shows the result by the light-cone QCD sum rule.
It is remarkable that the light-cone QCD sum rule give con-
sistent results with lattice QCD.
3 B% ρlν
Recently, UKQCD collaboration [ 13] and SPQcdR col-
laboration [ 14] started studies of B % ρlν form factors.
Both collaborations use O(a)-improved Wilson action for
the heavy quark and extrapolate the numerical results of
mQ ! mc towards the physical b quark mass. The lattice
spacings are a"1= 2.0 and 2.7 GeV for UKQCD and a"1 =
2.7 and 3.7 GeV for SPQcdR.
UKQCD fits the lattice data for q2 > 14 GeV2 to the fol-lowing form
1
|Vub|2d"
dq2=
G2Fq2[λ(q2)]1/2
192π3m3B
(a + b(q2 " q2max)), .
The fit coe#cients are a = 38+8"5 ± 5 GeV2 and b = 0 ± 2 ±
1, where the first error is statistical and the second is the
extrapolation error for both a and b .
SPQcdR collaboration obtains form factors for q2 > 10GeV2. They find the results which is consistent with the
light-cone QCD sum rule results.
4 B% D(#)lν
One can extract |Vcb| from the B % D(#)lν semileptonicdecay near zero recoil as
d"
dω(B% D(#)) & |Vcb|2|FB%D(#)(ω)|2, (9)
where ω $ v · v' and FB%D(#)lν are the linear combinationsof form factors h±, hA1,2,3 . One important outcome from theheavy quark symmetry is that the form factor FB%D(#)lν isequal to unity at zero recoil up to perturbatively calculable
7
Paul Mackenzie BaBar/Lattice QCD Workshop, Sept. 16, 2006
Ratio method: determine from a ratio that goes to 1 with vanishing errors in the symmetry limit.
B! D(!)l! decay
B(B! D(!)l!) " |Vcb|2|FB"D(!)(1)|2Zdw f (!)(w)
where w= vB · vD. Use double ratio (FNAL’99): CDV0B(t)CBV0D(t)
CDV0D(t)CBV0B(t)
! !D|V0|B"!B|V0|D"!D|V0|D"!B|V0|B"
B! Dl!
n f =2+1, FNAL/MILC
0 0.01 0.02 0.03
ml
1
1.1
F(1
)
Nf=2+1 (FNAL/MILC)
Nf=0 (FNAL’99)B!>D
FB"D(1) = 1.074 (18)sta(15)sys
Using HFAG’04 avg for |Vcb|F (1),|Vcb|Lat05=3.91(09)lat(34)exp"10#2
B! D!l!
S#PT calc. completed (Laiho’s talk)
0 0.025 0.05 0.075 0.1 0.125 0.15m_pi^2
0.89
0.9
0.91
0.92
0.93
h_A1
Cusp (in #PT) disappears in S#PT
n f = 2+1 calc. underway (FNAL)=# more precise |Vcb|
B➙Dlν
8
Hashimoto et al. (99),(Works for K➙πlν, too,Becirevic et al.)
B! D(!)l! decay
B(B! D(!)l!) " |Vcb|2|FB"D(!)(1)|2Zdw f (!)(w)
where w= vB · vD. Use double ratio (FNAL’99): CDV0B(t)CBV0D(t)
CDV0D(t)CBV0B(t)
! !D|V0|B"!B|V0|D"!D|V0|D"!B|V0|B"
B! Dl!
n f =2+1, FNAL/MILC
0 0.01 0.02 0.03
ml
1
1.1
F(1
)
Nf=2+1 (FNAL/MILC)
Nf=0 (FNAL’99)B!>D
FB"D(1) = 1.074 (18)sta(15)sys
Using HFAG’04 avg for |Vcb|F (1),|Vcb|Lat05=3.91(09)lat(34)exp"10#2
B! D!l!
S#PT calc. completed (Laiho’s talk)
0 0.025 0.05 0.075 0.1 0.125 0.15m_pi^2
0.89
0.9
0.91
0.92
0.93h_A1
Cusp (in #PT) disappears in S#PT
n f = 2+1 calc. underway (FNAL)=# more precise |Vcb|
B! D(!)l! decay
B(B! D(!)l!) " |Vcb|2|FB"D(!)(1)|2Zdw f (!)(w)
where w= vB · vD. Use double ratio (FNAL’99): CDV0B(t)CBV0D(t)
CDV0D(t)CBV0B(t)
! !D|V0|B"!B|V0|D"!D|V0|D"!B|V0|B"
B! Dl!
n f =2+1, FNAL/MILC
0 0.01 0.02 0.03
ml
1
1.1
F(1
)
Nf=2+1 (FNAL/MILC)
Nf=0 (FNAL’99)B!>D
FB"D(1) = 1.074 (18)sta(15)sys
Using HFAG’04 avg for |Vcb|F (1),|Vcb|Lat05=3.91(09)lat(34)exp"10#2
B! D!l!
S#PT calc. completed (Laiho’s talk)
0 0.025 0.05 0.075 0.1 0.125 0.15m_pi^2
0.89
0.9
0.91
0.92
0.93
h_A1
Cusp (in #PT) disappears in S#PT
n f = 2+1 calc. underway (FNAL)=# more precise |Vcb|
Fermilab/MILC 05.
B! D(!)l! decay
B(B! D(!)l!) " |Vcb|2|FB"D(!)(1)|2Zdw f (!)(w)
where w= vB · vD. Use double ratio (FNAL’99): CDV0B(t)CBV0D(t)
CDV0D(t)CBV0B(t)
! !D|V0|B"!B|V0|D"!D|V0|D"!B|V0|B"
B! Dl!
n f =2+1, FNAL/MILC
0 0.01 0.02 0.03
ml
1
1.1
F(1
)
Nf=2+1 (FNAL/MILC)
Nf=0 (FNAL’99)B!>D
FB"D(1) = 1.074 (18)sta(15)sys
Using HFAG’04 avg for |Vcb|F (1),|Vcb|Lat05=3.91(09)lat(34)exp"10#2
B! D!l!
S#PT calc. completed (Laiho’s talk)
0 0.025 0.05 0.075 0.1 0.125 0.15m_pi^2
0.89
0.9
0.91
0.92
0.93
h_A1
Cusp (in #PT) disappears in S#PT
n f = 2+1 calc. underway (FNAL)=# more precise |Vcb|
B! D(!)l! decay
B(B! D(!)l!) " |Vcb|2|FB"D(!)(1)|2Zdw f (!)(w)
where w= vB · vD. Use double ratio (FNAL’99): CDV0B(t)CBV0D(t)
CDV0D(t)CBV0B(t)
! !D|V0|B"!B|V0|D"!D|V0|D"!B|V0|B"
B! Dl!
n f =2+1, FNAL/MILC
0 0.01 0.02 0.03
ml
1
1.1
F(1
)
Nf=2+1 (FNAL/MILC)
Nf=0 (FNAL’99)B!>D
FB"D(1) = 1.074 (18)sta(15)sys
Using HFAG’04 avg for |Vcb|F (1),|Vcb|Lat05=3.91(09)lat(34)exp"10#2
B! D!l!
S#PT calc. completed (Laiho’s talk)
0 0.025 0.05 0.075 0.1 0.125 0.15m_pi^2
0.89
0.9
0.91
0.92
0.93
h_A1
Cusp (in #PT) disappears in S#PT
n f = 2+1 calc. underway (FNAL)=# more precise |Vcb|
Used in renormalization of the vector current.
Uncertainties cancel in ratio in the symmetry limit.
Okamoto, Lattice 2005
8
Paul Mackenzie BaBar/Lattice QCD Workshop, Sept. 16, 2006
B➙πlν
9
The players: F+, F
0
!ML(p!)|V µ|MH(p)" = F+(q2)(pµ + p!µ) + F"(q2)(pµ # p!µ)
= F+(q2)
!
pµ + p!µ #m2
H # m2L
q2qµ
"
+ F0(q2)
m2H # m2
L
q2qµ
• What are the measurable quantities ?
• What can we learn ?
2q
0 1 2
+F
0
1
2
3
4
BELLE
2q
0 10 20
+F
0
5
10 BABAR
We also perform model-independent one-dimensional (y) fits where the data in every of the100 q2/m2
! bins were fitted independently. The resulting distribution is shown in Fig.6. Thenormalization f+(0) = 1 is assumed. The visible non-linearity can be observed in Fig.7, wherethe ratio f+(t)/f+(0)/(1 + !+q2/m2
!) is presented. The parabolic curve represents the fit withthe quadratic non-linearity in the form-factor.
q2/m
2!
f(t)/f(0)
Figure 6: The value of f+(t)/f+(0) obtainedin the model-independent fits.
q2/m
2!
Figure 7: The value off+(t)/f+(0)/(1 + !+q2/m2
!). The fit withnon-linear contribution is shown.
This non-linearity can not be explained by a possible scalar contribution (that also resultsin the enhancement of the number of events at large values of q2). The row 4 of the Table1 represents a search for the scalar term with the vector form-factor set to be linear. Theresulting value of fS/f+(0) is compatible with zero.
We also perform a model-independent fit to extract simultaneously f+(t) and fS(t). Theresulting distribution for the value fS(t)/f+(0) is shown in Fig.8. The value of the scalarcontribution is compatible with zero with strong enhancement of the errors at small values oft. This enhancement is explained by the dependence of the scalar contributions (Eq. 2) on theDalitz variables. One can observe that the leading term |S|2 is proportional to t and vanishesat t ! 0.
The last row of the Table 1 represents a fit with both scalar contribution and the quadraticterm in the vector form-factor.
We also do not see any tensor contribution in our data (rows 3 and 5 in the Table 1).
6
!ISTRA
K ! !"# D ! !"# B ! !"#
Data has nontrivial shape.
Theory and experimental uncertainties are q2 dependent, severely so on the lattice.
Harder and more important to understand shape.
arX
iv:h
ep-p
h/0
30
92
25
v2
6
Oct
20
03
Workshop on the CKM Unitarity Triangle, IPPP Durham, April 2003
CKM03
Lattice Determination of Semileptonic Form Factors
T. Onogi
Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto, 606-8502, Japan
We report on the Lattice determination of the semileptonic form factors by lattice QCD. Comparison with the light-cone QCD sum rules
are made for B! !l", B! #l" semileptonic decays.
1 Introduction
Computation of the semileptonic form factors for B !!(#)l", B ! D(")l" and B ! K(")l+l# decays is a key to
the determination of the CKM matrix elements |Vub|, |Vcb|and |Vts| . In this report, I review the recent results by lat-
tice QCD methods. I will mainly focus on the semileptonic
decay B ! !l", because lattice calculations in various ap-
proaches are already at hand [ 1, 2, 3, 4, 5] and because
most of the problems which exists in this decay are com-
mon to other processes.
Lattice QCD calculations of B ! !l" form factors su!er
from (1) the error from the large heavy quark mass mQ in
lattice unit, (2) the chiral extrapolations in the light quark
mass mq using the simulation results with relatively large
mass region mq > ms/2, and (3) the limitations from the
accessible kinematic range of q2 from statistical and sys-
tematic errors.
In order to solve the first problem, two di!erent approaches
are made. One is to avoid the large discretization errors of
O(amQ) by computing the form factors with the conven-
tional relativistic quark action for charm quark mass re-
gion, and then extrapolate the results in 1/mQ. The other is
to use HQET e!ective theory with 1/mQ corrections. Since
both approaches have their own advantages and disadvan-
tages, it would be important to have both results and study
whether they give consistent results within quoted errors. I
will present some of the major calculations from these two
approaches and discuss the consistency of the results.
The second problem already gives a source of errors in the
quenched calculations but it would become even more se-
rious in unquenched QCD. Unfortunately, there are still no
results in the unquenched QCD. Since the chiral limit of
the form factors is ill-defined in the quenched QCD, all we
can do with the present lattice data is to discuss the light
quark mass dependence in the intermediate mass regime in
the quenched QCD. However, one may be able to give an
estimate of the low energy coe"cients of the chiral per-
turbation theory based on the present lattice data. I will
briefly review new studies of quenched chiral perturbation
theory (QChPT) and partially quenched chiral perturbation
theory (PQChPT) [ 6] for B ! !l" form factors, which
give a phenomenological estimate on the quenching errors
and chiral extrapolations. These studies will be even more
useful once new calculations in unquenched QCD will be
made.
The third problem arises because in semileptonic decays
large energies are released to the final states so that the lep-
ton pair invariant mass q2 can range from 0 to (mB # m!)2.
However, due to the discretization errors of O(aE) as well
as the statistical errors which grow as $ exp(const % (E #m!)t) where E is the energy of the pion, lattice QCD can
cover only large q2 region. The dispersion relation is a pos-
sible solution to give bounds for smaller q2 region [ 7]. The
light-cone QCD sum rule (LCSR) predicts form factors for
small q2, which is complementary to the lattice results. I
will give a comparison of the recent LCSR results with the
lattice results to see whether they will give consistent re-
sults.
I also review the form factors in other processes. Some
of the recent work on the lattice QCD calculations of the
B ! #l" form factors in relativistic formalism are pre-
sented. Very precise calculations of semileptonic form fac-
tors for B ! D(")l" at zero recoil and the calculations of
the slope of the Isgur Wise function are presented.
2 B! !l"
The exclusive semileptonic decay B ! !l" determines the
CKM matrix element |Vub| through the following formula,
d#
dq2=
G2F
24!3|k!|2|Vub|2| f +(q2)|2, (1)
where the form factor f + is defined as
&!(k!)|q̄$µb|B(pB)' = f +(q2)
!
"
"
"
"
#
(pB + k!)µ #
m2B # m
2!
q2qµ$
%
%
%
%
&
+ f 0(q2)m2B # m
2!
q2qµ, (2)
with q = pB # k! and q2 = m2B + m
2! # 2mBv · k!. The
following parameterization proposed by Burdman et al. [
9
Paul Mackenzie BaBar/Lattice QCD Workshop, Sept. 16, 2006
B➙πlν, quenched approximation
10
2 Workshop on the CKM Unitarity Triangle, IPPP Durham, April 2003
8]
!!(k!)|q̄"µb|B(v)" = 2!
f1(v · k!)vµ + f2(v · k!)kµ!
v · k!
"
, (3)
is also useful for discussing the heavy quark symmetry and
the chiral symmetry of the form factors in a transparent
way.
2.1 Lattice results
Lattice calculation is possible only in limited situations.
Spatial momenta must be much smaller than the cuto!, i.e.
|#pB|, |#k!| < 1 GeV. This means v·k! # E! < 1 GeV or equiv-alently q2 > 18 GeV2. Another limitation is that due tothe slowing down, simulations with very small light quark
masses are di"cult so that usual mass range for the light
quark masses in practical simulations is ms/3 $ mq $ ms
or m! = 0.4 % 0.8 GeV. Therefore in order to obtain phys-ical results chiral extrapolations in the light quark masses
are necessary.
So far all the lattice calculations of the form factors are
done only in quenched approximation. APE collaboration
[ 1] and UKQCD collaboration computed B & !l$ formfactors for a fine lattice with the inverse lattice spacing
a'1 % 2.7 GeV. They used relativistic formalism for the
heavy quark and extrapolated the results of heavy-lightme-
son around charm quark masses to the bottom quark mass.
Fermilab collaboration [ 3] used the Fermilab formalism
for the heavy quark and computed the form factors on three
lattices with a'1 = 1.2 % 2.6 GeV . JLQCD collaboration[ 4] computed the form factors using NRQCD formalism
for the heavy quark on a a'1 = 1.64 GeV. NRQCD col-
laboration [ 5] also used NRQCD formalism for the heavy
quark and an improved light quark action (D234 action)
on a anisotropic lattice with a'1 = 1.2 GeV (spatial), 3.3GeV (temporal). In all of these calculations the light pseu-
doscalar meson masses are 0.4 % 0.8 GeV. Fig. 1 showsthe result by di!erent lattice groups, f +(q2) agrees within
systematic errorswhile f 0(q2) shows deviations among dif-
ferent methods.
The reason for the discrepancies in f 0 can be attributed to
the systematic error in the chiral extrapolation and heavy
quark mass extrapolation (interpolation) error. In the fol-
lowing, we examine these errors in more detail. Light
quark mass mq dependence of form factors with fixed spa-
tial momenta ap = 2!16(1, 0, 0) is shown in Fig. 2. In con-
trast to the JLQCD data, Fermilab data shows a significant
increase towards the chiral limit. Large di!erence in Fer-
milab results and JLQCD results for f 0 in the chiral limit
arises from di!erent mq dependence, but the raw data for
similar quark masses are not so di!erent. Shigemitsu et al.
studied the mass dependence of f1 + f2 and find similar be-
havior as JLQCD. Further studies to clarify the light quark
mass dependence are required.
0 5 10 15 20 25 30
q2 (GeV
2)
0.0
1.0
2.0
f0,+(q
2)
UKQCD
APE
Fermilab
JLQCD
NRQCD
LCSR
LCSR
Figure 1. B& !l$ form factors by di!erent lattice groups.
Fig. 3 shows 1/MB dependence of form factors #0,+ #(mB f
0,+ at v · k! = 0.845 GeV for APE and JLQCD col-laboration data. It is found that the di!erence of APE
(UKQCD) vs JLQCD (NRQCD) for f 0 arises from the ex-
trapolation in 1/M. Linear extrapolation in 1/M is consis-
tent, while the quadratic extrapolation gives higher value.
The quadratic extrapolation 1/M is chosen for APE’s result,
since higher value gives better agreement with the soft pion
theorem. Simulations with static heavy quark may resolve
the problem.
The error of the form factors in the present calculations is
around 20%. Some of the major errors are the quenching
error, chiral extrapolation error statistical error in all cal-
culations. In addition, a large discretization error appears
in JLQCD results and a large 1/M extrapolation error is
contained in APE and UKQCD results.
There are several proposals to improve the form factor de-
termination. The quenching error can be resolved only
by performing the unquenched calculations. Recently,
JLQCD and UKQCD collaborations has accumulated n f =
2 unquenched lattice configurations with O(a)-improved
Wilson fermions and n f = 2+1 unquenched configurations
with improved staggered fermions have been produced by
the MILC collaboration. These unquenched QCD data
should be applied to form factor calculations.
In order to reduce the chiral extrapolation error, simulation
with even smaller light quark masses are necessary. For
Wilson type fermions, simulations with m! < 0.4 GeV willbe very slow and also appearance of exceptional configura-
tion may prevent the simulation for very light quark mass
range. On the other hand, MILC collaboration is now car-
rying out simulations with m! = 0.3' 0.5 GeV, which cor-responds tomq = 1/5ms'1/2ms [ 9]. Since n f = 2+1 sim-
ulations are performed by taking the square root or quar-
Onogi, CKM 2003.
10
Paul Mackenzie BaBar/Lattice QCD Workshop, Sept. 16, 2006
B➙πlν, unquenched
11
27
0 5 10 15 20 25
q2 [GeV
2]
0.0
1.0
2.0
3.0
UKQCD (1999)
Abada et al. (2000)
El-Khadra et al. (2001)
JLQCD (2001)
0 5 10 15 20 25
q2 [GeV
2]
0.0
1.0
2.0
3.0
Fermilab (2004)
HPQCD (2004)
Nf = 0
Nf = 2+1
0 5 10 15 20 25
q2 [GeV
2]
0
0.5
1
1.5
2
2.5N
f=2+1 (HPQCD)
Nf=2+1 (FNAL/MILC)
B!>!l"
f+
f0
Onogi, Lattice 2006.
Results agree well with quenched results. Probably not significant; not true for all quantities.
27
0 5 10 15 20 25
q2 [GeV
2]
0.0
1.0
2.0
3.0
UKQCD (1999)
Abada et al. (2000)
El-Khadra et al. (2001)
JLQCD (2001)
0 5 10 15 20 25
q2 [GeV
2]
0.0
1.0
2.0
3.0
Fermilab (2004)
HPQCD (2004)
Nf = 0
Nf = 2+1
0 5 10 15 20 25
q2 [GeV
2]
0
0.5
1
1.5
2
2.5N
f=2+1 (HPQCD)
Nf=2+1 (FNAL/MILC)
B!>!l"
f+
f0
11
Paul Mackenzie BaBar/Lattice QCD Workshop, Sept. 16, 2006
B➙πlν, finite range of q2
12
27
0 5 10 15 20 25
q2 [GeV
2]
0.0
1.0
2.0
3.0
UKQCD (1999)
Abada et al. (2000)
El-Khadra et al. (2001)
JLQCD (2001)
0 5 10 15 20 25
q2 [GeV
2]
0.0
1.0
2.0
3.0
Fermilab (2004)
HPQCD (2004)
Nf = 0
Nf = 2+1
0 5 10 15 20 25
q2 [GeV
2]
0
0.5
1
1.5
2
2.5N
f=2+1 (HPQCD)
Nf=2+1 (FNAL/MILC)
B!>!l"
f+
f0
The players: F+, F
0
!ML(p!)|V µ|MH(p)" = F+(q2)(pµ + p!µ) + F"(q2)(pµ # p!µ)
= F+(q2)
!
pµ + p!µ #m2
H # m2L
q2qµ
"
+ F0(q2)
m2H # m2
L
q2qµ
• What are the measurable quantities ?
• What can we learn ?
2q
0 1 2
+F
0
1
2
3
4
BELLE
2q
0 10 20
+F
0
5
10 BABAR
We also perform model-independent one-dimensional (y) fits where the data in every of the100 q2/m2
! bins were fitted independently. The resulting distribution is shown in Fig.6. Thenormalization f+(0) = 1 is assumed. The visible non-linearity can be observed in Fig.7, wherethe ratio f+(t)/f+(0)/(1 + !+q2/m2
!) is presented. The parabolic curve represents the fit withthe quadratic non-linearity in the form-factor.
q2/m
2!
f(t)/f(0)
Figure 6: The value of f+(t)/f+(0) obtainedin the model-independent fits.
q2/m
2!
Figure 7: The value off+(t)/f+(0)/(1 + !+q2/m2
!). The fit withnon-linear contribution is shown.
This non-linearity can not be explained by a possible scalar contribution (that also resultsin the enhancement of the number of events at large values of q2). The row 4 of the Table1 represents a search for the scalar term with the vector form-factor set to be linear. Theresulting value of fS/f+(0) is compatible with zero.
We also perform a model-independent fit to extract simultaneously f+(t) and fS(t). Theresulting distribution for the value fS(t)/f+(0) is shown in Fig.8. The value of the scalarcontribution is compatible with zero with strong enhancement of the errors at small values oft. This enhancement is explained by the dependence of the scalar contributions (Eq. 2) on theDalitz variables. One can observe that the leading term |S|2 is proportional to t and vanishesat t ! 0.
The last row of the Table 1 represents a fit with both scalar contribution and the quadraticterm in the vector form-factor.
We also do not see any tensor contribution in our data (rows 3 and 5 in the Table 1).
6
!ISTRA
K ! !"# D ! !"# B ! !"#
Lattice data extend over only a fraction of the q2 range on the physical B➙πlν decay.
With standard methods, discretization errors go like O(ap)2,
signal goes like exp(-Eπt).
Proposals to address:
*) Moving NRQCD (Davies, Lepage, et al.)
*) Calculate in charm region, extrapolate to bottom (Abada et al.)
*) Gibbons: global simultaneous fit of all experimental and lattice data.
*) Unitarity and analyticity (Lellouch, Fukunaga-Onogi, Arnesen et al., Becher-Hill, ...)
12
Paul Mackenzie BaBar/Lattice QCD Workshop, Sept. 16, 2006
B➙πlν, unitarity fits
13
2q
0 10 20
+F
0
5
10 BABAR
B!!
• Experiment has yet to observe more than a normalization and a slope
• What is the significance of this slope?
-z
-0.2 0 0.2
+ F!
P
-1
0
1
2
3
Pronounced q2 dependence in form factor is due to calculable effects. When those are factored out, two parameters suffice to describe the current experimental data.(Just like B➙Dlν, K➙πlν?!!)
“Arbitrary” analytic function -- choice only affects particular
values of coefficients (a’s)
Vanishes at subthreshold (e.g. B*) poles
P (t) !(t, t0) f(t) =!!
k=0
ak(t0)z(t, t0)k
13
Paul Mackenzie BaBar/Lattice QCD Workshop, Sept. 16, 2006
B➙πlν, unitarity fits
14
-0.2 -0.1 0 0.1 0.2z(t)
0
0.05
0.1
P(t
)!(t
,t0)f
(t)
P(t)!(t,t0)f
0(t)
P(t)!(t,t0)f
+(t)
B->" form factor data normalized by P(t) x !(t,t0) vs. z(t)
q2
max
Coefficients in z expansion are compatible with experiment.
0.0257228 +- 0.003 0.020256 +- 0.068
0.152234 +- 0.41
a0:a1:a2:
14
Paul Mackenzie BaBar/Lattice QCD Workshop, Sept. 16, 2006
B➙πlν, unitarity fits
15
0 5 10 15 20
q2(GeV
2)
0
0.5
1
1.5
2
2.5
3
3.5
f 0(q
2)
and
f+(q
2)
3 param. fit constrained such that f+(0)=f
0(0) -- !
2/d.o.f. = 0.35
f+(q
2) from constrained fit
unconstrained 3 parameter fit-- !2/d.o.f. = 0.35
f+(q
2) from unconstrained fit
B->" semileptonic form factors vs. q2
- Raw lattice data,- Not extrapolated in m or a,- Momentum dependent discretization errors not yet included.
Combined fits of f+ and f0 may give surprisingly good prediction for form factors well beyond the range of lattice data.
How can the results of such fits best be compared with experiment?
15
Paul Mackenzie BaBar/Lattice QCD Workshop, Sept. 16, 2006
Not covered, but interesting
• B➙ρlν, B➙ωlν, etc.
• Honest methods for treating unstable particles on the lattice exist (Lüscher,...) but they are much more demanding.
• B➙Klν, non-Standard Model effects
• Lattice calculation are no more difficult as long as effective operators are local.
16
16
Paul Mackenzie BaBar/Lattice QCD Workshop, Sept. 16, 2006
Summary and to-do list
• For lattice theorists: how well do lattice methods agree?
• staggered vs. clover vs. overlap, etc.
• Will Moving NRQCD allow calculation of the form factors for B➙πlν in the whole decay region?
• For theorists and experimentalists: how should lattice data be reported; how should lattice and experiment be compared?
• Raw lattice data in large global fit (Gibbons)
• Normalization and slope in the z expansion
• Form factor and slope at several fiducial points (Becher and Hill)
• All of the above
17
17