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本論文

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(格子上の非相対論的 QCDを用いた B→rrlvセミレプトニック崩壊の研究

1冊

公表論文

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→π セミレプトニック崩壊(非相対論的格子 QCDを用いた格子上の B

1冊の研究
共著者: ShojiHashimoto(橋本省二)， K
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参考論文

[
1
] HeavyQuarkE
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f
e
c
t
i
v
eTheoryandt
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eI
s
g
u
r
‑WiseF
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i
o
nont
h
eL
a
t
t
i
c
e
(格子上における重いクオークの有効理論と I
s
g
u
r
‑
W
i
s
e函 数 ) .
共著者: ShojiHasrumoto(橋本省二)
学会誌:P
r
o
g
r
e
s
so
fT
h
e
o
r
e
t
i
c
a
lP
h
y
s
i
c
sSupplement122(
1
9
9
6
)1
3
9
‑
1
4
4

[
2
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a
t
t
i
c
eheavyquarke
f
f
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o
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yandt
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‑Wisef
u
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c
t
i
o
n
(格子上の重いクオークの有効理論と I
s
g
u
r
‑
W
i
s
e函数

1冊

共著者: ShojiHasrumoto(橋本省二)
学会誌:P
h
y
s
i
c
a
lReviewD 56(
1
9
9
7
)7
0
2
8
‑
7
0
3
8

[
3
] fBw
i
t
hl
a
t
t
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eNRQCDi
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gO(l/~吋 ) c
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s

(O(l/m~) 補正を含む格子 NRQCD によるか

1 冊

共著者:S
h
o
j
iHasrumoto(橋本省二)， K
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n
‑
I
c
h
iI
s
h
i
k
a
w
a
(石川健一)，

TetsuyaOnogi(大野木哲也)， NorikazuYamada(山田憲和)
学会誌:P
h
y
s
i
c
a
lReviewD 56(
1
9
9
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1冊

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Study of B --+ 1rlv semileptonic decay
using lattice nonrelativistic QeD

Hideo Matsufuru

Department of Physics, Hiroshima University

Thesis,March 1998

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CONTENTS

1

Contents
1 INTRODUCTION
1.1 Lattice B Physics
1.2 B meson decays and basic theoretical ideas . . . . . . . . . . . .

2

The Lattice Simulations for the semileptonic decay form factors

6

1.3

2 LATTICE FORMULATION
2.1 Lattice NRQCD action .
2.2 Correlation functions ..

2
4

10
10

Perturbative corrections

12
14

3 SIMULATION DETAILS

18

2.3

Simulation parameters
Light-light meson . . . .
Heavy-light meson .. .
Three-point function and matrix elements
Form factors . . . . .
Chiral extrapolation . . . . . . . . . . . .

18
19
22

4 RESULTS AND PHYSICAL IMPLICATIONS

32

3.1
3.2
3.3
3.4
3.5
3.6

25

29
30

4.1

11mB dependence . . . . . . . . .

32

4.2

q2-dependence of the form factors
Soft pion theorem . . . . . . . . .

40

4.3

47

5 EXAMINATION OF SYSTEMATIC ERRORS

49

6

50

CONCLUSION

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INTRODUCTION

1.1

Lattice B Physics

INTRODUCTION

Study of the B meson systems, which contain single b-quark ( mass mb "'"' 4.1-4.5 Ge V), is referred as B physics, and is one of the most important subjects of current particle
physics. In the framework of the standard model, weak decays of B mesons are governedby the Cabibbo-Kobayashi-Maskawa (CKM) matrix, the unitary matrix which causes
the mixing among quark generations [1, 2]. Thus precise measurements of the CKM
matrix elements are necessary to test a validity of the standard model and to search a
new physics beyond the standard model [3]. Detection of CP violation in B 13 system
will give us detailed information on the complex phases of the CKM matrix, which
would shed light on the baryon asymmetry problem of the universe [3]. For this aim, B
Factories, dedicated accelerators to study B decay modes, are under construction andwill start to work near future [4].
To extract the CKM matrix element precisely from experimental data, it is crucial
to evaluate relevant hadronic matrix elements which suffer nonperturbative effect of thequantum chromo dynamics (QCD). On this point, recent progress of the heavy quark
effective theory (HQET) has much proceeded the understanding of the dynamics ofthe heavy-light systems [6]. However there still remains large uncertainties, since the
HQET predicts only relations among the matrix elements. The lattice simulation enables
us evaluation of these matrix elements incorporating the nonperturbative effect fromthe first principle. In fact, much effort has been payed for the lattice studies of the
weak matrix elements, as well as the mass spectrum and the decay constants of B
mesons. For example, on the decay constant of B meson, fE, current lattice results inthe quenched approximation agree within 20 %, and successive works with and without
the dynamical quarks will give more reliable answer [14]. On the other hand, calculationof the semileptonic form factors still remains a challenging subject, and requires moreworks with development of computational procedures [12,13, 14]. This work deals witha lattice calculation of the hadronic matrix element of the semileptonic weak decay
B -t- 7rlD (Fig. 1).
In principle, lattice simulations can compute any hadronic matrix elements fromthe first principle. There is, however, a problem in treating a heavy quark with mass
mQ on the lattice with currently available computer power. A typical lattice cutoff
accessible in current simulations, a- 1 "'"' 1 - 3 Ge V, is much less than the b-quark mass,
mb "'"' 4 Ge V. This causes unacceptably large systematic errors of order of amQ. Thus theearly lattice calculations of the B systems using the standard Wilson or clover actions
involved an extrapolation in the heavy quark mass from the charm quark mass regime

1.1

Lattice B Physics

3

l

Pp.

B

Figure 1: Semileptonic decay process B -t- rriD.
to the b-quark mass assuming a heavy quark mass scaling law, which could introduce
a potential systematic error. An alternative approach is to use effective theories, suchas the nonrelativistic QCD [24] and the Ferrnilab action [35J, in which large mass scaleof. order mQ is removed from the dynamics. These theories are not models but thesystematic expansions of the original QCD in a sense that one can reproduce the same
results as in QCD within a given order of, e.g. 1/mQ in NRQCD. In this work, we use
the lattice nonrelativistic QCD (NRQCD) up to O(l/mQ) corrections, thus it enables
direct simulations at the physical b-quark mass.
As an calculation of the weak decay form factors, this work is the first applicationof NRQCD. As mentioned above, previous calculations involve an extrapolation fromthe charm quark region to the B meson scale with a help of HQET scaling law. Since
this extrapolation leads potential uncertainties, direct simulation on the b-quark massis required to study the matrix element in detail. One of the goals of this exploratory
study is to test an applicability of NRQCD to these semileptonic processes and to clarify
the problems which should be considered in future calculations with higher statistics andimprovements. The procedures used in this work is easily applicable to calculations ofother processes, such as B -+ p and B -+ K·, which should also play important roles infuture B physics.
This thesis is organized as follows. The rest of this section contains a brief surveyof theoretical ideas concerning B -+ rr decays, and current lattice works on this process.
In the next section, we summarize the NRQCD formulation and numerical method to
calculate the matrix elements. The details of our numerical simulation is described in

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1 INTRODUCTION

Section 3. There we point out the subtleties in extracting the form factors: the definitionof the heavy meson energy, the choice of the two independent matrix elements to determine f+ and fa, and the procedure of chiral extrapolation. We study the uncertainties
due to the extraction and explain what is the best procedure. Physical implications of
numerical results are discussed in Section 4. We study 11mB dependence of the matrix
elements and q2 dependence of the form factors. The prediction from the soft pion theorem is compared with our data. In Section 5, we examine the systematic uncertainties
contained in this work. Section 6 is devoted to our conclusion.

1.2

B meson decays and basic theoretical ideas

In this subsection we briefiy summarize basic quantities and notions in B physics. We
also include ideas of heavy quark effective theory and heavy meson effective theory inthis subsection for later use.
Form factors The hadronic matrix elements of the B ---7 -rr semileptonic decay are
expressed in terms of two form factors f+ and fa as(1)
where qp. = Pp. - kp.- These two form factors correspond to the scalar and the vector
particles exchange modes respectively. The covariant normalization of the meson fields
is employed in this paper:

1.2

B meson decays and basic theoretical ideasof O((ap)2). In order to determine IVubl model independently, therefore, it is essential tocalculate f+(q2) in a q2 region where the experimental data will become available andthe systematic error does not spoil the reliability of a simulation.
Recently CLEO Collaboration reported the first measurement of this decay mode,
which gave the branching fraction of this process (1.8 ± 0.4 ± 0.3 ± 0.2) x 10- 4 [5].
Though current statistics of the experimental data (2.84 x 10 6 BE pairs) is not enoughto determine IVubl precisely, B Factories will provide enough statistics (lOB BB pairs).
Heavy quark effective theory Heavy quark effective theory (HQET) has muchadvanced our understanding of the heavy-light systems [6]. Especially in the case of
semileptonic decays B ---7 D, D*, six form factors are expressed in terms of single universal function known as the Isgur-Wise function. In a system containing single heavy
quark, the momentum transfer between the heavy quark and the light degrees of freedom are of order AQCD and much less than M Q , hence the velocity of the heavy quark isalmost unchanged. In the limit of mQ ---7 00, the dynamics does not depend on the fiavor(mass) and spin of the heavy quark ( 'heavy quark symmetry' ). HQET makes use ofthis fact to remove the large mass scale mQ from the dynamics and to incorporate thel/mQ power correction. The nonrelativistic QCD, which we use in this work to describe
the heavy quark, can realized as a special case of HQET with the vanishing heavy quarkclassical velocity.
In the case of B ---7 -rr decay, HQET predicts that the properly normalized matrix
element has a static limit, which depends only on the velocity of the heavy meson,

vJl. = pJl.lmB, and the mass and the momentum of the pion [7]:(-rr(k)IVJIB(p)) = e1(v . k)vJl. + e2 (v . k)~.

(2)

JmrrmB

From the condition that the matrix element is not singular at q2 = 0, the form factors
satisfy f+(O) = fO(O). The kinematical end point q!ax = (mB -m rr )2 corresponds to thezero recoil limit, where the lattice simulation works most efficiently.
For the massless lepton cases, f+(q2) dominates the decay rate, so that

(3)
where Vub is the Cabibbo-Kobayashi-Maskawa matrix element, G F the Fermi weak coupling constant, and(4)
The decay rate vanishes at q2 = q!ax' where the lattice simulation is most reliable. For thelarger momentum transfer the lattice calculation suffer from the larger systematic error

5

(5)

V· k

The finite mass correction enters into this expression as the power series of the inverse
heavy meson mass, 11mB. Evaluation of this correction is one of main topics of thiswork.
heavy meson effective theory In HQET, the fundamental degree of freedom is thatof the heavy quark. One can use, alternatively, the meson degrees of freedom such as B
and B* , in the same manner as in the chiral effective theory [8, 9, 10]. This heavy meson
effective theory, together with the chiral effective theory for the light mesons, can givethe form factors of various matrix elements with several phenomenological parameters.
Applying it to the case of B ---7 -rr leads [10]f±(q2) = ~ fB
2 f-rr[1 _fB·fB {A (1 + v2mb. k) + (AI _ A2)~}2m b(: . k~ mE)].
mb
q - mB·

(6)

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INTRODUCTION

where v . k = (m~ + m'; - q2)/2mB' and mb is the b-quark mass. A, AI, A2, and A are
phenomenological parameters to be fixed comparing with experiments. 1+ and 1 - aredefined as(7)
Eq. (6) implies that near the zero recoil point , dominant contribution to 1+(q2) comesfrom the B* pole. In Section 4.2, we discuss that to what extent this pole behavior isobserved in our numerical result. It is generally believed that the heavy meson theory
is applicable only in the region where the momentum transfer is order of AQcn , and notreliable in the hard scattering region.
Soft pion theorem In the chirallimit, in which the pion mass vanishes, the soft pion
theorem relate 1° at the zero recoil to the B meson decay constant, IB [9, 10, 11]:2) IB
f O( qmax= f7r'(8)

where 17r = 131 Me V is the pion decay constant. This relation should be satisfied if thenumerical simulations correctly reproduce the chirallimit. As mentioned below, we use
the standard Wilson action to describe the light quark, which breaks chiral symmetry
explicitly. Thus this relation plays an important role to examine proper incorporationof the chiral symmetry on the lattice. For the finite light quark mass case, with a helpof the heavy meson effective theory, the relation between 10( q!.J and IB becomes [10]

I

O( 2)
qmax

mB

IB

= mB + m7r f7r(9)This expression is helpful to argue the chiral extrapolation of the results obtained at
finite light quark masses.

1.3

The Lattice Simulations for the semileptonic decay form
factors; status and the present work

Here we briefly survey lattice works of semileptonic form factors which have been done
and are now under investigation and then state our simulation condition in comparison
with them. Several remarks on the notation used in this thesis is also given.
Status of lattice works Here we summarize the current status of works on thesemileptonic B -+(light meson) decays. Calculations of these processes, such as B -+7r P K* can be treated in almost same manner, while photodecay b -+ S'"" channels",
I
requires the values at q2 = 0 for physical applications. Table 1 summarize the works

1.3

The Lattice Simulations for the semileptonic decay form factors

7

performed previously and now under going. Until now all calculations are in the quenchedapproximation, in which virtual quark polarization effect is not included.
The early works include an extrapolation from the charm mass region to the b-quark
mass scale. ELC [15], APE [16], and UKQCD [19, 20, 21] Collaborations make use of
the HQET prediction, Eq. 5, to assume the extrapolation form. On the other hand,
Wuppertal group [22] first fit their results to several pole-type functions in the charm
region, then extrapolate I( q2 = 0) to the B meson mass with several functional form.
These extrapolation inevitably cause systematic uncertainties, thus the direct simulation
at b-quark mass is essentially important to get reliable answer.
To carry out direct simulations at b-quark with currently available computer power,
certain effective theory is called for. In this work, we describe the heavy quark using
NRQCD up to O(I/mQ). NRQCD has been applied to the heavy-heavy systems [28]and the heavy-light systems [14, 29], and obtained successful results .
An alternative approach is use of the clover action with realization as an effective
theory, i.e. the Fermilab action [35]. This procedure is employed by FNAL [12] andJLQCD [23] collaborations. Fermilab action is realized as the expansion of QCD in thequark's momentum, and its leading order and next-to-leading order expressions correspond to the Wilson and the clover actions respectively. An advantage of the Fermilab
action is applicability to an arbitrary mass values, so that it is convenient for the calculations around c-quark mass. On the other hand, further improvement beyond theclover action is much more difficult than in NRQCD, where O(I/m6) correction is easily
incorporated. Of course the results from these approaches should be consistent if thelattice simulations reliably reproduce the continuum physics.
The effective theory approaches are no longer necessary when sufficiently large computer is available to deal with a lattice cutoff much larger than mb. However, even if
such calculations are possible for quenched lattice, present lattice algorithms to simulate
dynamical quarks require further enormous computer power for such a lattice as large
as we could not reach in forthcoming decade. Therefore the use of effective theories
is essential to make use of the lattice results, at least with dynamical quarks, for the
analysis with experimental data from the B Factories. In addition, the effective theory
approaches enable us to carry out efficient calculations on rather small computers.
Summary of the lattice setup Now we turn to the summary of our lattice setup in
the present investigation. We calculate the matrix elements of B -+ 7r process using theheavy quark described by NRQCD up to O(I/mQ) terms and the Wilson light quark.
The simulation is carries out on a lattice of size 16 3 x 32 at f3 = 5.8, in the quenchedapproximation. The lattice cutoff at this f3 is a-l = 1. 7 Ge V, which is obtained using p

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8

INTRODUCTION
1.3

Group
ELC
APE
LANL
UKQCD
Wuppertal
FNAL
Hiroshima
JLQCD

f3
6.4
6.0
6.0
6.2
6.3
5.9
5.8
5.9

Actions (heavy-light)
Wilson-Wilson
clover-clover
Wilson-Wilson
clover-clover
Wilson-Wilson
clover-clover
NRQCD-Wilson
clover-clover

mQ

mq

me
me
rv me
rv me
rv me

chirallimit
chirallimit
chirallimit

I'V

rv

me, mb, 00
me-mb
mc-mb

rv

mil

chirallimitrv

milchiral limit
chirallimit

Refs.
[15]
[16,17][18][19, 20, 21]
[22]
[12]
this work[23]

Table 1: Current works on semileptonic decays of B -t(light meson). The results fromJLQCD Collaboration is preliminary.
meson mass. This lattice spacing is somewhat coarse compared with currently available
lattices, but convenient for an exploratory study at various masses and various momenta.
Instead the p meson mass, a-I from the string tension amounts a-I rv 1.4 Ge V, whichshows an scale ambiguity caused by the use of quenched approximation and the use ofrather coarse lattice.
We investigate the heavy quark mass dependence of the form factors, by taking themass of the heavy quark to cover a range of 1.5-8 Ge V. In the range around the charm
quark mass, O(l/m~) correction would be large. The use of the Wilson light quark
leads a O( aA qCD ) uncertainty, which is one of most significant systematic error of thissimulation. In future calculation, the clover action is called for to reduce this effectto O( a 2 A~cD).F To investigate the whole feature of the form factors, we measure morevalues of initial and final state momenta than previous works, which amount 20 pointsof q2.
To obtain the physical values of the matrix elements, we need two additional procedures, the renormalization correction and the chiral extrapolation. The former isperturbatively evaluated in Section 2.3. For reasons mentioned in Section 2.3, this is
only for a estimation of the size of the correction. The chiral extrapolation is discussed
at the end of Section 3. It is found that these two procedures are crucial to obtain
the physics results. Besides these, there are several subtle problems, which should beclarified in future calculations.

Notations

We denote the heavy-light and the light-light pseudoscalar mesons as B
and 7r mesons, respectively, regardless of actual masses of their heavy and light quarks
for simplicity. The momenta of B meson and pion is always expressed by the symbols p
and k respectively.

The Lattice Simulations for the semileptonic decay form factors

9

Without specifying, from now on we describe the all quantities in the lattice unit.
For example, the number '2.6' in the expression 'mQ = 2.6' is in the lattice unit anda-I must be multiplied to express in the physical unit. When we denote them in thephysical unit, the unit, e.g. 'Ge ~, is used explicitly. To convert lattice results into thephysical unit, we use the inverse lattice spacing a - I = 1.714(63) Ge V determined fromthe p meson mass on our lattice.

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2.1

LATTICE FORMULATIONIn this work, we employ the lattice NRQCD action including the O(l/mQ) terms 1

This section describes NRQCD formulation of the heavy quark, and how we evaluate thematrix element in the numerical simulation.
The lattice actions for the gauge field and the light quark field are most standard
ones [34]. The gauge field action is
Sgauge = f3

L:: (1 - ~ReTrUJ.Lv(x)) ,X,I.tV

SNRQCD1
1 Ho ) -n Q(x + 4) - (1 - 5H)Q(x) 1,= ~ Qt(x) [ (1 - 2nHo) -n U4 ( 1- 2n
(14)

where
__
1_~(2)

Ho

(10)

2mQ
1

3

5H =

where UJ.Lv(x) = UJ.L(x)Uv(x + f-t)US(x + v)U~(x). The coupling parameter f3
adjusts the lattice spacing implicitly.
The light quark is described with the Wilson action:
Slight11

Lattice NRQCD action= f3L::q(x)K(x,y)q(y),xy---O"·B
2mQ

(15)

'

(16)

'

and Q( x) is the effective two component spinor field, which describes the heavy quark.
~ (2) denotes the three dimensional Laplacian,(11)~(2)Q(x) = ~ [Ui(x)Q(x + i) + U}(x - i)Q(x - i) - 2Q(x)] .

( 17)

t

K(x,y) = 5xy - K.

L [(1 - 'J.L)UJ.L(x)5 +p.,y + (1 + IJ.L)U~(X - f-t)5 -p. ,y] ,
x

J.L

x

(12)

where I'\, is the hoping parameter which controls the light quark mass. For the light quark
field normalization, we adopt tadpole improved form /1 - 3K./4I'\,c [37], where I'\,c is thecritical hopping parameter which corresponds to the vanishing quark mass limit.
Present calculation is performed in the quenched approximation, in which the dynamical quark effect is neglected. In this case, the expectation value of an operator F[U]i& expressed as(F) =J'DUF[U]exp(-Sgauge).

0"is the Pauli matrices, and B is the standard clover-leaf type chromomagnetic field,

Bi = EijkFjk/2, with(18)
where P( x, f-Lv) means four plaquette which open at the site x and lie in (f-L, v) plane.
The operator I(M) make a matrix (-iM) hermitian and traceless:I(M) =(13)

In the simulation, this expectation values are evaluated on the configurations which are
generated in the weight exp( -Sgauge).

Lattice NRQCD action1)n u1 (1- 2nHo
1)n GQ(;v,t = 0) ,(1- 2nHo(1 The lattice NRQCD [24] has been extensively used for the investigations of the heavyheavy systems [28] and hadrons containing a single heavy quark [14, 29]. It is designed
to remove the large mass scale mQ from the theory using the l/mQ expansion and toreproduce the same results as of the relativistic QCD up to a given order of l / mQ. The
NRQCD action is derived by applying the Foldy-Wouthuysen-Tani transformation to theDirac spin or field. One of the best advantages of the NRQCD is that the l/mQ corrections can be easily incorporated. In fact , NRQCD including the O(I/m~) corrections
has been applied to the spectroscopy and the decay constant of the heavy-light system,
as well as to those of the quarkonium system.

(19 )This action generates the following evolution equation

GQ(;v,t = 1)

2.1

M - Mt1
2i- 3"Im[TrM].

2~ Ho ) u1 (1 - 2~ Ho) (1 - 5H) GQ(;v , t) .
n

n

(20)
(21 )

To avoid the singular behavior of high frequency modes in the evolution equation, thestabilizing parameter n is chosen to satisfy the condition 1
1- Ho/2nl < 1, which leads ton > 3/2mQ. From a viewpoint of the perturbation theory, further constraint, H o/ 2n < 1,is necessary to avoid singularities in some of vertices derived from the action (14). This
point is again discussed in the last part of this section in connection with our choice ofn in the simulation and the perturbative calculation.
IThis action differ from that we used in our previous study of fB [30]. which is organized to remove
O(aAQCD / mQ) error , at the cost of simulation speed.

.-

- .

.

-

-

-0

-.~.

_

-....

_

_.

.

_

~

•

_

.

•

_

.

,

0

•

,

_

"

•

•

.

.

2

LATTICE FORNIULATION

2.2

We apply the tadpole improvement procedure to this evolution equation according

and

12

to [36],u,..( x) -t U,..( x )/uo,(22)

where Uo is plaquette-defined mean field value of link variable, Uo = (TrUplaq /3)1/4 .
The four component spinor field 'lj;( x) of the heavy quark is expressed in terms of two
component spinor field Q( x) as

13

Correlation function s

ZB(p) = (010 B(0) IB(p) ),(28)

are their matrix elements.
In calculating Eq. (25) we vary t f with fixed ti and ts in order to find out the region
where the correlation functions are dominated by the ground state. The fixed ts is chosen
so that the pion two-point correlation function is dominated by the ground state, as is(23)

shown in Section 3.
Using the quark propagators, Eq. (25) is expressed as

t t f.) =C,..(3) ( p, k ;f,,,,'-iwhere Ll is the symmetric lattice covariant derivative ,~~ e-iP'~1 e-i(k-p).:l!/J~~(24)
O(l/mQ) correction appears in the lower component of 'lj;( x), which affects the heavylight current.

2.2

Correlation functions

We employ the standard simulation technique to calculate the hadronic matrix elementsof the semileptonic decay. We calculate the three-point correlation functions

C~3)(p,k;tf,ts,~) = LLe-iP'~/e-i(k-p) ' :l!6 (OIOB(:l!f,tf)VJ(:l!$)ts)O~(O,~) I O ),:l! fwhere spinor and color indices are suppressed. Sq is the light quark propagator, which
is obtained by solving the equation K(y, x)Sq(x, xd = ox ,x; with suitable iterative solver
algorithm. The heavy quark propagator, SQ, has 2 X 2 components in spinor space, hence
relevant components of Sq are selected .
The computational steps are as follows: (i) determine the light quark propagatorwith the source (O,ti), (ii) at the time slice ts, obtain b(x) by multiplying the current
matrix, heavy quark correction operator, and momentum transfer factor to the light(25)

quark propagator Sq(x,,)::l!.

b( x" ) -- e -i(k-p).:l!.

where OB and O-rr are the interpolating operators for Band 7r mesons, respectively, and(1 + 1 Dt)
--, .

2mQ

I,.. Sq ( x s , Xi ) ,(30 )V,.. = ql"''lj; is the heavy-light vector current. Figure 2 shows our setup of the source , the(iii) then using b(x s ) as the source, solve the heavy quark evolution equation, (iv) finallycurrent, and the sink operators.
As mentions in Section 1.3, we denote the heavy-light and the light-light pseudoscalarat the sink (:l! f' t f) combine the heavy with the light qaurk propagator and multiply

mesons as Band 7r, respectively, regardless of their mass parameters K and mQ for
simplicity. For tf » ts » ~ the correlation function Eq. (25) becomes

To obtain EqQ(p), E-rr (k), ZB(p) and Z-rr(k)' we also calculate the two-point correla-exp( -ip . :l! f) , and sum up.
tion functions with finite momentae1') (p; t j, til~ e-ip'" (OB (Xj )01 (x;)) -+ ~;~7~; exp( - E,Q(p) (t j - till ,(31)t

Z (k)2where EB(p) and E-rr(k) denote the energy of B meson and pion, respectively. The
exponent EqQ (p) is not the total energy but the binding energy of the B meson, because
the heavy quark mass mQ is subtracted in the NRQCD. We use the local interpolatingC~2) (k;tf,ti) = ~e-ik ' :l!/(07r(Xf)O~(Xi )) -t 2;-rr(k) exp(-E-rr(k)(t f - ti))'

Combining Eqs.(26), (31), and (32), one can easily see that the matrix element isexpressed as

operators for both of Band 7r,

(27)

(32)

f

.

. -

-.

- -..- . -..
~-- -. -

-

-. -

.

-

..

.. -

~

.

.

- .

14

2 LATTICE FORMULATIONq

Figure 2: The three-point correlation function to obtain the matrix element.
for tf » t3 » it, where Z = Z/-J2E. As expressed in Eq. (33), we use the twopoint correlation function itself to cancel the exponentially decaying factor of pion, while
One reason of thisasymmetric procedure is that the pion two-point function is constructed from the lightquark propagator with a point source at ti = 4, which is what we used to calculate thethree-point function (25), and then we expect the statistical fluctuation mostly cancels
between (25) and (32), while for the B meson exponential function, such a cancellation
is not expected. In addition, as we mention in the next section, the two-point correlation
function of B meson with the point source (31) requires larger time separation to reachthe plateau than the three-point function (25), for which the heavy quark source iseffectively 'smeared' at t3'

2.3

15

Perturbative corrections(mQ, n)A(5.0,1)(2.6,2)(2 .1 ,2)
(1.5,3)(1.2,3)(0.9 , 6)

0.07590.06680.06230.05280.04460.0309B
0.0124(4)0.0353(3)0.0449(3)0.0623(2)0.0757(1)0.0933(1 )

GV4

GVi

0.0210 (11)0.0004( 9)-0. 0068 (9)-0.0192(8)-0.0283(8)-0.0428(8)-0.0790(10)-0 .0 780(7)-0.0757(7)-0 .0 734(6)-0.0707(6)-0.0687(5)Table 2: The tadpole improved one-loop coefficients for the perturbative corrections Eo,
Zm, ZV4' and ZVi' Quoted errors represent the numerical uncertainties in the evaluationof loop integrals. The uncertainty of A is less than 10- 4 •Diriclet Boundary

use the values of EqQ obtained by fits to cancel the B meson's.

2.3

where q is the Wilson light quark and 'Ij; is defined in Eq. (23). Zv is the ratio of the
on-shell S-matrix elements in the continuum theory with MS scheme and that in thelattice theory. In our definition, Zv contains the leading logarithmi c term, cdog(mQa),
which comes from the continuum renormalization factor.
In calculating Zv we use the massless Wilson quark and the external momenta aretaken to be zero.

We did not take into account the one-loop operator mixing withhigher derivative operators, since there are already O( a) errors at tree-level from the
Wilson quark action. The one-loop coefficient is modified with the tadpole improvement[36]. For the mean link variable we use Uo = (TrUp1aQ /3)1/4 except for the light quark
wave function renorrnalization, for which we use another possible definition, Uo = 1/8",c[37]. Their one-loop perturbative expressions are used to determine the perturbative
coefficients of Zv:po .
The results for the one-loop coefficient G vpo in(35)

Perturbative corrections

Beyond the tree level, certain renormalization prescription is required to obtain a physicalquantity from the result of lattice calculation. We evaluate the perturbative corrections
to the current and the heavy quark mass, using the lattice perturbation theory at the
one-loop level.
To relate the matrix element in the lattice theory to that in the continuum QeD,
operator matching is required [26]. We have calculated the perturbative renormalization
factor ZVpo for the vector current [31]are presented in Table 2 for several values of (mQ, n). These values contain the leading
logarithmic contribution, log( mQa) / 47r 2 • The values of ZVpo with two choices of the latticecoupling constant g~(7r/a) = 2.19 and g~(1/a) = 3.80 are plotted as a function of 1/mQ
in Fig. 3. ( These coupling values are explained at the end of this section. ) We observethat the spatial component of the vector current receives larger perturbative corrections
than the temporal one. On the other hand, the l/mQ dependence is rather stronger for

Zv. than for Zv,.
(34)

When we discuss the l i mB dependence of the renormalized matrix elements in Sec-

-

.-

-

• -

-

~ -

,

•

'0'....

_.

_.

_..

_~

~.

_

_

_

.

~

~.

_

,

,

~

~

•

•

16

2

LATTICE FORlvIULATION

Perturbative corrections

17

The tadpole improved coefficients A and B are also given in Table 2. These coefficientsare determined by the numerical procedure proposed in [27].
For a historical reason , the stabilizing parameter we have used does not always satisfy

1.2
1.0

o

t he condition n > 3/mQ, which is necessary to avoid divergent tree level vertices, while
the simulation itself is stable with the condition n > 3/2mQ. We, therefore, quote theresults at tree level in the later sections as our main results. We estimate the size ofthe renormalization effect with the one-loop coefficients obtained with the combinationsof mQ and n, for which n's are larger than those we have used in the simulation andthe perturbation theory exists. Although this estimation is certainly incorrect, it gives
some idea for the one-loop effect, especially because the n-dependence of the simulation
results is observed to be very small (Section 3.4).
The values of coupling constant are determined as follows. Lepage and Mackenzie[36] pointed out that when we use appropriately defined coupling constant, the lattice
perturbatation theory works better than as it had been considered. According to them,
we adopt one of such definitions, av, which is defined with the expectation value of theplaquette using its two-loop perturbative expression ,

o

6

• •

••

•

0.8

~

N O.6
ZV4 (q~=rc/a)

0.4

o

•

ZvJ (q • =rc/a)

0.2

6

ZV4 (q =1/a)

A

ZVj

0.0

2.3

0.0

0 .2

0.6

0.4

0.8

(q'=1 /a)

1.0

1.2

1.4

1/mo
Figure 3: Renormalization constant for the vector current with two scales for the coupling constant, q* = 1r / a and 1/ a. The open and filled symbols represent Zv" and ZVj'
respectively.

/

2/11(PhYS)) _mB m B-(av(mB)( ( P h Y')) )
ay mB

3

3

(40)

In this work, we evaluate the perturbative corrections at two scales, 1r / a and 1/ a, to which
av is made run using the two-loop (3 function. Using our numerical value, (Up1aq /3) =tion 4, we multiply the leading logarithmic factor

8(

1
41r
-In( -Up1aq ) = -av(3.41 /a)[ 1 - 1.19av ].

(36)

to cancel the logarithmic divergence in the infinite heavy quark mass limit due to theanomalous dimension of the heavy-light current.
The calculation of heavy quark self-energy leads to the energy shift Eo, heavy quark
mass renormalization Zm, and the heavy quark field renormalization [25]. The firsttwo quantities appeared in the mass relation of heavy-light meson, and the last one isnecessary to calculate ZVJ." B meson mass is given through the binding energy of theheavy-light meson, EqQ(p = 0), as(37)
where Eo and Zm are obtained perturbatively

(38)
(39 )

0.567633(35), we obtain aV(1r/a) = 0.1742, and av(l/a) = 0.3026, which give above
val ues of

gJ.

.

~

-

.

- . - - .... -.

""-- -.. - - - -.....

- - --

"

.

.

.

-

..
"

.

. . . . ..

.

3

18

SIMULATION DETAILSSIMULATION DETAILS

3

Simulation parameters

The numerical simulations are performed on a 163 x 32 lattice with 120 quenched gauge
configurations generated with the standard plaquette gauge action at /3=5.8. Each configuration is separated by 2000 pseudo-heat-bath sweeps [38] after 20000 sweeps forthermalization and fixed to the Coulomb gauge [39]. The Wilson quark action is used
for the light quark at three K, values 0.1570, 0.1585 and 0.1600, which roughly lie in the -"range [m", 2m 3 ], and the critical hopping parameter is K,c=0.16346(7). The boundary
condition for the light quark is periodic and Dirichlet for spatial and temporal directions, respectively. We obtain the light quark propagator using the BiCGStab algorithm[40] with the Incomplete LV decomposition preconditioning [41]. The light quark field
3K,/4K,c according to [37]. The tadis normalized with the tadpole improved form
pole improvement is also applied for both the NRQCD action and the current operator
with the replacement of Up. ---t Up./ Uo using the average value of a single plaquette
Uo = (TrUp1aq /3)1/4 = 0.867994(13).
The lattice scale is determined from the p meson mass as a- 1=1.71(6) GeV, although
we expect a large O(a) error for mp with the unimproved Wilson fermion. The results
for the 7r and the p meson masses and the pion decay constant are summarized in Table

VI -

3.
The heavy quark mass mQ and the stabilizing parameter n used in our simulation= 0.1570
0.5677(30)0.6747(54)0.1496( 46)

K,

m-rr

0.15850.4933(33)0.6214(72)0.1380( 49)Light-light meson

19

are

In this section, we describe the numerical simulation in detail apart from discussions on
physical implications of the results , which will be discussed in the next section. Aftersummarizing the simulation parameters, two-point correlation functions of 7r and B
mesons with finite momenta are discussed. We describe how to extract the matrix
elements and the form factors from the three-point correlation functions. Finally, thechiral extrapolation of the matrix element is discussed .

3.1

3.2

0.16000.4118(37)0.567(11)0.1270(53)0.448 (17)0.1019(64)Table 3: The values of m-rr, m p , and pion decay constant without renorrnalization. Fitting
range is t = 14 - 24.

( r:Q) (5iO) ,( 2i6) ,( 2/ ) , ( 221 ) , ( \5 ) ,( 122 ) , ( \2 ) , ( °29) ,(41 )where mQ = 2.6 and 0.9 roughly correspond to b- and c-quark masses, respectively.
For mQ=2.1 and 1.2 we performed two sets of simulations with different values ofn, though the statistics is lower (=60) for (mQ,n) = (2.1,2) and (1.2,3). Since thedifferent choice of n introduces the different higher order terms in a in the evolution
equation, the choice of n should not affect the physical results for sufficiently small a.
The small dependence of the numerical results on n is also crucial for our estimation of
the perturbative corrections.
The spatial momentum of the B meson (p) and the pion (k) is taken up to Ipl,

Ikl ~.J3. 27r/16, which corresponds to the maximum momentum off"'V

1.2 GeV in thephysical unit. We measure the three-point correlation function at 20 different momentum
configurations (p, k) as listed in Table 4. The momentum configurations which areequivalent under the lattice rotational symmetry are averaged, and the number of such
equivalent sets are also shown in Table 4.
The light quark propagator is solved with a local source at ti=4, which is commonlyused for the two-point and three-point functions. The heavy-light vector current is placed
at t3 = 14, which is chosen so that the pion correlation function is dominated by theground state signal. The position of the B meson interpolating operator is varied in arange t f = 23 - 28, where we observe a good plateau as shown later.

3.2

Light-light meson

In order to obtain the form factors reliably, it is crucial to extract the ground state of
the B meson and the pion involving finite momentum properly. In Fig. 4 we show theeffective mass plot of pions with finite momentum at K, = 0.1570 and 0.1600. The spatial
momentum k = (k;x, ky, kz ) is understood with the unit of 27r /16. This notation will be
used throughout this paper. Although higher momentum states are rather noisy, we canobserve a plateau beyond t = 14. We fit the data with the single exponential function
to obtain the energy E-rr (k) shown by the horizontal solid lines in Fig. 4.
Figure 5 shows the energy momentum dispersion relation of pion, where the solidlines represent the relation in the continuum E-rr( k)2 = m; + k 2. We observe a small
discrepancy between the above relation and the data, which indicates the discretization
error. However the disagreement is about 1-1.5 standard deviation and only a few
percent.

3 SIMULATION DETAILS

20

3.2

Ligl1t-ligbt meson21"- = 0.1570

1.0
:t:

!

0.9
tq

1
2
3
4
5
6
7
8

k2

q2

° °

°

p2

°

1(~)

1
2
3
1
2

l( tt)l(tt )

°

1
2
3

1

9

2(~)

10
11
12
13
14
15
16
17
18
19
20

2
3
3
2

°1.)
1(
1

2(~)

2( tt)
2
2
3

°

4
3
1
2
6
2

3
1
4

°
2
6
3

P
( 0, 0,°)( 0, 0, 1 )(0,1,0)( 0, 0, 1 )( 0, 0, -1 )( 1, 0,( 0, 0, 1 )( 0, 0, 1 )( 0,0, -1 )( 0, 1, 1 )(1,1,0)( 0, 1, 1 )(0,1,-1)( 0, 1, 1 )( 1, 1,( 1, -1,( 1, 1, 1 )

°)

°°))

k

°)

( 0, 0,( 0, 0, 1 )( 0, 1, 1 )( 1, 1, 1 )( 0, 0,( 0, 0, 1 )( 0, 0, 1 )( 0, 0, 1 )( 0, 1, 1 )(0,1,1)( 1, 1, 1 )(1,1,1)°)°)( 0, 0,( 0, 0, 1 )( 0, 0, 1 )(0,1,1)(0,1,1)(0,1,1)( 0, 1, 1 )( 0, 0,°)-q = k - p~ (p, k)( 0, 0,( 0, 0, 1 )(0,1,1)(1,1,1)( 0, 0, -1 )( 0, -1, 1 )( 0, 0,(0,0,2)( - 1, 1, 1 )(0,1,0)(1,1,0)( 1, 1, 2 )( 0, -1, -1 )( -1, -1, 1 )( 0, - 1,( 0, 0, 2 )( 0, 0,(-1,0,1)(-1,2,1)( -1, -1, -1 )

1
6
12
8

°)

°)

°)
°)

6

24
6
2
24
24
24
8
12
24
24
4
12
48
16
8

Table 4: The momentum combinations (p, k) used in the simulation. In this table, thevalues of p, k, and q are expressed in the unit of 27f /16. The set which is equivalent withanother under the lattice rotational symmetry is identified with the same iq-number , anda representative is shown in the fifth through seventh columns . The last column shows
the numbers of equivalent combinations. The symbols in the third column denote thedirection of k against p as follows: ~: orthogonal, tt: parallel, tt : anti-parallel, andoblique for others. The set iq = 12 gives the minimum q2 value among the sets in thistable.

!

CD[Ii;:- 0.8
+
~
U

~
~

:J:

0.7

ill

x

:J:

! ! f
~

m

:J:

X

ill

CD

U

g; 0.6CDCD

CDo k=(O,O , O)

...

0.5

(0,0,1)(0,1,1)(1,1,1)

0

0.4

•

20

16

12

8

24

t
"- = 0.1600

1.0
0.9

III II ············· · · · · · ·· ·

0.8
+ 0.7

ill

~

U

~

U
OJ

m

~ ~ ~ ...........................
..~.:::.:::.: .. :::::[ .... p [

X

A

A

A

~

~

CD

~~
~. m
~._
..~
~~
. ..~~~~~~~-~.~_. _~..~_

4

0.6:~::::: ..:::::i:::::~,,::.:: .. :::::~ :::: ~

0.5

0

CD

0.4

o k=(O , O, O)
J.

0.3

o
•

0.2

8

(0,0, 1 )( 0, 1,1 )(1 , 1, 1)

12

16

20

24

t

Figure 4: Effective mass plot of pion at "- = 0.1570 and 0.1600 . The horizontal solidlines represent the fitted values and the fitting range with the statistical errors (dottedlines) .

3

22

SIMULATION DETAIL S

3.3

23

Heavy-light meson

mQ =

0.8

I

0.75

o K=0 .1570
0.15850.16006.
~

u¢.} .} T~ ~

---

.--.

C\J

....-...
.:::t:.

~

u

0.4

~ 111
1~

-........

en
0

I I 1
+++++

0.65

0.2
0.60
0.0

0.1

0.2

0.3

0.4

(0,1,1)( 1, 1, 1 )0\l¢4~~III+ 0.70

0.6

0.0
-0 .1

0 p = (0,0, 0)(0 , 0,1)
6.

4~ ! I

•

.,-

--w

2.6, K, = 0.1570

12

8

20

16

0.5

t

k2
mQ

Figure 5: Dispersion relation for pion. The solid lines represent the relation E;( k) =m; + k 2 with m7r the rest mas s obtained in the simulation. For K, = 0.1585 and 0.1600 ,symbols are slightly shifted in horizontal direction for clarity.

= 2.6, = 0.1600
K,

0 p = ( 0, 0, 0)( 0, 0,1 )
6.

4~ I

0.70

( 0,1, 1 )( 1 , 1, 1 )

0

\l

3.3

Heavy-light meson¢t ~ T~.,-+ 0.65~

To compute the B meson two-point correlation functions, we employ the smeared sourcefor heavy quark as well as the local source, with the local sink for both cases. The
smearing function for the heavy quark is obtained with a prior measurement of the wave
function with the local source. In Fig. 6 we plot the effective mass for both the locallocal and the smeared-local correlation functions at mQ = 2.6 and K, = 0.1 570, 0.1600.
The plateau is reached beyond t = 16 for the local-local, while the smeared-local exhibits
clear plateau from even earlier time slices.
We obtain the binding energy with a fit range [16,24] for both types of the correlation
functions and for all momenta, and the results are consistent in all cases. The binding
energy averaged over the results fitted from the local and the smeared sources are listed inTable 5 together with the values in the chirallimit. In Table 5, we also listed the binding
energy for the vector meson B· measured with the local-local correlation function , which
are used in later discussions on the B· pole contribution to the form factors. It is also
worth to note that the values of EqQ obtained with different stabilizing parameter n isconsistent with each other within their statistical errors .

24

U::::::..
~

u

Ol

o

0.60

0.55

;
7
7

·::::II1
8

12

16
t

7
[

[P

-'

(P (
L

.

[0 I
If!

" ::1'::-<D( D "'i'"

~

LP.

(~
~

-: --

7

r

... ..
~C~.......

20

24

Figure 6: Effective mass plot of B meson at mQ = 2.6 and", = 0.1570 , 0.1600. R esultswith the smeared source (filled symbols) are shown for Ipl2 = 0,1 as well as the resultswith the local source (open symbols). The horizontal solid lines express t h e averagevalues over the results of single exponential fit of the local-local and the smeared-local
correlation functions. The statistical errors of the fitted values are displayed at t he ri ght
end of the lines . For all mQ , "' , and momentum , the fi t ranges are set to t = 16 - 24.

3

24

SIMULATION DETAILS

3.4

Three-point function and matrix elements

Pseudoscalar meson binding energy: EqQ(p = 0)

= 0.1570(mQ, n)K,(5.0,1)(2.6,1)(2.1,1)(2.1 ,2)(1.5,2)(1.2,2)(1.2,3)(0.9,2)0.6304(69)0.6268(48)0.6247(45)0.6279(53)0.6180(42)0.6135( 40)
0.6142(51)0.6058(39)0.15850.6084(83)0.6041(56)0.6014(52)0.6056(62)0.5940(48)0.5889( 46)
0.5899(56)0.5805(43)0.16000.585 (11)0.5809(71)0.5777(65)0.5834(80)0.5696(59)0.5640(56)0.5655(68)0.5551(51)

25

0.75
I\, c

o ma =2.6 , K=O .1570
0 ma =2.6, K=O.1600

0.535 (15)0.530 (10)0.5260(91)0.534 (11)0.5162(81)0.5095(75)0.5117(92)0.4991(68)

0.70

0::

0.65
---w
CD

0.60

Vector meson binding energy: EqQ' (p = 0)(mQ, n)K,(5.0,1)(2.6,1)(2.1,1)(1.5 ,2)(1.2,2)(0.9 ,2)0.649 (12)0.6502 (62)0.6501 (56)0.6488 (52)0.6484 (51)0.6470 (50)

= 0.1570

0.15850.628 (14)0.6287 (76)0.6279 (68)0.6257 (61)0.6249 (59)0.6231 (57)0.16000.604 (19)0.6065 (99)0.6047 (88)0.6014 (79)0.6002 (76)0.5982 (73)0.55

0.555 (27)0.559 (14)0.555 (13)0.550 (11)0.547 (11)0.545 (10)

0.0

0.1

0.2

0.3

0.4

0.5

2

P

Figure 7: Dispersion relation for the B meson at mQ = 2.6 and K, = 0.1570, 0.1600. The
solid lines represent the relation EqQ(p) = EqQ(O) +p2 /2mB, for which mB is determinedwith the tree level formula mB = mQ + EgQ(O). Dashed lines represent the same relation
with the renormalized mB at the scale q* = 1/ a.

Table 5: The binding energy of the pseudoscalar and vector heavy-light mesons. The
single exponential fit is applied with the fitting range t = 16 - 24. For the pseudoscalar
we average the values obtained from the local-local and the smeared-local correlation
functions. For the vector mesons we use the local-local only, and there is no data availablefor (mQ,n) = (2.1,2) and (1.2,3).
The dispersion relation for the B meson takes the following nonrelativistic form

(42)

where the kinetic mass mkin should agree with the rest mass mB (37) in the continuum
limit. Since we use the NRQCD action correct up to O(I/mQ), including higher orderterms in l/mB in Eq. (42) does not make sense. In Fig. 7 EgQ (p) is shown as a functionof p2 at mQ = 2.6. The solid lines represent the relation (42) with mkin = mB determinedthrough the tree level relation mE = mQ + EijQ(O), which reproduce the data quite well.
With the one-loop correction (37) the agreement becomes even better as presented withthe dashed lines in the figure.

3.4

Three-point function and matrix elements

Figure 8 is the effective mass plot of the three-point function at mQ = 2.6 and K, = 0.1570,
0.1600. The horizontal axis represents the time slice on which the B meson interpolating
operator is put, and the vertical axis corresponds to the binding energy of the B meson.
The horizontal solid lines represent the binding energy EqQ (p) determined from the twopoint correlation functions. The figures display that the three-point correlation functions
are dominated by the ground states beyond t = 23, and there they give the consistentvalues for EqQ (p) with ones extracted from the two-point functions. Therefore, in thisregion we can use Eq. (33) together with the results of the two-point correlation functions
to extract the matrix elements.

It is useful to define the quantity V~ as(43 )because it is defined only through the residue of the two- and three-point correlation
functions without the knowledge how one defines the meson energies. Since there areuncertainties in the light-light and heavy-light meson dispersion relations, it is better to

3

26

1,q

q2

i'4

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20

7.071 (20)
6.280 (19)5.609 (19)5.017 (18)7.044 (20)
6.247 (19)6.555 (19)5.938 (19)5.571 (19)5.880 (19)5.283 (18)4.666 (18)7.017 (20)
6.214 (19)6.523 (19)5.534 (19)6.151 (19)5.842 (19)5.225 (19)6.990 (20)1.014 (34)0.844 (26)0.754 (50)0.612 (87)0.999 (36)0.832 (28)0.930 (30)0.750 (34)0.742 (49)0.827 (55)
0.66 (10)0.544 (68)0.992 (42)0.825 (34)0.923 (38)0.757 (76)0.920 (67)0.820 (58)0.669 (52)0.968 (58)SIMULATION DETAILS

3.4

x

Up

mQ = 2.6, K, = 0.1570,

Uk
0.878 (41)0.695 (61)0.57 (10)0.0475(28)0.0366( 4 7)
1.009 (46)-0.702 (48)0.040 (12)0.790 (68)
0.65 (12)-0.39 (12)0.0467(30)0.0360( 48)
0.517 (26)0.052 (53)0.863 (77)0.412 (36)-0.266 (41)0.0454 (33)27

Three-point function and matrix elementso p=(O ,O,O) , k=(O,O,O)
0.70

•

o

0.860 (41)1.009 (46)0.702 (48)0.674 (59)0.767 (66)
0.63 (11)0.477 (82)

6 0.65m

0::::.
.t:...
0

ill

Ol

m ~

0

I

... , ...

~__ .J •••

mmill--.- ...

20

22

~

~

m

I

m

0.848 (45)0.997 (51)0.670 (82)0.863 (77)0.758 (68)0.587 (61)

rn

ill

0.60

0.55

ID

14

m
16

18

24

(3)

o p=(O,O,O), k=(O,O,1)
0.70

•

~

Z 0.650I!

2:deal with the quantity which is free from the ambiguity. Moreover, Vt-' is the quantity
which has the infinite mass limit in the heavy quark effective theory. When the perturbative correction is incorporated, e(mBlm~h1')) given by Eq. (36) is multiplied to Vw
Therefore V~ is suitable quantity to study the 1/mQ dependence.
For the spatial components of V~, we also define the scalar products

~

Ol

0

0.60

0.55

14

C (3)

p=(O,O,-1), k=(O ,O,1)
p=(1,1,O), k=(O,O ,1).,.....

0

~

~ ~

4

~ ~ ~

. __....

! t ! ! !
~ ~

~ ~ 2

18

20

••. 1. _-. .........

~

16

22

24

t

r\(p,k) =k· V(p,k)k2

28mQ = 2.6, K, = 0.1570, 0 4

Table 6: V4 , Up, and Uk in the lattice unit at mQ = 2.6 and K, = 0.1570. iq denotes theset of momentum (p, k) summarized in Table 4. In the evaluation of q2, the B meson
mass is determined through the tree level relation mB = mQ + EqQ(O).

p. V(p,k)
,
p2

26

t

o

Up( p , k) =

)

4

rn
m m
I

3

C (3)

p=(O ,O,1) , k=(O,O,O)
p=(O,1 ,1), k=(O,O,O).,.....

ci(44)

In Table 6 we list the values of 114 , Up, and Uk for all momentum configurations (p, k)
at mQ = 2.6 and K, = 0.1570. In this table, we also list the values of q2 determined withthe tree level mass relation (37) for the B meson.

Figure 8: ( continued. )

26

28

3

28

rnQ

o
0.70

•

o

= 2.6,/'i,

SIlVIULATION DETAILS

p=(O,O, O), k=(O,O,1)
p= (O,O,-1), k=(O,O,1)
p=(1,1, O), k=(O,O,1)

kC(3)6 0.65
o

z-

o

CJ)

o

0.60

m

t
0.55

16

14

m
!

!! ~

3.5

~ ~
~

ill
20

18

24

22

26

28

t
mQ = 2.6, /'i, = 0.1600,
0

•

0.65

0

60

m

0.60
ill

:::::::..
~

0
CJ)

0

ill

0.55

~

~

ID
~

~

ill

I

14

16

~

~

I

ill ill

ill

(46)

ill

and f +(q2) is similarly obtained from (p + k)J.LVJ.L substituting the value of fa determinedID

18

To convert 114, Up , and Uk to the form factors, we need to assume certain dispersion
relations for EB(p) and E-rr(k). One method is to use the values obtained from thedispersion relation measured in the simulation. This, however, suffers from the large
statistical error for the finite spatial momenta. Alternatively, we adopt the following
relativistic dispersion relations for both the B meson and the pion.

where the measured rest mass is used for m-rr and mB' These relations well reproduce
the measured data as shown in Figs. 5 and 7 for light-light and heavy-light mesons ,respectively.
U sing the relations Eq. (45), the form factors are easily constructed from Vw First,
we calculate fO( q2) with

m ID
0.50

Form fact ors

)

4

ill m
ill m

29

(45)
3

C (3)

p=(O ,O,O), k=(O ,O,O)
p=(O ,O,1) , k=(O,O,O)
p=(O ,1,1) , k~(O,O,O)r-ci

Form factors

We have investigated the n-dependence of VJ.L at mQ = 2.1 with n = 1 and 2 and at
rnQ = 1.2 with n = 2 and 3, using the first 60 configurations on which (mQ,n) = (2.1,2)
and (1.2,3) data are measured 2 • For both of the heavy quark masses we observed small
dependence on n, which is at most 1%, 8% and 2% for V41 Up and Uk respectively, andsmaller than their statistical error. In the present work, therefore, we regard them tobe sufficiently small to estimate the size of the renormalization effect in the manner
described in Section 2.3.

= 0.1570, k·C (3)

or-

3.5

20

22

24

26

28

t

above.
For p =I 0 and k =I 0, fa and f+ are not uniquely determined from \%, Up, and Uk.
In this case there is an additional relation among VI-' 's, which should be satisfied whenthe Lorentz symmetry is restored. For p 1.. k this relation reads(47)
Figure 8: Effective mass plot for the three-point functions at rnQ = 2.6 and /'i, = 0.1570.
The horizontal lines express the values obtained from the two-point correlation functions.
Top and middle figures are for C~3), and bottom figure is for k . C(3).

2We note that n -dependence should be studied on the same configurations. In some of the figures,
there appear large deviations for the data with different n but the same mQ. However , in these graphs
only the results for (mQ,n) = (2.1,2),(1.2,3) are obtained from the first 60 configurations and theresults for the other combinations of (mQ, n) are obtained from the entire 120 configurations. It seems
that these large deviations see m to arise from the statistical fluctuation caused by the remaining 60
configurations for which there is no the data with (mQ , n) = (2.1,2), (1.2,3) .

330SIMULATION DETAILS

3.6

31

Chiral extrapolation

1.0

3.0

0.8
CJ)
.....

~

___ 0.6(l)0...
..........

(l)

v-

> 0.4

x

K=0.1S70, Treep=(0,1 ,0), k=(0,0,1)

~

cr;

0.0

2.0
ma =2.6, Tree

E

0.2

o v4
o v

(p=O,k=O)(p=O,k=1)
4.. kVI!k!2 (p=O ,k= 1)

1.5

0.0

0.2

0.4

0.6

0.8

6.10

11mB
Figure 9: Comparison of

6.20

6.30

6.40

11K

f4 to (EBUp + E7r Uk ) for iq = 6 at = 0.1570.
I),

We examine this condition for iq = 6,9,14 and 16 (iq is referred in Table 4). Figure
9 compares LHS and RHS of Eq. (47) at 11, = 0.1570 for i q =6, with the tree level
dispersion relation for E B . This figure exhibits a difference of about 15% . In other casesof i q , similar amount of the discrepancy is observed. The size of this systematic effect isFigure 10: Chiral extrapolation of the matrix elements for mQ = 2.6. V4 and k . V /J kJ2
are shown for two momentum configurations iq = I, 2. The solid and the dashed lines
represent the linear fit.
theory gives the following relations for the matrix element and the form factor:(48)

consistent with the naive expectation for O( a) error.

3.6

2.5

E

.::s:.

Chiral extrapolation

To obtain the form factors at the physical pion and B meson masses, it is necessary
to extrapolate the results to the chiral limit. There is, however, still a subtlety in thechiral extrapolation, because the light quark mass dependence of the matrix elementsor the form factors are not well understood. In principle, the chirallimit of the matrix
elements or the form factors must be taken using the result of the chiral effective theory
as a guide for its functional form. For the B -+ 7r semileptonic decay the heavy meson
effective theory with chiral Lagrangian gives such an example [8,9, 10j.
At least the heavy meson effective theories tell us that the matrix elements or theform factors depend on v . k, where vJ.l is the 4-velocity of the B meson. At the zeropion momentum, the quantity v . k could potentially give linear dependence in m which
could result in a ~ dependence. The zero recoil limit in the heavy meson effective
7r ,Assuming the linear dependence of fB' f7r' and mB on m q , at least in the zero recoil limit
the matrix element should have linear dependence on m q • In the following analysis, wetake the chiral limit of the matrix elements assuming the linear dependence on mq inany case of (p, k), although there is no proof.
Figure 10 shows the chiral extrapolation of the matrix element with the form
(49)
where mq = 1/21), - 1/211,c. The data itself do not show any sign of nonlinear behavior atleast around the strange quark mass. The form factors f+ (q2) and fO( q2) at the physicalpion mass are extracted after extrapolating the matrix elements to the chirallimit using
Eq .(49).

4 RESULTS AND PHYSICAL INIPLICATIONS

32

4

4.1

33

11mB dependence

RESULTS AND PHYSICAL IMPLICATIONSIn this section we discuss the physical implications of our results, which include the 11mB
dependence of the B -7 11" matrix elements and the q2 dependence of the form factors.

1.2

K=0 .1570, Tree

The prediction from the soft pion theorem is compared with our data.
-------

1.0

4.1---- ---

11mB dependence:Q' 0.8

The heavy quark effective theory predicts that the properly normalized B -7 11" matrix
element has a static limit, hence it can be described by an expansion in the inverse
heavy meson mass 11mB whose leading order is a function of the heavy meson velocity

~

a

II

-->

0...

0.6

f

v

0.4

0

...

vJ.l = pJ.llmB,

0 .2\1I"(k)IVJIB(p)) =()l(v.k)v +()2(v.k) kJ.l.
y'm m B

J.l

7r

0

.,.

(50)

v .k

k=(O,O,O)(0,0,1)(0,1,1)(1,1,1)0.0
0.0

Similar arguments for the heavy-light decay constant suggested that the quantity IB y'm.B
has the static limit while numerical simulations have shown that the 11mB correction
is very large. On the other hand, the 11mB dependence of the form factors have beenstudied only in the D meson region. Therefore it is important to study the 11mB
dependence of the matrix elements at fixed values of v . k.
Except for p = 0, fixing p is not quite identical to fixing v . k, since the velocity
vJ.l changes depending on the heavy meson mass . Thus it is awkward to use the matrix
elements with nonzero p. In the special case of p = 0, LHS of Eq. (50) is nothing but
the matrix elements~, Up and Uk, defined in Eqs . (43), and (44), multiplied by the mB

0.2

0.4

0.6

0.8

11mB
Figure 11: "C4 at K- = 0.1570. For k = 0 and Ik l = 1, the solid and the dashed lines
represent the results of linear and quadratic fits, respectively.

independent factor.
In the following analysis, we confine ourselves to examine the following quantities for

1<:=0.1570, Tree

1.0

the sake of simplicity:
0.8

V4(O)

(

(h(p = 0, k)

U~O)

(

Up(p = 0, k)

lim Up(p, k)
p 2 -to1.
("C4(p = 0, k)

(1)

(2)

mB

mB

)

1 + ~ + C4 2 + . . .

,

(51 )

:Q'
a 0.6
II

(1)

1 + ~ + C 2 + . ..
mB

_U/(O)

mB

(2)

p

k

--0...

)

(52)

mB

. :£

:::J 0.4... k=(0,0,1 )

( 53)
0.2
e(l)

e(2)

l+-P-+~+ ...
mB
mB

)

,

(54 )

for which we explicitly show the form of the 11mB expansion. All of the coefficients inthese expansions are a function of k.

0

.,.

0.0
0.0

(0,1,1)(1,1,1)
0.2

0.4

0.6

0.8

11m B
Figure 12: Uk at K- = 0.1570 . The solid and the dashed lines represent the results oflinear and quadratic fits, respectively.

4 RESULTS AND PHYSICAL IMPLICATIONS34

In Figs. 11 and 12 we show the 11mB dependence of 114 and Uk, respectively, at K, =0.1570. The limB correction is not significant for these quantities and almost negligible
around the B meson mass. This result exhibits a sharp contrast to the mass dependenceof the heavy-light decay constant fBJmB' for which the large 11mB correction to thestatic limit is observed. Results of the linear and quadratic fit in limB are listed inTable 7 for V4 and in Table 8 for Uk.

quadratic

linearK,~q0.1570

1
2
3
4
1
2
3
4
1
2
3
4

0.15850.1600• (0)~0.965(35)0.826(29)0.757(51)0.624(80)0.982( 42)0.807(35)0.758(76)0.62 (12)1.003(53)0.768( 46)
0.78 (14)0.70 (27)(1)

C4

0.184(55)0.080( 47)-0.038(59)-0.25 (11)0.165(63)0.075(57)-0.071(73)-0.40 (15)0.150(76)0.088(76)-0.17 (10)-0.64 (25)

• (0)

~
1.003( 4 7)
0.851(41)0.799(57)
0.79 (10)1.016(55)0.830( 48)
0.830(81)0.89 (19)1.023(66)0.788(58)0.96 (17)1.22 (55)

(1 )

(2)

C4

C4

-0.01(20)-0.06(17)-0.30(20)-1.29(36)-0.00(23)-0 .06 (20)-0.51(26)-1.83(50)0.05(27)-0.04(26)-1.13( 40)-2.45(80)0.21(18)
0.15(17)0.31(22)1.25(42)0.18(21)0.14(19)0.51(29)
1.75(60)
0.10(25)0.14(25 )1.13( 46)
. 2.26(94)4.111mB dependence

We note here that x2 /dof are less than unity for most cases of "\14, Uk, and also Up,
which will be mentioned in the next paragraph, though they do not exactly judge thegoodness of the fits for such data, which are correlated for different mQ.
In order to do the same discussion for Up, which is defined in the p2 ~ 0 limit, we
extrapolate the finite p results to the vanishing p point as shown in Fig. 13. There islittle p2 dependence observed and we employ a linear extrapolation in p2. In Fig. 14 weplot mBUp as a function of limB at K, = 0.1570. In contrary to the other matrix elements
we observe a sizable limB dependence. Table 9 summarizes the results of linear andquadratic fit of mBUp •Here we briefly discuss the effect of one-loop correction to these quantities. Figure
15 shows the renormalized values of V4(iq = 1), Uk(i q = 2), and mBUp(iq = 1) atK, =0.1570. As mentioned at the end of Section 2, the leading logarithmic factor ismultiplied to Vw As mentioned at the end of Section 2, the leading logarithmic factor
Eq. (36) is multiplied to V/-L' We also list the results of linear fits of the~ ip._T3:bl~ 10. A_~.
we discussed previously, the 1/mQ dependence of the one-loop coefficient is significant

0.12
K=0.1570
0.10

Table 7: Parameters for the linear and quadratic :fits of V4 (p = 0, k).

K,

0.15700.15850.1600

• (0)

0.
a.

quadratic

'l,q

Uk

(1)
Ck

2
3
4
2
3
4
2
3
4

0.945(39)0.762(56)0.655(88)1.004(52)0.808(92)0.72 (15)1.064(73)0.92 (20)0.94 (37)-0.194(44)-0.257(53)-0.364(91)-0.198(50)-0.242(64)-0.34 (14)-0 .214( 62)-0.219(90)-0.23 (26)

• (0)

(2)

(1)

Uk

Ck

ck

0.967( 47)
0.750(54)0.600(81 )1.023(58)0.769(80)0.58 (12)1.063(77)0.80 (16)0.55 (23)-0.30(19)-0.17(22)0.08(43)-0.28(22)0.00(30)0.77(74)-0.21 (29)0.47(50)
3.3 (23)

0.13(19)-0.10(24)- 0.54( 49)0.10(23)-0 .29( 32)-1.34(80)0.00(30)-0 .81(53)-4.1 (24)

Table 8: Parameters for the linear and quadratic :fits of Uk(p = 0, k).

----::J

0.06
0.04

0

6-

I

f

~

~

t

1

1;

f
!

~

~

~

~

r

0.08

?
linear35rk=(O,O,O)
k=(O,O,1 )

!

I

I

mo=1.5~mo=2.6

0.02
0.00
0.0

0.1

0.2

0.3

0.4

0.5

2

P

Figure 13: Extraction of Up(p = 0, k) is shown for mQ = 2.6 and 1.5 at K, = 0.1570. The
extrapolation is carried out linearly in p2. For k = 0, iq = 5,13,20 are used. For k = 1,
Up(O, k) is determined using iq = 6,14, for which p and k are perpendicular.

4 RESULTS AND PHYSICAL IMPLICATIONS

36

4.1

limB dependence

37

only for ~ and almost negligible for Vi. As a result, the limB dependence of ~ is largely
affected by the renormalization effect, and it even changes the sign of the slope in 11m B.
The limB dependence of ~ is still mild after the renormalization effect is included. For
Uk and mBUp the 11mB dependence is not affected by the one-loop correction, while
1.2

their amplitudes decrease by at most 30%.

1.0

K=0.1570, k=O

0.4

o Tree... q.=n/ao q =1/a

0.2

0.0

0.0

0.2

0.6

0.8

K=0.1570, k=(0,0,1)

1.0
0 .8

Figure 14: Up multiplied by mB at K, = 0.1570 as a function of limB' The values of mB
are determined with the tree level formula. The solid and the dashed lines represent thelinear and the quadratic fits, respectively.

linearc(1)(;,(0)

quadratic
c(1)

0 .2

c(2)

fi,

2q

(1'(0)
P

P

P

P

p

0.1570

1
2
1
2
1
2

0.0887(80)0.089 (14)0.0872(94)0.093 (20)0.088 (12)0.104 (33)

2.61(39)
1.29(38)2.65(47)0.98( 42)2.72(59)
0.67(47)

0.0717(95)0.072 (13)0.066 (11)0.080 (18)0.059 (15)0.097 (27)

4.5(12)2.7(11)
5.3(17)2.0(12)6.7(27)1.1(15)-1.55(76)-1.31(88)-2.1 (10)- 1.0 (11)
-3 .1 (17)-0.4 (15)0.15850.1600o Tree... q.=n/ao q =1/a

Table 9: Parameters for the linear and quadratic fits of Up(p = 0, k).

0.0~~~~~~~~~~~~~~~~

0.0

0.2

0.4

11m B

Figure 15: ( continued. )

0.6

0.8

~

•

_ .

.....

•

-

-~

•

_

.

_

•

•

r

4

38

•

•

•

•

RESULTS AND PHYSICAL IMPLICATIONS

4.1

39

11mB dependencelI4(p = O,h = 0) ( iq = 1 )q* = 1/ aq* = 7f / a

K=0.1570, k=Oo TreeJ. q'=nla

0.25

0--. 0.20:::t:.
0IIa.
----...-

(1 )

K,

Vt(O)

0.15700.15850.16001.002(36)1.019( 44)1.039(55)C40.052(55)0.039(63)0.030(77)C4

1.088(39)1.105( 46)
1.126(58)-0.209( 47)-0.216(55)-0.219(66)[h(p = 0, Ihl = 1) ( iq = 2 )q* = 1/ aq* = 7f / aq'=1/a

(0)

A

/Ii,

0.15700.15850.1600

0.15

a.

:::>

OJ

E 0.10

(1 )

(0)

A

(1)

Uk

Ck

Uk

Ck

0.732(31)0.778(42)0.826(59)0.013( 61)
0.005(68)-0.019(84)0.609(27)0.649(36)0.689(50)0.081(70)0.070(78)0.043(96)Up(p = 0, h = 0) ( iq = 1 )0.05

q* = 1/ aq* = 7f / a

0.00

(1 )

V}O)

0.0

0.2

0.4

0.6

0.8

11mB
Figure 15: One-loop renormalized V4 , [h, and Up as a function of 1/ mB. The solid, thedashed, and the long dashed lines represent the linear fits.

A

(0)

K,

Up

0.15700.15850.16000.0466( 66)
0.0453(77)0.045 (10)

c(1)
p

(0)
Up

c(1)

6.3(12)6.5(15)6.7(19)0.0268(58)0.0256(68)0.0248(87)11.3(30)
11.8(38)12.5(53)

A

p

Table 10: Parameters for the linear fit of the renormalized matrix elements lI4(p = 0, h =0), (h(p = 0, Ihl = 1), and Up(p = 0, h = 0).

4 RESULTS AND PHYSICAL IMPLICATIONS

40

4.2

4.2

q2-dependence of the form factors

41

q2-dependence of the form factors

First we study for which q2 region our present statistics allow us to compute the form
factors with reasonable statistical errors. The q2 dependence of the form factors f+ andf O are shown in Figs. 16 and 17 at mQ=2.6 and 1.5, respectively. We find that for
K, = 0.1570(mq '" 2m.,), the range of q2 in which the form factors have good signal covers
almost the entire kinematic region for D meson and one third of the kinematic region for

B meson. For K, = 0.1600 (m q rv m a ), the signal becomes much noisier, but still the form
factors have marginally good signal for half and one fourth of the kinematic region forD meson and B meson, respectively. Although our present results are very noisy after
the chiral extrapolation, this will be improved by future high statistics studies. This isencouraging in view of the fact that the future B Factories can produce 10 8 B-B pairs
and the branching fraction of B -7 7rlv from CLEO is (1.8 ± 0.4 ± 0.3 ± 0.2) x 10 - 4 [5] .
It is reasonable to expect that there is a possibility of observing B -7 7rlv events in theq2 regime which the present lattice calculation can cope with.
Secondly we study the q2 dependence to see whether the contribution from the B*
resonance to the form factor can actually be observed in the simulation data. At the chirallimit, unfortunately, the results are too noisy to discuss their q2 dependence, therefore
we use the finite mass results only in the following analysis of the q2 dependence. As
shown in Figs. 16 and 17, the lattice results are available only in the large q2 region, at
which the recoil momentum of pion is small enough. Therefore it is justified to express
the functional form of the form factors by an expansion around the zero recoil limit. For
this purpose we use the inverse form factors 1/ f+(q2) and 1/ fO(q2):

2.0NFigure 18 shows the inverse form factors at mQ = 2.6 as well as their fitted functions
with this form. The numerical results of the fit with and without the condition C2 = 0
are given in Table 11 for mQ = 2.6, 1.5, and 0.9.
The pole dominance model corresponds to a special case C2 = 0, which seems todescribe the data very well as shown in Fig. 18. The mass of the intermediate state

cr
'--'"

0

fO (q2 )0

t (q2)

1.5

- 1.0
0.5
0.0

0

2.0

-

cr

10
15
2
q [ GeV 2 ]

5

20

25

20

25

ma=2 .6, 1<:=0.1600. Tree

2.5

N

(55 )

mQ =2 .6, 1<:=0.1570. Tree

2.5ofO(q2)ot (q2)

1.5

'--'"

1.0
0.5
0.0

0

5

is given by m;ole = q!ax + 1/ (Cl f+ (q!aJ), which corresponds to the vector (B*) meson
mass in the pole dominance model. Precisely speaking, the more consistent analysis is to
impose the condition m po1e = mE- for the fit by Eq. (55). This constrained fit is shownwith the long dashed line in Fig. 18. It is found that now the fit do not quite agree withthe data, but the deviation is about 10 %.
In Fig. 19 we also compare m po1e and the measured vector meson mass as a function ofmE· Again we find that there is a discrepancy between mpole from the unconstrained fit

Figure 16: Form factors at mQ = 2.6 and K, = 0.1570, 0.1600. The solid curves represent
the fit to single pole functions.

4

42

RESULTS AND PHYSICAL IMPLICATIONS

4.2

43

q2-dependence of the form factors

2.5
2.5

2.0

mo =1.5 , K=O .1570, Tree

2.0ofO(q2)ot (q2)

mo=2.6, K=0.1570, Treeo 1/fo (q2)o1/( (q2)

18

20

1.5
C\J

IT

1.5

~

1.0

'+C\J

--- 1.0U...

:..:.::.- - . : . :..::.: -.

'+-

0.5

0.5

0.0
0.0

0.0

2.0

4.0
6.0
q2 [GeV 2 ]

8.0

10

14

12

q2

10.0

16

22

[GeV 2 ]Figure 18: 1/ f O and 1/ f+ as a function of q2 at mQ = 2.6 and K, = 0.1570. The solid
and the dotted curves represent the linear and the quadratic fits, respectively. The long
dashed line represents the linear fit with the constraint m po1e = mE", where mE is theB* meson mass obtained from the two-point correlation function.

2.5
ma=1.5, K=O.1600, Treeo fO(q2 )2.0

ot (q2)
6.0

1.5
C\J

5.0

--- 1.0
U

'+-

Q)

4.0

'0
a.

0.5

e

E 3.0

e

0.0

0.0

2.0

4.0
6.0
q2 [GeV 2 ]

8.0

e
e¢

2.0

10.0

Figure 17: Form factors at mQ = 1.5 and K, = 0.1570, 0.1600.

K=0.1570
Tree

:¢

CD

E

:¢

¥~

0

1.0
0.0

0

0.0

1.0

2.0

3.0

4.0

ms
mpo1e
5.0

6.0

ms
Figure 19: mE" obtained from the two-point correlation function of the B* meson andthe pole mass from the linear fit of 1/ f+.

4 RESULTS AND PHYSICAL IlVIPLICATIONS

44

quadratic fit

linear fit(mQ,n)

K,(2.6, 1)0.15700.15850.1600(1.5,2)0.15700.15850.1600(0.9,2)0.15700.15850.1600f-l(q:aJCl

1.373(54)0.480(21)1.436 (70)0.445(24)
1.531(94)0.407(27)1.167(38)0.597(25)
1.213(50)0.559(28)1.283( 67)
0.516(32)1.011(28)0.685(28)1.041(36)0.647(33)1.090(49)0.599(37)0.126(70)0.264(38)0.109(88)0.272(47)0.09 (11)0.276( 61)
0.209(81)0.472(64)0.19 (10)0.493(78)0.17 (14)0.52 (10)0.360(88)0.753(90)0.35 (11)0.79 (11)0.33 (15)0.85 (14)f-l(q: aJ
1.386(52)0.470(20)
1.438(64 )0.434(22)1.512(86)0.395(26)1.185(37)0.587(22)1.224( 4 7)
0.545(24)1.279(62)0.496(29)1.027(27)0.690(26)1.056(35)0.640(28)1.096( 48)
0.577(32)Cl

C2

o.058( 64)
0.335(40)0.098(91)0.366(59)0.22 (16)0.44 (10)0.086(80)0.548(60)0.10 (12)0.623(92)0.21 (20)0.77 (17)0.208(85)0.713(75)0.19 (13)0.86 (12)0.26 (24)
1.12 (23)0.046(53)- 0.051 (37)0.007(81)- 0.068(54)- 0.09 (14)-0.115(86)0.119 (87)
- 0.075 (78)
0.0 8 (14)- 0.13 (12)- 0.04 (24)
- 0.26 (20)
0.19 (13)0.05 (13)0.20 (21)- 0.09 (21)0.10 (35)- 0.36 (36)

Table 11: Parameters for the fit f- 1(q2) = f- 1(q:aJ + Cl (q!ax - q2) + C2( q;ax - q2) 2, where
C2 is set to zero for the linear fit. For each (mQ,n) and K" numbers in upper and lower
rows correspond fO and f+, respectively. In all the cases, X 2/ dof are less than unity.
and the measured mE', which is around few hundred MeV. Nevertheless, it is remarkablethat the deviation remains the same order and the mass dependence of m p o1e has the same
trend with mE'. We have not yet understood whether the above discrepancies can be
explained from the remaining systematic errors such as the discretization error . But atleast qualitatively judging from the size of the uncertainty in our calculation, our data
is not inconsistent with the picture that there is a sizable contribution from the B* poleto the form factor f+ near q;ax'
So far the discussion have been based on the tree level study. Let us now study how
one-loop renormalization changes the form factors. Because the one-loop correction isdifferent for V4 and Vi, the shape of the form factors may change significantly. Figure 20
shows the form factors for mQ = 2.6 and K, = 0.1570 with renormalization factors. The
leading logarithmic factor Eq.(36) is not multiplied in the present case. We find thatthe renormalized f + has stronger q2 dependence than that of at the tree level , while fO

4.2

q2-dependence of the form factors

45

receives only a small change. The renormalization makes the B* pole fit even worse. In
fact, the deviation of the constrained fit from our renormalized f + data is as large as 25% near q!ax' This is still within the typical size of O( a) errors. It is very important toperform the analysis with larger (3.

4

46

RESULTS AND PHYSICAL IMPLICATIONS

4.3

4.3

2.0
N

0

fO (q2 )0

t (q2)

g.

'+-

1.0
0.5
0

10
15
2
[
GeV
2 ]
q

5

Soft pion theorem( 56)-... 1.5

0.0

47

Applying the soft pion theorem to the B --t 7r matrix element, fO(q!aJ is related to the
B meson decay constant [9, 10, 11]ma=2 .6, K=0 .1570, q'=1t/a

2.5Soft pion theorem

20

25

in the massless pion limit. This relation is examined in Fig. 21. For the values of
fB' we refer our work on fB [30], which is obtained with an evolution equation of aslightly different form from that of the present work. We observe a large discrepancy
between fO and the decay constant both for the 11mB dependence and for the value
itself. fB increases rapidly toward heavier heavy quark masses, while fO( q!aJ almost
stays constant.
The discrepancy still remains significant when the renormalization effect is incorporated. In evaluating the renormalized values of fB' we use one-loop perturbative
coefficient obtained in the same manner as in Section 2.3 [31]. The leading logarithmic
factor Eq.(36) is multiplied to both fO(q!aJ and fB.
One may argue that the observed discrepancy can be explained by the uncertainty in

ma =2.6 , K=0.1570, q'=1/a

2.5
2.0

o

fO (q2 )ot (q2)
o fBmB 1121 f~

N

cr

4.0

1.5

'+-

!

1.0
0.5

~'::!::::."'

0

5

20

25

Figure 20: Renormalized form factors at mQ = 2.6 and K = 0.1570. Upper and lower
figures are obtained with g~(7rla) and g~(lla), respectively. The solid lines represent
the results of the linear fit. For f+ (q2), X 2 I dof are 1.9 and 4.1 for q* = 7r I a and II arespectively. X2 Idof for fO(q2) are less than 0.5 in the both cases.

•

fBmB 1 /2!f~(q'=1 /a)• fO(q2ma,.}mBll2

tIl !~! ~I(q'=1/a)m

2.0~i I ~ , ~ j

CD

0.0

o fO(q2ma.lm Bll2 (Tree)

f

----

(Tree)

ECD

Q2

'+-

~ -

1.0

0.0

0.0

0.1

0.2

0.3

0.4

0.5

11m B [GaV"]
Figure 21: Comparison of fO( q!aJ with fBI f-rr multiplying the factor yfmB in the chirallimit. Open and filled symbols are at the tree level and at the one-loop with g~(1 I a),
respectively.

4

48

RESULTS AND PHYSICAL IMPLICATIONS

49

5

In this section, we qualitatively discuss on the systematic uncertainties associated with

2.0

o f(q2 m•• ) : ma=26•

the lattice regularization. The following is a list of the main sources of systematic errors:(fe /f .) me/(me+m.l : ma =26/::,. fO(q2 m .. l : ma =1 .5

1.5

A (fe/f.) ma/(ma+m.l : ma=1. 5

I

E 1.0

• O( a) errors: The characteristic size of 0 (aA QCD ) error arising from the unimproved'Nilson quark action at (3 = 5.8 is 20-30%. This effect is large enough to explain!

the discrepancy between EB(p)Up + E71"(k)Uk and ~, mentioned in Section 3. Useof the O(a)-improved Clover action for the light quark will reduce this error to the&level of 5 %.

NO-

ci:'

m

• O( ap) error: The systems with finite momentum may suffer from the discretization
errors more seriously than that at the zero recoil point. The analytic estimate of

0.5

0.0

EXAMINATION OF SYSTEMATIC ERRORS

6.1

6.2

6.3

6.4

11K:

the momentum dependent error [12J shows that the effect is about 20 % at Ipi rv 1
Ge V even one uses the O( a)-improved current. In general, the use of the Wilson
action for the light quark could introduce such a error of order O( ap) in practicalnonperturbative case.

Figure 22: jO(q;,ax) and (jB/j71")mB/(mB +m71") for light quark masses. Two cases of
mQ, 2.6 and 1.5, are displayed at the tree level.

• Perturbative corrections: The one-loop correction could become significant especially for small (3 values. Strictly speaking, our calculation does not treat the
one-loop effects correctly, because the stabilizing parameter n does not have cor-the extrapolation procedure. To study this possibility, we compare fO(q!"J and fBI f71"
also in finite light quark mass cases, in the light of the heavy meson effective theory
which implies the relation (48). They are compared in Fig. 22 as a function of 1/K,. The

rect values. This problem must be removed in the future studies. In estimating thedifference between them are remarkable even for finite light quark mass cases.

included to obtain reliable results.

The reason why these differences occur is not clear. Since our present results sufferfrom various systematic uncertainties, as described in the next section, further study
with better control of systematic errors is necessary to clarify the origin of the problem.

one-loop corrections, we did not include the effect of the operator mixing, which
was reported to be significant in the case of fE [33]. This effect also should be• O(1/m8) effects: We described the heavy quark with the NRQCD action including
the order l/mQ terms. Further precise calculations may need to include O(l/m~)
corrections, although the effect was shown to be small[30, 32] for fB.
The finite volume effect may also be important.
Since the all above systematic errors can be large, there is no advantage of giving
quantitative estimates of each error at this stage. The use of the O( a)-improved (clover)action for light quark, as well as the simulation at higher {3 values will reduce mostof the above systematic errors. The simulation with dynamical quarks is also of great
importance for reliable predictions of the weak matrix elements.

~

-..",

-..

-. _...

-

. -

-

-

.-

...

.

6

50

6

.

CONCLUSIONCONCLUSION51ACKNOWLEDGMENT

In this paper, we present the results of the study of B --+ 7r form factors using NRQCD

This study has done in collaboration with Shoji Hashomoto of High Energy Accelerator

to describe the heavy quark with the Wilson light quark. Clear signal is observed forthe matrix element in a wide range of heavy quark mass containing the physical b-quark
mass. They are extrapolated to the chiral limit, although the result is so noisy for

Research Organization (KEK) (now he stays at Fermi National Accelerator Laboratory

quantitative conclusion.
The l/mB dependence of the matrix elements are studied and it is clarified thatthe temporal component and the part of the spatial component proportional to thepion momentum have fairly small dependencies on mQ. On the other hand, the partin U.S.A.), Tetsuya Onogi, Ken-Ichi Ishikawa, and Norikazu Yamada of Hiroshima University. First of all, I would like to thank them for fruitful discussions, valuable advise,and kind help. Especially the practical calculation of perturbative coefficients was largely
dlie to K-I. Ishikawa.
Numerical simulations were carried out on Intel Paragon XP /S at INSAM (Instituteof the spatial component proportional to the B momentum has a significant O(l/mB)

for Numerical Simulations and Applied Mathematics) in Hiroshima University. I am
grateful Shinji Hioki (now of Tezukayama university) and Yasuyuki Iwase for contin-

correction.
The q2 dependence of the form factors in the finite light quark masses are studied. We

ual support on the computational environment. I also thank Eiji Sugiura for his kindmanagement of our local computer system.

find that the q2 dependence of the form factor f+ ( q2) near q!"" becomes much stronger
for larger heavy quark mass. Model independent fit of 1/ f+(q2) near q!ax _shQws tha_L

S. Hioki also allowed us to use his program to generate gauge configurations. My
Colleagues, Nobuo Sasaki and Takashi Umeda, kindly helped me so that I could continue
the simulation even when I was absent in the University. I am grateful Professor Minoru
Yonezawa, Miho Kubokawa and all members of Hadron Physics Laboratory for their

the tree level results are consistent with the pole behavior for large q2 range, and thedifference of fitted pole mass and the measured mB. is around few hundred MeV for all
the heavy quark masses.
The values of f O at the zero recoil point are compared with the prediction of the softpion theorem, and the significant discrepancy is observed .
The size of the renormalization corrections are estimated by the one-loop perturbative
calculation. They almost does not affect their l/mQ dependence, but decrease Vi much
more than 114, which drastically change the shape of f+ . Our present result suffers fromlarge systematic uncertainties, and the most important one is O( a) error. It is very
important to study at higher {3 with improved actions .

continual support and encouragement.
I thank Shin-ichi Tominaga of KEK and other members of JLQCD Collaboration for
useful discussions.
This research was supported in part by a grant from the Japan Society for the Promotion of Science for Young Scientists.
Finally, I wish to express my sincere gratitude to Professor Osamu Miyamura ofHiroshima University for his constant encouragement and for reading the manuscript
and making a number of helpful suggestions.

REFERENCES

52

References

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