We report results (on an intermediate statistics sample) of a study of weak
semi-leptonic formfactors of $B$ and $D$ decays, addressing the uncertainties
from mass extrapolations to chiral and to heavy quarks. Moreover, we present a
nonperturbative test to the LMK current renormalization scheme for vector
current {\it transition} matrix elements and find remarkable agreement.Comment: 13 pages, uuencoded, updated table

Güsken, S.

Schilling, K.

Siegert, G.

English

arXiv

We report results (on an intermediate statistics sample) of a study of weak semi-leptonic formfactors of B and D decays, addressing the uncertainties from mass extrapolations to chiral and to heavy quarks. Moreover, we present a nonperturbative test to the LMK current renormalization scheme for vector current {\it transition} matrix elements and find remarkable agreement

Güsken, S

Siegert, G

Schilling, K

CERN Document Server

HLRZ 95-38
WUB 95-22
Weak Decays of Heavy-Light Mesons on the Lattice:
Semi-Leptonic Formfactors

S. Gusken
a
, G. Siegert
a;b
, K. Schilling
a;b y
a
Physics Department, Bergische Universitat, Gesamthochschule Wuppertal
Gaustrasse 20, D-42097 Wuppertal, Germany
b
HLRZ c/o KFA Julich, D-52425 Julich, Germany
and DESY, Hamburg, Germany
(July 4, 1995)
Abstract
We report results (on an intermediate statistics sample) of a study of weak
semi-leptonic formfactors of B and D decays, addressing the uncertainties
from mass extrapolations to chiral and to heavy quarks. Moreover, we present
a nonperturbative test to the LMK current renormalization scheme for vector
current transition matrix elements and nd remarkable agreement.
I. INTRODUCTION
The determination of weak formfactors for the semi-leptonic decays of heavy-light mesons
presents a big challenge for further exploration of the avordynamics within the Standard
Model, both experimentally and theoretically. The reason is that the study of weak in-
teraction amplitudes like the Kobayashi-Maskawa parameters calls for good control of the
hadronic binding eects in the analysis.
Heavy quark eective theory provides a suitable framework of understanding the 'ge-
ometric' part of the spin-avor structure of heavy quark decay in the innite mass limit
y
Invited talk given at the German-Japanese workshop Physics on Parallel Computers, Yamagata
1995

Work supported by DFG grant Schi 257/3-2 and 3-3
1
Mh
! 1, but cannot predict the 1=M
h
corrections in detail which are likely to be impor-
tant in the mass regions of D- and B-mesons. For this reason great eort has been spent
ever since the pioneering work of the El-Khadra et al [1] to compute QCD eects on these
decays by means of lattice gauge theory.
In this contribution, we present a status report about our ongoing work on 24
3
 64
lattices with quenched Wilson fermions at  = 6:3, based on 60 congurations (out of a
total sample of 100 congurations) that we are carrying out on the 32 node connection
machine CM5 with 40 Gbytes of parallel disc space SDA, at Wuppertal University. The
light quarks were represented by the following set of hopping parameters 
l
: .1511, .1507,
.1490, .1450, while the heavy quarks were varied over the set of 
h
: .1400, .1350, .1300, and
.1200. The initial meson was chosen at rest, while the nal meson could carry 11 dierent
momenta (in lattice units): (0,0,0), (1,0,0), (1,1,0), (1,1,1), (2,0,0), and permutations.
II. MASS EXTRAPOLATIONS
Let us for the moment consider the momentum transfer dependence of the formfactors,
leaving aside the issue of the lattice renormalization of these quantities.
Chiral extrapolation Fig.1 gives an idea on the quality of our data, in terms of the
conventional formfactors f
+
and f
0
for the transition PS ! PS + leptons as well as the
formfactors A
1
, A
2
, and V for the decay PS ! V + leptons. For the formfactor denitions,
see Ref. [3]. Kinematically, this gure refers to a heavy quark (
i
= :1350 decaying into a
light one ( = :1490), with a spectator quark, 
spec
= :1490. The full lines represent the
pole dominance predictions, with pole masses from the lattice two-point functions.
In Fig. 2 we demonstrate, for comparison the eects of chiral extrapolation for 
f
and

spec
. Physically this gure relates to the decays D !  and D ! .
Unfortunately, our lattice allows only for a small variation in the momentum transfer
between the initial and nal state hadrons, q
2
. Generally, we nd consistency with (but no
compelling evidence for) pole dominance.
Extrapolation to B Meson It is well known that currently accessible lattice sizes
do not allow for a direct simulation of B decays and therefore require a substantial mass
extrapolation, from the D to the B meson regime. While the transitions B ! D;D

lend
themselves rather easily to a heavy quark eective theory (HQET) like analysis [7], the
situation is much more involved in the decays B ! ; . One option is to carry out this
extrapolation in a manner suggested by HQET, i.e. at xed value of ! = v
i
v
f
[3,8]. This
extrapolation implies a big change in q
2
, which has to be compensated by a substantial
counter-extrapolation to q
2
= 0, the conventional point of comparison. Given the uncertain-
ties in the formfactors this procedure introduces sizeable systematic errors. We therefore
prefer to carry out the extrapolations at xed q
2
= 0, using dierent functional forms of
M
PS
dependence. This appears to be a better controlled way to judge the systematic errors
on the nal numbers.
In Fig.3 we display four of the formfactors F (q
2
= 0) versus the inverse mass of the
decaying meson, M
 1
PS
. The nal meson corresponds to K and K

, respectively. The dier-
ent curves represent dierent types of linear extrapolations to the B meson mass (leftmost
points): full lines ts to F itself, dashed lines ts to F
p
M
PS
, dotted lines ts to F
p
M
PS
 1
.
2
The plots illustrate that the data is not sensitive yet to discriminate between the M depen-
dencies of the ansatze.
III. RENORMALIZATION
In order to relate our results to the continuumwe need to renormalize the matrix elements
of the local (non-conserved) vector and axial vector currents. It has been repeatedly pointed
out that { on top of their short distance part { the renormalization factors may receive large
O(ma) contributions at large ma.
An obvious starting point is to take these contributions into account by approximately
adjusting the normalization of the lattice quark propagator (t; ~p =
~
0) to the one of the
continuum quark propagator [2] which has been advocated and rened more recently by
Lepage, Kronfeld and Mackenzie [5], and will hence be quoted as LMK scheme. From the
free case relation (t; ~p =
~
0)
cont
= 2(1 +ma)(t; ~p =
~
0)
latt
one uses mean eld arguments
to obtain
(t; ~p =
~
0)
cont
' 2N
LMK
()(t; ~p =
~
0)
latt
(1)
for the interacting case. As pointed out in ref. [6]. the normalization factor N
LMK
() =
1
2
(1 
3
4


c
) can be \eaten up" by an additional time hop of the lattice quark propagator
(t; ~p =
~
0)
cont
' (t+ 1; ~p =
~
0)
latt
: (2)
In this section we will measure the mass dependence of the renormalization constant of
the time component of the vector current and compare the results to the LMK predictions
as well as to the naive expectations, in which the LMK factor is absent.
In order to improve on statistical accuracy, we will work with ratios of 3-point functions,
rather than those functions themselves. We emphasize that the mass dependence of such
ratios directly measures lattice artefacts, as it disappears in the continuum limit.
We consider the ratio
1
R
H
0
H
(~p) =
hPS(m
H
0
;m
l
)jV
cons
0
jPS(m
H
;m
l
)i
~p
hPS(m
H
0
;m
l
)jV
loc
0
jPS(m
H
;m
l
)i
~p
: (3)
The ratio R
HH
(
~
0) yields directly the renormalization factor of V
loc
0
, as V
cons
0
is unaltered
in this instance. For the study of decays we are rather interested in the ratios R
hH
(
~
0) and
R
HH
(~p), however.
In order to derive predictions for the mass dependence of the ratios R above we have to
study the 3-point functions
G
cons(loc)
HH
0
l
(0; y
0
; x
0
) =
X
~x;~y
hP
y
H
0
l
(
~
0; 0)V
cons(loc)
0
(~y; y
0
)P
Hl
(~x; x
0
)i ; (4)
where P denotes the operator of the pseudoscalar meson and
1
A similar analysis has been carried out in ref. [9].
3
Vloc
0
(~y; y
0
) =

	
H
0
(~y; y
0
)
0
	
H
(~y; y
0
) (5)
V
cons
0
(~y; y
0
) =
1
2
h

	
H
0
(~y; y
0
)(
0
  1)U
0
(~y; y
0
)	
H
(~y; y
0
+ 1)
+

	
H
0
(~y; y
0
+ 1)(
0
+ 1)U
y
0
(~y; y
0
)	
H
(~y; y
0
)
i
: (6)
For 0  y
0
 x
0
one nds R
HH
0
= G
cons
HH
0
l
=G
loc
HH
0
l
. In terms of quark propagators, , one
easily obtains
G
loc
HH
0
l
(0; y
0
; x
0
) = h
X
~x;~y
Tr
n

H
(~y; y
0
; ~x; x
0
)
5

l
(~x; x
0
;
~
0; 0)
y
H
0
(~y; y
0
;
~
0; 0)
5

0
o
i (7)
G
cons
HH
0
l
(0; y
0
; x
0
) = h
1
2
X
~x;~y
[ (8)
Tr
n

H
(~y; y
0
+ 1; ~x; x
0
)
5

l
(~x; x
0
;
~
0; 0)
y
H
0
(~y; y
0
;
~
0; 0)
5
(
0
  1)U
0
(~y; y
0
)
o
+ Tr
n

H
(~y; y
0
; ~x; x
0
)
5

l
(~x; x
0
;
~
0; 0)
y
H
0
(~y; y
0
+ 1;
~
0; 0)
5
(
0
+ 1)U
y
0
(~y; y
0
)
oi
i :
Inserting the relations (1) and (2) for the quark propagators one can now read o the
LMK predictions for the mass dependence of the ratios R(
~
0). Note that the second term in
eq.(8) is expected to be negligible within our mass range
2
. Because of y
0
  x
0
< 0 one gets

H
(y
0
; x
0
)  (1   
0
) for very heavy quarks, and the second term becomes proportional to
(1  
0
)(1 + 
0
) = 0. The LMK predictions for the ratios therefore read
R
HH
(
~
0) =
~
Z
loc
V
1  
3
4

H

c
2
H
R
hH
(
~
0) =
~
Z
loc
V
~
Z
cons
V
1  
3
4

H

c
2
H
; (9)
where
~
Z
loc
V
and
~
Z
cons
V
are constants. We emphasize that R
hH
(
~
0) only depends on m
H
but
not on m
h
. This is a consequence of the cancellation of the second term in eq.(8). The naive
prediction simply reads R
HH
(
~
0) = Z
V
= R
hH
.
In g.4 we compare our numerical results for R
HH
(
~
0) with the LMK and the naive
predictions. We have inserted the (1-loop) perturbative values
~
Z
V
loc
= 1 0:82=4g
2
, Z
loc
V
=
1   0:174081g
2
, using the boosted coupling and assumed
~
Z
cons
V
= Z
cons
V
= 1. Throughout
our mass range of m
H
we nd the LMK prediction to approximate our data within 5%. The
naive prediction, however, fails drastically.
For actual physics applications it is more interesting to study the situation, where initial
and nal nonspectator quarks carry dierent masses. For that matter, g.4 shows the ratio
R
hH
(
~
0) as a function of m
h
at xed m
H
. Again we nd good agreement with the LMK
prediction. This nding has important bearing on the reliability of our calculations with
light nal states.
There remains the nal issue, whether LMK holds as well in the case of unequal momenta
of initial and nal state hadrons. To this end, g.4 displays the ratio R
HH
(~p) on the
2
This has also been veried numerically from our data.
4
momentum set j~paj = 2=64  f0; 1;
p
2; 2g. The solid line represents the above LMK
prediction, for ~p =
~
0. We nd that the data moves only marginally with increasing ~p and
is therefore well approximated by the LMK prediction over the entire ~p range. This result
is not at all trivial, as one would expect the normalization factor to be aected by the
momentum.
IV. PHYSICS RESULTS
In table I we have collected our results for the B decays into charmed particles, while
the tables III and II contain our results on decays into light hadrons and on D decays,
respectively, together with the data of other lattice investigations, as well as model and sum
rule calculations.
We nd good agreement with experimental data for theD ! K;K

decays, with possible
exception of A
2
(0), where the errors are large. Note that the kinematical constraint f
+
(0) =
f
0
(0) is well fullled by our data!
While our results for D ! ;  are generally fairly consistent with the results from
other lattice investigations, there is still rather large scatter in the results for the decays
B ! ; . This is related to the heavy mass extrapolations. We have not imposed the
constraint f
+
(0) = f
0
(0) in our analysis. Therefore the dierence in these quantities reects
systematic uncertainties. The extrapolative situation for these formfactors can be traced
from the upper plots in g.3. f
0
is determined with higher precision, due to the kinematical
point ~p
f
=
~
0. A t using the constraint f
+
(0) = f
0
(0) along the mass trajectory would
favour the value 0.20, of course. A similar kind of uncertainty is borne out by the range
quoted recently by the UKQCD collaboration [8] for f
+
(0) = f
0
(0). They have used the
latter constraint within an HQET inspired extrapolation procedure, using several variants
for the q
2
-dependence of the formfactors.
At this stage, one should note, that reliable knowledge obout the Z-factors is urgently
needed, as they enter directly into the lattice predictions for the formfactors. It is therefore
of great interest to continue studies about nonperturbative lattice renormalization.
5
REFERENCES
[1] A. El-Khadra et al, Nucl. Phys. B(Proc.Suppl.)9(1989) 186.
[2] S. Gusken et al., Phys.Lett.B227(1989)266.
[3] As. Abada et al., Nucl. Phys. B 416 (1994) 675.
[4] A. Abada et al, Nucl. Phys. B(Proc.Suppl.)34(1994)477.
[5] P. Lepage, Nucl. Phys. B(Proc.Suppl.)42(1992) 45;
P.B. Mackenzie, Nucl. Phys. B(Proc.Suppl.)30(1993) 35.
[6] C.W. Bernard, Nucl. Phys. B(Proc.Suppl.)34 (1994) 47.
[7] K.C Bowler et al., Southampton preprint SHEP 95-06.
[8] D.R. Burford et al. , Southampton Preprint SHEP 95-09.
[9] A. Vladikas, Nucl. Phys. B(Proc.Suppl.)42(1995) 400.
[10] M. Witherell, AIP Conf. Proceedings 302, Cornell Univ., 1993.
[11] M. Crisafulli et al., Phys. Lett. B 223 (1989) 90; Nucl. Phys. B 356 (1991) 1301; Phys.
Lett. B 274 (1992) 415.
[12] C.W. Bernard, A.X. El-Khadra, A. Soni, Nucl. Phys. B (Proc. Suppl.) 20 (1991) 439
and 26 (1992) 204; Brookhaven preprint BNL{45157; Phys. Rev.D43 (1992) 2140 and
D45 (1992) 869.
[13] R. Gupta et al. LA UR{93{3580, submitted to Phys. Rev. D.
[14] C.R. Allton et al., ROME preprint 94/1050.
[15] K.C. Bowler et al., Edinburgh Preprint 94/546.
[16] D. Scora, N. Isgur, Phys. Rev. D40 (1989) 1491 and D39 (1989) 799.
[17] M. Bauer et al. Z. Phys. C29 (1985) 637 and C34 (1987) 103 and C42 (1989) 671.
[18] J.G. Korner, G.A. Schuler, Z. Phys. C38 (1988) 511 and C46 (1990) 93 and Phys. Lett.
B231 (1989) 306.
[19] F.J. Gilman, R.R. Singleton, Phys. Rev. D 41 (1990) 142.
[20] P. Ball, V.M. Braun, H.G. Dosch, Phys. Rev. D 44 (1991) 3567.
[21] T.M. Aliev et al., Sov. J. Nucl. Phys. 40 (1984) 527; C.A. Dominguez, N. Paver, Phys.
Lett. B 207 (1988) 499.
[22] P. Ball, Phys. Rev. D 48 (1993) 3190.
6
FIGURES
0
1
2
3
-0.3 -0.2 -0.1 0 0.1
q2
f+
f+
0
1
2
3
-0.3 -0.2 -0.1 0 0.1
q2
f0
fo
0
1
2
3
-0.3 -0.2 -0.1 0 0.1
q2
A
1
A1
0
1
2
3
-0.3 -0.2 -0.1 0 0.1
q2
A
2
A2
0
1
2
3
-0.3 -0.2 -0.1 0 0.1
q2
V
V
FIG. 1. q
2
dependence of the formfactors, at 
i
= :1350, and 
f
= 
spec
= :1490. The curves
refer to ts: pole dominance with mass from two-point function (full line), with mass tted to
formfactor (dashed line), and linear t (dotted line).
7
01
2
3
-0.2 0 0.2 0.4
q2
f+
f+ k_c
0
1
2
3
-0.2 0 0.2 0.4
q2
f0
f0 k_c
0
1
2
3
-0.2 0
q2
A
1
A1 k_c
0
1
2
3
-0.2 0
q2
A
2
A2 k_c
0
1
2
3
-0.2 0
q2
V
V k_c
FIG. 2. Same as Fig.1, but extrapolated to the chiral limit.
8
00.5
1
0 0.2 0.4 0.6 0.8
1/MPs
f +
 
(q
2 =
0)
0
0.5
1
0 0.2 0.4 0.6 0.8
1/MPs
f 0 
(q
2 =
0)
0
0.5
1
0 0.2 0.4 0.6 0.8
1/MPs
A
1 
(q
2 =
0)
0
1
2
0 0.2 0.4 0.6 0.8
1/MPs
V 
(q
2 =
0)
FIG. 3. Extrapolation of formfactors in the mass of the decaying pseudoscalar particle [in
GeV ], with nal state hadron K and K, respectively. The data are included as crosses, and the
extrapolated points are plotted at the inverse B-mass, M
 1
PS
= :2. All ts are linear: (a) to F itself
(full line), (b) to FM
1=2
PS
(dashed line), (c) to FM
 1=2
PS
(dotted lines). The interpolated results at
the D-mass are included for convenience (open symbols).
9
FIG. 4. The ratios R
HH
(
~
0), R
hH
(
~
0) and R
HH
(~p) as a function of the quarkmass. Solid lines
represent the LMK and dashed lines the naive prediction.
10
TABLES
TABLE I. Formfactors for B ! D, D

und B
s
! D
s
, D

s
.
B ! D, D

f
+
(0) f
0
(0) V (0) A
1
(0) A
2
(0)
1.01(60) 0.88(29) 1.84(1.10) 0.97(32) 2.76(1.58)
B
s
! D
s
, D

s
f
+
(0) f
0
(0) V (0) A
1
(0) A
2
(0)
0.99(42) 0.85(24) 1.84(1.10) 0.91(24) 2.25(1.12)
11
TABLE II. Formfactors at q
2
= 0 for the decays D! K und D ! K

.
Method f
+
(0) f
0
(0) V (0) A
1
(0) A
2
(0)
World av. [10] Exp't 0.77(4) 1.16(16) 0.61(5) 0.45(9)
This work 0.71(12)
+10
 7
0.67(6)
+9
 7
1.34(24)
+19
 14
0.61(6)
+9
 7
0.83(20)
+12
 8
LMMS [11] Latt. 0.63(8) 0.86(10) 0.53(3) 0.19(21)
BKS [12] Wilson 0.90
+8+21
 8 21
0.70
+8+24
 8 24
1.43
+45+48
 45 49
0.83
+14+28
 14 28
0.59
+14+24
 14 23
ELC [3] 0.60(15)(7) 0.86(24) 0.64(16) 0.40(28)(4)
BG [13] 0.73(5) 0.73(4) 1.24(8) 0.66(3) 0.42(17)
APE [14] Latt. 0.72(9) 1.0(2) 0.64(11) 0.46(34)
UKQCD [15] Clover 0.67
+7
 8
0.65(7) 1.01
+30
 13
0.70
+7
 10
0.66
+10
 15
ISGW [16] 0.76 { 0.82 1.1 0.8 0.8
WSB [17] Quark- 0.76 1.27 0.88 1.15
KS [18] Models 0.76 0.8 0.82 0.8
GS [19] 0.69 1.5 0.73 0.55
BBD [20] 0.60
+15
 10
1.10(25) 0.50(15) 0.60(15)
AEK [21] Sumrules 0.60(15)
DP [21] 0.75(5)
12
TABLE III. Formfactors at q
2
= 0 for the decays D and B to  and .
D !  und D ! 
Method f
+
(0) f
0
(0) V (0) A
1
(0) A
2
(0)
This work 0.68(13)
+10
 7
0.65(5)
+9
 7
1.31(25)
+18
 13
0.59(7)
+8
 6
0.83(20)
+12
 8
LMMS [11] Latt. 0.58(9) 0.77(9) 0.47(7) -0.07(42)
BKS [12] Wilson 0.84(12)(35) 0.70(8)(24) 1.07(49)(35) 0.65(15)
+24
 23
0.59
+28
 25
UKQCD [15] Clover 0.61
+12
 11
0.95
+29
 14
0.63
+6
 9
0.51
+10
 15
ISGW [16] 0.51 0.51
WSB [17] Quark- 0.69 1.23 0.78 0.92
KS [18] Models 0.69 1.23 0.78 0.92
AEK, DP [21] Sumrules 0.6{0.75 0.6{0.75
B !  und B ! 
Method f
+
(0) f
0
(0) V (0) A
1
(0) A
2
(0)
This work Latt. 0.50(14)
+7
 5
0.20(3)
+2
 3
0.61(23)
+9
 6
0.16(4)
+22
 16
0.72(35)
+10
 7
ELC a [3] Wilson 0.28(14) 0.37(14) 0.24(6) 0.39(24)
ELC b [3] 0.33(17) 0.40(16) 0.21(5) 0.47(28)
APE a [14] Latt. 0.29(6) 0.45(22) 0.29(16) 0.24(56)
APE b [14] Clover 0.35(8) 0.53(31) 0.24(12) 0.27(80)
UKQCD [8] .49 - .24 .49 - .24
ISGW [16] Quark- 0.09 0.27 0.05 0.02
WSB [17] Models 0.33 0.33 0.28 0.28
Ball [22] Sumrules 0.26(2) 0.6(2) 0.5(1) 0.4(2)
13


Weak Decays of Heavy-Light Mesons on the Lattice: Semi-Leptonic Formfactors

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