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Exclusive semileptonic decays of heavy mesons in the quark model
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27 February 1997
2s e
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PHYSICS
ELSEVlER
LEITERS
B
Physics Letters B 394 (1997) 385-394
Exclusive semileptonic decays of heavy mesons in the quark model Dmitri Melikhov Nuclear Physics Institute, Moscow State University, Moscow, 119899, Russia
Received 5 December 1996 Editor: PV. Landshoff
Abstract Semileptonic decays D + K, K*, r, p, D, --f q, T’, 4, and B --+ D, D”, n-, p are analyzed within the dispersion formulation of the quark model. The form factors at timelike ~7are derived by the analytical continuation from spacelike CJ in the form factors of the relativistic light-cone quark model. The resulting double spectral representations allow a direct calculation of the form factors in the timelike region. The results of the model are shown to be in good agreement with all available experimental data. From the analysis of the B -+ D, D* decays we find IV&l = 0.037 f 0.004, and the B 4 rr, p decays give l&l = 0.004 f 0.001. PACS: 13.20.-v; 12.39.Hg; 12.39X Keywords: Semileptonic decays; Heavy quarks; Relativistic constituent quark model
The interest in exclusive semileptonic decays of heavy mesons lies in a possibility to obtain the most accurate values of the quark mixing angles and test various approaches to the description of the internal hadron structure. The decay rates are expressed through the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements and hadronic form factors of the weak currents which contain the information on hadron structure. New accurate data on B -+ D, D* and first measurements of the B + rr, p decays open a possibility to determine V& and VUb with high accuracy and require reliable theoretical predictions on the form factors and decay rates. A nonperturbative theoretical study should give these form factors in the whole kinematical region of momentum transfers 0 5 q” 5 (Ml - Mz)~, Ml and Ma the initial and final meson masses, respectively. The amplitudes of meson decays induced by the quark transition qi + qf through the vector VP = &yPqi and axial-vector A, = qfypy5qi currents have the following structure [ 1] : (P(M2,P2)I~(O)IP(Ml,pl))
= .f+Gm,
~V(M2,pz,~)IV,(O)IP(MI,pl)) (v(M2,172.~)
IA,(O)
IP(M1,m
+ f-(q2)q,,
=w12)$w&*VPfY~2P~ )> = ibid
[ .mz2k/m
+ a+(q2)m,P,
1Electronic address: [email protected]. 0370-2693/97/$17.00 0 1997 Published by Elsevier Science B.V. All rights reserved. PZZ SO370-2693(97)00048-S
+ ~-(q2kwp
11
(1)
D. Melikhou / Physics Letters B 394 (19971385-394
386
with q = pt - ~2, P = p1 + ~2. We use the notations: y5 = iy”y1y2y3, .eo123= -1, ,Sp(~~~r”~rP) = 4i@‘“fi. In general all the form factors are independent functions to be calculated within a nonperturbative approach. Considerable simplifications occur when both the parent and the daughter quarks active in the weak transition are heavy. Due to the heavy quark symmetry (HQS) [l] all the form factors which are in general functions of q2 and the masses, depend on the one dimensionless variable w = plpz/MlM2. It is convenient to introduce dimensionless ‘heavy quark’ form factors
g(q2) =
l 2dXK
In the leading l/m~ Isgur-Wise function h+(w)
= &(a)
&(W) 3
a+(q2) = -
1 hPmG
ha(w),
order all the form factors h can be expressed .$ as follows: = h,(w)
= hs(w)
=5(w),
h_(w)
f(q2)
=&i&%&l
+w)hf(co).
through the single universal
(2)
form factor, the
= 0.
(3)
The normalization of the Isgur-Wise function at zero recoil is known, e( 1) = 1. In contrast to meson decays induced by the heavy-to-heavy quark transitions, the general case of the transitions between hadrons with arbitrary masses, and in particular meson decays induced by a heavy-to-light quark transitions are not well-understood. To date, theoretical predictions on semileptonic decays induced by heavy-to-light quark transitions coming from the quark model [ 2-81, QCD sum rules [ 9-131, and lattice calculations [ 14-171 differ significantly. Recently, Stech noticed [ 181 that new relations between the form factors of meson transition can be derived if use is made of the constituent quark picture. These relations are based on the observation that if the meson wave function in terms of its quark constituents is strongly peaked in the momentum space with the width of order A N 0.5 GeV then for a heavy parent quark a small parameter A/m, appears in the picture and one can derive the leading-order expressions for the form factors of interest which turn out to be independent of subtle details of the meson structure. Although these relations give a guideline for the analysis of the decay processes, they cannot substitute calculations of the form factors in a more detailed dynamical model. The kinematically accessible q2-interval in the meson decay induced by the heavy-to-light transition is O(mi) so a relativistic treatment is necessary. A relativistic light-cone quark model (LCQM) [ 51 is an adequate framework for considering decay processes. The model is formulated at spacelike momentum transfers and the direct application of the model at timelike momentum transfer is hampered by pair-creation subprocesses which at q2 > 0 cannot be killed by an appropriate choice of the reference frame. In [5] the form factors at q’ > 0 were obtained by numerical extrapolation from the region q” < 0. As the relevant q2-interval is large, the accuracy of such a procedure is not high. In [ 7,8] the nonpartonic contribution of pair-creation subprocesses is neglected and only the partonic part of the form factor is calculated in the whole kinematical interval of q”. Unfortunately, the nonpartonic contribution is under control only at q2 = 0 where it vanishes. The dispersion formulation of the LCQM proposed in [ 191 overcomes these difficulties as it allows the analytical continuation to the timelike q. We apply this approach to calculating the transition form factors of the semileptonic decays. The transition of the initial meson Q (mz) Q( m3) with the mass Ml to the final meson Q(mz)Q( ms) with the mass M2 induced by the quark transition m2 ---f ml is given by the diagram of Fig. 1. The constituent quark structure of the initial and final mesons are described by the vertices It and I?*, respectively. The initial pseudoscalar meson the vertex has the spinorial structure It = iysGl/&; the final meson vertex has the structure I2 = iy5G2/ fi for a pseudoscalar state and the structure lYzP = [ Ay, + B ( kl - k3 ) p 1 G2 /fi for the vector state. The values A = -1, B = I/ (Js2 + ml + m3) correspond to an S-wave vector meson, and A = l/d,
B = [2&t
ml + ms] / [ fi(
s2 + (ml + m3)2) ] correspond
to a D-wave vector particle.
D. Melikhov/Physics
Letters B 394 (1997) 385-394
387
Fig. 1. One-loop graph for a meson decay.
At q2 < 0 the form factors of the LCQM [5] can be represented as double spectral representations over the invariant masses of the initial and final qq pairs as follows [ 191:
(4)
where sfYs2,
q2) =
sz(rnt + rni - q2) + q2(m: + rnz) - (mf - m$) (rn: - rni) 2rny
is the triangle function. Here fit ( q2) is the form factor of the and h(sl,s2,s3) = (Sl + $2 - S3j2 - 4sls2 constituent quark transition m2 + ml. In what follows we set f2t (8) = 1. The double spectral densities fi( st, s2,$) of the form factors have the following form: ~~++-=4[m~m2c~i-m2m3~i+mim3(l-cr~)-m~(l-a~)+~~2~2],
(5)
~+-f.-=4[m~m2a2-mlm3a2+m2m3(1-a2.)-m~(l-a~)+a~s~],
(6)
g~=--2A[m~a~++~a~+m3(1-a~-a~)]-4B/3,
(7)
ii++ii_
=-4A
[2m2all
ii+ - ii- = -4A[-mla2 f””
=
+2m3(al -mz(al
-a11)1
+4B[Clal
-ms(l-
-2a12)
+C3cull],
LYE -a:!f2ad]
(8) +4B
sfD + (” 1%; ” _ Mf,;, - s3) ;
[Cpl
+C3cyn],
(9)
(10)
M2c+,
where ELI = -4A[mlmzm3
+ z(s2 2
- rn: -m~)+~(s1-m~-m~)-~(s3-m~-m~)+2~(m2-m3)]
+ 4B C3P, ai = [(Si
+s2-
(11) s3jC.72 -mf+mg>
-2s2(s1
-mg+mg)]
/h(sl,s2,s3),
(12)
~2=[(~i+s2-s3)(s1-m~+m~)-2s1(~2-m~+m~)]/~(s~,s~,s3),
(13)
P=a
(14)
[2m:-al(sl
-m~+m~)
-a2(s2--mf+m:)],
D. Melikhov/Physics
388 all
= af
+ 4@2/h(Sl,
S2, S3),
Ci=S2-((mi+m3)2,
Letters B 394 (1997) 385-394
a12=a1a2-2P(sl+s2--3)/A(sl,s2,s3),
C2=s1-(m2-m~i)~,
C3=~3-(ml+m2)~-C~-C~.
(15)
(16)
Let us underline that the representation (4) with the spectral densities (5)~( 10) are just the dispersion form of the corresponding light-cone expressions from [ 51. It is important that double spectral representations without subtractions are valid for all the form factors except f which requires subtractions. In the LCQM the particular form of such a representation for the form factor f depends on the choice of the current component used for its determination and cannot be fixed uniquely. In [20] the behaviour of the form factors of the vector, axial-vector and tensor current has been studied in the limits of heavy-to-heavy and heavy-to-light quark transitions. The analysis of the behavior of the form factor f in the case of a heavy-to-light quark transition suggests another expression [ 201: (17) We shall use both of these prescriptions in the numerical analysis of semileptonic decays. For a pseudoscalar or vector meson with the mass M built up of the constituent quarks mq and mQ, the function G is normalized as follows [ 191:
J
G2(s)ds
A’/2(s,m$mi)
lr( s - h42)2
The quark-meson
G(s)
= gJ
87rs
(s - (m, - mq)2) = 1.
(18)
vertex G can be written as [5]
s2 - (ml - m$2 s - M2 -----w(k), s -
(ml - m2)2
s3/4
k = A’12(s, m:, ms) 2fi ’
(19)
where w(k) is the ground-state S-wave radial wave function. As the analytical continuation of the form factors (4) to the timelike region is performed, in addition to the normal contribution which is just the expression (4) taken at 4” > 0 the anomalous contribution emerges. The corresponding expression is given in [ 191. The normal contribution dominates the form factor at small timelike 4 and vanishes as q2 = (m2 - ml )2 while the anomalous contribution is negligible at small q2 and steeply in the region rises as q2 + (m2 - ml) 2. It should be emphasized that we derive the analytical continuation q2 < (m2 - ml)2. For the constituent quark masses used in the quark models this allows a direct calculation of the form factors of the P -+ V transitions in the whole kinematical decay region 0 I q’ 2 (Mp - Mv)~, as Mp - Mv < m2 - ml. For the P --f P’ transition this is not the case: normally, Mp - Mp > m2 - ml. For the P --+ P’ decays we directly calculate the form factors in the region 0 I q” < (m2 - ml)2 and perform numerical extrapolation in (m2 - ml)2 5 q” 5 (Mp - Mp) 2. 2 Numerical analysis shows the accuracy of this extrapolation procedure to be very high. We would like to notice that the direct calculation shows the derivative of the form factor f+ to be positive at the point q” = (1~12- ml) 2. This suggests that the maximum of the form factor f+ at q2 = (m2 - m1)2 observed in [7,8] is just an artifact of neglecting the nonpartonic contribution to the form factor. For calculating the form factors of semileptonic decays we assume that the wave function w can be approximated by a simple exponential function w(k) = exp( -k2/2p2) and adopt the numerical parameters of the ISGW2 model [ 211 shown in Table 1. * The analytical continuation to q* > (1112- ml)* is also possible. However one should be careful when applying the constituent quark model for such q*: we approach the unphysical q4 threshold q2 = (ml + rn~)~ which is obviously absent in the amplitudes of hadronic processes. This is a sign that we are coming to the region where the constituent quark picture is not adequate.
D. Melikhou / Physics Letters B 394 (1997) 38.5-394
389
Table 1 Parameters of the quark model Ref. ISCWZ
[21]
0.33
0.55
1.82
5.2
Table 2 Parameters of the fits to the calculated D -
f(O) cyr[GeVe2] q [GeVe4]
fk&ax)
Table 3 Decay rates for the D -
0.41
0.44
0.53
0.45
0.56
0.43
0.30
0.33
0.37
K, K” transition form factors
D-+K
D + K*
f+
8
0.781 0.201 0.0086 1.2
0.28 0.24 0.0135 0.35
K, K* transition in 10” s-’
a+ -0.168 0.189 0.001 -0.205
fLC
fHQ
1.747 0.0971 0.001 1.92
1.733 0.0767 0.001 1.86
using IV,,1= 0.975
This work
WSB [2)
ISGW2 1211
Jaus [5]
BBD [9]
EXP. [=I
T(D + K) lJ(D--tK*) VK*)/UK)
8.7 5.38 0.62
7.56 7.73
10.0 5.4
9.6 5.5
6.5 f 1.3 3.7 f 1.2
9.0 + 0.5 5.1 It 0.5
rL/rT
1.31
0.57 1.33
0.57 z!I0.15 0.86 -f 0.06
0.57 xt 0.08 1.15 xlz0.17
The results of calculating f(q2)
= f(O)/]
1 - W2
0.38
1.02
0.54 0.94
the form factors are fitted by the functions + a2q41
better than 0.5% accuracy, and for the form factor f+ this formula is used for numerical extrapolation to the region (ml - ~22)~ 5 q2 5 (Ml - Mz) 2. The decay rates are calculated from the form factors via the formulas from [ 41. Decay rate calculations are performed using the two prescriptions for the form factor f given by the relations (10) and (17); the corresponding results are labelled as LC and HQ, respectively. The decay D + K, K*. These CKM-favoured decays extend the widest possibility for detailed verification of the model. The parameters of the fits to the form factors are given in Table 2. Using the value V,, = 0.975 [25] the decay rates are found to be with
T(D + K) = 8.7 x 10’“s-’ T(D -+ K*) =
5 58 x 101os-l 5:38 x lotOs-1:
rL,lr* r&T
= 1.34 = 1.34
(LC) (HQ)
(20)
Table 3 compares the results of the model with the HQ prescription for the form factor f with the experimental data. One can observe perfect agreement with the data. The results of other approaches which give predictions for a wide set of the semileptonic decay modes are also shown. 77~ decay D -+ v, p. Table 4 present the parameters of the fits to the calculated form factors. Using the value Vcd= 0.22 [ 251 yields the following decay rates: I-(Do --) n--) = 0.62 x 101os-l,
D. Melikhov/Physics
390
Letters B 394 (1997) 385-394
Table 4 Parameters of the fits to the calculated Do -+ T-, p- transition form factors
f(O) cq [GeV-*I cq [ GeVe4] f(4Llx)
Table 5 Decay rates for the Do -+ (n-,
lY(D +n-)
UD +
PI
JTP)lUrr) rL/rT
D+?T
D+P
f+
g
0.681 0.225 0.010 1.63
0.252 0.274 0.017 0.36
p-)e+v
a+ -0.139 0.211 0.012 -0.18
1.326 0.110 0.002 1.52
1.257 0.071 0.003 1.37
transition in lOlo s-l using Iv&, = 0.22
This work
WSB [2]
ISGW2 [21]
Jaus [5]
Ball [ 101
Exp.
0.62 0.26 0.41 1.27
0.68 0.67 0.98 091
0.24 0.12 0.51 0.67
0.8 0.33 0.41 1.22
0.39 f 0.08 0.12 f 0.03 0.3 z!z0.1 1.31 f 0.11
0.92 f 0.45 0.45 f 0.22 0.5 f 0.35 -
Table 6 Parameters of the fits to the calculated D, -+ sS, 4 transition form factors
f(O) LYI[ GeVe2] a2[GeVv4]
T(D”
-+p_)
=
D, -+ SS
D, -+ 4
fi
g
0.800 0.192 0.008
0.266 0.246 0.015
0 30 x 1o’O s-t 0:26x 101os-‘:
a+ -0.149 0.201 0.009
rL/rT = 1.32
IL/IT
= 1.27
(LC), (HQ).
f”:
fUU
1.806 0.111 -0.003
1.710 0.085 0.013
(21)
Table 5 presents the rates for the HQ prescription of the model and the results of other approaches versus experimental data. The experimental results are obtained by combining the decay rates of the D ---f K, K* transition [22] with the following ratios measured by CLEO [ 231 Br( Do --+ r-e+v)/Br( Do + K-e+v) = = 0.088 f 0.062 f 0.028. The 0.103 f 0.039 f 0.013 and E653 [ 241 Br( Do -+ P-&Y) /Br( Do --f K*-e+y) calculated rates seem to be a bit small but nevertheless agree with the experimental values within large errors. The decay D, -+ 77,$, #J. Table 6 presents the results on the form factors. The calculated decay rates depend on the content of q and 7’ mesons and with IV,,l = 0.975 read I(D,
-+ 7) = 0.111 sin2(q)
T(D,
-+ 77’) =0.030
T(D,
+ 4) =
ps-‘,
cos2((p) ps-‘,
0.047 ps-’ , 0.040 ps-1 )
rL/rT rL/rT
= 1.30 = 1.28
(LC), (HQ).
(22)
Here 9 = 0p + arcsin(2/&) [25]. The decay rate of D, -+ q5 calculated with the (HQ) prescription agrees well with the results of the analysis [21] I’( D, -+ qhe+v) = (0.035 f O.O05)ps-‘. Table 7 compares the results on ratios of branching fractions with recent CLEO measurements 1261 and the ISGW2 model. The results for
D. Melikhov / Physics Letters B 394 (1997) 385-394
391
Table I Ratio of the decay rates for the D, + 7, v’. q5 transitions
Ur))lJ--(4) Url’)lUd) r(+)/r(v)
This work 8p = -100
ep = -140
ep =
1.45 0.4 0.27
1.24 0.46 0.37
0.9 0.6 0.67
-200
ISGW2 [21] @p= -100
ep = -200
1.2 0.5
0.8 0.7
Exp.[26]
1.24 f 0.27 0.43 ZII0.18 0.35 f 0.16
Table 8 Parameters of the fits to the calculated B -+ D, D* transition form factors B + D*
B-+D f-
f+
f(O) cy1[GeV-2] cu2[GeVp41 .fMlax)
0.684 0.0386 0.00042 1.12
a+
&z
-0.337 0.039 0.00038 -0.56
0.093 0.0416 0.00048 0.15
-0.0764 0.0387 0.00043 -0.12
f””
fHQ
4.533 0.02193 0.00007 5.85
4.729 0.02195 0.00007 6.10
Table 9 Parameters of the heavy-Quarkform factors for the B + D, D’ transition B 4
B-D h+
h(l) :2
0.96 0.91 0.37
h-
D*
hg
-0.04
0.99 0.54 I .04
0.79 0.63 1.20
h a-
hLC f
Hy hf
0.92 0.50 1.08
0.90 0.55 1.10
0.94 0.53 1.06
aI1 tip in the range -18O 5 BP 5 -10” compare favourably with the data, but the best agreement is observed for ep = -14O. The decay B ---f D, D*. This is a very interesting mode as it allows measuring corrections to the HQS limit. Table 8 shows the fit parameters of the form factors and Table 9 presents the parameters of the fit to the heavy-quark form factors in the form hi(w)
= hi( 1) [l - p;(w
- 1) + S(w - 1)2] .
(23)
We find h+( 1) = 0.96 and h_( 1) = -0.04 which compare favourably with the size of corrections to the HQS limit [30]. The values hyQ( 1) = 0.94 and hLfC( 1) = 0.9 both agree with the estimate of Neubert [30] hf( 1) = 0.93 f0.03. For the ratios of the heavy quark form factors Z?l = h,( l)/hf( 1) and R2 = h,, (l)/hf( 1) we obtain RyQ = 1.05 [Rtc = 1.11 and R,HQ = 0.84 [ Rkc = 0.881 to be compared with a recent CLEO result RI = 1.18f0.15&0.16and R~=0.71f0.22f0.07andpredictionsoftheISGW2model [21] RI = 1.27, R2= 1.01, Neubert [31] RI = 1.35, R2 = 0.79, and Close and Wambach [32] RI = 1.15, RZ = 0.91. CLEO [33] reported the value (&b(hf( 1) = 0.0351 + 0.0019 f 0.0018 + 0.0008. Combining this value with our result hyQ( 1) = 0.94 yields j&l
= 0.0373 A=0.0053
For the decay rates we find
[lepton
endpoint
region in B -+ D*lF] .
D. Meli?dwv / Physics Letters B 394 (1997) 385-394
392
Table 10 Decay rates for the B + D, D* transition in ps-’ This work r(B --+ D) r(B + D*) UD*)/r(D)
WSB [2]
8.7111,61*
Jaus [5]
11.91~b~*
8.11&i*
21.91V&[* 2.71
23.21V&12 2.65 1.28
rL/rT
ISGW2 [21]
Exp. 1.27 2.99 2.35 1.24 0.85
9.61&d*
25.31V&12 2.64
24.81&b/* 2.08 1.04
f zk zk f f
0.3 x IO-* [25] 0.66 x IO-* [27] 1.3 0.16 1281 0.45 [29]
Table 11 Parameters of the fits to the calculated B” -+ rr+, p+ transition form factors B-i%. a+
.f+
f(O)
0.2927 0.0511 0.00068 2.30
GeVp2] cy2[GeVw4] f (&ax) rwl
[
T(B --+ 0)
-0.0256
0.0635 0.0012 0.17
0.0567 0.0010 -0.097
fHQ
1.025 0.032 0.00028 2.19
1.098 0.0316 0.00038 2.12
= 8.712 x 10’2]V,#-1,
T(B -+ D*) =
21.0 x 10’2~v,~~W, C 23.2 x 10’21v,b12 s-l,
Table 10 compares the calculated result with a CLEO measurement yields l&l
0.0356
fLc
= 0.036 zt 0.004
rL/rr rL/rT
= 1.17 = 1.28
(LC), (HQ).
(24)
decay rates for the HQ prescription with other approaches. Combining our [27] f’(B -+ D*Z?) = [29.9 f 1.9(stat) f 2.7(syst) f 2.0(lifetime)]ns-’
[decay rate B -+ D*lii]
.
The branching ratio Br(B’ ---f D-Z+v) = ( 1.9 f OS)% and the B” lifetime rso = (1.56 f 0.06) ps [ 251 give the experimental decay rate I( B” -+ D-Z+v) = ( 1.22 I!Z0.3) 101o~-l and comparing with our (HQ) result yields l&l
= 0.038 f 0.004
All the three estimates excl = 0.037 l&b1
[decay rate B” -+ D-Z+v]
.
of V& agree with each other and the average value is found to be
It 0.004.
(25)
This is in perfect agreement with the updated values [ 341 I&, lexc’ = 0.0373 + 0.0045( exp) & 0.0065( th) and j&blinC1= 0.0398 f O.O008(exp) f O.O004(th). The decay B ---f v, p. This mode allows a determination of (&, I and hence reliability of theoretical predictions is very important. The form factors are given in Table 11. For the decay rates we find r(B”
3 r-)
T(BO -+ p-)
= 7.2 X io121vub12 s-l, =
8.44 x 10’2(v,b12 s-l, 9.64 x 10’2jv,b\2 s-l, C
rJrr rL/rT
= 0.95 = 1.13
(Lc), (HQ).
(26)
D. Melikhov/Physics
Lerters B 394 (1997) 385-394
393
Table 12 Decay rates for the B” -+ ( vf, p+)eF transition in ps-’ This work T(B --f ?r)
7.21&b/’
9.64/&,12 1.34 1.13
UB --t P) UP)lU~) rL/rT
WSB [2] 7.41v,b)*
26.01v,b/2 3.5 1.34
ISGW2 [21] 9.61v,bi2
14.21v,b[* 1.48 0.3
Jaus [5]
Ball
[ 10,131
1&01v,b[*
5.1 f
l.lli’,b1*
19.1~v,~~* 1.91 0.82
12 f 4( 14.5 f 4.5) Iv&l2 2.35 f 1.2 0.06 f 0.02 (0.52 f 0.1)
Exp. [35] 1.2 f 0.6 x 1O-4 1.67 dz 1.0 x 1O-4 -
This calculation is in agreement with our previous analysis of this decay mode using other sets of the quark model parameters [ 191. The calculated decay rates are compared with other theoretical predictions and first measurements by CLEO [ 351 in Table 12. The experimental values are obtained by combining the CLEO results [ 351 Br(B’ -+ r-Z+Y) = ( 1.8 f 0.4 f 0.3 f 0.2) low4 and Br( B” --+ p-Z+V) = (2.5 & 0.4 f 0.7 & 0.5) lop4 with the B” lifetime ~~0 = ( 1.56 & 0.06) ps [ 251. Comparing our results with the experimental values gives l&l
= 0.004 f 0.001
l&l = 0.00407 f 0.001
A good agreement branching
fractions
[B--+gl, [B +
PI-
of these values with each other shows that we have predicted correctly the ratio of the B -+ rr and B -+ p. The average value obtained from the two modes reads
j&J = 0.004 It 0.001. Taking the I&,
from (25) we find
(Vu&&( = 0.108 f 0.02, which is perhaps a bit large but nevertheless agrees with the PDG value I&b/V&l = 0.08 f 0.02. In conclusion, we have analysed semileptonic decays of heavy mesons within dispersion formulation of the constituent quark model and found agreement with all available data. The extracted values of the CKM matrix elements Vcb and Vub are also in agreement with estimates of other models. Nevertheless, we would like to briefly outline possible sources of uncertainties in the predictions of the model which should be taken into account in further analyses: 1. We used the parameters of the ISGW2 quark model and a simplified exponential ansatz for the wave function. It should be noticed however that the ISGW2 model does not calculate the form factors through the wave functions; rather a special prescription for constructing the form factors is formulated. As the analysis of [ 71 shows, the form factors of the heavy-to-light transition can be rather sensitive to the wave function shape. 2. We have taken into account only the leading process and neglected the O(cu,) corrections. Although the analysis of such corrections in the elastic pion form factor at low and high momentum transfers within the LCQM [ 361 found only a few % contribution even at w 11 lo-20 a numerical consideration of this contribution is plausible. 3. We have neglected the constituent quark transition form factor which has a complicated structure at timelike momentum transfers. In particular the quark transition form factor should contain a pole at q2 = M:e, with M,,, the mass of a resonance with appropriate quantum numbers. 4. We have identified the form factors obtained within the constituent quark model with the form factors of the full theory. However, the relationship between these two quantities is nontrivial: e.g. the form factors of the full theory acquire logarithmic corrections because of renormalization of the quark currents, which are absent in the quark model form factors. Analysing the l/rn~ expansion Scora and Isgur [21] performed a special matching procedure for obtaining the form factors of the full theory from their quark-model form factors. Although the
394
D. Melikhou/Physics
Letters B 394 (1997) 385-394
1/me behaviour of the LCQM form factors studied in [ 201 is better than that of the generically nonrelativistic form factors in the ISGW model [ 31, the relationship between the LCQM form factors and the form factors of the full theory should be studied in more detail. I am grateful to V. Anisovich, I. Narodetskii, K. Ter-Martirosyan, and B. Stech for discussions and interest in this work. The work was supported by the Russian Foundation for Basic Research under grant 96-02-18121a.
References [l] [2] [ 31 [4] [S] [6] [7] [S] [9] [lo] [ 111 [ 121 [ 131 [ 141 [ 151 [ 161 [ 171
[ 181 [ 191 [ 201 1211 [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [ 331 [34] [35] [ 361
N. Isgur and M. Wise, Phys. Lett. B 232 (1989) 113; B 237( 1990) 527. M. Wirbel, B. Stech and M. Batter, Z. Phys. C 29 (1985) 637. N. Isgur, D. Scora, B. Grinstein and M. Wise, Phys. Rev. D 39 (1989) 799. F. Gilman and R. Singleton, Phys. Rev. D 41 (1990) 142. W. Jaus, Phys. Rev. D 41 (1990) 3394; D 53 (1996) 1349. R.N. Faustov,V.O. Galkin, Yu.A. Mishurov, Phys. Lett. B 356 (1995) 516. I. Grach, I. Narodetskii and S. Simula, Phys. Lett. B 385 (1996) 317. H.Y. Cheng, C.Y. Cheung, C.W. Hwang, hep-ph/9607332 (1996). P Ball, V. Braun and H. Dosch, Phys. Lett. B 273 (1991) 316; Phys. Rev. D 44 (1991) 3567. P Ball, Phys. Rev. D 48 (1993) 3190. S. Narison, Phys. Lett. B 345 (1995) 166. A. Khodjamirian, R. Riickl, Nucl. Instmm. Meth. A 368 (1995) 28. P Ball, hep-ph/9605233 (1996). V. Lubicz, G. Martinelli, M. McCarthey and C. Sachrajda, Phys. Lett. B 274 (1992) 415. ELC Collaboration, A. Abada et al., Nucl. Phys. B 416 (1994) 675. APE Collaboration, C. Allton et al., Phys. L&t. B 345 (1995) 513. UKQCD Collaboration, K. Bowler et al., Nucl. Phys. B 447 (1995) 425; J. Flynn et al., Nucl. Phys. B 461 (1996) 327. B. Stech, Phys. I.&t. B 354 (1995) 447; hep_ph/9608297. D. Melikhov, Phys. Rev. D 53 (1996) 2460; Phys. Lett. B 380 (1996) 363. D. Melikhov and N. Nikitin, hep-ph/9609503 (1996). D. Scora and N. Isgur, Phys. Rev. D 52 (1995) 2783. M. Witherell, in: Lepton and photon interactions, Proceedings of the XVI International Symposium, Ithaca, New York, 1993, edited by P. Drell and D. Rubin, AIP Conf. Proc. No. 302 (AIP, New York, 1994). CLEO Collaboration, E Butler et al., Phys. Rev. D 52 (1995) 2656. E653 Collaboration, K. Kodama et al., Phys. Lett. B 316 (1993) 455. Particle Data Group, Phys. Rev. D 54 (1996) 1. CLEO Collaboration, G. Brandenburg et al., Phys. Rev. Lett. 75 (1995) 3804. CLEO Collaboration, B. Barish et al., Phys. Rev. D 51 (1995) 1014. CLEO Collaboration, S. Sanghera et al., Phys. Rev. D 47 (1993) 791. ARGUS Collaboration, H. Albrecht et al., Phys. Lett. B 324 (1994) 249. M. Neubert, Phys. Lett. B 338 (1994) 84. M. Neubert, Phys. Rep. 245 (1994) 259. F. Close and A. Wambach, Phys. Lett. B 348 (1995) 207. CLEO Collaboration, J. Duboscq et al., Phys. Rev. Lett. 76 (1996) 3898. T. Skwarnicki, HEPSY-95-05, 1995, Proceedings of the International Symposium on Lepton Photon Interactions (IHEP), Beijing, P.R. China, IO-15 Aug. 1995, p. 238. CLEO Collaboration, J.P. Alexander et al., Report No. CLNS-9611419 (1996). V. Anisovich, D. Melikhov and V. Nikonov, Phys. Rev. D 52 (1995) 5295.
27 February 1997
2s e
__
PHYSICS
ELSEVlER
LEITERS
B
Physics Letters B 394 (1997) 385-394
Exclusive semileptonic decays of heavy mesons in the quark model Dmitri Melikhov Nuclear Physics Institute, Moscow State University, Moscow, 119899, Russia
Received 5 December 1996 Editor: PV. Landshoff
Abstract Semileptonic decays D + K, K*, r, p, D, --f q, T’, 4, and B --+ D, D”, n-, p are analyzed within the dispersion formulation of the quark model. The form factors at timelike ~7are derived by the analytical continuation from spacelike CJ in the form factors of the relativistic light-cone quark model. The resulting double spectral representations allow a direct calculation of the form factors in the timelike region. The results of the model are shown to be in good agreement with all available experimental data. From the analysis of the B -+ D, D* decays we find IV&l = 0.037 f 0.004, and the B 4 rr, p decays give l&l = 0.004 f 0.001. PACS: 13.20.-v; 12.39.Hg; 12.39X Keywords: Semileptonic decays; Heavy quarks; Relativistic constituent quark model
The interest in exclusive semileptonic decays of heavy mesons lies in a possibility to obtain the most accurate values of the quark mixing angles and test various approaches to the description of the internal hadron structure. The decay rates are expressed through the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements and hadronic form factors of the weak currents which contain the information on hadron structure. New accurate data on B -+ D, D* and first measurements of the B + rr, p decays open a possibility to determine V& and VUb with high accuracy and require reliable theoretical predictions on the form factors and decay rates. A nonperturbative theoretical study should give these form factors in the whole kinematical region of momentum transfers 0 5 q” 5 (Ml - Mz)~, Ml and Ma the initial and final meson masses, respectively. The amplitudes of meson decays induced by the quark transition qi + qf through the vector VP = &yPqi and axial-vector A, = qfypy5qi currents have the following structure [ 1] : (P(M2,P2)I~(O)IP(Ml,pl))
= .f+Gm,
~V(M2,pz,~)IV,(O)IP(MI,pl)) (v(M2,172.~)
IA,(O)
IP(M1,m
+ f-(q2)q,,
=w12)$w&*VPfY~2P~ )> = ibid
[ .mz2k/m
+ a+(q2)m,P,
1Electronic address: [email protected]. 0370-2693/97/$17.00 0 1997 Published by Elsevier Science B.V. All rights reserved. PZZ SO370-2693(97)00048-S
+ ~-(q2kwp
11
(1)
D. Melikhou / Physics Letters B 394 (19971385-394
386
with q = pt - ~2, P = p1 + ~2. We use the notations: y5 = iy”y1y2y3, .eo123= -1, ,Sp(~~~r”~rP) = 4i@‘“fi. In general all the form factors are independent functions to be calculated within a nonperturbative approach. Considerable simplifications occur when both the parent and the daughter quarks active in the weak transition are heavy. Due to the heavy quark symmetry (HQS) [l] all the form factors which are in general functions of q2 and the masses, depend on the one dimensionless variable w = plpz/MlM2. It is convenient to introduce dimensionless ‘heavy quark’ form factors
g(q2) =
l 2dXK
In the leading l/m~ Isgur-Wise function h+(w)
= &(a)
&(W) 3
a+(q2) = -
1 hPmG
ha(w),
order all the form factors h can be expressed .$ as follows: = h,(w)
= hs(w)
=5(w),
h_(w)
f(q2)
=&i&%&l
+w)hf(co).
through the single universal
(2)
form factor, the
= 0.
(3)
The normalization of the Isgur-Wise function at zero recoil is known, e( 1) = 1. In contrast to meson decays induced by the heavy-to-heavy quark transitions, the general case of the transitions between hadrons with arbitrary masses, and in particular meson decays induced by a heavy-to-light quark transitions are not well-understood. To date, theoretical predictions on semileptonic decays induced by heavy-to-light quark transitions coming from the quark model [ 2-81, QCD sum rules [ 9-131, and lattice calculations [ 14-171 differ significantly. Recently, Stech noticed [ 181 that new relations between the form factors of meson transition can be derived if use is made of the constituent quark picture. These relations are based on the observation that if the meson wave function in terms of its quark constituents is strongly peaked in the momentum space with the width of order A N 0.5 GeV then for a heavy parent quark a small parameter A/m, appears in the picture and one can derive the leading-order expressions for the form factors of interest which turn out to be independent of subtle details of the meson structure. Although these relations give a guideline for the analysis of the decay processes, they cannot substitute calculations of the form factors in a more detailed dynamical model. The kinematically accessible q2-interval in the meson decay induced by the heavy-to-light transition is O(mi) so a relativistic treatment is necessary. A relativistic light-cone quark model (LCQM) [ 51 is an adequate framework for considering decay processes. The model is formulated at spacelike momentum transfers and the direct application of the model at timelike momentum transfer is hampered by pair-creation subprocesses which at q2 > 0 cannot be killed by an appropriate choice of the reference frame. In [5] the form factors at q’ > 0 were obtained by numerical extrapolation from the region q” < 0. As the relevant q2-interval is large, the accuracy of such a procedure is not high. In [ 7,8] the nonpartonic contribution of pair-creation subprocesses is neglected and only the partonic part of the form factor is calculated in the whole kinematical interval of q”. Unfortunately, the nonpartonic contribution is under control only at q2 = 0 where it vanishes. The dispersion formulation of the LCQM proposed in [ 191 overcomes these difficulties as it allows the analytical continuation to the timelike q. We apply this approach to calculating the transition form factors of the semileptonic decays. The transition of the initial meson Q (mz) Q( m3) with the mass Ml to the final meson Q(mz)Q( ms) with the mass M2 induced by the quark transition m2 ---f ml is given by the diagram of Fig. 1. The constituent quark structure of the initial and final mesons are described by the vertices It and I?*, respectively. The initial pseudoscalar meson the vertex has the spinorial structure It = iysGl/&; the final meson vertex has the structure I2 = iy5G2/ fi for a pseudoscalar state and the structure lYzP = [ Ay, + B ( kl - k3 ) p 1 G2 /fi for the vector state. The values A = -1, B = I/ (Js2 + ml + m3) correspond to an S-wave vector meson, and A = l/d,
B = [2&t
ml + ms] / [ fi(
s2 + (ml + m3)2) ] correspond
to a D-wave vector particle.
D. Melikhov/Physics
Letters B 394 (1997) 385-394
387
Fig. 1. One-loop graph for a meson decay.
At q2 < 0 the form factors of the LCQM [5] can be represented as double spectral representations over the invariant masses of the initial and final qq pairs as follows [ 191:
(4)
where sfYs2,
q2) =
sz(rnt + rni - q2) + q2(m: + rnz) - (mf - m$) (rn: - rni) 2rny
is the triangle function. Here fit ( q2) is the form factor of the and h(sl,s2,s3) = (Sl + $2 - S3j2 - 4sls2 constituent quark transition m2 + ml. In what follows we set f2t (8) = 1. The double spectral densities fi( st, s2,$) of the form factors have the following form: ~~++-=4[m~m2c~i-m2m3~i+mim3(l-cr~)-m~(l-a~)+~~2~2],
(5)
~+-f.-=4[m~m2a2-mlm3a2+m2m3(1-a2.)-m~(l-a~)+a~s~],
(6)
g~=--2A[m~a~++~a~+m3(1-a~-a~)]-4B/3,
(7)
ii++ii_
=-4A
[2m2all
ii+ - ii- = -4A[-mla2 f””
=
+2m3(al -mz(al
-a11)1
+4B[Clal
-ms(l-
-2a12)
+C3cull],
LYE -a:!f2ad]
(8) +4B
sfD + (” 1%; ” _ Mf,;, - s3) ;
[Cpl
+C3cyn],
(9)
(10)
M2c+,
where ELI = -4A[mlmzm3
+ z(s2 2
- rn: -m~)+~(s1-m~-m~)-~(s3-m~-m~)+2~(m2-m3)]
+ 4B C3P, ai = [(Si
+s2-
(11) s3jC.72 -mf+mg>
-2s2(s1
-mg+mg)]
/h(sl,s2,s3),
(12)
~2=[(~i+s2-s3)(s1-m~+m~)-2s1(~2-m~+m~)]/~(s~,s~,s3),
(13)
P=a
(14)
[2m:-al(sl
-m~+m~)
-a2(s2--mf+m:)],
D. Melikhov/Physics
388 all
= af
+ 4@2/h(Sl,
S2, S3),
Ci=S2-((mi+m3)2,
Letters B 394 (1997) 385-394
a12=a1a2-2P(sl+s2--3)/A(sl,s2,s3),
C2=s1-(m2-m~i)~,
C3=~3-(ml+m2)~-C~-C~.
(15)
(16)
Let us underline that the representation (4) with the spectral densities (5)~( 10) are just the dispersion form of the corresponding light-cone expressions from [ 51. It is important that double spectral representations without subtractions are valid for all the form factors except f which requires subtractions. In the LCQM the particular form of such a representation for the form factor f depends on the choice of the current component used for its determination and cannot be fixed uniquely. In [20] the behaviour of the form factors of the vector, axial-vector and tensor current has been studied in the limits of heavy-to-heavy and heavy-to-light quark transitions. The analysis of the behavior of the form factor f in the case of a heavy-to-light quark transition suggests another expression [ 201: (17) We shall use both of these prescriptions in the numerical analysis of semileptonic decays. For a pseudoscalar or vector meson with the mass M built up of the constituent quarks mq and mQ, the function G is normalized as follows [ 191:
J
G2(s)ds
A’/2(s,m$mi)
lr( s - h42)2
The quark-meson
G(s)
= gJ
87rs
(s - (m, - mq)2) = 1.
(18)
vertex G can be written as [5]
s2 - (ml - m$2 s - M2 -----w(k), s -
(ml - m2)2
s3/4
k = A’12(s, m:, ms) 2fi ’
(19)
where w(k) is the ground-state S-wave radial wave function. As the analytical continuation of the form factors (4) to the timelike region is performed, in addition to the normal contribution which is just the expression (4) taken at 4” > 0 the anomalous contribution emerges. The corresponding expression is given in [ 191. The normal contribution dominates the form factor at small timelike 4 and vanishes as q2 = (m2 - ml )2 while the anomalous contribution is negligible at small q2 and steeply in the region rises as q2 + (m2 - ml) 2. It should be emphasized that we derive the analytical continuation q2 < (m2 - ml)2. For the constituent quark masses used in the quark models this allows a direct calculation of the form factors of the P -+ V transitions in the whole kinematical decay region 0 I q’ 2 (Mp - Mv)~, as Mp - Mv < m2 - ml. For the P --f P’ transition this is not the case: normally, Mp - Mp > m2 - ml. For the P --+ P’ decays we directly calculate the form factors in the region 0 I q” < (m2 - ml)2 and perform numerical extrapolation in (m2 - ml)2 5 q” 5 (Mp - Mp) 2. 2 Numerical analysis shows the accuracy of this extrapolation procedure to be very high. We would like to notice that the direct calculation shows the derivative of the form factor f+ to be positive at the point q” = (1~12- ml) 2. This suggests that the maximum of the form factor f+ at q2 = (m2 - m1)2 observed in [7,8] is just an artifact of neglecting the nonpartonic contribution to the form factor. For calculating the form factors of semileptonic decays we assume that the wave function w can be approximated by a simple exponential function w(k) = exp( -k2/2p2) and adopt the numerical parameters of the ISGW2 model [ 211 shown in Table 1. * The analytical continuation to q* > (1112- ml)* is also possible. However one should be careful when applying the constituent quark model for such q*: we approach the unphysical q4 threshold q2 = (ml + rn~)~ which is obviously absent in the amplitudes of hadronic processes. This is a sign that we are coming to the region where the constituent quark picture is not adequate.
D. Melikhou / Physics Letters B 394 (1997) 38.5-394
389
Table 1 Parameters of the quark model Ref. ISCWZ
[21]
0.33
0.55
1.82
5.2
Table 2 Parameters of the fits to the calculated D -
f(O) cyr[GeVe2] q [GeVe4]
fk&ax)
Table 3 Decay rates for the D -
0.41
0.44
0.53
0.45
0.56
0.43
0.30
0.33
0.37
K, K” transition form factors
D-+K
D + K*
f+
8
0.781 0.201 0.0086 1.2
0.28 0.24 0.0135 0.35
K, K* transition in 10” s-’
a+ -0.168 0.189 0.001 -0.205
fLC
fHQ
1.747 0.0971 0.001 1.92
1.733 0.0767 0.001 1.86
using IV,,1= 0.975
This work
WSB [2)
ISGW2 1211
Jaus [5]
BBD [9]
EXP. [=I
T(D + K) lJ(D--tK*) VK*)/UK)
8.7 5.38 0.62
7.56 7.73
10.0 5.4
9.6 5.5
6.5 f 1.3 3.7 f 1.2
9.0 + 0.5 5.1 It 0.5
rL/rT
1.31
0.57 1.33
0.57 z!I0.15 0.86 -f 0.06
0.57 xt 0.08 1.15 xlz0.17
The results of calculating f(q2)
= f(O)/]
1 - W2
0.38
1.02
0.54 0.94
the form factors are fitted by the functions + a2q41
better than 0.5% accuracy, and for the form factor f+ this formula is used for numerical extrapolation to the region (ml - ~22)~ 5 q2 5 (Ml - Mz) 2. The decay rates are calculated from the form factors via the formulas from [ 41. Decay rate calculations are performed using the two prescriptions for the form factor f given by the relations (10) and (17); the corresponding results are labelled as LC and HQ, respectively. The decay D + K, K*. These CKM-favoured decays extend the widest possibility for detailed verification of the model. The parameters of the fits to the form factors are given in Table 2. Using the value V,, = 0.975 [25] the decay rates are found to be with
T(D + K) = 8.7 x 10’“s-’ T(D -+ K*) =
5 58 x 101os-l 5:38 x lotOs-1:
rL,lr* r&T
= 1.34 = 1.34
(LC) (HQ)
(20)
Table 3 compares the results of the model with the HQ prescription for the form factor f with the experimental data. One can observe perfect agreement with the data. The results of other approaches which give predictions for a wide set of the semileptonic decay modes are also shown. 77~ decay D -+ v, p. Table 4 present the parameters of the fits to the calculated form factors. Using the value Vcd= 0.22 [ 251 yields the following decay rates: I-(Do --) n--) = 0.62 x 101os-l,
D. Melikhov/Physics
390
Letters B 394 (1997) 385-394
Table 4 Parameters of the fits to the calculated Do -+ T-, p- transition form factors
f(O) cq [GeV-*I cq [ GeVe4] f(4Llx)
Table 5 Decay rates for the Do -+ (n-,
lY(D +n-)
UD +
PI
JTP)lUrr) rL/rT
D+?T
D+P
f+
g
0.681 0.225 0.010 1.63
0.252 0.274 0.017 0.36
p-)e+v
a+ -0.139 0.211 0.012 -0.18
1.326 0.110 0.002 1.52
1.257 0.071 0.003 1.37
transition in lOlo s-l using Iv&, = 0.22
This work
WSB [2]
ISGW2 [21]
Jaus [5]
Ball [ 101
Exp.
0.62 0.26 0.41 1.27
0.68 0.67 0.98 091
0.24 0.12 0.51 0.67
0.8 0.33 0.41 1.22
0.39 f 0.08 0.12 f 0.03 0.3 z!z0.1 1.31 f 0.11
0.92 f 0.45 0.45 f 0.22 0.5 f 0.35 -
Table 6 Parameters of the fits to the calculated D, -+ sS, 4 transition form factors
f(O) LYI[ GeVe2] a2[GeVv4]
T(D”
-+p_)
=
D, -+ SS
D, -+ 4
fi
g
0.800 0.192 0.008
0.266 0.246 0.015
0 30 x 1o’O s-t 0:26x 101os-‘:
a+ -0.149 0.201 0.009
rL/rT = 1.32
IL/IT
= 1.27
(LC), (HQ).
f”:
fUU
1.806 0.111 -0.003
1.710 0.085 0.013
(21)
Table 5 presents the rates for the HQ prescription of the model and the results of other approaches versus experimental data. The experimental results are obtained by combining the decay rates of the D ---f K, K* transition [22] with the following ratios measured by CLEO [ 231 Br( Do --+ r-e+v)/Br( Do + K-e+v) = = 0.088 f 0.062 f 0.028. The 0.103 f 0.039 f 0.013 and E653 [ 241 Br( Do -+ P-&Y) /Br( Do --f K*-e+y) calculated rates seem to be a bit small but nevertheless agree with the experimental values within large errors. The decay D, -+ 77,$, #J. Table 6 presents the results on the form factors. The calculated decay rates depend on the content of q and 7’ mesons and with IV,,l = 0.975 read I(D,
-+ 7) = 0.111 sin2(q)
T(D,
-+ 77’) =0.030
T(D,
+ 4) =
ps-‘,
cos2((p) ps-‘,
0.047 ps-’ , 0.040 ps-1 )
rL/rT rL/rT
= 1.30 = 1.28
(LC), (HQ).
(22)
Here 9 = 0p + arcsin(2/&) [25]. The decay rate of D, -+ q5 calculated with the (HQ) prescription agrees well with the results of the analysis [21] I’( D, -+ qhe+v) = (0.035 f O.O05)ps-‘. Table 7 compares the results on ratios of branching fractions with recent CLEO measurements 1261 and the ISGW2 model. The results for
D. Melikhov / Physics Letters B 394 (1997) 385-394
391
Table I Ratio of the decay rates for the D, + 7, v’. q5 transitions
Ur))lJ--(4) Url’)lUd) r(+)/r(v)
This work 8p = -100
ep = -140
ep =
1.45 0.4 0.27
1.24 0.46 0.37
0.9 0.6 0.67
-200
ISGW2 [21] @p= -100
ep = -200
1.2 0.5
0.8 0.7
Exp.[26]
1.24 f 0.27 0.43 ZII0.18 0.35 f 0.16
Table 8 Parameters of the fits to the calculated B -+ D, D* transition form factors B + D*
B-+D f-
f+
f(O) cy1[GeV-2] cu2[GeVp41 .fMlax)
0.684 0.0386 0.00042 1.12
a+
&z
-0.337 0.039 0.00038 -0.56
0.093 0.0416 0.00048 0.15
-0.0764 0.0387 0.00043 -0.12
f””
fHQ
4.533 0.02193 0.00007 5.85
4.729 0.02195 0.00007 6.10
Table 9 Parameters of the heavy-Quarkform factors for the B + D, D’ transition B 4
B-D h+
h(l) :2
0.96 0.91 0.37
h-
D*
hg
-0.04
0.99 0.54 I .04
0.79 0.63 1.20
h a-
hLC f
Hy hf
0.92 0.50 1.08
0.90 0.55 1.10
0.94 0.53 1.06
aI1 tip in the range -18O 5 BP 5 -10” compare favourably with the data, but the best agreement is observed for ep = -14O. The decay B ---f D, D*. This is a very interesting mode as it allows measuring corrections to the HQS limit. Table 8 shows the fit parameters of the form factors and Table 9 presents the parameters of the fit to the heavy-quark form factors in the form hi(w)
= hi( 1) [l - p;(w
- 1) + S(w - 1)2] .
(23)
We find h+( 1) = 0.96 and h_( 1) = -0.04 which compare favourably with the size of corrections to the HQS limit [30]. The values hyQ( 1) = 0.94 and hLfC( 1) = 0.9 both agree with the estimate of Neubert [30] hf( 1) = 0.93 f0.03. For the ratios of the heavy quark form factors Z?l = h,( l)/hf( 1) and R2 = h,, (l)/hf( 1) we obtain RyQ = 1.05 [Rtc = 1.11 and R,HQ = 0.84 [ Rkc = 0.881 to be compared with a recent CLEO result RI = 1.18f0.15&0.16and R~=0.71f0.22f0.07andpredictionsoftheISGW2model [21] RI = 1.27, R2= 1.01, Neubert [31] RI = 1.35, R2 = 0.79, and Close and Wambach [32] RI = 1.15, RZ = 0.91. CLEO [33] reported the value (&b(hf( 1) = 0.0351 + 0.0019 f 0.0018 + 0.0008. Combining this value with our result hyQ( 1) = 0.94 yields j&l
= 0.0373 A=0.0053
For the decay rates we find
[lepton
endpoint
region in B -+ D*lF] .
D. Meli?dwv / Physics Letters B 394 (1997) 385-394
392
Table 10 Decay rates for the B + D, D* transition in ps-’ This work r(B --+ D) r(B + D*) UD*)/r(D)
WSB [2]
8.7111,61*
Jaus [5]
11.91~b~*
8.11&i*
21.91V&[* 2.71
23.21V&12 2.65 1.28
rL/rT
ISGW2 [21]
Exp. 1.27 2.99 2.35 1.24 0.85
9.61&d*
25.31V&12 2.64
24.81&b/* 2.08 1.04
f zk zk f f
0.3 x IO-* [25] 0.66 x IO-* [27] 1.3 0.16 1281 0.45 [29]
Table 11 Parameters of the fits to the calculated B” -+ rr+, p+ transition form factors B-i%. a+
.f+
f(O)
0.2927 0.0511 0.00068 2.30
GeVp2] cy2[GeVw4] f (&ax) rwl
[
T(B --+ 0)
-0.0256
0.0635 0.0012 0.17
0.0567 0.0010 -0.097
fHQ
1.025 0.032 0.00028 2.19
1.098 0.0316 0.00038 2.12
= 8.712 x 10’2]V,#-1,
T(B -+ D*) =
21.0 x 10’2~v,~~W, C 23.2 x 10’21v,b12 s-l,
Table 10 compares the calculated result with a CLEO measurement yields l&l
0.0356
fLc
= 0.036 zt 0.004
rL/rr rL/rT
= 1.17 = 1.28
(LC), (HQ).
(24)
decay rates for the HQ prescription with other approaches. Combining our [27] f’(B -+ D*Z?) = [29.9 f 1.9(stat) f 2.7(syst) f 2.0(lifetime)]ns-’
[decay rate B -+ D*lii]
.
The branching ratio Br(B’ ---f D-Z+v) = ( 1.9 f OS)% and the B” lifetime rso = (1.56 f 0.06) ps [ 251 give the experimental decay rate I( B” -+ D-Z+v) = ( 1.22 I!Z0.3) 101o~-l and comparing with our (HQ) result yields l&l
= 0.038 f 0.004
All the three estimates excl = 0.037 l&b1
[decay rate B” -+ D-Z+v]
.
of V& agree with each other and the average value is found to be
It 0.004.
(25)
This is in perfect agreement with the updated values [ 341 I&, lexc’ = 0.0373 + 0.0045( exp) & 0.0065( th) and j&blinC1= 0.0398 f O.O008(exp) f O.O004(th). The decay B ---f v, p. This mode allows a determination of (&, I and hence reliability of theoretical predictions is very important. The form factors are given in Table 11. For the decay rates we find r(B”
3 r-)
T(BO -+ p-)
= 7.2 X io121vub12 s-l, =
8.44 x 10’2(v,b12 s-l, 9.64 x 10’2jv,b\2 s-l, C
rJrr rL/rT
= 0.95 = 1.13
(Lc), (HQ).
(26)
D. Melikhov/Physics
Lerters B 394 (1997) 385-394
393
Table 12 Decay rates for the B” -+ ( vf, p+)eF transition in ps-’ This work T(B --f ?r)
7.21&b/’
9.64/&,12 1.34 1.13
UB --t P) UP)lU~) rL/rT
WSB [2] 7.41v,b)*
26.01v,b/2 3.5 1.34
ISGW2 [21] 9.61v,bi2
14.21v,b[* 1.48 0.3
Jaus [5]
Ball
[ 10,131
1&01v,b[*
5.1 f
l.lli’,b1*
19.1~v,~~* 1.91 0.82
12 f 4( 14.5 f 4.5) Iv&l2 2.35 f 1.2 0.06 f 0.02 (0.52 f 0.1)
Exp. [35] 1.2 f 0.6 x 1O-4 1.67 dz 1.0 x 1O-4 -
This calculation is in agreement with our previous analysis of this decay mode using other sets of the quark model parameters [ 191. The calculated decay rates are compared with other theoretical predictions and first measurements by CLEO [ 351 in Table 12. The experimental values are obtained by combining the CLEO results [ 351 Br(B’ -+ r-Z+Y) = ( 1.8 f 0.4 f 0.3 f 0.2) low4 and Br( B” --+ p-Z+V) = (2.5 & 0.4 f 0.7 & 0.5) lop4 with the B” lifetime ~~0 = ( 1.56 & 0.06) ps [ 251. Comparing our results with the experimental values gives l&l
= 0.004 f 0.001
l&l = 0.00407 f 0.001
A good agreement branching
fractions
[B--+gl, [B +
PI-
of these values with each other shows that we have predicted correctly the ratio of the B -+ rr and B -+ p. The average value obtained from the two modes reads
j&J = 0.004 It 0.001. Taking the I&,
from (25) we find
(Vu&&( = 0.108 f 0.02, which is perhaps a bit large but nevertheless agrees with the PDG value I&b/V&l = 0.08 f 0.02. In conclusion, we have analysed semileptonic decays of heavy mesons within dispersion formulation of the constituent quark model and found agreement with all available data. The extracted values of the CKM matrix elements Vcb and Vub are also in agreement with estimates of other models. Nevertheless, we would like to briefly outline possible sources of uncertainties in the predictions of the model which should be taken into account in further analyses: 1. We used the parameters of the ISGW2 quark model and a simplified exponential ansatz for the wave function. It should be noticed however that the ISGW2 model does not calculate the form factors through the wave functions; rather a special prescription for constructing the form factors is formulated. As the analysis of [ 71 shows, the form factors of the heavy-to-light transition can be rather sensitive to the wave function shape. 2. We have taken into account only the leading process and neglected the O(cu,) corrections. Although the analysis of such corrections in the elastic pion form factor at low and high momentum transfers within the LCQM [ 361 found only a few % contribution even at w 11 lo-20 a numerical consideration of this contribution is plausible. 3. We have neglected the constituent quark transition form factor which has a complicated structure at timelike momentum transfers. In particular the quark transition form factor should contain a pole at q2 = M:e, with M,,, the mass of a resonance with appropriate quantum numbers. 4. We have identified the form factors obtained within the constituent quark model with the form factors of the full theory. However, the relationship between these two quantities is nontrivial: e.g. the form factors of the full theory acquire logarithmic corrections because of renormalization of the quark currents, which are absent in the quark model form factors. Analysing the l/rn~ expansion Scora and Isgur [21] performed a special matching procedure for obtaining the form factors of the full theory from their quark-model form factors. Although the
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D. Melikhou/Physics
Letters B 394 (1997) 385-394
1/me behaviour of the LCQM form factors studied in [ 201 is better than that of the generically nonrelativistic form factors in the ISGW model [ 31, the relationship between the LCQM form factors and the form factors of the full theory should be studied in more detail. I am grateful to V. Anisovich, I. Narodetskii, K. Ter-Martirosyan, and B. Stech for discussions and interest in this work. The work was supported by the Russian Foundation for Basic Research under grant 96-02-18121a.
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