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Charmed semileptonic B-meson exclusive decays in a relativistic potential model
Physics Letters B 272 ( 1991 ) 344-352 North-Holland
PHYSICS LETTERS B
Charmed semileptonic B-meson exclusive decays in a relativistic potential model P. C o l a n g e l o ", G . N a r d u l l i ,,b a n d L. T e d e s c o b " INFN, Sezione di Bari, 1-70126 Bari, Italy u Dipartimento di Fisica dell'Universit?t di Bari, 1-70126 Bari, Italy Received 8 May 1991
A QCD-inspired relativistic potential model is applied to the computation of the form factors of the decays B~D£v and B--,D*~v. The value of the Kobayashi-Maskawa matrix element Vcb is determined (V¢b--~0.04), as well as the branching ratios and the asymmetry parameter a = 2FL( B~ D*l~v) ~IT ( B--*D*~v) - 1. The results are in agreement with the experimental data.
The semileptonic exclusive decays o f h e a v y - l i g h t (QO) mesons are o f particular interest since they allow to investigate a n u m b e r o f aspects o f the quark interactions ~. By way o f example, they can be used to d e t e r m i n e elements o f the K o b a y a s h i - M a s k a w a matrix; moreover, if studied by potential models, these decays can shed light on the precise form o f the interquark forces, on the accuracy o f the instantaneous potential interaction and on the possible role o f the relativistic effects due to the lightness o f one o f the quarks. In this letter we shall study the c h a r m e d semileptonic B meson decays
B ~ D~v~,
( 1)
B-. D*~v~,
(2)
in a Q C D - i n s p i r e d relativistic potential m o d e l [2,3]. The analysis o f these decays is performed in terms o f the following amplitudes: G ,~/(B(p) ~ D ( p ' ) ~ ( p , )v~(p2)) = ~ V c b l ~ ( D ( P ' ) I V~ I B ( p ) ) ,
(3)
G d ( B ( p ) ~ D * ( p ' , e)~(p, )v~(P2)) = ~ VcbP'(D*(P', ~)IJu I B ( p ) ) ,
(4)
where G is the F e r m i constant, Vcb is a K o b a y a s h i - M a s k a w a matrix element, l u is the leptonic current /~ = a T . ( 1 - 75)v •
(5)
F o r the h a d r o n i c matrix elements appearing in eqs. ( 3 ), ( 4 ) we a d o p t the following form factors d e c o m p o s i t i o n [4]: (D(p')IVulB(P))=
(P+P')~
MB - - M D M~ - M ~ q2 q~ F t ( q 2 ) + q2 q~ F o ( q 2 ) '
( D * ( p ' , e ) l J ~ l B ( p ) ) = ( D * ( p ' , e)l V , - A u IB ( p ) ) ,
(6) (7)
~ For a review see ref. [ 1]. 344
0370-2693/91/$ 03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved.
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with ( D * ( p ' , e)l V~, IB(p) )
=%,,,p,~.*"pPp'°
2V(q 2)
(8)
MB + MD*
and ( D * ( p ' , e)IA,, ]B(p) ) =-i
[
. . ,{ (p+p'), e*~(MB+MD.)A,(q2)--(e .q)k~A2(q2)+
In eqs. ( 6 ) - ( 9 ) and
MB--MD. A2(q2) .
2MD.
F(q2 ) _
(9)
q=p-p' and the form factors Fj(q2), Aj(q 2) satisfy the relations Fo(0) =F~ (0), A3(0 ) = A o ( 0 )
A3(q2) - MB+MD. Aj(q2) As for the
2MD. )] ~u-q~[A3(q2)-Ao(q2)] .
(10)
2MD.
q2 dependence of such form factors, one can assume the general formula E(0)
l _q2 / m 2,
(11)
which is justified by single pole dominated dispersion relations ~2 On this basis the current matrix elements appearing in eqs. ( 6 ) - ( 9 ) depend on eight parameters: four couplings [Fo(0), V(O),A~(O),Ao(O)] and four pole masses: the 1 (be) mesonmassinF~(q 2) and V(q2),the 0 + (be) meson mass in Fo(q2), the 0 - (be) meson mass inAo(q 2) and the 1 + (be) meson mass in Aj(q 2) 0=1,2,3). Present models [4,5,7,8 ] give predictions for these form factors assuming potential interaction between the quarks composing the mesons. Harmonic oscillator wave functions in the infinite meson m o m e n t u m frame are adopted in refs. [4,5 ], while in ref. [7] approximations in the treatment of the relativistic quark kinematics produce an exponentially d u m p e d q2 behaviour of the form factors. The aim of this letter is to try to improve such computations by discussing a dynamical calculation that takes into account the relativistic kinematics o f the QCl pair and a realistic description of the interquark interaction. We use the model developed in refs. [2,3], where one considers meson wave functions ~' that are solutions of the equation { ~ / ( k + x p ) 2+m~2 + x / [ - k +
(1 - x ) p l Z + m 2 2 - ~ } ~ ' ( k + x p ,
+ f dk' V(p, k, k')~(k',p-k') = 0 .
- k + (1
-xp)) (12)
Eq. ( 12 ) is valid in a moving frame where the meson o f mass M has m o m e n t u m p; x is the fraction of the meson m o m e n t u m carried by the quark of mass m~. V is the instantaneous potential that in the meson rest frame coincides with the Richardson potential ~3 [ 9 ]; in r-space it takes the form
8n
V(r)=33_2n~-~
A(Ar_f(Ar)~ \ Ar } '
13)
where A is a parameter, nr is the number of flavours, and ~2 A different choice (i.e., polar or dipolar behaviour) is made in ref. [5], where the q2 dependence of the form factors is inspired to QCD counting rules [6 ]. ~3 This approximate form of the potential does not take into account spin terms, which should be a good approximation since we only deal with heavy mesons. 345
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oo
1 f(t)-=_4 f dq sin(qt) ( - ~) . ~z q ln(1 + q 2 )
(14)
0
This potential behaves linearly when r--+oo, whereas at small distances it follows perturbative Q C D predictions. For r near the origin one assumes a constant potential
V(r)=V(ri),
47t
r<~rM=23M,
15)
in order to avoid unphysical singularities [ 10 ]; 2 is a parameter whose fitted value is 2 = 0.6. The values of the other parameters are m u = m d = 3 8 MeV, m s = 1 1 5 MeV, m c = 1452 MeV, m b = 4 8 9 0 MeV and A = 3 9 7 MeV; using these parameters, the mass spectrum for the Qel system has been derived [ 2 ]. The calculation of the form factors requires the meson wave functions with p C 0; in other terms we have to boost the solutions 0 (k) = ~u(k, - k ) obtained in the rest frame by eq. ( 12 ). Here we encounter two difficulties. First, we do not know the transformation properties of the potential V, given its approximate and phenomenological nature. Second, we cannot expect full relativistic invariance since the potential model arises from the fully-fledged relativistic quantum field theory by restricting the Fock space to quark-antiquark pairs, and by assuming an instantaneous interaction. The problem o f finding solutions of relativistic wave equations in a general (p • 0) reference frame is an old one [ 11 ]. But for special cases (for instance the Lorentz boost can be performed for the relativistic harmonic oscillator wave functions), one can only find ad hoc solutions assuming transformation properties of the potential and the wave functions [ 12 ]. This procedure is unsatisfactory due to its arbitrariness; therefore we choose to follow a different strategy. First of all we do not attempt to predict the wave function for any value of p C 0, but we limit ourselves to an infinitesimal Lorentz boost, so that we shall neglect terms O ( p 2) in the following. As a consequence, we shall write
~'(k+ xp, - k + ( 1 - x ) p ) ~-(/)(k) +p.k g(k, x ) ,
(16)
where the ~a(k) have been determined in refs. [2,3 ]; on the other hand g(k, x) is an unknown function. Instead of trying to derive it from transformation properties of the potential V, we choose to minimize its effect by an appropriate choice of the parameter x ~4. As a matter of fact if we put
x=X,
(17)
where ~ is the value which minimizes the function g ( x ) = max Ig(k,x) l, k
(18)
then g(k, x) will be uniformly small in k and we shall not make a large error in neglecting it altogether in the final formulae. As a consequence physical quantities will be expressed only in terms o f the known rest frame wave functions ~ (k). Clearly this procedure is justified provided it is compatible with (at least) approximately relativistic invariance; in other words one has to check that scalar quantities computed in two different reference frames by this method are approximately equal. In order to determine )~ for B and D mesons we proceed as follows. We observe that, since in our approximation the potential V(r) does not contain spin terms, both D and D* (B and B*) have the same )?, namely xD (Xn). Let us now consider the current particle matrix elements ( 0 IA ij(0) I M ( p ) ) = ip~fe,
(19)
(01V~y(O) IV(p, e) ) = eZfv,
(20)
~4 This procedure has a counterpart in the non-relativistic case where x = m~/(m~+m2) so that the Schr6dinger equation can be separated. 346
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where the pseudoscalar and vector meson states are given by [ M ( p ) ) = i F o . 6<<1~ / ~ x fi.s ~ f dk~u(k+xp,-k+(1-x)p)b~-(k+xp,
r,a)d~-(-k+(1-x)p,s,
fl)[O),
(21)
IV(p, e) > =F,j d.p ( - e~a~),.~ f dk ~u(h÷xp, - k + ( 1 - x ) p ) b + (k+xp, r, a ) d + ( - k + ( 1 - x ) p , s, fl)[0) (22) where F,: is the flavour matrix, a and fl are colour indices, r and s are spin indices. The normalization of the wave function ~u(k+xp, - k + (1 - x ) p ) is
'I
(2~.)3
dk [~'12=2x/p2÷M2 .
(23)
On the other hand the vector and axial vector currents are given by
V/,=a~:j# f d q d q ' (
.1/2
m,m:
(~)3 kE,(q)E,(q')]
× : [/~i(q, r)b + (q, r, a ) + Oi(q, r)d,(q, r, a ) ] y/' [u:(q', s)bj(q', s, fl)+ vj(q', s)d + (q', s, fl)]: ,
A ~ _ . ~ . f dqdq' ( mimj -~'"~a° J (2~z)3 \E,(q)Ej(q')J
(24a)
1/2
: [a,(q, r)b + (q, r, ce) + v,(q, r)d,(q, r, a) ]~,q,5 [uj(q', s)b~(q', s, fl) + v:(q', s)cl 7 (q', s, fl) ]: ,
(24b)
where Ej(q) = x/q5 + m } and the sum over repeated indices is understood. We can now compute eqs. (19) and (20) in two different frames: the meson rest frame ( p = 0 ) and a moving frame with infinitesimal [p[. By using canonical anticommutation relations for the annihilation and creation operators appearing in ( 2 1 ) - ( 2 4 ) we get, in the rest frame [2,3]
f,=~
dkka(k)N'/2(k)
1-
,
(25)
'
(26)
0
k2 dkk~(k)NWZ(k)
fv=~
1+ 3(Ei+mi)(Ej+mj)
0
where ~(k)=k~(k)/x/27~ , M is the meson mass, ~ = T r ( J F )
is
a
Clebsch-Gordan coefficient and
N(k)={[E,(k)+mj]/Ei(k)}{[Ej(k)+mj]/Ej(k)}. On the other hand, in the moving frame with infinitesimal JPl we get fv = E ( x ) + ~)E(x) ,
(27a)
fv - E ( x ) + 8E'(x) ,
(27b)
Mv
where E(x)=~3fdkkgt(k)N,/2 k2(
1
1
F 1-x + x kEi + mi Ej + mj ~(mi(1-x)
+6 E/+mj E,+m,J\-E2~-E~-~mi)
k2
1 --x
x
3 ( E , ( E i + m , ) 2 + E : ( E j + m j ) 2) mix
~]
E}(~,+mj):_l
(28)
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and 5 E ( x ) , 8E'(x) are correcting factors containing the integral of the unknown function g(k, x). We now impose that g(k, x) is computed for x = £ , or, in other terms, that [ S E ( x ) ] 2+ [ S E ' ( x ) ] 2 has a minimum; in this way we obtain for the fractional m o m e n t u m carried by the heavy quark in the (c0) and (bfi) system respectively XD =0.82,
Xa = 0 . 8 8 .
(29)
We note that these large values of the fractional m o m e n t a are reasonable, since we expect that in heavy-light quark mesons the meson m o m e n t u m is carried on the average more by the heavy than by the light quark. If we now neglect 8E(xD ), 6E' (XD), 5E(xB), 5E' (XB), given their expected smallness, we obtain the results
fD ~--E(xD), fD. ~-MD.E(xo) ,
(30a)
fB ~--E(xB), fB. ~-M~.E(xB).
(30b)
Numerically we find E(xD)=0.222 GeV, E(xB)=0.257 GeV; by comparing these findings with the results obtained in the meson rest frame, i.e., from eqs. (25), (26) ( f o = 0 . 1 7 GeV, fD. = 0 . 5 3 GeV2, f a = 0 . 2 3 GeV, fB. = 1.51 GeV 2) [3], we get deviations of 10% in the B case and of 18% in the D case. These are the typical errors introduced by neglecting the function g(k, x) in eq. ( 16); we expect similar errors when this procedure is applied to other matrix elements, as in the evaluation of eqs. ( 6 ) - ( 9 ) . We note explicitly that the errors decrease when mQ increases ~5. Using eqs. (21 ) - ( 2 4 ) one can write the current matrix elements ( 6 ) - ( 9 ) as overlaps of the B and D (or D*) wave functions computed for different values of their argument. There is a particular frame where these expressions become simpler and both wave functions are expressed as ~,(k + xp, - k + ( 1 - x ) p ) , where p is the meson m o m e n t u m and x the active quark fractional momentum. This happens when ( 1- x a ) P = ( 1--
XD)p'
(31)
,
with p u = (x/m~8 + p 2, p ) and p '" = ( x / m 2. + p , 2, p, ). In this reference frame we can use the approximation ( 16 ) and the results (29) for the fractional m o m e n t a xB and xD carried by the active quarks in the B and D (or D*) mesons. By considering the limit ]p], ]p'[ --,0 we obtain the following results for the form factors computed at 2 = ( M B - M D ) 2 = t or qmax 2 = (MB__MD.)2=t* for the transitions B ~ D and B - , D * respecq 2 = q .2. . . where qma× tively (we neglect lepton masses): Fo(t) = 2(MB 1+ M D )
Fl(t)
( 1+ --11 -- xXDn
i d k ~ a ( k ) ~ , ( k ) N 1/2( l + 0 Xa--XD MB 1--XD MB
+Fo(t)
k2
),
(Eb+mb)(Ec+mc)
MB-MD
li
1--XD --2
(32)
dk~B(k)~(k)N1/ZG+(u)
,
(33)
0 oo
MB + M D . ! dk ft.(k)~t~(k)N t/2(k)G_ (k) , V ( t * ) = 4{MD.-- [ ( 1 - - x a ) / ( 1 - - X D ) ] m , }
A~(t*)-- 2 ( M . +MD.)I
; ( dk ~B(k)Ft~(k)N ~/2 1 0
k2 3(Eb + m b ) ( E c + m c )
(34)
) ,
(35)
,5 As a further check of this procedure we have calculated the matrix element ( 0 t qau.QIv (p, E) ) in two frames (p = 0 and p :~0) and we obtain relativistic invariance with an accuracy similar to eq. (30). 348
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PHYSICS LETTERS B
MDD
A2(t*)+ MB--MD,
5 December 1991
[A3(t*)--Ao(t*) ]
l l--XD -- 2 I--XB --J d k ~ B ( k ) ~ ( k ) N ~ / 2 G + ( k ) '
(36)
0
where xB + 1 -xB XD k 2( XB 1-XB XD G+_ - Eb..t..m b _ I--ND E c + m c -- 3 - \ ~ E b ( E b + m b ) 2 -+ I--XD E c ( E ~ T m c ) 2]
omc 6
~
+
- Ec+m~
\E~,(Eu+mb)
+ -1 --x D E ~ ( E ~ + m c ) J
(37)
"
The results ( 3 2 ) - ( 3 7 ) are valid for the transition b--, e; for the transition b-~ c all the form factors except V(q2) change their sign. We also observe that the couplings F o ( t ) a n d AI (t*) can be c o m p u t e d both in the frame p = p ' = 0 and in the frame defined by eq. (31 ), a n d they have the same expressions up to O (p2). Therefore they are on a sounder theoretical basis than the other ones; in particular we expect that the theoretical error for the Fo (q2) and A ~(q2) form factors should not be larger than 10%. F r o m eqs. ( 3 2 ) - ( 3 6 ) we get five relations between the eight unknown couplings and masses o f form factors. The r e m a i n i n g p a r a m e t e r s can be fixed a d o p t i n g the general criterion that the pole masses are the ones predicted by single pole d o m i n a t e d relations, with meson masses c o m p u t e d in our model. In particular, the condition F o ( 0 ) = F t ( 0 ) poses a strong constraint on the pole masses o f the form factors Fo and Ft. This constraint is satisfied for masses that differ by 15% from the quark model predictions. All the results are displayed in table 1. We now c o m p a r e our results with experimental data. The c o m p u t e d branching ratio for the decay B - , D e v is given by (we use rB= ( 1.24 + 0.09) × 10- ~2 s [ 13 ] ) B R ( B - . D e v ) = 10.6 I Vcbl 2 ,
(38)
to be c o m p a r e d with [ 14,15 ] BR(B °~D+e-ge)=
(1.75+0.42+0.35) N 10-2,
(39a)
BR ( B - ~ D ° e - g e ) = ( 1 . 6 0 + 0 . 6 + 0 . 2 ) × 10 -2 .
(39b)
This gives, from the first channel, ] gcb I =0.041 + 0 . 0 0 6
(40a)
Table 1 Parameters of the form factors for the transitions B--,D and B--,D*. Form factor
Value at q2=O
Pole mass (GeV)
Fo(q 2)
0.69 0.69 0.84 0.81 0.65 0.45 0.81
7.60 5.54 5.54 6.30 6.73 6.73 6.73
Fl (q2) V(q ~) Ao(q 2) A~(q 2) Az(q 2)
A3(q 2)
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and, from the second channel, [ rcb [ =0.039+_0.008.
(40b)
Let us consider the B ~ D*~v decays. Using standard formulae [ 16 ] and the values of Vcu in eq. (40) we get BR(B~D*~v) = 5.2× 10 z,
(41)
to be compared with the experimental results [ 13-15 ] B R ( B 0 ~ D * + ~ - 9 ) = ( 4 . 8 _ + 0 . 4 + _ 0 . 7 ) × 1 0 -2,
BR(B-~D*°~-9)=(4.1_+0.8_+0.8)×10 -2.
(42)
Moreover, we can also compute the asymmetry parameter ~ = 2 -FL(B°~D*+I~-9) - 1 , Fx(B° ~ D * + ~ - 9 )
(43)
where Fc and Fv refer to the longitudinal and transverse polarization of D*; we get c~=0.71
(44)
taking into account the experimental cutoffp~ > 1.4 GeV for the charged lepton momentum. The experimental results are c~=0.65 _+0.66_+0.25 [ 15], and c~=0.7_+0.9 [ 14] (with p~> 1.0 GeV). We can draw our conclusions. We have computed the form factors for the charmed semileptonic exclusive B decays using a QCD relativistic potential model and we have obtained results in agreement with the experimental data. Our results are also in general agreement with the findings of other potential models and with the calculations based on QCD sum rules [ 17 ]. We shall present a detailed comparison elsewhere. As for the predictions of the present model in the infinite heavy quark limit mb = mc = M-, oo, they turn out to be in agreement with general results [ 18,19 ], as discussed in the appendix. An advantage of the model calculation of the form factors we have presented in this letter is that the mass corrections to this relevant limit can be explicitly computed. Finally, it is worth observing that the value Vcb 2 0.04 obtained by the exclusive channels ( 1 ), (2) is sensibly lower (although compatible within the errors) than the value l / c b = 0 . 0 4 7 _+0.004 given by the analysis of the semileptonic inclusive B~Xcev spectrum by a QCD calculation which includes B meson wave function effects [ 3 ]. This result reflects a common trend [ 13 ] whose origin should be further investigated. U seful discussions with N. Paver are gratefully acknowledged. One of us (P.C.) would like to thank Professor V.K. Kadyshevsky for warm hospitality at the Laboratory of Theoretical Physics, JINR, Dubna, USSR.
Appendix. We wish to consider eqs.
( 3 2 ) - ( 3 7 ) in the S U ( 2 ) × S U ( 2 ) spin-flavour symmetry [18] charac-
terized by mb =
m c = M--,oc
(A. 1 )
MB =MD =MD. = M .
(A.2)
and
From the wave equation (12) one observes that, for each value of the momentum M-. oo. Moreover, using the method of section III of ref. [3 ], it can be shown that
k, k2gt(k)/M 2-,0 when
f dk~t~(k)~D(k)=2~[l+O(~---S)], 0
which, in the symmetry limit, reduces to the wave function normalization condition [see eq. (23) ]
350
(A.3)
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i dk t~2 ( k ) = 2 M .
(A.4)
0
Therefore, f r o m eqs. ( 3 2 ) - ( 3 5 ) o n e o b t a i n s
Fo(t) =
MB+MD
(A.5)
Aj ( t * ) -
2MB ~+ M D . I 1 q- 0(~------~)1 = 1 4- O ( ~ 2 )
(A.6)
whereas the l e a d i n g O ( 1 / M ) t e r m s are a b s e n t since there are no k/M corrections to the heavy q u a r k limit in these e q u a t i o n s . Let us observe that t = t*=qZmax in the s y m m e t r y limit. M o r e o v e r , f r o m the exact r e l a t i o n Fo ( 0 ) = F~ ( 0 ) a n d f r o m eq. ( 11 ) o n e gets F, ( t ) = l
+ O(~--52 ) .
(A.7)
Eqs. ( A . 5 ) - ( A . 7 ) satisfy the g e n e r a l i z a t i o n o f the A d e m o l l o - G a t t o t h e o r e m [20 ] valid for the heavy q u a r k s y m m e t r y . As for the other f o r m factors, we observe that, in the s y m m e t r y limit, xB = X D ~ 1 because o f the results ( A . 5 ) - ( A . 7 ) a n d eq. ( 3 3 ) . Therefore, f r o m eqs. ( 3 4 ) , ( 3 6 ) o n e gets i m m e d i a t e l y
v(t*) =
1,
A3(t*) = A o ( t * )
(A.8)
= 1,
(A.9)
in a g r e e m e n t with the i n f i n i t e h e a v y q u a r k l i m i t results [ 19]; the p r e d i c t i o n for A 2 follows f r o m eq. ( 1 0 ) .
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[ 16 ] J.G. K6rner and G.A. Schuler, Z. Phys. C. 46 (1990) 93; M. Bauer and M. Wirbel, Z. Phys. C 42 (1989) 671. [ 17] A.A. Ovchinnikov and V.A. Slobodenyuk, Z. Phys. C. 44 (1989) 433; P. Colangelo, G. Nardulli, A.A. Ovchinnikov and N. Paver, Phys. Lett. B 269 ( 1991 ) 201. [ 18] M.B. Voloshin and M.A. Shifman, Yad. Fiz. 47 (1988) 810 [Sov. J. Nucl. Phys. 47 (1988) 511 ]; N. lsgur and M. Wise, Phys. Lett. B 232 ( 1989 ) 113; B 237 (1990) 527. [ 19] M. Luke, Phys. Lett. B 252 (1990) 447; C. Glenn Boyd and D.E. Brahm, Phys. Lett. B 257 ( 1991 ) 393; R.F. Lebed and M. Suzuki, Phys. Rev. D 44 ( 1991 ) 829; M. Neuberl and V. Riecken, preprint HD-THEP-91-6 ( 1991 ). [20] M. Ademollo and R. Gatto, Phys. Rev. Lelt. 13 (1964) 264.
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PHYSICS LETTERS B
Charmed semileptonic B-meson exclusive decays in a relativistic potential model P. C o l a n g e l o ", G . N a r d u l l i ,,b a n d L. T e d e s c o b " INFN, Sezione di Bari, 1-70126 Bari, Italy u Dipartimento di Fisica dell'Universit?t di Bari, 1-70126 Bari, Italy Received 8 May 1991
A QCD-inspired relativistic potential model is applied to the computation of the form factors of the decays B~D£v and B--,D*~v. The value of the Kobayashi-Maskawa matrix element Vcb is determined (V¢b--~0.04), as well as the branching ratios and the asymmetry parameter a = 2FL( B~ D*l~v) ~IT ( B--*D*~v) - 1. The results are in agreement with the experimental data.
The semileptonic exclusive decays o f h e a v y - l i g h t (QO) mesons are o f particular interest since they allow to investigate a n u m b e r o f aspects o f the quark interactions ~. By way o f example, they can be used to d e t e r m i n e elements o f the K o b a y a s h i - M a s k a w a matrix; moreover, if studied by potential models, these decays can shed light on the precise form o f the interquark forces, on the accuracy o f the instantaneous potential interaction and on the possible role o f the relativistic effects due to the lightness o f one o f the quarks. In this letter we shall study the c h a r m e d semileptonic B meson decays
B ~ D~v~,
( 1)
B-. D*~v~,
(2)
in a Q C D - i n s p i r e d relativistic potential m o d e l [2,3]. The analysis o f these decays is performed in terms o f the following amplitudes: G ,~/(B(p) ~ D ( p ' ) ~ ( p , )v~(p2)) = ~ V c b l ~ ( D ( P ' ) I V~ I B ( p ) ) ,
(3)
G d ( B ( p ) ~ D * ( p ' , e)~(p, )v~(P2)) = ~ VcbP'(D*(P', ~)IJu I B ( p ) ) ,
(4)
where G is the F e r m i constant, Vcb is a K o b a y a s h i - M a s k a w a matrix element, l u is the leptonic current /~ = a T . ( 1 - 75)v •
(5)
F o r the h a d r o n i c matrix elements appearing in eqs. ( 3 ), ( 4 ) we a d o p t the following form factors d e c o m p o s i t i o n [4]: (D(p')IVulB(P))=
(P+P')~
MB - - M D M~ - M ~ q2 q~ F t ( q 2 ) + q2 q~ F o ( q 2 ) '
( D * ( p ' , e ) l J ~ l B ( p ) ) = ( D * ( p ' , e)l V , - A u IB ( p ) ) ,
(6) (7)
~ For a review see ref. [ 1]. 344
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with ( D * ( p ' , e)l V~, IB(p) )
=%,,,p,~.*"pPp'°
2V(q 2)
(8)
MB + MD*
and ( D * ( p ' , e)IA,, ]B(p) ) =-i
[
. . ,{ (p+p'), e*~(MB+MD.)A,(q2)--(e .q)k~A2(q2)+
In eqs. ( 6 ) - ( 9 ) and
MB--MD. A2(q2) .
2MD.
F(q2 ) _
(9)
q=p-p' and the form factors Fj(q2), Aj(q 2) satisfy the relations Fo(0) =F~ (0), A3(0 ) = A o ( 0 )
A3(q2) - MB+MD. Aj(q2) As for the
2MD. )] ~u-q~[A3(q2)-Ao(q2)] .
(10)
2MD.
q2 dependence of such form factors, one can assume the general formula E(0)
l _q2 / m 2,
(11)
which is justified by single pole dominated dispersion relations ~2 On this basis the current matrix elements appearing in eqs. ( 6 ) - ( 9 ) depend on eight parameters: four couplings [Fo(0), V(O),A~(O),Ao(O)] and four pole masses: the 1 (be) mesonmassinF~(q 2) and V(q2),the 0 + (be) meson mass in Fo(q2), the 0 - (be) meson mass inAo(q 2) and the 1 + (be) meson mass in Aj(q 2) 0=1,2,3). Present models [4,5,7,8 ] give predictions for these form factors assuming potential interaction between the quarks composing the mesons. Harmonic oscillator wave functions in the infinite meson m o m e n t u m frame are adopted in refs. [4,5 ], while in ref. [7] approximations in the treatment of the relativistic quark kinematics produce an exponentially d u m p e d q2 behaviour of the form factors. The aim of this letter is to try to improve such computations by discussing a dynamical calculation that takes into account the relativistic kinematics o f the QCl pair and a realistic description of the interquark interaction. We use the model developed in refs. [2,3], where one considers meson wave functions ~' that are solutions of the equation { ~ / ( k + x p ) 2+m~2 + x / [ - k +
(1 - x ) p l Z + m 2 2 - ~ } ~ ' ( k + x p ,
+ f dk' V(p, k, k')~(k',p-k') = 0 .
- k + (1
-xp)) (12)
Eq. ( 12 ) is valid in a moving frame where the meson o f mass M has m o m e n t u m p; x is the fraction of the meson m o m e n t u m carried by the quark of mass m~. V is the instantaneous potential that in the meson rest frame coincides with the Richardson potential ~3 [ 9 ]; in r-space it takes the form
8n
V(r)=33_2n~-~
A(Ar_f(Ar)~ \ Ar } '
13)
where A is a parameter, nr is the number of flavours, and ~2 A different choice (i.e., polar or dipolar behaviour) is made in ref. [5], where the q2 dependence of the form factors is inspired to QCD counting rules [6 ]. ~3 This approximate form of the potential does not take into account spin terms, which should be a good approximation since we only deal with heavy mesons. 345
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oo
1 f(t)-=_4 f dq sin(qt) ( - ~) . ~z q ln(1 + q 2 )
(14)
0
This potential behaves linearly when r--+oo, whereas at small distances it follows perturbative Q C D predictions. For r near the origin one assumes a constant potential
V(r)=V(ri),
47t
r<~rM=23M,
15)
in order to avoid unphysical singularities [ 10 ]; 2 is a parameter whose fitted value is 2 = 0.6. The values of the other parameters are m u = m d = 3 8 MeV, m s = 1 1 5 MeV, m c = 1452 MeV, m b = 4 8 9 0 MeV and A = 3 9 7 MeV; using these parameters, the mass spectrum for the Qel system has been derived [ 2 ]. The calculation of the form factors requires the meson wave functions with p C 0; in other terms we have to boost the solutions 0 (k) = ~u(k, - k ) obtained in the rest frame by eq. ( 12 ). Here we encounter two difficulties. First, we do not know the transformation properties of the potential V, given its approximate and phenomenological nature. Second, we cannot expect full relativistic invariance since the potential model arises from the fully-fledged relativistic quantum field theory by restricting the Fock space to quark-antiquark pairs, and by assuming an instantaneous interaction. The problem o f finding solutions of relativistic wave equations in a general (p • 0) reference frame is an old one [ 11 ]. But for special cases (for instance the Lorentz boost can be performed for the relativistic harmonic oscillator wave functions), one can only find ad hoc solutions assuming transformation properties of the potential and the wave functions [ 12 ]. This procedure is unsatisfactory due to its arbitrariness; therefore we choose to follow a different strategy. First of all we do not attempt to predict the wave function for any value of p C 0, but we limit ourselves to an infinitesimal Lorentz boost, so that we shall neglect terms O ( p 2) in the following. As a consequence, we shall write
~'(k+ xp, - k + ( 1 - x ) p ) ~-(/)(k) +p.k g(k, x ) ,
(16)
where the ~a(k) have been determined in refs. [2,3 ]; on the other hand g(k, x) is an unknown function. Instead of trying to derive it from transformation properties of the potential V, we choose to minimize its effect by an appropriate choice of the parameter x ~4. As a matter of fact if we put
x=X,
(17)
where ~ is the value which minimizes the function g ( x ) = max Ig(k,x) l, k
(18)
then g(k, x) will be uniformly small in k and we shall not make a large error in neglecting it altogether in the final formulae. As a consequence physical quantities will be expressed only in terms o f the known rest frame wave functions ~ (k). Clearly this procedure is justified provided it is compatible with (at least) approximately relativistic invariance; in other words one has to check that scalar quantities computed in two different reference frames by this method are approximately equal. In order to determine )~ for B and D mesons we proceed as follows. We observe that, since in our approximation the potential V(r) does not contain spin terms, both D and D* (B and B*) have the same )?, namely xD (Xn). Let us now consider the current particle matrix elements ( 0 IA ij(0) I M ( p ) ) = ip~fe,
(19)
(01V~y(O) IV(p, e) ) = eZfv,
(20)
~4 This procedure has a counterpart in the non-relativistic case where x = m~/(m~+m2) so that the Schr6dinger equation can be separated. 346
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where the pseudoscalar and vector meson states are given by [ M ( p ) ) = i F o . 6<<1~ / ~ x fi.s ~ f dk~u(k+xp,-k+(1-x)p)b~-(k+xp,
r,a)d~-(-k+(1-x)p,s,
fl)[O),
(21)
IV(p, e) > =F,j d.p ( - e~a~),.~ f dk ~u(h÷xp, - k + ( 1 - x ) p ) b + (k+xp, r, a ) d + ( - k + ( 1 - x ) p , s, fl)[0) (22) where F,: is the flavour matrix, a and fl are colour indices, r and s are spin indices. The normalization of the wave function ~u(k+xp, - k + (1 - x ) p ) is
'I
(2~.)3
dk [~'12=2x/p2÷M2 .
(23)
On the other hand the vector and axial vector currents are given by
V/,=a~:j# f d q d q ' (
.1/2
m,m:
(~)3 kE,(q)E,(q')]
× : [/~i(q, r)b + (q, r, a ) + Oi(q, r)d,(q, r, a ) ] y/' [u:(q', s)bj(q', s, fl)+ vj(q', s)d + (q', s, fl)]: ,
A ~ _ . ~ . f dqdq' ( mimj -~'"~a° J (2~z)3 \E,(q)Ej(q')J
(24a)
1/2
: [a,(q, r)b + (q, r, ce) + v,(q, r)d,(q, r, a) ]~,q,5 [uj(q', s)b~(q', s, fl) + v:(q', s)cl 7 (q', s, fl) ]: ,
(24b)
where Ej(q) = x/q5 + m } and the sum over repeated indices is understood. We can now compute eqs. (19) and (20) in two different frames: the meson rest frame ( p = 0 ) and a moving frame with infinitesimal [p[. By using canonical anticommutation relations for the annihilation and creation operators appearing in ( 2 1 ) - ( 2 4 ) we get, in the rest frame [2,3]
f,=~
dkka(k)N'/2(k)
1-
,
(25)
'
(26)
0
k2 dkk~(k)NWZ(k)
fv=~
1+ 3(Ei+mi)(Ej+mj)
0
where ~(k)=k~(k)/x/27~ , M is the meson mass, ~ = T r ( J F )
is
a
Clebsch-Gordan coefficient and
N(k)={[E,(k)+mj]/Ei(k)}{[Ej(k)+mj]/Ej(k)}. On the other hand, in the moving frame with infinitesimal JPl we get fv = E ( x ) + ~)E(x) ,
(27a)
fv - E ( x ) + 8E'(x) ,
(27b)
Mv
where E(x)=~3fdkkgt(k)N,/2 k2(
1
1
F 1-x + x kEi + mi Ej + mj ~(mi(1-x)
+6 E/+mj E,+m,J\-E2~-E~-~mi)
k2
1 --x
x
3 ( E , ( E i + m , ) 2 + E : ( E j + m j ) 2) mix
~]
E}(~,+mj):_l
(28)
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and 5 E ( x ) , 8E'(x) are correcting factors containing the integral of the unknown function g(k, x). We now impose that g(k, x) is computed for x = £ , or, in other terms, that [ S E ( x ) ] 2+ [ S E ' ( x ) ] 2 has a minimum; in this way we obtain for the fractional m o m e n t u m carried by the heavy quark in the (c0) and (bfi) system respectively XD =0.82,
Xa = 0 . 8 8 .
(29)
We note that these large values of the fractional m o m e n t a are reasonable, since we expect that in heavy-light quark mesons the meson m o m e n t u m is carried on the average more by the heavy than by the light quark. If we now neglect 8E(xD ), 6E' (XD), 5E(xB), 5E' (XB), given their expected smallness, we obtain the results
fD ~--E(xD), fD. ~-MD.E(xo) ,
(30a)
fB ~--E(xB), fB. ~-M~.E(xB).
(30b)
Numerically we find E(xD)=0.222 GeV, E(xB)=0.257 GeV; by comparing these findings with the results obtained in the meson rest frame, i.e., from eqs. (25), (26) ( f o = 0 . 1 7 GeV, fD. = 0 . 5 3 GeV2, f a = 0 . 2 3 GeV, fB. = 1.51 GeV 2) [3], we get deviations of 10% in the B case and of 18% in the D case. These are the typical errors introduced by neglecting the function g(k, x) in eq. ( 16); we expect similar errors when this procedure is applied to other matrix elements, as in the evaluation of eqs. ( 6 ) - ( 9 ) . We note explicitly that the errors decrease when mQ increases ~5. Using eqs. (21 ) - ( 2 4 ) one can write the current matrix elements ( 6 ) - ( 9 ) as overlaps of the B and D (or D*) wave functions computed for different values of their argument. There is a particular frame where these expressions become simpler and both wave functions are expressed as ~,(k + xp, - k + ( 1 - x ) p ) , where p is the meson m o m e n t u m and x the active quark fractional momentum. This happens when ( 1- x a ) P = ( 1--
XD)p'
(31)
,
with p u = (x/m~8 + p 2, p ) and p '" = ( x / m 2. + p , 2, p, ). In this reference frame we can use the approximation ( 16 ) and the results (29) for the fractional m o m e n t a xB and xD carried by the active quarks in the B and D (or D*) mesons. By considering the limit ]p], ]p'[ --,0 we obtain the following results for the form factors computed at 2 = ( M B - M D ) 2 = t or qmax 2 = (MB__MD.)2=t* for the transitions B ~ D and B - , D * respecq 2 = q .2. . . where qma× tively (we neglect lepton masses): Fo(t) = 2(MB 1+ M D )
Fl(t)
( 1+ --11 -- xXDn
i d k ~ a ( k ) ~ , ( k ) N 1/2( l + 0 Xa--XD MB 1--XD MB
+Fo(t)
k2
),
(Eb+mb)(Ec+mc)
MB-MD
li
1--XD --2
(32)
dk~B(k)~(k)N1/ZG+(u)
,
(33)
0 oo
MB + M D . ! dk ft.(k)~t~(k)N t/2(k)G_ (k) , V ( t * ) = 4{MD.-- [ ( 1 - - x a ) / ( 1 - - X D ) ] m , }
A~(t*)-- 2 ( M . +MD.)I
; ( dk ~B(k)Ft~(k)N ~/2 1 0
k2 3(Eb + m b ) ( E c + m c )
(34)
) ,
(35)
,5 As a further check of this procedure we have calculated the matrix element ( 0 t qau.QIv (p, E) ) in two frames (p = 0 and p :~0) and we obtain relativistic invariance with an accuracy similar to eq. (30). 348
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PHYSICS LETTERS B
MDD
A2(t*)+ MB--MD,
5 December 1991
[A3(t*)--Ao(t*) ]
l l--XD -- 2 I--XB --J d k ~ B ( k ) ~ ( k ) N ~ / 2 G + ( k ) '
(36)
0
where xB + 1 -xB XD k 2( XB 1-XB XD G+_ - Eb..t..m b _ I--ND E c + m c -- 3 - \ ~ E b ( E b + m b ) 2 -+ I--XD E c ( E ~ T m c ) 2]
omc 6
~
+
- Ec+m~
\E~,(Eu+mb)
+ -1 --x D E ~ ( E ~ + m c ) J
(37)
"
The results ( 3 2 ) - ( 3 7 ) are valid for the transition b--, e; for the transition b-~ c all the form factors except V(q2) change their sign. We also observe that the couplings F o ( t ) a n d AI (t*) can be c o m p u t e d both in the frame p = p ' = 0 and in the frame defined by eq. (31 ), a n d they have the same expressions up to O (p2). Therefore they are on a sounder theoretical basis than the other ones; in particular we expect that the theoretical error for the Fo (q2) and A ~(q2) form factors should not be larger than 10%. F r o m eqs. ( 3 2 ) - ( 3 6 ) we get five relations between the eight unknown couplings and masses o f form factors. The r e m a i n i n g p a r a m e t e r s can be fixed a d o p t i n g the general criterion that the pole masses are the ones predicted by single pole d o m i n a t e d relations, with meson masses c o m p u t e d in our model. In particular, the condition F o ( 0 ) = F t ( 0 ) poses a strong constraint on the pole masses o f the form factors Fo and Ft. This constraint is satisfied for masses that differ by 15% from the quark model predictions. All the results are displayed in table 1. We now c o m p a r e our results with experimental data. The c o m p u t e d branching ratio for the decay B - , D e v is given by (we use rB= ( 1.24 + 0.09) × 10- ~2 s [ 13 ] ) B R ( B - . D e v ) = 10.6 I Vcbl 2 ,
(38)
to be c o m p a r e d with [ 14,15 ] BR(B °~D+e-ge)=
(1.75+0.42+0.35) N 10-2,
(39a)
BR ( B - ~ D ° e - g e ) = ( 1 . 6 0 + 0 . 6 + 0 . 2 ) × 10 -2 .
(39b)
This gives, from the first channel, ] gcb I =0.041 + 0 . 0 0 6
(40a)
Table 1 Parameters of the form factors for the transitions B--,D and B--,D*. Form factor
Value at q2=O
Pole mass (GeV)
Fo(q 2)
0.69 0.69 0.84 0.81 0.65 0.45 0.81
7.60 5.54 5.54 6.30 6.73 6.73 6.73
Fl (q2) V(q ~) Ao(q 2) A~(q 2) Az(q 2)
A3(q 2)
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and, from the second channel, [ rcb [ =0.039+_0.008.
(40b)
Let us consider the B ~ D*~v decays. Using standard formulae [ 16 ] and the values of Vcu in eq. (40) we get BR(B~D*~v) = 5.2× 10 z,
(41)
to be compared with the experimental results [ 13-15 ] B R ( B 0 ~ D * + ~ - 9 ) = ( 4 . 8 _ + 0 . 4 + _ 0 . 7 ) × 1 0 -2,
BR(B-~D*°~-9)=(4.1_+0.8_+0.8)×10 -2.
(42)
Moreover, we can also compute the asymmetry parameter ~ = 2 -FL(B°~D*+I~-9) - 1 , Fx(B° ~ D * + ~ - 9 )
(43)
where Fc and Fv refer to the longitudinal and transverse polarization of D*; we get c~=0.71
(44)
taking into account the experimental cutoffp~ > 1.4 GeV for the charged lepton momentum. The experimental results are c~=0.65 _+0.66_+0.25 [ 15], and c~=0.7_+0.9 [ 14] (with p~> 1.0 GeV). We can draw our conclusions. We have computed the form factors for the charmed semileptonic exclusive B decays using a QCD relativistic potential model and we have obtained results in agreement with the experimental data. Our results are also in general agreement with the findings of other potential models and with the calculations based on QCD sum rules [ 17 ]. We shall present a detailed comparison elsewhere. As for the predictions of the present model in the infinite heavy quark limit mb = mc = M-, oo, they turn out to be in agreement with general results [ 18,19 ], as discussed in the appendix. An advantage of the model calculation of the form factors we have presented in this letter is that the mass corrections to this relevant limit can be explicitly computed. Finally, it is worth observing that the value Vcb 2 0.04 obtained by the exclusive channels ( 1 ), (2) is sensibly lower (although compatible within the errors) than the value l / c b = 0 . 0 4 7 _+0.004 given by the analysis of the semileptonic inclusive B~Xcev spectrum by a QCD calculation which includes B meson wave function effects [ 3 ]. This result reflects a common trend [ 13 ] whose origin should be further investigated. U seful discussions with N. Paver are gratefully acknowledged. One of us (P.C.) would like to thank Professor V.K. Kadyshevsky for warm hospitality at the Laboratory of Theoretical Physics, JINR, Dubna, USSR.
Appendix. We wish to consider eqs.
( 3 2 ) - ( 3 7 ) in the S U ( 2 ) × S U ( 2 ) spin-flavour symmetry [18] charac-
terized by mb =
m c = M--,oc
(A. 1 )
MB =MD =MD. = M .
(A.2)
and
From the wave equation (12) one observes that, for each value of the momentum M-. oo. Moreover, using the method of section III of ref. [3 ], it can be shown that
k, k2gt(k)/M 2-,0 when
f dk~t~(k)~D(k)=2~[l+O(~---S)], 0
which, in the symmetry limit, reduces to the wave function normalization condition [see eq. (23) ]
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(A.3)
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i dk t~2 ( k ) = 2 M .
(A.4)
0
Therefore, f r o m eqs. ( 3 2 ) - ( 3 5 ) o n e o b t a i n s
Fo(t) =
MB+MD
(A.5)
Aj ( t * ) -
2MB ~+ M D . I 1 q- 0(~------~)1 = 1 4- O ( ~ 2 )
(A.6)
whereas the l e a d i n g O ( 1 / M ) t e r m s are a b s e n t since there are no k/M corrections to the heavy q u a r k limit in these e q u a t i o n s . Let us observe that t = t*=qZmax in the s y m m e t r y limit. M o r e o v e r , f r o m the exact r e l a t i o n Fo ( 0 ) = F~ ( 0 ) a n d f r o m eq. ( 11 ) o n e gets F, ( t ) = l
+ O(~--52 ) .
(A.7)
Eqs. ( A . 5 ) - ( A . 7 ) satisfy the g e n e r a l i z a t i o n o f the A d e m o l l o - G a t t o t h e o r e m [20 ] valid for the heavy q u a r k s y m m e t r y . As for the other f o r m factors, we observe that, in the s y m m e t r y limit, xB = X D ~ 1 because o f the results ( A . 5 ) - ( A . 7 ) a n d eq. ( 3 3 ) . Therefore, f r o m eqs. ( 3 4 ) , ( 3 6 ) o n e gets i m m e d i a t e l y
v(t*) =
1,
A3(t*) = A o ( t * )
(A.8)
= 1,
(A.9)
in a g r e e m e n t with the i n f i n i t e h e a v y q u a r k l i m i t results [ 19]; the p r e d i c t i o n for A 2 follows f r o m eq. ( 1 0 ) .
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[ 16 ] J.G. K6rner and G.A. Schuler, Z. Phys. C. 46 (1990) 93; M. Bauer and M. Wirbel, Z. Phys. C 42 (1989) 671. [ 17] A.A. Ovchinnikov and V.A. Slobodenyuk, Z. Phys. C. 44 (1989) 433; P. Colangelo, G. Nardulli, A.A. Ovchinnikov and N. Paver, Phys. Lett. B 269 ( 1991 ) 201. [ 18] M.B. Voloshin and M.A. Shifman, Yad. Fiz. 47 (1988) 810 [Sov. J. Nucl. Phys. 47 (1988) 511 ]; N. lsgur and M. Wise, Phys. Lett. B 232 ( 1989 ) 113; B 237 (1990) 527. [ 19] M. Luke, Phys. Lett. B 252 (1990) 447; C. Glenn Boyd and D.E. Brahm, Phys. Lett. B 257 ( 1991 ) 393; R.F. Lebed and M. Suzuki, Phys. Rev. D 44 ( 1991 ) 829; M. Neuberl and V. Riecken, preprint HD-THEP-91-6 ( 1991 ). [20] M. Ademollo and R. Gatto, Phys. Rev. Lelt. 13 (1964) 264.
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