- Email: [email protected]
Reliable predictions in exclusive rare B decays
Physics Letters B 270 ( 1991 ) 55-60 North-Holland
PHYSICS LETTERS B
Reliable predictions in exclusive rare B decays Gustavo Burdman and John F. Donoghue Departmentof PhysicsandAstronomy, Universityof Massachusetts,Amherst,MA 01003, USA Received 13 August 1991
We discuss the relations between form factors for B--,K*7 and B--,pev which follow from static heavy quark symmetry methods plus SU (3). We discuss carefully the hard perturbative QCD corrections which could invalidate these results and find that the dominant perturbative corrections are ones which respect the static symmetry predictions, and in addition that the perturbative corrections are small compared to soft contributions. We show that there is a special kinematic point in B--,pev that involves the same combination of form factors as in B--,K*y,thus allowing a prediction of the ratio of rates which is largely free from hadronic uncertainties.
1. Introduction
X
e2 x)-txThe interest in studying rare B decays lies with the hope that one can measure the flavor changing processes which occur at one loop. T h e decay b--,s'y is p r e d i c t e d to occur at a rate such that it should soon be measurable. Unfortunately, one m u s t observe exclusive channels, such as B--, K ~ , instead o f inclusive b o sT rates, a n d the h a d r o n i c m a t r i x elements o f the exclusive channel are so uncertain as to m a k e it difficult to extract any useful information. T h e branching ratio for B-,K*T can vary b y an o r d e r o f magnitude in different quark m o d e l calculations [ 1,2 ]. We here show how one m a y o v e r c o m e this difficulty by studying the ratio o f B - , K*y to B---,p e r at a p a r t i c u l a r kinematic point. The p h o t o n i c decay occurs with the effective operator
Hv=rlgaU~[mb(1-Ts)+rns(l+Ts)]bFu~.
e
.
2
In
R e p r e s e n t a t i v e values o f F2 ( x ) are given in table 1. In eq. ( 1 ) the term p r o p o r t i o n a l to mb produces a left-handed s quark, while that p r o p o r t i o n a l to ms yields a right-handed s quark. In b decays, these can interfere only slightly such that the influence o f the latter t e r m is suppressed by at least ( m s × 1 G e V ) / m 2 ~< 10 -2. This leads us to d r o p this t e r m from now on a n d only consider the left-handed s quark. The semileptonic decay b ~ u ~ 9 involves the hamiltonian HSL= ~
Vub
7U(l+?s)bf?U(l+Ts)v.
(3)
T h e C K M matrix element V,b is at present roughly known from the end p o i n t o f the electron energy Table 1 F2 for different values of the top mass. Includes the quark-loop diagram plus one-loop QCD corrections.
2
rl= ~7~-~2 Vtb VtsmbFz( m t / M w ) , F2(x) =r-'6/23{p2(x) +~76[~(r'°/23--1)+ l(r28/23--1)]),
3X2 ~c
(1)
Here q includes the Q C D corrections which have been heavily studied [ 1,3,4]. Its analytic form is given by GF
(
(2)
where r - ors (mb) /czs ( M w ) a n d 0370-2693/91/$ 03.50 © 1991 Elsevier Science Publishers B.V. All fights reserved.
m, (GeV)
F2(m2t/MZw )
90 120 150 180 210
0.55 0.59 0.63 0.66 0.68
55
Volume 270, number 1
PHYSICS LETTERS B
spectrum in inclusive b decays [ 5,6 ], and we assume that it will be better known in the future from this source. The goal of this work is to relate the decay rates for these two processes in as model independent fashion as possible, in order to test quantitatively the standard model prediction, or perhaps find evidence for interactions beyond the standard model.
between many but not all of the above form factors. This follows since the Dirac matrices satisfy yiy 5 -~ ( 0 l and i 0 ~) : (__ o.i
Both transitions make left-handed light quarks. However, the Lorentz structure of the matrix elements is different. For the photonic transitions, there • are three form factors of the ga"~b operator,
,
(4)
where A, B, C are in general functions of q2 = ( p _ k ) 2. These are evaluated at q2=0 in the case of the real photon decay B~K*y. Because of the Dirac matrix identity (~o~23= + 1 )
a""ys = - ½it ~'~"aao
(5)
the matrix elements of ~a~'"?sb are given by the same form factors:
( f(*( ~, k) ISa~'~,sbl B ( p ) ) =i[A(~*C,p,'_e*,'p,) + B(~*~,k,'_e*,'ku) + C e*.p ( p ~ ' k " - p " k u) ] .
(6)
Isospin symmetry implies that the ~ o ~ g . o and B - ~ K*- amplitudes are equal. In the case of B ~ per, there are four form factors: ( p + (~, k) I~ b [ ~O(p) ) =iD~,,,,~#p,,~,ka, ( p + (E, k) luy~ysbl f3(p) ) (7)
Here again the form factors are functions of q2= ( p _ k)2, and the physical region encompasses a range of values of q 2, i.e., m ~ < q 2 < ( rnB - mp) 2. For our purposes, q2 = m ~ is negligible close to q 2= 0. Isgur and Wise [ 7 ] have already remarked that, if one assumes that only the upper two components of the b quark field contribute, there will be relations 56
oi) •
the two pairs in eqs. (8), (9) will have identical matrix elements. This leads to the relations
(R*(~, k)Iga""blf3(p) )
=E¢*l'+ F ¢*.p p~' + G ¢*.p k ~' .
o)
Therefore, if the heavy b quark has only upper components,
2. Relating form factors
=~'~#(A~*pa+B~*k#+C~*.pp,~ka)
7 November 1991
A=
- E + mnkoD , B=-rnBD , mB
D+ G C=-mB
(11)
We note that the heavy quark methodology cannot yield these relations as rigorous predictions of QCD without further work. The use of static heavy quark techniques is only justified when the momentum transfer is small compared to the heavy quark mass. In the case of B--,K*7, the momentum transfer is of order ½mB. There can then exist hard perturbative corrections in which the heavy quark need not be static. Indeed, the inclusive QCD formalism developed by Lepage and Brodsky [8 ] says that at high mass the perturbative diagrams are dominant, thus seemingly invalidating the conclusion drawn by Isgur and Wise. However, we will show that the leading perturbative contribution in factors of rnB will also respect the relations of eq. ( I 1 ) so that these can be considered as valid predictions of QCD. As a byproduct, we also note that the magnitude of the perturbative form factors seem to be smaller than the soft quark model contributions when evaluated at the B mass. The inclusive QCD techniques have been applied to heavy quark decays in ref. [ 9 ]. The relevant diagrams are given in fig. 1. (For simplicity we will study pseudoscalar mesons. ) Of these, fig. 1a will respect the static quark relations because the b quark in the wavefunction will not receive large momentum from the gluon exchange. In fig. lb however, the b quark at the vertex of the external current will not in gen-
Volume 270, number l
PHYSICS LETTERS B
7 November 1991
B meson with the m o m e n t u m fraction x having a mean value ( 1 - x ) ~ E, with
/m2+ (k±)2
e'~N/
~n
'
(14)
where m is the mass of the light quark. This yields e N 0 . 0 5 ~ 0 . 1 in B decays. The kinematic features which appear in the denominators of hard scattering amplitudes are (a)
xoB
Q2=(1-x)(1-y)(qE-mE)+(1-x)Em2, k2=(l-x)(qE-m2)+(1-x)2m 2,
(15b)
k 2-m E= (1--y)(qE--mE).
(15C)
Because of the dominance o f x ~ 1 we follow ref. [9 ] and neglect the ( 1 - x ) 2 factors in eq. (15a), (15b). The denominators then are DI a = (m 2 _q2)
2(1
-x)2( 1-y),
D~b = ( m E _ q 2 ) 2( 1 - - x ) ( 1 _ y ) 2 .
(b) Fig. 1. Perturbative diagrams for B--,n. eral be static because the hard gluon can give to it a m o m e n t u m of order ms. Thus this diagram could upset the relations of eq. ( l 1 ). The key feature that we need to study is the propagator associated with mom e n t u m k2 in fig: lb. Since k2 =PB-- ( 1 where y is the m o m e n t u m fraction carried by the light quark in the pion, the propagator is
--y)p~,
/~2 + m B
~ +mB- (1-y)~
k~-m~- (l-y)(q2-m~)
(15a)
(16)
Explicit calculation of the numerators for B-~ n (displayed below) show that fig. 1a contains a factor of 1 - x in the numerator, but that both numerators are finite as y ~ I. Thus both diagrams have the same powers of 1 - x in the denominator. The asymptotic form of the pion wavefunction is ¢~(y) ~ y ( l - y ) , so that diagram I a has the form l
I
f
1
Mla = 3 dy ¢~(y) ~ ' ~ _~f~a(Y) ~ 0
fdyYfla(Y) ( 17) 0
w h e r e f ( y ) is some smooth function o f y . This diagram is then equally sensitive to all values o f y . Diagram lb, however, has the form I
(12)
1
Mlb= f dy~(y)
(l-T~y)2flb(y)
0
where q2_. (pB_p~)2. If, for the moment, we focus on the n u m e r a t o r we see that as y ~ 1 the propagator assumes only an upper component, since
l
0
(13)
This is singular as y--, 1. In ref. [ 9 ] the integral is cut off at y = 1 - e under the argument of the existence of a Sudakov form factor in the endpoint region a~.
Thus if the y ~ l limit dominates, fig. lb will also satisfy the static quark relations. The b quark carries most of the m o m e n t u m of the
#~ The dominance of the y-~ 1 limit may raise problems about the applicability of the perturbative formalism to heavy meson decays. We hope to return to this topic in the future.
~a+mB=mB(yo+l)=2mB(lo 00).
57
Volume 270, number 1
PHYSICSLETTERSB
However, the integral remains dominated by the y--, 1 - ~ end, with the leading behavior of In _1 = In MB e x/m2+ ( k 2 ) "
Because of the propagator relation ofeqs. (12), (16) this leading behavior will obey the static quark assumption, and hence also generate the form factor relations ofeq. ( 11 ). We may also look at the magnitudes of the perturbative contributions. For example, in the B ~ ~ transition we find
<~(p~)l V~IB(pB)> 1--E
I
-3(mE_q2) 2
dx 0
dy l - x
×[(pB+p,~,u('(2g~__)m2 __qU((2g-1)m~\ ~Ly
l-y
0
+ m ~ - 2 q 2)
__rnZ)] ,
(19,
where g = 1 corresponds to the static quark approximation to the B wavefunction. The wavefunctions are normalized to 1
I_L_F ~o dx Oi(x)= 2x/~ i,
(20)
7 November 1991
for the rates are 1 to 2 orders of magnitude below those of quark models. It is also consistent with the analysis of ref. [ 11 ]. Our overall conclusions on the effects of the perturbative corrections are that the leading perturbative effect also respects the form factor relations of eq. ( 11 ), and that the perturbative magnitude appears smaller than the estimates of soft contributions.
3. P h e n o m e n o l o g y
The existence of relations between some of the form factors is not immediately useful in relating the decay rates. Other unknown form factors will in general enter, and BH--,pev and B~K*7 do not necessarily involve the same combinations of form factors. We will show that, rather remarkably, there exists a corner of the B~pev Dalitz plot which involves exactly the same combination of form factors as appear in B~K*7. In calculating the decay rate for B~K*'/, we first note that the C form factor does not contribute to the transition involving the on shell photon. The decay rate calculation is straightforward and only involves the combinations of form factors A + B: F ( B ~ K * T ) - IqI2(A+B)2 ( r n 2 - m ~ . ) 3 4/~m 3
(24)
where F~ = 92 MeV and we will use Fn = F~. Using 1
q~~y(1-y),
0,~
[~2/(l_x)+l/x_l]2,
(21)
the leading contribution to the form factors is then (with g = 1 ) f+(q2)=_f_(q2)=
32asm~ fBf~ln 1 " 3(m~-q2) 2 e
(22)
With ot~= 0.4, ~=0.1, this yields f÷ (0) = - f _ (0) =0.033.
(23)
Quark model calculations yield larger values for these form factors. For instance, the Bauer-Stech-Wirbel [10] model gives f ÷ ( 0 ) = 0 . 3 3 . Thus, unless these quark model results are far wrong, the perturbative contributions are not yet dominant. This conclusion is also visible in the results of ref. [9] (i.e., fig. 3) where the perturbative predictions 58
The B ~ pev rate is kinematically more complicated. The allowed region of electron and p energies is shown in fig. 2. The Dalitz plot distribution can be worked out to be dZF F ( B - . p e v ) = j dE dE e p dE e dEp, d2F I Vub 12G2 dEe dEo - 16n3ma
× { [E2+m~k2D2-2rnBED(mB-Ep-2Ee) ]qZ +fl[4Ee(rna-Ep-Ee) _q2] }, (25) where
Volume 270, number 1
PHYSICS LETTERS B
7 November 1991
Ep Fig. 3. Kinematic configuration along curve C in fig. 2. tineutrino is always right handed ( R H ) . To form a spin zero B °, the p must be of left handed helicity. In B ~ K*7, the photon is either LH or R H helicity and the K* must likewise be LH or RH in correlation with the photon's helicity. However, the weak interactions eq. ( 1 ) produces a left handed s quark, so only a LH K* is possible. The combination of form factors in eq. (27) corresponds to the production of a LH K* or p. The K* emission in B ~ K * y occurs at fixed energy. This corresponds to maximum recoil in the B ~ p e v case, i.e., to the point labeled MxR in fig. 2. Thus the form factor needs to be compared at the same value of q2. This is accomplished by forming the ratio of rates and taking the limit towards the point M,~R. In particular our main result is
Eo Fig. 2. Dalitz plot for B~peg. 2
m§ 1~2..1..1 ~ 2 ~ 2/-,h2 f l - 2m~ "-" " 2"'B~/ z / + 2_ m~ m2 (mBEp-m~)E(F+G) m2
+ (F+G) 2~
k2"
1
The combination of form factors at an arbitrary point on the Dalitz plot is generally far different from the combination relevant for B--.K*7. It is unrealistic to expect that each of the form factors could be individually measured. Thus the heavy quark symmetry methods appear difficult to apply. However, a fortunate feature emerges if one considers the decay distribution along the right hand side of the Dalitz plot, labeled by curve C in fig. 2. In this case the distribution simplifies to d2F dEodEe-
I Vub }2G2 16/r3ma
(E+mBkD)2q2.
(26)
This is precisely the combination of form factors which occurs in B--,K*'/decay since
E+mBkD mB
=A+B.
(27)
It is in fact possible to understand in a heuristic fashion why the same combination of form factors emerges. This side of the Dalitz plot region corresponds to the kinematic configuration of fig. 3. In the limit that the electron mass is neglected the electron helicity is completely left handed ( L H ) and the an-
clF
--1
= 4 n ~ It/[ 2 ( m ~ - m ~ , ) 3 G 2 IVub[ 2 m4
(28)
independent of hadronic form factors. The form factor relations and hence eq. (28) has involved the use of SU(3) invariance to relate K* and p amplitudes. In practice there may be some SU (3) breaking. We have studied this in the BauerStech-Wirbel model [ 10 ]. SU (3) breaking enters both in trivial kinematic factors, due to mp # rng., and in a modification of the wavefunction overlap integral, J. The kinematic effect in the form factors D and E is 4%: DK,(0) mb--ms m~--m~ JacK* JacK* Do(0) - mb--mu m~--mE, Js~p =0.96 ~ , EK*(0) __ m mb ~+
E~(0)
JB-K* _ 1.04 JB-K* m b + m u Jn~p JB~p
(29)
Our estimate of the wavefunction overlap effect is JB~K* - 1 . 1 0 . JB~
(30)
59
Volume 270, number 1
PHYSICS LETTERS B
Thus overall the SU (3) breaking is quite small. We would expect that the residual model dependence in the SU (3) breaking of this ratio of rates would be very much smaller than the original hadronic uncertainty in either of the rates alone. Since neither of these processes has been measured yet, this test is clearly not practical at present. However, it is expected that both reactions should occur at rates close to present sensitivity, so that one may look forward to their observation soon. It will require a significant increase in sensitivity in order to make a meaningful test, but this should be possible with the expected high luminosity B factories now being planned. The kinematic configuration required, that where both the electron and rho are very energetic, is favorable for experimental searches. The measurement of the ratio of rates should be an experimental goal as it would allow exclusive rare B decays to be useful as tests of the standard model.
Acknowledgement J.F.D. acknowledges stimulating conversation with
60
7 November 1991
Daniel Wyler about this topic, and the hospitality of the CERN theory group during early stages of the work. This work was supported in part by the US National Science Foundation.
References [ 1] N.G. Deshpande et al., Phys. Rev. Lett. 59 ( 1987 ) 183. [2 ] T. Altomari, Phys. Rev. D 37 (1988) 677; P.G. O'Donnell and H.K.K. Tung, University of Toronto Report UTPT-91-06 ( 1991 ). [ 3 ] S. Bertolini et al., Phys. Rev. Lett. 59 ( 1987 ) 180. [4 ] B. Grinstein, R. Springer and M.B. Wise, Nucl. Phys. B 339 (1990) 269. [ 5 ] CLEO Collab., R. Fulton et al., Phys. Rev. Lett. 64 (1990) 16. [6 ] ARGUS Collab., H. Albrecht et al., Phys. Lett. B 234 (1990) 409. [7] N. Isgur and M.B. Wise, Phys. Rev. D 42 (1990) 2388. [8] G.P. Lepage and Sd. Brodsky, Phys. Lett. B 87 (1979) 959; Phys. Rev. D 22 (1980) 2157. [9 ] A. Szczepaniak, E.M. Henley and S.J. Brodsky, Phys. Lett. B243 (1990) 287. [ 10] M. Wirbel, B. Stech and M. Bauer, Z. Phys. C 29 (1987) 637. [ 11 ] N. Isgur, Phys. Rev. D 43 ( 1991 ) 810.
PHYSICS LETTERS B
Reliable predictions in exclusive rare B decays Gustavo Burdman and John F. Donoghue Departmentof PhysicsandAstronomy, Universityof Massachusetts,Amherst,MA 01003, USA Received 13 August 1991
We discuss the relations between form factors for B--,K*7 and B--,pev which follow from static heavy quark symmetry methods plus SU (3). We discuss carefully the hard perturbative QCD corrections which could invalidate these results and find that the dominant perturbative corrections are ones which respect the static symmetry predictions, and in addition that the perturbative corrections are small compared to soft contributions. We show that there is a special kinematic point in B--,pev that involves the same combination of form factors as in B--,K*y,thus allowing a prediction of the ratio of rates which is largely free from hadronic uncertainties.
1. Introduction
X
e2 x)-txThe interest in studying rare B decays lies with the hope that one can measure the flavor changing processes which occur at one loop. T h e decay b--,s'y is p r e d i c t e d to occur at a rate such that it should soon be measurable. Unfortunately, one m u s t observe exclusive channels, such as B--, K ~ , instead o f inclusive b o sT rates, a n d the h a d r o n i c m a t r i x elements o f the exclusive channel are so uncertain as to m a k e it difficult to extract any useful information. T h e branching ratio for B-,K*T can vary b y an o r d e r o f magnitude in different quark m o d e l calculations [ 1,2 ]. We here show how one m a y o v e r c o m e this difficulty by studying the ratio o f B - , K*y to B---,p e r at a p a r t i c u l a r kinematic point. The p h o t o n i c decay occurs with the effective operator
Hv=rlgaU~[mb(1-Ts)+rns(l+Ts)]bFu~.
e
.
2
In
R e p r e s e n t a t i v e values o f F2 ( x ) are given in table 1. In eq. ( 1 ) the term p r o p o r t i o n a l to mb produces a left-handed s quark, while that p r o p o r t i o n a l to ms yields a right-handed s quark. In b decays, these can interfere only slightly such that the influence o f the latter t e r m is suppressed by at least ( m s × 1 G e V ) / m 2 ~< 10 -2. This leads us to d r o p this t e r m from now on a n d only consider the left-handed s quark. The semileptonic decay b ~ u ~ 9 involves the hamiltonian HSL= ~
Vub
7U(l+?s)bf?U(l+Ts)v.
(3)
T h e C K M matrix element V,b is at present roughly known from the end p o i n t o f the electron energy Table 1 F2 for different values of the top mass. Includes the quark-loop diagram plus one-loop QCD corrections.
2
rl= ~7~-~2 Vtb VtsmbFz( m t / M w ) , F2(x) =r-'6/23{p2(x) +~76[~(r'°/23--1)+ l(r28/23--1)]),
3X2 ~c
(1)
Here q includes the Q C D corrections which have been heavily studied [ 1,3,4]. Its analytic form is given by GF
(
(2)
where r - ors (mb) /czs ( M w ) a n d 0370-2693/91/$ 03.50 © 1991 Elsevier Science Publishers B.V. All fights reserved.
m, (GeV)
F2(m2t/MZw )
90 120 150 180 210
0.55 0.59 0.63 0.66 0.68
55
Volume 270, number 1
PHYSICS LETTERS B
spectrum in inclusive b decays [ 5,6 ], and we assume that it will be better known in the future from this source. The goal of this work is to relate the decay rates for these two processes in as model independent fashion as possible, in order to test quantitatively the standard model prediction, or perhaps find evidence for interactions beyond the standard model.
between many but not all of the above form factors. This follows since the Dirac matrices satisfy yiy 5 -~ ( 0 l and i 0 ~) : (__ o.i
Both transitions make left-handed light quarks. However, the Lorentz structure of the matrix elements is different. For the photonic transitions, there • are three form factors of the ga"~b operator,
,
(4)
where A, B, C are in general functions of q2 = ( p _ k ) 2. These are evaluated at q2=0 in the case of the real photon decay B~K*y. Because of the Dirac matrix identity (~o~23= + 1 )
a""ys = - ½it ~'~"aao
(5)
the matrix elements of ~a~'"?sb are given by the same form factors:
( f(*( ~, k) ISa~'~,sbl B ( p ) ) =i[A(~*C,p,'_e*,'p,) + B(~*~,k,'_e*,'ku) + C e*.p ( p ~ ' k " - p " k u) ] .
(6)
Isospin symmetry implies that the ~ o ~ g . o and B - ~ K*- amplitudes are equal. In the case of B ~ per, there are four form factors: ( p + (~, k) I~ b [ ~O(p) ) =iD~,,,,~#p,,~,ka, ( p + (E, k) luy~ysbl f3(p) ) (7)
Here again the form factors are functions of q2= ( p _ k)2, and the physical region encompasses a range of values of q 2, i.e., m ~ < q 2 < ( rnB - mp) 2. For our purposes, q2 = m ~ is negligible close to q 2= 0. Isgur and Wise [ 7 ] have already remarked that, if one assumes that only the upper two components of the b quark field contribute, there will be relations 56
oi) •
the two pairs in eqs. (8), (9) will have identical matrix elements. This leads to the relations
(R*(~, k)Iga""blf3(p) )
=E¢*l'+ F ¢*.p p~' + G ¢*.p k ~' .
o)
Therefore, if the heavy b quark has only upper components,
2. Relating form factors
=~'~#(A~*pa+B~*k#+C~*.pp,~ka)
7 November 1991
A=
- E + mnkoD , B=-rnBD , mB
D+ G C=-mB
(11)
We note that the heavy quark methodology cannot yield these relations as rigorous predictions of QCD without further work. The use of static heavy quark techniques is only justified when the momentum transfer is small compared to the heavy quark mass. In the case of B--,K*7, the momentum transfer is of order ½mB. There can then exist hard perturbative corrections in which the heavy quark need not be static. Indeed, the inclusive QCD formalism developed by Lepage and Brodsky [8 ] says that at high mass the perturbative diagrams are dominant, thus seemingly invalidating the conclusion drawn by Isgur and Wise. However, we will show that the leading perturbative contribution in factors of rnB will also respect the relations of eq. ( I 1 ) so that these can be considered as valid predictions of QCD. As a byproduct, we also note that the magnitude of the perturbative form factors seem to be smaller than the soft quark model contributions when evaluated at the B mass. The inclusive QCD techniques have been applied to heavy quark decays in ref. [ 9 ]. The relevant diagrams are given in fig. 1. (For simplicity we will study pseudoscalar mesons. ) Of these, fig. 1a will respect the static quark relations because the b quark in the wavefunction will not receive large momentum from the gluon exchange. In fig. lb however, the b quark at the vertex of the external current will not in gen-
Volume 270, number l
PHYSICS LETTERS B
7 November 1991
B meson with the m o m e n t u m fraction x having a mean value ( 1 - x ) ~ E, with
/m2+ (k±)2
e'~N/
~n
'
(14)
where m is the mass of the light quark. This yields e N 0 . 0 5 ~ 0 . 1 in B decays. The kinematic features which appear in the denominators of hard scattering amplitudes are (a)
xoB
Q2=(1-x)(1-y)(qE-mE)+(1-x)Em2, k2=(l-x)(qE-m2)+(1-x)2m 2,
(15b)
k 2-m E= (1--y)(qE--mE).
(15C)
Because of the dominance o f x ~ 1 we follow ref. [9 ] and neglect the ( 1 - x ) 2 factors in eq. (15a), (15b). The denominators then are DI a = (m 2 _q2)
2(1
-x)2( 1-y),
D~b = ( m E _ q 2 ) 2( 1 - - x ) ( 1 _ y ) 2 .
(b) Fig. 1. Perturbative diagrams for B--,n. eral be static because the hard gluon can give to it a m o m e n t u m of order ms. Thus this diagram could upset the relations of eq. ( l 1 ). The key feature that we need to study is the propagator associated with mom e n t u m k2 in fig: lb. Since k2 =PB-- ( 1 where y is the m o m e n t u m fraction carried by the light quark in the pion, the propagator is
--y)p~,
/~2 + m B
~ +mB- (1-y)~
k~-m~- (l-y)(q2-m~)
(15a)
(16)
Explicit calculation of the numerators for B-~ n (displayed below) show that fig. 1a contains a factor of 1 - x in the numerator, but that both numerators are finite as y ~ I. Thus both diagrams have the same powers of 1 - x in the denominator. The asymptotic form of the pion wavefunction is ¢~(y) ~ y ( l - y ) , so that diagram I a has the form l
I
f
1
Mla = 3 dy ¢~(y) ~ ' ~ _~f~a(Y) ~ 0
fdyYfla(Y) ( 17) 0
w h e r e f ( y ) is some smooth function o f y . This diagram is then equally sensitive to all values o f y . Diagram lb, however, has the form I
(12)
1
Mlb= f dy~(y)
(l-T~y)2flb(y)
0
where q2_. (pB_p~)2. If, for the moment, we focus on the n u m e r a t o r we see that as y ~ 1 the propagator assumes only an upper component, since
l
0
(13)
This is singular as y--, 1. In ref. [ 9 ] the integral is cut off at y = 1 - e under the argument of the existence of a Sudakov form factor in the endpoint region a~.
Thus if the y ~ l limit dominates, fig. lb will also satisfy the static quark relations. The b quark carries most of the m o m e n t u m of the
#~ The dominance of the y-~ 1 limit may raise problems about the applicability of the perturbative formalism to heavy meson decays. We hope to return to this topic in the future.
~a+mB=mB(yo+l)=2mB(lo 00).
57
Volume 270, number 1
PHYSICSLETTERSB
However, the integral remains dominated by the y--, 1 - ~ end, with the leading behavior of In _1 = In MB e x/m2+ ( k 2 ) "
Because of the propagator relation ofeqs. (12), (16) this leading behavior will obey the static quark assumption, and hence also generate the form factor relations ofeq. ( 11 ). We may also look at the magnitudes of the perturbative contributions. For example, in the B ~ ~ transition we find
<~(p~)l V~IB(pB)> 1--E
I
-3(mE_q2) 2
dx 0
dy l - x
×[(pB+p,~,u('(2g~__)m2 __qU((2g-1)m~\ ~Ly
l-y
0
+ m ~ - 2 q 2)
__rnZ)] ,
(19,
where g = 1 corresponds to the static quark approximation to the B wavefunction. The wavefunctions are normalized to 1
I_L_F ~o dx Oi(x)= 2x/~ i,
(20)
7 November 1991
for the rates are 1 to 2 orders of magnitude below those of quark models. It is also consistent with the analysis of ref. [ 11 ]. Our overall conclusions on the effects of the perturbative corrections are that the leading perturbative effect also respects the form factor relations of eq. ( 11 ), and that the perturbative magnitude appears smaller than the estimates of soft contributions.
3. P h e n o m e n o l o g y
The existence of relations between some of the form factors is not immediately useful in relating the decay rates. Other unknown form factors will in general enter, and BH--,pev and B~K*7 do not necessarily involve the same combinations of form factors. We will show that, rather remarkably, there exists a corner of the B~pev Dalitz plot which involves exactly the same combination of form factors as appear in B~K*7. In calculating the decay rate for B~K*'/, we first note that the C form factor does not contribute to the transition involving the on shell photon. The decay rate calculation is straightforward and only involves the combinations of form factors A + B: F ( B ~ K * T ) - IqI2(A+B)2 ( r n 2 - m ~ . ) 3 4/~m 3
(24)
where F~ = 92 MeV and we will use Fn = F~. Using 1
q~~y(1-y),
0,~
[~2/(l_x)+l/x_l]2,
(21)
the leading contribution to the form factors is then (with g = 1 ) f+(q2)=_f_(q2)=
32asm~ fBf~ln 1 " 3(m~-q2) 2 e
(22)
With ot~= 0.4, ~=0.1, this yields f÷ (0) = - f _ (0) =0.033.
(23)
Quark model calculations yield larger values for these form factors. For instance, the Bauer-Stech-Wirbel [10] model gives f ÷ ( 0 ) = 0 . 3 3 . Thus, unless these quark model results are far wrong, the perturbative contributions are not yet dominant. This conclusion is also visible in the results of ref. [9] (i.e., fig. 3) where the perturbative predictions 58
The B ~ pev rate is kinematically more complicated. The allowed region of electron and p energies is shown in fig. 2. The Dalitz plot distribution can be worked out to be dZF F ( B - . p e v ) = j dE dE e p dE e dEp, d2F I Vub 12G2 dEe dEo - 16n3ma
× { [E2+m~k2D2-2rnBED(mB-Ep-2Ee) ]qZ +fl[4Ee(rna-Ep-Ee) _q2] }, (25) where
Volume 270, number 1
PHYSICS LETTERS B
7 November 1991
Ep Fig. 3. Kinematic configuration along curve C in fig. 2. tineutrino is always right handed ( R H ) . To form a spin zero B °, the p must be of left handed helicity. In B ~ K*7, the photon is either LH or R H helicity and the K* must likewise be LH or RH in correlation with the photon's helicity. However, the weak interactions eq. ( 1 ) produces a left handed s quark, so only a LH K* is possible. The combination of form factors in eq. (27) corresponds to the production of a LH K* or p. The K* emission in B ~ K * y occurs at fixed energy. This corresponds to maximum recoil in the B ~ p e v case, i.e., to the point labeled MxR in fig. 2. Thus the form factor needs to be compared at the same value of q2. This is accomplished by forming the ratio of rates and taking the limit towards the point M,~R. In particular our main result is
Eo Fig. 2. Dalitz plot for B~peg. 2
m§ 1~2..1..1 ~ 2 ~ 2/-,h2 f l - 2m~ "-" " 2"'B~/ z / + 2_ m~ m2 (mBEp-m~)E(F+G) m2
+ (F+G) 2~
k2"
1
The combination of form factors at an arbitrary point on the Dalitz plot is generally far different from the combination relevant for B--.K*7. It is unrealistic to expect that each of the form factors could be individually measured. Thus the heavy quark symmetry methods appear difficult to apply. However, a fortunate feature emerges if one considers the decay distribution along the right hand side of the Dalitz plot, labeled by curve C in fig. 2. In this case the distribution simplifies to d2F dEodEe-
I Vub }2G2 16/r3ma
(E+mBkD)2q2.
(26)
This is precisely the combination of form factors which occurs in B--,K*'/decay since
E+mBkD mB
=A+B.
(27)
It is in fact possible to understand in a heuristic fashion why the same combination of form factors emerges. This side of the Dalitz plot region corresponds to the kinematic configuration of fig. 3. In the limit that the electron mass is neglected the electron helicity is completely left handed ( L H ) and the an-
clF
--1
= 4 n ~ It/[ 2 ( m ~ - m ~ , ) 3 G 2 IVub[ 2 m4
(28)
independent of hadronic form factors. The form factor relations and hence eq. (28) has involved the use of SU(3) invariance to relate K* and p amplitudes. In practice there may be some SU (3) breaking. We have studied this in the BauerStech-Wirbel model [ 10 ]. SU (3) breaking enters both in trivial kinematic factors, due to mp # rng., and in a modification of the wavefunction overlap integral, J. The kinematic effect in the form factors D and E is 4%: DK,(0) mb--ms m~--m~ JacK* JacK* Do(0) - mb--mu m~--mE, Js~p =0.96 ~ , EK*(0) __ m mb ~+
E~(0)
JB-K* _ 1.04 JB-K* m b + m u Jn~p JB~p
(29)
Our estimate of the wavefunction overlap effect is JB~K* - 1 . 1 0 . JB~
(30)
59
Volume 270, number 1
PHYSICS LETTERS B
Thus overall the SU (3) breaking is quite small. We would expect that the residual model dependence in the SU (3) breaking of this ratio of rates would be very much smaller than the original hadronic uncertainty in either of the rates alone. Since neither of these processes has been measured yet, this test is clearly not practical at present. However, it is expected that both reactions should occur at rates close to present sensitivity, so that one may look forward to their observation soon. It will require a significant increase in sensitivity in order to make a meaningful test, but this should be possible with the expected high luminosity B factories now being planned. The kinematic configuration required, that where both the electron and rho are very energetic, is favorable for experimental searches. The measurement of the ratio of rates should be an experimental goal as it would allow exclusive rare B decays to be useful as tests of the standard model.
Acknowledgement J.F.D. acknowledges stimulating conversation with
60
7 November 1991
Daniel Wyler about this topic, and the hospitality of the CERN theory group during early stages of the work. This work was supported in part by the US National Science Foundation.
References [ 1] N.G. Deshpande et al., Phys. Rev. Lett. 59 ( 1987 ) 183. [2 ] T. Altomari, Phys. Rev. D 37 (1988) 677; P.G. O'Donnell and H.K.K. Tung, University of Toronto Report UTPT-91-06 ( 1991 ). [ 3 ] S. Bertolini et al., Phys. Rev. Lett. 59 ( 1987 ) 180. [4 ] B. Grinstein, R. Springer and M.B. Wise, Nucl. Phys. B 339 (1990) 269. [ 5 ] CLEO Collab., R. Fulton et al., Phys. Rev. Lett. 64 (1990) 16. [6 ] ARGUS Collab., H. Albrecht et al., Phys. Lett. B 234 (1990) 409. [7] N. Isgur and M.B. Wise, Phys. Rev. D 42 (1990) 2388. [8] G.P. Lepage and Sd. Brodsky, Phys. Lett. B 87 (1979) 959; Phys. Rev. D 22 (1980) 2157. [9 ] A. Szczepaniak, E.M. Henley and S.J. Brodsky, Phys. Lett. B243 (1990) 287. [ 10] M. Wirbel, B. Stech and M. Bauer, Z. Phys. C 29 (1987) 637. [ 11 ] N. Isgur, Phys. Rev. D 43 ( 1991 ) 810.