- Email: [email protected]
Light cone QCD sum rule analysis of B → Kℓ+ℓ− decay
8 May 1997
PHYSICS ELSEVIER
LElTERS
B
Physics Letters B 400 ( 1997) 194-205
Light cone QCD sum rule analysis of B + K.&T decay T.M. Aliev a,1, H. Koru b, A. ozpinecia,
M. Savcl a
a Physics Department, Middle East Technical University, 06.531 Ankara, Turkey b Physics Department, Gazi University, 06460 Ankara, Turkey
Received 10 February 1997 Editor: L. Alvarez-GaumB
Abstract We calculate the transition formfactors for the B -+ K!+!?- decay in the framework of the light cone QCD sum rules. The invariant dilepton mass distribution and the final lepton longitudinal polarization asymmetry are investigated. The comparison analysis of our results with traditional sum rules method predictions on the formfactors is performed. @ 1997 Elsevier Science B.V.
1. Introduction Experimental observation [l] of the inclusive and exclusive radiative decays B -+ X,y and B -+ Ky stimulated the study of rare B decays on a new footing. These Flavor Changing Neutral Current (FCNC) b 4 s transitions in the Standard Model (SM) do not occur at the tree level and appear only at the loop level. Therefore the study of these rare B-meson decays can provide a means of testing the detailed structure of the SM at the loop level. These decays are also very useful for extracting the values of the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements [ 21, as well as for establishing new physics beyond the SM [ 31. Currently, the main interest on rare B-meson decays is focused on decays for which the SM predicts large branching ratios and can be potentially measurable in the near future. The rare B -+ Kl+e-and B -+ K*l+ldecays are such decays. The experimental situation for these decays is very promising [4], with e+e- and hadron colliders focusing only on the observation of exclusive modes with t = e, /_Land r final states, respectively. At quark level the process b -+ st+fT takes place via electromagnetic and Z penguin and W box diagrams and are described by three independent Wilson coefficients C7, Cg and Cto. Investigations allow us to study different structures, described by the above mentioned Wilson coefficients. In the SM, the measurement of the forward-backward asymmetry and invariant dilepton mass distribution in b -+ qL’+C-( q = s, d) provide information on the short distance contributions dominated by the top quark loops and are essential in separating the short distance FCNC process from the contributing long distance effects [ 51 and also are very sensitive to 1E-mail: [email protected]. 0370-2693/97/$17.00
0 1997 Elsevier Science B.V. All rights reserved.
PII SO370-2693(97)00327-4
TN. Alieu et al./ Physics Letters B 400 (19971 194-205
the contributions
from new physics
[ 61. Recently
it has been emphasized
195
by Hewett
b + &I!- will be measurable with the high statistics available at the B-factories currently under construction. However, in calculating the Branching ratios and other observables in hadron level, i.e. for B + KL?+C-decay, we have the problem of computing the matrix element of the effective Hamiltonian, XFl,~, between the states B and K. This problem is related to the non-perturbative sector of QCD. These matrix elements, in the framework of different approaches such as chiral theory [ 81, three point QCD sum rules method [ 91, relativistic quark model by the light-front formalism [ lo,11 1, have been investigated. The aim of this work is the calculation of these matrix elements in light cone QCD sum rules method and to study the lepton polarization asymmetry for the exclusive B --+Ke+C-decays. The effective Hamiltonian for the b --f sW_ decay, including QCD corrections [ 12-141 can be written as
(1) which is evolved from the electroweak scale down to ,u N mg by the renormalization group equations. Here I$j represent the relevant CKM matrix elements, and Oi are a complete set of renormalized dimension 5 and 6 operators involving light fields which govern the b + s transitions and Ci( ,LL) are the Wilson coefficients for the corresponding operators. The explicit forms of Ci( /_L)and Oi(p) can be found in [ 12-141. For b -+ se? decay, this effective Hamiltonian leads to the matrix element
where p2 is the invariant
dilepton mass, and L(R)
= [ 1 - (+)~5]
/2 are the projection
operators. The coefficient
C,““(,u, p2) s Cg(p) + Y( ,u, p2), where the function Y contains the contributions from the one loop matrix element of the four-quark operators, can be found in [ 12-141. Note that the function Y(,u, p2) contains both real and imaginary parts (the imaginary part arises when the c-quark in the loop is on the mass shell). The B ---f Ke+e-decay also receives large long distance contributions from the cascade process B A KJ/@($‘) + Kl+e-. These contributions are taken into account by introducing a Breit-Wigner form of the resonance
--
propagator
;c
v=Jl*>
and this procedure mvr(~
v
--)
leads to an additional
contribution
to Cg”” of the form [ 151
e+e-)
( q2 - m2,) - imvrv ’
As we noted earlier, in order to calculate the branching ratios for the exclusive B + KP-decays, the matrix must be calculated. These matrix elements can be elements (KISyF( 1 - y5)qjB) and (KISirr,,p’( 1 + ~5)qlB) parametrized in terms of the formfactors as follows (see also [9] ): (K(q)
ISy,(l
- ys)ql B(P f 4)) = 2q,f+(P2)
where p + q and q are the momentum 2. Sum rules for the I3 --f
+ [f’(p2)
+ f-(p2)]
PP 7
(3)
of B and K and Pp = (p + 2q),.
K transition formfactors
According to the QCD sum rules ideology, in order to calculate these formfactors, we start by considering the representation of a suitable correlator function in hadron and quark-gluon languages. For this purpose, we
196
T.M. Alieu et al/Physics
consider the following K-meson: II(‘)(p p
’ q) = i
II(*)(p,q)
= i
fi
matrix elements
of the T-product
d4x eipX(K(q)lT{WyFU
J
d4x @‘(K(q)lT{-(
J
Letters B 400 (1997) 194-205
of two currents
- ys)Wx)
between
~UWw(O)}~O)
8 x ) i~fiYPP(l +Ys)b(x)
the vacuum
7
~(O)~Y54(0))10)
(5) >
where q is K-meson momentum and p is the transfer momentum. The hadronic (physical) part of Eqs. (5) and (6) is obtained by inserting a complete the B-meson ground state, and higher states with B-meson quantum number:
$%,q)
mi- (P’ +
= (K(q)l%4y,U - ys)b(x)lB(p +q))
h (P + q)2)q,
+ F2(P2,
rIc2’(p fi ’ q) = (K(q)~%W+.wp”(1
(P + q)*)P,
set of states including
+ 4) l&Ww(O) IO) IO)
9
(7)
+~s)b(x)lB(p
+q))
+?‘dWlh(P+q))m;
+~(K(q)I3(x)iqw~~(l
(6)
- (P’ +q12(MP + 4) lmPY54(o)
+~(K(q)lWy,U - ys)Wx)lh(P+drni = Fl (p2,
(B(P
q12
state and the
1 m’B- (P + q)*
_
(B(P + q) l~(OM5q(O) IO)
,~+q)2@(P+q);~(0)~Y~q~O)~O)
h
= F3((P + q)21P2) [&LP” - P,(PP)] where PN = (p + 2q),. Then, for the invariant B meson momentum squared, (p + q>2, as
pl(p2,
s)
= a(~
-
m$)2f+(p2)
p2(p2,
$1 = a(s
-
m2,>
p3(p2,s)
[f+(p2>
fr(p2)
=S(s-m2,)
mB-kmK
m&fB mb +
F;:, one can write a general dispersion
amplitudes
+ &P2>
f-(p2>]
mkfB
(8)
9
s)
I p3h(p2
(10)
, P~(P2~~) 1
+
*
3
s)
.
.fBd mb
’
(11) (12)
mb
The first terms in Eqs. (lo)-( 12) represent the ground state B-meson contribution and (6) by inserting the matrix elements given in Eqs. (3)) (4) and by replacing (B16iy5q10) = -
relation in the
and follow from Eqs. (5)
(13)
in which we neglect the mass of the light quarks. In Eqs. (lo)-( 12), pf(p2, s) represent the spectral density of the higher resonances and of the continuum of states. In accordance with the QCD sum rules method we invoke the quark-hadron duality prescription and replace the spectral density p” by
T.M. Alieu et al./Physics
Letters B 400 (1997) 194-205
197
(14) where ImFgCD (p* , s > is obtained from the imaginary part of the correlator functions calculated in QCD starting from some threshold so. In order to suppress the higher states and continuum contributions, we follow the standard procedure in the QCD sum rules method and apply Bore1 transformation 8 on the variable (p +q) 2 to the dispersion integral to get
Fi(p2, M*) = &(p*,
(p + q)*)
=
pi(p2, s)eCSiMZds.
(1%
s m; Using Eqs. (lo)-(
F, tp*&‘*)
12) and (14) we get
= V+(P*)~~
F2(p2, M2) = [f+(p*)
-r&M2
m2,fB
f f-(p*)]
f
FpcD(p2,
/Im
k
~evm~iM2
s)e-SfM2ds,
+ a ~ImFr”(p”,s)e-‘/~‘ds,
(16)
(17)
SO
Fdp*,
M*)
fdp2>
=
mkfk?
,_&jM2
+
i
Tim
FyD(p2,
s)e-“jM2ds.
mb
MB+“‘%
The main problem is then the calculation of the correlator functions (5) and (6) in QCD. After applying Bore1 transformation, the result can be written in the following form:
(18) the
Co
Fi(p2,M2)
= h $
ImF~cD(p2,s)e-S~M2ds.
Equating the F;: in Eq. (19) to the corresponding formfactors, which describe the B -+ K transition:
ff(p2)
= 2Tyim2
(19)
E in Eqs. (16)-(
~ImF~cD(p2,~)~-(‘-ni,,“2ds, B
18), we arrive at the sum rules for the
(20)
4 so
f+(p*)
+ _r(p2>
Im
= ---!3--
rft&
F,QcD(p2, S)e-(S-“‘~)/M2dS,
s
(21)
4 fT(P2> = $?&(mg+mK)
41mF~cD(p2,s)e~(J~n’~~~M2d~. s In;
(22)
One can calculate ImFQCD (p2, s) , m ’ the deep Euclidean region, where both p2 and (p + q)2 are negative and large. The leading contribution to the operator product expansion comes from the contraction of the b-quark operators expressed in Eqs. (5) and (6) to the free b-quark propagator
198
T.M. Alieu et al/Physics
(OIT{b(x)6(0)}~0) =
Letters B 400 (1997) 194-205
-i/-&eP@$. b
Then we have
III”‘(p P
d4x d4k ’
q)
=i
(2~-)~(rn$ - k2)
s
Ic2’(p P ’ q)
e”fP-k)*(Ktq)I
d4x d4 k = -
(2~)~(rn;
J’
- k2>
[-rnbSY,(l
ei(“-k)xp”(K(q)/
y5y5)4+
-
[-?U&!(l
WJU
+ Y5)q]
-Y5)q+mbfgpv(1
IO) 9
+Y5)q]
(23)
10).
(24) Note that, as can be seen from Eqs. (23) and (24>, the problem is reduced to the calculation of the matrix elements of the gauge-invariant, nonlocal operators sandwiched in between the vacuum and the K meson states. These matrix elements define the K meson light cone wave functions. Following [ 16,171, we define the K-meson wave functions as
(K(q) l3X)Y,( 1 - Y5)4(0) IO) 1
du ehfqx [P&U) +x2&t)]
= &_fK
- f~ (xP - +&)
J
Id.
(2.5)
eiu@g2(u).
0
0
In Eq. (25) V)K(U) is the leading twist two, gt (u) and g2( u) are the twist four K meson wave functions, respectively. The second matrix element of Eq. (23)) can be split into two matrix elements using the identity Y/LYV= g,uv - l(+/LV’and the result can be evaluated using the twist three wave functions defined as [ 16-191 I
_fc4 (K(q) lftx)inqtO) IO) = ~ ms + mq
J
(26)
,
du eiU@qDp (u)
0
1
(K(q) l~tx)qw t 1 +
y5)4(0)
IO)= itq,h - wp)
s
du eiUqXp,,( u) 6t$+mkq,
.
(27)
0
The matrix elements
in Eq. (24) can be easily calculated
using the identities
i ffPv = -2
9
$wpp@pY5
- Yvg,,)
cPyYp = i(Y,g,,
+ $WppYPY5 )
to express it in terms of the wave functions in Eq. (25). Substituting the matrix elements (25)-( 27) into the Eqs. (23) x and k we get the following expressions for Fi (p2, (p + q) 2, : 1
Sm2 k(u)
FI (p*, (p + q)2) = mb.fK
+
and (24) and integrating
G(u)I +
A2 0
1
+ fK,UK
J 0
du a
W,(U) [
+ $X%(u)
1+ (
2rng - 2upq - 2q2u2 A >I
2ug2(u)
A
9
over the variables
1 (28)
T.M. Alieu et al./Physics
199
Letters B 400 (1997) I94-205
(29)
F3(p2, (p+s)2)
=-fKj-{ [ du A
4 ‘f’Ktu)
4 [gl(u)
-
+ Gz(u>l A
(1+T)] +mb,UKF} , (30)
0
where
q” = rni,
A=m~-(p+q~)~,
4
,UK=m,
s
Gz(u>=-
4
s 0
gz(u)du.
There are also contributions to the above considered wave functions from multi-particle meson wave functions. Here we consider only the operator gGq, which gives the main contribution and corresponds to the quarkantiquark-gluon components in the kaon (for more detail see [ 201) . In this approximation, the b-quark propagator is defined as (see [ 181)
=
d4k
i$j(x> - ig,
- ikx
1.
GpY(ux)~ P””+ ~uxpGpLY ’ rni - k2
@pe
(ux) yv
0
(31)
It is clear that when we take into account the second term in the right hand side of Eq. (3 1) , new matrix elements appear. Substituting Eq. (31) into Eqs. (5) and (6)) and using the identity YPYP~P~ = (a,&,
- UP&n)
one can express the resulting
+ i(gPAgUP - g,,g,n)
new matrix elements
- ~~~~~~~ - i~V,~&P~PcLpy5,
in terms of the three particle wave functions
[ 16-201
(K(q) 13x)gsGpv(ux>~agYsq(O) IO>
= if3K[
(qfiq&p
-
qvqa&P)
(qpqp&
- qvqPkk)
%$p3K
1
(ai)
eiqxo
,
(32)
s
(K(q) IS(x)y,ysg,G,p(ux)q(O) + .fKz
-
(%x/3
-
q@&z)
10) = fK [qp (gap Dcrgppl~ (q)
-
q&)
F)
-
qLu (m
-
F)]
/~~iip~(cri)eiqxw
)
(33)
s
(~(q)ls(x)Y,gs~,,(U~)q(O)/oj + ifK$(%&3
eQxo
-
=i.fK
ZhpQ s
[qp
(cq)
(&
eiqxxo
-
,
2)
-
qol (&
-
F)]
~~~iPl(ai)eiq””
(34)
where Gap = ;~~p~hG PA, Dtri = daldcqdcqEi( 1 - a1 - cq - arg), and w = CX~+ ucx3. Here qsK(Czi) is a twist three wave function and the remaining functions ~1, ~11, 61 and 911 are all twist four wave functions. When we substitute Eqs. (32)-( 34) into Bqs. (5) and (6), perform integration over x and k, add (28)) (29)) (30)) apply the Bore1 transformation over (p + q)2 and equate the obtained results to ( 16)) ( 17) and ( 18)) we get the following sum rules for the B + K transition formfactors:
200
T.M. Alieu et al./ Physics Letters B 400 (1997) 194-205
1
+
8mg [gl (u)
2 +
rnz + p2
- q2u2
UM2
Dcr~F!~(w- 8) exp
f3K
l+%(u)
+ G(u)
-
mi--p’(l-w)+q2w(1-ti) wM2
0 (2U
-
1 )p3K
.fK ,
rng - p2 - q2w2
3q2 + 2Ups~ wM2
w3M2
mb
i-q
+f-(p2)
f+(p2>
++,
+ +,,)
(35)
w2M4
f3K
_mi-p2tl -u> +q22t(l -u)
(
= se5
LlM2
* >
u
-$$$(d-p2+q2u2))] 1
rng-p2(1-w)+q2w(1-w)
DaiB(w-6)exp
+
wM2
0
(36)
rnz-p2(1-z4)+q2z4(1-z4)
&2M2
-
57
I
1
+
JS du
rni-p2(1-w)+q2w(1-0)
D,cui@(w-S)exp
oil@
II 3
0
X
u (q
where 6 = s.
‘ht’h)
- 244 + 4011+ 911- 2sol- 245-L. W2M2
w2M2
The functions
=-
Ul(@l),
LI r
Jql(u)tGl(u))du, 0
Ull(@ll)
(37)
in Eqs. (35)-(
*,,t+,,) = -
37) are defined in the following
way:
U r
J
(PII tu)t911tu) )du.
0
Note that the formfactor f+(p2) is investigated in [ 181 and [ 191 (In [ 181, it is necessary to make the simple replacement GT---f K), in light cone sum rules. Our results coincide with theirs if we let $- = 0.
TM. Alieu et d/Physics
At the end of this section final leptons and we obtain
we calculate
the differential
where A= l+r2fs2-2r-2s-2sr,
r=m$/mi,
me and D =
A = 2CGfff+ - C7 For the dileptonic defined by
4mbfT
C = 2ClO.f
decays of the B mesons,
$(8 =-1) &,(P*)
= dI --@(5=
s=p2/mg,
- $(8
+
,
polarization
,
of the
(38)
5 is the longitudinal
are its mass and velocity, respectively.
mB-i-mK’
201
decay rate with the longitudinal
(1 - Y - s)] - t Re(A*C)m&$h
+ 8mim$ [r ICI2 + s jDl2 + Re(C*D)
final lepton, as follows:
Letters B 400 (1997) 194-205
polarization
of the
In Eq. (38) A, C and D are defined
D = C~o(f+ + f-j.
the longitudinal
polarization
asymmetry
PL of the final state e, is
= 1) (40)
-1)
+ $(5=
1) .
where 5 = - 1 (+l ) corresponds to the left (right) handed lepton. If in Eq. (40)) we let me = 0, our results coincide with the results in [21] and if me # 0, they coincide with the ones in [ 111.
3. Numerical analysis The main input parameters in the sum rules (35)-( 37) are the kaon wave functions kaon wave functions we use the results of Refs. [ 16,17,19] : $DK= 6u( 1 - u) {I+
0.52 [5(2zf - 1)2-
401, = 1,
p, N 6u(l - u) , g1 (U) 21 $V(
1 - U)2 )
g2(u) 2: $%%( 1 - U) (2U - 1) , (~3dai)
= 360wza:,
pJol(cui) N lO&xt
- cy2)Ly~)
on the light cone. For
11 + 0.34 [21(2U - 1)4 - 14(2u - 1)2 + 11) ,
202
T.M. Alieo et ul./Physics Letters B 400 (1997) 194-205
here a*( ,ub) N 0.17 GeV2 at ,Ub N Jm2,-z (for more detail see [ 18]), Goldstone bosons PK
-=
Pu,
mb = 2.4 GeV*, which follows from the QCD sum rules analysis
E(P~) N 0.36. In order to estimate ,!.&K,we use the PCAC relation
(~~~)+(Ss))f~
=O(j2
2(qdf~
for the pseudo
tqcuord)
.
Here we use fr rv 133 MeV, fK N 160 MeV and we assume (44) f (Ss) = 1 + 0.8, which follows from QCD sum rules for strange hadrons [ 221. As a result we have
PK N 1 GeV when ,uu, = J
m*
1: 1.6 GeV.
mu+md
For the values of the other parameters, we choose: _fs N 0.14 GeV, which is obtained from 2-point QCD sum rules analysis [ 18,231, mb N 4.7 GeV and so N 35 GeV*, and II&cl N 0.045. Before giving numerical results on the formfactors, we must first determine the region for the Bore1 mass parameter M*, for which the sum rules yields reliable results. The lower limit of this region is determined by the requirement that, the terms proportional to Mm2”(n > 1) remain subdominant. The upper limit of M* is determined by requiring the higher resonance and continuum contributions to be less than N 30% of the total result. Our numerical results show that both requirements are satisfied in the region 8 GeV* < M* 5 16 GeV* and for the numerical analysis we use M 2 = 10 GeV*. When p* approaches the region rng - 0( 1 GeV*) a breakdown of the stability is expected, similar to the B -+ TTcase (see [ 18,191). In Fig. 1, we present the p* dependence of the formfactors f+(p*), f -(p*) and fr(p*). At zero momentum transfer, the QCD prediction for the formfactors are f+(p2
= 0) = 0.29,
f-(p2
= 0) = -0.21,
f&J* = 0) = -0.31.
In Fig. 2a(b) we present the p2 dependence of the differential Branching ratios for the B + K,ufpu- and (B -+ Kr+r-) decay with and without long distance effects. In both cases summation over the final lepton polarization is performed. Note that p2 dependence of the differential branching ratio B -+ KT+~- is analysed in [ 111 and [ 241 using the light front formalism and heavy meson chiral theory, respectively. In [ 111 both short and long distance contributions are considered while in [ 241 only the short distance contributions are taken into account and in both works the obtained spectrum is fully symmetric while in the present work the spectrum we obtain seems to be slightly asymmetric as a result of a highly asymmetric resonance-type behaviour due to the nonperturbative contributions. Performing the integration over p* in EZq. (38) and using the values of the life times r& = (1.56 f 0.06) x lo-l2 s, r~, = (1.62 f 0.06) x lo-‘* s [25], for the branching ratios, including only the short distance contributions, we get
B(Bd -+ Kc,u+,u-)
= (3.1 f 0.9) x 10-7,
B(&
-+ &+T-)
B(B,
= (3.2 =k0.8) x 10-7,
B(B,
+ K+T+T-)
+ Kf,u+p-)
= (1.7 f 0.4) X 10-7, = (1.77 dz 0.40) x 10-7,
where theoretical and experimental errors have been added quadratically. In Fig. 3 we display the lepton longitudinal polarization asymmetry PL as a function Kp+,u- and B --f KT + r - decays, at m, = 176 GeV, with and without the long distance this figure one can see that Pr vanishes at the threshold due to the kinematical factor u for the B -+ Kp+y- decay varies in the region (0 + -0.7) and (0 + -0.1) for the B -+ long distance effects are excluded.
of p* for the contributions. and the value Krirdecay,
B + From of PL, when
T.M. Alieu et al./ Physics Letters B 400 (1997) 194-205
p2 (GeV’)
p* (GeV’)
(a)
@I
_
203
-0.50
%
z
-0.E
-1.25
I
0
4
8
12
16
p’ (GeV’)
Fig. 1. The p2 dependence
of the fomfactors
ft(p*),
f- (p2) and fr(p*).
4. Conclusion In this work, we calculate the transition formfactors for the exclusive B + K-&?- (e = ,u, T) decay in the framework of the light cone QCD sum rules, and investigate the longitudinal polarization asymmetries of the muon and tau in this decay. From a comparison of our results with the traditional QCD sum rule predictions (see [ 91) , we observe that the behaviour of the formfactors are similar and the value of f + in both approaches coincides, while fr differs two times than that of [9] at p2 = 0. It is important to note that for a more refined analysis, it is necessary to take into account the SU(3) breaking effects: the differences between f~, ,LLKand jr, ,!L, and the differences of pion and kaon wave functions. These SU(3) breaking terms can lead to differences between B + n- and B + K formfactors. But we expect that these effects can change the results about 1520% and this lies at the accuracy level of the sum rules method. Few words about the possibility of the experimental observation of this decay are in order. Experimentally, to observe an asymmetry PL of a decay with the branching ratio B at the nu level, the required number of events is N = I$$
(see [ 111) . For example, to observe the r lepton polarization
at the exclusive
channel
B + KT+Y
204
TM. Aliev et d/Physics
p2(GeV’)
Letters B 400 (1997) 194-205
p’ (GeV*)
(a)
p* (GeV’)
Fig. for the the and
2. (a) Invariant mass squared distribution of the lepton pair the decay B + Kp+,u-. (b) The same as in a), but for decay B ----fKT+~-. Here and in all of the following figures solid line corresponds to the short distance contributions only the dashed line to the sum of both short and long distance
Fig. 3. (a) The longitudinal polarization asymmeny PL for the B -+ Kp+,u-decay. (b) The same as in (a), but for the B + Kr+r-decay.
contributions. at the 3a level, one needs at least N cv 5 x lo9 B B decays. Since in the future B-factories, it is expected that N lo9 B-mesons would be created per year, it is possible to measure the longitudinal polarization asymmetry of the T lepton.
References [ 1] MS. Alam et. al., CLEO Collaboration, Phys. Rev. Lett. 74 (1995) 2885; R. Ammar et. al., CLEO Collaboration, Phys. Rev. Lett. 71 (1993) 674. [2] Z. Ligeti and M. Wise, Phys. Rev. D 53 (1996) 4937. ]3] J.L. Hewett, in: Proc. of the 21s’ Annual SLAC Summer Institute, eds. L. De Parcel and C. Dunwoodie, SLAC-PUB 6521. [4] C. Anway-Wiese, CDF Collaboration, in: Proc. of the 8 r/r Meeting of the Division of Particle and Fields of the American Physical Society, Albuquerque, New Mexico (1994), ed. S. Seidel (World Scientific, Singapore, 1995). [5] N.G. Deshpande, J. Trampetic and K. Panose, Phys. Rev. D 39 (1989) 1461; C.S. Lim, T. Morozumi and A.I. Sanda, Phys. Len. B 218 ( 1989) 343. [6] A. Ali, T. Mannel and T. Morozumi, Phys. Lett. B 273 (1991) 505; A. Ali, G.F. Giudice and T. Mannel, Z. Phys. C 67 (1995) 417;
T.M. Aliev et al./ Physics Letten
[7] [8] [9] [IO] [ 111 [ 121 [ 131 [ 141
[ 151
[ 161 [ 171 [ 181 [ 191 [20] I21 ] [22] 1231 1241 [25]
B 400 (1997) 194-205
G. Burdman, Phys. Rev. D 52 (1995) 6400; E Kruger and L.M. Sehgal, Phys. Lett. B 380 (1996) 199. J.L. Hewett, Phys. Rev. D 53 (1996) 4964. R. Casalbuoni, A. Deandra, N. Di Btiolemo, R. Gatto and G. Nardulli, Phys. L.ett. B 312 (1993) 315. P. Colangelo, F. De Fazio, P. Sanntorelli and E. Scrimieri, Phys. Rev. D 53 (1996) 3672. W. Jaus and D. Wyler, Phys. Rev. D 41 (1990) 3405. C.Q. Ceng and C.P. Kao, Phys. Rev. D 54 (1996) 3656. B. Grinstein, M.J. Savage and M.B. Wise, Nucl. Phys. B 319 (1989) 271. M. Misiak, Nucl. Phys. B 398 (1993) 23; B 439 (1995) 461 (E). A.J. Buras and M. Miinz, Phys. Rev. D 52 (1995) 186; M. Ciuchini, E. France, G. MartinelB, L. Reina and L. Silvestrini. Phys. L&t. B 316 (1993) 127; M. Ciuchini, E. France, G. Martinelli and L. Reina, Nucl. Phys. B 415 (1994) 403; G. Cella, G. Curci, R. Ricciardi and A. Vicere, Nucl. Phys. B 421 (1994) 41; Phys. Lett. B 325 (1994) PJ. O’Donnell and H.K.K. Tung, Phys. Rev. D 43 (1991) R2067; A.I. Vainshtein, V.I. Zakharov, L.B. Okun and M.A. Shifman, Sov. J. Nucl. Phys. 24 ( 1976) 427. V.L. Chemyak and I.R. Zhitnitsky, Phys. Rep. C 112 (1984) 173. V.M. Braun and LB. Filyanov, Z. Phys. C 44 (1989) 157; C 48 (1990) 239. V.M. Belyaev, V.M. Braun, A. Khodjamirian and R. Riickl, Phys. Rev. D 51 (1995) 6177. V.M. Belyaev, A. Khodjamirian and R. Riickl, Z. Phys. C 60 (1993) 349. 1.1. Balitsky, V.M. Braun and A.V. Kolesnichenko, Nucl. Phys. B 312 (1989) 509. C. Greub, A. ioannissian and D. Wyler, Phys. Lett. B 346 ( 1995) 149. S. Narison, QCD spectral sum rules (World Scientific, Singapore, 1989). T.M. Aliev, V.L. Eletsky, Sov. J. Nucl. Phys. 38 (1983) 936. Dong-Sheng Du, C. Liu, Da-Xin Zhang, Phys. L.&t. B 317 (1993) 179. Particle Data Group, Phys. Rev. D 50 (1996).
205
227.
PHYSICS ELSEVIER
LElTERS
B
Physics Letters B 400 ( 1997) 194-205
Light cone QCD sum rule analysis of B + K.&T decay T.M. Aliev a,1, H. Koru b, A. ozpinecia,
M. Savcl a
a Physics Department, Middle East Technical University, 06.531 Ankara, Turkey b Physics Department, Gazi University, 06460 Ankara, Turkey
Received 10 February 1997 Editor: L. Alvarez-GaumB
Abstract We calculate the transition formfactors for the B -+ K!+!?- decay in the framework of the light cone QCD sum rules. The invariant dilepton mass distribution and the final lepton longitudinal polarization asymmetry are investigated. The comparison analysis of our results with traditional sum rules method predictions on the formfactors is performed. @ 1997 Elsevier Science B.V.
1. Introduction Experimental observation [l] of the inclusive and exclusive radiative decays B -+ X,y and B -+ Ky stimulated the study of rare B decays on a new footing. These Flavor Changing Neutral Current (FCNC) b 4 s transitions in the Standard Model (SM) do not occur at the tree level and appear only at the loop level. Therefore the study of these rare B-meson decays can provide a means of testing the detailed structure of the SM at the loop level. These decays are also very useful for extracting the values of the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements [ 21, as well as for establishing new physics beyond the SM [ 31. Currently, the main interest on rare B-meson decays is focused on decays for which the SM predicts large branching ratios and can be potentially measurable in the near future. The rare B -+ Kl+e-and B -+ K*l+ldecays are such decays. The experimental situation for these decays is very promising [4], with e+e- and hadron colliders focusing only on the observation of exclusive modes with t = e, /_Land r final states, respectively. At quark level the process b -+ st+fT takes place via electromagnetic and Z penguin and W box diagrams and are described by three independent Wilson coefficients C7, Cg and Cto. Investigations allow us to study different structures, described by the above mentioned Wilson coefficients. In the SM, the measurement of the forward-backward asymmetry and invariant dilepton mass distribution in b -+ qL’+C-( q = s, d) provide information on the short distance contributions dominated by the top quark loops and are essential in separating the short distance FCNC process from the contributing long distance effects [ 51 and also are very sensitive to 1E-mail: [email protected]. 0370-2693/97/$17.00
0 1997 Elsevier Science B.V. All rights reserved.
PII SO370-2693(97)00327-4
TN. Alieu et al./ Physics Letters B 400 (19971 194-205
the contributions
from new physics
[ 61. Recently
it has been emphasized
195
by Hewett
b + &I!- will be measurable with the high statistics available at the B-factories currently under construction. However, in calculating the Branching ratios and other observables in hadron level, i.e. for B + KL?+C-decay, we have the problem of computing the matrix element of the effective Hamiltonian, XFl,~, between the states B and K. This problem is related to the non-perturbative sector of QCD. These matrix elements, in the framework of different approaches such as chiral theory [ 81, three point QCD sum rules method [ 91, relativistic quark model by the light-front formalism [ lo,11 1, have been investigated. The aim of this work is the calculation of these matrix elements in light cone QCD sum rules method and to study the lepton polarization asymmetry for the exclusive B --+Ke+C-decays. The effective Hamiltonian for the b --f sW_ decay, including QCD corrections [ 12-141 can be written as
(1) which is evolved from the electroweak scale down to ,u N mg by the renormalization group equations. Here I$j represent the relevant CKM matrix elements, and Oi are a complete set of renormalized dimension 5 and 6 operators involving light fields which govern the b + s transitions and Ci( ,LL) are the Wilson coefficients for the corresponding operators. The explicit forms of Ci( /_L)and Oi(p) can be found in [ 12-141. For b -+ se? decay, this effective Hamiltonian leads to the matrix element
where p2 is the invariant
dilepton mass, and L(R)
= [ 1 - (+)~5]
/2 are the projection
operators. The coefficient
C,““(,u, p2) s Cg(p) + Y( ,u, p2), where the function Y contains the contributions from the one loop matrix element of the four-quark operators, can be found in [ 12-141. Note that the function Y(,u, p2) contains both real and imaginary parts (the imaginary part arises when the c-quark in the loop is on the mass shell). The B ---f Ke+e-decay also receives large long distance contributions from the cascade process B A KJ/@($‘) + Kl+e-. These contributions are taken into account by introducing a Breit-Wigner form of the resonance
--
propagator
;c
v=Jl*>
and this procedure mvr(~
v
--)
leads to an additional
contribution
to Cg”” of the form [ 151
e+e-)
( q2 - m2,) - imvrv ’
As we noted earlier, in order to calculate the branching ratios for the exclusive B + KP-decays, the matrix must be calculated. These matrix elements can be elements (KISyF( 1 - y5)qjB) and (KISirr,,p’( 1 + ~5)qlB) parametrized in terms of the formfactors as follows (see also [9] ): (K(q)
ISy,(l
- ys)ql B(P f 4)) = 2q,f+(P2)
where p + q and q are the momentum 2. Sum rules for the I3 --f
+ [f’(p2)
+ f-(p2)]
PP 7
(3)
of B and K and Pp = (p + 2q),.
K transition formfactors
According to the QCD sum rules ideology, in order to calculate these formfactors, we start by considering the representation of a suitable correlator function in hadron and quark-gluon languages. For this purpose, we
196
T.M. Alieu et al/Physics
consider the following K-meson: II(‘)(p p
’ q) = i
II(*)(p,q)
= i
fi
matrix elements
of the T-product
d4x eipX(K(q)lT{WyFU
J
d4x @‘(K(q)lT{-(
J
Letters B 400 (1997) 194-205
of two currents
- ys)Wx)
between
~UWw(O)}~O)
8 x ) i~fiYPP(l +Ys)b(x)
the vacuum
7
~(O)~Y54(0))10)
(5) >
where q is K-meson momentum and p is the transfer momentum. The hadronic (physical) part of Eqs. (5) and (6) is obtained by inserting a complete the B-meson ground state, and higher states with B-meson quantum number:
$%,q)
mi- (P’ +
= (K(q)l%4y,U - ys)b(x)lB(p +q))
h (P + q)2)q,
+ F2(P2,
rIc2’(p fi ’ q) = (K(q)~%W+.wp”(1
(P + q)*)P,
set of states including
+ 4) l&Ww(O) IO) IO)
9
(7)
+~s)b(x)lB(p
+q))
+?‘dWlh(P+q))m;
+~(K(q)I3(x)iqw~~(l
(6)
- (P’ +q12(MP + 4) lmPY54(o)
+~(K(q)lWy,U - ys)Wx)lh(P+drni = Fl (p2,
(B(P
q12
state and the
1 m’B- (P + q)*
_
(B(P + q) l~(OM5q(O) IO)
,~+q)2@(P+q);~(0)~Y~q~O)~O)
h
= F3((P + q)21P2) [&LP” - P,(PP)] where PN = (p + 2q),. Then, for the invariant B meson momentum squared, (p + q>2, as
pl(p2,
s)
= a(~
-
m$)2f+(p2)
p2(p2,
$1 = a(s
-
m2,>
p3(p2,s)
[f+(p2>
fr(p2)
=S(s-m2,)
mB-kmK
m&fB mb +
F;:, one can write a general dispersion
amplitudes
+ &P2>
f-(p2>]
mkfB
(8)
9
s)
I p3h(p2
(10)
, P~(P2~~) 1
+
*
3
s)
.
.fBd mb
’
(11) (12)
mb
The first terms in Eqs. (lo)-( 12) represent the ground state B-meson contribution and (6) by inserting the matrix elements given in Eqs. (3)) (4) and by replacing (B16iy5q10) = -
relation in the
and follow from Eqs. (5)
(13)
in which we neglect the mass of the light quarks. In Eqs. (lo)-( 12), pf(p2, s) represent the spectral density of the higher resonances and of the continuum of states. In accordance with the QCD sum rules method we invoke the quark-hadron duality prescription and replace the spectral density p” by
T.M. Alieu et al./Physics
Letters B 400 (1997) 194-205
197
(14) where ImFgCD (p* , s > is obtained from the imaginary part of the correlator functions calculated in QCD starting from some threshold so. In order to suppress the higher states and continuum contributions, we follow the standard procedure in the QCD sum rules method and apply Bore1 transformation 8 on the variable (p +q) 2 to the dispersion integral to get
Fi(p2, M*) = &(p*,
(p + q)*)
=
pi(p2, s)eCSiMZds.
(1%
s m; Using Eqs. (lo)-(
F, tp*&‘*)
12) and (14) we get
= V+(P*)~~
F2(p2, M2) = [f+(p*)
-r&M2
m2,fB
f f-(p*)]
f
FpcD(p2,
/Im
k
~evm~iM2
s)e-SfM2ds,
+ a ~ImFr”(p”,s)e-‘/~‘ds,
(16)
(17)
SO
Fdp*,
M*)
fdp2>
=
mkfk?
,_&jM2
+
i
Tim
FyD(p2,
s)e-“jM2ds.
mb
MB+“‘%
The main problem is then the calculation of the correlator functions (5) and (6) in QCD. After applying Bore1 transformation, the result can be written in the following form:
(18) the
Co
Fi(p2,M2)
= h $
ImF~cD(p2,s)e-S~M2ds.
Equating the F;: in Eq. (19) to the corresponding formfactors, which describe the B -+ K transition:
ff(p2)
= 2Tyim2
(19)
E in Eqs. (16)-(
~ImF~cD(p2,~)~-(‘-ni,,“2ds, B
18), we arrive at the sum rules for the
(20)
4 so
f+(p*)
+ _r(p2>
Im
= ---!3--
rft&
F,QcD(p2, S)e-(S-“‘~)/M2dS,
s
(21)
4 fT(P2> = $?&(mg+mK)
41mF~cD(p2,s)e~(J~n’~~~M2d~. s In;
(22)
One can calculate ImFQCD (p2, s) , m ’ the deep Euclidean region, where both p2 and (p + q)2 are negative and large. The leading contribution to the operator product expansion comes from the contraction of the b-quark operators expressed in Eqs. (5) and (6) to the free b-quark propagator
198
T.M. Alieu et al/Physics
(OIT{b(x)6(0)}~0) =
Letters B 400 (1997) 194-205
-i/-&eP@$. b
Then we have
III”‘(p P
d4x d4k ’
q)
=i
(2~-)~(rn$ - k2)
s
Ic2’(p P ’ q)
e”fP-k)*(Ktq)I
d4x d4 k = -
(2~)~(rn;
J’
- k2>
[-rnbSY,(l
ei(“-k)xp”(K(q)/
y5y5)4+
-
[-?U&!(l
WJU
+ Y5)q]
-Y5)q+mbfgpv(1
IO) 9
+Y5)q]
(23)
10).
(24) Note that, as can be seen from Eqs. (23) and (24>, the problem is reduced to the calculation of the matrix elements of the gauge-invariant, nonlocal operators sandwiched in between the vacuum and the K meson states. These matrix elements define the K meson light cone wave functions. Following [ 16,171, we define the K-meson wave functions as
(K(q) l3X)Y,( 1 - Y5)4(0) IO) 1
du ehfqx [P&U) +x2&t)]
= &_fK
- f~ (xP - +&)
J
Id.
(2.5)
eiu@g2(u).
0
0
In Eq. (25) V)K(U) is the leading twist two, gt (u) and g2( u) are the twist four K meson wave functions, respectively. The second matrix element of Eq. (23)) can be split into two matrix elements using the identity Y/LYV= g,uv - l(+/LV’and the result can be evaluated using the twist three wave functions defined as [ 16-191 I
_fc4 (K(q) lftx)inqtO) IO) = ~ ms + mq
J
(26)
,
du eiU@qDp (u)
0
1
(K(q) l~tx)qw t 1 +
y5)4(0)
IO)= itq,h - wp)
s
du eiUqXp,,( u) 6t$+mkq,
.
(27)
0
The matrix elements
in Eq. (24) can be easily calculated
using the identities
i ffPv = -2
9
$wpp@pY5
- Yvg,,)
cPyYp = i(Y,g,,
+ $WppYPY5 )
to express it in terms of the wave functions in Eq. (25). Substituting the matrix elements (25)-( 27) into the Eqs. (23) x and k we get the following expressions for Fi (p2, (p + q) 2, : 1
Sm2 k(u)
FI (p*, (p + q)2) = mb.fK
+
and (24) and integrating
G(u)I +
A2 0
1
+ fK,UK
J 0
du a
W,(U) [
+ $X%(u)
1+ (
2rng - 2upq - 2q2u2 A >I
2ug2(u)
A
9
over the variables
1 (28)
T.M. Alieu et al./Physics
199
Letters B 400 (1997) I94-205
(29)
F3(p2, (p+s)2)
=-fKj-{ [ du A
4 ‘f’Ktu)
4 [gl(u)
-
+ Gz(u>l A
(1+T)] +mb,UKF} , (30)
0
where
q” = rni,
A=m~-(p+q~)~,
4
,UK=m,
s
Gz(u>=-
4
s 0
gz(u)du.
There are also contributions to the above considered wave functions from multi-particle meson wave functions. Here we consider only the operator gGq, which gives the main contribution and corresponds to the quarkantiquark-gluon components in the kaon (for more detail see [ 201) . In this approximation, the b-quark propagator is defined as (see [ 181)
=
d4k
i$j(x> - ig,
- ikx
1.
GpY(ux)~ P””+ ~uxpGpLY ’ rni - k2
@pe
(ux) yv
0
(31)
It is clear that when we take into account the second term in the right hand side of Eq. (3 1) , new matrix elements appear. Substituting Eq. (31) into Eqs. (5) and (6)) and using the identity YPYP~P~ = (a,&,
- UP&n)
one can express the resulting
+ i(gPAgUP - g,,g,n)
new matrix elements
- ~~~~~~~ - i~V,~&P~PcLpy5,
in terms of the three particle wave functions
[ 16-201
(K(q) 13x)gsGpv(ux>~agYsq(O) IO>
= if3K[
(qfiq&p
-
qvqa&P)
(qpqp&
- qvqPkk)
%$p3K
1
(ai)
eiqxo
,
(32)
s
(K(q) IS(x)y,ysg,G,p(ux)q(O) + .fKz
-
(%x/3
-
q@&z)
10) = fK [qp (gap Dcrgppl~ (q)
-
q&)
F)
-
qLu (m
-
F)]
/~~iip~(cri)eiqxw
)
(33)
s
(~(q)ls(x)Y,gs~,,(U~)q(O)/oj + ifK$(%&3
eQxo
-
=i.fK
ZhpQ s
[qp
(cq)
(&
eiqxxo
-
,
2)
-
qol (&
-
F)]
~~~iPl(ai)eiq””
(34)
where Gap = ;~~p~hG PA, Dtri = daldcqdcqEi( 1 - a1 - cq - arg), and w = CX~+ ucx3. Here qsK(Czi) is a twist three wave function and the remaining functions ~1, ~11, 61 and 911 are all twist four wave functions. When we substitute Eqs. (32)-( 34) into Bqs. (5) and (6), perform integration over x and k, add (28)) (29)) (30)) apply the Bore1 transformation over (p + q)2 and equate the obtained results to ( 16)) ( 17) and ( 18)) we get the following sum rules for the B + K transition formfactors:
200
T.M. Alieu et al./ Physics Letters B 400 (1997) 194-205
1
+
8mg [gl (u)
2 +
rnz + p2
- q2u2
UM2
Dcr~F!~(w- 8) exp
f3K
l+%(u)
+ G(u)
-
mi--p’(l-w)+q2w(1-ti) wM2
0 (2U
-
1 )p3K
.fK ,
rng - p2 - q2w2
3q2 + 2Ups~ wM2
w3M2
mb
i-q
+f-(p2)
f+(p2>
++,
+ +,,)
(35)
w2M4
f3K
_mi-p2tl -u> +q22t(l -u)
(
= se5
LlM2
* >
u
-$$$(d-p2+q2u2))] 1
rng-p2(1-w)+q2w(1-w)
DaiB(w-6)exp
+
wM2
0
(36)
rnz-p2(1-z4)+q2z4(1-z4)
&2M2
-
57
I
1
+
JS du
rni-p2(1-w)+q2w(1-0)
D,cui@(w-S)exp
oil@
II 3
0
X
u (q
where 6 = s.
‘ht’h)
- 244 + 4011+ 911- 2sol- 245-L. W2M2
w2M2
The functions
=-
Ul(@l),
LI r
Jql(u)tGl(u))du, 0
Ull(@ll)
(37)
in Eqs. (35)-(
*,,t+,,) = -
37) are defined in the following
way:
U r
J
(PII tu)t911tu) )du.
0
Note that the formfactor f+(p2) is investigated in [ 181 and [ 191 (In [ 181, it is necessary to make the simple replacement GT---f K), in light cone sum rules. Our results coincide with theirs if we let $- = 0.
TM. Alieu et d/Physics
At the end of this section final leptons and we obtain
we calculate
the differential
where A= l+r2fs2-2r-2s-2sr,
r=m$/mi,
me and D =
A = 2CGfff+ - C7 For the dileptonic defined by
4mbfT
C = 2ClO.f
decays of the B mesons,
$(8 =-1) &,(P*)
= dI --@(5=
s=p2/mg,
- $(8
+
,
polarization
,
of the
(38)
5 is the longitudinal
are its mass and velocity, respectively.
mB-i-mK’
201
decay rate with the longitudinal
(1 - Y - s)] - t Re(A*C)m&$h
+ 8mim$ [r ICI2 + s jDl2 + Re(C*D)
final lepton, as follows:
Letters B 400 (1997) 194-205
polarization
of the
In Eq. (38) A, C and D are defined
D = C~o(f+ + f-j.
the longitudinal
polarization
asymmetry
PL of the final state e, is
= 1) (40)
-1)
+ $(5=
1) .
where 5 = - 1 (+l ) corresponds to the left (right) handed lepton. If in Eq. (40)) we let me = 0, our results coincide with the results in [21] and if me # 0, they coincide with the ones in [ 111.
3. Numerical analysis The main input parameters in the sum rules (35)-( 37) are the kaon wave functions kaon wave functions we use the results of Refs. [ 16,17,19] : $DK= 6u( 1 - u) {I+
0.52 [5(2zf - 1)2-
401, = 1,
p, N 6u(l - u) , g1 (U) 21 $V(
1 - U)2 )
g2(u) 2: $%%( 1 - U) (2U - 1) , (~3dai)
= 360wza:,
pJol(cui) N lO&xt
- cy2)Ly~)
on the light cone. For
11 + 0.34 [21(2U - 1)4 - 14(2u - 1)2 + 11) ,
202
T.M. Alieo et ul./Physics Letters B 400 (1997) 194-205
here a*( ,ub) N 0.17 GeV2 at ,Ub N Jm2,-z (for more detail see [ 18]), Goldstone bosons PK
-=
Pu,
mb = 2.4 GeV*, which follows from the QCD sum rules analysis
E(P~) N 0.36. In order to estimate ,!.&K,we use the PCAC relation
(~~~)+(Ss))f~
=O(j2
2(qdf~
for the pseudo
tqcuord)
.
Here we use fr rv 133 MeV, fK N 160 MeV and we assume (44) f (Ss) = 1 + 0.8, which follows from QCD sum rules for strange hadrons [ 221. As a result we have
PK N 1 GeV when ,uu, = J
m*
1: 1.6 GeV.
mu+md
For the values of the other parameters, we choose: _fs N 0.14 GeV, which is obtained from 2-point QCD sum rules analysis [ 18,231, mb N 4.7 GeV and so N 35 GeV*, and II&cl N 0.045. Before giving numerical results on the formfactors, we must first determine the region for the Bore1 mass parameter M*, for which the sum rules yields reliable results. The lower limit of this region is determined by the requirement that, the terms proportional to Mm2”(n > 1) remain subdominant. The upper limit of M* is determined by requiring the higher resonance and continuum contributions to be less than N 30% of the total result. Our numerical results show that both requirements are satisfied in the region 8 GeV* < M* 5 16 GeV* and for the numerical analysis we use M 2 = 10 GeV*. When p* approaches the region rng - 0( 1 GeV*) a breakdown of the stability is expected, similar to the B -+ TTcase (see [ 18,191). In Fig. 1, we present the p* dependence of the formfactors f+(p*), f -(p*) and fr(p*). At zero momentum transfer, the QCD prediction for the formfactors are f+(p2
= 0) = 0.29,
f-(p2
= 0) = -0.21,
f&J* = 0) = -0.31.
In Fig. 2a(b) we present the p2 dependence of the differential Branching ratios for the B + K,ufpu- and (B -+ Kr+r-) decay with and without long distance effects. In both cases summation over the final lepton polarization is performed. Note that p2 dependence of the differential branching ratio B -+ KT+~- is analysed in [ 111 and [ 241 using the light front formalism and heavy meson chiral theory, respectively. In [ 111 both short and long distance contributions are considered while in [ 241 only the short distance contributions are taken into account and in both works the obtained spectrum is fully symmetric while in the present work the spectrum we obtain seems to be slightly asymmetric as a result of a highly asymmetric resonance-type behaviour due to the nonperturbative contributions. Performing the integration over p* in EZq. (38) and using the values of the life times r& = (1.56 f 0.06) x lo-l2 s, r~, = (1.62 f 0.06) x lo-‘* s [25], for the branching ratios, including only the short distance contributions, we get
B(Bd -+ Kc,u+,u-)
= (3.1 f 0.9) x 10-7,
B(&
-+ &+T-)
B(B,
= (3.2 =k0.8) x 10-7,
B(B,
+ K+T+T-)
+ Kf,u+p-)
= (1.7 f 0.4) X 10-7, = (1.77 dz 0.40) x 10-7,
where theoretical and experimental errors have been added quadratically. In Fig. 3 we display the lepton longitudinal polarization asymmetry PL as a function Kp+,u- and B --f KT + r - decays, at m, = 176 GeV, with and without the long distance this figure one can see that Pr vanishes at the threshold due to the kinematical factor u for the B -+ Kp+y- decay varies in the region (0 + -0.7) and (0 + -0.1) for the B -+ long distance effects are excluded.
of p* for the contributions. and the value Krirdecay,
B + From of PL, when
T.M. Alieu et al./ Physics Letters B 400 (1997) 194-205
p2 (GeV’)
p* (GeV’)
(a)
@I
_
203
-0.50
%
z
-0.E
-1.25
I
0
4
8
12
16
p’ (GeV’)
Fig. 1. The p2 dependence
of the fomfactors
ft(p*),
f- (p2) and fr(p*).
4. Conclusion In this work, we calculate the transition formfactors for the exclusive B + K-&?- (e = ,u, T) decay in the framework of the light cone QCD sum rules, and investigate the longitudinal polarization asymmetries of the muon and tau in this decay. From a comparison of our results with the traditional QCD sum rule predictions (see [ 91) , we observe that the behaviour of the formfactors are similar and the value of f + in both approaches coincides, while fr differs two times than that of [9] at p2 = 0. It is important to note that for a more refined analysis, it is necessary to take into account the SU(3) breaking effects: the differences between f~, ,LLKand jr, ,!L, and the differences of pion and kaon wave functions. These SU(3) breaking terms can lead to differences between B + n- and B + K formfactors. But we expect that these effects can change the results about 1520% and this lies at the accuracy level of the sum rules method. Few words about the possibility of the experimental observation of this decay are in order. Experimentally, to observe an asymmetry PL of a decay with the branching ratio B at the nu level, the required number of events is N = I$$
(see [ 111) . For example, to observe the r lepton polarization
at the exclusive
channel
B + KT+Y
204
TM. Aliev et d/Physics
p2(GeV’)
Letters B 400 (1997) 194-205
p’ (GeV*)
(a)
p* (GeV’)
Fig. for the the and
2. (a) Invariant mass squared distribution of the lepton pair the decay B + Kp+,u-. (b) The same as in a), but for decay B ----fKT+~-. Here and in all of the following figures solid line corresponds to the short distance contributions only the dashed line to the sum of both short and long distance
Fig. 3. (a) The longitudinal polarization asymmeny PL for the B -+ Kp+,u-decay. (b) The same as in (a), but for the B + Kr+r-decay.
contributions. at the 3a level, one needs at least N cv 5 x lo9 B B decays. Since in the future B-factories, it is expected that N lo9 B-mesons would be created per year, it is possible to measure the longitudinal polarization asymmetry of the T lepton.
References [ 1] MS. Alam et. al., CLEO Collaboration, Phys. Rev. Lett. 74 (1995) 2885; R. Ammar et. al., CLEO Collaboration, Phys. Rev. Lett. 71 (1993) 674. [2] Z. Ligeti and M. Wise, Phys. Rev. D 53 (1996) 4937. ]3] J.L. Hewett, in: Proc. of the 21s’ Annual SLAC Summer Institute, eds. L. De Parcel and C. Dunwoodie, SLAC-PUB 6521. [4] C. Anway-Wiese, CDF Collaboration, in: Proc. of the 8 r/r Meeting of the Division of Particle and Fields of the American Physical Society, Albuquerque, New Mexico (1994), ed. S. Seidel (World Scientific, Singapore, 1995). [5] N.G. Deshpande, J. Trampetic and K. Panose, Phys. Rev. D 39 (1989) 1461; C.S. Lim, T. Morozumi and A.I. Sanda, Phys. Len. B 218 ( 1989) 343. [6] A. Ali, T. Mannel and T. Morozumi, Phys. Lett. B 273 (1991) 505; A. Ali, G.F. Giudice and T. Mannel, Z. Phys. C 67 (1995) 417;
T.M. Aliev et al./ Physics Letten
[7] [8] [9] [IO] [ 111 [ 121 [ 131 [ 141
[ 151
[ 161 [ 171 [ 181 [ 191 [20] I21 ] [22] 1231 1241 [25]
B 400 (1997) 194-205
G. Burdman, Phys. Rev. D 52 (1995) 6400; E Kruger and L.M. Sehgal, Phys. Lett. B 380 (1996) 199. J.L. Hewett, Phys. Rev. D 53 (1996) 4964. R. Casalbuoni, A. Deandra, N. Di Btiolemo, R. Gatto and G. Nardulli, Phys. L.ett. B 312 (1993) 315. P. Colangelo, F. De Fazio, P. Sanntorelli and E. Scrimieri, Phys. Rev. D 53 (1996) 3672. W. Jaus and D. Wyler, Phys. Rev. D 41 (1990) 3405. C.Q. Ceng and C.P. Kao, Phys. Rev. D 54 (1996) 3656. B. Grinstein, M.J. Savage and M.B. Wise, Nucl. Phys. B 319 (1989) 271. M. Misiak, Nucl. Phys. B 398 (1993) 23; B 439 (1995) 461 (E). A.J. Buras and M. Miinz, Phys. Rev. D 52 (1995) 186; M. Ciuchini, E. France, G. MartinelB, L. Reina and L. Silvestrini. Phys. L&t. B 316 (1993) 127; M. Ciuchini, E. France, G. Martinelli and L. Reina, Nucl. Phys. B 415 (1994) 403; G. Cella, G. Curci, R. Ricciardi and A. Vicere, Nucl. Phys. B 421 (1994) 41; Phys. Lett. B 325 (1994) PJ. O’Donnell and H.K.K. Tung, Phys. Rev. D 43 (1991) R2067; A.I. Vainshtein, V.I. Zakharov, L.B. Okun and M.A. Shifman, Sov. J. Nucl. Phys. 24 ( 1976) 427. V.L. Chemyak and I.R. Zhitnitsky, Phys. Rep. C 112 (1984) 173. V.M. Braun and LB. Filyanov, Z. Phys. C 44 (1989) 157; C 48 (1990) 239. V.M. Belyaev, V.M. Braun, A. Khodjamirian and R. Riickl, Phys. Rev. D 51 (1995) 6177. V.M. Belyaev, A. Khodjamirian and R. Riickl, Z. Phys. C 60 (1993) 349. 1.1. Balitsky, V.M. Braun and A.V. Kolesnichenko, Nucl. Phys. B 312 (1989) 509. C. Greub, A. ioannissian and D. Wyler, Phys. Lett. B 346 ( 1995) 149. S. Narison, QCD spectral sum rules (World Scientific, Singapore, 1989). T.M. Aliev, V.L. Eletsky, Sov. J. Nucl. Phys. 38 (1983) 936. Dong-Sheng Du, C. Liu, Da-Xin Zhang, Phys. L.&t. B 317 (1993) 179. Particle Data Group, Phys. Rev. D 50 (1996).
205
227.