- Email: [email protected]
Rare B → ννγ decay in light cone QCD sum rule
6 February 1997
PHYSICS
LETTERS B
Physics Letters B 393 (1997) 143- 148
Rare B -+ vcy decay in light cone QCD sum rule T.M. Aliev ’ , A. ozpineci 2, M. Savcl 3 Physics Department, Middle East Technical University, 06531 Ankara, Turkey
Received 16 October 1996; revised mattuscript received 20 November 1996 Editor: R. Gatto
Abstract Using the light cone QCD Sum Rules Method, we study the rare B + vVy decay and find that the branching ratios are, B( B, -+ vVy) N 7.5 x lo-*, B( Bd ---f vfi’y) N 4.2 x lo-‘. A comparison of our results on branching ratio with constituent quark and pole dominance model predictions are presented.
1. Introduction
The Flavour Changing Neutral Current (FCNC) process is one of the most promising field for testing the Standard Model (SM) predictions at loop level and for establishing new physics beyond that (for a review see [ l] and references therein). At the same time the rare decays provide a direct and reliable tool for extracting information about the fundamental parameters of the Standard Model (SM), such as the Cabibbo-KobayashiMaskawa (CKM) matrix elements &d, KS, &d and Vub[ 21. After the experimental observation of the b --t sy [3] and B -+ X,y [4] processes, the interest is focused on the other possible rare B-meson decays, which are expected to be observed at future B-meson factories and fixed target machines. In addition to being used in the determination of the CKM matrix elements, the rare B-meson decays could play an important role in extracting information about some hadronic parameters, such as the leptonic decay constants f~, and fs,. Pure leptonic decays B, ---)pi,uu- and B, --) e+e- are not useful for this purpose, since these decays are helicity suppressed and as a result they have branching ratios B(B,y ---) ,u+,u-) N 1.8 x 1O-9 and B(B, --) e+e-) 11 4.2 x lo-l4 [5]. For Bd meson case the situation becomes worse due to the smaller CKM angle. Although the process B, j T+T-, whose branching ratio in the SM is B( B, 4 T+T-) = 8 x 10B7 [6], is free of helicity suppression, its observability is expected to be compatible with the observability of the B, ---)p+p- decay only when its efficiency is better than 10W2. When a photon is emitted in addition to the lepton pair, no helicity suppression exists anymore and larger branching ratios are expected. For that reason, the investigation of the B,(d) 4 l+I- y decay becomes interesting. ’ E-mail: [email protected]. 2 E-mail: [email protected]. 3 E-mail: [email protected]. 0370-2693/97/$17.00 PII SO370-2693(96)0
Copyright 0 1997 Elsevier Science B.V. All rights reserved. 1598-S
144
T.M. Alieu et al./Physics
Letters B 393 (1997) 143-148
Note that in the SM, the decay Bs(d) -+ ~5 is forbidden by the helicity conservation. However, similar to the Bs(d) + r+r- decay, the photon radiation process Bs(d) -+ r+y takes place without any helicity suppression. The branching ratios of these processes depend quadratically on the leptonic decay constants of B mesons and hence it could be a possible alternate in determining f~, and fn,,. In [7], these decays are investigated in the SM using the constituent quark and pole model approaches and it is shown that the diagrams with a photon radiation from the light quark give the dominant contribution to the decay amplitude which is inversely proportional to the constituent light quark mass. However the concept of the “constituent quark mass” is itself poorly understood. Therefore, any prediction on the branching ratios, in the framework of the above mentioned approaches, is strongly model dependent. Note that a similar problem exists for the Bs(d) -+ Z+Z-y decays as well [8]. In this work, we investigate the Bs(d) --) z@y processes practically in a model independent way, namely, within the framework of the light cone QCD sum rules method (more about the method and its applications can be found in a recent review [9] ). The paper is organized as follows: In Section 2 we give the relevant effective Hamiltonian for the b -+ qvt decay. In Section 3 we derive the sum rules for the transition formfactors. Section 4 is devoted to the numerical analysis of the formfactors, where we calculate the differential and total decay width for the B + vly and compare our results with those of [7]. Our calculations show that the constituent quark model and sum rules predictions are equal for the constituent quark mass 1)29(md) N 250 MeV.
2. Effective Hamiltonian We start by considering the quark level process b -+ qvt (q = s, d). This process is described by the box and Z-mediated penguin diagrams. The effective Hamiltonian for this process is calculated in [6,10] to yield 3-1 eff=C4yp(l
c=
-
ys)bfiyp(l
-
(1)
y5)v
1’
GF~
2&7~ sin* BW
with x = m:/m2,, a = & is the fine structure constant at mb scale, sin* 8~ N 0.23 is the Weinberg angle and q E s,d. In our calculations we shall neglect the QCD corrections to the coefficient C, since they are negligible (see for example [ 61) . At quark level the process Bs(d) --) vFy is described by the same diagrams as the b --+qvF decay, in which a photon is emitted from any charged particle. Here we should note the following peculiarities of this process: a) The Wilson coefficient C is the same for the processes b + qvFy and b --+qvP, as a consequence of the extension of the Low’s low energy theorem (for more details see [ 111) . b) When a photon is emitted from internal charged particles (W and top quark), the above mentioned process will be suppressed by a factor mi/m* w (see [ 7]), in comparison to the process b -+ qvF, so that one can neglect the contributions of such diagrams. So, the main contribution to the process b -+ qvVy comes from the diagrams with a photon line attached to the initial quark lines. The matrix element for the process Bs(d) + vSy is given by (~l3-IefilB) = -Y,(
1 - ~s)~(~lcry,(
1 - ys)blB).
(2)
The transition amplitude (y I@, ( I- ys ) blB) can be written in terms of the two gauge invariant and independent structures,
T.M. Aliev et al./Physics
Letters B 393 (1997) 143-148
(Y(q)lcrY,( 1 - ys)Wp + 9)) = J4lrcyrFpdCppquy
+ +i
(eL(pq)
145
-
(e*p)q,)
f (P2)
-
m2B
1
1 .
(3)
Here, eP and qp stand for the polarization vector and momentum of the photon, p + q is the momentum of the B meson, g(p*) and f (p*) correspond to parity conserving and parity violating formfactors for the Bscdj --+ vDy decays. The main problem is to calculate the formfactors g(p*) and f(p*), including their dependence on momentum. 3. QCD sum rules for the transition formfactors Derivation of the effective Hamiltonian, Bq. ( l), is the first step in the analysis of the B, --+ v5y decay. The next step is to calculate the transition formfactors using the effective Hamiltonian. But in this step we immediately face with the problem of long distance (non-perturbative) effects. Therefore we need some reliable theoretical scheme which takes into account these non-perturbative effects. Among all of the non-perturbative methods the sum rules method occupies a special place since it is based on the fundamental QCD Lagrangian. In our work we will use the light cone QCD sum rules method. In order to calculate the transition amplitude for the B, + vfi,y decay within the framework of the QCD sum rules method, we must start by writing the representation of a suitable correlator function in the hadron and quark-gluon language. We begin by considering the following correlator:
Q(P9 4) = i
J
d4XeiPX(y(q)lT [4yp(1- y5)b(x)hq]
IO).
(4)
The general Lorentz decomposition of the above correlator is &(P94)
= K{ 7i-a EPvnpe&qpIIt
+ i [ei(pq)
- (e*p)sN]
n2}
,
(5)
with II, and II2 corresponding to the parity conserving and parity violating parts, pa is the momentum of the neutrino pair, e, and qp are the four-vector polarization and momentum of the photon, respectively. The formidable task here, is to calculate II1 and II*. This problem can be solved in the deep Euclidean region, where p* and (p + q) * are negative and large. The correlator function (4) in the framework of the light cone sum rules method is calculated in this deep region in [ 121 (see also [ 13-151). We have recalculated this correlator and our final answer is in confirmation with the results of [ 121. After the Bore1 transformation for the formfactors g(p*> and f (p*> we have (the details of the calculation can be found in [ 151) :
(6)
(7)
T.M. Aliev et al./Physics
146
Leiters B 393 (1997) 143-148
Here @( u)and gl (u) are the leading twist-2, while g(t) and g(*) are the twist-4 photon wave functions, x is the magnetic susceptibility, f = 2 fpm,, with f,, = 200MeV [ 121, ii = 1 - u, e4 and C?bare the electric charges of the light and beauty quarks, fs is the leptonic decay constant, and 6 = (mi - p*)/( SO- p*). The terms without the photon wave functions correspond to the perturbative contributions, when photon is emitted from heavy and light quark lines in the loop diagrams. The asymptotic form of the wave function Q(u) is well known [ 16-191: Q(u) =6uii. The twist-4 wave functions entering Eqs. (6) and (7) are given in [ 131 as = +2
P(u)
+ ii) )
g’qu)
=_1-2
4U
’
4. Numerical analysis The main issue concerning Eqs. (6) and (7), is the determination of g(p*) and f(p2). of the parameters entering Eqs. (6) and (7) : (a) d = -(0.24 mb = 4.7 GeV,
GeV)3 [ 191,
(gq),
g, = 5.5 [ 121,
=
0.8(qq), [20],
]~bv,;j = 0.045,
_f~ =0.14 GeV [21] , I&di(;l
= 0.010 [22].
We first give a list
SO= 35 GeV*,
(8)
The value of x in the presence of external field is determined in [23,24]: x(p2 = 1 GeV*) = -4.4 GeV-* . If we include the anomalous dimension of the current c@,pq, which is -& at fi = mb, we get x(,u2 = mi) = -3.4 GeV2 . Following [ 121, we shall take gl (u) = 1, to the leading twist accuracy. The Bore1 parameter M* has been varied in the region from 8 GeV* < M* < 20 GeV*. We have found that, within the variation limits of M* in this region, the results change by less than 8%. The sum rules for g(p*) and f(p*) is in the region rng -p2 N (few GeV2), which is smaller than the maximum available value p* = m$ To extend our results to the whole region of p*, we will use the extrapolation formulas. The best agreement is achieved with the dipole formulas (for more detail, see [ 121 and [ 251)
g(p2) =
“$,* ’
(1
f(P2)=
Cl
“$*
’
with hl 11 1.0 GeV,
ml N 5.6 GeV,
h2 z 0.8 GeV,
m2 N 6.5 GeV.
Using Eqs. (2) and (3) for the total decay rate, we get aC*m’ I-= LI, 2561r*
(9)
where I
Id-4
Jo
dX(1 -X13X{f2(X)
+g+)}
.
TN. Alieu et d/Physics
147
Letters B 393 (1997) 143-148
I
Table
B(h) B(b)
Sum rules
Constituent
quark model
7.50 x 10-s 0.42 x 1O-8
1.93 x 10-8(fe/o.2)* 2.26 x 10-9(f~/0.2)2
Pole dominance
Pole dominance
1.79 x lo-*(fp/o.2)* 2.10 x lo-‘(fp/O.2)2
0.94 x Io-X(o.2/fs)? 0.52 x 10-9(0.2/fB)*
Here x = 1 - 2 is the normalized photon energy. Let us compare our results with the ones that are obtained within the framework of the constituent quark and pole dominance models [ 71. (Note that Eqs. (6)) (7), ( 15) and ( 16) in [ 71, are all misprinted and all these equations must be multiplied by the factor 3). The corrected results are as follows: dI-
2m5,
C*af;
dx=
-dx(l-X)) rni (4&r>*
dI
C2ag2 f&m&mL(l
dx = 128~~
3C2a f$m”,
r=
-x>~x
(m;* _ xm;)2
(1447r)* rni
(Constituent
(10)
quark model)
C*a f$ mi.g2 )
I-=
768r2m3
(Pole dominance J%
model)
(11)
‘I
where f(y)=.-17y3+42y2-24y-6(4-y)(l-y)*ln(l-y). The coupling
constant
for B,B,*y transition
in the constituent
quark model is given by [ 261,
g=+$
( 12)
This coupling
constant
in the light cone QCD sum rules is calculated
in [ 151 to give
0.1 g=-
fBfB*mB’
(13)
Using the values of the input parameters and the lifetimes r( B,) = 1.34x lo-‘* S, r( Bd) = 1.50x lo-‘* s [ 241, we calculated the branching ratios of the decays, B, + vVy and Bd --+ vi+. The results are presented in Table 1. The results in the table for the third and fourth columns are obtained using the values for the coupling constant g given by Eqs. (12) and (13), respectively. Note that for the constituent masses, we used rnd N 0.35 GeV and m, N 0.51 GeV. We find out that Bqs. (10) and (11) yield results that are numerically close to Eq. (9), with the constituent quark masses mq N fq&. If we set f, N 200 MeV we get my N 250 MeV. If we use this value of the constituent quark mass, the branching ratios for the B, -+ vGy and Bd ---fvVy decays increase by a factor of 4 and 2.5, respectively. Also, for a comparison, we have calculated the photon spectra using the constituent quark, pole dominance models and QCD sum rules method, and found that the photon spectra for the constituent quark and pole dominance models are fully symmetrical. But, as a result of the balance between a typical highly asymmetric resonance-type behaviour given by the non-perturbative contributions and a perturbative photon emission, the sum rules method yields a slightly asymmetrical prediction. In conclusion, we calculate the branching ratios for the processes B, -+ vi;y and Bd --f vVy, in the SM within the framework of the light QCD sum rules method and obtain that B(B, --+ VP-~) N 7.5 x lo-* and B(Bd 4 vFy) 2 4.2 x 10mg. Within this range of branching ratios, it is possible to detect these processes in the future B factories and LHC.
148
T.M. Alieu et al./Physics
Letters B 393 (1997) 143-148
References [ 11 A. Ali Preprint DESY 96-106 ( 1996). to appear in: Proc. XXX Nathiagali Summer College on Physics and Contemporary Needs, Nova Science Publ., NY, Eds., Riazuddin, K.A. Shoaib et al.; A.J. Bums and M.K. Harlanger, Heavy Flavors, Ed%, A.J. Buras and M. Lindner (World Scientific, Singapore) pp. 58-201; A. Ali, Nucl. Phys. B Proc. Supp. 39 BC ( 1995) 408; S. Playfer and S. Stone, Int. J. Mod. Phys. A 10 (1995) 4107. [ 2 ] Z. Ligeti and M. Wise, Preprint CALT-68-2029, hep-ph/95 12225 ( 1995). [3] R. Ammar et al., CLEO Collaboration, Phys. Rev. I&t. 71 ( 1993) 674. [4] MS. Alam et al., CLEG Collaboration, Phys. Rev. Lett. 74 (1995) 2885. [5] B.A. Campbell and P.J. O’Donnell, Phys. Rev. D 52 (1982) 1989; A. Ali, in: B decays, Ed., S. Stone (World Scientific, Singapore) p. 67. [6] G. Buchalla and A.J. Buras, Nucl. Phys. B 400 (1993) 225. [7] C.D. Lil and D.X. Zhang, Phys. Lett. B 381 (1996) 348. [ 81 G. Eliam, C.D. Ltl and D.X. Zhang, Preprint Technion-PH-96-12, hep-ph/9606444 ( 1996). [9] V.M. Braun, Preprint NORDITA-95-69-P( 1995). hep-phi9510404 (1995). to appear in: Proc. of the Int. Europhys. Conf. on High Energy Physics, Brussels, Belgium, 1995. [ 101 T. Inami and C.S. Lim, Prog. Theor. Phys. 65 (1981) 297; 65 (1981) 1772(E). [ 1l] G.L. Lin, J. Liu and Y.P. Yao, Phys. Rev. D 42 (1990) 2314. [ 121 G. Eliam, I. Helperin and R. Mendel, Phys. Lett. B 361 (1995) 137. [ 131 A. Ah and V.M. Braun, Phys. I&t. B 359 (1995) 223. [ 141 A. Khodjamirian, G. St011and D. Wyler, Phys. Lett. B 358 (1995) 129. [ 151 T.M. Aliev, D.A. Demir, E, iltan and N.K. Pak, Phys. Rev. D 54 (1996) 857. [ 161 1.1.Bahtsky, V.M. Braun and A.V. Kolesnicheko, Nucl. Phys. B 312 (1989) 509. [ 171 1.1.Balitsky and V.M. Braun, Nucl. Phys. B 311 (1988) 541. [ 181 V.M. Braun and I. Filyanov, Z. Phys. C 44 ( 1989) 157; C 48 ( 1990) 239. [ 191 M.A. Shifman, A.I. Vainstein and V.I. Zakharov, Nucl. Phys. 147 (1979) 385. [20] V.M. Belyaev and B.L. Ioffe, Sov. JETP 83 ( 1982) 876. [21] T.M. Aliev and V.L. Eletsky, Sov. Nucl. Phys. 38 (1983) 936. [22] Particle Data Group, R.M. Barnett et al., Phys. Rev. D 54 ( 1996) 1. [23] V.M. Belyaev and Y.I. Kogan, Sov. Nucl. Phys. 40 (1984) 659. [24] 1.1.Balitsky, A.V. Kolesnicheko and A.Y. Yung, Sov. Nucl. Phys. 41 (1985) 178. [25] A. Ali, V.M. Braun and H. Simma, Z. Phys. C 63 ( 1994) 437. [26] H.Y. Cheng et al., Phys. Rev. D 5 1 1199.
PHYSICS
LETTERS B
Physics Letters B 393 (1997) 143- 148
Rare B -+ vcy decay in light cone QCD sum rule T.M. Aliev ’ , A. ozpineci 2, M. Savcl 3 Physics Department, Middle East Technical University, 06531 Ankara, Turkey
Received 16 October 1996; revised mattuscript received 20 November 1996 Editor: R. Gatto
Abstract Using the light cone QCD Sum Rules Method, we study the rare B + vVy decay and find that the branching ratios are, B( B, -+ vVy) N 7.5 x lo-*, B( Bd ---f vfi’y) N 4.2 x lo-‘. A comparison of our results on branching ratio with constituent quark and pole dominance model predictions are presented.
1. Introduction
The Flavour Changing Neutral Current (FCNC) process is one of the most promising field for testing the Standard Model (SM) predictions at loop level and for establishing new physics beyond that (for a review see [ l] and references therein). At the same time the rare decays provide a direct and reliable tool for extracting information about the fundamental parameters of the Standard Model (SM), such as the Cabibbo-KobayashiMaskawa (CKM) matrix elements &d, KS, &d and Vub[ 21. After the experimental observation of the b --t sy [3] and B -+ X,y [4] processes, the interest is focused on the other possible rare B-meson decays, which are expected to be observed at future B-meson factories and fixed target machines. In addition to being used in the determination of the CKM matrix elements, the rare B-meson decays could play an important role in extracting information about some hadronic parameters, such as the leptonic decay constants f~, and fs,. Pure leptonic decays B, ---)pi,uu- and B, --) e+e- are not useful for this purpose, since these decays are helicity suppressed and as a result they have branching ratios B(B,y ---) ,u+,u-) N 1.8 x 1O-9 and B(B, --) e+e-) 11 4.2 x lo-l4 [5]. For Bd meson case the situation becomes worse due to the smaller CKM angle. Although the process B, j T+T-, whose branching ratio in the SM is B( B, 4 T+T-) = 8 x 10B7 [6], is free of helicity suppression, its observability is expected to be compatible with the observability of the B, ---)p+p- decay only when its efficiency is better than 10W2. When a photon is emitted in addition to the lepton pair, no helicity suppression exists anymore and larger branching ratios are expected. For that reason, the investigation of the B,(d) 4 l+I- y decay becomes interesting. ’ E-mail: [email protected]. 2 E-mail: [email protected]. 3 E-mail: [email protected]. 0370-2693/97/$17.00 PII SO370-2693(96)0
Copyright 0 1997 Elsevier Science B.V. All rights reserved. 1598-S
144
T.M. Alieu et al./Physics
Letters B 393 (1997) 143-148
Note that in the SM, the decay Bs(d) -+ ~5 is forbidden by the helicity conservation. However, similar to the Bs(d) + r+r- decay, the photon radiation process Bs(d) -+ r+y takes place without any helicity suppression. The branching ratios of these processes depend quadratically on the leptonic decay constants of B mesons and hence it could be a possible alternate in determining f~, and fn,,. In [7], these decays are investigated in the SM using the constituent quark and pole model approaches and it is shown that the diagrams with a photon radiation from the light quark give the dominant contribution to the decay amplitude which is inversely proportional to the constituent light quark mass. However the concept of the “constituent quark mass” is itself poorly understood. Therefore, any prediction on the branching ratios, in the framework of the above mentioned approaches, is strongly model dependent. Note that a similar problem exists for the Bs(d) -+ Z+Z-y decays as well [8]. In this work, we investigate the Bs(d) --) z@y processes practically in a model independent way, namely, within the framework of the light cone QCD sum rules method (more about the method and its applications can be found in a recent review [9] ). The paper is organized as follows: In Section 2 we give the relevant effective Hamiltonian for the b -+ qvt decay. In Section 3 we derive the sum rules for the transition formfactors. Section 4 is devoted to the numerical analysis of the formfactors, where we calculate the differential and total decay width for the B + vly and compare our results with those of [7]. Our calculations show that the constituent quark model and sum rules predictions are equal for the constituent quark mass 1)29(md) N 250 MeV.
2. Effective Hamiltonian We start by considering the quark level process b -+ qvt (q = s, d). This process is described by the box and Z-mediated penguin diagrams. The effective Hamiltonian for this process is calculated in [6,10] to yield 3-1 eff=C4yp(l
c=
-
ys)bfiyp(l
-
(1)
y5)v
1’
GF~
2&7~ sin* BW
with x = m:/m2,, a = & is the fine structure constant at mb scale, sin* 8~ N 0.23 is the Weinberg angle and q E s,d. In our calculations we shall neglect the QCD corrections to the coefficient C, since they are negligible (see for example [ 61) . At quark level the process Bs(d) --) vFy is described by the same diagrams as the b --+qvF decay, in which a photon is emitted from any charged particle. Here we should note the following peculiarities of this process: a) The Wilson coefficient C is the same for the processes b + qvFy and b --+qvP, as a consequence of the extension of the Low’s low energy theorem (for more details see [ 111) . b) When a photon is emitted from internal charged particles (W and top quark), the above mentioned process will be suppressed by a factor mi/m* w (see [ 7]), in comparison to the process b -+ qvF, so that one can neglect the contributions of such diagrams. So, the main contribution to the process b -+ qvVy comes from the diagrams with a photon line attached to the initial quark lines. The matrix element for the process Bs(d) + vSy is given by (~l3-IefilB) = -Y,(
1 - ~s)~(~lcry,(
1 - ys)blB).
(2)
The transition amplitude (y I@, ( I- ys ) blB) can be written in terms of the two gauge invariant and independent structures,
T.M. Aliev et al./Physics
Letters B 393 (1997) 143-148
(Y(q)lcrY,( 1 - ys)Wp + 9)) = J4lrcyrFpdCppquy
+ +i
(eL(pq)
145
-
(e*p)q,)
f (P2)
-
m2B
1
1 .
(3)
Here, eP and qp stand for the polarization vector and momentum of the photon, p + q is the momentum of the B meson, g(p*) and f (p*) correspond to parity conserving and parity violating formfactors for the Bscdj --+ vDy decays. The main problem is to calculate the formfactors g(p*) and f(p*), including their dependence on momentum. 3. QCD sum rules for the transition formfactors Derivation of the effective Hamiltonian, Bq. ( l), is the first step in the analysis of the B, --+ v5y decay. The next step is to calculate the transition formfactors using the effective Hamiltonian. But in this step we immediately face with the problem of long distance (non-perturbative) effects. Therefore we need some reliable theoretical scheme which takes into account these non-perturbative effects. Among all of the non-perturbative methods the sum rules method occupies a special place since it is based on the fundamental QCD Lagrangian. In our work we will use the light cone QCD sum rules method. In order to calculate the transition amplitude for the B, + vfi,y decay within the framework of the QCD sum rules method, we must start by writing the representation of a suitable correlator function in the hadron and quark-gluon language. We begin by considering the following correlator:
Q(P9 4) = i
J
d4XeiPX(y(q)lT [4yp(1- y5)b(x)hq]
IO).
(4)
The general Lorentz decomposition of the above correlator is &(P94)
= K{ 7i-a EPvnpe&qpIIt
+ i [ei(pq)
- (e*p)sN]
n2}
,
(5)
with II, and II2 corresponding to the parity conserving and parity violating parts, pa is the momentum of the neutrino pair, e, and qp are the four-vector polarization and momentum of the photon, respectively. The formidable task here, is to calculate II1 and II*. This problem can be solved in the deep Euclidean region, where p* and (p + q) * are negative and large. The correlator function (4) in the framework of the light cone sum rules method is calculated in this deep region in [ 121 (see also [ 13-151). We have recalculated this correlator and our final answer is in confirmation with the results of [ 121. After the Bore1 transformation for the formfactors g(p*> and f (p*> we have (the details of the calculation can be found in [ 151) :
(6)
(7)
T.M. Aliev et al./Physics
146
Leiters B 393 (1997) 143-148
Here @( u)and gl (u) are the leading twist-2, while g(t) and g(*) are the twist-4 photon wave functions, x is the magnetic susceptibility, f = 2 fpm,, with f,, = 200MeV [ 121, ii = 1 - u, e4 and C?bare the electric charges of the light and beauty quarks, fs is the leptonic decay constant, and 6 = (mi - p*)/( SO- p*). The terms without the photon wave functions correspond to the perturbative contributions, when photon is emitted from heavy and light quark lines in the loop diagrams. The asymptotic form of the wave function Q(u) is well known [ 16-191: Q(u) =6uii. The twist-4 wave functions entering Eqs. (6) and (7) are given in [ 131 as = +2
P(u)
+ ii) )
g’qu)
=_1-2
4U
’
4. Numerical analysis The main issue concerning Eqs. (6) and (7), is the determination of g(p*) and f(p2). of the parameters entering Eqs. (6) and (7) : (a) d = -(0.24 mb = 4.7 GeV,
GeV)3 [ 191,
(gq),
g, = 5.5 [ 121,
=
0.8(qq), [20],
]~bv,;j = 0.045,
_f~ =0.14 GeV [21] , I&di(;l
= 0.010 [22].
We first give a list
SO= 35 GeV*,
(8)
The value of x in the presence of external field is determined in [23,24]: x(p2 = 1 GeV*) = -4.4 GeV-* . If we include the anomalous dimension of the current c@,pq, which is -& at fi = mb, we get x(,u2 = mi) = -3.4 GeV2 . Following [ 121, we shall take gl (u) = 1, to the leading twist accuracy. The Bore1 parameter M* has been varied in the region from 8 GeV* < M* < 20 GeV*. We have found that, within the variation limits of M* in this region, the results change by less than 8%. The sum rules for g(p*) and f(p*) is in the region rng -p2 N (few GeV2), which is smaller than the maximum available value p* = m$ To extend our results to the whole region of p*, we will use the extrapolation formulas. The best agreement is achieved with the dipole formulas (for more detail, see [ 121 and [ 251)
g(p2) =
“$,* ’
(1
f(P2)=
Cl
“$*
’
with hl 11 1.0 GeV,
ml N 5.6 GeV,
h2 z 0.8 GeV,
m2 N 6.5 GeV.
Using Eqs. (2) and (3) for the total decay rate, we get aC*m’ I-= LI, 2561r*
(9)
where I
Id-4
Jo
dX(1 -X13X{f2(X)
+g+)}
.
TN. Alieu et d/Physics
147
Letters B 393 (1997) 143-148
I
Table
B(h) B(b)
Sum rules
Constituent
quark model
7.50 x 10-s 0.42 x 1O-8
1.93 x 10-8(fe/o.2)* 2.26 x 10-9(f~/0.2)2
Pole dominance
Pole dominance
1.79 x lo-*(fp/o.2)* 2.10 x lo-‘(fp/O.2)2
0.94 x Io-X(o.2/fs)? 0.52 x 10-9(0.2/fB)*
Here x = 1 - 2 is the normalized photon energy. Let us compare our results with the ones that are obtained within the framework of the constituent quark and pole dominance models [ 71. (Note that Eqs. (6)) (7), ( 15) and ( 16) in [ 71, are all misprinted and all these equations must be multiplied by the factor 3). The corrected results are as follows: dI-
2m5,
C*af;
dx=
-dx(l-X)) rni (4&r>*
dI
C2ag2 f&m&mL(l
dx = 128~~
3C2a f$m”,
r=
-x>~x
(m;* _ xm;)2
(1447r)* rni
(Constituent
(10)
quark model)
C*a f$ mi.g2 )
I-=
768r2m3
(Pole dominance J%
model)
(11)
‘I
where f(y)=.-17y3+42y2-24y-6(4-y)(l-y)*ln(l-y). The coupling
constant
for B,B,*y transition
in the constituent
quark model is given by [ 261,
g=+$
( 12)
This coupling
constant
in the light cone QCD sum rules is calculated
in [ 151 to give
0.1 g=-
fBfB*mB’
(13)
Using the values of the input parameters and the lifetimes r( B,) = 1.34x lo-‘* S, r( Bd) = 1.50x lo-‘* s [ 241, we calculated the branching ratios of the decays, B, + vVy and Bd --+ vi+. The results are presented in Table 1. The results in the table for the third and fourth columns are obtained using the values for the coupling constant g given by Eqs. (12) and (13), respectively. Note that for the constituent masses, we used rnd N 0.35 GeV and m, N 0.51 GeV. We find out that Bqs. (10) and (11) yield results that are numerically close to Eq. (9), with the constituent quark masses mq N fq&. If we set f, N 200 MeV we get my N 250 MeV. If we use this value of the constituent quark mass, the branching ratios for the B, -+ vGy and Bd ---fvVy decays increase by a factor of 4 and 2.5, respectively. Also, for a comparison, we have calculated the photon spectra using the constituent quark, pole dominance models and QCD sum rules method, and found that the photon spectra for the constituent quark and pole dominance models are fully symmetrical. But, as a result of the balance between a typical highly asymmetric resonance-type behaviour given by the non-perturbative contributions and a perturbative photon emission, the sum rules method yields a slightly asymmetrical prediction. In conclusion, we calculate the branching ratios for the processes B, -+ vi;y and Bd --f vVy, in the SM within the framework of the light QCD sum rules method and obtain that B(B, --+ VP-~) N 7.5 x lo-* and B(Bd 4 vFy) 2 4.2 x 10mg. Within this range of branching ratios, it is possible to detect these processes in the future B factories and LHC.
148
T.M. Alieu et al./Physics
Letters B 393 (1997) 143-148
References [ 11 A. Ali Preprint DESY 96-106 ( 1996). to appear in: Proc. XXX Nathiagali Summer College on Physics and Contemporary Needs, Nova Science Publ., NY, Eds., Riazuddin, K.A. Shoaib et al.; A.J. Bums and M.K. Harlanger, Heavy Flavors, Ed%, A.J. Buras and M. Lindner (World Scientific, Singapore) pp. 58-201; A. Ali, Nucl. Phys. B Proc. Supp. 39 BC ( 1995) 408; S. Playfer and S. Stone, Int. J. Mod. Phys. A 10 (1995) 4107. [ 2 ] Z. Ligeti and M. Wise, Preprint CALT-68-2029, hep-ph/95 12225 ( 1995). [3] R. Ammar et al., CLEO Collaboration, Phys. Rev. I&t. 71 ( 1993) 674. [4] MS. Alam et al., CLEG Collaboration, Phys. Rev. Lett. 74 (1995) 2885. [5] B.A. Campbell and P.J. O’Donnell, Phys. Rev. D 52 (1982) 1989; A. Ali, in: B decays, Ed., S. Stone (World Scientific, Singapore) p. 67. [6] G. Buchalla and A.J. Buras, Nucl. Phys. B 400 (1993) 225. [7] C.D. Lil and D.X. Zhang, Phys. Lett. B 381 (1996) 348. [ 81 G. Eliam, C.D. Ltl and D.X. Zhang, Preprint Technion-PH-96-12, hep-ph/9606444 ( 1996). [9] V.M. Braun, Preprint NORDITA-95-69-P( 1995). hep-phi9510404 (1995). to appear in: Proc. of the Int. Europhys. Conf. on High Energy Physics, Brussels, Belgium, 1995. [ 101 T. Inami and C.S. Lim, Prog. Theor. Phys. 65 (1981) 297; 65 (1981) 1772(E). [ 1l] G.L. Lin, J. Liu and Y.P. Yao, Phys. Rev. D 42 (1990) 2314. [ 121 G. Eliam, I. Helperin and R. Mendel, Phys. Lett. B 361 (1995) 137. [ 131 A. Ah and V.M. Braun, Phys. I&t. B 359 (1995) 223. [ 141 A. Khodjamirian, G. St011and D. Wyler, Phys. Lett. B 358 (1995) 129. [ 151 T.M. Aliev, D.A. Demir, E, iltan and N.K. Pak, Phys. Rev. D 54 (1996) 857. [ 161 1.1.Bahtsky, V.M. Braun and A.V. Kolesnicheko, Nucl. Phys. B 312 (1989) 509. [ 171 1.1.Balitsky and V.M. Braun, Nucl. Phys. B 311 (1988) 541. [ 181 V.M. Braun and I. Filyanov, Z. Phys. C 44 ( 1989) 157; C 48 ( 1990) 239. [ 191 M.A. Shifman, A.I. Vainstein and V.I. Zakharov, Nucl. Phys. 147 (1979) 385. [20] V.M. Belyaev and B.L. Ioffe, Sov. JETP 83 ( 1982) 876. [21] T.M. Aliev and V.L. Eletsky, Sov. Nucl. Phys. 38 (1983) 936. [22] Particle Data Group, R.M. Barnett et al., Phys. Rev. D 54 ( 1996) 1. [23] V.M. Belyaev and Y.I. Kogan, Sov. Nucl. Phys. 40 (1984) 659. [24] 1.1.Balitsky, A.V. Kolesnicheko and A.Y. Yung, Sov. Nucl. Phys. 41 (1985) 178. [25] A. Ali, V.M. Braun and H. Simma, Z. Phys. C 63 ( 1994) 437. [26] H.Y. Cheng et al., Phys. Rev. D 5 1 1199.