fulltext

Radiative decays of excited meson in the light cone sum rules

T. BARAKAT

Civil Engineering Department, Near East University - Lefkos¸a, Mersin, Turkey (ricevuto il 31 Ottobre 1997; approvato il 7 Gennaio 1998)

Summary. — The contribution of the long-distance effects, namely parity-conserved

and parity-violated form factors in the radiative decays of excited B** Krg and D** Krg are calculated in QCD light cone sum rule.

PACS 13.20.He – Decays of bottom mesons.

1. – Introduction

The experimental and theoretical investigation of the heavy-flavored hadron physics is still an important theme in the existing literature. The observations of the inclusive decay B KXsg as well as the exclusive decay B KK* g [1] have placed the

study of rare B decays on a new footing. These flavor-changing neutral-current (FCNC) transitions represents an important class for testing the Standard Model (SM) at loop level [2] and a powerful tool for establishing “new physics” beyond it [3]. The heavy-flavor decays are also useful in the determination of Cabibbo-Kobayashi-Maskawa (CKM) matrix elements, and play an exceptional role in understanding the dynamics of QCD.

As is well known, the inclusive heavy-flavor decays like b Ksg can be easily evaluated within perturbative methods, but are more difficult to detect experimentally. Exclusive heavy-flavor decays, on the other hand, are more easy to detect, but require difficult non-perturbative calculations of matrix elements in order to yield useful results. This means that the dynamics of heavy-flavored hadron decays (exclusive decays) included the long-distance effects and their calculations are only possible in a non-perturbative framework.

In the radiative decays of the heavy-flavored meson decays, there are two different types of long-distance effects. The first is the penguin loops, which is responsible for b Ks( d ) transition and runs out at low momentum. Many papers in the literature are devoted to the analysis of such long-distance effect [4]. The main result of these works is that the long-distance contributions are very small in agreements with the general assumption that the short-distance penguin mechanism dominates in the exclusive decays. The second one is the weak annihilation mechanism (WA), which does not involve loops of heavy quarks, but takes place via weak annihilation (WA) of the initial

quarks. The way for estimating the long-distance effect via WA is to use a constituent quark model in which the photon and gluon are emitted from the light quark in the initial state [5]. In other words, the radiative B KMg decays is modelled by the sum of all possible hadronic intermediate states [6]. Many non-perturbative approaches have been suggested for computing the matrix elements appearing in these long-distance effects: among them are lattice theory, heavy-quark effective theory, chiral perturbative theory, quark models, 1 ONc expansion and QCD sum rule method. The resulting theoretical predictions are very strongly model dependent. However, the QCD sum rule method occupies a special place, since it is based on the first principles of QCD, and the non-perturbative parameters describing the long-distance effects have universal perspective.

Recently, a new alternative and practically model-independent method has been suggested for the calculation of the long-distance effects in the radiative heavy-meson decays [7-9]. This method is based on the light cone version of the QCD sum rules. It provides a reliable estimation of the long-distance effects, and is a quite successful for the investigation of the exclusive decays.

In the framework of the light cone QCD sum rules, the strong couplings gvrp,

gpNN[10], pAg * transition form factor [11], S Kpg decay [12], gB* Bp, gD* Dp coupling constants [13], B- and D-meson semileptonic decay form factors [14], radiative B- and D-meson decays [7, 8, 15] and the radiative B K

l

ng decay [8, 9] are successfully investigated.

In this work we estimate the long-distance effects in the excited B- and D-meson ( 01_{) decays, in the framework of the light cone QCD sum rules.}

The paper is organized as follows. In sect. 2 we derive the light cone QCD sum rules. In sect. 3 we carry out the numerical analysis and determine the transition form factors.

2. – Light cone sum rule

The relevant effective Hamiltonian for B**1

K r1g consists of two operators [16]: HW4

G

k2 VubV *ud]c1dgm( 1 2g5) uugm( 1 2g5) b 1 (1)

1c2ugm( 1 2g5) udgm( 1 2g5) b( ,

where c1and c2are the Wilson coefficients. Using the Fierz rearrangement formula the effective Hamiltonian, which describes B**1

K r1g decay, can be rewritten in the

following form:

HW4

G

k2 VubV *uda1dgm( 1 2g5) uugm( 1 2g5) b , (2)

where a1_{4 c}11 c2O3 . The value of the coefficient a1 is extracted from two-body non-leptonic decay, using the factorization hypothesis [17]. The fits of the experimental results indicate that the value of a1[18] is close to its perturbative one. Therefore we can conclude that for B**1

Using the factorization hypothesis the matrix element for B**1 K r1g decay can be written as (3) ar1 NHWNBbg4 G k2 VubV *uda1. ]arNugm( 1 2g5) N0ba0Nugm( 1 2g5) bNBbg1 1arNdgm( 1 2g5) uN0bga 0 Nugm( 1 2g5) bNBb( ,

and using

a 0 Nugm( 1 2g5) bNB(p)b 4fBpmB,

(4)

one can easily see that the second term is proportional to the mass of the light quark. We will neglect the mass of the lights in our further discussions.

Hence the second term in eq. (3) can be neglected. This means that the main contribution to the B**1

K r1g decay gives the diagrams, where photon is emitted

from the initial b and u quark lines. Using the definition ar1

Ndgm( 1 2g5) uN0b 4

frmremthe matrix element in eq. (3) can be written in the following form:

ar1 NHWNBbg4 G k2 VubV *uda1frmre r ma 0 Nugm( 1 2g5) bNB(p1q)bg, (5) where er

m is the r-meson vector polarization.

The matrix element describing the annihilation of B** to the current, ugm( 1 2

g5) b , which in turn emits the real photon, can be written in terms of the following two gauge-invariant form-factors T1and T2:

(6) _{ag(q) Nug}m( 1 2g5) bNB **(p1q)b 4

4 iemnrse *n prqsT1(p2) 1

(

e *m (pq) 2 (e * p) qm

)

T2(p2) , where e * , q are the vector polarization and four momenta of the photon, respectively, and p is the r-meson four-momentum.

Our aim now is to calculate the transition form factors T1 and T2as a function of momentum transfer. In order to obtain this dependence we use the light cone version of QCD sum rules. According to the QCD sum rule method, in order to calculate these form factors we start by considering the representation of a suitable correctors function in terms of hadronic and quark-gluon states. For this aim we consider the following correlation function:

Tm(p , q) 4i



d4x eipxagNT]u(x) gm( 1 2g5) b(x) b( 0 ) u( 0 )(N0b . (7)

The hadronic (physical) part of this correlator is obtained by inserting a complete set of states including the B**-meson ground state and higher states with ( 01_{) quantum} numbers.

Saturating eq. (5) by B**-meson state, we get Tm4 m2 B**fB** mb

(

mB**2 2 (p 1 q)2

)

]iemnrse *n prqsT11 (e *m(pq) 2 (e* p) qm) T2( . (8)

In deriving this equation we have used the definition

a 0 NugmbNB **(p1q)b 4fB**(p 1q)m.

(9)

Let us next calculate the theoretical part of the correlator function. The main idea of the method is to apply the operator product expansion (OPE) to the T-product of the currents in eq. (5) at spacelike momentum (p 1q)2

E 0 such that the b-quark is highly virtual and perturbative theory can be applied. Then the light quark propagator can be described by means of the local OPE, where the leading term is the free propagation and in the next order includes the interaction of quark and quark-gluon vacuum condensates. Therefore contracting the heavy-quark lines in eq. (3), we get

Tm(p , q) 4



d4_{x d}4_k ( 2 p)4(mb22 k2)

ei(p 2k)x_{ag(q) Nug}

m( 1 2g5)(k

4

1 mb) uN0b . (10)

To take into account the long-distance effects, we need to introduce additional non-perturbative parameters, describing the light-quark propagation in the presence of slowly varying external electromagnetic field. From eq. (8) it follows that the problem is reduced to the calculation of the matrix elements of the gauge-invariant non-local operators with a light cone separation x2

4 0 , sandwiched in between the vacuum and the photon state. These matrix elements define the photon light cone wave functions. Following [7-9] we define the two-particle photon wave functions in the following way:

(11) _{agNu(x) s}mnu( 0 ) N0b 4ieuauub



0 1

eiqux

du ](emqn2 enqm)

(

xf(u) 1x2g( 1 )(u)

)

1

1g( 2 )(u)[qx(emxn2 enxm) 1ex(xmqn2 xnqm) 2x2(emqn2 enqm) ]( ;

(12) _{ag(q) Nu(x) g}mg5uN0b 4 1 4 emnrsenqrxsf



0 1 du eiqux_{g»(u) .}

Here f(u), g»(u) are the leading twist-2 photon wave functions, and g( 1 ) _{and g}( 2 ) _are the two-particle twist-4 wave functions, x is the magnetic susceptibility of the quark condensate in the presence of external electromagnetic field. These parameters and x are chosen in a such way that the functions entering eq. (9) and eq. (10) are normalized to unity:



0 1 f (u) du 41 , (13)

the functions T1and T2, we get the following results: (14) T14 euauub



0 1 du A

k

xf(u) 2 4 A

g

1 1 2 m2 b A

h

(g12 g2)

l

2 mb 2 f



0 1 du A2 g»(u) , (15) T24 euauub



0 1 du A

k

xf(u) 2 4 g1 A

g

1 1 2 m2 b A

hl

, where A 4m2 b2 (p 1 qu)2.

The important contributions come from the perturbative loop diagrams, and the quark condensate diagrams is when the photon is emitted from the heavy quark. Omitting the details of the calculations (details of calculations can be found in [15]) for the above-mentioned contributions we get

(16) T1pert4 3 mb 4 p2



0 1 1 m2 b2 (p 1 qu)2

y

(eu1 eb) (mb22 p 2 ) u m2 b2 up2 2 ebln m 2 b2 up2 um2 b

z

2 2ebauub 1 (mb22 p2)

(

mb22 (p 1 q)2

)

, (17) T2pert4 3 m3 b 4 p2



0 1 1 mb22 (p 1 qu)2

{

y

(eu_{1 e}b)

g

2 u 211 p2 mb2 2 p 2_u2 mb22 up2

h

2 2(eu2 eb) u p 2 m2 b

z

(m2 b2 p2) u m2 b2 up2 2 eb( 2 u 21) ln m 2 b2 up2 um2 b

}

2 2ebauub 1 (m2 b2 p2)

(

mb22 (p 1 q)2

)

.

Sum of eqs. (12) and (14), (13) and (15) gives final answer for the theoretical part of the sum rules. Equating the theoretical and physical part of sum rules for Ti, and

performing the Borel transformation on the variable (p 1q)2 and subtracts the continuum and the higher state contributions, invoking quark-hadron duality (details of this procedure can be found in [7, 13, 14]), after this procedure we finally get the following sum rules for the form factors T1and T2:

(18) T₁₄ mb m2 B**fB**

{



D 1 du u e (mB**2 OM22 (mb22 up2) OuM2)_Q

Q

y

euauub

g

xf(u) 24

(

g1(u) 2g2(u)

)

g

mb21 uM2 u2_M4

hh

2 mbfg»(u) 2 uM2 1 13 mb 4 p2

g

(eu1 eb) (m2 b2 p2) u m2 b2 up2 2 ebln m 2 b2 up2 um2 b

h

z

2 ebauub m2 b2 p2 e(mB**2 2 m 2 b) OM2

}

_,

(19) T24 mb mB**2 fB**

{



D 1 du u e (mB**2 OM22 (m 2 b2 up2) OuM2)_Q

Q

y

euauub

g

xf(u) 24g1(u)

m2 b1 uM2 u2M4

h

1 3 m3 b 4 p2_(m2 b2 p2) Q Q

u

g

(eu1 eb)

g

2 u 211 p2 m2 b 2 p 2_u2 m2 b2 up2

h

2 (eu2 eb) u p 2 m2 b

h

(m2 b2 p2) u m2 b2 up2 2 2eb

g

2 u 211 p2 mb2

h

ln m 2 b2 up2 umb2

v

z

2 ebauub mb22 p2 e(mB**2 2 m 2 b) OM2

}

_. Here the D 4 (m2

b2 p2) O(s02 p2), s0is the threshold, and u 412u.

At the end of this section we would like to make the following remark. As we noted, the function g1(u) and g2(u) represent twist t 44 wave functions. Up to this accuracy, in eqs. (16) and (17) we must take into account other twist t 44 photon wave functions (see [19]). Using the equation of motion, one can relate them to the three-particle wave functions of t 44 with an additional gluon from the heavy quark [19]. In [19] it was shown that the contributions of these three-particle wave functions, in general are small and therefore we will neglected them. We also neglect the d 45 quark condensate, since according the hierarchy problem of the QCD sum rules method it is smaller than d 43 operators.

3. – Numerical analysis

In evaluating the form factors T1and T2we need to input parameters. Following [9] we choose for g»(u) and for f(u) the asymptotic value of the photon wave function that is

f(u) 4g»(u) 46uu . (20)

In [7, 9], for the twist-4 wave function, the following results was obtained:

g_{1(u) 42}1 8 u( 3 2u) , (21) g2(u) 42 1 4 u 2 . (22)

For the other parameters entering the sum rules, we have used: mb4 4.7 GeV, mB**4 5732 65620 MeV [19], s0C 40 GeV2, fB**4 (0.18 60.03) GeV [20]. The parameter f was obtained in [9], f C (euOgr) frmr, frC 200 MeV . The magnetic susceptibility x was

determined in [21], by the QCD sum rules method, and at m2

4 1 GeV2is given as

x 4./ ´

24.4 GeV22, _{r , r 8, r9 are included ,} 23.3 GeV22, only r is included .

Fig. 1. – a) The dependence of the form factors T1 in the B**1K r1g decay on the Borel

parameter M2_{; b) the dependence of the form factors T}

2in the B**1K r1g decay on the Borel

parameter M2_.

In the numerical calculations we take into account the anomalous dimension of the operator, which is equal to (24O27) and the anomalous dimension of the quark condensate ( 4 O9). With these parameters, the dependence of the form factors T1and T2 on Borel parameter M2_{at p}2

4 mr2is presented in fig. 1.

From the figure we can see that in the region 6 GeV2

E M2E 20 GeV2of the Borel parameter M2, the predictions for T1(p2_{4 m}r2) and T2(p24 mr2) are very stable and

the predictions are practically independent of the choice of the parameters mb, s0 and

fB**(with a change of about 4%). In this region of M2, the contributions of the twist-4 operators give small contribution of about 2%. We demand that the higher state contributions is less than 30% of the total result. Under this condition the form factor

T1and T2which describes the weak annihilation decay for B**1K r1g can be obtained and we get

T1(p2_{4 m}r2) 44Q1022GeV21,

(23)

T2(p24 mr2) 43Q1022GeV21.

(24)

The corresponding prediction for the amplitude of the neutral B**0_{K r}0g decay

can easily be obtained by replacing uKd in the sum rules. The prediction for the above-mentioned form factors in the same Borel parameter region are:

T1( B**0K r0g) 422Q1022GeV21, (25)

T2( B**0K r0g) 421.9Q1022GeV21. (26)

From eqs. (21), (22), (23) and (24) we see that as in B Krg decay (see also [8]), there is also strong isospin violation, in contrast to the short-distance penguin effects, which are independent of the flavor of the spectator quark.

From eq.(16) we can immediately determine the corresponding D-meson decay form factors. For this aim it is sufficient to make in the sum rules the following replacements: mbK mc, fB**K fD**, (s0)BK (s0)Dand ebK ec.

In the Borel parameter range, 2 GeV2

E M2E 8 GeV2, which satisfy all these requirements, we obtain T1( D**1K r1g) 429Q1022 GeV21, (27) T2( D**1K r1g) 422Q1022 GeV21, (28) T1( D**0K r0g) 41.8Q1021GeV21, (29) T2( D**0K r0g) 41.3Q1021GeV21. (30)

Finally, it is worthy at this stage to calculate the branching ratio for the corresponding channels. Using eqs. (3), (5) and (6) for branching ratio we obtain (31) BR( B**1 K r1g) 4 4 (G 2 O64 p) NVubVudN2a12fr2

(

(m 2 B**2 mr2) OmB**

)

3

[

NT1N21NT2N2

]

Gtot( B** ) . The branching ratio for the D**1

K r1g can be obtained easily from eq. (31) by the

replacements of mB**K mD**and Gtot( B** ) KGtot( D** ). As an example, we estimate the branching ratios for B**1

K r1g and B**0 K r0g

decays. In estimating these branching ratios we use the following input parameters:

frC 0.2 GeV , a14 1 , mr4 0.77 GeV , NVudN4 1 , NVubN4 0.0035 . For the total decay width we assume that G_{tot( B** ) CG}tot( B ) and we take ttot( B ) C 1.56 Q 10212_{s [22]. On substituting these values into eq. (31) the branching ratios are} estimated and we get

Br ( B**1 K r1g) 43.7Q1027_, (32) Br ( B**0 K r0g) 41.19Q1027. (33) * * *

The author thanks T. M. ALIEV for many useful discussions and encouragement

during this work.

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