fulltext

On heavy-to-heavy meson transitions in the effective quark model

with chiral symmetry

A. N. IVANOVand N. I. TROITSKAYA(*)

Institut für Kernphysik, Technische Universität Wien Wiedner Hauptstrasse 8-10, A-1040 Vienna, Austria

(ricevuto il 12 Settembre 1996; revisionato il 4 Novembre 1996; approvato il 14 Novembre 1996)

Summary. — Heavy-quark effective theory (HQET) and chiral perturbation theory at the quark level (CHPT)q, based on the extended Nambu-Jona-Lasinio model with linear realization of chiral U( 3 ) 3U(3) symmetry, are applied to the calculation of form factors of heavy-to-heavy meson transitions. The application of such an effective quark model has led to the appearance of the additional factor 1O2 distinguishing the normalization of form factors given within QCD under the admission concerning the unrenormalizability of HQET by strong low-energy interactions. One can explain this factor as a strong influence of heavy-antiquark degrees of freedom, which die out in the infinite mass limit in constituent quark loops describing in such an effective quark model strong low-energy interactions at energies restricted by the spontaneous breaking of chiral symmetry scale LxC 1 GeV. This should evidence the renormalization of HQET within such an effective quark model by strong interactions at scales p ELxE MQ, where MQ is the heavy Q-quark mass. Next-to-leading corrections in chiral and heavy-quark mass expansion are computed. The violation of Luke8s theorem within such an effective quark model is observed. The obtained results are confirmed within the effective quark model suggested by Bardeen and Hill.

PACS 12.90 – Miscellaneous theoretical ideas and models. PACS 12.39.Ba – Bag model.

PCAS 13.40.Ks – Electromagnetic corrections to strong- and weak-interaction processes.

1. – Introduction

Heavy-quark effective theory (HQET) [1-3], when the heavy Q-quark mass MQ tends to infinity (MQK Q), supplemented by chiral perturbation theory at the quark

(*) E-mail: ivanovHkph.tuwien.ac.at

Permanent address: State Technical University, Department of Theoretical Physics, 195251 St. Petersburg, Russian Federation.

level (CHPT)q[4], based on the extended Nambu-Jona-Lasinio model with a linear realization of chiral U( 3 ) 3U(3) symmetry [5-8], has been successfully applied to the description of properties of charmed mesons [9-14]. The extension of HQET by (CHPT)qhas been of use for the computation: i) the form factors of semileptonic decays of charmed pseudoscalar mesons both in the chiral limit [8, 10] and up to the first-order corrections in current s-quark mass expansion [11, 12], ii) the fine structure of mass spectra of charmed mesons up to the first-order corrections in current quark mass expansion [13], and iii) partial widths and probabilities of the strong and electromagnetic decays of D*-mesons [14]. All of theoretical predictions have been good compared with experimental data.

In this paper we apply HQET and (CHPT)qto the computation of form factors of the B

–

K D( D* ) transitions in semileptonic B–K D( D* )

l

n_l decays and the form factors of

the D KD( D* ) transitions. We show that the form factors of

heavy-to-heavy meson transitions can be expressed in terms of the universal structure function j(w) dependent on w 4vQv 8, where v (v 8) is the 4-velocity of the initial (final) heavy meson, and normalized to unity at zero recoil, i.e., j( 1 ) 41 at w41.

The form factors obtained within such an effective quark model coincide with the prediction given by the Isgur-Wise theorem [15] up to the common factor 1O2. Tracing the source of this factor one can conclude that it is connected with the limit MQK Q eliminating virtual heavy-antiquark degrees of freedom from heavy-light constituent quark loops describing contributions of strong low-energy interactions to the form factors of heavy-to-heavy meson transitions in (CHPT)q. The virtual momenta of constituent quarks in the loops are restricted by LxA 1 GeV, a scale of spontaneous breaking of chiral symmetry (SBxS). The appearance of the factor 1O2 can be considered as an evidence of the renormalization of HQET by strong interactions at scales p ELxE MQ within such an effective quark model. This is in contradiction with the assumption by Isgur and Wise, that HQET is not renormalized by «the physics at scales p EMQ» [15].

The effective quark model of light and heavy-light mesons, based on the Nambu–Jona-Lasinio model and describing strong low-energy interactions in terms of heavy-light constituent quark loops, has been suggested by Bardeen and Hill [16]. We find only an insignificant distinction between our approach and the Bardeen-Hill model. Indeed, in the Bardeen-Hill model heavy mesons are considered like partners of light mesons, whereas in HQET, supplemented by (CHPT)q, heavy mesons are external states with respect to the light ones. This distinction influences only the redefinition of parameters that are input into the models. Of course, all results obtained within HQET and (CHPT)q should be fully valid in the Bardeen-Hill model. Therefore, the factor 1O2 is also available in the form factors of heavy-to-heavy meson transitions computed in the Bardeen-Hill model (see sect. 9).

Due to the factor 1O2, the form factors of heavy-to-heavy meson transitions become normalized to 1O2 at wK1 instead of being normalized to unity. At first glance this provides a substantail disagreement with the normalization of form factors, computed at finite values of heavy-quark masses, at q2

K 0, where q24 (pB–2 pX)24 M 2 B –

1 MX22 2 MB–MXw is the squared momentum transfer of the B

–

K X (X 4 D or D*) transition. However, on second thought q2

4 Mb21 Mc22 2 MbMcw taken in HQET ignoring «differences between the heavy-quark mass and the mass of the hadron of which it is part» [16] (see p. 417), i.e., MB–C Mb, MD*C MDC Mc, is a poor established quantity at

q2

K Q at any value of w. Then, setting Mc4 Mb4 MQK Q, that is required by heavy-quark symmetry [15], one reduces q2_{to the form q}2

4 2 2 MQ2(w 21) [16] (see p. 417), that shows q2_{is infinitely large at w c 1 and q}2

4 2Q Q 0 4 ? at w 4 1. This entails the conclusion that at MQK Q the limit q2K 0 does not exist from the mathematical point of view, whereas the limit w K1 is well established. Therefore, at MQK Q one cannot say anything definite about the normalization of the form factors at q2

K 0, if the normalization of the structure functions at w K1 is fixed. In our case it is fixed at 1O2. These limits are not connected and non-interchangeable in the infinite heavy-quark mass limit MQK Q (see sect. 5).

An important role of the factor 1O2 can be illustrated by example of the radiative decays of charged D*-mesons, i.e. D*1

K D1g and Ds*1K D1s g. We show that, due to the additional factor 1O2 in the definition of the form factor of the D1

K D1transition, the amplitudes of the D*1

K D1g and Ds*1K D1s g decay, calculated in next-to-leading order in large Mc expansion and in the chiral limit, turn out to be suppressed. This agrees reasonably well with the experimental data.

For completeness we give the computation of the form factors of heavy-to-heavy meson transitions up to next-to-leading order corrections in both chiral and large heavy-quark mass expansion. These corrections are fully defined and do not contain arbitrary parameters. The main problem we encounter here is the violation of Luke’s theorem [17] demanding vanishing of next-to-leading corrections in heavy-quark mass expansion at zero recoil w 41. We explain the violation of Luke’s theorem as a result of a non-trivial dynamics of virtual heavy antiquark degrees of freedom at energies restricted by the SBxS scale LxA 1 GeV. This means that the violation of Luke’s theorem has the same nature as the factor 1O2 and occurs due to the renormalization of HQET by strong interactions at scales p ELxE MQ.

The paper is organized as follows. In sect. 2 we compute form factors of the B–_K D( D* ) and D KD( D*) transitions in leading order in both chiral and heavy-quark mass expansion. We show that these form factors can be expressed in terms of the universal structure function j(w) depending on w, i.e. the Isgur-Wise structure function [15]. We obtain the additional factor 1O2. Note that in HQET and (CHPT)q strong low-energy interactions are described by constituent quark loops. In the infinite heavy-quark mass limit MQK Q heavy-antiquark degrees of freedom are removed from the loops. This leads to the appearance of the factor 1O2 that evidences the renormalization of HQET by strong interactions at scales p ELxE MQ. In sect. 3 we trace the agreement of heavy-quark symmetry with our approach. We show that momentum integrals defining form factors of heavy-to-heavy meson transitions can be derived in terms of an effective Lagrangian distinctly invariant under heavy-quark symmetry [15]. In sect. 4 we discuss in detail the problem of the contribution of virtual heavy-antiquark degrees of freedom at energies restricted by the SBxS scale Lx. To this aim we give the elaborate computation of the contribution of virtual light-antiquark degrees of freedom to the matrix element of the p1

K p1 transition, governed by the vector current operator u–( 0 ) gmu( 0 ). In sect. 5 we illustrate the

non-changeability of the limits q2

K 0 and w K 1 at MQK Q by example of

electromagnetic form factor of the D1_{-meson within Vector Dominance Model. In} sect. 6 we calculate the amplitudes of electromagnetic and strong decays of D*-mesons in next-to-leading order in large Mcexpansion and in the chiral limit. We show that in this approximation the D*1

K D1g decay is suppressed in accordance to experimental data. We predict an analogous suppression for the D*1

K D1g decay. In sect. 7 we compute first-order corrections in current quark mass expansion to the form factors of

heavy-to-heavy meson transitions. These corrections are fully defined and do not contain any arbitrary parameters. In sect. 8 the form factors of heavy-to-heavy meson transitions are calculated to next-to-leading order in large heavy-quark mass expansion. The violation of Luke’s theorem is discussed. In sect. 9 we compute the form factors of the heavy-to-heavy meson transitions in the Bardeen-Hill model. We evidence the appearance of the factor 1O2 and the renormalization of HQET by strong interactions at scales p ELxE MQ. In sect. 10 we show that the contribution of the Isgur-Wise function to the amplitude of the D* KDg decays should be of order

O( 1 OMc) but not O( 1 OMc2) as has been assumed by Amundsen et al. [18]. In sect. 11 we trace out the derivation of the Isgur-Wise theorem within our and Bardeen-Hill effective quark models. In the conclusion we discuss the obtained results and the estimate of NVcbN, the CKM-matrix element, corrected in accordance with the factor 1O2.

2. – Form factors to leading order in both chiral and large heavy-quark mass expansion

The matrix elements of the B–_{K D( D* ) transitions describing the contributions of} strong interactions to the semi-leptonic B–_{K D( D* )}

_l

n_l decays are given by

.

`

`

/

`

`

´

M_m(B–_{K D ) 4 ( 2 E}Kp8V 2 EKpV )1 O2aD(p 8)Nc – ( 0 ) gmb( 0 ) NB – (p)b 4 4 f1(q 2 )(p 1p8)m1 f2(q 2 )(p 2p8)m, M_m( V )(B–_{K D* ) 4 ( 2 E}Kp8V 2 EKpV )1 O2aD * (p8)Nc – ( 0 ) gmb( 0 ) NB – (p)b 4 4 b(q2) emnabe *n(p8)(p1p8)a(p 2p8)b (e01234 1 ) , Mm( A )(B – K D* ) 4 ( 2 EKp8V 2 EKpV )1 O2aD * (p 8)Nc–( 0 ) gmg5b( 0 ) NB – (p)b 4 4 ia1(q2) e *m (p8)2ia2(q2)

(

e * (p8)Qp

)

(p 1p8)m2 ia3(q2)

(

e * (p8)Qp

)

(p 2p8)m, (2.1) where f₆(q2_{), b(q}2_{) and a}

i(q2)(i 41, 2, 3) are the form factors depending on the

squared mass of the leptonic pair q2_{4 (p 2 p 8)}2. The 4-momenta of B–, D and D*-mesons

pm _{and p 8}m _{can be expressed in terms of masses and 4-velocities of these mesons, i.e.}

pm

4 MB–vm, p 8m4 MDv 8m and p 8m4 MD*v 8m, respectively. Then e *m(p8) is the 4-polarization vector of the D*-meson. EKp and EKp8 are the energies of B

– and D( D* )-mesons, and V is a normalization volume.

In the infinite heavy-quark mass limit when the masses of the c- and b-quarks tend to infinity, i.e. Mc, MbK Q, the matrix elements (2.1) can be reduced to the form [15, 17, 19-25]

.

`

/

`

´

(MB–MD)21 O 2Mm(B – K D ) 4 j1(w)(v 1v8)m1 j2(w)(v 2v8)m, (MB–MD*)21 O 2Mm(B – K D* ) 4 2 z(w) emnabe *n(v8) vav 8b, (MB–MD*)21 O 2Mm(B – K D* ) 4 4 ij1(w)( 1 1w) e *m (v8)2ij2(w)

(

e * (v8)Qv

)

vm2 ij3(w)

(

e * (v8)Qv

)

v 8m, (2.2)

where ji(w) (i 46, 1, 2, 3) and z(w) are the structure functions depending on the

referring to heavy-quark symmetry and the assumption that HQET is not renormalized by strong interactions at scales p EMQ, these structure functions can be expressed in terms of the universal structure function j(w) such as [15]

. / ´ j1(w) 4z(w) 4j1(w) 4j3(w) 4j(w) , j2(w) 4j2(w) 40 . (2.3)

Due to heavy-quark symmetry the structure function j(w) should be normalized to unity at zero recoil w 41, i.e. j(1) 41.

Below we argue the change of relation (2.6) and predict

.

/

´

j1(w) 4z(w) 4j1(w) 4j3(w) 4 1 2j(w) , j2(w) 4j2(w) 40 . (2.4)

The appearance of the factor 1O2 is connected with the limit MQK Q eliminating virtual heavy-antiquark degrees of freedom from heavy-light constituent quark loops describing contributions of strong low-energy interactions to the form factors of heavy-to-heavy meson transitions in (CHPT)q. This might mean that in such an effective quark model HQET becomes renormalized by strong interactions at scales

p ELxE MQ. The factor 1O2 also appears in the Bardeen-Hill model [16] (see sect. 9), where strong interactions at scales p ELxE MQare described in terms of constituent quark loops.

Let us proceed to the computation of the structure functions ji(w)(i 46, 1, 2, 3)

and z(w) we carry out in leading order in large heavy-quark mass and large-N expansions, where N is the number of colours, keeping to the chiral limit.

It is convenient to illustrate the procedure of the computation by example of the matrix element _aD(p8)Nc–( 0 ) gmb( 0 ) NB

–

(p)b. Following [9-14] and applying the reduction technique we reduce the matrix element to the form

(2.5) _aD(p8)Nc–( 0 ) gmb( 0 ) NB – (p)b 4 4 lim p2_{K M}2 B – p 82 K MD2

i2



_d4_{x d}4_yexp [2ipQx1ip 8Qy] ( 2 EKp8V 2 EKpV )1 O2 (px1 M 2 B –_)(p y1 MD2) 3 3a 0 NT

(

WD(y) c – ( 0 ) gmb( 0 ) W † B –_(x)

₎

N0 bconn ., where W†B–(x) and WD(y) are the interpolating fields of B

–

and D-mesons, respectively.

In order to analyse the r.h.s. of (2.5) at the quark level we assume that the operators

W†B–(x) and WD(x) satisfy the equations of motion [9-14]

.

/

´

(px1 M 2 B –_{) W}† B –_{(x) 4g} B –_b–_{(x) ig}5_{q(x) ,} (px1 MD2) WD(x) 4gDq–(x) ig5c(x) , (2.6)

where q(x) is u(x), d(x) and s(x) for the corresponding heavy mesons. The coupling constants gD and gB– are phenomenological and should be fixed from the additional

conditions, for example, [13, 14]

.

/

´

M2 D4 McaD(p) Nc – ( 0 ) c( 0 ) ND(p)b(2EKpV ) , M_B–2 4 MbaB – (p) Nb–( 0 ) b( 0 ) NB–(p)b( 2 EKpV ) . (2.7)

This assumption allows one to define the masses of D and B–-mesons by analogy with the masses of light pseudoscalar mesons, for example

M2 p4 m×0ap1(p) Nu–( 0 ) u( 0 ) 1d – ( 0 ) d( 0 ) Np1_{(p)b( 2 E} p KV), etc . , (2.8) where m×

04 (m0 u1 m0 d) O2, and m0 uand m0 dare the masses of current u- and d-quarks. As a result the masses of light and heavy pseudoscalar mesons get defined by the matrix elements of the operator

(2.9) _{O(x) 4m}0 uu–(x) u(x) 1m0 dd – (x) d(x) 1m0 ss–(x) s(x) 1 1Mcc–(x) c(x) 1Mbb – (x) b(x) 1Mtt – (x) t(x), that is,

.

/

´

M2 p4 ap1(p) N O(x)Np1(p)b( 2 EKpV ) , M2 D4 aD(p) N O(x) ND(p)b( 2 EKpV ) , M_B–2_{4 aB}–_{(p) N O(x)NB}–_{(p)b( 2 E} p KV ) , etc . (2.10)

Note that O(x) describes the current quark mass term in QCD Lagrangian.

The computation of the matrix elements (2.11) gives the coupling constants gD and

gB–[14] gD4 2 pk2 kN

u

M2 D Mcv – 8

v

1 O2 , gB–4 2 pk2 kN

u

M_B–2 Mbv – 8

v

1 O2 , (2.11)

where v–_{8 4 4 L 4 2 .66 GeV [9-14] and L is the cut-off in Euclidean 3-momentum space,} connected with the SBxS scale Lxin (CHPT)q via the relation L 4LxOk2 40.67 GeV at Lx4 0.94 GeV [4]. The cut-off L appears due to the computation of heavy-light constituent quark loops describing the matrix elements (2.11) in (CHPT)q.

By substituting (2.15) to the r.h.s. of (2.5) one obtains (2.12) ( 2 EKp8V 2 EKpV )1 O2aD(p8)NB

–

c( 0 ) gmb( 0 ) NB

– (p)b 4

4 i2gDgB–



d4x d4y exp [2ipQx1ip 8Qy]3 3a 0 NT

(

q–(y) ig5_{c(y) c}– ( 0 ) gmb( 0 ) b – (x) ig5_q(x)

₎

N0 bconn ., where p2 4 MB–2 and p 824 MD2.

By analogy with the matrix element aD(p8)Nc–( 0 ) gmb( 0 ) NB

–

(p)b and following the procedure expounded above one can get the representation of the matrix elements of

the B–_{K D* transitions}

(2.13) ( 2 EKp8V 2 EKpV )1 O2aD * (p 8)Nc

–

( 0 ) Gb( 0 ) NB–(p)b 4

4 2 i2gD*gB–



d4x d4y exp [2ipQx1ip 8Qy]3 3a 0 NT

(

q–(y) gnc(y) c–( 0 ) Gb( 0 ) b

–

(x) ig5_q(x)

₎

N0 bconn .e *n(p8) , where G 4gm or gmg5, then e *n(p8) is the 4-polarization vector of the D*-mesons, and

p2_4MB–2 _{and p 8}2

4MD*2 . The coupling constant gD* can be fixed from the condition

M2 D*4 McaD * (p) Nc – ( 0 ) c( 0 ) ND *(p)b(2EKpV ) (2.14) and is given by [14] gD*4 2 pk2 kN

u

M2 D* Mcv–8

v

1 O2 . (2.15)

The formula (2.14) has been of use for the computation of the fine structure of the mass spectrum of D*-mesons caused by first-order corrections in chiral expansion [13]. Within the framework of HQET and (CHPT)qand keeping to leading terms in large

N and MQ (Q 4c, b) expansions, the r.h.s. of the formulas (2.12) and (2.13) can be represented by the following momentum integrals [9-14]:

.

`

`

`

`

/

`

`

`

`

´

M_m(B–_{K D ) 4 ig}B–igD

g

2 N 16 p2

h

3 3



d 4_k p2itr

m

g 5 1 m 2k× g 5

g

1 1v×8 2

h

gm

g

1 1v× 2

h

1 k Q v 1i0 1 k Q v 81 i0

n

, MV m (B – K D* ) 4 2 igB–gD*

g

2 N 16 p2

h

e * n_(v8)3 3



d 4 k p2_itr

m

g 5 1 m 2k× gn

g

1 1v×8 2

h

gm

g

1 1v× 2

h

1 k Q v 1i0 1 k Q v 81 i0

n

, MA m (B – K D* ) 4 2 igB–gD*

g

2 N 16 p2

h

e * n_(v8)3 3



d 4_k p2_itr

m

g 5 1 m 2k× gn

g

1 1v×8 2

h

gmg 5

g

1 1v× 2

h

1 k Q v 1i0 1 k Q v 81 i0

n

. (2.16)

Here m 40.33 GeV is the mass of the light constituent quark calculated in the chiral limit [4].

Putting McC MDC MD*and MbC MB–[15, 16] the coupling constants gD, gD*and gB– are given by gD4 2 pk2 kN

u

MD v – 8

v

1 O2 , gD*4 2 pk2 kN

u

MD* v – 8

v

1 O2 , gB–4 2 pk2 kN

u

MB– v – 8

v

1 O2 . (2.17)

(MB–MD*)1 O2 to the l.h.s. we reduce the r.h.s. of (2.16) to the expressions

.

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`

`

`

/

`

`

`

`

´

(MB–MD)21 O 2Mm(B – K D ) 4 1 2 v–₈ 3 3



d 4_k p2_itr

m

g 5 1 m 2k× g 5

g

1 1v×8 2

h

gm

g

1 1v× 2

h

1 k Q v 1i0 1 k Qv 81 i0

n

, (MB–MD*)21 O 2Mm( V )(B – K D* ) 4 i 2 v–₈ e *n_(v8)3 3



d 4_k p2_itr

m

g 5 1 m 2k× gn

g

1 1v×8 2

h

gm

g

1 1v× 2

h

1 k Q v 1i0 1 k Q v 81 i0

n

, (MB–MD*)21 O 2Mm( A )(B – K D* ) 4 i 2 v–₈ e *n_(v8)3 3



d 4 k p2_itr

m

g 5 1 m 2k× gn

g

1 1v×8 2

h

gmg 5

g

1 1v× 2

h

1 k Q v 1i0 1 k Q v 81 i0

n

. (2.18)

The momentum integrals in the r.h.s. of (2.18) depend on the 4-velocities of B– and D( D* ) mesons only. This corroborates the main result of the Isgur-Wise theorem that in the infinite heavy-quark mass limit the structure functions of heavy-to-heavy meson transitions should depend on w only [15].

The momentum integrals in (2.18) should define the explicit form of the Isgur-Wise structure function within our effective quark model. The same integrals define the Isgur-Wise structure function in the Bardeen-Hill model (see sect. 9).

2.1. The j6(w) structure functions. – In order to obtain the structure functions describing the matrix element Mm(B

–

K D ) we have to compute the momentum integral (2.19) (MB–MD)21 O 2Mm(B – K D ) 4 2 1 2 v–₈ 3 3



d 4_k p2i ( 1 2w) km1 (k Q v8) vm1 (k Q v) v 8m2 m(v 1 v8)m [m2 2 k22 i0 ][k Q v 1 i0 ][k Q v 8 1 i0 ] . It is seen that M_m(B–_{K D ) is expressed in terms of the integrals}



d4_k p2_i 1 m2 2 k22 i0 1 k Q v 1i0 , (2.20)



d4_k p2i k_m m2_{2 k}2_{2 i0} 1 k Q v 1i0 1 k Q v 81 i0 , (2.21)



d4k p2_i 1 m2 2 k22 i0 1 k Q v 1i0 1 k Q v 81 i0 . (2.22)

The integrals (2.20) and (2.21) diverge linearly whereas the integral (2.22) is a logarithmically divergent. Holding the leading divergent contributions [9-14,16] only,

which are linear in the case of integrals (2.20) and (2.21), we neglect the contribution of the integral (2.22) being logarithmically divergent.

Following [9-14] we compute (2.20) and (2.21) in the form



d4_k p2i 1 m2_{2 k}2_{2 i0} 1 k Q v 1i0 4 2 v – 8 , (2.23)



d4_k p2i k_m m2_{2 k}2_{2 i0} 1 k Q v 1i0 1 k Q v 81 i0 4 2 v – 8 1 w 11(v 1v8)m. (2.24)

Substituting (2.23) and (2.24) in (2.19) we obtain (MB–MD)21 O 2Mm(B

–

K D ) 4 1

w 11(v 1v8)m.

(2.25)

Comparing (2.25) with (2.2) one obtains the structure functions

.

/

´

j1(w) 4 1 2j(w) 4 1 w 11 , j2(w) 40 . (2.26)

Beneath the structure function j(w) given by the expression

j(w) 4 2 w 11

(2.27)

we identify with the Isgur-Wise universal structure function [15]. The same structure functions (2.26) can be computed in the Bardeen-Hill model [16] (see sect. 7).

2.2. The z(w) structure function. – The structure z(w) function describing the matrix element Mm( V )(B

–

K D* ) is defined by the momentum integral (2.28) (MB–MD*)21 O 2Mm( V )(B – K D* ) 4 4 1 2 v–₈ e *n_(v8)



d 4_k p2_i emnab(kav 8b1 vakb) [m2 2 k22 i0 ][k Q v 1 i0 ][k Q v 8 1 i0 ] . The r.h.s. of (2.28) is expressed in terms of the integral (2.24). As a result one obtains (MB–MD*)21 O 2Mm( V )(B – K D* ) 4 2 1 w 11emnabe * n_{(v8) v}a_{v 8}b_. (2.29)

The comparison of (2.29) with (2.2) gives the expression of the z(w) function

z(w) 4 1

2j(w) . (2.30)

Picking up the relation (2.26) one can find that that z(w) 4j1(w). This agrees with the prediction of heavy-quark symmetry.

2.3. The ji(w) (i 41, 2, 3) structure functions. – The structure functions ji(w) (i 4

1 , 2 , 3 ) describing the matrix element M( A )

m (B

–

K D* ) are defined by the momentum integral (2.31) (MB–MD*)21O2Mm(A)(B – KD*)4 4 i 2 v–₈ e *n_(v8)



d 4 k p2_i 2(k Q v 1 k Q v8) gmn1 knv 8m1 v 8ukm2 knvm1 kmvn [m2 2 k22 i0 ][k Q v 1 i0 ][k Q v 8 1 i0 ] . The r.h.s. of (2.31) is expressed in terms of the momentum integrals (2.23) and (2.24). Thereby one obtains

(MB–MD*)21 O 2Mm( A )(B – K D* ) 4 ie *m (v8)2i 1 w 11

(

e * (v8)Qv

)

v 8m. (2.32)

By comparing (2.32) with (2.2) we obtain the structure functions ji(w) (i 4

1 , 2 , 3 )

.

/

´

j1(w) 4j3(w) 4 1 2j(w) , j2(w) 40 . (2.33)

Summarizing the results given by relations (2.26), (2.30) and (2.33) we arrive at relation (2.4). We conclude that the structure functions of the B–_{K D( D* ) transitions computed} within HQET and (CHPT)qare subject to the requirements of heavy-quark symmetry. The additional factor 1O2 can be considered as a renormalization of HQET by strong interactions at scales p ELxE MQ within the effective quark model describing strong low-energy interactions in terms of constituent quark loop diagrams.

Due to the additional factor 1O2 the structure functions within our approach are normalized to 1O2 at zero recoil w41, i.e.

j1( 1 ) 4z(1) 4j1( 1 ) 4j3( 1 ) 4 1 2 (2.34)

instead of to unity [15, 17, 19-25].

The inclusion of the contributions diverging logarithmically leaves relations (2.4) and (2.34) unchanged. Up to the logarithmically divergent contributions the universal structure function j(w) is given by

j(w) 4 2 w 11

u

1 2g1 1 2g

o

w 11 w 21ln

y

k_{w 11 1}k_{w 21} k_{w 11 2}k_{w 21}

z

v

, (2.35) where g 4 4 m v – 8 ln

u

v – 8 4 m

vy

1 1 4 m v – 8 ln

u

v – 8 4 m

vz

21 C 1 4 . (2.36)

Here we have also taken into account the logarithmically divergent contributions to the coupling constants gD, gD* and gB–.

It is seen that the structure function (2.35) equals unity at zero recoil. Thereby the normalization condition (2.34) is retained.

2._{4. The structure functions of the D KD and DKD* transitions. – By applying}

the procedure having been expounded above one can show that in the limit McK Q the matrix elements of the D KD and DKD* transitions can be expressed in terms of the universal structure function j(w), i.e.

.

`

/

`

´

M_{m( D KD ) 4M}D 1 2j(w)(v 1v8)m, M_{m( D KD*) 42}kMD*MD 1 2j(w) emnabe * n_{(v8) v}a_{v 8}b_, (2.37)

where j(w) is given by (2.27). The matrix element of the transition D KD* gives the contribution to the amplitude of the D* KDg decay of order O(1OMc) and can be dropped out at leading order in large Mc expansion [14].

The computation of next-to-leading order corrections in large-Mc expansion to the amplitudes of the D*1

K D1g and D*1

s K D1s g decays is given in sect. 4. It is shown that the factor 1O2 appearing in the form factor of the D KD* transition plays an important role for the theoretical explanation of the experimental suppression of the D*1

K D1g: B(D*1

K D1g)exp4 1.1 6 1.4 6 1.5% [26, 27]. In the chiral limit we predict the analogous suppression for the Ds*1K D1s g decay. In sect. 8 we show that the contribution of the Isgur-Wise structure function to the amplitude of the D*qK Dqg is of order O( 1 OMc) but not O( 1 OMc2) as has been assumed by Amundsen et

al. [18].

3. – Heavy-quark symmetry

In this section we show that momentum integrals like eq. (2.16) and so on, defining form factors of the heavy-to-heavy meson transitions, can be derived by using the effective Lagrangian of quark-meson interactions invariant under heavy-quark symmetry [15]. The r.h.s. of the equations of motion (2.6) can be obtained from the ef-fective heavy-light quark Lagrangian

(3.1) _Leff(x) 4 c–(x)(i ¯×2Mc) c(x) 1b – (x)(i ¯×_2Mb) b(x) 1

!

q 4u, d, s3 3[ gDc–(x) ig5q(x) Dq–c(x) 1gD*c–(x) gmq(x) D *q–cm(x) 1 1gB–b – (x) ig5 q(x) B–q–b(x) 1gB–*b – (x) gmq(x) B – *m q –_{b(x) 1h.c. 1q}–_{(x)(i ¯}×_{2m) q(x) ] ,} where the subscripts (q–c) and (q–b) describe the quark structure of heavy mesons, i.e.

Du–c4 D0, B – u

–_{b4 B}2_{etc. The coupling constant gB}–

*is defined by analogy with gD*. The heavy meson fields are included in the effective Lagrangian (3.1) as external fields.

Now let us transform the effective Lagrangian (3.1) to the distinct heavy-quark symmetry form. For this aim we have to proceed to the infinite heavy-quark mass limit

Mc, MbK Q and carry out the change [3]

.

`

/

`

´

c(x) 4

g

1 1v× 2

h

exp [2iMcv Q x] cv(x) , b(x) 4

g

1 1v× 2

h

exp [2iMbv Q x] bv(x). (3.2) Note that

g

1 1v× 2

h

cv(x) 4cv(x) ,

g

1 1v× 2

h

bv(x) 4bv(x) , (3.3)

i.e., the fields cv(x) and bv(x) always carry an implicit factor of ( 1 1v×)O2 [3, 16].

Substituting (3.2) to (3.1) we reduce the effective Lagrangian to the form (3.4) _Leff(x) 4 c–v(x) iv Q ¯cv(x) 1b – v(x) iv Q ¯bv(x) 1 1

!

q 4u, d, s

k

gDc–v(x)

g

1 1v× 2

h

ig 5 q(x) Dq–c(x) 1gD*c–v(x)

g

1 1v× 2

h

gmq(x) D* m q –_{c (x) 1} 1gB–b – v(x)

g

1 1v× 2

h

ig 5 q(x) B–q–b(x) 1gB–*b – v(x)

g

1 1v× 2

h

gmq(x) B – *m q –_{b(x) 1} 1h.c. 1 q–(x)(i ¯×_{2m) q(x)}

l

. Here we have denoted

.

`

/

`

´

Dq–c(x) 4exp [iMcv Q x] D–qc(x) Cexp [iMDv Q x] Dq–c(x) , D*q–c(x) 4exp [iMcv Q x] D *–qc(x) Cexp [iMD*v Q x] D *q–c(x) , B – q–b(x) 4exp [iMbv Q x] B – q –_{b(x) Cexp [iMbv Q x] B}–_q–_{b(x) ,} B – *m q –_{b(x) 4exp [iMbv Q x] B}–_*m q –_{b(x) Cexp [iMB*v Q x] B}–_*m q –_{b(x) ,} (3.5)

where vm _{has taken the sense of the fixed 4-velocity of a heavy meson [3,16].}

Using the coupling constants gD, gD*, gB– and gB–* given by

.

`

/

`

´

gD4 2 p kN

o

2 MD v – 8 , gD*4 2 p kN

o

2 MD* v – 8 , gB–4 2 p kN

o

2 MB– v – 8 , gB–*4 2 p kN

o

2 MB–* v – 8 , (3.6)

we can rewrite the effective Lagrangian (3.4) as follows: (3.7) _Leff(x) 4 c – v(x) iv Q ¯cv(x) 1b – v(x) iv Q ¯bv(x) 1 1 2 p kN q 4u, d, s

!

y

c – v(x)

g

1 1v× 2

h

ig 5

o

2 MD v – 8 Dq–c(x) q(x) 1 1 c–v(x)

g

1 1v× 2

h

gm

o

2 MD* v – 8 D* m q –_{c (x) q(x) 1b}–_v(x)

g

1 1v× 2

h

ig 5

o

2 MB– v–₈ B – q –_{b(x) q(x) 1} 1 b–v(x)

g

1 1v× 2

h

gm

o

2 MB–* v – 8 B – *m q –_{b(x) q(x) 1h.c. 1q}–_{(x)(i ¯}×_{2m) q(x)}

z

_. In order to implement heavy-quark symmetry to the effective Lagrangian (3.7) it is convenient to introduce new heavy-quark and heavy-meson fields [3, 16], which are

ha

v(x) and Hq–a(x , v) (a 4c, b), respectively, with components

.

`

/

`

´

hvc(x) 4cv(x) , h b v(x) 4bv(x) , Hq–c(x , v) 4

g

1 1v× 2

h

y

ig 5

o

2 MD v – 8 D q –_{c(x) 1g} m

o

2 MD* v – 8 D* m q –_{c (x)}

z

_, Hq–b(x , v) 4

g

1 1v× 2

h

y

ig 5

o

2 MB– v – 8 B – q –_{b(x) 1g} m

o

2 MB–* v – 8 B – *m q –_b(x)

z

_. (3.8)

As a result the effective Lagrangian (3.7) reads (3.9) _Leff(x) 4 h –_a v(x) iv Q ¯hva(x) 1 2 p kN q 4u, d, s

!

h –_a v(x) Hq–a(x , v) q(x) 1h.c.1 1

!

q 4u, d, s q –_{(x)(i ¯}× 2m) q(x) . The effective Lagrangian (3.9) has a form being distinctly invariant under SU( 2 )HQ heavy-quark symmetry. Thereby the results obtained within HQET supplemented by (CHPT)q cannot contradict to heavy-quark symmetry predictions.

In order to understand the appearance of the additional factor 1O2 in the structure functions of the B–_{K D( D* ) and D KD(} D* ) transitions we suggest to trace the derivation of the Isgur-Wise theorem [15].

For the starting point of the derivation of the Isgur-Wise theorem it is important to note that in reality heavy-quark masses MQ (Q 4c, b) are finite and energies of quark-gluon fluctuations describing contribution of strong interactions to the matrix element of H1(v) KH2(v8) heavy-to-heavy meson transition run over the region restricted from above by the ultra-violet cut-off L c MQ and from below by QCD parameter LQCDbMQ. In order to apply the infinite heavy-quark mass limit MQK Q we have to deal with quark-gluon fluctuations interacting at energies restricted by the scale m EMQ. This can be reached by the integration over all quark-gluon fluctuations interacting at energies exceeding the scale m. The integration can be performed within

perturbative QCD and the result of the integration can be represented by the factor

CQCD(m) [15]. As a result the energies of quark-gluon interactions, contributing to the matrix element of H1(v) KH2(v8) heavy-to-heavy meson transition, got restricted by the scale m that less than heavy-quark masses. Now in accordance to the Appelquist-Carazzone theorem [28] one can apply the infinite heavy-quark mass limit

MQK Q. For the completion of the computation of the matrix element of the H1(v) K

H2(v8) transition one should integrate over quark-gluon fluctuations contributing to the matrix element of the H1(v) KH2(v8) heavy-to-heavy meson transition at energies below the scale m.

The result of the integration has been reduced to the factor 1 by the assumption: «The vector current V 9n in this effective theory is not renormalized by the physics at

scales p EMQ, since at such scales both quarks are effectively static» [15], where V 9n

has the structure Q–2gnQ1. The operator

s

d3x V 90(x) then has been associated with a generator of the SU( 2 )HQ heavy-quark symmetry. The former has led trivially to the constraint aH2NV 90NH1b 41, where H2and H1are heavy-light mesons containing Q2and

Q1 heavy quarks, respectively.

Thus one can see that just the assumption, regarding the unrenormalizability of HQET by strong interactions at scales p EMQ, lays in the foundation of eq. (2.3) that has set up the proportionality between j(w) and structure functions of H1(v) KH2(v8) heavy-to-heavy meson transition with a factor 1. Indeed, if one assumes that the vector current V 9n can be renormalized by strong interactions at scales p EmEMQ, one should redefine V 9n as follows, i.e. V 9n4 C(m E MQ) VRn . Then the operator

s

d3xVR0 (x) has to be associated with a generator of the SU( 2 )HQ heavy-quark symmetry. The former entails the normalization aH2NV R0NH1b 41, defining the normalization of the universal structure function j(w), i.e. j( 1 ) 41. Thereby, the normalization of the structure functions of heavy-to-heavy meson transitions differs by a factor C(m EMQ) from the normalization of the Isgur-Wise structure function j(w). For the computation of C(m E

MQ) one should apply a phenomenological model based on QCD and describing strong low-energy interactions of hadrons at the quark level. In our approach such a model is (CHPT)q.

(CHPT)q gives the possibility to describe the contribution of quark-gluon fluctuations at energies below m, i.e. at p EmEMQ, in terms of constituent quark degrees of freedom. The scale m is fixed and equal to the scale of SBxS, i.e. m 4Lx4 0.94 GeV [27]. The matrix element of H1(v) KH2(v8) heavy-to-heavy meson transition can be given in terms of heavy-light constituent quark loops with the momenta of virtual constituent quarks restricted by Lx4 0.94 GeV.

What happens with the heavy-light constituent quark loops in the infinite heavy-quark mass limit MQK Q?

Taking the integrands in the limit MQK Q one removes virtual heavy-antiquark degrees of freedom contributing to the heavy-quark Green functions. This is to halve virtual heavy-flavour degrees of freedom. Halving virtual heavy-flavour degrees of freedom one obtains the result being half as much as that should be obtained with the complete set of virtual heavy-antiquark degrees of freedom. This is the reason of the appearance of the additional factor 1O2 that should take the meaning of the renormalization of HQET by strong interactions at scales p ELxE MQ.

Thus summing up the contributions of strong interactions at scales both with pDLx

and p ELxE MQ, we arrive at HQET renormalized with a factor ( 1 O2) CQCD(Lx). Up to this factor the structure functions of heavy-to-heavy meson transitions have to be

normalized to unity at zero recoil in accordance with requirement of heavy-quark symmetry. For the detailed computation see sect. 11.

4. – Heavy-antiquark degrees of freedom

In this section we investigate the influence of virtual heavy-antiquark degrees of freedom by example of the matrix element ap1_(p8)Nu–_{( 0 ) g}

mu( 0 ) Np1(p)b being good

established within (CHPT)q. For the computation of this matrix element we apply the procedure having been expounded above. By using the reduction technique we get (4.1) ap1_(p8)Nu–_{( 0 ) g} mu( 0 ) Np1(p)b 4 4 lim p2_{K M}2 p p 82K Mp2

i2



_d4_{x d}4_y exp [2ipQx1ip 8Qy] ( 2 E 8pV 2 EpV )1 O2

(px1 Mp2)(py1 Mp2) 3

3a 0 NT

(

Wp1(y) u–( 0 ) g_mu( 0 ) W†_p1(x)

)

N0 b_{conn .},

where W†p1(x) and W_p1(y) are the interpolating fields of p1-mesons satisfying the

equations of motion

.

`

/

`

´

(px1 Mp2) W†p1(x) 4 gp k2 u – (x) ig5_{d(x) ,} (py1 Mp2) Wp1(y) 4 gp k2 d – (y) ig5_{u(y) .} (4.2)

The coupling constant gp describes the quark-pion interactions and obeys the constraint [4]

gp2I2(m) 41 , (4.3)

where I2(m) is a logarithmically divergent integral

I2(m) 4 N 16 p2



d4_k p2i 1 (m2 2 k2)2 4 N 16 p2ln

g

L2 x m2

h

1 O( 1 ) (4.4)

(see Kikkawa [5]). E 8p and Ep are the energies of pions.

Substituting (4.2) in (4.1) we express the matrix element

ap1_(p8)Nu–_{( 0 ) g}

mu( 0 ) Np1(p)b in terms of the vacuum expectation value of the

time-ordered product of current quark densities, i.e. (4.5) ( 2 E 8pV 2 EpV )1 O2ap1(p8)Nu–( 0 ) gmu( 0 ) Np1(p)b 4

4 21 2g

2

p



d4x d4y exp [2ipQx1ip 8Qy]3 3a 0 NT(d–(y) ig5_{u(y) u}–

( 0 ) gmu( 0 ) u–(x) ig5d(x)

)

N0 bconn .. Applying the formulas of quark conversion [4] we express the vacuum expectation

value (4.5) in terms of constituent quark Green functions (4.6) ( 2 E 8pV 2 EpV )1 O2ap1(p8)Nu – ( 0 ) gmu( 0 ) Np1(p)b 4 4 2 N1 2 g 2

pi



d4x d4y exp [2ipQx1ip 8Qy] tr ]SF(x 2y)g5SF(y) gmSF(2x) g5( .

The Green function of the free constituent quark SF(z) can be given in the form of the

decomposition iSF(z) 4u(z0)



d3_q ( 2 p)3 m 1q× 2 EKq exp [2iqQz]1u(2z0₎



d 3_q ( 2 p)3 m 2q× 2 EKq exp [iq Q z] , (4.7) where q Q z 4EKqz02 q K

Q zKand Eq4

k

K2q 1 m2. The terms in the r.h.s. of (4.7) are due to

the contributions of quark and antiquark degrees of freedom, respectively.

If the Green functions in the integrand of (4.6) contain the contributions of both quark and antiquark degrees of freedom, the momentum representation of the matrix element under consideration reads [4]

(4.8) ( 2 E 8pV 2 EpV )1 O2ap1(p8)Nu–( 0 ) gmu( 0 ) Np1(p)b 4 4 1 2 g 2 p N 16 p2



d4_k p2_itr

m

g 5 1 m 2k× g 5 1 m 2k×2 p×8gm 1 m 2k×2 p×

n

. Readers can find in [4] bulky examples of the computation of the momentum integrals like that in (4.8), performed within (CHPT)q. Here we give the result

( 2 E 8pV 2 EpV )1 O2ap1(p8)Nu–( 0 ) gmu( 0 ) Np1(p)b 4gp2I2(m)(p 1p8)m4 (p 1 p8)m,

(4.9)

that corresponds to the computation of the momentum integral in leading order in chiral expansion [4]. Note that in the r.h.s. of (4.9) we have used the constraint (4.3). The matrix element of the time component of the current u–( 0 ) gmu( 0 ) calculated at

p

K

8 4 pK4 0 and E 8p4 Ep4 Mp is given by

( 2 MpV)ap1(Mp) Nu†( 0 ) u( 0 ) Np1(Mp)b 42Mp. (4.10)

For the subsequent analysis it is convenient to differentiate this relation with respect to Mp

ˇ ˇMp

[ ( 2 MpV)ap1(Mp) Nu†( 0 ) u( 0 ) Np1(Mp)b] 42 . (4.11)

Thus one can see that, if the constituent quark Green functions in the integrand of (4.6) contain the contributions of both quark and antiquark degrees of freedom, we get the relation (4.11).

Let us show that the r.h.s. of the relation (4.11) will be halved if we remove the contributions of antiquarks to the constituent quark Green functions in the environment of gm. This realizes the situation we face with the computation of the

matrix elements of the B–_{K D , D* and D K D , D* transitions in the infinite} heavy-quark mass limit. Now the matrix element ap1_(p8)Nu†

defined by the following momentum integral: (4.12) ( 2 E 8pV 2 EpV)1 O2ap1(p8)Nu†( 0 ) u( 0 ) Np1(p)b 4 1 2g 2 p N 16 p2 3 3



d 4_k p2_i 1 k0 2 EKp8 1 kK1 E 8p1 i0 1 k0 2 EKp1k1 Ep1 i0 3 3tr

{

g5 1 m 2k×2i0 g 5 m 1g 0 EKp8 1 kK2 g K Q(pK8 1 k K ) 2 EKp8 1 kK g0 3m 1g 0 EKp1 kK2 g K Q(pK1 k K ) 2 EKp1 kK

}

. The r.h.s. of (4.12) contains a linearly divergent term which does not depend on the energies and momenta of pions. One can show that this term should be cancelled by the contribution of antiquark degrees of freedom. The linearly divergent term appears as a

result of the violation of Lorentz covariance of the matrix element

ap1_(p8)Nu–_{( 0 ) g}

mu( 0 ) Np1(p)b caused by the elimination of contributions of virtual

antiquarks with finite masses. In order to get rid this linearly divergent term, that is irrelevant to the problem of the appearance of the factor 1O2, we suggest to differentiate the matrix element ( 2 MpV)ap1(Mp) Nu†( 0 ) u( 0 ) Np1(Mp)b with respect to Mp. Therefore, we perform the subsequent calculation putting p

K

8 4 pK4 0 and E 8p4

Ep4 Mp, differentiating with respect to Mp and keeping to the leading terms in Mp expansion. As result we get

ˇ ˇMp [ ( 2 MpV)ap1(Mp) Nu†( 0 ) u( 0 ) Np1(Mp)b] 4 (4.13) 4 2 gp2 N 8 p2



d4_k p2_i 1 [k0 2 EKk1 i0 ]3 1 k0 1 EKk2 i0 . The integration over k0 gives

ˇ ˇMp [ ( 2 MpV )ap1(Mp) Nu†( 0 ) u( 0 ) Np1(Mp)b] 4gp2 N 16 p2 1 2 p



d3k EK_k3 . (4.14)

By using the relation 1 2 p



d3_k EK_k3 4



d4k p2_i 1 (m2 2 k22 i0 )2 , (4.15)

we reduce the r.h.s. of (4.14) to the form

(4.16) ˇ ˇMp [ ( 2 MpV )ap1(Mp) Nu†( 0 ) u( 0 ) Np1(Mp)b] 4 4 gp2 N 16 p2



d4_k p2i 1 (m2_{2 k}2_{2 i0 )}2 4 g 2 pI2(m) 41 . Thus we have shown that the elimination of the antiquark contributions to the constituent quark Green functions surrounding the vertex gm in the constituent quark

loop, describing the matrix element ap1_(p8)Nu–_{( 0 ) g}

mu( 0 ) Np1(p)b, diminishes the

result of the integration over the virtual momentum twice as many as that obtained with the account of virtual antiquark contributions. This computation corroborates our assertion concerning the nature of the additional factor 1O2 in the form factors of the B

–

K D , D* and D K D , D* transitions calculated in the infinite heavy-quark mass limit.

5. – Normalization and non-interchangeability of limits q2

K 0 and w K 1 We suggest to follow up the non-interchangeability of the limits q2

K 0 and w K 1 at

MQK Q by example of the electromagnetic form factor of the D1-meson, defined by the matrix element

aD1_{(p 8 Nj}el

m ( 0 ) ND1(p)b 4fDel1(q2)(p 1p8)_m,

(5.1) where jel

m ( 0 ) is the operator of the electromagnetic quark current. It is well known that,

due to the electric charge conservation, the form factor fel

D1(q2) has to be normalized by the condition lim q2K 0 fel D1(q2) 4f_Del1( 0 ) 41 , (5.2)

whereas in our approach, naively putting w 41, one obtains lim McK Q fel D1(q2) 4 1 2j(w) K 1 2 (5.3)

at w K1. The resolution of this paradox lays in the non-interchangeability of limits

q2_{K 0 and w K 1 at M}cK Q. We argue that one cannot compare normalizations of form factors at q2

4 0 with the normalization of structure functions at w 4 1, when the limit

MQK Q has been taken. It is due to the limit q24 2 2 Mc2(w 21) K0 does not exist at

McK Q, i.e. q24 2 Q Q 0 4 ?

In order to illustrate the non-interchangeability of the limits q2

K 0 and w K 1 at

McK Q we suggest to consider the electromagnetic form factor of the D1-meson calculated within Vector Dominance Model (VDM) [30]. The simplest form of fel

D1(q2)

can be obtained admitting the validity of SU( 4 )-symmetry of VPP-interactions where V and P are standard 15-plets of vector and pseudoscalar mesons (see, «Quark Model» in [26]). As a result fel D1(q2) reads [31] fel D1(q2) 4 2 3 M2 J Oc M_{J Oc}2 _{2 q}2 1 1 2 Mr2 Mr22 q2 2 1 6 Mv2 Mv22 q2 , (5.4)

where Mr, Mvand MJ Ocare the masses of the r, v and J Oc mesons. It is seen that at

q2

4 0 the form factor (5.4) is normalized by the condition (5.2). In order to pass to the limit McK Q we put MJ Oc4 2 Mc and q24 2 2 Mc(w 21) [16] (p. 417). It is then seen that in the limit McK Q the contributions of the r- and v-mesons should disappear and only the contribution of the J Oc-meson survives. As a result we get

lim McK Q fel D1(q2) 4 2 3 2 w 11 4 2 3j(w) . (5.5)

Now let us take the limit w K1, and get lim w K1MlimcK Qf el D1(q2) 4 2 3 . (5.6)

This limit should be compared with lim

McK Qqlim2

K 0

fDel1(q2) 41 .

(5.7)

The relations (5.6) and (5.7) confirm the non-interchangeability if limits q2

K 0 and w K 1 at McK Q. The factor 2 O 3 in the r.h.s. of (5.6) is compared with the factor 1O2 given in our approach eq. (5.3).

The comparison of the limits (5.6) and (5.7) evidences that the normalization of the form factors of any H1(v) KH2(v8) heavy-to-heavy meson transitions, calculated in the limit MQK Q(Q 4 c , b) and taken at w 4 1, can tell nothing about the normalization of the form factors, obtained at q2

4 0 and at finite values of heavy-quark masses, i.e.

MQKO Q( Q 4 c , b).

6. – 1 OMc corrections to amplitudes of electromagnetic and strong decays of D*-mesons

The appearance of the additional factor 1O2 in the form factors of the heavy-to-heavy meson transitions should influence the matrix elements of physical processes. As an example of an interesting influence of this factor we suggest to consider the computation of the amplitude of the D*1

K D1g decay in next-to-leading order in large Mc expansion and in the chiral limit.

In [14] we have calculated the amplitudes of strong and electromagnetic decays of the D*-mesons in the chiral limit and keeping to leading order in large Mcand large N expansions. Most of theoretical predictions have been compared with experimental data. Only discrepancy between theoretical results and experimental data have been observed for the electromagnetic decay of the D*1_{-meson, i.e., D*}1

K D1g. There has been obtained [14]

G( D*1

K D1g)th4 0.2 keV(3%) , B( D*1K D1g)exp4 1.1 6 1.4 6 1.5% . (6.1)

The experimental magnitude of the probability, given by CLEO Collaboration [30], implies the suppression of the D*1

K D1g decay. However the theoretical magnitude of the probability obtained in leading order in large-Mc expansion agrees with the experimental data from our point of view only qualitatively.

In order to reconcile the theoretical and experimental data on the probability of the D*1

K D1g decay we suggest to take into account next-to-leading order cor-rections in large Mc expansion. Indeed this is the only possibility to gain an additional non-perturbative contribution within the framework of the accepted approach. In terms of non-relativistic quark model this should mean the account of the effective magnetic moment of the c-quark [32]. We obtain the contribution of the Isgur-Wise structure function to the amplitude of the D*1

K D1g decay of order O( 1 OMc) but not

O( 1 OM2

c) as has been asserted in [18]. This problem we discuss in sect. 10.

element of the D* KDg transition reads

aout ; g(k) D(p) ND *(Q); inb , (6.2)

where Q, p and k are the 4-momenta of D*, D and a photon, respectively. By applying the reduction technique we reduce the matrix element (6.2) to the form

(6.3) _{aout ; g(k) D(p) ND *(Q); inb 4} 4 lim k2_{K 0}(2i)



_d4_yexp [2ikQy] ( 2 vKkV)1 O2 pyaD(p) NAm(y) ND *(Q)b e *m(k) ,

where Am(y) is the interpolating photon field, and e *m(k) is the 4-vector of a photon

polarization, vKk is the energy of a photon and V is a normalization volume. The photon

interpolating field A_m(y) satisfies the equation of motion pyAm(y) 4jmel(y) 4

!

q 4u, d, s

eqq–(y) gmq(y) 1ecc–(y) gmc(y) 1R,

(6.4)

where q(y) and c(y) are light and charmed current quark fields with N colour degrees of freedom, and eu4 ec4 2 e O 3 and ed4 es4 2 e O 3, e is the charge of a positron. The ellipses describe the contribution of the quark fields (b , t) being irrelevant to the problem under consideration. Substituting (6.4) in (6.3) one obtains

aout ; g(k) D(p) ND *(Q); inb 4i(2 p)4

d( 4 ) (Q 2p2k) M

(

D* (Q) KD(p) g(k)

)

( 2 EQKV2 EKpV2 vKkV)1 O2 , (6.5) where ( 2 EQKV 2 EKpV )21 O 2M

(

D * (Q) KD(p) g(k)

)

4 2 aD(p) Njmel( 0 ) ND *(Q)b e *m(k) . (6.6)

Then EQK and EKp are the energies of the D* and D-mesons.

It is convenient to represent the matrix element M

(

_{D* (Q) KD(p) g(k)}

)

in the form of decomposition

M

(

_{D* (Q) KD(p) g(k)}

)

_{4 M}(c)

(

D* (Q) KD(p) g(k)

)

1 M(q)

(

D* (Q) KD(p) g(k)

)

, (6.7)

where we have denoted

(6.8) ( 2 EQKV 2 EKpV )21 O 2M(c)

(

D* (Q) KD(p) g(k)

)

4

4 2 ecaD(p) Nc–( 0 ) gmc( 0 ) ND *(Q)b e *m(k) ,

(6.9) ( 2 EQKV 2 EKpV )21 O 2M(q)

(

D* (Q) KD(p) g(k)

)

4

4 2 eqaD(p) Nq–( 0 ) gmq( 0 ) ND *(Q)b e *m(k) ,

where q( 0 ) 4u(0) or d(0) for D*0_{( D}0_{) or D*}1_{( D}1_{), respectively.}

Now let us proceed to the computation of the r.h.s. of (6.8) and (6.9). First we consider the matrix element M( c )

(

D* (Q) KD(p) g(k)

)

. This matrix element is of order

O( 1 OMc) and can be expressed in terms of the Isgur-Wise structure function j(w) (2.34). Within HQET and (CHPT)qthe matrix element M(c)

(

D* (Q) KD(p) g(k)

)

can be

described by the momentum integral (6.10) M( c )

(

D* (Q) KD(p) g(k)

)

4 N 16 p2igD*gD(2ec)



d4

_l

p2_i 3 3tr

{

g5

g

1 1v×8 2

h

gm

g

1 1v× 2

h

gn 1 m 2

l

× 1

l

Q v 1i0 1

l

Q v 81 i0

}

e * m_{(k) e}n_{(v) ,}

where en_{(v) is the 4-vector of meson polarization, v and v 8 are 4-velocities of}

D*-and D-mesons, respectively. After some algebra the r.h.s. of (6.10) reduces to the form (6.11) M( c )

(

D* (Q) KD(p) g(k)

)

4 2 ecgD*gD N 16 p2e * m_{(k) e}n_{(v) 3} 3



d 4

l

p2_i emnab(

l

avb1 v 8a

l

b) [m2 2

l

22 i0 ][

l

Q v 1i0][

l

Q v 81 i0] . Integration over

_l

gives



d4

l

p2i

l

a [m2₂

_l

2_{2 i0 ]} 1 [

_l

_{l Q v 1i0]} 1 [

_l

_{Qv 81 i0]} 4 2 v – 8 2 j(w)(v 1v 8) a_, (6.12)

where j(w) is given by the formula (2.27).

In terms of the Isgur-Wise structure function the matrix element M( c )

(

D* (Q) K D(p) g(k)

)

is defined as follows: (6.13) M( c )

(

D* (Q) KD(p) g(k)

)

4 ec 1 2kMDMD*j(w) emnabe * m_{(k) e}n_{(v) v 8}a_vb 4 4 2 ec( 2 EKQV 2 EKpV )1 O2aD(p) Nc–( 0 ) gmc( 0 ) ND *(Q)b e *m(k) ,

where the matrix element aD(p) Nc–( 0 ) gmc( 0 ) ND *(Q)b reads

(6.14) ( 2 EQKV2 EKpV)1 O2aD(p) Nc – ( 0 ) gmc( 0 ) ND *(Q)b 4 4 21 2 kMDMD*j(w) emnabe n_{(v) v 8}a_vb_.

We refer readers to eq. (2.37).

Comparing expression (6.14) with the formerly defined [15, 17, 19-25]

(6.15) ( 2 EQKV 2 EKpV )1 O2aD(p) Nc–( 0 ) gmc( 0 ) ND *(Q)b42kMDMD*j(w) emnaben(v) v 8avb,

we argue the appearance of the additional factor 1O2 caused by the elimination of heavy-antiquark degrees of freedom in the infinite heavy-quark mass limit.

Since vm_{C v 8}m, which is due to MD*C MD, we get j(w) Cj(1) 41. As a result the matrix element M( c )

(

D* (Q) KD(p) g(k)

)

reads

M( c )

(

D* (Q) KD(p) g(k)

)

4kMDMD* M( c )( D* KDg) , (6.16)

where M( c )( D* KDg) 42 ec 2 Mc emnabe *m(k) en(v) kavb. (6.17)

Here we have used the relation MDC Mc accepted in HQET.

At leading order in large-Mc expansion the matrix element M( q )

(

D* (Q) K D(p) g(k)

)

has been calculated in [14]

M( q )

(

D* (Q) KD(p) g(k)

)

4kMDMD* M(q)( 0 )( D* KDg) , (6.18) where M( 0 ) ( q )( D* KDg) 42 eq 2 v – 8 ln

u

v – 8 4 m

v

emnabe * m_{(k) e}n_{(v) k}a_vb_. (6.19)

In order to pick up next-to-leading order corrections in large Mc expansion to the matrix element M( q )

(

D* (Q) KD(p) g(k)

)

we suggest the following procedure. First we have to apply the reduction technique to the r.h.s. of (6.9). This gives

(6.20) M( q )

(

D* (Q) KD(p) g(k)

)

4 4 lim

p2K MD2

Q2_{K M}2 D*

(2eq)



d4x d4y exp [ip Q x 2iQQy](px1 MD2) 3

3(py1 MD*2 )a 0 NT

g

WD(x) q–( 0 ) gmq( 0 ) W†nD*(y)

h

N0 bconn .e *m(k) en(Q) ,

where WD(x) and W†nD*(y) are the interpolating fields of D and D*-mesons, respectively. In order to analyse the r.h.s. of (6.20) at the quark level we assume that the operators WD(x) and W†nD*(y) satisfy the equations of motion

.

/

´

(px1 MD2) WD(x) 4gDq – (x) ig5_{c(x) ,} (py1 MD*2 ) W†nD*(y) 4gD*c–(y) gnq(y) . (6.21)

By substituting (6.21) in (6.20) one obtains

(6.22) M( q )

(

D* (Q) KD(p) g(k)

)

4 (2eq) gDgD*



d4x d4y exp [ip Q x 2iQQy]3

3a 0 NT

(

q–( 0 ) gmq( 0 ) q–(x) ig5c(x) c–(y) gnq(y)

)

N0 bconn .e *m(k) en(Q) , where Q2

Following eqs. (2.11) and (2.15) we define M( q )

(

D* KDg): (6.23) _M( q )

(

D* KDg) 4 (2eq)

8 p2

N v–₈



_d4_{x d}4

y exp [2ikQx1iQQ (x2y) ]3

3a 0 NT

(

q–( 0 ) gmq( 0 ) q–(x) ig5c(x) c–(y) gnq(y)

)

N0 bconn .e *m(k) en(Q) . The convolution of the c-quark field operators leads to the appearance of the Green function of the c-quark field, which in leading order in large-N expansion coincides with the Green function of the free c-quark field [1]. By applying then the formulas of quark conversion [5] to the r.h.s. of eq. (6.23) we represent the vacuum expectation value in terms of the momentum integral

(6.24) _M( q )

(

D* KDg) 42 eq i 2 v–₈ e *m_{(k) e}n_{(v) 3} 3



d 4 q p2_itr

m

1 m 2q× gm 1 m 2q× 2k× g 5 1 Mc2 q× 2Q× g n

n

.

Since in our approach the integral over q is restricted by LxbMc, so in accordance to the Appelquist-Carazzone theorem [26] we can expand the c-quark Green function in the integrand in powers of q OMc. Holding only the terms of order O(q OMc), we get (6.25) 1 Mc2 q× 2Q× 4 2

g

1 1v× 2

h

1 v Q q 1i0 2 2 1 2 Mc

y

q × v Q q 1i0 2

g

1 1v× 2

h

q2 (v Q q 1i0)2

z

1 O

g

1 Mc2

h

,

where, as usually, we have set Qm

4 MD*vmC Mcvm.

Thus the amplitude M( q )

(

D* KDg) is given by the expansion M( q )

(

D* KDg) 4 M( q )( 0 )

(

D* KDg)1 M( q )( 1 )

(

D* KDg) , (6.26) where (6.27) _M( 0 ) ( q )

(

D* KDg) 4eq i 2 v–₈ e *m(k) en_{(v) 3} 3



d 4 q p2_itr

m

1 m 2q×gm 1 m 2q× 2k× g 5

g

1 1v× 2

h

gn

n

1 v Q q 1i0 4 4 2 eq 2 v – 8 ln

u

v – 8 4 m

v

emnabe * m_{(k) e}n_{(v) k}a_vb_,

(6.28) _M( 1 ) ( q )( D* KDg) 4 eq 2 Mc i 2 v–₈ e *m(k) en_{(v) 3}



d4_q p2_itr

{

1 m 2q× gm 1 m 2q× 2k× g 5

y

q× v Q q 1i0 2

g

1 1v× 2

h

q2 (v Q q 1i0)2

z

gn

}

4 4 2 eq 2 Mc

y

1 2 4 m v – 8 ln

u

v – 8 4 m

vz

emnabe * m_{(k) e}n_{(v) k}a_vb_.

Summing up the contributions we arrive at the total amplitude of the D* KDg decay, which reads

M

(

_{D* (Q) KD(p) g(k)}

)

_{4 2 eg}D* Dgemnabe *m(k) en(Q) kaQb,

(6.29)

where we have defined

egD* Dg4

o

MD MD*

y

eq 2 v – 8 ln

u

v – 8 4 m

v

g

1 2 m Mc

h

1 eq1 ec 2 Mc

z

. (6.30)

The coupling constants of the D*0

K D0g, D*1 K D1g and D*1 s K D1s g decay channels are given by

.

`

`

`

/

`

`

`

´

gD*0_D0_g4

o

MD MD*

y

4 3 1 v – 8 ln

u

v – 8 4 m

v

g

1 2 m Mc

h

1 2 3 1 Mc

z

4 4

o

MD MD* 8 3 1 v – 8 ln

u

v – 8 4 m

vy

1 2 m v – 8 ln

u

v – 8 4 m

vz

4 0.62 GeV 21_, gD*1_D1_g4 g_D*1 s D1s g4

o

MD MD*

y

22 3 1 v – 8 ln

u

v – 4 m

v

g

1 2 m Mc

h

1 1 6 1 Mc

z

4 4 2

o

MD MD* 1 3 1 v–₈ ln

u

v – 8 4 m

vy

1 2 4 m v – 8 ln

u

v – 8 4 m

vz

4 2 0.06 GeV 21_. (6.31)

Here we have used the numerical relation 1 Mc 4 2 v – 8 ln

u

v – 8 4 m

v

. (6.32)

The partial widths of the D*0

K D0g, D*1 K D1g and D*1 s K D1s g decays read

.

`

/

`

´

G

(

D*0 K D0g) 4 a 3 g 2 D*0_D0_gN k K N34 2 .36 keV , G

(

D*1 K D1g) 4 a 3 g 2 D*1_D1_gN k K N34 0.02 keV , G

(

D*1 s K D1s g) 4 a 3 g 2 D*1 s D1s gN k K N34 0 .02 keV . (6.33)