Semileptonic decays of Bmeson transition intoD-wave charmed meson doublets

Long-Fei Gan* and Ming-Qiu Huang

Department of Physics, National University of Defense Technology, Hunan 410073, China(Received 14 January 2009; published 20 February 2009)

We use QCD sum rules to estimate the leading-order universal form factors describing the semileptonic

B decay into orbital excitedD-wave charmed doublets, including the (1�, 2�) states (D�1,D

02) and the (2

�,3�) states (D2, D

�3). The decay rates we predict are �B!D�

1‘ �� ¼ �B!D0

2‘ �� ¼ 2:4� 10�18 GeV, �B!D2‘ �� ¼

6:2� 10�17 GeV, and �B!D�3‘ �� ¼ 8:6� 10�17 GeV. The branching ratios are BðB ! D�

1‘ ��Þ ¼ BðB !D0

2‘ ��Þ ¼ 6:0� 10�6, BðB ! D2‘ ��Þ ¼ 1:5� 10�4, and BðB ! D�3‘ ��Þ ¼ 2:1� 10�4, respectively.

DOI: 10.1103/PhysRevD.79.034025 PACS numbers: 14.40.�n, 11.55.Hx, 12.38.Lg, 12.39.Hg

I. INTRODUCTION

Higher excitations thanDð�Þ play an important role in theunderstanding of semileptonic B decays. Knowledge ofthese processes is important to reduce the uncertainties ofthe measurements on other semileptonic B decays, andthus the determination of the Cabibbo-Kobayashi-Maskawa matrix elements, such as jVcbj. Theoretically,the semileptonic decay processes are described by someform factors. The challenge for theory is the calculation ofthese decay form factors. Fortunately, the heavy quarkeffective theory (HQET) [1], with an expansion in termsof 1=mQ for hadrons containing a single heavy quark,

provides a systematic method for investigating such pro-cesses. In HQET the approximate symmetries allow one toorganize the spectrum of heavy mesons according to parityP and total angular momentum sl of the light degree offreedom. Coupling the spin of the light degrees of freedomsl with the spin of a heavy quark sQ ¼ 1=2 yields a doubletof meson states with a total spin s ¼ sl � 1=2. Forcharmed mesons, the lowest lying states ð0�; 1�Þ doubletðD;D�Þ are S-wave states with the spin of light degreessl ¼ 1=2. The P-wave excitation corresponds to two seriesof states, one is the sl ¼ 1=2 series, the ð0þ; 1þÞ doubletðD�

0; D01Þ; the other is the sl ¼ 3=2 series, the ð1þ; 2þÞ

doublet ðD1; D�2Þ. For D-wave states, those are ð1�; 2�Þ

and ð2�; 3�Þ doublets (ðD�1; D

02Þ and ðD2; D

�3Þ), correspond-

ing to the spin of light degrees of freedom sl ¼ 3=2 andsl ¼ 5=2. The early study of the heavy-light mesons can befound in Ref. [2]. The S-wave and P-wave charmed stateshave been observed so far. The properties of these stateshave been extensively studied using different approachesduring the past few years, including masses [3,4], decayconstants [5–7], and decay widths [8–11]. For the D-wavecharmed mesons, their properties were investigated withthe potential model [10] and QCD sum rules [12].

Semileptonic B decay into an excited heavy meson hasbeen observed in experiments [13,14]. Recently, BABARhas measured semileptonic B decays into orbitally excitedcharmed mesons D1ð2420Þ and D�

2ð2460Þ [15]. They also

reported two newDs statesDsJð2860Þ andDsJð2690Þ in theDK channel, which may fit in the D-wave charm-strangedoublets [16]. A similar state DsJð2715Þ has also beenobserved by Belle [17]. It is expected that the nonstrangeD-wave charmed mesons will be found, and the measure-ments of the semileptonic B decays into these states be-come available in the near future. To this end we study thepredictions of HQET for semileptonic B decays toD-wavecharmed mesons.The semileptonic decay rate of a Bmeson transition into

an charmed meson is determined by the correspondingmatrix elements of the weak axial-vector and vector cur-rents. In the heavy quark limit these elements are de-scribed, respectively, by one universal Isgur-Wisefunction at the leading order of heavy quark expansion[18]. The universal Isgur-Wise function is a nonperturba-tive parameter. It must be calculated in some nonperturba-tive approaches. The main theoretical approaches are QCDsum rules [19], constituent quark models, and lattice QCD.The investigations of semileptonic B decays into charmedmesons can be found in Refs. [5,18,20–22] with differentmethods. In this work, we estimate the leading-order Isgur-Wise functions describing the decays B ! ðD�

1; D02ޑ �� and

B ! ðD2; D�3Þ‘ �� and give a prediction for the widths of the

decays.The remainder of this paper is organized as follows. In

Sec. II we present the formulas of weak current matrixelements and decay rates. In Sec. III we give the relevantsum rules for two-point correlators, and then deduce thethree-point sum rules for the Isgur-Wise functions.Section IV is devoted to numerical results and discussions.

II. ANALYTIC FORMULATIONS FORSEMILEPTONIC DECAYAMPLITUDESB ! ðD�

1; D02Þ‘ �� AND B ! ðD2; D

�3ޑ ��

The heavy-light meson doublets can be expressed con-veniently by effective operators [23]. For the ground dou-blet, the operator is

Ha ¼ 1þ v62

½D���

� �D�5�: (1)*lfgan@nudt.edu.cn

PHYSICAL REVIEW D 79, 034025 (2009)

1550-7998=2009=79(3)=034025(7) 034025-1 � 2009 The American Physical Society

The effective operators describing the meson doubletsDð1�; 2�Þ and Dð2�; 3�Þ are given by

X� ¼ 1þ v62

��D

0��2 �5�� �D�

1�

ffiffiffi3

2

s �g�� � 1

3��ð�� þ v�Þ

��;

(2)

and

H�� ¼ 1þv62

�D����

3 �� �ffiffiffi3

5

s�5D

��2

��g��g�� �

��

5g��ð�� �v�Þ���

5g�� ð�� �v�Þ

��:

(3)

In these operators, D��, D, D0��

2 , D�1�, D

����3 , and D��

2

separately represent annihilation operators of the Q �q me-sons with appropriate quantum numbers and v6 ¼ v � �, vis the heavy meson velocity. The theoretical description ofsemileptonic decays involves the matrix elements of vectorand axial-vector currents (V� ¼ �c��b and A� ¼ �c���5b)between B mesons and excited D mesons. For the pro-cesses B ! ðD�

1; D02Þ‘ �� and B ! ðD2; D

�3ޑ ��, these matrix

elements can be parametrized through applying the traceformalism as follows [23]:

hD�1ðv0; "ÞjðV � AÞ�jBðvÞi ¼

ffiffiffi3

2

s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimBmD�

1

p�1ðyÞ

�"� � v

�v� � yþ 2

3v0�

�� i

1� y

3�����"��v0

�v�

�; (4)

hD02ðv0; "ÞjðV � AÞ�jBðvÞi ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

mBmD02

p�1ðyÞ"��v½g��ðy� 1Þ � v�v0� þ i�����v0

�v��; (5)

and

hD2ðv0; "ÞjðV � AÞ�jBðvÞi ¼ffiffiffi5

3

s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimBmD2

p�2ðyÞ"���v�

�2ð1� y2Þ

5g�� � v�v� þ 2y� 3

5v�v0� þ i

2ð1þ yÞ5

���vv0

�;

(6)

hD�3ðv0; "ÞjðV � AÞ�jBðvÞi ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

mBmD�3

p�2ðyÞ"���v�v�½g�ð1þ yÞ � vv0� þ i���vv

0��; (7)

where ðV � AÞ� ¼ �c��ð1� �5Þb is the weak current, y ¼ v � v0 and �1ðyÞ, �2ðyÞ are the universal form factors, and "��,"���, "

��� are the polarization tensors of these mesons. The differential decay rates are calculated by making use of the

formulas (4)–(7) given above:

d�

dyðB ! D�

1‘ ��Þ ¼G2

FV2cbm

2Bm

3D�

1

72�3ð�1ðyÞÞ2ðy� 1Þ5=2ðyþ 1Þ3=2½ð1þ r21Þð2yþ 1Þ � 2r1ðy2 þ yþ 1Þ�; (8)

d�

dyðB ! D0

2‘ ��Þ ¼G2

FV2cbm

2Bm

3D0

2

72�3ð�1ðyÞÞ2ðy� 1Þ5=2ðyþ 1Þ3=2½ð1þ r22Þð4y� 1Þ � 2r2ð3y2 � yþ 1Þ�; (9)

d�

dyðB ! D2‘ ��Þ ¼

G2FV

2cbm

2Bm

3D2

360�3ð�2ðyÞÞ2ðy� 1Þ5=2ðyþ 1Þ7=2½ð1þ r23Þð7y� 3Þ � 2r3ð4y2 � 3yþ 3Þ�; (10)

d�

dyðB ! D�

3‘ ��Þ ¼G2

FV2cbm

2Bm

3D�

3

360�3ð�2ðyÞÞ2ðy� 1Þ5=2ðyþ 1Þ7=2½ð1þ r24Þð11yþ 3Þ � 2r4ð8y2 þ 3yþ 3Þ�; (11)

with ri ¼ mDi

mB(Di ¼ D�

1; D02; D2; D

�3 for i ¼ 1; 2; 3; 4). In the equations above, we have presented the decay rates of B

semileptonic decay processes B ! ðD�1; D

02Þ‘ �� and B ! ðD2; D

�3Þ‘ �� in terms of the universal form factors �1ðyÞ and �2ðyÞ,

respectively. The only unknown factors in these equations are �1ðyÞ and �2ðyÞ, which need to be determined bynonperturbative methods.

LONG-FEI GAN AND MING-QIU HUANG PHYSICAL REVIEW D 79, 034025 (2009)

034025-2

III. SUM RULES FOR ISGUR-WISE FUNCTIONS

In the calculation of Isgur-Wise functions in HQET bymeans of QCD sum rule, the interpolating currents arepotentially important. In Ref. [4], two series of interpolat-ing currents with nice properties were proposed:

Jy�1...�j

j;P;i ¼ �hvðxÞ�f�1...�jgj;P;i ðDxtÞqðxÞ (12)

or

J0y�1...�j

j;P;i ¼ �hvðxÞ�f�1...�jgj;P;i ðDxtÞð�iÞ 6DxtqðxÞ; (13)

where i ¼ 1; 2 corresponding to two series of doublets of

the spin-parity ½jð�1Þjþ1; ðjþ 1Þð�1Þjþ1� and ½jð�1Þj ; ðjþ

1Þð�1Þj�, respectively. Dt� ¼ D� � v�ðv �DÞ is the trans-

verse component of the covariant derivative with respect tothe velocity of the meson and

�f�1...�jgðDxtÞ ¼ symmetrizef��1...�jðDxtÞ� 1

3g�1�2t gt�0

1�02��0

1�02�3����jg (14)

with the transverse metric g��t ¼ g�� � v�v�. For the

doublets of spin-parity ½jð�1Þjþ1; ðjþ 1Þð�1Þjþ1� and

½jð�1Þj ; ðjþ 1Þð�1Þj�, the expressions for ��1...�jðDxtÞ havebeen explicitly given in [4] as

�ðDxtÞ ¼8<:

ffiffiffiffiffiffiffiffi2jþ12jþ2

q�5ð�iÞjD�2

xt � � �D�jxt ðD�1

xt � j2jþ1�

�1t 6DxtÞ; for jð�1Þjþ1

1ffiffi2

p ��1t ð�iÞjD�2

xt � � �D�jxt ; for ðjþ 1Þð�1Þjþ1

�ðDxtÞ ¼8<:

1ffiffi2

p �5ð�iÞj��1t D�2

xt � � �D�jþ1xt ; for ðjþ 1Þð�1Þjffiffiffiffiffiffiffiffi

2jþ12jþ2

qð�iÞjD�2

xt � � �D�jxt ðD�1

xt � j2jþ1�

�1t 6DxtÞ; for jð�1Þj ;

where �t� ¼ �� � v6 v� is the transverse component of ��

with respect to the heavy quark velocity.For the D-wave meson doublets with sl ¼ 3

2� and sl ¼

52�, where j ¼ 1 and j ¼ 2, the currents are given by the

following expressions:

Jy�1;�;3=2 ¼ �iffiffi34

q�hvðD�

t � 13�

�t 6DtÞq; (15)

Jy��2;�;3=2 ¼ �i1ffiffiffi2

p T��;�� �hv�5�t�Dt�q; (16)

and

Jy��2;�;5=2 ¼ �ffiffi56

qT��;�� �hv�5ðDt�Dt� � 2

5Dt��t� 6DtÞq;(17)

Jy��3;�;5=2 ¼ � 1ffiffiffi2

p T��;��� �hv�t�Dt�Dt�q; (18)

which correspond to Eq. (12), and corresponding toEq. (13) are

Jy�1;�;3=2 ¼ �ffiffi34

q�hvðD�

t � 13�

�t 6DtÞ 6Dtq; (19)

Jy��2;�;3=2 ¼ � 1ffiffiffi2

p T��;�� �hv�5�t�Dt� 6Dtq; (20)

and

Jy��2;�;5=2 ¼�ffiffi56

qT��;�� �hv�5ðDt�Dt� � 2

5Dt��t� 6DtÞð�iÞ 6Dtq;

(21)

Jy��3;�;5=2 ¼ � 1ffiffiffi2

p T��;��� �hv�t�Dt�Dt�ð�iÞ 6Dtq; (22)

where hv is the generic velocity-dependent heavy quarkeffective field in HQET and q denotes the light quark field.The tensors T��;�� and T��;��� are used to symmetrizeindices and are given by [4]

T��;�� ¼ 12ðg��t g��t þ g��t g��t Þ � 1

3g��t g��

t ; (23)

T��;��� ¼ 1

6ðg��t g��t g�t þ g

��t g��t g�t þ g��t g

��t g�t þ g��t g��t g

�t þ g��t g��t g

�t þ g��t g

��t g�t Þ

� 1

15ðg��t g��

t g�t þ g��t g��t g�t þ g��t g��t g�t þ g�t g��

t g��t þ g�t g��t g��t þ g�t g��t g��t þ g�t g��

t g��t

þ g�t g��t g��t þ g�t g��t g��t Þ: (24)

SEMILEPTONIC DECAYS OF B MESON TRANSITION . . . PHYSICAL REVIEW D 79, 034025 (2009)

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Usually the currents with derivatives of the lowest order(12) are used in the QCD sum rule approach. However,currents with derivatives of one order higher (13) are alsoused in some conditions because in the nonrelativisticquark model there is a corresponding relation betweenthe orbital angular momenta and the orders of derivativesin the space wave functions. As for the orbital D-wavemesons, which corresponding to derivatives of order two, itis reasonable to use the currents (17)–(20).

These currents have nice properties, they have nonvan-ishing projection only to the corresponding states of theHQET in the mQ ! 1 limit, without mixing with states of

the same quantum number but different sl. Thus we candefine one-particle-current couplings as follows:

JP ¼ 1�: hD�1ðv; "ÞjJ�j0i ¼ f1

ffiffiffiffiffiffiffiffiffimD�

1

p"��; (25)

JP ¼ 2�: hD02ðv; "ÞjJ��j0i ¼ f02

ffiffiffiffiffiffiffiffiffimD0

2

p"���; (26)

JP ¼ 2�: hD2ðv; "ÞjJ��j0i ¼ f2ffiffiffiffiffiffiffiffiffimD2

p"���; (27)

JP ¼ 3�: hD�3ðv; "ÞjJ��j0i ¼ f3

ffiffiffiffiffiffiffiffiffimD�

3

p"���: (28)

The couplings fi are low-energy parameters which aredetermined by the dynamics of the light degree of freedom.Since the pairs ðf1; f02Þ and ðf2; f3Þ are related by the spinsymmetry, we will consider f1 and f2 hereafter. The decayconstants fi can be estimated from two-point sum rules,therefore we list the sum rules after the Borel transforma-tion. For the ground-state heavy mesons, the sum rule forthe correlator of two heavy-light currents is well known. Itis [20]

f2�;1=2e�2 ���;1=2=T ¼ 3

16�2

Z !c0

0!2e�!=Td!

� 1

2h �qqi

�1� m2

0

4T2

�: (29)

For the sPl ¼ 32� doublet, when the currents (19) and (20)

are used, the corresponding sum rule is

f2�;3=2e�2 ���;3=2=T ¼ 1

28�2

Z !c1

0!6e�!=Td!

� 5

3� 28

Z !c1

0!2e�!=Td!

��s

�GG

�:

(30)

For the sPl ¼ 52� doublet, when the currents (17) and (18)

are used, the corresponding sum rule is

f2�;5=2e�2 ���;5=2=T ¼ 1

5� 27�2

Z !c2

0!6e�!=Td!

� 5

3� 26

Z !c2

0!2e�!=Td!

��s

�GG

�:

(31)

As we have just mentioned, for the amplitudes of thesemileptonic decays into excited states in the infinite masslimit, the only unknown quantities in (8)–(11) are theuniversal functions �1ðyÞ and �2ðyÞ. In Ref. [24], theform factors �1ðyÞ and �2ðyÞ were estimated throughQCD sum rule by using currents with derivatives of lowerorder, (15)–(18). Considering that the corresponding rela-tion between the orbital angular momentum and the orderof the derivative mentioned above, we use the currents (19)and (20) instead of (15) and (16) for the ðD�

1; D02Þ doublet.

As for the ðD2; D�3Þ doublet, we also use the currents (17)

and (18).In order to calculate these two form factors by QCD sum

rules, we study the analytic properties of three-point cor-relators:

i2Z

d4xd4zeiðk0�x�k�zÞh0jT½J�1;�ðxÞJ�ðv;v0ÞV;A ð0ÞJy0;�ðzÞj0i

¼ �ð!;!0; yÞL��V;A; (32)

i2Z

d4xd4zeiðk0�x�k�zÞh0jT½J��2;�ðxÞJ�ðv;v0ÞV;A ð0ÞJy0;�ðzÞj0i

¼ �0ð!;!0; yÞL���V;A ; (33)

where J�ðv;v0ÞV ¼ hðv0Þ��hðvÞ and J�ðv;v0Þ

A ¼hðv0Þ���5hðvÞ. The variables kð¼ P�mbvÞ andk0ð¼ P0 �mcv

0Þ denote residual ‘‘off-shell’’ momenta ofthe initial and final meson states, respectively. For heavyquarks in bound states they are typically of order�QCD and

remain finite in the heavy quark limit. �ð!;!0; yÞ and�0ð!;!0; yÞ are analytic functions in the ‘‘off-shell’’ ener-gies ! ¼ 2v � k and !0 ¼ 2v0 � k0 with discontinuities forpositive values of these variables. They also depend on thevelocity transfer y ¼ v � v0, which is fixed in a physicalregion. LV;A are Lorentz structures.

Following the standard QCD sum rule procedure, thecalculations of �ð!;!0; yÞ and �0ð!;!0; yÞ are straightfor-ward. First, we saturate Eqs. (32) and (33) with physicalintermediate states in HQET and find the hadronic repre-sentations of the correlators as follows:

�hadronð!;!0; yÞ ¼ f�;1=2f�;jl�iðyÞð2 ���;1=2 �!� i"Þð2 ���;jl �!0 � i"Þþ higher resonances; (34)

where f�;jl are the decay constants defined in Eqs. (25) and

(27), ���;jl ¼ m�;jl �mQ. Second, the functions can be

approximated by a perturbative calculation supplementedby nonperturbative power corrections proportional to thevacuum condensates which are treated as phenomenologi-cal parameters. The perturbative contribution can be rep-resented by a double dispersion integral in � and �0 pluspossible subtraction terms. So the theoretical expressionfor the correlator has the form

LONG-FEI GAN AND MING-QIU HUANG PHYSICAL REVIEW D 79, 034025 (2009)

034025-4

�theoð!;!0; yÞ ’Z

d�d�0 pertð�; �0; yÞð��!� i"Þð�0 �!0 � i"Þ

þ subtractionsþ �condð!;!0; yÞ: (35)

The perturbative part of the spectral density can be calcu-lated straightforward. Confining us to the leading order ofperturbation, the perturbative spectral densities of the twosum rules for �1ðyÞ and �2ðyÞ are

pertð�; �0; yÞ ¼ 3

28�2

1

ðyþ 1Þ3=2ðy� 1Þ5=2� �0½ð3�2 � ð1þ 2yÞð2��0 � �02Þ���ð�Þ�ð�0Þ�ð2y��0 � �2 � �02Þ; (36)

and

pertð�; �0; yÞ ¼ 3

28�2

1

ðyþ 1Þ7=2ðy� 1Þ5=2� ½ð5�� 12y�0 þ 3�0Þ�2 þ ð3�þ �0Þð2y2� 2yþ 1Þ�02��ð�Þ�ð�0Þ��ð2y��0 � �2 � �02Þ: (37)

Following the arguments in Refs. [5,25], the perturbativeand the hadronic spectral densities cannot be locally dualto each other, the necessary way to restore duality is tointegrate the spectral densities over the ‘‘off-diagonal’’variable �� ¼ �� �0, keeping the ‘‘diagonal’’ variable

�þ ¼ �þ�02 fixed. It is in �þ that the quark-hadron duality

is assumed for the integrated spectral densities. The inte-gration region can be expressed in terms of the variables�� and �þ and we choose the triangular region defined by

the bounds: 0 � �þ � !c, �2ffiffiffiffiffiffiffiy�1yþ1

q�þ � �� � 2

ffiffiffiffiffiffiffiy�1yþ1

q�þ.

As discussed in Refs. [5,25], the upper limit !c for �þ in

the region 12 ½ðyþ 1Þ � ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

y2 � 1p �!c0 � !c � 1

2 �ð!c0 þ!c2Þ is reasonable. A double Borel transformationin ! and !0 is performed on both sides of the sum rules, inwhich for simplicity we take the Borel parameters equal[5,20,24]: T1 ¼ T2 ¼ 2T. In the calculation, we have con-sidered the operators of dimension D � 5 in OPE. Afteradding the nonperturbative parts, we obtain the sum rulesfor �1 and �2 as follows:

�1ðyÞf�;1=2f�;3=2e�ð ���;1=2þ ���;3=2Þ=T

¼ 1

24�2

1

ð1þ yÞ3Z !0

c

0d�þe�ð�þ=TÞ�4þ � T

3� 25

� 2yþ 3

ðyþ 1Þ2��s

�GG

�; (38)

�2ðyÞf�;1=2f�;5=2e�ð ���;1=2þ ���;5=2Þ=T

¼ 3

8�2

1

ð1þ yÞ4Z !c

0d�þe�ð�þ=TÞ�4þ � T

3� 24

� 1

ðyþ 1Þ3��s

�GG

�: (39)

We also derive the sum rule for �2 by using the currents(21) and (22), which appears to be

�2ðyÞf�;1=2f�;5=2e�ð ���;1=2þ ���;5=2Þ=T

¼ 21

5� 24�2

1

ð1þ yÞ4Z !c

0d�þe�ð�þ=TÞ�5þ

þ T2

3� 244y� 25

ðyþ 1Þ3��s

�GG

�: (40)

IV. NUMERICAL RESULTS AND DISCUSSIONS

We now evaluate the sum rules numerically. For theQCD parameters entering the theoretical expressions, wetake the standard values: h �qqi ¼ �ð0:24Þ3 GeV3,h�sGGi ¼ 0:04 GeV4, and m2

0 ¼ 0:8 GeV2. In the nu-

merical calculations, we take 2.83 GeV [2,10] for themass of the sl ¼ 5=2 doublet and 2.78 GeV for the sl ¼3=2 doublet. For mass of initial B meson, we use mB ¼5:279 GeV [26].In order to obtain information of �1ðyÞ and �2ðyÞ with

less systematic uncertainties in the calculation, we dividethe three-point sum rules by the square roots of relevanttwo-point sum rules, as many authors did [5,20,24], toreduce the number of input parameters and improve stabil-ities. Then we obtain expressions for the �1ðyÞ and �2ðyÞ asfunctions of the Borel parameter T and the continuumthresholds. Imposing usual criteria for the upper and lowerbounds of the Borel parameter, we found they have acommon sum rule ‘‘window’’: 0:7 GeV< T < 1:5 GeV,which overlaps with those of two-point sum rules (29)–(31) (see Fig. 1). Notice that the Borel parameter in the sumrules for three-point correlators is twice the Borel parame-ter in the sum rules for the two-point correlators. In theevaluation we have taken 2:0 GeV<!c0 < 2:4 GeV[5,20], 2:8 GeV<!c1 < 3:2 GeV, and 3:2 GeV<!c2 <3:6 GeV. The regions of these continuum thresholds arefixed by analyzing the corresponding two-point sum rules.According to the discussion in Sec. III, we can fix !0

c and!c in the regions 2:3 GeV<!0

c < 2:6 GeV and2:5 GeV<!c < 2:7 GeV. The results are showed inFig. 2. The resulting curves for �1ðyÞ and �2ðyÞ can beparametrized by the linear approximation

�1ðyÞ ¼ �1ð1Þ½1� 2�1ðy� 1Þ�;

�1ð1Þ ¼ 0:14� 0:03;

2�1 ¼ 0:13� 0:02;

(41)

SEMILEPTONIC DECAYS OF B MESON TRANSITION . . . PHYSICAL REVIEW D 79, 034025 (2009)

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�2ðyÞ ¼ �2ð1Þ½1� 2�2ðy� 1Þ�;

�2ð1Þ ¼ 0:57� 0:09;

2�2 ¼ 0:78� 0:13:

(42)

The errors mainly come from the uncertainty due to !c’sand T. It is difficult to estimate these systematic errorswhich are brought in by the quark-hadron duality. The

maximal values of y are yD�

1max ¼ y

D02

max ¼ ð1þ r21;2Þ=2r1;2 �1:213 and yD2

max ¼ yD�

3max ¼ ð1þ r23;4Þ=2r3;4 � 1:201. By us-

ing the parameters Vcb ¼ 0:04, GF ¼ 1:166�10�5 GeV�2, we get the semileptonic decay rates of B !ðD�

1; D02Þ‘ �� and B ! ðD2; D

�3Þ‘ ��. Consider that �B ¼

1:638 ps [26], we get the branching ratios, respectively.All these results are listed in Table I.

Because of the large background from B ! Dð�Þ‘ �� de-cays, there is no experimental data available so far. As wecan see from Table I, the rates of semileptonic B decay intothe sPl ¼ 3

2� doublet are tiny and our results are larger than

those predicted by Ref. [24] in the B to sPl ¼ 52� charmed

doublet channels. The difference comes because the way inwhich we choose the parameters is different from theirs.They chose the parameters according to other theoreticalapproaches. In contrast, we choose the parameters follow-ing the way of Ref. [5]. In addition, we also estimate theuniversal form factor �2ðyÞ with the sum rule (40) and weget almost the same result as (42). When trying to estimatethe �1ðyÞ by using the currents (15) and (16), we find thatafter the quark-hadron duality are assumed the integralover the perturbative spectral density becomes zero. Asfor the P-wave and the F-wave mesons, similar results canbe obtained after the calculations above have been care-fully repeated.The semileptonic and leptonic B decay rate is about

10.9% of the total B decay rate, in which the S-wavecharmed mesons D and D� contribute about 8.65% [26]and the P-wave charmed mesons contribute about 0.9%[20]. Our results then suggest that the D-wave charmedmesons contribute about 0.04% of the total B decay rate.Sum up the branching ratios of these semileptonic B decay

1 1.05 1.1 1.15 1.2 1.25 1.3 1.350

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

y

Isgu

r−W

ise

Fun

ctio

ns

T=1.0GeV

τ2(y)

τ1(y)

FIG. 2 (color online). Prediction for the Isgur-Wise functions�1ðyÞ and �2ðyÞ.

TABLE I. Predictions for the decay widths and branchingratios.

Decay

mode

Decay width

� (GeV)

Branching

ratio

Branching ratio

of Ref. [24]

B ! D�1‘ �� 2:4� 10�18 6:0� 10�6

B ! D02‘ �� 2:4� 10�18 6:0� 10�6

B ! D2‘ �� 6:2� 10�17 1:5� 10�4 1� 10�5

B ! D�3‘ �� 8:6� 10�17 2:1� 10�4 1� 10�5

0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50.05

0.1

0.15

0.2

T ( GeV )

τ 1(y=

1)ω

c0 = 2.0 GeV

ωc1

= 3.2 GeV

( a )

ω’c =2.6 GeV

ω’c =2.5 GeV

ω’c =2.4 GeV

0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50.3

0.4

0.5

0.6

0.7

0.8

0.9

T ( GeV )

τ 2(y=

1)

ωc0

=2.0GeVω

c2=3.6GeV

( b )

ωc=2.7 GeV

ωc=2.6 GeV

ωc=2.5 GeV

FIG. 1. Dependence of �1ðyÞ and �2ðyÞ on Borel parameter T at y ¼ 1.

LONG-FEI GAN AND MING-QIU HUANG PHYSICAL REVIEW D 79, 034025 (2009)

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processes, the eight lightest charmed mesons contributeabout 9.59% of the B decay rate. Therefore, semileptonicdecays into higher excited states and nonresonant multi-body channels should be about 1.31% of the B decay rate.Whatsoever, our result is just a leading-order estimate ofthe contribution of the D-wave charmed mesons channelsto the semileptonic B decay.

In summary, we estimate the leading-order universalform factors describing the B meson of ground-state tran-sition into orbital excitedD-wave charmed resonances, theð1�; 2�Þ states ðD�

1; D02Þ, which belong to the sPl ¼ 3

2�

heavy quark doublet and the ð2�; 3�Þ states ðD2; D�3Þ,

which belong to the sPl ¼ 52� heavy quark doublet, by use

of QCD sum rules within the framework of HQET. Thesemileptonic decay widths as well as the branching ratioswe get are shown in Table I. The predictions are larger thanthose predicted by Ref. [24]. This needs future experiments

for clarification. We also prove that when sPl ¼ 52� the

interpolating currents (12) and (13) proposed in Ref. [4]are really equivalent. It is worth noting that in the estimateof the semileptonic B decay form factors when the currents(12) with quantum numbers of light degree of freedomsPl ¼ 1

2þ, 32

�, 52þ are used for the excited charmed mesons,

we find the perturbative contributions vanish after thequark-hadron duality are assumed. In this case we shoulduse the currents (13) which contain derivatives of one orderhigher.

ACKNOWLEDGMENTS

L. F. Gan thanks M. Zhong for useful discussions. Thiswork was supported in part by the National NaturalScience Foundation of China under ContractNo. 10675167.

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