1/mQ corrections to the Bethe-Salpeter equation for ΛQ in the diquark picture

Guo, Xin-Heng; Thomas, Anthony William; Williams, Anthony Gordon

Review D, 2000; 61(11):116015

© 2000 American Physical Society

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27

^th

March 2013

1 Õ m

_Q

corrections to the Bethe-Salpeter equation for ⌳

Q

in the diquark picture

X.-H. Guo*

Department of Physics and Mathematical Physics, and Special Research Center for the Subatomic Structure of Matter, University of Adelaide, Adelaide SA 5005, Australia

and Institute of High Energy Physics, Academia Sinica, Beijing 100039, China A. W. Thomas^†

Department of Physics and Mathematical Physics, and Special Research Center for the Subatomic Structure of Matter, University of Adelaide, Adelaide SA 5005, Australia

A. G. Williams^‡

Department of Physics and Mathematical Physics, and Special Research Center for the Subatomic Structure of Matter, University of Adelaide, Adelaide SA 5005, Australia

and Department of Physics and SCRI, Florida State University, Tallahassee, Florida 32306-4052 共Received 30 November 1999; published 11 May 2000兲

Corrections of order 1/m_Q (Q⫽b or c) to the Bethe-Salpeter共BS兲 equation for⌳Q are analyzed on the assumption that the heavy baryon⌳Qis composed of a heavy quark and a scalar, light diquark. It is found that in addition to the one BS scalar function in the limit mQ→⬁, two more scalar functions are needed at the order 1/mQ. These can be related to the BS scalar function in the leading order in our model. The six form factors for the weak transition⌳b→⌳care expressed in terms of these wave functions and the results are consistent with HQET to order 1/m_Q. Assuming the kernel for the BS equation in the limit m_Q→⬁to consist of a scalar confinement term and a one-gluon-exchange term we obtain numerical solutions for the BS wave functions, and hence for the⌳b→⌳cform factors to order 1/mQ. Predictions are given for the differential and total decay widths for⌳b→⌳cl¯ , and also for the nonleptonic decay widths for␯ ⌳b→⌳cplus a pseudoscalar or vector meson, with QCD corrections being also included.

PACS number共s兲: 11.10.St, 12.39.Hg, 14.20.Lq, 14.20.Mr I. INTRODUCTION

Heavy flavor physics provides an important area within which to study many important physical phenomena in par- ticle physics, such as the structure and interactions inside heavy hadrons, the heavy hadron decay mechanism, and the plausibility of present nonperturbative QCD models. Heavy baryons have been studied much less than heavy mesons, both experimentally and theoretically. However, more ex- perimental data for heavy baryons is being accumulated 关1–6兴and we expect that the experimental situation for them will continue to improve in the near future. On the theoreti- cal side, heavy quark effective theory共HQET兲 关7兴provides a systematic way to study physical processes involving heavy hadrons. With the aid of HQET heavy hadron physics is simplified when m_QⰇ⌳QCD. In order to get the complete physics, HQET is usually combined with some nonperturba- tive QCD models which deal with dynamics inside heavy hadrons.

As a formally exact equation to describe the hadronic bound state, the Bethe-Salpeter共BS兲equation is an effective method to deal with nonperturbative QCD effects. In fact, in combination with HQET, the BS equation has already been applied to the heavy meson system 关8–10兴. The Isgur-Wise

function was calculated 关8,10兴 and 1/m_Q corrections were also considered 关8兴. In previous work 关11–13兴, we estab- lished the BS equations in the heavy quark limit (m_Q→⬁) for the heavy baryons⌳Q and␻Q

(*⁾ _共where␻⫽⌶, ⌺ or ⍀ and Q⫽b or c). These were assumed to be composed of a heavy quark, Q, and a light scalar and axial-vector diquark, respectively. We found that in the limit m_Q→⬁, the BS equations for these heavy baryons are greatly simplified. For example, only one BS scalar function is needed for ⌳Q in this limit. By assuming that the BS equation’s kernel consists of a scalar confinement term and a one-gluon-exchange term we gave numerical solutions for the BS wave functions in the covariant instantaneous approximation, and consequently applied these solutions to calculate the Isgur-Wise functions for the weak transitions⌳b→⌳c and⍀b

(*⁾_→⍀c⁽*⁾.

In reality, the heavy quark mass is not infinite. Therefore, in order to give more exact phenomenological predictions we have to include 1/m_Q corrections, especially 1/m_c correc- tions. It is the purpose of the present paper to analyze the 1/m_Qcorrections to the BS equation for⌳Qand to give some phenomenological predictions for its weak decays. As in the previous work关11–14兴, we will still assume that⌳Qis com- posed of a heavy quark and a light, scalar diquark. In this picture, the three body system is simplified to a two body system.

In the framework of HQET, the eigenstate of HQET La- grangian 兩⌳Q典HQET has 0^⫹ light degrees of freedom. This leads to only one Isgur-Wise function␰⁽␻⁾⁽␻is the velocity transfer兲for⌳b→⌳cin the leading order of the 1/m_Qexpan-

*Email address: xhguo@physics.adelaide.edu.au

†Email address: athomas@physics.adelaide.edu.au

‡Email address: awilliam@physics.adelaide.edu.au

0556-2821/2000/61共11兲/116015共11兲/$15.00 61 116015-1 ©2000 The American Physical Society

sion 关15–20兴. When 1/m_Q corrections are included, another form factor in HQET and an unknown flavor-independent parameter which is defined as the mass difference m_⌳

⫺m_Q in the heavy quark limit are involved关19兴. This pro-Q

vides some relations among the six form factors for ⌳b

→⌳c to order 1/m_Q. Consequently, if one form factor is determined, the other five form factors can be obtained.

Here we extend our previous work to solve the BS equa- tion for⌳Qto order 1/m_Q, in combination with the results of HQET. It can be shown that two BS scalar functions are needed at the order 1/m_Q, in addition to the one scalar func- tion in the limit m_Q→⬁. The relationship among these three scalar functions can be found. Therefore, our numerical re- sults for the BS wave function in the order m_Q→⬁ can be applied directly to obtain the 1/m_Q corrections to the form factors for the weak transition⌳b→⌳c. It can be shown that the relations among all the six form factors for ⌳b→⌳c in the BS approach are consistent with those from HQET to order 1/m_Q. We also give phenomenological predictions for the differential and total decay widths for⌳b→⌳cl␯¯ , and for the nonleptonic decay widths for ⌳b→⌳c plus a pseudo- scalar or vector meson. Since the QCD corrections are com- parable with the 1/m_Q corrections, we also include QCD corrections in our predictions. Furthermore, we discuss the dependence of our results on the various input parameters in our model, and present the comparison of our results with those of other models.

The remainder of this paper is organized as follows. In Sec. II we discuss the BS equation for the heavy quark and light scalar diquark system to order 1/m_Q and introduce the two BS scalar functions appearing at this order. We also discuss the constraint on the form of the kernel. In Sec. III we express the six form factors for⌳b→⌳c in terms of the BS wave function. The consistency of our model with HQET is discussed. We also present numerical solutions for these form factors. In Sec. VI we apply the solutions for the ⌳b

→⌳cform factors, with QCD corrections being included, to the semileptonic decay ⌳b→⌳cl¯ , and the nonleptonic de-␯ cays⌳b→⌳c plus a pseudoscalar or vector meson. Finally, Sec. VI contains a summary and discussion.

II. THE BS EQUATION FOR⌳QTO 1ÕmQ

Based on the picture that⌳Qis a bound state of a heavy quark and a light, scalar diquark, its BS wave function is defined as关11兴

␹共x₁,x₂, P兲⫽具⁰兩T␺Q共x₁兲␸共x₂兲兩⌳Q共P兲典^, 共1兲 where␺Q(x₁) and␸^(x2) are the field operators for the heavy quark Q and the light, scalar diquark, respectively, P

⫽m_⌳

Qvis the total momentum of⌳Q andv is its velocity.

Let m_Q and m_D be the masses of the heavy quark and the light diquark in⌳Q, p be the relative momentum of the two constituents, and define ␭1⫽m_Q/(m_Q⫹m_D),␭2⫽m_D/(m_Q

⫹m_D). The BS wave function in momentum space is defined as

␹共x₁,x₂, P兲⫽e^{i PX}

冕

_共2^d␲^4p兲⁴e^{i px}␹P共p兲, 共2兲 where X⫽␭1x₁⫹␭2x₂is the coordinate of the center of mass and x⫽x₁⫺x₂. The momentum of the heavy quark is p₁

⫽␭1P⫹p and that of the diquark is p₂⫽⫺␭2P⫹p. ␹P( p) satisfies the following BS equation关21兴:

␹P共p兲⫽S_F共␭1P⫹p兲

冕

_共2^d4␲^q兲⁴K共P, p,q兲␹P共q兲

⫻S_D共⫺␭2P⫹p兲, 共3兲 where K( P, p,q) is the kernel, which is defined as the sum of all the two particle irreducible diagrams with respect to the heavy quark and the light diquark. For convenience, in the following we use the variables

p_l⬅v•p⫺␭2m_⌳

Q, p_t⬅p⫺共v•p兲v. 共4兲

It should be noted that p_land p_t are of the order⌳QCD. The mass of⌳Qcan be written in the following form with respect to the 1/m_Qexpansion共from HQET兲:

m_⌳

Q⫽m_Q⫹m_D⫹E₀⫹ 1

m_QE₁⫹O共1/m_Q²兲, 共5兲 where E₀ and E₁/m_Qare binding energies at the leading and first order in the 1/m_Qexpansion, respectively. m_D, E₀ and E₁ are independent of m_Q.

Since we are considering 1/m_Q corrections to the BS equation, we expand the heavy quark propagator S_F(␭1P

⫹p) to order 1/m_Q. We find S_F⫽S_0F⫹ 1

m_QS_1F, 共6兲 where S_0F is the propagator in the limit m_Q→⬁ 关11兴

S_0F⫽i 1⫹v”

2共p_l⫹E₀⫹m_D⫹i⑀兲, 共7兲 and

S_1F⫽i

冋

^{2共⫺共pEl⫹1⫹E0p⫹t2/2m兲共D1⫹⫹i⑀v”兲兲2}

⫹ p”t

2共p_l⫹E₀⫹m_D⫹i⑀兲⫺1⫺v”

4

册

^{. 共8兲}

It can be shown that the light diquark propagator to 1/m_Q still keeps its form in the limit m_Q→⬁,

S_D⫽ i

p_l²⫺W_p²⫹i⑀^{, 共9兲} where W_p⬅

冑

^pt

2⫹m_D² 共we have defined p_t²⬅⫺p_t•p_t).

Similarly to Eq.共6兲, we write␹P( p) and K( P, p,q) in the following form 共to order 1/m_Q):

␹P共p兲⫽␹0 P共p兲⫹ 1

m_Q␹1 P共p兲,

共10兲 K共P, p,q兲⫽K₀共P, p,q兲⫹ 1

m_QK₁共P, p,q兲, where␹1 P( p) and K₁( P, p,q) arise from 1/m_Q corrections.

As in our previous work, we assume the kernel contains a scalar confinement term and a one-gluon-exchange term.

Hence we have

⫺iK₀⫽I_丢IV₁⫹v_␮丢共p₂⫹p₂⬘兲^␮V₂,

共11兲

⫺iK₁⫽I_丢IV₃⫹␥_␮丢共p₂⫹p₂⬘兲^␮V₄,

wherev_␮in K₀appears because of the heavy quark symme- try.

Substituting Eqs.共6兲and共10兲into the BS equation共3兲we have the integral equations for␹0 P( p) and␹1 P( p)

␹0 P共p兲⫽S_0F共␭1P⫹p兲

冕

_共2^d␲^4q兲⁴K₀共P, p,q兲

⫻␹0 P共q兲S_D共⫺␭2P⫹p兲, 共12兲 and

␹1 P共p兲⫽S_0F共␭1P⫹p兲

冕

_共2^d␲^4q兲⁴K₁共P, p,q兲

⫻␹0 P共q兲S_D共⫺␭2P⫹p兲⫹S_1F共␭1P⫹p兲

⫻

冕

_共2^d4␲^q兲⁴K₀共P, p,q兲␹0 P共q兲S_D共⫺␭2P⫹p兲

⫹S_0F共␭1P⫹p兲

冕

_共2^d␲^4q兲⁴K₀共P, p,q兲

⫻␹1 P共q兲S_D共⫺␭2P⫹p兲. 共13兲 Equation 共12兲 is what we obtained in the limit m_Q→⬁, which together with Eq.共7兲gives

v”␹0 P共p兲⫽␹0 P共p兲, 共14兲 since v”v”⫽v²⫽1 and so v”S_0F⫽S_0F. Therefore, S_0F(␭1P

⫹p)␥␮␹0 P(q)⫽S_0F(␭1P⫹p)v_␮␹0 P(q) in the first term of Eq. 共13兲. So to order 1/m_Q, the Dirac matrix ␥_␮ ^{from the} one-gluon-exchange term in K₁( P, p,q) can still be replaced byv_␮.

We divide␹1 P( p) into two parts by defining

␹1 P共p兲⫽␹1 P⫹共p兲⫹␹1 P⫺共p兲, v”␹1 P⫾共p兲⫽⫾␹1 P⫾共p兲, 共15兲 i.e., ␹1 P⫹( p)⬅¹2关␹1 P( p)⫹v”␹1 P( p)兴 and␹1 P⫺( p)⬅¹2关␹1 P( p)

⫺v”␹1 P( p)兴. Multiplying Eq. 共13兲 with either (1⫹v”)/2 or

(1⫺v”)/2 and using Eqs.共7兲,共8兲,共9兲,共11兲and共14兲, we obtain the following integral equations for␹1 P⫹( p) and␹1 P⫺( p):

␹1 P⫹共p兲⫽⫺ 1

共p_l⫹E₀⫹m_D⫹i⑀兲共p_l²⫺W_p²⫹i⑀兲

⫻

冕

_共2^d4␲^q兲⁴K₀共P, p,q兲␹1 P⫹共q兲

⫺ 1

共p_l⫹E₀⫹m_D⫹i⑀兲共p_l²⫺W_p²⫹i⑀兲

⫻

冕

_共2^d4␲^q兲⁴

冋

^{K1共P, p,q兲⫹pl⫹pEt20/2⫹⫺mED1⫹i⑀}

⫻K₀共P, p,q兲

册

^{␹0 P共q兲, 共16兲}

␹1 P⫺共p兲⫽⫺ p”^t

2共p_l⫹E₀⫹m_D⫹i⑀兲共p_l²⫺W_p²⫹i⑀兲

⫻

冕

_共2^d␲^4q兲⁴K₀共P, p,q兲␹0 P共q兲. 共17兲 After writing down all the possible terms for␹0 P( p) and

␹1 P⫾( p), and considering the constraints on them, Eqs. 共14兲 and共15兲, we obtain that

␹0 P共p兲⫽␾0 P共p兲u_⌳

Q共v,s兲,

␹1 P⫹共p兲⫽␾1 P共p兲u_⌳

Q共v,s兲, 共18兲

␹1 P⫺共p兲⫽␾2 P共p兲p”tu_⌳

Q共v,s兲,

where␾0 P( p), ␾1 P( p) and␾2 P( p) are Lorentz scalar func- tions.

Substituting Eq. 共18兲 into Eqs. 共12兲,共16兲,共17兲 and using Eqs. 共7兲,共8兲,共9兲,共11兲and共14兲we have

␾0 P共p兲⫽⫺ 1

共p_l⫹E₀⫹m_D⫹i⑀兲共p_l²⫺W_p²⫹i⑀兲

⫻

冕

_共2^d␲^4q兲⁴K₀共P, p,q兲␾0 P共q兲, 共19兲

␾1 P共p兲⫽⫺ 1

共p_l⫹E₀⫹m_D⫹i⑀兲共p_l²⫺W_p²⫹i⑀兲

⫻

冕

_共2^d␲^4q兲⁴K₀共P, p,q兲␾1 P共q兲

⫺ 1

共p_l⫹E₀⫹m_D⫹i⑀兲共p_l²⫺W_p²⫹i⑀兲

⫻

冕

_共2^d␲^4q兲⁴

冋

^{K1共P, p,q兲⫹pl⫹pEt20/2⫹⫺mED1⫹i⑀}

⫻K₀共P, p,q兲

册

^{␾0 P共q兲, 共20兲}

and

␾2 P共p兲⫽⫺ 1

2共p_l⫹E₀⫹m_D⫹i⑀兲共p_l²⫺W²_p⫹i⑀兲

⫻

冕

_共2^d␲^4q兲⁴K₀共P, p,q兲␾0 P共q兲. 共21兲

Equations共19兲and共21兲lead to

␾2 P共p兲⫽1

2␾0 P共p兲. 共22兲

␾0 P( p) is the BS scalar function in the leading order of the 1/m_Qexpansion, which was calculated in关11兴. From Eq.

共22兲␾2 P( p) can be given in terms of␾0 P( p). The numerical solutions for ␾0 P( p) and ␾1 P( p) can be obtained by dis- cretizing the integration region into n pieces 共with n suffi- ciently large兲. In this way, the integral equations become matrix equations and the BS scalar functions ␾0 P( p) and

␾1 P( p) become n dimensional vectors. Thus␾0 P( p) is the solution of the eigenvalue equation (A⫺I)␾0⫽0, where A is an n⫻n matrix corresponding to the right hand side of Eq.

共19兲. In order to have a unique solution for the ground state, the rank of (A⫺I) should be n⫺1. From Eq.共20兲,␾1 P( p) is the solution of (A⫺I)␾1⫽B, where B is an n dimensional vector corresponding to the second integral term on the right hand side of Eq.共20兲. In order to have solutions for␾1 P( p), the rank of the augmented matrix (A⫺I,B) should be equal to that of (A⫺I), i.e., B can be expressed as linear combi- nation of the n⫺1 linearly independent columns in (A⫺I).

This is difficult to guarantee if B⫽0, since the way to divide (A⫺I) into n columns is arbitrary. Therefore, we demand the following condition in order to have solutions for␾1 P( p)

冕

_共2^d4␲^q兲⁴

冋

^{K1共P, p,q兲⫹pl⫹pEt20/2⫹⫺mED1⫹i⑀}

⫻K₀共P, p,q兲

册

^{␾0 P共q兲⫽0. 共23兲}

Equation共23兲is a constraint we impose on the O(1) and O(1/m_Q) parts of the BS equation kernel, K₁( P, p,q) and K₀( P, p,q), under which we have solutions for␾1 P( p). In fact, K₀( P, p,q) and K₁( P, p,q) are the sum of two particle irreducible diagrams and both of them are determined by the complicated nonperturbative interactions between the heavy quark and the light diquark. As we cannot solve these kernels from the first principles of QCD we have to make a phenom- enological model. The form assumed for K₀( P, p,q) was given in 关11兴in the covariant instantaneous approximation.

Equation 共23兲constraints the form of K₁( P, p,q). The sim- plest K₁( P, p,q) which satisfies Eq.共23兲 is 关(a⫺p_t²/2)/( p_l

⫹E₀⫹m_D⫹i⑀⁾兴K₀( P, p,q), where a is a parameter in K₁( P, p,q) which should be equal to E₁. However, as will be seen later, once Eq. 共23兲is satisfied, the physical results do not depend on the explicit form of K₁( P, p,q).

We would like to stress that Eq.共23兲is just one possibility we have found which guarantees that we have solutions for

␾1 P( p). There may be other possible forms for K₁( P, p,q) which can also lead to solutions for ␾1 P( p), and whether or not Eq. 共23兲is a reasonable hypothesis should be tested by experiments.

With Eq.共23兲,␾1 P( p) satisfies the same eigenvalue equa- tion as␾0 P( p). Therefore, we have

␾1 P共p兲⫽␴␾0 P共p兲, 共24兲 where ␴ is a constant of proportionality, with mass dimen- sion, which can be determined by Luke’s theorem关22兴at the zero-recoil point in HQET. We will discuss it in the next section.

Since both␾1 P( p) and␾2 P( p) can be related to␾0 P( p), we can calculate the 1/m_Q corrections without explicitly solving the integral equations for␾1 P( p) and␾2 P( p). In the previous work关11兴␾0 P( p) was solved by assuming that V₁ and V₂ in Eq. 共11兲 arise from linear confinement and one- gluon-exchange terms, respectively. In the covariant instan- taneous approximation, V˜_i⬅V_i兩p_l⫽q_l, i⫽1,2, we find

V˜

1⫽ 8␲␬

关共p_t⫺q_t兲²⫹␮²兴²⫺共2␲兲³␦³共p_t⫺q_t兲

⫻

冕

_共2^d␲^3k兲³ 8␲␬

共k²⫹␮²兲²,

共25兲 V˜₂⫽⫺16␲

3

␣s (eff)2

Q₀²

关共p_t⫺q_t兲²⫹␮²兴关共p_t⫺q_t兲²⫹Q₀²兴, where␬ ^and␣s

(eff) are coupling parameters related to scalar confinement and the one-gluon-exchange diagram, respec- tively. They can be related to each other when we solve the eigenvalue equation for ␾0 P( p). The parameter␮ ^{is intro-} duced to avoid the infrared divergence in numerical calcula- tions, and the limit ␮→0 is taken in the end. It should be noted that in V˜₂ we introduced an effective form factor, F(Q²)⫽␣s

(eff)Q₀²/(Q²⫹Q₀²), to describe the internal struc- ture of the light diquark 关23兴.

Defining ␾^˜0 P( p_t)⫽兰(dp_l/2␲⁾␾0 P( p) the BS equation for ␾^˜0 P( p_t) is关11兴

␾^˜0 P共p_t兲⫽⫺ 1

2共E₀⫺W_p⫹m_D兲W_p

冕

_共^d2³␲^q兲^t3

⫻共V˜₁⫺2W_pV˜₂兲␾^˜0 P共q_t兲, 共26兲

in the covariant instantaneous approximation. The numerical results for ␾^˜0 P(k_t) can be obtained from Eq.共26兲, with the overall normalization constant being fixed by the normaliza- tion of the Isgur-Wise function at the zero-recoil point关11兴. Furthermore,␾0 P( p) is expressed in terms of␾^˜0 P(q_t):

␾0 P共p兲⫽ i

共p_l⫹E₀⫹m_D⫹i⑀兲共p_l²⫺W_p²⫹i⑀兲

⫻

冕

_共^d2³␲^q兲^t3共V˜₁⫹2 p_lV˜₂兲␾^˜0 P共q_t兲. 共27兲

III.⌳b\⌳cFORM FACTORS TO 1ÕmQ

In this section we will express the six form factors for the

⌳b→⌳c weak transition in terms of the BS wave function and show the consistency between our model and HQET.

On the grounds of Lorentz invariance, the matrix element for ⌳b→⌳c can be expressed as

具⌳c共v⬘兲兩J_␮兩⌳b共v兲典⫽¯u

⌳_c共v⬘兲关F₁共␻兲␥_␮⫹F₂共␻兲v_␮

⫹F₃共␻兲v_␮⬘⫺„G₁共␻兲␥_␮⫹G₂共␻兲v_␮

⫹G₃共␻兲v_␮⬘…␥5兴u_⌳

b共v兲, 共28兲

where J_␮is the V⫺A weak current,vandv⬘ are the veloci- ties of ⌳b and⌳c, respectively, and␻⫽v⬘•v.

The form factors F_i and G_i(i⫽1,2,3) are related to each other by the following equations, to order 1/m_Q, when HQET is applied关19兴:

F₁⫽G₁

冋

^1⫹

冉

^m1c⫹m1b

冊

^1⫹⌳¯␻

册

^,

F₂⫽G₂⫽⫺G₁ 1 m_c

⌳¯

1⫹␻, 共29兲

F₃⫽⫺G₃⫽⫺G₁ 1 m_b

⌳¯ 1⫹␻^,

where ⌳¯ is an unknown parameter which is defined as the mass difference m_⌳

Q⫺m_Q in the limit m_Q→⬁.

On the other hand, the transition matrix element of ⌳b

→⌳cis related to the BS wave functions of ⌳b and⌳c by the following equation:

具^⌳c共v⬘^兲兩J␮兩⌳b共v兲典

⫽

冕

_共2^d4␲^p兲⁴␹^¯P⬘共p⬘兲␥␮共1⫺␥5兲␹P共p兲S_D^⫺1共p₂兲, 共30兲 where P( P⬘) is the momentum of ⌳b (⌳c) and p⬘ ^{is the} relative momentum defined in the BS wave function of

⌳c(v⬘^), ␹^¯P⬘( p⬘), which can also be expressed in terms of the three BS scalar functions␾0 P( p),␾1 P( p) and␾2 P( p) in Eq. 共18兲

␹

¯_P共p兲⫽¯u_⌳

Q共v,s兲

再

^{␾0 P共p兲⫹m1Q关␾1 P共p兲⫹␾2 P共p兲p”t兴}

冎

^.

共31兲 Substituting Eqs.共18兲and共31兲into Eq.共30兲and using the relations in Eq. 共29兲 we find the following results by com- paring the ␥_␮^, ␥_␮␥5, v_␮(1⫺␥5) andv_␮⬘⁽¹⫹␥5) terms, re- spectively:

G₁

冋

^1⫹

冉

^m1c⫹m1b

冊

^1⫹⌳¯␻

册

^⫽⫺i

冕

共2^d␲^4k兲⁴

再

^{␾0 P⬘共k⬘兲␾0 P共k兲共kl2⫺Wk2兲⫹m1c关␾1 P⬘共k⬘兲⫺共kl⬘⫹mD兲␾2 P⬘共k⬘兲兴}

⫻␾0 P共k兲共k_l²⫺W_k²兲⫹ 1

m_c共⫺f₁⫹f₂⫹2m_DF兲⫹ 1

m_b共f₁⫺f₂兲⫹ 1

m_b␾0 P⬘共k⬘兲

⫻关␾1 P共k兲⫺共k_l⫹m_D兲␾2 P共k兲兴共k_l²⫺W_k²兲

冎

^{⫹O共1/mQ2兲, 共32兲}

G₁⫽⫺i

冕

_共2^d␲^4k兲⁴

再

^{␾0 P⬘共k⬘兲␾0 P共k兲共kl2⫺Wk2兲⫹m1c共f1⫹f2兲⫹m1c关␾1 P⬘共k⬘兲}

⫺共k_l⬘⫹m_D兲␾2 P⬘共k⬘兲兴␾0 P共k兲共k_l²⫺W_k²兲⫹ 1

m_b共f₁⫹f₂兲⫹ 1

m_b␾0 P⬘共k⬘兲关␾1 P共k兲

⫺共k_l⫹m_D兲␾2 P共k兲兴共k_l²⫺W_k²兲

冎

^{⫹O共1/mQ2兲, 共33兲}

1

m_c

冋

^{iG11⫹⌳¯␻⫹2共f1⫺mDF兲}

册

^{⫽O共1/mQ2兲, 共34兲}

1

m_b

冋

^{iG11⫹⌳¯␻⫹2 f2}

册

^{⫽O共1/mQ2兲, 共35兲}

where we have defined f₁, f₂ and F by the following equations, on the grounds of Lorentz invariance:

冕

_共2^d␲^4k兲⁴␾2 P⬘共k⬘兲␾0 P共k兲共k_l²⫺W_k²兲⫽F, 共36兲

冕

_共2^d␲^4k兲⁴␾2 P⬘共k⬘兲␾0 P共k兲k^␮共k_l²⫺W_k²兲⫽f₁v^␮⫹f₂v⬘^␮. 共37兲 Equation共37兲leads to

f₁⫹f₂⫽ 1

1⫹␻

冕

_共2^d␲^4k兲⁴␾2 P⬘共k⬘兲␾0 P共k兲共k_l²⫺W_k²兲共v•k⫹v⬘•k兲. 共38兲 Equations共32兲and共33兲give the expression for G₁to order 1/m_Q. From Eqs.共34兲and共35兲we can see that Eq.共32兲is the same as Eq.共33兲. Therefore, we can calculate G₁to 1/m_Qfrom either of these two equations. This indicates that our model is consistent with HQET to order 1/m_Q.

Substituting Eq.共38兲into Eq.共33兲and using Eq. 共22兲we have

G₁⫽⫺i

冕

_共2^d␲^4k兲⁴

再

^{␾0 P⬘共k⬘兲␾0 P共k兲共kl2⫺Wk2兲⫹m1c}

冋

^{␾1 P⬘共k⬘兲⫺12共kl⬘⫹mD兲␾0 P⬘共k⬘兲}

册

^{␾0 P共k兲共kl2⫺Wk2兲⫹m1b␾0 P⬘共k⬘兲}

⫻

冋

^{␾1 P共k兲⫺12共kl⫹mD兲␾0 P共k兲}

册

^{共kl2⫺Wk2兲⫹}

冉

^m1c⫹m1b

冊

^2共11⫹␻兲␾0 P⬘共k⬘兲␾0 P共k兲共k_l²⫺W_k²兲共v•k⫹v⬘•k兲

冎

^{. 共39兲}

The first term in Eq. 共39兲 gives the Isgur-Wise function which was calculated in our earlier work 关11兴. In order to obtain the 1/m_Q corrections, we have to fix ␾1 P(k). Fortu- nately, this can be done by applying Luke’s theorem 关22兴. The conservation of vector current in the case of equal masses for the initial and final heavy quarks leads to

G₁共␻⫽1兲⫽1⫹O共1/m_Q²兲. 共40兲 Now we consider Eq.共39兲at the zero-recoil point,␻⫽1, at which P⬘⫽P and k⬘⫽k. Since the first term in Eq.共39兲is the Isgur-Wise function, this term is 1 when␻⫽1. From Eq.

共40兲 the 1/m_c and 1/m_b terms in Eq.共39兲should be zero at the zero-recoil point. Substituting Eq.共24兲into Eq.共39兲and noticing that v•k⫹v⬘•k⫽2(k_l⫹m_D)⫹O(1/m_Q) when ␻

⫽1 关see Eq.共4兲兴, we can show that

␴⫽0. 共41兲

Therefore,␾1 P(k) does not contribute to G₁.

Now we calculate G₁through Eq.共39兲. Since in the weak transition the diquark acts as a spectator, its momentum in the initial and final baryons should be the same, p₂⫽p₂⬘^. Then we can show that to order 1/m_Q

k_l⬘^v⬘^⫹kt⬘⫽k_lv⫹k_t. 共42兲 From Eq. 共42兲 we can obtain relations between k_l⬘^{, k}t⬘ ^and k_l, k_t straightforwardly:

k_l⬘⫽k_l␻⫺k_t

冑

␻²⫺1cos␪^,

k_t⬘²⫽k_t²⫹k_t²共␻²⫺1兲cos²␪⫹k_l²共␻²⫺1兲

⫺2k_lk_t␻

冑

␻²⫺1cos␪^, 共43兲 where␪ is defined as the angle between k_tandv_t⬘^.

Substituting the relation between ␾0 P( p) and ␾^˜0 P( p_t) 关Eq. 共27兲兴 into Eq. 共39兲, using the BS equation 共26兲, and integrating the k_l component by selecting the proper contour we have

G₁共␻兲⫽␰共␻兲⫹ 1

m_cA_c共␻兲⫹ 1

m_bA_b共␻兲, 共44兲 where

␰共␻兲⫽⫺

冕

_共^d2³␲^k兲^t3F共␻^,kt兲, 共45兲 A_c共␻兲⫽⫺

冕

_共^d2³␲^k兲^t3

共␻²⫺1兲W_k⫹␻^kt

冑

␻²⫺1cos␪ 2共␻⫹1兲

⫻F共␻^,kt兲, 共46兲

A_b共␻兲⫽

冕

_共^d2³␲^k兲^t3

k_t

冑

␻²⫺1cos␪

2共␻⫹1兲 F共␻^,kt兲, 共47兲

and F(␻^,kt) is defined as F共␻^,kt兲⫽ ␾^˜0 P共k_t兲

E₀⫹m_D⫺␻^Wk⫺k_t

冑

␻²⫺1cos␪

⫻

冕

_共2^d3_␲^r_兲^{t3␾˜0 P⬘共rt兲关V˜1共kt⬘⫺rt兲}

⫺2共␻^Wk⫹k_t

冑

␻²⫺1cos␪兲V˜₂共k_t⬘⫺r_t兲兴兩k_l⫽⫺W_k. 共48兲 The three dimensional integrations in Eqs.共45兲–共47兲can be reduced to one dimensional integrations by using the fol- lowing identities:

冕

_共^d2³␲^q兲^t3

␳共q_t²兲 关共p_t⫺q_t兲²⫹␮²兴²

⫽

冕

^q4^t2␲^dq2t

2␳共q_t²兲

共p_t²⫹q_t²⫹␮²兲²⫺4 p_t²q_t², 共49兲 and

冕

_共^d2³␲^q兲^t3

␳共q_t²兲 共p_t⫺q_t兲²⫹␦²

⫽

冕

^q4^t2␲^dq2t

␳共q_t²兲

2兩p_t兩兩q_t兩ln共兩p_t兩⫹兩q_t兩兲²⫹␦² 共兩p_t兩⫺兩q_t兩兲²⫹␦^{2, 共50兲} where␳^(qt

2) is some arbitrary function of q_t². In our model we have several parameters,␣s

(eff)

, ␬^{, Q}0 2, m_D, E₀ and E₁. The parameter Q₀² can be chosen as 3.2 GeV² from the data for the electromagnetic form factor of the proton关23兴. As discussed in Ref.关11兴, we let␬ ^{vary in} the region between 0.02 GeV³and 0.1 GeV³. In HQET, the binding energies should satisfy the constraint Eq. 共5兲. Note that m_D⫹E₀ and E₁ are independent of the flavor of the heavy quark. From the BS equation solutions in the meson case, it has been found that the values m_b⫽5.02 GeV and m_c⫽1.58 GeV give predictions which are in good agreement with experiments 关8兴. Since in the b-baryon case the O(1/m_b²) corrections are very small, we use the following equation to discuss the relations among m_D, E₀ and E₁,

m_D⫹E₀⫹ 1

m_bE₁⫽0.62 GeV, 共51兲 where we have used m_⌳

b⫽5.64 GeV. The parameter m_D cannot be determined, although there are suggestions from the analysis of valence structure functions that it should be around 0.7 GeV for non-strange scalar diquarks关24兴. Hence we let it vary within some reasonable range, 0.65–0.75 GeV.

In the expansion with respect to the heavy quark mass, we roughly expect 关(1/m_b)E₁兴/E₀⬃⌳QCD/m_b. Therefore, E₁ should be of the order ⌳QCDE₀. In our numerical calcula- tions, we let␤⁽⫽E₁/E₀) change between 0.2 and 1.0. Then

for some values of m_D and␤ we can determine E₀. Using Eqs. 共44兲–共50兲and Eq.共29兲we obtain numerical results for the weak decay form factors F_i, G_i(i⫽1,2,3) to order 1/m_Q. It turns out that the numerical results are very insen- sitive to the value of ␤, so we ignore this dependence. We also find that the dependence of F_i, G_i (i⫽1,2,3) on the diquark mass m_Dis not strong. In Fig. 1 we plot the numeri- cal results for F_i(i⫽1,2,3) for ␬⫽0.02 GeV³ and ␬⫽0.10 GeV³, respectively, with m_D⫽0.7 GeV.

IV. APPLICATIONS TO⌳b\⌳cl␯¯ AND⌳b\⌳cP„V… With the numerical results for F_i, G_i(i⫽1,2,3) to 1/m_Q obtained in Sec. III, we can predict the⌳b→⌳csemileptonic and nonleptonic weak decay widths to order 1/m_Q. Since the QCD corrections to these form factors are comparable with the 1/m_Qeffects, we will include both of them to give phe- nomenological predictions.

Neubert 关25兴 has shown that the QCD corrections to the weak decay form factors can be written in the following form 共up to corrections of the order ␣s⌳¯ /m_Q):

⌬F₁⫽␰␣s共m¯兲

␲ ^{v1, ⌬G1⫽}␰␣s共m¯兲

␲ ^a1,

共52兲

⌬F_i⫽⫺␰␣s共m¯兲

␲ ^{vi, ⌬Gi⫽⫺}␰␣s共m¯兲

␲ ^{ai, 共i⫽2,3兲,} wherevi⫽vi(␻^{) and a}i⫽a_i(␻⁾⁽ⁱ⫽1,2,3) are the QCD cor- rections calculated from the next-to-leading order renormal- ization group improved perturbation theory. The scale m¯ is chosen such that higher-order terms (␣sln(m_b/m_c))ⁿ(n

⬎1) do not contribute. Consequently, it is not necessary to apply a renormalization group summation as far as only nu- merical evaluations are concerned. It is shown that m¯ can be chosen as 2m_bm_c/(m_b⫹m_c)⯝2.3 GeV. The detailed formu- las forv_iand a_ican be found in关25兴, which also includes a discussion on the infrared cutoff employed in the calculation of the vertex corrections. As in关25兴, we choose this cutoff to FIG. 1. The numerical results for F_i(i⫽1,2,3) for ␬⫽0.02 GeV³共solid lines兲and␬⫽0.10 GeV³共dotted lines兲, with m_D⫽0.7 GeV. From top to bottom we have F1, F3, and F2, respectively.

be 200 MeV which is a fictitious gluon mass. Furthermore, we use ⌳QCD⫽200 MeV in our numerical calculations.

A. Semileptonic decays⌳b\⌳cl␯¯

Making use of the general kinematical formulas by Ko¨ner and Kra¨mer关26兴, we find for the differential decay width of

⌳b→⌳cl␯^¯ 关14兴 d⌫

d␻^⫽ 2 3m_⌳

c 4 m_⌳

bAF₁²

冑

␻²⫺1

再

^{3␻共␩⫹␩⫺1兲⫺2⫺4␻2}

⫺⌳¯

冉

^m1c⫹m1b

冊

^{关3共␩⫹␩⫺1兲⫺4⫺2␻兴⫹␣␲sv1共␻⫺1兲}

⫻关3共␩⫹␩^⫺1兲⫹2⫺4␻兴⫹␣s

␲^a1共␻⫹1兲关3共␩⫹␩^⫺1兲

⫺2⫺4␻兴⫺␣s

␲^共␻²⫺1兲关v2共1⫹␩兲⫹v3共1⫹␩^⫺1兲

⫹a₂共1⫺␩兲⫹a₃共␩^⫺1⫺1兲兴

冎

^{, 共53兲}

where␩⫽m_⌳

c/m_⌳

b and A⫽关G_F²/(2␲⁾³兴兩V_cb兩²B(⌳c→ab), with 兩V_cb兩 being the Kobayashi-Maskawa matrix element.

B(⌳c→ab) is the branching ratio for the decay ⌳c

→a(¹₂^⫹)⫹b(0^⫺) through which ⌳c is detected, since the structure for such decays is already well known. It should be noted that in Eq. 共53兲 O(␣s⌳¯ /mQ) correction