application/xmlSubleading Isgur–Wise function of Λb→Λc1 using QCD sum rulesJong-Phil LeeGye T. ParkPhysics Letters B 552 (2003) 185-197. doi:10.1016/S0370-2693(02)03161-1journalPhysics Letters BCopyright © 2002 Elsevier Science B.V. All rights reserved.Elsevier B.V.0370-26935523-423 January 20032003-01-23185-19718519710.1016/S0370-2693(02)03161-1http://dx.doi.org/10.1016/S0370-2693(02)03161-1doi:10.1016/S0370-2693(02)03161-1http://vtw.elsevier.com/data/voc/oa/OpenAccessStatus#Full2014-01-01T00:14:32ZSCOAP3 - Sponsoring Consortium for Open Access Publishing in Particle Physicshttp://vtw.elsevier.com/data/voc/oa/SponsorType#FundingBodyhttp://creativecommons.org/licenses/by/3.0/

Journals

S300.2

PLB19451S0370-2693(02)03161-110.1016/S0370-2693(02)03161-1Elsevier Science B.V.Phenomenology

Fig. 1

Feynman diagrams for the three-point function with derivative interpolating fields. Double line denotes the heavy quark.

Fig. 2

Three-dimensional plot of τ as a function of y and T in units of GeV. The continuum threshold is chosen to be ωc(y)=1.4 GeV.

Fig. 3

τ(1) as a function of the Borel parameter T. Each graph corresponds to ω0=1.2,1.3,1.4,1.5,1.6 GeV, respectively, from the top.

Fig. 4

τ(y) at a fixed Borel parameter T=0.34. Each graph corresponds to ω0=1.2,1.3,1.4,1.5,1.6 GeV, respectively, from the top.

Subleading Isgur–Wise function of Λb→Λc1 using QCD sum rules

Jong-Phil

Lee

jplee@phya.yonsei.ac.kr

Gye T.

Park

gtpark@phya.yonsei.ac.kr

Department of Physics and IPAP, Yonsei University, Seoul 120-749, South Korea

Editor: T. Yanagida

Abstract

Subleading Isgur–Wise form factor τ(v·v′) at O(1/mQ) for Λb→Λc11/2,3/2 weak transition is calculated by using the QCD sum rules in the framework of the heavy quark effective theory (HQET), where Λ1/2c1 and Λ3/2c1 are the orbitally excited charmed baryon doublet with JP=(1−/2,3−/2). We consider the subleading contributions from the weak current matching in the HQET. The interpolating currents with transverse covariant derivative are adopted for Λ1/2c1 and Λ3/2c1 in the analysis. The slope parameter ρ2 in linear approximation of τ is obtained to be ρ2=2.76 and the interception to be τ(1)=−1.27 GeV.

Introduction

The ground state bottom baryon Λb weak decays [1] provide a testing ground for the standard model (SM). They reveal some important features of the physics of bottom quark. The experimental data on these decays have been accumulated to wait for reliable theoretical calculations. With the discovery of the orbitally excited charmed baryons Λc(2593) and Λc(2625) [2], it would be of great interest for one to investigate the Λb semileptonic decays into these baryons.From the phenomenological point of view, these semileptonic transitions are interesting since in principle they may account for a sizeable fraction of the inclusive semileptonic rate of Λb decay. In addition, the properties of excited baryons have attracted attention in recent years. Investigation on them will extend our ability in the application of QCD. It can also help us foresee any other excited heavy baryons that have not been discovered yet.The heavy quark symmetry [3] is a useful tool to classify the hadronic spectroscopy containing a heavy quark Q. In the infinite mass limit, the spin and parity of the heavy quark and that of the light degrees of freedom are separately conserved. Coupling the spin of light degrees of freedom jℓ with the spin of heavy quark sQ=1/2 yields a doublet with total spin J=jℓ±1/2 (or a singlet if jℓ=0). This classification can be applied to the ΛQ-type baryons. For the charmed baryons the ground state Λc contains light degrees of freedom with spin-parity jℓP=0+, being a singlet. The excited states with jℓP=1− are spin symmetry doublet with JP(1−/2,3−/2). The lowest states of such excited charmed states, Λ1/2c1 and Λ3/2c1, have been observed to be identified with Λc(2593) and Λc(2625), respectively [2].However, the difficulties in the SM calculations are mainly due to the poor understanding of the nonperturbative aspects of the strong interaction (QCD). The heavy quark effective theory (HQET) based on the heavy quark symmetry provides a model-independent method for analyzing heavy hadrons containing a single heavy quark [3]. It allows us to expand the physical quantity in powers of 1/mQ systematically, where mQ is the heavy quark mass. Within this framework, the classification of the Λb exclusive weak decay form factors has been greatly simplified. The decays such as Λb→Λclν̄[4], Λb→Σc(∗)lν̄[5], Λb→Σc(∗)πlν̄[6], Λb→p(Λ) [7] have been studied.To obtain detailed predictions for the hadrons, at this point, some nonperturbative QCD methods are also required. We have adopted QCD sum rules [8] in this Letter. QCD sum rule is a powerful nonperturbative method based on QCD. It takes into account the nontrivial QCD vacuum which is parametrized by various vacuum condensates in order to describe the nonperturbative nature. In QCD sum rule, hadronic observables can be calculated by evaluating two- or three-point correlation functions. The hadronic currents for constructing the correlation functions are expressed by the interpolating fields. In describing the excited heavy baryons, transverse covariant derivative is included in the interpolating field. The static properties of Λb and Λc1 (Λc1 denotes the generic jℓP=1− charmed state) have been studied with QCD sum rules in the HQET in Refs. [9–11], respectively. Recently, the leading order Isgur–Wise (IW) function is also calculated in the HQET QCD sum rule in Ref. [12].In Λb→Λc1 decay, 1/mQ corrections are very important. At the heavy quark limit of mQ→∞, the transition matrix elements should vanish at zero recoil since the light degrees of freedom change their configurations. Nonvanishing contribution to, say, B(Λb→Λc1ℓν̄) at zero recoil appears at 1/mQ order. Since both Λb and Λc1 are heavy enough, the behavior of the matrix elements near the zero recoil is very important. That explains why people pay attention to the next-to-leading order (NLO) contributions. The same situation occurs in heavy mesons. As for B→D1(D2∗)ℓν̄ decay, leading and subleading Isgur–Wise (IW) functions have been computed using QCD sum rule in Refs. [13–17]. They showed that the branching ratio is enhanced considerably when the subleading contributions are included.In HQET, 1/mQ corrections appear in a two-fold way. At the Lagrangian level, subleading terms are summarized in λ1 and λ2. λ1 parametrizes the kinetic term of higher derivative, while λ2 represents the chromomagnetic interaction which explicitly breaks the heavy quark spin symmetry. At the current level, 1/mQ corrections come from the small portion of the heavy quark fields which correspond to the virtual motion of the heavy quark. In this Letter, the subleading IW function from the latter case, i.e., at the current level, is analyzed in the HQET QCD sum rules.In Section 2, the weak transition matrix elements are parametrized by the leading and subleading IW functions. By evaluating the three-point correlation function, we give the subleading IW function in Section 3. We present, in Section 4, the numerical analysis and discussions. The summary is given in Section 5.2

Weak transition matrix elements and the subleading Isgur–Wise functions

The weak transition matrix elements for Λb→Λc1 are parametrized by the 14-form factors as (1a)〈Λc11/2(v′,s′)|Vμ|Λb(v,s)〉4MΛc1(1/2)MΛb=ūΛc1(v′,s′)F1γμ+F2vμ+F3v′μγ5uΛb(v,s),(1b)〈Λc11/2(v′,s′)|Aμ|Λb(v,s)〉4MΛc1(1/2)MΛb=ūΛc1(v′,s′)G1γμ+G2vμ+G3v′μuΛb(v,s),(1c)〈Λc13/2(v′,s′)|Vμ|Λb(v,s)〉4MΛc1(3/2)MΛb=ūαΛc1(v′,s′)vα(K1γμ+K2vμ+K3v′μ)+K4gαμuΛb(v,s),(1d)〈Λc13/2(v′,s′)|Aμ|Λb(v,s)〉4MΛc1(3/2)MΛb=ūαΛc1(v′,s′)vα(N1γμ+N2vμ+N3v′μ)+N4gαμγ5uΛb(v,s), where v (v′) and s (s′) are the four-velocity and spin of Λb (Λc1), respectively. And the form factors Fi, Gi, Ki and Ni are functions of y≡v·v′. In the limit of mQ→∞, all the form factors are related to one independent universal form factor ξ(y) called Isgur–Wise (IW) function. A convenient way to evaluate hadronic matrix elements is by introducing interpolating fields in HQET developed in Ref. [18] to parametrize the matrix elements in Eqs. (1). With the aid of this method the matrix element can be written as [19](2)c̄Γb=h̄(c)v′Γh(b)v=ξ(y)vαψ̄αv′Γψv at leading order in 1/mQ and αs, where Γ is any collection of γ-matrices. The ground state field, ψv, destroys the Λb baryon with four-velocity v; the spinor field ψαv is given by (3)ψαv=ψ3/2αv+13γα+vαγ5ψ1/2v , where ψ1/2v is the ordinary Dirac spinor and ψ3/2αv is the spin-32 Rarita–Schwinger spinor, they destroy Λ1/2c1 and Λ3/2c1 baryons with four-velocity v, respectively. To be explicit, (4)F1=13(y−1)ξ(y),G1=13(y+1)ξ(y),F2=G2=−23ξ(y),K1=N1=ξ(y),(others)=0. In general, the IW form factor is a decreasing function of the four velocity transfer y. Since the kinematically allowed region of y for heavy to heavy transition is very narrow around unity, (5)1⩽y⩽MΛb2+MΛc122MΛbMΛc1≃1.3, and hence it is convenient to approximate the IW function linearly as (6)ξ(y)=ξ(1)1−ρξ2(y−1), where ρξ2 is the slope parameter which characterizes the shape of the leading IW function.The ΛQCD/mQ corrections come in two ways. One is from the subleading Lagrangian of the HQET while the other comes from the small portion of the heavy quark field to modify the effective currents. We only consider the latter case here.Including ΛQCD/mb and ΛQCD/mc, the weak current is given by (7)c̄Γb=h̄(c)v′Γ−i2mcD̷Γ+i2mbΓD/h(b)v. Keeping the Lorentz structure, the subleading terms are expanded in general as (8)h̄(c)v′iD̷Γh(b)v=ψ̄αv′τ1(c)vαv̷+τ2(c)vαv̷′+τ3(c)γαΓΛv,h̄(c)v′ΓiD/h(b)v=ψ̄αv′Γτ1(b)vαv̷+τ2(b)vαv̷′+τ3(b)γαΛv, where τi(Q) are the subleading IW functions to be evaluated.The matrix elements of these currents modify Eq. (4) as (9)3F1=(y−1)ξ−ϵc(y−1)−τ1(c)+τ2(c)+3τ3(c)+ϵb(y−1)τ1(b)−τ2(b)−τ3(b),3F2=−2ξ+ϵc2yτ1(c)+2τ2(c)+ϵb−2τ1(b)+2τ2(b),3F3=−2ϵb(1+y)τ2(b)+τ3(b),3G1=(y+1)ξ−ϵc(y+1)τ1(c)+τ2(c)+3τ3(c)+ϵb(y+1)τ1(b)+τ2(b)+τ3(b),3G2=−2ξ+ϵc2yτ1(c)+2τ2(c)−2ϵbτ1(b)+τ2(b),3G3=2ϵb(y−1)τ2(b)+τ3(b),K1=ξ+ϵcτ1(c)−τ2(c)+ϵbτ1(b)−τ2(b),N1=ξ−ϵcτ1(c)+τ2(c)+ϵbτ1(b)+τ2(b),K2=N2=−2ϵcτ1(c),K3=−N3=2ϵbτ2(b),K4=−N4=2ϵbτ3(b), where ϵQ≡1/2mQ. It is quite convenient to define (10a)Ωαβ(cΓ)≡γα+v′αγ51+v̷′2γβΓ1+v̷2,(10b)Ωαβ(bΓ)≡γα+v′αγ51+v̷′2Γγβ1+v̷2. Possible contractions of Ωαβ are listed in Appendix A. From the Eqs. (3) and (8), Eqs. (1) can be reexpressed in terms of τi(Q) and Ωαβ: (11)〈Λc11/2(v′,s′)|Γ|Λb(v,s)〉4MΛc1(1/2)MΛb=13ūΛc1(v′,s′)ξvαv′αΩ(cΓ)αβ−ϵcτ1(c)vαvβ+τ2(c)vαv′β+τ3(c)gαβΩ(cΓ)αβ+ϵbτ1(b)vαvβ+τ2(b)vαv′β+τ3(b)gαβΩ(bΓ)αβuΛb(v,s). A similar expression can be obtained for the spin-32 final states (12)〈Λc13/2(v′,s′)|Γ|Λb(v,s)〉4MΛc1(3/2)MΛb=ūαΛc1(v′,s′)ξvαΓ−ϵcτ1(c)vαvβ+τ2(c)vαv′β+τ3(c)gαβγβΓ+ϵbτ1(b)vαvβ+τ2(b)vαv′β+τ3(b)gαβΓγβuΛb(v,s).3

QCD sum rule evaluation

As a starting point of QCD sum rule calculation, let us consider the interpolating field of heavy baryons. The heavy baryon current is generally expressed as (13)jvJ,P(x)=ϵijkqiT(x)CΓJ,Pτqj(x)Γ′J,Phkv(x), where i,j,k are the color indices, C is the charge conjugation matrix, and τ is the isospin matrix while q(x) is a light quark field. ΓJ,P and Γ′J,P are some gamma matrices which describe the structure of the baryon with spin-parity JP. Usually Γ and Γ′ with least number of derivatives are used in the QCD sum rule method. The sum rules then have better convergence in the high energy region and often have better stability. For the ground state heavy baryon, we use Γ1/2,+=γ5, Γ′1/2,+=1. In the previous work [10], two kinds of interpolating fields are introduced to represent the excited heavy baryon. In this Letter, we find that only the interpolating field of transverse derivative is adequate for the analysis. Nonderivative interpolating field results in a vanishing perturbative contribution. The choice of Γ and Γ′ with derivatives for the Λc11/2 and Λc13/2 is then (14)Γ1/2,−=(a+bv̷)γ5,Γ′1/2,−=iD̷tMγ5,Γ3/2,−=(a+bv̷)γ5,Γ′3/2,−=13MiDμt+iD̷tγμt, where a transverse vector Aμt is defined to be Aμt≡Aμ−vμv·A, and M in Eq. (14) is some hadronic mass scale. a, b are arbitrary numbers between 0 and 1.The baryonic decay constants in the HQET are defined as follows, (15a)〈0|jv1/2,+|Λb〉=fΛbψv,(15b)〈0|jv1/2,−Λc11/2=f1/2ψ1/2v,(15c)〈0|jvμ3/2,−Λc13/2=13f3/2ψ3/2μv, where f1/2 and f3/2 are equivalent since Λc11/2 and Λc13/2 belong to the same doublet with jℓP=1−. The QCD sum rule calculations give [9](16)fΛb2e−Λ̄/T=120π4∫0ωcdωω5e−ω/T+16〈q̄q〉2e−m02/8T2+〈αsGG〉32π3T2, and [10](17)M2f1/22e−Λ̄′/T′=∫0ωc′dω3Nc!4π4·7!ω724a2+40b2e−ω/T′+〈αsGG〉32π3T′4−a2+b2+Nc!2π2〈q̄q〉T′5(16ab)−〈q̄gσ·Gq〉T′3ab−〈q̄gσ·Gq〉4π2T′3(3ab). In the above equations, T(′) are the Borel parameters and ωc(′) are the continuum thresholds, and Nc=3 is the color number. In the heavy quark limit, the mass parameters Λ̄ and Λ̄′ are defined as (18)Λ̄′=MΛQ1−mQ,Λ̄=MΛQ−mQ. The main point in QCD sum rules for the IW function is to study the analytic properties of the 3-point correlators, (19a)Ξ1/2(ω,ω′,y)=i2∫d4xd4zei(k′·x−k·z)〈0|Tjv′1/2,−(x)h̄(c)v′(0)Γh(b)v(0)j̄v1/2,+(z)|0〉=Ξhadron(ω,ω′,y)3ξvαv′αΩ(cΓ)αβ−ϵcτ1(c)vαvβ+τ2(c)vαv′β+τ3(c)gαβΩ(cΓ)αβ+ϵbτ1(b)vαvβ+τ2(b)vαv′β+τ3(b)gαβΩ(bΓ)αβ,(19b)Ξ3/2μ(ω,ω′,y)=i2∫d4xd4zei(k′·x−k·z)〈0|Tjv′α3/2,−(x)h̄(c)v′(0)Γh(b)v(0)j̄v3/2,+(z)|0〉=Ξhadron(ω,ω′,y)Λ+μαξvαΓ−ϵcτ1(c)vαvβ+τ2(c)vαv′β+τ3(c)gαβγβΓ+ϵbτ1(b)vαvβ+τ2(b)vαv′β+τ3(b)gαβΓγβ1+v̷2. The variables k, k′ denote residual “off-shell” momenta which are related to the momenta P of the heavy quark in the initial state and P′ in the final state by k=P−mQv, k′=P′−mQ′v′, respectively.The coefficient Ξ(ω,ω′,y)hadron in Eq. (20) is an analytic function in the “off-shell energies” ω=v·k and ω′=v′·k′ with discontinuities for positive values of these variables. It furthermore depends on the velocity transfer y=v·v′, which is fixed at its physical region for the process under consideration. By saturating with physical intermediate states in HQET, one finds the hadronic representation of the correlators as following (20)Ξhadron(ω,ω′,y)=f1/2f∗Λb(Λ̄′−ω′)(Λ̄−ω)+higher resonances. In obtaining the above expression the Dirac and Rartia–Schwinger spinor sums (21)Λ+=∑s=12u(v,s)ū(v,s)=1+v̷2,Λ+μν=∑s=14uμ(v,s)ūν(v,s)=−gtμν+13γtμγtν1+v̷2, have been used, where gμνt=gμν−vμvν.In the quark–gluon language, Ξ(ω,ω′,y)1/2,3/2 in Eq. (20) is written as (22)Ξ(ω,ω′,y)1/2,3/2=∫0∞dνdν′ρpert(ν,ν′,y)(ν−ω)(ν′−ω′)+(subtraction)+Ξcond(ω,ω′,y), where the perturbative spectral density function ρpert(ν,ν′,y) and the condensate contribution Ξcond are related to the calculation of the Feynman diagrams depicted in Fig. 1

. In Eq. (22), the γ-structures of spin-12 and 32 are the same as those in Eq. (20), respectively. Subleading IW functions, τi(Q), obtained from spin-12 and 32 are, therefore, identical.

The six τi(Q) (Q=c,b, i=1,2,3) are not independent. From the fact that (23)i∂αh̄(c)v′Γh(b)v=h̄(c)v′iDαΓ+ΓiDαh(b)v=Λ̄vα−Λ̄′v′αh̄(c)v′Γh(b)v,Eq. (8) implies (24)τ1(c)+τ1(b)vαvβ+τ2(c)+τ2(b)vαv′β+τ3(c)+τ3(b)gαβ=Λ̄vβ−Λ̄′v′βvαξ(y). The above expression relates τi(c) with τi(b) as (25a)τ1(c)+τ1(b)=Λ̄ξ,(25b)τ2(c)+τ2(b)=−Λ̄′ξ,(25c)τ3(c)+τ3(b)=0. Other relations are obtained from the equation of motion of the heavy quark, v·Dh(Q)v=0: (26a)h̄(c)v′iv·DΓh(b)v=ψ̄αv′yτ1(c)+τ2(c)ΓΛv=0,(26b)h̄(c)v′Γiv·Dh(b)v=ψ̄αv′Γτ1(b)+yτ2(b)+τc(b)Λv=0. From the above 5 equations in Eqs. (25), (26), all the six subleading IW functions are reduced to only one independent form factor. We just pick up τ1(b)(y)≡τ(y), then others are (27a)τ1(c)=Λ̄ξ−τ,(27b)τ2(c)=−yΛ̄ξ+yτ,(27c)τ3(c)=yyΛ̄−Λ̄′ξ−y2−1τ,(27d)τ2(b)=yΛ̄−Λ̄′ξ−yτ,(27e)τ3(b)=−yyΛ̄−Λ̄′ξ+y2−1τ.Now that all the subleading IW functions are related to τ(y), we have only to extract the coefficient of vαvβΩ(bΓ)αβ (or Λμα+vαvβΓγβ for spin-32) in Eqs. (19) and (22).The QCD sum rule is obtained by equating the phenomenological and theoretical expressions for Ξ. In doing this the quark–hadron duality needs to be assumed to model the contributions of higher resonance part of Eq. (20). Generally speaking, the duality is to simulate the resonance contribution by the perturbative part above some thresholds ωc and ω′c, that is (28)res.=∫ωc∞∫ω′c∞dνdν′ρpert(ν,ν′,y)(ν−ω)(ν′−ω′). In the QCD sum rule analysis for B semileptonic decays into ground state D mesons, it was argued by Neubert in [20], and Blok and Shifman in [21] that the perturbative and the hadronic spectral densities cannot be locally dual to each other, and therefore the necessary way to restore duality is to integrate the spectral densities over the “off-diagonal” variable ν−=y+1y−1(ν−ν′)/2, keeping the “diagonal” variable ν+=(ν+ν′)/2 fixed. It is in ν+ that the quark–hadron duality is assumed for the integrated spectral densities. The same prescription shall be adopted in the following analysis. On the other hand, in order to suppress the contributions of higher resonance states a double Borel transformation in ω and ω′ is performed to both sides of the sum rule, which introduces two Borel parameters T1 and T2.Combining Eqs. (20), (22), our duality assumption and making the double Borel transformation, one obtains the sum rule for ξ(y) as follows; (29)Mf1/2f∗Λbe−Λ̄′/2T′e−Λ̄/2T1+v̷′2CΓ1+v̷2=2y−1y+11/2∫0ωc(y)dν+∫−ν+ν+dν−exp−ν+−y−1y+1ν−2T′−ν++y−1y+1ν−2Tρ(ν+,ν−;y)+Bω′2T′Bω2TΞcond, where ν=ν++y−1y+1ν−, ν′=ν+−y−1y+1ν−, and (30)CΓ=13ξvαv′αΩ(cΓ)αβ−ϵcτ1(c)vαvβ+τ2(c)vαv′β+τ3(c)gαβΩ(cΓ)αβ+ϵbτ1(b)vαvβ+τ2(b)vαv′β+τ3(b)gαβΩ(bΓ)αβ(for spin-12),ξvαΓ−ϵcτ1(c)vαvβ+τ2(c)vαv′β+τ3(c)gαβγβΓ+ϵbτ1(b)vαvβ+τ2(b)vαv′β+τ3(b)gαβΓγβ(for spin-32).Now the remaining thing is to evaluate the relevant diagrams in Fig. 1. The leading contributions are given in [12]. For the subleading corrections to the perturbative spectral density function ρ(ω,ω′;y), we have (31)ρω,ω′;y=B−z′1/ω′B−z1/ωBω′1/z′Bω1/zΞpert=6Nc!aiπ4Ωαβ12sinh7θΘ(ω)Θ(ω′)Θ2yω′ω−ω2−ω′22vαv′βsinh2θ2coshθA3B33!3!−e−θA2B42!4!−eθA4B24!2!+2vαvβsinh2θe2θA4B24!2!+e−2θA2B42!4!−2A3B33!3!−gαβA3B33!3!, from the perturbative diagram Fig. 1(a), where (32a)Ωαβ≡−iϵcΩ(cΓ)αβ+iϵbΩ(bΓ)αβ,(32b)A≡ω′−ωe−θ,B≡ωeθ−ω′,(32c)eθ≡y+y2−1.For the condensate contributions we just give results when T′=T for simplicity; (33a)Bω′2TBω2TΞ〈q̄q〉=−ibgαβΩαβ2π2(1+y)264〈q̄q〉T5−13〈q̄gσ·Gq〉T3(4y+5/2)−ibvαΩαβ4π2(1+y)3−128〈q̄q〉T5(3v+2v′)β+43〈q̄gσ·Gq〉(6y+7/2)vβ+(y−3/2)v′β,(33b)Bω′2TBω2TΞ〈q̄gσ·Gq〉=−ib〈q̄gσ·Gq〉T312(1+y)3Ωαβ−2gαβ2y2+3y+1+(10y+6)vαvβ+4yvαv′β,(33c)Bω′2TBω2TΞ〈αsGG〉=ia〈αsGG〉T4192π3(1+y)5Ωαβ8(y+1)2(y−2)−gαβ+5vα(v+v′)β+24(y−1)vαv′β−16(y+1)(y+4)vαvβ−ia〈αsGG〉T4512π3(1+y)4Ωαβ−2(1+y)gαβ+6vα(v+v′)β. Note that these results are from Λc11/2. If Λc13/2 were the final state, Ωαβ would be replace by a proper γ-structure, leaving all the other things unchanged.4

Results and discussions

For the numerical analysis, the standard values of the condensates are used; (34)〈q̄q〉=−(0.23 GeV)3,〈αGG〉=0.04 GeV4,〈q̄gσ·Gq〉≡m02〈q̄q〉,m02=0.8 GeV2. There are many parameters engaged in the QCD sum rule calculations. The key point in the numerical analysis is to find a reasonable parameter space where the QCD sum rule results are stable. First, the continuum threshold ωc′ in f1/2(3/2) (Λ̄′) can differ from that in fΛb (Λ̄). However, it is expected that the values of ωc and 9ωc′ would not be different significantly. This is because the mass difference Λ̄′−Λ̄ is fairly small [10], Λ̄′−Λ̄≃0.2 GeV. Indeed, the central values of them were close to each other in the sum rules analysis for f1/2(3/2) (Λ̄′) and fΛb (Λ̄). One more thing to be noticed here is that the continuum threshold ωc in Eq. (29) can be a function of y in general. But for simplicity, we take it to be a constant ωc(y)=ωc=ωc′=ω0 in the numerical analysis. In this sense, we use only one constant continuum threshold throughout the analysis. An alternative choice of ωc(y)=(1+y)ω0/2y is suggested in Ref. [20]. We find that this choice yields almost no numerical differences. This is because the kinematically allowed region is very narrow around the zero recoil.Second, there are input parameters of a and b in the interpolating fields in Eq. (14). They are the parameters that generalize pseudoscalar or axial-vector nature of the light degrees of freedom (Γ1/2,3/2 in Eq. (14)). In Ref. [10], a particular choice of (a,b)=(1,0) gives the best stability for the mass parameter Λ̄′. We adopt the same choice of (a,b)=(1,0) in the present analysis.Third, there are two Borel parameters T1 and T2 distinct in general, corresponding to ω and ω′ in Ξ(ω,ω′,y), respectively. We have taken T1=T2 in the analysis. In Ref. [16] for B into excited charmed meson transition, the authors found a 10% increase in the leading IW function at zero recoil when T2/T1=1.5 as compared to the case when T1=T2. It seems quite reasonable for one to expect that in the case of heavy baryon, the numerical results should be similar for the small variations around T2/T1=1.In short, we adopt the same parameters used in [10,12] where the mass parameter and the leading IW function are calculated. It makes sense because the observables involved are directly related to the subleading IW function τ(y) through Eq. (29).In Fig. 2

, τ is plotted as a function of (y,T). Fig. 3

shows the stability of τ(y=1) for the Borel parameter. The sum rule window is (35)0.1⩽T⩽1.0 (GeV). The upper and lower bounds are fixed such that the pole contribution amounts to 50% while the condensate one to 12%. One notes that the window given in Eq. (35) overlaps those obtained in the Refs. [9,10,12]. Of course, this reflects the self-consistency of the sum rule analysis. In Fig. 4

, we present the shape of τ(y) for a fixed Borel parameter. We found that (36)τ(y)=τ(1)1−ρ2(y−1),τ(1)=−1.27−0.17+0.18 GeV,for ω0=1.4±0.1 GeV,ρ2=2.76−0.004+0.008,for ω0=1.4±0.1 GeV.

Summary

Subleading contributions of O(1/mQ) to the Λb→Λc1 weak form factors are important because some of the form factors do not survive at the heavy quark limit, and other remaining form factors vanish at zero recoil. Using the QCD sum rules, we calculate the subleading IW function τ(y) which appears in the current matching in the HQET at O(1/mQ). We obtain τ(y) given by (37)τ(y)=−1.271−2.76(y−1) GeV. The best stability is attained when the continuum threshold ω0=1.4 GeV. The parameter space for the analysis is the same as previous one for the leading IW function. The fact that by using the same set of parameters the present sum rule window for the mass parameter, leading and NLO IW function overlaps the previous ones ensures the self-consistency of the QCD sum rules. Our results can be applied directly to the decay mode Λb→Λc1ℓν̄, along with the use of the previous LO IW function, but a complete analysis at O(1/mQ) requires the information on another NLO contributions from the HQET Lagrangian.

Acknowledgements

This work was supported by the BK21 Program of the Korean Ministry of Education. The work of G.T.P. was supported in part by Yonsei University Research Fund of 2000.Appendix A

Contractions of Ωαβ

After a simple algebra, possible contractions for Ωαβ are given by (A.1)v′αΩαβ=0,vαvβΩαβ(cV)=1+v̷′2−2yvμγ51+v̷2,vαv′βΩαβ(cV)=1+v̷′2(y−1)γμγ5−2vμγ51+v̷2,gαβΩαβ(cV)=1+v̷′23γμγ51+v̷2,vαvβΩαβ(cA)=1+v̷′2−2yvμ+2γμ1+v̷2,vαv′βΩαβ(cA)=1+v̷′2(y+1)γμ−2vμ1+v̷2,gαβΩαβ(cA)=1+v̷′23γμ1+v̷2,vαvβΩαβ(bV)=1+v̷′2(y−1)γμγ5−2vμγ51+v̷2,vαv′βΩαβ(bV)=1+v̷′2(1−y)γμγ5+2vμγ5−2(y+1)v′μγ51+v̷2,gαβΩαβ(bV)=1+v̷′2−γμγ5−2v′μγ51+v̷2,vαvβΩαβ(bA)=1+v̷′2(y+1)γμ−2vμ1+v̷2,vαv′βΩαβ(bA)=1+v̷′2(y+1)γμ−2vμ+2(y−1)v′μ1+v̷2,gαβΩαβ(bA)=1+v̷′2γμ+2v′μ1+v̷2, where V(A)≡γμ(γμγ5).

References

[1]

D.E.

Groom

Particle Data GroupEur. Phys. J. C

2000

[2]

Albrecht

ARGUS Collaboration

Phys. Lett. B

317

1993

227

P.L.

Frabetti

E687 Collaboration

Phys. Rev. Lett.

1994

961

K.W.

Edwards

CLEO Collaboration

Phys. Rev. Lett.

1995

3331

J.P.

Alexander

CLEO Collaboration

Phys. Rev. Lett.

1999

3390

[3]

Isgur

M.B.

WisePhys. Lett. B

232

1989

113

Isgur

M.B.

WisePhys. Lett. B

237

1990

527

E.V.

ShuryakPhys. Lett. B

1980

134

GeorgiPhys. Lett. B

240

1990

447

Eichten

HillPhys. Lett. B

234

1990

511

M.B.

Voloshin

M.A.

ShifmanYad. Fiz.

1987

463

M.B.

Voloshin

M.A.

ShifmanYad. Fiz.

1988

801

Nussinov

WetzelPhys. Rev. D

1987

130

A.F.

Falk

Georgi

Grinstein

M.B.

WiseNucl. Phys. B

343

1990

[4]

Isgur

M.B.

WiseNucl. Phys. B

348

1991

276

GeorgiNucl. Phys. B

348

1991

293

J.-P.

Lee

Liu

H.S.

SongPhys. Rev. D

1998

014013

[5]

Mannel

Roberts

RyzakPhys. Lett. B

271

1991

421

Y.-B.

Dai

X.-H.

Guo

C.-S.

HuangNucl. Phys. B

421

1994

277

[6]

ChoPhys. Lett. B

285

1992

145

J.-P.

Lee

Liu

H.S.

SongPhys. Rev. D

1999

014006

[7]

C.-S.

Huang

C.-F.

Qiao

H.-G.

YanPhys. Lett. B

437

1998

403

C.-S.

Huang

H.-G.

YanPhys. Rev. D

1999

114022

[8]

Shifman

Vainshtein

ZakharovNucl. Phys. B

147

1979

385

Shifman

Vainshtein

ZakharovNucl. Phys. B

147

1979

448

[9]

A.G.

Grozin

O.I.

ZakovlevPhys. Lett. B

285

1992

254

E.V.

ShuryakNucl. Phys. B

198

1982

Bagan

Chabab

H.G.

Dosch

NarisonPhys. Lett. B

301

1993

243

Y.-B.

Dai

C.-S.

Huang

Liu

C.-D.

LüPhys. Lett. B

371

1996

Y.-B.

Dai

C.-S.

Huang

M.-Q.

Huang

LiuPhys. Lett. B

387

1996

379

Groote

J.G.

Körner

O.I.

YakovlevPhys. Rev. D

1997

3016

Groote

J.G.

Körner

O.I.

YakovlevPhys. Rev. D

1997

3943

[10]

J.-P.

Lee

Liu

H.S.

SongPhys. Lett. B

476

2000

303

[11]

S.-L.

ZhuPhys. Rev. D

2000

114019

C.-S.

Huang

Zhang

S.-L.

ZhuPhys. Lett. B

492

2000

288

[12]

M.-Q.

Huang

J.-P.

Lee

Liu

H.S.

SongPhys. Lett. B

502

2000

133

[13]

A.K.

Leibovich

Ligeti

I.W.

Stewart

M.B.

WisePhys. Rev. Lett.

1997

3995

A.K.

Leibovich

Ligeti

I.W.

Stewart

M.B.

WisePhys. Rev. D

1997

308

[14]

Colangelo

De Fazio

PaverPhys. Rev. D

1998

116005

[15]

M.-Q.

Huang

Y.-B.

DaiPhys. Rev. D

1999

034018

[16]

M.-Q.

Huang

C.-Z.

Y.-B.

DaiPhys. Rev. D

2000

054010

M.-Q.

Huang

Y.-B.

DaiPhys. Rev. D

2001

014034

[17]

W.Y.

Wang

Y.L.

WuInt. J. Mod. Phys. A

2001

2505

[18]

A.F.

FalkNucl. Phys. B

378

1992

[19]

A.K.

Leibovich

I.W.

StewartPhys. Rev. D

1998

5620

[20]

NeubertPhys. Rev. D

1992

2451

[21]

Blok

ShifmanPhys. Rev. D

1993

2949