Strong and Electromagnetic Decays of The D-wave Heavy Mesons

Strong and Electromagnetic Decays of The D-wave Heavy Mesons

Abstract

We calculate the , , , and coupling constants between the heavy meson doublets and / within the framework of the light-cone QCD sum rule at the leading order of heavy quark effective theory. Most of the sum rules are stable with the variations of the Borel parameter and the continuum threshold. Then we calculate the strong and electromagnetic decay widths of the -wave heavy mesons. Their total widths are around several tens of MeV, which is helpful in the future experimental search.

Heavy quark effective theory, Light-cone QCD sum rule
pacs:
12.39.Hg, 12.38.Lg

I Introduction

Heavy quark effective theory (HQET) hqet () is a framework which is widely used to study the spectra and transition amplitudes of heavy hadrons containing one heavy quark. In HQET, the expansion is performed in terms of , where is the mass of the heavy quark involved. At the leading order of , the HQET Lagrangian respects the heavy quark flavor-spin symmetry, therefore heavy hadrons form a series of degenerate doublets. The two members in a doublet share the same quantum number , the angular momentum of the light components. The -wave doublet is conventionally denoted as and the -wave doublets are conventionally denoted as . We denote the -wave doubtlets as .

Shifman-Vainshtein-Zakharov (SVZ) sum rules svz () is a nonperturbative approach used to determine hadronic parameters such as the hadron mass. The vacuum expectation value of the product of two interpolating currents is considered in this approach. After performing the operator product expansion (OPE), one obtains sum rules which relate the hadronic parameters to expressions containing vacuum condensates parameterizing the QCD nonperturbative effect. In the late 1980s, light-cone QCD sum rules (LCQSR) light-cone () was developed to calculate various hadronic transition form factors. Now the OPE of the product of two interpolating currents sandwiched between the vacuum and an hadronic state is performed near the light-cone rather than at a small distance as in the conventional SVZ sum rules.

The coupling constants and were calculated with LCQSR in full QCD in Ref. rhodecayaliev (). The couplings , , , and were calculated in full QCD in Ref. rhodecayli (). Their values in the limit are also discussed in this paper. The coupling constants between the three doublets , , and within the two doublets , are systematically studied with LCQSR at the leading order of HQET in Ref. rhodecayzhu (). The coupling constants between the -wave and -wave heavy mesons have been studied using QCD sum rules or/and LCQSR in Ref. pidecaycolangelo (). The coupling constants between / and // are calculated with LCQSR at the leading order of HQET in Ref. pidecaydaizhu (); pidecaywei (). The radiative decay between , , and are studied using the light-cone QCD sum rule at the leading order of HQET in Ref. radiativedecayzhu (). In Ref. radiativedecaycolangelo (), the radiative decays of and are studied using LCQSR approach.

In this work, we use LCQSR to calculate the , , , and coupling constants between the doublets and /. Because of the covariant derivative in the interpolating currents of the doublet, the contribution from the 3-particle light-cone distribution amplitudes of , , , and have to be included. We work in HQET to differentiate the two states with the same value and yet quite different decay widths. The interpolating currents adopted in our work have been properly constructed in Ref. huang (). They satisfy

 ⟨0|Jα1⋯αjj,P,jl(0)|j′,P′,j′l⟩ = fPjlδjj′δPP′δjlj′lηα1⋯αj, (1) i⟨0|T{Jβ1⋯βj′j,P,jl(x)J†α1⋯αjj′,P′,j′l(0)}|0⟩ = δjj′δPP′δjlj′l(−1)jSgα1β1t⋯gαjβjt∫dtδ(x−vt)ΠP,jl(x), (2)

in the limit . Here is the polarization tensor for the spin state, is the velocity of the heavy quark, , denotes symmetrizing the indices and subtracting the trace terms separately in the sets and .

Ii Sum Rules for the π coupling constants

We shall perform the calculation at the leading order of HQET. According to Ref. huang (), the interpolating currents for the doublets , , and read as

 J†0,−,12 = √12¯hvγ5q, J†α1,−,12 = √12¯hvγαtq, J†0,+,12 = √12¯hvq, J†α1,+,12 = √12¯hvγ5γαtq, J†α1,+,32 = √34¯hvγ5(−i){Dαt−13γαt^Dt}q, J†α1α22,+,32 = √18¯hv(−i){γα1tDα2t+γα2tDα1t−23gα1α2t^Dt}q, J†α1,−,32 = √34¯hv(−i){Dαt−13γαt^Dt}q, J†α1α22,−,32 = √18¯hv(−i)γ5{γα1tDα2t+γα2tDα1t−23gα1α2t^Dt}q, (3)

where is the heavy quark field in HQET, , , , and is the velocity of the heavy quark.

We consider the decay of to to illustrate our calculation. Here the subscript of () indicates the spin of the meson involved. Owing to the conservation of the angular momentum of the light components in the limit , there is only one independent coupling constant between doublets and . We denote it as where and the number following it indicate the orbital and total angular momentum of the final meson respectively. can be defined in terms of the decay amplitude as

 M(M2→H1+π) = Iηα1α2[ϵ∗α1tqα2t−13gα1α2t(ϵ∗⋅qt)]gp1M2H1π, (4)

where and denote the polarization tensors of the initial and final heavy mesons respectively, is the momentum of the meson. The transversal tensor are defined as , , and . for the charged and neutral meson, respectively.

To obtain the sum rules for the coupling constants , we consider the correlation function

 ∫d4xe−ik⋅x⟨π(q)|T{Jβ1,−,12(0)J†α1α22,−,32(x)}|0⟩ = I[12(gα1βtqα2t+gα2βtqα1t)−13gα1α2tqβt]Gp1M2H1π(ω,ω′), (5)

where , . At the leading order of HQET, the heavy quark propagator reads as

 ⟨0|T{hv(0)¯hv(x)}|0⟩=1+^v2∫dtδ4(−x−vt). (6)

The correlation function can now be expressed as

 −i4∫dxe−ik⋅x∫∞0dtδ(−x−vt)Tr{γβt1+^v2γ5[γα1tDα2t+γα2tDα1t−23gα1α2t^Dt]⟨π(q)|q(x)¯q(0)|0⟩}. (7)

It can be further calculated using the light-cone wave functions of the meson. To our approximation, we need the 2- and 3-particle light-cone wave functions. Their definitions are collected in Appendix A.

At the hadron level, in (5) has the following pole terms

 Gp1M2H1π(ω,ω′)=f−,1/2f−,3/2gp1M2H1π(2¯Λ−,1/2−ω′)(2¯Λ−,3/2−ω)+c2¯Λ−,1/2−ω′+c′2¯Λ−,3/2−ω, (8)

where , , , etc. are the overlap amplitudes of their interpolating currents with the heavy mesons.

can now be expressed by the meson light-cone wave functions. After the Wick rotation and the double Borel transformation with and , the single-pole terms in (8) are eliminated. We arrive at

 gp1M2H1πf−,12f−,32e−¯Λ−,3/2+¯Λ−,1/2T =−148fπ{−12[ϕ′π(¯u0)−(uϕπ)′(¯u0)]T2f1(ωcT) =−4m2πmu+md[6T[1,0](u0)+6ϕp(¯u0)−6(uϕp)(¯u0)+ϕσ(¯u0)]Tf0(ωcT) (9)

where is the continuum subtraction factor, and is the continuum threshold, , , and . and are the two Borel parameters. We have employed the Borel transformation to obtain (II). In the above expressions, we have used the functions and which are defined in Appendix B.

The coupling constant between doublets and can be defined similarly:

 M(M2→S1+π) = Iiηα1α2ϵ∗βεβα1qvqα2tgd2M2S1π, M(M1→T1+π) = I(η⋅ϵ∗t)gs0M1T1π+I[(η⋅qt)(ϵ∗⋅qt)−q2t3(η⋅ϵ∗t)]gd2M1T1π, M(M2→M1+π) = 2Iηα1α2[ϵ∗α1tqα2t−13gα1α2t(ϵ∗⋅qt)]gp1M2M1π (10) +Iηα1α2{qα1tqα2t(ϵ∗⋅qt)−q2t5[2ϵ∗α1tqα2t+gα1α2t(ϵ∗⋅qt)]}gf3M2M1π.

Here the vector notations in the Levi-Civita tensor come from an index contraction between the Levi-Civita tensor and the vectors, for example, . The Levi-Civita tensor is defined as . The sum rules for the coupling constants in Eq. (II) are

 gd2M2S1πf+,12f−,32e−¯Λ−,3/2+¯Λ+,1/2T =−124fπ{12[ϕπ(¯u0)−(uϕπ)(¯u0)]Tf0(ωcT)+4m2πmu+md[6T[0,0](u0)−ϕσ(¯u0)+(uϕσ)(¯u0)] gs0M1T1πf+,32f−,32e−¯Λ−,3/2+¯Λ+,3/2T =fπ48(u(1−u)ϕπ)(3)(¯u0)T4f3(ωcT)+fπm2π24mud[m2π−m2ud6m2πϕ(2)σ(¯u0)+(u(1−u)ϕP)(2)(¯u0)+(1−α2T)[3,0](u0) =−(α3T)[3,1](u0)]T3f2(ωcT)−fπm2π12[38A′(¯u0)+116(u(1−u)A)(3)(¯u0)+B[1](¯u0)+(uB−12B)(¯u0) =−12(u(1−u)B)′(¯u0)+(u(1−u)ϕπ)′(¯u0)+(2A∥+2A⊥+V∥+V⊥)[1,0](u0)+((1−α2)(A∥+V⊥))[2,0](u0) =−(α3(A∥+V⊥))[2,1](u0)]T2f1(ωcT)−fπm4π6mud[(u(1−u)ϕP)(¯u0)+m2π−m2ud6m2πϕσ(¯u0)+((1−α2)T)[1,0](u0) =−(α3T)[1,1](u0)]Tf0(ωcT)+fπm4π3[116(u(1−u)A)′(¯u0)+(uB−12B)[2](¯u0)−12(u(1−u)B)[1](¯u0) =+B[3](¯u0)+(A∥+A⊥−V∥−V⊥)[−1,0](u0)−((1−α2)(A∥+V⊥))[0,0](u0)+(α3(A∥+V⊥))[0,1](u0)], gd2M1T1πf+,32f−,32e−¯Λ−,3/2+¯Λ+,3/2T =−fπ8(u(1−u)ϕπ)′(¯u0)T2f1(ωcT)−fπm2π4mud[(u(1−u)ϕP)(¯u0)+m2π−m2ud6m2πϕσ(¯u0)+((1−α2)T)[1,0](u0) =−(α3T)[1,1](u0)]Tf0(ωcT)+fπm2π2[116(u(1−u)A)′(¯u0)−12(u(1−u)B)[1](¯u0)+B[3](¯u0)+((u−12)B)[2](¯u0) =+(A∥+A⊥+2V∥+2V⊥)[−1,0](u0)−((1−α2)(A∥+V⊥))[0,0](u0)+(α3(A∥+V⊥))[0,0](u0)], gp1M2M1πf2−,32e−2¯Λ−,3/2T =fπmud80√6(u(1−u)ϕπ)(2)(¯u0)T3f2(ωcT)−fπm2π240√6mud[m2π−m2udm2π(u(1−u)ϕσ)(2)(¯u0)+6((1−α2)T)[2,0](u0) =−6(α3T)[2,1](u0)]T2f1(ωcT)+fπm2π20√6[−58A(¯u0)−116(u(1−u)A)(2)(¯u0)−(u(1−u)ϕπ)(¯u0) =−4(A∥+A⊥)[0,0](u0)+((1−α2)(A∥+A⊥))[1,0](u0)−(α3(A∥+A⊥))[1,1](u0)−5(V∥+V⊥)[0,0](u0)]× =Tf0(ωcT)+fπm4π60√6mud[m2π−m2udm2π(u(1−u)ϕσ)(¯u0)−6((1−α2)T)[0,0](u0)+6(α3T)[0,1](u0)]+fπm4π5√6× =[116(u(1−u)A)(¯u0)−(A∥+A⊥)[−2,0](u0)+((1−α2)(A∥+A⊥))[−1,0](u0)−(α3(A∥+A⊥))[−1,1](u0)]1T, gf3M2M1πf2−,32e−2¯Λ−,3/2T =−√6fπ4(u(1−u)ϕπ)(¯u0)Tf0(ωcT)+√6fπm2π2mud[m2π−m2ud6m2π(u(1−u)ϕσ)(¯u0)−((1−α2)T)[0,0](u0) =+(α3T)[0,1](u0)]+√6fπm2π[116(u(1−u)A)(¯u0)−(A∥+A⊥)[−2,0](u0)+((1−α2)(A∥+A⊥))[−1,0](u0) =−(α3(A∥+A⊥))[−1,1](u0)]1T, (11)

with . The coupling constants of the other decay channels are defined as

 M(M1→H0+π) = I(η⋅qt)gp1M1H0π, M(M1→H1+π) = Iiεηϵ∗qvgp1M1H1π, M(M1→S1+π) = I[(η⋅qt)(ϵ∗⋅qt)−13(η⋅ϵ∗t)q2t]gd2M1S1π, M(M2→S0+π) = Iηα1α2[qα1tqα2t−13gα1α2tq2t]gd2M2S0π, M(M1→T2+π) = 2Iiϵ∗β1β2εβ1ηqvqβ2tgd2M1T2π, M(M2→T1+π) = 2Iiηα1α2εα1ϵ∗qvqα2tgd2M2T1π, M(M2→T2+π) = 2Iηα1α2ϵ∗β1β2[gα1β1tgα2β2t−13gα1α2tgβ1β2t]gs0M2T2π +Iηα1α2ϵ∗β1β2{qα1tqα2tgβ1β2t+qβ1tqβ2tgα1α2t M(M1→M1+π) = Iiεηϵ∗qvgp1M1M1π, M(M2→M1+π) = 2Iηα