The Vector Meson And Heavy Meson Strong Interaction

# The Vector Meson And Heavy Meson Strong Interaction

## Abstract

We calculate the coupling constants between the light vector mesons and heavy mesons within the framework of the light-cone QCD sum rule in the leading order of heavy quark effective theory. The sum rules are very stable with the variations of the Borel parameter and the continuum threshold. The extracted couplings will be useful in the study of the possible heavy meson molecular states. They may also helpful in the interpretation of the proximity of X(3872), Y(4260) and Z(4430) to the threshold of two charmed mesons through the couple-channel mechanism.

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

## I Introduction

A number of hadronic states which can not be easily accommodated in the conventional quark model have been observed experiementally in recent years, such as 3872 (), Y(4260) 4260 () and 4430 (). Their masses are very close to the thresholds of , and respectively. It was speculated that the coupled-channel effect may play an important role because of the attraction between these D mesons. Alternatively, they were considered to be possible candidates of the heavy molecular states composed of two mesons. These loosely bound states are formed by exchanging light mesons such as , , and etc. Up to now, the pion heavy meson strong interaction is relatively known due to chiral symmetry. However, the vector meson heavy meson strong interaction has not be extensively studied yet, which accounts for the relatively short distance interaction between two heavy mesons.

Heavy quark effective theory (HQET) hqet () is a systematic approach to study the spectra and transition amplitudes of heavy hadrons. In HQET, the expansion is performed in term of , where is the mass of the heavy quark involved. In the limit , heavy hadrons form a series of degenerate doublets due to heavy quark symmetry. The two states in a doublet share the same quantum number , the angular momentum of the light components. The meson doublets , and are conventionally denoted as , and .

Light-cone QCD sum rules (LCQSR) light-cone () is a very useful non-perturbative approach to determine various hadronic transition form factors. One considers the T-product of the two interpolating currents sandwiched between the vacuum and an hadronic state in this framework. Now the operator product expansion (OPE) is performed near the light-cone rather than at small distance as in the conventional SVZ sum rules svz (). The double Borel transformation is always invoked to suppress the excited state and the continuum contribution.

The coupling constant between and was calculated with LCQSR in full QCD in Ref. aliev (). The couplings , , and were calculated in full QCD in Ref. li (). Their values in the limit are also discussed in this paper. The coupling between doublets and are studied in the leading order of HQET in Ref. zhu () .

In this work we use LCQSR to calculate the coupling constants between three doublets , , and within the two doublets , . Due to the covariant derivative in the interpolating currents of doublet, the contribution from the 3-particle light-cone distribution amplitudes of the meson has to be included when dealing with the decay between doublets and (). We work in HQET to differentiate the two states with the same value 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{Jα1⋯αjj,P,jl(x)J†β1⋯βj′j,P,jl(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 ρ coupling constants

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

 J†0,−,12 = √12¯hvγ5q, (3) J†α1,−,12 = √12¯hvγαtq, (4) J†0,+,12 = √12¯hvq, (5) J†α1,+,12 = √12¯hvγ5γαtq, (6) J†α1,+,32 = √34¯hvγ5(−i){Dαt−13γαt^Dt}q, (7) J†α1α22,+,32 = √18¯hv(−i){γα1tDα2t+γα2tDα1t−23gα1α2t^Dt}q, (8)

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 light components in the limit , there are three independent coupling constants between doublets and . We denote them as , and where and the number following them indicates the orbital and total angular momentum of the final meson respectively. All of these three coupling constants appear in the decay process under consideration. The decay amplitude can now be written as

 M(T1→H1+ρ) = Ii{ϵηϵ∗e∗vgs1T1H1ρ+[ϵηϵ∗qv(e∗⋅qt)−13ϵηϵ∗e∗vq2t]gd1T1H1ρ (9) +[ϵηe∗qv(ϵ∗⋅qt)+ϵϵ∗e∗qv(η⋅qt)]gd2T1H1ρ},

where , and denote the polarization vector of , and respectively, is the momentum of the meson, and . for the charged and neutral meson respectively. The vector notations in Levi-Civita tensor come from an index contraction between Levi-Civita tensor and the vectors, for example, .

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

 ∫d4xe−ik⋅x⟨ρ(q)|T{Jβ1,−,12(0)J†α1,+,32(x)}|0⟩ = Ii{ϵαβe∗vGs1T1H1ρ(ω,ω′) (10) +[ϵαβqv(e∗⋅qt)−13ϵαβe∗vq2t]Gd1T1H1ρ(ω,ω′) +[ϵαe∗qvqβt+ϵβe∗qvqαt]Gd2T1H1ρ(ω,ω′)},

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

 ⟨0|T{hv(0)¯hv(x)}|0⟩=1+^v2∫dtδ4(−x−vt). (11)

The correlation function can now be expressed as

 −√38∫dxe−ik⋅x∫∞0dtδ(−x−vt)Tr{γβt1+^v2(−iγ5)(Dαt−13γαt^Dt)⟨ρ(q)|q(x)¯q(0)|0⟩}. (12)

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

At the hadron level, the ’s in (10) has the following pole terms

 GT1H1ρ(ω,ω′)=f−,1/2f+,3/2gT1H1ρ(2¯Λ−,1/2−ω′)(2¯Λ+,3/2−ω)+c2¯Λ−,1/2−ω′+c′2¯Λ+,3/2−ω, (13)

where , . etc is 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 (13) are eliminated. We arrive at

 √3gs1T1H1ρf−,12f+,32e−¯Λ+,3/2+¯Λ−,1/2T = 13fTρm4ρhs[1]∥(¯u0)1T−13fTρm4ρ(uhs∥)[1](¯u0)1T−23fTρm4ρS[−1,0](u0)1T+124fTρm4ρAT(¯u0)1T (14) −124fTρm4ρAT(¯u0)¯u01T+23fTρm4ρB[3]T(¯u0)1T−14fTρm2ρC[1]T(¯u0)Tf0(ωcT)−14fTρm2ρhs∥(¯u0)Tf0(ωcT) −112fTρm2ρhs(1)∥(¯u0)Tf0(ωcT)+112fTρm2ρ(uhs∥)′(¯u0)Tf0(ωcT)+13fTρm2ρS[1,0](u0)Tf0(ωcT) −16fTρm2ρφ⊥(¯u0)Tf0(ωcT)+16fTρm2ρφ⊥(¯u0)¯u0Tf0(ωcT)−196fTρm2ρA(2)T(¯u0)Tf0(ωcT) +196fTρm2ρ(uAT)(2)(¯u0)Tf0(ωcT)−16fTρm2ρB[1]T(¯u0)Tf0(ωcT)+124fTρφ(2)⊥(¯u0)T3f2(ωcT) −124fTρ(φ⊥u)(2)(¯u0)T3f2(ωcT)+13fTρm4ρT[−1,0](u0)1T−43fTρm4ρT[−1,0]2(u0)1T +112fTρm2ρT[1,0](u0)Tf0(ωcT)+16fTρm2ρT[1,0]2(u0)Tf0(ωcT)−23fTρm4ρT[−1,0]3(u0)1T −16fTρm2ρT[1,0]3(u0)Tf0(ωcT)+112fρm5ρA[2](¯u0)1T2−112fρm5ρA[1](¯u0)1T2+112fρm5ρ(uA)[1](¯u0)1T2 +43fρm5ρC[4](¯u0)1T2−23fρm5ρC[3](¯u0)1T2+23fρm5ρ(uC)[3](¯u0)1T2+124fρm3ρA(¯u0) +148fρm3ρA′(¯u0)−148fρm3ρ(uA)′(¯u0)−13fρm3ρA[0,0](u0)+16fρm3ρC[2](¯u0)+16fρm3ρC[1](¯u0) −16fρm3ρ(uC)[1](¯u0)+112fρm3ρg(a)[1]⊥(¯u0)−112fρm3ρg(a)⊥(¯u0)+112fρm3ρg(a)⊥(¯u0)¯u0 +13fρm3ρg(v)[2]⊥(¯u0)−112fρmρA[2,0](u0)T2f1(ωcT)+12fρm3ρV[0,0](u0)+16fρmρV[2,0](u0)T2f1(ωcT) −13fρm3ρφ[2]∥(¯u0)+13fρm3ρφ[1]∥(¯u0)−13fρm3ρ(uφ∥)[1](¯u0)+43fρm5ρΨ[−2,0](u0)1T2+16fρm3ρV[0,0](u0) −13fρm3ρΨ[0,0](u0)+43fρm5ρ~Φ[−2,0](u0)1T2−13fρm3ρ~Φ[0,0](u0)+124fρmρg(a)(1)⊥(¯u0)T2f1(ωcT) +148fρmρg(a)(2)⊥(¯u0)T2f1(ωcT)−148fρmρ(g(a)⊥u)(2)(¯u0)T2f1(ωcT)+16fρmρg(v)⊥(¯u0)T2f1(ωcT) −16fρmρφ∥(¯u0)T2f1(ωcT)−112fρmρφ′∥(¯u0)T2f1(ωcT)+112fρmρ(uφ∥)′(¯u0)T2f1(ωcT),
 √3gd1T1H1ρf−,12f+,32e−¯Λ+,3/2+¯Λ−,1/2T = −fTρm2ρhs[1]∥(¯u0)1T+fTρm2ρ(uhs∥)[1](¯u0)1T+2fTρm2ρS[−1,0](u0)1T+116fTρm2ρAT(¯u0)1T (15) −116fTρm2ρAT(¯u0)¯u01T−2fTρm2ρB[3]T(¯u0)1T−14fTρφ⊥(¯u0)Tf0(ωcT)+14fTρφ⊥(¯u0)¯u0Tf0(ωcT) −fTρm2ρT[−1,0](u0)1T+fTρm2ρT[−1,0]2(u0)1T−fTρm2ρT[−1,0]3(u0)1T−14fρm3ρA[2](¯u0)1T2 +14fρm3ρA[1](¯u0)1T2−14fρm3ρ(uA)[1](¯u0)1T2−4fρm3ρC[4](¯u0)1T2+2fρm3ρC[3](¯u0)1T2 −2fρm3ρ(uC)[3](¯u0)1T2−12fρmρA[0,0](u0)−14fρmρg(a)[1]⊥(¯u0)−18fρmρg(a)⊥(¯u0) +18fρmρg(a)⊥(¯u0)¯u0−fρmρg(v)[2]⊥(¯u0)+fρmρV[0,0](u0)+fρmρφ[2]∥(¯u0)−fρmρφ[1]∥(¯u0) +fρmρ(uφ∥)[1](¯u0)−4fρm3ρΨ[−2,0](u0)1T2−4fρm3ρ~Φ[−2,0](u0)1T2,
 √3gd2T1H1ρf−,12f+,32e−¯Λ+,3/2+¯Λ−,1/2T = 316fTρm2ρAT(¯u0)1T−316fTρm2ρAT(¯u0)¯u01T−34fTρφ⊥(¯u0)Tf0(ωcT)+34fTρφ⊥(¯u0)¯u0Tf0(ωcT) (16) −3fTρm2ρT[−1,0]2(u0)1T−3fTρm2ρT[−1,0]3(u0)1T+32fρmρA[0,0](u0)−38fρmρg(a)⊥(¯u0)+38fρmρg(a)⊥(¯u0)¯u0,

where is the continuum subtraction factor, and is the continuum threshold, , and . and are the two Borel parameters. We have used the Borel transformation to obtain (II),(II) and (II). In the above expressions we have used the following functions and . They are defined as

 F[n](¯u0) ≡ ∫¯u00⋯∫x30∫x20F(x1)dx1dx2⋯dxn, (17) F[0,0](u0) ≡ ∫u00∫1−α2u0−α2F(1−α2−α3,α2,α3)α3dα3dα2, (18) F[1,0](u0) ≡ ∫u00F(1−u0,α2,u0−α2)u0−α2dα2−∫1−u00F(u0,1−u0−α3,α3)α3dα3, (19) F[2,0](u0) ≡ (20) F[−1,0](u0) ≡ ∫u00∫u0−α20F(1−α2−α3,α2,α3)dα3dα2 (21) +∫u00∫1−α2u