\rho Meson Decays of Heavy Hybrid Mesons

# ρ Meson Decays of Heavy Hybrid Mesons

Liang Zhang School of Physics, Beihang University, Beijing 100191, China    Peng-Zhi Huang Department of Physics and State Key Laboratory of Nuclear Physics and Technology
Peking University, Beijing 100871, China
Theoretical Physics Center for Science Facilities, CAS, Beijing 100049, China
###### Abstract

We calculate the meson couplings between the heavy hybrid doublets and the ordinary doublets in the framework of the light-cone QCD sum rule. The sum rules obtained rely mildly on the Borel parameters in their working regions. The resulting coupling constants are rather small in most cases.

Heavy hybrid meson, QCD sum rule, Heavy quark effective theory
###### pacs:
12.39.Mk, 12.38.Lg, 12.39.Hg

## I Introduction

Hadron states that do not fit into the constituent quark model have been studied widely in the past several decades. In recent years, the discovery of a number of unexpected exotic resonances such as the so called XYZ mesons has revitalized the research of the existence of unconventional hadron states and their nature.

Theoretically, Quantum Chromodynamics (QCD), the fundamental theory of the strong interaction, may allow a far richer spectrum than the conventional quark model. Fox example, hybrids (), glueballs (, , ), and multi-quark states (, , ) may not be prohibited by QCD. Those with are called “exotic” states. They have attracted much interest because they are not allowed by the constituent quark model and do not mix with the ordinary mesons.

Evidence of exotic mesons with , e.g. 1400 , 1600 , have emerged in the last few years. They are usually considered as candidates of hybrid mesons and have been studied extensively in various frameworks such as QCD sum rules, lattice QCD, AdS/QCD, the flux tube model, etc. The masses and decay properties of the states have been studied in the framework of QCD sum rules QSRmass ; QSRdecay .

Based on the accumulated evidence of these light hybrid mesons, it is plausible to assume the existence of heavy quarkonium hybrids () and heavy hybrid mesons containing one heavy quark () which may be not exotic. Govaerts et al. have studied these states in several works Govaerts . In zhu1 , the masses of were calculated at the leading order of heavy quark effective theory (HQET) hqet . In zhu2 , the masses of and their pionic couplings to ordinary heavy mesons were calculated.

In the heavy quark limit, the binding energy and the pionic couplings of to were worked out in HeavyHybridparameter by Shifman-Vainshtein-Zakharov (SVZ) sum rules svz . HQET describes the large mass () asymptotics. At the leading order of this theory, the Lagrangian is endowed with the heavy quark flavor-spin symmetry, and the spectrum of consists of degenerate doublets. The components of a doublet share the same , the angular momentum of the light degrees of freedom. For example, we denote the doublet as , which consists of two -wave . Similarly, the -wave doublets are denoted as and the -wave doublets as . We denote the two doublets with parity and as and , respectively. Similarly, we use and to denote the two doublets with positive parity and negative parity, respectively.

In this work, we adopt the light-cone QCD sum rules (LCQSR) approach light-cone to investigate the meson couplings between and . We derive the sum rules for the meson couplings between doublets and () in Sec. 2. The numerical analysis is given in Sec. 3, followed by a brief conclusion in Sec. 4. The details of the partial amplitudes of these decay channels are presented in Appendix A. The light-cone wave functions of the meson involved in our calculation are listed in Appendix B.

## Ii ρ meson couplings

The interpolating currents for and adopted in our calculation can be written as

 J†Hh0=√12¯hvigsγ5σt⋅Gq, J†αHh1=√12¯hvigsγαtσt⋅Gq, J†αMh1=¯hvgs[3Gαβtγβ+iγαtσt⋅G]q, J†α1α2Mh2=√32¯hvgsγ5[Gα1βtγβγα2t+Gα2βtγβγα1t−23igα1α2tσt⋅G]q, (1)

where and . The subscript means that the corresponding Lorentz tensor is perpendicular to , the 4-velocity of the heavy quark. . For any asymmetric tensor , we may define

 Aα1α2⋯αnt=Aα1α2⋯αn−n∑i=1(Aα1⋯αi−1ααi+1⋯αnvα)vαi. (2)

We define the overlapping amplitudes between the these interpolating currents and the corresponding hybrids as

 ⟨0|JHh0(0)|Hh0(v)⟩=fHh0, ⟨0|JαHh1(0)|Hh1(v,λ)⟩=fHh1ηαHh1(v,λ), ⟨0|JαMh1(0)|Mh1(v,λ)⟩=fMh1ηαMh1(v,λ), ⟨0|Jα1α2Mh2(0)|Mh2(v,λ)⟩=fMh2ηα1α2Mh2(v,λ), (3)

where denotes the polarization of the heavy hybrid. These symmetric traceless tensors are perpendicular to , namely .

We obtain the interpolating currents for the doublets and by simply inserting into the currents in Eq. (II):

 J†Sh0=√12¯hvigsσt⋅Gq, J†αSh1=√12¯hvigsγ5γαtσt⋅Gq, J†α1α2Th2=√32¯hvgs[Gα1βtγβγα2t+Gα2βtγβγα1t−23igα1α2tσt⋅G]q. (4)

The corresponding overlapping amplitudes and projection operators can be defined similarly to Eq. (II).

The interpolating currents for doublets and read:

 J†H0 = √12¯hvγ5q, J†αH1 = √12¯hvγαtq, J†S0 = √12¯hvq, J†αS1 = √12¯hvγαtγ5q. (5)

The amplitudes between the ordinary heavy mesons and the states created by these currents acting on the vacuum state are

 ⟨0|JH0(0)|H0(v)⟩=fH0, ⟨0|JαH1(0)|H1(v,λ)⟩=fH1ϵαH1(v,λ), ⟨0|JS0(0)|S0(v)⟩=fS0, ⟨0|JαS1(0)|S1(v,λ)⟩=fS1ϵαS1(v,λ). (6)

Here we outline the deduction of the sum rules for and , where is the orbital angular momentum of the meson, the superscript ‘0’ and ‘1’ are the total angular momentum of the meson. We define and in term of the decay amplitude of the process :

 M(Hh1→H1+ρ) = I[(e∗⋅ηt)(ϵ∗⋅qt)−(e∗⋅ϵ∗t)(η⋅qt)]gp1Hh1H1ρ+I(e∗⋅qt)(ϵ∗⋅ηt)gp0Hh1H1ρ, (7)

where , and are the polarization of , and , respectively, and denotes the momentum of the . For the charged meson, , and if the final meson is neutral.

We consider the following correlation function:

 i∫dx e−ik⋅x⟨ρ(q)|JβH1(0)J†αHh1(x)|0⟩=I[eαtqβt−qαteβt]Gp1Hh1H1ρ(ω,ω′)+Igαβt(e⋅qt)Gp0Hh1H1ρ(ω,ω′), (8)

where and , and we have the following dispersion relation

 Gp1Hh1H1ρ(ω,ω′)=∫∞0ds1∫∞0ds2ρp1Hh1H1ρ(s1,s2)(s1−ω−iϵ)(s2−ω′−iϵ)+∫∞0ds1ρp11(s1)s1−ω−iϵ+∫∞0ds2ρp12(s2)s2−ω′−iϵ+⋯, (9)

with

 ρp1Hh1H1ρ(s1,s2)=fHhfHgp1Hh1H1ρδ(s1−2ΛHh)δ(s2−2ΛH)+⋯. (10)

The case of is similar. can be worked out by OPE near the light-cone when , and be formulated with the meson light-cone wave functions

 Gp0Hh1H1ρ(ω,ω′) = −14∫∞0dt∫Dα–– eit(¯u2ω+u2ω′)m2(q⋅v)3 [−2fρm3˜Ψ(α––)−2fTρm2T(α––)(q⋅v)+fρm[6˜Φ(α––)+2˜Ψ(α––)+A(α––)](q⋅v)2 +2fTρ[T(α––)+2T1(α––)+2T2(α––)](q⋅v)3], Gp1Hh1H1ρ(ω,ω′) = 14∫∞0dt∫Dα–– eit(¯u2ω+u2ω′)mq⋅v (11) [fρm2[V(α––)+A(α––)]−2fTρm[T1(α––)−T2(α––)+S(α––)](q⋅v)−2fρ[V(α––)+A(α––)](q⋅v)2].

in which and .

The double Borel transformation eliminates the terms on the right side of Eq. (9), except the first one which is a double dispersion relation. Now we arrive at

 fHhfHgp0Hh1H1ρe−2¯u0ΛHh/T−2u0ΛH/T =m2ρ{−12fρm3ρ˜Ψ{−3}−12fTρm2ρT{−2}−fρmρ[6˜Φ[−1](u0)+2˜Ψ[−1](u0)+A[−1](u0)] =+fTρ[T[0](u0)+2T[0]1(u0)+2T[0]2(u0)]Tf0(ω′cT)}, fHhfHgp1Hh1H1ρe−2¯u0ΛHh/T−2u0ΛH/T =mρ{fρm2ρ[V[−1](u0)+A[−1](u0)]+fTρmρ[T[0]1(u0)−T[0]2(u0)+S[0](u0)]Tf0(ω′cT) =−12fρ[V[1](u0)+A[1](u0)]T2f1(ω′cT)}, (12)

with

 T=T1T2T1+T2,    u0=T1T1+T2,    fn(x)=1−e−xn∑i=0xii!. (13)

Here we employ functions to subtract the contribution of the continuum.

s are defined as

 F[0](u0) ≡ ∫u00F(¯u0,α2,u0−α2)dα2, F[1](u0) ≡ F(¯u0,u0,0)−∫u00dα2∂F(1−α2−α3,α2,α3)∂α3∣∣∣α3=u0−α2, F[2](u0) ≡ ∂F(1−α2,α2,0)∂α2∣∣∣α2=u0+∂F(¯u0−α3,u0,α3)∂α3∣∣∣α3=0−∫u00dα2∂2F(1−α2−α3,α2,α3)∂α23∣∣∣α3=u0−α2, F[−1](u0) ≡ ∫10∫1−α20F(1−α2−α3,α2,α3)dα3dα2−∫u00∫u0−α20F(1−α2−α3,α2,α3)dα3dα2, F[−2](u0) ≡ ∫10∫1−α20∫α30F(1−α2−x,α2,x)dxdα3dα2−∫u00∫u0−α20∫α30F(1−α2−x,α2,x)dxdα3dα2 (14) −¯u0∫10∫1−α20F(1−α2−α3,α2,α3)dα3dα2.

Using the above mentioned method, we obtain the sum rules of other meson coupling constants as follows. Their definitions are presented in Appendix A.

 fHhfSgs1Hh1S1ρe−2¯u0ΛHh/T−2u0ΛS/T =16mρ{12fρm4ρ[4Φ−2˜Φ−2˜Ψ−A]{−2} =+4fTρm3ρ[T[−1](u0)−2T[−1]1(u0)+2T[−1]2(u0)−2T[−1]4(u0)+˜S[−1](u0)] =+fρm2ρ[−4V[0](u0)−4Φ[0](u0)+2˜Φ[0](u0)+2˜Ψ[0](u0)−3A[0](u0)]Tf0(ω′cT) =−fTρmρ[T[1](u0)+4T[1]1(u0)+2T[1]2(u0)−2T[1]4(u0)−2˜S[1](u0)]T2f1(ω′cT)−fρ[V[2](u0)+A[2](u0)]T3f2(ω′cT)}, fHhfSgd1Hh1S1ρe−2¯u0ΛHh/T−2u0ΛS/T =14mρ{fρm2ρ[4Φ−2˜Φ−2˜Ψ−A]{−2} =+8fTρmρ[T[−1](u0)+T[−1]1(u0)+2T[−1]2(u0)+T[−1]4(u0)+˜S[−1](u0)]+4fρ[V[0](u0)+A[0](u0)]Tf0(ω′cT)}, fMhfHgp1Mh1H1ρe−2¯u0ΛMh/T−2u0ΛH/T =14√2mρ{2fρm2ρ[A[−1](u0)−2V[−1](u0)] =−2fTρmρ[T[0]2(u0)−T[0]1(u0)+2S[0](u0)]Tf0(ω′cT)−fρ[A[1]2(u0)−2V[1](u0)]T2f1(ω′cT)}, fMhfHgp2Mh1H1ρe−2¯u0ΛMh/T−2u0ΛH/T =−340√2mρ{4fρm4ρ˜Ψ{−3}−2fTρm3ρT{−2}−4fρm2ρ[A[−1](u0)−4˜Ψ[−1](u0)] =+4fTρmρ[T[0](u0)+5T[0]1(u0)+5T[0]2(u0)]Tf0(ω′cT)+6fρA[1](u0)T2f1(ω′cT)}, fMhfHgf2Mh1H1ρe−2¯u0ΛMh/T−2u0ΛH/T =−34√2mρ{2fρm2ρ˜Ψ{−3}−fTρmρT{−2}+8fρA[−1](u0)}, fMhfSgs1Mh1S1ρe−2¯u0ΛMh/T−2u0ΛS/T =112√2mρ{fρm4ρ[2Φ+2˜Φ+2˜Ψ+A]{−2} =+4fTρm3ρ[T[−1](u0)−2T[−1]1(u0)+2T[−1]2(u0)−2T[−1]4(u0)−2˜S[−1](u0)] =−2fρm2ρ[2V[0](u0)+2Φ[0](u0)+2˜Φ[0](u0)+2˜Ψ[0](u0)−3A[0](u0)]Tf0(ω′cT) =−fTρmρ[T[1](u0)+4T[1]1(u0)+2T[1]2(u0)−2T[1]4(u0)+4˜S[1](u0)]T2f1(ω′cT)−fρ[V[2](u0)−2A[2](u0)]T3f2(ω′cT)}, fMhfSgd1Mh1S1ρe−2¯u0ΛMh/T−2u0ΛS/T =−14√2mρ{fρm2ρ[2Φ+2˜Φ+2˜Ψ+A]{−2} =+4fTρmρ[T[−1](u0)+2T[−1]1(u0)+2T[−1]2(u0)+T[−1]4(u0)−2˜S[−1](u0)]+2fρ[V[0](u0)−2A[0](u0)]Tf0(ω′cT)}, fMhfSgd2Mh1S1ρe−2¯u0ΛMh/T−2u0ΛS/T =32√2mρ{2fTρmρ[T[−1]1(u0)+T[−1]4(u0)]+fρV[0](u0)Tf0(ω′cT)}, fShfHgs1Sh1H1ρe−2¯u0ΛSh/T−2u0ΛH/T =−16mρ{12fρm4ρ[4Φ−2˜Φ−2˜Ψ−A]{−2} =−4fTρm3ρ[T[−1](u0)−2T