$\rho$ Meson Decays of Heavy Hybrid Mesons

0.21 MB·English

by  Liang Zhang

Submittedto‘ChinesePhysicsC’
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 and
Theoretical Physics Center for Science Facilities, CAS, Beijing 100049, China
6 We calculate the ρ meson couplings between the heavy hybrid doublets Hh/Sh/Mh/Th and the
1 ordinary qQ¯ doublets in the framework of the light-cone QCD sum rule. The sum rules obtained
0 rely mildly on the Borel parameters in their working regions. The resulting coupling constants are
2 rathersmall in most cases.

r
a PACSnumbers: 12.39.Mk,12.38.Lg,12.39.Hg
M Keywords: Heavyhybridmeson,QCDsumrule,Heavyquarkeffectivetheory

6
1

I. INTRODUCTION
]
h
p
Hadronstatesthatdonotfitintotheconstituentquarkmodelhavebeenstudiedwidelyinthepastseveraldecades.
p- In recent years, the discovery of a number of unexpected exotic resonances such as the so called XYZ mesons has
e revitalized the researchof the existence of unconventional hadron states and their nature.
h Theoretically, Quantum Chromodynamics (QCD), the fundamental theory of the strong interaction, may allow a
[
far richer spectrum than the conventionalquark model. Fox example, hybrids (qq¯g, ), glueballs (gg, ggg, ), and
3 multi-quark states (qqq¯q¯, qqqqq¯, ) may not be prohibited by QCD. Those with··J·PC = 0−−,0+−,1−+,·2·+· −,
··· ···
v are called “exotic” states. They have attracted much interest because they are not allowed by the constituent quark
4 model and do not mix with the ordinary mesons.
6 Evidence of exotic mesons with JPC = 1−+, e.g. π (1400) [1], π (1600) [2], have emerged in the last few years.
1 1
8
They are usually consideredas candidates of hybridmesons and have been studied extensively in variousframeworks
2
such as QCD sum rules, lattice QCD, AdS/QCD, the flux tube model, etc. The masses and decay properties of the
0
. 1−+ states have been studied in the framework of QCD sum rules [3, 4].
1
Based on the accumulated evidence of these light hybrid mesons, it is plausible to assume the existence of heavy
0 quarkonium hybrids (QQ¯g) and heavy hybrid mesons containing one heavy quark (qQ¯g) which may be not exotic.
5
Govaertsetal. havestudiedthesestatesinseveralworks[5]. In[6], the massesofQQ¯g werecalculatedatthe leading
1
: order of heavy quark effective theory (HQET) [7]. In [8], the masses of qQ¯g and their pionic couplings to ordinary
v
heavy mesons were calculated.
i
X In the heavy quark limit, the binding energy and the pionic couplings of qQ¯g to qQ¯ were worked out in [9] by
r Shifman-Vainshtein-Zakharov (SVZ) sum rules [10]. HQET describes the large mass (mQ) asymptotics. At the
a leadingorderofthis theory,the Lagrangianisendowedwiththe heavyquarkflavor-spinsymmetry,andthe spectrum
of qQ¯ consists of degeneratedoublets. The components ofa doublet sharethe same j, the angularmomentum of the
l
lightdegrees offreedom. For example, we denote the doublet (0−,1−) as H, which consistsof two j = 1 S-waveqQ¯.
l 2
Similarly, the P-wave doublets (0+,1+)/(1+,2+) are denoted as S/T and the D-wave doublets (1−,2−)/(2−,3−) as
M/N. We denote the two j = 1 qQ¯g doublets with parity P =+ and P = as Sh and Hh, respectively. Similarly,
l 2 −
we use Th and Mh to denote the two j = 3 doublets with positive parity and negative parity, respectively.
l 2
In this work,we adopt the light-cone QCD sum rules (LCQSR) approach[11] to investigate the ρ meson couplings
betweenqQ¯gandqQ¯. WederivethesumrulesfortheρmesoncouplingsbetweendoubletsDhandD(D =H/S/T/M)
in Sec. 2. The numericalanalysisis givenin Sec. 3, followedby a brief conclusionin Sec. 4. The details of the partial
amplitudes of these ρ decay channels are presented in Appendix A. The light-cone wave functions of the ρ meson
involved in our calculation are listed in Appendix B.
∗Electronicaddress: [email protected]
2
II. ρ MESON COUPLINGS
The interpolating currents for Hh and Mh adopted in our calculation can be written as
1
J† = h¯ ig γ σ Gq,
H0h r2 v s 5 t·
1
J†α = h¯ ig γασ Gq,
H1h r2 v s t t·
J†α =h¯ g 3Gαβγ +iγασ G q,
M1h v s t β t t·
(cid:20) (cid:21)
3 2
J†α1α2 = h¯ g γ Gα1βγ γα2 +Gα2βγ γα1 igα1α2σ G q, (1)
M2h r2 v s 5(cid:20) t β t t β t − 3 t t· (cid:21)
where Gαβ = Gnαβλn/2 and hv(x) = eimQv·x1+2v/Q(x). The subscript t means that the corresponding Lorentz tensor
is perpendicular to v, the 4-velocity of the heavy quark. gαβ = gαβ vαvβ. For any asymmetric tensor α1α2···αn,
t − A
we may define
n
α1α2···αn = α1α2···αn ( α1···αi−1ααi+1···αnv )vαi. (2)
At A − A α
i=1
X
We define the overlapping amplitudes between the these interpolating currents and the corresponding hybrids as
0J (0)Hh(v) =f ,
h | H0h | 0 i H0h
0Jα (0)Hh(v,λ) =f ηα (v,λ),
h | H1h | 1 i H1h H1h
0Jα (0)Mh(v,λ) =f ηα (v,λ),
h | M1h | 1 i M1h M1h
0Jα1α2(0)Mh(v,λ) =f ηα1α2(v,λ), (3)
h | M2h | 2 i M2h M2h
where η(v,λ) denotes the polarization of the heavy hybrid. These symmetric traceless tensors are perpendicular to
v, namely η vα =0.
αα2···αn
Weobtainthe interpolatingcurrentsforthe doubletsSh andTh bysimplyinsertingγ intothecurrentsinEq.(1):
5
1
J† = h¯ ig σ Gq,
S0h r2 v s t·
1
J†α = h¯ ig γ γασ Gq,
S1h r2 v s 5 t t·
J†α =h¯ g γ 3Gαβγ +iγασ G q,
T1h v s 5 t β t t·
(cid:20) (cid:21)
3 2
J†α1α2 = h¯ g Gα1βγ γα2 +Gα2βγ γα1 igα1α2σ G q. (4)
T2h r2 v s(cid:20) t β t t β t − 3 t t· (cid:21)
The corresponding overlapping amplitudes and projection operators can be defined similarly to Eq. (3).
The interpolating currents for qQ¯ doublets H and S read:
1
J† = h¯ γ q,
H0 2 v 5
r
1
J†α = h¯ γαq,
H1 2 v t
r
1
J† = h¯ q,
S0 2 v
r
1
J†α = h¯ γαγ q. (5)
S1 2 v t 5
r
3
The amplitudes between the ordinary heavy mesons and the states created by these currents acting on the vacuum
state are
0J (0)H (v) =f ,
h | H0 | 0 i H0
0Jα (0)H (v,λ) =f ǫα (v,λ),
h | H1 | 1 i H1 H1
0J (0)S (v) =f ,
h | S0 | 0 i S0
0Jα (0)S (v,λ) =f ǫα (v,λ). (6)
h | S1 | 1 i S1 S1
Here we outline the deduction of the sum rules for gp0 and gp1 , where p is the orbital angular momentum
H1hH1ρ H1hH1ρ
of the ρ meson, the superscript ‘0’ and ‘1’ are the total angular momentum of the ρ meson. We define gp0 and
H1hH1ρ
gp1 in term of the decay amplitude of the process Hh H +ρ:
H1hH1ρ 1 → 1
(Hh H +ρ) = I (e∗ η )(ǫ∗ q ) (e∗ ǫ∗)(η q ) gp1 +I(e∗ q )(ǫ∗ η )gp0 , (7)
M 1 → 1 · t · t − · t · t H1hH1ρ · t · t H1hH1ρ
h i
where η, ǫ∗ and e∗ are the polarizationof Hh, H andρ, respectively,and q denotes the momentum of the ρ. For the
1 1
charged ρ meson, I =1, and I =1/√2 if the final ρ meson is neutral.
We consider the following correlation function:
i dx e−ik·x ρ(q)Jβ (0)J†α(x)0 =I eαqβ qαeβ Gp1 (ω,ω′)+Igαβ(e q )Gp0 (ω,ω′), (8)
h | H1 H1h | i t t − t t H1hH1ρ t · t H1hH1ρ
Z h i
where ω =2k v and ω′ =2(k q) v, and we have the following dispersion relation
· − ·
Gp1 (ω,ω′)= ∞ds ∞ds ρpH11hH1ρ(s1,s2) + ∞ds ρp11(s1) + ∞ds ρp21(s2) + , (9)
H1hH1ρ Z0 1Z0 2(s1−ω−iǫ)(s2−ω′−iǫ) Z0 1s1−ω−iǫ Z0 2s2−ω′−iǫ ···
with
ρp1 (s ,s )=f f gp1 δ(s 2Λ )δ(s 2Λ )+ . (10)
H1hH1ρ 1 2 Hh H H1hH1ρ 1− Hh 2− H ···
The case of Gp0 (ω,ω′) is similar. Gp1 (ω,ω′) can be worked out by OPE near the light-cone when ω,ω′ 0,
H1hH1ρ H1hH1ρ ≪
and be formulated with the ρ meson light-cone wave functions
Gp0 (ω,ω′) = 1 ∞dt α eit(u2¯ω+u2ω′) m2
H1hH1ρ −4Z0 Z D (q·v)3
2f m3Ψ(α) 2fTm2 (α)(q v)+f m[6Φ(α)+2Ψ(α)+ (α)](q v)2
− ρ − ρ T · ρ A ·
(cid:20)
+2fT[ (αe)+2 (α)+2 (α)](q v)3 , e e
ρ T T1 T2 ·
(cid:21)
Gp1 (ω,ω′) = 1 ∞dt α eit(u2¯ω+u2ω′) m
H1hH1ρ 4Z0 Z D q·v
f m2[ (α)+ (α)] 2fTm[ (α) (α)+S(α)](q v) 2f [ (α)+ (α)](q v)2 . (11)
ρ V A − ρ T1 −T2 · − ρ V A ·
(cid:20) (cid:21)
in which u α +α and u¯ 1 u.
2 3
The doub≡le Borel transfor≡mat−ion BωT1BωT2′ eliminates the terms on the right side of Eq. (9), except the first one
which is a double dispersion relation. Now we arrive at
fHhfHgHp01hH1ρe−2u¯0ΛHh/T−2u0ΛH/T
1 1
=m2 f m3Ψ{−3} fTm2 {−2} f m [6Φ[−1](u )+2Ψ[−1](u )+ [−1](u )]
ρ −2 ρ ρ − 2 ρ ρT − ρ ρ 0 0 A 0
(cid:26)
ω′
+fT[ [0](u )e+2 [0](u )+2 [0](u )]Tf ( c)e , e
ρ T 0 T1 0 T2 0 0 T
(cid:27)
4
fHhfHgHp11hH1ρe−2u¯0ΛHh/T−2u0ΛH/T
ω′
=m f m2[ [−1](u )+ [−1](u )]+fTm [ [0](u ) [0](u )+S[0](u )]Tf ( c)
ρ ρ ρ V 0 A 0 ρ ρ T1 0 −T2 0 0 0 T
(cid:26)
1 ω′
f [ [1](u )+ [1](u )]T2f ( c) , (12)
ρ 0 0 1
− 2 V A T
(cid:27)
with
T T T n xi
T = 1 2 , u = 1 , f (x)=1 e−x . (13)
0 n
T +T T +T − i!
1 2 1 2
i=0
X
Here we employ functions f (x) to subtract the contribution of the continuum.
n
[αi]s are defined as
F
u0
[0](u ) (u¯ ,α ,u α )dα ,
0 0 2 0 2 2
F ≡ F −
Z0
u0 ∂ (1 α α ,α ,α )
[1](u ) (u¯ ,u ,0) dα F − 2− 3 2 3 ,
0 0 0 2
F ≡ F − ∂α
Z0 3 (cid:12)α3=u0−α2
[2](u ) ∂F(1−α2,α2,0) + ∂F(u¯0−α3,u0,α3) (cid:12)(cid:12)(cid:12) u0dα ∂2F(1−α2−α3,α2,α3) ,
F 0 ≡ ∂α ∂α − 2 ∂α2
2 (cid:12)α2=u0 3 (cid:12)α3=0 Z0 3 (cid:12)α3=u0−α2
(cid:12) (cid:12) (cid:12)
1 1−α2 (cid:12) u(cid:12)0 u0−α2 (cid:12)
[−1](u ) (1 α(cid:12) α ,α ,α )dα dα (cid:12) (1 α α ,α ,α )dα dα , (cid:12)
0 2 3 2 3 3 2 2 3 2 3 3 2
F ≡ F − − − F − −
Z0 Z0 Z0 Z0
1 1−α2 α3 u0 u0−α2 α3
[−2](u ) (1 α x,α ,x)dxdα dα (1 α x,α ,x)dxdα dα
0 2 2 3 2 2 2 3 2
F ≡ F − − − F − −
Z0 Z0 Z0 Z0 Z0 Z0
1 1−α2
u¯ (1 α α ,α ,α )dα dα . (14)
0 2 3 2 3 3 2
− F − −
Z0 Z0
Using the above mentionedmethod, we obtainthe sum rules of other ρ meson coupling constants as follows. Their
definitions are presented in Appendix A.
fHhfSgHs11hS1ρe−2u¯0ΛHh/T−2u0ΛS/T
1 1
= m f m4[4Φ 2Φ 2Ψ ]{−2}
6 ρ 2 ρ ρ − − −A
(cid:26)
+4fTm3[ [−1](u ) e2 [−e1](u )+2 [−1](u ) 2 [−1](u )+S[−1](u )]
ρ ρ T 0 − T1 0 T2 0 − T4 0 0
ω′
+fρm2ρ[−4V[0](u0)−4Φ[0](u0)+2Φ[0](u0)+2Ψ[0](u0)−3A[0](eu0)]Tf0(Tc)
ω′ ω′
fTm [ [1](u )+4 [1](u )+2 [e1](u ) 2 [e1](u ) 2S[1](u )]T2f ( c) f [ [2](u )+ [2](u )]T3f ( c) ,
− ρ ρ T 0 T1 0 T2 0 − T4 0 − 0 1 T − ρ V 0 A 0 2 T
(cid:27)
fHhfSgHd11hS1ρe−2u¯0ΛHh/T−2u0ΛS/T e
1
= m f m2[4Φ 2Φ 2Ψ ]{−2}
4 ρ ρ ρ − − −A
(cid:26)
ω′
+8fTm [ [−1](u )e+ [−e1](u )+2 [−1](u )+ [−1](u )+S[−1](u )]+4f [ [0](u )+ [0](u )]Tf ( c) ,
ρ ρ T 0 T1 0 T2 0 T4 0 0 ρ V 0 A 0 0 T
(cid:27)
fMhfHgMp11hH1ρe−2u¯0ΛMh/T−2u0ΛH/T e
1
= m 2f m2[ [−1](u ) 2 [−1](u )]
4√2 ρ ρ ρ A 0 − V 0
(cid:26)
ω′ ω′
2fTm [ [0](u ) [0](u )+2S[0](u )]Tf ( c) f [ [1](u ) 2 [1](u )]T2f ( c) ,
− ρ ρ T2 0 −T1 0 0 0 T − ρ A2 0 − V 0 1 T
(cid:27)
fMhfHgMp21hH1ρe−2u¯0ΛMh/T−2u0ΛH/T
3
= m 4f m4Ψ{−3} 2fTm3 {−2} 4f m2[ [−1](u ) 4Ψ[−1](u )]
−40√2 ρ ρ ρ − ρ ρT − ρ ρ A 0 − 0
(cid:26)
e e
5
ω′ ω′
+4fTm [ [0](u )+5 [0](u )+5 [0](u )]Tf ( c)+6f [1](u )T2f ( c) ,
ρ ρ T 0 T1 0 T2 0 0 T ρA 0 1 T
(cid:27)
fMhfHgMf21hH1ρe−2u¯0ΛMh/T−2u0ΛH/T
3
= m 2f m2Ψ{−3} fTm {−2}+8f [−1](u ) ,
−4√2 ρ ρ ρ − ρ ρT ρA 0
(cid:26) (cid:27)
fMhfSgMs11hS1ρe−2u¯0ΛeMh/T−2u0ΛS/T
1
= m f m4[2Φ+2Φ+2Ψ+ ]{−2}
12√2 ρ ρ ρ A
(cid:26)
+4fTm3[ [−1](u ) 2e[−1](eu )+2 [−1](u ) 2 [−1](u ) 2S[−1](u )]
ρ ρ T 0 − T1 0 T2 0 − T4 0 − 0
ω′
−2fρm2ρ[2V[0](u0)+2Φ[0](u0)+2Φ[0](u0)+2Ψ[0](u0)−3A[0](u0e)]Tf0(Tc)
ω′ ω′
fTm [ [1](u )+4 [1](u )+2 e[1](u ) 2 e[1](u )+4S[1](u )]T2f ( c) f [ [2](u ) 2 [2](u )]T3f ( c) ,
− ρ ρ T 0 T1 0 T2 0 − T4 0 0 1 T − ρ V 0 − A 0 2 T
(cid:27)
fMhfSgMd11hS1ρe−2u¯0ΛMh/T−2u0ΛS/T e
1
= m f m2[2Φ+2Φ+2Ψ+ ]{−2}
−4√2 ρ ρ ρ A
(cid:26)
ω′
+4fTm [ [−1](u )+2 e[−1](ue)+2 [−1](u )+ [−1](u ) 2S[−1](u )]+2f [ [0](u ) 2 [0](u )]Tf ( c) ,
ρ ρ T 0 T1 0 T2 0 T4 0 − 0 ρ V 0 − A 0 0 T
(cid:27)
fMhfSgMd21hS1ρe−2u¯0ΛMh/T−2u0ΛS/T e
3 ω′
= m 2fTm [ [−1](u )+ [−1](u )]+f [0](u )Tf ( c) ,
2√2 ρ ρ ρ T1 0 T4 0 ρV 0 0 T
(cid:26) (cid:27)
fShfHgSs11hH1ρe−2u¯0ΛSh/T−2u0ΛH/T
1 1
= m f m4[4Φ 2Φ 2Ψ ]{−2}
−6 ρ 2 ρ ρ − − −A
(cid:26)
4fTm3[ [−1](u ) 2e [−1]e(u )+2 [−1](u ) 2 [−1](u )+S[−1](u )]
− ρ ρ T 0 − T1 0 T2 0 − T4 0 0
ω′
+fρm2ρ[−4V[0](u0)−4Φ[0](u0)+2Φ[0](u0)+2Ψ[0](u0)−3A[0](eu0)]Tf0(Tc)
ω′ ω′
+fTm [ [1](u )+4 [1](u )+2 [e1](u ) 2 [e1](u ) 2S[1](u )]T2f ( c) f [ [2](u )+ [2](u )]T3f ( c) ,
ρ ρ T 0 T1 0 T2 0 − T4 0 − 0 1 T − ρ V 0 A 0 2 T
(cid:27)
fShfHgSd11hH1ρe−2u¯0ΛSh/T−2u0ΛH/T e
1
= m f m2[4Φ 2Φ 2Ψ ]{−2}
−4 ρ ρ ρ − − −A
(cid:26)
ω′
8fTm [ [−1](u )+e [−1]e(u )+2 [−1](u )+ [−1](u )+S[−1](u )]+4f [ [0](u )+ [0](u )]Tf ( c) ,
− ρ ρ T 0 T1 0 T2 0 T4 0 0 ρ V 0 A 0 0 T
(cid:27)
fShfSgSp01hS1ρe−2u¯0ΛSh/T−2u0ΛS/T e
1 1
=m2 f m3Ψ{−3} fTm2 {−2}+f m [6Φ[−1](u )+2Ψ[−1](u )+ [−1](u )]
ρ 2 ρ ρ − 2 ρ ρT ρ ρ 0 0 A 0
(cid:26)
ω′
+fT[ [0](ue)+2 [0](u )+2 [0](u )]Tf ( ec) , e
ρ T 0 T1 0 T2 0 0 T
(cid:27)
fShfSgSp11hS1ρe−2u¯0ΛSh/T−2u0ΛS/T
ω′
= m f m2[ [−1](u )+ [−1](u )] fTm [ [0](u ) [0](u )+S[0](u )]Tf ( c)
− ρ ρ ρ V 0 A 0 − ρ ρ T1 0 −T2 0 0 0 T
(cid:26)
1 ω′
f [ [1](u )+ [1](u )]T2f ( c) ,
ρ 0 0 1
− 2 V A T
(cid:27)
fThfHgTs11hH1ρe−2u¯0ΛTh/T−2u0ΛH/T
6
1
= m f m4[2Φ+2Φ+2Ψ+ ]{−2}
−12√2 ρ ρ ρ A
(cid:26)
4fTm3[ [−1](u ) 2 [e−1](ue)+2 [−1](u ) 2 [−1](u ) 2S[−1](u )]
− ρ ρ T 0 − T1 0 T2 0 − T4 0 − 0
ω′
−2fρm2ρ[2V[0](u0)+2Φ[0](u0)+2Φ[0](u0)+2Ψ[0](u0)−3A[0](u0e)]Tf0(Tc)
ω′ ω′
+fTm [ [1](u )+4 [1](u )+2 e[1](u ) 2 e[1](u )+4S[1](u )]T2f ( c) f [ [2](u ) 2 [2](u )]T3f ( c) ,
ρ ρ T 0 T1 0 T2 0 − T4 0 0 1 T − ρ V 0 − A 0 2 T
(cid:27)
fThfHgTd11hH1ρe−2u¯0ΛTh/T−2u0ΛH/T e
1
= m f m2[2Φ+2Φ+2Ψ+ ]{−2}
−4√2 ρ ρ ρ A
(cid:26)
ω′
4fTm [ [−1](u )+2 e[−1](ue)+2 [−1](u )+ [−1](u ) 2S[−1](u )]+2f [ [0](u ) 2 [0](u )]Tf ( c) ,
− ρ ρ T 0 T1 0 T2 0 T4 0 − 0 ρ V 0 − A 0 0 T
(cid:27)
fThfHgTd21hH1ρe−2u¯0ΛTh/T−2u0ΛH/T e
3 ω′
= m 2fTm [ [−1](u )+ [−1](u )] f [0](u )Tf ( c) ,
2√2 ρ ρ ρ T1 0 T4 0 − ρV 0 0 T
(cid:26) (cid:27)
fThfSgTp11hS1ρe−2u¯0ΛTh/T−2u0ΛS/T
1
= m 2f m2[ [−1](u ) 2 [−1](u )]
−4√2 ρ ρ ρ A 0 − V 0
(cid:26)
ω′ ω′
+2fTm [ [0](u ) [0](u )+2S[0](u )]Tf ( c) f [ [1](u ) 2 [1](u )]T2f ( c) ,
ρ ρ T2 0 −T1 0 0 0 T − ρ A2 0 − V 0 1 T
(cid:27)
fThfSgTp21hS1ρe−2u¯0ΛTh/T−2u0ΛS/T
3
= m 4f m4Ψ{−3}+2fTm3 {−2} 4f m2[ [−1](u ) 4Ψ[−1](u )]
40√2 ρ ρ ρ ρ ρT − ρ ρ A 0 − 0
(cid:26)
ω′ ω′
4fTm [ [0](u )e+5 [0](u )+5 [0](u )]Tf ( c)+6f [1](ue)T2f ( c) ,
− ρ ρ T 0 T1 0 T2 0 0 T ρA 0 1 T
(cid:27)
fThfSgTf12hS1ρe−2u¯0ΛTh/T−2u0ΛS/T
3
= m 2f m2Ψ{−3} fTm {−2}+8f [−1](u ) . (15)
4√2 ρ ρ ρ − ρ ρT ρA 0
(cid:26) (cid:27)
e
III. NUMERICAL ANALYSIS
The parametersin the distributionamplitudes ofthe ρ mesontake their valuesfrom[12]. Inthis work,we takethe
values with µ = 1 GeV, realizing that the heavy quark behaves almost as a spectator of the decay processes in our
discussion at the leading order of HQET:
fk[MeV] f⊥[MeV] ak a⊥ ζk ω˜k ωk ω⊥ ζk ω˜k ζ⊥ ζ˜⊥
ρ ρ 2 2 3ρ 3ρ 3ρ 3ρ 4 4 4 4
216(3) 165(9) 0.15(7) 0.14(6) 0.030(10) 0.09(3) 0.15(5) 0.55(25) 0.07(3) 0.03(1) 0.03(5) 0.08(5)
− − − −
For the mass sum rules of H and S, the workingregion of the Borel parameter T is about 0.8<T <1.1 GeV [13],
which is in the vicinity of that of the mass sum rules for Dh (D = H/S/M/T) [9]. So we choose u = 1/2 in our
0
calculation. The continuumcontributioncanbe subtractedcleanly with this choice. Asymmetric choice ofu , onthe
0
other hand, would result in a fuzzy continuum substraction [14].
The binding energy and the overlapping amplitudes of doublets H/S [13] and Hh/Mh, Sh/Th [9] involved in our
numerical analysis are as follows.
TheworkingregionofT isdeterminedbythe insensitivityofthecouplingconstanttothe variationofT andbythe
requirement that the pole contribution should be not less than 40%, We display the sum rules for these ρ couplings
7
H S Hh/Mh Sh/Th
Λ [GeV] 0.50 1.15 2.0 2.5
f 0.25 GeV3/2 0.40 GeV3/2 1.1 GeV7/2 1.6 GeV7/2
with ω′ =2.8,3.0,3.2 GeV in Fig. 1.
c
1.0 1.0
ω'=2.8 GeV
0.8 ω'c=2.8 GeV 0.8 ω'c=3.0 GeV
/0−1pGeVghHHρ 11 00..46 ωω'c'c==33..20 GGeeVV /1−1pGeVghHHρ 11 00..46 ω'cc=3.2 GeV
0.2 0.2
0.0 0.0
0 1 2 3 4 5 0 1 2 3 4 5
T/GeV T/GeV
(a) (b)
FIG. 1: The sum rules for (a) gp0 and (b) gp1 with continuum threshold ω′ =2.8,3.0,3.2GeV.
H1hH1ρ H1hH1ρ c
The following relations arise naturally in our calculation
gp0 gp0 = gp0 ,
HhHρ ≡ H1hH1ρ − H0hH0ρ
gp1 gp1 =gp1 =gp1 ,
HhHρ ≡ H1hH1ρ H1hH0ρ H0hH1ρ
gs1 gs1 =gs1 =gs1 ,
HhSρ ≡ H1hS1ρ H0hS1ρ H1hS0ρ
gd1 gd1 =gd1 =gd1 ,
HhSρ ≡ H1hS1ρ H0hS1ρ H1hS0ρ
1 √6
gp1 gp1 = gp1 = gp1 ,
MhHρ ≡ M1hH1ρ −2 M1hH0ρ 3 M2hH1ρ
√6
gp2 gp2 = gp2 = √6gp2 ,
MhHρ ≡ M1hH1ρ 2 M2hH0ρ − M2hH1ρ
√6
gf2 gf2 = gf2 = √6gf2 ,
MhHρ ≡ M1hH1ρ 2 M2hH0ρ − M2hH1ρ
1 √6
gs1 gs1 = gs1 = gs1 ,
MhSρ ≡ M1hS1ρ −2 M1hS0ρ 3 M2hS1ρ
1 √6
gd1 gd1 = gd1 = gd1 ,
MhSρ ≡ M1hS1ρ 2 M1hS0ρ − 3 M2hS1ρ
√6
gd2 gd2 = gd2 =√6gd2 ,
MhSρ ≡ M1hS1ρ 2 M2hS0ρ M2hS1ρ
gs1 gs1 = gs1 =gs1 ,
ShHρ ≡ S1hH1ρ − S0hH1ρ S1hH0ρ
gd1 gd1 = gd1 =gd1 ,
ShHρ ≡ S1hH1ρ − S0hH1ρ S1hH0ρ
gp0 gp0 =gp0 ,
ShSρ ≡ S1hS1ρ S0hS0ρ
gp1 gp1 =gp1 = gp1 ,
ShSρ ≡ S1hS1ρ S1hS0ρ − S0hS1ρ
1 √6
gs1 gs1 = gs1 = gs1 ,
ThHρ ≡ T1hH1ρ −2 T1hH0ρ − 3 T2hH1ρ
1 √6
gd1 gd1 = gd1 = gd1 ,
ThHρ ≡ T1hH1ρ −2 T1hH0ρ − 3 T2hH1ρ
8
√6
gd2 gd2 = gd2 = √6gd2 ,
ThHρ ≡ T1hH1ρ − 2 T2hH0ρ − T2hH1ρ
1 √6
gp1 gp1 = gp1 = gp1 ,
ThSρ ≡ T1hS1ρ −2 T1hS0ρ − 3 T2hS1ρ
√6
gp2 gp2 = gp2 =√6gp2 ,
ThSρ ≡ T1hS1ρ − 2 T2hS0ρ T2hS1ρ
√6
gf2 gf2 = gf2 =√6gf2 . (16)
ThSρ ≡ T1hS1ρ − 2 T2hS0ρ T2hS1ρ
These simple proportionalrelations among the obtainedcouplings result fromthe heavy quark flavor-spinsymmetry.
Theyalsojustifyourconstructionoftheinterpolatingcurrentsforheavyhybridmesons. Thespinoftheinterpolating
currents can be deduced from the symmetry of their Lorentz indices. The P parity can be obtained directly from
the P-transformation property of these currents. The tensor structure of the correlation functions considered above
verifies their JP quantum numbers. For example, if the JP quantum number of J†α =h¯ g γ [3Gαβγ +iγασ G]q
T1h v s 5 t β t t·
is not 1+, the tensor structure of the correlationfunction
i dx e−ik·x ρ(q)Jβ (0)J†α(x)0 (17)
h | H1 T1h | i
Z
cannot include (only) s1, d1 and d2.
Furthermore, if J†α =J†α +λJ†α (λ=0), where J†α and J†α are pure interpolating currents with j =3/2 and
T1h T1h0 S1h0 6 T1h0 S1h0 l
j =1/2, respectively, we have
l
i dx e−ik·x ρ(q)Jβ (0)J†α(x)0 = Iǫαβe∗vGs1 +I ǫαβqv(e∗ q ) 1ǫαβe∗vq2 Gd1
h | H1 T1h0 | i T1h0H1ρ · t − 3 t T1h0H1ρ
Z h i
+I ǫαeqvqβ +ǫβeqvqα) Gd2 ,
t t T1h0H1ρ
i dx e−ik·x ρ(q)Jβ (0)J†α(x)0 = Iǫαβhe∗vGs1 +I ǫαβiqv(e∗ q ) 1ǫαβe∗vq2 Gd1 ,
h | H1 S1h0 | i S1h0H1ρ · t − 3 t S1h0H1ρ
Z h i
and
1
i dx e−ik·x ρ(q)J (0)J†α(x)0 = Ie∗αGs1 +I qα(e∗ q ) e∗αq2 Gd1 ,
h | H0 T1h0 | i t T1h0H0ρ t · t − 3 t t T1h0H0ρ
Z h i
1
i dx e−ik·x ρ(q)J (0)J†α(x)0 = Ie∗αGs1 +I qα(e∗ q ) e∗αq2 Gd1 .
h | H0 S1h0 | i t S1h0H0ρ t · t − 3 t t S1h0H0ρ
Z h i
Now let us focus on the s1 part. It is straightforwardthat
Gs1 =c Gs1 , Gs1 =c Gs1 ,
T1h0H1ρ 1 T1h0H0ρ S1h0H1ρ 2 S1h0H0ρ
due to the heavy quark flavor-spin symmetry, therefore
Gs1 =c Gs1 +λc Gs1 ,
T1hH1ρ 1 T1h0H0ρ 2 S1h0H0ρ
Gs1 =Gs1 +λGs1 ,
T1hH0ρ T1h0H0ρ S1h0H0ρ
c =Gs1 /Gs1 =Gs1 /Gs1 =gs1 /gs1 ,
1 T1h0H1ρ T1h0H0ρ T1hH1ρ T1hH0ρ T1hH1ρ T1hH0ρ
c =Gs1 /Gs1 =Gs1 /Gs1 =gs1 /gs1 .
2 S1h0H1ρ S1h0H0ρ S1hH1ρ S1hH0ρ S1hH1ρ S1hH0ρ
When Gs1 is proportionalto Gs1 , we have c =c , namely,
T1hH1ρ T1hH0ρ 1 2
gs1 /gs1 =gs1 /gs1 .
T1hH1ρ T1hH0ρ S1hH1ρ S1hH0ρ
This is inconsistent with the results (see Eq. (16)) we just obtained: gs1 /gs1 = gs1 /gs1 . This implies
T1hH1ρ T1hH0ρ 6 S1hH1ρ S1hH0ρ
J†α = J†α. In other words, the interpolating current J†α carries j = 3/2. The J, P and j quantum numbers of
Th Th Th l l
1 10 1
other interpolating currents can be verified in a similar way.
The final values of these couplings are listed in Table I. In most channels they are rather small, which may be
attributed to the fading of the gluon degree of freedom in the decay.
9
gp0 gp1 gs1 gd1 gp1 gp2 gf2 gs1 gd1 gd2
H1hH1ρ H1hH1ρ H1hS1ρ H1hS1ρ M1hH1ρ M1hH1ρ M1hH1ρ M1hS1ρ M1hS1ρ M1hS1ρ
0.2 0.5 −1.0 0.09 0.12 0.21 −0.12 −0.01 0.02 0.07
gs1 gd1 gp0 gp1 gs1 gd1 gd2 gp1 gp2 gf2
S1hH1ρ S1hH1ρ S1hS1ρ S1hS1ρ T1hH1ρ T1hH1ρ T1hH1ρ T1hS1ρ T1hS1ρ T1hS1ρ
1.4 −0.4 0.14 −0.22 −0.18 0.09 −0.09 −0.02 −0.15 0.06
TABLE I: The absolute values of the coupling constants. The units of the P-, D-wave and F-wave coupling
constants are GeV−1, GeV−2 and GeV−3 respectively.
IV. CONCLUSION
At the heavy quark limit, we have constructed interpolating currents respecting the flavor-spin symmetry for qQ¯g
andqQ¯. Withthesecurrents,theρmesoncouplingsbetweenqQ¯g andqQ¯ havebeenworkedoutbymeansofLCQSR.
The derivedsumrules relymildly onthe Borelparametersintheir workingregions. The resultingcoupling constants
are rather small in most cases.
The main error of our calculation originate from the inaccuracy of LCQSR: truncation of the OPE near the light-
cone, the uncertainty of the parameters in the light-cone wave functions, the dependence of the coupling constant on
the continuumthresholdω andthe Borelparameterin the workingregion,the uncertaintyofthe binding energyΛ¯’s
c
andthe overlappingamplitudes f’s. As far as the charmquarkis concerned, the 1/m correctionmay be significant,
Q
while the correction from the finite mass of the bottom quark should be negligible.
We hope that our calculation may be helpful to experimental searches for these heavy hybrid mesons and the
understandingoftheirstronginteractionwithconventionalheavymesons. Moreover,thecouplingconstantscalculated
in our work might shed further light on the nature of the XYZ mesons.
Acknowledgments
This work is supported by the National Natural Science Foundation of China under Grants No. 11105007.
[1] D.Aldeet al.,Phys.Lett.B205,397(1988); H.Aoyagiet al.,Phys.Lett.B314,246(1993);D.R.Thompsonetal.[E852
Collaboration], Phys. Rev. Lett. 79, 1630 (1997); A. Abele et al. [The Crystal Barrel Collaboration], Phys. Lett. B423,
175 (1998).
[2] Yu. P. Gouz et al. [VES Collaboration.], AIP Conf. Proc. 272, 572 (1993); Yu A. Khokholov et al. [VES Collaboration],
Nucl.Phys.A663, 596 (2000); A.Zaitsev et al.[VES Collaboration], Nucl.Phys.A675,155c, (2000); G. S.Adamset al.
[E852 Collaboration], Phys.Rev.Lett.81, 5760 (1998); S.U.Chunget al. [E852 Collaboration], Phys.Rev.D65, 072001
(2002); E. I. Ivanov et al. [E852 Collaboration], Phys. Rev. Lett. 86, 3977 (2001); J. Kuhn et al. [E852 Collaboration],
Phys. Lett. B595, 109 (2004); M. Lu et al. [E852 Collaboration], Phys. Rev. Lett. 94, 032002 (2005); C. A. Baker et al.,
Phys. Lett. B563, 140 (2003); M. Alekseev et al. [The COMPASS Collaboration], Phys. Rev. Lett. 104, 241803 (2010);
B. Grube et al. [The COMPASS Collaboration], arXiv:1002.1272 [hep-ex](2010).
[3] I. I. Balitsky, D. I. DYakonov, and A. V. Yung, Phys. Lett. B112, 71 (1982); J. Govaerts et al., Phys. Lett. B128, 262
(1983); J. I. Latorre et al., Phys. Lett. B147, 169 (1984); J. Govaerts et al., Nucl. Phys. B248, 1 (1984); I. I. Balitsky,
D.I.DYakonov,and A. V.Yung,Z. Phys.C33, 265 (1986); J. I. Latorre, S. Narison, and P.Pascual, Z. Phys.C 34, 347
(1987); K. G. Cheyrkin and S. Narison, Phys. Lett. B485, 145 (2000); H. Y. Jin, J. G. Korner, and T. G. Steele, Phys.
Rev.D67, 014025 (2003); K. C. Yang, Phys.Rev.D76, 094001 (2007); S.Narison, Phys. Lett.B675, 319 (2009).
[4] F. De Viron and J. Govaerts, Phys. Rev. Lett. 53, 2207 (1984); J. I. Latorre, P. Pascual, and S. Narison, Z. Phys. C34,
347 (1987); S. L. Zhu, Phys. Rev. D60, 097502 (1999); H. X. Chen, Z. X. Cai, P. Z. Huang, and S. L. Zhu, Phys. Rev.
D83, 014006 (2011); P. Z. Huang,H. X.Chen, and S.L. Zhu,Phys. Rev.D83, 014021 (2011).
[5] J.Govaerts et al.,Nucl. Phys.B258, 215 (1985); Nucl.Phys. B262, 575 (1985); Nucl. Phys.B284, 674 (1987).
[6] S.L. Zhu,Phys. Rev.D60, 031501 (1999).
[7] B.Grinstein, Nucl.Phys.B339, 253 (1990); E.Eichten andB. Hill,Phys.Lett.B234, 511 (1990); A.F.Falk, H.Georgi,
B. Grinstein, and M. B. Wise, Nucl. Phys.B343, 1 (1990).
[8] S.L. Zhu,Phys. Rev.D60, 014008 (1999).
[9] P.Z. Huang and S. L.Zhu, arXiv: 1103.0602
[10] M.A. Shifman, A.I. Vainshtein,and V.I. Zakharov,Nucl. Phys. B147, 385, 448, 519 (1979).
10
[11] I.I.Balitsky,V.M.Braun,andA.V.Kolesnichenko,Nucl.Phys.B312,509(1989);V.M.BraunandI.E.Filyanov,Z.Phys.
C44, 157 (1989); V.L. Chernyak and I.R.Zhitnitsky,Nucl. Phys.B345, 137 (1990).
[12] P.Ball and V.M. Braun, Phys.Rev.D58, 094016 (1998); P. Ball and R.Zwicky, Phys.Rev.D71, 014029 (2005).
[13] E. Bagen, P. Ball, V. M. Braun, and H. G. Dosch, Phys. Lett. B278, 457 (1992); M. Neubert, Phys. Rev. D45, 2451
(1992); D. J. Broadhurst and A. G. Grozin, Phys. Lett. B274, 421 (1992); Y. B. Dai and S. L. Zhu, Phys. Rev. D58,
074009 (1998).
[14] I. I. Balitsky, V. M. Braun, and A. V. Kolesnichenko, Nucl. Phys. B312, 509 (1989); V. M. Braun and I. E. Filyanov, Z.
Phys.C44, 157 (1989); V. M. Belyaev, V.M. Braun, A. Khodjamirian, and R. Ruckl,Phys.Rev. D51, 6177 (1995).
[15] P.Ball and G. W. Jones, JHEP 0703, 069 (2007); P. Ball, V.M. Braun, and A. Lenz, JHEP 0708, 090 (2007).
Appendix A: The ρ decay amplitudes of heavy hybrid mesons
The ρ decay amplitudes considered in the text are as follows.
(Hh H +ρ) = I(e∗ q )gp0 , (A1)
M 0 → 0 · t H0hH0ρ
(Hh H +ρ) = Iǫǫ∗e∗qvgp1 , (A2)
M 0 → 1 H0hH1ρ
(Hh H +ρ) = Iǫηe∗qvgp1 , (A3)
M 1 → 0 H1hH0ρ
(Hh H +ρ) = I(e∗ q )(ǫ∗ η )gp0 +I (e∗ η )(ǫ∗ q ) (e∗ ǫ∗)(η q ) gp1 , (A4)
M 1 → 1 · t · t H1hH1ρ · t · t − · t · t H1hH1ρ
h 1 i
(Hh S +ρ) = I(e∗ ǫ∗)gs1 +I (ǫ∗ q )(e∗ q ) (e∗ ǫ∗)q2 gd1 , (A5)
M 0 → 1 · t H0hS1ρ · t · t − 3 · t t H0hS1ρ
h 1 i
(Hh S +ρ) = I(e∗ η )gs1 +I (η q )(e∗ q ) (e∗ η )q2 gd1 , (A6)
M 1 → 0 · t H1hS0ρ · t · t − 3 · t t H1hS0ρ
(Hh S +ρ) = Iǫηǫ∗e∗vgs1 +I hǫηǫ∗qv(e∗ q ) 1ǫηǫ∗e∗vq2 gdi1 , (A7)
M 1 → 1 H1hS1ρ · t − 3 t H1hS1ρ
(Mh H +ρ) = Iǫηe∗qvgp1 , h i (A8)
M 1 → 0 M1hH0ρ
(Mh H +ρ) = Ii[(η e∗)(ǫ∗ q ) (η q )(ǫ∗ e∗)]gp1
M 1 → 1 · t · t − · t · t M1hH1ρ
2
+Ii (η e∗)(ǫ∗ q )+(η q )(ǫ∗ e∗) (η ǫ∗)(e∗ q ) gp2
· t · t · t · t − 3 · t · t M1hH1ρ
(cid:20) (cid:21)
q2
+Ii (η q )(ǫ∗ q )(e∗ q ) t [(η ǫ∗)(e∗ q )+(η e∗)(ǫ∗ q )+(η q )(ǫ∗ e∗)] gf2 ,
· t · t · t − 5 · t · t · t · t · t · t M1hH1ρ
(cid:26) (cid:27)
(A9)
1
(Mh H +ρ) = 2Iiη e∗α1qα2 gα1α2(e∗ q ) gp2
M 2 → 0 α1α2 t t − 3 t · t M2hH0ρ
(cid:20) (cid:21)
q2
+Iiη qα1qα2(e∗ q ) t [gα1α2(e∗ q )+2e∗α1qα2] gf2 , (A10)
α1α2 t t · t − 5 t · t t t M2hH0ρ
(cid:26) (cid:27)
(Mh H +ρ) = 2Iη εα1e∗qvǫ∗α2 + 1gα1α2εǫ∗e∗qv gp1
M 2 → 1 α1α2 − t 3 t M2hH1ρ
(cid:20) (cid:21)
+2Iη εα1ǫ∗e∗vqα2 +εα1ǫ∗qve∗α2 gp2
α1α2 t t M2hH1ρ
+2Iη hεα1ǫ∗qvqα2(e∗ q ) qt2 εiα1ǫ∗qve∗α2 +εα1ǫ∗e∗vqα2 gf2 , (A11)
α1α2 t · t − 5 t t M2hH1ρ
(cid:26) h i(cid:27)
1
(Mh S +ρ) = Ii(η e∗)gs1 +Ii (η q )(e∗ q ) (η e∗)q2 gd1 , (A12)
M 1 → 0 · t M1hS0ρ · t · t − 3 · t t M1hS0ρ
(cid:20) (cid:21)
(Mh S +ρ) = Iǫηǫ∗e∗vgs1 +I ǫηǫ∗qv(e∗ q ) 1ǫηǫ∗e∗vq2 gd1
M 1 → 1 M1hS1ρ · t − 3 t M1hS1ρ
(cid:20) (cid:21)
+I ǫηe∗qv(ǫ∗ q )+ǫǫ∗e∗qv(η q ) gd2 , (A13)
· t · t M1hS1ρ
(Mh S +ρ) = 2Iηh ǫα1e∗qvqα2gd2 , i (A14)
M 2 → 0 α1α2 t M2hS0ρ