Exploring the $\Upsilon(6S)\to \chi_{bJ}\phi$ and $\Upsilon(6S)\to \chi_{bJ}\omega$ hidden-bottom hadronic transitions

3.7 MB·

by  Qi Huang

ExploringtheΥ(6S) → χ φandΥ(6S) → χ ωhidden-bottomhadronictransitions
bJ bJ
Qi Huang,∗ Bo Wang,† and Xiang Liu‡
School of Physical Science and Technology, Lanzhou University, Lanzhou 730000, China
ResearchCenterforHadronandCSRPhysics,LanzhouUniversityandInstituteofModernPhysicsofCAS,Lanzhou730000,China
Dian-Yong Chen§
Department of Physics, Southeast University, Nanjing 210094, China
Takayuki Matsuki¶
Tokyo Kasei University, 1-18-1 Kaga, Itabashi, Tokyo 173-8602, Japan
Theoretical Research Division, Nishina Center, RIKEN, Wako, Saitama 351-0198, Japan
In this work, we investigate the hadronic loop contributions to the Υ(6S) → χ φ (J = 0,1,2) along
bJ
7 with Υ(6S) → χ ω (J = 0,1,2) transitions. We predict that the branching ratios of Υ(6S) → χ φ,
bJ b0
1 Υ(6S) → χ φandΥ(6S) → χ φare(0.68 ∼ 4.62)×10−6,(0.50 ∼ 3.43)×10−6 and(2.22 ∼ 15.18)×10−6,
b1 b2
0 respectively and those of Υ(6S) → χ ω, Υ(6S) → χ ω and Υ(6S) → χ ω are (0.15 ∼ 2.81)×10−3,
b0 b1 b2
2 (0.63 ∼ 11.68)×10−3 and (1.08 ∼ 20.02)×10−3, respectively. Especially, some typical ratios, which re-
flect the relative magnitudes of the predicted branching ratios, are given, i.e., for Υ(6S) → χ φ transi-
b bJ
tions, Rφ = B[Υ(6S)→χ φ]/B[Υ(6S)→χ φ] ≈ 0.74, Rφ = B[Υ(6S)→χ φ]/B[Υ(6S)→χ φ] ≈
e 10 b1 b0 20 b2 b0
F 3.28, and Rφ = B[Υ(6S)→χ φ]/B[Υ(6S)→χ φ] ≈ 4.43, and for Υ(6S) → χ ω transitions, Rω =
21 b2 b1 bJ 10
8 BRω[Υ(=6SB)[→Υ(6χSb1)ω→]/Bχ[Υω(6]/SB)[→Υ(6χSb0)ω→] χ≈ ω4.]11≈, 1R.72ω02.=WiBth[Υth(6eSr)un→ninχgb2ωof]/BBe[lΥle(I6ISin)→theχnb0eωar]fu≈tur7e.,0e6x,paenrid-
2 21 b2 b1
mentalmeasurementofthesetwokindsoftransitionswillbeapotentialresearchissue.

]
h PACSnumbers:14.40.Pq,13.25.Gv
p
-
p I. INTRODUCTION hadronicloopmechanism,whichisanequivalentdescription
e ofthecoupledchanneleffect[4–6,8–10,12–16]. Byanalyz-
h
ingthesetransitions,therelativedecayratesofΥ(6S)→χ φ
[ Asaninterestingresearchissue,experimentalstudiesofthe bJ
(J = 0,1,2) and Υ(6S) → χ ω (J = 0,1,2), which are a
2 hadronic transitions of Υ(5S) have been focused on by the bJ
typical physical quantity given by our calculation, are deter-
v Belle Collaboration in the past decade. When surveying the
mined. Especially, our results show that these relative decay
4 reported hadronic transitions of Υ(5S), we found their gen-
rates are weakly dependent on the model parameters. Thus,
9 eral property, i.e., their observed hadronic transitions have
8 large branching ratios. For example, Belle observed anoma- experimental measurement of these rates can be a crucial
0 lous decay widths of the Υ(5S) → Υ(nS)π+π− [1], and test of the hadronic loop mechanism in the Υ(6S) → χbJφ
0 and Υ(6S) → χ ω decays. In addition, we also estimate
Υ(5S) → χ ω (J = 0,1,2) transitions[2]. Inaddition, two bJ
. bJ the typical values of the branching ratios of Υ(6S) → χ φ
1 bottomonium-like states Z (10610) and Z (10650) were ob- bJ
0 served in Υ(5S) → Υ(nS)bπ+π− [3]. As inbdicated in a serial and Υ(6S) → χbJω, which can be measured experimentally
7 in near future. Anyway, we would like to inspire experi-
of theoretical studies [4–10], the puzzling phenomena hap-
1 menlists’ interest in searching for the Υ(6S) → χ φ and
peningonΥ(5S)transitionsreflectanunderlyingmechanism bJ
v: mediatedbyacoupledchanneleffectsinceΥ(5S)isabovethe Υ(6S)→χbJωdecaysbyourresultspresentedinthiswork.
Xi thresholdsofB((∗s))B¯((∗s))[11]. weTphrisespenaptetrheisdeotragialendizecdalcauslaftoiollnowosf.Υ(A6fSte)r→introχduφctiaonnd,
bJ
r In the bottomonium family, the Υ(6S) has the similar sit- Υ(6S) → χ ω via the hadronic loop mechanism in Sec. II.
a uation to that of Υ(5S). We have a reason to believe that ThenumericbaJlresultsarepresentedinSec.III.Thepaperends
the coupled channel effect is still important to the hadronic withashortsummary.
transitionsofΥ(6S),whoseexplorationis,thus,anintriguing
topic.Thisthemecanprovideusavaluableinformationofthe
coupled-channeleffectonthesedecays. II. Υ(6S)→χ φANDΥ(6S)→χ ωTRANSITIONSVIA
bJ bJ
Inthiswork, wecalculatetheΥ(6S) → χ φ(J = 0,1,2) HADRONICLOOPMECHANISM
bJ
along with Υ(6S) → χ ω (J = 0,1,2) processes via the
bJ
Under the hadronic loop mechanism, the Υ(6S)→χ φ
bJ
transitionsoccurviathetriangleloopscomposedofB(∗)0 and
s
B¯(∗)0, which play a role of the bridge to connect the initial
∗Electronicaddress:[email protected] staste Υ(6S) and final states φ and χ . In Figs. 1-3, we list
†Electronicaddress:[email protected] thetypicaldiagramsdepictingtheΥ(b6JS)→χ φ(J =0,1,2)
‡Electronicaddress:[email protected] bJ
§Electronicaddress:[email protected] transitions. For the Υ(6S)→χbJω transitions, due to very
¶Electronicaddress:[email protected] differentquarkcontentsbetweenφandω,thebridgeschange
2
toB(∗) andB¯(∗) andthediagramschangesimultaneouslyasin
Figs. 4-6.
To calculate these diagrams at the hadron level, we adopt
theeffectiveLagrangianapproach,inwhichwefirstintroduce
theLagrangiansrelevanttoourcalculation.
For the interactions between a heavy quarkonium and two
heavy-lightmesons,theLagrangiansareconstructedbasedon
theheavyquarkeffectivetheory. Intheheavyquarklimit,the
lightdegreesoffreedom s isagoodquantumnumber. Thus,
(cid:96)
eachvalueof s isassignedtoadoubletformedbythestates
(cid:96)
with a total angular momentum J = s ±1/2, while for the
(cid:96)
heavyquarkonium,sincethedegeneracyisexpectedunderthe
rotationsoftwoheavyquarkspins,thereisamultipletformed
byheavyquarkoniawiththesameangularmomentum(cid:96).
FIG.3:SchematicdiagramsdepictingtheΥ(6S)→χ φprocessvia
b2
thehadronicloopmechanism.
FIG.1:SchematicdiagramsdepictingtheΥ(6S)→χ φprocessvia
b0
thehadronicloopmechanism.
FIG.4:SchematicdiagramsdepictingtheΥ(6S)→χ ωprocessvia
b0
thehadronicloopmechanism.
FIG.2:SchematicdiagramsdepictingtheΥ(6S)→χ φprocessvia
b1
thehadronicloopmechanism.
FIG.5:SchematicdiagramsdepictingtheΥ(6S)→χ ωprocessvia
b1
thehadronicloopmechanism.
3
For the interaction between a light vector meson and two
heavy-lightmesons,thegeneralformoftheLagrangianreads
as[18,22–26]
L =iβTr[Hjvµ(−ρ )iH¯ ]+iλTr[HjσµνF (ρ)H¯ ], (5)
V µ j i µν i
where
g
ρ = i√V V , (6)
µ µ
2
F (ρ) = ∂ ρ −∂ ρ +[ρ ,ρ ], (7)
µν µ ν ν µ µ ν
andavectoroctetVhastheform
V =  √12(Kρρ0∗−−+ω) √12(−Kρ¯ρ+∗00+ω) KKφ∗∗+0 . (8)
ByexpandingtheLagrangiansinEqs. (1)and(5),thefol-
lowingconcreteexpressionsareobtained
L
ΥB(∗)B(∗)
=−igΥBBΥµ(∂µBB†−B∂µB†)
↔ ↔
FIG.6:SchematicdiagramsdepictingtheΥ(6S)→χb2ωprocessvia +gΥB∗Bεµναβ∂µΥν(B∗α ∂β B†−B ∂β B∗α†)
thehadronicloopmechanism. ↔
+igΥB∗B∗Υµ(B∗ν∂νB∗µ†−∂νB∗µB∗ν†−B∗ν ∂µ B∗ν†), (9)
Therefore, under the framework of heavy quark symme-
L
try,generalformsofcouplingsbetweenanS-waveorP-wave χbJB(∗)B(∗)
heavy quarkonium and two heavy-light mesons can be con- =−g χ BB†−g χ B∗B∗µ†
structedas[18] χb0BB b0 χb0B∗B∗ b0 µ
+ig χµ (B∗B†−BB∗†)
(cid:20) ↔ (cid:21) χc1BB∗ b1 µ µ
L = igTr R(QQ¯)H¯(Q¯q)γµ ∂ H¯(Qq¯) +H.c., −g χµν∂ B∂ B†
s µ χb2BB b2 µ ν
Lp = ig1Tr(cid:104)P(QQ¯)µH¯(Q¯q)γµH¯(Qq¯)(cid:105)+H.c., (1) +gχb2B∗B∗χµb2ν(B∗µB∗ν†+B∗νB∗µ†)
−ig ε ∂αχµρ(∂ B∗ν∂βB†−∂βB∂ B∗ν†),(10)
in which R(QQ¯) and P(QQ¯) denote multiplets formed by bot- χb2B∗B µναβ b2 ρ ρ
tomoniawith(cid:96)=0and(cid:96)=1,andtheirdetailedexpressions,
asinRef. [21],canbewrittenas LB(∗)B(∗)V
↔µ
R(QQ¯) = 1+v/(cid:104)Υµγ −η γ (cid:105)1−v/, (2) =−igBBVB†i ∂ Bj(Vµ)ij
2 µ b 5 2 ↔α ↔α
−2f ε (∂µVν)i(B† ∂ B∗βj−B∗β† ∂ Bj)
B∗BV µναβ j i i
P(QQ¯)µ = 1+2 v/(cid:104)χµb2αγα+ √12εµαβγvαγβχb1γ +igB∗B∗VB∗iν† ↔∂µ B∗νj(Vµ)ij
+4if B∗†(∂µVν−∂νVµ)iB∗j. (11)
+√1 (cid:0)γµ−vµ(cid:1)χ +hµγ (cid:105)1−v/, (3) B∗B∗V iµ j ν
3 b0 b 5 2 With the above effective Lagrangians, we can write out
the amplitudes of hadronic loop contributions to Υ(6S) →
respectively. H(Qq¯)representsadoubletformedbyheavy-light
χ φ(J = 0,1,2). FortheΥ(6S) → χ φtransition, theam-
pseudoscalarandvectormesons[18–21] bJ b0
plitudescorrespondingtoFig. 1are
1+v/(cid:104) (cid:105)
H(Qq¯) = 2 B∗µγµ−Bγ5 , (4) M(0−1) = (cid:90) (d2π4q)4[−igΥBsBs(cid:15)Υµ((ik1)µ−(ik2)µ)]
with definitions B(∗)† = (B(∗)+,B(∗)0,B(s∗)0) and B(∗) = ×[−ig (cid:15)∗ ((−ik )λ−(iq)λ)][−g ]
(B(∗)−,B¯(∗)0,B¯(∗)0)T as in Ref. [13]. H(Q¯q) corresponds to a BsBsφ φλ 1 BsBsχb0
s
1 1 1
doubletformedbyheavy-lightanti-mesons,whichcanbeob- × F2(q2), (12)
tainedbyapplyingthechargeconjugationoperationtoH(Qq¯). k2−m2 k2−m2 q2−m2
1 Bs 2 Bs Bs
4
Then,thepartialdecaywidthreads
(cid:90)
d4q 1 1 |(cid:126)p |
M(0−2) = (2π)4[−gΥB∗sBsεµναβ(cid:15)Υµ(−ip1)ν((ik2)β−(ik1)β)] ΓΥ(6S)→χbJφ = 38πm2φ |MTΥo(6taSl)→χbJφ|2, (17)
Υ(6S)
×[2fB∗sBsφελρδσ(cid:15)φ∗λ(ip2)ρ((−ik1)δ−(iq)δ)][−gBsBsχb0] where the overline indicates the sum over polarizations of
×−gσαk+2k−1αmk21σ/m2B∗s k2−1m2 q2−1m2 F2(q2), (13) eΥr(a6gSe)o,vφe,rathnedpχobl1ar(iozratχiobn2)oafnindittihaelΥfa(c6tSor).31 denotes the av-
1 B∗s 2 Bs Bs InthecaseofΥ(6S) → χ ω,theexpressionofthepartial
bJ
decaywidthisgivenby
(cid:90)
d4q
M(0−3) = (2π)4[gΥBsB∗sεµναβ(cid:15)Υµ(−ip1)ν((ik2)β−(ik1)β)] ΓΥ(6S)→χbJω = 3181πm|2(cid:126)pω| |ATΥo(6taSl)→χbJω|2, (18)
×[−2fBsB∗sφελρδσ(cid:15)φ∗λ(ip2)ρ((−ik1)δ−(iq)δ)] Υ(6S)
with a general expression of the total amplitude of
×[−gB∗sB∗sχb0]k2−1m2 −gζαk+2k−2αmk22ζ/m2B∗s Υ(6S)→χbJωas (cid:88)
1 Bs 2 B∗s ATotal =4 A (19)
−gσ+qσq /m2 Υ(6S)→χbJω (J−j)
× ζ ζ B∗sF2(q2), (14) j
q2−m2
B∗s byconsideringtheisospinandchargesymmetry.Thedetailed
expressionsofA arecollectedinAppendix.
(J−j)
(cid:90)
d4q
M(0−4) = (2π)4[igΥB∗sB∗s(cid:15)Υµ(gναgµβ(ik2)ν−gµαgνβ(ik1)ν
III. NUMERICALRESULTS
−g ((ik ) −(ik ) ))]
αβ 2 µ 1 µ
×[(cid:15)φ∗λ(igB∗sB∗sφgδσ((−ik1)λ−(iq)λ) timWatiethththeehafodrrmonuilcasloloisptecdonintriSbeuct.ioInIsatnodthAepΥpe(6nSdi)x,→weχesφ-
+4ifB∗sB∗sφ(ip2)ρ(gλδgρσ−gλσgρδ))][−gB∗sB∗sχb0] together with Υ(6S) → χ ω (J = 0,1,2) transitions. BbJe-
bJ
×−gαδ +k1δk1α/m2B∗s −gβζ +k2βk2ζ/m2B∗s sides the masses taken from the Particle Data Book [11], all
k2−m2 k2−m2 theotherinputparametersweneedarethecouplingconstants.
1 B∗s 2 B∗s Since the Υ(6S) is above the threshold of B(∗)B¯(∗), the cou-
−g +q q /m2 (s) (s)
× σζ σ ζ B∗sF2(q2), (15) pling constants between Υ(6S) and B((∗s))B¯((∗s)) can be evaluated
q2−m2B∗s by the partial decay widths of Υ(6S) → B((∗s))B¯((∗s)). In Table I
where p , p and p are momenta of Υ(6S), φ/ω and χ , welisttherelevantpartialdecaywidthsgiveninRef. [27]as
1 2 3 bJ
andk ,k ,andqaremomentaofinternal B(∗) andexchanged wellasthecorrespondingextractedcouplingconstants.
1 2 (s)
B(∗), respectively. In these expressions of the decay am-
(s)
plitudes, the monopole form factor is introduced, by which TABLEI:ThecouplingconstantsofΥ(6S)interactingwithB(∗)B¯(∗).
(s) (s)
the inner structure of interaction vertices is reflected and the Here,wealsolistthecorrespondingpartialdecaywidthsprovidedin
off-shell effect of the exchanged bottom-strange mesons is Ref.[27].
compensated. Here, the adopted form factor is taken as Finalstate Decaywidth(MeV) Couplingconstant
F(q2) = (m2 − Λ2)/(q2 − Λ2), with m being the mass of BB¯ 1.32 0.654
the exchangeEd boson and the cutoff Λ Ebeing parameterized BB¯∗ 7.59 0.077GeV−1
as Λ = mE + αΛΛQCD with ΛQCD = 0.22 GeV as in Refs. BB∗BB¯¯∗ 1.315.×8910−3 00..601413
[14–16]. We need to specify that the monopole behavior of s s
B B¯∗ 0.136 0.023GeV−1
theadoptedformfactorwassuggestedbytheQCDsumrule s s
B∗B¯∗ 0.310 0.354
studiedinRef. [17]. Inaserialofpublishedpapers(seeRefs. s s
[4–6, 8–10, 12–16, 21, 29]), the monopole form factor was
Thecouplingconstantsrelevanttotheinteractionsbetween
adoptedtostudythetransitionsofcharmoniaandbottomonia,
χ and B(∗)B¯(∗) in the heavy quark limit are related to one
and B decays. Thus, this approach has been tested by these bJ (s) (s)
gaugecouplingg giveninEq. (1),i.e.,
successfulstudies. 1
In a similar way, we can further write out the decay am- √ √ 2 √
plitudes of Υ(6S) → χb1φ and Υ(6S) → χb2φ, which are gχb0BB = 2 3g1 mχb0mB, gχb0B∗B∗ = √3g1 mχb0mB∗,
collectedinAppendix. Byconsideringtheisospinsymmetry, √
waigtehnJer=al0e,x1p,r2esissiownriottfenthaestotalamplitudeofΥ(6S)→χbJφ gχb1BB∗ = 2√2g1√mχb1mBmB∗, gχb2BB =2g1 mmBχb0,
(cid:115)
(cid:88) m √
MTotal =2 M . (16) g = g χb2 , g =4g m m ,
Υ(6S)→χbJφ (J−j) χb2BB∗ 1 m3 m χb2B∗B∗ 1 χb2 B∗
j B∗ B
5
(cid:113)
whereg = − mχb0 1 [21]and f = 175±55MeVisthe
decayco1nstantof3χ fχ[b028]. χb0 20
B((∗sS))Bi¯m((∗si))lacralny,betheextrcaobc0utpedlinfrgomcoEnqst.a(n5t)s, between φ or ω and 3-10)16 ΥΥΥ(((666SSS)))→→→χχχbbb012ωωω
×
gBsBsφ = gB∗sB∗sφ = β√g2V, →χω](bJ12
fBsB∗sφ = fmB∗sBB∗s∗sφ = λ√g2V, B6S[Υ() 48
βg
g = g = V,
BBω B∗B∗ω
2
fBB∗ω = fBm∗B∗ω = λg2V, 0.5 0.6 0.7 αΛ 0.8 0.9 1.0
B∗
with β = 0.9 and λ = 0.56 GeV−1. Additionally, we have FofIGΥ.(68S:)(C→olχoroωn.line). TheαΛ dependenceofthebranchingratios
g = m /f along with the pion decay constant f = 132 bJ
V ρ π π
MeV[22–25].
Withtheabovepreparation,wecanevaluatethebranching
ratios of the Υ(6S) → χ φ and Υ(6S) → χ ω transitions.
bJ bJ
However,inourmodel,therestillexistsafreeparameterαΛ,
which is introduced to parameterize the cutoff Λ. Since the 4
cutoff Λ should not be too far from the physical mass of the
exchangedmesons[29],inthisworkwesettherange0.65 ≤ 3
αΛ ≤1.15forΥ(6S)→χbJφtransitionsandset0.45≤αΛ ≤ ϕRij
1.15 for Υ(6S) → χ ω transitions to present the numerical Rϕ
bJ 2 10
results. Rϕ
20
In Figs. 7 and 8, we illustrate the αΛ dependence of the Rϕ
branching ratios of Υ(6S) → χbJφ and Υ(6S) → χbJω, re- 1 21
spectively, and in Figs. 9 and 10 we present the αΛ depen-
denceoftherelativemagnitudesamongthebranchingwidths
ofΥ(6S)→χ φandΥ(6S)→χ ω,respectively. 0.7 0.8 0.9 1.0 1.1
bJ bJ αΛ
FIG. 9: (Color online). The αΛ dependence of the ratios R1φ0 =
B[Υ(6S) → χ φ]/B[Υ(6S) → χ φ], Rφ = B[Υ(6S) →
b1 b0 20
χ φ]/B[Υ(6S)→χ φ]andRφ =B[Υ(6S)→χ φ]/B[Υ(6S)→
14 b2 b0 21 b2
Υ(6S)→χb0ϕ χb1φ].
6-0)12 ΥΥ((66SS))→→χχbb12ϕϕ
1
×10
ϕ](
χbJ 8 8
→
S) 6
6
Υ( Rω
B[ 4 6 R1ω0
20
Rω
2 21
ωRij
4
0.7 0.8 0.9 1.0 1.1
αΛ
2
FIG.7: (Coloronline). TheαΛ dependenceofthebranchingratios
ofΥ(6S)→χ φ.
bJ
0.5 0.6 0.7 0.8 0.9 1.0
αΛ
FIG. 10: (Color online). The αΛ dependence of the ratios R1ω0 =
B[Υ(6S) → χ ω]/B[Υ(6S) → χ ω], Rω = B[Υ(6S) →
b1 b0 20
χ ω]/B[Υ(6S)→χ ω]andRω =B[Υ(6S)→χ ω]/B[Υ(6S)→
b2 b0 21 b2
χ ω].
b1
6
Varying αΛ between 0.65 and 1.15 in Υ(6S) → χbJφ, we BB¯ threshold,wherethecoupled-channeleffectmaybecome
havefromFig. 7, important,whichwastestedbythestudiesinRefs. [4–10,12,
13]. Itisobviousthatthisisnottheendofthewholestory.
B[Υ(6S)→χb0φ] = (0.68∼4.62)×10−6, If the hadronic loop mechanism is a universal mechanism
B[Υ(6S)→χ φ] = (0.50∼3.43)×10−6, existinginhigherbottomoniumtransitions,wehaveareason
b1
B[Υ(6S)→χ φ] = (2.22∼15.18)×10−6, to believe that this mechanism also plays an important role
b2
in higher bottomonium transitions. Considering the similar-
and for αΛ varying from 0.45 to 1.15 in Υ(6S) → χbJω, we ity between Υ(6S) and Υ(5S), where Υ(6S) is the third bot-
havefromFig. 8 tomomium with open-bottom channels, we have focused on
Υ(6S) → χ φ and Υ(6S) → χ ω hadronic decays. Using
bJ bJ
B[Υ(6S)→χ ω] = (0.15∼2.81)×10−3, thehadronicloopmechanism,wehaveestimatedthebranch-
b0
B[Υ(6S)→χb1ω] = (0.63∼11.68)×10−3, ing ratios of Υ(6S) → χbJφ and Υ(6S) → χbJω, which can
reach up to 10−6 and 10−3, respectively. In the near future,
B[Υ(6S)→χ ω] = (1.08∼20.02)×10−3.
b2 BelleIIwillberunningneartheenergyrangeofΥ(6S),which
makes BelleII have a great opportunity to find the χ φ and
Inaddition,sometypicalvaluesfortherelativemagnitudes bJ
χ ω decay modes of Υ(6S). If these rare decays are ob-
of the predicted branching ratios are obtained from Figs. 9 bJ
served,thehadronicloopeffectscanbefurthertested.
and 10, which are weakly dependent on the free parameter
αΛ,i.e., In this work, we have especially obtained six almost sta-
ble ratios Rφ , Rφ and Rφ in addition to Rω, Rω and Rω
Rφ = B[Υ(6S)→χb1φ] ≈0.74, reflecting th1e0rela2t0ive mag2n1itudes of the Υ(6S10) →20χbJφ an2d1
10 B[Υ(6S)→χ φ] Υ(6S) → χ ω decays, which are weakly dependent on our
b0 bJ
Rφ = B[Υ(6S)→χb2φ] ≈3.28, model parameter αΛ. Thus, these obtained ratios are impor-
20 B[Υ(6S)→χ φ] tant observable quantities. We have also suggested their ex-
b0
B[Υ(6S)→χ φ] perimental measurement, which is also a crucial test of our
Rφ = b2 ≈4.43, model.
21 B[Υ(6S)→χ φ]
b1 We notice therecent talk of the status ofSuperKEKB and
B[Υ(6S)→χ ω]
Rω = b1 ≈4.11, the future plan of taking data at the BelleII experiment [30].
10 B[Υ(6S)→χb0ω] Since the collision data on Υ(6S) will be taken, we need to
B[Υ(6S)→χ ω] explore the possible interesting research issues about Υ(6S).
Rω = b2 ≈7.06,
20 B[Υ(6S)→χ ω] Ourpresentworkisonlyonesteptowardthelongmarch.
b0
B[Υ(6S)→χ ω]
Rω = b2 ≈1.72.
21 B[Υ(6S)→χ ω]
b1
Acknowledgments
As shown in numerical results on the Υ(6S) → χ φ de-
bJ
cays,thepartialdecaywidthsofΥ(6S)→χ φandΥ(6S)→
b0 This project is supported by the National Natural Sci-
χ φ are the same order of magnitude, while the partial de-
b1 ence Foundation of China under Grants No. 11222547,
caywidthofΥ(6S) → χ φisoneorderofmagnitudelarger
b2 No.11175073,No.11375240,andNo.11035006,andbyChi-
than those of Υ(6S) → χ φ and Υ(6S) → χ φ. On the
b0 b1 nese Academy of Sciences under the funding Y104160YQ0
other hand for the Υ(6S) → χ ω decays, the partial decay
bJ andtheagreementNo.2015-BH-02. XLisalsosupportedby
widths of Υ(6S) → χ ω and Υ(6S) → χ ω are nearly the
b1 b2 theNationalProgramforSupportofYoungTop-notchProfes-
same order of magnitude, while the partial decay width of
sionals.
Υ(6S) → χ φ is one order of magnitude smaller than those
b0
ofΥ(6S)→χ ωandΥ(6S)→χ ω.
b1 b2
Appendix
IV. SUMMARY
AsfortheΥ(6S) → χ φtransition,theamplitudescorre-
b1
In the past years, the anomalous hadronic transitions like spondingtoFig. 2are
Υ(5S) → Υ(nS)π+π− (n = 1,2,3) [1] and Υ(5S) → χ ω
bJ
(J = 0,1,2) [2] were reported by Belle, which has stimu- (cid:90) d4q
latedtheorists’interestinrevealingtheunderlyingmechanism M(1−1) = (2π)4[−igΥBsBs(cid:15)Υµ((ik1)µ−(ik2)µ)]
behind these phenomena [4–10]. As a popular and accepted
opinion,thehadronicloopmechanismhasbeenappliedtoex- ×[−2fBsB∗sφελρδσ(cid:15)φ∗λ(ip2)ρ((−ik1)δ−(iq)δ)]
pInlaainddwithioynt,hmeroereexpisrteadnicotmioanlsouresletrvaannstittioonthsefoΥr(Υ5S(5)Str)a[n4s–i1ti0o]n. ×[igBsB∗sχb1(cid:15)χ∗bζ1]k2−1m2 k2−1m2
1 Bs 2 Bs
weregiveninRefs. [12,13].
−gσ+q qσ/m2
Themainreasontointroducethehadronicloopmechanism × ζ ζ B∗sF2(q2), (20)
isthatΥ(5S)isthesecondobservedbottomoniumabovethe q2−m2
B∗s
7
(cid:90)
d4q
M(1−2) = (2π)4[−gΥB∗sBsεµναβ(cid:15)Υµ(−ip1)ν((ik2)β−(ik1)β)] (cid:90) d4q
×[(cid:15)φ∗λ(igB∗sB∗sφgδσ((−ik1)λ−(iq)λ) M(2−3) = (2π)4[−gΥB∗sBsεµναβ(cid:15)Υµ(−ip1)ν((ik2)β−(ik1)β)]
+4ifB∗sB∗sφ(ip2)ρ(gλδgρσ−gλσgρδ))] ×[2fB∗sBsφελρδσ(cid:15)φ∗λ(ip2)ρ((−ik1)δ−(iq)δ)]
−g +k k /m2 ×[−g (cid:15)∗ζη(−iq) (−ik ) ]
×[igBsB∗sχb1(cid:15)χ∗bζ1] αδk12−1αm2B1δ∗s B∗s ×−gσαBs+Bskχb12αkχ1σb2/m2B∗s ζ 1 2 η 1 F2(q2),(26)
× 1 −gσζ +qσqζ/m2B∗sF2(q2), (21) k12−m2B∗s k22−m2Bs q2−m2Bs
k2−m2 q2−m2
2 Bs B∗s
(cid:90)
d4q
M(1−3) = (2π)4[gΥBsB∗sεµναβ(cid:15)Υµ(−ip1)ν((ik2)β−(ik1)β)] (cid:90) d4q
×[−igBsBsφ(cid:15)φ∗λ((−ik1)λ−(iq)λ)] M(2−4) = (2π)4[−gΥB∗sBsεµναβ(cid:15)Υµ(−ip1)ν((ik2)β−(ik1)β)]
×[−igB∗sBsχb1(cid:15)χ∗bζ1]k12−1m2Bs ×+[4(cid:15)iφ∗fλB(∗siBg∗sBφ∗(sBip∗sφ2g)ρδσ(g((λ−δgikρσ1)−λ−gλ(σiqg)ρλδ)))]
×−gαζ +k2αk2ζ/m2B∗s 1 F2(q2), (22) ×[−igBsB∗sχb2εζωκξ(cid:15)χ∗bζ2η(ip3)κ(−iq)η(−ik2)ξ]
k22−m2B∗s q2−m2Bs ×−gαδ−k1αk1δ/m2B∗s 1
k2−m2 k2−m2
M(1−4) = (cid:90) (d2π4q)4[igΥB∗sB∗s(cid:15)Υµ(gναgµβ(ik2)ν−gµαgνβ(ik1)ν ×−gωσ+1qσqωB/∗sm2B∗sF22(q2), Bs (27)
q2−m2
−gαβ((ik2)µ−(ik1)µ))] B∗s
×[2fB∗sBsφελρδσ(cid:15)φ∗λ(ip2)ρ((−ik1)δ−(iq)δ)]
−gασ+kαkσ/m2
×[−igB∗sBsχb1(cid:15)χ∗bζ1] k2−1m21 B∗s
1 B∗s (cid:90)
×−gβζ +k2ζk2β/m2B∗s 1 F2(q2). (23) M(2−5) = (d2π4q)4[gΥBsB∗sεµναβ(cid:15)Υµ(−ip1)ν((ik2)β−(ik1)β)]
k2−m2 q2−m2
2 B∗s Bs ×[−igBsBsφ(cid:15)φ∗λ((−ik1)λ−(iq)λ)]
spoAnsdifnogrtthoeFΥig(.63S)ar→e χb2φtransition,theamplitudescorre- ×[igB∗sBsχb2εζωκξ(cid:15)χ∗bζ2η(ip3)κ(−iq)ξ(−ik2)η]
(cid:90) d4q × 1 −gωα +k2αk2ω/m2B∗s
M(2−1) = (2π)4[−igΥBsBs(cid:15)Υµ((ik1)µ−(ik2)µ)] k12−m2Bs k22−m2B∗s
1
×[−ig (cid:15)∗ ((−ik )λ−(iq)λ)] × F2(q2), (28)
BsBsφ φλ 1 q2−m2
×[−g (cid:15)∗ζη(−iq) (−ik ) ] Bs
BsBsχb2 χb2 ζ 2 η
1 1 1
× F2(q2), (24)
k2−m2 k2−m2 q2−m2
1 Bs 2 Bs Bs
(cid:90)
(cid:90) d4q
M(2−2) = (d2π4q)4[−igΥBsBs(cid:15)Υµ((ik1)µ−(ik2)µ)] M(2−6) = (2π)4[gΥBsB∗sεµναβ(cid:15)Υµ(−ip1)ν((ik2)β−(ik1)β)]
×[−2fBsB∗sφελρδσ(cid:15)φ∗λ(ip2)ρ((−ik1)δ−(iq)δ)] ×[−2fBsB∗sφελρδσ(cid:15)φ∗λ(ip2)ρ((−ik1)δ−(iq)δ)]
××[−ig1BsB∗sχb2εζω1κξ(cid:15)χ∗bζ2η(ip3)κ(−iq)η(−ik2)ξ] ××[gB∗s1B∗sχb2(cid:15)χ∗−bζ2ηg(καg+ζκgkη2ξαk+2κ/gmηκ2Bg∗sζξ)]
k2−m2 k2−m2 k2−m2 k2−m2
1 Bs 2 Bs 1 Bs 2 B∗s
−gσω+qσqω/m2 −gσξ+qσqξ/m2
× B∗sF2(q2), (25) × B∗sF2(q2), (29)
q2−m2 q2−m2
B∗s B∗s
8
M(2−7) = (cid:90) (d2π4q)4[igΥB∗sB∗s(cid:15)Υµ(gναgµβ(ik2)ν−gµαgνβ(ik1)ν A(0−4) = (cid:90) (d2π4q)4[igΥB∗B∗(cid:15)Υµ(gναgµβ(ik2)ν−gµαgνβ(ik1)ν
−gαβ((ik2)µ−(ik1)µ))] −gαβ((ik2)µ−(ik1)µ))]
××[[2igfBB∗s∗sBBssχφbε2ελρζδωσκξ(cid:15)(cid:15)φ∗χλ∗bζ(2ηi(pi2p)3ρ)(κ((−−iikq1))ξδ(−−i(ki2q))ηδ])]] +×4[−(cid:15)iωg∗fλBα(∗+iBg∗ωBk∗(Bip∗kω2αg)/ρδm(σg2(λ(δ−−giρkgσ1β)ζ−λ+−gkλ(σβikqgζ)ρ/λδ)m))]2[−gB∗B∗χb0]
−gασ+kαkσ/m2 −gβω+kβkω/m2 × δ 1δ 1 B∗ 2 2 B∗
× 1 1 B∗s 2 2 B∗s k2−m2 k2−m2
k2−m2 k2−m2 1 B∗ 2 B∗
1 B∗s 2 B∗s −g +q q /m2
1 × σζ σ ζ B∗F2(q2). (35)
× F2(q2), (30) q2−m2
q2−m2 B∗
Bs AsfortheΥ(6S) → χ ωtransition,theamplitudescorre-
b1
spondingtoFig. 5are
(cid:90)
M(2−8) = (d2π4q)4[igΥB∗sB∗s(cid:15)Υµ(gναgµβ(ik2)ν−gµαgνβ(ik1)ν A(1−1) = (cid:90) (d2π4q)4[−igΥBB(cid:15)Υµ((ik1)µ−(ik2)µ)]
−g ((ik ) −(ik ) ))]
αβ 2 µ 1 µ ×[−2f ε (cid:15)∗λ(ip )ρ((−ik )δ−(iq)δ)]
×[(cid:15)φ∗λ(igB∗sB∗sφgδσ((−ik1)λ−(iq)λ) BB∗ω λρδσ ω1 2 1 1
+4ifB∗sB∗sφ(ip2)ρ(gλδgρσ−gλσgρδ))] ×[igBB∗χb1(cid:15)χ∗bζ1]k12−m2Bk22−m2B
××[−ggBαδ∗sB+∗sχbk21(cid:15)δχ∗kbζ2η1α(/gmζκ2Bg∗sη−ξg+βκgη+κgkζ2βξk)2κ]/m2B∗s ×−gσζq+2−qζmqσ2B/∗m2B∗F2(q2), (36)
k2−m2 k2−m2
1 B∗s 2 B∗s (cid:90)
×−gξσ+qσqξ/m2B∗sF2(q2). (31) A(1−2) = (d2π4q)4[−gΥB∗Bεµναβ(cid:15)Υµ(−ip1)ν((ik2)β−(ik1)β)]
q2−m2
B∗s ×[(cid:15)∗ (ig gδσ((−ik )λ−(iq)λ)
ωλ B∗B∗ω 1
AsfortheΥ(6S) → χb0ωtransition,theamplitudescorre- +4ifB∗B∗ω(ip2)ρ(gλδgρσ−gλσgρδ))]
spondingtoFig. 5are −g +k k /m2
×[ig (cid:15)∗ζ ] αδ 1α 1δ B∗
A(0−1) = (cid:90) (d2π4q)4[−igΥBB(cid:15)Υµ((ik1)µ−(ik2)µ)] × B1B∗χb1 −χbg1σζ +qkσ12qζ−/mm2B2B∗∗F2(q2), (37)
×[−ig (cid:15)∗ ((−ik )λ−(iq)λ)][−g ] k2−m2 q2−m2
BBω ωλ 1 BBχb0 2 B B∗
1 1 1
× F2(q2), (32)
(cid:90)
k2−m2 k2−m2 q2−m2 d4q
1 B 2 B B A(1−3) = (2π)4[gΥBB∗εµναβ(cid:15)Υµ(−ip1)ν((ik2)β−(ik1)β)]
(cid:90) d4q ×[−igBBω(cid:15)ω∗λ((−ik1)λ−(iq)λ)]
A(0−2) = (2π)4[−gΥB∗Bεµναβ(cid:15)Υµ(−ip1)ν((ik2)β−(ik1)β)] ×[−igB∗Bχb1(cid:15)χ∗bζ1]k2−1m2
×[2f ε (cid:15)∗λ(ip )ρ((−ik )δ−(iq)δ)][−g ] 1 B
×−gσαB∗+Bωk1λαρkδσ1σ/ωm2B∗ 2 1 1 1 F2(q2)B,B(χ3b03) ×−gαζk+2k−2αmk22ζ/m2B∗ q2−1m2F2(q2), (38)
k2−m2 k2−m2 q2−m2 2 B∗ B
1 B∗ 2 B B
(cid:90)
d4q
(cid:90) d4q A(1−4) = (2π)4[igΥB∗B∗(cid:15)Υµ(gναgµβ(ik2)ν−gµαgνβ(ik1)ν
A(0−3) = (2π)4[gΥBB∗εµναβ(cid:15)Υµ(−ip1)ν((ik2)β−(ik1)β)] −gαβ((ik2)µ−(ik1)µ))]
×[−2fBB∗ωελρδσ(cid:15)ω∗λ(ip2)ρ((−ik1)δ−(iq)δ)] ×[2fB∗Bωελρδσ(cid:15)ω∗λ(ip2)ρ((−ik1)δ−(iq)δ)]
×[−gB∗B∗χb0]k2−1m2 −gζαk+2k−2αmk22ζ/m2B∗ ×[−igB∗Bχb1(cid:15)χ∗bζ1]−gασk2+−k1αmk21σ/m2B∗
1 B 2 B∗ 1 B∗
×−gσζ +qσqζ/m2B∗F2(q2), (34) ×−gβζ +k2ζk2β/m2B∗ 1 F2(q2). (39)
q2−m2 k2−m2 q2−m2
B∗ 2 B∗ B
9
AsfortheΥ(6S) → χ ωtransition,theamplitudescorre-
b2
spondingtoFig. 6are
(cid:90)
d4q
(cid:90) d4q A(2−6) = (2π)4[gΥBB∗εµναβ(cid:15)Υµ(−ip1)ν((ik2)β−(ik1)β)]
A(2−1) = (2π)4[−igΥBB(cid:15)Υµ((ik1)µ−(ik2)µ)] ×[−2f ε (cid:15)∗λ(ip )ρ((−ik )δ−(iq)δ)]
BB∗ω λρδσ ω 2 1
×[−igBBω(cid:15)ω∗λ((−ik1)λ−(iq)λ)] ×[gB∗B∗χb2(cid:15)χ∗bζ2η(gζκgηξ+gηκgζξ)]
×[−gBBχb2(cid:15)χ∗bζ2η(−iq)ζ(−ik2)η] × 1 −gκα+k2αk2κ/m2B∗
× 1 1 1 F2(q2), (40) k12−m2B k22−m2B∗
k12−m2Bk22−m2Bq2−m2B −gσξ+qσqξ/m2
× B∗F2(q2), (45)
q2−m2
(cid:90) B∗
d4q
A(2−2) = (2π)4[−igΥBB(cid:15)Υµ((ik1)µ−(ik2)µ)] (cid:90)
d4q
×[−2fBB∗ωελρδσ(cid:15)ω∗λ(ip2)ρ((−ik1)δ−(iq)δ)] A(2−7) = (2π)4[igΥB∗B∗(cid:15)Υµ(gναgµβ(ik2)ν−gµαgνβ(ik1)ν
×[−igBB∗χb2εζωκξ(cid:15)χ∗bζ2η(ip3)κ(−iq)η(−ik2)ξ] −gαβ((ik2)µ−(ik1)µ))]
× 1 1 ×[2fB∗Bωελρδσ(cid:15)ω∗λ(ip2)ρ((−ik1)δ−(iq)δ)]
k−12g−σωm+2Bqkσ22q−ω/mm2B2 ×[igB∗Bχb2εζωκξ(cid:15)χ∗bζ2η(ip3)κ(−iq)ξ(−ik2)η]]
× B∗F2(q2), (41) −gασ+kαkσ/m2 −gβω+kβkω/m2
q2−m2 × 1 1 B∗ 2 2 B∗
B∗ k2−m2 k2−m2
1 B∗ 2 B∗
1
(cid:90) × F2(q2), (46)
A(2−3) = (d2π4q)4[−gΥB∗Bεµναβ(cid:15)Υµ(−ip1)ν((ik2)β−(ik1)β)] q2−m2B
×[2fB∗Bωελρδσ(cid:15)ω∗λ(ip2)ρ((−ik1)δ−(iq)δ)] (cid:90) d4q
×[−gBBχb2(cid:15)χ∗bζ2η(−iq)ζ(−ik2)η] A(2−8) = (2π)4[igΥB∗B∗(cid:15)Υµ(gναgµβ(ik2)ν−gµαgνβ(ik1)ν
×−gσα +k1αk1σ/m2B∗ 1 1 F2(q2), (42) −gαβ((ik2)µ−(ik1)µ))]
k2−m2 k2−m2 q2−m2 ×[(cid:15)∗ (ig gδσ((−ik )λ−(iq)λ)
1 B∗ 2 B B ωλ B∗B∗ω 1
+4if (ip ) (gλδgρσ−gλσgρδ))]
B∗B∗ω 2 ρ
A(2−4) = (cid:90) (d2π4q)4[−gΥB∗Bεµναβ(cid:15)Υµ(−ip1)ν((ik2)β−(ik1)β)] ×[−ggBα∗B+∗χbk2(cid:15)χ∗kbζ2ηα(/gmζκ2gη−ξg+βκgη+κgkζβξk)κ]/m2
×[(cid:15)ω∗λ(igB∗B∗ωgδσ((−ik1)λ−(iq)λ) × δk2−1δm12 B∗ k2−2m22 B∗
1 B∗ 2 B∗
+4if (ip ) (gλδgρσ−gλσgρδ))]
B∗B∗ω 2 ρ −gξ +q qξ/m2
×[−ig ε (cid:15)∗ζη(ip )κ(−iq) (−ik )ξ] × σ σ B∗F2(q2). (47)
BB∗χb2 ζωκξ χb2 3 η 2 q2−m2B∗
×−gαδ−k1αk1δ/m2B∗ 1
k2−m2 k2−m2
1 B∗ 2 B
−gω+q qω/m2
× σ σ B∗F2(q2), (43)
q2−m2
B∗
(cid:90)
d4q
A(2−5) = (2π)4[gΥBB∗εµναβ(cid:15)Υµ(−ip1)ν((ik2)β−(ik1)β)]
×[−ig (cid:15)∗ ((−ik )λ−(iq)λ)]
BBω ωλ 1
×[ig ε (cid:15)∗ζη(ip )κ(−iq)ξ(−ik ) ]
B∗Bχb2 ζωκξ χb2 3 2 η
× 1 −gωα +k2αk2ω/m2B∗
k2−m2 k2−m2
1 B 2 B∗
1
× F2(q2), (44)
q2−m2
B
10
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