Light axial vector mesons

0.41 MB·English

by  Kan Chen

Lightaxialvectormesons
Kan Chen1,2, Cheng-Qun Pang1,2, Xiang Liu1,2 , and Takayuki Matsuki3,4,1
∗ † ‡§ ¶
1School of Physical Science and Technology, Lanzhou University, Lanzhou 730000, China
2ResearchCenterforHadronandCSRPhysics,LanzhouUniversity&InstituteofModernPhysicsofCAS,Lanzhou730000,China
3Tokyo Kasei University, 1-18-1 Kaga, Itabashi, Tokyo 173-8602, Japan
4Theoretical Research Division, Nishina Center, RIKEN, Saitama 351-0198, Japan
Inspired by the abundant experimental observation of axial-vector states, we study whether the observed
axial-vectorstatescanbecategorizedintotheconventional axial-vectormesonfamily. Inthispaperwecarry
outananalysisbasedonthemassspectraandtwo-bodyOkubo-Zweig-Iizuka-alloweddecays. Besidestesting
the possible axial-vector meson assignments, we also predict abundant information for their decays and the
propertiesofsomemissingaxial-vectormesons,whicharevaluableforfurtherexperimentalexplorationofthe
5
observedandpredictedaxial-vectormesons.
1
0
PACSnumbers:14.40.Be,12.38.Lg,13.25.Jx
2

r
p
I. INTRODUCTION Thispaperis organizedasfollows. In Sec. II, we present
A
the phenomenologicalanalysis by combiningour theoretical
results with the correspondingexperimentaldata; the Regge
8 Among the light unflavored mesons listed in the Particle
1 trajectory analysis is adopted to study mass spectra of the
Data Group (PDG) [1], there are abundantlight axial-vector
axial-vectormesonfamilyandthequark-paircreation(QPC)
mesons with a spin-parityquantum number JP = 1+, which
] modelisappliedtocalculatetheirOZI-allowedstrongdecay
h form a P-wave meson family. Usually, we adopt h , b , f ,
1 1 1 behavior. Finally,thediscussionandconclusionare givenin
p and a to expressthe correspondingstates with the quantum
- numb1ers IG(JPC) = 0 (1+ ), 1+(1+ ), 0+(1++), and 1 (1++), Sec. III.
p − − − −
e respectively. InTableI,wecollecttheexperimentalinforma-
h tion on the observed h , b , f , and a states, as well as the
1 1 1 1
[ correspondingresonanceparametersand the observeddecay II. PHENOMENOLOGICALSTUDYOFOBSERVED
AXIAL-VECTORSTATES
channels.
3
v Facingsomanyaxial-vectorstatesinthePDG,weneedto
A Regge trajectory analysis is an effective approach to
6 examinewhetherallofthesestatescanbecategorizedintothe
6 axial-vectormesonfamily,whichiscrucialforrevealingtheir studya meson spectrum[31], especially a light-mesonspec-
7 trum. The masses and radial quantum numbers of light
underlyingstructures. We also notice that most axial-vector
7 mesonswiththesamequantumnumbersatisfythefollowing
states are either omitted by the PDG or are recent findings
0 relation:
. needingconfirmation. Duetotheunclearexperimentalstatus
1
of light axial-vector states, we need to carry out a quantita-
0 M2 = M2+(n 1)µ2, (1)
tiveinvestigationofthem,whichwouldbehelpfulforfurther 0 −
5
experimental studies, especially of those axial-vector states
1 where M and M are the masses of the groundstate and the
: either omitted by the PDG or unconfirmed by other experi- 0
v correspondingradial excitation with radial quantum number
ments.
i n, respectively. µ2 denotes the slope of the trajectory with a
X Inthis work, we carryouta systematic studyof theaxial- universalµ2 =1.25 0.15GeV2[31].
r vector states by analyzing mass spectra and Okubo-Zweig- InFig.1,weprese±nttheReggetrajectoryanalysis,inwhich
a
Iizuka(OZI)-allowedtwo-bodystrongdecaybehaviors. Our
weconsideralloftheaxial-vectorstateslistedinthePDGas
investigationsarebasedonthe assumptionthatalloftheax-
showninTableI.Besidestheobservedones,wealsopredict
ial mesons can be explainedwithin the conventionalqq¯ pic-
some missing states and show them in Fig. 1. Additionally,
ture. Comparingournumericalresultswiththeexperimental
we notice that there are two possible candidates for the a
1
data,wecanfurthertestthepossibleassignmentsofthestates meson with quantum number n2s+1J = 33P , i.e., a (1930)
L 1 1
in the axial-vector meson family. In addition, information
anda (2095). Ontheotherhand,both f (1420)and f (1510)
1 1 1
onthepredicteddecaysoftheaxial-vectorstatesobservedor
canbeanss¯partnerof f (1285)byanalyzingonlytheRegge
1
stillmissinginexperimentsisvaluabletofurtherexperimental
trajectory. Thus,afurtherstudyoftheirstrongdecaybehav-
studyofaxial-vectormeson.
iors would help to test these possible assignmentsto the ob-
servedaxial-vectorstatesandcouldprovidemorepredictions
oftheobservedandstill-missingaxial-vectormesons,which
arevaluableforthe futureexperimentalexplorationofaxial-
‡Correspondingauthor vectormesons.
Electronicaddress:chenk˙[email protected]
∗ To obtain the decay behaviorsof the axial-vectormesons,
Electronicaddress:[email protected]
†
weadopttheQPCmodel,whichwasfirstproposedbyMicu
Electronicaddress:[email protected]
§
Electronicaddress:[email protected] [32]andfurtherdevelopedbytheOrsaygroup[33–37]. This
¶
2
TABLEI:Resonanceparametersandstrongdecaychannelsoftheaxial-vectorstatescollectedinthePDG[1]. Themassesandwidthsare
averagevaluestakenfromthePDG.ThestatesomittedfromthePDGsummarytablearemarkedbyasuperscript♮,whilethestateslistedas
furtherstatesinthePDGaremarkedbyasuperscript♭.
IG(JPC) State Mass(MeV) Width(MeV) Theobserveddecaychannels
a (1260) 1230 40 250 600 3π[2],πρ[3],σπ[4]
1 ± ∼
a (1640)♮ 1647 22 254 27 3π[5],πρ[4,6],σπ[5], f (1270)π[5]
1 ± ± 2
1 (1++) a (1930)♭ 1930+30 155 45 3π0[7]
− 1 70 ±
−
a (2095)♭ 2096 17 121 451 41 81 π+π π [8]
1 ± ± ± ± − −
a (2270)♭ 2270+55 305+70 3π0[7]
1 40 40
− −
b (1235) 1229.5 3.2 142 9 ωπ[9–11]
1 ± ±
1+(1+ ) b (1960)♭ 1960 35 345 75 ωπ0[12]
− 1 ± ±
b (2240)♭ 2240 35 320 85 ωπ0[12]
1 ± ±
f (1285) 1282.1 0.6 24.2 1.1 ρ0ρ0[13],ηππ[14–16],a π[15–17],KK¯π[15,16,18]
1 ± ± 0
f (1420) 1426.4 0.9 54.9 2.6 KK¯π[19,20],KK¯ (892)+c.c[18–20]
1 ± ± ∗
0+(1++) f (1510)♮ 1518 5 73 25 KK¯ (892)+c.c[21,22],π+π η[23]
1 ± ± ∗ − ′
f (1970)♭ 1971 15 240 45 ηπ0π0[24]
1 ± ±
f (2310)♭ 2310 60 255 70 ηπ0π0[24]
1 ± ±
h (1170) 1170 20 360 40 πρ[25–27]
1 ± ±
h (1380)♮ 1386 19 91 30 KK¯ (892)+c.c[21,28]
1 ± ± ∗
0 (1+ ) h (1595)♮ 1594 15+10 384 60+70 ωη[29]
− − 1 ± 60 ± 100
−
h (1965)♭ 1965 45 345 75 ωη[30]
1 ± ±
h (2215)♭ 2215 40 325 55 ωη[30]
1 ± ±
modelwaswidelyappliedtostudytheOZI-allowedtwo-body canbetreatedseparately. Inaddition,iand jdenotethecolor
strongdecayofhadrons[38–59]. Inthefollowing,webriefly indicesofaqq¯ pair.
introducetheQPCmodel. By the Jacob-Wick formula [60], the decay amplitude is
ForadecayprocessA B+C,wecanwrite expressedas
→
hBC|T|Ai=δ3(PB+PC)MMJAMJBMJC, (2) JL(P) = √4π(2L+1) L0;JM J M
wherePB(C)isathree-momentumofamesonB(C)intherest M 2JA+1 MXJBMJCh JA| A JAi
afrnamorebiotaflammagensoetnicAm. oAmesnutbusmcr.ipTthMetJria(ins=itioAn,Bop,Cer)atdoernoteiss ×hJBMJB;JCMJC|JAMJAiMMJAMJBMJC, (4)
T
introduced to describe a quark-antiquark pair creation from andthegeneraldecaywidthreads
vacuum, which has the quantumnumber JPC = 0++, i.e.,
T
canbeexpressedas π P
Γ = | | JL(P)2, (5)
4m2 X|M |
A J,L
= 3γ 1m;1 m00 dp dp δ3(p +p )
T − Xm h − | iZ 3 4 3 4 wheremA isthemassofaninitialstate A. Weusethesimple
p p harmonicoscillatorwavefunctiontodescribethespacewave
×Y1m(cid:18) 3−2 4(cid:19)χ314,−mφ304(cid:16)ω304(cid:17)ijb†3i(p3)d4†j(p4), (3) functionofmesons,whichhasthefollowingexpression:
which is constructedin a completelyphenomenologicalway Ψ (R,p)= (R,p) (p), (6)
nlm nl lm
toreflectthecreationofaquark-antiquarkpairfromvacuum, R Y
where the quark and antiquark are denoted by indices 3 and wheretheconcretevaluesoftheparameterRinvolvedinour
4, respectively. As a dimensionlessparameter, γ depicts the calculationaregiveninRef. [61]forthegroundstates. How-
strength of the creation of qq¯ from vacuum, where γ = 8.7 ever,itsvalueistobefixedforeachexcitedstate.
and 8.7/√3 [51] correspondsto the uu¯/dd¯and ss¯ creations, With the above preparation, we further discuss the OZI-
respectively. (p) = pℓY (p)isthesolidharmonic. χ,φ, alloweddecaybehaviorsoftheaxial-vectormesons. The al-
ℓm ℓm
Y | |
andωdenotethespin,flavor,andcolorwavefunctions,which loweddecaymodesarelistedinTablesIIandIII.
3
6 6
a(2270)
1
5 5
a1(2095) b1(2240)
4 4
2GeV) 3 a1(1930) 2GeV) 3 b1(1960)
2M( a1(1640) 2M( b1(1640)
2 2
a(1260) 1−1+ b(1235) 1+1+
1 1 1 1
0 1 2 3 4 5 0 1 2 3 4 5
n n
6 6
f1(2310) h1(2340)
5 5
f1(2110) f1(2210) h1(2120) h1(2215)
4 4
2GeV) 3 f(1510) f1(1800) f1(1970) 2GeV) 3 h1(1780) h1(1965)
2M( 1 f1(1640) 2M( h(1380)
2 2 1 h(1595)
f(1420) 1
1
1 f1(1285) 0+1++ 1 h(1170) 0−1+−
1
0 1 2 3 4 5 0 1 2 3 4 5
n n
FIG.1: (Coloronline)Reggetrajectoryanalysisfora ,b , f ,andh withtypicalµ2 =1.30GeV2,1.19GeV2,1.10GeV2,and1.19GeV2,
1 1 1 1
respectively,whichcanbecoveredbyµ2 =1.25 0.15GeV2giveninRef. [31]. Theexperimentalerrorsofdiscussedaxial-vectorstatesare
±
given,whicharetakenfromthePDG[1].Here, and denotetheoreticalandexperimentalvalues,respectively.
◦ •
A. a states
1
500 40
The Regge trajectory analysis indicates that a (1260) can
1 σπ
be regardedasa groundstate. Theobtainedtotalandpartial
30
decay widths of a1(1260) are listed in Fig. 2, which shows 450 CMD−2
thatπρisthedominantchannel. InFig. 2,wegivethepartial
decaywidthsofa1(1260) πρfromtheS-waveandD-wave Total Width 20 (πρ)
contributions. Here, the →D-wave/S-wave amplitude ratio in D
the decay a (1260) πρ is 0.248 with a typical value of 400
1
bRle=w3i.t8h4t6heGBeV85−21 [d6a→1ta]i(n0o.u1r4−calc0u.0la4tion0,.0w7h)ic[4h].isOcuormrpeasrual-t (πρ)S 10 πf0(980)
− ± ±
alsoshowsthata1(1260) f0πisasubordinatedecaymode 350
with the partial decay wid→th 1.82 MeV, which explains why 3.3 3.6 3.9 4.2 4.5 3.3 3.6 3.9 4.2 4.5
therehasbeennoevidenceofa (1260) f πinexperiments R (GeV−1) R(GeV−1)
1 0
→
[3]. AsshowninFig. 2,thecalculatedtotalwidthcanrepro-
ducetheCMD2datagiveninRef. [62]. Inaddition,wealso FIG. 2: (Color online) Total and partial decay widths of a (1260)
1
give some typical ratios relevant to the partial decay and to- depending on R. Here, the dot-dashed line with the band is taken
talwidthstogetherwiththecorrespondingexperimentaldata fromtheexperimentaldatafromRef.[62].TheS-waveandD-wave
contributions to the decay width of a (1260) πρ are also given
inTableIV.Insummary,ourresultsarecomparablewiththe 1 →
separately.AllresultsareinunitsofMeV.
experimentalvaluesandsupporta (1260)asagroundstatein
1
thea mesonfamily.
1
If a (1640) is the first radial excitation of a (1260), its
1 1
decay behavior depending on the R value is shown in Fig.
3. We use the experimental total width [5] and the ratio
Γ(f (1270)π)/Γ(σπ)=0.24 0.07[5]togetR=(4.30 4.64)
2
± ∼
4
TABLEII:OZI-allowedtwo-bodydecaychannelsfora andh statesmarkedby√. Here,ρ,ω,andηdenoteρ(770),ω(782), andη(548),
1 1
respectively.Theaxial-vectorstatespredictedbytheReggetrajectoryanalysisaremarkedbyasuperscript♮.
Channel a1(1260)a1(1640)a1(1930)a1(2095)a1(2270) Channel h1(1170)h1(1380)h1(1595)h♮1(1780)h1(1965)h♮1(2120)h1(2215)h♮1(2340)
πρ √ √ √ √ √ πρ √ √ √ √ √ √ √ √
σπ √ √ √ √ √ KK √ √ √ √ √ √ √
∗
πf √ √ √ √ √ ηω √ √ √ √ √ √ √
0
πf (1420) √ √ √ √ ωσ √ √ √ √ √ √
1
πρ(1450) √ √ √ √ KK (1270) √ √ √ √ √
1
ρω √ √ √ √ ωη(958) √ √ √ √ √
′
ηa (980) √ √ √ √ ωf √ √ √ √ √
0 0
KK √ √ √ √ ρa (980) √ √ √ √ √
∗ 0
πb (1235) √ √ √ √ πρ(1450) √ √ √ √ √
1
πf (1270) √ √ √ √ KK (1400) √ √ √ √
2 1
πf (1285) √ √ √ √ KK (1410) √ √ √ √
1 ∗
ρa (980) √ √ √ KK (1430) √ √ √ √
0 0∗
KK (1400) √ √ √ KK (1430) √ √ √ √
1 2∗
ηa (1260) √ √ √ K K √ √ √ √
1 ∗ ∗
πρ(1700) √ √ √ ηω(1420) √ √ √ √
KK (1270) √ √ √ σh (1170) √ √ √ √
1 1
KK (1410) √ √ √ πρ(1700) √ √ √ √
∗
KK (1430) √ √ √ ρa (1320) √ √ √
0∗ 2
KK (1430) √ √ √ ωf (1270) √ √ √
2∗ 2
K K √ √ √ σω(1420) √ √ √
∗ ∗
ηa (1320) √ √ √ ωf (1285) √ √ √
2 1
σa (1260) √ √ √ ρπ(1300) √ √ √
1
σa (1320) √ √ √ ρa (1260) √ √ √
2 1
ρπ(1300) √ √ K K (1270) √ √
∗ 1
ηa (1450) √ √ f h (1170) √ √
0 0 1
ωb (1235) √ √ KK (1680) √ √
1 ∗
ρh (1170) √ √ ωf (1420) √ √
1 1
ρa (1260) √ √ K K (1400) √
1 ∗ 1
K K (1270) √
∗ 1
ρa (1320) √
2
ρω(1420) √
ρa (1450) √
0
KK (1680) √
∗
η(958)a (980) √
′ 0
GeV 11.Themaindecaymodesofa (1640)areπρ,πρ(1450), TableV.
− 1
πf (1270),πf (1285),andρω. Additionally,we providefur-
2 1 Therearetwopossiblecandidatesforthesecondradialex-
ther information on the typical ratios of a (1640) decays in
1 citation of a (1260). In the following, we discuss the decay
1
behaviorsof a (1930)anda (2095)by combiningthe corre-
1 1
spondingexperimentaldata. InFigs. 4and5,wepresentthe
R dependence of the decay behaviors of these a ’s, respec-
1
1Using the experimental total width [5], we find that overlap exists be- tively.Thatis,theobtainedtotalwidthofa (1930)canbefit-
tween ourtheoretical andexperimental results whentakingR = 4.26 1
4.92 GeV−1. Then, we can further constrain the R values by the rati∼o tedwiththedatafromRef. [7]whenR=4.58∼4.92GeV−1,
Γ(f2(1270)π)/Γ(σπ)=0.24 0.07[5],wheretheconstrainedR=(4.30 whilethatofa1(2095)canoverlapwiththeexperimentaldata
4.64)GeV−1,whichisadopt±edtopresentothertypicalratiosofa1(1640)∼. [7] when R = (4.78 5.16) GeV−1. Thus, it is difficult to
∼
distinguish which a is more suitable as a candidate for the
1
5
TABLEIII:OZI-allowedtwo-bodydecaychannelsforb and f statesmarkedby√. Here,ρ,ω,andηdenoteρ(770),ω(782),andη(548),
1 1
respectively.Theaxial-vectorstatespredictedbytheReggetrajectoryanalysisaremarkedbyasuperscript♮.
Channel b1(1235)b♮1(1640)b1(1960)b1(2240) Channel f1(1285)f1(1420)f1(1510)f1♮(1640)f1♮(1800)f1(1970)f1♮(2110)f1♮(2210)f1(2310)
πω √ √ √ √ πa (980) √ √ √ √ √ √ √ √ √
0
πa (980) √ √ √ √ ση √ √ √ √ √ √ √ √ √
0
πa (1260) √ √ √ πa (1260) √ √ √ √ √ √ √ √
1 1
πa (1320) √ √ √ KK √ √ √ √ √ √ √ √
2 ∗
πω(1420) √ √ √ πa (1320) √ √ √ √ √ √ √
2
πa (1450) √ √ √ πa (1450) √ √ √ √ √ √
0 0
ηρ √ √ √ ηf √ √ √ √ √ √
0
ρρ √ √ √ ρρ √ √ √ √ √ √
KK √ √ √ ωω √ √ √ √ √ √
∗
σρ √ √ √ ση √ √ √ √ √ √
′
η(958)ρ √ √ KK (1270) √ √ √ √ √
′ 1
ρf √ √ K K √ √ √ √ √
0 ∗ ∗
ωa (980) √ √ ωh (1170) √ √ √ √
0 1
KK (1270) √ √ η(958)f √ √ √ √
1 ′ 0
KK (1400) √ √ ηf (1270) √ √ √ √
1 2
KK (1410) √ √ KK (1400) √ √ √ √
0∗ 1
KK (1430) √ √ KK (1410) √ √ √ √
0∗ ∗
KK (1430) √ √ KK (1430) √ √ √ √
2∗ 0∗
σb (1235) √ √ KK (1430) √ √ √ √
1 2∗
K K √ √ ηf (1285) √ √ √ √
∗ ∗ 1
ωa (1320) √ σf (1270) √ √ √ √
2 2
ωπ(1300) √ σf (1285) √ √ √ √
1
K K (1270) √ σf (1420) √ √ √
∗ 1 1
ηρ(1450) √ ρb (1235) √ √ √
1
KK (1680) √ ηf (1420) √ √ √
∗ 1
a (980)h (1170) √ ωω(1420) √ √
0 1
ρf (1270) √ K K (1270) √ √
2 ∗ 1
ρf (1285) √ KK (1680) √
1 ∗
ρf (1420) √ f f (1270) √
1 0 2
ρb (1235) √ f f (1285) √
1 0 1
ωa (1260) √ a (980)a (1260) √
1 0 1
a (980)π(1300) √
0
a (980)a (1320) √
0 2
η(958)f (1270) √
′ 2
ρρ(1450) √
η(958)f (1285) √
′ 1
K K (1400) √
∗ 1
secondradialexcitationofa (1260)bystudyingonlythetotal whiletheπf (1285)andσπmodesalsohavesizablecontribu-
1 1
decay widths. Besides, we can learn from the Regge trajec- tions. The decays of a (1930) into KK (1430), KK (1430),
1 0∗ 2∗
tory analysis that there is only one state for the 33P state, and K (896)K (896) have tiny decay widths, which are not
1 ∗ ∗
and it is doubtful that both a (1930) and a (2095) exist, as listedinFig. 4. Asfora (2095),itsdominantdecaychannels
1 1 1
mentionedin Ref. [7]. However,thereexistdifferentbehav- areπb (1235),πρ,andπρ(1450)andareshowninFig. 5. The
1
iors of the partial decay widths of these a ’s. The a (1930) other decay channels—like ρa (980), πρ(1700), πf (1285),
1 1 0 1
mainly decaysinto finalstates πρ, πρ(1450),and πb (1235), πf , and σπ—also haveconsiderablecontributionsto the to-
1 0
6
thefollowing,theexperimentalconfirmationofa (1930)and
1
TABLE IV: Some typical ratiosof decay widths of a (1260). The
1 a (2095) will be crucial for identifying the candidate of the
Γ(πρ) representtheS(D)-wavedecaywidthofa (1260) πρ. 1
S(D) 1 → 33P1 state in the a1 mesonfamily. If a1(1930)and a1(2095)
Ourwork Experimentaldata cannot be established in experiments, we suggest an experi-
Γ((πρ) )/Γ 0.86 0.60[3] mentalsearchfora (33P );theresultsfora (33P )predicted
S Total 1 1 1 1
Γ((πρ) )/Γ 5.3 10 2 (1.30 0.60 0.22) 10 2 [3] inthisworkwouldbehelpfulforsuchasearch.
D Total × − ± ± × −
Γ /Γ 8.2 10 2 (18.76 4.29 1.48) 10 2 [3]
πσ Total × − ± ± × −
Γ /Γ 0.09 0.06 0.05[1]
σπ (ρπ)S ± TABLEVI:Typicalratiosfora (1930)anda (2095). TheRranges
1 1
are(4.58 4.92)GeV 1 and(4.78 5.16)GeV 1fora (1930)and
∼ − ∼ − 1
30 a1(2095),respectively.
360 Total width
310 SPEC 25 ρω Ratio a1(1930) a1(2095)
260 20 Γπρ/ΓTotal 0.151∼0.162 0.139∼0.176
210 πρ(1450) 15 Γπb1(1235)/ΓTotal 0.092∼0.1600.206∼0.2542
160 Γ /Γ 0.005 0.0240.039 0.0529
11600 πρ(770) 150 πf2(1270) πf1(1285) Γσπρπ(/17Γ0T0)otalTotal 0.088∼∼0.097 0.058∼∼0.073
Γ /Γ 0.339 0.347 0.189 0.253
10 0 πρ(1450) Total ∼ ∼
Γ /Γ 0.271 0.462 0.348 1.813
πb1(1235) πρ(1450) ∼ ∼
40 πb1(1235) 1.6 πf(1420) Γηa1(1260)/ΓKK∗(892) 0.629∼0.719 1.141∼1.742
32 1 Γ Γ 0.705 0.850 0.188 0.451
σπ 1.2 ρω πf2(1270) ∼ ∼
24 KK*(892) Γηa0(980)/Γπρ(1700) 0.317∼0.809 0.239∼0.279
0.8 Γ /Γ 0.508 0.553 1.693 4.846
16 KK1(1400) ηa2(1320) ∼ ∼
8 πf0(980) 0.4 a0η ΓKK1(1400)/Γρa0(980) − 0.145∼0.184
Γ /Γ 0.206 0.838
ηa0(1450) KK0∗(1430) − ∼
0 0
4 4.2 4.4 4.6 4.8 5 4 4.2 4.4 4.6 4.8 5
R (GeV−1) R (GeV−1)
300
FIG. 3: (Color online) R dependence of the decay behaviors of 10
Total Width 20
a (1640).Here,thedot-dashedlinewiththebandistheexperimental
to1talwidthfromRef.[5].AllresultsareinunitsofMeV. 200 8 KK*(892) 16 π f(1285)
SPEC 6 f(1420)π 12 1
100 πρ(1450) 4 1 8 π ρ(170a01)(1260)σ
tal decay width. In Table VI, we also list some typical ra- 2 ηa0(980) 4 ρ ω
tios relevantto theirdecays. We still needto emphasizeone 0 π b1(1235) 0 0
plioshinetd. iAntepxrpeesreinmt,enat1s(.19T3h0e)aauntdhoar1s(o2f09R5e)f.ar[e7]nointdwicealtleedsttahba-t 12.5 KK*(1410) 0.6 ηa2(1320) 40
a2(1950)anda1(1930)arenotsecurelyidentifiedinmassand 10 0.4
width,thoughsomesuchcontributionsaredefinitelyrequired 7.5 KK1(1400) π ρ σ π
[7]. However, when consideringthe Regge trajectory analy- 5 η a1(1260) 0.2 20
sis,onefindsthatthe33P1 stateinthea1 mesonfamilyhasa 2.5 KK(1270) π f0(980) ρa0(980)
ma1a(s2s09ar5o)usntadte2s0c0o0ulMdebVe.cTanhdeidtwatoesufnocrotnhfier3m3Ped1sat1a(t1e9i3n0th)eanad1 04.5 4.7π f24(.19270)5.1 04.5 4.7 41.9 5.1 04.5 4.7 4.9 5.1
mesonfamily,sincetheirmassesareclosetothatofthe33P R (GeV−1) R (GeV−1) R (GeV−1)
1
stateinthea mesonfamily. Thus,theexperimentalstudyof
1
FIG. 4: (Color online) R dependence of the calculated partial and
thepartialdecaywidthsofa (1930)anda (2095)willhelpto
1 1 total decay widths of thea (1930). Here, thedot-dashed linewith
reducethetwopossiblecandidates—a (1930)anda (2095)— 1
1 1 bandistheexperimentaltotalwidthfromRef. [7]. Allresultsarein
of the second radial excitation of the a (1260) to one. In
1 unitsofMeV.
In Fig. 6, we discuss the decay behavior of a (2270)
1
TABLE V: Typical ratios of the decay widths of a (1640) corre- as the third radial excitation of a (1260). We find that the
1 1
spondingtotheRrange(4.30∼4.64)GeV−1. maindecaymodeincludesthedecaychannelsπb1(1235),πρ,
Ratio Value Ratio Value πρ(1450),andπρ(1700). Inaddition,KK∗(1410),ρh1(1170),
KK (1680), πσ, and σa (1260) have important contribu-
Γ /Γ 0.216 0.227 Γ /Γ 0.473 0.474 ∗ 1
πρ Total ∼ πρ(1450) Total ∼ tions to the total decay width. ρa (1320), η(958)a (980),
Γ /Γ 0.014 0.059 Γ /Γ 0.166 0.221 2 ′ 0
πb1(1235) Total ∼ KK∗ σπ ∼ andK K (1270)aresubordinatedecaymodes,whicharenot
Γ /Γ 0.523 0.855 Γ /Γ 0.089 0.094 ∗ 1
πf2(1270) ρω ∼ πf1(1420) πf1(1285) ∼ showninFig. 6. InTableVII,wealsolistthetypicalratiosof
Γ 0.166 0.221
πf0 ∼ thedecaysofthea1(2270).
7
7 20
800 6 πf(980)
Total Width 16 πf(1285) 0 2
5 KK(1270) 1
1
600 4 12 1.5 ηa0(1450)
B852
400 3 K*K* 8 σa(1320) 1 KK*(1430)
2 0
2 ηa(1320) KK*(1430)
200 π b1(1235) 1 2 4 0.5 2
πρ KK(1400)
0 0 0 1 0
15
120 ρπ(1300) 70 70 KK*(892)
100 πρ(1450) 60 ρ h(1170) 60 ωb(1235) 12 ηa1(1260)
50 1 50 1
80 πρ(1700) 9
40 40 σa(1260)
4600 σ π 2300 a0(980)ρ 2300 ωρ 1 6πf1(1420)
20 a(1260)ρ 10πf2(1270) 10 KK*(1410) 3 ηa(980)
1 0
0 0 0 0
3.7 4 4.3 4.6 4.9 5.2 5.5 3.7 4 4.3 4.6 4.9 5.2 5.5 3.7 4 4.3 4.6 4.9 5.2 5.5 3.7 4 4.3 4.6 4.9 5.2 5.5
R (GeV−1) R (GeV−1) R (GeV−1) R (GeV−1)
FIG.5: (Coloronline)Rdependenceofthecalculatedpartialandtotaldecaywidthsofa (2095). Here,thedot-dashedlinewiththebandis
1
theexperimentaltotalwidthfromRef.[7].AllresultsareinunitsofMeV.
800 30
700 40 25 12 ηa(980)
600 3305 ρπ(1300) 1250 KK*(1410) 8 ηa1(1260) 0
450000 Total Width 2205 ρa1(1260) 150 ρh1(1170) ωb(1235) 4 πf(1420)
300 SPEC 15 0 1 0 1
200 πb1(1235) πρ 10 15 πf0(980) 4
1000 πρ(1450) 05 KK1(1400) 192 πf1(1285) KK*(892) 3 σa2(1320) ηa0(1450)
2
35 120 6 KK0(1430)
30 100 ρω 3 KK(1270) 1 KK2(1430)
0 1 0
25
80 50
1250 ρa(1450π)f2(1270) 60 3400 KK*(1680)σa1(1260) 8 K*K*
150 0 ρω(1420) 2400 πρ(1700) ρa0(980) 1200 πσ 4 ηa2(1320)
0 0 0 0 a0η(958)
4.7 5 5.3 5.6 5.9 4.7 5 5.3 5.6 5.9 4.7 5 5.3 5.6 5.9 4.7 5 5.3 5.6 5.9
R (GeV−1) R (GeV−1) R (GeV−1) R (GeV−1)
FIG.6: (Coloronline)Rdependenceofthecalculatedpartialandtotaldecaywidthsofa (2270). Here,thedot-dashedlinewiththebandis
1
theexperimentaltotalwidthfromRef.[7].AllresultsareinunitsofMeV.
B. b states tainedtotalwidth overlapswith experimentaldata fromRef.
1
[63]. Since b ωπ occurs via S and D waves, we obtain
1
→
the D-wave/S-waveamplituderatiooftheb ωπprocess,
The Regge trajectory analysis indicates that b (1235), 1 →
1 whichis0.465inourwork;thisisconsistentwiththeCrystal
b (1960),andb (2240)arethegroundstate,secondradialex-
1 1 Barreldata(0.45 0.04)[10]. On the otherhand, the decay
citation, and third radial excitation in the b meson family, ±
1 channelπf hasapartialdecaywidththatislessthan1MeV.
respectively. Inaddition,we alsopredicta missingb (1640) 0
1
asthe firstradialexcitation. Inthe following,we studytheir
decays.
Asforb (1235),therearetwoalloweddecaychannels:πω As a predicted b state, b (1640) has the decay behavior
1 1 1
andπa (980). Theresultshownin Fig. 7showsthattheob- listed in Fig. 8, where we take the same R range as that
0
8
300 70 18
TABLEVII:Calculatedratiosofthedecays ofa (2270). Here, all
1
theresultscorrespondtotheRrange(5.12∼5.32)GeV−1. 250 Total Width 60 πω(1420) 15 KK*(892)
Ratio Value Ratio Value 50
200 12
Γπρ/ΓTotal 0.164∼0.184 Γπf1(1285)/Γπσ 0.313∼0.435 150 40 a(1320)π ωπ 9
Γ /Γ 0.247 0.264 Γ /Γ 0.313 0.487 2
Γππbρ117(10023/5Γ)TotTalotal 0.052∼∼0.056 ΓKπKf1∗((1849220))/Γηρaa10((1928600)) 0.404∼∼0.469 100 πa0(980) 2300 ηρ 6 πa1(1260)
Γ /Γ 0.064 0.070 Γ /Γ 0.532 0.612
Γσπ To/taΓl 0.134∼0.157 ηΓa0(980) π/fΓ2(1270) 0.099∼0.131 50 ρρ 10 3 σρ πa0(1450)
πρ(1450) Total ∼ ηa2(1320) πf0 ∼
Γ /Γ 0.789 0.926Γ /Γ 0.236 0.273 0 0 0
ρa1(1260) ωb1(1235) ∼ ηa0(1450) ωb1(1235) ∼ 4 4.3 4.6 4.9 4 4.3 4.6 4.9 4 4.3 4.6 4.9
ΓKK1(1400)/Γρπ1300 0.352∼0.446Γηa1(1260)/ΓKK∗(1680)0.256∼0.297 R(GeV−1) R(GeV−1) R(GeV−1)
Γ /Γ 0.573 0.639Γ /Γ 0.633 0.638
ρh1(1170) KK1(1400) ∼ KK∗(1410) σa1(1260) ∼
FIG. 8: (Color online) R dependence of the calculated partial and
totaldecaywidthsofb (1640).AllresultsareinunitsofMeV.
1
TABLEVIII:Typicalratiosfordecaysofb (1640)correspondingto
1
R=4.20 4.90GeV 1.
−
130 ∼
Ratio Value Ratio Value
Γ /Γ 0.352 0.368 Γ /Γ 0.324 0.347
125 πa0(980) Total ∼ KK∗ ωπ ∼
Γ /Γ 0.164 0.263 Γ /Γ 0.565 0.681
ηρ πω(1420) ∼ πa2(1320) ρρ ∼
120 ToTtoaTtl aowlt aiwdl itwdht ih(dπ(tωπhω))
Assuming that b (1960) is the second radial excitation of
115 1
b (1235),wepresentitstotalandpartialdecaywidthsinFig.
1
AAASSSTTTEEE
9. Ourcalculatedtotalwidthcancovertheexperimentaldata
110
giveninRef.[12].Itsmaindecaychannelsareπa (1450),πω,
0
105 πa0(980),andπω(1420),whilethepartialdecaywidthsofthe
decay modes πa (1260), ρη, and πa (1320) are also consid-
1 2
100 erable. Wealsoobtainsomeratiosofpartialdecaywidthsof
b (1960),whicharelistedinTableIX.
3 3.4 3.8 4.2 1
R(GeV−1)
60 30
400
FIG. 7: (Color online) R dependence of the calculated total decay Total Width πa0(980) 25 σb1(1235)
width of b1(1235). Here, the dot-dashed line with the band is the 300 40 20
experimentaltotalwidthfromRef. [63]. Thetotaldecaywidthisin SPEC
200 15
unitsofMeV. ρη
100 πa0(1450) πω 20 πa1(1260) 150 KK*
0 0 KK*(1410) 0 ρη(958)
0.6 40 0.7
0.6
for a1(1640). 2 Its main decay channel is πa0(980), while 0.4 ρf0(980) 2300 πω(1420) 000...345 KK0*(1K4K310()1270)
πa2(1320), ρρ, πω(1420), KK∗, and ωπ also have consider- 0.2 KK(1400) ωa0(980) 10 πa2(1320) ρρ 0.2 K*K*
able contributions to the total decay width. The total decay 1 σρ 0.1
0 0 0
width is predicted to be 200 232 MeV. Table VIII shows 4.3 4.6 4.9 5.2 5.5 4.3 4.6 4.9 5.2 5.5 4.3 4.6 4.9 5.2 5.5
someratiosthatarerelevantto∼thedecaysofb (1640),which R(GeV−1) R (GeV−1) R (GeV−1)
1
is valuable for further experimental searches for this axial-
vectorstate. FIG. 9: (Color online) R dependence of the calculated partial and
total decay widths of b (1960). Here, the dot-dashed linewiththe
1
bandistheexperimentaltotalwidthfromRef. [12].Sincethewidth
of the KK mode is tiny, we do not list its contribution here. All
2∗
resultsareinunitsofMeV.
2Sinceb1(1640)isapredicted state, wetakethesameRrangeasthatof
a1(1640)topredictthedecaybehaviorofb1(1640). Thistreatmentisdue
In Fig. 10, we show the decay behavior of b (2240) as
tothefact thatb1(1640)isthe isospinpartner ofa1(1640), whichhas a 1
similarRrange. thethirdradialexcitationofb1(1235). Additionally,itsmain
9
In addition, there exist relations among f (1970), the pre-
1
TABLEIX:Obtainedratiosfordecaysofb (1960). Allresultscor-
1 dicted f (2110)and f (2210),and f (2310),i.e.,
respondtoR=4.66 5.16GeV 1. 1 1 1
−
∼
Ratio Value Ratio Value f (1970) cosφ sinφ nn¯
| 1 i = 2 − 2 | i , (9)
ΓΓππaω01(492800/)Γ/ΓToTtoatlal 00..108868∼∼00..213057 ΓΓωρρπ//ΓΓTToottaall 00..002787∼∼00..013612 |f1(2110)i  sinφ2 cosφ2  |ss¯i 
Γ /Γ 0.572 0.624 Γ /Γ 0.648 0.736 and
ρρ πa2(1320) ∼ KK∗(892) ηρ ∼
Γ /Γ 0.029 0.030 Γ /Γ 0.249 0.316
ρf0 πa1(1260) ∼ KK1(1400) KK1(1270) ∼ f (2210) cosφ sinφ nn¯
| 1 i = 3 − 3 | i , (10)
 f (2310)   sinφ cosφ  ss¯ 
decay modes are ωπ, πω(1420), πa0(980), and πa0(1450). | 1 i  3 3  | i 
Of course, the decay modes ρρ, ρb1(1235), πa2(1320), and Here,themixinganglesφi (i = 1,2,3)cannotbeconstrained
πa (1260)also have obviouscontributionsto the total decay byouranalysis.Inthefollowingdiscussions,wetakeatypical
1
width. For the convenience of further experimental studies valueφi =φ=24◦togivethequantitativeresults.
of this state, we provide informationon typical ratios of the Asfor f1(1285),weshowitspartialandtotaldecaywidths
partialwidthofb (2240)inTableX. inFig. 11,wherethecalculatedtotaldecaywidthisinagree-
1
ment with the experimental data from Ref. [67]. However,
wenoticethatthecalculatedbranchingratioforΓ /Γ =
πa0 total
TABLE X: Calculated ratios for b1(2240) corresponding to R = 0.67 0.68, corresponding to R = (3.00 4.00) GeV−1,
5.20 5.54GeV−1. which∼is a little bit larger than (36 7)% lis∼ted in the PDG
∼
±
Ratio Value Ratio Value [1]. The PDG data also shows that the branching ratio of
Γ /Γ 0.097 0.128 Γ /Γ 0.131 0.232 its decay ηππ can reach up to (52.4+1.9)% [1], which is the
πa0(980) Total ∼ πa0(1450) Total ∼ 2.2
ΓΓπω1420/Γ/TΓotal 00..006785∼00..017012 ΓΓωπ/ΓTotal 00..107597∼00..109696 mwoarink,cownetrsibtuudtiyonthteotphreoctoetsaslesdefc1a(y12w8−i5d)thofηf1σ(1285)η.πIπnathnids
πa2(1320) Total ∼ πa1(1260) ∼ f (1285) πa (980) ηππ,whichcan→becalcu→latedusing
Γηρ(1450)/ΓTotal 0.055∼0.064 Γηρ/ΓTotal 0.050∼0.062 th1eQPCm→odel0. Thus,→thedecaywidthof f (1285) ησ
ΓKK∗(1680)/ΓTotal 0.042∼0.057Γρf1(1285)/Γωa1(1260)0.678∼0.819 ηππcanbewrittenas[44] 1 → →
Γ /Γ 0.112 0.173Γ /Γ 0.254 0.317
ρf0 ρb1(1235) ∼ ρf2(1270) KK1(1400) ∼
ΓΓKK1(1270)//ΓΓKK0∗(1430)00..293033∼00..398530ΓΓρη′(958)//ΓΓKK∗(1410) 00..316548∼00..328014 Γ(f1 →η+σ→η+ππ)
KK∗(1410) KK∗(892) ∼ ωa0(980) ωa1(1260) ∼ = 1 (mf1−mη)2dr√rΓf1→η+σ(r)·Γσ→ππ(r), (11)
πZ (r m2)2+(m Γ )2
4m2π − σ σ σ
wheretheinteractionofσwithtwopionscanbedescribedby
C. f1states theeffectiveLagrangian
Whendiscussing f1 states,weneedtoconsidertheadmix- Lσππ =gσσ(2π+π−+π0π0). (12)
turesoftheflavorwavefunctions nn¯ =(uu¯ + dd¯ )/√2and
| i | i | i Thecouplingconstantg = 2.12 2.81GeV is determined
|ss¯i. f1(1285)and f1(1420)/f1(1510)satisfy by the total width Γ =σ400 70∼0 MeV [1], and the decay
σ
∼
widthreadsas
f (1285) cosφ sinφ nn¯
| 1 i = − | i , (7)
|f1(1420)/f1(1510)i  sinφ cosφ  |ss¯i  Γσ→ππ(r)= g82σπλr2[(r−(22m√πr)2)r]1/2, (13)
where both f (1420) and f (1510) are partners of f (1285).
1 1 1
(Wepresenttheirdecaybehaviorsbelow.)φdenotesamixing whereλ= √2and1forπ+π andπ0π0,respectively.
−
angle. This mixing angle was determined in a phenomeno-
Theprocess f (1285) πa (980) ηππiscalculatedina
logical way [64] and is given by φ = (20 30) , which is 1 → 0 →
consistentwithφ = (24+3.2) reportedbythe−LHC◦bCollabo- similarway,andtheequationisgivenby
2.7 ◦
ration [65] and φ = (21− 5) from the updatedlattice QCD
◦ Γ(f a +π η+ππ)
± 1 0
analysis [66]. When calculating the decays of f (1285) and → →
f1(1420)/f1(1510),wetaketheLHCbvalueφ=214◦. = 1 (mf1−mπ)2dr√rΓf1→π+a0(r)·Γa0→ηπ(r), (14)
In Fig. 1, we have predicted that f1(1640) is the first ra- πZ(mπ+mη)2 (r−m2a0)2+(ma0Γa0)2
dial excitation of f (1285) and that f (1800) is a partner of
1 1
f1(1640);thesetwopredictedaxial-vectormesonsarerelated wherethedecaywidthfora0(980)→ηπis
by
Γ (r)
a0(980)→ηπ
f (1640) cosφ sinφ nn¯ g2 [(r (m +m )2)(r (m m ))2]1/2
| 1 i = 1 − 1 | i , (8) = a0 − η π − η− π , (15)
|f1(1800)i  sinφ1 cosφ1  |ss¯i  8πr 2√r
10
800 60 100 2.5
KK*(1430)
2
600 Total Width 45 πa(980) 80 πa2(1320) 2
0 60 πa0(1450) 1.5 ρf0(980)
400 30
SPEC πω(1420) 40 1
ρf(1420)
200 ωπ 15 ρb(1235) ωπ(1300) 20 σb1(1235)0.5 ρf2(1270) 1
ρρ 1
π a(1260)
0 0 0 1 0
4 18 ωa1(1260) 50 20
15
40 ρη
3 KK*0(1430) KK1(1400) 12 30 15
2 KK1(1270) 69 KK*(1410)a0h1(1170) 20 KK*(1680) ηρ(1450) 10 σρ KK*(892)
01 K*K* 03 ωa2(1320)100 ρη(958) 05 ρf1(1ω2a805()980)
4.8 5.1 5.4 5.7 6 4.8 5.1 5.4 5.7 6 4.8 5.1 5.4 5.7 6 4.8 5.1 5.4 5.7 6
R (GeV−1) R (GeV−1) R (GeV−1) R (GeV−1)
FIG.10: (Coloronline)Rdependenceofthecalculatedpartialandtotaldecaywidthsofb (2240). Here,thedot-dashedlinewiththebandis
1
theexperimentaltotalwidthfromRef. [12]. WedonotpresenttheK K (1270)contributionsincethisdecayhasatinywidth. Allresultsare
∗ 1
inunitsofMeV.
45 25 f1(1420) and f1(1510) as partners of f1(1285). As for
f (1420),theobtainedtotaldecaywidthcanoverlapwiththe
1
36 Total width 20 DM2result[68],asshowninFig. 12. Itsmaindecaychannel
is KK . Thus, the present study of decay of f (1420) sup-
∗ 1
27 MPS2 15 portsthepredictionthat f1(1420)isapartnerof f1(1285). As
for f (1510), its partial and total decay widths are listed in
1
Fig.13, whichshowsthatthecalculatedtotaldecaywidthis
18 πa (980) 10 ηππ(total)
0 larger than the experimentaldata [68]. Thus, the possibility
that f (1510)isapartnerof f (1285)canbeexcluded.
9 5 ηππ(ησ) 1 1
ησ
0 0 350 0.5
3 3.4 3.8 4.2 3 3.4 3.8 4.2
R (GeV−1) R (GeV−1) 300 Total Width
0.4
250
FIG.11: (Coloronline)Rdependenceofthetotalandpartialdecay
widthsof f (1285). Wealsopresentthedecaywidthof f (1285) 200 KK* 0.3
1 1 →
aηnπdπovnialythfreominttehremiendtieartmeecdhiaanteneclhsanηnσelanηdσπ(pai0n(k98b0a)n(dg)r.eeHnerbea,ntdh)e, 150 LASS 0.2 ση
experimentaltotalwidthfromRef.[67]isdenotedbythedot-dashed 100
linewiththeband.AllresultsareinunitsofMeV. 0.1 πa (1260)
50 πa 1
0
0 0
wishedreeterthmeinceodupblyingthceontosttaanltwgiad0th=of1.a26(2980∼) (2Γ.524 GeV= 2 2.4R (G2e.8V−13).2 3.6 2 2.4R (G2e.8V−13).2 3.6
0 a0(980)
50 100 MeV). The final result of the width of
∼
f (1285) πa (980) ηππincludesthecontributionsfrom FIG.12: (Coloronline)Rdependenceofthetotalandpartialdecay
1 0
bothηπ0π→0andηπ+π .→ widths of f (1420). Here, the experimental total width from Ref.
− 1
Thedecaywidthof f (1285) ηππviaboththeinterme- [68]isshownbythedot-dashedlinewiththeband.Allresultsarein
1
diateησandπa (980)channelsa→ndonlytheintermediateησ unitsofMeV.
0
channel are shown in Fig. 11. In addition, the decay width
of f (1285) ηππ from the intermediate πa (980) chan- In Figs. 14 and 15, we further illustrate the decay prop-
1 0
→
nel is comparable with the corresponding experimental data erties of the two predicted states f (1640) and f (1800). In
1 1
[(16 7)%]inthePDG[1]. addition, we also list some of their typical ratios, which are
±
In the following, we discuss the decay behaviors of weakly dependent on the R value (see Table XI), where we