Higher bottom and bottom-strange mesons

0.25 MB·

by  Yuan Sun

Higherbottom andbottom-strange mesons
Yuan Sun1,2, Qin-Tao Song1,3,6, Dian-YongChen1,3, Xiang Liu1,2 , and Shi-Lin Zhu4,5
∗† ‡§
1ResearchCenterforHadronandCSRPhysics,LanzhouUniversity&InstituteofModernPhysicsofCAS,Lanzhou730000,China
2School of Physical Science and Technology, Lanzhou University, Lanzhou 730000, China
3Nuclear Theory Group, Institute of Modern Physics of CAS, Lanzhou 730000, China
4DepartmentofPhysicsandStateKeyLaboratoryofNuclearPhysicsandTechnology,PekingUniversity,Beijing100871,China
5Collaborative Innovation Center of Quantum Matter, Beijing 100871, China
6University of Chinese Academy of Sciences, Beijing 100049, China
Motivated by the recent observation of the orbital excitation B(5970) by the CDF Collaboration, we have
performedasystematicstudyofthemassspectrumandstrongdecaypatternsofthehigher Band B mesons.
s
Hopefully the present investigation may provide valuable clues to further experimental exploration of these
4 intriguingexcitedheavymesons.
1
0 PACSnumbers:14.40.Nd,12.38.Lg,13.25.Hw
2

r
a I. INTRODUCTION available experimental data and other theoretical results. In
M
Sec. III, we discuss the two-body strong decay behavior of
the higherbottomand bottom-strangemesons. The last sec-
The past decade has witnessed the discovery of many
1
tionisdevotedtothediscussionandConclusion.
charmed and charmed-strange states such as D (2317) [1–
2 sJ
3], Ds(2460)[2,3], DsJ(2632)[4], DsJ(2860)[5], DsJ(2715)

] [6], D (3040) [7], D(2550) [8]/D (2580) [9], D (2600)
sJ J ∗
h II. THEMASSSPECTRUM
[8]/D (2650) [9], D(2750) [8]/D (2740) [9], D (2760)
p ∗J J ∗
p- [ri8c]h/Ded∗J(t2h7e6f0a)m[i9l]y,oanfdthDe(Jc∗)h(a3r0m0e0d)[a9n]d,wchhaicrmhhedav-setrnaontgoensltyateens-, We first calculate the mass spectrumof the higher bottom
e andbottom-strangemesonsintheframeworkoftherelativis-
but also stimulated extensive discussions of their properties
h tic quark model [25], where the total Hamiltonian H˜ de-
(seeRef. [10]foraminireviewoftheresearchstatusofthese 1
[
newlyobservedcharmedandcharmed-strangestates). scribes the interaction between quark and anti-quark in the

3 meson
The present situation of the experimental exploration of
v
5 bottomand bottom-strangestates is strikinglysimilar to that H˜ = p2+m2 1/2+ p2+m2 1/2+H˜conf+H˜so+H˜hyp, (1)
of charmed and charmed-strange states in 2003; i.e., some 1 1 2 12 12 12
9
(cid:16) (cid:17) (cid:16) (cid:17)
5 candidates for the P-wave bottom and bottom-strange me- whereH˜confdenotestheconfinementtermandH˜soisthespin-
1 sonwereannounced[11–19]whiletheradiallyexcitedstates 12 12
orbittermwhichcanbedecomposedintothesymmetricpart
. seemwhitinreach[20].Veryrecently,theCDFCollaboration
1 H˜so andtheantisymmetricpartH˜so . Inaddition,H˜hypisthe
0 studied the orbitally excited B mesons, and reportedthat the (12) [12] 12
sumofthetensorandcontactterms,i.e.,
4 new B(5790) state could be the radially excited state in the
1 bottommesonfamily[20]. Thereareseveraltheoreticalstud- H˜hyp = H˜tensor+H˜c . (2)
: iesofthebottomandbottom-strangemesonsbefore[21,22] 12 12 12
v
i andafter[23,24]theexperimentalobservationofthesestates. TheconcreteformsofthesetermscanbefoundinAppendix
X
Nowisagoodtimetocarryoutacomprehensivetheoretical AofRef. [25].
ar study on higher bottom and bottom-strange mesons. In this In the bases n2S+1LJ , the antisymmetricpartof the spin-
work, we will calculate the mass spectrum of higher bottom orbittermH˜so |andtheitensortermH˜tensorhavenonvanishing
[12] 12
and bottom-strangemesonsand the correspondingtwo-body off-diagonalelements,whichresultinthemixingofthestates
strongdecaybehavior.Wehopethepresentinvestigationmay withquantumnumbers3L and1L orwith3L and3(L 2) .
J J J J
notonly shed lighton the propertiesof the observedbottom Thus,thetotalHamiltonianH˜ canbedividedintotwo±parts
1
and bottom-strangestates, but also providevaluableclues to in this bases, which include the diagonal part H and off-
diag
furtherexperimentalexplorationof the radiallyand orbitally diagonalpartH withtheform
off
excitedbottomandbottom-strangestates.
This paper is organized as follows. After the Introduc- H = H˜so + H˜tensor , (3)
tion,wepresenttheanalysisofthemassspectrumofthebot- off [12] (cid:16) 12 (cid:17)off
tomandbottom-strangemesonfamilyincomparisonwiththe where H˜tensor denotestheoff-diagonalpartsof H˜tensor. In
12 off 12
the fol(cid:16)lowing,(cid:17)we first diagonalize H in the simple har-
diag
monicoscillatorbasesand obtaintheeigenvaluesand eigen-
vectorscorrespondingtothewavefunctionofthemeson.One
Correspondingauthor
∗ alsoneedstodiagonalizetheoff-diagonalpartH inthebases
Correspondingauthor off
‡ n2S+1L , which is treated as the perturbative term. We ne-
†Electronicaddress:[email protected] | Ji
§Electronicaddress:[email protected] glecttheperturbativetermHoff inthepresentcalculation.
2
Thefreeparametersoftheadoptedrelativisticquarkmodel
are listed in Table II of Ref. [25], which include the quark
masses and the coefficients in the effective potential. Here, 6600
welistthequarkmasses 23D1 23D3
6400 33S1 21D2 23D2 13F2 13F4
m =m =220MeV, m =4977MeV, m =419MeV, (4) 31S0 23P0 23P2 11F3 13F3
u d b s V) 6200 21P1 23P1 13D1 13D3
whicharealsoappliedinthefollowingtwo-bodystrongdecay Me 6000 23S1 11D2 13D2
B(5970)
cthaelWcbuioltahtttiotohmne.aabnodvbeopttroempa-rsattriaonng,ewmeeosbotnainfatmheilimesasasssspheocwtrnaoinf Mass( 5800 21S011P113P013P113P2 BB2∗1((55774271))
5600
Fig. I,wherethemassesofthe1S,2S,3S,1P,2P,1D,2D,
and 1F states are given. Godfrey et al. calculated the mass 5400 13S1
spectrum of some of the bottom and bottom-strangemesons 11S0 Bottommeson CDF
5200
longago[25].Wealsonoticethatthereexistpredictionsofthe
massspectraofthebottomandbottom-strangemesonsusing
other theoretical models. For example, Ebert et al. adopted
ptqchorueorararreckechltmai[ot2oinv6di]esoltwfiwch1ii/qtlmheuatbthrhkieenamDuthiotrhedaoecprlosHbtoaeafmnsRetiidlaetlfoo.nins[i2ata7nhl]sepouoqstieuendancsttliihuapedlo,ertwdeenl.hatetiIiarvnelisaatthdpiec-- V) 666246000000 3313SS0121P123P023P123P21211DD221233DD111233DD221233DD3311F313F213F313F4
docoiftriDroen/sD,psion/nBdR/Biensfgsw.ree[sr2ue8lpt,sr2el9ids]itcettdheedin,mwTaahsbiscleehsIao.rfetchoemrapdairaalbelexcwitiathtiothnes ass(Me 56800000 2213SS0111P113P013P113P2 BBss∗12((55883400))
M
InFig. 1andTableI,welistourresultsofthemassspectra
5600
ofthebottomandbottom-strangemesonfamilies,theresults
from othergroups, and comparethem with the experimental 13S1
5400
data. 11S0 Bottomstrangemeson
Until now, there has been only some limited experimen- 5200 CDF
tal information on the low-lying bottom and bottom-strange
mesons, which are shown in the fifth and ninth columns.
Thetheoreticalresultscanreproducetheseexperimentaldata FIG. 1: (color online). The comparison of the mass spectrum of
well. WhencomparingourresultswiththosegivenbyRefs. bottomandbottom-strangemesoncalculatedbytherelativisticquark
[26, 27], we notice that the discrepancy of the values of the model(seeTableIfortheconcretevalues)withtheexperimentaldata
massfromdifferentmodelsisabout50MeVforthe2S,3S, [11–20].
and 1D states, while the mass difference for the 1F, 2P and
2D states from differentmodelcalculationsmay reach up to
ThetransitionoperatorT iswrittenas[46]
about100 200 MeV. Thus, we will consider the mass de-
∼
pendenceofthestrongdecaywidthofthehigherbottomand
T = 3γ 1m;1 m00 dk dk δ3(k +k )
bottom-strange mesons, which will be discussed in the next − h − | iZ 3 4 3 4
Xm
section. In the following, we will employ our numericalre-
k k
shuiglthsetrobosttutodmy athnedbtwoott-obmod-sytrastnrgoengmedseocnasy.behavior of these ×Y1m 3−2 4!χ314,−mφ304ω304d3†i(k3)b†4j(k4), (5)
where the flavor and color wave functions have the forms
φ34 = (uu¯ + dd¯+ ss¯)/√3 and ω34 = δ /√3, respectively.
0 0 ij
III. TWO-BODYSTRONGDECAYBEHAVIOR Thecolorindicesaredenotedbyiand j. Thesolidharmonic
polynomialis (k) = klY (k). Thespinwavefunctionis
lm lm
In this work, we will study the two-body strong decay of χ34 with anYangular m|om| entum quantum number (1, m).
1, m −
thehigherbottomandbottom-strangemesonsallowedbythe γ d−enotesthe modelparameter,whichdescribesthe strength
Okubo-Zweig-Iizuka (OZI) rule, where the quark pair cre- ofthequark-antiquarkpaircreationfromthevacuum. Inour
ation(QPC)modelisadoptedinourcalculation. calculation,theγvalueischosentobe6.3and6.3/√3forthe
The QPC model[31–37] is an effective approachto study creationsofu/dquarkandsquark[41],respectively.
the strong decay of hadrons, which has been widely applied Thehelicityamplitude MJAMJBMJC(K)ofthe A B+C
M →
to study the decay behaviors of the newly observed hadron decayisdefinedas
states [38–45]. In the QPC model, the meson OZI-allowed
decay occursvia the creationof a quark-antiquarkpair from hBC|T|Ai=δ3(KB+KC−KA)MMJAMJBMJC(K), (6)
thevacuum. Thecreatedquark-antiquarkpairisaflavorand which can be further expressed as the product of the flavor,
color singlet with the vacuum quantum number JPC = 0++. spinandspatialmatrixelements,wherethecalculationofthe
3
TABLEI: Theobtainedmassesofthebottomandbottom-strangemesonsinthisworkincomparisonwiththeexperimentalvalues[30]and
othertheoreticalresults[26,27]. Inthesecondandsixthcolumns, thevalueslistedinthebracketsaretheβvaluesinthesimpleharmonic
oscillator(SHO)wavefunction,whichwillbeadoptedinstudyingthestrongdecaysofthesehigherbottomandbottom-strangemesons.Here,
theβvalueintheSHOwavefunctionisobtainedbyreproducingtherealisticroot-mean-square(rms)radiusgivenbythiswork. Inaddition,
wewanttoemphasizethattheresultsfromRefs.[26,27]considerthemixingofthestateswiththesameJPquantumnumber.
Bottommeson Bottom-strangemeson
n2S+1L
J
Thiswork Ref.[26] Ref.[27] Expt.[30] Thiswork Ref.[26] Ref.[27] Expt.[30]
11S 5309(0.63) 5280 5279 5279.55 0.26 5390(0.69) 5372 5373 5366.7 0.4
0 ± ±
13S 5369(0.57) 5326 5324 5325.2 0.4 5447(0.63) 5414 5421 5415.8 1.5
1 ± ±
21S 5904(0.51) 5890 5886 – 5985(0.54) 5976 5985 –
0
23S 5934(0.50) 5906 5920 – 6013(0.53) 5992 6019 –
1
31S 6334(0.47) 6379 6320 – 6409(0.49) 6467 6421 –
0
33S 6355(0.46) 6387 6347 – 6429(0.48) 6475 6449 –
1
13P 5796(0.49) 5741 5714 5743 5 5875(0.52) 5842 5820 5839.96 0.20
2 ± ±
13P 5756(0.56) 5749 5706 – 5830(0.59) 5833 5804 –
0
13P 5782(0.53) 5723 5700 5723.5 2.0 5859(0.56) 5831 5805 5828.7 0.4
1 ± ±
11P 5779(0.52) 5774 5742 – 5858(0.55) 5865 5842 –
1
23P 6213(0.46) 6260 6188 – 6295(0.49) 6359 6292 –
2
23P 6214(0.49) 6221 6163 – 6279(0.51) 6318 6292 –
0
23P 6219(0.48) 6281 6175 – 6291(0.50) 6345 6278 –
1
21P 6206(0.48) 6209 6194 – 6284(0.50) 6321 6296 –
1
13D 6105(0.46) 6091 5933 – 6178(0.48) 6191 6103 –
3
13D 6110(0.50) 6119 6025 – 6181(0.52) 6209 6127 –
1
13D 6113(0.48) 6121 5985 – 6185(0.50) 6218 6095 –
2
11D 6108(0.48) 6103 6025 – 6180(0.50) 6189 6140 –
2
23D 6459(0.44) 6542 – – 6534(0.46) 6637 – –
3
23D 6475(0.47) 6534 – – 6542(0.48) 6629 – –
1
23D 6472(0.46) 6554 – – 6542(0.47) 6651 – –
2
21D 6464(0.46) 6528 – – 6536(0.47) 6625 – –
2
13F 6364(0.44) 6380 6226 – 6431(0.45) 6475 6337 –
4
13F 6387(0.47) 6412 6264 – 6453(0.48) 6501 6369 –
2
13F 6380(0.45) 6420 6220 – 6446(0.47) 6515 6332 –
3
11F 6375(0.45) 6391 6271 – 6441(0.47) 6468 6376 –
3
spatial matrix element is the most crucial part in the whole Finally,thedecaywidthrelevanttothepartialwaveamplitude
QPC calculation (see [46] for more details). In order to ob- is
tain the spatial matrix element, we adopt the mock state to
describethemesons[47]. Inourcalculation,thespatialradial
wave function of the meson is represented by a SHO wave Γ=π2|K| (K)2, (8)
function [46] with a parameter β. The β values in the SHO M2 |MLS |
A XLS
wavefunctionsaredeterminedbyreproducingthermsradius
ofthecorrespondingstatescalculatedintherelativisticquark
model. InTableI,welisttheseadoptedβvaluesforthebot- where M isthemassoftheinitialmesonA. Wetaketheex-
A
tomandbottom-strangemesonsdiscussed. perimentalmassasinputifthecorrespondingdecayinvolves
Converting the helicity amplitude MJAMJBMJC(K) to the the observed state. Otherwise, we take the predicted mass
M
partialwaveamplitude SL(K)throughtheJacob-Wickfor- fromtherelativisticquarkmodeinTableIwhenstudyingthe
M | |
mula[48],weget strongdecayofthebottomandbottom-strangemesons.
√2L+1 Inthefollowingsubsections,weillustratetheOZI-allowed
(K) = L0SM J M
MLS 2J +1 h JA| A JAi strong decay behaviors of the bottom and bottom-strange
A MJXB,MJC mesons in detail. We use B and Bs to represent bottom and
×hJBMJBJCMJC|SMJAiMMJAMJBMJC(K). (7) bottom-strangemesons,respectively.
4
A. Bottommeson easilydistinguishthese two1+ states; i.e., the1+ state in the
(0+,1+)doubletisbroadwhiletheother1+stateinthe(1+,2+)
1. 1Pand2Pstates doubletis narrow. In Eq. (9), the 1P(1+) and 1P′(1+) states
correspondthe1+ statesinthe(0+,1+)and(1+,2+)doublets,
respectively. The following quantitative study of the decay
behaviorswillconfirmtheabovequalitativeobservation.
TABLE II: The calculated partial and total decay widths (in units In 2007, D0 Collaboration reported the observation of
of MeV)of 2Pstatesinthe Bmeson family. Theforbidden decay B (5721)0 in the B +π channel, whereits measuredmass is
channelsaremarkedby–. Forthe2P(1+)/2P′(1+)states,weuse(cid:3) 57120.6 2.4 1.4M∗eV−[15].Later,CDFconfirmedB (5721)0
tomarkthealloweddecaychannels. with th±e mas±s 5725.3+1.6+1.4 MeV [16]. Very recent1ly, CDF
2.2 1.5
Channels 23P0 23P2 2P(1+)/2P′(1+) [20] studied the orbita−lly−excited B mesonsby using the full
πB 47 9.7 10−2 – CDFRunIIdatasample. BesidesconfirmingB1(5721),CDF
πB – 1×.4 (cid:3) reported the width of B1(5721) for the first time, where the
∗
measured width is 20 2 5 MeV and 42 11 13 MeV
πB(13P0) – – (cid:3) fortheneutralandchar±ged±B (5721)states[2±0],re±spectively.
πB(13P2) – 4.6 (cid:3) Since B (5721)hasa narrow1width, B (5721)isa goodcan-
1 1
πB(1P) 10 16 (cid:3) didateofthe1P(1+)stateintheBmesonfamily.
′
πB(1P) 51 1.3 (cid:3)
′
ηB 5.6 0.11 –
ηB∗ – 0.45 (cid:3) 250 (a)
ρB – 4.3 (cid:3) V)200
e
ρωBB∗ 2–7 12.63 (cid:3)(cid:3) Width(M110500 −54.7◦ BB((11PP(′(11++))))
50
ωB 8.8 9.3 (cid:3)
∗
0
KB 3.6 1.3 10 2 – −100 −50 0 50 100
KBs – 0×.11 − (cid:3) θ1P (degree)
∗s 40
K∗Bs – 2.7×10−2 – V)30 (b)
Total 154 64 – e
M
(20
h CDF
Althoughthe predictedmass ofthe 13P0 bottommesonis Widt10 B(1P′(1+))
above both Bπ and B π thresholds, B(13P ) can only decay
∗ 0 0
intoBπ.ItsB πdecayisforbiddenduetotheparityconserva- −80 −75 −70 −65 −60 −55 −50 −45 −40 −35 −30
∗
tion. Thepartialdecaywidthof B(13P ) Bπcanreachup θ1P (degree)
0
→
to225MeV.Thisstateisverybroad,whichprovidesanatural
explanation of why B(13P ) is still missing experimentally. FIG.2: (coloronline). (a)Thedependenceofthetotaldecaywidths
0
There were other theoreticalcalculations of the decay width ofB(1P(1+))(dashedcurve)andB(1P′(1+))(solidcurve)onthemix-
of the B(13P0) state. For example, the authorsused the chi- ingangleθ1P. (b)Thedependence ofthetotalwidthof B(1P′(1+))
ralquarkmodelandobtainedΓ(B(13P0) Bπ) = 272MeV onthemixingangleθ1P = (−80 ∼ −30)◦ andthecomparisonwith
→ theCDFdata[20]. Here,theverticaldashedlinein(a)corresponds
[23],whichisconsistentwithourresult.
totheidealmixingangleθ = 54.7 .
The two JP = 1+ states are the mixture of the 11P1 and 1P − ◦
13P states
1
For B (5721), its dominant decay is B π. With the QPC
1 ∗
11PP((11++)) = cosisnθθ1P csoinsθθ1P 1113PP1 , (9) pmeonddeeln,cweeofcathlceuclaatlecuBla1t(e5d7w21id)0th→of BB1∗(π5.72In1)F0iogn. th2e, tmheixdineg-
whereθ1P den′otesthem−ixinga1nPgle. In1Pgeneral,i1ntheheavy tahneglceorθr1ePspisongdivinegn.wIidftthakoifngB1th(5e7i2d1e)a0livsa1lu.8eMθ1PeV=, w−h5ic4h.7i◦s,
quarklimitonehasθ = 54.7 [49]. However,theobtained far smaller than the experimental data. We find that the
1P ◦
θ1P valuedeviatesfromthi−sidealvaluewhentheheavyquark calculated width of B1(5721)0 overlaps with the experimen-
mass is finite in the case of the bottom and bottom-strange tal value when θ1P is in the range of ( 77◦ 70◦) or
mesons. Wewilldiscussthisissuelater. ( 40◦ 33◦). Wealsopredictthewidth−ofB(1∼P(1−+))state,
Inheavyquarkeffectivetheory,theheavy-lightmesonsys- w−here∼is−Γ(B(1P(1+))) = 170 197 MeV correspondingto
∼
tem can be categorized into different doublets by the angu- θ1P = ( 77◦ 70◦) or ( 40◦ 33◦), which confirms
lar momentum of the light component j , which is a good B(1P(1+−))isa∼bro−adstate. − ∼ −
ℓ
quantum number in the limit of mQ . There are two B∗2(5747)was first observedby D0 in its B∗+π− and B+π−
1+ states in the(0+,1+) and(1+,2+) P→-wa∞ve doublets,which channels[15]. Other experimentalinformationof B∗2(5747)0
correspondto j = 1/2+ and j = 3/2+,respectively. Wecan given by D0 includes: M(B (5747)0) = 5746.8 2.4 1.7
ℓ ℓ ∗2 ± ±
5
MeVandΓ(B (5747)0 B +π )/Γ(B (5747)0 B( )+π ) =
∗2 → ∗ − ∗2 → ∗ − 150 B(2P(1+)) 150 B(2P′(1+))
0.475 0.095 0.069 [15]. In 2009, CDF announced the
confirm±ationof±B (5747)0withmassM =5740.2+1.7+0.9MeV Total −54.7◦ Total
∗2 1.8 0.8 100 100
and width Γ = 22.7+3.8+3.2 MeV [16]. In Ref.−[2−0], CDF −54.7◦
3.2 10.2
aargeaiMnca=rr5ie7d36o.u6tthe1−.s2tu−dy1.o2f B∗20(.527/4577)3.7T.1hem1.a1ssa0n.d9wid0t.h2 50 πB∗ πB(13Pρ2B)∗ πB(1P′(1+)) 50 ρB∗ πB′(1P(1+)) πB(13P2)
± ± ± ± ± ± πB∗
MeVandΓ=26 3 3/17 6 8MeV,respectivelyforthe
neutral/chargedB±(57±47)sta±tes[±20]. 0 0
∗2 15 15
As a JP = 2+ B meson, B∗2(5747) decays into the Bπ πB(13P0) πB(13P0)
and B π channels. In the QPC model, the total decay width
of B∗2(∗5747)0 is around 3.7 MeV, which is composed of (MeV) 10 10 πB(1P(1+))
1Γ.(8B∗2M(5e7V4.7W)0e→noBtiπce) =tha1t.9ouMrerVesuanltdoΓf(tBh∗2e(5to7t4a7l)d0e→cayBw∗πi)dt=h Width 5 ωB∗ πB(1P(1+)) ρB 5 ωB∗ ρB
of B (5747)0 is smaller than the experimental central value
∗2
[16, 20]. As shown in Refs. [16, 20], there exist large ex- 0 0
perimentalerrorsforthemeasuredwidthof B (5747). Thus, 10 10
∗2
further experimentalmeasurementof the resonance parame- 8 8
ter of B∗2(5747) will be helpful to clarify this difference. In 6 ωB∗ 6 ωB∗
addition,wealsoobtaintheratio ηB∗
4 4
B(B (5747) B +π ) ηB∗
∗2 → ∗ − =0.95, (10) 2 2
B(B (5747) B+π )
∗2 → − KBs∗ KBs∗
0 0
−100 −50 0 50 100−100 −50 0 50 100
whichisconsistentwiththeexperimentalvalue1.10 0.42
± ± θ2P(Degree) θ2P(Degree)
0.31[15].
Forthe2Pstatesinthe Bmesonfamily,moredecaychan-
nels are open. However, therehas been no experimentalob-
FIG. 3: (color online). The dependence of the decay behavior of
servationuptonow. Welistourpredictionsofthepartialand
B(2P(1+))(thefirstcolumn)andB(2P(1+))(thesecondcolumn)on
′
totaldecaywidthsinTableII.
themixingangleθ .
2P
For the B(23P ) meson, the dominant decay modes are
0
πB(1P), πB, and ρB . In addition, B(23P ) can decay into
′ ∗ 0
ωB∗, KBs, ηB, and πB(1P). The sum of all partial decay πB(1P(1+)). Inaddition,theotherdecaymodesofB(2P(1+))
widths of B(23P0) reaches up to 154 MeV, which indicates andB(2P(1+))arealsopresentedinFig. 3.
′
B(23P ) is a broad state. Among the dominant decay chan-
0
nelsofB(23P ),πBisthemostsuitablechannelforsearching
0
B(23P ).
0 2. 2S and3S states
B(23P ) mainly decays into ρB , πB(1P), ωB , and
2 ∗ ∗
πB(13P ), where the partial width of B(23P ) ρB is the
2 2 ∗
largestsince B(23P ) ρB occursviaS-wave→. Incontrast, Veryrecently,theCDFCollaborationreportedtheevidence
the D-wavedecaym2od→es B(∗23P ) πB, KB , K B can be ofanewresonanceB(5970)inanalyzingbothB0π+andB+π−
2 s ∗ s
neglected.ThewidthofB(23P )is→64MeV. massdistributions[20]. ThemassandwidthofB(5970)are
2
Asthemixtureofthe21P and23P states,thetwo1+states
1 1
B(2P(1+))andB(2P′(1+))satisfythefollowingrelation (M,Γ)B(5970)0 = (5978±5±12, 70±18±31)MeV,
(M,Γ) = (5961 5 12, 60 20 40)MeV,
B(5970)+
2P(1+) cosθ sinθ 21P ± ± ± ±
= 2P 2P 1 , (11)
2P(1+)  sinθ cosθ  23P  whichcorrespondtotheneutralandchargedB(5970),respec-
 ′  − 2P 2P  1  tively. Thecomparisonofthemassof B(5970)andthemass
where θ denotes the mixing angle. The allowed decay spectruminTableIindicatesthatthemassofB(5970)isclose
2P
modesofB(2P(1+))andB(2P(1+))areshowninTableII. tothe estimatedmassesof the21S and23S states ofthe B
′ 0 1
We obtain the total and partial decay widthsof B(2P(1+)) mesonfamily. Since B(5970)candecayinto Bπ,we canex-
and B(2P(1+)), which depend on the mixing angle θ (see cludetheB(21S )assignmentofB(5970).
′ 2P 0
Fig. 3 for details). Since the experimental information of TheobtainedtotalwidthofB(23S )is47MeV,whichisin
1
B(2P(1+)) and B(2P(1+)) is absent, in the followingdiscus- agreementwiththeexperimentalwidthofB(5970)ifonecon-
′
sionwetakethetypicalvalueθ = 54.7 . Wefindthatthe siderstheexperimentalerror.Thecalculationofthepartialde-
2P ◦
total decay width of B(2P(1+)) (Γ(B−(2P(1+))) = 153 MeV) caywidthsof B(23S )indicatesthatπB ,πBandπB(1P(1+))
1 ∗
is largerthan thatof B(2P(1+)) (Γ(B(2P(1+))) = 70MeV), are its main decay channels. The results of the other partial
′ ′
which is consistent with the estimate under the heavy quark decay widths of B(23S ) are listed in Table III. At present,
1
effectivetheory. For B(2P(1+)) and B(2P(1+)), its main de- B(5970)wasonlyobservedintheπBchannel. Ourstudyin-
′
cay channels include πB , πB(13P ), πB(1P(1+)), ρB and dicates that the B(5970) is very probably B(23S ). We also
∗ 2 ′ ∗ 1
6
TABLEIII:Thepartialandtotaldecaywidths(inunitsofMeV)of TABLEIV:Thepartialandtotaldecaywidths(inunitsofMeV)of
2S and3S statesintheBmesonfamily. Here,weadopt–todenote 1Dand2DstatesintheBmesonfamily. Theforbiddendecaychan-
theforbiddendecaychannels. nelsaremarkedby–. Forthe1D(2 )/1D(2 )and2D(2 )/2D(2 )
− ′ − − ′ −
states,weuse(cid:3)tomarkthealloweddecaychannels.Here,thevalue
Channels 21S 23S 31S 33S
0 1 0 1 a 10 bisabbreviatedasa[b].
−
πB – 9.1 – 2.6 ×
1D(2 ) 2D(2 )
πB∗ 33 23 8.8 6.1 Channels 13D1 13D3 23D1 23D3 1D(2−) 2D(2−)
′ − ′ −
πB(13P ) 3.9 – 3.7 –
0
πB 69 4.9 27 1.2 – –
πB(13P ) 2.0 10 3 1.0 10 2 11.2 4.9
2 × − × − πB 34 6.2 12 0.49 (cid:3) (cid:3)
πB(1P(1+)) – 10 – 3.9 ∗
πB(13P ) – – – – (cid:3) (cid:3)
πB(1P(1+)) – 0.2 – 2.8 0
′ πB(13P ) 1.6 0.74 5.3 0.21 (cid:3) (cid:3)
2
ηB – 1.2 – 0.49
πB(1P(1+)) 6.8[2] 9.0[2] 8.1 6.4 (cid:3) (cid:3)
ηB 0.62 2.0 1.0 0.87
∗ πB(1P(1+)) 147 0.17 28 7.5[2] (cid:3) (cid:3)
ηB(13P ) – – 0.16 – ′
0
ηB 12 0.21 4.3 5.8[2] – –
ηB(13P ) – – 7.2 10 2 8.8 10 2
2 × − × − ηB 5.6 0.20 1.6 1.2[3] (cid:3) (cid:3)
ηB(1P(1+)) – – – 0.28 ∗
ηB(13P ) – – – – – (cid:3)
ηB(1P(1+)) – – – 0.11 0
′ ηB(13P ) – – 0.62 0.19 – (cid:3)
2
ρB – – 1.3 1.3
ηB(1P(1+)) – – 0.74 0.57 – (cid:3)
ρB – – 9.4 7.0
∗ ηB(1P(1+)) – – 0.65 2.1[2] – (cid:3)
ωB – – 0.49 5.7 10 2 ′
× − ρB 8.3 1.8[2] 0.12 2.0 (cid:3) (cid:3)
ωB – – 3.3 1.7
∗ ρB 0.93 1.3 23 6.1 (cid:3) (cid:3)
∗
KB – 0.43 – 0.24
s ωB 2.4 3.7[3] 3.2[2] 0.67 (cid:3) (cid:3)
KB – 0.58 0.45 0.41
∗s ωB 0.10 – 7.5 2.0 (cid:3) (cid:3)
∗
K B – – 0.46 0.27
∗ s KB 7.4 5.4[2] 2.5 4.4[2] – –
s
K B – – 0.49 0.90
∗ ∗s KB 3.3 4.5[2] 0.86 8.6[3] (cid:3) (cid:3)
KB (13P ) – – 4.0 10 2 – ∗s
s 0 × − K B – – 0.25 1.9[3] – (cid:3)
KB (13P ) – – – 2.8 10 3 ∗ s
s 2 × − K B – – 0.65 0.75 – (cid:3)
KB (1P(1+)) – – – 0.10 ∗ ∗s
s KB(13P ) – – – – – (cid:3)
KB (1P(1+)) – – – 3.7 10 3 s 0
s ′ × − KB(13P ) – – 0.20 4.5[2] – (cid:3)
s 2
Total 38 47 41 33
KB(1P(1+)) – – 0.19 0.12 – (cid:3)
s
KB(1P(1+)) – – 0.14 4.2[3] – (cid:3)
s ′
Total 294 14 122 23 – –
suggest the experimental search for B(5970) via its πB de-
∗ 3. 1Dand2Dstates
cay.
Thedecaybehaviorsof1D and 2Dstates are givenin Ta-
AsthepartnerofB(23S ),B(21S )dominantlydecaysinto
1 0 bleIV,wherewefirstlistthealloweddecaychannelsforthe
πB . In addition, there existsa considerablepartialwidth of
∗ 1D(2 )/1D(2 )2D(2 )/2D(2 )states.
B(21S ) πB(13P ). ComparedwiththeπB andπB(13P ), − ′ − − ′ −
the rem0a→ining two d0ecay channels ηB and π∗B(13P ) can0be B(13D1) is very broad and its total width reaches up to
∗ 2 294 MeV. B(13D ) dominantly decays into πB(1P(1+)), πB
ignored. Thus,πB istheidealchanneltosearchforB(21S ) 1 ′
∗ 0 and πB . These channels are suitable to search for B(13D )
infutureexperiments. ∗ 1
in future experiments. B(23D ) is the first radial excita-
1
tion of B(13D ). We notice that πB(1P(1+)), πB, ρB ,
1 ′ ∗
The masses of the two 3S states B(31S ) and B(33S ) are and πB are the main decay modes of B(23D ), which ren-
0 1 ∗ 1
6334 MeV and 6355 MeV, respectively. As shown in Table ders B(23D ) broad. Different from the broad B(13D ) and
1 1
III, more decay channels are open for B(31S ) and B(33S ). B(23D ), B(13D )and B(23D )arerathernarrowstatessince
0 1 1 3 3
The main decay modes of B(31S ) are πB(13P ), ρB , πB , thetotaldecaywidthsof B(13D )andB(23D )are14and23
0 2 ∗ ∗ 3 3
ωB . For B(33S ), mainly ρB , πB , πB(13P ), πB(1P(1+)), MeV,respectively.ForB(13D ),thereexisttwodominantde-
∗ 1 ∗ ∗ 2 3
πB(1P(1+)), πB, ωB , and ρB contribute to the total de- cay channels, πB and πB . Althoughmore decay modesare
′ ∗ ∗
cay width. The partial widths of the modes ηB(13P ), ωB, included for B(23D ), its total decay width is not obviously
2 3
KB (13P )andKB (1P(1+))arequitesmall. enhanced.TheπB(1P(1+)),ρB ,ρB,ωB andπBarethemain
s 2 s ′ ∗ ∗
7
300 300
250 B(1D(2−)) 250 B(1D′(2−)) Total 110200 B(2D(2−)) 110200 B(2D′(2−))
125000πB(13P2)−50.8◦ Total 125000 −50.8◦ πB(13P2) 468000 πB∗ −50.8◦ Total ρB∗ 468000 −50ρ.B8◦∗ πB∗TotalπB(13P2)
100 100 20 πB(13P2) 20
50 πB∗ ηB∗ ρB 50 ρB ηB∗πB∗ 0 0
12 12
0 0 10 πB(1P′(1+)) 10 πB(1P′(1+))
10 10 8 πB(1P(1+)) 8 πB(13P0)
MeV) 68 KB∗ ωB 68 ωB KBs∗ V) 246πB(13P0) ρB 246 πB(1P(ρ1B+))
Width( 4 πB(1P′(1+)) ρB∗ 4 ρB∗ πB(1P′(1+)) Width(Me 06 06
02 02 4 ωB ηB∗ ωB∗ 4 ωηBB∗∗ ωB
2 2
πB(1P(1+))
ωB∗ πB(1P(1+)) ωB∗ 0 KBs∗ 0 KBs∗
0.1 0.1 1.5 1.5
K∗B∗ K∗Bs∗
1 1
0.05 πB(13P0) 0.05
πB(13P0) 0.5 K∗Bs ηB(13P2) 0.5 ηB(13P2) K∗Bs
0 0
−100 −50 0 50 100 −100 −50 0 50 100
θ1D(Degree) θ1D(Degree) −0100 −50 0 50 100−0100 −50 0 50 100
θ2D(Degree) θ2D(Degree)
FIG. 4: (color online). The dependence of the decay behavior of
B(1D(2 ))(thefirstcolumn)andB(1D(2 ))(thesecondcolumn)on FIG.5:(coloronline).Thedependenceofthepartialandtotaldecay
− ′ −
themixingangleθ . widthsof B(2D(2 ))(thefirstcolumn) and B(2D(2 ))(thesecond
1D − ′ −
column)onthemixingangleθ .
2D
decaymodesofB(23D ). Theconcreteinformationofthede-
3
caypatternofB(13D1),B(23D1),B(13D3)andB(23D3)canbe 112and40MeV,respectively,correspondingtoθ2D =−50.8◦.
foundinTableIV.
B(nD(2 ))andB(nD(2 ))arethemixtureofthen3D and
− ′ − 2
n1D states,whichsatisfythefollowingrelation: 4. 1Fstates
2
nD(2−) = cosθnD sinθnD n1D2 (12) There are four 1F states. Their decay pattern is listed in
withn=1n,2D.′(2−) −sinθnD cosθnD  n3D2  TqoufabiBtlee(1bV3rFo.a2Wd),,eπannBdo(1tciPcaen′(1trhe+aa)t)chtihsuetphteototma2lo5ds4etcMiamyepVwo.riAtdatmnhtoocnhfgaBna(ln1le3dlF.ec2T)ahyiess
InFig. 4,wepresentthedependenceofthepartialandto- other main decay modes of B(13F2) are πB, πB∗, πB(13P2),
tal decay widths of B(1D(2−)) and B(1D′(2−)) on the mix- ηB(1P′(1+)),ρB,andρB∗. ForB(13F4),itstotalwidthisesti-
ing angle θ1D. In the heavy quark limit, the mixing angle matedas103MeV,whereρB∗andωB∗areitsdominantdecay
θ = 50.8 was given in Ref. [49]. πB(13P ) and πB mode.
1D ◦ 2 ∗
are the−dominant decays for B(1D(2−)) while ρB, πB∗, and In the following, we focus on B(1F(3+)) and B(1F′(3+)),
ωB are the main contribution to the width of B(1D′(2−)). whicharethemixingofthe13F3and11F3states,
The QPC calculationfurtherindicatesthat the total width of
B(1D(2−)) is quite broad. Compared with the total decay 1F(3+) = cosθ1F sinθ1F 11F3 . (13)
width of B(1D(2 )), the total decay width of B(1D(2 )) is 1F (3+)  sinθ cosθ 13F 
narrow. Here, we−need to specify that the above con′cl−usion  ′  − 1F 1F  3 
is obtained with the typical value θ = 50.8 . The de- withθ = 49.1 determinedinheavyquarklimit[26].InFig.
1D ◦ 1F ◦
− −
cay behaviors of the partial decay widths of B(2D(2 )) and 6,theθ dependenceofthepartialandtotaldecaywidthsof
− 1F
B(2D(2 )) depend on the concrete value of θ (see Fig. 4 B(1F(3+))andB(1F (3+))ispresented.Ifwetakethetypical
′ − 1D ′
formoredetails). valueθ = 49.1 , we noticethat B(1F(3+)) is a quite broad
1F ◦
−
The variation of the total and partial decay widths of resonance. From Fig. 6, we can easily distinguish the main
B(2D(2 ))and B(2D(2 ))withθ isshowninFig. 5,from decay mode among all allowed decay channels, and we can
− ′ − 2D
whichwecanobtainthemainandsubordinatedecaysofthese distinguishwhichchannelisvaluabletofurtherexperimental
twostates. ThetotalwidthsofB(2D(2 ))andB(2D(2 ))are searchofthe1F stateintheBmesonfamily.
− ′ −
8
250 250
TABLEV:Thepartialandtotaldecaywidths(inunitsofMeV)of1F B(1F(3+)) B(1F′(3+))
statesinBmesonfamily. Theforbiddendecaychannelsaremarked 200 200
by–.For1F(3+)/1F′(3+)states,weuse(cid:3)tomarkthealloweddecay 150 −49.1◦ Total 150 −49.1◦ Total
channels. πB(13P2)
Channels 13F2 13F4 1F(3+)/1F′(3+) 100 πB∗ πB(13P2) ρB 100 ρB πB∗
πB 30 3.7 – 50 50
πB∗ 21 5.3 (cid:3) 0 0
35 35
πB(13P ) – – (cid:3)
0 30 30
πB(13P ) 14 2.5 (cid:3)
2 V) 25 ρB∗ 25
ππBB((11PP′((11++)))) 102.51 01..428 (cid:3)(cid:3) Width(Me 1250 ωB∗ ωB 1250 ωB ρωBB∗∗
ηB 5.5 0.31 – 10 10
ηB 3.9 0.41 (cid:3) 5 ηB∗ 5 ηB∗
∗
ηB(13P ) – – (cid:3) 0 0
0 10 10
ηηηBBB(((1113PPP((121)++)))) 1.30×1.11100−3 274...449××111000−−333 (cid:3)(cid:3)– 68 KBs∗ηB(13P2)πB(1P′(1+)) 68 πB(1P′(1+)) KBs∗ηB(13P2)
′ −
×
ρB 15 1.1 (cid:3) 4 4
ρB 15 56 (cid:3)
∗ 2 2
ωB 4.7 0.33 (cid:3)
ωB 2.2 31 (cid:3) −0100 −50 0 50 100−0100 −50 0 50 100
∗
KB 2.8 7.2 10 2 – θ1F(Degree) θ1F(Degree)
s × −
KB 2.0 9.2 10 2 (cid:3)
∗s × −
K B 0.75 1.3 10 2 (cid:3)
∗ s × − FIG.6:(coloronline).Thedependenceofthepartialandtotaldecay
K∗B∗s 8.8×10−2 0.24 – widthsof B(1F(3+)) (thefirst column) and B(1F′(3+)) (the second
KBs(13P0) – – (cid:3) column)onthemixingangleθ1F.
KB (13P ) 1.1 10 3 1.2 10 4 –
s 2 × − × −
KB (1P(1+)) 5.4 10 4 2.0 10 4 (cid:3)
s × − × −
KB (1P(1+)) 3.3 4.9 10 5 (cid:3)
s ′ × −
Total 254 103 –
120
B. Bottom-strangemeson 100 (a)
MeV) 80 Bs(1P′(1+))
( 60
1. 1Pand2Pstates h −54.7◦
Widt 40 Bs(1P(1+))
20
Thedecaymodesof B (13P )is BK. Thepartialwidthof
s 0 0
B (13P ) BK obtainedfrom the QPC modelis 225 MeV, −100 −50 0 50 100
s 0 → θ1P (degree)
whichisalmostthesameasthatfromthechiralquarkmodel
2
in Ref. [23] (Γ = 227MeV). B (13P ) is also a broadstate,
verysimilar to B(13P0). Generasllysp0eaking,it is difficultto V)1.5 Bs(1P′(1+)) (b)
detectaverybroadstateexperimentally. Me
( 1
TheobservedB (5830)andB (5840)aregoodcandidates h
s1 ∗s2 dt
of B (1P(1+)) and B (13P ), respectively. Theexperimental Wi0.5 CDF
s ′ s 2
information on B (5830) and B (5840) from the CDF, D0,
s1 ∗s2 0
andLHCbCollaborations[17–20]iscollectedinTableVI. −64 −62 −60 −58 −56 −54 −52 −50 −48 −46
If we take the mass of B (5840) as input, B (13P ) can θ1P (degree)
∗s2 s 2
decayintoBKandB K. WiththeQPCmodel,thetotaldecay
∗
width of B (13P ) is around Γ(B (13P )) = 0.26 MeV and FIG.7: (coloronline). (a)Thedependenceofthetotaldecaywidths
s 2 s 2
slightlysmallerthantheexperimentalcentralvaluewhichhas of Bs(1P(1+)) (dashed curve) and Bs(1P′(1+)) (solid curve) on the
mixingangleθ .(b)ThevariationofthetotalwidthofB(1P(1+))
alargeerror[19,20]. However,theobtainedratio 1P s ′
withthemixingangleθ =( 64 46) andthecomparisonwith
Γ(B B +K ) theCDFdata[20]. Here1P,the−vertic∼al−dash◦edlinein(a)corresponds
∗s2→ ∗ − =0.088 (14) totheidealmixingangleθ = 54.7 .
Γ(B∗s2→ B+K−) 1P − ◦
is in good agreement with the experimental measurement
9
TABLEVI:ThesummaryofexperimentalinformationofB (5830)andB (5840).
s1 ∗s2
State Collaboration Mass Width Observeddecays Γ(B∗s2→B∗+K−)
Γ(B∗s2→B+K−)
CDF[17] 5829.4 0.7MeV – B+K –
∗ −
±
Bs1(5830) LHCb[19] 5828.40 0.04 0.04 0.41 – B∗+K− –
± ± ±
CDF[20] 5828.3 0.1 0.1 0.4 0.7 0.3 0.3 B+K –
∗ −
± ± ± ± ±
CDF[17] 5839.6 0.7 – B+K –
−
±
D0[18] 5839.6 1.1 0.7 – B+K –
−
B (5840) ± ±
∗s2
LHCb[19] 5839.99 0.05 0.11 0.17 1.56 0.13 0.47 B+K ,B+K 0.093 0.013 0.012
∗ − −
± ± ± ± ± ± ±
CDF[20] 5839.7 0.1 0.1 0.2 2.0 0.4 0.2 B+K ,B+K 0.11 0.03
∗ − −
± ± ± ± ± ±
givenbyLHCb[19]andCDF[20],asshowninTableVI.
In analogy to B(1P(1+)) and B(1P′(1+)), Bs(1P(1+)) and 150 Bs(2P(1+)) 150 Bs(2P′(1+))
Bs(1P′(1+))arethemixtureofthe13P1and11P1statesinthe Total Total
Bs mesonfamily,which also satisfy Eq. (9). Thereonlyex- 100 −54.7◦ K∗B∗ K∗B 100 −54.7◦ K∗B∗
ists the B K decay channel for B (1P(1+)) and B (1P(1+)). K∗B
Thus, we∗give the dependence ofsthe total decayswid′ths of 50 KB∗ 50 KB∗
o7B)rs.(−1W5P2h(.1e0n+◦)t∼)hae−nmd49iBx.0isn(◦1,gPtah′ne(1gt+loe)ta)islodtnaektcheaenymawsiixdθi1tnhPgo=afn−Bg6sl(e01.θ5P1◦′P(∼1(s+−e)e5o7Fv.ei5gr◦-. Width(MeV) 22050 KB(13P2) 22050 KB(13P2)
lapstheobservedwidthofB (5830)givenbyCDF[20]. We
s1
notice thatthere exists a differencebetween the realistic and 15 15
KB(1P′(1+)) KB(1P′(1+))
ideal value of the mixing angle θ , which is similar to the
1P 10 10
situationof B1(5721). Ifadoptingθ1P = 60.5◦ 57.5◦ or KB(1P(1+))
52.0 49.0 ,thetotaldecaywidthof−B (1P(∼1+−)isabout 5 5
◦ ◦ s
1−10Me∼V.−Hence, B (1P(1+)isabroadresonance,consistent 0 0 KB(1P(1+))
s −100 −50 0 50 100−100 −50 0 50 100
withtheroughestimateinheavyquarklimit. θ2P(Degree) θ2P(Degree)
FIG. 8: (color online). The variation of the decay width of
TABLE VII: The partial and total decay widths (in units of MeV)
B(1P(1+))(thefirstcolumn) and B (1P(1+))(thesecond column)
of2PstatesintheB mesonfamily. Theforbiddendecaychannels s s ′
s
aremarkedby–. For2P(1+)/2P(1+)states, weuse(cid:3)tomarkthe with the mixing angle θ2P. Here, we only listed their main decay
′
channels.
alloweddecaychannels.
Channels 23P 23P 2P(1+)/2P(1+)
0 2 ′
KB 65 2.6 – 8 leads to the total decay width around 142 MeV. Thus,
KB – 7.3 (cid:3) Bs(22P0) is a broad Bs meson. For Bs(23P2), there is only
∗
one dominant decay channel K B , where the branching ra-
K B – 1.6 (cid:3) ∗ ∗
∗ tio of B (23P ) K B is 0.84. At present, B (23P ) and
K∗B∗ 36 69 (cid:3) B (23P )s are2stil→l mis∗sin∗g. Thus, the obtained ms ain0decay
s 2
KB(13P0) – – (cid:3) modesofBs(23P0)andBs(23P2)maybeusefultotheexperi-
KB(13P0) – – (cid:3) mentalsearchofB(23P0)andB(23P2).
KB(1P(1+)) 5.3 0.70 (cid:3) The dependence of the total and partial decay widths of
B (2P(1+)) and B (2P(1+)) on θ are presented in Fig. 8.
KB(1P(1+) 32 3.5 10 3 (cid:3) s s ′ 2P
′ × − If θ takes the typical value, the main decay modes are
ηBs 1.7 5.3×10−2 – Bs(22PP(1+))/Bs(2P′(1+)) K(∗)B(∗) (see Fig. 8 for de-
ηB∗s – 0.14 (cid:3) tailed information). We c→onclude that both Bs(2P(1+)) and
Total 142 82 – Bs(2P′(1+))arebroadBsstates,
The decay behavior of the four 2P states are listed in Ta-
ble VII, where B (2P(1+)) and B (2P(1+)) are the mixed C. 2S and3S states
s s ′
states satisfying Eq. (11). The three main decay channels
for B (23P ) are KB, K B , and KB(1P(1+)). The sum of In the following, we illustrate the decay behavior of
s 0 ∗ ∗ ′
all partial decay widths listed in the second column of Fig. B (21S ),B (23S ),B (31S ),andB (33S )(seeTableVIII).
s 0 s 1 s 0 s 1
10
TABLEVIII:Thepartialandtotaldecaywidths(inunitsofMeV)of TABLEIX:Thepartialandtotaldecaywidths(inunitsofMeV)of
2S and3S statesintheB mesonfamily.Here,weadopt–todenote 1Dand2DstatesintheB mesonfamily.Theforbiddendecaychan-
s s
theforbiddendecaychannels. nelsaremarkedby–. Forthe1D(2 )/1D(2 )and2D(2 )/2D(2 )
− ′ − − ′ −
states,weuse(cid:3)tomarkthealloweddecaychannels.Here,thevalue
Channels 21S 23S 31S 33S
0 1 0 1 a 10 bisabbreviatedasa[b].
−
KB – 17 – 5.8 ×
1D(2 ) 2D(2 )
KB∗ 44 34 15 12 Channels 13D1 13D3 23D1 23D3 1D(2−) 2D(2−)
′ − ′ −
K B – – 3.0 0.41
∗
KB 125 5.2 49 0.61 – –
K B – – 21 12
∗ ∗ KB 58 5.7 21 1.4[2] (cid:3) (cid:3)
KB(13P ) – – 9.8 10 4 – ∗
0 × − K B 1.1 3.0[5] 0.99 5.5 (cid:3) (cid:3)
KB(13P ) – – 14 7.5 ∗
2 K B – – 41 13 – (cid:3)
KB(1P(1+)) – – – 0.95 ∗ ∗
KB(13P ) – – – – – (cid:3)
KB(1P(1+)) – – – 6.0 0
′ KB(13P ) – – 9.7 4.3 – (cid:3)
2
ηB – 0.29 – 0.17
s KB(1P(1+)) – – 12 11 – (cid:3)
ηB 0.14 0.30 0.28 0.26
∗s KB(1P(1+)) – – 33 0.38 – (cid:3)
ηB (13P ) – – 0.11 – ′
s 0 ηB 4.8 5.3[2] 1.7 1.2[2] – –
ηB (13P ) – – 5.4 10 3 1.2 10 2 s
s 2 × − × − ηB 2.0 4.5[2] 0.60 2.2[5] (cid:3) (cid:3)
ηB (1P(1+)) – – – 0.15 ∗s
s ηB(13P ) – – – – – (cid:3)
ηB (1P(1+)) – – – 1.3 10 2 s 0
s ′ × − ηB(13P ) – – 0.16 5.8[2] – (cid:3)
s 2
φB – – 0.26 0.31
s ηB(1P(1+)) – – 0.16 0.14 – (cid:3)
s
Total 44 51 54 46
ηB(1P(1+)) – – 0.13 6.3[3] – (cid:3)
s ′
φB – – 0.38 6.7[2] – (cid:3)
s
Accordingto the numericalresults in Table VIII, we con- φB∗s – – 0.70 1.7 – (cid:3)
clude: Total 191 11 171 37 – –
B (21S ) dominantly decays into KB . The contribu-
s 0 ∗
• tionfromηBs(0+)isnegligible. B (23D ) mainly decaysinto KB, K B , KB(1P(1+)),
s 1 ∗ ∗ ′
KB andKBarethetwomaindecaysofB (23S ). The • KB∗,KB(1P(1+)),andKB(13P2).
∗ s 1
• contributionfrombothηB andηB (0+)decaymodesis
∗s s ForB (13D ),therearetwodominantdecaymodes,KB
quitesmall. • and KsB . F3or B (23D ), K B , KB(1P(1+)), K B, and
∗ s 3 ∗ ∗ ∗
As the radial excitation of Bs(21S0), the total decay KB(13P2) are its main decay channels, while the re-
• widthof B (31S )isΓ(B (31S )) = 54MeV.Here,the mainingdecaychannelshavesmallpartialdecaywidths
s 0 s 0
branchingratiosof B (31S )) KB ,K B ,KB(13P ) (seeTableIXformoredetails).
s 0 ∗ ∗ ∗ 2
→
are0.28,0.39,and0.26,respectively.
As the mixed states B (1D(2 ))/B (1D(2 )) and
s − s ′ −
• TKh∗eB∗m,KaiBn,KdeBc(a1y3Pm2)oadnedsKofB(B1Ps(′3(31S+)1)).) include KB∗, aBnsd(2BD((22D−)()2/−B))s/(2BD(2′(D2′−()2)−a)r)einsimEqil.ar(1to2)B,(i1nDF(i2g−s).)9/Ba(n1dD1′(02w−)e)
thusshowthevariationoftheirpartialandtotaldecaywidths
withθ /θ . Withthetypicalθ = θ = 50.8 ,wehave
1D 2D 1D 2D ◦
D. 1Dand2Dstates thefollowingobservations: −
In Table IX, we list the numerical results of For Bs(1D(2−))/Bs(1D′(2−)), its KB∗ mode is very
•
B (13D ), B (13D ), B (23D ) and B (23D ) and the al- important. The variation of the partial width
s 1 s 3 s 1 s 3
lowed decay channels of Bs(1D(2−))/Bs(1D′(2−)) and Bs(1D(2−))/Bs(1D′(2−)) → KB∗ with the mixing an-
B (2D(2 ))/B (2D(2 )). gle is verysimilar to thatof thetotaldecaywidth(see
s − s ′ −
Ourcalculationshows Fig. 9).
Both B (13D ) and its radial excitation B (23D ) are Bs(2D(2−)) is broad and mainly decays into KB∗,
s 1 s 1 •
• verybroad,whilebothB (13D )andB (23D )arequite KB(13P2), K∗B∗, and K∗B. As the parter of
s 3 s 3
B (2D(2 )), the total decay width of B (2D(2 ))
narrow. s − s ′ −
is smaller than that of B (2D(2 )). The main de-
s −
The dominant decay modes of B (13D ) are KB and cay mode of B (2D(2 )) is K B . While the sum
s 1 s ′ − ∗ ∗
•
KB with the branching ratios 0.65 and 0.30, respec- of the partial decay widths of the decay modes
∗
tively. KB(1P(1+)),KB(13P ),KB(1P(1+)),K B is compara-
0 ′ ∗