Search for $χ_{c_J}(2P)$ from Higher Charmonim E1 Transitions and X,Y,Z States

0.23 MB·English

by  Bai-Qing Li

Search for χc (2P) from Higher Charmonim E1 Transitions and X,Y,Z States
J
Bai-Qing Lia, Ce Mengb and Kuang-Ta Chaob,c
aDepartment of Physics, Huzhou Teachers College, Huzhou 313000, China;
bDepartment of Physics and State Key Laboratory of Nuclear
Physics and Technology, Peking University, Beijing 100871, China;
cCenter for High Energy physics, Peking University, Beijing 100871, China
We calculate the E1 transition widths of higher vector charmonium states into the spin-triplet
2P states in three typical potential models, and discuss the possibility to detect these 2P states
via these E1 transitions. We attempt to clarify the nature of some recently observed X,Y,Z states
by comparing them with these 2P charmonium states in these E1 transitions. In particular, the
′
calculated branching ratios of ψ(4040),ψ(4160) → χc γ (J=0,1,2) are found to be in the range of
2 10−4-10−3, and sensitive to the 3S-2D mixing of ψ(40J40) and ψ(4160). The mixing angle may be
′
1 constrained bymeasuring ψ(4040),ψ(4160)→Z(3930)γ, ifZ(3930) is identifiedwith theχc2 state,
20 andthenbeusedinmeasuringχ′c0,1 states. Theseprocessescanbestudiedexperimentallyate+e−
colliders such as BEPCII/BESIII and CESR/CLEO.

n
a PACSnumbers: 12.39.Jh,13.20.Gd,14.40.Pq
J

9
1 INTRODUCTION theBaBarCollaboration[13]separatelyareanothercan-
′
didates for χ . The X(3940), which was found by the
] BelleCollabocr1a,0tion[14]in the recoilingspectrumofJ/ψ
h Since the discovery of J/ψ in 1974 [1, 2], a lot of
p in the e+e− annihilation process e+e− J/ψ+X and
- charmonium states had been found in the last cen- e+e− J/ψ+DD¯∗, seemsnotto be a2→Pstate[15,16].
p tury. Among them, the vector states ψ(4040), ψ(4415), →
Another well known state, the X(3872), which was first
e and ψ(4160), which are commonly assigned as ψ(3S),
1 [h ψdthu(u4cseSd)ctaahnnrdobueψgh(2ree3a+Dde1i−l)yraednsenptieheccitltaievtdeiolyna,ticnaetno+eob−neecdpoihrlleoidctteolrnys,palrinkode- fBmouimgnhedtsiobnnetdihneetceJar/ypψraerπtoe+udπna−dsi3tnh8v7ea2rDiMa∗n0etDVm0[+a1s7cs].dcw.isimtthroiblJeucPtuCiolen=dinu1e+th+toe,
v theclosenessofitsmasstotheD∗0D0 threshold. Butits
BEPCII/BESIII [3] and CESR/CLEO.
5 large production rate in p p¯collisions at the Tevatron
5 Aside from these vector resonances themselves, it is and some properties seem−to support that it could be a
1 alsointeresting to detect the products via E1transitions ′
compact bound state, such as the 2P charmonium χ ,
1.4 oEfsptheecsiaellvye,cttohreredseocnaaynccehsainnntoellsowψe(r4c0h4a0r,m41o6n0i,u4m41s5t)ates. or a mixture of the χ′c1 with the D∗0D0+c.c. molecucle1,
0 χ′ γ, J = 0,1,2 can be used to detect the 2P cha→r- despite of its lower mass. In fact, the mass of χc1(2P)
2 cJ ′ can be lowered to below 3.9 GeV if the color screening
monia χ and to study the properties of these missing
1 cJ effects andcoupledchanneleffects areconsidered[4,15].
states. Theimportance ofexperimentalestablishmentof
: So it is interesting to examine in the E1 transitions of
v
these 2P charmonia consists in at least two aspects. On
higher charmonia if the X(3872) can be seen by having
i ′
rX tohneecsaidlceu,lathtieonpsroinpevratriieosuosfpχotceJntairaelmimopdoerltsaanntdtcooculpalreidfy- the χ′c1 component in its wave function.
a channel models (see, e.g. [4] and references therein). On
Because of the phenomenological importance of the
′
the other side, the χ could be the candidates of some ′
cJ χcJ states mentioned above, we will study the produc-
of the recently observed charmonium-like states, the so-
tion of these states in the E1 transitions of higher vec-
called ”X,Y,Z” states (for a review see e.g. [5]).
tor charmonia, say, ψ(4040), ψ(4160) and ψ(4415). The
According to potential model estimates, the spin- E1 transition width can be estimated by potential mod-
triplet 2P states lie between 3.9 and 4.0 GeV in the els. Various potential models predict various transition
charmonium family [6–8]. Experimentally, five charmo- widths. However, the transition widths of ψ(4415)
′ →
nium(like)resonancesaround3940MeVhavebeenfound χ γ are usually small because of the smallness of the
cJ
recently. The Z(3930) [9] observed by the Belle Col- overlap integral between the wave functions of 4S and
laboration in 2006 in the γγ fusion experiment with 2P states. On the other hand, the transition widths
′ ′
a mass 3929 5 2 MeV is identified with the χ . of ψ(4040,4160) χ γ, are relatively large (tens to
± ± c2 → cJ
The X(3915), which was also produced in the γγ fu- hundreds of KeV as those shown in Table I and Ta-
sion experiment[10] and detected in the J/ψω channel ble II) andthe correspondingbranching ratiosareabout
with a mass about 3915 MeV, is possibly the χ′ [11]. 10−4 10−3. Sotheχ′ maybedetectedattheupgraded
c0 − cJ
The Y(3940) and Y(3915), which were detected in the BEPCII/BESIII through these channels. Note that the
B J/ψωK processbytheBelleCollaboration[12]and E1transitionwidth depends onthe phasespacewhichis
→
2
′ ′
determined by the mass of χ . Since the χ has been in the cc¯ system gradually becomes neutralized by the
cJ c2
identified with Z(3930), we can compare various poten- produced light quark pair, and string breaking emerges.
′
tial model predictions with the experimental data of χ The potentials in Model II [6] are:
c2
toseeifitcanbedetectedintheE1transitionsofhigher
4α
charmonium states. This comparison may provide some V (r)= C, V (r)=λr, (3)
V S
′ −3 r
hints in searching for the other two χ states.
cJ
We introduce three typicalpotentialmodels in section with model parameters taken similar to Ref. [7]:
II, and calculate in section III the E1 transition widths
′ α =0.5461, λ=0.1425GeV2, m =1.4794GeV
of ψ(4040,4160,4415) χ γ with both lowest- and C c
→ cJ
first-order wave functions in the non-relativistic expan-
The potentials in Model III [18, 19] are
sion within these models. We discuss the possibility for
′
producingχ fromE1transitionsofhigherexcitedchar- 8π 1 1 Λr
monium stactJes and compare them with those ”X,Y,Z” VV(r)= 25ln(Λr)1+−Λr, VS(r)=λr − C, (4)
states in section IV, where the effects of S-D mixing of
with parameters
ψ(4040)andψ(4160)ontheE1transitionwidthsarealso
considered. A summary is given in section V. Λ=0.47GeV, λ=0.22GeV2,
m =1.84GeV, C = 0.975GeV. (5)
c
−
THE POTENTIAL MODELS
The potentials above are used to calculate the lowest-
order and the first-order non-relativistic wave functions.
There are many phenomenologically successful poten- For the first-order relativistic corrections to the wave
tial models in the literature. Among them the Cornell functions, we include the spin-dependent part of H ,
SS
model [6](here we mark it by Model II) is well known, H , H and the spin-independent part H as pertur-
LS T SI
whichdescribesthecharmoniumsystemquitewell. How- bations.
ever, the predicted masses of higher charmonium states The spin-spin contact hyperfine interaction is
seem to be larger than their experimental values [15]. A
′ 2
MdisetVinclatrgexeramthpalne itshetheexpmearismseonftaχlc2vawluhei.chTihseasbcoreuetn4ed0 HSS = 3m2cS~1·S~2▽2VV. (6)
potentialmodel(seeRef[15]andreferencestherein)(here
For Coulombic vector potential, 2(1) δ3(~r) which
wemarkitbyModelI)wasproposedtolowerthemasses ∇ r ∝
gives too large a mass splitting of J/ψ η , so we make
of higher charmonia. So we take it here to estimate the − c
′ a substitution as in Ref. [7] for Model I and Model II:
E1transitionproductionofχ . Thethirdmodelwetake
cJ
was proposed by Ding et al. [18, 19](here we mark it by 32πα
Model III), in which the Coulomb potential has a run- HSS = 9m2C δ˜σ(r)S~c·S~c¯, (7)
c
ningcouplingconstant. TheHamiltonianinthesemodels
hisavveectthoerfpoormtenotfiaHl a=nd−VmP~2c(+r)ViVs(src)a+laVrSp(ort)e,nwtihaelraenVdVm(r) wMhoedreel Iδ˜σa(nrd) σ==(σ1/.0√94π6)G3ee−Vσ2irn2ManoddelσII=. 1.362GeV in
S c
is the mass of charm quark. The spin-orbit term is
The potentials in Model I [15] are:
H = 1 (3V′(r) V′(r))L~ S~, (8)
4α 1 e−µr LS 2m2cr V − S ·
C
V (r)= , V (r)=λ( − ), (1)
V S
−3 r µ and the tensor force term is
rwahnegreesµcailsarthpearstcrVeSe(nri)ngbefacoctmoer wflahticwhhmenakres≫theµ1 laonndg HT = 121m2c(1rVV′(r)−VV′′(r))S12, (9)
ssltoilplel,inaenadrlαyCrisisintghewchoeenffirci≪entµ1o,fλthisetChoeullionmeabrppootteennttiaial.l whTerheeSs1p2in=-in3d(σe~p1e·nrˆd)(eσn~2t·prˆa)r−t iσs~1a·σb~2it. complicated. We
The model parameters are chosen following Ref [15]: take the form as Ref. [20]:
αC =0.5007, λ=0.21GeV2, H = P~4 + 1 2V (r)
µ=0.0979GeV, m =1.4045GeV, (2) SI −4m3 4m2 ▽ V
c c c
1
where αC αs(mcvc) is essentially the strong coupling −2m2 P~1·VV(r)ℑ·P~2
≈ c
constant at the scale mcvc. Here µ is the characteristic nn ′ oo
1 V (r)
sthcaalteafotrdciostloarnsccerseleanrignegr,tahnadn11//µµistahbeosuttat2icfmco,loimr spolyuirncge +2m2c ((P~1·~r Vr ~r·P~2)), (10)
3
where P~1 and P~2 are momenta of c and c¯ quarks in the predictions for the widths Γ(ψ(4040,4160)→ χ′cJγ)
the rest frame of charmonium, respectively, which sat- are large and insensitive to model details, resulting in
isfy P~1 = P~2 = P~, is the unit second-order tensor, quite steady values in different models.
− ℑ
and is the Gromes’s notation The results obtained with first-order wave functions
{{ }}
are listed in Table II. From (7-10), one can see that the
1
A~ B~ = (A~B~ : +A~ B~+B~ A~+ :A~B~), (11) corrections to the non-relativistic potential involve some
{{ ·ℜ· }} 4 ℜ ·ℜ ·ℜ ℜ
derivative terms, which make the potential to be more
where is a second-order tensor. attractivetowardsthe origin. Asaresult,thewavefunc-
Noteℜthat we do not include the contributions from tions with relativistic corrections will be thinner than
the scalarpotentialinH since itis stillunclearhow to those without relativistic corrections. This effect usually
SI
deal with the spin-independent corrections arising from reduces the spatial matrix elements < f ri > defined
| |
the scalar potential theoretically. in (13), which can be seen directly by comparison be-
′
tween the results of Γ(ψ(4040,4160) χ γ) listed in
→ cJ
Table I and in Table II. However, the relativistic correc-
E1 TRANSITION WIDTHS tions can also change the node structures of the wave
functions of higher exited states, such as ψ(4415), and
E1 transitions of higher excited S- and D-wave char- makethe cancelationinthe overlapintegralbetween the
monium states are of interest here because they can be wavefunctionsof4Sand2Pstatesmoremodest,andthis
used to produce and identify P-wave states. For the E1 can be seen in Table II. On the other hand, the transi-
transition width for charmonium, we use the formula of tion width is proportionalto the factor E3, which favors
γ
Ref. [21]: initialstateswithhighermasses. Thus,thedecaywidths
′
Γ(ψ(4415) χ γ)listedinTableIIbecomelarger. But
ΓE1(n2S+1LJ n′2S′+1L′J′ +γ) weshouldm→entcioJnthattheseresultsarenotveryreliable
→
4 andaremoresensitiveto the modeldetailsthanthoseof
= 3CfiδSS′e2cα|hf|r|ii|2E3γ (12) Γ(ψ(4040,4160)→χ′cJγ).
We calculate the E1 transition branching ratios of
where E is the emitted photon energy.
γ ψ(4040), ψ(4160), and ψ(4415) with their total width
The spatial matrix element
takenfromPDG(2010)[22]. Since the errorsofthe total
∞ widths are relatively small for these states, we only take
<f ri>= Rf(r)Ri(r)r3dr, (13) the central values of the total widths in calculating the
| |
Z0 branching ratios and do not consider the errors.
involves the initial and final state radial wave functions,
and the angular matrix element C is
fi
DISCUSSIONS ON XYZ STATES
′ ′ 2
L J S
′ ′
C =max(L, L)(2J +1) . (14)
fi
J L 1 In this section, we fucus on the implication of the re-
(cid:26) (cid:27)
′
sultsofBr(ψ(4040,4160,4415) χ γ)onsearchingfor
We are only interested in initial states with JPC = → cJ
XYZ states in these channels. The 3S-2D mixing effects
−−
1 , i.e., ψ(4040), ψ(4415) and ψ(4160), since they can
of ψ(4040) and ψ(4160) are also considered in details.
be easily produced in e+e− annihilation and can transit
into spin-triplet 2P states by emitting a photon.
Forthe massesofinitialandfinalstatesusedto calcu-
Z(3930)
lateE inabovethreemodels,wetakethecentralvalues
γ
from PDG(2010) [22] if the states are well established
′ The Z(3930) was found by the Belle Collaboration [9]
experimentally. The mass of χc0 is supposed to be 3915 in the process γγ DD¯ with
MeV, while for the mass of χ′ we choose two different →
c1
values: 3872 MeV and 3915 MeV. M(Z(3930)) = 3929 5 2 MeV, (15)
The calculated results with lowest-order wave func- ± ±
Γ(Z(3930)) = 29 10 2 MeV, (16)
tionsarelistedinTableI.TheresultsofBarnes,etal.[7], ± ±
Γ (Z(3930) DD¯) = 0.18 0.05 0.03 KeV,(17)
whicharesimilartotheModelIIandtheresultsofGod- γγ
B → ± ±
frey, et al. [8], with a relativized Cornell model, are also
and confirmed by the BaBar Collaboration [23] with
listed in Table I for comparison. From Table I, one can
′
see that the widths Γ(ψ(4415) χ γ), are very small
→ cJ M(Z(3930)) = 3926.7 2.7 1.1 MeV,(18)
due to large cancelation in the overlap integral between ± ±
wave functions of 4S and 2P states, and therefore are Γ(Z(3930)) = 21.3 6.8 3.6 MeV, (19)
± ±
very sensitive to the model details. On the other hand, Γ (Z(3930) DD¯) = 0.24 0.05 0.04 KeV.(20)
γγ
B → ± ±
4
TABLE I: E1 transition widths and branching ratios of charmonium states with the lowest-order wave functions in various
potentialmodels. ThemassesandtotalwidthsoftheinitialstatesusedinthecalculationarethePDG[22]centralvalues,while
the masses of the final states in the Model I-III calculations are denoted by the numbers in the parentheses. For comparison,
theresults of Refs. [7] and [8] are also listed.
Process <f|r|i> (GeV−1) k(MeV) Γ (keV) Br (×10−4)
thy thy
Intial Final I II III Ours Ref.[7] Ref.[8] I II III Ref.[7] Ref.[8] I II III Ref.[7] Ref.[8]
′
ψ(4040) χc2(3929) -4.9 -4.4 -3.4 109 67 119 74 59 36 14 48 9.3 7.4 4.5 1.8 6.0
′
χc1(3872) -4.9 -4.4 -3.4 164 113 145 151 122 74 39 43 18.9 15.3 9.3 4.9 5.4
′
χc1(3915) -4.9 -4.4 -3.4 122 113 145 63 51 31 39 43 7.9 6.4 3.9 4.9 5.4
′
χc0(3915) -4.9 -4.4 -3.4 122 184 180 21 17 10 54 22 2.6 2.1 1.3 6.8 2.8
′
ψ(4160) χc2(3929) 5.0 4.6 3.8 218 183 210 12 10 7.3 5.9 6.3 1.2 0.97 0.71 0.57 0.61
′
χc1(3872) 5.0 4.6 3.8 271 227 234 355 299 210 168 114 34.5 29.0 20.4 16.3 11.1
′
χc1(3915) 5.0 4.6 3.8 231 227 234 219 185 132 168 114 21.3 18.0 12.8 16.3 11.1
′
χc0(3915) 5.0 4.6 3.8 231 296 269 292 247 173 483 191 28.3 24.0 16.8 46.9 18.5
′
ψ(4415) χc2(3929) -0.013 0.093 -0.028 465 421 446 0.04 2.1 0.2 0.62 15 0.006 0.34 0.03 0.1 2.4
′
χc1(3872) -0.013 0.093 -0.028 515 423 469 0.04 1.7 0.2 0.49 0.92 0.006 0.27 0.03 0.08 0.15
′
χc1(3915) -0.013 0.093 -0.028 477 423 469 0.03 1.4 0.1 0.49 0.92 0.005 0.23 0.02 0.08 0.15
′
χc0(3915) -0.013 0.093 -0.028 477 527 502 0.01 0.45 0.04 0.24 0.39 0.002 0.07 0.006 0.04 0.06
TABLE II: E1 transition widths and branching ratios of charmonium states in various potential models with the first-order
wavefunctions. Themassesandtotalwidthsoftheinitial statesusedinthecalculation arethePDG[22]centralvalues,while
themasses of thefinal states in theModel I-IIIcalculations are denoted by thenumbersin the parentheses.
Process <f|r|i> (GeV−1) k(MeV) Γ (keV) Br (×10−4)
thy thy
′ ′ ′ ′ ′ ′ ′ ′ ′
Initial Final I II III I II III I II III
′
ψ(4040) χc2(3929) -4.3 -3.9 -3.1 109 56 47 29 7.0 5.9 3.6
′
χc1(3872) -3.7 -3.4 -2.8 164 88 72 50 11.0 9.0 6.3
′
χc1(3915) -3.7 -3.4 -2.8 122 37 30 21 4.6 3.8 2.6
′
χc0(3915) -3.0 -2.7 -2.5 122 7.9 6.2 5.3 0.99 0.78 0.66
′
ψ(4160) χc2(3929) 4.3 4.1 3.4 218 9.2 8.2 5.9 0.89 0.80 0.57
′
χc1(3872) 3.6 3.4 3.2 271 189 169 147 18.3 16.4 14.3
′
χc1(3915) 3.6 3.4 3.2 231 117 105 91 11.4 10.2 8.8
′
χc0(3915) 2.7 2.6 2.9 231 89 81 97 8.6 7.9 9.4
′
ψ(4415) χc2(3929) -0.42 -0.13 -0.19 465 42 4.1 8.7 6.8 0.66 1.4
′
χc1(3872) -1.1 -0.77 -0.39 515 219 116 30 35.3 18.7 4.8
′
χc1(3915) -1.1 -0.77 -0.39 477 174 92 24 28.1 14.8 3.9
′
χc0(3915) -1.8 -1.4 -0.65 477 164 105 22 26.5 16.9 3.5
The production rate and the angular distribution in From Table I and Table II, we can see the transition
′
theγγ center-of-massframesuggestthatthisstateisthe width of ψ(4040) χ γ is 36-74 KeV with the lowest-
′ → c2
previously unobserved χ [9, 23]. Its mass, however, is order wave functions and 29-56 KeV with the first-order
c2
about 40-50 MeV larger than the commonly predicted wave functions in our calculations within Models I-III
value in the quenched potential model (see, e.g. [7]). andinRef.[8]. Thecorrespondingbranchingratiois4.5-
A lower mass can be obtained by considering the color 9.3 10−4and3.6-7.0 10−4,respectively. Thebranching
screening effect described in Model I [15] in which the rat×io of order of 10−×4 is encouraging to detect χ′ in
c2
predicted mass is 3937 MeV. ψ(4040) E1 transitions. Note that the results of Ref. [7]
isnotablysmallthanourresults. Thisismainlybecause,
′
Since Z(3930) is established as the candidate of the mass of χ used in Ref. [7] is larger than ours and
′ c2
χ , searching for Z(3930) in the E1 transitions of the corresponding energy of the emitted photon is much
c2
ψ(4040,4160,4415)isimportanttofurtherverifythisas- smaller than ours.
signment and can also be a good criterion in searching
′ ′
for and identifying other χ states in these transitions. The branching ratio of ψ(4160) χ γ is about one-
cJ → c2
5
′
fifth of that of ψ(4040) χ γ but is still close to with masses
1 10−4. The branching→ratioc2of ψ(4415) χ′ γ is
× → c2
sensitive to the model details. So it is difficulty to pre-
′ M(X(3915)) = 3915 3 2 MeV, (24)
dict how large is the branching ratio of ψ(4415) χ γ ± ±
→ c2 M(Y(3940)) = 3943 11 13 MeV, (25)
in potential models.
± ±
M(Y(3915)) = 3914.6+3.8 2.0 MeV, (26)
−3.4±
X(3915),Y(3940),Y(3915)
total widths
The X(3915) [10], Y(3940) [12] and Y(3915) [13] are
allobservedin the invariancemass distribution of J/ψω Γ(X(3915)) = 17 10 3 MeV, (27)
in processes ± ±
Γ(Y(3940)) = 87 22 26 MeV, (28)
± ±
γγ X(3915) J/ψω, (21) Γ(Y(3915)) = 34+−182 5 MeV, (29)
→ → ±
B KY(3940), Y(3940) J/ψω, (22)
→ →
B KY(3915), Y(3915) J/ψω, (23) and partial widths
→ →
61 17 8 eV, JP =0+
Γγγ × B(X(3915)→J/ψω) = 18±5 ±2 eV, JP =2+,helicity-2 (30)
(cid:26) ± ±
(B KY(3940)) (Y(3940) J/ψω) =(7.1 1.3 3.1) 10−5, (31)
B → × B → ± ± ×
(B+ K+Y(3915)) (Y(3915) J/ψω) =(3.5 0.2 0.4) 10−4, (32)
B → × B → ± ± ×
(B0 K0Y(3915)) (Y(3915) J/ψω) =(3.1 0.6 0.3) 10−4. (33)
B → × B → ± ± ×
Although the differences of masses and total widths tential model predictions [7, 8], and they have positive
of the three signals are within 2σ, especially those of charge parity since they decay to J/ψω.
X(3915) and Y(3915) are less than one σ, it is not clear The X(3915), which is observed in γγ fusion, may be
whetherthese threesignalscomefromthe sameparticle. χ′ orχ′ . ButthemassofX(3915)isabout2σ different
c0 c2
BaBar [13] considers Y(3915) and Y(3940) as the same from that of Z(3930) and from (17) and (30). Further-
state since the smaller values of both the mass and to- more,if X(3915)is the same mesonas Z(3930),we would
tal width of Y(3915) derived from fitting data by BaBar get (χ′ J/ψω)/ (χ′ ) DD¯) 0.1 0.04, which
can partially be attributed to larger data sample used seemBstoc2b→e toolargeBforcch2ar→monium≈. SoX±(3915)isun-
by BaBar,whichenable them to use smaller J/ψω mass likely to be χ′ and we tend to regard it as a candidate
c2
bin in their analysis [24]. Y(3940) and X(3915) are also ′
for χ .
considered to be the same state in Refs. [10, 24–26]. c0
The Y(3915), which is produced in B decays, has so
Besides whether they are the same particle or not,
close mass and total width to the X(3915) that we sus-
therearenodecisiveinterpretationsofthesestates. That
pect they are the same state. However, if Y(3915) is
they are detected in the J/ψω channel but not in the ′
DD¯ or DD¯∗ channel makes people suspect they are notX(3915),thenitmaybe χc1. Since theE1transition
′
ratefromψ(4040)toχ isthreetofourtimeslargerthan
not conventional charmonium states. Ref. [10] argues c1
′
that X(3915) is not an excited charmonium state but that to χc0 with both lowest-order and first-order wave
favors the prediction of D∗D¯∗ bound state model [27]. functions if they have the same mass,we candistinguish
The Y(3915) and Y(3940) are also interpreted as D∗D¯∗ between them by measuring these E1 transitions.
′ ′
molecular states by [28, 29]. The Y(3940) is unlikely to be χ . If it is χ , its
However,Liuetal.[11]pointedoutthatthenodeeffect maindecaymodeoughttobeDD¯∗. Hc1owever,Bellec1gives
ofexcitedwavefunctionsmaychangetheopencharmde- (Y(3940) J/ψω)/ (Y(3940) D0D¯∗0) > 0.71 at
B → B →′
caywidthsdramatically. Theycalculatedtheopencharm 90% CL [30], which disfavor the χ assignment. But
decay of χ′cJ and′χ′c′J in the 3P0 model, and argued that Y(3940) might still be a candidate foc1r χ′c0.
X(3915) is the χc0 state. We may detect and identify them in the E1 transi-
If they are conventionalcharmonia, they are probably tions of higher charmonium states. We can see from
′
the χ states since their masses are consistent with po- Table I and Table II that the transition widths are
cJ
6
′
O(10) KeV for ψ(4040) χ γ and O(100) KeV for We are interested here in detecting X(3872) in the E1
→ c0,1
ψ(4160) χ′ γ, corresponding to branching ratios of transitionsofhighercharmonia,ifX(3872)isthe2Pchar-
order 10−→4 ancd0,110−3, in all listed potential models. The moniumχ′c1orcontainssome2Pcharmoniumcomponent
in its wave function. One can see from Table I that the
effect of corrections of wave functions is moderate for
branchingratioforψ(4040) χ′ γis(0.93 1.89) 10−3
ψra(t4io0s40m)aaynmdaψk(e41th6e0)detrtaecntsiiotnionosf.χ′cT0′,h1epolasrsgibeleb.ranching aonudrctahlcautlfaotrioψn(w41it6h0z)e→ro-χor′c1d→eγriwsca1(v2e.0f4un−ct3io.0−n5s).×Th1×e0−la3rgine
The transitions of ψ(4415) → χc0,1γ are model de- branching ratios may enable us to find χ′ in the e+e−
pendent and sensitive to the node structure, just like c1
′ machinesandcompareitwithX(3872). NotethatRef.[7]
ψ(4415) χ γ as we have remarked.
→ c2 andRef.[8]givesmallerbranchingratios. Itispartlybe-
′
cause they take a larger mass for χ in calculations. If
c1
′
X(3872) they take the mass of χc1 to be the same as us, the dif-
ferences will diminish. This means the branching ratios
The X(3872) was first observed by Belle [17] in are not sensitive to models. The calculated results with
the J/ψ π+π− invariant mass distribution in B+ relativisticallycorrectedwavefunctions area bit smaller
K+J/ψ π+π− decay as a very narrow peak (Γ < 2→.3 but still quite large (see Table II). It indicates that the
X
results are not sensitive to the nodes of wave functions
MeV) around 3872 MeV. The mass of X(3872) in the
J/ψ π+π− mode was recently updated by CDF Collab- and should be reliable.
oration [31] as
M(X(3872))=3871.61 0.16 0.19 MeV, (34) 3S-2D Mixing
± ±
whichisveryclosetothe D0D¯∗0 thresholdm(D0D¯∗0)= The ψ(4040) and ψ(4160) are commonly assigned as
3871.81 0.36 MeV [32]. Moreover, analyzes both by ψ(33S1) and ψ(23D1) respectively. Therefore, the above
±
Belle [33] and CDF [34] favor the quantum number results of E1 transitions are all based on these sim-
JPC = 1++. The mass seems to be too small for a ple assignments. However, the observed leptonic width
JPC =1++ χ′ charmonium, but the color-screeningef- Γee(4040) Γee(4160) is inconsistent with this picture.
c1 ≈
fects and the coupled channel effects may lower its mass The simplest explanation is that they are roughly 1:1
towards3872Mev[4,15],ashasbeendeclaredinSection mixtures of ψ(33S1) and ψ(23D1). Neither the ten-
I. sor force nor the coupled channel effects can cause such
There are lots of possible explanations for strong mixing (see Ref. [44] and references therein) and
X(3872)(see [15, 24] for a review). Aside from the themixingmechanismremainsunknown. Herewedonot
most popular one, i.e., the D0D¯∗0 molecular state, considerthe mixing mechanism, andsimply assume that
the 1++ charmonium [15, 35, 36] or a mixed 1++ they are mixtures of ψ(33S1) and ψ(23D1) with a mix-
charmonium-D0D¯∗0 state [37, 38] for X(3872) was also ing angle θ, and calculate how the E1 transition widths
proposed. More data samples are needed to distinguish vary with θ. In this case, the ψ(4040) and ψ(4160) are
between various explanations. expressed as
Recently, an analysis of the ω π+π−π0 spectrum in
the decayB KX KJ/ψω p→erformedbyBaBar[39] ψ(4040) = 33S1 cosθ+ 23D1 sinθ, (35)
claimedthat→the JPC→ofX(3872)mightdisfavor1++, as |ψ(4160)i=| 33Si1 sinθ+| 23Di1 cosθ. (36)
had widely been accepted, but favor 2−+. If this result | i −| i | i
is confirmed, the natural assignment is the 1D2 char- Using the data of leptonic decay widths of ψ(4040) and
monium ηc2. However, the mass of this D-wave state is ψ(4160) [22] as inputs, one can determine the mixing
about3.80-3.84GeVinthepotentialmodels[6–8],which angle θ in the potential models. It is about 35o or
−
istoolowtobethe candidateofX(3872). Besides,some +55o in all the three models used in Sec. II. And this is
recenttheoreticalstudiesonthepropertiesofηc2indicate consistent with earlier estimates of the mixing angle[45,
thatitisnotapttothethecandidateofX(3872)[40–43]. 46]
Therefore, we will ignore this possibility in the following ThecorrespondingE1transitionwidthsparameterized
analysis. by the mixing angle are
7
Γ(ψ(4040)→χ′c0γ) = 247αe2ck3(cos2θ <23P0|r|33S1 >2 −2√2cosθsinθ <23P0|r|33S1 ><23P0|r|23D1 >
+2sin2θ <23P0 r23D1 >2), (37)
| |
Γ(ψ(4040)→χ′c1γ) = 94αe2ck3(cos2θ <23P1|r|33S1 >2 +√2cosθsinθ <23P1|r|33S1 ><23P1|r|23D1 >
1
+ sin2θ <23P1 r23D1 >2), (38)
2 | |
Γ(ψ(4040)→χ′c2γ) = 2207αe2ck3(cos2θ <23P2|r|33S1 >2 −√52cosθsinθ <23P2|r|33S1 ><23P2|r|23D1 >
1
+ sin2θ <23P2 r23D1 >2), (39)
50 | |
Γ(ψ(4160)→χ′c0γ) = 247αe2ck3(2cos2θ <23P0|r|23D1 >2 +2√2cosθsinθ <23P0|r|23D1 ><23P0|r|33S1 >
+sin2θ <23P0 r33S1 >2, (40)
| |
Γ(ψ(4160)→χ′c1γ) = 49αe2ck3(12cos2θ <23P1|r|23D1 >2 −√2cosθsinθ <23P1|r|23D1 ><23P1|r|33S1 >
+sin2θ <23P1 r33S1 >2), (41)
| |
Γ(ψ(4160)→χ′c2γ) = 2207αe2ck3(510cos2θ <23P2|r|23D1 >2 +√52cosθsinθ <23P2|r|23D1 ><23P2|r|33S1 >
+sin2θ <23P2 r33S1 >2), (42)
| |
We use the lowest-order wave functions calculated in one of the two channels must be enhanced by the 3S-2D
model I-III and impose mχ′ = 3929MeV, mχ′ = mixing if the mixing mechanism just affect the leptonic
c2 c1
3872MeVandmχ′ =3915MeV. Theresultsaresimilar decaysandtheE1transitionsbysimplymixingthewave
c0 functions.
in the three models, i.e., the E1 transition widths reach
′
The decay width of ψ(4040) χ (3915)γ reaches its
their maximum or minimum values almost at the same → c0
maximum of about 52 KeV corresponding to a branch-
mixing angle in the three models. As an example, we
ing ratio of 6.5 10−4 near 55o, which is about 2.5
display the results in Model I and Model II in Figs. 1-
×
times larger than that of non-mixing and reaches its
4, and do not show the similar results in Model III for
minimum (zero) near 35o. While the decay width of
simplicity.
′ −
ψ(4160) χ (3915)γ reaches its minimum (zero) near
We see that the decaywidth ofψ(4040)→χ′c2(3929)γ 55o and →reachc0es its maximum of about 450 KeV corre-
reachesitsmaximumofabout75KeVcorrespondingtoa sponding to a branching ratio of about 4.5 10−3 near
branchingratioof9.3 10−4nearθ =10o. Whilethede- 35o. ×
caywidthofψ(4160) × χ′ (3929)γ reachesitsminimum − ′
(zero)near10o andit→s macx2imumofabout600KeVnear Since the decay width of ψ(4160) → χc′2(3929)γ
reaches its minimum (zero) and ψ(4040) χ (3929)γ
−co8r0reosaptonwdhiincghbψr(a4n1c6h0i)ngisraaltmioosistaabpouurte6ψ(313S0−1)3.andthe reachesits maximum near 100, which is no→t farc2from the
× non-mixing case, in our model calculations, we may use
′
The decay width of ψ(4040) χ (3872)γ reaches its these two channels to check whether there is substantial
→ c1 ′
maximum of about 250 KeV corresponding to a branch- 3S-2Dmixingbetweenψ(4040)andψ(4160). Ifχ (3929)
ing ratio of 3.1 10−3 near 35o, which is about 2 can be detected in the E1 transitions from ψ(40c420) but
× −
times larger than that of non-mixing, and reaches its notfromψ(4160),thenthemixingangleshouldbesmall.
minimum (zero) near 55o. While the decay width of Ingeneral,sincetheZ(3930)isidentifiedwithχ′ ,theob-
′ c2
ψ(4160) χ (3872)γ reaches its minimum (zero) near served E1 transition rates to Z(3930) from ψ(4040) and
→ c1
35o and reaches its maximum of about 1050 KeV cor- ψ(4160) will be useful to constrain the value of 3S-2D
−
respondingtoabranchingratioofabout1%near55o. It mixing angle by comparing the measurements and the
is very interesting to note that the two values happen to theoretical predictions. When we have a better control
becorrespondingtothemixinganglesdeterminedbylep- over the value of mixing angle, we will be in a position
tonicdecaywidthsofψ(4040)andψ(4160). Thatmeans, to study the properties of X(3872) and X(3915), and to
8

250 1100
1000
V) 200 V) 900
e B e
h(K h(K 800 B
Widt 150 Widt 700
on on 600 A
nsiti nsiti 500
1 Tra 100 C 1 Tra 400 C
E E
A 300
50 200
100
0 0
-80 -60 -40 -20 0 20 40 60 80 -80 -60 -40 -20 0 20 40 60 80
Mixing Angle Mixing Angle
FIG. 1: E1 transition widths of ψ(4040) varying with 3S- FIG. 3: E1 transition widths of ψ(4160) varying with 3S-
′ ′
2D mixing angle in Model I. A: ψ(4040) → χc2(3929)γ, B: 2D mixing angle in Model I. A: ψ(4160) → χc2(3929)γ, B:
′ ′ ′ ′
ψ(4040)→χc1(3872)γ, C: ψ(4040)→χc0(3915)γ. ψ(4160)→χc1(3872)γ, C: ψ(4160)→χc0(3915)γ.

900
200 V) 800
V) Ke 700
dth(Ke 150 B Width( 600 B
Wi n
ansition 100 Transitio 450000 A C
1 Tr E1 300
E
C
50 A 200
100
0 0
-80 -60 -40 -20 0 20 40 60 80 -80 -60 -40 -20 0 20 40 60 80
Mixing Angle Mixing Angle
FIG. 2: E1 transition widths of ψ(4040) varying with 3S- FIG. 4: E1 transition widths of ψ(4160) varying with 3S-
′ ′
2D mixing angle in Model II. A: ψ(4040) → χc2(3929)γ, B: 2D mixing angle in Model II. A: ψ(4160) → χc2(3929)γ, B:
′ ′ ′ ′
ψ(4040)→χc1(3872)γ, C: ψ(4040)→χc0(3915)γ. ψ(4160)→χc1(3872)γ, C: ψ(4160)→χc0(3915)γ.
′
see if they can be respectively the χ (or partially be) for these X,Y,Z states by measuring the E1 transition
and χ′ by comparing the observed cE11 transition rates rates of ψ(4040) and ψ(4160) to χ′ (J=2,1,0) charmo-
c0 cJ
with theoreticalpredictions for ψ(4040)and ψ(4160)de- nium states.
′
cays to χ γ. However, we must keep in mind that the We also calculate the E1 transition widths of
c1,0
3S-2D mixing model for ψ(4040) and ψ(4160) is only a ψ(4040,4160) χ (1P)γ in order to see whether these
→ cJ
simplificationfortherealsituation,sincethemixingwith can be helpful in determining the 3S-2D mixing angle.
4S state and the charm meson pairs are all neglected in The results are listed in Table III. Although the calcu-
the 3S-2D mixing model. Nevertheless, with this simple latedwidths areexpectedly small,the transitionbranch-
model we hope some useful information can be obtained ingratioofχ J/ψγ arerelativelylarge(about20%
c1,2 →
9
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10
TABLEIII:E1transitionwidthsandbranchingratiosofcharmoniumstatesinvariouspotentialmodelswithlowest-orderwave
functions. The masses and total widths of theinitial and final states used in thecalculation are the PDG[22] central values.
Process <f|r|i> (GeV−1) k(MeV) Γ (keV) Br (×10−4)
thy thy
Intial Final I II III Ours Ref.[7] Ref.[8] I II III Ref.[7] Ref.[8] I II III Ref.[7] Ref.[8]
ψ(4040) χC2(3556) 0.026 0.078 -0.011 454 455 508 0.16 1.4 0.03 0.70 12.7 0.02 0.18 0.004 0.09 1.6
χC1(3511) 0.026 0.078 -0.011 493 494 547 0.12 1.1 0.02 0.53 0.85 0.02 0.14 0.003 0.07 0.11
χC0(3415) 0.026 0.078 -0.011 576 577 628 0.06 0.6 0.01 0.27 0.63 0.01 0.08 0.001 0.03 0.08
ψ(4160) χC2(3556) 0.43 0.34 0.23 554 559 590 1.5 0.9 0.4 0.79 0.027 0.15 0.09 0.04 0.08 0.003
χC1(3511) 0.43 0.34 0.23 592 598 628 28 17 7.9 14 3.4 2.7 1.7 0.77 1.4 0.33
χC0(3415) 0.43 0.34 0.23 672 677 707 55 33 15 27 35 5.3 3.2 1.5 2.6 3.4
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