Remarks on the quantum numbers of X(3872) from the invariant mass distributions of the rho J/psi and omega J/psi final states

0.14 MB·English

by  C. Hanhart

Remarks on the quantum numbers of X(3872) from the invariant mass distributions of
the ρJ/ψ and ωJ/ψ final states
C. Hanhart
Forschungszentrum Ju¨lich, Institute for Advanced Simulation,
Institut fu¨r Kernphysik (Theorie) and Ju¨lich Center for Hadron Physics, D-52425 Ju¨lich, Germany
Yu. S. Kalashnikova, A. E. Kudryavtsev, and A. V. Nefediev
Institute for Theoretical and Experimental Physics,
117218, B.Cheremushkinskaya 25, Moscow, Russia
We re-analyse the two- and three-pion mass distributions in the decays X(3872) → ρJ/ψ and
X(3872) → ωJ/ψ and argue that the present data favour the 1++ assignment for the quantum
numbersof theX.
2
1
PACSnumbers: 14.40.Pq,13.25.Gv,14.40.Rt
0
2
n I. INTRODUCTION 0 =[4.3 1.2(stat) 0.4(syst)] 10−6,
B2π ± ± ×
a
J for the charged B+ J/ψρK+ and neutral B0
Since its discovery in 2003 [1], the X(3872) charmo- J/ψρK0 mode, respect→ively, with being the produ→ct
5 2π
nium issubject ofmanyexperimentalandtheoreticalef- branchingfractionBr(B KX(38B72)) Br(X(3872)
] forts aimed at disclosing its nature—for a recent review π+π−J/ψ) in the corresp→onding mode.×The number→of
h see [2]. The most recentdata onthe mass of the X is [3] events in the background-subtracted combined distribu-
p
tion is
- MX =(3871.85 0.27(stat) 0.19(syst)) MeV, (1)
p ± ±
e with a width of ΓX < 1.2 MeV. However, the problem N2π =196.0+−1185..92.
h of the quantum numbers for the X is not fully resolved
[ yet: while the analysis of the π+π−J/ψ decay mode of For the decay X(3872)→π+π−π0J/ψ Babar reports[5]
v2 [t3h,e4X],(t3h8e7r2e)ceynietldasnaeliythsiesro1f+t+heoπr+2π−−+πq0uJa/nψtummodneumsebeemrss B30+π ==[[00..66±00..32((ssttaatt))±00..11((ssyysstt))]]×1100−−55,,
1 tofavourthe 2−+ assignment[5],thoughthe 1++ option B3π ± ± ×
4 is not excluded. This question is clearly very central, for the charged mode B+ J/ψωK+ and for the neu-
2 tral mode B0 J/ψωK→0, respectively. Similarly to
for the most promising explanations for the X in the S-
6 wave DD¯∗ molecule model [6] as well as in the coupled- the two-pion ca→se above, 3π stands for the product
. B
1 channelmodel [7] requirethe quantum numbers 1++. In branchingfractionBr(B KX(3872)) Br(X(3872)
1 addition, the X cannot be a naive cc¯2−+ state, for its π+π−π0J/ψ) in the corr→esponding mod×e. The numb→er
1 large branching fraction for the D0D¯0π0 mode [8] is not of events in the combined distribution is
1
: compatible with the quark-model estimates for the 2−+ Nsig+bg =34.0 6.6, Nbg =8.9 1.0, (2)
v charmonium[9]. So,forthe2−+ quantumnumbers,very 3π ± 3π ±
Xi exotic explanations for the X would have to be invoked. andweassumeaflatbackground. Note,thespectrumre-
Theaimofthepresentpaperistoperformacombined portedin [5]and usedbelow appearsnot to be efficiency
r
a analysis of the data on the π+π− and π+π−π0 mass dis- corrected. However, since only the shape of this spec-
tribution in the π+π−J/ψ and π+π−π0J/ψ mode, re- trum plays a role for the analysis (see number-of-event
spectively. We findthatthe S-waveamplitudes fromthe distributions (11) below) and we can reproduce the the-
decay of a 1++ state provide a better overalldescription oretical spectra of [5], which have the efficiency of the
of the data than the P-wave ones from the 2−+, espe- detectorconvolutedinviaaMonteCarlosimulation,the
cially when the parameter range is restricted to realistic invariantmassdependenceoftheefficiencycorrectionsis
values. We conclude then that the existing data favour expected to be mild and therefore should not affect our
1++ quantumnumbersoftheX,however,improveddata analysis significantly.
in the π+π−π0J/ψ mode are necessary to allow for defi- Thus the updated ratio of branchings reads [3]
nite conclusions regarding the X quantum numbers.
Br(X(3872) π+π−π0J/ψ)
3π
B = → =0.8 0.3. (3)
Br(X(3872) π+π−J/ψ) ±
2π
B →
II. EXPERIMENTAL SITUATION
In our analysis we use the ratio (3) as well as
RecentlyBelleannounced[3]theupdatedresultsofthe N2π =196, N3π =25.1, (4)
measurements for the reaction X(3872) π+π−J/ψ:
→ and he corresponding bin sizes are ∆E2π =20 MeV and
+ =[8.61 0.82(stat) 0.52(syst)] 10−6, ∆E =7.4 MeV.
B2π ± ± × 3π
2
III. THEORETICAL π+π− AND π+π−π0 The transition amplitudes for the decays X J/ψV
→
INVARIANT MASS DISTRIBUTIONS are parameterised in the standard way, namely,
A =g f (p), (8)
As in previous analyses, we assume that the two-pion X→J/ψV XV lX
finalstateismediatedbytheρintheintermediatestate, with the Blatt-Weisskopf “barrier factor”
while the three-pion final state is mediated by the ω. It
was shown in [4] that the description of the π+π−J/ψ f0X(p)=1, and f1X(p)=(1+r2p2)−1/2, (9)
spectrum with the 2−+ assumption is improved dras- for the 1++ and 2−+ assignment, respectively. Here p
tically if the isospin-violating ρ-ω mixing is taken into
denotes the J/ψ momentum in the X rest frame. The
account. Theoretical issues of the ρ-ω mixing are dis-
“radius” r is not well understood. If one associates it
cussed, for example, in [10]. Here we include this effect
with the size of the X ρ(ω)J/ψ vertex, it might be
with the help of the prescription used in [11], where the →
related to the range of force. In the quark model this
transition amplitude is described by the real parameter radius is r 0.2 fm = 1 GeV−1. This is also in line
A =A† =ǫ. Thus, the amplitudes for the decays ∼
ω→ρ ρ→ω with the inverse mass of the lightest exchange particle
X π+π−J/ψ and X π+π−π0J/ψ take the form
allowed between J/ψ and ρ/ω, namely, f (980). On the
→ → 0
other hand, a larger value r = 5 GeV−1 is used in the
A =A G A
2π X→J/ψρ ρ ρ→2π
experimental analysis of [3]. Therefore, in the analysis
+A G ǫG A ,
X→J/ψω ω ρ ρ→2π presentedbelow we use both values r =1 GeV−1 as well
A3π =AX→J/ψωGωAω→3π as r = 5 GeV−1, keeping in mind that smaller values of
+A G ǫG A , r are preferred by phenomenology.
X→J/ψρ ρ ω ω→3π
The theoretical invariant mass distributions for the
where the vector meson propagators are π+π− and π+π−π0 final state take the form:
G−V1 =m2V −m2−imVΓV(m), V =ρ,ω, (5) dBdmr2π =BmρΓρ→2πp2l+1fl2X(p)|RXGρ+ǫGρGω|2,
with m = m (m = m ) being the π+π− (π+π−π0) (10)
ππ πππ dBr
invariantmassintheπ+π−J/ψ(π+π−π0J/ψ)finalstate. 3π = m Γ p2l+1f2 (p) G +ǫR G G 2,
dm B ω ω→3π lX | ω X ω ρ|
Masses of the vector mesons used below are [12]
where R =g /g and the parameter absorbs the
X Xρ Xω
mρ =775.5 MeV, mω =782.65 MeV. detailsoftheshort-rangeddynamicsoftheXB production.
The theoretical number-of-event distributions read
The complex mixing amplitude multiplying the ω prop-
agator used, for example, in [4] to analyse the two-pion N2π∆E2π dBr2π
N (m)= ,
2π
spectrum, in our notation reads as ǫGρ(mω); in partic- B2π × dm (11)
ular we reproduce naturally the phase of 95o quoted in N ∆E dBr
3π 3π 3π
[4]. Note, as we shall only study the invariant mass dis- N3π(m)= .
B × dm
tributions ofthe two final states,we do not needto keep 3π
explicitly the vector nature of the intermediate states. The ρ-ω mixing parameter ǫ is extracted from the
For the “running” ρ meson width we use ω 2π decay width (Br(ω 2π)=1.53%). The corre-
→ →
sponding amplitude reads
2
m q(m) f (q(m))
Γ (m) Γ (m)=Γ(0) ρ 1ρ , A =εG A , (12)
ρ ≈ ρ→2π ρ mq(m ) f (q(m )) ω→2π ρ ρ→2π
ρ (cid:20) 1ρ ρ (cid:21)
and we find that
where q(m) = m2 4m2/2, f (q) = (1+r2q2)−1/2,
with r = 1.5 GeV−−1 andπ with1ρthe nominalρρ meson ǫ mωmρΓρΓω→2π 3.4 10−3 GeV2. (13)
ρ p ≈ ≈ ×
width Γρ(0) =146.2 MeV. IV. pFITTING STRATEGY AND RESULTS
For the ω meson “running” width (the nominal width
(0)
being Γ =8.49MeV),the 3π andπγ decaymodesare
ω The number-of-event distributions (11) possess 3 free
summed, with the branchings
parameters: the “barrier”factorr, the ratioofcouplings
Br(ω 3π)=89.1%, Br(ω πγ)=8.28%. (6) RX and the overall normalisation parameter . As out-
→ → lined above, we perform the analysis for twoBvalues of
In particular, r, namely, the preferred value of 1 GeV−1 and a signifi-
cantly larger value of 5 GeV−1 used in earlier analyses.
Γω→πγ(m)=Γω(0→)πγ mmω((mm22−mm22π)) 3, (7) Sofintcheethtweonobrrmanaclihsiantgios,nwfaecteoxrtrBacdtrothpesorauttiofroRmthdeireracttliyo
(cid:20) ω − π (cid:21) X
from the integrated data, that is from the relation
while,fortheΓ (m),weresorttotheexpressionsde-
ω→3π
rivedin[13],withareducedcontacttermwhichprovides dBr3π dBr2π
dm dm = / , (14)
thecorrectnominalvalueoftheω 3πdecaywidth[14]. dm dm B3π B2π
→ (cid:18)Z (cid:19)(cid:30)(cid:18)Z (cid:19)
3
TABLE I. Sets of parameters and the quality of the combined fits to the data. The last column refers to the presentation of
thedifferent fits in Fig. 1.
Fit Wave JPC r, GeV−1 RX =gXρ/gXω χ2/Ndof CL Curve/Markers in Fig. 1
S S 1++ − 0.26+0.08 1.07 37% Solid (red)/Triangles
−0.05
P5 P 2−+ 5 0.15+−00..0053 1.33 14% Dash-dotted (blue)/Squares
P1 P 2−+ 1 0.09+−00..0032 2.77 10−5 Dashed (green)/Diamonds
where the value of the ratio on the right-hand side is besignificantmeson-loopcontributions[15],theestimate
fixed by Eq. (3), and in the integration above we have (15) is to be regardedas a conservative upper bound for
cut off the π+π− invariant mass at 400 MeV, as in [3], the isospin violation strength for compact charmonia.
and the π+π−π0 invariant mass at 740 MeV, as in [5]. Incontrasttothis,intheS-wavemolecularpicturefor
Thereforethenorm isouronlyfittingparameterwhich the X, isospin violation is enhanced significantly com-
B
governs the overall strength of the signal in both chan- paredtothestrength(15)foritproceedsviaintermediate
nelssimultaneously,whiletheshapeofthecurvesisfully DD¯∗ statesandisthereforedrivenbythemassdifference
determined from other sources. ∆ 8MeVoftheneutralandchargedDD¯∗ threshold—
≈
InTableI,welisttheparametersofthe3combinedfits see, for example, [16, 17]. An order-of-magnitude esti-
tothedata,foundforthe2valuesoftheBlatt-Weisskopf mate is provided by the expression
parameter r. The corresponding line shapes and the re-
I (M ) I (M ) √m ∆
sult of the integration in bins are shown in Fig. 1. Rmol 0 X − c X D 0.13, (16)
One can see from Table I and Fig. 1 that the best X ∼(cid:12)I0(MX)+Ic(MX)(cid:12)∼ β ∼
overall description of the data for the two channels un- (cid:12) (cid:12)
where m (cid:12) is the D meson (cid:12)mass, while I (M ) and
der consideration is provided by the S-wave fit. The D(cid:12) (cid:12) 0 X
I (M ) denote the amplitudes corresponding to loop
P-wave fit is capable to provide the description of the c X
diagrams with neutral and charged DD¯∗ intermediate
data of a comparable (howeversomewhat lower)quality,
states, respectively, evaluated at the X mass. They are
only for largevalues of the Blatt-Weisskopfparameter r,
r =5 GeV−1. The P-wave fit becomes poorer when the composed of two terms, the strongly channel-dependent
analytic continuation of the unitarity cut, proportional
Blatt-Weisskopf parameter is decreased, and for values
of r of order 1 GeV−1, the quality of the P-wave fit is to the typical momentum of the meson pair, and the
weakly channel-dependent principle value term, whose
unsatisfactory,whichis theresultofaverypoordescrip-
size is identified with the inverse range of forces of order
tion of the two-pion spectrum—see the dashed (green)
of 1 GeV (see above). This estimate is within a factor
curvein Fig.1. Varyingthe ratio ofbranchings /
3π 2π
B B of 2 consistentwith the value R 0.26found from our
around its central value within the experimental uncer- X ∼
S-wave fit—see Table I.
tainty interval [see Eq. (3)] leads only to minor changes
Ontheotherhand,iftheX hasthequantumnumbers
in the fits and does not affect the conclusions.
2−+, one should expect the isospin violation in the X
wavefunction to be ofthe naturalcharmoniumsize,and
thus ofthe orderof10−2—see discussionbelowEq.(15),
V. DISCUSSION
since the DD¯∗ loop effects are suppressed by the ad-
ditional centrifugal barrier: the estimate analogous to
SincenochargedpartnersoftheX(3872)areobserved
Eq. (16) now reads Rmol (√M ∆/β)3 2 10−3.
experimentally, it is supposed to be (predominantly) an Thus, for the state withXth∼e quantuDm numb∼ers 2×−+, one
isoscalar. Then the ratio R = g /g measures the
X Xρ Xω expects values of at most R 10−2, that are signifi-
strength of the isospin violation in the X VJ/ψ de- X ∼
→ cantlysmallerthanthosefollowingfromthedata(seeTa-
cay vertex. As discussed above, this ratio is extracted
bleI).OneisledtoconcludethereforethatforR &0.1,
directlyfromthe dataonthe ratioofthe branchings(3). X
needed for the quantum numbers 2−+ to be consistent
Anisospin-violatingobservableforacompactcharmo-
with the data on the X decays, a new, yet unknown,
niumistheratioofthebranchingfractionsfortheψ(2S)
isospin violation mechanism would have to be invoked.
decays into ηJ/ψ and π0J/ψ final states as
gπ0J/ψ Br(ψ(2S) π0J/ψ) kη3 VI. CONCLUSIONS
R = = → 0.03,
ψ(2S) gηJ/ψ s Br(ψ(2S)→ηJ/ψ) kπ30 ≈
(15) Weconcludethereforethat,althoughthepresentqual-
where kπ0 and kη are the center-of-mass momenta of π0 ity of the data in the X π+π−π0J/ψ channel is not
→
andη,respectively,andtheψ(2S)branchingfractionsare sufficient to draw a definite conclusion concerning the
taken from [12]. However, since here also the denomina- quantum numbers of the X(3872), the combined anal-
torviolatesasymmetry,namely,SU(3), andtheremight ysis of the existing two- and three-pion spectra favours
4
40
V V
Me Me
30
(cid:144)nts20 òà òà òà (cid:144)nts7.410 ò
ve20 à ve ò
e ò ò e à à
berof ìà òìà òìà ì ì ì ì berof 5ìà ìà òì ì
Num10òìà òìà òìà òìà òìà òìà òìà òìà òìà òìà òìà ò ìà Num ò ò ìà
500 600 700 800 745 750 755 760 765 770 775
EHMeVL EHMeVL
FIG. 1. Comparison of the theoretical predictions with the data. Experimental data, taken from [3, 5] for the two-pion (left
plot) andthree-pion (right plot) spectrum,respectively,aregiven byblack circles with errorbars, while thetheoretical results
areshownbothasthecontinuousnumber-of-eventdistributions(11)aswellasbymarkersforthecorrespondingintegralsover
thebins. Fit S isshown asthe(red)solid lineandtriangles, fitP1 isshown asthe(green)dashedlineanddiamonds,while fit
P5 is shown as the(blue) dash-dotted line and squares. See Table I for thedetails.
the S-wave fit, related to the 1++ assignment for the ACKNOWLEDGMENTS
X(3872),over the P-wave fit, related to the 2−+ assign-
ment. WenoticethatanacceptableP-wavefitcallsfora WeacknowledgeusefuldiscussionswithE.Braaten,S.
largerangeparameterintheBlatt-Weisskopfformfactor Eidelman,F.-K.Guo,andR.Mizuk. The workwassup-
which meets certain difficulties with its phenomenolog- ported in parts by funds provided from the Helmholtz
ical interpretation. In addition, while the value RX = Association (Grant Nos VH-NG-222 and VH-VI-231),
gXρ/gXω canbeunderstoodtheoreticallyforthe1++ as- by the DFG (Grant Nos SFB/TR 16 and 436 RUS
signment, the value extracted for the 2−+ assignment is 113/991/0-1),bytheEUHadronPhysics2project,bythe
too largeto be explainedfromknownmechanisms ofthe RFFI (Grant Nos RFFI-09-02-91342-NNIOa and RFFI-
isospin violation. 09-02-00629a),and by the State Corporation of Russian
Federation “Rosatom.”
[1] S.-K. Choi et al. (Belle Collaboration), Phys. Rev. Lett. [10] Ulf-G. Meißner, Phys. Rept. 161, 213 (1988); A. Ku-
91, 262001 (2003). cukarslan, Ulf-G. Meißner, Mod. Phys. Lett. A 21, 1423
[2] N.Brambilla et al.,Eur. Phys.J. C 71, 1534 (2011). (2006); Ya.I. Azimov, Eur. Phys.J. A 16, 209 (2003).
[3] S.-K.Choietal.(BelleCollaboration), Phys.Rev.D84, [11] J. F. de Troconiz, F. J. Yndurain, Phys. Rev. D 65 ,
052004 (2011). 093001 (2002).
[4] A. Abulencia et al. (CDF Collaboration), Phys. Rev. [12] K.Nakamuraetal.[ParticleDataGroup],J.Phys.G37,
Lett.98, 132002 (2007). 075021 (2010).
[5] P.delAmoSanchezetal.(BABARCollaboration),Phys. [13] E. A. Kuraev and Z. K. Silagadze, Yad. Fiz. 58, 1687
Rev.D 82, 011101(R) (2010). (1995) [Phys. At.Nucl. 58, 1589 (1995)].
[6] M. B. Voloshin and L. B. Okun, Pisma Zh. Eksp. Teor. [14] O.Kaymakcalan,S.Rajeev,andJ.Schechter,Phys.Rev.
Fiz. 23, 369 (1976) [JETP Lett. 23, 333 (1976)]; N. A. D 30, 594 (1984).
Tornqvist, Phys. Rev. Lett. 67, 556 (1991); N. A. Torn- [15] F.-K. Guo, C. Hanhart, and Ulf-G. Meißner, Phys. Rev.
qvist, Phys. Lett. B 590, 209 (2004); E. S. Swanson, Lett.103,082003 (2009); F.-K.Guo,C.Hanhart,G.Li,
Phys.Lett. B 588, 189 (2004). Ulf-G. Meißner, and Q. Zhao, Phys. Rev. D 83, 034013
[7] Yu.S.Kalashnikova,Phys.Rev.D72, 034010 (2005); I. (2011).
V. Danilkin and Yu. A. Simonov, Phys. Rev. Lett. 105, [16] M. Suzuki,Phys.Rev.D 72, 114013 (2005).
102002 (2010). [17] D. Gamermann and E. Oset, Phys. Rev. D 80, 014003
[8] G.Gokhrooetal.(BelleCollaboration),Phys.Rev.Lett. (2009); D.Gamermann, J. Nieves,E.Oset, and E.R. Ar-
97,162002 (2006); B.Aubertet al.(BABARCollabora- riola, Phys.Rev. D 81, 014029 (2010).
tion), Phys. Rev. D 77, 011102 (2008); T. Aushev et al.
(Belle Collaboration), Phys. Rev.D 81, 031103 (2010).
[9] Yu. S. Kalashnikova and A. V. Nefediev, Phys. Rev. D
82, 097502 (2010).