YITP-11-68 KANAZAWA-11-12 X(3872) and Its Iso-Triplet Partners Kunihiko Terasaki Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan Institute for Theoretical Physics, Kanazawa University, Kanazawa 920-1192, Japan DecaysofX(3872) anditspartnersashidden-charmaxial-vectortetra-quarkmesonsarestudied. Astheresult, it isseen thattheiso-triplet partnersofX(3872) can bebroad,andtherefore, higher statistics will beneeded tofind them. Tetra-quark mesons are classified into the following four groups qqq¯q¯ =[qq][q¯q¯] (qq)(q¯q¯) [qq](q¯q¯) (qq)[q¯q¯] , (q =u,d,s,c) (1) 2 { } ⊕ ⊕{ ⊕ } 1 in a quark model [1] as in the MIT bag model [2], where parentheses and square brackets denote symmetry and 0 anti-symmetry,respectively,offlavorwavefunctionsunderexchangeofflavorsbetweenthem. Eachtermintheright- 2 hand-side of Eq. (1) is again classified into two groups [2] with ¯3c×3c and 6cׯ6c of the color SUc(3). However, n the former is expected to be lower in the case of heavy mesons, because forces [3] between two quarks are attractive a when they are of ¯3c, while repulsive when 6c, and because a possible mixing between these two states is expected J to be small at the scale of heavy meson mass [4]. Regarding with the spin (J), the [qq] and (qq) have J = 0 and 1, 0 respectively,andhencethespinandparityJP of[qq][q¯q¯]and [qq](q¯q¯) (qq)[q¯q¯] areJP =0+ and1+,respectively,in 2 { ⊕ } the flavorsymmetrylimit. Inthe realworld,however,the flavorsymmetryis brokenandhencethe abovetetra-quark states can have J =0, 1 and 2, in general. Nevertheless, no indication of tetra-quark meson with JP =2+ has been ] h observed, so that the above assignment seems to be favored in the real world. Thus, we treat the [qq][q¯q¯] mesons as p scalar ones [1] and the [qq](q¯q¯) (qq)[q¯q¯] as axial-vector ones [5], in contrast with Ref. [6] in which X(3872) has p- been assignedto [cn][c¯n¯{] with a la⊕rge violat}ionof isospinsymmetry. The charmstrange scalarDs+0(2317)observedin e e+e− annihilation [7] as well as in B decays [8] and the ηπ0 peak at 3.2 GeV in two photon collision[9], respectively, h are good candidates [1, 4, 10] of [cn][s¯n¯] Fˆ and [cn][c¯n¯] δˆc, (n = u, d). In addition, an indication of I=1 I I=1 2 [ tahneotihsoe-rscinhgalremt psatrratnnegre[s1c,al4a]r[cwnh]i[cs¯hn¯]has beeFnˆ+o∼bosferDv+ed(2in31t7h)e=Ds∗Fˆ++γ.c∼Hhaonwneevleor,f Bthedehciadydsen[8]chisaramsu[i1t1a]bXle(c3a8n7d2i)dahtaesoaf v massmuchhigher thanthatofthe abI=ov0e∼can0didate os0fthe hidden-Icharmscalarδˆc, andits spin-parityis favoredto be 8 JP =1+ or2− byexperiments[12,13]. Therefore,weassign[5]X(3872)to [cn](c¯n¯)+(cn)[c¯n¯] (butnotto[cn][c¯n¯]) 6 with JP =1+. However,we ignore [14] the (qq)(q¯q¯) mesons in this short no{te. } 8 5 Now we review very briefly X(3872) for later discussions. A recent analysis [15] in X(3872) π+π−J/ψ provides → . its mass and width as 7 0 m =3871.56 0.22 MeV and Γ <1.2 MeV (90%CL). (2) 1 X ± X 1 Because it decays into γJ/ψ state, its charge conjugation parity ( ) is even [16], and hence, we here take [12] JPC = : v 1++. Although the X(3872) π+π−J/ψ decay proceeds throughC the ρ0J/ψ intermediate state [17, 18], its isospin i → X quantum number is favoredto be I =0, because no indication of chargedpartner of X(3872)has been observed[19]. In addition, the aboveisospinassignmentis consistentwith the observationsofthe X(3872) ωJ/ψ π+π−π0J/ψ ar decay [13, 17]. Thus the X(3872) π+π−J/ψ decay is isospin non-conserving. Here, it sh→ould be→noted that the rate for the X(3872) π+π−J/ψ d→ecay is nearly equal [20] to that for the X(3872) π+π−π0J/ψ, → → Br(X(3872) π+π−π0J/ψ) → =0.8 0.3. (3) Br(X(3872) π+π−J/ψ) ± → Comparing the above ratio with the measured ratio Γ(ω 3π)/Γ(ω 2π) 60, one might feel that Eq. (3) is → → ∼ strange. However, these two ratios are not necessarily parallel to each other, as seen below. The X(3872) ωJ/ψ ρ0J/ψ π+π−J/ψ decay in the denominator of Eq. (3) is extraordinarily enhanced, because of doub→le pole co→ntribution→of ω and ρ0 with m2 m2 m2, where the broad width [16] Γ 150 MeV of ρ has to be taken into account, and because the kinemωat−icalρco≪nditωion of the former in which ω dρec≃ays into π+π−π0 in the energy region lower than m m m Γ /2 is different from that of the latter (on the mass-shell of ω), so X(3872) J/ψ ω ω that the rate for the X(3872−) π+π≃−π0J/−ψ decay might be sensitive to a mechanism of ω 3π and hence that of X(3872) π+π−π0J/ψ. N→evertheless, the mechanism of ω 3π is still uncertain. To→see this, we consider the ω 3π→decay and the radiative ω γπ0 in addition to ρ±,0→ γπ±,0 under the vector meson dominance [21] (VMD→). By taking the measured rate [→16] Γ(ω γπ0) = 701 →25 keV as the input data, our calculated rate, exp Γ(ρ γπ) 72 73 keV reproduces considera→blywell the measu±red rates [16], Γ(ρ± γπ±) 67 8 keV and th exp → ≃ − → ≃ ± 2 Γ(ρ0 γπ0) 90 12 keV, although the measured rates still have large ambiguities. From the above, it is seen exp that t→he VMD w≃orks±in these decays, at least in the ω γπ0 and ρ± γπ± decays. Next, we determine the ωρπ → → coupling strengthfrom the aboveΓ(ω γπ0) andapply it to the ω ρπ 3π decay. Howeverthe resulting rate exp → → → Γ(ω ρπ 3π) 5 MeV fails to reproduce the measured one [16], Γ(ω 3π) =7.57 0.09 MeV. It suggests th exp → → ≃ → ± that some extra contribution(s) are needed, because the contribution of ρ meson pole is sizable but insufficient, i.e., the mechanism of the ω 3π decay and hence that of the X(3872) ωJ/ψ π+π−π0J/ψ are still uncertain and not simple. For this reas→on, we have considered the X(3872) γJ/ψ→decay in→place of the X(3872) π+π−π0J/ψ in Ref. [22]. Although the measured ratio [17, 23] of the ra→tes Γ(X(3872) γJ/ψ)/Γ(X(3872) →π+π−J/ψ) is → → less than unity against the well-known hierarchy of hadron interactions [4], isospin conserving int. ( O(1)) | ∼ | ≫ electromagnetic int. ( O(√α)) isospin non-conserving int. ( O(α) [24]), it has been reproduced [14, 22] in | ∼ | ≫ | ∼ | our scheme that X(3872) is the tetra-quark meson given before and the ωρ0 mixing is the origin of the isospin non- conservation. It suggests again that the X(3872) π+π−J/ψ is extraordinarily enhanced. In contrast, if X(3872) → were a charmonium, the ratio of decay rates under consideration could not be reproduced [14, 22], because such an enhancement cannot work in this case. Analyses in the D0D¯∗0+c.c. ( D0D¯0π0 and D0D¯0γ) channels also have reported observations [25] of X(3875). → Recent results [26] on its mass and width are m =3872.6+0.6+0.4 MeV and Γ =3.9+2.8+0.2 MeV. (4) X(3875) −0.4−0.5 X(3875) −1.4−1.1 If the numerical results in Eqs. (2) and (4) were literally accepted, X(3875) and X(3872) would be different states. However, it is unnatural to assign X(3875) and X(3872) to different states as will be discussed later, and therefore, we here presume that the narrow X(3875)and X(3872)are identical. In this case,the averagedratio of rates for the X(3872)=X(3875) D0D¯∗0+c.c. decay to the X(3872) π+π−J/ψ has been given by [20] → → Γ(X(3872) D0D¯∗0+c.c.) → =9.5 3.1. (5) Γ(X(3872) π+π−J/ψ) ± → Now, we study possible decay modes of X(3872)and its partners. In the present scheme, hidden-charm iso-singlet axial-vector tetra-quark mesons with = are given by X( ) = X ( )+X ( ) /√2, where X ( ) and X ( ) u d u d C ± ± { ± ± } ± ± are provided by 1 3s X ( )= [cu]1s(c¯u¯)3s (cu)3s[c¯u¯]1s , (6) u ± 2√2n ¯3c 3c ± ¯3c 3co1c X ( )= 1 [cd]1s(c¯d¯)3s (cd)3s[c¯d¯]1s 3s. (7) d ± 2√2n ¯3c 3c ± ¯3c 3co1c Here the subscripts 1 , ¯3 , 3 denote the color multiplets, and the superscripts 1 and 3 the spin multiplets. The c c c s s above X ( ) can be decomposed as u ± X (+)= 1 1 √2(cc¯)3s(uu¯)3s +(cu¯)1s(uc¯)3s +(cu¯)3s(uc¯)1s 3s + u 2r6 1c 1c 1c 1c 1c 1c 1c ··· (cid:8) (cid:9) 1 1 (uc¯)1s(cu¯)3s +(uc¯)3s(cu¯)1s +√2(uu¯)3s(cc¯)3s 3s + , (8) −2r6 1c 1c 1c 1c 1c 1c 1c ··· (cid:8) (cid:9) X ( )= 1 1 (cc¯)1s(uu¯)3s +(cc¯)3s(uu¯)1s +√2(cu¯)3s(uc¯)3s 3s + u − 2r6 1c 1c 1c 1c 1c 1c 1c ··· (cid:8) (cid:9) 1 1 √2(uc¯)3s(cu¯)3s +(uu¯)1s(cc¯)3s +(uu¯)3s(cc¯)1s 3s + . (9) −2r6 1c 1c 1c 1c 1c 1c 1c ··· (cid:8) (cid:9) where denotesacolor-singletsumofproductsofcolor-octet qq¯ pairs. DecompositionsofX ( )areobtainedby d replaci·n·g·ubydintheaboveequations. Replacementofthecolo{rsin}glet qq¯ pairs, uu¯ dd¯ 1s/√±2, uu¯+dd¯ 1s/√2, { } { − }1c { }1c cc¯ 1s, etc. by the ordinary π0, η , η , etc., respectively, leads to { }1c 0 c 1 1 X(+)= 2(J/ψω ωJ/ψ)+(D0D¯∗0 D¯∗0D0)+(D∗0D¯0 D¯0D∗0) 4r3 − − − (cid:8) +(D+D¯∗− D∗−D+)+(D∗+D− D−D∗+) + , (10) − − ··· (cid:9) 1 2 X( )= (η ω ωη )+(J/ψη η J/ψ) − 4r3 c − c 0− 0 (cid:8) +(D∗0D¯∗0 D¯∗0D∗0)+(D∗+D∗− D∗−D∗+) + . (11) − − ··· (cid:9) 3 Their iso-triplet neutral partners X0( ) also can be decomposed as I ± 1 1 X0(+)= 2(J/ψρ0 ρ0J/ψ)+(D0D¯∗0 D¯∗0D0)+(D¯0D∗0 D∗0D¯0) I 4r3 − − − (cid:8) (D+D¯∗− D∗−D+) (D∗+D− D−D∗+) + , (12) − − − − ··· (cid:9) 1 2 X0( )= (η ρ0 ρ0η )+(J/ψπ0 π0J/ψ) I − 4r3 c − c − (cid:8) +(D∗0D¯∗0 D¯∗0D∗0) (D∗+D∗− D∗−D∗+) + . (13) − − − ··· (cid:9) FromEqs.(10)–(13),wecansee(i)X(+),towhichX(3872)isassigned,couplestoωJ/ψandD0D¯∗0+c.c.Therefore, itcandecayintoπ+π−π0J/ψ (andπ+π−J/ψ)throughω pole(withtheωρ0 mixing)andintoD0D¯0π0(or γ)through D0D¯∗0 +c.c., as have been observed. It is also seen that rates for these decays are small because of small overlap of flavor, color and spin wave functions as the Fˆ+ = D+(2317) D+π0 decay[4], isospin non-conservation in the π+π−J/ψ decay and very small phase space vIolumes isn0 the D0→D¯0π0sdecay through the D0D¯∗0 +c.c. and the π+π−π0J/ψ decay through the ωJ/ψ. (ii) Its opposite -parity partner X( ) couples to J/ψη , η ω and D∗D¯∗. 0 c Nevertheless,the thresholdof D∗D¯∗ decay is beyondmX(C3872), and hence prob−ably beyondmX(−). Therefore, X( ) might be observed in the ηJ/ψ channel. (iii) The iso-triplet partners X (+)’s of X(3872) couple to ρJ/ψ and DD−¯∗ I (and D¯D∗). Therefore, one might identify X0(+) with X(3875) observed in the D0D¯∗0+c.c. channel. However, it I should be noted that the rate for the isospin conserving X0(+) ρ0J/ψ π+π−J/ψ decay will be much larger than that for the isospin non-conserving X(3872) ωJ/ψI ρ0→J/ψ π+→π−J/ψ, as will be explicitly seen later. Therefore,X0(+)canbebroad,andhenceitseemst→obeunna→turaltoas→signthenarrowX(3875)tothehypothetically I broad X0(+), as noted before. (iv) The iso-triplet partners X ( )’s with negative -parity couple to πJ/ψ and η ρ. I I − C c If the spatial wave function of X ( ) is not very much different from that of X (+), the rate for the X ( ) πJ/ψ I I I would be much larger than that fo−r the X0(+) ρ0J/ψ π+π−J/ψ, because of much larger phase spac−e v→olume. We here study the rate for the X0(+) I ρ0J→/ψ π+π→−J/ψ decay to see why X0(+) has not been observed. In Eq. (22) of Ref. [22], we have calcuIlated→the rate f→or the X(3872) ωJ/ψ ρ0J/Iψ π+π−J/ψ decay with the ωρ0 mixing. The rate for the abovedecay ofX0(+) canbe obtained→by replac→ingX(387→2)by X0(+) andeliminating I I the contribution of ω pole with the ωρ0 mixing in the equation. Taking m m (because both of them XI(+) ≃ X(3872) consistofthe samequarksandtheirflavorwavefunctionsareofthe sametype, asinthe caseofFˆ+ =D+(2317)and I s0 Fˆ+ which have been observed as signal and indication, respectively, at the same mass in B decays [8] as discussed 0 before), we get the ratio of rates Γ(X0(+) ρ0J/ψ π+π−J/ψ) I → → 200, (14) Γ(X(3872) π+π−J/ψ) ∼ → whereithasbeenassumedthatthesizeoftheX0(+)ρ0J/ψcouplingisapproximatelyequaltothatoftheX(+)ωJ/ψ, I because the spatial wave functions of X (+) and X(+) are expected to be not very much different from each other, I as in the case [4] of Fˆ+ and Fˆ+. I 0 To estimate the denominator of Eq. (14), we assume that the full width of X(3872) is approximately saturated as Γ Γ(X(3872) π+π−J/ψ+Γ(X(3872) π+π−π0J/ψ) X(3872) ≃ → → +Γ(X(3872) D0D¯∗0+c.c.). (15) → By using Eqs. (3) and (5), Γ(X(3872) π+π−J/ψ) can be given by Γ . We have listed two different values, X(3872) → Γ in Eq. (2) and Γ in Eq. (4), where X(3872) and X(3875) are now identified. The latter is consistent X(3872) X(3875) with the measured width [11] 2.5 0.5 MeV of X(3872). However, this is narrower than the experimental energy ± resolution, and therefore, some corrections might be needed. Such a corrected width has been given in Eq. (2). Taking Γ in Eq. (4), as an example, we obtain 0.1.Γ(X(3872) π+π−J/ψ).0.9 MeV, and therefore, X(3875) → Γ(X0(+) ρ0J/ψ π+π−J/ψ) (20 200) MeV. (16) I → → ∼ − The above rate is much larger than that for the near threshold X (+) D0D¯∗0+c.c. decay, because Γ(X (+) I I D0D¯∗0+c.c.)≃Γ(X(+)→D0D¯∗0+c.c.) will be obtained, if mXI(+) ≃→mX(3872) and |gXI(+)D0D¯∗0|≃|gX(+)D0D¯∗→0| as discussed before. Therefore, the full width of X (+) would be dominated by Γ(X (+) ρJ/ψ ππJ/ψ), and I I → → hence X (+) would be much broader than X(3872). In this case, it is expected that the broad enhancement of the I π+π−J/ψ mass distribution from X0(+) would be behind the background of the narrow X(3872) peak, unless the I production rates of X (+)’s are much larger than that of X(3872). Although the existing search for the charged I 4 partnerofX(3872)mentionedbeforehasreported[19]noindicationofanarrowπ−π0J/ψ resonancearoundthemass of X(3872), this does not necessarily exclude existence of iso-triplet partners, because their production rate has not been knownyet and, in addition, the presentstatisticalaccuracymight be insufficient to observethe broadX0(+) in I the ππJ/ψ mass distribution. On the other hand, if Γ in Eq. (2) as another example is taken, a small Γ X(3872) XI(+) would be possible. In this case, X (+)’s could have been observed in the ππJ/ψ channels. However, the negative I resultonthesearchforthe(ππ)−J/ψmightimplythatthetruewidthofX(3872)isneartheupperboundofΓ X(3872) in Eq.(2), andtherefore X (+)wouldbe considerablybroad,if its productionrateis of the same orderofmagnitude I as thatofX(3872). Therefore,moreprecise determinationofintrinsic widths ofX(3872)andits partnersinaddition to their production rates will provide important informations to search for these partners. In summary we have studied X(3872) and its partners, assigning these axial-vector mesons to [cn](c¯n¯) { ± (cn)[c¯n¯] . As the results, we have discussed their possible decay modes, and pointed out that the iso-triplet I=1,0 } partners of X(3872) can be considerably broad and therefore higher statistics will be needed to find them. Acknowledgments The author would like to thank Professor H. J. Lipkin for discussions by which this work was motivated. He also would like to appreciate Professor K. Miyabayashifor informing the current status of X(3872)experiments. [1] K.Terasaki, Phys. Rev.D 68 (2003), 011501(R); K. Terasaki, AIPConf. Proc. 717 (2004), 556; hep-ph/0309279. [2] R.L. Jaffe, Phys.Rev.D 15 (1977), 267 and 281. [3] M. Y.Han and Y. Nambu,Phys.Rev.139 (1965), B 1006; S. Hori, Prog. Theor. Phys.36 (1966), 131. [4] A. Hayashigaki and K. Terasaki, Prog. Theor. Phys. 114 (2005), 1191; hep-ph/0410393; K. Terasaki, Invited talk at the workshop on Resonances in QCD, July 11 – 15, 2005, ECT*, Trento, Italy; hep-ph/0512285. [5] K.Terasaki, Prog. Theor. Phys. 118 (2007), 821; arXiv:0706.3944[hep-ph]. [6] L. Maiani, F. Piccinini, A. D.Polosa and V. Riquer,Phys.Rev. D 71 (2005), 014028. [7] B. Aubert et al., Babar Collaboration, Phys. Rev. Lett. 90 (2003), 242001; D. Besson et al., CLEO Collaboration, Phys. Rev.D 68 (2003), 032002; [8] P.Krokovnyet al., Belle Collaboration, Phys.Rev.Lett. 91 (2003), 262002. [9] S.Uehara et al., Belle Collaboration, Phys. Rev.D 80 (2009), 032001. [10] K.Terasaki, Prog. Theor. Phys. 121 (2009), 211; arXiv:0805.4460 [hep-ph]. [11] S.K. Choi et al., Belle Collaboration, Phys.Rev.Lett. 91 (2003), 262001. [12] K.Abeet al., Belle Collaboration, hep-ex/0505038. [13] P.del Amo Sanchezet al., Babar Collaboration, Phys.Rev. D 82 (2010), 011101. [14] K.Terasaki, Prog. Theor. Phys. Suppl.No.186 (2010), 141; arXiv:1005.5573 [hep-ph]. [15] S.K. Choi et al., arXiv: 1107.0163 [hep-ex], and references quoted therein. [16] K.Nakamura et al., Particle Data Group, J. Phys. G 37 (2010), 075021. [17] K.Abeet al., Belle Collaboration, hep-ex/0505037. [18] A.Abulencia et al., CDF Collaboration, Phys.Rev. Lett.96 (2006), 102002. [19] B. Aubert et al., Babar Collaboration, Phys.Rev.D 71 (2005), 031501. [20] S.-K.Choi, arXiv: 1101.5691 [hep-ex], and references quoted therein. [21] M. Gell-Mann and F. Zachariasen, Phys. Rev. 124, 953 (1961); J. J. Sakurai, Currents and Mesons (Chicago, Ill., 1969); K.Terasaki, Lett. NuovoCim. 31, 457 (1981); NuovoCim. 66A, 475 (1981). [22] K.Terasaki, Prog. Theor. Phys. 122 (2009), 1285; arXiv:0904.3368 [hep-ph]. [23] B. Aubert et al., Babar Collaboration, arXiv:0809.0042 [hep-ex]. [24] R.H. Dalitz and F. Von Hippel, Phys. Lett.10 (1964), 153. [25] G. Gokhroo et al., Belle Collaboration, hep-ex/0606055; B. Aubert et al., Babar Collaboration, Phys. Rev. D 77 (2008), 011102(R). [26] I.Adachi et al., Belle Collaboration, arXiv:0810.0358 [hep-ex].