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Production of the Y(4260) state in B meson decay R.M. Albuquerque,1,∗ M. Nielsen,2,† and C.M. Zanetti3,‡ 1Instituto de F´ısica Teórica, Universidade Estadual Paulista (IFT-UNESP) R.Dr.BentoTeobaldoFerraz, 271 - Bl.II Sala 207, 01140-070 São Paulo/SP - Brasil 2Instituto de F´ısica, Universidade de São Paulo, C.P. 66318, 05389-970 São Paulo, SP, Brazil 3Faculdade de Tecnologia, Universidade do Estado do Rio de Janeiro, Rod. Presidente Dutra Km 298, Po´lo Industrial, 27537-000, Resende/RJ - Brasil We calculate the branching ratio for the production of the meson Y(4260) in the decay B− → Y(4260)K−. We use QCD sum rules approach and we consider the Y(4260) to be a mixture between charmonium and exotic tetraquark, [c¯q¯][qc], states with JPC = 1−−. Using the value of the mixing angle determined previously as: θ = (53.0±0.5)◦, we get the branching ratio B(B → Y(4260)K)=(1.34±0.47)×10−6,whichallowsustoestimateanintervalonthebranchingfraction 3.0×10−8 <B < 1.8×10−6 in agreement with the experimental upper limit reported by Babar Y Collaboration. 5 The Y(4260) state was first observed by BaBar col- be completely ruled out. However,there are some calcu- 1 laboration in the e+e− annihilation through initial state lations, within the QCD sum rules (QCDSR) approach 0 radiation [1], and it was confirmed by CLEO and Belle [5, 21–23], that can not explain the mass of the Y(4260) 2 collaborations [2]. The Y(4260) was also observed in supposing it to be a tetraquark state [24], or a D D, 1 n the B− Y(4260)K− J/Ψπ+π−K− decay [3], and D D∗ hadronicmolecule[24],oraJ/ψf (980)molecular 0 0 a CLEOre→portedtwoaddi→tionaldecaychannels: J/Ψπ0π0 state [25]. J and J/ΨK+K− [2]. The Y(4260) is one of the many In the framework of the QCDSR the mass and the 1 charmonium-like state, called X, Y and Z states, re- decay width, in the channel J/ψππ, of the Y(4260)were 3 cently observed in e+e− collisions by BaBar and Belle computedwithgoodagreementwithdata,consideringit collaborations that do not fit the quarkonia interpreta- asa mixing betweentwoand four-quarkstates [26]. The ] h tion. The production mechanism, masses, decay widths, mixing is done at the level of the hadronic currents and, p spin-parity assignments and decay modes of these states physically,thiscorrespondstoafluctuationoftheccstate - p have been discussed in some reviews [4–7]. The Y(4260) where a gluon is emitted and subsequently splits into a e is particularly interesting because some new states have lightquark-antiquarkpair,whichlivesforsometimeand h beenidentifiedinthe decaychannelsofthe Y(4260),like behaves like a tetraquark-like state. The same approach [ the Z+(3900). The Z+(3900) was first observed by the was applied to the X(3872) state and good agreement c c 1 BESIII collaboration in the (π±J/ψ) mass spectrum of with the data were obtained for its mass and the decay v the Y(4260) J/ψπ+π− decay channel [8]. This struc- width into J/ψππ [27], its radiative decay [28], and also → 9 ture, was also observed at the same time by the Belle in the X(3872)production rate in B decay [29]. 1 collaboration [9] and was confirmed by the authors of In this work we will focus on the production of the 1 Ref. [10] using CLEO-c data. Y(4260),usingthe mixedtwo-quarkandfour-quarkpre- 0 The decay modes of the Y(4260) into J/ψ and other scription of Ref. [26] to perform a QCDSR analysis of 0 charmonium states indicate the existence of a c¯c in the processB− Y(4260)K−. The experimentalupper . 2 → its content. However, the attempts to classify this limitonthebranchingfractionforsuchaproductioninB 0 state in the charmonium spectrum have failed since the meson decay has been reported by BaBar Collaboration 5 Ψ(3S), Ψ(2D) and Ψ(4S) cc¯ states have been assigned [3], with 95% C.L., 1 : to the well established Ψ(4040), Ψ(4160), and Ψ(4415) v mesonsrespectively,andthepredictionfromquarkmod- <2.9 10−5 (1) i BY × X els for the Ψ(3D) state is 4.52GeV. Therefore, the mass r of the Y(4260) is not consistent with any of the 1−− cc¯ where BY ≡B(B−→K−Y(4260),Y(4260)→J/ψπ+π−). a states [4, 5]. Y Some theoretical interpretations for the Y(4260) are: tetraquark state [11], hadronic D D, D D∗ molecule q 1 0 [12], χ ω molecule [13], χ ρ molecule [14], J/ψf (980) c1 c1 0 B molecule [15], a hybrid charmonium [16], a charm- baryonium[17],acusp[18–20], etc. Withinthe available p O experimental information, none of these suggestions can 2 p’ K ∗ [email protected] † [email protected] FIG.1. TheprocessforproductionoftheY(4260)stateinB ‡ [email protected] meson decay,mediated by an effective vertex operator O2. 2 The process B Y(4260)K occurs via weak decay of meson pole Q2 = m2, using the QCDSR approach for → − Y the b quark, while the u quark is a spectator. The Y the three-point correlator [30]: meson as a mixed state of tetraquark and charmonium interacts via c¯c component of the weak current. In effec- Π (p,p′)= d4xd4yei(p′·x−p·y) 0T JW(0) tive theory, at the scale µ m m , the weak decay µ Z h | { µ × b W is treated as a four-quark∼local i≪nteraction described by J (x)J†(y) 0 , (9) × K B }| i the effective Hamiltonian (see Fig. 1): wheretheweakcurrent,JW,isdefinedin(3)andthein- µ G C (µ) terpolatingcurrentsoftheB andK pseudoscalarmesons = FV V∗ C (µ)+ 1 + , (2) HW √2 cb cs(cid:20)(cid:18) 2 3 (cid:19)O2 ···(cid:21) are: J =iu¯ γ s , J =iu¯ γ b . (10) where V are CKM matrix elements, C (µ) and C (µ) K a 5 a B a u a ik 1 2 are short distance Wilson coefficients computed at the The obtained result for the form factor was [29]: renormalization scale µ (m ). The four-quark effec- b ∼ O tive operator is O2 =Jµ(c¯c)JµW, with f (Q2)= (17.55±0.04) GeV2 . (11) + (105.0 1.8) GeV2+Q2 JW =s¯Γ b, J(c¯c) =c¯Γ c, (3) ± µ µ µ µ For the decay width calculation, we need the value of anUdsΓinµg=faγcµto(r1iz−atγi5o)n.,thedecayamplitudeoftheprocess tthheeYfo(r4m26f0a)ctmoresaotnQ. U2 s=in−gmm2Y,=wh(4er2e51mY9)isMtheeVm[3a1s]swoef Y is calculated from the Hamiltonian (2), by splitting the get: ± matrix element in two pieces: G C f+(Q2)|Q2=−m2Y =0.206±0.004. (12) =i FV V∗ C + 1 M √2 cb cs(cid:18) 2 3 (cid:19) The parameter λ can also be determined using the W B(p)JW K(p′) Y(q)Jµ(c¯c) 0 , (4) QCDSR approachfor the two-point correlator: ×h | µ | ih | | i wherep=p′+q. FollowingRef.[29],thematrixelements Π (q)=i d4y eiq·y 0T JY(y)J(c¯c)(0) 0 , (13) in Eq. (4) are parametrized as: µν Z h | { µ ν }| i Y(q)J(c¯c) 0 =λ ǫ∗(q), (5) wherethecurrentJν(c¯c) isdefinedin(3). FortheY meson h | µ | i W µ we will follow [26] and consider a mixed charmonium- and tetraquark current: hB(p)|JµW|K(p′)i=f+(q2)(pµ+p′µ)+f−(q2)(pµ−p′µ). JµY =sinθJµ(4)+cosθJµ(2), (14) (6) where The parameterλ in(5) givesthe coupling betweenthe W dcuesrcrerinbteJtµ(hc¯ec)waenadktthreanYsitsitoanteB. TheKf.orHmenfcaectworescfa±n(qse2e) Jµ(4) = ǫab√cǫ2dech(qaTCγ5cb)(q¯dγµγ5Cc¯Te)+ → thatthefactorizationofthematrixelementdescribesthe +(qTCγ γ c )(q¯ γ Cc¯T) , (15) a 5 µ b d 5 e decay as two separated sub-processes. i The decay width for the process B− Y(4260)K− is → given by J(2) = 1 q¯q c¯ γ c 1 q¯q J′(2) . (16) 2 µ √2h i(cid:16) a µ a(cid:17) ≡ √2h i µ Γ(B YK)= |M| λ(m2,m2 ,m2), (7) → 16πm3Bq B K Y InEq.(14), θ isthemixinganglethatwasdeterminedin [26] to be: θ =(53.0 0.5)0. with λ(x,y,z) = x2 +y2 +z2 2xy 2xz 2yz. The Insertingthecurren±ts(3)and(14)inthecorrelatorwe − − − invariant amplitude squared can be obtained from (4), have in the OPE side of the sum rule using (5) and (6): q¯q ΠOPE(q)=sinθΠ4,2(q)+ h icosθΠ2,2(q), (17) G2 C 2 µν µν √2 µν 2 = F V V 2 C + 1 |M| 2m2 | cb cs| (cid:18) 2 3 (cid:19) where Y λ(m2,m2 ,m2)λ2 f2 . (8) Π4,2(q)=i d4y eiq·y 0T J(4)(y)J (0) 0 × B K Y W + µν Z h | { µ ν(c¯c) }| i The coupling constant f+ was determined in Ref.[29] Π2,2(q)=i d4y eiq·y 0T J′(2)(y)J (0) 0 .(18) through extrapolation of the form factor f+(Q2) to the µν Z h | { µ ν(c¯c) }| i 3 Thus we can calculate the decay width in Eq. (7) by TABLE I.QCD inputparameters. using the values of f ( M2) and λ , determined in + − Y W Eqs. (12) and (23). The branching ratio is evaluated Parameters Values dividing the result by the total width of the B meson mc (1.23−1.47) GeV Γtot =4.280 10−4eV: hq¯qi −(0.23±0.03)3 GeV3 × hgs2G2i (0.88±0.25) GeV4 B(B →Y(4260)K)=(1.34±0.47)×10−6, (24) m20 ≡hq¯Gqi/hq¯qi (0.8±0.1) GeV2 where we have used the CKM parameters V = 1.023, cs V = 40.6 10−3 [31], and the Wilson coefficients cb × Only the vector part of the current Jν(c¯c) contributes to Can1d(µΛ¯) = =1.028225,MC2e(Vµ)[3=2].−0.185, computed at µ = mb the correlators in Eq. (18). Therefore, these correlators MS In order to compare the branching ratio in Eq. (24) are the same as the ones calculated in Ref. [26] for the with the branching fraction obtained experimentally in mass of the Y(4260). Eq. (1), we might use the results found in Ref. [26]: To evaluate the phenomenologicalside we insertinter- mediate states of the Y: (Y(4260) J/ψπ+π−)=(4.3 0.9) 10−2, (25) B → ± × i Πphen(q)= 0JY Y(q) Y(q)J(c¯c) 0 , µν q2 m2 h | µ| ih | ν | i and then, considering the uncertainties, we can estimate − Y > 3.0 10−8. However, it is important to notice iλ λ q q BY × = Y W g µ ν (19) that the authors in Ref. [26] have considered two pions Q2+m2Y (cid:18) µν − m2Y (cid:19) in the final state coming only from intermediate states, whereq2 = Q2,andwehaveusedthedefinition(5)and e.g. σ andf0(980)mesons,whichcouldindicatethatthe − result in Eq. (25) can be underestimated. In this sense, 0JY Y(q) =λ ǫ (q). (20) consideringthatthemaindecaychannelobservedforthe h | µ| i Y µ Y(4260)stateisintoJ/ψπ+π−,wewouldnaivelyexpect Theparameterλ ,thatdefinesthecouplingbetweenthe Y that the branching ratio into this channel could also be currentJµY andtheY meson,wasdeterminedinRef.[26] (Y(4260) J/ψπ+π−) 1.0,whichwouldleadtothe to be: λY =(2.00±0.23)×10−2 GeV5. Bfollowing re→sult, BY <1.8∼×10−6. Therefore, we obtain As usualinthe QCDSRapproach,weperforma Borel an interval on the branching fraction transformtoQ2 M2 toimprovethematchingbetween both sides of the→sumBrules. After performing the Borel 3.0 10−8 < <1.8 10−6 (26) × BY × transforminbothsides ofthe sumrulewe getinthe g µν which is in agreementwith the experimental upper limit structure: reported by Babar Collaboration given in Eq. (1). In λWλYe−Mm2YB2 = s√in2θ Π4,2(MB2)+ h√q¯q2icosθΠ2,2(MB2)(21) gfreancetrioanl tthaekeesxpinetroimaecnctoaulnetvaaludadtitioionnaolf ftahcetorbsrarneclahtinedg to the numbers of reconstructed events for the final where the invariant functions Π2,2(MB2) and Π4,2(MB2) state (J/ψ π+π− K), for the reference process (B are written in terms of a dispersion relation, → Y(4260)K), and for the respective reconstruction effi- ∞ ciencies. However, since such information has not been Π(M2)= ds e−s/MB2 ρ(s) , (22) B Z 4m2c withtheirrespectivespectraldensitiesρ2,2(s)andρ4,2(s) given in Appendix. We perform the calculation of the coupling parameter λ using the same values for the masses and QCD W condensates as in Ref. [26] which are listed in Table I. To be consistent with the calculation of λ we also Y use the same region in the threshold parameter s as in 0 Ref. [26]: √s0 = (4.70 0.10) GeV. As one can see in ± Fig. 2, the regionwhere we get M2-stability is given by: B (8.0 M2 25.0) GeV2. ≤ B ≤ Taking into account the variation in the Borel mass parameter,inthecontinuumthreshold,inthequarkcon- densate, in the coupling constant λ and in the mixing angle θ, the result for the λ paramYeter is: FIG.2. ThecouplingparameterλW asafunctionofMB2,for W different values of thecontinuum threshold. λ =(0.90 0.32) GeV2. (23) W ± 4 provided in Ref. [3], we have neglected these factors in the Y(4260), we may say that both components are ex- the calculation of the branching fraction . Therefore, tremely important, and that, in our approach, it is not Y B thecomparisonofourresultwiththeexperimentalresult possible to explain all the experimental data about the could be affected by these differences. Y(4260) with only one component. In conclusion, we have used the QCDSR approach to evaluate the productionof the Y(4260)state,considered as a mixed charmonium-tetraquark state, in the decay B YK. Using the factorization hypothesis, we find → that the sum rules result in Eq. (24), is compatible with ACKNOWLEDGMENT the experimental upper limit. Our result can be inter- preted as a lower limit for the branching ratio, since we This work has been partially supported by Saõ Paulo did not considered the non-factorizable contributions. Research Foundation (FAPESP), grant n.2012/22815-3, Ourresultwasobtainedbyconsideringthe mixingan- andNationalCounselofTechnologicalandScientificDe- gle in Eq.(14) in the rangeθ =(53.0 0.5)0. This angle velopment (CNPq-Brazil). ± wasdeterminedinRef.[26]wherethemassandthedecay width of the Y(4260) in the channel J/ψπ+π− were determined in agreementwith experimentalvalues. There- fore, since there is no new free parameter in the present analysis, the result presented here strengthens the con- Appendix A: Spectral Densities for the Two-point clusion reached in [26] that the Y(4260) is probably a Correlation Function mixture between a cc¯state and a tetraquark state. As discussed in [29], it is not simple to determine the We list the spectral densities for the invariant functions charmoniumandthetetraquarkcontributiontothestate relatedtothecouplingbetweenthecurrentJ(c¯c) andthe described by the current in Eq. (14). From Eq. (14) one µ Y(4260)state. WeconsidertheOPEcontributionsupto can see that, besides the sinθ, the cc¯component of the dimension-fivecondensatesandkeeptermsatleadingor- current is multiplied by a dimensional parameter, the derinα . Inordertoretaintheheavyquarkmassfinite, quark condensate, in order to have the same dimension s we use the momentum-space expression for the heavy of the tetraquark part of the current. Therefore, it is quark propagator. We calculate the light quark part not clear that only the angle in Eq. (14) determines the of the correlation function in the coordinate-space and percentageofeachcomponent. Onepossiblewaytoeval- usethe Schwingerparametrizationto evaluatethe heavy uate the importance of each part of the current it is to quark part of the correlator. For the d4y integration in analyzewhat one wouldget forthe productionrate with Eq.(13),weuseagaintheSchwingerparametrization,af- each component, i.e., using θ = 0 and 90◦ in Eq. (14). ter a Wick rotation. Finally, the result of these integrals Doing this we get respectively for the pure tetraquark aregivenintermsoflogarithmicfunctionsthroughwhich and pure charmonium: weextractthespectraldensities. Thesametechniquecan (B Y K)=(1.25 0.23) 10−6, (27) be used for evaluating the condensate contributions. tetra B → ± × Then, in the g structure, we evaluate the spectral (B Y K)=(1.14 0.20) 10−5. (28) µν B → c¯c ± × densities for the Π2,2(M2) function, B Comparing the results for the pure states with the one m2 1 g2G2 v 1 for the mixed state (24), we can see that the branching ρ2,2(s)= c v 2+ + h s i 4 1 ratio for the pure tetraquark is one order smaller, while 4π2 (cid:16) x(cid:17) 48π2 MB2(cid:20) (cid:16) − x(cid:17) thepurecharmoniumislarger. Fromtheseresultswesee m2 5 m2 2 1 c 11 + c 3 ,(A1) that the cc¯part of the state plays a very important role − MB2 x(cid:16) − x(cid:17) (cid:16)MB2 x(cid:17) (cid:16) − x(cid:17)(cid:21) inthedeterminationofthebranchingratio. 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