φφ and J/ψφ mass spectra in decay B0 J/ψφφ. s → A. A. Kozhevnikov1,2,∗ 1Laboratory of Theoretical Physics, S. L. Sobolev Institute for Mathematics, Novosibirsk, Russian Federation 2Novosibirsk State University, Novosibirsk, Russian Federation (Dated: January 10, 2017) ThemassspectraoftheφφandJ/ψφstatesinthedecayB0 →J/ψφφrecentlyobservedbyLHCb s are calculated in the model which takes into account the JP =0+,0−,2+ intermediate resonances R1,R2 intheφφchannelandtheJP =1+ ones,X1,X2,intheJ/ψφchannel. Whenobtainingthe expressions for theeffectiveamplitudes and mass spectra, theapproximatethreshold kinematics of thedecay is used essentially. The R1−R2 and X1−X2 mixings arising due to thecommon decay modes φφ and J/ψφ, respectively, are also taken into account. The obtained expressions for the massspectraareappliedforextractingtheinformation aboutmassesandcouplingconstantsofthe 7 resonances in theφφ and J/ψφ final states. 1 0 PACSnumbers: 13.25.Hw,14.40.Cs,14.40.Nd 2 n a I. INTRODUCTION masses and coupling constants of these resonances. It J should be emphasized that the presentation of the re- 8 Recently,theLHCbcollaborationhasreportedtheob- sults in terms of coupling constants is more informative servation of the decay B0 J/ψφφ [1]. The interest in than the popular representation in terms of the partial h] this decay is related to tshe→possible existence of the ex- widths. p otic glueball state decaying into the φφ pair [2–5]. The The paper is organized as follows. Section II contains p- spin-parityquantumnumbersoftheresonancestatesde- the expressionsfor the lowestmomenta effective vertices he c[6a,y7in].g into φφ are reported to be JP = 0+,0−, and 2+ oXf1,t2he BJs0/ψ→φ Rtr1a,n2Jsi/tψio,nsBas0ss→umXin1g,2Jφ,PR=1,20+→,0−φ,φ2,+afnodr [ The LHCb collaboration has also reported the obser- the R→1,2 resonancesandJP =1+ for the X1,2 ones. The v1 Jva/tψioφnKo+f,foinurthreesomnaasnscerasntgruec4tu14re0s–in47th0e0 dMeceaVy, Bde+ca→y- pgiavretnialthdeercea.ySwecidtitohns IRII1,i2s→devφoφtedantdo tXh1e,2de→rivja/tiψonaroef 4 ing into J/ψφ [8, 9]. This group of resonances is widely the Bs0 →J/ψφφ decay amplitudes upon taking into ac- 3 discussed because of their possible exotic nature [10– count the R1 R2 mixing due to the common φφ decay 9 12], side by side with the explanations based on the dy- channel, and X−1 X2 mixing due to the J/ψφ one. The 01 namical rescattering effects [13]. Two of them, X(4140) modulus squared−of the Bs0 → J/ψφφ decay amplitude and X(4274), have masses in the range 4100 – 4350 andtheexpressionsfortheφφandJ/ψφmassspectraare . 1 MeV attained in the J/ψφ mass spectrum in the decay calculatedin Sec. IV. In Sec. V, these expressionare ap- 0 B0 J/ψφφ [1]. The preferable spin-parity quantum pliedtothedescriptionoftheLHCbdata[1]. SectionVI 7 nusm→bers of these resonances are JP =1+ [8]. containsthebriefdiscussionoftheobtainedresults. Sec- 1 : The data of Ref. [1] were plotted against the phase tion VII servesas a conclusion. Some details used in the v spacedistributionwhichwasshowntobeinadequatebe- derivation of expressions in the main text are given in Xi causetheresonancebumps wereseeninboththeφφand the Appendices. J/ψφmassspectra[1]. Theaimofthepresentworkisto r a write down the amplitudes and the mass spectra of the above final states in the decay B0 J/ψφφ upon tak- II. EFFECTIVE VERTICES AND PARTIAL s → DECAY WIDTHS ing into account possible intermediate resonance states in the φφ and J/ψφ channels, irrespective of the model assumptions about their nature. The task of obtaining the effective amplitudes is We assume the existence of the resonances at m = φφ greatly simplified when one takes into account the near- 2.07and2.2GeVinthe φφ massspectrumwhichwillbe threshold kinematics of the decay B0 J/ψφφ. Then calledas R , and the resonancesX(4140)andX(4274) s → 1,2 one can neglect the higher partial waves in the decay with masses in the range 4100 – 4350 MeV, in the J/ψφ amplitudes. The expressions for the φφ and J/ψφ mass mass spectrum [14, 15] which will be called X . One 1,2 spectraobtainedundersuchanapproximationupontak- needs the effective vertices B0 J/ψR, R φφ, B0 s → → s → ing into account the intermediate resonances R φφ φX, X J/ψφ. See Fig. 1. Since all particles in the 1,2 and X J/ψφ are used to extract from the fit→s the final stat→e of the decay B0 J/ψφφ have unit spin, the 1,2 → s → numberofeffective contributionsto the decayamplitude isfrighteninglylarge,especiallywhentakingintoaccount the fact that space parity is not conserved because the ∗ [email protected] B0 meson decays due to the weak interactions, so that s 2 only the angular momentum conservation, not the space Alleffectiveverticesareassumedtobereal,andthepos- parity,restrictsthenumberofpossibleLorenzstructures sibledependenceonthemomentumsquaredisneglected. in the effective decay amplitudes. The couplings of R Letusgivetheeffectiveamplitudesfortheweakdecays with φφ and X with J/ψφ are considered to be due to oftheB0 meson. Thelowestmomentaamplitudesofthe s the strong interactions, hence they conserve parity, but transitions B0 J/ψR for different quantum numbers s → again the unit spin admits many independent effective of the resonance R are contributions. The situation can be greatly simplified if onetakesintoaccountthefactthatthekinematicsofthe MB0→J/ψR(0±) =2gB0J/ψR(ǫ(J/ψ) k), s s · decay MBs0→J/ψR(2+) =gBs0J/ψRTµαǫ(µJ/ψ)qα, (2.6) Bs0(Q)→J/ψ(q)+φ(k1)+φ(k2) (2.1) where k = k1 + k2. Note that, in the first expression in Eq. (2.6), the amplitude conserves (breaks) the space is such that all particles in the final state have relatively parity in the case of R with JP = 0− (JP = 0+), re- low momenta. Indeed, the invariant mass of the φφ pair spectively. The expression of the decay amplitude to varies in the range the tensor resonance in the second line of Eq. (2.6) breaks parity. The parity-conserving amplitude is 2mφ ≤mφφ ≤mBs0 −mJ/ψ, (2.2) T ε q ǫ(J/ψ)k q , hence it has the D wave form t∝o µα µνλσ α ν λ σ that is, 2.04 m 2.27GeV. The maximum momen- beneglectedsidebysidewiththeparity-breakingDwave φφ tum of the φ≤is reac≤hed when the final J/ψ meson is at expression Tµαqµqα(ǫ(J/ψ) k). The lowestmomentum restwhiletwoφmesonsmoveinoppositedirections,and B0 φX t∝ransition amplitu·de, s → this gives k /m = 0.5. Analogously, the invariant 1 max φ mass of th|e J|/ψφ state varies in the range MB0→φX(1+) =gB0Xφ(ǫ(φ) ǫ(X)), (2.7) s s · m +m m m m , (2.3) breaks space parity. φ J/ψ J/ψφ B0 φ ≤ ≤ s − Ourgoalhereistotakeintoaccounttheenergydepen- that is, 4.12 mJ/ψφ 4.35 GeV. The maximum mo- denceofthe partialwidthsoftheresonancesinvolved,as ≤ ≤ mentum of the J/ψ meson is reached when one of the φ well as their mixing due to the common decay modes mesonsisatrest,whileothermovesoppositelytoJ/ψre- (if any). The Particle Data Group (PDG) gives the φφ, sulting in |q|max/mJ/ψ = 0.2. However, these relatively KK¯ decay modes for the tensor f2(2010), f2(2300), and small ratios are not in fact reached because the above f (2340)resonancesand the ηη one for the f (2340)res- 2 2 kinematic situations taking place at the border of phase onance [6]. Again, because of the low statistics of the space are suppressed by the final state phase space fac- available data we will take into account only the φφ de- tors. SeeEqs.(4.7)and(4.9)below. Thispermitsoneto cay mode relevant in the context of the data presented take into account only the effective decay vertices with in Ref. [1]. The same assumption will be adopted for the lowest nonvanishing powers of momenta. the scalarandpseudoscalaronesobservedbythe BESIII Let us start with the parity-conserving effective ver- collaboration[7]. tices of R φφ and X J/ψφ. The required lowest Thestandardcalculationgivesthepartialdecaywidths → → momenta R φφ vertices for various quantum numbers for the resonances with the given quantum numbers. → are Taking into account only the lowest nonvanishing mo- menta, one gets, using the effective vertices Eq. (2.4), MR(0+)→φφ =gRφφ(ǫ(φ1)·ǫ(φ2)), the following expressions: MR(0−)→φφ =gRφφεµνλσk1µǫν(φ1)k2λǫσ(φ2), 3g2 MR(2+)→φφ =gRφφTµνǫµ(φ1)ǫν(φ2), (2.4) ΓR(0+)→φφ(m2)= 3R2(π0m+)3φφλ1/2(m2,m2φ,m2φ) where T is the polarization tensor of the spin two res- g2 onance,µǫν(φ1,2) and k1,2 stand for the polarization four- ΓR(0−)→φφ(m2)= 6R4(π0−m)φ3φλ3/2(m2,m2φ,m2φ) vectorandfour-momentumofthe φ1,2 meson,andεµνλσ g2 is the totally antisymmetric unit Levy-Civita` tensor. ΓR(2+)→φφ(m2)= 3R2(π2+m)φ3φλ1/2(m2,m2φ,m2φ). (2.8) The quantum numbers of the X(4140) and X(4274) resonancesarenow established: JPC =1++ [8, 9]. Then Hereafter, the effective lowest momentum X J/ψφ vertex looks like → λ(x,y,z)=x2+y2+z2 2xy 2xz 2yz (2.9) − − − MX(1+)→J/ψφ =gXJψφεµνλσpµǫ(νX)ǫ(λJ/ψ)ǫ(σφ). (2.5) is the standard K¨all´en function. Correspondingly, the X(1+) J/ψφ partial width calculated from Eq. (2.5) Intheaboveexpressions,ǫ(X),ǫ(J/ψ) standforthepolar- → is ization four-vectors of the X, J/ψ mesons, respectively, and p is the four-momentum of the X resonance. The g2 justificµationofthisexpressionisgivenintheAppendixB. ΓX→J/ψφ(m2)= X8πJ/mψφλ1/2(m2,m2J/ψ,m2φ). (2.10) 3 J/ψ(q) φ(k1) The LHCb data [1] visibly demonstrate the appearance Bs0(Q) Bs0(Q) Bs0(Q) φ(k2) of two enhancements in the φφ and J/ψφ mass spectra. + + So, it will be assumed in what followsthat there are two R φ(k1) X φ(k2) X φ(k1) resonancesR ,R withthespinzeroand(or)spintwoin 1 2 J/ψ(q) the φφ systemmassrangefrom2.0 to2.4 GeV[6,7]and φ(k2) J/ψ(q) tworesonancesX ,X withJP =1+intheJ/ψφsystem 1 2 FIG. 1. The diagrams of thedecay B0 →J/ψφφ. mass range from 4.14 to 4.35 GeV [8, 9]. Their masses s areclose,and the resonanceshavecommondecaymodes like φφ in the case of R or J/ψφ in the case of X . 1,2 1,2 In what follows, the B0 J/ψφφ decay amplitudes to They can mix inside each group. Using Refs. [16, 17] we s → take into account the R R and X X mixing by be derived below will include only the resonance contri- 1 2 1 2 − − bution with specific JP. By this reason, all resonances introducing the nondiagonalpolarizationoperators Π(1R2) Rwithdifferentquantumnumberswillbelabeledbythe and Π(X), respectively. Then the simplest Breit-Wigner 12 single letter R in the coupling constants, without point- resonance contribution in the case of, say, R , 1,2 ing to the JP quantum numbers. g g B0J/ψR Rφφ BW s , (3.1) ∝ m2 m2 im Γ III. AMPLITUDES OF THE DECAY B0 →J/ψφφ R− − R R s shouldbe generalizedto include the effects of energyde- pendent widths and mixing by means of introducing the According to the diagrams shownin Fig. 1, let us rep- amplitudeG(R) withamputatedkinematicfactors(tobe 12 resent the B0 J/ψφφ decay amplitude as the sum of included below). Taking into account the effect of reso- the R and Xsr→esonance contributions, nancemixingingeneralcaseisoutlinedinAppendixBby Eq. (B1). In the case of two mixed resonances Eq. (B1) M =M +M . reduces to R X D Π(R) g 1 G1(R2) ≡G(1R2)(m2)=(cid:0)gBs0J/ψR1, gBs0J/ψR2 (cid:1) Π(1RR22) D1R21 !(cid:18)gRR12φφφφ (cid:19)DR1DR2 −Π(1R2)2. (3.2) Here, the common φφ mode is Π(R) Π(R)(m2)=ReΠ(R)(m2)+ DRi ≡DRi(m2)=m2Ri −m2−imΓRi→φφ(m2), (3.3) 12 ≡im12ΓR1→φφ(m2)ggR122φφ. (3.4) R1φφ with i = 1,2 where ΓRi→φφ(m2) for different quantum Analogously, the mixing of X resonances is taken into numbers of the R resonances are given by Eq. (2.8), account by introducing the amplitude with amputated 1,2 andthenondiagonalpolarizationoperatorwhichincludes kinematical factors: G1(X2) ≡G(1X2)(m2)=(cid:0)gBs0φX1, gBs0φX2 (cid:1) ΠD(1XX22) ΠD(1XX21) !(cid:18)ggXX12JJ//ψψφφ (cid:19)DX1DX21−Π(1X2)2. (3.5) The inverse propagator of the X resonances (i = 1,2) mixing is written analogously to Eq. (3.4): i that appears in Eq. (3.5) is Π(X) Π(X)(m2)=ReΠ(X)(m2)+ 12 ≡ 12 12 g DXi ≡DXi(m2)=m2Xi−m2−imΓXi→J/ψφ(m2), (3.6) imΓX1→J/ψφ(m2)gXX21JJ//ψψφφ. (3.7) The partial width of the X decay to J/ψφ is given by i where m is the mass of the X resonance. The non- Eq. (2.10). Equations (3.2) and (3.5) reduce to the sim- Xi i diagonal polarization operator responsible for X X ple sum of two Breit-Wigner contributions in the case of 1 2 − 4 vanishing mixing Π(R,X) 0. The explicit alternative where 12 → expression for the mixing amplitude valid in the case of 1 1 the two resonances mixed via the single common decay Pµν,λσ = (PµλPνσ +PµσPνλ) PµνPλσ, (3.12) 2 − 3 channel is given by Eq. (B2) in Appendix B. As for the Eqs. (3.3) and (3.4), their imaginary parts with originated from the φφ loop contribution are fixed by k k µ ν the unitarity relation. The realpartsare divergentwhen P P (k)= η + (3.13) µν ≡ µν − µν k2 calculated from the dispersion relation upon neglecting the vertex form factors, stands for the result of summation over the polar- izations of the intermediate tensor resonance; η = Π(1R2)(s)= πggR2φφ ∞ √ss′′ΓR1s→φφi(0s′)ds′. (3.8) idsiaugs(e1fu,−l t1o,−d1o,t−h1is).inWthheenresctalrceuflearteinncgemfraasmsespofectthµreνa,φiφt R1φφ Z4m2φ − − pair, k =(m ,0,0,0), where 12 In the case of JP = 0+ and 2+, when restricting to the m2 =(k +k )2 (3.14) S-waveapproximation,thedivergenceislogarithmicand 12 1 2 can be regularized by the subtraction at the resonance is the invariant mass squared of the φφ state. In this mass. However, there is the D wave contribution ne- frame, P reduces to the three-dimensional form ex- µν,λσ glected in Eq. (2.8) which makes the divergence much pressed through the Kronecker delta: stronger, and there are no model independent ways to fix the subtraction constants. The same holds for the 1 1 P = (δ δ +δ δ ) δ δ . (3.15) P wave decay of the JP = 0− resonance. Alternatively, mn,ls 2 ml ns ms nl − 3 mn ls one may insert the vertex form factors to make the loop The second necessary ingredient for obtaining the φφ integrations finite, but this again requires the fixing of and J/ψφ mass spectra in the decay B0 J/ψφφ is the additionalfreeparameterscharacterizingthe aboveform s → contribution of the X exchange schematically depicted factors. The same refers to the X resonances whose de- as the second and third Feynman diagrams in Fig. 1. It cay width contains the D-wave contribution essential in looks like the dispersion integral at large momenta. Inpractice,one canadoptthe followingwayofaction. M =ε ǫ(φ1)ǫ(J/ψ)ǫ(φ2) G(X)(m2 )(q+k ) Inthe presentcase,there areno sharpenergydependen- X µνλσ ν λ σ 12 13 1 µ− h cies of the loop effects like those observed in the case of G(X)(m2 )(q+k ) , (3.16) the difference of the K+K− and K0K¯0 loop contribu- 12 23 2 µ i tions [18, 19]. The decay kinematics is such that it in- where volves relatively narrowintervals of the invariant masses m2 =(q+k )2 (3.17) of the φφ and J/ψφ states. See Eqs. (2.2) and (2.3). 13 1 Hence,onecanignorethepossibledependenceonenergy and of the really incalculable smooth real parts of the loop contributions and take them as constants. In the case m2 =(q+k )2 (3.18) 23 2 of the diagonal polarization operators,the constants are absorbedinthe massesof the resonancesmR1,2, while in stand for the invariant mass squared of the J/ψφ1,2 thenondiagonalone,Π(R,X),itisincludedasthefreepa- states, respectively. Note that the amplitude (3.16) is 12 even under parity reflection and symmetric under per- rameter a(R,X) ReΠ(R,X), all these to be determined 12 ≡ 12 mutation of two φ mesons. Equation (3.16) can be sim- from the fit. plified when taking into account the small momenta of After these remarks,one can write the contributionof final particles. It is composed as the difference of two the mixed resonances in the φφ state. Lorenz-invariantexpressionseach of which canbe evalu- (a) R =0+. 1,2 ated in its respective rest reference frame: MR =2G(1R2)(ǫ(J/ψ)·k)(ǫ(φ1)·ǫ(φ2)), (3.9) MX =m13G(1X2)(m213) [ǫ1×ǫ2]·ǫ(J/ψ) 13− which is odd under the parity inversion. Hereafter k = m G(X)(m2 )(cid:16)[ǫ ǫ ] ǫ(J/ψ)(cid:17) , (3.19) k1+k2. 23 12 23 1× 2 · 23 − (cid:16) (cid:17) (b) R1,2 =0 . where indices 13 or 23 at the vector structures point to the rest reference frame of the state J/ψφ or J/ψφ . 1 2 MR =2G1(R2)(ǫ(J/ψ)·k)εµνλσk1µǫν(φ1)k2λǫσ(φ2), (3.10) Now, the three-dimensional polarization vector ǫ of the vector meson with the four-momentum (E,p) is ex- which is even under the parity inversion. pressed through its counterpart ξ in the rest frame: (c) R =2+. 1,2 p(ξ p) MR =2G(1R2)Pµν,λσǫ(µJ/ψ)qνǫλ(φ1)ǫσ(φ2), (3.11) ǫ=ξ+ m(E+· m). (3.20) 5 Hence, both three-dimensionalpolarizationstructures in are subjected to the constraint Eq.(3.19)canbe representedin the formwhichincludes only the polarization three-vectors in the rest frame, m212+m213+m223 =m2Bs0 +m2J/ψ+2m2φ ≡Σ, (4.1) one should takeinto accountthe approximatelynonrela- [ǫ ǫ ] ǫ(J/ψ) [ξ ξ ] ξ(J/ψ), 1 2 1 2 tivisticcharacteroftheproblemandkeeponlythelowest × · ≈ × · powers of the final particle momenta. In this case, the because they differ by the terms squared in momenta four-dimensional scalar product of two four-momenta is which can be neglected in the considered case. Under 1 p p 2 this approximation the amplitude of X exchange looks p p m m + 1 2 1 2 1 2 like · ≈ 2(cid:18)m1 − m2(cid:19) ≈ p2 m m +O . (4.2) MX ≈ G(1X2)(m213)m13−G(1X2)(m223)m23 × 1 2 m21,2! [hξ ξ ] ξ(J/ψ). i (3.21) Thenoneobtainstherequiredexpressionsforthespecific 1 2 × · quantumnumbersoftheresonanceintheφφstate. They are the following. TheexpressionsfortheφφandJ/ψφmassspectrainthe (a) R = 0+. In this case M is odd while M is 1,2 R X decay Bs0 →J/ψφφ are given in the next section. even under the space parity reflection hence they do not interfere. The modulus squared of the decay amplitude looks like IV. AMPLITUDES SQUARED AND MASS M 2 3 G(R)(m2 ) 2 λ(m2Bs0,m2J/ψ,m212) + SPECTRA IN THE DECAY B0→J/ψφφ | | ≈ 12 12 m2J/ψ s (cid:12) (cid:12) M(cid:12)(cid:12)X 2. (cid:12)(cid:12) (4.3) | | − (b) R = 0 . Here both R and X contributions 1,2 When calculating the modulus squaredof the relevant are even under parity reflection, hence the interference amplitude, M 2 M(m2 ,m2 ,m2 )2, where the in- is nonzero. The expression for the modulus squared of | | ≡ | 12 13 23 | variant masses squared in Eqs. (3.14), (3.17), and (3.18) the decay amplitude looks like 2 G(R)(m2 ) 12 12 M 2 λ(m2 ,m2 ,m2 )λ(m2 ,m2,m2)+ M 2+ | | ≈ (cid:12)(cid:12) 2m2J/ψ (cid:12)(cid:12) × Bs0 J/ψ 12 12 φ φ | X| (cid:12) (cid:12) 2m212(m2 m2 )Re G(R)∗(m2 ) G(X)(m2 )m G(X)(m2 )m . (4.4) m 13− 23 12 12 12 13 13− 12 23 23 J/ψ n h io 2 Note also that the lowest order even parity R contribu- 6 G(X)(m2 )m G(X)(m2 )m (4.6) tionresultsintheφmesonsinP wave,andbythisreason 12 13 13− 12 23 23 itcontainsadditionalfactorλ(m212,m2φ,m2φ)proportional stands for the c(cid:12)(cid:12)ontribution of the intermediate(cid:12)(cid:12)X res- (cid:12) (cid:12) to the momentum squared of the final φ meson. onance with quantum numbers JP = 1+. It should be (c) R1,2 = 2+. Similar to case (a) above, here R and emphasizedonceagainthatonlythelowestnonvanishing X contributions do not interfere because they have op- powersoftheparticlemomentaaretakenintoaccountin posite space parity. The modulus squared of the decay Eqs. (4.3), (4.4), (4.5), and (4.6). amplitude is The spectra of interest in the decay B0 J/ψφφ are s → given by the following expressions. The φφ spectrum is 2 5 G(R)(m2 ) |M|2 ≈ M(cid:12)(cid:12)(cid:12) 3122m.2J/ψ12 (cid:12)(cid:12)(cid:12) ×λ(m2Bs0,m2J/ψ,m212)+(4.5) dmdΓ12 = λ1/2(m2Bs0,m(22Jπ/)ψ3,×m62124)mλ13B/s02m(m12212,m2φ,m2φ) × | X| 1 M(m2 ,m2 ,m2 )2dx, (4.7) In the above expressions, Z−1| 12 13 23 | M 2 M 2(m2 ,m2 ) where the explicit expression for m2 to be inserted into | X| ≡| X| 13 23 ≈ 13 6 Eq. (4.7), in the rest frame of the φφ pair, is J/ψφ mass spectra are not normalized so that the mag- nitudes ofg , g , g ,andg have 1 x Bs0J/ψR1 Bs0J/ψR2 Bs0φX1 Bs0φX2 m2 = (Σ m2 ) λ1/2(m2 ,m2 ,m2 ) no absolute values. Hence one canobtainonly the ratios 13 2 − 12 − 2m212 Bs0 J/ψ 12 × of all except one to, say, gBs0J/ψR1. Here we fix the nor- λ1/2(m2 ,m2,m2). (4.8) malizationof the LHCb datain sucha waythatfor each 12 φ φ spectrum, φφ or J/ψφ, the plotted is the quantity Here, x is the cosine of the angle between the directions n (m) of one of the φ mesons, say φ , and the J/ψ meson, in bin 1 f (m)= , (5.1) exptl the rest frame of the φφ pair. The expression for m223 is binsnbin(m)∆m obtained from Eq. (4.8) by inverting the sign of x. where n (m)∆m is pProportional to the number of The expression for the J/ψφ mass spectrum is given bin events in the bin. Correspondingly, we plot the quan- by the expression tities dmdΓ23 = λ1/2(m2Bs0,m(22φπ,)m3×223)6λ41m/23B(s0mm22233,m2J/ψ,m2φ) × ftheor(mφφ)=Γ−to1tdmdΓφφ (5.2) 1 and M(m2 ,m2 ,m2 )2dx∗, (4.9) Z−1| 12 13 23 | f (m )=Γ−1 dΓ , (5.3) theor J/ψ totdm where one should insert J/ψφ where m2 =2(m2 +E E k k x∗), (4.10) 12 φ 1 2−| 1|| 2| Γ = mBs0−mJ/ψ dΓ dm = with tot dm φφ Z2mφ φφ E1 = m2Bs0 −2mm22φ3−m223, ZmmφB+s0m−Jm/ψφ dmdJΓ/ψφdmJ/ψφ, (5.4) k = λ1/2(m2Bs0,m2φ,m223), against the renormalized LHCb data. Notice that Γtot | 1| 2m23 is proportional to the Bs0 → J/ψφφ decay width. As compared to the notations adopted in Eqs. (4.7) and m2 +m2 m2 E2 = 23 2mφ2−3 J/ψ, (m4J.9/)ψ,φherme 2a3n.d in the figures one has mφφ ≡ m12 and ≡ λ1/2(m2 ,m2,m2 ) The results of fitting the normalized data [1] are rep- k2 = 23 φ J/ψ (4.11) resented in Table I. When fitting, we first take into ac- | | 2m 23 count the real parts of the polarization operators of the being the energy and momentum of the φ mesons in the mixing a(R) ReΠ(R) and a(X) ReΠ(X) as free pa- restreference frame ofthe J/ψφ system; x∗ is the cosine rameters.12Ho≡wever,1t2he fit ch1o2ose≡s zero v1a2lues of them, of the angle between the momenta of the φ mesons in and their inclusion does not result in the lowering of χ2, this frame. The direct numerical evaluation shows that hence they are set to zero. The corresponding curves the integrations of Eqs. (4.7) and (4.9) over the invari- are shown in Figs. 2 and 3 in the case of the scalar res- antmass intervalsEqs.(2.2) and (2.3), respectively,give onances R = 0+, in Figs. 4 and 5 in the case of the 1,2 coincident results. pseudoscalar resonances R =0−, and in Figs. 6 and 7 1,2 in the case of the tensor resonances R = 2+. Figures 1,2 8 and 9 demonstrate the comparison of the curves ob- V. APPLICATION tained in the framework of the above three models with the LHCb data [1]. In all these cases, the X resonance in the J/ψφ mass spectrum is considered to have the Let us apply the theoretical spectra obtained in the quantum numbers JP =1+ [8]. previous section to the description of available data [1]. When fitting experimental data, the so-called back- As it is pointed out in the Introduction, the presenta- ground contribution is sometimes included. Its form is tion of results in terms of masses andcoupling constants arbitrary. In the present work we have attempted to in- ofresonanceswithdifferentchannelsismoreinformative cludesuchbackgroundbyaddingthepointlikeamplitude thanthatintermsofmassesandpartialwidths. Inprin- with the lowest power of momenta. It looks like ciple, the model includes 14 free parameters which are M =c ε ǫJ/ψǫ ǫ (k k ) + m ,g ,g ,m ,g ,g ,a(R),m , point 1 µνλσ µ 1µ 2λ 1− 2 σ gaB(1RR2s0,1φXX)1B=,s0gJRX/eψ1ΠRJ/1(1ψR2φ,X,R)1mφaφrXe2t,aRkge2Bns0φtBXos02bJ,e/ψgcRXo2n1Js/tψaRnφ2tφ,,φaa(1sX2e1)x2,plwahinXeer1de cc32[[((ǫǫJJ//ψψ ··ǫǫ11))((ǫǫ22q·)k+1)(+ǫJ(/ǫψJ/·ψǫ2·)ǫ(2ǫ1)(·ǫq1)·]k+2)]+ earlier in this paper. However, the experimental φφ and c (ǫJ/ψ k)(ǫ ǫ ), (5.5) 4 1 2 · · 7 TABLEI.Theresonance parameters found from fittingthedataon theφφand J/ψφ mass spectraof thedecayB0 →J/ψφφ s [1]. Parameter/model R1,2 =0+ R1,2 =0− R1,2 =2+ m [GeV] 2.089±0.004 2.088±0.003 2.081±0.001 R1 g 3.6±0.4 GeV −10.5±0.6 GeV−1 −6.1±0.9 GeV R1φφ m [GeV] 2.191±0.006 2.209±0.003 2.211±0.001 R2 gBs0J/ψR2 1.1±0.2 0.6±0.2 1.5±0.2 ggBs0J/ψR1 2.1±0.5 GeV 3.3±0.5 GeV−1 3.7±0.2 GeV R2φφ m [GeV] 4.146±0.004 4.151±0.002 4.151±0.001 X1 gBs0φX1 [GeV] 0.9±0.2 0.8±0.1 1.5±0.3 ggBs0J/ψR1 −1.1±0.2 −0.7±0.2 −0.6±0.4 X1J/ψφ m [GeV] 4.247±0.002 4.248±0.001 4.247±0.001 X2 gBs0φX2 [GeV] 0.9±0.2 0.9±0.2 1.6±0.3 ggBs0J/ψR1 0.20±0.10 0.37±0.15 0.20±0.11 X2J/ψφ χ2/ndof 19/21 16/21 14/21 12 LHCb normalized LHCb normalized 12 R (0 +)+X R (0 +)+X 1,2 1,2 1,2 1,2 R (0 +) 10 R (0 +) 1,2 1,2 10 X X 1,2 1,2 8 8 1] 1] -V -V Ge e G [ 6 [ 6 myfJ/ mff d d ot tot G/t 4 G/ 4 Gd Gd 2 2 0 0 4.10 4.15 4.20 4.25 4.30 4.35 2.05 2.10 2.15 2.20 2.25 m [GeV] J/yf m [GeV] ff FIG.3. TheJ/ψφ mass spectrum in thedecayB0 →J/ψφφ s FIG. 2. The φφ mass spectrum in the decay Bs0 → J/ψφφ obtainedinthemodelwiththeR1,2 resonancequantumnum- in the model with the R1,2 resonance quantum numbers bers JP =0+. The designations of curvesare thesame as in JP =0+(solidcurve). ThecontributionsoftheR1,2andX1,2 Fig. 2. resonances are shown with dashed and dotted lines, respec- tively. LHCb data [1] are normalized to the unity in accord with Eq. (5.1). ting appears unsatisfactory. The reason for this, in the where the term with c is even under the space reflec- present case, seems to be the fact that, as one may ob- 1 tion while those with c are odd. We try to include serve in Figs. 2 – 7, the R φφ decay process in 2,3,4 1,2 → thiscontribution(uponneglectingtheterm c because Fig. 1 serves as the backgroundfor the J/ψφ mass spec- 1 ∝ it contains additional power of momentum as compared trum very much like the X J/ψφ decay plays the 1,2 → with the terms c ). However, the obtained fit- same role in the φφ mass spectrum. 2,3,4 ∝ 8 14 12 LHCb normalized R (0 -)+X +interf LHCb normalized 1,2 1,2 12 RX1,2(0 -) 10 RR1,2((22 ++))+X1,2 1,2 1,2 10 interf X1,2 8 8 1] 1] -V -V e e G G [ 6 [ 6 mff mff d d ot ot t 4 t G/ G/ 4 Gd Gd 2 2 0 -2 0 2.05 2.10 2.15 2.20 2.25 2.05 2.10 2.15 2.20 2.25 m [GeV] m [GeV] ff ff FIG. 4. The same as in Fig. 2, but in the model with the FIG. 6. R1,2 resonance quantum numbers JP = 0−. Also shown The same as in Fig. 2, but in the model with theR1,2 (dot-dashed line) is the contribution of the interference term resonance quantum numbersJP =2+. Eq. (4.4). 18 16 LHCb normalized LHCb normalized R (0-)+X +interf 16 R (2 +)+X 1,2 1,2 1,2 1,2 14 R (0-) R (2 +) X1,2 14 X1,2 12 1,2 1,2 interf 12 10 -1V] -1V] 10 e 8 e G G m [yfJ/ 6 m [yfJ/ 8 d d tot G/ 6 G/ 4 Gd Gd 4 2 2 0 0 -2 4.10 4.15 4.20 4.25 4.30 4.35 4.15 4.20 4.25 4.30 4.35 mJ/yf [GeV] mJ/yf [GeV] FreIsGon.a5n.cTehqeusaanmtuemasniunmFbiegr.s3JbPut=in0−th.emodelwiththeR1,2 FreIsGon.a7n.cTehqeusaanmtuemasniunmFbiegr.s3JbPut=in2+th.emodelwiththeR1,2 9 14 VI. DISCUSSION LHCb normalized R (0 +)+X 1,2 1,2 12 R (0 -)+X 1,2 1,2 Using the coupling constantsfound in fits one cancal- R (2 +)+X culate the central values of the partial decay widths of 1,2 1,2 the resonances R φφ and X J/ψφ. They are 1,2 1,2 10 the following. → → (A) R =0+. 1,2 1] -V 8 ΓR1→φφ =40 MeV, e [G ΓR2→φφ =22 MeV, mff ΓX1→J/ψφ =21 MeV, dtot 6 ΓX2→J/ψφ =9 MeV. G/ (6.1) Gd 4 (B) R =0−. 1,2 ΓR1→φφ =50 MeV, 2 ΓR2→φφ =33 MeV, ΓX1→J/ψφ =9 MeV, 0 ΓX2→J/ψφ =6 MeV. 2.05 2.10 2.15 2.20 2.25 (6.2) m [GeV] ff (C) R =2+. 1,2 tF(rsIuoGlmi.d8il.ninTet)hh,ee0c−momo(ddpeaalssrhiseoodfnltoohnfeed)Re,s1ac,n2ridprte2is+oonn(sadnoofctetteshdewlφiintφhe)m.JaPss=spe0c+- ΓΓRR21→→φφφφ ==1336 MMeeVV,; ΓX1→J/ψφ =7 MeV, ΓX2→J/ψφ =3 MeV. 16 (6.3) LHCb normalized 14 R (0 +)+X The accuracy of the R1,2 width evaluation is about 50 1,2 1,2 percent. As for the X resonances, their evaluated R (0 -)+X 1,2 1,2 1,2 widths spread from 7 to 21 MeV in the case of X and 1 R (2 +)+X 12 1,2 1,2 from 3 to 9 MeV in the case of X2. Such wide inter- vals are obtained upon evaluation with the parameters extracted from the fits with different assumptions about 10 1] spin parity of the R1,2 resonances. In some sense such -V largespreadinΓ canbeinterpretedasthemodelun- e X1,2 G 8 certaintysothatthe X widthevaluationis validupto [ the factor of 3. 1,2 myfJ/ One can observe in Figs. 4 and 5 that in the variant dtot 6 of R1,2 with JP = 0− the contribution of the R − X G/ interference term is relatively small. This is natural due Gd 4 to the different quantum numbers of the R and X 1,2 1,2 resonances: inthelimitoftheirvanishingwidthstheydo not interfere at all. 2 A few words about spectroscopic identification of the R andX resonancesconsideredinthepresentwork. 1,2 1,2 0 The masses of the R and R resonances obtained from 1 2 thefitsfallclosetothemassesoftheη(2100)andη(2225) 4.15 4.20 4.25 4.30 4.35 resonances observed by the BESIII collaboration [7] in m [GeV] the decay J/ψ γφφ but the central values of the cal- J/yf → culated widths are lower than those given in Ref. [7]. Weattributethistothe oversimplifiedassumptionofthe FIG. 9. The same as in Fig. 8 but for the J/ψφ mass spec- trum. single φφ decay mode of the R1,2 resonance made in the course of the present work. Also, one should have in 10 mind a rather largeuncertainty of the widths inRef. [7]. areclosetothemassesoftheX(4140)andX(4274)reso- The same refers to the cases of 0+ and 2+ with the pos- nances reportedin Refs. [8, 9]. More precise data on the sible identification R f (2100)(but without R ), and decay B0 J/ψφφ when (and if) appeared, together R f (2010) and R1 ≡ f0(2300), respectively. 2 with thesda→ta on the decay B+ J/ψφK+ [8, 9], could 1 2 2 2 ≡ ≡ → AsfortheX andX resonancesintheJ/ψφchannel, resolve the issue. The comparison of descriptions in the 1 2 their masses obtained here from the fits fall close to the models with different spin parity of the R resonances 1,2 masses of the X(4140) and X(4274) resonances cited in presented in Figs. 8 and 9 shows no essential difference, Ref. [8]. The central values of the evaluated widths are however, the χ2 value is lower in the case of JP = 2+. also lower than those given in [8]. However, taking into HavinginmindtherestrictedstatisticsoftheLHCbdata account the large model uncertainty up to the factor of [1], the inclusion of the sum of the contributions of the 3 of the evaluated widths, it seems that their values are R resonanceswithallpossiblespin-parityassignments 1,2 not in contradiction with the results of Ref. [8]. JP =0+,0−,2+ seems to be premature. The currenttheoreticalinterpretationsofthe X(4140) and X(4274)resonances as the exotic states [10–12] rely mainly on the masses and spin-parity assignments. Fur- Appendix A: Effective low momentum X →J/ψφ vertex ther information on their nature could be obtained from the model predictions for the coupling constants of the considered resonances to the pertinent final states to be Let us justify the expression (2.5) for effective low compared, in turn, with their magnitudes obtained from momentum vertex X(1+) J/ψ(1−)φ(1−). There are → the data fits presented here. three possibilities to get JP = 1+ for the X resonance, (S,L) = (1,0),(1,2), and (2,2), from the final state quantum numbers of spin S and angular momentum L. VII. CONCLUSION Hence, there should be three independent Lorenz struc- tures in the effective Lagrangian: 1 = ε g F(J/ψ)F(φ)F(X)+ In the present work, the attempt is made to describe Leff 2 µνλσ 1 µν λα ασ the LHCb data [1] on the φφ and J/ψφ mass spectra g F(φ)Fh(X)F(J/ψ)+ of the decay B0 J/ψφφ in the resonance model which 2 µν λα ασ takesintoaccousn→ttheR resonanceswithJP =0+,0−, g F(X)F(J/ψ)F(φ) , (A1) 1,2 3 µν λα ασ or 2+ in the φφ state and ones X , with JP = 1+, in 1,2 i the J/ψφ state, irrespective of their nature. Taken into where F(A) = ∂ V(A) ∂ V(A) stands for the field µν µ ν ν µ − account are the energy dependence of the partial widths strength of the vector field V(A) corresponding to the µ andthemixinginsideeachsectorarisingduetothecom- vector meson A. Doing this in the usual way, one can mon decay modes. Note that the popular parametriza- obtain effective vertex from the Lagrangian (A1). Ne- tionsoftheamplitudesneglectthiskindofmixing. How- glecting the D waves,one finds in the X rest frame that ever,webelievethatthemixingduetothecommondecay cshhoaunlndelbsebetiankgenthienmtoanacifceosutanttiobnecoafuthseeltohoispecffonectrtibisutdioicn- MX→J/ψφ = m4X "m4X −(mm2J/2Xψ −m2φ)2(g1+g2)− tated by unitarity and should exist in any effective the- ory. Thenear-thresholdkinematicshaspermittedoneto g (m2 m2 m2) 3 X − J/ψ− φ × restrict the large number of independent Lorenz struc- tures by a few, with the lowestpowers of momenta. The (ξ(X) [ξ(J/ψ) ξ(φ)]),i (A2) · × dataarestillnotpreciseenoughtomakefirmstatements where m and ξ(A) are the mass and the rest frame po- A aboutthe spectroscopyofthe resonancesR φφ and 1,2 → larizationthree-vectorofthe mesonA,respectively. One X J/ψφ. For example, as compared with the fits 1,2 → can denote the factors in front of the polarization struc- showninFig.3,5,7,and9,inwhichthepeakintheJ/ψφ ture as g and use the expression XJ/ψφ massspectrumislocatedatm 4.25GeV,thereare J/ψφ ≈ efiqtsuawlliythgothoedpχe2akvallouceast.edTahtemloJw/eψrφe≈xp4er.2im6eGnteaVl,pwoiitnht MX→J/ψφ =gXJ/ψφεµνλσpµǫ(νX)ǫJλ/ψǫ(σφ) (A3) located between the higher points at the above masses as the effective vertex of the X J/ψφ decay in the → doesnotexcludethecaseoftwonarrowresonanceswhile low momentum approximation adopted throughout the it does not permit one to attribute the higher points to paper. the different resonancesor to the different shoulders of a single wider resonance. Nevertheless, the masses of the R and R resonances in the φφ mass spectrum found Appendix B: Mixing of resonances 1 2 from the fits are close to the values cited in the litera- ture [6, 7]. The same refers to the resonances X and Let us make some remarks concerning the above ex- 1 X in the J/ψφ mass spectrum whose extracted masses pressions. As for Eqs. (3.2) and (3.5), one can include 2