EExcited ρ mesons in B c → ψ ( (cid:48) ) KK S decays A. V. Luchinsky ∗ Institute for High Energy Physics, Protvino, Russia
In the presented paper exclusive decays B c → J/ψKK S and B c → ψ (2 S ) KK S are analyzed. Itis shown that contributions of the excited ρ mesons should be taken into account to describe thesedecays. It is also shown that, unlike the corresponding τ lepton decays, peaks in m KK S distributionscaused by these resonances are clearly seen and can be easily separated. Theoretical predictions forthe branching fractions of the reactions and m ψK distributions are also presented. I. INTRODUCTION
The lightest vector hadron, i.e. ρ (770) meson, has been studied in details. One cannot say the same, however, aboutit’s excited partners, ρ (1450), ρ (1570), and ρ (1700). For these mesons only neutral states were observed, mainly in ee and ππ channels. Their decays into KK pair is hard to detect.One of the reactions that can be used to observe KK decay of the charged excited ρ meson is the τ lepton decay τ → ν τ KK S . This process was first studied experimentally by the CLEO collaboration in 1996 [1]. Recently amore detailed result, obtained by BaBar collaboration, appeared in [2, 3]. According to papers [4] CEO data can beexplained theoretically using Flatte formalism [5] and taking into account contributions of three ρ mesons. It shouldbe interesting to check this approach on new BaBar data.There is, however, a fundamental problem with using τ lepton decays to analyze contributions of the excited ρ mesons. It is evident that in this reaction the available energy is limited by the mass of τ lepton, m τ = 1 . ρ (1570) can hardly be observed. It is clear, on the other hand, that a larger energy rangeis available in the decays of the heavier particles, e.g. B c meson. In a series of papers (see, for example, [6–11])it was shown how the QCD factorization theorem can be used to connect differential branching fraction of lightmesons’ production in exclusive τ lepton and B c meson decays. Predictions presented in these article are in goodagreement with experimental results [12–16]. It could be interesting to try such an approach for B c → J/ψKK S and B c → ψ (2 S ) KK S decays.The rest of the paper is organized as follows. In the next section we use data on τ → ν τ KK decay obtained byCLEO collaboration to determine the coupling constants of the excited ρ mesons decays into KK S pair. In sectionIII these results are used to make theoretical predictions for the branching fractions of B c → ψ ( (cid:48) ) KK S decays anddistributions over different kinematical variables. Short discussion is presented in the last section. Figure 1. Feynman diagram for τ → ν τ KK S decay ∗ [email protected] a r X i v : . [ h e p - ph ] D ec II. τ → ν τ KK S DECAYS
Let us first consider KK S pair production in τ lepton decay τ → ν τ KK S . The Feynman diagram describing thisprocess is shown in Fig. 1 and the corresponding amplitude can be written in the form M τ = G F √ u ν ( k ) γ µ (1 + γ ) u τ ( P ) F ( q )( p − p ) µ (1)where P , k , p , are the momenta of the initial lepton, τ neutrino and final K mesons respectively (in the following wewill neglect the difference in K and K S masses), q = p + p is the momentum of the virtual W boson, and F ( q ) isthe form factor of W → KK S transition. It is clear, that the quantum numbers of final KK S pair should be equal to G I ( J P ) = 1 + (1 − ), so this transition should be saturated by contributions of the charged ρ meson and its excitations.It is convenient to use the Flatte parametrization of the form factor [5] and write it in the form F ( s ) = (cid:88) i c Ki BW i ( s ) , (2)where the summation is performed over the intermediate ρ mesons (in the following we will take into account onlycontributions of the ground state ρ (770) and two excited mesons ρ (cid:48) = ρ (1450) and ρ (cid:48)(cid:48) = ρ (1700)), c Ki are the couplingconstants, BW i ( s ) = m i m i − s − i √ s Γ i ( s ) , (3) m i is the mass of the corresponding particle, and Γ i ( ρ ) is the energy dependent width of ρ → π decay. Since final π mesons in these decays are in P wave state, the latter width can be calculated asΓ i ( s ) = m i s (cid:18) − m π /s − m π /m i (cid:19) / Γ i (4)where Γ i = Γ i ( m i ) is the decay widths of the corresponding meson on its mass shell.The model parameters m i , Γ i , and c Ki can be determined from analysis of the experimental data, especially q -distributions in the considered decay. If we are interested only in q distributions of the considered decays, we can useformalism described in [17]. In this framework the differential width of τ → ν τ KK S decay can be written as d Γ( τ → ν τ KK S ) d (cid:112) q = 2 (cid:112) q G πm τ ( m τ − q ) m τ ( m τ + 2 q ) ρ T ( q ) , (5)where ρ T ( q ) = (cid:18) − m K q (cid:19) / (cid:12)(cid:12) F ( q ) (cid:12)(cid:12) π (6)is the transversal spectral function of W → KK S transition. Experimental analysis of the considered decay wasperformed, for example, by CLEO [1] and BaBar [2, 3] collaborations. In paper [4] obtained by CLEO collaborationresults were used to determine the values of the model parameters m i , Γ i , and c Ki . According to this paper in orderto describe CLEO results the following values of the parameters should be used m ρ = 775 MeV , Γ ρ = 150 MeV , c Kρ = 1 . ± .
009 (7) m ρ (cid:48) = 1465 MeV , Γ ρ (cid:48) = 400 MeV , c Kρ (cid:48) = − . ± .
010 (8) m ρ (cid:48)(cid:48) = 1720 MeV , Γ ρ (cid:48)(cid:48) = 250 MeV , c Kρ (cid:48)(cid:48) = − . ± . . (9)In the left panel of Fig. 2 we show the resulting q -dependence of the the differential width in comparison with obtainedby CLEO and BaBar collaboration experimental results and It is clear that the agreement with this results is prettygood. The contributions of the exited ρ mesons (especially ρ (cid:48)(cid:48) one), however, can hardly be seen since these mesons liealmost on the upper limit of the allowed phase space. Indeed, the relation (5) is universal and only spectral functiondepends on the final hadronic state, so this relation can be rewritten in the form d Γ( τ → ν τ KK S ) d (cid:112) q = d Γ( τ → ν τ µν µ ) d (cid:112) q ρ T ( q ) ρ µνT ( q ) , (10) ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ⅆ B r ( τ → ν τ KK S ) / ⅆ q , - / G e V ( a ) ρ ' ρ '' 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.70.00.51.01.5 q, GeV ⅆ B r ( τ → ν τ μ ν μ ) / dq , - / G e V ( b ) ρ ' ρ '' 1.0 1.2 1.4 1.6 1.8 2.00.000.050.100.150.20 q, GeV ρ T KK ( q ) , - ( c ) ρ ' ρ '' Figure 2. Differential width of τ → ν τ KK S decayFigure 3. Feynman diagram for B c → KK S decay where the transverse spectral function of the leptonic pair is ρ µνT ( q ) = 1 / (16 π ). Transferred momentum distributionof the semileptonic τ decay is shown in figure 2 and it is clearly seen that in the region of excited ρ mesons is stronglysuppressed. That is why it could be interesting to study production of KK S pair in some other experiments. In thenext section we will perform the calculation of B c → ψ ( (cid:48) ) KK S decays and show that in this case the contributions ofthe excited states are much more clear. III. B c → ψ ( (cid:48) ) KK S DECAYS
The decay B c → ψ ( (cid:48) ) KK S is described by shown in Fig. 3 Feynman diagram. The corresponding matrix elementcan be written as M ( B c → ψ ( (cid:48) ) KK S ) = G F V cb √ a (cid:104) V − A (cid:105) µ F ( q )( p − p ) µ , (11)where a is the Wilson coefficient, that describe the effect of soft gluon interaction [18], B c → ψ ( (cid:48) ) W transition isdescribed by the matrix element (cid:104) V − A (cid:105) µ = (cid:104) M ψ A ( q ) q µ q ν q + ( M B c − M ψ ) A ( q ) (cid:18) g µν − q µ q ν q (cid:19) − A ( q ) q ν (cid:32) P µ + k µ − M B c − M ψ q q µ (cid:33) − i V ( q ) M B c + M ψ e µναβ P α k β (cid:105) (cid:15) ν , (12)where P , k , p , are the momenta of the B c meson, final vector charmonium, and K mesons respectively, (cid:15) µ is thepolarization vector of ψ ( (cid:48) ) , q = P − k is the momentum of virtual weak boson, M B c and M ψ are the masses of thecorresponding particles, and V ( q ), A , , ( q ) are dimensionless form factors, whose numerical values will be discussedlater.If we are interested in q distribution only, we can use the formalism of the spectral functions and the corresponding ( q ) A ( q ) A ( q ) Figure 4. B c → J/ψW form factors. Solid blue and dashed red lined correspond to SR and PM form factor sets respectively decay width is equal to d Γ( B c → ψ ( (cid:48) ) KK S ) dq = G V cb a ρ T ( q )128 πM B c M ψ ( M B c + M ψ ) (cid:115) − ( M ψ + q ) M B c (cid:115) − ( M ψ − q ) M B c × (cid:104) ∆ (cid:0) ∆ A (cid:0) q (cid:1) + 8 M ψ q V (cid:0) q (cid:1)(cid:1) +∆ ( M B c + M ψ ) A (cid:0) q (cid:1) − ( M B c + M ψ ) A (cid:0) q (cid:1) A (cid:0) q (cid:1) (cid:105) , (13)where ∆ = M B c − M B c (cid:0) M ψ + q (cid:1) + (cid:0) M ψ − q (cid:1) (14)∆ = M B c − M B c (cid:0) M ψ + q (cid:1) + M ψ + 10 M ψ q + q , (15)∆ = M B c − M B c (cid:0) M ψ + q (cid:1) + M B c (cid:0) M ψ + 2 M ψ q + 3 q (cid:1) − (cid:0) M ψ − q (cid:1) (cid:0) M ψ + q (cid:1) (16)Let us discuss the parameterizations of the B c → J/ψKK S decay first. It is clear that the corresponding formfactors are essentially non-perturbative, so some other methods such as QCD sum rules of Potential Models shouldbe used for their calculation. This topic is widely discussed in the literature. In the following we will use the resultspresented in works [19] (QCD sum rules were used in this work, in the following we will refer to it as SR) and [20](in this case the author use potential model, PM in the following). It is clear that A ( q ) form factor does not givecontributions to the process under consideration. Transferred momentum dependence of all other form factors formodels used in our work is shown in figure 4. Using these values it is easy to see that the branching fractions of thedecay in different form factors models are equal toBr SR ( B c → J/ψKK S ) = (6 . ± . × − , (17)Br P M ( B c → J/ψKK S ) = (3 . ± . × − , (18)where the uncertainty is caused by the experimental error in τ → ν τ KK S branching fractions [2, 3]. The corresponding (cid:112) q distributions are shown in figure 5(a). One can see that, unlike τ → ν τ KK S decay, the contributions of theexcited ρ mesons are clearly seen and can be easily separated. It is because in the case of B c meson decay thebranching fraction of the semileptonic reaction B c → J/ψµν is not suppressed in q ∼ m ρ (cid:48) region [see figure 5(b)]. It isalso interesting to note that form of the distributions produced by different form factor sets is almost the same withthe only difference in overall normalization. The reason is that, as it can be seen from the left panel of the Figure. 4,in the significant for our task energy region SR and PM form factors are almost proportional to each other.The distribution of the considered branching fraction over the invariant mass of J/ψK pair can also be observedexperimentally. It is clear, that this distribution cannot be obtained using spectral function formalism, so we need to ⅆ B r ( B c → J / ψ KK S ) / ⅆ q , - / G e V ( a ) ρ ' ρ '' 0.0 0.5 1.0 1.5 2.0 2.5 3.0051015202530 q, GeV ⅆ B r ( B c → J / ψ μ ν ) / ⅆ q , - / G e V ( b ) ρ ' ρ '' 4.0 4.5 5.0 5.50.000.010.020.030.040.05 m ψ K , GeV ⅆ B r ( B c → J / ψ KK S ) / ⅆ m ψ K , - / G e V ( c ) Figure 5. B c → J/ψµν and B c → J/ψKK S distributions. Solid blue and dashed red lines correspond to SR and PM formfactor sets respectively. Vertical dashed lines show the position of excited ρ resonances ( q ) A ( q ) A ( q ) Figure 6. B c → ψ (2 S ) W form factors. Notations are the same as in figure 4 calculate the corresponding squared matrix element. As a result we have d Γ( B c → ψ ( (cid:48) ) KK S ) dq dm ψ = G V cb a | F ( q ) | π M B c M ψ ( M B c + M ψ ) (cid:110) − M B c + M ψ ) ( m ψ − m ψ ) A A ( M B c − M ψ − q ) − M ψ V ( M B c (4 m K − q ) − M B c (4 m K − q )( M ψ + q )+4 m K ( M ψ − q ) + q ( m ψ − m ψ m ψ + m ψ − ( M ψ − q ) ))+( m ψ − m ψ ) A ( M B c − M B c ( M ψ + q ) + ( M ψ − q ) )+( M B c + M ψ ) A ( − (16 m K M ψ − ( m ψ − m ψ ) − M ψ q )) (cid:111) , (19)where m ψ , = ( k + p , ) are the corresponding Dalitz variables (according to momentum conservation q + m ψ + m ψ = M B c + M ψ + 2 m K ). The corresponding distribution is shown in Fig. 5(c). It should be noted that two peaksin these distributions do not correspond to any resonances, but come from the form of B c → ψ ( (cid:48) ) µν matrix element.The form factors of B c → ψ (2 S ) W transition were also studied, for example, in papers [19, 20] and we show themin figure 6. Using these form factors it is easy to calculate the branching fractions of B c → ψ (2 S ) KK S decay indifferent models: Br SR ( B c → J/ψKK S ) = (2 . ± . × − , (20)Br P M ( B c → ψ (2 S ) KK S ) = (1 . ± . × − . (21)The distributions over KK S and ψ (2 S ) K invariant masses are shown in Fig. 7. Note that in this case the forms of q distributions for different form factor sets are quite different from each other. IV. CONCLUSION
In the presented article production of KK S pair in exclusive τ and B c decays is discussed. It is clear that thisfinal state can be produced only from decay of virtual vector charged particle, i.e. ρ meson and its excitations. As a ⅆ B r ( B c → ψ ( S ) KK S ) / ⅆ q , - / G e V ( a ) ρ ' ρ '' 0.0 0.5 1.0 1.5 2.0 2.50.00.10.20.30.40.5 q, GeV ⅆ B r ( B c → ψ ( S ) μ ν ) / ⅆ q , - / G e V ( b ) ρ ' ρ '' 4.5 5.0 5.50.000.050.100.150.200.25 m ψ K , GeV ⅆ B r ( B c → ψ ( S ) KK S ) / ⅆ m ψ K , - / G e V ( c ) Figure 7. B c → ψ (2 S ) µν and B c → ψ (2 S ) KK S distributions. Notations are the same as in Fig. 5 result, experimental investigation of the decays can give us additional information about masses and widths of theseparticles and the coupling constants of ρ ( (cid:48) ) → KK S decays.The decay τ → KK S ν τ was studied experimentally, for example, in the recent BaBar papers [2, 3]. According toanalysis presented in [4], these results can be explained by taking into account contributions of ρ (770) meson and itstwo excitations, ρ (1450) and ρ (1700). It is clear, however, that τ lepton’s mass is not very large, so peak caused bythe last resonance peak cannot be seen in m KK S distribution. For this reason it could be interesting to study KK S pair production in decays of a heavier particle, e.g. B c meson.In our paper we perform such an analysis and give theoretical description of B c → J/ψKK S and B c → ψ (2 S ) KK S decays. It is clear, that the form factors of B c → ψ ( (cid:48) ) transitions are required for calculations of these decays, so in ourwork we used two different sets of these form factors, obtained using QCD sum rules and Potential models. Accordingto our results, peaks caused both by ρ (1450) and ρ (1700) resonances are clearly seen in m KK S distributions and canbe easily separated. The branching fractions of the considered decays are also calculated.It is clear, that the final K S meson will be detected in K S → ππ decay, so observed state of the considered heredecays will be ψ ( (cid:48) ) Kππ . According to [11] the same final state can be produced also in the decay chain B c → ψ ( (cid:48) ) K → ψ ( (cid:48) ) Kρ → ψ ( (cid:48) ) Kππ and the branching fractions of these reactions are significantly larger than the branching fractionsof the decays considered in our article. It should be noted, however, that the same can also be said about thecorresponding τ lepton decays, but both decays modes were observed.The author would like to thank A.K. Likhoded and Dr. Filippova for fruitful discussions. The work was carriedout with the financial support of RFFBR (grant 19-02-00302). [1] T. E. Coan et al. (CLEO), Phys. Rev. D53 , 6037 (1996).[2] J. P. Lees et al. (BaBar), Phys. Rev.
D98 , 032010 (2018), arXiv:1806.10280 [hep-ex].[3] S. I. Serednyakov (BaBar), in (2018) arXiv:1810.06242 [hep-ex].[4] C. Bruch, A. Khodjamirian, and J. H. Kuhn, Eur. Phys. J.
C39 , 41 (2005), arXiv:hep-ph/0409080 [hep-ph].[5] S. M. Flatte, Phys. Lett. , 224 (1976).[6] A. K. Likhoded and A. V. Luchinsky, Phys. Rev.
D81 , 014015 (2010), arXiv:0910.3089 [hep-ph].[7] A. K. Likhoded and A. V. Luchinsky, Phys. Rev.
D82 , 014012 (2010), arXiv:1004.0087 [hep-ph].[8] A. V. Berezhnoy, A. K. Likhoded, and A. V. Luchinsky, (2011), arXiv:1104.0808 [hep-ph].[9] Z.-G. Wang, Phys. Rev.
D86 , 054010 (2012), arXiv:1205.5317 [hep-ph].[10] A. V. Luchinsky, Phys. Rev.
D86 , 074024 (2012), arXiv:1208.1398 [hep-ph].[11] A. V. Luchinsky, (2013), arXiv:1307.0953 [hep-ph].[12] R. Aaij et al. (LHCb), Phys. Rev.