Production and mixing of scalar mesons in η c and χ c1 decays
V. R. Debastiani, Wei-Hong Liang, Ju-Jun Xie, E. Oset, M. Bayar
aa r X i v : . [ h e p - ph ] J a n Production and mixing of scalar mesons in η c and χ c decays V. R. Debastiani ∗ Departamento de Física Teórica and IFIC, Centro Mixto Universidad de Valencia-CSICInstitutos de Investigación de Paterna, Aptdo. 22085, 46071 Valencia, SpainE-mail: [email protected]
Wei-Hong Liang
Department of Physics, Guangxi Normal University, Guilin 541004, ChinaE-mail: [email protected]
Ju-Jun Xie
Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, ChinaE-mail: [email protected]
E. Oset
Departamento de Física Teórica and IFIC, Centro Mixto Universidad de Valencia-CSICInstitutos de Investigación de Paterna, Aptdo. 22085, 46071 Valencia, SpainE-mail: [email protected]
M. Bayar
Department of Physics, Kocaeli University, 41380, Izmit, TurkeyEmail: [email protected]
We briefly discuss how the chiral unitary approach in coupled channels and SU ( ) symmetry canbe used to describe the production of f ( ) , f ( ) and a ( ) in the χ c → ηπ + π − reaction,recently measured by the BESIII collaboration. In this reaction a very strong peak for the a ( ) can be seen in the ηπ invariant mass, while clear signals for the f ( ) and f ( ) appear inthe one of π + π − . Next, we show the predictions made with the same model for the analogousdecay η c → ηπ + π − , which could also be measured experimentally. We discuss the differencesof these two reactions which are interesting to test the picture where these scalar mesons aredynamically generated from the interaction of pairs of pseudoscalars. Furthermore, we commenton a new recent work where the same model was used to study the a ( ) − f ( ) mixingin the χ c → π π η and χ c → π π + π − reactions, showing that quantitative agreement with theexperimental measurement of this mixing, also performed by BESIII, can be obtained, revealinginteresting aspects of the dynamics of this process and the importance of coupled channels. XVII International Conference on Hadron Spectroscopy and Structure - Hadron201725-29 September, 2017University of Salamanca, Salamanca, Spain ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/ roduction and mixing of scalar mesons in η c and χ c decays V. R. Debastiani
1. Introduction
The experiment on the χ c → ηπ + π − decay performed with high statistics by the BESIIIcollaboration [1], and previously by the CLEO collaboration [2], presents an interesting opportunityto test the picture where the scalar mesons f ( ) , f ( ) and a ( ) are dynamically generatedfrom the final state interaction of meson pairs π + π − and ηπ ± . Indeed, it is found that the mostdominant two-body structure comes from a ( ) ± π ∓ , with a ( ) ± → ηπ ± .First we will briefly discuss the work of Refs. [3, 4] where the chiral unitary approachand SU ( ) symmetry were used to describe the production of these three scalars in the BESIIIexperiment and to make predictions for the analogous reaction with η c instead of χ c . We willmake a short discussion on SU ( ) scalars and compare the treatment of the amplitude and massdistribution used to describe each decay. In the end we also comment on the recent work of Ref. [5],where the same model was used to study the a ( ) − f ( ) mixing in the χ c → π π η and χ c → π π + π − reactions, which was suggested in Ref. [6] and later measured by BESIII [7].
2. Common Formalism
As in Ref. [8] we start by considering that the charmonium states c ¯ c behave as a SU ( ) scalar,and use the following φ matrix to get the weight of every trio of pseudoscalar mesons created inthe χ c or η c decay φ ≡ √ π + √ η + √ η ′ π + K + π − − √ π + √ η + √ η ′ K K − ¯ K − √ η + q η ′ . (2.1)If we think of φ as a q ¯ q matrix, as discussed in Ref. [3], it is natural to build a SU ( ) scalar bytaking SU ( )[ scalar ] ≡ Trace ( φφφ ) , whereTrace ( φφφ ) = √ ηπ + π − + √ ηπ π + √ ηηη + π + K K − + π − K + ¯ K , where we have neglected the terms that cannot make a transition to the final state ηπ + π − , and alsothe terms containing η ′ , which plays only a marginal role in the building of the f ( ) , f ( ) , a ( ) resonances, because of its large mass and small couplings.In fact, there are four SU ( ) scalars: Trace ( φφφ ) , Trace ( φ ) Trace ( φφ ) , [ Trace ( φ )] andDet ( φ ) . But by the Cayley-Hamilton relation,2Trace ( φφφ ) − ( φ ) − ( φ ) Trace ( φφ ) + [ Trace ( φ )] = , (2.2)only three of them are independent. In Refs. [4, 8] we discussed other possibilities and concludedthat the best choice is indeed Trace ( φφφ ) , since it yields results in good agrement with the recentexperiment of BESIII [1] on the χ c → ηπ + π − decay. Indeed, in Ref. [5] we have also added thatthis is in fact expected from large N c counting, since each time one takes a trace a factor 1 / N c isintroduced [9, 10]. Besides, if one does not include the η − which we do through the inclusionof η − η ′ mixing, in order to relate the φ matrix with the q ¯ q matrix [3] − but instead take η → η and no η ′ , then Trace ( φ ) = ( φφφ ) .1 roduction and mixing of scalar mesons in η c and χ c decays V. R. Debastiani
Next, we use the chiral unitary approach to describe how the scalar mesons are dynamicallygenerated from the interaction of pairs of pseudoscalars in coupled channels. We follow theframework of Ref. [11], using an effective chiral Lagrangian where mesons are the degrees offreedom L = f π Trace [ ( ∂ µ φφ − φ∂ µ φ ) + M φ ] , (2.3)where φ is the matrix in Eq. (2.1), f π is pion decay constant and M = m π m π
00 0 2 m K − m π . (2.4)From this Lagrangian we extract the kernel of each channel, which in charge basis are: 1) π + π − , 2) π π , 3) K + K − , 4) K ¯ K , 5) ηη , 6) π η and can be found in Refs. [12, 13]. Thesekernels are used to build the V matrix which is then inserted into the Bethe-Salpeter equation,summing the contribution of every meson-meson loop. T = ( − V G ) − V , (2.5)where G is the meson-meson loop function, which we regularize with a cutoff using q max ∼ q and cos θ we have G = Z q max q dq ( π ) ω + ω ω ω [( P ) − ( ω + ω ) + i ε ] , (2.6)with ω i = q q + m i , P = s . Each kernel is projected in S -wave and a normalization factor isincluded when identical particles are present, which later needs to be restored. Finally, the T matrixwill give us the scattering and transition amplitudes between each channel, and isospin symmetryis used to obtain the amplitude of channels with different charges [3].
3. Theoretical description of χ c → ηπ + π − Following the assumption that c ¯ c behaves as a SU ( ) scalar, we look at the quantum numbersof the initial and final states, combining them in two cases: η leaves in P -wave while π + π − gothrough final state interaction with I = f ( ) and f ( ) in S -wave; and π − (or π + ) leaves in P -wave while ηπ + (or ηπ − ) go through final state interaction with I = a ± ( ) in S -wave.To illustrate our method, we will describe the case where η leaves in P -wave and π + π − interact. In this case we will consider the diagrams of Fig. 1. Then from the SU ( ) scalar in Eq.(2.2), we select the terms in which we can isolate one η and let the other pairs rescatter, since ourcoupled channels approach allows them to make a transition to π + π − final state, η √ π + π − + √ π π + √ ηη ! . (3.1)2 roduction and mixing of scalar mesons in η c and χ c decays V. R. Debastiani χ c χ c η π + π − η π + π − + Figure 1:
Diagrams considered in the description of f ( ) and f ( ) production in χ c → ηπ + π − reaction: tree-level (left) and rescattering of π + π − pair (right). Then we will have the sum of tree-level and rescattering: t η = V P (cid:0) ~ ε χ c · ~ p η (cid:1) h π + π − + ∑ i h i S i G i [ M inv ( π + π − )] t i , π + π − [ M inv ( π + π − )] ! , (3.2)where h i are the weights of Eq. (3.1), S i are symmetry and combination factors for the identicalparticles and V P provides a global normalization factor, which is adjusted to the data in the a ( ) peak. Finally, we can write the differential mass distribution for π + π − d Γ dM inv ( π + π − ) = ( π ) M χ c p η p η ˜ p π | t η | , (3.3)where p η is the η momentum in the χ c rest frame and ˜ p π is the pion momentum in the π + π − rest frame. Using this simple picture one can obtain a fair agreement with the experimental data ofRef. [1], as shown in Ref. [3] and further discussed in Refs. [4, 8].
4. Predictions for η c → ηπ + π − In the analogous reaction η c → ηπ + π − the dominant structure will be the one where everyfinal state meson goes out in S -wave. Therefore one must consider the interference between eachterm in the amplitude, then t = t tree + t η + t π + + t π − , t tree = V P h ηπ + π − . (4.1)Each of the later three terms is a function of an invariant mass, analogous to Eq. (3.2). Weselect M inv ( π + π − ) and M inv ( π + η ) as variables and the third one is determined by the relation: M = M η c + m π + m η − M − M . It is also necessary to consider the double differential massdistribution [14] d Γ dM inv ( π + π − ) dM inv ( π + η ) = ( π ) M η c M inv ( π + π − ) M inv ( π + η ) | t | , (4.2)where we need to integrate in one of the invariant masses to get the distribution of the other one.This way the background of π + η appears naturally in the π + π − mass distribution and vice-versa.3 roduction and mixing of scalar mesons in η c and χ c decays V. R. Debastiani
Since our approach is valid only for energies up to 1.2 GeV, we need to introduce a cut ineach amplitude to perform the integration. To do that we evaluate Gt ( M inv ) combinations up to M inv = M cut . From there on, we multiply Gt by a smooth factor to make it gradually decrease atlarge M inv , Gt ( M inv ) = Gt ( M cut ) e − α ( M inv − M cut ) , for M inv > M cut . (4.3)
650 700 750 800 850 900 950 1000 1050 1100 1150 12000246810121416182022 d / d M ( a . u . ) M (MeV) M cut = 1100 MeV no background
300 400 500 600 700 800 900 1000 1100 1200012345678910 d / d M ( a . u . ) M (MeV) M cut = 1100 MeV no background Figure 2:
Predictions from Ref. [4] for the mass distribution of πη (left) and π + π − (right) in η c → ηπ + π − ,using M cut = α = − , which reduce Gt by a factor 3, 5 and 10,respectively, at M cut +
300 MeV. The “no background” curve is obtained by keeping only the tree-level andthe main rescattering amplitude.
In Fig. 2 we show the predictions for the production of f ( ) , f ( ) and a ( ) in the η c → ηπ + π − decay. To see the effect of the background and interference introduced by consideringall the amplitudes in S -wave, we show with the solid curves, denoted by “no background”, theresults obtained by keeping only the tree-level and the main rescattering amplitude t π − [ M inv ( π + η )] in the case of a ( ) and t η [ M inv ( π + π − )] in the case of the f ( ) and f ( ) . a ( ) − f ( ) mixing in χ c → π π η ( π + π − ) This same model was recently used to study the a ( ) − f ( ) mixing in the χ c → π π η and χ c → π π + π − reactions in Ref. [5], showing that quantitative agreement with theexperimental measurement of this mixing, also performed by BESIII [7], can be obtained, revealinginteresting aspects of the dynamics of this process and the importance of the coupled channelsapproach. It was shown that the neutral a ( ) can be produced in the isospin-allowed mode χ c → π a ( ) → π π η while the isospin-violating production of f ( ) can be seen in the χ c → π f ( ) → π π + π − mode, where the proximity of both scalar resonances to the K ¯ K threshold, and the fact that both couple to the K ¯ K channel is responsible for the mixing.The difference in the mass of the charged and neutral kaons is the dominant cause of theisospin violation. The f ( ) production appears between the thresholds of K ¯ K and K + K − ,and there are two important process, K ¯ K → π + π − and π η → π + π − , where the latter one appearsdue to the coupled channels approach, and both sum up constructively. This latter one is possibleonly when different masses for the kaons are also considered in the propagators that go insidethe Bethe-Salpeter equation (2.5), and it was also found that the agreement with the experimentalmeasurement of the mixing is much better when this is included.4 roduction and mixing of scalar mesons in η c and χ c decays V. R. Debastiani
Acknowledgments
We would like to thank N. Kaiser for information concerning SU ( ) invariants and Feng-Kun Guo for the information about the 1 / N c factor. V. R. Debastiani wishes to acknowledgethe organizers of the event and the support from the Programa Santiago Grisolia of GeneralitatValenciana (GRISOLIA/2015/005). E. Oset wishes to acknowledge the support from the ChineseAcademy of Science in the Program of Visiting Professorship for Senior International Scientists(Grant No. 2013T2J0012). This work is partly supported by the National Natural ScienceFoundation of China under Grants No. 11565007, No. 11547307 and No. 11475227. It is alsosupported by the Youth Innovation Promotion Association CAS (No. 2016367). This work is alsopartly supported by the Spanish Ministerio de Economia y Competitividad and European FEDERfunds under the contract number FIS2014-57026-REDT, FIS2014-51948-C2-1-P and FIS2014-51948-C2-2-P, and the Generalitat Valenciana in the program Prometeo II-2014/068. References [1] M. Ablikim et al. [BESIII Collaboration], Phys. Rev. D , no. 3, 032002 (2017).[2] G. S. Adams et al. [CLEO Collaboration], Phys. Rev. D , 112009 (2011).[3] W. H. Liang, J. J. Xie and E. Oset, Eur. Phys. J. C , no. 12, 700 (2016).[4] V. R. Debastiani, W. H. Liang, J. J. Xie and E. Oset, Phys. Lett. B , 59 (2017).[5] M. Bayar and V. R. Debastiani, Phys. Lett. B , 94 (2017).[6] J. J. Wu and B. S. Zou, Phys. Rev. D , 074017 (2008).[7] M. Ablikim et al. [BESIII Collaboration], Phys. Rev. D , 032003 (2011).[8] V. R. Debastiani, W. H. Liang, J. J. Xie and E. Oset, arXiv:1707.07228 [hep-ph].[9] A. V. Manohar, arXiv:hep-ph/9802419[10] F. K. Guo, L. Liu, U. G. Meißner and P. Wang, Phys. Rev. D , 074506 (2013).[11] J. A. Oller and E. Oset, Nucl. Phys. A , 438 (1997); Erratum: [Nucl. Phys. A , 407 (1999)].[12] W. H. Liang and E. Oset, Phys. Lett. B , 70 (2014).[13] J. J. Xie, L. R. Dai and E. Oset, Phys. Lett. B , 363 (2015).[14] C. Patrignani et al. [Particle Data Group Collaboration], Chin. Phys. C , no. 10, 100001 (2016)., no. 10, 100001 (2016).