The "molecular" charmonium models predict both I = 0 and I = 1 states. The closeness of one of these (4.028 GeV) to one of its main thresholds and the large electromagnetic mass splitting of its daughters may induce a large mixing between states of different isospin. This would manifest itself in deviations from unity of the ratio of charged to neutral decay modes. © 1977 The American Physical Society

Bander, M

Bander, Myron

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UC IrvineUC Irvine Previously Published WorksTitleIsospin mixing in charmonium statesPermalinkhttps://escholarship.org/uc/item/5pn8t1n2JournalPhysical Review D, 16(9)ISSN2470-0010AuthorBander, MyronPublication Date1977-11-01DOI10.1103/physrevd.16.2900Copyright InformationThis work is made available under the terms of a Creative Commons Attribution License, availalbe at https://creativecommons.org/licenses/by/4.0/ Peer reviewedeScholarship.org Powered by the California Digital LibraryUniversity of CaliforniaPH YSICAL REVIEW D VOLUME 16, N UMBER 9Isospin mixing in charmoninm states*1 NOVEMBER 1977Myron BanderDepartment of Physics, University of California, Irvine, California 92717(Received 21 June 1977)The "molecular" charmonium models predict both I = 0 and I = 1 states. The closeness of one of these(4.028 GeV) to one of its main thresholds and the large electromagnetic mass splitting of its daughters mayinduce a large mixing between states of different isospin. This would manifest itself in deviations from unityof the ratio of charged to neutral decay modes.The rich structure observed in e'e annihilationin the energy region between the g' and 4.4 GeV(Ref. 1) has led to the hypothesis that at least someof these states are four-quark composites, ' ' ormore specifically bound states of DD, DD*, andand D D*."' The appellation "molecular"charmonium has been given to these states. Theisospin of these states can be zero or one. Thedetermination of the l=1 character of any of thesestates would be the cleanest way of distinguishingthem from cc or "atomic" charmonium states,whose isospin is zero.Two tests for the existence of I= 1 states wereproposed in Ref. 2: (1) a search for predominantlyeven m decay modes of any of these resonancesand (2) peaks in inclusive charged-v distributionsresulting from the decay of a higher-lying res-onance into a charged one with lower mass. Thefirst method is, to a large extent, moot as thedominant decay of these resonances is to charmedmesons and not directly to v's. The second methodis valid. In this paper we wish to propose anothertechnique, which, even after the hypothetical dis-covery of such states, will have an interest of itsown.One of the most prominent resonances is the oneat 4.028 QeV. ' The method proposed depends onthe existence of a resonance of opposite isospinwith a mass near the above one. We shall makethis more precise below. The aforementionedrich structure and other estimates' make thishypothesis plausible. If the above is true then thecloseness of 4.028 GeV to the D D~ threshold andthe D~'-D*' mass difference will induce a sizablemixing of the I= 0 and I= 1 states. This wonldmanifest itself in the deviation of the ratioI'(4.028 D D )I'(4.028 —D'D )The masses of the charmed mesons are 1IIDo =1865+3 MeV, MD =1875+5 MeV, MD+o-2005. 7+1.5MeV, and MD*+ —-2010+3 MeV. In spite of thecloseness of the (4.028) to the D D* thresholds,this resonance decays predominantly into thischannel; I'(4.028 —D* D*'): I'(4.028 —D D*+ DOD 0): r(4.028 -D D ) = (0.48 + 0.8): (0.47 + 0.08):(0.05+0.03). Though not precisely established,the widths of the resonances above the g appearto be between 30 and 50 MeV. ' For subsequentestimates we shall use 40 MeV as a typical width;larger values would enhance the deviation of R.For the sake of discussion let us assume thatthe resonance at 4.028 GeV has isospin zero andthere is an isospin-one state nearby. None of ourresults would change if the isospin situation werereversed. The Wigner-Weisskopf techniques, asdiscussed in Ref. 6, applied to this two-level sys-tem mix isospins in the (4.028) state:14.028) = 11=0)+ (2)The states ~1) are eigenstates of isospin. WithH, the interaction responsible for the decay ofthis resonance we obtain-&0)H, [k, o)&k, o [H, [I&Mo —Eq+ iE (3)where M, = 4.028 QeV. Restricting the sum overo.' in (3) to D*D* intermediate states, we can easilyfind q, the imaginary part of the off-diagonal ele-ment of the mass matrix:4 is the difference of the diagonal elements of themass matrix,(4)andI' =2gg dk 6(M —E )[&i ~H, ~k, )~from unify (in addition to the P wave phase-space-ratio of 1.1). Such mixings have been discussed inconnection with the K'-K' and co- p systems. 'Before discussing the mixing formalism, wewill review some pertinent experimental facts. '2Mq =go&i 1 — M 02M *.1—Mo(6)16 2900IS Q SPIN MI XIN 6 IN CHAR MQ N I U M STATES 2901g„g, (Ref. 6) entail all parts of the matrix elementsof H, to the I=O and I=1 states except for the P-wave phase space. In the same notation the par-tial width of the I= 0 state (aside from small cor-rections due to mixing) to D~D* is1",(D*D*)= 2g,' 1—Mo2MD +++0Ass"m '"g' le I = lgo I and using the masses andwidths discussed previously, we obtainI~l = r, (D*D*)/4=6 MeV.More relevant for experimental consequences ofthis interference is the real part, p, of the mix-ing. This value is difficult to obtain without a de-tailed model for the off-shell effects in the decay.%'e may estimate it by evaluating a crude disper-sion integral in M, of the imaginary part. Forthis absorptive part we take the values implied byEg. (6) from the D*'D*' threshold to Mo+ 1,. Atthis value we set the absorptive part to zero (theartificial oscillations due to this sharp cutoff dono affect the real part at M,). With this procedurelpl=5 Mev. (9)[One often expects real parts to vanish at a res-onance peak. However, in this case the imagin-ary part, Eq. (6), has not reached a maximum atM„but is still rising, and we obtain a finite realpart. ]Returning to Eq. (2) and Eq. (1) we obtain1-(p-ie)/& '1+ (p —uI)/~=1 —4p/(M -M ) . (10)Valuable discussions with Dr. T. Curtright, Dr.S. Meshkov, and Dr. G. Shaw are gratefully ack-nowledged.In the above we have assumed that M, -M,&,'-(1', —I'o). If the splitting of the I= 1 and I= 0resonances turns out to be of the order of the&width this ratio cvould deviatefrom unity by asmuch as SOVo.This effect would likewise show up in the anal-ogous decays into the DD + DD* modes. Thoughthe branching ratio into this channel is much high-er than into DD the latter one is subject to fewerexperimental ambiguities. Should a resonanceexist just above the DD* threshold, a similarargument could be repeated for it. Admixturesfrom nonresonant DD backgrounds are not expectedto be significant, as recent results from thedirect electron counter (DELCO) group'o indicatethat such nonresonant production is very small.This analysis provides a method for testing thehypothesis that states of opposite isospin are lo-cated in the 4 GeV to 4.1 GeV region. If thesestates are found by other means, the mixing dis-cussed here will still have other interests, es-pecially in studying the properties of the decayHamiltonian. To base the parameters of thismixing on firmer theoretical arguments we need,in addition to the existence of both isospin statesand a knowledge of their masses and width, abetter determination of the real part. "*%ork supported in part by the National Science Foun-dation.~J. Siegrist et al. , Phys. Rev. Lett. 36, 700 (1976);H. Meyer, DESY Report No. 77/19 (unpublished).~M. Bander, G. L. Shaw, P. Thomas, and S. Meshkov,Phys. Bev. Lett. 36, 695 (1976).3C. Rosenzweig, Phys. Rev. Lett. 36, 697 (1976).4M. B.Voloshin and I . B.Okun, Zh. Eksp. Teor. Fix.Pis'ma Red. 23, 369 (1976) [JETP Lett. 23, 333 (1976)l.5A. De RGjula, H. Georgi, and S. Glashow, Phys. Rev.Lett. 38, 317 (1977).6There is an extensive literature on this subject. Threerecent sources containing references to earlier arti-cles are P. K. Kabir, Tge CP Purple (Academic, NewYork, 1968); K. Gottfried, in Proceedings 29T1 Inter-national Symposium on Electron and Photon Interactionsat High Energies, edited by N. B. Mi.stry (Cornell Univ.Press, Ithaca, 1972};F. M. Renard, in Springer Tractsin Modern Physics, edited by G. H'ohler (Springer,New York, 1972), Vol. 63, p. 98.J.Kiss, presented at the Fifth International Conferenceon Meson Spectroscopy, 1977 (unpublihhed; G. J. Feld-man et al ., Phys. Bev. Lett. 38, 1313 (1977).%'e have assumed isospin invariance in the decay matrixelements. A violation of isospin in these matrix ele-ments would yield another source of mixing.~The fact that all the widths above 4 GeV are of compar-able magnitude justifies this assumption.~OP rivate communication from R. Burns.~'Although the Q value of these decays is very small,Coulomb effects are not expected to be important dueto the P-wave character of this decay.

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Isospin mixing in charmonium states

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Isospin mixing in charmonium states

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