First, recent work on light scalar mesons, which is of possible interest in connection with the strong coupling region of QCD is briefly discussed. Then a very short highlighting of a paper concerned with an application to the eta -\u3e 3 pi problem is presented

Abdel-Rehim, Abdou

Black, Deirdre

Fariborz, Amir H.

Schechter, Joseph

English

Syracuse University Research Facility and Collaborative Environment

Syracuse University SURFACE Physics College of Arts and Sciences 4-17-2003 The Scalar Sector and the Eta -> 3 Pi Problem Joseph Schechter Department of Physics, Syracuse University, Syracuse, NY Abdou Abdel-Rehim Syracuse University Deirdre Black Jefferson Laboratory Amir H. Fariborz State University of New York Institute of Technology Follow this and additional works at: https://surface.syr.edu/phy  Part of the Physics Commons Recommended Citation Schechter, Joseph; Abdel-Rehim, Abdou; Black, Deirdre; and Fariborz, Amir H., "The Scalar Sector and the Eta -> 3 Pi Problem" (2003). Physics. 280. https://surface.syr.edu/phy/280 This Article is brought to you for free and open access by the College of Arts and Sciences at SURFACE. It has been accepted for inclusion in Physics by an authorized administrator of SURFACE. For more information, please contact surface@syr.edu. arXiv:hep-ph/0304177v1  17 Apr 2003February 1, 2008 20:54 WSPC/Trim Size: 9in x 6in for Proceedings schechterTHE SCALAR SECTOR AND THE η → 3pi PROBLEMA. ABDEL-REHIMA, D. BLACKB , A. H. FARIBORZC AND J.SCHECHTERD∗(A) Physics Department,Syracuse University,Syracuse, NY 13244-1130,USA.E-mail: abdou@phy.syr.edu(B) Theory Group, Jefferson Laboratory,12000 Jefferson Avenue,Newport News, Virginia, 23606, USA.E-mail: dblack@jlab.org(C) Department of Mathematics/Science,State University of New York Institute of Technology,Utica, NY 13504-3050, USA.E-mail: fariboa@sunyit.edu(D)Physics Department,Syracuse University,Syracuse, NY 13244-1130, USA.E-mail: schechte@phy.syr.eduFirst, recent work on light scalar mesons, which is of possible interest in connectionwith the strong coupling region of QCD, is briefly discussed. Then a very shorthighlighting of a paper concerned with an application to the η → 3pi problem ispresented.1. IntroductionAt very large energy scales, the asymptotic freedom of QCD guarantees thata controlled perturbation expansion is a practical tool. At very low energyscales, for example close to the threshold of ππ scattering. the running QCDcoupling constant is expected to be large and perturbation theory is not∗Speaker1February 1, 2008 20:54 WSPC/Trim Size: 9in x 6in for Proceedings schechter2expected to work. Fortunately, a controlled expansion based on an effectivetheory with the correct symmetry structure- Chiral Perturbation Theory1-seems to work reasonably well. The new information about Strong Inter-actions which this approach reveals is closely related to the spectrum andflavor ”family” properties of the lowest lying pseudoscalar meson multipletand was, in fact, essentially known before QCD.Clearly it is important to understand how far in energy above thresholdthe Chiral Perturbation Theory program will take us. To get a roughestimate consider the experimental data for the real part of the I = J = 0ππ scattering amplitude, R00 displayed in Fig. 1. The chiral perturbationseries should essentially give a polynomial fit to this shape, which up toabout 1 GeV is crudely reminiscent of one cycle of a sine curve.Roos     (GeV)Figure 1. Illustration of the real part of the pi pi scattering amplitude extracted fromexperimental data.Now consider polynomial approximations to one cycle of the sine curvewith various numbers of terms. These are illustrated in Fig. 2. Note thateach succesive term departs from the true sine curve right after the preced-ing one. It is clear that something like eight terms are required for a decentfit. This would correspond to seven loop order of chiral perturbation theoryand seems presently impractical.2. Need for light scalar mesonsThus an alternative approach is indicated for going beyond threshold ofpi pi scattering up to about 1 GeV. The data itself suggests the presenceof s-wave resonances, the lowest of which is denoted the ”sigma”. Physi-cally, one then expects the practical range of chiral perturbation theory toFebruary 1, 2008 20:54 WSPC/Trim Size: 9in x 6in for Proceedings schechter3approximate_sine_curves–1.5–1–0.500.511.5y1 2 3 4 5 6xFigure 2. Polynomial approximations to one cycle of the sine curve.be up to about 450-500 MeV, just before the location of this lowest res-onance. In the last few years there have been studies 2 by many authorswhich advance this picture. All of them are ”model dependent” but this isprobably inevitable for the strongly coupled regime of QCD. For example3, in a framework where the amplitude is computed from a non linear chiralLagrangian containing explicit scalars as well as vectors and pseudoscalars,the fit shown in Fig. 1 emerges as a sum of four pieces: i. the currentalgebra ”contact” term, ii. the ρ exchange diagram iii. a non Breit Wignerσ(560) pole diagram and exchange, iv. an f0(980) pole in the backgroundproduced by the other three. It is not just a simple sum of Born graphs butincludes the approximate unitarization features of the non Breit Wignershape of the sigma and a Ramsauer Townsend mechanism which reversesthe sign of the f0(980). Also note that i. and ii. provide very substantialbackground to the sigma pole, partially explaining why the sigma does not”jump right out” of various experimental studies. Qualitative agreementwith this approach is obtained by K-matrix unitarization of the two flavorlinear sigma model 4 and three flavor linear sigma model 5 amplitudes.Workers on scalar mesons entertain the hope that, after the revelationsabout the vacuum structure of QCD confirmed by the broken chiral sym-metric treatment of the pseudoscalars, an understanding of the next layerof the ”strong interaction onion” will be provided by studying the lightscalars. An initial question is whether the light scalars belong to a flavorSU(3) multiplet as the underlying quark structure might suggest. Apartfrom the σ(560), the f0(980) and the isovector a0(980) are fairly well es-tablished. This leaves a gap concerning the four strange- so called kappa-states. This question is more controversial than that of the sigma state .February 1, 2008 20:54 WSPC/Trim Size: 9in x 6in for Proceedings schechter4In the unitarized non linear chiral Lagrangian framework one must thusconsider pi−K scattering. In this case the low energy amplitude is taken6to correspond to the sum of a current algebra contact diagram, vector ρand K∗ exchange diagrams and scalar σ(560), f0(980) and κ(900) exchangediagrams. The situation in the interesting I = 1/2 s-wave channel turnsout to be very analogous to the I = 0 channel of s-wave ππ scattering. Nowa non Breit Wigner κ is required to restore unitarity; it plays the role ofthe σ(560) in the ππ case. It was found that a satisfactory description ofthe 1-1.5 GeV s-wave region is also obtained by including the well knownK∗0 (1430) scalar resonance, which plays the role of the f0(980) in the ππcalculation. As in the case of the sigma, the light kappa seems hiddenby background and does not jump right out of the initial analysis of theexperimental data.Thus the nine states associated with the σ(560), κ(900), f0(980) anda0(980) seem to be required in order to fit experiment in this chiral frame-work. What would their masses and coupling constants suggest about theirquark substructure if they were assumed to comprise an SU(3) nonet 7?Clearly the mass ordering of the various states is inverted compared to the”ideal mixing”8 scenario which approximately holds for most meson nonets.This means that a quark structure for the putative scalar nonet of the formN ba ∼ qaq¯b is unlikely since the mass ordering just corresponds to countingthe number of heavier strange quarks. Then the nearly degenerate f0(980)and a0(980) which must have the structure N11 ± N22 would be lightestrather than heaviest. However the inverted ordering will agree with thiscounting if we assume that the scalar mesons are schematically constructedas N ba ∼ TaT¯ b where Ta ∼ ǫacdq¯cq¯d is a ”dual” quark (or anti diquark).This interpretation is strengthened by consideration 7 of the scalars’ cou-pling constants to two pseudoscalars. Those couplings depend on the valueof a mixing angle, θs between N33 and (N11 −N22 )/√2). Fitting the couplingconstants to the treatments of ππ and Kπ scattering gives a mixing anglesuch that σ ∼ N33+ ”small”; σ(560) is thus a predominantly non-strangeparticle in this picture. Furthermore the states N11 ± N22 now would eachpredominantly contain two extra strange quarks and would be expected tobe heaviest. Four quark pictures of various types have been sugggested asarising from spin-spin interactions in the MIT bag model9, unitarized quarkmodels10 and meson-meson interaction models11.There seems to be another interesting twist to the story of the lightscalars. The success of the phenomenological quark model suggests thatthere exists, in addition, a nonet of “conventional” p-wave qq¯ scalars in theFebruary 1, 2008 20:54 WSPC/Trim Size: 9in x 6in for Proceedings schechter5energy region above 1 GeV. The experimental candidates for these statesare a0(1450)(I = 1), K∗0 (1430)(I = 1/2) and for I = 0, f0(1370), f0(1500)and f0(1710). These are enough for a full nonet plus a glueball. Howeverit is puzzling that the strange K∗0 (1430) isn’t noticeably heavier than thenon strange a0(1450) and that they are not lighter than the correspondingspin 2 states. These and another puzzle may be solved in a natural way12if the heavier p-wave scalar nonet mixes with a lighter qqq¯q¯ nonet of thetype mentioned above. The mixing mechanism makes essential use of the”bare” lighter nonet having an inverted mass ordering while the heavier”bare” nonet has the normal ordering. A rather rich structure involvingthe light scalars seems to be emerging. At lower energies one may consideras a first approximation, ”integrating out” the heavier nonet and retainingjust the lighter one.3. Effect of light scalars in η → 3piHistorically, this isospin violating process has been important as a relativelyclean test of the effective chiral Lagrangian approach and as a source ofinformation on the quark mass difference md−mu. As for the experimentalstatus, the shape of the Dalitz plot for the π+π−π0 mode agrees with chiralmodels. The experimental width for this mode 13 is 267 ± 25 eV. On theother hand, the tree level chiral result (which might be expected to beaccurate to within 25 per cent or so) is 106 eV while the one loop theoreticalnumber 14 is 160± 50 eV. A correction due to final state interactions, butoutside the chiral perturbation expansion, yields 15 209± 20 eV,In ref.16 we studied the effects of light scalars at tree level for this stillinteresting reaction. The calculation used a chiral Lagrangian of pseu-doscalars, vectors and scalars, with the scalar masses and coupling con-stants taken from ref.7 mentioned above and from17. We included nonminimal derivative terms of pseudoscalars but will only describe here theresults for the minimal pseudoscalar model with scalars and neglect of vec-tors. The 16 Feynman diagrams are shown in Fig. 3 and Fig. 4. The”driving mechanism” for this decay is the non zero value of md−mu whichresults in isospin violating bilinear pseudoscalar terms η, η′ − π0, bilinearscalar terms f0.f′0 − a00 as well as the quadrilinear term in (a) of Fig.3.The pseudoscalar diagrams in Fig. 3 (with the minimal chiral La-grangian) predict a width of 106 eV for the π+π−π0 mode. A priori onemight expect important contributions from the sigma exchange diagramsin (a),(b) and (c) of Fig. 4, since the σ(560) propagators will not be tooFebruary 1, 2008 20:54 WSPC/Trim Size: 9in x 6in for Proceedings schechter6ηpipipi(a) (b)ηpipipiηpipipi(c).. +−+−−+000piη,η0/Figure 3. Feynman diagrams representing the pseudoscalar meson contribution to thedecay η → pi+pi−pi0 .η pipipipiσ, f000+−.(a).η,η /σ, f0pi 0pi+pi−η(b)pi +pi 0−a0+−pi−+η.η,η /(c). σ, f00api 0pi+pi−η(d)Figure 4. Feynman diagrams representing the scalar meson contributions to the decayη → pi+pi−pi0 .far from their mass shells. However the signs are such that these threeleading scalar contributions almost cancel each other. The net result is atotal increase of the tree width by about 13 per cent to about 120 eV. Thisis based on the fits we obtained for the coupling constants which appear.As discussed in section IV of 16 there may be some room to increase this,owing to the delicate partial cancellations. The result is in the right di-rection but does illustrate that some care is needed in treating this simplelooking reaction.We are grateful to M. Harada, P. Herrera-Siklody and F. Sannino forvery helpful discussions. We would like to thank Professors K. Yamawaki,Y. Kikukawa and M. Harada for all their hard work in organizing thisstimulating conference. The work of A. A-R. and J.S. has been supportedin part by the US DOE under contract DE-FG-02-85ER 40231. D.B. wishesto acknowledge support from the Thomas Jefferson National AcceleratorFebruary 1, 2008 20:54 WSPC/Trim Size: 9in x 6in for Proceedings schechter7Facility operated by the Southeastern Universities Research Association(SURA) under DOE Contract No. DE-AC05-84ER40150. The work ofA.H.F. has been supported by grants from the State of New York/UUPProfessional Development Committee, and the 2002 Faculty Grant fromthe School of Arts and Sciences, SUNY Institute of Technology.References1. J. Gasser and H. Leutwyler, Ann. Phys. 158, 142(1984); Nucl. Phys. B250,465 (1985).2. See the dedicated conference proceedings, S. Ishida et al ”Possible existenceof the sigma meson and its implication to hadron physics”, KEK Proceedings2000-4, Soryyushiron Kenkyu 102, No.5 2001. Additional points of view areexpressed in the proceedings, D. Amelin and A. M. Zaitsev ”Hadron spec-troscopy”, Ninth international conference on hadron spectroscopy, Protvino,Russia (2001).3. M. Harada, F. Sannino and J. Schechter, Phys. Rev. D54, 1991 (1996).4. N. N. Achasov and G. N. Shestakov, Phys. Rev. D49,5779 (1984). See also S.Ishida, M. Y. Ishida, H. Takahashi, T. Ishida, K. Takamatsu and T. Tsuru,Prog. Theor. Phys. 95, 745 (1966).5. See for example D. Black, A. H. Fariborz, S. Moussa, S. Nasri and J. Schechter,Phys. Rev. D64, 014031 (2001). A more complete list of references is givenhere.6. D. Black, A. H. Fariborz, F. Sannino and J. Schechter, Phys. Rev.D58, 054012(1998).7. D. Black, A. H. Fariborz, F. Sannino and J. Schechter, Phys. Rev.D59, 074026(1999).8. S. Okubo, Phys. Lett. 5, 165 (1963).9. R. L. Jaffe, Phys. Rev. D15, 267 (1977).10. N. A. Tornqvist, Z. Phys. C 68, 647 (1995); E. van Beveren and G. Rupp,Eur. Phys. J. C 10, 469 (1999).11. J. D. Weinstein and N. Isgur, Phys. Rev. Lett. 48, 659 (1982).12. D. Black, A. H. Fariborz and J. Schechter, Phys. Rev. D 61, 074001 (2000);D. Black et al, 5 above; T. Teshima, I. Kitamura and N. Morisita, J. Phys. G28, 1391 (2002); F. Close and N. Tornqvist, arXiv:hep-ph/0204205.13. K. Hagiwara et al, Phys. Rev. D66,010001 (2002).14. J. Gasser and H. Leutwyler, Nucl. Phy. B250, (1985) 539. Recent reviewsinclude B. R. Holstein hep-ph/0112150 and J. Bijnens and J.Gasser hep-ph/0202242.15. J. Kambor, C. Wiesendanger and D. Wyler, Nucl. Phys. B465 (1996) 215.See also A. Anisovich and H. Leutwyler, Phys. Lett. B375, 335 (1996).16. A. Abdel-Rehim, D. Black, A. H. Fariborz and J. Schechter, Phys. Rev. D67,054001 (2003). Many other references as well as a short history are containedhere.17. A. H. Fariborz and J. Schechter, Phys. Rev. D 60, 034002 (1999).

The Scalar Sector and the Eta -\u3e 3 Pi Problem

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The Scalar Sector and the Eta -\u3e 3 Pi Problem

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