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The qqbar S-wave Axial-Vector Mesons in the Covariant U ~ (12)-Scheme

The qqbar S-wave Axial-Vector Mesons in the Covariant U ~ (12)-Scheme. T. MAEDA 　（ Nihon Univ. ）. in collaboration with S. Ishida, K. Yamada (Nihon Univ.) and M. Oda (Kokushikan Univ.). 8 October 2007. XII International Conference on Hadron Spectroscopy - Hadron07.

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The qqbar S-wave Axial-Vector Mesons in the Covariant U ~ (12)-Scheme

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  1. The qqbar S-wave Axial-Vector Mesons in the Covariant U~(12)-Scheme T. MAEDA　（Nihon Univ.） in collaboration with S. Ishida, K. Yamada (Nihon Univ.) and M. Oda (Kokushikan Univ.) 8 October 2007 XII International Conference on Hadron Spectroscopy - Hadron07 Laboratori Nazionali di Frascati (Rome)

  2. 1.Introduction P-Wave excited states? chiral partner S-Wave ground states SU(2)σpartner The a1 Problems • The sigma meson as chiral partner of the pi meson will be different from 3P0 state in the conventional non-relativistic (NR) classification scheme. • From this viewpoint, it is worthy to reconsider whether the a1 meson, as a chiral partner of the rho meson, is really simply-identified 3P1 state in the NR classification. Still not defined yet!

  3. 1.Introduction U~(12)-scheme S. Ishida, M. Ishida, and T.M., PTP104 (2000) • In recent years, we have proposed the U~(12)-scheme, a relativistic covariant level-classification scheme of hadrons. • In this scheme, the light scalar sigma meson has been identified as ``the relativistic qqbar S-wave state'' as well as pi meson, and they play mutually the role of chiral partners. chiral partner These denote the spin wave-function (WF) of sigma- and pi- meson in U(12)-scheme, respectively. More about them will be discussed in later.

  4. 1.Introduction The other `` Relat. qqbar S-wave states'' • U~(12)-scheme leads a just appropriate candidate of the axial vector meson as chiral partner of the rho meson. • Here it is notable that these new-type of ``S-wave states’’ are expected to have the light mass compared with those of the P-wave states. chiral partner • U~(12)-scheme also includes the another axial vector state with JPC=1+- , corresponding to b1 meson. chiral partner

  5. 1.Introduction Purpose of this work • In this theoretical analysis, we identify the experimentally known states, a1(1260), b1(1235), to our relativistic S-wave states, a1(N), b1(N), respectively. • Then, by using the wave function determined by U~(12)- scheme，we study the following decay properties of a1 and b1 mesons． • Radiative decay : • Strong decay : • Through the above analyses, we will examine whether or not these identifications arepromising.

  6. Outline of this talk • 1. Introduction 2. Wave Functions in the U~(12)-Scheme 3. Radiative Decays of a1 and b1 Mesons 4. Pion Emissions of a1 and b1 Mesons 5. Summary and Future Subjects

  7. 2. Wave Functions in the U~(12)-Scheme Basic framework of our level-classification scheme is what is called `boosted LS-coupling scheme’. The WF of qqbar ground state meson are given by the following ( bi-local Klein-Gordon ) field with one each upper and lower indices. Ground State 4-Dim. Oscillator WF C.M. Coordinate Flavor WF Relative Coordinate Spin Space-Time Here A relativistic extension of conventional NRQM by separately boosting! denotes Dirac spinor / flavor indices A remarkable point is that WFs of hadron are described as the direct product of spin part and space-time part.

  8. 2. Wave Functions in the U~(12)-Scheme -Spin Degree of Freedom Important feature of U~(12)-scheme is that it contains extra SU(2) spin degree of freedom, called ‘‘rho-spin’’ . Chiral Group As a result, spin WFs are classified to the representation of the chiral group. Expansion bases of spinor WF are given by Dirac spinor with hadron on-shell 4-velocity , as Here, u+ corresponds to conventional constituent quark degree of freedom, while u- represents relativistic effect for confined system. They form the chiral partner in basic rep. of chiral group. Chirality : Parity :

  9. 2. Wave Functions in the U~(12)-Scheme Complete sets of qqbar Spin WF Basis of qqbar meson WF is defined as, ,which is expanded by complete sets of totally 16 components as follows. Ps ×2 V ×2 S ×2 A× 2 Relativistic S-wave states Describing with at least one Dirac spinors with negative ρ3 value, which never appeared in non-relativistic quark model.

  10. 2. Wave Functions in the U~(12)-Scheme Explicitly, we use the following form for spin WFs for the relevant application in this analyses. Spin WFs for the Relevant Applications Hereinafter, we restrict the chiral symmetry in case of the SU(2). (1) Relat. S-Wave Axial-Vector Mesons Candidate ① a1(1260) ② b1(1235)

  11. 2. Wave Functions in the U~(12)-Scheme Note that the is just coincide with NRQM WF at rest. Similarly for the case of (2) S-Wave Vector Mesons Candidate ③ ④ Physical states are expected to be mixing states of them in equal weight. ρ(770) ρ(1300) Likewise for omega meson

  12. 2. Wave Functions in the U~(12)-Scheme (3) Relat. S-Wave Pseudo-Scalar Mesons Candidate ⑤ π(140) ⑥ π(1300) From the consideration on linear realization of chiral symmetry, the WF of pion should be reflecting its Goldstone-boson nature.

  13. ３. Radiative Decays of a1 and b1 Mesons In the following section, I will present some associated model predictions. At first, we will consider the radiative decay of a1 and b1 mesons. Due to the our assignments of a1, and b1 to the S-wave states, following gauge invariant spin-type coupling are considered. Here we introduced 2 independent coupling parameters g and g’. The g term contributes only quark chirality conserving transitions, while the g’ term does chirality non-conserving one.

  14. ３. Radiative Decays of a1 and b1 Mesons Meson current matrix elements are given by the following formulas. 2 parameters g, g’ (A) Chirality Conserving Spin Current Term (B) Chirality Non-Conserving Spin Current Term ：Spin WF of Mesons ( (+) : annihilation part) ：Momentum of Emitted Photon

  15. ３. Radiative Decays of a1 and b1 Mesons Overlapping Integral of Space-time WF Overlapping Integral of Space-time Oscillator function gives a Lorentz Invariant Form Factor. : Ground State 4-Dim.Oscillator WF ：　Regge Slope Inverse Parameter ：Mass of Initial (Final) Meson

  16. ３. Radiative Decays of a1 and b1 Mesons Remarkable Features of the Current 1. Covariant treatment for CM motion Conserved current 2. It contains the electric dipole (with rho3 flip) transition, in addition to the conventional magnetic (without rho3 flip) transition. magnetic Parity change electric Parity inv. 3. The chirality non-conservingspin current (g’ term ) is first included.

  17. ３. Radiative Decays of a1 and b1 Mesons Fixing the Parameters Pion mass in the spin FF : 0.78 (GeV) otherwise : Physical Meson Mass (from PDG)

  18. ３. Radiative Decays of a1 and b1 Mesons Numerical Results The estimated widths are in comparison with experiment. in (keV) Process Our Results Exp 68 (input) 68±7 604 640±246 230 (input) 230±60 Result for this calculation is consistent with experiments, but we should check the validity of decay interaction, by applying for various other transitions.

  19. 4. Pion Emissions of a1 and b1 Mesons Decay Model for One Pion Emissions Next we consider strong decays with one pion emission. Relevant decay amplitudes T are obtained by following 2-types of effective quark-pion interactions. ps coupling : pv coupling : Note that here, pi ( and sigma ) meson is described as a external local-field (s) in this work.

  20. 4. Pion Emissions of a1 and b1 Mesons Resultant matrix elements are described as sum of two term; T = Tps + Tpv Tps = gps < W (v’) (- iγ5π) W (v) ivγ > + C.C. q (-iγ5π) q ps coupling Tpv = gpv < W (v’) (-γ5γμqμπ) W (v) ivγ > + C.C. pv coupling q (-iγ5 γμ)q　∂μπ

  21. 4. Pion Emissions of a1 and b1 Mesons Explicit forms of the relevant amplitude are given as It should be included at least two coupling type ( expressed `f1 and f2’ in the above ) to reproduce the experimental data of D/S amplitude ratios.

  22. 4. Pion Emissions of a1 and b1 Mesons Fixing the Parameters In this simple decay model, there are two independent coupling parameters, gps and gpv, which are commonly-applied to all quark-pion vertices. These are determined from the experimental data of D/S amplitude ratio and total width of b1 meson. Physical Meson Mass (from PDG)

  23. 4. Pion Emissions of a1 and b1 Mesons Numerical Results Partial Width ( in MeV ) Process Our Results Exp Data Our Results Exp Data (input) (input) The method to calculate the hadronic decay adopted in this work can be also applied to some other process. See, K. Yamada’s talk (Oct. 11, Thursday )!

  24. 5. Summary and Future Subjects Summary of this work We investigate the decay properties of relativistic S-wave a1 and b1 meson in the U~(12)-scheme, by assigning them a1(1260) and b1(1235) mesons respectively. 1． Radiative decay : By inputting the and , predicted width of is consistent with experiment. 2． Strong decay : We input the total width of b1 and D/S amplitude ratio of , to a simple decay model, and determine the 2 independent coupling parameters in it. As a result, the sign of D/S amplitude ratio of is coincide with the experiments, but its absolute value is about 3 time larger than experiment. Partial width of is predicted as .

  25. 5. Summary and Future Subjects Future Subjects The naïve interaction adopted in this work for the radiative / strong decays should be tested by applying to other various decay process. It might be extended to in the future.

  26. Experimental Data (PDG06) Appendices 1

  27. Appendices 2 K. Yamada, arXiv: hep-ph/06012337 Experimental Candidates (Ground States)

  28. Appendices 3 K. Yamada, arXiv: hep-ph/06012337 Experimental Candidates (Excited States)

  29. Appendices 4 Expansion of Spin WF Boost op. complete set of SU(2)σ×SU(2)ρ Boost op. From the above representations, one can easily check the symmetry properties of respective mesons! Chirality : Charge conjugation : Parity :

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