View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by UNT Digital Library DOE/ER/41048−10 THEORETICAL STUDIES OF HADRONS AND NUCLEI Final Progress Report June 1997 - July 2006 Stephen R. Cotanch Department of Physics North Carolina State University Raleigh, North Carolina 27695-8202 PREPARED FOR THE U. S. DEPARTMENT OF ENERGY UNDER GRANT NUMBER DE-FG02-97ER41048 2 INTRODUCTION This report details final research results obtained during the 9 year period from June 1, 1997 through July 15, 2006. The research project, entitled ”Theoretical Studies of Hadrons and Nuclei”, was supported by grant DE-FG02-97ER41048 between North Carolina State University [NCSU] and the U. S. Department of Energy [DOE]. In compliance with grant requirements the Principal Investigator [PI], Professor Stephen R. Cotanch, conducted a theoretical research program investigating hadrons and nuclei and devoted to this program 50% of his time during the academic year and 100% of his time in the summer. Highlights of new, significant research results are briefly summarized in the following three sections corresponding to the respective sub-programs of this project. Recent progress is also discussed in a recent renewal/supplemental grant proposal submitted to DOE. Finally, full detailed descriptions of completed work can be found in the publications listed at the end of this report. A. Hadron Structure 1. Many-Body Effective Hamiltonian Approach The focus of this project is the study of the the quark and gluon structure of hadrons, especially exotic systems (non qqq or qq¯ components). The long term goal is to obtain a complete description of hadrons from fundamental Quantum Chromodynamics [QCD] and to obtain a dynamic understanding of color confinement. Before detailing significant results it is appropriate to provide a brief synopsis of this project and related sub-projects. This program approaches hadron structure from a nuclear physics many-body perspective and approximately diagonalizes an effective QCD Hamiltonian. There are two sub-programs; 1) obtain from the exact QCH Hamiltonian, H, an effective Hamiltonian, Heff , that describes color confinement; 2) use many-body techniques and large-scale computations to diagonalize Heff for the hadron states including the ground state (QCD vacuum). We have specifically chosen the Coulomb gauge since it is amendable to a Hamiltonian formulation and also is rigorously renormalizable in this gauge. Further, and as detailed by Zwanziger, formulation in this gauge is free from the Gribov problem. A few years ago, using the similarity transformation technique of Glazek and Wilson, we were able to renormalize our Hamiltonian perturbatively to remove sensitivity to the regularization (truncation) schemes that were necessary to obtain a calculable Hamiltonian [4, 7]. We also applied flow equations, which are a series of continuous similarity transformations, to obtain a Heff [13, 15]. We initially used a linear confining potential with slope predetermined from lattice simulations for the energy of heavy static quarks as a function of distance. More recently, we have used an improved, self-consistent effective confining interaction. For a given Heff we then use many-body techniques to approximately diagonalize for the ground 3 state (QCD model vacuum) and hadron states having quantum numbers J PC . For the ground state vacuum we employed the Bardeen, Cooper and Schreiffer [BCS] variational approach in which the free (current) quarks and (zero mass) gluons are transformed into dressed quasiparticles by first performing a similarity transformation (BCS rotation) of the Fock operators and then finding the rotation (gap) angle by minimizing the ground state energy with respect to this variational parameter. This yields the quark and gluon gap equations (formally equivalent to the Dyson-Schwinger propagator equation in the rainbow approximation). The quasiparticles acquire constituent masses, roughly 100 and 250 MeV for the u/d and s quarks, respectively, and 800 MeV for the dressed gluon. Further, the respective quark and gluon condensates (Cooper pairs) 3 4 are -(100 MeV) and (433 MeV) . With this quasiparticle basis Heff is then diagonalized for the hadron states using the Tamm-Dancoff approximation [TDA] truncated at the 1 parton, 1 anti- parton level [26]. For comparison we also diagonalized using the random phase approximation [RPA] because it conserves chiral symmetry, unlike the TDA [11]. The RPA is formally equivalent to the Bethe-Salpeter equation using an instantaneous kernel. Strengths of this approach are that Heff has only a few (1 or 2) dynamical parameters, in contrast to typical quark models, and contains the same symmetries and parton (quark, gluon) field structure as H. Equally significant, it provides a unifying framework for both quarks and gluons with controllable approximations. Our first application reproduced the lattice glueball spectra and subsequently we correctly predicted the observed meson spectra as well as glueball and hybrid meson (a three-body quark, antiquark and constituent gluon) masses in good agreement with lattice gauge calculations. 2. Application in the Gluon Sector We, with F. Llanes-Estrada (PI’s former student now tenured at the Univ. Complutense in Madrid) and P. Bicudro (Instituto Superior Tecnico in Lisbon), obtained significant new results in our application of Heff to the gluon sector to study glueballs. As detailed further in our key publication [48], we performed a large scale BCS/TDA variational relativistic three-body calculation for the |ggg > glueball spectrum. We specifically wished to investigate the odd J −− states (oddballs) to determine if there were Regge trajectories, document the odderon, and also determine the intercept. Table I lists our predicted masses and compares to other approaches, especially lattice. To provide insight and document theoretical model uncertainty, we also calculated the same glueball states with a simpler nonrelativistic constituent gluon potential model having several model parameters. Using Jacobi coordinates and a large harmonic oscillator expansion, we exactly diagonalized this simpler Hamiltonian. The model parameters were taken from previous quark studies (used the qq¯ funnel potential) and, only adjusting the spin-spin potential strength to optimize comparison with lattice J ++ states, obtained the even signature Regge trajectory, M αP (t) = 0.23t + 1.0, which is consistent with the pomeron and also our trajectory using the 4 TABLE I: Glueball quantum numbers and masses in MeV. Model J PC 0−+ 1−− 2−− 3−− 5−− 7−− our work color f d d d d d S 0 1 2 3 3 3 L 0 0 0 0 2 4 Heff 3900 3950 4150 4150 5050 5900 HM 3400 3490 3660 3920 5150 6140 lattice (Morningstar/Peardon) 3640 3850 3930 4130 lattice (Meyer/Teper) 3250 3100 3550 4150 Wilson loops (Simonov) 3770 3490 3710 4030 more sophisticated Heff approach. The |ggg > glueball states predicted by this model (HM ) are also listed in the table. The difference between approaches is an overall indication of theoretical uncertainty which is at the 10 to 15% level. Both of our model calculations yielded an odderon, eff M αO (t) = 0.23t − 0.880 and αO (t) = 0.18t + 0.25, having slopes similar to the pomeron but much lower intercepts, even below the ω meson intercept value of 0.5. All results are pictorially summarized in Fig. 1. This is one of our important findings and indicates that it will not be possible to observe the odderon in high energy scattering processes in which the pomeron contributes since the cross section scales as s1−α(0) and is dominated by the trajectory with the largest intercept. It may be possible to still observe the odderon in reactions having C parity = - 1 exchange where the pomeron is not present. Oddball Chew-Frautschi plot C=P= -1 pure glue states 8 7 6 5 J 4 3 Morningstar and Peardon Meyer and Teper 2 Kaidalov and Simonov ω trajectory H M (this work) 1 Coulomb gauge H eff (this work) 0 0481216 20 24 28 32 36 40 2 2 M (GeV ) FIG. 1: Odderon trajectory from three-gluon spectroscopy and lattice compared to the ω meson Regge trajectory. 5 3. Application in the Quark Sector In our project involving Heff applications to the quark sector, several papers [11, 26, 39, 40, 42] were published detailing significant, new findings. We have utilized a confining interaction that was determined self-consistently from our model QCD vacuum and also included a hyperfine interaction that was an effective vector-vector potential with generic form following from QCD. The hyperfine interaction significantly increased both the quark constituent mass and condensate yielding improved results for each and, for the first time, simultaneously described the small, ≈ 100 MeV, J/ψ spin splitting and very large π-ρ mass difference with a simple interaction. Table II summarizes our predictions for the light and heavy pseudoscalar and vector meson spectra. We have also predicted the vector meson leptonic decay widths, ΓV →e+e− , using the same Heff as shown in the table. The predicted masses are in reasonable agreement especially since no fit was attempted (dynamic parameters and quark masses were not fine-tuned). For comparison, two other theoretical calculations are also presented. It is noteworthy that our few parameter approach provides a description which is comparable to these multi-parameter models. In typical quark model formulations, a complicated multi-parameter hyperfine interaction is used to reproduce the the π-ρ mass differ, however applying the same interaction to excited π-ρ states, as well as to J/ψ spin splittings, produces unphysically large mass differences. This is because chiral symmetry is generally not incorporated which our RPA approach does (produces a small pion mass, Goldstone boson). By comparing with our TDA π-ρ result, which does not TABLE II: Observed and predicted vector meson masses and widths compared to other calculations marked 1 (Godfrey and Isgur S-D wave results without mixing) and 2 (Ebert, Faustov and Galkin).