5-5-qr-l I VECTOR MESON MODELS OF' STROI{GLY INTERACTING SYSTEMS Tunsrs pon DocroR oF Purr,osopuy DEPARTMENT OF PHYSICS AND MATHBMATICAL PHYSICS By Heath B. O'Connell B.Sc. (Hons) Supervisors: Prof. A.W. Thomas and Dr. A.G. lVilliams October 1996 Abstract The consequences of current conservation for vector meson models are examined. As an example of this, the p-a mixing model for the isospin violation seen in the pion electromagnetic form factor is studied in detail. Assuming current conservation, we predict a strong momentum dependence for vector mixing. As this result also applies to photon- vector meson mixing, in contradiction to traditional photon-hadron models, we describe an equivalent model which includes a momentum dependent coupling of the photon to vector mesons. To ensure that the information obtained previously from the pion form- factor is consistent with conserved current picture, we redo the fit to data and find a considerable model dependence to the quantities of interest that is not a consequence of momentum dependence. nI ]V VI Acknowledgements I would like to thank my supervisors, Tony Thomas and Tony Williams, for their guid- ance and close attention to my work. I also extend thanks to the staff at Adelaide Uni- versity who have helped me during my PhD candidature, Rod Crewther, Lindsay Dodd, Jim McCarthy and Bruce Pearce. In addition, my work has benefitted from discussions with Maurice Benayoun, Reg Cahill, Susan Gardner, Kim Maltman, Andy Rawlinson, Craig Roberts and Peter Tandy. I also thank the hospitality of the Physics departments of Argonne National Laboratory, Indiana University, Kent State University, Los Alamos National Laboratory, Ohio State University, TRIUMF, the Technical University of Munich and the University of Maryland. Thanks also to my family for their support and the Australian Government for financial assistance. vil vilr Contents Abstract lll Statement V Acknowledgements vii 1 Introduction 1 1.1 Historical development of VMD 2 1.2 Gauge invariance and VMD 4 1.3 The electromagnetic form-factor of the pion 7 I.4 Summary I 2 Standard treatment of p-a mixing 11 2.I The theoretical beginning of p-a mixing . 11 2.2 The experimental discovery of p-u mixing L2 2.3 p-o rrixing and F"(q,) 13 2.4 Intrinsic a.r¡ decay in e*e- -- r*r- 15 2.5 The use of. p-u mixing in nuclear physics T7 3 Behaviour of p-u mixing 2L 3.1 Models for p-a mixing . 22 3.2 QCD Sum Rules 24 3.3 Chiral Perturbation Theory 27 Discussion 3.4 28 4 Effective models and momentum dependence 29 4.1 The node theorem 29 4.2 The consequences of momentum dependence for VMD 33 4.3 Sakurai and the two representations of VMD 35 4.4 The use of VMD1 39 1X 4.5 A VMDl-like model 42 4.6 Discussion 46 5 fsospin violation in F"(qz) with ChpT 47 5.1 An introduction to ChPT 48 5.2 The standard ChPT treatment of F,Q2) 51 5.3 The (rn, - *o) contribution 52 5.4 Discussion 55 6 New analysis of the pion form-factor 57 6.1 An S-matrix approach 57 6.2 Mixing Formalism 58 6.3 Contributions to the pion form-factor 61 6.4 Numerical results 65 6.5 A word on transformations between bases 67 6.6 Conclusion 68 7 Conclusion 7T Bibliography 73 A Field current identity 79 B Appendix 83 8.1 Chiral Perturbation Theory expressions 83 8.2 Useful Integrals 84 --+ 8.3 Calculation for A8 rlr- 85 C Historical perspective on p-ø mixing 89 C.1 Historical perspective 89 C.1.1 The eariiest work on p-a mixing 89 C.1.2 CSV and p-a.' mixing 91 D Published work 95 X List of Figures 1.1 One-particle-irreducible QCD contribution to the photon propagator. This is the complete QCD contribution to the photon self-energy 2 7.2 A simple VMD-picture representation of the hadronic contribution to the photon propagator. The heavier vector mesons are included in generalised VMD models. 3 1.3 Electron-positron pair annihilating to form a photon which then decays to a pion pair. 7 2.L Electromagnetic contribution to the (r-resonance of ere- -- ¡rtr- 12 2.2 The cross-section for the reaction e*e- -- r*r- from the data of Ref. [27] in the p-0 resonance region. 13 2.3 The contribution of p-u mixing to the pion form-factor. I4 2.4 Physical intermediate states contributin g to p-a mixing. 16 2.5 Contribution of a pion loop to the p self-energy 16 2.6 p-u: mixing contribution to the CSV l/1ú interaction. 19 4.r vMD dressing of the photon propagator by a series of intermediate p prop- agators. .....35 4.2 Contributions to the pion form-factor in the two representations of vector meson dominance a) VMD1 b) VMD2. 39 4.3 Cross-section of e*e- -- r*r- plotted as a function of the energy in the centre of mass. The experimental data is from Refs. ]I7,ZI] 4I 4.4 Cross-section for e*e- -- r*r- in the region around the resonance where p-ar mixing is most noticeable. The experimental data is from Refs. [17,21). 42 4.5 The PW model prediction for the mixing amplitude is related to the tra- ditional vMD coupling gok") using the central result of trq. (a.5b ). The resulting behaviour of gllar versus q = JF is then plotted in the timelike region for this model. Shown for comparison are a typical pair of results (2.27 +.0.23 at Q:0 and2.20 + 0.06 at q: Tnp) see text) taken from a traditional vMD based analysis of cross section data in Ref. [zJ] 45 XI 5.1 The chiral contributions to .y - r*r- 53 6.1 The allowed values of G : g,,nnf gp,nn and n(rnl) (in MeV2) are plotterl as a function of the Orsay phase, /. The vertical (dotted) lines indicate the experimental uncertainty in ó (: It6.Z + 5.8)" and the uncertainty in the amplitude,4 (0.0109+0.0011) (see text) gives rise to the spread of possible values of G at each value of /. 66 xlt Chapter 1 Introduction Before proceeding with our discussion of p-a mixing, it will be helpful to introduce the concept of Vector Meson Dominance (VMD), the model within which p-a.' mixing is usually described. To motivate this, it is useful to provide the historical development of the subject, which will then lead naturally into the consideration of. p-u mixing and its theoretical description. Where possible, the connection between the original hadronic models and our present understanding will be discussed. The physics of hadrons was a topic of intense study long before the gauge field the- ory Quantum Chromodynamics (QCD), now believed to describe it completely, was in- vented [1]. Hadronic physics was described using a variety of models and incorporating approximate symmetries. It is a testimony to the insight behind these models (and the inherent difficulties in solving non-perturbative QCD) that they still play an important role in our understanding. One particularly important aspect of hadronic physics which concerns us here is the interaction between the photon and hadronic matter [2]. This has to date been remarkably well described using the vector meson dominance (VMD) model. This assumes that the hadronic components of the vacuum polarisation of the photon consist exclusively of the known vector mesons. This is certainly an approximation, but in the resonance region around the vector meson masses, it appears to be a particularly good one. As vector mesons are believed to be bound states of quark-antiquark pairs, it is tempting to try to establish a connection between the old language of VMD and the Standard Model. In the Standard Model, quarks, being charged, couple to the photon and so the strong sector contribution to the photon propagator arises, in a manner analogous to the electron- positron loops in QED, as shown in Fig. 1.1. The diagram contains dressed quark propagators and the proper (i.e., one-particle irreducible) photon-quark vertex (the shaded circles include one-particle-reducible parts, while the empty circles are one-particle-irreducible [3]). This diagram contains a// strong 1 2 CHAPTER 1, INTRODUCTION interaction contributions to the photon self-energy. In QED, because the fine structure constant (a - 11137) is small, we can approximate the photon self-energy reasonably well using bare propagators and vertices without worrying about higher-order dressing. However, in QCD, the dressing of these quark prcrpaga[ors and the quark-photon proper vertex cannot be so readily dismissed as being of higher order in a perturbative expan- sion (although for the heavier quarks, higher order effects can be ignored as a conse- quence of asymptotic freedo- [4]). Recent work [5], using numerical solutions to the non-perturbative Dyson-Schwinger equations of QCD [3], has succeeded in producing a spectrum of quark-antiquark bound states, but the mass distribution does not yet corre- spond to the observed spectrum. q v v -q Figure 1.1: One-particle-irreducible QCD contribution to the photon propagator This is the complete QCD contribution to the photon self- energy. 1.1 Historical development of VMD The seeds of VMD were sown by Nambu [6] in 1957 when he suggested that the charge distribution of the proton and neutron, as determined by electron scattering [7], could be accounted for by a heavy neutral vector meson contributing to the nucleon form factor.