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A Study of Dual Models in the Theory of Strong Interactions

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A Study of Dual Models in the Theory of Strong Interactions DOKTORSAVHANDLINGAR vid CHALMERS TEKNISKA HOGSKOLA Mr l>0 A Study of Dual Models in the Theory of Strong Interactions by LARS BRINK G5TEBORG 1973 DOKTORSAVHANDLINGAR vid CHALMERS TEKNISKA HOGSKOLA '•& A Study of Dual Models in the Theory of Strong Interactions by LARS BRINK GOTEBORG 1973 The thesis consists or the following publications: I A one-pion exchange model applied to the reaction pp-'pir (A )pir at 19 GeV/c Nucl. Phys. B26, 611 (1971) (together with S.O. Holmgren) II Regge-pole fit to the Deck-peak in the reaction pp-*A pir at 19 GeV/c Physica Scripta in press (together with S.O. Holmgren) III A comment on nucleon-antinucleon annihilation and the possibility to determine the spin for resonances in the direct channel of this process Physica Scripta £, 117 (1972) IV Some consequences of resonance production according to the scaling hypo- thesis Phys. Letters 37B, 192 (1971) (together with W.N. Cottingham and S. Nussinov) V The physical state projection operator in dual resonance models for the critical dimension of space-time Nucl. Phys. B56, 253 (1973) (together with D. Olive) VI Recalculation of the unitary single planar dual loop in the critical dimension of space-time Nucl. Phys. B5j9, 237 (1973) (together with D. Olive) VII Unitary single non-planar dual loops in 26 dimensions of space-time CERN TH-1773 (together with D. Olive) VIII The gauge properties of the dual model pomeron-reggeon vertex: their derivation and their consequences Nucl. Phys. in press (together with D. Olive and J. Scherk) IX The missing gauge conditions for the dual fermion emission vertex and their consequences Phys. Letters i+5B, 379 (1973) (together with D. Olive, C. Rebbi and J. Scherk X A physical interpretation of the Jacobi imaginary transformation and the critical dimension in dual models Phys. Letters k3B, 319 (1973) (together with H.B. Nielsen) XI A simple physical interpretation of the critical dimension of space-time in dual models Phys. Letters UjjB, 332 (1973) (together with H.B. Nielsen) XII Dual models with SL(2,c) symmetry Nucl. Phys. Bft6, 505 (1972) (together with A. Kihlberg) XIII The nucleon form factors in a field theory on a homogeneous space of the Poincare group II Nuovo Cimento JOA, 533 (1972) (together with R. Marnelius) XIV A study of the Hiltoert space properties of the Veneziano model operator formalism to be published in J. Math. Phys. (together with P.H. Frampton and H.B. Nielsen) Contents 1. Introduction page 1 2. Basic features of strong interaction 5 2.1 Nuclear reactions 5 2.2 Elementary particle reactions 6 3. Possible ways to construct a theory for the strong inter- actions 10 3.1 Field theoretic approach 10 3.2 S-matrix approach 12 3.2.1 Postulates 13 h. Some simple phenomenological tests of the theory of strong interaction 21 5. Dual Resonance Models 30 5.1 Four-particle Amplitudes 30 5.2 N-particle Amplitudes 36 5.3 The operator formalism of the Veneziano Model U0 5.^ The algebraic properties of the Veneziano model with unit intercept k$ 5-5 Unitarity corrections 51 5.6 The Neveu-Schwarz model 60 5.7 Fermions in dual models 65 5.8 The string interpretation of dual models 71 6. Attempts to construct new models 8k 7. Conclusions and speculations 92 Acknowledgement 95 -1- 3 <f» 1. Introduction The human curiousity and interest in Nature led already the ancient Greek philosophers to speculate about the existence of fundamental constituents of matter. In modern philosophy and science this still remains one of the basic questions; a question that hopefully can be answered in a near future. This hope, however, did also permeate earlier generations of scientists. During the 19th century it was believed that the atom played this role, as ths existing chemistry could be understood fairly well on the basis of this concept. It was not until 1911 that Rutherford did reveal that the atom in itself is built up as a nucleus surrounded by a cloud of electrons. In the early 1930's thc> nuclei 2) had been understood to be built up by protons and neutrons . Beside these e_ljem£nt_ary_pa,rt_i£l£_s one had found the electron and its antiparticle, the posi- tron and some far-sighted scientists could also foreshadow the antiparticles of the proton and the neutron even if the antiproton was not discovered until some twenty years later . The goal then seemed to be reached. The electron, proton and neutron together with the photon, the mediatior of the electromagnetic interaction, were the natural candidates for the basic entities of matter. How- ever, puzzling questions remained. The interaction between the proton and the neutron was still to be understood and to offer an explanation of this, Yukawa'' postulated the existence of another elementary particle, the pion. Through ex- tensive studies of cosmic rays the pion was eventually found , but, unexpected- ly, other particles were also found. All these particles could be distinguish- ed from the proton, neutron and electron in that they were unstable, decaying with a life-time of about 10 seconds. With the advent of the high energy accelerator physics some twenty years later new phenomena were discovered. A seemingly endless number of very shortlived resonance states with a life-time -2- -23 7) of about 10 seconds occur in elementary particle processes . In many cases they can be thought of as excited states of other more stable par- ticles but still the fact remains that the study of the smallest distances in Nature has revealed an enormous complexity. Even if one by now has found many symmetries and regularities among all these particles, the concept of elementary particles as the basic building block in Nature that seemed on such a firm ground kO years ago is now an open question. A natural way out would be to say that the particles that have been found are not the fundamental ones, but are built up by another set of more fundamental entities. In the late 195O's when the concept of internal sym- metries grew out, one tried anew to find this set. The first attempt was made by Sakata , who tried the proton, neutron and theA-particle as fun- damental entities, but he was not very successful. These attempts culmina- ted when Gell-Mann and Zweig found that all existing particles can be thought of as bound states of three new particles and their antiparticles. These postulated particles were named quarks. This is a beautiful idea and an extensive search for quarks began immediately. However, so far one has not found any of these particles and if they exist as free particles their masses have to be quite large. The situation today is hence somewhat con- fusing. We do not know if Nature can be built up from a sma?l number of fundamental entities or if the set of possibly infinitely many particles that are seen in high-energy physics today constitute the fundamental set. Closely related to the physical states that we see in Nature is the in- teraction between these states. As a matter of fact it is through their interactions that they reveal their often elusive existence. The gravita- tional and the electromagnetic interactions were found centuries ago and were thought to be the only ones in Nature, but with the increasing insight in the physics of nuclei it became clear that none of these interactions could bo responsible for the bindings in the nuclei. It was shown already in 1933 i\ by Wignor that the nuclear forces must have a very short range of action ] a 3 1i "- ] and that they necessarily are very strong within this range. These forces have .1 hence no classical analogue as do the electromagnetic and gravitational forces. 1 Furthermore already in 1896 Becquerel had found a new phenomenon, radioactive I decay of atoms and in particular thef&-decay, indicating another kind of inter- • 13) -< action. Not until some 35 years later Fermi tried to formulate a theory li for this process. With the increasing knowledge of elementary particles it ;1 was found that this interaction is responsible for a huge number of processes. 5 J The difference between this interaction and the nuclear one is first of all I ;i that the nuclear forces are much stronger. Hence one obtained the division u /! into strong and weak interactions. Secondly and of tantamount importance was q 1 ^) j the discovery of Lee and Yang 1956 that the weak interaction did not con- >l serve parity, i.e. it distinguishes between left and right. It was soon found : that this interaction violates most of the known conservation laws that the •} strong interaction obeys. Since then one distinguishes clearly the four types of interaction present in Nature, the strong, the electromagnetic, the weak and the gravitational ones. It is to be remembered that while elemen- , tary particle physicists today divide their field according to the four types •>] of interaction only 20 years ago many physicists believed that the various •.,'j d physical phenomena observed were only different manifestations of one general J interaction. jI With the concept of elementary particles one has usually associated a I particle with no extension in space, however difficult this concept might be -;| to understand from a quantum mechanical point of view. If we believe that all 4 elementary particles are point-like then the ratio of the strengths between the strong, the electromagnetic, the weak and the gravitational interactions is approximately 1 : 10 : 10 : 10 . We are here comparing dimension- less constants. As the nature of the weak interaction as well as the strong interaction is not completely known, these numbers are somewhat model depen- dent.
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