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A dynamical model for the meson spectrum
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up the evollltion of a star. The particular process of importance here is the plasma neutrino process of Adams, Ruderman, and Woo, which depends critically on the existence of t,he (Ed,) (Ed.) interact,ion. We have found that’ the evolution of stars in t’he mass range O.i1.1 M, (Chandrasekhar mass limit for white dwarfs = 1.4 M,) is accelerated by a factor of 10 to 100, depending on the assumed chemical composition, if plasma neutrino emission is included; below 0.1 M, , neutrino emission is unimportant. The more massive Gap stars may hi ideniified observationally on the basis of their very blue color. An upper limit to their evolution time is t.hen obt,ained from a variety of observational arguments; this upper limit is 5 X 10” years, to be compared with the theoretical lifetimes of 2 X lo5 years with netltrino emission and 2 X lo6 years without it. From these results we can draw the following conclusions: (1) the presently known rate of plasma neutrino energy loss roughly reprodllces the astronomical data; (2) t,he weak interaction coupling constant for the (ev,.~ (cv,) interaction has a lower limit close to the currently accepted value. Further evidence from the knowtl observat)ional lifetime of central stars and from some crude stellar models indicates that neutrino emission (photoneutrino process) may also be accelerating evollltion in the central stars. iVorlr/ of /hc Optical Model. N. A~s~~EEN, A. PRAKASH, University of Pittsburgh, Pittsburgh. Pennsylvania; and R. M. I)ILISKO, Universit,y of Pittsburgh, Pittsburgh, Pennsylvania, :md Oak Ridge National Laboratory, Oak Ridge, Tennessee. At1 approximate parameterization of an exact expression for the reflection coefficietl~ ?Y, that describes elastic scattering by a spirl-independent complex potential, yields a p:~rameterized phase-shift model of elastic scattering. The parameterization is based on :t detailed strtdy of the rote played by the nuclear surface region; the nuclear interior is regarded as a region of uniform complex potent,ial. Surface and volume effects are represented by distinct parameters. In all, nine parameters are used. These include the dept,hs of the real alld imaginary potentials in the interior, the radins at which the interior and exterior regions join, alit1 six “surface parameters” that modify the effetts caused by the interior regiolr. These nine parameters describe the same physics that is described by an optical potential that includes an arbitrary mixtllre of surface and volume illt,eractions. This work is relaied to earlier work by Kikuchi, by Peaslee, and by Vogt The Ilsllal resldts of the strong-absorption case are derived by suitable specialization of the model. A nlunerical example that illustrates the application of the full model is given. T’eclor Mnson Emission by Leplons in A’uciear Coulomb Fields. C. M. ANDEHSON, A. 11.\1,tattIN, alld Il. PHIM.\KOFF, ljepartment of Physics, Prlrdlle University, Lafayet,te, Indiarla. It is proposed that experiments be lmdertaken which study the emission of ‘p, W, atrd p[~ by very high-energy electrons or mnons incident on the Coulomb fields of high-% nllceli. A determiluttioll of the cross section for this process would provide an indirect way of measuring decays of the type I’ + e+ + P- where ‘p, w, and p0 and t* = e+, pi. The cross section for the reactiotl I+ + % - P* + V + Z, where I’ is emitted from a lepton-lepton-meson vertex, is derived, and some estimates are made of hackgrormd processes. ,d Dynnmiccrl
Jfodel for the iVeson Spectrum. C. WILKEK and CHAK 1lo.v~.Mo, Brookhavc~ll Nalional Laboratory, Upton, ?Jew York. 4 model for the meson spectrum is constnlcted under the following dynamical ass,lmp. tions: (A) Domillance of processes with low thresholds; (B) dominance of one-particle exchange forces; CC) Sri3 symmetry; (I>) dominance of quasi-two-particle states in production mocesses.
554
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A bootstrap system is sought which satisfies the following conditions: (i) “Consistency” in that all exchanged particles are bound as poles in the appropriate direct channels; (ii) “closure” in that no further low-lying states are bound other than those whose exchange contributes to the forces; and (iii) “completeness ” in that all “low” energy processes involving any of the particles in the spectrum are taken into account. Starting from the O- octet (r, K, 7) and the I- octet (p, R*, $), these consistency considerations lead to the meson supermultiplets listed in Table I. Interesting regularities in the predicted spectrum are noted. In particular one sees that apart from “Regge” recrtrrences, the spect,rum happens to coincide with that contained in the (6*, 6)-(6, 6*) representat,ion of the recently proposed US X US supersymmetry. A tentative assignment, is attempted of known meson states into the scheme. Apart from the K+-r+4 enhancement at 1270 MeV, all reasonably well-established resonances to-date fit, comfortably into the spectrum. Further empirical regularities are noted in the assignment, which include one in the Bronzan-Low quantum number A. Wherever feasible, the dynamical parameters (masses, coupling constants) are computed by requiring numerical “bootstrap” consistency using a method proposed by Zachariasen and Zemach. These values are listed in Table II, where it is seen that qualitative agreement 1s obtained for all parameters so far calculated. Analytical Expressions for Inelastic Scattering Cross Sections. Part II. S. T’ARMA and A. DAR, Department of Nuclear Physics, The Weizmann Institute of Science, Rehovoth, Israel. Closed-form expressions are derived for t,he differential cross section for inelastic scattering via mutual and double collective excitations and under strong absorption conditions. “Blair Phase Rules.” The predictions are comThe expressions result in the generalized pared with experimental data, and good agreement is obtained. The N-Quantum Approximation in Static Models. ANTONIO PAGNAMENT~, Rlltgers, The State University, New Brunswick, New Jersey. The N-quantum approximation (N&A) is a method designed to construct approximate operator solutions to the field equations. It is shown in detail how this method works and can be made compatible with the renormalization program in t,he Lee model and for t,he scalar field interacting with a fixed fermion source. The terminated in-field expansions can still be applied successfully in the presence of bound states, for which one will introduce new in-fields, or if one deals with resonances. This is discussed on the examples of the V-Sbound state and the unstable V-particle. Theory of Alpha Matter. JOHN W. CLARK and TSO-PIN WANG, Depart,ment of Physics, Washington University, St. Louis, Missouri. The usual concept of nuclear matter is discussed, and a generalization of this concept which allows a qualitative understanding of the effects of clustering phenomena is prothe properties of the ground state posed. As an example of “generalized nuclear matter,” of a uniform extended system of structureless OL particles are studied with the method developed by Wu and Feenberg for liquid He4. The Wu-Feenberg method requires the use of (i) a Bijl-Dingle-Jastrow wave function, and (ii) the Kirkwood superposition approximation. The assumed 0l-a interactions fit the S-wave phase shifts up to a c.m. energy of 10 MeV. The results, in comparison with results from theoretical nuclear-matter calculations which entertain no clustering (both expressed in terms of the energy per nucleon as a func-
OF
PAPERS
TO
SPPEAR
IN
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553
up the evollltion of a star. The particular process of importance here is the plasma neutrino process of Adams, Ruderman, and Woo, which depends critically on the existence of t,he (Ed,) (Ed.) interact,ion. We have found that’ the evolution of stars in t’he mass range O.i1.1 M, (Chandrasekhar mass limit for white dwarfs = 1.4 M,) is accelerated by a factor of 10 to 100, depending on the assumed chemical composition, if plasma neutrino emission is included; below 0.1 M, , neutrino emission is unimportant. The more massive Gap stars may hi ideniified observationally on the basis of their very blue color. An upper limit to their evolution time is t.hen obt,ained from a variety of observational arguments; this upper limit is 5 X 10” years, to be compared with the theoretical lifetimes of 2 X lo5 years with netltrino emission and 2 X lo6 years without it. From these results we can draw the following conclusions: (1) the presently known rate of plasma neutrino energy loss roughly reprodllces the astronomical data; (2) t,he weak interaction coupling constant for the (ev,.~ (cv,) interaction has a lower limit close to the currently accepted value. Further evidence from the knowtl observat)ional lifetime of central stars and from some crude stellar models indicates that neutrino emission (photoneutrino process) may also be accelerating evollltion in the central stars. iVorlr/ of /hc Optical Model. N. A~s~~EEN, A. PRAKASH, University of Pittsburgh, Pittsburgh. Pennsylvania; and R. M. I)ILISKO, Universit,y of Pittsburgh, Pittsburgh, Pennsylvania, :md Oak Ridge National Laboratory, Oak Ridge, Tennessee. At1 approximate parameterization of an exact expression for the reflection coefficietl~ ?Y, that describes elastic scattering by a spirl-independent complex potential, yields a p:~rameterized phase-shift model of elastic scattering. The parameterization is based on :t detailed strtdy of the rote played by the nuclear surface region; the nuclear interior is regarded as a region of uniform complex potent,ial. Surface and volume effects are represented by distinct parameters. In all, nine parameters are used. These include the dept,hs of the real alld imaginary potentials in the interior, the radins at which the interior and exterior regions join, alit1 six “surface parameters” that modify the effetts caused by the interior regiolr. These nine parameters describe the same physics that is described by an optical potential that includes an arbitrary mixtllre of surface and volume illt,eractions. This work is relaied to earlier work by Kikuchi, by Peaslee, and by Vogt The Ilsllal resldts of the strong-absorption case are derived by suitable specialization of the model. A nlunerical example that illustrates the application of the full model is given. T’eclor Mnson Emission by Leplons in A’uciear Coulomb Fields. C. M. ANDEHSON, A. 11.\1,tattIN, alld Il. PHIM.\KOFF, ljepartment of Physics, Prlrdlle University, Lafayet,te, Indiarla. It is proposed that experiments be lmdertaken which study the emission of ‘p, W, atrd p[~ by very high-energy electrons or mnons incident on the Coulomb fields of high-% nllceli. A determiluttioll of the cross section for this process would provide an indirect way of measuring decays of the type I’ + e+ + P- where ‘p, w, and p0 and t* = e+, pi. The cross section for the reactiotl I+ + % - P* + V + Z, where I’ is emitted from a lepton-lepton-meson vertex, is derived, and some estimates are made of hackgrormd processes. ,d Dynnmiccrl
Jfodel for the iVeson Spectrum. C. WILKEK and CHAK 1lo.v~.Mo, Brookhavc~ll Nalional Laboratory, Upton, ?Jew York. 4 model for the meson spectrum is constnlcted under the following dynamical ass,lmp. tions: (A) Domillance of processes with low thresholds; (B) dominance of one-particle exchange forces; CC) Sri3 symmetry; (I>) dominance of quasi-two-particle states in production mocesses.
554
ABSTRACTS
OF
PAPERS
TO
APPEAR
IN
FUTURE
ISSUES
A bootstrap system is sought which satisfies the following conditions: (i) “Consistency” in that all exchanged particles are bound as poles in the appropriate direct channels; (ii) “closure” in that no further low-lying states are bound other than those whose exchange contributes to the forces; and (iii) “completeness ” in that all “low” energy processes involving any of the particles in the spectrum are taken into account. Starting from the O- octet (r, K, 7) and the I- octet (p, R*, $), these consistency considerations lead to the meson supermultiplets listed in Table I. Interesting regularities in the predicted spectrum are noted. In particular one sees that apart from “Regge” recrtrrences, the spect,rum happens to coincide with that contained in the (6*, 6)-(6, 6*) representat,ion of the recently proposed US X US supersymmetry. A tentative assignment, is attempted of known meson states into the scheme. Apart from the K+-r+4 enhancement at 1270 MeV, all reasonably well-established resonances to-date fit, comfortably into the spectrum. Further empirical regularities are noted in the assignment, which include one in the Bronzan-Low quantum number A. Wherever feasible, the dynamical parameters (masses, coupling constants) are computed by requiring numerical “bootstrap” consistency using a method proposed by Zachariasen and Zemach. These values are listed in Table II, where it is seen that qualitative agreement 1s obtained for all parameters so far calculated. Analytical Expressions for Inelastic Scattering Cross Sections. Part II. S. T’ARMA and A. DAR, Department of Nuclear Physics, The Weizmann Institute of Science, Rehovoth, Israel. Closed-form expressions are derived for t,he differential cross section for inelastic scattering via mutual and double collective excitations and under strong absorption conditions. “Blair Phase Rules.” The predictions are comThe expressions result in the generalized pared with experimental data, and good agreement is obtained. The N-Quantum Approximation in Static Models. ANTONIO PAGNAMENT~, Rlltgers, The State University, New Brunswick, New Jersey. The N-quantum approximation (N&A) is a method designed to construct approximate operator solutions to the field equations. It is shown in detail how this method works and can be made compatible with the renormalization program in t,he Lee model and for t,he scalar field interacting with a fixed fermion source. The terminated in-field expansions can still be applied successfully in the presence of bound states, for which one will introduce new in-fields, or if one deals with resonances. This is discussed on the examples of the V-Sbound state and the unstable V-particle. Theory of Alpha Matter. JOHN W. CLARK and TSO-PIN WANG, Depart,ment of Physics, Washington University, St. Louis, Missouri. The usual concept of nuclear matter is discussed, and a generalization of this concept which allows a qualitative understanding of the effects of clustering phenomena is prothe properties of the ground state posed. As an example of “generalized nuclear matter,” of a uniform extended system of structureless OL particles are studied with the method developed by Wu and Feenberg for liquid He4. The Wu-Feenberg method requires the use of (i) a Bijl-Dingle-Jastrow wave function, and (ii) the Kirkwood superposition approximation. The assumed 0l-a interactions fit the S-wave phase shifts up to a c.m. energy of 10 MeV. The results, in comparison with results from theoretical nuclear-matter calculations which entertain no clustering (both expressed in terms of the energy per nucleon as a func-