Mixing Effect between Particles and Resonances

Abstract

The system of pion, nucleon, and (3-3) particle acting mutually through the Yukawa interaction is investigated by means of the static meson theory. It is assumed that these particles (including the (3-3) particle) can be treated as elementary ones although they are equally constructed from urbaryons. An integral equation for the scattering amplitude is solved in some reasonable approximation. Since the Yukawa interaction is strong enough to produce resonances between pion and nucleon, one may expect that two resonances (or bound states) exist in the (3 3) state of pion-nucleon scattering. In fact, the solution with two resonances is obtained in case the mixing energy is small. It is shown, howevel", that one o£ them disappears when the mixing energy increases. § l. lntrodu.ctionRecently much progress has been made in elementary particle physics. The 5'U(6) theory 1 > has gained success in systematizing the low-lying octet and decuplet baryons into a 56-plet. This suggests that these baryons are equally constructed from urbaryons mainly by spin-unitary spin independent force.On the other hand, from the analysis of nuclear force, it has been well known that the Yukawa interaction exists between mesons and baryons. It has also been pointed out that the Yukawa interaction is not compatible with the SU (6) -symmetry.To get over this trouble, Ohnuki and Toyoda 2 > suggested from the standpoint of the composite model that the interaction Hamiltonian of the urbaryons concerned can be divided into two parts as follows :where FI1 is SU(6) -invariant and Fin is not. The Yukawa interaction, should be considered to be derived from the SU (6) -breaking part 1--In. Therefore, we can infer from this that, at first, both octet and decuplet baryons are equally constructed from urbaryons according to 11 1 , and subsequently the Yukawa interaction switches on between these baryons and n1esons, and the baryons come to wear meson cloud. Here it should be noted that the Yukawa interaction is not necessarily weak. For instance, using the static meson theory, Chew 3 > at University of Bath, Library on June 21, 2015 http://ptp.oxfordjournals.org/ Downloaded from iVlLring Eflect bet·ween Particles and Resonances *> Since yAjD 0 L)O as E-':>±=, there occurs only the odd num.ber of intersections. When ])A has three zero points, one of them corresponds to the energy at which the phase shift goes downward across n:/2. at University of Bath, Library on June 21, 2015 http://ptp.oxfordjournals.org/ Downloaded from *l Since ~-A/ D 0 A represents self-energy and D 0 A can be considered to be proportional to the propagator of a resonance (or particle) of type TT, yA corresponc1s to the mixing energy between resonances of types I and TI. at University of Bath, Library on June 21, 2015 http://ptp.oxfordjournals.org/ Downloaded from ()) .T. L. Gammel, Phys. l{ev. 95

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“…By the use of the invariant functions the self-energy part is represented such that Σ(p) = (1 − A(p 2 ))p / + B(p 2 ). It has been known for a long time [1,2,3] that Eq. (1) allows a non-vanishing solution for B(p 2 ) with A(p 2 ) = 1 if α > α c = π/3.…”

“…In particular it plays an important role when we deal with the early stage of the universe. It has been known for a long time that Abelian gauge theories at vanishing temperature subject to the chiral symmetry breaking in the strong coupling region and hence the massless fermions acquire a mass in a dynamical way [1,2,3,4]. This phenomenon has been clarified mostly by using the Schwinger-Dyson equation [5].…”

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“…while the set of second-class constraints is ς 1 = π θ and ς 2 = σ a π a − σ 2 a θ. Since the second-class constraints are assumed to be equal to zero in a strong way [37] the Dirac brackets are constructed as…”

“…This allows us to compute the primary Dirac brackets. Due to the Maskawa-Nakajima theorem [37], the primary Dirac brackets among the phase space fields are canonical. To demonstrate this, the brackets are computed explicitly.…”

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