A concept of asymptotic symmetry is formulated and applied to the determination of mixing parameters between the 7r°, TJ°, and r)'°(958) in broken SU(3) and SU(2) symmetries. One of the solutions gives rise to a rather large violation of the | Al |=\ rule in the K lZ decays which is not inconsistent with present experiment.This paper aims to discuss the particle mixing effects by taking the view of asymptotic symmetry which assumes that the SU(3) and SU(2) symmetries are well realized among particles of extremely high momenta. A similar assumption has been used in deriving several successful sum rules from the chiral SU(3)®SU(3) algebra. 1 ' 2 To derive our result by a short-cut (but instructive) computation, we express the requirement of asymptotic SU(3) symmetry in a simple form. Let us consider the pseudoscalar nonet and denote their annihilation operators as a a (k) 1 where a stands for TT*' 0 , K + »°, K + *°, r]°, and 7?' 0 (958), and k denotes their momenta. It should be noted that a a (k) are not the Heisenberg operators but the operators of the physical (i.e., incoming) particles with mass m a . Denoting the SU(3) generators by V i9 the transformation of physical particles in broken SU(3) symmetry can be expressed
In the framework of the chiral SU(3)@SU(3) charge algebra and asymptotic SU(3), a scheme of asymptotic algebraic realization of SU(3) is proposed. For the axial-vector couplings, it yields D/F =+ but g,(O) =to, f being the universal fractional contribution (=50%) to the sum rules coming from the ground-state baryons. Neither the assumption of saturation by lowlying states nor the introduction of large configuration mixing is necessary.We propose a possible scheme of algebraic realization of SU(3) in the framework of the chiral SU(3)B SU(3) charge algebra and present a new derivation of the D / F ratio of hyperon axial-vector semileptonic couplings and g,(O). Our proposed scheme may provide new insight in hadron physics. Some years ago, Gersteinl observed that, by saturating the algebra s o l e l y by the $+ octet and $+ decuplet, the exact SU(6) value2 of D/F can be reproduced without assuming SU(6) o r SU(6) currentcommutation relations. However, though certainly remarkable. Gerstein's result cannot be taken toc seriously for the following reasons: (i) The saturation argument used i s hardly justified. Actually the calculation yields, at the same time, a bad value gA(0) = 9 , a s is also the case with exact SU(6).' On the contrary, the Adler-Weisberger c a l c~l a t i o n ,~ which is based on the same algebra but does not use the saturation argument, gives a correct value of gA(0). (ii) Exact SU (3) is assumed for the matrix elements involved. There is no a priori justification for this assumption.In this paper, we point out that there i s a way out of these difficulties. Our new points a r e a s follows: (a) The saturation argument of Gerstein can be replaced by our scheme of asymptotic algebraic realization of SU(3). (b) Exact SU(3) used in Ref. 1 can be simply replaced by our asymptotic SU(3) proposed bef01-e.~ The degree of accuracy of our asymptotic Su(3) can be best seen a s foll o w~.~ If the basic (not effective) SU(3)-breakingHamiltonian belongs to an octet, in the framework of asymptotic SU(3) the Gell-Mann-Okuko mass formulas (including the effect of particle mixing) become exact mass formulas (rather than firstorder formulas5).We have already applied our proposed scheme and asymptotic SU(3) to bosons and found an encouraging result."here, the generalized boson nonet coupling scheme has emerged a s a consequence which reduces to the ideal nonet scheme [also obtained2 from exact SU (6)], whenever one of the constraints of ideal nonet (such a s mu-mp) i s realized. We denote the baryons by B,,,. CY stands for the -(physical) SU(3) multiplets, i.e., (N,, A,, Z, , z,) for octet and (A,, Z ; : , Z,*, as) for decuplet. s denotes the J~ and other quantum numbers. We consider the algebra sandwiched between the oneparticle baryon state with infinite momentum. The algebra [V,, 51 = ifilkVk will then be automatically satisfied by our asymptotic SU(3). The algebra [&,A,] =ifil,Ak yields the following important information when combined with our asymptotic SU(3): Although SU(3) i s broken, the matrix elements of t...
Resonance saturation of axial charge commutators taken between states of the pseudoscalar-meson octet is examined and found to be strongly suggestive of the existence of a 0 + octet and a 0 + singlet in the 500-1000-MeV region. The vector-meson contribution to the matrix element of the vector current taken between K and ir states is shown to give only 50% or so of the experimental value, in contrast to the conclusions of other authors, but in agreement with general SU(3) considerations and the estimate of Adler of the p contribution to the sum rule for the TT-T scattering. The approach to T e s decay based on dispersion theory and charge-current commutators already used by Marshak et al. for K e z decay is shown to give good agreement with experiment if vector-meson dominance is assumed. It is pointed out that in both cases the resulting expressions for the form factors at zero momentum transfer differ exactly by a factor of 2 from the corresponding ones obtained in the approach using only commutators of charges.
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