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Of course, this doesn’t make much sense in the limit in which the symmetry breaking goes tozero.When the quark mass differences become very small compared to typical momenta in thebaryon bound state, the distinction between different energy and momentum redistributions in thedecay disappears. Then the coefficients of the extra terms level off so that they can approach finitebut large limits as the symmetry breaking vanishes.The result of all this is what we expect, in the baryon theory, the operators constructed outof (6.5.17) to appear with large coefficients. Four of these contribute independently to thep–wavedecays. This eliminates all relations for thep–wave amplitudes except the isospin relations, whichare well satisfied.In a sufficiently explicit quark model, it should be possible to find relations among some ofthese extra parameters and test these ideas in detail. For now, we will content ourselves with thenegative result that thep–wave amplitudes cannot be simply predicted.Problems6-1.Derive (6.3.6).6-2.WithLB=LB0+LB1, calculate the axial vector currents, and calculate their matrix ele-ments between baryon states. Derive the Goldberger-Treiman relations. Use the meson Lagrangianincluding the chiral symmetry-breaking term.6-3.Find the quark-current contribution to the vector and axial-vector currents fromLq=Lq0+Lq1, (6.3.2) and (6.3.11). Derive a Goldberger-Treiman relation for constituent quark masses.6-4.Find all theSU(3)×SU(3) invariant terms in the chiral-quark Lagrangian with twoderivatives or oneμM, and no more than two quark fields.6-5.If we include only the spin-dependent relativistic corrections, the masses of the ground-state baryons have the formXimi+κXi,j~Si·~SjmimjShow that (6.4.3) gives a good fit to the octet and decuplet masses.6-6.Use the chiral-quark model to findd/fin the chiral-baryon Lagrangian (6.5.7). Note thatin the nonrelativistic limit, the matrix elements ofjμ5aare nonzero only for the space components,and these have the formXquarkTa~σ6-7.Calculate thes-wave andp-wave amplitudes for the observed hyperon nonleptonic decaysfrom (6.5.7) and compare with the data in the “Review of Particle Properties.” You will need tounderstand their sign conventions. Find the four operators built from (6.5.17) that contribute tothe observed decays. Calculate their contributions and find the coefficients such that together with(6.5.7) and (6.5.9) they give a reasonable picture of boths-wave andp-wave amplitudes.
Chapter 6a- Anomalies6a.1Electromagnetic Interactions andπ0→2γTheπ0decay into two photons obviously involves electromagnetic interactions, not weak inter-actions. Nevertheless, I cannot resist discussing it here. It affords a beautiful illustration of thechiral Lagrangian technique together with a good excuse to study some of the history of the axialanomaly (which we used in Section 5.4 to argue away the axialU(1) symmetry).