Of course this doesnt make much sense in the limit in which the symmetry

# Of course this doesnt make much sense in the limit in

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Of course, this doesn’t make much sense in the limit in which the symmetry breaking goes to zero. When the quark mass differences become very small compared to typical momenta in the baryon bound state, the distinction between different energy and momentum redistributions in the decay disappears. Then the coefficients of the extra terms level off so that they can approach finite but large limits as the symmetry breaking vanishes. The result of all this is what we expect, in the baryon theory, the operators constructed out of (6.5.17) to appear with large coefficients. Four of these contribute independently to the p –wave decays. This eliminates all relations for the p –wave amplitudes except the isospin relations, which are well satisfied. In a sufficiently explicit quark model, it should be possible to find relations among some of these extra parameters and test these ideas in detail. For now, we will content ourselves with the negative result that the p –wave amplitudes cannot be simply predicted. Problems 6-1 . Derive (6.3.6). 6-2 . With L B = L B 0 + L B 1 , calculate the axial vector currents, and calculate their matrix ele- ments between baryon states. Derive the Goldberger-Treiman relations. Use the meson Lagrangian including the chiral symmetry-breaking term. 6-3 . Find the quark-current contribution to the vector and axial-vector currents from L q = L q 0 + L q 1 , (6.3.2) and (6.3.11). Derive a Goldberger-Treiman relation for constituent quark masses. 6-4 . Find all the SU (3) × SU (3) invariant terms in the chiral-quark Lagrangian with two derivatives or one μM , 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 form X i m i + κ X i,j ~ S i · ~ S j m i m j Show that (6.4.3) gives a good fit to the octet and decuplet masses. 6-6 . Use the chiral-quark model to find d/f in the chiral-baryon Lagrangian (6.5.7). Note that in the nonrelativistic limit, the matrix elements of j μ 5 a are nonzero only for the space components, and these have the form X quark T a 6-7 . Calculate the s -wave and p -wave amplitudes for the observed hyperon nonleptonic decays from (6.5.7) and compare with the data in the “Review of Particle Properties.” You will need to understand their sign conventions. Find the four operators built from (6.5.17) that contribute to the 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 both s -wave and p -wave amplitudes.
Chapter 6a - Anomalies 6a.1 Electromagnetic Interactions and π 0 2 γ The π 0 decay into two photons obviously involves electromagnetic interactions, not weak inter- actions. Nevertheless, I cannot resist discussing it here. It affords a beautiful illustration of the chiral Lagrangian technique together with a good excuse to study some of the history of the axial anomaly (which we used in Section 5.4 to argue away the axial U (1) symmetry).