Standard Model and Beyond · Problem 3.14

Gauge Theories

Physics — Particle Physics · spurion-isospin-symmetry-breaking

Problem

It is sometimes useful to introduce a spurion, which is a fictitious or composite field with the right quantum numbers to parametrize the effects of symmetry breaking, and which can therefore be used as a bookkeeping device. For example, it is known that the effective operator for strangeness-changing nonleptonic weak transitions, which could in principle involve both ΔI=12\Delta I = \frac{1}{2} and ΔI=32\Delta I = \frac{3}{2} components, is strongly dominated by ΔI=12\Delta I = \frac{1}{2}. Therefore, the transitions Σ+→π0p\Sigma^+ \to \pi^0 p, Σ+→π+n\Sigma^+ \to \pi^+ n, and Σ−→π−n\Sigma^- \to \pi^- n should be related in the same way as the isospin-conserving amplitudes for Σ+S→π0p\Sigma^+ S \to \pi^0 p, Σ+S→π+n\Sigma^+ S \to \pi^+ n, and Σ−S→π−n\Sigma^- S \to \pi^- n, where SS is a spin-0 spurion carrying I=1/2I = 1/2 and I3=−1/2I_3 = -1/2. Show that the amplitudes (using the Condon-Shortley convention for π+\pi^+ and Σ+\Sigma^+) should be related by M(Σ+→π+n)−M(Σ−→π−n)=−2M(Σ+→π0p).M(\Sigma^+ \to \pi^+ n) - M(\Sigma^- \to \pi^- n) = -\sqrt{2}M(\Sigma^+ \to \pi^0 p). This relation, which holds separately for the SS and PP wave amplitudes (Problem 2.26), works reasonably well. (This simple example could be handled easily using the Wigner-Eckart theorem, but the spurion formalism is more convenient in more complicated cases.)

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