A. Zee, Quantum Field Theory in a Nutshell · Problem VIII.3.3

Field Theory Redux

Physics — QFT

Problem

Let A,B,C,DA, B, C, D denote four spin 12\frac{1}{2} fields and label their handedness by a subscript: γ5Ah=hAh\gamma_5 A_h = h A_h with h=±1h = \pm 1. Thus, A+A_+ is right handed, A−A_- left handed, and so on. Show that (AhBh)(C−hD−h)=−12(AhγμD−h)(C−hγμBh)(A_h B_h)(C_{-h} D_{-h}) = -\frac{1}{2}(A_h \gamma^\mu D_{-h})(C_{-h} \gamma_\mu B_h) This is an example of a broad class of identities known as Fierz identities (some of which we will need in discussing supersymmetry.) Argue that if proton decay proceeds in lowest order from the exchange of a heavy vector particle then only the terms (lLCqL)(uRCdR)(l_L C q_L)(u_R C d_R) and (eRCuR)(qLCqL)(e_R C u_R)(q_L C q_L) are allowed in the Lagrangian.

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