Question

The helicity operator is defined as h = sigma middot p/|p| where sigma = (sigma 0 0 sigma). Lets also define positive and negative helicity projection operators P^h_plusminus = (1 plusminus h)/2. Use natural units, h = c = 1. (a) By using the Dirac equation for a fermions with spinor u(p, s), mass m and energy E, show that P^h_plusminus = 1/2 (1 + gamma^5_|P| (E - beta m)). (b) For massless spin-1/2 particles like neutrino, show that the above helicity projection operator equals to the so-called chiral projection operators (the left and right chiral operators), P_L = 1/2 (1 - gamma^5), P_R = 1/2 (1 + gamma^5). (c) In the massless limit show explicitly that the P_L u(p, s) defines a fermion with a negative helicity. (d) In the massless limit show explicitly that the P_R u(p, s) defines a fermion with a positive helicity. (e) Is the handedness Lorentz invariant. Explain your reasoning. (f) Now consider massive (m) spin-1/2 particles electron. Decompose the helicity projection operators into the chiral ones. Under what condition, they are equivalent to each other. (g) Use the result in part (f) to explain why the decay mode pi rightarrow e + v_e is much suppressed as compared to pi rightarrow mu + v_mu.

Answer 1

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