An Introduction To Quantum Field Theory - M. Peskin, D. Schroeder (Perseus, 1995) · Problem 19.2

Perturbation Theory Anomalies — M. Peskin, D. Schroeder (Perseus, 1995)

Physics — QFT · perturbation-theory-anomalies

Problem

19.2 Weak decay of the pion. (a) In the effective Lagrangian for semileptonic weak interactions (18.28), the hadronic part of the operator is a left-handed current involving the uu and dd quarks. Show that this current is related to the quark currents of Section 19.3 as follows: uˉLγμdL=12(jμ1+ijμ2−jμ51−ijμ52),\bar{u}_L \gamma^\mu d_L = \frac{1}{2}(j^{\mu 1} + ij^{\mu 2} - j^{\mu 51} - ij^{\mu 52}), where 1, 2 are isospin indices. Using this identification and (19.88), show that the amplitude for the decay π+→ℓ+ν\pi^+ \to \ell^+ \nu is given by iM=GFfπuˉ(q)p̸(1−γ5)v(k),i\mathcal{M} = G_F f_\pi \bar{u}(q) \not{p}(1-\gamma^5) v(k), where pp, kk, qq are the momenta of the π+\pi^+, ℓ+\ell^+, ν\nu. (b) Compute the decay rate of the pion. Show that this rate vanishes in the limit of zero lepton mass, and that the relative rate of pion decay to muons and electrons is given by Γ(π+→e+ν)Γ(π+→μ+ν)=(memμ)2(1−me2/mπ2)2(1−mμ2/mπ2)2=10−4.\frac{\Gamma(\pi^+ \to e^+ \nu)}{\Gamma(\pi^+ \to \mu^+ \nu)} = \left(\frac{m_e}{m_\mu}\right)^2 \frac{(1 - m_e^2/m_\pi^2)^2}{(1 - m_\mu^2/m_\pi^2)^2} = 10^{-4}. From the measured pion lifetime, τπ=2.6×10−8\tau_\pi = 2.6 \times 10^{-8} sec, and the pion and muon masses, mπ=140m_\pi = 140 MeV, mμ=106m_\mu = 106 MeV, determine the value of fπf_\pi.

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