Srednicki — Quantum Field Theory · Problem 48.5

Canonical Quantization of Scalar Fields — Mark Srednicki

Physics — QFT · spin-averaged-cross-sections-pion-decay

Problem

The charged pion π−\pi^- is represented by a complex scalar field φ\varphi, the muon μ−\mu^- by a Dirac field M\mathcal{M}, and the muon neutrino νμ\nu_\mu by a spin-projected Dirac field PLNP_{\mathrm{L}}\mathcal{N}, where PL=12(1−γ5)P_{\mathrm{L}} = \frac{1}{2}(1-\gamma_5). The charged pion can decay to a muon and a muon antineutrino via the interaction L1=2c1GFfπ∂μφ M‾γμPLN+h.c.  ,(48.30)\mathcal{L}_1 = 2c_1 G_{\mathrm{F}} f_\pi \partial_\mu \varphi\, \overline{\mathcal{M}}\gamma^\mu P_{\mathrm{L}}\mathcal{N} + \mathrm{h.c.} \;, \tag{48.30} where c1c_1 is the cosine of the Cabibbo angle, GFG_{\mathrm{F}} is the Fermi constant, and fπf_\pi is the pion decay constant. a) Compute the charged pion decay rate Γ\Gamma. b) The charged pion mass is mπ=139.6 MeVm_\pi = 139.6\,\mathrm{MeV}, the muon mass is mμ=105.7 MeVm_\mu = 105.7\,\mathrm{MeV}, and the muon neutrino is massless. The Fermi constant is measured in muon decay to be GF=1.166×10−5 GeV−2G_{\mathrm{F}} = 1.166\times10^{-5}\,\mathrm{GeV}^{-2}, and the cosine of the Cabibbo angle is measured in nuclear beta decays to be c1=0.974c_1 = 0.974. The measured value of the charged pion lifetime is 2.603×10−8 s2.603\times10^{-8}\,\mathrm{s}. Determine the value of fπf_\pi in MeV. Your result is too large by 0.8\%, due to neglect of electromagnetic loop corrections.
Intuitive explanation

The key to this problem is recognizing that the derivative coupling ∂μφ\partial_\mu \varphi acts like a momentum factor in the decay amplitude, and that you must carefully set up the two-body phase space for a massive parent decaying to a massive daughter and a massless neutrino. The amplitude squared will depend on the pion momentum transfer, which is fixed by kinematics in the rest frame; then integrate over the muon–neutrino phase space to find the total decay rate, and invert to extract fπf_\pi from the measured lifetime.

Intuitive explanation

The key to this problem is recognizing that the derivative coupling ∂μφ\partial_\mu \varphi acts like a momentum factor in the decay amplitude, and that you must carefully set up the two-body phase space for a massive parent decaying to a massive daughter and a massless neutrino. The amplitude squared will depend on the pion momentum transfer, which is fixed by kinematics in the rest frame; then integrate over the muon–neutrino phase space to find the total decay rate, and invert to extract fπf_\pi from the measured lifetime.

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