Srednicki — Quantum Field Theory · Problem 11.3

Canonical Quantization of Scalar Fields — Mark Srednicki

Physics — QFT · muon-decay-rate

Problem

Consider the process of muon decay, μ−→e−νˉeνμ\mu^- \to e^-\bar{\nu}_e\nu_\mu. In section 88, we will compute ∣T∣2|\mathcal{T}|^2 for this process (summed over the possible spin states of the decay products, and averaged over the possible spin states of the initial muon), with the result ∣T∣2=64GF2(k1⋅k2′)(k1′⋅k3′),(11.52)|\mathcal{T}|^2 = 64G_F^2(k_1 \cdot k_2')(k_1' \cdot k_3') , \tag{11.52} where GFG_F is the Fermi constant, k1k_1 is the four-momentum of the muon, and k1,2,3′k_{1,2,3}' are the four-momenta of the νˉe\bar{\nu}_e, νμ\nu_\mu, and e−e^-, respectively. In the rest frame of the muon, its decay rate is therefore Γ=32GF2m∫(k1⋅k2′)(k1′⋅k3′) dLIPS3(k1),(11.53)\Gamma = \frac{32G_F^2}{m}\int (k_1 \cdot k_2')(k_1' \cdot k_3')\, d\mathrm{LIPS}_3(k_1) , \tag{11.53} where k1=(m,0)k_1 = (m,\mathbf{0}), and mm is the muon mass. The neutrinos are massless, and the electron mass is 200 times less than the muon mass, so we can take the electron to be massless as well. To evaluate Γ\Gamma, we perform the following analysis. a) Show that Γ=32GF2m∫dk~3′ k1μk3ν′∫k2′μk1′ν dLIPS2(k1−k3′).(11.54)\Gamma = \frac{32G_F^2}{m}\int \widetilde{dk}_3'\, k_{1\mu}k_{3\nu}'\int k_2^{\prime\mu} k_1^{\prime\nu}\, d\mathrm{LIPS}_2(k_1 - k_3') . \tag{11.54} b) Use Lorentz invariance to argue that, for m1′=m2′=0m_{1'} = m_{2'} = 0, ∫k1′μk2′ν dLIPS2(k)=Ak2gμν+Bkμkν,(11.55)\int k_1^{\prime\mu} k_2^{\prime\nu}\, d\mathrm{LIPS}_2(k) = Ak^2 g^{\mu\nu} + Bk^\mu k^\nu , \tag{11.55} where AA and BB are numerical constants. c) Show that, for m1′=m2′=0m_{1'} = m_{2'} = 0, ∫dLIPS2(k)=18π.(11.56)\int d\mathrm{LIPS}_2(k) = \frac{1}{8\pi} . \tag{11.56} d) By contracting both sides of eq. (11.55) with gμνg_{\mu\nu} and with kμkνk_\mu k_\nu, and using eq. (11.56), evaluate AA and BB. e) Use the results of parts (b) and (d) in eq. (11.54). Set k1=(m,0)k_1 = (m,\mathbf{0}), and compute dΓ/dEed\Gamma/dE_e; here Ee≡E3′E_e \equiv E_3' is the energy of the electron. Note that the maximum value of EeE_e is reached when the electron is emitted in one direction, and the two neutrinos in the opposite direction; what is this maximum value? f) Perform the integral over EeE_e to obtain the muon decay rate Γ\Gamma. g) The measured lifetime of the muon is 2.197×10−6 s2.197 \times 10^{-6}\,\mathrm{s}. The muon mass is 105.66 MeV105.66\,\mathrm{MeV}. Determine the value of GFG_F in GeV−2\mathrm{GeV}^{-2}. (Your answer is too low by about 0.2\%, due to loop corrections to the decay rate.) h) Define the energy spectrum of the electron P(Ee)≡Γ−1dΓ/dEeP(E_e) \equiv \Gamma^{-1}d\Gamma/dE_e. Note that P(Ee)dEeP(E_e)dE_e is the probability for the electron to be emitted with energy between EeE_e and Ee+dEeE_e + dE_e. Draw a graph of P(Ee)P(E_e) vs.Ee/mμE_e/m_\mu.

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