Srednicki — Quantum Field Theory · Problem 66.4

Canonical Quantization of Scalar Fields — Mark Srednicki

Physics — QFT · value-of-alpha-at-mw

Problem

The value of α(MW)\alpha(M_{\mathrm{W}}). The solution of eq. (66.12) is 1α(MW)=1α(μ)−23π∑iQi2ln⁡(MW/μ),(66.30)\frac{1}{\alpha(M_{\mathrm{W}})} = \frac{1}{\alpha(\mu)} - \frac{2}{3\pi} \sum_i Q_i^2 \ln(M_{\mathrm{W}}/\mu) , \tag{66.30} where the sum is over all quarks and leptons (each color of quark counts separately), and we have chosen the W±W^\pm boson mass MWM_{\mathrm{W}} as a reference scale. We can define a different renormalization scheme, modified decoupling subtraction or DS‾\overline{\mathrm{DS}}, where we imagine integrating out a field when μ\mu is below its mass. In this scheme, eq. (66.30) becomes 1α(MW)=1α(μ)−23π∑iQi2ln⁡[MW/min⁡(mi,μ)],(66.31)\frac{1}{\alpha(M_{\mathrm{W}})} = \frac{1}{\alpha(\mu)} - \frac{2}{3\pi} \sum_i Q_i^2 \ln[M_{\mathrm{W}}/\min(m_i,\mu)] , \tag{66.31} where the sum is now over all quarks and leptons with mass less than MWM_{\mathrm{W}}. For μ<me\mu < m_e, the DS‾\overline{\mathrm{DS}} scheme coincides with the OS scheme, and we have 1α(MW)=1α−23π∑iQi2ln⁡(MW/mi),(66.32)\frac{1}{\alpha(M_{\mathrm{W}})} = \frac{1}{\alpha} - \frac{2}{3\pi} \sum_i Q_i^2 \ln(M_{\mathrm{W}}/m_i) , \tag{66.32} where α=1/137.036\alpha = 1/137.036 is the fine-structure constant in the OS scheme. Using mu=md=ms∼300 MeVm_u = m_d = m_s \sim 300\,\mathrm{MeV} for the light quark masses (because quarks should be replaced by hadrons at lower energies), and other quark and lepton masses from sections 83 and 88, compute α(MW)\alpha(M_{\mathrm{W}}).

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