UNITARY SYMMETRY AND LEPTONIC DECAYS

UNITARY SYMMETRY AND LEPTONIC DECAYS

15 JUNE 1963 | Nicola Cabibbo
The chapter discusses the optical model of nuclear reactions, focusing on the effects of varying parameters such as \( V_2 \), \( V_1 \), \( R_b \), \( W_1 \), \( W_2 \), and \( a \) on the curve's location and width. Increasing \( V_2 \) shifts the curve to larger angles, while increasing \( V_1 \) and \( |\eta| \) shifts it to smaller angles. The overall width is primarily determined by \( R_b \), with larger \( R_b \) values decreasing the width and increasing the cross-section magnitude at the curve's center. The effects of increasing \( W_1 \) and \( W_2 \) are minor, and the ratio of peak heights changes depending on whether \( V_2 \) or both \( V_1 \) and \( V_2 \) are increased. The physical implications suggest that distorted-wave analysis of $(p, 2 p)$ experiments provides accurate single-particle wave functions. The rms radius of the charge distribution in \(\mathrm{C}^{12}\) is found to be 2.5 F, close to the experimental value of 2.4 F. The chapter also mentions ongoing investigations into the rms radius for \( s \)-state protons and other light nuclei. The second part of the chapter, authored by Nicola Cabibbo, presents an analysis of leptonic decays based on unitary symmetry for strong interactions and the \( V-A \) theory for weak interactions. The analysis assumes that the weak current \( J_\mu \) transforms according to the eightfold representation of \( SU_4 \) and that the vector part of \( J_\mu \) is in the same octet as the electromagnetic current. The parameters \( a \) and \( b \) are determined to ensure universality, and the angle \( \theta \) is calculated using experimental data for \( K^+ \rightarrow \mu^+ + \nu \) and \( \pi^+ \rightarrow \mu^+ + \nu \). The leptonic decays of baryons are discussed, and the predictions for electron modes with \( \Delta S = 1 \) are compared with experimental data, showing good agreement. The vector coupling constant for \( \beta \)-decay is corrected by \( 6.6\% \), but the discrepancy between \( \Omega^4 \) and muon lifetimes remains unexplained. The final section, by V. Hagopian and W. Selove, reports preliminary results on \(\pi-\pi\) scattering near the \(\rho\) and \(f^0\) resonances. The data suggests a significant contribution from the one-pion-exchange mechanism for low nucleon recoil momentum, and the spin of the \( f^0 \) is greater than zero. The chapter concludes with a brief summary of the findingsThe chapter discusses the optical model of nuclear reactions, focusing on the effects of varying parameters such as \( V_2 \), \( V_1 \), \( R_b \), \( W_1 \), \( W_2 \), and \( a \) on the curve's location and width. Increasing \( V_2 \) shifts the curve to larger angles, while increasing \( V_1 \) and \( |\eta| \) shifts it to smaller angles. The overall width is primarily determined by \( R_b \), with larger \( R_b \) values decreasing the width and increasing the cross-section magnitude at the curve's center. The effects of increasing \( W_1 \) and \( W_2 \) are minor, and the ratio of peak heights changes depending on whether \( V_2 \) or both \( V_1 \) and \( V_2 \) are increased. The physical implications suggest that distorted-wave analysis of $(p, 2 p)$ experiments provides accurate single-particle wave functions. The rms radius of the charge distribution in \(\mathrm{C}^{12}\) is found to be 2.5 F, close to the experimental value of 2.4 F. The chapter also mentions ongoing investigations into the rms radius for \( s \)-state protons and other light nuclei. The second part of the chapter, authored by Nicola Cabibbo, presents an analysis of leptonic decays based on unitary symmetry for strong interactions and the \( V-A \) theory for weak interactions. The analysis assumes that the weak current \( J_\mu \) transforms according to the eightfold representation of \( SU_4 \) and that the vector part of \( J_\mu \) is in the same octet as the electromagnetic current. The parameters \( a \) and \( b \) are determined to ensure universality, and the angle \( \theta \) is calculated using experimental data for \( K^+ \rightarrow \mu^+ + \nu \) and \( \pi^+ \rightarrow \mu^+ + \nu \). The leptonic decays of baryons are discussed, and the predictions for electron modes with \( \Delta S = 1 \) are compared with experimental data, showing good agreement. The vector coupling constant for \( \beta \)-decay is corrected by \( 6.6\% \), but the discrepancy between \( \Omega^4 \) and muon lifetimes remains unexplained. The final section, by V. Hagopian and W. Selove, reports preliminary results on \(\pi-\pi\) scattering near the \(\rho\) and \(f^0\) resonances. The data suggests a significant contribution from the one-pion-exchange mechanism for low nucleon recoil momentum, and the spin of the \( f^0 \) is greater than zero. The chapter concludes with a brief summary of the findings
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