Identities between strong and weak interactions

Abstract

Models of the axial vector weak interaction proposed byGell-Mann andLevy are further investigated. The conditionP μ,μ α (x)=iaπ α (x) , on the weak axial vector, under which the identity\({{ - G_A } \mathord{\left/ {\vphantom {{ - G_A } {G = \left( {{{\mu _0^2 } \mathord{\left/ {\vphantom {{\mu _0^2 } {m_\pi ^2 }}} \right. \kern-\nulldelimiterspace} {m_\pi ^2 }}} \right)}}} \right. \kern-\nulldelimiterspace} {G = \left( {{{\mu _0^2 } \mathord{\left/ {\vphantom {{\mu _0^2 } {m_\pi ^2 }}} \right. \kern-\nulldelimiterspace} {m_\pi ^2 }}} \right)}}\sqrt {Z_3 } \left( {{{f_1 } \mathord{\left/ {\vphantom {{f_1 } {f_0 }}} \right. \kern-\nulldelimiterspace} {f_0 }}} \right)d_\pi \left( 0 \right)F_\pi \left( 0 \right)\) holds, is somewhat weakened. It is also demonstrated that the « generalized Ward identity » derived byBernsteinet al. should really be replaced by the equation (2.20) in Section2. The pion decay process into leptons is discussed by introducing the direct coupling between them, in addition to the four-fermion interaction. It is pointed out that the presence of the direct coupling between pion and leptons does not alter the above relation for the ratioG A /G, its value being independent also of the constant for direct coupling.