On the Mass and Width of the Z-boson and Other Relativistic Quasistable Particles

The ambiguity in the definition for the mass and width of relativistic resonances is discussed, in particular for the case of the Z-boson. This ambiguity can be removed by requiring that a resonance’s width $\mathrm{\Gamma }$ (defined by a Breit-Wigner lineshape) and lifetime $\tau$ (defined by the exponential law) always and exactly fulfill the relation $\mathrm{\Gamma }=\mathrm{\hslash }\mathrm{/}\tau$. To justify this one needs relativistic Gamow vectors which in turn define the resonance’s mass M_R as the real part of the square root $\mathrm{R}\mathrm{e}\sqrt{s_R}$ of the S-matrix pole position s_R. For the Z-boson this means that $M_R \approx M_Z - 26{MeV}$and$\Gamma_R \approx \Gamma_Z-1.2{MeV}$where{M}_{Z}and$\Gamma_Z$ are the values reported in the particle data tables.