Time Asymmetric Boundary Conditions and the Definition of Mass and Width for Relativistic Resonances

The definition of mass and width of relativistic resonances and in particular of the $Z$-boson is discussed. For this we use the theory based on time asymmetric boundary conditions given by Hardy class spaces ${\mathbf \Phi}_-$ and ${\mathbf \Phi}_+$ for prepared in-states and detected out-states respectively, rather than time symmetric Hilbert space theory. This Hardy class boundary condition is a mathematically rigorous form of the singular Lippmann-Schwinger equation. In addition to the rigorous definition of the Lippmann-Schwinger kets $|[j,{\mathsf s}]^{\pm}>$ as functionals on the spaces ${\mathbf \Phi}_{\mp}$, one obtains Gamow kets $|[j,{\mathsf s}_R]^- >$ with complex centre-of-mass energy value ${\mathsf s}_R=(M_R-i\Gamma_R/2)^2$. The Gamow kets have an exponential time evolution given by $\exp{(-iM_Rt-\Gamma_Rt/2)}$ which suggests that $(M_R,\Gamma_R)$ is the right definition of the mass and width of a resonance. This is different from the two definitions of the $Z$-boson mass and width used in the Particle Data Table and leads to a numerical value of $M_R=(91.1626\pm 0.0031) {\rm GeV}$ from the $Z$-boson lineshape data.