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

Abstract

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 $\exp (-i{M}_{R}t-{\Gamma }_{R}t/2)$ 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 $M_R=(91.1626\pm 0.0031)\mathrm{GeV}$ from the $Z$-boson lineshape data.

21 pages revtex file