Time asymmetric quantum theory and the ambiguity of the Z-boson mass and width

Abstract

Relativistic Gamow vectors emerge naturally in a time asymmetric quantum theory as the covariant kets associated to the resonance pole \(s=s_R\) in the second sheet of the analytically continued S-matrix. They are eigenkets of the self-adjoint mass operator with complex eigenvalue \(\sqrt{s_R}\) and have exponential time evolution with lifetime \(\tau = - \hbar/2\mathrm{Im}\sqrt{s_R}\). If one requires that the resonance width \(\Gamma\) (defined by the Breit-Wigner lineshape) and the resonance lifetime \(\tau\) always and exactly fulfill the relation \(\Gamma=\hbar/\tau\), then one is lead to the following parameterization of \(s_R\) in terms of resonance mass \(M_R\) and width \(\Gamma_R\): \(s_R = (M_R - i\Gamma/2)^2\). Applying this result to the \(Z\)-boson implies that \(M_R \approx M_Z - 26\mbox{MeV}\) and $\Gamma_R \approx \Gamma_Z-1.2\mbox{MeV}$ are the mass and width of the {\it Z}-boson and not the particle data values \((M_Z,\Gamma_Z)\) or any other parameterization of the Z-boson lineshape. Furthermore, the transformation properties of these Gamow kets show that they furnish an irreducible representation of the causal Poincare semigroup, defined as a semi-direct product of the homogeneous Lorentz group with the semigroup of space-time translations into the forward light cone. Much like Wigner's unitary irreducible representations of the Poincare group which describe stable particles, these irreducible semigroup representations can be characterized by the spin-mass values \((j,s_R=(M_R-i\Gamma/2)^2)\).