Relativistic resonances: Their masses, widths, lifetimes, superposition, and causal evolution

Whether one starts from the analytic S-matrix definition or the requirement of gauge-parameter independence in renormalization theory, a relativistic resonance is given by a pole at a complex value $s_R$ of the energy squared $s$. The complex number $s_R$ does not define the mass and the width separately, and the pole definition alone is also not sufficient to describe the interference of two or more Breit-Wigner resonances as observed in experiments. To achieve this and obtain a unified theory of relativistic resonances and decay, we invoke the decaying particle aspect of a resonance and associate to each pole a space of relativistic Gamow kets. The Gamow kets transform irreducibly under causal Poincaré transformations and have an exponential time evolution. Therefore one can choose of the many possible width parameters, the width ${\Gamma }_R$ of the relativistic resonance such that the lifetime $\tau =\hslash /{\Gamma }_R$. This leads to the parametrization $s_R=(M_R-i{\Gamma }_R/2{)}^2$ and uniquely defines these $(M_R,{\Gamma }_R)$ as the mass and width parameters for a resonance. Further it leads to the following new results: Two poles in the same partial wave are given by the sum of two Breit-Wigner amplitudes and by a superposition of two Gamow vectors with each Gamow vector corresponding to one Breit-Wigner amplitude. In addition to the sum of Breit-Wigner amplitudes, the scattering amplitude contains a background amplitude representing direct production of the final state (contact terms). This contact amplitude is associated to a background vector representing the nonexponential energy continuum; omitting it gives the two interfering exponentials of the Weisskopf-Wigner methods. To accomplish all this required a minor modification in the foundation of quantum theory, which led to a quantum theory that contains the time asymmetry of causality.

Figure 1

The two sheeted S-matrix. The $j$th partial S-matrix $S_j(s)$ is an analytic function on a Riemann energy surface cut along the positive real axis from $m_0^2\le s<\infty$ indicated in (a). The integration in Eq. (59) is along the cut in (a), either on the lower edge of the “physical sheet” or along the upper edge of the second sheet. The contour of integration can be deformed into the lower half plane of the second sheet, and ultimately into the contours around the two resonance poles indicated by $\times$ and into an integral from $m_0^2$ to $-{\infty }_{II}$ along the upper edge of the second sheet. This is shown in (b); the arrows indicate the direction of integration. Thus we have the equality of the integral Eqs. (59, 64). For the nonrelativistic case the picture is similar, except that one has to identify $E=E_0=0$ with $s=m_0^2$.