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When calculating characteristics of various physical processes involving particle interactions, one must deal with the production and decay of unstable particles participating in these processes, such as W and Z bosons, the top quark, the Higgs boson, and others. Unlike stable particles, unstable particles are characterized not only by their mass but also by partial and total decay widths. The lifetime of an unstable particle is equal to the inverse of its total width.
To describe the contribution of an unstable particle to the amplitude of a physical process, the well-known Breit-Wigner formula is most commonly employed. The maximum of the scattering cross section, proportional to the square of the amplitude, corresponds to the resonance mass of the particle. Additionally, background processes involving the same initial and final state particles contribute to both the amplitude and the cross section of processes involving unstable particles. For example, in electron-positron annihilation into a muon pair, two Feynman diagrams contribute to the amplitude: one involving a virtual photon and another involving a Z boson. Consequently, in this case the cross section contains several terms: the background contribution (virtual photon), the signal contribution (Z boson production), and the interference between them.
The literature discusses that the Breit-Wigner formula can be refined by including the square of the particle's width in the denominator of the propagator—a refinement that may be important when achieving high experimental precision or when calculating processes involving hypothetical new particles with potentially large decay widths. The purpose of this note is to study the impact of such corrections on the integrated cross section and to compare it with the frequently used Breit-Wigner formula, in which this correction is neglected.
