The declared single-channel resonance model: pole-shaped channel factor and its cross-section reading

(a) Framework data. Adopt the declared short-range scattering model The declared short-range scattering model: state space, potential classes, assigned energy observables, asymptotic vocabulary and units with mass $m>0$, spin zero and action constant $\mathrm{\hslash }>0$, and its partial-wave and phase-shift reading The model's partial-wave and phase-shift vocabulary for a declared central potential for a declared central short-range potential. Fix a partial-wave label $l\in \{0,1,2,\dots \}$ in the sense of clause (b) of that reading. Here a single channel means the whole degree-$l$ partial-wave sector: $\mathrm{span}\{Y_{l\mu }:-l\le \mu \le l,\mu \in \mathbb{Z}\}$, with all $2l+1$ angular labels $(l,\mu )$ sharing the same scalar factor $S_l$. The symbol $\mu$ is the angular index; $m$ remains the positive mass. This convention agrees with that reading's ${\sigma }_l$, which includes all these angular labels. Units are those of Dimensions, SI unit conventions and quantity types of the framework.

(b) The declared single-channel resonance model. As an additional model assumption of the pair, fix real resonance parameters $E_r$ and $\mathrm{\Gamma }>0$, both read in joules, and declare the channel factor of the label $l$ to be $S_l(E):=\frac{E-E_r-i\mathrm{\Gamma }\mathrm{/}2}{E-E_r+i\mathrm{\Gamma }\mathrm{/}2}(E>0),$ read for positive energies only, with the phase shift of the channel read from the relation $S_l=e^{2i{\delta }_l}$ modulo $\pi$ and the channel cross-section reading ${\sigma }_l(E):=\frac{4\pi }{k(E{)}^2}(2l+1){\sin }^2{\delta }_l(E),k(E):=\frac{\sqrt{2mE}}{\mathrm{\hslash }},$ read in ${\mathrm{m}}^2$. The model calls $E_r$ the resonance energy and $\mathrm{\Gamma }$ the width parameter; both are declared model parameters and neither is identified with a measured quantity. The parameter $E_r$ may be zero or negative; calling it a resonance energy does not assign a physical cross section at that parameter value unless $E_r>0$. Auxiliary real-axis algebraic comparisons of the Lorentzian factor do not extend $k(E)$ or ${\sigma }_l(E)$ beyond the positive-energy domain.

(c) Status and scope. Clauses (a)-(b) declare a conditional one-channel model. The pair asserts no analytic continuation, no pole of the resolvent and no property of any potential that would produce the declared factor $S_l$; the model is used only to compute the induced phase shift and channel cross section of the declared factor. In particular, no exponential decay law, no lifetime $\mathrm{\hslash }\mathrm{/}\mathrm{\Gamma }$ and no Lorentzian line shape of a measured distribution is asserted or derived; those would be separate models with their own assumptions and their own uncertainty qualifications.

• Relation to the full cross section. Substituting the declared channel factor into the phase-shift series of Partial-wave expansion, unit-modulus channel factors and the phase-shift cross-section series produces a conditional total cross section for an idealized situation in which a single channel dominates; the pair does not assert that any declared potential has that behavior.
• Width versus rate. The width parameter $\mathrm{\Gamma }$ is an energy parameter of the declared factor, not a decay rate; the identification of $\mathrm{\Gamma }$ with a rate times $\mathrm{\hslash }$ requires a separate decay model and is not made here.