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The conversion from a measured beam peak to a particle resonance mass is fixed, but the missing graph supplies neither the peak position nor its width. Let $T_R$ be the incoming pion kinetic energy at the peak and $\Delta T$ the full width at half maximum after allowing for instrumental resolution. With a stationary proton and natural units,Thus the peak invariant mass isThe width must also be transformed to the centre-of-mass frame. To first order about the peak,For a broad feature, transform the two half-height positions separately: if they are $T_{-}$ and $T_{+}$, then the corresponding width in invariant mass is $\sqrt{(m_p+m_{\pi }{)}^2+2m_p{T}_{+}}-\sqrt{(m_p+m_{\pi }{)}^2+2m_p{T}_{-}}$. An energy-dependent numerator or width can also displace the observed maximum from the resonance pole parameter.

The low-energy ${\pi }^{+}p$ channel is normally dominated by ${\Delta }^{++}(1232)$, a Delta baryon with $Q=2$, $B=1$ and $I=3/2$. As an external reference example, not a reading of the absent plot, $M=1232\mathrm{MeV}$ and $\Gamma =117\mathrm{MeV}$ give $T_R\simeq 190\mathrm{MeV}$, $\Delta T\simeq 154\mathrm{MeV}$ and $\tau \simeq 5.63\times 10^{-24}\mathrm{s}$.^[ref] The actual graphical estimates remain unavailable. In particular, using the laboratory width directly in $\tau =\hslash /\Delta T$ misses the factor $m_p/M$.