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For the selected muon channel, finite curvature resolution in the magnetic tracking system broadens the reconstructed transverse momentum spectrum. A high-$p_T$ track has small sagitta, so spatial measurement errors, alignment errors and magnetic-field calibration errors become important; at high momentum an approximately fixed uncertainty in $1/p_T$ gives $\delta p_T\propto p_T^2$. Multiple Coulomb scattering contributes at lower momenta. If the transverse mass is fitted, finite hadronic-recoil resolution additionally smears the inferred missing transverse momentum. These are experimental effects, distinct from the physical recoil and width in ii.

The sagitta is the maximum distance between the circular track and its chord. The perpendicular bisector of a chord of length $L$ gives a right triangle with hypotenuse $R$ and sides $L/2$ and $R-s$. HenceAt fixed electric charge, magnetic field and lever arm, $p=|q|B{L}^2/(8s)$, so differentiation and propagation of uncertainty giveThus a fixed spatial measurement uncertainty produces a relative momentum uncertainty growing linearly with $p$. If momentum is expressed in $\mathrm{GeV}/c$, $B$ in tesla, $L,s$ in metres and $z=q/e$, restore the usual conversion $p=0.2998|z|BR$.

The magnetic bending angle is $\varphi \simeq |q|BL/p$. For a momentum estimate based on this angle, propagation of uncertainty gives $\sigma (p)/p=\sigma (\varphi )/\varphi$. Comparing the total Multiple Coulomb scattering angle with the bend therefore gives the commonly intended characteristic estimateIts independence of $p$ is the key result for relativistic particles: the scattering angle and magnetic bend both scale as $1/p$.

A precise standard deviation requires specifying how tracks are measured. If $\varphi$ is obtained from independently measured entrance and exit directions and all the material lies between them, the bending-plane projected root mean square is $\mathrm{RMS}(\theta )/\sqrt{2}$. For that estimator the equality isA three-point sagitta fit weights scatter positions differently again. The paper does not specify these measurement details, so it fixes the scaling and characteristic estimate, but not one universal geometric coefficient for all track fits.

The maximum distance between a circular arc and its chord of length $L$. For radius $R$ and $L\ll R$, expansion gives $s\simeq L^2/(8R)$. In a uniform transverse magnetic field, a charged track has $R=p/(|q|B)$, so the sagitta measures its momentum.

For a short circular track of chord length $L$, sagitta $s$ and fixed magnetic field $B$, $p=|q|B{L}^2/(8s)$. Differentiating this inverse relation gives $\sigma (p)/p=\sigma (s)/s$. A fixed spatial measurement uncertainty therefore makes the relative momentum uncertainty grow linearly with momentum.