= Sagitta momentum uncertainty
{title2=$\sigma(p_\perp)/p_\perp=8p_\perp\sigma(s)/(|q|BL^2)$}
A <particle track> projected onto the plane perpendicular to a uniform <magnetic field> $B$ is a circular arc with <radius> $R=p_\perp/(|q|B)$. Here $p_\perp$ is the <momentum> component perpendicular to the <magnetic field>; it is the collider <transverse momentum> when the field follows the beam axis. For chord length $L$ and <sagitta> $s$, circle geometry gives
$$
(R-s)^2+(L/2)^2=R^2,\qquad R=\frac{L^2}{8s}+\frac{s}{2}.
$$
For a short arc with $s\ll L$, this yields
$$
p_\perp\simeq\frac{|q|BL^2}{8s},\qquad \frac{\sigma(p_\perp)}{p_\perp}\simeq\frac{\sigma(s)}{s}=\frac{8p_\perp\sigma(s)}{|q|BL^2}.
$$
The <measurement uncertainty> follows by differentiating the inverse relation, treating $B$ and $L$ as known. A fixed spatial <measurement uncertainty> therefore makes fractional <momentum> uncertainty grow with <momentum>, whereas increasing the lever arm improves it quadratically in this idealization. <Multiple Coulomb scattering>, field uncertainties and alignment add other contributions; a helical <particle track> also requires its pitch to recover the full <momentum>.<Review of Particle Physics by Particle Data Group (2022)>[Particle detectors at accelerators, section 35.12.1, equation 35.49: sagitta and magnetic momentum analysis]{ref}
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