 Sagitta momentum uncertainty

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 givesFor a short arc with $s\ll L$, this yieldsThe 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.^[ref]