Tracking curvature and momentum resolution

In a uniform field along the positive z direction, the Lorentz force bends the transverse projection into a circle while ideal longitudinal motion remains uniform. Let R be the positive radius of that circle and z q the signed charge in units of the positive elementary charge. The momentum magnitude perpendicular to the field is proportional to absolute charge, field magnitude, and radius. Curvature sign can determine charge only after the field and track-direction conventions have been specified.

Using momentum in GeV per c, field in tesla, and radius in metres gives the familiar numerical conversion near 0.300. This conversion already accounts for the SI charge and unit change; it is not an instruction to set a tesla equal to an energy squared. A general detector has a mapped nonuniform field, energy loss, and finite material. Its track fit propagates a state through that environment rather than fitting every trajectory to one perfect circle.

Consider the sagitta s, the maximum separation between a circular arc and its chord of length L in the bending plane. Circle geometry gives the radius as a sum of a term proportional to inverse sagitta and a small sagitta correction. When s is much smaller than L, the leading term yields momentum inversely proportional to s. The lever arm enters quadratically: spreading measurements across a larger baseline can matter more than adding nearby hits.

Differentiate the approximate inverse-sagitta momentum relation while holding the field and lever arm fixed. A small sagitta error produces the same fractional momentum error in magnitude, with the opposite sign in the change. This linear propagation requires uncertainty small compared with measured sagitta. If a fitted curvature is consistent with zero, transforming a Gaussian curvature distribution into momentum creates strongly asymmetric tails; a symmetric momentum error bar then becomes a poor description.

Finite hit precision gives an approximately momentum-independent absolute curvature uncertainty, so its fractional momentum contribution grows with momentum. Multiple Coulomb scattering adds random angular deflections whose effects are more prominent at lower momentum and depend on material and speed. Energy-loss fluctuations can also matter. Covariances between hits, constraints, and track parameters must be carried through a fit; treating every hit residual as independent after fitting the same trajectory would underestimate uncertainty.

Alignment and field-scale errors can shift the inferred momentum even in a sample with very narrow residuals. Some coherent distortions leave local residuals small while biasing curvature, so resonance masses, cosmic tracks, field reversals where available, and external survey information offer complementary tests. Resolution describes dispersion around a model expectation, whereas calibration addresses its scale and bias. A published momentum response should specify kinematic region, track selection, tails, and data-based validation rather than advertise one universal percentage.