Silicon Tracker Curvature

What is modelled

A charged particle in a magnetic field moves on a circle whose radius is proportional to its transverse momentum. Thin silicon layers record where it crossed, each with a small position error, and each layer scatters it by a small random angle. Fitting a parabola to the hits gives the curvature, and so the momentum. At high momentum the track is nearly straight and the hit error dominates; at low momentum scattering in the material dominates.

Equations

• Bending radius R = pT / (0.3 B) (m, GeV/c, T).
• Sagitta of the point at distance L s = L² / (2R).
• Parabola fit y = a + b x + (κ/2) x² with κ = 0.3 B / pT; its variance follows from the least-squares covariance, close to Gluckstern's σκ = (σ / L²) √(720 / (N + 4)).
• Scattering per layer θ0 = (13.6 MeV / p) √(x / X0) (simple form, β = 1).

Checks run in the page

• The Lorentz force integrated along the path against the bending radius.
• A track stepped out to radius L against the exact sagitta.
• Thousands of simulated tracks fitted with a parabola against the least-squares curvature error and Gluckstern's approximation.
• The sum of random scattering kicks against the square-root rule.

Honest simplifications

The field is uniform and exactly axial, the layers are infinitely thin cylinders with a single Gaussian hit error, scattering uses the simple Highland form without its logarithmic correction or tails, the particle is taken as relativistic, there is no energy loss, no vertex constraint, no pattern recognition and no pile-up. The fit assumes the starting direction is known to a few degrees. The fitted momentum can come out far from the truth for stiff tracks; the histogram shows that tail. The 3D cutaway is fictional and its sliders carry few units.

• Choose Stiff track: the sagitta shrinks to a few times the hit error and the momentum error approaches 100 percent.
• Choose Strong field, long lever arm: the error falls with the square of the arm length.
• Choose Low momentum, much scattering: the scattering, not the pixels, then limits the precision.