Fundamental relationship: q*pT=B*r.
This gives pT=0.3*B*r when pT is in GeV/c, B is in T, R in m, and q=e.
Momentum measurement from sagitta.
See also pages 11-14 here.
If you have 3 hits:
you measure L (distance P1-P3) and s (distance P2-L).
L/2 is the sine of the opening angle of the big triangle: this implies that tgθ=L/[2(r-s)]. But when s is small (s<<r), also θ is small and tgθ~θ. So, θ~L/2r.
But the sagitta is given by s=r(1-cosθ) (*). In the approximation of small θ, this leads to s~rθ2/2.
Putting this together with θ~L/2r, we get s~L2/8r, therefore pT=0.3*B*L2/8s.
Error: since L is given by the lever arm of the spectrometer (which is given by the geometry of the experiment) and B is usually well mapped, we have essentially δpT/pT=δs/s.
But δs is fixed (given by the hit resolution in the tracker), while s depends on pT.
From this, we demonstrate that δpT∝pT2.
(*) Consider the triangle with sides r, r-s and L/2. Pythagoras' theorem gives s=r-sqrt(r2-L2/4)=r(1-cosθ).