Momentum Resolution

In the case of no multiple scattering, the momentum resolution for a tracking detector with (n+1) equally spaced sampling points is given by Eq.B.9 through

$$\begin{array} {lll} \sigma_\kappa^M = \sigma_{P_T} / P_T^2.\end{array}$$ (104)

We plotted $\sigma_\kappa^M$ in Fig.3.1-a) as a function of lever arm length for different numbers of sampling points. Notice that we set $B=1~{\rm T}$ and $\sigma_x = 100~\mu{\rm m}$there to ease the conversion to different B or $\sigma_x$ values: $\sigma_\kappa^M$ is inversely proportional to B, while it is proportional to $\sigma_x$.

The multiple scattering in the tracking volume adds $\sigma_\kappa^{MS}$ given by Eq.3.2.56 to the above measurement error in quardrature (see Eq.3.2.48). Fig.3.1-b) shows $\sigma_\kappa^{MS}/\kappa$ as a function of the lever arm length for different values of material thickness in the tracking device.

 

Figure 3.1:   (a) The reciprocal transverse momentum resolution as a function of lever arm length for $B=1~{\rm T}$, $\sigma_x = 100~\mu{\rm m}$, and $N_{sample} \equiv (n+1) = 40, 60, 80, \cdots, 200$. (b) The multiple scattering contribution as a function of lever arm length for $B=1~{\rm T}$ and $X_0^{-1} = 0.1, 0.2, \cdots, 1.0\%/{\rm m}$.