1 Proposed Implementation of Energy Calibration

This is a short not to outline a proposed method of improving the energy calibration at momenta other than at the decay positron endpoint momentum (p_{edge}). Recall that in our first measurement of ρ and δ we used the corrected momentum p_{ec} given by:

p_{ec} = \frac{p_{rec}}{1 + \frac{β}{p_{edge}}} + \frac{α}{| cos θ|}  (1)

where p_{rec} is the reconstructed momentum cos(θ) is the reconstructed cosine of the decay positron angle, β is the shift in the spectrum from the theoretical endpoint momentum, and α = (α_{u}, α_{d}) depending on whether the track is upstream (u) or downstream (d) of the muon stop.

Rob’s studies of michel spectrum momenta positron tracks that traverse the detector can be used to look at momenta that differ from the endpoint momenta. Specifically he has made plots of the difference between the upstream and downstream reconstructed momenta times cos θ versus the momentum (Δ̄p cos θ versus p). These plots show a clear almost linear dependence that can be written with the form:

Δ̄p cos θ = α_{0} + α_{1}p = α_{tot}(p)  (2)

This momentum dependent α_{tot}(p) should be related by a constant scaling to our endpoint momentum calibration α_{sum} = α_{u} + α_{d} at the endpoint. We obtain momentum dependent α_{u}(p) and α_{d}(p):

α_{u}(p) = \frac{α_{u}(p_{edge})}{α_{tot}p_{edge}}(α_{0} + α_{1}p)  (3)

α_{d}(p) = \frac{α_{d}(p_{edge})}{α_{tot}p_{edge}}(α_{0} + α_{1}p)  (4)