## Setting Length Scales in HEP Experiments

Jim Pivarski who contributes to the Everything Seminar from Cornell University, is an expert in the problem of alignment in high-energy physics experiments. I like to discuss alignment issues with him, and the following problem came up for discussion.

Imagine we are trying to reconstruct events recorded at the LHC. We have the hits in the tracking detectors, and we associate them to form tracks for individual charged particles – a pair of oppositely-charged muons, for example. From fitting those tracks to a model of the trajectory, in crude terms, a helix in a homogeneous magnetic field, we measure the momentum vector of each particle at the point of production. We can then calculate the invariant mass of the muon pair, and reconstruct known resonances such as the Z boson, the J/ψ meson, etc.

Suppose, now, that all length scales were re-scaled by some small amount:

r → (1+α) r

where α is a small number compared to one, and r is a position vector in coordinate space. Ideally, we would know that α is zero. What are the bounds on α and how do we set them?

Let us recall how the curvature of a track is ‘converted’ to a momentum. In most HEP and nuclear physics experiments, charged particles are made to pass through a calibrated magnetic field. Their deflection is proportional to the magnetic field and inversely proportional to the component of the momentum perpendicular to the field. For small deflections, the vector difference between the initial and final momentum vectors is proportional to the integral of B⋅dL. Ideally, there is no change in the magnitude of the momentum vector, so the only relevant quantity is the angle of deflection, which we can measure. Hence, knowing that angle and also B⋅dL, we can infer the momentum vector.

Notice that everything hinges on the product B⋅dL. So if we re-scaled, perhaps willfully, L by (1+α) and and the same time re-scaled the magnetic field by the factor (1+α)-1, the momentum would not change, and the peaks of the Z and J/ψ resonances will remain in place. There is a kind of scale freedom here, apparently.

In reality, we have little freedom to re-scale B beyond a factor of 10-4 or so. We use Hall probes to give an absolute value for the magnitude of B, and these probes cannot be arbitrarily re-calibrated. Ultimately, the values we record from our Hall probes relate to fundamental constants of Nature through a variety of atomic physics experiments.

Furthermore, we know what the meter is, as well. When we design and construct our tracking devices, we use the standard meter and measure actual dimensions to small fractions of a millimeter – there is little danger that all such dimensions are off by a common factor of (1+α). So the task of aligning detectors reduces to the rather difficult task of reconciling the `as-built and installed’ dimensions and distances to the `as-designed and ideal’ ones. As I said before, Jim is one of the world’s experts on this, and has some interesting ideas that eventually will come to light.

It always amuses me to see that in the use of the bending of a charged particle trajectory in a magnetic field links ‘position’ with ‘momentum’ with knowledge of the value of the magnetic field as the fulcrum; this echoes in my mind the profound connection between coordinate space and momentum space with Plank’s constant as the link. Of course, there is no deep connection here…

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Entry filed under: Particle Physics.