How do cross-sections respond to changes of the dimensions and impact parameters of the wave-form envelopes of two colliding beams?

Question

The de Broglie wavelength of a 1 Gev (1 TeV) electron beam is approximately \(10^{-15}\) meters ( \(10^{-18}\) meters), whereas the dimensions of the wave-form envelope is of the order of millimeters (and there is a subsidiary sense in which the envelope is kilometers in length).

Are there experiments that investigate with precision how cross-sections change in response to changes of the dimensions and impact parameters of the wave-form envelopes of two colliding beams (not necessarily of electrons), given constant de Broglie wavelengths? If so, what terminology do experimentalists use for this kind of investigation?

I'm not clear that there are any theoretical reasons to think that precision studies of this might be interesting, but are there any?

Comments

What is exactly meant by the wave-form envelope of a beam, can it be described as the superposition of all the waves corresponding to the single electrons present in the beam?

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@Dilaton At a textbook QFT level, we most usually talk about a test function \(\exp{(-\mathrm{i}k\cdot x)}\) as a waveform perfectly evenly spread through all space, but real accelerator beams, while close to that over scales quite large relative to the de Broglie wavelength, nonetheless have a finite width in all four dimensions. To a first approximation, there is a moving peak intensity, so one might have \(\exp{(-\mathrm{i}k\cdot x +...)}\), where the "..." is a Gaussian that describes the motion of the center of the beam's peak intensity (with different widths in all four dimensions). Certainly this can be modeled as a superposition of wave numbers, but a Gaussian factor in real space is in some ways more natural. One might also construct arbitrary wave-form envelopes, where the multiplicative factor "..." is not Gaussian, as one does for Bessel beams for light, for example.

Sorry not to know the right experimentalist language for this.

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Ok thanks @PeterMorgan for these explanations, I think I now understand a bit better what you are asking (+1), but for an answer we maybe need a good experimentalist such as @annav.

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This needs a beam specialist :). I have never seen these type of expansions in calculating interaction crossections. I am afraid experimentalists are much more results oriented. We use luminosity which is given by the accelerator people, and count the number of interactions at our detectors to get crossections. Luminosity is given to the experiments by the beam people and if you look how they determine it, it is against well known theoretical curves and crossections. 

I would suppose that the  analysis you are talking about is already taken into account, by using measurements of known crossections and theoretical ones. A beam specialist would be able to answer if there would be extra insight or merit in the way you propose; sorry not to know anybody to ask them to come here and answer.

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Thanks for the links @annav. It looks like I'm asking whether details of the micron-to-millimeter scale geometry of the bunches affect relative cross-sections for different processes in nontrivial ways (whereas the only number usually exchanged between beam physicists and detector physicists is the relatively crude measure of luminosity, which in any case is superfluous insofar as there is a count of the total number of events). I guess it would be curious and perhaps it might be news if the effective dynamics depends on bunch properties.

Less fundamentally, Bessel Beams, for example, can be useful when working with light, so I'm curious whether it might also be useful to engineer exotic bunch geometries.

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Beams from accelerators are hardly coherent, they are mixed states of $e^{i k x}$ states with "classical" distributions of $k(b)$, with $b$ the beam parameter

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@CharlesJQuarra, although I'm sure beam physics can make such a description work out, it's problematic for a mathematician that an \(\mathrm{e}^{-\mathrm{i}k\cdot x}\) state is an improper state of infinite norm in the non-interacting Fock space, which is perfectly evenly spread over all space-time; as also must be a mixture of such states.

Answer 1

I will attempt a qualitative answer. In your last comment you state:

It looks like I'm asking whether details of the micron-to-millimeter scale geometry of the bunches affect relative cross-sections for different processes in nontrivial ways (whereas the only number usually exchanged between beam physicists and detector physicists is the relatively crude measure of luminosity, which in any case is superfluous insofar as there is a count of the total number of events

The mm and micron scale is orders of magnitude larger than the scale where the interactions of interest, protons on protons for the sake of the argument,  can have a measurable effect, order of fermi and less , 10^-15m. The goal is to have a high energy proton hitting an opposite high energy proton and measure the apparent  cross sectional  area. The difference in scales is such that any modulation in the shape of the beams cannot play a role except in the gross probability of beam meeting beam, and this is what the luminosity supplied takes care of. BTW luminosity  is necessary to be provided because just counting the number of interactions would not give us a cross section, i.e. area, measurement to relate to calculated crossections from the theories. It is a  kind of calibration.

Another way of looking at this is to suppose that the proton crossection is a fixed number , protons are little billiard balls. The number of deviations from the beam direction of the classical scatter, given the luminosity , would give us this fixed crossection. Any beam modulation would reflect in the rates per unit time  and it will be only the errors in the measurement that would be affected by the modulations.

Comments

Thanks, +1. I take your answer to be "No, there are no precision studies because no-one would expect there to be effects of significant interest", I guess. As you say, the length scales of the bunches of the beam and of the range of the de Broglie wavelengths associated with the beam are separated by more than 10 orders of magnitude.

As an aside, to your BTW, my guess (a philosophically prejudiced view) is that precision characterizations of beam luminosity use interactions for which cross-sections have been accurately measured in the past, so that new cross-sections are effectively relative to the whole historical record of cross-sections established by earlier accelerator physics.

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Yes, to your comment on BTW, we build up the structure we use for measurements too. It is not entirely circular though, there exist luminosity independent measurements.