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Determining Momentum of a Particle from Radius, Distance, and Angle? 
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#1
Jan2113, 12:35 PM

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I didn't know whether to put this in homework or in here because it is not homework but it is a question, so sorry in advance! I like to learn about quantum physics as a hobby, and I found this great link about working with actual particle accelerator data by an organization called Quarknet. Here is the link:
http://quarknet.fnal.gov/projects/su...nt/index.shtml I thought I would try and use this data(after clicking on "project context") to do what the site says and determine the mass and lifetime of the Z boson. I understand I need to use the E^{2} = m^{2}c^{4} + pc^{2} formula, and I've found the energies, but the only problem is I don't know how to determine momentum. They give you the XY radius as well as the Z component of the particle's path, as I think it travels in a helical path because of the electromagnetic field. They also give you the cosine of the x, y, and z angles. I am totally lost on how to solve for momentum and I'd really like to understand how one would go about doing this. I know that momentum for relativistic particles is not simply mass times velocity, so could someone please help? Thank you! 


#2
Jan2213, 11:33 AM

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P: 11,589

##r=\frac{p_t}{qB}## with the transversal momentum p_{t} (see here, for example). Geometry allows to get the total momentum p just as in the nonrelativistic case.



#3
Jan2213, 02:32 PM

P: 9

http://quarknet.fnal.gov/projects/su...t/theta2.shtml And also, these are electronpositron collisions, so am I going to have to worry about charge, or do I just use 1.6*10^{19} which simplifies the equation when put into electronvolts. 


#4
Jan2213, 04:37 PM

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Determining Momentum of a Particle from Radius, Distance, and Angle?
I don't know which θ you mean.



#5
Jan2213, 09:19 PM

P: 9

I'm using the total energy for a single event as E, in GeV (I already adjusted for the calibration). For pc I'm using .3Br where B is the magnetic field in Teslas and r is the XY radius in meters. I'm putting it into E^{2} = m^{2}c^{4} + (pc)^{2} and solving for m, does that seem right? I'm not getting what the Z boson mass should be. I'm basically just subtracting the momentum term from the energy squared term and square rooting it to get the mass in GeV/c^{2}. 


#6
Jan2313, 09:40 AM

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P: 11,589

Wait... the Z boson is not measured in the detector at all.
Using an assumption for the particle mass, you can get the total energy of the particle. Calorimeter data could be used, too, but in real data the uncertainty there is problematic. Once you have the 4momenta of the Z decay products, you can add them to get the 4momentum of the Zboson candidate and calculate its mass via your formula. 


#7
Jan2313, 06:02 PM

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#8
Jan2413, 11:25 AM

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P: 11,589

You are mixing two different steps here.
Starting from physics: A Zboson is created and immediately decays into two muons (or electrons, I'll consider muons here). Those muons fly through your detector and get detected. You cannot see the Zboson itself, so you have to work backwards: You have to detect both muons in the detector. For each individual muon, calculate its 4momentum (start with that). If an event just contains a single muon, you might have missed one, or there was no Z boson. In any way, you can do nothing here. If an event contains at least one positive and one negative muon, they might come from the decay of a single particle. If that is true, their added 4momenta must be the 4momentum of the decaying particle. So you add the calculated 4momenta, and calculate the mass of that (possible) particle. If the muons really come from a Z boson, the calculated mass will be the Zmass. If they come from a different particle, the calculation will give the mass of this other particle. If they were produced independently, the calculated mass is random. If you collect a lot of those muon pairs, you will see events with the Zmass, plus contributions from other particles, and some background. 


#9
Jan2413, 05:49 PM

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#10
Jan2513, 10:34 AM

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#11
Jan2813, 05:35 PM

P: 9

http://en.wikipedia.org/wiki/Fourvector#Scalar_product And then that'll give me the momentum to plug into the energy momentum equation? But what do I use for E in that equation, the combined energy of the muons? 


#12
Jan2913, 06:00 AM

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P: 11,589

Right
$$\vec{p_{\mu_1}}+\vec{p_{\mu_2}} = \vec{p_Z}$$ In components: $$\begin{pmatrix} E_1 \\ p_{x1} \\ p_{y1} \\ p_{z1} \end{pmatrix} + \begin{pmatrix} E_2 \\ p_{x2} \\ p_{y2} \\ p_{z2} \end{pmatrix} = \begin{pmatrix} E_1+E_2 \\ p_{x1}+p_{x2} \\ p_{y1}+p_{y2} \\ p_{z1}+p_{z2} \end{pmatrix} = \begin{pmatrix} E_Z \\ p_{xZ} \\ p_{xZ} \\ p_{xZ} \end{pmatrix}$$ 


#13
Jan3013, 09:31 PM

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#14
Feb113, 10:46 AM

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P: 11,589

For other particles, use the mass of other particles ;). Combining the energy measurement from the calorimeters with the momentum measurement of the tracking system would lead to large errors. If tracking does not give a reliable momentum measurement (for highenergetic particles, for example), you can use the measured energy and direction to reconstruct the momentum, again with the known mass of the particle.
If the particle type cannot be identified*, the usual approach is to test all hypotheses: Treat it as a possible pion (with the pion mass), look if it could fit to some interesting process with a pion. Treat it as a possible muon, look if it could fit to some interesting process with a muon, and the same for kaons (, electrons) and protons. *if particle identification is crucial, the detectors have special parts for that. 


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