Posted by dorigo in personal, physics, science.
Tags: CMS, momentum scale, PDF, Z boson
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Tonight I feel accomplished, since I have completed a crucial update of the cornerstone of the algorithm which provides the calibration of the CMS momentum scale. I have no time to discuss the details tonight, but I will share with you the final result of a complicated multi-part calculation (at least, for my mediocre standards): the probability distribution function of measuring the Z boson mass at a certain value 
, using the quadrimomenta of two muon tracks which correspond to an estimated mass resolution 
, when the rapidity of the Z boson is 
.
The above might -and should, if you are not a HEP physicist- sound rather meaningless, but the family of two-dimensional functions 
is needed for a precise calibration of the CMS tracker. They can be derived by convoluting the production cross-section of Z bosons at a given rapidity 
with the proton’s parton distribution functions using a factorization integral, and then convoluting the resulting functions with a smearing Gaussian distribution of width .
Still confused ? No worry. Today I will only show one sample result – the probability distribution as a function of and for Z bosons produced at a rapidity 
, and tomorrow I will explain in simple terms how I obtained that curve and the other 39 I have extracted today.
In the three-dimensional graph above, one axis has the reconstructed mass of muon pairs (from 71 to 111 GeV), the other has the expected mass resolution (from 0 to 10 GeV). The height of the function is the probability of observing the mass value , if the expected resolution is . On top of the graph one also sees in colors the curves of equal probability displayed on a projected plane. It will not escape to the keen eye that the function is asymmetric in mass around its peak: that is entirely the effect of the parton distribution functions…
It would be interesting to see the effect of the PDF uncertainty on this plot (and how sensitive it is the the uncertainties). Indeed, around the Z peak, the CTEQ6.1 uncertainty is ~5%.
Hi James,
indeed, these are CTEQ6 PDF, and my program does use the 40 eigenvector variations for an estimate of the uncertainty. However, so far I have not ran the full grid of points for the systematics, because it is very time-expensive. I however think that while the normalization does have a sizable uncertainty, the shape is not affected too much.
Another thing to note is that this is a LO calculation of the factorization integral, and so I must use LO PDF. The eigenvectors are known for the NLO sets. They can be used for a eyeballing of the uncertainty all the same, but the right thing to do is rather to compare the results I get to those I get with MRST sets. I will post about that when I get there…
Cheers,
T.
Ah that’s interesting. So, I take it then that your method is sensitive to shape and not to normalisation then?
Well, yes, I only care about the shapes. In fact, the probability distribution shown in the plot is normalized such that the integral over the mass range, for any given resolution, is unity.
I just checked a few things about the MRST04 set I want to use for some comparisons, in http://xxx.lanl.gov/PS_cache/hep-ph/pdf/9907/9907231v1.pdf , and indeed they speak of 5% differences in cross-section; but they also admit they had some error in earlier calculations.
Cheers,
T.
[Aesthetic question]
Hey! How do you do, using ROOT, to draw that contour projection on the top plane of your 3D box frame????
TH2D * A = new TH2D (“A”, “Goofy”, 100, 0., 100., 100, 0., 100.);
….
A->Draw(“SURF3″);
Cheers,
T.
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