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Guess the function: results January 21, 2009

Posted by dorigo in physics, science.
Tags: , , , , , ,

Thanks to the many offers for help received a few days ago, when I asked for hints on possible functional forms to interpolate a histogram I was finding hard to fit, I have successfully solved the problem, and can now release the results of my study.

The issue is the following one: at the LHC, Z bosons are produced by electroweak interactions, through quark-antiquark annihilation. The colliding quarks have a variable energy, determined by probability density functions (PDF) which determine how much of the proton’s energy they carry; and the Z boson has a resonance shape which has a sizable width: 2.5 GeV, for a 91 GeV mass. The varying energy of the center of mass, determined by the random value of quark energies due to the PDF, “samples” the resonance curve, creating a distortion in the mass distribution of the produced Z bosons.

The above is not the end of the story, but just the beginning: in fact, there are electromagnetic corrections (QED) due to the radiation of photons, both “internally” and by the two muons into which the Z decays (I am focusing on that final state of Z production: a pair of high-momentum muons from Z \to \mu^+ \mu^-). Also, electromagnetic interactions cause a interference with Z production, because a virtual photon may produce the same final state (two muons) by means of the so-called “Drell-Yan” process. All these effects can only be accounted for by detailed Monte Carlo simulations.

Now, let us treat all of that as a black box: we only care to describe the mass distribution of muon pairs from Z production at the LHC, and we have a pretty good simulation program, Horace (developed by four physicists in Pavia University: C.M. Carloni Calame, G. Montagna, O. Nicrosini and A. Vicini), which handles the effects discussed above. My problem is to describe with a simple function the produced Z boson lineshape (the mass distribution) in different bins of Z rapidity. Rapidity is a quantity connected to the momentum of the particle along the beam direction: since the colliding quarks have variable energies, the Z may have a high boost along that direction. And crucially, depending on Z rapidity, the lineshape varies.

In the post I published here a few days ago I presented the residual of lineshape fits which used the original resonance form, neglecting all PDF and QED effects. By fitting those residuals with a proper parametrized function, I was trying to arrive at a better parametrization of the full lineshape.

After many attempts, I can now release the results. The template for residuals is shown below, interpolated with the function I obtained from an advice by Lubos Motl:

After multiplying that function by the original Breit-Wigner resonance function, I could fit the 24 lineshapes extracted from a binning in rapidity. This produced additional residuals, which are of course much smaller than the first-order ones above, and have a sort of parabolic shape this time. A couple of them are shown on the right.

I then interpolated those residuals with parabolas, and extracted their fit parameters. Then, I could parametrize the parameters, as the graph below shows: the three degrees of freedom have roughly linear variations with Z rapidity. The graphs show the five parameter dependences on Z rapidity (left column) for lineshapes extracted with the CTEQ set of parton PDF; for MRST set (center column); and the ratio of the two parametrization (right column), which is not too different from 1.0.

Finally, the 24 fits which use the f(m,y) shape, with now all of the rapidity-dependent parameters fixed, are shown below (the graph shows only one fit, click to enlarge and see all of them together).

The function used is detailed in the slide below:

I am rather satisfied by the result, because the residuals of these final fits are really small, as shown on the right: they are certainly smaller than the uncertainties due to PDF and QED effects. The f(m,y) function above will now be used to derive a parametrization of the probability that we observe a dimuon pair with a given mass m at a rapidity y, as a function of the momentum scale in the tracker and the muon momentum resolution.



1. anonCMT - January 21, 2009

Maybe someone said thys before, but I have not read all comments to the previous post… Anyway, to me the residuals look a lot like a Fano resonance (http://en.wikipedia.org/wiki/Fano_resonance) with negative Fano parameter. It’s similar to Lubos’ polynomial ratio guess, but it has a physical origin and its known to describe interference effects.

2. dorigo - January 22, 2009

Yes anon, in the other thread Maarten suggested an interference shape, but it does not capture the subtleties of the combination of QCD and QED effects….


3. JJ - January 22, 2009

Does your function? Have you looked at the Z differential cross section when you apply both QCD and QED NLO (and higher…) corrections? We used HORACE + MC@NLO to look at this in the high invariant mass region, but I’m not sure what the effects are at the Z peak though. There is a paper that looked at it somewhere from a year or two ago…

4. Amos - January 22, 2009

Maybe the answer to this is obvious, but… Why does the lineshape vary with rapidity?

5. dorigo - January 22, 2009

Amos, no, the answer is not obvious.
The parton distribution functions determine the Z rapidity with their value (it depends on the difference x_1-x_2, if the two partons have momentum fractions x1 and x2 of their parents’ momenta), as much as they determine the produced mass, M^2=x_1 x_2 s (s is the square of the collider energy).

Since the PDF are complicated functions, it would be strange if the mass distribution were invariant with x_1 - x_2.

The matter is explained in some more detail in some earlier post on Z production.


6. dorigo - January 22, 2009

Hi JJ,

well, how much the function I gave above works depends on the level of accuracy one wants. I am satisfied with percent differences, because resolution effects will wash those away. I will however look more into the QCD NLO effects now.


7. D R Lunsford - January 23, 2009

Nice to see physics at work!


8. Luboš Motl - January 27, 2009

Sorry, Tommaso, this is the first time I see it, thanks for leading me here. Czech TV didn’t report on this pile of data, and even if it has, I didn’t see the TV news last night. 😉

Well, yes, the very last residuals are pretty parabolic – more generally, Taylor expansion would probably always be right for this kind of “last fixes” as long as the initial rough approximations are smooth, natural, and fine-tuned to the best fit.

Did you say 5-parameter fits? I see slightly more but I guess that you don’t count some of those that I do (like the mass?).

dorigo - January 28, 2009

Yes, mass and rapidity are not parameters in my fit, as I see them. There are more than 5 parameters, but once some are fixed (the 60, the 60-1, the 1/4 and the 1/4 squared, and the two numbers close to those you suggested) the rest can be studied more easily.


9. Luboš Motl - January 27, 2009

One more preemptive comment.

If a colleague of yours will tell you that you should have known the Padé approximants, especially because you work in Padova, it is no justification for not acknowledging your sources.

You can still call it Padé-Motl approximations because the coefficients had to be eyeballed sufficiently well to convince you that Padé is a good concept, hadn’t they? 😉

10. Luboš Motl - January 31, 2009

Understood, fitting a few parameters is surely a bit easier than fitting dozens of them. But if you were talking about the rigidity of the interpolating functions, I don’t see how you could not count 1/4 squared and similar things as parameters.

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