A look inside the Tevatron Mtop combination March 10, 2008Posted by dorigo in physics.
They say that people who follow the law and people who eat sausage should not ask how these are made. If you are going to trust the top mass average just released by the Tevatron Electroweak Working Group, you should instead educate yourself on what is behind it: the reason is that it is in fact a very instructive case study of a combination of different measurements of the same quantity.
Before I try to do my little bit to discuss some details of the Tevatron combination, there is one outstanding -if a bit silly- preliminary question to answer. Is it the same quantity, after all, the one we are averaging ?
If you studied Measurement Theory in your undergrad courses, you will surely remember that one of the most easily overlooked sources of systematic uncertainty in a measurement is frequently the ill-defined nature of the quantity one is attempting to determine. As an example, take a determination of the g constant. g is the downward acceleration of a body on the surface of the Earth due to the latters’ gravitational field. Depending how you measure it, you might find different values, which could be significantly at odds with one another. What is downward, for instance ? And what is the Earth’s surface ? If downward is defined as the direction of the center of the Earth, one needs to keep in mind that the Earth is not spherical; one instrument could be more sensitive to this feature than another.
A quark is an elementary particle, but elementary does not mean “simple”. The top quark decays in a time so short that it does not have time to emit radiation – both a good and a bad thing if you want to measure its mass, because if the absence of radiation helps you “picture” the quark decay and measure it well from its decay products, on the other hand the shortness of its life demands you to assume a lifetime to model the probability of exhibiting one particular mass value. Here is an example of how the imprecise definition of the the mass -given a unknown lifetime , or unknown width – is usually treated as a systematic uncertainty: experimentalists will just bracket the lifetime within reasonable bounds, and check the variation of their results on the variable assumption on .
[As a side note, there are further issues with the top quark mass, connected with the nature of QCD, but let us forget about those… They will be an issue only when we will reach a 200-300 MeV uncertainty in the top mass, maybe at a 500 GeV linear collider…]
As I opened this bracket on what is it that the Tevatron experts are averaging, however, I was thinking of something more basic than the imprecise definition of the top mass: not the unknown value of , but the quite exotic speculation that the top mass might not be one single value. I heard it claimed several times in the past, mostly at dinner tables, with a glass of alcoholic beverage in hand, that there was a intriguing “jet effect” in the top mass measurement. Indeed, by putting side-by-side different determinations of the top mass based on different final states, one observed that as the number of jets in the final state increased, so did the measured top quark mass. The effect has never been large enough to cause more than a shrug of shoulders in who, like me, believes that statistical fluctuations are a necessary player in experimental physics. However, it is nice to have a look at the situation today, for a start.
The graph above shows the Tevatron average of the top quark mass in three bins, corresponding to the dilepton, single lepton, and all-hadronic final state of top pair decay. In the dilepton decay you have two b-jets and two leptonic W decays; in the single-lepton decay you have two b-jets plus two more from a W; and in the all-hadronic decay you get six jets. The graph shows the present state of the “effect”. As you see, a straight constant fit to the three points produces a very nice (2.27/2), and the effect is a non-issue.
Ok. Now that we have put the issue of a dependence of the top mass on the final state behind our back, let me discuss the averaging of 12 different results obtained by CDF and D0 in Run I and Run II. First of all, those 12 results are only a small selection of the determinations that the Tevatron experiments have produced in the last 14 years – from the “evidence” paper by CDF in 1994 (quoting based on the analysis of 19 inverse picobarns of data) to this winter’s precise results obtained with 110 times more statistics. The selection leaves out a few results that have too much correlation with those included, or ones whose inclusion would have added no statistical power to the combination. Alas, it also excludes one of those I was a main author of – it but includes many more I have contributed to, fortunately.
The list of 12 included results is best shown graphically. Please see the plot below:
Marvelous display of science, is it not ? As you see, there are determinations for all final states by CDF both in Run I and Run II data, while D0’s determinations in the all-hadronic final state have been left out. Included, instead, is one measurement which is quite different from all others: it is the last, quite wide, error bar, which is obtained by fitting the decay length distribution of b-quarks in top-decay b-jets. The decay length of b-quarks is in fact a function of the b-quark boost, which is in turn a function of the top mass, since the latter decays to two bodies in the process , and there is an approximate linearity between top quark mass and the energy -and then the boost- of the produced b-quark. The measurement is valuable because it only uses tracking information, unlike all others, and is thus subjected to very different systematic uncertainties. Most notably, it is totally unaffected by the jet energy scale uncertainty, the single largest source of uncertainty of most if not all other determinations.
The combination of the 12 results above is not just a weighted average, of course. The authors took in consideration several sources of systematic uncertainty, and accounted for the cross-correlation of each result with the others. In particular, the treatment of the systematics due to the jet energy scale is exceedingly accurate: there are no less than six different factors related to the measurement of jets energy by the two experiments; some of these are totally uncorrelated, and some are partly or fully correlated, through the assumptions on the fragmentation properties of light-quark jets, those of b-jets, the decay properties of the latter, and other modeling issues such as out-of-cone energy. It is also to be noted that CDF measurements in Run II using the in situ calibration of the jet energy scale through the fitting of the hadronic W signal in the top pair decay no longer include an external constraint from the jet energy scale determination in QCD samples, since that input is no longer contributing appreciably to the uncertainty.
The last comments were a bit too technical for such a long post, I gather. What I still need to do is to mention a few of the heroes of these many excellent measurements. Really, too many people have contributed to making top mass measurements the trademark of the Tevatron, and it would take a week to mention all of them duly; besides, I have no notion of who are the principal actors in D0 and so I have to avoid discussing that side of the coin. I only want to cite two CDF colleagues here: Lina Galtieri and Doug Glenzinski.
Lina has been in CDF ever since the sun rose on that orange building, and she is one of the most striking examples of continuity and devotion to top quark physics in our experiment. She is a professor at Berkeley and has directed her group in Run I and Run II to top physics for two decades. I think she is one of the few people to whom CDF owes more than the other way around.
Doug is one of the members of the TeV Electroweak Working Group. He joined our experiment in 1999, and he has directed the effort of CDF in top quark physics for several years. He is now the physics coordinator in CDF, and several of the successful measurements in the top sector by CDF have his fingerprints on them.
Finally, I have to apologize to the many brilliant D0 colleagues who are behind the D0 measurements shown in the plot above. I can only invite any one of them willing to discuss the subtleties between their top mass measurements to contribute here with a guest post…