A Quantum Diaries Survivor

I am very happy to post a link to my friend David’s blog, where he has a streaming video from an event in Rome, at the point where he poses a tough and interesting question to Al Gore on the accelerating debate on the Climate Crisis. Please have a look!

Since I am currently preparing the slides of the talk I will give next week at PPC2008, a conference being held in Albuquerque on the interconnection between particle physics and cosmology, I have my hands full with material that would be perfect for this blog. The talk is a review of new results from the CDF experiment, and there is literally a ton of them! What makes it hard for me is to sort out the stuff that is really the very best and most worthy of being shown at the particular event.

I am however not posting here direct CDF results, but rather a plot that my friend Sven Heinemeyer was kind to produce according to my directives, which I meant to allow me to summarize in my talk the status of electroweak fits by separating the main contributions of experimental measurements. In the graph below, showing the dependence of the Higgs boson mass on the value of W boson and top quark masses, you can see several different regions highlighted with black, blue, and magenta lines. The black lines bracket the LEP II determination of the W mass; the blue ellipse describes the Tevatron measurements of the two parameters, and the magenta hatched “wing profile” area shows the allowed values of the two quantities according to electroweak fits performed using LEP I and SLD determinations of electroweak parameters.

Also shown in the plot is the SM-allowed range (in red), where the Higgs boson has a mass varying between the lower LEP II limit of 114.4 GeV (upper border of the red hatched area) and 400 GeV (lower border), and the SUSY allowed region (hatched green), which shows the zone allowed by different choices of some of the many SUSY parameters, in particular the mass of supersymmetric particles.

Now, let me make a few points concerning the plot above.

1. Although precise, the indirect experimental input shown in the plot is still incapable of discriminating between SM and SUSY - and it probably never will by itself, since LHC will soon rule out or find SUSY before it shrinks the ellipse sizably (ok, ok, I am neglecting the possibility of split SUSY…)
2. the celebrated LEP I / SLD data looks obsolete from this particular vantage point, in light of the more recent direct measurements; this would however be an unfair interpretation, given that electroweak fits have many more parameters than just W and top quark masses.
3. The LEP I / SLD data is obsolete as far as the top quark is concerned: in the plot it does not even appear to constrain it if compared with the ultra-precise (+-0.8%) Tevatron determination!
4. The top quark mass has been bouncing up and down a bit, although always well within errors, in the last 5 years, from 178 to 170 to 172.4 GeV. This has slightly moved up and down the preferred value of fit Higgs mass in the SM. However, as the ellipse shrinks, this is becoming less of an issue. In fact, to justify the effort of producing the best possible top mass measurement, we used to say that a 1 GeV precision on the top mass was equivalent to a 7 MeV precision on the W mass as far as the knowledge we would obtain on MH was concerned, based on the slope of the Higgs contours in the plot above. Now that the error on top mass is well below 2 GeV, however, it becomes clear that we will not gain much knowledge by increasing the precision much further. The W mass has become one of the main players in the game of precision SM fits now!
5. The ellipse includes 68% of the area of the two-dimensional gaussian centered on the Mw-Mt determination, just as much as the black bars do, but being two-dimensional it is deceiving: the single most precise determination of the W boson mass is in fact from CDF, and Tevatron and LEP II are basically at the same level of precision on that quantity!

Why is the comparison deceiving ? Because if you have one single quantity, you determine the 68% interval by integrating a gaussian distribution from its center outwards, until you “cover” 68% of its total integral (from -inf to +inf). If you add a dimension to your single-variable gaussian, and make it a two-dimensional gaussian shape, the 68% bounds remain the same unless you integrate by expanding an ellipse, rather than a band, around the center. The ellipse encompasses values of the 2-dimensional distribution which have the same “probability”, but in so doing it “cuts the corners”, and to total a 68% of the 2-dim integral it now has to extend past the one-dimensional 68% boundaries in each of the two variables. A sketch will clarify matters:

Well, not exactly “clarified”… But I have no time to make the graph easier to understand. The point is that the ellipse “cuts” only a part of the band in each direction, and so the integral of the 2-dimensional curve it comprises is much smaller than the band. To make the ellipse include 68% of the 2-dimensional distribution constructed with the two gaussian curves, one has to widen it to a roughly double size.

So, paradoxically: if LEP II had a determination of the top quark mass too, the band bracketed by the two black lines in the plot by Heinemeyer would convert into an ellipse which would be about as wide in the vertical direction as the Tevatron blue ellipse.

Not convinced ? Oh well. Think at it this way: with a single measurement of the W mass, you say “the probability that the mass is between 80.35 and 80.45 GeV is 68%, because I determined it to be 80.4 and I have an error of 0.05 GeV”. Fine: the gaussian distribution, if integrated from -1-sigma to +1-sigma, provides 68% of its total normalization. The same goes if you claim that, having measured the top mass at $172.4 \pm 1.4 GeV$, there is a 68% chance that it lies in the interval 171-173.8 GeV. However, if you ask what is the probability that the W mass is between 80.35 and 80.45 GeV AND the top mass is between 171 and 173.8 GeV, this is much smaller than 68%, because independent probabilities multiply each other: it is, in fact, only 46.2%; but this corresponds to the square drawn around the circle in the graph! The probability that the two values lie in the ellipse with major axes equal to 1-sigma band widths is (if I recall correctly) about 37%.

The bottomline? Whenever you look at a plot with two measurements, one described by an ellipse and the other by a band, always regard the ellipse with more respect than it seems to deserve!