The mass of the Higgs boson must be between 114 GeV and 1000 GeV. If it were lighter than 114 GeV, it would have already been detected by LEP, the Large Electron-Positron collider at CERN that lived in the same tunnel that is currently being seized by the LHC. Unless one of those strange models is right which would allow the Higgs to be as light as 105 GeV.
The God particle must also be lighter than a TeV or so because otherwise its self-interaction would have to be too strong. Because the quartic coupling is growing stronger at higher energies, we would encounter a lethal Landau pole too soon: the interaction strength would become infinite and most field theories don't like that.
The mass of the top quark is essentially (approximately, pretty accurately) determined by the RG flow as we have repeatedly mentioned. The right value depends on the electroweak couplings i.e. essentially on the W mass as well as the Higgs mass. A precise measurement of the top quark mass and the W mass can therefore be used to indirectly derive the Higgs mass. That's what was done to create the picture above.
The allowed range from 114 GeV (left) to 1000 GeV is the green band on the picture. Older measurements from LEP 1 and SLD at SLAC have led to the interior of the red squashed ellipse as the 68 percent confidence level region. Newer combined data from LEP 2 and the Tevatron suggest a somewhat lighter top quark as well as a significantly heavier W boson - a jump from 80.35 to 80.40 GeV. The blue dotted ellipse is again the 68 percent confidence level band according to these newer measurements.
You see that the new, blue ellipse doesn't quite overlap with the green band. Of course it doesn't have to. But statistically speaking, it now seems more likely that the Higgs is quite light. If you remember, LEP at CERN has had some unconvincing Higgs signals around 115 GeV just before they shut it down. It was not enough to claim a Higgs discovery but it was suggestive. The recent high precision data can revive the speculations that the 115 GeV Higgs signal was actually a real effect caused by a real God particle.
Recall that supersymmetry tends to predict a light Higgs boson, too. Those of us who are convinced that quantum field theory is a correct description up to pretty high energies where characteristic effects of string theory take over also believe the calculation that implies that the Higgs should be lighter than 180 GeV or so, regardless of the fate of supersymmetry - a much stronger bound than those 1000 GeV above. The new precision data is consistent with this expectation.
Appendix: Confidence levels
Let me recall some basic properties of the normal distribution. For one variable "x", the infinitesimal probability to get a value between "x" and "x+dx" is
- dP = dx . exp(-x2/2) / sqrt(2 pi).
In that case, we say that "x" is described by a normal (Gaussian) distribution with mean value of zero and a standard deviation of one. Note that the normalization is chosen so that the "integral dP" from "-infinity" to "+infinity" equals one. Everyone should be able to move the distribution to a different mean value - replace "x" by "x-x0". Also, everyone should know how to change the standard deviation to "sigma" - replace "x" by "(x-x0)/sigma" and divide "dP" by "sigma" to keep the integral being equal to one.
Fine. Return to the simply normalized formula for "dP". Its integral from -1 to +1 is equal to 0.683 or so. This means that there is a 68 percent probability for the normal distribution to give a result that differs from the mean value by less than 1 standard deviation. For plus minus two standard deviations, you get about 95 percent, and for plus minus three standard deviations, you get about 99.7 percent.
You must redo this calculation of the relation between the "radius" and the "confidence level" if you have "n" variables instead of one but the results only differ quantitatively, not qualitatively.
