After the issuing of new top mass results by the Tevatron experiments, it is time for another look at global electroweak fits of standard model observables. The Gfitter group has produced new fits for the standard model in search of the most probable value of the Higgs boson mass, given the new measurements of top quark mass and other quantities, and the huge amount of existing information on sensitive observables from the standard model.

Unfortunately, I could find no update including the new Higgs search results yet. I guess such a fit will be ready in a few weeks... But the new released information is already interesting enough that we may meaningfully spend a few words around some figures here.

(A note: while also the LEP electroweak working group is in the game, and has also produced updated plots here, I prefer the GFitter group's view; the biggest difference is that GFitter includes the likelihood of direct Higgs searches in the fits).

Two Preliminary Explanations

Before I delve into very technical details accessible only to enlightened amateurs, graduate students, and professionals, I wish to however provide the rest of you with some background on the whole idea behind "global electroweak fits". What is a fit ? And what is an electroweak fit then ? Let me try to explain this -my homework for today.

1 - A "fit" is the adaptation of a model to a set of experimental measurements, in search of the value of the model parameters which best agree with the measurements.

Still in the dark ? Okay, then imagine you are walking in a foreign town in search for a bakery. It is very early in the morning, and bakeries are working at full steam to produce the bread of the day. The town has a maze of small narrow streets, and the smell of fresh bread diffuses around quickly. You have a map, and you mark on it a few locations with the intensity of the fresh bread scent you have felt there. Then you go home, and try to interpret the data. Your "model" consists in a single bakery diffusing the scent around, in a way defined by the wind and the topography of the town. The data allow you to determine the most likely place where the bakery is located, if the model turns out to be a good approximation of reality. Note that if there are several bakeries in the surroundings your results will be wrong! Your model is false, so the results you get for it are also wrong.

In the example above, the coordinates of the bakery are the model parameters which we try to fit. The experimental measurements are locations on the map and corresponding intensities of the fresh bread scent. The scent diffusion model connects the two sets of variables. The fit is whatever numerical procedure allows you to find the most probable place from where the scent was diffusing, given the model and the data.

Now, I am sure that if you do not like the smell of fresh bread, you can find a similar example for what a fit is and what it does to your data and your model. But in a nutshell, the idea is quite trivial: to "fit" the model to the data. The parameters shape the model such that it "fits", hence the very un-cryptic naming.

2 - So what are global electroweak fits ? This is rather a physics question, and a tough one at that. It so happens that there are many quantities in the standard model which are tightly related among each other: the knowledge of the value some of them allow one to infer the value of the remaining ones... More or less. A global fit may take in consideration ("parametrize") all the nuisances of the standard model that connect the value of electroweak parameters, blend in all experimental measurements of electroweak physics (quantities measured in Z boson decays by the LEP experiments, quantities derived by neutrino scattering experiments, top quark mass and W boson mass measurements), and proceed to determine the best value of all parameters: both those measured and the one yet basically unknown, the Higgs boson mass.

Please note that there is no conceptual difference between determining the most likely value of a parameter which we have measured very well (say, the Z boson mass) and one which we know "nothing" about -the Higgs mass. This is a subtle but important point!

A few results from GFitter

Armed with the understanding of what a "global electroweak fit" is, you will certainly be able of appreciating the meaning of three figures I stole from the GFitter web site. The first figure I wish to paste here shows what value of the Higgs mass is returned by the global fit when in turn the knowledge of each of the most important experimentally measured quantities is ignored.

The figure shows a green vertical band on an axis describing the Higgs mass: that is the one-sigma result on the Higgs mass value allowed by all measurements together. And then you see a bunch of points with error bars, each associated with one parameter of the standard model. Take the first point at the top: its meaning is that if we were to ignore the knowledge of the Z boson mass in the global fit, we could still fit for the Higgs boson mass; and we would then get a very low value, 53+43-22 GeV.

Or take the last one: if we were to ignore the precise measurements of the top quark mass obtained at the Tevatron, all other measurements would point to a Higgs mass of 116 GeV, with a large error bar. In practice, the larger the error bar in this graph is, the more important is the relative variable in fitting for the Higgs mass!

The graph therefore allows to visually "pick up" which are the variables that are pulling one way or the other, and how strongly they do so. The experimental value of the Z mass pulls up, since if we remove it we fit a very low Higgs mass; the same goes for Delta_alpha_had, the sixth bar from the bottom (no, I am not going to explain what this is, but just mention that it is the variation of a coupling constant with energy). Top mass and W mass instead pull the unknown Higgs mass down significantly, since their removal brings the prediction up.

You can also enjoy the figure by observing how the bars betray the "irrelevance" of individual measurements. Take A_c, the bar next to the middle. The fits does not change by an inch if we include or exclude its measured value! This means that that quantity is not sensitive to the Higgs mass at the level of precision with which we know other quantities. This is actually a property shared by many of the electroweak variables in the graph: it is due to the fact that many of them are very tightly connected: measure N-1 of them, and the value of the Nth is almost already well defined.

A second figure I wish to discuss is shown below. Do not be scared by the complicated graphics- there is nothing particularly difficult to understand here. On the horizontal axis we have the top quark mass, and on the vertical axis the W boson mass. These have been directly measured with very high precision by LEP II and the Tevatron, as shown by the 1-sigma green bands. The additional electroweak measurements can be displayed in this graph together as the blue ellipsoids: the ellipsoids show what value of top and W masses would be consistent with all other parameters together.

The W and top masses are totally free parameters of the standard model, and yet they can be inferred by the others ? This is only apparently a contradiction: these masses influence the value of the electroweak phenomena through what we call "radiative corrections": quantum diagrams that involve the exchange of virtual top and W bosons in processes which do not seem to involve those particles.

The constraint, however, is quite loose, as you can see by a sheer comparison of the size of the rectangle consistent with the two green bands with the smaller, darker-blue ellipse.

The three different shading of blue describes the regions allowed at 68% confidence level (what we call "one-sigma"), 95%, and 99%. The same convention is used for a different set of ellipsoids, the yellow-orange ones. These show what electroweak fits imply for top and W masses if we include in the fit the results of the searches for the Higgs boson! As you see, the fact that the Higgs boson is heavier than 114.4 GeV amounts to cutting away the upper part of the blue ellipses. Aha! This agrees with our previous observation that both top and W mass measurements "pull" the Higgs mass toward smaller values.

A third set of ellipsoids, the green ones, finally includes all information together, id est also the direct W and top mass measurements. Not surprisingly, these last ellipsoids are very close to the rectangle produced by the intersection of the direct measurement bands; yet it lies below it: a further hint that the direct W and top measurements are pointing to a lighter Higgs than what direct searches for the Higgs demand.

All in all, a very interesting figure, from which one can size up visually the interplay of electroweak measurements and top and W mass determinations.

Finally, let me discuss a third figure. This one has the Higgs mass on the horizontal axis and the "delta chisquared" on the vertical axis. It basically informs us on the relative likelihood (loosely speaking!) of different Higgs mass values, as they result from global electroweak fits that include the recent 2010 Tevatron measurements of the top mass. Note that the exclusion band around 165 GeV is the one published by the Tevatron in 2009: this figure has not been redone by including the effect of direct Higgs searches in 2010. The figure will change significantly when the new likelihood released by the Tevatron a few days ago is included.

Let us concentrate on the blue dashed line: it includes everything we know from electroweak observables and Higgs search likelihood results. This curve has a sharp minimum at 116 GeV, and it rises very quickly for masses above 135 GeV, indicating that the probability of a high-mass Higgs boson is small.

This figure might be criticized for its too strong message -a lot of subtleties are being swept under the carpet here. By looking at this plot alone one could observe that the likelihood of the Higgs being close to the "brick wall" of LEP II exclusion at 115 GeV is significant: based on this plot, one might bet on even odds that the Higgs boson is lighter than about 130 GeV or so... I do not know whether that conclusion is actually well founded. But we will know soon!