Adam Para's discussion on simulation software, on software needs of the Higgs studies, and on a workshop on b/c tagging (to be held at Fermilab June 14-15) was postponed until 17 May 2000.

Consider the well-known ``blue-band'' plot produced by the LEP Electroweak Working Group.  This plot is often shown to argue that experimental data (from LEP, SLD, the Tevatron collider, and deep inelastic scattering) point to a light Higgs boson, in the Standard Model.  The point of today's discussion is see how much depends on that last qualifier, ``in the Standard Model.''

The blue band is obtained by fitting various observables to the SM formulae with an unknown Higgs mass.  The data are precise enough to require two-loop accuracy.  Also included is the direct exclusion limit.  As of Winter 2000, the upper limit from the fits to radiative corrections require m_H < 188 GeV or so, at 95 % CL.  It is important to emphasize, however, that the theoretical curve is obtained in the Standard Model with the known quarks, leptons, W and Z plus one Higgs doublet to break electroweak symmetry breaking, and nothing else.  If one changes the model to include other (heavy) particles, the radiative corrections change, and the bound need not hold.

A convenient framework for exploring other models is the S-T-U formalism of Peskin and Takeuchi.  These parameters measure the radiative corrections in W and Z propagators; they are most helpful when most of the new radiative corrections come from the propagators, and the vertex and box corrections are small.

The basic formulae are shown here in two slides taken from a talk by Bruce Schumm at Physics in Collision last year:

(Bruce's write-up is hep-ex/9909053.)  The radiative corrections to any observable contain the polarization functions Pi₁₁, etc, and can, thus, be re-expressed in terms of S, T, and U.  In this way the data can be used to determine S, T, and U, and these measured values can be compared to the model predictions.  In many cases heavy particles make negligible contributions to vertex and box diagrams, so their major effect on present-day experiments is solely through S, T, and U.

Schumm's second slide, above, shows contributions from the top quark and the single Higgs of the Standard Model.  In non-standard models of EWSB, the logarithmic terms with m_H remain, with the same coefficient, but there may be additional terms depending not on the (lightest) Higgs mass, but on the masses of heavier particles.

A comparison, from e-print hep-ex/9912026, by Morris Swartz, of S and T from data and the Standard Model is shown here:

(This is based on Swartz's talk at Lepton-Photon 1999.)  The ellipses are 67 and 95 % confidence bands from the data.  The four-sided reddish-brown region is that of the one-doublet model, with 100 GeV < m_H < 1000 GeV; its height comes from the uncertainty in m_t, as measured at the Tevatron.  As one can see, the contours cut the SM region closer to the small m_H end; that is the same result shown in the blue band plot.

One can deduce what is needed to evade the low-mass constraint by following the m_H axis of the reddish-brown region out of the ellipse.  To remain consistent with the data, the Higgs may still have a high mass, as high as 700-1000 TeV, provided additional particles beyond the SM give new contributions to S and T.  Evidently, what is needed is either a negative contribution to S or a positive contribution to T.  Experience in model building suggests that the former is difficult, but the latter is simple.  Indeed, according to Michael Peskin, there is only one published model providing a negative contribution to S; when that model has a high mass Higgs boson, however, there are other light states.

On the other hand, if there are additional SU(2)-singlets Dirac fermions, a positive contribution to T is straighforward.  Chivukula, Evans, and Hölbling (hep-ph/0002022) have plotted, model independently, the region in the m_H-T plane allowed by the data.  The contour for 95 % CL extends well above the theoretical upper limit from triviality.  (This version of the plot omits the triviality bounds shown by CEH.)

The ``triviality'' bound stems from the cutoff dependence of the scalar field self-coupling.  In the Higgs context, the cutoff should be thought of as the scale of new physics.  One finds that a high Higgs mass requires a low cutoff; when the two are comparable, one should expect new physics to accompany the Higgs.  For example, in hep-ph/9303215 Neuberger et al. find cutoff pi×m_H at m_H = 710 ± 60 GeV.  See also Hambye and Riesselmann, hep-ph/9610272.  Because the cutoff enters logarithms, the difference between a cutoff M and 2piM is related to renormalization conventions.  Therefore, triviality bounds are not sharp, neither for the Higgs mass or for the mass of the accompanying new physics. 

A positive contribution to T is obtained in a simple, concrete model, the top-quark see-saw model of Dobrescu and Hill.  This model consists of the Standard Model fields, with one additional colored field, chi, whose left- and right-handed components are both singlets under SU(2).  (In models with extra dimension, this chi field could be a Kaluza-Klein excitation of top.)  The positive contribution from the chi particle can balance the negative pull on T of the Higgs.  Collins, Grant, and Georgi have analyzed the implications of the precision electroweak data to this model.  They find that the data allow anything from a light Higgs with a super-heavy chi (m_chi > 10 TeV) to a heavy Higgs (m_H ~ 1-2 TeV) with a still-heavy chi (m_chi ~ 5-6 TeV).  To see their plot, click here.  Because the chi always has a high mass, the properties of the Higgs boson in this model should be very similar to those of a heavy SM Higgs boson.  Thus, we have a simple, well motivated example that, with our current understanding of the precision data, supports a heavy SM-like Higgs.  This is a quite sobering conclusion: not only is a heavy Higgs allowed in this model, the accompanying new physics is at about 2pi m_H.

We should note that the stance of the precision data has changed qualitatively in the last year.  Since Summer 1998, LEP's value of m_W has increased slightly, coming closer to the Tevatron measurements, and the peak hadronic cross section went up by +0.9sigma.  (For details, see Swartz.)  The latter effect was half due to improvements in the treatment of radiative corrections (and presumably half due to statistics and experimental effects).  These changes moved the ellipse up and to the right.  The old picture did not support models with new, positive contributions to T.  The picture looked qualitatively like this:

This plot is obtained with Summer '99 data, determining sin²theta_W from leptonic observables only.  In this fit m_H would be constrained somewhat more model-independently: the highest masses would require new negative contributions to S; see Peskin's comment above.  The physics motivation to consider only the leptonic determinations of sin²theta_W is a 3sigma peculiarity, shown here:
The blue (five left-most) points are leptonic determinations, and the red (three right-most) are hadronic; for more details, see Swartz.  If one chooses to ascribe the discrepancy to new physics in the Zbb vertex, say, then one would be justified in omitting A_FB^b in the S-T analysis.  (Recall, that S-T analyses are relevant when new physics affects vertices and boxes in only slightly.)  Indeed, the last sentence on Schumm's second slide, above, suggests such a choice.

Finally, we note that work by Pierce, Bagger, Matchev, and Zhang, and by Hagiwara and Cho indicates that precision fits to the MSSM favor high masses for the additional Higgs bosons and the superpartners.  (The MSSM has m_H < 140 GeV for theoretical reasons.)

Some conclusions:

• one should interpret the precision electroweak fits with caution
• a low-mass Higgs boson, while plausible, is not guaranteed
• more Z-pole running would be valuable, e.g., in an LC validating new technology
• high energies will be important, to study either a heavy Higgs or other higher mass particles