In the minimal SM, we introduce only one Higgs doublet field. The
Higgs potential is given by
In particular, the formula suggests that the mass of the Higgs boson reflects the strength of the electroweak symmetry breaking dynamics. The heavy Higgs boson implies the strongly-interacting dynamics and the light Higgs boson is consistent to the weakly interacting scenario such as grand unified theory (GUT) or supersymmetric (SUSY) unified models. Although the Higgs boson mass is a free parameter within the minimal SM, we can determine the upper and lower bound of its mass, if we require that the SM is valid up to some cut-off scale, beyond which the SM should be replaced by a more fundamental theory. If the cut-off scale is taken to be the Planck scale GeV, the possible mass range is 135 - 180 GeV. For a larger mass the Higgs self-coupling constant blows up below the Planck scale, and for a lower mass the vacuum stability is not guaranteed.
There are many possibilities to extend the Higgs sector of the minimal SM. Because the parameter determined from the electroweak measurements is close to unity, the dominant contribution to the electroweak breaking should come from weak-doublet fields. Two Higgs doublet model is one of the simplest extensions. In this case, there are two CP-even Higgs bosons (h, H), a CP-odd Higgs boson (A) and a pair of charged Higgs bosons ). If we require the two Higgs doublet model is valid up to the cut-off scale, the mass range of the lighter CP-even Higgs boson (h) is determined in a similar way to the SM case. For the case that the Planck scale is the cut-off scale, the upper-bound is 180 GeV, just like the SM. The lower bound can be smaller than the corresponding mass bound in the SM. In particular, the lower-bound is about 100 GeV in the case that the only one SM-like Higgs boson becomes light compared to other Higgs states. Since the present electroweak measurements are precise enough to be sensitive to virtual loop effects, we can put a useful bound on the Higgs boson mass, independent of the theoretical assumption on the validity of the theory up to some high energy scale. Within the minimal SM, the 95 % upper bound on the Higgs boson mass is about 210 GeV using the direct measurements of the W boson and top quark mass in addition to various electroweak data at LEP and SLD experiments. For more general model, for example in the two Higgs doublet model, this kind of strong bound is not obtained only from the electroweak data.
Since the end of 1970's, it has been known that supersymmetry (SUSY) can be a solution of the naturalness problem in the SM. If we consider the cut-off scale of the SM is the Planck or the GUT scale, we need extremely precise fine-tuning in the renormalization of the Higgs mass term to keep the weak scale much smaller than the cut-off scale. There is no such problem in SUSY models because the problematic quadratic divergence in the scalar mass renormalization is absent. Motivated by this observation, SUSY extensions of the SM and the GUT were proposed, and many phenomenological studies have been done. For the last ten years, SUSY models has become the most promising candidate of physics beyond the SM, because the gauge coupling constants determined at LEP and SLD experiments turned out to be consistent with the prediction of the SUSY GUT scenario.
The simplest SUSY extension of the SM is called the minimal
supersymmetric standard model (MSSM). The Higgs sector of the
MSSM is the type-II two Higgs doublet model, where Higgs doublet fields
H1 and H2 are introduced for the down-type quark/lepton Yukawa coupling
and the up-type quark Yukawa coupling, respectively. We define two
angle variables to parameterize the Higgs sector. One is the vacuum
mixing angle given by
The other is the mixing angel between two CP-even Higgs bosons
h and H, (mh<mH).
|hWW, hZZ||HWW, HZZ|
In the MSSM, we can derive the upper-bound of the the lightest CP-even
Higgs boson mass (mh) without reference to the cut-off scale of the theory.
This is because the self-coupling of the Higgs field is completely
determined by the gauge coupling constants at the tree level.
Although h has to be lighter than Z0 boson at the
tree level, contribution from the loop effects by the top quark and stop
squark can extend the possible mass region .
Taking into account the top and stop one-loop corrections,
mh is given by
Besides the Higgs potential, there is a case where radiative correction
becomes potentially important for the MSSM Higgs boson phenomenology.
For a large value of
SUSY correction can generate
contributions to the bottom-higgs Yukawa coupling which is not present
at the tree level. The top and bottom Yukawa couplings
with the neutral Higgs fields are given by
In extended versions of SUSY model, the upper-bound of the lightest CP-even Higgs boson can be determined only if we require that any of dimensionless coupling constants of the model does not blow up below some cut-off scale. For the SUSY model with an extra gauge singlet Higgs field, called the next-to-minimal supersymmetric standard model (NMSSM), the bound is a slightly larger than the upper-bound for the MSSM case. Because there is a new tree level contribution to the Higgs mass formula, the maximum value corresponds to a lower value of , which is quite different from the MSSM case where the Higgs mass becomes larger for large . The upper-bound for the lightest CP-even Higgs boson in the NMSSM is shown in Fig 2.1. In Ref.  the upper-bound of the lightest CP-even Higgs boson was calculated for SUSY models with gauge-singlet or gauge-triplet Higgs field and the maximal possible value was studied in those extensions of the MSSM. It was concluded that the mass bound can be as large as 210 GeV for a specific type model with a triplet-Higgs field for a stop mass of 1 TeV. The mass bound was also studied for the SUSY model with extra matter fields. In this model the upper-bound becomes larger due to loop corrections of extra matter multiplets. If the extra fields have representations in SU(5) GUT symmetry, the maximum value of the lightest CP-even Higgs boson mass becomes 180 GeV for the case that the squark mass is 1 TeV .
As we show above, it is very likely that the scalar boson associated with the electroweak symmetry breaking exists below 200 GeV, as long as we take a scenario that the Higgs sector remains weakly-interacting up to the GUT or the Planck scale, where the unification of gauge interactions, or gauge and gravity interactions may take place. In particular, there is a strict theoretical mass bound for the lightest CP-even Higgs boson in the MSSM. The precise determination on properties of a light Higgs boson is one of the most important tasks of the LC experiment.