2.2 Theoretical Overview

In the minimal SM, we introduce only one Higgs doublet field. The Higgs potential is given by

$$V=-\mu^2\vert H\vert^2 + \lambda\vert H\vert^4.$$ (2.1)

Through the electroweak symmetry breaking, the gauge bosons, fermions and the Higgs boson receive their masses. At the tree level, the mass formulae are given by $m_W= \frac{1}{2}g v$ for the W boson, $m_Z= \frac{1}{2}\sqrt{g+g'}v$ for the Z boson, $m_f= \frac{y_f}{\sqrt2}v$ for fermions and $m_H= \sqrt{\lambda}v$, where $v (\simeq$ 246 GeV) is the Higgs vacuum expectation value, g and g' are SU(2) and U(1) gauge coupling constants, and y_f is the Yukawa coupling constant for the fermion f. These expressions imply that a particle mass is determined by the strength of its interaction to the Higgs field, therefore the measurement of the coupling constants related to the Higgs boson is an important check of the mass generation mechanism in the SM.

In particular, the formula $m_H= \sqrt{\lambda}v$ 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 $(\sim 10^{19})$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 $\rho$ 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 $(H^\pm$). 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[8] 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 $\tan{\beta}\equiv <H_2^0>/<H_1^0>$. The other is the mixing angel between two CP-even Higgs bosons h and H, (m_h<m_H).

$$(v \cos{\beta} - h \sin{\alpha} + H \cos{\alpha})/\sqrt{2},$$ Re H₁⁰ = (2.2)

$$(v \sin{\beta} + h \cos{\alpha} + H \sin{\alpha})/\sqrt{2},$$ Re H₂⁰ = (2.3)

where H₁⁰ and H₂⁰ are the neutral components of two Higgs doublet fields. The ratio of various tree level coupling constants in the MSSM and those in the SM are listed in table 2.1.

 

Table: 2.1 The coupling constant in the MSSM normalized by the corresponding coupling constant in the SM. In the Yukawa coupling for the CP-odd Higgs boson (A), the fermion current is of the pseudo-scalar type.

hWW, hZZ	HWW, HZZ	$ht\bar{t}$	$hb\bar{b}, h\tau\bar{\tau}$	$Ht\bar{t}$	$Hb\bar{b}, H\tau\bar{\tau}$	$At\bar{t}$	$Ab\bar{b}, A\tau\bar{\tau}$
$\sin{(\beta-\alpha)}$	$\cos{(\beta-\alpha)}$	$\frac{\cos{\alpha}}{\sin{\beta}}$	$-\frac{\sin{\alpha}}{\cos{\beta}}$	$\frac{\sin{\alpha}}{\sin{\beta}}$	$\frac{\cos{\alpha}}{\cos{\beta}}$	$\cot{\beta}$	$\tan{\beta}$

In the MSSM, we can derive the upper-bound of the the lightest CP-even Higgs boson mass (m_h) 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 [23]. Taking into account the top and stop one-loop corrections, m_h is given by

$$m_h^2 \leq m_Z^2 \cos^2{2\beta}+\frac{3}{2\pi^2}\frac{m_t^4}{v^2} ln{\frac{m_{stop}^2}{{m_t}^2}},$$ (2.4)

where we assume that two stop mass states have the same mass. More precise formula is available in the literature. It is concluded that m_h is bounded by about 130 GeV, even if we take the stop mass to be a few TeV. The upper-bound of m_h is shown in Fig 2.1.

  

Figure: 2.1 The upper bounds of the mass of the lightest CP-even Higgs boson in MSSM and NMSSM calculated by Y. Okada et al [11]. (Left) Upper bounds as a function of the top quark mass in the MSSM, with various MSSM parameters as shown in the plot. (Right) Upper bound for MSSM and NMSSM with additional constraints of the finite Higgs coupling up to GUT scale. The top quark mass measured at Tevatron is indicated by the grey bands.

Eps files of left figure and right figure

The radiative correction can also change the mass formulas of the heaver Higgs bosons. The Higgs potential is parametrized by three mass parameters, gauge coupling constants and the parameters of the top and stop sector through the one-loop correction. The independent parameters can be taken as $\tan{\beta}$, and one of Higgs boson masses, which is usually taken as the CP-odd Higgs boson mass (m_A) and the top and the stop masses. More precisely, the parameters appearing in the formula of the radiative correction are not just one stop mass, but two stop masses, trilinear coupling constant for stop sector (A_t), the higgsino mass parameter $\mu$, and sbottom masses, etc. (If we use more precise formula, we need to specify more input parameters.) Once these parameters are specified, we can calculate the mass and the mixing of the Higgs sector. The Higgs boson masses are shown as a function of the CP-odd Higgs mass in Figure 2.2.

  

Figure 2.2: Higgs boson masses as a function of the CP-odd Higgs boson mass in MSSM.

Eps file of this figure

It is important to distinguish two regions in this figure. Namely, when m_A is much larger than 150 GeV, H, A and ${H}^{\pm}$states become approximately degenerate and the mass of h approaches to its upper-bound value for each $\tan{\beta}$. This limit is called the decoupling limit. In this limit, h has properties similar to the SM Higgs boson, and the coupling of the heavy Higgs bosons to two gauge-boson states is suppressed. On the other hand, if m_Ais less than 150 GeV, the lightest CP-even Higgs boson has sizable components of the SM Higgs field and the other doublet field.

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 $\tan{\beta}$, SUSY correction can generate contributions to the bottom-higgs Yukawa coupling which is not present at the tree level[24]. The top and bottom Yukawa couplings with the neutral Higgs fields are given by

$$L_{Yukawa}=y_t\bar{t_L}t_R H_2^0 + y_b\bar{b_L}b_R H_1^0 + \epsilon_b y_b\bar{b_L}b_R H_2^{0*} + h.c. \nonumber$$

The $\epsilon_b$ term is induced by loop diagrams with internal sbottom-gluino and stop-chargino. The bottom mass is then expressed by $m_b=y_b(1+\epsilon_b\tan{\beta})v\cos{\beta}/\sqrt{2}$. Although $\epsilon_b$ is typically O(10^-2), the correction to m_b enters with a combination of $\epsilon_b\tan{\beta}$ which can be close to O(1) for a large value of $\tan{\beta}$. Because of this correction, the ratio of $B(h \rightarrow \tau ^+ \tau ^-)$ and $B(h \rightarrow b\bar{b})$ is modified to

$$R_{\tau \tau/bb}\equiv \frac{B(h \rightarrow \tau^+\tau^-)} {... ...}} {1-\epsilon_b/\tan{\alpha}} \right)^2 R_{\tau \tau/bb}(SM),$$ (2.5)

where $R_{\tau \tau/bb}(SM)$ is the same ratio evaluated in the SM. $R_{\tau \tau/bb}$ is the same as $R_{\tau \tau/bb}(SM)$ if the SUSY loop effect to the $b\bar{b}H_2^{0*}$ vertex is negligible. Notice that in the decoupling limit with $m_A \rightarrow \infty$, $\tan{\alpha}$ is approaching to $-1/\tan{\beta}$, so that the ratio reduces to the SM prediction. In actual evaluation, however, the approach to the asymptotic form is slow for large $\tan{\beta}$, so that deviation from the SM prediction can be sizable for $\tan{\beta}\gsim 30$.

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 $\tan{\beta}$, which is quite different from the MSSM case where the Higgs mass becomes larger for large $\tan{\beta}$. The upper-bound for the lightest CP-even Higgs boson in the NMSSM is shown in Fig 2.1. In Ref. [12] 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 $\bar{5}+10+5+\bar{10}$ 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 [25].

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.