sector. The introduction of higher representation fields generally causes a tree-level deviation in the $\rho$- parameter from unity. For example, in the model with a triplet field $\Delta(1,2)$ with the VEV $v_\Delta$, $\rho \sim 1 - 2(v_\Delta/v)^2$ is given, so that in such a model a tuning $(v_\Delta/v)^2 \ll 1$ is required to satisfy the data. We note that there are exceptional Higgs sectors with larger isospin representations which predict $\rho = 1$ at the tree level. In the model proposed by Georgi and Machacek [253], the Higgs sector is composed of an isospin doublet field with additional a complex (1,2) and a real (1,0) triplet fields, which satisfies $\rho = 1$ at the tree level. Addition of the septet field (3,2) to the SM Higgs sector also predicts $\rho = 1$ at the tree level.

Extended Higgs sectors with a multi-doublet structure, in general, receive a severe constraint from the results of FCNC experiments. The data show that FCNC processes such as $K^0 \to \mu^+\mu^-$, $B^0 - \bar{B}^0$ and so on are highly suppressed [252]. In the SM with a doublet Higgs field, the suppression of FCNC processes is perfectly explained by the so-called Glashow-Illiopoulos-Miani mechanism [254]. On the other hand, in general multi Higgs doublet models where multiple Higgs doublets couple to a quark or a charged lepton, Higgs boson mediated FCNC processes can easily occur at the tree level. In these models, in order to avoid such dangerous FCNC processes, it is required that these Higgs doublet fields have different quantum numbers [255].

In Sect. 2.6.2, we discuss properties of the two Higgs doublet model (2HDM), and its phenomenology at the LHC and the ILC. The physics of the model with the Higgs sector with a triplet is discussed in Sect. 2.6.3. The possibility of more exotic extended Higgs sectors are shortly discussed in Sect. 2.6.4.

## 2.6.2 Two Higgs doublet models

The 2HDM is one of the simplest extensions of the standard Higgs sector with one scalar doublet field. The model has many typical characteristics of general extended Higgs sectors, such as the existence of additional neutral Higgs states, charged scalar states, and the source of CP violation. In fact, the 2HDM often appears in the low energy effective theory of various new physics models which try to solve problems in the SM such as the minimal supersymmetric SM (MSSM), to some models of neutrino masses, dark matter, and electrowak baryogenesis. Therefore, it is useful to study properties of 2HDMs with their collider phenomenology.

In the 2HDM, two isospin doublet scalar fields $\Phi_1$ and $\Phi_2$ are introduced with a hypercharge $Y = 1$. The Higgs potential under the standard gauge symmetry is given by [79]

$$ 
\begin{aligned}
V = m_1^2 |\Phi_1|^2 &+ m_2^2 |\Phi_2|^2 - (m_3^2 \Phi_1^\dagger \Phi_2 + \text{h.c.}) \\
&+ \frac{\lambda_1}{2} |\Phi_1|^4 + \frac{\lambda_2}{2} |\Phi_2|^4 + \lambda_3 |\Phi_1|^2 |\Phi_2|^2 + \lambda_4 |\Phi_1^\dagger \Phi_2|^2 \\
&+ \left[ \frac{\lambda_5}{2} (\Phi_1^\dagger \Phi_2)^2 + \left\{ \lambda_6 (\Phi_1^\dagger \Phi_1) + \lambda_7 (\Phi_2^\dagger \Phi_2) \right\} \Phi_1^\dagger \Phi_2 + \text{h.c.} \right],
\end{aligned} 
\quad (33) $$

where $m_1^2, m_2^2$ and $\lambda_{1-4}$ are real while $m_3^2$ and $\lambda_{5-7}$ are complex. We here discuss the case of CP conservation with taking these complex as real. The doublet fields can be parameterized as

$$ \Phi_i = \left[ \frac{w_i^+}{\sqrt{2}} (v_i + h_i + i z_i) \right], \quad (i=1,2), \quad (34) $$

where $v_1$ and $v_2$ are the VEVs of $\Phi_1$ and $\Phi_2$, which satisfy $v = \sqrt{v_1^2 + v_2^2}$. The ratio of the two VEVs is a parameter written as $\tan \beta = v_2/v_1$. The mass eigenstates for the scalar bosons are obtained by

$$ 
\begin{aligned}
\begin{pmatrix} w_1^\pm \\ w_2^\pm \end{pmatrix} &= R(\beta) \begin{pmatrix} G^\pm \\ H^\pm \end{pmatrix}, \quad \begin{pmatrix} z_1 \\ z_2 \end{pmatrix} = R(\beta) \begin{pmatrix} G^0 \\ A \end{pmatrix}, \\
\begin{pmatrix} h_1 \\ h_2 \end{pmatrix} &= R(\alpha) \begin{pmatrix} H \\ h \end{pmatrix}, \text{ with } R(\theta) = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix},
\end{aligned} 
\quad (35) $$

where $G^\pm$ and $G^0$ are the Nambu-Goldstone bosons absorbed by the longitudinal component of $W^\pm$ and $Z$, respectively. As the physical degrees of freedom, consequently, we have two CP-even Higgs bosons $h$ and $H$, a CP-odd Higgs boson $A$ and a pair of singly-charged Higgs boson $H^\pm$. We define $h$ as the SM-like Higgs boson with the mass of about 125 GeV.

As already mentioned, in general 2HDMs, FCNCs can appear via tree-level Higgs-mediated diagrams, which are not phenomenologically acceptable. The simple way to avoid such dangerous FCNCs is to impose a discrete $Z_2$ symmetry, under which the two doublets are transformed as $\Phi_1 \to +\Phi_1$ and $\Phi_2 \to -\Phi_2$ [255–258]. Then, each quark or lepton can couple with only one of the two doublets, so that the Higgs mediated FCNC processes are forbidden at the tree level.

We hereafter concentrate on the case with the discrete symmetry. Under this symmetry, $\lambda_6$ and $\lambda_7$ in the Higgs potential in Eq. (33) are zero. On the other hand, the soft-breaking mass $m_3^2$ of the discrete symmetry can be allowed, because the discrete symmetry is introduced just to suppress FCNC interactions. As we consider the CP-conserving scenario, $m_3^2$ and $\lambda_5$ are real. Eight parameters in the potential are rewritten as the following eight physical parameters; the
<table><caption>**Table 19** Four possible $Z_2$ charge assignments of scalar and fermion fields to forbid tree-level Higgs-mediated FCNCs [260].</caption><thead><tr><th></th><th>Φ<sub>1</sub></th><th>Φ<sub>2</sub></th><th>u<sub>R</sub></th><th>d<sub>R</sub></th><th>ℓ<sub>R</sub></th><th>Q<sub>L</sub></th><th>L<sub>L</sub></th></tr></thead><tbody><tr><td>Type-I</td><td>+</td><td>-</td><td>-</td><td>-</td><td>-</td><td>+</td><td>+</td></tr><tr><td>Type-II</td><td>+</td><td>-</td><td>-</td><td>+</td><td>+</td><td>+</td><td>+</td></tr><tr><td>Type-X</td><td>+</td><td>-</td><td>-</td><td>-</td><td>+</td><td>+</td><td>+</td></tr><tr><td>Type-Y</td><td>+</td><td>-</td><td>-</td><td>+</td><td>-</td><td>+</td><td>+</td></tr></tbody></table>

masses of $h$, $H$, $A$ and $H^\pm$, two mixing angles $\alpha$ and $\beta$ appearing in Eq. (35), the VEV $v$ and the soft-breaking parameter $M^2$ defined by

$$M^2 = \frac{m_3^2}{\sin \beta \cos \beta}. \qquad (36)$$

In terms of these parameters, the quartic coupling constants in the Higgs potential are expressed as [259]

$$\lambda_1 = \frac{1}{v^2 \cos^2 \beta} (-M^2 \sin^2 \beta + m_h^2 \sin^2 \alpha + m_H^2 \cos^2 \alpha), \qquad (37a)$$

$$\lambda_2 = \frac{1}{v^2 \sin^2 \beta} (-M^2 \cos^2 \beta + m_h^2 \cos^2 \alpha + m_H^2 \sin^2 \alpha), \qquad (37b)$$

$$\lambda_3 = \frac{1}{v^2} \left[ -M^2 - \frac{\sin 2\alpha}{\sin 2\beta} (m_h^2 - m_H^2) + 2m_{H^\pm}^2 \right], \qquad (37c)$$

$$\lambda_4 = \frac{1}{v^2} (M^2 + m_A^2 - 2m_{H^\pm}^2), \qquad (37d)$$

$$\lambda_5 = \frac{1}{v^2} (M^2 - m_A^2). \qquad (37e)$$

Under the softly-broken discrete symmetry, the Yukawa interactions of the 2HDM can be written as

$$\mathcal{L}_{\text{Yukawa}}^{\text{2HDM}} = -\bar{Q}_L Y_u \tilde{\Phi}_u u_R - \bar{Q}_L Y_d \Phi_d d_R - \bar{L}_L Y_\ell \Phi_\ell \ell_R + \text{h.c.}, \qquad (38)$$

where $R$ and $L$ are the right-handed and left-handed chirality of fermions, respectively, and $\Phi_{f=u,d,\ell}$ are chosen from $\Phi_1$ or $\Phi_2$. There are four types of Yukawa interactions depending on the parity assignment of the discrete symmetry for fermions [261] shown in Table 19. Type-I is the case that all the quarks and charged leptons obtain the masses from $v_2$, while Type-II is that up-type quark masses are generated by $v_2$ but the masses of down-type quarks and charged leptons are generated by $v_1$. In Type-X, both up- and down-type quarks couple to $\Phi_2$ while charged leptons couple to $\Phi_1$. In Type-Y, up-type quarks and charged leptons couple to $\Phi_2$ while up-type quarks couple to $\Phi_1$. Because of these variations in types of Yukawa interaction, the 2HDM with the discrete symmetry can provide rich phenomenology. We note that Type-I is for example used in the neutrino-philic mode [262] approximately, Type-II is predicted in the context of the minimal supersymmetric SM (MSSM) [79, 263]

and that Type-X is used for example in some of radiative seesaw models [264].

Yukawa interaction in Eq. (38) is rewritten in terms of the mass eigenstates as

$$\mathcal{L}_{\text{Yukawa}}^{\text{2HDM}} = - \sum_{f=u,d,\ell} \left[ \begin{aligned} & \frac{m_f}{v} \xi_h^f \bar{f} f h + \frac{m_f}{v} \xi_H^f \bar{f} f H \\ & - i \frac{m_f}{v} \xi_A^f \gamma_5 \bar{f} f A \end{aligned} \right] \\ & - \left\{ \begin{aligned} & \frac{\sqrt{2} V_{ud}}{v} \bar{u} \left[ m_u \xi_A^u P_L + m_d \xi_A^d P_R \right] d H^+ + \\ & \frac{\sqrt{2} m_\ell}{v} \xi_A^\ell \bar{v}_L \ell R H^+ + \text{h.c.} \end{aligned} \right\}, \qquad (39)$$

where $P_{R,L}$ are the chiral projection operators. The coefficients $\xi_h^f$ are summarized in Table 20.

There are two possibilities to explain the current LHC data, which show that the Higgs sector is approximately SM-like. When $M^2 \gg v^2$, the additional Higgs bosons $H$, $A$ and $H^\pm$ are as heavy as $\sqrt{M^2}$, and only $h$ stays at the electroweak scale, behaving as the SM-like Higgs boson. The effective Lagrangian is

$$\mathcal{L}_{\text{eff}} = \mathcal{L}_{\text{SM}} + \frac{1}{M^2} \mathcal{O}^{(6)}. \qquad (40)$$

Another case is for $\sqrt{M^2} \sim v$. In the limit where the $hWW$ coupling takes the same value as the SM prediction $\sin(\beta - \alpha) = 1$, all the Yukawa couplings and the self-coupling for $h$ take the SM values, while $HWW$ is zero. In this case, $h$ behaves as the SM-like Higgs boson. Contrary, $H$, $A$ and $H^\pm$ do not couple to gauge bosons, and they only couple to the SM particles via Yukawa interaction. When $\sin(\beta - \alpha)$ is slightly smaller than unity, the couplings $hVV$ ($V = W, Z$) and $hff$ ($f = t, b, c, \dots$) deviate from the SM predictions depending on the type of Yukawa interaction. By detecting the pattern of the deviation in each coupling of $h$ at future experiments, we can distinguish the type of Yukawa coupling in the 2HDMs even without directly discovering the additional Higgs bosons.

The decay widths and branching ratios of additional Higgs bosons can be calculated for given values of $\tan\beta$, $\sin(\beta - \alpha)$ and the masses for each type of Yukawa interaction. We refer to Ref. [260] where the total decay widths are discussed in details for $\sin(\beta - \alpha) \simeq 1$. Explicit formulae for all the partial decay widths can be found, e.g., in Ref. [260].

In Fig. 64, decay branching ratios of additional Higgs bosons $H$, $A$, and $H^\pm$ are plotted in each type of Yukawa interaction as a function of $\tan\beta$ for the masses of 250 GeV. For simplicity, the SM-like limit $\sin(\beta - \alpha) = 1$ is taken. In this limit, the decay modes of $H \to W^+W^-$, $ZZ$, $hh$ as well as $A \to Zh$ are absent. In this limit, decay branching ratios of the SM-like Higgs boson are completely the same as those in the SM at the tree level, so that we cannot distinguish
<table><caption>**Table 20** The coefficients for different type of Yukawa interactions [260]. $c_{\theta} = \cos \theta$, and $s_{\theta} = \sin \theta$ for $\theta = \alpha, \beta$.</caption><thead><tr><th></th><th>ξ<sup>u</sup><sub>h</sub></th><th>ξ<sup>d</sup><sub>h</sub></th><th>ξ<sup>ℓ</sup><sub>h</sub></th><th>ξ<sup>u</sup><sub>H</sub></th><th>ξ<sup>d</sup><sub>H</sub></th><th>ξ<sup>ℓ</sup><sub>H</sub></th><th>ξ<sup>u</sup><sub>A</sub></th><th>ξ<sup>d</sup><sub>A</sub></th><th>ξ<sup>ℓ</sup><sub>A</sub></th></tr></thead><tbody><tr><td>Type-I</td><td>c<sub>α</sub>/s<sub>β</sub></td><td>c<sub>α</sub>/s<sub>β</sub></td><td>c<sub>α</sub>/s<sub>β</sub></td><td>s<sub>α</sub>/s<sub>β</sub></td><td>s<sub>α</sub>/s<sub>β</sub></td><td>s<sub>α</sub>/s<sub>β</sub></td><td>cot β</td><td>-cot β</td><td>-cot β</td></tr><tr><td>Type-II</td><td>c<sub>α</sub>/s<sub>β</sub></td><td>-s<sub>α</sub>/c<sub>β</sub></td><td>-s<sub>α</sub>/c<sub>β</sub></td><td>s<sub>α</sub>/s<sub>β</sub></td><td>c<sub>α</sub>/c<sub>β</sub></td><td>c<sub>α</sub>/c<sub>β</sub></td><td>cot β</td><td>tan β</td><td>tan β</td></tr><tr><td>Type-X</td><td>c<sub>α</sub>/s<sub>β</sub></td><td>c<sub>α</sub>/s<sub>β</sub></td><td>-s<sub>α</sub>/c<sub>β</sub></td><td>s<sub>α</sub>/s<sub>β</sub></td><td>s<sub>α</sub>/s<sub>β</sub></td><td>c<sub>α</sub>/c<sub>β</sub></td><td>cot β</td><td>-cot β</td><td>tan β</td></tr><tr><td>Type-Y</td><td>c<sub>α</sub>/s<sub>β</sub></td><td>-s<sub>α</sub>/c<sub>β</sub></td><td>c<sub>α</sub>/s<sub>β</sub></td><td>s<sub>α</sub>/s<sub>β</sub></td><td>c<sub>α</sub>/c<sub>β</sub></td><td>s<sub>α</sub>/s<sub>β</sub></td><td>cot β</td><td>tan β</td><td>-cot β</td></tr></tbody></table>

<figure><img src="image_66.png" /><figcaption>**Fig. 64** The decay branching ratios of H, A and $H^{\pm}$ in 2HDMs for Type I, Type II, Type X and Type Y as a function of $\tan\beta$ with $m_H = m_A = m_{H^{\pm}} = 250$ GeV and $\sin(\beta - \alpha) = 1$ [269].</figcaption></figure>

models by precision measurements of the couplings of the SM-like Higgs boson $h^{16}$.

*Constraints on the Higgs potential from perturbative unitarity and vacuum stability*

The condition of tree-level unitarity requires the scattering amplitudes to be perturbative [270]; i.e. $|a_i^0| < 1/2$ [79], where $a_i^0$ are the eigenvalues of the S-wave amplitudes of the elastic scatterings of the longitudinal component of weak gauge bosons and the Higgs boson. In the 2HDM with the softly-broken $Z_2$ symmetry, this condition gives constraints on the quartic couplings in the Higgs potential [271–273]. The eigenvalues for $14 \times 14$ scattering matrix for neutral

<footnote n="16">¹⁶The decay branching ratios of $h$ can be different from the SM prediction at the next-to-leading order [259, 265–268].</footnote>
states are given as [271],

$$a_1^{\pm} = \frac{1}{16\pi} \left[ \frac{3}{2}(\lambda_1 + \lambda_2) \pm \sqrt{\frac{9}{4}(\lambda_1 - \lambda_2)^2 + (2\lambda_3 + \lambda_4)^2} \right], \quad (41a)$$

$$a_2^{\pm} = \frac{1}{16\pi} \left[ \frac{1}{2}(\lambda_1 + \lambda_2) \pm \sqrt{\frac{1}{4}(\lambda_1 - \lambda_2)^2 + \lambda_4^2} \right], \quad (41b)$$

$$a_3^{\pm} = \frac{1}{16\pi} \left[ \frac{1}{2}(\lambda_1 + \lambda_2) \pm \sqrt{\frac{1}{4}(\lambda_1 - \lambda_2)^2 + \lambda_5^2} \right], \quad (41c)$$

$$a_4 = \frac{1}{16\pi}(\lambda_3 + 2\lambda_4 - 3\lambda_5), \quad a_5 = \frac{1}{16\pi}(\lambda_3 - \lambda_5), \quad (41d)$$

$$a_6 = \frac{1}{16\pi}(\lambda_3 + 2\lambda_4 + 3\lambda_5), \quad a_7 = \frac{1}{16\pi}(\lambda_3 + \lambda_5), \quad (41e)$$

$$a_8 = \frac{1}{16\pi}(\lambda_3 + \lambda_4), \quad (41f)$$

and for singly charged states, one additional eigenvalue is added [272];

$$a_9 = \frac{1}{16\pi}(\lambda_3 - \lambda_4). \quad (42a)$$

The condition of vacuum stability that the Higgs potential must be bounded from below gives [274–276]

$$\lambda_1 > 0, \quad \lambda_2 > 0, \quad \sqrt{\lambda_1 \lambda_2 + \lambda_3 + \text{Min}(0, \lambda_4 - |\lambda_5|)} > 0. \quad (43)$$

The parameter space of the model is constrained by these conditions on the coupling constants in the Higgs potential.

### Constraints on the Higgs potential from electroweak precision observables

Further constraints on the Higgs sector of the 2HDM are from the electroweak precision measurements. The S, T and U parameters [277] are sensitive to the loop effects of Higgs bosons [278,279]. The T parameter corresponds to the electroweak $\rho$ parameter, which is severely constrained by experimental observations as has been discussed. The mass splitting between the additional Higgs bosons are strongly bounded [280, 281]. This implies that the Higgs potential has to respect the custodial SU(2) symmetry approximately.

### Flavour constraints on $m_{H^{\pm}}$ and $\tan\beta$

Flavour experiments provide strong constraints on the 2HDMs through the $H^{\pm}$ contribution to the flavour mixing observables at the tree level or at the loop level [260, 282, 283]. Because the amplitudes of these processes necessarily contain the Yukawa interaction, constraints on the 2HDM strongly depends on the type of Yukawa interaction. In Ref. [284], the limits on the general couplings from flavour physics are translated into those on the ($m_{H^{\pm}}, \tan\beta$) plane for all four types of Yukawa interaction in the 2HDM: see Fig. 65, where Type III and Type IV correspond to Type

Y and Type X, respectively. See also the more recent studies [285–287].

A strong exclusion limit is given from the result for the branching ratio of the $B \to X_s\gamma$ process [288]. For Type-II and Type-Y, a $\tan\beta$-independent lower limit of $m_{H^{\pm}} \gtrsim 380$ GeV is obtained [289] by comparing with the NNLO calculation [290]. For Type-I and Type-X, on the other hand, $\tan\beta \lesssim 1$ is excluded for $m_{H^{\pm}} \lesssim 800$ GeV, while no lower bound on $m_{H^{\pm}}$ is obtained.

By the results for the $B^0_d\bar{B}^0_d$ mixing, lower $\tan\beta$ regions ($\tan\beta \le 1$) are excluded for $m_{H^{\pm}} \lesssim 500$ GeV for all types of Yukawa interaction [288].

Constraints in larger $\tan\beta$ regions are obtained only for Type-II, which come from the results for leptonic meson decay processes [288], $B \to \tau\nu$ [291] and $D_s \to \tau\nu$ [292]. Upper bounds on $\tan\beta$ are obtained at around 30 for $m_{H^{\pm}} \simeq 350$ GeV and around 60 for $m_{H^{\pm}} \simeq 700$ GeV [284]. On the other hand, the other types do not receive any strong constraint for large $\tan\beta$ values, because the relevant couplings behave $\xi_A^d\xi_A^l = \tan^2\beta$ for Type-II while $\xi_A^d\xi_A^l = -1$ ($\cot^2\beta$) for Type-X and Type-Y (Type-I).

### Constraint from the data at LEP/SLC, Tevatron and also from the current LHC data

At the LEP direct search experiments, lower mass bounds on H and A have been obtained as $m_H > 92.8$ GeV and $m_A > 93.4$ GeV in the CP-conservation scenario [293, 294]. Combined searches for $H^{\pm}$ give the lower mass bound $m_{H^{\pm}} > 80$ GeV, by assuming $\mathcal{B}(H^+ \to \tau^+\nu) + \mathcal{B}(H^+ \to c\bar{c}) = 1$ [295–297].

At the Fermilab Tevatron, CDF and D0 collaborations have studied the processes of $p\bar{p} \to b\bar{b}H/A$, followed by $H/A \to b\bar{b}$ or $H/A \to \tau^+\tau^-$ [298–300]. By using the $\tau^+\tau^-$ (b\bar{b}) decay mode, which can be sensitive for the cases of Type-II (Type-II and Type-Y), upper bounds on $\tan\beta$ have been obtained to be from about 25 to 80 (40 to 90) for $m_A$ from 100 GeV to 300 GeV, respectively. For the direct search of $H^{\pm}$, the decay modes of $H^{\pm} \to \tau\nu$ and $H^{\pm} \to c\bar{c}$ have been investigated by using the production from the top quark decay $t \to bH^{\pm}$ [301–303]. Upper bounds on $\mathcal{B}(t \to bH^{\pm})$ have been obtained, which can be translated into the bound on $\tan\beta$ in various scenarios. For Type-I with $H^{\pm}$ heavier than the top quark, upper bounds on $\tan\beta$ have been obtained to be from around 20 to 70 for $m_{H^{\pm}}$ from 180 GeV to 190 GeV, respectively [301].

At the LHC, additional Higgs boson searches have been performed by using currently accumulated events at the experiments with a center-of-mass energy of 7 TeV with the integrated luminosity of 4.9 fb$^{-1}$ in 2011 and also 8 TeV with 19.7 fb$^{-1}$ in 2012. The CMS Collaboration has searched H and A which decay into the $\tau^+\tau^-$ final state, and upper limits on $\tan\beta$ have been obtained in the MSSM (or in the Type-II 2HDM) from 4 to 60 for $m_A$ from 140 GeV to
<figure><img src="image_67.png" /><figcaption>**Fig. 65** The constraint on the parameter space in the 2HDM for Type I, Type II, Type IV (Type X) and Type III (Type Y) from various flavor experiments [284].</figcaption></figure>

900 GeV, respectively [304]. By the ATLAS Collaboration similar searches have also been done [305]. In the Type-II and Type-Y 2HDMs, CMS has also searched the bottom-quark associated production process of H or A which decays into the $b\bar{b}$ final state [306], and has obtained the upper bounds on $\tan\beta$: i.e., $\tan\beta \gtrsim 16$ (28) is excluded at $m_A = 100$ GeV (350 GeV). ATLAS has reported the $H^{\pm}$ searches via the $\tau$+jets final state [307, 308]. In the Type-II 2HDM with $m_{H^{\pm}} \lesssim m_t$, wide parameter regions have been already excluded by the data for 100 GeV $\lesssim m_{H^{\pm}} \lesssim 140$ GeV with $\tan\beta \gtrsim 1$. Moreover, the parameter regions of $\tan\beta \gtrsim 50$ at $m_{H^{\pm}} = 200$ GeV and $\tan\beta \gtrsim 65$ at $m_{H^{\pm}} = 300$ GeV have been excluded for $m_{H^{\pm}} \gtrsim 180$ GeV, respectively. The searches for $H^{\pm}$ in the $\text{cs}$ final-state have been performed by ATLAS [309], and the upper limit on the branching ratio of $t \to bH^{\pm}$ decay is obtained assuming the 100% branching ratio of $H^{\pm} \to \text{cs}$. For $\sin(\beta - \alpha) < 1$, searches for $H \to W^{+}W^{-}$, $hh$ and $A \to Zh$ give constraints on the 2HDMs with Type-I and Type-II Yukawa interactions [310, 311].

### Prospect of extra Higgs boson searches at the LHC (13-14 TeV)

At the LHC experiments with the collision energy of 13-14 TeV and the integrated luminosity of $L = 300 \text{ fb}^{-1}$ and also $3000 \text{ fb}^{-1}$, the expected discovery potential for additional Higgs bosons have been studied in the 2HDM in Refs. [269, 313, 314], by using the signal and background analysis for various channels given in Ref. [312]. Processes available for the searches for additional Higgs bosons are [269]

- $H/A(+b\bar{b})$ inclusive and associated production followed by the $H/A \to \tau^{+}\tau^{-}$ decay [316].

- $H/A+b\bar{b}$ associated production followed by the $H/A \to b\bar{b}$ decay [316-318].

- $gb \to tH^{\pm}$ production followed by the $H^{\pm} \to tb$ decay [319, 320].

- $q\bar{q} \to HA \to 4\tau$ process [315, 321].

For the production cross sections, the tree-level cross sections have been convoluted with the CTEQ6L parton distribution functions [322]. The scales of the strong coupling

constant and the parton distribution function are chosen to the values used in Ref. [323]. For details, see Ref. [269], where the latest recommendations from the LHC Higgs Cross Section Working Group for 2HDM cross section (and branching ratio) evaluations can be found in Ref. [324].

In Fig. 66, the contour plots of the expected exclusion regions [2$\sigma$ confidence level (CL)] in the $(m_\phi, \tan\beta)$ plane are shown at the LHC $\sqrt{s} = 14$ TeV with the integrated luminosity of 300 fb$^{-1}$ (thick solid lines) and 3000 fb$^{-1}$ (thin dashed lines), where $m_\phi$ represents common masses of additional Higgs bosons. From the left panel to the right panel, the results for Type-I, Type-II, Type-X and Type-Y are shown separately. Following the analysis in Ref. [312], the reference values of the expected numbers of signal and background events are changed at the several values of $m_\phi$ [269, 314], which makes sharp artificial edges of the curves in Figs. 66.

For Type-I, $H/A$ production followed by the decay into $\tau^{+}\tau^{-}$ can be probed for $\tan\beta \lesssim 3$ and $m_{H,A} \le 350$ GeV, where the inclusive production cross section is enhanced by the relatively large top Yukawa coupling with the sizable $\tau^{+}\tau^{-}$ branching ratio. The $tH^{\pm}$ production decaying into $H^{\pm} \to tb$ can be used to search $H^{\pm}$ in relatively smaller $\tan\beta$ regions. $H^{\pm}$ can be discovered for $m_{H^{\pm}} < 800$ GeV and $\tan\beta \lesssim 1$ (2) for the integrated luminosity of 300 fb$^{-1}$ (3000 fb$^{-1}$).

For Type-II, the inclusive and the bottom-quark-associated production processes of $H/A$ with the decay into $\tau^{+}\tau^{-}$ or the $b\bar{b}$ can be used to search $H$ and $A$ for relatively large $\tan\beta$. They can also be used in relatively small $\tan\beta$ regions for $m_{H,A} \lesssim 350$ GeV. $H^{\pm}$ can be searched by the $tH^{\pm}$ production with $H^{\pm} \to tb$ decay for $m_{H^{\pm}} \gtrsim 180$ GeV for relatively small and large $\tan\beta$ values. The region of $m_{H^{\pm}} \gtrsim 350$ GeV (500 GeV) could be excluded with the 300 fb$^{-1}$ (3000 fb$^{-1}$) data.

For Type-X, $H$ and $A$ can be searched via the inclusive production and $HA$ pair production by using the $\tau^{+}\tau^{-}$ decay mode, which is dominant. The inclusive production could exclude the region of $\tan\beta \lesssim 10$ with $m_{H,A} \lesssim 350$ GeV. Regions up to $m_{H,A} \simeq 500$ GeV (700 GeV) with $\tan\beta \gtrsim 10$ could be excluded by using the pair production with the
<figure><img src="image_68.png" /><figcaption>**Fig. 66** Expected exclusion regions (2σ CL) in the plane of $\tan\beta$ and the mass scale $m_\phi$ of the additional Higgs bosons at the LHC. Curves are evaluated by using the signal and background analysis given in Ref. [312] for each process, where the signal events are rescaled to the prediction in each case [313,314], except the $4\tau$ process for which we follow the analysis in Ref. [315]. Thick solid lines are the expected exclusion contours by $L = 300 \text{ fb}^{-1}$ data, and thin dashed lines are for $L = 3000 \text{ fb}^{-1}$ data. For Type-II, the regions indicated by circles may not be excluded by $H/A \to \tau^+\tau^-$ search by using the 300 fb$^{-1}$ data due to the large SM background.</figcaption></figure>

300 fb$^{-1}$ (3000 fb$^{-1}$) data. The search for $H^\pm$ is the similar to that for Type-I.

Finally, for Type-Y, the inclusive production of $H$ and $A$ followed by $H/A \to \tau^+\tau^-$ can be searched for the regions of $\tan\beta \lesssim 2$ and $m_{H,A} \le 350$ GeV. The bottom-quark associated production of $H$ and $A$ with $H/A \to b\bar{b}$ can be searched for the regions of $\tan\beta \gtrsim 30$ up to $m_{H,A} \approx 800$ GeV. The search of $H^\pm$ is similar to that for Type-II.

For Type-II and Type-Y (Type-X), if all the curves are combined by assuming that all the masses of additional Higgs bosons are the same, the mass below 400 GeV (350 GeV) could be excluded by the 300 fb$^{-1}$ data for all value of $\tan\beta$, and with 3000 fb$^{-1}$, the mass below 550 GeV (400 GeV) could be excluded. On the other hand, for Type-I, the regions with $\tan\beta \gtrsim 5$ (10) cannot be excluded by 300 fb$^{-1}$ (3000 fb$^{-1}$) data. In the general 2HDM, however, the mass spectrum of additional Higgs boson has more degrees of freedom, so that we can still find allowed parameter regions where $m_H$ is relatively light but $m_A(\simeq m_{H^\pm})$ are heavy. Thus, the overlaying of these exclusion curves for different additional Higgs bosons may only be applied to the case of $m_H = m_A = m_{H^\pm}$.

If $H^\pm$ is discovered at the LHC, its mass could be determined immediately [312,325]. Then the determination of the type of the Yukawa interaction becomes important. At the LHC, however, we would not completely distinguish the types of Yukawa interaction, because the Type-I and Type-X, or Type-II and Type-Y have a common structure for the $tbH^\pm$ interaction. In addition, as seen in Fig. 66, there can be no complementary process for the neutral Higgs boson searches in some parameter regions; e.g., $m_{H,A} \gtrsim 350$ GeV with relatively small $\tan\beta$, depending on the type of the Yukawa interaction. At the ILC, on the other hand, as long as $m_{H,A} \lesssim 500$ GeV, the neutral Higgs bosons can be produced and investigated almost independent of $\tan\beta$. Therefore, it is quite important to search for the additional Higgs bosons with the mass of 350-500 GeV, and to determine the models and parameters at the ILC, even after the LHC.

Notice that the above results are obtained in the SM-like limit, $\sin(\beta - \alpha) = 1$. A deviation from the SM-like limit causes appearance of additional decay modes such as $H \to W^+W^-$, ZZ, $hh$ as well as $A \to Zh$ [79, 326-328]. Especially, for Type-I with a large value of $\tan\beta$, branching ratios of these decay modes can be dominant even with a small deviation from the SM-like limit [260, 327]. Therefore, searches for additional Higgs bosons in these decay modes can give significant constraints on the deviation of $\sin(\beta - \alpha)$ from the SM-like limit [310, 311], which is independent of coupling constants of $hVV$.

### Prospect for the searches for the additional Higgs bosons at the ILC

At LC's the main production mechanisms of additional Higgs bosons in the 2HDM are $e^+e^- \to HA$ and $e^+e^- \to H^+H^-$, where a pair of additional Higgs bosons is produced via gauge interactions as long as kinematically allowed. For energies below the threshold, the single production processes, $e^+e^- \to H(A)\bar{f}\bar{f}$ and $e^+e^- \to H^\pm f\bar{f}'$ are the leading contributions [329]. They are enhanced when the relevant Yukawa couplings $\phi f\bar{f}'(l)$ are large. The cross sections of these processes have been studied extensively [196,329-331], mainly for the MSSM or for the Type-II 2HDM.

Here, we discuss the result in the general 2HDMs but with softly-broken discrete symmetry. The following processes are considered:

$$e^+e^- \to \tau^+\tau^-H, \quad \tau^+\tau^-A, \qquad (44a)$$

$$e^+e^- \to b\bar{b}H, \quad b\bar{b}A, \qquad (44b)$$

$$e^+e^- \to t\bar{t}H, \quad t\bar{t}A, \qquad (44c)$$

$$e^+e^- \to \tau^-\nu H^+, \quad \tau^+\bar{\nu} H^-, \qquad (44d)$$

$$e^+e^- \to \bar{t}bH^+, \quad \bar{b}tH^-. \qquad (44e)$$

For energies above the threshold of the pair production, $\sqrt{s} > m_H + m_A$, the contribution from $e^+e^- \to HA$ can be significant in the processes in Eqs. (44a-44c). Similarly for $\sqrt{s} > 2m_{H^\pm}$, the contribution from $e^+e^- \to H^+H^-$ can be
<figure><img src="image_69.png" /><figcaption>**Fig. 67** Contour plots of the four-particle production cross sections through the H/A production and $H^{\pm}$ production process at the ILC with $\sqrt{s} = 500$ GeV in the $(m_{H^{\pm}}, \tan \beta)$ plane. Contour of $\sigma = 0.1$ fb is drawn for each signature [269].</figcaption></figure>

significant in the processes in Eqs. (44d, 44e). Below the threshold, the processes including diagrams of $e^{+}e^{-} \to f\bar{f}^{*}$ and $e^{+}e^{-} \to f^{*}\bar{f}$ dominate.

Both the pair and single production processes of additional Higgs bosons mostly result in four-particle final-states (including neutrinos). In Ref. [269], the cross sections of various four-particle final-states are studied for given masses of additional Higgs bosons and $\tan\beta$ with setting $\sin(\beta - \alpha) = 1$, and draw contour curves where the cross sections are 0.1 fb. This value is chosen commonly for all processes as it could be regarded as a typical order of magnitude of the cross section of the additional Higgs boson production [331]. In addition, this value can also be considered as a criterion for observation with the expected integrated luminosity at the ILC [49, 196]. Certainly, the detection efficiencies are different for different four-particle final-states, and the decay of unstable particles such as tau leptons and top quarks have to be considered if they are involved. We here restrict ourselves to simply compare the various four-particle production processes in four types of Yukawa interaction in the 2HDMs with taking the criterion of 0.1 fb as a magnitude of the cross sections. Expected background processes and a brief strategy of observing the signatures are discussed in Ref. [269].

In Fig. 67, contour plots of the cross sections of four-particle production processes through H and/or A are shown in the $(m_{H/A}, \tan\beta)$ plane (upper figures), and those through $H^{\pm}$ are shown in the $(m_{H^{\pm}}, \tan\beta)$ plane (lower figures) for the collision energy to be $\sqrt{s} = 500$ GeV. From left to right, the figures correspond to the results in Type 1, Type II,

Type X and Type Y. We restrict ourselves to consider the degenerated mass case, $m_H = m_A$.

In Fig. 68, contour plots of the cross sections of four-particle production processes through H and/or A are shown in the $(m_{H/A}, \tan\beta)$ plane (upper figures), and those through $H^{\pm}$ are shown in the $(m_{H^{\pm}}, \tan\beta)$ plane (lower figures) for the collision energy to be $\sqrt{s} = 1$ TeV. From left to right, the figures correspond to the results in Type 1, Type II, Type X and Type Y. We restrict ourselves to consider the degenerated mass case, $m_H = m_A$.

We here give a comment on the SM background processes and their cross sections [269]. In general, for the four-particle production processes, the SM background cross sections are larger for $\sqrt{s} = 250$ GeV, but decrease with the collision energy. The typical orders of cross sections are of the order of 1 fb to 10 fb for the Z/γ mediated processes, and of the order of 10 to 100 fb for the processes which are also mediated by $W^{\pm}$. For the four-quark production processes, gluon exchange diagrams also contribute. In order to reduce the background events, efficient kinematical cuts are required.

The cross section of the $4t$ production is very small in the SM. Therefore, a clean signature can be expected to be detected in this mode. Detailed studies on the signal and background processes for $tbtb$ production can be found in Ref. [330], and the signal-to-background analysis for the $4\tau$ production can be found in Ref. [332] with the reconstruction method of the masses of additional Higgs bosons.

Finally, we discuss some concrete scenarios to show the complementarity of direct searches for the additional Higgs bosons in the 2HDMs at the LHC and the ILC. As bench-
<figure><img src="image_70.png" /><figcaption>**Fig. 68** Contour plots of the four-particle production cross sections through the H/A production and $H^{\pm}$ production process at the ILC with $\sqrt{s} = 1$ TeV in the $(m_{H^{\pm}}, \tan \beta)$ plane. Contour of $\sigma = 0.1$ fb is drawn for each signature [269].</figcaption></figure>

<table><caption>**Table 21** Expected signatures to be observed at the LHC and ILC for the benchmark scenarios with $m_{\phi} = 220$ GeV [269]. Observable final-states are listed as the signatures of additional Higgs bosons, H, A and $H^{\pm}$. LHC300, LHC3000, ILC500 represent the LHC run of 300 fb$^{-1}$, 3000 fb$^{-1}$ luminosity, ILC run of 500 GeV, respectively.</caption><thead><tr><th>(m<sub>φ</sub>, tan β)</th><th></th><th colspan="2">Type-I</th><th colspan="2">Type-II</th><th colspan="2">Type-X</th><th colspan="2">Type-Y</th></tr><tr><th></th><th></th><th>H,A</th><th>H<sup>±</sup></th><th>H,A</th><th>H<sup>±</sup></th><th>H,A</th><th>H<sup>±</sup></th><th>H,A</th><th>H<sup>±</sup></th></tr></thead><tbody><tr><td rowspan="3">(220 GeV, 20)</td><td>LHC300</td><td>—</td><td>—</td><td>ττ, bb</td><td>tb</td><td>4τ</td><td>—</td><td>bb</td><td>tb</td></tr><tr><td>LHC3000</td><td>—</td><td>—</td><td>ττ, bb</td><td>tb</td><td>4τ</td><td>—</td><td>bb</td><td>tb</td></tr><tr><td>ILC500</td><td>4b, 2b2τ, 4g,<br>2b2g, 2τ2g</td><td>tbtb</td><td>4b, 2b2τ,<br>4τ</td><td>tbtb, tbτν,<br>τντν</td><td>4τ</td><td>tbτν,<br>τντν</td><td>4b</td><td>tbtb, tbcb</td></tr><tr><td rowspan="3">(220 GeV, 7)</td><td>LHC300</td><td>—</td><td>—</td><td>ττ</td><td>tb</td><td>4τ</td><td>—</td><td>—</td><td>tb</td></tr><tr><td>LHC3000</td><td>—</td><td>tb</td><td>ττ</td><td>tb</td><td>ττ, 4τ</td><td>—</td><td>—</td><td>tb</td></tr><tr><td>ILC500</td><td>4b, 2b2τ, 4g,<br>2b2g, 2τ2g</td><td>tbtb</td><td>4b, 2b2τ,<br>4τ</td><td>tbtb, tbτν,<br>τντν</td><td>2b2τ, 4τ</td><td>tbtb, tbτν,<br>τντν</td><td>4b</td><td>tbtb, tbcb</td></tr><tr><td rowspan="3">(220 GeV, 2)</td><td>LHC300</td><td>—</td><td>tb</td><td>ττ</td><td>tb</td><td>ττ, 4τ</td><td>tb</td><td>—</td><td>tb</td></tr><tr><td>LHC3000</td><td>ττ</td><td>tb</td><td>ττ</td><td>tb</td><td>ττ, 4τ</td><td>tb</td><td>—</td><td>tb</td></tr><tr><td>ILC500</td><td>4b, 2b2τ, 4g,<br>2b2g, 2τ2g</td><td>tbtb</td><td>4b, 2b2τ,<br>4τ, 2b2g</td><td>tbtb,<br>tbτν</td><td>4b, 2b2τ,<br>4τ</td><td>tbtb,<br>tbτν</td><td>4b, 2b2τ,<br>2b2g</td><td>tbtb</td></tr></tbody></table>

mark scenarios, three cases $\tan\beta = 2, 7$ and 20 are considered for $m_{\phi} = 220$ GeV and $\sin(\beta - \alpha) = 1$, where $m_{\phi}$ represents the common mass of H, A and $H^{\pm}$. In Table 21, the expected signatures of H/A and $H^{\pm}$ are summarized to be observed at the LHC with 300 fb$^{-1}$, 3000 fb$^{-1}$ and at the ILC with $\sqrt{s} = 500$ GeV.

First, for the case of $(m_{\phi}, \tan \beta) = (220 \text{ GeV}, 20)$, no signature is predicted for Type-I, while different signatures are predicted for Type-II, Type-X and Type-Y at the LHC with 300 fb$^{-1}$ and 3000 fb$^{-1}$. Therefore those three types could be discriminated at the LHC. On the other hand, at the ILC

with $\sqrt{s} = 500$ GeV, all the four types of the Yukawa interaction including Type-I predict signatures which are different from each other. Therefore, complete discrimination of the type of Yukawa interaction could be performed at the ILC.

Next, we turn to the second case with $(m_{\phi}, \tan \beta) = (220 \text{ GeV}, 7)$. At the LHC with 300 fb$^{-1}$, Type-I cannot be observed, while Type-II, Type-X and Type-Y are expected to be observed with different signatures. At the LHC with 3000 fb$^{-1}$, the signature of Type-I can also be observed with the same final state as Type-Y. Type-I and Type-Y can be basically separated, because for Type-Y the signals can be
observed already with 300 fb⁻¹ while for Type-I that can be observed only with 3000 fb⁻¹. Therefore, at the LHC with 3000 fb⁻¹, the complete discrimination can be achieved. At the ILC, the four types of Yukawa interaction can also be separated by a more variety of the signatures for both channels with the neutral and charged Higgs bosons.

Finally, for the case of $(m_\phi, \tan\beta) = (220 \text{ GeV}, 2)$, signals for all the four types of Yukawa interaction can be observed at the LHC with 300 fb⁻¹. However, the signatures of Type-I and Type-Y are identical, so that the two types cannot be discriminated. With 3000 fb⁻¹, the difference between the Type-I and Type-Y emerges in the H/A signature. Therefore the two types can be discriminated at this stage. Again, at the ILC, the four types can also be separated with a more variety of the signatures for both channels with the neutral and charged Higgs bosons.

### Fingerprinting the type of the 2HDM by precision measurement of the Higgs couplings at the ILC

Extra Higgs bosons in extended Higgs sectors can be discovered as long as their masses are not too large as compared to the electroweak scale. On the other hand, at the ILC [333], these extended Higgs sectors can also be tested by accurately measuring the coupling constants with the discovered Higgs bosons $h$. This is complementary with the direct searches at the LHC.

In the extended Higgs sectors, the gauge couplings and Yukawa interactions of $h$ are parameterized by

$$ \mathcal{L}^{\text{int}} = +\kappa_W \frac{2m_W^2}{v} hW^{+\mu}W_{\mu}^{-} + \kappa_Z \frac{m_Z^2}{v} hZ^{\mu}Z_{\mu}^{-} - \sum_f \kappa_f \frac{m_f}{v} \bar{f}f h + \dots, \quad (45) $$

where $\kappa_V$ ($V = W$ and $Z$) and $\kappa_f$ ($f = t, b, c, \dots$) are the scaling factors measuring the deviation from the SM predictions. In the SM, we have $\kappa_V = \kappa_f = 1$. According to Refs. [245, 313, 333], the $hVV$ couplings are expected to be measured with about 4% accuracy at the LHC with 300 fb⁻¹ (although requiring some theory input). The accuracy for the $h\bar{t}\bar{t}$, $hb\bar{b}$ and $h\tau\tau$ couplings are supposed to be about 16%, 14% and 11%, respectively. At the ILC250 (ILC500) where the collision energy and the integrated luminosity are 250 GeV (500 GeV) and 250 fb⁻¹ (500 fb⁻¹) combining with the results assuming 300 fb⁻¹ at the LHC, the $hWW$ and $hZZ$ couplings are expected to be measured by about 1.9% (0.2%) and about 0.4% (0.3%), respectively. The $hc\bar{c}$, $hb\bar{b}$ and $h\tau\tau$ couplings are supposed to be measured by about 5.1% (2.6%), 2.8% (1.0%) and 3.3% (1.8%) at the ILC250 (ILC500). For the $h\bar{t}\bar{t}$ coupling, it will be measured with 12.0% and 9.6% at the ILC250 and ILC500, respectively.

In the 2HDM, the scaling factors $\kappa_V$ are given by $\kappa_V = \sin(\beta - \alpha)$, while those for the Yukawa interactions are given depending on the type of Yukawa interaction [260]. For the

<figure><img src="image_71.png" /><figcaption>**Fig. 69** Left: The scaling factors in 2HDM with four types of Yukawa interactions. Right: The scaling factors in models with universal Yukawa couplings. The current LHC bounds and the expected LHC and ILC sensitivities are also shown at the 68.27 % C.L.. For details, see Ref. [313, 314]</figcaption></figure>

SM-like limit $\kappa_V = 1$, all the scaling factors $\kappa_f$ become unity. In Fig. 69 (left), the scale factors $\kappa_f$ in the 2HDM with the softly-broken symmetry are shown on the $\kappa_f$-$\kappa_d$ plane for various values of $\tan\beta$ and $\kappa_V$ ($= \sin(\beta - \alpha)$). The points and the dashed curves denote changes of $\tan\beta$ by steps of one. $\kappa_V$ ($= \kappa_W = \kappa_Z$) is taken as $\kappa_V^2 = 0.99, 0.95$ and $0.90$. The current LHC constraints as well as the expected LHC and ILC sensitivities for $\kappa_d$ and $\kappa_\ell$ are also shown at the 68.27 % Confidence Level (C.L.). For the current LHC constraints (LHC30), we take the numbers from the universal fit in Eq. (18) of Ref. [334]. For the future LHC sensitivities (LHC300 and LHC3000), the expectation numbers are taken from the Scenario 1 in Tab. 1 of Ref. [335]. The central values and the correlations are assumed to be the same as in LHC30. The ILC sensitivities are taken from Table. 2.6 in Ref. [333]. The same central value without correlation is assumed for the ILC sensitivity curves. For more details see Refs. [313], and for some revisions see Ref. [314].

The analysis including radiative corrections has been done recently [268]. We show the one-loop results for the Yukawa couplings in the planes of fermion scale factors. In
<figure><img src="image_72.png" /></figure>

<figure><img src="image_73.png" /><figcaption>**Fig. 70** Predictions of various scale factors on the $\kappa_{\tau}$ vs $\kappa_b$ (upper panel), and $\kappa_{\tau}$ vs $\kappa_c$ (bottom panel) in four types of Yukawa interactions in the cases with $\cos(\beta - \alpha) < 0$ [268]. Each black dot shows the tree level result with $\tan\beta=1, 2, 3$ and $4$. One-loop corrected results are indicated by red for $\sin^2(\beta - \alpha) = 0.99$ and blue for $\sin^2(\beta - \alpha) = 0.95$ regions where $m_{\Phi}$ and $M$ are scanned over from 100 GeV to 1 TeV and 0 to $m_{\Phi}$, respectively. All the plots are allowed by the unitarity and vacuum stability bounds.</figcaption></figure>

Fig. 70, predictions of various scale factors are shown on the $\kappa_{\tau}$ vs $\kappa_b$ (upper panels), and $\kappa_{\tau}$ vs $\kappa_c$ (bottom panels) planes. When we consider the case with $\sin(\beta - \alpha) \neq 1$, the sign dependence of $\cos(\beta - \alpha)$ to $\kappa_f$ is also important. We here show the both cases with $\cos(\beta - \alpha) < 0$. The value of $\tan\beta$ is discretely taken as $\tan\beta=1, 2, 3$ and $4$. The tree level predictions are indicated by the black dots, while the one-loop corrected results are shown by the red for $\sin^2(\beta - \alpha) = 0.99$ and blue for $\sin^2(\beta - \alpha) = 0.95$ regions where the values of $m_{\Phi}$ and $M$ are scanned over from 100 GeV to 1 TeV and 0 to $m_{\Phi}$, respectively. All the plots are allowed by the unitarity and vacuum stability bounds.

Even when we take into account the one-loop corrections to the Yukawa couplings, this behavior; i.e., predictions are well separated among the four types of THDMs, does not so change as we see the red and blue colored re-

<figure><img src="image_74.png" /><figcaption>**Fig. 71** Contour plots of the deviation in the $hhh$ coupling in the $(m_{\Phi}, M)$ plane for $m_h = 125$ GeV and $\sin(\beta - \alpha) = 1$. The red line indicates $\phi_c/T_c = 1$, above which the strong first order phase transition occurs ($\phi_c/T_c > 1$) [337].</figcaption></figure>

gions. Therefore, we conclude that all the 2HDMs can be distinguished from each other by measuring the charm, bottom and tau Yukawa couplings precisely when the gauge couplings $hVV$ are deviated from the SM prediction with $\mathcal{O}(1)\%$<fnref n="17" />.

The Higgs boson couplings $h\gamma\gamma$ and $hgg$ are absent at the tree level but are produced at the one-loop level via the higher dimensional operators

$$ \frac{1}{M^2} |\Phi|^2 F^{\mu\nu} F_{\mu\nu}, \quad \frac{1}{M^2} |\Phi|^2 G^{(a)}_{\mu\nu} G^{(a)}_{\mu\nu}, \qquad (46) $$

where $F^{\mu\nu}$ and $G^{(a)}_{\mu\nu}$ are the field strength tensors of $U(1)_{EM}$ and $SU(3)_C$, and $M$ is a dimensionful parameter. In the 2HDM, the coupling can deviate from the SM due to the mixing effect of neutral scalar bosons and, for $h\gamma\gamma$, also due to the loop contributions of additional Higgs bosons $H$, $A$ and $H^{\pm}$. The latter effect can be significant even in the SM-like limit where $\sin(\beta - \alpha) = 1$ as long as $M$ is not too large. At the LHC (300 fb⁻¹), the HL-LHC (3000 fb⁻¹), and the ILC (1 TeV-up) [245,313], $\kappa_{\gamma}$ is expected to be measured with 5-7 %, 2-5 % and 2.4 %, respectively. If deviations in $\kappa_{\gamma}$ and $\kappa_g$ are detected in future precision measurements at the LHC and the ILC, we can directly extract information of new particles in the loop such as their mass scales.

The triple Higgs boson coupling $hhh$ is essentially important to be measured to obtain the information of the Higgs potential. The tree-level behavior of the $hhh$ coupling constant has been discussed in the 2HDM in Ref. [338]. The deviation from the SM predictions are sensitive to the mix-

<footnote n="17">¹⁷We here give a comment on the radiative correction to the $hVV$ couplings in the THDMs. Although the tree level deviations in the $hVV$ couplings are described by the factor $\sin(\beta - \alpha)$, these values can be modified at the one-loop level. In Ref. [336], the one-loop corrected $hZZ$ vertex has been calculated in the softly-broken $Z_2$ symmetric 2HDM. It has been found that for the fixed value of $\sin(\beta - \alpha)$, the one-loop corrections to the $hZZ$ vertex are less than a few %.</footnote>
ing parameters tan α and sin( β − α). In the SM-like limit sin( β − α) = 1, the value of the hhh coupling coincide with that in the SM. At the one-loop level, even when the SM-like limit, the hhh coupling can deviate from the SM prediction due to the quantum loop effects of H , A and H ± [336, 339]. For the SM-like limit sin( β − α) = 1, the one-loop corrected effective hhh coupling in the 2HDM can be expressed as

$$
\lambda_{hhh}^{eff} = \frac{3m_h^2}{v} \left\{ 1 + \frac{m_H^4}{12\pi^2 m_h^2 v^2} \left(1 - \frac{M^2}{m_H^2}\right)^3 \right. \\
\left. \qquad + \frac{m_A^4}{12\pi^2 m_h^2 v^2} \left(1 - \frac{M^2}{m_A^2}\right)^3 + \frac{m_{H^\pm}^4}{6\pi^2 m_h^2 v^2} \left(1 - \frac{M^2}{m_{H^\pm}^2}\right)^3 \right. \\
\left. \qquad - \frac{N_{c_i} m_t^4}{3\pi^2 m_h^2 v^2} + \mathcal{O} \left( \frac{p_i^2 m_\Phi^2}{m_h^2 v^2}, \frac{m_\Phi^2}{v^2}, \frac{p_i^2 m_t^2}{m_h^2 v^2}, \frac{m_t^2}{v^2} \right) \right\}, \quad (47)
$$

where m$_{Φ}$ and p$_{i}$ represent the mass of H , A or H ± and the momenta of external Higgs lines, respectively. The de- viation from the SM prediction can be O (100) % under the constraint from perturbative unitarity and vacuum stability as well as the current LHC results , in the non-decoupling case v 2 ∼ M $^{2}$. For M 2 ≫ v $^{2}$, such a large quantum effect de- couples in the hhh coupling because of the decoupling theo- rem.

It is well known that such a large non-decoupling loop effect on the triple Higgs boson coupling is related to the strong first-order phase transition of the electroweak gauge symmetry [340], which is required for successful electroweak baryogenesis [341, 342] $^{18}$. In the scenario of electroweak baryogenesis, one of the Sakharov’s conditions of the depar- ture from thermal equilibrium is satisfied when φ$_{c}$/T$_{c}$ > 1, where T$_{c}$ is the critical temperature and φ$_{c}$ is the order pa- rameter at T$_{c}$ . With the mass of the discovered Higgs bo- son to be 125 GeV, the SM cannot satisfy this condition. On the other hand, in the extended Higgs sector, the condition φ$_{c}$/T$_{c}$ > 1 can be satisfied without contradicting the current data. In Fig. 71, the correlation between the large deviation in the hhh coupling and the first order phase transition is shown [337, 340]. This results show that we may be able to test the scenario of electroweak baryogenesis by measuring the hhh coupling by the 13% accuracy [313]. Such a preci- sion measurement can be achieved at the ILC.

2.6.3 Higgs Triplet models

We here discuss the Higgs boson properties in the mini- mal Higgs triplet model (HTM). A motivation to study this model is that tiny neutrino masses can be explained via the so-called type-II seesaw mechanism [345]. The Higgs sec- tor of the HTM is composed of one isospin doublet field Φ with hypercharge Y = 1 and the triplet field Δ with Y = 2.

The Higgs fields can be parameterized by

$$
\Phi = \begin{bmatrix} \phi^+ \\ \frac{1}{\sqrt{2}}(\phi + v_{\phi} + i\chi) \end{bmatrix}, \quad \Delta = \begin{bmatrix} \frac{\Delta^+}{\sqrt{2}} & \Delta^{++} \\ \Delta^0 & -\frac{\Delta^+}{\sqrt{2}} \end{bmatrix} \\
\text{with } \Delta^0 = \frac{1}{\sqrt{2}}(\delta + v_{\Delta} + i\eta), \tag{48}
$$

where v$_{φ}$ and v$_{Δ}$ are the VEVs of the neutral components of doublet Higgs field φ 0 and the triplet Higgs field δ $^{0}$, re- spectively, which satisfy v 2 ≡ v 2 φ + 2 v 2 Δ ≃ (246 GeV) $^{2}$. The masses of the W boson and the Z boson are obtained at the tree level as

$$
m_W^2 = \frac{g^2}{4} (v_{\phi}^2 + 2v_{\Delta}^2), \quad m_Z^2 = \frac{g^2}{4 \cos^2 \theta_W} (v_{\phi}^2 + 4v_{\Delta}^2). \quad (49)
$$

One of the striking features of the HTM is the prediction that the electroweak ρ - parameter ρ deviates from unity at the tree level due to the non-zero VEV of the triplet field v$_{Δ}$ . From Eq. (32), we obtain

$$
\rho \equiv \frac{m_W^2}{m_Z^2 \cos^2 \theta_W} = \frac{1 + \frac{2v_{\phi}^2}{v_{\phi}^2}}{1 + \frac{4v_{\Delta}^2}{v_{\phi}^2}}. \quad (50)
$$

The experimental value of the ρ -parameter is quite close to unity, so that v$_{Δ}$ has to be less than about 8 GeV from the tree level formula given in Eq. (50).

The Yukawa interaction for neutrinos [345] is given by

$$
\mathcal{L}_Y = h_{ij} \overline{L_L^{ic}} i \tau_2 \Delta L_L^j + \text{h.c.}, \quad (51)
$$

where h$_{ij}$ is the 3 × 3 complex symmetric Yukawa matrix. Notice that the triplet field Δ carries the lepton number of − 2. The mass matrix for the left-handed neutrinos is ob- tained as

$$
(\mathcal{M}_V)_{ij} = \sqrt{2} h_{ij} v_{\Delta}. \qquad (52)
$$

Current neutrino oscillation data can be explained in the HTM [346–348]. It is seen from Eq. (52) that the neutrino mixing pattern is simply determined by the h$_{ij}$ matrix. Since the decay rate of H ±± into the same-sign dilepton is propor- tional to | h$_{ij}$ | $^{2}$, the type-II seesaw scenario can be tested by looking at the same-sign dilepton decay mode of H ±± [346– 348].

The Higgs potential of the HTM is given by

$$
\begin{align}
V(\Phi, \Delta) ={}& m^2 \Phi^\dagger \Phi + M^2 \operatorname{Tr}(\Delta^\dagger \Delta) + [\mu \Phi^T i \tau_2 \Delta^\dagger \Phi + \text{h.c.}] \nonumber \\
& + \lambda_1 (\Phi^\dagger \Phi)^2 + \lambda_2 [\operatorname{Tr}(\Delta^\dagger \Delta)]^2 + \lambda_3 \operatorname{Tr}[(\Delta^\dagger \Delta)^2] \nonumber \\
& + \lambda_4 (\Phi^\dagger \Phi) \operatorname{Tr}(\Delta^\dagger \Delta) + \lambda_5 \Phi^\dagger \Delta \Delta^\dagger \Phi, \tag{53}
\end{align}
$$

where m and M are the dimension full real parameters, µ is the dimension full complex parameter which violates the lepton number, and λ$_{1}$ - λ$_{5}$ are the coupling constants which are real. We here take µ to be real.

<footnote n="18">18 See also Ref. [343].</footnote>
<figure><img src="image_75.png" /><figcaption>Fig. 72 Constraints from the unitarity and vacuum stability bounds for $\lambda_1 = m_h^2/(2v^2) \simeq 0.13$ in the $\lambda_4-\lambda_5$ plane. We take $\lambda_\Delta = 1.5$ for the left panel and $\lambda_\Delta = 3$ for the right panel with $\lambda_\Delta = \lambda_2 = \lambda_3$ [344].</figcaption></figure>

<figure><img src="image_76.png" /><figcaption>Fig. 73 Decay branching ratio of $H^{++}$ as a function of $v_\Delta$. In the left figure, $m_{H^{++}}$ is fixed to be 300 GeV, and $\Delta m$ is taken to be zero. In the middle figure, $m_{H^{++}}$ is fixed to be 320 GeV, and $\Delta m$ is taken to be 10 GeV. In the right figure, $m_{H^{++}}$ is fixed to be 360 GeV, and $\Delta m$ is taken to be 30 GeV.</figcaption></figure>

The potential respects additional global symmetries in some limits. First, there is the global $U(1)$ symmetry in the potential in the limit of $\mu = 0$, which conserves the lepton number. As long as we assume that the lepton number is not spontaneously broken, the triplet field does not carry the VEV; i.e., $v_\Delta = 0$. Next, an additional global $SU(2)$ symmetry appears in the limit where $\mu = \lambda_5 = 0$. Under this $SU(2)$ symmetry, $\Phi$ and $\Delta$ can be transformed with the different $SU(2)$ phases. All the physical triplet-like Higgs bosons are then degenerate in mass.

The mass matrices for the scalar bosons can be diagonalized by rotating the scalar fields as

$$ 
\begin{pmatrix} \phi^{\pm} \\ \Delta^{\pm} \end{pmatrix} = \begin{pmatrix} \cos \beta & -\sin \beta \\ \sin \beta & \cos \beta \end{pmatrix} \begin{pmatrix} G^{\pm} \\ H^{\pm} \end{pmatrix}, \\
\begin{pmatrix} \chi \\ \eta \end{pmatrix} = \begin{pmatrix} \cos \beta' & -\sin \beta' \\ \sin \beta' & \cos \beta' \end{pmatrix} \begin{pmatrix} G^0 \\ A \end{pmatrix}, \\
\begin{pmatrix} \phi \\ \delta \end{pmatrix} = \begin{pmatrix} \cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{pmatrix} \begin{pmatrix} h \\ H \end{pmatrix}, \qquad (54)
$$

with the mixing angles

$$ 
\begin{gathered}
\tan \beta = \frac{\sqrt{2}v_{\Delta}}{v_{\phi}}, \quad \tan \beta' = \frac{2v_{\Delta}}{v_{\phi}}, \\
\tan 2\alpha = \frac{v_{\Delta}}{v_{\phi}} \frac{2v_{\phi}^2(\lambda_4 + \lambda_5) - 4M_{\Delta}^2}{2v_{\phi}^2\lambda_1 - M_{\Delta}^2 - 2v_{\Delta}^2(\lambda_2 + \lambda_3)}.
\end{gathered}
\qquad (55)
$$

In addition to the three Nambu-Goldstone bosons $G^\pm$ and $G^0$ which are absorbed by the longitudinal components of the W boson and the Z boson, there are seven physical mass eigenstates; i.e., a pair of doubly-charged (singly-charged) Higgs bosons $H^{\pm\pm}(H^\pm)$, a CP-odd Higgs boson A and CP-even Higgs boson H and h, where h is taken as the SM-like Higgs boson. The six parameters $\mu$ and $\lambda_1-\lambda_5$ in the Higgs potential in Eq. (53) can be written in terms of the physical scalar masses, the mixing angle $\alpha$ and VEVs $v_\phi$ and $v_\Delta$.

As required by the $\rho$-parameter data, when the triplet VEV $v_\Delta$ is much less than the doublet VEV $v_\phi$, there is relationships among the masses of the triplet-like Higgs bosons by neglecting $\mathcal{O}(v_\Delta^2/v_\phi^2)$ terms as

$$ m_{H^{++}}^2 - m_{H^+}^2 = m_{H^+}^2 - m_A^2 \left( = -\frac{\lambda_5}{4} v^2 \right), \qquad (56) $$

$$ m_A^2 = m_H^2 (= M_\Delta^2). \qquad (57) $$
In the limit of $v_Δ/v_φ \to 0$, the four mass parameters of the triplet-like Higgs bosons are determined by two parameters. Eqs. (56) and (57) can be regarded as the consequence of the global symmetries mentioned above.

The condition for the vacuum stability bound has been derived in Ref. [349], where we require that the Higgs potential is bounded from below in any direction of the large scalar fields region. The unitarity bound in the HTM has been discussed in Ref. [349]. In Fig. 72, the excluded regions by the unitarity bound and the vacuum stability condition are shown for $\lambda_1 = m_H^2/(2v^2) \simeq 0.13$ in the $\lambda_4$-$\lambda_5$ plane [344]. We take $\lambda_Δ = 1.5$ (3) in the left (right) panel. Excluded regions by the unitarity and vacuum stability bounds are shown.

The most interesting feature of the HTM is the existence of doubly charged Higgs bosons $H^{\pm\pm}$. Their discovery at colliders can be a direct probe of the exotic Higgs sectors. The doubly charged Higgs bosons $H^{\pm\pm}$ can decay into $\ell^\pm\ell^\pm$, $H^\pm W^\pm$ and $W^\pm W^\pm$ depending on the magnitude of $v_Δ$ [350]. In Fig. 73, the branching ratios are shown as a function of the vacuum expectation value of the triplet field, $v_Δ$, for the cases with the mass difference $\Delta m = m_{H^{++}} - m_{H^+} = 0, 10$ GeV and $30$ GeV [351]. The decay branching ratio of $H^{\pm\pm}$ is shown in Fig. 74 assuming all the elements in $(M_V)_{ij}$ to be $0.1$ eV. The dominant decay mode changes from the same-sign dilepton mode to the same-sign diboson mode at $v_Δ = 0.1-1$ MeV.

When the triplet-like Higgs bosons are degenerate in mass or $H^{\pm\pm}$ is the lightest of all of them, the main decay mode of $H^{\pm\pm}$ is the same-sign dilepton (diboson) in the case where $v_Δ$ is less (larger) than about 1 MeV. The signal directly shows the existence of the doubly charged scalar boson with lepton number 2, which can be a strong evidence for the neutrino mass generation via Eq. (51). At the LHC, $H^{\pm\pm}$ are produced by the Drell-Yan process $pp \to Z^*/\gamma^* \to H^{++}H^{--}$ and the associated process $pp \to W^* \to H^{\pm\pm}H^{\mp}$. The search for $H^{\pm\pm}$ in the dilepton decay scenario has been performed at the LHC. The scenario based on the same-sign dilepton decay of $H^{\pm\pm}$ has been studied in Refs. [346–348]. The strongest lower limit on $m_{H^{++}}$ has been given by 459 GeV [352] at the 95% CL assuming the 100% decay of $H^{\pm\pm} \to \mu^+\mu^+$ from the 7 TeV and $4.9$ fb$^{-1}$ data. This bound becomes weaker as 395 GeV [352] when we only use the pair production process. However, when $H^{\pm\pm}$ mainly decay into the same-sign diboson, this bound can no longer be applied.

When $v_Δ$ is sufficiently larger than $10^{-3}$ GeV, the diboson decay $H^{\pm\pm} \to W^\pm W^\pm$ becomes dominant. In this case, the signal can also be same sign four leptons, but its rate is reduced by the branching ratios of leptonic decays of Ws. The scenario for the same sign diboson decay of $H^{\pm\pm}$ has been discussed in Refs. [347, 354]. The discovery potential of $H^{\pm\pm}$ at the LHC has also been investigated in Ref. [354]

<figure><img src="image_77.png" /><figcaption>Fig. 74 Decay branching ratio of $H^{++}$ as a function of $v_Δ$ with $m_{H^+} = m_{H^{++}}$. The solid, dashed and dotted curves respectively show the results in the case of $m_{H^{++}} = 150, 300$ and $500$ GeV [344].</figcaption></figure>

<figure><img src="image_78.png" /><figcaption>Fig. 75 The signal cross section as a function of $m_{H^{++}}$ with the collision energy to be 7 TeV from Ref. [353]. The light (dark) shaded band shows the 95% CL (expected) upper bound for the cross section from the data with the integrate luminosity to be $4.7$ fb$^{-1}$ ($20$ fb$^{-1}$).</figcaption></figure>

in the HTM and also the Georgi-Machacek model [253]. In Ref. [353], the lower bound on $m_{H^{++}}$ has been obtained by using the same-sign dilepton event measured at the LHC with 7 TeV and $4.7$ fb$^{-1}$ data [355]. In Fig. 75, the sum of the cross sections of the processes

$$ \begin{aligned} & pp \to H^{++}H^{--} \to W^{+(*)}W^{+(*)}H^{--} \to \mu^{+}\mu^{+}E_{\text{miss}}H^{--}, \\ & pp \to H^{++}H^{-} \to W^{+(*)}W^{+(*)}H^{-} \to \mu^{+}\mu^{+}E_{\text{miss}}H^{-}, \end{aligned} \qquad (58) $$

are shown as a function of $m_{H^{++}}$ assuming $m_{H^+} = m_{H^{++}}$. We can see that $m_{H^{++}}$ smaller than about 60 GeV is excluded at the 95% CL. The bound is much relaxed as compared to that in the dilepton decay scenario. By the extrapolation of the data to $20$ fb$^{-1}$ with the same collision energy, the lower limit is obtained as 85 GeV. Therefore, a light $H^{\pm\pm}$ such as around 100 GeV is still allowed by the current data at the LHC, and in this case the ILC may be able to discover
<figure><img src="image_79.png" /><figcaption>**Fig. 76** Production cross section of the $e^{+}e^{-} \to H^{++}H^{--}$ process as a function of $m_{H^{++}}$. The black, blue and red curves are respectively the results with the collision energy $\sqrt{s}=250, 500$ and $1000$ GeV.</figcaption></figure>

the doubly charged Higgs boson. See also recent progress in Ref. [356].

At the ILC, doubly charged Higgs bosons are produced via the pair production $e^{+}e^{-} \to H^{++}H^{--}$. In the diboson decay scenario, the final state is the same-sign dilepton, missing energy and multi-jets; i.e., $e^{+}e^{-} \to H^{++}H^{--} \to \ell^{+}\ell^{+}E_{\text{miss}}jjjj$, where $\ell = e, \mu$ [344]. The background comes from the four W bosons production; i.e., $e^{+}e^{-} \to W^{+}W^{+}W^{-}W^{-} \to \ell^{\pm}\ell^{\pm}E_{\text{miss}}jjjj$. For example, when $\sqrt{s} = 500$ GeV and the $m_{H^{++}} = 230$ GeV is taken, the signal (background) cross section of the final state $\ell^{\pm}\ell^{\pm}E_{\text{miss}}4j$ is obtained to be 1.07 fb ($2.37 \times 10^{-3}$ fb) (Fig. 76) [344]. The above numbers are obtained after taking the following basic kinematic cuts

$$p_T^\ell \ge 15 \text{ GeV}, \quad |\eta^\ell| \le 2.5, \qquad (59)$$

where $p_T^\ell$ and $\eta^\ell$ are the transverse momentum and pseudo rapidity for $\ell$, respectively. Therefore, this process is almost background free. In Fig. 77, the invariant mass $M_{\ell^+\ell^+}$ for the $\ell^+\ell^+$ system (left panel) and the transverse mass $M_T$ (right panel) distributions for $\ell^+\ell^+E_{\text{miss}}$ system are shown. The red and black curves denote the distribution from the signal and background, respectively. Around 230 GeV, there is an endpoint in the $M_T$ distribution that corresponds to $m_{H^{++}}$. The $M_T$ distribution is useful to measure $m_{H^{++}}$.

If the triplet-like Higgs bosons are light enough, the direct detection of them at the LHC and the ILC is the most important probe of the HTM as already discussed. On the other hand, they can also be indirectly tested by measuring the deviations from the SM in the Higgs boson couplings for the SM-like Higgs boson $h$, such as the coupling constants with the weak gauge bosons $hVV$, the Yukawa couplings $h\bar{f}\bar{f}$ and the triple Higgs boson coupling $hhh$, where $V$ represents gauge bosons, and $f$ does quarks and leptons.

The indirect searches can be useful even when no new particles is directly found. At the ILC, the Higgs boson couplings are expected to be precisely measured. For example, the Higgs boson couplings with the weak gauge bosons ($hZZ$ and $hWW$) and the Yukawa couplings ($hb\bar{b}, h\tau\bar{\tau}$ and $ht\bar{t}$) are expected to be measured with $\mathcal{O}(1)\%$ accuracy [245, 313, 333, 357]. In the HTM, the loop induced $h\gamma\gamma$ coupling has been calculated in Refs. [358]. The one-loop corrections to the $hWW$, $hZZ$ and $hhh$ vertices have also been calculated in Refs. [359, 360]. In Ref. [360], it has been found that there is a correlation among the deviation in the Higgs boson couplings. For example, when the decay rate of $h \to \gamma\gamma$ deviates by 30% (40%) from the SM prediction, deviations in the one-loop corrected $hVV$ and $hhh$ vertices are predicted to be about $-0.1\%$ ($-2\%$) and $-10\%$ (150%), respectively<fnref n="19" />. By comparing these deviations with the precisely measured value at the ILC, we can discriminate the HTM from the other models.

### 2.6.4 Other exotic models

Precision measurements for the couplings of the SM-like Higgs boson $h$ at the ILC can also discriminate exotic Higgs sectors. According to Refs. [313, 314], we here consider various extended Higgs sectors which satisfy $\rho = 1$ at the tree level; i.e., the model with an additional singlet scalar field with $Y = 0$, the 2HDM (Type I), the model with a septet scalar field with $Y = 4$ [361], and the Georgi Machacek model where a complex ($Y = 2$) and a real ($Y = 0$) triplet scalar fields are added to the SM-like Higgs doublet [253]. In these models, all quark and leptons receive their masses from only one scalar doublet. Consequently, the Yukawa coupling constants with respect to the SM-like Higgs boson $h\bar{f}\bar{f}$ from the SM values are commonly suppressed due to the mixing between the two (or more) neutral states. In a, we have a universal suppression on the coupling constants, $\kappa_F = \kappa_V = \cos\theta$ with $\theta$ being the mixing angle between the doublet field and the singlet field. However, $\kappa_F \neq \kappa_V$ is usually predicted in more complicated Higgs sectors such as the 2HDM (Type I), the Georgi-Machacek model [253] and the doublet-septet model [361]. Notice that in exotic models with higher representation scalar fields such as the Georgi-Machacek model and doublet-septet model, $\kappa_V$ can be greater than 1. This can be a signature of exotic Higgs sectors. From Eq. (32), a VEV from these additional scalar multiplets do not change $\rho = 1$ at the tree level. All the VEVs $v_{\text{ext}}$ of these additional Higgs multiplets except for that of the singlet partially contribute to the spontaneous breaking of the electroweak gauge symmetry. The VEVs satisfy $v^2 = v_0^2 + (\eta_{\text{ext}} v_{\text{ext}})^2$, where $v_\phi$ is the VEV of the SM-like Higgs doublet $\Phi$ and $\eta_{\text{ext}} = 1$ and 4 in the Type-I THDM

<footnote n="19">¹⁹ In the HTM, deviations in $h\bar{f}\bar{f}$ couplings are small because of $v_\Delta \ll v_\phi$.</footnote>
<figure><img src="image_80.png" /><figcaption>**Fig. 77** The invariant mass distribution (left panel) and the transverse mass distribution (right panel) for the $\ell^+\ell^+$ and $\ell^+\ell^+ E_{\text{miss}}$ systems, respectively, in the case of $m_{H^{++}} = 230$ GeV and $\sqrt{s} = 500$ GeV [344]. The integrated luminosity is assumed to be 500 fb$^{-1}$.</figcaption></figure>

<table><caption>**Table 22** The fraction of the VEVs $\tan \beta$ and the scaling factors $\kappa_f$ and $\kappa_V$ in the extended Higgs sectors with universal Yukawa couplings [314].</caption><thead><tr><th></th><th>tan β</th><th>κ<sub>f</sub></th><th>κ<sub>V</sub></th></tr></thead><tbody><tr><td>Doublet-Singlet Model</td><td>—</td><td>cos α</td><td>cos α</td></tr><tr><td>Type-I THDM</td><td>v<sub>0</sub>/v<sub>ext</sub></td><td>cos α / sin β = sin(β - α) + cot β cos(β - α)</td><td>sin(β - α)</td></tr><tr><td>GM Model</td><td>v<sub>0</sub>/(2√2v<sub>ext</sub>)</td><td>cos α / sin β</td><td>sin β cos α - 2√6/3 cos β sin α</td></tr><tr><td>Doublet-Septet Model</td><td>v<sub>0</sub>/(4v<sub>ext</sub>)</td><td>cos α / sin β</td><td>sin β cos α - 4 cos β sin α</td></tr></tbody></table>

and the model with the septet, respectively. It is convenient to define the ratio of the VEVs as $\tan \beta = v_0/(\eta_{\text{ext}} v_{\text{ext}})$ [314]. In tab. 22, the scaling factors $\kappa_f$ and $\kappa_V$ are listed in terms of $\alpha$ and $\beta$ in the four models.

In Fig. 78, the predictions for the scale factors of the universal Yukawa coupling $\kappa_F$ and the gauge coupling $\kappa_V$ are plotted in exotic Higgs sectors for each set of mixing angles. The current LHC bounds, expected LHC and ILC sensitivities for $\kappa_F$ and $\kappa_V$ are also shown at the 68.27 % C.L.. Therefore, exotic Higgs sectors can be discriminated by measuring $\kappa_V$ and $\kappa_F$ precisely. For details, see Refs. [313, 314].

### 2.6.5 Summary

Although the Higgs boson with the mass 125 GeV was found at the LHC, knowledge about the structure of the Higgs sector is very limited. Since there are no theoretical principle for the minimal Higgs sector with one Higgs doublet, there are many possibilities of non-minimal Higgs sectors. Such extended Higgs sectors appear in many new physics models beyond the SM. Therefore, the Higgs sector is a window to new physics, and we can explore new physics from clarifying the structure of the Higgs sector by coming collider experiments. At the LHC, direct discovery of additional Higgs bosons can be expected as long as they are not too heavy. On the other hand, the Higgs sector can also be explored by precisely measuring the properties of the discovered Higgs boson $h$ accurately. The precision measurements will be performed partially at the high luminosity LHC with 3000 fb$^{-1}$. Using high ability of the ILC for measuring the Higgs boson couplings, we can further test extended Higgs sectors, and consequently narrow down the new physics models.

<figure><img src="image_81.png" /><figcaption>**Fig. 78** The scaling factors in models with universal Yukawa couplings. The current LHC bounds and the expected LHC and ILC sensitivities are also shown at the 68.27 % C.L.. For details, see Ref. [314]</figcaption></figure>
## 2.7 Higgs physics in strong-interaction scenarios<fnref n="20" />

The Higgs mechanism [1–4], which has been introduced to provide masses for the fermions and gauge bosons without violating gauge principles, can describe electroweak symmetry breaking (EWSB) but fails to explain it. Within the Standard Model (SM) there is no dynamics leading to the typical mexican hat shape of the Higgs potential. Moreover, in order to keep the Higgs boson mass at the experimentally measured value of 125 GeV [55, 87] in the presence of high scales at which the SM will eventually has to be amended, a substantial amount of finetuning is necessary unless the mass is protected from higher order corrections due to some symmetry. Such a symmetry must act non-linearly on the Higgs field. Besides supersymmetry a prominent example is given by a global symmetry when the Higgs boson appears as a pseudo Nambu–Goldstone boson. A Higgs boson is needed to ensure the proper decoupling of the longitudinal polarizations of the massive EW gauge bosons at high energy. Indeed, these longitudinal modes of $W^±$ and $Z$ can be described by Nambu–Goldstone bosons associated to the coset $SU(2)_L \times SU(2)_R/SU(2)_{\text{isospin}}$. Their kinetic term corresponds to the gauge boson mass terms,

$$ \frac{1}{2} m_Z^2 Z_\mu Z^\mu + m_W^2 W_\mu^\dagger W^{-\mu} = \frac{v^2}{4} \text{Tr}(D_\mu \Sigma^\dagger D^\mu \Sigma) \quad (60) $$

with $\Sigma = e^{i\sigma^a\pi^a/v}$, where $\sigma^a$ ($a=1,2,3$) are the usual Pauli matrices. Due to the Goldstone boson equivalence theorem the non-trivial scattering of the longitudinal gauge bosons $V$ ($V = W^±, Z$) is controlled by the contact interactions among four pions from the expansion of the Lagrangian Eq. (60), leading to amplitudes growing with the energy,

$$ \mathcal{A}(V_L^a V_L^b \rightarrow V_L^c V_L^d) = \mathcal{A}(s)\delta^{ab}\delta^{cd} + \mathcal{A}(t)\delta^{ac}\delta^{bd} \\ + \mathcal{A}(u)\delta^{ad}\delta^{bc} \quad \text{with} \quad \mathcal{A}(s) \approx \frac{s}{v^2}. \quad (61) $$

Here $s, t, u$ denote the Mandelstam variables, and $v$ represents the vacuum expectation value (VEV) with $v \approx 246$ GeV. The amplitude grows with the center-of-mass (c.m.) energy squared $s$, and therefore perturbative unitarity will be lost around $4\pi v \sim 3$ TeV, unless there is a new weakly coupled elementary degree of freedom. The simplest realization of new dynamics restoring perturbative unitarity is given by a single scalar field $h$, which is singlet under $SU(2)_L \times SU(2)_R/SU(2)_{\text{isospin}}$ and couples to the longitudinal gauge bosons and fermions as [362–364],

$$ \begin{aligned} \mathcal{L}_{\text{EWSB}} = & \frac{1}{2}(\partial_{\mu}h)^2 - V(h) + \\ & \frac{v^2}{4} \text{Tr}(D_{\mu}\Sigma^{\dagger}D^{\mu}\Sigma) \left(1 + 2a\frac{h}{v} + b\frac{h^2}{v^2} + \sum_{n \ge 3} b_n \frac{h^n}{v^n} + \dots\right) \\ & - \frac{v}{\sqrt{2}}(\bar{u}_L^i d_L^i)\Sigma \left(1 + c\frac{h}{v} + \sum_{n \ge 2} c_n \frac{h^n}{v^n} + \dots\right) \left(\begin{matrix} y_{ij}^\mu u_R^j \\ y_{ij}^d d_R^j \end{matrix}\right) \\ & + \text{h.c.} \end{aligned} \quad (62) $$

with

$$ V(h) = \frac{1}{2} m_h^2 h^2 + \frac{d_3}{6} \left( \frac{3m_h^2}{v} \right) h^3 + \frac{d_4}{24} \left( \frac{3m_h^2}{v^2} \right) h^4 + \dots \quad (63) $$

For $a=1$ the scalar exchange cancels the piece growing with the energy in the $V_L V_L$ amplitude. If in addition $b=a^2$ then also in the inelastic amplitude $V_L V_L \rightarrow hh$ unitarity is maintained, while for $ac=1$ the $V_L V_L \rightarrow f\bar{f}^\dagger$ amplitude remains finite. The SM Higgs boson is defined by the point $a=b=c=1$ and $d_3=d_4=1$, $c_{n \ge 2}=b_{n \ge 3}=0$. The scalar resonance and the pions then combine to from a doublet which transforms linearly under $SU(2)_L \times SU(2)_R$. The Lagrangian Eq. (62) describes either an elementary or a composite Higgs boson. For $a \ne 1$ the Higgs boson alone cannot fully unitarize the $V_L V_L$ scattering, with the breakdown of perturbative unitarity pushed to a higher scale now, which is of the order $4\pi v/\sqrt{1-a^2}$. The residual growth of the scattering amplitude $\mathcal{A}(s) \approx (1-a^2)s/v^2$ will finally be cancelled by the exchange of other degrees of freedom. The Lagrangian Eqs. (62), (63) introduces deviations in the Higgs boson phenomenology [362, 365] away from the SM point by rescaling all Higgs couplings through the modifiers $a, b$ and $c$,

$$ g_{hVV} = a g_{hVV}^{\text{SM}}, \quad g_{hhVV} = b g_{hhVV}^{\text{SM}}, \quad g_{hf\bar{f}^\dagger} = c g_{hf\bar{f}^\dagger}^{\text{SM}}, \quad (64) $$

while keeping the same Lorentz structure. With $c$ being flavor-universal, minimal flavor violation is built in and the only source of flavor violation are the usual SM Yukawa couplings. There are additional new couplings as, e.g., the $c_2$ coupling between two Higgs bosons and two fermions, which contributes to multi-Higgs production [362–364].

In composite Higgs models, the deviations from the SM point $a=b=1$ are controlled by the ratio of the weak scale over the compositeness scale $f$. In these models the Higgs boson is a composite bound state which emerges from a strongly interacting sector [366–371]. The good agreement with the electroweak precision data is achieved by a mass gap that separates the Higgs scalar from the other resonances of the strong sector. This mass gap arises dynamically in a natural way if the strongly interacting sector has a global symmetry $G$, which is spontaneously broken at a scale $f$ to a subgroup $H$ so that the coset $G/H$ contains a fourth

<footnote n="20">²⁰Christoph Grojean, M. Margarete Mühlleitner</footnote>
Nambu-Goldstone boson which is identified with the Higgs boson. Composite Higgs models can be viewed as a continuous interpolation between the SM and technicolor type models. With the compositeness scale of the Higgs boson given by the dynamical scale $f$, the limit $\xi \equiv v^2/f^2 \to 0$ corresponds to the SM where the Higgs boson appears as an elementary light particle and the other resonances of the strong sector decouple. In the limit $\xi \to 1$ the Higgs boson does not couple to the $V_L$'s any longer and other (heavy) resonances are necessary to ensure unitarity in the gauge boson scattering. The $\xi \to 1$ limit corresponds to the technicolor paradigm [83, 84] where the strong dynamics directly breaks the electroweak symmetry down to the electromagnetism subgroup.

### 2.7.1 Effective Lagrangian and Higgs couplings

Independently of its dynamical origin, the physics of a strongly interacting light Higgs (SILH) boson can be captured in a model-independent way by an effective Lagrangian which involves two classes of higher dimensional operators: (i) those being genuinely sensitive to the new strong force and which will qualitatively affect the Higgs boson phenomenology and (ii) those being sensitive only to the spectrum of the resonances and which will simply act as form factors. The size of the various operators is controlled by simple rules and the effective Lagrangian can be cast into the generic form [362]

$$ \mathcal{L}_{\text{SILH}} = \frac{c_H}{2f^2} (\partial_\mu |H|^2)^2 + \frac{c_T}{2f^2} (H^\dagger D^\mu H)^2 - \frac{c_6 \lambda}{f^2} |H|^6 \\ + \left( \frac{c_y y_f}{f^2} |H|^2 \bar{f}_L H f_R + \text{h.c.} \right) + \frac{i c_W g}{2 m_\rho^2} (H^\dagger \sigma^i D^\mu H) \times \\ (D^\nu W_{\mu\nu})^i + \frac{i c_B g'}{2 m_\rho^2} (H^\dagger D^\mu H) (\partial^\nu B_{\mu\nu}) + \dots \quad (65) $$

with the SM electroweak (EW) couplings $g, g'$, the SM Higgs quartic coupling $\lambda$ and the SM Yukawa coupling $y_f$ to the fermions $f_{LR}$. The coefficients in Eq. (65) are expected to be of order one unless protected by some symmetry. The SILH Lagrangian gives rise to oblique corrections at tree-level. The coefficient $c_T$ vanishes in case the strong sector is assumed to respect custodial symmetry. The form factor operators induce a contribution to the $\hat{S}$ parameter, $\hat{S} = (c_W + c_B)m_W^2/m_\rho^2$, where $m_\rho$ denotes the mass scale of the heavy strong sector resonances, which imposes a lower bound $m_\rho \ge 2.5$ TeV. Since the Higgs couplings to the SM vector bosons receive corrections of the order $v^2/f^2$ the cancellation between the Higgs and the gauge boson contributions taking place in the SM, is spoiled and the $\hat{S}$ and $\hat{T}$ parameters become logarithmically divergent [372] when all the low energy degrees of freedom are considered. This infrared (IR) contribution imposes an upper bound of $\xi \lesssim 0.1$ [373–376] which can be relaxed by a factor of 2

<table><caption>**Table 23** Higgs coupling values of the effective Lagrangian Eq. (62), in the SILH set-up Eq. (65) and in explicit SO(5)/SO(4) composite Higgs models built in warped 5D space-time, MHCM4 and MHCM5. From Ref. [86].</caption><thead><tr><th>Parameters</th><th>SILH</th><th>MCHM4</th><th>MCHM5</th></tr></thead><tbody><tr><td>a</td><td>1 - c<sub>H</sub>ξ/2</td><td>√1 - ξ</td><td>√1 - ξ</td></tr><tr><td>b</td><td>1 - 2c<sub>H</sub>ξ</td><td>1 - 2ξ</td><td>1 - 2ξ</td></tr><tr><td>b<sub>3</sub></td><td>-4/3 ξ</td><td>-4/3 ξ √1 - ξ</td><td>-4/3 ξ √1 - ξ</td></tr><tr><td>c</td><td>1 - (c<sub>H</sub>/2 + c<sub>y</sub>)ξ</td><td>√1 - ξ</td><td>1 - 2ξ / √1 - ξ</td></tr><tr><td>c<sub>2</sub></td><td>-(c<sub>H</sub> + 3c<sub>y</sub>)ξ/2</td><td>-ξ/2</td><td>-2ξ</td></tr><tr><td>d<sub>3</sub></td><td>1 + (c<sub>6</sub> - 3c<sub>H</sub>/2)ξ</td><td>√1 - ξ</td><td>1 - 2ξ / √1 - ξ</td></tr><tr><td>d<sub>4</sub></td><td>1 + (6c<sub>6</sub> - 25c<sub>H</sub>/3)ξ</td><td>1 - 7ξ/3</td><td>1 - 28ξ(1 - ξ)/3 / (1 - ξ)</td></tr></tbody></table>

if a partial cancellation of $\mathcal{O}(50\%)$ with contributions from other states is allowed. Light top partners, as required to generate the Higgs mass, also contribute to the EW oblique parameters and can change the range of value of $\xi$ preferred by EW precision data [377]. The Higgs kinetic term, which receives a correction from the operator $c_H$, can be brought back to its canonical form by rescaling the Higgs field. This induces in the Higgs couplings a universal shift by a factor $1 - c_H \xi / 2$. For the fermions, it adds up to the modified Yukawa interactions.

The effective Lagrangian Eq. (65) represents the first term in an expansion in $\xi = v^2/f^2$. For large values of $\xi \sim \mathcal{O}(1)$ the series has to be resummed, examples of which have been given in explicit models such as those constructed in 5D warped space based on the coset SO(5)/SO(4) [378–380]. In the MCHM4 [379], where the SM fermions transform as spinorial representations of SO(5), all SM Higgs couplings are suppressed by the same modification factor as function of $\xi$, so that the branching ratios are unchanged and only the total width is affected. In the MCHM5 [380] with the fermions in the fundamental representation of SO(5) on the other hand the Higgs couplings to gauge bosons and to fermions are modified differently inducing non-trivial changes both in the branching ratios and the total width. The relations between the couplings in the effective Lagrangian Eq. (62), the SILH Lagrangian Eq. (65) and the MCHM4 and MCHM5 models is summarized in Table 23, see also Ref. [381].

The Higgs anomalous couplings affect both the Higgs production and decay processes. The Higgs boson branching ratios of a 125 GeV Higgs boson are shown in Fig. 79 for MCHM5. For $\xi = 0.5$ the Higgs boson becomes fermiophobic and the branching ratios into fermions and gluons vanish while the ones into gauge bosons become enhanced. As
explained above, in MCHM4 the branching ratios are un-
changed. The modified production cross sections can easily
be obtained from the corresponding SM results by rescaling
with the appropriate coupling modification factors squared.
As the QCD couplings are not affected the higher order QCD
corrections can be taken over from the SM, while the EW
corrections would change and have to be omitted as they are
not available so far.

The anomalous couplings can be tested by a measure-
ment of the Higgs interaction strengths. In case of a uni-
versal coupling modification as, e.g., in MCHM4 the pro-
duction rates and the total width have to be tested. At an
e<sup>+</sup>e<sup>-</sup> linear collider an accuracy of a few percent can be
achieved in the measurement of the SM Higgs couplings
to gauge bosons and fermions [49]. For an investigation of
the prospects for the determination of &xi; at the LHC, see
Ref. [382]. In Ref. [383] a study of Higgs couplings per-
formed in the context of genuine dimension-six operators
showed that a sensitivity of up to 4&pi;f ~ 40 TeV can be
reached for a 120 GeV Higgs boson already at 500 GeV
with 1 ab<sup>-1</sup> integrated luminosity. At the high-energy phase
of the CLIC project, i.e., at 3 TeV with 2 ab<sup>-1</sup> integrated
luminosity, the compositeness scale of the Higgs boson will
be probed up to 60 - 90 TeV [384]. Also the total width of
a 125 GeV Higgs boson can be measured at a few percent
precisely already at the low-energy phase of the ILC pro-
gramme.

2.7.2 Strong processes

If no new particles are discovered at the LHC, deviations
from the SM predictions for production and decay rates can
point towards models with strong dynamics. It is, however,
only the characteristic signals of a composite Higgs bo-
son in the high energy region which unambiguously imply
the existence of new strong interactions. Since in the com-
posite Higgs scenario the V_L V_L scattering amplitude is not
fully unitarized the related interaction necessarily becomes

strong and eventually fails tree-level unitarity at the cutoff
scale. The VV scattering therefore becomes strong at high
energies. As the transversely polarized vector boson scatter-
ing is numerically large in the SM, the test of the energy
growth in longitudinal gauge boson scattering is difficult at
the LHC [363]. Another probe of the strong dynamics at the
origin of EWSB is provided by longitudinal vector boson
fusion in Higgs pairs which also grows with the energy. For
the test of strong double Higgs production the high luminos-
ity upgrade of the LHC would be needed, however [363].
Besides testing the high-energy behaviour in strong dou-
ble Higgs production, new resocances unitarizing the scat-
tering amplitudes can be searched for. The ILC has been
shown to be able to test anomalous strong gauge couplings
up to a scale ~ 3 TeV and exclude p-like resonances below
2.5 TeV [49].

2.7.3 Non-linear Higgs couplings

Vertices involving more than one Higgs boson could also
provide a way to test the composite nature of the Higgs.
Double Higgs production is a process that depends on the
Higgs self-coupling and on the coupling between two Higgs
bosons and two massive gauge bosons. At a low energy
e<sup>+</sup>e<sup>-</sup> collider, double Higgs production proceeds mainly
via double Higgs-strahlung off Z bosons, e<sup>+</sup>e<sup>-</sup> &rarr; ZHH,
and WW boson fusion to Higgs pairs, e<sup>+</sup>e<sup>-</sup> &rarr; HH&nu;&#773;&nu; [72].

<figure><img src="image_82.png" /><figcaption>**Fig. 80** Generic Feynman diagrams contributing to Higgs pair produc-
tion via Higgs-strahlung off Z bosons.</figcaption></figure>

Generic diagrams are shown in Fig. 80 for double Higgs-
strahlung and Fig. 81 for WW boson fusion.

<figure><img src="image_83.png" /><figcaption>**Fig. 81** Generic Feynman diagrams contributing to Higgs pair produc-
tion via W boson fusion.</figcaption></figure>

<figure><img src="image_84.png" /><figcaption>**Fig. 79** Higgs boson branching ratios in MCHM5 as a function of ξ for M_h = 125 GeV.</figcaption></figure>

The double Higgs-strahlung process dominates at low
energies, and in the MCHM4 and MCHM5 it is always
smaller than in the SM, which is due to the suppressed
Higgs-gauge couplings. On the other hand, the WW fusion
process, which becomes important for higher c.m. energies,
<figure><img src="image_85.png" /><figcaption>**Fig. 82** The ZHH (upper two) and WW fusion (lower two) cross sections in the SM (red) and the MCHM5 for $\xi = 0.2$ (blue), $\xi = 0.5$ (black) and $\xi = 0.8$ (green) divided by the cross section of the corresponding model at $\kappa=1$, as a function of $\kappa$, for $\sqrt{s} = 500$ GeV and $\sqrt{s} = 1$ TeV.</figcaption></figure>

is enhanced compared to the SM for non-vanishing values of $\xi$ [385, 386]. This is due to interference effects related to the anomalous Higgs couplings. Furthermore, the amplitude growths like the c.m. energy squared contrary to the SM where it remains constant. The sensitivity of double Higgs-strahlung and gauge boson fusion processes to the trilinear Higgs self-coupling of the corresponding model

can be studied by varying the Higgs trilinear coupling in terms of the respective self-interaction of the model in consideration, hence $\lambda_{HHH}(\kappa) = \kappa \lambda_{HHH}^{MCHM4,5}$. This gives an estimate of how accurately the Higgs pair production process has to be measured in order to extract $\lambda_{HHH}$ within the investigated model with a certain precision. Note, however, that this does not represent a test of models beyond the actually investigated theory. Figure 82 shows for the SM and for the MCHM5 with three representative $\xi$ values ($\xi = 0.2, 0.5, 0.8$) the normalized double Higgs production cross sections for Higgs-strahlung and gauge boson fusion, respectively, at two c.m. energies, $\sqrt{s} = 500$ GeV and 1 TeV, as a function of the modification factor $\kappa$. The cross sections are normalized with respect to the double Higgs production cross sections at $\kappa = 1$ of the respective model. As can be inferred from the figure, both Higgs-strahlung and double Higgs production are more sensitive to $\lambda_{HHH}$ at lower c.m. energies. This is due to the suppression of the propagator in the diagrams which contain the trilinear Higgs self-coupling with higher energies. In addition in WW fusion the t- and u-channel diagrams, insensitive to this coupling, become more important with rising energy. A high energy $e^{+}e^{-}$ collider can exploit the WW fusion process to study the deviations in the coupling between two Higgs bosons and two gauge bosons by looking at the large $m_{HH}$ invariant mass distribution [384]. The sensitivity obtained on $\xi$ via this process is almost an order of magnitude better than the one obtained from the study of double Higgs-strahlung [384].

The parton level analysis in Refs. [385, 386] showed that both double Higgs-strahlung and WW fusion have, in the 4b final state from the decay of the two 125 GeV Higgs bosons, sensitivity to a non-vanishing $\lambda_{HHH}$ at the $5\sigma$ level in almost the whole $\xi$ range, with the exception of $\xi = 0.5$ in MCHM5, where the trilinear Higgs coupling vanishes, cf. Table 23.

### 2.7.4 Top sector

The fermionic sector of composite Higgs models, in particular the top sector, also shows an interesting phenomenology. With the fermion coupling strengths being proportional to their masses the top quark has the strongest coupling to the new sector and is most sensitive to new physics. It is hence natural to consider one of the two top helicities to be partially composite. The top quark mass then arises through linear couplings to the strong sector. ATLAS and CMS already constrained the top partners to be heavier than 600 - 700 GeV at 95% confidence level [387]. The associated new heavy top quark resonances have been shown to influence double Higgs production through gluon fusion [388, 389]. At $e^{+}e^{-}$ colliders these new resonances can be searched for either in single or in pair production [390].
### 2.7.5 Summary

Composite Higgs models offer a nice possibility to solve the hierarchy problem by introducing a Higgs boson which emerges as pseudo Nambu–Goldstone boson from a strongly interacting sector. The phenomenology of these models is characterized by a light Higgs resonance which is separated through a mass gap from the other resonances of the strong sector, and which has modified couplings to the SM fermions and gauge bosons. At an $e^+e^-$ collider these couplings can be tested at high accuracy, and interactions with more than one Higgs boson, among which the Higgs self-interactions, will also be accessible. Genuine probes of the strong sector are provided by strong double Higgs production through gauge boson fusion and longitudinal gauge boson scattering, which both rise with the energy. A high-energy $e^+e^-$ collider like CLIC can also become sensitive to the tails of the spin-1 resonance contributions to the $WW \to WW$ and $WW \to HH$ amplitudes. Assuming partial compositeness in the top sector, new top resonances arise which can be also searched for at a future linear collider above the current LHC bound around 700 GeV. Figure 83 summarizes the sensitivities at the LHC and CLIC for observing non-SM signatures from the composite nature of the Higgs boson in the plane of $\xi$ and $m_\rho$, the typical mass scale of the strong sector resonances.

<figure><img src="image_86.png" /><figcaption>Fig. 83 Summary plot of the current constraints and prospects for direct and indirect probes of Higgs compositeness. The dark brown region shows the current LHC limit from direct search for vector resonance. The dark (medium light) horizontal purple bands indicate the sensitivity on $\xi$ expected at the LHC from double (single) Higgs production with $300 \text{ fb}^{-1}$ of integrated luminosity. The pink horizontal band reports the sensitivity reach on $\xi$ from the study of double Higgs processes alone at CLIC with $1 \text{ ab}^{-1}$ of integrated luminosity at 3 TeV while the light blue horizontal band shows the sensitivity reach on $\xi$ when considering single Higgs processes. Finally, experimental electroweak precision tests (EWPT) favor the region below the orange thick line with and without additional contributions to $\Delta\rho$. From Ref. [384].</figcaption></figure>

### 2.8 The Higgs portal$^{21}$

A large fraction of matter in the Universe is dark and not incorporated in the Standard Model (SM). Nevertheless, this new kind of invisible matter is expected to interact with the SM fields, naturally by gravitational interaction. However, another path could be opened by a Higgs portal which connects the SM Higgs field with potential Higgs fields in the dark sector, respecting all symmetry principles and well founded theoretical SM concepts like renormalizability.

Even though the particles of the novel sector are invisible, the portal nevertheless induces observable signals in the SM, in the Higgs sector in particular. Mixings among Higgs bosons of the SM and of the dark sector modify Higgs couplings to the SM particles and give rise to invisible Higgs decays (beyond the cascades to neutrinos).

Crucial to an extraction of the $m_H \simeq 125$ GeV Higgs boson candidate's couplings to known matter is a good understanding of Higgs production $p$ and decay mechanisms $d$, which can be constrained by measuring

$$ \sigma_p \times \text{BR}_d \sim \frac{\Gamma_p \Gamma_q}{\Gamma_{\text{tot}}} \sim g_p^2 g_d^2 / \left( \sum_{\text{modes}} g_i^2 \right), \qquad (66) $$

where $\sigma_p$, $BR_d$, and $g_i$ denote the involved production cross sections, branching ratios and couplings, as usual. Precisely reconstructing these underlying parameters is systematically hindered by the unavailable measurement of the total decay width $\Gamma_{\text{tot}}$. As a matter of fact, un-adapted search strategies at LHC miss certain non-SM decay modes, which naturally arise in models beyond the SM [391–395] and which would then manifest as an invisible branching ratio [396] in global fits. The expected constraint on such an invisible Higgs boson decay at the LHC is $\text{BR}(H \to \text{invisible}) \simeq 10\%$ [397], a bound too loose to efficiently constrain physics beyond the SM, especially models where the Higgs field provides a portal to a hidden sector [73, 74, 398], which can provide a viable dark matter candidate [399].

At a linear collider (LC) it is straightforward to derive the total width of the Higgs boson by combining the model-independent measurement of the partial width $\Gamma(ZZ^*)$ in semi-inclusive Higgs-strahlung with the measurement of the branching ratio $\text{BR}(ZZ^*)$:

$$ \Gamma_{\text{tot}}(H) = \Gamma(ZZ^*) / \text{BR}(ZZ^*) . \qquad (67) $$

Subsequently $\text{BR}(H \to \text{invisible})$ can be determined in a model-independent way [400].

From Eq. (66), we need to interpret the strong Higgs exclusion for heavy Higgs masses as a sign of a highly sup-

<footnote>$^{21}$Christoph Englert</footnote>
<figure><img src="image_87.png" /><figcaption>Fig. 84 Scan over the Higgs portal potential Eq. 68. We include the constraints from electroweak precision measurements.</figcaption></figure>

pressed production cross section for heavier Higgs-like resonances. That heavy Higgs copies need to be weakly coupled in simple model-building realizations is already known from the investigation of electroweak precision measurements performed during the LEP era. This complements the requirement to include unitarizing degrees of freedom for longitudinal gauge boson scattering $V_L V_L \to V_L V_L$ ($V = W^\pm, Z$), and, to less constraint extent, massive quark annihilation to longitudinal gauge bosons $q\bar{q} \to V_L V_L$. Saturating all three of these requirements fixes key characteristics of the phenomenological realization of the Higgs mechanism, and does not allow dramatic modifications of the couplings $\{g_i\}$ in Eq. (66) away from the SM expectation of a light Higgs — the common predicament of electroweak scale model building. In this sense gaining additional sensitivity to invisible Higgs decays (or the Higgs total width in general) beyond the limitations of the LHC’s hadronic environment is crucial to the understanding of electroweak physics at the desired level, before the picture will be clarified to the maximum extent possible at a LC.

The aforementioned Higgs portal model [73, 74, 398] provides a theoretically well-defined, renormalizable, and yet minimal framework to explore both effects in a consistent way [402]: the influence of $\Gamma_{\text{inv}}$ on the Higgs phenomenology is captured while heavier Higgs boson-like particles with suppressed couplings are naturally incorporated. Therefore, the Higgs portal model not only provides a well-motivated SM Higgs sector extension in the context of dark matter searches<fnref n="22" /> and current data, but it represents an ideal model to generalize the SM in its phenomenologically unknown parameters to facilitate the SM’s validation by constraining the additional portal parameters beyond introducing biases (e.g. $\Gamma_H^{\text{tot}} \equiv \Gamma_H^{\text{SM}}$).

In its simplest form, leading to both a modified electroweak phenomenology and an invisible Higgs decay channel, the Higgs portal is given by the potential

$$ \mathcal{Y} = \mu_s^2 |\phi_s|^2 + \lambda_s |\phi_s|^4 + \mu_h^2 |\phi_h|^2 + \lambda_h |\phi_h|^4 + \eta_\chi |\phi_s|^2 |\phi_h|^2, \quad (68) $$

where $\phi_{s,h}$ are the SM and the hidden Higgs doublet fields, respectively, i.e. the Higgs sector is mirrored [407]. The visible sector communicates to the hidden world via the additional operator $\eta_\chi |\phi_s|^2 |\phi_h|^2$, which exploits the fact that both $|\phi_s|^2$ and $|\phi_h|^2$ are singlet operators under both the SM and the invisible gauge groups.

After symmetry breaking which is triggered by the Higgs fields acquiring vacuum expectation values $|\phi_{s,h}| = v_{s,h}/\sqrt{2}$, the would-be-Nambu Goldstone bosons are eaten by the $W^\pm$, Z fields, and correspondingly in the hidden sector. The only effect (formulated here in unitary gauge) is a two-dimensional isometry which mixes the visible and the hidden Higgs bosons $H_{s,h}$:

$$ H_1 = \cos \chi H_s + \sin \chi H_h, \quad (69) $$

$$ H_2 = -\sin \chi H_s + \cos \chi H_h, \quad (69) $$

with the mixing angle

$$ \tan 2\chi = \frac{\eta_{\chi} v_s v_h}{\lambda_s v_s^2 - \lambda_h v_h^2}. \quad (70) $$

The masses of the two Higgs fields are given by

$$ M_{1,2}^2 = [\lambda_s v_s^2 + \lambda_h v_h^2] \mp |\lambda_s v_s^2 - \lambda_h v_h^2| \sqrt{1 + \tan^2 2\chi}. \quad (71) $$

We assume $M_1 \simeq 125$ GeV in the following. The inverse phenomenological situation $M_1 < M_2 \simeq 125$ GeV, i.e. a Higgs field hiding below the upper LEP2 bound, is obviously reconciled by $\chi \to \pi - \chi$ since the potential has a $Z_2$ symmetry. Consistency with electroweak precision data and an efficient unitarization of the $V_L V_L$ scattering amplitudes relies in this case on $\cos^2 \chi$ being close to unity.

As a consequence of the mixing we have universally suppressed cross sections of the SM-Higgs

$$ \begin{aligned} \sigma_1 &= \cos^2 \chi \sigma_1^{\text{SM}} \\ \sigma_2 &= \sin^2 \chi \sigma_2^{\text{SM}}, \end{aligned} \quad (72a) $$

and

$$ \begin{aligned} \Gamma_{1,2}^{\text{vis}} &= \cos^2 \chi \{\sin^2 \chi\} \Gamma_{1,2}^{\text{SM}} + \Delta_2^{\text{vis}} \Gamma_2^{\text{HH}} \\ \Gamma_{1,2}^{\text{inv}} &= \sin^2 \chi \{\cos^2 \chi\} \Gamma_{1,2}^{\text{hid}} + \Delta_2^{\text{inv}} \Gamma_2^{\text{HH}}, \end{aligned} \quad (72b) $$

where $\Delta_2^{\text{vis}\{\text{inv}\}} = \zeta^2 \{[1-\zeta]^2\} \neq 0$ and $\zeta = 1/[1+\tan^2 \chi \Gamma_1^{\text{hid}}/\Gamma_{\text{tot},1}^{\text{SM}}]$. We understand the index in $\Delta_2$ such that this contribution only arises for the heavier state labelled with index=2.

<footnote n="22">²²In fact, there are only two other possibilities to couple the SM to a hidden sector: U(1) mixing [403, 404] and mixing with a right-handed sterile neutrino [405, 406]. The Higgs portal model is least constrained amongst these possibilities.</footnote>
<figure><img src="image_88.png" /><figcaption>Fig. 85 95% confidence level contours for a measurement of $\Gamma_1^{\text{hid}}/\Gamma_1^{\text{SM}}$ at the LHC and a 350 GeV LC. We use SFITTER [401] for the LHC results and we adopt the linear collider uncertainties of reference [400].</figcaption></figure>

<figure><img src="image_89.png" /><figcaption>Fig. 86 Measurement of a hypothetical portal model at a 350 GeV linear collider, uncertainties are adopted from Ref. [400]. A measurement of $\mathcal{R}_2$ at the LHC, with only an upper 95% confidence level bound on $J_2$ does not constrain the region $\Gamma_2^{\text{hid}}/\Gamma_{\text{tot},2}^{\text{SM}}$ below the $J_2$ curve. This degeneracy is lifted with a measurement at a linear collider.</figcaption></figure>

We have also included cascade decays $\Gamma_2^{\text{HH}}$ (if they are kinematically allowed for $M_2 \ge 2M_1$) and the possibility for a hidden partial decay width in Eq. (72b). The latter naturally arise if the hidden sector has matter content with $2m \le m_{H_1}$, i.e. in models with light dark matter candidates. Weak coupling of the heavier Higgs-like state is made explicit when correlating the Higgs portal model with electroweak precision constraints [402].

Generically, the branching ratio of the heavier Higgs boson to two light Higgs states is small (Fig. 84) and kinematically suppressed, so that a direct measurement of the cascade decay at the LHC is challenging. Measurement strategies targeting invisible Higgs boson decays at the LHC [408] are based on measurements in weak boson fusion [409] and associated production [410, 411]. Recent re-analysis of the monojet+Higgs production [395, 412], however, suggest that additional sensitivity can be gained in these channels, at least for the 7 and 8 TeV data samples.

The production of multiple final state Higgs particles is another strong test of this model, since it predicts resonant

contributions which can be large, see Fig. 84. A measurement of the involved trilinear coupling $H_2H_1H_1$ is challenging at the LHC [413, 414] and can be achieved more straightforwardly at a high luminosity LC [72]. Especially because we can separate the different final states of the light Higgs decay at the latter experiment, we can use the prediction of the various trilinear couplings that arise from Eq. (68) to reconstruct the potential.

The precision to which invisible decays can be studied at the LHC is ultimately limited by the machine's systems which will saturate at luminosities $\mathcal{L} \simeq 300 \text{ fb}^{-1}$, see Fig. 85. Bounds on visible decays are typically expressed as ratios to the SM expectation, which, for the lighter $M_1 \simeq 125$ GeV state, can be rephrased in the portal model for either $i = pp$ or $e^+e^-$ initial states

$$ \frac{\sigma[i \to H_1 \to F]}{\sigma[i \to H_1 \to F]^{\text{SM}}} = \frac{\cos^2 \chi}{1 + \tan^2 \chi [\Gamma_1^{\text{hid}}/\Gamma_{\text{tot},1}^{\text{SM}}]} \le \mathcal{R}_1, \quad (73) $$
where $\mathcal{R}_1$ denotes the observed exclusion limit (signal strength). An identical quantity can be derived from future constraints on invisible decays

$$ \frac{\sigma[i \to H_1 \to inv]}{\sigma[i \to H_1]^{\text{SM}}} = \frac{\sin^2 \chi [\Gamma_1^{\text{hid}}/\Gamma_{\text{tot},1}^{\text{SM}}]}{1 + \tan^2 \chi [\Gamma_1^{\text{hid}}/\Gamma_{\text{tot},1}^{\text{SM}}]} \le \mathcal{J}_1. \quad (74) $$

Similar relations hold for $H_2$, and there are portal-specific sum rules which facilitate the reconstruction of the mixing angle from measurements of $\mathcal{J}_{1,2}$ and $\mathcal{R}_{1,2}$,

$$ \begin{aligned} \mathcal{R}_1 + \mathcal{J}_1 &= \cos^2 \chi, \\ \mathcal{R}_2 + \mathcal{J}_2 &= \sin^2 \chi. \end{aligned} \quad (75) $$

While the LHC running at 14 TeV will eventually probe small visible production cross sections $\mathcal{R}_2$ (Eq. (74)) becomes an equality), the invisible decay searches at the LHC will most likely yield a 95% confidence level bound [415] on $\mathcal{J}_{1,2}$ [408] rather than a statistically significant observation. The bounds can be vastly improved by performing by performing precision spectroscopy of the 125 GeV Higgs candidate in the associated production channel $e^+e^- \to HZ$ at, e.g., a 350 GeV LC (see also Ref. [416]). Still, invisible Higgs searches that solely provide upper limits on both $\mathcal{J}_{1,2}$ are not enough to fully reconstruct the portal model if a second Higgs-like state is discovered as a result of Eq. (75). Only the precise measurement, which is impossible at the LHC, solves this predicament, but an LC is the perfect instrument to pursue such an analysis in the associated production channel.

In Fig. 86 we show a hypothetical situation, where $H_2$ is discovered at the LHC with $\mathcal{R}_2 = 0.4$; the error is given by a more precise measurement at a 350 GeV LC, see Fig. 84. The measurement of $\mathcal{J}_2 = 0.4$ allows to reconstruct $\sin^2 \chi$, which can be seeded to a reconstruction algorithm [402] that yields the full Higgs portal potential Eq. (68).

From Eq. (75) we also obtain the sum rule

$$ \mathcal{R}_1 + \mathcal{J}_1 + \mathcal{R}_2 + \mathcal{J}_2 = 1. \quad (76) $$

which provides a strong additional test of the portal model Eq. (75) when a measurement of the invisible branching ratios via $\mathcal{J}_{1,2}$ becomes available at a future linear collider.

To summarize, the Higgs portal can open the path to the dark sector of matter and can allow crucial observations on this novel kind of matter in a global way. While first hints may be expected from LHC experiments, high-precision analyses of Higgs properties and the observation of invisible decays at LC can give rise to a first transparent picture of a new world of matter.

## 2.9 The NMSSM$^{23}$

In the Next-to-Minimal Supersymmetric Standard Model (NMSSM) the Higgs sector of the MSSM is extended by an additional gauge singlet superfield $\hat{S}$ [82]. It is the simplest supersymmetric extension of the Standard Model with a scale invariant superpotential; the $\mu$-term $\mu\hat{H}_u\hat{H}_d$ in the superpotential $W_{\text{MSSM}}$ of the MSSM is replaced by

$$ W_{\text{NMSSM}} = \lambda \hat{S} \hat{H}_u \hat{H}_d + \frac{\kappa}{3} \hat{S}^3. \quad (77) $$

Once the scalar component $S$ of the superfield $\hat{S}$ assumes a vacuum expectation value $s$, the first term in the superpotential (77) generates an effective $\mu$-term with

$$ \mu_{\text{eff}} = \lambda s. \quad (78) $$

In addition to the NMSSM-specific Yukawa couplings $\lambda$ and $\kappa$, the parameter space of the NMSSM contains soft supersymmetry breaking trilinear couplings $A_\lambda$, $A_\kappa$ and soft supersymmetry breaking mass terms $m_{\hat{H}_u}^2$, $m_{\hat{H}_d}^2$ and $m_{\hat{S}}^2$. The order of $s$ and hence of $\mu_{\text{eff}}$ is essentially determined by $A_\kappa$ and $m_{\hat{S}}^2$, hence $\mu_{\text{eff}}$ is automatically of the order of the soft supersymmetry breaking terms.

The physical states in the Higgs sector of the NMSSM (assuming CP-conservation) consist in three neutral CP-even states $H_i$ (ordered in mass), two neutral CP-odd states $A_i$ and charged Higgs bosons $H^\pm$. The CP-even states $H_i$ are mixtures of the real components of the weak eigenstates $H_u$, $H_d$ and $S$:

$$ H_i = S_{1,d} H_d + S_{1,u} H_u + S_{1,s} S, \quad (79) $$

where the mixing angles $S_{i,j}$ depend on the a priori unknown parameters in the Higgs potential. Similarly, the two CP-odd states $A_i$ are mixtures of the imaginary components of the weak eigenstates $H_u$, $H_d$ and $S$ without the Goldstone boson. In addition, the fermionic component of the superfield $\hat{S}$ leads to a fifth neutralino, which mixes with the four neutralinos of the MSSM.

In view of the mass of 125-126 GeV of the at least approximately Standard Model-like Higgs boson $H_{\text{SM}}$ measured at the LHC, the NMSSM has received considerable attention: In contrast to the MSSM, no large radiative corrections to the Higgs mass (implying a fine-tuning in parameter space) are required in order to obtain $M_{H_{\text{SM}}}$ well above $M_Z$, the upper bound on $M_{H_{\text{SM}}}$ at tree level in that model. In the NMSSM, additional tree level contributions to $M_{H_{\text{SM}}}$ originate from the superpotential Eq. (77) [82]. Also a mixing with a lighter mostly singlet-like Higgs boson can increase the mass of the mostly Standard-Model-like Higgs boson [417], in which case one has to identify $H_{\text{SM}}$ with $H_2$.

<footnote>$^{23}$Ulrich Ellwanger</footnote>
Both effects allow to obtain $M_{HSM} \sim 125-126$ GeV without fine-tuning and, moreover, such a mixing could easily explain an enhanced branching fraction of this Higgs boson (from now on denoted as $H_{125}$) into $\gamma\gamma$ [236, 418-427].

Depending on the mixing angles, on the masses of the additional Higgs bosons and on their branching fractions, the LHC can be blind to the extended Higgs sector of the NMSSM beyond the mostly Standard Model-like state. Then the detection of the additional states will be possible only at a LC. Also if hints for such an extended Higgs sector are observed at the LHC, only a LC will be able to study its properties in more detail. Earlier studies of the detection of NMSSM Higgs bosons at $e^+e^-$ colliders can be found in [428-433].

The dominant production modes of CP-even Higgs bosons at a LC (associate ZH production and VBF) depend on the Higgs couplings to the electroweak gauge bosons. Denoting the coupling of $H_{SM}$ to electroweak gauge bosons by $g_{SM}$, the couplings $g_i$ of the CP-even states $H_i$ satisfy the sum rule

$$ \sum_i g_i^2 = g_{SM}^2. \qquad (80) $$

If a measurement of the coupling $g_i$ of the 125 GeV Higgs boson at the LC gives a value significantly below $g_{SM}$, one can deduce the presence of additional Higgs states. The scenario where $H_{125} = H_2$ is particularly natural in the parameter space of the NMSSM. Then the coupling $g_1$ of the lightest Higgs boson $H_1$ must satisfy constraints from LEP II, if its mass is below $\sim 114$ GeV.

The allowed gauge couplings$^2 \times$ branching fractions into $bb$ of $H_1$ and $H_2$ have been studied as function of $M_{H_1}$, once $M_{H_2} \sim 125$ GeV is imposed, in the parameter space of the semi-constrained NMSSM in [424]. (In the semi-constrained NMSSM, squark and slepton masses at the GUT scale are given by a common value $m_0$, gaugino masses by a common value $M_{1/2}$, but the NMSSM-specific soft Higgs masses and trilinear couplings are left free.) The results for the allowed values of $R_i^{bb} = \frac{g_i^2 \times BR(H_i \to bb)}{g_{SM} \times BR(H_{SM} \to bb)}$ are shown in Figs. 87 and 88. Since here $BR(H_i \to bb) \approx BR(H_{SM} \to bb)$, one has $R_i^{bb} \approx \frac{g_i^2}{g_{SM}^2}$.

For $M_{H_1} < 114$ GeV, the upper bounds on $R_1^{bb}$ in Fig. 87 follow from the LEP II constraints in [294]. Still, even for $M_{H_1} < 110$ GeV, a detection of $H_1$ at a LC is possible (but difficult at the LHC within the semi-constrained NMSSM). From Fig. 88 one finds that, if $M_{H_1} > 114$ GeV, $R_1^{bb}$ can assume all possible values from 0 to 1. Note that $R_1^{bb}$ and $R_2^{bb}$ satisfy approximately $R_1^{bb} + R_2^{bb} \sim 1$.

For $M_{H_1} \sim 100$ GeV and $R_1^{bb} \sim 0.1-0.25$, $H_1$ can explain the $\sim 2\sigma$ excess in the $bb$ final state for this range of Higgs masses at LEP II [294]. Properties of such points in the parameter space of the semi-constrained NMSSM have

<figure><img src="image_90.png" /><figcaption>Fig. 87 The reduced signal cross section $R_1^{bb}$ at a $e^+e^-$ collider as defined in the text, as function of $M_{H_1}$ in the semi-constrained NMSSM (from [424]).</figcaption></figure>

<figure><img src="image_91.png" /><figcaption>Fig. 88 The reduced signal cross section $R_2^{bb}$ as function of $M_{H_1}$ in the semi-constrained NMSSM (from [424]).</figcaption></figure>

been studied in [434], amongst others the production cross sections of the various Higgs bosons in various channels at a LC.

For a typical point with $M_{H_1} \sim 99$ GeV, $M_{H_2} \sim 124$ GeV (and an enhanced signal rate in the $\gamma\gamma$ final state at the LHC), $M_{H_3} \sim 311$ GeV, $M_{A_1} \sim 140$ GeV, $M_{A_2} \sim 302$ GeV and $M_{H^\pm} \sim 295$ GeV, the production cross sections in the channels $ZH_1$, $ZH_2$, $H^+H^-$ and $H_iA_j$ are shown in Fig. 89 as function of $\sqrt{s}$ of a LC (from [434]). Note that, for suitable mixing angles of $H_i$ and $A_j$, also $H_iA_j$ production via $e^+ + e^- \to H_iA_j$ is possible as in the MSSM. However, an additional CP-even Higgs boson with sizeable coupling $g_i$ can also be heavier than 125 GeV; such a scenario is motivated by best fits to present LHC and Tevatron data [435].
<figure><img src="image_92.png" /><figcaption>**Fig. 89** Higgs production cross sections at a $e^{+}e^{-}$ collider in the channels $ZH_1$, $ZH_2$, $H^{+}H^{-}$ and $H_{t}A_{j}$ for a point in the parameter space of the semi-constrained NMSSM with Higgs masses as indicated in the text, from [434].</figcaption></figure>

Other NMSSM-specific scenarios are possible Higgs-to-Higgs decays (see, e.g., [436]). For the 125 GeV Higgs boson, the measured Standard Model-like decay modes at the LHC indicate that Higgs-to-Higgs decays are not dominant for this state, but branching fractions of $\mathcal{O}(10\%)$ are allowed. In the NMSSM, $H_{125}$ could decay into pairs of lighter CP-even or CP-odd states (if kinematically possible). If these states are heavier than ~10 GeV and decay dominantly into $bb$, such decay modes of $H_{125}$ into $4b$ (or $2b2\tau$) would be practically invisible at the LHC. At a LC, using the leptonic decays of $Z$ in the $ZH$ Higgs production mode and/or VBF, such unconventional decays can be discovered [432].

In addition, more Higgs-to-Higgs decays involving all three CP-even states $H$ and both CP-odd states $A$ (omitting indices for simplicity) like $H \to HH$, $H \to AA$, $H \to ZA$, $A \to AH$, $A \to ZH$, $H^{\pm} \to W^{\pm}H$ and $H^{\pm} \to W^{\pm}A$ are possible whenever kinematically allowed, and visible whenever the “starting point” of the cascade has a sufficiently large production cross section (see, e.g., Fig. 89) and the involved couplings are not too small. Even if a mostly Standard Model-like Higgs boson at 125 GeV is imposed, the remaining unknown parameters in the Higgs sector of the NMSSM allow for all of these scenarios.

The relevance of a $\gamma\gamma$ collider for the study of Higgs-to-Higgs decays in the NMSSM has been underlined in [437]. Astonishingly, also pure singlet-like states $H$ and $A$ can be produced in the $\gamma\gamma$ mode of a LC. In the Standard Model, a $H\gamma\gamma$-vertex is loop-induced with mainly $W^{\pm}$ bosons and

<figure><img src="image_93.png" /><figcaption>**Fig. 90** The reduced coupling $R^{\gamma\gamma}$, as defined in Eq. (81), as function of $M_{\text{chargino}_1}$ for $M_{A_S} \sim M_{H_S} \sim 260$ GeV, for a scenario explaining a 130 GeV photon line from dark matter annihilation in the galactic center.</figcaption></figure>

top-quarks circulating in the loops. In the case of the NMSSM and dominantly singlet-like states $H_S$ and $A_S$ (without couplings to $W^{\pm}$ bosons or top-quarks), higgsino-like charginos can circulate in the loops. The corresponding couplings of $H_S$ and $A_S$ to higgsino-like charginos originate from the term $\lambda \hat{S}H_u\hat{H}_d$ in the superpotential (77) and are absent for the MSSM-like CP-even and CP-odd Higgs states.

Possible values of the reduced couplings $R^{\gamma\gamma}$ of such nearly pure singlet-like states $H_S$ and $A_S$ are shown in Fig. 90, where we define

$$R^{\gamma\gamma} = \frac{\Gamma(H/A \to \gamma\gamma)}{\Gamma(H_{\text{SM}} \to \gamma\gamma)} \qquad (81)$$

for a Standard Model-like $H_{\text{SM}}$ of the same mass as $H_S$ or $A_S$. The production cross sections of these states in the $\gamma\gamma$ mode of a LC are given by the production cross section of $H_{\text{SM}}$ multiplied by same ratio $R^{\gamma\gamma}$.

The values of $R^{\gamma\gamma}$ shown in Fig. 90 correspond to a region in the parameter space of the NMSSM where the Standard Model-like $H_{\text{SM}}$ has a mass of ~125 GeV and, simultaneously, dark matter annihilation in the galactic center can give rise to a 130 GeV photon line [438]. Hence the LSP mass is 130 GeV, $M_{A_S}(\equiv M_{A_1}) \sim 260$ GeV in order to produce two photons from LSP annihilation with $A_S$ exchange in the s-channel, and $M_{H_S}(\equiv M_{H_2}) \approx 260$ GeV such that $H_S$ exchange in the s-channel gives a relic density compatible with WMAP. $\lambda$ varies between 0.6 and 0.65, the wino mass parameter is fixed to $M_2 = 300$ GeV, but $\mu_{\text{eff}}$ varies from 250-350 GeV. The nature of the chargino$_1$ varies slightly with $\mu_{\text{eff}}$, but is always ≈ 50% wino and higgsino-like. The values shown in Fig. 90 have been obtained using the code NMSSMTools [439, 440]. We see in Fig. 90 that notably $R^{\gamma\gamma}(A_S)$ can assume values close to 0.3, leading to a significant production cross section in the $\gamma\gamma$ mode of a LC.
<figure><img src="image_94.png" /><figcaption>**Fig. 91** The reduced coupling $R^{\gamma\gamma}$ as function of $M_{A_S}$, for points in the semi-constrained NMSSM where $H_S$ with $M_{H_S} \sim 100$ GeV explains the excess in *bb* at LEP II (from [434]; orange diamonds satisfy the WMAP constraint on the dark matter relic density).</figcaption></figure>

<figure><img src="image_95.png" /><figcaption>**Fig. 92** The reduced coupling $R^{\gamma\gamma}$ as function of $M_{H_S}$, for points in the semi-constrained NMSSM where $H_S$ with $M_{H_S} \sim 100$ GeV explains the excess in *bb* at LEP II (from [434]; orange diamonds satisfy the WMAP constraint on the dark matter relic density).</figcaption></figure>

Returning to the semi-constrained NMSSM with $M_{H_1} = M_{H_S} \sim 100$ GeV and $M_{H_2} \sim 125$ GeV, scatter plots for $R^{\gamma\gamma}(A_S)$ and $R^{\gamma\gamma}(H_S)$ as function of $M_{A_S}$ and $M_{H_S}$ are shown in Figs. 91 and 92 (from [434]). Again we see that the prospects for $A_S/H_S$ discovery are quite promising for sufficiently large luminosity, since the production cross sections are typically about 10% (possibly larger) than those of a SM-like Higgs boson of a corresponding mass.

Finally the NMSSM differs from the MSSM also due to the presence of a fifth neutralino, the fermionic component of the superfield $\hat{S}$. Phenomenological analyses of pair production of neutralinos in the NMSSM at $e^+e^-$ colliders at higher energies have been performed in [39, 441–445]. Since the information on the neutralino sector from the LHC will be quite limited, a $e^+e^-$ collider can be crucial to distinguish the NMSSM neutralino sector from the one of the MSSM [444], although it cannot be guaranteed that the difference is visible if one is close to the decoupling limit

$\lambda, \kappa \to 0$. This question has also been addressed in the radiative production of the lightest neutralino pair, $e^+e^- \to \tilde{\chi}_1^0 \tilde{\chi}_1^0 \gamma$, at a LC with $\sqrt{s} = 500$ GeV in [445].

To summarize, the NMSSM is a well motivated supersymmetric extension of the Standard Model, notably in view of the discovery of a Higgs boson at 125 GeV and a potentially enhanced branching fraction into $\gamma\gamma$. Due to their reduced couplings to electroweak gauge bosons it is not clear, however, whether the LHC will be able to verify the extended Higgs and neutralino sectors of the NMSSM. Only a LC will be able to perform measurements of such reduced couplings, correspondingly reduced production cross sections, and possible unconventional decay modes. These incompass both possible Higgs-to-Higgs cascade decays, as well as cascades in the neutralino sector.

## 2.10 Little Higgs$^{24}$

The Little Higgs (LH) model [447–449] is known to be one of the attractive scenarios for physics beyond the standard model (SM). In this subsection, we review the physics of the model at future linear collider experiments by referring to several studies reported so far.

### 2.10.1 About the LH model

The cutoff scale of the standard model (SM) is constrained by electroweak precision measurements: If we assume the existence of a $\sim 125$ GeV SM Higgs-like resonance, the cutoff scale should be higher than roughly 5 TeV [450, 451]. However, such a relatively high cutoff scale requires a fine-tuning in the Higgs potential because the Higgs potential receives the quadratic divergent radiative correction.

In LH models, the Higgs boson is regarded as a pseudo Nambu-Goldsone (NG) boson which arises from a global symmetry breaking at high energy, $\sim 10$ TeV. Although Yukawa and gauge couplings break the global symmetry explicitly, some global symmetry is not broken by one of these couplings: in LH models, the breaking of such a symmetry is achieved only by two or more couplings, which is called “collective” symmetry breaking. Because of the collective symmetry breaking, the quadratic divergence from SM loop diagrams is canceled by new particle diagrams at the one-loop level.

As a bottom-up approach, specifying a coset group, we investigate the phenomenology of such a scenario by a nonlinear sigma model. In particular, the littlest Higgs (LLH) model [449] described by an SU(5)/SO(5) symmetry breaking and the simplest little Higgs (SLH) model [452] described by an $[SU(3) \times U(1)]^2 / [SU(2) \times U(1)]^2$ symmetry

<footnote>$^{24}$Masaki Asano and Shigeki Matsumoto
Both M.A. and S.M. would like to thank all the members of the ILC physics subgroup [446] for useful discussions.</footnote>
breaking have been studied about its expected phenomenol-
ogy well so far. Here we review the ILC physics mainly fo-
cusing on the LLH model.

The LLH model is based on a non-linear sigma model describing an SU(5)/SO(5) symmetry breaking with the vacuum expectation value $f \sim \mathcal{O}(1)$ TeV. An $[\text{SU}(2)_L \times \text{U}(1)_Y]^2$ subgroup of the SU(5) is gauged and broken down to the SM $\text{SU}(2)_L \times \text{U}(1)_Y$. Fourteen NG bosons arise and it can be decomposed into $\mathbf{1}_0 \oplus \mathbf{3}_0 \oplus \mathbf{2}_{\pm 1/2} \oplus \mathbf{3}_{\pm 1}$ under the electroweak gauge group. The $\mathbf{1}_0 \oplus \mathbf{3}_0$ are eaten by heavy gauge bosons $A_H, Z_H, W_H^{\pm}$, and $\mathbf{2}_{\pm 1/2} \oplus \mathbf{3}_{\pm 1}$ are the SM Higgs field $h$ and new triplet Higgs field $\mathbf{\Phi}$, respectively. To realize the collective symmetry breaking, SU(2) singlet vector-like top quark partners, $T_L$ and $T_R$, are also introduced. These heavy particles have masses which are proportional to $f$ and depend also on the gauge coupling, charges and Yukawa couplings. The Higgs potential is generated radiatively and it depends also on parameters of UV theory at the cutoff scale $\Lambda \sim 4\pi f$.

Even in the model, the new particle contributions are
strongly constrained at precision measurements.

Pushing new particle masses up to avoid the constraint, the
fine-tuning in the Higgs potential is reintroduced. To avoid
the reintroducing the fine-tuning, implementing of the $Z_2$
symmetry called T-parity has been proposed [453–455].<fnref n="25" />

In the LLH model, the T-parity is defined as the in-
variance under the exchanging gauged $[SU(2) \times U(1)]_1$ and
$[SU(2) \times U(1)]_2$. Then, for all generation of lepton and
squark sector, new heavy fermions are introduced to imple-
ment this symmetry. Under the parity, the new particles are
assigned to be a minus charge (T odd), while the SM parti-
cles have a plus charge (T even). Thus, heavy particles are
not mixing with SM particles. Then, the tree level new parti-
cle contribution to electroweak precision measurements are
forbidden and the new particle masses can be light.

It has been suggested that the T-parity is broken by anoma-
lies in the typical strongly coupled UV theory [458, 459]
and the possibilities of the conserved T-parity scenario and
another parity are also studied [460–464]. If the T-parity is
an exact symmetry, the lightest T-odd particle, heavy pho-
ton in the LLH model, is stable and provides a dark mat-
ter candidate. Even if the T-parity is broken by anomalies,
contribution to electroweak precision measurements are still
suppressed while the lightest T-odd particle would decay at
colliders [465, 466].

As described above, top quark partner, new gauge bosons
and additional scalar bosons are expected in LH models while
its details strongly depend on models. In the model with T-
parity, T-odd quark partners and lepton partners are intro-
duced additionally. The Higgs boson phenomenology would
be different from the SM prediction due to the new particle
contributions as well as deviations from the SM coupling

<figure><img src="image_96.png" /><figcaption>**Fig. 93** Accessible range of $\Gamma(h \to gg)$ and $\Gamma(h \to \gamma\gamma)$ normalized to the SM value in the LLH model (from [467]).</figcaption></figure>

which would be appeared from higher dimensional opera-
tors.

2.10.2 Higgs phenomenology in LH

In LH models, parameters of the Higgs potential cannot be
estimated without calculating the contribution of a specify-
ing UV theory. As a phenomenological approach, we con-
sider these parameters as free parameters and these are de-
termined by observables, e.g., Higgs mass. As described here,
there are possibilities to change the Higgs boson phenomenol-
ogy from the SM prediction and it may be checked at the
ILC.

*Higgs decay from loop diagram*
One of the possibility to
change the Higgs phenomenology is contributions from top
partner as well as the deviation from the SM couplings. It
leads to deviations in the decay branching ratios of the $h \to$
$gg$ (also indicating deviations in the main Higgs production
channel at the LHC) and $h \to \gamma\gamma$ modes, via the top partner-
loop diagrams. The extra gauge bosons and charged scalar
bosons also contribute to the $h \to \gamma\gamma$ decay.

Fig.93 shows the range of partial decay widths, $\Gamma(h \to gg)$ and $\Gamma(h \to \gamma\gamma)$, in the LLH model varying model parameters [467]. In the model, the deviation of the top Yukawa coupling suppresses the $\Gamma(h \to gg)$ while contributions from top partner and mixing in the top sector enhance the partial decay width. Totally, these additional top sector contributions suppresses the $\Gamma(h \to gg)$ in Fig.93. On the other hand, it enhances the $\Gamma(h \to \gamma\gamma)$ because the W boson loop contribution is dominant in the SM and the fermion-loop contributions have a minus sign. The contribution from the heavy gauge bosons suppresses the $\Gamma(h \to \gamma\gamma)$ as well as the deviation of the gauge boson coupling and mixing in the gauge boson sector due to the sign of the $W_H W_H h$ coupling. The charged Higgs contribution leads to an enhancement. The doubly-charged Higgs contribution is small because the

<footnote n="25">²⁵ As the other possibility, for example, the model decoupling the new gauge bosons have been also proposed [456, 457].</footnote>
<figure><img src="image_97.png" /><figcaption>**Fig. 94** The $\Gamma(h \to gg)$ normalized to the SM value (from [471]). The $f_{min}$ is defined as the smallest value allowed by electroweak precision measurements and the values are 1.2 TeV for the LLH model, 500 GeV for T-parity case, 700 GeV for custodial littlest Higgs model and 500 GeV for minimal composite Higgs model, respectively (for details, see [471]).</figcaption></figure>

coupling to the Higgs boson is suppressed, then, it is neglected here [467]. In a similar way the $\gamma Z$ decay would be affected [468].

In the model with T-parity, there is also the contribution from T-odd heavy fermions and the contribution is negative to $\Gamma(h \to gg)$ and positive to $\Gamma(h \to \gamma\gamma)$ [469]. Furthermore, in the model with T-parity case (and also in a decoupling gauge partner case, e.g., [470]), the new particle can be light consisting with electroweak precision measurements, thus, the deviation could be greater than the case without T-parity. For example, in the littlest Higgs model with T-parity (LHT), the $\Gamma(h \to gg)$ normalized to the SM value can be around 60% at $f = 500$ GeV case [471] (see Fig. 94).

The expected precision for measurements of the Higgs coupling including $h \to \gamma\gamma$ and $h \to gg$ branch at ILC are summarized in Sect.2.3. One of the possibilities to measure the deviation of the $\Gamma(h \to \gamma\gamma)$ is the $\gamma\gamma \to h \to b\bar{b}$ mode in photon collider option [472,473].

**Higgs decay at tree level** The deviation of the SM coupling and new particles would also change the Higgs phenomenology at tree level. The deviation of $h\bar{t}\bar{t}$ and top partner change the cross section of $h\bar{t}\bar{t}$ production [474–476]. In LHT, production cross section of the $e^+e^- \to h\bar{t}\bar{t}$ normalized to the SM value is about 90% at $f = 1$ TeV [475].

The deviation of $hWW$ and $hZZ$ couplings (e.g. [477] in LLH model) also change the cross sections of the Higgs boson production as well as the decay branching ratio $^{26}$. In some case, the deviation rates of partial decay widths are

the same, then, the branching ratio of the Higgs decay can be close to the SM prediction [469].

However, the down-type Yukawa coupling has model dependence and the couplings could be significantly suppressed in some case of the LHT [469]. Thus, the decay branching ratio of a light Higgs boson ($m_h < 2m_W$) could significantly change because the dominant decay width, $h \to b\bar{b}$ is suppressed. Fig. 95 shows the correction of the branching ratio from the SM prediction [469].

**Higgs decay to new particles** Another possibility is additional decay branches of Higgs boson into new particles. For example, the lightest new particle in the LHT is the heavy photon which mass is $\sim 60$ GeV with $f = 400$ GeV. If it kinematically possible, the Higgs boson also decays into two heavy photons and the value of the branching ratio could be large ($> 80\%$) in the 125 GeV Higgs boson case because it decays via the gauge coupling [481,482]. If the T-parity is an exact symmetry, it is the invisible decay. On the other hand, the produced heavy photon decays mainly into SM fermions in such a light Higgs boson case if the T-parity is broken by anomaly. The decay width is about $10^{-1}-10^{-2}$ eV [465,466].

**Additional scalar bosons** In some models, e.g., simple group models, there could be a pseudo scalar, $\eta$, although the mass depends on the models. The Higgs boson could also decay into $\eta\eta$ and $Z\eta$ [483] if it is kinematically possible. Furthermore, because the $Z-h-\eta$ coupling cannot appear in product group models, the measurement at ILC helps to distinguish the kind of LH models [484]. Other phenomenology studies for $\eta$ can be found in Refs. [485,486]. As another example of additional scalars, there is the triplet Higgs boson in the LLH model, although these mass is proportional to $f$ [487–490].

**Higgs self coupling** The measurement of Higgs self coupling is one of the important test for the Higgs boson. In the LH models, the triplet and quartet coupling could slightly change from the SM expectation. Study for Zhh process in LLH [491] and the one-loop correction to the hhh coupling from vector-like top quarks [492] have been studied.

### 2.10.3 Other direct LH signals

Since the LH model is discussed only in this subsection, we also mention here other signals of the model at future linear collider experiments. The signals can be divided into two categories; direct and indirect signals. The direct signals means the direct productions of new particles predicted by the LH model. The indirect signals are, on the other hand, the LH contributions to the processes whose final states are composed only of SM particles. We consider only the direct signals, while we omit to discuss the indirect ones for

<footnote>$^{26}$ For the deviation of vector boson fusion process at ILC, see [478–480].</footnote>
<figure><img src="image_98.png" /><figcaption>**Fig. 95** The (a) shows the total decay width normalized to the SM value in the LHT (from [469]). The difference between case A and case B comes from the definition of the down-type Yukawa term (for details, see [469]). The (b) shows the partial Higgs branching ratios normalized to the SM value (from [469]).</figcaption></figure>

want of space. Please see references [477, 493–514] for the indirect signals.

The direct signals can future be divided into two subcategories; the direct productions of colored particles and non-colored ones. This is because the LH model requires the cancellation of quadratically divergent corrections to the Higgs mass term from top loop and those of electroweak gauge bosons at one-loop level, and thus the model inevitably predicts both colored and non-colored new particles. When the T-parity (or some other $Z_2$-symmetry distinguishing SM and new particles) is not imposed on the model like the littlest or the simplest Higgs model, non-colored new particles will be produced by following two processes; single productions (i.e., $e^+e^- \to V_H$) [515–523] and associate productions (i.e., $e^+e^- \to V_H + \gamma(Z)$) [524–528], where $V_H$ is the LH partner of the weak gauge boson (heavy gauge boson). On the other hand, when the T-parity is imposed like the case of the LHT, non-colored new particles must be produced in pair (i.e., $e^+e^- \to V_H V_H$) [529–534]. For the productions of colored new particles, associate productions (i.e., $e^+e^- \to T+t$) and pair productions (i.e. $e^+e^- \to f_H f_H$) are frequently considered to find LH signals [535–537], where T is the LH partner of the top quark (top partner) and $f_H$ is the new colored fermion like the top partner or heavy fermions which are introduced by imposing the T-parity on the model.

We first consider the productions of non-colored new particles. Among several relevant studies reported so far, the most comprehensive one involving realistic numerical simulations has been performed in reference [534]. They have considered following five pair production processes in the framework of the LHT; $e^+e^- \to Z_H Z_H$, $Z_H A_H$, $W_H^+ W_H^-$, $e_H^+ e_H^-$, and $\nu_{eH} \bar{\nu}_{eH}$, which are followed by the decays $Z_H \to A_H h$, $W_H^\pm \to A_H W_\pm$, $e_H^\pm \to Z_H e_\pm$, $\nu_{eH} \to W_H^+ e^-$ ($\bar{\nu}_{eH} \to W_H^- e^+$), where $e_H^\pm (e_H^\pm)$ and $\nu_{eH} (\bar{\nu}_{eH})$ are the T-parity partners of electron (positron) and electron neutrino

(anti-neutrino), respectively. The mass spectrum of the non-colored new particles used in this study is the following (to be taken as a representative example),

<table><thead><tr><th></th><th>M<sub>AH</sub></th><th>M<sub>WH</sub></th><th>M<sub>ZH</sub></th><th>M<sub>eH</sub></th><th>M<sub>νeH</sub></th></tr></thead><tbody><tr><th>Mass (GeV)</th><td>81.9</td><td>368</td><td>369</td><td>410</td><td>400</td></tr></tbody></table>

The above mass spectrum has been obtained by choosing the vacuum expectation value of the global symmetry $f$ and the Yukawa coupling of the heavy electron $\kappa_e$ to be 580 GeV and 0.5, respectively.<fnref n="27" /> Flavor-changing effects caused by the heavy lepton Yukawa couplings are implicitly assumed to be negligibly small.

By measuring the energy distribution of visible (SM) particles emitted in each production process, the masses of the non-colored new particles can be precisely extracted. This is because the initial energy of electron (positron) is completely fixed at the $e^+e^-$ colliders and thus measuring the energy distribution allow us to reconstruct the process accurately without any assumption of the LHT model. With assuming the integrated luminosity of 500 fb⁻¹ at $\sqrt{s} = 1$ TeV running and use of the four processes, $e^+e^- \to Z_H Z_H$, $W_H^+ W_H^-$, $e_H^+ e_H^-$, and $\nu_{eH} \bar{\nu}_{eH}$, the resultant accuracies of the mass extractions turns out to be as follows [534].

<table><thead><tr><th></th><th>M<sub>AH</sub></th><th>M<sub>WH</sub></th><th>M<sub>ZH</sub></th><th>M<sub>eH</sub></th><th>M<sub>νeH</sub></th></tr></thead><tbody><tr><th>Accuracy</th><td>1.3%</td><td>0.20%</td><td>0.56%</td><td>0.46%</td><td>0.1%</td></tr></tbody></table>

Since the relevant physics of the LHT model is described with only two model parameters $f$ and $\kappa_e$, the masses of non-colored new particles are also given by the parameters. Performing these model-independent mass measurements

<footnote n="27">²⁷The Higgs mass is assumed to be 134 GeV, because this analysis has been performed before the discovery of the Higgs-like boson. The result of the analysis is not changed significantly even if the Higgs mass is set to be around 125 GeV.</footnote>
therefore provides strong evidence that the discovered new particles are indeed LHT particles. The parameters $f$ and $\kappa_e$ are eventually extracted from the measurements very accurately; $f$ and $\kappa_e$ are extracted at accuracies of 0.16% and 0.01%.

More interestingly, by assuming the vertex structures of the LHT model (i.e. the Lorentz structure, the ratio of right- and left-handed couplings, etc.), it is possible to extract the couplings concerning heavy gauge bosons/heavy leptons through cross section measurements. There are a total of eight vertices concerning the five pair production processes. Extracting all the couplings is therefore possible by measuring the total cross sections of the five processes and the angular distribution (the difference cross section) of the produced heavy gauge boson for appropriate three processes. See reference [534] for more detailed strategy to extract the couplings. Though numerical simulations for the three differential cross sections are not performed yet, the measurement accuracies for the five total cross sections have already been obtained as follows.

<table><thead><tr><th></th><th>A<sub>H</sub>Z<sub>H</sub></th><th>Z<sub>H</sub>Z<sub>H</sub></th><th>e<sup>+</sup><sub>H</sub>e<sup>-</sup><sub>H</sub></th><th>ν<sub>eH</sub>ν<sub>eH</sub></th><th>W<sup>+</sup><sub>H</sub>W<sup>-</sup><sub>H</sub></th></tr></thead><tbody><tr><td>Accuracy</td><td>7.70%</td><td>0.859%</td><td>2.72%</td><td>0.949%</td><td>0.401%</td></tr></tbody></table>

Only $Z_H A_H$ process has been analyzed with $500 \text{ fb}^{-1}$ data at $\sqrt{s} = 500$ GeV running, while others have been done with the same luminosity at 1 TeV running.

We next consider the direct productions of colored new particles. Among several colored new particles, the most important one is the top partner $T$ (and its T-parity partner $T_-$), because it is responsible for the cancellation of the quadratically divergent correction to the Higgs mass term from top loop. Since the top partner has a color-charge, it is expected to be constrained by the LHC experiment when its mass is not heavy. Thus we summarize the current status of the constraint before going to discuss the physics of the top partner at future linear collider experiments.

The most severe limit on the mass of the top partner comes from its pair production process followed by the decay $T \to bW$ [538]. The limit is $m_T > 650$ GeV at 95% C.L. with assuming $\text{BR}(T \to bW) = 1$. Since the top partner has other decay channels like $T \to tZ/T \to th$ and the branching fraction to $bW$ is typically about 40%, the actual limit on the mass is $m_T > 500$ GeV. On the other hand, the T-parity partner of the top partner $T_-$ decays into $tA_H$ with $\text{BR}(T_\to tA_H) \simeq 1$. The most severe limit on its mass again comes from its pair production process, which gives $m_{T_-} > 420$ GeV at 95% C.L. when $A_H$ is light enough [539].

The physics of the top partner at future linear collider experiments has been discussed in some details in reference [537]. When $m_T \simeq 500$ GeV, the cross section of its pair production process ($e^+e^- \to t\bar{T}$) is $\mathcal{O}(100)$ fb, while that of the associate production process ($e^+e^- \to t\bar{T} + \bar{t}T$) is $\mathcal{O}(1-10)$ fb with appropriate center of mass energy. It

has been shown that the Yukawa coupling of the top partner and the coupling of the interaction between $h$, $t$, and $T$ can be precisely measured with use of the threshold productions of these processes. Since these couplings are responsible for the cancellation of the quadratically divergent correction to the Higgs mass term from top loop, these measurements will give a strong test of the LH model.

The physics of the T-parity partner $T_-$ at future linear collider experiments has been discussed in some details in reference [536]. When $m_{T_-} \simeq 500$ GeV, the cross section of its pair production process ($e^+e^- \to T_-\bar{T}_-$) is $\mathcal{O}(100)$ fb with appropriate center of mass energy. Since $T_-$ decays into $tA_H$, the masses of both $T_-$ and $A_H$ can be precisely measured using the energy distribution of reconstructed top quarks, which will provide an excellent test of the LHT model by comparing this signal with those of non-colored new particles. Furthermore, it has also been pointed out that the process can be used to discriminate new physics models at the TeV scale. This is because many new physics models predict similar processes, a new colored particle decaying into $t$ and an invisible particle like a squark decaying into $t$ and a neutralino in the MSSM.

As a recent review and recent studies for current status of new particles and dark matter in LHT, please see also [246, 540–542].

## 2.11 Testing Higgs physics at the photon linear collider 28

A photon collider (hereafter we use abbreviation PLC – photon linear collider) is based on photons obtained from laser light back-scattered from high-energy electrons of Linear Collider (LC). Various high energy gamma-gamma and electron-gamma processes can be studied here. With a proper choice of electron beam and laser polarization, the high-energy photons with high degree polarization (dependent on energy) can be obtained. The direction of this polarization can be easily changed by changing the direction of electron and laser polarization. By converting both electron beams to the photon beams one can study $\gamma\gamma$ interactions in the energy range up to $\sqrt{s_{\gamma\gamma}} \sim 0.8 \cdot \sqrt{s_{ee}}$, whereas by converting one beam only the $e\gamma$ processes can be studied up to $\sqrt{s_{e\gamma}} \sim 0.9 \cdot \sqrt{s_{ee}}$ [543].

In a nominal LC option, i.e. with the electron-beam energy of 250 GeV, the geometric luminosity $L_{geom} = 12 \cdot 10^{34} \text{cm}^{-2} \text{s}^{-1}$ can be obtained, which is about four times higher than the expected $e^+e^-$ luminosity. Still, the luminosity in the high energy $\gamma\gamma$ peak (see Fig. 96) corresponds to about $1/3$ of the nominal $e^+e^-$ luminosity – so we expect $L_{\gamma\gamma}(\sqrt{s_{\gamma\gamma}} > 0.65 \cdot \sqrt{s_{ee}})$ equal to about $100 \text{ fb}^{-1}$ per year ($400 \text{ fb}^{-1}$ for a whole energy range) [544, 545]. Adjusting

<footnote n="28">28 Maria Krawczyk and Ilya Ginzburg: We are grateful to Filip Żarnecki for clarification of old analyses as well as to Jan Kalinowski.</footnote>
<figure><img src="image_99.png" /><figcaption>**Fig. 96** The distribution of $\gamma\gamma$ and $e\gamma$ center-of-mass energy $W$ with respect to the $e^+e^-$ energy (2$E_0$) from simulation of the PLC luminosity spectra [544]. Contributions of various spin states of produced system are shown.</figcaption></figure>

the initial electron beam energy and direction of polarizations of electrons and laser photons at fixed laser photon energy one can vary a shape of the $\gamma\gamma$ effective mass spectrum.

At a $\gamma\gamma$ collider the neutral C-even resonance with spin 0 can be produced, in contrast to C-odd spin 1 resonances in the $e^+e^-$ collision. Simple change of signs of polarizations of incident electron and laser photon for one beam transforms PLC to a mode with total helicity 2 at its high-energy part. It allows to determine degree of possible admixture of state with spin 2 in the observed Higgs state. The s-channel resonance production of $J^{PC} = 0^{++}$ particle allows to perform precise measurement of its properties at PLC.

In summer 2012 a Higgs boson with mass about 125 GeV has been discovered at LHC [87]. We will denote this particle as $\mathcal{H}$. The collected data [546, 547] allow to conclude that the SM-like scenario, suggested e.g. in [548,549], is realized [550]: all measured $\mathcal{H}$ couplings are close to their SM-values in their absolute value. Still following interpretations of these data are discussed: A) $\mathcal{H}$ is Higgs boson of the SM. B) We deals with phenomenon beyond SM, with $\mathcal{H}$ being some other scalar particle (e.g. one of neutral Higgs bosons of Two Higgs Doublet Model (2HDM) – in particular MSSM, in the CP conserving 2HDM that are $h$ or $H$). In this approach following opportunities are possible: 1) Measured couplings are close to SM-values, however some of them (especially the $t\bar{t}H$ coupling) with a "wrong" sign. 2) In addition some new heavy charged particles, like $H^\pm$ from 2HDM, can contribute to the loop couplings. 3) The observed signal is not due to one particle but it is an effect of two or more particles, which were not resolved experimentally

– the degenerated Higgses. Each of these opportunities can lead to the enhanced or suppressed, as compared to the SM predictions, $\mathcal{H}\gamma\gamma$, $\mathcal{H}gg$ and $\mathcal{H}Z\gamma$ loop-coupling.

The case with the observed Higgs-like signal being due to degenerated Higgses $h_i$ demands a special effort to diagnose it. In this case the numbers of events with production of some particle $x$ are proportional to sums like $\sum_i (\Gamma_i^x / \Gamma_i^{tot}) \Gamma_i^{gg}$. Data say nothing about couplings of the individual Higgs particles and there are no experimental reasons in favor of the SM-like scenario for one of these scalars. In such case each of degenerated particles have low total width, and there is a hope that the forthcoming measurements at PLC can help to distinguish different states due to much better effective mass resolution. The comparison of different production mechanisms at LHC, $e^+e^-$ LC and PLC will give essential impact in the problem of resolution of these degenerated states. Below we do not discuss the case with degenerated Higgses with masses $\sim$125 GeV in more details, concentrating on the case when observed is one Higgs boson $\mathcal{H}$, for which the SM-like scenario is realized.

In the discussion we introduce useful relative couplings, defined as ratios of the couplings of each neutral Higgs boson $h^{(i)}$ from the considered model, to the gauge bosons $W$ or $Z$ and to the quarks or leptons ($j = V(W,Z), u, d, l$...), to the corresponding SM couplings: $\chi_j^{(i)} = g_j^{(i)} / g_j^{SM}$. Note that all couplings to EW gauge bosons $\chi_V^{(i)}$ are real, while the couplings to fermions are generally complex. For CP-conserving case of 2HDM we have in particular $\chi_j^h, \chi_j^H, \chi_j^A$ (with $\chi_V^A = 0$), where couplings of fermions to $h$ and $H$ are real while couplings to $A$ are purely imaginary.

The SM-like scenario for the observed Higgs $\mathcal{H}$, to be identified with some neutral $h^{(i)}$, corresponds to $|\chi_j^\mathcal{H}| \approx 1$. Below we assume this scenario is realized at present.

It is known already from a long time that the PLC is very good observatory of the scalar sector of the SM and beyond SM, leading to important and in many cases complementary to the $e^+e^-$ LC case tests of the EW symmetry breaking mechanism [551]-[552]. The $e^+e^-$ LC, together with its PLC options ($\gamma\gamma$ and $e\gamma$), is very well suited for the precise study of properties of this newly discovered $\mathcal{H}$ particle, and other scalars. In particular, the PLC offers a unique opportunity to study resonant production of Higgs bosons in the process $\gamma\gamma \to$ Higgs which is sensitive to charged fundamental particles of the theory. In principle, PLC allows to study also resonant production of heavier neutral Higgs particles from the extension of the SM. Other physics topic which could be studied well at PLC is the CP property of Higgs bosons. Below we discuss the most important aspects of the Higgs physics which can be investigated at PLC. Our discussion is based on analyses done during last two decades
and takes into account also some recent "realistic" simula-
tions supporting those results.

2.11.1 *Studies of 125 GeV Higgs* ℏ

The discussion in this section is related to the case when
$\mathcal{H}$ is one of the Higgs bosons $h^{(i)}$ of 2HDM. In the CP
conserving case of 2HDM it can be either $h$ or $H$.

<figure><img src="image_100.png" /><figcaption>**Fig. 97** Distributions of the corrected invariant mass, $W_{corr}$, for selected $b\bar{b}$ events; contributions of the signal, for $M_{HSM} = 120$ GeV, and of the different background processes, are shown separately [553].</figcaption></figure>

Several NLO analyses of the production at the PLC of a
light SM-Higgs boson *H*<sub>SM</sub> decaying into *b*&#x0304;*b* final state were
performed, including the detector simulation, eg. [554]–
[555]. These analyses demonstrate a high potential of this
collider to measure accurately the Higgs two-photon width.
By combining the production rate for γγ → *H*<sub>SM</sub> → *b*&#x0304;*b* (Fig.
97), to be measured with 2.1 % accuracy, with the measure-
ment of the BR(*H*<sub>SM</sub> → *b*&#x0304;*b*) at *e*<sup>+</sup>*e*<sup>-</sup> LC, with accuracy ~
1 %, the width Γ(*H*<sub>SM</sub> → γγ) for *H*<sub>SM</sub> mass of 120 GeV can
be determined with precision ~ 2 %. This can be compared
to the present value of the measured at LHC signal strength
for 125 GeV ℏ particle, which ratio to the expected signal
for SM Higgs with the same mass (approximately equal to
the ratio of |g<sub>γγHSM</sub>|<sup>2</sup> / |g<sub>γγHSM</sub>|<sup>2</sup>), are 1.17±0.27 and 1.14<sup>+0.26</sup><sub>-0.23</sub>
from ATLAS [94] and CMS [556], respectively.

The process $\gamma\gamma \to \mathcal{H} \to \gamma\gamma$ is also observable at the
PLC with reasonable rate [555]. This measurement allows
to measure directly two-photon width of Higgs without as-
sumptions about unobserved channels, couplings, etc.

Neutral Higgs resonance couples to photons via loops
with charged particles. In the Higgs $\gamma\gamma$ coupling the heavy

charged particles, with masses generated by the Higgs mech-
anism, do not decouple. Therefore the $\mathcal{H} \to \gamma\gamma$ partial width
is sensitive to the contributions of charged particles with
masses even far beyond the energy of the $\gamma\gamma$ collision. This
allows to recognize which type of extension of the minimal
SM is realized. The $H^+$ contribution to the $\mathcal{H}\gamma\gamma$ loop cou-
pling is proportional to $\mathcal{H}H^+H^-$ coupling, which value and
sign can be treated as free parameters of model$^{29}$. The sim-
plest example gives a 2HDM with type II Yukawa interac-
tion (2HDM II). For a small $m_{12}^2$ parameter, see Sect. 2.6,
the contribution of the charged Higgs boson $H^+$ with mass
larger than 400 GeV leads to 10% suppression in the $\mathcal{H} \to
\gamma\gamma$ decay width as compare to the SM one, for $M_{\mathcal{H}}$ around
120 GeV [548, 549], Table 24 (solution A). The enhance-
ment or decreasing of the $\mathcal{H}\gamma\gamma$ coupling is possible, as dis-
cussed for 2HDM with various Yukawa interaction models
in [557]–[558] as well in the Inert Doublet Model$^{30}$ [559,
560].

In the Littlest Higgs model a 10% suppression of the $\gamma\gamma$ decay width for $M_{\mathcal{H}} \approx 120$ GeV is expected due to the new heavy particles with mass around 1 TeV at the suitable scale of couplings for these new particles [467, 502], see Fig. 98.

<figure><img src="image_101.png" /><figcaption>**Fig. 98** Ratio $\frac{\Gamma(h \rightarrow \gamma\gamma)}{\Gamma(h \rightarrow \gamma\gamma)^{\text{SM}}}$ as a function of the mass scale of the new physics $f$ in the Littlest Higgs model [467], for different Higgs boson masses. “Accessible” indicates the possible variation of the rate for fixed $f$.</figcaption></figure>

<footnote n="29"><sup>29</sup>Except if some additional symmetry is present in the model.</footnote>

<footnote n="30">30 That is the $Z_2$ symmetric 2HDM where one Higgs doublet plays a role of SM Higgs field $\phi_S$, interacting with fermions as in Model I, with the SM-like Higgs boson $h$ and another Higgs doublet $\phi_D$, having no v.e.v.. The latter one contains four scalars $D, D^A, D^\pm$, the lightest among them $D$ (analog of $H$ of 2HDM) can be DM particle, scalars $D^A$ and $D^\pm$ (analog of $A$ and $H^\pm$, respectively).</footnote>
The Higgs $\gamma\gamma$ loop coupling is sensitive to the relative signs of various contributions. For example, in 2HDM II sign of some Yukawa couplings may differ from the SM case, still strength (ie. absolute value) of all squared direct Higgs couplings to WW/ZZ and fermions being as in the SM. This may lead to the enhancement of the $\mathcal{H} \to \gamma\gamma$ decay-width with respect to the SM predictions, up to 2.28 for a "wrong" sign of the $\mathcal{H}tt$ for $M_{\mathcal{H}} = 120$ GeV (1.28 for $\mathcal{H} \to gg$ and 1.21 for $\mathcal{H} \to Z\gamma$, respectively) coupling, Table 24 (solution $B_{\mathcal{H}t}$), [548]$^{31}$. The "wrong" sign of $\mathcal{H}bb$ coupling (solution $B_{\mathcal{H}b}$ in Table 24) could lead to an enhancement in the $\mathcal{H} \to gg$, and in the corresponding rate for gluon fusion of Higgs at LHC, similarly as the "wrong" sign of $\mathcal{H}tt$ coupling. Such solution is still considered as a possible for 125 GeV $\mathcal{H}$ particle [546].

<table><caption>**Table 24** SM-like realizations in the 2HDM II [548], [549] together with ratios of loop-induced partial widths to their SM values at $M_{\mathcal{H}} = 120$ GeV, $M_{H^{\pm}} = 800$ GeV, $|m_{t2}| \le 40$ GeV$^2$.</caption><thead><tr><th>solution</th><th>basic couplings</th><th>|χ<sub>gg</sub>|<sup>2</sup></th><th>|χ<sub>γγ</sub>|<sup>2</sup></th><th>|χ<sub>Zγ</sub>|<sup>2</sup></th></tr></thead><tbody><tr><td>A<sub>ħ</sub></td><td>χ<sub>V</sub> ≈ χ<sub>b</sub> ≈ χ<sub>t</sub> ≈ ±1</td><td>1.00</td><td>0.90</td><td>0.96</td></tr><tr><td>B<sub>ħb</sub></td><td>χ<sub>V</sub> ≈ −χ<sub>b</sub> ≈ χ<sub>t</sub> ≈ ±1</td><td>1.28</td><td>0.87</td><td>0.96</td></tr><tr><td>B<sub>ħt</sub></td><td>χ<sub>V</sub> ≈ χ<sub>b</sub> ≈ −χ<sub>t</sub> ≈ ±1</td><td>1.28</td><td>2.28</td><td>1.21</td></tr></tbody></table>

The observed Higgs particle can have definite CP parity or can be admixture of states with different CP parity (CP-mixing). In the latter case the PLC provides the best among all colliders place for the study of such mixing. Here, the opportunity to simply vary polarization of photon beam allows to study this mixing via dependence of the production cross section on the incident photon polarization [562–568]. In particular, the change of sign of circular polarization (++↔−−) results in variation of production cross section of the 125 GeV Higgs in 2HDM by up to about 10%, depending on a degree of CP-admixture. Using mixed circular and linear polarizations of photons gives opportunity for more detailed investigations [569].

The important issue is to measure a Higgs selfcoupling, $\mathcal{H}\mathcal{H}\mathcal{H}$. In the SM this selfcoupling is precisely fixed via Higgs mass (and v.e.v. $v = 246$ GeV), while deviations from it's SM value would be a clear signal of more complex Higgs sector. Both at the $e^{+}e^{-}$ collider and at the $\gamma\gamma$ collider the two neutral Higgs bosons are produced in processes both with and without selfinteraction, namely

$$ \begin{gathered} e^{+}e^{-} \to Z \to \mathcal{H}(Z \to Z\mathcal{H}) \oplus e^{+}e^{-} \to Z \to Z(\mathcal{H} \to \mathcal{H}\mathcal{H}) \\ \gamma\gamma \to \text{loop} \to \mathcal{H}\mathcal{H} \oplus \gamma\gamma \to \text{loop} \to \mathcal{H} \to \mathcal{H}\mathcal{H}. \end{gathered} $$

In the SM case the cross sections for above processes are rather low but measurable, so that coupling under interest

can be extracted, both in the $e^{+}e^{-}$ and $\gamma\gamma$, modes of $e^{+}e^{-}$ LC, see [570]-[571]. The feasibility of this measurement at a PLC has been performed recently in [572]. For Higgs mass of 120 GeV and the integrating luminosity 1000 fb$^{-1}$ the statistical sensitivity as a function of the $\gamma\gamma$ energy for measuring the deviation from the SM Higgs selfcoupling $\lambda = \lambda_{SM}(1 + \delta\kappa)$ has been estimated. The optimum $\gamma\gamma$ collision energy was found to be around 270 GeV for a such Higgs mass, assuming that large backgrounds due to WW/ZZ and bbbb production can be suppressed for correct assignment of tracks. As a result, the Higgs pair production can be observed with a statistical significance of 5 $\sigma$ by operating the PLC for 5 years.

The smaller but interesting effects are expected in $e\gamma \to e\mathcal{H}$ process with $p_{\perp e} > 30$ GeV, where $\mathcal{H}Z\gamma$ vertex can be extracted with reasonable accuracy [573].

### 2.11.2 Studies of heavier Higgses, for 125 GeV $\mathcal{H} = h^{(1)}$

A direct discovery of other Higgs bosons and measurement of their couplings to gauge bosons and fermions is necessary for clarification the way the SSB is realized. In this section we consider the case when observed 125 GeV Higgs is the lightest neutral Higgs, $\mathcal{H} = h^{(1)}$ (in particular in the CP-conserving case this means $\mathcal{H} = h$). A single Higgs production at $\gamma\gamma$ collider allows to explore roughly the same mass region for neutral Higgs bosons at the parent $e^{+}e^{-}$ LC but with higher cross section and lower background. The $e\gamma$ collider allows in principle to test wider mass region in the process $e\gamma \to eH, eA$ however with a lower cross section.

Before general discussion, we present some properties of one of the simplest Higgs model beyond the minimal SM, namely 2HDM (in particular, also the Higgs sector of MSSM), having in mind that the modern data are in favour of a SM-like scenario. Let us enumerate here some important properties of 2HDM for each neutral Higgs scalar $h^{(i)}$ in the CP conserving case $h^{(1)} = h, h^{(2)} = H, h^{(3)} = A$:

(i) For an arbitrary Yukawa interaction there are sum rules for coupling of different neutral Higgses to gauge bosons $V = W, Z$ and to each separate fermion $f$ (quark or lepton)

$$ \sum_{i=1}^{3} (\chi_{V}^{(i)})^{2} = 1. \qquad \sum_{i=1}^{3} (\chi_{f}^{(i)})^{2} = 1. \qquad (82) $$

The first sum rule (to the gauge bosons) was discussed e.g. in [574]–[575]. The second one was obtained only for Models I and II of Yukawa interaction [576], however in fact it holds for any Yukawa sector [577].

In the first sum rule all quantities $\chi_V^{(i)}$ are real. Therefore, in SM-like case (i.e. at $|\chi_V^{(1)}| \approx 1$) both couplings $|\chi_V^{2,3}|$

<footnote>$^{31}$ The recent analysis of the LHC data leads to constraints of the relative $\mathcal{H}tt$ coupling $\chi_t^\mathcal{H}$ [561].</footnote>
are small. The couplings entering the second sum rule (for fermions) are generally complex. Therefore this sum rule shows that for $|\chi_f^{(1)}|$ close to 1, either $|\chi_f^{(2)}|^2$ and $|\chi_f^{(3)}|^2$ are simultaneously small, or $|\chi_f^{(2)}|^2 \approx |\chi_f^{(3)}|^2$.

(ii) For the 2HDM I there are simple relations, which in the CP conserved case are as follows

$$ \chi_u^{(h)} = \chi_d^{(h)}, \quad \chi_u^{(H)} = \chi_d^{(H)}. \qquad (83) $$

(iii) In the 2HDM II following relations hold:

a) The pattern relation among the relative couplings for each neutral Higgs particle $h^{(i)}$ [578,579]:

$$ (\chi_u^{(i)} + \chi_d^{(i)})\chi_V^{(i)} = 1 + \chi_u^{(i)}\chi_d^{(i)}. \qquad (84a) $$

b) For each neutral Higgs boson $h^{(i)}$ one can write a horizontal sum rule [580]:

$$ |\chi_u^{(i)}|^2 \sin^2 \beta + |\chi_d^{(i)}|^2 \cos^2 \beta = 1. \qquad (84b) $$

Below, in Table 25, we present benchmark points for the SM-like $h$ scenario in the CP conserving 2HDM II. The total widths for H and A for various $\chi_t^A = 1/\tan\beta$ are shown assuming with $\chi_V^h \approx 0.87$, $|\chi_V^H| = 0.5$ and $|\chi_t^h| = 1$ for H and A. 32

<table><caption>**Table 25** Total width (in MeV) of H, A in some benchmark points for the SM-like h scenario (M<sub>h</sub>=125 GeV) in the 2HDM ($\chi_V^h \approx 0.87$, $|\chi_V^H| = 0.5$ and $|\chi_t^h| = 1$). Results for tan β = 1/7, 1 and 7 are shown.</caption><thead><tr><th></th><th>Γ<sub>H</sub>, Γ<sub>A</sub></th><th></th><th>Γ<sub>H</sub>, Γ<sub>A</sub></th><th></th><th>Γ<sub>H</sub>, Γ<sub>A</sub></th></tr></thead><tbody><tr><td>M<sub>H,A</sub></td><td>tan β = 1/7</td><td></td><td>tan β = 1</td><td></td><td>tan β = 7</td></tr><tr><td>200</td><td>0.35</td><td>8 · 10<sup>-5</sup></td><td>0.35</td><td>4 · 10<sup>-3</sup></td><td>0.4</td><td>0.2</td></tr><tr><td>300</td><td>2.1</td><td>1.2 · 10<sup>-4</sup></td><td>2.1</td><td>6 · 10<sup>-3</sup></td><td>0.75</td><td>0.3</td></tr><tr><td>400</td><td>138</td><td>132</td><td>8.8</td><td>2.7</td><td>2.5</td><td>0.45</td></tr><tr><td>500</td><td>537</td><td>524</td><td>22.8</td><td>10.7</td><td>6.1</td><td>0.7</td></tr></tbody></table>

In the SM-like $h$ scenario it follows from sum rule (82) that the W-contribution to the Hγγ width is much smaller than that of would-be heavy SM Higgs, with the same mass, $M_{HSM} \approx M_H$. At the large tan β also H → tt, A → tt decay widths are extremely small, so that the total widths of H, A become very small 33.

Let us compare properties of heavy H, A in 2HDM with a would-be heavy SM Higgs-boson with the same mass. The cross section for production of such particles in the main gluon-gluon fusion channel, being $\propto \Gamma_{H,A}^{gg}\Gamma_{H,A}/M_H^3$, is lower than that in SM. At large tan β resonances H, A become very narrow, as discussed above, besides, the two-gluon decay width become about 1/ tan$^2$ β smaller. Consequently, these main at LHC production channels cross section are suppressed by roughly 1/ tan$^4$ β w.r.t. the would-be SM Higgs boson with the same mass and H and A can escape observation in these channels at the LHC. (The same is valid for e$^+$e$^-$ LC due to small value of $\chi_V^H$ for H and $\chi_V^A = 0$.)

Moreover, in MSSM with $M_h = 125$ GeV we can have heavy and degenerate H and A, $M_H \approx M_A$. At large tan β the discovery channel of H/A at LHC is $gg \to b\bar{b} \to b\bar{b}H/A$. Nevertheless, in some region of parameters, at intermediate tan β, these H and A are elusive at LHC. That is so called LHC wedge region [583], see the latest analysis [584]. The PLC allows to diminish this region of elusiveness, since here the H and A production is generally not strongly suppressed and the $b\bar{b}$ background is under control [250,581,582,585]. Figure 99 show that PLC allows to observe joined effect of H, A within this wedge region. Precision between 11% to 21 % for $M_A$ equal to 200-300 GeV, tan β = 7 of the Higgs-boson production measurement (μ =200 GeV (the Higgs mixing parameter) and A<sub>f</sub>=1500 GeV (the trilinear Higgs-sfermion couplings)) can be reached after one year [582]. To separate these resonances even in the limiting case $\chi_V^H = 0$ is a difficult task, since the total number of expected events is small.

At $\chi_V^H \neq 0$, taking $\chi_V^H \sim 0.3-0.4$ as an example (what is allowed by current LHC measurement of couplings of $\mathcal{H} = h$ to ZZ), an observation of H → ZZ decay channel can be good method for the H discovery in 2HDM. The signal γγ → H → WW, ZZ interferes with background of γγ → WW, ZZ, what results in irregular structure in the effective-mass distribution of products of reaction γγ → WW, ZZ (this interference is constructive and destructive below and above resonance, respectively). The study of this irregularity seems to be the best method for discovery of heavy Higgs, decaying to WW, ZZ [586], and to measure the corresponding $\phi_{\gamma\gamma}$ phase, provided it couples to ZZ/WW reasonably strong$^{34}$.

Just as it was described above for the observed 125 GeV Higgs, PLC provides the best among colliders place for the study of spin and the CP properties of heavy $h^{(2)}, h^{(3)}$. That are CP parity in the CP conserved case (with ($h^{(2)}, h^{(3)}$ = (H, A)), and (complex) degree of the admixtures of states with different CP parity, if CP is violated. This admixture

<footnote n="32">32 The total width $\Gamma_H$ differs from the total width $\Gamma_A$ by the W/Z contribution, since $\chi_V^A = 0$.</footnote>

<footnote n="33">33 At tan β ≪ 1 we obtain the strong interaction in the Higgs sector mediated by t-quarks, what is signalizing by the fact that the calculated in standard approach total widths of heavy H, A is becoming close to or even higher than the corresponding masses. Of course, in this case such tree-level estimates become inadequate. In the same manner at tan β > 70 corresponds to the region of a strong interaction in the Higgs sector mediated by b-quarks. We don't consider such scenarios.</footnote>

<footnote n="34">34 Similar calculations given in [587] demonstrate this opportunity for a 2HDM version $B_{hu}$.</footnote>
<figure><img src="image_102.png" /><figcaption>**Fig. 99** Top: Production of A and H, with parameters corresponding to the LHC wedge, at the γγ collider. Exclusion and discovery limits obtained for NLC collider for $\sqrt{ee} = 630$ GeV, after 2 or 3 years of operation [581]. Bottom: The case $M_H = M_A = 300$ GeV at $\chi_V^H \approx 0$ in the MSSM. Distributions of the corrected invariant mass $W_{corr}$ for selected $b\bar{b}$ events at $\tan\beta = 7$ [582].</figcaption></figure>

determines dependence on the Higgs production cross section on direction of incident photon polarization [563, 565–568,589]. These polarization measurements are useful in the study of the case when the heavy states $h^{(2)}$, $h^{(3)}$ ($H, A$) are degenerated in their masses. A study [569] shows that the 3-years operation of PLC with linear polarization of photons, the production cross-section of the H and A corresponding to the LHC wedge for MSSM (with mass $\sim 300$ GeV) can be separately measured with precision 20%. Pure scalar versus

<figure><img src="image_103.png" /><figcaption>**Fig. 100** The specific decay angular distributions $\Sigma_i$ in the $\gamma\gamma \to h^{(i)} \to t\bar{t}$ process in dependence on the $t\bar{t}$ invariant mass for the scalar (dashed) and pseudoscalar (thick solid) $h^{(i)}$ with $M_H = 400$ GeV [588].</figcaption></figure>

pure pseudoscalar states can be distinguished at the $\sim 4.5\sigma$ level.

We point out on important difference between the CP mixed and the mass-degenerate states. In the degeneracy of some resonances A and B one should distinguish two opportunities:

a) instrumental degeneracy when $|M_B - M_A| > \Gamma_B + \Gamma_A$, with mass difference within a mass resolution of detector. This effect can be resolved with improving of a resolution of the detector

b) physical degeneracy when $|M_B - M_A| < \Gamma_B + \Gamma_A$.

In the CP conserving case for both types of degeneracy the overlapping of H, A resonances does not result in their mixing, and the production of a resonante state cannot vary with change of sign of photon beam polarization. In the CP violating case, the overlapping of resonances results in additional mixing of incident $h^{(2)}$, $h^{(3)}$ states, and the production cross-section varies with the change of polarization direction of incident photons.

Another method for study of CP content of a produced particle provides the measurement of angular distribution of decay products [562, 590, 591]. In the $t\bar{t}$ decay mode one can perform a study of the CP-violation, exploiting fermion polarization. The interference between the Higgs exchange and the continuum amplitudes can be sizable for the polarized photon beams, if helicities of the top and anti top quarks are measured. This enables to determine the CP property of the Higgs boson completely [588, 592], Fig. 100.

The discovery of charged Higgses $H^{\pm}$ will be a crucial signal of the BSM form of Higgs sector. These particles can be produced both at the LC ($e^{+}e^{-} \to H^{+}H^{-}$) and at the PLC ($\gamma\gamma \to H^{+}H^{-}$). These processes are described well by QED. The $H^{+}H^{-}$ production process at PLC has worse energy-threshold behaviour than the corresponding process at the LC, but higher cross section. On the other hand, the
process $e^{+}e^{-} \to H^{+}H^{-}$ can be analysed at LC better by measurements of decay products due to known kinematics. At the PLC the variation of an initial-beam polarization could be used for checking up spin of $H^{\pm}$ [593]. See also analysis for flavour violation models in [594, 595].

After a $H^{\pm}$ discovery, the observation of processes $e^{+}e^{-} \to H^{+}H^{-}h$ and $\gamma\gamma \to H^{+}H^{-}h$, $H^{+}H^{-}H$, $H^{+}H^{-}A$ may provide direct information on a triple Higgs ($H^{+}H^{-}h$) coupling $\lambda$, with cross sections in both cases $\propto \alpha^2\lambda^2$. The $\gamma\gamma$ collisions are preferable here due to a substantially higher cross section and opportunity of study polarization effects in the production process via variation of initial photon polarizations.

Synergy of LHC, LC and PLC colliders may be useful in determination of Higgs couplings, as different production processes dominating at these colliders lead to different sensitivity to gauge and Yukawa couplings. For example LC Higgstrahlung leads to large sensitivity to the Higgs coupling to the EW gauge bosons, while at PLC $\gamma\gamma$ and $Z\gamma$ loop couplings depend both on the Higgs gauge and Yukawa couplings, as well as on coupling with $H^{+}$, see results both for CP conservig/CP violating in e.g. [591, 596, 597].
# 3 Top and QCD<fnref n="35" />

## 3.1 Introduction

The experimental studies of electron-positron annihilation into hadrons were historically essential to establish Quantum Chromodynamics (QCD) as the theory of strong interaction: from the measurement of the R-ratio $\sigma_{had}/\sigma_t$, the number of colors could be determined, the discovery of three-jet events at PETRA provided the first direct indication of the gluon, and the measurement of the Bengtson-Zerwas and Nachtmann-Reiter angles illustrated the non-abelian gauge structure of QCD —to name only a few milestones on the road to develop the theory of strong interactions.

At the Large Electron Positron Collider (LEP) the experimental tests of QCD were further refined. Three, four, and even five-jet rates were measured with unprecedented accuracy. These measurements provided important input to constrain the structure constants of the underlying non-abelian gauge group and to determine the QCD coupling constant $\alpha_s$ with high precision. The R-ratio and the forward-backward asymmetry were studied in detail including precise investigations of the flavor (in-)dependence. At SLD the measurements were extended to polarized electrons in the initial state. The tremendous experimental effort has been complemented over the time by a similar effort on the theory side: Next-to-leading order (NLO) calculations have been performed for event-shape observables and jet-rates involving jets originating from massless as well as massive quarks. New jet-algorithms with an improved theoretical behavior were developed. Very recently theoretical predictions for three-jet rates have been extended to next-to-next-to-leading order (NNLO) accuracy. For inclusive hadron production the theoretical predictions have been extended to N³LO accuracy in QCD. Beyond fixed order perturbation theory also power corrections and soft gluon resummation have been considered. All this effort has paved the way to establish QCD as the accepted theory of strong interaction.

Today QCD is a mature theory and no longer the primary target of experimental studies. Assuming QCD as the underlying theory of strong interaction the precision measurements possible in $e^{+}e^{-}$ annihilation can be used to determine fundamental parameters like coupling constants and particle masses. For example three-jet rates at LEP have been used to measure the QCD coupling constant and the b-quark mass. Since the small b-quark mass leads only to effects of the order of 5% at the Z-resonance (compared to massless b-quarks), this example nicely illustrates the impressive theoretical and experimental precision reached. The steadily increasing experimental accuracy together with LHC as a “QCD machine” and the perspective of a future linear collider have kept QCD a very active field, where significant

progress has been achieved in the last two decades. Conceptually effective field theories have been further developed with specific realizations for dedicated applications. For example, soft collinear effective theory (SCET) is nowadays used to systematically improve the quality of the perturbative expansion through the resummation of logarithmically enhanced contributions. SCET may also help to deepen our current understanding of factorization of QCD amplitudes. Applications to the production of top-quark pair production have also demonstrated the power of this approach to assess the impact of non-perturbative corrections. Non-relativistic QCD (NRQCD) provides the well established theoretical framework to analyze the threshold production of top-quark pair production where binding effects between top quarks are important. The theoretical description of unstable particles in the context of effective field theories have demonstrated another successful application of effective field theories. Theoretical predictions for a future Linear Collider will profit from the improved theoretical understanding in terms of an increased precision. Recently we have witnessed a major breakthrough in the development of technologies for one-loop calculations. One-loop calculations involving multiplicities of five or even more particles in the final state —which were a major bottleneck over several years in the past— are today regularly performed for a variety of different processes. The new techniques have also led to an increased automation of the required calculations. Various programs are now publicly available to generate NLO matrix elements. Furthermore a standardized interface allows the phase space integration within MC event generators like for example Sherpa. Also the two-loop technology has seen important progress and is now a continuously growing field. The description of threshold effects in the production of heavy particles notably heavy quarks has been further improved to include higher order corrections in the perturbative expansion.

The detailed understanding of QCD achieved today has been proven essential for the current interpretation of LHC results and the very precise measurements performed so far. Evidently LHC data can also be used for QCD studies in the TeV regime. However owing to the complicated hadronic environment it will be difficult to reach accuracies at the per cent level or even below. In contrast $e^{+}e^{-}$ Linear Colliders allows to test QCD at the sub per cent level at energies above the Z resonance. The reachable precision of any measurement involving strongly interacting particles will depend on the ability of making accurate predictions within QCD. QCD studies will thus continue to play an important role at a future Linear Collider. Since non-perturbative effects are intrinsically difficult to assess, the highest accuracy —and thus the most precise tests of the underlying theory— can be reached for systems, where these effects are believed to be small or even negligible. A particular interesting example

<footnote n="35">³⁵Authors: Frank Simon, Peter Uwer, Kiyo Yuichiro</footnote>
is provided by top-quark physics. With a mass almost as heavy as a Gold atom the top quark is the heaviest elementary fermion discovered so far.

Top-quarks have unique properties making them a highly interesting research topic on their own right. The large mass leads to an extremely short life time such that top quarks decay before they can form hadronic bound states. This simple observation has several important consequences. First of all the finite width essentially cuts off non-perturbative physics such that top-quark properties can be calculated with high accuracy in perturbative QCD. Top-quark physics thus allows to study the properties of a 'bare quark'. In the Standard Model top quarks decay almost exclusively through electroweak interactions into a W-boson and a b-quark. The parity violating decay offers the possibility to study the polarization of top-quarks through the angular distribution of the decay products. Polarization studies, which are difficult in the case of the lighter quarks since hadronization usually dilutes the spin information, offer an additional opportunity for very precise tests of the underlying interaction. This is of particular interest since top-quark physics is controlled in the Standard Model by only 'two parameters': The top-quark mass and the relevant Cabbibo-Kobayashi-Maskawa matrix elements. Once these parameters are known top-quark interactions are predicted through the structure of the Standard Model. In particular all the couplings are fixed through local gauge invariance. Top-quark physics thus allows to test the consistency of the Standard Model with high precision. A prominent example is the relation between the top-quark mass and the mass of the W-boson. Obviously the accuracy of such tests is connected to the precision with which the top-quark mass —as most important input parameter— can be determined. While the LHC achieved already an uncertainty in the mass measurements of one GeV, it is expected that a Linear Collider will improve this accuracy by an order of magnitude down to 100 MeV or even below. Using top-quarks to test the Standard Model with high precision and search for new physics is very well motivated. In addition to the high experimental and theoretical accuracy achievable in top-quark measurements, top-quarks provide a particular sensitive probe to search for Standard Model extensions. Due to its large mass top-quarks are very sensitive to the mechanism of electroweak symmetry breaking. In many extensions of the Standard Model which aim to present an alternative mechanism of electroweak symmetry breaking top-quarks play a special role. It is thus natural to ask whether the top-quark mass, being so much larger than the masses of the lighter quarks, is indeed produced by the Englert-Brout-Higgs-Guralnik-Hagen-Kibble mechanism. A detailed measurement of the top-quark Yukawa coupling to the Higgs boson, which is very difficult to assess at a hadron collider, will provide a crucial information to answer this question. In the past top-

quarks have been extensively studied at the Tevatron and the LHC. With exception of the forward-backward charge asymmetry studied at the Tevatron the measurements are in very good agreement with the Standard Model predictions. However it should be noted that due to the complex environment at a hadron collider the accuracy is often limited. The top-quark mass which is now measured with sub per cent accuracy represents an important exception. While the measurements at the Tevatron and the LHC are perfectly consistent the precise interpretation of the measured mass value in terms of a renormalized parameter in a specific scheme is still unclear. The mass which is determined from a kinematical reconstruction of the top-quark decay products is assumed to be close to the pole mass. Since precise theoretical predictions for the measured observable are lacking the exact relation between the measured mass and the pole mass has not been quantified so far. An alternative method in which the mass is determined from cross section measurements where the renormalization is uniquely fixed through a higher order calculation gives consistent results. However the experimental uncertainties of this method are quite large owing to the weak sensitivity of the total cross section with respect to the top-quark mass. A new method using top-quark pair production in association with an additional jet represents an interesting alternative but will most likely also be limited in precision to one GeV. Although not better in precision the advantage of this method lies in the fact that the method gives a clear interpretation of the measured value in a specific renormalization scheme. Given the importance of a precise determination of the top-quark mass, going significantly below one GeV may remain the task of a future Linear Collider.

In the following we shall briefly describe in Section 3.2 recent progress in QCD with a special emphasis on $e^{+}e^{-}$ annihilation. In Section 3.3 we summarize new developments in top-quark physics in particular concerning the theoretical understanding of top-quark production at threshold. In the last Section we briefly comment on the physics potential of a future linear collider with respect to QCD and top-quark physics. In particular the prospects of a precise measurement of the top-quark mass are discussed.

## 3.2 Recent progress in QCD

### 3.2.1 Inclusive hadron production

The inclusive cross section for the production of hadrons in $e^{+}e^{-}$ annihilation or alternatively the R-ratio is a fundamental observable to be studied at any $e^{+}e^{-}$ collider. For hadrons originating from the fragmentation of massless quarks substantial progress has been obtained over the last 10 years. Starting from the $n_f^2\alpha_s^4$ contribution presented in Ref. [598] more than ten years ago the full N$^3$LO result including all
color structures have been derived over the last decade in a ground breaking calculation [599–601]. Using $\sin^2 \theta_W = 0.231$ for the sine squared of the weak mixing angle the result for the hadronic decay width of the Z-boson reads [601]:

$$ \Gamma_Z = \frac{G_F M_Z^3}{24\pi\sqrt{2}} R^{nc} \qquad (85) $$

$$ R^{nc} = 20.1945 + 20.1945 a_s + (28.4587 - 13.0575 + 0) a_s^2 + (-257.825 - 52.8736 - 2.12068) a_s^3 + (-1615.17 + 262.656 - 25.5814) a_s^4, \qquad (86) $$

with $a_s = \alpha_s(M_Z)/\pi$. The three terms inside the brackets display the non-singlet, axial singlet and vector singlet contributions. An important application of the improved theoretical description is the determination of the QCD coupling constant. It is thus interesting to investigate the impact of the newly calculated correction on the determined $\alpha_s$ value. For $\alpha_s(M_Z) = 0.1190$ the impact of the 4-loop correction on the extracted $\alpha_s$ value is found to be very small. A shift $\delta\alpha_s = -0.00008$ in the $\alpha_s$ value when extracted from the hadronic cross section is expected. For the quality of the perturbative expansion not only the size of the corrections is important but also the residual renormalization scale dependence. In Ref. [601] it has been shown that the scale dependence is also improved by including the four-loop contributions. As far as the order in the QCD coupling constant is concerned the R ratio is certainly one of the best known QCD observables.

### 3.2.2 Three-jet production at NNLO

Jet production in $e^+e^-$ annihilation is another classical QCD observable. The underlying physical picture explaining the outgoing bundles of hadrons called jets, is the production of colored high energetic partons in a short distance process. The partons are then assumed to fragment into uncolored hadrons. As a consequence, the naive expectation is, that the fragmentation products somehow share the momentum of the mother parton. This simple picture is reflected in iterative jet algorithms which try to bridge the gap between the experimentally observed hadrons and the partonic final states used in the theoretical predictions. To make contact between theory and experiment, in both analysis the same jet algorithms are applied and the results are compared. In Born approximation the number of partons is equal to the number of jets. In this case each jet is thus ‘modeled’ by a single parton. Including additional real radiation in higher order predictions allows for the recombination of two or even more partons into one jet and gives thus an improved theoretical description of the jets. Three jet production in $e^+e^-$ annihilation is of particular interest since the three jet rate is directly proportional to the coupling constant of strong interaction. Until recently the precision of $\alpha_s$ extracted from

three jet rates was limited due to the unknown NNLO corrections. The main problems which had to be overcome were the evaluation of the two-loop amplitudes for the process $e^+e^- \to (Z^*, \gamma^*) \to q\bar{q}g$ and the systematic cancellation of mass and infrared singularities present in individual contributions. The former problem was solved in Refs. [602–604]. The highly non-trivial combination of virtual corrections, real emission at one-loop order, and double real emission took another five years until completion. Predictions for different observables at NNLO accuracy in QCD have been presented in Refs. [605–612] by two competing groups. The fixed order NNLO calculation lead to a 10 % smaller central value for $\alpha_s$ [613]. In addition the inclusion of the NNLO corrections reduce the variation in $\alpha_s$ extracted from different event-shape observables. The NNLO corrections thus lead to a more coherent description of the data. Furthermore the scale uncertainty is reduced by a factor of two compared to the NLO calculation. However the scale uncertainty still dominates the extraction of $\alpha_s$ when compared to uncertainties due to finite statistics and hadronization. The scale uncertainty is roughly three times larger than the uncertainty due to hadronization. In Ref. [614] the fixed order NNLO predictions have been extended by resumming large logarithmic corrections due to multiple soft gluon emission at next-to-leading logarithmic accuracy (NLLA). It turns out, that the resummation has very little impact on the central value of $\alpha_s$ determined from different event shapes. However the theoretical uncertainties due to scale variations are slightly increased. As a final result

$$ \alpha_s(M_Z) = 0.1224 \pm 0.0009 (\text{stat.}) \pm 0.0009 (\text{exp.}) \pm 0.0012 (\text{had.}) \pm 0.0035 (\text{theo.}) \qquad (87) $$

is quoted [614]. Evidently the NNLO predictions will also find their application at a future linear collider. Even with a limited statistics a future measurement above the Z resonance will be interesting due to the possibility to further constrain $\alpha_s$ at high scale. It is also conceivable that the theoretical uncertainties are slightly reduced at higher energies due to the smaller value of $\alpha_s$.

### 3.2.3 NLO QCD corrections to 5-jet production and beyond

At the LEP experiments exclusive production of jet multiplicities up to five jets were studied experimentally. However until very recently only NLO results for four-jet production were available due to the tremendous growth in complexity of the theoretical calculations. In Ref. [615] the NLO QCD corrections to five-jet production are presented.

The virtual corrections were calculated using generalized unitarity (for more details about this method we refer to the next Section 3.2.4), relying to a large extend on amplitudes calculated in Ref. [616] where one-loop corrections to $W^+ + 3$-jet production in hadronic collisions were studied.
The real corrections are calculated using MadFKS [617]— an implementation of the Frixione-Kunszt-Signer (FKS) subtraction scheme [618] into Madgraph. The Durham jet algorithm is used to define the jets. Results for the five-jet rate, differential with respect to the parameter $y_{45}$, which determines the $y_{cut}$-value at which a five-jet event becomes a four-jet event, are shown. Furthermore the five-jet rate as function of the jet resolution parameter $y_{cut}$ is presented. In addition hadronization corrections are analyzed using the Sherpa event generator. At fixed order in perturbation theory it is found that the scale uncertainty is reduced from about [-30%, +45%] in LO to about [-20%, +25%] in NLO. In this analysis the renormalization scale has been chosen to be $\mu = 0.3\sqrt{s}$ and variations up and down by a factor of two were investigated. The central scale is chosen smaller than what is usually used for lower jet multiplicities. The reasoning behind this is that for increasing multiplicities the average transverse momentum per jet becomes smaller. This is taken into account by using $\mu = 0.3\sqrt{s}$ instead of the more common setting $\mu = \sqrt{s}$. It would be interesting to compare with a dynamical scale like $H_T$, the sum of the ‘transverse energies’, which has been proven in four- and five-jet production at hadron colliders to be a rather useful choice [619–621]. Using in LO $\alpha_s=0.130$ and in NLO $\alpha_s=0.118$ NLO corrections of the order of 10–20% are found. It is noted that using the same value of $\alpha_s$ in LO and NLO would amount to corrections at the level of 45–60%. Including hadronization corrections through Sherpa the theoretical results are used to extract $\alpha_s$ from the experimental data. As final result $\alpha_s(M_Z) = 0.1156_{-0.0034}^{+0.0041}$ is quoted which is well consistent with the world average and also shows the large potential of $\alpha_s$ measurements using jet rates for high multiplicities: The uncertainty is similar to the $\alpha_s$ determinations from three-jet rates using NNLO+NLLA predictions [614]. As an interesting observation it is also pointed out in Ref. [615] that hadronization corrections calculated with standard tools like HERWIG, PYTHIA and ARIADNE are typically large and uncertain unless the tools are matched/tuned to the specific multi-jet environment. It is suggested to use in such cases event generators like SHERPA which incorporates high-multiplicity matrix elements through CKKW matching.

Recently an alternative method to calculate one-loop corrections has been used to calculate the NLO corrections for six- and seven-jet production. The method developed in [622–629] combines the loop integration together with the phase space integration. Both integrations are done together using Monte Carlo integration. Since the analytic structure of the one-loop integrand is highly non-trivial special techniques have to be developed to enable a numerical integration. In Ref. [630] this technique has been applied to the NLO calculation of the six- and five-jet rate in leading color approximation.

No phenomenological studies are presented. It is however shown that the method offers a powerful alternative to existing approaches.

### 3.2.4 Progress at NLO

An essential input for NLO calculations are the one-loop corrections. Four momentum conservation at each vertex attached to the loop does not fix the momentum inside the loop. As a consequence an additional integration over the unconstrained loop momentum is introduced. Since the loop momenta appears not only in the denominator through the propagators but also in the numerator in general tensor integrals have to be evaluated. The traditional method to deal with these tensor integrals is the so-called Passarino-Veltman reduction which allows to express the tensor integrals in terms of a few basic scalar one-loop integrals [631]. All relevant scalar integrals have been calculated and can be found for example in Refs. [632–634]. In practical applications the Passarino-Veltman reduction procedure may lead to large intermediate expressions when applied analytically to processes with large multiplicities or many different mass scales. An alternative to overcome this problem is to apply the reduction procedure numerically. In this case however numerical instabilities may appear in specific phase space regions where the scalar one-loop integrals degenerate for exceptional momentum configurations. Approaching these exceptional momentum configurations the results behave as “0/0”. Evaluating the limit analytically one finds a well defined result. The numerical evaluation however will typically lead to instabilities unless special precautions are taken to deal with these configurations. In the past various approaches have been developed to stabilize the numerical evaluation of exceptional momentum configurations. Details can be found for example in Refs. [635–644] and references therein. With the steadily increasing computing power of modern CPU’s today an alternative approach is frequently used: Instead of stabilizing the numerical evaluation it is checked during the numerical evaluation whether instabilities were encountered. If this is the case the numerical evaluation of the respective phase space point is repeated using extended floating point precision. The price to pay in this approach is a slightly increase of computing time which is however affordable as long as the fraction of points needed to be recomputed remains small.

Beside the numerical evaluation of tensor integrals the significant increase in complexity when studying virtual corrections for processes with large multiplicities is another major bottleneck of one-loop calculations. Here the recently developed method of generalized unitarity may provide a solution. The starting point of this method is the observation that any one-loop amplitude can be written in terms of scalar one-point, two-point, three-point and four-point one-
loop integrals. No higher point scalar integrals are required. This observation is a direct consequence of the Passarion-Veltman reduction procedure. Starting from this observation one can reformulate the problem of one-loop calculations: How do we calculate most efficiently the coefficients in this decomposition? One answer to this question is the method proposed by Ossola, Papadopoulos, Pittau (OPP) [645]. The idea of this method is to perform a decomposition at the integrand level: The integrand is decomposed into contributions which integrate to zero or lead to scalar integrals. To derive the decomposition at integrand level internal propagators are set on-shell. As a consequence the integrand factorizes into a product of on-shell tree amplitudes. For more details about the method of generalized unitarity we refer to the recent review of Ellis, Kunszt, Melnikov and Zanderighi [646]. From the practical point of view the important result is that the algorithm can be implemented numerically and requires as input only on-shell tree amplitudes. For on-shell tree amplitudes very efficient methods to calculate them, like for example the Berends-Giele recursion, exist [647]. In principle it is also possible to use analytic results for the tree-level amplitudes or apply on-shell recursions à la Britto, Cachazo, Feng, and Witten ((BCFW) see for example Ref. [648]). Using tree amplitudes instead of individual Feynman diagrams helps to deal with the increasing complexity of processes for large multiplicities. It may also lead to numerically more stable results since the tree amplitudes are gauge invariant and gauge cancellation —usually occuring in Feynman diagrammatic calculations— are avoided. The enormous progress made recently is well documented in the increasing number of publicly available tools to calculate one-loop amplitudes, see for example Refs. [649–654]. As can be seen from recent work e.g. Refs. [630, 655, 656] further progress can be expected in the near future (for the method discussed in Ref. [630] see also the discussion at the end of the previous section). As mentioned already the calculation of real emission processes can be considered as a solved problem since very efficient algorithms to calculate the required Born matrix elements are available. In principle also the cancellation of the infrared and collinear singularities appearing in one-loop amplitudes as well as in the real emission processes can be considered as solved. General algorithms like Catani-Seymour subtraction method [657] or FKS subtraction [618] exist to perform the required calculation. Also here significant progress has been obtained in the recent past towards automation. The required subtractions can now be calculated with a variety of publicly available tools [617, 658–661]. While most of the aforementioned tools have been applied recently to LHC physics it is evident that an application to $e+e^-$ annihilation is also possible. It can thus be assumed that for a future Linear Collider all relevant NLO QCD corrections will be available.

### 3.3 Recent progress in top-quark physics

In the Standard Model the top quark appears in the third family as up-type partner of the bottom quark. As missing building block of the third family the existence of the top quark was predicted long before its discovery in 1994. Top-quark interactions are fixed through the gauge structure of the Standard Model. The coupling strengths follow from the local $SU(3) \times SU(2)_L \times U(1)_Y$ gauge invariance. In particular the QCD coupling to the gluons is the same as for the lighter quarks. The coupling to the Z-boson involves vector and axial vector couplings while the coupling to the W-boson is of $V-A$ type. The couplings can be expressed in terms of the third component of the weak iso-spin $T_3$, the hyper charge $Y$ (or alternatively the electric charge $Q$) and the weak mixing angle $\theta_W$. For example the coupling to the Z-boson reads:

$$ -i \frac{e}{\sin \theta_W \cos \theta_W} \left( T_3 \gamma^\mu \frac{1}{2} (1 - \gamma_5) - \sin^2 \theta_W Q \gamma_\mu \right). \quad (88) $$

As a matter of fact top-quark specific aspects or more general flavor dependencies enter only through the top-quark mass and the Cabbibo-Kobayashi-Maskawa (CKM) matrix which relates the mass eigenstates and the eigenstates of the weak interaction. Assuming three families and unitarity the CKM matrix elements are highly constrained from indirect measurements. A global fit of available flavor data gives [662]:

$$ V = \begin{pmatrix} 0.97427 \pm 0.00015 & 0.22534 \pm 0.00065 & 0.00351^{+0.00015}_{-0.00014} \\ 0.22520 \pm 0.00065 & 0.97344 \pm 0.00016 & 0.0412^{+0.0011}_{-0.0005} \\ 0.00867^{+0.00029}_{-0.00031} & 0.0404^{+0.0011}_{-0.0005} & 0.999146^{+0.000021}_{-0.000046} \end{pmatrix} \quad (89) $$

Very recently $V_{tb}$ has been determined also in direct measurements using single-top-quark production at Tevatron and LHC. Combining the various measurements the Particle Data Group quotes [662]:

$$ |V_{tb}| = 0.89 \pm 0.07. \quad (90) $$

The result is consistent with the indirect measurements. However, the complicated experimental environment leads to large uncertainties. Further improvements can be expected from future measurements at the LHC. The top-quark mass has been measured at the Tevatron and the LHC with various techniques. At the Tevatron a combination [663] of various D0 and CDF measurements gives

$$ M_t = 173.18 \pm 0.56 \text{ (stat.)} \pm 0.75 \text{ (syst.) GeV.} \quad (91) $$