Composite Higgs models based on

Article funded by SCOAP3

0$ and a negative $A$ close to the boundary $A \gtrsim -4 B$. In that region, the mass of the Higgs can be computed from the second derivative of the above potential, and it can be expressed as \be \label{eq:higgsmass} m_h^2 = 64\, B \, v^2 c_\theta^2\,. \ee This simple analysis unveils the two tunings that are needed in the potential: on the one hand, obtaining a small misalignment requires $A \sim -4 B$; on the other hand, the correct value of the Higgs mass requires a small coefficient $B$: \be B = \frac{1}{c_\theta^2} \frac{m_h^2}{64 v^2} \sim 0.004\ \frac{1}{c_\theta^2}\,. \ee While the above issues are common to all composite pNGB Higgs models, we anticipate that, analysing the contribution of the top to the potential, another issue arises in the specific $\SU(5)/\SO(5)$ model, namely one needs to avoid a misalignment of the vacuum along the custodial triplet, which would generate a tree level contribution to the $\rho$ parameter. In the rest of the section, we will analyse in detail the contribution to the potential from the gauge, underlying HF mass and top couplings. ]]>

0$: the assumption on the sign of the coefficient relies on the fact that gauge loops typically tend not to break the gauge symmetry itself. Note that the tadpole for the Higgs $h$ vanishes at the minimum. ]]>

0$ and for positive masses, the minimum from this term is also at $\theta=0$: this is again expected, as the mass of the underlying fermions should simply give a mass to the pNGBs, if defined around the correct vacuum. \looseness=-1 It is tantalising that, by changing the sign of the mass terms, the alignment of the theory may change: for instance, if we keep $\mu_d>0$, turning the other mass negative will change the sign of the potential for $\mu_s < -\mu_d$ and thus push the minimum at $\theta=\pi/2$. The breaking of the EW symmetry by HF mass terms alone is, however, only a consequence of the inappropriate choice of the EW preserving vacuum, as we prove in appendix~\ref{app:massvacuum}. It is in fact enough to define the theory around the second inequivalent vacuum, defined in eq.~\eqref{vacuum2}, to flip all the signs in front of $\mu_s$. Thus, the theory becomes equivalent to the one with positive $\mu_s$ defined around $\Sigma_0$, with minimum at $\theta=0$. In the appendix we also show that for negative $\mu_s$ the singlet $\eta$ acquires a tachyonic mass around the wrong vacuum, and that the connection between the two vacua is given by a misalignment along the singlet direction. This example shows how important it is to study the theory on the correct vacuum, and that the presence of tachyonic mass terms cannot simply be cured by assuming a VEV for the corresponding pNGB but needs a change of vacuum. Examples of this sort have also been pointed out in the $\SU(4)/\SP(4)$ CH models in ref.~\cite{Alanne-ml-2018wtp}. ]]>

4 C_L\,, \qquad \frac{3}{4\pi} C_L > - C_g (3 g^2 + g'^2)\,. \end{equation} Assuming that the correct vacuum is attained, the two coefficients $C_{L/R}$ can be fixed by imposing the minimisation condition and the value of the Higgs mass $m_h$: \begin{align} C_L & = \pi \frac{c_{2\theta}}{6 c_\theta^2} \frac{m_h^2}{v^2} - \frac{4 \pi}{3} C_g (3 g^2 + g'^2)\,, \nonumber \\ C_R & = \pi \frac{8 c_{\theta}^2 -1}{30 c_\theta^2} \frac{m_h^2}{v^2} - \frac{16 \pi}{15} C_g (3 g^2 + g'^2)\,. \end{align} The expressions above can be used to define Yukawa-free ratios of coefficients, as explained in the previous section, which in the present case amounts to a single ratio \be \label{eq:ratioDD} R_\text{D} \equiv \frac{ C_t ^2}{ C_L C_R } = \ \frac{576 v^2 c_{\theta }^2 m_t^2}{c_{2 \theta }^3 \left(4 c_{2 \theta }+3\right) m_h^4} \ \approx \frac{576}{7} \frac{v^2 m_t^2}{m_h^4} \left( 1 + \frac{43}{7} s_\theta^2 + \mathcal{O} (s_\theta^4) \right)\,. \ee Numerically $R_D \sim 600$, thus showing that the form factors associated to the potential operators are required to be significantly smaller than the NDA estimate. \begin{figure} \centering \includegraphics[width=.48\textwidth]{figs/spectrumDDneut.pdf}\hfill \includegraphics[width=.48\textwidth]{figs/spectrumDDcharg.pdf} ]]>

0$, the potential has the correct minimum. However, in the absence of HF masses, there is always a tachyonic state in the pNGB spectrum, meaning that the vacuum misalignment is not stable. Therefore, in this case it is necessary to include the contribution of the HF masses, in such a way to give a positive contribution to the pNGB masses and remove tachyons. We remark that this is not an ad-hoc choice, as the HF masses are always present in all models. For simplicity, we also set the coefficient $C_m$ to one, as this is equivalent to reabsorbing it into the definition of the mass parameters $\mu_{d,s}$. We also define the quantity $\displaystyle \delta \equiv \frac{\mu_s - \mu_d}{f}$, which measures the amount of explicit $\SO(5)$ breaking due to the HF mass term, and an average mass $\displaystyle \mu \equiv \frac{ \mu_d + \mu_s}{2}$. The minimum and Higgs mass conditions are easily enforced by solving for two of the free parameters: in the following, we chose to solve for $C_R$-$\mu$ and $C_R$-$C_g$, alternatively, in order to give a broader picture of the available parameter space. \begin{figure} \centering \includegraphics[width=.4\textwidth]{figs/ALDRtac2.pdf}\qquad \includegraphics[width=.4\textwidth]{figs/ALDRtac3.pdf} ]]>

0$ (left-panel) and a range of $\mu$ (right-panel) in order to avoid tachyons.]]>

0$ as a necessary condition (see discussion in section~\ref{sec:gaugeloops}), the absence of tachyons implies that $\delta > 0$ and shows that, for each value of $\delta$, there is an upper limit on $C_g$, as we already observed in the $D_R$-$D_L$ case. The blue shading in the left panel indicates $\mu>0$, thus showing that negative values of the average mass are also allowed. The right panel of figure~\ref{fig:ALDRexcl} shows the same parameter space in terms of $\mu$-$\delta$, where $C_g>0$ corresponds to the blue shaded area. ]]>

0$ for $\pi/4 < \theta < \pi/2$, and that it is symmetric under change of sign of $\zeta$. The mass of the top quark is also modified in the new vacuum, but its explicit form depends on the chosen spurions. Here we will focus on the anti-symmetric irrep of $\SU(5)$ as in section~\ref{sec:antisym}, for which it reads: \be m_t \ = \ \frac{C_{t, A_L A_R}}{4\pi} f \st c_\zeta \left( \ct - \st s_\zeta\right)\,. \ee Besides the tree level correction to the $\rho$ parameter, the new vacuum alignment also modifies the couplings of the would-be Higgs to $W$, $Z$ and tops, in a non-custodial way. We define the modified couplings with respect to the SM ones as \be \kappa^\phi_V = \frac{v}{2 m_V^2} g_{\phi VV}\,, \quad \kappa^\phi_t = \frac{v}{m_t} g_{\phi \bar{t} t}\,, \ee where $v$ is defined by matching the expression of the $W$ mass to the SM formula, yielding \be v = f \st \sqrt{1+s_\zeta^2}\,, \ee and $\phi = h, \eta_3^0$. For the $W$, we obtain \be \kappa_W^h = c_\zeta \frac{\ct (1+s_\zeta^2) - s_\zeta^2}{\sqrt{1+s_\zeta^2}}\,, \quad \kappa_W^\eta = - s_\zeta \frac{\ct (1+s_\zeta^2) + c_\zeta^2}{\sqrt{1+s_\zeta^2}}\,. \ee The corresponding expressions for $Z$ and top are more complicated, and we report them in appendix~\ref{app:vacuum}. In the limit $\zeta \to 0$, where the vacuum is misaligned along the Higgs direction only, we recover the familiar expression $\kappa_W^h = \ct$ while the coupling of the triplet vanishes. On the contrary, for $\zeta = \pi/2$, where the vacuum is only misaligned along the triplet, $\kappa_W^\eta = - \sqrt{2} \ct$ while the coupling of $h$ vanishes. This confirms our expectation that, in the chosen basis, $h$ matches the doublet and $\eta_3^0$ the custodial triplet. From eq.~\eqref{eq:rho} we observed that $\delta \rho$ vanishes for $\theta=\pi/4$, besides the obvious region near vanishing triplet misalignment $\zeta = 0$. It is, therefore, tantalising to think that such a large $\theta$ region may be still allowed by the Higgs data and by precision tests. The most recent determination of the $\rho$ parameter gives $\rho = 1.0005 \pm 0.0005$~\cite{pdg2018}, where the fit is marginalised on contributions to other oblique observable such as the $S$ parameter. Thus, we will impose a bound on $\delta \rho$ at $3\sigma$, which gives the following range: $-0.1\% < \delta \rho < 0.2\%$. This is but a rough estimate of the impact of electroweak physics in this model, because at one loop level additional contributions will emerge coming from the modifications of the Higgs couplings and from loops of heavier resonances. For examples of this kind of calculations in other models, we refer the reader to refs.~\cite{Carena-ml-2006bn,Barbieri-ml-2007bh,Arbey-ml-2015exa}. A detailed analysis of this issue is, however, beyond the scope of this paper, because it heavily relies on the details of the model. One interesting feature is that corrections to the $S$ parameters are typically positive~\cite{Peskin-ml-1990zt,Hirn-ml-2006nt,Contino-ml-2010rs,Panico-ml-2015jxa}, and due to the correlation in the EW fit, a positive contribution to $\delta \rho$ (i.e.\ $T$) is welcome as it may push the model back into the allowed region~\cite{Grojean-ml-2013qca}. In the model under study, this happens for $\theta > \pi/4$. It is the deviation on the couplings of the Higgs that will tell us if such a region is still viable. \begin{figure} \centering \includegraphics[width=.32\textwidth]{figs/coup11.pdf}\hfill \includegraphics[width=.32\textwidth]{figs/coup12.pdf}\hfill \includegraphics[width=.32\textwidth]{figs/coup13.pdf} ]]>

- \mu_d$, the minimum is at $\theta=0$, and it generates the following masses for the Higgs doublet, triplets and singlet $\eta$: \begin{equation} \label{eq:spectrumHF} m_{\rm Higgs}^2 = 8 C_m f (\mu_d + \mu_s)\,, \quad m_{\rm triplets}^2 = 16 C_m f \mu_d\,, \quad m_{\eta}^2 = 16 C_m f (\mu_d + 4 \mu_s)\,. \end{equation} We clearly see that for $\mu_s < - \mu_d$, the Higgs doublet squared mass turns negative, thus justifying the expectation that the minimum of the potential would be moved away fro zero. However, the above expressions show that the singlet $\eta$ becomes tachyonic for less negative values of $\mu_s$, i.e.\ $\mu_s < - \mu_d/4$. This implies that the vacuum needs to be misaligned along the singlet even before the Higgs direction is destabilised. We will show now that this destabilisation along the singlet, that occurs for negative $\mu_s$, calls for choosing the second inequivalent EW preserving vacuum \be \label{vacuum2} \Sigma_0'=\left( \begin{array}{cc|c} & i \sigma_2 & \\ -i \sigma_2 & & \\ \hline & & -1 \end{array} \right)\,, \ee which, compared to the one in eq.~\eqref{vacuum}, has a minus sign corresponding to the negative singlet squared mass. One can go from the vacuum $\Sigma_0$ to the new one $\Sigma_0'$ with a $\SU(5)$ transformation by the generator associated to $\eta$ (times an overall phase shift): \be \Omega_s = \left. e^{- i \alpha/2} e^{i \sqrt{10} X_\eta\ \alpha} \right|_{\alpha = \pi/5} = \left( \begin{array}{ccccc} 1 & & & & \\ & 1 & & & \\ & & 1 & & \\ & & & 1 & \\ & & & & - i \end{array} \right)\,. \ee The relation is, therefore, \be \Sigma_0' = \Omega_s \Sigma_0 \Omega_s^T\,. \ee Let us now consider a theory with an HF mass term with $\mu_d > 0$ and $\mu_s < 0$. If we define the EW vacuum $\Sigma_0'$, all the equations for the vacuum alignment eq.~\eqref{eq:cm} and~\eqref{eq:spectrumHF} would be the same but for a flipped sign in front of $\mu_s$. That is, the results are the same as we would obtain for a theory with $\mu_d>0$ and $\mu_s > 0$ around the vacuum $\Sigma_0$. Thus, also for $\mu_s<0$, once the correct EW preserving vacuum is chosen, the theory is well defined at the minimum $\theta=0$. This analysis proves that the HF term alone cannot trigger EWSB. As a final remark, we would like to remind that the broken generator bases in the two vacua are not the same, thus one needs to define a new pNGB basis on $\Sigma_0'$, which is related to the basis for $\Sigma_0$ in eq.~\eqref{eq:Pimatrix} by \be \Pi' = \Omega_s \Pi \Omega_s^\dagger\,. \ee ]]>