We provide a unified description, both at the effective and fundamental Lagrangian
level, of models of composite Higgs dynamics where the Higgs itself can emerge,
depending on the way the electroweak symmetry is embedded, either as a
pseudo-Goldstone boson or as a massive excitation of the condensate. We show that,
in general, these states mix with repercussions on the electroweak physics and
phenomenology. Our results will help clarify the main differences, similarities,
benefits and shortcomings of the different ways one can naturally realize a
composite nature of the electroweak sector of the Standard Model. We will
analyze the minimal underlying realization in terms of fundamental strongly
coupled gauge theories supporting the flavor symmetry breaking pattern
Article funded by SCOAP3
2 |X_m|$, in which case it is indeed the minimum of the potential. In the other case, the only allowed solution is $\theta=0$. The masses for the scalar fields are given by \begin{align} m_h^2 &= \frac{f^2}{4} \left(X_t \left(1-2 c_\theta^2\right) + 2 X_m c_\theta\right)\,, \\ m_\eta^2 &= \frac{f^2}{4} \left(X_t \left(1- c_\theta^2\right) + 2 X_m c_\theta\right)\,. \label{mass-ESB} \end{align} On the solution $c_\theta = 2 X_m/X_t$, the masses read \begin{align} m_h^2 &= \frac{f^2}{4} \frac{X_t^2 - 4 X_m^2}{X_t} = \frac{f^2}{4} s_\theta^2 X_t\,, \\ m_\eta^2 &= \frac{f^2}{4} X_t\,. \end{align} We see that $m_h^2 > 0$ for $X_t > 2 |X_m|$, for which there is always a solution for $c_\theta$. Furthermore, we recover the relation $m_\eta^2 = m_h^2/s_\theta^2$~\cite{Galloway-ml-2010bp}. On the other solution $\theta=0$, where the electroweak symmetry is unbroken, the masses~read \beq m_h^2 = \frac{f^2}{4} (2 X_m - X_t)\,, \qquad m_\eta^2 = \frac{f^2}{2} X_m\,. \eeq This solution is a stable minimum for $X_t < 2 X_m$. \paragraph{More on the Higgs mass and fine tuning.} Using the expressions for the top and gauge masses, the composite GB Higgs can be rewritten as \begin{equation} m_h^2 \!=\! \frac{f^2 s_\theta^2}{4} \left( {y'_t}^2 C_t \!-\! \frac{3 g^2 \!+\! {g'}^2}{2} C_g \right) \!=\! \frac{C_t m_t^2}{4} \left( 1 \!-\! \frac{2 m_W^2 \!+\! m_Z^2}{4 m_t^2} \frac{C_g}{C_t} \right) \!\sim\! \frac{C_t m_t^2}{4} \left( 1 \!-\! 0.18 \frac{C_g}{C_t} \right). \end{equation} This shows explicitly that the contribution of the gauge loops is typically smaller than the top one, even assuming $C_g \sim C_t$. Also, neglecting the gauge contribution, to fit the observed Higgs mass $m_h = 125$\,GeV, we would need \beq C_t \sim 2\,. \eeq Another interesting point is related to the value of $\theta$: in fact, this angle parametrizes the corrections to the Higgs couplings to the gauge bosons, which are by now well constrained by LHC data. Therefore, a realistic model would require $\theta$ to be small. In order to achieve this, we would need: \beq c_\theta \sim 1 \quad \Rightarrow \quad X_t \sim 2 X_m\,. \eeq In other words, a realistic model would require a non-trivial fine tuning between the contribution of the top loops, and the contribution of the explicit breaking of \SU(4), which is induced by a completely different mechanism. ]]>