The Composite Higgs and Light Resonance Connection

TL;DR

This work addresses how a light Higgs around $m_h\approx125\ \mathrm{GeV}$ constrains the spectrum of fermionic resonances in composite Higgs scenarios. By applying Weinberg sum rules in the large-$N$ limit and saturating correlators with the lightest resonances, the authors compute the Higgs mass as a function of fermionic resonance masses in the MCHM and its extensions, showing that a 125 GeV Higgs generically requires resonances below the TeV scale and yields detectable deviations in Higgs couplings to SM fermions. The gauge sector cannot by itself induce EWSB, while the top sector dominates the Higgs potential; explicit results for MCHM$_5$ (and extensions to MCHM$_{10}$ and other embeddings) give lower bounds and, in two-resonance saturation, precise expressions (e.g., Eq. mh22) that tie $m_h$ to light fermionic states around $0.5$–$0.7$ TeV, with lighter custodians often implied by sizable top compositeness. The findings have important LHC implications, predicting accessible fermionic resonances and characteristic Higgs-coupling patterns, while noting that special cancellations or alternative embeddings can soften these bounds in specific scenarios.

Abstract

Weinberg sum-rules have been used in the past to successfully predict the electromagnetic contribution to the charged-pion mass as a function of the meson masses. Following the same approach we calculate in the minimal composite Higgs model (MCHM) the Higgs mass as a function of the fermionic resonance masses. The simplicity of the method allows us to study several versions of the MCHM and show that a Higgs with a mass around 125 GeV requires, quite generically, fermionic resonances below the TeV, and therefore accessible at the LHC. We also examine the couplings of the Higgs to the SM fermions and calculate their deviation from the SM value.

Figures (3)

• Figure 1: Masses of the two lightest fermion resonances for $m_h=125\,\mathrm{GeV}$ (taking $\xi=0.2$ and $m_t=160\,\mathrm{GeV}$ (the running top mass at $\sim$ TeV)). In blue we plot the $MCHM_5$ result; the solid line corresponds to Eq. (\ref{['mh22']}) calculated in the approximation $\epsilon^2\ll 1$, while the dashed line is the exact result (always $\Delta F^2=0$). In solid red we plot the result for the $MCHM_{10}$ ($\epsilon^2\ll 1$ and $\Delta F^2=0$) with ${Q_1}\rightarrow {Q_6}$. The black solid line is for ${\bf r_L=5}$ and ${\bf r_R=1}$ (denoted MCHM$_{5+1}$), fixing for illustration $F_{Q_1}^L=\sqrt{2}\tilde{F}_{Q_1}^R$.
• Figure 2: Predictions of a generic MCHM in the ($g_{hff}/g^{\rm SM}_{hff}$, $g_{hWW}/g^{\rm SM}_{hWW}$)-plane. The different curves corresponds to different values of $n$, going downwards from n=0 to $n=5$. The red part of the curves is for $0<\xi<0.25$ and the blue one for $0.25<\xi<1$. The contours are the $68\%$, $95\%$ and $99\%$ CL for a 125 GeV Higgs as obtained in Ref. Azatov:2012bz from the CMS data.
• Figure 3: Upper bound for the mass of the lightest fermionic resonance in various composite Higgs models, for $\xi=0.2$ and $m_t=160\,\mathrm{GeV}$ (the running top mass at $\sim$ TeV). The orange line corresponds to Eq. (\ref{['lowerbound']}) that is the upper bound for the MCHM$_{5,10,14}$. The bound Eq. (\ref{['lowerbound2']}) is shown in dashed blue, while Eq. (\ref{['MH2']}) is shown in dotted red for $n=1$ and $m'=6$. A comparison with the Holographic MCHM$_{5}$ model of Ref. Contino:2006qr is shown in grey. The vertical line is for $m_h=125\,\mathrm{GeV}$.