Composite Higgs model at a conformal fixed point

We propose to construct a chirally broken model based on the infrared fixed point of a conformal system by raising the mass of some flavors while keeping the others massless. In the infrared limit, the massive fermions decouple, and the massless fermions break chiral symmetry. The running coupling of this system “walks,” and the energy range of walking can be tuned by the mass of the heavy flavors. Renormalization group considerations predict that the spectrum of such a system shows hyperscaling. We have studied a model with four light and eight heavy flavors coupled to $SU(3)$ gauge fields and verified the above expectations. We determined the mass of several hadronic states and found that some of them are in the 2–3 TeV range if the scale is set by the pseudoscalar decay constant $F_{\pi }\approx 250\mathrm{GeV}$. The $0^{++}$ scalar state behaves very differently from the other hadronic states. In most of our simulations, it is nearly degenerate with the pion, and we estimate its mass to be less than half of the vector resonance mass.

Figure 1

Top: The expected running gauge coupling of conformal and mass-split systems. The solid blue curve sketches the evolution of the gauge coupling in a conformal system. The dashed blue curve shows the change in a mass-split system, while the red long-dashed curve describes a situation where the fermions decouple before the gauge coupling could approach the conformal IRFP. Bottom: Numerical results for the running coupling constant $g^2$ for different values of ${\widehat{m}}_h$ and in the ${\widehat{m}}_h=\infty$ (four-flavor limit) with ${\widehat{m}}_{\ell }$ extrapolated to the chiral limit. The emergence of the walking regime is evident as ${\widehat{m}}_h\to 0$. The dashed sections of the lines indicate where we suspect cutoff effects may be significant.

Figure 2

The pion, rho, isosinglet $0^{++}$ and isomultiplet $a_0$ scalar, axial, and nucleon mass of the light flavor spectrum in units of $F_{\pi }$. The first narrow panel shows the experimental values for QCD [32] normalized by $F_{\pi }=94\mathrm{MeV}$, while the last one corresponds to average values obtained from $N_f=12$-flavor simulations [7, 16, 33, 34]. The four wider panels show the $N_f=4+8$ spectrum as the function of the light quark mass $a_{\star }{m}_{\ell }$ for $a{m}_h=0.100$, 0.080, 0.060, and 0.050. If the chirally broken $N_f=4+8$ system triggered EWSB, $F_{\pi }\approx 250\mathrm{GeV}$ would set the correct electroweak scale.

Figure 3

Combining all four $a{m}_h$ values from Fig. 2. The left panel shows the pion, rho, and $a_1$, and the right panel shows the $0^{++}$, $a_0$, and nucleon states in units of $F_{\pi }$.