QCD with an infrared fixed point and a dilaton

Following previous work we further explore the possibility that the chirally broken phase of gauge theories admits an infrared fixed point interpretation. The slope of the $\beta$ function, ${\beta }_{*}^{\prime }$, is found to vanish at the infrared fixed point which has several attractive features such as logarithmic running. We provide a more in-depth analysis of our previous result that the mass anomalous assumes ${\gamma }_{*}=1$ at the fixed point. The results are found to be consistent with $N=1$ supersymmetric gauge theories. In a second part the specific properties of a dilaton, the (pseudo-) Goldstone, due to spontaneous symmetry breaking are investigated. Dilaton soft theorems indicate that a soft dilaton mass can only originate from an operator of scaling dimension two. In the gauge theory this role is taken on by the $\overline{q}q$-operator. The quantum chromodynamics (QCD) dilaton candidate, the $\sigma =f_0\left(500\right)$ meson is investigated and the singlet-octet mixing is found to be relevant. We briefly discuss the dilaton as a candidate for the Higgs boson, which relies on the ratio of dilaton to pion decay constant being close to unity. In QCD this is approximately satisfied but it is remains unclear if this is accidental or whether there is yet to be uncovered principle behind it.