Techni-dilaton at Conformal Edge

Techni-dilaton (TD) was proposed long ago in the technicolor (TC) near criticality/conformality. To reveal the critical behavior of TD, we explicitly compute the nonperturbative contributions to the scale anomaly $<θ^μ_μ>$ and to the techni-gluon condensate $<G_{μν}^2>$, which are generated by the dynamical mass m of the techni-fermions. Our computation is based on the (improved) ladder Schwinger-Dyson equation, with the gauge coupling $α$ replaced by the two-loop running one $α(μ)$ having the Caswell-Banks-Zaks IR fixed point $α_*$: $α(μ) \simeq α= α_*$ for the IR region $m < μ< Λ_{TC}$, where $Λ_{TC}$ is the intrinsic scale (analogue of $Λ_{QCD}$ of QCD) relevant to the perturbative scale anomaly. We find that $-<θ^μ_μ>/m^4\to const \ne 0$ and $<G_{μν}^2>/m^4\to (α/α_{cr}-1)^{-3/2}\to\infty$ in the criticality limit $m/Λ_{TC}\sim\exp(-π/(α/α_{cr}-1)^{1/2})\to 0$ ($α=α_* \to α_{cr}$) ("conformal edge"). Our result precisely reproduces the formal identity $<θ^μ_μ>=(β(α)/4 α) <G_{μν}^2>$, where $β(α)=-(2α_{cr}/π) (α/α_{cr}-1)^{3/2}$ is the nonperturbative beta function corresponding to the above essential singularity scaling of $m/Λ_{TC}$. Accordingly, the PCDC implies $(M_{TD}/m)^2 (F_{TD}/m)^2=-4<θ_μ^μ>/m^4 \to const \ne 0$ at criticality limit, where $M_{TD}$ is the mass of TD and $F_{TD}$ the decay constant of TD. We thus conclude that at criticality limit the TD could become a "true (massless) Nambu-Goldstone boson" $M_{TD}/m\to 0$, only when $m/F_{TD}\to 0$, namely getting decoupled, as was the case of "holographic TD" of Haba-Matsuzaki-Yamawaki. The decoupled TD can be a candidate of dark matter.

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