Techni-dilaton at Conformal Edge

Abstract

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 $<{\theta }_{\mu }^{\mu }>$ and to the techni-gluon condensate $<G_{\mu \nu }^2>$, 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 $\alpha$ replaced by the two-loop running one $\alpha (\mu )$ having the Caswell-Banks-Zaks IR fixed point ${\alpha }_{\ast }$: $\alpha (\mu )\simeq \alpha ={\alpha }_{\ast }$ for the IR region $m<\mu <{\Lambda }_{TC}$, where ${\Lambda }_{TC}$ is the intrinsic scale (analogue of ${\Lambda }_{QCD}$ of QCD) relevant to the perturbative scale anomaly. We find that $-<{\theta }_{\mu }^{\mu }>/m^4\to const\ne 0$ and $<G_{\mu \nu }^2>/m^4\to (\alpha /{\alpha }_{cr}-1{)}^{-3/2}\to \infty$ in the criticality limit $m/{\Lambda }_{TC}\sim \exp (-\pi /(\alpha /{\alpha }_{cr}-1{)}^{1/2})\to 0$ ($\alpha ={\alpha }_{\ast }\to {\alpha }_{cr}$) ("conformal edge"). Our result precisely reproduces the formal identity $<{\theta }_{\mu }^{\mu }>=(\beta (\alpha )/4\alpha )<G_{\mu \nu }^2>$, where $\beta (\alpha )=-(2{\alpha }_{cr}/\pi )(\alpha /{\alpha }_{cr}-1{)}^{3/2}$ is the nonperturbative beta function corresponding to the above essential singularity scaling of $m/{\Lambda }_{TC}$. Accordingly, the PCDC implies $(M_{TD}/m{)}^2(F_{TD}/m{)}^2=-4<{\theta }_{\mu }^{\mu }>/m^4\to const\ne 0$ 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.

17 pages, 14 figures; discussions clarified, references added, to appear in Phys.Rev.D

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