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We first quantize the action proposed by Casalbuoni and Gomis in [Phys. Rev.D \textbf{90}, 026001 (2014)], an action that describes two masslessrelativistic scalar particles interacting via a conformally invariantpotential. The spectrum is a continuum of massive states that may beinterpreted as unparticles. We then obtain in a similar way the mass operatorfor a deformed action in which two terms are introduced that break theconformal symmetry: a mass term and an extra position-dependent couplingconstant. A simple Ansatz for the latter leads to a mass operator with linearconfinement in terms of an effective string tension $\sigma\,$. The quantizedmodel is confining when $\sigma\neq0$ and its mass spectrum shows Reggetrajectories. We propose a tensionless limit in which highly excited confinedstates reduce to (gapped) unparticles. Moreover, the low-lying confined boundstates become massless in the latter limit as a sign of conformal symmetryrestoration and the ratio between their masses and $\sqrt\sigma$ staysconstant. The originality of our approach is that it applies to both confiningand conformal phases via an effective interacting model.

A First-Quantized Model For Unparticles and Gauge Theories Around Conformal Window