We give a new perspective on the dynamics of conformal theories realized in the SU(N) gauge theory, when the number of flavors N_f is within the conformal window. Motivated by the RG argument on conformal theories with a finite IR cutoff Lambda_{IR}, we conjecture that the propagator of a meson G_H(t) on a lattice behaves at large t as a power-law corrected Yukawa-type decaying form G_H(t) = c_H exp{(-m_H t)}/t^{alpha_H} instead of the exponentially decaying form c_Hexp{(-m_H t)}, in the small quark mass region where m_H le c Lambda_{IR}: m_H is the mass of the ground state hadron in the channel H and c is a constant of order 1. The transition between the conformal region and the confining region is a first order transition. Our numerical results verify the predictions for the N_f=7 case and the N_f=16 case in the SU(3) gauge theory with the fundamental representation.
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