Conformal Windows of SU(N) Gauge Theories, Higher Dimensional Representations and The Size of The Unparticle World

TL;DR

The paper defines and computes conformal windows for SU(N) gauge theories with vector-like matter in higher representations, across both ${\cal N}=1$ supersymmetric and nonsupersymmetric cases. It introduces a theory-space measure to quantify the fraction of asymptotically free theories that possess infrared fixed points (the conformal region) versus those that do not (the nonconformal region). A striking result is the universal 50% conformal fraction in the SUSY case, exact and representation-independent, contrasted with about 25% in the nonsupersymmetric case, derived under ladder approximations. Extending to multiple representations shows the conformal region can dominate the landscape, with up to ~70% in four-representation nonsupersymmetric scenarios. Collectively, these findings imply the unparticle sector, as a conformal gauge-theory origin, could be substantial within theoretical particle-space and have implications for walking technicolor and unparticle phenomenology.

Abstract

We present the conformal windows of SU(N) supersymmetric and nonsupersymmetric gauge theories with vector-like matter transforming according to higher irreducible representations of the gauge group. We determine the fraction of asymptotically free theories expected to develop an infrared fixed point and find that it does not depend on the specific choice of the representation. This result is exact in supersymmetric theories while it is an approximate one in the nonsupersymmetric case. The analysis allows us to size the unparticle world related to the existence of underlying gauge theories developing an infrared stable fixed point. We find that exactly 50 % of the asymptotically free theories can develop an infrared fixed point while for the nonsupersymmetric theories it is circa 25 %. When considering multiple representations, only for the nonsupersymmetric case, the conformal regions quickly dominate over the nonconformal ones. For four representations, 70 % of the asymptotically free space is filled by the conformal region. According to our theoretical landscape survey the unparticle physics world occupies a sizable amount of the particle world, at least in theory space, and before mixing it (at the operator level) with the nonconformal one.

Figures (2)

• Figure 1: Phase diagram for supersymmetric theories with fermions in the: i) fundamental representation (blue), ii) two-index antisymmetric representation (purple), iii) two-index symmetric representation (red), iv) adjoint representation (green) as a function of the number of flavors and the number of colors. The shaded areas depict the corresponding conformal windows. Above the upper solid curve the theories are no longer asymptotically free. In between the upper and the lower solid curves the theories develop an infrared fixed point. The dashed curve represents the change of sign in the second coefficient of the beta function.
• Figure 2: Phase diagram for nonsupersymmetric theories with fermions in the: i) fundamental representation (blue), ii) two-index antisymmetric representation (purple), iii) two-index symmetric representation (red), iv) adjoint representation (green) as a function of the number of flavors and the number of colors. The shaded areas depict the corresponding conformal windows. Above the upper solid curve the theories are no longer asymptotically free. In between the upper and the lower solid curves the theories are expected to develop an infrared fixed point. The dashed curve represents the change of sign in the second coefficient of the beta function. Diagram appeared first in Dietrich:2006cm.