Abstract
We investigate the conformal window of four-dimensional gauge theories with fermionic matter fields in multiple representations. Of particularly relevant examples are the ultra-violet complete models with fermions in two distinct representations considered in the context of composite Higgs and top partial-compositeness. We first discuss various analytical approaches to unveil the lower edge of the conformal window and their extension to the multiple matter representations. In particular, we argue that the scheme-independent series expansion for the anomalous dimension of a fermion bilinear at an infrared fixed point, $\gamma_{\bar{\chi}\chi,\,{\rm IR}}$, combined with the conjectured critical condition, $\gamma_{\bar{\chi}\chi,\,{\rm IR}} = 1$ or equivalently $\gamma_{\bar{\chi}\chi,\,{\rm IR}} (2-\gamma_{\bar{\chi}\chi,\,{\rm IR}})=1$, can be used to determine the boundary of conformal phase transition on fully physical grounds. In illustrative cases of $SU(2)$ and $SU(3)$ theories with $N_R$ Dirac fermions in various representations, we assess our results by comparing to other analytical or lattice results.
Highlights
The existence of a nonzero infrared (IR) fixed point in the renormalization-group (RG) beta function of asymptotically free gauge theories in four dimensions with a sufficient number of massless fermions Nf for a given number of colors Nc has been of particular interests recent years because of its potential application to phenomenological model buildings in the context of physics beyond the standard model (BSM), as well as its distinctive feature of conformal phase in contrast to the nonconformal phase as in quantum chromodynamics (QCD)
While various analytical proposals are made in the literature [63,64,65] besides the traditional Schwinger-Dyson analysis, we propose in this paper to use the critical condition on the anomalous dimension of a fermion bilinear operator at an IR fixed point, γIR 1⁄4 1 or equivalently γIRð2 − γIRÞ 1⁄4 1, for the conformal phase transition to occur
III we briefly review the scheme-independent calculation of γIR for gauge theories with fermions in one or two different representations, and determine the lower bound of the conformal window in the exemplified cases of SUð2Þ and SUð3Þ gauge theories with NR Dirac fermions in various representations
Summary
The existence of a nonzero infrared (IR) fixed point in the renormalization-group (RG) beta function of asymptotically free gauge theories in four dimensions with a sufficient number of massless fermions Nf for a given number of colors Nc has been of particular interests recent years because of its potential application to phenomenological model buildings in the context of physics beyond the standard model (BSM), as well as its distinctive feature of conformal phase in contrast to the nonconformal phase as in quantum chromodynamics (QCD). A finite range of the number of flavors for which the theory has a nonzero IR fixed point is called conformal window, and the chirally-broken theories near the phase transition are expected to have quite different IR dynamics, compared to QCD-like theories. We instead emphasize that it becomes an alternative method to map out the conformal window in a schemeindependent way if we adopt the series expansion of γIR recently developed by Ryttov and Shrock [67,68,69,70,71,72,73] We find that this method is useful to discuss the sequential condensates of fermions in different representations, which are expected in the near-conformal theories..
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