Abstract
We study the conformal window of asymptotically free gauge theories containing ${N}_{f}$ flavors of fermion matter transforming to the vector and two-index representations of $SO(N),SU(N)$ and $Sp(2N)$ gauge groups. For $SO(N)$ we also consider the spinorial representation. We determine the critical number of flavors ${N}_{f}^{\mathrm{cr}}$, corresponding to the lower end of the conformal window, by using the conjectured critical condition on the anomalous dimension of the fermion bilinear at an infrared fixed point, ${\ensuremath{\gamma}}_{\overline{\ensuremath{\psi}}\ensuremath{\psi},\mathrm{IR}}=1$ or equivalently ${\ensuremath{\gamma}}_{\overline{\ensuremath{\psi}}\ensuremath{\psi},\mathrm{IR}}(2\ensuremath{-}{\ensuremath{\gamma}}_{\overline{\ensuremath{\psi}}\ensuremath{\psi},\mathrm{IR}})=1$. To compute ${\ensuremath{\gamma}}_{\overline{\ensuremath{\psi}}\ensuremath{\psi},\mathrm{IR}}$ we employ the (scheme-independent) Banks-Zaks conformal expansion up to the 4th order in ${\mathrm{\ensuremath{\Delta}}}_{{N}_{f}}={N}_{f}^{\mathrm{AF}}\ensuremath{-}{N}_{f}$ with ${N}_{f}^{\mathrm{AF}}$ corresponding to the onset of the loss of asymptotic freedom, where we show that the latter critical condition provides a better performance along with this conformal expansion. To quantify the uncertainties in our analysis, which potentially originate from nonperturbative effects, we propose two distinct approaches by assuming the large order behavior of the conformal expansion separately, either convergent or divergent asymptotic. In the former case, we take the difference in the Pad\'e approximants to the two definitions of the critical condition, whereas in the latter case the truncation error associated with the singularity in the Borel plane is taken into account. Our results are further compared to other analytical methods as well as lattice results available in the literature. In particular, we find that $SU(2)$ with six and $SU(3)$ with ten fundamental flavors are likely on the lower edge of the conformal window, which are consistent with the recent lattice results. We also predict that $Sp(4)$ theories with fundamental and antisymmetric fermions have the critical numbers of flavors, approximately ten and five, respectively.
Highlights
Since it was discovered that SUðNÞ gauge theories with Nf flavors of fundamental fermions could exhibit an interacting conformal phase at an infrared (IR) fixed point with a nonzero coupling constant [1,2], a substantial amount of work has been devoted to investigate its properties as well as near-conformal behavior in the vicinity of the phase boundary
II B, we briefly review the conformal expansion of γψψ;IR defined at an IR fixed point
The recent computations of the perturbative beta function at the 5th order in the gauge coupling [40,41,42,43], along with the results for the anomalous dimension at the 4th order [44,45], within the modified minimal subtraction (MS) scheme enable to determine the coefficients cl to the 4th order in ΔNf, where the explicit results in terms of group invariants for fermions transforming according to the representation R of a generic gauge group G are presented in Ref. [10]
Summary
Since it was discovered that SUðNÞ gauge theories with Nf flavors of fundamental fermions could exhibit an interacting conformal phase at an infrared (IR) fixed point with a nonzero coupling constant [1,2], a substantial amount of work has been devoted to investigate its properties as well as near-conformal behavior in the vicinity of the phase boundary. The purpose of this work is to estimate the critical number of flavors Ncfr, corresponding to the phase boundary between the chirally broken and the IR conformal, in G 1⁄4 SUðNÞ, Spð2NÞ and SOðNÞ gauge theories with fermion matter content in a single representation R. We describe our strategy to determine the lower edge of the conformal window in a scheme independent way in Sec. II C: we apply the critical condition, which is responsible for the chiral phase transition, to γψψ;IR computed from the conformal expansion at finite order.
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