Conformal window 2.0: The large Nf safe story

Abstract

We extend the phase diagram of SU(N) gauge-fermion theories as a function of the number of flavors and colors to the region in which asymptotic freedom is lost. We argue, using large N f results, for the existence of an ultraviolet interacting fixed point at a sufficiently large number of flavors opening up to a second ultraviolet conformal window in the number of flavors vs colors phase diagram. We first review the state-ofthe-art for the large N f beta function and then estimate the lower boundary of the ultraviolet window. The theories belonging to this new region are examples of safe non-Abelian quantum electrodynamics, termed here safe QCD. Therefore, according to Wilson, they are fundamental. An important critical quantity is the fermion mass anomalous dimension at the ultraviolet fixed point that we determine at leading order in 1=N f . We discover that its value is comfortably below the bootstrap bound. We also investigate the Abelian case and find that at the potential ultraviolet fixed point the related fermion mass anomalous dimension has a singular behavior suggesting that a more careful investigation of its ultimate fate is needed. DOI: 10.1103/PhysRevD.97.116007 The discovery of asymptotic freedom [1,2] has been a landmark in our understanding of fundamental interactions. By fundamental we mean that, following Wilson [3,4], these theories are valid at arbitrary short and long distance scales. Asymptotic freedom has therefore guided a great deal of Standard Model (SM) extensions. Likewise the discovery of four-dimensional asymptotically safe field theories [5] constitutes an important alternative to asymptotic freedom. It has opened the door to new ways to generalize the Standard Model [6][7][8][9][10][11] with impact in dark matter physics and cosmology [10]. The essential feature of an asymptotically safe completion of the SM is that it tames its high energy behavior dynamically uplifting to the status of a truly fundamental field theory. In practice this means that theory does not have a physical cutoff and that the UV theory is mapped into an interacting conformal field theory. What kind of theories can be asymptotically safe and what are their fundamental features? We know already that scalar field theories (Higgs-like) are unsafe, and that quantum electrodynamics is unsafe as well, at least in perturbation theory.In the original construction [5] elementary scalars and their induced Yukawa interactions played a crucial role in helping make the overall gauge-Yukawa theory safe. Here we will investigate, instead, the ultraviolet fate of gaugefermion theories at a finite number of colors but a very large number of flavors of both Abelian and non-Abelian nature.We start by considering an SUðN c Þ gauge theory with N f fermions transforming according to a given representation of the gauge group. We will assume that asymptotic freedom is lost, meaning that the number of flavors is larger than N AF f > 11C G =ð4T R Þ, where the first coefficient of the beta function changes sign. We do not need to ...

Safety versus triviality on the lattice

“…Even though the large N f beta function develops a non-trivial zero, it was demonstrated in Ref. [31], that at the alleged UV fixed point the fermion mass anomalous dimension violates the unitarity bound and hence the UV fixed point is unphysical. At this order in 1/N f we conclude that QED is trivial.…”

“…For the fundamental representation, the leading 1/N f expansion is applicable only for N c ∼ < N f /10, while for the adjoint representation we find N f ∼ > 7 for any N c . Following [31], it is sensible to use these values as first rough estimate of the lower boundary of the Safe QCD region. Altogether, these constraints allowed to draw the corresponding phase diagrams in [31].…”

“…The paper is organised as follows: In Section II we briefly review the analytical results for (non)-abelian gauge theories at leading order in 1/N f . We compare the large N f beta function with two-loop perturbation theory and discuss the conformal window 2.0 [31]. It is straightforward to specialise the results of this section to the case of SU(2) gauge theory investigated on the lattice.…”

“…Interesting hints come from the knowledge of the beta function for abelian and non-abelian gauge-fermion theories at leading order in 1/N f [27][28][29][30]. This result suggests, as we shall review below, the potential existence of short distance conformality [30,31]. However due to the fact that the UV zero in the beta function stems from a logarithmic singularity and more generally that the beta function is not a physical quantity, careful consistency checks of these results are crucial [32].…”

The β-function for Yukawa theory at large Nf

“…refs. [1][2][3]. In particular, gauge-Yukawa theories with a large number of fermion flavours, N f , provide interesting candidates within the asymptotic-safety framework as opposed to the traditional asymptotic-freedom paradigm [4,5].…”