Abstract
We present the first numerical study of the ultraviolet dynamics of non-asymptotically free gauge-fermion theories at large number of matter fields. As testbed theories we consider non-abelian SU(2) gauge theories with 24 and 48 Dirac fermions on the lattice. For these number of flavors asymptotic freedom is lost and the theories are governed by a gaussian fixed point at low energies. In the ultraviolet they can develop a physical cutoff and therefore be trivial, or achieve an interacting safe fixed point and therefore be fundamental at all energy scales. We demonstrate that the gradient flow method can be successfully implemented and applied to determine the renormalized running coupling when asymptotic freedom is lost. Additionally, we prove that our analysis is connected to the gaussian fixed point as our results nicely match with the perturbative beta function. Intriguingly, we observe that it is hard to achieve large values of the renormalized coupling on the lattice. This might be an early sign of the existence of a physical cutoff and imply that a larger number of flavors is needed to achieve the safe fixed point. A more conservative interpretation of the results is that the current lattice action is unable to explore the deep ultraviolet region where safety might emerge. Our work constitutes an essential step towards determining the ultraviolet fate of non asymptotically free gauge theories.
Highlights
Free theories [1,2] are fundamental according to Wilson [3,4] since they are well defined from low to arbitrary high energies
The UV behavior of gauge-fermion theories as a function of the number of matter fields poses a fundamental problem on our understanding of quantum field theory, both perturbatively and nonperturbatively
We have discussed in detail the categorization of gauge-fermion theories based on their possible UV behaviors
Summary
Free theories [1,2] are fundamental according to Wilson [3,4] since they are well defined from low to arbitrary high energies. 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] For these reasons, and because it is important to uncover the phase diagram of four-dimensional quantum field theories, we initiate here a consistent lattice investigation of the ultraviolet fate of non-Abelian gauge-fermion theories at a small number of colors but at a large number of flavors where asymptotic freedom is lost. [11] to the case in which asymptotic freedom is lost (Nf ≥ 3Nc) for super QCD with(out) a meson [11,63], one can show that these theories are unsafe for any number of matter fields This is in agreement with the 1=Nf studies of the supersymmetric beta functions [63,68] as explained in Ref. These are the two theories we choose to investigate on the lattice, i.e., an SU(2) gauge theory with either 24 or 48 flavors transforming according to the fundamental representation of the underlying gauge group
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