Abstract
To explore the possibility of self-organized criticality, we look for CFTs without any relevant scalar deformations (a.k.a dead-end CFTs) within power-counting renormalizable quantum field theories with a weakly coupled Lagrangian description. In three dimensions, the only candidates are pure (Abelian) gauge theories, which may be further deformed by Chern-Simons terms. In four dimensions, we show that there are infinitely many non-trivial candidates based on chiral gauge theories. Using the three-loop beta functions, we compute the gap of scaling dimensions above the marginal value, and it can be as small as $\mathcal{O}(10^{-5})$ and robust against the perturbative corrections. These classes of candidates are very weakly coupled and our perturbative conclusion seems difficult to refute. Thus, the hypothesis that non-trivial dead-end CFTs do not exist is likely to be false in four dimensions.
Highlights
This game is designed to examine the possibility of self-organized criticality [1] in quantum field theories
Using the three-loop beta functions, we compute the gap of scaling dimensions above the marginal value, and it can be as small as O(10−5) and robust against the perturbative corrections
A clever use of the S-matrix and its analyticity properties has led to many important results in quantum field theories such as the proof of the a-theorem in four dimensions [3], enhancement of scale invariance to conformal invariance [4, 5], convexity properties of large twist operators in general CFTs [6], and so on
Summary
This game is designed to examine the possibility of self-organized criticality [1] (see e.g. [2] for a review) in quantum field theories. We begin with the matter content of renormalizable quantum field theories that have a weakly coupled Lagrangian description in three dimensions It consists of a certain number of bosonic spin-zero scalar fields and fermionic spin-half spinor fields charged under gauge groups that have suitable kinetic terms. As in the scalar case, one can use the same bilinear form to construct gauge invariant Majorana mass terms (or real mass terms) for the fermions proportional to gabψaψb These are relevant deformations with the power-counting scaling dimension ∆ = 2. Renormalizable field theories in four dimensions with a weakly coupled Lagrangian description can have bosonic spin-zero scalar fields and fermionic spin-half spinor fields, charged under the gauge group, with finite kinetic terms. One can use the same bilinear form to construct gauge invariant mass terms for the scalars proportional to gIJ φI φJ See e.g. [13] and references therein for further discussions on the non-perturbative renormalizability
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