Abstract
In the paper [L. Fei etal., J. High Energy Phys. 09 (2015) 076JHEPFG1029-847910.1007/JHEP09(2015)076] a cubic field theory of a scalar field σ and two anticommuting scalar fields, θ and θ[over ¯], was formulated. In 6-ε dimensions it has a weakly coupled fixed point with imaginary cubic couplings where the symmetry is enhanced to the supergroup OSp(1|2). This theory may be viewed as a "UV completion" in 2<d<6 of the nonlinear sigma model with hyperbolic target space H^{0|2} described by a pair of intrinsic anticommuting coordinates. It also describes the q→0 limit of the critical q-state Potts model, which is equivalent to the statistical mechanics of spanning forests on a graph. In this Letter we generalize these results to a class of OSp(1|2M) symmetric field theories whose upper critical dimensions are d_{c}(M)=2[(2M+1)/(2M-1)]. They contain 2M anticommuting scalar fields, θ^{i}, θ[over ¯]^{i}, and one commuting one, with interaction g(σ^{2}+2θ^{i}θ[over ¯]^{i})^{(2M+1)/2}. Ind_{c}(M)-ε dimensions, we find a weakly coupled IR fixed point at an imaginary value of g. We propose that these critical theories are the UV completions of the sigma models with fermionic hyperbolic target spaces H^{0|2M}. Of particular interest is the quintic field theory with OSp(1|4) symmetry, whose upper critical dimension is 10/3. Using this theory, we make a prediction for the critical behavior of the OSp(1|4) lattice system in three dimensions.
Highlights
Introduction.—This Letter builds on the paper [1] where the field theory was studied with the Euclidean action
The global Sp(2) symmetry of this model becomes enhanced to the supergroup OSpð1j2Þ because at the IR fixed point in 6 − ε dimensions the two coupling constants are imaginary and related by gÃ2 1⁄4 2gÃ1
The beta function of this theory is the same as for the OðNÞ. nonlinear sigma model continued to N 1⁄4 −1, and the theory is asymptotically free for negative g2
Summary
Ddxð∂μθ∂μθ − θθ∂μθ∂μθÞ; ð3Þ and it is important to take g2 < 0 so that the model is asymptotically free in d 1⁄4 2 [4,11,12,13]. the beta function of this theory is the same as for the OðNÞ. nonlinear sigma model continued to N 1⁄4 −1, and the theory is asymptotically free for negative g2. We are led to consider theories with interactions of order 2M þ 1, i.e., σ2Mþ1 plus terms involving the anticommuting fields [20] Such theories have the upper critical dimensions dcðMÞ. We find OSpð1j2MÞ invariant IR fixed points where the interaction term is proportional to ðσ þ 2θiθiÞð2Mþ1Þ=2 with an imaginary coefficient These critical theories appear to be nonperturbatively well defined, and it would be very interesting to compare the continuum results with those in the OSpð1j2MÞ lattice systems. Scaling dimensions for the OSp ð1j2Þ model.—The oneloop beta functions and anomalous dimensions for the theory (22) are [1] These results can be extended to the four-loop order using the formulae from [24] for the OðNÞ invariant cubic model [25,26], and setting N 1⁄4 −2 [27]. The important operator in the sigma model has dimension [30]
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