Abstract
A tensorial representation of $\phi^4$ field theory introduced in Phys. Rev. D. 93, 085005 (2016) is studied close to six dimensions, with an eye towards a possible realization of an interacting conformal field theory in five dimensions. We employ the two-loop $\epsilon$-expansion, two-loop fixed-dimension renormalization group, and non-perturbative functional renormalization group. An interacting, real, infrared-stable fixed point is found near six dimensions, and the corresponding anomalous dimensions are computed to the second order in small parameter $\epsilon=6-d$. Two-loop epsilon-expansion indicates, however, that the second-order corrections may destabilize the fixed point at some critical $\epsilon_c <1$. A more detailed analysis within all three computational schemes suggests that the interacting, infrared-stable fixed point found previously collides with another fixed point and becomes complex when the dimension is lowered from six towards five. Such a result would conform to the expectation of triviality of $O(2)$ field theories above four dimensions.
Highlights
The question of the existence of conformally invariant interacting field theories in dimensions higher than four has recently stimulated efforts in two closely related directions
II, we extend the ε expansion to the two-loop level and analyze the ensuing renormalization group (RG) flow equations
As discussed in the Introduction, not directly violating them, this finding seems to go against the intuition based on proofs [4,5] that state that at least for N 1⁄4 1 and N 1⁄4 2 the standard φ4 theory must be trivial for any d > 4
Summary
The question of the existence of conformally invariant interacting field theories in (spacetime) dimensions higher than four has recently stimulated efforts in two closely related directions. A perturbative one-loop analysis [3] of Eq (1) in dimension d 1⁄4 6 − ε identified a nontrivial real IR-stable fixed point in certain ranges of small values of N that include, most interestingly, the physically relevant cases of N 1⁄4 2 and N 1⁄4 3. Examining the nontrivial roots of the beta function, one finds that the fixed point is real only for ε ≪ 1, being rendered complex on its way to the physical ε 1⁄4 1 by the collision and annihilation with another fixed point This is less surprising after recalling that the same occurs at some critical ε at every even order of expansion around d 1⁄4 4 in the canonical φ4 theory as well [6]. Their evaluation in d 1⁄4 6 − ε dimensions is a straightforward exercise in combinatorics and standard momentum
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