Challenge to theaTheorem in Six Dimensions

Abstract

The possibility of a strong a theorem in six dimensions is examined in multiflavor ϕ^{3} theory. Contrary to the case in two and four dimensions, we find that, in perturbation theory, the relevant quantity a[over ˜] increases monotonically along flows away from the trivial fixed point. a[over ˜] is a natural extension of the coefficient a of the Euler term in the trace anomaly, and it arises in any even spacetime dimension from an analysis based on Weyl consistency conditions. We also obtain the anomalous dimensions and beta functions of multiflavor ϕ^{3} theory to two loops. Our results suggest that some new intuition about the a theorem is in order.

The a-function in six dimensions

“…This is the technique we shall apply to obtain A (4) , A (5) later in this paper. We should note that A has no explicit two-loop contributions; however there is a one-loop (free-field) contribution [19,20]. 3 At this point we have all the information required for our computation of A (4) , but we shall postpone this to section 4 where we shall explain the general method we shall use to compute both A (4) and A (5) .…”

“…Our purpose here is to extend the results of ref. [19,20] to higher orders, again for the multiflavor φ 3 theory. However, we shall do this by using the β-functions together with eq.…”

“…[14] which differs by a factor of 6. In six dimensions, G IJ has recently been computed to be negative definite at leading order for a multiflavor φ 3 theory [19,20]. The six-dimensional case has also been considered in more general terms in ref.…”

The a-function for gauge theories

“…Here G IJ = G JI ,ρ I and Q J may all be computed perturbatively within the theory extended to curved spacetime and x-dependent g I ; for weak couplings G IJ can be shown to be positive definite in four dimensions (in six dimensions, G IJ has recently been computed to be negative definite at leading order [13]). Eq.…”

Nonlocal correction to the energy-momentum tensor for $$\phi ^{3}$$ ϕ 3 theory in six dimensions

“…Outside of perturbation theory, only a weak version of the conjecture has been shown [9]. However, a challenge to this proposal in six dimensions has been published recently in [10].…”