Musings

The general strategy is to look for a dilatation current of the form $$D^\mu(x) = x^\nu \tensor{T}{_\nu_^\mu}(x) -V^\mu(x)$$ where the Virial current, $V^\mu(x)$, is supposed to be gauge-invariant and to have no explicit dependence on the spacetime coordinates. Scale invariance is obtained if $$\tensor{T}{_\mu_^\mu}= \partial_\mu V^\mu$$ The stress tensor can be improved to a traceless one (and hence scale invariance promoted to conformal invariance), provided

with $L^{\mu\nu}$ a symmetric tensor, and $\J^\mu$ a conserved current. So, to achieve their goal, they must find a Virial current which cannot be written in the form (1). They study a general renormalizable gauge theory, with scalars and Weyl fermions, writing a candidate Virial current of the form

where the matrix $Q_{a b}$ is real anti-symmetric and $P_{i j}$ is anti-Hermitian. Scale invariance, then requires that a particular linear relation hold between (on the one hand) the $\beta$-function coefficients for the scalar quartic couplings and Yukawa couplings and (on the other) the $Q$s and the $P$s. (The $\beta$-function for the gauge coupling, mercifully, is still supposed to vanish.)

The trick, then, is to write a sufficiently complicated Lagrangian such that, at some loop order, the $\beta$-function constraints no longer force the Virial (2) (and hence the $\beta$-function coefficients themselves) to vanish. The simplest theory seems to involve 2 scalars, 2 fermions, and 11 coupling constants (5 scalar self-couplings and 6 Yukawa couplings). And one needs to go to 3-loop order in dimensional regularization (2-loops in some scheme that I don’t really understand), to see the violation of conformal invariance.

It strikes me that this is way too complicated for what (if it’s real) should be a robust phenomenon. But, hey, the week is yet young …