## March 14, 2012

### Spring Break

I could be back in Austin, chillin’ at SXSW. Instead, I’m here in College Station, at a workshop on higher-dimensional CFTs.

I’ll be giving a talk, on Thursday, which will be a lot like the one I gave in Munich, two weeks ago. The only advantage will be that the paper, that I alluded to previously, is finally be out.

There are, I’ll admit, some compensations. Andreas Stergiou gave a very nice talk, today, about his work with Ben Grinstein and Jean-François Fortin on theories with scale but not conformal invariance. They have to work in $4-\epsilon$ dimensions, and need to go to 3 loops to find a violation of conformal invariance. Aside from their complexity, (interacting, unitary and Poincaré-invariant) scale invariant QFTs are weird. They are manifested as Renormalization Group limit cycles (or, even crazier, quasi-periodic motion on a torus), rather than fixed points. Fixed-point theories really are conformally-invariant.

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

(1)$V^\mu = J^\mu +\partial_\nu L^{\nu\mu}$

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

(2)$V_\mu = Q_{a b} \phi^a D_\mu \phi^b - i P_{i j} \overline{\psi}^i \overline{\sigma}_\mu \psi^j$

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 …

Posted by distler at March 14, 2012 1:04 AM

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