Skip to the Main Content

Note:These pages make extensive use of the latest XHTML and CSS Standards. They ought to look great in any standards-compliant modern browser. Unfortunately, they will probably look horrible in older browsers, like Netscape 4.x and IE 4.x. Moreover, many posts use MathML, which is, currently only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser.

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 4ϵ4-\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 μ(x)=x νTν μ(x)V μ(x) D^\mu(x) = x^\nu \tensor{T}{_\nu_^\mu}(x) -V^\mu(x) where the Virial current, V μ(x)V^\mu(x), is supposed to be gauge-invariant and to have no explicit dependence on the spacetime coordinates. Scale invariance is obtained if Tμ μ= μV μ \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 μ=J μ+ νL νμV^\mu = J^\mu +\partial_\nu L^{\nu\mu}

with L μνL^{\mu\nu} a symmetric tensor, and J μ\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 μ=Q abϕ aD μϕ biP ijψ¯ iσ¯ μψ jV_\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 abQ_{a b} is real anti-symmetric and P ijP_{i j} is anti-Hermitian. Scale invariance, then requires that a particular linear relation hold between (on the one hand) the β\beta-function coefficients for the scalar quartic couplings and Yukawa couplings and (on the other) the QQs and the PPs. (The β\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 β\beta-function constraints no longer force the Virial (2) (and hence the β\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

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/2506

1 Comment & 0 Trackbacks

Re: Spring Break

Thanks for posting a link to your talk. It’s always interesting to read your posts, and to see a talk by you.

Take care,
mike

Posted by: mike on March 14, 2012 9:25 PM | Permalink | Reply to this

Post a New Comment