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April 28, 2004


One of my favourite young physicists, Nima Arkani-Hamed, was in town today. He gave two talks.

One was about his work on spontaneously-broken diffeomorphism invariance. Specifically, consider a theory in which spatial diffeomorphisms are preserved, but time-translations are spontaneously-broken. There’s a scalar field which, in a really horrible pun, they call the ghostino, whose expectation value satisfies ϕ˙=M 2\langle\dot\phi\rangle=M^2 Expanding the field about its VEV,

(1)ϕ(x,t)=M 2t+π(x,t), \phi(x,t) = M^2 t + \pi(x,t),

under an infinitesimal diffeomorphism,

(2)δx μ =ξ μ(x,t) δπ =ξ μ μπM 2ξ 0\array{\arrayopts{\padding{0}\colalign{right left}} \delta x^\mu &= \xi^\mu(x,t)\\ \delta \pi &= \xi^\mu\partial_\mu \pi - M^2\xi^0 }

so π\pi transforms as a scalar under spatial diffeomorphisms, but transforms inhomogeneously under temporal ones, as befits a Goldstone boson. We also impose a shift symmetry under phiϕ+constphi\to \phi +\text{const}. The naive time-translation symmetry of a static spacetime is broken in this background, but the combination

(3)tt+c,ππc/M 2 t \to t+c,\quad \pi\to \pi-c/M^2

is unbroken.

If you write out a general symmetry-breaking effective Lagrangian for ϕ\phi (compatible with the shift symmetry and, for simplicity, with ϕϕ\phi\to-\phi), and expand it about a minimum, you find something like (after rescaling π\pi to give it a canonically-normalized kinetic energy)

(4)L eff=12π˙ 2β2( 2π) 2+ L_{\text{eff}} = \frac{1}{2} \dot\pi^2 - \frac{\beta}{2} (\nabla^2\pi)^2 +\dots

The dispersion relation is a nonrelativistic one (unsurprising, since the symmetry-breaking has picked out a preferred Lorentz frame) and power-counting is a bit unconventional. tt should have mass dimension 1-1, xx should have mass dimension 1/2-1/2 and π\pi should have mass dimension 1/41/4. The leading interaction term is

(5)L int=γM 1/4π˙(π) 2+ L_{\text{int}} = \frac{\gamma}{M^{1/4}} \dot\pi (\nabla\pi)^2 +\dots

and is irrelevant in the infrared, so there’s a good perturbative effective field theory description.

Anyway, if you take M10 3M\sim 10^{-3}eV, the coupling of this theory to gravity modifies gravity at cosmological distance scales, with interesting ramifications for cosmology.

There’s a bit of a swindle here, since the theory just described breaks down above the scale MM, and requires some ultraviolet completion there. However, if ϕ\phi couples only gravitationally, they argue that it doesn’t really matter what the ultraviolet completion is. While there remains a challenge to embed this in a “real” theory, their effective Lagrangian analysis indicates that it’s not completely crazy to try to do so. You might not have expected it, but the long-distance physics makes sense.

Nima’s other talk was about “high energy” supersymmetry, some as yet unpublished work of his with Savas Dimopoulos, in which supersymmetry is broken at a relatively high scale and, of the superpartners, only the gauginos are light.

I’ll talk about that in more detail some other time…

Posted by distler at April 28, 2004 1:14 AM

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