## April 28, 2004

### Nima2

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 $\langle\dot\phi\rangle=M^2$ Expanding the field about its VEV,

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

under an infinitesimal diffeomorphism,

(2)\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 $\pi$ 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 $phi\to \phi +\text{const}$. The naive time-translation symmetry of a static spacetime is broken in this background, but the combination

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

is unbroken.

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

(4)$L_{\text{eff}} = \frac{1}{2} \dot\pi^2 - \frac{\beta}{2} (\nabla^2\pi)^2 +\dots$

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. $t$ should have mass dimension $-1$, $x$ should have mass dimension $-1/2$ and $\pi$ should have mass dimension $1/4$. The leading interaction term is

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

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

Anyway, if you take $M\sim 10^{-3}$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 $M$, and requires some ultraviolet completion there. However, if $\phi$ 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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