Musings

Choptuik and the Pomeron

One of the most intriguing papers of 2006 was by Álvarez-Gaumé,Gómez and Vázquez-Mozo. They noticed a remarkable numerical coincidence between the Choptuik exponent for the formation of 5D blackholes, and a certain critical exponent associated to the BFKL Pomeron in 4D gauge theories.

The latter emerges in perturbative gauge theories, in the regime $s\gg -t \gg \Lambda_{\text{QCD}}^2$, from resumming a set of diagrams to all orders in ${\left(\alpha_s \log(s/t)\right)}^n$, $n\geq 0$.

For $t\sim 0$, the scattering amplitude of two hadrons can be written in terms of impact functions, $\Phi_i(\vec{k}_i^2)$, which are functions of the 2D transverse momenta, $\vec{k}_i$, and a kernel, $K(\vec{k}_1^2,\vec{k}_2^2,s)$

In the leading-log approximation, this BFKL kernel is

where we’ve defined the dimensionless parameters $$y = \frac{g^2 N}{(2\pi)^2}\log\left(\frac{s}{\mu^2}\right),\quad \tau_i = \log\left(\frac{\vec{k}_i^2}{\mu^2}\right)$$ with $\mu$ some characteristic energy scale for the process, and the function $\chi_0$ is written in terms of digamma functions $$\chi_0(\eta) =2\psi(1)-\psi(\eta)-\psi(1-\eta)$$

The BFKL amplitude rises faster than $s\log^2(s)$, and eventually violates the unitarity bound. The idea is to define a critical exponent by evaluating the above integral in a saddle point approximation, with the saddle point located at $\eta=\eta_*$, given by

One then finds the value of $y$, as a function of $\tau_1-\tau_2$, which saturates the unitarity bound.

Numerically,

They would like to identify this critical exponent with Choptuik exponent

for 5D gravitational collapse. The space of initial value data for the gravitational collapse has two basins of attraction¹, corresponding to empty space and to the formation of a blackhole. These two basins of attraction are separated by a critical surface of codimension-1. By tuning a single parameter ($p$) in the initial value data, one can tune to this critical surface. For $p\gt p_*$, a blackhole forms, and its mass is universally determined by this scaling exponent, $\gamma$. Numerically, in 5D, $\gamma=0.408\pm2\%$.

Why should these two exponents be related?

Álvarez-Gaumé et al give some arguments, but I don’t find them terribly convincing.

1. The next-to-leading-order corrections to the BFKL kernel are large. In a conformally-invariant theory, they are not as large as in QCD, but they are still substantial enough to make one wonder whether the apparent agreement between $\gamma_{\text{BFKL}}$ and $\gamma$ is spurious.
2. Leaving that aside, one wonders why a weak-coupling calculation of the BFKL exponent has anything to do with a computation in classical 5D GR. Weak coupling in the gauge theory corresponds to a small radius, highly-curved AdS space, where $\alpha'$ corrections to GR are substantial. Now, it’s true (as they argue) that blackhole creation at threshold involves regions of large curvature (where the corrections to GR are, presumably, substantial). But the whole interest of Choptuik’s result is that it is supposed to be robust against such corrections. The scaling behaviour, (6), for instance, continues to hold reasonably far from the critical surface, where the curvature at the apparent horizon is small enough that GR is trustworthy.
3. The BFKL Pomeron has been studied at strong 't Hooft coupling. But the resulting kernel does not look much (2).

I find it very plausible that there’s a relation between the bulk gravitational collapse problem and Pomeron physics. But I don’t (yet) see how the particulars of the leading-log BFKL calculation connects to the physics of Choptuik scaling. I hope 2007 brings some insight into whether their observation is a mere coincidence, or the first hint of something very deep.