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January 2, 2007

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 stΛ QCD 2s\gg -t \gg \Lambda_{\text{QCD}}^2, from resumming a set of diagrams to all orders in (α slog(s/t)) n{\left(\alpha_s \log(s/t)\right)}^n, n0n\geq 0.

For t0t\sim 0, the scattering amplitude of two hadrons can be written in terms of impact functions, Φ i(k i 2)\Phi_i(\vec{k}_i^2), which are functions of the 2D transverse momenta, k i\vec{k}_i, and a kernel, K(k 1 2,k 2 2,s)K(\vec{k}_1^2,\vec{k}_2^2,s)

(1)𝒜(s,0)=sd 2k 1d 2k 2k 1 2k 2 2Φ 1(k 1 2)K(k 1 2,k 2 2,s)Φ 2(k 2 2) \mathcal{A}(s,0) = s \int \frac{\mathrm{d}^2 k_1 \mathrm{d}^2 k_2}{\vec{k}_1^2 \vec{k}_2^2} \Phi_1(\vec{k}_1^2)K(\vec{k}_1^2,\vec{k}_2^2,s)\Phi_2(\vec{k}_2^2)

In the leading-log approximation, this BFKL kernel is

(2)K(k 1 2,k 2 2,s)=1πk 2 2 12i 12=idη2πie yξ 0(η)η(τ 1τ 2) K(\vec{k}_1^2,\vec{k}_2^2,s) = \frac{1}{\pi \vec{k}_2^2} \int_{\frac{1}{2}-i\infty}^{\frac{1}{2}=i\infty} \frac{\mathrm{d}\eta}{2\pi i} e^{y\xi_0(\eta) -\eta(\tau_1-\tau_2)}

where we’ve defined the dimensionless parameters y=g 2N(2π) 2log(sμ 2),τ i=log(k i 2μ 2) 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 μ\mu some characteristic energy scale for the process, and the function χ 0\chi_0 is written in terms of digamma functions χ 0(η)=2ψ(1)ψ(η)ψ(1η) \chi_0(\eta) =2\psi(1)-\psi(\eta)-\psi(1-\eta)

The BFKL amplitude rises faster than slog 2(s)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 η=η *\eta=\eta_*, given by

(3)η *χ 0(η *)=χ 0(η *) \eta_* \chi'_0(\eta_*)=\chi_0(\eta_*)

One then finds the value of yy, as a function of τ 1τ 2\tau_1-\tau_2, which saturates the unitarity bound.

(4)y c=2χ 0(η *)12(τ 1τ 2)γ BFKL12(τ 1τ 2) y_c = \frac{2}{\chi'_0(\eta_*)} \tfrac{1}{2}(\tau_1-\tau_2)\equiv \gamma_{\text{BFKL}} \tfrac{1}{2}(\tau_1-\tau_2)


(5)γ BFKL0.409552 \gamma_{\text{BFKL}}\sim 0.409552

They would like to identify this critical exponent with Choptuik exponent

(6)M BH(pp *) γ M_{\text{BH}} \propto (p-p_*)^\gamma

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 (pp) in the initial value data, one can tune to this critical surface. For p>p *p\gt p_*, a blackhole forms, and its mass is universally determined by this scaling exponent, γ\gamma. Numerically, in 5D, γ=0.408±2%\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 γ BFKL\gamma_{\text{BFKL}} and γ\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 α\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.

1 In many “realistic” systems, there’s a third basin of attraction, corresponding to the formation of stable stars.

Posted by distler at January 2, 2007 3:33 AM

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Re: Choptuik and the Pomeron

Dear Jacques,

I would think that since there is a cosmological constant in the gravity dual of QCD, there is a natural scale. So I am not sure that the scaling phenomena in the gravitational side would exhibit the usual Choptuik scaling (Type 2 critical phenomenon) and not a mass gap (Type 1). When there is a natural scale in the gravitational system (like when the scalar field is massive) this is what one usually finds from numerical simulations.

Am I wrong in thinking that? The numerical simulations of Sorkin and Oren for the 5D black hole (hep-th/0502034), from which Alvarez-Gaume et al cite their result, is done for no cosmological constant.

Any comment would be highly appreciated. The idea that saturation of bound objects in QCD is dual to black hole processes strikes me as very plausible, so I tried to look at the literature on the Next-to-Leading Log (NLL) BFKL and saturation effects when the Alvarez-Gaume et al’s paper came out, and talked to some people who are in the business of doing these things. But the coincidence of the numbers at leading log seems A LOT like an artifact of the formalism. I was hoping that the NLL corrections would somehow reproduce the scale-echoing, but literature (and other people who know better) tell me that the saturation exponent changes drastically as well.

Hope everything is well in Austin,
Chethan Krishnan.

Posted by: chethan on January 3, 2007 6:51 AM | Permalink | Reply to this

Re: Choptuik and the Pomeron

I have been trying to convince myself that black hole formation is in some sense “local” so that the cosmological constant is dominated by local curvature effects. But I am not sure how much I believe it.

Posted by: chethan on January 3, 2007 7:12 AM | Permalink | Reply to this

Re: Choptuik and the Pomeron

You’re right that Sorkin and Oren (and, indeed, most everyone else) work in flat space.

The presence of a cosmological constant drastically affects the behaviour of large blackholes. But small blackholes (the subject of interest) in AdS behave exactly as in flat space. So it’s plausible that the Choptuik exponent is unchanged.

Posted by: Jacques Distler on January 3, 2007 8:50 AM | Permalink | PGP Sig | Reply to this

Re: Choptuik and the Pomeron

There is a qualitative difference between the two cases. When there is a length scale, I worry that there will be a mass-gap (the mass of that critical black hole will be detremined by that length scale). When there is no length scale, the black hole at criticality has zero-mass (a naked singularity), and there it makes sense to talk about the Choptuik scaling exponent.

We are not talking about the formation of ONE small black hole, but about a certain universal behavior in the space of initial value data, which is defined on some Cauchy surface - and that is NOT a local construction. My concern is that the universality class is affected by the presence or absence of the cosmological constant.


Posted by: chethan on January 3, 2007 10:26 AM | Permalink | Reply to this

Re: Choptuik and the Pomeron

Yes, there’s a small parameter: the cosmological constant in Planck (or string) units. When that parameter is zero, the transition is 2nd order. You are worried that, when you turn the parameter on, the transition becomes 1st order. Physically, I don’t see why that should be. If you are close to the transition, all the physics of the blackhole formation is happening on length scales much shorter than the radius of curvature of the AdS space.

Posted by: Jacques Distler on January 3, 2007 11:08 AM | Permalink | PGP Sig | Reply to this

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