Tuesday, March 30, 2021

Pinball, heavy strings, black holes, saddles, and chaos

The new hep-th preprints mention the word "string" 27 times today – in 4 abstracts of new hep-th papers, 2 new cross-listed ones, and 2 replacements. I think that the most interesting paper in this "string set" is
Chaotic scattering of highly excited strings
by (QCD co-father) David Gross and Vladimir Rosenhaus. It is about some high-energy scattering, in this case that of heavily excited strings, the the chaos that it exhibits.



Note that Gross (who was 80 last month, congratulations, David) has been investigating similar high-energy behavior of amplitudes for decades, see e.g. Gross-Mende. Here, they are calculating the tree-level perturbative string amplitudes for heavily excited string modes at high energies. Note that highly excited string states are "progenitors" of black hole microstates. And because the latter are very efficient in quickly and effectively mixing and scrambling the information, the excited string modes might be able to do it, too, and indeed, they do find out that the strings do behave in this chaotic way, too.



Their mental model of chaos is a pinball with lots of collisions of balls. The elastic collisions finely depend on the precise angle and even the qualitatively character of the collisions has many possibilities how to evolve. I think that billiard is good enough for this concept – and I think that "success" in billiard becomes a matter of good luck when the number of ball collisions becomes large, like 5 or more.

In this paper, they calculate the Veneziano-like (or Virasoro-Shapiro-like) scattering amplitudes at a fixed angle, asymptotically high energies, and they realize that even for low-lying states, there are the so-called scattering equations that determine the positions of the saddles in the world sheet plane (or half-plane) as a function of the string momenta \(p^\mu\). The appearance of the saddle approximation is cool and sort of unavoidable. The number of such saddles tends to be \((n-3)!\) and the saddles provide us with a "dual" (and particle-like or pinball-like) reorganization of the whole calculation.

You know, the integrals that produce the Veneziano amplitude may look like some "totally chaotic functions with contributions from everywhere". But in the extreme limit of high energies, the integrand is highly non-uniform, being much larger at some positions of the vertex operators \(\{z_i\}\) than others. That's why the high-energy limit is also one where the integrals may be approximated by the saddle points. You just find out where the relevant saddle is located and what is the Gaussian-like approximation of the integrand around that point.

Some special properties occur when they consider particular highly excited strings instead of the low-lying string excitations and the mathematical structures they find generalize both the scattering equations and CHY (Cachazo-He-Yuan, two influential papers from 2014), who used some KLT decompositions to find some link between the world sheet and spacetime expressions.

Some of the Gross-Rosenhaus formulae are compact and precise and they demonstrate the chaotic, pinball-like behavior of the high-mass, high-energy perturbative (tree-level) scattering amplitudes.

Unfortunately, the economy of the more nontrivial explanations doesn't add up so I reduce the old-fashioned texts about nontrivial topics by 80-90 percent and avoid proofreading.

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