### Froissart Follies

I’ve been meaning, for some time, to write something about string scattering at high energies. There’s an excellent review by Veneziano and a recent paper by Giddings, Gross and Maharana.

But, I have to get something off my chest, first.

In the Discussion section of their paper, Giddings *et al* note that the cross section for blackhole formation

grows as a power of $s$, and hence violates the Froissart bound which, in $D$-dimensions is

from which they would like to gather

While the basic assumptions needed to derive these bounds, particularly the existence of a gap, are not strictly satisfied in gravitational scattering, it is tempting to conclude that violation of these bounds is associated with some essential nonlocality associated with non-perturbative gravitational dynamics.

Whatever the nature of non-perturbative gravitational dynamics, the violation of the Froissart bound is far more prosaic in origin.

First of all, the total cross section doesn’t even *exist* (the integral, over solid angle, of the differential cross section diverges) for $D\lt 7$. For $D\geq 7$, the elastic cross section grows as
$\sigma_{\text{el}}(s) \sim s^{\frac{D-2}{D-4}}$
which dominates over (1) at high energies and, of course, it violates (2).

The massless graviton means that elastic scattering already violates Froissart, and so it’s not particularly surprising that inelastic blackhole formation does too. At least, the latter is not persuasive evidence for any sort of stringy nonlocality.

Perhaps it might be helpful to disgress and explain how these bounds are proven.

The starting point is to do a partial wave expansion of the scattering amplitude. Let
$\cos\theta = 1 +\tfrac{2t}{s -4m^2}$
be the center-of-mass scattering angle and write^{1}

The crucial features that one, then, relies upon are

- $\tilde{f}(s,x)$ is analytic in $x$, and the series (3) converges in some ellipse in the complex $x$-plane, whose foci are $x=\pm 1$.
- For $x\gt1$, the Legendre polynomials (in higher dimensions, the Gegenbauer Polynomials, $C^\lambda_l(x) = \frac{\Gamma(2\lambda+l)}{\Gamma(l+1)\Gamma(2\lambda)} \multiscripts{_2}{F}{_1}\left(2\lambda+l,-1,\lambda+\tfrac{1}{2},\tfrac{1-x}{2}\right)$ with $\lambda= (D-3)/2$) are positive definite. In fact, for any $x\gt 1$, The $C^\lambda_l(x)$ are an monotonically increasing function of $l$, growing like $x^l$, for large $l$.

Demanding that the series (3), nonetheless, converge everywhere in the Lehmann ellipse, puts an upper bound on the (imaginary part and magnitude of) the amplitude at $x=1$ ($t=0$). By the Optical Theorem, $\sigma_{\text{tot}}= \frac{Im \tilde{f}(s, \cos\theta=1)}{2\sqrt{s(s-4m^2)}}$ the former translates into an upper bound on the total cross section.

What happens when you have massless particles is that the $t$-channel cut extends all the way down to $t=0$, and (3) doesn’t converge for $|x|\gt 1$ and you don’t get the concomitant constraint on the forward scattering amplitude. Perhaps some bound weaker than (2) continues to hold. Indeed, calculations seem to indicate that may be the case. But you would need different methods to prove it. And, to my knowledge, no one’s done it.

Anyway, all of this has to do with scattering at large impact parameter, where the eikonal approximation, pioneered by Amati *et al* is valid. The really interesting physics (ultimately, the physics of blackhole formation) is at small impact parameter.

But I guess that will have to be the subject of another post.

^{1}I’ll stick to the 4D case; see, *e.g.* Chaichian and Fischer for the higher dimensional generalization.