Tuesday, January 09, 2007

Zeros of scattering amplitudes

Take the combined Veneziano amplitude for four U(1) open string bosonic tachyons, or the Virasoro-Shapiro amplitude for four closed string bosonic tachyons. Write it as a function of s,t,u, the Mandelstam variables.

In both cases, the amplitude factorizes
  • M(s,t,u) = f(s) f(t) f(u)

where the function f(s) is something like

  • f(s) = zeta(s) / zeta(1-s)
  • f(s) = Gamma(s) / Gamma(1-s)

for open string tachyons and closed string tachyons, respectively. I've simplified the linear relations, so "s,t,u" really mean some standardized linear functions of the usual "s,t,u" variables, but the qualitative message is not changed. You see that in both cases,

  • the amplitude factorizes to functions of s,t,u separately
  • the amplitude gets inverted if s,t,u are replaced by 1-s,1-t,1-u

Is that a coincidence? Concerning the first point, the only allowed poles of a four-point function are those that depend on either s, or t, or u, corresponding to particles in the intermediate channels. This would still allow the amplitude to be redefined by an arbitrary exponential

  • M' = M exp(g(s,t,u))

For example, the amplitude could look like a sum of F(s)+F(t)+F(u) that would also inherit poles as functions of s,t,u.

It seems that the function "g" in the exponent is just zero in all cases and the factor from the exponential is one. It is not used. And the amplitude is a product and not a sum. Why? Moreover, the amplitude seems to care about the position of the zeros. This clearly can't work for non-scalar amplitudes. No one has a full authoritative answer here.

Also, the second property, the inversion of the amplitude when s,t,u are mapped to minus themselves (with an additive shift), may be more general. The zeroes of the amplitude may correspond to some "antistates" for physical vertex operators. In 2D CFT, take an operator O without the exp(ipx) factor of dimension D. The derivative of the Lagrangian with respect to O formally has a dimension of 2-D if it exists. Combined with an exp(ipx) factor, it gives a vertex operator of an "antistate". This antistate could still be interpreted as zeroes of the amplitude as a function of s,t,u.

Four-graviton scattering decreases as exp(-#.S) where S is the entropy of the black hole that you can create in the s-channel and # is a numerical constant of order one. The inversion property explained above would imply that in the unphysical region where s is large and negative while t,u are positive, the amplitude should be exponentially large, exp(+#.S). This simplified Ansatz assumes that the gravitons are scalars. Their spin should only modify the qualitative result by some subleading prefactors.

I wonder whether someone thinks that these results are more general, beyond the tree level amplitudes, and whether there is a conceptual explanation of the factorization and the inversion properties of the four-point functions.

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