# Why can the Euler beta function be interpreted as a scattering amplitude?

The Wikipedia article on the Veneziano Amplitude claims that the Euler beta function can be interpretted as a scattering amplitude. Why is this?

In another word, when the Euler beta function is interpreted as a scattering amplitude, what features does it have that make it able to explain strong force of mesons?

What properties of (or why) the Euler beta function (when interpreted as scattering amplitude) has string behavior?

• Susskind's lecture 6 on string theory discusses the scattering interpretation semi-informally, offset at about 45 minutes into the video, stretched to a length of about 30 minutes. – ccorn Mar 10 '14 at 17:15
• I refer you to Schwarz and Witten's textbook 'Superstring Theory,' which provides an excellent analysis of the beta function, and its interpretation as a scattering amplitude. – JamalS Apr 2 '14 at 9:35
• Some more details about the calculation of for example the scattering amplitude of four open string tachyons and why it corresponds to the Euler-Beta function are outlined here. – Dilaton Jul 11 '14 at 21:07
• @JamalS No love for Green? – Ryan Unger Apr 9 '15 at 15:47
• @0celo7 Couldn't be bothered typing :) – JamalS Apr 9 '15 at 15:47

A function can be interpretable as a scattering amplitude if that function satisfies the axioms of relativistic S-matrix theory [1]:

1. Lorentz invariance
2. Unitarity (Not realized by the beta function, but may be dropped if the function is interpreted as a Born approximation to the exact amplitude)
3. T, C, P invariance (only for strong nucl. interactions)
4. Analyticity: singularities in invariant energy complex plane correspond to particle poles or thresholds, in a way that doesn't violate causality.
5. Crossing symmetry.
6. Power boundedness
7. Well-behaved particle poles (stable particle masses are positive, and their residues should be negative)
8. Analyticity of the second kind: analytic in the complex angular momentum plane.

Then the function should also match some of the empirical facts gathered by experimentalists. When the beta function was proposed the following list would be conjured:

1. All poles in the complex angular momentum plane move to the right linearly with increasing energy at a universal rate.
2. Diffractive scattering (not realized by the Euler beta function, but Virasoro's amplitude does realize this)
3. Inclusive high energy reaction exhibit scaling (also not realized by Euler beta)
4. Particle spectrum (determined by the poles) in agreement with the quark model (not realized either, but Chan and Paton's method get close)
5. Reproduces data from weak and electromagnetic processes (not realized.)

References: John Schwarz "Dual Resonance Theory" Phys Rep 8, no. 4, (1973) 269-335

I certainly don't have a complete answer, but what I do know is the following. The Veneziano amplitude is the following: $$\mathcal{A}^{(4)} \propto \lambda\left(B(-\kappa s -1, -\kappa t -1) + B(-\kappa s -1, -\kappa u -1) + B(-\kappa t -1, -\kappa u -1)\right),$$ where $s, t, u$ are the Mandelstem variables and $B(x,y)$ the Euler-Beta function. Note that the above formula is symmetric under exchange of $s,t,u$. I think I read somewhere that physicists expected the strong interaction to be symmetric under such momentum exchanges, but I do not know precisely why. Note that the above formula is what one obtains in string theory for the scattering of open string tachyons, with $\lambda = g_c$ (closed string coupling constant) and $\kappa = \alpha'$.