Positivity of the Veneziano Amplitude in $D=4$

Understand

The Veneziano amplitude was put forward as a solution to the axioms of the S-matrix bootstrap.

• However, unitarity, reflected in the positivity of the coefficients in the Gegenbauer expansion of the amplitude is not obvious.
• In this note we compute the generating function of these coefficients in terms of the Appell hypergeometric function.
• We use this to read off an exact form of this coefficient on the leading Regge trajectory in $D=4$.