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This is veneziano amplitude: $$B(-a(s),-a(t))$$ where the $a(s)$ and $a(t)$ are a kind of leading trajectories (regge trajectory). which is born of strings theory.

  1. What is the reason that Euler's beta function shows strings behavior?

  2. What is the reason that Veneziano uses Euler's beta function?

  3. What is derivation of Veneziano amplitude?

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Maybe one should ask instead, "What is the reason that string theory uses Euler's beta function?" (I'm not being sarcastic here). –  MPW Mar 10 at 13:50
    
Susskind's lecture 6 on string theory discusses this semi-informally, offset at about 45 minutes into the video, stretched to a length of about 30 minutes. –  ccorn Mar 10 at 17:01
    
Related older question on Physics SE: Why can the Euler beta function be interpreted as a scattering amplitude?. No detailed answers yet. –  ccorn Mar 10 at 17:12

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