String theory was discovered from the study of Regge trajectories in the scattering of mesons and baryons. Veneziano came up with a beta function amplitude to explain this phenomena. Nambu and Goto showed that such an amplitude can be obtained from a bosonic String theory. Very crudely, the scattering of four open string tachyons can be given at tree level by the S matrix amplitude:

 

[math]

S(k_1 ;k_2 ;k_3 ;k_4 ) \propto B( - \alpha 's - 1, - \alpha 't - 1) + B( - \alpha 's - 1, - \alpha 'u - 1) + B( - \alpha 't - 1, - \alpha 'u - 1)

[/math]

 

where the Euler Beta function is given by:

 

[math]

B\left( { - \alpha 'x - 1, - \alpha 'y - 1} \right) = \frac{{\Gamma \left( { - \alpha 'x - 1} \right)\Gamma \left( { - \alpha 'y - 1} \right)}}

{{\Gamma \left( { - \alpha 'x - \alpha 'y - 2} \right)}}

[/math]

 

Where gamma is the Euler gamma function:

[math]

\Gamma (x) = \int\limits_0^\infty {e^{ - u} } u^{x - 1} du

[/math]

 

Using the conventions found in Polchinski's book

 

[math]

s + t + u = - \frac{4}

{{\alpha '}}

[/math]

 

Now the Regge limit is when [math]s \to \infty[/math] and t is fixed

 

Using the Stirling approximation of the the gamma function for large s

 

[math]

\Gamma \left( {( - \alpha 's - 2) + 1} \right) \approx ( - \alpha 's - 2)^{( - \alpha 's - 2)} \exp \left( {\alpha 's + 2} \right)\left( {2\pi ( - \alpha 's - 2)} \right)^{1/2}

[/math]

 

So the Beta function for the s channel is approximately

 

[math]

B( - \alpha 's - 1, - \alpha 't - 1) \approx \frac{{\left( { - \alpha 's - 2} \right)^{ - \alpha 's - 2} \exp ( - \alpha 't - 1)(2\pi ( - \alpha 's - 2))^{1/2} }}

{{\left( { - \alpha 's - \alpha 't - 3} \right)^{ - \alpha 's - \alpha 't - 3} (2\pi ( - \alpha 's - \alpha 't - 3))^{1/2} }}\Gamma ( - \alpha 't - 1)

[/math]

 

Doing a little algebra

 

[math]

\frac{{\left( { - \alpha 's - 2} \right)^{ - \alpha 's - 2} }}

{{\left( { - \alpha 's - \alpha 't - 3} \right)^{ - \alpha 's - \alpha 't - 3} }} = \left( {1 + \frac{{\alpha 't + 1}}

{{\alpha 's + 2}}} \right)^{\alpha 's + 2} \left( { - \alpha 's - \alpha 't - 3} \right)^{\alpha 't + 1}

[/math]

 

and

 

[math]

\mathop {\lim }\limits_{s \to \infty } \left( {1 + \frac{{\alpha 't + 1}}

{{\alpha 's + 2}}} \right)^{\alpha 's + 2} \left( { - \alpha 's - \alpha 't - 3} \right)^{\alpha 't + 1} = \exp (\alpha 't + 1)\left( { - \alpha 's - \alpha 't - 3} \right)^{\alpha 't + 1}

[/math]

 

So

 

[math]

B( - \alpha 's - 1, - \alpha 't - 1) \approx \mathop {\lim }\limits_{s \to \infty } \left( { - \alpha 's - \alpha 't - 3} \right)^{\alpha 't + 1} \left( {\frac{{ - \alpha 's - 2}}

{{ - \alpha 's - \alpha 't - 3}}} \right)^{1/2} \Gamma \left( { - \alpha 't - 1} \right)

[/math]

 

Using L' Hopital's rule you can show the square root term approaches 1 in the limit so for the s channel your just left with:

 

[math]

B( - \alpha 's - 1, - \alpha 't - 1) \approx \left( { - \alpha 's} \right)^{\alpha 't + 1} \Gamma \left( { - \alpha 't - 1} \right)

[/math]

 

I can do a similar thing for the t channel and show I get

 

[math]

B( - \alpha 's - 1, - \alpha 'u - 1) \approx \left( {\alpha 's} \right)^{\alpha 't + 1} \Gamma \left( { - \alpha 't - 1} \right)

[/math]

 

but for the u channel I get a mess.

 

According to Polchinski's book the answer should be something like:

 

[math]

S(k_1 ;k_2 ;k_3 ;k_4 ) \propto s^{\alpha 't + 1} \Gamma \left( { - \alpha 't - 1} \right)

[/math]

 

Any ideas or proofs of this would be greatly appreciated.

Polchinski's books are good, but lack enough examples and through calculations. This makes it difficult to compare what you have done to his writings. I had a quick scan through some of the other books I have here on string theory and they don't go into enough detail for me to help.

 

This is a result I am aware of but never followed it through myself. Sorry.