Regge Trajectories and the Veneziano amplitude

[Jim Kata]

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.

[ajb]

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.