Calculating Tree Amplitude for Spin J Exchange

[gamma5]

I am wondering something about the tree amplitude for exchange of a spin J resonance-- the basic calculation that exhibits the famous regge behavior. I see that one way to do the calculation is simply to note that the 4-point function for spin J exchange amongst massless particles arises from an interaction term with J derivatives, and hence J powers of momenta, from which we immediately infer that the t-channel amplitude is of the form
s^J/(t-M^2) + ... . However another method appears in the literature, which seems to be a partial wave expansion, noting that the spin J term of the expansion for the scattering amplitude is of the form (2J+1) f(t) P_J(cos \theta), where P_J (\cos(\theta) is the Legendre polynomial of spin J and \theta is the angle between the incoming t-channel particles. The asymptotic large s behavior can then be obtained by noting that cos(\theta) --> s/4m^2, and then that P_J (\cos (theta)) --> (cos \theta)^J --> s^J.

My question is why the two methods? Why does anyone use partial waves if it is so much simpler to get the result from field theory? And this is a very basic question, but what is the validity of a partial wave expansion in a field theory context?

 

[AndyW]

Thank you for your question regarding the calculation of the tree amplitude for exchange of a spin J resonance. I can explain the reasoning behind the two methods you mentioned and discuss the validity of a partial wave expansion in a field theory context.

The first method you mentioned, using field theory, is based on the interaction term with J derivatives in the 4-point function for spin J exchange. This method is commonly used because it is relatively straightforward and can provide a direct calculation of the amplitude. However, it is important to note that this method assumes that the particles involved are massless, and therefore may not be applicable in all scenarios.

The second method, using a partial wave expansion, is based on the idea that the scattering amplitude can be expressed as a sum of terms with different spin projections. This method is commonly used in the context of quantum field theory because it allows for a more general treatment of the problem, including cases where the particles involved have non-zero masses. It also provides a way to analyze the behavior of the amplitude at different angles, which can be useful in certain scenarios.

Now, why do some scientists choose to use the partial wave expansion method instead of the field theory method? This can depend on the specific problem they are trying to solve. In some cases, the field theory method may be more appropriate and simpler, as you noted. However, in other cases, the partial wave expansion method may provide a more general and flexible approach, allowing for a better understanding of the underlying physics.

Regarding the validity of a partial wave expansion in a field theory context, it is important to note that this method is based on the principles of quantum field theory. As such, it is a valid approach that has been widely used in many different contexts and has been shown to provide accurate results. However, as with any theoretical framework, it is important to consider the assumptions and limitations of the method and to use it appropriately.

In summary, both the field theory method and the partial wave expansion method have their strengths and limitations. Scientists may choose to use one or the other depending on the specific problem they are trying to solve. The validity of the partial wave expansion in a field theory context is supported by its use in many successful calculations and its consistency with the principles of quantum field theory. I hope this helps to answer your questions.