Old vs new unitarity cut methods

[earth2]

Hey folks,

I've been stumbeling recently about new unitarity methods to obtain one-loop amplitudes by cutting them in all possible channels thereby reducing the full amplitude to products of tree amplitudes (pioneered by Bern, Dixon, Kosower, Dunbar).

From what I understand from my QFT classes cutting a diagram will give you the imaginary part of the amplitude and by dispersion relations one will get the full amplitude. But why don't these guys need to do the dispersion integrals? From what I understand they get the full amplitude through these cuts from the start...no detour via imaginary parts and doing dispersion integrals...

How is this possible when both methods rely on simple unitarity? Where does the difference lie? I don't get it and would be happy if someone could enlighten me!

Thanks
earth2

 

[AndyW]

sky

Hi earth2sky,

Thank you for bringing up this interesting topic. The method you are referring to is called the unitarity method or on-shell recursion, and it has indeed been pioneered by a group of physicists including Bern, Dixon, Kosower, and Dunbar.

To understand how this method works, it is important to first clarify the difference between the dispersion relation approach and the unitarity approach. In the dispersion relation approach, we start with the full amplitude and use the fact that it can be expressed as a sum over all possible cuts of the diagram. This allows us to write the full amplitude as an integral over the imaginary part of the amplitude, which can then be evaluated using the Cauchy's residue theorem. This is where the dispersion integrals come in.

On the other hand, in the unitarity approach, we start with the tree-level amplitudes and use the fact that they satisfy certain on-shell recursion relations. These relations allow us to write the full amplitude as a sum over products of tree amplitudes, with each term corresponding to a different cut of the diagram. This is where the difference lies - instead of integrating over the imaginary part of the amplitude, we are summing over products of tree amplitudes.

So, to answer your question, the unitarity method does not require the dispersion integrals because it bypasses the need for them by directly working with the tree amplitudes. This approach is particularly useful for calculating one-loop amplitudes, where the number of diagrams and cuts can become quite large. By using the on-shell recursion relations, we can greatly simplify the calculation and reduce it to a sum over a much smaller number of terms.

I hope this helps clarify the difference between the two methods. Both rely on unitarity, but the unitarity method takes a more direct approach by working with the tree amplitudes. If you would like to learn more, I recommend checking out some papers by the physicists I mentioned earlier, as well as the book "Scattering Amplitudes in Gauge Theories" by Elvang and Huang.

Best of luck in your studies!