Calculation of an amplitude using the Grassmannian approach (amplituhedron)

[kdv]

In the following review paper on scattering amplitudes, by Elvang and Huang:
https://arxiv.org/abs/1308.1697

they calculate the amplitude for 6 particles with half of positive helicity and half of negative helicity in section 9.3.2. Their matrix C (a point in the relevant Grassmannian) is parametrized as in Eq. 9.35.
They then use a contour of integration giving three contributions (after applying the residue theorem). The three contributions are given in 9.45 and 9.46.

Now, Arkani-Hamed et al do the same calculation using a slightly different approach, see https://arxiv.org/pdf/1212.5605.pdf. They use plabic graphs to represent different contributions to an amplitude. The plabic graphs corresponding to the amplitude of interest are given in Figure 16.6, with the total amplitude given just below, in Eq. 16.7.

Continuing with Arkani-Hamed et al, the contribution from *one* of these three plabic graphs is calculated (without providing much details) in Eqs 8.6 and 8.7. They give now the matrix C once the integral is done. Now, they don't show it but they do mention (just below 8.7) that they use a parametrization "making the residue about the pole (123)=0 easy to read off" (here, (123) is the minor involving the first three columns of C). So the matrix C they give in (8.6) is the result obtained after doing the integral.

Now I can ask my question: How are the two approaches related? In the case of Elvang and Huang, they have a single integral which generates all three terms in the full amplitude. In the case of Arkani-Hamed et al, they have three plabic graphs that each apparently contribute a single term in the amplitude.

So the initial integral used by Elvang and Huang must differ from the integral used by Arkani-Hamed et al. It is difficult to see because of the different notation used and the different ways of writing the delta functions. I am hoping that someone who knows this material will be able to point out how the two approaches differ, at a basic level. A related question is: then it does not seem that one can associate to the intergal a plabic graph to the approach followed by Elvang and Huang since there single integral gives all three plabic graphs. Is that right?

Thanks in advance

 

[AndyW]

Thank you for bringing up this interesting question. The two approaches, as described in the papers by Elvang and Huang and Arkani-Hamed et al, are indeed different but are related in the sense that they both use different strategies to calculate the same physical quantity.

In the paper by Elvang and Huang, they use a contour integral approach to calculate the amplitude for 6 particles with half of positive helicity and half of negative helicity. They parametrize their matrix C, which is a point in the relevant Grassmannian, and use a contour of integration to get three contributions (after applying the residue theorem). These three contributions correspond to the three terms in the full amplitude.

On the other hand, Arkani-Hamed et al use a different approach, where they use plabic graphs to represent different contributions to the amplitude. The plabic graphs corresponding to the amplitude of interest are given in Figure 16.6, with the total amplitude given just below, in Eq. 16.7. They then calculate the contribution from one of these three plabic graphs using a different parametrization of the matrix C. This parametrization makes the residue about the pole (123)=0 easy to read off, as mentioned in the paper.

So, in summary, the two approaches are different in terms of the parametrization of the matrix C and the method of integration, but they both lead to the same physical result. To answer your related question, yes, it is not possible to associate a plabic graph to the approach followed by Elvang and Huang, as their single integral gives all three plabic graphs.

I hope this helps to clarify the relationship between the two approaches. Thank you for your interest in this research and for your question.