Calculation of amplitude with n=5, k=3 using the book by Arkani-Hamed et al

[nrqed]

This is a pretty technical question so before I post all the details, let me first ask if someone might be able to help.

I am going over the book Grassmannian Geometry of Scattering Amplitudes and I am trying to calculate the anti MHV amplitude with five particles, at tree level (or, equivalently, the NHMV). The answer if the three particles of negative helicities are the first three, should be [tex] \frac{[45]^4}{[12][23][34][45][51] } [/tex] (all that times the usual four momentum delta function, of course).

I have tried to get this using an approach similar to what they do on page 90 to obtain the amplitude for n=6,k=3. I have a matrix in the style of to their matrix in Eq.(8.6) which obeys all the condition imposed by the delta functions. However, when I plug that into the formula (8.4) to get the amplitude and I do some manipulations, I end up with almost the correct expression, the only problem is that I have an extra factor of [itex] \langle 2 4 \rangle [/itex] in the denominator. I get a factor of [itex] \langle 2 4 \rangle [/itex] for each of the five minors appearing in the denominator and I get a factor of [itex] \langle 2 4 \rangle^4 [/itex] from the integration over the Grassmann variables when choosing the first three particles to have [itex]h=-1[/itex].

I will provide more details if someone knows a bit about this topic. This is driving me crazy since it should be straightforward. There is obviously something very simple that I am completely missing.

 

[AndyW]

Hello,

Thank you for reaching out for help with your calculation. I am a scientist with experience in Grassmannian geometry and scattering amplitudes, so I may be able to assist you.

Firstly, it is important to double check that your matrix obeys all the conditions imposed by the delta functions. Sometimes, a small mistake in the matrix can lead to discrepancies in the final result. It may also be helpful to compare your matrix to the one used in Eq.(8.6) in the book and see if there are any differences.

Secondly, it is possible that the extra factor of \langle 2 4 \rangle in the denominator is due to the integration over the Grassmann variables. Have you tried using a different approach to integrate the Grassmann variables? Sometimes, using different methods can lead to different results.

If you could provide more details about your calculation, such as the specific values of the minors in the denominator and the integration over the Grassmann variables, I may be able to pinpoint the exact issue. It may also be helpful to see the steps you took to obtain your result.

I hope this helps and I look forward to hearing back from you with more details. Don't hesitate to ask for further clarification or assistance.