A technical question about the amplituhedron

[nrqed]

One of the ways to introduce the amplituhedron is through the equation $Y = C Z^T$ where $Z \in M^+(k+m,n)$, $C \in Gr_{\leq 0} (k,n)$, $Y \in Gr(k,k+m)$.

I am trying to understand what the parameter $k$ represents. For a while, I thought that this $k$ was counting the number of particles with negative helicities but that it seems to be incorrect. Can someone help? Thanks.

 

[MathematicalPhysicist]

If Gr means Grassmannian, then take a look here:
https://en.wikipedia.org/wiki/Grassmannian

otherwise, I don't know.

 

[nrqed]

Thank you. Yes, Gr is for Grassmannian. I know the math,my questions is about how the value $k$ is related to the number of negative helicity particles in the amplitude.
Thanks for your feedback.

 

[physicsworks]

I am trying to understand what the parameter $k$ represents

Originally, $k$ is the R-charge of the $SU(4)$ symmetry of the amplitude, so physically it represents the number of negative helicity gluons. Note, however, that the authors later (already in "Into the Amplituhedron" paper, for example) redefine this label according to $k \equiv K = k-2$. This is why the 4 particle case they consider corresponds to $n=4$, $k=0$ (and an arbitrary number of loops $L$). Here $k$ is the new $k$, i.e. the number of negative helicity gluons minus two. For $n=4$ this is the only non-trivial amplitude to consider since all the other helicity assignments correspond to amplitudes with vanishing kinematical support.