How does one actually compute the amplituhedron?

I was watching Nima's very popular talk (download if you're using chrome) (also mirrored at youtube here) about the "Amplituhedron", which has suddenly become very popular recently.

He talks all about how the amplituhedron computes the same result for the scattering amplitudes as ordinary peturbation theory in a simple and elegant way, but I fail to understand how one actually computes the amplituhedron for a certain scattering process anyway?

As per the recent TRF posts about amplituhedron and why they don't wear diapers, I can understand that one may calculate the scattering amplitudes by simply taking the volume of the amplituhedrons (ignoring constants, I guess), but how does one actually calculate the amplituhedron?

I'm especially stunned by the image (looks like a sort of a concrete example, don't know how they constructed the amplituhedron):

To summarise my question, how does one actually figure out, or construct, the amplituhedron based on the specific scattzering process?

• Wait for the paper. It's a geometric object in some space whose dimension depends on the number of external particles, number of loops, and number of "helicity flips". The volume form, the integrand, is a simple form roughly scaling like $1/x$ where $x$ is the distance from a face, and the faces are given by inequalities of the type "determinants of a submatrix are zero". These inequalities depend on the external momenta and/or twistor variables, sort of linearly or simply. The scattering amplitude is the single simple integral of the volume form over the polytope. – Luboš Motl Sep 22 '13 at 9:23
• Maybe you will enjoy watching, "Scattering without space time". He elaborates on these ideas, in a 3 talk series – Prathyush Sep 23 '13 at 21:20

To begin with, The Amplituhedron formalism only works for a specific theory, N=4 SYM in the planar limit (only planar Feynmann diagrams are considered). Because of supersymmetry, you can classify scattering processes with two parameters: $n$ and $k$. n is the number of particles involved, and k is, roughly speaking, the number of spin flips in the process.

In addition, there is also the number of loop $L$ at which you want to perform your calculation. Be aware that at loop level you are not computing the (super)amplitude, but rather the integrand of the amplitude. That is, thanks to the planar limit, you can uniquely define a function dependent on external data + virtual momenta that has to be integrated in order to obtain the amplitude. To clarify, is what you obtain if you swap $\sum_{diagrams} \int_{loops} = \int_{loops} \sum_{diagrams}$, you can do that with no ambiguity because the planarity allows to choose a scheme to fix the loop variables. This integrand is what is produced by the Amplituhedron at loop level.

Now, for any $n,k$ and $L$ and for fixed external data $Z$ (a $(k+4) \times n$ matrix which ultimately encodes the external momenta and supermomenta) you build an amplituhedron $A_{n,k,L}$ in a standard way. At tree level ($L=0$) it is the subset of $G_{+}(k,k+4)$ of points $Y$ written as $Y = C . Z$, for any $C$ in $G_{+}(k,n)$. At loop level is a little bit more complicate, but nothing terrible.

An important feature of the Amplituhedron is that you can "triangulate" it in a very interesting way. (By triangulate we mean divide it in zones that have in common only their boundary.) In fact, in a prievous work it was shown that the amplitude/integrand for $n,k,L$ is written as a sum over certain on-shell diagrams, which in turn label certain cells in the Grassmannian $G_{+}(k,n)$. (The BCFW rules are used to obtain this expression of the amplitudes.) If you consider the image of these cells under the map $Y = C . Z$ you obtain a triangulation for the Amplituhedron!

Now these Amplituhedra have codimension one "boundaries". These are zero-loci of non-negative functions defined on the Amplituhedra. A crucial point will be that these boundaries may be written as product of other Amplituhedra. You can define a (unique?) volume form on the Amplituhedron by the condition that it should have logarithmic singularities (i.e. like 1/x) at these codimension one boundaries.

A concrete way to do so is to consider a triangulation of the Amplituhedra, for example the one deriving from BCFW recursion relations as above. On every "triangle" you are already given a form that has log singularities at the face of the "triangle". This is so because the cells of $G_{+}(k,n)$ used in this triangulation are equipped with simple positive charts $(\alpha_1, ... , \alpha_d)$ that allows to reach a boundary of the cell by setting certain $\alpha$ to zero. Therefore the forms $\Pi d\alpha/\alpha$ have the desired property cell-wise. You just sum these forms over the cells: the "spurious" poles, associated to cell-boundaries that are not amplituhedron-boundaries cancels (they are shared by two cells) and you are left with the right log singularities.

But you can also obtain this volume form in other ways, following directing by the definition, as explained in the paper.

Finally, you evaluate the volume form obtained above in a particular point of the amplituhedra, and what you obtain is the amplitude/integrand.

The procedure seems a little bit complicate and abstract, but actually the idea is quite simple. The geometry of the amplituhedron captures the intricate factorization properties of amplitudes/integrands required by Locality and Unitarity: The boundaries of an amplituhedron factorizes in the corresponding smaller amplituhedra. Therefore a form with logarithmic singularities at these boundaries will have the right poles and factorizations as well.

Since this question, Arkani-Hamed among others has published the text, Grassmanian Geometry of Scattering Amplitudes which elucidates in a pedagogic fashion the amplituhedron approach.

To fully understand the derivation, there are many preliminaries to be understood, but I will try to present a relatively self-contained computation of an amplitude, but this requires accepting some results a priori. The book Scattering Amplitudes by Elvang et al is also a good reference.

The key figure in the amplituhedron approach is the Grassmannian $\mathrm{Gr}(k,n)$ which is the space of all $k$-planes in an $n$-dimensional space. One has as elements the $n\times k$ matrices $C_{al}$ which are $k$ $n$-component vectors that define the plane in $\mathbb C^n$.

However, note in the spinor helicity formalism, by specifying momenta as $p_{\alpha \dot \alpha} = \lambda_\alpha \bar{\lambda}_{\dot \alpha}$, we can make the rescaling $\lambda_\alpha \to t_\alpha \lambda_\alpha$ and describe the same momenta, so we have a $GL(1)$ redundancy. For the Grassmannian, this means we consider any non-degenerate linear transformation to describe the same plane, and so we have a $GL(k)$ redundancy.

The key to evaluating amplitudes is the unique cyclically invariant integral that generates all Yangian invariants, given by,

$$\mathcal L_{n,k}([i|, |i\rangle, \eta_i) = \int\frac{d^{n \times k} C}{GL(k) \prod_{j=1}^n M_j} \left[ \prod_{a=1}^k \delta^2 \left(\sum_i C_{ai}[i| \right) \delta^{(4)} \left(\sum_{i}^n C_{ai}{\eta_i}_A\right)\right] \times$$ $$\times \left[ \prod_{a'=k+1}^n \delta^2 \left(\sum_i \tilde C_{a' i} \langle i|\right)\right]$$

where $M_j$ are the minors, and we specify the external data as $([i|, |i\rangle, \eta_i)$ in momentum space where $\eta_i$ are the superpartners and $\tilde C$ is the complement of $C$. It turns out this $\mathcal L_{n,k}$ is the cyclic invariant integral over all $k$-planes in the Grassmannian.

The delta function constraints geometrically signify that the $2$-plane spanned by $[i|$ is orthogonal to the $k$-plane $C$ and similarly the plane defined by $|i\rangle$ is orthogonal to the $(n-k)$-plane $\tilde C$. Since $C,\tilde C$ are complements, they each respectively contain the planes spanned by $[i|$ and $|i\rangle$. From the geometry, we end up getting momentum conservation for free: $$\sum_{i=1}^n |i\rangle [i| = 0.$$

Let's now compute $\mathcal L_{n,2}$. For $k=2$ we identify the planes specified by the matrices $C$ as those spanned by the momenta data. We have,

$$\begin{pmatrix} C_{11} &C_{12} & \dots & C_{1n}\\ C_{21} & C_{22} & \dots & C_{2n} \end{pmatrix} = \begin{pmatrix} |1\rangle^i &|2\rangle^i & \dots & |n\rangle^i\\ |1\rangle^{\dot 2} & |2\rangle^{\dot 2} & \dots & |n\rangle^{\dot 2} \end{pmatrix}.$$

The bosonic delta function encodes $\delta^{(4)}(P) = 0$ and the Grassmann delta function encodes conservation of the supermomenta, $\delta^{(8)}(\tilde Q) = 0$. The minors are,

$$M_i = \epsilon^{ab}C_{ai}C_{b,i+1} = -\epsilon_{\dot a \dot b} |i\rangle^{\dot a} |i+1\rangle^{\dot b} = -\langle i, i+1 \rangle$$

and combining all the factors yields,

$$\mathcal L_{n,2} = (-1)^n \frac{\delta^{(8)}(\tilde Q) \delta^{(4)}(P)}{\prod_{i=1}^n \langle i, i+1 \rangle}$$

which is the tree-level MHV amplitude, $\mathcal A_{n}^{\mathrm{MHV}}$, written in the compact spinor-helicity formalism.