^{1}

^{2}

^{3}

^{2}

^{3}.

In this Letter we compute a canonical set of cuts of the integrand for maximally helicity violating amplitudes in planar

The past decade has revealed a variety of surprising mathematical and physical structures underlying particle scattering amplitudes, providing, with various degrees of completeness, reformulations of this physics where the normally foundational principles of locality and unitarity are derivative from ultimately combinatoric-geometric origins. An example is the amplituhedron

There has also been an ongoing effort to use the amplituhedron picture to make all-loop order predictions for loop integrands. This effort was initiated in Ref.

The original definition of the amplituhedron refers to the auxiliary space of extended kinematical variables constrained by positivity conditions. Recently, an equivalent definition was provided directly in the momentum twistor space using the conditions on

For the

At loop level, in addition to the external momentum twistors

The amplituhedron geometry for the MHV case when

In Ref.

Prior to any detailed investigation, the geometry of the amplituhedron makes an amazing prediction for the structure of this cut. Owing to the trivialization of the mutual positivity by setting all the

It is easy to show that we have to impose

We now proceed to solving the geometry problem for the four point case where all lines pass through the same point

Since the

In the four point case the all-in-plane solution can be extracted from the all-in-point solution. Instead of the common point

At four points the original amplituhedron picture is identical to the new sign flip definition. However, for higher point MHV amplitudes while still equivalent the sign flip picture is much more suitable for actually solving the geometry. Here we provide the final

While for the four point case the forms

Given the integrand for the

As we have stressed, the deepest cut is sensitive to the most complicated topologies for Feynman diagrams and on-shell processes that can contribute to the amplitude. It is interesting to see this more quantitatively at four points. For the four-point

We have to stress that this counting corresponds to graphs which in principle can contribute on the cut (counting the number of

Our formulas for

Any analytic comparison with standard local expressions for the cut would have to proceed by algebraically canceling the spurious poles. This immediately leads to an explosion of complexity: while the formula for

The expressions get even more complicated if we rewrite the numerators using

In this Letter we studied the deepest cut in the planar

We thank Lance Dixon, Enrico Herrmann, Thomas Lam, and Hugh Thomas for useful discussions and comments. N. A-H. is supported by DOE Grant No. SC0009988. C. L. and J. T. are supported by DOE Grant No. DE-SC0009999 and by the funds of University of California.