PHYSICAL REVIEW LETTERS 122, 051601 (2019) Editors' Suggestion Deep Into the Amplituhedron: Amplitude Singularities at All Loops and Legs Nima Arkani-Hamed,1 Cameron Langer,2 Akshay Yelleshpur Srikant,3 and Jaroslav Trnka2 1School of Natural Sciences, Institute for Advanced Study, Princeton, New Jersey 08540, USA 2Center for Quantum Mathematics and Physics (QMAP), University of California, Davis, California 95616, USA 3Department of Physics, Princeton University, New Jersey 08544, USA (Received 23 October 2018; revised manuscript received 12 December 2018; published 7 February 2019) In this Letter we compute a canonical set of cuts of the integrand for maximally helicity violating amplitudes in planar N ¼ 4 supersymmetric Yang-Mills theory, where all internal propagators are put on shell. These “deepest cuts” probe the most complicated Feynman diagrams and on-shell processes that can possibly contribute to the amplitude, but are also naturally associated with remarkably simple geometric facets of the amplituhedron. The recent reformulation of the amplituhedron in terms of combinatorial geometry directly in the kinematic (momentum-twistor) space plays a crucial role in understanding this geometry and determining the cut. This provides us with the first nontrivial results on scattering amplitudes in the theory valid for arbitrarily many loops and external particle multiplicities. DOI: 10.1103/PhysRevLett.122.051601 Introduction.—The past decade has revealed a variety of There has also been an ongoing effort to use the surprising mathematical and physical structures underlying amplituhedron picture to make all-loop order predictions particle scattering amplitudes, providing, with various for loop integrands. This effort was initiated in Ref. [10], degrees of completeness, reformulations of this physics which calculated certain all-loop order cuts for four point where the normally foundational principles of locality amplitudes that were impossible to obtain using any other and unitarity are derivative from ultimately combina- methods. In this Letter, we go much further and use the toric-geometric origins. An example is the amplituhedron new topological definition of the amplituhedron [4] to [1], a geometric picture for scattering amplitudes in planar calculate a particular cut of n-point MHV amplitudes to N ¼ 4 supersymmetric Yang-Mills (SYM) theory. All all loops. This cut places on-shell internal propagators, tree-level amplitudes and loop integrands in this theory which are arbitrarily deep in the interior of contributing correspond to the differential forms with logarithmic Feynman diagrams. Thus, we aptly refer to this as the singularities on boundaries of the amplituhedron geometry. “deepest cut.” It appears hopelessly difficult to calculate The original definition was based on a generalization of this cut using standard unitarity-based methods, as almost the positive Grassmannian, centrally connected to on-shell all diagrams contribute as we increase the loop order. diagrams [2] and loop level recursion relations [3]. More However, we will show that the new topological formu- recently, a more intrinsic definition of the amplituhedron lation of the amplituhedron allows us to easily understand was found [4], directly in momentum-twistor space, using the geometry of the facet associated with this cut and certain topological notions—winding numbers and sign leads to a strikingly simple, one-line expression for the flip patterns—associated with projections of the momen- cut valid to all loops and all multiplicities. This is the tum-twistor data. The amplituhedron has been extensively first calculation giving us nontrivial access to the regime studied from many angles including mathematical aspects of arbitrarily large loop order and particle multiplicities in [5–7], positive geometry and volume interpretation [8,9], the theory. triangulations [10–13], connections to on-shell diagrams Amplitudes from sign flips.—The original definition of [14,15], and geometric structures in the final amplitudes the amplituhedron refers to the auxiliary space of extended [16,17]. Some early steps in extending this circle of ideas kinematical variables constrained by positivity conditions. well beyond the planar N ¼ 4 SYM theory have been Recently, an equivalent definition was provided directly in taken in Refs. [18–20]. the momentum twistor space using the conditions on sign flips [4]. For the n-point amplitude, the kinematics is given by n I momentum twistors Za, a ¼ 1; 2; …;n, I ¼ 1; …; 4. The Published by the American Physical Society under the terms of SLð4Þ dual conformal symmetry acts on the I index. We the Creative Commons Attribution 4.0 International license. I J K L define the SLð4Þ invariants habcdi ≡ ϵIJKLZaZbZc Zd . Further distribution of this work must maintain attribution to k the author(s) and the published article’s title, journal citation, The space for the N MHV amplituhedron is described 3 and DOI. Funded by SCOAP . by the set of Za for which all hiiþ1jjþ1i ≥ 0 [with twisted 0031-9007=19=122(5)=051601(7) 051601-1 Published by the American Physical Society PHYSICAL REVIEW LETTERS 122, 051601 (2019) kþ1 Z Z positivity for ð−1Þ hn1jjþ1i > 0], and the following i iþ1. Here we consider an opposite case where none of sequence: the external propagators hðABÞαii þ 1i are cut while all internal propagators are on shell: fh123iig for i ¼ 4; …;n has k sign flips: ð1Þ hðABÞαðABÞβi¼0 for all α; β ¼ 1; …; l: ð5Þ For example, for n ¼ 6 the sequence has three terms, Prior to any detailed investigation, the geometry of the fh1234i; h1235i; h1236ig; amplituhedron makes an amazing prediction for the struc- ture of this cut. Owing to the trivialization of the mutual and possible sign sequences fþ þ þg, fþ þ −g, fþ −−g, positivity by setting all the hðABÞαðABÞβi → 0, the only and fþ − þg. The first sequence corresponds to k ¼ 0, the remaining constraint of the geometry is that all the lines next two to k ¼ 1, and the last to k ¼ 2 kinematics. In ðABÞα live in the one-loop amplituhedron. This leads us to general the k ¼ 0 MHV amplitude has the sign pattern expect that the all-loop geometry should be expressible as l fþþþgand zero sign flips, while for higher k we have independent copies of the one-loop geometry, associated various sign patterns which have k sign flips in total. with a formula for the cut with the structure of a product At loop level, in addition to the external momentum over independent pieces determined by this one-loop twistors Za, we also have lines Lα ¼ðABÞα corresponding problem. We will see this expectation borne out perfectly to loop momenta. For each line we can write ðABÞα ¼ in our analysis. AαBα, where Aα, Bα are two points on that line. The It is easy to show that we have to impose 2l − 3 line ðABÞα is in the one-loop amplituhedron if all conditions in order to satisfy Eq. (5). There are two hðABÞαii þ 1i > 0 [again with twisted positivity solutions to this problem, each with a different geometrical kþ1 ð−1Þ hðABÞαn1i > 0] and if the sequence meaning: in the first solution, all-in-point, all lines intersect in a common point A while in the second solution, all-in- P fhðABÞα1iig for i ¼ 2; …;n has ðk þ 2Þ sign flips: plane, all lines lie on the same plane . ð2Þ The collection of l lines, fðABÞαg is in the l-loop amplituhedron if each line satisfies Eq. (2) and, in addition, the mutual positivity conditions (independent of n, k) hðABÞαðABÞβi > 0 for all α; β ¼ 1; …; l: ð3Þ The n-point NkMHV l-loop integrand is given by a degree These configurations are mapped into each other by the 4 k l differential form on Za; AB space, deter- ð þ Þ f ð Þαg usual projective duality interchanging points and planes, mined by the property of having logarithmic singularities which also reflects parity; had we been discussing MHV on boundaries of the intersection of this space with a amplitudes the cuts associated with the pictures would be canonical (4 × k)-dimensional affine subspace in the con- reversed. In the first solution the lines ðABÞα can be figuration space of momentum twistors fZag. parametrized The amplituhedron geometry for the MHV case when k ¼ 0 is especially simple. The space is simply that of l ðABÞα ¼ ABα; ð6Þ lines ðABÞα in momentum twistor space subject to Eq. (3). The sign-flip conditions (2) can be rewritten in an appa- where Bα and A are arbitrary. The common intersection rently different but equivalent form, in terms of inequalities point A has three degrees of freedom (d.o.f.) while each Bα AB on each ð Þα: has two d.o.f. as the geometry only depends on the lines AB 2l 3 AB i − 1 ii 1 ∩ j − 1 jj 1 > 0 i; j: α. As a result, the configuration is ð þ Þ dimen- hð Þαð þ Þ ð þ Þi for all sional. We now have a simple geometry problem. Given a ð4Þ point A, we want to identify all the lines ðABÞ passing through A, which lie in the one-loop amplituhedron. This This condition is the same for every loop—we can say carves out some subset in the two-dimensional space of that this just demands that each ðABÞα lives in the one-loop possible points B. The shape of this region can change as amplituhedron. The interaction between different loops we move around in A space. Thus we are led to find a joint is then captured by the mutual positivity properties “triangulation” in A, B space, specified by breaking up the hðABÞαðABÞβi > 0. three-dimensional A space into regions for which the Definition of the deepest cut.—In Ref. [10] we focused corresponding two-dimensional geometry in B has a uni- on cuts where Lα ¼ðABÞα passed through Zi or cut lines form shape.