Anatomy of the amplituhedron

Received: February Anatomy of the amplituhedron Sebastian Franco 1 Daniele Galloni 1 Alberto Mariotti 1 Jaroslav Trnka 0 Open Access c The Authors. 0 Walter Burke Institute for Theoretical Physics, California Institute of Technology 1 Institute for Particle Physics Phenomenology, Department of Physics, Durham University We initiate a comprehensive investigation of the geometry of the amplituhedron, a recently found geometric object whose volume calculates the integrand of scattering amplitudes in planar N = 4 SYM theory. We do so by introducing and studying its stratification, focusing on four-point amplitudes. The new stratification exhibits interesting combinatorial properties and positivity is neatly captured by permutations. As explicit examples, we find all boundaries for the two and three loop amplitudes and related geometries. We recover the stratifications of some of these geometries from the singularities of the corresponding integrands, providing a non-trivial test of the amplituhedron/scattering amplitude correspondence. We finally introduce a deformation of the stratification with remarkably simple topological properties. Supersymmetric gauge theory; Scattering Amplitudes Contents 1 Introduction The amplituhedron Tree-level amplituhedron Loop geometry The full amplituhedron The scattering amplitude The degrees of freedom of C Extended positivity and boundaries Mini stratification Full stratification Simple examples: basic properties Non-minimal minors Combinatorial stratification Perfect matchings and the stratification of G+(k, n) Multi-loop geometry and hyper perfect matchings The combinatorics of extended positivity Further thoughts on extended positivity Hyper perfect matchings: good, bad and neutral Extended positivity and the return of permutations Two loops Mini stratification The amplituhedron The log of the amplitude Gluing the amplitude to its Log Full stratification Three loops Mini stratification The amplitude The log of the amplitude 10.1 Examples 10.1.1 1-loop 10.1.2 2-loops 10.1.3 3-loops 10.1.4 4-loops 11 Conclusions and outlook Introduction the tools we introduce, will nicely fit into this trend. The amplituhedron to [39, 40] for further details. Tree-level amplituhedron all scattering amplitudes in planar N given by Y = C Z , G+(k, n) Z G(k, k + 4) . L(i) = D(i) Z , L(2) = . D(2) Loop geometry vertices, which lies in P3: number of matrices D(i): D(ij) = D(ijk) = D(j) , etc. The full amplituhedron loop space into a larger matrix L(2) L(L) .. .. . = . Z D(2) D(L) Y = C Z , or more neatly least as many columns as rows, i.e. that The scattering amplitude for systematically constructing the integrand form. its boundary structure. the D(i) matrices: C = for L = 1. 4 Even question in its own right. The degrees of freedom of C Extended positivity and boundaries i.e. that form a (i1,...,im). For each of these choices, there are 2m n J For a given number of loops L, there are ways of choosing m matrices D(i) to ways of choosing the set J of mn/2 minors, although they do not fully specify them. do not fully specify boundaries. explicit examples of both of these occurrences. to generate an Eulerian poset [44]. Mini stratification solutions for a given positivity-preserving label. At this point in our discussion, it is classes as the boundaries of the mini stratification. Full stratification 7By this we mean configurations in which some minors vanish. When all follows: full stratification uses extended labels. turned into the convenient form 12 13 13 12 13 34 34 13 23 13 13 23 13 14 14 13 = 0, we obtain 12 13 13 12 13 34 34 13 , which is the full set of Region 1: Region 2: 12 13 13 12 12 13 13 12 > 0 and < 0 and 13 34 34 13 13 34 34 13 y 13 34 34 13 and k 12 13 13 (122) , (1) (2) (1) xy k (k > 0) x, y < 0. the only non-minimal minor. hAB34ihCD12i + hAB12ihCD34i hABCDihAB12ihAB34ihCD12ihCD14ihCD23ihCD34i for the loop amplituhedron.9 d = NI Nrel L , dimensional region associated to it. 10 I 0. 0. stratification, it is implemented slightly differently. D(i), i.e. of each G+(2, n). 0 and are thus removed. The surviving collections of (i) represent I stratification will be introduced in section 5. 11 arising from this step. 12 degree of freedom by rescaling the coordinates, we get n(n 1) 2 2 2n + 3 1 = 2(n 2) Some remarks are already in order: 13 (12,13,23) (12,13,24,34) (13,14,23,24) (14,24,34) one coordinate turned on and has dimension 0. issue, it is the violation of the Plucker relation. poset is Eulerian, i.e. X(1)dN(d) = 1 , 14 section 3.5. in section 3.5. coordinates. minors, which in this particular case do not exist. factors within them. in figure 4.14 according to the following rules: external nodes or not. 16 disk, whose boundary is shown in gray. 17 consideration we obtain the lattice shown in figure 6. unified object in its own right. 18 hABCDi = hAB24ihCD24i hAB24ihCD24i and integrand language. 40 to set hABCDi 0 as late as possible. The log of the amplitude hAB13ihCD24i + hAB24ihCD13i set hABCDi 0 as late as possible. amplituhedron conjecture. The deformed G+(0, n; L) can be made. 41 might exhibit a remarkably simple topology. 42 The 4 4 minors become the form neously positive. of the form19 Examples the topology. (1,2) (1,3) (2,3) NM,deformed E = Total contribution N (i,j)s. We can determine this by just collapsing the various types of 3(Ii,j)s in table 8 I additional contributions as before. 45 NM,deformed 1 061 154 found using the methods of section 6.3. Keeping only those boundaries which satisfy boundaries are added: = 15 boundaries = 63 boundaries. number equal to 1: Conclusions and outlook positivity is beautifully captured by permutations. come from factorizing non-minimal minors. from the square of G+(2, 4). 47 very limited data. data, respectively. handle on it. 48 significance. lated to ours. Acknowledgments of G+(2, 4). 49 Dimension 8. to automatically vanish. Constraints on AB and CD Plucker coordinates turned on Hyper perfect matchings present hAB34ihCD12i+hAB23ihCD14i+hAB14ihCD23i+hAB12ihCD34i hABCDihAB12ihAB14ihAB23ihAB34ihCD12ihCD14ihCD23ihCD34i P1,1, P1,2, P1,3, P1,4, P1,5, P1,6, P2,1, P2,2, P2,3, P2,4, P2,5, P2,6, P3,1, P3,2, P3,3, P3,4, P3,5, P3,6, P4,1, P4,2, P4,3, P4,4, P4,5, P4,6, P5,1, P5,2, P5,3, P5,4, P5,5, P5,6, P6,1, P6,2, P6,3, P6,4, P6,5, P6,6 present all of them below. hAB34ihCD12i+hAB23ihCD14i+hAB14ihCD23i+hAB12ihCD34i hAB12ihAB14ihAB23ihAB34ihCD12ihCD14ihCD23ihCD34i hAB34ihCD12i+hAB23ihCD14i+hAB14ihCD23i hABCDihAB14ihAB23ihAB34ihCD12ihCD14ihCD23ihCD34i hABCDi 0 hAB12i 0 P1,1, P1,2, P1,3, P1,4, P1,5, P1,6, P2,1, P2,2, P2,3, P2,4, P1,1, P1,2, P1,3, P1,4, P1,5, P1,6, P3,1, P3,2, P3,3, P3,4, P2,5, P2,6, P3,1, P3,2, P3,3, P3,4, P3,5, P3,6, P4,1, P4,2, P3,5, P3,6, P4,1, P4,2, P4,3, P4,4, P4,5, P4,6, P5,1, P5,2, P4,3, P4,4, P4,5, P4,6, P5,1, P5,2, P5,3, P5,4, P (...truncated)