PHYSICAL REVIEW LETTERS 122, 051601 (2019)

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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. The regions in Bα space are the same for all α.

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Thus for each such piece we take the product of the A form are irrelevant while other diagrams with more internal and the Bα forms, and then sum over all terms in the propagators contribute, such as triangulation. Because there are no mutual positivity con- ditions left for the given point A, the form in Bα is given by a simple product. The final result can then be written as

X Yl ðlÞ ðnÞ ðnÞ Ωn ¼ ωj ðAÞ ∧ κj ðBαÞ: ð7Þ j α¼1

In the second solution all lines ðABÞα lie in the same plane P and we denote them where the diagram has DCI numerator hAB14iL−2, with AB ðABÞα ¼ Lα: ð8Þ being the upper loop. In general, one has to check if the numerator of a given diagram reduces the degree of The plane P has three d.o.f. while each line Lα has two the singularity in the denominator, and consider it in the d.o.f., which is 2l þ 3 in total. Again, we imagine fixing counting. Furthermore, it should be also checked explicitly the plane P and looking for all the lines L in P that lie in the that a given diagram indeed gives nonzero residue on the one-loop amplituhedron. This breaks up P, L space into deepest cut. pieces where, given P in a certain region, the corresponding Four point case.—We now proceed to solving the geometry in L space has a uniform shape. The differential geometry problem for the four point case where all lines form for this space can be written as pass through the same point A. The positivity of external data at four points is a single condition h1234i > 0 while l X Y the sign flip conditions for lines Lα ¼ ABα turn into the set Ω˜ ðlÞ ω˜ ðnÞ P ∧ κ˜ðnÞ L : n ¼ j ð Þ j ð αÞ ð9Þ of inequalities j α¼1

hABα12i > 0; hABα23i > 0; hABα34i > 0; hABα14i > 0; The various differential forms ωðAÞ; κðBαÞ; ωðPÞ and κðLÞ are proportional to universal measure factors dμA; hABα13i < 0; hABα24i < 0: ð14Þ dμB;dμP;dμL associated with the free points, planes, and lines characterizing the geometries. The measures for the As there are no mutual inequalities between different lines point A and points Bα are ABα in this problem once we solve the inequalities for one B it is automatically solved for all of them, and the 3 IJKL I J K L dμA ¼hAd Ai ≡ ϵ A dA ∧ dA ∧ dA ; differential form is given by the simple product (7).As 2 IJKL I J K L all inequalities involve the point A we can project through dμB ¼hABd Bi ≡ ϵ A B dB ∧ dB : ð10Þ that point to a two-dimensional plane that contains pro- 0 0 0 0 0 IJK IJK IJKL jected points Z1, Z2, Z3, Z4, and Bk For the plane P ≡ P , we can define P ≡ ϵ pL and then the measure for P is

3 IJKL dμP ¼hpd pi ≡ ϵ pIdpJ ∧ dpK ∧ dpL: ð11Þ

I The plane P can be parametrized using three points pi , i ¼ 1, 2, 3 up to a GLð3Þ transformation on the i index. IJ Then the line ðABÞ is related to the line Lk on P as

IJ ijk IJ ðABÞ ¼ ϵ ðpipjÞ Lk; ð12Þ

ijk where the ϵ acts on the labels of points pi on the plane P. Finally, the measure of the line L is

2 ijk dμL ¼hLLd Li ≡ ϵ LidLj ∧ dLk: ð13Þ We omit the primes in the following. These inequalities cut The richness of the deepest cut is revealed when compared out an allowed region for A. The task is to triangulate this to the amplitude written as a sum of Feynman integrals. region such that for each term in the triangulation, there is a Contributing diagrams must have at least 2l − 3 internal corresponding allowed region for B whose shape does not propagators. Ladder diagrams with only l − 1 propagators change within this A region. For the four point case the

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3 situation is quite simple: A can be written as linear dμAh1234i A c Z c Z c Z c Z ωðAÞ¼ ; ð16Þ combination ¼ 1 1 þ 2 2 þ 3 3 þ 4 4 and the hA123ihA124ihA134ihA234i choice of signs of cj are equivalent to the choice of signs of the set, and K is the number of minus signs in Eq. (15). The B forms depend on the shape of the allowed region in the two- fhA123i; hA124i; hA134i; hA234ig: ð15Þ dimensional plane. The triangle in the picture above is bounded by the lines (23),(34),(14) and the form is There are 24 ¼ 16 possible choices corresponding to 16 A dμBhA134ihA234i regions for which we have to find the B geometry. The two- ωðBÞ¼ : ð17Þ hAB23ihAB34ihAB14i dimensional configuration of the projected Zi points must respect given hAijki inequalities. In particular, hAijki > 0 ijk Z Z Z If the configuration was more complicated, e.g., a penta- if the triangle ð Þ made of points i, j, k is oriented gon, we would have to decompose it into triangles but that Aijk < 0 clockwise. The other sign h i corresponds to does not happen in the four point case. Putting all pieces counterclockwise orientation. As a result, we get two types together we can write the result as of configurations, ðlÞ Ω4 ¼ ωðAÞ ∧ κðBÞ; ð18Þ

where ωðAÞ is given by Eq. (16) and κðBÞ is a sum of four triangles with signs coming from different A regions,

X4 Y dμBhAj−1jjþ1ihAjjþ1jþ2i κðBÞ¼ ð−1Þj ; ABαj−1j ABαjj 1 ABαj 1j 2 j¼1 α h ih þ ih þ þ i ð19Þ where the first corresponds to fþ; þ; þ; þg and the second where the sum over j includes the cyclic twist: n þ k → k. to fþ; þ; −; þg. Adding all possible permutations of Zi gives us eight quadrilateral configurations and eight tri- In the four point case the all-in-plane solution can be extracted from the all-in-point solution. Instead of the com- angle configurations with one point inside. In the second A P P P P step we have to find the space of all B points which satisfy mon point we have a common plane ¼ð 1 2 3Þ on which all Lα live. The space of P planes is bounded by inequalities (14). Geometrically, the inequality hABiji > 0 points Z1, Z2, Z3, Z4 and the form in P up to a sign is means that B must be on the right side of the line ZiZj or, given by alternatively, the triangle ðBijÞ must be oriented clockwise, for hABiji < 0 counterclockwise. The first configuration dμPh1234i has no allowed B region, while for the second we get ω˜ ðPÞ¼ : ð20Þ hP1ihP2ihP3ihP4i

On each plane P we have two-forms on the space of lines Lα. The triangulation is analogous to the all-in-line case ˜ ðlÞ and the result can be written as Ω4 ¼ ω˜ ðPÞ ∧ κ˜ðABÞ, where

X4 Yl κ˜ ¼ ð−1Þj j¼1 α¼1

dμL hj − 1jj þ 1j þ 2ihPjihPjþ1i × α ; ð21Þ which is a triangle given by the lines (23)(34)(14). There hðABÞαj − 1jihðABÞαjjþ1ihðABÞαjþ1jþ2i are three more A configurations which give nonempty B regions. All these regions are triangles bounded by lines where ðABÞα are related to the lines Lα restricted to P via (12)(23)(34), (12)(34)(14), and (12)(23)(14). (12). Note that hj − 1jj þ 1j þ 2i¼ð−1Þjh1234i. Since the A and B geometry are factorized, so are the Higher point formulas.—At four points the original corresponding logarithmic volume forms. For the A part for amplituhedron picture is identical to the new sign flip all possible signs in Eq. (15) the boundaries are the same; definition. However, for higher point MHV amplitudes therefore the form is given for all possible regions by while still equivalent the sign flip picture is much more ð−1ÞKωðAÞ, where suitable for actually solving the geometry. Here we provide

051601-4 PHYSICAL REVIEW LETTERS 122, 051601 (2019) the final n-point expressions for the residues on the deepest cut for both solutions; detailed derivations and a number of results for further nontrivial cuts will be provided in Ref. [21]. The basic strategy is the same as in the four point case: for the all-in-point solution, triangulate the A space and find the corresponding Bα geometry. The main difference is that the boundaries of the different pieces in the triangulation of the A space are now different. The result can be schematically drawn as the tetrahedron × where the point Xik ≡ ði − 1iÞ ∩ ðA; i þ k − 2;iþ k − 1Þ. polygon, The expression for the integrand takes the product form

Xn−1 Xn Yl dμ N i ðlÞ Bα k-gonð Þ Ωn ¼ ½i − 1;i;iþ k − 2;iþ k − 1 ∧ ; ð22Þ ABαi − 1i ABαii 1 ABαi k − 2;i k − 1 k¼1 i¼1 α¼1 h ih þ ih þ þ i where ½a; b; c; d is the canonical form in A space for the the corresponding differential forms. Note that in the tetrahedron with vertices Za, Zb, Zc, Zd triangulation the A space is given just by a simple tetrahe- dron while the B space is a more complicated polygon. 3 ðlÞ ˜ ðlÞ dμAhabcdi While for the four point case the forms Ω4 and Ω4 ½abcd¼ : ð23Þ hAabcihAabdihAacdihAbcdi were related for higher points they are different. Following a similar geometric procedure we have to first triangulate P The Bα part is a form on the polygon bounded by the lines the space; here the general term in the triangulation has ði − 1iÞ, ðii þ 1Þ,…, ði þ k − 2;iþ k − 1Þ. The numerator six boundaries, and the corresponding space of lines on the N P k-gon was given in Ref. [22]. Alternatively, one can plane has three boundaries for each line. The expression ˜ ðlÞ triangulate the polygon as a sum of triangles and collect for Ωn is then given by the product form

Xn−2 Xn−1 Xn Yl dμ ⟪P; i; j; k⟫ Ω˜ ðlÞ i; j; k ∧ Lα ; n ¼ f g AB ii 1 AB jj 1 AB kk 1 ð24Þ i¼1 j¼iþ1 k¼jþ1 α¼1 hð Þα þ ihð Þα þ ihð Þα þ i where the bracket fi; j; kg is defined by the deepest cuts Ω and Ω˜ can be straightforwardly computed by taking residues. Explicit expressions for dμP⟪P; i; j; k⟫ fi; j; kg¼ ; ð25Þ the MHV integrand are available in the literature up to hPiihPi þ 1ihPjihPj þ 1ihPkihPk þ 1i ten loops for n ¼ 4 [23–26], and up to three loops for any n [27–29]. We have verified our cut up to five loops for n ¼ 4 with and for general n up to three loops. I I J J K K As we have stressed, the deepest cut is sensitive to the ⟪P; i; j; k⟫ PI1J1K1 PI2J2K2 Z 1 Z 2 Z 1 Z 2 Z 1 Z 2 ¼ i iþ1 j jþ1 k kþ1 most complicated topologies for Feynman diagrams and ¼hðiiþ1ÞðPÞ ∩ ðjjþ1ÞðPÞ ∩ ðkkþ1Þi: ð26Þ on-shell processes that can contribute to the amplitude. It is interesting to see this more quantitatively at four points. For Note that ⟪P; i; j; k⟫ is completely symmetric in i, j, k, the four-point l-loop integrand the number of dual con- though the representation in terms of four brackets does not formal invariant integrals contributing on the cut can be manifest this symmetry. Thinking for convenience dually of counted from the Mathematica code provided in Ref. [26] l 10 P as a point and the Za as planes, fi; j; kg is the canonical up to ¼ . In the Table we provide the number of form of a cube with opposing facets associated with topologies; the complete set of integrals also involves Z ;Z Z ;Z Z ;Z ð i iþ1Þ, ð j jþ1Þ, ð k kþ1Þ. permutation over all loop momenta and cycling in external Exceptional efficiency.—Given the integrand for the labels. The number of contributing diagrams is the same for n-point, l-loop MHV integrand, the differential form for both solutions of the cut.

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The expressions get even more complicated if we rewrite l Total # of topologies Contributing on cut % the numerators using hABαiiþ1i to match the Feynman 48 450integral expansion. For N 3 it would be given by 24 terms 5 34 20 58.8 corresponding to tennis court diagrams at three loops. The 6 229 146 63.8 expressions obviously get more complicated at higher 7 1873 1248 66.6 loops and the number of terms matches the number of 8 19 949 13 664 68.5 contributing Feynman integrals. In the numerator N l all 9 247 856 172 471 69.6 L 10 3 586 145 2 530 903 70.6 lines α are completely entangled and there is no product structure. Again, from this point of view the amazingly simple product form (18) is a total surprise, and without the We have to stress that this counting corresponds to geometric picture one would never discover it. graphs which in principle can contribute on the cut Conclusion.—In this Letter we studied the deepest cut in (counting the number of hABαABβi in the numerator and the planar N ¼ 4 SYM theory using the new topological denominator. One has to further check that there are no definition of the amplituhedron geometry using sign flips, more subtleties and the integrand has a support on the which allowed us to easily find explicit triangulations and residue. In any case, we see the monotonic increase in the concrete expressions for this cut of the n-point MHV percentage of diagrams potentially contributing as a func- amplitudes (22) and (24). When compared to the Feynman tion of l. We expect this percentage should approach 100% diagram expansion, the deepest cut probes the most compli- for l → ∞. cated diagrams. We expect that for l → ∞ this cut captures ˜ Our formulas for Ω and Ω are remarkably simple and some of the essential properties of the full integrand. Indeed, compact, while the complete loop integrand gets more and the deepest cut can be used as a new jumping off point for more complicated for higher l. A notable feature of our approaching the determination of the full geometry by expressions is their representation as a sum over pieces gradually relaxing the mutual intersection properties in steps each having a trivial product structure over loops. However, [21]. It should also be possible to compute the deepest cut for this simplicity comes at the cost of introducing spurious all k; as with the MHV case, in the topological picture this poles. These are all the poles in Ω involving only the should again reduce to merely finding a precise characteri- intersection point “A” rather than the lines ABα. This zation of the one-loop amplituhedron geometry, but the phenomenon is by now a familiar one in the on-shell associated analogs of our “tetrahedral” and “polygonal” approach to scattering amplitudes, but occurs here in a regions are expected to be more nontrivial and interesting. novel setting. Of course the spurious poles cancel non- trivially in the sum. The existence of such a strikingly We thank Lance Dixon, Enrico Herrmann, Thomas Lam, simple and unusual representation for this cut is completely and Hugh Thomas for useful discussions and comments. mysterious from any conventional point of view (Feynman N. A-H. is supported by DOE Grant No. SC0009988. C. L. diagrams, all-loop BCFW recursion or Wilson loops). and J. T. are supported by DOE Grant No. DE-SC0009999 However, as we stressed even before embarking on any and by the funds of University of California. detailed calculation, the existence of such a picture is made almost trivially obvious from the topological picture of the amplituhedron geometry. Any analytic comparison with standard local expressions [1] N. Arkani-Hamed and J. Trnka, J. High Energy Phys. 10 for the cut would have to proceed by algebraically cancel- (2014) 030. ing the spurious poles. This immediately leads to an [2] N. Arkani-Hamed, J. Bourjaily, F. Cachazo, A. Goncharov, explosion of complexity: while the formula for Ω has A. Postnikov, and J. Trnka, Grassmannian Geometry of Scattering Amplitudes (Cambridge University Press, Cam- the same form for any l, when canceling spurious poles the l bridge, 2016), DOI: 10.1017/CBO9781316091548. result gets more complicated for higher . Even at four [3] N. Arkani-Hamed, J. L. Bourjaily, F. Cachazo, S. Caron- points, when all spurious poles are canceled we get Huot, and J. Trnka, J. High Energy Phys. 01 (2011) 041. [4] N. Arkani-Hamed, H. Thomas, and J. Trnka, J. High Energy Yl dμB Phys. 01 (2018) 016. ΩðlÞ dμ N α ; 4 ¼ A l [5] T. Lam, Commun. Math. Phys. 343, 1025 (2016). hABα12ihABα23ihABα34ihABα14i k¼1 [6] S. N. Karp and L. K. Williams, arXiv:1608.08288. l 3 B [7] P. Galashin and T. Lam, arXiv:1805.00600. where for ¼ we get up to symmetrization in α [8] N. Arkani-Hamed, Y. Bai, and T. Lam, J. High Energy Phys. 2 3 A124 2 AB 13 AB 23 AB 34 11 (2017) 039. h i h 1 ih 2 ih 3 i [9] L. Ferro, T. Lukowski, and M. Parisi, J. Phys. A 52, 045201 6 2 7 N 3 ¼ 4 þhA234i hAB112ihAB213ihAB314i 5: (2019). A234 A124 AB 13 AB 13 AB 24 [10] N. Arkani-Hamed and J. Trnka, J. High Energy Phys. 12 þh ih ih 1 ih 2 ih 3 i (2014) 182.

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