Abstract
In [1], two of the present authors along with P. Raman attempted to extend the Amplituhedron program for scalar field theories [2] to quartic scalar interactions. In this paper we develop various aspects of this proposal. Using recent seminal results in Representation theory [3, 4], we show that projectivity of scattering forms and existence of kinematic space associahedron completely capture planar amplitudes of quartic interaction. We generalise the results of [1] and show that for any n-particle amplitude, the positive geometry associated to the projective scattering form is a convex realisation of Stokes polytope which can be naturally embedded inside one of the ABHY associahedra defined in [2, 5]. For a special class of Stokes polytopes with hyper-cubic topology, we show that they have a canonical convex realisation in kinematic space as boundaries of kinematic space associahedra.
We then use these kinematic space geometric constructions to write world-sheet forms for 𝜙4 theory which are forms of lower rank on the CHY moduli space. We argue that just as in the case of bi-adjoint 𝜙3 scalar amplitudes, scattering equations can be used as diffeomorphisms between certain \( \frac{n-4}{2} \) forms on the world-sheet and \( \frac{n-4}{2} \) forms on ABHY associahedron that generate quartic amplitudes.
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References
P. Banerjee, A. Laddha and P. Raman, Stokes polytopes: the positive geometry for 𝜙4interactions, JHEP08 (2019) 067 [arXiv:1811.05904] [INSPIRE].
N. Arkani-Hamed, Y. Bai, S. He and G. Yan, Scattering Forms and the Positive Geometry of Kinematics, Color and the Worldsheet, JHEP05 (2018) 096 [arXiv:1711.09102] [INSPIRE].
Y. Palu, V. Pilaud and P.-G. Plamondon, Non-kissing and non-crossing complexes for locally gentle algebras, arXiv:1807.04730.
A. Padrol, Y. Palu, V. Pilaud and P.-G. Plamondon, Associahedra for finite cluster type algebra and minimal relations between g-vectors, arXiv:1906.06861.
V. Bazier-Matte, G. Douville, K. Mousavand, H. Thomas and E. Yildirim, ABHY Associahedra and Newton polytopes of F -polynomials for finite type cluster algebras, arXiv:1808.09986 [INSPIRE].
N. Arkani-Hamed and J. Trnka, The Amplituhedron, JHEP10 (2014) 030 [arXiv:1312.2007] [INSPIRE].
N. Arkani-Hamed and J. Trnka, Into the Amplituhedron, JHEP12 (2014) 182 [arXiv:1312.7878] [INSPIRE].
S. Franco, D. Galloni, A. Mariotti and J. Trnka, Anatomy of the Amplituhedron, JHEP03 (2015) 128 [arXiv:1408.3410] [INSPIRE].
N. Arkani-Hamed, C. Langer, A. Yelleshpur Srikant and J. Trnka, Deep Into the Amplituhedron: Amplitude Singularities at All Loops and Legs, Phys. Rev. Lett.122 (2019) 051601 [arXiv:1810.08208] [INSPIRE].
N. Arkani-Hamed, H. Thomas and J. Trnka, Unwinding the Amplituhedron in Binary, JHEP01 (2018) 016 [arXiv:1704.05069] [INSPIRE].
S. He, G. Yan, C. Zhang and Y. Zhang, Scattering Forms, Worldsheet Forms and Amplitudes from Subspaces, JHEP08 (2018) 040 [arXiv:1803.11302] [INSPIRE].
L. de la Cruz, A. Kniss and S. Weinzierl, Properties of scattering forms and their relation to associahedra, JHEP03 (2018) 064 [arXiv:1711.07942] [INSPIRE].
P. Raman, The positive geometry for 𝜙pinteractions, JHEP10 (2019) 271 [arXiv:1906.02985] [INSPIRE].
P.B. Aneesh, M. Jagadale and N. Kalyanapuram, Accordiohedra as positive geometries for generic scalar field theories, Phys. Rev.D 100 (2019) 106013 [arXiv:1906.12148] [INSPIRE].
F. Cachazo, S. He and E.Y. Yuan, Scattering Equations and Matrices: From Einstein To Yang-Mills, DBI and NLSM, JHEP07 (2015) 149 [arXiv:1412.3479] [INSPIRE].
C. Baadsgaard, N.E.J. Bjerrum-Bohr, J.L. Bourjaily and P.H. Damgaard, Scattering Equations and Feynman Diagrams, JHEP09 (2015) 136 [arXiv:1507.00997] [INSPIRE].
C. Baadsgaard, N.E.J. Bjerrum-Bohr, J.L. Bourjaily and P.H. Damgaard, String-Like Dual Models for Scalar Theories, JHEP12 (2016) 019 [arXiv:1610.04228] [INSPIRE].
T. Manneville and V. Pilaud, Geometric realizations of the accordion complex of a dissection, Discrete Comput. Geom.61 (2019) 507 [arXiv:1703.09953].
J. Stasheff, Homotopy Associativity of H-Spaces. I, Trans. Am. Math. Soc.108 (1963) 275.
J. Stasheff, Homotopy Associativity of H-Spaces. II, Trans. Am. Math. Soc.108 (1963) 293.
Y. Baryshnikov, On Stokes sets, in New Developments in Singularity Theory, Nato Science Series II, volume 21, Springer (2001), pp. 65–86.
F. Chapoton, Stokes posets and serpent nest, Discret Math. Theor. Comput. Sci.18 (2016) 1 [arXiv:1505.05990].
S.L. Devadoss, T. Heath and W. Vipismakul, Deformations of bordered surfaces and convex polytopes, Notices Am. Math. Soc.58 (2011) 530 [arXiv:1002.1676].
N. Kalyanapuram, Stokes Polytopes and Intersection Theory, arXiv:1910.12195 [INSPIRE].
P. Etingof, A. Henriques, J. Kamnitzer and E.M. Rains, The cohomology ring of the real locus of the moduli space of stable curves of genus 0 with marked points, Ann. Math.171 (2010) 731.
S. Devadoss, Tessellations of Moduli Spaces and the Mosaic Operad, Contemp. Math.239 (1999) 91 [math.AG/9807010].
S. Mizera, Aspects of Scattering Amplitudes and Moduli Space Localization, Ph.D. Thesis, University of Waterloo, Waterloo Ontario Canada (2019), arXiv:1906.02099 [INSPIRE].
S. Mizera, Scattering Amplitudes from Intersection Theory, Phys. Rev. Lett.120 (2018) 141602 [arXiv:1711.00469] [INSPIRE].
F. Cachazo, S. He and E.Y. Yuan, Scattering of Massless Particles: Scalars, Gluons and Gravitons, JHEP07 (2014) 033 [arXiv:1309.0885] [INSPIRE].
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Aneesh, P., Banerjee, P., Jagadale, M. et al. On positive geometries of quartic interactions: Stokes polytopes, lower forms on associahedra and world-sheet forms. J. High Energ. Phys. 2020, 149 (2020). https://doi.org/10.1007/JHEP04(2020)149
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DOI: https://doi.org/10.1007/JHEP04(2020)149