Abstract
In this paper, we study a novel behavior developed by certain tree-level scalar scattering amplitudes, including the biadjoint, NLSM, and special Galileon, when a subset of kinematic invariants vanishes without producing a singularity. This behavior exhibits properties which we call smooth splitting and semi-locality. The former means that an amplitude becomes the product of exactly three amputated Berends-Giele currents, while the latter means that any two currents share one external particle. We call these smooth splittings 3-splits. In fact, there are exactly \( \left(\begin{array}{c}n\\ {}3\end{array}\right)-n \) such 3-splits of an n-particle amplitude, one for each tripod in a polygon; as they cannot be obtained from standard factorization, they are a new phenomenon in Quantum Field Theory. In fact, the resulting splitting is analogous to the one first seen in Cachazo-Early-Guevara-Mizera (CEGM) amplitudes which generalize standard cubic scalar amplitudes from their Tr G(2, n) formulation to Tr G(k, n), where Tr G(k, n) is the tropical Grassmannian. Along the way, we show how smooth splittings naturally lead to the discovery of mixed amplitudes in the NLSM and special Galileon theories and to novel BCFW-like recursion relations for NLSM amplitudes.
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References
L.J. Dixon, Calculating scattering amplitudes efficiently, in Theoretical Advanced Study Institute in Elementary Particle Physics (TASI 95): QCD and beyond, (1996), p. 539 [hep-ph/9601359] [INSPIRE].
R. Britto, F. Cachazo, B. Feng and E. Witten, Direct proof of tree-level recursion relation in Yang-Mills theory, Phys. Rev. Lett. 94 (2005) 181602 [hep-th/0501052] [INSPIRE].
R. Britto, F. Cachazo and B. Feng, New recursion relations for tree amplitudes of gluons, Nucl. Phys. B 715 (2005) 499 [hep-th/0412308] [INSPIRE].
R. Britto, F. Cachazo and B. Feng, Generalized unitarity and one-loop amplitudes in N = 4 super-Yang-Mills, Nucl. Phys. B 725 (2005) 275 [hep-th/0412103] [INSPIRE].
N. Arkani-Hamed and J. Kaplan, On tree amplitudes in gauge theory and gravity, JHEP 04 (2008) 076 [arXiv:0801.2385] [INSPIRE].
P. Benincasa and F. Cachazo, Consistency conditions on the S-matrix of massless particles, arXiv:0705.4305 [INSPIRE].
H. Elvang and Y.-T. Huang, Scattering amplitudes in gauge theory and gravity, Cambridge University Press (2015).
C.R. Mafra, Berends-Giele recursion for double-color-ordered amplitudes, JHEP 07 (2016) 080 [arXiv:1603.09731] [INSPIRE].
N. Early, Planar kinematic invariants, matroid subdivisions and generalized Feynman diagrams, arXiv:1912.13513 [INSPIRE].
K. Kampf, J. Novotny and J. Trnka, Tree-level amplitudes in the nonlinear sigma model, JHEP 05 (2013) 032 [arXiv:1304.3048] [INSPIRE].
F. Cachazo, S. He and E.Y. Yuan, Scattering equations and matrices: from Einstein to Yang-Mills, DBI and NLSM, JHEP 07 (2015) 149 [arXiv:1412.3479] [INSPIRE].
F. Cachazo, P. Cha and S. Mizera, Extensions of theories from soft limits, JHEP 06 (2016) 170 [arXiv:1604.03893] [INSPIRE].
C. Cheung, K. Kampf, J. Novotny, C.-H. Shen and J. Trnka, On-shell recursion relations for effective field theories, Phys. Rev. Lett. 116 (2016) 041601 [arXiv:1509.03309] [INSPIRE].
I.M. Gelfand, M.I. Graev and A. Postnikov, Combinatorics of hypergeometric functions associated with positive roots, in The Arnold-Gelfand mathematical seminars, Springer (1997), p. 205.
N. Early, Planarity in generalized scattering amplitudes: PK polytope, generalized root systems and worldsheet associahedra, arXiv:2106.07142 [INSPIRE].
N. Early, Generalized permutohedra in the kinematic space, arXiv:1804.05460 [INSPIRE].
X. Gao, S. He and Y. Zhang, Labelled tree graphs, Feynman diagrams and disk integrals, JHEP 11 (2017) 144 [arXiv:1708.08701] [INSPIRE].
F. Cachazo, S. He and E.Y. Yuan, Scattering equations and Kawai-Lewellen-Tye orthogonality, Phys. Rev. D 90 (2014) 065001 [arXiv:1306.6575] [INSPIRE].
F. Cachazo, S. He and E.Y. Yuan, Scattering of massless particles in arbitrary dimensions, Phys. Rev. Lett. 113 (2014) 171601 [arXiv:1307.2199] [INSPIRE].
F. Cachazo, S. He and E.Y. Yuan, Scattering of massless particles: scalars, gluons and gravitons, JHEP 07 (2014) 033 [arXiv:1309.0885] [INSPIRE].
S.G. Naculich, CHY representations for gauge theory and gravity amplitudes with up to three massive particles, JHEP 05 (2015) 050 [arXiv:1501.03500] [INSPIRE].
M. Gell-Mann and M. Levy, The axial vector current in beta decay, Nuovo Cim. 16 (1960) 705 [INSPIRE].
S.L. Adler, Consistency conditions on the strong interactions implied by a partially conserved axial-vector current. II, Phys. Rev. 139 (1965) B1638 [INSPIRE].
L. Susskind and G. Frye, Algebraic aspects of pionic duality diagrams, Phys. Rev. D 1 (1970) 1682 [INSPIRE].
N. Arkani-Hamed, F. Cachazo and J. Kaplan, What is the simplest quantum field theory?, JHEP 09 (2010) 016 [arXiv:0808.1446] [INSPIRE].
C. Cheung, K. Kampf, J. Novotny and J. Trnka, Effective field theories from soft limits of scattering amplitudes, Phys. Rev. Lett. 114 (2015) 221602 [arXiv:1412.4095] [INSPIRE].
K. Hinterbichler and A. Joyce, Hidden symmetry of the Galileon, Phys. Rev. D 92 (2015) 023503 [arXiv:1501.07600] [INSPIRE].
K. Hinterbichler, Theoretical aspects of massive gravity, Rev. Mod. Phys. 84 (2012) 671 [arXiv:1105.3735] [INSPIRE].
G.R. Dvali, G. Gabadadze and M. Porrati, 4D gravity on a brane in 5D Minkowski space, Phys. Lett. B 485 (2000) 208 [hep-th/0005016] [INSPIRE].
K. Kampf and J. Novotny, Unification of Galileon dualities, JHEP 10 (2014) 006 [arXiv:1403.6813] [INSPIRE].
N. Arkani-Hamed, S. He, G. Salvatori and H. Thomas, Causal diamonds, cluster polytopes and scattering amplitudes, arXiv:1912.12948 [INSPIRE].
S. He and Q. Yang, An etude on recursion relations and triangulations, JHEP 05 (2019) 040 [arXiv:1810.08508] [INSPIRE].
G. Salvatori and S. Stanojevic, Scattering amplitudes and simple canonical forms for simple polytopes, JHEP 03 (2021) 067 [arXiv:1912.06125] [INSPIRE].
F. Cachazo, N. Early, A. Guevara and S. Mizera, Scattering equations: from projective spaces to tropical Grassmannians, JHEP 06 (2019) 039 [arXiv:1903.08904] [INSPIRE].
F. Cachazo, Combinatorial factorization, arXiv:1710.04558 [INSPIRE].
F. Cachazo, N. Early, A. Guevara and S. Mizera, ∆-algebra and scattering amplitudes, JHEP 02 (2019) 005 [arXiv:1812.01168] [INSPIRE].
N. Early and V. Reiner, On configuration spaces and Whitehouse’s lifts of the Eulerian representations, J. Pure Appl. Alg. 223 (2019) 4524.
N. Early, Honeycomb tessellations and canonical bases for permutohedral blades, arXiv:1810.03246 [INSPIRE].
N. Early, Combinatorics and representation theory for generalized permutohedra I: simplicial plates, arXiv:1611.06640.
N. Arkani-Hamed, S. He and T. Lam, Stringy canonical forms, JHEP 02 (2021) 069 [arXiv:1912.08707] [INSPIRE].
N. Early, Weighted blade arrangements and the positive tropical Grassmannian, arXiv:2005.12305 [INSPIRE].
N. Early, From weakly separated collections to matroid subdivisions, arXiv:1910.11522 [INSPIRE].
F. Borges and F. Cachazo, Generalized planar Feynman diagrams: collections, JHEP 11 (2020) 164 [arXiv:1910.10674] [INSPIRE].
S. He, L. Ren and Y. Zhang, Notes on polytopes, amplitudes and boundary configurations for Grassmannian string integrals, JHEP 04 (2020) 140 [arXiv:2001.09603] [INSPIRE].
B. Sturmfels and S. Telen, Likelihood equations and scattering amplitudes, arXiv:2012.05041 [INSPIRE].
D. Agostini, T. Brysiewicz, C. Fevola, L. Kühne, B. Sturmfels and S. Telen, Likelihood degenerations, arXiv:2107.10518 [INSPIRE].
F. Cachazo and N. Early, Planar kinematics: cyclic fixed points, mirror superpotential, k-dimensional Catalan numbers, and root polytopes, arXiv:2010.09708 [INSPIRE].
F. Santos, C. Stump and V. Welker, Noncrossing sets and a Grassmann associahedron, Forum Math. Sigma 5 (2017) 1.
N. Arkani-Hamed, S. He, T. Lam and H. Thomas, Binary geometries, generalized particles and strings, and cluster algebras, arXiv:1912.11764 [INSPIRE].
L. Dolan and P. Goddard, Proof of the formula of Cachazo, He and Yuan for Yang-Mills tree amplitudes in arbitrary dimension, JHEP 05 (2014) 010 [arXiv:1311.5200] [INSPIRE].
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Cachazo, F., Early, N. & Umbert, B.G. Smoothly splitting amplitudes and semi-locality. J. High Energ. Phys. 2022, 252 (2022). https://doi.org/10.1007/JHEP08(2022)252
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DOI: https://doi.org/10.1007/JHEP08(2022)252