Abstract
We present a new formula for the biadjoint scalar tree amplitudes m(α|β) based on the combinatorics of dual associahedra. Our construction makes essential use of the cones in ‘kinematic space’ introduced by Arkani-Hamed, Bai, He, and Yan. We then consider dual associahedra in ‘dual kinematic space.’ If appropriately embedded, the intersections of these dual associahedra encode the amplitudes m(α|β). In fact, we encode all the partial amplitudes at n-points using a single object, a ‘fan,’ in dual kinematic space. Equivalently, as a corollary of our construction, all n-point partial amplitudes can be understood as coming from integrals over subvarieties in a toric variety. Explicit formulas for the amplitudes then follow by evaluating these integrals using the equivariant localisation formula. Finally, by introducing a lattice in kinematic space, we observe that our fan is also related to the inverse KLT kernel, sometimes denoted \( {m}_{\alpha^{\prime }}\left(\alpha \Big|\beta \right) \).
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
N. Arkani-Hamed, Y. Bai, S. He and G. Yan, Scattering forms and the positive geometry of kinematics, color and the worldsheet, JHEP 05 (2018) 096 [arXiv:1711.09102] [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].
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].
S. Mizera, Scattering amplitudes from intersection theory, Phys. Rev. Lett. 120 (2018) 141602 [arXiv:1711.00469] [INSPIRE].
K. Matsumoto, Intersection numbers for logarithmic k-forms, Osaka J. Math. 35 (1998) 873.
C. Ceballos, F. Santos and G.M. Ziegler, Many non-equivalent realizations of the associahedron, Combinatorica 35 (2015) 513.
N. Arkani-Hamed, Y. Bai and T. Lam, Positive geometries and canonical forms, JHEP 11 (2017) 039 [arXiv:1703.04541] [INSPIRE].
Z. Bern, J.J.M. Carrasco and H. Johansson, New relations for gauge-theory amplitudes, Phys. Rev. D 78 (2008) 085011 [arXiv:0805.3993] [INSPIRE].
S. Stieberger and T.R. Taylor, Closed string amplitudes as single-valued open string amplitudes, Nucl. Phys. B 881 (2014) 269 [arXiv:1401.1218] [INSPIRE].
C.R. Mafra and O. Schlotterer, Non-Abelian Z-theory: Berends-Giele recursion for the α ′ -expansion of disk integrals, JHEP 01 (2017) 031 [arXiv:1609.07078] [INSPIRE].
H. Kawai, D.C. Lewellen and S.-H. Henry Tye, A relation between tree amplitudes of closed and open strings, Nucl. Phys. B 269 (1986) 1 [INSPIRE].
S. Mizera, Inverse of the string theory KLT kernel, JHEP 06 (2017) 084 [arXiv:1610.04230] [INSPIRE].
S.L. Devadoss, Tessellations of moduli spaces and the mosaic operad, math.AG/9807010.
S. Mizera, Combinatorics and topology of Kawai-Lewellen-Tye relations, JHEP 08 (2017) 097 [arXiv:1706.08527] [INSPIRE].
M. Kita and M. Yoshida, Intersection theory for twisted cycles II — degenerate arrangements, Math. Nachr. 168 (1994) 171.
M.M. Kapranov, The permutoassociahedron, Mac Lane’s coherence theorem and asymptotic zones for the KZ equation, J. Pure Appl. Alg. 85 (1993) 119.
V. Reiner and G.M. Ziegler, Coxeter-associahedra, Mathematika 41 (1994) 364.
A. Postnikov, Permutohedra, associahedra, and beyond, math.CO/0507163.
A. Postnikov, V. Reiner and L. Williams, Faces of generalized permutohedra, math.CO/0609184.
N. Early, Canonical bases for permutohedral plates, arXiv:1712.08520 [INSPIRE].
M.F. Atiyah, Convexity and commuting hamiltonians, Bull. Lond. Math. Soc. 14 (1982) 1.
E. Witten, A new look at the path integral of quantum mechanics, arXiv:1009.6032 [INSPIRE].
E. Frenkel, A. Losev and N. Nekrasov, Instantons beyond topological theory. I, hep-th/0610149 [INSPIRE].
T. Oda, Convex bodies and algebraic geometry: an introduction to the theory of toric varieties, 1st edition, Springer-Verlag, Berlin, Heidelberg, Germany, (1988).
A. Barvinok and J.E. Pommersheim, An algorithmic theory of lattice points in polyhedra, in New perspectives in geometric combinatorics, Math. Sci. Res. Inst. Publ. 38, Cambridge Univ. Press, Cambridge, U.K., (1999), pg. 91.
M. Brion, Points entiers dans les polyèdres convexes (in French), Ann. Sci. École Norm. Sup. 21 (1988) 653.
N. Berline and M. Vergne, Local Euler-Maclaurin formula for polytopes, math.CO/0507256.
M. Brion and M. Vergne, Residue formulae, vector partition functions and lattice points in rational polytopes, J. Amer. Math. Soc. 10 (1997) 797.
M. Brion and M. Vergne, Lattice points in simple polytopes, J. Amer. Math. Soc. 10 (1997) 371.
A. Barvinok, Integer points in polyhedra, Amer. Mathematical Society, Zürich, Switzerland, August 2008.
W. Fulton, Introduction to toric varieties, Princeton University Press, Princeton, NJ, U.S.A., July 1993.
J.J. Duistermaat and G.J. Heckman, On the variation in the cohomology of the symplectic form of the reduced phase space, Invent. Math. 69 (1982) 259.
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1802.03384
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Frost, H. Biadjoint scalar tree amplitudes and intersecting dual associahedra. J. High Energ. Phys. 2018, 153 (2018). https://doi.org/10.1007/JHEP06(2018)153
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP06(2018)153