Built on

J. Golden, A. J. McLeod, M. Spradlin and A. Volovich, The Sklyanin Bracket and Cluster Adjacency at All Multiplicity , Journal of High Energy Physics 2019
Original
1902
Earlier work this paper cites.
S. Caron-Huot, L. J. Dixon, F. Dulat, M. von Hippel, A. J. McLeod and G. Papathanasiou, Six-Gluon amplitudes in planar 𝒩 \mathcal{N} = 4 super-Yang-Mills theory at six and seven loops , Journal of High Energy Physics 2019
Original
1903
Earlier work this paper cites.
F. Cachazo, N. Early, A. Guevara and S. Mizera, Scattering equations: from projective spaces to tropical grassmannians , Journal of High Energy Physics 2019
Original
1903
Earlier work this paper cites.
J. Mago, A. Schreiber, M. Spradlin and A. Volovich, Yangian invariants and cluster adjacency in 𝒩 \mathcal{N} = 4 Yang-Mills , Journal of High Energy Physics 2019
Original
1906
Earlier work this paper cites.
S. Caron-Huot, L. J. Dixon, F. Dulat, M. Von Hippel, A. J. McLeod and G. Papathanasiou, The Cosmic Galois Group and Extended Steinmann Relations for Planar 𝒩 = 4 \mathcal{N}=4 SYM Amplitudes , Journal of High Energy Physics 2019
Original
1906
Earlier work this paper cites.
F. Cachazo and J. M. Rojas, Notes on Biadjoint Amplitudes, Trop G ( 3 , 7 ) {\rm Trop}\,G(3,7) and X ⁡ ( 3 , 7 ) X(3,7) Scattering Equations , 1906.05979
Original
1906
Earlier work this paper cites.
J. Drummond, J. Foster, Ö. Gürdoğan and C. Kalousios, Tropical Grassmannians, cluster algebras and scattering amplitudes , 1907.01053
Original
1907
Earlier work this paper cites.
T. Łukowski, M. Parisi, M. Spradlin and A. Volovich, Cluster Adjacency for m = 2 m=2 Yangian Invariants , Journal of High Energy Physics 2019
Original
1908
Earlier work this paper cites.
D. García Sepúlveda and A. Guevara, A Soft Theorem for the Tropical Grassmannian , 1909.05291
Original
1909
Earlier work this paper cites.
F. Borges and F. Cachazo, Generalized Planar Feynman Diagrams: Collections , 1910.10674
Original
1910
Earlier work this paper cites.
N. Early, From weakly separated collections to matroid subdivisions , 1910.11522
Original
1910
Earlier work this paper cites.
F. Cachazo, B. Umbert and Y. Zhang, Singular Solutions in Soft Limits , 1911.02594
Original
1911
Earlier work this paper cites.
C. Zhang, Z. Li and S. He, Two-loop Octagons, Algebraic Letters and Q ¯ \bar{Q} Equations , 1911.01290
Original
1911
Earlier work this paper cites.
V. Del Duca, S. Druc, J. M. Drummond, C. Duhr, F. Dulat, R. Marzucca et al., All-order amplitudes at any multiplicity in the multi-Regge limit , 1912.00188
Original
1912
Earlier work this paper cites.
J.M. Drummond, J. Foster, Ö. Gürdoğan, C. Kalousios, Algebraic singularities of scattering amplitudes from tropical geometry , 1912.08217
Original
1912
Earlier work this paper cites.
N. Arkani-Hamed, T. Lam and M. Spradlin, Non-perturbative geometries for planar 𝒩 = 4 \mathcal{N}=4 SYM amplitudes , 1912.08222
Original
1912
Earlier work this paper cites.
D. Maclagan, Introduction to tropical algebraic geometry , 1207.1925
Original
1925
Earlier work this paper cites.
A. Sabry, Fourth order spectral functions for the electron propagator , Nuclear Physics 33
1962
Earlier work this paper cites.
K.-T. Chen, Iterated path integrals , Bull. Amer. Math. Soc. 83
1977
Earlier work this paper cites.
V. P. Nair, A Current Algebra for Some Gauge Theory Amplitudes , Phys. Lett. B214
1988
Earlier work this paper cites.
A. Zamolodchikov, On the thermodynamic Bethe ansatz equations for reflectionless ADE scattering theories , Physics Letters B 253
1991
Earlier work this paper cites.
Graduate texts in Mathematics. Springer-Verlag, 1995
G. Ziegler, Lectures on Polytopes · 1995
Earlier work this paper cites.
S. Fomin and A. Zelevinsky, Cluster Algebras I: Foundations , Journal of the American Mathematical Society 15
2002
Earlier work this paper cites.
F. Chapoton, S. Fomin and A. Zelevinsky, Polytopal Realizations of Generalized Associahedra , Canadian Mathematical Bulletin 45
2002
Earlier work this paper cites.
S. Fomin and A. Zelevinsky, Cluster algebras II: Finite type classification , Inventiones mathematicae 154
2003
Earlier work this paper cites.
A. Berenstein, S. Fomin and A. Zelevinsky, Cluster algebras III: Upper bounds and double Bruhat cells , Duke Mathematical Journal 126
2003
Earlier work this paper cites.
D. Speyer and B. Sturmfels, The Tropical Grassmannian , Advances in Geometry 4
2003
Earlier work this paper cites.
S. Fomin and A. Zelevinsky, Y-systems and generalized associahedra , Annals of Mathematics 158
2003
Earlier work this paper cites.
A. B. Goncharov, Galois symmetries of fundamental groupoids and noncommutative geometry , Duke Math. J. 128
2005
Earlier work this paper cites.
Z. Bern, L. J. Dixon and V. A. Smirnov, Iteration of planar amplitudes in maximally supersymmetric Yang-Mills theory at three loops and beyond , Phys. Rev. D72
2005
Earlier work this paper cites.
D. Speyer and L. Williams, The Tropical Totally Positive Grassmannian , Journal of Algebraic Combinatorics 22
2005
Earlier work this paper cites.
G. Mikhalkin, Tropical geometry and its applications , International Congress of Mathematicians 2
2006
Earlier work this paper cites.

Similar

J. S. Scott, GRASSMANNIANS AND CLUSTER ALGEBRAS , Proceedings of the London Mathematical Society 92
2006
Cited alongside, same era.
S. Fomin, M. Shapiro and D. Thurston, Cluster algebras and triangulated surfaces. Part I: Cluster complexes , arXiv Mathematics e-prints (Aug., 2006) math/0608367, [ math/0608367 ]
2006
Cited alongside, same era.
S. Fomin and A. Zelevinsky, Cluster algebras IV: Coefficients , Compositio Mathematica 143
2007
Cited alongside, same era.
J. Drummond, J. Henn, V. Smirnov and E. Sokatchev, Magic identities for conformal four-point integrals , Journal of High Energy Physics 2007
2007
Cited alongside, same era.
Z. Bern, M. Czakon, L. J. Dixon, D. A. Kosower and V. A. Smirnov, The Four-Loop Planar Amplitude and Cusp Anomalous Dimension in Maximally Supersymmetric Yang-Mills Theory , Phys. Rev. D75
2007

Then

G. Papathanasiou, Evaluating the six-point remainder function near the collinear limit , Int. J. Mod. Phys. A29
Original
2014
Later among the works it cites.
B. Basso, A. Sever and P. Vieira, Space-time S-matrix and Flux-tube S-matrix IV. Gluons and Fusion , Journal of High Energy Physics 2014
Original
2014
Later among the works it cites.
L. J. Dixon, J. M. Drummond, C. Duhr and J. Pennington, The four-loop remainder function and multi-Regge behavior at NNLLA in planar 𝒩 = 4 \mathcal{N}=4 super-Yang-Mills theory , Journal of High Energy Physics 2014
Original
2014
Later among the works it cites.
L. J. Dixon, J. M. Drummond, C. Duhr, M. von Hippel and J. Pennington, Bootstrapping six-gluon scattering in planar 𝒩 = 4 \mathcal{N}=4 super-Yang-Mills theory , PoS LL2014
Original
2014
Later among the works it cites.

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2019

How tropical are seven- and eight-particle amplitudes?

Henke, Niklas, Papathanasiou, Georgios

Understand

We study tropical Grassmanians Tr$(k,n)$ in relation to cluster algebras, and assess their applicability to $n$-particle amplitudes for $n=7,8$.

In $\mathcal{N}=4$ super Yang-Mills theory, we first show that while the totally positive part of Tr$(4,7)$ may encompass the iterated discontinuity structure of the seven-point Maximally Helicity Violating (MHV) amplitude, it is too small for the Next-to-MHV helicity configuration.
Then, using Tr$(4,8)$ we propose a finite set of 356 cluster $\mathcal{A}$-coordinates expected to contain the rational symbol letters of the eight-particle MHV amplitude, and discuss how the remaining square-root letters may be obtained from limits of infinite mutation sequences.
Finally, we use a triangulation of the totally positive part of Tr$(3,8)$ to obtain the associated generalised biadjoint scalar amplitude in a form containing a near-minimal amount of spurious poles.

Built on

J. Golden, A. J. McLeod, M. Spradlin and A. Volovich, The Sklyanin Bracket and Cluster Adjacency at All Multiplicity , Journal of High Energy Physics 2019
Original
1902
Earlier work this paper cites.
S. Caron-Huot, L. J. Dixon, F. Dulat, M. von Hippel, A. J. McLeod and G. Papathanasiou, Six-Gluon amplitudes in planar 𝒩 \mathcal{N} = 4 super-Yang-Mills theory at six and seven loops , Journal of High Energy Physics 2019
Original
1903
Earlier work this paper cites.
F. Cachazo, N. Early, A. Guevara and S. Mizera, Scattering equations: from projective spaces to tropical grassmannians , Journal of High Energy Physics 2019
Original
1903
Earlier work this paper cites.
J. Mago, A. Schreiber, M. Spradlin and A. Volovich, Yangian invariants and cluster adjacency in 𝒩 \mathcal{N} = 4 Yang-Mills , Journal of High Energy Physics 2019
Original
1906
Earlier work this paper cites.
S. Caron-Huot, L. J. Dixon, F. Dulat, M. Von Hippel, A. J. McLeod and G. Papathanasiou, The Cosmic Galois Group and Extended Steinmann Relations for Planar 𝒩 = 4 \mathcal{N}=4 SYM Amplitudes , Journal of High Energy Physics 2019
Original
1906
Earlier work this paper cites.
F. Cachazo and J. M. Rojas, Notes on Biadjoint Amplitudes, Trop G ( 3 , 7 ) {\rm Trop}\,G(3,7) and X ⁡ ( 3 , 7 ) X(3,7) Scattering Equations , 1906.05979
Original
1906
Earlier work this paper cites.
J. Drummond, J. Foster, Ö. Gürdoğan and C. Kalousios, Tropical Grassmannians, cluster algebras and scattering amplitudes , 1907.01053
Original
1907
Earlier work this paper cites.
T. Łukowski, M. Parisi, M. Spradlin and A. Volovich, Cluster Adjacency for m = 2 m=2 Yangian Invariants , Journal of High Energy Physics 2019
Original
1908
Earlier work this paper cites.
D. García Sepúlveda and A. Guevara, A Soft Theorem for the Tropical Grassmannian , 1909.05291
Original
1909
Earlier work this paper cites.
F. Borges and F. Cachazo, Generalized Planar Feynman Diagrams: Collections , 1910.10674
Original
1910
Earlier work this paper cites.
N. Early, From weakly separated collections to matroid subdivisions , 1910.11522
Original
1910
Earlier work this paper cites.
F. Cachazo, B. Umbert and Y. Zhang, Singular Solutions in Soft Limits , 1911.02594
Original
1911
Earlier work this paper cites.
C. Zhang, Z. Li and S. He, Two-loop Octagons, Algebraic Letters and Q ¯ \bar{Q} Equations , 1911.01290
Original
1911
Earlier work this paper cites.
V. Del Duca, S. Druc, J. M. Drummond, C. Duhr, F. Dulat, R. Marzucca et al., All-order amplitudes at any multiplicity in the multi-Regge limit , 1912.00188
Original
1912
Earlier work this paper cites.
J.M. Drummond, J. Foster, Ö. Gürdoğan, C. Kalousios, Algebraic singularities of scattering amplitudes from tropical geometry , 1912.08217
Original
1912
Earlier work this paper cites.
N. Arkani-Hamed, T. Lam and M. Spradlin, Non-perturbative geometries for planar 𝒩 = 4 \mathcal{N}=4 SYM amplitudes , 1912.08222
Original
1912
Earlier work this paper cites.
D. Maclagan, Introduction to tropical algebraic geometry , 1207.1925
Original
1925
Earlier work this paper cites.
A. Sabry, Fourth order spectral functions for the electron propagator , Nuclear Physics 33
1962
Earlier work this paper cites.
K.-T. Chen, Iterated path integrals , Bull. Amer. Math. Soc. 83
1977
Earlier work this paper cites.
V. P. Nair, A Current Algebra for Some Gauge Theory Amplitudes , Phys. Lett. B214
1988
Earlier work this paper cites.
A. Zamolodchikov, On the thermodynamic Bethe ansatz equations for reflectionless ADE scattering theories , Physics Letters B 253
1991
Earlier work this paper cites.
Graduate texts in Mathematics. Springer-Verlag, 1995
G. Ziegler, Lectures on Polytopes · 1995
Earlier work this paper cites.
S. Fomin and A. Zelevinsky, Cluster Algebras I: Foundations , Journal of the American Mathematical Society 15
2002
Earlier work this paper cites.
F. Chapoton, S. Fomin and A. Zelevinsky, Polytopal Realizations of Generalized Associahedra , Canadian Mathematical Bulletin 45
2002
Earlier work this paper cites.
S. Fomin and A. Zelevinsky, Cluster algebras II: Finite type classification , Inventiones mathematicae 154
2003
Earlier work this paper cites.
A. Berenstein, S. Fomin and A. Zelevinsky, Cluster algebras III: Upper bounds and double Bruhat cells , Duke Mathematical Journal 126
2003
Earlier work this paper cites.
D. Speyer and B. Sturmfels, The Tropical Grassmannian , Advances in Geometry 4
2003
Earlier work this paper cites.
S. Fomin and A. Zelevinsky, Y-systems and generalized associahedra , Annals of Mathematics 158
2003
Earlier work this paper cites.
A. B. Goncharov, Galois symmetries of fundamental groupoids and noncommutative geometry , Duke Math. J. 128
2005
Earlier work this paper cites.
Z. Bern, L. J. Dixon and V. A. Smirnov, Iteration of planar amplitudes in maximally supersymmetric Yang-Mills theory at three loops and beyond , Phys. Rev. D72
2005
Earlier work this paper cites.
D. Speyer and L. Williams, The Tropical Totally Positive Grassmannian , Journal of Algebraic Combinatorics 22
2005
Earlier work this paper cites.
G. Mikhalkin, Tropical geometry and its applications , International Congress of Mathematicians 2
2006
Earlier work this paper cites.

Similar

J. S. Scott, GRASSMANNIANS AND CLUSTER ALGEBRAS , Proceedings of the London Mathematical Society 92
2006
Cited alongside, same era.
S. Fomin, M. Shapiro and D. Thurston, Cluster algebras and triangulated surfaces. Part I: Cluster complexes , arXiv Mathematics e-prints (Aug., 2006) math/0608367, [ math/0608367 ]
2006
Cited alongside, same era.
S. Fomin and A. Zelevinsky, Cluster algebras IV: Coefficients , Compositio Mathematica 143
2007
Cited alongside, same era.
J. Drummond, J. Henn, V. Smirnov and E. Sokatchev, Magic identities for conformal four-point integrals , Journal of High Energy Physics 2007
2007
Cited alongside, same era.
Z. Bern, M. Czakon, L. J. Dixon, D. A. Kosower and V. A. Smirnov, The Four-Loop Planar Amplitude and Cusp Anomalous Dimension in Maximally Supersymmetric Yang-Mills Theory , Phys. Rev. D75
2007