Cluster Algebras, Steinmann Relations, and Amplitudes in planar N = - PowerPoint PPT Presentation

Cluster Algebras, Steinmann, and the Lie Cobracket Andrew McLeod Cluster Algebras, Steinmann Relations, and Amplitudes in planar N = 4 the Lie Cobracket in planar N = 4 sYM · Symmetries and Simplifications · Infrared and Helicity Structure Cluster Algebras and Polylogarithms Andrew McLeod · Polylogarithms, the Coaction, and the Lie cobracket (Niels Bohr Institute) · Cluster-Algebraic Structure Subalgebra Galileo Galilei Institute Constructibility November 8, 2018 · Decomposing the Remainder Function Conclusions Based on work in collaboration with John Golden 1810.12181, 190x.xxxxx

Cluster Algebras, Outline Steinmann, and the Lie Cobracket Andrew McLeod Amplitudes in planar N = 4 ◦ Planar N = 4 supersymmetric Yang-Mills (sYM) theory · Symmetries and Simplifications · Infrared and Helicity • Symmetries and Simplifications Structure Cluster Algebras • Infrared and Helicity Structure and Polylogarithms · Polylogarithms, the Coaction, and the ◦ Polylogarithms and Cluster-Algebraic Structure Lie cobracket · Cluster-Algebraic Structure • Polylogarithms, the Coaction, and the Lie Cobracket Subalgebra Constructibility • Cluster-Algebraic Structure in N = 4 sYM · Decomposing the Remainder Function Conclusions ◦ Subalgebra Constructibility • Decomposing the Remainder Function

Amplitudes in N = 4 sYM Cluster Algebras, Steinmann, and the Lie Cobracket Andrew McLeod Amplitudes in planar N = 4 · Symmetries and Simplifications SUSY Ward identities ⇒ many relations among amplitudes · Infrared and Helicity Structure with different helicity structure Cluster Algebras and Polylogarithms Conformal theory ⇒ no running of the coupling · Polylogarithms, the Coaction, and the or UV divergences Lie cobracket · Cluster-Algebraic Structure AdS 5 × S 5 dual theory ⇒ multiple ways to calculate Subalgebra quantities of interest Constructibility · Decomposing the Remainder Function Supersymmetric ⇒ the types of functions that Conclusions version of QCD show up here also appear in QCD

Cluster Algebras, Planar Limit and Dual Conformal Symmetry Steinmann, and the Lie Cobracket Andrew McLeod We work with in the N c → ∞ limit with fixed g 2 = g 2 Amplitudes in YM N c / (16 π 2 ) planar N = 4 · Symmetries and Simplifications · Infrared and Helicity ◦ All non-planar graphs are suppressed in this limit, giving rise to a Structure natural ordering of external particles Cluster Algebras and Polylogarithms ◦ This ordering can be used to define a set of dual coordinates · Polylogarithms, the Coaction, and the Lie cobracket x 3 · Cluster-Algebraic p µ i σ α ˙ α = λ α i ˜ λ ˙ α i = x α ˙ α − x α ˙ α Structure x 2 p 2 µ i i +1 p 3 Subalgebra Constructibility x 4 · Decomposing the ◦ The coordinates x µ i label the cusps of p 1 Remainder Function a light-like polygonal Wilson loop in the p 4 Conclusions dual theory, which respects a superconformal p 5 x 1 symmetry in this dual space x 5 [Alday, Maldacena; Drummond, Henn, Korchemsky, Sokatchev]

Cluster Algebras, Helicity and Infrared Structure Steinmann, and the Lie Cobracket Andrew McLeod Since we are working with all massless particles, our amplitudes A n Amplitudes in planar N = 4 must be renormalized in the infrared · Symmetries and Simplifications · Infrared and Helicity ◦ Infrared divergences are universal and entirely accounted for by Structure the ‘BDS Ansatz’ [Bern, Dixon, Smirnov] Cluster Algebras and Polylogarithms ◦ In the dual theory, the BDS Ansatz constitutes a particular · Polylogarithms, the Coaction, and the solution to an anomalous conformal Ward identity that determines Lie cobracket · Cluster-Algebraic the Wilson loop up to a function of dual conformal invariants Structure [Drummond, Henn, Korchemsky, Sokatchev] Subalgebra Constructibility · Decomposing the Remainder Function finite function of dual conformal invariants Conclusions � �� � � � + P N 2 MHV A n = A BDS 1 + P NMHV + · · · + P MHV × exp( R n ) × n n n n � �� � � �� � IR structure helicity structure

Cluster Algebras, Dual Conformal Invariants Steinmann, and the Lie Cobracket Andrew McLeod ◦ We can construct dual conformally invariant cross ratios out of Amplitudes in combinations of Mandelstam invariants planar N = 4 ij = ( x i − x j ) 2 = ( p i + p i +1 + · · · + p j − 1 ) 2 x 2 · Symmetries and Simplifications · Infrared and Helicity Structure that remain invariant under the dual inversion generator Cluster Algebras and Polylogarithms x 2 ) = x α ˙ α I ( x α ˙ α i I ( x 2 ij ⇒ ij ) = · Polylogarithms, the i x 2 x 2 i x 2 Coaction, and the Lie cobracket i j · Cluster-Algebraic Structure ◦ These can first be constructed for n = 6 since x 2 i,i +1 = p 2 i = 0 Subalgebra Constructibility x 3 x 2 · Decomposing the Remainder Function Conclusions u = x 2 13 x 2 v = x 2 24 x 2 w = x 2 35 x 2 46 51 62 , , x 4 x 1 x 2 14 x 2 x 2 25 x 2 x 2 36 x 2 36 41 52 x 5 x 6 ◦ In general, we can form 3 n − 15 independent ratios

Loops and Legs in Planar N = 4 Cluster Algebras, Steinmann, and the Lie Cobracket Andrew McLeod ∞ . . Amplitudes in . planar N = 4 8 · Symmetries and Simplifications 7 MHV Legs · Infrared and Helicity Structure 6 NMHV Cluster Algebras 5 Ω and Polylogarithms 4 · Polylogarithms, the Coaction, and the Lie cobracket 1 2 3 4 5 6 . . . ∞ · Cluster-Algebraic Structure Loops Subalgebra Constructibility [Bern, Caron-Huot, Dixon, Drummond, Duhr, Foster, G¨ urdo˘ gan, He, Henn, von Hippel, Golden, Kosower, AJM, Papathanasiou, Pennington, Roiban, Smirnov, Spradlin, Vergu, Volovich, . . . ] · Decomposing the Remainder Function ◦ Unexpected and striking structure exists in the the direction of Conclusions both higher loops and legs • Galois Coaction Principle • Cluster-Algebraic Structure ◦ This talk will focus on using these polylogarithmic amplitudes (especially the two-loop MHV ones) as a data mine

Cluster Algebras, Polylogarithms Steinmann, and the Lie Cobracket Andrew McLeod ◦ Loop-level contributions to MHV (and NMHV) amplitudes are Amplitudes in expected to be multiple polylogarithms of uniform transcendental planar N = 4 · Symmetries and weight 2 L , meaning that the derivatives of these functions satisfy Simplifications · Infrared and Helicity � Structure F s i d log s i dF = Cluster Algebras i and Polylogarithms for some set of ‘symbol letters’ { s i } , where F s i is a multiple · Polylogarithms, the Coaction, and the Lie cobracket polylogarithm of weight 2 L − 1 · Cluster-Algebraic Structure [Arkani-Hamed, Bourjaily, Cachazo, Goncharov, Postnikov, Trnka] Subalgebra ◦ The symbol letters { s i } can in general be algebraic functions of Constructibility dual conformal invariants · Decomposing the Remainder Function ◦ Examples of such functions (and their special values) include Conclusions log( z ) , iπ , Li m ( z ) , and ζ m . The classical polylogarithms Li m ( z ) involve only the symbol letters { z, 1 − z } � z Li m − 1 ( t ) Li 1 ( z ) = − log(1 − z ) , Li m ( z ) = dt t 0

Cluster Algebras, The Coaction Steinmann, and the Lie Cobracket Andrew McLeod Amplitudes in ◦ Multiple polylogarithms are endowed with a coaction that maps planar N = 4 functions to a tensor space of lower-weight functions [Goncharov; Brown] · Symmetries and Simplifications · Infrared and Helicity � ∆ Structure H p ⊗ H dr H w − → q Cluster Algebras p + q = w and Polylogarithms · Polylogarithms, the ◦ If we iterate this map w − 1 times we will arrive at a function’s Coaction, and the Lie cobracket ‘symbol’, in terms of which all identities reduce to familiar · Cluster-Algebraic Structure logarithmic identities Subalgebra ◦ The location of branch cuts is determined by the ∆ 1 ,w − 1 Constructibility · Decomposing the component of the coproduct, up to terms involving Remainder Function transcendental constants Conclusions ◦ The derivatives of a function are encoded in the ∆ w − 1 , 1 component of its coproduct ∆ 1 ,..., 1 Li m ( z ) = − log(1 − z ) ⊗ log z ⊗ · · · ⊗ log z

Cluster Algebras, Symbol Alphabets and Discontinuities Steinmann, and the Lie Cobracket Andrew McLeod The symbol exposes the discontinuity structure of polylogarithms ◦ In six-particle kinematics there are only 9 symbol letters: Amplitudes in planar N = 4 · Symmetries and S 6 = { u, v, w, 1 − u, 1 − v, 1 − w, y u , y v , y w } Simplifications · Infrared and Helicity Structure s 12 s 45 s i...k = ( p i + · · · + p k ) 2 , u = Cluster Algebras s 123 s 345 and Polylogarithms · Polylogarithms, the � (1 − u − v − w ) 2 − 4 uvw Coaction, and the y u = 1 + u − v − w − Lie cobracket � · Cluster-Algebraic (1 − u − v − w ) 2 − 4 uvw 1 + u − v − w + Structure Subalgebra ◦ Only letters whose vanishing locus coincides with internal Constructibility propagators going on shell can appear in the first symbol entry · Decomposing the Remainder Function ◦ In seven-particle kinematics there are 42 analogous symbol letters, Conclusions 14 of which are parity odd ◦ For more than seven particles, symbol alphabets not as well understood • algebraic roots appear in symbol letters even at one loop in N 2 MHV amplitudes [Prlina, Spradlin, Stankowicz, Stanojevic, Volovich]