Cluster Algebra Structures in

Scattering AmplitudesAnastasia VolovichBrown University

Miami, December 2014

Golden, Goncharov, Paulos, Spradlin, Vergu

Scattering Amplitudes

• The bread and butter of quantum fields theories, such as QCD, both theoretically and experimentally are scattering amplitudes.

• The last few years have seen a lot of progress in our understanding of the structure of scattering amplitudes and in our ability to do computations both for theoretical and phenomenological purposes.

• Remarkable results range from precision predictions in QCD which are important for understanding the LHC data to the discovery of new symmetries in gauge and gravity theories.

Amplitudeology Scattering amplitudes play important role in many interrelated subjects.

They bring together the community of Amplitudeologists. Amplitudes 2009, Durham; Amplitudes 2010, London; Amplitudes 2011, Ann Arbor; Amplitudes 2012 Hamburg; Amplitudes 2013 Tegernsee; Amplitudes 2014 Paris; Amplitudes 2015 Zurich

The goals of this program

• to explore the rich mathematical structures of scattering amplitudes

• to exploit these structures as much as possible to compute amplitudes

Many have contributed to recent developments including:

Alday, Arkani-Hamed, Basso, Badger, Bargheer, Beisert, Belitski, Bern, Berkovits, Boels, Bourjaily, Brandhuber, Brodel, Bjerrum-Bohr, Bullimore, Britto, Cachazo, Caron-Huot, Carrasco, Cheung, Damgaard, Dennen, Del Duca, Duhr, Dixon, Dolan, Duhr, Drummond, Eden, Ellis, Elvang, Fend, Ferarra, Ferro, Forde, Forini, Franko, Freedman, Gangl, Gaiotto, Goddard, Goncharov, Green, He, Henn, Heslop, Hodges, Huang, Huber, Ita, Johnannson, Kallosh, Kaplan, Khoze, Kiermaier, Kraimer, Kristjansen, Korchemsky, Kosower, Kuntz, Lipatov, Loebbert, Melnikov, Maitre, Mafra, Maldacena, Mason, Monteiro, Naculich, O’Connell, Nastase, Papathanasiou, Plefka, Prygarin, Paulos, Roiban, Rosso, Sabio Vera, Schnizer, Schloterer, Schwab, Sever, Skinner, Smirnov, Sokatchev, Spence, Spradlin, Staudacher, Stelle, Stieberger, Svrcek, Taylor, Travaglini, Trnka, Tye, Vanhove, Vergu, Vieira, Yang, Wen, Weinzel, Witten, Zanderighi

Goal of My Talk• explore cluster algebra structure of scattering

amplitudes in planar N=4 Yang-Mills • explain how to use it to compute 2 and 3 loop

amplitudes

Planar 2-loop 6-point MHV N=4 SYM

Arkani-Hamed: N=4 YM is 21st century harmonic oscillator

[Bern, Dixon, Kosower, Roiban, Spradlin, Vergu, AV]

Goncharov, Spradlin, Vergu, AV

Is there a structure? How to generalize this formula?Why do these particular arguments appear?Why only classical polylog functions?

Cluster algebraic structure is the key

Cluster Algebras

• Cluster algebras were first discovered and developed by Fomin and Zelevinski (2002).

• Very informally: commutative algebras constructed from distinguished generators (cluster variables) grouped into disjoint sets of constant cardinality (clusters) which are constructed recursively from the initial cluster by mutations.

• Cluster algebra portal: http://www.math.lsa.umich.edu/~fomin/cluster.html

Cluster Algebra • Consider a collection of variables subject to

the exchange relation

• Seed exchange relation with initial cluster

• -- cluster coordinates forming cluster algebra• -- mutation

represent using quivers

Quivers and Mutations

• We can define cluster algebra by a quiver:oriented graph without loops and 2-cycles.• Given a quiver, get a new one by mutation rule:

For vertex 1:

Quivers and Cluster Coordinates

We can encode a quiver by a skew-symmetric matrix

To each vertex i associate variable

Use matrix b to define mutation relation at vertex k

In practice to construct all cluster variables from a given quiver see Keller Java program

--Amplitudes are functions on --Scott (2003) classified all Grassmannian cluster algebras of finite type. 3 x (n-5) initial quiver.

Grassmanian cluster algebras

cluster algebra• Start with quiver. Generate all coordinates by

mutations. Mutation generates 14 clusters.

• 15 A-coordinates: 6 fixed <ii+1>; 9 unfixed <ij>• 15 X-coordinates:

• Note: The top 9/15 are exactly the arguments in 2-loop 6-point amplitudewith Golden, Goncharov, Spradlin, Vergu

cluster algebra

• 49 A-coordinates: 35 Plucker brackets + cyclic =14

• Mutations generate 833 clusters• Stasheff polytope: 833 V, 2499 E, 2856

F2 (1785S+ 1071P), 1547 F3 • Analyzing all quivers: 385 X-coordinates

by D. Parker

Cluster Structure in Amplitudes

• Symbols: all n-point amplitudes in SYM theory have symbol alphabet with subset of cluster A-coordinates on Gr(4,n)Cluster Bootstrap in Mark’s talk

• Coproduct: -- for two-loop MHV amplitudes, only cluster X-coordinates appear (with particular Poisson brackets) -- for higher-loop or non-MHV, not yet understood• Functions: there is a particular class of natural

``cluster polylogarithm’’ functions which exhibit these properties

2. Coproduct: 2-loop 7-point MHV

2. Coproduct: 2-loop 7-point MHV

• All and in coproduct for 2-loop 7-points amplitude are cluster X-coordinates for cluster algebra

• Out of 385 only 231 appear in the amplitude. What is the criterion?? [Note: 9/15=231/385!]• For each are in the same

cluster. Appear in pairs with zero Poisson bracket.• is a sum of 42 squares of Stasheff

polytope.with Golden, Goncharov, Spradlin, Vergu

3. Function: Cluster Polylogarithm

In order to find the corresponding function, we need to find a function whose coproduct can be expressed entirely in terms of cluster coordinates

3. Function: Cluster Polylogarithm

• To find functions, all one has to do is solve • There is a unique solution for cluster algebra

3. Function: Cluster Polylogarithm

Recall that derived from the amplitude side only has pairs with Poisson bracket zero, which is an additional constraint, and leads to a particular combination of pentagon functions.

2. Function: other Cluster Polylogarithms

All non-trivial degree 4 cluster functions for are linear combinations of functions

With D. Parker and A. Scherlis

2. Function: 2-loop 7-point amplitude

Cluster polylog functions are building blocks necessary to write down all-n function for 2-loop MHV.

with Golden, Spradlin, Paulos

Conclusion• We have advocated the study of cluster structure of

N=4 YM amplitudes.• We can use this structure for advancing computations:

2 and 3 loop MHVs.• Many questions remain: cluster structure @ higher

loops, other helicities, strong coupling….. very impressive explicit results by Dixon, Drummond, Duhr, Henn, Pennington, von Hippel & Basso, Sever, Vieira

• Connection to the integrands: cluster structure also appeared in on-shell diagrams

[Arkahi-Hamed, Bourjaily, Cachazo, Goncharov, Postnikov, Trnka] [Paulos, Schwab]