New Approaches to Scattering Amplitudes

From On-Shell Diagrams to the Amplituhedron

Based  on:  arXiv:1207.0807,      S.  Franco  arXiv:1408.3410,      S.  Franco,  D.  Galloni,  A.  Mario6,  J.  Trnka  arXiv:1502.02034,  S.  Franco,  D.  Galloni,  B.  Penante,  C.  Wen  

Sebastián Franco

The City College of New York

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Outline

�  Introduction and Motivation

�  On-shell diagrams

�  From on-shell diagrams to the Grassmannian

v  Canonical variables

v  Towards a combinatorial classification of general on-shell diagrams

v  Non-planar boundary measurement

�  The Amplituhedron

�  Conclusions

�  A systematic approach to non-planar on-shell diagrams

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Introduction and Motivation

�  The standard approach to it is based on Feynman diagrams, which make locality and unitarity manifest

�  Our description of the microscopic world is based on Quantum Field Theory

�  Not all symmetries are manifest in the Feynman Diagram approach ⇒ inefficient

�  For concreteness, let us focus on N=4 SYM:

Ø  Expectation: some of the lessons can be extrapolated to “more standard” QFTs

Ø  Very symmetric example of a 4d interacting QFT

Towards a New Formulation of QFT

�  Goals: v  Study a new formulation of QFT that makes all such symmetries manifest

v  Initiate a systematic investigation of the non-planar case, developing the technology that is necessary for doing so

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�  The new formulation is based on on-shell diagrams Arkani-­‐Hamed,  Bourjaily,  Cachazo,  Goncharov,  Postnikov,  Trnka  

�  The elementary building blocks are the 3-point MHV and MHV super-amplitudes

�  In the planar limit, scattering amplitudes can be constructed in terms of planar on-shell diagrams

On-Shell Diagrams

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Which can be glued using on-shell lines to give rise to more complicated diagrams

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Points in G(k,n):

Ø  Rows: n-dimensional vectors spanning the planes

n = # scattered particles k = # negative helicity

The  Grassmannian  G(k,  n)          :    space of k-dimensional planes in n dimensions

On-Shell Diagrams and the Grassmannian

C = ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

n

k Ø  Up to a GL(k) redundancy

1) Pick a perfect orientation:

Franco,  Galloni,  MarioM  (multiple boundaries) Franco,  Galloni,  Penante,  Wen  (arbitrary diagrams)

�  On-shell diagrams provide a powerful parametrization of the Grassmanian Postnikov  (disk) Gekhtman,  Shapiro,  Vainshtein  (annulus)

From On-Shell Diagrams to G(k,n)

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2) The entries in the matrix C are given by the boundary measurement:

1 2 3 4 5 6 7 8 1 1 C12 C13 0 C15 0 0 C18

4 0 C42 C43 1 C45 0 0 C48

6 0 C62 C63 0 C65 1 0 C68 7 0 C72 C73 0 C75 0 1 C78

C = Example:  

Plücker coordinates = k × k minors

�  Every on-shell diagram is associated to a differential form:

The On-Shell Form

on-shell form Ω

Arkani-­‐Hamed,  Bourjaily,  Cachazo,  Goncharov,  Postnikov,  Trnka  

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�  There is an infinite number of on-shell diagrams

Moves and Reductions

�  A  pressing  quesVon: find global diagnostics for equivalence and reductions that apply to non-planar graphs

Equivalence

Bubble Reduction

Merger Square  move  (every diagram can be made bipartite)

�  They can be organized into equivalence classes and connected by reductions

­  More generally, deleting and edge while covering the same region of the Grassmannian

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�  It is possible to introduce GL(1)-invariant generalized face variables

Generalized face variables

�  Automatically dlog form:

Embedding the Diagram

�  It is useful to consider the embedding of the diagram into a bordered Riemann surface

For F faces, B boundaries and genus g:

v  Faces: they are subject to ∏ f i =1

v  Cuts: B-1 paths connecting pairs of boundaries

v  Fundamental  cycles:  αm and βm pairs, m=1,…,g

d = #edges - #nodes = F + B + 2g - 2 = F – χembed.

Volume Form

Degrees of Freedom

Franco,  Galloni,  Penante,  Wen

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v  Plücker matching: when two on-shell diagrams give rise to the same sets of Plücker coordinates and relations

A General Classification of On-Shell Diagrams

v  Reduction: an on-shell diagram B is a reduction of an on-shell diagram A, if it is obtained from A by deleting edges and it covers the same region of the Grassmannian

v  Reduced graph: it is impossible to remove edges while covering the same region of the Grassmannian

�  How do we organize the infinite number of on-shell diagrams? It is desirable to develop a classification of generic on-shell diagrams

Franco,  Galloni,  Penante,  Wen  (arbitrary) Arkani-­‐Hamed,  Bourjaily,  Cachazo,  Goncharov,  Postnikov,  Trnka  (planar)

the only thing we care about for leading singularities

�  For planar diagrams, it can be combinatorially phrased in terms of the properties of zig-zag paths and permutations

General Definitions

not enough for non-planar diagrams

Ø  Two necessary conditions for equivalence are Plücker matching and having the same number of degrees of freedom

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A Useful Tool: Perfect Matchings

�  Perfect matchings play a central role in connecting bipartite on-shell diagrams to combinatorics and geometry

�  (Almost)  Perfect  Matching: p is a subset of the edges in the graph such that:

Ø  Every internal node is the endpoint of exactly on edge in p

Ø  Every external point belongs to either one or zero edges in p

�  Finding the perfect matchings reduces to calculating the determinant of an adjacency matrix of the graph (Kasteleyn matrix), and some generalizations Franco  

p1   p2   p3   p4  

p5   p6   p7  

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Polytopes from On-Shell Diagrams Franco,  Galloni,  Penante,  Wen

1. Generalized Matching Polytope

Piµ = 1 if Xi ∈ pµ

0 if Xi ∈ pµ

Xi = ∏ pµ P iµ

µ

�  Let us define the (#edges × #perfect matchings)-dimensional matrix P:

�  The following map between edge weights and perfect matchings automatically satisfies all relations between edge weights:

Xi : edge pµ : perfect matching

�  Generalized  Matching  Polytope:   there is a distinct point in the polytope for every perfect matching pµ, with a position vector given by the corresponding column in the matrix P

�  We can associate two types of polytopes to every on-shell diagram

Useful for?

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¤  4 degrees of freedom ¤  7 perfect matchings 7 points

4d polytope

A  Simple  Example:  the  Top-­‐Cell  of  G(2,4)  

Linear relations between positions of points  

Relations between edge weights  

�  The dimensionality of the generalized matching polytope is equal to the number of degrees of freedom in the diagram

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�  Projection of the matching polytope that only preserves information on the external legs

Perfect matchings which coincide on all

external legs

Plücker coordinate

Generalized Matroid Polytope

�  Multiple perfect matchings can be projected to the same point in this polytope

Useful for?

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Example:  the  Top-­‐Cell  of  G(2,4)  

�  Every point in the generalized matching polytope corresponds to a Plücker coordinate

�  All relations among them are encoded in the positions of the points

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Combinatorial Classification

v  Plücker matching: two on-shell diagrams give rise to the same sets of Plücker coordinates and relations if their generalized matroid polytopes coincide

v  Reduction: an on-shell diagram B is a reduction of an on-shell diagram A, if it is obtained from A by deleting edges and it has the same generalized matroid polytope of A

v  Reduced graph: it is impossible to remove edges while preserving the generalized matroid polytope

�  The classification of general on-shell diagrams we introduced above has a powerful implementation in terms of generalized matching and matroid polytopes

Franco,  Galloni,  Penante,  Wen

�  It is interesting to compare this global approach with one for planar graphs based on permutations. For example, permutations generically change under reduction

�  A novel feature of non-planar diagrams: new constraints between Plucker coordinates beyond Plucker relations can arise Systematically captured by the polytopes

Franco,  Galloni,  Penante,  Wen Arkani-­‐Hamed,  Bourjaily,  Cachazo,  Postnikov,  Trnka

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Genus-1 Examples

�  None of these diagrams admits a genus-0 embedding

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�  Both diagrams have the same number of degrees of freedom

1 0 0 0 0 0 0 1 1 0 1 1 0 1 1 1 1 0 0 1 1 2 0 1 0 1 0 1 0 1 0 1 0 0 0 1 0 1 0 1 1 1 3 0 1 0 1 1 0 0 1 1 0 0 0 0 1 1 0 1 1 0 1 4 0 0 1 1 1 0 0 0 0 0 1 0 1 1 1 0 1 0 1 1 5 0 0 1 1 0 1 0 0 0 0 1 1 0 1 0 1 1 1 1 0 6 0 0 0 0 0 0 1 1 1 0 1 1 0 1 1 1 1 1 0 0 6 2 2 1 2 2 2 1 2 2 1 2 2 1 1 1 1 1 1 1

G1 =

1 0 0 0 0 0 0 1 1 0 1 1 0 1 1 1 1 0 0 1 1 2 0 1 0 1 0 1 0 1 0 1 0 0 0 1 0 1 0 1 1 1 3 0 1 0 1 1 0 0 1 1 0 0 0 0 1 1 0 1 1 0 1 4 0 0 1 1 1 0 0 0 0 0 1 0 1 1 1 0 1 0 1 1 5 0 0 1 1 0 1 0 0 0 0 1 1 0 1 0 1 1 1 1 0 6 0 0 0 0 0 0 1 1 1 0 1 1 0 1 1 1 1 1 0 0 5 3 3 2 3 3 1 1 2 2 1 2 2 2 1 1 2 2 2 2

G2 =

Only the multiplicities change

Generalized Matroid Polytopes

(1)   (2)  

in fact, they are equivalent

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The Non-Planar Boundary Measurement

�  Order external nodes using boundaries and cuts (for B>1)

Ø  Consistency with the combinatorial classification

�  ObjecVves:  

Ø  The general expression must reduce to the one for planar graphs (positivity) Postnikov

Higher genus do this within unit cell

A Proposal

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Franco,  Galloni,  Penante,  Wen

�  Close flows into loops by using boundaries and cuts

4 5 1.  Overall sign for Cij entry: (-1)

n ij

nij = number of sources between i and j

2.  Additional sign for each contribution: (-1)r+1 r = rotation number = clock - counterclock

Plücker coordinates are sums of contributions from all perfect matchings at the corresponding point in the generalized matroid polytope

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An Example

�  A B=2, g=1 example:

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Beyond Plücker Relations

�  Non-planar diagrams can exhibit a new type of pole, at which no Plücker coordinate vanishes

�  However, it is possible to remove an edge while preserving the matroid polytope (i.e. the non-vanishing Plücker coordinates) a relation is generated

Franco,  Galloni,  Penante,  Wen Arkani-­‐Hamed,  Bourjaily,  Cachazo,  Postnikov,  Trnka

�  The relations are systematically accounted for by the matching and matroid polytopes, when expressing Plücker coordinates in terms of perfect matchings

�  The dimension of the graph is equal to 9, i.e. equal to the one of G(3,6)

Combinatorial Manifestation

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A  G(3,6)  example:  

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�  The amplituhedron is a new algebraic geometric construction that maximally geometrizes scattering amplitudes in planar N=4 SYM in terms of a single object:

The Amplituhedron

v  Amplitude = Volume

v  Different representations of the amplitude from different “triangulations”

Arkani-­‐Hamed,  Trnka  

Beyond the Positive Grassmannian

L(1)

⋮ L(L)

Y

D(1)

⋮ D(L)

C

= · Z Loops

Tree-level

Amplituhedron

Z ∈ M+(4+k,n) C ∈ G+(k,n) D(i) ∈ G(2,n)

k = 0 : MHV

(Z) D(i)

C

D(i)

D(j)

C (C) , , , ,

⋯

�  Extended positivity:

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�  Ideas that are similar to the ones presented in this talk can used for stratifying the amplituhedron

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Anatomy of the Amplituhedron

�  Precisely these 1232 boundaries can alternatively be determined from the integrand of the scattering amplitude, providing an exquisite test of the amplituhedron/scattering amplitude correspondence

1 9 44 140 274 330 264 136 34

Example:  2 loops, 4 particles, k=0 χ = 2

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Conclusions

�  We initiated a systematic study of non-planar on-shell diagrams in N=4 SYM

Ø  Canonical variables

Ø  Combinatorial classification (i.e. region matching and reductions) in terms of generalized matching and matroid polytopes

�  We carried out an extensive study of the geometry of the amplituhedron

Ø  Boundary measurement for completely general on-shell diagrams

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�  We developed powerful technology to do so, including:

Ø  We stratified the 4-particle MHV amplituhedra for L=2 and 3, finding remarkably simple topologies and exact agreement with the singularity structure of the amplitude. We also developed a combinatorial implementation of it.

Ø  We introduced and studied a class of deformations of the amplituhedron. For 4-particle MHV, which have χ=2 at least up to L=4

Ø  It might be possible to exploit these simple deformed geometries to constraint the integrand

The Future Se

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�  There seems to exist an intriguing relation, yet to be explored, with recently studied deformations of the Grassmannian

On-Shell Diagrams and Quivers

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�  Interesting connection between on-shell diagrams and a general class of quiver gauge theories, known as bipartite field theories (BFTs)

Ferro,  Lukowski,  Meneghelli,  Ple]a,  Staudacher Beisert,  Broedel,  Rosso

Franco    Yamazaki,  Xie  

�  Classify non-equivalent reduced diagrams for top-dimensional cells of G(k,n). How can they be generated? Bourjaily,  Franco,  Galloni,  Wen  (in  progress)    

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