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On-shell diagrams in SYM beyond the planar limit
Brenda PenanteQueen Mary University of LondonHumboldt University of Berlin 12/01/16
Total Positivity: a bridge between Representation Theory and Physics
- Franco, Galloni, BP, Wen hep-th/1502.02034 based on:
- University of Kent -
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Overview
Scattering amplitudes in planar N=4 SYM admit adescription in terms of the positive Grassmannian.
Loop leading singularities are residuesof a Grassmannian integral.
All-loop integrand determined via theBCFW recursion relation.Britto, Cachazo, Feng, Witten / Arkani-Hamed, Bourjaily, Cachazo, Caron-Huot, Trnka
Arkani-Hamed, Cachazo, Cheung, Kaplan / Mason, Skinner
... but how to go beyond planar N=4 SYM?
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No ordering = No positivity
fixed , with Planar limit:
( Finite N corrections multiple traces )
Partial amplitude(colour ordered)
gauge group
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Tree-level amplitudes enjoy Yangian symmetry
[[ Yangian = Superconformal + Dual Superconformal ]]Drummond, Henn, Plefka
Drummond, Henn, Korchemsky, Sokatchev
Loop level:Yangian symmetry broken due to IR divergencesLoop integrand
Rational function ofexternal and loop momenta
Planar loop integrand
dummy variables, but must be defined consistently among various terms
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Planar loop integrandAmbiguities:
Planar loop integrand well defined: dual variables
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State of the art in the planar limitAll-loop integrand determined by the all-loop recursion relation
Arkani-Hamed, Bourjaily, Cachazo, Caron-Huot, Trnka
Dual variables allow different terms in recursion relationto be combined in a non-ambiguous way
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State of the art in the planar limit
Unavailable for non-planar integrands
??Non-planar integrand not well-defined
Can still study non planar Leading SingularitiesEden, Landshoff, Olive, Polkinghorne / Britto, Cachazo, Feng
All-loop integrand determined by the all-loop recursion relationArkani-Hamed, Bourjaily, Cachazo, Caron-Huot, Trnka
Dual variables allow different terms in recursion relationto be combined in a non-ambiguous way
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"Cut" propagators:
Compute residue of the integrand Leading singularities: Maximal number of propagators cut (4xL)Ex: 1-loop
solutions
cut 4 propagators
Leading Singularities[[ Eden, Landshoff, Olive, Polkinghorne / Britto, Cachazo, Feng ]]
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Planar
In the planar limit basis ofdual conformal integrands with "unit leading singularity"
LS are sufficient to determine the all-loop integrand!
All planar LS are residues of a(positive) Grassmannian integral
Positive Grassmannian parametrised by planar on-shell diagrams
[[ Arkani-Hamed, Bourjaily, Cachazo, Trnka - 2010 ]]
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Non-planar integrand not well defined
Planar Non-Planar
Consider non-planar LS
Residues of a Grassmannian integral
Parametrised by non-planar on-shell diagrams
In the planar limit basis ofdual conformal integrands with "unit leading singularity"
LS are sufficient to determine the all-loop integrand!
All planar LS are residues of a(positive) Grassmannian integral
Positive Grassmannian parametrised by planar on-shell diagrams
[[ Arkani-Hamed, Bourjaily, Cachazo, Trnka - 2010 ]]
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Motivation
Grassmannian formulation is linked toon-shell diagrams
Region of the Grasmmannian + dlog on-shell form
Arkani-Hamed, Bourjaily, Cachazo, Goncharov, Postnikov, Trnka
[[ Arkani-Hamed, Bourjaily, Cachazo, Trnka ]]Conjecture: Non-planar ampshave only log singularities and no poles at infinity.
Non-planar on-shell diagrams are the natural objects to study
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Outline1. Grassmannian formulation for amplitudes
2. On-shell diagrams
3. Conclusions + more possible applications
Review of planarNon-planar corrections
Review of planarGeneralised face variablesGeneral boundary measurementA new type of singularityEquivalence, reductions, and polytopes
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NotationSU(N) colour generators
Polarisation vectorsKinematics
Spinor variables
Partial amplitude:
(kinematics and polarisation)
Auxiliary fermionic variables(helicity)
(colour ordered)
Spinor products
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Grassmannian Formulation
[[ Arkani-Hamed, Cachazo, Cheung, Kaplan - 2009 ]][[ Mason, Skinner - 2009 ]]
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and a fermionic 4-plane :
The geometry of momentum conservation
Organise them in the , 2-planes in :
External data: for each particle
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The geometry of momentum conservation
(Super) momentum conservation:
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DEF: Grassmannian is the space of -planes in
Element of : choose -vectors:
gauge redundancy
Grassmannian formulation[[ Arkani-Hamed, Cachazo, Cheung, Kaplan - 2009 ]]
Scattering of gluons For us:: total number of gluons: number of gluons
Recall:
k=2k=n-2
MHVMHV
k=0, 1, n-1, n Amp = 0
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Grassmannian formulation[[ Arkani-Hamed, Cachazo, Cheung, Kaplan - 2009 ]]
Coordinates in Maximal minors (Plücker coords.)
Positive Grassmannian
Plücker relations: Ex:
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Grassmannian formulation[[ Arkani-Hamed, Cachazo, Cheung, Kaplan - 2009 ]]
Planar LS are residues of the following integral over
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Gauge fix entries of
Grassmannian formulation[[ Arkani-Hamed, Cachazo, Cheung, Kaplan - 2009 ]]
Planar LS are residues of the following integral over
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Gauge fix entries of
Ensure
Grassmannian formulation[[ Arkani-Hamed, Cachazo, Cheung, Kaplan - 2009 ]]
Planar LS are residues of the following integral over
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Gauge fix entries of
Ensure
consecutive minors of
Ex:
Grassmannian formulation[[ Arkani-Hamed, Cachazo, Cheung, Kaplan - 2009 ]]
Planar LS are residues of the following integral over
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Non-zero only if the plane is orthogonal to , and contains
Momentum conservation
Note:For impossible to have For impossible to have
Amplitudes automatically zero!
Grassmannian formulation[[ Arkani-Hamed, Cachazo, Cheung, Kaplan - 2009 ]]
Planar LS are residues of the following integral over
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Grassmannian formulation[[ Arkani-Hamed, Cachazo, Cheung, Kaplan - 2009 ]]
Non-planarPoles when consecutive minors vanish
invariance: cross ratio of minors
Ex: k=3
No notion of ordering or positivity in non-planar case
[[ Galloni, Franco, BP, Wen - 2015 ]]
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On-shelldiagrams
[[ Arkani-Hamed, Bourjaily, Cachazo, Goncharov, Postnikov, Trnka - 2012 ]]
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On-shell formulation[[ Arkani-Hamed, Bourjaily, Cachazo, Goncharov, Postnikov, Trnka - 2012 ]]
Bi-coloured graphs made of the building blocks:
Edges:
Nodes
on-shell momentum
MHV amplitude
MHV amplitude
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Constructing on-shell diagramsTo connect two nodes, integrate over on-shell phase spaceof edge in common:
Little group
Can construct more complicated diagramsEach node has two degrees of freedom
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Constructing on-shell diagramsExamples:
13
2
4 5
13
54
2
1
2
5
43
4
1 2
3
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Constructing on-shell diagramsExamples:
Planar:Can be embedded on a disk
13
2
4 5
13
54
2
1
2
5
43
4
1 2
3
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Constructing on-shell diagramsExamples:
Planar:Can be embedded on a disk
Non planar:Can be embedded on a surface withmultiple boundaries/ higher genus13
54
2
1
2
5
43
13
2
4 5
4
1 2
3
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Equivalence MovesOn-shell diagrams are equivalent if they are related by thefollowing moves:Merger:
Square move:
Note that every on-shell diagram can be made bipartiteNote that every on-shell diagram can be made bipartite
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Fusing Grassmannians
On-shell diagram
Boundary measurement
Ex: 4
12
3
An on-shell diagram with black nodes, white nodesand internal edges is associated to , where:
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Bipartite technology
Perfect matching Choice of edges such that every internal node is the endpoint of only one edge
Perfect orientation
4
12
34
12
34
12
3
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Bipartite technology
Perfect matching Choice of edges such that every internal node is the endpoint of only one edge
Perfect orientation
4
12
34
12
34
12
3
Sources
Sinks
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4
12
3 4
12
34
12
34
12
34
12
34
12
3 4
12
3
Example4
12
3
Perfect matchings:
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4
12
3 4
12
34
12
34
12
34
12
34
12
3 4
12
3
Oriented perfect matchings:
Example4
12
3
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4
12
3 4
12
34
12
34
12
34
12
34
12
3 4
12
3
Oriented perfect matchings:
Example4
12
3
Flows:4
12
3 4
12
34
12
34
12
34
12
34
12
3 4
12
3
Reference
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Boundary measurement
4
12
3 4
12
34
12
34
12
34
12
34
12
3 4
12
3
Flows:
Map between on-shell diagram and element of the Grassmannian
Reference
Flows from to 41 2 31
2 Sign prescription
4
12
3
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Boundary measurement
Plücker coordinates are positive in planar case and are a sum of flows with corresponding source set.
Ex:
41 2 3
1
2
4
12
3 4
12
34
12
34
12
34
12
34
12
3 4
12
3
Sign prescription
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ImportantOn-shell diagrams parametrise regions of the Grassmannian
4
12
3
Equivalent diagrams parametrise the same region:
4
12
3
Graph is reducible if possible to delete edges while preserving region.4
12
3
Otherwise graph is reduced.This changes for non-planar graphs!Planar: set of non-zero minors preserved
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Parametrising on-shell diagramsPlanar:
# degrees of freedom of a planar on-shell diagram is
# faces
4
12
3
Face variables: ?General for non-planar
4
12
3
Edge variables:
On-shell dlog form: variables unfixed by delta-functions mapped to loop integration variables.
Bases for expressing flows: Edges and Faces
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Generalised face variables
Faces
Paths connecting different boundaries
Fundamental cycles
1
3
2
4
6
5
1
3
2
4
6
5
1
3
2
4
6
5
α β b
Ex: Genus 1
F = # facesB = # boundariesg = genus
[[ Galloni, Franco, BP, Wen - 2015 ]]
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Faces
Paths connecting different boundaries
Fundamental cycles
dlog on-shell form:
F = # facesB = # boundariesg = genus
Generalised face variables[[ Galloni, Franco, BP, Wen - 2015 ]]
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Some properties of planar graphs
If it is impossible to remove an edge of a graph without sending some Plücker coord to zero, the graph is reduced.
(Positroid stratification of )
[[ Arkani-Hamed, Bourjaily, Cachazo, Goncharov, Postnikov, Trnka - 2012 ]]
Boundaries of the regions in the positive Grassmannian are always
Recall:
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A non-planar novelty:
2
3
4
56
1
It is possible to remove an edge of a reduced graph withoutsending any Plücker coord to zero!
Recall: Deformation from planar Grassmannian integrand
Non-planar novelties[[ Arkani-Hamed, Bourjaily, Cachazo, Postnikov, Trnka - 2014, Galloni, Franco, BP, Wen - 2015 ]]
Method for determining : generalisation of [[ Arkani-Hamed, Bourjaily, Cachazo, Postnikov, Trnka - 2014 ]]
from k=2 leading singularities to higher kOBS:
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Removal of an edge does not set anyto zero, but gives rise to the relation
2
3
4
56
1
A non-planar novelty:It is possible to remove an edge of a reduced graph withoutsending any Plücker coord to zero!
Recall: Deformation from planar Grassmannian integrand
Reducibility & Equivalence: Non-planar[[ Arkani-Hamed, Bourjaily, Cachazo, Postnikov, Trnka - 2014, Galloni, Franco, BP, Wen - 2015 ]]
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2
3
4
56
1
A non-planar novelty:It is possible to remove an edge of a reduced graph withoutsending any Plücker coord to zero!
Recall: Deformation from planar Grassmannian integrand
Reducibility & Equivalence: Non-planar[[ Arkani-Hamed, Bourjaily, Cachazo, Postnikov, Trnka - 2014, Galloni, Franco, BP, Wen - 2015 ]]
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Polytopes
Sum of flows with source set with coefficients
[[ Gekhtman, Shapiro, Vainshtein - 2013 ]] Annulus[[ Franco, Galloni, Mariotti - 2013 ]] Arbitrary B, genus zero[[ Franco, Galloni, BP, Wen - 2015 ]] Any graph
Point in matching polytope
Point in matroid polytope
Notions of equivalence/reduction can be rephrased in terms of polytopes:
Plücker coord.
[[ Postnikov, Speyer, Williams - 2008, Franco, Galloni, Mariotti - 2013 ]]
Non-vanishing minors and relations can be seen from the polytopes
on-shell diagram
Perfect matching
Perfect matchings with same source set
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Characterisation of on-shell diagrams
Equivalent graphs have the same matroid polytope
A graph is reduced if it is impossible to remove edges while preserving the matroid polytope.
Cell of the Grassmannian characterised by a zig zag path
Non-planar without extra constraintson Plücker coordinates
planar / reduced
Cell of the Grassmannian characterised by matroid polytope
Reduction of a diagram removes 1 dof while preserving the matroid polytope
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Constraints and polytopes: example[[ Galloni, Franco, BP, Wen - 2015 ]]
2
3
4
56
1
Before removal: 40 perfect matchings
Before and after removal: After removal disappears
After removal: 33 perfect matchings
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Concluding remarks & Outlook1) Physical interpretation:
All tree level amplitudes and loop integrandsvia BCFW recursion relation.
Planar:
dlog form of the loop integrand:
Arkani-Hamed, Bourjaily, Cachazo, Goncharov, Postnikov, Trnka
Britto, Cachazo, Feng, Witten / Arkani-Hamed, Bourjaily, Cachazo, Caron-Huot, Trnka
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Concluding remarks & Outlook
Leading singularities of the loop integrandNon planar:Non-planar loop integrand?
[[ Arkani-Hamed, Bourjaily, Cachazo, Trnka - 2014 ]]Conjecture: Non-planar ampshave only log singularities and no poles at infinity.
Non-planar Grassmannian formulation?
1) Physical interpretation:All tree level amplitudes and loop integrandsvia BCFW recursion relation.
Planar:
dlog form of the loop integrand:
Arkani-Hamed, Bourjaily, Cachazo, Goncharov, Postnikov, Trnka
Britto, Cachazo, Feng, Witten / Arkani-Hamed, Bourjaily, Cachazo, Caron-Huot, Trnka
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Concluding remarks & Outlook
2) Non-planar diagrams parametrise regions of withhidden relations between Plücker coordinates.
? Method for finding representative graph given a constraint
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Concluding remarks & Outlook
2) Non-planar diagrams parametrise regions of withhidden relations between Plücker coordinates.
? Method for finding representative graph given a constraint
3) MHV non-planar leading singularities are sums of planar ones.
Same not true for Non-MHV, however similar method can beused to find the deformation of the integrand .
[[ Arkani-Hamed, Bourjaily, Cachazo, Postnikov, Trnka - 2014 ]]
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Concluding remarks & Outlook
2) Non-planar diagrams parametrise regions of withhidden relations between Plücker coordinates.
? Method for finding representative graph given a constraint
3) MHV non-planar leading singularities are sums of planar ones.
Same not true for Non-MHV, however similar method can beused to find the deformation of the integrand .
[[ Arkani-Hamed, Bourjaily, Cachazo, Postnikov, Trnka - 2014 ]]
4) Positive Grassmannian Non-planar generalisation?
Amplituhedron[[ Arkani-Hamed, Trnka - 2013 ]]
[[ Evidence: Bern, Herrmann, Litsey, Stankowicz, Trnka - 2015 ]]