Amplitudes et périodes 3-7 December 2012
Niels Emil Jannik Bjerrum-BohrNiels Bohr International Academy,
Niels Bohr Institute
Amplitude relations in Yang-Mills theory and
Gravity
2
Introduction
3
Amplitudes in Physics
• Important concept: Classical and Quantum Mechanics
Amplitude square = probability
3
Large Hadron Collider
…
LHC ’event’
Proton
Proton
Jets
JetsJets:Reconstruction complicated..
Calculations necessary:Amplitude
4
How to compute amplitudes
Field theory: write down Lagrangian (toy model):
Quantum mechanics:
Write down Hamiltonian
Kinetic term Mass term Interaction term
E.g. QED Yukawa theory Klein-Gordon QCD Standard Model
5
Solution to Path integral -> Feynman diagrams!
6
How to compute amplitudes
Method: Permutations over all possible outcomes (tree + loops (self-interactions))
Field theory: Lagrange-function
Feature: Vertex functions, Propagator (gauge fixing)
6
7
General 1-loop amplitudes
Vertices carry factors of loop momentum
n-pt amplitude
(Passarino-Veltman) reduction
Collapse of a propagator
p = 2n for gravityp=n for YM
Propagators
8
Unitarity cuts• Unitarity methods are building on the
cut equation
Singlet Non-Singlet
9
Computation of perturbative amplitudes
Complex expressions involving e.g. (pi pj) (no manifest symmetry (pi εj) (εI ε j) or simplifications)
Sum over topological different diagrams
Generic Feynman amplitude
# Feynman diagrams: Factorial Growth!
10
Amplitudes
Simplifications
Spinor-helicity formalism
Recursion
Specifying external polarisation tensors (ε I ε j)
Loop amplitudes:(Unitarity,Supersymmetric decomposition)
Colour ordering
Tr(T1 T2 .. Tn)
Inspirationfrom
String theory
Symmetry
11
Helicity states formalismSpinor products :
Momentum parts of amplitudes:
Spin-2 polarisation tensors in terms of helicities, (squares of those of YM):
(Xu, Zhang, Chang)
Different representations of the Lorentz group
12
Scattering amplitudes in D=4Amplitudes in YM theories and gravity
theories can hence be expressed via The external helicies
e.g. : A(1+,2-,3+,4+, .. )
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MHV Amplitudes
14
Yang-Mills MHV-amplitudes(n) same helicities vanishes
Atree(1+,2+,3+,4+,..) = 0
(n-1) same helicities vanishes
Atree(1+,2+,..,j-,..) = 0
(n-2) same helicities:
Atree(1+,2+,..,j-,..,k-,..) =
1) Reflection properties: An(1,2,3,..,n) = (-1)n An(n,n-1,..,2,1)2) Dual Ward: An(1,2,..,n) + An(1,3,2,..n)+..+An(1,perm[2,..n]) = 03) Further identities as we will see….
Tree amplitudes
First non-trivial example: One single term!!
Many relations between YM amplitudes, e.g.
15
Gravity AmplitudesExpand Einstein-Hilbert Lagrangian :
Features:Infinitely many verticesHuge expressions for vertices!No manifest cancellations norsimplifications
(Sannan)
45 terms + sym
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Simplifications from Spinor-Helicity
Vanish in spinor helicity formalismGravity:
Huge simplifications
Contractions
45 terms + sym
17
String theory
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String theoryDifferent form for amplitude
Feynman
diagrams sums separat
e kinematic poles
String theory adds
channels up..
<->
xx
xx
. .
12
3
M
...+ +=
1
2
1 M 12
3
s12 s1M s123
19
Notion of color ordering
String theory
1
2
s12
Color ordered Feynman rules
xx
xx
. .
12
3
M
20
…a more efficient way
Gravity Amplitudes
21
Closed StringAmplitude
Left-movers Right-moversSum over
permutations
Phase factor (Kawai-Lewellen-Tye)
Not Left-Right
symmetric
22
Gravity Amplitudes
(Link to individual Feynman diagrams lost..)
Certain vertex relations possible
(Bern and Grant; Ananth and Theisen;
Hohm)
xx
xx
. .
12
3
M
...+ +=
1
2
1 M 12
3
s12 s1M s123
Concrete Lagrangian formulation possible?
23
Gravity AmplitudesKLT explicit representation:
’ -> 0ei -> Polynomial (sij)
No manifest crossing symmetry
Double poles x
xx
x
. .
1
23
M
...+ +=
1
2
1 M 12
3
s12 s1M s123
Sum gauge invariant
(1)
(2)(4)
(4)
(s124)
Higher point expressions quite bulky ..
Interesting remark: The KLT relations work independently of external polarisations
(Bern et al)
24
Gravity MHV amplitudes• Can be generated from KLT via YM
MHV amplitudes.
(Berends-Giele-Kuijf) recursion formula
Anti holomorphic Contributions
– feature in gravity
25
New relationsfor Yang-Mills
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New relations for amplitudes
• NewKinematic structure in Yang-Mills: (Bern, Carrasco, Johansson)
Relations between amplitudes
Kinematic analogue – not unique ??
n-pt
4pt vertex??
27
New relations for amplitudes
(n-3)!
5 points
Nice new way to do gravity
Double-copy gravity from YM!
(Bern, Carrasco, Johansson;Bern, Dennen, Huang,
Kiermeier)
Basis where 3 legs are fixed
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Monodromy
29 29
xx
xx
. .
1 3
M
...+ +=
1
2
1 M 12
3
s12 s1M s123
2
String theory
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Monodromy relations
31
Monodromy relations
FT limit-> 0
(NEJBB, Damgaard, Vanhove; Stieberger)
New relations (Bern, Carrasco, Johansson)
KK relations
BCJ relations
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Monodromy relations
Monodromy related
(Kleiss – Kuijf) relations
(n-2)! functions in basis
(BCJ) relations
(n-3)! functions in basis
Real part :
Imaginary part :
Monodromy relations
34
Gravity
35
Gravity AmplitudesPossible to monodromy relations to rearrange KLT
•
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Gravity Amplitudes
More symmetry but can do better…
BCJ monodromy!!
Monodromy and KLTAnother way to express the BCJ monodromy relations
using a momentum S kernel
Express ‘phase’ difference between orderings in sets
38
Monodromy and KLT(NEJBB, Damgaard, Feng, Sondergaard;NEJBB, Damgaard,
Sondergaard,Vanhove)
String Theory also a natural interpretation via
Stringy BCJ monodromy!!
KLT relationsRedoing KLT using S kernels leads to…
Beautifully symmetric form for (j=n-1) gravity…
40
SymmetriesString theory may trivialize certain symmetries (example monodromy)
Monodromy relations between different orderings of legs gives reduction of basis of amplitudes
Rich structure for field theories:Kawai-Lewellen-Tye gravity relations
41
Vanishing relations
Also new ‘vanishing identities’ for YM amplitudes possible
Related to R parity violations
(NEJBB, Damgaard, Feng, Sondergaard
(Tye and Zhang; Feng and He; Elvang and Kiermeier) Gives link between amplitudes in YM
42
Jacobi algebra relations
Monodromy and Jacobi relations
• NewKinematic structure in Yang-Mills: (Bern, Carrasco, Johansson)
Monodromy -> (n-3)! reduction <- Vertexkinematic structures
3pt vertex only… natural in string theory
YM in lightcone gauge (space-cone) (Chalmers and Siegel, Congemi)
Direct have spinor-helicity formalism foramplitudes via vertex rules
Monodromy and Jacobi relations
45
Algebra for amplitudesSelf-dual sector:
(O’Connell and Monteiro)
Light-cone coordinates:
(Chalmers and Siegel, Congemi, O’Connell and Monteiro)
Simple vertex rules
Gauge-choice + Eq. of motion
46
Algebra for amplitudes
Jacobi-relations
47
Algebra for amplitudes
Self-dual vertex e.g.
...+ +1
2
2
3s12 s1Ms123
vertex
•
48
Algebra for amplitudes
self-dual
full action
49
Algebra for amplitudes
Have to do two algebras, and
Pick reference frame thatmakes 4pt vertex -> 0
(O’Connell and Monteiro)
Algebra for amplitudes
Jacobi-relations
MHV case: Still only cubic vertices – one needed
51
Algebra for amplitudes
MHV vertex as self-dual case… with now
(O’Connell and Monteiro)
vertex
•
on one reference vertex
...+ +1
2
2
3s12 s1Ms123
52
Algebra for amplitudesGeneral case:
Possible to do something similar for generalnon-MHV amplitudes??
Problem to make 4pt interaction go away
53
Algebra for amplitudesInspiration from self-dual theories
•
Work out result for amplitude….Jacobi works… so ????
54
Algebra for amplitudesTry something else…
Pick (n-3)! scalar theories (different Y)
•
different scalar theories
(n-3)! basis for YM
YM (colour ordered)
•
(NEJBB, Damgaard, O’Connell and
Monteiro)
55
Algebra for amplitudes
Full amplitude
Now we have (manifest Jacobi YM amplitudes):
56
Color-dual formsYM amplitude
YM dual amplitude(Bern, Dennen)
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Relations for loop amplitudes
Jacobi relations for numerators also exist at loop level.. but still an open question to developdirect vertex formalism (scalar amplitudes??)
Especially in gravity computations – such relations can be crucial testing UV behaviour (see Berns talk)
Monodromy relations for finite amplitudes (A(++++..++) and A(-+++..++) (NEJBB, Damgaard,Johansson, Søndergaard)
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Conclusions
59
ConclusionsMuch more to learn about amplitude relations…
Presented explicit way of generating numerator factors satisfying Jacobi.
Useful for better understanding of Yang-Mills and gravity!
Open question: which Lie algebras are best?
60
ConclusionsMore to learn from String theory??…loop-level? pure spinor formalism (Mafra, Schlotterer, Stieberger)
Many applications for gravity, N=8, N=4, (double copy) computations impossible otherwise.
Inspiration from self-dual/MHV – can we do better?
More investigation needed…
Higher derivative operators? (Dixon, Broedel)