amplitude relations in yang-mills theory and gravity
DESCRIPTION
Amplitude relations in Yang-Mills theory and Gravity. Amplitudes et périodes 3-7 December 2012 Niels Emil Jannik Bjerrum -Bohr Niels Bohr International Academy, Niels Bohr Institute. Introduction. Amplitudes in Physics. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Amplitude relations in Yang-Mills theory and Gravity](https://reader036.vdocuments.net/reader036/viewer/2022062302/56816621550346895dd97861/html5/thumbnails/1.jpg)
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
![Page 2: Amplitude relations in Yang-Mills theory and Gravity](https://reader036.vdocuments.net/reader036/viewer/2022062302/56816621550346895dd97861/html5/thumbnails/2.jpg)
2
Introduction
![Page 3: Amplitude relations in Yang-Mills theory and Gravity](https://reader036.vdocuments.net/reader036/viewer/2022062302/56816621550346895dd97861/html5/thumbnails/3.jpg)
3
Amplitudes in Physics
• Important concept: Classical and Quantum Mechanics
Amplitude square = probability
3
![Page 4: Amplitude relations in Yang-Mills theory and Gravity](https://reader036.vdocuments.net/reader036/viewer/2022062302/56816621550346895dd97861/html5/thumbnails/4.jpg)
Large Hadron Collider
…
LHC ’event’
Proton
Proton
Jets
JetsJets:Reconstruction complicated..
Calculations necessary:Amplitude
4
![Page 5: Amplitude relations in Yang-Mills theory and Gravity](https://reader036.vdocuments.net/reader036/viewer/2022062302/56816621550346895dd97861/html5/thumbnails/5.jpg)
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!
![Page 6: Amplitude relations in Yang-Mills theory and Gravity](https://reader036.vdocuments.net/reader036/viewer/2022062302/56816621550346895dd97861/html5/thumbnails/6.jpg)
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
![Page 7: Amplitude relations in Yang-Mills theory and Gravity](https://reader036.vdocuments.net/reader036/viewer/2022062302/56816621550346895dd97861/html5/thumbnails/7.jpg)
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
![Page 8: Amplitude relations in Yang-Mills theory and Gravity](https://reader036.vdocuments.net/reader036/viewer/2022062302/56816621550346895dd97861/html5/thumbnails/8.jpg)
8
Unitarity cuts• Unitarity methods are building on the
cut equation
Singlet Non-Singlet
![Page 9: Amplitude relations in Yang-Mills theory and Gravity](https://reader036.vdocuments.net/reader036/viewer/2022062302/56816621550346895dd97861/html5/thumbnails/9.jpg)
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!
![Page 10: Amplitude relations in Yang-Mills theory and Gravity](https://reader036.vdocuments.net/reader036/viewer/2022062302/56816621550346895dd97861/html5/thumbnails/10.jpg)
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
![Page 11: Amplitude relations in Yang-Mills theory and Gravity](https://reader036.vdocuments.net/reader036/viewer/2022062302/56816621550346895dd97861/html5/thumbnails/11.jpg)
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
![Page 12: Amplitude relations in Yang-Mills theory and Gravity](https://reader036.vdocuments.net/reader036/viewer/2022062302/56816621550346895dd97861/html5/thumbnails/12.jpg)
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+, .. )
![Page 13: Amplitude relations in Yang-Mills theory and Gravity](https://reader036.vdocuments.net/reader036/viewer/2022062302/56816621550346895dd97861/html5/thumbnails/13.jpg)
13
MHV Amplitudes
![Page 14: Amplitude relations in Yang-Mills theory and Gravity](https://reader036.vdocuments.net/reader036/viewer/2022062302/56816621550346895dd97861/html5/thumbnails/14.jpg)
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.
![Page 15: Amplitude relations in Yang-Mills theory and Gravity](https://reader036.vdocuments.net/reader036/viewer/2022062302/56816621550346895dd97861/html5/thumbnails/15.jpg)
15
Gravity AmplitudesExpand Einstein-Hilbert Lagrangian :
Features:Infinitely many verticesHuge expressions for vertices!No manifest cancellations norsimplifications
(Sannan)
45 terms + sym
![Page 16: Amplitude relations in Yang-Mills theory and Gravity](https://reader036.vdocuments.net/reader036/viewer/2022062302/56816621550346895dd97861/html5/thumbnails/16.jpg)
16
Simplifications from Spinor-Helicity
Vanish in spinor helicity formalismGravity:
Huge simplifications
Contractions
45 terms + sym
![Page 17: Amplitude relations in Yang-Mills theory and Gravity](https://reader036.vdocuments.net/reader036/viewer/2022062302/56816621550346895dd97861/html5/thumbnails/17.jpg)
17
String theory
![Page 18: Amplitude relations in Yang-Mills theory and Gravity](https://reader036.vdocuments.net/reader036/viewer/2022062302/56816621550346895dd97861/html5/thumbnails/18.jpg)
18
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
![Page 19: Amplitude relations in Yang-Mills theory and Gravity](https://reader036.vdocuments.net/reader036/viewer/2022062302/56816621550346895dd97861/html5/thumbnails/19.jpg)
19
Notion of color ordering
String theory
1
2
s12
Color ordered Feynman rules
xx
xx
. .
12
3
M
![Page 20: Amplitude relations in Yang-Mills theory and Gravity](https://reader036.vdocuments.net/reader036/viewer/2022062302/56816621550346895dd97861/html5/thumbnails/20.jpg)
20
…a more efficient way
![Page 21: Amplitude relations in Yang-Mills theory and Gravity](https://reader036.vdocuments.net/reader036/viewer/2022062302/56816621550346895dd97861/html5/thumbnails/21.jpg)
Gravity Amplitudes
21
Closed StringAmplitude
Left-movers Right-moversSum over
permutations
Phase factor (Kawai-Lewellen-Tye)
Not Left-Right
symmetric
![Page 22: Amplitude relations in Yang-Mills theory and Gravity](https://reader036.vdocuments.net/reader036/viewer/2022062302/56816621550346895dd97861/html5/thumbnails/22.jpg)
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?
![Page 23: Amplitude relations in Yang-Mills theory and Gravity](https://reader036.vdocuments.net/reader036/viewer/2022062302/56816621550346895dd97861/html5/thumbnails/23.jpg)
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)
![Page 24: Amplitude relations in Yang-Mills theory and Gravity](https://reader036.vdocuments.net/reader036/viewer/2022062302/56816621550346895dd97861/html5/thumbnails/24.jpg)
24
Gravity MHV amplitudes• Can be generated from KLT via YM
MHV amplitudes.
(Berends-Giele-Kuijf) recursion formula
Anti holomorphic Contributions
– feature in gravity
![Page 25: Amplitude relations in Yang-Mills theory and Gravity](https://reader036.vdocuments.net/reader036/viewer/2022062302/56816621550346895dd97861/html5/thumbnails/25.jpg)
25
New relationsfor Yang-Mills
![Page 26: Amplitude relations in Yang-Mills theory and Gravity](https://reader036.vdocuments.net/reader036/viewer/2022062302/56816621550346895dd97861/html5/thumbnails/26.jpg)
26
New relations for amplitudes
• NewKinematic structure in Yang-Mills: (Bern, Carrasco, Johansson)
Relations between amplitudes
Kinematic analogue – not unique ??
n-pt
4pt vertex??
![Page 27: Amplitude relations in Yang-Mills theory and Gravity](https://reader036.vdocuments.net/reader036/viewer/2022062302/56816621550346895dd97861/html5/thumbnails/27.jpg)
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
![Page 28: Amplitude relations in Yang-Mills theory and Gravity](https://reader036.vdocuments.net/reader036/viewer/2022062302/56816621550346895dd97861/html5/thumbnails/28.jpg)
28
Monodromy
![Page 29: Amplitude relations in Yang-Mills theory and Gravity](https://reader036.vdocuments.net/reader036/viewer/2022062302/56816621550346895dd97861/html5/thumbnails/29.jpg)
29 29
xx
xx
. .
1 3
M
...+ +=
1
2
1 M 12
3
s12 s1M s123
2
String theory
![Page 30: Amplitude relations in Yang-Mills theory and Gravity](https://reader036.vdocuments.net/reader036/viewer/2022062302/56816621550346895dd97861/html5/thumbnails/30.jpg)
30
Monodromy relations
![Page 31: Amplitude relations in Yang-Mills theory and Gravity](https://reader036.vdocuments.net/reader036/viewer/2022062302/56816621550346895dd97861/html5/thumbnails/31.jpg)
31
Monodromy relations
FT limit-> 0
(NEJBB, Damgaard, Vanhove; Stieberger)
New relations (Bern, Carrasco, Johansson)
KK relations
BCJ relations
![Page 32: Amplitude relations in Yang-Mills theory and Gravity](https://reader036.vdocuments.net/reader036/viewer/2022062302/56816621550346895dd97861/html5/thumbnails/32.jpg)
32
Monodromy relations
Monodromy related
(Kleiss – Kuijf) relations
(n-2)! functions in basis
(BCJ) relations
(n-3)! functions in basis
![Page 33: Amplitude relations in Yang-Mills theory and Gravity](https://reader036.vdocuments.net/reader036/viewer/2022062302/56816621550346895dd97861/html5/thumbnails/33.jpg)
Real part :
Imaginary part :
Monodromy relations
![Page 34: Amplitude relations in Yang-Mills theory and Gravity](https://reader036.vdocuments.net/reader036/viewer/2022062302/56816621550346895dd97861/html5/thumbnails/34.jpg)
34
Gravity
![Page 35: Amplitude relations in Yang-Mills theory and Gravity](https://reader036.vdocuments.net/reader036/viewer/2022062302/56816621550346895dd97861/html5/thumbnails/35.jpg)
35
Gravity AmplitudesPossible to monodromy relations to rearrange KLT
•
![Page 36: Amplitude relations in Yang-Mills theory and Gravity](https://reader036.vdocuments.net/reader036/viewer/2022062302/56816621550346895dd97861/html5/thumbnails/36.jpg)
36
Gravity Amplitudes
More symmetry but can do better…
![Page 37: Amplitude relations in Yang-Mills theory and Gravity](https://reader036.vdocuments.net/reader036/viewer/2022062302/56816621550346895dd97861/html5/thumbnails/37.jpg)
BCJ monodromy!!
Monodromy and KLTAnother way to express the BCJ monodromy relations
using a momentum S kernel
Express ‘phase’ difference between orderings in sets
![Page 38: Amplitude relations in Yang-Mills theory and Gravity](https://reader036.vdocuments.net/reader036/viewer/2022062302/56816621550346895dd97861/html5/thumbnails/38.jpg)
38
Monodromy and KLT(NEJBB, Damgaard, Feng, Sondergaard;NEJBB, Damgaard,
Sondergaard,Vanhove)
String Theory also a natural interpretation via
Stringy BCJ monodromy!!
![Page 39: Amplitude relations in Yang-Mills theory and Gravity](https://reader036.vdocuments.net/reader036/viewer/2022062302/56816621550346895dd97861/html5/thumbnails/39.jpg)
KLT relationsRedoing KLT using S kernels leads to…
Beautifully symmetric form for (j=n-1) gravity…
![Page 40: Amplitude relations in Yang-Mills theory and Gravity](https://reader036.vdocuments.net/reader036/viewer/2022062302/56816621550346895dd97861/html5/thumbnails/40.jpg)
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
![Page 41: Amplitude relations in Yang-Mills theory and Gravity](https://reader036.vdocuments.net/reader036/viewer/2022062302/56816621550346895dd97861/html5/thumbnails/41.jpg)
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
![Page 42: Amplitude relations in Yang-Mills theory and Gravity](https://reader036.vdocuments.net/reader036/viewer/2022062302/56816621550346895dd97861/html5/thumbnails/42.jpg)
42
Jacobi algebra relations
![Page 43: Amplitude relations in Yang-Mills theory and Gravity](https://reader036.vdocuments.net/reader036/viewer/2022062302/56816621550346895dd97861/html5/thumbnails/43.jpg)
Monodromy and Jacobi relations
• NewKinematic structure in Yang-Mills: (Bern, Carrasco, Johansson)
Monodromy -> (n-3)! reduction <- Vertexkinematic structures
![Page 44: Amplitude relations in Yang-Mills theory and Gravity](https://reader036.vdocuments.net/reader036/viewer/2022062302/56816621550346895dd97861/html5/thumbnails/44.jpg)
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
![Page 45: Amplitude relations in Yang-Mills theory and Gravity](https://reader036.vdocuments.net/reader036/viewer/2022062302/56816621550346895dd97861/html5/thumbnails/45.jpg)
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
![Page 46: Amplitude relations in Yang-Mills theory and Gravity](https://reader036.vdocuments.net/reader036/viewer/2022062302/56816621550346895dd97861/html5/thumbnails/46.jpg)
46
Algebra for amplitudes
Jacobi-relations
![Page 47: Amplitude relations in Yang-Mills theory and Gravity](https://reader036.vdocuments.net/reader036/viewer/2022062302/56816621550346895dd97861/html5/thumbnails/47.jpg)
47
Algebra for amplitudes
Self-dual vertex e.g.
...+ +1
2
2
3s12 s1Ms123
vertex
•
![Page 48: Amplitude relations in Yang-Mills theory and Gravity](https://reader036.vdocuments.net/reader036/viewer/2022062302/56816621550346895dd97861/html5/thumbnails/48.jpg)
48
Algebra for amplitudes
self-dual
full action
![Page 49: Amplitude relations in Yang-Mills theory and Gravity](https://reader036.vdocuments.net/reader036/viewer/2022062302/56816621550346895dd97861/html5/thumbnails/49.jpg)
49
Algebra for amplitudes
Have to do two algebras, and
Pick reference frame thatmakes 4pt vertex -> 0
(O’Connell and Monteiro)
![Page 50: Amplitude relations in Yang-Mills theory and Gravity](https://reader036.vdocuments.net/reader036/viewer/2022062302/56816621550346895dd97861/html5/thumbnails/50.jpg)
Algebra for amplitudes
Jacobi-relations
MHV case: Still only cubic vertices – one needed
![Page 51: Amplitude relations in Yang-Mills theory and Gravity](https://reader036.vdocuments.net/reader036/viewer/2022062302/56816621550346895dd97861/html5/thumbnails/51.jpg)
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
![Page 52: Amplitude relations in Yang-Mills theory and Gravity](https://reader036.vdocuments.net/reader036/viewer/2022062302/56816621550346895dd97861/html5/thumbnails/52.jpg)
52
Algebra for amplitudesGeneral case:
Possible to do something similar for generalnon-MHV amplitudes??
Problem to make 4pt interaction go away
![Page 53: Amplitude relations in Yang-Mills theory and Gravity](https://reader036.vdocuments.net/reader036/viewer/2022062302/56816621550346895dd97861/html5/thumbnails/53.jpg)
53
Algebra for amplitudesInspiration from self-dual theories
•
Work out result for amplitude….Jacobi works… so ????
![Page 54: Amplitude relations in Yang-Mills theory and Gravity](https://reader036.vdocuments.net/reader036/viewer/2022062302/56816621550346895dd97861/html5/thumbnails/54.jpg)
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)
![Page 55: Amplitude relations in Yang-Mills theory and Gravity](https://reader036.vdocuments.net/reader036/viewer/2022062302/56816621550346895dd97861/html5/thumbnails/55.jpg)
55
Algebra for amplitudes
Full amplitude
Now we have (manifest Jacobi YM amplitudes):
![Page 56: Amplitude relations in Yang-Mills theory and Gravity](https://reader036.vdocuments.net/reader036/viewer/2022062302/56816621550346895dd97861/html5/thumbnails/56.jpg)
56
Color-dual formsYM amplitude
YM dual amplitude(Bern, Dennen)
![Page 57: Amplitude relations in Yang-Mills theory and Gravity](https://reader036.vdocuments.net/reader036/viewer/2022062302/56816621550346895dd97861/html5/thumbnails/57.jpg)
57
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)
![Page 58: Amplitude relations in Yang-Mills theory and Gravity](https://reader036.vdocuments.net/reader036/viewer/2022062302/56816621550346895dd97861/html5/thumbnails/58.jpg)
58
Conclusions
![Page 59: Amplitude relations in Yang-Mills theory and Gravity](https://reader036.vdocuments.net/reader036/viewer/2022062302/56816621550346895dd97861/html5/thumbnails/59.jpg)
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?
![Page 60: Amplitude relations in Yang-Mills theory and Gravity](https://reader036.vdocuments.net/reader036/viewer/2022062302/56816621550346895dd97861/html5/thumbnails/60.jpg)
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)