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+, .. )

13

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

16

Simplifications from Spinor-Helicity

Vanish in spinor helicity formalismGravity:

Huge simplifications

Contractions

45 terms + sym

17

String theory

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

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

26

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

28

Monodromy

29 29

xx

xx

. .

1 3

M

...+ +=

1

2

1 M 12

3

s12 s1M s123

2

String theory

30

Monodromy relations

31

Monodromy relations

FT limit-> 0

(NEJBB, Damgaard, Vanhove; Stieberger)

New relations (Bern, Carrasco, Johansson)

KK relations

BCJ relations

32

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

•

36

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)

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)

58

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)