Amplitudes et périodes­ - PowerPoint Presentation

Slide1

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

GravitySlide2

2IntroductionSlide3

3Amplitudes in Physics

Important concept:

Classical and Quantum Mechanics

Amplitude square = probability

3Slide4

Large Hadron Collider

…

LHC ’event’

Proton

Proton

Jets

Jets

Jets:

Reconstruction complicated..

Calculations necessary:

Amplitude

4Slide5

How to compute amplitudesField theory: write down Lagrangian (toy model):Quantum mechanics:Write down HamiltonianKinetic term Mass term Interaction term E.g. QED Yukawa theory Klein-Gordon QCD Standard Model5Solution to Path integral -> Feynman diagrams!Slide6

6How to compute amplitudes

Method: Permutations over all possible outcomes (tree + loops (self-interactions))

Field theory: Lagrange-function

Feature: Vertex functions, Propagator (gauge fixing)

6Slide7

7General 1-loop amplitudes

Vertices carry factors of loop momentum

n-pt amplitude

(

Passarino-Veltman

)

reduction

Collapse of a propagator

p = 2n for gravity

p=n for YM

PropagatorsSlide8

8Unitarity cutsUnitarity methods are building on the cut equationSingletNon-SingletSlide9

9Computation of perturbative amplitudesComplex 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!Slide10

10AmplitudesSimplificationsSpinor-helicity formalismRecursionSpecifying external polarisation tensors (ε

I



ε

j

)

Loop

amplitudes:

(Unitarity,

Supersymmetric

decomposition)

Colour ordering

Tr(T

1

T

2

.. T

n

)

Inspiration

from

String theory

SymmetrySlide11

11Helicity 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 groupSlide12

12Scattering amplitudes in D=4Amplitudes in YM theories and gravity theories can hence be expressed via The external heliciese.g. : A(1+,2-,3+,4+, .. ) Slide13

13MHV AmplitudesSlide14

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

:

A

tree

(1

+

,2

+

,..,j

-

,..,k

-

,..)

=

Reflection properties:

A

n

(1,2,3,..,n) = (-1)

n

A

n(n,n-1,..,2,1)Dual Ward: An(1,2,..,n) + A

n

(1,3,2,..n)+..+An(1,perm[2,..n]) = 0

Further identities

as we will see….

Tree amplitudes

First

non-trivial

example:

One

single term!!

Many relations between YM amplitudes, e.g.Slide15

15Gravity AmplitudesExpand Einstein-Hilbert Lagrangian : Features:Infinitely many verticesHuge expressions for vertices!

No manifest

cancellations

nor

simplifications

(Sannan)

45 terms + symSlide16

16Simplifications from Spinor-HelicityVanish in spinor helicity formalismGravity:Huge simplificationsContractions

45 terms + symSlide17

17String theorySlide18

String theoryDifferent form for amplitude18Feynman diagrams sums separate kinematic polesString theory adds channels up.. <->

x

x

x

x

.

.

1

2

3

M

...

+

+

=

1

2

1

M

1

2

3

s

12

s

1M

s

123Slide19

Notion of color ordering19String theory

1

2

s

12

Color ordered

Feynman rules

x

x

x

x

.

.

1

2

3

MSlide20

20…a more efficient waySlide21

Gravity Amplitudes21Closed StringAmplitudeLeft-movers

Right-movers

Sum over

permutations

Phase factor

(

Kawai-Lewellen-Tye

)

Not

Left-Right

symmetricSlide22

22Gravity Amplitudes(Link to individual Feynman diagrams lost..)

Certain vertex

relations possible

(Bern and Grant;

Ananth

and

Theisen

;

Hohm

)

x

x

x

x

.

.

1

2

3

M

...

+

+

=

1

2

1

M

1

2

3

s

12

s

1M

s

123

Concrete

Lagrangian

formulation possible?Slide23

23Gravity AmplitudesKLT explicit representation:’ -> 0ei -> Polynomial (sij)

No

manifest

crossing symmetry

Double poles

x

x

x

x

.

.

1

2

3

M

...

+

+

=

1

2

1

M

1

2

3

s

12

s

1M

s

123

Sum gauge invariant

(1)

(2)

(4)

(4)

(s

124

)

Higher point expressions quite bulky ..

Interesting remark

: The KLT relations work

independently

of external

polarisations

(Bern et al)Slide24

24Gravity MHV amplitudesCan be generated from KLT via YM MHV amplitudes.(Berends-Giele-Kuijf) recursion formulaAnti holomorphic Contributions – feature in gravitySlide25

25New relationsfor Yang-MillsSlide26

26New relations for amplitudesNewKinematic structure in Yang-Mills:(Bern, Carrasco, Johansson)Relations between amplitudes

Kinematic analogue

– not unique ??

n

-pt

4pt vertex??Slide27

27New relations for amplitudes(n-3)!5 pointsNice new way to do gravity Double-copy gravity from YM!

(Bern,

Carrasco

, Johansson;

Bern,

Dennen

, Huang,

Kiermeier

)

Basis where 3 legs are fixedSlide28

28MonodromySlide29

2929x

x

x

x

.

.

1

3

M

...

+

+

=

1

2

1

M

1

2

3

s

12

s

1M

s

123

2

String theorySlide30

Monodromy relations30Slide31

Monodromy relations31FT limit-> 0(NEJBB, Damgaard, Vanhove

;

Stieberger

)

New relations

(Bern,

Carrasco

, Johansson)

KK relations

BCJ relations Slide32

32Monodromy relationsMonodromy related(Kleiss – Kuijf) relations(n-2)! functions in basis(BCJ) relations

(n-3)!

functions

in basisSlide33

Real part :Imaginary part :Monodromy relationsSlide34

34GravitySlide35

35Gravity AmplitudesPossible to monodromy relations to rearrange KLT Slide36

36Gravity AmplitudesMore symmetry but can do better… Slide37

BCJ monodromy!!Monodromy and KLTAnother way to express the BCJ monodromy relations using a momentum S kernelExpress ‘phase’ difference between orderings in setsSlide38

38Monodromy and KLT(NEJBB, Damgaard, Feng, Sondergaard;NEJBB, Damgaard, Sondergaard,Vanhove)

String Theory also a natural

interpretation via

Stringy BCJ

monodromy

!!Slide39

KLT relationsRedoing KLT using S kernels leads to… Beautifully symmetric form for (j=n-1) gravity…Slide40

Symmetries String 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 relations40Slide41

41Vanishing relationsAlso new ‘vanishing identities’ for YM amplitudes possibleRelated to R parity violations

(NEJBB, Damgaard,

Feng

,

Sondergaard

(

Tye

and

Zhang

;

Feng

and He;

Elvang

and

Kiermeier

)

Gives link between amplitudes in YMSlide42

42Jacobi algebra relationsSlide43

Monodromy and Jacobi relationsNewKinematic structure in Yang-Mills:(Bern, Carrasco, Johansson)Monodromy -> (n-3)! reduction <-

Vertex

k

inematic structures Slide44

3pt vertex only… natural in string theoryYM in lightcone gauge (space-cone) (Chalmers and Siegel, Congemi)Direct have spinor-helicity formalism foramplitudes via vertex rulesMonodromy and Jacobi relationsSlide45

45Algebra 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 motionSlide46

46Algebra for amplitudesJacobi-relationsSlide47

47Algebra for amplitudesSelf-dual vertex e.g.

...

+

+

1

2

2

3

s

12

s

1M

s

123

vertex

Slide48

48Algebra for amplitudesself-dualfull actionSlide49

49Algebra for amplitudesHave to do two algebras, and Pick reference frame thatmakes 4pt vertex -> 0(O’Connell and Monteiro)Slide50

Algebra for amplitudesJacobi-relationsMHV case: Still only cubic vertices – one neededSlide51

51Algebra for amplitudesMHV vertex as self-dual case… with now(O’Connell and Monteiro)vertex

on one reference vertex

...

+

+

1

2

2

3

s

12

s

1M

s

123Slide52

52Algebra for amplitudesGeneral case:Possible to do something similar for generalnon-MHV amplitudes??Problem to make 4pt interaction go awaySlide53

53Algebra for amplitudesInspiration from self-dual theories Work out result for amplitude….Jacobi works… so ????Slide54

54Algebra for amplitudesTry something else…Pick (n-3)! scalar theories (different Y) different scalar theories(n-3)! basis for YMYM (colour ordered)

(NEJBB, Damgaard,

O’Connell

and

Monteiro

)Slide55

55Algebra for amplitudesFull amplitudeNow we have (manifest Jacobi YM amplitudes):Slide56

56Color-dual formsYM amplitudeYM dual amplitude(Bern, Dennen)Slide57

57Relations for loop amplitudesJacobi 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) Slide58

58ConclusionsSlide59

59ConclusionsMuch 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?Slide60

60ConclusionsMore 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

)