structure of amplitudes in gravity i lagrangian formulation of gravity, tree amplitudes, helicity...

Structure of Amplitudes in Gravity

I

Lagrangian Formulation of Gravity, Tree amplitudes, Helicity Formalism, Amplitudes

in Twistor Space, New techniques

Playing with Gravity - 24th Nordic Meeting

Gronningen 2009Niels Emil Jannik Bjerrum-Bohr

Niels Bohr International AcademyNiels Bohr Institute

Outline

Outline

• Quantum Gravity and General Relativity

• Lagrangian formulation of Gravity– Tree Amplitudes

• Helicity Formalism– Twistor Space

• New Techniques for tree AmplitudesGronningen 3-5 Dec 2009

Playing with Gravity 3

Quantum

Gravity

Quantum Gravity

Gronningen 3-5 Dec 2009


We desire a quantum theory with an interacting particle the graviton

• It should obey an attractive inverse square law (graviton mass-less)

• It should couple with equal strength to all matter sources (graviton tensor field)

No observed or ‘experimental’ effects of a

quantum theory for gravity so far…

Einstein-Hilbert Lagrangian

L E H =R

d4xhp

¡ gRi



Features:

• Consistent with General Relativity (gives trees)

• Action: Non-renormalisable! – Not valid beyond tree-level / one-loop

• Explicit one-loop divergence with matter (t’ Hooft and Veltman)

• Explicit two-loop divergence! (Goroff, Sagnotti; van de Van)

GN = 1=M 2P lanck

(dimensionful)

Quantum Gravity

• Still waiting on a fundamental theory for Gravity..

• String theory: – a natural candidate– not point like theory– however still not a string theory model

fully consistent with field theory….



Quantum Gravity

• Effective field theory description

– Consistent with String theory– Low energy predictions unique and fit

General Relativity– The simplest extension of Einstein-Hilbert

we can think of– Including supersymmetry: easy and

excludes certain higher derivative terms



Effective Lagrangian

L E f f =Z

d4xhp

¡ gR (Einstein-Hilbert)

+ c1R2¹ º + c2R2 : : :(higher derivatives)

+ L matter

i(matter couplings)



Features:

• Derivative terms consistent with symmetry

• Action: valid till Planck scale by construction

Quantising Gravity



g¹ º ´ ¹g¹ º + h¹ º = ¹g¹ ®[±®º + h®

º ]

g¹ º = ¹g¹ º ¡ h¹ º + h¹®h®º + O(h3)

L =p

¡ ¹gh2¹R

· 2 +1·

£h®

®¹R ¡ 2¹R®

º hº®

¤+ ¹R

·14[h®

®]2 ¡12h®

¯ h¯®

¸

¡ h®®hº

¯¹R¯

º + 2hº¯ h¯

®¹R®

º + h®®;º hº;¯

¯ ¡ hº;®¯ h¯

®;º +12h¯

®;º h®;º¯ ¡

12h®;º

® h¯¯ ;º

i

¡ °® ! : : :

R ! : : :R¹ º ! : : :

p¡ gL matter =

12

·@¹ Á@¹ Á¡ m2Á2

¸

¡·2

h¹ º·@¹ Á@º Á¡

12´¹ º

©@®Á@®Á¡ m2Á2ª

¸

+ · 2·

12h¹ ¸ h º

¸ ¡14hh¹ º

¸@¹ Á@º Á¡

· 2

8

·h® h® ¡

12h2

¸¡@° Á@° Á¡ m2Á2¢

Gravity with backgroundfield

Scalar field coupling to Gravity

Features:• Infinitely many and huge

vertices!

• No manifest simplifications

(Sannan)

45 terms + sym



Pure graviton verticesGravity with flat field

Very messy!!

P ® ° ± =12

iq2 + i²

£´®° ´¯ ±+ ´¯ ° ´®±¡ ´® ´° ±¤

Perturbative amplitudes

...+ +

1

2

1 M 12

3

s12 s1M s123

Standard textbook way:

Feynman rules

1) Lagrangian (easy) 2) Vertices (easy) (3-vertex gravity over 100 terms..) 3) Diagrams (increasing difficult) 4) Sum Diagrams over all contractions

(hard) 5) Loops (integrations) more about this lecture II

(close to impossible / impossible)


12Playing with Gravity

X

¾(1;:::M )

( )

Computation of perturbative amplitudes



Complex expressions involving e.g. (no manifest

symmetry or simplifications)

Sum over topological different diagrams

Generic Feynman amplitude

# Feynman diagrams: Factorial Growth!

pi ¢pj

pi ¢²j ²i ¢²j

Amplitudes



Simplifications

Spinor-helicity formalism

Recursion

Specifying external polarisation tensors

Loop amplitudes:(Unitarity,Supersymmetric decomposition)

Colour ordering

InspirationfromString theory

²i ¢²j

Tr(T1T2T3 : : :Tn)

Helicity states formalismSpinor products :

Momentum parts of amplitudes:

Spin-2 polarisation tensors in terms of helicities, (squares of YM):

(Xu, Zhang, Chang)

Different representations of the Lorentz group

Gronningen 3-5 Dec 2009 15Playing with Gravity

Simplifications from Spinor-Helicity

Vanish in spinor helicity formalismGravity:

Huge simplifications

Contractions

45 terms + sym



Scattering amplitudes in D=4

Amplitudes in gravity theories as well as Yang-Mills can hence be expressed completely specifying– The external helicies

e.g. : A(1+,2-,3+,4+, .. )

– The spinor variables

Spinor Helicity formalism



Note on notationWe will use the notation:

Traces...Gronningen 3-5 Dec 2009


Amplitudes via String

Theory

Gravity Amplitudes



Closed StringAmplitude

Left-movers Right-moversSum over permutations

Phase factor

Open amplitudes: Sum over different factorisations(Link to individual Feynman diagrams lost..)

Sum gauge invariant

Certain vertex relations possible(Bern and

Grant)

(Kawai-Lewellen-Tye)

Not Left-Right symmetric

xx

xx

. .

12

3

M

...+ +=

1

2

1 M 12

3

s12 s1M s123

Gravity Amplitudes



KLT explicit representation:’ ! 0ei ! ,’ (n-3, ij) sij

= Polynomial (sij)

No manifest crossing symmetry

Double poles

xx

xx

. .

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

22

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 0

Atree MHV Given by the formula (Parke and Taylor) and proven by (Berends and Giele)

Tree amplitudes

First non-trivial example, (M)aximally (H)elicity (V)iolating (MHV) amplitudes

One single term!!




Examples of KLT relations

M (1¡ ;2¡ ;3+) »h12i4

h12ih23ih31i£

h12i4

h12ih23ih31i=

h12i6

h23i2h31i2


M (1¡ ;2¡ ;3+;4+) »h12i4

h12ih23ih34ih41i£

h12i4

h12ih24ih43ih31i= ¡

h12i6

h23ih24ih34i2h13ih14i

Gravity MHV amplitudes Can be generated from KLT via YM

MHV amplitudes.

Berends-Giele-Kuijf recursion formula



Anti holomorphic Contributions – feature in gravity

Recent work: (Elvang, Freedman: Nguyen, Spradlin, Volovich, Wen)

KLT for NMHV

• KLT hold independent of helicity

• NMHV amplituder are more complicated

but KLT can still be used

• NMHV amplitudes change much by Helicity structure

• In Lecture II we will see how KLT is very useful in cuts as well…



26

Twistor space

Duality

Proposal that N=4 super Yang-Mills is dual to a string theory in twistor

space? (Witten)

TopologicalString Theorywith twistor target space CP3

PerturbativeN=4 superYang-Mills



$

Twistor space• Transformation of amplitudes

into twistor space (Penrose)

• In metric signature ( + + - - ) :

2D Fourier transform

• In twistor space : plane wave function is a line:

Tree amplitudes in YM on degenerate algebraic curves

Degree : number of negative helicities

(Witten)

Degree : N-1+L



Review: CSW expansion of Yang-Mills amplitudes

• In the CSW-construction : off-shell MHV-amplitudes building blocks for more complicated amplitude expressions (Cachazo, Svrcek and Witten)

• MHV vertices:



Example of how this works



Example of A6(1-,2-,3-,4+,5+,6+)

31

Twistor space properties

• Twistor-space properties of gravity: More complicated!

Derivatives of - functions

-functionsSignature of non-locality typical in gravity

N=4

Anti-holomorphic pieces in gravity amplitudes



32

Collinear and Coplanar Operators



33

Twistor space properties• For gravity : Guaranteed that

• Five-point amplitude. (Giombi, Ricci, Rables-Llana and Trancanelli; Bern, NEJBB and Dunbar)

Tree amplitudes :

Acting with differential operators F and K



(Bern, NEJBB and Dunbar)

34

Recursion

35

BCFW Recursion for trees

Shift of the spinors :

Amplitude transforms as

We can now evaluate the contour integral over A(z)

Complex momentum space!!

a and b will remain on-shell even after shift

(Britto, Cachazo, Feng, Witten)



36

BCWF Recursion for trees Given that

• A(z) vanish for • A(z) is a rational function• A(z) has simple poles

Residues : Determined by factorization properties

Tree amplitude : Factorise in product of tree amplitudes

• in z

(Britto, Cachazo, Feng, Witten)



z ! 1 C1 = 0

4pt Example

A B

¸2 ! ¸2 + z¸1

¹1 ! ¹

1 ¡ z¹2

[2 p] unaffected by shift so non-zero so <2p> must vanish!

A4(1¡ ;2¡ ;3+;4+) » A3(1¡ ; P ¡14;4

+) £1

s14£ A3(2¡ ;3+;¡ P +

14)

A4(1¡ ;2¡ ;3+;4+) »[3P14]3

[23][P142]£

1s14

£h1P14i3

h41ihP144i

P14 = P14 ¡ z[2h1; z =[24][41]

»[3P141i3

[23][2P144ih41ih41i[14]=

[34]3

[12][23][41]

3pt vertex defined in complex momentum

38

MHV vertex expansion for gravity tree amplitudes

• CSW expansion in gravity• Shift (Risager)

Reproduce CSW for Yang-Mills

Shift : Correct factorisation

CSW vertex

(NEJBB, Dunbar, Ita, Perkins, Risager)



39


• Negative legs shifted in the following way

• Analytic continuation of amplitude into the complex plane

• If Mn(z), 1) rational, 2) simple poles at points z, and 3) vanishes (justified assumption) :

Mn(0) = sum of residues (as in BCFW),



C1

40

• All poles : Factorise as :

• vanishes linearly in z :

• Spinor products : not z dependent (normal CSW)




41

• For gravity : Substitutions

MHV amplitudes on the pole MHV vertices!– MHV vertex expansion for gravity


Contact term!

non-locality



!

Matter MHV expansion considered by (Bianchi, Elvang, Friedman) problem with expansion beyond 12pt..

Conclusions lecture I• Considered Lagrangian Formulation

of Quantum Gravity– Einstein-Hilbert / Effective Lagrangian – Tree amplitudes

• Helicity Formalism – Amplitudes in Twistor Space,

• New techniques– Amplitudes via KLT– Amplitudes via Recursion, BCFW and

CSWGronningen 3-5 Dec 2009


Outline of lecture II

• Outline af lecture II– In Lecture II we will consider how the

tree results can be used to derive results for loop amplitudes

– We will see how simple results for tree amplitudes makes it possible to derive simple loop results

– Also we will see how symmetries of trees are carried over to loop amplitudes



Simplicity…SUSY N=4, N=1,QCD, Gravity..

Loops simple and symmetric

Unitarity

Cuts

Trees (Witten)Twistor

s

Trees simple and symmetric

Hidden Beauty!New simple analytic expressions



l

structure of amplitudes in gravity i lagrangian formulation of gravity, tree amplitudes, helicity...

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