niels emil jannik bjerrum-bohr niels bohr international academy niels bohr institute

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 Playing with Gravity 5

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 = Rd4xhp ¡ gR

i


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 termsGronningen 3-5 Dec 2009 Playing with Gravity 8

Effective LagrangianL E f f =

Zd4x

hp ¡ gR (Einstein-Hilbert)

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

+ L matteri

(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¯ ¡ 1

2h®;º® h¯

¯ ;ºi

¡ °® ! : : :

R ! :: :R¹ º ! : : :

p ¡ gLmatter = 12·@¹ Á@¹ Á¡ m2Á2

¸

¡ ·2h¹ º

·@¹ Á@º Á¡ 1

2´¹ º©@®Á@®Á¡ m2Á2ª¸

+ · 2· 1

2h¹ ¸ h º¸ ¡ 1

4hh¹ º¸

@¹ Á@º Á¡ · 2

8·h® h® ¡ 1

2h2¸ ¡@° 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)Gronningen 3-5 Dec 2009 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 ¢pjpi ¢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 18Playing with Gravity

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

x x 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

x x xx. .

12

3

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

One single term!!


Playing with Gravity 23

Examples of KLT relations

M (1¡ ;2¡ ;3+) » h12i4

h12ih23ih31i £ h12i4

h12ih23ih31i = h12i6

h23i2h31i2

Gronningen 3-5 Dec 2009

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

complicatedbut 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…Gronningen 3-5 Dec 2009 25Playing with Gravity

26

Twistor space

DualityProposal 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 ExampleA B

¸2 ! ¸2 + z¸1¹ 1 ! ¹ 1 ¡ z¹ 2

[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] £ 1

s14£ 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 factorisationCSW 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 42Playing with Gravity

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

niels emil jannik bjerrum-bohr niels bohr international academy niels bohr institute

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