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Structure of Amplitudes in Gravity I Lagrangian Formulation of Gravity, Tree amplitudes, Helicity Formalism, Amplitudes in Twistor Space, New techniques Playing with Gravity - 24 th Nordic Meeting Gronningen 2009. Niels Emil Jannik Bjerrum-Bohr Niels Bohr International Academy - PowerPoint PPT Presentation
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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
Gronningen 3-5 Dec 2009 Playing with Gravity 6
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….
Gronningen 3-5 Dec 2009 Playing with Gravity 7
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)
Gronningen 3-5 Dec 2009 Playing with Gravity 9
Features:• Derivative terms consistent with symmetry• Action: valid till Planck scale by
construction
Quantising Gravity
Gronningen 3-5 Dec 2009 Playing with Gravity 10
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
Gronningen 3-5 Dec 2009 Playing with Gravity 11
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
Gronningen 3-5 Dec 2009 Playing with Gravity 13
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
Gronningen 3-5 Dec 2009 Playing with Gravity 14
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
Gronningen 3-5 Dec 2009 16Playing with Gravity
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
Gronningen 3-5 Dec 2009 Playing with Gravity 17
Note on notationWe will use the notation:
Traces...Gronningen 3-5 Dec 2009 18Playing with Gravity
Amplitudes via String
Theory
Gravity Amplitudes
Gronningen 3-5 Dec 2009 Playing with Gravity 20
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
Gronningen 3-5 Dec 2009 Playing with Gravity 21
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!!
Gronningen 3-5 Dec 2009 22Playing with Gravity
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
Gronningen 3-5 Dec 2009 Playing with Gravity 24
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
Gronningen 3-5 Dec 2009 27Playing with Gravity
$
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
Gronningen 3-5 Dec 2009 28Playing with Gravity
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:
Gronningen 3-5 Dec 2009 29Playing with Gravity
Example of how this works
Gronningen 3-5 Dec 2009 30Playing with Gravity
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
Gronningen 3-5 Dec 2009 31Playing with Gravity
32
Collinear and Coplanar Operators
Gronningen 3-5 Dec 2009 32Playing with Gravity
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
Gronningen 3-5 Dec 2009 33Playing with Gravity
(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)
Gronningen 3-5 Dec 2009 35Playing with Gravity
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)
Gronningen 3-5 Dec 2009 36Playing with Gravity
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)
Gronningen 3-5 Dec 2009 38Playing with Gravity
39
MHV vertex expansion for gravity tree amplitudes
• 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),
Gronningen 3-5 Dec 2009 39Playing with Gravity
C1
40
• All poles : Factorise as :
• vanishes linearly in z :
• Spinor products : not z dependent (normal CSW)
MHV vertex expansion for gravity tree amplitudes
Gronningen 3-5 Dec 2009 40Playing with Gravity
41
• For gravity : Substitutions
MHV amplitudes on the pole MHV vertices!– MHV vertex expansion for gravity
MHV vertex expansion for gravity tree amplitudes
Contact term!
non-locality
Gronningen 3-5 Dec 2009 41Playing with Gravity
!
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
Gronningen 3-5 Dec 2009 43Playing with Gravity
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
Gronningen 3-5 Dec 2009 44Playing with Gravity
l