Transcript Structure of Amplitudes in Gravity I

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 2009
Niels Emil Jannik Bjerrum-Bohr
Niels Bohr International Academy
Niels Bohr Institute
Outline
Outline
• Quantum Gravity and General Relativity
• Lagrangian formulation of Gravity
– Tree Amplitudes
• Helicity Formalism
– Twistor Space
• New Techniques for tree Amplitudes
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Quantum
Gravity
Quantum Gravity
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…
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Einstein-Hilbert Lagrangian
LEH =
R
d4 x
hp
i
¡ gR
Features:
• Consistent with General Relativity (gives trees)
• Action: Non-renormalisable!
– Not valid beyond tree-level / one-loop
2
GN = 1=M Planck
(dimensionful)
• Explicit one-loop divergence with matter (t’ Hooft and
Veltman)
• Explicit two-loop divergence! (Goroff, Sagnotti; van de Van)
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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….
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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
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Effective Lagrangian
Z
LEf f =
d4 x
hp
¡ gR
(Einst ein-Hilbert )
+ c1 R ¹2 º + c2 R 2 : : : (higher derivat ives)
i
+ L m at t er
(mat t er couplings)
Features:
• Derivative terms consistent with symmetry
• Action: valid till Planck scale by construction
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Quantising Gravity
g¹ º ´ g
¹ ¹ º + h¹ º =
g
¹ ¹ ® [±º®
+
h®
º ]
g¹ º = g
¹ ¹ º ¡ h¹ º + h¹®h®º + O(h3 )
¡ °®¯ ! : : : Gravity with background
field
R ! :::
R¹ º ! : : :
·
¸
h2R
¹
¤
1£ ®¹
1
1
2
¯
¹ ® hº + R
¹
L = ¡ g
¹ 2 +
h® R ¡ 2R
[h®
h®
º ®
®] ¡
¯ h®
·
·
4
2
1 ¯ ®;º
1 ®;º ¯ i
º ;¯
º ;® ¯
® º ¹¯
º ¯ ¹®
®
¡ h® h¯ Rº + 2h¯ h® Rº + h®;º h¯ ¡ h¯ h®;º + h®;º h¯ ¡ h® h¯ ;º
2
2
p
p
¡ gL mat t er
·
¸
1
Scalar field coupling to Gravity
=
@¹ Á@¹ Á ¡ m 2 Á2
2
·
¸
©
ª
· ¹º
1
¡ h
@¹ Á@º Á ¡ ´ ¹ º @® Á@® Á ¡ m 2 Á2
2
2
·
¸
·
¸
2
¡
¢
1
1
·
1
2
¹¸
º
¹º
®¯
2
°
2 2
+ ·
h h¸ ¡ hh
@¹ Á@º Á ¡
h h®¯ ¡ h
@° Á@ Á ¡ m Á
2
4
8
2
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Pure graviton vertices
Gravity with flat field
(Sannan)
45 terms
+ sym
Features:
P
®¯ ° ±
£ ®° ¯ ±
¤
1 i
¯ ° ®±
®¯ ° ±
=
´ ´ + ´ ´ ¡ ´ ´
2 q2 + i ²
• Infinitely many and huge vertices!
Very messy!!
• No manifest simplifications
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Perturbative amplitudes
Standard textbook way:
Feynman rules
X
¾( 1;:::M )
1)
2)
3)
4)
(
1
s12
2
1
+
M
s1M
2
+
1
s123
...
3
Lagrangian
Vertices
Diagrams
Sum Diagrams
(easy)
(easy) (3-vertex gravity over 100 terms..)
(increasing difficult)
over all contractions
(hard)
5) Loops (integrations) more about this lecture II
(close to impossible / impossible)
)
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Computation of perturbative
amplitudes
# Feynman diagrams:
Factorial Growth!
Sum over topological
different diagrams
Generic Feynman amplitude
Complex expressions involving e.g.
pi ¢pj
(no manifest symmetry
pi ¢² j
² i ¢² j
or simplifications)
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Amplitudes
Specifying external
polarisation tensors
² i ¢² j
Colour ordering
Simplifications
Tr(T1 T2T3 : : : Tn )
Recursion
Spinor-helicity Loop amplitudes:
(Unitarity,
formalism
Supersymmetric
decomposition)
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Inspiration
from
String theory
14
Helicity states formalism
Spinor products :
Different representations of
the Lorentz group
Momentum parts of amplitudes:
Spin-2 polarisation tensors in terms of helicities, (squares of YM):
(Xu, Zhang,
Chang)
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Simplifications from SpinorHelicity
Huge simplifications
45 terms
+ sym
Vanish in spinor helicity formalism
Gravity:
Contractions
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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
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Note on notation
We will use the notation:
Traces...
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Amplitudes via
String Theory
Gravity Amplitudes
Not Left-Right
symmetric
Closed String
Phase
Amplitude Sum over factor
permutations
Left-movers
Right-movers
(Kawai-Lewellen-Tye)
Sum gauge invariant
1x
2
x
x
x
M
.
1
3
1
=
.
s12
+
M
s1M
2
2
+
1
s123
...
3
Open amplitudes: Sum over different factorisations
(Link to individual Feynman diagrams lost..)
Certain vertex
relations possible
(Bern and Grant)
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Gravity Amplitudes
KLT explicit representation:
’ ! 0
ei ! ,’ (n-3, ij) sij
= Polynomial (sij)
No manifest
crossing symmetry
Higher point
expressions quite
bulky ..
(1)
1x
Double poles
(4)
3
x
1
s12
x =
(2)
2
x
.
Sum gauge invariant
1
+
M
s1M
2
+
1
(s124)
s123
...
3
(4)
Interesting remark: The KLT relations work independently of external polarisations
M
.
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Yang-Mills MHV-amplitudes
(n) same helicities vanishes
Tree amplitudes
Atree(1+,2+,3+,4+,..) = 0
(n-1) same helicities vanishes
Atree(1+,2+,..,j-,..) = 0
First non-trivial
example,
(M)aximally
(H)elicity (V)iolating
(MHV) amplitudes
(n-2) same helicities:
Atree(1+,2+,..,j-,..,k-,..) ¹ 0
One single term!!
Atree MHV Given by the formula
(Parke and Taylor) and proven
by (Berends and Giele)
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Examples of KLT relations
h12i 4
h12i 4
h12i 6
M (1 ; 2 ; 3 ) »
£
=
h12i h23i h31i
h12i h23i h31i
h23i 2 h31i 2
¡
¡
+
h12i 4
h12i 4
h12i 6
M (1 ; 2 ; 3 ; 4 ) »
£
= ¡
h12i h23i h34i h41i h12i h24i h43i h31i
h23i h24i h34i 2 h13i h14i
¡
¡
+
+
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Gravity MHV amplitudes
Can be generated from KLT via YM
MHV amplitudes.
Anti holomorphic
Contributions
– feature in gravity
Berends-Giele-Kuijf recursion formula
Recent work: (Elvang, Freedman: Nguyen, Spradlin, Volovich, Wen)
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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…
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Twistor space
26
Duality
Proposal that N=4 super Yang-Mills is dual
to a string theory in twistor space?
(Witten)
Topological
String Theory
with twistor
target space CP3
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Perturbative
N=4 super
Yang-Mills
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Twistor space
• Transformation of amplitudes
into twistor space (Penrose)
Tree amplitudes in YM on
degenerate algebraic curves
Degree : N-1+L
• In metric signature ( + + - - ) :
2D Fourier transform
• In twistor space : plane wave
function is a line:
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(Witten)
Degree : number of negative
helicities
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Review: CSW expansion of YangMills amplitudes
• In the CSW-construction : off-shell MHV-amplitudes
building blocks for more complicated amplitude expressions
(Cachazo, Svrcek and Witten)
• MHV vertices:
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Example of how this works
Example of A6(1-,2-,3-,4+,5+,6+)
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Twistor space properties
• Twistor-space properties of gravity:
More complicated!
N=4
-functions
Anti-holomorphic
pieces in gravity
amplitudes
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Signature of non-locality  typical in gravity
Derivatives of - functions
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Collinear and Coplanar Operators
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Twistor space properties
• For gravity : Guaranteed that
Acting with differential operators F and K
• Five-point amplitude. (Giombi, Ricci, Rables-Llana and
Trancanelli; Bern, NEJBB and Dunbar)
Tree amplitudes :
(Bern, NEJBB and Dunbar)
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Recursion
34
BCFW Recursion for trees
Complex
momentum
space!!
Shift of the spinors :
a and b will remain on-shell even after shift
Amplitude transforms as
(Britto, Cachazo, Feng, Witten)
We can now evaluate the contour integral over A(z)
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BCWF Recursion for trees
Given that
• A(z) vanish for z ! 1
• A(z) is a rational function
• A(z) has simple poles
C1 = 0
(Britto, Cachazo, Feng, Witten)
Residues : Determined by factorization properties
Tree amplitude : Factorise in product of tree amplitudes
• in z
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4pt Example
A
B
[2 p] unaffected by shift so non-zero so <2p> must vanish!
¸ 2 ! ¸ 2 + z¸ 1
¸¹ 1 ! ¸¹ 1 ¡ z¸¹ 2
3pt vertex defined in complex momentum
1
¡
+
¡
+
^
^
A 4 (1 ; 2 ; 3 ; 4 ) » A 3 ( 1 ; P14 ; 4 ) £
£ A 3 ( ^2¡ ; 3+ ; ¡ P14
)
s14
[3P^14 ]3
1
h^1P^14 i 3
¡
¡
+
+
A 4 (1 ; 2 ; 3 ; 4 ) »
£
£
^
^
^
s
[23][P14 2]
14
h4^1i hP^14 4i
¡
¡
+
+
[3P^14 1i 3
[34]3
»
=
^
[12][23][41]
[23][2P14 4i h41i h41i [14]
[24]
P^14 = P14 ¡ z[2h1; z =
[41]
MHV vertex expansion for
gravity tree amplitudes
• CSW expansion in gravity
• Shift (Risager) (NEJBB, Dunbar, Ita, Perkins, Risager)
Shift : Correct factorisation
CSW vertex
Reproduce CSW for Yang-Mills
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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) C1
vanishes (justified assumption) :
Mn(0) = sum of residues (as in BCFW),
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MHV vertex expansion for
gravity tree amplitudes
• All poles : Factorise as :
•
vanishes linearly in z :
• Spinor products : not z dependent (normal
CSW)
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MHV vertex expansion for
gravity tree amplitudes
• For gravity : Substitutions
non-locality
MHV amplitudes on the pole ! MHV vertices!
– MHV vertex expansion for gravity
Contact term!
Matter MHV expansion considered by (Bianchi, Elvang, Friedman) problem with
expansion beyond 12pt..
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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 CSW
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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
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Simplicity…
SUSY N=4, N=1,
QCD, Gravity..
Trees
Twistors
(Witten)
Hidden Beauty!
New simple analytic
expressions
Trees simple and symmetric
l
Cuts
Loops simple and symmetric
Unitarity
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