Valentin V. Khoze- Gauge Theory Amplitudes, Scalar Graphs and Twistor Space
Structure of Amplitudes in Gravity I Lagrangian Formulation of Gravity, Tree amplitudes, Helicity...
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Transcript of 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
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 =R
d4xhp
¡ gRi
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 terms
Gronningen 3-5 Dec 2009
Playing with Gravity 8
Effective Lagrangian
L E f f =Z
d4xhp
¡ gR (Einstein-Hilbert)
+ c1R2¹ º + c2R2 : : :(higher derivatives)
+ L matter
i(matter couplings)
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Playing with Gravity 9
Features:
• Derivative terms consistent with symmetry
• Action: valid till Planck scale by construction
Quantising Gravity
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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¯ ¡
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
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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)
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12Playing with Gravity
X
¾(1;:::M )
( )
Computation of perturbative amplitudes
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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 ¢pj
pi ¢²j ²i ¢²j
Amplitudes
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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
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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
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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
xx
xx
. .
12
3
M
...+ +=
1
2
1 M 12
3
s12 s1M s123
Gravity Amplitudes
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Playing with Gravity 21
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!!
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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 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…
Gronningen 3-5 Dec 2009
25Playing with Gravity
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
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
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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:
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29Playing with Gravity
Example of how this works
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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
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31Playing with Gravity
32
Collinear and Coplanar Operators
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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
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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)
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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)
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36Playing with Gravity
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)
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),
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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
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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
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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
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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