Transcript Amplitudes in Gravity and Yang- Mills Theories University of Sussex

Amplitudes in Gravity and YangMills Theories
Workshop on Continuum and Lattice
Approaches to Quantum Gravity
University of Sussex
Niels Emil Jannik Bjerrum-Bohr
Includes work in collaboration with
Z. Bern, D.C. Dunbar, H. Ita, W. Perkins, K. Risager and P. Vanhove
Introduction
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Twistor space / New insights
Amplitudes N=4,
N=1, QCD
at NLO, Gravity..
Trees
Twistors
(Witten)
Hidden Beauty!
Loop amplitudes
Simple expressions
for amplitudes
Unitarity
Cuts
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Yang-Mills vs Gravity
Yang-Mills
Gravity
• Twistor space
structure??!
• Yang-Mills provide Inspiration
•Twistor theory for gravity?
• Much progress
•SUSY theories
•QCD
• Progress
•SUSY theories esp N = 8 SUGRA
•Less for pure gravity
• Many new results both
•Trees
•One-loop
•n-Loops
•Planar theories
•Conformal invariance
•New results
•Trees
•One-loop (SUGRA)
•n-Loop (3-loop counterterms)
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pp ! jets
Motivation QCD / SUSY
Signals of new physics
The LHC collider approaching
Smaller scales / higher energies..
New physics?? Supersymmetry?
Theory
versus
Experiment
Higgs?…
Precision calculations:
QCD background at NLO
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Motivation gravity
• Spinor – Helicity / Twistor space methods
– Analytic structure of amplitudes
– MHV rules for gravity
Witten
(Witten)
• Gravity from (Yang-Mills)2
– KLT / String based rules
– Recursion / MHV rules
• Extra cancellations in gravity / finiteness of N=8???
– Amplitudes at multi-loop level
– Factorisation of amplitudes
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Calculation of perturbative
amplitudes
# Feynman diagrams:
Factorial Growth!
Momentum vectors :
(pi ¢ pj)
Generic Feynman amplitudes
Sum over topological
different diagrams
External polarisation
tensors :
(pi ¢ εj) (ε i ¢ ε j) 7
Amplitudes
Specifying external
polarisation tensors (ε i , ε j)
Colour ordering
Simplifications
Tr(T1 T2 .. Tn)
Recursion
Spinor-helicity
formalism
Loop amplitudes
(Unitarity,
Supersymmetric
decomposition)
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Gravity Trees
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Quantum theory for gravity
• Gravity as a theory of point-like interactions
• Non-renormalisable theory!
Dimensionful
GN=1/M2planck
• Traditional belief : – no known symmetry can remove higher
derivative divergences.. String theory can by introducing new length scale
• Focus: N=8 supergravity – maximal supersymmetry
(Cremmer,Julia, Scherk;
Cremmer, Julia)
– Also cancellations in pure gravity as well..
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Gravity Amplitudes
Expand Einstein-Hilbert Lagrangian :
Infinitely
many
vertices
Vertices: 3pt, 4pt, 5pt,..n-pt
Feynman diagrams:
Complicated expressions for vertices!
not attractive...!
45 terms
+ sym
(Sannan)
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Gravity Amplitudes
KLT relationship (Kawai, Lewellen and Tye)
The KLT relationship relates open and
closed strings
Not manifest
crossing symmetry
(Bern, Carrasco,
Johansson
Better understanding
of KLT /
organisation of
amplitudes
KLT not manifestly crossing symmetric – explicit representation :
KLT not the simplest
form but better
than
Feynman diagrams
Momentum prefactors cancel double poles
Simplicity of YM amplitudes!!12
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 those of YM):
(Xu, Zhang,
Chang) 13
Scattering amplitudes in D=4

Amplitudes in gravity theories as well as
YM 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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Yang-Mills
Trees
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Yang-Mills MHV-amplitudes
(n) same helicities vanishes
Atree(1+,2+,3+,4+,..) = 0
(n-1) same helicities vanishes
Atree(1+,2+,..,j-,..) = 0
Tree amplitudes
First non-trivial
example,
(M)aximally
(H)elicity (V)iolating
(MHV) amplitudes
(n-2) same helicities:
Atree(1+,2+,..,j-,..,k-,..)  0
One single term!!
Atree MHV Given by the formula
(Parke and Taylor) and proven
by (Berends and Giele)
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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
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Simplifications from SpinorHelicity
Huge simplifications
45 terms
+ sym
Vanish in spinor helicity formalism
Gravity:
Contractions
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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:
(Witten)
Degree : number of negative
helicities
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Review: CSW expansion of YM
amplitudes
• In the CSW-construction : off-shell MHV-amplitudes building
blocks for more complicated amplitude expressions
(Cachazo, Svrcek and Witten) Vertex construction !
spin off of twistor support properties
• MHV vertices:
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CSW expansion of amplitudes
• Example of A6(1-,2-,3-,4+,5+,6+)
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Twistor space properties for gravity
• Twistor-space properties N=8 Supergravity:
More complicated!
N=4
-functions
Anti-holomorphic
pieces in gravity
amplitudes
Signature of non-locality ! typical in gravity
Derivatives of - functions
N=8
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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 :
Gravity
(Bern, NEJBB and Dunbar)
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BCFW
Recursion
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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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Recursion for tree amplitudes
• Tree-level : No other factorizations in complex plane
Interesting Fact
Only 3pt
amplitudes
needed
Generating gravity amplitudes from IR and recursion
(Bedford, Brandhuber, Spence and Travaglini; Cachazo and Svrcek;
NEJBB, Dunbar, Ita; Arkani-Hamed, Cachazo, Kaplan)
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MHV vertex
construction
gravity
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MHV vertex expansion for
gravity tree amplitudes
• CSW expansion in gravity
(NEJBB, Dunbar, Ita, Perkins, Risager;
• Shift (Risager)
Bianchi ,Elvang, Friedman)
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,
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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!
Some issue for amplitudes beyond 12pt .. Unresolved (Bianchi, Elvang, Friedman)
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Gravity
Trees
Gravity
tree properties
MHV rules for gravity
Recursion
(Bedford,
Brandhuber,Spence,
Travaglini; Cachazo,
Svrtec; NEJBB, Dunbar,
Ita; Arkani-Hamed,
Kaplan; Hall; Cheung,
Arkani-Hamed, Cachazo,
Kaplan)
MHV
(NEJBB,Dunbar,Ita,Perkins,Risager;
Bianchi, Elvang, Freedman; Mason,
Skinner; Boels, Larsen, Obers, Vonk)
Gravity scaling behaviour: Unexpected!!
A(z) » 1/z2
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One Loop
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Unitarity cuts
• Unitarity methods are building on the
cut equation
Singlet
Non-Singlet
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General 1-loop amplitudes
n-pt amplitude
Vertices
carry factors
of loop
momentum
p = 2n for gravity
p=n for YM
Propagators
(Passarino-Veltman) reduction
Collapse of a propagator
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Supersymmetric decomposition
The three types of multiplets are:
• Linked by :
• QCD amplitudes for gluons :
Combine:
N=4 : vector multiplet
N=1 : chiral multiplet
+ extra A[0] contribution
(that may contain rational non cut contributions)
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Supersymmetric
decomposition in YM
• Super-symmetry imposes a simplicity of the expressions for loop
amplitudes.
– For N=4 YM only scalar boxes appear.
– For N=1 YM scalar boxes, triangles and bubbles appear.
• One-loop amplitudes are built up from a linear combination of
terms (Bern, Dixon, Dunbar, Kosower).
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Quadruple cuts in complex
momenta
• Observation : Quadruple cuts of N = 4 box coefficients
) Coefficients of box functions by algebra
(Britto, Cachazo and Feng)
Solving the on-shell conditions
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One-loop YM
• Many new results
(Britto, Cachazo, Feng)
N=4 SYM
N=1 SYM
• Many new techniques
– Recursion
(Berger, Bern, NEJBB, Dixon, Dunbar
Forde, Ita, Kosower, Su, Xiao, Yang, Zhu)
– Direct extraction in complex plane (Forde)
– Improved cutting techniques / better results for trees
• Phenomelogically interesting amplitudes
2p ! (Z,W..) + · 4 jets
– NLO is needed for precision at LHC
A key research task!!
Issues:
Rational pieces
Bulky results
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No-Triangle Hypothesis
Justified suggestion…….
Factorisation suggests this is true for all
one-loop amplitudes
Consequence: N=8 supergravity same one-loop
structure as N=4 SYM
Evidence?
True for 4pt
Direct
evaluation
of cuts
(Green,Schwarz,Brink)
n-point MHV
(Bern,Dixon,Perelstein,Rozowsky)
6pt NMHV (IR)
(Bern, NEJBB, Dunbar,Ita)
6pt Proof
7pt evidence
n-pt proof
(NEJBB, Dunbar,Ita, Perkins, Risager;
Bern, Carrasco, Forde, Ita, Johansson)
(NEJBB, Vanhove; Arkani-Hamed,
Cachazo, Kaplan)
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No-Triangle Hypothesis by
Cuts
Attack different parts of amplitudes 1) .. 2) .. 3) ..
(1)
Look at soft divergences (IR)
! 1m and 2m triangles
Check that boxes gives the correct IR divergences
(2)
Explicit unitary cuts
! bubble and 3m triangles
(3)
Factorisation
! rational terms.
(NEJBB, Dunbar,Ita, Perkins, Risager;
Arkani-Hamed, Cachazo, Kaplan) 42
Boxes or Quadruple cuts in
complex momenta
• Observation : Quadruple cuts of N = 4 box coefficients
) Coefficients of box functions by algebra
(Britto, Cachazo and Feng)
Solving the on-shell conditions
(l1)2=0, (l2)2=0, (l3)2=0, (l4)2=0
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Supergravity boxes / amplitudes
KLT
N=4 YM results can be recycled
into results for N=8 supergravity
(Bern, NEJBB, Dunbar)
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Supergravity boxes /
amplitudes
Box Coefficients
(Bern, NEJBB, Dunbar)
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String based
analysis
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No-triangle hypothesis
(NEJBB, Vanhove)
Generic loop amplitude
Passarino-Veltman
Naïve counting!!
Tensor integrals derivatives in Qn
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No-triangle hypothesis
String based formalism natural basis of integrals is
Amplitude takes the form
Constraint from SUSY
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No-triangle hypothesis
Now if we look at integrals
Typical expressions
Use
+ integration by parts
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No-triangle hypothesis
N=8 Maximal Supergravity
(r = 2 (n – 4), s = 0)
(NEJBB, Vanhove)
(r = 2 (n – 4) - s, s >0)
Higher dimensional contributions
– vanish by amplitude gauge
invariance
Proof of No-triangle hypothesis
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No-triangle hypothesis
Generic gravity theories:
N · 3 theories
constructable from
cuts
• Prediction N=4 SUGRA
• Prediction pure gravity
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Gravity Multiloop
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No-triangle for multiloops
•
No-triangle hypothesis 1-loop
• Consequences for powercounting
arguments above one-loop..
Possible to obtain YM bound??
D < 6/L + 4 for gravity???
D < 10/L + 2
Two-particle cut might miss certain cancellations
Bound might be too conservative!!
Iterated two-particle cut
Three/N-particle cut
Explicitly possible to
see extra cancellations!
(Bern, Dixon, Perelstein, Rozowsky; Bern, Dixon,
Roiban)
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Two-Loop SYM/ Supergravity
Explicit at two loops :
‘No-triangle hypothesis’ holds at two-loops 4pt
Two-loop 5pt would be
interesting to know
(Bern,Rozowsky,Yan)
(Bern,Dixon,Dunbar,Perelstein,Rozowsky)
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Three-Loop SYM/ Supergravity
• Three-loop four-point amplitude of N=8 supergravity
directly constructed via unitarity.
• The amplitude is ultraviolet finite in four dimensions.
• Degree of divergence in D dimensions at three loop
to be no worse than that of N=4 super-Yang-Mills
theory. Confirms ‘no-triangle hypothesis’ for three
loops.
– Remark: Surprising extra cancellations between diagrams
which are not just ‘triangle-type’..
(Bern, Carrasco, Dixon,
Johansson, Kosower, Roiban)
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Outlook
• Wittens conjecture ! very inspiring time
– Further investigations needed to fully grasp..1-loop??
• Trees solved by
– MHV rules
– BCFW recursion
• Analytic properties of amplitudes
– Complex analysis methods
– Recursion results extended (double shifts etc)
– Development of new techniques
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Conclusions YM
Important further investigations..
SYM theories with massless particles : Almost a closed
chapter at one-loop
Multi-loop N=4 SYM:
– real phenomenology at NLO a challenge!!
•
•
–
Further push for phenomenological results
Automatisation
(Bern et al)
1. Methods are well-developed
2. Possible to construct automatic computer code
More investigations in analytic results
Key: Simplicity / Twistor support..
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Conclusions Gravity
• Graviton amplitudes $ much benefit from recent progress
(..twistor / helicity structure, hidden simplicity,
string based formalism..)
• Gravity $ much simpler – than Lagrangian / power counting
indicate (no-triangle property $ extra simplicity..)
• Unordered amplitudes might be even simpler than
ordered amplitudes (due to lack of boundary terms..)
Consequences at higher loop order
Finiteness??!
• String based / helicity formalism is very helpful
– however better ways to deal with gravity amplitudes
still important to focus on..
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