University%20of%20Sussex

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Amplitudes in Gravity and Yang-Mills Theories

• 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

Workshop on Continuum and Lattice Approaches to
Quantum Gravity
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Introduction
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Twistor space / New insights
Twistors
Trees

• Amplitudes N4, N1, QCD
• at NLO, Gravity..

(Witten)
Hidden Beauty!
Loop amplitudes
Simple expressions for amplitudes
Cuts
Unitarity
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Yang-Mills vs Gravity

• Yang-Mills
• Twistor space structure??!
• Much progress
• SUSY theories
• QCD
• Many new results both
• Trees
• One-loop
• n-Loops
• Planar theories
• Conformal invariance

• Gravity
• Yang-Mills provide Inspiration
• Twistor theory for gravity?
• Progress
• SUSY theories esp N 8 SUGRA
• Less for pure gravity
• New results
• Trees
• One-loop (SUGRA)
• n-Loop (3-loop counterterms)

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Motivation QCD / SUSY
pp ! jets
Signals of new physics
The LHC collider approaching
Smaller scales / higher energies..
New physics??
Higgs?
Supersymmetry?
Precision calculations QCD background at NLO
Theory versus Experiment
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Motivation gravity

• Spinor Helicity / Twistor space methods
• Analytic structure of amplitudes
• MHV rules for gravity
• Gravity from (Yang-Mills)2
• KLT / String based rules
• Recursion / MHV rules
• Extra cancellations in gravity / finiteness of
  N8???
• Amplitudes at multi-loop level
• Factorisation of amplitudes

Witten
(Witten)
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Calculation of perturbative amplitudes

• Momentum vectors (pi pj)
• External polarisation tensors
• (pi ej) (e i e j)

Feynman diagrams Factorial Growth!
Generic Feynman amplitudes
Sum over topological different diagrams
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Amplitudes
Colour ordering
Specifying external polarisation tensors (e i ,
e j)
Tr(T1 T2 .. Tn)
Simplifications
Recursion
Loop amplitudes (Unitarity, Supersymmetric
decomposition)
Spinor-helicity formalism
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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!
• Traditional belief no known symmetry can
  remove higher derivative divergences..
• Focus N8 supergravity maximal supersymmetry
• Also cancellations in pure gravity as well..

Dimensionful GN1/M2planck
String theory can by introducing new length scale
(Cremmer,Julia, Scherk Cremmer, Julia)
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Gravity Amplitudes

• Expand Einstein-Hilbert Lagrangian
• Vertices 3pt, 4pt, 5pt,..n-pt

Infinitely many vertices
Feynman diagrams not attractive...!
Complicated expressions for vertices!
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
Simplicity of YM amplitudes!!
Momentum prefactors cancel double poles
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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)
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Scattering amplitudes in D4

• 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
Tree 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)

First non-trivial example, (M)aximally
(H)elicity (V)iolating (MHV) amplitudes
One single term!!
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Gravity MHV amplitudes

• Can be generated from KLT via YM MHV amplitudes.
• (Berends-Giele-Kuijf) recursion formula

Anti holomorphic Contributions feature in
gravity
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Simplifications from Spinor-Helicity
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)
• 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

Degree N-1L
(Witten)
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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)
• MHV vertices

Vertex construction ! spin off of twistor
support properties
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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 N8 Supergravity
• More complicated!

N4
?-functions
Signature of non-locality ! typical in gravity
Derivatives of ?- functions
Anti-holomorphic pieces in gravity amplitudes
N8
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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
Gravity
(Bern, NEJBB and Dunbar)
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BCFW Recursion
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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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BCWF Recursion for trees

• Given that
• A(z) vanish for z ! 1
• 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

(C1 0)
(Britto, Cachazo, Feng, Witten)
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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 constructiongravity
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MHV vertex expansion for gravity tree amplitudes

• CSW expansion in gravity
• Shift (Risager)
• Reproduce CSW for Yang-Mills

(NEJBB, Dunbar, Ita, Perkins, Risager Bianchi
,Elvang, Friedman)
Shift Correct factorisation
CSW vertex
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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
• MHV amplitudes on the pole -gt MHV vertices
• MHV vertex expansion for gravity

non-locality
Contact term!
Some issue for amplitudes beyond 12pt ..
Unresolved (Bianchi, Elvang, Friedman)
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Gravity tree properties
Gravity Trees

• MHV rules for gravity
• Recursion

MHV
(NEJBB,Dunbar,Ita,Perkins,Risager Bianchi,
Elvang, Freedman Mason, Skinner Boels, Larsen,
Obers, Vonk)
(Bedford, Brandhuber,Spence, Travaglini Cachazo,
Svrtec NEJBB, Dunbar, Ita Arkani-Hamed, Kaplan
Hall Cheung, Arkani-Hamed, Cachazo, Kaplan)
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
p 2n for gravity pn for YM
n-pt amplitude
Vertices carry factors of loop momentum
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
• extra A0 contribution
• (that may contain rational non cut contributions)

N4 vector multiplet N1 chiral multiplet
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Supersymmetric decomposition in YM

• Super-symmetry imposes a simplicity of the
  expressions for loop amplitudes.
• For N4 YM only scalar boxes appear.
• For N1 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
(Britto, Cachazo, Feng)

• Many new results
• Many new techniques
• Recursion
• 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

N4 SYM
N1 SYM
(Berger, Bern, NEJBB, Dixon, Dunbar Forde, Ita,
Kosower, Su, Xiao, Yang, Zhu)
A key research task!!
Issues Rational pieces Bulky results
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No-Triangle Hypothesis
Factorisation suggests this is true for all
one-loop amplitudes
Justified suggestion.
Consequence N8 supergravity same one-loop
structure as N4 SYM
Evidence?
True for 4pt n-point MHV 6pt NMHV (IR)
6pt Proof 7pt evidence n-pt proof
(Green,Schwarz,Brink)
(Bern,Dixon,Perelstein,Rozowsky)
(Bern, NEJBB, Dunbar,Ita)
Direct evaluation of cuts
(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
• (2) Explicit unitary cuts
• ! bubble and 3m triangles
• (3) Factorisation
• ! rational terms.

Check that boxes gives the correct IR divergences
(NEJBB, Dunbar,Ita, Perkins, Risager
Arkani-Hamed, Cachazo, Kaplan)
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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)20, (l2)20, (l3)20, (l4)20
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Supergravity boxes / amplitudes
KLT
N4 YM results can be recycled into results for
N8 supergravity
(Bern, NEJBB, Dunbar)
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Supergravity boxes / amplitudes
Box Coefficients
(Bern, NEJBB, Dunbar)
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String basedanalysis
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No-triangle hypothesis
(NEJBB, Vanhove)
Generic loop amplitude
Naïve counting!!
Passarino-Veltman
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
N8 Maximal Supergravity
(r 2 (n 4), s 0)
(NEJBB, Vanhove)
(r 2 (n 4) - s, s gt0)
Higher dimensional contributions vanish by
amplitude gauge invariance
Proof of No-triangle hypothesis
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No-triangle hypothesis
N 3 theories constructable from cuts

• Generic gravity theories
• Prediction N4 SUGRA
• Prediction pure gravity

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Gravity Multi-loop
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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 lt 6/L 4 for gravity???
D lt 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
(Bern,Rozowsky,Yan)
Two-loop 5pt would be interesting to know
(Bern,Dixon,Dunbar,Perelstein,Rozowsky)
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Three-Loop SYM/ Supergravity

• Three-loop four-point amplitude of N8
  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 N4
  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 N4 SYM
• real phenomenology at NLO a challenge!!
• Further push for phenomenological results
• Automatisation
• Methods are well-developed
• Possible to construct automatic computer code
• More investigations in analytic results
• Key Simplicity / Twistor support..

(Bern et al)
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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..)
• String based / helicity formalism is very helpful
  
• however better ways to deal with gravity
  amplitudes
• still important to focus on..

Consequences at higher loop order Finiteness??!