Twistors and Perturbative Gravity

1
Twistors and Perturbative Gravity
Dave Dunbar, Swansea University
Steve Bidder
Harald Ita
Warren Perkins
Emil Bjerrum-Bohr
Zvi Bern (UCLA) and Kasper Risager (NBI)
UK Theory Institute 20/12/05
2
Plan

• Recently a duality between Yang-Mills and
  twistor string theory has inspired a variety of
  new techniques in perturbative Yang-Mills
  theories. First part of talk will review these
• Look at Gravity Amplitudes
• -which, if any, features apply to gravity
• Application Loop Amplitudes
• N4 Yang Mills
• N8 Supergravity
• Consequences and Conclusions

3
Duality with String Theory

• Witten (2003) proposed a Weak-Weak duality
  between
• A) Yang-Mills theory ( N4 )
• B) Topological String Theory with twistor target
  space

-Since this is a weak-weak duality perturbative
S-matrix of two theories should be identical
order by order
-True for tree level scattering
Rioban, Spradlin,Volovich
4
Featutures of Duality

• Topological String Theory with twistor target
  space CP3
• -open string instantons correspond to Yang-Mills
  states
• -theory has conformal symmetry, N4 SYM
• -closed string states correspond to N4
  superconformal gravity
• - N lt 4 ??

BerkovitsWitten, Berkovits
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Is the duality useful?
Theory A hard, interesting
Theory B easy
-duality may be useful indirectly
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Twistor Definitions

• Consider a massless particle with momenta
• We can realise as
• So we can express
• where are two component Weyl
  spinors

7

• This decomposition is not unique but

We can also turn polarisation vector into
fermionic objects, Spinor Helicity formalism
Xu, Zhang,Chang 87
-Amplitude now a function of spinor variables
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Transform to Twistor Space
Penrose
-note we make a choice which to transform
new coordinates
Twistor Space is a complex projective (CP3) space
n-point amplitude is defined on (CP3)n
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Twistor Structure

• Conjecture (Witten) amplitudes have non-zero
  support on curves in twistor space
• support should be a curve of degree
• (number of ve helicities)(loops) -1

Carrying out the transform is problematic,
instead we can test structure by acting with
differential operators
10
We test collinearity and coplanarity by acting
with differential operators Fijk and Kijkl
-action of F is obtained using fact that points
Zi collinear if
Allows us to test without determining
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Collinearity of MHV amplitudes

• We organise gluon scattering amplitudes according
  to the number of negative helicities
• Amplitude with no or one negative helicities
  vanish
• for supersymmetric theories to all order
• for non-supersymmetric true for tree
  amplitudes
• Amplitudes with exactly two negative helicities
  are refered to as MHV amplitudes

Parke-Taylor, Berends-Giele
(amplitudes are color-ordered)
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Collinearity of MHV amplitudes

• MHV amplitudes only depend upon
• So, for Yang-Mills, FijkAn0 trivially
• MHV amplitudes have collinear support when
  transforming to a function in twistor space since
• Penrose transform yields a ? function after
  integration .

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MHV amplitudes have suppport on line only
Curve of degree 1 ( 02-1)
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NMHV amplitudes in twistor space

• amplitudes with three ve helicity known as NMHV
  amplitudes
• remarkably NMHV amplitudes have coplanar support
  in
• twistor space
• prove this not directly but by showing

- time to look at techniques motivated by duality
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TechniquesI MHV-vertex construction

• Promotes MHV amplitude to fundamental object by
  off-shell continuation

Cachazo Svrcek Witten,
Nair

• Works for gluon scattering tree amplitudes
• Works for (massless) quarks
• Works for Higgs and Ws
• Works for photons
• -No known derivation from a
  Lagrangian
• (but Khoze, Mason,
  Mansfield)

Wu,Zhu Su,Wu Georgiou Khoze
Badger, Dixon, Glover, Forde, Khoze, Kosower
Mastrolia
OzerenStirling

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A MHV diagram
_
_

_

_
_

_

_


_

-three point vertices allowed
-number of vertices (number of -) -1
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eg for NMHV amplitudes
k
k1
q

-

1-
3-
2-
2(n-3) diagrams
Topology determined by number of ve helicity
gluons
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Coplanarity-byproduct of MHV vertices
-NMHV amplitudes is sum of two MHV vertices
Two intersecting lines in twistor space define
the plane
Curve is a degenerate curve of degree 2
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Techniques2 Recursion Relations
Britto,Cachazo,Feng and Witten

• Return of the analytic S-matrix!
• Shift amplitude so it is a complex function of z

Amplitude becomes an analytic function of z, A(z)
Full amplitude can be reconstructed from analytic
properties
Within the amplitude momenta containing only one
of the pair are z-dependant q(z)
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-results in recursive on-shell relation
( cf Berends-Giele off-shell recursive technique
)
q
(three-point amplitudes must be included)
Amplitude has poles ? Amplitude is poles
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MHV vs BCF recursion

• Difference
• MHV asymmetric between helicity sign
• BCF chooses two special legs
• For NMHV MHV expresses as a product of
  two MHV
• BCF uses (n-1)-pt NMHV
• Similarities-
• both rely upon analytic structure
• both for trees but Loops
• MHV Bedford,
  Brandhuber,Spence, Travaglini
• Recursive Bern,Dixon Kosower
• Bern,
  Bjerrum-Bohr, Dunbar, Ita, Perkins

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Gravity-Strategy

• 1) Try to understand twistor structure
• 2) Develop formalisms

- a priori we might expect Einstein gravity to
contain no knowledge of twistor structure since
duality contains conformal gravity
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..Perturbative Quantum Gravityfirst some review
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• Feynman diagram approach to perturbative quantum
  gravity is extremely complicated
• Gravity (Yang-Mills)2
• Feynman diagrams for Yang-Mills horrible mess
• How do we deal with (horrible mess)2

Using traditional techniques even the four-point
tree amplitude is very difficult
Sannan,86
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Kawai-Lewellen-Tye Relations
Kawai,Lewellen Tye, 86
-pre-twistors one of few useful
techniques -derived from string theory
relations -become complicated with increasing
number of legs -involves momenta prefactors -MHV
amplitudes calculated using this
Berends,Giele, Kuijf
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Recursion for Gravity

• Gravity, seems to satisfy the conditions to use
  recursion relations
• Allows (re)calculation of MHV gravity tree amps
• Expression for six-point NMHV tree

Bedford, Brandhuber, Spence, Travaglini Cachazo,Sv
rcek
Bedford, Brandhuber, Spence, Travaglini
Cachazo,Svrcek
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Gravity MHV amplitudes

• For Gravity Mn is polynomial in with degree
  (2n-6), eg
• Consequently
• In fact..
• Upon transforming Mn has a derivative of ?
  function support

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Coplanarity
NMHV amplitudes in Yang-Mills have coplanar
support
For Gravity we have verified
n5 by Giombi, Ricci, Robles-Llana Trancanelli
n6,7,8 Bern, Bjerrum-Bohr,Dunbar
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MHV construction for gravity

• Need the correct off-shell continuation
• Proved to be difficult
• Resolution involves continuing the of the
  negative
• helicity legs
• The ri are chosen so that
• a) momentum is conserved
• b) multi-particle poles q2(ri) are on-shell
• -this fixes them uniquely

Shift is the same as that used by Risager to
derive MHV rules using analytic structure
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Eg NMHV amplitudes
k
k1

-

1-
3-
2-
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Loop Amplitudes

• Loop amplitudes perhaps the most interesting
  aspect of gravity calculations
• UV structure always interesting
• Chance to prove/disprove our prejudices
• Studying Amplitudes may uncover symmetries not
  obvious in Lagrangian
• Loop amplitudes are sensitive to the entire
  theory
• For loops we must be specific about which theory
  we are studying

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Tale of two theories, N4 SYM vs N8
Supergravity
Cremmer, Julia, Scherk
N4 SYM is maximally supersymmetric gauge theory
(spin 1 ) N8 Supergravity is maximal theory
with gauged supersymmetry (spin 2 )
-both appear in low energy limit of superstring
theory -S-matrix of both theories is constrained
by a rich set of symmetries -N4 key in Weak-Weak
duality -in D4 YM has dimensionless coupling
constant wheras gravity has a dimensionful
coupling constant -both theories are extremelly
important models toy or otherwise
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General Decomposition of One- loop n-point
Amplitude
pn Yang-Mills p2n Gravity
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Passarino-Veltman reduction
Decomposes a n-point integral into a sum of (n-1)
integral functions obtained by collapsing a
propagator

• process continues until we reach four-point
  integral functions with (in yang-mills up to
  quartic numerators) In going from 4-gt 3 scalar
  boxes are generated
• similarly 3 -gt 2 also gives scalar triangles. At
  bubbles process ends. Quadratic bubbles can be
  rational functions involving no logarithms.
• so in general, for massless particles

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N4 Susy Yang-Mills

• In N4 Susy there are cancellations between the
  states of different spin circulating in the loop.
• Leading four powers of loop momentum cancel (in
  well chosen gauges..)
• N4 lie in a subspace of the allowed amplitudes
  
• 
  (Bern,Dixon,Dunbar,Kosower, 94)
• Determining rational ci determines amplitude
• 4pt. Green, Schwarz, Brink
• MHV,6pt 7pt,gluinos Bern, Dixon, Del Duca
  Dunbar, Kosower
• Britto,
  Cachazo, Feng Roiban Spradlin Volovich
• Bidder,
  Perkins, Risager

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Basis in N4 Theory
easy two-mass box
hard two-mass box
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Box Coefficients-Twistor Structure

• Box coefficients has coplanar support for NMHV
  1-loop
• amplitudes

-true for both N4 and QCD!!!
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N8 Supergravity

• Loop polynomial of n-point amplitude of degree
  2n.
• Leading eight-powers of loop momentum cancel (in
  well chosen gauges..) leaving (2n-8)
• Beyond 4-point amplitude contains
  triangles..bubbles
• Beyond 6-point amplitude is not cut-constructible

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No-Triangle Hypothesis
-against this expectation, it might be the case
that.
Evidence?
true for 4pt n-point MHV 6pt NMHV
Green,Schwarz,Brink
Bern,Dixon,Perelstein,Rozowsky
Bjerrum-Bohr, Dunbar,Ita
-factorisation suggests this is true for all
one-loop amplitudes
consequences?

• One-Loop amplitudes N8 SUGRA look just like N4
  SYM

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Beyond one-loops
Two-Loop Result obtained by reconstructing
amplitude from cuts
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Two-Loop SYM/ Supergravity
Bern,Rozowsky,Yan
IPs,t planar double box integral
Bern,Dixon,Dunbar,Perelstein,Rozowsky (BDDPR)
-N8 amplitudes very close to N4
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Beyond 2-loops UV pattern (98)
Honest calculation/ conjecture (BDDPR)
N8 Sugra
N4 Yang-Mills
Based upon 4pt amplitudes
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Pattern obtained by cutting
Beyond 2 loop , loop momenta get caught
within the integral functions Generally, the
resultant polynomial for maximal supergravity of
the square of that for maximal super
yang-mills Eg in this case YM P(li)(l1l2)2
SUGRA P(li)((l1l2)2)2
l1
l2
I P(li)
BUT..
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on the three particle cut..
For Yang-Mills, we expect the loop to yield a
linear pentagon integral
For Gravity, we thus expect a quadratic pentagon
However, a quadratic pentagon would give
triangles which are not present in an on-shell
amplitude
-indication of better behaviour in entire
amplitude
? relations to work of Green and Van Hove
45

• Does no-triangle hypothesis imply
  perturbative expansion of N8 SUGRA more similar
  to that of N4 SYM than power counting/field
  theory arguments suggest????
• If factorisation is the key then perhaps yes.
  Four point amplitudes very similar
• Is N8 SUGRA perturbatively finite?????

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Conclusions

• Perturbation theory is interesting and still
  contains many surprises
• Recent discoveries are interesting and useful
• Studying on-shell amplitudes can give information
  not obvious in the Lagrangian
• Gravity calculations amenable to many of the new
  twistor inspired techniques
• -both recursion and MHV vertex formulations
  exist
• -perturbative expansion of N8 seems to be
  surprisingly simple. This may have consequences
  for the UV behaviour
• Consequences for the duality?