twistors and perturbative gravity emil bjerrum-bohr uk theory institute 20/12/05 steve bidder harald...
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Twistors and Perturbative Gravity
Emil Bjerrum-Bohr
UK Theory Institute 20/12/05UK Theory Institute 20/12/05
Steve Bidder
Harald Ita
Warren Perkins
+Zvi Bern (UCLA) and Kasper Risager (NBI)
Dave Dunbar, Swansea University
D Dunbar UK Inst 05 2/46
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 N=4 Yang –Mills N=8 Supergravity
• Consequences and Conclusions
D Dunbar UK Inst 05 3/46
Duality with String Theory
Witten (2003) proposed a Weak-Weak duality between
• A) Yang-Mills theory ( N=4 )• 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
D Dunbar UK Inst 05 4/46
Featutures of Duality Topological String Theory with twistor target space CP3
-open string instantons correspond to Yang-Mills states
-theory has conformal symmetry, N=4 SYM
-closed string states correspond to N=4 superconformal gravity
- N < 4 ??
Berkovits+Witten, Berkovits
D Dunbar UK Inst 05 5/46
Is the duality useful?
Theory A :Theory A :
hard, hard, interestinginteresting
Theory B: Theory B:
easyeasy
Perturbative Perturbative Gauge Theories,Gauge Theories,hard, interestinghard, interesting
TopologicalTopologicalString TheoryString Theory::
harder, uninteresting harder, uninteresting
-duality may be useful -duality may be useful indirectlyindirectly
D Dunbar UK Inst 05 6/46
Twistor Definitions
• Consider a massless particle with momenta
• We can realise as
• So we can express
where are two component Weyl spinors
D Dunbar UK Inst 05 7/46
•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
D Dunbar UK Inst 05 8/46
Transform to Twistor Space
Twistor Space is a complex projective (CP3) space
n-point amplitude is defined on (CP3)n
new coordinates
-note we make a choice which to transform
Penrose+
D Dunbar UK Inst 05 9/46
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
D Dunbar UK Inst 05 10/46
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
D Dunbar UK Inst 05 11/46
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)
D Dunbar UK Inst 05 12/46
Collinearity of MHV amplitudes• MHV amplitudes only depend upon
• So, for Yang-Mills, FijkAn=0 trivially
• MHV amplitudes have collinear support when transforming to a function in twistor space since
Penrose transform yields a function after integration .
D Dunbar UK Inst 05 13/46
MHV amplitudes have suppport on line only
Curve of degree 1 (= 0+2-1)
D Dunbar UK Inst 05 14/46
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
D Dunbar UK Inst 05 15/46
Techniques:I MHV-vertex construction
• Works for gluon scattering tree amplitudes• Works for (massless) quarks• Works for Higgs and W’s
• Works for photons
-No known derivation from a Lagrangian (but…… Khoze, Mason, Mansfield)
Ozeren+Stirling
Badger, Dixon, Glover, Forde, Khoze, Kosower Mastrolia
Wu,Zhu; Su,Wu; Georgiou Khoze
Cachazo Svrcek Witten,
Nair
• Promotes MHV amplitude to fundamental object by off-shell continuation
D Dunbar UK Inst 05 16/46
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-three point vertices allowed-number of vertices = (number of -)
-1
A MHV diagram
D Dunbar UK Inst 05 17/46
eg for NMHV amplitudes
3-1-
k+
2-
k+1+
2(n-3) diagrams
+
Topology determined by number of –ve helicity gluons
- +q
D Dunbar UK Inst 05 18/46
Coplanarity-byproduct of MHV vertices
Two intersecting lines in twistor space define the plane
-NMHV amplitudes is sum of two MHV vertices
Curve is a degenerate curve of degree 2
D Dunbar UK Inst 05 19/46
Techniques:2 Recursion Relations
• 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
Britto,Cachazo,Feng and Witten
Within the amplitude momenta containing only one of the pair are z-dependant q(z)
D Dunbar UK Inst 05 20/46
-results in recursive on-shell relation
(three-point amplitudes must be included)
1 2
( cf Berends-Giele off-shell recursive technique )
q
Amplitude has poles Amplitude is poles
D Dunbar UK Inst 05 21/46
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
D Dunbar UK Inst 05 22/46
Gravity-Strategy
1) Try to understand twistor structure2) Develop formalisms
- a priori we might expect Einstein gravity to contain no knowledge of twistor structure since duality contains conformal gravity
D Dunbar UK Inst 05 23/46
…..Perturbative Quantum Gravity…first some review
D Dunbar UK Inst 05 24/46
• 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
D Dunbar UK Inst 05 25/46
Kawai-Lewellen-Tye Relations
-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
Kawai,Lewellen Tye, 86
Berends,Giele, Kuijf
D Dunbar UK Inst 05 26/46
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,Svrcek
Bedford, Brandhuber, Spence, Travaglini
Cachazo,Svrcek
D Dunbar UK Inst 05 27/46
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
D Dunbar UK Inst 05 28/46
CoplanarityNMHV amplitudes in Yang-Mills have coplanar support
For Gravity we have verified
n=5 by Giombi, Ricci, Robles-Llana Trancanelli
n=6,7,8 Bern, Bjerrum-Bohr,Dunbar
D Dunbar UK Inst 05 29/46
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
D Dunbar UK Inst 05 30/46
Eg NMHV amplitudes
3-1-
k+
2-
k+1+
+- +
D Dunbar UK Inst 05 31/46
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
D Dunbar UK Inst 05 32/46
Tale of two theories, N=4 SYM vs N=8 Supergravity
N=4 SYM is maximally supersymmetric gauge theory (spin · 1 )
N=8 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
-N=4 key in Weak-Weak duality
-in D=4 YM has dimensionless coupling constant wheras gravity has a dimensionful coupling constant
-both theories are extremelly important models: toy or otherwise
Cremmer, Julia, Scherk
D Dunbar UK Inst 05 33/46
General Decomposition of One- loop n-point Amplitude
Vertices involve loop momentumpropagators
p
degree p in l
p=n : Yang-Mills
p=2n Gravity
D Dunbar UK Inst 05 34/46
Passarino-Veltman reduction
•process continues until we reach four-point integral functions with (in yang-mills up to quartic numerators) In going from 4-> 3 scalar boxes are generated•similarly 3 -> 2 also gives scalar triangles. At bubbles process ends. Quadratic bubbles can be rational functions involving no logarithms. •so in general, for massless particles
Decomposes a n-point integral into a sum of (n-1) integral functions obtained by collapsing a propagator
D Dunbar UK Inst 05 35/46
N=4 Susy Yang-Mills• In N=4 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..)
• N=4 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
D Dunbar UK Inst 05 36/46
Basis in N=4 Theory‘‘easy’ two-mass easy’ two-mass boxbox
‘‘hard’ two-mass hard’ two-mass boxbox
D Dunbar UK Inst 05 37/46
Box Coefficients-Twistor Structure
• Box coefficients has coplanar support for NMHV 1-loop
• amplitudes
-true for both N=4 and QCD!!!
D Dunbar UK Inst 05 38/46
N=8 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
D Dunbar UK Inst 05 39/46
No-Triangle Hypothesis-against this expectation, it might be the case that…….
Evidence?true for 4pt
n-point MHV
6pt NMHV
-factorisation suggests this is true for all one-loop amplitudes
Bern,Dixon,Perelstein,Rozowsky
Bjerrum-Bohr, Dunbar,Ita
Green,Schwarz,Brink
consequences?• One-Loop amplitudes N=8 SUGRA look just like N=4
SYM
D Dunbar UK Inst 05 40/46
Beyond one-loops
Two-Loop Result obtained by reconstructing amplitude from cuts
D Dunbar UK Inst 05 41/46
Two-Loop SYM/ Supergravity
Bern,Rozowsky,Yan
Bern,Dixon,Dunbar,Perelstein,Rozowsky (BDDPR)
-N=8 amplitudes very close to N=4
IPs,t planar double box integral
D Dunbar UK Inst 05 42/46
Beyond 2-loops: UV pattern (98)
D=11
0 #/
D=10
0(!) #/
D=9 0 #/
D=8 #/ #’/+#”/
D=7 0 #/
D=6 0 0
D=5 0 0 0
D=4 0 0 0 0
L=1 L=2 L=3 L=4 L=5 L=6
N=4 Yang-Mills
Honest calculation/ conjecture (BDDPR)
N=8 Sugra
Based upon 4pt amplitudes
D Dunbar UK Inst 05 43/46
Pattern obtained by cuttingBeyond 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)=(l1+l2)2
SUGRA :P(li)=((l1+l2)2)2
I[ P(li)]
l1
l2
BUT…………..
D Dunbar UK Inst 05 44/46
on the three particle cut..
For Yang-Mills, we expect the loop to yield a linear pentagon integralFor 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
D Dunbar UK Inst 05 45/46
• Does ``no-triangle hypothesis’’ imply perturbative expansion of N=8 SUGRA more similar to that of N=4 SYM than power counting/field theory arguments suggest????
• If factorisation is the key then perhaps yes. Four point amplitudes very similar
• Is N=8 SUGRA perturbatively finite?????
D Dunbar UK Inst 05 46/46
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 N=8 seems to be
surprisingly simple. This may have consequences for the UV behaviour
• Consequences for the duality?