twistor inspired techniques in perturbative gauge theories including work with z. bern, s bidder, e...
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Twistor Inspired techniques in Perturbative Gauge Theories
including work with Z. Bern, S Bidder, E
Bjerrum-Bohr, L. Dixon, H Ita, W Perkins K. Risager
KIAS-KIAST KIAS-KIAST 2005
David Dunbar,
Swansea University, Wales
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Outline-Twistor basics
-Weak-Weak Duality
-Cachazo-Svercek-Witten MHV–vertex construction for gluon Scattering
-Britto-CachazoFeng recursive techniques
-Gravity
- Loop amplitudes
- N=4 amplitudes
- twistor structure
- QCD amplitudes
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Twistor Definitions
Consider a massless particle with momenta
We can realise as
With
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Definitions: continued
For a massless particles
Where are two component Weyl spinors
or twistors. This decomposition is not unique but
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Scattering Amplitudes For Gluons
Textbook approach yields amplitude
We rewrite this in terms of twistors in two steps
1) Replacing momentum p 2) replacing polarisation
NB two notations : traditional methods+twistor
Some notation:
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Step2:Spinor HelicityXu, Zhang,Chang 87
Gluon Momenta
Reference Momenta
-extremely useful technique which produces relatively compact expressions for amplitudes
-amplitude now entirely in terms of spinorial variables
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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
-transform like a x-p transform
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Duality with String Theory
Witten’s proposed of 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
-True for tree level gluon scattering
Rioban, Spradlin,Volovich
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Colour-Ordering
Gauge theory amplitudes depend upon colour indices of gluons.
We can split colour from kinematics by colour decomposition
The colour ordered amplitudes have cyclic symmetric rather than full crossing symmetry
Colour ordering is not text-book in field theory books but is in string theory texts
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Twistor Support
Look at simple Yang-Mills Amplitudes in Twistor Space
Look at helicity colour ordered amplitudes,
(all legs outgoing )
-known as MHV amplitude
Parke-Taylor, Berends-Giele
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MHV amplitudes in Twistor Space
Wavefunction of MHV amplitude only depends upon
via factor
So fourier transform gives
Corresponding to amplitude being non-zero only upon a line in twistor space
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MHV amplitudes have suppport on line only
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We can test collinearity without transforming by action with differential operator F
implies A has non-zero support on line defined by points i,j,k
-action of F upon MHV amplitudes is trivial
(useful since Fourier/Penrose transform difficult)
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Similarly there is a coplanarity operator Kijkl
Implies amplitude has non-zero support only in the plane defined by point i,j,k and l
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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
-expected from duality
-support should be a curve of degree n+l-1
Witten
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Is the duality useful?
Theory A :Theory A :
hard, hard, interestinginteresting
Theory B: Theory B:
easyeasy
Perturbative QCD,Perturbative QCD,hard, interestinghard, interesting
TopologicalTopologicalString TheoryString Theory::
harder, uninteresting harder, uninteresting
-duality may be useful indirectly-duality may be useful indirectly
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Inspired by duality – the CSW/MHV-vertex construction
Promotes MHV amplitude to fundamental object by
-Off-shell continuation
-MHV amplitudes have no multi-particle factorisation
Cachazo Svercek Witten 04, (Nair)
Parke-Taylor, Berends-Giele(colour ordered amplitudes)
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-three point vertices allowed-number of vertices = (number of -)
-1
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For NMHV amplitudes
3-1-
k+
2-
k+1+
2(n-3) diagrams
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Topology determined by number of –ve helicity gluons
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Coplanarity
Two intersecting lines in twistor space define the plane
-Points on one MHV vertex
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MHV-vertex constructionWorks for gluon scattering tree amplitudes
Works for (massless) quarks
Works for Higgs and W’s
Works for photons
Works for gravity
Bjerrum-Bohr,DCD,Ita,Perkins, Risager
Ozeren+Stirling
Badger, Dixon, Glover, Forde, Khoze, Kosower Mastrolia
Wu,Zhu; Su,Wu; Georgiou Khoze
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Inspired by duality –BCFW construction
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 P(z)
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Provided,
then
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-proof
Czi
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Use this with f(z)=A(z)/z
Provided A(z) vanishes at infinity the contour integral vanishes.
The function A(z)/z has a pole at z=0 with residue A(0) which is just the unshifted amplitude
Residues occur when amplitude factorises on multiparticle pole (including two-particles)
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-results in recursive on-shell relation
(three-point amplitudes must be included)
1 2
NB Berends-Giele recursive techniques
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CSW vs BCF
Difference
CSW asymmetric between helicity sign
BCF chooses two special legs
For NMHV : CSW expresses as a product of two MHV
: BCF uses (n-1)-pt NMHV
Similarities-
both rely upon analytic structure
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CSW can be derived from a type of analytic shiftRisager; Bjerrum-Bohr,Dunbar,Ita,Perkins and Risager, 05
gives a the CSW expansion of NMHV
-this a combination of three shifts
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Gravity Amplitudes -very little known for graviton scattering amplitude
-Kawai Llewellen Tye relations can be used which express
Gravity amplitudes as a product of YM tree e.g.
No concept of colour ordering although spinor helicity can be used for spin-2
particles
Momentum prefactorreordering
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Gravity MHV
dependace upon
Gravity MHV amplitudes are polynomial
in and rational in
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-twistor structure of gravity amplitudes not so clear…
-for MHV transforming to twistor space yields support
on ``derivative of delta-function of line’’
-this implies that
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Loop Amplitudes
-lots of work for tree
-how about loops?
-which theory? QCD/N=4 Super-Yang-Mills
-tree level gluon amplitudes are the same in N=4 and pure Yang-Mills
f
f
-duality for N=4 SYM
-makes a difference at 1-loop
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MHV vertices at 1-loop
-MHV vertices were shown to work for N=4 (and N=1)
-specific computation was (repeat) of N=4
MHV amplitudes
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Bedford,Brandhuber, Spence and Travaglini;
Qigley,Rozali
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-looks very much like unitary cut of amplitude
-but continuing away from li2=0
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MHV construction fails?
One loop amplitude A(++++..++)
-vanishes in supersymmetric theory
-non-zero in non-supersymmetric theory
-however it is rational function with no cuts-no possible MHV diagrams!
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N=4 One-Loop Amplitudes –solved!
Amplitude is a a sum of scalar box functions with rational coefficients (BDDK,1994)
Coefficients are ``cut-constructable’’ (BDDK,1994)
Quadruple cuts turns calculus into algebra (Britto,Cachazo,Feng,2005)
Box Coefficients are actually coefficients of terms like
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Conclusions-perturbation theory holds many symmetries which
lead
to surprisingly simple results
-duality has inspired alternate perturbative expansions for tree amplitudes in gauge theories
-underlying these are old concepts of unitarity and factorisation i.e the physical singularities of an amplitude
- N=4 one-loop amplitudes well understood now`
-we will apply some of this to QCD next time
-limited understanding of string theory side of duality