Twistor Inspired techniques in Perturbative Gauge Theories

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

D Dunbar, KIAS-KIAST 05

2

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


3

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:


6

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


7

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


8

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


9

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


12

MHV amplitudes have suppport on line only


13

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)


14

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


15

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

+

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


22

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)


23

Provided,

then


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-proof

Czi


25

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)


26

-results in recursive on-shell relation

(three-point amplitudes must be included)

1 2

NB Berends-Giele recursive techniques


27

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


28

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


29

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


30

Gravity MHV

dependace upon

Gravity MHV amplitudes are polynomial

in and rational in


31

-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


32

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


33

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

+

-

-

+

-

+

+

+

-

+

+

+

+

+

+

+

-

+

+

-

+

+

-

-

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


37

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

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Twistor Inspired techniques in Perturbative Gauge Theories

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