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

D Dunbar, KIAS-KIAST 05 3 Twistor Definitions Consider a massless particle with momenta We can realise as With

D Dunbar, KIAS-KIAST 05 4 Definitions: continued For a massless particles Where are two component Weyl spinors or twistors. This decomposition is not unique but

D Dunbar, KIAS-KIAST 05 5 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:

D Dunbar, KIAS-KIAST 05 6 Step2:Spinor Helicity Xu, 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

D Dunbar, KIAS-KIAST 05 7 Transform to Twistor Space Twistor Space is a complex projective (CP 3 ) space n-point amplitude is defined on (CP 3 ) n new coordinates -note we make a choice which to transform -transform like a x-p transform

D Dunbar, KIAS-KIAST 05 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

D Dunbar, KIAS-KIAST 05 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

D Dunbar, KIAS-KIAST 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

D Dunbar, KIAS-KIAST 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

D Dunbar, KIAS-KIAST MHV amplitudes have suppport on line only

D Dunbar, KIAS-KIAST 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)

D Dunbar, KIAS-KIAST Similarly there is a coplanarity operator K ijkl Implies amplitude has non-zero support only in the plane defined by point i,j,k and l

D Dunbar, KIAS-KIAST 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

D Dunbar, KIAS-KIAST Is the duality useful? Theory A : hard, interesting hard, interesting Theory B: easy Perturbative QCD, hard, interesting Topological String Theory : harder, uninteresting -duality may be useful indirectly

D Dunbar, KIAS-KIAST 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)

D Dunbar, KIAS-KIAST _ _ _ _ _ _ _ _ -three point vertices allowed -number of vertices = (number of -) -1

D Dunbar, KIAS-KIAST For NMHV amplitudes k+k k+1 + 2(n-3) diagrams + Topology determined by number of –ve helicity gluons -+

D Dunbar, KIAS-KIAST Coplanarity Two intersecting lines in twistor space define the plane -Points on one MHV vertex

D Dunbar, KIAS-KIAST MHV-vertex construction Works 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

D Dunbar, KIAS-KIAST 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)

D Dunbar, KIAS-KIAST Provided, then

D Dunbar, KIAS-KIAST proof C zizi

D Dunbar, KIAS-KIAST 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)

D Dunbar, KIAS-KIAST results in recursive on-shell relation (three-point amplitudes must be included) 12 NB Berends-Giele recursive techniques

D Dunbar, KIAS-KIAST 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

D Dunbar, KIAS-KIAST CSW can be derived from a type of analytic shift Risager; Bjerrum-Bohr,Dunbar,Ita,Perkins and Risager, 05 gives a the CSW expansion of NMHV -this a combination of three shifts

D Dunbar, KIAS-KIAST 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 prefactor reordering

D Dunbar, KIAS-KIAST Gravity MHV dependace upon Gravity MHV amplitudes are polynomial in and rational in

D Dunbar, KIAS-KIAST 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

D Dunbar, KIAS-KIAST 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

D Dunbar, KIAS-KIAST 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

D Dunbar, KIAS-KIAST looks very much like unitary cut of amplitude -but continuing away from l i 2 =0

D Dunbar, KIAS-KIAST 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!

D Dunbar, KIAS-KIAST 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

D Dunbar, KIAS-KIAST 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