Twistor Inspired techniques in Perturbative Gauge Theories

1. Twistor Inspired techniques in Perturbative Gauge Theories David Dunbar, Swansea University, Wales including work with Z. Bern, S Bidder, E Bjerrum-Bohr, L. Dixon, H Ita, W Perkins K. Risager KIAS-KIAST 2005

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

4. Definitions: continued • For a massless particles • Where are two component Weyl spinors • or twistors. This decomposition is not unique but

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:

6. Gluon Momenta Reference Momenta Step2:Spinor Helicity Xu, Zhang,Chang 87 -extremely useful technique which produces relatively compact expressions for amplitudes -amplitude now entirely in terms of spinorial variables

7. Transform to Twistor Space -transform like a x-p transform -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

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

10. 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

11. 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

16. Is the duality useful? Theory A : hard, interesting Theory B: easy Topological String Theory: harder, uninteresting Perturbative QCD, hard, interesting -duality may be useful indirectly

17. Inspired by duality – the CSW/MHV-vertex construction Cachazo Svercek Witten 04, (Nair) • Promotes MHV amplitude to fundamental object by • -Off-shell continuation • -MHV amplitudes have no multi-particle factorisation (colour ordered amplitudes) Parke-Taylor, Berends-Giele

18. + _ _ + _ + _ _ + _ + _ + + _ -three point vertices allowed -number of vertices = (number of -) -1

19. For NMHV amplitudes k+ k+1+ + - + 1- 3- 2- 2(n-3) diagrams Topology determined by number of –ve helicity gluons

20. Coplanarity -Points on one MHV vertex Two intersecting lines in twistor space define the plane

21. 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 Wu,Zhu; Su,Wu; Georgiou Khoze Badger, Dixon, Glover, Forde, Khoze, Kosower Mastrolia Ozeren+Stirling Bjerrum-Bohr,DCD,Ita,Perkins, Risager

22. Inspired by duality –BCFW construction 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 P(z)

23. Provided, then

24. -proof C zi

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. 1 2 -results in recursive on-shell relation NB Berends-Giele recursive techniques (three-point amplitudes must be included)

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

29. Momentum prefactor reordering 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

30. dependace upon Gravity MHV amplitudes are polynomial in and rational in Gravity MHV

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. f f 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 -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

34. -looks very much like unitary cut of amplitude -but continuing away from li2=0

35. + + + + + + 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!

36. 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