Recursive Approaches to QCD Matrix Elements including work with Z. Bern, S Bidder, E Bjerrum-Bohr, L. Dixon, H Ita, D Kosower W Perkins K. Risager RADCOR RADCOR 2005 David Dunbar, Swansea University, Wales

D Dunbar, RADCOR 05 2 QCD Matrix Elements -QCD matrix elements are an important part of calculating the QCD background for processes at LHC -NLO calculations (at least!) are needed for precision -One-Loop n-point on-shell amplitudes unknown for n>5 -In this talk we discuss how recent (> Dec 03) ideas of a “weak-weak” duality might help, - apply to 2g  4g 2  4 talks of Binoth,Dittmaier

D Dunbar, RADCOR 05 3 Duality with String Theory In Dec 03 Witten proposed a Weak-Weak duality between A) Yang-Mills theory ( N=4 ) B) Topological String Theory S-matrix of two theories should be identical - True for tree level gluon scattering Rioban, Spradlin,Volovich proposal constructive for N=4 but… N<4..??

D Dunbar, RADCOR 05 4 Duality Details Amplitude a function Spinor helicity replaces polarisation by fermionic variables Use fermionic coordinates and replace bosonic momenta by `twistors` Amplitude now function entirely of fermionic variables Fourier (Penrose transform) one of spinors, Amplitude in twistor space is closest to string theory Xu, Zhang,Chang 87

D Dunbar, RADCOR 05 5 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, RADCOR 05 6 Inspired by duality – the CSW/MHV-vertex construction Promotes MHV amplitude to fundamental object by -Off-shell continuation -makes YM perturbative expansion match that of B-instantons Cachazo Svercek Witten 04, (Nair) Parke-Taylor, Berends-Giele (colour ordered amplitudes)

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

D Dunbar, RADCOR 05 8 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 -Works for one-loops in SUPERSYMMETRIC theories -In trouble for non-SUSY theories Bjerrum-Bohr,DCD,Ita,Perkins, Risager Ozeren+Stirling Badger, Dixon, Glover, Forde, Khoze, Kosower Mastrolia Wu,Zhu; Su,Wu; Georgiou Khoze Bedford,Brandhuber, Spence and Travaglini; Qigley,Rozali A(+++…++)

D Dunbar, RADCOR 05 9 Inspired by duality –BCFW construction for tree amplitudes 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)

D Dunbar, RADCOR Provided, Residues occur when amplitude factorises on multiparticle pole (including two- particles) then

D Dunbar, RADCOR results in recursive on-shell relation (three-point amplitudes must be included) 12

D Dunbar, RADCOR CSW vs BCF Difference CSW asymmetric between helicity sign BCF chooses two special legs Similarities- both rely upon analytic structure CSW can be derived from a type of analytic shift Risager; Bjerrum-Bohr,Dunbar,Ita,Perkins and Risager, 05

D Dunbar, RADCOR One-Loop Amplitudes One Loop Gluon Scattering Amplitudes in QCD -Four Point : Ellis+Sexton -Five Point : Bern, Dixon,Kosower -Six-Point and beyond--- present problem n-point MHV amplitudes supersymmetric theories Bern,Dixon,Dunbar and Kosower 94/95 Six-point N=4 amplitudes

D Dunbar, RADCOR Supersymmetric Decomposition Supersymmetric gluon scattering amplitudes are the linear combination of QCD ones+scalar loop -this can be inverted

D Dunbar, RADCOR 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, RADCOR Box-Coefficients -works for massless corners (complex momenta) S Britto,Cachazo,Feng or signature (--++) -works for non-supersymmetric Bjerrum-Bohr,Bidder,DCD,Perkins

D Dunbar, RADCOR N=1 One-Loop Amplitudes -???? Important to choose a good basis of functions A) choose chiral multiplet B) use D=6 boxes Amplitude also cut constructible -six gluon amplitudes now obtained using unitarity Bidder,Bjerrum-Bohr,Dixon, Dunbar, Perkins Britto, Buchbinder Cachazo, Feng, 04/05

D Dunbar, RADCOR The Final Pieces : scalar contributions -last component of QCD amplitudes - R is rational and not cut constructible (to O(  )) cut construcible recursive? -can we avoid direct integration?

D Dunbar, RADCOR Recursion for Rational terms -can we shift R and obtain it from its factorisation? 1)Function must be rational 2)Function must have simple poles 3)We must understand these poles

D Dunbar, RADCOR understanding poles -multiparticle factorisation theorems Bern,Chalmers

D Dunbar, RADCOR Complication, Either R or the coefficients of integral functions may contain Spurious Singularities which are not present in the full amplitude It is important and non-trivial to find shift(s) which avoid these spurious singularities whilst still affecting the full R/coefficient

D Dunbar, RADCOR Example of Spurious singularities Collinear Singularity Multi-particle pole Co-planar singularity

D Dunbar, RADCOR Splitting Amplitude into C and R is not unique The integral functions can be defined to include rational pieces, e.g rather than avoids a spurious singularity as r  1 (r=s/s’) Spurious singularities spoil understanding of residues – can we avoid them?

D Dunbar, RADCOR Results: It has been demonstrated, using A) (-++….+++), (+++++….++) and B) (--++++) and (--++….+++) that shifts can be found which allow calculation of rational parts recursively Bern, Dixon Kosower 1) 2) Shifts can be found which allow the integral coefficients to be computed recursively A( …++) Bern, Bjerrum-Bohr, Dunbar, Ita Forde, Kosower

D Dunbar, RADCOR State of Play Six Gluon Scattering X XX X X X X X X X X X X X X X X 2g-  4g

D Dunbar, RADCOR Conclusions -Reasons for optimism in computing one-loop QCD matrix elements -Recent progress uses UNITARITY and FACTORISATION as key features of on-shell amplitudes -Inspired by Weak-Weak duality but not dependant upon it -after much progress in highly super-symmetric theories the (harder) problem of QCD beginning to yield results -first complete result for a partial 2g  ng amplitude! -also fermions, masses, multiloop……..???