Twistor Inspired techniques in Perturbative Gauge Theories-II 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 Seminar II Hadron Colliders, LHC Need for NLO computations Pieces of NLO computations QCD calculations D Dunbar, KIAS-KIAST 05 3 Hadron Colliders LHC D Dunbar, KIAS-KIAST 05 4 5 6 7 LHC Physics -hadron machines are DISCOVERY machines (SPS:W+Z,Tevatron: t) -LHC will hunt the higgs -hunt SUSY -hunt new physics D Dunbar, KIAS-KIAST 05 8 Higgs Production and Decay H g g W/Z -four final state particles end-point of Higgs production -very often four jets eg D Dunbar, KIAS-KIAST 05 9 10 D Dunbar, KIAS-KIAST most decay end-points of new physics can be simulated by background standard model processes -very important to have robust accurate predictions for background decay rates/event shapes/angular distribution based upon known physics -jets are inclusive processess : experimentally we cannot distinguish colour, helicity, spin. D Dunbar, KIAS-KIAST 05 12 D Dunbar, KIAS-KIAST Pieces of Theoretical Prediction Probability of producing final state = Structure Functions Matrix Elements Hadronisation -piece that twistors may help with D Dunbar, KIAS-KIAST Need for NLO Matrix Elements calculations for jets Consider 2g -> 2g +g 4 2 g2g2 2 2 g3g3 D Dunbar, KIAS-KIAST g2g2 +g 4 +g g3g3 g4g4 +g 5 NNLO D Dunbar, KIAS-KIAST Wny is NLO neccessary -accurancy, QCD is strong(ish) -scale dependance -cone-size dependance D Dunbar, KIAS-KIAST 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 -Five and Six-Point mixed procecess n-point MHV amplitudes supersymmetric theories Bern,Dixon,Dunbar and Kosower 94/95 Six-point N=4 amplitudes D Dunbar, KIAS-KIAST General Decomposition of One- loop Amplitude Linear in loop momentum propagators n degree n in l D Dunbar, KIAS-KIAST Passerino-Veltman reduction Decomposes a n-point integral into a sum of (n-1) integral functions obtained by collaspsing a propagator k l l-k D Dunbar, KIAS-KIAST process continues until we reach four-point integral functions with (in yang-mills up to quartic numerators) -similarly 3-> 2 also gives scalar triangles. At bubbles process ends. Quadratic bubbles can be rational functions involving no logarithms. -so in general, for massless particles Functions of a single kinematic invariant, ln(s) D Dunbar, KIAS-KIAST Supersymmetric Decomposition Supersymmetric gluon scattering amplitudes are the linear combination of QCD ones+scalar loop -this can be inverted 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 N=4 Susy In N=4 susy there are cancelations between the states of different spin circulating in the loop. Leading four-powers of loop momentum cancel (in well chosen gauges..) N=4 lie in a small subspace of the allowed possible amplitudes D Dunbar, KIAS-KIAST Basis in N=4 Theory easy two-mass box hard two-mass box 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 -is unitarity the key? D Dunbar, KIAS-KIAST 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, KIAS-KIAST Box Coefficients-Twistor Structure Box coefficients has coplanar support for NMHV 1-loop amplitudes -true for both N=4 and QCD!!! D Dunbar, KIAS-KIAST 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, KIAS-KIAST 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, KIAS-KIAST 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, KIAS-KIAST understanding poles -multiparticle factorisation theorems Bern,Chalmers D Dunbar, KIAS-KIAST 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, KIAS-KIAST Example of Spurious singularities Collinear Singularity Multi-particle pole Co-planar singularity D Dunbar, KIAS-KIAST 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, KIAS-KIAST 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, KIAS-KIAST 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, KIAS-KIAST 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! -NNLO is the goal for LHC -analytic vs. numerical? -fermions, masses, multi-loops,..
