Twistor Inspired techniques in Perturbative Gauge Theories-II

1. Twistor Inspired techniques in Perturbative Gauge Theories-II 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. Seminar II • Hadron Colliders, LHC • Need for NLO computations • Pieces of NLO computations • QCD calculations

3. Hadron Colliders LHC

7. LHC Physics • -hadron machines are DISCOVERY machines (SPS:W+Z,Tevatron: t) • -LHC will “hunt the higgs” • -hunt SUSY • -hunt new physics

8. Higgs Production and Decay eg g W/Z H g W/Z -four final state particles end-point of Higgs production -very often four jets

11. -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.

13. Matrix Elements Hadronisation Structure Functions -piece that twistors may help with Pieces of Theoretical Prediction Probability of producing final state =

14. 2 2 2 g2 +g4 g3 Need for NLO Matrix Elements calculations for jets • Consider 2g -> 2g

15. 2 2 2 +g6 g3 g2 +g4 +g5 g4 NNLO

16. Wny is NLO neccessary • -accurancy, QCD is strong(ish) • -scale dependance • -cone-size dependance

17. 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 Six-point N=4 amplitudes Bern,Dixon,Dunbar and Kosower 94/95

18. degree n in l n Linear in loop momentum propagators General Decomposition of One-loop Amplitude

19. l-k l k Passerino-Veltman reduction Decomposes a n-point integral into a sum of (n-1) integral functions obtained by collaspsing a propagator

20. Functions of a single kinematic invariant, ln(s) • -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

21. Supersymmetric Decomposition Supersymmetric gluon scattering amplitudes are the linear combination of QCD ones+scalar loop -this can be inverted

22. 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

23. 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

24. Basis in N=4 Theory ‘easy’ two-mass box ‘hard’ two-mass box

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

26. Box Coefficients-Twistor Structure • Box coefficients has coplanar support for NMHV 1-loop • amplitudes -true for both N=4 and QCD!!!

27. 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

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

29. Recursion for Rational terms -can we shift R and obtain it from its factorisation? • Function must be rational • Function must have simple poles • We must understand these poles

30. -understanding poles -multiparticle factorisation theorems Bern,Chalmers

31. Complication, Either R or the coefficients of integral functions may containSpurious Singularitieswhich 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

32. Collinear Singularity Multi-particle pole Co-planar singularity Example of Spurious singularities

33. Spurious singularities spoil understanding of residues – can we avoid them? 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’)

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

35. State of Play Six Gluon Scattering 2g- 4g X X X X X X X X X X X X X X X X X

36. 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,…..