Computation of Multi-Jet QCD Amplitudes at NLO

1. Carola Berger (SLAC), Zvi Bern (UCLA), Lance Dixon (SLAC), Darren Forde (SLAC), David Kosower (Saclay), Daniel Maitre (SLAC), YorgosSofianatos (SLAC). Computation of Multi-Jet QCD Amplitudes at NLO

2. Overview

3. What’s the problem? • Precise QCD amplitudes are needed to maximise the discovery potential of the LHC (2008). •  NLO amplitudes  1-loop amplitudes.

4. “Famous” Les Houches experimentalist wish list, (2005) What do we need? Six or more legs, until recently a bottleneck

5. What's the hold up? • Calculating using Feynman diagrams is Hard! • Factorialgrowth in the number of Feynman diagrams. • Known results much simpler than would be expected!

6. A<n R<n Rn The unitarity bootstrap • Use the most efficient approach for each piece, Unitarity cuts K3 On-shell recurrence relations A3 A2 A1 K1 K2 Recycle results of amplitudes with fewer legs “Glue” together trees to produce loops

7. Recursion relations • On-shell recursion relations originally developed for massless tree amplitudes (Phys.Rev.Lett.94-Britto,Cachazo,Feng,Witten) • Very general, proof relies only on Factorization properties of amplitudes and Cauchy’s theorem. • Extended to massive particles, (JHEP 0507-Badger,Glover,Khoze,Svrcek) • All-plus and single-minus all-multiplicity amplitudes for a pair of massive scalars, An(φ,+,…,±,…,+, φ). (Phys.Rev.D73-Forde,Kosower) • Extended to one-loop amplitudes with no cut pieces. All-plus and single-minus helicity amplitudes, An(±,+,+,…,+), • Just gluons,(Phys.Rev.D71-Bern, Dixon, Kosower) • Both quarks and gluons with an arbitrary number of legs,(Phys.Rev.D71-Bern, Dixon, Kosower)

8. Including Unitarity Cuts • Amplitudes with two or more negativity helicity legs contain cut terms. • Apply unitarity bootstrap; cut terms previously calculated (Nucl.Phys.B435&B425-Bern,Dixon,Dunbar,Kosower) • Adjacent 2-minus with 6 legs, (Phys.Rev.D73-Bern, Dixon, Kosower) • Minimal growth in “complexity” of solution with arbitrary numbers of legs, An(-,-,+,…,+),(Phys.Rev.D73 -Forde, Kosower)

9. Further Applications • Non-adjacent 2-minus amplitude, An(-,+,…,-,…,+), (Phys.Rev.D75-Berger, Bern, Dixon, Forde, Kosower) • Three minus adjacent amplitude, An(-,-,-,+,…,+), (Phys.Rev.D74-Berger, Bern, Dixon, Forde, Kosower) • Important contributions to the recently derived complete six gluon amplitude.(Bern,Dixon,Kosower) (Berger,Bern,Dixon,Forde,Kosower) (Xiao,Yang,Zhu) (Bedford,Brandhuber,Spence,Travaglini) (Britto,Feng,Mastrolia) (Bern,Bjerrum-Bohr,Dunbar,Ita). • A Higgs boson plus arbitrary numbers of gluons or a pair of quarks for the all-plus and one-minus helicity combinations, An(φ,+,…,±,…,+). (Phys.Rev.D74-Berger, Del Duca, Dixon)

10. Increasing Efficiency • Do better - use generalised unitarity for cut terms, • New techniques produce “compact” results in a direct manner. • Generally applicable, including “wish list” processes. Quadruple cuts, give box coefficients (Nucl.Phys.B725-Britto, Cachazo, Feng) Two-particle and triple cuts, give bubble and triangle coefficients (Phys.Rev.D-Forde)

11. Conclusion