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

Why do we need new techniques? Overview Why do we need new techniques? Recurrence relations, unitarity and the “Unitarity bootstrap technique”. Results. 2

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

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

What's the hold up? Calculating using Feynman diagrams is Hard! Factorial growth in the number of Feynman diagrams. Known results much simpler than would be expected! 6 gluons ~10,000 diagrams. 7 gluons ~150,000 diagrams. 8 gluons more than 3 million. n gluons ∞ diagrams.

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

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)

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)

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)

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)

Working towards producing processes on the “Les Houches” wish list. Conclusion New tools for NLO calculations, Increased efficiency of production of one-loop amplitudes. Growing list of results. Working towards producing processes on the “Les Houches” wish list.