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).

Precise QCD amplitudes are needed to maximise the discovery potential of the LHC (2008).

NLO amplitudes 1-loop amplitudes.

“Famous” Les Houches experimentalist wish list, (2005)

Six or more legs, until recently a bottleneck

Calculating using Feynman diagrams is Hard!

Factorial growth in the number of Feynman diagrams.

Known results much simpler than would be expected!

Unitarity cutsK3

K2K1

A3

A2

A1

On-shell recurrence relations

ji

“Glue” together trees to produce loops

Recycle results of amplitudes with fewer legs

Use the most efficient approach for each piece,

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)

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)

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)

Do better - use generalised unitarity for cut terms,

New techniques produce “compact” results in a direct manner.

Generally applicable, including “wish list” processes.

i ij ijki ij ijk

b c d

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)