1. 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

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

3. 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.

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

5. 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.

6. 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

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. 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.