BlackHat: NLO QCD for the LHC

1. BlackHat:  NLO QCD for the LHC Darren Forde Work in collaboration with C. Berger (MIT), Z. Bern (UCLA), L. Dixon (SLAC), F. Febres Cordero (UCLA), H. Ita (UCLA), D. Kosower (Saclay), D. Maître (SLAC). SLAC Annual Program Review

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

3. What do we need? • One-loop high multiplicity processes, Newest Les Houches list, (2007)

4. 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!

5. A<n R<n Rn The Unitarity Bootstrap • Use the most efficient approach for each piece, (Bern, Dixon, Kosower) (Berger, Bern, Dixon, Forde, Kosower) 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

6. Glue together tree amplitudes One-loop integral basis • A one-loop amplitude decomposes into • Compute the coefficients from unitarity by taking cuts • Apply multiple cuts, generalised unitarity.(Bern, Dixon, Kosower) (Britto, Cachazo, Feng) Want these coefficients Rational terms, from recursion. 1-loop scalar integrals

7. Box Coefficients • Quadruple cuts freeze the integral  coefficient (Britto, Cachazo, Feng) In 4 dimensions 4 integrals  4 delta functions Box coefficient l l1 l3 Spinor helicity notation, (Mathematica implementation “S@M” (Maître, Mastrolia)) l2

8. Bubbles & Triangles • Compute the coefficients using different numbers of cuts • Analytically examining the large value behaviour of the integrand in these components gives the coefficients (Phys.Rev.D75-Forde)(technique widely applicable e.g. analysis of gravity amplitudes (Phys.Rev.D77-Bern, Carrasco, Forde, Ita, Johansson)) • Straightforward modification for a numerical implementation. Quadruple cuts, gives box coefficients Depends upon unconstrained components of loop momenta.

9. Analytic Results • 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. Automation • For the LHC large number of processes to calculate, • Automatic procedure highly desirable. • We want to go from • Implement Unitarity bootstrap numerically. An(1-,2-,3+,…,n+), An(1-,2-, An(1-,2-,3+,…,n+) An(1-,2-,3+,..

11. (To appear in Phys Rev D.- Berger, Bern, Dixon, Febres Cordero, Forde, Ita, Kosower, Maître) BlackHat “Compact” On-shell inputs Rational building blocks • Numerical implementation of the unitarity bootstrap approach in c++, Much fewer terms to compute & no large cancelations compared with Feynman diagrams.

12. Numerical Stability • Maximise efficiency by using 16 digits of precision for majority of points  good final precision of amplitude. • For a small number of exceptional points use up to 32 or 64. • Detect exceptional points, where we must switch, using 3 tests: • Bubble coefficients in the cut must satisfy, • The sum of all bubbles must be zero for each spurious pole, zs • Large cancellation between cut and rational terms. • Box and Triangle terms feed into bubble  test all pieces.

13. MHV results • Precision tests using 100,000 phase space points with cuts. • ET>0.01√s. • Pseudo-rapidity η>3. • ΔR>4, Log10 number of points No tests Apply tests Recomputed higher precision Precision

14. NMHV results • Other 6-pt amplitudes are similar Log10 number of points Precision

15. More MHV results • Again similar results when increasing the number of legs Log10 number of points Precision

16. Timing • Efficient, e.g. on a 2.33GHz Xenon processor

17. Future Work • Go beyond just gluons, for phenomologically more interesting processes, including • Fermions (Quarks & Leptons). • Z & W bosons. • Combine into full NLO results, • Deal with Infra red (IR) singularities, automated programs exist (e.g. implemented within the SHERPA framework (Gleisberg, Krauss))

18. Conclusion