1. On-Shell Meets Observation or, the Rubber Meets the Road
David A. Kosower Institut de Physique Théorique, CEA–Saclay
on behalf of the BlackHat Collaboration
Carola Berger, Z. Bern, L. Dixon, Fernando Febres Cordero, Darren Forde, Harald Ita, DAK, Daniel Maître, Tanju Gleisberg
at the Hidden Structures in Field Theory Amplitudes Workshop
Niels Bohr Institute, Copenhagen, Denmark
August 12, 2009

2. November 16? Campbell, Huston & Stirling ‘06
On-Shell Meets Observation, NBI Workshop on Hidden Structures in Field Theory Amplitudes, August 12–14, 2009

3. On-Shell Meets Observation, NBI Workshop on Hidden Structures in Field Theory Amplitudes, August 12–14, 2009

4. On-Shell Meets Observation, NBI Workshop on Hidden Structures in Field Theory Amplitudes, August 12–14, 2009

5. Backgrounds at the LHC Basic tools:
Parton shower programs (Pythia, Herwig, Sherpa)
Leading-order parton-level codes (MADGRAPH; CompHEP; AMEGIC++; ALPGEN; HELAC; O’MEGA)
Parton shower programs with matching to leading-order matrix elements
These give a basic description of a given background
A process with n jets starts at O(αsn), and has an expansion,
σLO αsn(μ) + σNLO(μ) αsn+1(μ) + σNNLO(μ) αsn+2(μ) + …
but at leading order the description is not quantitative because of the renormalization-scale dependence
On-Shell Meets Observation, NBI Workshop on Hidden Structures in Field Theory Amplitudes, August 12–14, 2009

6. Why NLO? QCD at LO is not quantitative
CDF, PRD 77:011108
QCD at LO is not quantitative
renormalization scale enters into the definition of the coupling
physical quantities are independent of it
computations to fixed order are not
LO: large dependence, no quantitative prediction
NLO: reduced dependence, first quantitative prediction
NNLO: precision prediction
…to get quantitative predictions; for W+3 jets too
 NLO
W+2 jets
 PS+LO matching
 PS+LO matching
On-Shell Meets Observation, NBI Workshop on Hidden Structures in Field Theory Amplitudes, August 12–14, 2009

7. Backgrounds to New Physics
Ströhmer, WIN ‘07
On-Shell Meets Observation, NBI Workshop on Hidden Structures in Field Theory Amplitudes, August 12–14, 2009

8. Ingredients for NLO Leading-order contributions: tree-level
Real-emission contributions: tree-level with one more parton
Virtual corrections: one-loop amplitude 2
On-Shell Meets Observation, NBI Workshop on Hidden Structures in Field Theory Amplitudes, August 12–14, 2009

9. Ingredients for NLO Calculations
Tree-level matrix elements for LO and real-emission terms known since ’80s 
Singular (soft & collinear) behavior of tree-level amplitudes, integrals, initial-state collinear behavior known since ’90s 
NLO parton distributions known since ’90s 
General framework for numerical programs known since ’90s  Catani, Seymour (1996) [Giele, Glover, DAK (1993); Frixione, Kunszt, Signer (1995)]
Automating it for general processes Gleisberg, Krauss; Seymour, Tevlin; Hasegawa, Moch, Uwer; Frederix, Gehrmann, Greiner (2008)
Bottleneck: one-loop amplitudes
W+2 jets (MCFM)  W+3 jets
Bern, Dixon, DAK, Weinzierl (1997–8); Campbell, Glover, Miller (1997) 2
On-Shell Meets Observation, NBI Workshop on Hidden Structures in Field Theory Amplitudes, August 12–14, 2009

10. One-Loop Amplitudes From the point of view of the final user , want
Accurate and reliable codes
Many processes
Several suppliers, to cross-check results
Easy integration into existing tools (MCFM, SHERPA, ALPGEN, HELAC PYTHIA,…)
, want
On-Shell Meets Observation, NBI Workshop on Hidden Structures in Field Theory Amplitudes, August 12–14, 2009

11. Traditional Approach Feynman diagrams Tensor loop integrals
Tensor reductions (Brown–Feynman, Passarino–Veltman)
Loop integral reductions (Melrose)
 Huge intermediate expressions
On-Shell Meets Observation, NBI Workshop on Hidden Structures in Field Theory Amplitudes, August 12–14, 2009

12. Why Feynman Diagrams Won’t Get You There
Huge number of diagrams in calculations of interest — factorial growth
2 → 6 jets: tree diagrams, ~ 2.5 ∙ 107 terms
~2.9 ∙ loop diagrams, ~ 1.9 ∙ 1010 terms
But answers often turn out to be very simple
Vertices and propagators involve gauge-variant off-shell states
Each diagram is not gauge invariant — huge cancellations of gauge-noninvariant, redundant, parts in the sum over diagrams
Simple results should have a simple derivation — Feynman (attr)
Is there an approach in terms of physical states only?
On-Shell Meets Observation, NBI Workshop on Hidden Structures in Field Theory Amplitudes, August 12–14, 2009

13. Focus on Amplitudes Differential cross sections
Use helicities, replace momenta by spinors
Use spinor notation
On-Shell Meets Observation, NBI Workshop on Hidden Structures in Field Theory Amplitudes, August 12–14, 2009

14. New Technologies: On-Shell Methods
Use only information from physical states
Use properties of amplitudes as calculational tools
Factorization → on-shell recursion relations
Unitarity → unitarity method
Underlying field theory → integral basis
Formalism
Known integral basis:
On-shell Recursion; D-dimensional unitarity via ∫ mass
Unitarity
On-Shell Meets Observation, NBI Workshop on Hidden Structures in Field Theory Amplitudes, August 12–14, 2009

15. Unitarity: Prehistory
General property of scattering amplitudes in field theories
Understood in ’60s at the level of single diagrams in terms of Cutkosky rules
obtain absorptive part of a one-loop diagram by integrating tree diagrams over phase space
obtain dispersive part by doing a dispersion integral
In principle, could be used as a tool for computing 2 → 2 processes
No understanding
of how to do processes with more channels
of how to handle massless particles
of how to combine it with field theory: false gods of S-matrix theory
On-Shell Meets Observation, NBI Workshop on Hidden Structures in Field Theory Amplitudes, August 12–14, 2009

16. Unitarity as a Practical Tool
Bern, Dixon, Dunbar, & DAK (1994)
Key idea: sew amplitudes not diagrams
Compute cuts in a set of channels
Compute required tree amplitudes
Reconstruct corresponding Feynman integrals
Perform algebra to identify coefficients of master integrals
Assemble the answer, merging results from different channels
On-Shell Meets Observation, NBI Workshop on Hidden Structures in Field Theory Amplitudes, August 12–14, 2009

17. Generalized Unitarity
Corresponds to requiring two propagators
Can require more  sew together more than two tree amplitudes
Similar to ‘leading singularities’ in older language
Isolates contributions of a smaller set of integrals: only integrals with propagators at cuts will show up
Bern, Dixon, DAK (1997)
Example: in triple cut, only boxes and triangles will contribute
Combine with use of complex momenta to determine box coeffs directly in terms of tree amplitudes
Britto, Cachazo, & Feng (2004)
No integral reductions needed
On-Shell Meets Observation, NBI Workshop on Hidden Structures in Field Theory Amplitudes, August 12–14, 2009

18. Remaining Integrals In QCD, we also need triangle and bubble integrals
Triangle coefficients can be extracted from triple cuts
But boxes have triple cuts too  need to isolate triangle from them
Subtract box integrands (Ossola, Papadopoulos, Pittau)
Isolate using different analytic behavior (Forde)
Spinor residue extraction (Britto, Feng, Mastrolia)
On-Shell Meets Observation, NBI Workshop on Hidden Structures in Field Theory Amplitudes, August 12–14, 2009

19. “One of the most remarkable discoveries in elementary particle physics has been the existence of the complex plane”
On-Shell Meets Observation, NBI Workshop on Hidden Structures in Field Theory Amplitudes, August 12–14, 2009

20. Triangle Cuts
Unitarity leaves one degree of freedom in triangle integrals.
Coefficients are the residues at  Forde 2007 1 2 3
On-Shell Meets Observation, NBI Workshop on Hidden Structures in Field Theory Amplitudes, August 12–14, 2009

21. Direct Rational Contributions
Badger (2008)
Replace D-dimensional components of gluon by massive scalar, integrate over scalar’s mass
Careful parametrization of loop momentum isolates rational coefficient as high-power behavior of integrand in the scalar mass, or via pole at 
On-Shell Meets Observation, NBI Workshop on Hidden Structures in Field Theory Amplitudes, August 12–14, 2009

22. A Contour Integral Consider the contour integral
Determine A(0) in terms of other residues
On-Shell Meets Observation, NBI Workshop on Hidden Structures in Field Theory Amplitudes, August 12–14, 2009

23. On-Shell Recursion Relation=
On-Shell Meets Observation, NBI Workshop on Hidden Structures in Field Theory Amplitudes, August 12–14, 2009

24. Rational Terms Consider contour integral  Spurious singularities
Britto, Cachazo, Feng, & Witten; Bern, Dixon, DAK (2005)
Consider contour integral 
Spurious singularities
If there’s a surface term, use auxiliary recursion
On-shell recursion
On-Shell Meets Observation, NBI Workshop on Hidden Structures in Field Theory Amplitudes, August 12–14, 2009

25. Computational Complexity
Basic object: color-ordered helicity amplitude
Differential cross section needs sum over color orderings and helicities
Use phase-space symmetry to reduce sum over color orderings to polynomial number (or do sum by Monte Carlo)
Sum over helicities by Monte Carlo
What is the complexity of a helicity amplitude? C exp n vs C np
Asymptotically, only overall behavior really matters
We’re interested in moderate n
Brute-force calculations have huge prefactors as well as exp or worse behavior
Want polynomial behavior
Purely analytic answers for generic helicities are exponential
Require a (partly) numerical approach for maximal common-subexpression elimination, with analytic building blocks (e.g. ∫s)
On-Shell Meets Observation, NBI Workshop on Hidden Structures in Field Theory Amplitudes, August 12–14, 2009

26. Berends–Giele Recursion Relations
J5 appearing inside J10 is identical to J5 appearing inside J17
Compute once numerically  maximal reuse
Polynomial complexity per helicity
Recursion with caching
On-Shell Meets Observation, NBI Workshop on Hidden Structures in Field Theory Amplitudes, August 12–14, 2009

27. Aaron Bacall
“Rather than learning how to solve that, shouldn’t I be learning how to write software that can solve that?”
On-Shell Meets Observation, NBI Workshop on Hidden Structures in Field Theory Amplitudes, August 12–14, 2009

28. Intelligent Automation
To date: bespoke calculations
Need industrialization  automation
Easiest to do this numerically
Numerical approach overall: worry about numerical stability
Do analysis analytically
Do algebra numerically
Only the unitarity method combined with a numerically recursive approach for the cut trees can yield a polynomial-complexity calculation at loop level
On-Shell Meets Observation, NBI Workshop on Hidden Structures in Field Theory Amplitudes, August 12–14, 2009

29. BlackHat
Carola Berger (MIT), Z. Bern, L. Dixon, Fernando Febres Cordero (UCLA), Darren Forde (SLAC), Harald Ita (UCLA), DAK, Daniel Maître (SLAC→Durham); Tanju Gleisberg (SLAC)
Written in C++
Framework for automated one-loop calculations
Organization in terms of integral basis (boxes, triangles, bubbles)
Assembly of different contributions
Library of functions (spinor products, integrals, residue extraction)
Tree amplitudes (ingredients)
Caching
Thus far: implemented gluon amplitudes; V + one/two quark pairs + gluon amplitudes
On-Shell Meets Observation, NBI Workshop on Hidden Structures in Field Theory Amplitudes, August 12–14, 2009

30. Other groups pursuing complementary lines of attack for numerical calculations within the unitarity framework: Ossola, Papadopoulos, Pittau, Mastrolia, Draggiotis, Garzelli, van Hameren Ellis, Giele, Kunszt, et al. Anastasiou, Britto, Feng, Mastrolia
Internal masses: Britto, Feng, Mastrolia; Ellis, Giele, Kunszt, et al.; Badger
On-Shell Meets Observation, NBI Workshop on Hidden Structures in Field Theory Amplitudes, August 12–14, 2009

31. Numerical Stability
Usually ordinary double precision is sufficient for stability
Detect instabilities dynamically, event by event
Check correctness of 1/ε coefficient
Check cancellation of spurious singularities in bubble coefficients
For exceptional points (1-2%), use quad precision
On-Shell Meets Observation, NBI Workshop on Hidden Structures in Field Theory Amplitudes, August 12–14, 2009

32. Numerical Studies
Check MHV amplitudes (− −++…+), against all-n analytic expressions; check other six-point amplitudes against high-precision values
Check W + 3 jet amplitudes against ultra high-precision values
On-Shell Meets Observation, NBI Workshop on Hidden Structures in Field Theory Amplitudes, August 12–14, 2009

33. To compute a physical cross-section, also need
real-emission contributions
subtraction terms
Integration over phase space
Analysis package
Use SHERPA for these
Standard CDF jet cuts
SIScone (Salam & Soyez), with R = 0.4
On-Shell Meets Observation, NBI Workshop on Hidden Structures in Field Theory Amplitudes, August 12–14, 2009

34. On-Shell Meets Observation, NBI Workshop on Hidden Structures in Field Theory Amplitudes, August 12–14, 2009

35. Run at 14 TeV, use “generic” LHC cuts:
Again use SIScone (Salam & Soyez), with R = 0.4
On-Shell Meets Observation, NBI Workshop on Hidden Structures in Field Theory Amplitudes, August 12–14, 2009

36. On-Shell Meets Observation, NBI Workshop on Hidden Structures in Field Theory Amplitudes, August 12–14, 2009

37. Choosing Scales More important for LO calculations than NLO
Can affect shapes of distributions!
No perfect solution in processes with high jet multiplicity
Total partonic transverse energy appears to be a good choice, ETW isn’t
On-Shell Meets Observation, NBI Workshop on Hidden Structures in Field Theory Amplitudes, August 12–14, 2009

38. W Polarization
Polarization of low-pT Ws is well-known  dilution in charged-lepton rapidity distribution asymmetry at Tevatron
Ws also appear to be polarized at high pT  ET dependence of e+/e− ratio and missing ET in W+/W−
Useful for distinguishing “prompt” Ws from daughter Ws in top decay?
On-Shell Meets Observation, NBI Workshop on Hidden Structures in Field Theory Amplitudes, August 12–14, 2009

39. Summary
It’s been a long road, but an entirely new approach to amplitude calculations — on-shell methods — has made contact with experimental data
It promises to deliver important theoretical support for the LHC experimental program
Future insights and enhancements will have a role to play too!
On-Shell Meets Observation, NBI Workshop on Hidden Structures in Field Theory Amplitudes, August 12–14, 2009

40. On-Shell Meets Observation, NBI Workshop on Hidden Structures in Field Theory Amplitudes, August 12–14, 2009