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

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

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

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

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

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

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

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

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

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

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

Why Feynman Diagrams Won’t Get You There Huge number of diagrams in calculations of interest — factorial growth 2 → 6 jets: 34300 tree diagrams, ~ 2.5 ∙ 107 terms ~2.9 ∙ 106 1-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

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

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

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

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

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

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

“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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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