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On-Shell Methods in QCD and N=4 Super-Yang-Mills Theory

On-Shell Methods in QCD and N=4 Super-Yang-Mills Theory. Lance Dixon (CERN & SLAC) DESY Theory Workshop 21 Sept. 2010. The S matrix reloaded.

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On-Shell Methods in QCD and N=4 Super-Yang-Mills Theory

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  1. On-Shell Methods in QCD and N=4 Super-Yang-Mills Theory Lance Dixon (CERN & SLAC) DESY Theory Workshop 21 Sept. 2010

  2. The S matrix reloaded • Almost everything we know experimentally about gauge theory is based on scattering processes with asymptotic, on-shell states, evaluated in perturbation theory. • Nonperturbative, off-shell information very useful, but in QCD it is often more qualitative (except for lattice). • All perturbative scattering amplitudes can be computed with Feynman diagrams – but that is not necessarily the best way, especially if there is hidden simplicity. • N=4 super-Yang-Mills theory has lots of simplicity, both manifest and hidden. A particularly beautiful application of on-shell methods L. Dixon On-Shell Methods

  3. On-shell methods in QCD L. Dixon On-Shell Methods

  4. Need precise understanding of “old physics” that looks like new physics LHC is a multi-jet environment • Every process also comes with one more jet at ~ 1/5 the rate • Understand not only SM production of X but also of • X + n jets • where • X = W, Z, tt, WW, • H, … • n = 1,2,3,… LHC @ 7 TeV • new physics? L. Dixon On-Shell Methods

  5. n n c c  nn Backgrounds to Supersymmetry at LHC • Cascade from gluino to neutralino • (dark matter, escapes detector) • Signal: missing energy + 4 jets • SM background from Z + 4 jets, • Z neutrinos Current state of art for Z + 4 jets based on LO tree amplitudes (matched to parton showers)  normalization still quite uncertain • Motivates goal of L. Dixon On-Shell Methods

  6. One-loop QCD amplitudes via Feynman diagrams For V + n jets (maximum number of external gluons only) # of jets # 1-loop Feynman diagrams L. Dixon On-Shell Methods

  7. Remembering a Simpler Time... • In the 1960s there was no QCD, • no Lagrangian or Feynman rules • for the strong interactions L. Dixon On-Shell Methods

  8. Poles • Branch cuts The Analytic S-Matrix Bootstrap program for strong interactions: Reconstruct scattering amplitudes directly from analytic properties (on-shell information): Chew, Mandelstam; Eden, Landshoff, Olive, Polkinghorne; Veneziano; Virasoro, Shapiro; …(1960s) Analyticity fell out of favor in 1970s with the rise of QCD & Feynman rules Now resurrected for computing amplitudes for perturbativeQCD – as alternative to Feynman diagrams! Important: perturbative information now assists analyticity. Works even better in theories with lots of SUSY, like N=4 SYM L. Dixon On-Shell Methods

  9. Generalized unitarity Ordinary unitarity: Im T = T†T put 2 particles on shell Generalized unitarity: put 3 or 4 particles on shell L. Dixon On-Shell Methods

  10. coefficients are all rational functions – determine algebraically from products of trees using (generalized) unitarity known scalar one-loop integrals, same for all amplitudes rational part One-loop amplitudes reduced to trees When all external momenta are in D = 4, loop momenta in D = 4-2e (dimensional regularization), one can write: Bern, LD, Dunbar, Kosower (1994) L. Dixon On-Shell Methods

  11. Generalized Unitarity for Box Coefficients di Britto, Cachazo, Feng, hep-th/0412308 No. of dimensions = 4 = no. of constraints  discrete solutions (2, labeled by ±) Easy to code, numerically very stable L. Dixon On-Shell Methods

  12. Box coefficients di (cont.) Solutions simplify (and are more stable numerically) when all internal lines massless, at least one external line (K1) massless: BH, 0803.4180; Risager 0804.3310 L. Dixon On-Shell Methods

  13. Unitarity method – numerical implementation Each box coefficient uniquely isolated by a “quadruple cut” given simply by a product of 4 tree amplitudes Britto, Cachazo, Feng, hep-th/0412103 Ossola, Papadopolous, Pittau, hep-ph/0609007; Mastrolia, hep-th/0611091; Forde, 0704.1835; Ellis, Giele, Kunszt, 0708.2398; Berger et al., 0803.4180;… triangle coefficients come from triple cuts, product of 3 tree amplitudes, but these are also “contaminated” by boxes bubble coefficients come from ordinary double cuts, after removing contributions of boxes and triangles L. Dixon On-Shell Methods

  14. Box-subtracted triple cut has poles only at t = 0, ∞ Triangle coefficient c0 plus all other coefficients cj obtained by discrete Fourier projection, sampling at (2p+1)throots of unity Triangle coefficients Forde, 0704.1835; BH, 0803.4180 Triple cut solution depends on one complex parameter, t Solves for suitable definitions of Bubble similar L. Dixon On-Shell Methods

  15. Several Recent Implementations of On-Shell Methods for 1-Loop Amplitudes Method for Rational part: CutTools: Ossola, Papadopolous, Pittau, 0711.3596 NLO WWW, WWZ, ... Binoth+OPP, 0804.0350 NLO ttbb, tt + 2 jetsBevilacqua, Czakon, Papadopoulos, Pittau, Worek, 0907.4723; 1002.4009 specialized Feynman rules _ _ _ D-dim’l unitarity Rocket: Giele, Zanderighi, 0805.2152 Ellis, Giele, Kunszt, Melnikov, Zanderighi, 0810.2762 NLO W + 3 jets in large Nc approx./extrapolation EMZ, 0901.4101, 0906.1445; Melnikov, Zanderighi, 0910.3671 D-dim’l unitarity + on-shell recursion SAMURAI: Mastrolia, Ossola, Reiter, Tramontano, 1006.0710 Blackhat: Berger, Bern, LD, Febres Cordero, Forde, H. Ita, D. Kosower, D. Maître; T. Gleisberg, 0803.4180, 0808.0941, 0907.1984, 0912.4927, 1004.1659  + Sherpa NLO production of W,Z + 3 (4) jets L. Dixon On-Shell Methods

  16. Virtual Corrections • Divide into leading-color terms, such as: and subleading-color terms, such as: The latter include many more terms, and are much more time-consuming for computer to evaluate. But they are much smaller (~ 1/30 of total cross section) so evaluate them much less often. L. Dixon On-Shell Methods

  17. Recent analytic application: One-loop amplitudes for a Higgs boson + 4 partons Unitarity for cut parts, on-shell recursion for rational parts (mostly) H = f + f† Badger, Glover, Risager, 0704.3914 Glover, Mastrolia, Williams, 0804.4149 Badger, Glover, Mastrolia, Williams, 0909.4475 Badger, Glover, hep-ph/0607139 LD, Sofianatos, 0906.0008 Badger, Campbell, Ellis, Williams, 0910.4481 by parity L. Dixon On-Shell Methods

  18. 5-point – still analytic BGMW DS L. Dixon On-Shell Methods

  19. Besides virtual corrections, also need real emission • General subtraction methods for integrating real-emission contributions developed in mid-1990s Frixione, Kunszt, Signer, hep-ph/9512328; Catani, Seymour, hep-ph/9602277, hep-ph/9605323 • Recently automated by several groups Gleisberg, Krauss, 0709.2881; Seymour, Tevlin, 0803.2231; Hasegawa, Moch, Uwer, 0807.3701; Frederix, Gehrmann, Greiner, 0808.2128; Czakon, Papadopoulos, Worek, 0905.0883; Frederix, Frixione, Maltoni, Stelzer, 0908.4272 Infrared singularities cancel L. Dixon On-Shell Methods

  20. Les Houches Experimenters’ Wish List Feynman diagram methods now joined by on-shell methods BCDEGMRSW; Campbell, Ellis, Williams Berger table courtesy of C. Berger L. Dixon On-Shell Methods

  21. Tevatron W + n jets Data CDF, 0711.4044 [hep-ex] n = 1 NLO parton level (MCFM) LO matched to parton shower MC with different schemes n = 2 n = 3 only LO available in 2007 L. Dixon On-Shell Methods

  22. W + 3 jets at NLO at Tevatron Ellis, Melnikov, Zanderighi, 0906.1445 Berger et al., 0907.1984 Rocket Leading-color adjustment procedure Exact treatment of color L. Dixon On-Shell Methods

  23. W + 3 jets at LHC • LHC has much greater dynamic range • Many events with jet ETs >> MW • Must carefully choose appropriate renormalization + factorization scale • Scale we used at the Tevatron, • also used in several other LO studies, • is not a good choice: • NLO cross section can even dive negative! L. Dixon On-Shell Methods

  24. Better Scale Choices Q: What’s going on? A: Powerful jets and wimpy Ws • If (a) dominates, then is OK • But if (b) dominates, then the scale ETW is too low. • Looking at large ET for the 2nd jet forces configuration (b). • Better: total (partonic) transverse energy • (or fixed fraction of it, or sum in quadrature?); gets large properly for both (a) and (b) Bauer, Lange 0905.4739 • Another reasonable scale is invariant mass of the n jets L. Dixon On-Shell Methods

  25. Compare Two Scale Choices logs not properly cancelled for large jet ET – LO/NLO quite flat, also for many other observables L. Dixon On-Shell Methods

  26. Total Transverse Energy HT at LHC often used in supersymmetry searches 0907.1984  flat LO/NLO ratio due to good choice of scale m = HT L. Dixon On-Shell Methods

  27. NLO pp  W+ 4 jetsnow available C. Berger et al., 1009.2338 Virtual terms: leading-color (including quark loops); omitted terms only ~ few % L. Dixon On-Shell Methods

  28. One indicator of NLO progress pp  W + 0 jet 1978 Altarelli, Ellis, Martinelli pp  W + 1 jet 1989 Arnold, Ellis, Reno pp  W + 2 jets 2002 Arnold, Ellis pp  W + 3 jets 2009 BH+Sherpa; EMZ pp  W + 4 jets 2010 BH+Sherpa L. Dixon On-Shell Methods

  29. NLO Parton-Level vs. Shower MCs • Recent advances on Les Houches NLO Wish List all at parton level: no parton shower, no hadronization, no underlying event. • Methods for matching NLO parton-level results to parton showers, maintaining NLO accuracy • MC@NLO Frixione, Webber (2002), ... • POWHEG Nason (2004); Frixione, Nason, Oleari (2007); ... • POWHEG in SHERPA Höche, Krauss, Schönherr, Siegert, 1008.5339 • GenEvA Bauer, Tackmann, Thaler (2008) • However, none is yet implemented for final states with multiple light-quark & gluon jets • NLO parton-level predictions generally give best normalizations for total cross sections (unless NNLO available!), and distributions away from shower-dominated regions. • Right kinds of ratios will be considerably less sensitive to shower + nonperturbative effects L. Dixon On-Shell Methods

  30. On-shell methods in N=4 SYM L. Dixon On-Shell Methods

  31. Why N=4 SYM? • Dual to gravity/string theory on AdS5 x S5 • Very similar in IR to QCD talk by Magnea • Planar (large Nc) theory is integrabletalk by Beisert • Strong-coupling limit a minimal area problem (Wilson loop) Alday, Maldacena • Planar amplitudes possess dual conformal invariance Drummond, Henn, Korchemsky, Sokatchev • Some planar amplitudes “known” to all orders in coupling Bern, LD, Smirnov + AM + DHKS • More planar amplitudes “equal” to expectation values of light-like Wilson loops talk by Spradlin • N=8 supergravity closely linked bytree-level Kawai-Lewellen-Tye relation and more recent “duality” relations Bern, Carrasco, Johansson • More recent Grassmannian developments Arkani-Hamed et al. • Excellent arena for testing on-shell & related methods L. Dixon On-Shell Methods

  32. N=4 SYM “states” all states in adjoint representation, all linked by N=4 supersymmetry • Interactions uniquely specified by gauge group, say SU(Nc), 1 coupling g • Exactly scale-invariant (conformal) field theory: b(g) = 0 for all g L. Dixon On-Shell Methods

  33. Planar N=4 SYM and AdS/CFT • In the ’t Hooft limit, fixed, planar diagrams dominate • AdS/CFT duality suggests that weak-coupling perturbation series in lfor large-Nc(planar) N=4 SYM should have special properties, because large l limit  weakly-coupled gravity/string theory on AdS5 x S5 Maldacena; Gubser, Klebanov, Polyakov; Witten L. Dixon On-Shell Methods

  34. AdS/CFT in one picture L. Dixon On-Shell Methods

  35. Scattering at strong coupling Alday, Maldacena, 0705.0303 [hep-th] • Use AdS/CFT to compute an appropriate scattering amplitude • High energy scattering in string theory is semi-classical Gross, Mende (1987,1988) r Evaluated on the classical solution, action is imaginary  exponentially suppressed tunnelling configuration Can also do with dimensional regularization instead of L. Dixon On-Shell Methods

  36. Dual variables and strong coupling • T-dual momentum variables introduced by Alday, Maldacena • Boundary values for world-sheet • are light-like segments in : for gluon with momentum • For example, • for gg gg 90-degree scattering, • s = t = -u/2, the boundary looks like: Corners (cusps) are located at – same dual momentum variables appear at weak coupling (in planar theory) L. Dixon On-Shell Methods

  37. Generalized unitarity for N=4 SYM Found long ago that one-loop N=4 amplitudes contain only boxes, due to SUSY cancellations of loop momenta in numerator: Bern, LD, Dunbar, Kosower (1994) More recently, L-loop generalization of this property conjectured: All (important) terms determined by “leading-singularities” – imposing 4L cuts on the L loop momenta in D=4Cachazo, Skinner, 0801.4574; Arkani-Hamed, Cachazo, Kaplan, 0808.1446 L. Dixon On-Shell Methods

  38. Multi-loop generalized unitarity at work Allowing for complex cut momenta, one can chop an amplitude entirely into 3-point trees  maximal cuts or ~ leading singularities Bern, Carrasco, LD, Johansson, Kosower, Roiban, hep-th/0702112; Bern, Carrasco, Johansson, Kosower, 0705.1864 These cuts are maximally simple, yet give an excellent starting point for constructing the full answer. (No conjectures required.) In planar (leading in Nc) N=4 SYM, maximal cuts find all terms in the complete answer for 1, 2 and 3 loops L. Dixon On-Shell Methods

  39. Recent supersum advances to evaluate more complicated cuts Drummond, Henn, Korchemsky, Sokatchev, 0808.0491; Arkani-Hamed, Cachazo, Kaplan, 0808.1446; Elvang, Freedman, Kiermaier, 0808.1720; Bern, Carrasco, Ita, Johansson, Roiban, 2009 Finding missing terms Maximal cut method: Allowing one or two propagators to collapse from each maximal cut, one obtains near-maximal cuts These near-maximal cuts are very useful for analyzing N=4 SYM (including nonplanar) and N=8 SUGRA at 3 loops BCDJKR, BCJK (2007); Bern, Carrasco, LD, Johansson, Roiban, 0808.4112 • Maximal cut method is completely systematic • not restricted to N=4 SYM • not restricted to planar contributions L. Dixon On-Shell Methods

  40. 2 loops: Bern, Rozowsky, Yan (1997); Bern, LD, Dunbar, Perelstein, Rozowsky (1998) 4-gluon amplitude in N=4 SYM at 1 and 2 Loops • 1 loop: Green, Schwarz, Brink; Grisaru, Siegel (1981) L. Dixon On-Shell Methods

  41. x1 x4 x2 invariant under inversion: x5 x3 Dual Conformal Invariance Broadhurst (1993); Lipatov (1999);Drummond, Henn, Smirnov, Sokatchev, hep-th/0607160 A conformal symmetry acting in momentum space, on dual (sector) variables xi First seen in N=4 SYM planar amplitudes in the loop integrals k L. Dixon On-Shell Methods

  42. Dual conformal invariance at 4 loops • Simple graphical rules: • 4 (net) lines into inner xi • 1 (net) line into outer xi • Dotted lines are for • numerator factors 4 loop planar integrals all of this form BCDKS, hep-th/0610248 also true at 5 loops BCJK, 0705.1864 L. Dixon On-Shell Methods

  43. Insight from string theory • As a property of full (planar) amplitudes, rather than integrals, • dual conformal invariance follows, at strong coupling, from bosonic T duality symmetry of AdS5 x S5. • Also, strong-coupling calculation ~ equivalent to computation of Wilson line for n-sided polygon with vertices at xi Alday, Maldacena, 0705.0303 Wilson line blind to helicity formalism – doesn’t know MHV from non-MHV. Some recent attempts to go beyond this Alday, Eden, Maldacena, Korchemsky, Sokatchev, 1007.3243; Eden, Korchemsky, Sokatchev, 1007.3246, 1009.2488 L. Dixon On-Shell Methods

  44. Leads to “rung rule” for easily computing all contributions which can be built by iterating 2-particle cuts The rung rule Many higher-loop contributions to gg gg scattering deduced from a simple property of the 2-particle cuts at one loop Bern, Rozowsky, Yan (1997) L. Dixon On-Shell Methods

  45. 3 loop cubic graphs Nine basic integral topologies Seven (a-g) were already known (2-particle cuts  rung rule) BDDPR (1998) Two new ones (h,i) have no 2-particle cuts BCDJKR (2007); BCDJR (2008) L. Dixon On-Shell Methods

  46. Omit overall N=4 numerators at 3 loops manifestly quadratic in loop momentum L. Dixon On-Shell Methods

  47. Four loops:full color N=4 SYMas input for N=8 SUGRA Bern, Carrasco, LD, Johansson, Roiban, 1008.3327 BCDJR, 0905.2326 L. Dixon On-Shell Methods

  48. 4 loop 4 point amplitude in N=4 SYM Number of cubic 4-point graphs with nonvanishing Coefficients and various topological properties L. Dixon On-Shell Methods

  49. 3 2 4 1 Twist identity • If the diagram contains a four-point tree subdiagram, can use a Jacobi-like identity to relate it to other diagrams. Bern, Carrasco, Johansson, 0805.3993 • Relate non-planar topologies to planar, etc. • For example, at 3 loops, (i) = (e) – (e)T [ + contact terms ] - = L. Dixon On-Shell Methods

  50. Box cut Bern, Carrasco, Johansson, Kosower, 0705.1864 • If the diagram contains a box subdiagram, can use the simplicity of the 1-loop 4-point amplitude to compute the numerator very simply • Planar example: • Only five 4-loop cubic topologies • do not have box subdiagrams. • But there are also “contact terms” • to determine. L. Dixon On-Shell Methods

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