1 / 64

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.

aquene
Download Presentation

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

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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

More Related