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 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 DESY Workshop 21 Sept 2010 L. Dixon On-Shell Methods 2

DESY Workshop 21 Sept 2010 L. Dixon On-Shell Methods 3 On-shell methods in QCD

DESY Workshop 21 Sept 2010 L. Dixon On-Shell Methods 4 LHC is a multi-jet environment  new physics? Need precise understanding of “old physics” that looks like new physics 7 TeV 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,…

5 Cascade from gluino to neutralino (dark matter, escapes detector) Signal: missing energy + 4 jets SM background from Z + 4 jets, Z  neutrinos Backgrounds to Supersymmetry at LHC Current state of art for Z + 4 jets based on LO tree amplitudes (matched to parton showers)  normalization still quite uncertain   DESY Workshop 21 Sept 2010 L. Dixon On-Shell Methods Motivates goal of  

6 One-loop QCD amplitudes via Feynman diagrams For V + n jets (maximum number of external gluons only) # of jets# 1-loop Feynman diagrams DESY Workshop 21 Sept 2010 L. Dixon On-Shell Methods

DESY Workshop 21 Sept 2010 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

DESY Workshop 21 Sept 2010 L. Dixon On-Shell Methods 8 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 perturbative QCD – as alternative to Feynman diagrams! Important: perturbative information now assists analyticity. Poles Branch cuts Works even better in theories with lots of SUSY, like N=4 SYM

DESY Workshop 21 Sept 2010 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

10 One-loop amplitudes reduced to trees rational part When all external momenta are in D = 4, loop momenta in D = 4-2  (dimensional regularization), one can write: Bern, LD, Dunbar, Kosower (1994) known scalar one-loop integrals, same for all amplitudes coefficients are all rational functions – determine algebraically from products of trees using (generalized) unitarity DESY Workshop 21 Sept 2010 L. Dixon On-Shell Methods

DESY Workshop 21 Sept 2010 L. Dixon On-Shell Methods 11 Generalized Unitarity for Box Coefficients d i Britto, Cachazo, Feng, hep-th/ No. of dimensions = 4 = no. of constraints  discrete solutions (2, labeled by ±) Easy to code, numerically very stable

DESY Workshop 21 Sept 2010 L. Dixon On-Shell Methods 12 Box coefficients d i (cont.) Solutions simplify (and are more stable numerically) when all internal lines massless, at least one external line (K 1 ) massless: BH, ; Risager

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/ bubble coefficients come from ordinary double cuts, after removing contributions of boxes and triangles DESY Workshop 21 Sept 2010 L. Dixon On-Shell Methods triangle coefficients come from triple cuts, product of 3 tree amplitudes, but these are also “contaminated” by boxes Ossola, Papadopolous, Pittau, hep-ph/ ; Mastrolia, hep-th/ ; Forde, ; Ellis, Giele, Kunszt, ; Berger et al., ;…

DESY Workshop 21 Sept 2010 L. Dixon On-Shell Methods 14 Triangle coefficients Solves for suitable definitions of Box-subtracted triple cut has poles only at t = 0, ∞ Triangle coefficient c 0 plus all other coefficients c j obtained by discrete Fourier projection, sampling at (2p+1) th roots of unity Forde, ; BH, Triple cut solution depends on one complex parameter, t Bubble similar

15 Several Recent Implementations of On-Shell Methods for 1-Loop Amplitudes CutTools: Ossola, Papadopolous, Pittau, NLO WWW, WWZ,... Binoth+OPP, NLO ttbb, tt + 2 jets Bevilacqua, Czakon, Papadopoulos, Pittau, Worek, ; Rocket: Giele, Zanderighi, Ellis, Giele, Kunszt, Melnikov, Zanderighi, NLO W + 3 jets in large N c approx./extrapolation EMZ, , ; Melnikov, Zanderighi, Blackhat: Berger, Bern, LD, Febres Cordero, Forde, H. Ita, D. Kosower, D. Maître; T. Gleisberg, , , , , Sherpa  NLO production of W,Z + 3 (4) jets Method for Rational part: D-dim’l unitarity + on-shell recursion specialized Feynman rules DESY Workshop 21 Sept 2010 L. Dixon On-Shell Methods _ _ _ SAMURAI: Mastrolia, Ossola, Reiter, Tramontano, D-dim’l unitarity

DESY Workshop 21 Sept 2010 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.

17 Recent analytic application: One-loop amplitudes for a Higgs boson + 4 partons DESY Workshop 21 Sept 2010 L. Dixon On-Shell Methods Badger, Glover, Risager, Glover, Mastrolia, Williams, Badger, Glover, Mastrolia, Williams, Badger, Glover, hep-ph/ LD, Sofianatos, Badger, Campbell, Ellis, Williams, by parity H =  +  † Unitarity for cut parts, on-shell recursion for rational parts (mostly)

18 5-point – still analytic DESY Workshop 21 Sept 2010 L. Dixon On-Shell Methods DS BGMW

Besides virtual corrections, also need real emission 19 General subtraction methods for integrating real-emission contributions developed in mid-1990s Frixione, Kunszt, Signer, hep-ph/ ; Catani, Seymour, hep-ph/ , hep-ph/ Recently automated by several groups Gleisberg, Krauss, ; Seymour, Tevlin, ; Hasegawa, Moch, Uwer, ; Frederix, Gehrmann, Greiner, ; Czakon, Papadopoulos, Worek, ; Frederix, Frixione, Maltoni, Stelzer, DESY Workshop 21 Sept 2010 L. Dixon On-Shell Methods Infrared singularities cancel

Les Houches Experimenters’ Wish List DESY Workshop 21 Sept 2010 L. Dixon On-Shell Methods 20 Feynman diagram methods now joined by on-shell methods Berger table courtesy of C. Berger BCDEGMRSW; Campbell, Ellis, Williams

DESY Workshop 21 Sept 2010 L. Dixon On-Shell Methods 21 W + n jets Data NLO parton level (MCFM) n = 1 n = 2 n = 3 only LO available in 2007 LO matched to parton shower MC with different schemes CDF, [hep-ex] Tevatron

W + 3 jets at NLO at Tevatron 22 DESY Workshop 21 Sept 2010 L. Dixon On-Shell Methods Berger et al., Ellis, Melnikov, Zanderighi, Leading-color adjustment procedure Exact treatment of color Rocket

DESY Workshop 21 Sept 2010 L. Dixon On-Shell Methods 23 W + 3 jets at LHC LHC has much greater dynamic range Many events with jet E T s >> M W 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!

DESY Workshop 21 Sept 2010 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 E T W is too low. Looking at large E T for the 2 nd 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) Another reasonable scale is invariant mass of the n jets Bauer, Lange

DESY Workshop 21 Sept 2010 L. Dixon On-Shell Methods 25 Compare Two Scale Choices logs not properly cancelled for large jet E T – LO/NLO quite flat, also for many other observables

DESY Workshop 21 Sept 2010 L. Dixon On-Shell Methods 26 Total Transverse Energy H T at LHC often used in supersymmetry searches flat LO/NLO ratio due to good choice of scale  = H T

27 NLO pp  W + 4 jets now available DESY Workshop 21 Sept 2010 L. Dixon On-Shell Methods C. Berger et al., Virtual terms: leading-color (including quark loops); omitted terms only ~ few %

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 DESY Workshop 21 Sept 2010 L. Dixon On-Shell Methods 28

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 Frixione, Webber (2002),... –POWHEG Nason (2004); Frixione, Nason, Oleari (2007);... –POWHEG in SHERPA Höche, Krauss, Schönherr, Siegert, –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 DESY Workshop 21 Sept 2010 L. Dixon On-Shell Methods 29

DESY Workshop 21 Sept 2010 L. Dixon On-Shell Methods 30 On-shell methods in N=4 SYM

DESY Workshop 21 Sept 2010 L. Dixon On-Shell Methods 31 Why N=4 SYM? Dual to gravity/string theory on AdS 5 x S 5 Very similar in IR to QCD talk by Magnea Planar (large N c ) theory is integrable talk 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 by tree-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

DESY Workshop 21 Sept 2010 L. Dixon On-Shell Methods 32 N=4 SYM “states” Interactions uniquely specified by gauge group, say SU(N c ), 1 coupling g Exactly scale-invariant (conformal) field theory:  (g) = 0 for all g all states in adjoint representation, all linked by N=4 supersymmetry

DESY Workshop 21 Sept 2010 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 for large-N c (planar) N=4 SYM should have special properties, because large limit  weakly-coupled gravity/string theory on AdS 5 x S 5 Maldacena; Gubser, Klebanov, Polyakov; Witten

DESY Workshop 21 Sept 2010 L. Dixon On-Shell Methods 34 AdS/CFT in one picture

DESY Workshop 21 Sept 2010 L. Dixon On-Shell Methods 35 Scattering at strong coupling Use AdS/CFT to compute an appropriate scattering amplitude High energy scattering in string theory is semi-classical Evaluated on the classical solution, action is imaginary  exponentially suppressed tunnelling configuration Alday, Maldacena, [hep-th] Gross, Mende (1987,1988) Can also do with dimensional regularization instead of r

DESY Workshop 21 Sept 2010 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)

DESY Workshop 21 Sept 2010 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=4 Cachazo, Skinner, ; Arkani-Hamed, Cachazo, Kaplan,

DESY Workshop 21 Sept 2010 L. Dixon On-Shell Methods 38 Multi-loop generalized unitarity at work These cuts are maximally simple, yet give an excellent starting point for constructing the full answer. (No conjectures required.) Allowing for complex cut momenta, one can chop an amplitude entirely into 3-point trees  maximal cuts or ~ leading singularities In planar (leading in N c ) N=4 SYM, maximal cuts find all terms in the complete answer for 1, 2 and 3 loops Bern, Carrasco, LD, Johansson, Kosower, Roiban, hep-th/ ; Bern, Carrasco, Johansson, Kosower,

DESY Workshop 21 Sept 2010 L. Dixon On-Shell Methods 39 Finding missing terms These near-maximal cuts are very useful for analyzing N=4 SYM (including nonplanar) and N=8 SUGRA at 3 loops Maximal cut method: Allowing one or two propagators to collapse from each maximal cut, one obtains near-maximal cuts BCDJKR, BCJK (2007); Bern, Carrasco, LD, Johansson, Roiban, Recent supersum advances to evaluate more complicated cuts Drummond, Henn, Korchemsky, Sokatchev, ; Arkani-Hamed, Cachazo, Kaplan, ; Elvang, Freedman, Kiermaier, ; Bern, Carrasco, Ita, Johansson, Roiban, 2009 Maximal cut method is completely systematic not restricted to N=4 SYM not restricted to planar contributions

DESY Workshop 21 Sept 2010 L. Dixon On-Shell Methods 40 1 loop: 4-gluon amplitude in N=4 SYM at 1 and 2 Loops Bern, Rozowsky, Yan (1997); Bern, LD, Dunbar, Perelstein, Rozowsky (1998) 2 loops: Green, Schwarz, Brink; Grisaru, Siegel (1981)

DESY Workshop 21 Sept 2010 L. Dixon On-Shell Methods 41 Dual Conformal Invariance A conformal symmetry acting in momentum space, on dual (sector) variables x i First seen in N=4 SYM planar amplitudes in the loop integrals Broadhurst (1993); Lipatov (1999); Drummond, Henn, Smirnov, Sokatchev, hep-th/ x5x5 x1x1 x2x2 x3x3 x4x4 k invariant under inversion :

DESY Workshop 21 Sept 2010 L. Dixon On-Shell Methods 42 Dual conformal invariance at 4 loops Simple graphical rules: 4 (net) lines into inner x i 1 (net) line into outer x i Dotted lines are for numerator factors 4 loop planar integrals all of this form BCDKS, hep-th/ BCJK, also true at 5 loops

DESY Workshop 21 Sept 2010 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 AdS 5 x S 5. Also, strong-coupling calculation ~ equivalent to computation of Wilson line for n-sided polygon with vertices at x i Alday, Maldacena, 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, ; Eden, Korchemsky, Sokatchev, ,

DESY Workshop 21 Sept 2010 L. Dixon On-Shell Methods 44 Many higher-loop contributions to gg  gg scattering deduced from a simple property of the 2-particle cuts at one loop The rung rule Bern, Rozowsky, Yan (1997) Leads to “rung rule” for easily computing all contributions which can be built by iterating 2-particle cuts

DESY Workshop 21 Sept 2010 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) Two new ones (h,i) have no 2-particle cuts BDDPR (1998) BCDJKR (2007); BCDJR (2008)

DESY Workshop 21 Sept 2010 L. Dixon On-Shell Methods 46 N=4 numerators at 3 loops Omit overall manifestly quadratic in loop momentum

DESY Workshop 21 Sept 2010 L. Dixon On-Shell Methods 47 Four loops: full color N=4 SYM as input for N=8 SUGRA BCDJR, Bern, Carrasco, LD, Johansson, Roiban,

DESY Workshop 21 Sept 2010 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

DESY Workshop 21 Sept 2010 L. Dixon On-Shell Methods 49 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, Relate non-planar topologies to planar, etc. For example, at 3 loops, (i) = (e) – (e) T [ + contact terms ] =

DESY Workshop 21 Sept 2010 L. Dixon On-Shell Methods 50 Box cut 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. Bern, Carrasco, Johansson, Kosower,

DESY Workshop 21 Sept 2010 L. Dixon On-Shell Methods 51 Vacuum cubic graphs at 4 loops To decorate with 4 external legs cannot generate a nonvanishing (no-triangle) cubic 4-point graph only generate rung rule topologies the most complex

DESY Workshop 21 Sept 2010 L. Dixon On-Shell Methods 52 Dual conformal (pseudoconformal) invariance, acting on dual or sector variables x i Greatly limits the possible numerators No such guide for the nonplanar terms Planar terms well known Bern, Czakon, LD, Kosower, Smirnov, hep-th/ Drummond, Henn, Smirnov, Sokatchev, hep-th/

53 Simplest (rung rule) graphs N=4 SYM numerators shown DESY Workshop 21 Sept 2010 L. Dixon On-Shell Methods

54 Most complex graphs N=4 SYM numerators shown [N=8 SUGRA numerators much larger] DESY Workshop 21 Sept 2010 L. Dixon On-Shell Methods

DESY Workshop 21 Sept 2010 L. Dixon On-Shell Methods 55 Checks on final N=4 result Lots of different products of MHV tree amplitudes. NMHV 7 * anti-NMHV 7 and MHV 5 * NMHV 6 * anti-MHV 5 – evaluated by Elvang, Freedman, Kiermaier,

DESY Workshop 21 Sept 2010 L. Dixon On-Shell Methods 56 N=4 SYM in the UV Want the full color dependence of the UV divergences in N=4 SYM in the critical dimension. BCDJR (April `09) D c = 8 (L = 1) D c = 4 + 6/L (L = 2,3) For G = SU(N c ), divergences organized in terms of color structures: Found absence of double-trace terms, later studied by Bossard, Howe, Stelle, , ; Berkovits, Green, Russo, Vanhove, ; Bjornsson, Green,

DESY Workshop 21 Sept 2010 L. Dixon On-Shell Methods 57 N=4 in UV at 4 loops By injecting external momentum in right place, can rewrite as 4-loop propagator integrals that factorize into product of -1-loop propagator integral with UV pole - finite 3-loop propagator integral Do this in multiple ways  Either “gluing relations” or cross-check. Need UV poles of 4-loop vacuum graphs (doubled propagators represented by blue dots). Only 3 vacuum integrals required D c = 4 + 6/4 = 11/2

L. Dixon On-Shell Methods DESY Workshop 21 Sept UV behavior of N=8 at 4 loops All 50 cubic graphs have numerator factors composed of terms loop momenta l external momenta k Maximum value of m turns out to be 8 in every integral, vs. 4 for N=4 SYM In order to show that need to show that all cancel Integrals all have 13 propagators, so But they all do  N=8 SUGRA still no worse than N=4 SYM in UV at 4 loops!

DESY Workshop 21 Sept 2010 L. Dixon On-Shell Methods 59 Conclusions On-shell methods at one loop have many practical applications to LHC physics – analytically, but especially in numerical implementations On-shell methods in (planar) N=4 SYM have led to BDS ansatz and information about its violation at 6 points. Very recently used to construct the planar NMHV 2 loop 6 point amplitude Kosower, Roiban, Vergu, And the full-color 4 point 4 loop amplitude in N=4 SYM Latter result was used to construct the 4 point 4 loop amplitude in N=8 supergravity, which showed that it is still as well-behaved as N=4 super-Yang-Mills theory through this order Wealth of IR information in gauge theory & gravity is also available … once technology is developed for doing non-planar 4-point integrals (even numerically) in D = 4 – 2  at L = 3,4

DESY Workshop 21 Sept 2010 L. Dixon On-Shell Methods 60 Extra Slides

DESY Workshop 21 Sept 2010 L. Dixon On-Shell Methods 61 N=4 SYM in UV at one loop Box integral in D c =  with color factor  where Corresponds to counterterms such as and (no extra derivatives)

DESY Workshop 21 Sept 2010 L. Dixon On-Shell Methods 62 N=4 SYM in UV at two loops Planar and nonplanar double box integrals in D c =  [BDDPR 1998] with color factors  Corresponds to counterterms such as and (two extra derivatives)

DESY Workshop 21 Sept 2010 L. Dixon On-Shell Methods 63 N=4 SYM in UV at four loops Combining UV poles of integrals with color factors  Again corresponds to type counterterms. Absence of double-trace terms at L = 3 and 4.

L. Dixon On-Shell Methods DESY Workshop 21 Sept Cancellations between integrals Cancellation of k 4 l 8 terms [vanishing of coefficient of ] simple: just set external momenta k i  0, collect coefficients of 2 resulting vacuum diagrams, observe that the 2 coefficients cancel. Cancellation of k 5 l 7 [and k 7 l 5 ] terms is trivial: Lorentz invariance does not allow an odd-power divergence. UV pole cancels in D=5-2  N=8 SUGRA still no worse than N=4 SYM in UV at 4 loops! Cancellation of k 6 l 6 terms [vanishing of coefficient of ] more intricate: Expand to second subleading order in limit k i  0, generating 30 different vacuum integrals. Evaluating UV poles for all 30 integrals (or alternatively deriving consistency relations between them), we find that