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

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 2010L. Dixon On-Shell Methods 2


On-shell methods in QCD


LHC is a multi-jet environment

new physics?

Need precise understandingof “old physics”that looks likenew physics

LHC

@ 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 jetswhere

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 2010L. 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



Remembering a Simpler Time...

• In the 1960s there was no QCD,no Lagrangian or Feynman rulesfor the strong interactions


The Analytic S-MatrixBootstrap 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


Generalized unitarityOrdinary unitarity: Im T = T†Tput 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 algebraicallyfrom products of trees using (generalized) unitarity



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


Box coefficients di (cont.)

Solutions simplify (and are morestable numerically) when all internal lines massless, at least oneexternal line (K1) massless:

BH, 0803.4180; Risager 0804.3310

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

bubble coefficients come from ordinary double cuts, after removing contributions of boxes and triangles


triangle coefficients come from triple cuts, product of 3 tree amplitudes, but these are also “contaminated” by boxes

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

Solves

for suitable definitions of

Box-subtracted triple cut has polesonly at t = 0, ∞

Triangle coefficient c0

plus all other coefficients cj

obtained by discrete Fourierprojection, sampling at (2p+1)th roots of unity

Forde, 0704.1835; BH, 0803.4180

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, 0711.3596

NLO WWW, WWZ, ... Binoth+OPP, 0804.0350

NLO ttbb, tt + 2 jets Bevilacqua, Czakon, Papadopoulos, Pittau, Worek, 0907.4723; 1002.4009

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

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

Method forRational part:

D-dim’lunitarity+ on-shellrecursion

specializedFeynmanrules


_ _ _

SAMURAI: Mastrolia, Ossola, Reiter, Tramontano, 1006.0710

D-dim’lunitarity


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


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

H = + †

Unitarity for cut parts, on-shell recursion for rational parts (mostly)

18

5-point – still analytic


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/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 singularitiescancel

Les Houches Experimenters’ Wish List


Feynmandiagrammethods

now joinedby

on-shellmethods

Berger

table courtesy ofC. Berger

BCDEGMRSW; Campbell, Ellis, Williams


W + n jets Data

NLO parton level(MCFM)

n = 1

n = 2

n = 3 only LOavailablein 2007

LO matched to parton shower MC with differentschemes

CDF, 0711.4044 [hep-ex]

Tevatron

W + 3 jets at NLO at Tevatron

22DESY Workshop 21 Sept 2010L. Dixon On-Shell Methods

Berger et al., 0907.1984Ellis, Melnikov, Zanderighi, 0906.1445

Leading-color adjustment procedure Exact treatment of color

Rocket


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!


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) • Another reasonable scale is invariant mass of the n jets

Bauer, Lange0905.4739


Compare Two Scale Choices

logs not properlycancelled for large jet ET

– LO/NLO quite flat,also for many other observables


Total Transverse Energy HT at LHCoften used in supersymmetry searches

0907.1984

flat LO/NLO ratiodue to good choice ofscale = HT

27

NLO pp W + 4 jets now available


C. Berger et al., 1009.2338

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


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



On-shell methods in N=4 SYM


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 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


N=4 SYM “states”

• Interactions uniquely specified by gauge group, say SU(Nc), 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


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-Nc (planar) N=4 SYM should have special properties, because

large limit weakly-coupled gravity/string theory

on AdS5 x S5

Maldacena; Gubser, Klebanov, Polyakov; Witten


AdS/CFT in one picture


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, 0705.0303 [hep-th]

Gross, Mende (1987,1988)

Can also do with dimensional regularization instead of

r


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 variablesappear at weak coupling (in planar theory)


Generalized unitarity for N=4 SYMFound 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, 0801.4574; Arkani-Hamed, Cachazo, Kaplan, 0808.1446


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 Nc) 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/0702112; Bern, Carrasco, Johansson, Kosower, 0705.1864


Finding missing terms

These near-maximal cuts are veryuseful for analyzing N=4 SYM (including nonplanar)and N=8 SUGRA at 3 loops

Maximal cut method:Allowing one or two propagatorsto collapse from each maximal cut, one obtains near-maximal cuts

BCDJKR, BCJK (2007); Bern, Carrasco, LD, Johansson, Roiban, 0808.4112

Recent supersum advances to evaluate more complicated cutsDrummond, Henn, Korchemsky, Sokatchev, 0808.0491; Arkani-Hamed, Cachazo, Kaplan, 0808.1446; Elvang, Freedman, Kiermaier, 0808.1720; Bern, Carrasco, Ita, Johansson, Roiban, 2009

Maximal cut method is completely systematic• not restricted to N=4 SYM• not restricted to planar contributions


• 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)


Dual Conformal Invariance

A conformal symmetry acting in momentum space, on dual (sector) variables xi

First seen in N=4 SYM planar amplitudes in the loop integrals

Broadhurst (1993); Lipatov (1999); Drummond, Henn, Smirnov, Sokatchev, hep-th/0607160

x5

x1

x2

x3

x4

kinvariant under inversion:


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 integralsall of this form

BCDKS, hep-th/0610248

BCJK, 0705.1864

also true at 5 loops


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 thisAlday, Eden, Maldacena, Korchemsky, Sokatchev,1007.3243; Eden, Korchemsky, Sokatchev, 1007.3246,1009.2488


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


3 loop cubic graphsNine basic integraltopologies

Seven (a-g) werealready known(2-particle cuts rung rule)

Two new ones (h,i)have no 2-particle cuts

BDDPR (1998)

BCDJKR (2007); BCDJR (2008)


N=4 numerators at 3 loopsOmit overall

manifestly quadratic in loop momentum


Four loops:

full color N=4 SYM

as input for N=8 SUGRABCDJR, 0905.2326

Bern, Carrasco, LD, Johansson, Roiban, 1008.3327


4 loop 4 point amplitude in N=4 SYM

Number of cubic 4-point graphs with nonvanishingCoefficients and various topological properties


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 ]

= -

2 3

1 4


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, 0705.1864


Vacuum cubic graphs at 4 loops

To decorate with 4 external legs

cannot generatea nonvanishing(no-triangle)cubic 4-pointgraph

only generaterung ruletopologies

the most complex


Dual conformal (pseudoconformal) invariance, acting on dual or sector variables xi

Greatly limits the possible numeratorsNo such guide for the nonplanar terms

Planar terms well knownBern, Czakon, LD, Kosower, Smirnov, hep-th/0610248

Drummond, Henn, Smirnov, Sokatchev, hep-th/0607160

53

Simplest (rung rule) graphsN=4 SYM numerators shown


54

Most complex graphsN=4 SYM numerators shown

[N=8 SUGRA numerators much larger]



Checks on final N=4 result• Lots of different products of MHV tree amplitudes.

• NMHV7 * anti-NMHV7 and MHV5 * NMHV6 * anti-MHV5

– evaluated by Elvang, Freedman, Kiermaier, 0808.1720


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)

Dc = 8 (L = 1) Dc = 4 + 6/L (L = 2,3)• For G = SU(Nc), divergences organized in terms of color structures:

• Found absence of double-trace terms, later studied by Bossard, Howe, Stelle, 0901.4661, 0908.3883;

Berkovits, Green, Russo, Vanhove, 0908.1923; Bjornsson, Green, 1004.2692


N=4 in UV at 4 loops

• By injecting external momentum in right place, can rewrite as 4-loop propagator integralsthat 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 propagatorsrepresented byblue dots).

Only 3 vacuumintegrals required Dc = 4 + 6/4 = 11/2

L. Dixon On-Shell MethodsDESY Workshop 21 Sept 2010 58

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

allcancel

• 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!


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, 1009.1376• 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


Extra Slides


N=4 SYM in UV at one loop

• Box integral in Dc = 8 - 2 with color factor

where• Corresponds to counterterms such as and (no extra derivatives)


N=4 SYM in UV at two loops

• Planar and nonplanar double box integrals in Dc = 7 - 2 [BDDPR 1998] with color factors

• Corresponds to counterterms such as and (two extra derivatives)


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 MethodsDESY Workshop 21 Sept 2010 64

Cancellations between integrals• Cancellation of k4 l8 terms [vanishing of coefficient of ]

simple: just set external momenta ki 0,collect coefficients of 2 resulting vacuum diagrams,observe that the 2 coefficients cancel.

• Cancellation of k5 l7 [and k7 l5] terms is trivial: Lorentz invariance does not allow an odd-power divergence.

UV pole cancels in D=5-2N=8 SUGRA still no worse than N=4 SYM in UV at 4 loops!

• Cancellation of k6 l6 terms [vanishing of coefficient of ]

more intricate: Expand to second subleading order in limit ki 0, generating 30 different vacuum integrals.• Evaluating UV poles for all 30 integrals (or alternatively derivingconsistency relations between them), we find that

Documents

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