Multi-Parton QCD Amplitudes For The LHC.

Harald Ita, UCLA

Based on: arXiv:0803.4180 ; arXiv:0807.3705; arXiv:0808.0941

In collaboration with:Carola Berger, Zvi Bern, Fernando Febres Cordero, Lance Dixon, Darren Forde,

Daniel Maitre and David Kosower. Recently joined by Tanju Gleisberg

Content:

1. Simulation of scattering processes.2. LO versus NLO.3. The Wish-List.4. On-Shell Methods.5. Example technicalities.6. Amplitudes from BlackHat.7. Conclusions.

Scattering processes at hadron colliders: A multi-layered problem

Tevatron graphic taken from Rick Field

…+ multi-parton scattering +...

proton remnants.

hard scattering quark/gluon.

Tevatron graphic taken from Rick Field

• Understanding the underlying event.• Control over parton distribution functions.• The hard scattering:

• LO• NLO (Virtual <- BlackHat, Real)• Beyond

• Showering (matching).• Hadronization.

Layers of simulation:

Some random set of tools.1. Parton-level LO matrix element generators:

• MADGRAPH; Maltoni, Stelzer• …

2. LO ME + shower MCs + …:• ALPGEN; Mangano, Moretti, Piccinini, Pittau, Polosa• HERWIG; Marchesini, Webber, Abbiendi, Corcella, Knowles, Moretti, Odagiri,

Richardson, Seymour, Stanco • PYTHIA; Sjostrand, Mrenna, Skands• SHERPA; Gleisberg, Hoeche, Krauss, Schoenherr, Schumann, Siegert, Winter• …

3. NLO PL matrix element generators:• MCFM; Campbell, Ellis; (max 6 partons)• (BlackHat …,6,7,..(?) partons) • …

4. MC@NLO,POWHEG-method; Frixione, Webber; +Nason, Ridolfi, Oleari

high-multiplicities + automation

Leading Order versus Next to Leading Order.

QCD for :

LO versus NLO.

• Tree-level (LO) predictions qualitative, due to poor convergence of expansion in strong coupling,

• large NLO corrections can be 30% - 80% of LO.

implies scale uncertainty: •1 jet about 12%•2 jets about 24%•3 jets about 36%

K-factors?

Tevatron: Single Top ProductionTevatron: Single Top Productionsingle top productionsingle top production

Matrix element method uses full information of LO matrixelements to pull the signal out of background.

It should be possible to do better by using NLO matrix elements (about 20% gain in significance).

T. Aaltonen et al. [CDF Collaboration], arXiv:08092581

the LHC: an example of discovery

[ATLAS Collaboration]S. Vahsen

M L Mangano [arXiv:0809.1567]

Producing heavy colored particles

Backgrounds:

• Irreducible: •Z(->neutrinos)+4 jets

• Reducible:•W(->tau+neutrino)+3 jets•W(->undetected leptons)

+4 jets• top pairs

• Instrumental:• Multijets

How good are our tools?

SMPR-model: Mrena, Richardson, 2004.MLM-model: Alwall et al. arXiv:0706.2569. LO,

MCFM; parton level; including Bern, Dixon, Kosower, Weinzierlmatrix elements.

NLO.

Wanted: LHC studies with extra jets:

T. Aaltonen et al. [CDF Collaboration], arXiv:0711.4044

+ . . .

Wish-List.

QCD: “Experimenters’ Wish List”QCD: “Experimenters’ Wish List”Les Houches 2007

• Five-particle processes under good control with Feynman diagram based approaches.

• Six-particle processes still difficult challenge.

• Most physics results done from Feynman diagram approach:– QCD corrections to vector boson pair production (W+W-, W±Z & ZZ) via vector

boson fusion (VBF). (Jager, Oleari, Zeppenfeld)+(Bozzi)

– QCD and EW corrections to Higgs production via VBF. (Ciccolini, Denner, Dittmaier)

– pp → Higgs+2 jets. (via gluon fusion Campbell, Ellis, Zanderighi), (via weak interactions Ciccolini, Denner, Dittmaier). pp → Higgs+3 jets (leading contribution) (Figy, Hankele, Zeppenfeld).

– pp→ . (Beenakker, Dittmaier, Krämer, Plümper, Spira, Zerwas), (Dawson, Jackson, Reina, Wackeroth)

– pp → ZZZ, (Lazopoulos, Petriello, Melnikov) pp → +(McElmurry)

– pp → WWZ, WWW (Hankele, Zeppenfeld, Campanario, Oleari, Prestel)

– pp→WW+j+X. (Campbell, Ellis, Zanderighi). (Dittmaier, Kallweit, Uwer)

– pp →W/Z (Febres Cordero, Reina, Wackeroth),

– pp → +jet (Dittmaier,Uwer,Weinzierl),

– (Bredenstein,Denner,Dittmaier,Pozzorini),

ttZ

ttH

bbtt

qq tt bb

What Has Been Done?

What we need.What we need.

• Scalable to larger numbers of external partons.• A general solution that applies to any process.• Numerical stability and efficiency.• Automation.

On-Shell Methods.

• Calculating using Feynman diagrams is Hard! • A Factorial growth in the number of terms, particularly bad

for large number of gluons.

• Example of difficulty:

Gauge dependantquantities, large cancellationsbetween terms.

Economic techniques?

Note: this is trivial on modern computer. Non-trivial for larger numbers of external particles.

Result of performing the integration gives:

Calculations explode for larger numbers of particlesor loops. There should be a better way.

• Calculated ON-SHELL, amplitudes much simpler than expected.

• For example: some tree level all-multiplicity gluon amplitudes can fit on a page:

On-shell simplifications.

Park, Taylor

• Vertices and propagators involve unphysical gauge-dependent off-shell states.

• Feynman diagram loops have to be off-shell because they encode the uncertainty principle.

• Keep particles on-shell in intermediate steps of calculation, not in final results.

Think off-shell, work on-shell!

Bern, Dixon, Dunbar, Kosower

Fact: Off-shellness is essential for getting the correct answer.

Would like to reconstruct amplitude using only on-shell information.

Exploit universal physical properties for decomposition in terms of on-shell objects:

1. Unitarity relation.2. Universal Factorization.3. Complex momenta through spinor variables.

On-shell methods: opening a gate to new computational possibilities...

• Past year progress using unitarity and related techniques,– gg → gggg amplitude. (Bern,Dixon,Kosower), (Britto,Feng,Mastrolia), (Bern,Bjerrum-

Bohr,Dunbar,H.I.), (Berger,Bern,Dixon,Forde,Kosower), (Bedford,Brandhuber,Spence,Travaglini) (Xiao,Yang,Zhu) ,(Berger,Bern,Dixon,Forde,Kosower), (Giele,Kunszt,Melnikov)

– Lots of gluons (Giele,Zanderighi), (Berger, Bern, Dixon, Febres Cordero, Forde,H.I., Kosower, Maître)

– 6 photons (Nagy, Soper), (Ossola, Papadopoulos, Pittau), (Binoth, Heinrich, Gehrmann, Mastrolia)

– pp → ZZZ, WZZ, WWZ, ZZZ (Binoth, Ossola, Papadopoulos, Pittau),

– gg → using D-Dimensional Unitarity (Ellis,Giele,Kunszt,Melnikov)

• Numerical packages under construction:– BlackHat Berger, Bern, Dixon, Febres Cordero, Forde, H.I., Kosower, Maître

– CutTools Ossola, Papadopoulos, Pittau– Rocket Ellis, Giele, Kunszt, Melnikov, Zanderighi

ttg

What Has Been Done?

Example technicalities.

The result: one-loop basis.All external momenta in D=4, loop momenta in D=4-2ε

(dimensional regularization).

• Cut Part from unitarity cuts in 4 dimensions.• Rational part from on-shell recurrence relations.

See Bern, Dixon, Dunbar, Kosower, hep-ph/9212308.

Rational part Cut part

Process dependent D=4 rational integral coefficients

Unitarity Method.• Unitarity Approach:

– Bern, Dixon, Dunbar, Kosower, hep-ph/9403226, hep-th/9409265.

– Recent Advances using spinorial integration techniques:• Cachazo, Svrcek, Witten; Britto, Feng, Cachazo; Britto, Feng, Mastrolia

• Generalized Unitarity: – Bern, Dixon, Kosower, hep-ph/9708239, hep-ph/0001001.– Britto, Cachazo, Feng, hep-th/0412103.– Recent Advances: classification of surface terms.

• del Aguila and Pittau, hep-ph/0404120.• Ossola, Papadopoulos and Pittau, hep-ph/0609007.• Forde, 0704.1835; Badger, 0806.4600, 0807.1245 .• Ellis, Giele, Kunszt, 0708.2398; Giele, Kunszt and Melnikov, 0801.2237;

Ellis, Giele, Kunszt, Melnikov, 0806.3467;

Unitarity: an on-shell method of calculation. Bern, Dixon, Kosower

Cutting loops = sewing trees: Sewing:

Cutting: 2x

And NOT:

Equation:

Number of diags.

Generalized Unitarity: isolate the leading discontinuity.

More cuts, more trees, less algebra:•Two-particle cut: product of trees contains subset of box-, triangle- and bubble-integrals. (Bern, Dixon, Kosower, Dunbar)

•Triple-cut: product of three trees contains triangle- and box-integrals. (Bern, Dixon, Kosower)

•Quadruple-cut: read out single box coefficient. (Britto, Cachazo, Feng)

Cutting: n x

Boxes: the simplest cuts.

Un-physical (=spurious) singularities from parameterization.Have to cancel eventually: role of rational term R.

Berger, Bern, Dixon, Febres Cordero, Forde,H.I., Kosower, Maitre 0803.4180; Risager 0804.3310.

Amplitudes From BlackHat.

BlackHat: A C++ implementation of on-shell techniques for 1-loop amplitudes

• Portability (standard libraries for unix systems)• Modularity (object oriented)• Malleability (to accept several routines – numerics and

analytics)• Numerical precision and efficiency• Ready to use with existing Monte Carlo programs

– Work in progress with automated real dipole subtraction from Sherpa (with T. Gleisberg)

Gluon amplitudes: The Tails. Double-precision numerical computation.

Dynamic multi-precision computation. Reference: analytic targets from Bern, Dixon, Dunbar, Kosower, hep-ph/9403226, hep-

ph/9409265, hep-ph/0507005.

Natural tail

Multi-precision

Natural tail

(III)Multi-precision

100 000 PS points, ET>0.01 s, pseudo rapidity<3, separation cut >0.4

Watch Instabilities.• Monitor using known IR/UV pole structure of

amplitudes.

• Generalization for rational part. (A consistency condition of spurious residues.)

• Avoid instabilities with analytic tricks: – Use good loop momentum parametrizations & spinor variables.

Cross SectionsBlackHat: a snapshot…

Multiprecision arithmetic gives excellent control over numerical stability…double

double doublequadruple double

One Loop Helicity Amplitude

Rational PartCut Part

boxes triangles bubbles

Trees

Recursive diagrams

Spurious poles

One Loop Helicity Amplitude

Rational PartCut Part

boxes triangles bubbles

Trees

Recursive diagrams

Spurious poles

One Loop Helicity Amplitude

Rational PartCut Part

boxes triangles bubbles

Trees

Recursive diagrams

Spurious poles

Progress Report.

October 08

Z+3jets: Leading Color Stability StudyBerger, Bern, Dixon, FFC, Forde, Ita, Kosower, Maître, arXiv:0808.0941[hep-ph]100 000 PS points, ET>0.01 s,

pseudo rapidity<3, separation cut >0.4

Timing: Z+3 jets.

Same order as 6pt gluon amplitudes ~50ms: Quite an improvement compared to other numerical methods!

Helicity:

+++: 4.01ms/4.06ms 10.6ms/10.7ms (now: ~4ms)

-++: 6.41ms/6.64ms 41.7ms/42.9ms

++-: 6.47ms/6.67ms 28.0ms/28.4ms (now~6.5 ms)

-+-: 7.70ms/7.92ms 39.2ms/40.5ms

Double precision Dynamic multi precision

Full amplitude4-D cut-part

Conclusions

• NLO QCD corrections to hard cross sections will be an important tool for LHC analyses.

• On-shell methods have opened a new gate to computational power in QFTs.

• BlackHat-implementation has proven good precision properties.

• We expect first important phenomenological results, for example in V+3jets processes and beyond!