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### 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:

- Simulation of scattering processes.
- LO versus NLO.
- The Wish-List.
- On-Shell Methods.
- Example technicalities.
- Amplitudes from BlackHat.
- 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.

Some random set of tools.

- Parton-level LO matrix element generators:
- MADGRAPH; Maltoni, Stelzer
- …
- 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
- …
- NLO PL matrix element generators:
- MCFM; Campbell, Ellis; (max 6 partons)
- (BlackHat …,6,7,..(?) partons)
- …
- MC@NLO,POWHEG-method; Frixione, Webber; +Nason, Ridolfi, Oleari

high-multiplicities + automation

LOversusNLO.

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

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

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

K-factors?

Tevatron: Single Top Production

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

single top production

Matrix element method uses full information of LO matrix

elements to pull the signal out of background.

It should be possible to do better by using

NLO matrix elements (about 20% gain in significance).

the LHC: an example of discovery

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

[ATLAS Collaboration]

S. Vahsen

Wanted: LHC studies with extra jets:

How good are our tools?T. Aaltonen et al. [CDF Collaboration], arXiv:0711.4044

+ . . .

SMPR-model: Mrena, Richardson, 2004.

MLM-model: Alwall et al. arXiv:0706.2569.

LO,

MCFM; parton level; including Bern,

Dixon, Kosower, Weinzierl

matrix elements.

NLO.

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+2jets.(via gluon fusion Campbell, Ellis, Zanderighi), (via weak interactions Ciccolini, Denner, Dittmaier).pp → Higgs+3jets (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),

What we need.

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

Calculating using Feynman diagrams is Hard!

A Factorial growth in the number of terms, particularly bad for large number of gluons.

Example of difficulty:

Economic techniques?

Gauge dependant

quantities,

large cancellations

between terms.

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 particles

or 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

Think off-shell, work on-shell!

- 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.

Would like to reconstruct amplitude using only

on-shell information.

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

Bern, Dixon, Dunbar, Kosower

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

- Exploit universal physical properties for decomposition in terms of on-shell objects:
- Unitarity relation.
- Universal Factorization.
- Complex momenta through spinor variables.

- 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
- CutToolsOssola, Papadopoulos, Pittau
- Rocket Ellis, Giele, Kunszt, Melnikov, Zanderighi

The result: one-loop basis.

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

All external momenta in D=4, loop momenta inD=4-2ε (dimensional regularization).

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

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

Sewing:

Cutting loops = sewing trees:

Cutting: 2x

Equation:

And NOT:

Number

of diags.

Generalized Unitarity: isolate theleading discontinuity.

Cutting: n x

- 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)

Boxes: the simplest cuts.

Berger, Bern, Dixon, Febres Cordero, Forde,

H.I., Kosower, Maitre 0803.4180; Risager0804.3310.

Un-physical (=spurious) singularities from parameterization.

Have to cancel eventually: role of rational term R.

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

Natural tail

Multi-

precision

(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.

BlackHat: a snapshot…

Trees

Trees

Trees

One Loop Helicity Amplitude

One Loop Helicity Amplitude

One Loop Helicity Amplitude

Cut Part

Rational Part

Cut Part

Cut Part

Rational Part

Rational Part

Recursive diagrams

Spurious poles

Recursive diagrams

Recursive diagrams

Spurious poles

Spurious poles

boxes

boxes

boxes

triangles

triangles

triangles

bubbles

bubbles

bubbles

quadruple double

double double

double

Multiprecision arithmetic gives excellent control over numerical stability…

Z+3jets: Leading Color Stability Study

Berger, 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

October 08

Timing: Z+3 jets.

Double precision

Dynamic multi precision

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

Helicity:

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

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

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

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

4-D cut-part

Full amplitude

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!

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