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Peter Uwer *)

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Graduiertenkolleg “Physik an Hadronbeschleunigern”, Freiburg 07.11.07

ppttj and ppWWj at next-to-leading order in QCD

Peter Uwer*)

Universität Karlsruhe

Work in collaboration with S.Dittmaier, S. Kallweit and S.Weinzierl

*) Financed through Heisenberg fellowship and SFB-TR09

- Introduction
- Methods
- Results
- Conclusion / Outlook

Technicalities

Physics

Non-Experts

Experts

Outline of the main problems/issues/challenges with only brief description of methods used

Why do we need to go beyond the Born approximation

?

Quantum corrections lead to scale dependence of the coupling constants, i.e:

Large residual scale dependence of the Born approximation

In particular, if we have high powers of as:

40 %

30 %

20 %

10 %

1

3

2

4

n

22

QCD

23

QCD

24

QCD

For m≈ mt and a variation of a factor 2 up and down:

In addition we have also the factorization scale...

Born approximation gives only crude estimate!

Need loop corrections to make quantitative predictions

Top-quark pair production at LHC:

[Dawson, Ellis, Nason ’89,

Beenakker et al ’89,’91,

Bernreuther, Brandenburg, Si, P.U. ‘04]

~30-40%

m/mt

Scale independent corrections are also important !

WW production via gluon fusion:

[Duhrssen, Jakobs, van der Bij, Marquard 05Binoth, Ciccolini,Kauer Krämer 05,06]

tot = no cuts, std = standard LHC cuts, bkg = Higgs search cuts

30 % enhancement due to an “NNLO” effect (as2)

NLO corrections are needed because

- Large scale dependence of LO predictions…

- New channels/new kinematics in higher orders can have important impact in particular in the presence of cuts

- Impact of NLO corrections very difficult to predict without actually doing the calculation

Shall we calculate NLO corrections

for everything

?

Higgs search:

- For 155 GeV < mh < 185 GeV, H WW is important channel
- In mass range 130 ―190 GeV, VBF dominates over ggH

[Han, Valencia, Willenbrock 92

Figy, Oleari, Zeppenfeld 03,

Berger,Campbell 04, …]

NLO corrections for VBF known

Signal:

two forward tagging jets + Higgs

Background reactions:

WW + 2 Jets, WW + 1 Jet

Top of the

Les Houches list 07

NLO corrections

unknown

If only leptonic decay of W´s and 1 Jet is demanded

(improved signal significance)

LHC is as top quark factory

- Important signal process
- Top quark physics plays important role at LHC
- Large fraction of inclusive tt are due to tt+jet
- Search for anomalous couplings
- Forward-backward charge asymmetry (Tevatron)
- Top quark pair production at NNLO ?
- New physics ?
- Also important as background (H via VBF)

Methods

1

1

1

n

n

n+1

*)

*

Experimentally soft and collinear partons cannot be resolved due to finite detector resolution

Real corrections have to be included

The inclusion of real corrections also solves the problem

of soft and collinear singularities*)

Regularization needed dimensional regularisation

*) For hadronic initial state additional term from factorization…

1

1

1

1

1

1

n

n

n

n

n+1

n+1

Many diagrams,

complicated structure,

Loop integrals (scalar and tonsorial)

divergent (soft and mass sing.)

*

Combination procedure to add

virtual and real corrections

+

Many diagrams,

divergent (after phase space integ.)

Consider toy example:

Phase space slicing method:

[Giele,Glover,Kosower]

[Frixione,Kunszt,Signer ´95, Catani,Seymour ´96, Nason,Oleari 98, Phaf, Weinzierl, Catani,Dittmaier,Seymour, Trocsanyi ´02]

Subtraction method

[Frixione,Kunszt,Signer ´95, Catani,Seymour ´96, Nason,Oleari 98, Phaf, Weinzierl, Catani,Dittmaier,Seymour, Trocsanyi ´02]

How it works in practise:

Requirements:

in all single-unresolved regions

Due to universality of soft and collinear factorization,

general algorithms to construct subtractions exist

Recently: NNLO algorithm

[Daleo, Gehrmann, Gehrmann-de Ridder, Glover, Heinrich, Maitre]

Universality of soft and coll. Limits!

Universal structure:

Generic form of individual dipol:

Leading-order amplitudes

Vector in color space

universal

!

!

Color charge operators,

induce color correlation

Spin dependent part,

induces spin correlation

6 different colorstructures in LO,

36 (singular) dipoles

Exampleggttgg:

LO – amplitude,

with colour information,

i.e. correlations

List of dipoles we

want to calculate

2

1

3

4

5

0

reduced kinematics,

“tilde momenta” + Vij,k

Dipole di

1

n+1

LO amplitudes enter in many places…

Many different methods to calculate LO amplitudes exist

(Tools: Alpgen [MLM et al], Madgraph [Maltoni, Stelzer], O’mega/Whizard [Kilian,Ohl,Reuter],…)

We used:

- Berends-Giele recurrence relations
- Feynman-diagramatic approach
- Madgraph based code

Helicity bases

Issues:

Speed and numerical stability

1

1

*

n

n

Scalar integrals

Issues:

- Scalar integrals
- How to derive the decomposition?

Traditional approach: Passarino-Veltman reduction

Large expressions numerical implementation

Numerical stability and speed are important

[Passarino, Veltman 79]

?

Four and lower-point tensor integrals:

Reduction à la Passarino-Veltman,

withspecial reductionformulae insingular regions,

two complete independent implementations !

Five-point tensor integrals:

- Apply4-dimensional reductionscheme, 5-point tensor integrals are reduced to 4-point tensor integrals

No dangerous Gram determinants!

[Denner,

Dittmaier 02]

Based on the fact that in 4 dimension 5-point integrals can be reduced to 4 point integrals

[Melrose ´65, v. Neerven, Vermaseren 84]

- Reduction à la Giele and Glover

[Duplancic, Nizic 03, Giele, Glover 04]

Use integration-by-parts identities to reduce loop-integrals

nice feature: algorithm provides diagnostics and rescue system

- For tree amplitudes no advantage compared to Berends-Giele like techniques (numerical solution!)
- In one-loop many open questions
- Spurious poles
- exceptional momentum configurations
- speed

My opinion:

- For tree amplitudes tune Berends-Giele for stability and speed taking into account the CPU architecture of the LHC periode: x86_64
- For one-loop amplitudes have a look at cut inspired methods

Results

Sample diagrams (LO):

Partonic processes:

related by crossing

One-loop diagrams (~ 350 (100) for gg (qq)):

Most complicated 1-loop diagramspentagons of the type:

LHC

Tevatron

- Assume top quarks as always tagged
- To resolve additional jet demand minimum kt of 20 GeV

Observable:

- Strong scale dependence of LO result
- No dependence on jet algorithm
- Cross section is NOT small

Note:

- Leading-order amplitudes checked with Madgraph
- Subtractions checked in singular regions
- Structure of UV singularities checked
- Structure of IR singularities checked

Most important:

- Two complete independent programs using a complete different tool chain and different algorithms, complete numerics done twice !

Feynarts 1.0 — Mathematica — Fortran77

Virtual corrections:

QGraf — Form3 — C,C++

[Dittmaier, P.U., Weinzierl PRL 98:262002,’07]

Tevtron

LHC

- Scale dependence is improved
- Sensitivity to the jet algorithm
- Corrections are moderate in size
- Arbitrary (IR-safe) obserables calculable

work in progress

[Dittmaier, P.U., Weinzierl PRL 98:262002,’07]

Effect appears already in

top quark pair production

[Kühn, Rodrigo]

- Numerics more involved due to cancellations, easy to improve
- Large corrections, LO asymmetry almost washed out
- Refined definition (larger cut, different jet algorithm…) ?

Preliminary *)

*) Virtual correction cross checked, real corrections underway

LHC

Tevtron

Corrections of the oder of 10-20 %,

again scale dependence is improved

Tevtron

LHC

Asymmetry is washed out by the NLO corrections

The K-factor is not

a constant!

Phase space dependence, dependence on the observable

Tevtron

Leading-order – sample diagrams

Next-to-leading order – sample diagrams

Next-to-leading order – sample diagrams

Many different channels!

Similar to those made in tt + 1 Jet

Main difference:

Virtual corrections were cross checked using LoopTools

[T.Hahn]

[Dittmaier, Kallweit, Uwer 07]

Cross section defined as in tt + 1 Jet

[NLO corrections have been calculated also by Ellis,Campbell, Zanderighi t0+1d]

[Dittmaier, Kallweit, Uwer 07]

Note: shown results independent from the decay of the W´s

General lesson:

- NLO calculations are important for the success of LHC
- After more than 30 years (QCD) they are still difficult
- Active field, many new methods proposed recently!
- Many new results

Top quark pair + 1-Jet production at NLO:

- Two complete independent calculations
- Methods used work very well
- Cross section corrections are under control
- Further investigations for the FB-charge asymmetry necessary (Tevatron)
- Preliminary results for distributions

WW + 1-Jet production at NLO:

- Two complete independent calculations
- Scale dependence is improved (LHC jet-veto)
- Corrections are important

[Gudrun Heinrich ]

- Proper definition of FB-charge asymmetry
- Further improvements possible
- (remove redundancy, further tuning, except. momenta,…)
- Distributions
- Include decay
- Apply tools to other processes

The End