Peter Uwer*)

Universität Karlsruhe

*) Financed through Heisenberg fellowship and SFB-TR09

Graduiertenkolleg “Physik an Hadronbeschleunigern”, Freiburg 07.11.07

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

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

2

Contents

1. Introduction

2. Methods

3. Results

4. Conclusion / Outlook

3

Preliminaries

Technicalities Physics

Experts Non-Experts

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

4

Why do we need to go beyond the Born approximation

?

5

Residual scale dependence

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 s:

6

Scale dependence

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

1 2 3 4 n

20 %

40 %

30 %

10 %

22QCD

23QCD

24QCD

In addition we have also the factorization scale...

Need loop corrections to make quantitative predictions

Born approximation gives only crude estimate!

7

Corrections are not small...

Top-quark pair production at LHC:

~30-40%

/mt

[Dawson, Ellis, Nason ’89, Beenakker et al ’89,’91,Bernreuther, Brandenburg, Si, P.U. ‘04]

Scale independent corrections are also important !

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...and difficult to estimate

WW production via gluon fusion:

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

30 % enhancement due to an “NNLO” effect (s2)

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

9

To summarize:

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

10

Shall we calculate NLO correctionsfor everything

?

11

WW + 1 Jet ― Motivation

● For 155 GeV < mh < 185 GeV, H WW is important channel

● In mass range 130 ―190 GeV, VBF dominates over ggH

NLO corrections for VBF known [Han, Valencia, Willenbrock 92Figy, Oleari, Zeppenfeld 03,Berger,Campbell 04, …]

Signal:

Background reactions:

WW + 2 Jets, WW + 1 Jet

If only leptonic decay of W´s and 1 Jet is demanded(improved signal significance)

Higgs search:

two forward tagging jets + Higgs

NLO corrections unknown

Top of the Les Houches list 07

12

t t + 1 Jet ― Motivation

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)

LHC is as top quark factory

13

Methods

14

Next-to leading order corrections

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 problemof soft and collinear singularities*)

Regularization needed dimensional regularisation

1

n

1

n

* 1

n+1

*) For hadronic initial state additional term from factorization…

*)

15

Ingredients for NLO

1

n

1

n

*

1

n+1

1

n+1

1

n

1

n

+

Many diagrams, complicated structure,

Loop integrals (scalar and tonsorial)divergent (soft and mass sing.)

Many diagrams,divergent (after phase space integ.)

Combination procedure to addvirtual and real corrections

16

How to do the cancellation in practice

Consider toy example:

Phase space slicing method:

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

[Giele,Glover,Kosower]

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Dipole subtraction method (1)

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

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

Recently: NNLO algorithm [Daleo, Gehrmann, Gehrmann-de Ridder, Glover, Heinrich, Maitre]

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Dipole subtraction method (2)

Universal structure:

Generic form of individual dipol:Leading-order amplitudes

Vector in color space

Color charge operators,induce color correlation

Spin dependent part,induces spin correlation

universal

Example ggttgg: 6 different colorstructures in LO,

36 (singular) dipoles

! !

Universality of soft and coll. Limits!

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Dipole subtraction method — implementation

LO – amplitude, with colour information,

i.e. correlations

List of dipoles we want to calculate

0

1234

5

reduced kinematics,“tilde momenta” + Vij,k

Dipole di

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1

n+1

LO amplitudes enter in many places…

21

Leading order amplitudes ― techniques

Many different methods to calculate LO amplitudes exist

We used:

● Berends-Giele recurrence relations

● Feynman-diagramatic approach

● Madgraph based code

Issues:

Speed and numerical stability

Helicity bases

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

22

nn

*1 1

23

Virtual corrections

Issues:

● Scalar integrals

● How to derive the decomposition?

Traditional approach: Passarino-Veltman reduction

Scalar integrals

Large expressions numerical implementation

Numerical stability and speed are important

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Passarino-Veltman reduction

?

[Passarino, Veltman 79]

25

Reduction of tensor integrals — what we did…

Reduction à la Passarino-Veltman,with special reduction formulae in singular regions, two complete independent implementations !

Five-point tensor integrals:

Four and lower-point tensor integrals:

● Apply 4-dimensional reduction scheme, 5-point tensor

integrals are reduced to 4-point tensor integrals

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

No dangerous Gram determinants!

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

[Denner, Dittmaier 02]

● Reduction à la Giele and Glover [Duplancic, Nizic 03, Giele, Glover 04]

Use integration-by-parts identities to reduce loop-integralsnice feature: algorithm provides diagnostics and rescue system

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What about twistor inspired techniques ?

● 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

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Results

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tt + 1-Jet production

Sample diagrams (LO):

Partonic processes:

related by crossing

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

Most complicated 1-loop diagrams pentagons of the type:

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Leading-order results — some features

Observable: ● Assume top quarks as always tagged

● To resolve additional jet demand minimum kt of 20 GeV

Note:● Strong scale dependence of LO result

● No dependence on jet algorithm

● Cross section is NOT small

LHCTevatron

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Checks of the NLO calculation

● 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 !

Virtual corrections:

QGraf — Form3 — C,C++

Feynarts 1.0 — Mathematica — Fortran77

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Top-quark pair + 1 Jet Production at NLO[Dittmaier, P.U., Weinzierl PRL 98:262002, ’07]

● Scale dependence is improved

● Sensitivity to the jet algorithm

● Corrections are moderate in size

● Arbitrary (IR-safe) obserables calculable

Tevtron LHC

work in progress

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Forward-backward charge asymmetry (Tevatron)

● Numerics more involved due to cancellations, easy to improve

● Large corrections, LO asymmetry almost washed out● Refined definition (larger cut, different jet algorithm…) ?

Effect appears already in top quark pair production

[Kühn, Rodrigo]

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

33

Differential distributions

Preliminary *)

*) Virtual correction cross checked, real corrections underway

34

pT distribution of the additional jet

Corrections of the oder of 10-20 %,again scale dependence is improved

LHCTevtron

35

Pseudo-Rapidity distribution

LHCTevtron

Asymmetry is washed out by the NLO corrections

36

Top quark pt distribution

The K-factor is nota constant!

Phase space dependence, dependence on the observable

Tevtron

37

WW + 1 Jet

Leading-order – sample diagrams

Next-to-leading order – sample diagramsNext-to-leading order – sample diagrams

Many different channels!

38

Checks

Similar to those made in tt + 1 Jet

Main difference:

Virtual corrections were cross checked using LoopTools[T.Hahn]

39

Scale dependence WW+1jet

Cross section defined as in tt + 1 Jet

[Dittmaier, Kallweit, Uwer 07]

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

40

Cut dependence[Dittmaier, Kallweit, Uwer 07]

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

41

Conclusions

● 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

General lesson:

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Conclusions

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

43

Conclusions

WW + 1-Jet production at NLO:

● Two complete independent calculations

● Scale dependence is improved (LHC jet-veto)

● Corrections are important

[Gudrun Heinrich ]

44

Outlook

● Proper definition of FB-charge asymmetry

● Further improvements possible

(remove redundancy, further tuning, except. momenta,…)

● Distributions

● Include decay

● Apply tools to other processes

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The End