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R JETS Predictions at NLO with MCFM. Hugo Beauchemin Renaud Bruneliere James Buchanan. Introduction. R Jets Definition and motivation for measurement. Parton Level Theoretical Uncertainties MCFM and some basics of it’s use. Results for 0 and 1 Jets: Scale Uncertainties

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R jets predictions at nlo with mcfm

RJETS Predictions at NLO with MCFM

Hugo Beauchemin

Renaud Bruneliere

James Buchanan


Introduction
Introduction

  • RJets Definition and motivation for measurement.

  • Parton Level Theoretical Uncertainties

  • MCFM and some basics of it’s use.

  • Results for 0 and 1 Jets:

    • Scale Uncertainties

    • PDF Uncertainties

  • Conclusions and Further Work


R jets definition and motivation
RJETS - Definition and Motivation

  • RJets is defined as the ratio of the W+n jets cross section and the Z+n jets cross section:

  • Experimentally taking such a ratio has benefits due to the cancellation of certain factors, in this case notably the Luminosity and factors in the acceptance x efficiency due to the jets.

  • Largest uncertainties due to different backgrounds and different acceptance x efficiency for 2 leptons vs. lepton + MET in final state.


R jets definition and motivation1
RJETS - Definition and Motivation

  • The intention is to perform this measurement as a function of the kinematic of the jets. Constructing the measurement this way allows:

    • A study of lepton efficiencies, fake rates, scale and resolution uncertainties at different scales

    • Low energy deviations indicate poor understanding of backgrounds and efficiencies

    • Deviations in higher kinematic regions would indicate new physics – binning in terms of number of jets give sensitivity to different scenarios

    • Possible to constrain some theory uncertainties.


Theoretical uncertainties
Theoretical Uncertainties

  • A parton level cross section is given via the QCD factorization theorem as:

  • At this level uncertainties arise due to:

    • Uncertainties in the fit to the parton distribution functions f(xa,µF).

    • Uncertainties due to the choice of renormalization and factorization scale. At higher orders of calculation the dependence on these parameters decreases but at any given fixed order a specific choice must be made and this affects the prediction.

  • To what extent do these cancel in the ratio?


MCFM

  • To answer this question we used MCFM, a parton level Next to Leading Order Monte Carlo written by John Cambell and Keith Ellis.

  • Produces weighted events. Cross sections are found by summing weights of events in appropriate regions of phase space:

( Somewhat simplified – ignores importance sampling)


Counterterm subtraction
Counterterm Subtraction

  • Both real and virtual contributions at NLO contain infrared divergences which formally cancel.

  • To numerically evaluate these contributions they need to be separately rendered finite in such a way that their sum is unchanged, this is done by Counterterm subtraction.

  • Additional terms are generated with the same divergence structure as the Real emission terms but which are simple enough to analytically integrate and which, on integration, formally render the Virtual contribution finite.

  • The resulting Virtual terms can be calculated analytically for each event. The real emission counterterms can have different kinematics to a given real event and are generated separately  for each real emission event there is a set of corresponding ‘counter-events’.


  • Events and counter events not independent  not possible to quote statistical uncertainty using variance of weight distribution within a bin. A given event can have a large +ve weight which is compensated by a large -ve weight for a counter event

  • Given events and counter events can have different kinematics it is possible to have large fluctuations in adjacent bins when they are separated by insufficiently coarse binning. This can be smoothed by searching for adjacent bins with values that deviate from a local fit in opposite directions.

  • Due to the simplified structure of the counter-terms the counter events are generated with LO kinematics and this can cause instabilities at the jet PT threshold, notably in the Boson PT distribution.


Upper left and right: Removing spikes due to events and counter events in adjacent bins.

Left: Systemic asymptotic behaviour in Boson PT distribution for Z + 1 jet due to separation of events and counter events at jet PT threshold.


Scale uncertainties
Scale Uncertainties counter events in adjacent bins.

  • It is hoped that quantifying the scale variation will provide a measure of the effect of unknown higher order corrections.

  • Standard Model Cross Section Task Force prescription for evaluating scale uncertainties:

    • Choose appropriate central scale µ based on kinematics of events under consideration.

    • Central cross section is then calculated as σ (µ, µ)

    • Error bounds given by (max / min ) σ(µR, µF) :

      • µ/2 < (µR, µF) < 2 µ

      • 0.5 < µR / µF < 2

  • You can create a ‘grid’ of predictions for all fixed scales and apply this prescription to obtain an uncertainty but this is resource intensive and time consuming.


  • Alternatively MCFM allows one to choose each scale dynamically on an event by event basis:

    • µ2=P2T(V) + M2(V)

    • µ2=<PT(Jets)>2

    • µ=HT= ΣiPT i ε (unclustered final state objects)

    • µ=HT(hat)= PT(V) + ΣiPT(Jets)

  • We chose HT/2 as a central scale and quoted uncertainties using the previous prescription but only using (µR, µF) ε (HT/4, HT/2, HT).


Selections
Selections: dynamically on an event by event basis:

  • Generator Level: Jet PT(min) = 30 GeV, Jet Eta Max = 6, Jet Algorithm: AntiKt6. Min Z invariant Mass = 40 GeV. PDF = CTEQ66.

    • C.f. https://twiki.cern.ch/twiki/bin/view/AtlasProtected/WZJetsTheoSys#MCFM

  • Analysis selections: ‘Electron A’ reference selections used:

    • Lepton PT > 20 GeV

    • 0 < Lepton η< 1.37 || 1.52 < Lepton η< 2.47

    • 80 GeV < MZ < 100 GeV , MT(W) > 40 GeV

    • Neutrino PT ( W MET ) > 25 GeV

    • Jet Eta < 2.8


Summary 0 jets
Summary – 0 Jets dynamically on an event by event basis:

MSTW :-15.918%, -15.744%, +0.208%

Red = Z, Blue = W, Gold = Ratio


1 jet distributions
1 Jet Distributions dynamically on an event by event basis:


1 jet distributions1
1 Jet Distributions dynamically on an event by event basis:

Uncertainty on cross section increases roughly linearly with jet PT after 40 GeV. Upper error bound due to correlated upward variation (both scales = HT) at high PT, Lower due to downward correlated variation (both scales = HT/4).


Ratio n 1 jet
Ratio (n=1 Jet) dynamically on an event by event basis:

The uncertainty on the ratio is ~ 2% but is almost constant with increasing Jet kinematic unlike the uncertainty on the cross section.


Mstw and k factors
MSTW and K Factors dynamically on an event by event basis:

K factor for ratio is constant at just over 1 for all Jet PT’s up to 100 GeV. The deviations in cross section for both W and Z when using MSTW2008 as opposed to CTEQ66 are ~4% while the deviation on the ratio is ~1%


In the one jet case changing the scale does appear to distort the eta distributions with lower renormalization scale leading to an enhancement of the distributions at high eta and vice versa. Variation of the factorization scale seems to have no noticeable effect.


Summary 1 jet
Summary – 1 Jet distort the eta distributions with lower renormalization scale leading to an enhancement of the distributions at high eta and vice versa. Variation of the factorization scale seems to have no noticeable effect.

MSTW :+4.192%, +3.443%, -0.719%

Red = Z, Blue = W, Gold = Ratio


Pdf uncertainties lo
PDF Uncertainties - LO distort the eta distributions with lower renormalization scale leading to an enhancement of the distributions at high eta and vice versa. Variation of the factorization scale seems to have no noticeable effect.

From above R1 Jet seems more robust against PDF uncertainties than the cross sections. For the 0 Jets case the quadrature uncertainty from PDFs was found to be 5.05% for the Z cross section but only 1.08% for the Ratio.


Pdf uncertainties nlo
PDF uncertainties - NLO distort the eta distributions with lower renormalization scale leading to an enhancement of the distributions at high eta and vice versa. Variation of the factorization scale seems to have no noticeable effect.

Spread of values for Ratio does appear more confined at low scales but statistics rapidly degrade – fairly inconclusive, need higher statistics ntuples.


Conclusions
Conclusions distort the eta distributions with lower renormalization scale leading to an enhancement of the distributions at high eta and vice versa. Variation of the factorization scale seems to have no noticeable effect.

  • 0 Jets:

    • Systematic uncertainty on cross sections ~10% while that on R ~ 1%.

    • Deviation between CTEQ66 and MSTW2008 ~15% for cross sections, ~ 0.2% for R.

    • While the 1 Jet predictions display only a weak dependence on µF, this is not the case for 0 Jet, the dependence is comparable to that for µR (except opposite direction!) – not really surprising.

  • 1 Jet:

    • Systematic uncertainty on cross sections ~4% while that on R ~ 1.5%.

    • Deviation between CTEQ66 and MSTW2008 ~4% for cross sections, ~ 0.7% for R.

    • For cross sections, relative uncertainty increases roughly linearly with Jet PT, for R the relative uncertainty is constant at ~ 1.5%

  • R is far more immune to uncertainties in the details of the theoretical prediction than the cross sections are!


Further work
Further Work distort the eta distributions with lower renormalization scale leading to an enhancement of the distributions at high eta and vice versa. Variation of the factorization scale seems to have no noticeable effect.

  • Calculate statistical uncertainties using multiple pseudo-experiments.

    • 9 Samples now ready for central scales.

  • Produce Ntuples with all PDF error weights and higher statistics.

  • DiJet Predictions

    • Low statistics samples prepared, need to be analysed, kinematic reach likely to be very poor.

  • Correct Fluctuations at Jet threshold???


Backup
Backup distort the eta distributions with lower renormalization scale leading to an enhancement of the distributions at high eta and vice versa. Variation of the factorization scale seems to have no noticeable effect.


More counterterm subtraction
More Counterterm Subtraction distort the eta distributions with lower renormalization scale leading to an enhancement of the distributions at high eta and vice versa. Variation of the factorization scale seems to have no noticeable effect.

  • (Courtesy of J. Campbell )

  • V = Virtual Contribution, R = Real Emission

  • S = V + Integral( R )

  • V is analytic and formally divergent. R diverges on integration over soft / collinear partons.

  • Introduce Counterterm CR which is simple enough to integrate analytically which has compensating divergence to Real emission contribution. Use Dimensional Regularization D4-2ε and perform integration.

  • S =[ V + Integral(CR) ] + Integral( R – CR )

  • Divergences in V and Int(CR) cancel on taking limit ε0


Counterterm subtraction1
Counterterm Subtraction distort the eta distributions with lower renormalization scale leading to an enhancement of the distributions at high eta and vice versa. Variation of the factorization scale seems to have no noticeable effect.

  • While V and Integral(CR) separately diverge their sum is finite and analytically calculable -> This is used to generate virt events.

  • Integral (R-CR) is NOT analytically calculable – 2 sets of events are generated ‘real events’ – correspond to R and ‘Counter-events’ – correspond to CR.

  • The divergence structure of these contributions are the same and should mostly cancel – the residual is an important contribution to S.

  • To Calculate a statistical uncertainty one needs to be able to identify the counter events corresponding to a given event and bin them together. At present this is not possible.


Mstw pdf errors nlo
MSTW PDF ERRORS - NLO distort the eta distributions with lower renormalization scale leading to an enhancement of the distributions at high eta and vice versa. Variation of the factorization scale seems to have no noticeable effect.

0 Jet Z quadrature deviation : 2.65%

0 Jet R quadrature deviation : 0.235%


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