Monte Carlo for the LHC - UCL HEP Group · Monte Carlo for the LHC Christian Gütschow...

Monte Carlo for the LHC

Christian Gütschow

Intercollegiate Postgraduate Course

20 November 2017

CHRISTIAN GÜTSCHOW

MONTE CARLO FOR THE LHC

Disclaimer

Ü Monte Carlo simulation is a cornerstone of the LHC programme

Ü Monte Carlo samples used to calibrate the detectors, compare data to theoryand search for new physics phenomena

Ü these slides are merely meant to touch upon some of the key conceptsin the world of LHC Monte Carlo and are by no means exhaustive

Ü some material has been stolen borrowed from Yu Bai, Andy Buckley, Stefan Höche, FrankKrauss, Josh McFayden, John Morris, Marek Schönherr, Torbjörn Sjöstrand, Peter Skands

Ü Many thanks!

Intercollegiate Postgraduate Course, 20 Nov 2017 [email protected] 2/29

CHRISTIAN GÜTSCHOW


Monte Carlo – randomness to the rescue

Ü expectation value of observable O obtained byintegrating over it weighted by probability P toproduce point Φn in n-particle phase space

〈O〉 =∑

n

∫dΦnP(Φn)O(Φn)

Ü need numerical approximations whenanalytic calculation becomes too hard

Ü use Monte Carlo methods to integratenumerically using pseudorandom numbers and theacceptance–rejection method (‘hit or miss’)


CHRISTIAN GÜTSCHOW


Overview: Simulating LHC collisions using Monte Carlo

Ü hard scatter – matrix elements fromfirst principles


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Ü incoming partons fromparton-distribution functions (PDFs)


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Ü radiative corrections – resumminglogarithms to all orders


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Ü multiple parton interactions –additional interactions between protonremnants


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Ü hadronisation – going colourless


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Ü hadron decays – from excited statesto final-state particles


CHRISTIAN GÜTSCHOW








Ü hadron decays – from excited statesto final-state particles

Ü photon radiation – QED correctionsarXiv:1411.4085 [hep-ph]


https://arxiv.org/abs/1411.4085

CHRISTIAN GÜTSCHOW


Born-level cross sections

dσa,b→n ∼∫ 1

0dxadxb f (xa, µf)f (xb, µf)

∫dΦn

∣∣Mpa,pb→n(Φn;µf , µr)∣∣2

Ü parton density functions f (x , µf)

Ü e.g. MMHT14, CT14, NNPDF3.0(and many more in LHAPDF)

Ü phase-space element dΦnfor n-particle final state

Ü squared matrix elementMpa,pb→n(summed/averaged over polarisations)

Ü depends on factorisation scale µf andrenormalisation scale µr

Results.

Above procedure completely determines parton distributions at present. Alsodetermines αS(M2

Z) = 0.118± 0.0015 (expt) – as good as most other determinations.Partons and their uncertainties essential input to all LHC and Tevatron studies.

QCD Partons 2016 50

Eur.Phys.J. C75 (2015) no.5, 204


https://www.hep.ucl.ac.uk/mmht/grids.shtml

http://hep.pa.msu.edu/cteq/public/ct14.html

http://nnpdf.mi.infn.it/nnpdf3-0/

https://lhapdf.hepforge.org/


CHRISTIAN GÜTSCHOW


Higher-order corrections

σNLO =

∫ndσLO +

∫ndσvirt +

∫n+1

dσreal

Ü including higher-order corrections reducestheory uncertainty (in general)

Ü calculations at NLO in the strong couplingstandard for most processes these days

Ü NNLO becoming more and more available(but computationally expensive)

Ü higher-order corrections in the electroweakcoupling often large and negative

Ü formally suppressed by α/αs,but NNLO QCD ∼ NLO EWsince O(α2

s) ∼ O(α)

14

Z(`+`�)+ jet

W(`n)+ jet

g+ jet

Z(`+`�)+ jet

10 x W(`n)+ jet

100 x g+ jet

LONLO QCDNNLO QCD

10�1010�910�810�710�610�510�410�310�210�1

110 110 210 310 4

(N)NLO QCD for V+jet @ 13 TeV

ds

/dp T

,V[p

b/G

eV]

0.40.50.60.70.80.91.01.11.2

ds

/ds N

LO

QC

D

0.40.50.60.70.80.91.01.11.2

ds

/ds N

LO

QC

D

100 200 500 1000 30000.40.50.60.70.80.91.01.11.2

pT,V [GeV]

ds

/ds N

LO

QC

D

Fig. 6: Higher-order QCD predictions and uncertainties for Z(`+`�)+jet, W±(`⌫)+jet, and �+jet production at13 TeV. Absolute predictions at LO, NLO and NNLO QCD are displayed in the main frame. The ratio plotsshow results for individual processes normalised to NLO QCD. The bands correspond to the combination (inquadrature) of the three types of QCD uncertainties, �(i)KNkLO, i.e. scale uncertainties according to Eq. (33),shape uncertainties according to Eq. (35), and process-correlation uncertainties according to Eq. (38).

ratios (see also Figure 19). However, one should keepin mind that an additional analysis-dependent photon-isolation uncertainty (see Section 3.1) has to be consid-ered for these ratios.

In general, comparing QCD predictions at differentorders we observe a good convergence of the perturba-tive expansion, and the fact that process ratios receivevery small corrections both at NLO and NNLO providesstrong evidence for the universality of QCD dynamicsis all V + jet processes. Results at NNLO provide alsoa crucial test of the goodness of the proposed approachfor the estimate of QCD uncertainties and their correla-tions. In particular, the remarkable consistency betweenNNLO and NLO predictions in Figure 8 confirms that

QCD uncertainties for process ratios are as small as1–2%.

4.2 Electroweak corrections

For EW higher-order corrections we use the notation,d

dx�

(V )NLO EW =

d

dx�

(V )LO QCD +

d

dx��

(V )NLO EW, (40)

d

dx�

(V )nNLO EW =

d

dx�

(V )NLO EW +

d

dx��

(V )NNLO Sud,

where ��(V )NLO EW denotes exact O(↵2↵S) contributions,

and ‘NNLO Sud’ stands for O(↵3↵S) EW Sudakov loga-rithms in NLL approximation (see below). Their combi-nation is dubbed nNLOEW as it accounts for the dom-inant EW effects at NNLO. While our power counting

arXiv:1705.04664 [hep-ph]



CHRISTIAN GÜTSCHOW


Loops and legs

X (3) X + 1(3)

X (2) X + 1(2) X + 2(2)

X (1) X + 1(1) X + 2(1) X + 3(1)

Born X + 1(0) X + 2(0) X + 3(0) X + 4(0)

legs (additional resolvable parton emissions)

loop

s(v

irtua

lcor

rect

ions

)

Ü Born process: X + 0j@LO


CHRISTIAN GÜTSCHOW


Loops and legs

X (3) X + 1(3)

X (2) X + 1(2) X + 2(2)

X (1) X + 1(1) X + 2(1) X + 3(1)

Born X + 1(0) X + 2(0) X + 3(0) X + 4(0)


loop

s(v

irtua

lcor

rect

ions

)

Ü X + 2j@LO


CHRISTIAN GÜTSCHOW


Loops and legs

X (3) X + 1(3)

X (2) X + 1(2) X + 2(2)

X (1) X + 1(1) X + 2(1) X + 3(1)

Born X + 1(0) X + 2(0) X + 3(0) X + 4(0)


loop

s(v

irtua

lcor

rect

ions

)

Ü X + 0j@NLO (includes X + 1j@LO)


CHRISTIAN GÜTSCHOW


Loops and legs

X (3) X + 1(3)

X (2) X + 1(2) X + 2(2)

X (1) X + 1(1) X + 2(1) X + 3(1)

Born X + 1(0) X + 2(0) X + 3(0) X + 4(0)


loop

s(v

irtua

lcor

rect

ions

)

Ü X + 1j@NLO (includes X + 2j@LO)


CHRISTIAN GÜTSCHOW


Loops and legs

X (3) X + 1(3)

X (2) X + 1(2) X + 2(2)

X (1) X + 1(1) X + 2(1) X + 3(1)

Born X + 1(0) X + 2(0) X + 3(0) X + 4(0)


loop

s(v

irtua

lcor

rect

ions

)

Ü X + 1j@NNLO (includes X + 1j@NLO +2j@LO)


CHRISTIAN GÜTSCHOW


Matching fixed order calculations to a parton shower

Ü use parton showers to evolve partons from a starting scale down to small momenta,subject to additonal parton splittings/emissions

Ü no-emission probability given by Sudakov form factor ∆(K)n (t , t0) = exp

[−∫ t

t0dΦKn(Φ)

]with splitting kernel Kn for n-particle final state (somewhat similar to radioactive decay)

Ü different matching procedures exist (e.g. CKKW, MLM)

A shower Monte Carlo program alone will generate a transverse momentum distributionthat is accurate only for small transverse momenta, since dσ(MC) is reliable only in thecollinear approximation. For small transverse momenta, however, rather than having thesingular behaviour of an NLO calculation, it is well behaved, with the Sudakov form factordamping the small pT singularity of the tree level result. Many event generators are capableof adding a matrix-element correction (MEC), such that for large transverse momentum theshower result matches the fixed-order result [10]. This is achieved, in essence, by replacingσ(MC) with σ(NLO) in Equation 10. Assuming, for the moment, that we are dealing with ashower algorithm ordered in transverse momentum, the generation of the first emission inMEC is given by

dσ(MEC) = BdΦB

[∆(Q0) + ∆(pT)

R

BdΦrad

], ∆(pT) = exp

[−∫

R

Bδ(pT(ΦR) − pT)dΦrad

].

(13)The notation used in Equation 13 deserves some explanation. We write in a compact nota-tion a fully differential cross section that can have different final states as a single formula.The first term in the square bracket represents the production of an event with the Bornkinematics, and phase space ΦB. In the Higgs example, it represents a Higgs boson with zerotransverse momentum. The second term represents the full real process, with production ofa Higgs and a parton, balanced in transverse momentum. The above formula represents theprobability that either event is produced.

The shower unitarity Equation 11 is then written in the general form

∆(Q0) +

∫∆(pT)

R

BdΦrad = 1 , (14)

where it is intended that the dΦrad integration is limited to the region where pT(ΦR) ≥ Q0.In Figure 2 we give a pictorial representation of the distribution of the transverse mo-

Figure 2: Transverse momentum distribution of the Higgs at NLO, in a shower algorithm,and in a MEC shower.

mentum of the Higgs boson at fixed rapidity at NLO order (i.e. O(α3S)), from the shower

algorithm, and from a MEC shower algorithm. For the NLO result, one should imagine thatthe NLO curve diverges at small pT up to a tiny cutoff, and that a tiny bin with a verylarge, negative value is located at pT = 0. The resummation of collinear and soft singulari-ties performed by the shower algorithm using the exact real emission cross section starts todiffer from the LO one at pT around 40 GeV, and for smaller pT it tames the divergence ofthe NLO cross section. The shower approximation has the same behaviour for moderate tosmall pT, but it drops rapidly as pT approaches the maximum scale of radiation allowed bythe shower algorithm (an exact implementation of Equation 10 would imply that the crosssection vanishes exactly for pT ≥ Q. Subsequent emissions in the shower process will tendto smear the region of pT ≈ Q). The area under the two shower curves equals the Borncross section.

The main objective of a NLO+PS implementation is to improve the shower approxima-tion, in such away that it achieves NLO accuracy for inclusive quantities. Thus, referring to

7

cross section. In fact, the second line of Equation 19 gives only a minor contribution tothe integral, since it is non-vanishing only in the large transverse momentum region wherethe collinear approximation differs substantially from the tree-level matrix element. Byincluding the full round bracket on the first line of Equation 19, we increase only the Scontribution by a K factor depending on the Born kinematics ΦB. We thus see that in theMC@NLO case the K factor acts uniformly over the pT region that is dominated by Sevents, which typically extends out to pT values of the order of the hard scale of the processin question, while the region of harder emissions is not affected by it.

The practical implementation of the MC@NLO method proceeds as follows. The NLOpart first performs the integrations necessary to determine the weights of the S events andthen generates the S and H events (i.e. Born-like and Born+one-parton configurations,respectively) to serve as the starting-points of the corresponding MC generator. These mayremain weighted or can be unweighted as described above. For the purposes of partonshowering and hadronization, each event has to be assigned a unique colour flow, whichcorresponds to the large-Nc limit of QCD, where Nc is the number of colours. Therefore acolour flow is selected according to their relative probabilities in the large-Nc limit of thecorresponding Born or real emission matrix element. However, it should be emphasisedthat the sum of all colour flows reproduces the full NLO result, including all subleading Nc

dependence. Each coloured external line of the event can then be processed by the showergenerator in the normal way. In particular, there is no restriction that shower emissionsfrom H events should be softer than the extra parton emitted at the NLO level, becausethe weights of those events are computed assuming unrestricted showering. However, Hevents are not singular in the collinear and soft regions, and thus their contribution to thoseregions is phase-space suppressed, so that the cross section for producing events in whichthe shower generates radiation much harder than that generated in the H configuration ispower suppressed.

Figure 3: Transverse momentum (upper left plot) and rapidity (upper right) distributionsof the top quark, and transverse momentum (lower right) and relative azimuth (lower left)distributions of the tt̄ pair at the LHC (14 TeV), obtained at NLO, with HERWIG, andwith MC@NLO. Figures from [11].

Figure 3 shows some MC@NLO results on top quark pair production at the LHC,compared with those obtained at NLO and with HERWIG. The MC@NLO and HERWIG

10

Ann.Rev.Nucl.Part.Sci. 62 (2012) 187-213Intercollegiate Postgraduate Course, 20 Nov 2017 [email protected] 18/29

https://arxiv.org/abs/hep-ph/0109231

https://arxiv.org/abs/hep-ph/0602031


CHRISTIAN GÜTSCHOW


Multijet merging

Ü merging of additional matrixelements allows to capture(usually dominant) effect ofhigher-order QCD correctionsby including more resolvable jetemissions in the calculation

Ü gets complicated quickly: overlapbetween X + nj@NLO andX + (n + 1)j@(N)LO andparton-shower emissions

Ü different merging proceduresexist, e.g. MEPS@NLO, FxFx




CHRISTIAN GÜTSCHOW


Welcome to the real world

Ü final states based on individualpartons are unphysical

Ü colour-charge carrying particles havenever been observed in isolation

Ü perturbation theory breaks down at low energieswhere αs becomes large (≈ 1 GeV)

Ü colour confinement realisedusing non-perturbative modelse.g. Lund string model (top)or cluster model (bottom)

Ü typically yields resonanceswhich are then decayed furtherto arrive at the final state


https://inspirehep.net/record/189583?ln=en

https://inspirehep.net/record/193786/

CHRISTIAN GÜTSCHOW


LHC baseline generators

Ü SHERPA (Sherpa 2.2, using external loop provider, such as OPENLOOPS)

Ü multiple matrix-element providers and parton showers in one package

Ü anything from LO to NNLOPS with focus on multi-jet merging at NLO accuracy (MEPS@NLO)

Ü MADGRAPH (aMC@NLO_MG5)

Ü parton-level only, needs interfacing with separate parton shower

Ü anything from LO to NLO, also merging of NLO matrix elements using FxFx

Ü POWHEG BOX (v2)

Ü parton-level only, needs interfacing with separate parton shower

Ü anything from LO to NNLO, but no merging

Ü HERWIG (Herwig 7)

Ü mainly known for its angular-ordered parton shower

Ü matrix-element providers interfaced through Matchbox steering

Ü PYTHIA (Pythia 8.2)

Ü general purpose parton shower with focus on hadronisation, MPI and underlying-event modelling


https://sherpa.hepforge.org/trac/wiki

http://openloops.hepforge.org/

http://madgraph.physics.illinois.edu/

http://powhegbox.mib.infn.it/

https://herwig.hepforge.org/

http://home.thep.lu.se/~torbjorn/Pythia.html

CHRISTIAN GÜTSCHOW


Analysing generator output

Christian Gütschow | Software Tutorial | 6 Oct 2016 1

MC generator chain

Ü Monte Carlo generators stop with set of stable final-state particles (‘particle level’)

Ü complete 4-vector information and decay chain relations known and stored (‘truth record’)

Ü event tree will look very different for different generators

Ü Is there a way to analyse the output in a generator-independent waywithout having to manually navigate the truth record?


CHRISTIAN GÜTSCHOW


Analysis prototyping with Rivet

Ü Robust Independent Validation of Experiment and Theory

Ü generator-agnostic, efficient and fast

Ü quick, easy and powerful way to get physics plots from lots of MC generators

Ü only requirement: use HepMC event record (standard for all LHC baseline generators)

Ü lightweight way to exchanging analysis details and ideas

Ü Rivet has become the LHC standard for archiving LHC data analyses

Ü focus on unfolded measurements more than searches,but fast detector simulation now also intrinsic to Rivet

Ü key input to MC validation and tuning – increasingly comprehensive coverage

Ü also “recasting” of SM and BSM data results on to new/more general new-physics models


http://rivet.hepforge.org/

CHRISTIAN GÜTSCHOW


Analysis preservation with Rivet

Ü currently ∼430 analyses total(∼ 230 LHC analyses alone)

Ü until recently only 27 dedicatedBSM searches and BSM-sensitiveSM measurements

Ü SM focus on unfolded observables,not sufficient for most BSM studies

2007 2009 2011 2013 2015 2017Year

0

100

200

300

400

# an

alys

es

Ü Rivet 2.5.0 introduced detector smearing machinery

Ü added many real-world examples of how to write BSM routines

Ü also added tools to help with object filtering, cutflows, etc.

Ü Rivet is in good shape for preserving new physics searches!Intercollegiate Postgraduate Course, 20 Nov 2017 [email protected] 24/29

CHRISTIAN GÜTSCHOW


Analysis folding with Rivet

Ü explicit fast detector simulation vs. smearing/efficiencies

MC truth

detector readout

reconstruction level

detector hitsdigitisation

trigger

reconstruction??

triggers

efficiencies

smearing

Ü explicit fast sim takes the “long way round”

Ü reconstruction already reverses most detector effects!

Ü reco calibration to MC truth: smearing is a few-percent effect

Ü (lepton) efficiency & mis-ID functions dominate – and are tabulated in both approaches

Ü smearing is more flexible: efficiencies change with phase-space, reco version, run, . . .need to guarantee stability for preservation


CHRISTIAN GÜTSCHOW


The long way round: from particle level to detector level

particle collider(e.g. LHC)

event generator(e.g. Sherpa)

detector, DAQ(e.g. ATLAS)

detector simulation(e.g. GEANT4)

event reconstruction(e.g. Athena)

physics analysis(e.g. ROOT)

Ü Monte Carlo generators stop with setof stable final-state particles(‘particle level’)

Ü complete 4-vector information anddecay chain relations knownand stored (‘truth record’)

Ü pass through full simulation of detectorresponse (5mins/event), overlay pileupand digitise to arrive at (simulated) rawdetector output

Ü use same reconstruction algorithmsfor data and simulation(‘reconstruction level’)

Ü pileup, digitisation & reconstructionanother 1–1.5mins/event


CHRISTIAN GÜTSCHOW


Room for improvement: reweighting Monte Carlo samples

Ü Monte Carlo samples usually producedwell in advance of data taking

Ü need to assume pileup profile, i.e. guessthe distribution of the number of primaryvertices in the dataset prior to collecting it

Ü perform aposteriori reweighting of thepileup profile to correct the simulation

Ü similar reweightings usually performed forvarious object-reconstruction efficiencies

Ü individual reconstructed objects smearedto improve description of momentumscales and resolutions as well

9Intercollegiate Postgraduate Course, 20 Nov 2017 [email protected] 27/29

CHRISTIAN GÜTSCHOW


Theory uncertainties: rules of thumb

Ü PDF uncertainties

Ü estimate standard deviation from toys (‘replicas’) or use some form of eigenvector decomposition

Ü try different PDF families when performing a search, stick to one family for a measurement

Ü perturbative uncertainties – estimate impact of missing higher orders

Ü scale dependency conventionally estimated using up and down variations by a factor of 2

Ü no good reason for this choice other than tradition,does not even reflect a ‘Gaussian 68 % probability’ (!)

Ü interplay between different scales typically probed through multi-point variations,e.g. (µf , µr)× {(0.5, 0.5), (0.5, 1.0), (1.0, 0.5), (1.0, 1.0), (1.0, 2.0), (2.0, 1.0), (2.0, 2.0)}

Ü uncertainty typically quoted as difference between envelope and nominalif scales probe similar source of uncertainty

Ü should assign scale uncertainties from varying µf and µr in the parton shower as well

Ü additional (matching-)scale uncertainty from variations of the parton-shower starting scale

Ü non-perturbative uncertainties

Ü probe hadronisation, MPI and underlying event (do not necessarily factorise)

Ü can vary parameters of the nominal model or try a different model altogether


CHRISTIAN GÜTSCHOW


Summary

Ü the LHC programme relies heavily on Monte Carlo simulation

Ü Monte Carlo predictions are not the same as LHC collisions

Ü the calculation can be incorrect, incomplete, inaccurate,subject to higher-order corrections or mismodelling, . . .

Ü detector simulation might not account for all detector effects

Ü some mismodelling effects can be corrected for,e.g. through reweighting and smearing

Ü comparing predictions from different models is important (but not always feasible)

Ü some disagreement can be expected due to limited accuracy of the calculation

Ü check if mismodelling is covered by theory uncertainties before panicking


Monte Carlo for the LHC - UCL HEP Group · Monte Carlo for the LHC Christian Gütschow...

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