Monte Carlo Monte Carlo tools tools

for the LHCfor the LHCMichelangelo ManganoMichelangelo Mangano

TH Division, CERNTH Division, CERN

Nov 6, 2002Nov 6, 2002

Final states at the LHCFinal states at the LHCProcessProcess Evts/secEvts/sec

((LL=10=103333))

Evts/yrEvts/yr

Jet, EJet, ETT>100GeV>100GeV 101033 10101010

Jet, EJet, ETT>1TeV>1TeV 1.5x101.5x10-2-2 1.5x101.5x1055

WWl l 2020 2x102x1088

bbbb 5x105x1055 5x105x101212

tttt 11 101077

WW WW l l l l 6x106x10-3-3 6x106x1044

Goal of MC development for the LHC is to provide a description as accurate as possible of these events (and more),

as well as of the features of new physics processes:rates, distributions, fine details of the final states (overall

multiplicities, heavy-quark content)

Use and abuse of MC Use and abuse of MC simulationsimulation

o Use:Use:o benchmarks for the design of detectors, trigger benchmarks for the design of detectors, trigger

and analysis strategies and analysis strategies o tests and measurements of SMtests and measurements of SMo study of properties of new particles (masses, study of properties of new particles (masses,

cross-sections, couplings)cross-sections, couplings)o Abuse: claims of discoveries!Abuse: claims of discoveries!

o top and SUSY discovery in UA1top and SUSY discovery in UA1o RRbb at LEP at LEPo quark compositeness at CDFquark compositeness at CDF

o Only the benchmarking against real data Only the benchmarking against real data can turn MC simulation into powerful study can turn MC simulation into powerful study toolstools

Example: HExample: Hbb in qq bb in qq HqqHqq

mmHH procsprocs

120 GeV120 GeV 140 GeV140 GeV

SignalSignal 2.8 x 103

1.1 x 103

bbjjbbjj 8.0 x 105

5.7 x 105

jjjjb b jj jjbb 7.9 x 103

9.0 x 103

• bbjj bg is ≈102 times the signal, but can be extracted from data (smooth behaviour under the signal peak)• bg from multiple collisions (jjjjb b jj jjbb) ≈ signal, but peak under the signal! Much more sensitive to MC simulation uncertainties!

MLM, Moretti, Piccinini, Pittau, Polosa

Factorization TheoremFactorization Theorem

€

dσ

dX= f j (x1,Qi ) fk (x2 ,Qi )

d ˆ σ jk(Qi ,Q f )

d ˆ X F( ˆ X → X;Qi ,Q f )

ˆ X

∫j,k

∑

€

ˆ σ f(x,Qi)

€

ˆ X

€

XF

€

F( ˆ X → X;Qi ,Q f )

transition from partonic final state to the hadronic observable (hadronization, fragm. function, jet definition, etc) Sum over all histories with X in them

€

f j (x,Q)

sum over all initial state histories leading, at the scale Q, to:

€

rp j = x

r P proton

The possible histories of initial and final The possible histories of initial and final state, and their relative probabilities, are state, and their relative probabilities, are in principle independent of the hard in principle independent of the hard process (they only depend on the flavours process (they only depend on the flavours of partons involved and on the scales of partons involved and on the scales QQ))

Once an algorithm is developed to describe Once an algorithm is developed to describe IS and FS evolution, it can be applied to IS and FS evolution, it can be applied to partonic IS and FS arising from the partonic IS and FS arising from the calculation of an arbitrary hard processcalculation of an arbitrary hard process

Depending on the extent to which different Depending on the extent to which different possible FS and IS histories affect the possible FS and IS histories affect the value of the observable value of the observable XX, different , different realizations of the factorization theorem realizations of the factorization theorem can be usedcan be used

`Cross-section evaluators’`Cross-section evaluators’ Only some component of the final state is singled out for the Only some component of the final state is singled out for the

measurement, all the rest being ignored (i.e. integrated over). measurement, all the rest being ignored (i.e. integrated over). E.g. ppE.g. ppee++ee-- + X + X

No ‘events’ are ‘generated’, only cross-sections are evaluated:No ‘events’ are ‘generated’, only cross-sections are evaluated:

Experimental selection criteria (e.g. jet definition or Experimental selection criteria (e.g. jet definition or acceptance) are applied on parton-level quantities. Provided acceptance) are applied on parton-level quantities. Provided these are infrared/collinear finite, it therefore doesn’t matter these are infrared/collinear finite, it therefore doesn’t matter what what F(X)F(X) is, as we assume ( is, as we assume (fact. theorem)fact. theorem) that: that:

Thanks to the inclusiveness of the result, it is Thanks to the inclusiveness of the result, it is `straightforward’ to include higher-order corrections, as well `straightforward’ to include higher-order corrections, as well as to resum classes of dominant and subdominant logsas to resum classes of dominant and subdominant logs

€

σ pp → Z 0( ),

dσ

dM (e+e−) dy(e+e−), K

€

F( ˆ X ,X ) =1X

∑ ∀ ˆ X

State of the artState of the art

NLO available for:NLO available for: jet and heavy quarks productionjet and heavy quarks production prompt photon productionprompt photon production gauge boson pairsgauge boson pairs most new physics processes (e.g. SUSY)most new physics processes (e.g. SUSY)

NNLO available for:NNLO available for: W/Z/DY productionW/Z/DY production Higgs production Higgs production

€

(qq → W )

€

(gg → H )

Parton-level (Parton-level (akaaka matrix-elementmatrix-element) ) MC’sMC’s

Parton level configurations (i.e. sets of Parton level configurations (i.e. sets of quarks and gluons) are generated, with quarks and gluons) are generated, with probability proportional to the respective probability proportional to the respective perturbative M.E. perturbative M.E.

Transition function between a final-state Transition function between a final-state parton and the observed object (jet, parton and the observed object (jet, missing energy, lepton, etc) is unitymissing energy, lepton, etc) is unity

No need to expand No need to expand f(x)f(x) or or F(X)F(X) in terms of in terms of histories, since they all lead to the same histories, since they all lead to the same observableobservable

Experimentally, equivalent to assumingExperimentally, equivalent to assuming `smart’ jet clustering (`smart’ jet clustering (parton parton jet jet) ) linear detector responselinear detector response

Codes available for:Codes available for:

W/Z/gamma + N jets (NW/Z/gamma + N jets (N6)6) W/Z/gamma + Q Qbar + N jets (NW/Z/gamma + Q Qbar + N jets (N4)4) Q Qbar + N jets (NQ Qbar + N jets (N4)4) Q Qbar Q’ Q’bar + N jets (NQ Qbar Q’ Q’bar + N jets (N2)2) Q Qbar H + N jets (NQ Qbar H + N jets (N3)3) nW + mZ + kH + N jets (n+m+k+N nW + mZ + kH + N jets (n+m+k+N 8, N8, N2)2)

N jets (NN jets (N5)5)

Shower Monte CarloShower Monte Carlo After the generation of a given parton-level After the generation of a given parton-level

configuration (typically LO, 2configuration (typically LO, 21 or 21 or 22) , 2) , each possible IS and FS parton-level each possible IS and FS parton-level history (`history (`showershower’) is generated, with ’) is generated, with probability defined by the shower probability defined by the shower algorithm algorithm (unitary evolution).(unitary evolution).

`̀AlgorithmAlgorithm’: numerical, Markov-like ’: numerical, Markov-like evolution, implementing within a given evolution, implementing within a given appoximation scheme the QCD dynamics:appoximation scheme the QCD dynamics: branching probabilitiesbranching probabilities infrared cutoff schemeinfrared cutoff scheme hadronization model hadronization model

Herwig, Pythia, IsajetHerwig, Pythia, Isajet

Complementarity of the 3 approachesComplementarity of the 3 approachesShower MC’sShower MC’sX-sect evaluatorsX-sect evaluatorsME MC’sME MC’s

Full information Full information available at the available at the hadron levelhadron level

Limited access Limited access to final state to final state structurestructure

Hard partons Hard partons jets. jets. Describes geometry, Describes geometry, correlations, etccorrelations, etc

Final state Final state descriptiondescription

Approximate, Approximate, incomplete phase incomplete phase space at large anglespace at large angle

Straighforward Straighforward to implement, to implement, when availablewhen available

Included, up to Included, up to high orders high orders (multijets)(multijets)

Higher order Higher order effects: hard effects: hard emissionsemissions

Unitary Unitary implementation implementation (i.e. correct shapes, (i.e. correct shapes, but not total rates)but not total rates)

Possible, when Possible, when availableavailable

????

ResummatioResummation of large n of large logslogs

Included as vertex Included as vertex corrections corrections (Sudakov FF’s)(Sudakov FF’s)

Straighforward Straighforward to implement, to implement, when availablewhen available

Hard to implement, Hard to implement, require introduction require introduction of negative of negative probabilities probabilities

Higher order Higher order effects: loop effects: loop correctionscorrections

w=-∞ dw=-∞ w=-∞ dw=-∞

2’ guide to shower MC’s2’ guide to shower MC’s Evaluate parton-level probability, from Feynman rules + phase space. E.g.:

q

q

q’

q’

As a result of acceleration, q’ will emit radiation The probability that radiation will (or will not) be emitted is evaluated as a function of the acceleration of the colour charges: Q2 = (q’- q)2 = - t

q q’

SudakoSudakovv

1)1) Generate Generate 11

2)2) If If 1 1 < P(Q , Q< P(Q , Q00)) no radiation, no radiation, q’ goes directly on-shell at q’ goes directly on-shell at scale Qscale Q00≈GeV≈GeV

3)3) ElseElsea)a) calculate calculate QQ11 / P(Q / P(Q11,Q,Q00)= )= 11

b)b) emission at scale Qemission at scale Q11

4)4) Go back to 1) and reiterate, Go back to 1) and reiterate, until shower stops in 2). At until shower stops in 2). At each step the probability of each step the probability of emission gets smaller and emission gets smaller and smallersmaller

Q0 QQ1

1€

P(Q,Q0 )∝ exp −dq

qα s q( )

Q0

Q

∫ ⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥

P

11

Q2

22

Q1= relative momentum

prob. of no radiation between Q and Q0

Problems (1):Problems (1): Quantum coherence Quantum coherence

+

2 2

+2

€

Φ x( ) =e

r R +

r x

+−e

r R −

r x

=e

r x

r R

3if

r R >>

r x , coherent

e min( x1 , x2 ) ifr R ≈

r x , incoherent

⎧

⎨ ⎪

⎩ ⎪

Solution Solution (a.k.a. angular (a.k.a. angular

ordering)ordering)

2

2

1

2

(1

(2

2

= +

lack of hard, large-angle emission poor description of multijet events

no emission outside C1 C2:C1

C2

loss of accuracy for intrajet radiation

incoherent emission inside C1 C2:

Drawbacks:

HadronizationHadronizationAt the end of the perturbative evolution, the final state consists of quarks and gluons, forming, as a result of angular-ordering, low-mass clusters of colour-singlet pairs:

p

p

N

N

ExamplExample: e:

Wbb+jeWbb+jets ts

Proc Nj=2

Nj=3

Nj=4 Nj=5

Nj=6

q qbarW bb g..g 2.6 0.6 0.14 0.04

0.01

q g W bb q g..g 3.0 2.1 1.1 0.47

q qbarW bb qq g..g

0.29 0.24

0.13

q gW bb qqq g..g 0.03

0.03

Tot (pb) 2.6 3.6 2.5 1.4 0.64

Issues to be addressed for the Issues to be addressed for the evaluation of multiparton matrix evaluation of multiparton matrix

elementselements

nnjj 22 33 44 55 66 77 88

diag’sdiag’s

44 2255

222200

24824855

3430343000

5x105x1055

101077

For example, for ggnj gluons:

Complexity of multiparton amplitudesComplexity of multiparton amplitudes

Evaluation of probabilities for configurations with given colour flows

Colour-flow decompositionColour-flow decompositionThe angular ordering prescription can be extended to cases with higher parton multiplicity. To enforce it, we need to be able to associate probabilities to colour-flow configurations. String theory taught us how to do it:

€

A(g1,g2 ,K ,gn ) = tr λa1λ

a2 L λan( )

P(1,2,L ,n−1)

∑ m( p1, p2 ,K , pn )

€

A(g1,L ,gn )colors

∑2

= Nn−2(N2 −1) m( p1, p2 ,L , pn )Zn/Z2

∑2

+ O(1/N2 ) ⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥

m(p1,p2,…,pn) =

The ALPHA algorithm for the The ALPHA algorithm for the computation of multi-parton processes computation of multi-parton processes

(Caravaglios, M.Moretti)(Caravaglios, M.Moretti)1. A multiparton amplitude can be obtained from:

€

Z J[ ] = −Γ φ[ ] + J(x) φ(x)

€

J =δΓ

δφ

€

δnZ J[ ]δJ1δJ2 L δJn

where and (x) is a classical field, solution of:

€

J p( ) = a j δ p − p j( )j=1

n

∑

€

x( ) = bm δ p − Pm( )m

∑ , Pm = cml pl

l=1

n

∑ , cml = 0,1( )

2. In the case of tree-level scattering, J(x) is a trivial source:

and the solutions for (x) must be of the simple form:

3. E.g. for a 3 theory, we have:

€

Z ai[ ] = Δlm blbm +l,m

∑ Dlmk blbmbk +l,m,k

∑ δli blail,i

∑

where

€

Δlm =1 Pl2 if Pl + Pm = 0 and Dlmk = λ if Pl + Pm + Pk = 0

4. Minimization w.r.t. bl gives:

€

Δlm bm + Dlmk bmbk = δli ai

5. Since

€

δnZ J[ ]δJ1δJ2 L δJn

∝δ nZ J[ ]

δa1δa2 L δan ai =0

only the truncation of Z[J] multilinear in ai

is required finite iterative solution of the above quadratic system!!

Alpha, Alpha, continuecontinue

No need to explicitly evaluate Feynman graphsNo need to explicitly evaluate Feynman graphs Technique extended to QCD: allows calculations of both dual Technique extended to QCD: allows calculations of both dual

and full amplitudesand full amplitudes Numerical complexity Numerical complexity O(aO(ann)) with with a~2-3a~2-3, , instead of instead of n!n! Achieved calculation of processes with up to 10 final-state Achieved calculation of processes with up to 10 final-state

gluons -- over 5x10gluons -- over 5x1099 Feynman diagrams Feynman diagrams (Maltoni et al)(Maltoni et al)

Complete evaluation of multijet processes requires inclusion of Complete evaluation of multijet processes requires inclusion of quarks quarks extra complexity, due to all possible flavour extra complexity, due to all possible flavour combinations.combinations.

nn # distinct # distinct ampsamps 55 3,8533,853

22 3535 66 31,08731,087

33 123123 77 200,45200,4555

44 777777 88 1,676,81,676,88585

Enumeration of independent amplitudes vs number of jets. Asymptotic estimates related to partition function of 0-dim field theories. (Kleiss&Draggiotis)

Problems (2):Problems (2): Q Q22 choice for choice for evolutionevolution

The choice is almost unambiguous for final states with 1 or 2 partons:

Ex:Ex: q

qZ, W Q2 = s

Ex:Ex: q

q

g

g

Q2 pT2 – t

the factorization theorem is easily implemented, due to the existence of a single scale

The choice is more difficult in more complex cases

q

q

g2

gn

g1 If pT1 << pT2 << … << pTn , or (pi+pj)2 varying significantly for different (i,j) Ambiguous implementation of the factorization theorem Potential problem of double counting:

g3

q

q

g1 (from shower evolution)

g4 (from matrix element)

g2

versusversus

g3

q

q

g4 (from shower evolution)

g1 (from matrix element)

g2

with with ppT1T1 << p << pT4 T4 << p<< pT2T2,, ppT3T3

Leading vs subleading double countingLeading vs subleading double countingExample: corrections to 3-parton final statesExample: corrections to 3-parton final states

p1

p2

p3

p4which gives a contribution to σ3-jet of order

€

αs logp2 + p3( )

2

ET jet2

≈ α s logpT

max

pTmin

+ log1

ΔR

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

Double counting is sub-leading provided ΔR and are not too large

€

pTmax

pTmin

p1

p2

p3

p4

is of αs relative to the LO process

p1

p2

p3unless:

Example -- W+3 jet Example -- W+3 jet events events

Bottom line:Bottom line:

Implementation of quantum coherence in Implementation of quantum coherence in shower MC’s is possible, in the limit of shower MC’s is possible, in the limit of large-Nlarge-Ncc and for soft and collinear emission. and for soft and collinear emission.

Large-angle, hard emission cannot be Large-angle, hard emission cannot be described accuratelydescribed accurately

Possible cure requires starting the shower Possible cure requires starting the shower with “seed” multi-parton configurations, with “seed” multi-parton configurations, evaluated using exact (possibly tree-level evaluated using exact (possibly tree-level only) matrix elements.only) matrix elements.

Potential problems, however, due to double Potential problems, however, due to double counting for extra jet emission counting for extra jet emission

Progress towards Progress towards solutions solutions

(I) matrix element (I) matrix element correctionscorrections

Ex: Z03 jets

x1=2E1/MZ

x2=2E2/MZ

≤ x x2≤2

x10

x1 =

1 gq

2

I2

I1

x2

1

1x2=1 gq1

I1: ph.space

covered by angular-ordered emission

I2: ph.space NOT

covered by angular-ordered emission

Algorithm: (M.Seymour)

• generate events in I2 with (finite!) probability:

€

=M (Z → qq g)

I2∫

2

σ (I1)+σ (I2 )

and distributions given by

€

M (Z → qq g) 2

• Use (qqg) matrix element to correct MC weights in I1

Drawback:• requires analytic representation of the phase-space domain generated by the angular-ordering prescription

Progress towards solutions (II) vetoed Progress towards solutions (II) vetoed showersshowers

((Catani, Krauss, Kuhn, WebberCatani, Krauss, Kuhn, Webber))

€

yij =2 min Ei

2 ,E j2

{ } 1− cosθij( )

s≥ ycut =

Qcut2

s

1

2

3

4y34 > ycut

From the sample of 4-hard-parton events

: Sudakov correction

(splitting rejected if y45<ycut )

1

2

453

From the sample of 3-hard-parton events

• Generate samples of different jet multiplicities according to exact tree-level ME’s, with Njet defined using a kperp algorithm

• Reweight the matrix elements by vertex Sudakov form factors, assuming jet clustering sequence defines the colour flow• Remove double counting by vetoing shower histories (i.e. yij sequences already generated by the matrix elements)• Fully successfull for e+e- collisions, being extended to hadronic collisions

MM((ontecarloontecarlo) ) oo((ff) ) EE((verythingverything))

Shower MC’sShower MC’s

Matrix Element Matrix Element MC’sMC’s

Cross-Section Cross-Section Evaluators Evaluators

• Parton Level generators at NLO• KLN negative-wgt events• Formalism for extension to NNLO

• Implementation of NNLO• Implementaiton of resummation corrections to X-sections

• Formalism for extraction of colour flows• Common standards for event coding

• Implementation of double-counting removal in hadronic collisions

available

in progress

• Better treatment of radiation off heavy quarks• Full treatment of spin correlations in production and decay• Better description of underlying event• Better decay tables• …………..

• Formalism for inclusion of NLO• Applications to WW and QQ

• Implementaiton of resummation corrections to X-sections• NLO accuracy in shower evolution• Inclusion of power corrections

Urgent items, nobody working on them to my Urgent items, nobody working on them to my knowledge:knowledge:

Parton-level NNLO MC for W and Z:Parton-level NNLO MC for W and Z: matrix elements available, simplest and most matrix elements available, simplest and most

useful system where to test NNLO formalismuseful system where to test NNLO formalism should allow should allow OO(1%) accuracy in determination (1%) accuracy in determination

of cross-section of cross-section best possible luminometer best possible luminometer at the LHC: at the LHC:

σσ(W)(W)MRSTMRST

20002000MRSTMRST

20012001

NLONLO

FnalFnal 2.392.39 2.412.41

LHCLHC 20.520.5 20.620.6

NNLONNLO

FnalFnal 2.512.51 2.502.50

LHCLHC 19.919.9 20.020.0

NLL description of `jet shapes’, and NLL description of `jet shapes’, and inclusion of power corrections (see LEP)inclusion of power corrections (see LEP)

formalism established and tested with great formalism established and tested with great success at LEP, where it provides an essential tool success at LEP, where it provides an essential tool for the high-accuracy determination of for the high-accuracy determination of ss essential to extend the formalism to hadronic essential to extend the formalism to hadronic collisions, to exploit the lever arm in collisions, to exploit the lever arm in Q Q in the in the measurement of measurement of ss

Power Power correctionscorrections• Classes of non-perturbative effects

linked to the dominant power-like (1/Q) corrections can be parametrised in terms of a single quantity, formally given by:

€

α0 μ 0( ) =1

μ 0

dq0

μ0

∫ α s q( )

• In the case of 1st moments of shape variables, for example:F=FPT + Fnon-PT

Fnon-PT=cF P, withandP=P0 [α0()-α0( S)] /S

F 1-T

M2

H

BT BW

cF 2 2 1 1/2

• The impact of these effects at LEP is very large, and their understanding is essential for any quantitative QCD study

Their effect is expected to be very large even at the Tevatron, and in general for LHC events with jets in the few-hundred GeV energy range.

Final remarksFinal remarks A lot of progress has taken place in the recent A lot of progress has taken place in the recent

years, but years, but 30 yrs after QCD, still a lot of work to be done to 30 yrs after QCD, still a lot of work to be done to

achieve a satisfactory description of all high-Qachieve a satisfactory description of all high-Q22 processes accessible at LHCprocesses accessible at LHC

most of the key conceptual difficulties have been most of the key conceptual difficulties have been recently, or are being, solved, and their recently, or are being, solved, and their implementation into concrete MC schemes should implementation into concrete MC schemes should be achievable in the next 5 yearsbe achievable in the next 5 years

forthcoming data from Tevatron will help improving forthcoming data from Tevatron will help improving our tools, but the final test will need real LHC dataour tools, but the final test will need real LHC data

there is plenty of room for creative and rewarding there is plenty of room for creative and rewarding work for young phenomenologists!work for young phenomenologists!

Workshop on MCs for the LHC, July 7 - Aug 2 Workshop on MCs for the LHC, July 7 - Aug 2 2003, at CERN2003, at CERN