Introduction to Event Generators

Introduction to Event Generators Stephen Mrenna CD/Simulations Group Fermilab Email: mrenna@fnal.gov Stephen MRENNA  Fermilab

Motivation • Experiments rely on Monte Carlo programs which calculate physical observables • Correct for finite detector acceptance • Find efficiency of isolation cuts • Jet Energy (out of cone) corrections • Connect particles to partons • Determine promising signatures of “new” physics • Optimize cuts for discovery/limit • Planning of future facilities • . . . Stephen MRENNA  Fermilab

Field Theory Trinity • Many different calculational schemes from same basic principles • Tree level (lowest order) • Many partons • All spin correlations • Full color structure • NNLO • Smaller theoretical errors • More inclusive kinematics • “All” orders in towers of logarithms • Leading Logarithm, NLL, … • Analytic resummation (soft gluons integrated out) • Parton showers (soft gluons at leading log) How to make sense of it all? How to use the best parts of each? Stephen MRENNA  Fermilab

Many computer programs • CompHEP / Madgraph / Whizard / … • MCFM / DYRad / JetRad / … • Pythia / Herwig / Isajet / Ariadne / … • Often treated as Black Boxes • Purpose of the Lectures: Open the Box • Separate the regions of validity • Determine overlap • Merge (use the best of each) • Special role of Event Generators Stephen MRENNA  Fermilab

An Important Topic! • Uncertainties in how events should be generated are significant or most important errors for: • Top mass determination • Precision W-mass extraction • Together, a window to new physics • NNLO jet predictions with kT-algorithm • … • Our ignorance limits what we can learn about Nature Stephen MRENNA  Fermilab

Event Generators: Introduction • Most theorists make predictions about Partons • Valid to a specific order in perturbation theory • The asymptotic states are not the physical ones • Quarks & gluons confined within hadrons • Some predictions have peculiar properties • Slicing of phase space with cutoffs • Negative weights cancelling positive ones • Experimentalists measure Objects in detector • No distinction between Perturbative and Non-Perturbative physics • In- and Out-states are quasi-stable particles • Multiplicities can be large • Observe (positive) integer number of events Stephen MRENNA  Fermilab

Event Generators Connect “Theory” to “Experiment” • Describe the complicated Experimental Observable in terms of a chain of simpler, sequential processes • Some components are perturbative • hard scattering, parton showering, some decays, … • Others are non-perturbative and require modelling • hadronization, underlying event, kT smearing, … • models are not just arbitrary parametrizations, but have semi-classical, physical pictures • Sometimes as important as the perturbative pieces • The Chain contains complicated integrals over probability distributions • Positive Definite • Rely heavily on Monte Carlo techniques to choose a history • Final Output is E,p,x,t of stable and quasi-stable particles • Ready to Interface with Detector Simulations Stephen MRENNA  Fermilab

Interconnection Bose-Einstein Particle Decay Partial Event Diagram Remnant Hadronization “Underlying Event” FSR ISR Hard Scatter Resonance Decay Stephen MRENNA  Fermilab

Hard Scattering • Characterizes the rest of the event • Sets a high energy scale Q • Fixes a short time scale where partons are free objects • Allows use of perturbation theory • External partons can be treated as on the mass-shell • Valid to 1/Q • Properties at scales below Q are swept into PDFs and fragmentation functions • This is the Factorization Theorem • Sets flow of Quantum numbers (particularly Color) • Note: Parton shower and hadronization models work in 1/NC approximation • Gluon replaced by color-anticolor lines • All color flows can be drawn on a piece of paper Stephen MRENNA  Fermilab

Examples of color flows • Can influence: • Pattern of additional soft gluon radiation • Fragmentation/Hadronization Stephen MRENNA  Fermilab

Tree Level Calculation of Hard Scatter • Read Feynman rules from iLint • Use Wave Functions from Relativistic QM • Propagators (Green functions) for internal lines • Specify initial and final states • Track spins/colors/etc. if desired • Draw all valid graphs connecting them • Tedious, but straight-forward • Algorithm can be coded in a computer program • MadGraph / CompHEP / … • Calculate (Matrix Element)2 • Evaluate Amplitudes, Add them, and Square (MadGraph) • Symbolically Square, Evaluate (CompHEP) • Do something trickier (Alpha) • (Monte Carlo) Integrate over Phase Space • VEGAS … Number of graphs grows quickly with number of partons Efficiency decreases with number of internal lines Stephen MRENNA  Fermilab

CompHEP Diagrams for udW+bb Vcb Vub Naïve application of rules leads to many diagrams! Vtd Vcd Stephen MRENNA  Fermilab

Higher Order = Higher Topology How sensitive is Mbb to additional gluon radiation? Both diagrams have 6 colored lines Amplitudes diverge in soft/collinear limit Stephen MRENNA  Fermilab

Leading order matrix element calculations describe explicit, many- particle topologies Well-separated partons Full spin correlations Color flow Many computer programs Different approaches to the same problem Analytic vs Numeric Matrix Element vs Phase Space CompHep SM + MSSM + editable models Symbolic evaluation of squared matrix element 2  4-6 processes with all QCD and EW contributions color flow information outputs cross sections/plots/etc. Grace similar to CompHep Madgraph SM + MSSM helicity amplitudes “unlimited” external particles (12?) color flow information not much user interface (yet) Alpha + O’Mega does not use Feynman diagrams gg10 g (5,348,843,500 diagrams) Tree Level Overview Stephen MRENNA  Fermilab

Why Go Beyond Tree Level ? • Tree level (lowest order) prediction X has a large dependence on the scale in couplings • “the” hard scale  is ambiguous • Ideally dX/d=0, but not possible if X ~ g()N • More likely if X= a g()N + b g()N+1 • No clear way to merge different topologies • Some W/Z+N parton events will be reconstructed as W/Z + N-1 jet events • Some W/Z+N-1 parton events will be W+N-1 jet events • No way to avoid soft and/or collinear singularities • In fact, multiple gluon emission occurs in these kinematic configurations • No direct connection to hadronization Stephen MRENNA  Fermilab

NLO: Looking Beyond the Trees Example: hb+X gs gs2 gs3 Stephen MRENNA  Fermilab

NLO: Some Improvement • Scale dependence greatly reduced • Shape of some distributions similar up to scale (K) factor • But kinematics are inclusive • Separation of different topologies depends on cutoff • Multiple, soft-collinear gluon emissions are not included Stephen MRENNA  Fermilab

Wbb and Zbb Stephen MRENNA  Fermilab

W+jj also to NLO • W+jj can still be used to normalize W+bb • Overall scale dependence of W/Z+jj reduced • Program MCFM (Campbell, R.K. Ellis) Stephen MRENNA  Fermilab

The need for even higher order… • As long as observables are inclusive enough, this is extremely important and useful • Beware of correlations between kinematics of different objects • These can be sensitive to multiple, soft gluon emission Stephen MRENNA  Fermilab

Consider W production At LO in pQCD, the rapidity Y and transverse momentum QT of the W are fixed by incoming partons At NLO, single gluon emission occurs with QT>0 Cross sections at large QT or QT averaged are described well by fixed order in S However, some observables are sensitive to region QT« Q For W/Z production, this is most of the data! Solution: Reorganize perturbative expansion N lnM(Q2/QT2) Sums up infinite series of soft gluon emissions kT dependent PDF’s Resummation: Beyond Fixed Order Stephen MRENNA  Fermilab

Sudakov Effect Multiple soft and collinear gluon emissions included, but integrated out • Solid: resummed • Superior at lower QT • Dot/Dash-dot: W+1j/W+2j • Superior at high QT • Ln(Q/Q)=0 Stephen MRENNA  Fermilab

Higher Order vs All Order • In the lower QT region, significantly different predictions from NLO • Decay products retain information about W production • Important for MW measurement in Run II NLO prediction depends on cutoff Stephen MRENNA  Fermilab

Tree Level Predictions Pros Full spin and color correlations “Easy” to calculate Good for large PT’s and angular separations Cons Rate not reliable Not clear how to merge different topologies (N)NLO Predictions Pros Reduced scale dependence Merging of topologies Cons Also requires large PT’s and angular separations Resummation Pros (N)NLO accuracy with all orders accuracy in kinematics Cons No information on soft gluons All of these approaches ignore details of physics below hard scale Not yet connected to hadronization Color must be screened Review of “Hard Scattering” Stephen MRENNA  Fermilab

Monte Carlo Event Generator • Calculate the probability of a hard scattering at scale Q treating in and out partons as on-shell • Rest of the event can be described by positive probability distributions • Prior to and after hard scatter, evolution of partons is sensitive to quantum fluctuations below scale Q • Cancellation of Virtual (-) and Real (+) effects occurs at scales too small to resolve • For color evolution, scale is typically QCD • In-partons evolved from some parents with P=1 • Out-partons evolve into daughters with P=1 • Final state partons hadronize with P=1 • Beam particle remnant also hadronizes with P=1 • The Factorization Theorem is essential for this to work Stephen MRENNA  Fermilab

“Monte Carlo” “Event Generator” • Improving the Physics complicates the Numerics • Difficult Integrand in Many dimensions • Well-suited to Monte Carlo methods • Integrands are positive definite • Normalize to be probability distributions • Hit-or-Miss • Test integrand to find maximum weight WMAX (or just guess) • Calculate weight W at some random point • If W > r WMAX, then keep it, otherwise pick new W • Sample enough points to keep error small • Can generate events like they will appear in an experiment • N = [Xb] L[Xb-1] • NNLO QCD programs are not event generators • Not positive definite (Cancellations between N and N+1) • Superior method for calculating suitable observables • Tree level programs are not event generators • Only limited topologies • Can follow spins/color exactly Stephen MRENNA  Fermilab

Parton Shower • Hard Scattering sets scale Q • Structure f(x,Q2) or fragmentation D(x,Q2) functions, couplings S(Q2), etc. are evaluated at Q • Asymptotic states have a scale Q0~1 GeV • Incoming/Outgoing partons are highly virtual • How do incoming partons acquire mass2 ~ -Q2 ? • INITIAL STATE RADIATION (ISR) • How do outgoing partons approach the mass shell ? • FINAL STATE RADIATION (FSR) • Typically, resolving smaller scales generates many partons with lower virtuality • Virtualities on the order of QCD are expected for partons bound in hadrons • “traditional” calculations based on a small number of Feynman diagrams are incomplete • Parton Showering Monte Carlos are an approximation to high-order, perturbative QCD Stephen MRENNA  Fermilab

Parton Showering: More Motivation Semi-classical description • Accelerated charges radiate • Color is a charge, and thus quarks also radiate • Gluon itself has charge (=q-q* pair to 1/Nc) • Field Theory • Block and Nordsieck (QED) • Must include virtual and real (emission) corrections to obtain IR finite cross section • Electron is ALWAYS accompanied by cloud of quanta (photons) Stephen MRENNA  Fermilab

Example: gluon emission in * events u Q2 s t z1 when gluon is Soft, collinear or both t 0 when gluon is Soft, collinear or both • Factorization of Mass Singularities • Probability of one additional soft emission proportional to rate without emission • dN+1 = NS/2dt/tdz P(z) Stephen MRENNA  Fermilab

Tower of emissions described by Sudakov Form Factor • Series of subsequent showers “exponentiate” • Shower of resolvable emissions q*(p)  q(zp) + g([1-z]p) • Emission RESOLVED if zC < z < 1 - zC • Sudakov built from Probability of no resolvable emission for small t • Prob(tmax,t) = S(tmax)/S(t) = random r • Pick random r and solve for new t • Resolvable emission at the end of “nothing”: dS/dt • Continue picking new t’s down to tmin ~ QCD • Stop shower & begin hadronization Stephen MRENNA  Fermilab

Virtuality-Ordered PS Highly virtual Nearly on-shell Stephen MRENNA  Fermilab

Initial State Radiation • Similar picture, but solving DGLAP for PDFs Increasing parton virtuality Parent has more momentum Stephen MRENNA  Fermilab

Backwards Showering Sjostrand • -Q02>-Q12>…>-Qn2 • showering added after hard scatter with unit probability • Something happens, even if not resolvable Marchesini/Webber PRIMORDIAL KT Stephen MRENNA  Fermilab

Analytic Resummation soft gluon emissions exponeniate into Sudakov form factor kT conserved Total rate at (N)NLO modified PDF's corrections for hard emission soft gluons are integrated out Predicts observables for a theoretical W Needs modelling of non-perturbative physics Parton Showering DGLAP evolution generates a shower of partons LL with some N-LL Exact gluon kinematics LO event rates underestimates single, hard emissions Explicit history of PS More closely related to object identified with a W natural transition to hadronization models Follow color flow down to small scales Parton Shower is a Resummation Similar physics, but different approach with different regimes of applicability Stephen MRENNA  Fermilab

Comparison of Predictions Analytic Resum Example of Treating Kinematics differently in shower Example of Matrix Element Corrections to Parton Showering Stephen MRENNA  Fermilab

In previous discussion of PS, interference effects were ignored, but they can be relevant Add a soft gluon to a shower of N almost collinear gluons incoherent emission: couple to all gluons |M(N+1)|2 ~ N S NC coherent emission: soft means long wavelength resolves only overall color charge (that of initial gluon) |M(N+1)|2 ~ 1 S NC Color Coherence Stephen MRENNA  Fermilab

Showers should be Angular-Ordered  = pI • pJ / EI EJ = (1 - cosIJ) ~ IJ2/2 1 > 2 > 3 … Running coupling depends on kT2 z(1-z)Q2 Dead Cone for Emissions Q2 = E2  < Q2max Q2max = z2 E2  < 1 [not 2]  < /2 No emission in backwards hemisphere Angular-Ordered PS Stephen MRENNA  Fermilab

Color Coherence in Practice • Emission is restricted inside cones defined by the color flow Large N picture Partonic picture Enhanced emission Beam line Stephen MRENNA  Fermilab

Essential to Describe Data • 3 Jet Distributions in Hadronic Collisions No Coherence Full Coherence Soft emissions know about beam line (large Y) Partial Coherence Pseudorapidity of Gluon Jet Stephen MRENNA  Fermilab

The Programs (Pyt/Isa/Wig/Aria) • ISAJET • Q2 ordering with no coherence • large range of hard processes • PYTHIA • Q2 ordering with veto of non-ordered emissions • large range of hard processes • HERWIG • complete color coherence & NLO evolution for large x • smaller range of hard processes • ARIADNE • complete color dipole model (best fit to HERA data) • interfaced to PYTHIA/LEPTO for hard processes Stephen MRENNA  Fermilab

Parton Shower Summary • Accelerated color charges radiate gluons • Gluons are also charged • Showers of partons develop • IMPORTANT effect for experiments • Showering is a Markov process and is added to the hard scattering with P=1 • Derived from factorization theorems of full gauge theory • Performed to LL and some sub-LL accuracy with exact kinematics • Color coherence leads to angular ordering • Modern PS models are very sophisticated implementations of perturbative QCD • Still need hadronization models to connect with data • Shower evolves virtualities of partons to a low enough values where this connection is possible Stephen MRENNA  Fermilab

Strings (Pythia) PRODUCTION of HADRONS is non-perturbative, collective phenomena Careful Modelling of non-perturbativedynamics Improving data hasmeant successively refining perturbativephase of evolution Clusters (Herwig) PERTURBATION THEORY can be applied down to low scales if the coherence is treated correctly There must be non-perturbative physics, but it should be very simple Improving data has meant successively making non-pert phase more stringlike See Bill Gary’s lectures Comparison STRING model includes some non-perturbative aspect of color coherence Stephen MRENNA  Fermilab

The Programs • ISAJET • Independent fragmentation & incoherent parton showers • JETSET (now PYTHIA) • THE implementation of the Lund string model • Excellent fit to e+ e- data • HERWIG • THE implementation of the cluster model • OK fit to data, but problems in several areas • String effect a consequence of full angular-ordering Stephen MRENNA  Fermilab

Good testing ground for parton showers, LO,NLO Large Scale dependence at LO Good agreement with NLO W + Jet(s) at the Tevatron Stephen MRENNA  Fermilab

W + N jet Rates • VECBOS for N hard partons • HERWIG for additional gluon radiation and hadronization • VECBOS for N-1 hard partons • HERWIG for 1 “hard” parton plus … Stephen MRENNA  Fermilab

Start with W + N jets from VECBOS + HERWIG Start with W + (N-1) jets from VECBOS + HERWIG W + N jet Shapes Stephen MRENNA  Fermilab

#events >1 jet >2 jets >3 jets >4 jet pT>10 GeV/c Data 920 213 42 10 VECBOS + HERPRT (Q=<pT>) W + 1jet 920 178 21 1 W + 2jet ----- 213 43 6 W + 3jet ----- ----- 42 10 VECBOS+HERPRT(Q=mW) W + 1jet 920 176 24 2 W + 2jet ---- 213 46 6 W + 3jet ---- ----- 42 7 When good Monte Carlos go bad CDF Run0 Data VECBOS starting point More jets generated by HERWIG parton shower Results are cut dependent • PS only has collinear part of matrix element • PS has ordering in angles Normalize 1st bin to data Stephen MRENNA  Fermilab

Correcting the Parton Shower • PS is an accurate description for soft/collinear kinematics • Most of the data for a given process! • Underestimates wide angle emissions • Also, no 1/NC suppressed color flows • Tails of kinematic distributions are often most interesting • PS with a single emission can be reweighted to behave like fixed-order result • Correct all or hardest-so-far emissions this way • Populate kinematic regions not included in PS • Delicate matching between different regions • Actual correction is generator dependent • No attempt to generate NLO rate Sjostrand/Miu/Seymour/Gorcella … Stephen MRENNA  Fermilab

Parton Showering and Heavy Quarks • Heavy Quarks look like light quarks at large angles but are sterile at small angles • Naïve -ordered shower has a cutoff • > 0 = mQ/EQ = r • Creates ‘dead cone’ • Virtuality-ordered shower also needs a special treatment Stephen MRENNA  Fermilab

Pythia Corrections to Top Decay Angle btw. Quark and Gluon • Relatively easy for Q2 ordered showering • Rewrite Parton Shower weight in terms of Matrix Element kinematics • Modify PS probability by WME / WPS Not Significant Hadron Level Mtop Parton Level Mtop Stephen MRENNA  Fermilab

Introduction to Event Generators