Introduction to Event Generators Lecture 1 of 4 Peter Skands Monash University (Melbourne, Australia) 1 1 T H M C N E T S C H O O L O N M O N T E C A R L O E V E N T G E N E R A T O R S F O R L H C ( J U L Y 2 0 1 7 , L U N D )
MELBOURNE? Monash Australia’s 4 University: 70,000 students deadliest (Australia’ s largest uni) animals: ~ 20km SE of Melbourne City Horses (7/yr) Centre Cows (3/yr) School of Physics & Dogs (3/yr) Astronomy ; 4 HEP theorists Roos (2/yr) + post docs & students Melbourne Physics Lab 2 Peter Skands Monash University
DISCLAIMER ๏ This course covers: Lecture 1: Foundations of MC Generators ๏ Lecture 2: Parton Showers ๏ Lecture 3: Jets and Confinement ๏ Lecture 4: Physics at Hadron Colliders ๏ Supporting Lecture Notes (~80 pages) : “Introduction to QCD” , arXiv:1207.2389 + MCnet Review: “General-Purpose Event Generators” , Phys.Rept.504(2011)145 ๏ It does not cover: Simulation of BSM physics → Lectures by V Hirschi ๏ Matching and Merging → Lectures by S Höche ๏ Heavy Ions and Cosmic Rays → Lectures by K Werner ๏ Event Generator Tuning → Lecture by H Schulz ๏ + many other (more specialised) topics such as: heavy quarks, hadron and τ ๏ decays, exotic hadrons, lattice QCD, spin/polarisation, low-x, elastic, … 3 Peter Skands Monash University
CONTENTS 1. Foundations of MC Generators 2. Parton Showers 3. Jets and Confinement 4. Physics at Hadron Colliders 4 Peter Skands Monash University
MAKING PREDICTIONS Scattering LHC detector source Experiments: Cosmic-Ray detector Neutrino detector ∆Ω X-ray telescope … → Integrate differential cross sections over specific phase-space regions d Ω = d cos θ d φ d Ω d σ Predicted number of counts Z N count ( ∆Ω ) ∝ = integral over solid angle d Ω ∆Ω In particle physics: Integrate over all quantum histories (+ interferences) 5 Peter Skands Monash University
d σ /d Ω ; how hard can it be? ๏ If event generators could talk: • Someone hold my drink while I approximate the amplitude (squared) for this … (to all orders, … integrate it + non- over a ~300- perturbative dimensional effects) phase space Candidate t ¯ tH event ATLAS-PHOTO-2016-014-13 … and estimate the detector response 6 Peter Skands Monash University
Q C D in Event Generators
q ψ qi − 1 q ( i γ µ )( D µ ) ij ψ j L = ¯ q − m q ¯ ψ i ψ i 4 F a µ ν F aµ ν Covariant Derivative ๏ Quark fields D µ ij = δ ij ∂ µ − ig s t a ij A µ ψ 1 ψ j L invariant under q = ψ 2 a ψ → U ψ a ∈ [1,8] ⇒ Feynman rules i,j ∈ [1,3] ψ 3 SU(3) Local Gauge Symmetry j i Gell-Mann Matrices (t a = ½λ a ) (Traceless and Hermitian) 0 1 0 1 0 1 0 1 0 1 0 0 − i 0 1 0 0 0 0 1 λ 1 = A , λ 2 = A , λ 3 = A , λ 4 = 1 0 0 i 0 0 0 − 1 0 0 0 0 @ @ @ @ A 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 1 0 1 0 1 0 0 − i 0 0 0 0 0 0 √ 3 λ 5 = A , λ 6 = A , λ 7 = A , λ 8 = 1 0 0 0 0 0 0 0 1 0 0 − i B C √ @ @ @ 3 @ A − 2 i 0 0 0 1 0 0 i 0 0 0 √ 3 8 Peter Skands Monash University
INTERACTIONS IN COLOUR SPACE ๏ A quark-gluon interaction q ( i γ µ )( D µ ) ij ψ j ¯ ψ i q − • (= one term in sum over colours) D µ ij = δ ij ∂ µ − ig s t a ij A µ A 1 µ ¯ − i λ 1 2 g s ψ qR ψ qG ∝ 0 1 0 1 0 1 0 0 − i � � = 1 0 0 1 0 0 1 2 g s @ A @ A 0 0 0 0 ψ qG ψ qR gluon (adjoint) colour index ∈ [1,8] gluon Lorentz-vector index ∈ [0,3] ij γ µ ij γ µ − i g s t 1 αβ A 1 − i g s t 2 αβ A 2 µ − . . . µ fermion spinor indices ∈ [1,4] fermion colour indices ∈ [1,3] Amplitudes Squared summed over colours → traces over t matrices → Colour Factors (see literature, or back of these slides) 9 Peter Skands Monash University
INTERACTIONS IN COLOUR SPACE qi − 1 ๏ A gluon-gluon interaction 4 F a µ ν F aµ ν • (no equivalent in QED) F a µ ν = ∂ µ A a ν − ∂ ν A a + g s f abc A b µ A c . µ A 4 ν ν ( k 2 ) | {z } | {z } Abelian non − Abelian − g s f 246 [( k 3 − k 2 ) ρ g µ ν ∝ +( k 2 − k 1 ) µ g νρ +( k 1 − k 3 ) ν g ρ µ ] if abc = 2Tr { t c [ t a , t b ] } } | {z } A 6 A 2 Structure Constants of SU(3) ρ ( k 1 ) µ ( k 3 ) (14) f 123 = 1 f 147 = f 246 = f 257 = f 345 = 1 (15) 2 f 156 = f 367 = − 1 (16) (there is also a 4- 2 √ 3 gluon vertex, (17) f 458 = f 678 = 2 proportional to g s 2 ) Antisymmetric in all indices All other f abc = 0 Amplitudes Squared summed over colours → traces over t matrices → Colour Factors (see literature, or back of these slides) 10 Peter Skands Monash University
COLOUR VERTICES IN EVENT GENERATORS ๏ MC generators use a simple set of rules for “colour flow” 2 • Based on “Leading Colour” ( ➾ valid to ~ 1/N C ~ 10%) 8 = 3 ⌦ 3 1 LC: represent gluons as outer products of triplet and antitriplet g → q ¯ q → qg q Illustrations from PDG Review on MC Event Generators g → gg ➜ Lecture 2 “Strong Ordering”, + Mass effects: t, b, (c?) quarks, coloured resonances; α s (p ⊥ ), “Coherence”, Spin effects (J cons; polarisation; spin correlations); Corrections beyond LC (or LL) “Recoils” [(E,p) cons.] 11 Peter Skands Monash University
COLOUR FLOW ๏ Showers (can) generate lots of partons, 𝒫 (10-100). • Colour Flow used to determine between which partons confining potentials arise Example: Z 0 → qq 1 1 3 2 5 4 7 1 1 4 5 5 3 3 4 7 6 2 2 System #1 System #2 System #3 Coherence of pQCD cascades → suppression of “overlapping” systems → Leading-colour approximation pretty good ( LEP measurements in e + e - → W + W - → hadrons confirm this (at least to order 10% ~ 1/N c2 ) ) Note : (much) more color getting kicked around in hadron collisions. Signs that LC approximation is breaking down? → Lecture 4 12 Peter Skands Monash University
THE STRONG COUPLING Bjorken scaling: Q 2 ∂α s ∂ Q 2 = β ( α s ) ๏ Asymptotic Freedom • To first approximation, QCD is ) = − α 2 s ( b 0 + b 1 α s + b 2 α 2 s + . . . ) SCALE INVARIANT (a.k.a. conformal) b 0 = 11 C A − 2 n f 1-Loop β function Landau Pole at > 0 coefficient: Λ QCD ~200 MeV 12 π for n f ≤ 16 Jets inside jets inside jets … ๏ 0.5 Fluctuations (loops) inside April 2012 ๏ α s (Q) τ decays (N 3 LO) fluctuations inside fluctuations … F Lattice QCD (NNLO) u l l DIS jets (NLO) 0.4 m c Heavy Quarkonia (NLO) – e + e jets & shapes (res. NNLO) 2 If the strong coupling didn’t - L Z pole fit (N 3 LO) o ๏ o p pp –> jets (NLO) “run”, this would be absolutely 0.3 1-Loop true (e.g., N=4 Supersymmetric Yang-Mills) m b Since α s only runs slowly Large values, 0.2 ๏ α s ( m Z ) ∼ 0 . 118 fast running at (logarithmically) → can still gain low scales insight from fractal analogy 0.1 ( → lecture 2 on showers) QCD α ( Μ ) = 0.1184 ± 0.0007 s Z ๏ 1 10 100 Q [GeV] Note: I use the terms “conformal” and “scale invariant” interchangeably Strictly speaking, conformal (angle-preserving) symmetry is more restrictive than just scale invariance 13 Peter Skands Monash University
MANY WAYS TO SKIN A CAT ๏ The strong coupling is (one of) the main perturbative parameter(s) in event generators. It controls: The overall amount of QCD initial- and final-state radiation MCs: get value Strong-interaction cross sections (and resonance decays) from: PDG? PDFs? Fits to The rate of (mini)jets in the data (tuning)? underlying event 0.5 0.5 similar infrared limits April 2012 α (Q) s Example (for Final-State Radiation): α s (Q) τ decays (N 3 LO) ๏ Lattice QCD (NNLO) DIS jets (NLO) 0.4 0.4 Heavy Quarkonia (NLO) – e + e SHERPA : uses PDF or PDG value, with “CMW” translation jets & shapes (res. NNLO) Z pole fit (N 3 LO) alphaS(m Z ) default = 0.118 (pp) or 0.1188 (LEP) pp –> jets (NLO) 0.3 0.3 running order: default = 3-loop (pp) or 2-loop (LEP) PYTHIA is ~ 10% higher Blue CMW scheme translation: default use ~ alphaS(p T /1.6) than SHERPA due to tuning band → roughly 10% increase in the effective value of α s to LEP 3-jet rate illustrates 0.2 0.2 factor-2 scale will undershoot LEP 3-jet rate by ~ 10% (unless combined with NLO 3-jet ME) variation; relative to PYTHIA PYTHIA : tuning to LEP 3-jet rate; requires ~ 20% increase 0.1 0.1 TimeShower:alphaSvalue default = 0.1365 QCD α ( Μ ) = 0.1184 ± 0.0007 s Z TimeShower:alphaSorder default = 1 1 10 100 Q [GeV] TimeShower:alphaSuseCMW default = off (also note: definitions of Agrees with LEP 3-jet rate “out of the box”; but no guarantee tuning is universal. Q=p T not exactly the same) 14 Peter Skands Monash University
USING SCALE VARIATIONS TO ESTIMATE UNCERTAINTIES ๏ Scale variation ~ uncertainty; why? • Scale dependence of calculated orders must be canceled by contribution from uncalculated ones (+ non-pert) ๏ 1 α s ( Q 2 ) = α s ( m 2 b 0 = 11 N C − 2 n f Z ) 1 + b 0 α s ( m Z ) ln Q 2 Z + O ( α 2 s ) 12 π m 2 → α s ( Q 2 1 ) − α s ( Q 2 2 ) = α 2 s b 0 ln( Q 2 2 /Q 2 1 ) + O ( α 3 s ) → Generates terms of higher order, proportional to what you already have (|M| 2 ) → a first naive * way to estimate uncertainty *warning: some believe it is the only way … but be agnostic! Really a lower limit. There are other things than scale dependence … 15 Peter Skands Monash University
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