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Bryan Webber Cavendish Laboratory University of Cambridge QCD and Monte Carlo Tools Bryan Webber Cavendish Laboratory University of Cambridge Bryan Webber 1 CERN-Fermilab HCP School 2017 Monte Carlo Event Generation • Aim is to produce simulated (particle-level) datasets like those from real collider events ✤ i.e. lists of particle identities, momenta, ... ✤ simulate quantum effects by (pseudo)random numbers • Essential for: ✤ Designing new experiments and data analyses ✤ Correcting for detector and selection effects ✤ Testing the SM and measuring its parameters ✤ Estimating new signals and their backgrounds Bryan Webber 2 CERN-Fermilab HCP School 2017 Monte Carlo Basics Bryan Webber 3 CERN-Fermilab HCP School 2017 Monte Carlo Integration • Basis of all Monte Carlo methods: b 1 N I = f(x) dx (b a)f(x ) I ⇡ N − i ⌘ N Za i=1 X weight wi where x i are randomly (uniformly) distributed on [a,b]. Then I =limIN = E[w] ,σI = Var[w]/N • N !1 p where Var[w]=E[(w E[w])2]=E[w2] (E[w])2 − − b b 2 2 =(b a) f(x) dx f(x) dx V − − ⌘ Za Za ⇥ ⇤ ⇥ Central limit⇤ theorem: I = I V/N N ± P (< 1σ) = 68% p Bryan Webber 4 CERN-Fermilab HCP School 2017 Convergence • Monte Carlo integrals governed by Central Limit Theorem: error c.f. trapezium rule only if derivatives exist and are finite, e.g. Simpson’s rule 1 ⇡ I = 1 x2 dx = =0.785 0 − 4 Z p 2 ⇡2 pV = =0.223 r3 − 16 0.223 I =0.785 ± pN Bryan Webber 5 CERN-Fermilab HCP School 2017 Importance Sampling • Convergence is improved by putting more points in regions where integrand is largest • Corresponds to a Jacobian transformation • Variance is reduced (weights “flattened”) 2/3 x3 ⇢ = x − 3 0 2 1 4 ⇡ 2 − ⇡2 Bryan Webber r 6 CERN-Fermilab HCP School 2017 Hit-and-Miss • Accept points with probability = wi/wmax (provided all wi > 0) • Accepted points are distributed like real events • MC efficiency eMC = E[w]/wmax improved by importance sampling 1 2 "MC =1/I =2/⇡ = 64% " = dx(1 x )/I =3/⇡ = 95% MC − Z0 " (1 " ) 0.48 " (1 " ) 0.21 σ = MC − MC = σ = MC − MC = r N pN r N pN Bryan Webber 7 CERN-Fermilab HCP School 2017 Multi-dimensional Integration • Formalism extends trivially to many dimensions • Particle physics: very many dimensions, e.g. phase space = 3 dimensions per particles, LHC event ~ 250 hadrons. • Monte Carlo error remains • Trapezium rule • Simpson's rule Bryan Webber 8 CERN-Fermilab HCP School 2017 Monte Carlo: Summary Disadvantages of Monte Carlo: • Slow convergence in few dimensions. Advantages of Monte Carlo: • Fast convergence in many dimensions. • Arbitrarily complex integration regions (finite discontinuities not a problem). • Few points needed to get first estimate (“feasibility limit”). • Every additional point improves accuracy (“growth rate”). • Easy error estimate. • Hit-and-miss allows unweighted event generation, i.e. points distributed in phase space just like real events. Bryan Webber 9 CERN-Fermilab HCP School 2017 Phase Space Generation Phase space: Two-body easy: Bryan Webber 10 CERN-Fermilab HCP School 2017 Phase Space Generation Other cases by recursive subdivision: Or by ‘democratic’ algorithms: RAMBO, MAMBO Can be better, but matrix elements rarely flat. Bryan Webber 11 CERN-Fermilab HCP School 2017 Particle Decays ` Simplest example W t ⌫ e.g. top quark decay: b p p p p t · ⇥ b · ν Breit-Wigner peak of W very strong - but can be removed by importance sampling: m2 M 2 m2 arctan W − W (prove it!) W ⇥ Γ M W W ⇥ Bryan Webber 12 CERN-Fermilab HCP School 2017 Associated Distributions e.g. lepton momentum in top decays: Big advantage of Monte Carlo integration: Simply histogram any ` associated quantities. Almost any other technique requires new ⌫ integration for each observable. Can apply arbitrary cuts/ smearing. Bryan Webber 13 CERN-Fermilab HCP School 2017 Hadron-Hadron Cross Sections `+ pp p pq¯ = x2pp¯ Z _ p pq = x1pp p p¯ `− 2 s = Ecm =4pppp¯ sˆ =4pqpq¯ = x1x2s 0 + Consider e.g. pp¯ Z ` `− • ! ! • Integrations over incoming parton momentum distributions: 1 1 σ(s)= dx1f(x1) dx2f(x2)ˆσ(x1x2s) Z0 Z0 • Hard process cross section σ ˆ (ˆ s ) has strong peak, due to Z 0 resonance: needs importance sampling (like W in top decay) Bryan Webber 14 CERN-Fermilab HCP School 2017 Drell-Yan production • Large number of measurements o Total and differential cross-sections in boson * (lepton) kinematics (y, pT, " ,…) o Cross-section ratios (W+/W-, W/Z, …) o Jet distributions • ATLAS/CMS and LHC-b complementary • Low & high x accessed by off-shell data • Large statistics, clean signature, excellent detector calibration à typical experimental systematics +~1% • ppLuminosity ` systematics ` − cross (2-3%) andsection also other contributions! cancel in ratios 9 orders in cross-section • Understand proton structure: constrain PDFs • Probe higher-order pQCD Full spin correlations calculations • Study non-perturbative efffects, soft gluon resummation, parton shower modelling • Test Monte Carlo tools M=psˆ [GeV/c2] 2 CMS-PAS-SMP-16-009 M=psˆ [GeV/c ] • Essential for precision physics at LHC G. Pásztor: Hard QCD @ LHC, LP2017 27 4⇡sˆ Γ`Γq σˆqq¯ Z0 `+` = ! ! − 3M 2 (ˆs M 2 )2 +Γ2 M 2 Z − Z Z Z + “Background” is qq¯ γ⇤ ` `− • ! ! Bryan Webber 15 CERN-Fermilab HCP School 2017 Parton-Level Monte Carlo Calculations Now we have everything we need to make parton-level cross section calculations and distributions Can be largely automated… • MADGRAPH • GRACE • COMPHEP • AMEGIC++ • ALPGEN But… • Fixed parton/jet multiplicity • No control of large higher-order corrections • Parton level Need hadron level event generators Bryan Webber 16 CERN-Fermilab HCP School 2017 Monte Carlo Event Generation Bryan Webber 17 CERN-Fermilab HCP School 2017 Monte Carlo Event Generation • Monte Carlo event generation: ✤ theoretical status and limitations • Recent improvements: ✤ perturbative and non-perturbative • Overview of results: ✤ W, Z, top, Higgs, BSM (+jets) Bryan Webber 18 CERN-Fermilab HCP School 2017 4 4 Limits 0.0045 Quark-Quark Quark-Gluon 0.004 Gluon-Gluon 0.0035 CMS Simulation Preliminary 0.003 s = 8 TeV, WideJets |η| < 2.5 & |∆η | < 1.3 jj 0.0025 Normalized yield/GeV 0.002 0.0015 0.001 0.0005 0 1000 2000 3000 4000 5000 6000 Dijet Mass (GeV) Figure 2: The reconstructed resonance mass spectrum generated with the PYTHIA MC simula- tion and Tune D6T for qq G qq, qg q qg, gg G gg for resonance masses of ⇥ ⇥ ⇥ ∗ ⇥ ⇥ ⇥ 1.0, 2.0, 3.0, 4.0, and 5.0 TeV. A high-mass dijet event Figure 3: The event with the highest invariant mass: 3D view (left) and 2D view (right). The invariant massM of the= two 5.15 wide TeV jets is 5.15 TeV. • jj CMS PAS EXO-12-059 Bryan Webber 19 CERN-Fermilab HCP School 2017 LHC dijet p p Hard process Bryan Webber 20 CERN-Fermilab HCP School 2017 LHC dijet p p Hard process Parton showers Bryan Webber 21 CERN-Fermilab HCP School 2017 LHC dijet p p Hard process Parton showers Underlying event Bryan Webber 22 CERN-Fermilab HCP School 2017 LHC dijet p p Hard process Parton showers Underlying event Confinement Bryan Webber 23 CERN-Fermilab HCP School 2017 quark jet LHC dijet ) p beam ) jet beam p jet ) Hard process Parton showers Underlying event Confinement ) Hadronization quark jet Bryan Webber 24 CERN-Fermilab HCP School 2017 Theoretical Status Exact fixed-order perturbation theory p p Hard process Bryan Webber 25 CERN-Fermilab HCP School 2017 Theoretical Status Exact fixed-order perturbation theory Approximate all-order perturbation theory p p Hard process Parton showers Bryan Webber 26 CERN-Fermilab HCP School 2017 quark jet Theoretical Status) Exact fixed-order perturbation theory Approximate all-order perturbation theory Semi-empirical local models only p beam ) jet beam p jet ) Hard process Parton showers Underlying event Confinement ) Hadronization quark jet Bryan Webber 27 CERN-Fermilab HCP School 2017 QCD Factorization 1 2 2 2 2 2 σpp X (Epp)= dx1 dx2 fi(x1,µ ) fj(x2,µ )ˆσij X (x1x2Epp,µ ) ! ! Z0 ( ( momentum parton hard process fractions distributions cross section at scale µ2 • Jet formation and underlying event take place over a much longer time scale, with unit probability • Hence they cannot affect the cross section • Scale dependences of parton distributions and hard process cross section are perturbatively calculable, and cancel order by order Bryan Webber 28 CERN-Fermilab HCP School 2017 Parton Shower Approximation • Keep only most singular parts of QCD matrix elements: ↵S d⇠i dφi Collinear dσn+1 Pii(zi,φi) dzi dσn ⇠i =1 cos ✓i • ⇡ 2⇡ ⇠i 2⇡ i − ↵ X p p dφ Soft dσ S ( T T ) i · j !d!d⇠ i dσ • n+1 ⇡ 2⇡ − i · j p kp k i 2⇡ n i,j i j X · · ↵ ⇠ d! dφ = S ( T T ) ij d⇠ i dσ 2⇡ − i · j ⇠ ⇠ ! i 2⇡ n i,j i j X ↵ d! d⇠ S ( T T )⇥(⇠ ⇠ ) i dσ ⇡ 2⇡ − i · j ij − i ! ⇠ n i,j i X j ✓ij >✓i ! =(1 z )E − i i i ✓i ziEi Angular-ordered parton shower (or dipoles) Bryan Webber 29 CERN-Fermilab HCP School 2017 Parton Shower Evolution ✓ ✓ Ei+1 hard 0 1 ✓3 z = ✓ ✓4 E process E0 E1 2E2 E3 E4 i ✓ 0 2 H 4 1+z A Pq q(z)= D ! 3 1 z R − (E ,✓ ) O 1 1 N I Z (E2,✓2) A T (E ,✓ ) I 3 3 (E4,✓4) O N E✓< Qmin E0 Bryan Webber 30 CERN-Fermilab HCP School 2017 Parton Shower Evolution ✓ ✓ Ei+1 hard 0 1 ✓3 z = ✓ ✓4 E process E0 E1 2E2 E3 E4 i ✓ 0 2 H 4 1+z A Pq q(z)= D ! 3 1 z R − (E E ,✓ ) (E1,✓1) O 0 − 1 1 N Pg g(z)= ! I 1+z4 +(1 z)4 ✓ Z (E2,✓2) 3 − A z(1 z) T (E ,✓ ) − I 3 3 (E4,✓4) 1 2 2 O Pg q = z +(1 z) N ! 2 − ⇥ ⇤ E✓< Qmin E E0 Bryan Webber 31 CERN-Fermilab HCP School 2017 Sudakov Factor • Di(Q,Qmin) = probability for parton i to evolve from Q to Qmin without any resolvable emissions: ↵ E0 d! ✓0 d✓ ∆ (Q ,Q ) 1 C S 2 ⇥(!✓ Q )+..
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