CTEQ-MCnet School 2010 Lauterbad, Germany 26 July - 4 August 2010 Introduction to Monte Carlo Event Generators Torbj¨ orn Sj¨ ostrand Lund University 1. (today) Introduction and Overview; Monte Carlo Techniques 2. (today) Matrix Elements; Parton Showers I 3. (tomorrow) Parton Showers II; Matching Issues 4. (tomorrow) Multiple Parton–Parton Interactions 5. (Wednesday) Hadronization and Decays; Generator Status
Matrix Elements and Their Usage L ⇒ Feynman rules ⇒ Matrix Elements ⇒ Cross Sections + Kinematics ⇒ Processes ⇒ . . . ⇒ (Higgs simulation in CMS)
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Next-to-leading order (NLO) calculations I. Lowest order, O ( α em ) : qq → Z 0 d σ/ d p ⊥ lowest order finite σ 0 p ⊥
Next-to-leading order (NLO) calculations I. Lowest order, II. First-order real, O ( α em ) : O ( α em α s ) : qq → Z 0 qq → Z 0 g etc. d σ/ d p ⊥ d σ/ d p ⊥ lowest order real, + ∞ finite σ 0 p ⊥ p ⊥
Next-to-leading order (NLO) calculations I. Lowest order, II. First-order real, III. First-order virtual, O ( α em ) : O ( α em α s ) : O ( α em α s ) : qq → Z 0 with loops qq → Z 0 qq → Z 0 g etc. d σ/ d p ⊥ d σ/ d p ⊥ d σ/ d p ⊥ lowest order real, + ∞ finite σ 0 p ⊥ p ⊥ p ⊥ virtual, −∞
� � � σ NLO = n d σ LO + n +1 d σ Real + n d σ Virt Simple one-dimensional example: x ∼ p ⊥ /p ⊥ max , 0 ≤ x ≤ 1 Divergences regularized by d = 4 − 2 ǫ dimensions, ǫ < 0 � 1 x 1+ ǫ M ( x ) + 1 d x σ R+V = ǫ M 0 0 KLN cancellation theorem: M (0) = M 0 Phase Space Slicing : Introduce arbitrary finite cutoff δ << 1 (so δ ≫ | ǫ | ) � 1 � δ d x x 1+ ǫ M ( x ) + 1 d x σ R+V = x 1+ ǫ M ( x ) + ǫ M 0 δ 0 � 1 � δ d x x 1+ ǫ M 0 + 1 d x x M ( x ) + ≈ ǫ M 0 δ 0 � 1 d x x M ( x ) + 1 � 1 − δ − ǫ � = M 0 ǫ δ � 1 d x ≈ x M ( x ) + ln δ M 0 δ
Alternatively Subtraction : � 1 � 1 � 1 d x d x x 1+ ǫ M 0 + 1 d x σ R+V = x 1+ ǫ M ( x ) − x 1+ ǫ M 0 + ǫ M 0 0 0 0 � 1 M ( x ) − M 0 − 1 ǫ + 1 � � = d x + M 0 x 1+ ǫ ǫ 0 � 1 M ( x ) − M 0 ≈ d x + O (1) M 0 x 0 NLO provides a more accurate answer for an integrated cross section: K(pp → H+X) √ s = 14 TeV 3 Warning! 2.5 Neither approach operates 2 with positive definite quantities 1.5 No obvious event-generator implementation 1 No trivial connection to LO NNLO 0.5 NLO physical events 0 100 120 140 160 180 200 220 240 260 280 300 M H [GeV]
Cross sections and kinematics s = ( p 1 + p 2 ) 2 u (1) u (3) ˆ t = ( p 1 − p 3 ) 2 = − ˆ s (1 − cos ˆ g ˆ θ ) / 2 u = ( p 1 − p 4 ) 2 = − ˆ s (1 + cos ˆ d (2) d (4) ˆ θ ) / 2 s 2 + ˆ u 2 dˆ σ t = π 4 ˆ qq ′ → qq ′ 9 α 2 : ( ∼ Rutherford) s s 2 t 2 dˆ ˆ ˆ s = ( p A + p B ) 2 p ( A ) x 1 ≈ E 1 /E A 1 x 2 ≈ E 2 /E B 2 p ( B ) s = x 1 x 2 s ˆ ( x 2 , Q 2 ) dˆ σ ij ��� t f ( A ) ( x 1 , Q 2 ) f ( B ) � d x 1 d x 2 dˆ σ = i j dˆ t i,j Factorization: proven for a few processes, assumed for more!
Parton Distribution/Density Functions (PDFs) http://durpdg.dur.ac.uk/hepdata/pdf.html Initial conditions nonperturbative; evolution perturbative (DGLAP): � 1 d f b ( x, Q 2 ) d z z f a ( x ′ , Q 2 ) α s z = x � � � = 2 π P a → bc d(ln Q 2 ) x ′ x a
Peaking of PDF’s at small x and of QCD ME’s at low p ⊥ = ⇒ most of the physics is at low transverse momenta . . . (nb/GeV) CDF Run II Preliminary 2 10 Integrated L = 177 pb -1 10 JetClu Cone R = 0.7 1 η Uncorrected d -1 10 T η / dE 0.1 < | | < 0.7 Det -2 η 10 0.7 < | | < 1.4 Det σ η 2 1.4 < | | < 2.1 -3 10 d Det η 2.1 < | | < 2.8 Det -4 10 -5 10 -6 10 -7 10 -8 10 0 100 200 300 400 500 600 Inclusive Jet Measured E (GeV) T . . . but New Physics likely to show up at large masses/ p ⊥ ’s
At NLO PDFs are not physical objects and not required positive definite: σ = ˆ σ ⊗ PDF , and both can be negative. Dangerous for LO MCs: recently introduce new MC-adapted PDFs � 1 0 xf i ( x, Q 2 ) > 1 as “built-in K factor” • allow � i • use NLO-calculated pseudodata as target for tunes Current usage: • conventional: CTEQ 5L, CTEQ 6L, CTEQ 6L1, MSTW 2008 LO • MC-adapted: MRST LO* and LO**; CT09 MC1, MC2 and MCS
Colour flow in hard processes One Feynman graph can correspond to several possible colour flows, e.g. for qg → qg : ✘ ✘ ✏ ✏ r r br gb ☎ ✑ ☎ ✙ ✆ ✆ ✙ ✑ while other qg → qg graphs only admit one colour flow: ✘ ✘ ✏ ✏ r r br gb ☎ ✑ ☎ ✙ ✆ ✆ ✙ ✑
s, ˆ so nontrivial mix of kinematics variables (ˆ t ) and colour flow topologies I , II : t ) | 2 t ) | 2 s, ˆ s, ˆ s, ˆ |A (ˆ = |A I (ˆ t ) + A II (ˆ t ) | 2 + |A II (ˆ t ) | 2 + 2 R e t ) A ∗ s, ˆ s, ˆ s, ˆ s, ˆ � A I (ˆ � = |A I (ˆ II (ˆ t ) � � t ) A ∗ s, ˆ s, ˆ with R e A I (ˆ II (ˆ t ) � = 0 ⇒ indeterminate colour flow, while • showers should know it (coherence), • hadronization must know it (hadrons singlets). Normal solution: interference 1 ∝ N 2 total C − 1 so split I : II according to proportions in the N C → ∞ limit, i.e. t ) | 2 t ) | 2 t ) | 2 s, ˆ s, ˆ s, ˆ |A (ˆ = |A I (ˆ mod + |A II (ˆ mod t ) | 2 � s, ˆ � |A I (ˆ t ) | 2 t ) | 2 s, ˆ s, ˆ s, ˆ |A I (ˆ = |A I (ˆ t ) + A II (ˆ mod t ) | 2 + |A II (ˆ t ) | 2 s, ˆ s, ˆ |A I (ˆ N C →∞ t ) | 2 s, ˆ |A II (ˆ = . . . mod
Process Libraries Traditionally generators come each with its own subprocess library, handcoded since before the days of automatic code generation. Subprocess lists with hundreds of entries look impressive, and are useful to rapidly get going, but: ⋆ Processes usually only in lowest nontrivial order ⇒ need programs that include HO loop corrections to cross sections, alternatively do ( p ⊥ , y ) -dependent rescaling by hand? ⋆ No multijet topologies (except in SHERPA) ⇒ have to trust shower to get it right, alternatively match to HO (non-loop) ME generators ⋆ Spin correlations often absent or incomplete (in PYTHIA) e.g. top produced unpolarized, while t → bW + → b ℓ + ν ℓ decay correct ⇒ have to use external programs when important ⋆ New physics scenarios appear at rapid pace ⇒ need to have a bigger class of “one-issue experts” contributing code ⇒ The Les Houches Accord =
The Les Houches Accord Some Specialized Generators: Specialized Generator • AcerMC: ttbb , . . . = ⇒ Hard Process • ALPGEN: W / Z+ ≤ 6j , n W + m Z + k H+ ≤ 3j , . . . • CalcHEP: generic LO Les Houches Interface • Comix: generic LO (event file, or commonblock) • CompHEP: generic LO • GRACE+Bases/Spring: generic LO+ some NLO loops HERWIG or PYTHIA • HELAC–PHEGAS: generic LO (Resonance Decays) • MadCUP: W / Z+ ≤ 3j , ttbb Parton Showers • MadGraph+HELAS: generic LO Underlying Event • MCFM: NLO W / Z+ ≤ 2j , Hadronization WZ , WH , H+ ≤ 1j Ordinary Decays • O’Mega+WHIZARD: generic LO Apologies for all unlisted programs
Do it yourself MadGraph, CompHEP and CalcHEP can easily be run interactively: • user specifies process, e.g. gg → W + ud , and cuts • program finds all contributing lowest-order Feynman graphs, • the required amplitudes/cross sections are calculated, • phase-space is sampled and unweighted to give parton-level events, • parton-level properties can be histogrammed, • Les Houches Accord = ⇒ complete events. CompHEP/CalcHEP (matrix-elements-based, good for ∼≤ 4 outgoing): http://theory.sinp.msu.ru/comphep/ http://theory.sinp.msu.ru/ ∼ pukhov/calchep.html MadGraph (amplitude-based, can handle ∼≤ 7 outgoing): http://madgraph.physics.uiuc.edu/ Comix (in Sherpa): powerful new framework based on recursion relations . . . but • stiff price to pay for each additional parton = ⇒ optimized LO libraries, • confined to lowest-order processes = ⇒ NLO libraries.
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