Parton Show er Monte C arl os P e t e r S k a n d s ( C E R N T h e o r e t i c a l P h y s i c s D e p t ) L H C p h e n o n e t S u m m e r S c h o o l C r a c o w , P o l a n d , S e p t e m b e r 2 0 1 3
Scattering Experiments LHC detector source Cosmic-Ray detector Neutrino detector ∆Ω X-ray telescope … → Integrate differential cross sections over specific phase-space regions d Ω d σ Z Predicted number of counts N count ( ∆Ω ) ∝ = integral over solid angle d Ω ∆Ω In particle physics: Lots of dimensions? Complicated integrands? Integrate over all quantum histories → Use Monte Carlo (+ interferences) 2 P. S k a n d s
General-Purpose Event Generators Reality is more complicated Calculate Everything ≈ solve QCD → requires compromise! Improve lowest-order perturbation theory, by including the ‘most significant’ corrections → complete events (can evaluate any observable you want) The Workhorses PYTHIA : Successor to JETSET (begun in 1978) . Originated in hadronization studies: Lund String. HERWIG : Successor to EARWIG (begun in 1984) . Originated in coherence studies: angular ordering. SHERPA : Begun in 2000 . Originated in “matching” of matrix elements to showers: CKKW-L. + MORE SPECIALIZED: ALPGEN, MADGRAPH, ARIADNE, VINCIA, WHIZARD, MC@NLO, POWHEG, … 3 P. S k a n d s
Divide and Conquer Factorization → Split the problem into many (nested) pieces + Quantum mechanics → Probabilities → Random Numbers P event = P hard ⊗ P dec ⊗ P ISR ⊗ P FSR ⊗ P MPI ⊗ P Had ⊗ . . . Hard Process & Decays: Use (N)LO matrix elements → Sets “hard” resolution scale for process: Q MAX ISR & FSR (Initial & Final-State Radiation): Altarelli-Parisi equations → differential evolution, dP/dQ 2 , as function of resolution scale; run from Q MAX to ~ 1 GeV (More later) MPI (Multi-Parton Interactions) Additional (soft) parton-parton interactions: LO matrix elements → Additional (soft) “Underlying-Event” activity (Not the topic for today) Hadronization Non-perturbative model of color-singlet parton systems → hadrons 4 P. S k a n d s
(PYTHIA) PYTHIA anno 1978 (then called JETSET) LU TP 78-18 November, 1978 A Monte Carlo Program for Quark Jet Generation T. Sjöstrand, B. Söderberg A Monte Carlo computer program is presented, that simulates the fragmentation of a fast parton into a jet of mesons. It uses an iterative scaling scheme and is compatible with the jet model of Field and Feynman. Note : Field-Feynman was an early fragmentation model Now superseded by the String (in PYTHIA) and Cluster (in HERWIG & SHERPA) models. 5 P. S k a n d s
(PYTHIA) PYTHIA anno 2013 ~ 100,000 lines of C++ What a modern MC generator has inside: (now called PYTHIA 8) • Hard Processes (internal, inter- LU TP 07-28 (CPC 178 (2008) 852) faced, or via Les Houches events) October, 2007 • BSM (internal or via interfaces) A Brief Introduction to PYTHIA 8.1 • PDFs (internal or via interfaces) T. Sjöstrand, S. Mrenna, P. Skands • Showers (internal or inherited) The Pythia program is a standard tool • Multiple parton interactions for the generation of high-energy • Beam Remnants collisions, comprising a coherent set of physics models for the evolution • String Fragmentation from a few-body hard process to a complex multihadronic final state. It • Decays (internal or via interfaces) contains a library of hard processes • Examples and Tutorial and models for initial- and final-state parton showers, multiple parton-parton • Online HTML / PHP Manual interactions, beam remnants, string fragmentation and particle decays. It • Utilities and interfaces to also has a set of utilities and external programs interfaces to external programs. […] 6 P. S k a n d s
(some) Physics cf. equivalent-photon approximation Weiszäcker, Williams ~ 1934 Charges Stopped or kicked Radiation Radiation a.k.a. Bremsstrahlung Synchrotron Radiation Associated field The harder they stop, the harder the fluctations that continue to become radiation (fluctuations) continues 7
Jets = Fractals Most bremsstrahlung is 1 driven by divergent i a ∝ 2( p a · p b ) propagators → simple structure j b Amplitudes factorize in k singular limits ( → universal “conformal” or “fractal” structure) Partons ab → P(z) = Altarelli-Parisi splitting kernels, with z = energy fraction = E a /(E a +E b ) “collinear”: P ( z ) |M F +1 ( . . . , a, b, . . . ) | 2 a || b → g 2 2( p a · p b ) |M F ( . . . , a + b, . . . ) | 2 s C Gluon j Coherence → Parton j really emitted by (i,k) “colour antenna” → “soft”: ( p i · p k ) |M F +1 ( . . . , i, j, k. . . ) | 2 j g → 0 → g 2 ( p i · p j )( p j · p k ) |M F ( . . . , i, k, . . . ) | 2 s C + scaling violation: g s2 → 4 πα s (Q 2 ) Can apply this many times → nested factorizations See: PS, Introduction to QCD , TASI 2012, arXiv:1207.2389 8 P. S k a n d s
Bremsstrahlung d σ X = ✓ For any basic process (calculated process by process) dσ X+1 & dσ X+2& d d σ X$ σ X+2 ds i 1 ds 1 j d σ X +1 ∼ N C 2 g 2 ✓ d σ X s s i 1 s 1 j & ds i 2 ds 2 j d σ X +2 ∼ N C 2 g 2 ✓ d σ X +1 s s i 2 s 2 j ds i 3 ds 3 j d σ X +3 ∼ N C 2 g 2 . . . d σ X +2 s s i 3 s 3 j Factorization in Soft and Collinear Limits P(z) : “Altarelli-Parisi Splitting Functions” (more later) s C P ( z ) i || j | M ( . . . , p i , p j . . . ) | 2 | M ( . . . , p i + p j , . . . ) | 2 → g 2 s ij s C 2 s ik j g → 0 | M ( . . . , p i , p j , p k . . . ) | 2 | M ( . . . , p i , p k , . . . ) | 2 g 2 → s ij s jk “Soft Eikonal” : generalizes to Dipole/Antenna Functions (more later) 9 P. S k a n d s
Bremsstrahlung d σ X = ✓ For any basic process (calculated process by process) dσ X+1 & dσ X+2& d d σ X$ σ X+2 ds i 1 ds 1 j d σ X +1 ∼ N C 2 g 2 ✓ d σ X s s i 1 s 1 j & ds i 2 ds 2 j d σ X +2 ∼ N C 2 g 2 ✓ d σ X +1 s s i 2 s 2 j ds i 3 ds 3 j d σ X +3 ∼ N C 2 g 2 . . . d σ X +2 s s i 3 s 3 j Singularities: mandated by gauge theory Non-singular terms: process-dependent SOFT COLLINEAR 2 s ik ✓ s ij |M ( Z 0 → q i g j ¯ q k ) | 2 ◆� 1 + s jk q K ) | 2 = g 2 s 2 C F + |M ( Z 0 → q I ¯ s ij s jk s IK s jk s ij 2 s ik ✓ s ij |M ( H 0 → q i g j ¯ q k ) | 2 ◆� 1 + s jk q K ) | 2 = g 2 s 2 C F + + 2 |M ( H 0 → q I ¯ s ij s jk s IK s jk s ij SOFT COLLINEAR +F 10 P. S k a n d s
Bremsstrahlung d σ X = ✓ For any basic process (calculated process by process) dσ X+1 & dσ X+2& d d σ X$ σ X+2 ds i 1 ds 1 j d σ X +1 ∼ N C 2 g 2 ✓ d σ X s s i 1 s 1 j & ds i 2 ds 2 j d σ X +2 ∼ N C 2 g 2 ✓ d σ X +1 s s i 2 s 2 j ds i 3 ds 3 j d σ X +3 ∼ N C 2 g 2 . . . d σ X +2 s s i 3 s 3 j Iterated factorization Gives us a universal approximation to ∞ -order tree-level cross sections. Exact in singular (strongly ordered) limit. Finite terms (non-universal) → Uncertainties for non-singular (hard) radiation But something is not right … Total σ would be infinite … 11 P. S k a n d s
Loops and Legs Coefficients of the Perturbative Series The corrections from X (2) X+1 (2) … Quantum Loops are missing s p o X+1 (1) X+2 (1) X+3 (1) X (1) … o L Universality Born X+1 (0) X+2 (0) X+3 (0) (scaling) … Jet-within-a-jet-within-a-jet-... L e g s 12 P. S k a n d s
Evolution Q ∼ Q X Leading Order “Experiment” 100 100 75 75 % % 50 50 of LO of σ tot 25 25 0 0 Born +1 +2 Born (exc) +1 (exc) +2 (inc) Exclusive = n and only n jets Inclusive = n or more jets 13 P. S k a n d s
Evolution Q X Q ∼ “A few” Leading Order “Experiment” 100 100 75 75 % % 50 50 of LO of σ tot 25 25 0 0 Born +1 +2 Born (exc) +1 (exc) +2 (inc) Exclusive = n and only n jets Inclusive = n or more jets 14 P. S k a n d s
Evolution UNITARIT Y Q ⌧ Q X Leading Order “Experiment” ✓ 400 100 300 75 % % 200 50 of LO of σ tot 100 25 0 0 Born +1 +2 Born (exc) + 1 (exc) + 2 (inc) Cross Section Remains = Born (IR safe) Cross Section Diverges Number of Partons Diverges (IR unsafe) 15 P. S k a n d s
Unitarity → Evolution Imposed by Event evolution : Unitarity When (X) branches to (X+1): Kinoshita-Lee-Nauenberg: Gain one (X+1). Loose one (X). (sum over degenerate quantum states = finite) → evolution equation with kernel d σ X +1 Loop = - Int(Tree) + F d σ X Parton Showers neglect F Evolve in some measure of resolution ~ hardness, 1/time … ~ fractal scale → Leading-Logarithmic (LL) Approximation → includes both real (tree) and virtual (loop) corrections ► Interpretation: the structure evolves! (example: X = 2-jets) • Take a jet algorithm, with resolution measure “Q”, apply it to your events • At a very crude resolution, you find that everything is 2-jets • At finer resolutions some 2-jets migrate 3-jets = σ X+1 (Q) = σ X;incl – σ X;excl (Q) • Later, some 3-jets migrate further, etc σ X+n (Q) = σ X;incl – ∑σ X+m<n;excl (Q) • This evolution takes place between two scales, Q in ~ s and Q end ~ Q had ► σ X;tot = Sum ( σ X+0,1,2,3,…;excl ) = int( d σ X ) 16 P. S k a n d s
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