D I S C O V E R Y S e m i n a r, S e p 2 7 2 0 1 2 , N B I , C o p e n h a g e n Sol vi ng the LH C Confinement Showers h Amplitudes | M (0) H | 2 Elementary Fields Jets Hadrons P e t e r S k a n d s ( C E R N T H )
Why? July 4 th 2012: “Higgs- + huge amount of other like” stuff at CERN physics studies: # of journal papers: 144 ATLAS, 116 CMS, 51 LHCb, 27 ALICE Some of these are already, or will ultimately be, theory limited Precision = Clarity , in our vision of the Terascale Searching towards lower cross sections, the game gets harder + Intense scrutiny (after discovery) requires high precision Theory task: invest in precision This talk: a new formalism for highly accurate collider- physics predictions, and future perspectives 2 P. S k a n d s
How? Fixed Order Perturbation Theory: Problem: limited orders Parton Showers: Problem: limited precision “Matching”: Best of both Worlds? Problem: stitched together, slow Markovian Perturbation Theory → Infinite orders, high precision, fast 3 P. S k a n d s
Bremsstrahlung Radiation Radiation Accelerated Charges Associated field The harder they get kicked, the harder the fluctations that continue to become strahlung (fluctuations) continues 4 P. S k a n d s
Bremsstrahlung Most bremsstrahlung is emitted by particles that are almost on shell Divergent propagators → Bad fixed-order convergence (would need very high orders to get reliable answer) + Would be infinitely slow to carry out separate phase- space integrations for N, N+1, N+2, etc … 5 P. S k a n d s
Jets = Fractals PS, Introduction to QCD , TASI 2012, arXiv:1207.2389 Most bremsstrahlung is 1 driven by Divergent i a ∝ 2( p a · p b ) propagators → simple structure j b Gauge amplitudes factorize k in singular limits ( → universal “conformal” or “fractal” structure) Partons ab P(z) = Altarelli-Parisi splitting kernels, with z = 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 Coherence → Parton j really emitted by (i,k) “antenna” Gluon j → 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 6 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 7 P. S k a n d s
Last Ingredient: Loops PS, Introduction to QCD , TASI 2012, arXiv:1207.2389 Kinoshita-Lee-Nauenberg: Unitarity (KLN): Loop = - Int(Tree) + F Singular structure at loop level must Neglect F → Leading-Logarithmic (LL) be equal and opposite to tree level Approximation → Virtual (loop) correction: ✓ 2 s ik 2 Z d s ij d s jk ◆ � � 2Re[ M (0) F M (1) ∗ � M (0) ] ⊃ − g 2 = + less singular terms s N C � � F F 16 ⇡ 2 s ijk s ij s jk � that cancels the divergence coming from equation (52) itself. Further, since this is universally Realized by Event evolution in Q = fractal scale (virtuality, p T , formation time, …) d N F ( t ) = − d σ F +1 Resolution scale N F ( t ) t = ln(Q 2 ) d σ F dt = Approximation to Real Emissions Z d σ F +1 Probability to remain ✓ ◆ N F ( t ) “unbranched” from t 0 to t N F ( t 0 ) = ∆ F ( t 0 , t ) = exp − d σ F → The “Sudakov Factor” = Approximation to Loop Corrections 8 P. S k a n d s
Bootstrapped Perturbation Theory → All Orders (resummed) Born + Shower X (2) X+1 (2) … Unitarity s p o X+1 (1) X+2 (1) X+3 (1) X (1) … o L Exponentiation Born X+1 (0) X+2 (0) X+3 (0) Universality (scaling) … Jet-within-a-jet-within-a-jet-... L e g s But ≠ full QCD ! Only LL Approximation. 9 P. S k a n d s
→ Jack of All Orders, Master of None? Good Algorithm(s) → Dominant all-orders structures But what about all these unphysical choices? Renormalization Scales (for each power of α s ) The choice of shower evolution “time” ~ Factorization Scale(s) The radiation/antenna/splitting functions (finite terms arbitrary) The phase space map (“recoils”, d Φ n+1 /d Φ n ) The infrared cutoff contour (hadronization cutoff) Nature does not depend on them → vary to estimate uncertainties Problem : existing approaches vary only one or two of these choices 2. Higher-Order Corrections 1. Systematic Variations → Systematic Reduction of → Comprehensive Theory Uncertainty Estimates Uncertainties 10 P. S k a n d s
VINCIA Virtual Numerical Collider with Interleaved Antennae Written as a Plug-in to PYTHIA 8 C++ (~20,000 lines) Giele, Kosower, Skands, PRD 78 (2008) 014026, PRD 84 (2011) 054003 Gehrmann-de Ridder, Ritzmann, Skands, PRD 85 (2012) 014013 i 1 1 j i I k Based on antenna factorization j I k K - of Amplitudes (exact in both soft and collinear limits) m+1 m+1 - of Phase Space (LIPS : 2 on-shell → 3 on-shell partons, with (E,p) cons) K 1.0 Resolution Time 0.6 0.8 0.6 Infinite family of continuously deformable Q E y jk 1.0 p T 0.4 0.8 0.2 0.8 0.2 0.2 Special cases: transverse momentum, invariant mass, energy ⌦ ↵ 0.4 0.6 0.8 0.4 0.0 y jk 0.0 0.2 0.4 0.6 0.8 1.0 m D 0.4 0.8 1.0 y ij 0.6 0.6 + Improvements for hard 2 → 4: “smooth ordering” 0.8 0.2 0.4 0.2 0.6 0.0 0.0 0.2 0.4 0.6 0.8 1.0 y jk y ij 0.4 E g 0.6 Radiation functions 0.2 0.2 0.8 0.0 0.4 0.0 0.2 0.4 0.6 0.8 1.0 y ij Written as Laurent-series with arbitrary coefficients, ant i ∗ √ 2 (c) Special cases for non-singular terms: Gehrmann-Glover, MIN, MAX + Massive antenna functions for massive fermions (c,b,t) Kinematics maps Formalism derived for infinitely deformable κ 3 → 2 Special cases: ARIADNE, Kosower, + massive generalizations vincia.hepforge.org 11 P. S k a n d s
Changing Paradigm Ask: Is it possible to use the all-orders structure that the shower so nicely generates for us, as a substrate, a stratification, on top of which fixed-order amplitudes could be interpreted as corrections, which would be finite everywhere? Answer: Used to be no. (Though first order worked out in the eighties (Sjöstrand), expansions rapidly became too complicated) For multileg amplitudes, people then resorted to slicing up phase space (fixed-order amplitude goes here , shower goes there ), generated many different cookbook recipes and much bookkeeping 12 P. S k a n d s
Solution: (MC) 2 “Higher-Order Corrections To Timelike Jets” GeeKS: Giele, Kosower, Skands, PRD 84 (2011) 054003 Idea: Start from quasi-conformal all-orders structure (approximate) Impose exact higher orders as finite corrections Truncate at fixed scale (rather than fixed order) Bonus: low-scale partonic events → can be hadronized Problems: Traditional parton showers are history-dependent (non-Markovian) → Number of generated terms grows like 2 N N! Parton- (or Catani-Seymour) Shower: + Highly complicated expansions After 2 branchings: 8 terms After 3 branchings: 48 terms After 4 branchings: 384 terms Solution: (MC) 2 : Monte-Carlo Markov Chain Markovian Antenna Showers (VINCIA) Markovian Antenna Shower: After 2 branchings: 2 terms → Number of generated terms grows like N After 3 branchings: 3 terms + extremely simple expansions After 4 branchings: 4 terms 13 P. S k a n d s
New: Markovian pQCD * * ) pQCD : perturbative QCD Start at Born level Loops Cutting Edge: Embedding virtual amplitudes | M F | 2 = Next Perturbative Order → Precision Monte Carlos Generate “shower” emission +2 | M F +1 | 2 LL X a i | M F | 2 ∼ +1 i ∈ ant X Correct to Matrix Element ∈ +0 | M F +1 | 2 P a i | M F | 2 a i → a i → t +0 +1 +2 +3 Legs a e p P | | Unitarity of Shower e R Z Virtual = − Real + Z The VINCIA Code PYTHIA 8 Correct to Matrix Element Z | M F | 2 → | M F | 2 + 2Re[ M 1 “Higher-Order Corrections To Timelike Jets” F M 0 F ] + Real GeeKS: Giele, Kosower, Skands, PRD 84 (2011) 054003 14 P. S k a n d s
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