# Presentation: 090690

## Towards Improved Event Generators

- Peter Skands
- Fermilab / Particle Physics Division / Theoretical Physics

## Overview

- Introduction
- The structure of high-energy collisions
- Event Generators
- Combining higher-order matrix elements with parton showers
- The problem of double counting  matching
- The VINCIA code
- The Underlying Event
- Multiple Perturbative Interactions (multiple parton interactions)
- A study of string reconnections and the top mass at the Tevatron

- Event Generator Status

- Peter Skands

## Main Tool
Matrix Elements in perturbative Quantum Field Theory

Example:

- Main Tool
- Matrix Elements in perturbative Quantum Field Theory
- Example:

- QuantumChromoDynamics

- Reality is more complicated

- Event Generator Status

- Peter Skands

## Monte Carlo Generators

**Large-dimensional phase spaces **
- Monte Carlo integration
- Stratified sampling + stochastic error ~ N1/2 independent of dimension  ‘events’
**+ Markov Chain formulation of fragmentation:**

| Parton showers: iterative application of universal and pertubatively calculable kernels for n  n+1 partons ( = resummation of soft/collinear Sudakov logarithms)
2. Hadronization: iteration of X  X’ + hadron, at present according to phenomenological models based on known properties of nonperturbative QCD, lattice studies, and fits to data. |
| --- |

- Event Generator Status

- Peter Skands

## Classic Example: Number of tracks

- UA5 @ 540 GeV, single pp, charged multiplicity in minimum-bias events

- Simple physics models ~ Poisson
- Can ‘tune’ to get _average_ right, but much too small _fluctuations_
-   inadequate physics model

- Morale (will return to the models later):
- It is not possible to ‘tune’ anything better than the underlying physics model allows
- Failure of a physically motivated model usually points to more, interesting physics

- Event Generator Status

- Peter Skands

## Collider Energy Scales

- Non-perturbative
- hadronisation, colour reconnections, beam remnants, non-perturbative fragmentation functions, pion/proton ratio, kaon/pion ratio, Bose-Einstein correlations ...

- Soft Jets + Jet Structure
- Multiple collinear/soft emissions (initial and final state brems radiation), Underlying Event (multiple perturbative 22 interactions + ... ?), semi-hard separate brems jets

- Resonance Masses ...

- Hard Jet Tail
- High-pT wide-angle jets

-  & Widths

- + “UNPHYSICAL” SCALES:
-  QF , QR : Factorisation(s) & Renormalisation(s)

- s

**Inclusive**

**Exclusive**

- Hadron Decays

- Event Generator Status

- Peter Skands

## The Bottom Line 

- The S matrix is expressible as a series in gi, gin/tm, gin/xm, gin/mm, gin/fπm , ...
**To do precision physics**:
- _Solve more of QCD_
- Combine approximations which work in different regions: matching
- _Control it_
- Good to have comprehensive understanding of uncertainties
- Even better to have a way to systematically improve
- _Non-perturbative effects_
- don’t care whether we know how to calculate them

- FO

- DGLAP

- BFKL

- HQET

- χPT

- Event Generator Status

- Peter Skands

## Cross Sections and Kinematics

- Starting point 2  X hard scattering perturbative matrix element

- Fold with parton distribution functions  pp  X cross section

- MC integration  22 ‘events’ defined at hard scale QF2: starting point for Markov Chain

- From here on, unitarity  total sigma not changed

- Event Generator Status

- Peter Skands

## QuantumChromoDynamics

- to Landau Pole

- QuantumChromoDynamics

- Problem 1: QCD becomes non-perturbative at scales below ~ 1 GeV

- Problem 2: bremsstrahlung corrections singular for soft and collinear configurations

- To connect this with ‘real’ final states, 2 fundamental problems:

- e+e-  3 jets

- Event Generator Status

- Peter Skands

## Starting observation: forward singularity of bremsstrahlung is universal 
 Leading contributions to all radiation processes (QED & QCD) can be worked out to all orders once and for all
 exponentiated (Altarelli-Parisi) integration kernels

Iterative (Markov chain) formulation = parton shower
Generates the leading “collinear” parts of QED and QCD corrections to any process, to infinite order in the coupling
The chain is ordered in an “evolution variable”: e.g. parton virtuality, jet-jet angle, transverse momentum, ...
 a series of successive factorizations the lower end of which can be matched to a hadronization description at some fixed low hadronization scale ~ 1 GeV

- Starting observation: forward singularity of bremsstrahlung is universal
-  Leading contributions to all radiation processes (QED & QCD) can be worked out to _all orders_ once and for all
-  exponentiated (Altarelli-Parisi) integration kernels
- Iterative (Markov chain) formulation = _parton shower_
- Generates the leading “collinear” parts of QED and QCD corrections to any process, to infinite order in the coupling
- The chain is ordered in an “evolution variable”: e.g. parton virtuality, jet-jet angle, transverse momentum, ...
-  a series of successive factorizations the lower end of which can be matched to a hadronization description at some fixed low hadronization scale ~ 1 GeV

- Bremsstrahlung: Parton Showers

- dσn+1 = dσn dΠnn+1 Pnn+1
-  dσn+2 = dσn (dΠnn+1 Pnn+1)2   and so on ...  exp[]

- Schematic: Forward (collinear) factorization of QCD amplitudes  exponentiation

- Event Generator Status

- Peter Skands

## A Problem

- The best of both worlds? We want:
- A description which accurately predicts hard additional jets
- + jet structure and the effects of multiple soft emissions
- How to do it?
- Compute emission rates by parton showering?
- Misses relevant terms for hard jets, rates only correct for strongly ordered emissions pT1  >> pT2 >> pT3 ...
- (common misconception that showers are soft, but that need not be the case. They can err on *either side* of the right answer.)
- Compute emission rates with matrix elements?
- Misses relevant terms for soft/collinear emissions, rates only correct for well-separated individual partons
- Quickly becomes intractable beyond one loop and a handfull of legs

- Event Generator Status

- Peter Skands

## Double Counting

- Combine different multiplicites  *inclusive *sample?
- In practice – Combine
- [X]ME + showering
- [X + 1 jet]ME + showering
- ...
-  Double Counting:
- [X]ME + showering produces some X + jet configurations
- The result is X + jet in the shower approximation
- If we now add the *complete* [X + jet]ME *as well*
- the total rate of X+jet  is now _approximate + exact_ ~ double !!
- some configurations are generated *twice*.
- and the total inclusive cross section is also not well defined
- When going to X, X+j, X+2j, X+3j, etc, this problem gets worse

****

- Event Generator Status

- Peter Skands

## Matching

- Matching of up to one hard additional jet
- PYTHIA-style (reweight shower: ME = w*PS)
- HERWIG-style (add separate events from ME: weight = ME-PS)
- MC@NLO-style (ME-PS subtraction similar to HERWIG, but NLO)
- Matching of generic (multijet) topologies (at tree level)
- ALPGEN-style (MLM)
- SHERPA-style (CKKW)
- ARIADNE-style (Lönnblad-CKKW)
- PATRIOT-style (Mrenna & Richardson)
- Brand new approaches (still in the oven)
- Refinements of MC@NLO (Nason)
- CKKW-style at NLO (Nagy, Soper)
- SCET approach (based on SCET – Bauer, Schwarz)
- VINCIA (based on QCD antennae – Giele, Kosower, PS)

- Event Generator Status

- Peter Skands

## Improved Perturbative Monte Carlo

- Zero’th order approach to matching:
- Improve the parton showers themselves
- E.g. improved soft limit of ordering variable in HERWIG++
- Completely new pT-ordered hybrid parton-dipole showers in PYTHIA 8
- Step 1: A comprehensive look at the uncertainty
- Vary the evolution variable (~ factorization scheme)
- Vary the antenna function
- Vary the kinematics map (angle around axis perp to 23 plane in CM)
- Vary the renormalization scheme (argument of αs)
- Vary the infrared cutoff contour (hadronization cutoff)
- Step 2: Systematically improve on it
- Understand how each variation could be cancelled when
- Matching to fixed order matrix elements
- Higher logarithms are included
- Step 3: Write a generator
- Make Step 1 explicit in a Markov Chain  Parton Shower MC algorithm ‘with error bands’
- Include Step 2  Matched Parton Shower algorithm, ‘with smaller error band’

- Event Generator Status

- Peter Skands

## VINCIA

- _VINCIA Dipole shower_
- C++ code for gluon showers
- Standalone since ~ half a year
- Plug-in to PYTHIA 8 (C++ PYTHIA) since ~ last week
- Most results presented here use the plug-in version
- So far:
- 2 different shower evolution variables:
- pT-ordering (~ ARIADNE, PYTHIA 8)
- Virtuality-ordering (~ PYTHIA 6, SHERPA)
- For each: an infinite family of antenna  functions
- shower functions = leading singularities plus arbitrary polynomials (up to 2nd order in sij)
- Shower cutoff contour: independent of evolution variable
-  IR factorization “universal”  less wriggle room for non-pert physics?
- Phase space mappings: 3 choices implemented
- ARIADNE angle, Emitter + Recoiler, or “DK1” (+ ultimately smooth interpolation?)

- Dipoles – a dual description of QCD

- 1

- 3

- 2

- VIRTUAL NUMERICAL COLLIDER WITH INTERACTING ANTENNAE

- Giele, Kosower, PS : in progress

- Event Generator Status

- Peter Skands

## Checks: Analytic vs Numerical vs Splines

- Calculational methods
- Analytic integration over resolved region (as defined by evolution variable) – obtained by hand, used for speed and cross checks
- Numeric: antenna function integrated directly (by nested adaptive gaussian quadrature)  can put in any function you like
- In both cases, the generator constructs a set of natural cubic splines of the given Sudakov (divided into 3 regions linearly in QR – coarse, fine, ultrafine)
- Test example
- Precision target: 10-6
- ggggg Sudakov factor (with nominal αs = unity)

- ggggg: Δ(s,Q2)
-  Analytic
-  Splined

- pT-ordered Sudakov factor
- = no-branching probability, generating function for shower

-  Numeric / Analytic
-  Spline (3x1000 points) / Analytic

- Ratios

- Spline off by a few per mille at scales corresponding to less than a per mille of all dipoles  global precision ok ~ 10-6

- VINCIA 0.010
- (Pythia8 plug-in version)

- (a few experiments with single & double logarithmic splines  not huge success. So far linear ones ok for desired speed & precision)

- Event Generator Status

- Peter Skands

## Why Splines?

- Example: mH = 120 GeV
- Hgg + shower
- Shower start: 120 GeV. Cutoff = 1 GeV
- Speed (2.33 GHz, g++ on cygwin)
- Tradeoff: small downpayment at initialization  huge interest later &v.v.
- (If you have analytic integrals, that’s great, but must be hand-made)
- Aim to eventually handle any function & region  numeric more general

|  | Initialization 
(PYTHIA 8 + VINCIA) | 1 event |
| --- | --- | --- |
| Analytic, no splines | 2s | (< 10-3s ?) |
| Analytic + splines | 2s | < 10-3s |
| Numeric, no splines | 2s | 6s |
| Numeric + splines | 50s | < 10-3s |

- Numerically integrate the antenna function (= branching probability) over the resolved 2D branching phase space for *every single Sudakov trial evaluation*

- Have to do it only once for each spline point during initialization

- Event Generator Status

- Peter Skands

## Expanding the Shower

- Start from Sudakov factor
- = No-branching probability: (n or more  n and only n)
- Decompose inclusive cross section
- Simple example (sufficient for matching through NLO):

**NB: simplified notation! **
- Differentials are over entire respective phase spaces
- Sums run over all possible branchings of all antennae

- Event Generator Status

- Peter Skands

## Matching at NLO: tree part

- NLO real radiation term from parton shower
- Add extra tree-level X + jet (at this point arbitrary)
- Correction term is given by matching to fixed order:
-  variations (or dead regions) in |a|2 canceled by matching at this order
- (If |a| too hard, correction can become negative  constraint on |a|)
- Subtraction can be automated from ordinary tree-level ME’s
- + no dependence on unphysical cut or preclustering scheme (cf. CKKW)
- - not a complete order: normalization changes (by integral of correction), but still LO

**NB: simplified notation! **
- Differentials are over entire respective phase spaces
- Sums run over all possible branchings of all antennae
- Twiddles = finite (subtracted) ME corrections
- Untwiddled = divergent (unsubtracted) MEs

- Event Generator Status

- Peter Skands

## Matching at NLO: loop part

- NLO virtual correction term from parton shower
- Add extra finite correction (at this point arbitrary)
- Have to be slightly more careful with matching condition (include unresolved real radiation) but otherwise same as before:
- Probably more difficult to fully automate, but |a|2 not shower-specific
- Currently using Gehrmann-Glover (global) antenna functions
- Will include also Kosower’s (sector) antenna functions
- Any other subtraction function used in NLO is just a finite part away  use the polynomial terms

- Tree-level matching just corresponds to using zero

-  (This time, too small |a|  correction negative)

- Event Generator Status

- Peter Skands

## Matching at NNLO: tree part

- Adding more tree-level MEs is straightforward
- Example: second emission term from NLO matched parton shower
- X+2 jet tree-level ME correction term and matching equation

**Matching equation looks identical to 2 slides ago ******
- If all indices had been shown: sub-leading colour structures not derivable by nested 23 branchings do not get subtracted

- Event Generator Status

- Peter Skands

## Under the Rug

- The simplified notation allowed to skip over a few issues we want to understand slightly better, many of them related
- Start and re-start scales for the shower away from the collinear limit
- Evolution variable: global vs local definitions
- How the arbitrariness in the choice of phase space mapping is canceled by matching
- How the arbitrariness in the choice of evolution variable is canceled by matching
- Constructing an exactly invertible shower (sector antenna functions)
- Matching in the presence of a running renormalization scale
- Dependence on the infrared factorization (hadronization cutoff)
- Degree of automation and integration with existing packages
- To what extent negative weights (oversubtraction) may be an issue
- None of these are showstoppers as far as we can tell

- Event Generator Status

- Peter Skands

## VINCIA Example: H  gg  ggg

- VINCIA Example: H  gg  ggg

- VINCIA 0.008
**Unmatched**
- “soft” |A|2

- VINCIA 0.008
**Unmatched**
- “hard” |A|2

- VINCIA 0.008
**Matched**
- “soft” |A|2

- VINCIA 0.008
**Matched**
- “hard” |A|2

- y12

- y23

- y23

- y23

- y23

- y12

- First Branching ~ first order in perturbation theory
- Unmatched shower varied from “soft” to “hard” : soft shower has “radiation hole”. Filled in by matching.

- radiation hole in high-pT region

_**Outlook:**_
- Immediate Future:
- Paper about gluon shower
- Include quarks  Z decays
- Matching
- Then:
- Initial State Radiation
- Hadron collider applications

- Event Generator Status

- Peter Skands

## The Underlying Event

- Towards a complete picture of hadron collisions

## Domain of fixed order and parton shower calculations: hard partonic scattering, and bremsstrahlung associated with it.

But hadrons are not elementary 
+ QCD diverges at low pT 
 multiple perturbative parton-parton collisions should occur

Normally omitted in explicit perturbative expansions

+ Remnants from the incoming beams
+ additional (non-perturbative / collective) phenomena?
Bose-Einstein Correlations
Non-perturbative gluon exchanges / colour reconnections ?
String-string interactions / collective multi-string effects ?
Interactions with “background” vacuum / with remnants / with active medium?

- But hadrons are not elementary
- + QCD diverges at low pT
-  multiple _perturbative_ parton-parton collisions should occur
- Normally omitted in explicit perturbative expansions
- + Remnants from the incoming beams
- + additional (non-perturbative / collective) phenomena?
- Bose-Einstein Correlations
- Non-perturbative gluon exchanges / colour reconnections ?
- String-string interactions / collective multi-string effects ?
- Interactions with “background” vacuum / with remnants / with active medium?

- e.g. 44, 3 3, 32

- Additional Sources of Particle Production

- Event Generator Status

- Peter Skands

## Basic Physics

- Sjöstrand and van Zijl (1987):
- First serious model for the underlying event
- Based on resummation of _perturbative_ QCD 22 scatterings at successively smaller scales  multiple parton-parton interactions
- Dependence on impact parameter crucial to explain Nch distributions.
- Peripheral collisions  little matter overlap  few interactions
- Central collisions  many interactions (+ jet pedestal effect)
-  wider than Poissonian!
- Colour correlations also essential
- Determine between which partons hadronizing strings form (each string  log(mstring) hadrons)
- Important ambiguity: what determines how strings form *between* the different interactions?

- Event Generator Status

- Peter Skands

## The ‘New’ Model

- Sjöstrand and PS (2005):
- ‘Interleaved’ evolution of event structure, all ordered in pT

- Sjöstrand & PS : JHEP03(2004)053, EPJC39(2005)129

- multiparton
- PDFs derived
- from sum rules

- Beam remnants
- Baryon Junctions
- Fermi motion /
- Primordial kT

- Fixed order
- matrix elements

- pT-ordered
- parton shower
- (matched to ME
- for W/Z/H/G + jet)

- perturbative
- “intertwining”?
- (incomplete)

- Pythia 6.4: available via PYTUNE (6.408+)
- Pythia 8: default

- Event Generator Status

- Peter Skands

## Underlying Event and Colour

- Not much was known about the colour correlations, so some “theoretically sensible” default values were chosen
- Rick Field (CDF) noted that the default model produced too soft charged-particle spectra.

- The same is seen at RHIC:
- For ‘Tune A’ etc, Rick noted that <pT> increased when he increased the colour correlation parameters
- But needed ~ 100% correlation. So far not explained
- Virtually all ‘tunes’ now used by the Tevatron and LHC experiments employ these more ‘extreme’ correlations
- Tune A is now also the default in PYTHIA

- M. Heinz, nucl-ex/0606020; nucl-ex/0607033

- Event Generator Status

- Peter Skands

## Searched for at LEP 
Major source of W mass uncertainty
Most aggressive scenarios excluded
But effect still largely uncertain  Preconnect ~ 10%

Prompted by CDF data and Rick Field’s studies to reconsider. 	         	What do we know?
Non-trivial initial QCD vacuum
A lot more colour flowing around, not least in the UE
String-string interactions? String coalescence? 
Collective hadronization effects?
More prominent in hadron-hadron collisions?
What is <pT>(Nch) telling us?
What (else) is RHIC, Tevatron telling us? 
Implications for Top mass? Implications for LHC?

**toy model for all final states: colour annealing**

- Searched for at LEP
- Major source of W mass uncertainty
- Most aggressive scenarios excluded
- But effect still largely uncertain  Preconnect ~ 10%
- Prompted by CDF data and Rick Field’s studies to reconsider. 	         	What do we know?
- Non-trivial initial QCD vacuum
- A lot more colour flowing around, not least in the UE
- String-string interactions? String coalescence?
- Collective hadronization effects?
- More prominent in hadron-hadron collisions?
- What is <pT>(Nch) telling us?
- What (else) is RHIC, Tevatron telling us?
*Implications for Top mass? Implications for LHC?*

**OPAL**, Phys.Lett.B453(1999)153 & **OPAL**, hep-ex0508062

- Sjöstrand, Khoze, Phys.Rev.Lett.72(1994)28 & Z. Phys.C62(1994)281 + more ...

- Color Reconnections

- Sandhoff + PS, in Les Houches ’05 SMH Proceedings, hep-ph/0604120

- Baryons
- Strangeness
- Neutral/Charged

- Event Generator Status

- Peter Skands

## Colour Annealing

- Sandhoff + PS, in Les Houches ’05 SMH Proceedings, hep-ph/0604120

- Toy model of (non-perturbative) color reconnections, applicable to any final state
- at hadronisation time, each string piece has a probability to interact with the vacuum / other strings:
- 				Preconnect = 1 – (1-χ)n
- χ = strength parameter: fundamental reconnection probability (free parameter)
- n = # of multiple interactions in current event ( ~ counts # of possible interactions)
- For the interacting string pieces:
- New string topology determined by annealing-like minimization of  ‘Lambda measure’
- Similar to area law for fundamental strings: Lambda ~ potential energy ~ string length ~ log(m) ~ N
-  good enough for order-of-magnitude

- Event Generator Status

- Peter Skands

## A First Study

- Using Tevatron min-bias as constraint
- Those were the distributions that started it all
- High-multiplicity tail should be somewhat similar to top  less extrapolation required
- Why not use LEP? Again, since the extrapolation might not be valid.
- No UE in ee, no beam remnants, less strings, no ‘bags’ in initial state.
- The comparison would still be interesting and should be included in a future study
- As a baseline, all models were tuned to describe Nch and <pT>(Nch)

- No CR

- Field’s Tunes & new models

- Improved Description of Min-Bias
- Effect _Still_ largely uncertain
- Worthwhile to look at top etc

- Tevatron Run II min-bias

- PYTHIA 6.408

- PYTHIA 6.408

- A few weeks ago: D. Wicke + PS, **hep-ph/0703081**

- Event Generator Status

- Peter Skands

## Preliminary Conclusions

- Delta(mtop) ~ 1 GeV from parton shower
- To some extent already accounted for by HERWIG – PYTHIA, should still be investigated
- Match to hard matrix elements for top + jets + further constrain shower parameters

- Delta(mtop) ~ 0.5 GeV from infrared effects
- Early days. May be under- or overestimated. Models are crude, mostly useful for reconnaissance
- Pole mass does have infrared sensitivity. Can we figure out some different observable which is more stable?
- It may be difficult to derive one from first principles, given the complicated environment, but proposals could still be tested on models
- Infrared physics ~ universal?  use complimentary samples to constrain it. Already used a few min-bias distributions, but more could be included

- A few weeks ago: D. Wicke + PS, **hep-ph/0703081**

- Event Generator Status

- Peter Skands

## Words of Warning

- Still far from a complete description of hadron collisions
- We are really looking at just the first few terms in large expansions
- Non-perturbative physics not well understood

*J. D. Bjorken*

- from a talk given at the 75th anniversary celebration of the Max-Planck Institute of Physics, Munich, Germany, **December 10th, 1992**. As quoted in: Beam Line, Winter 1992, Vol. 22, No. 4

*[...] The Monte Carlo simulation has become the major means of visualization of not only detector performance but also of physics phenomena. So far so good. But it often happens that the physics simulations provided by the Monte Carlo generators carry the authority of data itself. **They look like data and feel like data**, and if one is not careful they are accepted as if they were data.*
*[...] They do allow one to look at, indeed visualize, the problems in new ways. But I also fear a kind of **“terminal illness”**, perhaps traceable to the influence of television at an early age. There the way one learns is simply to **passively stare into a screen and wait for the truth to be delivered. **A number of physicists nowadays seem to do just this.*

- Event Generator Status

- Peter Skands