# Presentation: 656718

## The Theory of Hadron CollisionsPast and Present

- Fermi National Accelerator Laboratory

- Peter Skands
- Theoretical Physics Dept

- CEA Saclay, SPhT, 27 November 2006

## Introduction
Collider energy scales
QCD in hadron collisions
Hard Scattering and Bremsstrahlung 
Monte Carlo Event Generators
Matching with QCD antennae
The Underlying Event
Multiple Parton Interactions and Beam Remnants
Strings and Colour

- Collider energy scales
- QCD in hadron collisions
- Hard Scattering and Bremsstrahlung
- Monte Carlo Event Generators
- Matching with QCD antennae
- The Underlying Event
- Multiple Parton Interactions and Beam Remnants
- Strings and Colour

- Overview

## Hadron Colliders

- Tevatron (2 TeV, Fermilab)
- 4 – 8 fb-1 by LHC turn-on
- Large Z, W, and ttbar samples
- Huge QCD samples
- Always: Potential discoveries...

**pp at 200 GeV,**
**+ 450GeV?**
****** ****help to determine energy scaling!**

- ( > 1fb-1 on tape
- per experiment)

## No Free Lunch

- Not all discovery channels produce dramatic signatures  Need theoretical control of shapes, backgrounds, uncertainties, ...
- Scattering at LHC≠ rescaled scattering at Tevatron/RHIC
- Aiming for percent level measurements, PDFs, luminosities, jets etc 
- solid understanding of QCD in hadron collisions, both **perturbative** and **non-perturbative**, is crucial

## QuantumChromoDynamics

- QuantumChromoDynamics

## The (QCD) Landscape

- D. B. Leinweber, hep-lat/0004025

- In reality, this all happens on top of each other.
- (only possible exception: long-lived colour singlet)

## Introduction: Summary

- To get high precision at hadron colliders:
- _Solve more of QCD_
- Over all of phase space, including
- non-perturbative region
- phase space borders
- more than 2 incoming partons
- _Control it more exactly_
- reliable uncertainties
- systematically improvable

## Introduction
Collider energy scales
QCD in hadron collisains
Hard Scattering and Bremsstrahlung 
Monte Carlo Event Generators
Matching with QCD antennae
The Underlying Event
Perturbative definition
Non-perturbative aspects: strings and colour

- Collider energy scales
- QCD in hadron collisains
- _Hard Scattering and Bremsstrahlung_
- Monte Carlo Event Generators
- Matching with QCD antennae
- The Underlying Event
- Perturbative definition
- Non-perturbative aspects: strings and colour

- Overview

## QuantumChromoDynamics

- Known Gauge Group and Lagrangian
- Rich variety of dynamical phenomena, not least _confinement_.

- Large coupling constant also means _perturbative_ expansion tricky.

- To calculate higher _perturbative_ orders, 2 approaches:
- Feynman Diagrams
- Complete _**matrix elements**_ order by order ****
- Complexity rapidly increases + unstable in soft/collinear region ****
- Resummation
- In certain limits, we are able to sum the entire perturbative series to infinite order **** e.g. _**parton showers**_** **
- Exact only in the relevant limits ****

## 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

## Monte Carlo Event Generators

**Large-dimensional final state phase spaces **
-  Monte Carlo integration
**+ 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)
**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.

- Main virtues
- Error is stochastic O(N-1/2) and independent of dimension
- Fully exclusive final states (for better or worse – cf. the name ‘Pythia’ ... )
- Only need to redo part of calculation for each different observable.
- Have proven essential for detailed experimental studies: can compute detector response event by event

## Parton Showers

- Starting observation: collinear limit of perturbative QCD is universal (process-independent)
- QCD corrections can be worked out to all orders once and for all
-  exponentiated (Altarelli-Parisi) integration kernels
- Iterative (Markov chain) formulation = parton shower
- can be used to generate the collinear singular parts of QCD corrections to any process to infinite order in the coupling
- ordered in a measure of resolution  a series of successive factorizations the lower end of which can be matched to a non-perturbative description at some fixed low scale
- Limitations
- misses interference terms relevant in the deep non-singular region
- kinematic ambiguities and double counting between fixed order part and resummed part

## Nuclear Decay (naïve approach ~ fixed order):
Suppose N1 nuclei at time t = t1   
Decay probability per unit time = |A|2
dN/dt = |A|2  N(t) = N1 (1 - |A|2t ) < 0 for late times ! 

Nuclear Decay (“resummed” approach)
Reason: only first term in expansion. 
For late times must include each nucleus can only decay once:
dN(t)/dt = |A|2   N(t) = N1 exp(-|A|2 t) 

- Suppose N1 nuclei at time t = t1
- Decay probability per unit time = |A|2
- dN/dt = |A|2  N(t) = N1 (1 - |A|2t ) **< 0** for late times !
- Nuclear Decay (“resummed” approach)
- Reason: only first term in expansion.
- For late times must include each nucleus can only decay once:
- dN(t)/dt = |A|2   N(t) = N1 exp(-|A|2 t)

**Δ****(Q****1****2****,Q****2****2****;s,Q****0****2****)**

- The Sudakov Form Factor

-  Sudakov = generating function for parton shower

## Bremsstrahlung Example:Supersymmetry at the LHC

- Problem: Need to get both
- soft and hard emissions
- “right”  “Matching”

- Bremsstrahlung Example:Supersymmetry at the LHC

- Perturbation Theory:
- Fixed Order Truncation with  explicit jets.
- Parton Showers / Resummation to infinite order in singular limits

## Matching – state of the art

- X=Anything (e.g. ttbar)
- PS=Parton Shower

- [ + FEHiP: NNLO (no PS) for pp hhγγ + jets ]

- Matching – state of the art

-   + new: single top

- See e.g. hep-ph/0602031

## New Approaches – Why Bother?

- MC@NLO:
- Used to think it was impossible! 
- But complicated  much work needed for each process 
- “Only” gets first jet right (rest is PS) 
- Hardwired to HERWIG 
- CKKW & MLM:
- Best approach when multiple hard jets important.
- Relatively straightforward (but still very time-consuming)
- Retains LO normalization 
- Dependence on matching scale 

**Not easy to control theoretical uncertainty on exponentiated part ******

**MC@NLO**

**MLM**

**CKKW**

- Improved MC@NLO: Nason, Ridolfi, Frixione, ...
- “CKKW@NLO”: Nagy & Soper  ...
- MC with SCET: Bauer & Schwartz  ...

## Giele, Kosower, PS
Markov chain algorithm for gluon cascade
Dipole shower with 2 different resolution/ordering variables:

- Giele, Kosower, PS
- Markov chain algorithm for _gluon cascade_
- Dipole shower with 2 different resolution/ordering variables:

- Illustration with quarks, sorry

- 1

- 3

- 2

- Dipole phase space

- Antenna Showers

- RII(m12,m23) 	= 4 min(s212,s223)/s
- 		~ virtuality ordering

- RI(m12,m23) 	= 4 s12s23/s
- 		= p2T;ARIADNE

## Antenna Showers

| Antenna Function | IR QCD singularities plus arbitrary finite terms
|A(sa,sb;s)|2    
 (a.k.a. Radiation Function, a.k.a. Splitting Kernel) |
| --- | --- |
| Evolution / Ordering Variable | Perturbative factorization / resolution scale
Type I : QR ≡ 2 mamb/m = 2pT;ARIADNE
Type II : QR = 2mmin = 2 min(ma,mb) |
| Renormalization Scheme | Scale choice as function of invariants
μR2(sa,sb;s) = s  or
μR2(sa,sb;s) = QR2  |
| Kinematics
nn+1 | p’jμ(piμ,sa,sb,φ,ψ,θ) ; j[1,n+1] ; i[1,n] 
φ: “roll”, ψ: “heading”, θ: “pitch” (θ=0 fixed by dipole axis) |
| Hadronization Cutoff | QH(sa,sb;s) < Q0 : “unresolved”
Can be chosen independent of QR  “universal” |

- _Definitions:_

## Antenna Showers

- For _each evolution variable_:
- an infinite family of antenna  functions, all with correct collinear and soft behaviour:

- 1

- 3

- 2

- Changes to Gehrman-Glover:
-  ordinary DGLAP limit
-  First parton shower with systematic possibility for variation 		  (+ note: variation absorbed by matching!)

-  Gehrmann-De Ridder, Gehrmann, Glover PLB612(2005)49

## Antenna Showers

**Δ****(Q****1****2****,Q****2****2****;s,Q****0****2****)**

- Δ is the probability for no branching
- in the region bounded by Q1 > QR(sa,sb;s) > Q2
- for a dipole with mass s,
- restricted to the perturbatively resolved region QH(sa,sb;s)  > Q0 :

## Antenna Showers

-  **Sudakov Form Factor integral over PS:**

-  Can also be integrated numerically  easy to implement any Antenna function and any Phase space ordering.
- Sudakovs then splined, so only need to evaluate once  fast.

-  Compact analytical solutions (here without Cmn pieces)

## Antenna Branching Markov Chain

- Given a set of N momenta {pi}N
- colour-ordered dipole masses {sij}N,
- defined at the factorization scale QN,
- the shower-improved distribution of the observable O is:

## Perturbative expansion for some observable J, 			dm=0dm	;	dm= dm|M|2(J-J(k1,k2,...,km))
Assume 
We calculate some Matrix Elements d0 , d1 , ... dn  (w or w/o loops) 
And we have the shower approximation dn+1 ~ Tn n+1 dn 
A ‘best guess’ cross section for J is then: 				dd0 + d1 + ... + dn (1 + Tn n+1 + Tn n+1Tn+1n+2 + ... )	            	dd0 + d1 + ... + dn Sn	; Sn= 1 + Tn n+1 Sn+1
Since the Tn n+1 have the correct singularities the logarithmically enhanced terms are correctly resummed
Now reorder this series in a useful way ...

- Assume
- We calculate some Matrix Elements d0 , d1 , ... dn  (w or w/o loops)
- And we have the shower approximation dn+1 ~ Tn n+1 dn
- A ‘best guess’ cross section for J is then: 				dd0 + d1 + ... + dn (1 + Tn n+1 + Tn n+1Tn+1n+2 + ... )	            	dd0 + d1 + ... + dn Sn	; Sn= 1 + Tn n+1 Sn+1
- Since the Tn n+1 have the correct singularities the logarithmically enhanced terms are correctly resummed
- Now reorder this series in a useful way ...

- Matching to Results obtained in Fixed Order Perturbation Theory

## Assume we know Hgg and Hggg. Then reorder:

d~ dgg + dggg Sggg					  	= Sggdgg + Sggg (dggg – Tgggggdgg)			= Sggdgg + Sggg dggg		(generalises to n gluons) 
I.e shower off gg and subtracted ggg matrix element. 
Double counting avoided since singularities (shower) subtracted in dggg .
The shower kernels, Tgg, are precisely the singular subtraction terms used in HO perturbative calculations. Due to the simple forms of their analytical integrals we have so far taken the Gehrman-Glover “global subtraction” ones, but we plan to extend this to sector decomposition as well.

- d~ dgg + dggg Sggg					  	= Sggdgg + Sggg (dggg – Tgggggdgg)			= Sggdgg + Sggg dggg		(generalises to n gluons)
- I.e shower off gg and subtracted ggg matrix element.
- Double counting avoided since singularities (shower) subtracted in dggg .
- The shower kernels, Tgg, are precisely the singular subtraction terms used in HO perturbative calculations. Due to the simple forms of their analytical integrals we have so far taken the Gehrman-Glover “global subtraction” ones, but we plan to extend this to sector decomposition as well.

- Use 1=Sn-Tnn+1Sn+1

- Subtraction Example: “H” gluons

## Antenna ShowersControlling the Uncertainties

| Antenna Function | IR singularities plus arbitrary finite terms
|A(sa,sb;s)|2    
 (a.k.a. Radiation Function, a.k.a. Splitting Kernel) |
| --- | --- |
| Evolution / Ordering Variable | Perturbative factorization / resolution scale
Type I : QR ≡ 2 mamb/m = 2pT;ARIADNE
Type II : QR = 2mmin = 2 min(ma,mb) |
| Renormalization Scheme | Scale choice as function of invariants
μR2(sa,sb;s) = s  or
μR2(sa,sb;s) = QR2  |
| Kinematics
nn+1 | p’jμ(piμ,sa,sb,φ,ψ,θ) ; j[1,n+1] ; i[1,n] 
φ: “roll”, ψ: “heading”, θ: “pitch” (θ=0 fixed by dipole axis) |
| Hadronization Cutoff | QH(sa,sb;s) < Q0 : “unresolved”
Can be chosen independent of QR  “universal” |

- Antenna ShowersControlling the Uncertainties

- Systematically improved by matching

- Systematically improved by matching (?)

- Systematically improved by loops

- Systematically
- Improved by
- ???

## Each phase space ordering gives a different Sudakov 
Cij gives “error band” for each Sudakov
NB: Negative Cij can lead to negative antenna function (unphysical)
Dealt with by explicit zeroing in numerical integration

(+variation of Cij canceled (to fixed order) by matching)

- Cij gives “error band” for each Sudakov
- NB: Negative Cij can lead to negative antenna function (unphysical)
- Dealt with by explicit zeroing in numerical integration
- (+variation of Cij canceled (to fixed order) by matching)

- Type 2

- Type 1
- C00 = -1, 0, 1

- analytic

-  **Sudakov Form Factor**

**Δ****(Q****1****2****,Q****2****2****;s,Q****0****2****)**

- Antenna ShowersControlling the Uncertainties

## Antenna ShowersControlling the Uncertainties

| Antenna Function | IR singularities plus arbitrary finite terms
|A(sa,sb;s)|2    
 (a.k.a. Radiation Function, a.k.a. Splitting Kernel) |
| --- | --- |

- Antenna ShowersControlling the Uncertainties

- Systematically improved by matching

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

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

- First Branching

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

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

## Antenna ShowersControlling the Uncertainties

| Evolution / Ordering Variable | Perturbative factorization / resolution scale
Type I : QR ≡ 2 mamb/m = 2pT;ARIADNE
Type II : QR = 2mmin = 2 min(ma,mb) |
| --- | --- |

- Antenna ShowersControlling the Uncertainties

- Systematically improved by matching (?)

- VINCIA 0.008
- Type I (pT-ordering)
- More Collinear, Less Soft

- VINCIA 0.008
- Type II (m2-ordering)
- Less Collinear, More Soft

- We need to study better how this ambiguity is addressed by matching

## Introduction
Collider energy scales
QCD in hadron collisains
Hard Scattering and Bremsstrahlung 
Monte Carlo Event Generators
Matching with QCD antennae
The Underlying Event
Perturbative definition
Non-perturbative aspects: strings and colour

- Collider energy scales
- QCD in hadron collisains
- Hard Scattering and Bremsstrahlung
- Monte Carlo Event Generators
- Matching with QCD antennae
- _The Underlying Event_
- Perturbative definition
- Non-perturbative aspects: strings and colour

- Overview

## 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) phenomena?
Bose-Einstein Correlations
Non-perturbative gluon exchanges 
String-string interactions
Interactions with “background” vacuum

- 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) phenomena?
- Bose-Einstein Correlations
- Non-perturbative gluon exchanges
- String-string interactions
- Interactions with “background” vacuum

- PS: I’m not talking about *pile-up* here

- Additional Sources of Particle Production

- e.g. 44, 3 3, 32

## Balancing Minijets

- Look for additional balancing jet pairs “under” the hard interaction.
- Several studies performed, most recently by Rick Field at CDF  ‘lumpiness’ in the underlying event.

- example (old Run I)

- angle between 2 ‘best-balancing’ pairs

- CDF, PRD 56 (1997) 3811

## Interleaved Evolution

-  Underlying Event
- (note: interactions correllated in colour: hadronization _not_ independent)

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

- multiparton
- PDFs derived
- from sum rules

- Beam remnants
- Fermi motion /
- primordial kT

- Fixed order
- matrix elements

- pT-ordered PS
- matched to ME
- for W/Z/H/G + jet

- perturbative
- “intertwining”?

## MC4LHC summer 2003: 
“Complete” model
Still old showers
not yet interleaved
One ambiguity:

Comparisons with data were encouraging
Except for one distribution: 

- MC4LHC summer 2003:
- “Complete” model
- Still old showers
- not yet interleaved
- One ambiguity:
- Comparisons with data were encouraging
- Except for one distribution:

- Does it work?

- “Tune A” ~ data

- “By the failure of even the Δλ ordering of the colour lines in the initial state to describe the <pT>(Nch) distribution, it appears that the colour flow in physical events cannot be correctly described by merely arranging the colour lines present in the initial state.”

- Sjöstrand & PS : JHEP03(2004)053

## Fragmentation
Nch ~ log(mstring)
More strings  more hadrons, but average pT stays same
Flat <pT>(Nch) spectrum ~ ‘uncorrellated’ underlying event

Space-time “Area” of string system is large ~ potential energy

- Fragmentation
- Nch ~ log(mstring)
- More strings  more hadrons, but _average_ pT stays same
- Flat <pT>(Nch) spectrum ~ ‘uncorrellated’ underlying event
- Space-time “Area” of string system is large ~ potential energy

- But if MPI interactions correlated in colour
- each scattering does not produce an independent string,
- average pT  not flat
- Multiplicity vs pT correllation probes color correllations!
- What’s so special about Tune A anyway?
- It (and all other realistic ‘tunes’ made) need to go to the very most extreme end of the parameter range, with ~100% color correlation in final state.

- Underlying Event and Color

- Sjöstrand & v Zijl  : Phys.Rev.D36:2019,1987  “Old” Pythia model

## Searched for at LEP 
Major source of W mass uncertainty
Most aggressive scenarios excluded
But effect still largely uncertain  P ~ 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 
More prominent in hadron-hadron collisions?
What is <pT>(Nch) telling us?
Top mass? 
Implications for LHC?

Problem: existing models only for e+e- WW

- Color Reconnections

- Searched for at LEP
- Major source of W mass uncertainty
- Most aggressive scenarios excluded
- But effect still largely uncertain  P ~ 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
- More prominent in hadron-hadron collisions?
- What is <pT>(Nch) telling us?
- Top mass?
- Implications for LHC?
- Problem: existing models only for e+e- WW

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

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

## 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:

String formation for interacting string pieces determined by annealing-like minimization of  ‘Lambda measure’ (~string length~log(m)~N)
Similar to area law for fundamental strings

 good enough for order-of-magnitude

- 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:
- String formation for interacting string pieces determined by annealing-like minimization of  ‘Lambda measure’ (~string length~log(m)~N)
- Similar to area law for fundamental strings
-  good enough for order-of-magnitude

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

- cf. e.g. J. Rathsman, PLB452(1999)364

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

- First Results

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

## D. Wicke (DØ)

- D. Wicke (DØ)

## Work Remaining ... 

- To get high precision at hadron colliders:
- _Solve more of QCD_
- Over all of phase space, including
- non-perturbative region
- phase space borders
- more than 2 incoming partons
- _Control it more exactly_
- reliable uncertainties
- systematically improvable

## Low-pT phenomena

- LEP 
- Final state fragmentation properties  fragmentation models (cluster and string) well constrained
- E.g. colour reconnections ruled out in WW (to ~ 10%)
- Measurements at hadron colliders
- Different vacuum, colour in initial state  ???
- Underlying Event and Beam Remnants
- Fermi motion, “intrinsic kT”
- Lots of min-bias. Fragmentation tails  fakes!

- Example Problem:
- Can non-perturbative effects
- impact top mass measurement?

- Sjostrand+Khoze (1994), Sandhoff+PS (2005), PS+Wicke (in prep.)

## (Remnant PDFs)

- Used to select flavor and x for each parton-parton interaction, and for the interleaved evolution of the initial-state shower. Dynamically evaluated.

- Sjöstrand & PS : JHEP03(2004)053

## Scales involved in calculating properties of high energy collision events

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

- 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