sigma (mb)

(1)

Syddansk Universitet Odense, Denmark 27 - 28 March 2008

Event Generators for LHC

Torbj ¨orn Sj ¨ostrand

Lund University

1. (yesterday) Introduction and Overview;

Parton Showers; Matching Issues 2. (today) Multiple Interactions;

Hadronization; Generators & Conclusions

(2)

Multiple Interactions

(3)

What is multiple interactions?

Cross section for 2 → 2 interactions is dominated by t-channel gluon exchange, so diverges like dσ/dp

²_⊥

≈ 1/p

⁴_⊥

for p

_⊥

→ 0.

integrate QCD 2 → 2

qq

⁰

→ qq

⁰

qq → q

⁰

q

⁰

qq → gg qg → qg gg → gg gg → qq

with CTEQ 5L PDF’s

_0.01

0.1 1 10 100 1000 10000

0 5 10 15 20 25 30 35 40 45 50

sigma (mb)

pTmin (GeV)

Integrated cross section above pTmin for pp at 14 TeV jet cross section total cross section

(4)

So σ _int (p _⊥min ) > σ _tot for p _⊥min ∼ 5 ^< GeV

Half a solution: many interactions per event σ

_tot

=

∞ X

n=0

σ

_n

σ

_int

=

∞ X

n=0

n σ

_n

σ

_int

> σ

_tot

⇐⇒ hni > 1

n P

_n

hni = 2

0 1 2 3 4 5 6 7

If interactions occur independently then Poissonian statistics

P

_n

= hni

ⁿ

n! e

^−hni

but energy–momentum conservation

⇒ large n suppressed

(5)

perturbative QCD not valid at small p

_⊥

since q, g not asymptotic states (confinement!).

Naively breakdown at p

_⊥min

' ¯ h

r

p

≈ 0.2 GeV · fm

0.7 fm ≈ 0.3 GeV ' Λ

_QCD

. . . but better replace r

p

by (unknown) colour screening length d in hadron

r r

d resolved

r r

d

screened

λ ∼ 1/p

_⊥

(6)

so modify dˆ σ

dp

²_⊥

∝ α

²_s

(p

²_⊥

)

p

⁴_⊥

→ α

²_s

(p

²_⊥

)

p

⁴_⊥

θ (p

_⊥

− p

_⊥min

) (simpler)

or → α

²_s

(p

²_⊥0

+ p

²_⊥

)

(p

²_⊥0

+ p

²_⊥

)

²

(more physical)

p

²_⊥

dˆ σ/dp

²_⊥

0 where p

_⊥min

or p

_⊥0

are free parameters, empirically of order 2 GeV

Typically 2 – 3 interactions/event at the Tevatron, 4 – 5 at the LHC, but may be more

in “interesting” high-p

_⊥

ones.

(7)

Modelling multiple interactions

T. Sj ¨ ostrand, M. van Zijl, PRD36 (1987) 2019: first model(s) for event properties based on perturbative multiple interactions (1) Simple scenario:

• Sharp cut-off at p

_⊥min

main free parameter

• Is only a model for nondiffractive events, i.e. for σ

_nd

' (2/3)σ

_tot

• Average number of interactions is hni = σ

_int

(p

_⊥min

)/σ

_nd

• Interactions occur almost independently, i.e.

Poissonian statistics P

_n

= hni

ⁿ

e

^−hni

/n!

with fraction P

₀

= e

^−hni

pure low-p

_⊥

events

• Interactions generated in ordered sequence p

_⊥1

> p

_⊥2

> p

_⊥3

> . . . by “Sudakov” trick (what happens “first”?)

dP

dp

_⊥i

= 1 σ

_nd

dσ

dp

_⊥

exp

"

−

Z _p

⊥(i−1)

p_⊥

1 σ

_nd

dσ

dp

⁰_⊥

dp

⁰_⊥

#

• Momentum conservation in PDF’s ⇒ P

_n

narrower than Poissonian

• Simplify after first interaction: only gg or qq outgoing, no showers, . . .

(8)

(2) More sophisticated scenario:

• Smooth turn-off at p

_⊥0

scale

• Require ≥ 1 interaction in an event

• Hadrons are extended,

e.g. double Gaussian (“hot spots”):

ρ

_matter

(r) = N

₁

exp − r

²

r

₁²

!

+ N

₂

exp − r

²

r

₂²

!

where r

₂

6= r

₁

represents “hot spots”

• Events are distributed in impact parameter b

• Overlap of hadrons during collision O(b) =

Z

d

³

x _{dt ρ}

^boosted_1,matter

( x _{, t)ρ}

^boosted_2,matter

( x _{, t)}

• Average activity at b proportional to O(b)

⇒ central collisions normally more active

⇒ P

_n

broader than Poissonian

• More time-consuming (b, p

_⊥

) generation

• Need for simplifications remains

0.01 0.1 1 10

0 0.5 1 1.5 2 2.5

ρ(r) total r₁ = 1 r₂ = 0.4

p p

b

b hni

1

(9)

(3) HERWIG

Soft Underlying Event (SUE), based on UA5 Monte Carlo

l

y

v v

• Distribute a (∼ negative binomial) number of clusters independently in rapidity and transverse momentum according to parametrization/extrapolation of data

• modify for overall energy/momentum/flavour conservation

• no minijets; correlations only by cluster decays (4) Jimmy (HERWIG add-on)

• similar to PYTHIA (2) above; but details different

• matter profile by electromagnetic form factor

• no p

_⊥

-ordering of emissions, no rescaling of PDF:

abrupt stop when (if) run out of energy (5) Phojet/DTUjet

• comes from “historical” tradition of soft physics

of “cut Pomerons” ≈ p

_⊥

→ 0 limit of multiple interactions

• extended also to “hard” interactions similarly to PYTHIA

(10)

without multiple interactions

(11)

with multiple interactions

(12)

Evidence for multiple interactions

• Width of multiplicity distribution: UA5, E735 (previous slides)

• Forward–backward correlations: UA5 (previous slides)

• Minijet rates: UA1

No. jets UA1 no MI simple double

(%) Gaussian

1 9.96 14.30 11.51 8.88

2 3.45 2.45 2.45 2.67

3 1.12 0.22 0.32 0.74

4 0.22 0.01 0.04 0.25

5 0.05 0.00 0.00 0.07

(13)

• Direct observation: AFS, (UA2,) CDF

Order 4 jets p

_⊥1

_> p

_⊥2

_> p

_⊥3

_> p

_⊥4

and define ϕ as angle between p

_⊥1

− p

_⊥2

and p

_⊥3

− p

_⊥4

Double Parton Scattering

1 2

3

4 | p

_⊥1

+ p

_⊥2

_{| ≈ 0}

| p

_⊥3

+ p

_⊥4

_{| ≈ 0} dσ/dϕ flat

Double BremsStrahlung

1 2

3 4

| p

_⊥1

+ p

_⊥2

_{| 0}

| p

_⊥3

+ p

_⊥4

_{| 0} dσ/dϕ peaked at ϕ ≈ 0 AFS 4-jet analysis (pp at 63 GeV);

double bremsstrahlung subtracted:

observed 6 in arbitrary units

no MI 0

simple MI 1

double Gaussian 3.7

(14)

CDF 3-jet + prompt photon analysis Yellow region = double parton scattering (DPS) The rest =

PYTHIA showers

σ

_DPS

= σ

_A

σ

_B

σ

_eff

for A 6= B =⇒ σ

_eff

= 14.5 ± 1.7

^+1.7_−2.3

mb

Strong enhancement relative to naive expectations!

(15)

• Jet pedestal effect: UA1, H1, CDF

Events with hard scale (jet, W/Z, . . . ) have more underlying activity!

Events with n interactions have n chances that one of them is hard, so “trigger bias”: hard scale ⇒ central collision

⇒ more interactions ⇒ larger underlying activity.

Centrality effect saturates at p

_⊥hard

∼ 10 GeV.

Studied in detail by Rick Field, comparing with CDF data:

Rick Field December 1, 2004

TeV4LHC Meeting Page 4 of 27

“MAX/MIN Transverse” Densities

x Define the MAX and MIN “transverse” regions on an event-by-event basis with MAX (MIN) having the largest (smallest) density.

x The “transMIN” region is very sensitive to the “beam-beam remnant” and multiple parton interaction components of the “underlying event”.

x The difference, “transMAX” minus “transMIN”, is very sensitive to the “hard scattering” component of the “underlying event” (i.e. hard initial and final- state radiation).

Jet #1 Direction 'I

“Toward”

“TransMAX” “TransMIN”

“Away”

Jet #1 Direction

'I

“TransMAX” “TransMIN”

“Toward”

“Away”

“Toward-Side” Jet

“Away-Side” Jet Jet #3

“TransMIN” very sensitive to the “beam-beam remnants”!

(16)

MC Tools for the LHC CERN July 31, 2003

Rick Field - Florida/CDF Page 24

Old PYTHIA default (more initial-state radiation)

0.5 0.5

PARP(83)

0.4 0.4

PARP(84)

0.25 0.25

PARP(90)

0.95 1.0

PARP(86)

1.8 TeV 1.8 TeV

PARP(89)

4.0 0.9 2.0 GeV

4 1 Tune A

1.0 PARP(67)

1.0 PARP(85)

1.9 GeV PARP(82)

4 MSTP(82)

1 MSTP(81)

Tune B Parameter

Tuned PYTHIA 6.206 Tuned PYTHIA 6.206

¨

Plot shows the “Transverse” charged particle density versus P_T(chgjet#1) compared to the QCD hard

scattering predictions of two tuned versions of

PYTHIA 6.206 (CTEQ5L, Set B (PARP(67)=1) and Set A (PARP(67)=4)).

"Transverse" Charged Particle Density: dN/dKdI

0.00 0.25 0.50 0.75 1.00

0 5 10 15 20 25 30 35 40 45 50

PT(charged jet#1) (GeV/c)

"Transverse" Charged Density

1.8 TeV |K|<1.0 PT>0.5 GeV

CDF Preliminary

data uncorrected theory corrected

CTEQ5L

PYTHIA 6.206 (Set A) PARP(67)=4

PYTHIA 6.206 (Set B) PARP(67)=1

0.5 0.5

PARP(83)

0.4 0.4

PARP(84)

0.25 0.25

PARP(90)

0.95 1.0

PARP(86)

1.8 TeV 1.8 TeV

PARP(89)

4.0 0.9 2.0 GeV

4 1 Tune A

1.0 PARP(67)

1.0 PARP(85)

1.9 GeV PARP(82)

4 MSTP(82)

1 MSTP(81)

Tune B Parameter

PYTHIA 6.206 CTEQ5L

New PYTHIA default (less initial-state radiation)

Double Gaussian

Old PYTHIA default (more initial-state radiation)

Tune A CDF Run 2 Default!

(17)

MC Tools for the LHC CERN July 31, 2003

Tuned PYTHIA 6.206 Tuned PYTHIA 6.206

“Transverse” P

“Transverse” P _T _T Distribution Distribution

"Transverse" Charged Particle Density: dN/dKdI

0.00 0.25 0.50 0.75 1.00

0 5 10 15 20 25 30 35 40 45 50

PT(charged jet#1) (GeV/c)

"Transverse" Charged Density

1.8 TeV |K|<1.0 PT>0.5 GeV CDF Preliminary

CTEQ5L

PYTHIA 6.206 (Set A) PARP(67)=4

PYTHIA 6.206 (Set B) PARP(67)=1

PARP(67)=4.0 (old default) is favored over PARP(67)=1.0 (new default)!

P_T(charged jet#1) > 30 GeV/c

"Transverse" Charged Particle Density

1.0E-06 1.0E-05 1.0E-04 1.0E-03 1.0E-02 1.0E-01 1.0E+00

0 2 4 6 8 10 12 14

PT(charged) (GeV/c) Charged Density dN/dKdIdPT (1/GeV/c)

CDF Data

1.8 TeV |K|<1 PT>0.5 GeV/c PT(chgjet#1) > 5 GeV/c

PT(chgjet#1) > 30 GeV/c

PYTHIA 6.206 Set A PARP(67)=4

PYTHIA 6.206 Set B PARP(67)=1

¨

Compares the average “transverse” charge particle density (|K|<1, P_T>0.5 GeV) versus P_T(charged jet#1) and the P_T distribution of the “transverse” density, dN_chg/dKdIdP_T with the QCD Monte-Carlo predictions of two tuned versions of PYTHIA 6.206 (P_T(hard) > 0, CTEQ5L, Set B (PARP(67)=1) and Set A (PARP(67)=4)).

(18)

Rick Field December 1, 2004

TeV4LHC Meeting Page 5 of 27

Leading Jet: “MAX & MIN Transverse” Densities

PYTHIA Tune A HERWIG

"MAX/MIN Transverse" Charge Density: dN/dKdI

0.0 0.4 0.8 1.2 1.6

0 50 100 150 200 250

ET(jet#1) (GeV)

"Transverse" Charge Density CDF Preliminary

data uncorrected theory + CDFSIM

PYTHIA Tune A 1.96 TeV

Charged Particles (|K|<1.0, PT>0.5 GeV/c)

"MAX"

"MIN"

"AVE"

Leading Jet

"MAX/MIN Transverse" Charge Density: dN/dKdI

0.0 0.4 0.8 1.2 1.6

0 50 100 150 200 250

ET(jet#1) (GeV)

"Transverse" Charge Density CDF Preliminary

HERWIG 1.96 TeV

"MAX"

"MIN"

"AVE"

Leading Jet

"MAX/MIN Transverse" PTsum Density: dPT/dKdI

0.0 0.5 1.0 1.5 2.0 2.5

0 50 100 150 200 250

ET(jet#1) (GeV)

"Transverse" PTsum Density (GeV/c)

CDF Preliminary

PYTHIA Tune A 1.96 TeV

"MAX"

"MIN"

"AVE"

Leading Jet

"MAX/MIN Transverse" PTsum Density: dPT/dKdI

0.0 0.5 1.0 1.5 2.0 2.5

0 50 100 150 200 250

ET(jet#1) (GeV)

"Transverse" PTsum Density (GeV/c)

CDF Preliminary

HERWIG 1.96 TeV

"MAX"

"MIN"

"AVE"

Leading Jet

Charged particle density and PTsum density for “leading jet” events versus ET(jet#1) for PYTHIA Tune A and HERWIG.

(19)

KITP Collider Workshop February 17, 2004

Back Back - - to to - - Back Back “Associated” “Associated”

Charged Particle Densities Charged Particle Densities

'I

Jet#1 Region

PTmaxT Direction

Jet#2 Region

¨

Shows the 'I dependence of the “associated” charged particle density, dNchg/dKdI, p_T > 0.5 GeV/c, |K| < 1, PTmaxT > 2.0 GeV/c (not including PTmaxT) relative to PTmaxT (rotated to 180^o) and the charged particle density, dN^chg/dKdI, p_T > 0.5 GeV/c, |K| < 1, relative to jet#1 (rotated to 270^o) for “back-to-back events” with 30 < E_T(jet#1) < 70 GeV.

Jet #1 Direction 'I

“Toward”

“Transverse” “Transverse”

“Away”

Jet #2 Direction

Charged Particle Density: dN/dKdI

2

6 10 14

18 22

26 30

34 38

42 46

50 54

58

62

66

70

74

78

82

86

90

94

98

102

106

110

114

118

122

126 130

134 138 142 146 150 154 158 162 166 174 170 178 182 190 186 194 198 202 206 210 214 218 222 226 230 234 238 242 246 250 254 258 262 266 270 274 278 282 286

290 294

298 302

306 310

314 318

322 326

330 334

338 342

346 350 354 358

CDF Preliminary

data uncorrected

30 < ET(jet#1) < 70 GeV Back-to-Back

"Transverse"

Region "Transverse"

Region Jet#1

Associated Density PTmaxT > 2 GeV/c

(not included) PTmaxT

Polar Plot

“Back-to-Back”

“associated” density

“Back-to-Back”

charge density

0.5

1.0

1.5

2.0

(20)

“ “ Associated” Charge Density Associated” Charge Density PYTHIA Tune A

PYTHIA Tune A vs vs HERWIG HERWIG

Associated Particle Density: dN/dKdI

0.1 1.0 10.0

0 30 60 90 120 150 180 210 240 270 300 330 360

'I (degrees)

Associated Particle Density

PTmaxT > 2.0 GeV/c PY Tune A

Back-to-Back 30 < ET(jet#1) < 70 GeV Charged Particles

(|K|<1.0, PT>0.5 GeV/c)

PTmaxT

CDF Preliminary

data uncorrected

theory + CDFSIM PTmaxT not included

"Jet#1"

Region

Associated Particle Density: dN/dKdI

0.1 1.0 10.0

0 30 60 90 120 150 180 210 240 270 300 330 360

'I (degrees)

Associated Particle Density

PTmaxT > 2.0 GeV/c HERWIG

Back-to-Back 30 < ET(jet#1) < 70 GeV Charged Particles

(|K|<1.0, PT>0.5 GeV/c)

PTmaxT

CDF Preliminary

data uncorrected

theory + CDFSIM PTmaxT not included

"Jet#1"

Region

Data - Theory: Associated Particle Density dN/dKdI

-1.6 -0.8 0.0 0.8 1.6

0 30 60 90 120 150 180 210 240 270 300 330 360

'I (degrees)

Data - Theory

CDF Preliminary

Back-to-Back 30 < ET(jet#1) < 70 GeV PYTHIA Tune A

PTmaxT "Jet#1"

Region PTmaxT > 2.0 GeV/c (not included)

Data - Theory: Associated Particle Density dN/dKdI

-1.0 -0.5 0.0 0.5 1.0

0 30 60 90 120 150 180 210 240 270 300 330 360

'I (degrees)

Data - Theory

CDF Preliminary

Back-to-Back 30 < ET(jet#1) < 70 GeV HERWIG

PTmaxT "Jet#1"

Region PTmaxT > 2.0 GeV/c (not included)

For PTmaxT > 2.0 GeV both PYTHIA and HERWIG produce

slightly too many “associated”

particles in the direction of PTmaxT!

But HERWIG (without multiple parton interactions) produces

too few particles in the direction opposite of PTmaxT!

PTmaxT > 2 GeV/c

(21)

Colour correlations

hp

_⊥

i(n

_ch

) is very sensitive to colour flow

p p

long strings to remnants ⇒ much n

_ch

/interaction ⇒ hp

_⊥

i(n

_ch

) ∼ flat

p p

short strings (more central) ⇒ less

n

_ch

/interaction ⇒ hp

_⊥

i(n

_ch

) rising

(22)

“ “ Transverse” < Transverse” < p p _T _T > versus > versus

“Transverse”

“Transverse” N N chg chg

Jet #1 Direction 'I

“Toward”

“Away”

Jet #1 Direction 'I

“Toward”

“Away”

Jet #2 Direction

¨

Shows <p_T> versus N^chg in the “transverse” region (p_T > 0.5 GeV/c, |K| < 1) for

“Leading Jet” and “Back-to-Back” events with 30 < E_T(jet#1) < 70 GeV compared with

“min-bias” collisions.

“Leading Jet”

“Back-to-Back”

¨

Look at the <p_T> of particles in the “transverse” region (p_T > 0.5 GeV/c, |K| < 1) versus the number of particles in the “transverse” region: <p_T> vs N^chg.

Min-Bias

"Transverse" Average PT versus Nchg

0.5 1.0 1.5 2.0

0 2 4 6 8 10 12 14 16 18 20 22

Number of Charged Particles

Average PT (GeV/c)

CDF Run 2 Preliminary

Charged Particles (|K|<1.0, PT>0.5 GeV/c) PYTHIA Tune A 1.96 TeV

Min-Bias

Leading Jet 30 < ET(jet#1) < 70 GeV

Back-to-Back 30 < ET(jet#1) < 70 GeV

(23)

Initiators and Remnants

p

g u

s s u d

initiators:

in to hard interaction

beam remnants

Need to assign:

• correlated flavours

• correlated x

_i

= p

_zi

/p

_ztot

• correlated primordial k

_⊥i

• correlated colours

• correlated showers

• PDF after preceding MI/ISR activity:

0) Squeeze range 0 < x < 1 into 0 < x < 1 −

^P

x

_i

(ISR: i 6= i

_current

) 1) Valence quarks: scale down by number already kicked out

2) Introduce companion quark q/q to each kicked-out sea quark q/q, with x based on assumed g → qq splitting

3) Gluon and other sea: rescale for total momentum conservation

(24)

Interleaved Multiple Interactions

interaction number

p

_⊥

hard int.

1

mult. int.

2

mult. int.

3

mult int.

4

p_⊥max

p_⊥min

p_⊥1

p_⊥2

p_⊥3

p_⊥4

p_⊥23

ISR

p⁰_⊥1

(25)

Extrapolation to LHC

Energy dependence of p

_⊥min

and p

_⊥0

:

Larger collision energy

⇒ probe parton ( ≈ gluon) density at smaller x

⇒ smaller colour screening length d

⇒ larger p

_⊥min

or p

_⊥0

Post-HERA PDF fits steeper at small x

⇒ stronger energy dependence

Current PYTHIA default (Tune A, old model), tied to CTEQ 5L, is

p

_⊥min

(s) = 2.0 GeV s

(1.8 TeV)

²

!_0.08

(26)

5^thNovember 2004 Minimum-bias and the Underlying Event at the LHC

A. M. Moraes

LHC predictions: pp collisions at ¥s = 14 TeV

0 2 4 6 8 10

10² 10³ 10⁴ 10⁵

PYTHIA6.214 - tuned PYTHIA6.214 - default PHOJET1.12

pp interactions-

UA5 and CDF data

dN chg/dȘatȘ=0

¥s (GeV)

•PYTHIAmodels favour ln²(s);

•PHOJETsuggests a ln(s)dependence.

LHC

2 4 6 8 10 12

0 10 20 30 40 50

CDF data

PYTHIA6.214 - tuned

PHOJET1.12

LHC

Tevatron

x1.5 x 3

dN_chg/dȘ ~ 30

dN_chg/dȘ ~ 15

Central Region

(min-bias dN_chg/dȘ ~ 7)

Transverse < N chg>

P_t(leading jet in GeV)

(27)

5^thNovember 2004 Minimum-bias and the Underlying Event at the LHC

A. M. Moraes

LHC predictions: JIMMY4.1 Tunings A and B vs.

PYTHIA6.214 – ATLAS Tuning (DC2)

5 10 15 20

0 10 20 30 40 50

CDF data

JIMMY4.1 - Tuning A JIMMY4.1 - Tuning B

PYTHIA6.214 - ATLAS Tuning

Transverse < N chg>

P

_t

(leading jet in GeV) Tevatron LHC

x 4

x 5

x 3

(28)

18 PTJIM=4.9

PTJIM=4.9

= 2.8

= 2.8 x (14 / 1.8)x (14 / 1.8)^0.27^0.27

x3

x2.7 LHC

Tevatron

•energy dependent PTJIM •energy dependent PTJIM generates UE predictions generates UE predictions similar to the ones

similar to the ones generated by PYTHIA6.2 generated by PYTHIA6.2 –– ATLAS.

ATLAS.

UE tunings: Pythia vs. Jimmy

(29)

Hadronization/Fragmentation models

Perturbative → nonperturbative =⇒ not calculable from first principles!

Model building = ideology + “cookbook”

Common approaches:

1) String Fragmentation (most ideological)

2) Cluster Fragmentation (simplest?)

3) Independent Fragmentation (most cookbook)

4) Local Parton–Hadron Duality (limited applicability)

Best studied in

e

⁺

e

⁻

→ γ

^∗

/Z

⁰

→ qq

DELPHI Interactive Analysis

Run: 39265 Evt: 4479

Beam: 45.6 GeV Proc: 4-May-1994

DAS : 5-Jul-1993 14:16:48 Scan: 3-Jun-1994

TD TE TS TK TV ST PA

Act

Deact 95 (145)

0 ( 0)

173 (204)

0 ( 20)

0 ( 0)

38 ( 38)

0 ( 42)

0 ( 0)

X Y Z

(30)

The Lund String Model

In QED, field lines go all the way to infinity

+

...

.... ... ... ... ...

...

.. ...

...

... ...

...

....

...

....

...

....

...

.... ...

...

... ...

...

. ... ... ...

... ...

...

... ...

...

... ...

...

....

...

. ...

...

− +

since photons cannot interact with each other.

Potential is simply additive:

V ( x _{) ∝}

^X

i

1 | x − x

_i

|

(31)

In QCD, for large charge separation, field lines seem to be compressed to tubelike region(s) ⇒ string(s)

r r

...

....

.. . ... ... ... ... ... ... ... ...

...

... ... ... ...

... ...

...

... ... ... ... ...

... .... ... ... ...^...^{.... ...}...^{.... ...}.... ..............................................

...... ...... .... .................

...

......

.......

......

...

.... ...

...

............................................

. ...

...

......

.......

...

.............

...

.................

... ... ... ... ... ... ...

...

. ...

...

... ... ... ... ... ... ... ...

by self-interactions among soft gluons in the “vacuum”.

(Non-trivial ground state with quark and gluon “condensates”.

Analogy: vortex lines in type II superconductor) Gives linear confinement with string tension:

F (r) ≈ const = κ ≈ 1 GeV/fm ⇐⇒ V (r) ≈ κr Separation of transverse and longitudinal degrees of freedom

⇒ simple description as 1+1-dimensional object – string –

with Lorentz invariant formalism

(32)

Linear confimenent confirmed e.g. by quenched lattice QCD

MC for LHC 3 Mike Seymour

Interquark potential

Can measure from quarkonia spectra:

or from lattice QCD:

String tension

V (r)

r linear part

Coulomb part

total

V (r) ≈ − 4 3

α

_s

r + κr ≈ − 0.13

r + r (for α

_s

≈ 0.5, r in fm and V in GeV)

V (0.4 fm) ≈ 0: Coulomb important for internal structure of hadrons,

not for particle production (?)

(33)

Real world (??, or at least unquenched lattice QCD)

=⇒ nonperturbative string breakings gg . . . → qq V (r)

r quenched QCD

full QCD

Coulomb part

simplified colour representation:

r r

...

... ... ... ...

⇓

r r

...

... ... ... ...

r r

⇓

r r

. ...

... ... ... ... ...

r r

...

... ...

(34)

Repeat for large system ⇒ Lund model which neglects Coulomb part:

dE dz

=

dp

_z

dz

=

dE dt

=

dp

_z

dt

= κ

Motion of quarks and antiquarks in a qq system:

z q t

q

gives simple but powerful picture of hadron production

(with extensions to massive quarks, baryons, . . . )

(35)

How does the string break?

q q

⁰

q

⁰

q

m

_⊥q0

= 0

q q

⁰

q

⁰

q

d = m

_⊥q

/κ m

_⊥q0

> 0

String breaking modelled by tunneling:

P ∝ exp





− πm

²_⊥q

κ





= exp





− πp

²_⊥q

κ





exp − πm

²_q

κ

!

1) common Gaussian p

_⊥

spectrum

2) suppression of heavy quarks uu : dd : ss : cc ≈ 1 : 1 : 0.3 : 10

⁻¹¹

3) diquark ∼ antiquark ⇒ simple model for baryon production

Hadron composition also depends on spin probabilities, hadronic wave functions, phase space, more complicated baryon production, . . .

⇒ “moderate” predictivity (many parameters!)

(36)

Fragmentation starts in the middle and spreads outwards:

z q t

q m

²_⊥

m

²_⊥

2 1

but breakup vertices causally disconnected

⇒ can proceed in arbitrary order

⇒ left–right symmetry

P(1, 2) = P(1) × P(1 → 2)

= P(2) × P(2 → 1)

⇒ Lund symmetric fragmentation function

f (z) ∝ (1 − z)

^a

exp(−bm

²_⊥

/z)/z

₀

0.5 1 1.5 2 2.5 3

0 0.2 0.4 0.6 0.8 1 f(z), a = 0.5, b= 0.7

m_T² = 0.25 m_T² = 1 m_T² = 4

(37)

The iterative ansatz

q

₁

q

₁

q

₂

q

₂

q

₃

q

₃

q

₀

, p

_⊥0

, p

₊

q

₀

q

₁

, p

_⊥0

₋ p

_⊥1

_{, z}

₁

_p

₊

q

₁

q

₂

, p

_⊥1

− p

_⊥2

, z

₂

(1 − z

₁

)p

₊

q

₂

q

₃

, p

_⊥2

₋ p

_⊥3

_{, z}

₃

_{(1 − z}

₂

_{)(1 − z}

₁

_)p

₊

and so on until joining in the middle of the event

Scaling in lightcone p

_±

= E ± p

_z

(for qq system along z axis) implies flat central rapidity plateau + some endpoint effects:

y dn/dy

hn

_ch

i ≈ c

₀

+ c

₁

ln E

_cm

, ∼ Poissonian multiplicity distribution

(38)

The Lund gluon picture

q (r)

g (rb) The most characteristic feature of the Lund model

q (b)

snapshots of string position

strings stretched

from q (or qq) endpoint via a number of gluons to q (or qq) endpoint

Gluon = kink on string, carrying energy and momentum

Force ratio gluon/ quark = 2, cf. QCD N

_C

/C

_F

= 9/4, → 2 for N

_C

→ ∞ No new parameters introduced for gluon jets!, so:

• Few parameters to describe energy-momentum structure!

• Many parameters to describe flavour composition!

(39)

Independent fragmentation

Based on a similar iterative ansatz as string, but

q

q g

= q +

q

+ g

+

minor

corrections in middle

String effect (JADE, 1980)

≈ coherence in nonperturbative context

Further numerous and detailed tests at LEP favour string picture . . .

. . . but much is still uncertain when moving to hadron colliders.

(40)

The HERWIG Cluster Model

“Preconfinement”:

colour flow is local

in coherent shower evolution

●

subprocess

underlying event p

jet jet

p hard

●

+

Z

0

e

⁻

! !" "

# #

$ $

%%&&

''((

)*

++,,

l

1) Introduce forced g → qq branchings 2) Form colour singlet clusters

3) Clusters decay isotropically to 2 hadrons according to phase space weight ∼ (2s

₁

+ 1)(2s

₂

+ 1)(2p

^∗

/m)

simple and clean, but . . .

(41)

1) Tail to very large-mass clusters (e.g. if no emission in shower);

if large-mass cluster → 2 hadrons then

incorrect hadron momentum spectrum, crazy four-jet events

=⇒ split big cluster into 2 smaller along “string” direction;

daughter-mass spectrum ⇒ iterate if required;

∼ 15% of primary clusters are split, but give ∼ 50% of final hadrons 2) Isotropic baryon decay inside cluster

=⇒ splittings g → qq + qq

3) Too soft charm/bottom spectra

=⇒ anisotropic leading-cluster decay 4) Charge correlations still problematic

=⇒ all clusters anisotropic (?) 5) Sensitivity to particle content

=⇒ only include complete multiplets

(42)

String vs. Cluster

c g g b

D⁻_s Λ⁰

n η

π⁺ K^∗−

φ K⁺ π⁻ B⁰

e⁺e⁻ Event Generator

• hard scattering

• (QED) initial/final state radiation

• partonic decays, e.g.

t → bW

• parton shower evolution

• nonperturbative gluon splitting

• colour singlets

• colourless clusters

• cluster fission

• cluster→ hadrons

• hadronic decays

Bryan Webber, QCD Simulation for LHC and Herwig++, KEK, 6 April 2004 2

program PYTHIA HERWIG

model string cluster

energy–momentum picture powerful simple

predictive unpredictive

parameters few many

flavour composition messy simple

unpredictive in-between

parameters many few

“There ain’t no such thing as a parameter-free good description”

(43)

Local Parton–Hadron Duality

Analytic approach:

Run shower down to to Q ≈ Λ

_QCD

(or m

_hadron

, if larger)

“Hard Line”: each parton ≡ one hadron

“Soft Line”: local hadron density

∝ parton density

describes momentum spectra dn/dx

_p

and semi-inclusive particle flow, but fails for identified particles + “renormalons” (power corrections) h1 − T i = a α

s

(E

cm

) + b α

²_s

(E

cm

)

+c/E

cm

arbitrary units

E_cm [GeV]

<1-T>

<ρ>

<B_W>

<B_T>

<C>

O(α²_s)+1/Q O(α_s²)*MC corr.

TASSO PLUTO JADE CELLO HRS MARKII

AMY TOPAZ L3 DELPHI

ALEPH

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

25 50 75 100 125 150 175 200

Not Monte Carlo, not for arbitrary quantities

(44)

Decays

Unspectacular/ungrateful but necessary:

this is where most of the final-state particles are produced!

Involves hundreds of particle kinds and thousands of decay modes.

e.g.

B

^∗0

γ

B

⁰

→ B

⁰

D

^∗+

ν

e

⁻

π

⁺

D

⁰

K

⁻

ρ

⁺

π

⁺

π

⁰

e

⁺

e

⁻

γ

• B

^∗0

→ B

⁰

γ : electromagnetic decay

• B

⁰

→ B

⁰

mixing (weak)

• B

⁰

→ D

^∗+

ν

_e

e

⁻

: weak decay, displaced vertex, |M|

²

∝ (p

_B

p

_ν

)(p

_e

p

_D^∗

)

• D

^∗+

→ D

⁰

π

⁺

: strong decay

• D

⁰

→ ρ

⁺

K

⁻

: weak decay, displaced vertex, ρ mass smeared

• ρ

⁺

→ π

⁺

π

⁰

: ρ polarized, |M|

²

∝ cos

²

θ in ρ rest frame

• π

⁰

→ e

⁺

e

⁻

γ : Dalitz decay, m(e

⁺

e

⁻

) peaked

Dedicated programs, with special attention to polarization effects:

• EVTGEN: B decays

• TAUOLA: τ decays

(45)

sigma (mb)

Event Generators for LHC

Torbj ¨orn Sj ¨ostrand

Lund University

1. (yesterday) Introduction and Overview;

Parton Showers; Matching Issues 2. (today) Multiple Interactions;

Hadronization; Generators & Conclusions

Multiple Interactions

What is multiple interactions?

Cross section for 2 → 2 interactions is dominated by t-channel gluon exchange, so diverges like dσ/dp

≈ 1/p

for p

→ 0.

integrate QCD 2 → 2

qq

→ qq

qq → q

q

qq → gg qg → qg gg → gg gg → qq

with CTEQ 5L PDF’s

So σ int (p ⊥min ) > σ tot for p ⊥min ∼ 5 < GeV

Half a solution: many interactions per event σ

=

σ

σ

=

n σ

σ

> σ

⇐⇒ hni > 1

n P

hni = 2

0 1 2 3 4 5 6 7

If interactions occur independently then Poissonian statistics

P

= hni

n! e

but energy–momentum conservation

⇒ large n suppressed

Other half of solution:

perturbative QCD not valid at small p

since q, g not asymptotic states (confinement!).

Naively breakdown at p

' ¯ h

r

≈ 0.2 GeV · fm

0.7 fm ≈ 0.3 GeV ' Λ

. . . but better replace r

by (unknown) colour screening length d in hadron

r r

d resolved

r r

screened

λ ∼ 1/p

so modify dˆ σ

dp

∝ α

(p

)

p

→ α

(p

)

p

θ (p

− p

) (simpler)

or → α

(p

+ p

)

(p

+ p

)

(more physical)

p

dˆ σ/dp

0

where p

or p

So σ _int (p _⊥min ) > σ _tot for p _⊥min ∼ 5 ^< GeV

x _{dt ρ}

( x _{, t)ρ}

( x _{, t)}