Monte Carlo 2. Matching, MPI’s and Hadronization

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1 of 1 29/05/2013 14:23

Monte Carlo

2. Matching, MPI’s and Hadronization

Torbj¨orn Sj¨ostrand

Department of Astronomy and Theoretical Physics Lund University

S¨olvegatan 14A, SE-223 62 Lund, Sweden

TRIUMF, Vancouver, Canada, 5 July 2013

(2)

Event Generators Reminder

An event consists of many different physics steps, which have to be modelled by event generators:

Torbj¨orn Sj¨ostrand Monte Carlo 2 slide 2/1

(3)

Matrix elements vs. parton showers

ME : Matrix Elements

+ systematic expansion in α_s (‘exact’) + powerful for multiparton Born level + flexible phase space cuts

− loop calculations very tough

− negative cross section in collinear regions

⇒ unpredictive jet/event structure

− no easy match to hadronization PS : Parton Showers

− approximate, to LL (or NLL)

− main topology not predetermined

⇒ inefficient for exclusive states + process-generic ⇒ simple multiparton + Sudakov form factors/resummation

⇒ sensible jet/event structure + easy to match to hadronization

(4)

PS matching to MEs: realistic hard default

(5)

Matrix Elements and Parton Showers

Recall complementary strengths:

• ME’s good for well separated jets

• PS’s good for structure inside jets Marriage desirable! But how?

Very active field of research; requires a lecture series of its own Reweight first PS emission by ratio ME/PS (simple POWHEG) Combine several LO MEs, using showers for Sudakov weights

CKKW: analytic Sudakov – not used any longer CKKW-L: trial showers gives sophisticated Sudakovs MLM: match of final partonic jets to original ones Match to NLO precision of basic process

MC@NLO: additive ⇒ LO normalization at high p_⊥ POWHEG: multiplicative ⇒ NLO normalization at high p_⊥ Combine several orders, as many as possible at NLO

MENLOPS

UNLOPS (U = unitarized = preserve normalizations)

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Simple POWHEG (a.k.a. merging)

= cover full phase space with smooth transition ME/PS.

Want to reproduce W^ME= 1 σ(LO)

dσ(LO + g) d(phasespace) by shower generation + correction procedure

wanted

z }| { W^ME=

generated

z }| { W^PS

correction

z }| { W^ME

W^PS

• Exponentiate ME correction by shower Sudakov form factor:

W_actual^PS (Q²)=W^ME(Q²)exp − Z _Q_max²

Q²

W^ME(Q⁰²)dQ⁰²

!

• Do not normalize W^ME to σ(NLO) (error O(α²_s) either way)

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PYTHIA final-state shower merging

PYTHIA performs merging with generic FSR a → bcg ME, in SM:γ^∗/Z⁰/W^±→ qq, t → bW⁺, H⁰ → qq,

and MSSM:t → bH⁺, Z⁰ → ˜q˜q, ˜q → ˜q⁰W⁺, H⁰ → ˜q˜q, ˜q → ˜q⁰H⁺, χ → q˜q, χ → q˜q, ˜q → qχ, t → ˜tχ, ˜g → q˜q, ˜q → q˜g, t → ˜t˜g g emission for different R₃^bl(y_c): mass effects

colour, spin and parity: in Higgs decay:

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Vetoed Parton Showers – 1

Generic method to combine ME’s of several different orders to NLL accuracy; has become a standard tool for many studies Basic idea:

consider (differential) cross sections σ0, σ1, σ2, σ3, . . ., corresponding to a lowest-order process (e.g. W production), with more jets added to describe more complicated topologies, in each case to the respective leading order

σi, i ≥ 1, are divergent in soft/collinear limit

absent virtual corrections would have ensured “detailed balance”, i.e. an emission that adds to σ_{i +1} subtracts from σ_i such virtual corrections correspond (approximately) to the Sudakov form factors of parton showers

so use shower routines to provide missing virtual corrections

⇒ rejection of events (especially) in soft/collinear regions

S. Catani, F. Krauss, R. Kuhn, B.R. Webber, JHEP 0111 (2001) 063; L. L¨onnblad, JHEP0205 (2002) 046;

F. Krauss, JHEP 0208 (2002) 015; S. Mrenna, P. Richardson, JHEP0405 (2004) 040;

M.L. Mangano et al., JHEP0701 (2007) 013

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Vetoed Parton Showers – 2

Summary of CKKW(-L)/MLM:

use ME for real emissions and PS for virtual corrections (to ∼ restore unitarity).

CKKW-L:

1 generate n-body by ME mixed in proportionsR dσ_n above p⊥min cut

2 reconstruct fictitious p⊥-ordered PS

p⊥0 p⊥1 p⊥2 p⊥3 p⊥min

4 reject from fix αs= αs(p⊥min) to αs(p⊥i)

5 run trial shower between each p⊥i and p⊥i +1

6 reject if shower branching ⇒ Sudakov factor

7 regular shower below p⊥min (or below p⊥n for n = n_max)

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Vetoed Parton Showers – 3

) [pb]jet N≥) + -l+ l→*(γ(Z/σ

10-3

10-2

10-1

1 10 102

103

104

105

106

= 7 TeV) s Data 2011 ( ALPGEN SHERPA MC@NLO

+ SHERPA HAT BLACK

ATLAS Z/γ*(→ l⁺l^-)+jets (l=e,µ) L dt = 4.6 fb-1

∫ _t jets, R = 0.4 anti-k

| < 4.4 > 30 GeV, |yjet jet pT

≥0 ≥1 ≥2 ≥3 ≥4 ≥5 ≥6 ≥7

NLO / Data 0.6 0.8 1 1.2

1.4 _B_LACK_H_AT_{+ SHERPA}

0

≥ ≥1 ≥2 ≥3 ≥4 ≥5 ≥6 ≥7

MC / Data 0.6 0.8 1 1.2 1.4 _ALPGEN

Njet 0

≥ ≥1 ≥2 ≥3 ≥4 ≥5 ≥6 ≥7

MC / Data

0.6 0.8 1 1.2 1.4 _SHERPA

[1/GeV]jet T/dpσ) d-l+ l→*γZ/σ(1/

10-7

10-6

10-5

10-4

10-3

10-2

10-1

= 7 TeV) s Data 2011 ( ALPGEN SHERPA MC@NLO

+ SHERPA HAT BLACK

ATLAS Z/γ*(→ l⁺l^-)+ ≥ 1 jet (l=e,µ) L dt = 4.6 fb-1

∫ _t jets, R = 0.4 anti-k

| < 4.4 > 30 GeV, |yjet jet pT

100 200 300 400 500 600 700

NLO / Data 0.6 0.8 1 1.2

1.4 _B_LACK_H_AT_{+ SHERPA}

100 200 300 400 500 600 700

MC / Data 0.6 0.8 1 1.2 1.4 _ALPGEN

(leading jet) [GeV]

jet

pT

100 200 300 400 500 600 700

MC / Data

0.6 0.8 1 1.2 1.4 _SHERPA

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MC@NLO

Total rate should be accurate to NLO.

NLO results are obtained for all observables when (formally) expanded in powers of α_s.

Hard emissions are treated as in the NLO computations.

Soft/collinear emissions are treated as in shower MC.

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POWHEG

dσ = B(v )dΦ¯ _v R(v , r ) B(v ) exp

− Z

p⊥

R(v , r⁰) B(v ) dΦ⁰_r

dΦ_r

, B(v )¯ = B(v ) + V (v ) +

Z

dΦr[R(v , r ) − C (v , r )] .

v , dΦv Born-level n-body variables and differential phase space r , dΦ_r extra n + 1-body variables and differential phase space B(v ) Born-level cross section

V (v ) Virtual corrections

R(v , r ) Real-emission cross section

C (v , r ) Conterterms for collinear factorization of parton densities.

Note thatRB(v )dΦ¯ _v ≡ σ_NLO andR [· · · dΦ_r] ≡ 1.

So pick the real emission with largest p⊥ according to complete ME’s + ME-based Sudakov, with NLO normalization, and

let showers do subsequent evolution downwards from this p⊥ scale.

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Interpolation between POWHEG and MC@NLO

Master formula for meaningful NLO implementations:

dσ =dσ_R,hard+(σ_B + σ_R,soft+ σ_V) dσ_R,soft σB

exp

−

Z dσ_R,soft σB

ordered in “p⊥”, with shower from selected “p⊥” downwards POWHEG: σR,hard= 0

MC@NLO: σ_R,soft= σ_R,MC

“Best” choice process-dependent (guess NLO behaviour of σ_R)

S.Alioli, P. Nason, C. Oleari, E. Re, JHEP 00904 (2009) 002

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ME–PS Matching: Legs and Loops

Issue: ? multileg does not guaranteeP

n=0σ_{H+n jet}= σ_H;

? loop only guarantees σ_H to NLO and σ_{H+1 jet} to LO.

Recent progress: σH to NLO, σH+1 jet to NLO, σH+≥2 jet to LO.

Requires careful unitarity considerations, expansion of PDF’s, . . .

⇒ lengthy formulae and even lengthier procedures:

this means that once we want to add a one-jet NLO calculation, we have to subtract its integrated version from zero-jet events. Similarly we need to remove the O!α⁰_s(µ_R)"and O!α¹_s(µR)"in the UMEPS tree-level weights, not only for the one-jet events but also for the corresponding subtracted zero-jet events. In this way we ensure that the inclusive zero-jet cross section is still given by the NLO calculation and we will also improve the O(α²s)−term of exclusive zero-jet observables.

The UNLOPS prediction for an observable O, when simultaneously merging inclusive NLO calculations for m=0, . . . , M jets, and including up to N tree-level calculations, is given by

"O# =

M −1#

m=0

$ dφ₀

$

· · ·

$ O(S+mj)

% B_m+&

' B_m(

−m,m+1

−

#M

i=m+1

$

s

Bi→m−

#M

i=m+1

) $

s

' Bi→m

*

−i,i+1−

#N

i=M +1

$

s

' Bi→m

+

$ dφ₀

$

· · ·

$ O(S+M j)

% B_M+&

' B_M(

−M,M +1−

#N

i=M +1

$

s

' B_i→M

+

#N

n=M +1

$ dφ0

$

· · ·

$ O(S+nj)

% ' Bn−

#N

i=n+1

$

s

' Bi→n

+

(3.10)

Here we see, in the first line, the addition of the Bmand the removal of the O (α^ms (µR)) and O!

α^m+1_s (µR)"

terms of the original UMEPS 'Bmcontribution. On the second line we see the subtracted integrated B_m+1term to make the m-parton NLO-calculation exclusive and the corresponding O (α^ms(µR))- and O!α^m+1_s (µR)"-subtracted UMEPS term together with subtracted terms from higher multiplicities where intermediate states in the clustering were below the merging scale. The third line is the special case of the highest multiplicity corrected to NLO, and the last line is the standard UMEPS treatment of higher multiplicities.

The full derivation of this master formula is given in appendix D, where we also discuss the case of exclusive NLO samples and explain the necessity for subtraction terms from higher multiplicities. We will limit ourselves to including only zero- and one-jet NLO calculations in the results section. For the sake of clarity we will thus only discuss this special case here. For this case, the UNLOPS prediction (when including only up to two tree-level jets) is

"O# =

$ dφ0

% O(S+0j)

, B0 −

$

s

B1→0 − ) $

s

'B1→0

*

−1,2−

$

s

'B2→0

-

+

$ O(S+1j)

. B1+&

' B1

(

−1,2−

$

s

' B2→1

/

+

$ $

O(S+2j)'B2 (3.11)

– 16 –

(UNLOPS, L. L¨onnblad, S. Prestel)

Future: further complications or new synthesis?

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Event topologies

Expect and observe high multiplicities at the LHC.

What are production mechanisms behind this?

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What is minimum bias (MB)?

MB ≈ “all events, with no bias from restricted trigger conditions”

σtot=

σ_elastic+σsingle−diffractive+σdouble−diffractive+ · · · +σnon−diffractive

Schematically:

Reality: can only observe events with particles in central detector:

no universally accepted, detector-independent definition σ_min−bias≈σnon−diffractive+σdouble−diffractive≈ 2/3 × σ_tot

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What is underlying event (UE)?

In an event containing a jet pair or another hard process, how much further activity is there, that does not have its origin in the hard process itself, but in other physics processes?

Pedestal effect: the UE contains more activity than a normal MB event does (even discarding diffractive events).

Trigger bias: a jet ”trigger” criterion E⊥jet > E⊥min is more easily fulfilled in events with upwards-fluctuating UE activity, since the UE E⊥ in the jet cone counts towards the E⊥jet. Not enough!

(18)

What is pileup?

hni = L σ

where L is machine luminosity per bunch crossing, L ∼ n1n2/A and σ ∼ σtot≈ 100 mb.

Current LHC machine conditions ⇒ hni ∼ 10 − 20.

Pileup introduces no new physics, and is thus not further considered here, but can be a nuisance.

However, keep in mind concept of bunches of hadrons leading to multiple collisions.

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The divergence of the QCD cross section

Cross section for 2 → 2 interactions is dominated by t-channel gluon exchange, so diverges like dˆσ/dp²_⊥≈ 1/p_⊥⁴ for p_⊥→ 0.

Integrate QCD 2 → 2 qq⁰ → qq⁰

qq → q⁰q⁰ qq → gg qg → qg gg → gg gg → qq

(with CTEQ 5L PDF’s)

(20)

What is multiple partonic interactions (MPI)?

Note that σint(p⊥min), the number of (2 → 2 QCD) interactions above p⊥min, involves integral over PDFs,

σ_int(p⊥min) = Z Z Z

p⊥min

dx₁dx₂dp_⊥² f₁(x₁, p_⊥²) f₂(x₂, p_⊥²) dˆσ dp_⊥² withR dx f (x, p²_⊥) = ∞, i.e. infinitely many partons.

So half a solution to σ_int(p⊥min) > σ_tot is

many interactions per event: MPI (historically MI or MPPI)

σtot =

∞

X

n=0

σn

σ_int =

∞

X

n=0

n σ_n

σint > σtot⇐⇒ hni > 1

(21)

Colour screening

Other half of solution is that perturbative QCD is not valid at small p⊥ since q, g are not asymptotic states (confinement!).

Naively breakdown at p⊥min' ~

rp

≈ 0.2 GeV · fm

0.7 fm ≈ 0.3 GeV ' Λ_QCD

. . . but better replace r_p by (unknown) colour screeninglength d in hadron:

(22)

Regularization of low-p

_⊥

divergence

so need nonperturbative regularization for p⊥ → 0 , e.g.

dˆσ

dp_⊥² ∝ α²_s(p²_⊥)

p⁴_⊥ → α²_s(p²_⊥)

p_⊥⁴ θ (p⊥− p_⊥min) (simpler) or → α²_s(p²_⊥0+ p_⊥²)

(p_⊥0² + p_⊥²)² (more physical) wherep⊥min orp⊥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.

(23)

MPI effects

By now several direct tests of back-to-back jet pairs and similar.

However, only probes high-p⊥ tail of effects.

More direct and dramatic are effects on multiplicity distributions:

(24)

MPI and event generators

All modern general-purpose generators are built on MPI concepts PYTHIA implementation main points:

MPIs are gererated in a falling sequence of p⊥ values;

recall Sudakov factor approach to parton showers.

Energy, momentum and flavour are subtracted from proton by all “previous” collisions.

Protons modelled as extended objects, allowing both central and peripheral collisions, with more or less activity.

(Partons at small x more broadly spread than at large x .) Colour screening increases with energy, i.e. p⊥0 = p⊥0(Ecm), as more and more partons can interact.

(Rescattering: one parton can scatter several times.) Colour connections: each interaction hooks up with colours from beam remnants, but also correlations inside remnants.

Colour reconnections: many interaction “on top of” each other ⇒ tightly packed partons ⇒ colour memory loss?

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Interleaved evolution

• Transverse-momentum-ordered parton showers for ISR and FSR

• MPI also ordered in p_⊥

⇒ Allows interleaved evolution for ISR, FSR and MPI:

dP dp⊥

=

dP_MPI dp⊥

+XdP_ISR dp⊥

+XdP_FSR dp⊥

× exp

−

Z p⊥max

p⊥

dP_MPI

dp⁰_⊥ +XdP_ISR

dp⁰_⊥ +XdP_FSR dp⁰_⊥

dp_⊥⁰

Ordered in decreasing p⊥ using “Sudakov” trick.

Corresponds to increasing “resolution”:

smaller p_⊥ fill in details of basic picture set at larger p_⊥. Start from fixed hard interaction ⇒ underlying event No separate hard interaction ⇒ minbias events Possible to choose two hard interactions, e.g. W⁻W⁻

(26)

Colour correlations and hp

_⊥

i(n

_ch

) – 1

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Colour correlations and hp

_⊥

i(n

_ch

) – 2

Comparison with data, generators before and after LHC data input:

N ch

0 50 100

[GeV]〉 T p〈

0.6 0.8 1 1.2 1.4 1.6

ATLAS Pythia 8 Pythia 8 (no CR)

7000 GeV pp Soft QCD (mb,diff,fwd)

mcplots.cern.ch 200k events≥Rivet 1.8.2, Pythia 8.175

ATLAS_2010_S8918562 > 0.5 GeV/c) > 1, pT (Nch vs Nch Average pT

0 50 100

0.5 1

1.5 Ratio to ATLAS

see also A. Buckley et al., Phys. Rep. 504 (2011) 145 [arXiv:1101.2599[hep-ph]]

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Jet pedestal effect – 1

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.

Studied in particular by Rick Field, with CDF/CMS data:

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Jet pedestal effect – 2

Cosmic QCD 2013 LPTHE Paris, May 15, 2013

Rick Field – Florida/CDF/CMS Page 47

“ “ Tevatron Tevatron ” ” to the LHC to the LHC

"Transverse" Charged Particle Density: dN/d d

0.0 0.4 0.8 1.2

0 5 10 15 20 25 30 35

PTmax (GeV/c)

"Transverse" Charged Density

RDF Preliminary

Corrected Data

Charged Particles (PT>0.5 GeV/c) 1.96 TeV

300 GeV

900 GeV

PYTHIA Tune Z1 7 TeV

(30)

Jet pedestal effect – 3

Cosmic QCD 2013 LPTHE

Paris, May 15, 2013 Rick Field – Florida/CDF/CMS Page 58

MB versus the UE MB versus the UE

Charged Particle Density

0.0 0.2 0.4 0.6 0.8 1.0

0.1 1.0 10.0

Center-of-Mass Energy (TeV)

Charged Density

RDF Preliminary Corrected Data

Charged Particles (| |<0.8, PT>0.5 GeV/c) CMS squares

CDF dots

Overall

At least 1 charged particle

"Transverse"

5 < PTmax < 6 GeV/c Overall Charged Particle Density

0.20 0.30 0.40 0.50

0.1 1.0 10.0

Charged Density

RDF Preliminary Corrected Data

Charged Particles (| |<0.8, PT>0.5 GeV/c) CMS red squares

CDF blue dots

0.2 0.4 0.6 0.8 1.0

0.1 1.0 10.0

"Transverse" Charged Density RDF Preliminary Corrected Data

Charged Particles (| |<0.8, PT>0.5 GeV/c) 5.0 < PTmax < 6.0 GeV/c CMS red squares

CDF blue dots

CorrectedCDF and CMS dataon the overall density of charged particles with pT

> 0.5 GeV/c and | | < 0.8 for events with at least one charged particle with pT> 0.5 GeV/c and | | < 0.8 and on the charged particle density, in the “transverse” region as defined by the leading charged particle (PTmax) for charged particles with p_T> 0.5 GeV/c and | | < 0.8 with 5 < PTmax < 6 GeV/c. The data are plotted versus the center-of-mass energy (log scale).

0.2 0.4 0.6 0.8 1.0

0.1 1.0 10.0

"Transverse" Charged Density

Charged Particles (| |<0.8, PT>0.5 GeV/c) 5.0 < PTmax < 6.0 GeV/c RDF Preliminary

Corrected Data Tune Z1 Generator Level

PYTHIA Tune Z1 CMS red squares

CDF blue dots Overall Charged Particle Density

0.20 0.30 0.40 0.50 0.60

0.1 1.0 10.0

Charged Density

Charged Particles (| |<0.8, PT>0.5 GeV/c) CMS red squares

CDF blue dots

At least 1 charged particle RDF Preliminary

PYTHIA Tune Z1

Charged Particle Density

0.0 0.2 0.4 0.6 0.8 1.0

0.1 1.0 10.0

Charged Density

Charged Particles (| |<0.8, PT>0.5 GeV/c) CMS squares

CDF dots

Overall

"Transverse"

5 < PTmax < 6 GeV/c RDF Preliminary

PYTHIA Tune Z1

Amazing!

Conclusion: “transMIN” (MPI+BBR) increases much faster with E_cm than “transDIF” (ISR+FSR), proportionately speaking.

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Diffraction

Ingelman-Schlein: Pomeron as hadron with partonic content Diffractive event = (Pomeron flux) × (IPp collision)

Diffraction

Ingelman-Schlein: Pomeron as hadron with partonic content Diffractive event = (Pomeron flux) × (IPp collision)

p p

IP p

Used e.g. in POMPYT POMWIG PHOJET

1) σ_SDand σ_DDtaken from existing parametrization or set by user.

2) Shape of Pomeron distribution inside a proton, f_IP/p(x_IP, t) gives diffractive mass spectrum and scattering p_⊥of proton.

3) At low masses retain old framework, with longitudinal string(s).

Above 10 GeV begin smooth transition to IPp handled with full pp machinery: multiple interactions, parton showers, beam remnants, . . . . 4) Choice between 5 Pomeron PDFs.

Free parameter σ_IPpneeded to fix #ninteractions$ = σ_jet/σ_IPp. 5) Framework needs testing and tuning, e.g. of σ_IPp.

1) σ_SDand σ_DD taken from existing parametrization or set by user.

2) f_IP/p(xIP, t) ⇒ diffractive mass spectrum, p⊥ of proton out.

3) Smooth transition from simple model at low masses to IPp with full pp machinery: multiple interactions, parton showers, etc.

4) Choice between 5 Pomeron PDFs.

5) Free parameter σIPp needed to fix hninteractionsi = σ_jet/σIPp.

(32)

Hadronization

Hadronization/confinement is nonperturbative ⇒ only models.

Begin with e⁺e⁻→ γ^∗/Z⁰→ qq and e⁺e⁻→ γ^∗/Z⁰ → qqg:

Y

Z X

Y

Z X

(33)

The QCD potential – 1

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

Gives force/potential between a q and a q:

F (r ) ≈ const = κ ⇐⇒ V (r ) ≈ κr

κ ≈ 1 GeV/fm≈ potential energy gain lifting a 16 ton truck.

Flux tube parametrized by center location as a function of time

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

(34)

The QCD potential – 2

Linear confinement confirmed e.g. by lattice QCD calculation of gluon field between a static colour and anticolour charge pair:

At short distances also Coulomb potential, important for internal structure of hadrons, but not for particle production (?).

(35)

The QCD potential – 3

Full QCD = gluonic field between charges (“quenched QCD”) plus virtual fluctuations g → qq (→ g)

=⇒ nonperturbative string breakings gg . . . → qq

(36)

String motion

The Lund Model: starting point Use only linear potential V (r ) ≈ κr to trace string motion, and let string fragment by repeated qq breaks.

Assume negligibly small quark masses.

Then linearity between space–time and energy–momentum gives

dE dz

=

dp_z dz

=

dE dt

=

dp_z dt

= κ

(c = 1) for a qq pair flying apart along the ±z axis.

But signs relevant: the q moving in the +z direction has dz/dt = +1 but dp_z/dt = −κ.

B. Andersson et a!., Patton fragmentation and string dynamics 41

____

^-L/2 ^L12 ^X

____

^-p ^p~

<V

Fig. 2.1. The motion of q and ~ in the CM frame. The hatched areas Fig. 2.2. The motion of q and ~ in a Lorentz frame boosted relative to

show where the field is nonvanishing. the CM frame.

M2. In fig. 2.2 the same motion is shown after a Lorentz boost /3. The maximum relative distance has been contracted to

L’ = Ly(1 ^—

/3)

^L

e~and the time period dilated to

T’ = TI’y = T

cosh(y) where y

is the rapidity difference between the two frames.

In this model the “field” corresponding to the potential energy carries no momentum, which is a consequence of the fact that in 1

⁺

1 dimensions there is no Poynting vector. Thus all the momentum is carried by the endpoint quarks. This is possible since the turning points, where q and 4 have zero momentum, are simultaneous only in the CM frame. In fact, for a fast-moving q4 system the q4-pair will most of the time move forward with a small, constant relative distance (see fig. 2.2).

In the following we will use this kind of yo-yo modes as representations both of our original q4 jet system and of the final state hadrons formed when the system breaks up. It is for the subsequent work necessary to know the level spectrum of the yo-yo modes. A precise calculation would need a knowledge of the quantization of the massless relativistic string but for our purposes it is sufficient to use semi-classical considerations well-known from the investigations of Schrodinger operator spectra.

We consider the Hamiltonian of eq. (2.14) in the CM frame with

q =

x

1

^—

x2

H=IpI+KIql

(2.18)

and we note that our problem is to find the dependence on n of the nth energy level

E~.

If the spatial size of the state is given by 5~then the momentum size of such a state with n

^—

1 nodes is

IpI=nI& (2.19)

and the energy eigenvalue

E~

corresponds according to variational principles to a minimum of

H(6~)=

n/&,

+ Kô~

(2.20)

i.e.

2Vttn. (2.21)

(37)

The Lund Model

Combine yo-yo-style string motion with string breakings!

Motion ofquarksandantiquarks with intermediatestring pieces:

Aqfrom onestring break combines with aqfrom an adjacent one.

Gives simple but powerful picture of hadron production.

(38)

Where does the string break?

Fragmentation starts in the middle and spreads outwards:

Corresponds to roughly same invariant time of all breaks, τ²= t²− z² ∼ constant,

with breaks separated by hadronic area m²_⊥= m²+ p_⊥². Hadrons at outskirts are more boosted.

Approximately flat rapidity distribution, dn/dy ≈ constant

⇒ total hadron multiplicity in a jet grows like ln E_jet.

(39)

How does the string break?

String breaking modelled by tunneling:

P ∝ exp −πm²_⊥q κ

!

= exp −πp_⊥q² κ

!

exp −πm_q² κ

!

• Common Gaussian p_⊥ spectrum, hp⊥i ≈ 0.4 GeV.

• Suppression of heavy quarks,

uu : dd : ss : cc ≈ 1 : 1 : 0.3 : 10⁻¹¹.

• Diquark ∼ antiquark ⇒ simple model for baryon production.

String model unpredictive in understanding of hadron mass effects

⇒ many parameters, 10–20 depending on how you count.

(40)

The Lund gluon picture – 1

Gluon = kink on string Force ratio gluon/ quark = 2,

cf. QCD NC/CF = 9/4, → 2 for NC → ∞ No new parameters introduced for gluon jets!

(41)

The Lund gluon picture – 2

Energy sharing between two strings makes hadrons in gluon jets softer, more and broader in angle:

Jetset 7.4 Herwig 5.8 Ariadne 4.06 Cojets 6.23

OPAL

(1/Nevent ) dnch. /dxE

x_E uds jet gluon jet k_⊥ definition:

y_cut=0.02

0. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.

10^-3 10^-2 10^-1 1 10 10²

0.5 1.

1.5

Correction factors 0

0.05 0.1 0.15

0 5 10 15 20 25 30 35

g_incl. jets uds jets Jetset 7.4 Herwig 5.9 Ariadne 4.08 AR-2 AR-3

n_ch.

P(nch.)

(a) OPAL

|y| 2

0 0.05 0.1 0.15 0.2

0 5 10 15 20

gincl. jets uds jets Jetset 7.4 Herwig 5.9 Ariadne 4.08 AR-2 AR-3

n_ch.

P(nch.)

(b) OPAL

|y| 1

Jetset 7.4 Herwig 5.8 Ariadne 4.06 Cojets 6.23

0. 10. 20. 30. 40. 50. 60.

0.

0.02 0.04 0.06 0.08

OPAL

(1/Ejet) dEjet/dχ

χ (degrees) uds jet gluon jet

k_⊥ definition:

ycut=0.02 0.5

1.

1.5

Correction factors

(42)

The Lund gluon picture – 3

Particle flow in the qqg event plane depleted in q–q region owing to boost of string pieces in q–g and g–q regions:

(43)

String vs. Cluster

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”

(44)

Colour flow in hard processes – 1

One Feynman graph can correspond to several possible colour flows, e.g. for qg → qg:

while other qg → qg graphs only admit one colour flow:

(45)

Colour flow in hard processes – 2

so nontrivial mix of kinematics variables (ˆs, ˆt) and colour flow topologies I, II:

|A(ˆs, ˆt)|² = |AI(ˆs, ˆt) + AII(ˆs, ˆt)|²

= |AI(ˆs, ˆt)|²+ |AII(ˆs, ˆt)|²+ 2 Re AI(ˆs, ˆt)A^∗_II(ˆs, ˆt) with Re AI(ˆs, ˆt)A^∗_II(ˆs, ˆt) 6= 0

⇒ indeterminate colour flow, while

• showers should know it (coherence),

• hadronization must know it (hadrons singlets).

Normal solution:

interference

total ∝ 1

N_C²− 1

so split I : II according to proportions in the NC→ ∞ limit, i.e.

|A(ˆs, ˆt)|² = |AI(ˆs, ˆt)|²_mod+ |AII(ˆs, ˆt)|²_mod

|AI(II)(ˆs, ˆt)|²_mod = |AI(ˆs, ˆt) + AII(ˆs, ˆt)|² |A_I(II)(ˆs, ˆt)|²

|AI(ˆs, ˆt)|²+ |AII(ˆs, ˆt)|²

!

NC→∞

(46)

Colour Reconnection Revisited

Colour rearrangement well established e.g. in B decay.

Introduction

(V.A. Khoze & TS, PRL72 (1994) 28, ZPC62 (1994) 281, EPJC6 (1999) 271;

L. L ¨onnblad & TS, PLB351 (1995) 293, EPJC2 (1998) 165)

Γ_W, Γ_Z, Γ_t ≈ 2 GeV

Γ_h > 1.5 GeV for m_h > 200 GeV Γ_SUSY ∼ GeV (often)

τ = 1

Γ ≈ 0.2 GeV fm

2 GeV = 0.1 fm # rhad ≈ 1 fm

⇒ hadronic decay systems overlap, between pairs of resonances

⇒ cannot be considered separate systems!

Three main eras for interconnection:

1. Perturbative: suppressed for ω > Γ by propaga- tors/timescales ⇒ only soft gluons.

2. Nonperturbative, hadronization process:

colour rearrangement.

B⁰

d

b c

W⁻ c

s

!

"

!

"

B⁰

d b

c W⁻

c gs ^!

"

K⁰_S

!

"

J/ψ

3. Nonperturbative, hadronic phase:

Bose–Einstein.

Above topics among unsolved problems of strong in- teractions: confinement dynamics, 1/N

_C²

effects, QM interferences, . . . :

• opportunity to study dynamics of unstable parti- cles,

• opportunity to study QCD in new ways, but

• risk to limit/spoil precision mass measurements.

So far mainly studied for m

_W

at LEP2:

1. Perturbative: !δm

W

" ∼ 5 MeV.

^<

2. Colour rearrangement: many models, in general

!δm

W

" ∼ 40 MeV.

^<

e⁻ e⁺

W⁻ W⁺

q3

q4

q2

q1

!

"

!

"

π⁺ π⁺

#

$BE

3. Bose-Einstein: symmetrization of unknown am- plitude, wider spread 0–100 MeV among models, but realistically !δm

W

" ∼ 40 MeV.

^<

In sum: !δm

W

"

tot

< m

_π

, !δm

W

"

tot

/m

_W

∼ 0.1%; a

^<

small number that becomes of interest only because we aim for high accuracy.

At LEP 2 search for effects in e⁺e⁻→ W⁺W⁻→ q_qq₂q₃q₄: perturbative hδM_Wi . 5 MeV : negligible!

nonperturbative hδM_Wi ∼ 40 MeV : signs but inconclusive.

Bose-Einstein hδM_Wi . 100 MeV : full effect ruled out.

Hadronic collisions with MPI’s: many overlapping colour sources.

Reconnection established by hp⊥i(n_ch), but details unclear.

(47)

The Mass of Unstable Coloured Particles – 1

MC: close to pole mass, in the sense of Breit–Wigner mass peak.

t, W, Z: cτ ≈ 0.1 fm < rp .

t

W b

At the Tevatron:mt= 173.20 ± 0.51 ± 0.71 GeV = PMAS(6,1) At the LHC: mt= 173.4 ± 0.4 ± 0.9 GeV (CMS)= 6:m0 ? Need better mass definition for coloured particles?

(48)

The Mass of Unstable Coloured Particles – 2

DependenceofTopMassonEvent

Kinema2cs*

10*

!  First#top#mass#measurement#binned#in#kinema3c#observables.#

!  Addi2onal*valida2on*for*the*top*mass*measurements.**

!  With*the*current*precision,*no*mis^modelling*eﬀect*due*to*

"  color*reconnec2on,*ISR/FSR,**b^quark*kinema2cs,*diﬀerence*

between*pole*or*MS~*masses.*

color*recon.*

ISR/FSR*

b^quark*kin.*

Global*χ²/ndf*=*0.9*based*on*

mt1D*and*JES*which*are*

independent*(comparing*data*

and*MadGraph)*neglec2ng*

correla2ons*between*

observables.*

CMS^PAS^TOP^12^029*

NEW*

0 50 100 150 200 250 300

> [GeV]2Dt - <m

2D tm

-4 -2 0 2 4

-1) Data (5.0 fb MG, Pythia Z2 MG, Pythia P11 MG, Pythia P11noCR MC@NLO, Herwig = 7 TeV, lepton+jets s

CMS preliminary,

[GeV]

T,t,had p

0 50 100 150 200 250 300

data - MG Z2

-5 0 5

E. Yazgan

(Moriond 2013)

(49)

QCD and BSM physics

BNV ⇒ junction topology

⇒ special handling of showers and hadronization

Hidden valleys:

showers potentially interleaved with normal ones;

hadronization in hidden sector;

decays back to normal sector R-hadron formation

Squark fragmenting to meson or baryon

Gluino fragmenting to baryon or glueball

Most hadronization properties by analogy with normal string fragmentation, but

glueball formation new aspect, assumed⇠ 10% of time(or less).

Torbj¨orn Sj¨ostrand QCD Aspects of BSM Physics slide 12/18

R-hadrons: long-lived ˜g or ˜q;

new: hadronization of massive object “inside” the string