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Nonperturbative models in PYTHIA

Torbj¨ orn Sj¨ ostrand

Department of Astronomy and Theoretical Physics Lund University

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

Taming the accuracy of event generators, 23-27 August 2021

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Contents

Event structure Some current topics

Strangeness enhancement Charm baryon enhancement Beam drag effects

Lund model reminder Some recent studies

Space–time evolution Hadronic rescattering Cosmic ray cascades Forward modelling Summary

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The structure of an event

An event consists of many different physics steps to be modelled:

Parton Distributions Matrix Elements Initial-State Radiation Final-State Radiation Match and Merge Multiparton Interactions Beam–Beam Remnants Colour Reconnection Fragmentation

Collectivity (shove? ropes?) Decays

Rescattering Bose–Einstein

tot= elastic + di↵ractive + · · · Unknown?

Many/most require nonperturbative modelling!

Evolution in dialogue with experimental observations.

Torbj¨orn Sj¨ostrand Nonperturbative models in PYTHIA slide 3/23

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A breakdown of jet universality

|< 0.5 η

|

η

ch/d N

d 10

INEL>0

pp )π) / (h/π(h/

0.5 1 1.5 2

S K0

Λ Ξ ALICE

= 7 TeV s pp,

= 5.02 TeV sNN p-Pb,

|< 0.5 η

|

η

ch/d N

d 10

Baryon to meson ratio

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

PYTHIA8 DIPSY EPOS LHC ALICE

= 7 TeV s pp,

= 5.02 TeV sNN p-Pb,

×2) π ( p/

S /K0 Λ

Significant strangeness enhancement;

the more the merrier.

Minimal baryon enhancement.

Not described by the Lund string fragmentation model.

Partly addressed by colour ropes or by core–corona models.

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Rope hadronization (Dipsy model)

Dense environment⇒ several intertwined strings ⇒ rope.

Sextet example:

3⊗ 3 = 6 ⊕ 3 C2(6) = 52C2(3)

q2

q4

q1

q3

space time

quark antiquark pair creation Atfirststring break κeff ∝ C2(6)− C2(3) ⇒κeff = 32κ.

Atsecond string break κeff ∝ C2(3) ⇒κeff =κ.

Multiple∼parallel strings ⇒ random walk in colour space.

Largerκeff ⇒ larger exp

πmκeff2q

⇒ more strangeness and baryons mainly agrees with ALICE (butp/π overestimated)

Bierlich, Gustafson, L¨onnblad, Tarasov, JHEP 1503, 148 Alternative: close-packing of strings⇒ smaller string area

⇒ (continuously) larger κ ⇒ “thermodynamical” fragmentation N. Fischer, TS, JHEP 1701, 140

Torbj¨orn Sj¨ostrand Nonperturbative models in PYTHIA slide 5/23

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Charm hadron composition – 1

EPS-HEP 2021 | Highlights from the ALICE experiment | K. Reygers

Charm hadronization in pp (1):

26

More charm quarks in baryons in pp than in e+e and ep collisions

Charm quarks hadronize into baryons 40% of the time

~ 4 times more than in e+e arXiv:2105.06335 talk Luigi Dello Stritto

K. Reygers, EPS-HEP 2021

EPS-HEP 2021 | Highlights from the ALICE experiment | K. Reygers

0 5 10 15 20 25

) (GeV/c pT

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0/D+ cΛ ALICE| < 0.5y|

= 5 TeV pp, s

= 13 TeV pp, s

PYTHIA 8.243, Monash 2013 PYTHIA 8.243, CR-BLC:

Mode 0 Mode 2

Mode 3

SHM+RQM Catania QCM

ALI-DER-493847

Charm hadronization in pp (3)

28

ratio in pp significantly different than in e+e Λ+c/D0

arXiv:2011.06079

Charm quark fragmentation not universal!

e+e

Standard PYTHIA 8 below data Fair description by

PYTHIA 8 with CR

Coalescence + fragmentation (Catania)

SH mode + RQM

(T = 170 MeV, additional states crucial) Measurement of charmed hadrons down to unprecedentedly low pT at midrapidity

Λ+c(udc) → pKπ+

→ pK0s

arXiv:2106.08278

Λ+c/D0 four times higher than in e+e!

Bute+e result recovered at large p.

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Charm hadron composition – 2

Possible reconnections

Ordinary string reconnection

(qq: 1/9, gg: 1/8, model: 1/9)

Triple junction reconnection

(qq: 1/27, gg: 5/256, model: 2/81)

Double junction reconnection

(qq: 1/3, gg: 10/64, model: 2/9)

Zipping reconnection

(Depends on number of gluons)

Jesper Roy Christiansen (Lund) Non pertubative colours November 3, MPI@LHC 10 / 15

Christiansen, Skands: CR-BLC:

Colour Reconnection Beyond Leading Colour JHEP 08 (2015) 003

Mode 0, 2, 3: different causality restrictions, 0 = none

. . . but Ξ+c/D0 still not described

EPS-HEP 2021 | Highlights from the ALICE experiment | K. Reygers

Charm hadronization in pp (4):

29

not described by models that get right!

Ξ0c/D0 Λ+c/D0

Ξ0c(dsc) → Ξe+νe

→ Ξπ+

arXiv:2105.05187

Coalescence model comes closest to data talk Luigi Dello Stritto

PYTHIA 8 with CR (mode 2) below data, even though this model describes Λ+c/D0

Torbj¨orn Sj¨ostrand Nonperturbative models in PYTHIA slide 7/23

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Beam drag effects

(E. Norrbin & TS, 2000)

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Bottom asymmetries

uncertainties on the Pythia models shown here are only due to the limited sample size of about 12.5 million events. The results of the Pythia hadronisation model describing the data best, along with the predictions of the heavy-quark recombination model are presented in Fig. 11. The uncertainties on the heavy-quark recombination model are the systematic uncertainties given in Ref. [5]. Overall, the predictions from the heavy-quark recombination model are consistently higher than the 8 TeV measurements, but remain within uncertainties. For Pythia, only the model CR1 shows a good agreement with theps = 7 TeV measurements but it is also consistently higher at 8 TeV. The two other tested settings predict asymmetries that are too large, exhibiting the strongest deviation at low transverse momentum.

2 2.5 3 3.5 4

0y Λb 0

2 4 6 8 10 12 14

[%]Aprod 16 Data 1fb-1

Pythia8 (CR1) Pythia8 (CR2) Pythia8 (Monash) = 7 TeV s LHCb

0 10 20

] c [GeV/

pT 0

Λb 0

2 4 6 8 10 12 14

[%]prodA -1fbData 1

Pythia8 (CR1) Pythia8 (CR2) Pythia8 (Monash) = 7 TeV s LHCb

2 2.5 3 3.5 4

0y Λb 0

2 4 6 8 10 12 14

[%]Aprod 16 Data 2fb-1

Pythia8 (CR1) Pythia8 (CR2) Pythia8 (Monash) = 8 TeV s LHCb

0 10 20

] c [GeV/

pT 0

Λb 0

2 4 6 8 10 12 14

[%]prodA -1fbData 2

Pythia8 (CR1) Pythia8 (CR2) Pythia8 (Monash) = 8 TeV s LHCb

Figure 10: Comparison of the ⇤0b production asymmetry predicted by the various Pythia models, where CR1 refers to the QCD-inspired model and CR2 refers to the gluon-move model, and the measured production asymmetries. Results versus ⇤0b(left) rapidity y and (right) pTare shown for centre-of-mass energies of (top)p

s = 7 TeV and (bottom)p

s = 8 TeV. Uncertainties on the predictions are due to limited simulation sample sizes.

9 Conclusions

The most precise measurements of the ⇤0bproduction asymmetry inps = 7 TeV and 8 TeV proton-proton collisions have been presented. A new method to estimate asymmetries in the interaction of protons and antiprotons with the detector material has been developed.

21 LHCb, 2107.09593

A = σ(Λ0b)− σ(Λ0b) σ(Λ0b) +σ(Λ0b) CR1 = CR-BLC, no enhancement at low p.

Enhanced Λb production at low p, like for Λc, dilutes asymmetry?

Asymmetries observed also for other charm and bottom hadrons.

Revived field of study?

Torbj¨orn Sj¨ostrand Nonperturbative models in PYTHIA slide 9/23

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The Lund Model – 1

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

space time

quark antiquark pair creation

Aqfrom one string break combines with aqfrom an adjacent one.

String tensionκ≈ 1 GeV/fm relates (t, x) and (E , p).

Gives simple but powerful picture of hadron production.

(11)

The Lund Model – 2

The most characteristic feature of the Lund model:

quark

antiquark gluon

string motion in the event plane (without breakups)

Generalizes to multiple intermediate gluons, closed gluon loops, junction topologies.

In principle always unique relationship (t, x)↔ (E , p), but in practice can become quite complicated string motion.

Torbj¨orn Sj¨ostrand Nonperturbative models in PYTHIA slide 11/23

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Space–time evolution

PYTHIA can now calculate production vertex of each particle, e.g. number of hadrons as a function of time for pp at 13 TeV:

time(fm/c) 1 10 102 103 104 105 106 107 108 109 1010 1011 1012 1013 1014 1015

hadn

0 20 40 60 80 100 120 140 160

180 Total number of hadrons

Primary hadrons Secondary hadrons Total number of final hadrons

S. Ferreres-Sol´e, TS, EPJC 78, 983

(13)

Hadronic rescattering

13 TeV nondiffractivepp events:

0 2 4 6 8 10

τ (fm) 0

5 10 15 20 25 30 35 40

dN/dτ

Invariant production time, rescattered or not rescattered not rescattered sum both rescattering off

PYTHIA now contains framework for hadronic rescattering:

1)Space–time motion and scattering opportunities 2)Cross section for low-energy hadron–hadron collisions

3)Final-state topology in such collisions

M. Utheim, TS, EPJC80, 907

In principle already covered by other programs like UrQMD or SMASH, but then interfacing issues limits usefulness.

Torbj¨orn Sj¨ostrand Nonperturbative models in PYTHIA slide 13/23

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Rescattering effects in pp and AA – 1

Softening of pion spectrum in pp (and AA) in right direction, but more complicated for other hadrons.

Generally, observable consequences inpp minor, but important for AA modelling

0 1 2 3 4 5

p (GeV)

102 101 100 101 102

d/dp

Transverse momentum, rescattered or not, pp @ 5.02 TeV , rescattering on , rescattering off K, rescattering on K, rescattering off N, rescattering on N, rescattering off D, rescattering on D, rescattering off

0 1 2 3 4 5

p (GeV)

100 101 102 103 104

d/dp

Transverse momentum, rescattered or not, PbPb @ 5.02 TeV , rescattering on , rescattering off K, rescattering on K, rescattering off N, rescattering on N, rescattering off D, rescattering on D, rescattering off

C. Bierlich, M. Utheim, TS, EPJA57, 227

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Rescattering effects in pp and AA – 2

Significant contribution to collective flow in AA:

b b b bb b b b bbb b bb b b b b bbbbb

b bb bb

b bb bb

b bbbbbbbbb

bb

b bb bb

b bb bb b bb b b b b ALICE Data

Pythia 8/Angantyr + rescattering

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

Pb-Pbs

nn=5.02 TeV, v2{2,|∆η| >1.4}

v2{2,|η|>1.4} b b b bb b b b bbb b bb b b b b bbbbbb b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b

102 103

0.50.6 0.70.8 0.91 1.11.2 1.31.4

Nch(|η| <0.8)

MC/Data b b b b b b b b b b b b bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb

b ALICE Data Pythia 8/Angantyr

+ rescattering

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14Pb-Pbs

nn=5.02 TeV, v2{8}

v2{8} b b b b b b b b b b b b bbbbbbbb b b b b b b b b b b b b b b b b b b b b b b b

102 103

0.50.6 0.70.8 0.91 1.11.2 1.31.4

Nch(|η| <0.8)

MC/Data

but need more, e.g. shove:

t = t1 t = t2 t = t3 t = t4

by bx

Figure 1: Cartoon in impact parameter space showing strings overlapping at time t = t1, and as time progresses (t1< t2< t3< t4), they move apart, picking up pas indicated with arrows.

transverse coordinate space (b). Colour-connected partons separated by a distance ∆b are also given opposite transverse momenta p≈ ∆b/(∆b)2. The initial state is two Lorentz contracted pancakes colliding at z = 0, and the string segments are then stretched out mainly along the z direction. The distribution of gluons is approximately boost invariant, and to visualize the effect of the transverse repulsion, it is most easy to study a string segment stretched between two gluons in a system where they have rapidities

±∆y/2. The endpoints of this string segment will then move out with longitudinal velocities vL=± tanh(∆y/2), and the length of the segment in coordinate space, at time t, is consequently t·tanh(∆y). The repulsive transverse force between two strings is proportional to the length of the overlapping region, and is therefore proportional to f·t ·∆y, where f is the force per unit string length.

The cartoon in figure 1 represents in a schematic way a ”slice” in rapidity4. The result of the repulsion will be a transverse velocity for the string, which might be represented by very many very soft gluons. The breakup of such a string state cannot be handled current implementations of string hadronization, as in e.g. Pythia8. As the DIPSY gen- erator interfaces to the Pythia8 hadronization implementation, this must be remedied. A transverse gluon will give momentum to hadrons within one unit of rapidity on either side of the gluon. It is therefore possible to simulate the effect of the continuous distribution of infinitely soft gluons by finite gluons separated by at most one rapidity unit. In our calculations we cut the event into many rapidity slices, and in each slice we let the strings

“shove” each other apart. The mechanism for shoving is to add a small excitation (i.e. a gluon) to each string in each slice. In each time–step δt a string within a slice δy (and thus length δl = t δy) will get a kick in the transverse direction δp= f t δy δt. As the mass of the string piece is≈ κ δl = κ t δy also is proportional to the time t, we note that the factors t drop out in the result for the transverse velocity boost. When the strings no longer overlap, the many small kicks are added to a set of gluons, which can be handled

4In reality the strings are, of course, not distributed symmetrically, instead there are large fluctuations in the transverse positions of the strings.

3

Bierlich, Gustafson, L¨onnblad, PLB 779, 58 + ongoing

Torbj¨orn Sj¨ostrand Nonperturbative models in PYTHIA slide 15/23

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Cosmic ray cascades

high-energy cosmic ray in atmosphere,

not with PYTHIA, but could have been

Have implemented components needed for hadronic cascade

total and partial cross sections for hp/hn, from threshold to high energies

PDFs for different hadrons, for MPIs handling

rapid switching between hadrons and energies

atmosphere = nuclear targets by poor man’s Angantyr M. Utheim, TS, 2108.03481

Missing: incoming nuclei or photons, electromagnetic cascades

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Sample distributions

10-1 100 101 102 103 104

ECM (GeV) 0

20 40 60 80 100

σ (mb)

π+p cross sections as a function of energy total

LEHE NDLE HEel LEHE D/ELE HE

0.0 0.2 0.4 0.6 0.8 1.0

x

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

xv(x,Q

2 0)

Valence contents for some mesons at Q20= 0.26 qv

sKv qKv cDv qDv bBv qBv

10.0 7.5 5.0 2.5 0.0 2.5 5.0 7.5 10.0

y

0 2 4 6 8 10 12 14

(1/nev)dN/dy

y spectrum

+ K+ D0 J/B+

0 250 500 750 1000 1250 1500 1750

X (g/cm2) 100

101 102 103 104

Probability

Atmospheric depth of interactions

Uniform p/n atmosphere Uniform nitrogen Exponential nitrogen Exponential nitrogen at 45

Torbj¨orn Sj¨ostrand Nonperturbative models in PYTHIA slide 17/23

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Forward data

1000 2000 3000 4000 5000 6000

]-1 dN/dE [GeVine1/N

10

10

9

10

8

10

7

10

6

10

5

10

=13TeV photon s

LHCf

° φ=180

> 10.94, η

Ldt=0.191nb-1

Data QGSJET II-04 EPOS-LHC DPMJET 3.06 SIBYLL 2.3 PYTHIA 8.212

Energy [GeV]

1000 2000 3000 4000 5000 6000

MC/Data

0 1 2 3

4 1000 2000 3000 4000 5000 6000

]-1 dN/dE [GeVine1/N

10

10

9

10

8

10

7

10

6

10

5

10

=13TeV photon s

LHCf

° φ=20

<8.99, η 8.81<

Ldt=0.191nb-1

Energy [GeV]

1000 2000 3000 4000 5000 6000

MC/Data

0 1 2 3 4

Figure 4: Comparison of the photon spectra obtained from the experimental data and MC predictions. The top panels show the energy spectra, and the bottom panels show the ratio of MC predictions to the data. The hatched areas indicate the total uncertainties of experimental data including the statistical and the systematic uncertainties.

Acknowledgments

We thank the CERN staff and the ATLAS Collaboration for their essen- tial contributions to the successful operation of LHCf. This work was partly

290

supported by JSPS KAKENHI Grant Numbers JP26247037, JP23340076 and the joint research program of the Institute for Cosmic Ray Research (ICRR), University of Tokyo. This work was also supported by Istituto Nazionale di Fisica Nucleare (INFN) in Italy. Parts of this work were performed using the computer resource provided by ICRR (University of Tokyo), CERN and CNAF

295

(INFN).

References

[1] A. Aab et al. (Pierre Auger Collaboration), Nucl. Instrum. Methods Phys.

Res., Sect. A 798 (2015) 172.

14

V. Kireyeu et al.: Hadron production in elementary nucleon-nucleon reactions from low to ultra-relativistic energies 9

0.98 0.99 1 1.01 1.02 1.03 1.04 1.05 1.06

102 103

a) -

FSIon / FSIo

sNN, GeV

p+pp+n n+n

0.98 0.99 1 1.01 1.02 1.03 1.04 1.05 1.06

102 103

b) +

FSIon / FSIo

sNN, GeV

p+pp+n n+n

0.98 0.99 1 1.01 1.02 1.03 1.04 1.05 1.06

102 103

c) -

FSIon / FSIo

sNN, GeV

p+pp+n n+n

0.98 0.99 1 1.01 1.02 1.03 1.04 1.05 1.06

102 103

d) +

FSIon / FSIo

sNN, GeV

p+pp+n n+n

0.98 0.99 1 1.01 1.02 1.03 1.04 1.05 1.06

102 103

e) p

FSIon / FSIo

sNN, GeV

p+pp+n n+n

0.98 0.99 1 1.01 1.02 1.03 1.04 1.05 1.06

102 103

f) p

FSIon / FSIo

sNN, GeV

p+pp+n n+n

0.98 0.99 1 1.01 1.02 1.03 1.04 1.05 1.06

102 103

g) s0

FSIon / FSIo

sNN, GeV

p+pp+n n+n

0.98 0.99 1 1.01 1.02 1.03 1.04 1.05 1.06

102 103

h) + 0

FSIon / FSIo

sNN, GeV

p+pp+n n+n

Fig. 6. Ratios of total multiplicities with FSI (’FSIon’) and without FSI (’FSIo↵’) of ⇡±, K±, p , ¯p, Ks0and ⇤ + ⌃0produced in N + N collisions: the red lines correspond to p + p, blue lines – to p + n and green lines – to n + n reactions.

0 0.2 0.4 0.6 0.8 1

xF 0.2

0.4 0.6 0.8 1 1.2 1.4 1.6

FdN/dx

= 17.3 GeV sNN

p, PHSD PYTHIA NA49 data

0 0.2 0.4 0.6 0.8 1

xF 0.2

0.3 0.4 0.5 0.6 0.7 0.8

> (GeV/c)T<p

= 17.3 GeV sNN

p, PHSD PYTHIA NA49 data

0 0.2 0.4 0.6 0.8 1

xF 0.2

0.3 0.4 0.5 0.6 0.7 0.8

> (GeV/c)T<p

= 17.3 GeV sNN +, π

PHSD PYTHIA NA49 data

Fig. 7. Proton xF distribution (left plot) in p + p collisions atpsN N = 17.3 GeV. Mean transverse momentum < pT > of protons (middle plot) and ⇡+(right plot) as a function of xF in p + p collisions atpsN N= 17.3 GeV. The experimental data are taken from the NA49 Collaboration [40, 38].

V. Kireyeu et al., arXiv:2006.14739 LHCf, PLB 78, 233

Need mechanism for protons to take more energy (from pions)?

Diffractive-related or not?

Forward region also important for cosmic-ray physics.

Torbj¨orn Sj¨ostrand Nonperturbative models in PYTHIA slide 18/23

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Forward modelling

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 xF

0.0 0.5 1.0 1.5 2.0 2.5 3.0

dn/dxF

Nucleon Feynman-x distribution diquark default no popcorn + hard frag

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 xF

10-3 10-2 10-1 100 101

dn/dxF

π±,0 Feynman-x distribution single quark default no popcorn + hard frag

Two “improvements”:

• Forbid popcorn mechanism for remnant diquarks only;

i.e. baryon always produced at end of string, never meson

• Set a and b parameters separately in Lund fragmentation f (z)∝ 1

z(1− z)aexp



−bm2 z



with a = 0.68→ 0 (+. . . ) and b = 0.98 → 2

Torbj¨orn Sj¨ostrand Nonperturbative models in PYTHIA slide 19/23

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Comparison with LHCf

b b b b b b b b b b b b b b

b Data

default modified

0 0.0001 0.0002 0.0003 0.0004 0.0005ppn ats

=7 TeV with η>10.76

dσ/dE[mb/GeV] b b b b b b b b b b b b b b

500 1.0 ·103 1.5 ·103 2.0 ·103 2.5 ·103 3.0 ·103 3.5 ·103 0.50.6

0.70.8 0.91 1.11.2 1.31.4

Energy [GeV]

MC/Data b b b b b b b b b b b b b b b b b b b b b

b Data

default modified

10−4 10−3 10−2 10−1

ppπ0ats

=7 TeV for 0.0<pT[GeV/c] <0.2

1/σinelEd3σ/dp3[GeV2] b b b b b b b b b b b b b b b b b b b b b

1.0 ·103 1.5 ·103 2.0 ·103 2.5 ·103 3.0 ·103 3.5 ·103 0.50.6

0.70.8 0.91 1.11.2 1.31.4

Longitudinal momentum pz[GeV/c]

MC/Data

courtesy Max Fieg

Warning: limited acceptance; for baryons only at per cent level;

some additional (modest) tuning of primordial k also helped.

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Some topics not discussed

Tuning of fragmentation parameters, with new PSs and M&M Consequences of NLO (negative) PDFs in showers & MPIs Multi-parton PDFs (modelled in PYTHIA, but not tested) Partonic rescattering (3→ 3 etc. in MPIs)

Initial-state impact-parameter picture, e.g. Dipsy dipoles Colour reconnection and the top mass

Differences between quark and gluon jets Heavy-flavour production and hadronization Jet quenching in high-multiplicitypp systems (?)

Transition showers to hadronization; e.g. scale MPI, ISR, FSR Bose–Einstein (and Fermi–Dirac) effects

Deuteron, tritium, helium, tetraquark, pentaquark coalescence Diffraction; rapidity gaps and jets

Real and virtual photons, e.g. in ultraperipheral collisions . . .

Torbj¨orn Sj¨ostrand Nonperturbative models in PYTHIA slide 21/23

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Summary

Goodby jet universality, welcome new mechanisms!

Strangeness enhancement⇒ ropes?

Charm baryon enhancement⇒ junction reconnection?

Bottom asymmetries ⇒ beam drag+ above?

Hadronic p spectra⇒ rescattering + more?

Hadronic flow ⇒ shove+ rescattering?

Forward hadrons ⇒ differentremnant rules?

Also other issues:

Relation to AA: is there a Quark–Gluon Plasma already in pp?

Applications to other fields, like cosmic rays . . .

Before LHC we used to think the picture was messy, but now will we ever find back to such a “simple” description?

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Appendix: PYTHIA collaboration status

New organization as of May this year:

New home page: https://pythia.org New mail address: authors@pythia.org

• Spokesperson: Peter Skands

• Code master: Philip Ilten

• Web master: Christian Bierlich

Current authors:

Christian Bierlich Nishita Desai Leif Gellersen Ilkka Helenius Philip Ilten Leif L¨onnblad Stephen Mrenna Stefan Prestel Christian Preuss Torbj¨orn Sj¨ostrand Peter Skands Marius Utheim Rob Verheyen

Torbj¨orn Sj¨ostrand Nonperturbative models in PYTHIA slide 23/23

References

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