[PPT] - Colour Evolution Patrick Kirchgaeer (KIT) with S. Gieseke (KIT), PowerPoint Presentation

SLIDE 1 Lund, 27.2.19 Patrick Kirchgaeßer Patrick Kirchgaeßer (KIT) with S. Gieseke (KIT), S.Plätzer (Vienna), A. Siodmok (Cracow)

Colour Evolution

Based on JHEP11(2018)149

SLIDE 2 Patrick Kirchgaeßer Context/Motivation/Goals How do we decide which quarks to connect? Interplay of hard & soft QCD [Pre-confinement as a property of pQCD, Amati, Veneziano]

SLIDE 3 Patrick Kirchgaeßer QCD scattering amplitudes are vectors in spin and colour space The bare amplitude can be related to the renormalized amplitude as and the renormalization constant Z is an operator in the space of colour structures.

|M({p}, µ2)i = Z−1({p}, µ2, ✏)| ˜ M({p}, ✏)i

Formalism: Perturbative colour evolution Interplay of hard & soft QCD With the colour flow basis

SLIDE 4 Patrick Kirchgaeßer Where the pre factor is called soft anomalous dimension The evolution equation can be solved by

Γ({p}, µ2) = −Z−1({p}, µ2, ✏)µ2 @ @µ2 Z({p}, µ2, ✏) |M({p}, µ2)i = U({p}, µ2, {M 2

ij})|H({p}, Q2, {M 2 ij})i where Final colour structure depends on the soft anomalous dimension matrix

U = exp (Z M 2

αβ µ2

dq2 q2 Γ({p}, q2) )

Formalism: Perturbative colour evolution Interplay of hard & soft QCD The structure of Z governed by RGE

µ2 d dµ2 |M({p}, µ2)i = Γ({p}, µ2)|M({p}, µ2)i

SLIDE 5 Patrick Kirchgaeßer Conjecture for soft anomalous dimension matrix [Becher, Neubert, Phys. Rev. Lett. 102 (2009)]

Γ = X

{i,j}

Ti · Tj 2 γcusp(αs) ln( µ2 −sij ) + X

i

γi(αs)

At 1-loop: γcusp = αs/2π Neglect

γi

(does not change the color structure)

Γ = X

i≤j

Ωi¯

jTi · Tj +

X

i<j

ΩijTi · Tj + X

i<j

Ω¯

i¯ jT¯ i · T¯ j With

Ωαβ = Z M 2

αβ µ2

dq2 q2 αs 2π ln M 2

αβ

q2 − iπ ! = αs 2π 1 2 ln2 M 2

αβ

q2 − iπ ln M 2

αβ

µ2 !

Formalism: Perturbative colour evolution Interplay of hard & soft QCD

SLIDE 6 Patrick Kirchgaeßer

U

{p}, µ2, {M 2

ij}

= exp

@X

i6=j

Ti · Tj αs 2π 1 2 ln2 M 2

ij

µ2 − iπ ln M 2

ij

µ2 !1 A

Aτ→σ = hσ|U

{p}, µ2, {M 2

ij}

|τi .

Pτ→σ = |Aτ→σ|2 P

ρ |Aτ→ρ|2 Putting everything together Starting point for evolution of a colour flow Define reconnection probability Formalism: Perturbative colour evolution Interplay of hard & soft QCD

SLIDE 7 Patrick Kirchgaeßer Evolution in colour flow basis (compact notation)

|σi =

¯

1 ¯ 2 ... ¯ n 1 2 ... n

= δ

¯ 1 1 δ ¯ 2 2 ... δ¯ n n 2 cluster system (4 legs) 2 different color flows 1 1 2 2 1 1 2 2 1 1 1 1 2 2 1 1

¯

1 ¯ 2 1 2

= |1 2i

|2 1i

States in color flow notation

|12i = ✓1 ◆ |21i = ✓0 1 ◆

[Plätzer, EPJC 74 (2014) 6] [Martinez, De Angelis, Forshaw, Plätzer, Seymour, JHEP 05 (2018) 044] Each index runs over N colours Example: Two cluster evolution Interplay of hard & soft QCD

SLIDE 8 Patrick Kirchgaeßer Start evolution with initial colour flow |12i Project out all possible color flows

h12|τi = U11h12|12i + U12h12|21i h21|τi = U11h21|12i + U12h21|21i hσ|τi = N m−#transpositions(σ,τ)

Where

P = |h21|τi|2 |h12|τi|2 + |h21|τi|2

Define probability for alternative colour flow (reconnection probability)

|τi = U|12i = ✓U11 U21 U12 U22 ◆ ✓1 ◆ = U11|12i + U12|21i

[Martinez, De Angelis, Forshaw, Plätzer, Seymour, 1802.08531] Example: Two cluster evolution Interplay of hard & soft QCD

SLIDE 9 Patrick Kirchgaeßer In short

Study small systems (2-5 clusters = 4-10 coloured legs) (toy mc)
Consider iterated soft gluon exchange between any two legs to all orders
Evolve colour structure of legs to decide which quarks to connect
Different input for hadronization (cluster model)

1 Interplay of hard & soft QCD

SLIDE 10 Patrick Kirchgaeßer

Toy Monte Carlo for up to 5 clusters
2 phase space algorithms (RAMBO,UA5 type model) to create quark kinematics

RAMBO high mass clusters (unphysical) 500 1000 1500 2000 M[GeV] 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 N Initial clusters Final clusters 500 1000 1500 2000 M[GeV] 10−4 10−3 10−2 10−1 N Initial clusters Final clusters UA5 with random initial connections Numerical results Interplay of hard & soft QCD

SLIDE 11 Patrick Kirchgaeßer 1 2 3 4 5 6 ∆Y 200 400 600 800 N Initial clusters Final clusters Algorithm produces results attributed to properties of CR (e.g reduction of invariant cluster masses, connects quarks which are closer in spacetime) RAMBO UA5 with random initial connections 1 2 3 4 5 6 7 ∆Y 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 N Initial clusters Final clusters

Change in Delta Y between the constituent quarks

0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 ∆Y 0.00 0.01 0.02 0.03 0.04 0.05 N Initial clusters Final clusters Numerical results Interplay of hard & soft QCD

SLIDE 12 Patrick Kirchgaeßer

Physical cluster masses O(GeV)

Colour length drop

Ωαβ = α 2π 1 2 ln2 M 2

αβ

µ2 − iπ ln M 2

αβ

µ2 ! ∆if = 1 − P M 2

f

P M 2

i 1 2 3 4 5 6 7 Mass[GeV] 0.0 0.1 0.2 0.3 0.4 N initial µ = 1 GeV µ = 0.01 GeV −2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 ∆if 10−3 10−2 10−1 100 101 N µ = 1 GeV µ = 0.01 GeV Numerical results Interplay of hard & soft QCD

SLIDE 13 Patrick Kirchgaeßer Explicit form of soft anomalous dimension for the two cluster evolution Needs to be exponentiated Exponentiated matrix here Rowland, Todd and Weisstein, Eric W. "Matrix Exponential." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/MatrixExponential.html For 5 clusters 120x120 matrix n! colourflows Bottleneck for full colour flow evolution of a LHC event Interplay of hard & soft QCD

SLIDE 14 Patrick Kirchgaeßer Baryonic configurations Can construct a baryonic state (3 quarks, 3 antiquarks) Calculate baryonic reconnection probability as before where the amplitude is

|[ijk]i = 1 K ✏ijk✏¯

i¯ j¯ k = 1

Severe underestimation of produced baryons (strangeness also) at LHC

(ALICE, ATLAS, CMS)

Improved description with new production mechanisms possible

[Pythia, (Christiansen, Skands) JHEP08 (2015) 003] [Herwig, (Gieseke, PK, Plätzer) Eur.Phys.J. C78 (2018) 99]

Herwig: Baryonic clusters

Aτ→Bijk⊗˜

σijk = hBijk| ⌦ h˜

σijk|U

{p}, µ2, {M 2

ij}

|τi

Interplay of hard & soft QCD

SLIDE 15 Patrick Kirchgaeßer Probabilities Reasonable values with clear dependence on number of clusters 3 4 5

Number of Clusters

0.02 0.04 0.06 0.08 0.10 0.12

PBaryonic

RAMBO RAMBO Random UA5 UA5 Random Interplay of hard & soft QCD

SLIDE 16 Patrick Kirchgaeßer Baryonic reconnection probabilities in detail Tail towards higher values -> preferred kinematic configuration? 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

PBaryonic

10−5 10−4 10−3 10−2 10−1

N

3 clusters 4 clusters 5 clusters Interplay of hard & soft QCD

SLIDE 17 Patrick Kirchgaeßer Baryonic reconnection probability for 3 cluster evolution 1.0 1.5 2.0 2.5 3.0 3.5 4.0 (h∆RBi + h∆R ¯ Bi)/2 0.0 0.1 0.2 0.3 0.4 0.5 PBaryonic median Clear kinematic dependence Interplay of hard & soft QCD

SLIDE 18 Patrick Kirchgaeßer Independent subsystems 1 2 3 transpositions 2 4 6 8 10 12 14 N 4 clusters 1 2 3 4 transpositions 2 4 6 8 N 5 clusters

No 1/N dependence -> independent subsystems

Effects of perturbative colour flow evolution may be factorizable

> allows to implement in MCEG

Interplay of hard & soft QCD

SLIDE 19 Patrick Kirchgaeßer Summary

Toy Monte Carlo for full Colour Flow Evolution for up to 5 clusters
Evolution into baryonic states possible
Strong support for geometric/kinematic models
Future: Study evolution of independent subsystems in Herwig and see where this

approach is applicable Interplay of hard & soft QCD

SLIDE 20 Patrick Kirchgaeßer Backup

Backup

SLIDE 21 Patrick Kirchgaeßer Backup Semi-hard MPI event only, random evolution of 10 clusters, B(10,5) = 252 50 100 150 200 250 iterations 500 1000 1500 2000 2500 3000 3500 4000 sum cluster mass [GeV] 3 cluster evo 4 cluster evo 5 cluster evo No veto of colour flows Interplay of hard & soft QCD

SLIDE 22 Patrick Kirchgaeßer Backup Extreme LHC event, iterated random evolution of 82 (very light) clusters, B(82,5) = 27285336 50 100 150 200 250 iterations 400 600 800 1000 1200 1400 1600 1800 sum cluster mass [GeV] 3 cluster evo 4 cluster evo 5 cluster evo No veto of colour flows Interplay of hard & soft QCD

SLIDE 23 Patrick Kirchgaeßer Perturbative colour flow evolution Colour charge operators T_i : [Catani, Seymour, Nucl. Phys. B485 (1997) 291-419 ] Colour charge products given by

Ti · Tj = 1 2(δi0

j δj0 i − 1

N δi0

i δj0 j ) In the large N limit corresponds to exchange of colour between legs i,j In color flow basis: Matrices in colour space which change the colour structure

SLIDE 24 Patrick Kirchgaeßer Consider evolution of an amplitude in colour space

M = UM0 Γ = X

{i,j}

Ti · Tj 2 αs 2π ln M 2

αβ

q2 − iπ ! U = exp Z M 2

αβ µ2

dq2 q2 Γ({p}, q2) !

Colour structure of an event determined by the 1-loop soft anomalous dimension Colour Evolution 1 1 2 2 Swaps/crosses colour lines between legs

SLIDE 25 Patrick Kirchgaeßer

|σi = δi

¯ i...δj ¯ j Colour state can be represented as product of kronecker deltas Possible Colour flows of a 2 cluster system 1 1 2 2 1 1 1 1 2 2 1 1 Calculate and exponentiate soft anomalous dimension matrix (2x2) for 2 clusters kinematics -> matrix elements of U

M 2

i¯ i = (pi + p¯ i)2 Colour Evolution for a 2 cluster system Radiating dipole

SLIDE 26 Patrick Kirchgaeßer Perturbative colour flow evolution