Colour Evolution Patrick Kirchgaeßer (KIT) with S. Gieseke (KIT), S.Plätzer (Vienna), A. Siodmok (Cracow) Lund, 27.2.19 Based on JHEP11(2018)149 Patrick Kirchgaeßer
Context/Motivation/Goals How do we decide which quarks to connect? [Pre-confinement as a property of pQCD, Amati, Veneziano] Interplay of hard & soft QCD Patrick Kirchgaeßer
Formalism: Perturbative colour evolution QCD scattering amplitudes are vectors in spin and colour space With the colour flow basis The bare amplitude can be related to the renormalized amplitude as |M ( { p } , µ 2 ) i = Z − 1 ( { p } , µ 2 , ✏ ) | ˜ M ( { p } , ✏ ) i and the renormalization constant Z is an operator in the space of colour structures. Interplay of hard & soft QCD Patrick Kirchgaeßer
Formalism: Perturbative colour evolution The structure of Z governed by RGE µ 2 d d µ 2 |M ( { p } , µ 2 ) i = Γ ( { p } , µ 2 ) |M ( { p } , µ 2 ) i Where the pre factor is called soft anomalous dimension Γ ( { p } , µ 2 ) = − Z − 1 ( { p } , µ 2 , ✏ ) µ 2 @ @ µ 2 Z ( { p } , µ 2 , ✏ ) The evolution equation can be solved by |M ( { p } , µ 2 ) i = U ( { p } , µ 2 , { M 2 ij } ) |H ( { p } , Q 2 , { M 2 ij } ) i where (Z M 2 ) dq 2 αβ q 2 Γ ( { p } , q 2 ) U = exp µ 2 Final colour structure depends on the soft anomalous dimension matrix Interplay of hard & soft QCD Patrick Kirchgaeßer
Formalism: Perturbative colour evolution Conjecture for soft anomalous dimension matrix [Becher, Neubert, Phys. Rev. Lett. 102 (2009)] γ cusp ( α s ) ln( µ 2 T i · T j X X γ i ( α s ) Γ = ) + 2 − s ij { i,j } i At 1-loop: γ cusp = α s / 2 π γ i Neglect (does not change the color structure) X X X Γ = Ω i ¯ Ω ij T i · T j + Ω ¯ j T i · T j + j T ¯ i · T ¯ i ¯ j i ≤ j i<j i<j Z M 2 ! M 2 dq 2 αβ α s αβ Ω αβ = ln − i π With q 2 q 2 2 π µ 2 ! 2 ln 2 M 2 M 2 1 = α s αβ αβ − i π ln q 2 µ 2 2 π Interplay of hard & soft QCD Patrick Kirchgaeßer
Formalism: Perturbative colour evolution Putting everything together 0 !1 2 ln 2 M 2 µ 2 − i π ln M 2 1 α s @X ij ij � { p } , µ 2 , { M 2 � ij } T i · T j U = exp A µ 2 2 π i 6 = j Starting point for evolution of a colour flow { p } , µ 2 , { M 2 � � A τ → σ = h σ | U ij } | τ i . Define reconnection probability |A τ → σ | 2 P τ → σ = P ρ |A τ → ρ | 2 Interplay of hard & soft QCD Patrick Kirchgaeßer
Example: Two cluster evolution Evolution in colour flow basis (compact notation) 1 1 � ¯ ¯ � 1 2 ¯ ... n ¯ ¯ � 1 2 ... δ ¯ 2 | σ i = = δ 1 δ n � 1 2 n ... n � Each index runs over N colours 2 cluster system (4 legs) 2 di ff erent color flows 1 1 1 1 2 2 1 1 1 1 2 2 States in color flow notation 2 2 ✓ 1 ◆ ✓ 0 ◆ � ¯ ¯ � 1 2 | 12 i = | 21 i = � | 2 1 i = | 1 2 i � 0 1 1 2 � [Plätzer, EPJC 74 (2014) 6] [Martinez, De Angelis, Forshaw, Plätzer, Seymour, JHEP 05 (2018) 044] Interplay of hard & soft QCD Patrick Kirchgaeßer
Example: Two cluster evolution Start evolution with initial colour flow | 12 i ✓ U 11 ◆ ✓ 1 ◆ U 21 | τ i = U | 12 i = = U 11 | 12 i + U 12 | 21 i U 12 U 22 0 Project out all possible color flows h 12 | τ i = U 11 h 12 | 12 i + U 12 h 12 | 21 i h 21 | τ i = U 11 h 21 | 12 i + U 12 h 21 | 21 i Where [Martinez, De Angelis, Forshaw, Plätzer, Seymour, 1802.08531] h σ | τ i = N m − #transpositions( σ , τ ) Define probability for alternative colour flow (reconnection probability) | h 21 | τ i | 2 P = | h 12 | τ i | 2 + | h 21 | τ i | 2 Interplay of hard & soft QCD 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 • Di ff erent input for hadronization (cluster model) 1 Interplay of hard & soft QCD Patrick Kirchgaeßer
Numerical results • Toy Monte Carlo for up to 5 clusters • 2 phase space algorithms (RAMBO,UA5 type model) to create quark kinematics 0 . 035 Initial clusters Initial clusters Final clusters Final clusters 10 − 1 0 . 030 0 . 025 10 − 2 0 . 020 N N 0 . 015 10 − 3 0 . 010 10 − 4 0 . 005 0 . 000 0 500 1000 1500 2000 0 500 1000 1500 2000 M [GeV] M [GeV] RAMBO high mass clusters (unphysical) UA5 with random initial connections Interplay of hard & soft QCD Patrick Kirchgaeßer
Numerical results • Change in Delta Y between the constituent quarks Initial clusters 0 . 05 Initial clusters Initial clusters 0 . 8 Final clusters Final clusters Final clusters 800 0 . 7 0 . 04 0 . 6 600 0 . 5 0 . 03 N N N 0 . 4 400 0 . 02 0 . 3 0 . 2 0 . 01 200 0 . 1 0 . 00 0 . 0 0 . 0 2 . 5 5 . 0 7 . 5 10 . 0 12 . 5 15 . 0 17 . 5 0 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 ∆ Y ∆ Y ∆ Y RAMBO UA5 with random initial connections Algorithm produces results attributed to properties of CR (e.g reduction of invariant cluster masses, connects quarks which are closer in spacetime) Interplay of hard & soft QCD Patrick Kirchgaeßer
Numerical results • Physical cluster masses O(GeV) initial µ = 1 GeV 10 1 µ = 1 GeV µ = 0 . 01 GeV 0 . 4 µ = 0 . 01 GeV 10 0 0 . 3 10 − 1 N N 0 . 2 10 − 2 0 . 1 10 − 3 0 . 0 0 1 2 3 4 5 6 7 − 2 . 0 − 1 . 5 − 1 . 0 − 0 . 5 0 . 0 0 . 5 1 . 0 ∆ if Mass[GeV] Colour length drop ! 2 ln 2 M 2 M 2 1 P M 2 Ω αβ = α αβ αβ − i π ln µ 2 µ 2 f 2 π P M 2 ∆ if = 1 − i Interplay of hard & soft QCD Patrick Kirchgaeßer
Bottleneck for full colour flow evolution of a LHC event Explicit form of soft anomalous dimension for the two cluster evolution Needs to be exponentiated For 5 clusters 120x120 matrix n! colourflows Exponentiated matrix here Rowland, Todd and Weisstein, Eric W. "Matrix Exponential." From MathWorld --A Wolfram Web Resource. http://mathworld.wolfram.com/MatrixExponential.html Interplay of hard & soft QCD Patrick Kirchgaeßer
Baryonic configurations • 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 Can construct a baryonic state (3 quarks, 3 antiquarks) | [ ijk ] i = 1 k = 1 K ✏ ijk ✏ ¯ K ( | ijk i � | ikj i � | jik i + | jki i + | kij i � | kji i ) j ¯ i ¯ Calculate baryonic reconnection probability as before where the amplitude is { p } , µ 2 , { M 2 � � A τ → B ijk ⊗ ˜ σ ijk = h B ijk | ⌦ h ˜ σ ijk | U ij } | τ i Interplay of hard & soft QCD Patrick Kirchgaeßer
Probabilities RAMBO 0 . 12 RAMBO Random UA5 UA5 Random 0 . 10 P Baryonic 0 . 08 0 . 06 0 . 04 0 . 02 3 4 5 Number of Clusters Reasonable values with clear dependence on number of clusters Interplay of hard & soft QCD Patrick Kirchgaeßer
Baryonic reconnection probabilities in detail 3 clusters 10 − 1 4 clusters 5 clusters 10 − 2 N 10 − 3 10 − 4 10 − 5 0 . 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 P Baryonic Tail towards higher values -> preferred kinematic configuration? Interplay of hard & soft QCD Patrick Kirchgaeßer
Baryonic reconnection probability for 3 cluster evolution median 0 . 5 0 . 4 P Baryonic 0 . 3 0 . 2 0 . 1 0 . 0 1 . 0 1 . 5 2 . 0 2 . 5 3 . 0 3 . 5 4 . 0 ( h ∆ R B i + h ∆ R ¯ B i ) / 2 Clear kinematic dependence Interplay of hard & soft QCD Patrick Kirchgaeßer
Independent subsystems • No 1/N dependence -> independent subsystems 4 clusters 5 clusters 14 8 12 10 6 N 8 N 6 4 4 2 2 0 0 0 1 2 3 0 1 2 3 4 transpositions transpositions E ff ects of perturbative colour flow evolution may be factorizable -> allows to implement in MCEG Interplay of hard & soft QCD 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 Patrick Kirchgaeßer
Backup Backup Backup Backup Backup Backup Patrick Kirchgaeßer
Backup Semi-hard MPI event only, random evolution of 10 clusters, B(10,5) = 252 3 cluster evo No veto of 4000 4 cluster evo colour flows 5 cluster evo 3500 sum cluster mass [GeV] 3000 2500 2000 1500 1000 500 0 0 50 100 150 200 250 iterations Interplay of hard & soft QCD Patrick Kirchgaeßer
Backup Extreme LHC event, iterated random evolution of 82 (very light) clusters, B(82,5) = 27285336 No veto of 3 cluster evo 1800 colour flows 4 cluster evo 5 cluster evo 1600 sum cluster mass [GeV] 1400 1200 1000 800 600 400 0 50 100 150 200 250 iterations Interplay of hard & soft QCD Patrick Kirchgaeßer
Perturbative colour flow evolution Colour charge operators T_i : [Catani, Seymour, Nucl. Phys. B485 (1997) 291-419 ] Colour charge products given by T i · T j = 1 i − 1 j δ j 0 i δ j 0 2( δ i 0 N δ i 0 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 Patrick Kirchgaeßer
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