Parton Branching Algorithms & Improved Parton Showers Simon - PowerPoint PPT Presentation

Parton Branching Algorithms & Improved Parton Showers Simon Plätzer Particle Physics — University of Vienna at the CERN QCD Lunch Remote locations | 3 April 2020 Mainly based on recent work with Forshaw & Holguin

QCD, Event Generators & Phenomenology QCD description of collider reactions: Complexity challenges precision. Hard partonic scattering: NLO QCD routinely Jet evolution — parton showers: NLL sometimes, mostly unclear Multi-parton interactions Hadronization

Bottlenecks Parton shower algorithms Lack a systematic expansion, obstruct fully differential NNLO for the hard process, open questions regarding mass effects and unstable particles. Hadronization models Lack constraints from perturbative evolution: Hiding perturbative corrections? Genuine uncertainties/constraints? Rethink foundations of parton showers.

QCD Coherence Resummation of observables which globally measure deviations from n-jet limit. Use for analytic resummation and basis of parton shower algorithms.

QCD Coherence Parton branching algorithm: Parton shower: Formulates iterative structure of Numerical implementation of a contributions to cross sections. parton branching algorithm. Used to Could mean iterative structure of solve evolution equations stemming amplitudes or ‘density operators’. from a parton branching algorithm.

Non-global Observables No global measure of deviation from jet configuration: Coherent branching fails, full complexity of amplitudes strikes back. Even with a dipole approach 1/N effects possibly become comparable to subleading logarithms, and intrinsically 10% effects. Cannot ignore in the quest for higher precision.

Parton Branching at Amplitude Level Formulate iterative structures quite generally, with the goal of systematically approximating the iteration, not “iterating an approximation”. Similar in spirit to Nagy & Soper Theoretical control Actual predictions Guiding principle to Explore new methods incremental improvements and paradigms in their of existing algorithms. own right. Non-global observables and accuracy for global observables both set the level of complexity.

Parton Branching at Amplitude Level [Angeles, De Angelis, Forshaw, Plätzer, Seymour – JHEP 05 (2018) 044] [Forshaw, Holguin, Plätzer – JHEP 1908 (2019) 145] density operator observable phase space integration Z d R n V q ⊥ ,q n ⊥ D n A n − 1 ( q n ⊥ ; { p } n − 1 ) D † n V † A n ( q ⊥ ; { p } n ) = q ⊥ ,q n ⊥ Θ ( q ⊥  q n ⊥ ) . Z b ✓ ◆ d q ⊥ A 2 V a,b = Pexp � Γ n ( q ⊥ ) q ⊥ A 1 a A 0 a a p } n − 1 ) O D † D n ( q n ⊥ ; q n ∪ { ˜ n ( q n ⊥ ; q n ∪ { ˜ p } n − 1 ) = 2 1 Z Z |M i h M| 1 2 δ q ( i n ,j n ) δ q ( i n , ~ n ) X X n O S j n † n O C i n † ( q n ⊥ ) S i n ( q n ⊥ ) C i n + n , b n n ⊥ n ⊥ b i n ,j n i n Γ † D † D † D 2 Γ 1 D 1 H 0 1 2 collinear soft contributions contributions

Non-global Observables and Large-N [Angeles, De Angelis, Forshaw, Plätzer, Seymour – JHEP 05 (2018) 044] Primary application: Non-global observables E ∂ G n ( E ) = − Γ G n ( E ) − G n ( E ) Γ † + D µ nµ u ( E, ˆ n G n − 1 ( E ) D † k n ) . ∂ E Utilise colour flow basis, and expand around large-N: 1 1 ¯ 1 2 l 2 � ¯ X Leading ( l ) 2 τσ [ A ] = A τσ 1 /N k δ # transpositions ( τ , σ ) ,l − k � 3 3 � ¯ k =0 4 3 | 123 i | 213 i | 312 i h 123 | 123 i h 123 | 213 i h 123 | 312 i � � � � Re-derive BMS equation: Prototype of constructing a dipole shower 0 1 2 � � h i λ j W ( n ) λ i ¯ X V ( n ) Leading (0) � V ( n ) Leading (0) V n A n V † = exp @ − N = δ τσ τσ [ A n ] � � σ ij A n τσ σ � i,j c.c. in σ h i λ j R ( n ) Leading (0) λ i ¯ X Leading (0) D n A n − 1 D † = δ τσ τ \ n, σ \ n [ A n − 1 ] n ij τσ i,j c.c. in σ \ n colour connected dipoles

Beyond Leading Colour [Angeles, De Angelis, Forshaw, Plätzer, Seymour – JHEP 05 (2018) 044] virtuals reals Γ 3 [ τ | Γ | σ i = N δ τσ Γ σ + Σ τσ + 1 ( t [ ... ] t | 0 flips ) r − 1 t [ ... ] t | 2 flips × 1 (0 flips) × 1 × ( α s N ) n ( t [ ... ] t | 0 flips ) r − 1 t [ ... ] s | 1 flip × N − 1 N δ τσ ρ N 3 ( t [ ... ] t | 0 flips ) r − 1 s [ ... ] s | 0 flips × N − 2 Γ 2 ΣΓ 2 ( t [ ... ] t | 0 flips ) r (1 flip) × α s × ( α s N ) n ( t [ ... ] t | 0 flips ) r − 1 t [ ... ] s | 1 flip × N − 1 [Plätzer – EPJ C 74 (2014) 2907] N 2 ρ Γ 2 Γ ΣΓ (0 flips) × α s N − 1 × ( α s N ) n ( t [ ... ] t | 0 flips ) r N 1 Σ 2 Γ Systematically sum colour (0 flips) × α 2 ( t [ ... ] t | 0 flips ) r s × ( α s N ) n ( t [ ... ] t | 0 flips ) r − 1 t [ ... ] t | 2 flips (2 flips) × α 2 s × ( α s N ) n ρ Γ ρ ΣΓ Σ 1 enhanced terms N 0 Σ 2 Σ 3 ρ 2 Γ ρ 1 ρ Σ ρ Σ 2 N − 1 | σ i � 1 X V LC + NLC | σ i = V ( n ) δ # transpositions ( τ , σ ) , 1 Σ ( n ) ng α s N ∼ 1 . στ | τ i ρ 2 1 ρ 2 Σ n σ N τ easing powers N − 2 ρ 3 1 στ = N e − W ( n ) � e − W ( n ) σ τ Σ ( n ) ⇥ N − 3 W ( n ) � W ( n ) σ τ α 0 α 1 α 2 α 3 ⇣ ⌘ λ i λ j W ( n ) λ l W ( n ) λ l W ( n ) λ k λ j W ( n ) X + ¯ λ k ¯ � λ i ¯ � ¯ s s s s δ i,l c.c. in τ ij kl il kj k,j c.c. in τ i,k c.c. in σ dipole flips at next-to-leading colour j,l c.c. in σ

Collinear Subtractions [Forshaw, Holguin, Plätzer – JHEP 1908 (2019) 145] Identify and subtract collinear singularities in soft evolution softness Z d 3 k Z b 2 d q 2 1 ln W ab = ↵ s X T g i · T g ordering for soft evolution j q 2 ⇡ ( S · k ) 2 2 ⇡ 2 E a 2 i<j n i · n j K 2 ( p i , p j ; k ) � ( q 2 − K 2 ( p i , p j ; k )) ✓ ij ( k ) ordering for n i · n n · n j collinear evolution ! � ( q 2 − K 2 ( p i ; k )) ✓ i ( k ) − K 2 ( p j ; k ) − K 2 ( p i ; k ) � ( q 2 − K 2 ( p j ; k )) ✓ j ( k ) n i · n n j · n (Dipole) pt ordering Energy ordering Z d y d φ Z b d k ⊥ Z d Ω = α s Z b � X T g i · T g d E ✓ n i · n j − n i · n − n j · n ◆ energy = ↵ s ln W ab ( θ ij ( k ) − θ i ( k )) � � X T g i · T g ln W ab j � k ⊥ 2 π π � k T j E 4 ⇡ n i · n n · n j ⇡ � a a i<j i<j Z b Z Z = ↵ s d E ln n i · n j X T g i · T g Z d φ Z b Z 1 j E 2 ⇡ d k ⊥ d z = α s a � i<j ( T g X j ) 2 ln K ab � 2 π k ⊥ 1 − z + α 2 π � k T a α i Z d φ Z b Z 1 − α Z d Ω Z b d k ⊥ d z = α s d E 2 energy = ↵ s X ( T g � j ) 2 X ( T g i ) 2 ln K ab � 2 π k ⊥ 1 − z 2 π E 4 ⇡ n i · n ⇡ � a 0 a i i

Collinear Subtractions [Forshaw, Holguin, Plätzer – JHEP 1908 (2019) 145] Identify and subtract collinear singularities in soft evolution softness Z d 3 k Z b 2 d q 2 1 ln W ab = ↵ s X T g i · T g ordering for soft evolution j q 2 ⇡ ( S · k ) 2 2 ⇡ 2 E a 2 i<j n i · n j K 2 ( p i , p j ; k ) � ( q 2 − K 2 ( p i , p j ; k )) ✓ ij ( k ) ordering for n i · n n · n j collinear evolution ! � ( q 2 − K 2 ( p i ; k )) ✓ i ( k ) − K 2 ( p j ; k ) − K 2 ( p i ; k ) � ( q 2 − K 2 ( p j ; k )) ✓ j ( k ) n i · n n j · n Q p 2 ? 2 h Q | h M ( Q ) | p 1 ? 1 3 p 3 ? Q

Parton Showers from Amplitude Evolution [Forshaw, Holguin, Plätzer – arXiv:2003:06400] Start from amplitude evolution equations ∂ A n ( q ⊥ ; { p } n ) = − Γ n ( q ⊥ ) A n ( q ⊥ ; { p } n ) − A n ( q ⊥ ; { p } n ) Γ † n ( q ⊥ ) q ⊥ ∂ q ⊥ Z d R n D n ( q n ⊥ ) A n − 1 ( q n ⊥ ; { p } n − 1 ) D † + n ( q n ⊥ ) q ⊥ δ ( q ⊥ − q n ⊥ ) . n ! Z X Y d σ n ( µ ) = d Π i Tr A n ( µ ) , Σ ( µ ; { p } 0 , { v } ) = d σ n ( µ ) u ( { p } n , { v } ) , i =1 n Combine insight from soft evolution, large-N expansions and collinear subtractions: • Can we reproduce existing algorithms as well-defined limits of amplitude evolution? • Can we use this to obtain an ideal combination of coherent and dipole branching?

Collinear Subtractions & Angular Ordering [Forshaw, Holguin, Plätzer – arXiv:2003:06400] Collinear subtractions within a dipole? Recall angular ordering and coherent branching: n i n · n j n 2 P i n j n = n i n · n j n � n i n · n 1 n i n · n n j n · n = P i n j n + P j n i n , where + n i n · n n j n · n n i n · n Azimuthal average will result in angular ordering and simplify colour structures. j n φ n |M n | 2 u ( { p } n ) |M n | 2 ↵ ⌦ ↵ ⌦ 1 ,...,n = 1 ,...,n h u ( { p } n ) i 1 ,...,n θ n,j n n X |M n | 2 ↵ |M n | 2 ↵ X ⌦ ⌦ + σ m ( ⌦ ↵ 1 ,...,n ) σ m ( h u ( { p } n ) i 1 ,...,n ) Cor m ( 1 ,...,n , h u ( { p } n ) i 1 ,...,n ) θ i n ,j n m =1 + higher order correlations , i n irrelevant for global observables at NLL