A shower algorithm based on the dipole formalism Stefan Weinzierl Universität Mainz in collaboration with M. Dinsdale and M. Ternick Introduction: I.: II: III: Event generators and perturbative calculations Parton showers Showers from dipoles Numerical results
Event generators pdf s hard scattering parton shower hadronization and decay Underlying event: Multiple interactions: Pile-up events: Interactions of the proton remnants. more than one pair of partons undergo hard scattering more than one hadron-hadron scattering within a bunch crossing
Exact perturbative calculations Leading order (LO) and next-to-leading order (NLO): At leading order only Born amplitudes contribute: At next-to-leading order: One-loop amplitudes and Born amplitudes with an additional parton. 2 Re +
Infrared divergences and the Kinoshita-Lee-Nauenberg theorem In addition to ultraviolet divergences, loop integrals can have infrared divergences. For each IR divergence there is a corresponding divergence with the opposite sign in the real emission amplitude, when particles becomes soft or collinear (e.g. unresolved). The Kinoshita-Lee-Nauenberg theorem: Any observable, summed over all states degenerate according to some resolution criteria, will be finite.
General methods at NLO Fully differential NLO Monte Carlo programs need a general method to handle the cancelation of infrared divergencies. Phase space slicing e + e : W. Giele and N. Glover, (992) initial hadrons: W. Giele, N. Glover and D.A. Kosower, (993) massive partons, fragmentation: S. Keller and E. Laenen, (999) Subtraction method residue approach: S. Frixione, Z. Kunzst and A. Signer, (995) dipole formalism: S. Catani and M. Seymour, (996) massive partons: L. Phaf and S.W. (2), S. Catani, S. Dittmaier, M. Seymour and Z. Trócsányi, (22)
The dipole formalism The dipole formalism is based on the subtraction method. The NLO cross section is rewritten as Z Z σ NLO = = n+ Z n+ dσ R + n dσ V ( dσ R dσ A) + Z n dσ V + The approximation dσ A has to fulfill the following requirements: Z dσ A dσ A must be a proper approximation of dσ R such as to have the same pointwise singular behaviour in D dimensions as dσ R itself. Thus, dσ A acts as a local counterterm for dσ R and one can safely perform the limit ε. Analytic integrability in D dimensions over the one-parton subspace leading to soft and collinear divergences.
The subtraction terms The approximation term dσ A is given as a sum over dipoles: dσ A = pairs i, j D i j,k. k i, j Each dipole contribution has the following form: i j k D i j,k = A n () 2p i p j ( p,..., p (i j),..., p k,... ) T k T i j ( T 2 V i j,k A n () p,..., p (i j),..., p k,... ). i j Colour correlations through T k T i j. Spin correlations through V i j,k. The dipoles have the correct soft and collinear limit.
The physical origin of the correlations In the soft limit, amplitudes factorize completely in spin space, but colour correlations remain. In the collinear limit, amplitudes factorize completely in colour space, but spin correlations remain. Complete factorization after average over azimuthal angle.
Basics of shower algorithm Starting point: Collinear factorization. Probability for particle a to split into particles b and c: dp a = α s 2π P a bc(z)dtdz, b,c ( ) t = ln Q 2 Λ 2 Splitting kernels: P q qg (z) = C F + z 2 z, ( z( z)) 2 P g gg (z) = C A, z( z) P g q q (z) = T R ( z 2 + ( z) 2). a b c P q qg has a soft singularity for z, P g gg has a soft singularity for z and z.
The Sudakov factor Probability that a branching occurs during a small range of t: di (t) = dt z + (t) Z z (t) dz b,c α s 2π P a bc(z), Sudakov factor: Probability that no branching occurs between t and t : (t,t ) = exp Z t t dt z Z + (t) dz b,c z (t) α s 2π P a bc(z)
A typical shower algorithm Choose the next scale t according to the Sudakov factor. Choose the momentum fraction z according to P a bc (z). Choose the azimuthal angle uniform or according to spin-dependent splitting functions. Insert the new particle. If t > t min goto first step, otherwise stop.
Angular ordering Amplitude for the emission of a soft gluon from a q- q-antenna: dσ g = dσ α s C F π dk k dφ 2π d cosθ cosθ q q ( cosθ qg )( cosθ g q ) cosθ q q ( cosθ qg )( cosθ g q ) = W q +W q, W q = [ cosθ g q cosθ q q 2 ( cosθ qg )( cosθ g q ) + [ W q = 2 ( cosθ qg ) cosθ qg cosθ q q ( cosθ qg )( cosθ g q ) + ( cosθ g q ) ], ]. Z dφ 2π W q = { cosθ qg, if θ qg < θ q q, otherwise Angular ordering: No emission if θ qg > θ q q!
Momentum conservation. Momentum conservation: p a = p b + p c a b c 2. Momenta are on-shell, for massless particles: p 2 a = p 2 b = p 2 c =. 3. Momenta are real. For 2 splittings it is not possible to satisfy all three requirements.
Recent developments Rewriting and improvement of Pythia, Herwig and Ariadne, Sherpa as a new event generator Sjöstrand, Skands; Gieseke, Stephens, Webber; Lönnblad, Krauss, Kuhn, Schälicke, Soff; Uncertainties of parton showers Gieseke; Stephens, van Hameren; Bauer, Tackmann Matching of parton showers with fixed-order tree level matrix elements Catani, Krauss, Kuhn, Webber; Mangano, Moretti, Pittau; Mrenna and Richardson; Matching of parton showers with NLO Frixione, Gieseke, Laenen, Latunde-Dada, Motylinski, Nason, Oleari, Ridolfi, Webber; Krämer, Mrenna, Soper; Odaka, Kurihara; Giele, Kosower, Skands; Parton shower based on the dipole formalism Proposal by Nagy, Soper; Implementation by Schumann, Krauss and Dinsdale, Ternick, SW.
Parton shower based on the dipole formalism 2 3 splittings: An emitter-spectator pair radiates off an additional particle. Can satisfy momentum conservation and on-shell conditions. Splitting kernels of the Sudakov factors are given by the dipole splitting functions. Correct behaviour in the collinear and the soft limit. No conceptional distinction between initial- and final-state shower. Natural choice to combine with NLO
Technical details 4 cases for emitter-spectator-pair: final-final, final-initial, initial-final, initial-initial. Only the singular terms of the dipole splitting functions are unique. Freedom to choose the finite terms. For a parton shower we would like to have a probabilistic interpretation: The splitting functions have to be non-negative everywhere. Adjust finite terms Rearrange terms between D i j,k and D k j,i
Technical details Sudakov factor: i j,k (t,t 2 ) = exp Z t t 2 dtcĩ, k Z dφ unres δ ( ) t Tĩ, k Pi j,k, t = ln k2 Q. 2 Dipole phase space: Z dφ unres = (p ĩ + p k )2 6π 2 Z dκ z + (κ) Z z (κ) dz 4z( z) ( ) κ, κ = 4 ( k2 ) 4z( z) (pĩ + p k) 2.
Numerical results: Electron-positron annihilation.6.4 Q min = GeV Q min = 2GeV 3.5 3 Q min = GeV Q min = 2GeV dσ dc C σ C.2.8.6.4.2 dσ dd D σ D 2.5 2.5.5..2.3.4.5 C.6.7.8.9..2.3.4.5 D.6.7.8.9 6 5 Q min = GeV Q min = 2GeV dσ d( T) T σ T 4 3 2.5..5.2.25 T.3.35.4.45.5
Numerical results: Four-jet angles.5.4 strict mod.9 strict mod dσ σ d cosθ NR.3.2. dσ σ d cosθksw.8.7.6.9.5.8..2.3.4.5.6 cosθ NR.7.8.9.4 - -.8 -.6 -.4 -.2.2 cosθ KSW.4.6.8 modified Nachtmann-Reiter angle Körner-Schierholz-Willrodt angle 3 2.5 strict mod.7.6.5 strict mod dσ σ d cos χbz 2.5 dσ σ d cosα34.4.3.2..5..2.3.4.5.6 cosχ BZ.7.8.9 - -.8 -.6 -.4 -.2 cosα 34.2.4.6.8 Bengtsson-Zerwas angle angle α 34
Z/γ -production at the Tevatron and at the LHC.45.3.4.35 Tevatron.25 Tevatron dσ σ d p [/GeV].3.25.2.5 dσ σ dy.2.5...5.5 2 4 6 p 8 [GeV] 2 4-6 -4-2 y 2 4 6.6.8 dσ σ d p [/GeV].4.2..8.6.4.2 LHC dσ σ dy.6.4.2..8.6.4.2 LHC 2 4 6 p 8 [GeV] 2 4-6 -4-2 y 2 4 6
Summary Implementation of a new parton shower algorithm based on the dipole formalism. Transverse momentum as evolution variable. Momentum conservation and angular ordering are inherent. Initial- and final-state partons are treated on the same footing. Natural choice to combine with NLO.