Lecture 2: Matching in e + e − collisions Why Matching? Present matching approaches Matching of Matrix Elements and Parton Showers CKKW matching in e + e − collisions Lecture 2: Matching in e + e − collisions The MLM procedure Johan Alwall Theoretical Physics, SLAC HELAS/MadGraph Workshop, KEK, 18-27 Oct 2006 1 / 29
Lecture 2: Why Matching? Matching in e + e − collisions Why Matching? Present matching approaches CKKW matching in Why Matching? 1 e + e − collisions The MLM procedure Present matching approaches 2 CKKW matching in e + e − collisions 3 The MLM procedure 4 2 / 29
Lecture 2: Why Matching? Matching in e + e − collisions Why Matching? Present matching approaches CKKW matching in e + e − collisions The MLM procedure Matrix element and Parton shower approaches complementary. Want to combine them without double-counting! 3 / 29
Lecture 2: Present matching approaches Matching in e + e − collisions Why Matching? Present matching approaches Why Matching? 1 The Pythia approach The Herwig approach Present matching approaches 2 CKKW matching in e + e − collisions The Pythia approach The MLM procedure The Herwig approach CKKW matching in e + e − collisions 3 The MLM procedure 4 4 / 29
Lecture 2: The Pythia approach Matching in e + e − collisions Pythia uses a virtuality-ordered shower, similar to the descriptions presented last lecture. In the Pythia shower, the first emission is corrected such that its Why Matching? kinematical distribution corresponds to the matrix element expression for a Present matching number of simple processes (1 → 2 or 2 → 1-processes). approaches The Pythia approach The procedure is as follows: The Herwig approach Choose a starting scale for the (forward final-state or backward 1 CKKW matching in e + e − collisions initial-state) showering corresponding to the total amount of available energy in the event s . The MLM procedure With the kinematics of the Pythia parton shower, this choice of starting 2 scale ensures that the first radiation will over-populate the whole phase-space (above the cut-off). The produced radiation is then kept with a probability 3 d σ ME / d σ PS If the radiation is rejected, the shower evolution is continued from the 4 rejected t value onwards (the veto algoritm) Examples of ME-improved processes in Pythia Final-state: Z 0 → q ¯ q , t → bW + , h → q ¯ q , ... q → Z , W + , γ ∗ and gg → h (in heavy top limit) Initial-state: q ¯ 5 / 29
Lecture 2: The Herwig approach Matching in e + e − collisions Why Matching? The Herwig shower uses the branching angle as evolution variable. This Present matching approaches corresponds to explicitly taking into account the coherence effects, due to The Pythia interference between soft gluon emissions from different legs, giving strict approach The Herwig angular ordering (towards smaller emission angles), while it is not ordered in k T approach or virtuality. CKKW matching in e + e − collisions In Pythia, coherence effects are taken into account “post-facto” by vetoing increasing The MLM angles in the emission. procedure In Herwig, this gives rise to “dead cones”, where there is no emission at all from the parton showers. The matrix element corrections of Herwig is applied to the dead cones, and to all emissions which are found to be the “hardest so far”, since there the first emission does not necessarily have the highest k T . Examples of ME-improved processes in Herwig Some e + e − processes, DIS, top decay, Drell-Yan, gg → h (in heavy top limit) 6 / 29
CKKW matching in e + e − collisions Lecture 2: Matching in e + e − collisions Why Matching? Why Matching? 1 Present matching approaches Present matching approaches CKKW matching in 2 e + e − collisions Overview of the CKKW procedure CKKW matching in e + e − collisions Clustering the 3 n -jet event Sudakov reweighting Overview of the CKKW procedure Vetoed parton showers Clustering the n -jet event Highest multiplicity treatment Sudakov reweighting Results of CKKW matching (Sherpa) Vetoed parton showers Difficulties with practical implementations Highest multiplicity treatment The MLM procedure Results of CKKW matching (Sherpa) Difficulties with practical implementations The MLM procedure 4 7 / 29
CKKW matching in e + e − collisions Lecture 2: Matching in e + e − collisions Previous methods for matching ME and PS are restricted to certain processes (or classes of processes) and only allowed the matching of the first emission (or Why Matching? a fixed number of emissions). Present matching approaches The CKKW (Catani-Krauss-Kuhn-Webber) algorithm for ME-PS matching in e + e − collisions is constructed to allow corrections of (in principle) any number CKKW matching in e + e − collisions of emissions. Overview of the CKKW procedure Clustering the The method is constructed to use matrix elements to describe the distribution n -jet event of particle with a phase-space separation y ij > y cut (using some jet separation Sudakov reweighting measure y ), and parton showers to describe particles with a smaller separation Vetoed parton showers then y cut . Highest multiplicity For e + e − , the procedure uses the infrared-safe Durham (or k T ) jet measure: treatment Results of CKKW matching (Sherpa) “ ” Difficulties with E 2 i , E 2 (1 − cos θ ij ) / E 2 y ij = 2 min practical j cm implementations The MLM procedure The procedure ensures that the matrix elements behave as parton showers close to the cut-off, while the parton shower is vetoed above the cut-off in such a way that there is no dependence on the chosen cut-off value to NLL order = ⇒ smooth passage between regions described by parton showers and matrix elements 8 / 29
Lecture 2: Overview of the CKKW procedure Matching in e + e − collisions The procedure to generate configurations in e + e − → n jets at a c.m. energy Q 0 = E CM and jet resolution y ini can be summarized as follows: Why Matching? Present matching Select the jet multiplicity n and parton identities i with probability 1 approaches σ n , i CKKW matching in P ( n , i ) = e + e − collisions P k = N σ k , j Overview of the k , j CKKW procedure Clustering the where σ n , i is the tree-level e + e − → n -jet cross section with the strong n -jet event coupling calculated as α s ( d ini ) ( d ini = Q 0 √ y ini ) Sudakov reweighting Vetoed parton Pick parton momenta according to the n -parton matrix elements squared showers 2 Highest |M n , i | 2 , still using fixed α s ( d ini ), with jet resolution cut-off y ini multiplicity treatment Cluster the event to find the parton-shower history corresponding to the 3 Results of CKKW event and the splitting node resolution values d j = Q 0 √ y j for each jet matching (Sherpa) Difficulties with ( j > 2) practical implementations For each strong node (vertex) d k , apply a coupling-constant weight of 4 The MLM procedure α s ( d k ) /α s ( d ini ) < 1. For each quark or gluon line running between two nodes d j > d k (where d k 5 can be the cut-off scale d ini ), apply a Sudakov weight factor ∆ i ( d j ) / ∆ i ( d k ) < 1. Accept or reject the configuration according to the combined weight. 6 If the configuration is accepted, perform parton showers starting from the 7 generation scale of the parton, but vetoed above y ini 9 / 29
Lecture 2: Clustering the n -jet event Matching in e + e − collisions Why Matching? Present matching approaches CKKW matching in e + e − collisions Overview of the CKKW procedure Clustering the n -jet event Sudakov reweighting Vetoed parton showers Highest multiplicity treatment Results of CKKW matching (Sherpa) Find the two partons with smallest jet separation y ij 1 Difficulties with practical If partons allowed to cluster by QCD splitting rules: combine partons to 2 implementations new particle (e.g. q ¯ q → g , qg → q ) The MLM procedure Iterate 1-2 until 2 → 2 process reached ( e + e − → q ¯ q ) 3 With the choice of the Durham jet measure, the jet separations d i = √ y i Q 0 at each branching corresponds closely to the k T of that branching, and is therefore suitable to use as argument for α s in the branching. 10 / 29
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