Merging Procedure of Matrix-element and Parton Showering

For example, a merging procedure in events with jets is simply described as follows [49].

• All relevant cross sections including jets are calculated for a process such aspp → X+ njetswithn= 0,1, ..., N.

• Hard parton samples are produced with a probability proportional to the respective total cross section. The kinematics of the hard partons is based on the configuration following

the matrix element.

• The kinematics configuration is accepted or rejected with a dynamical and kinematics-dependent probability that includes both effects of the running coupling constant and of the Sudakov form factor. If it is rejected, a new parton sample is selected.

• The parton shower is invoked with the suitable initial condition. In all cases of merging algorithms, the parton shower is constrained not to produce any extra jet. In other words, the configuration that would fall into the realm of the matrix element with a higher jet multiplicity is vetoed in the parton shower step.

A difference in the merging algorithms mainly originates from a jet definition used in the matrix element; a way of the acceptance or the rejection of jet configurations stemming from the matrix element is performed; the details concerning starting conditions of the parton showering and the jet vetoing inside the parton showering.

For the CKKW scheme, the merging procedure is roughly described as follows.

• The separation of the matrix element and the parton shower realms for different multiple jet processes is performed on a k_⊥ variable. The algorithm sets k_⊥0, which indicates the merging scale.

• The internal jet identification at the matrix element level is done by using

k_⊥²_ij = 2min{p_⊥i, p_⊥j}²(cosh (ηi−ηj)−cos (φi−φj))

D² , (5.1)

whereiandjindicate two final state partons. Ifk²_⊥ij is larger than the critical valuek²_⊥0, the two final state particles belong to two different jets. The transverse momentum of each jet is larger than the merging scalek_⊥0. TheDparameter should be the same as a cone size of a jetk_talgorithm used in the analysis.

• The acceptance or the rejection of jet configurations proceeds through a reweighting of the matrix element with the Sudakov form factor, the factor has different scales inα_s. The algorithm identifies a splitting point, then, it combines two partons into the mother parton according to their kinematics at the scale.

• Through the clustering procedure, partons are connected. The procedure continues until the algorithm finds a core process,2→2.

• A starting scale for the parton shower evolution of each parton is given by the scalek_⊥0.

• A vetoed parton shower algorithm is used to guarantee that no unwanted hard jets are pro-duced during the showor evolution.

The weight attached to the generated matrix element consists of two components, a string coupling weight and an Sudakov form factor weight. For their determinations, ak_tjet algorithm guided by only physically allowed parton combinations is applied on the initial matrix element configuration.

The scales of the strong coupling constants correspond to thekT values at the splitting points. The Sudakov form factors provide a probability of having no emission during the evolution from higher scale to lower scale. The sequence of clusterings stops after identifying a core process,2 → 2.

Figure 5.2: Schematic view of the merging procedure in the CKKW thechnique.

This2→2core process defines the starting condition for the vetoed shower. Figure 5.2 shows a schematic view of the merging procedure in the CKKW thechnique.

Through the clustering procedure, the scale of the strong coupling constant on each parton changes α(Q_M)toα(Q_i)withi= 0,1,2. On the showering procedure, we remove the earlier emissions with highp_T, which were previously emitted before the gluon emission from the matrix element, and the hardest emission at higher scales than the merging scale. When we denote the merging scale asQ_cut, the values of the scales areQ₀> Q₁ > Q₂ >> Q_cut> Q₃ > Q₄.

For the MLM scheme, the merging procedure in theW+jets process is roughly described as follows.

• The first step is the generation of parton level configurations for all final state parton multi-plicitiesnup to a givenN. For example, when we requireW boson +N partons events, the samples with each parton multiplicity,W+ 1,W+ 2, ...,W+Npartons, will be produced.

They are defined by the following kinematical cuts.

p^parton_⊥ > p^min_⊥ , |ηparton|< ηmax, ∆Rjj > Rmin, (5.2) wherep^parton_⊥ andη_partonare the transverse momentum and the pseudo rapidity of the final state partons, respectively, and∆Rjj is their minimal separation is the (η, φ) plane. The parameters p^min_⊥ , η_max and R_min are called generation parameters, and are the same for n= 1, ..., N.

• The renormalization scale is set according to the CKKW prescription. A necessary tree branching structure, which is consistent with the colour structure of the event is defined for each event. For a pair of final state partonsiandj, we calculate a selection value defined by

d_ij = ∆R²_ijmin (p²_⊥i, p²_⊥j), (5.3) where∆R²_ij = ∆η_ij² + ∆φ²_ij. For a pair of initial state or final state partons, thed_ij is the same as thep²_⊥of the initial state or final state one.

• The selection value at each vertex is used as a scale for the relative power of α_s. The factorization scale for the parton densities is given by the hard scale of the process,Q²₀ = m²_W +p²_⊥W.

• Events are then showered by using PYTHIA or HERWIG. The evolution for each parton starts at the scale determined by the default PYTHIA and HERWIG algorithms on the basis of the kinematics and colour connections of the event. The upper veto cutoff to the shower evolution is given by the hard scale of the process Q₀. After the evolution, a jet cone algorithm is applied to the partons produced in the perturbative phase of the shower. The jets are defined by matching parameters, which are a cone sizeR_clus, the minimum transverse energyE_⊥^clusand the maximum pseudo rapidityη_max^clus.

• Matching procedure starts from the hardest parton and the jet which are closest in theφ−η plane. If the distance between the parton and the jet is smaller than1.5×R_clus, the matching is successful. The matching jet is removed from a list of jets, then, the matching procedure for subsequent partons is performed. The event is fully matched if each parton matches to a jet. If the matching is failure due to the very close partons, which cannot generate independent jets, the event is rejected.

• The algorithm requires that the passing event withn partons,n < N, have no extra jets, which leads to the larger number of jets than the number of partons, because the event with npartons and additional jets should be explained by the event with n+ 1partons sample.

This requirement prevents the double counting of events. The extra jets are removed by replacing the Sudakov reweighting used in the CKKW algorithm. In the case ofn = N, events with extra jets can be kept since they will not be generated by samples with higher n. In order to avoid double counting, their transverse momentum of extra jets should be smaller than that of the softest of the matched jets.