From the previous discussion it should be clear that there are two different approaches for the calculation of observables in hadron collisions: the matrix element approach which relies on perturbative calculations performed at a given fixed order and the parton shower approach which includes all-order contributions in the collinear limit.
The two approaches are complementary in the sense that in the soft/collinear region, where the parton shower approach is valid, the matrix-element calculation breaks down due to the appearance of large logarithms, while in the hard/wide-angle emis-sion region, where the matrix-element calculation provides a good description, the approximations involved in the parton shower approach become invalid.
Combining the ME with the PS calculations offers the advantages of both ap-proaches, extending the validity of the perturbative calculations to the whole of the phase space, as well as allowing to make the ME predictions exclusive and interface them to hadronization generators. While combining LO matrix elements with parton showers poses no difficulty, from NLO onwards, ambiguities start to arise due to the multiple counting of certain configurations, as illustrated in Figure1.6. For instance a(N+1)-jet event can be obtained both from a NLO correction to aN-jet event and from extra emissions added by the parton shower to theN-jet event. The same prob-lem arises also when attempting to merge LO+PS calculations of different final state particle multiplicities.
Several schemes have been constructed that allow for the matching of NLO calcula-tions with parton showers and for the merging of LO+PS calculacalcula-tions with different final state particle multiplicities, avoiding the double counting problem, as explained in the following.
Figure 1.6:Schematic illustration of the double counting problem in the matching and merging of ME and PS calculations. Figure (a) illustrates a LO Feynman diagram of a pp→e+e−process. Figure (b) represents a possible configuration that may result from propagating the event in Figure (a) through a parton shower algorithm.
Figure (c) represents a real emission (NLO) correction to the diagram in Figure (a) and corresponds to the same configuration as the one obtained from the parton shower (Figure b). Therefore naïvely matching NLO computations to parton showers results in a double counting of configurations. Alternatively Figure (c) can be thought of as the LO contribution to the pp→e+e−+1 jet process, therefore merging LO+PS calculations with different final state particle multiplicities also results in the double counting of certain configurations.
Merging LO+PS calculations with different final state particle multiplicities
There are three schemes for merging LO matrix-element calculations interfaced to parton showers: CKKW [63], CKKW-L [64] and MLM [65]. The fundamental concept of these merging schemes is the partitioning of the phase space into two regions with the use of a transition scale yini defined by a given jet measure y. Ifycut is the resolution variable of a given jet algorithm, then forycut > yini the observables are taken from matrix elements modified by Sudakov form factors, while forycut <yini the observables are taken from the parton showers subjected to a veto procedure. In more detail, the different merging schemes follow the same basic procedure
1. definition of a jet measureyand calculation of the cross sections for the processes pp →X+n−jets withn =0, 1, . . . ,nmax. Typical examples include the jet radius used in the Alpgen implementation of the MLM matching and thekT distance used in CKKW matching.
2. generation of hard partons with a probability proportional to the total cross section and a kinematic configuration given by the matrix element
3. acceptance or rejection of the configuration with a probability that includes Sudakov and running coupling effects
4. parton showering with a ‘veto’ that rejects events with extra jets The differences between the merging schemes lie in
• the definition of the jet measure
• the way the acceptance/rejection of step (2) above is carried out
• the initial conditions for the parton shower algorithms and the application of the
‘veto’.
A comparative study of the aforementioned merging algorithms was performed in [66].
Matching NLO calculations with parton showers
There are two methods for matching NLO matrix elements with parton showers, dubbed MC@NLO [67] and POWHEG [67,68]. In both of these methods, the emission with the highest pT is taken from the NLO matrix element (with the shower approxi-mation subtracted) and the following emissions are taken from the parton shower and are thus only reliable in the collinear limit.
An observable calculated with NLO+PS accuracy can be schematically written as
hOiNLOPS =
In the above equations,B,V,Rare the Born, virtual and real emission matrix elements multiplied by the PDFs,Rsis the the real emission contribution in the soft and collinear
limit andΦn,Φr parametrize the phase space which hasnfinal state particles in the case of the Born and virtual contributions andn+1 particles in the case of the real emission contributions. The last term in (1.32) is the so-calledmatrix element correction, which provides the hard, large-angle contribution to the hardest emission calculated from the NLO matrix element.
Some comments are in order with respect to the specific implementation of the NLOPS formalism in the POWHEG and MC@NLO approaches. The first term of (1.32) ¯B(Φn)dΦB is what is called aS event (for Standard MC evolution) and the last term [R(Φn+1)−Rs(Φn+1)]dΦn+1 is called a H event (for Hard MC evolution). In MC@NLO the differenceR(Φn+1)−Rs(Φn+1)can become negative, thus leading to the appearance of events with a negative weight. In POWHEG on the other hand, one has a freedom to choose a parametrizationRs(Φn+1) = R(Φn+1)F(Φn+1)with 0 ≤ F(Φn+1) ≤1 and F(Φn+1) → 1 in the soft and collinear limit. ThusR(Φn+1)− Rs(Φn+1) = R(Φn+1)[1−F(Φn+1)] ≥ 0. Another difference with MC@NLO is that the part within the curly braces in (1.32), which corresponds to the hardest emission, is generated within POWHEG and is thus independent of the showering generator.
Finally, we note that although the two approaches are equivalent at NLO, differences may arise at NNLO.