<chapter name="Matching and Merging"> 
 
<h2>Matching and Merging</h2> 
 
Starting from a Born-level leading-order (LO) process, higher orders 
can be included in various ways. The three basic approaches would be 
<ul> 
<li>A formal order-by-order perturbative calculation, in each order 
higher including graphs both with one particle more in the final 
state and with one loop more in the intermediate state. This is 
accurate to the order of the calculation, but gives no hint of 
event structures beyond that, with more particles in the final state. 
Today next-to-leading order (NLO) is standard, while 
next-to-next-to-leading order (NNLO) is coming. This approach 
thus is limited to few orders, and also breaks down in soft and 
collinear regions, which makes it unsuitable for matching to 
hadronization. 
</li> 
<li>Real emissions to several higher orders, but neglecting the 
virtual/loop corrections that should go with it at any given order. 
Thereby it is possible to allow for topologies with a large and 
varying number of partons, at the prize of not being accurate to any 
particular order. The approach also opens up for doublecounting, 
and as above breaks down in soft and colliner regions. 
</li> 
<li>The parton shower provides an approximation to higher orders, 
both real and virtual contributions for the emission of arbitrarily 
many particles. As such it is less accurate than either of the two 
above, at least for topologies of well separated partons, but it 
contains a physically sensible behaviour in the soft and collinear 
limits, and therefore matches well onto the hadronization stage. 
</li> 
</ul> 
Given the pros and cons, much of the effort in recent years has 
involved the development of different prescriptions to combine 
the methods above in various ways. 
 
<p/> 
The common traits of all combination methods are that matrix elements 
are used to describe the production of hard and well separated 
particles, and parton showers for the production of soft or collinear 
particles. What differs between the various approaches that have been 
proposed are which matrix elements are being used, how doublecounting 
is avoided, and how the transition from the hard to the soft regime 
is handled. These combination methods are typically referred to as 
"matching" or "merging" algorithms. There is some confusion about 
the distinction between the two terms, and so we leave it to the 
inventor/implementor of a particular scheme to choose and motivate 
the name given to that scheme. 
 
<p/> 
PYTHIA comes with methods, to be described next, that implement 
or support several different kind of algorithms. The field is 
open-ended, however: any external program can feed in 
<aloc href="LesHouchesAccord">Les Houches events</aloc> that 
PYTHIA subsequently showers, adds multiparton interactions to, 
and hadronizes. These events afterwards can be reweighted and 
combined in any desired way. The maximum <ei>pT</ei> of the shower 
evolution is set by the Les Houches <code>scale</code>, on the one 
hand, and by the values of the <code>SpaceShower:pTmaxMatch</code>, 
<code>TimeShower:pTmaxMatch</code> and other parton-shower settings, 
on the other. Typically it is not possible to achieve perfect 
matching this way, given that the PYTHIA <ei>pT</ei> evolution 
variables are not likely to agree with the variables used for cuts 
in the external program. Often one can get close enough with simple 
means but, for an improved matching, 
<aloc href="UserHooks">User Hooks</aloc> can be inserted to control 
the steps taken on the way, e.g. to veto those parton shower branchings 
that would doublecount emissions included in the matrix elements. 
 
<p/> 
Zooming in from the "anything goes" perspective, the list of relevent 
approaches actively supported is as follows. 
<ul> 
 
<li>For many/most resonance decays the first branching in the shower is 
merged with first-order matrix elements <ref>Ben87, Nor01</ref>. This 
means that the emission rate is accurate to NLO, similarly to the POWHEG 
strategy (see below), but built into the 
<aloc href="TimelikeShowers">timelike showers</aloc>. 
The angular orientation of the event after the first emission is only 
handled by the parton shower kinematics, however. Needless to say, 
this formalism is precisely what is tested by <ei>Z^0</ei> decays at 
LEP1, and it is known to do a pretty good job there. 
</li> 
 
<li>Also the <aloc href="SpacelikeShowers">spacelike showers</aloc> 
contain a correction to first-order matrix elements, but only for the 
one-body-final-state processes 
<ei>q qbar &rarr; gamma^*/Z^0/W^+-/h^0/H^0/A0/Z'0/W'+-/R0</ei> 
<ref>Miu99</ref> and <ei>g g &rarr; h^0/H^0/A0</ei>, and only to 
leading order. That is, it is equivalent to the POWHEG formalism for 
the real emission, but the prefactor "cross section normalization" 
is LO rather than NLO. Therefore this framework is less relevant, 
and has been superseded the following ones. 
</li> 
 
<li>The POWHEG strategy <ref>Nas04</ref> provides a cross section 
accurate to NLO. The hardest emission is constructed with unit 
probability, based on the ratio of the real-emission matrix element 
to the Born-level cross section, and with a Sudakov factor derived 
from this ratio, i.e. the philosophy introduced in <ref>Ben87</ref>. 
<br/>While POWHEG is a generic strategy, the POWHEG BOX 
<ref>Ali10</ref> is an explicit framework, within which several 
processes are available. The code required for merging the PYTHIA 
showers with POWHEG input can be found in <code>examples/main31</code>,
and is further described on a 
<aloc href="POWHEGMerging">separate page</aloc>. 
</li> 
 
<li>The other traditional approach for NLO calculations is the 
MC@NLO one <ref>Fri02</ref>. In it the shower emission probability, 
without its Sudakov factor, is subtracted from the real-emission 
matrix element to regularize divergences. It therefore requires a 
analytic knowledge of the way the shower populates phase space. 
Currently there is no MC@NLO implementation for PYTHIA 8, but one is 
in preparation by Paolo Torrielli and Stefano Frixione, for the 
aMC@NLO program <ref>Fre11</ref>. The global-recoil option of the 
PYTHIA final-state shower has been constructed in anticipation of 
its use for the above-mentioned subtraction. 
</li> 
 
<li>Multi-jet merging in the CKKW-L approach <ref>Lon01</ref> 
is directly available. Its implementation, relevant parameters 
and test programs are documented on a 
<aloc href="CKKWLMerging">separate page</aloc>. 
</li> 
 
<li>Multi-jet matching in the MLM approach <ref>Man02, Man07</ref> 
is also available, either based on the ALPGEN or on the Madgraph 
variant, and with input events either from ALPGEN or from 
Madgraph. For details see 
<aloc href="JetMatching">separate page</aloc>. 
</li> 
 
<li>Unitarised matrix element + parton shower merging (UMEPS) 
is directly available. Its implementation, relevant parameters 
and test programs are documented on a 
<aloc href="UMEPSMerging">separate page</aloc>. 
</li> 
 
<li>Next-to-leading order multi-jet merging (in the NL3 and UNLOPS approaches) 
is directly available. Its implementation, relevant parameters 
and test programs are documented on a 
<aloc href="NLOMerging">separate page</aloc>. 
</li> 
 
</ul> 
 
 
</chapter> 
 
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