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<chapter name="A Second Hard Process">
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<h2>A Second Hard Process</h2>
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When you have selected a set of hard processes for hadron beams, the
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<aloc href="MultipartonInteractions">multiparton interactions</aloc>
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framework can add further interactions to build up a realistic
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underlying event. These further interactions can come from a wide
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variety of processes, and will occasionally be quite hard. They
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do represent a realistic random mix, however, which means one cannot
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predetermine what will happen. Occasionally there may be cases
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where one wants to specify also the second hard interaction rather
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precisely. The options on this page allow you to do precisely that.
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<flag name="SecondHard:generate" default="off">
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Generate two hard scatterings in a collision between hadron beams.
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The hardest process can be any combination of internal processes,
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available in the normal <aloc href="ProcessSelection">process
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selection</aloc> machinery, or external input. Here you must further
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specify which set of processes to allow for the second hard one, see
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<h3>Process Selection</h3>
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In principle the whole <aloc href="ProcessSelection">process
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selection</aloc> allowed for the first process could be repeated
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for the second one. However, this would probably be overkill.
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Therefore here a more limited set of prepackaged process collections
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are made available, that can then be further combined at will.
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Since the description is almost completely symmetric between the
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first and the second process, you always have the possibility
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to pick one of the two processes according to the complete list
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Here comes the list of allowed sets of processes, to combine at will:
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<flag name="SecondHard:TwoJets" default="off">
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Standard QCD <ei>2 -> 2</ei> processes involving gluons and
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<ei>d, u, s, c, b</ei> quarks.
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<flag name="SecondHard:PhotonAndJet" default="off">
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A prompt photon recoiling against a quark or gluon jet.
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<flag name="SecondHard:TwoPhotons" default="off">
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Two prompt photons recoiling against each other.
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<flag name="SecondHard:Charmonium" default="off">
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Production of charmonium via colour singlet and colour octet channels.
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<flag name="SecondHard:Bottomonium" default="off">
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Production of bottomonium via colour singlet and colour octet channels.
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<flag name="SecondHard:SingleGmZ" default="off">
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Scattering <ei>q qbar -> gamma^*/Z^0</ei>, with full interference
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between the <ei>gamma^*</ei> and <ei>Z^0</ei>.
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<flag name="SecondHard:SingleW" default="off">
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Scattering <ei>q qbar' -> W^+-</ei>.
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<flag name="SecondHard:GmZAndJet" default="off">
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Scattering <ei>q qbar -> gamma^*/Z^0 g</ei> and
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<ei>q g -> gamma^*/Z^0 q</ei>.
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<flag name="SecondHard:WAndJet" default="off">
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Scattering <ei>q qbar' -> W^+- g</ei> and
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<ei>q g -> W^+- q'</ei>.
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<flag name="SecondHard:TopPair" default="off">
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Production of a top pair, either via QCD processes or via an
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intermediate <ei>gamma^*/Z^0</ei> resonance.
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<flag name="SecondHard:SingleTop" default="off">
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Production of a single top, either via a <ei>t-</ei> or
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an <ei>s-</ei>channel <ei>W^+-</ei> resonance.
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A further process collection comes with a warning flag:
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<flag name="SecondHard:TwoBJets" default="off">
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The <ei>q qbar -> b bbar</ei> and <ei>g g -> b bbar</ei> processes.
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These are already included in the <code>TwoJets</code> sample above,
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so it would be doublecounting to include both, but we assume there
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may be cases where the <ei>b</ei> subsample will be of special interest.
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This subsample does not include flavour-excitation or gluon-splitting
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contributions to the <ei>b</ei> rate, however, so, depending
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on the topology if interest, it may or may not be a good approximation.
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<h3>Cuts and scales</h3>
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The second hard process obeys exactly the same selection rules for
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<aloc href="PhaseSpaceCuts">phase space cuts</aloc> and
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<aloc href="CouplingsAndScales">couplings and scales</aloc>
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as the first one does. Specifically, a <ei>pTmin</ei> cut for
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<ei>2 -> 2</ei> processes would apply to the first and the second hard
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process alike, and ballpark half of the time the second could be
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generated with a larger <ei>pT</ei> than the first. (Exact numbers
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depending on the relative shape of the two cross sections.) That is,
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first and second is only used as an administrative distinction between
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the two, not as a physics ordering one.
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Optionally it is possible to pick the mass and <ei>pT</ei>
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<aloc href="PhaseSpaceCuts">phase space cuts</aloc> separately for
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the second hard interaction. The main application presumably would
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be to allow a second process that is softer than the first, but still
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hard. But one is also free to make the second process harder than the
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first, if desired. So long as the two <ei>pT</ei> (or mass) ranges
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overlap the ordering will not be the same in all events, however.
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<h3>Cross-section calculation</h3>
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As an introduction, a brief reminder of Poissonian statistics.
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Assume a stochastic process in time, for now not necessarily a
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high-energy physics one, where the probability for an event to occur
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at any given time is independent of what happens at other times.
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Then the probability for <ei>n</ei> events to occur in a finite
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P_n = <n>^n exp(-<n>) / n!
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where <ei><n></ei> is the average number of events. If this
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number is small we can approximate <ei>exp(-<n>) = 1 </ei>,
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so that <ei>P_1 = <n></ei> and
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<ei>P_2 = <n>^2 / 2 = P_1^2 / 2</ei>.
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Now further assume that the events actually are of two different
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kinds <ei>a</ei> and <ei>b</ei>, occuring independently of each
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other, such that <ei><n> = <n_a> + <n_b></ei>.
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It then follows that the probability of having one event of type
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<ei>a</ei> (or <ei>b</ei>) and nothing else is
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<ei>P_1a = <n_a></ei> (or <ei>P_1b = <n_b></ei>).
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P_2 = (<n_a> + <n_b>)^2 / 2 = (P_1a + P_1b)^2 / 2 =
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(P_1a^2 + 2 P_1a P_1b + P_1b^2) / 2
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it is easy to read off that the probability to have exactly two
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events of kind <ei>a</ei> and none of <ei>b</ei> is
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<ei>P_2aa = P_1a^2 / 2</ei> whereas that of having one <ei>a</ei>
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and one <ei>b</ei> is <ei>P_2ab = P_1a P_1b</ei>. Note that the
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former, with two identical events, contains a factor <ei>1/2</ei>
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while the latter, with two different ones, does not. If viewed
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in a time-ordered sense, the difference is that the latter can be
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obtained two ways, either first an <ei>a</ei> and then a <ei>b</ei>
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or else first a <ei>b</ei> and then an <ei>a</ei>.
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To translate this language into cross-sections for high-energy
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events, we assume that interactions can occur at different <ei>pT</ei>
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values independently of each other inside inelastic nondiffractive
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(= "minbias") events. Then the above probabilities translate into
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<ei>P_n = sigma_n / sigma_ND</ei> where <ei>sigma_ND</ei> is the
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total nondiffractive cross section. Again we want to assume that
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<ei>exp(-<n>)</ei> is close to unity, i.e. that the total
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hard cross section above <ei>pTmin</ei> is much smaller than
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<ei>sigma_ND</ei>. The hard cross section is dominated by QCD
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jet production, and a reasonable precaution is to require a
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<ei>pTmin</ei> of at least 20 GeV at LHC energies.
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(For <ei>2 -> 1</ei> processes such as
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<ei>q qbar -> gamma^*/Z^0 (-> f fbar)</ei> one can instead make a
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similar cut on mass.) Then the generic equation
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<ei>P_2 = P_1^2 / 2</ei> translates into
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<ei>sigma_2/sigma_ND = (sigma_1 / sigma_ND)^2 / 2</ei> or
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<ei>sigma_2 = sigma_1^2 / (2 sigma_ND)</ei>.
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Again different processes <ei>a, b, c, ...</ei> contribute,
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and by the same reasoning we obtain
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<ei>sigma_2aa = sigma_1a^2 / (2 sigma_ND)</ei>,
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<ei>sigma_2ab = sigma_1a sigma_1b / sigma_ND</ei>,
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There is one important correction to this picture: all collisions
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do no occur under equal conditions. Some are more central in impact
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parameter, others more peripheral. This leads to a further element of
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variability: central collisions are likely to have more activity
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than the average, peripheral less. Integrated over impact
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parameter standard cross sections are recovered, but correlations
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are affected by a "trigger bias" effect: if you select for events
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with a hard process you favour events at small impact parameter
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which have above-average activity, and therefore also increased
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chance for further interactions. (In PYTHIA this is the origin
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of the "pedestal effect", i.e. that events with a hard interaction
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have more underlying activity than the level found in minimum-bias
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events.) When you specify a matter overlap profile in the
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multiparton-interactions scenario, such an enhancement/depletion factor
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<ei>f_impact</ei> is chosen event-by-event and can be averaged
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during the course of the run. As an example, the double Gaussian
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form used in Tune A gives approximately
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<ei><f_impact> = 2.5</ei>. The above equations therefore
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have to be modified to
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<ei>sigma_2aa = <f_impact> sigma_1a^2 / (2 sigma_ND)</ei>,
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<ei>sigma_2ab = <f_impact> sigma_1a sigma_1b / sigma_ND</ei>.
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Experimentalists often instead use the notation
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<ei>sigma_2ab = sigma_1a sigma_1b / sigma_eff</ei>,
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from which we see that PYTHIA "predicts"
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<ei>sigma_eff = sigma_ND / <f_impact></ei>.
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When the generation of multiparton interactions is switched off it is
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not possible to calculate <ei><f_impact></ei> and therefore
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When this recipe is to be applied to calculate
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actual cross sections, it is useful to distinguish three cases,
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depending on which set of processes are selected to study for
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the first and second interaction.
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(1) The processes <ei>a</ei> for the first interaction and
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<ei>b</ei> for the second one have no overlap at all.
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For instance, the first could be <code>TwoJets</code> and the
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second <code>TwoPhotons</code>. In that case, the two interactions
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can be selected independently, and cross sections tabulated
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for each separate subprocess in the two above classes. At the
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end of the run, the cross sections in <ei>a</ei> should be multiplied
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by <ei><f_impact> sigma_1b / sigma_ND</ei> to bring them to
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the correct overall level, and those in <ei>b</ei> by
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<ei><f_impact> sigma_1a / sigma_ND</ei>.
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(2) Exactly the same processes <ei>a</ei> are selected for the
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first and second interaction. In that case it works as above,
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with <ei>a = b</ei>, and it is only necessary to multiply by an
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additional factor <ei>1/2</ei>. A compensating factor of 2
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is automatically obtained for picking two different subprocesses,
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e.g. if <code>TwoJets</code> is selected for both interactions,
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then the combination of the two subprocesses <ei>q qbar -> g g</ei>
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and <ei>g g -> g g</ei> can trivially be obtained two ways.
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(3) The list of subprocesses partly but not completely overlap.
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For instance, the first process is allowed to contain <ei>a</ei>
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or <ei>c</ei> and the second <ei>b</ei> or <ei>c</ei>, where
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there is no overlap between <ei>a</ei> and <ei>b</ei>. Then,
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when an independent selection for the first and second interaction
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both pick one of the subprocesses in <ei>c</ei>, half of those
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events have to be thrown, and the stored cross section reduced
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accordingly. Considering the four possible combinations of first
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and second process, this gives a
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sigma'_1 = sigma_1a + sigma_1c * (sigma_2b + sigma_2c/2) /
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(sigma_2b + sigma_2c)
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with the factor <ei>1/2</ei> for the <ei>sigma_1c sigma_2c</ei> term.
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At the end of the day, this <ei>sigma'_1</ei> should be multiplied
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by the normalization factor
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f_1norm = <f_impact> (sigma_2b + sigma_2c) / sigma_ND
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here without a factor <ei>1/2</ei> (or else it would have been
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doublecounted). This gives the correct
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(sigma_2b + sigma_2c) * sigma'_1 = sigma_1a * sigma_2b
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+ sigma_1a * sigma_2c + sigma_1c * sigma_2b + sigma_1c * sigma_2c/2
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The second interaction can be handled in exact analogy.
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For the considerations above it is assumed that the phase space cuts
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are the same for the two processes. It is possible to set the mass and
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transverse momentum cuts differently, however. This changes nothing
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for processes that already are different. For two collisions of the
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same type it is partly a matter of interpretation what is intended.
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If we consider the case of the same process in two non-overlapping
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phase space regions, most likely we want to consider them as
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separate processes, in the sense that we expect a factor 2 relative
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to Poissonian statistics from either of the two hardest processes
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populating either of the two phase space regions. In total we are
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therefore lead to adopt the same strategy as in case (3) above:
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only in the overlapping part of the two allowed phase space regions
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could two processes be identical and thus appear with a 1/2 factor,
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elsewhere the two processes are never identical and do not
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include the 1/2 factor. We reiterate, however, that the case of
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partly but not completely overlapping phase space regions for one and
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the same process is tricky, and not to be used without prior
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The listing obtained with the <code>pythia.statistics()</code>
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already contain these corrections factors, i.e. cross sections
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are for the occurence of two interactions of the specified kinds.
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There is not a full tabulation of the matrix of all the possible
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combinations of a specific first process together with a specific
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second one (but the information is there for the user to do that,
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if desired). Instead <code>pythia.statistics()</code> shows this
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matrix projected onto the set of processes and associated cross
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sections for the first and the second interaction, respectively.
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Up to statistical fluctuations, these two sections of the
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<code>pythia.statistics()</code> listing both add up to the same
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total cross section for the event sample.
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There is a further special feature to be noted for this listing,
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and that is the difference between the number of "selected" events
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and the number of "accepted" ones. Here is how that comes about.
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Originally the first and second process are selected completely
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independently. The generation (in)efficiency is reflected in the
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different number of intially tried events for the first and second
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process, leading to the same number of selected events. While
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acceptable on their own, the combination of the two processes may
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be unacceptable, however. It may be that the two processes added
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together use more energy-momentum than kinematically allowed, or,
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even if not, are disfavoured when the PYTHIA approach to provide
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correlated parton densities is applied. Alternatively, referring
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to case (3) above, it may be because half of the events should
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be thrown for identical processes. Taken together, it is these
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effects that reduced the event number from "selected" to "accepted".
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(A further reduction may occur if a
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<aloc href="UserHooks">user hook</aloc> rejects some events.)
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It is allowed to use external Les Houches Accord input for the
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hardest process, and then pick an internal one for the second hardest.
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In this case PYTHIA does not have access to your thinking concerning
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the external process, and cannot know whether it overlaps with the
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internal or not. (External events <ei>q qbar' -> e+ nu_e</ei> could
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agree with the internal <ei>W</ei> ones, or be a <ei>W'</ei> resonance
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in a BSM scenario, to give one example.) Therefore the combined cross
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section is always based on the scenario (1) above. Corrections for
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correlated parton densities are included also in this case, however.
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That is, an external event that takes a large fraction of the incoming
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beam momenta stands a fair chance of being rejected when it has to be
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combined with another hard process. For this reason the "selected" and
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"accepted" event numbers are likely to disagree.
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In the cross section calculation above, the <ei>sigma'_1</ei>
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cross sections are based on the number of accepted events, while
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the <ei>f_1norm</ei> factor is evaluated based on the cross sections
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for selected events. That way the suppression by correlations
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between the two processes does not get to be doublecounted.
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The <code>pythia.statistics()</code> listing contains two final
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lines, indicating the summed cross sections <ei>sigma_1sum</ei> and
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<ei>sigma_2sum</ei> for the first and second set of processes, at
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the "selected" stage above, plus information on the <ei>sigma_ND</ei>
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and <ei><f_impact></ei> used. The total cross section
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generated is related to this by
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<f_impact> * (sigma_1sum * sigma_2sum / sigma_ND) *
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(n_accepted / n_selected)
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with an additional factor of <ei>1/2</ei> for case 2 above.
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The error quoted for the cross section of a process is a combination
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in quadrature of the error on this process alone with the error on
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the normalization factor, including the error on
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<ei><f_impact></ei>. As always it is a purely statistical one
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and of course hides considerably bigger systematic uncertainties.
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<h3>Event information</h3>
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Normally the <code>process</code> event record only contains the
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hardest interaction, but in this case also the second hardest
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is stored there. If both of them are <ei>2 -> 2</ei> ones, the
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first would be stored in lines 3 - 6 and the second in 7 - 10.
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For both, status codes 21 - 29 would be used, as for a hardest
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process. Any resonance decay chains would occur after the two
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main processes, to allow normal parsing. The beams in 1 and 2
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only appear in one copy. This structure is echoed in the
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full <code>event</code> event record.
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Most of the properties accessible by the
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<code><aloc href="EventInformation">pythia.info</aloc></code>
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methods refer to the first process, whether that happens to be the
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hardest or not. The code and <ei>pT</ei> scale of the second process
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are accessible by the <code>info.codeMPI(1)</code> and
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<code>info.pTMPI(1)</code>, however.
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The <code>sigmaGen()</code> and <code>sigmaErr()</code> methods provide
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the cross section and its error for the event sample as a whole,
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combining the information from the two hard processes as described
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above. In particular, the former should be used to give the
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weight of the generated event sample. The statitical error estimate
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is somewhat cruder and gives a larger value than the
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subprocess-by-subprocess one employed in
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<code>pythia.statistics()</code>, but this number is
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anyway less relevant, since systematical errors are likely to dominate.
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<!-- Copyright (C) 2012 Torbjorn Sjostrand -->
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xmldoc/ASecondHardProcess.xml
