## A Second Hard Process

When you have selected a set of hard processes for hadron beams, the multiparton interactions framework can add further interactions to build up a realistic underlying event. These further interactions can come from a wide variety of processes, and will occasionally be quite hard. They do represent a realistic random mix, however, which means one cannot predetermine what will happen. Occasionally there may be cases where one wants to specify also the second hard interaction rather precisely. The options on this page allow you to do precisely that.

`flag` SecondHard:generate (`default = off`)
Generate two hard scatterings in a collision between hadron beams. The hardest process can be any combination of internal processes, available in the normal process selection machinery, or external input. Here you must further specify which set of processes to allow for the second hard one, see the following.

### Process Selection

In principle the whole process selection allowed for the first process could be repeated for the second one. However, this would probably be overkill. Therefore here a more limited set of prepackaged process collections are made available, that can then be further combined at will. Since the description is almost completely symmetric between the first and the second process, you always have the possibility to pick one of the two processes according to the complete list of possibilities.

Here comes the list of allowed sets of processes, to combine at will:

`flag` SecondHard:TwoJets (`default = off`)
Standard QCD 2 → 2 processes involving gluons and d, u, s, c, b quarks.

`flag` SecondHard:PhotonAndJet (`default = off`)
A prompt photon recoiling against a quark or gluon jet.

`flag` SecondHard:TwoPhotons (`default = off`)
Two prompt photons recoiling against each other.

`flag` SecondHard:Charmonium (`default = off`)
Production of charmonium via colour singlet and colour octet channels.

`flag` SecondHard:Bottomonium (`default = off`)
Production of bottomonium via colour singlet and colour octet channels.

`flag` SecondHard:SingleGmZ (`default = off`)
Scattering q qbar → gamma^*/Z^0, with full interference between the gamma^* and Z^0.

`flag` SecondHard:SingleW (`default = off`)
Scattering q qbar' → W^+-.

`flag` SecondHard:GmZAndJet (`default = off`)
Scattering q qbar → gamma^*/Z^0 g and q g → gamma^*/Z^0 q.

`flag` SecondHard:WAndJet (`default = off`)
Scattering q qbar' → W^+- g and q g → W^+- q'.

`flag` SecondHard:TopPair (`default = off`)
Production of a top pair, either via QCD processes or via an intermediate gamma^*/Z^0 resonance.

`flag` SecondHard:SingleTop (`default = off`)
Production of a single top, either via a t- or an s-channel W^+- resonance.

A further process collection comes with a warning flag:

`flag` SecondHard:TwoBJets (`default = off`)
The q qbar → b bbar and g g → b bbar processes. These are already included in the `TwoJets` sample above, so it would be double-counting to include both, but we assume there may be cases where the b subsample will be of special interest. This subsample does not include flavour-excitation or gluon-splitting contributions to the b rate, however, so, depending on the topology if interest, it may or may not be a good approximation.

### Cuts and scales

The second hard process obeys exactly the same selection rules for phase space cuts and couplings and scales as the first one does. Specifically, a pTmin cut for 2 → 2 processes would apply to the first and the second hard process alike, and ballpark half of the time the second could be generated with a larger pT than the first. (Exact numbers depending on the relative shape of the two cross sections.) That is, first and second is only used as an administrative distinction between the two, not as a physics ordering one.

Optionally it is possible to pick the mass and pT phase space cuts separately for the second hard interaction. The main application presumably would be to allow a second process that is softer than the first, but still hard. But one is also free to make the second process harder than the first, if desired. So long as the two pT (or mass) ranges overlap the ordering will not be the same in all events, however.

### Cross-section calculation

As an introduction, a brief reminder of Poissonian statistics. Assume a stochastic process in time, for now not necessarily a high-energy physics one, where the probability for an event to occur at any given time is independent of what happens at other times. Then the probability for n events to occur in a finite time interval is
P_n = <n>^n exp(-<n>) / n!
where <n> is the average number of events. If this number is small we can approximate exp(-<n>) = 1 , so that P_1 = <n> and P_2 = <n>^2 / 2 = P_1^2 / 2.

Now further assume that the events actually are of two different kinds a and b, occurring independently of each other, such that <n> = <n_a> + <n_b>. It then follows that the probability of having one event of type a (or b) and nothing else is P_1a = <n_a> (or P_1b = <n_b>). From
P_2 = (<n_a> + <n_b>)^2 / 2 = (P_1a + P_1b)^2 / 2 = (P_1a^2 + 2 P_1a P_1b + P_1b^2) / 2
it is easy to read off that the probability to have exactly two events of kind a and none of b is P_2aa = P_1a^2 / 2 whereas that of having one a and one b is P_2ab = P_1a P_1b. Note that the former, with two identical events, contains a factor 1/2 while the latter, with two different ones, does not. If viewed in a time-ordered sense, the difference is that the latter can be obtained two ways, either first an a and then a b or else first a b and then an a.

To translate this language into cross-sections for high-energy events, we assume that interactions can occur at different pT values independently of each other inside inelastic nondiffractive (sometimes equated with "minbias") events. Then the above probabilities translate into P_n = sigma_n / sigma_ND where sigma_ND is the total nondiffractive cross section. Again we want to assume that exp(-<n>) is close to unity, i.e. that the total hard cross section above pTmin is much smaller than sigma_ND. The hard cross section is dominated by QCD jet production, and a reasonable precaution is to require a pTmin of at least 20 GeV at LHC energies. (For 2 → 1 processes such as q qbar → gamma^*/Z^0 (→ f fbar) one can instead make a similar cut on mass.) Then the generic equation P_2 = P_1^2 / 2 translates into sigma_2/sigma_ND = (sigma_1 / sigma_ND)^2 / 2 or sigma_2 = sigma_1^2 / (2 sigma_ND).

Again different processes a, b, c, ... contribute, and by the same reasoning we obtain sigma_2aa = sigma_1a^2 / (2 sigma_ND), sigma_2ab = sigma_1a sigma_1b / sigma_ND, and so on.

There is one important correction to this picture: all collisions do no occur under equal conditions. Some are more central in impact parameter, others more peripheral. This leads to a further element of variability: central collisions are likely to have more activity than the average, peripheral less. Integrated over impact parameter standard cross sections are recovered, but correlations are affected by a "trigger bias" effect: if you select for events with a hard process you favour events at small impact parameter which have above-average activity, and therefore also increased chance for further interactions. (In PYTHIA this is the origin of the "pedestal effect", i.e. that events with a hard interaction have more underlying activity than the level found in minimum-bias events.)

When you specify a matter overlap profile in the multiparton-interactions scenario, such an enhancement/depletion factor f_impact is chosen event-by-event and can be averaged during the course of the run. As an example, the double Gaussian form used in Tune A gives approximately <f_impact> = 2.5. In general, the more uneven the distribution the higher the <f_impact>. Also the pT0 parameter value has an impact, even if it is less important over a realistic range of values, although it implies that <f_impact> is energy-dependent. The origin of this effect is as follows. A lower pT0 implies more MPI activity at all impact parameters, so that the nondiffractive cross section sigma_ND increases, or equivalently the proton size. But if sigma_ND is fixed by data then the input radius of the matter overlap profile (not explicitly specified but implicitly adjusted at initialization) has to be shrunk so that the output value can stay constant. This means that the proton matter is more closely packed and therefore <f_impact> goes up.

The above equations therefore have to be modified to sigma_2aa = <f_impact> sigma_1a^2 / (2 sigma_ND), sigma_2ab = <f_impact> sigma_1a sigma_1b / sigma_ND. Experimentalists often instead use the notation sigma_2ab = sigma_1a sigma_1b / sigma_eff, from which we see that PYTHIA "predicts" sigma_eff = sigma_ND / <f_impact>. When the generation of multiparton interactions is switched off it is not possible to calculate <f_impact> and therefore it is set to unity.

When this recipe is to be applied to calculate actual cross sections, it is useful to distinguish three cases, depending on which set of processes are selected to study for the first and second interaction.

(1) The processes a for the first interaction and b for the second one have no overlap at all. For instance, the first could be `TwoJets` and the second `TwoPhotons`. In that case, the two interactions can be selected independently, and cross sections tabulated for each separate subprocess in the two above classes. At the end of the run, the cross sections in a should be multiplied by <f_impact> sigma_1b / sigma_ND to bring them to the correct overall level, and those in b by <f_impact> sigma_1a / sigma_ND.

(2) Exactly the same processes a are selected for the first and second interaction. In that case it works as above, with a = b, and it is only necessary to multiply by an additional factor 1/2. A compensating factor of 2 is automatically obtained for picking two different subprocesses, e.g. if `TwoJets` is selected for both interactions, then the combination of the two subprocesses q qbar → g g and g g → g g can trivially be obtained two ways.

(3) The list of subprocesses partly but not completely overlap. For instance, the first process is allowed to contain a or c and the second b or c, where there is no overlap between a and b. Then, when an independent selection for the first and second interaction both pick one of the subprocesses in c, half of those events have to be thrown, and the stored cross section reduced accordingly. Considering the four possible combinations of first and second process, this gives a
sigma'_1 = sigma_1a + sigma_1c * (sigma_2b + sigma_2c/2) / (sigma_2b + sigma_2c)
with the factor 1/2 for the sigma_1c sigma_2c term. At the end of the day, this sigma'_1 should be multiplied by the normalization factor
f_1norm = <f_impact> (sigma_2b + sigma_2c) / sigma_ND
here without a factor 1/2 (or else it would have been double-counted). This gives the correct
(sigma_2b + sigma_2c) * sigma'_1 = sigma_1a * sigma_2b + sigma_1a * sigma_2c + sigma_1c * sigma_2b + sigma_1c * sigma_2c/2
The second interaction can be handled in exact analogy.

For the considerations above it is assumed that the phase space cuts are the same for the two processes. It is possible to set the mass and transverse momentum cuts differently, however. This changes nothing for processes that already are different. For two collisions of the same type it is partly a matter of interpretation what is intended. If we consider the case of the same process in two non-overlapping phase space regions, most likely we want to consider them as separate processes, in the sense that we expect a factor 2 relative to Poissonian statistics from either of the two hardest processes populating either of the two phase space regions. In total we are therefore lead to adopt the same strategy as in case (3) above: only in the overlapping part of the two allowed phase space regions could two processes be identical and thus appear with a 1/2 factor, elsewhere the two processes are never identical and do not include the 1/2 factor. We reiterate, however, that the case of partly but not completely overlapping phase space regions for one and the same process is tricky, and not to be used without prior deliberation.

The listing obtained with the `pythia.statistics()` already contain these corrections factors, i.e. cross sections are for the occurrence of two interactions of the specified kinds. There is not a full tabulation of the matrix of all the possible combinations of a specific first process together with a specific second one (but the information is there for the user to do that, if desired). Instead `pythia.statistics()` shows this matrix projected onto the set of processes and associated cross sections for the first and the second interaction, respectively. Up to statistical fluctuations, these two sections of the `pythia.statistics()` listing both add up to the same total cross section for the event sample.

There is a further special feature to be noted for this listing, and that is the difference between the number of "selected" events and the number of "accepted" ones. Here is how that comes about. Originally the first and second process are selected completely independently. The generation (in)efficiency is reflected in the different number of initially tried events for the first and second process, leading to the same number of selected events. While acceptable on their own, the combination of the two processes may be unacceptable, however. It may be that the two processes added together use more energy-momentum than kinematically allowed, or, even if not, are disfavoured when the PYTHIA approach to provide correlated parton densities is applied. Alternatively, referring to case (3) above, it may be because half of the events should be thrown for identical processes. Taken together, it is these effects that reduced the event number from "selected" to "accepted". (A further reduction may occur if a user hook rejects some events.)

It is allowed to use external Les Houches Accord input for the hardest process, and then pick an internal one for the second hardest. In this case PYTHIA does not have access to your thinking concerning the external process, and cannot know whether it overlaps with the internal or not. (External events q qbar' → e+ nu_e could agree with the internal W ones, or be a W' resonance in a BSM scenario, to give one example.) Therefore the combined cross section is always based on the scenario (1) above. Corrections for correlated parton densities are included also in this case, however. That is, an external event that takes a large fraction of the incoming beam momenta stands a fair chance of being rejected when it has to be combined with another hard process. For this reason the "selected" and "accepted" event numbers are likely to disagree.

In the cross section calculation above, the sigma'_1 cross sections are based on the number of accepted events, while the f_1norm factor is evaluated based on the cross sections for selected events. That way the suppression by correlations between the two processes does not get to be double-counted.

The `pythia.statistics()` listing contains two final lines, indicating the summed cross sections sigma_1sum and sigma_2sum for the first and second set of processes, at the "selected" stage above, plus information on the sigma_ND and <f_impact> used. The total cross section generated is related to this by
<f_impact> * (sigma_1sum * sigma_2sum / sigma_ND) * (n_accepted / n_selected)
with an additional factor of 1/2 for case 2 above.

The error quoted for the cross section of a process is a combination in quadrature of the error on this process alone with the error on the normalization factor, including the error on <f_impact>. As always it is a purely statistical one and of course hides considerably bigger systematic uncertainties.

### Event information

Normally the `process` event record only contains the hardest interaction, but in this case also the second hardest is stored there. If both of them are 2 → 2 ones, the first would be stored in lines 3 - 6 and the second in 7 - 10. For both, status codes 21 - 29 would be used, as for a hardest process. Any resonance decay chains would occur after the two main processes, to allow normal parsing. The beams in 1 and 2 only appear in one copy. This structure is echoed in the full `event` event record.

Most of the properties accessible by the `pythia.info` methods refer to the first process, whether that happens to be the hardest or not. The code and pT scale of the second process are accessible by the `info.codeMPI(1)` and `info.pTMPI(1)`, however.

The `sigmaGen()` and `sigmaErr()` methods provide the cross section and its error for the event sample as a whole, combining the information from the two hard processes as described above. In particular, the former should be used to give the weight of the generated event sample. The statistical error estimate is somewhat cruder and gives a larger value than the subprocess-by-subprocess one employed in `pythia.statistics()`, but this number is anyway less relevant, since systematical errors are likely to dominate.