`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.

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.

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.

where

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.

`process`

event record only contains the
hardest interaction, but in this case also the second hardest
is stored there. If both of them are `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.