- Separation of soft diffraction into low and high masses
- Low-mass soft diffraction
- High-mass soft diffraction
- Hard diffraction

For ease of use (and of modelling), the Pomeron-specific parts of the
generation of inclusive ("soft") diffractive events are subdivided into
three sets of parameters that are rather independent of each other:

(i) the total, elastic and diffractive cross sections are
parametrized as functions of the CM energy, or can be set by the user
to the desired values, see the
Total Cross Sections page;

(ii) once it has been decided to have a diffractive process,
a Pomeron flux parametrization is used to pick the mass of the
diffractive system(s) and the *t* of the exchanged Pomeron,
also here see the
Total Cross Sections page;

(iii) a diffractive system of a given mass is classified either
as low-mass unresolved, which gives a simple low-*pT* string
topology, or as high-mass resolved, for which the full machinery of
multiparton interactions and parton showers are applied, making use of
Pomeron PDFs.

The parameters related to multiparton interactions, parton showers
and hadronization are kept the same as for normal nondiffractive events,
with only a few exceptions. This may be questioned, especially for the
multiparton interactions, but we do not believe that there are currently
enough good diffractive data that would allow detailed separate tunes.

The above subdivision may not represent the way "physics comes about". For instance, the total diffractive cross section can be viewed as a convolution of a Pomeron flux with a Pomeron-proton total cross section. Since neither of the two is known from first principles there will be a significant amount of ambiguity in the flux factor. The picture is further complicated by the fact that the possibility of simultaneous further multiparton interactions ("cut Pomerons") will screen the rate of diffractive systems. In the end, our set of parameters refers to the effective description that emerges out of these effects, rather than to the underlying "bare" parameters.

In the event record the diffractive system in the case of an excited
proton is denoted `p_diffr`

, code 9902210, whereas
a central diffractive system is denoted `rho_diffr`

,
code 9900110. Apart from representing the correct charge and baryon
numbers, no deeper meaning should be attributed to the names.

PYTHIA also includes a possibility to select hard diffraction. This
machinery relies on the same sets of parameters as described above,
for the Pomeron flux and PDFs. The main difference between the hard
and the soft diffractive frameworks is that the user can select any
PYTHIA hard process in the former case, e.g. diffractive *Z*'s
or *W*'s, whereas only QCD processes are generated in the latter.
These QCD processes are generated inclusively, which means that mostly
they occur in the low-*pT* region, even if a tail stretches to
higher *pT* scales, thus overlapping with hard diffraction.
Both hard and soft diffractive processes also include the normal PYTHIA
machinery, such as MPIs and showers, but for the former the MPI
framework can additionally be used to determine whether a hard process
survives as a diffractive event or not. The different diffractive types
- low mass soft, high mass soft and hard diffraction - are described
in more detail below.

which vanishes for the diffractive system mass

`parm `

** Diffraction:mMinPert **
(`default = `

; **10.**`minimum = 5.`

)

The abovementioned threshold mass *m_min* for phasing in a
perturbative treatment. If you put this parameter to be bigger than
the CM energy then there will be no perturbative description at all,
but only the older low-*pt* description.

`parm `

** Diffraction:mWidthPert **
(`default = `

; **10.**`minimum = 1.`

)

The abovementioned threshold width *m_width.*

`parm `

** Diffraction:probMaxPert **
(`default = `

; **1.**`minimum = 0.`

; `maximum = 1.`

)

The abovementioned maximum probability *P_max.*. Would
normally be assumed to be unity, but a somewhat lower value could
be used to represent a small nonperturbative component also at
high diffractive masses.

is assumed.

`parm `

** Diffraction:pickQuarkNorm **
(`default = `

; **5.0**`minimum = 0.`

)

The abovementioned normalization *N* for the relative quark
rate in diffractive systems.

`parm `

** Diffraction:pickQuarkPower **
(`default = `

)**1.0**

The abovementioned mass-dependence power *p* for the relative
quark rate in diffractive systems.

When a gluon is kicked out from the hadron, the longitudinal momentum
sharing between the the two remnant partons is determined by the
same parameters as above. It is plausible that the primordial
*kT* may be lower than in perturbative processes, however:

`parm `

** Diffraction:primKTwidth **
(`default = `

; **0.5**`minimum = 0.`

)

The width of Gaussian distributions in *p_x* and *p_y*
separately that is assigned as a primordial *kT* to the two
beam remnants when a gluon is kicked out of a diffractive system.

`parm `

** Diffraction:largeMassSuppress **
(`default = `

; **4.**`minimum = 0.`

)

The choice of longitudinal and transverse structure of a diffractive
beam remnant for a kicked-out gluon implies a remnant mass
*m_rem* distribution (i.e. quark plus diquark invariant mass
for a baryon beam) that knows no bounds. A suppression like
*(1 - m_rem^2 / m_diff^2)^p* is therefore introduced, where
*p* is the `diffLargeMassSuppress`

parameter.

For now, however, an attempt at the most general solution would carry too far, and instead we patch up the problem by using a larger Pomeron-proton total cross section, such that average activity makes more sense. This should be viewed as the main tunable parameter in the description of high-mass diffraction. It is to be fitted to diffractive event-shape data such as the average charged multiplicity. It would be very closely tied to the choice of Pomeron PDF; we remind that some of these add up to less than unit momentum sum in the Pomeron, a choice that also affect the value one ends up with. Furthermore, like with hadronic cross sections, it is quite plausible that the Pomeron-proton cross section increases with energy, so we have allowed for a power-like dependence on the diffractive mass.

`parm `

** Diffraction:sigmaRefPomP **
(`default = `

; **10.**`minimum = 2.`

; `maximum = 40.`

)

The assumed Pomeron-proton effective cross section, as used for
multiparton interactions in diffractive systems. If this cross section
is made to depend on the mass of the diffractive system then the above
value refers to the cross section at the reference scale, and
*
sigma_PomP(m) = sigma_PomP(m_ref) * (m / m_ref)^p
*

where *m* is the mass of the diffractive system, *m_ref*
is the reference mass scale `Diffraction:mRefPomP`

below and
*p* is the mass-dependence power `Diffraction:mPowPomP`

.
Note that a larger cross section value gives less MPI activity per event.
There is no point in making the cross section too big, however, since
then *pT0* will be adjusted downwards to ensure that the
integrated perturbative cross section stays above this assumed total
cross section. (The requirement of at least one perturbative interaction
per event.)

`parm `

** Diffraction:mRefPomP **
(`default = `

; **100.0**`minimum = 1.`

)

The *mRef* reference mass scale introduced above.

`parm `

** Diffraction:mPowPomP **
(`default = `

; **0.0**`minimum = 0.0`

; `maximum = 0.5`

)

The *p* mass rescaling pace introduced above.

Also note that, even for a fixed CM energy of events, the diffractive
subsystem will range from the abovementioned threshold mass
*m_min* to the full CM energy, with a variation of parameters
such as *pT0* along this mass range. Therefore multiparton
interactions are initialized for a few different diffractive masses,
currently five, and all relevant parameters are interpolated between
them to obtain the behaviour at a specific diffractive mass.
Furthermore, *A B → X B* and *A B → A X* are
initialized separately, to allow for different beams or PDF's on the
two sides. These two aspects mean that initialization of MPI is
appreciably slower when perturbative high-mass diffraction is allowed.

Diffraction tends to be peripheral, i.e. occur at intermediate impact
parameter for the two protons. That aspect is implicit in the selection
of diffractive cross section. For the simulation of the Pomeron-proton
subcollision it is the impact-parameter distribution of that particular
subsystem that should rather be modeled. That is, it also involves
the transverse coordinate space of a Pomeron wavefunction. The outcome
of the convolution therefore could be a different shape than for
nondiffractive events. For simplicity we allow the same kind of
options as for nondiffractive events, except that the
`bProfile = 4`

option for now is not implemented.

`mode `

** Diffraction:bProfile **
(`default = `

; **1**`minimum = 0`

; `maximum = 3`

)

Choice of impact parameter profile for the incoming hadron beams.
`option `

** 0** : no impact parameter dependence at all.
`option `

** 1** : a simple Gaussian matter distribution;
no free parameters.
`option `

** 2** : a double Gaussian matter distribution,
with the two free parameters *coreRadius* and
*coreFraction*.
`option `

** 3** : an overlap function, i.e. the convolution of
the matter distributions of the two incoming hadrons, of the form
*exp(- b^expPow)*, where *expPow* is a free
parameter.

`parm `

** Diffraction:coreRadius **
(`default = `

; **0.4**`minimum = 0.1`

; `maximum = 1.`

)

When assuming a double Gaussian matter profile, *bProfile = 2*,
the inner core is assumed to have a radius that is a factor
*coreRadius* smaller than the rest.

`parm `

** Diffraction:coreFraction **
(`default = `

; **0.5**`minimum = 0.`

; `maximum = 1.`

)

When assuming a double Gaussian matter profile, *bProfile = 2*,
the inner core is assumed to have a fraction *coreFraction*
of the matter content of the hadron.

`parm `

** Diffraction:expPow **
(`default = `

; **1.**`minimum = 0.4`

; `maximum = 10.`

)

When *bProfile = 3* it gives the power of the assumed overlap
shape *exp(- b^expPow)*. Default corresponds to a simple
exponential drop, which is not too dissimilar from the overlap
obtained with the standard double Gaussian parameters. For
*expPow = 2* we reduce to the simple Gaussian, *bProfile = 1*,
and for *expPow → infinity* to no impact parameter dependence
at all, *bProfile = 0*. For small *expPow* the program
becomes slow and unstable, so the min limit must be respected.

`SigmaDiffractive:PomFlux`

, see
here, and the partonic content
of the Pomeron is described by the Pomeron PDFs, see
here. From these components we can
evaluate the probability for the chosen hard process to have been coming
from a diffractively excited system, based on the ratio of the Pomeron flux
convoluted with Pomeron PDF to the inclusive proton PDF.
If the hard process is likely to have been created inside a diffractively
excited system, then we also evaluate the momentum fraction carried by the
Pomeron, *x_Pomeron*, and the momentum transfer, *t*, in
the process. This information can also be extracted in the main programs,
see eg. example `main325.cc`

.

Further, we distiguish between two alternative scenarios for the classification of hard diffraction. The first is based solely on the Pomeron flux and PDF, as described above. In the second an additional requirement is imposed, wherein the MPI machinery is not allowed to generate any extra MPIs at all, since presumably these would destroy the rapidity gap and thereby the diffractive nature. We refer to the former as MPI-unchecked and the latter as MPI-checked hard diffraction. The MPI-checked option is likely to be the more realistic one, but the MPI-unchecked one offers a convenient baseline for the study of MPI effects, which still are poorly understood.

Recently, a scenario for hard diffraction with *gamma* beams has
been introduced. Thus hard diffraction can be evaluated for both
*gamma + gamma* and *gamma + p* processes within the
usual photoproduction framework. A Pomeron can be taken from a
*gamma* beam only if the photon is resolved. Currently this photon
is then assumed always to be in a virtual *rho* state, thus
leaving behind a physical *rho* beam remnant. If the Pomeron
is taken from the proton, in the *gamma + parton* framework,
the photon is allowed to interact with the Pomeron with both its
resolved and unresolved components. If the Pomeron is taken from the
resolved *gamma*, the proton Pomeron flux is used but rescaled
by a factor of *sigma_tot^gamma+p/sigma_tot^pp*, as a very first
approximation to this unmeasured distribution. Otherwise all options are
available as for hard diffraction in *pp* processes, and all
limitations and cautions apply as for the photoproduction framework.

For the selected hard processes, the user can choose whether to generate the inclusive sample of both diffractive and nondiffractive topologies or diffractive only, and in each case with the diffractive ones distinguished either MPI-unchecked or MPI-checked.

`flag `

** Diffraction:doHard **
(`default = `

)**off**

Allows for the possibility to include hard diffraction tests in a run.

`mode `

** Diffraction:hardDiffSide **
(`default = `

; **0**`minimum = 0`

; `maximum = 2`

)

Side which diffraction is evaluated for. Especially useful for diffraction
in ep, where experiments only look for gaps on the proton side.
`option `

** 0** : Check for diffraction on boths sides A and B.
`option `

** 1** : Check for diffraction on side A only.
`option `

** 2** : Check for diffraction on side B only.

There is also the possibility to select only a specific subset of events in hard diffraction.

`mode `

** Diffraction:sampleType **
(`default = `

; **2**`minimum = 1`

; `maximum = 4`

)

Type of process the user wants to generate. Depends strongly on how an event
is classified as diffractive.
`option `

** 1** : Generate an inclusive sample of both diffractive and
nondiffractive hard processes, MPI-unchecked.
`option `

** 2** : Generate an inclusive sample of both diffractive and
nondiffractive hard processes, MPI-checked.
`option `

** 3** : Generate an exclusive diffractive sample, MPI-unchecked.
`option `

** 4** : Generate an exclusive diffractive sample, MPI-checked.

The Pomeron PDFs have not been scaled to unit momentum sum by the
H1 Collaboration, but instead they let the PDF normalization float
after the flux had been normalized to unity at *x_Pom=0.03*.
This means that the H1 Pomeron has a momentum sum that is about a half.
It could be brought to unit momentum sum by picking the parameter
`PDF:PomRescale`

to be around 2. In order not to change the
convolution of the flux and the PDFs, the flux then needs to be rescaled
by the inverse. This introduces a new rescaling parameter:

`parm `

** Diffraction:PomFluxRescale **
(`default = `

; **1.0**`minimum = 0.2`

; `maximum = 2.0`

)

Rescale the Pomeron flux by this uniform factor. It should be
`1 / PDF:PomRescale`

to preserve the convolution of Pomeron
flux and PDFs, but for greater flxibility the two can be set separately.

When using the MBR flux, the model requires a renormalization of the Pomeron flux. This suppresses the flux with approximately a factor of ten, thus making it incompatible with the MPI suppression of the hard diffraction framework. We have thus implemented an option to renormalize the flux. If you wish to use the renormalized flux of the MBR model, you must generate the MPI-unchecked samples, otherwise diffractive events will be suppressed twice.

`flag `

** Diffraction:useMBRrenormalization **
(`default = `

)**off**

Use the renormalized MBR flux.

The transverse matter profile of the Pomeron, relative to that of the proton, is not known. Generally a Pomeron is supposed to be a smaller object in a localized part of the proton, but one should keep an open mind. Therefore below you find three extreme scenarios, which can be compared to gauge the impact of this uncertainty.

`mode `

** Diffraction:bSelHard **
(`default = `

; **1**`minimum = 1`

; `maximum = 3`

)

Selection of impact parameter *b* and the related enhancement
factor for the Pomeron-proton subsystem when the MPI check is carried
out. This affects the underlying-event activity in hard diffractive
events.
`option `

** 1** : Use the same *b* as already assigned for the
proton-proton collision. This implicitly assumes that a Pomeron is
as big as a proton and centered in the same place. Since small
*b* values already have been suppressed, few events should
have high enhancement factors.
`option `

** 2** : Use the square root of the *b* as already
assigned for the proton-proton collision, thereby making the
enhancement factor fluctuate less between events. If the Pomeron
is very tiny then what matters is where it strikes the other proton,
not the details of its shape. Thus the variation with *b* is
of one proton, not two, and so the square root of the normal variation,
loosely speaking. Tecnhically this is difficult to implement, but
the current simple recipe provides the main effect of reducing the
variation, bringing all *b* values closer to the average.
`option `

** 3** : Pick a completely new *b*. This allows a broad
spread from central to peripheral values, and thereby also a more
varying MPI activity inside the diffractive system than the other two
options. This offers an extreme picture, even if not the most likely
one.