Diffraction is not well understood, and several alternative approaches
have been proposed. Here we follow a fairly conventional Pomeron-based
one, in the Ingelman-Schlein spirit Ing85,
but integrated to make full use of the standard PYTHIA machinery
for multiparton interactions, parton showers and hadronization
Nav10,Cor10a. This is the approach pioneered in the PomPyt
program by Ingelman and collaborators Ing97.
For ease of use (and of modelling), the Pomeron-specific parts of the
generation of inclusive ("soft") diffreactive 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,
see below;
(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 one exception. 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.
Pomeron flux
As already mentioned above, the total diffractive cross section is fixed
by a default energy-dependent parametrization or by the user, see the
Total Cross Sections page.
Therefore we do not attribute any significance to the absolute
normalization of the Pomeron flux. The choice of Pomeron flux model
still will decide on the mass spectrum of diffractive states and the
t spectrum of the Pomeron exchange.
Parametrization of the Pomeron flux f_Pom/p( x_Pom, t).
In options 3, 4, 6, and 7 above, the Pomeron Regge trajectory is
parametrized as
alpha(t) = 1 + epsilon + alpha' t
The epsilon and alpha' parameters can be set
separately in options 3 and 4, and additionally alpha'
is set in option 1:
The Pomeron trajectory intercept epsilon above for the 3 and 4
flux options. For technical reasons epsilon > 0 is necessary
in the current implementation.
The Pomeron trajectory slope alpha' above for the 1, 3 and 4
flux options.
Values are fixed in options 6 and 7.
When option 5 is selected, the following parameters of the MBR model
Cie12 are used:
the parameters of the Pomeron trajectory.
the Pomeron-proton coupling, and the total Pomeron-proton cross section.
the lowest value of the mass squared of the dissociated system.
the minimum width of the rapidity gap used in the calculation of
Ngap(s) (flux renormalization).
the minimum width of the rapidity gap used in the calculation of cross
sections, i.e. the parameter dy_S, which suppresses the cross
section at low dy (non-diffractive region).
the parameter sigma_S, used for the cross section suppression at
low dy (non-diffractive region).
Separation of soft diffraction into low and high masses
Preferably one would want to have a perturbative picture of the
dynamics of Pomeron-proton collisions, like multiparton interactions
provide for proton-proton ones. However, while PYTHIA by default
will only allow collisions with a CM energy above 10 GeV, the
mass spectrum of diffractive systems will stretch to down to
the order of 1.2 GeV. It would not be feasible to attempt a
perturbative description there. Therefore we do offer a simpler
low-mass description, with only longitudinally stretched strings,
with a gradual switch-over to the perturbative picture for higher
masses. The probability for the latter picture is parametrized as
P_pert = P_max ( 1 - exp( (m_diffr - m_min) / m_width ) )
which vanishes for the diffractive system mass
m_diffr < m_min, and is 1 - 1/e = 0.632 for
m_diffr = m_min + m_width, assuming P_max = 1.
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.
The abovementioned threshold width m_width.
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.
Low-mass soft diffraction
When an incoming hadron beam is diffractively excited, it is modeled
as if either a valence quark or a gluon is kicked out from the hadron.
In the former case this produces a simple string to the leftover
remnant, in the latter it gives a hairpin arrangement where a string
is stretched from one quark in the remnant, via the gluon, back to the
rest of the remnant. The latter ought to dominate at higher mass of
the diffractive system. Therefore an approximate behaviour like
P_q / P_g = N / m^p
is assumed.
The abovementioned normalization N for the relative quark
rate in diffractive systems.
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:
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.
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.
High-mass soft diffraction
The perturbative description need to use parton densities of the
Pomeron. The options are described in the page on
PDF Selection. The standard
perturbative multiparton interactions framework then provides
cross sections for parton-parton interactions. In order to
turn these cross section into probabilities one also needs an
ansatz for the Pomeron-proton total cross section. In the literature
one often finds low numbers for this, of the order of 2 mb.
These, if taken at face value, would give way too much activity
per event. There are ways to tame this, e.g. by a larger pT0
than in the normal pp framework. Actually, there are many reasons
to use a completely different set of parameters for MPI in
diffraction than in pp collisions, especially with respect to the
impact-parameter picture, see below. A lower number in some frameworks
could alternatively be regarded as a consequence of screening, with
a larger "bare" number.
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.
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.)
The mRef reference mass scale introduced above.
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.
Choice of impact parameter profile for the incoming hadron beams.
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.
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.
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.
Hard diffraction
When PYTHIA is requested to generate a hard process, by default it is
assumed that the full perturbative cross section is associated with
nondiffractive topologies. With the options in this section, PYTHIA
includes a possibility for creating a perturbative process diffractively,
however. This framework is denoted hard diffraction to distiguish it
from soft diffraction, but recall that the latter contains a tail of
high-pT processes that could alternatively be obtained as
hard diffraction. The idea behind hard diffraction is similar to soft
diffraction, as they are both based on the Pomeron model. The proton is
thus modelled as having a Pomeron component, described by the Pomeron
fluxes above, and the partonic content of the Pomeron is described by
the Pomeron PDFs, also described above. From these components we can
evaluate the probability for the chosen hard process to have been coming
from a diffractively exited 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 main61.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.
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.
Allows for the possibility to include hard diffraction tests in a run.
There is also the possibility to select only a specific subset of events
in hard diffraction.
Type of process the user wants to generate. Depends strongly on how an event
is classified as diffractive.
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:
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.
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.
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.