Diffraction
Introduction
Diffraction is not well understood, and several alternative approaches
have been proposed. Here we follw a fairly conventional Pomeron-based
one, but integrated to make full use of the standard PYTHIA machinery
for multiple interactions, parton showers and hadronization. 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 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
multiple interactions and parton showers are applied, making use of
Pomeron PDFs.
The parameters related to multiple 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
multiple 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 multiple 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.
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.
mode Diffraction:PomFlux
(default = 1; minimum = 1; maximum = 4)
Parametrization of the Pomeron flux f_Pom/p( x_Pom, t).
option 1 : Schuler and Sjöstrand [Sch94]: based on a
critical Pomeron, giving a mass spectrum roughly like dm^2/m^2;
a mass-dependent exponential t slope that reduces the rate
of low-mass states; partly compensated by a very-low-mass (resonance region)
enhancement. Is currently the only one that contains a separate
t spectrum for double diffraction and separate parameters
for pion beams.
option 2 : Bruni and Ingelman [Bru93]: also a critical
Pomeron giving close to dm^2/m^2, with a t distribution
the sum of two exponentials.
option 3 : a conventional Pomeron description, in the RapGap
manual [Jun95] attributed to Berger et al. and Streng
[Ber87a], but there (and here) with values updated to a
supercritical Pomeron with epsilon > 0 (see below),
which gives a stronger peaking towards low-mass diffractive states,
and with a mass-dependent (the alpha' below) exponential
t slope.
option 4 : a conventional Pomeron description, attributed to
Donnachie and Landshoff [Don84], again with supercritical Pomeron,
with the same two parameters as option 3 above, but this time with a
power-law t distribution.
In the last two options above, the Pomeron Regge trajectory is
parametrized as
alpha(t) = 1 + epsilon + alpha' t
The epsilon and alpha' parameters can be set
separately:
parm Diffraction:PomFluxEpsilon
(default = 0.085; minimum = 0.02; maximum = 0.15)
The Pomeron trajectory intercept epsilon above. For technical
reasons epsilon > 0 is necessary in the current implementation.
parm Diffraction:PomFluxAlphaPrime
(default = 0.25; minimum = 0.1; maximum = 0.4)
The Pomeron trajectory slope alpha' above.
Separation into low and high masses
Preferably one would want to have a perturbative picture of the
dynamics of Pomeron-proton collisions, like multiple 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 = 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.
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 = 0.)
The abovementioned threshold width m_width.
Low-mass 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.
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; minimum = 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 = 2.; 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.
High-mass 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 multiple 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 MI in
diffraction than in pp collisions, e.g. with respect to the
impact-parameter picture. 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.
parm Diffraction:sigmaPomP
(default = 10.; minimum = 2.; maximum = 40.)
The assumed Pomeron-proton effective cross section, as used for
multiple interactions in diffractive systems. A larger value gives
less MI 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.)
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 multiple
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 MI is
appreciably slower when perturbative high-mass diffraction is allowed.