<chapter name="Diffraction"> 
 
<h2>Diffraction</h2> 
 
Diffraction is not well understood, and several alternative approaches 
have been proposed, both for the cross section of diffractive events and 
for the particle production in these. For the latter we here follow a 
fairly conventional Pomeron-based one, in the Ingelman-Schlein spirit 
<ref>Ing85</ref>, but integrated to make full use of the standard PYTHIA 
machinery for multiparton interactions, parton showers and hadronization 
<ref>Nav10,Cor10a</ref>. This is the approach pioneered in the PomPyt 
program by Ingelman and collaborators <ref>Ing97</ref>. 
 
<p/> 
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: 
<br/>(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 
<aloc href="TotalCrossSections">Total Cross Sections</aloc> page; 
<br/>(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 <ei>t</ei> of the exchanged Pomeron, 
also here see the 
<aloc href="TotalCrossSections">Total Cross Sections</aloc> page; 
<br/>(iii) a diffractive system of a given mass is classified either 
as low-mass unresolved, which gives a simple low-<ei>pT</ei> string 
topology, or as high-mass resolved, for which the full machinery of 
multiparton interactions and parton showers are applied, making use of 
<aloc href="PDFSelection">Pomeron PDFs</aloc>. 
<br/>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. 
 
<p/> 
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. 
 
<p/> 
In the event record the diffractive system in the case of an excited 
proton is denoted <code>p_diffr</code>, code 9902210, whereas 
a central diffractive system is denoted <code>rho_diffr</code>, 
code 9900110. Apart from representing the correct charge and baryon 
numbers, no deeper meaning should be attributed to the names. 
 
<p/> 
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 <ei>Z</ei>'s 
or <ei>W</ei>'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-<ei>pT</ei> region, even if a tail stretches to 
higher <ei>pT</ei> 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. 
 
<h3>Separation of soft diffraction into low and high masses</h3> 
 
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 
<eq> 
P_pert = P_max ( 1 - exp( (m_diffr - m_min) / m_width ) ) 
</eq> 
which vanishes for the diffractive system mass 
<ei>m_diffr &lt; m_min</ei>, and is <ei>1 - 1/e = 0.632</ei> for 
<ei>m_diffr = m_min + m_width</ei>, assuming <ei>P_max = 1</ei>. 
 
<parm name="Diffraction:mMinPert" default="10." min="5."> 
The abovementioned threshold mass <ei>m_min</ei> 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-<ei>pt</ei> description. 
</parm> 
 
<parm name="Diffraction:mWidthPert" default="10." min="1."> 
The abovementioned threshold width <ei>m_width.</ei> 
</parm> 
 
<parm name="Diffraction:probMaxPert" default="1." min="0." max="1." > 
The abovementioned maximum probability <ei>P_max.</ei>. 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. 
</parm> 
 
<h3>Low-mass soft diffraction</h3> 
 
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 
<eq> 
P_q / P_g = N / m^p 
</eq> 
is assumed. 
 
<parm name="Diffraction:pickQuarkNorm" default="5.0" min="0."> 
The abovementioned normalization <ei>N</ei> for the relative quark 
rate in diffractive systems. 
</parm> 
 
<parm name="Diffraction:pickQuarkPower" default="1.0"> 
The abovementioned mass-dependence power <ei>p</ei> for the relative 
quark rate in diffractive systems. 
</parm> 
 
<p/> 
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 
<ei>kT</ei> may be lower than in perturbative processes, however: 
 
<parm name="Diffraction:primKTwidth" default="0.5" min="0."> 
The width of Gaussian distributions in <ei>p_x</ei> and <ei>p_y</ei> 
separately that is assigned as a primordial <ei>kT</ei> to the two 
beam remnants when a gluon is kicked out of a diffractive system. 
</parm> 
 
<parm name="Diffraction:largeMassSuppress" default="4." min="0."> 
The choice of longitudinal and transverse structure of a diffractive 
beam remnant for a kicked-out gluon implies a remnant mass 
<ei>m_rem</ei> distribution (i.e. quark plus diquark invariant mass 
for a baryon beam) that knows no bounds. A suppression like 
<ei>(1 - m_rem^2 / m_diff^2)^p</ei> is therefore introduced, where 
<ei>p</ei> is the <code>diffLargeMassSuppress</code> parameter. 
</parm> 
 
<h3>High-mass soft diffraction</h3> 
 
The perturbative description need to use parton densities of the 
Pomeron. The options are described in the page on 
<aloc href="PDFSelection">PDF Selection</aloc>. 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 <ei>pT0</ei> 
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. 
 
<p/> 
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 name="Diffraction:sigmaRefPomP" default="10." min="2." max="40."> 
The assumed Pomeron-proton effective cross section, as used for 
multiparton interactions in single and double 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 
<eq> 
sigma_PomP(m) = sigma_PomP(m_ref) * (m / m_ref)^p 
</eq> 
where <ei>m</ei> is the mass of the diffractive system, <ei>m_ref</ei> 
is the reference mass scale <code>Diffraction:mRefPomP</code> below and 
<ei>p</ei> is the mass-dependence power <code>Diffraction:mPowPomP</code>. 
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 <ei>pT0</ei> 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> 
 
<parm name="Diffraction:mRefPomP" default="100.0" min="1."> 
The <ei>mRef</ei> reference mass scale introduced above. 
</parm> 
 
<parm name="Diffraction:mPowPomP" default="0.0" min="0.0" max="0.5"> 
The <ei>p</ei> mass rescaling pace introduced above. 
</parm> 
 
<parm name="Diffraction:sigmaRefPomPom" default="2." min="0.5" max="10."> 
The assumed Pomeron-Pomeron effective cross section, as used for 
multiparton interactions in central diffractive systems. Closely 
parallels the <code>Diffraction:sigmaRefPomP</code> above, so see 
there for details. Given the absence of relevant data, there is no 
point in introducing a separate possibility for an energy dependence, 
so instead <code>Diffraction:mRefPomP</code> and 
<code>Diffraction:mPowPomP</code> are used also here. 
</parm> 
 
<p/> 
Also note that, even for a fixed CM energy of events, the diffractive 
subsystem will range from the abovementioned threshold mass 
<ei>m_min</ei> to the full CM energy, with a variation of parameters 
such as <ei>pT0</ei> 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, <ei>A B &rarr; X B</ei> and <ei>A B &rarr; A X</ei> 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. 
 
<p/> 
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 
<code>bProfile = 4</code> option for now is not implemented. 
 
<modepick name="Diffraction:bProfile" default="1" 
min="0" max="3"> 
Choice of impact parameter profile for the incoming hadron beams. 
<option value="0">no impact parameter dependence at all.</option> 
<option value="1">a simple Gaussian matter distribution; 
no free parameters.</option> 
<option value="2">a double Gaussian matter distribution, 
with the two free parameters <ei>coreRadius</ei> and 
<ei>coreFraction</ei>.</option> 
<option value="3">an overlap function, i.e. the convolution of 
the matter distributions of the two incoming hadrons, of the form 
<ei>exp(- b^expPow)</ei>, where <ei>expPow</ei> is a free 
parameter.</option> 
</modepick> 
 
<parm name="Diffraction:coreRadius" default="0.4" min="0.1" max="1."> 
When assuming a double Gaussian matter profile, <ei>bProfile = 2</ei>, 
the inner core is assumed to have a radius that is a factor 
<ei>coreRadius</ei> smaller than the rest. 
</parm> 
 
<parm name="Diffraction:coreFraction" default="0.5" min="0." max="1."> 
When assuming a double Gaussian matter profile, <ei>bProfile = 2</ei>, 
the inner core is assumed to have a fraction <ei>coreFraction</ei> 
of the matter content of the hadron. 
</parm> 
 
<parm name="Diffraction:expPow" default="1." min="0.4" max="10."> 
When <ei>bProfile = 3</ei> it gives the power of the assumed overlap 
shape <ei>exp(- b^expPow)</ei>. Default corresponds to a simple 
exponential drop, which is not too dissimilar from the overlap 
obtained with the standard double Gaussian parameters. For 
<ei>expPow = 2</ei> we reduce to the simple Gaussian, <ei>bProfile = 1</ei>, 
and for <ei>expPow &rarr; infinity</ei> to no impact parameter dependence 
at all, <ei>bProfile = 0</ei>. For small <ei>expPow</ei> the program 
becomes slow and unstable, so the min limit must be respected. 
</parm> 
 
<h3>Hard diffraction</h3> 
 
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-<ei>pT</ei> 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 switch <code>SigmaDiffractive:PomFlux</code>, see 
<aloc href="TotalCrossSections">here</aloc>, and the partonic content 
of the Pomeron is described by the Pomeron PDFs, see 
<aloc href="PDFSelection">here</aloc>. 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. 
 
<p/> 
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, <ei>x_Pomeron</ei>, and the momentum transfer, <ei>t</ei>, in 
the process. This information can also be extracted in the main programs, 
see eg. example <code>main325.cc</code>. 
 
<p/> 
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. 
 
<p/> 
Recently, a scenario for hard diffraction with <ei>gamma</ei> beams has 
been introduced. Thus hard diffraction can be evaluated for both 
<ei>gamma + gamma</ei> and <ei>gamma + p</ei> processes within the 
usual photoproduction framework. A Pomeron can be taken from a 
<ei>gamma</ei> beam only if the photon is resolved. Currently this photon 
is then assumed always to be in a virtual <ei>rho</ei> state, thus 
leaving behind a physical <ei>rho</ei> beam remnant. If the Pomeron 
is taken from the proton, in the <ei>gamma + parton</ei> 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 <ei>gamma</ei>, the proton Pomeron flux is used but rescaled 
by a factor of <ei>sigma_tot^gamma+p/sigma_tot^pp</ei>, as a very first 
approximation to this unmeasured distribution. Otherwise all options are 
available as for hard diffraction in <ei>pp</ei> processes, and all 
limitations and cautions apply as for the photoproduction framework. 
 
<p/> 
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 name="Diffraction:doHard" default="off"> 
Allows for the possibility to include hard diffraction tests in a run. 
</flag> 
 
<modepick name="Diffraction:hardDiffSide" default="0" min="0" max="2"> 
Side which diffraction is evaluated for. Especially useful for diffraction 
in ep, where experiments only look for gaps on the proton side. 
<option value="0">Check for diffraction on boths sides A and B. 
</option> 
<option value="1">Check for diffraction on side A only. 
</option> 
<option value="2">Check for diffraction on side B only. 
</option> 
</modepick> 
 
<p/> 
There is also the possibility to select only a specific subset of events 
in hard diffraction. 
 
<modepick name="Diffraction:sampleType" default="2" min="1" max="4"> 
Type of process the user wants to generate. Depends strongly on how an event 
is classified as diffractive. 
<option value="1">Generate an inclusive sample of both diffractive and 
nondiffractive hard processes, MPI-unchecked. 
</option> 
<option value="2">Generate an inclusive sample of both diffractive and 
nondiffractive hard processes, MPI-checked. 
</option> 
<option value="3">Generate an exclusive diffractive sample, MPI-unchecked. 
</option> 
<option value="4">Generate an exclusive diffractive sample, MPI-checked. 
</option> 
</modepick> 
 
<p/> 
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 <ei>x_Pom=0.03</ei>. 
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 
<code>PDF:PomRescale</code> 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 name="Diffraction:PomFluxRescale" default="1.0" min="0.2" max="2.0"> 
Rescale the Pomeron flux by this uniform factor. It should be 
<code>1 / PDF:PomRescale</code> to preserve the convolution of Pomeron 
flux and PDFs, but for greater flxibility the two can be set separately. 
</parm> 
 
<p/> 
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 name="Diffraction:useMBRrenormalization" default="off"> 
Use the renormalized MBR flux. 
</flag> 
 
<p/> 
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. 
 
<modepick name="Diffraction:bSelHard" default="1" min="1" max="3"> 
Selection of impact parameter <ei>b</ei> 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 value="1">Use the same <ei>b</ei> 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 
<ei>b</ei> values already have been suppressed, few events should 
have high enhancement factors. 
</option> 
<option value="2">Use the square root of the <ei>b</ei> 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 <ei>b</ei> 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 <ei>b</ei> values closer to the average. 
</option> 
<option value="3">Pick a completely new <ei>b</ei>. 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. 
</option> 
</modepick> 
 
</chapter> 
 
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