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2<head>
3<title>Diffraction</title>
4<link rel="stylesheet" type="text/css" href="pythia.css"/>
5<link rel="shortcut icon" href="pythia32.gif"/>
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7<body>
8
9<h2>Diffraction</h2>
10
11<h3>Introduction</h3>
12
13Diffraction is not well understood, and several alternative approaches
14have been proposed. Here we follow a fairly conventional Pomeron-based
15one, in the Ingelman-Schlein spirit [<a href="Bibliography.html" target="page">Ing85</a>],
16but integrated to make full use of the standard PYTHIA machinery
17for multiparton interactions, parton showers and hadronization
18[<a href="Bibliography.html" target="page">Nav10,Cor10a</a>]. This is the approach pioneered in the PomPyt
19program by Ingelman and collaborators [<a href="Bibliography.html" target="page">Ing97</a>].
20
21<p/>
22For ease of use (and of modelling), the Pomeron-specific parts of the
23generation are subdivided into three sets of parameters that are rather
24independent of each other:
25<br/>(i) the total, elastic and diffractive cross sections are
26parametrized as functions of the CM energy, or can be set by the user
27to the desired values, see the
28<a href="TotalCrossSections.html" target="page">Total Cross Sections</a> page;
29<br/>(ii) once it has been decided to have a diffractive process,
30a Pomeron flux parametrization is used to pick the mass of the
31diffractive system(s) and the <i>t</i> of the exchanged Pomeron,
32see below;
33<br/>(iii) a diffractive system of a given mass is classified either
34as low-mass unresolved, which gives a simple low-<i>pT</i> string
35topology, or as high-mass resolved, for which the full machinery of
36multiparton interactions and parton showers are applied, making use of
37<a href="PDFSelection.html" target="page">Pomeron PDFs</a>.
38<br/>The parameters related to multiparton interactions, parton showers
39and hadronization are kept the same as for normal nondiffractive events,
40with only one exception. This may be questioned, especially for the
41multiparton interactions, but we do not believe that there are currently
42enough good diffractive data that would allow detailed separate tunes.
43 
44<p/>
45The above subdivision may not represent the way "physics comes about".
46For instance, the total diffractive cross section can be viewed as a
47convolution of a Pomeron flux with a Pomeron-proton total cross section.
48Since neither of the two is known from first principles there will be
49a significant amount of ambiguity in the flux factor. The picture is
50further complicated by the fact that the possibility of simultaneous
51further multiparton interactions ("cut Pomerons") will screen the rate of
52diffractive systems. In the end, our set of parameters refers to the
53effective description that emerges out of these effects, rather than
54to the underlying "bare" parameters. 
55 
56<p/>
57In the event record the diffractive system in the case of an excited
58proton is denoted <code>p_diffr</code>, code 9902210, whereas
59a central diffractive system is denoted <code>rho_diffr</code>,
60code 9900110. Apart from representing the correct charge and baryon
61numbers, no deeper meaning should be attributed to the names.
62
63<h3>Pomeron flux</h3>
64
65As already mentioned above, the total diffractive cross section is fixed
66by a default energy-dependent parametrization or by the user, see the
67<a href="TotalCrossSections.html" target="page">Total Cross Sections</a> page.
68Therefore we do not attribute any significance to the absolute
69normalization of the Pomeron flux. The choice of Pomeron flux model
70still will decide on the mass spectrum of diffractive states and the
71<i>t</i> spectrum of the Pomeron exchange.
72
73<p/><code>mode&nbsp; </code><strong> Diffraction:PomFlux &nbsp;</strong> 
74 (<code>default = <strong>1</strong></code>; <code>minimum = 1</code>; <code>maximum = 5</code>)<br/>
75Parametrization of the Pomeron flux <i>f_Pom/p( x_Pom, t)</i>.
76<br/><code>option </code><strong> 1</strong> : Schuler and Sj&ouml;strand [<a href="Bibliography.html" target="page">Sch94</a>]: based on a
77critical Pomeron, giving a mass spectrum roughly like <i>dm^2/m^2</i>;
78a mass-dependent exponential <i>t</i> slope that reduces the rate
79of low-mass states; partly compensated by a very-low-mass (resonance region)
80enhancement. Is currently the only one that contains a separate
81<i>t</i> spectrum for double diffraction (along with MBR) and
82separate parameters for pion beams. 
83<br/><code>option </code><strong> 2</strong> : Bruni and Ingelman [<a href="Bibliography.html" target="page">Bru93</a>]: also a critical
84Pomeron giving close to <i>dm^2/m^2</i>,  with a <i>t</i> distribution
85the sum of two exponentials. The original model only covers single
86diffraction, but is here expanded by analogy to double and central
87diffraction. 
88<br/><code>option </code><strong> 3</strong> : a conventional Pomeron description, in the RapGap
89manual [<a href="Bibliography.html" target="page">Jun95</a>] attributed to Berger et al. and Streng
90[<a href="Bibliography.html" target="page">Ber87a</a>], but there (and here) with values updated to a
91supercritical Pomeron with <i>epsilon &gt; 0</i> (see below),
92which gives a stronger peaking towards low-mass diffractive states,
93and with a mass-dependent (the <i>alpha'</i> below) exponential
94<i>t</i> slope. The original model only covers single diffraction,
95but is here expanded by analogy to double and central diffraction.
96 
97<br/><code>option </code><strong> 4</strong> : a conventional Pomeron description, attributed to
98Donnachie and Landshoff [<a href="Bibliography.html" target="page">Don84</a>], again with supercritical Pomeron,
99with the same two parameters as option 3 above, but this time with a
100power-law <i>t</i> distribution. The original model only covers single
101diffraction, but is here expanded by analogy to double and central
102diffraction. 
103<br/><code>option </code><strong> 5</strong> :  the MBR (Minimum Bias Rockefeller) simulation of
104(anti)proton-proton interactions [<a href="Bibliography.html" target="page">Cie12</a>]. The event
105generation follows a renormalized-Regge-theory model, sucessfully tested
106using CDF data. The simulation includes single and double diffraction,
107as well as the central diffractive (double-Pomeron exchange) process (106).
108Only <i>p p</i>, <i>pbar p</i> and <i>p pbar</i> beam combinations
109are allowed for this option. Several parameters of this model are listed
110below.   
111   
112
113<p/>
114In options 3 and 4 above, the Pomeron Regge trajectory is
115parametrized as
116<br/><i>
117alpha(t) = 1 + epsilon + alpha' t
118</i><br/>
119The <i>epsilon</i> and <i>alpha'</i> parameters can be set
120separately:
121
122<p/><code>parm&nbsp; </code><strong> Diffraction:PomFluxEpsilon &nbsp;</strong> 
123 (<code>default = <strong>0.085</strong></code>; <code>minimum = 0.02</code>; <code>maximum = 0.15</code>)<br/>
124The Pomeron trajectory intercept <i>epsilon</i> above. For technical
125reasons <i>epsilon &gt; 0</i> is necessary in the current implementation.
126
127<p/><code>parm&nbsp; </code><strong> Diffraction:PomFluxAlphaPrime &nbsp;</strong> 
128 (<code>default = <strong>0.25</strong></code>; <code>minimum = 0.1</code>; <code>maximum = 0.4</code>)<br/>
129The Pomeron trajectory slope <i>alpha'</i> above.
130
131<p/>
132When option 5 is selected, the following parameters of the MBR model
133[<a href="Bibliography.html" target="page">Cie12</a>] are used:
134
135<p/><code>parm&nbsp; </code><strong> Diffraction:MBRepsilon &nbsp;</strong> 
136 (<code>default = <strong>0.104</strong></code>; <code>minimum = 0.02</code>; <code>maximum = 0.15</code>)<br/>
137<p/><code>parm&nbsp; </code><strong> Diffraction:MBRalpha &nbsp;</strong> 
138 (<code>default = <strong>0.25</strong></code>; <code>minimum = 0.1</code>; <code>maximum = 0.4</code>)<br/>
139the parameters of the Pomeron trajectory.
140
141<p/><code>parm&nbsp; </code><strong> Diffraction:MBRbeta0 &nbsp;</strong> 
142 (<code>default = <strong>6.566</strong></code>; <code>minimum = 0.0</code>; <code>maximum = 10.0</code>)<br/>
143<p/><code>parm&nbsp; </code><strong> Diffraction:MBRsigma0 &nbsp;</strong> 
144 (<code>default = <strong>2.82</strong></code>; <code>minimum = 0.0</code>; <code>maximum = 5.0</code>)<br/>
145the Pomeron-proton coupling, and the total Pomeron-proton cross section.
146
147<p/><code>parm&nbsp; </code><strong> Diffraction:MBRm2Min &nbsp;</strong> 
148 (<code>default = <strong>1.5</strong></code>; <code>minimum = 0.0</code>; <code>maximum = 3.0</code>)<br/>
149the lowest value of the mass squared of the dissociated system.
150
151<p/><code>parm&nbsp; </code><strong> Diffraction:MBRdyminSDflux &nbsp;</strong> 
152 (<code>default = <strong>2.3</strong></code>; <code>minimum = 0.0</code>; <code>maximum = 5.0</code>)<br/>
153<p/><code>parm&nbsp; </code><strong> Diffraction:MBRdyminDDflux &nbsp;</strong> 
154 (<code>default = <strong>2.3</strong></code>; <code>minimum = 0.0</code>; <code>maximum = 5.0</code>)<br/>
155<p/><code>parm&nbsp; </code><strong> Diffraction:MBRdyminCDflux &nbsp;</strong> 
156 (<code>default = <strong>2.3</strong></code>; <code>minimum = 0.0</code>; <code>maximum = 5.0</code>)<br/>
157the minimum width of the rapidity gap used in the calculation of
158<i>Ngap(s)</i> (flux renormalization).
159
160<p/><code>parm&nbsp; </code><strong> Diffraction:MBRdyminSD &nbsp;</strong> 
161 (<code>default = <strong>2.0</strong></code>; <code>minimum = 0.0</code>; <code>maximum = 5.0</code>)<br/>
162<p/><code>parm&nbsp; </code><strong> Diffraction:MBRdyminDD &nbsp;</strong> 
163 (<code>default = <strong>2.0</strong></code>; <code>minimum = 0.0</code>; <code>maximum = 5.0</code>)<br/>
164<p/><code>parm&nbsp; </code><strong> Diffraction:MBRdyminCD &nbsp;</strong> 
165 (<code>default = <strong>2.0</strong></code>; <code>minimum = 0.0</code>; <code>maximum = 5.0</code>)<br/>
166the minimum width of the rapidity gap used in the calculation of cross
167sections, i.e. the parameter <i>dy_S</i>, which suppresses the cross
168section at low <i>dy</i> (non-diffractive region).
169
170<p/><code>parm&nbsp; </code><strong> Diffraction:MBRdyminSigSD &nbsp;</strong> 
171 (<code>default = <strong>0.5</strong></code>; <code>minimum = 0.001</code>; <code>maximum = 5.0</code>)<br/>
172<p/><code>parm&nbsp; </code><strong> Diffraction:MBRdyminSigDD &nbsp;</strong> 
173 (<code>default = <strong>0.5</strong></code>; <code>minimum = 0.001</code>; <code>maximum = 5.0</code>)<br/>
174<p/><code>parm&nbsp; </code><strong> Diffraction:MBRdyminSigCD &nbsp;</strong> 
175 (<code>default = <strong>0.5</strong></code>; <code>minimum = 0.001</code>; <code>maximum = 5.0</code>)<br/>
176the parameter <i>sigma_S</i>, used for the cross section suppression at
177low <i>dy</i> (non-diffractive region).
178
179<h3>Separation into low and high masses</h3>
180
181Preferably one would want to have a perturbative picture of the
182dynamics of Pomeron-proton collisions, like multiparton interactions
183provide for proton-proton ones. However, while PYTHIA by default
184will only allow collisions with a CM energy above 10 GeV, the
185mass spectrum of diffractive systems will stretch to down to
186the order of 1.2 GeV. It would not be feasible to attempt a
187perturbative description there. Therefore we do offer a simpler
188low-mass description, with only longitudinally stretched strings,
189with a gradual switch-over to the perturbative picture for higher
190masses. The probability for the latter picture is parametrized as
191<br/><i>
192P_pert = P_max ( 1 - exp( (m_diffr - m_min) / m_width ) )
193</i><br/> 
194which vanishes for the diffractive system mass
195<i>m_diffr &lt; m_min</i>, and is <i>1 - 1/e = 0.632</i> for
196<i>m_diffr = m_min + m_width</i>, assuming <i>P_max = 1</i>.
197
198<p/><code>parm&nbsp; </code><strong> Diffraction:mMinPert &nbsp;</strong> 
199 (<code>default = <strong>10.</strong></code>; <code>minimum = 5.</code>)<br/>
200The abovementioned threshold mass <i>m_min</i> for phasing in a
201perturbative treatment. If you put this parameter to be bigger than
202the CM energy then there will be no perturbative description at all,
203but only the older low-<i>pt</i> description.
204 
205
206<p/><code>parm&nbsp; </code><strong> Diffraction:mWidthPert &nbsp;</strong> 
207 (<code>default = <strong>10.</strong></code>; <code>minimum = 0.</code>)<br/>
208The abovementioned threshold width <i>m_width.</i>
209 
210
211<p/><code>parm&nbsp; </code><strong> Diffraction:probMaxPert &nbsp;</strong> 
212 (<code>default = <strong>1.</strong></code>; <code>minimum = 0.</code>; <code>maximum = 1.</code>)<br/>
213The abovementioned maximum probability <i>P_max.</i>. Would
214normally be assumed to be unity, but a somewhat lower value could
215be used to represent a small nonperturbative component also at
216high diffractive masses.
217 
218
219<h3>Low-mass diffraction</h3>
220
221When an incoming hadron beam is diffractively excited, it is modeled
222as if either a valence quark or a gluon is kicked out from the hadron.
223In the former case this produces a simple string to the leftover
224remnant, in the latter it gives a hairpin arrangement where a string
225is stretched from one quark in the remnant, via the gluon, back to the   
226rest of the remnant. The latter ought to dominate at higher mass of
227the diffractive system. Therefore an approximate behaviour like
228<br/><i>
229P_q / P_g = N / m^p
230</i><br/> 
231is assumed.
232
233<p/><code>parm&nbsp; </code><strong> Diffraction:pickQuarkNorm &nbsp;</strong> 
234 (<code>default = <strong>5.0</strong></code>; <code>minimum = 0.</code>)<br/>
235The abovementioned normalization <i>N</i> for the relative quark
236rate in diffractive systems.
237 
238
239<p/><code>parm&nbsp; </code><strong> Diffraction:pickQuarkPower &nbsp;</strong> 
240 (<code>default = <strong>1.0</strong></code>)<br/>
241The abovementioned mass-dependence power <i>p</i> for the relative
242quark rate in diffractive systems.
243 
244
245<p/>
246When a gluon is kicked out from the hadron, the longitudinal momentum
247sharing between the the two remnant partons is determined by the
248same parameters as above. It is plausible that the primordial
249<i>kT</i> may be lower than in perturbative processes, however:
250
251<p/><code>parm&nbsp; </code><strong> Diffraction:primKTwidth &nbsp;</strong> 
252 (<code>default = <strong>0.5</strong></code>; <code>minimum = 0.</code>)<br/>
253The width of Gaussian distributions in <i>p_x</i> and <i>p_y</i> 
254separately that is assigned as a primordial <i>kT</i> to the two
255beam remnants when a gluon is kicked out of a diffractive system.
256 
257
258<p/><code>parm&nbsp; </code><strong> Diffraction:largeMassSuppress &nbsp;</strong> 
259 (<code>default = <strong>2.</strong></code>; <code>minimum = 0.</code>)<br/>
260The choice of longitudinal and transverse structure of a diffractive
261beam remnant for a kicked-out gluon implies a remnant mass
262<i>m_rem</i> distribution (i.e. quark plus diquark invariant mass
263for a baryon beam) that knows no bounds. A suppression like
264<i>(1 - m_rem^2 / m_diff^2)^p</i> is therefore introduced, where
265<i>p</i> is the <code>diffLargeMassSuppress</code> parameter.   
266 
267
268<h3>High-mass diffraction</h3>
269
270The perturbative description need to use parton densities of the
271Pomeron. The options are described in the page on
272<a href="PDFSelection.html" target="page">PDF Selection</a>. The standard
273perturbative multiparton interactions framework then provides
274cross sections for parton-parton interactions. In order to
275turn these cross section into probabilities one also needs an
276ansatz for the Pomeron-proton total cross section. In the literature
277one often finds low numbers for this, of the order of 2 mb.
278These, if taken at face value, would give way too much activity
279per event. There are ways to tame this, e.g. by a larger <i>pT0</i>
280than in the normal pp framework. Actually, there are many reasons
281to use a completely different set of parameters for MPI in
282diffraction than in pp collisions, especially with respect to the
283impact-parameter picture, see below. A lower number in some frameworks
284could alternatively be regarded as a consequence of screening, with
285a larger "bare" number.   
286
287<p/>
288For now, however, an attempt at the most general solution would
289carry too far, and instead we patch up the problem by using a
290larger Pomeron-proton total cross section, such that average
291activity makes more sense. This should be viewed as the main
292tunable parameter in the description of high-mass diffraction.
293It is to be fitted to diffractive event-shape data such as the average
294charged multiplicity. It would be very closely tied to the choice of
295Pomeron PDF; we remind that some of these add up to less than unit
296momentum sum in the Pomeron, a choice that also affect the value
297one ends up with. Furthermore, like with hadronic cross sections,
298it is quite plausible that the Pomeron-proton cross section increases
299with energy, so we have allowed for a powerlike dependence on the
300diffractive mass.
301
302<p/><code>parm&nbsp; </code><strong> Diffraction:sigmaRefPomP &nbsp;</strong> 
303 (<code>default = <strong>10.</strong></code>; <code>minimum = 2.</code>; <code>maximum = 40.</code>)<br/>
304The assumed Pomeron-proton effective cross section, as used for
305multiparton interactions in diffractive systems. If this cross section
306is made to depend on the mass of the diffractive system then the above
307value refers to the cross section at the reference scale, and
308<br/><i>
309sigma_PomP(m) = sigma_PomP(m_ref) * (m / m_ref)^p 
310</i><br/>
311where <i>m</i> is the mass of the diffractive system, <i>m_ref</i> 
312is the reference mass scale <code>Diffraction:mRefPomP</code> below and
313<i>p</i> is the mass-dependence power <code>Diffraction:mPowPomP</code>.
314Note that a larger cross section value gives less MPI activity per event.
315There is no point in making the cross section too big, however, since
316then <i>pT0</i> will be adjusted downwards to ensure that the
317integrated perturbative cross section stays above this assumed total
318cross section. (The requirement of at least one perturbative interaction
319per event.)
320 
321
322<p/><code>parm&nbsp; </code><strong> Diffraction:mRefPomP &nbsp;</strong> 
323 (<code>default = <strong>100.0</strong></code>; <code>minimum = 1.</code>)<br/>
324The <i>mRef</i> reference mass scale introduced above.
325 
326
327<p/><code>parm&nbsp; </code><strong> Diffraction:mPowPomP &nbsp;</strong> 
328 (<code>default = <strong>0.0</strong></code>; <code>minimum = 0.0</code>; <code>maximum = 0.5</code>)<br/>
329The <i>p</i> mass rescaling pace introduced above.
330 
331
332<p/> 
333Also note that, even for a fixed CM energy of events, the diffractive
334subsystem will range from the abovementioned threshold mass
335<i>m_min</i> to the full CM energy, with a variation of parameters
336such as <i>pT0</i> along this mass range. Therefore multiparton
337interactions are initialized for a few different diffractive masses,
338currently five, and all relevant parameters are interpolated between
339them to obtain the behaviour at a specific diffractive mass.
340Furthermore, <i>A B -&gt;X B</i> and <i>A B -&gt;A X</i> are
341initialized separately, to allow for different beams or PDF's on the
342two sides. These two aspects mean that initialization of MPI is
343appreciably slower when perturbative high-mass diffraction is allowed.
344
345<p/> 
346Diffraction tends to be peripheral, i.e. occur at intermediate impact
347parameter for the two protons. That aspect is implicit in the selection
348of diffractive cross section. For the simulation of the Pomeron-proton
349subcollision it is the impact-parameter distribution of that particular
350subsystem that should rather be modelled. That is, it also involves
351the transverse coordinate space of a Pomeron wavefunction. The outcome
352of the convolution therefore could be a different shape than for
353nondiffractive events. For simplicity we allow the same kind of
354options as for nondiffractive events, except that the
355<code>bProfile = 4</code> option for now is not implemented.
356
357<p/><code>mode&nbsp; </code><strong> Diffraction:bProfile &nbsp;</strong> 
358 (<code>default = <strong>1</strong></code>; <code>minimum = 0</code>; <code>maximum = 3</code>)<br/>
359Choice of impact parameter profile for the incoming hadron beams.
360<br/><code>option </code><strong> 0</strong> : no impact parameter dependence at all. 
361<br/><code>option </code><strong> 1</strong> : a simple Gaussian matter distribution;
362no free parameters. 
363<br/><code>option </code><strong> 2</strong> : a double Gaussian matter distribution,
364with the two free parameters <i>coreRadius</i> and
365<i>coreFraction</i>. 
366<br/><code>option </code><strong> 3</strong> : an overlap function, i.e. the convolution of
367the matter distributions of the two incoming hadrons, of the form
368<i>exp(- b^expPow)</i>, where <i>expPow</i> is a free
369parameter.   
370 
371
372<p/><code>parm&nbsp; </code><strong> Diffraction:coreRadius &nbsp;</strong> 
373 (<code>default = <strong>0.4</strong></code>; <code>minimum = 0.1</code>; <code>maximum = 1.</code>)<br/>
374When assuming a double Gaussian matter profile, <i>bProfile = 2</i>,
375the inner core is assumed to have a radius that is a factor
376<i>coreRadius</i> smaller than the rest.
377   
378
379<p/><code>parm&nbsp; </code><strong> Diffraction:coreFraction &nbsp;</strong> 
380 (<code>default = <strong>0.5</strong></code>; <code>minimum = 0.</code>; <code>maximum = 1.</code>)<br/>
381When assuming a double Gaussian matter profile, <i>bProfile = 2</i>,
382the inner core is assumed to have a fraction <i>coreFraction</i> 
383of the matter content of the hadron.
384   
385
386<p/><code>parm&nbsp; </code><strong> Diffraction:expPow &nbsp;</strong> 
387 (<code>default = <strong>1.</strong></code>; <code>minimum = 0.4</code>; <code>maximum = 10.</code>)<br/>
388When <i>bProfile = 3</i> it gives the power of the assumed overlap
389shape <i>exp(- b^expPow)</i>. Default corresponds to a simple
390exponential drop, which is not too dissimilar from the overlap
391obtained with the standard double Gaussian parameters. For
392<i>expPow = 2</i> we reduce to the simple Gaussian, <i>bProfile = 1</i>,
393and for <i>expPow -> infinity</i> to no impact parameter dependence
394at all, <i>bProfile = 0</i>. For small <i>expPow</i> the program
395becomes slow and unstable, so the min limit must be respected.
396   
397
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