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1<html>
2<head>
3<title>Spacelike Showers</title>
4<link rel="stylesheet" type="text/css" href="pythia.css"/>
5<link rel="shortcut icon" href="pythia32.gif"/>
6</head>
7<body>
8
9<h2>Spacelike Showers</h2>
10
11The PYTHIA algorithm for spacelike initial-state showers is
12based on the article [<a href="Bibliography.html" target="page">Sjo05</a>], where a
13transverse-momentum-ordered backwards evolution scheme is introduced,
14with the extension to fully interleaved evolution covered in
15[<a href="Bibliography.html" target="page">Cor10a</a>].
16This algorithm is a further development of the virtuality-ordered one
17presented in [<a href="Bibliography.html" target="page">Sj085</a>], with matching to first-order matrix
18element for <i>Z^0</i>, <i>W^+-</i> and Higgs (in the
19<i>m_t -> infinity</i> limit) production as introduced in
20[<a href="Bibliography.html" target="page">Miu99</a>].
21
22<p/>
23The normal user is not expected to call <code>SpaceShower</code> 
24directly, but only have it called from <code>Pythia</code>,
25via <code>PartonLevel</code>. Some of the parameters below,
26in particular <code>SpaceShower:alphaSvalue</code>,
27would be of interest for a tuning exercise, however.
28
29<h3>Main variables</h3>
30
31The maximum <i>pT</i> to be allowed in the shower evolution is
32related to the nature of the hard process itself. It involves a
33delicate balance between not doublecounting and not leaving any
34gaps in the coverage. The best procedure may depend on information
35only the user has: how the events were generated and mixed (e.g. with
36Les Houches Accord external input), and how they are intended to be
37used. Therefore a few options are available, with a sensible default
38behaviour.
39
40<p/><code>mode&nbsp; </code><strong> SpaceShower:pTmaxMatch &nbsp;</strong> 
41 (<code>default = <strong>0</strong></code>; <code>minimum = 0</code>; <code>maximum = 2</code>)<br/>
42Way in which the maximum shower evolution scale is set to match the
43scale of the hard process itself.
44<br/><code>option </code><strong> 0</strong> : <b>(i)</b> if the final state of the hard process
45(not counting subsequent resonance decays) contains at least one quark
46(<i>u, d, s, c ,b</i>), gluon or photon then <i>pT_max</i> 
47is chosen to be the factorization scale for internal processes
48and the <code>scale</code> value for Les Houches input;
49<b>(ii)</b> if not, emissions are allowed to go all the way up to
50the kinematical limit.
51The reasoning is that in the former set of processes the ISR
52emission of yet another quark, gluon or photon could lead to
53doublecounting, while no such danger exists in the latter case.
54 
55<br/><code>option </code><strong> 1</strong> : always use the factorization scale for an internal
56process and the <code>scale</code> value for Les Houches input,
57i.e. the lower value. This should avoid doublecounting, but
58may leave out some emissions that ought to have been simulated.
59(Also known as wimpy showers.)
60 
61<br/><code>option </code><strong> 2</strong> : always allow emissions up to the kinematical limit.
62This will simulate all possible event topologies, but may lead to
63doublecounting.
64(Also known as power showers.)
65 
66<br/><b>Note 1:</b> These options only apply to the hard interaction.
67Emissions off subsequent multiparton interactions are always constrainted
68to be below the factorization scale of the process itself. 
69<br/><b>Note 2:</b> Some processes contain matrix-element matching
70to the first emission; this is the case notably for single
71<i>gamma^*/Z^0, W^+-</i> and <i>H^0</i> production. Then default
72and option 2 give the correct result, while option 1 should never
73be used.
74 
75
76<p/><code>parm&nbsp; </code><strong> SpaceShower:pTmaxFudge &nbsp;</strong> 
77 (<code>default = <strong>1.0</strong></code>; <code>minimum = 0.25</code>; <code>maximum = 2.0</code>)<br/>
78In cases where the above <code>pTmaxMatch</code> rules would imply
79that <i>pT_max = pT_factorization</i>, <code>pTmaxFudge</code> 
80introduces a multiplicative factor <i>f</i> such that instead
81<i>pT_max = f * pT_factorization</i>. Only applies to the hardest
82interaction in an event, cf. below. It is strongly suggested that
83<i>f = 1</i>, but variations around this default can be useful to
84test this assumption.
85 
86
87<p/><code>parm&nbsp; </code><strong> SpaceShower:pTmaxFudgeMPI &nbsp;</strong> 
88 (<code>default = <strong>1.0</strong></code>; <code>minimum = 0.25</code>; <code>maximum = 2.0</code>)<br/>
89A multiplicative factor <i>f</i> such that
90<i>pT_max = f * pT_factorization</i>, as above, but here for the
91non-hardest interactions (when multiparton interactions are allowed).
92 
93
94<p/><code>mode&nbsp; </code><strong> SpaceShower:pTdampMatch &nbsp;</strong> 
95 (<code>default = <strong>0</strong></code>; <code>minimum = 0</code>; <code>maximum = 2</code>)<br/>
96These options only take effect when a process is allowed to radiate up
97to the kinematical limit by the above <code>pTmaxMatch</code> choice,
98and no matrix-element corrections are available. Then, in many processes,
99the fall-off in <i>pT</i> will be too slow by one factor of <i>pT^2</i>.
100That is, while showers have an approximate <i>dpT^2/pT^2</i> shape, often
101it should become more like <i>dpT^2/pT^4</i> at <i>pT</i> values above
102the scale of the hard process. Whether this actually is the case
103depends on the particular process studied, e.g. if <i>t</i>-channel
104gluon exchange is likely to dominate. If so, the options below could
105provide a reasonable high-<i>pT</i> behaviour without requiring
106higher-order calculations.
107<br/><code>option </code><strong> 0</strong> : emissions go up to the kinematical limit,
108with no special dampening.
109 
110<br/><code>option </code><strong> 1</strong> : emissions go up to the kinematical limit, 
111but dampened by a factor <i>k^2 Q^2_fac/(pT^2 + k^2 Q^2_fac)</i>,
112where <i>Q_fac</i> is the factorization scale and <i>k</i> is a
113multiplicative fudge factor stored in <code>pTdampFudge</code> below.
114 
115<br/><code>option </code><strong> 2</strong> : emissions go up to the kinematical limit,
116but dampened by a factor <i>k^2 Q^2_ren/(pT^2 + k^2 Q^2_ren)</i>,
117where <i>Q_ren</i> is the renormalization scale and <i>k</i> is a
118multiplicative fudge factor stored in <code>pTdampFudge</code> below.
119 
120<br/><b>Note:</b> These options only apply to the hard interaction.
121Emissions off subsequent multiparton interactions are always constrainted
122to be below the factorization scale of the process itself. 
123 
124
125<p/><code>parm&nbsp; </code><strong> SpaceShower:pTdampFudge &nbsp;</strong> 
126 (<code>default = <strong>1.0</strong></code>; <code>minimum = 0.25</code>; <code>maximum = 4.0</code>)<br/>
127In cases 1 and 2 above, where a dampening is imposed at around the
128factorization or renormalization scale, respectively, this allows the
129<i>pT</i> scale of dampening of radiation by a half to be shifted
130by this factor relative to the default <i>Q_fac</i> or <i>Q_ren</i>.
131This number ought to be in the neighbourhood of unity, but variations
132away from this value could do better in some processes.
133 
134
135<p/>
136The amount of QCD radiation in the shower is determined by
137<p/><code>parm&nbsp; </code><strong> SpaceShower:alphaSvalue &nbsp;</strong> 
138 (<code>default = <strong>0.137</strong></code>; <code>minimum = 0.06</code>; <code>maximum = 0.25</code>)<br/>
139The <i>alpha_strong</i> value at scale <code>M_Z^2</code>.
140Default value is picked equal to the one used in CTEQ 5L. 
141 
142
143<p/>
144The actual value is then regulated by the running to the scale
145<i>pT^2</i>, at which it is evaluated
146<p/><code>mode&nbsp; </code><strong> SpaceShower:alphaSorder &nbsp;</strong> 
147 (<code>default = <strong>1</strong></code>; <code>minimum = 0</code>; <code>maximum = 2</code>)<br/>
148Order at which <i>alpha_strong</i> runs,
149<br/><code>option </code><strong> 0</strong> : zeroth order, i.e. <i>alpha_strong</i> is kept
150fixed. 
151<br/><code>option </code><strong> 1</strong> : first order, which is the normal value. 
152<br/><code>option </code><strong> 2</strong> : second order. Since other parts of the code do
153not go to second order there is no strong reason to use this option,
154but there is also nothing wrong with it. 
155 
156
157<p/>
158QED radiation is regulated by the <i>alpha_electromagnetic</i>
159value at the <i>pT^2</i> scale of a branching.
160 
161<p/><code>mode&nbsp; </code><strong> SpaceShower:alphaEMorder &nbsp;</strong> 
162 (<code>default = <strong>1</strong></code>; <code>minimum = -1</code>; <code>maximum = 1</code>)<br/>
163The running of <i>alpha_em</i>.
164<br/><code>option </code><strong> 1</strong> : first-order running, constrained to agree with
165<code>StandardModel:alphaEMmZ</code> at the <i>Z^0</i> mass.
166 
167<br/><code>option </code><strong> 0</strong> : zeroth order, i.e. <i>alpha_em</i> is kept
168fixed at its value at vanishing momentum transfer. 
169<br/><code>option </code><strong> -1</strong> : zeroth order, i.e. <i>alpha_em</i> is kept
170fixed, but at <code>StandardModel:alphaEMmZ</code>, i.e. its value
171at the <i>Z^0</i> mass.
172   
173 
174
175<p/>
176The natural scale for couplings and PDFs is <i>pT^2</i>. To explore
177uncertainties it is possibly to vary around this value, however, in
178analogy with what can be done for
179<a href="CouplingsAndScales.html" target="page">hard processes</a>.
180
181<p/><code>parm&nbsp; </code><strong> SpaceShower:renormMultFac &nbsp;</strong> 
182 (<code>default = <strong>1.</strong></code>; <code>minimum = 0.1</code>; <code>maximum = 10.</code>)<br/>
183The default <i>pT^2</i> renormalization scale is multiplied by
184this prefactor. For QCD this is equivalent to a change of
185<i>Lambda^2</i> in the opposite direction, i.e. to a change of
186<i>alpha_strong(M_Z^2)</i> (except that flavour thresholds
187remain at fixed scales). Below, when <i>pT^2 + pT_0^2</i> is used
188as scale, it is this whole expression that is multiplied by the prefactor.
189 
190
191<p/><code>parm&nbsp; </code><strong> SpaceShower:factorMultFac &nbsp;</strong> 
192 (<code>default = <strong>1.</strong></code>; <code>minimum = 0.1</code>; <code>maximum = 10.</code>)<br/>
193The default <i>pT^2</i> factorization scale is multiplied by
194this prefactor.
195 
196
197<p/>
198There are two complementary ways of regularizing the small-<i>pT</i> 
199divergence, a sharp cutoff and a smooth dampening. These can be
200combined as desired but it makes sense to coordinate with how the
201same issue is handled in multiparton interactions.
202
203<p/><code>flag&nbsp; </code><strong> SpaceShower:samePTasMPI &nbsp;</strong> 
204 (<code>default = <strong>off</strong></code>)<br/>
205Regularize the <i>pT -> 0</i> divergence using the same sharp cutoff
206and smooth dampening parameters as used to describe multiparton interactions.
207That is, the <code>MultipartonInteractions:pT0Ref</code>,
208<code>MultipartonInteractions:ecmRef</code>,
209<code>MultipartonInteractions:ecmPow</code> and
210<code>MultipartonInteractions:pTmin</code> parameters are used to regularize
211all ISR QCD radiation, rather than the corresponding parameters below.
212This is a sensible physics ansatz, based on the assumption that colour
213screening effects influence both MPI and ISR in the same way. Photon
214radiation is regularized separately in either case.
215<br/><b>Warning:</b> if a large <code>pT0</code> is picked for multiparton
216interactions, such that the integrated interaction cross section is
217below the nondiffractive inelastic one, this <code>pT0</code> will
218automatically be scaled down to cope. Information on such a rescaling
219does NOT propagate to <code>SpaceShower</code>, however.
220   
221
222<p/>
223The actual <code>pT0</code> parameter used at a given CM energy scale,
224<i>ecmNow</i>, is obtained as
225<br/><i>
226    pT0 = pT0(ecmNow) = pT0Ref * (ecmNow / ecmRef)^ecmPow
227</i><br/>
228where <i>pT0Ref</i>, <i>ecmRef</i> and <i>ecmPow</i> are the
229three parameters below.
230
231<p/><code>parm&nbsp; </code><strong> SpaceShower:pT0Ref &nbsp;</strong> 
232 (<code>default = <strong>2.0</strong></code>; <code>minimum = 0.5</code>; <code>maximum = 10.0</code>)<br/>
233Regularization of the divergence of the QCD emission probability for
234<i>pT -> 0</i> is obtained by a factor <i>pT^2 / (pT0^2 + pT^2)</i>,
235and by using an <i>alpha_s(pT0^2 + pT^2)</i>. An energy dependence
236of the <i>pT0</i> choice is introduced by the next two parameters,
237so that <i>pT0Ref</i> is the <i>pT0</i> value for the reference
238cm energy, <i>pT0Ref = pT0(ecmRef)</i>.   
239 
240
241<p/><code>parm&nbsp; </code><strong> SpaceShower:ecmRef &nbsp;</strong> 
242 (<code>default = <strong>1800.0</strong></code>; <code>minimum = 1.</code>)<br/>
243The <i>ecmRef</i> reference energy scale introduced above.
244 
245
246<p/><code>parm&nbsp; </code><strong> SpaceShower:ecmPow &nbsp;</strong> 
247 (<code>default = <strong>0.0</strong></code>; <code>minimum = 0.</code>; <code>maximum = 0.5</code>)<br/>
248The <i>ecmPow</i> energy rescaling pace introduced above.
249 
250
251<p/><code>parm&nbsp; </code><strong> SpaceShower:pTmin &nbsp;</strong> 
252 (<code>default = <strong>0.2</strong></code>; <code>minimum = 0.1</code>; <code>maximum = 10.0</code>)<br/>
253Lower cutoff in <i>pT</i>, below which no further ISR branchings
254are allowed. Normally the <i>pT0</i> above would be used to
255provide the main regularization of the branching rate for
256<i>pT -> 0</i>, in which case <i>pTmin</i> is used  mainly for
257technical reasons. It is possible, however, to set <i>pT0Ref = 0</i> 
258and use <i>pTmin</i> to provide a step-function regularization,
259or to combine them in intermediate approaches. Currently <i>pTmin</i> 
260is taken to be energy-independent. 
261 
262
263<p/><code>parm&nbsp; </code><strong> SpaceShower:pTminChgQ &nbsp;</strong> 
264 (<code>default = <strong>0.5</strong></code>; <code>minimum = 0.01</code>)<br/>
265Parton shower cut-off <i>pT</i> for photon coupling to a coloured
266particle.
267 
268
269<p/><code>parm&nbsp; </code><strong> SpaceShower:pTminChgL &nbsp;</strong> 
270 (<code>default = <strong>0.0005</strong></code>; <code>minimum = 0.0001</code>)<br/>
271Parton shower cut-off mass for pure QED branchings.
272Assumed smaller than (or equal to) <i>pTminChgQ</i>.
273 
274
275<p/><code>flag&nbsp; </code><strong> SpaceShower:rapidityOrder &nbsp;</strong> 
276 (<code>default = <strong>off</strong></code>)<br/>
277Force emissions, after the first,  to be ordered in rapidity,
278i.e. in terms of decreasing angles in a backwards-evolution sense.
279Could be used to probe sensitivity to unordered emissions.
280Only affects QCD emissions.
281 
282
283<h3>Further variables</h3>
284
285These should normally not be touched. Their only function is for
286cross-checks.
287
288<p/>
289There are three flags you can use to switch on or off selected
290branchings in the shower:
291
292<p/><code>flag&nbsp; </code><strong> SpaceShower:QCDshower &nbsp;</strong> 
293 (<code>default = <strong>on</strong></code>)<br/>
294Allow a QCD shower; on/off = true/false.
295 
296
297<p/><code>flag&nbsp; </code><strong> SpaceShower:QEDshowerByQ &nbsp;</strong> 
298 (<code>default = <strong>on</strong></code>)<br/>
299Allow quarks to radiate photons; on/off = true/false.
300 
301
302<p/><code>flag&nbsp; </code><strong> SpaceShower:QEDshowerByL &nbsp;</strong> 
303 (<code>default = <strong>on</strong></code>)<br/>
304Allow leptons to radiate photons; on/off = true/false.
305 
306
307<p/>
308There are some further possibilities to modify the shower:
309
310<p/><code>flag&nbsp; </code><strong> SpaceShower:MEcorrections &nbsp;</strong> 
311 (<code>default = <strong>on</strong></code>)<br/>
312Use of matrix element corrections; on/off = true/false.
313 
314
315<p/><code>flag&nbsp; </code><strong> SpaceShower:MEafterFirst &nbsp;</strong> 
316 (<code>default = <strong>on</strong></code>)<br/>
317Use of matrix element corrections also after the first emission,
318for dipole ends of the same system that did not yet radiate.
319Only has a meaning if <code>MEcorrections</code> above is
320switched on.
321 
322
323<p/><code>flag&nbsp; </code><strong> SpaceShower:phiPolAsym &nbsp;</strong> 
324 (<code>default = <strong>on</strong></code>)<br/>
325Azimuthal asymmetry induced by gluon polarization; on/off = true/false.
326 
327
328<p/><code>flag&nbsp; </code><strong> SpaceShower:phiIntAsym &nbsp;</strong> 
329 (<code>default = <strong>on</strong></code>)<br/>
330Azimuthal asymmetry induced by interference; on/off = true/false.
331 
332
333<p/><code>parm&nbsp; </code><strong> SpaceShower:strengthIntAsym &nbsp;</strong> 
334 (<code>default = <strong>0.7</strong></code>; <code>minimum = 0.</code>; <code>maximum = 0.9</code>)<br/>
335Size of asymmetry induced by interference. Natural value of order 0.5;
336expression would blow up for a value of 1.
337 
338
339<p/><code>mode&nbsp; </code><strong> SpaceShower:nQuarkIn &nbsp;</strong> 
340 (<code>default = <strong>5</strong></code>; <code>minimum = 0</code>; <code>maximum = 5</code>)<br/>
341Number of allowed quark flavours in <i>g -> q qbar</i> branchings,
342when kinematically allowed, and thereby also in incoming beams.
343Changing it to 4 would forbid <i>g -> b bbar</i>, etc.
344 
345
346<h3>Technical notes</h3>
347
348Almost everything is equivalent to the algorithm in [1]. Minor changes
349are as follows.
350<ul>
351<li>
352It is now possible to have a second-order running <i>alpha_s</i>,
353in addition to fixed or first-order running.
354</li>
355<li>
356The description of heavy flavour production in the threshold region
357has been modified, so as to be more forgiving about mismatches
358between the <i>c/b</i>  masses used in Pythia relative to those
359used in a respective PDF parametrization. The basic idea is that,
360in the threshold region of a heavy quark <i>Q</i>, <i>Q = c/b</i>,
361the effect of subsequent <i>Q -> Q g</i> branchings is negligible.
362If so, then
363<br/><i>
364   f_Q(x, pT2) = integral_mQ2^pT2  dpT'2/pT'2 * alpha_s(pT'2)/2pi
365      * integral P(z) g(x', pT'2) delta(x - z x')
366</i><br/>
367so use this to select the <i>pT2</i> of the <i>g -> Q Qbar</i> 
368branching. In the old formalism the same kind of behaviour should
369be obtained, but by a cancellation of a <i>1/f_Q</i> that diverges
370at the theshold and a Sudakov that vanishes.
371<br/>
372The strategy therefore is that, once <i>pT2 &lt; f * mQ2</i>, with
373<i>f</i> a parameter of the order of 2, a <i>pT2</i> is chosen
374like <i>dpT2/pT2</i> between <i>mQ2</i> and <i>f * mQ2</i>, a
375nd a <i>z</i> flat in the allowed range. Thereafter acceptance
376is based on the product of three factors, representing the running
377of <i>alpha_strong</i>, the splitting kernel (including the mass term)
378and the gluon density weight. At failure, a new <i>pT2</i> is chosen
379in the same  range, i.e. is not required to be lower since no Sudakov
380is involved.
381</li>
382<li>
383The QED algorithm now allows for hadron beams with non-zero photon
384content. The backwards-evolution of a photon in a hadron is identical
385to that of a gluon, with <i>CF -> eq^2</i> and <i>CA -> 0</i>.
386Note that this will only work in conjunction with
387parton distribution that explicitly include photons as part of the
388hadron structure (such as the MRST2004qed set). Since Pythia's
389internal sets do not allow for photon content in hadrons, it is thus
390necessary to use the LHAPDF interface to make use of this feature. The
391possibility of a fermion backwards-evolving to a photon has not yet
392been included, nor has photon backwards-evolution in lepton beams.
393</li>
394</ul>
395
396</body>
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