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b584e2f5 | 1 | <chapter name="Spacelike Showers"> |
2 | ||
3 | <h2>Spacelike Showers</h2> | |
4 | ||
5 | The PYTHIA algorithm for spacelike initial-state showers is | |
6 | based on the recent article <ref>Sjo05</ref>, where a | |
7 | transverse-momentum-ordered backwards evolution scheme is introduced. | |
8 | This algorithm is a further development of the virtuality-ordered one | |
9 | presented in <ref>Sj085</ref>, with matching to first-order matrix | |
10 | element for <ei>Z^0</ei>, <ei>W^+-</ei> and Higgs (in the | |
11 | <ei>m_t -> infinity</ei> limit) production as introduced in | |
12 | <ref>Miu99</ref>. | |
13 | ||
14 | <p/> | |
15 | The normal user is not expected to call <code>SpaceShower</code> | |
16 | directly, but only have it called from <code>Pythia</code>, | |
17 | via <code>PartonLevel</code>. Some of the parameters below, | |
18 | in particular <code>SpaceShower:alphaSvalue</code>, | |
19 | would be of interest for a tuning exercise, however. | |
20 | ||
21 | <h3>Main variables</h3> | |
22 | ||
23 | The maximum <ei>pT</ei> to be allowed in the shower evolution is | |
24 | related to the nature of the hard process itself. It involves a | |
25 | delicate balance between not doublecounting and not leaving any | |
26 | gaps in the coverage. The best procedure may depend on information | |
27 | only the user has: how the events were generated and mixed (e.g. with | |
28 | Les Houches Accord external input), and how they are intended to be | |
29 | used. Therefore a few options are available, with a sensible default | |
30 | behaviour. | |
31 | ||
32 | <modepick name="SpaceShower:pTmaxMatch" default="0" min="0" max="2"> | |
33 | Way in which the maximum shower evolution scale is set to match the | |
34 | scale of the hard process itself. | |
35 | <option value="0"><b>(i)</b> if the final state of the hard process | |
36 | (not counting subsequent resonance decays) contains at least one quark | |
37 | (<ei>u, d, s, c ,b</ei>), gluon or photon then <ei>pT_max</ei> | |
38 | is chosen to be the factorization scale for internal processes | |
39 | and the <code>scale</code> value for Les Houches input; | |
40 | <b>(ii)</b> if not, emissions are allowed to go all the way up to | |
41 | the kinematical limit. | |
42 | The reasoning is that in the former set of processes the ISR | |
43 | emission of yet another quark, gluon or photon could lead to | |
44 | doublecounting, while no such danger exists in the latter case. | |
45 | </option> | |
46 | <option value="1">always use the factorization scale for an internal | |
47 | process and the <code>scale</code> value for Les Houches input, | |
48 | i.e. the lower value. This should avoid doublecounting, but | |
49 | may leave out some emissions that ought to have been simulated. | |
50 | (Also known as wimpy showers.) | |
51 | </option> | |
52 | <option value="2">always allow emissions up to the kinematical limit. | |
53 | This will simulate all possible event topologies, but may lead to | |
54 | doublecounting. | |
55 | (Also known as power showers.) | |
56 | </option> | |
57 | <note>Note 1:</note> These options only apply to the hard interaction. | |
58 | Emissions off subsequent multiple interactions are always constrainted | |
59 | to be below the factorization scale of the process itself. | |
60 | <note>Note 2:</note> Some processes contain matrix-element matching | |
61 | to the first emission; this is the case notably for single | |
62 | <ei>gamma^*/Z^0, W^+-</ei> and <ei>H^0</ei> production. Then default | |
63 | and option 2 give the correct result, while option 1 should never | |
64 | be used. | |
65 | </modepick> | |
66 | ||
67 | <parm name="SpaceShower:pTmaxFudge" default="1.0" min="0.25" max="2.0"> | |
68 | In cases where the above <code>pTmaxMatch</code> rules would imply | |
69 | that <ei>pT_max = pT_factorization</ei>, <code>pTmaxFudge</code> | |
70 | introduces a multiplicative factor <ei>f</ei> such that instead | |
71 | <ei>pT_max = f * pT_factorization</ei>. Only applies to the hardest | |
72 | interaction in an event, cf. below. It is strongly suggested that | |
73 | <ei>f = 1</ei>, but variations around this default can be useful to | |
74 | test this assumption. | |
75 | </parm> | |
76 | ||
77 | <parm name="SpaceShower:pTmaxFudgeMI" default="1.0" min="0.25" max="2.0"> | |
78 | A multiplicative factor <ei>f</ei> such that | |
79 | <ei>pT_max = f * pT_factorization</ei>, as above, but here for the | |
80 | non-hardest interactions (when multiple interactions are allowed). | |
81 | </parm> | |
82 | ||
83 | <modepick name="SpaceShower:pTdampMatch" default="0" min="0" max="2"> | |
84 | These options only take effect when a process is allowed to radiate up | |
85 | to the kinematical limit by the above <code>pTmaxMatch</code> choice, | |
86 | and no matrix-element corrections are available. Then, in many processes, | |
87 | the fall-off in <ei>pT</ei> will be too slow by one factor of <ei>pT^2</ei>. | |
88 | That is, while showers have an approximate <ei>dpT^2/pT^2</ei> shape, often | |
89 | it should become more like <ei>dpT^2/pT^4</ei> at <ei>pT</ei> values above | |
90 | the scale of the hard process. Whether this actually is the case | |
91 | depends on the particular process studied, e.g. if <ei>t</ei>-channel | |
92 | gluon exchange is likely to dominate. If so, the options below could | |
93 | provide a reasonable high-<ei>pT</ei> behaviour without requiring | |
94 | higher-order calculations. | |
95 | <option value="0">emissions go up to the kinematical limit, | |
96 | with no special dampening. | |
97 | </option> | |
98 | <option value="1">emissions go up to the kinematical limit, | |
99 | but dampened by a factor <ei>k^2 Q^2_fac/(pT^2 + k^2 Q^2_fac)</ei>, | |
100 | where <ei>Q_fac</ei> is the factorization scale and <ei>k</ei> is a | |
101 | multiplicative fudge factor stored in <code>pTdampFudge</code> below. | |
102 | </option> | |
103 | <option value="2">emissions go up to the kinematical limit, | |
104 | but dampened by a factor <ei>k^2 Q^2_ren/(pT^2 + k^2 Q^2_ren)</ei>, | |
105 | where <ei>Q_ren</ei> is the renormalization scale and <ei>k</ei> is a | |
106 | multiplicative fudge factor stored in <code>pTdampFudge</code> below. | |
107 | </option> | |
108 | <note>Note:</note> These options only apply to the hard interaction. | |
109 | Emissions off subsequent multiple interactions are always constrainted | |
110 | to be below the factorization scale of the process itself. | |
111 | </modepick> | |
112 | ||
113 | <parm name="SpaceShower:pTdampFudge" default="1.0" min="0.25" max="4.0"> | |
114 | In cases 1 and 2 above, where a dampening is imposed at around the | |
115 | factorization or renormalization scale, respectively, this allows the | |
116 | <ei>pT</ei> scale of dampening of radiation by a half to be shifted | |
117 | by this factor relative to the default <ei>Q_fac</ei> or <ei>Q_ren</ei>. | |
118 | This number ought to be in the neighbourhood of unity, but variations | |
119 | away from this value could do better in some processes. | |
120 | </parm> | |
121 | ||
122 | <p/> | |
123 | The amount of QCD radiation in the shower is determined by | |
124 | <parm name="SpaceShower:alphaSvalue" default="0.137" min="0.06" max="0.25"> | |
125 | The <ei>alpha_strong</ei> value at scale <code>M_Z^2</code>. | |
126 | Default value is picked equal to the one used in CTEQ 5L. | |
127 | </parm> | |
128 | ||
129 | <p/> | |
130 | The actual value is then regulated by the running to the scale | |
131 | <ei>pT^2</ei>, at which it is evaluated | |
132 | <modepick name="SpaceShower:alphaSorder" default="1" min="0" max="2"> | |
133 | Order at which <ei>alpha_strong</ei> runs, | |
134 | <option value="0">zeroth order, i.e. <ei>alpha_strong</ei> is kept | |
135 | fixed.</option> | |
136 | <option value="1">first order, which is the normal value.</option> | |
137 | <option value="2">second order. Since other parts of the code do | |
138 | not go to second order there is no strong reason to use this option, | |
139 | but there is also nothing wrong with it.</option> | |
140 | </modepick> | |
141 | ||
142 | <p/> | |
143 | QED radiation is regulated by the <ei>alpha_electromagnetic</ei> | |
144 | value at the <ei>pT^2</ei> scale of a branching. | |
145 | ||
146 | <modepick name="SpaceShower:alphaEMorder" default="1" min="-1" max="1"> | |
147 | The running of <ei>alpha_em</ei>. | |
148 | <option value="1">first-order running, constrained to agree with | |
149 | <code>StandardModel:alphaEMmZ</code> at the <ei>Z^0</ei> mass. | |
150 | </option> | |
151 | <option value="0">zeroth order, i.e. <ei>alpha_em</ei> is kept | |
152 | fixed at its value at vanishing momentum transfer.</option> | |
153 | <option value="-1">zeroth order, i.e. <ei>alpha_em</ei> is kept | |
154 | fixed, but at <code>StandardModel:alphaEMmZ</code>, i.e. its value | |
155 | at the <ei>Z^0</ei> mass. | |
156 | </option> | |
157 | </modepick> | |
158 | ||
159 | <p/> | |
160 | There are two complementary ways of regularizing the small-<ei>pT</ei> | |
161 | divergence, a sharp cutoff and a smooth dampening. These can be | |
162 | combined as desired but it makes sense to coordinate with how the | |
163 | same issue is handled in multiple interactions. | |
164 | ||
165 | <flag name="SpaceShower:samePTasMI" default="off"> | |
166 | Regularize the <ei>pT -> 0</ei> divergence using the same sharp cutoff | |
167 | and smooth dampening parameters as used to describe multiple interactions. | |
168 | That is, the <code>MultipleInteractions:pT0Ref</code>, | |
169 | <code>MultipleInteractions:ecmRef</code>, | |
170 | <code>MultipleInteractions:ecmPow</code> and | |
171 | <code>MultipleInteractions:pTmin</code> parameters are used to regularize | |
172 | all ISR QCD radiation, rather than the corresponding parameters below. | |
173 | This is a sensible physics ansatz, based on the assumption that colour | |
174 | screening effects influence both MI and ISR in the same way. Photon | |
175 | radiation is regularized separately in either case. | |
176 | <note>Warning:</note> if a large <code>pT0</code> is picked for multiple | |
177 | interactions, such that the integrated interaction cross section is | |
178 | below the nondiffractive inelastic one, this <code>pT0</code> will | |
179 | automatically be scaled down to cope. Information on such a rescaling | |
180 | does NOT propagate to <code>SpaceShower</code>, however. | |
181 | </flag> | |
182 | ||
183 | <p/> | |
184 | The actual <code>pT0</code> parameter used at a given CM energy scale, | |
185 | <ei>ecmNow</ei>, is obtained as | |
186 | <eq> | |
187 | pT0 = pT0(ecmNow) = pT0Ref * (ecmNow / ecmRef)^ecmPow | |
188 | </eq> | |
189 | where <ei>pT0Ref</ei>, <ei>ecmRef</ei> and <ei>ecmPow</ei> are the | |
190 | three parameters below. | |
191 | ||
192 | <parm name="SpaceShower:pT0Ref" default="2.0" | |
193 | min="0.5" max="10.0"> | |
194 | Regularization of the divergence of the QCD emission probability for | |
195 | <ei>pT -> 0</ei> is obtained by a factor <ei>pT^2 / (pT0^2 + pT^2)</ei>, | |
196 | and by using an <ei>alpha_s(pT0^2 + pT^2)</ei>. An energy dependence | |
197 | of the <ei>pT0</ei> choice is introduced by the next two parameters, | |
198 | so that <ei>pT0Ref</ei> is the <ei>pT0</ei> value for the reference | |
199 | cm energy, <ei>pT0Ref = pT0(ecmRef)</ei>. | |
200 | </parm> | |
201 | ||
202 | <parm name="SpaceShower:ecmRef" default="1800.0" min="1."> | |
203 | The <ei>ecmRef</ei> reference energy scale introduced above. | |
204 | </parm> | |
205 | ||
206 | <parm name="SpaceShower:ecmPow" default="0.0" min="0." max="0.5"> | |
207 | The <ei>ecmPow</ei> energy rescaling pace introduced above. | |
208 | </parm> | |
209 | ||
210 | <parm name="SpaceShower:pTmin" default="0.2" | |
211 | min="0.1" max="10.0"> | |
212 | Lower cutoff in <ei>pT</ei>, below which no further ISR branchings | |
213 | are allowed. Normally the <ei>pT0</ei> above would be used to | |
214 | provide the main regularization of the branching rate for | |
215 | <ei>pT -> 0</ei>, in which case <ei>pTmin</ei> is used mainly for | |
216 | technical reasons. It is possible, however, to set <ei>pT0Ref = 0</ei> | |
217 | and use <ei>pTmin</ei> to provide a step-function regularization, | |
218 | or to combine them in intermediate approaches. Currently <ei>pTmin</ei> | |
219 | is taken to be energy-independent. | |
220 | </parm> | |
221 | ||
222 | <parm name="SpaceShower:pTminChgQ" default="0.5" min="0.01"> | |
223 | Parton shower cut-off <ei>pT</ei> for photon coupling to a coloured | |
224 | particle. | |
225 | </parm> | |
226 | ||
227 | <parm name="SpaceShower:pTminChgL" default="0.0005" min="0.0001"> | |
228 | Parton shower cut-off mass for pure QED branchings. | |
229 | Assumed smaller than (or equal to) <ei>pTminChgQ</ei>. | |
230 | </parm> | |
231 | ||
232 | <flag name="SpaceShower:rapidityOrder" default="off"> | |
233 | Force emissions, after the first, to be ordered in rapidity, | |
234 | i.e. in terms of decreasing angles in a backwards-evolution sense. | |
235 | Could be used to probe sensitivity to unordered emissions. | |
236 | Only affects QCD emissions. | |
237 | </flag> | |
238 | ||
239 | <h3>Further variables</h3> | |
240 | ||
241 | These should normally not be touched. Their only function is for | |
242 | cross-checks. | |
243 | ||
244 | <p/> | |
245 | There are three flags you can use to switch on or off selected | |
246 | branchings in the shower: | |
247 | ||
248 | <flag name="SpaceShower:QCDshower" default="on"> | |
249 | Allow a QCD shower; on/off = true/false. | |
250 | </flag> | |
251 | ||
252 | <flag name="SpaceShower:QEDshowerByQ" default="on"> | |
253 | Allow quarks to radiate photons; on/off = true/false. | |
254 | </flag> | |
255 | ||
256 | <flag name="SpaceShower:QEDshowerByL" default="on"> | |
257 | Allow leptons to radiate photons; on/off = true/false. | |
258 | </flag> | |
259 | ||
260 | <p/> | |
261 | There are three further possibilities to simplify the shower: | |
262 | ||
263 | <flag name="SpaceShower:MEcorrections" default="on"> | |
264 | Use of matrix element corrections; on/off = true/false. | |
265 | </flag> | |
266 | ||
267 | <flag name="SpaceShower:phiPolAsym" default="on"> | |
268 | Azimuthal asymmetry induced by gluon polarization; on/off = true/false. | |
269 | </flag> | |
270 | ||
271 | <flag name="SpaceShower:phiIntAsym" default="on"> | |
272 | Azimuthal asymmetry induced by interference; on/off = true/false. | |
273 | </flag> | |
274 | ||
275 | <parm name="SpaceShower:strengthIntAsym" default="0.7" | |
276 | min="0." max="0.9"> | |
277 | Size of asymmetry induced by interference. Natural value of order 0.5; | |
278 | expression would blow up for a value of 1. | |
279 | </flag> | |
280 | ||
281 | <modeopen name="SpaceShower:nQuarkIn" default="5" min="0" max="5"> | |
282 | Number of allowed quark flavours in <ei>g -> q qbar</ei> branchings, | |
283 | when kinematically allowed, and thereby also in incoming beams. | |
284 | Changing it to 4 would forbid <ei>g -> b bbar</ei>, etc. | |
285 | </modeopen> | |
286 | ||
287 | <h3>Technical notes</h3> | |
288 | ||
289 | Almost everything is equivalent to the algorithm in [1]. Minor changes | |
290 | are as follows. | |
291 | <ul> | |
292 | <li> | |
293 | It is now possible to have a second-order running <ei>alpha_s</ei>, | |
294 | in addition to fixed or first-order running. | |
295 | </li> | |
296 | <li> | |
297 | The description of heavy flavour production in the threshold region | |
298 | has been modified, so as to be more forgiving about mismatches | |
299 | between the <ei>c/b</ei> masses used in Pythia relative to those | |
300 | used in a respective PDF parametrization. The basic idea is that, | |
301 | in the threshold region of a heavy quark <ei>Q</ei>, <ei>Q = c/b</ei>, | |
302 | the effect of subsequent <ei>Q -> Q g</ei> branchings is negligible. | |
303 | If so, then | |
304 | <eq> | |
305 | f_Q(x, pT2) = integral_mQ2^pT2 dpT'2/pT'2 * alpha_s(pT'2)/2pi | |
306 | * integral P(z) g(x', pT'2) delta(x - z x') | |
307 | </eq> | |
308 | so use this to select the <ei>pT2</ei> of the <ei>g -> Q Qbar</ei> | |
309 | branching. In the old formalism the same kind of behaviour should | |
310 | be obtained, but by a cancellation of a <ei>1/f_Q</ei> that diverges | |
311 | at the theshold and a Sudakov that vanishes. | |
312 | <br/> | |
313 | The strategy therefore is that, once <ei>pT2 < f * mQ2</ei>, with | |
314 | <ei>f</ei> a parameter of the order of 2, a <ei>pT2</ei> is chosen | |
315 | like <ei>dpT2/pT2</ei> between <ei>mQ2</ei> and <ei>f * mQ2</ei>, a | |
316 | nd a <ei>z</ei> flat in the allowed range. Thereafter acceptance | |
317 | is based on the product of three factors, representing the running | |
318 | of <ei>alpha_strong</ei>, the splitting kernel (including the mass term) | |
319 | and the gluon density weight. At failure, a new <ei>pT2</ei> is chosen | |
320 | in the same range, i.e. is not required to be lower since no Sudakov | |
321 | is involved. | |
322 | </li> | |
323 | <li> | |
324 | The QED algorithm now allows for hadron beams with non-zero photon | |
325 | content. The backwards-evolution of a photon in a hadron is identical | |
326 | to that of a gluon, with <ei>CF -> eq^2</ei> and <ei>CA -> 0</ei>. | |
327 | Note that this will only work in conjunction with | |
328 | parton distribution that explicitly include photons as part of the | |
329 | hadron structure (such as the MRST2004qed set). Since Pythia's | |
330 | internal sets do not allow for photon content in hadrons, it is thus | |
331 | necessary to use the LHAPDF interface to make use of this feature. The | |
332 | possibility of a fermion backwards-evolving to a photon has not yet | |
333 | been included, nor has photon backwards-evolution in lepton beams. | |
334 | </li> | |
335 | </ul> | |
336 | ||
337 | </chapter> | |
338 | ||
339 | <!-- Copyright (C) 2010 Torbjorn Sjostrand --> | |
340 |