1 <chapter name="Spacelike Showers">
3 <h2>Spacelike Showers</h2>
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
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.
21 <h3>Main variables</h3>
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
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.
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.)
52 <option value="2">always allow emissions up to the kinematical limit.
53 This will simulate all possible event topologies, but may lead to
55 (Also known as power showers.)
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
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
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).
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.
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.
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.
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.
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.
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.
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
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>
143 QED radiation is regulated by the <ei>alpha_electromagnetic</ei>
144 value at the <ei>pT^2</ei> scale of a branching.
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.
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.
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.
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.
184 The actual <code>pT0</code> parameter used at a given CM energy scale,
185 <ei>ecmNow</ei>, is obtained as
187 pT0 = pT0(ecmNow) = pT0Ref * (ecmNow / ecmRef)^ecmPow
189 where <ei>pT0Ref</ei>, <ei>ecmRef</ei> and <ei>ecmPow</ei> are the
190 three parameters below.
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>.
202 <parm name="SpaceShower:ecmRef" default="1800.0" min="1.">
203 The <ei>ecmRef</ei> reference energy scale introduced above.
206 <parm name="SpaceShower:ecmPow" default="0.0" min="0." max="0.5">
207 The <ei>ecmPow</ei> energy rescaling pace introduced above.
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.
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
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>.
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.
239 <h3>Further variables</h3>
241 These should normally not be touched. Their only function is for
245 There are three flags you can use to switch on or off selected
246 branchings in the shower:
248 <flag name="SpaceShower:QCDshower" default="on">
249 Allow a QCD shower; on/off = true/false.
252 <flag name="SpaceShower:QEDshowerByQ" default="on">
253 Allow quarks to radiate photons; on/off = true/false.
256 <flag name="SpaceShower:QEDshowerByL" default="on">
257 Allow leptons to radiate photons; on/off = true/false.
261 There are three further possibilities to simplify the shower:
263 <flag name="SpaceShower:MEcorrections" default="on">
264 Use of matrix element corrections; on/off = true/false.
267 <flag name="SpaceShower:phiPolAsym" default="on">
268 Azimuthal asymmetry induced by gluon polarization; on/off = true/false.
271 <flag name="SpaceShower:phiIntAsym" default="on">
272 Azimuthal asymmetry induced by interference; on/off = true/false.
275 <parm name="SpaceShower:strengthIntAsym" default="0.7"
277 Size of asymmetry induced by interference. Natural value of order 0.5;
278 expression would blow up for a value of 1.
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.
287 <h3>Technical notes</h3>
289 Almost everything is equivalent to the algorithm in [1]. Minor changes
293 It is now possible to have a second-order running <ei>alpha_s</ei>,
294 in addition to fixed or first-order running.
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.
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')
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.
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
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.
339 <!-- Copyright (C) 2010 Torbjorn Sjostrand -->