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.
51 <option value="2">always allow emissions up to the kinematical limit.
52 This will simulate all possible event topologies, but may lead to
55 <note>Note 1:</note> These options only apply to the hard interaction.
56 Emissions off subsequent multiple interactions are always constrainted
57 to be below the factorization scale of the process itself.
58 <note>Note 2:</note> Some processes contain matrix-element matching
59 to the first emission; this is the case notably for single
60 <ei>gamma^*/Z^0, W^+-</ei> and <ei>H^0</ei> production. Then default
61 and option 2 give the correct result, while option 1 should never
65 <parm name="SpaceShower:pTmaxFudge" default="1.0" min="0.5" max="2.0">
66 In cases where the above <code>pTmaxMatch</code> rules would imply
67 that <ei>pT_max = pT_factorization</ei>, <code>pTmaxFudge</code>
68 introduced a multiplicative factor <ei>f</ei> such that instead
69 <ei>pT_max = f * pT_factorization</ei>. Only applies to the hardest
70 interaction in an event. It is strongly suggested that <ei>f = 1</ei>,
71 but variations around this default can be useful to test this assumption.
74 <modepick name="SpaceShower:pTdampMatch" default="0" min="0" max="2">
75 These options only take effect when a process is allowed to radiate up
76 to the kinematical limit by the above <code>pTmaxMatch</code> choice,
77 and no matrix-element corrections are available. Then, in many processes,
78 the fall-off in <ei>pT</ei> will be too slow by one factor of <ei>pT^2</ei>.
79 That is, while showers have an approximate <ei>dpT^2/pT^2</ei> shape, often
80 it should become more like <ei>dpT^2/pT^4</ei> at <ei>pT</ei> values above
81 the scale of the hard process. Whether this actually is the case
82 depends on the particular process studied, e.g. if <ei>t</ei>-channel
83 gluon exchange is likely to dominate. If so, the options below could
84 provide a reasonable high-<ei>pT</ei> behaviour without requiring
85 higher-order calculations.
86 <option value="0">emissions go up to the kinematical limit,
87 with no special dampening.
89 <option value="1">emissions go up to the kinematical limit,
90 but dampened by a factor <ei>k^2 Q^2_fac/(pT^2 + k^2 Q^2_fac)</ei>,
91 where <ei>Q_fac</ei> is the factorization scale and <ei>k</ei> is a
92 multiplicative fudge factor stored in <code>pTdampFudge</code> below.
94 <option value="2">emissions go up to the kinematical limit,
95 but dampened by a factor <ei>k^2 Q^2_ren/(pT^2 + k^2 Q^2_ren)</ei>,
96 where <ei>Q_ren</ei> is the renormalization scale and <ei>k</ei> is a
97 multiplicative fudge factor stored in <code>pTdampFudge</code> below.
99 <note>Note:</note> These options only apply to the hard interaction.
100 Emissions off subsequent multiple interactions are always constrainted
101 to be below the factorization scale of the process itself.
104 <parm name="SpaceShower:pTdampFudge" default="1.0" min="0.25" max="4.0">
105 In cases 1 and 2 above, where a dampening is imposed at around the
106 factorization or renormalization scale, respectively, this allows the
107 <ei>pT</ei> scale of dampening of radiation by a half to be shifted
108 by this factor relative to the default <ei>Q_fac</ei> or <ei>Q_ren</ei>.
109 This number ought to be in the neighbourhood of unity, but variations
110 away from this value could do better in some processes.
114 The amount of QCD radiation in the shower is determined by
115 <parm name="SpaceShower:alphaSvalue" default="0.127" min="0.06" max="0.25">
116 The <ei>alpha_strong</ei> value at scale <code>M_Z^2</code>.
117 Default value is picked equal to the one used in CTEQ 5L.
121 The actual value is then regulated by the running to the scale
122 <ei>pT^2</ei>, at which it is evaluated
123 <modepick name="SpaceShower:alphaSorder" default="1" min="0" max="2">
124 Order at which <ei>alpha_strong</ei> runs,
125 <option value="0">zeroth order, i.e. <ei>alpha_strong</ei> is kept
127 <option value="1">first order, which is the normal value.</option>
128 <option value="2">second order. Since other parts of the code do
129 not go to second order there is no strong reason to use this option,
130 but there is also nothing wrong with it.</option>
134 QED radiation is regulated by the <ei>alpha_electromagnetic</ei>
135 value at the <ei>pT^2</ei> scale of a branching.
137 <modepick name="SpaceShower:alphaEMorder" default="1" min="-1" max="1">
138 The running of <ei>alpha_em</ei>.
139 <option value="1">first-order running, constrained to agree with
140 <code>StandardModel:alphaEMmZ</code> at the <ei>Z^0</ei> mass.
142 <option value="0">zeroth order, i.e. <ei>alpha_em</ei> is kept
143 fixed at its value at vanishing momentum transfer.</option>
144 <option value="-1">zeroth order, i.e. <ei>alpha_em</ei> is kept
145 fixed, but at <code>StandardModel:alphaEMmZ</code>, i.e. its value
146 at the <ei>Z^0</ei> mass.
151 There are two complementary ways of regularizing the small-<ei>pT</ei>
152 divergence, a sharp cutoff and a smooth dampening. These can be
153 combined as desired but it makes sense to coordinate with how the
154 same issue is handled in multiple interactions.
156 <flag name="SpaceShower:samePTasMI" default="on">
157 Regularize the <ei>pT -> 0</ei> divergence using the same sharp cutoff
158 and smooth dampening parameters as used to describe multiple interactions.
159 That is, the <code>MultipleInteractions:pT0Ref</code>,
160 <code>MultipleInteractions:ecmRef</code>,
161 <code>MultipleInteractions:ecmPow</code> and
162 <code>MultipleInteractions:pTmin</code> parameters are used to regularize
163 all ISR QCD radiation, rather than the corresponding parameters below.
164 This is a sensible physics ansatz, based on the assumption that colour
165 screening effects influence both MI and ISR in the same way. Photon
166 radiation is regularized separately in either case.
167 <note>Warning:</note> if a large <code>pT0</code> is picked for multiple
168 interactions, such that the integrated interaction cross section is
169 below the nondiffractive inelastic one, this <code>pT0</code> will
170 automatically be scaled down to cope. Information on such a rescaling
171 does NOT propagate to <code>SpaceShower</code>, however.
175 The actual <code>pT0</code> parameter used at a given cm energy scale,
176 <ei>ecmNow</ei>, is obtained as
178 pT0 = pT0(ecmNow) = pT0Ref * (ecmNow / ecmRef)^ecmPow
180 where <ei>pT0Ref</ei>, <ei>ecmRef</ei> and <ei>ecmPow</ei> are the
181 three parameters below.
183 <parm name="SpaceShower:pT0Ref" default="2.2"
184 min="0.5" max="10.0">
185 Regularization of the divergence of the QCD emission probability for
186 <ei>pT -> 0</ei> is obtained by a factor <ei>pT^2 / (pT0^2 + pT^2)</ei>,
187 and by using an <ei>alpha_s(pT0^2 + pT^2)</ei>. An energy dependence
188 of the <ei>pT0</ei> choice is introduced by the next two parameters,
189 so that <ei>pT0Ref</ei> is the <ei>pT0</ei> value for the reference
190 cm energy, <ei>pT0Ref = pT0(ecmRef)</ei>.
193 <parm name="SpaceShower:ecmRef" default="1800.0" min="1.">
194 The <ei>ecmRef</ei> reference energy scale introduced above.
197 <parm name="SpaceShower:ecmPow" default="0.16" min="0." max="0.5">
198 The <ei>ecmPow</ei> energy rescaling pace introduced above.
201 <parm name="SpaceShower:pTmin" default="0.2"
202 min="0.1" max="10.0">
203 Lower cutoff in <ei>pT</ei>, below which no further ISR branchings
204 are allowed. Normally the <ei>pT0</ei> above would be used to
205 provide the main regularization of the branching rate for
206 <ei>pT -> 0</ei>, in which case <ei>pTmin</ei> is used mainly for
207 technical reasons. It is possible, however, to set <ei>pT0Ref = 0</ei>
208 and use <ei>pTmin</ei> to provide a step-function regularization,
209 or to combine them in intermediate approaches. Currently <ei>pTmin</ei>
210 is taken to be energy-independent.
213 <parm name="SpaceShower:pTminChgQ" default="0.5" min="0.01">
214 Parton shower cut-off <ei>pT</ei> for photon coupling to a coloured
218 <parm name="SpaceShower:pTminChgL" default="0.0005" min="0.0001">
219 Parton shower cut-off mass for pure QED branchings.
220 Assumed smaller than (or equal to) <ei>pTminChgQ</ei>.
223 <flag name="SpaceShower:rapidityOrder" default="off">
224 Force emissions, after the first, to be ordered in rapidity,
225 i.e. in terms of decreasing angles in a backwards-evolution sense.
226 Could be used to probe sensitivity to unordered emissions.
227 Only affects QCD emissions.
230 <h3>Further variables</h3>
232 These should normally not be touched. Their only function is for
236 There are three flags you can use to switch on or off selected
237 branchings in the shower:
239 <flag name="SpaceShower:QCDshower" default="on">
240 Allow a QCD shower; on/off = true/false.
243 <flag name="SpaceShower:QEDshowerByQ" default="on">
244 Allow quarks to radiate photons; on/off = true/false.
247 <flag name="SpaceShower:QEDshowerByL" default="on">
248 Allow leptons to radiate photons; on/off = true/false.
252 There are three further possibilities to simplify the shower:
254 <flag name="SpaceShower:MEcorrections" default="on">
255 Use of matrix element corrections; on/off = true/false.
258 <flag name="SpaceShower:phiPolAsym" default="on">
259 Azimuthal asymmetry induced by gluon polarization; on/off = true/false.
263 <modeopen name="SpaceShower:nQuarkIn" default="5" min="0" max="5">
264 Number of allowed quark flavours in <ei>g -> q qbar</ei> branchings,
265 when kinematically allowed, and thereby also in incoming beams.
266 Changing it to 4 would forbid <ei>g -> b bbar</ei>, etc.
269 <h3>Technical notes</h3>
271 Almost everything is equivalent to the algorithm in [1]. Minor changes
275 It is now possible to have a second-order running <ei>alpha_s</ei>,
276 in addition to fixed or first-order running.
279 The description of heavy flavour production in the threshold region
280 has been modified, so as to be more forgiving about mismatches
281 between the <ei>c/b</ei> masses used in Pythia relative to those
282 used in a respective PDF parametrization. The basic idea is that,
283 in the threshold region of a heavy quark <ei>Q</ei>, <ei>Q = c/b</ei>,
284 the effect of subsequent <ei>Q -> Q g</ei> branchings is negligible.
287 f_Q(x, pT2) = integral_mQ2^pT2 dpT'2/pT'2 * alpha_s(pT'2)/2pi
288 * integral P(z) g(x', pT'2) delta(x - z x')
290 so use this to select the <ei>pT2</ei> of the <ei>g -> Q Qbar</ei>
291 branching. In the old formalism the same kind of behaviour should
292 be obtained, but by a cancellation of a <ei>1/f_Q</ei> that diverges
293 at the theshold and a Sudakov that vanishes.
295 The strategy therefore is that, once <ei>pT2 < f * mQ2</ei>, with
296 <ei>f</ei> a parameter of the order of 2, a <ei>pT2</ei> is chosen
297 like <ei>dpT2/pT2</ei> between <ei>mQ2</ei> and <ei>f * mQ2</ei>, a
298 nd a <ei>z</ei> flat in the allowed range. Thereafter acceptance
299 is based on the product of three factors, representing the running
300 of <ei>alpha_strong</ei>, the splitting kernel (including the mass term)
301 and the gluon density weight. At failure, a new <ei>pT2</ei> is chosen
302 in the same range, i.e. is not required to be lower since no Sudakov
309 <!-- Copyright (C) 2008 Torbjorn Sjostrand -->