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2<head>
3<title>A Second Hard Process</title>
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29
30<h2>A Second Hard Process</h2>
31
32When you have selected a set of hard processes for hadron beams, the
33<?php $filepath = $_GET["filepath"];
34echo "<a href='MultipartonInteractions.php?filepath=".$filepath."' target='page'>";?>multiparton interactions</a>
35framework can add further interactions to build up a realistic
36underlying event. These further interactions can come from a wide
37variety of processes, and will occasionally be quite hard. They
38do represent a realistic random mix, however, which means one cannot
39predetermine what will happen. Occasionally there may be cases
40where one wants to specify also the second hard interaction rather
41precisely. The options on this page allow you to do precisely that.
42
43<br/><br/><strong>SecondHard:generate</strong>  <input type="radio" name="1" value="on"><strong>On</strong>
44<input type="radio" name="1" value="off" checked="checked"><strong>Off</strong>
45 &nbsp;&nbsp;(<code>default = <strong>off</strong></code>)<br/>
46Generate two hard scatterings in a collision between hadron beams.
47The hardest process can be any combination of internal processes,
48available in the normal <?php $filepath = $_GET["filepath"];
49echo "<a href='ProcessSelection.php?filepath=".$filepath."' target='page'>";?>process
50selection</a> machinery, or external input. Here you must further
51specify which set of processes to allow for the second hard one, see
52the following.
53 
54
55<h3>Process Selection</h3>
56
57In principle the whole <?php $filepath = $_GET["filepath"];
58echo "<a href='ProcessSelection.php?filepath=".$filepath."' target='page'>";?>process
59selection</a> allowed for the first process could be repeated
60for the second one. However, this would probably be overkill.
61Therefore here a more limited set of prepackaged process collections
62are made available, that can then be further combined at will.
63Since the description is almost completely symmetric between the
64first and the second process, you always have the possibility
65to pick one of the two processes according to the complete list
66of possibilities.
67
68<p/>
69Here comes the list of allowed sets of processes, to combine at will:
70
71<br/><br/><strong>SecondHard:TwoJets</strong>  <input type="radio" name="2" value="on"><strong>On</strong>
72<input type="radio" name="2" value="off" checked="checked"><strong>Off</strong>
73 &nbsp;&nbsp;(<code>default = <strong>off</strong></code>)<br/>
74Standard QCD <i>2 -> 2</i> processes involving gluons and
75<i>d, u, s, c, b</i> quarks.
76 
77
78<br/><br/><strong>SecondHard:PhotonAndJet</strong>  <input type="radio" name="3" value="on"><strong>On</strong>
79<input type="radio" name="3" value="off" checked="checked"><strong>Off</strong>
80 &nbsp;&nbsp;(<code>default = <strong>off</strong></code>)<br/>
81A prompt photon recoiling against a quark or gluon jet.
82
83<br/><br/><strong>SecondHard:TwoPhotons</strong>  <input type="radio" name="4" value="on"><strong>On</strong>
84<input type="radio" name="4" value="off" checked="checked"><strong>Off</strong>
85 &nbsp;&nbsp;(<code>default = <strong>off</strong></code>)<br/>
86Two prompt photons recoiling against each other.
87
88<br/><br/><strong>SecondHard:Charmonium</strong>  <input type="radio" name="5" value="on"><strong>On</strong>
89<input type="radio" name="5" value="off" checked="checked"><strong>Off</strong>
90 &nbsp;&nbsp;(<code>default = <strong>off</strong></code>)<br/>
91Production of charmonium via colour singlet and colour octet channels.
92
93<br/><br/><strong>SecondHard:Bottomonium</strong>  <input type="radio" name="6" value="on"><strong>On</strong>
94<input type="radio" name="6" value="off" checked="checked"><strong>Off</strong>
95 &nbsp;&nbsp;(<code>default = <strong>off</strong></code>)<br/>
96Production of bottomonium via colour singlet and colour octet channels.
97
98<br/><br/><strong>SecondHard:SingleGmZ</strong>  <input type="radio" name="7" value="on"><strong>On</strong>
99<input type="radio" name="7" value="off" checked="checked"><strong>Off</strong>
100 &nbsp;&nbsp;(<code>default = <strong>off</strong></code>)<br/>
101Scattering <i>q qbar -> gamma^*/Z^0</i>, with full interference
102between the <i>gamma^*</i> and <i>Z^0</i>.
103 
104
105<br/><br/><strong>SecondHard:SingleW</strong>  <input type="radio" name="8" value="on"><strong>On</strong>
106<input type="radio" name="8" value="off" checked="checked"><strong>Off</strong>
107 &nbsp;&nbsp;(<code>default = <strong>off</strong></code>)<br/>
108Scattering <i>q qbar' -> W^+-</i>.
109 
110
111<br/><br/><strong>SecondHard:GmZAndJet</strong>  <input type="radio" name="9" value="on"><strong>On</strong>
112<input type="radio" name="9" value="off" checked="checked"><strong>Off</strong>
113 &nbsp;&nbsp;(<code>default = <strong>off</strong></code>)<br/>
114Scattering <i>q qbar -> gamma^*/Z^0 g</i> and
115<i>q g -> gamma^*/Z^0 q</i>.
116 
117
118<br/><br/><strong>SecondHard:WAndJet</strong>  <input type="radio" name="10" value="on"><strong>On</strong>
119<input type="radio" name="10" value="off" checked="checked"><strong>Off</strong>
120 &nbsp;&nbsp;(<code>default = <strong>off</strong></code>)<br/>
121Scattering <i>q qbar' -> W^+- g</i> and
122<i>q g -> W^+- q'</i>.
123 
124
125<br/><br/><strong>SecondHard:TopPair</strong>  <input type="radio" name="11" value="on"><strong>On</strong>
126<input type="radio" name="11" value="off" checked="checked"><strong>Off</strong>
127 &nbsp;&nbsp;(<code>default = <strong>off</strong></code>)<br/>
128Production of a top pair, either via QCD processes or via an
129intermediate <i>gamma^*/Z^0</i> resonance.
130 
131
132<br/><br/><strong>SecondHard:SingleTop</strong>  <input type="radio" name="12" value="on"><strong>On</strong>
133<input type="radio" name="12" value="off" checked="checked"><strong>Off</strong>
134 &nbsp;&nbsp;(<code>default = <strong>off</strong></code>)<br/>
135Production of a single top, either via a <i>t-</i> or
136an <i>s-</i>channel <i>W^+-</i> resonance.
137 
138
139<p/>
140A further process collection comes with a warning flag:
141
142<br/><br/><strong>SecondHard:TwoBJets</strong>  <input type="radio" name="13" value="on"><strong>On</strong>
143<input type="radio" name="13" value="off" checked="checked"><strong>Off</strong>
144 &nbsp;&nbsp;(<code>default = <strong>off</strong></code>)<br/>
145The <i>q qbar -> b bbar</i> and <i>g g -> b bbar</i> processes.
146These are already included in the <code>TwoJets</code> sample above,
147so it would be doublecounting to include both, but we assume there
148may be cases where the <i>b</i> subsample will be of special interest.
149This subsample does not include flavour-excitation or gluon-splitting
150contributions to the <i>b</i> rate, however, so, depending
151on the topology if interest, it may or may not be a good approximation.   
152 
153
154<h3>Cuts and scales</h3>
155
156The second hard process obeys exactly the same selection rules for
157<?php $filepath = $_GET["filepath"];
158echo "<a href='PhaseSpaceCuts.php?filepath=".$filepath."' target='page'>";?>phase space cuts</a> and
159<?php $filepath = $_GET["filepath"];
160echo "<a href='CouplingsAndScales.php?filepath=".$filepath."' target='page'>";?>couplings and scales</a>
161as the first one does. Specifically, a <i>pTmin</i> cut for
162<i>2 -> 2</i> processes would apply to the first and the second hard
163process alike, and ballpark half of the time the second could be
164generated with a larger <i>pT</i> than the first. (Exact numbers
165depending on the relative shape of the two cross sections.) That is,
166first and second is only used as an administrative distinction between
167the two, not as a physics ordering one.
168
169<p/>
170Optionally it is possible to pick the mass and <i>pT</i>
171<?php $filepath = $_GET["filepath"];
172echo "<a href='PhaseSpaceCuts.php?filepath=".$filepath."' target='page'>";?>phase space cuts</a> separately for
173the second hard interaction. The main application presumably would
174be to allow a second process that is softer than the first, but still
175hard. But one is also free to make the second process harder than the
176first, if desired. So long as the two <i>pT</i> (or mass) ranges
177overlap the ordering will not be the same in all events, however.
178
179<h3>Cross-section calculation</h3>
180
181As an introduction, a brief reminder of Poissonian statistics.
182Assume a stochastic process in time, for now not necessarily a
183high-energy physics one, where the probability for an event to occur
184at any given time is independent of what happens at other times.
185Then the probability for <i>n</i> events to occur in a finite
186time interval is
187<br/><i>
188P_n = &lt;n&gt;^n exp(-&lt;n&gt;) / n!
189</i><br/>
190where <i>&lt;n&gt;</i> is the average number of events. If this
191number is small we can approximate <i>exp(-&lt;n&gt;) = 1 </i>,
192so that <i>P_1 = &lt;n&gt;</i> and
193<i>P_2 = &lt;n&gt;^2 / 2 = P_1^2 / 2</i>.
194
195<p/>
196Now further assume that the events actually are of two different
197kinds <i>a</i> and <i>b</i>, occuring independently of each
198other, such that <i>&lt;n&gt; = &lt;n_a&gt; + &lt;n_b&gt;</i>.
199It then follows that the probability of having one event of type
200<i>a</i> (or <i>b</i>) and nothing else is
201<i>P_1a = &lt;n_a&gt;</i> (or <i>P_1b = &lt;n_b&gt;</i>).
202From
203<br/><i>
204P_2 = (&lt;n_a&gt; + &lt;n_b&gt)^2 / 2 = (P_1a + P_1b)^2 / 2 =
205(P_1a^2 + 2 P_1a P_1b + P_1b^2) / 2
206</i><br/>
207it is easy to read off that the probability to have exactly two
208events of kind <i>a</i> and none of <i>b</i> is
209<i>P_2aa = P_1a^2 / 2</i> whereas that of having one <i>a</i>
210and one <i>b</i> is <i>P_2ab = P_1a P_1b</i>. Note that the
211former, with two identical events, contains a factor <i>1/2</i>
212while the latter, with two different ones, does not. If viewed
213in a time-ordered sense, the difference is that the latter can be
214obtained two ways, either first an <i>a</i> and then a <i>b</i>
215or else first a <i>b</i> and then an <i>a</i>.
216
217<p/>
218To translate this language into cross-sections for high-energy
219events, we assume that interactions can occur at different <i>pT</i>
220values independently of each other inside inelastic nondiffractive
221(= "minbias") events. Then the above probabilities translate into
222<i>P_n = sigma_n / sigma_ND</i> where <i>sigma_ND</i> is the
223total nondiffractive cross section. Again we want to assume that
224<i>exp(-&lt;n&gt;)</i> is close to unity, i.e. that the total
225hard cross section above <i>pTmin</i> is much smaller than
226<i>sigma_ND</i>. The hard cross section is dominated by QCD
227jet production, and a reasonable precaution is to require a
228<i>pTmin</i> of at least 20 GeV at LHC energies.
229(For <i>2 -> 1</i> processes such as
230<i>q qbar -> gamma^*/Z^0 (-> f fbar)</i> one can instead make a
231similar cut on mass.) Then the generic equation
232<i>P_2 = P_1^2 / 2</i> translates into
233<i>sigma_2/sigma_ND = (sigma_1 / sigma_ND)^2 / 2</i> or
234<i>sigma_2 = sigma_1^2 / (2 sigma_ND)</i>.
235
236<p/>
237Again different processes <i>a, b, c, ...</i> contribute,
238and by the same reasoning we obtain
239<i>sigma_2aa = sigma_1a^2 / (2 sigma_ND)</i>,
240<i>sigma_2ab = sigma_1a sigma_1b / sigma_ND</i>,
241and so on.
242
243<p/>
244There is one important correction to this picture: all collisions
245do no occur under equal conditions. Some are more central in impact
246parameter, others more peripheral. This leads to a further element of
247variability: central collisions are likely to have more activity
248than the average, peripheral less. Integrated over impact
249parameter standard cross sections are recovered, but correlations
250are affected by a "trigger bias" effect: if you select for events
251with a hard process you favour events at small impact parameter
252which have above-average activity, and therefore also increased
253chance for further interactions. (In PYTHIA this is the origin
254of the "pedestal effect", i.e. that events with a hard interaction
255have more underlying activity than the level found in minimum-bias
256events.) When you specify a matter overlap profile in the
257multiparton-interactions scenario, such an enhancement/depletion factor
258<i>f_impact</i> is chosen event-by-event and can be averaged
259during the course of the run. As an example, the double Gaussian
260form used in Tune A gives approximately
261<i>&lt;f_impact&gt; = 2.5</i>. The above equations therefore
262have to be modified to
263<i>sigma_2aa = &lt;f_impact&gt; sigma_1a^2 / (2 sigma_ND)</i>,
264<i>sigma_2ab = &lt;f_impact&gt; sigma_1a sigma_1b / sigma_ND</i>.
265Experimentalists often instead use the notation
266<i>sigma_2ab = sigma_1a sigma_1b / sigma_eff</i>,
267from which we see that PYTHIA "predicts"
268<i>sigma_eff = sigma_ND / &lt;f_impact&gt;</i>.
269When the generation of multiparton interactions is switched off it is
270not possible to calculate <i>&lt;f_impact&gt;</i> and therefore
271it is set to unity.
272
273<p/>
274When this recipe is to be applied to calculate
275actual cross sections, it is useful to distinguish three cases,
276depending on which set of processes are selected to study for
277the first and second interaction.
278
279<p/>
280(1) The processes <i>a</i> for the first interaction and
281<i>b</i> for the second one have no overlap at all.
282For instance, the first could be <code>TwoJets</code> and the
283second <code>TwoPhotons</code>. In that case, the two interactions
284can be selected independently, and cross sections tabulated
285for each separate subprocess in the two above classes. At the
286end of the run, the cross sections in <i>a</i> should be multiplied
287by <i>&lt;f_impact&gt; sigma_1b / sigma_ND</i> to bring them to
288the correct overall level, and those in <i>b</i> by
289<i>&lt;f_impact&gt; sigma_1a / sigma_ND</i>.
290 
291<p/>
292(2) Exactly the same processes <i>a</i> are selected for the
293first and second interaction. In that case it works as above,
294with <i>a = b</i>, and it is only necessary to multiply by an
295additional factor <i>1/2</i>. A compensating factor of 2
296is automatically obtained for picking two different subprocesses,
297e.g. if <code>TwoJets</code> is selected for both interactions,
298then the combination of the two subprocesses <i>q qbar -> g g</i>
299and <i>g g -> g g</i> can trivially be obtained two ways.
300 
301<p/>
302(3) The list of subprocesses partly but not completely overlap.
303For instance, the first process is allowed to contain <i>a</i>
304or <i>c</i> and the second <i>b</i> or <i>c</i>, where
305there is no overlap between <i>a</i> and <i>b</i>. Then,
306when an independent selection for the first and second interaction
307both pick one of the subprocesses in <i>c</i>, half of those
308events have to be thrown, and the stored cross section reduced
309accordingly. Considering the four possible combinations of first
310and second process, this gives a
311<br/><i>
312sigma'_1 = sigma_1a + sigma_1c * (sigma_2b + sigma_2c/2) /
313(sigma_2b + sigma_2c)
314</i><br/>
315with the factor <i>1/2</i> for the <i>sigma_1c sigma_2c</i> term.
316At the end of the day, this <i>sigma'_1</i> should be multiplied
317by the normalization factor
318<br/><i>
319f_1norm = &lt;f_impact&gt; (sigma_2b + sigma_2c) / sigma_ND
320</i><br/>
321here without a factor <i>1/2</i> (or else it would have been
322doublecounted). This gives the correct
323<br/><i>
324(sigma_2b + sigma_2c) * sigma'_1 = sigma_1a * sigma_2b
325+ sigma_1a * sigma_2c + sigma_1c * sigma_2b + sigma_1c * sigma_2c/2
326</i><br/>
327The second interaction can be handled in exact analogy.
328
329<p/>
330For the considerations above it is assumed that the phase space cuts
331are the same for the two processes. It is possible to set the mass and
332transverse momentum cuts differently, however. This changes nothing
333for processes that already are different. For two collisions of the
334same type it is partly a matter of interpretation what is intended.
335If we consider the case of the same process in two non-overlapping
336phase space regions, most likely we want to consider them as
337separate processes, in the sense that we expect a factor 2 relative
338to Poissonian statistics from either of the two hardest processes
339populating either of the two phase space regions. In total we are
340therefore lead to adopt the same strategy as in case (3) above:
341only in the overlapping part of the two allowed phase space regions
342could two processes be identical and thus appear with a 1/2 factor,
343elsewhere the two processes are never identical and do not
344include the 1/2 factor. We reiterate, however, that the case of
345partly but not completely overlapping phase space regions for one and
346the same process is tricky, and not to be used without prior
347deliberation. 
348
349<p/>
350The listing obtained with the <code>pythia.statistics()</code>
351already contain these corrections factors, i.e. cross sections
352are for the occurence of two interactions of the specified kinds.
353There is not a full tabulation of the matrix of all the possible   
354combinations of a specific first process together with a specific
355second one (but the information is there for the user to do that,
356if desired). Instead <code>pythia.statistics()</code> shows this
357matrix projected onto the set of processes and associated cross
358sections for the first and the second interaction, respectively.
359Up to statistical fluctuations, these two sections of the
360<code>pythia.statistics()</code> listing both add up to the same
361total cross section for the event sample.
362
363<p/>
364There is a further special feature to be noted for this listing,
365and that is the difference between the number of "selected" events
366and the number of "accepted" ones. Here is how that comes about.
367Originally the first and second process are selected completely
368independently. The generation (in)efficiency is reflected in the
369different number of intially tried events for the first and second
370process, leading to the same number of selected events. While
371acceptable on their own, the combination of the two processes may
372be unacceptable, however. It may be that the two processes added
373together use more energy-momentum than kinematically allowed, or,
374even if not, are disfavoured when the PYTHIA approach to provide
375correlated parton densities is applied. Alternatively, referring
376to case (3) above, it may be because half of the events should
377be thrown for identical processes. Taken together, it is these
378effects that reduced the event number from "selected" to "accepted".
379(A further reduction may occur if a
380<?php $filepath = $_GET["filepath"];
381echo "<a href='UserHooks.php?filepath=".$filepath."' target='page'>";?>user hook</a> rejects some events.)
382
383<p/>
384It is allowed to use external Les Houches Accord input for the
385hardest process, and then pick an internal one for the second hardest.
386In this case PYTHIA does not have access to your thinking concerning
387the external process, and cannot know whether it overlaps with the
388internal or not. (External events <i>q qbar' -> e+ nu_e</i> could
389agree with the internal <i>W</i> ones, or be a <i>W'</i> resonance
390in a BSM scenario, to give one example.) Therefore the combined cross
391section is always based on the scenario (1) above. Corrections for
392correlated parton densities are included also in this case, however.
393That is, an external event that takes a large fraction of the incoming
394beam momenta stands a fair chance of being rejected when it has to be
395combined with another hard process. For this reason the "selected" and 
396"accepted" event numbers are likely to disagree.
397
398<p/>
399In the cross section calculation above, the <i>sigma'_1</i>
400cross sections are based on the number of accepted events, while
401the <i>f_1norm</i> factor is evaluated based on the cross sections
402for selected events. That way the suppression by correlations
403between the two processes does not get to be doublecounted.
404
405<p/>
406The <code>pythia.statistics()</code> listing contains two final
407lines, indicating the summed cross sections <i>sigma_1sum</i> and
408<i>sigma_2sum</i> for the first and second set of processes, at
409the "selected" stage above, plus information on the <i>sigma_ND</i>
410and <i>&lt;f_impact&gt;</i> used. The total cross section
411generated is related to this by
412<br/><i>
413&lt;f_impact&gt; * (sigma_1sum * sigma_2sum / sigma_ND) *
414(n_accepted / n_selected)
415</i><br/>
416 with an additional factor of <i>1/2</i> for case 2 above.
417
418<p/>
419The error quoted for the cross section of a process is a combination
420in quadrature of the error on this process alone with the error on
421the normalization factor, including the error on
422<i>&lt;f_impact&gt;</i>. As always it is a purely statistical one
423and of course hides considerably bigger systematic uncertainties.
424
425<h3>Event information</h3>
426
427Normally the <code>process</code> event record only contains the
428hardest interaction, but in this case also the second hardest
429is stored there. If both of them are <i>2 -> 2</i> ones, the
430first would be stored in lines 3 - 6 and the second in 7 - 10.
431For both, status codes 21 - 29 would be used, as for a hardest
432process. Any resonance decay chains would occur after the two
433main processes, to allow normal parsing. The beams in 1 and 2
434only appear in one copy. This structure is echoed in the
435full <code>event</code> event record.
436
437<p/>
438Most of the properties accessible by the 
439<code><?php $filepath = $_GET["filepath"];
440echo "<a href='EventInformation.php?filepath=".$filepath."' target='page'>";?>pythia.info</a></code>
441methods refer to the first process, whether that happens to be the
442hardest or not. The code and <i>pT</i> scale of the second process
443are accessible by the <code>info.codeMPI(1)</code> and
444<code>info.pTMPI(1)</code>, however.
445
446<p/>
447The <code>sigmaGen()</code> and <code>sigmaErr()</code> methods provide
448the cross section and its error for the event sample as a whole,
449combining the information from the two hard processes as described
450above. In particular, the former should be used to give the
451weight of the generated event sample. The statitical error estimate
452is somewhat cruder and gives a larger value than the
453subprocess-by-subprocess one  employed in
454<code>pythia.statistics()</code>, but this number is
455anyway less relevant, since systematical errors are likely to dominate.
456
457<input type="hidden" name="saved" value="1"/>
458
459<?php
460echo "<input type='hidden' name='filepath' value='".$_GET["filepath"]."'/>"?>
461
462<table width="100%"><tr><td align="right"><input type="submit" value="Save Settings" /></td></tr></table>
463</form>
464
465<?php
466
467if($_POST["saved"] == 1)
468{
469$filepath = $_POST["filepath"];
470$handle = fopen($filepath, 'a');
471
472if($_POST["1"] != "off")
473{
474$data = "SecondHard:generate = ".$_POST["1"]."\n";
475fwrite($handle,$data);
476}
477if($_POST["2"] != "off")
478{
479$data = "SecondHard:TwoJets = ".$_POST["2"]."\n";
480fwrite($handle,$data);
481}
482if($_POST["3"] != "off")
483{
484$data = "SecondHard:PhotonAndJet = ".$_POST["3"]."\n";
485fwrite($handle,$data);
486}
487if($_POST["4"] != "off")
488{
489$data = "SecondHard:TwoPhotons = ".$_POST["4"]."\n";
490fwrite($handle,$data);
491}
492if($_POST["5"] != "off")
493{
494$data = "SecondHard:Charmonium = ".$_POST["5"]."\n";
495fwrite($handle,$data);
496}
497if($_POST["6"] != "off")
498{
499$data = "SecondHard:Bottomonium = ".$_POST["6"]."\n";
500fwrite($handle,$data);
501}
502if($_POST["7"] != "off")
503{
504$data = "SecondHard:SingleGmZ = ".$_POST["7"]."\n";
505fwrite($handle,$data);
506}
507if($_POST["8"] != "off")
508{
509$data = "SecondHard:SingleW = ".$_POST["8"]."\n";
510fwrite($handle,$data);
511}
512if($_POST["9"] != "off")
513{
514$data = "SecondHard:GmZAndJet = ".$_POST["9"]."\n";
515fwrite($handle,$data);
516}
517if($_POST["10"] != "off")
518{
519$data = "SecondHard:WAndJet = ".$_POST["10"]."\n";
520fwrite($handle,$data);
521}
522if($_POST["11"] != "off")
523{
524$data = "SecondHard:TopPair = ".$_POST["11"]."\n";
525fwrite($handle,$data);
526}
527if($_POST["12"] != "off")
528{
529$data = "SecondHard:SingleTop = ".$_POST["12"]."\n";
530fwrite($handle,$data);
531}
532if($_POST["13"] != "off")
533{
534$data = "SecondHard:TwoBJets = ".$_POST["13"]."\n";
535fwrite($handle,$data);
536}
537fclose($handle);
538}
539
540?>
541</body>
542</html>
543
544<!-- Copyright (C) 2012 Torbjorn Sjostrand -->
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