<chapter name="Close-Packing"> 
 
<h2>Close-Packing</h2> 
 
<p> 
If strings are closely packed, e.g. as a consequence of MPIs, it is 
likely that they receive an increased string tension, which translates 
into a broader <ei>pT</ei> spectrum, see further 
<ref>Fis16</ref>. It also means an enhanced rate (or rather reduced 
suppression) of strange-quark (and optionally also diquark) production 
relative to up/down quark production. 
</p> 
<p> 
For each string undergoing fragmentation, close-packing effects, for 
each string break, scale with the effective number of 
nearby strings, determined according to rapidity measured along the 
z-axis of the system. This should be quite appropriate for multi-string 
systems produced by MPIs in pp collisions but would not make much sense 
to apply to processes such as ee&rarr;WW&rarr;hadrons, for which the 
z-axis does not play any special role; such studies would be 
interesting but would require a generalisation of the current 
implementation. 
</p> 
<p> 
For each string break, the numbers of 
overlapping strings with flux orientations parallel and 
antiparallel to the string on which the break occurs, are denoted by 
<ei>p</ei> and <ei>q</ei> respectively. The effective string tension 
is then assumed to scale like 
<eq> 
    &kappa;<sub>eff</sub> = (1 + (k<sub>p</sub>p + k<sub>q</sub>q) 
    /(1+p<sup>2</sup><sub>T had</sub>/p<sup>2</sup><sub>T0</sub>)) 
    <sup>2r</sup>&kappa;<sub>0</sub>. 
</eq> 
where <ei>p<sub>T had</sub></ei> is the transverse momentum of the 
hadron produced in the break. Close-packing effects are thus 
suppressed for particles with <ei>p<sub>T</sub></ei> values greater than the 
reference <ei>p<sub>T0</sub></ei> value. This reflects the expectation 
that the high-<ei>p<sub>T</sub></ei> parts of hard jets are not 
affected by close-packing. 
</p> 
<p> 
This scales the probability <code>StringFlav:probStoUD</code> by 
<eq> 
    P'(s:u/d) = P(s:u/d)<sup>(&kappa;<sub>0</sub>/&kappa;<sub>eff</sub>)</sup> 
</eq> 
</p> 
<code>StringFlav:probSQtoQQ</code> and <code>StringFlav:probQQ1toQQ0</code> 
also scale in this way. The width of the <ei>pT</ei> spectrum, given by 
<ei>&sigma;^2</ei> is scaled by 
<eq> 
    &sigma;'<sup>2</sup> = &sigma;<sup>2</sup> 
    &kappa;<sub>eff</sub>/&kappa;<sub>0</sub>. 
</eq> 
 
 
The diquark production probability <code>StringFlav:probQQtoQ</code> 
scales according to 
<eq> 
    P'(qq:q)=&alpha;'(P(qq:q)/&alpha;)<sup>&kappa;<sub>eff</sub></sup>, 
</eq> 
 
and <ei>&alpha;'</ei> is <ei>&alpha;</ei> recalculated given 
<ei>&kappa;<sub>eff</sub></ei>. 
 
We allow different parameters to set <ei>k<sub>p</sub></ei> depending 
on what the effective string tension is scaling. In principle if the 
close-packing mechanism were exact, <code>enhanceStrange</code>, 
<code>enhancePT</code>, and <code>enhanceDiquark</code> would be 
equal, however we have built in extra degrees of freedom in the model. 
 
<flag name="ClosePacking:doClosePacking" default="off"> 
Turns on the close-packing model. 
</flag> 
 
<parm name="ClosePacking:fluxRatio" default="0.5" min="0.0" max="1.0"> 
<ei>k<sub>q</sub>/k<sub>p</sub></ei> ratio. The default value of 0.5 
corresponds to the ratio obtained if the effective string tensions 
scale like effective colour charges (Casimir scaling). A value of zero 
would mean that antiparallel fluxes do not increase the tension at 
all, while a value of unity would make the increase independent of 
flux direction. 
</parm> 
 
<parm name="ClosePacking:PT0" default="2.0" min="0.0"> 
<ei>p<sub>T0</sub></ei> parameter above. 
</parm> 
 
<parm name="ClosePacking:enhanceStrange" default="0.08" min="0.0"> 
<ei>k<sub>p</sub></ei> parameter above used for scaling 
<code>StringFlav:probStoUD</code>, <code>StringFlav:probSQtoQQ</code> 
and <code>StringFlav:probQQ1toQQ0</code>. 
</parm> 
 
<parm name="ClosePacking:enhancePT" default="0.5" min="0.0"> 
<ei>k<sub>p</sub></ei> parameter above used for scaling 
<code>StringPT:sigma</code>. 
</parm> 
 
<parm name="ClosePacking:enhanceDiquark" default="0.5" min="0.0"> 
<ei>k<sub>p</sub></ei> parameter above used for scaling 
<code>StringFlav:probQQtoQ</code>. 
</parm> 
 
<flag name="ClosePacking:doEnhanceDiquark" default="on"> 
As probQQtoQ is scaled by &kappa; by &alpha;' regardless of 
<code>ClosePacking:enhanceDiquark</code> value, here we allow 
the scaling of probQQtoQ with an effective &kappa; to be switched off 
altogether to capture the ambiguity in the scaling of probQQtoQ given 
the popcorn mechanism for diquark creation. 
</flag> 
 
Parameters <code>ClosePacking:expNSP</code> and 
<code>ClosePacking:expMPI</code> are typically used for the thermal 
model of string breaks. Normally only one of the options below would 
be used, but technically both are allowed and then combine 
multiplicatively. 
 
<parm name="ClosePacking:expNSP" default="0.5" min="0.0" max="1.0"> 
<ei>r</ei> parameter above to allow non-linear scaling of 
<ei>&kappa;<sub>eff</sub></ei>. Default value is based on Gaussian string 
breaks. For the thermal model, the default value is 0.13. 
</parm> 
 
<parm name="ClosePacking:expMPI" default="0.0" min="0.0" max="1.0"> 
Allows scaling of width/temperature by number of MPI rather than number of 
near strings. The width/temparture will get the prefactor 
<i>N(MPI)<sup>expMPI</sup></i>. 
</parm> 
 
<h3>Popcorn destructive interference</h3> 
 
The popcorn mechanism creates diquarks via a step-wise procedure of 
two successive colour fluctuations on a string. By allowing the first 
of these colour fluctuaions to break strings given the correct colour 
configuration, the probability of forming a diquark is reduced in 
densely packed string systems. To modify 
<code>StringFlav:probQQtoQ</code>, the survival probability of a 
diquark is considered, with <ei>f<sub>q</sub></ei> 
and <ei>f<sub>p</sub></ei> being the probability of the colour fluctuation 
to connect with an antiparallel or parallel nearby string 
respectively. Connections with antiparallel strings is expected to be 
dominant due to the favoured colour orientation. These probabilities 
are <ei>pT</ei> suppressed the same way as 
<ei>&kappa;<sub>eff</sub></ei> is for close-packing, using parameter 
<code>ClosePacking:PT0</code>. 
 
As colour indices used in colour reconnections are currently not 
stored in the event, an equal distribution of colours is assumed 
instead. Considering the survival probability of the first fluctuation 
for a given colour configuration, the probability 
<code>StringFlav:probQQtoQ</code> is modified by 
 
<eq> 
    P'(qq:q) = ((1-f<sub>q</sub>)<sup>n<sub>q</sub>/9</sup> 
    (1-f<sub>p</sub>)<sup>n<sub>p</sub>/9</sup>) P(qq:q). 
</eq> 
 
<parm name="ClosePacking:baryonSup" default="0.5" min="0." max="1.0"> 
<ei>f<sub>q</sub></ei> parameter above. 
</parm> 
 
<parm name="ClosePacking:parallelBaryonSup" default="0." min="0." max="1.0"> 
<ei>f<sub>p</sub></ei> parameter above. 
</parm> 
 
</chapter> 
