<chapter name="Deuteron Production"> 
 
<h2>Deuteron Production</h2> 
 
After the final state particles of an event are produced, protons and 
neutrons are selected and combined into pairs which may form 
deuterons. A generalized version of the model in <ref>Dal15</ref> is 
implemented here. The cross-sections used to determine if a combination 
binds into a deuteron are differential in 
<ei>k &equiv; |p<sub>1</sub> - p<sub>2</sub>|</ei> where 
<ei>p<sub>i</sub></ei> are the momenta of the binding combination in 
the center-of-mass frame of the combination. Here all notation is used 
to specify deuteron production, but anti-deuteron production is also 
provided where the anti-particles for all relevant particles are used. 
 
<p/> 
To ensure conservation of momentum and energy, the final state for 
each deuteron production channel is required to have at least two 
final products, where one product is a deuteron. This is in contrast 
to the traditional coalescence model of <ref>Scd63</ref> and 
<ref>Kap80</ref> where typically only a deuteron is produced from a 
two-body initial state and so momentum and energy are not 
conserved. Here, a modified coalescence model can be used where the 
final state can be specified as a deuteron and a photon. 
 
<p/> 
The default deuteron production setup is taken from <ref>Dal15</ref>, 
where 8 production channels are defined: 
<ul> 
<li><ei>p+ n0 &rarr; gamma 2H  </ei></li> 
<li><ei>p+ n0 &rarr; pi0 2H    </ei></li> 
<li><ei>p+ n0 &rarr; pi0 pi0 2H</ei></li> 
<li><ei>p+ n0 &rarr; pi+ pi- 2H</ei></li> 
<li><ei>p+ p+ &rarr; pi+ 2H    </ei></li> 
<li><ei>p+ p+ &rarr; pi+ pi0 2H</ei></li> 
<li><ei>n0 n0 &rarr; pi- 2H    </ei></li> 
<li><ei>n0 n0 &rarr; pi- pi0 2H</ei></li> 
</ul> 
These channels, and the models used to calculate their differential 
cross-sections, can be fully specified by the user as detailed below. The 
<code>DeuteronProduction</code> class performs the deuteron 
production, first building all valid two-particle combinations, then 
determining whether the combinations bind, and finally performing an 
isotropic decay of the bound state into the specified final state. 
 
<h3>Main parameters</h3> 
 
Assuming you have set <code>HadronLevel:DeuteronProduction = on</code>, 
you can regulate the detailed behaviour with the following settings. 
 
<p/> 
<b>Warning</b>: changed <code>mvec</code> and <code>wvec</code> values 
must be provided as a comma-separated list enclosed in braces, 
<code>{...}</code>. 
 
<parm name="DeuteronProduction:norm" default="119.6" min="1"> 
An overall unitless multiplicative normalization scale to increase the 
maximum used when sampling the differential cross-sections. Increasing 
this scale decreases the cross-section for deuteron production. Using 
the default model, the maximum differential cross-section is <ei>3.2 
mb</ei> with the <ei>n0 n0 &rarr; pi- 2H</ei> and <ei>p+ p+ &rarr; pi+ 
2H</ei> channels. To convert the fit values from table VIII of 
<ref>Dal15</ref> into this multiplicative normalization factor, take 
<ei>1/(value from table * maximum cross-section)</ei>. For example, 
the ALICE 7 TeV deuteron fit with a <ei>1/&sigma;<sub>0</sub></ei> of 
2.63 inverse barn, the multiplicative normalization is 119.6. 
</parm> 
 
<wvec name="DeuteronProduction:channels" default="{2212 2112 > 22, 2212 2112 > 111, 2212 2112 > 111 111, 2212 2112 > 211 -211, 2212 2212 > 211, 2212 2212 > 211 111, 2112 2112 > -211, 2112 2112 > -211 111}"> 
List of the deuteron production channels to include of the form 
<ei>initial states &gt; final states </ei> where <ei>initial 
states</ei> are the PDG particle IDs of the initial state particles to 
combine (there must be two), and <ei>final states</ei> are the PIDs of 
the final state particles, excluding the deuteron. There must be one 
or more final state particles in addition to the deuteron. The 
products of each channel are space separated, while the channels are 
comma separated. 
</wvec> 
 
<mvec name="DeuteronProduction:models" default="{1, 2, 3, 3, 2, 3, 2, 3}"> 
List of the parameterization mode for each deuteron production channel 
specified by <code>DeuteronProduction:channels</code>. The parameters 
for each of the channels is specified with 
<code>DeuteronProduction:parms</code>. The length of this vector must 
be the same as <code>DeuteronProduction:channels</code>. 
<option value="0">the coalescence model, <ei>p<sub>1</sub> 
&theta;(p<sub>0</sub> - k)</ei>, a step function with two 
parameters: <ei>p<sub>0</sub></ei> the cut-off of the step function 
and <ei>p<sub>1</sub></ei> the normalization of the step 
function.</option> 
<option value="1">the differential cross-section model in 
<ei>&kappa;</ei> is given by the piece-wise function 
<ei>&sum;<sub>i=1</sub><sup>12</sup> p<sub>i</sub> 
&kappa;<sup>i-2</sup></ei> for <ei>&kappa; &lt; p<sub>0</sub></ei>, 
otherwise <ei>exp(-p<sub>13</sub> &kappa; - p<sub>14</sub> 
&kappa;<sup>2</sup>)</ei>. A total of 15 parameters are required: the 
function split followed by 12 polynomial coefficients for the first 
function, and 2 polynomial coefficients for the second exponential 
function. The function is fixed below <ei>0.1 GeV</ei> to its value 
at this <ei>&kappa;</ei> where <ei>&kappa;</ei> is defined as 
<ei>k/GeV</ei>.</option> 
<option value="2">the differential cross-section model in 
<ei>&eta;</ei> is given by <ei>p<sub>0</sub> 
&eta;<sup>p<sub>1</sub></sup>/((p<sub>2</sub> - exp(p<sub>3</sub> 
&eta;))<sup>2</sup> + p<sub>4</sub>)</ei>. A total of 5 parameters is 
required. Here, <ei>&eta;</ei> is the momentum of the non-deuteron 
final state divided by its mass in the COM. This model is intended for 
two-body final states containing a deuteron and a pion.</option> 
<option value="3"></option>the differential cross-section model in 
<ei>&kappa;</ei> is given by <ei>&sum;<sub>i=0</sub> p<sub>5*i+0</sub> 
&kappa;<sup>p<sub>5*i+1</sub></sup>/((p<sub>5*i+2</sub> - 
exp(p<sub>5*i+3</sub> &kappa;))<sup>2</sup> + p<sub>5*i+4</sub>)</ei> 
where the number of parameters can be any multiple of 5 and 
<ei>&kappa;</ei> is defined as <ei>k/GeV</ei>. 
</mvec> 
 
<wvec name="DeuteronProduction:parms" default="{1.28 2.3034605532591175 -93.663463313902028 2565.3904680353621 -25594.100560137995 143513.10872427333 -503572.89020794741 1149248.0196165806 -1723683.9119787284 1679348.7891145353 -1019888.5470232342 349840.35161061864 -51662.760038375141 -5.1885266705385051 2.9195632726211609, 85 1.34 1.77 0.38 0.096, 2.85519622e+06 1.31114126e+01 2.96145497e+03 5.57220777e+00 1.46051932e+06, 6.46455516e+06 1.05136338e+01 1.97924778e+03 5.36301369e+00 6.04534114e+05 2.54935423e+15 1.65669163e+01 2.32961298e+07 1.11937373e+01 2.86815089e+16, 170 1.34 1.77 0.38 0.096, 5.09870846e+15 1.65581228e+01 2.33337076e+07 1.13304315e+01 2.86815089e+16, 170 1.34 1.77 0.38 0.096, 5.09870846e+15 1.65581228e+01 2.33337076e+07 1.13304315e+01 2.86815089e+16}"> 
The parameters needed for the model of each channel. The parameters 
for each channel are space separated, while the channels are separated 
by commas. The length of this vector must be the same as 
<code>DeuteronProduction:channels</code>. Note, to match the parameters 
of <ref>Dal15</ref> these parameters give differential cross-sections in 
<ei>&mu;b</ei> not <ei>mb</ei>. 
</wvec> 
 
<h3>Technical parameters</h3> 
 
The maximum cross-section for each deuteron production channel is 
calculated using an initial grid search followed by the simplex algorithm. 
Assuming parameters similar to the default parameters above are used, the 
following technical parameters should not need to be changed. However, if 
the maximum finding fails for a given configuration, the following can be 
modified to ensure appropriate determination of the cross-section maximums. 
 
<parm name="DeuteronProduction:kMin" default="1e-1"> 
The minimum bracketing value of <ei>k</ei> (in <ei>GeV</ei>) when maximum 
finding. For the default <ei>p n &rarr; gamma 2H</ei> production model 
the cross-section diverges at low <ei>k</ei>, but the probability of 
producing two nucleons with low <ei>k</ei> suppresses this divergence. 
</parm> 
 
<parm name="DeuteronProduction:kMax" default="5"> 
The maximum bracketing value of <ei>k</ei> (in <ei>GeV</ei>) when maximum 
finding. 
</parm> 
 
<parm name="DeuteronProduction:kTol" default="1e-4"> 
The minimum relative tolerance in <ei>k</ei> required when maximum 
finding. 
</parm> 
 
<mode name="DeuteronProduction:kSteps" default="100"> 
The number of steps to sample in <ei>k</ei> when calculating the 
initial grid for maximum finding. 
</mode> 
 
</chapter> 
 
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