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<h1 class="title">Final State Particle Generation</h1>
<p>Sar<em>t</em>re generates processes of the form <em>e p</em> &rarr; <em>e'
p' V and </em><em>e A</em> &rarr; <em>e'
A' V</em> where V can be any vector meson. This process is unique in that it
allows to almost completely determine the final state particles (e', A' or p',
and V) given <em>t</em>, <em>W</em>, and <em>Q</em>. </p>
<p> The energies of the beam
particles (<em>e</em> and <em>p or A</em>)
are input to the generator and the event generator delivers <em>t</em>, <em>W</em>,
and <em>Q</em> and with that also <em>y</em> and <em>x</em>.
This input is sufficient to derive the 4-momenta of the final
state particles, <em>e'
p' and V. </em>Below we outline the underlying calculations
that are implemented in the
class ExclusiveFinalStateGenerator used by class <a href="refSartre.html">Sartre</a>.</p>
<p>In what follows the 4 <strong>vectors</strong> are written in bold. </p>
<h2>Input</h2>
<ul>
<li>Incoming electron: <em><strong>q</strong> = (E<sub>e</sub>, 0, 0, q<sub>z</sub>)</em></li>
<li>Incoming proton: <em><strong>p</strong> = (E<sub>p</sub>, 0, 0, p<sub>z</sub>)</em></li>
<li><em>s, t, x, y, Q, W</em></li>
</ul>
<p>The beam electron comes from the right, meaning <em>q<sub>z</sub></em> is <em>negative</em>, the proton beam is coming from the left, meaning it has <em>positive</em> <em>p<sub>z</sub></em>.</p>
<h2>Scattered Electron </h2>
<p>The scattered electron momentum <strong><em>q'</em></strong> can be calculated as:</p>
<p><img src="images/scatteredElectron.png" width="416" height="158" /></p>
<p>However, in order to generate a true 4-momentum we need to select an azimuthal angle. This doesn't change <em>q<sub>t</sub></em> or <em>q<sub>z</sub></em> so we have freedom here. Hence <em>&phi;</em> is chosen randomly (flat in [0, 2&pi;[).</p>
<p>So this gives us the final value: <em><strong>q'</strong> = (E'<sub>e</sub>, q'<sub>t</sub> &middot; </em>sin<em> &phi;, q'<sub>t</sub> &middot; </em>cos<em> &phi;, q<sub>z</sub>)</em>.</p>
<h2>Virtual Photon</h2>
<p>Also the 4-momentum of the virtual photon is straightforward:</p>
<p><img src="images/photon.png" width="334" height="152" /></p>
<h2>Scattered Proton</h2>
<p class="latex">This is not a as trivial and I could not find a pure analytical solution. Here is how it goes:</p>
<p class="latex">From the definition of <em>t = (<strong>p</strong>-<strong>p</strong>')<sup>2</sup></em> we get directly: </p>
<p class="latex"><img src="images/scatteredProton1.png" alt="" width="181" height="45" />.</p>
<p class="latex">We also have the more or less trivial equations (again using a random <em>&phi;</em>):</p>
<p class="latex"><img src="images/scatteredProton2.png" alt="" width="208" height="88" /></p>
<p class="latex">The issue is E'. Once it is known the others are determined as well. Here it helps that there is only one generated particle, the vector meson, with a well defined mass <em>m<sub>V</sub></em>. </p>
<p class="latex"><img src="images/scatteredProton3.png" alt="" width="164" height="22" /></p>
<p class="latex"><strong><em>p</em></strong><em><sub>&gamma;</sub></em> and <em><strong>p</strong></em> are
known. We have 3 unknowns (<em>E', p<sub>z</sub>',</em> and <em>p<sub>t</sub>'</em>)
and 3 equations. Unfortunately the equations are non-linear and have
no analytical solution. In the class ExclusiveFinalStateGenerator a root
finder algorithm (Brent method) is used to solve the equation. With <em>E'</em> known,
the momenta can be directly calculated. Note, that finding the correct <em>E'</em> is
not trivial and requires additional calculation to limit the search to the
kinematically valid.</p>
<h2 class="latex">Vector Meson</h2>
<p class="latex">The 4-momentum of the vector meson is simply: </p>
<p class="latex"><strong><em>p</em></strong><em><sub>V</sub></em> = (<em><strong>p</strong></em> + <em><strong>q</strong></em> - <em><strong>q'</strong></em> - <em><strong>p'</strong></em>)</p>
<p class="latex">&nbsp;</p>
<p class="latex">
<math></math>
The actual implementation is of course a bit more complex but the above
reflects the basic idea.
<p>&nbsp;</p>
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