Nuclear Physics

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## <a style="color: gray; font-size: 20pt;"> What are - amongst many - the objectives of </a>   <a style="color: darkgreen; font-size: 40pt;">nuclear physics?</a>

<a style="color: tomato; font-size: 20pt;">(I) Spectra:  </a>
<a style="color: dodgerblue;text-align: center;">Relative <strong>stability</strong> of <strong>composites</strong> of neutrons and protons</a>
<p style="text-align:center;"><img src=/images/nuclbinding.png style="text-align:center;border-radius: 8px; max-width:75%; margin-top:10px;"></p>

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<a style="color: tomato; font-size: 20pt;">(II) Reactions:  </a>
<a style="color: dodgerblue;text-align: center;">Response of nuclear <strong>composites</strong> to their like and unlike</a>

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<div class="grid-item" style="background-color: rgb(255, 255, 255);"><img src="./images/nuclcross.png" style="max-width: 85%;">
  <div class="l1"><a href="https://journals.aps.org/prc/abstract/10.1103/PhysRevC.69.044002">\(^4\text{He}\left(\gamma,p\right)^3\text{H}\)</a></div>
  <div class="l2" style="color:dodgerblue;">photo dissociation cross sections</div>
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<div class="grid-item" style="background-color: rgb(255, 255, 255);"><img src="./images/nucldiss.png" style="max-width: 95%;">
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<div style="text-align: center;">
<h3 style="color: black; font-size: 18px;"> <strong>Appropriate</strong> theory for low-energy processes:</h3>
<ul style="font-size: 14px;color: gray;">
<li> degrees of freedom: neutrons and protons <i>("What do we measure?")</i></li>
<li> symmetry: Galilei \(\approx\) low-energy Lorentz <i>(translational and rotationally invariant <strong>potentials</strong>)</i></li>
<li> particle statistics and identity: quantum mechanics</li>
</ul>

<div style="color: darkgreen;"> \(\Rightarrow i\partial_t\Psi(\tilde{r}_1,\ldots,\tilde{r}_A)=\hat{H}\Psi(\tilde{r}_1,\ldots,\tilde{r}_A)\)
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<div style="color: black;text-align: center;margin-top:5px">
with   \(\hat{H}=-\frac{\hbar^2}{2}\sum\limits_i\frac{\vec{\nabla}^2}{m_i}+\sum\limits_{i,j}v_{ij}+\sum\limits_{i,j,k}v_{ijk}\) 
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<h4 style="color: tomato;text-align: center;margin-top: 10px;">Example:</h4>
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\begin{eqnarray}
v_{ij}&=&\sum\limits_na_n\frac{e^{-n\mu r_{ij}}}{\mu r_{ij}}+\sum\limits_nc_n\frac{e^{-n\mu r_{ij}}}{\mu r_{ij}}\vec{L}\cdot\vec{S}\\
&&+\left\lbrace\frac{b_1}{\mu r_{ij}}\left[\left(\frac{1}{3}+\frac{1}{\mu r_{ij}}+\frac{1}{(\mu r_{ij})^2}\right)e^{-\mu r_{ij}}-\left(\frac{b_0}{\mu r_{ij}}+\frac{1}{(\mu r_{ij})^2}\right)e^{-b_0\mu r_{ij}}\right]+\sum\limits_nb_n\frac{e^{-n\mu r_{ij}}}{\mu r_{ij}}\right\rbrace \left[3\left(\vec{\sigma}_1\cdot\hat{\vec{r}}\right)\left(\vec{\sigma}_2\cdot\hat{\vec{r}}\right)-\vec{\sigma}_1\cdot\vec{\sigma}_2\right]
\end{eqnarray}
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<div style="color: black;text-align: center">But how can we <strong style="color:red">derive</strong> this from a <i>fundamental</i> theory?</div>

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relativistic quantum field theory <strong style="color:red">\(\to\)</strong> non-relativistic dynamics <i>(Schrödinger, Lippmann-Schwinger)</i> \(\oplus\) few-body potentials
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<div style="color: black;text-align: center;font-size:12px;margin-top:20px">
Understand the dynamical equation as a <strong style="font-size: 150%;color: dodgerblue;">wave equation</strong> resulting from a
Lagrangean/Hamiltonian density:
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<div style="color: black;text-align: center;margin-top:15px">
\(\mathcal{L}_0=\frac{i}{2}\Psi^*\partial_t\Psi-\frac{i}{2}\Psi\partial_t\Psi^*-\frac{\left(\nabla\Psi^*\right)\left(\nabla\Psi\right)}{2m}\)
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<div style="color: black;text-align: center;font-size:12px;margin-top:20px">
where the <i>field</i> \(\Psi(\vec{x},t)\) replaces the position \(\vec{x}(t)\) as the fundamental degree of freedom and
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\(\left[\Psi(\vec{x},t),\Psi^*(\vec{x}',t)\right]=\delta(\vec{x}-\vec{x}')\)
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<h3>The front:</h3>

<div style="color: black;text-align: center">
  <strong style="font-size: 150%;color: dodgerblue;">Renormalized form of an \(A\)-body contact theory</strong>.
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<div style="color: tomato;text-align: center;font-size:22px;margin-top:20px">
Challenges:
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<ul style="font-size: 20px;">
<li> Analytical guidance (Faddeev/Yakubovsky style)</li> 
<li> Accurate numerics</li>
<li> Efficient numerics</li>
<li> The appropriate expansion point (more in 
<a href="https://journals.aps.org/pra/abstract/10.1103/PhysRevA.109.022814"> Improved-action approach</a>)
</li>
</ul>