INT 24: From few to complex reactions.

<div style="text-align:center">
<h3>4-body reaction features <span style="color:tomato;">universally</span> correlated with 
<br>dimer & trimer subsystems<br>
 <p style="color: tomato;font-size: 40%;">February 25, 2025</h3>
</h3>
</div>

<div style="font-family: serif; display:flex ;flex-direction:column;vertical-align:bottom;margin-top: auto;">
<h4 style="color:grey;font-size: 15px;">
  Sourav Mondal, Rakshanda Goswami, Udit Raha -- <span style="color:black;font-size:80%">IIT Guwahati</span><br>
  J. Kirscher -- <span style="color:black;font-size:80%">SRM University AP</span>
</h4>
</div>

---

<div style="text-align:top">
<h3 style="color:gray;font-size: 25px;text-align: center;"> <span style="color: dodgerblue;"> Universal</span>
and <span style="color: tomato;"> peculiar</span> features of 
<span style="color: dodgerblue;"> short-distance-different</span> particles<br>
</div>

</div>
</div>

<p style="text-align: center; color: tomato;font-size: 90%;">
∅td's (NFL '13)$\approx 2.8$   
∅goals (UK '13)$\approx 2.8$   
∅tries (AUS '13)$\approx 5.7$ 
 </p>

<p style="text-align: center; color: dodgerblue;font-size: 90%;">
<img src="./images/fottball_b.png" width=26 height=26> $\approx 410(15)$g     
<img src="./images/soccer_b.png" width=26 height=26> $\approx 430(20)$g     
<img src="./images/rugby_b.png" width=26 height=26> $\approx 430(25)$g
 </p>

---

<div style="text-align:top">
<h3 style="color:gray;font-size: 25px;"> Universal <span style="color: dodgerblue;"> Bose</span> 
and <span style="color: tomato;"> Fermi</span> properties
  <br>
 
</div>

<img src="./images/phillips.png" style="max-height: 230px;">
<figcaption style="font-size:10px;color:gray;">
  <a href="http://arxiv.org/abs/nucl-th/9906032v4"><i style="font-size:14px;color:DodgerBlue" class="fa"> </i></a>
  P.F. Bedaque, H.-W. Hammer, U. van Kolck (2000)</figcaption>

<div class="l2" style="margin-top:20px">Phillips <i>line</i>:</div>
  <div class="l3" style="color:dodgerblue;">low-energy deuteron-neutron amplitude $=f(B_3)$</div>

</div>

<img src="./images/nalpha.png" style="max-height: 230px;">
<figcaption style="font-size:10px;color:gray;">
  <a href="https://arxiv.org/abs/2004.08474"><i style="font-size:14px;color:DodgerBlue" class="fa"> </i></a>
  K. Kravvaris et al. (2020)</figcaption>
  <div class="l2">neutron-$\alpha$:</div>
  <div class="l3" style="text-align: center; color:tomato;">resonance poles$=f(?)$</div>

</div>
</div>

<p style="text-align: center; color: dodgerblue;font-size: 90%;margin-top: 40px;">
\[a_{\text{dimer-dimer}}/a_{\text{atom-atom}}\approx0.6\]
<span style="font-size:10px;color:gray;">
  <a href="https://arxiv.org/abs/cond-mat/0309010v1"><i style="font-size:14px;color:DodgerBlue" class="fa"> </i></a>
  D.S. Petrov, C. Salomon, and G.V. Shlyapnikov (2003)</span>
</p>

---

<div style="text-align:top">
<h3 style="color:gray;font-size: 20px;"> Universal aspects of nuclear <span style="color: dodgerblue;"> fusion</span> 
and <span style="color: tomato;"> fission</span>
  <br>
 
</div>

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<div class="grid-container">
  <div class="grid-item"><img src="/images/fission.png"  style="display: block;margin-right: auto;max-height: 350px;">
<span style="font-size:10px;color:gray;">
  Mayer-Kuckuk, Theo. Kernphysik: eine Einführung. Germany, Teubner, 1994.</span>
  </div>
  <div class="grid-item" style="font-size: 14px;">Mass difference between<br> <strong>i</strong>nitial and <strong>f</strong>inal state:
$$Q=(m_i-m_f)\,c^2=T_f-T_a$$

<table class="tabular">
    <th>reaction</th>
    <th>$Q$ [MeV]</th>
<tr>
    <td class="r t li2">$d(d,n)^3\text{He}$</td>
    <td class="t">$3.27$</td></tr>
    <td class="r t li2">$d(d,p)t$</td>
    <td class="t">$4.03$</td></tr>
    <td class="r t li2">$d(d,\gamma)\alpha$</td>
    <td class="t">$23.8$</td></tr>
    <td class="r t li2">$t(p,n)^3\text{He}$</td>
    <td class="t">$-0.76$</td></tr>
    <td class="r t li2" style="color:tomato;">$t(d,n)\alpha$</td>
    <td class="t">$17.6$</td></tr>                
<tr>
</tbody>
</table>

</div>
</div>

<div class="grid-item" style="background-color: rgb(255, 255, 255);">
<p style="text-align: center; color: dodgerblue;font-size: 60%;margin-top: 40px;">
\[\sigma_{\text{max}}(t+d\to n+\alpha)\approx5000\,\text{mb}\]
<span style="font-size:20px;color:tomato;">
  Reaction-enhancement mechanism?
  </span>
</p>
</div>
<div class="grid-item", style="background-color: rgb(255, 255, 255);"><img src="/images/fusion.jpg"  style="display: block;margin-right: auto;max-height: 200px;"></div>
</div>

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<div style="text-align:top">
<h3 style="color:gray;font-size: 20px;"> Resonances in <span style="color:tomato;">neutron-$\alpha$</span> scattering and relevance for deuteron-triton <span style="color: dodgerblue;"> fusion</span> 
  <br>
 
</div>

---

<h3>The front:

<a style="color: black;text-align: center">
  <strong style="font-size: 100%;color: dodgerblue;">Renormalized form of an $A$-body contact theory</strong>
</a>

</h3>

<div style="color: tomato;text-align: left;font-size:22px;margin-top:20px">
Challenges:
</div>

<ul style="font-size: 20px;">
<li> Analytical guidance (Faddeev/Yakubovsky style)</li> 
<li> Accurate <strong>and</strong> efficient numerics for $A>4$</li>
<li> The appropriate expansion point<br> 
<span style="font-size:10px;color:gray;">
  <a href="https://journals.aps.org/pra/abstract/10.1103/PhysRevA.109.022814"><i style="font-size:14px;color:DodgerBlue" class="fa"> </i> Improved-action formulation</a>
  L. Contessi, <strong>M. Schäfer</strong>, and U. van Kolck (2024)</span>
</p>
</li>
</ul>

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<div style="text-align:top">
<h3 style="color:gray;font-size: 20px;"> The universal, statistical <span style="color: dodgerblue;"> particle-exchange </span>interaction:
</h3>
</div>

<p style="color: tomato;font-size: 90%; text-align: center;">i.e. <strong>identical-fermion</strong> swap between di-neutron states $\big|i,j,m,n\big\rangle$</p>

<p style="color: dodgerblue;font-size: 90%; text-align: center;">
$
\zeta\begin{pmatrix}i&m\\j&n\end{pmatrix}=
\int d^3(x_1,x_2,x_{1'},x_{2'})
\big\langle i\big\vert x_{1},x_{2'}\big\rangle
\big\langle j\big\vert x_{1'},x_{2}\big\rangle
\big\langle x_{1'},x_{2'}\big\vert n\big\rangle
\big\langle x_{1},x_{2}\big\vert m\big\rangle
$
</p>

<p style="color: black;font-size: 110%; text-align: center;margin-top: 50px;">
di-neutron <strong>m</strong> contains the <strong>same</strong> neutron as di-neutron <strong>i</strong>, e.g., $|\uparrow\,\rangle$<br>
</p>

---

<div style="text-align:top">
<h3 style="color:gray;font-size: 20px;"> The universal minimum $\hat{V}$: 
  <span style="color: dodgerblue;"> structure </span>  <strong>and</strong> 
   <span style="color: tomato;"> input </span><br>
 <p style="color: black;font-size: 70%;">
 \begin{align}
 \Psi(E)&=C+C^2\cdot \color{dodgerblue}{L_1(E)}+C^3\cdot \color{dodgerblue}{L_2(E)}+\ldots \\
     &=C(\lambda)+C(\lambda)^2\cdot L_{1,\lambda}(E)+C(\lambda)^3\cdot L_{2,\lambda}(E)+\ldots \\
     &\stackrel{!}{"="}\color{tomato}{B_{\text{exp}}}=\frac{\langle\Psi\vert\hat{H}\vert\Psi\rangle}{\langle\Psi\vert\Psi\rangle}
 \end{align}
</h3>
</div>

---

<div style="text-align:top">
<h3 style="color:gray;font-size: 20px;"> What do we know about <span style="color: dodgerblue;"> universal</span> 
  phenomena in the 4-<span style="color: dodgerblue;">boson</span> system?<br>
 <p style="color: tomato;font-size: 70%;">$a_2\to\infty$ i.e. <strong>no</strong> 2-body scale</h3>
</div>

<div style="color: gray;text-align: center;margin-top: 30px;font-size: 80%;o">
<img src="./images/efimov4.svg" style="max-height: 500px;">
<figcaption style="font-size:10px;color:gray;">
  <a href="https://arxiv.org/abs/1610.09805"><i style="font-size:14px;color:DodgerBlue" class="fa"> </i></a>
  P. Naidon, S. Endo (2016 review)</figcaption>

</div>

</div>
</div>

---

<div style="text-align:top">
<h3 style="color:gray;font-size: 20px;"> How does the <span style="color: dodgerblue;">minimal</span> 
  theory describe 4-<span style="color: dodgerblue;">component-fermion</span> reactions?  
  <a style="color: tomato;font-size: 110%;">$a_2<\infty$</a></h3>
</div>

</div>
</div>

---

<div style="text-align:top">
<h3 style="color:gray;font-size: 20px;"> <span style="color: darkgreen;">Phase-shift</span> parameters of the 2-channel 
  <span style="color: dodgerblue;"> S-matrix</span> 
 <a style="color: tomato;font-size: 110%;float: right;">$S_{if}=\eta_{if}\,e^{2i\delta_{if}}$</a></h3>
</div>

<div>
<img src="./images/phases.svg" style="max-height: 470px;">
<figcaption style="font-size:10px;color:gray;">
  <a href="https://arxiv.org/abs/2411.00386"><i style="font-size:14px;color:DodgerBlue" class="fa"> </i></a>
  S. Mondal, R. Goswami, U. Raha, JK (2024)</figcaption>
  <span style="float:right;">$\zeta=B_3/B_2$</span>
</div>

---

<div style="text-align:top">
<h3 style="color:gray;font-size: 20px;"> <span style="color: darkgreen;">Diagonal- and mixing-strength</span>
 parameters of the 2-channel 
  <span style="color: dodgerblue;"> S-matrix</span>
 <a style="color: tomato;font-size: 110%;float: right;">$S_{if}=\eta_{if}\,e^{2i\delta_{if}}$</a></h3>
</div>

<img src="./images/sigma.svg" style="max-height: 450px;">
<figcaption style="font-size:10px;color:gray;">
  <a href="https://arxiv.org/abs/2411.00386"><i style="font-size:14px;color:DodgerBlue" class="fa"> </i></a>
  S. Mondal, R. Goswami, U. Raha, JK (2024)</figcaption>
  <span style="float:right;">$\zeta=B_3/B_2$</span>
  
  
</div>
---

<div style="text-align:top">
<h3 style="color:gray;font-size: 20px;"> "Canonical" bosonic reaction rates in the 
  <span style="color: dodgerblue;"> unitary limit</span> <br>
 
</div>

<img src="./images/sigma_AD.png" style="max-height: 500px;">
<figcaption style="font-size:10px;color:gray;">
  <a href="https://arxiv.org/abs/1107.3956v1"><i style="font-size:14px;color:DodgerBlue" class="fa"> </i></a>
  A. Deltuva (2011)</figcaption>

</div>
</div>

---

<div style="text-align:top">
<h3 style="color:gray;font-size: 20px;"> Reaction vs. elastic strength with different 
  <span style="color: dodgerblue;"> atom-atom scattering lengths $a_2$</span> <br>
 
</div>

<img src="./images/crosssection_of_bd.svg" style="max-height: 500px;">
<figcaption style="font-size:10px;color:gray;">
  <a href="https://arxiv.org/abs/2411.00386"><i style="font-size:14px;color:DodgerBlue" class="fa"> </i></a>
  S. Mondal, R. Goswami, U. Raha, JK (2024)</figcaption>
  <span style="float:right;">$\zeta=B_3/B_2$</span>
</div>

---

<div style="text-align:top">
<h3 style="color:gray;font-size: 20px;"> Comparison with potential models: 
</div>

<img src="./images/lazauskas_fusion.png" style="max-height: 500px;">
<figcaption style="font-size:10px;color:gray;">
  <a href="https://arxiv.org/abs/2002.05876"><i style="font-size:14px;color:DodgerBlue" class="fa"> </i></a>
  Rimantas Lazauskas, Jaume Carbonell (2020)</figcaption>
</div>

---

<div style="text-align:top">
<h3 style="color:gray;font-size: 25px;"> <strong><i>An</i></strong> <span style="color: dodgerblue;"> algorithm</span>
for the analytic parametrization of <span style="color: dodgerblue;"> inter-cluster</span> 
potential with <span style="color: dodgerblue;"> 2-, 3-, &c. couplings</span><br>
 
</div>

<ul>
<li style="font-size: 50%;color:red">Expand in <i>realistic</i> DoF's
<p style="color: black;font-size: 80%;">
  \begin{align*}
\Psi=\hat{\mathcal{A}}\;{\Bigg\lbrace}&
\sum\limits_i\phi(A_i)\phi(B_i)F_i(\mathbf{R}_i)
+
\sum\limits_j\phi(A_j)\phi(B_j)\phi(C_j)F_j(\mathbf{R}_{1j},\mathbf{R}_{2j})\\
&+\ldots+\sum\limits_mc_m\chi_m\Bigg\rbrace\; Z(\mathbf{R}_{\text c.m.})
\end{align*}
</p>
</li>
<li style="font-size: 50%;color:red">
Obtain these states explicitly within "your" theory
<p style="color: black;font-size: 80%;">
$\hat{H}\phi^{(n)}_A=e_A^{(n)}\phi^{(n)}_A$
</p>  
</li>
<li style="font-size: 50%;color:red">
"Freeze" the asymptotic states
<p style="color: black;font-size: 80%;">
$\delta\Psi\stackrel{!}{=}\delta F_i$
</p>
<li style="font-size: 50%;color:red">
Integrate-out/average-over internal DoF's
<p style="color: black;font-size: 80%;">
\begin{gather*}
\left(\hat{T}_{\mathbf{R}}-E_\text{rel}+\mathbb{N}^{-1}\langle\phi_A\phi_B\vert\hat{V}\vert\phi_A\phi_B\rangle\right)\chi(\mathbf{R})\\
-\mathbb{N}^{-1}\int d\mathbf{R}'\left[\langle\phi_A\phi_B\vert\left(\hat{T}_{\mathbf{R}}-E_\text{rel}+
\hat{V}\right)\hat{A}\left\lbrace\vert\phi_A\phi_B\rangle\delta(\mathbf{R}-\mathbf{R}')\right\rbrace\right]\chi(\mathbf{R}')=0
\end{gather*}
</p>
</li>
<li style="font-size: 50%;color:red">
Obtain inter-cluster dynamics with a potential matrix parametrized with "microscopic" observables
<p style="color: black;font-size: 80%;">
\begin{gather*}
\sum_{n=1}^{N_{\text{loc}}}\hat{\eta}_n~e^{-w_n\mathbf{R}^2}\chi(\mathbf{R})-
\sum_{n=1}^{N_{\text{n-loc}}}\int\left\lbrace\hat{\zeta}_n\,e^{-a_n\mathbf{R}^2-b_n\mathbf{R}\cdot\mathbf{R}'-c_n\mathbf{R}'^2}\right\rbrace\chi(\mathbf{R}') d\mathbf{R}'=0\\
\color{dodgerblue}{\text{with$\;\;\;\hat{\eta}_n,\hat{\zeta}_n,w_n,a_n,b_n,c_n\;\;\;\text{dependent upon}\;\;\;C_{nn}(\lambda),C_{nnn}(\lambda),\color{red}{\alpha(\lambda)},E_{\text{rel}},A,B$}} 
\end{gather*}
</p>
</li>
</ul>

</div>
</div>

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<div style="text-align:top">
<h3 style="color:gray;font-size: 20px;"> 2-body contact-interaction strength's 
  <span style="color: dodgerblue;"> regulator dependence</span> <br>
 
</div>

---

<div style="text-align:top">
<h3 style="color:gray;font-size: 20px;"> dimer-atom scattering with an effective inter-cluster potential 
  <span style="color: dodgerblue;"> regulator dependence</span> <br>
 
</div>

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<h3 style="font-size:90%">SRM University campus at <a href="https://en.wikipedia.org/wiki/Mangalagiri">Mangalagiri</a>
  <img src="./images/in_flag.svg" width=18 height=18></h3>
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