INT 24: From few to complex reactions.

<div style="text-align:top">
<h3 style="color:gray;font-size: 20px;"> Quantum Few- and Many-Body Systems in <span style="color: dodgerblue;"> Universal</span> Regimes<br>
 <p style="color: tomato;font-size: 40%;">October 9 - November 8, 2024</h3>
</div>

<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%;">October 9, 2024</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 -- IIT Guwahati<br>
  J. Kirscher -- SRM University AP
</h4>
</div>

---

<div style="text-align:top">
<h3 style="color:gray;font-size: 20px;"> <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: 20px;"> 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;"> 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>
</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?<br>
 <p style="color: tomato;font-size: 90%;">$a_2<\infty$</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> <br>
 <p style="color: tomato;font-size: 90%;">$S_{if}=\eta_{if}\,e^{2i\delta_{if}}$</h3>
</div>

<img src="./images/phases.svg" style="max-height: 530px;">
<figcaption style="font-size:10px;color:gray;">
  <a href="http://arxiv.org/abs/nucl-th/?"><i style="font-size:14px;color:DodgerBlue" class="fa"> </i></a>
  S. Mondal, R. Goswami, U. Raha, JK (2024)</figcaption>

---

<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> <br>
 <p style="color: tomato;font-size: 90%;">$S_{if}=\eta_{if}\,e^{2i\delta_{if}}$</h3>
</div>

<img src="./images/sigma.svg" style="max-height: 450px;">
<figcaption style="font-size:10px;color:gray;">
  <a href="http://arxiv.org/abs/nucl-th/?"><i style="font-size:14px;color:DodgerBlue" class="fa"> </i></a>
  S. Mondal, R. Goswami, U. Raha, JK (2024)</figcaption>

</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="http://arxiv.org/abs/nucl-th/?"><i style="font-size:14px;color:DodgerBlue" class="fa"> </i></a>
  S. Mondal, R. Goswami, U. Raha, JK (2024)</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>

---

<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>

---
<div class="bg"></div>