INT 24: From few to complex reactions.

Quantum Few- and Many-Body Systems in Universal Regimes

October 9 - November 8, 2024

4-body reaction features universally correlated with
dimer & trimer subsystems

October 9, 2024

Sourav Mondal, Rakshanda Goswami, Udit Raha -- IIT Guwahati
J. Kirscher -- SRM University AP

1 / 14

Universal and peculiar features of short-distance-different particles

∅td's (NFL '13) $\approx 2.8$ ∅goals (UK '13) $\approx 2.8$ ∅tries (AUS '13) $\approx 5.7$

$\approx 410(15)$ g $\approx 430(20)$ g $\approx 430(25)$ g

2 / 14

Universal Bose and Fermi properties

P.F. Bedaque, H.-W. Hammer, U. van Kolck (2000)

Phillips line:

low-energy deuteron-neutron amplitude

$=f(B_3)$

K. Kravvaris et al. (2020)

neutron-

$\alpha$ :

resonance poles

$=f(?)$

$a_{\text{dimer-dimer}}/a_{\text{atom-atom}}\approx0.6$ D.S. Petrov, C. Salomon, and G.V. Shlyapnikov (2003)

3 / 14

The universal minimum $\hat{V}$ : structure and input

$\begin{align} \Psi(E)&=C+C^2\cdot \color{dodgerblue}{L_1(E)}+C^3\cdot \color{dodgerblue}{L_2(E)}+\ldots \\ &=C(\lambda)+C(\lambda)^2\cdot L_{1,\lambda}(E)+C(\lambda)^3\cdot L_{2,\lambda}(E)+\ldots \\ &\stackrel{!}{"="}\color{tomato}{B_{\text{exp}}}=\frac{\langle\Psi\vert\hat{H}\vert\Psi\rangle}{\langle\Psi\vert\Psi\rangle} \end{align}$

4 / 14

What do we know about universal phenomena in the 4-boson system?

$a_2\to\infty$ i.e. no 2-body scale

P. Naidon, S. Endo (2016 review)

5 / 14

How does the minimal theory describe 4-component-fermion reactions?

$a_2<\infty$

6 / 14

Phase-shift parameters of the 2-channel S-matrix

$S_{if}=\eta_{if}\,e^{2i\delta_{if}}$

S. Mondal, R. Goswami, U. Raha, JK (2024)

7 / 14

Diagonal- and mixing-strength parameters of the 2-channel S-matrix

$S_{if}=\eta_{if}\,e^{2i\delta_{if}}$

S. Mondal, R. Goswami, U. Raha, JK (2024)

8 / 14

"Canonical" bosonic reaction rates in the unitary limit

A. Deltuva (2011)

9 / 14

Reaction vs. elastic strength with different atom-atom scattering lengths $a_2$

S. Mondal, R. Goswami, U. Raha, JK (2024)

10 / 14

An algorithm for the analytic parametrization of inter-cluster potential with 2-, 3-, &c. couplings

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

11 / 14

2-body contact-interaction strength's regulator dependence

12 / 14

dimer-atom scattering with an effective inter-cluster potential regulator dependence

13 / 14

14 / 14

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Quantum Few- and Many-Body Systems in Universal Regimes October 9 - November 8, 2024

4-body reaction features universally correlated with dimer & trimer subsystems October 9, 2024

Sourav Mondal, Rakshanda Goswami, Udit Raha -- IIT Guwahati J. Kirscher -- SRM University AP

Universal and peculiar features of short-distance-different particles

Universal Bose and Fermi properties

What do we know about universal phenomena in the 4-boson system? a_2\to\inftya_2\to\infty i.e. no 2-body scale

How does the minimal theory describe 4-component-fermion reactions? a_2<\inftya_2<\infty

Phase-shift parameters of the 2-channel S-matrix S_{if}=\eta_{if}\,e^{2i\delta_{if}}S_{if}=\eta_{if}\,e^{2i\delta_{if}}

Diagonal- and mixing-strength parameters of the 2-channel S-matrix S_{if}=\eta_{if}\,e^{2i\delta_{if}}S_{if}=\eta_{if}\,e^{2i\delta_{if}}

"Canonical" bosonic reaction rates in the unitary limit

Reaction vs. elastic strength with different atom-atom scattering lengths a_2a_2

An algorithm for the analytic parametrization of inter-cluster potential with 2-, 3-, &c. couplings

2-body contact-interaction strength's regulator dependence

dimer-atom scattering with an effective inter-cluster potential regulator dependence

Universal and peculiar features of short-distance-different particles

Help

Quantum Few- and Many-Body Systems in Universal Regimes

October 9 - November 8, 2024

4-body reaction features universally correlated with
dimer & trimer subsystems

October 9, 2024

Sourav Mondal, Rakshanda Goswami, Udit Raha -- IIT Guwahati
J. Kirscher -- SRM University AP

What do we know about universal phenomena in the 4-boson system?

$a_2\to\infty$ i.e. no 2-body scale

How does the minimal theory describe 4-component-fermion reactions?

$a_2<\infty$

Phase-shift parameters of the 2-channel S-matrix

$S_{if}=\eta_{if}\,e^{2i\delta_{if}}$

Diagonal- and mixing-strength parameters of the 2-channel S-matrix

$S_{if}=\eta_{if}\,e^{2i\delta_{if}}$

Reaction vs. elastic strength with different atom-atom scattering lengths $a_2$