On the existence of bound states in SIMP dark sectors

TL;DR

The study demonstrates that in a QCD-like SIMP dark sector with Sp(4) gauge dynamics and Nf=2, a scalar isosinglet bound state formed from two dark pions emerges in a controlled region where $m_\pi/f_\pi$ is a few. By unitarizing the chiral amplitudes in a Lippmann–Schwinger framework, the authors locate an S-wave bound-state pole below the two-pion threshold and extract the bound-state wave function at the origin $|\Psi(0)|$, finding $|\Psi(0)| \sim \mathcal{O}(0.1)\,m_\pi^{3/2}$ for binding energies near the freeze-out temperature $E_B \sim m_\pi/20$. They compare the T-matrix result with variational estimates, observing good agreement for shallow binding, and show that the bound state can significantly enhance bound-state–assisted annihilation and DM self-interactions, with the latter potentially addressing small-scale structure. The work thus provides a dynamical basis for the sigma-like dark scalar in SIMP models and lays groundwork for lattice matching and extended phenomenology, including finite-temperature Boltzmann analyses and vector-resonance effects.

Abstract

In strongly interacting massive particle (SIMP) scenarios, dark matter is comprised of stable dark pions whose $3\to 2$ or $4\to 2$ reactions set the dark matter relic abundance. Recent work has shown that shallow two-pion bound states significantly affect the freeze-out, but did not establish whether such states actually form. In this work we demonstrate that a scalar isosinglet bound state does exist in a well-defined region of parameter space by solving an on-shell Lippmann--Schwinger equation in a chiral-unitary framework and analyzing the $S$-wave $ππ$ amplitude in the complex energy plane. We determine the range of $m_π/f_π$ for which a pole appears below the two-pion threshold, extract the corresponding residue, and, in the non-relativistic limit, obtain the bound-state wave function at the origin, $|Ψ(0)|$, which controls bound-state-assisted annihilation and decay rates relevant for catalyzed freeze-out. Comparing this T-matrix based result with variational estimates using simple finite-range potentials, we find agreement within order-one factors for shallow binding. For binding energies of order the freeze-out temperature, $E_B \sim m_π/20$, we obtain $|Ψ(0)|\sim \mathcal{O}(0.1)\,m_π^{3/2}$, thereby supporting the parametric assumptions used in previous phenomenological analyses.

Figures (4)

• Figure 1: The LO (a) and NLO (b--d) diagrams for pion-pion scattering. The solid blue circle is a vertex from $\mathcal{L}_2$, and the solid red square is a vertex from $\mathcal{L}_4$ .
• Figure 2: Left: Comparative contributions of NLO and NNLO T-matrix amplitudes relative to the one of lower chiral order, after taking $S$-wave projection for the flavor singlet channel at $s= (1.95m_\pi)^2$. Right: Real and imaginary components of the amplitudes at LO and NLO level, as a function of $s$ for $m_\pi/f_\pi =4$. In all figures, uncertainties of LECs at NLO (combination of NLO and NNLO) are included by adding independent Gaussian fluctuations to their central values with a standard deviation of $10^{-4}$ ($10^{-6}$), shown as pink (cyan) dots.
• Figure 3: Left: Binding energy $E_B$ normalized to pion mass $m_\pi$ of $S$-wave bound state as a function of effective coupling $m_\pi/f_\pi$. Pink dots show the range of variation under adding Gaussian noise to the LECs, as specified in the main text. Right: Estimated absolute values of the bound state wave function at $r=0$ for small binding energies, calculated via the T-matrix and variational methods with various trial functions as labeled.
• Figure 4: Two-quark and four-quark states. Left: the singlet $\eta_0'$ and quintuplet $\pi^{A,\dots,E}$ of two-quark states (circles in black). Right: four-quark states (circles in red) constructed from the two quintuplets: ${\bf 5}\; \otimes \; {\bf 5} = {\bf 1} \;\oplus {\bf 10} \;\oplus {\bf 14}$. Here each possible combination of two pion states (one circle of red) is depicted schematically by adding another pion (indicated by a blue arrow according to its weight vector), to a pre-existing one (labeled as $\pi^I$ in black).