Unitarity violation and restoration in radiative bound-state formation

Abstract

State-of-the-art calculations motivated by dark matter exhibit severe violation of partial-wave unitarity in the non-relativistic regime in radiative bound-state-formation processes. It has been recently shown, in a model-independent fashion, that unitarity is restored by the proper resummation of the inelastic contributions to the self-energy of the incoming state. In this work, we first derive Kramers-like formulae for individual partial waves, demonstrating that existing calculations of bound-state formation severely violate unitarity. We then discuss how unitarity is restored through the resummation of the absorptive contributions to the incoming-state self-energy, generated by bound-state formation processes, taking into account their analytic structure in the complex momentum plane. Our results can be generalized in a variety of theories and employed in phenomenological studies, such as dark-matter freeze-out, indirect detection and self-interactions.

Figures (12)

• Figure 1: The 2PI kernels generating long- and finite-range potentials for $XX^{\dagger}$ pairs (upper) and $XX$ or $X^{\dagger}X^{\dagger}$ pairs (lower), in the model of \ref{['eq:L_Abelian']}. The arrows denote the flow of the $U(1)_D$ charge. Figure adapted from Ref. Oncala:2019yvj.
• Figure 2: Bound-state formation via emission of a charged scalar, $X+ X \to {\cal B} (XX^{\dagger}) + \varrho$, in the model of \ref{['eq:L_Abelian']}. The arrows denote the flow of the $U(1)_D$ charge. Figure adapted from Ref. Oncala:2019yvj.
• Figure 3: The factors $r_{n\ell} \propto \sigma_\ell (k;n) / \sigma_\ell^U (k)$ vs $\zeta_{\mathsmaller{\cal B}}$, for $\alpha_{\mathsmaller{\cal S}}=0$ and different values of $n$, $\ell$, as denoted in the labels. We compare the original expression \ref{['eq:rnl_alphaS=0']}, with the analytic approximation \ref{['eq:rnl_alphaS=0_nggl']}. While the phase of the oscillations is different, both curves are enveloped by the same function. Larger $n$ implies more oscillations, higher maximum value of $r_{n\ell}$ and larger $\zeta_{\mathsmaller{\cal B}}$ value at which this maximum occurs.
• Figure 5: $r_\ell = \sum_{n=1+\ell}^\infty r_{n\ell}$ vs $\zeta_{\mathsmaller{\cal B}}$, for different values of $\ell$. At $\zeta_{\mathsmaller{\cal B}} \gg 1+\ell$, $r_\ell$ depends only mildly on $\ell$. We compare $r_\ell$ calculated numerically using the full expression \ref{['eq:rnl_alphaS=0']} for $r_{n\ell}$, with the analytic approximation \ref{['eq:rl_alphaS=0']}. The growth of $r_\ell$ with $\zeta_{\mathsmaller{\cal B}}$ implies that unitarity is violated at sufficiently low velocities, for arbitrarily low couplings, $\alpha_{\rm rad}$ [cf. \ref{['eq:BSF_sigma_lS_total']}].
• Figure 6: $r_{n\ell}$ vs $\zeta_{\mathsmaller{\cal B}}$, for $\ell=0$, and different values of $\lambda=\alpha_{\mathsmaller{\cal S}}/\alpha_{\mathsmaller{\cal B}}$ and $n$, as denoted in the labels. The asymptotic behavior at low velocities and large $n$, $\zeta_{\mathsmaller{\cal B}} , |\zeta_{\mathsmaller{\cal S}}| \gg n \gg \ell+1$ does not depend on $\ell$ [cf. \ref{['eq:rnl_LowVelocities_lambdaCases']}]. (This does not hold for $\lambda=0$.) Upper row:$\lambda \leqslant 0$; the excited levels contribute at lower velocities and at increasing strength. The Sommerfeld suppression for $\lambda<0$ gives rise to a sharp cutoff at low velocities. Lower row:$\lambda > 0$. Capture into excited levels is significant for $\lambda \lesssim 1$, but becomes exponentially suppressed for $\lambda\gg1$. (For $\lambda=1$, see \ref{['foot:lambda=1']}.) The Sommerfeld enhancement implies that $r_{n\ell}$ asymptotes to a constant value at low velocities, for $\lambda >0$ and any $\ell$. However, $\lambda\neq 1$ gives rise to non-monotonic dependence on $v_{\rm rel}$.