Towards a theory of dissipative Dark Matter I: the Born limit

Abstract

We derive the energy-differential cross section and energy loss rate for dissipative self-interacting dark matter (dSIDM) models within the Born regime using perturbative quantum field theory. Six dissipative scenarios are considered, incorporating the emission of particles that may be either massless or possess a kinematically allowed light mass. Both short-range and long-range force-mediated dSIDM interactions are examined. In the non-relativistic regime, we obtain closed-form expressions of the energy-differential cross sections by a controlled expansion in the initial relative dark matter velocity. Up to trivial factors, the leading-order squared emission amplitude is model-independent for massless emissions. Model dependence arises for massive particle emission and at the next-to-leading order. The latter reduces to three distinct cases. The derived analytical expressions exhibit excellent agreement with numerical computations, providing simple, ready-to-use formulas. Furthermore, we analyze the behavior of these processes in the soft emission limit. Our results show that additional corrections are necessary when applying factorization at the next-to-leading order in a velocity expansion to ensure consistency between the soft energy-differential cross section and the full counterparts across a broad energy range. Finally, we investigate the regime of perturbative validity in terms of the model parameters, identifying the conditions under which our results are applicable.

Figures (9)

• Figure 1: Emission diagrams for dissipative DM self-scattering. The orange dots show other possible emission points. Depending on the type of diagram, we introduce the following terminology: (a) "External leg diagrams" for emission from external legs producing four diagrams for distinguishable and eight diagrams (including crossed channels) for identical particles. These diagrams are relevant for all considered models. (b) "Vertex diagrams" for scalar DM candidates $S$ and $\tilde{S}$, where emission occurs at the mediator-exchange vertex. They constitute two diagrams for distinguishable particles and, including crossed channels, four diagrams for identical particles. (c) "Contact diagrams" due to quartic scalar DM interactions; there are four diagrams of this type. (d) "Virtual Internal Bremsstrahlung" (VIB) from a scalar mediator with scalar emission of $\phi$ particles, in principle relevant to all DM candidates; they lead to one (two) diagram(s) in the case of distinguishable (identical) particle scattering. The emitted particle $\phi$ or $V^{\mu}$ carries four-momentum $q$; the four-momentum assignment of propagators is emission-point dependent, and the full kinematic dependency is taken into account when evaluating the various amplitudes.
• Figure 2: Energy-differential cross section for dipole emission in distinguishable particle scattering. The solid brown lines correspond to massless emission. The dotted colored lines show massive emission with $m_{\phi,V}/\omega_{\text{max}} = 0.01$ (leftmost) and $m_{\phi,V}/\omega_{\text{max}} =0.1$ (rightmost). The black stars indicate values of the energy-differential cross section obtained from a fully numerical integration of unexpanded amplitudes. Left panel: Dipole, long-range mediation \ref{['dipolelongrange']}. Numerical integration parameters: $m_{2}=1$ MeV, $m_{1}=1$ GeV, and $v_{i} \approx 4 \times 10^{-3}$. Right panel: Dipole, short-range mediation \ref{['dipoleshortrange']}. Numerical integration parameters: $m_{2}=1$ MeV, $m_{1}=1$ GeV, $m_{\phi',V^{\prime}}=0.1$ GeV and $v_{i} \approx 4 \times 10^{-3}$.
• Figure 3: Quadrupole energy-differential cross section for the different dSIDM scenarios with long-range mediation. The massless emission limits are shown as solid colored lines, while the dotted colored lines represent massive emission with $m_{\phi,V}/\omega_{\text{max}} = 0.1$. The black stars indicate values of the energy-differential cross section obtained from a fully numerical integration of unexpanded amplitudes. The set of couplings $\{a, a_j\}$ depends on the dSIDM scenario: i) for vector emission, $\{g, g_j\}$; ii) for scalar emission with $\chi (\tilde{\chi})$, $\{y, y_j\}$; iii) for scalar emission with $S (\tilde{S})$, iii.a) in the limit ($(A/m_{S})^2 \gg \lambda_j$), $\{A/m_{S}, A_j/m_j\}$iii.b) in the limit ($(A/m_{S})^2\ll\lambda_j$), $(a/m_{\chi,S})^2a^{2}_{1}a_{2}^2 \to ((A_{2}/m_{2})\lambda_{1}/m_{1}+(A_{1}/m_{1})\lambda_{2}/m_{2})^2$ and $a^{6}\to (A/m_{S})^2\lambda^2$. In the numerical analysis, these two limits were implemented by setting $(A_{j}/m_j)^2 = 100\,\lambda_j$ and $(A_{j}/m_j)^2 = 0.01\,\lambda_j$, respectively. Left panel: Distinguishable particle scattering with massless (massive) emission derived in Sec. \ref{['quadrudisting']} (App. \ref{['Massiveemission']}) with equal coupling-to-mass ratio. Numerical integration parameters: $m_{1}=0.1$ GeV, $m_{2}=1$ GeV, $a_{1}=0.1\,a_{2}$ and $v_{i} \approx 4 \times 10^{-3}$. Right panel: Identical particle scattering (Sec. \ref{['identicalparticles']}). Numerical integration parameters: $m_{S,\chi} = 1$ GeV and $v_{i}\approx6 \times 10^{-3}$.
• Figure 4: Quadrupole energy-differential cross section for the different dSIDM scenarios mediated by a short-range force. The massless emission limits are shown as solid colored lines, while the dotted colored lines represent massive emission with $m_{\phi,V}/\omega_{\text{max}} = 0.1$. The black stars indicate values of the energy-differential cross section obtained from a fully numerical integration of unexpanded amplitudes. The relevant coupling combinations $\{a, a', a_j' \}$ vary by scenario: i) for $\chi\chi \to \chi\chi V$, $\{g, g', g_j' \}$; ii) for $SS \to SS V$, $\{g, C^{1/2}_{g',\lambda_S}, C^{1/2}_{g_j',\lambda_S} \}$; iii) for scalar emission from $\chi$($\tilde{\chi}$), $\{y, y', y_j' \}$; iv) for scalar emission with $S$($\tilde{S}$) iv.a) in the external-leg regime (see App. \ref{['scalarscalarquad']}), $\{A/m_S, C^{1/2}_{A^{\prime},\lambda_S}, C^{1/2}_{A^{\prime}_j,\lambda_S} \}$, iv.b) in the vertex-dominated regime (see App. \ref{['scalarscalarquad']}), $(a/m_{S,\chi})^2a^{\prime2}_{1}a^{\prime2}_{2}\to ((A'_{2}/m_{2})\lambda'_{1}/m_{1}+(A'_{1}/m_{1})\lambda'_{2}/m_{2})^2$. Left panel: Distinguishable particle scattering (massless emission in Sec. \ref{['quadrudisting']} and massive emission in App. \ref{['Massiveemission']}) with equal coupling-to-mass ratio $a/m$. Numerical integration parameters: $m_{1}=0.1$ GeV, $m_{2}=1$ GeV, $a_{1}=0.1\,a_{2}$, $v_{i} \approx 4 \times 10^{-3}$, and $m_{\phi',V'}=0.1$ GeV. Right panel: Identical particle scattering (Sec. \ref{['identicalparticles']}). Numerical integration parameters: $m_{S,\chi} = 1$ GeV, $v_{i} \approx 6 \times 10^{-3}$, and $m_{\phi',V'}=0.1$ GeV. Additionally, we assume $\lambda_S \sim g^{\prime}_j,A^{\prime}_{j}/m_{j}$.
• Figure 5: Dipole $(n=1)$ and quadrupole $(n=2)$ energy-differential cross sections for distinguishable particle scattering. Solid lines are the massless limits of our analytical energy-differential cross section expressions; dotted lines show the associated soft emission limits. A separation scale between "hard" and "soft" emission at half the available CM energy $\Lambda \sim \mu v_i^2/4$ is introduced; soft modes ($\omega \lesssim \Lambda$) are shaded light purple, hard modes ($\omega \gtrsim \Lambda$) light blue. Left: short-range mediation. The brown line shows the dipole result \ref{['dipoleshortrange']}, while the brown dotted line shows its soft limit \ref{['softdipole:short']}. Quadrupole lines follow \ref{['shortrangequad']} (full) and \ref{['softquadheay']} (soft). Right: long-range mediation. The brown line shows the dipole result \ref{['dipolelongrange']}, and the corresponding dotted line shows its soft limit \ref{['softdipole:long']}. For quadrupole emission, solid colored lines show the full results \ref{['crosssectionquadrupole']}, and the dotted colored lines the soft limit \ref{['softquadlight']}.