---
title: Semi-Annihilation in Dark Matter
url: https://www.emergentmind.com/topics/semi-annihilation
type: topic
---

# Semi-Annihilation in Dark Matter

Semi-annihilation is a dark-matter number-changing process in which two stable dark-sector particles react to produce one stable dark-sector particle plus an unstable state or a Standard Model particle, schematically \(\psi_i\psi_j\to\psi_k\phi\) or \(\chi\chi\to\chi X\). Unlike ordinary annihilation, which removes two dark particles from the thermal bath, semi-annihilation changes the total dark-matter number by one unit. It is forbidden in the standard \(\mathbb Z_2\)-stabilized WIMP setup but becomes allowed when the stabilizing symmetry is larger than \(\mathbb Z_2\), notably \(\mathbb Z_3\), \(\mathbb Z_4\), or more general hidden-sector “baryon” and “flavor” symmetries [1003.5912, 1101.5413].

## 1. Definition, symmetry origin, and distinction from related processes

The defining reaction is
\[
\psi_i\psi_j\to\psi_k\phi,
\]
with \(\psi_i,\psi_j,\psi_k\) stable dark-sector states and \(\phi\) an unstable state, either a Standard Model particle or a mediator that later decays to the Standard Model. In ordinary annihilation, by contrast, two dark particles disappear into non-dark final states,
\[
\psi\bar\psi\to \text{SM SM},
\]
while decay involves one unstable particle,
\[
\psi\to \text{lighter states}.
\]
Semi-annihilation is also distinct from conversion processes such as \(\psi_i\psi_j\to\psi_k\psi_m\), which reshuffle species without necessarily reducing the total dark-particle number by one, and from coannihilation in the usual Griest–Seckel sense, which still removes two dark-sector particles into visible states [1003.5912, 1101.5413].

Its symmetry origin is central. Under a simple \(\mathbb Z_2\), any allowed interaction contains an even number of dark fields, so processes with an odd number of external dark-sector legs are forbidden. Larger stabilizing symmetries permit them. The simplest example is a single-species \(\mathbb Z_3\) model, in which
\[
\chi\chi\to \bar\chi \phi
\]
is symmetry-allowed while \(\chi\) remains stable. More generally, semi-annihilation arises naturally in multicomponent sectors with conserved quantum numbers analogous to baryon number or flavor, including QCD-like hidden sectors and models of non-Abelian gauge-boson dark matter [1101.5413, 1202.2962].

Kinematic consistency requires that the process not open crossed decays of the stable states. The basic condition is
\[
m_k<m_i+m_j,
\]
together with crossed-channel analogues. This is the mechanism by which semi-annihilation can be present while all dark-sector states remain cosmologically stable [1210.7817, 1003.5912].

The early literature explicitly treated semi-annihilation as a distinct extension of standard relic-density lore. In that formulation, it was compared with the classic “exceptions” to the simplest freeze-out picture and described as a kind of “fourth exception” because it modifies both the Boltzmann structure and the indirect-detection phenomenology [1101.5413].

## 2. Boltzmann dynamics and freeze-out

For a single \(\mathbb Z_3\)-stabilized complex scalar \(\chi\), the number density obeys
\[
\frac{d n_{\chi}}{d t} + 3 H n_{\chi} = - \langle \sigma v \rangle_{\chi\bar{\chi}\rightarrow\phi\phi} \left[n_{\chi}^2 - (n_{\chi}^{\rm eq})^2\right] - \frac{1}{2} \langle \sigma v \rangle_{\chi\chi\rightarrow\bar{\chi}\phi} \left[n_{\chi}^2 - n_{\chi} n_{\chi}^{\rm eq}\right].
\]
The factor of \(1/2\) is characteristic: each semi-annihilation removes only one net dark particle. In the same setup, the relic density depends on the effective depletion combination
\[
\langle \sigma v \rangle_{\rm eff}\sim \langle \sigma v \rangle_{\chi\bar\chi\to\phi\phi}+\frac12\langle \sigma v \rangle_{\chi\chi\to\bar\chi\phi},
\]
so thermal production can be completely controlled by semi-annihilation [1101.5413].

In scalar \(Z_N\) models with \(N>2\), the same structure reappears in abundance form. Writing \(Y=n/s\), one may define
\[
\sigma_v \equiv  \langle v\sigma^{x x^* \rightarrow XX} \rangle + \frac{1}{2} \langle v \sigma^{xx\rightarrow x^* X} \rangle,
\qquad
\alpha=\frac{\frac{1}{2}\sigma_v^{x x\rightarrow x^* X}}{\sigma_v},
\]
so that
\[
\frac{dY}{dt}=-s \sigma_v \left(Y^2-\alpha Y \overline{Y} -(1-\alpha) \overline{Y}^2 \right).
\]
This modifies the freeze-out condition itself: the paper emphasizes that decoupling begins earlier and ends later than in the standard annihilation-only case [1202.2962].

In genuine multicomponent sectors there is generally no reduction to a single effective Lee–Weinberg equation. The full coupled Boltzmann system must be solved numerically because semi-annihilation competes with ordinary annihilation, species conversion, and, where relevant, dark-partner decays. An early numerical result was that semi-annihilation can remain efficient in regions where conversion is phase-space suppressed, so it is not merely another name for inter-species conversion [1003.5912, 1101.5413].

Later model studies made the same point in concrete settings. In the Majoron-coupled \(\mathbb Z_3\) scalar model, the dominant process is
\[
\chi\chi\to\bar\chi J,
\]
with
\[
\sigma_{\chi\chi}v_{\rm rel}\approx\frac{3\lambda^2}{128\pi m_\chi^2}
\]
for \(m_J\ll m_\chi\). There the total density obeys
\[
\frac{dn}{dt}+3Hn=-\frac{\langle\sigma_{\chi\chi}v_{\rm rel}\rangle}{4}\left(n^2-n n^\mathrm{eq}\right),
\]
and reproducing \(\Omega h^2\simeq 0.12\) requires roughly
\[
\sigma_{\chi\chi}v_\mathrm{rel}\sim 2\times10^{-25}~\mathrm{cm^3/s}.
\]
The relic abundance is therefore set by freeze-out through semi-annihilation rather than by ordinary annihilation [2201.05412].

A systematic model-building conclusion emerged for inert scalar multiplets: with one inert multiplet, efficient renormalizable semi-annihilation is not viable; with two multiplets, semi-annihilation can be efficient, but only a narrow class of technically natural models survives, centered on the \(\mathbb Z_3\) configuration with \(Y_1=0\), \(Y_2=-1/2\), and \(|n_1-n_2|=1\) [2403.01729].

## 3. Kinematics and indirect-detection signatures

Semi-annihilation changes not only the freeze-out equation but also the observable kinematics. For monochromatic gamma rays from
\[
\psi_i\psi_j\to\psi_k\gamma,
\]
the photon energy is
\[
E^{ij\to k}_\gamma = \frac{(m_i+m_j)^2-m_k^2}{2(m_i+m_j)}.
\]
This differs from ordinary annihilation into \(\gamma\gamma\), for which \(E_\gamma^i=m_i\). In the degenerate limit \(m_i=m_j=m_k=m_\psi\),
\[
E_\gamma^{\psi\psi\psi}=\frac34 m_\psi,
\]
so a \(130\) GeV semi-annihilation line implies
\[
m_\psi=\frac43\times 130~{\rm GeV}\simeq 173~{\rm GeV},
\]
whereas an ordinary \(\gamma\gamma\) interpretation would point to \(m_\psi=130\) GeV [1210.7817].

The parametric suppression is also different. For neutral dark matter,
\[
\sigma(\bar\psi_i \psi_i \to \gamma \gamma)\propto \alpha_{\rm EM}^2\,\alpha_i^2,
\qquad
\sigma(\psi_i\psi_j \to \psi_k\gamma)\propto \alpha_{\rm EM}\,\alpha_i\alpha_j\alpha_k.
\]
Semi-annihilation into a single photon is therefore enhanced relative to ordinary annihilation into photon pairs by replacing one power of \(\alpha_{\rm EM}\) with a dark-sector coupling [1210.7817].

A distinctive consequence is line multiplicity. With \(N\) dark species, ordinary annihilation gives \(N\) possible line energies through \(E_\gamma^i=m_i\), while semi-annihilation allows one line for each allowed \(ij\to k\) channel, up to
\[
\frac{N(N-1)(N-2)}{2},
\]
that is, parametrically \(\mathcal O(N^3)\). This was proposed as “dark sector spectroscopy.” In the simplest degenerate case, the identified smoking-gun signature is a strong \(130\) GeV semi-annihilation line accompanied by a weaker annihilation line at \(173\) GeV [1210.7817].

Semi-annihilation can also produce correlated boosted-dark-matter signals. In the solar process
\[
\chi\chi\to\bar\chi\nu,
\]
nonrelativistic initial states give
\[
E_\nu=\frac34 m_\chi,
\qquad
E_{\bar\chi}=\frac54 m_\chi,
\]
so the total flux from the Sun contains two narrow spectral features near the dark-matter mass. This “double peak” structure was identified as a distinctive signature for future large-volume neutrino detectors such as DUNE and Hyper-Kamiokande [2109.05911].

## 4. Representative model realizations

Semi-annihilation is realized in a wide range of ultraviolet and effective constructions. The following examples recur across the literature.

| Framework | Characteristic channel | Distinctive feature |
|---|---|---|
| \(\mathbb Z_3\) scalar portal | \(\chi\chi\to\bar\chi\phi\) | Simplest one-species realization |
| Gamma-line semi-annihilation | \(\psi_i\psi_j\to\psi_k\gamma\) | Multiple gamma lines and dark-sector spectroscopy |
| \(\mathbb Z_4\) scalar-plus-wino sector | \(\Psi\Psi\to\phi\,{\rm SM}\), \(\Psi\phi\to\Psi\,{\rm SM}\) | Semi-annihilation plus Sommerfeld dynamics |
| Majoron-coupled scalar DM | \(\chi\chi\to\bar\chi J\) | Halo self-heating and box-shaped neutrino spectrum |
| Two inert electroweak multiplets | \(\phi_1\phi_1\to \phi_2^\dagger H^\dagger\) | Only one technically natural class clearly survives |
| Topological freeze-out | \(\chi\chi\to\chi X_\mu\) | Gauged Skyrme current and purely \(p\)-wave semi-annihilation |

The \(2012\) gamma-line study constructed two explicit models. One was a non-Abelian vector dark-matter model with messenger fermions, in which the low-energy interaction is a non-Abelian Euler–Heisenberg-type operator and the leading semi-annihilation process \(VV\to V\gamma\) arises from box diagrams. The other was a retrofitted Rayleigh dark-matter model in which adding a dark vector \(Z'\) opens the semi-annihilation-like channel
\[
\chi\chi\to\gamma Z',
\qquad
M_\chi<M_{Z'}<2M_\chi,
\]
thereby reproducing a \(130\) GeV line with heavier dark matter and smaller couplings than the original annihilating RayDM setup [1210.7817].

A minimal gauge-charged fermionic realization is the \(\mathbb Z_4\)-symmetric scalar singlet plus wino-like \(SU(2)_L\) triplet. There the dark sector contains a real scalar \(\phi\) and a Dirac fermion triplet \(\psi\), with semi-annihilation channels
\[
\Psi\Psi\to \phi\,{\rm SM},
\qquad
\Psi\phi\to \Psi\,{\rm SM},
\]
and conversion
\[
\Psi\Psi\to\phi\phi.
\]
This model was used to show that semi-annihilation and dark-matter exchange can deplete the fermion relic density enough to allow \(m_\psi\gtrsim 3\) TeV, a region excluded for a pure wino [1510.02179].

The systematic inert-multiplet analysis reached a narrower conclusion. One inert multiplet never yields efficient renormalizable semi-annihilation. With two multiplets, the favored model is the \(\mathbb Z_3\) case with \(\phi_1\) an odd-dimensional \(Y=0\) multiplet, \(\phi_2\) an even-dimensional \(Y=-1/2\) multiplet, and \(|n_1-n_2|=1\). In that setup the unsuppressed renormalizable operator
\[
\lambda_1\,\phi_1^2\phi_2 H+\text{h.c.}
\]
drives the dominant Higgs-emission semi-annihilation channels [2403.01729].

Recent work has also embedded semi-annihilation into confining and neutrino-mass models. In “Topological Freeze-out by Semi-Annihilation,” gauging dark baryon number in a QCD-like dark sector produces the low-energy topological interaction
\[
\frac{e_B}{12\pi^2 f_\chi^3}\epsilon^{\mu\nu\rho\sigma}f^{abc}X_\mu \partial_\nu\chi^a\partial_\rho\chi^b\partial_\sigma\chi^c,
\]
which induces
\[
\chi\chi\to\chi X_\mu
\]
and dominates freeze-out [2506.05468]. In a \(2026\) radiative neutrino-mass model, a Dirac fermion \(\psi\) semi-annihilates through
\[
\psi\psi\to\overline\psi\,\nu_\alpha,
\]
with the same couplings entering a two-loop neutrino-mass diagram; the phenomenology favors an \(\mathcal O(1)\) MeV mediator and proton elastic-scattering cross sections of \(\mathcal O(10^{-36})~\mathrm{cm}^2\) for boosted-dark-matter searches [2606.02751].

## 5. Thermal, halo, and cosmological consequences

Semi-annihilation can continue to affect the dark sector after chemical freeze-out because it injects kinetic energy into the surviving dark particle. In the self-heating scenario based on
\[
\chi\chi\to\chi\phi,
\]
semi-annihilation alone can maintain kinetic equilibrium until nearly the end of freeze-out, and after freeze-out the dark-matter temperature scales as
\[
T_\chi\propto \frac{1}{a}
\]
as long as self-scattering remains efficient, rather than the standard nonrelativistic scaling \(T_\chi\propto a^{-2}\). This was proposed as a mechanism that suppresses structure formation at subgalactic scales like keV warm dark matter but with GeV-scale self-heating dark matter [1707.09238].

The Majoron-coupled \(\mathbb Z_3\) model developed this idea in a concrete particle-physics setting. There the process
\[
\chi\chi\to\bar\chi J
\]
injects recoil energy into the halo, and with only modest elastic self-interaction,
\[
\sigma_{\rm self}/m_\chi \sim 10^{-3}\,\mathrm{cm^2/g},
\]
can induce halo core formation. The same paper stresses that this mechanism is expected to be more effective in dwarf-sized halos than in larger halos on the same timescale [2201.05412].

Semi-annihilation can also create distinctive neutrino spectra through on-shell mediators. In the same Majoron framework, \(J\to\nu\nu\) yields a box-shaped neutrino spectrum because the Majoron is produced on shell and boosted. Hyper-Kamiokande can probe the relevant signal for light dark matter, roughly in the range
\[
25~\mathrm{MeV}\lesssim m_\chi\lesssim 35~\mathrm{MeV},
\]
and with a boost factor of \(10\) in the present-day semi-annihilation rate the reach can extend up to \(m_\chi\sim 200\) MeV [2201.05412].

Resonant enhancement adds another layer of cosmological structure. In models with an \(s\)-channel resonance near threshold, the late-time semi-annihilation signal can be enhanced by up to five orders of magnitude over the thermal relic cross section. The relic density then depends sensitively on the dark-matter temperature evolution, and self-heating allows number-changing processes to remain effective long after kinetic decoupling of the dark and visible sectors [1807.00832].

Not all semi-annihilation mechanisms share this behavior. In the topological freeze-out scenario, the process
\[
\chi\chi\to\chi X_\mu
\]
is purely \(p\)-wave. That removes the usual late-time indirect-detection problem: the relic-setting channel is velocity suppressed in the present universe while still efficient during freeze-out [2506.05468].

## 6. Effective-operator systematics and search constraints

A model-independent effective-operator analysis of \(2\to2\) semi-annihilation up to dimension \(6\), plus leading dimension-\(7\) terms, found that the dark-matter-only theory space is highly constrained. Under the assumptions of gauge-singlet scalar and/or fermion dark matter, there are \(15\) operators in total when only dark matter is light, and only \(2\) for single-component dark sectors. Once light unstable dark partners are included, the operator basis becomes much larger and all Standard Model final states become possible [1611.09360].

That same analysis emphasized a structural phenomenological point: semi-annihilation contributes to thermal freeze-out but is largely irrelevant for direct detection and collider searches in the dark-matter-only EFT, so the irreducible probes are indirect detection and astrophysical observations. For semi-annihilation to electrons and light quarks, the thermal relic cross sections can be excluded up to about \(m_{\rm DM}\sim 100\) GeV; for \(\tau\) final states the exclusion reaches roughly \(50\) GeV; for Higgs, gauge-boson, and neutrino final states the limits are generally weaker than the thermal relic contour except near threshold [1611.09360].

Light semi-annihilating dark matter in the MeV–GeV range is constrained by diffuse X-ray and gamma-ray observations. In the \(Z_3\) scalar model with
\[
SS\to S^*\phi,
\]
current data from COMPTEL, EGRET, INTEGRAL, and Fermi Gamma-ray Space Telescope, together with the projected e-ASTROGAM reach, were translated into bounds on the semi-annihilation cross section in the range
\[
10^{-28}\,\mathrm{cm}^3/\mathrm{s}\ \text{to}\ 10^{-22}\,\mathrm{cm}^3/\mathrm{s},
\]
depending on \(m_S\) and \(m_\phi\). EGRET provides the strongest current constraint in that analysis, while e-ASTROGAM could probe the whole parameter space studied [2302.06159].

For inert scalar multiplets, the indirect-detection picture is highly representation dependent. The dedicated analysis of semi-annihilation in these models found that all studied gauge combinations can reproduce the relic density, but for all cases except \((n_1,n_2)=(3,4)\), HESS excludes thermal relic solutions for cuspy Galactic profiles unless the profile contains a sufficiently large core. The exceptional \((3,4)\) model remains viable even for very cuspy halos because its semi-annihilation channel is Sommerfeld-suppressed rather than enhanced [2407.01096].

This suggests a broad contemporary picture. Semi-annihilation is no longer treated merely as a discrete-symmetry curiosity; it functions as a general organizing principle for dark sectors whose stabilization symmetries exceed \(\mathbb Z_2\). Its characteristic signatures include modified Boltzmann equations, kinematic decoupling of line energy from dark-matter mass, boosted-dark-matter final states, mediator-induced box spectra, and representation-dependent Sommerfeld behavior. At the same time, the surviving viable parameter space is strongly conditioned by the symmetry structure, the mediator spectrum, and the velocity dependence of the semi-annihilation channel itself [1210.7817, 1611.09360].

Source: https://www.emergentmind.com/topics/semi-annihilation