--- title: D-wave Annihilation in Dark Matter url: https://www.emergentmind.com/topics/d-wave-annihilation type: topic --- # D-wave Annihilation in Dark Matter Searching arXiv for recent papers on d-wave annihilation and related usages of the term. {"query":"all:(\"d-wave annihilation\" OR \"d wave annihilation\" OR \"velocity-dependent annihilation\" dark matter d-wave)","max_results":10,"sort_by":"relevance"} {"query":"ti:\"No evidence for p- or d-wave dark matter annihilation from local large-scale structure\" OR ti:\"Enhancements in velocity-dependent dark matter annihilation in Galactic subhalos\" OR ti:\"Effective J-factors for Milky Way dwarf spheroidal galaxies with velocity-dependent annihilation\"","max_results":10,"sort_by":"relevance"} D-wave annihilation most commonly denotes an annihilation channel whose nonrelativistic cross section is dominated by the partial wave with orbital angular momentum \(\ell=2\). In the velocity expansion used in indirect dark-matter phenomenology, this is written as \(\sigma v = a_\ell (v/c)^{2\ell}\), so the d-wave case obeys \(\sigma v = a_2 (v/c)^4\). The defining consequence is a very strong weighting toward environments with large relative velocities: signals are suppressed in dynamically cold systems such as dwarf spheroidals and are enhanced in massive halos, the smooth Galactic halo, and perturbed regions with an enlarged high-speed tail [2304.10301; 1909.13197; 2203.08853]. The expression also appears in other areas of physics with different meanings, including d-wave cold-atom scattering, annihilation of d-wave gap nodes in superconductors, and weak-annihilation topologies in heavy-flavor decays; those usages are conceptually distinct [1205.0644; 1608.05840; 1811.00392]. ## 1. Formal definition and astrophysical signal formalism For velocity-dependent annihilation models, the basic parametrization used in several dark-matter analyses is \[ \sigma_A v_{\rm rel} = (\sigma_A v_{\rm rel})_0 \left(\frac{v_{\rm rel}}{c}\right)^n, \] with \(n=4\) for d-wave annihilation, or equivalently \[ \sigma v = a_\ell \left(\frac{v}{c}\right)^{2\ell}, \] with \(\ell=2\) [2509.05519; 2304.10301]. In this notation, \(a_2\) is the velocity-independent normalization multiplying \((v/c)^4\) [2304.10301]. The formalism immediately shows why d-wave annihilation differs sharply from \(s\)-wave, \(p\)-wave, or Sommerfeld-enhanced cases: the rate is strongly weighted toward the high-velocity tail of the pairwise relative-velocity distribution. Because the cross section depends explicitly on velocity, the usual density-squared \(J\)-factor is not sufficient. The relevant quantity is an effective or generalized \(J\)-factor that folds the local velocity distribution into the line-of-sight integral. One form used for subhalos and the Milky Way halo is \[ {\mathcal J}(\theta)=\int d\ell \int d^3{\bf v}_{\rm rel}\, P_{\bf x}({\bf v}_{\rm rel}) \left(\frac{v_{\rm rel}}{c}\right)^n [\rho(r(\ell,\theta))]^2, \] while a moment-based form introduces \[ \mu_n(x)=\int d^3v_{\rm rel}\, P_{\bf x}(v_{\rm rel})\, v_{\rm rel}^n, \] so that the d-wave signal is governed by \(\mu_4/c^4\) [2509.13540]. In the dwarf-spheroidal literature, the same idea appears as an effective \(J\)-factor, \[ J_S(\theta)=\int d\ell \int d^3v_1\, d^3v_2\, S(|\boldsymbol v_1-\boldsymbol v_2|/c)\, f(r,v_1)\, f(r,v_2), \] with \(S(v/c)=(v/c)^4\) for d-wave annihilation [1909.13197]. The corresponding gamma-ray flux retains the standard factorized structure once the generalized \(J\)-factor is defined: \[ \frac{d\Phi_\gamma}{dE}=\frac{(\sigma_A v_{\rm rel})_0}{8\pi m_{\rm DM}^2}\frac{dN_\gamma}{dE}\,\mathcal J. \] What changes is entirely in the astrophysical weighting. This is the central methodological distinction between d-wave and velocity-independent annihilation analyses [2509.13540]. ## 2. Dwarf spheroidal galaxies and the first indirect limits The first systematic effective-\(J\) treatment of d-wave annihilation for Milky Way dwarf spheroidal galaxies computed effective \(J\)-factors for 25 dSphs under an NFW density profile, Eddington inversion for the dark-matter velocity distribution, and a spherical Jeans analysis of stellar kinematics [1909.13197]. The halo parameters \(\rho_s\), \(r_s\), and \(D\) were inferred using a Plummer stellar density profile, constant stellar anisotropy, Gaussian priors on the half-light radius, ellipticity, and distance, and Jeffreys priors on the halo parameters. The gamma-ray limits were then obtained with the MADHAT framework using nearly 11 years of Fermi-LAT Pass 8R3 data in the 1–100 GeV range [1909.13197]. For the d-wave case, this analysis explicitly presented the first indirect-detection bound on a \(v^4\)-suppressed annihilation cross section from dwarf gamma-ray data [1909.13197]. Its physical interpretation was equally important: because dwarf spheroidals are low-velocity systems, their effective \(J\)-factors are much smaller than in the \(s\)-wave case, and changing the annihilation model can alter the inferred \(J\)-factor by orders of magnitude. The paper therefore established that the astrophysical ranking of indirect-detection targets is model dependent once velocity weighting is included. This conclusion is reinforced by the scaling relation used in the dwarf analysis, \[ J_{S(n)}(\tilde\theta)=2\rho_s^2r_s\left(\frac{4\pi G_N\rho_sr_s^2}{c^2}\right)^{n/2}\tilde J_{S(n)}(\tilde\theta), \] which for \(n=4\) adds two more powers of the characteristic velocity scale than the \(p\)-wave case and four more than the \(s\)-wave case [1909.13197]. A direct implication is that dense but dynamically cold systems can be much less constraining for d-wave models than for velocity-independent annihilation. ## 3. Local large-scale structure as the dominant target class A major revision of the target hierarchy came from a full-sky search for velocity-dependent annihilation in local large-scale structure, based on the CSiBORG suite of 101 constrained \(N\)-body simulations derived from the BORG Bayesian reconstruction of the 2M++ galaxy catalogue [2304.10301]. The analysis constructed gamma-ray templates for galaxy- and cluster-mass halos within \(\sim 200\) Mpc, resolved the local halo field out to \(155\,h^{-1}\) Mpc, and compared the resulting templates with Fermi-LAT data from mission weeks 9–634 in the 500 MeV–50 GeV range, using PS3 SOURCEVETO events, 9 energy bins, HEALPix maps with \(nside=256\), and a Galactic-plane mask \(|\lambda|<30^\circ\) [2304.10301]. The d-wave-specific astrophysical factor in that work required the fourth velocity moment, \[ J^{(\ell=2)}_{\mu,p} = \int_{\mu,p} d\Omega\, d\mu\, \rho^2(\mathbf r)\left(2\left<\frac{\mathbf v^4}{c^4}\right>(\mathbf r)+\frac{10}{3}\left<\frac{\mathbf v^2}{c^2}\right>^2(\mathbf r)\right), \] implemented by assuming an ergodic distribution function with \(\beta=0\) and an NFW profile, and then coding the analytic result in CLUMPY [2304.10301]. The likelihood marginalized both over reconstruction uncertainties by averaging across all 101 CSiBORG realizations and over the non-dark-matter templates, namely isotropic background, Galactic diffuse emission, and point sources. The result was a null detection: the inferred template amplitudes were consistent with zero in every energy bin, and there was no evidence for d-wave annihilation for any channel over the mass range \(m_\chi = 2\)–\(500\,\mathrm{GeV}/c^2\) [2304.10301]. For the benchmark \(b\bar b\) channel at \(m_\chi=10\,\mathrm{GeV}/c^2\), the bound was \[ a_2 < 3.0\times 10^{-18}\,\mathrm{cm^3\,s^{-1}} \quad\text{at 95\% confidence}, \] and the paper concluded that the d-wave limits are about seven orders of magnitude tighter than dwarf-spheroidal limits [2304.10301]. The analysis further found that the constraints are dominated by the most massive halos, roughly \(M_h\sim 10^{14-16}\), because d-wave annihilation benefits from the large velocity dispersions of cluster-mass objects. The same work estimated the thermal-relic coefficient for d-wave annihilation as \(a_2\sim (4\text{–}9)\times10^{-25}\,\mathrm{cm^3\,s^{-1}}\), so the observational upper limits still do not exclude a thermal relic with d-wave annihilation [2304.10301]. The resulting observational hierarchy is summarized below. | Target class | Analysis | D-wave outcome | |---|---|---| | 25 Milky Way dSphs | Effective \(J\)-factors + Fermi-LAT Pass 8R3 | First indirect-detection bounds [1909.13197] | | Local large-scale structure within \(\sim 200\) Mpc | CSiBORG + BORG + Fermi-LAT | About seven orders tighter than dSph limits; null detection [2304.10301] | | Large Magellanic Cloud | Auriga MW–LMC analogue + 16.57 years Fermi-LAT | \(\sim 4\)–\(6\) orders more stringent than previous dwarf bounds [2509.13540] | ## 4. Smooth halo dominance, subhalo suppression, and host-halo overlap Cosmological simulations of Milky Way-like halos showed that d-wave annihilation is the most strongly smooth-halo-dominated of the commonly studied partial-wave models. Using six Auriga Milky Way analogues, each with hydrodynamical and dark-matter-only realizations, the generalized annihilation luminosity was written as \[ L_n=\int d^3x \int d^3v_{\rm rel}\, P_{\mathbf x}(\mathbf v_{\rm rel})\left(\frac{v_{\rm rel}}{c}\right)^n \rho^2(x), \] with \(n=4\) for d-wave annihilation [2203.08853]. Subhalo velocities were found to be well approximated by a Maxwell-Boltzmann form, and for d-wave emission the relevant factor is the fourth velocity moment \(\mu_4\) [2203.08853]. The main qualitative conclusion was that the smooth halo dominates the annihilation luminosity at all radii within \(r_{200}\) for \(s\)-wave, \(p\)-wave, and d-wave models, with d-wave subhalo emission being the weakest of all four cases considered [2203.08853]. In the d-wave subhalo luminosity function, the fitted power-law indices lie in the ranges \(a\simeq 1.38\)–\(1.49\) for Auriga hydrodynamical runs and \(a\simeq 1.35\)–\(1.44\) for dark-matter-only runs, implying that the total resolved subhalo luminosity is dominated by the brightest resolved subhalos rather than by a large population of faint objects [2203.08853]. For Au6, the fraction of d-wave subhalo luminosity coming from subhalos above \(10^8\,M_\odot\) is 1.000 in the hydrodynamical run and 0.999 in the dark-matter-only run, so extrapolating to much smaller unresolved masses has little effect [2203.08853]. A later Auriga-based study refined this picture by including the contribution of unbound dark-matter particles from the smooth Galactic halo that spatially overlap with subhalos [2509.05519]. The enhancement factor was defined as \(\mathcal J_{\rm BUB}/\mathcal J_{\rm B}\), comparing bound-plus-unbound to bound-only emission, and the maximal enhancement factor for d-wave models was found to be approximately \(37{,}000\) [2509.05519]. Across six Milky Way-like hosts, about 6 d-wave subhalos were above the smooth foreground when both bound and unbound particles were included, compared with only 3 when only bound particles were used [2509.05519]. The strongest boosts occurred in lower-mass subhalos closer to the Galactic center; representative examples include Au16, 9 with d-wave boost \(738.4\), Au16, 140 with d-wave boost \(13334.1\), Au24, 221 with d-wave boost \(3285.0\), and Au23, 7 with d-wave boost \(165.2\) [2509.05519]. These two results concern different observables rather than incompatible global conclusions. The 2022 Auriga study compared the total annihilation flux from subhalos with that from the smooth halo within the virial radius, whereas the 2025 analysis isolated individual subhalos and quantified how host-halo overlap modifies their \(\mathcal J\)-factors [2203.08853; 2509.05519]. A plausible implication is that large line-of-sight boosts for selected inner subhalos need not overturn the broader smooth-halo dominance of the total d-wave luminosity. ## 5. The Large Magellanic Cloud and dynamical reshaping of the Milky Way halo A further extension of the velocity-dependent framework examined the Large Magellanic Cloud using a Milky Way–LMC analogue drawn from the Auriga magneto-hydrodynamical simulations [2509.13540]. The analogue system consisted of Auriga halo 25 / re-simulated halo 13, with an LMC analogue of halo mass at infall \(3.2\times10^{11}\,M_\odot\), present-day-like separation \(\sim 50\) kpc, speed \(\sim 317\) km/s, and a Milky Way virial mass of \(\sim 1.2\times10^{12}\,M_\odot\) [2509.13540]. The d-wave model again took \[ \sigma_A v_{\rm rel} = (\sigma_A v_{\rm rel})_0 \left(\frac{v_{\rm rel}}{c}\right)^4, \] so the signal was especially sensitive to the high-speed component of the distribution. The study found that the gamma-ray signal from the LMC analogue rises above the Milky Way foreground by more than a factor of 100 not only for \(s\)-wave annihilation but also for Sommerfeld, \(p\)-wave, and d-wave models [2509.13540]. Using 16.57 years of Fermi-LAT data in a \(10^\circ\times 10^\circ\) region of interest centered on the LMC, and taking the simulation-derived \(\mathcal J\)-map as the dark-matter template, the authors derived upper limits on the annihilation cross section for all four velocity dependences. For d-wave annihilation, the LMC-derived upper limits were stated to be \(\sim 4\)–\(6\) orders of magnitude more stringent than previous dwarf-galaxy bounds [2509.13540]. The LMC also alters the Milky Way halo itself. According to the simulation analysis, it both brings in its own fast dark-matter particles and gravitationally accelerates native Milky Way halo particles, producing outer-halo overdensities and shifting the relative-speed distribution toward higher speeds [2509.13540]. For the d-wave case, the Milky Way \(\mathcal J\)-factor can be boosted by up to a factor of about 6 in the outer halo [2509.13540]. This matters because d-wave annihilation weights the high-speed tail very strongly. The paper therefore elevated the LMC from a mere target to a dynamical agent that changes the indirect-detection interpretation of the Milky Way outskirts. ## 6. Other meanings of “d-wave” and “annihilation” Outside dark-matter phenomenology, the same words label different physical mechanisms. In ultracold atoms, the relevant topic is control of d-wave scattering rather than particle annihilation. A magneto-optical theory with two strong photoassociation lasers and, optionally, an \(s\)-wave magnetic Feshbach resonance predicts that coherence between two excited ro-vibrational bound states can enhance elastic d-wave scattering and suppress inelastic scattering [1205.0644]. For \(^{174}\)Yb at \(100\,\mu\)K, the paper estimated an inelastic scattering rate reducible to \(20\,\mathrm{s}^{-1}\) and an elastic rate about two orders of magnitude larger [1205.0644]. This usage concerns \(\ell=2\) scattering in a continuum-bound optical control problem, not annihilation into final particles. In iron-based superconductors, “annihilation” refers to the merger and disappearance of displaced Dirac gap nodes in a d-wave superconducting state [1608.05840]. For two \(\Gamma\)-centered hole pockets derived from \(d_{xz}\) and \(d_{yz}\) orbitals, the d-wave order parameter in the orbital basis becomes a mixture of intra-band and inter-band pairing in the band basis, \[ \Delta_a=\Delta\cos 2\theta,\qquad \Delta_b=\Delta\sin 2\theta, \] which shifts the nodal points away from the normal-state Fermi surfaces [1608.05840]. If the two pockets are sufficiently close, the shifted nodes merge at a critical \(\Delta_{\rm cr}\) and annihilate below some \(T