SSemi-Annihilating Wino-Like Dark Matter
A. Spray ∗ University of MelbourneE-mail: [email protected]
Y. Cai
University of MelbourneE-mail: [email protected]
Semi-annihilation is a generic feature of dark matter theories with symmetries larger than Z . Weexplore a model based on a Z -symmetric dark sector comprised of a scalar singlet and a “wino”-like fermion SU ( ) L triplet. This is the minimal example of semi-annihilation with a gauge-charged fermion. We study the interplay of the Sommerfeld effect in both annihilation and semi-annihilation channels. The modifications to the relic density allow otherwise-forbidden regionsof parameter space and can substantially weaken indirect detection constraints. We perform aparameter scan and find that the entire region where the model comprises all the observed darkmatter is accessible to current and planned direct and indirect searches. ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). http://pos.sissa.it/ a r X i v : . [ h e p - ph ] O c t emi-Annihilating Wino-Like Dark Matter A. Spray
1. Introduction
The dark matter (DM) problem remains one of the most important questions in contemporaryparticle physics. Measurements across a range of scales, from galaxy rotation curves to fluctuationsin the cosmic microwave background, all point to the existence of a cold non-baryonic componentof matter in the Universe. However, no unambiguous non-gravitational signal has been found andthe microscopic properties of DM remain unknown. For this reason, it is important to consider asvaried a range of DM phenomenology as possible, to check the effectiveness of planned searches.Semi-annihilation (SA) is a generic feature of dark sector phenomenology that occurs when-ever the symmetry that stabilizes DM is larger than Z [1]. We illustrate it in Figure 1. For theusually-considered case, the only allowed 2 → χχ VV χ χ χ V χ χ χ χ Figure 1:
Three types of dark sector processes, where χ ( V ) is a dark (visible) sector field. (Left): DMannihilation to/from the SM; this is the only process possible when the dark matter is stabilised by a Z sym-metry. (Centre): Semi-annihilation, a non-decay process with an odd number of external visible particles.(Right): DM exchange, only possible when the dark sector is multicomponent. Previous studies of SADM have mostly focused on scalar or vector DM, see e.g. [1–5] (butsee Ref. [6] for an exception). This is because bosons can have renormalisable SA self-couplings, e.g. a cubic term in a Z theory. Equivalently, fermionic examples of SADM necessitate multi-component dark sectors, enriching the phenomenology. The dark sector must include bosons,to avoid an accidental Z forbidding fermionic SA. Thus the minimal model consists of a Diracfermion ψ and a real scalar singlet φ , with respective charges 1 and 2 under a Z symmetry. Unfortunately, ψ has no direct couplings to the SM which impedes the observation of fermionicSA. This leads us to consider a next-to-minimal “wino-like” model, where ψ is an SU ( ) L tripletwith zero hypercharge. There are three physical Dirac fermion states, two charged ( ψ ± ) and oneneutral ( ψ ); loop effects split them by δ m ψ ≡ m ψ ± − m ψ ≈
167 MeV [7]. Analogously with thewino, we expect the Sommerfeld effect (SE) to be important when m ψ ∼ m W / α ∼ . A Z symmetry would require a complex scalar singlet and would introduce additional interaction terms. emi-Annihilating Wino-Like Dark Matter A. Spray
Field Type G SM Z φ Real Scalar (1, 1, 0) 2 ψ ∼ ( ψ + , ψ , ψ − ) Dirac Fermion (1, 3, 0) 1
Table 1:
New particle content for the model we consider in this paper.
We summarise the new particle content in Table 1. The Lagrangian for this theory is L = L SM + ¯ ψ ( iD / − m ψ ) ψ + ( ∂ µ φ ) + ( m φ φ − λ h φ v )+ ( y φ ¯ ψ c ψ + h . c . ) + λ h φ H † H φ + λ φ φ . (1.1)There are five new parameters compared to the SM: the masses of the two dark sector particles m ψ and m φ , the Higgs portal coupling λ h φ , the semi-annihilation coupling y and the new scalarquartic λ φ . The last of these is phenomenologically unimportant, so we effectively have a four-dimensional parameter space. We may take y real and positive without loss of generality. In thelimit when y →
0, this model reduces to the combination of a Dirac wino-like fermion and of ascalar singlet coupled through a Higgs portal [10]. In these proceedings we discuss our study ofthis model from Ref. [11]. In Section 2, we consider the interplay of SA and the SE on the relicdensity and indirect detection. We then apply those results in Section 3 to a scan of the fermiontriplet parameter space.
2. Semi-Annihilation and the Sommerfeld Effect
The two aspects of DM phenomenology directly affected by SA and the SE are the relic den-sity and indirect detection. Both involve processes initiated by two dark sector states (so the SE isimportant) and potentially with dark sector final states (so SA is relevant). Other aspects of phe-nomenology, e.g. production at colliders or direct detection, are either unaffected or only indirectlysensitive through the modification to the relic density.The computation of the relic density in general SA models may be found in e.g.
Ref. [12]. Forour specific model, the coupled Boltzmann equations take the form dY Ψ dx = sZHx (cid:20)(cid:0) Y Ψ − ( Y eq Ψ ) (cid:1) (cid:104) σ v (cid:105) ( ΨΨ → SM ) + (cid:18) Y Ψ − Y φ ( Y eq Ψ ) Y eq φ (cid:19) (cid:104) σ v (cid:105) ( ΨΨ → φ SM )+ (cid:18) Y Ψ − Y φ ( Y eq Ψ ) ( Y eq φ ) (cid:19) (cid:104) σ v (cid:105) ( ΨΨ → φ φ ) (cid:21) , (2.1) dY φ dx = sZHx (cid:20)(cid:0) Y φ − ( Y eq φ ) (cid:1) (cid:104) σ v (cid:105) ( φ φ → SM ) + Y Ψ (cid:0) Y φ − Y eq φ (cid:1) (cid:104) σ v (cid:105) ( Ψ φ → Ψ SM )+ (cid:18) Y φ ( Y eq Ψ ) Y eq φ − Y Ψ (cid:19) (cid:104) σ v (cid:105) ( ΨΨ → φ SM ) + (cid:18) Y φ ( Y eq Ψ ) ( Y eq φ ) − Y Ψ (cid:19) (cid:104) σ v (cid:105) ( ΨΨ → φ φ ) (cid:21) . (2.2)Here, s is the entropy density, H the Hubble rate, x = T / T the inverse temperature, Y i = n i / s theabundance of species i , and Z ≈ Ψ denotes a sum over all fermion and antifermion species. Scattering off the SM maintains ther-mal equilibrium among the different fermion components. This suppresses Y ψ ± when T (cid:46) δ m ψ ,3 emi-Annihilating Wino-Like Dark Matter A. Spray affecting the SE which is relevant at these late times. The second term of Eq. (2.1), and the sec-ond and third terms of Eq. (2.2), are SA; while the last terms of both equations are DME. The SEmodifies all processes with two fermions in the initial state, specifically all terms in Eq. (2.1) andthe second line of Eq. (2.2). We approximate the SE as only applying to the s -wave piece of the(semi-)annihilation cross section, so that (cid:104) σ v (cid:105) = (cid:104) S (cid:105) σ + ( (cid:104) σ v (cid:105) − σ ) , (2.3)with (cid:104) σ v (cid:105) ( σ ) the unenhanced thermally averaged ( s -wave) cross section and (cid:104) S (cid:105) the thermallyaveraged SE factor.In our computation of the SE, we follow the formalism of Ref. [13]. We split the calculationinto independent subgroups by the unbroken quantum numbers: charge Q , angular momentum J = S and Z charge q . For each subspace, we solve a Schrödinger equation for a generally matrix-valued two-particle wavefunction Φ i j : − M Φ (cid:48)(cid:48) i j ( r ) + ∑ k V ik ( r ) Φ k j ( r ) = K Φ i j ( r ) , (2.4)with M the mass, K the centre of momentum frame kinetic energy at large separation, r the sep-aration and V i j ( r ) the long-range potential. The indices label different two-particle states. Thewavefunction satisfies the boundary conditions Φ i j ( ) = δ i j and lim r → ∞ Φ (cid:48) i j ( r ) Φ i j ( r ) = i (cid:112) M ( K − V ii ( ∞ )) (no sum) . (2.5)The enhancement matrix A i j is given by A i j = lim r → ∞ Φ i j ( r ) exp ( i ℜ (cid:112) M ( K − V ii ( ∞ )) r ) , (2.6)such that A i j = δ i j in the absence of the SE. The cross section for each two-body initial state is σ i = c i ( A · Γ · A † ) ii , (2.7)where Γ are annihilation matrices and c i = i contains distinct (identical) particles.The model-dependent elements of this calculation are the matrices V and Γ . Their diagonalentries have simple physical interpretations as the potential energies and annihilation cross sectionsof the associated two-body state. The off-diagonal elements are more opaque, but the formal defi-nitions are in terms of the real ( V ) and imaginary ( Γ ) parts of the generalised two-body propagator ( ΨΨ ) i → ( ΨΨ ) j . Explicit expressions for V and Γ in our model are given in Ref. [11]. We plot thethermally averaged annihilation and SA cross sections in Figure 2. We see that in the absence of theSE, SA effectively vanishes at low temperatures. This is due to the previously discussed thermalsuppression in the abundance of charged fermions when T < δ m ψ . In contrast to ψ ¯ ψ → W + W − ,the ψ ψ state can not semi-anihilate. It follows that the SE is relatively more important to the SAchannel than to annihilation.The calculation of the SE for indirect detection follows a very similar methodology. Themain difference is that all charged fermions have decayed at late times. This means that only the4 emi-Annihilating Wino-Like Dark Matter A. Spray
10 10 m ψ / T σ v ( ψψ → S M )( pb )
10 10 - - m ψ / T σ v ( ψψ → ϕ S M )( pb ) Figure 2:
Thermally-averaged cross sections for annihilation (left) and SA (right). These results are for m ψ = m φ =
158 GeV and values of y as labelled, except for the blue dashed line which shows thecross sections without including the SE. Q = ψ ψ state does not interact means that indirectsignals from SA are suppressed. This is also true for γ -ray lines from SA despite the fact that both ψ ¯ ψ → γγ and ψ ψ → φ γ have no tree-level contribution. Both processes appear at one-loop,but this is not the leading contribution to the annihilation channel: instead, the SE-mediated process ψ ¯ ψ → ψ ± ¯ ψ ∓ → γγ dominates [14]. The equivalent SA process ψ + ψ − → φ γ is a pure spin-1process, but the ψ ψ state can only exist in spin-0 by Fermi statistics. - - - - - m ψ ( GeV ) σ v ( c m s - ) y = = = = - - - - - m ψ ( GeV ) σ v ( c m s - ) y = = = = Figure 3:
Combined annihilation cross sections to WW , ZZ and Z γ (left) and γγ and Z γ (right), for m φ =
200 GeV and different values of y . Note that the resonant peak occurs at smaller values of m ψ as y increases. However, indirect signals do retain some dependence on the coupling y . When m φ (cid:28) m ψ ,the scalar contributes to the potential, changing the position of the resonance. We illustrate thisin Figures 3 and 4, where we plot the annihilation cross sections for different y and m φ . FromFigure 3, we see that as y increases the resonance becomes stronger and moves to smaller fermionmasses, while Figure 4 demonstrates how this effect varies with the scalar mass.
3. Fermion Triplet Phenomenology
We now use the results of Section 2 to explore the parameter space of the fermion triplet model.5 emi-Annihilating Wino-Like Dark Matter
A. Spray m ψ ( GeV ) m ϕ ( G e V ) y = m ψ ( GeV ) m ϕ ( G e V ) y = σ v ( cm s - ) - - - - - - - Figure 4:
Annihilation cross sections for fermions to massive gauge bosons for y = . m ψ as m φ decreases, and this effect isstronger for larger y . We have a strict lower bound m ψ >
480 GeV from LHC searches for disappearing tracks [15; 16].These constrain pp → W ± → ψ ± ¯ ψ followed by the displaced decay ψ ± → ψ π ± . The upperbound on m ψ comes from requiring the total relic density be no more than observations; we find m ψ (cid:46) < m φ <
63 GeV or m φ >
130 GeV. We choose λ h φ = . m φ > m ψ the scalar is unstable and the phenomenology essentially reduces to that of a purefermion triplet stabilised by a Z symmetry.We scan the m φ – m ψ plane of parameter space for y = .
1, 0.5, 0.7 and 1.0. For y = .
1, weexpect SA to be subdominant to fermion and scalar annihilation and the two DM particles to freezeout independently. Increasing y will reveal the effect of SA. We show our results in in Figure 5.The grey shaded region is excluded by a too large total DM relic density: Ω φ + ψ > . Ω cdm , wherewe have allowed for a 10% theoretical uncertainty. Exclusions from LHC, LUX, HESS γ -ray linesfrom the galactic centre [19] and Fermi diffuse photon fluxes [20; 21] are as marked. The HESS andFermi limits are respectively the strongest limits for an (optimistic) NFW [22] and (conservative)cored DM density profile. Prospective limits from LUX [23] (Xenon1T [24], LZ [25]) are shown byblue solid (dashed, dot-dashed) contours; and from CTA [26; 27], assuming an optimistic profile, bythe orange solid contour. Note that all direct detection limits come from scalar scattering mediatedby the Higgs portal; and for indirect detection from fermion annihilation to gauge bosons.We observe that for y = . Ω ψ is a function only of m ψ ; while Ω φ is a function only of m φ , unless m φ > m ψ . This is as expected when SA is negligible; the only effect of y comesfrom the DME process φ φ → ψ ¯ ψ when m φ > m ψ . Figure 6 shows us how this changes as SAbecomes important. For the scalar at large y , we see the relic density drop for m φ (cid:46) m ψ , due to ψφ → ¯ ψ + SM ; while it increases for heavy scalars, due to ψψ → φ + SM . The latter process isenhanced, so it can increase Ω φ by orders of magnitude. For the fermion, when m ψ > m φ / m φ ),SA (DME) depletes the relic density, though less dramatically as the annihilation is also enhanced.6 emi-Annihilating Wino-Like Dark Matter A. Spray m ϕ > m ψ Ω h > / Fermi500 1000 300010030010003000 m ψ ( GeV ) m ϕ ( G e V ) y = m ϕ > m ψ Ω h > / Fermi500 1000 300010030010003000 m ψ ( GeV ) m ϕ ( G e V ) y = m ϕ > m ψ Ω h > / Fermi500 1000 300010030010003000 m ψ ( GeV ) m ϕ ( G e V ) y = m ϕ > m ψ Ω h > / Fermi500 1000 300010030010003000 m ψ ( GeV ) m ϕ ( G e V ) y = Figure 5:
Slices of the fermion triplet parameter space in the m φ – m ψ plane, for λ h φ = . y as labelled.The grey shaded regions show regions where the total relic density is larger than observations. In the whitehatched region, the scalar decays to two fermions. The red (blue, orange) shaded regions and contours showcurrent and future bounds from the LHC (direct detection, indirect searches). See the text for more details. This motivates splitting our analysis of Figure 5 into two regions. When m φ < m ψ , the scalarfreezes out independently of the fermion. For small y , the fermion density is also set independently.The correct relic density occurs when the two components accidentally sum to the observed value.Due to the lower bounds on the DM masses, we find Ω φ ∼ Ω ψ in this case. For large y , SA andDME deplete the fermion abundance and shift the resonance as shown in Figure 4. This allowsfermion masses m ψ (cid:38) y , the full LUX data set canexclude this region as direct detection depends only on Ω φ .7 emi-Annihilating Wino-Like Dark Matter A. Spray
500 1000 30000.000.020.040.060.080.100.120.14 m ψ ( GeV ) Ω ψ h - m ϕ ( GeV ) Ω ϕ h Figure 6:
Fermion (left) and scalar (right) relic densities for different values of y as labelled, and m φ = . m ψ = σ measurement. When m φ (cid:38) m ψ , the correct relic density is typically produced for nearly equal masses. Thisis due to the SA and DME-induced rapid variation in Ω φ visible in Figure 6. Along this line wefind Ω φ ∼ Ω ψ . The exception occurs when m ψ ≈ . Ω ψ (cid:29) Ω φ andthe phenomenology mostly reduces to a pure fermion triplet. The presence of SA and DME allowpoints in the m φ – λ h φ plane that are excluded for a pure scalar singlet. The region in parameter spacewhere our model produces the full DM relic density can be excluded for optimistic DM galacticprofiles. For m ψ (cid:46) . m ψ (cid:38) . m ψ ≈ . m φ (cid:46)
300 GeV and m ψ ∼ y would be difficult. The best chance wouldbe if the fermion was either lighter or heavier than a wino. The former might offer evidence of theshifted position of the resonance, while the latter would point to the need of SA or DME to depletethe fermion relic abundance. In other situations, we would likely only observe one DM particlefor some time. The evidence for the existence of a second state would be difficulty in reconcilingthe measured DM properties with the observed relic abundance. However, direct evidence of bothDM states from a 100 TeV collider would probably be required to rule out alternative explanations,such as non-thermal production. Acknowledgments
We thank F. Gao and M. A. Schmidt for valuable discussions. YC and AS were supportedby the Australian Research Council. This work was supported by IBS under the project code,IBS-R018-D1. 8 emi-Annihilating Wino-Like Dark Matter
A. Spray
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