Indirect Searches

Abstract

There exist several ways of searching for dark matter in earth-bound or satellite experiments. All of them rely on the interaction of the dark matter particle with matter, which means they only work if the dark matter particles interacts more than only gravitationally. This is the main assumption of these lecture notes, and it is motivated by the fact that the weak gauge coupling and the weak mass scale happen to predict roughly the correct relic density, as described in Sect. 3.1.

There exist several ways of searching for dark matter in earth-bound or satellite experiments. All of them rely on the interaction of the dark matter particle with matter, which means they only work if the dark matter particles interacts more than only gravitationally. This is the main assumption of these lecture notes, and it is motivated by the fact that the weak gauge coupling and the weak mass scale happen to predict roughly the correct relic density, as described in Sect. 3.1.

The idea behind indirect searches for WIMPS is that the generally small current dark matter density is significantly enhanced wherever there is a clump of gravitational matter, as for example in the sun or in the center of the galaxy. In these regions dark matter should efficiently annihilate even today, giving us either photons or pairs of particles and anti-particles coming from there. Particles like electrons or protons are not rare, but anti-particles in the appropriate energy range should be detectable. The key ingredient to the calculation of these spectra is the fact that dark matter particles move only very slowly relative to galactic objects. This means we need to compute all processes with incoming dark matter particles essentially at rest. This approximation is even better than at the time of the dark matter freeze-out discussed in Sect. 3.2.

Indirect detection experiments search for many different particles which are produced in dark matter annihilation. First, this might be the particles that dark matter directly annihilated into, for example in a 2 → 2 scattering process. This includes protons and anti-protons if dark matter annihilates into quarks. Second, we might see decay products of these particles. An example for such signatures are neutrinos. Examples for dark matter annihilation processes are

$$\displaystyle \begin{aligned} \tilde{\chi}_1^0 \tilde{\chi}_1^0 &\to \ell^+ \ell^- \\ \tilde{\chi}_1^0 \tilde{\chi}_1^0 &\to q \bar{q} \to p \bar{p} + X \\ \tilde{\chi}_1^0 \tilde{\chi}_1^0 &\to \tau^+ \tau^-, W^+ W^-, b \bar{b} + X \to \ell^+ \ell^-, p\bar{p} + X \qquad \ldots \end{aligned} $$

(5.1)

The final state particles are stable leptons or protons propagating large distances in the Universe. While the leptons or protons can come from many sources, the anti-particles appear much less frequently. One key experimental task in many indirect dark matter searches is therefore the ability to measure the charge of a lepton, typically with the help of a magnetic field. For example, we can study the energy dependence of the antiproton–proton ratio or the positron–electron ratio as a function of the energy. The dark matter signature is either a line or a shoulder in the spectrum, with a cutoff

$$\displaystyle \begin{aligned} E_{e^+} \approx m_{\tilde{\chi}_1^0} \qquad \text{or} \qquad E_{e^+} < m_{\tilde{\chi}_1^0} \; . \end{aligned} $$

(5.2)

The main astrophysical background is pulsars, which produce for example electron–positron pairs of a given energy. There exists a standard tool to simulate the propagation of all kinds of particles through the Universe, which is called GALPROP. For example Pamela has seen such a shoulder with a positron flux pointing to a very large WIMP annihilation rate. An interpretation in terms of dark matter is inconclusive, because pulsars could provide an alternative explanation and the excess is in tension with PLANCK results from CMB measurements, as discussed in Sect. 3.4.

In these lecture notes we will focus on photons from dark matter annihilation, which we can search for in gamma ray surveys over a wide range of energies. They also will follow one of two kinematic patterns: if they occur in the direct annihilation process, they will appear as a mono-energetic line in the spectrum

$$\displaystyle \begin{aligned} \chi \chi \to \gamma \gamma \qquad \text{with} \quad E_\gamma \approx m_\chi \; , {} \end{aligned} $$

(5.3)

for any weakly interacting dark matter particle χ. This is because the massive dark matter particles are essentially at rest when colliding. If the photons are radiated off charged particles or appear in pion decays π ⁰ → γγ

$$\displaystyle \begin{aligned} \chi \chi \to \tau^+ \tau^-, b\bar{b}, W^+ W^- \to \gamma + \cdots \; , {} \end{aligned} $$

(5.4)

they will follow a fragmentation pattern. We can either compute this photon spectrum or rely on precise measurements from the LEP experiments at CERN (see Sect. 7.1 for a more detailed discussion of the LEP experiments). This photon spectrum will constrain the kind of dark matter annihilation products we should consider, as well as the mass of the dark matter particle.

The energy dependence of the photon flow inside a solid angle ΔΩ is given by

$$\displaystyle \begin{aligned} \frac{d \varPhi_\gamma}{d E_\gamma} = \frac{\left\langle \sigma v \right\rangle}{8 \pi m_{\tilde{\chi}_1^0}^2} \; \frac{d N_\gamma}{d E_\gamma} \; \int_{\varDelta \varOmega} d \varOmega \int_l\, d z \; \rho_\chi^2(z) \; , {} \end{aligned} $$

(5.5)

where E _γ is the photon energy, $\left \langle \sigma v \right \rangle$ is the usual velocity-averaged annihilation cross-section, N _γ is the number of photons produced per annihilation, and l is the distance from the observer to the actual annihilation event (line of sight). The photon flux depends on the dark matter density squared because it arises from the annihilation of two dark matter particles. A steeper dark matter halo profile, i.e. the dark matter density increasing more rapidly towards the center of the galaxy, results in a more stringent bound on dark matter annihilation. The key problem in the interpretation of indirect search results in terms of dark matter is that we cannot measure the dark matter distributions ρ _χ(l) for example in our galaxy directly. Instead, we have to rely on numerical simulations of the dark matter profile, which introduce a sizeable parametric or theory uncertainty in any dark-matter related result. Note that the dark matter profile is not some kind of multi-parameter input which we have the freedom to assume freely. It is a prediction of numerical dark matter simulations with associated error bars. Not all papers account for this uncertainty properly. In contrast, the constraints derived from CMB anisotropies discussed in Sect. 1.4 are largely free of astrophysical uncertainties.

There exist three standard density profiles; the steep Navarro-Frenk-White (NFW) profile is given by

$$\displaystyle \begin{aligned} \rho_{\text{NFW}}(r) = \frac{\rho_\odot}{\left( \dfrac{r}{R} \right)^\gamma \left( 1+ \dfrac{r}{R} \right)^{3 - \gamma}} \stackrel{\gamma=1}{=} \frac{\rho_\odot}{\dfrac{r}{R} \left( 1+ \dfrac{r}{R} \right)^2} \; , {} \end{aligned} $$

(5.6)

where r is the distance from the galactic center. Typical parameters are a characteristic scale R = 20 kpc and a solar position dark matter density ρ _⊙ = 0.4 GeV∕cm³ at r _⊙ = 8.5 kpc. In this form we can easily read off the scaling of the dark matter density in the center of the galaxy, i.e. r ≪ R; there we find ρ _NFW ∝ r ^−γ. The second steepest is the exponential Einasto profile ,

$$\displaystyle \begin{aligned} \rho_{\text{Einasto}}(r) = \rho_\odot~ \exp \left[-\frac{2}{\alpha} \left(\left( \frac{r}{R}\right)^\alpha -1\right) \right] \; , {} \end{aligned} $$

(5.7)

with α = 0.17 and R = 20 kpc. It fits micro-lensing and star velocity data best. Third is the Burkert profile with a constant density inside a radius R,

$$\displaystyle \begin{aligned} \rho_{\text{Burkert}}(r) = \frac{\rho_\odot}{\left( 1+ \dfrac{r}{R} \right) \left(1+ \dfrac{r^2}{R^2} \right)} \; , {} \end{aligned} $$

(5.8)

where we assume R = 3 kpc. Assuming a large core results in very diffuse dark matter at the galactic center, and therefore yields the weakest bound on neutralino self annihilation. Instead assuming R = 0.1 kpc only alters the dark matter annihilation constraints by an order-one factor. We show the three profiles in Fig. 5.1 and observe that the difference between the Einasto and the NFW parametrizations are marginal, while the Burkert profile has a very strongly reduced dark matter density in the center of the galaxy. One sobering result of this comparison is that whatever theoretical considerations lie behind the NFW and Einasto profiles, once their parameters are fit to data the possibly different underlying arguments play hardly any role. The impact on gamma ray flux of different dark matter halo profiles is conveniently parameterized by the factor

$$\displaystyle \begin{aligned} J \propto \int_{\varDelta \varOmega} d \varOmega \int_{\text{line of sight}} d z \; \rho_\chi^2(z) \qquad \text{with} \quad J(\rho_{\text{NFW}}) \equiv 1 \; . \end{aligned} $$

(5.9)

Also in Fig. 5.1 we quote the J factors integrated over the approximate HESS galactic center gamma ray search range, r = 0.05…0.15 kpc. As expected, the Burkert profile predicts a photon flow lower by almost two orders of magnitude. In a quantitative analysis of dark matter signals this difference should be included as a theory error or a parametric error, similar to for example parton densities or the strong coupling in LHC searches.

Fig. 5.1

Dark matter galactic halo profiles, including standard Einasto and NFW profiles along with a Burkert profile with a 3 kpc core. J factors are obtained assuming a spherical dark matter distribution and integrating over the radius from the galactic center from r ≃ 0.05 to 0.15 kpc. J factors are normalized so that J(ρ _NFW) = 1. Figure from Ref. [1]

While at any given time there is usually a sizeable set of experimental anomalies discussed in the literature, we will focus on one of them: the photon excess in the center of our galaxy, observed by Fermi, but discovered in their data by several non-Fermi groups. The excess is shown in Fig. 5.2 and covers the wide photon energy range

$$\displaystyle \begin{aligned} E_\gamma = 0.3~\ldots~5~\text{GeV} \; , \end{aligned} $$

(5.10)

and clearly does not form a line. The error bars refer to the interstellar emission model, statistics, photon fragmentation, and instrumental systematics. Note that the statistical uncertainties are dominated not by the number of signal events, but by the statistical uncertainty of the subtracted background events. The fact that uncertainties on photon fragmentation, means photon radiation off other Standard Model particles are included in the analysis, indicates, that for an explanation we resort to photon radiation off dark matter annihilation products, Eq. (5.4). This allows us to link the observed photon spectrum to dark matter annihilation, where the photon radiation off the final state particles is known very well from many collider studies. Two aspects of Fig. 5.2 have to be matched by any explanation. First, the total photon rate has to correspond to the dark matter annihilation rate. It turns out that the velocity-averaged annihilation rate has to be in the same range as the rate required for the observed relic density,

$$\displaystyle \begin{aligned} \langle \sigma_{\chi \chi} \, v \rangle = \frac{10^{-8}~\ldots~10^{-9}}{\text{GeV}^2} \; , {} \end{aligned} $$

(5.11)

but with a much lower velocity spectrum now. Second, the energy spectrum of the photons reflects the mass of the dark matter annihilation products. Photons radiated off heavier, non-relativistic states will typically have higher energies. This information is used to derive the preferred annihilation channels given in Fig. 5.3. The official Fermi data confirms these ranges, but with typically larger error bars. As an example, we quote the fit information under the assumption of two dark matter Majorana fermions decaying into a pair of Standard Model states [5]:

Fig. 5.2

Excess photon spectrum of the Fermi galactic center excess. Figure from Ref. [2], including the original data and error estimates from Ref. [3]

Fig. 5.3

Preferred dark matter masses and cross sections for different annihilation channels [4]. Figure from Ref. [5]

Channel	[fb]	m χ [GeV]
$q\bar {q}$	275 ± 45	24 ± 3
$b\bar {b}$	580 ± 85	49 ± 6
τ + τ −	110 ± 17	10 ± 1
W + W −	1172 ± 160	80 ± 1

For each of these annihilation channels the question arises if we can also generate a sizeable dark matter annihilation rate at the center of the galaxy today, while also predicting the correct relic density Ω _χ h ².

5.1 Higgs Portal

Similar to our calculation of the relic density, we will first show what range of annihilation cross sections from the galactic center can be explained by Higgs portal dark matter. Because the Fermi data prefers a light dark matter particle we will focus on the two velocity-weighted cross sections accounting for the observed relic density and for the galactic center excess around the Higgs pole m _S∕2 = m _H. First, we determine how large an annihilation cross section in the galactic center we can achieve. The typical cross sections given in Eq. (5.11) can be explained by m _S = 220 GeV and λ ₃ = 1∕10 as well as a more finely tuned m _S = m _H∕2 = 63 GeV with λ ₃ ≈ 10⁻³, as shown in Fig. 4.1.

We can for example assume that the Fermi excess is due to on-shell Higgs-mediated annihilation, while the observed relic density does not probe the Higgs pole. The reason we can separate these two annihilation signals based on the same Feynman diagram this way is that the Higgs width is smaller than the typical velocities, Γ _H∕m _H ≪ v. We start with the general annihilation rate of a dark matter scalar, Eq. (4.10) and express it including the leading relative velocity dependence from Eq. (3.20),

$$\displaystyle \begin{aligned} s = 4 m_S^2 + m_S^2 v^2 = 4 m_S^2 \, \left( 1 + \frac{v^2}{4} \right) \, . {} \end{aligned} $$

(5.12)

The WIMP velocity at the point of dark matter decoupling in the early universe we find roughly

$$\displaystyle \begin{aligned} x_{\text{dec}} := \frac{m_S}{T_{\text{dec}}} \stackrel{\text{Eq. (3.9)}}{=} 28 \quad \Leftrightarrow \quad T_{\text{dec}} \approx \frac{m_S}{28} = \frac{m_S}{2} v_{\text{ann}}^2 \quad \Leftrightarrow \quad v_{\text{ann}}^2 = \frac{1}{14} \; . {} \end{aligned} $$

(5.13)

Today the Universe is colder, and the WIMP velocity is strongly red-shifted. Typical galactic velocities today are

$$\displaystyle \begin{aligned} v_0 \approx 2.3 \cdot 10^{5} \, \frac{\text{m}}{\text{s}} \; \frac{1}{c} \approx \frac{1}{1300} \ll v_{\text{ann}} \; , \end{aligned} $$

(5.14)

This hierarchy in typical velocities between the era of thermal dark matter production and annihilation and dark matter annihilation today is what will drive our arguments below.

Only assuming m _b ≪ s the general form of the scalar dark matter annihilation rate is

$$\displaystyle \begin{aligned} & \sigma v \Bigg|{}_{SS \to b\bar{b}} = \frac{N_c}{2 \pi} \; \lambda_3^2 m_b^2 \; \frac{1}{m_S \, \sqrt{s}} \; \frac{s}{\left( s - m_H^2 \right)^2 + m_H^2 \varGamma_H^2} \\ &\quad = \left( 1 + \dfrac{v^2}{8} + \mathcal{O} (v^4) \right) \; \frac{N_c}{2 \pi} \; \lambda_3^2 m_b^2 \; \frac{1}{2 m_S^2} \; \frac{4 m_S^2}{\left( 4 m_S^2 - m_H^2 + m_S^2 v^2 \right)^2 + m_H^2 \varGamma_H^2} \\ &\quad = \left( 1 + \dfrac{v^2}{8} \right) \; \frac{N_c}{2 \pi} \; \lambda_3^2 m_b^2 \; \frac{1}{2 m_S^2} \; \frac{4 m_S^2}{\left( 4 m_S^2 - m_H^2 \right)^2 + 2 ( 4 m_S^2 - m_H^2 ) m_S^2 v^2 + m_H^2 \varGamma_H^2} \\ & \qquad + \mathcal{O} (v^4) \; . \end{aligned} $$

(5.15)

The typical velocity of the dark matter states only gives a small correction for scalar, s-wave annihilation. It includes two aspects: first, an over-all reduction of the annihilation cross section for finite velocity v > 0, and second a combined cutoff of the Breit-Wigner propagator,

$$\displaystyle \begin{aligned} \max \left[ 2 (4 m_S^2 - m_H^2 ) m_S^2 v^2, m_H^2 \varGamma_H^2 \right] = m_S^4 \; \max \left[ 8 v^2 \left( 1 - \frac{m_H^2}{4 m_S^2} \right), 16 \cdot 10^{-10} \right] \; . {} \end{aligned} $$

(5.16)

Close to but not on the on-shell pole m _H = m _S∕2 the modification of the Breit-Wigner propagator can be large even for small velocities, while the rate reduction can clearly not account for a large boost factor describing the galactic center excess. We therefore ignore the correction factor (1 + v ²∕8) when averaging the velocity-weighted cross section over the velocity spectrum. If, for no good reason, we assume a narrow Gaussian velocity distribution centered around $\bar {v}$ we can approximate Eq. (5.15) as [6]

(5.17)

with a fitted $\xi \approx 2 \sqrt {2}$. This modified on-shell pole condition shifts the required dark matter mass slightly below the Higgs mass $2 m_S \lesssim m_H$. The size of this shift depends on the slowly dropping velocity, first at the time of dark matter decoupling, $\bar {v} \equiv v_{\text{ann}}$, and then today, $\bar {v} \equiv v_0 \ll v_{\text{ann}}$. This means that during the evolution of the Universe the Breit-Wigner propagator in Eq. (5.17) is always probed above its pole, probing the actual pole only today.

We first compute the Breit–Wigner suppression of 〈vσ〉 in the early universe, starting with today’s on-shell condition responsible for the galactic center excess,

$$\displaystyle \begin{aligned} m_S \stackrel{!}{=} \frac{m_H}{2 \sqrt{1 + \dfrac{v_0^2}{\sqrt{2}}}} \approx \frac{m_H}{2} \quad &\Rightarrow \quad 4 m_S^2 - m_H^2 + \xi \, m_S^2 v_{\text{ann}}^2 \\ {} &= 4 m_S^2 \left( 1 + \frac{v_{\text{ann}}^2}{\sqrt{2}} \right) - m_H^2 \\ &= 4 m_S^2 \left( \dfrac{v_{\text{ann}}^2}{\sqrt{2}} - \dfrac{v_0^2}{\sqrt{2}} \right) \\ & \stackrel{v_{\text{ann}} \gg v_0}{\approx} \frac{m_S^2}{5} \; . {} \end{aligned} $$

(5.18)

This means that the dark matter particle has a mass just slightly below the Higgs pole. Using Eq. (5.17) the ratio of the two annihilation rates, for all other parameters constant, then becomes

$$\displaystyle \begin{aligned} \frac{\langle \sigma_0\,v \rangle}{\langle \sigma_{\text{ann}} v \rangle} = \frac{8 m_S^4 v_{\text{ann}}^4}{4 m_S^2 \varGamma_H^2} = \frac{2 v_{\text{ann}}^4}{16 \cdot 10^{-10}} = \frac{1}{8} \, \; \frac{1}{14^2} \; 10^{10} \gtrsim 10^6 \; . {} \end{aligned} $$

(5.19)

This is the maximum enhancement we can generate to explain Fermi’s galactic center excess. The corresponding Higgs coupling λ ₃ is given in Fig. 4.1.

We can turn the question around and compute the smallest annihilation cross section in the galactic center consistent with the observed relic abundance in the Higgs portal model. For this purpose we assume that unlike in Eq. (5.18) the pole condition is fulfilled in the early universe, leading to a Breit-Wigner suppression today of

$$\displaystyle \begin{aligned} m_S &\stackrel{!}{=} \frac{m_H}{2 \sqrt{1 + \dfrac{v_{\text{ann}}^2}{\sqrt{2}}}} \\ &\Rightarrow \quad 4 m_S^2 - m_H^2 + \xi \, m_S^2 v_0^2 = 4 m_S^2 \left( 1 + \frac{v_0^2}{\sqrt{2}} \right) - m_H^2 \stackrel{v_{\text{ann}} \gg v_0}{\approx} - \frac{m_S^2}{5} \; . {} \end{aligned} $$

(5.20)

This gives us a ratio of the two velocity-mediated annihilation rates

$$\displaystyle \begin{aligned} \frac{\langle \sigma_0\,v \rangle}{\langle \sigma_{\text{ann}}\, v \rangle} &= \frac{4 m_S^2 \varGamma_H^2}{8 m_S^4 v_{\text{ann}}^4} \stackrel{\text{Eq. (5.19)}}{\lesssim} 10^{-6} \\ &\quad \text{for} \quad m_S \approx \frac{m_H}{2} \left( 1 - \dfrac{v_{\text{ann}}^2}{2 \sqrt{2}} \right) \stackrel{\text{Eq. (5.13)}}{=} 62.91~\text{GeV} \; . \end{aligned} $$

(5.21)

The dark matter particle now has a mass further below the pole. This means that we can interpolate between the two extreme ratios of velocity-averaged annihilation rates using a very small range of m _S < m _H∕2. If we are willing to tune this mass relation we can accommodate essentially any dark matter annihilation rate today with the Higgs portal model, close to on-shell Higgs pole annihilation. The key to this result is that following Eq. (5.16) the Higgs width-to-mass ratio is small compared to v _ann, so we can decide to assign the on-shell condition to each of the two relevant annihilation processes. In between, none of the two processes will proceed through the on-shell Higgs propagator, which indeed gives 〈σ ₀ v〉≈〈σ _ann v〉. The corresponding coupling λ ₃ we can read off Fig. 4.1. Through this argument it becomes clear that the success of the Higgs portal model rests on the wide choice of scalings of the dark matter annihilation rate, as shown in Eq. (4.21).

5.2 Supersymmetric Neutralinos

An explanation of the galactic center excess has to be based on the neutralino mass matrix given in Eq. (4.39), defining a dark matter Majorana fermion as a mixture of the bino singlet, the wino triplet, and two higgsino doublets. Some of its relevant couplings are given in Eq. (4.43). Correspondingly, some annihilation processes leading to the observed relic density and underlying our interpretation of the Fermi galactic center excess are illustrated in Fig. 4.3. One practical advantage of the MSSM is that it offers many neutralino parameter regions to play with. We know that pure wino or higgsino dark matter particles reproducing the observed relic density are much heavier than the Fermi data suggests. Instead of these pure states we will rely on mixed states. A major obstacle of all MSSM interpretations are the mass ranges shown in Fig. 5.3, indicating a clear preference of the galactic center excess for neutralino masses $m_{\tilde {\chi }_1^0} \lesssim 60$ GeV. This does not correspond to the typical MSSM parameter ranges giving us the correct relic density. This means that in an MSSM analysis of the galactic center excess the proper error estimate for the photon spectrum is essential.

We start our discussion with the finely tuned annihilation through a SM-like light Higgs or through a Z-boson, i.e. $\tilde {\chi }_1^0 \tilde {\chi }_1^0 \to h^*,Z^* \to b\bar {b}$. The properties of this channel are very similar to those of the Higgs portal. On the left y-axes of Fig. 5.4 we show the (inverse) relic density for a bino-higgsino LSP, both for a wide range of neutralino masses and zoomed into the Higgs pole region. We decouple the wino to M ₂ = 700 GeV and vary M ₁ to give the correct relic density for three fixed, small higgsino mass values. We see that the $b\bar {b}$ annihilation channel only predicts the correct relic density in the two pole regions of the MSSM parameter space, with $m_{\tilde {\chi }_1^0} = 46$ GeV and $m_{\tilde {\chi }_1^0} = 63$ GeV. The width of both peaks is given by the momentum smearing through velocity spectrum rather than physical Higgs width and Z-width. The enhancement of the two peaks over the continuum is comparable, with the Z-funnel coupled to the velocity-suppressed axial-vector current and the Higgs funnel suppressed by the small bottom Yukawa coupling.

Fig. 5.4

Inverse relic density (solid, left axis) and annihilation rate in the galactic center (dashed, right axis) for an MSSM parameter point where the annihilation is dominated by $\tilde {\chi }_1^0 \tilde {\chi }_1^0 \to b\bar {b}$. Figure from Ref. [2]

On the right y-axis of Fig. 5.4, accompanied by dashed curves, we show the annihilation rate in the galactic center. The rough range needed to explain the Fermi excess is indicated by the horizontal line. As discussed for the Higgs portal, the difference to the relic density is that the velocities are much smaller, so the widths of the peaks are now given by the physical widths of the two mediators. The scalar Higgs resonance now leads to a much higher peak than the velocity-suppressed axial-vector coupling to the Z-mediator. This implies that continuum annihilation as well as Z-pole annihilation would not explain the galactic center excess, while the Higgs pole region could.

This is why in the right panel of Fig. 5.4 we zoom into the Higgs peak regime. A valid explanation of the galactic center excess requires the solid relic density curves to cross the solid horizontal line and at the same time the dashed galactic center excess lines to cross the dashed horizontal line. We see that there exist finely tuned regions around the Higgs pole which allow for an explanation of the galactic center excess via a thermal relic through the process $\tilde {\chi }_1^0 \tilde {\chi }_1^0 \to b\bar {b}$. The physics of this channel is very similar to scalar Higgs portal dark matter.

For slightly larger neutralino masses, the dominant annihilation becomes $\tilde {\chi }_1^0 \tilde {\chi }_1^0 \to WW$, mediated by a light t-channel chargino combined with chargino-neutralino co-annihilation for the relic density. Equation (4.43) indicates that in this parameter region the lightest neutralino requires either a wino content or a higgsino content. In the left panel of Fig. 5.5 we show the bino-higgsino mass plane indicating the preferred regions from the galactic center excess. The lightest neutralino mass varies from $m_{\tilde {\chi }_1^0} \approx 50$ GeV to more than 250 GeV. Again, we decouple the wino to M ₂ = 700 GeV, so the LSP is a mixture of higgsino, coupling to electroweak bosons, and bino. For this slice in parameter space an increase in |μ| compensates any increase in M ₁, balancing the bino and higgsino contents. The MSSM parameter regions which allow for efficient dark matter annihilation into gauge bosons are strongly correlated in M ₁ and μ, but not as tuned as the light Higgs funnel region with its underlying pole condition. Around M ₁ = |μ| = 200 GeV a change in shape occurs. It is caused by the on-set of neutralino annihilation to top pairs, in spite of a heavy Higgs mass scale of 1 TeV.

Fig. 5.5

Left: lightest neutralino mass based on the Fermi photon where $\tilde {\chi }_1^0 \tilde {\chi }_1^0 \to WW$ is a dominant annihilation channel. Right: lightest neutralino mass based on the Fermi photon spectrum for m _A = 500 GeV, where we also observe $\tilde {\chi }_1^0 \tilde {\chi }_1^0 \to t \bar {t}$. The five symbols indicate local best-fitting parameter points. The black shaded regions are excluded by the Fermi limits from dwarf spheroidal galaxies

To trigger a large annihilation rate for $\tilde {\chi }_1^0 \tilde {\chi }_1^0 \to t \bar {t}$ we lower the heavy pseudoscalar Higgs mass to m _A = 500 GeV. In the right panel of Fig. 5.5 we show the preferred parameter range again in the bino-higgsino mass plane and for heavy winos, M ₂ = 700 GeV. As expected, for $m_{\tilde {\chi }_1^0} > 175$ GeV the annihilation into top pairs follows the WW annihilation region in the mass plane. The main difference between the WW and $t\bar {t}$ channels is the smaller M ₁ values around |μ| = 200 GeV. The reason is that an increased bino fraction compensates for the much larger top Yukawa coupling. The allowed LSP mass range extends to $m_{\tilde {\chi }_1^0} \gtrsim 200$ GeV.

The only distinctive feature for m _A = 500 GeV in the M ₁ vs μ plane is the set of peaks around M ₁ ≈ 300 GeV. Here the lightest neutralino mass is around 250 GeV, just missing the A-pole condition. Because on the pole dark matter annihilation through a 2 → 1 process becomes too efficient, the underlying coupling is reduced by a smaller higgsino fraction of the LSP. The large-|M ₁| regime does not appear in the upper left corner of Fig. 5.5 because at tree level this parameter region features $m_{\tilde {\chi }_1^+} < m_{\tilde {\chi }_1^0}$ and we have to include loop corrections to revive it.

In principle, for $m_{\tilde {\chi }_1^0} > 126$ GeV we should also observe neutralino annihilation into a pair of SM-like Higgs bosons. However, the t-channel neutralino diagram which describes this process will typically be overwhelmed by the annihilation to weak bosons with the same t-channel mediator, shown in Fig. 4.3. From the annihilation into top pairs we know that s-channel mediators with m _A,H ≈ 2m _h are in principle available, and depending on the MSSM parameter point the heavy scalar Higgs can have a sizeable branching ratio into two SM-like Higgses. For comparably large velocities in the early universe both s-channel mediators indeed work fine to predict the observed relic density. For the smaller velocities associated with the galactic center excess the CP-odd mediator A completely dominates, while the CP-even H is strongly velocity-suppressed. On the other hand, only the latter couples to two light Higgs bosons, so an annihilation into Higgs pairs responsible for the galactic center excess is difficult to realize in the MSSM.

Altogether we see that the annihilation channels

$$\displaystyle \begin{aligned} \tilde{\chi}_1^0 \tilde{\chi}_1^0 \to b\bar{b}, WW, t\bar{t} \qquad \text{with} \quad m_{\tilde{\chi}_1^0} = 63~\ldots~250~\text{GeV} \end{aligned} $$

(5.22)

can explain the Fermi galactic center excess and the observed relic density in the MSSM. Because none of them correspond to the central values of a combined fit to the galactic center excess, it is crucial that we take into account all sources of (sizeable) uncertainties. An additional issue which we will only come to in Chap. 6 is that direct detection constraints in addition to requiring the correct relic density and the correct galactic center annihilation rate is a serious challenge to the MSSM explanations.

5.3 Next-to-Minimal Neutralino Sector

An obvious way out of the MSSM limitations is to postulate an addition particle in the s-channel of the neutralino annihilation process and a new, lighter dark matter fermion. This leads us to the next-to-minimal supersymmetric extension, the NMSSM. It introduces an additional singlet under all Standard Model gauge transformations, with its singlino partner. The singlet state forms a second scalar H ₁ and a second pseudo-scalar A ₁, which will appear in the dark matter annihilation process. As singlets they will only couple to gauge bosons through mixing with the Higgs fields, which guarantees that they are hardly constrained by many searches. The singlino will add a fifth Majorana state to the neutralino mass matrix in Eq. (4.39),

(5.23)

The singlet/singlino sector can be described by two parameters, the mass parameter $\tilde \kappa$ and the coupling for example to the other neutralinos $\tilde \lambda$ [7]. First, we need to include the singlino in our description of the neutralino sector. While the wino and the two higgsinos form a triplet or two doublets under SU(2)_L, the singlino just adds a second singlet under SU(2)_L. The only difference to the bino is that the singlino is also a singlet under U(1)_Y, which makes no difference unless we consider co-annihilation driven by hypercharge interaction. A singlet neutralino will therefore interact and annihilate to the observed relic density through its mixing with the wino or with the higgsinos, just like the usual bino.

What is crucial for the explanation of the galactic center excess is the s-channel dark matter annihilation through the new pseudoscalar,

$$\displaystyle \begin{aligned} \tilde{\chi}_1^0 \tilde{\chi}_1^0 \to A_1 \to b\bar{b} \qquad \text{with} \quad m_{\tilde{\chi}_1^0} &= \frac{m_{A_1}}{2} \approx 50~\text{GeV} \\ g_{A_1^0 \tilde{\chi}_1^0 \tilde{\chi}_1^0} &= \sqrt{2} \, g \tilde{\lambda} \; \left( N_{13} N_{14} - \tilde{\kappa} N_{15}^2 \right) \; . \end{aligned} $$

(5.24)

We can search for these additional singlet and singlino states at colliders. One interesting aspect is the link between the neutralino and the Higgs sector, which can be probed by looking for anomalous Higgs decays, for example into a pair of dark matter particles. Because an explanation of the galactic center excess requires the singlet and the singlino to be light and to mix with their MSSM counterparts, the resulting invisible branching ratio of the Standard-Model-like Higgs boson can be large.

5.4 Simplified Models and Vector Mediator

The discussion of the dark matter annihilation processes responsible for today’s dark matter density as well as a possible galactic center excess nicely illustrates the limitations of the effective theory approach introduced in Sect. 4.4. To achieve the currently observed density with light WIMPs we have to rely on an efficient annihilation mechanism, which can be most clearly seen in the MSSM. For example, we invoke s-channel annihilation or co-annihilation, both of which are not well captured by an effective theory description with a light dark matter state and a heavy, non-propagating mediator. In the effective theory language of Sect. 4.4 this means the mediators are not light compared to the dark matter agent,

$$\displaystyle \begin{aligned} m_\chi \lesssim m_{\text{med}} \; . \end{aligned} $$

(5.25)

In addition, the MSSM and the NMSSM calculations illustrate how one full model extending the Standard Model towards large energy scales can offer several distinct explanations, only loosely linked to each other. In this situation we can collect all the necessary degrees of freedom in our model, but ignore additional states for example predicted by an underlying supersymmetry of the Lagrangian. This approach is called simplified models. It typically describes the dark matter sector, including co-annihilating particles, and a mediator coupling the dark matter sector to the Standard Model. In that language we have come across a sizeable set of simplified models in our explanation of the Fermi galactic center excess:

• dark singlet scalar with SM Higgs mediator (Higgs portal, $SS \to b\bar {b}$);

• dark fermion with SM Z mediator (MSSM, $\tilde {\chi }_1^0 \tilde {\chi }_1^0 \to f\bar {f}$, not good for galactic center excess);

• dark fermion with SM Higgs mediator (MSSM, $\tilde {\chi }_1^0 \tilde {\chi }_1^0 \to b\bar {b}$);

• dark fermion with t-channel fermion mediator (MSSM, $\tilde {\chi }_1^0 \tilde {\chi }_1^0 \to WW$);

• dark fermion with heavy s-channel pseudo-scalar mediator (MSSM, $\tilde {\chi }_1^0 \tilde {\chi }_1^0 \to t\bar {t}$);

• dark fermion with light s-channel pseudo-scalar mediator (NMSSM, $\tilde {\chi }_1^0 \tilde {\chi }_1^0 \to b\bar {b}$).

In addition, we encountered a set of models in our discussion of the relic density in the MSSM in Sect. 4.3:

• dark fermion with fermionic co-annihilation partner and charged s-channel mediator (MSSM, $\tilde {\chi }_1^0 \tilde {\chi }_1^- \to \bar t b$);

• dark fermion with fermionic co-annihilation partner and SM W-mediator (MSSM, $\tilde {\chi }_1^0 \tilde {\chi }_1^- \to \bar u d$);

• dark fermion with scalar t-channel mediator (MSSM, $\tilde {\chi }_1^0 \tilde {\chi }_1^0 \to \tau \tau$);

• dark fermion with scalar co-annihilation partner (MSSM, $\tilde {\chi }_1^0 \tilde {\tau } \to \tau ^*$)

Strictly speaking, all the MSSM scenarios require a Majorana fermion as the dark matter candidate, but we can replace it with a Dirac neutralino in an extended supersymmetric setup.

One mediator which is obviously missing in the above list is a new, heavy vector V or axial-vector. Heavy gauge bosons are ubiquitous in models for physics beyond the Standard Model, and the only question is how we would link or couple them to a dark matter candidate. In principle, there exist different mass regimes in the m _χ − m _V mass plane,

$$\displaystyle \begin{aligned}\begin{array}{r*{20}l} m_V & > 2 m_\chi \qquad \qquad \qquad && \text{possible effective theory} \\ m_V &\approx 2 m_\chi && \text{on-shell, simplified model} \\ m_V & < 2 m_\chi && \text{light mediator, simplified model}\; . {} \end{array}\end{aligned} $$

(5.26)

To allow for a global analysis including direct detection as well as LHC searches, we couple the vector mediator to a dark matter fermion χ and the light up-quarks,

$$\displaystyle \begin{aligned} \mathcal{L} \supset g_u\ \bar{u}\ \gamma^\mu V_\mu \ u + g_\chi\ \bar{\chi}\ \gamma^\mu V_\mu \ \chi \; . {} \end{aligned} $$

(5.27)

The typical mediator width for m _χ ≪ m _V is

$$\displaystyle \begin{aligned} \frac{\varGamma_V}{m_V} \lesssim 0.4~\ldots~10\% \qquad \text{for} \quad g_u = g_\chi=0.2~\ldots~1 \; . \end{aligned} $$

(5.28)

Based on the annihilation process

$$\displaystyle \begin{aligned} \chi \chi \to V^* \to u \bar{u} \end{aligned} $$

(5.29)

we can compute the predicted relic density or the indirect detection prospects. While the χ − χ − V interaction also induces a t-channel process χχ → V ^∗ V ^∗, its contribution to the total dark matter annihilation rate is always strongly suppressed by its 4-body phase space. The on-shell annihilation channel

$$\displaystyle \begin{aligned} \chi \chi \to VV \end{aligned} $$

(5.30)

becomes important for m _V < m _χ, with a subsequent decay of the mediator for example to two Standard Model fermions. In that case the dark matter annihilation rate becomes independent of the mediator coupling to the Standard Model, giving much more freedom to avoid experimental constraints.

In Fig. 5.6 we observe that for a light mediator the predicted relic density is smaller than the observed values, implying that the annihilation rate is large. In the left panel we see the three kinematic regimes defined in Eq. (5.26). First, for small mediator masses the 2 → 2 annihilation process is $\chi \chi \to u\bar {u}$. The dependence on the light mediator mass is small because the mediator is always off-shell and the position of its pole is far away from the available energy of the incoming dark matter particles. Around the pole condition 2m _χ ≈ m _V ± Γ _V the model predicts the correct relic density with very small couplings. For heavy mediators the 2 → 2 annihilation process rapidly decouples with large mediator masses, as follows for example from Eq. (3.3). In the right panel of Fig. 5.6 we assume a constant mass ratio m _V∕m _χ ≳ 1, finding that our simplified vector model has no problems predicting the correct relic density over a wide range of model parameters.

Fig. 5.6

Relic density for the simplified vector mediator model of Eq. (5.27) as a function of the mediator mass for constant dark matter mass (left) and as a function of the dark matter mass for a constant ratio of mediator to dark matter mass (right). Over the shaded bands we vary the couplings g _u = g _χ = 0.2, …, 1. Figure from Ref. [8]

One issue we can illustrate with this non-MSSM simplified model is a strong dependence of our predictions on the assumed model features. The Lagrangian of Eq. (5.27) postulates a coupling to up-quarks, entirely driven by our goal to link dark matter annihilation with direct detection and LHC observables. From a pure annihilation perspective we can also define the mediator coupling to the Standard Model through muons, without changing any of the results shown in Fig. 5.6. Coupling to many SM fermions simultaneously, as we expect from an extra gauge group, will increase the predicted annihilation rate easily by an order of magnitude. Moreover, it is not clear how the new gauge group is related to the U(1)_Y × SU(2)_L structure of the electroweak Standard Model. All this reflects the fact that unlike the Higgs portal model or supersymmetric extensions a simplified model is hardly more than a single tree-level or loop-level Feynman diagram describing dark matter annihilation. It describes the leading effects for example in dark matter annihilation based on 2 → 2 or 2 → 1 kinematics or the velocity dependence at threshold. However, because simplified models are usually not defined on the full quantum level, they leave a long list of open questions. For new gauge bosons, also discussed in Sect. 4.2, they include fundamental properties like gauge invariance, unitarity, or freedom from anomalies.