WIMP Models

Abstract

If we want to approach the problem of dark matter from a particle physics perspective, we need to make assumptions about the quantum numbers of the weakly interacting state which forms dark matter. During most of these lecture notes we assume that this new particle has a mass in the GeV to TeV range, and that its density is thermally produced during the cooling of the Universe. Moreover, we assume that the entire dark matter density of the Universe is due to one stable particle.

If we want to approach the problem of dark matter from a particle physics perspective, we need to make assumptions about the quantum numbers of the weakly interacting state which forms dark matter. During most of these lecture notes we assume that this new particle has a mass in the GeV to TeV range, and that its density is thermally produced during the cooling of the Universe. Moreover, we assume that the entire dark matter density of the Universe is due to one stable particle.

The first assumption fixes the spin of this particle. From the Standard Model we know that there exist fundamental scalars, like the Higgs, fundamental fermions, like quarks and leptons, and fundamental gauge bosons, like the gluon or the weak gauge bosons. Scalars have spin zero, fermions have spin 1/2, and gauge bosons have spin 1. Because calculations with gauge bosons are significantly harder, in particular when they are massive, we limit ourselves to scalars and fermions.

When we construct particle models of dark matter we are faced with this wide choice of new, stable particles and their quantum numbers. Moreover, dark matter has to couple to the Standard Model, because it has to annihilate to produce the observed relic density Ω _χ h = 0.12. This means that strictly speaking we do not only need to postulate a dark matter particle, but also a way for this state to communicate to the Standard Model along the line of the table of states in Sect. 3.5. The second state is usually called a mediator.

4.1 Higgs Portal

An additional scalar particle in the Standard Model can couple to the Higgs sector of the Standard Model in a unique way. The so-called Higgs portal interactions is renormalizable, which means that the coupling constant between two Higgs bosons and two new scalars has a mass unit zero and can be represented by a c-number. All we do in such a model is extend the renormalizable Higgs potential of the Standard Model [1], which has a non-zero vacuum expectation value (VEV) for $\mu _H^2 < 0$,

$$\displaystyle \begin{aligned} V_{\text{SM}} &= \mu_H^2 \; \phi^\dagger \phi + \lambda_H ( \phi^\dagger \phi )^2 \supset \mu_H^2 \frac{(H+v_H)^2}{2} \\ & \quad + \lambda_H \frac{(H+v_H)^4}{4} \supset - \frac{m_H^2}{2} H^2 + \frac{m_H^2}{2v_H} H^3 + \frac{m_H^2}{8 v_H^2} H^4 \; , {} \end{aligned} $$

(4.1)

In the Standard Model this leads to the two observable mass scales

$$\displaystyle \begin{aligned} v_H \kern-1pt=\kern-1pt \sqrt{ \frac{-\mu_H^2}{\lambda_H}} \kern-1pt=\kern-1pt 246~\text{GeV} \quad \text{and} \quad m_H \kern-1pt=\kern-1pt \sqrt{2 \lambda_H} \, v_H \kern-1pt=\kern-1pt 2 \sqrt{ -\mu_H^2 } \kern-1pt=\kern-1pt 125~\text{GeV} \approx \frac{v_H}{2}. {} \end{aligned} $$

(4.2)

The last relation is a numerical accident. The general Higgs potential in Eq. (4.1) allows us to couple a new scalar field S to the Standard Model using a renormalizable, dimension-4 term (ϕ ^† ϕ)(S ^† S).

For any new scalar field there are two choices we can make. First, we can give it some kind of multiplicative charge, so we actually postulate a set of two particles, one with positive and one with negative charge. This just means that our new scalar field has to be complex values, such that the two charges are linked by complex conjugation. In that case the Higgs portal coupling includes the combination S ^† S. Alternatively, we can assume that no such charge exists, in which case our new scalar is real and the Higgs portal interaction is proportional to S ².

Second, we know from the case of the Higgs boson that a scalar can have a finite vacuum expectation value. Due to that VEV, the corresponding new state will mix with the SM Higgs boson to form two mass eigenstates, and modify the SM Higgs couplings and the masses of the W and Z bosons. This is a complication we neither want nor need, so we will work with a dark real scalar. The combined potential reads

$$\displaystyle \begin{aligned} V &= \mu_H^2 \; \phi^\dagger \phi + \lambda_H ( \phi^\dagger \phi )^2 + \mu_S^2 \; S^2 + \kappa \; S^3 + \lambda_S \; S^4 + \kappa_3 \phi^\dagger \phi S \\ & \quad + \lambda_3 \phi^\dagger \phi S^2 \supset -\frac{m_H^2}{2} H^2 + \frac{m_H^2}{2v_H} H^3 + \frac{m_H^2}{8 v_H^2} H^4 \\ &\quad - \mu_S^2 \, S^2 + \kappa \, S^3 + \lambda_S \, S^4 + \frac{\kappa_3}{2} (H+v_H)^2 S + \frac{\lambda_3}{2} (H+v_H)^2 S^2 \; . {} \end{aligned} $$

(4.3)

A possible linear term in the new, real field is removed by a shift in the fields. In the above form the new scalar S can couple to two SM Higgs bosons, which induces a decay either on-shell S → HH or off-shell S → H ^∗ H ^∗→ 4b. To forbid this, we apply the usual trick, which is behind essentially all WIMP dark matter models; we require the Lagrangian to obey a global $\mathbb {Z}_2$ symmetry

$$\displaystyle \begin{aligned} S \to -S, \qquad \qquad H \to +H, \quad \cdots \end{aligned} $$

(4.4)

This defines an ad-hoc $\mathbb {Z}_2$ parity + 1 for all SM particles and − 1 for the dark matter candidate. The combined potential now reads

$$\displaystyle \begin{aligned} V &\supset - \frac{m_H^2}{2} H^2 + \frac{m_H^2}{2v_H} H^3 + \frac{m_H^2}{8 v_H^2} H^4 - \mu_S^2 \, S^2 + \lambda_S \, S^4 + \frac{\lambda_3}{2} (H+v_H)^2 S^2 \\ &= - \frac{m_H^2}{2} H^2 + \frac{m_H^2}{2v_H} H^3 + \frac{m_H^2}{8 v_H^2} H^4 \\ &\quad - \left( \mu_S^2 - \lambda_3 \frac{v_H^2}{2} \right) \, S^2 + \lambda_S \, S^4 + \lambda_3 v_H \, H S^2 + \frac{\lambda_3}{2} H^2 S^2 \; . {} \end{aligned} $$

(4.5)

The mass of the dark matter scalar and its phenomenologically relevant SSH and SSHH couplings read

$$\displaystyle \begin{aligned} m_S = \sqrt{ 2 \mu_S^2 - \lambda_3 v_H^2 } \qquad g_{SSH} = - 2 \lambda_3 v_H \qquad g_{SSHH} = - 2 \lambda_3 \; . {} \end{aligned} $$

(4.6)

The sign of λ ₃ is a free parameter. Unlike for singlet models with a second VEV, the dark singlet does not affect the SM Higgs relations in Eq. (4.2). However, the SSH coupling mediates SS interactions with pairs of SM particles through the light Higgs pole, as well as Higgs decays H → SS, provided the new scalar is light enough. The SSHH coupling can mediate heavy dark matter annihilation into Higgs pairs. We will discuss more details on invisible Higgs decays in Chap. 7.

For dark matter annihilation, the $SSf \bar {f}$ transition matrix element based on the Higgs portal is described by the Feynman diagram

All momenta are defined incoming, giving us for an outgoing fermion and an outgoing anti-fermion

$$\displaystyle \begin{aligned} \mathcal{M} &= \bar{u}(k_2) \, \frac{-i m_f}{v_H} \, v(k_1) \; \frac{-i}{(k_1 + k_2)^2 - m_H^2 + i m_H \varGamma_H} \; (- 2 i \lambda_3 v_H ) \; . {} \end{aligned} $$

(4.7)

In this expression we see that v _H cancels, but the fermion mass m _f will appear in the expression for the annihilation rate. We have to square this matrix element, paying attention to the spinors v and u, and then sum over the spins of the external fermions,

(4.8)

In the sum over spin and color of the external fermions the averaging is not yet included, because we need to specify which of the external particles are incoming or outgoing. As an example, we compute the cross section for the dark matter annihilation process to a pair of bottom quarks

$$\displaystyle \begin{aligned} S S \to H^* \to b \bar{b} \; . \end{aligned} $$

(4.9)

This s-channel annihilation corresponds to the leading on-shell Higgs decay $H \to b\bar {b}$ with a branching ratio around 60%. In terms of the Mandelstam variable s = (k ₁ + k ₂)² it gives us

$$\displaystyle \begin{aligned} \overline{ \sum_{\text{spin,color}} |\mathcal{M}|{}^2 } &= N_c \; 8 \lambda_3^2 m_b^2 \; \frac{s - 4 m_b^2}{\left( s - m_H^2 \right)^2 + m_H^2 \varGamma_H^2} \\ \Rightarrow \qquad \sigma(SS \to b\bar{b}) &= \frac{1}{16 \pi s} \; \sqrt{ \frac{1 - 4 m_b^2/s}{1 - 4 m_S^2/s}} \; \overline{ \sum |\mathcal{M}|{}^2 } \\ &= \frac{N_c}{2 \pi \sqrt{s}} \; \lambda_3^2 m_b^2 \; \sqrt{ \frac{1 - 4 m_b^2/s}{s - 4 m_S^2}} \; \frac{s - 4 m_b^2}{\left( s - m_H^2 \right)^2 + m_H^2 \varGamma_H^2} \; . {} \end{aligned} $$

(4.10)

To compute the relic density we need the velocity-averaged cross section. For the contribution of the $b\bar {b}$ final state to the dark matter annihilation rate we find the leading term in the non-relativistic limit, $s = 4 m_S^2$

$$\displaystyle \begin{aligned} \langle \sigma v \rangle \Bigg|{}_{SS \to b\bar{b}} \equiv \sigma v \Bigg|{}_{SS \to b\bar{b}} &\stackrel{\text{Eq. (3.20)}}{=} v \; \frac{N_c \lambda_3^2 m_b^2}{2 \pi \sqrt{s}} \; \frac{\sqrt{1 - 4 m_b^2/s}}{m_S v} \; \frac{s - 4 m_b^2}{\left( s - m_H^2 \right)^2 + m_H^2 \varGamma_H^2} \\ &\stackrel{\text{threshold}}{=} \frac{N_c \lambda_3^2 m_b^2}{4 \pi m_S^2} \; \sqrt{1 - \frac{m_b^2}{m_S^2}} \; \frac{4 m_S^2 - 4 m_b^2}{\left( 4 m_S^2 - m_H^2 \right)^2 + m_H^2 \varGamma_H^2} \\ &\stackrel{m_S \gg m_b}{=} \frac{N_c \lambda_3^2 m_b^2}{\pi} \; \frac{1}{\left( 4 m_S^2 - m_H^2 \right)^2 + m_H^2 \varGamma_H^2} \; . {} \end{aligned} $$

(4.11)

This expression holds for all scalar masses m _S. In our estimate we identify the v-independent expression with the thermal average. Obviously, this will become more complicated once we include the next term in the expansion around v ≈ 0. The Breit–Wigner propagator guarantees that the rate never diverges, even in the case when the annihilating dark matter hits the Higgs pole in the s-channel.

The simplest parameter point to evaluate this annihilation cross section is on the Higgs pole. This gives us

(4.12)

with Γ _H ≈ 4 ⋅ 10⁻⁵ m _H. While it is correct that the self coupling required on the Higgs pole is very small, the full calculation leads to a slightly larger value λ ₃ ≈ 10⁻³, as shown in Fig. 4.1.

Fig. 4.1

Higgs portal parameter space in terms of the self coupling λ _hSS ∼ λ ₃ and the dark matter mass M _DM = m _S. The red lines indicate the correct relic density Ω _χ h ². Figure from Ref. [2]

Lighter dark matter scalars also probe the Higgs mediator on-shell. In the Breit-Wigner propagator of the annihilation cross section, Eq. (4.11), we have to compare to the two terms

$$\displaystyle \begin{aligned} m_H^2 - 4 m_S^2 = m_H^2 \left( 1 - \frac{4 m_S^2}{m_H^2} \right) \quad \Leftrightarrow \quad m_H \varGamma_H \approx 4 \cdot 10^{-5} \; m_H^2 \; . \end{aligned} $$

(4.13)

The two states would have to fulfill exactly the on-shell condition m _H = 2m _S for the second term to dominate. We can therefore stick to the first term for m _H > 2m _S and find for the dominant decay to $b\bar {b}$ pairs in the limit $m_H^2 \gg m_S^2 \gg m_b^2$

(4.14)

Heavier dark matter scalars well above the Higgs pole also include the annihilation channels

$$\displaystyle \begin{aligned} SS \to \tau^+ \tau^-, W^+ W^-, ZZ, HH, t\bar{t} \; . \end{aligned} $$

(4.15)

Unlike for on-shell Higgs decays, the $b\bar {b}$ final state is not dominant for dark matter annihilation when it proceeds through a 2 → 2 process. Heavier particles couple to the Higgs more strongly, so above the Higgs pole they will give larger contributions to the dark matter annihilation rate. For top quarks in the final state this simply means replacing the Yukawa coupling $m_b^2$ by the much larger $m_t^2$. In addition, the Breit-Wigner propagator will no longer scale like $1/m_H^2$, but proportional to $1/m_S^2$. Altogether, this gives us a contribution to the annihilation rate of the kind

$$\displaystyle \begin{aligned} \langle \sigma v \rangle \Bigg|{}_{SS \to t\bar{t}} = \frac{N_c \lambda_3^2 m_t^2}{\pi \left( 4 m_S^2 - m_H^2 \right)^2} \stackrel{2 m_S \gg m_H}{=} \frac{N_c \lambda_3^2 m_t^2}{16 \pi m_S^4} \; . {} \end{aligned} $$

(4.16)

The real problem is the annihilation to the weak bosons W, Z, because it leads to a different scaling of the annihilation cross section. In the limit of large energies we can describe for example the process SS → W ⁺ W ⁻ using spin-0 Nambu-Goldstone bosons in the final state. These Nambu-Goldstone modes in the Higgs doublet ϕ appear as the longitudinal degrees of freedom, which means that dark matter annihilation to weak bosons at large energies follows the same pattern as dark matter annihilation to Higgs pairs. Because we are more used to the Higgs degree of freedom we calculate the annihilation to Higgs pairs,

$$\displaystyle \begin{aligned} SS \to HH \; . \end{aligned} $$

(4.17)

The two Feynman diagrams with the direct four-point interaction and the Higgs propagator at the threshold $s = 4 m_S^2$ scale like

$$\displaystyle \begin{aligned} \mathcal{M}_4 &= g_{SSHH} = - 2 \lambda_3 \\ \mathcal{M}_H &= \frac{g_{SSH}}{s - m_H^2} \frac{3 m_H^2}{v_H} \stackrel{\text{threshold}}{=} - \frac{2 \lambda_3 v_H}{4 m_S^2 - m_H^2} \frac{3 m_H^2}{v_H} \stackrel{m_S \gg m_H}{=} - \frac{6 \lambda_3 m_H^2}{4 m_S^2} \ll \mathcal{M}_4 \; . \end{aligned} $$

(4.18)

This means for heavy dark matter we can neglect the s-channel Higgs propagator contribution and focus on the four-scalar interaction. In analogy to Eq. (4.11) we then compute the velocity-weighted cross section at threshold,

$$\displaystyle \begin{aligned} \sigma (SS \to HH) &= \frac{1}{16 \pi \sqrt{s}} \; \frac{\sqrt{1 - 4 m_H^2/m_S^2}}{\sqrt{s - 4 m_S^2}} \; 4 \lambda_3^2 \stackrel{\text{Eq. (3.20)}}{=} \frac{\lambda_3^2}{4 \pi \sqrt{s}} \sqrt{1 - \frac{4 m_H^2}{m_S^2}} \; \frac{1}{v m_S} \\ \sigma v \Bigg|{}_{SS \to HH} &\kern-1pt=\kern-1pt \frac{\lambda_3^2}{4 \pi m_S \sqrt{s}} \; \sqrt{1 \kern-1pt-\kern-1pt \frac{4 m_H^2}{m_S^2}} \kern-1pt\stackrel{\text{threshold}}{=}\kern-1pt \frac{\lambda_3^2}{8 \pi m_S^2} \; \sqrt{1 \kern-1pt-\kern-1pt \frac{4 m_H^2}{m_S^2}} \kern-1pt\stackrel{m_S \gg m_H}{=}\kern-1pt \frac{\lambda_3^2}{8 \pi m_S^2} \end{aligned} $$

(4.19)

For m _S = 200 GeV we can derive the coupling λ ₃ which we need to reproduce the observed relic density,

(4.20)

The curve in Fig. 4.1 shows two thresholds related to four-point annihilation channels, one at m _S = m _Z and one at m _S = m _H. Starting with m _S = 200 GeV and corresponding values for λ ₃ the annihilation to Higgs and Goldstone boson pairs dominates the annihilation rate.

One lesson to learn from our Higgs portal considerations is the scaling of the dark matter annihilation cross section with the WIMP mass m _S. It does not follow Eq. (3.3) at all and only follows Eq. (3.35) for very heavy dark matter. For our model, where the annihilation is largely mediated by a Yukawa coupling m _b, we find

(4.21)

It will turn out that the most interesting scaling is on the Higgs peak, because the Higgs width is not at all related to the weak scale.

4.2 Vector Portal

Inspired by the WIMP assumption in Eq. (3.3) we can use a new massive gauge boson to mediate thermal freeze-out production. The combination of a free vector mediator mass and a free dark matter mass will allow us to study a similar range scenarios as for the Higgs portal, Eq. (4.21). A physics argument is given by the fact that the Standard Model has a few global symmetries which can be extended to anomaly-free gauge symmetries.

The extension of the Standard Model with its hypercharge symmetry U(1)_Y by an additional U(1) gauge group defines another renormalizable portal to dark matter. Since U(1)-field strength tensors are gauge singlets, the kinetic part of the Lagrangian allows for kinetic mixing ,

$$\displaystyle \begin{aligned} \mathcal{L}_{\text{gauge}} & = -\frac{1}{4}\hat B^{\mu\nu}\hat B_{\mu\nu} -\frac{s_\chi}{2}\hat V^{\mu\nu} \hat B_{\mu\nu} -\frac{1}{4}\hat V^{\mu\nu}\hat V_{\mu\nu} \\ & = -\frac{1}{4} \begin{pmatrix} \hat{B}_{\mu \nu} & \hat{V}_{\mu \nu} \end{pmatrix} \begin{pmatrix} 1 & s_\chi \\ s_\chi & 1 \end{pmatrix} \begin{pmatrix} \hat{B}_{\mu \nu} \\ \hat{V}_{\mu \nu} \end{pmatrix} \; , {} \end{aligned} $$

(4.22)

where $s_\chi \equiv \sin \chi$ is assumed to be a small mixing parameter. In principle it does not have to be an angle, but for the purpose of these lecture notes we assume that it is small, s _χ ≪ 1, so we can treat it as a trigonometric function and write $c_\chi \equiv \sqrt {1-s_\chi }$ and t _χ ≡ s _χ∕c _χ. Even if the parameter s _χ is chosen to be zero at tree-level, loops of particles charged under both U(1)_X and U(1)_Y introduce a non-zero value for it. Similar to the Higgs portal, there is no symmetry that forbids it, so we do not want to assume that all quantum corrections cancel to a net value s _χ = 0.

The notation $\hat B_{\mu \nu }$ indicates that the gauge fields are not yet canonically normalized, which means that the residue of the propagator it not one. In addition, the gauge boson propagators derived from Eq. (4.22) are not diagonal. We can diagonalize the matrix in Eq. (4.22) and keep the hypercharge unchanged with a non-orthogonal rotation of the gauge fields

$$\displaystyle \begin{aligned} \begin{pmatrix} \hat{B}_\mu \\\hat W^3_\mu \\\hat{V}_\mu \end{pmatrix} = G(\theta_{V}) \, \begin{pmatrix} B_\mu \\ W^3_\mu\\V_\mu \end{pmatrix} = \begin{pmatrix} 1 & 0&-s_\chi/c_\chi \\ 0&1&0\\0 &0&1/c_\chi \end{pmatrix} \begin{pmatrix} B_\mu \\ W^3_\mu\\V_\mu \end{pmatrix} \; . {} \end{aligned} $$

(4.23)

We now include the third component of the SU(2)_L gauge field triplet $W_\mu =(W^1_\mu , W^2_\mu , W^3_\mu )$ which mixes with the hypercharge gauge boson through electroweak symmetry breaking to produce the massive Z boson and the massless photon. Kinetic mixing between the SU(2)_L field strength tensor and the U(1)_X field strength tensor is forbidden because $\hat V^{\mu \nu } \hat A^a_{\mu \nu }$ is not a gauge singlet. Assuming a mass $\hat m_V$ for the V -boson we write the combined mass matrix as

$$\displaystyle \begin{aligned} \mathcal{M}^2 \stackrel{\text{Eq. (4.23)}}{=}\frac{v^2}{4} \begin{pmatrix} g^{\prime2} & -g\,g^{\prime} &-{g^{\prime}}^2 s_\chi \\ {} -g\,g^{\prime} & g^2 & g\,g^{\prime} \,s_\chi\\ -{g^{\prime}}^2 s_\chi \quad & \quad g\,g^{\prime} s_\chi \quad & \quad \dfrac{4\hat m_V^2}{v^2}(1+s_\chi^2)+g^{\prime 2} s_\chi^2 \end{pmatrix} +\mathcal{O}(s_\chi^3) \; . {} \end{aligned} $$

(4.24)

This mass matrix can be diagonalized with a combination of two block-diagonal rotations with the weak mixing matrix and an additional angle ξ,

$$\displaystyle \begin{aligned} R_1(\xi)R_2(\theta_w)=\begin{pmatrix} 1&0 & 0\\ 0&c_\xi & s_\xi \\ 0&-s_\xi & c_\xi \end{pmatrix}\, \begin{pmatrix} c_w& s_w & 0\\ -s_w& c_w&0\\ 0&0&1\end{pmatrix} \; , \end{aligned} $$

(4.25)

giving

$$\displaystyle \begin{aligned} R_1(\xi)R_2(\theta_w) \mathcal{M}^2\,R_2(\theta_w)^TR_1(\xi)^T= \begin{pmatrix} m_\gamma^2&0&0\\ 0&m_Z^2&0\\ 0&0&m_V^2\end{pmatrix}\; , {} \end{aligned} $$

(4.26)

provided

$$\displaystyle \begin{aligned} \tan 2 \xi = \frac{2s_\chi s_w}{1- \dfrac{\hat m_V^2}{\hat m_Z^2} }+\mathcal{O}(s_\chi^2) \; . \end{aligned} $$

(4.27)

For this brief discussion we assume for the mass ratio

$$\displaystyle \begin{aligned} \frac{\hat m_V^2}{\hat m_Z^2} = \frac{2 \hat m_V^2}{(g^2+g^{\prime2}) v^2} \ll 1\,, \end{aligned} $$

(4.28)

and find for the physical masses

$$\displaystyle \begin{aligned} m_\gamma^2=0\,, \qquad m_Z^2 = \hat m_Z^2 \left[ 1+s_\chi^ 2s_w^2 \left( 1+ \frac{\hat m_V^2}{\hat m_Z^2} \right) \right]\,, \qquad m_{V}^2 = \hat m_V^2 \left[ 1+s_\chi^2c_w^2\right] \; . \end{aligned} $$

(4.29)

In addition to the dark matter mediator mass we also need the coupling of the new gauge boson V  to SM matter. Again, we start with the neutral currents for the not canonically normalized gauge fields and rotate them to the physical gauge bosons defined in Eq. (4.26),

$$\displaystyle \begin{aligned} &\left(ej_{\text{EM}} , \frac{e}{\sin \theta_w \cos \theta_w} j_Z, g_{D}j_{D}\right) \begin{pmatrix}\hat A\\ \hat Z\\\hat A'\end{pmatrix} \\ & \quad =\left(ej_{\text{EM}} , \frac{e}{\sin \theta_w \cos \theta_w} j_Z, g_{D}j_{D}\right) \,K\,\begin{pmatrix}A\\ Z\\ V\end{pmatrix} \,, \end{aligned} $$

(4.30)

with

$$\displaystyle \begin{aligned} K=\left[ R_1(\xi)R_2(\theta_w)G^{-1}(\theta_\chi)R_2(\theta_w)^{-1}\right]^{-1} \approx \begin{pmatrix} 1 & 0 & -s_\chi c_w \\ 0 & 1& 0 \\ 0 & s_\chi s_w& 1 \end{pmatrix} \; . {} \end{aligned} $$

(4.31)

The new gauge boson couples to the electromagnetic current with a coupling strength of − s _χ c _w e, while to leading order in s _χ and $\hat m_V/\hat m_Z$ its coupling to the Z-current vanishes. It is therefore referred to as hidden photon. This behavior changes for larger masses, $\hat m_V/\hat m_Z \gtrsim 1$, for which the coupling to the Z-current can be the dominating coupling to SM fields. In this case the new gauge boson is called a Z′-boson. For the purpose of these lecture notes we will concentrate on the light V -boson, because it will allow for a light dark matter particle.

There are two ways in which the hidden photon could be relevant from a dark matter perspective. The new gauge boson could be the dark matter itself, or it could provide a portal to a dark matter sector if the dark matter candidate is charged under U(1)_X. The former case is problematic, because the hidden photon is not stable and can decay through the kinetic mixing term. Even if it is too light to decay into the lightest charged particles, electrons, it can decay into neutrinos $V\to \nu \bar \nu$ through the suppressed mixing with the Z-boson and into three photons V → 3γ through loops of charged particles. For mixing angles small enough to guarantee stability on time scales of the order of the age of the universe, the hidden photon can therefore not be thermal dark matter.

If the hidden photon couples to a new particle charged under a new U(1)_X gauge group, this particle could be a dark matter candidate. For a new Dirac fermion with U(1)_X charge Q _X, we add a kinetic term with a covariant derivative to the Lagrangian,

$$\displaystyle \begin{aligned} \mathcal{L}_{\text{DM}}=\bar \chi i\gamma^\mu D_\mu \chi -m_\chi \bar\chi\chi \qquad \text{with} \quad D_\mu=\partial_\mu-ig_D Q_\chi V_\mu \; . \end{aligned} $$

(4.32)

Through the U(1)_X-mediator this dark fermion is in thermal contact with the Standard Model through the usual annihilation shown in Eq. (3.1). If the dark matter is lighter than the hidden photon and the electron m _V > m _χ > m _e, the dominant s-channel Feynman diagram contributing to the annihilation cross section is shown on the left of Fig. 4.2. This diagram resembles the one shown above Eq. (4.7) for the case of a Higgs portal and the cross section can be computed in analogy to Eq. (4.10),

$$\displaystyle \begin{aligned} \sigma(\chi\bar\chi \to e^+e^-) &=\frac{1}{12\pi}(s_\chi c_w e g_D Q_\chi)^2 \left(1+\frac{2m_e^2}{s}\right) \left(1+\frac{2m_\chi^2}{s}\right) \\ & \quad \times \frac{s}{(s-m_V^2)^2+m_V^2\varGamma_V^2}\, \frac{\sqrt{1-4\dfrac{m_e^2}{m_V^2}}}{\sqrt{1-4\dfrac{m_\chi^2}{m_V^2}}} \; , \end{aligned} $$

(4.33)

Fig. 4.2

Feynman diagrams contributing to the annihilation of dark matter coupled to the visible sector through a hidden photon

with Γ _V the total width of the hidden photon V . For the annihilation of two dark matter particles $s=4m_\chi ^2$, and assuming m _V ≫ Γ _V, the thermally averaged annihilation cross section is given by

$$\displaystyle \begin{aligned} \langle \sigma v\rangle &= \frac{1}{4\pi}(s_\chi c_w e g_D Q_\chi)^2\sqrt{1-\frac{m_e^2}{m_\chi^2}}\left(1+\frac{m_e^2}{2m_\chi^2}\right)\frac{4m_\chi^2}{(4m_\chi^2-m_V^2)^2} \\ &\quad \stackrel{m_\chi\gg m_e}{\approx}\frac{m_\chi^2}{\pi m_V^4}(s_\chi c_w e Q_f g_D Q_\chi)^2\; . \end{aligned} $$

(4.34)

It exhibits the same scaling as in the generic WIMP case of Eq. (3.3). In contrast to the WIMP, however, the gauge coupling is rescaled by the mixing angle s _χ and for very small mixing angles the hidden photon can in principle be very light. In Eq. (4.34) we assume that the dark photon decays into electrons. Since the hidden photon branching ratios into SM final states are induced by mixing with the photon, for masses m _V > 2m _μ the hidden photon also decays into muons and for hidden photon masses above a few 100 MeV and below 2m _τ, the hidden photon decays mainly into hadronic states. For m _V > m _χ, the PLANCK bound on the DM mass in Eq. (3.44) implies m _V > 10 GeV and Eq. (4.34) would need to be modified by the branching ratios into the different kinematically accessible final states. Instead, we illustrate the scaling with the scenario Q _χ = 1, m _χ = 10 MeV and m _V = 100 MeV that is formally excluded by the PLANCK bound, but only allows for hidden photon decays into electrons. In this case, we find the observed relic density given in Eq. (3.32) for a coupling strength

(4.35)

In the opposite case of m _χ > m _V > m _e, the annihilation cross section is dominated by the diagram on the right of Fig. 4.2, with subsequent decays of the hidden photon. The thermally averaged annihilation cross section then reads

$$\displaystyle \begin{aligned} \langle \sigma v\rangle=\frac{g_D^4 Q_\chi^4}{8\pi}\frac{1}{m_\chi^2}\frac{\left(1-\dfrac{m_V^2}{m_\chi^2}\right)^{\frac{3}{2}}}{\left(1-\dfrac{m_V^2}{2m_\chi^2}\right)^2} \stackrel{m_\chi \gg m_V}{\approx} \frac{g_D^4 Q_\chi^4}{8\pi m_\chi^2}\; . \end{aligned} $$

(4.36)

The scaling with the dark matter mass is the same as for a WIMP with m _χ > m _Z, as shown in Eq. (3.35). The annihilation cross section is in principle independent of the mixing angle s _χ, motivating the name secluded dark matter for such models, but the hidden photon needs to eventually decay into SM particles. Again assuming Q _χ = 1, and m _χ = 10 GeV, we find

(4.37)

4.3 Supersymmetric Neutralinos

Supersymmetry is a (relatively) fashionable model for physics beyond the Standard Model which provides us with a very general set of dark matter candidates. Unlike the portal model described in Sect. 4.1 the lightest supersymmetric partner (LSP) is typically a fermion, more specifically a Majorana fermion. Majorana fermions are their own anti-particles. An on-shell Dirac fermion, like an electron, has four degrees of freedom; for the particle e ⁻ we have two spin directions, and for the anti-particle e ⁺ we have another two. The Majorana fermion only has two degrees of freedom. The reason why the minimal supersymmetric extension of the Standard Model, the MSSM , limits us to Majorana fermions is that the photon as a massless gauge boson only has two degrees of freedom. This holds for both, the bino partner of the hypercharge gauge boson B and the wino partner of the still massless SU(2)_L gauge boson W ³. Just like the gauge bosons in the Standard Model mix to the photon and the Z, the bino and wino mix to form so-called neutralinos. The masses of the physical state can be computed from the bino mass parameter M ₁ and the wino mass parameter M ₂.

For reasons which we do not have to discuss in these lecture notes, the MSSM includes a non-minimal Higgs sector: the masses of up-type and down-type fermions are not generated from one Higgs field. Instead, we have two Higgs doublets with two vacuum expectation values v _u and v _d. Because both contribute to the weak gauge boson masses, their squares have to add to

$$\displaystyle \begin{aligned} v_u^2 + v_d^2 &= v_H^2 = (246~\text{GeV} )^2 \\ \Leftrightarrow \qquad v_u &= v_H \cos \beta \qquad v_d = v_H \sin \beta \qquad \Leftrightarrow \qquad \tan \beta = \frac{v_u}{v_d} \; . \end{aligned} $$

(4.38)

Two Higgs doublets include eight degrees of freedom, out of which three Nambu-Goldstone modes are needed to make the weak bosons massive. The five remaining degrees of freedom form a light scalar h ⁰, a heavy scalar H ⁰, a pseudo-scalar A ⁰, and a charged Higgs H ^±. Altogether this gives four neutral and four charged degrees of freedom. In the Standard Model we know that the one neutral (pseudo-scalar) Nambu-Goldstone mode forms one particle with the W ³ gauge bosons. We can therefore expect the supersymmetric higgsinos to mix with the bino and wino as well. Because the neutralinos still are Majorana fermions, the eight degrees of freedom form four neutralino states $\tilde {\chi }_i^0$. Their mass matrix has the form

(4.39)

The mass matrix is real and therefore symmetric. In the upper left corner the bino and wino mass parameters appear, without any mixing terms between them. In the lower right corner we see the two higgsino states. Their mass parameter is μ, the minus sign is conventional; by definition of the Higgs potential it links the up-type and down-type Higgs or higgsino fields, so it has to appear in the off-diagonal entries. The off-diagonal sub-matrices are proportional to m _Z. In the limit s _w → 0 and $\sin \beta = \cos \beta = 1/\sqrt {2}$ a universal mixing mass term $m_Z/\sqrt {2}$ between the wino and each of the two higgsinos appears. It is the supersymmetric counterpart of the combined Goldstone-W ³ mass m _Z.

As any symmetric matrix, the neutralino mass matrix can be diagonalized through a real orthogonal rotation,

$$\displaystyle \begin{aligned} & N \; \mathcal{M} \; N^{-1} = \text{diag} \left( m_{\tilde{\chi}_j^0} \right) \qquad j=1,2 \end{aligned} $$

(4.40)

It is possible to extend the MSSM such that the dark matter candidates become Dirac fermions, but we will not explore this avenue in these lecture notes.

Because the SU(2)_L gauge bosons as well as the Higgs doublet include charged states, the neutralinos are accompanied by chargino states. They cannot be Majorana particles, because they carry electric charge. However, as a remainder of the neutralino Majorana property they do not have a well-defined fermion number, like electrons or positrons have. The corresponding chargino mass matrix will not include a bino-like state, so it reads

$$\displaystyle \begin{aligned} \mathcal{M} = \begin{pmatrix} M_2 & \sqrt{2} m_W \sin \beta \\ \sqrt{2} m_W \cos \beta & \mu \end{pmatrix} {} \end{aligned} $$

(4.41)

It includes the remaining four degrees of freedom from the wino sector and four degrees of freedom from the higgsino sector. As for the neutralinos, the wino and higgsino components mix via a weak mass term. Because the chargino mass matrix is real and not symmetric, it can only be diagonalized using two unitary matrices,

$$\displaystyle \begin{aligned} U^* \; \mathcal{M} \; V^{-1} = \text{diag} \left( m_{\tilde{\chi}_j^\pm} \right) \qquad j=1,2 \end{aligned} $$

(4.42)

For the dark matter phenomenology of the neutralino–chargino sector it will turn out that the mass difference between the lightest neutralino(s) and the lightest chargino are the relevant parameters. The reason is a possible co-annihilation process as described in Sect. 3.3

We can best understand the MSSM dark matter sector in terms of the different SU(2)_L representations. The bino state as the partner of the hypercharge gauge boson is a singlet under SU(2)_L. The wino fields with the mass parameter M ₂ consist of two neutral degrees of freedom as well as four charged degrees of freedom, one for each polarization of W ^±. Together, the supersymmetric partners of the W boson vector field also form a triplet under SU(2)_L. Finally, each of the two higgsinos arise as supersymmetric partner of an SU(2)_L Higgs doublet. The neutralino mass matrix in Eq. (4.39) therefore interpolates between singlet, doublet, and triplet states under SU(2)_L.

The most relevant couplings of the neutralinos and charginos we need to consider for our dark matter calculations are

$$\displaystyle \begin{aligned} g_{Z \tilde{\chi}_1^0 \tilde{\chi}_1^0} &= \frac{g}{2 c_w} \; \left( |N_{13}|{}^2 - |N_{14}|{}^2 \right) \\ g_{h \tilde{\chi}_1^0 \tilde{\chi}_1^0} &= \left( g' N_{11} - g N_{12} \right) \; \left( \sin \alpha \; N_{13} + \cos \alpha \; N_{14} \right) \\ g_{A \tilde{\chi}_1^0 \tilde{\chi}_1^0} &= \left( g' N_{11} - g N_{12} \right) \; \left( \sin \beta \; N_{13} - \cos \beta \; N_{14} \right) \\ g_{\gamma \tilde{\chi}_1^+ \tilde{\chi}_1^-} &= e \\ g_{W \tilde{\chi}_1^0 \tilde{\chi}_1^+} &= g \; \left( \frac{1}{\sqrt{2}} N_{14} V_{12}^* - N_{12} V_{11}^* \right) \; , {} \end{aligned} $$

(4.43)

with e = gs _w, $s_w^2 \approx 1/4$ and hence $c_w^2 \approx 3/4$. The mixing angle α describes the rotation from the up-type and down-type supersymmetric Higgs bosons into mass eigenstates. In the limit of only one light Higgs boson with a mass of 126 GeV it is given by the decoupling condition $\cos (\beta - \alpha ) \to 0$. The above form means for those couplings which contribute to the (co-) annihilation of neutralino dark matter

• neutralinos couple to weak gauge bosons through their higgsino content

• neutralinos couple to the light Higgs through gaugino–higgsino mixing

• charginos couple to the photon diagonally, like any other charged particle

• neutralinos and charginos couple to a W-boson diagonally as higgsinos and gauginos

Finally, supersymmetry predicts scalar partners of the quarks and leptons, so-called squarks and sleptons. For the partners of massless fermions, for example squarks $\tilde {q}$, there exists a $q \tilde {q} \tilde {\chi }_j^0$ coupling induced through the gaugino content of the neutralinos. If this kind of coupling should contribute to neutralino dark matter annihilation, the lightest supersymmetric scalar has to be almost mass degenerate with the lightest neutralino. Because squarks are strongly constrained by LHC searches and because of the pattern of renormalization group running, we usually assume one of the sleptons to be this lightest state. In addition, the mixing of the scalar partners of the left-handed and right-handed fermions into mass eigenstates is driven by the corresponding fermion mass, the most attractive co-annihilation scenario in the scalar sector is stau–neutralino co-annihilation. However, in these lecture notes we will focus on a pure neutralino–chargino dark matter sector and leave the discussion of the squark–quark–neutralino coupling to Chap. 7 on LHC searches.

Similar to the previous section we now compute the neutralino annihilation rate, assuming that in the 10–1000 GeV mass range they are thermally produced. For mostly bino dark matter with

$$\displaystyle \begin{aligned} M_1 \ll M_2, |\mu| {} \end{aligned} $$

(4.44)

the annihilation to the observed relic density is a problem. There simply is no relevant 2 → 2 Feynman diagram, unless there is help from supersymmetric scalars $\tilde {f}$, as shown in Fig. 4.3. If for example light staus appear in the t-channel of the annihilation rate we find

$$\displaystyle \begin{aligned} \sigma(\tilde{B} \tilde{B} \to f \bar{f}) \approx \frac{g^4 m_{\tilde{\chi}_1^0}^2}{16 \pi m_{\tilde{f}}^4} \qquad \text{with} \quad m_{\tilde{\chi}_1^0} \approx M_1 \ll m_{\tilde{f}} \; . \end{aligned} $$

(4.45)

The problem with pure neutralino annihilation is that in the limit of relatively heavy sfermions the annihilation cross section drops rapidly, leading to a too large predicted bino relic density. Usually, this leads us to rely on stau co-annihilation for a light bino LSP. Along these lines it is useful to mention that with gravity-mediated supersymmetry breaking we assume M ₁ and M ₂ to be identical at the Planck scale, which given the beta functions of the hypercharge and the weak interaction turns into the condition M ₁ ≈ M ₂∕2 at the weak scale, i.e. light bino dark matter would be a typical feature in these models.

Fig. 4.3

Sample Feynman diagrams for the annihilation of supersymmetric binos (left), winos (center), and higgsinos

If M ₁ becomes larger than M ₂ or μ we can to a good approximation consider the limit

$$\displaystyle \begin{aligned} M_2 \ll M_1 \to \infty \qquad \text{and} \qquad |\mu| \ll M_1 \to \infty \; , \end{aligned} $$

(4.46)

In that case we know that independent of the relation of M ₂ and μ there will be at least the lightest chargino close in mass to the LSP. It appears in the t-channel of the actual annihilation diagram and as a co-annihilation partner. To avoid a second neutralino in the co-annihilation process we first consider wino dark matter,

$$\displaystyle \begin{aligned} M_2 \ll \mu, M_1, m_{\tilde{f}} \; . \end{aligned} $$

(4.47)

From the list of neutralino couplings in Eq. (4.43) we see that in the absence of additional supersymmetric particles pure wino dark matter can annihilate through a t-channel chargino, as illustrated in Fig. 4.3. Based on the known couplings and on dimensional arguments the annihilation cross section should scale like

(4.48)

The scaling with the mass of the dark matter agent does not follow our original postulate for the WIMP miracle in Eq. (3.3), which was $\sigma _{\chi \chi } \propto m_{\tilde {\chi }_1^0}^2/m_W^4$. If we only rely on the direct dark matter annihilation, the observed relic density translated into a comparably light neutralino mass,

$$\displaystyle \begin{aligned} \langle \sigma v \rangle \Bigg|{}_{\tilde{W} \tilde{W} \to W^+ W^-} = \frac{g^4 s_w^4}{16 \pi c_w^4 m_{\tilde{\chi}_1^0}^2} \approx \frac{0.7^4}{450 \, m_{\tilde{\chi}_1^0}^2} &\stackrel{\text{Eq. (3.35)}}{=} 1.7 \cdot 10^{-9} \frac{1}{\text{GeV}^2} \\ \Leftrightarrow \qquad m_{\tilde{\chi}_1^0} &\approx 560~\text{GeV} \; . \end{aligned} $$

(4.49)

However, this estimate is numerically poor. The reason is that in contrast to this lightest neutralino, the co-annihilating chargino can annihilate through a photon s-channel diagram into charged Standard Model fermions,

$$\displaystyle \begin{aligned} \sigma( \tilde{\chi}_1^+ \tilde{\chi}_1^- \to \gamma^* \to f\bar{f} ) \approx \sum_f \frac{N_c e^4}{16 \pi m_{\tilde{\chi}_1^\pm}^2} = \sum_f \frac{N_c g^4 s_w^2}{16 \pi m_{\tilde{\chi}_1^\pm}^2} \; . {} \end{aligned} $$

(4.50)

For light quarks alone the color factor combined with the sum over flavors adds a factor 5 × 3 = 15 to the annihilation rate. In addition, for $\tilde {\chi }_1^+ \tilde {\chi }_1^-$ annihilation we need to take into account the Sommerfeld enhancement through photon exchange between the slowly moving incoming charginos, as derived in Sect. 3.5. This gives us the correct values

(4.51)

In Fig. 4.4 and in the left panel of Fig. 4.5 this wino LSP mass range appears as a horizontal plateau in M ₂, with and without the Sommerfeld enhancement. In the right panel of Fig. 4.5 we show the mass difference between the lightest neutralino and the lighter chargino. Typical values for a wino-LSP mass splitting are around Δm = 150 MeV, sensitive to loop corrections to the mass matrices shown in Eqs. (4.39) and (4.41).

Fig. 4.4

Combinations of neutralino mass parameters M ₁, M ₂, μ that produce the correct relic abundance, accounting for Sommerfeld-enhancement, along with the LSP mass. The relic surface without Sommerfeld enhancement is shown in gray. Figure from Ref. [3]

Fig. 4.5

Left: combinations of neutralino mass parameters M ₁, M ₂, μ that produce the correct relic abundance, not accounting for Sommerfeld-enhancement, along with the leading annihilation product. Parameters excluded by LEP are occluded with a white or black box. Right: mass splitting between the lightest chargino and lightest neutralino. Parameters excluded by LEP are occluded with a white or black box. Figures from Ref. [4]

Finally, we can study higgsino dark matter in the limit

$$\displaystyle \begin{aligned} |\mu| \ll M_1, M_2, m_{\tilde{f}} \; . \end{aligned} $$

(4.52)

Again from Eq. (4.43) we see that in addition to the t-channel chargino exchange, annihilation through s-channel Higgs states is possible. Again, the corresponding Feynman diagrams are shown in Fig. 4.3. At least in the pure higgsino limit with N _i3 = N _i4 the two contributions to the $\tilde {H} \tilde {H} Z^0$ coupling cancel, limiting the impact of s-channel Z-mediated annihilation. Still, these channels make the direct annihilation of higgsino dark matter significantly more efficient than for wino dark matter. The Sommerfeld enhancement plays a sub-leading role, because it mostly affects the less relevant chargino co-annihilation,

(4.53)

The higgsino LSP appears in Fig. 4.4 as a vertical plateau in μ. The corresponding mass difference between the lightest neutralino and chargino is much larger than for the wino LSP; it now ranges around a GeV.

Also in Fig. 4.4 we see that a dark matter neutralino in the MSSM can be much lighter than the pure wino and higgsino results in Eqs. (4.51) and (4.53) suggest. For a strongly mixed neutralino the scaling of the annihilation cross section with the neutralino mass changes, and poles in the s-channels appear. In the left panel of Fig. 4.5 we add the leading Standard Model final state of the dark matter annihilation process, corresponding to the distinct parameter regions

• the light Higgs funnel region with $2 m_{\tilde {\chi }_1^0} = m_h$ . The leading contribution to dark matter annihilation is the decay to b quarks. As a consequence of the tiny Higgs width the neutralino mass has to be finely adjusted. According to Eq. (4.43) the neutralinos couple to the Higgs though gaugino-higgsino mixing. A small, $\mathcal {O}(10\%)$ higgsino component can then give the correct relic density. This very narrow channel with a very light neutralino is not represented in Fig. 4.5. Decays of the Higgs mediator to lighter fermions, like tau leptons, are suppressed by their smaller Yukawa coupling and a color factor;

• the Z-mediated annihilation with $2 m_{\tilde {\chi }_1^0} \approx m_Z$, with a final state mostly consisting out of light-flavor jets. The corresponding neutralino coupling requires a sizeable higgsino content. Again, this finely tuned low-mass channel in not shown in Fig. 4.5;

• s-channel annihilation through the higgsino content with some bino admixture also occurs via the heavy Higgs bosons A ⁰, H ⁰, and H ^± with their large widths. This region extends to large neutralino masses, provided the Higgs masses follows the neutralino mass. The main decay channels are $b\bar {b}$, $t\bar {t}$, and $t\bar {b}$. The massive gauge bosons typically decouple from the heavy Higgs sector;

• with a small enough mass splitting between the lightest neutralino and lightest chargino, co-annihilation in the neutralino–chargino sector becomes important. For a higgsino-bino state there appears a large annihilation rate to $\tilde {\chi }_1^0 \tilde {\chi }_1^0 \to W^+ W^-$ with a t-channel chargino exchange. The wino-bino state will mostly co-annihilate into $\tilde {\chi }_1^0 \tilde {\chi }_1^\pm \to W^\pm \to q\bar {q}'$, but also contribute to the W ⁺ W ⁻ final state. Finally, as shown in Fig. 4.5 the co-annihilation of two charginos can be efficient to reach the observed relic density, leading to a W ⁺ W ⁺ final state;

• one channel which is absent from our discussion of purely neutralino and chargino dark matter appears for a mass splitting between the scalar partner of the tau lepton, the stau, and the lightest neutralino of few per-cent or less the two states can efficiently co-annihilate. In the scalar quark sector the same mechanism exists for the lightest top squark, but it leads to issues with the predicted light Higgs mass of 126 GeV.

In the right panel of Fig. 4.5 we show the mass difference between the lightest chargino and the lightest neutralino. In all regions where chargino co-annihilation is required, this mass splitting is small. From the form of the mass matrices shown in Eqs. (4.39) and (4.41) this will be the case when either M ₂ or μ are the lightest mass parameters. Because of the light higgsino, the two higgsino states in the neutralino sector lead to an additional level separation between the two lightest neutralinos, the degeneracy of the lightest chargino and the lightest neutralino masses will be less precise here. For pure winos the mass difference between the lightest chargino and the lightest neutralino can be small enough that loop corrections matter and the chargino becomes long-lived.

Note that all the above listed channels correspond to ways of enhancing the dark matter annihilation cross section, to allow for light dark matter closer to the Standard Model masses. In that sense they indicate a fine tuning around the generic scaling $\sigma _{\chi \chi } \propto 1/m_{\tilde {\chi }_1^0}^2$ which in the MSSM predicts TeV-scale higgsinos and even heavier winos.

4.4 Effective Field Theory

As another, final theoretical framework to describe the dark matter relic density we introduce an effective theory of dark matter [5]. We will start from the MSSM description and show how the heavy mediator can decouple from the annihilation process. This will put us into a situation similar to the description for example of the muon decay in Fermi’s theory. Next, we will generalize this result to an effective Lagrangian. Finally, we will show how this effective Lagrangian describes dark matter annihilation in the early universe.

Let us start with dark matter annihilation mediated by a heavy pseudoscalar A in the MSSM, as illustrated in the right panel of Fig. 4.3. The $A\tilde {\chi }_1^0 \tilde {\chi }_1^0$ coupling is defined in Eq. (4.43). If we assume the heavy Higgs to decay to two bottom quarks, the 2 → 2 annihilation channel is

$$\displaystyle \begin{aligned} \tilde{\chi}_1^0 \tilde{\chi}_1^0 \to A^* \to b\bar{b} \; , {} \end{aligned} $$

(4.54)

This description of dark matter annihilation includes two different mass scales, the dark matter mass $m_{\tilde {\chi }_1^0}$ and a decoupled mediator mass $m_A \gg m_{\tilde {\chi }_1^0}$. The matrix element for the dark matter annihilation process includes the A-propagator. From Sect. 3.2 we know that for WIMP annihilation the velocity of the incoming particles is small, v ≪ 1. If the energy of the scattering process, which determines the momentum flowing through the A-propagator is much smaller than the A-mass, we can approximate the intermediate propagator as

$$\displaystyle \begin{aligned} \frac{1}{q^2 - m_A^2} \to - \frac{1}{m_A^2} \qquad \Leftrightarrow \qquad \sigma( \tilde{\chi}_1^0 \tilde{\chi}_1^0 \to b\bar{b}) \propto g_{A\tilde{\chi}_1^0 \tilde{\chi}_1^0}^2 g_{A bb}^2\,\frac{m_b^2}{m_A^4} \; . \end{aligned} $$

(4.55)

The fact that the propagator of the heavy scalar A does not include a momentum dependence is equivalent of removing the kinetic term of the A-field from the Lagrangian. We remove the heavy scalar field from the propagating degrees of freedom of our theory. The only actual particles we can use in our description of the annihilation process of Eq. (4.54) are the dark matter fermions $\tilde {\chi }_1^0$ and the bottom quarks. Between them we observe a four-fermion interaction.

On the Lagrangian level, such a four-fermion interactions mediated by a non-propagating state is given by an operator of the type

$$\displaystyle \begin{aligned} g_{\text{ann}} \; \overline{\psi}_{\tilde{\chi}_1^0} \varGamma^\mu\psi_{\tilde{\chi}_1^0} \overline{\psi}_b \varGamma_\mu\psi_b \; , \end{aligned} $$

(4.56)

where Γ ^μ = {1, γ ₅, γ _μ, γ _μ γ ₅, [γ _μ, γ _ν]} represents some kind of Lorentz structure. We know that a Lagrangian has mass dimension four, and a fermion spinor has mass dimension 3/2. The four-fermion interaction then has mass dimension six, and has to be accompanied by a mass-dependent prefactor,

$$\displaystyle \begin{aligned} \mathcal{L} \supset \frac{g_{\text{ann}}}{\varLambda^2} \; \overline{\psi}_{\tilde{\chi}_1^0} \varGamma^\mu \psi_{\tilde{\chi}_1^0} \overline{\psi}_b \varGamma_\mu\psi_b \; . {} \end{aligned} $$

(4.57)

Given this Lagrangian, the question arises if we want to use this interaction as a simplified description of the MSSM annihilation process or view it as a more general structure without a known ultraviolet completion. For example for the muon decay we nowadays know that the suppression is given by the W-mass of the weak interaction. Using our derivation of Eq. (4.57) we are inspired by the MSSM annihilation channel through a heavy pseudoscalar. In that case the scale Λ should be given by the mass of the lightest particle we integrate out. This defines, modulo order-one factors, the matching condition

$$\displaystyle \begin{aligned} \varLambda = m_A \qquad \qquad \text{and} \qquad \qquad g_{\text{ann}} = g_{A \tilde{\chi}_1^0 \tilde{\chi}_1^0} \; g_{Abb} \; . \end{aligned} $$

(4.58)

From Eq. (4.59) we see that all predictions by the effective Lagrangian are invariant under a simultaneous scaling of the new physics scale Λ and the underlying coupling g _ann. Moreover, we know that the annihilation process $\tilde {\chi }_1^0 \tilde {\chi }_1^0 \to f \bar {f}$ can be mediated by a scalar in the t-channel. In the limit $m_f \ll m_{\tilde {\chi }_1^0} \ll m_{\tilde {f}}$ this defines essentially the same four-fermion interaction as given in Eq. (4.57).

Indeed, the effective Lagrangian is more general than its interpretation in terms of one half-decoupled model. This suggests to regard the Lagrangian term of Eq. (4.57) as the fundamental description of dark matter, not as an approximation to a full model. For excellent reasons we usually prefer renormalizable Lagrangians, only including operators with mass dimension four or less. Nevertheless, we can extend this approach to examples including all operators up to mass dimension six. This allows to describe all kinds of four-fermion interactions. From constructing the Standard Model Lagrangian we know that given a set of particles we need selection rules to choose which of the possible operators make it into our Lagrangian. Those rules are given by the symmetries of the Lagrangian, local symmetries as well as global symmetries, gauge symmetries as well as accidental symmetries. This way we define a general Lagrangian of the kind

(4.59)

where the operators $\mathcal {O}_j$ are organized by their dimensionality. The c _j are couplings of the kind shown in Eq. (4.57), called Wilson coefficients, and Λ is the new physics scale.

The one aspect which is crucial for any effective field theory or EFT analysis is the choice of operators contributing to a Lagrangian. Like for any respectable theory we have to assume that any interaction or operator which is not forbidden by a symmetry will be generated, either at tree level or at the quantum level. In practice, this means that any analysis in the EFT framework will have to include a large number of operators. Limits on individual Wilson coefficients have to be derived by marginalizing over all other Wilson coefficients using Bayesian integration (or a frequentist profile likelihood).

From the structure of the Lagrangian we know that there are several ways to generate a higher dimensionality for additional operators,

• external particles with field dimensions adding to more than four. The four-fermion interaction in Eq. (4.57) is one example;

• an energy scale of the Lagrangian normalized to the suppression scale, leading to corrections to lower-dimensional operators of the kind v ²∕Λ ²;

• a derivative in the Lagrangian, which after Fourier transformation becomes a four-momentum in the Feynman rule. This gives corrections to lower-dimensional operators of the kind p ²∕Λ ².

For dark matter annihilation we usually rely on dimension-6 operators of the first kind. Another example would be a $\tilde {\chi }_1^0 \tilde {\chi }_1^0 WW$ interaction, which requires a dimension-5 operator if we couple to the gauge boson fields and a dimension-7 operator if we couple to the gauge field strengths. The limitations of an EFT treatment are obvious when we experimentally observe poles, for example the A-resonance in the annihilation process of Eq. (4.54). In the presence of such a resonance it does not help to add higher and higher dimensions—this is similar to Taylor-expanding a pole at a finite energy around zero. Whenever there is a new particle which can be produced on-shell we have to add it to the effective Lagrangian as a new, propagating degree of freedom. Another limiting aspect is most obvious from the third kind of operators: if the correction has the form p ²∕Λ ², and the available energy for the process allows for p ² ≳ Λ ², higher-dimensional operators are no longer suppressed. However, this kind of argument has to be worked out for specific observables and models to decide whether an EFT approximation is justified.

Finally, we can estimate what kind of effective theory of dark matter can describe the observed relic density, Ω _χ h ² ≈ 0.12. As usual, we assume that there is one thermally produced dark matter candidate χ. Two mass scales given by the propagating dark matter agent and by some non-propagating mediator govern our dark matter model. If a dark matter EFT should ever work we need to require that the dark matter mass is significantly smaller than the mediator mass,

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

(4.60)

In terms of one coupling constant g governing the annihilation process we can use the usual estimate of the WIMP annihilation rate, similar to Eq. (3.3),

$$\displaystyle \begin{aligned} \langle \sigma_{\chi \chi} \, v \rangle \approx \frac{g^4 m_\chi^2}{4 \pi \, m_{\text{med}}^4} \stackrel{\text{Eq. (3.32)}}{=} \frac{1.7 \cdot 10^{-9}}{\text{GeV}^2} \; . {} \end{aligned} $$

(4.61)

We know that it is crucial for this rate to be large enough to bring the thermally produced dark matter rate to the observed level. This gives us a lower limit on the ratio $m_\chi /m_{\text{med}}^2$ or alternatively an upper limit on the mediator mass for fixed dark matter mass. As a rough relation between the mediator and dark matter masses we find

$$\displaystyle \begin{aligned}\begin{array}{r*{20}l} \frac{m_{\text{med}}^2}{g^2 m_\chi} = 6.8~\text{TeV} &\qquad \stackrel{m_\chi = 10~\text{GeV}}{\Rightarrow} \qquad &\frac{m_{\text{med}}}{g} &= 260~\text{GeV} \gg m_\chi \\ &\qquad \stackrel{m_\chi = m_{\text{med}}/2}{\Rightarrow} \qquad &\frac{m_{\text{med}}}{g} &= 3.4~\text{TeV} = m_\chi \; . {} \end{array}\end{aligned} $$

(4.62)

The dark matter agent in the EFT model can be very light, and the mediator will typically be significantly heavier. An EFT description of dark matter annihilation seems entirely possible.

Going back to our two models, the Higgs portal and the MSSM neutralino, it is less clear if an EFT description of dark matter annihilation works well. In part of the allowed parameter space, dark matter annihilation proceeds through a light Higgs in the s-channel on the pole. Here the mediator is definitely a propagating degree of freedom. For neutralino dark matter we discuss t-channel chargino-mediated annihilation, where $m_{\tilde {\chi }_1^\pm } \approx m_{\tilde {\chi }_1^0}$. Again, the chargino is clearly propagating at the relevant energies.

Finally, to fully rely on a dark matter EFT we need to make sure that all relevant processes are correctly described. For our WIMP models this includes the annihilation predicting the correct relic density, indirect detection and possibly the Fermi galactic center excess introduced in Chap. 5, the limits from direct detection discussed in Chap. 6, and the collider searches of Chap. 7. We will comment on the related challenges in the corresponding sections.