(1) the condensate $\langle s^{2}\rangle\simeq
0.13\,{H^{2}\over\sqrt{\lambda_{s}}}$ forms during inflation, (2) after
inflation, the field starts oscillating around $s=0$ with a decreasing
amplitude $\propto a^{-1}$, (3) the field enters a non–relativistic regime,
where the mass term dominates the potential. Since $s$ is stable and
non–relativistic, it plays the role of cold dark matter. If the coupling is
not too small, the scalar can also thermalize due to self–interactions and its
final abundance would then be dictated by the usual freeze–out. The scalar
condensate can account for all of the dark matter for a range of the couplings
and masses. In particular, if no thermalization occurs, simple scaling
arguments show that the correct relic density of DM is reproduced for [141]

$\lambda_{s}^{5/8}~{}{{\rm GeV}\over m_{s}}\sim 10^{7}\times\left({H_{\rm
end}\over M_{\rm Pl}}\right)^{3/2}\;,$ (197)

where $H_{\rm end}$ is the Hubble rate at the end of inflation. An analogous
relation holds for thermal $s$ as well, although the coupling dependence of
the left hand side becomes $\lambda_{s}^{3/8}$. An important feature of this
scenario is the presence of isocurvature perturbations since the inflaton
fluctuations are not correlated with those of $s$ [144]. The PLANCK
observations [38] put strong constraints on such perturbations, yet tiny,
$\lesssim{\cal O}(10^{-19})$, as well as quite large, ${\cal O}(1)$, couplings
remain allowed [141]. Similarly, one finds that a massive scalar with no
self–interaction can also constitute all of the dark matter [145].

##  8 Postinflationary Higgs and dark matter production

Particle production after inflation constitutes an essential ingredient in our
understanding of the Early Universe physics. It is necessary in order to
explain how the Universe became radiation–dominated and how dark matter was
created. In general, the SM fields and dark matter can be produced at
different stages of the evolution via different mechanisms. If the relevant
couplings are strong, DM thermalizes erasing the memory of these mechanisms.
This possibility will be discussed in Sections 9 and 10. On the other hand, at
weak couplings, the total number of DM quanta created shortly after inflation
remains approximately constant throughout the history of the Universe and the
system retains some memory of the DM production mechanism. In what follows, we
focus on this possibility of non–thermal dark matter.

The only renormalizable inflaton couplings to the Standard Model are

$V_{\phi h}={1\over 4}\lambda_{\phi h}\phi^{2}h^{2}+{1\over 2}\sigma_{\phi
h}\phi h^{2}\;.$ (198)

Therefore, these interactions are expected to play the leading role in
reheating the Universe. The minimal option to account for dark matter is to
introduce a real scalar $s$ of mass $m_{s}$ with the interactions

$V_{\phi s}={1\over 4}\lambda_{\phi s}\phi^{2}s^{2}+{1\over 2}\sigma_{\phi
s}\phi s^{2}\;.$ (199)

Since we are interested in non–thermal DM, we may neglect possible Higgs–DM
interaction as well as its self–interaction. The above couplings are assumed
sufficiently small so that they do not induce large loop corrections to the
inflaton potential and DM self–coupling. On the other hand, if they are to
represent the main DM production mechanism, these couplings cannot be too
small: the scalar condensate $\langle s^{2}\rangle$ and other sources of dark
matter should be subleading.

The above interactions are sufficient to $fully$ describe both reheating and
dark matter production. Depending on the model, different combinations of the
couplings can play the leading role. To avoid stable or long–lived inflaton
relics, it is natural to assume that $\sigma_{\phi h}$ is significant and
responsible for producing the SM particles. Then, dark matter can be generated
primarily either via $\lambda_{\phi s}$ or $\sigma_{\phi s}$ couplings, both
of which yield viable options to be studied below. The results are sensitive
to the inflaton potential during the inflaton oscillation epoch, which can be
very different from the potential at large field values.

Particle production can take place in different regimes: it can be purely
perturbative or semiclassical via parametric or tachyonic resonance. Resonant
particle production was discussed in Section 5, so let us now consider the
perturbative mechanisms.

###  8.1 Particle production by an oscillating background

A time–varying classical field can lead to particle production. This applies,
in particular, to an oscillating inflaton background [116],[146]. Let us
consider perturbative production of the Higgs quanta following [147] and
extend that analysis to the massive Higgs case. We take the Higgs–inflaton
coupling to be sufficiently small so that it is outside the broad resonance
regime. To focus on the essentials of the mechanism, we also neglect the
Universe expansion at the initial stage.

####  8.1.1 Quartic Higgs–inflaton interaction

Let us start with the quartic Higgs–inflaton interaction

$\Delta V={1\over 4}\lambda_{\phi h}\phi^{2}h^{2}\;,$ (200)

with a single Higgs d.o.f. and write

$\phi^{2}(t)=\sum_{n=-\infty}^{\infty}\zeta_{n}e^{-in\omega t}\;,$ (201)

where the coefficients $\zeta_{n}$ are time–independent. We are interested in
the amplitude of creating two Higgs quanta with momenta $p,q$ from the
background, i.e. the transition $|0\rangle\rightarrow|p,q\rangle$. Using the
Peskin–Schroeder conventions [148] for the creation and annihilation operators
as well as the state normalization, we have

$-i\int_{-\infty}^{\infty}dt\langle f|V(t)|i\rangle=-i\,{\lambda_{\phi h}\over
2}\,(2\pi)^{4}\delta({\bf{p}}+{\bf{q}})\sum_{n=1}^{\infty}\zeta_{n}\delta(E_{p}+E_{q}-n\omega).$
(202)

This corresponds to the invariant amplitude for the $n$-th mode decay ${\cal
M}_{n}=-\lambda_{\phi h}\zeta_{n}/2$. Taking into account the symmetry of the
final state and integrating over the phase space $\Pi$ produces the usual
factor $|{\bf p}|/(8\pi E_{\rm cm})\times\theta(E_{\rm cm}-2m_{h})$, where
$E_{\rm cm}=n\omega$. The reaction rate per unit volume is

$\Gamma=\sum_{n=1}^{\infty}\Gamma_{n}=\sum_{n=1}^{\infty}{1\over 2}\int|{\cal
M}_{n}|^{2}d\Pi_{n}={\lambda_{\phi h}^{2}\over
64\pi}\sum_{n=1}^{\infty}|\zeta_{n}|^{2}\sqrt{1-\left({2m_{h}\over
n\omega}\right)^{2}}\;\theta(n\omega-2m_{h})\;.$ (203)

In our approximation, the coefficients $\zeta_{n}$ are constant, so the decay
is neglected. Nevertheless, one can estimate the inflaton decay rate by
resorting to energy conservation. A unit volume loses $\Gamma\Delta t\langle
E\rangle$ of energy in infinitesimal time period $\Delta t$, where $\langle
E\rangle$ is the average energy of the decay products,

$\langle E\rangle={\sum_{n}n\omega\Gamma_{n}\over\sum_{n}\Gamma_{n}}\;.$ (204)

By virtue of energy conservation, this energy loss can also be expressed in
terms of the inflaton decay rate
$\Gamma_{\phi}=-{1\over\rho_{\phi}}\,{d\rho_{\phi}\over dt}$ and the inflaton
energy density $\rho_{\phi}$,

$\rho_{\phi}\Gamma_{\phi}\Delta t=\langle E\rangle\Gamma\Delta t\;.$ (205)

Hence,

$\Gamma_{\phi}={\lambda_{\phi h}^{2}\omega\over
64\pi\rho_{\phi}}\;\sum_{n=1}^{\infty}n|\zeta_{n}|^{2}\sqrt{1-\left({2m_{h}\over
n\omega}\right)^{2}}\;\theta(n\omega-2m_{h})\;.$ (206)

In the massless limit, one recovers the result of [147]. Let us analyze what
this implies for specific inflaton potentials.

Quadratic inflaton potential. Suppose $V(\phi)={1\over
2}m_{\phi}^{2}\phi^{2}$, so that $\phi(t)=\phi_{0}\cos m_{\phi}t$, and neglect
the Higgs mass, $m_{h}\rightarrow 0$. Since $\rho_{\phi}={1\over
2}m_{\phi}^{2}\phi_{0}^{2}$ and only one term in the sum contributes
($n=1,\omega=2m_{\phi}$), we get171717We find a factor of 2 discrepancy with
the corresponding result in [147].

$\Gamma_{\phi}={\lambda_{\phi h}^{2}\phi_{0}^{2}\over 256\pi m_{\phi}}\;.$
(207)

This formula is trivially extended to the massive Higgs case, where the decay
is only allowed for $m_{\phi}>m_{h}$.

Quartic inflaton potential. Take $V(\phi)={1\over 4}\lambda_{\phi}\phi^{4}$.
The equation of motion for the inflaton with the initial condition
$\phi(0)=\phi_{0}$ is solved by the Jacobi cosine,

$\phi(t)=\phi_{0}\,{\rm
cn}\left(\sqrt{\lambda_{\phi}}\phi_{0}\,t,{1\over\sqrt{2}}\right)={\sqrt{\pi}\Gamma\left({3\over
4}\right)\over\Gamma\left({5\over
4}\right)}\,\phi_{0}\,\sum_{n=1}^{\infty}\left(e^{i(2n-1)\omega
t}+e^{-i(2n-1)\omega t}\right)\,{e^{-(\pi/2)(2n-1)}\over 1+e^{-\pi(2n-1)}}\;,$
(208)

where

$\omega={1\over 2}\sqrt{\pi\over 6}{\Gamma\left({3\over
4}\right)\over\Gamma\left({5\over 4}\right)}\,m_{\phi}^{\rm eff}$ (209)

and

$m_{\phi}^{\rm eff}=\sqrt{3\lambda_{\phi}}\phi_{0}\;.$ (210)

The above sum is dominated by the first term with $n=1$, while the second term
is suppressed by $e^{-\pi}$. Keeping just the leading term, one finds that
$\phi^{2}(t)$ has frequency $2\omega$ and in the limit $m_{h}\rightarrow
0\;$181818We find a factor of 2 discrepancy with the corresponding result in
[147].,

$\Gamma_{\phi}=C\;{\lambda_{\phi h}^{2}\over
64\pi}\,{\phi_{0}\over\sqrt{\lambda_{\phi}}}\,,$ (211)

with $C\simeq 0.4$. This result is similar to that for the quadratic potential
if one assigns the inflaton an effective mass $m_{\phi}^{\rm eff}$.

In case of a heavy Higgs, $m_{h}>\omega$, the decay is exponentially
suppressed. The production is dominated by the $n$–th mode, where $n$ is the
closest integer to $m_{h}/\omega$ from above. Taking $n\sim m_{h}/\omega$, the
decay width scales approximately as

$\Gamma_{\phi}\sim\Gamma_{\phi}\Bigl{|}_{m_{h}=0}\;\left(m_{h}\over\omega\right)^{2}e^{-2\pi\left({m_{h}\over\omega}-1\right)}\;.$
(212)

####  8.1.2 Trilinear Higgs–inflaton interaction

Let us extend the above results to the trilinear Higgs–inflaton interaction

$\Delta V={1\over 2}\sigma_{\phi h}\phi h^{2}\;,$ (213)

with a single Higgs d.o.f. and expand

$\phi(t)=\sum_{n=-\infty}^{\infty}\xi_{n}e^{-in\omega t}\;.$ (214)

Repeating the decay rate calculation, one finds

$\Gamma_{\phi}={\sigma_{\phi h}^{2}\omega\over
16\pi\rho_{\phi}}\;\sum_{n=1}^{\infty}n|\xi_{n}|^{2}\sqrt{1-\left({2m_{h}\over
n\omega}\right)^{2}}\;\theta(n\omega-2m_{h})\;.$ (215)

Consider now the quadratic and quartic inflaton potentials, as before.

Quadratic inflaton potential. For $V(\phi)={1\over 2}m_{\phi}^{2}\phi^{2}$ and
neglecting the Higgs mass, one finds

$\Gamma_{\phi}={\sigma_{\phi h}^{2}\over 32\pi m_{\phi}}\;.$ (216)

This is identical to the to the decay width for the quantum process
$\phi\rightarrow hh$. Again, this process is kinematically forbidden for
$m_{\phi}<2m_{h}$.

Quartic inflaton potential. For $V(\phi)={1\over 4}\lambda_{\phi}\phi^{4}$ in
the massless Higgs limit, we have

$\Gamma_{\phi}=\tilde{C}\,{\sigma_{\phi h}^{2}\over
32\pi}\,{1\over\sqrt{\lambda_{\phi}}\phi_{0}}\;,$ (217)

where $\tilde{C}\simeq 1.6$. Here we have included only the dominant
contribution from the $n=1$ mode. We observe that the decay rate is similar to
that of the fully quantum reaction $\phi\rightarrow hh$, where $\phi$ is
assigned an effective mass $m_{\phi}^{\rm
eff}=\sqrt{3\lambda_{\phi}}\phi_{0}$.

The decay is allowed even for $\omega<2m_{h}$, although it is exponentially
suppressed. The rate scales as

$\Gamma_{\phi}\sim\Gamma_{\phi}\Bigl{|}_{m_{h}=0}\;{2m_{h}\over\omega}\,e^{-\pi\left({2m_{h}\over\omega}-1\right)}\;,$
(218)

where we have included the dominant contribution of the mode with $n\sim
2m_{h}/\omega$.

The effect of the Universe expansion in the above considerations can be taken
into account adiabatically by including time dependence into the oscillation
amplitude $\phi_{0}\rightarrow\phi_{0}(t)$. The equations of motion for the
inflaton in an expanding Universe can be solved exactly and one finds that the
amplitude after a few oscillations evolves according to [109]

$\displaystyle\phi^{2}~{}{\rm
potential:}~{}~{}\phi_{0}\rightarrow\phi_{0}/a(t)^{3/2}~{}~{},~{}~{}a(t)=a_{0}\,t^{2/3}\;,$
$\displaystyle\phi^{4}~{}{\rm
potential:}~{}~{}\phi_{0}\rightarrow\phi_{0}/a(t)~{}~{}~{}~{}~{}\,,~{}~{}a(t)=a_{0}\,t^{1/2}\;.$
(219)

Note that, in the latter case, this also implies a time dependent frequency
$\omega$ according to (209).

The expansion also has the effect of redshifting the momenta of the decay
products. Consequently, the narrow resonance resulting from the Bose
enhancement of the decay amplitude becomes inefficient and often can be
neglected (see e.g. [101]).

Finally, if the Higgs VEV can be neglected, the rates should be multiplied by
4 to account for all of the Higgs degrees of freedom.

###  8.2 Effect of the non–minimal inflaton–gravity coupling

The inflaton potential during the oscillation epoch can be more complicated
than $\phi^{2}$ or $\phi^{4}$. This is the case, in particular, in the
presence of a non–minimal inflaton–gravity coupling. Nevertheless, the
particle production rates can be extrapolated to this case as well.

Consider inflation driven by a non–minimal inflaton–gravity coupling, as in
Section 3. Let us focus on a single field case,

$\Omega=1+\xi_{\phi}\phi^{2}\;,$ (220)

where we use the Planck units ${\rm M_{Pl}}=1$, and treat the Higgs as a heavy
spectator ($\xi_{h}=0$). After inflation, $\phi$ takes on values below the
Planck scale and we are interested in its full field range. The inflaton
kinetic function, in general, is given by
$(\Omega+6\xi_{\phi}^{2}\phi^{2})/\Omega^{2}$, so the canonically normalized
$\chi$ satisfies

${d\chi\over
d\phi}=\sqrt{1+\xi_{\phi}(1+6\xi_{\phi})\phi^{2}\over(1+\xi_{\phi}\phi^{2})^{2}}\;.$
(221)

The solution reads [126]

$\,\chi(\phi)=\sqrt{1+6\xi_{\phi}\over\xi_{\phi}}\,\sinh^{-1}\left(\sqrt{(1+6\xi_{\phi})\xi_{\phi}}\,\phi\right)-\sqrt{6}\,\sinh^{-1}\left({\sqrt{6}\xi_{\phi}\phi\over\sqrt{1+\xi_{\phi}\phi^{2}}}\right)\;.$
(222)

This expression applies for all $-\infty<\phi<\infty$ and can be simplified in
various limits using $\sinh^{-1}x=\ln\left(x+\sqrt{x^{2}+1}\right)$. In the
inflationary regime, i.e. for $\sqrt{\xi_{\phi}}\phi\gg 1$, we recover
$\chi={\sqrt{1+6\xi_{\phi}\over\xi_{\phi}}}\ln\sqrt{\xi_{\phi}}\phi$.

Let us consider in more detail the large non–minimal coupling limit. For
$6\xi_{\phi}\gg 1$, we have

$|\chi|=\sqrt{3\over
2}\ln(1+\xi_{\phi}\phi^{2})~{}~{},~{}~{}V_{E}(\chi)={\lambda_{\phi}\over
4\xi_{\phi}^{2}}\,\left(1-e^{\sqrt{2\over 3}|\chi|}\right)^{2}$ (223)

in the entire field range of $\phi$ as long as $1/(6\xi_{\phi})$ can be
neglected. Since the exact expression assures that the sign of $\chi$
coincides with the sign of $\phi$, in the limit $\xi_{\phi}\phi^{2}\ll 1$, we
have $\chi\simeq\pm\sqrt{3\over 2}\xi_{\phi}\phi^{2}$. At very small $\phi$,
however, terms of order $1/(6\xi_{\phi})$ become important and the above
approximation breaks down. In this case, one finds instead $\chi\simeq\phi$
which can also be seen directly from (221). Thus, after inflation we have

$\chi\simeq\left\\{\begin{tabular}[]{ccc}$\phi$&for&$\phi^{2}\ll{1\over
6\xi_{\phi}^{2}}~{},$\\\ $\pm\sqrt{3\over 2}\xi_{\phi}\phi^{2}$&for&${1\over
6\xi_{\phi}^{2}}\ll\phi^{2}\ll{1\over\xi_{\phi}}~{}.$\end{tabular}\right.$
(224)

The corresponding scalar potential in these regimes is

$V_{E}(\chi)\simeq\left\\{\begin{tabular}[]{ccc}${\lambda_{\phi}\over
4}\chi^{4}$&for&$|\chi|\ll{1\over 2\xi_{\phi}}~{},$\\\ ${\lambda_{\phi}\over
6\xi_{\phi}^{2}}\chi^{2}$&for&${1\over 2\xi_{\phi}}\ll|\chi|\ll
1~{}.$\end{tabular}\right.$ (225)

The potential shape appears quite peculiar: at larger field values, it is
quadratic, while at smaller values, it is quartic. This entails different
particle production regimes at early and late times. Our previous results for
the production rates, therefore, apply to the case at hand as long as the
oscillation amplitude falls into an appropriate field range.

It is also possible that dark matter itself has a substantial non–minimal
coupling to gravity. Since the curvature oscillates in the inflaton
oscillation epoch, a tachyonic instability ensues resulting in efficient dark
matter production even in the absence of the direct inflaton–DM coupling
[149].

###  8.3 Resonant vs non–resonant dark matter production and reheating

Dark matter and SM radiation can be produced by entirely different mechanisms.
Consider the set–up where the dominant Higgs–inflaton coupling is trilinear,
while the leading DM–inflaton coupling is quartic,

$V_{\phi h}={1\over 2}\sigma_{\phi h}\,\phi h^{2}~{}~{},~{}~{}V_{\phi
s}={1\over 4}\lambda_{\phi s}\phi^{2}s^{2}\;,$ (226)

where $s$ is a scalar DM candidate. Suppose that the Higgs–DM and DM
self–interaction are negligible such that it never thermalizes. In this case,
the DM abundance is determined by a short period of DM production right after
inflation, while the Universe reheats via perturbative inflaton decay into the
Higgses. Depending on the couplings, the dynamics of the system can be quite
complicated due a non–trivial resonance structure and rescattering effects.
Thus, lattice simulations are often necessary to make reliable predictions
[150]. Related studies have appeared in [122], while much of the following
material relies on the analysis of [151]. Further improvements can be
facilitated by the advanced computational tools of [152],[153].

####  8.3.1 $\phi^{2}$ inflaton potential

Let us assume the quadratic inflaton potential during preheating,
$V_{\phi}={1\over 2}m_{\phi}^{2}\phi^{2}$, where $m_{\phi}$ is a free
parameter (not directly related to inflationary predictions), and consider
light dark matter, $m_{\phi}\gg m_{s}$. For small enough $\sigma_{\phi h}$ and
significant $\lambda_{\phi s}$, the leading particle production mechanism
after inflation is the parametric resonance whose strength is controlled by
(see Section 5.2.2)

$q={\lambda_{\phi s}\Phi^{2}\over 2m_{\phi}^{2}}\gg 1\;,$ (227)

where $\Phi$ is the time–dependent amplitude of inflaton oscillations,
$\phi(t)\simeq\Phi(t)\cos\,m_{\phi}t$ and
$\Phi(t)=\Phi_{0}/(m_{\phi}t)=\Phi_{0}/a^{3/2}$. The resonance is active until
either $q$ reduces to about 1 or the produced DM induces a large inflaton mass
(“backreaction”) [107],

$\displaystyle q$ $\displaystyle\simeq$ $\displaystyle
1~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}({\rm weak~{}coupling})\;,$
$\displaystyle\sqrt{\lambda_{\phi s}\langle s^{2}\rangle\over 2m_{\phi}^{2}}$
$\displaystyle\sim$ $\displaystyle 1~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}({\rm
strong~{}coupling})\;.$ (228)

In the first case, the energy density of the Universe is dominated by the
non–relativistic inflaton and the total number of the DM quanta remains
constant after the resonance has ended. In the second case, the system can be
highly relativistic with the inflaton and DM contributing roughly 50% each to
the energy balance. Due to rescattering, the total number of DM particles can
still grow significantly after the end of the resonance, although the growth
rate is much slower than that during the resonance. To evaluate the resulting
DM output, one has to resort to lattice simulations. Since the inflaton is
heavy, the system becomes non–relativistic and inflaton–dominated some time
after the end of rescattering. Subsequently, the perturbative inflaton decay
into the Higgses reheats the Universe.

The dark matter abundance is normally parametrized by $Y=n/s_{\rm SM}$, where
$n$ is the DM number density and $s_{\rm SM}$ is the SM entropy density,
$s_{\rm SM}=2\pi^{2}g_{*s}T^{3}/45$, with $g_{*s}$ being the number of degrees
of freedom contributing to the entropy. The observed DM density corresponds to
$Y_{\infty}=4.4.\times 10^{-10}\;{\rm GeV}/m_{s}$. The abundance $Y$ remains
constant after reheating and thus can be evaluated at the reheating stage
defined by

$\Gamma(\phi\rightarrow h_{i}h_{i})\simeq
H_{R}~{}~{}~{},~{}~{}~{}\Gamma(\phi\rightarrow h_{i}h_{i})={\sigma_{\phi
h}^{2}\over 8\pi m_{\phi}}\;,$ (229)

where $H_{R}$ is the Hubble rate at reheating and 4 Higgs d.o.f. have been
taken into account. At this point, the inflaton energy density converts into
radiation,

$H_{R}=\sqrt{\rho\over 3}=\sqrt{\pi^{2}g_{*}\over 90}T_{R}^{2}\;$ (230)

in Planck units, $M_{\rm Pl}=1$, and $g_{*}$ being the number of d.o.f.
contributing to the energy density.

In order to cover both the weak and strong coupling regimes, let us express
$Y$ directly in terms of the simulation output. The result depends on the
duration of the relativistic phase after the resonance. Denoting by
$a_{e},a_{*}$ and $a_{R}$ the scale factors at the end of the simulation, at
the beginning of the non–relativistic epoch and at reheating, respectively,
the system evolves according to

$a_{e}\stackrel{{\scriptstyle\rm{rel}}}{{\longrightarrow}}a_{*}\stackrel{{\scriptstyle\rm
nrel}}{{\longrightarrow}}a_{R}\;,$ (231)

where in the first period the equation of state is $w\simeq 1/3$, while in the
second $w\simeq 0$.191919In reality, the equation of state and its evolution
are more complicated [154], yet this does not affect the results in a
significant way. During the relativistic phase, the DM contribution to the
total energy density can be substantial, up to about 50%, whereas in the
second phase it is negligible since DM remains relativistic. Denoting

$\rho_{e}(s)=\delta\;\rho_{e}(\phi)\;,$ (232)

where $\rho_{e}(\phi)$ and $\rho_{e}(s)$ are the energy densities of the
inflaton and dark matter at the end of the simulation, the corresponding
Hubble rate is given by $H_{e}=\sqrt{1+\delta}\,\sqrt{\rho_{e}(\phi)/3}$.
Since only the inflaton energy gets converted into SM radiation, the Hubble
rate scaling $H\propto a^{-3(w+1)/2}$ implies

$H_{R}={H_{e}\over\sqrt{1+\delta}}\;{a_{e}^{2}\over
a_{*}^{2}}\;{a_{*}^{3/2}\over a_{R}^{3/2}}\;.$ (233)

Solving for $a_{R}$, one finds that the coupling $\sigma_{\phi h}$ required by
the correct DM abundance is given by

$\sigma_{\phi h}\simeq 1.6\times
10^{-8}\;\sqrt{m_{\phi}}\;{H_{e}^{2}\over(1+\delta)\;n_{e}}\,{a_{e}\over
a_{*}}~{}~{}\left({{\rm GeV}\over m_{s}}\right)$ (234)

in Planck units, assuming $g_{*}\simeq 107$. Here $H_{e}$ and $n_{e}$ are the
simulation output, while $a_{e}/a_{*}$ can be estimated by considering the
average energy of the inflaton quantum: for example, one may require
$p_{e}/(a_{*}/a_{e})\sim m_{\phi}$, where $p_{e}$ is the typical particle
energy or momentum at the end of the simulation. At weak coupling,
$a_{e}/a_{*}$ is close to 1, while at stronger coupling this ratio is about
1/few or below. Since it is computationally challenging to observe the system
over a long period, $a_{e}/a_{*}$ can only be estimated and thus introduces
uncertainty.

When the coupling is sufficiently strong, the system reaches the state of
quasi–equilibrium by the end of the simulation. In this case, the above
formula simplifies and takes on a $universal$ form. Since the energy density
is distributed almost equally between the inflaton and dark matter,
$\delta\simeq 1$ and $3H_{e}^{2}/\left[(1+\delta)n_{e}\right]$ gives an
average energy of the DM quantum at the end of the simulation. In
quasi–equilibrium, it is similar to the average energy of the inflaton
quantum. The latter multiplied by $a_{e}/a_{*}$ gives $m_{\phi}$ by
definition, so

$\sigma_{\phi h}\simeq 5\times 10^{-9}\;m_{\phi}^{3/2}~{}~{}\left({{\rm
GeV}\over m_{s}}\right)\;.$ (235)

The result is independent of the coupling and initial conditions, and also
applies to other inflaton potentials, e.g. $\phi^{4}$. This is natural since
memory of the initial condition is erased in equilibrium. Note that the
crucial assumption is that the inflaton and DM reach a quasi–equilibrium state
in the relativistic regime, so the coupling must be strong enough and the
system must contain sufficient energy to make the inflaton and DM
relativistic.

At weaker coupling, an order of magnitude of the required $\sigma_{\phi h}$
can be estimated using the theory of broad parametric resonance. Since DM
makes only a small contribution to the energy density, the Hubble rate is
determined by the inflaton potential, $H={1\over\sqrt{6}}m_{\phi}\Phi$. The
number density of $s$ produced by the resonance can be approximated by [107]

$n\sim{k_{*}^{3}\over 64\pi^{2}a^{3}\sqrt{\pi\mu m_{\phi}t}}\;e^{2\mu
m_{\phi}t}\;,$ (236)

where $k_{*}=\left(\sqrt{\lambda_{\phi s}/2}\;m_{\phi}\Phi_{0}\right)^{1/2}$
and $\mu$ is the effective Floquet exponent. The scale factor and the
corresponding $m_{\phi}t$ at the end of the resonance are found by requiring
$q\simeq 1$. Noting that $H^{2}/n$ remains constant after that, one finds

$\sigma_{\phi h}\sim 5\times 10^{-7}\;{\sqrt{m_{\phi}\over\lambda_{\phi
s}}}\,\Phi_{0}\;e^{-2\mu\sqrt{\lambda_{\phi s}\over 2}\,{\Phi_{0}\over
m_{\phi}}}~{}~{}\left({{\rm GeV}\over m_{s}}\right)\;.$ (237)

in Planck units. This result is sensitive to the exact value of $\mu$ and thus
only yields a ballpark estimate. For the special case of $\phi^{2}$–inflation,
$\Phi_{0}\simeq 1$ and $m_{\phi}\simeq 5\times 10^{-6}$, while the typical
value of $\mu$ is between 0.15 and 0.175. This gives, for instance,
$\sigma_{\phi h}\sim 10^{-13}-10^{-12}$ at $\lambda_{\phi s}=10^{-7}$,
$m_{s}=1$ GeV. The resulting reheating temperature is relatively low,
$T_{R}\sim 10^{2}\sigma_{\phi h}\ll m_{\phi}$.

Figure 12:  $\sigma_{\phi h}$ vs $\lambda_{\phi s}$ producing the correct DM
relic abundance for a $\phi^{2}$ potential, based on LATTICEEASY simulations.
$\sigma_{\phi h}$ is given in units of $M_{\rm Pl}=2.4\times 10^{18}$ GeV and
the initial inflaton value is chosen to be $\Phi_{0}\simeq M_{\rm Pl}$. The
area above the curve is excluded by overabundance of dark matter.

Fig. 12 shows the couplings producing the right amount of dark matter with
mass 1 GeV based on Eq. 234 and lattice simulations. The results for other DM
masses are obtained by a simple rescaling as in Eq. 234. The weak coupling
regime is reasonably well described by (237), while for $\lambda_{\phi s}$
above $10^{-7}$ the behaviour becomes qualitatively different. The inflaton
zero mode gets destroyed through rescattering and the energy balance is shared
almost equally between the inflaton and dark matter. Thus, increasing
$\lambda_{\phi s}$ does not increase the DM output leading to a plateaux in
the figure. This result is consistent with Eq. 235.

The trend shown in the figure is clear: a smaller $\sigma_{\phi h}$ leads to
delayed inflaton decay and thus dilutes the DM density. Therefore, the area
above the curve leads to over–abundance of dark matter and is therefore ruled
out.

The above considerations neglect certain subleading effects which could, in
general, affect the DM abundance. In particular, at weak coupling, the
inflaton zero mode continues oscillating after the resonance and hence leads
to further DM production. However, as discussed in the next subsection, this
effect is modest and can be neglected for our purposes. Also, a substantial
$\sigma_{\phi h}$ results in Higgs production via tachyonic resonance at early
times. For the values of $\sigma_{\phi h}$ shown in the figure, this does not
affect the DM abundance in any significant way.

The range of allowed couplings is limited by the following factors. Large
values of $\lambda_{\phi s}\gtrsim{\cal O}(10^{-3})$ generally lead to
significant radiative corrections to the inflaton potential, while yet
stronger couplings can result in dark matter thermalization via induced DM
self–coupling of order $\lambda_{\phi s}/(4\pi)^{2}$. On the other hand, very
small couplings $\lambda_{\phi s}<10^{-8}$ do not lead to (broad) resonant DM
production. While this possibility is viable, it requires a perturbative
instead of semiclassical treatment, which we discuss next.

Non–resonant dark matter production. At weaker couplings, $q\lesssim 1$, there
is no resonant enhancement of dark matter production. Nevertheless, $s$ is
still produced by a time–varying inflaton background and can account for all
of the observed DM. According to Eq. 203, the process is allowed kinematically
when the induced DM mass $\sqrt{\lambda_{\phi s}/2}\,\Phi$ is below $m_{\phi}$
and the production rate per unit volume is

$\Gamma(\phi\phi\rightarrow ss)\simeq{\lambda_{\phi s}^{2}\over
1024\pi}\Phi^{4}\;,$ (238)

where we have used $\zeta_{1}=\Phi^{2}/4$, $\omega=2m_{\phi}$, and neglected
the narrow resonance effects which are insignificant in an expanding Universe.
The corresponding $s$ number density is found via the Boltzmann equation,

$\dot{n}+3Hn=2\,\Gamma(\phi\phi\rightarrow ss)\;.$ (239)

At weak coupling, the energy density is dominated by the non–relativistic
inflaton and $\Phi=\Phi_{0}\,a^{-3/2}$. The Boltzmann equation is then easily
integrated given an appropriate boundary condition. The solution has the form
$a^{3}\,n(t)=\alpha-\beta\,a^{-3/2}$, with constant $\alpha,\beta$.

Let us consider two coupling regimes. If the coupling is substantial, $s$ is
produced initially via the resonance and after that, when $q\lesssim 1$, it is
produced perturbatively. On the other hand, if the coupling is very weak,
$q(0)<1$, no resonance gets activated and perturbative DM production takes
place from the start. In the first case, the boundary condition is $n=0$ at
$q=1$ and one finds

$n(t)={\sqrt{2}\lambda_{\phi s}^{3/2}\Phi_{0}^{3}\over 512\pi a^{3}}$ (240)

at late times. In our convention $a=(mt)^{2/3}$ with $a=1$ at the start of the
inflaton oscillation epoch. One can verify that the above density is much
smaller than that created by the resonance, so it can indeed be neglected in
our analysis.

In the second case, $n=0$ at $a=1$, so

$n(t)={\lambda_{\phi s}^{2}\Phi_{0}^{4}\over 512\pi m_{\phi}a^{3}}$ (241)

for $a\gg 1$. As before, the reheating scale factor is found via
$H_{R}\simeq\Gamma$ and the observed abundance of dark matter is reproduced
for

$\sigma_{\phi h}\simeq 4.4\times 10^{-6}\;{m_{\phi}^{7/2}\over\lambda_{\phi
s}^{2}\Phi_{0}^{2}}~{}~{}\left({{\rm GeV}\over m_{s}}\right)$ (242)

in Planck units.

Depending on $m_{s}$, the required $\sigma_{\phi h}$ may be large,
$\sigma_{\phi h}\Phi_{0}/m_{\phi}^{2}\gg 1$, such that the Higgs production
via tachyonic resonance becomes important. In particular, this can destroy the
classical inflaton background and invalidate our calculation. However, dark
matter production peaks right after the inflaton starts oscillating, while the
Higgs backreaction sets in significantly later. Therefore, the total number of
produced DM quanta does not change significantly, while the system turns
relativistic for some time introducing a factor analogous to $a_{e}/a_{*}$ in
(234).

We note that as long as $\lambda_{\phi s}\gtrsim 10^{-10}$, the effective mass
of $s$ during $\phi^{2}$–inflation is greater than the Hubble rate. As a
result, it behaves as a heavy field frozen at the origin in field space and no
significant condensate $\langle s^{2}\rangle$ gets generated. In this regime,
$\lambda_{\phi s}$ is indeed the main source of dark matter production. For
other inflationary potentials, the corresponding bound on $\lambda_{\phi s}$
is model dependent.

####  8.3.2 $\phi^{4}$ inflaton potential

Due to approximate conformal invariance, particle production in the potential
$V_{\phi}={\lambda_{\phi}\over 4}\phi^{4}$ exhibits qualitatively different
features [108]. Although a massless inflaton is not a phenomenologically
viable option, its small mass can be neglected at the initial stages. In this
case, $\phi$ satisfies the equation of motion

$\ddot{\phi}+3H\dot{\phi}+\lambda_{\phi}\,\phi^{3}=0\;.$ (243)

The solution oscillates with a decreasing amplitude. After a few oscillations,
it can be written in a closed form in terms of the conformal time $\eta$
defined by $d\eta=a(t)dt$. Introducing a rescaled field $\varphi=a\phi$, one
finds that $\varphi$ satisfies a simple “Minkowski space” equation
$\varphi^{\prime\prime}_{\eta\eta}+\lambda_{\phi}\,\varphi^{3}=0$, where a
negligible term $a^{\prime\prime}_{\eta\eta}/a$ has been omitted. This is an
equation for an elliptic function, which implies

$\phi(t)={\Phi_{0}\over a(t)}\;{\rm cn}\left(x,{1\over\sqrt{2}}\right)\;$
(244)

where $x=(48\lambda_{\phi})^{1/4}\sqrt{t}$. The scale factor satisfies
$a(0)=1$ and $a(t)\propto\sqrt{t}$ shortly thereafter. The oscillation period
in $x$ is $T=\Gamma^{2}(1/4)/\sqrt{\pi}\simeq 7.4$ and the Jacobi cosine is
dominated by the first term in its expansion, $\propto\cos{2\pi\over T}x$ .

The dark matter field $s$ can be quantized treating $\phi$ as a classical
background, as in Section 5.2.1. In particular, the expansion (139) applies to
$s$ as well. Denoting by $s_{k}$ the corresponding momentum $k$–modes, it is
convenient to define a “comoving” mode $X_{k}(t)\equiv a(t)s_{k}(t)$ in
analogy to $\varphi=a\phi$. The equation of motion for $X_{k}$ in the
$\phi$–background can be approximated by

$X_{k}^{\prime\prime}+\left[\kappa^{2}+{\lambda_{\phi s}\over
2\lambda_{\phi}}\;{\rm
cn}^{2}\left(x,{1\over\sqrt{2}}\right)\right]\,X_{k}=0\;,$ (245)

where the prime denotes differentiation with respect to $x$ and

$\kappa^{2}={k^{2}\over\lambda_{\phi}\Phi_{0}^{2}}\;.$ (246)

This belongs to the class of Lamé equations. Since the effective mass term is
periodic, the Floquet analysis applies and the $X_{k}$ evolution is determined
by a stability chart in terms of $\kappa^{2}$ and $\tilde{q}={\lambda_{\phi
s}/(2\lambda_{\phi}})$, an example of which is shown in [108]. It is important
that the growth of $|X_{k}|$ is controlled by the ratio of the couplings
$\tilde{q}$, unlike in the case of the Mathieu equation. For ${\lambda_{\phi
s}/(2\lambda_{\phi}})>1$, the Floquet exponent is significant, ${\cal
O}(10^{-1})$, leading to fast amplitude growth. For ${\lambda_{\phi
s}/(2\lambda_{\phi}})<1$, the Floquet exponent scales as
$10^{-1}\,{\lambda_{\phi s}/(2\lambda_{\phi}})$ and becomes negligible at
small $\lambda_{\phi s}$.

The conformal nature of the system leads to qualitatively different particle
production features compared to those in the $\phi^{2}$ case. The ratio
${\lambda_{\phi s}/(2\lambda_{\phi}})$, which plays the role of the
$q$–parameter of the Mathieu equation, does not decrease in time and therefore
the resonance always ends for a different reason, namely, the backreaction of
produced particles [108]. The latter transfer the energy from the coherently
oscillating field thereby reducing its amplitude and, furthermore, induce a
mass term. The $\phi^{4}$ resonance can also be narrow, yet efficient. Indeed,
at ${\lambda_{\phi s}/(2\lambda_{\phi}})\ll 1$, the Lamé equation is very well
approximated by the Mathieu equation with $q\ll 1$. Thus, even in an expanding
space, the particle number grows exponentially, although the effect is only
seen on a relatively long timescale. In contrast, the narrow resonance for a
$\phi^{2}$ potential is inefficient due to the redshifting of the produced
particle momenta or reduction of $q$.

Another important difference compared to the $\phi^{2}$ case is that inflaton
oscillations lead to production of the $\phi$–quanta. Quantizing the
fluctuations above the $\phi$–background, one finds that the corresponding
$k$–modes satisfy the Lamé equation (245) with a special value ${\lambda_{\phi
s}/(2\lambda_{\phi}})=3$. Although the Floquet exponent is quite small in this
case, the $\phi$–quanta production is more efficient than production of DM
quanta for ${\lambda_{\phi s}/(2\lambda_{\phi}})\lesssim 1/3$. This process
stops due to backreaction roughly when $\langle\phi^{2}\rangle\simeq
0.05\,\Phi_{0}^{2}$, making the DM output suppressed.

In realistic constructions, the inflaton mass is non–zero and the resonance
structure becomes more complicated [108]. For small enough $m_{\phi}$, the
$\phi^{4}$–potential approximation is applicable until the oscillation
amplitude reaches some limiting value, typically around
$m_{\phi}/\sqrt{\lambda_{\phi}}$.

Figure 13:  $\sigma_{\phi h}$ vs $\lambda_{\phi s}$ producing the correct DM
relic abundance for a $\phi^{4}$ potential, based on LATTICEEASY simulations.
$\sigma_{\phi h}$ is given in units of $M_{\rm Pl}=2.4\times 10^{18}$ GeV and
the initial inflaton value is chosen to be $\Phi_{0}\simeq 1.7\,M_{\rm Pl}$.
The area above the curve is excluded by overabundance of dark matter.

In practice, the dynamics of the DM production are complicated by various
effects such as backreaction and rescattering, hence lattice simulations are
often necessary for a reliable analysis. The simulations can be trusted
outside the narrow resonance regime so that the occupation numbers are
sufficiently large. This is the case roughly for $\lambda_{\phi s}\gtrsim
0.5\,\lambda_{\phi}$. Then, about half of the inflaton energy gets transferred
to dark matter and the total number of the DM quanta remains constant at the
end of the simulation. The zero inflaton mode, i.e. the classical background,
gets largely destroyed by that time, while the characteristic energy of the DM
and inflaton quanta becomes of order
$\sqrt{\lambda_{\phi}}\phi\times(\lambda_{\phi s}/2\lambda_{\phi})^{1/4}$.

An example of the coupling set producing the correct DM relic abundance for
$m_{\phi}=1$ TeV is shown in Fig. 13. To calculate the required $\sigma_{\phi
h}$ in terms of the simulation output, one can again use Eq. 234, which for
substantial couplings simplifies to Eq. 235. As before, the system goes
through the relativistic and non–relativistic stages. During the first period,
the energy density scales as $a^{-4}$. Even though DM acquires large induced
mass $\propto\sqrt{\lambda_{\phi s}}\phi$, far above its momentum, it
redshifts just like the momentum does. As a result, the total energy density
scales as if the system were relativistic. When the characteristic momentum
redshifts to small enough values, the bare inflaton mass becomes relevant and
the system turns non–relativistic. The relativistic period lasts much longer
than it does in the $\phi^{2}$ case and one finds $a_{e}\ll a_{*}\ll a_{R}$.
Dark matter does not get diluted in the long first stage, which generally
leads to a dark Universe. Therefore, a subsequent, rather long
non–relativistic period is required implying a very small $\sigma_{\phi h}$,
of order eV for the parameters of Fig. 13. The corresponding reheating
temperature is low, $50$ MeV, but still consistent with the cosmological data
[157].

Fig. 13 shows that $\sigma_{\phi h}$ is almost independent of $\lambda_{\phi
s}$, at least for ${\lambda_{\phi s}/(2\lambda_{\phi}})>1$. This is expected
from Eq. 235, since the system enters the state of quasi–equilibrium. One may
verify that these results are also independent of $\lambda_{\phi}$. For lower
couplings, the simulations are less reliable due to a limited running time and
the apparent dip around $\lambda_{\phi s}\sim 10^{-12}$ may in fact be an
artefact of the numerical approximations.

The trend shown in Fig. 13 persists at larger values of the inflaton–DM
coupling until $\lambda_{\phi s}\sim 4\pi\sqrt{\lambda_{\phi}}$, where it
starts inducing a significant radiative correction to the inflaton
self–coupling. Also, at large $\lambda_{\phi s}$, the effect of a non–zero
$m_{\phi}$ on the resonance becomes tangible [108]. For weak inflaton–DM
couplings, the system enters a narrow resonance regime and the classical
simulation becomes unreliable. At $\lambda_{\phi s}/2\lambda_{\phi}\ll 1$, a
perturbative analysis is expected to apply, at least on a moderate time scale,
i.e. when the Bose enhancement is not very strong.

The values of $\sigma_{\phi h}$ lying above the curve in Fig. 13 are excluded
since they lead to a “dark Universe” with overabundant dark matter. In this
case, the non–relativistic inflaton–dominated period is too short and does not
provide for enough dilution of the DM density. It is remarkable that even tiny
values of $\sigma_{\phi h}$ above ${\cal O}$(eV) are ruled as long as
$m_{\phi}=1$ TeV and ${\lambda_{\phi s}/(2\lambda_{\phi}})>1$.

Finally, dark matter can be considered heavy during inflation as long as
$\sqrt{\lambda_{\phi}}\phi\lesssim\sqrt{\lambda_{\phi s}}$, so that no
appreciable DM condensate forms. In this case, the effect described in Section
7 is insignificant. If, however, $s$ is light, the condensate contribution to
the DM abundance can be neglected if the self–coupling $\lambda_{s}$ is not
too small (but still insufficient for thermalization).

###  8.4 Matter production via standard inflaton decay

Perhaps the simplest possibility to reheat the Universe and produce dark
matter is to employ the trilinear couplings

$V_{\phi h}={1\over 2}\sigma_{\phi h}\,\phi h^{2}~{}~{},~{}~{}V_{\phi
s}={1\over 2}\sigma_{\phi s}\,\phi s^{2}\;,$ (247)

which induce the processes $\phi\rightarrow ss$, $\phi\rightarrow h_{i}h_{i}$.
At small enough couplings, the tachyonic resonance effects can be neglected
and the decay is perturbative (see [155] for a recent discussion). For a
quadratic inflaton potential, $V_{\phi}={1\over 2}m_{\phi}^{2}\phi^{2}$, the
tachyonic resonance is weak if $\sigma_{\phi h}\Phi_{0}/m_{\phi}^{2}\ll 1$,
$\sigma_{\phi s}\Phi_{0}/m_{\phi}^{2}\ll 1$. This also implies that the
inflaton decay into $hh$ and $ss$ is kinematically allowed. In more
complicated systems, e.g. with complex scalars, the DM abundance may be
determined by an interplay of different effects [156], one of which is the
direct inflaton decay.

The DM abundance is controlled by $\phi\rightarrow ss$, while the dominant
inflaton decay mode is $\phi\rightarrow h_{i}h_{i}$,

$\Gamma(\phi\rightarrow ss)={\sigma_{\phi s}^{2}\over 32\pi
m_{\phi}}~{}~{},~{}~{}\Gamma_{\rm tot}\simeq{\sigma_{\phi h}^{2}\over 8\pi
m_{\phi}}\;,$ (248)

with $\sigma_{\phi s}\ll\sigma_{\phi h}$. $Y$ is given by the number of the DM
quanta accumulated up to reheating, where the inflaton energy converts into
that of SM radiation. The DM number density $n$ in this period is found from
the Boltzmann equation

$\dot{n}+3Hn=2\Gamma(\phi\rightarrow ss)\;n_{\phi}\;,$ (249)

where $n_{\phi}$ is the inflaton number density, $n_{\phi}={1\over
2}m_{\phi}\phi^{2}$. The effect of the decay on the inflaton amplitude can be
neglected until the reheating stage, so the DM number grows linearly with
time,

$n(t)={1\over a^{3}}\;\int dt\;2\Gamma(\phi\rightarrow
ss)\;n_{\phi}\;a^{3}=2\Gamma(\phi\rightarrow ss)\,n_{\phi}t\;,$ (250)

with the boundary condition $n(0)=0$. Trading the time variable for the scale
factor, $m_{\phi}t=a^{3/2}$ for $a\gg 1$, and requiring
$H_{R}\simeq\Gamma_{\rm tot}$, one finds that at the reheating point $Y\simeq
0.3\,{\Gamma(\phi\rightarrow ss)\Phi_{0}\over\Gamma^{1/2}_{\rm tot}m_{\phi}}$.
The correct DM relic density is then reproduced for

$\sigma_{\phi h}\simeq 4\times 10^{7}\;{\sigma_{\phi s}^{2}\Phi_{0}\over
m_{\phi}^{3/2}}\;{m_{s}\over{\rm GeV}}$ (251)

in Planck units.

To get a feeling for the numerical values, consider the input parameters
typical for quadratic inflation, $\Phi_{0}\sim 1$, $m_{\phi}\sim 5\times
10^{-6}$ and take $m_{s}=1$ GeV. Then $\sigma_{\phi h}\simeq 4\times
10^{15}\sigma_{\phi s}^{2}$, implying that the Higgs decay mode dominates for
$\sigma_{\phi s}>3\times 10^{-16}\sim$ 1 TeV. The reheating temperature is
roughly $T_{R}\sim 10^{2}\sigma_{\phi h}$, far above the required few MeV. To
avoid strong tachyonic resonance, one must ensure $\sigma_{\phi
h}\Phi_{0}/m_{\phi}^{2}\ll 1$, which for the above parameters implies
$\sigma_{\phi h}\ll 3\times 10^{-11}$. Therefore, there is a viable range of
$\sigma_{\phi h}$ between $10^{-15}$ and $10^{-11}$ for our parameter choice.

It is important to note that we require the absence of tachyonic resonance in
order for simple perturbative estimates to be applicable. Nevertheless, large
$\sigma$–terms can lead to viable models, yet a dedicated lattice study would
normally be needed in this case.

Finally, in all of the above considerations, we have assumed that the portal
couplings are the main actors in producing (dark) matter. In addition, there
are ever–present gravity contributions which may or may not be significant.
Indeed, the Universe expansion itself creates a setting for particle
production [114],[158]. If the expansion is fast enough, it can result in
strong non–adiabaticity generating field modes for which
$\dot{\omega}/\omega^{2}>1$. In the scalar case, the two relevant quantities
are the particle mass and the non–minimal coupling to gravity. Both of them
lead to particle production and the effect is particularly strong for
non–conformally coupled ($\xi_{s}\not=-1/6$) or heavy states, which can, for
example, constitute superheavy dark matter (WIMPzillas) [159]–[162]. However,
only a tiny fraction of the inflaton energy gets converted into dark matter
[159], unlike for DM production via strong parametric or tachyonic resonance,
and the contribution to the observed DM abundance can be completely neglected
for a light conformally coupled scalar. Recent analyses show that the
gravitational production mechanism is also efficient for vector dark matter
[163],[164] as well as very light scalars [143]. It should be noted that, in
this class of models, dark matter density perturbations are not directly
correlated with those of the inflaton, hence they are often subject to
significant isocurvature perturbation constraints. After inflation, dark
matter can also be efficiently produced via its non–minimal coupling to
gravity $\xi_{s}$ [149], when the curvature oscillations induce tachyonic
resonance. This mechanism can account for all of the DM in a wide range of
masses, depending on $\xi_{s}$, and should in general be superimposed with the
Higgs portal DM production.

To summarize this section, the simplest renormalizable Higgs portal and
inflaton–DM couplings are sufficient to describe both reheating and
non–thermal dark matter production. This mechanism can be very efficient such
that even tiny couplings can produce the right amount of dark matter and SM
radiation. On one hand, it raises a theoretical question as to how such small
couplings can be justified. This issue can only be addressed in a more
fundamental theory, presumably incorporating gravity. On the other hand, it
also shows that such couplings should not be ignored and treated carefully
since even their tiny values can make a difference. The prospects of directly
observing dark matter in this class of models are clearly rather dim.

##  9 Higgs portal dark matter

In this section, we focus on the Higgs coupling to dark matter, which was
taken negligible in our previous considerations. Such a coupling can be very
significant and responsible for production of thermal dark matter as well as
its observable signatures.

The Higgs field plays a special role in probing the dark sector. Already at
the renormalizable level, it can couple to dark scalar states which carry no
Standard Model quantum numbers [7]. The reason is that $H^{\dagger}H$ is a
unique SM dim-2 operator which is both gauge and Lorentz invariant. The lowest
dimension interactions between the Standard Model fields and the SM singlets
$\phi,V_{\mu},\chi$ of spin 0, 1, 1/2, respectively, possessing $Z_{2}$ parity
are given by

$\displaystyle-\Delta{\cal L}_{\rm scal}={1\over 2}\,\lambda_{\phi
h}H^{\dagger}H\,\phi^{2}\;,$ (252) $\displaystyle-\Delta{\cal L}_{\rm
vect}={1\over 2}\,\lambda_{Vh}H^{\dagger}H\,V_{\mu}V^{\mu}\;,$ (253)
$\displaystyle-\Delta{\cal L}_{\rm ferm}={1\over 2\Lambda}\,\lambda_{\chi
h}H^{\dagger}H\,\left(\bar{\chi}\chi+c\;\bar{\chi}i\gamma_{5}\chi\right)\;,$
(254)

where $\phi$ is real, $\chi$ can be either Dirac or Majorana; $\lambda_{\phi
h},\lambda_{Vh},\lambda_{\chi h},c$ are real dimensionless couplings,
$\Lambda$ is a scale, and we have allowed for CP violation in the fermion
interactions. The assumed parity symmetry $f_{\alpha}\rightarrow-f_{\alpha}$
($f_{\alpha}=\phi,V_{\mu},\chi$) makes these fields stable and, thus, dark
matter candidates. In addition to the above Higgs portal terms, the dark
fields generally possess self–interaction as well as mass terms.

Among these couplings, only the scalar one is renormalizable. The vector and
fermion interactions arise from integrating out heavy states. In the fermion
case, this is manifested in the interaction being dimension 5. The vector
coupling violates gauge invariance, so the corresponding coupling must be
proportional to the gauge field mass [165]. This makes the corresponding Higgs
portal interaction higher dimensional, in fact, dimension 6. Therefore, the
couplings (253),(254) should be treated within effective field theory.

Dark matter phenomenology can be studied in terms of the effective couplings.
In this case, one must make sure that, in the corresponding parameter range,
the effective field theory approximation is legitimate. In particular, the
energy of the relevant processes must be below the unitarity cuf–off
[165],[166]. This implies, for example, that the vector cannot be too light,
otherwise the LHC scattering processes as well as $h\rightarrow VV$ would
“blow up”.

The effective approach to dark matter of different spins was pursued
originally in [167],[168]. In particular, Ref. [168] has found that light
thermal dark matter ($m_{\rm DM}<60$ GeV) is in conflict with the LHC Higgs
measurements due to the efficient invisible decay mode. The most recent
analysis from GAMBIT [169] concludes that all the DM candidates remain viable,
although the available parameter space has shrunk considerably owing to the
improved direct DM detection limits. In the vector case, the viable window
corresponds to the dark matter mass above a few TeV with ${\cal O}(1)$ Higgs
portal coupling, apart from the narrow resonance strip with $m_{\rm DM}\simeq
m_{h_{0}}/2$, where $m_{h_{0}}$ is the SM Higgs mass. In the fermion case, the
allowed DM mass spans a wide range from the resonance, $m_{\rm DM}\simeq
m_{h_{0}}/2$, to multi–TeV, as long as sufficient CP violation is present
[170],[171]. The scalar case will be discussed below in detail.

In what follows, instead of the effective couplings, we will consider the
simplest UV complete models leading to the above interactions. This has some
advantages. For example, it uncovers the non–trivial origin of the parity
symmetry in the vector case. Also, it allows for DM annihilation channels
which are absent in the effective theory, yet essential for model viability
(see Section 9.2.5). One finds that phenomenology of the UV complete models is
healthier, while the results in the effective field theory limit are obtained
by decoupling the heavy states.

A review of various aspects of Higgs portal dark matter including collider
phenomenology can be found in [172]. An effective field theory approach to
dark matter of different spins is discussed in [173],[174].

###  9.1 Scalar dark matter

The minimal model of Higgs portal dark matter is a scalar extension of the
Standard Model with $Z_{2}$ parity [7]. We will consider both a real and a
complex scalar options, with the latter being particular interesting
phenomenologically. The simplest models and their close relatives have been
studied extensively in the literature [175]-[188] and remain viable to date,
although their allowed parameter shrivels due to constant pressure from direct
DM detection experiments.

####  9.1.1 Minimal real scalar dark matter

The simplest option to account for dark matter is to add a single new degree
of freedom, that is, a real scalar $\phi$ [7]. To ensure stability of DM, one
imposes a $Z_{2}$ symmetry,

$\phi\rightarrow-\phi\;.$ (255)

This is a discrete symmetry which may be a remnant of gauge symmetry at high
energy. For instance, when a field with even charge under a gauged U(1)
develops a VEV, a $Z_{2}$ subgroup remains unbroken. Examples of this type
appear in realistic string constructions [189]. For our purposes, the origin
of the $Z_{2}$ is unimportant and, in fact, it may not even be exact: a tiny
violation of this parity would be compatible with the data. The corresponding
scalar potential is given by Eq. 14, which we reproduce here for convenience,

$\displaystyle V(\phi,h)={1\over 4}\lambda_{h}h^{4}+{1\over 4}\lambda_{\phi
h}h^{2}\phi^{2}+{1\over 4}\lambda_{\phi}\phi^{4}+{1\over
2}m_{h}^{2}h^{2}+{1\over 2}m_{\phi}^{2}\phi^{2}\;,$

except now $\phi$ does not develop a VEV in order to preserve $Z_{2}$. The
potential minimization is trivial and the dark matter mass $m$ is given
by202020In principle, the dark matter mass is temperature dependent, e.g. via
$v=v(T)$. However, typically its effect on the relic abundance is
insignificant: the freeze–out temperature is roughly $m/20$. For heavy DM, it
may be above the EW transition temperature, but in this case
$m_{\phi}^{2}\gg\lambda_{\phi h}v^{2}/4$.

$m^{2}=m_{\phi}^{2}+{1\over 4}\lambda_{\phi h}v^{2}\;,$ (256)

where $v\simeq 246$ GeV. Shifting $h\rightarrow v+h$ in the above potential,
we obtain the Higgs–DM interaction terms

$V_{\phi h}={1\over 2}\lambda_{\phi h}v\,h\phi^{2}+{1\over 4}\lambda_{\phi
h}h^{2}\phi^{2}\;,$ (257)

where we now denote the physical Higgs boson by $h$.

As long as DM self–interaction can be neglected, the model is described by 2
parameters:

$m~{}~{},~{}~{}\lambda_{\phi h}~{}~{}.$ (258)

This means the model is very predictive and a priori it would be difficult to
fit all the data with just these two parameters. The most important
constraints on the model are imposed by requiring the correct DM relic
density, absence of the signal in direct detection experiments and the LHC
Higgs measurements. In addition, for light enough DM, indirect DM detection
places a further significant constraint. Let us consider these aspects of DM
phenomenology separately.

DM relic density. If the Universe temperature is high enough and the Higgs
portal coupling is sufficiently strong, dark matter reaches thermal
equilibrium. In this case, the relic DM density can be calculated in terms of
$m$ and $\lambda_{\phi h}$. Note, however, that this approach relies on
thermalization of dark matter which may have never been achieved for low
reheating temperatures. In fact, the standard cosmology only requires
temperatures to be above a few MeV for successful nucleosynthesis.
Furthermore, the DM density calculation can be substantially different in
non–standard cosmologies [190].

Assuming the standard cosmological history and a thermal momentum distribution
for DM, its number density $n(t)$ evolves according to the Boltzmann equation
[191]

${dn\over dt}+3Hn=-\langle\sigma v_{r}\rangle\,\left[n^{2}-n^{2}_{\rm
eq}\right]\,.$ (259)

Here, $H$ is the Hubble rate determined by the effective number of degrees of
freedom $g_{*}$ at temperature $T$,

$H=\sqrt{\pi^{2}g_{*}\over 90}\,{T^{2}\over M_{\rm Pl}}\;,$ (260)

$\langle\sigma v_{r}\rangle$ is the thermal average of the annihilation cross
section times the relative velocity $v_{r}$, and $n_{\rm eq}$ is the
equilibrium number density at $T$. The thermal average in the non–relativistic
limit is defined by

$\langle\sigma v_{r}\rangle={\int{d^{3}{\bf p_{1}}}{d^{3}{\bf
p_{2}}}\,f(p_{1})f(p_{2})\;\sigma v_{r}\over\int{d^{3}{\bf p_{1}}}{d^{3}{\bf
p_{2}}}\,f(p_{1})f(p_{2})}~{}~{},~{}~{}f(p)=e^{-(E-\mu)/T}\;,$ (261)

where $\mu$ is the time–dependent effective chemical potential and

$n=\int{d^{3}{\bf p}\over(2\pi)^{3}}\,f(p)\;.$ (262)

For a given $T(t)$ which is dictated by entropy conservation, the evolution of
$n(t)$ is determined by the time dependence of $\mu$. The equilibrium value
corresponds to $\mu=0$ and

$n_{\rm eq}=\left({mT\over 2\pi}\right)^{3/2}\;e^{-{m/T}}\;.$ (263)

The above Boltzmann equation is valid for dark matter consisting of one
species. There is no phase space symmetry factor associated with identical
particles in the initial state since it gets cancelled by a factor of 2
stemming from the DM number change in a single reaction. For DM consisting of
particles and anti–particles, the Boltzmann equation for the total number
density contains an extra factor of 1/2 in front of the cross section [192].

The solution to the above Boltzmann equation is well known: $n(t)$ tracks
closely $n_{\rm eq}(t)$ until the reaction rate becomes slower than the Hubble
rate. At this point, the effective chemical potential becomes significant and
the DM ensemble falls out of (chemical) equilibrium with the SM thermal bath.
For WIMPs, this occurs at temperature $T\sim m/20$. The total number of DM
particles remains approximately constant and is conveniently expressed in
terms of the DM abundance $Y$,

$Y={n\over s_{\rm SM}}~{}~{},~{}~{}s_{\rm SM}={2\pi^{2}\over
45}\,g_{*s}\,T^{3}\;,$ (264)

where $s_{\rm SM}$ is the SM entropy density and $g_{*s}$ is the effective
number of degrees of freedom contributing to the entropy. As long as the DM
contribution can be neglected, the SM thermal bath entropy is conserved,
$s_{\rm SM}a^{3}=\,$const, which determines the temperature evolution.

The cosmological and astrophysical data [334] require that in the present
epoch

$Y_{\infty}=4.4\times 10^{-10}\;\left({{\rm GeV}\over m}\right)\;.$ (265)

The number density at freeze–out is determined by the the annihilation cross
section, so for a given $m$ the relic abundance constraint fixes the Higgs
portal coupling $\lambda_{\phi h}$.

Figure 14:  Leading diagrams for scalar DM annihilation.

There are a number of annihilation channels $\phi\phi\rightarrow$ SM as shown
in Fig. 14. For light dark matter, the $s$–channel Higgs exchange dominates
with the relevant DM coupling being ${1\over 2}\lambda_{\phi h}v\,h\phi^{2}$.
For example, the annihilation cross section into fermions of mass $m_{f}$ with
$N_{c}^{f}$ colors is given by

$\sigma v_{r}=N_{c}^{f}\;{\lambda_{\phi h}^{2}m_{f}^{2}\over
16\pi}\;{1\over(4m^{2}-m_{h_{0}}^{2})^{2}}\;\left(1-{m_{f}^{2}\over
m^{2}}\right)^{3/2}\;.$ (266)

This expression exhibits resonant enhancement of DM annihilation close to
$2m\simeq m_{h_{0}}$, where $m_{h_{0}}$ is the SM Higgs mass. In this case,
the pole is regularized by including the Higgs decay width in the propagator:
$(4m^{2}-m_{h_{0}}^{2})^{2}\rightarrow(4m^{2}-m_{h_{0}}^{2})^{2}+m_{h_{0}}^{2}\Gamma_{h}^{2}$,
which can otherwise be neglected. Even a small $\lambda_{\phi h}\ll 1$ is
sufficient to obtain the right relic abundance.

Extension to all final states in the $s$–channel can be done by subsuming the
corresponding contributions into the modified Higgs width. Indeed, both the
decay width and $\sigma v_{r}$ include the same final state contributions and
phase space integral, while differing in the center-of-mass energy. One finds
[176]

$\sigma v_{r}={\lambda_{\phi h}^{2}v^{2}\over
m\,(4m^{2}-m_{h_{0}}^{2})^{2}}\,\Gamma(m_{h_{0}}^{*}=2m)\;,$ (267)

where $\Gamma(m_{h_{0}}^{*}=2m)$ is the decay width of the SM Higgs boson with
mass $2m$.

Practical calculations are often performed with the software package
micrOMEGAs [194]. It takes into account all the annihilation channels and
performs numerical integration of the Boltzmann equation. A parameter space
scan shows that, away from the resonance region, EW/TeV masses are consistent
with the relic DM abundance constraint for $\lambda_{\phi h}$ of order ${\cal
O}(10^{-1})$ to ${\cal O}(1)$ (see e.g., [168]). Close to the resonance, the
coupling can be much smaller, $10^{-4}-10^{-3}$, however the relic density
computation becomes more involved due to kinetic decoupling of DM from the SM
plasma [195].212121Different stages of the WIMP freeze–out have been analyzed
in detail in [196].

Direct DM detection. Dark matter can potentially be detected via its
interactions with nuclei. Such scattering is mediated by the Higgs boson,
whose interactions are spin–independent. The signal rate is determined by a
coherent sum of the nucleon–DM scattering amplitudes and grows with the size
of the nucleus. The result is conveniently parametrized in terms of the
spin–independent nucleon–DM scattering cross section.

Scattering of dark matter off nucleons proceeds via Higgs exchange in the
$t$–channel. The corresponding momentum transfer is suppressed by the DM
velocity $v_{\rm DM}\sim 10^{-3}$, so it does not exceed ${\cal O}(100$ MeV)
and can be neglected. In this case, the Higgs boson can be “integrated out”
resulting in the following contact interaction with light quarks $q$ and
gluons,222222This result assumes the standard Yukawa couplings, which is not
necessarily the case for light quarks. With Higgs–dependent Yukawa couplings
[197], the amplitude can be significantly enhanced [198].

$\Delta{\cal L}^{\rm eff}={\lambda_{\phi h}\over
2m_{h_{0}}^{2}}\;\phi^{2}\;\left(\sum_{q}m_{q}\bar{q}q-{\alpha_{s}\over
4\pi}G_{\mu\nu}^{a}G^{\mu\nu\,a}\right)\;,$ (268)

where $G^{a}_{\mu\nu}$ is the gluon field strength and $\alpha_{s}$ is the
SU(3) fine structure constant. Here, the heavy quarks have been integrated out
using the conformal anomaly according to Shifman, Vainshtein and Zakharov
[199]: a heavy quark contribution $m_{Q}\bar{Q}Q$ gets replaced with
$-\alpha_{s}/12\pi\;G_{\mu\nu}^{a}G^{\mu\nu\,a}$. To get the dark matter
interaction with nucleons, one takes the matrix element of the quark and gluon
operators over the nucleon state,

$\displaystyle\langle N|m_{q}\bar{q}q|N\rangle=m_{N}f_{q}$
$\displaystyle\langle N|G_{\mu\nu}^{a}G^{\mu\nu\,a}|N\rangle=-{8\pi\over
9\alpha_{s}}m_{N}f_{TG}\;,$ (269)

where $f_{q},f_{TG}$ are some parameters determined from low energy
experiments [200],[201],[202] and lattice simulations [203]. This results in
the effective DM–nucleon interaction

$\Delta{\cal L}^{\rm eff}_{\phi N}={\lambda_{\phi h}m_{N}f_{N}\over
2m_{h_{0}}^{2}}\;\phi^{2}\;\bar{N}N\;,$ (270)

with

$f_{N}=\sum_{u,d,s}f_{q}+{2\over 9}f_{TG}\simeq 0.3\;.$ (271)

Numerically, the sum over the light quarks contributes about 0.1 to $f_{N}$,
while the gluon term contributes about 0.2. Thus, the nucleon–Higgs coupling
is dominated by the gluons or, more precisely, the heavy quarks.

The resulting DM–nucleon scattering cross section is [167]

$\sigma^{\rm SI}_{\phi-N}={\lambda_{\phi h}^{2}f_{N}^{2}\over 4\pi
m_{h_{0}}^{4}}\,{m_{N}^{4}\over(m+m_{N})^{2}}\;.$ (272)

The current XENON1T upper bound on $\sigma^{\rm SI}$ is of order $10^{-46}$
cm2 for electroweak DM masses and reaches the minimum $4\times 10^{-47}$ cm2
at $m\simeq 30$ GeV [204]. This imposes a strong constraint on the model. In
particular, apart from the very narrow resonance region, it pushes the thermal
DM to a multi–TeV range. More generally, it rules out the Higgs portal
couplings above $10^{-2}-10^{-1}$ for $m$ between 100 GeV and 1 TeV [172].

Invisible Higgs decay. A strong constraint on light dark matter is imposed by
the LHC bound on invisible Higgs decay [168]. The SM Higgs decay width is
dominated by the small $b$–quark Yukawa coupling which makes it very narrow,
$\Gamma_{h}\sim 4$ MeV. Thus, the decay branching ratios are sensitive to
non–standard modes including $h\rightarrow\phi\phi$, when kinematically
allowed. In our case, the invisible decay width is given by

$\Gamma(h\rightarrow\phi\phi)={\lambda_{\phi h}^{2}v^{2}\over 32\pi
m_{h_{0}}}\;\sqrt{1-{4m^{2}\over m_{h_{0}}^{2}}}\;.$ (273)

Close to the kinematic threshold $m\simeq m_{h_{0}}/2$, collider DM production
via Higgs decay should include a momentum–dependent width as described in
[205].

A recent detailed analysis of the relevant LHC constraints can be found in
[172]. The direct bound on the invisible Higgs decay branching ratio is ${\rm
BR}(h\rightarrow{\rm inv})<0.24$ at 95% CL [206]. The indirect bound is
stronger: the value adopted in [172] is 0.2, while the bound based on the
overall Higgs signal strength $\mu$ used in [207] is close to 0.1. The reason
is that the number of produced Higgs bosons which subsequently decay into
known 2–body final states agrees very well with the Standard Model prediction,
within 10%. A significant ${\rm BR}(h\rightarrow{\rm inv})$ would spoil this
agreement by reducing $\mu$, which leads to a stronger bound.

In case of thermal dark matter, the required Higgs portal coupling is above
$10^{-1}$ which leads a large invisible branching ratio, typically above 80%.
Thus, for either choice of the ${\rm BR}(h\rightarrow{\rm inv})$ constraint,
light dark matter is ruled out [168]. For non–thermal DM, this possibility
remains and the invisible decay provides one with a sensitive channel to probe
parameter space with $m<m_{h_{0}}/2$.

Indirect DM detection. Dark matter can be detected indirectly via products of
its annihilation in objects with large DM density, for instance the Galactic
Center or dwarf galaxies. Non–observation of a significant signal places
further bounds on the model. Excellent reviews of indirect DM detection can be
found in [208],[209].

The main constraint comes from gamma and X–rays produced in the annihilation
either directly or through cascade decays of the final state. The relevant
observable is the photon flux at the detector within the energy range $dE$ and
solid angle $d\Omega$,

${dN_{\gamma}\over dE\,dt\,d\Omega}={A\over 4\pi}\;\left({dN_{\gamma}\over
dE}\right)_{0}\;\kappa\,{\langle\sigma v_{r}\rangle\over
m^{2}}\;\int_{0}^{\infty}\rho^{2}_{\rm DM}dr\;.$ (274)

Here $N_{\gamma}$ is the photon number, $A$ is the detector area,
$(dN_{\gamma}/dE)_{0}$ is the photon number within the energy range $dE$
emitted per annihilation, and $\kappa=1/2$ for self–conjugate DM (as in our
case) or $1/4$ for DM consisting of particles and anti–particles. The last
factor is the DM density squared integral along the line of sight defined by
the angular variables $(\varphi,\theta)$, often called the $J$–factor,

$J(\varphi,\theta)\equiv\int_{0}^{\infty}\rho^{2}_{\rm
DM}(\varphi,\theta,r)\,dr\;,$ (275)

where $r$ is the distance from the detector. It is a function of the DM
density profile, e.g. Navarro-Frenk-White (NFW) [210] or Einasto [211].

Although the above formulae are quite elementary, it is worth clarifying them
in order to understand their limitations. The annihilation rate per unit
volume is given by the average cross section and the DM number density,
$\Gamma=\langle\sigma v_{r}\rangle\,n_{\rm DM}^{2}$. Since the photon emission
is isotropic, the fraction $A/(4\pi r^{2})$ of the photons reaches the
detector. Each elementary volume $dV=r^{2}drd\Omega$ thus contributes
$\Gamma\,dV\,A/(4\pi r^{2})\,(dN_{\gamma}/dE)_{0}$ detector photons per unit
time and $dE$. Integrating over $r$ along the line of sight and recalling that
in the non–relativistic limit $\rho_{\rm DM}=m\,n_{\rm DM}$, one obtains the
above results. The $\kappa$–factor takes into account the phase space symmetry
of the initial state as well as the difference between the total DM density
and the density of its annihilating components. For identical particles, there
is just one component and $\kappa=1/2$. For DM consisting of particles and
anti–particles, there is no symmetry factor, but the annihilation rate is
$\Gamma=\langle\sigma v_{r}\rangle\,n_{+}n_{-}$, where $n_{+}=n_{-}=n_{\rm
DM}/2$ are the particle and antiparticle densities, respectively. Hence,
$\kappa=1/4$ in this case.

Although the $J$–factor is larger in the Galactic Center, the dwarf spheroidal
galaxies impose a stronger constraint on DM annihilation [212]. The latter
contain fewer baryons leading to a smaller background and better control over
a possible signal. The resulting bounds on the model parameters have been
discussed in [213],[214],[215],[169], yet the XENON1T direct detection
constraints and the LHC bound supersede those from indirect detection. In
practice, the indirect detection constraints are conveniently incorporated
using micrOMEGAs [194].

Combined constraints. The latest comprehensive analysis of all of the
constraints has been performed within the GAMBIT framework [216]. It assumes
thermal production of DM and takes into account the LHC, direct and indirect
DM detection bounds. In addition, it requires perturbativity of the couplings
at least up to the scale max$(m,m_{t})$, which bounds $\lambda_{\phi h}$ from
above by $\sqrt{4\pi}$ in this energy range. At $m\gg m_{t}$, the Higgs portal
coupling necessary for the right DM relic abundance becomes large, so the
requirement of perturbativity of $\lambda_{\phi h}$ and $\lambda_{h}$ sets the
limit $m<4.5$ TeV [216]. Among the other constraints, the most prominent role
is played by the direct DM detection bound from XENON1T, which rules out much
of the parameter space. It gets weaker at $m>1$ TeV [172] such that the
allowed parameter space is skewed towards large DM masses. The constraint is
also loose around $m\simeq m_{h_{0}}/2$, where a very small $\lambda_{\phi h}$
is sufficient to obtain the correct relic DM abundance.

Figure 15:  Dark matter mass vs the Higgs portal coupling in the scalar DM
model subject to all of the constraints. The color coding corresponds to the
profile likelihood, with the light regions being more likely. The white
contours indicate $1\sigma$ and $2\sigma$ confidence regions. In terms of the
convention in the text, $m_{S}\equiv m$, $\lambda_{hS}\equiv\lambda_{\phi h}$.
The figure is from Ref. [216] © CC–BY.

Fig. 15 shows two notable (light–colored) regions where all of the constraints
are satisfied [216]. The first one is the resonant annihilation region at
$m\simeq m_{h_{0}}/2$, while the other corresponds to multi–TeV DM with order
one coupling. The constraints are also satisfied for $m$ close to $m_{h_{0}}$,
although the allowed parameter range is small. The profile likelihood has been
computed by varying the model parameters as well as the nuisance parameters,
e.g. the DM local density and mean velocity, nuclear matrix elements,
etc.232323Ref. [216] also treats the relic DM density constraint as an upper
bound: $Y(t\rightarrow\infty)\leq Y_{\rm obs}$. The lower edges of the allowed
regions correspond to the observed value (see also [215]).  The scanning
procedure is described in detail in [216].

The conclusion is that, apart from rather contrived possibilities $m\simeq
m_{h_{0}}/2$ and $m\simeq m_{h_{0}}$, the scalar DM mass is pushed into the
multi–TeV range by the XENON1T bounds. Yet, it cannot be too heavy due to
perturbativity constraints such that the allowed parameter range will largely
be probed by upcoming direct detection experiments. On the other hand, the LHC
prospects for observing scalar DM are not very promising: its production via
an off–shell Higgs is impeded by a small cross section ($<1$ fb) [168]. The
on–shell Higgs decay can be a useful probe, e.g. via monojet events [217], but
its kinematic reach is limited.

####  9.1.2 Pseudo–Goldstone dark matter

As we have seen, the minimal Higgs portal model favors a multi–TeV dark matter
mass. This is no longer the case if one adds one more degree of freedom and
makes the scalar $\phi$ complex. In this Section, we consider an interesting
possibility of pseudo–Goldstone dark matter, which naturally satisfies the
direct DM detection constraints even for light masses [218].

Consider a scalar potential invariant under a global U(1): $\phi\rightarrow
e^{i\xi}\phi$. Upon spontaneous symmetry breaking, this entails a massless
Goldstone boson. To make the model phenomenologically viable, let us add soft
(dim–2) explicit symmetry breaking resulting in a massive pseudo–Goldstone
boson. The potential then consists of two pieces [218],

$\displaystyle
V_{0}=m_{h}^{2}|H|^{2}+m_{\phi}^{2}|\phi|^{2}+\lambda_{h}|H|^{4}+\lambda_{\phi
h}|H|^{2}|\phi|^{2}+\lambda_{\phi}|\phi|^{4}\;,$ $\displaystyle V_{\rm
soft}=-{\tilde{m}_{\phi}^{2}\over 4}\,\phi^{2}+{\rm h.c.}$ (276)

The absence of higher dimensional U(1) breaking terms can be justified by
treating the couplings as spurions, as we discuss below. All the parameters
except for $\tilde{m}_{\phi}^{2}$ are real, while $\tilde{m}_{\phi}^{2}$ can
be made real and positive by a U(1) rotation. In this case, the potential
depends on the phase of $\phi$ as $-\cos(2{\rm Arg}\,\phi)$ and is minimized
at ${\rm Arg}\,\phi=0$.

An important feature of the above potential is that after spontaneous symmetry
breaking it retains the CP–like symmetry,

$\phi\rightarrow\phi^{*}\;.$ (277)

This makes the imaginary part of $\phi$ stable and, thus, a dark matter
candidate. Expressing

$\phi={1\over\sqrt{2}}\left(\varphi+i\chi+w\right)\;,$ (278)

where the VEV $w$ is real, one finds that $\varphi$ mixes with the SM Higgs
producing two CP–even mass eigenstates $h_{1},h_{2}$ as detailed in Section
2.3. The formulae for the mixing angle and the mass eigenvalues (21),(22)
remain the same. The pseudoscalar component $\chi$ does not mix with the
scalars and plays the role of dark matter with mass

$m_{\chi}=\tilde{m}_{\phi}\;.$ (279)

The interactions relevant to DM–nucleon scattering calculations are

$\Delta{\cal L}_{\chi-f}={1\over
2w}\,\chi^{2}\;\left(h_{1}\;m_{1}^{2}\,\sin\theta-
h_{2}\;m_{2}^{2}\,\cos\theta\right)-\sum_{f}{m_{f}\over
v}\;\bar{f}f\,\left(h_{1}\cos\theta+h_{2}\sin\theta\right)\;.$ (280)

The scattering proceeds via the $t$–channel $h_{1}$ and $h_{2}$ exchange. One
immediately concludes that the two contributions cancel each other in the
limit of zero momentum transfer,

${\cal A}\propto\sin\theta\,\cos\theta\;\left({m_{2}^{2}\over
t-m_{2}^{2}}-{m_{1}^{2}\over t-m_{1}^{2}}\right)\approx 0\;,$ (281)

where the neglected terms are of order
$t/m_{1,2}^{2}\sim{\bf{q}}^{2}/m_{1,2}^{2}$, where ${\bf{q}}$ is the momentum
transfer. This remarkable cancellation has a simple explanation. The
pseudo–Goldstone $\chi$ inherits the Goldstone boson property that the
trilinear vertex vanishes in the limit of zero momentum transfer, as long as
the massive $\chi$ is on–shell. This is most easily seen in polar coordinates,

$\phi={1\over\sqrt{2}}\,\rho\,e^{i\chi}\;,$ (282)

such that the $\chi\,\chi\,\rho\,(q=0)$ vertex vanishes when
$(\Box+m_{\chi}^{2})\,\chi=0$, as shown in Fig. 16. This property, however, is
not respected by higher dimensional U(1)–breaking operators, e.g. $\phi^{4}$.
As long as the latter are suppressed, the above amplitude is vanishingly
small.

Figure 16:  The pseudo–Goldstone boson vertex vanishes for $\chi$ on–shell and
zero momentum transfer.

The main contribution to the direct detection amplitude is generated at 1–loop
level. Indeed, radiative corrections induce dim–4 U(1)–breaking operators

$\Delta V=\lambda_{\phi
h}^{\prime}|H|^{2}\phi^{2}+\lambda_{\phi}^{\prime}\phi^{4}+\lambda_{\phi}^{\prime\prime}|\phi|^{2}\phi^{2}+{\rm
h.c.}$ (283)

The couplings are all real so that the symmetry $\phi\rightarrow\phi^{*}$ is
unbroken and vanish as $m_{\chi}\rightarrow 0$. Their explicit expressions can
be found in [218], while for our purposes it suffices to know that they are
bounded by ${\cal O}(\lambda_{\phi h}\lambda_{\phi}/16\pi^{2})$ or ${\cal
O}(\lambda_{\phi}^{2}/16\pi^{2})$. The resulting DM–nucleon scattering cross
section is of order [218]

$\sigma_{\chi-N}^{\rm SI}\sim{\sin^{2}\theta\over
64\pi^{5}}\,{m_{N}^{4}f_{N}^{2}\over m_{1}^{4}v^{2}}\,{m_{2}^{8}\over
m_{\chi}^{2}w^{6}}$ (284)

for EW scale masses, while for light $\chi$ it is suppressed further. It does
not exceed $10^{-49}$ cm2, whereas the current XENON1T upper bound is $4\times
10^{-47}$ cm2 at the most sensitive point. Similar results are obtained by a
careful analysis of $\chi-N$ scattering at 1 loop [219],[220],[221]. As a
result, absence of direct DM detection does not impose any significant bound
on the model parameters. In fact, a more stringent constraint is set by
perturbative unitarity: the process $h_{2}h_{2}\rightarrow h_{2}h_{2}$
requires $\lambda_{\phi}\lesssim 4\pi/3$ [29], which can be translated into a
bound on $w$ for a fixed $m_{2}$.

The DM annihilation amplitude does not exhibit the same suppression since the
momentum transfer in this case is significant, $2m_{\chi}$. Therefore, it is
easy to reconcile the direct DM detection bounds with the required DM
annihilation cross section. While $m_{\chi}<m_{1}/2$ is essentially excluded
by the Higgs invisible width measurements, the indirect detection constraint
is insignificant [207] such that most of the remaining mass range is
available.

Figure 17:  Constraints on pseudo–Goldstone dark matter [218]. The red band
corresponds to the correct thermal relic abundance as measured by PLANCK. The
purple shaded region is excluded by invisible Higgs decay, while in the grey
region perturbative unitarity is violated. In terms of the convention in the
text, $v_{s}\equiv w$. Figure credit: reprinted figure with permission from C.
Gross, O. Lebedev and T. Toma, Phys. Rev. Lett. 119, no.19, 191801 (2017);
doi:10.1103/PhysRevLett.119.191801. Copyright (2017) by the American Physical
Society.

An example of the allowed parameter space is shown in Fig. 17. The vertical
axis $v/w$ represents the strength of the DM coupling to $h_{1},h_{2}$ (see
Eq. 280). The red band is consistent with the thermal DM relic abundance
[334]. It features the characteristic dips associated with resonant
annihilation through $h_{1}$ and $h_{2}$. Most points within the band satisfy
all the other constraints such that the allowed DM mass ranges from about 60
GeV to 10 TeV.

Some of the parameter space can be probed at HL–LHC [207]. The most promising
channel appears to be VBF production of $h_{1}$ and $h_{2}$ with their
subsequent (on–shell) decay into a pair of $\chi$’s. For the $h_{2}$ mode, the
invisible decay branching ratio can be large and its kinematic reach extends
further than that for $h_{1}$ does, making it a particularly interesting probe
of pseudo–Goldstone dark matter. While the thermal DM relic appears elusive to
the LHC searches (apart from the resonance regions), the non–thermal option
can be probed efficiently for $\sin\theta\gtrsim 0.1$ and
$m_{2}\,,\,m_{\chi}<m_{2}/2$ in the electroweak range. The indirect DM
detection constraint plays a significant role in this case [207], while future
missions can become a sensitive probe of the model [222].

Finally, let us discuss how this set–up can arise from a more fundamental
theory. We have neglected the U(1) breaking terms of dimension higher than 2.
This can be justified if the underlying, possibly gauge, U(1) symmetry is
broken at a high scale by a VEV of another scalar $\Phi$ with even U(1) charge
$q_{\Phi}$, while the charge of $\phi$, $q_{\phi}$, is odd. The residual
parity symmetry $\phi\rightarrow-\phi$ requires that the low energy effective
theory contain only even powers of $\phi$. The higher dimensional operators
are suppressed by a higher power of the effective theory scale $\Lambda$.
Indeed, the allowed operators have the form

${\phi^{2k}\,\Phi^{l}\over\Lambda^{2k+l-4}}\;,$ (285)

with integer $k,l$. Define

$\epsilon\equiv{\langle\Phi\rangle\over\Lambda}~{}~{}~{},~{}~{}~{}n\equiv-2\,{q_{\phi}\over
q_{\Phi}}\;,$ (286)

where $n$ can be chosen integer and $\epsilon\ll 1$. Then

$\tilde{m}_{\phi}^{2}\sim\langle\Phi\rangle^{2}\,\epsilon^{n-2}~{}~{},~{}~{}\lambda_{\phi
h}^{\prime}\sim\lambda_{\phi}^{\prime\prime}\sim\epsilon^{n}~{}~{},~{}~{}\lambda_{\phi}^{\prime}\sim\epsilon^{2n}\;.$
(287)

Clearly, for small enough $\epsilon$, the leading term is
$\tilde{m}_{\phi}^{2}$, while the others are suppressed. The higher
dimensional terms can violate CP leading to DM decay. However, a very small
$\epsilon$ makes the decay time much longer than the age of the Universe such
that dark matter can be treated as stable for all practical purposes [218].

Various aspects of the pseudo–Goldstone dark matter and its generalizations
have been studied in [223]–[233]. In particular, the gravitational wave output
during a phase transition in such setups was considered in [224],[228],[232].
In [230],[231] it was noted that the U(1) can be identified with the B–L
symmetry. Possible venues to generalize the set–up to non–Abelian symmetries
were explored in [225].

###  9.2 Vector dark matter

The concept of gauge symmetry plays a central role in the Standard Model. It
is thus reasonable to assume that it also features in the hidden sector. As we
show below, stability of dark matter can be a natural consequence of gauge
symmetry.

Vector dark matter stability stems from symmetries of the Lie groups,
including their outer automorphisms. When combined with the minimal field
content, they result in “custodial” symmetries of the Lagrangian which
stabilize dark matter [234],[165],[235]. This phenomenon is analogous to the
appearance of the baryon number symmetry in the SM: SU(3) $\times$ SU(2)
$\times$ U(1) gauge symmetry and the SM field content automatically imply a
conserved baryon number (apart from the anomaly) which makes the proton
stable. In our case, some of the gauge fields are stable and play the role of
dark matter.

Suppose the hidden sector is endowed with gauge symmetry. To avoid appearance
of light or massless states, let us introduce the minimal number of hidden
“Higgs” multiplets $\phi_{i}$ to break this gauge symmetry
completely.242424Massless states can also be avoided via confinement in the
hidden sector, see e.g. [236]. Such multiplets couple naturally to the
Standard Model through the Higgs portal. At the renormalizable level, the
coupling has to be quartic since the trilinear terms are forbidden by gauge
symmetry. For the U(N) dark sector symmetry and $\phi_{i}$ in fundamental
representation, the Higgs portal couplings have the form

$V_{\phi_{i}h}=\lambda_{\phi_{i}h}\,H^{\dagger}H\phi^{\dagger}_{i}\phi_{i}\;,$
(288)

where $i$ labels the hidden Higgs multiplets. If $\phi_{i}$ with different $i$
have the same quantum numbers under other possible symmetries, the mixed terms
involving $\phi_{i}^{\dagger}\phi_{j}$ are also allowed. One general
consequence of these couplings is that the Higgs mixes with the hidden scalars
because both $H$ and $\phi_{i}$ develop non–zero VEVs. These states then
mediate interactions between dark and observable matter.

In what follows, we focus on unitary gauge groups in analogy with the Standard
Model, although other possibilities can be equally viable. Let us start with
the simplest case of an Abelian symmetry group.

####  9.2.1 U(1) gauge symmetry

Consider a U(1) gauge theory with a charged scalar $\phi$ [165],

${\cal L_{\rm dark}}=-{1\over
4}F_{\mu\nu}F^{\mu\nu}+(D_{\mu}\phi)^{\dagger}D^{\mu}\phi-V\;,$ (289)

where $F_{\mu\nu}$ is the field strength tensor of the gauge field $A_{\mu}$
and $\phi$ is assigned charge +1/2 (for an easier comparison with the
non–Abelian case). Suppose $\phi$ develops a VEV,

$\langle\phi\rangle={1\over\sqrt{2}}~{}\tilde{v}\;,$ (290)

which can always be taken real. The imaginary part of $\phi$ gets absorbed by
the gauge field that now becomes massive,

$m_{A}={\tilde{g}\tilde{v}\over 2}\;,$ (291)

where $\tilde{g}$ is the gauge coupling. The real part of $\phi$ is a physical
scalar, which in the unitary gauge can be written as

$\phi={1\over\sqrt{2}}\;\bigl{(}\rho+\tilde{v}\bigr{)}\;,$ (292)

such that $\rho$ is canonically normalized. This theory has the following
gauge–scalar interactions:

$\displaystyle\Delta{\cal L}_{\rm s-g}={\tilde{g}^{2}\over
4}\tilde{v}\rho\;A_{\mu}A^{\mu}+{\tilde{g}^{2}\over
8}\rho^{2}\;A_{\mu}A^{\mu}\;.$ (293)

We observe that the system possesses the $Z_{2}$ symmetry

$A_{\mu}\rightarrow-A_{\mu}~{},$ (294)

which corresponds to the usual charge conjugation. In the Lie group language,
it acts as an outer automorphism by conjugating the U(1) group elements. In
terms of the original fields, the symmetry transforms
$\phi\rightarrow\phi^{*}$ and $A_{\mu}\rightarrow-A_{\mu}$, which leaves both
the Lagrangian and the vacuum invariant.

Since the vertices always contain pairs of $A_{\mu}$, the massive gauge field
is stable and can constitute dark matter. It interacts with the SM fields via
the scalar mediators: due to the Higgs portal coupling (288), $\rho$ mixes
with the Higgs $h$ producing mass eigenstates $h_{1}$,$h_{2}$ as in Section
2.3,

$\displaystyle\rho=-h_{1}\;\sin\theta+h_{2}\;\cos\theta\;,$ $\displaystyle
h=h_{1}\;\cos\theta+h_{2}\;\sin\theta\;.$ (295)

Interactions mediated by $h_{1}$ and $h_{2}$ allow for the usual DM freeze–out
and also produce a direct detection signal, which we discuss in Section 9.2.4.
If the mixing angle is sufficiently small and $h_{2}$ can be integrated out,
we recover the effective Higgs–vector interaction (253).

The above considerations assume that the U(1) does not mix with the SM
hypercharge, that is, no $Z_{2}$–violating $F_{\mu\nu}F^{\mu\nu}_{Y}$ term is
present in the Lagrangian. This is justified if the U(1) is secluded, i.e.
there are no fields charged under both the SM gauge group and the dark sector
symmetry. In this case, if the U(1) and U(1)Y are orthogonal initially so that
there is no tree level kinetic mixing, such a mixing will not be generated
radiatively. Settings of this type are common in string theory, e.g. in the
$E_{8}\times E_{8}$ heterotic string [237], where the SM gauge group
originates from one $E_{8}$, while the dark symmetry comes from the other
$E_{8}$. Similar considerations apply to field theoretic constructions based
on group products $G_{1}\times G_{2}$ which contain no fields transforming
under both groups, but possess a Higgs portal–like coupling
$|s_{1}|^{2}|s_{2}|^{2}$, where $s_{1}$ and $s_{2}$ are scalars transforming
under $G_{1}$ and $G_{2}$, respectively. Appropriate symmetry breaking within
each sector brings us to the framework considered here.

The model at hand can be viewed as an effective field theory limit of a more
fundamental framework. In general, higher dimensional operators may affect
stability of dark matter. This depends on whether or not the UV completion
preserves the charge conjugation symmetry ${\cal C}$. The effective theory is
built out of gauge invariant operators

${O}\left(F_{\mu\nu},D_{\mu},\phi,\phi^{*}\right)\;.$ (296)

If ${\cal C}$ is violated, operators like

$c\;\left[\phi^{*}D_{\mu}\phi\right]\,\left[\phi^{*}D^{\mu}\phi\right]+{\rm
h.c.}$ (297)

with complex $c$ induce linear couplings of the gauge field to $\rho$. In this
case, $A_{\mu}$ decays via $\rho-h$ mixing, although its lifetime can be very
long depending on the suppression scale of the dim-6 operator. This leads to
an interesting option of long–lived dark matter.

Also, if the matter sector at low energies is enlarged, the stabilizing
$Z_{2}$ can be broken. For instance, introduction of the second charged scalar
generally entails both the presence of a linear in $A_{\mu}$ coupling and a
scalar–pseudo-scalar mixing. In this case, stability of dark matter is lost.

Our conclusion here is that a very simple renormalizable model with a secluded
U(1) and a single dark Higgs leads to stable dark matter. The stability is
guaranteed by gauge symmetry and the minimal field content.

####  9.2.2 SU(2) gauge symmetry

The above considerations are extended to the SU(2) case [234] in a
straightforward manner. It is sufficient to introduce a single dark Higgs
doublet $\phi$ in order to break the symmetry completely. The relevant
Lagrangian is

${\cal L_{\rm dark}}=-{1\over
4}F_{\mu\nu}^{a}F^{a\;\mu\nu}+(D_{\mu}\phi)^{\dagger}D^{\mu}\phi-V\;,$ (298)

where $a=1,2,3$ is the adjoint SU(2) index. At the minimum of the potential,
$\phi$ develops a VEV $\tilde{v}$ and the field can be written (in the unitary
gauge) as

$\phi={1\over\sqrt{2}}\;\left(\begin{matrix}0\\\
\rho+\tilde{v}\end{matrix}\right)~{},$ (299)

where $\rho$ is real. As in the U(1) case, the vector field mass in terms of
the gauge coupling $\tilde{g}$ is given by $m_{A}=\tilde{g}\tilde{v}/2$. The
SU(2) gauge interactions can be split into two terms,

$\displaystyle\Delta{\cal L}_{\rm s-g}={\tilde{g}^{2}\over
4}\tilde{v}\rho\;A_{\mu}^{a}A^{a\;\mu}+{\tilde{g}^{2}\over
8}\rho^{2}\;A_{\mu}^{a}A^{a\;\mu}\;,$ $\displaystyle\Delta{\cal L}_{\rm
g-g}=-\tilde{g}\epsilon^{abc}(\partial_{\mu}A_{\nu}^{a})A^{\mu\;b}A^{\nu\;c}-{\tilde{g}^{2}\over
4}\left((A_{\mu}^{a}A^{\mu\;a})^{2}-A_{\mu}^{a}A_{\nu}^{a}\;A^{\mu\;b}A^{\nu\;b}\right)~{}.$
(300)

Although the triple gauge vertex breaks the $Z_{2}$ of the previous section,
the Lagrangian preserves a custodial SO(3),

$A_{\mu}\rightarrow{\cal O}A_{\mu}\;,$ (301)

with ${\cal O}\in$ SO(3). This symmetry contains analogs of the $Z_{2}$ parity
which reflect two of the gauge fields. Since only the gauge fields transform
under SO(3) and are degenerate in mass, they are stable. In many respects,
such dark matter behaves similarly to the Abelian vector DM, except it
experiences self–interaction. An advantage over the U(1) case is that the
kinetic mixing with the SM gauge bosons is forbidden by gauge invariance.

An interesting feature of non–Abelian dark sectors is that some DM components
remain stable in the presence of arbitrary non–renormalizable operators. As
long as gauge invariance is preserved by integrating out heavy states, the
effective theory is built out of operators

${O}\left(F_{\mu\nu}^{a},D_{\mu},\phi,\phi^{\dagger}\right)\;.$ (302)

SU(2) invariance requires the number of $\phi$’s to be equal to the number of
$\phi^{\dagger}$’s. Thus, the system has a global U(1) symmetry

$\phi\rightarrow e^{i\zeta}\phi\;.$ (303)

A combination of this U(1) and gauge transformation ${\cal U}={\rm
diag}(e^{i\zeta},e^{-i\zeta})$ remains unbroken by the vacuum. Therefore,
after spontaneous symmetry breaking, the system possesses a global U(1)′
$\sim$ SO(2) and, generally, the custodial symmetry gets reduced by higher-
dimensional operators:

${\rm SO(3)\rightarrow SO(2)}\;.$ (304)

Under this SO(2),

$\displaystyle A_{1}\rightarrow A_{1}\;\cos 2\zeta+A_{2}\;\sin 2\zeta\;,$
$\displaystyle A_{2}\rightarrow-A_{1}\;\sin 2\zeta+A_{2}\;\cos 2\zeta\;,$
(305)

while all other fields are invariant. As a result, $A_{1,2}$ remain stable and
constitute dark matter. On the other hand, $A_{3}$ decays, for instance, via
an analog of operator (297) which breaks SO(3). Decaying SU(2) dark matter can
produce an interesting gamma ray signature [238] .

####  9.2.3 SU(3) and larger groups

The SU(3) case [235] is more complicated as it involves multiple dark Higgses.
One may instead use a larger group representation, e.g. a 6–plet, to break the
symmetry, however, the corresponding scalar potential is too constrained and
the breaking is not complete. The minimal option to break SU(N) to nothing
employs N-1 dark scalar multiplets in the fundamental representation [235].

Consider an SU(3) gauge theory with 2 scalar triplets $\phi_{1}$, $\phi_{2}$,

${\cal L}_{\rm
dark}=-\frac{1}{2}\textrm{tr}\\{G_{\mu\nu}G^{\mu\nu}\\}+|D_{\mu}\phi_{1}|^{2}+|D_{\mu}\phi_{2}|^{2}-V\,.$
(306)

The potential $V=V_{\rm portal}+V_{\rm dark}$ includes the Higgs portal terms,

$V_{\rm
portal}=\lambda_{H11}\,|H|^{2}|\phi_{1}|^{2}+\lambda_{H22}\,|H|^{2}|\phi_{2}|^{2}-(\lambda_{H12}\,|H|^{2}\phi_{1}^{\dagger}\phi_{2}+{\rm
h.c.})\;,$ (307)

as well as the purely dark scalar potential,

$\displaystyle V_{\rm dark}$
$\displaystyle=m_{11}^{2}|\phi_{1}|^{2}+m_{22}^{2}|\phi_{2}|^{2}-(m_{12}^{2}\phi_{1}^{\dagger}\phi_{2}+{\rm
h.c.})$
$\displaystyle+\frac{\lambda_{1}}{2}|\phi_{1}|^{4}+\frac{\lambda_{2}}{2}|\phi_{2}|^{4}+\lambda_{3}|\phi_{1}|^{2}|\phi_{2}|^{2}+\lambda_{4}|\phi_{1}^{\dagger}\phi_{2}|^{2}$
$\displaystyle+\left[\frac{\lambda_{5}}{2}(\phi_{1}^{\dagger}\phi_{2})^{2}+\lambda_{6}|\phi_{1}|^{2}(\phi_{1}^{\dagger}\phi_{2})+\lambda_{7}|\phi_{2}|^{2}(\phi_{1}^{\dagger}\phi_{2})+{\rm
h.c.}\right]\,.$ (308)

SU(3) gauge freedom allows one to remove 5 degrees of freedom of $\phi_{1}$
and 3 degrees of freedom of $\phi_{2}$, so that they can be put in the form

$\phi_{1}={1\over\sqrt{2}}\,\left(\begin{array}[]{c}0\\\ 0\\\
v_{1}+\varphi_{1}\end{array}\right)\,,\quad\phi_{2}={1\over\sqrt{2}}\,\left(\begin{array}[]{c}0\\\
v_{2}+\varphi_{2}\\\
(v_{3}+\varphi_{3})+i(v_{4}+\varphi_{4})\end{array}\right)~{},$ (309)

where $v_{i}$ are real VEVs and $\varphi_{1-4}$ are real scalars. This gauge
choice is analogous to the unitary gauge in the Higgs sector of the Standard
Model, $H^{T}=(0,v+h)/\sqrt{2}$.

The Lagrangian possesses an accidental global U(1),

$\phi_{i}\rightarrow e^{i\xi}\phi_{i}\;.$ (310)

Even though it gets broken by the VEVs, a combination of this U(1) and a gauge
transformation

${\cal U}={\rm diag}\left(e^{2i\xi},e^{-i\xi},e^{-i\xi}\right)$ (311)

remains preserved. This global U(1)′ symmetry guarantees that some of the
gauge bosons are stable. Indeed, the gauge fields transform as

$A^{\mu}\rightarrow{\cal U}\;A^{\mu}\;{\cal U}^{\dagger}\;,$ (312)

so, in the Gell–Mann basis for the SU(3) generators, $A^{\mu}_{3,6,7,8}$
remain invariant while $A^{\mu}_{1,2,4,5}$ transform non–trivially,

$\displaystyle A_{1}\rightarrow A_{1}\;\cos 3\xi+A_{2}\;\sin 3\xi\;,$
$\displaystyle A_{2}\rightarrow-A_{1}\;\sin 3\xi+A_{2}\;\cos 3\xi\;,$ (313)

where the Lorentz index has been suppressed. The same relation holds for the
$(A_{4},A_{5})$ pair up to the replacement $1\rightarrow 4$, $2\rightarrow 5$.
The scalars are all neutral under U(1)′, so the lighter pair out of
$(A_{1},A_{2})$ and $(A_{4},A_{5})$ has to be stable. Note that U(1)′ ensures
$m_{A_{1}}=m_{A_{2}}$ and $m_{A_{4}}=m_{A_{5}}$.

The potential is quite complicated. To simplify its analysis, let us assume
that CP is conserved in the dark sector: the couplings and the VEVs of
$\phi_{1,2}$ are real ($v_{4}=0$). In this case, the system possesses an extra
$Z_{2}$ analogous to charge conjugation, i.e. it conjugates the group elements
and the scalar fields. Under the $Z_{2}$,

$\displaystyle A_{1,3,4,6,8}$ $\displaystyle\rightarrow$ $\displaystyle-
A_{1,3,4,6,8}\;,$ $\displaystyle\varphi_{4}$ $\displaystyle\rightarrow$
$\displaystyle-\varphi_{4}\;,$ (314)

while the other fields are invariant. This symmetry ensures that additional
states are stable. Their identity depends on the mass spectrum which defines
which decays are allowed kinematically.

A complication in the SU(3) case is that, in our gauge, some of the scalars
mix kinetically with the gauge bosons, i.e. terms of the type
$\kappa_{ai}A^{\mu}_{a}\partial_{\mu}\phi_{i}$ appear in the Lagrangian. These
are eliminated by a “gauge–like” transformation $A^{\mu}_{a}\rightarrow
A^{\mu}_{a}+\omega_{ai}\partial^{\mu}\phi_{i}$, where $\omega_{ai}$ are
constants depending on the VEVs. Such a transformation does not affect the
gauge field masses and kinetic terms, so we will keep the same notation for
the transformed fields, $A^{\mu}_{a}$.

Let us make a further simplification and set $v_{3}=0$. Then, the gauge boson
masses read

$m^{2}_{A^{1}}=m^{2}_{A^{2}}=\frac{\tilde{g}^{2}}{4}v_{2}^{2}\;,\quad
m^{2}_{A^{4}}=m^{2}_{A^{5}}=\frac{\tilde{g}^{2}}{4}v_{1}^{2}\;,\quad
m^{2}_{A^{6}}=m^{2}_{A^{7}}=\frac{\tilde{g}^{2}}{4}(v_{1}^{2}+v_{2}^{2})\;.$
(315)

The components corresponding to the diagonal generators mix. The mass matrix
is diagonalized by

$\displaystyle\left(\begin{array}[]{c}A^{\prime 3}_{\mu}\\\ A^{\prime
8}_{\mu}\end{array}\right)=\left(\begin{array}[]{c}\cos\alpha\,A^{3}_{\mu}+\sin\alpha\,A^{8}_{\mu}\\\
\cos\alpha\,A^{8}_{\mu}-\sin\alpha\,A^{3}_{\mu}\end{array}\right)\,,\quad\textrm{where}\quad\tan
2\alpha=\frac{\sqrt{3}v_{2}^{2}}{2v_{1}^{2}-v_{2}^{2}}\;,$ (320)

and the eigenvalues are

$m^{2}_{A^{\prime
3}}=\frac{\tilde{g}^{2}v_{2}^{2}}{4}\Big{(}1-\frac{\tan\alpha}{\sqrt{3}}\Big{)}\,,\qquad
m^{2}_{A^{\prime
8}}=\frac{\tilde{g}^{2}v_{1}^{2}}{3}\frac{1}{1-\frac{\tan\alpha}{\sqrt{3}}}\,,$
(321)

For $v_{1}>v_{2}$, the lightest gauge bosons are

$A_{1},A_{2},A_{3}^{\prime}\;.$ (322)

In the simplest case, these constitute dark matter. As long as
$A_{3}^{\prime}$ is lighter than the pseudoscalar $\varphi_{4}$, their
stability is ensured by the U(1)${}^{\prime}\times Z_{2}$ symmetry. Note that
although $A_{1,2}$ are somewhat heavier than $A_{3}^{\prime}$, their decay is
forbidden by U(1)′.

Dark matter can also be composed of stable components with different spins: if
$\varphi_{4}$ is light, we have multicomponent DM consisting of
$A_{1},A_{2},\varphi_{4}$. While the scalars all generally mix with the Higgs,
$\varphi_{4}$ does not as long as CP is conserved in the dark sector. In all
cases, DM interacts with the SM via exchange of the Higgs–like scalars which
are mixtures of $h$ and $\varphi_{1-3}$. The list of the interaction vertices
for the canonically normalized fields can be found in [235]. The resulting
phenomenology is rather involved [239].

In the most general case, the scalar potential contains complex couplings and
CP is thus broken. Then, only U(1)′ remains and dark matter consists of

$A_{1},A_{2}\;.$ (323)

This scenario is quite interesting phenomenologically since it allows for pair
annihilation of $A_{1}$ and $A_{2}$ into the lighter $A_{3}^{\prime}$. The
latter then decays into the SM states, facilitating the “secluded dark matter”
scenario which we discuss later. In this case, the direct DM detection rate is
effectively decoupled from the DM annihilation rate. It is interesting that
this scenario is realized regardless of the relation between $v_{1}$ and
$v_{2}$: one of the diagonal generator fields is always the lightest, so the
above argument applies to the other regimes up to the field relabelling [239].

Higher dimensional operators do not affect stability of dark matter since
U(1)′ remains unbroken. The argument parallels that for the SU(2) case (cf.
Eq. 302).

Generalization to the SU(N) case is quite straightforward [235]: to break
SU(N) completely, one introduces N-1 fundamentals. They can be parametrized in
a way similar to (309) such that N-1 components of the first N–plet are
rotated away and so on. One row of all of the multiplets can be set to zero,
which always entails an unbroken global U(1). The lightest gauge fields are
associated with an SU(2) block that couples to the smallest VEV. In the
simplest case, these states constitute dark matter whose phenomenology is
similar to that of SU(3) dark matter.

Finally, it should be noted that breaking SU(N) to nothing is not strictly
speaking necessary: the unbroken gauge group can confine at low energies and
thus lead to acceptable phenomenology. The logarithmic coupling running
creates a scale hierarchy in the dark sector such that one encounters “dark
radiation” [240] and/or heavy non–WIMP dark matter [241],[242].

####  9.2.4 Phenomenology of vector dark matter

Abelian dark matter. Let us start with the U(1) case. The relevant terms in
the Lagrangian are

$\displaystyle{\cal L}$ $\displaystyle\supset{1\over
2}m_{A}^{2}A_{\mu}A^{\mu}+{\tilde{g}\,m_{A}\over
2}\left(-h_{1}\sin\theta+h_{2}\cos\theta\right)A_{\mu}A^{\mu}$
$\displaystyle+{\tilde{g}^{2}\over
8}\left(h_{1}^{2}\sin^{2}\theta-2h_{1}h_{2}\sin\theta\;\cos\theta+h_{2}^{2}\cos^{2}\theta\right)A_{\mu}A^{\mu}\;.$
(324)

The scalars $h_{1},h_{2}$ with masses $m_{1},m_{2}$ couple to the SM fields
just like the Higgs does, up to the suppression factors $\cos\theta$ and
$\sin\theta$, respectively.

It is instructive to highlight the differences between the scalar and vector
DM. For many applications one may use the the non–relativistic limit $|{\bf
p}_{A}|\ll m_{A}$, in which case vector DM behaves as 3 distinct massive
scalars with no inter–species coupling. Compared to single species scalar DM
with the same density, the vector DM annihilation cross section is therefore
smaller by a factor of 3 while the scattering rate off nuclei remains the
same. On the other hand, the collider phenomenology can be very different:
production of the longitudinal components of the vector at high energies is
enhanced by the derivative (Goldstone) couplings. Thus, the decay
$h_{2}\rightarrow AA$ with $m_{2}\gg m_{A}$ is much more efficient for the
vectors.

The DM scattering on nucleons is mediated by the $t$–channel exchange of
$h_{1,2}$. The corresponding spin–independent cross is [235]

$\sigma^{\rm SI}_{A-N}={\tilde{g}^{2}\,\sin^{2}2\theta\over
16\pi}\;{m_{N}^{4}f_{N}^{2}\over v^{2}}\;{(m_{2}^{2}-m_{1}^{2})^{2}\over
m_{1}^{4}m_{2}^{4}}\;,$ (325)

where $m_{N}$ is the nucleon mass and $f_{N}\simeq 0.3$ parametrizes the
Higgs–nucleon coupling. Although the $h_{1}$ and $h_{2}$ contributions come
with opposite signs, normally there is no significant cancellation between
them and the amplitude is dominated by the $h_{1}$ exchange. For
electroweak/TeV masses and $\theta\sim 10^{-1}$, the stringent direct DM
detection bounds constrain the dark gauge coupling to be ${\cal O}(10^{-1})$
or below. These relax for much heavier DM, in which case the experimental
limits are loose.

The DM relic abundance depends strongly on its production mechanism. Let us
assume that DM reaches thermal equilibrium with the SM thermal bath, while
keeping in mind that non–thermal production can also be viable. In this case,
the abundance is dictated by the annihilation cross section, which is
characteristic of WIMP dark matter. The annihilation process is mediated by
$h_{1},h_{2}$ or produces $h_{1}$ and $h_{2}$ in the final state. The leading
diagrams are shown in Fig. 18. The main $s$–channel becomes less efficient as
$\sin\theta$ decreases. For instance, the annihilation cross section into
fermions having $N_{c}^{f}$ colors is given by

$\sigma
v_{r}\Bigl{|}_{AA\rightarrow\bar{f}f}=\frac{\tilde{g}^{2}N_{c}^{f}}{48\pi
v^{2}}\;\sin^{2}{2\theta}\;\frac{{m_{f}^{2}m_{A}^{2}\left(m_{1}^{2}-m_{2}^{2}\right)}^{2}}{{\left(m_{1}^{2}-4m_{A}^{2}\right)}^{2}{\left(m_{2}^{2}-4m_{A}^{2}\right)}^{2}}\;{\left(1-\frac{m_{f}^{2}}{m_{A}^{2}}\right)}^{3/2}\;,$
(326)

while the results for the other final states can be found in [239]. Thus, in
order to reconcile the required cross section with the direct detection
constraints, one has to resort to resonant enhancement. Indeed, for
$2m_{A}\sim m_{2}$, the $s$–channel annihilation becomes very efficient even
at small couplings. For a heavy $h_{2}$, the resonance is broad and no
significant fine–tuning is necessary to satisfy the bounds [235]. On the other
hand, the $h_{1}$ resonance is narrow and the resonant enhancement is at work
for $2m_{A}$ very close to $m_{1}$. Another interesting regime is
$m_{A}>m_{2}$, in which case pair annihilation into $h_{2}$’s can dominate,
while the direct detection amplitude gets suppressed by small $\theta$. This
“secluded DM” option will be discussed in the next section.

Indirect DM detection bounds do not impose any significant additional
constraints unless DM is light, $m_{A}\lesssim m_{W}$ [235]. For $m_{A}<62$
GeV, the invisible Higgs decay sets a strong bound on the Higgs–DM coupling.
At $m_{A}\ll m_{1},m_{2}$, it requires roughly $\tilde{g}\,\theta\lesssim
10^{-4}\;m_{A}/{\rm GeV}$, which rules out thermal DM [168].

Figure 18:  Leading diagrams for vector DM annihilation [235].

Altogether, one finds that for electroweak/TeV masses, (thermal) vector dark
matter is viable for $2m_{A}\sim m_{2}$ and $\tilde{g}\,\theta\sim{\cal
O}(10^{-2})$. Smaller mixing angles are allowed if the secluded option is
realized.

Both thermal and non–thermal DM options can to some extent be probed at the
LHC. The DM production is mediated by $h_{1}$ and $h_{2}$, and manifests
itself as missing energy, $E_{T}$. The most promising channels are the monojet
and $h_{1,2}$–VBF production with missing energy,

$\displaystyle pp\rightarrow h_{1,2}\;j~{}\rightarrow AA\;j~{},$
$\displaystyle pp\rightarrow h_{1,2}\;jj\rightarrow AA\;jj\;.$ (327)

These are efficient if the scalars decay into DM on–shell. The $h_{2}$ decay
appears more interesting as it has a larger kinematic reach. The corresponding
decay width is

$\Gamma(h_{2}\rightarrow AA)={\tilde{g}^{2}\cos^{2}\theta\;m_{2}^{3}\over
128\pi m^{2}_{A}}\left(1-{4m_{A}^{2}\over m_{2}^{2}}+{12m_{A}^{4}\over
m_{2}^{4}}\right)\sqrt{1-{4m_{A}^{2}\over m_{2}^{2}}}\;.$ (328)

The consequent decay branching ratio is calculated using
$\Gamma(h_{2}\rightarrow{\rm SM})=\sin^{2}\theta\;\Gamma_{\rm
SM}(m_{h}=m_{2})$ as well as the $h_{2}\rightarrow h_{1}h_{1}$ width given by
Eq. 30a. To evaluate the signal strength, one also needs the $h_{2}$
production cross section, which is easily obtained by rescaling the SM result
with $m_{h}=m_{2}$ by $\sin^{2}\theta$. The cut–based monojet study [243]
concludes that only the light $h_{2}$ range with $m_{2}\lesssim 300$ GeV can
be probed for a realistic $\sin\theta$. Similar results are obtained in the
VBF channel [244], while the corresponding analysis in the decoupling limit
$m_{2}\rightarrow\infty$ can be found in [245]. The prospects, however, are
likely to improve when modern statistical techniques, e.g. machine learning,
are employed (see the discussion below Eq. 329).

Various aspects of Abelian vector DM, including phenomenology of models with
extended scalar sectors, have been studied in [246]-[251]. For example, in
addition to the usual constraints, one may require the Higgs portal coupling
to stabilize the Higgs potential [248]. Another variation is to relax the
assumption of thermalized DM and focus on its freeze–in production [249]. Loop
corrections to the direct detection amplitude have been computed in [252].

Non–Abelian dark matter. Many properties of Abelian DM are retained up to the
summation over the group indices, $A_{\mu}A^{\mu}\rightarrow
A_{\mu}^{a}A^{\mu,a}$, $a=1,..,N^{2}-1$. DM contains multiple components which
can only annihilate with its own species resulting in a smaller by a factor of
$1/({N^{2}-1})$ annihilation cross section into the SM final states. Yet, DM
self–interaction brings in new “semi–annihilation” channels [234],[253], where
the DM number reduces by one, Fig. 19. However, their effect is not very
significant in the allowed parameter space.

Figure 19:  Semi–annihilation of vector DM [235].

DM self–interaction also allows for $3\rightarrow 2$ processes which can be
efficient if the coupling is large enough [254],[255]. A scalar analog of this
phenomenon will be discussed in Section 10.2. For yet larger coupling values,
the dark sector undergoes confinement and the massive vectors can be described
by interpolating operators of the type
$\phi^{\dagger}T^{a}i\overleftrightarrow{D}_{\mu}\phi$, whose phenomenology is
discussed in [236].

At weak coupling, the main difference from the Abelian case is that there are
gauge fields with different masses which generally leads to a broader spectrum
of reactions. Some of them allow for very efficient DM annihilation even at
tiny $\theta$ [256], which we discuss in the next section. Regarding collider
physics, larger non–Abelian sectors generally entail more complicated cascade
decays, including those with multi–Higgs final states and missing $E_{T}$
[257]. For example, the processes

$\displaystyle pp\rightarrow h_{n}\;\rightarrow\tilde{A}\tilde{A}\rightarrow
AA\;h_{1}h_{1}~{},$ $\displaystyle pp\rightarrow
h_{n}\;\rightarrow\tilde{A}\tilde{A}\rightarrow AA\;h_{1}h_{1}h_{1}$ (329)

are only possible for dark groups larger than SU(2). Here $h_{n}$ and
$\tilde{A}$ denote collectively the scalars that mix with the Higgs and
unstable gauge fields, respectively. Again, these channels are efficient if
$h_{n}$ decays into gauge bosons on–shell. At the HL–LHC, the two Higgs mode
can be studied with the help of the Boosted Decision Trees technique. For
$h_{1}$’s decaying to $\bar{b}b+\gamma\gamma$ and $\bar{b}b+WW$, the model can
be probed in the regime $m_{h_{n}}\gg m_{A}$ with $m_{h_{n}}\lesssim 600$ GeV
and substantial $\sin\theta\sim 0.3$ [257]. For a fully hadronic $h_{1}$
decay, $h_{1}h_{1}\rightarrow\bar{b}b\bar{b}b$, and heavy $h_{n}$, one may use
jet substructure analysis to probe $m_{h_{n}}$ up to 1 TeV or so [258]. The
three Higgs final state, on the other hand, is likely to be of little
relevance at the LHC [257].

Various aspects as well as extensions of the SU(2) DM model have been studied
in [238],[259]-[268]. For example, adding non–renormalizable operators
generally makes some of DM unstable, whose decay can produce gamma ray lines
[238]. One may also impose additional constraints on the model, e.g. classical
scale invariance [259],[260],[261],[265]. Finally, variations of the SU(3)
vector DM has been studied in [269],[270].

A recent update of the parameter space analysis of vector dark matter,
including the XENON1T constraint as well as the prospects for observing
invisible Higgs decay, has appeared in [271].

####  9.2.5 “Secluded” dark matter: evading direct detection

The main idea of secluded DM is to decouple direct DM detection from DM
annihilation using metastable mediators [272],[273]. This allows one to obtain
the correct DM relic density with no pressure from the ever-improving direct
detection bounds.

In our case, the direct detection amplitude goes to zero as $\theta\rightarrow
0$, while the annihilation can still be efficient if there are states lighter
than DM. In particular, the channel [256]

$A_{1}A_{1}\,,\,A_{2}A_{2}\rightarrow A_{3}^{\prime}A_{3}^{\prime}$ (330)

is always open kinematically for SU(3) and larger groups. Here the group
indices refer to the SU(2) block which couples to the smallest VEV (cf. Eqs.
315, 321). The diagonal component mixes with the other Cartan generators
resulting in a lighter mass eigenstate $A_{3}^{\prime}$. If CP is broken in
the dark sector, $A_{3}^{\prime}$ is unstable and decays into (on– or
off–shell) scalars which mix with the Higgs. The relevant vertices are listed
in the Appendix of [235]. The above annihilation process is controlled by the
gauge coupling and the mass difference between $A_{1,2}$ and $A_{3}^{\prime}$.
It is insensitive to $\theta$, so the correct relic abundance can be obtained
without violating the direct detection bounds as long as $\theta\ll 1$.

Figure 20:  SU(3) vector dark matter constraints in the “secluded” regime. The
red band indicates the correct DM relic density. The other curves mark the
following constraints: grey – perturbativity, purple – invisible Higgs decay
(${\rm BR}_{\rm inv}<0.11$), dark red – Higgs couplings ($\sin\theta<0.33$),
green and orange – direct DM detection constraints. The blue line represents
the direct detection event rate corresponding to the background neutrino
scattering off nuclei. The figure is from Ref. [256].

In addition to the above channel, there may exist further analogous processes
which survive at $\theta\ll 1$. If $h_{2}$ is lighter than DM, the channel

$A_{1}A_{1}\,,\,A_{2}A_{2}\rightarrow h_{2}h_{2}$ (331)

is available even for U(1) and SU(2) dark gauge groups. The corresponding
cross section in the limit $\theta\ll 1$ and $m_{1}\ll m_{A},m_{2}$ is given
by [256]

$\displaystyle\sigma{v_{r}}=\frac{\tilde{g}^{4}}{576\pi
m_{A}^{2}}\;\frac{11m_{2}^{8}-80m_{2}^{6}m_{A}^{2}+240m_{2}^{4}m_{A}^{4}-320m_{2}^{2}m_{A}^{6}+176m_{A}^{8}}{\left(4m_{A}^{2}-m_{2}^{2}\right)^{2}\left(2m_{A}^{2}-m_{2}^{2}\right)^{2}}\;\sqrt{1-\frac{m_{2}^{2}}{m_{A}^{2}}}\;,$
(332)

while the total DM annihilation cross section is rather complicated. This
process is sufficient to reconcile the relic density and direct detection
constraints in much of the parameter space with small $\theta$ [256].

A representative summary of the constraints for the SU(3) case is shown in
Fig. 20. In addition to $AA\rightarrow h_{2}h_{2}$, another channel
$AA\rightarrow\chi\chi$, where $\chi$ is the mostly pseudoscalar state,
contributes to DM annihilation when kinematically allowed. On the other hand,
the channel $AA\rightarrow A_{3}^{\prime}A_{3}^{\prime}$ is always open. The
red band produces the correct (thermal) DM relic density and exhibits
characteristic dips at $m_{A}\simeq m_{2}/2,m_{2},m_{\chi}$ corresponding to
the resonance and kinematic thresholds. Further constraints include
perturbativity of scalar and gauge couplings, bounds on the Higgs couplings
and direct detection constraints [274],[275].252525Here, the XENON1T
“prospect” constraint assumes $\sigma_{A-N}^{\rm SI}\lesssim 10^{-46}$ cm2 at
the most sensitive point $m_{A}\simeq 40$ GeV, which is not far from the
current bound.  We conclude that all of the above constraints can be satisfied
for $\theta\ll 1$ in a broad range of DM masses. The plot does not include,
however, indirect detection constraints which are relevant for light DM and
exclude some of the parameter space (see e.g [270]). Finally, such secluded DM
is notoriously difficult to observe at colliders: the small $\theta$ makes its
production cross section highly suppressed.

Our conclusion is that vector dark matter provides a viable WIMP dark matter
candidate. Within the Higgs portal framework, its stability is a natural
consequence of gauge symmetry and the minimal matter content.

###  9.3 Fermion dark matter

Dark matter can be composed of fermion fields. A simple and viable option is
provided by an SM singlet fermion $\chi$ [276],[277]. Although a pair of such
DM particles cannot couple directly to the Higgs field at the renormalizable
level, it can couple to a scalar singlet which mixes with the Higgs due to the
usual Higgs portal interaction. Redefining the scalar field and performing a
chiral rotation on the fermion, the relevant Lagrangian can be brought to the
form

$-\Delta{\cal
L}=m_{\chi}\bar{\chi}\chi+\lambda_{\chi}\,\phi\,\bar{\chi}\chi+\lambda_{\chi}^{\rm
cp}\,\phi\,\bar{\chi}i\gamma_{5}\chi\;,$ (333)

where $\phi$ has a zero VEV and $m_{\chi}$, $\lambda_{\chi}$,
$\lambda_{\chi}^{\rm cp}$ are real. The renormalizable interactions, in
general, contain a CP violating coupling [170], which in our convention we
identify with $\lambda_{\chi}^{\rm cp}$, while the scalar potential can
contain odd powers of $\phi$. The explicit form of the above transformations
can be found in [170],[214]. For our purposes, it is not necessary and we may
parametrize the observables in terms of
$m_{\chi},\lambda_{\chi},\lambda_{\chi}^{\rm cp}$ as well as the scalar sector
parameters: the Higgs–singlet mixing angle $\theta$, and the scalar masses
$m_{1},m_{2}$ as in Section 6.2. Decoupling $h_{2}$ at small enough $\theta$,
we recover the effective Higgs–fermion interaction (254).

In the fermion case, stability of dark matter stems from a conserved fermion
number corresponding to the symmetry

$\chi\rightarrow e^{i\kappa}\chi\;,$ (334)

which can be either global or gauge. If there is no conserved number or
parity, a coupling to the leptons $H^{c}\bar{l}\chi$ would be allowed making
$\chi$ unstable although possibly long–lived. It is thus natural to take
$\chi$ to be a Dirac fermion. In what follows, we will consider both the Dirac
and Majorana fermion options, with the latter possessing a Majorana parity
$\chi\rightarrow-\chi$.

####  9.3.1 Dirac fermion dark matter

Consider the Dirac fermion case. DM annihilation into SM fermions proceeds via
$s$–channel exchange of the two scalars. The amplitudes proportional to
$\lambda_{\chi}$ and $\lambda_{\chi}^{\rm cp\;}$ do not interfere and the
corresponding cross section is given by

$\sigma v_{r}=N_{c}^{f}\,{m_{\chi}^{2}m_{f}^{2}\sin^{2}2\theta\over 8\pi
v^{2}}\,\left[v_{r}^{2}\,\lambda_{\chi}^{2}+\lambda_{\chi}^{\rm
cp\;2}\right]\,{(m_{2}^{2}-m_{1}^{2})^{2}\over(4m_{\chi}^{2}-m_{1}^{2})^{2}(4m_{\chi}^{2}-m_{2}^{2})^{2}}\,\left(1-{m_{f}^{2}\over
m_{\chi}^{2}}\right)^{3/2}\;,$ (335)

where subleading terms in DM velocity $v_{r}$ have been dropped. This
expression agrees with the corresponding result in [277]. The contribution of
the $\lambda_{\chi}$–coupling is velocity suppressed, $v_{r}^{2}\sim T_{\rm
fo}/m_{\chi}\sim 1/20$. This is easily understood from parity or CP
considerations: a fermion–antifermion pair has parity $(-1)^{l+1}$, where $l$
is the orbital angular momentum. The $s$–wave state, therefore, cannot
annihilate into a parity–even scalar $\phi$ if parity is conserved. The
coupling $\lambda_{\chi}^{\rm cp}$ lifts this condition. The same
considerations apply to other SM final states.

While the CP–even contribution to DM annihilation is suppressed, for direct DM
detection the situation is reversed: the CP–violating coupling can be
neglected. The bilinear $\bar{\chi}i\gamma_{5}\chi$ vanishes in the static
limit, so the contribution of the $\lambda_{\chi}^{\rm cp}$ coupling to the
DM–nucleon scattering cross section is suppressed by the current dark matter
velocity squared $\sim 10^{-6}$. Thus, the result is completely dominated by
the $\lambda_{\chi}$–coupling,

$\sigma^{\rm
SI}_{\chi-N}={\lambda_{\chi}^{2}\sin^{2}2\theta\,m_{N}^{2}f_{N}^{2}\over 4\pi
v^{2}}\,{m_{\chi}^{2}m_{N}^{2}\over(m_{\chi}+m_{N})^{2}}\,{(m_{2}^{2}-m_{1}^{2})^{2}\over
m_{1}^{4}m_{2}^{4}}\;.$ (336)

This agrees with the cross section quoted in [278]. The presence of the
CP–violating term has a beneficial effect: it increases the annihilation cross
section while having no impact on the direct DM detection [170]. As a result,
for $\lambda_{\chi}^{\rm cp}>\lambda_{\chi}$, much of the parameter space is
consistent with the thermal DM constraints. A similar interplay is observed
for the axial DM coupling $\bar{\chi}\gamma_{\mu}\gamma_{5}\chi$ to a
$Z^{\prime}$ [279].

For light DM, the invisible Higgs decay imposes an important constraint on the
model. One finds

$\Gamma(h_{1}\rightarrow\chi\chi)={\sin^{2}\theta\,m_{1}\over
8\pi}\;\left[\lambda_{\chi}^{\rm
cp\;2}+\lambda_{\chi}^{2}\,\left(1-{4m_{\chi}^{2}\over
m_{1}^{2}}\right)\right]\,\sqrt{1-{4m_{\chi}^{2}\over m_{1}^{2}}}\,.$ (337)

An analogous expression holds for $h_{2}\rightarrow\chi\chi$ up to the
replacement $m_{1}\rightarrow m_{2}$, $\sin\theta\rightarrow\cos\theta$. We
observe that close to the resonance regions, $m_{\chi}\simeq m_{1,2}/2$, the
contribution of $\lambda_{\chi}$ to the invisible scalar decays is velocity
suppressed and the CP–violating coupling dominates. Such regions are of
particular interest since the relic density constraint is easily compatible
with the direct detection bound near the resonances.

Phenomenology of the CP–conserving limit of the model has been studied in
[277],[278]. Most results carry over to the Majorana case, as discussed below.
All the annihilation channels are included via the micrOMEGAs package [194].
The thermal DM relic abundance is compatible with the direct detection bounds
around the resonances $m_{\chi}\simeq m_{1,2}/2$ and when $\chi\chi\rightarrow
h_{2}h_{2}$ is efficient. In the latter case, $\theta\ll 1$ realizes the
“secluded” DM option. The presence of the CP–violating coupling
$\lambda_{\chi}^{\rm cp}>\lambda_{\chi}$ widens significantly the allowed
parameter space making the constraints easily compatible [170]. The conclusion
is that the EW/TeV DM masses and EW cross sections are generally allowed such
that $\chi$ behaves as a WIMP.

The LHC Higgs measurements strongly constrain and essentially rule out thermal
DM with $2m_{\chi}<125$ GeV [168], unless the DM annihilation proceeds via
$\chi\chi\rightarrow h_{2}h_{2}$ [170] with $\theta\ll 1$. In this case, one
requires a light $h_{2}$ which normally entails an efficient $h_{1}\rightarrow
h_{2}h_{2}$ decay. This mode can however be suppressed by tuning
$\lambda_{\phi h}$ for a given $\sin\theta$ [23].

####  9.3.2 Majorana fermion dark matter

If $\chi$ is a Majorana fermion with mass $m_{\chi}$, one must replace in Eq.
333,

$m_{\chi}\rightarrow{1\over 2}m_{\chi}\;.$ (338)

Keeping the interaction terms the same as in the Dirac case, one finds the
following modifications.

The annihilation rate increases by a factor of 4 since the amplitude receives
two identical contributions,262626It appears that $\sigma v_{r}$ quoted in
[168] applies to the Dirac instead of Majorana DM.

$\sigma v_{r}\Bigl{|}_{\rm Maj}=4\;\sigma v_{r}\Bigl{|}_{\rm Dir}\;.$ (339)

One should keep in mind, however, that in the reaction rate calculation, the
integral over the initial state phase space gets halved due to 2 identical
particles and each reaction reduces the fermion number by 2.

The velocity suppression of the CP–even amplitude still holds: while the
fermion and anti-fermion intrinsic parities are opposite, the Majorana
intrinsic parity is $\pm i$. Thus, the $s$-wave is still P–odd and its
annihilation into $\phi$ via $\lambda_{\chi}$ is forbidden.

Figure 21:  Profile likelihood for Majorana dark matter in terms of the
CP–violating phase and DM mass. The CP–phase is defined by
$\xi\equiv\arctan\left(\lambda_{\chi}^{\rm cp}/\lambda_{\chi}\right)$ and the
effective field theory limit $m_{2}\rightarrow\infty$ is assumed. The
constraints and scanning procedure are as those for Fig. 15. The figure is
from Ref. [169] © CC–BY.

The direct detection cross section also increases by a factor of 4 since the
amplitude doubles,

$\sigma^{\rm SI}_{\chi-N}\Bigl{|}_{\rm Maj}=4\;\sigma^{\rm
SI}_{\chi-N}\Bigl{|}_{\rm Dir}\;.$ (340)

Concerning the invisible width, the amplitude increase is partially
compensated by the phase space symmetry factor such that

$\Gamma(h_{1}\rightarrow\chi\chi)\Bigl{|}_{\rm
Maj}=2\;\Gamma(h_{1}\rightarrow\chi\chi)\Bigl{|}_{\rm Dir}\;.$ (341)

These results agree with those quoted in [167] in the limit $m_{2}\gg m_{1}$,
$\lambda_{\chi}^{\rm cp\;}\rightarrow 0$.

Altogether, the effect of replacing a Dirac fermion with a Majorana one is
non–trivial and does not simply amount to a rescaling of the couplings. In
general, if DM consists of particles and distinct from them antiparticles, the
annihilation cross section required to obtain the same total number density as
in the case of identical particles gets doubled (see Section 9.1.1) [192].
Thus, for Majorana particles it is easier to reconcile the relic density
constraint with the direct detection bound.

Phenomenology of Majorana DM has been studied extensively in
[167],[169],[170],[280],[281]. Most results of these analyses apply to the
Dirac case as well. The conclusion is that the minimal fermionic Higgs portal
dark matter is a viable WIMP candidate, especially if CP violation is allowed
in the dark sector. This is illustrated in Fig. 21, which shows the profile
likelihood in terms of the CP–violating phase
$\xi\equiv\arctan\left(\lambda_{\chi}^{\rm cp}/\lambda_{\chi}\right)$ and
$m_{\chi}$ in the effective field theory limit $m_{2}\rightarrow\infty$ [169].
It assumes thermal dark matter subject to all the relevant constraints as in
Fig. 15. The color coding shows how likely a particular parameter choice is.
We observe that $m_{\chi}$ is allowed to be anywhere above the resonance up to
about 10 TeV. Ref. [169] concludes that future direct and indirect DM
detection experiments will probe large portions of the favored parameter
space.

To summarize, all of the Higgs portal DM candidates presented here remain
viable, especially if one goes beyond the effective field theory framework.
The UV complete models have certain favorable features, some of which are not
captured by the effective approach. These include, for instance, new efficient
DM annihilation channels into metastable states as well as natural
cancellations among different contributions to the direct DM detection
amplitude. In most cases, we find that the framework will further be explored
by, first and foremost, the ever–improving direct DM detection experiments.
Certain regions of parameter space, especially for light dark matter, can also
be probed at the LHC and via indirect DM detection.

##  10  Feebly interacting dark scalars

While WIMPs may offer a natural explanation for the observed dark matter relic
abundance, the lack of their direct detection signal motivates one to explore
further options. Dark matter may be very weakly interacting and its density
could be determined by mechanisms other than the standard freeze–out. In this
section, we discuss the cosmology of scalars with a feeble Higgs portal
coupling

$V_{\phi h}={1\over 2}\,\lambda_{\phi h}H^{\dagger}H\,\phi^{2}\;,$ (342)

$\lambda_{\phi h}\ll 1$. Here, $\phi$ can play the role of (non-WIMP) dark
matter if $\langle\phi\rangle=0$. For a non–zero VEV, it is an unstable scalar
which may couple to dark matter. In the latter case, there are both quartic
and trilinear Higgs–singlet interactions. Although, in general, one expects an
independent $H^{\dagger}H\,\phi$ term in $V_{\phi h}$, its effect can be
mirrored by that of $\lambda_{\phi h}\langle\phi\rangle\,H^{\dagger}H\,\phi$,
so we will not consider it separately.

The dark scalars may be produced by the SM thermal bath via the Higgs portal
coupling. If the coupling is very small, the dark sector never thermalizes and
the relic abundance is determined by “freeze–in” rather than freeze–out. This
applies as well to unstable dark scalars which could subsequently decay into
DM or SM states. It is also possible that $\lambda_{\phi h}$ is completely
negligible, in which case the singlet must be produced shortly after
inflation, for instance, through its small coupling to the inflaton. In this
limit, the dark matter thermodynamics is determined by the self–coupling
$\phi^{4}$ [282] and its initial density.

Unlike in the WIMP case, much of the relevant dynamics can take place at
relativistic energies, which requires going beyond the standard
non–relativistic approach.

###  10.1 Relativistic treatment

Most dark matter studies employ the non–relativistic approximation to the
reaction rates. This approach is often justified. For example, the WIMP
freeze–out occurs at temperatures far below its mass such that the
corresponding relic abundance can be computed in the non–relativistic regime.
However, for feebly interacting scalars, this is not the case and interesting
physics can occur at high temperatures where the non–relativistic
approximation breaks down. It is therefore necessary to develop a fully
relativistic approach to dark matter dynamics.

The main ingredient in these studies is the relativistic $a\rightarrow b$
reaction rate per unit volume [191],

$\Gamma_{a\rightarrow b}=\int\left(\prod_{i\in a}{d^{3}{\bf
p}_{i}\over(2\pi)^{3}2E_{i}}f(p_{i})\right)~{}\left(\prod_{j\in b}{d^{3}{\bf
p}_{j}\over(2\pi)^{3}2E_{j}}(1+f(p_{j}))\right)|{\cal M}_{a\rightarrow
b}|^{2}~{}(2\pi)^{4}\delta^{4}(p_{a}-p_{b}).$ (343)

Here $p_{i}$ and $p_{j}$ are the initial and final state momenta,
respectively. ${\cal M}_{a\rightarrow b}$ is the QFT $a\rightarrow b$
transition amplitude. For convenience, we absorb in $|{\cal M}_{a\rightarrow
b}|^{2}$ both the initial and final state phase space symmetry factors. $f(p)$
is the momentum distribution function, which for scalars in kinetic
equilibrium takes the Bose–Einstein form [283],[284],[191]

$f(p)={1\over\exp^{E-\mu\over T}-1}\;,$ (344)

where $\mu$ is the effective chemical potential. This distribution maximizes
entropy for a given energy and particle number. The final state factor
$1+f(p_{j})$ accounts for the Bose enhancement of the reaction rate due to
degenerate states.

In this section, we are primarily interested in scalar systems with no
conserved quantum numbers and no CP–violation. The latter implies, in
particular, $|{\cal M}_{a\rightarrow b}|=|{\cal M}_{b\rightarrow a}|$. In full
thermal equilibrium, the absence of conserved numbers means that the effective
chemical potential vanishes and

$\prod_{i\in a}f(p_{i})\prod_{j\in b}\bigl{(}1+f(p_{j})\bigr{)}=\prod_{j\in
b}f(p_{j})\prod_{i\in a}\bigl{(}1+f(p_{i})\bigr{)}$ (345)

by virtue of energy conservation. Therefore,

$\Gamma_{a\rightarrow b}=\Gamma_{b\rightarrow a}\;,$ (346)

which signifies detailed balance. In this situation, the particle number is
conserved. More generally, at non–zero effective chemical potential,

$\Gamma_{a\rightarrow b}=\exp\left(\sum_{i\in a}\mu_{i}/T-\sum_{j\in
b}\mu_{j}/T\right)\;\Gamma_{b\rightarrow a}\;.$ (347)

This applies, in particular, to scalars in kinetic equilibrium and leads to
particle production or absorption. For example, the dark matter freeze–out
regime belongs to this category.

The evolution of the number density $n=\int{d^{3}{\bf
p}\over(2\pi)^{3}}\,f(p)$ of a given species is described by the Boltzmann
equation. In case the number change is due to the reactions $a\rightarrow b$
and $b\rightarrow a$, it reads

${dn\over dt}+3Hn=\Delta N\;(\Gamma_{a\rightarrow b}-\Gamma_{b\rightarrow
a})\;,$ (348)

where $\Delta N$ is the particle number change in the reaction $a\rightarrow
b$ and $H$ is the Hubble rate. This equation is to be supplemented with the
entropy conservation condition for a closed system,

$s\,a^{3}={\rm const}\;,$ (349)

where $s$ is the entropy density and $a$ is the scale factor. The two