NORDITA-2022-018 UPPSALA-19/22

Radiation reaction for spinning black-hole scattering
Francesco Alessioa,b and Paolo Di Vecchiaa,c

a NORDITA, KTH Royal Institute of Technology and Stockholm University,
Hannes Alfvéns väg 12, SE-11419 Stockholm, Sweden
b Department of Physics and Astronomy, Uppsala University,
Box 516, SE-75120 Uppsala, Sweden

c The Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen, Denmark

Starting from the leading soft term of the $5$-point amplitude, involving a
graviton and two Kerr black holes, that factorises into the product of the
elastic amplitude without the graviton and the leading soft factor, we compute
the infrared divergent contribution to the imaginary part of the two-loop
eikonal. Then, using analyticity and crossing symmetry, we determine the
radiative contribution to the real part of the two-loop eikonal and from it
the radiative part of the deflection angle for spins aligned to the orbital
angular momentum, the loss of angular momentum and the zero frequency limit of
the energy spectrum for any spin and for any spin orientation. For spin one we
find perfect agreement with recent results obtained with the supersymmetric
worldline formalism.

##  1 Introduction

The recent detection of gravitational waves (GW) by the LIGO and Virgo
collaborations [1, 2, 3, 4, 5, 6, 7] emitted by binary black holes or neutron
stars mergers is shedding new light on the nature of strongly gravitating
systems existing in the universe. The increasing sensitivity of the detectors
demands for an effort to develop new theoretical tools capable of high-
precision predictions of GW waveform templates. For these reasons, recently
there has been considerable interest in the use of amplitude-based techniques
for studying the inspiral phase of binary non-spinning massive objects
coalescence, extracting the conservative part of the Effective-One-Body (EOB)
potential [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], classical
observables [13, 22, 23, 24, 25], waveforms and radiation [26, 21, 27, 28, 20,
29, 30]. Astrophysical objects are also characterized by their spin.
Therefore, the post-Minkowskian (PM) expansion in the Newton’s constant $G$
for studying the scattering of spinning objects has also received attention,
using amplitudes methods and classical General Relativity (GR) analytical
techniques [31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46,
47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64].
Furthermore, observables computed for scattering events in the PM expansion
can be compared with those computed in the more traditional post-Newtonian
(PN) [65] perturbation theory (for spin effects in the PN scheme see e.g. [66,
67, 68, 69, 70, 71, 72, 73, 74, 75]), which is more suitable for describing
bound orbits, using the so-called Bound-to-Boundary (B2B) [16, 17, 18]
correspondence.

The contribution of the radiation reaction in the scattering of two spinless
black holes at 3PM has been crucial [76] to get rid of the divergence at high
energy that appears in the conservative part of the deflection angle [77, 78,
79]. This has been first shown in massive ${\cal{N}}=8$ supergravity by
performing the explicit calculation in the soft region [76] instead of just
the conservative region [80]. Immediately after the radiation reaction was
also computed in GR with two different methods. The first, discussed in Ref.
[81], is based on the computation of the loss of angular momentum that is then
inserted in a formula derived in Ref. [82] that gives the radiative part of
the deflection angle. The second one [83] uses instead the leading soft part
of the $5$-point amplitude for the emission of a graviton that allows to
compute the infrared divergent contribution to the imaginary part of the
eikonal at 3PM order. Then, using considerations based on analyticity and
crossing symmetry, the real part of the radiative eikonal at 3PM is also
extracted and from it the radiation reaction part of the deflection angle at
3PM has been obtained. The two approaches give the same result that, at high
energy, is also in agreement with the old result of Ref. [84]. This result is
also confirmed [85, 86, 87] by extracting from the explicit two-loop
calculations the radial action and from it compute the deflection angle.

In this paper we generalise the approach of Ref. [83] to the case of Kerr
black holes obtaining a closed expression for the radiation reaction part of
the deflection angle in the aligned-spin case and to all order in spin.
Assuming that the logarithmic divergence appearing in the radiation reaction
contribution to the scattering angle in the ultra-relativistic limit cancels
with that appearing in its conservative part, we conjecture a formula
describing the high-energy limit of the latter. Furthermore, using the Bini-
Damour formula, [82, 81, 88], we find expressions for the 2PM radiated angular
momentum for any spin configuration and to all orders in spin. Our results for
spin one agree with recent explicit calculations in Ref. [55, 62].

The paper is organized as follows. We start in section 2 by introducing the
kinematical setup describing the elastic $(2\rightarrow 2)$ and inelastic
$(2\rightarrow 3)$ scattering processes under considerations and by showing
the corresponding tree-level four-point and soft five-point amplitudes. We
approximate the latter by using Weinberg’s soft graviton theorem, that allows
to factorize it in terms of the four-point amplitude and the leading soft
factor. In section 3 we introduce the main ideas of our computation and then
we proceed to explicitly calculate the infrared divergent contribution to the
imaginary part of two-loop eikonal $\delta_{2}$ using the three-particle
unitarity cut. Then from it, using analyticity and crossing symmetry we derive
the radiation reaction piece of the real part of $\delta_{2}$. Our result
shows that all the spin dependence is encoded in a single vector field,
denoted by $\mathbf{f}$ in (3.14). We discuss how this feature can be
understood in terms of a Newman-Janis shift [89], which relates the Kerr to
the Schwarzschild solution. We end section 3 by displaying the zero-frequency
limit of the emitted energy spectrum. In section 4 we show explicit
expressions for the radiative contribution of the 3PM scattering angle for an
aligned-spin configuration and the emitted components of the 2PM angular
momentum in any spin configuration. Our final equations (4.6), (4.8) and (4.9)
hold to all orders in spin. We close in section 5 with a short summary and an
outlook. Throughout the rest paper we will use the mostly plus signature
$\eta_{\mu\nu}=\mathrm{diag}(-1,1,1,1)$ and we will mainly follow the
conventions used in [83].

##  2 Tree-level four-point and soft five-point amplitudes

We start by considering the tree-level $4$-point amplitude for elastic
scattering of two Kerr black holes with masses $m_{1}$ and $m_{2}$, rescaled
spin vectors $a_{1}^{\mu}=S_{1}^{\mu}/m_{1}$ and
$a_{2}^{\mu}=S_{2}^{\mu}/m_{2}$, that can be written in an exponentiated form
as [37, 39, 41]

$\displaystyle\mathcal{A}^{\mathrm{tree}}_{4}(q)=-\frac{\kappa^{2}m^{2}_{1}m^{2}_{2}}{t}\sigma^{2}\sum_{\pm}(1\pm
v)^{2}\mathrm{exp}\bigg{(}\pm
i\frac{\epsilon_{\mu\nu\alpha\beta}q^{\mu}a^{\nu}p_{1}^{\alpha}p_{2}^{\beta}}{m_{1}m_{2}\sigma
v}\bigg{)}+\mathcal{O}(t^{0}),$ (2.1)

where in the amplitude with one-graviton exchange we keep only terms non-
analytic for $t=-q^{2}\sim 0$ that are relevant for long-range effects. Here
$\kappa=\sqrt{8\pi G}$ where $G$ is the Newton constant, $a=a_{1}+a_{2}$,
$p_{1}$ and $p_{2}$ are the incoming momenta, $q=-p_{1}-p_{4}=p_{2}+p_{3}$ is
the transferred momentum and
$\sigma=\frac{1}{\sqrt{1-v^{2}}}=-\frac{p_{1}\cdot p_{2}}{m_{1}m_{2}}$. The
amplitude in (2.1), which is the main building block of our analysis, can be
obtained by gluing two massive spinning three-point amplitudes at minimal
coupling [31, 34, 39], as shown in Fig.1.

Figure 1: t-channel for the one-graviton exchange.

Note that we treat all vectors as formally ingoing. We work in the center-of-
mass frame, defined by

$\displaystyle p_{1}=(E_{1},\mathbf{p}),\qquad
p_{2}=(E_{2},-\mathbf{p}),\qquad\left|\mathbf{p}\right|=p,$ (2.2)

where $E_{i}=\sqrt{m^{2}_{i}+p^{2}}$ and the total energy,
$E=E_{1}+E_{2}=\sqrt{s}$ ($s$ being the Mandelstam variable
$s=-(p_{1}+p_{2})^{2}$), satisfies $Ep=m_{1}m_{2}\sqrt{\sigma^{2}-1}$. The
spin vectors are given by

$\displaystyle
a_{1}=\bigg{(}\frac{\mathbf{p}\cdot\mathbf{a}_{1}}{m_{1}},\mathbf{a}_{1}+\frac{\mathbf{p}\cdot\mathbf{a}_{1}}{m_{1}(m_{1}+E_{1})}\mathbf{p}\bigg{)},\quad
a_{2}=\bigg{(}-\frac{\mathbf{p}\cdot\mathbf{a}_{2}}{m_{2}},\mathbf{a}_{2}+\frac{\mathbf{p}\cdot\mathbf{a}_{2}}{m_{2}(m_{2}+E_{2})}\mathbf{p}\bigg{)},$
(2.3)

so that $a_{i}^{2}=\mathbf{a}_{i}^{2}$. We choose the spatial component of the
momenta and of the impact parameter to be along the $x^{3}$ and $x^{1}$
directions, respectively, i.e. $\mathbf{p}=p\,\mathbf{e}_{3}$ and
$\mathbf{b}=b\,\mathbf{e}_{1}$, so that the asymptotic scattering plane is the
$x^{1}x^{3}$ plane. Here $\mathbf{e}_{i}$ are the Cartesian spatial unit
vectors. So far we have not specified the directions of the spatial spin
vectors $\mathbf{a}_{i}$. If we choose them to be perpendicular to the
$x^{1}x^{3}$ plane we are considering an aligned-spin configuration. However
we will leave these directions arbitrary in the following, unless otherwise
specified.

The exponential factor appearing in the amplitude (2.1) in the center-of-mass
frame can be expressed in terms of spatial vectors only as

$\displaystyle\pm
i\frac{\epsilon_{\mu\nu\alpha\beta}q^{\mu}a^{\nu}p_{1}^{\alpha}p_{2}^{\beta}}{m_{1}m_{2}\sigma
v}=\pm
i\mathbf{q}\cdot(\hat{\mathbf{p}}\times\mathbf{a}),\qquad\hat{\mathbf{p}}=\frac{\mathbf{p}}{p},$
(2.4)

where $\mathbf{q}$ is the $2$-dimensional spatial transferred momentum
satisfying $\mathbf{q}\cdot\mathbf{p}=0$ ($\epsilon_{0123}=1$). We now
consider the inelastic $(2\rightarrow 3)$ process with the emission of an
additional graviton of momentum $k$ and we denote the corresponding $5$-point
amplitude by $\mathcal{A}_{5}^{\mu\nu}(q,k)$. We use the following
parametrisation for the momenta

$\displaystyle p_{1}\simeq\bar{p}_{1}-\frac{1}{2}q,\quad
p_{2}\simeq\bar{p}_{2}+\frac{1}{2}q,\quad
p_{3}\simeq-\bar{p}_{2}+\frac{1}{2}q,\quad
p_{4}\simeq-\bar{p}_{1}-\frac{1}{2}q,$ (2.5)

where $p_{1}^{2}=m^{2}_{1}\simeq\bar{p}^{2}_{1}$,
$p_{2}^{2}=m_{2}^{2}\simeq\bar{p}^{2}_{2}$ and $\bar{p}_{1}\cdot q\simeq 0$,
$\bar{p}_{2}\cdot q\simeq 0$ hold up to terms proportional to $k$, which are
negligible in the soft $k\rightarrow 0$ limit. In this regime, the five point
amplitude for such process factorizes according to Weinberg soft graviton
theorem [90] as

$\displaystyle\mathcal{A}_{5}^{\mu\nu}(q,k)=\kappa\sum_{i=1}^{4}\frac{p_{i}^{\mu}p_{i}^{\nu}}{k\cdot
p_{i}}\mathcal{A}^{\mathrm{tree}}_{4}(q)+\mathcal{O}(k^{0}).$ (2.6)

The multiplicative factor is the so-called Weinberg pole, that diverges as
$k^{-1}$ for small $k$. In general, one could go further in the soft expansion
and consider subleading soft terms with higher powers of $k$ [91, 92].
However, since we shall only be concerned in the infrared divergent part of
the imaginary part of the two-loop eikonal $\delta_{2}$, which is responsible
for radiation reaction effects [83, 30], we can neglect all such terms and
consider contributions that come only from those diagrams where the graviton
line is attached to an external leg.

By taking the classical limit, corresponding to $q\rightarrow 0$, and
therefore expanding the soft factor in powers of $q$ and keeping only the
terms linear in $q$, equation (2.6) becomes

$\displaystyle\mathcal{A}_{5}^{\mu\nu}(q,k)=\kappa\bigg{(}\frac{\bar{p}_{1}^{\mu}\bar{p}_{1}^{\nu}}{(\bar{p}_{1}\cdot
k)^{2}}(q\cdot
k)-\frac{(\bar{p}_{1}^{\mu}q^{\nu}+\bar{p}_{1}^{\nu}q^{\mu})}{\bar{p}_{1}\cdot
k}-1\leftrightarrow
2\bigg{)}\mathcal{A}_{4}^{\mathrm{tree}}(q)+\mathcal{O}(q^{2},k^{0}).$ (2.7)

where we have dropped higher powers of $q$ that are negligible in the
classical limit.

##  3 Infrared divergence of the two-loop eikonal

In this section we first go to impact parameter space and then we compute the
infrared divergent contribution to the unitarity relation given by the three-
particle cut that provides the imaginary part of the two-loop eikonal. Then
following the analysis based on crossing symmetry and analyticity discussed in
Refs. [76, 83, 30] we can extract the radiation reaction contribution to the
real part of the eikonal by means of the relation

$\displaystyle\mathrm{Re}\,2\delta_{2}^{\mathrm{rr}}=-\lim_{\epsilon\to
0}[\pi\epsilon(\mathrm{Im}\,2\delta_{2})],$ (3.1)

where $\epsilon=\frac{4-D}{2}$. Equation (3.1) has been confirmed by explicit
two-loop calculations in massive ${\cal{N}}=8$ supergravity and in in GR [30]
and is believed to hold at all energies and for all gravitational theories
involving two derivatives. As a consequence of (3.1), the infrared divergent
part of the tree-level, on-shell, inelastic amplitude for the $(2\rightarrow
3)$ process under consideration, which is simply fixed in (2.7) by the leading
Weinberg soft graviton theorem, is the only necessary ingredient to study
radiation reactions effects [83, 93]. This method provides a powerful shortcut
to get information about $\mathcal{O}(G^{3})$ classical observables, which are
usually obtained by a more involved two-loop computation. In particular, the
two observables that one can extract from Eq. (3.1) are the zero-frequency
limit of the emitted energy spectrum [94] and the radiative contribution to
the 3PM scattering angle given by

$\displaystyle\left.\frac{dE^{\mathrm{rad}}}{d\omega}\right|_{\omega\to
0}=\lim_{\epsilon\to
0}[-4\hbar\epsilon(\mathrm{Im}\,2\delta_{2})],\qquad\theta^{\mathrm{rad}}_{3}=-\frac{\hbar}{p}\frac{\partial\mathrm{Re}\,2\delta_{2}^{\mathrm{rr}}}{\partial
b}.$ (3.2)

Furthermore, having determined $\theta_{3}^{\mathrm{rad}}$ from Eq. (3.2) and
assuming the Bini-Damour [82, 81, 88, 95, 96] formula that gives it in terms
of total radiated angular momentum [DiVecchia:2022] and the conservative
deflection angle $\theta^{\mathrm{cons}}$,

$\displaystyle\theta^{\mathrm{rad}}=-\frac{1}{2}\frac{\partial\theta^{\mathrm{cons}}}{\partial
E}E^{\mathrm{rad}}-\frac{1}{2}\frac{\partial\theta^{\mathrm{cons}}}{\partial
J}J^{\mathrm{rad}},$ (3.3)

one can extract the total radiated angular momentum $J^{\mathrm{rad}}$ where
$J=bp$ is the total angular momentum in the center-of-mass frame. In
particular, inverting this formula and discarding the radiated energy
contribution, that starts at $\mathcal{O}(G^{3})$, one gets the radiated
angular momentum at $\mathcal{O}(G^{2})$, denoted here by
$J^{\mathrm{rad}}_{2}$.

Note that the scattering angle can be only defined in the aligned-spin
configuration. Indeed, when the two spin vectors are not aligned, the
scattering dynamics is not planar and the asymptotic scattering planes at
$t=-\infty$ and $t=\infty$ will not coincide and there are rather two
scattering angles [49]. In this case, the angular momentum in the center-of-
mass frame is radiated in both directions orthogonal to $\mathbf{p}$ [59]. In
our kinematics setup described after equations (2.3), it means that the
radiated angular momentum has non-vanishing components along $x^{1}$ and
$x^{2}$. In the rest of this section we compute the infrared divergent part of
$\mathrm{Im}\,2\delta_{2}$.

We first take the leading soft $5$-point amplitude in Eq. (2.7) and we go to
impact parameter space by the following formula:

$\displaystyle\tilde{\mathcal{A}}_{5}^{\mu\nu}(b,k)=\int\frac{d^{4-2\epsilon}q}{(2\pi)^{4-2\epsilon}}e^{iq\cdot
b}\hat{\delta}(2q\cdot\bar{p}_{1})\hat{\delta}(2q\cdot\bar{p}_{2})\mathcal{A}^{\mu\nu}_{5}(q,k),$
(3.4)

where $\hat{\delta}(2q\cdot\bar{p}_{i})\equiv 2\pi\delta(2q\cdot\bar{p}_{i})$
enforces the orthogonality conditions $q\cdot\bar{p}_{i}=0$. Because of the
two $\delta$-functions the integral over the two longitudinal components
spanned by $\bar{p}_{i}$ can be easily performed and one arrives at

$\displaystyle\tilde{\mathcal{A}}^{\mu\nu}_{5}(b,k)=\frac{1}{4m_{1}m_{2}\sqrt{\sigma^{2}-1}}\int\frac{d^{2-2\epsilon}\mathbf{q}}{(2\pi)^{2-2\epsilon}}e^{i\mathbf{q}\cdot\mathbf{b}}\mathcal{A}^{\mu\nu}_{5}(\mathbf{q},k).$
(3.5)

Inserting the amplitude (2.7) in the above integral and using the
representation in (2.4) for the spin exponential factor in the center-of-mass
frame we see that the entire spin dependence of
$\tilde{\mathcal{A}}^{\mu\nu}(\mathbf{b},k)$ is encoded in the shift
$\mathbf{b}\rightarrow\mathbf{b}\pm\hat{\mathbf{p}}\times\mathbf{a}$. This
property had been already noticed in [42, 51]111See also [97, 98] for earlier
connected works. and it comes from the nature of minimally coupled amplitudes
introduced in [31] and from the Fourier factor
$e^{i\mathbf{q}\cdot\mathbf{b}}$ relating classical observables in impact
parameter space to amplitudes [13, 40]. Ultimately, it admits an
interpretation in terms of a Newman-Janis shift [89], relating the spacetime
of a spinning black hole to that of a spinless one.

This allows to easily perform the above integral for all spins, yielding the
following amplitude in impact parameter space

$\displaystyle\tilde{\mathcal{A}}_{5}^{\mu\nu}(b,k)\simeq\frac{i\kappa^{3}}{8\pi}\frac{m_{1}m_{2}\sigma^{2}}{\sqrt{\sigma^{2}-1}}\sum_{\pm}\frac{(1\pm
v)^{2}}{\mathbf{b}_{\pm}^{2}}\bigg{(}\frac{\bar{p}_{1}^{\mu}\bar{p}_{1}^{\nu}}{(\bar{p}_{1}\cdot
k)^{2}}(\mathbf{k}\cdot\mathbf{b}_{\pm})-\frac{\bar{p}_{1}^{\mu}b_{\pm}^{\nu}+\bar{p}_{1}^{\nu}b_{\pm}^{\mu}}{\bar{p}_{1}\cdot
k}-1\leftrightarrow 2\bigg{)},$ (3.6)

where we defined for convenience
$b^{\mu}_{\pm}\equiv(0,\mathbf{b}\pm\hat{\mathbf{p}}\times\mathbf{a})$. Note
that this result is valid to all orders in spin and in any spin configuration.
The multipole expansion in the spin vector is recovered from the expansion of
$\mathbf{b}_{\pm}^{-2}$ around $\mathbf{a}=0$ as

$\displaystyle\frac{1}{\mathbf{b}_{\pm}^{2}}=\frac{1}{b^{2}}\mp\frac{\mathbf{b}\cdot(\hat{\mathbf{p}}\times\mathbf{a})}{b^{4}}-\frac{(\hat{\mathbf{p}}\times\mathbf{a})^{2}}{b^{4}}+2\frac{[\mathbf{b}\cdot(\hat{\mathbf{p}}\times\mathbf{a})]^{2}}{b^{6}}+\mathcal{O}(\mathbf{a}^{3}).$
(3.7)

We are now ready to compute the infrared divergent part of
$\mathrm{Im}\,2\delta_{2}$ using the three-particle unitarity cut [99, 76, 83]
depicted in Fig.2 as

$\displaystyle\mathrm{Im}\,2\delta(b,\sigma)=\frac{1}{2}\int\frac{d^{3-2\epsilon}\mathbf{k}}{2\left|\mathbf{k}\right|(2\pi)^{3-2\epsilon}}\tilde{\mathcal{A}}_{5}^{\mu\nu}(b,k)P_{\mu\nu;\rho\sigma}\tilde{\mathcal{A}}_{5}^{*\rho\sigma}(b,k),$
(3.8)

where $\omega=\left|\mathbf{k}\right|$ and we use dimensional regularisation
to capture the infrared divergence in $\mathrm{Im}\,2\delta_{2}$.

Figure 2: Three-particle cut yielding the infrared divergent part of
$\mathrm{Im}\,2\delta_{2}$ in (3.8)

The de Donder projector
$P_{\mu\nu;\rho\sigma}=\frac{1}{2}\eta_{\mu\rho}\eta_{\nu\sigma}+\frac{1}{2}\eta_{\mu\sigma}\eta_{\nu\rho}-\frac{1}{2}\eta_{\mu\nu}\eta_{\rho\sigma}$
has been inserted in (3.8) in order to single out only the physical,
transverse-traceless projection of $\tilde{\mathcal{A}}_{5}^{\mu\nu}$.

We parametrize the momentum of the emitted graviton222In our conventions $k$
is ingoing. as $k=-\omega(1,\hat{\mathbf{x}})$ where
$\hat{\mathbf{x}}=(\sin\theta,\sin\theta\sin\varphi,\cos\theta)$ and hence
$\left|\mathbf{k}\right|=\omega$. The various scalar product are

$\displaystyle\mathbf{k}\cdot\mathbf{b}_{\pm}=-\omega\sin\theta[b\cos\varphi\pm(a^{1}\sin\varphi-a^{2}\cos\varphi)],$
(3.9) $\displaystyle\bar{p}_{1}\cdot
k=\omega(E_{1}-p\cos\theta),\qquad\bar{p}_{2}\cdot
k=\omega(E_{2}+p\cos\theta),$ (3.10)
$\displaystyle\mathbf{b}_{\pm}^{2}=b^{2}+(a^{1})^{2}+(a^{2})^{2}\mp 2ba^{2}.$
(3.11)

where $\mathbf{a}=a^{i}\mathbf{e}_{i}$. Note that the result will not depend
on $a^{3}$, component of $\mathbf{a}$ along $\mathbf{p}$. Integrating over
$\omega$ and over $\varphi$ and keeping only the term proportional to
$\epsilon^{-1}$ we get for the infrared divergent part of the imaginary part
of the two-loop eikonal333In equation (3.12) we restored $\hbar$, that was set
equal to one so far.

$\displaystyle\mathrm{Im}\,2\delta_{2}(\sigma,b)$
$\displaystyle\simeq-\frac{1}{2\epsilon}\frac{\pi}{2\hbar(2\pi)^{3}}\bigg{(}\frac{\kappa^{3}m_{1}m_{2}\sigma^{2}}{8\pi\sqrt{\sigma^{2}-1}}\bigg{)}^{2}\bigg{(}\sum_{\pm}\frac{(1\pm
v)^{2}\mathbf{b}_{\pm}}{\mathbf{b}^{2}_{\pm}}\bigg{)}^{2}\mathcal{I}(\sigma),$
(3.12)

where [81]444Our definition of $\mathcal{I}(\sigma)$ has an additional factor
$2$ with respect to the one in [55].

$\displaystyle\mathcal{I}(\sigma)=$
$\displaystyle\int_{-1}^{1}\text{d}x\bigg{\\{}\frac{1-x^{2}}{4}\bigg{[}\frac{m_{1}^{4}}{(E_{1}-\bar{p}x)^{4}}+\frac{m_{2}^{4}}{(E_{2}+\bar{p}x)^{4}}-\frac{2m_{1}^{2}m_{2}^{2}(2\sigma^{2}-1)}{(E_{1}-\bar{p}x)^{2}(E_{2}+\bar{p}x)^{2}}\bigg{]}$
$\displaystyle-2\bigg{[}\frac{m_{1}^{2}}{(E_{1}-\bar{p}x)^{2}}+\frac{m_{2}^{2}}{(E_{2}+\bar{p}x)^{2}}-\frac{2m_{1}m_{2}\sigma}{(E_{1}-\bar{p}x)(E_{2}+\bar{p}x)}\bigg{]}\bigg{\\}}$
$\displaystyle=\frac{2}{\sigma^{2}-1}\bigg{[}\frac{8-5\sigma^{2}}{3}-\frac{\sigma(3-2\sigma^{2})}{\sqrt{\sigma^{2}-1}}\cosh^{-1}(\sigma)\bigg{]}.$
(3.13)

Introducing the notation $\beta(\sigma)=2m_{1}m_{2}G(2\sigma^{2}-1)$ as in
[83, 30] and the spatial vector $\mathbf{f}$ as

$\displaystyle\sum_{\pm}\frac{(1\pm
v)^{2}\mathbf{b}_{\pm}}{\mathbf{b}_{\pm}^{2}}\equiv\frac{2(2\sigma^{2}-1)}{\sigma^{2}b}\mathbf{f}(a,b,\sigma),$
(3.14)

the final result for the infrared divergent part of $\mathrm{Im}\,2\delta_{2}$
can be simply written as

$\displaystyle\mathrm{Im}\,2\delta_{2}(\sigma,b)\simeq-\frac{1}{2\epsilon}\frac{G\beta^{2}(\sigma)}{2\pi\hbar(\sigma^{2}-1)b^{2}}\mathcal{I}(\sigma)\mathbf{f}^{2}(a,b,\sigma).$
(3.15)

When the spins are parallel to the orbital angular momentum the vector
$\mathbf{f}$ is

$\displaystyle\mathbf{f}(a,b,\sigma)\equiv
f(a,b,\sigma)\frac{\mathbf{b}}{b},\qquad
f(a,b,\sigma)=\frac{1+\frac{2\sigma\sqrt{\sigma^{2}-1}}{2\sigma^{2}-1}\frac{a}{b}}{1-\left(\frac{a}{b}\right)^{2}}.$
(3.16)

For arbitrary spin orientation, $\mathbf{f}$ has non-vanishing components also
along $\hat{\mathbf{p}}\times\mathbf{a}$ and explicit expressions for
$\mathbf{f}$ and $\mathbf{f}^{2}$ can be found in Appendix A.

Using Eq. (3.1) we can extract the radiative contribution to the real part of
the two-loop eikonal

$\displaystyle\mathrm{Re}\,2\delta_{2}^{\mathrm{rr}}(\sigma,b)=\frac{G\beta^{2}(\sigma)}{4\hbar(\sigma^{2}-1)b^{2}}\mathcal{I}(\sigma)\mathbf{f}^{2}(a,b,\sigma)=\mathrm{Re}\,2\delta_{2}^{\mathrm{rr}}(\sigma,b)\big{|}_{\mathbf{a}=0}\mathbf{f}^{2}(a,b,\sigma),$
(3.17)

This equation shows that the radiation reaction part of the eikonal of two
Kerr black-holes scattering is simply obtained from that of two Schwarzschild
black holes by means of a Newman-Janis shift, as stressed after (3.5). In
particular, we find a simple rule that allows to get the result for spinning
black holes from that of spinless ones. We start to notice that, for the
latter, the combination $[2(2\sigma^{2}+1)\mathbf{b}/\sigma^{2}b^{2}]^{2}$
that appears in $\mathrm{Re}\,2\delta_{2}$ can be rewritten as

$\displaystyle\left(\frac{2(2\sigma^{2}-1)}{\sigma^{2}b^{2}}\mathbf{b}\right)^{2}=\bigg{(}\frac{(1+v)^{2}\mathbf{b}}{\mathbf{b}^{2}}+\frac{(1-v)^{2}\bar{\mathbf{b}}}{\bar{\mathbf{b}}^{2}}\bigg{)}^{2},$
(3.18)

where we introduced an auxiliary vector $\bar{\mathbf{b}}$ such that
$\bar{\mathbf{b}}\stackrel{{\scriptstyle\mathbf{a}=0}}{{=}}\mathbf{b}$. Then
to get the result for $\mathbf{a}\neq 0$ we just need to do the replacements

$\displaystyle\mathbf{b}\stackrel{{\scriptstyle\mathbf{a}\neq
0}}{{\longrightarrow}}\mathbf{b}_{+},\qquad\bar{\mathbf{b}}\stackrel{{\scriptstyle\mathbf{a}\neq
0}}{{\longrightarrow}}\mathbf{b}_{-}.$ (3.19)

We complete this section by using the first equation of (3.2) for computing
the zero frequency limit of the energy spectrum

$\displaystyle\frac{dE^{\mathrm{rad}}}{d\omega}\bigg{|}_{\omega\to
0}=\frac{4G^{3}m_{1}^{2}m_{2}^{2}(2\sigma^{2}-1)^{2}}{\pi
b^{2}(\sigma^{2}-1)}\mathcal{I}(\sigma)\mathbf{f}^{2}(a,b,\sigma),$ (3.20)

that can be expanded in the static (PN) limit $\sigma\rightarrow 1$ yielding,

$\displaystyle\frac{dE^{\mathrm{rad}}}{d\omega}\bigg{|}_{\omega\to
0}\simeq\frac{32G^{3}m_{1}^{2}m_{2}^{2}}{5\pi}\frac{b^{2}}{(\mathbf{b}+\hat{\mathbf{p}}\times\mathbf{a})^{2}(\mathbf{b}-\hat{\mathbf{p}}\times\mathbf{a})^{2}},$
(3.21)

that gives the all orders in spin correction to the original Smarr formula
[100, 94]. Further expanding up to quadratic order in spin we find

$\displaystyle\frac{dE^{\mathrm{rad}}}{d\omega}\bigg{|}_{\omega\to
0}\simeq\frac{32G^{3}m_{1}^{2}m_{2}^{2}}{5\pi
b^{2}}\left(1+2\frac{(a^{2})^{2}-(a^{1})^{2}}{b^{2}}\right).$ (3.22)

When the spins are orthogonal to the scattering plane, Eq. (3.21) reduces to

$\displaystyle\frac{dE^{\mathrm{rad}}}{d\omega}\bigg{|}_{\omega\to
0}\simeq\frac{32G^{3}m_{1}^{2}m_{2}^{2}}{5\pi
b^{2}}\frac{1}{\left[1-\left(\frac{a}{b}\right)^{2}\right]^{2}}.$ (3.23)

##  4 Radiative scattering angle and Radiated angular momentum to all orders
in spin

In this section we start studying the case in which the two spins are aligned
i.e. we set $a^{1}=0$ and we define $a\equiv a^{2}$ to simplify the notation
and from the real part of the eikonal in (3.17) we extract the radiative part
of the scattering angle to all orders in spin. In this case the vector
$\mathbf{f}$ is given in (3.16) so that the real part of the radiation
reaction eikonal becomes

$\displaystyle\mathrm{Re}\,2\delta_{2}^{\mathrm{rr}}(\sigma,b)=\frac{G\beta^{2}(\sigma)}{4\hbar(\sigma^{2}-1)b^{2}}\mathcal{I}(\sigma)f^{2}(a,b,\sigma),$
(4.1)

and $\theta^{\mathrm{rad}}_{3}$, using (3.2) is

$\displaystyle\theta^{\mathrm{rad}}_{3}(\sigma,b)=\frac{G\beta^{2}(\sigma)}{2(\sigma^{2}-1)pb^{3}}\mathcal{I}(\sigma)\frac{(1+\frac{2\sigma\sqrt{\sigma^{2}-1}}{2\sigma^{2}-1}\frac{a}{b})\left[1+\frac{4\sigma\sqrt{\sigma^{2}-1}}{2\sigma^{2}-1}\frac{a}{b}+\left(\frac{a}{b}\right)^{2}\right]}{\left[1-\left(\frac{a}{b}\right)^{2}\right]^{3}}.$
(4.2)

In the case of spin $1$ that contains only terms up to the order $a^{2}$ the
previous expression is equal to

$\theta^{\mathrm{rad}}_{3}(\sigma,b)=\frac{G\beta^{2}(\sigma)}{2(\sigma^{2}-1)pb^{3}}\mathcal{I}(\sigma)\Bigg{(}1+\frac{6\sigma\sqrt{\sigma^{2}-1}}{2\sigma^{2}-1}\frac{a}{b}+4\frac{6\sigma^{4}-6\sigma^{2}+1}{(2\sigma^{2}-1)^{2}}\frac{a^{2}}{b^{2}}\Bigg{)},$
(4.3)

that agrees with Eq. (19) of [62] for $s_{E,+}=0$. For spin $1$ the authors of
Ref. [62] checked that the deflection angle is not divergent at high energy
because the logarithmic divergence that one finds in the conservative part is
cancelled by the logarithmic divergence appearing in the radiative part in
analogy with what happens for spin zero. Assuming that the same cancellation
also happens for arbitrary spin, from the high energy behaviour of the real
part of the two-loop eikonal given by

$\displaystyle\mathrm{Re}\,2\delta_{2}^{\mathrm{rr}}(\sigma,b)\sim\frac{16G^{3}m_{1}^{2}m_{2}^{2}}{(1-\frac{a}{b})^{2}}\sigma^{2}\log\sigma,\qquad\sigma\to\infty,$
(4.4)

we can deduce the high energy limit of the conservative part of the real part
of the two-loop eikonal

$\displaystyle\mathrm{Re}\,2\delta_{2}^{\mathrm{cons}}(\sigma,b)\sim-\frac{16G^{3}m_{1}^{2}m_{2}^{2}}{(1-\frac{a}{b})^{2}}\sigma^{2}\log\sigma,\qquad\sigma\to\infty.$
(4.5)

This implies that the high energy limit of the conservative scattering angle
must be equal to

$\displaystyle\theta^{\mathrm{cons}}_{3}(\sigma,b)\simeq-\frac{32G^{3}Em_{1}m_{2}}{(1-\frac{a}{b})^{3}}\sigma\log\sigma,\qquad\sigma\to\infty,$
(4.6)

The previous equations are valid for any spin and in particular Eq. (4.6)
agrees with the high energy limit of Eqs. (16a), (16b) and (16c) of Ref. [62].

Then we want to use Eq. (3.3) to determine the radiated angular momentum from
the radiative scattering angle. To do so we need the 1PM conservative
scattering angle $\theta_{1}^{\mathrm{cons}}$ [37, 41]

$\displaystyle\theta_{1}^{\mathrm{cons}}(\sigma,b)=\frac{\beta(\sigma)}{pb\sqrt{\sigma^{2}-1}}f(a,b,\sigma)=\frac{\beta(\sigma)}{J\sqrt{\sigma^{2}-1}}\left(\frac{1+\frac{2\sqrt{\sigma^{2}-1}}{2\sigma^{2}-1}\frac{pa}{J}}{1-\left(\frac{pa}{J}\right)^{2}}\right).$
(4.7)

that is given by $-\frac{1}{p}\frac{\partial\chi_{1}}{\partial b}$, where
$\chi_{1}$ is the 1PM eikonal, obtained by inserting the tree-level $4$-point
amplitude of (2.1) in Eq. (3.5).

We find that the radiated angular momentum in the center-of-mass frame is
equal to

$\displaystyle\mathbf{J}^{\mathrm{rad}}_{2}(\sigma,b)=J\frac{G\beta(\sigma)}{b^{2}\sqrt{\sigma^{2}-1}}\mathcal{I}(\sigma)\hat{\mathbf{p}}\times\left(f(a,b,\sigma)\frac{\mathbf{b}}{b}\right)=J\frac{G\beta(\sigma)}{b^{2}\sqrt{\sigma^{2}-1}}\mathcal{I}(\sigma)f(a,b,\sigma)\mathbf{e}_{2}.$
(4.8)

Note that the angular momentum is radiated only along the $\mathbf{e}_{2}$
direction, perpendicular to the scattering plane. This is due to the fact that
in the aligned-spin case the scattering dynamics is planar, just as in the
spinless scenario.

We note that (4.8) admits a natural generalisation in the case of non-aligned
spin as555We have also confirmed equation (4.9) by an explicit computation for
the radiated angular momentum using the results in [101].

$\displaystyle\mathbf{J}^{\mathrm{rad}}_{2}(\sigma,b)=J\frac{G\beta(\sigma)}{b^{2}\sqrt{\sigma^{2}-1}}\mathcal{I}(\sigma)\hat{\mathbf{p}}\times\mathbf{f}(a,b,\sigma).$
(4.9)

Contrarily to (3.16), in this case $\mathbf{f}$ does not lie entirely along
$\mathbf{b}$, but has also one non-vanishing component along
$\hat{\mathbf{p}}\times\mathbf{a}$, see Appendix A. Therefore, as already
mentioned, the angular momentum in the center-of-mass frame is radiated in
both directions orthogonal to $\hat{\mathbf{p}}$. Even if (4.9) has been
obtained by generalising (4.8) straightforwardly, it admits an interpretation
as follows. Introducing the 1PM classical momentum transfer as
$Q^{\mu}_{1}=-\frac{\partial\chi_{1}}{\partial b_{\mu}}$, we find
$Q^{0}_{1}=0$ and666We thank Carlo Heissenberg and Justin Vines for
discussions about this part of the paper.

$\displaystyle\mathbf{Q}_{1}(\sigma,b)=\frac{\beta(\sigma)}{b\sqrt{\sigma^{2}-1}}\mathbf{f}(a,b,\sigma),\qquad\hat{\mathbf{p}}\cdot\mathbf{Q}_{1}=0,$
(4.10)

so that the radiated 2PM angular momentum can be more conveniently rewritten
as

$\displaystyle\mathbf{J}_{2}^{\mathrm{rad}}(\sigma,b)=G\,\mathcal{I}(\sigma)\,\mathbf{p}\times\mathbf{Q}_{1}(\sigma,b)=\mathbf{L}_{2}^{\mathrm{rad}}(\sigma,b)+\mathbf{S}_{2}^{\mathrm{rad}}(\sigma,b),$
(4.11)

where

$\displaystyle\mathbf{L}^{\mathrm{rad}}_{2}=G\,\mathcal{I}(\sigma)c_{o}(a,b,\sigma)\mathbf{p}\times\mathbf{b}\qquad\mathbf{S}^{\mathrm{rad}}_{2}(a,b,\sigma)=G\,\mathcal{I}(\sigma)c_{s}(a,b,\sigma)\mathbf{a},$
(4.12)

where $c_{o}$ and $c_{s}$ have been defined in Appendix A. Equation (4.11)
admits an interpretation. The 2PM radiated angular momentum receives
contributions from the variation of directions of the momenta and of the spin
vectors of the particles involved in the scattering process. Note that, as
stressed in the beginning of this section, at $\mathcal{O}(G^{2})$ the
dynamics of the system is conservative, and therefore the 2PM angular momentum
can only carry information about directions of the spatial vectors.
Expanding equations (4.8) and (4.9) up to quadratic order in the spin we find
respectively

$\displaystyle\mathbf{J}^{\mathrm{rad}}_{2}(\sigma,b)=J\frac{G\beta(\sigma)}{b^{2}\sqrt{\sigma^{2}-1}}\mathcal{I}(\sigma)\left[1+\frac{2\sigma\sqrt{\sigma^{2}-1}}{2\sigma^{2}-1}\frac{a}{b}+\left(\frac{a}{b}\right)^{2}\right]\mathbf{e}_{2},$
(4.13)

for aligned spins and

$\displaystyle\mathbf{J}^{\mathrm{rad}}_{2}(\sigma,b)$
$\displaystyle=J\frac{G\beta(\sigma)}{b^{2}\sqrt{\sigma^{2}-1}}\mathcal{I}(\sigma)\left[\bigg{(}-\frac{2\sigma\sqrt{\sigma^{2}-1}}{2\sigma^{2}-1}\frac{a^{1}}{b}-2\frac{a^{1}a^{2}}{b^{2}}\bigg{)}\mathbf{e}_{1}\right.$
$\displaystyle+\left.\bigg{(}1+\frac{2\sigma\sqrt{\sigma^{2}-1}}{2\sigma^{2}-1}\frac{a^{2}}{b}+\frac{(a^{2})^{2}-(a^{1})^{2}}{b^{2}}\bigg{)}\mathbf{e}_{2},\right],$
(4.14)

for non-aligned spins. These equations exactly match equation (18) of [62] and
equation (29) of [55] obtained with supersymmetric worldline formalism. Note
that equation (4.2), (4.8) and (4.9) extend these results to all orders in
spin.

##  5 Conclusions and outlook

From the leading soft term of a five-point amplitude involving two particles
with spin minimally coupled to gravity that scatter producing also a low-
energy graviton, we have computed the infrared divergent piece of the
imaginary part of the two-loop eikonal, $\delta_{2}$, generalising the
procedure followed for spin zero in Ref. [83]. Using then analyticity and
crossing symmetry [83] we have derived the radiation reaction contribution to
the real part of the $\delta_{2}$ and from it the radiation reaction
contribution to the deflection angle at 3PM ($\mathcal{O}(G^{3})$).

Using the Bini-Damour formula in Eq. (3.3), we have determined the radiated
angular momentum at $\mathcal{O}(G^{2})$ for arbitrary spin configurations and
to all orders in spin.

Our results agree for spin zero with those in Refs. [81, 83] and for spin one
with the complete calculation performed in Ref. [62].

An interesting aspect of our results is that spin effects for soft
bremsstrahlung are entirely encoded in the vector $\mathbf{f}$ in (3.14), that
comes from a Newman-Janis shift in impact parameter space, as explained in
(3.19). The origin of this unexpected feature is that the entire calculation
we have performed has the tree-level four-point amplitude in (2.1) as starting
point. In the latter, the spin structure appears as compactly organised in a
single exponential factor, because of the nature of minimally coupled
amplitudes. It would be interesting to understand to what extent such a simple
criterion could be used to gather additional information about the
conservative part of $\delta_{2}$, whose ultra-relativistic behaviour appears
to be already fixed by the present analysis.

In equation (3.20) we have shown the zero frequency limit of the emitted
energy spectrum at $\mathcal{O}(G^{3})$. The reason why we could do this is
that we have only used the Weinberg’s soft graviton theorem to fix the leading
term of the inelastic five-point amplitude, because it was the necessary
ingredient to fix the infrared divergence in $\mathrm{Im}\,2\delta_{2}$. In
principle, one could go further in the soft expansion [91, 92], therefore
considering also terms containing the total angular momentum operator, and
understand how they contribute to the emitted energy and angular momentum
spectrum beyond the leading soft limit.

### Acknowledgements

We thank Luca Buoninfante, Alessandro Georgudis, Kays Haddad, Carlo
Heissenberg, Henrik Johansson, Alexander Ochirov, Paolo Pichini, Rodolfo
Russo, Ali Seraj and Justin Vines for many very useful discussions. The
research of FA (PDV) is fully (partially) supported by the Knut and Alice
Wallenberg Foundation under grant KAW 2018.0116. Nordita is partially
supported by Nordforsk.

##  Appendix A Expressions for $\mathbf{f}(a,b,\sigma)$

The vector $\mathbf{f}$ introduced in (3.14) can be written as

$\displaystyle\mathbf{f}(a,b,\sigma)=c_{o}(a,b,\sigma)\mathbf{b}-c_{s}(a,b,\sigma)\hat{\mathbf{p}}\times\mathbf{a},$
(A.1)

where

$\displaystyle
c_{o}(a,b,\sigma)=\frac{b[(\sigma+\sqrt{\sigma^{2}-1})^{2}\left|\mathbf{b}-\hat{\mathbf{p}}\times\mathbf{a}\right|^{2}+(\sigma-\sqrt{\sigma^{2}-1})^{2}\left|\mathbf{b}+\hat{\mathbf{p}}\times\mathbf{a}\right|^{2}]}{2(2\sigma^{2}-1)\left|\mathbf{b}-\hat{\mathbf{p}}\times\mathbf{a}\right|^{2}\left|\mathbf{b}+\hat{\mathbf{p}}\times\mathbf{a}\right|^{2}},$
(A.2) $\displaystyle
c_{s}(a,b,\sigma)=\frac{(\sigma-\sqrt{\sigma^{2}-1})^{2}\left|\mathbf{b}+\hat{\mathbf{p}}\times\mathbf{a}\right|^{2}]-b[(\sigma+\sqrt{\sigma^{2}-1})^{2}\left|\mathbf{b}-\hat{\mathbf{p}}\times\mathbf{a}\right|^{2}}{2(2\sigma^{2}-1)\left|\mathbf{b}-\hat{\mathbf{p}}\times\mathbf{a}\right|^{2}\left|\mathbf{b}+\hat{\mathbf{p}}\times\mathbf{a}\right|^{2}}.$
(A.3)

The components of $\mathbf{f}$ are explicitly given by

$\displaystyle
f^{1}(a,b,\sigma)=\frac{\left(1+\frac{(a^{1})^{2}+(a^{2})^{2}}{b^{2}}\right)\left(1-\frac{2\sigma\sqrt{\sigma^{2}-1}}{2\sigma^{2}-1}\frac{a^{2}}{b}\right)+\frac{2a^{2}}{b}\left(\frac{2\sigma\sqrt{\sigma^{2}-1}}{2\sigma^{2}-1}-\frac{a^{2}}{b}\right)}{(1-\frac{2a^{2}}{b}+\frac{(a^{1})^{2}+(a^{2})^{2}}{b^{2}})(1+\frac{2a^{2}}{b}+\frac{(a^{1})^{2}+(a^{2})^{2}}{b^{2}})},$
(A.4) $\displaystyle
f^{2}(a,b,\sigma)=\frac{\frac{2\sigma\sqrt{\sigma^{2}-1}}{2\sigma^{2}-1}\frac{a^{1}}{b}\left(1+\frac{2\sigma^{2}-1}{\sigma\sqrt{\sigma^{2}-1}}\frac{a^{2}}{b}+\frac{(a^{1})^{2}+(a^{2})^{2}}{b^{2}}\right)}{(1-\frac{2a^{2}}{b}+\frac{(a^{1})^{2}+(a^{2})^{2}}{b^{2}})(1+\frac{2a^{2}}{b}+\frac{(a^{1})^{2}+(a^{2})^{2}}{b^{2}})},$
(A.5) $\displaystyle f^{3}(a,b,\sigma)=0.$ (A.6)

Therefore we have

$\displaystyle\mathbf{f}^{2}(a,b,\sigma)=\frac{1+\frac{4\sigma\sqrt{\sigma^{2}-1}}{2\sigma^{2}-1}\frac{a^{2}}{b}+\frac{4\sigma^{2}(\sigma^{2}-1)}{(2\sigma^{2}-1)^{2}}\frac{(a^{1})^{2}+(a^{2})^{2}}{b^{2}}}{(1-\frac{2a^{2}}{b}+\frac{(a^{1})^{2}+(a^{2})^{2}}{b^{2}})(1+\frac{2a^{2}}{b}+\frac{(a^{1})^{2}+(a^{2})^{2}}{b^{2}})}.$
(A.7)

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Dark Matter

Marco Cirelli

Laboratoire de Physique Théorique et Hautes Énergies (LPTHE),

CNRS & Sorbonne Université, 4 Place Jussieu, Paris, France

Alessandro Strumia

Dipartimento di Fisica dell'Università di Pisa, Italia

Jure Zupan

Department of Physics, University of Cincinnati, Cincinnati, Ohio 45221,USA

We review observational, experimental and theoretical results related to Dark Matter.

Version 0, November 11, 2011 – Version 1, August 27, 2024

Dark Matter basics

Dark Matter detection

Currently, the existence of Dark Matter is based on observations that only probe its gravitational coupling, see section <ref>. This gives us information about the total mass of DM in the Universe, as well as the amount of DM in a particular astrophysical region, including, to some extent, its spatial  distribution. However, in order to really understand what Dark Matter is, we also need to observe its other interactions with ordinary matter.

The next question is: how?

There are three main avenues of investigation
(traditionally schematized as in figure <ref>):

$\circ$ Direct Detection (Chapter <ref>) aims at detecting the recoil event produced by a passing DM particle hitting one of the nuclei or electrons in a highly shielded and closely monitored underground detector, made of ultrapure semiconductors, noble gasses, pristine crystals, etc.

$\circ$ Indirect Detection (Chapter <ref>) aims at detecting DM annihilations or DM decays, in our Galaxy or in other astrophysical environments, by searching for signatures in cosmic rays (in a broad sense: charged particles, photons or neutrinos arriving at Earth).

$\circ$ Accelerator-based Detection (Chapter <ref>) aim at producing DM particles in a controlled environment: e.g., in $pp$ collisions at the Large Hadron Collider at CERN, in $e^+e^-$ collisions at lower energy machines or in beam dump experiments[The same SM SM $\to$ DM DM processes also occur in astrophysical systems featuring energetic SM particles. However, the rates are expected to be too low to be phenomenologically relevant.], and then detecting the presence of DM via missing energy or other signatures.

In each of the above approaches there are many ongoing experimental efforts, with a good fraction of them represented in figure <ref>.  They are distributed over many continents and are performed in very different environments. Most, if not all, of the experiments are run by international collaborations of varying sizes, from merely tens up to several thousands of scientists, from many institutions. The search for DM is a truly global effort of the international scientific community.

Chapters <ref>, <ref> and <ref> introduce the phenomenology of different search strategies as well as the computations relevant for predicting the expected signals,
and discuss the current status.
Chapter <ref> discusses some of the recent observational anomalies.

[The three main DM searches]The three main DM search strategies, from left to right:
direct detection (the plot specifies the DM/nucleus kinematics used in chapter <ref>);
indirect detection (chapter <ref>; one of the DM lines is dashed, indicating that both DM decays and DM annihilations are relevant);
production at particle accelerators (chapter <ref>).
The “SM” label stands for the Standard Model particle, or it can be viewed to stand for the “Standard Matter”.

[Experimental efforts]
The experiments trying to detect particle DM are performed all over the Earth (on surface as well as below ground and in orbit).

Dark Matter theories