capbtabboxtable[][]

# The Semi-Classical Regime for Dark Matter Self-Interactions

Brian Colquhoun<EMAIL_ADDRESS>Department of Physics and Astronomy, York
University,
Toronto, Ontario M3J 1P3, Canada  Saniya Heeba<EMAIL_ADDRESS>Institute for Theoretical Particle Physics and Cosmology (TTK), RWTH Aachen
University, 52056 Aachen, Germany  Felix Kahlhoefer<EMAIL_ADDRESS>aachen.de Institute for Theoretical Particle Physics and Cosmology (TTK), RWTH
Aachen University, 52056 Aachen, Germany  Laura Sagunski<EMAIL_ADDRESS>frankfurt.de Institute for Theoretical Physics, Goethe University, 60438
Frankfurt am Main, Germany  Institute for Theoretical Particle Physics and
Cosmology (TTK), RWTH Aachen University, 52056 Aachen, Germany  Sean Tulin
<EMAIL_ADDRESS>Department of Physics and Astronomy, York University,
Toronto, Ontario M3J 1P3, Canada

###### Abstract

Many particle physics models for dark matter self-interactions—motivated to
address long-standing challenges to the collisionless cold dark matter
paradigm—fall within the semi-classical regime, with interaction potentials
that are long-range compared to the de Broglie wavelength for dark matter
particles. In this work, we present a quantum mechanical derivation and new
analytic formulas for the semi-classical momentum transfer and viscosity cross
sections for self-interactions mediated by a Yukawa potential. Our results
include the leading quantum corrections beyond the classical limit and allow
for both distinguishable and identical dark matter particles. Our formulas
supersede the well-known formulas for the momentum transfer cross section
obtained from the classical scattering problem, which are often used in
phenomenological studies of self-interacting dark matter. Together with
previous approximation formulas for the cross section in the quantum regime,
our new results allow for nearly complete analytic coverage of the parameter
space for self-interactions with a Yukawa potential. We also discuss the
phenomenological implications of our results and provide a new velocity-
averaging procedure for constraining velocity-dependent self-interactions. Our
results have been implemented in the newly released code CLASSICS.

††preprint: TTK-20-39

##  I Introduction

Collisional interactions among dark matter (DM) particles arise from the
underlying microphysics of DM. If governed solely by weak-scale physics, the
scattering rate is sufficiently small that DM behaves as a collisionless fluid
during cosmic structure formation. On the other hand, if the hidden forces
between DM particles are comparable to the nuclear forces between protons and
neutrons—with self-scattering cross section per unit mass of $\sigma/m\sim{\rm
barn\,GeV^{-1}}$—then DM self-interactions affect the inner structure of
galactic halos on kiloparsec scales. In this case, longstanding tensions
between observations and $N$-body simulations for cold collisionless DM can be
brought into accord through the effect of self-interactions Spergel and
Steinhardt (2000) (see Ref. Tulin and Yu (2018) for a review).111We also refer
to Refs. Bullock and Boylan-Kolchin (2017); Salucci (2019) for further recent
reviews of small scale structure issues for dark matter. DM self-interactions
may be mediated by new dark sector particles at or below the GeV scale, which
are actively being searched for in particle physics experiments worldwide
Essig  _et al._ (2013); Alexander  _et al._ (2016); Beacham  _et al._ (2020).

Self-scattering in DM halos is non-relativistic and the cross section is
calculated by solving the Schrödinger equation, which provides a non-
perturbative resummation of multiple mediator exchanges during a single
scattering Buckley and Fox (2010); Tulin  _et al._ (2013a, b). By analogy with
nuclear forces, the most widely considered interaction is the Yukawa potential
Feng  _et al._ (2010a, b); Loeb and Weiner (2011), expressed in natural units
($\hbar=c=1$) as

$U(r)=\pm\frac{\alpha_{\chi}}{r}e^{-m_{\phi}r}\,,$ (1)

where $\alpha_{\chi}$ is the dark sector analog of the fine structure
constant. The potential $U(r)$ can be attractive ($-$) or repulsive ($+$)
depending on the nature of the DM particle(s) $\chi$ and mediator particle
$\phi$ (their masses are denoted $m_{\chi}$ and $m_{\phi}$, respectively). The
latter can be a fundamental scalar or vector boson in a weakly-interacting
theory Buckley and Fox (2010); Feng  _et al._ (2010a, b); Loeb and Weiner
(2011) or a composite state exchanged as a residual force in a strongly-
interacting theory Boddy  _et al._ (2014).

In general, analytic solutions to the Schrödinger equation are not possible
for the Yukawa potential outside the Born (perturbative) regime where
$\alpha_{\chi}m_{\chi}/m_{\phi}\ll 1$. However, for momentum $k\ll m_{\phi}$,
$S$-wave scattering dominates and accurate approximations have been obtained
for the cross section, both for the Yukawa potential (using the Hulthén
approximation) Tulin  _et al._ (2013b) and in a general framework Chu  _et
al._ (2020). On the other hand, for $k\gtrsim m_{\phi}$, the partial wave
analysis requires a sum over higher modes $\ell>0$, so the calculation of the
cross section becomes numerically intensive for $k\gg m_{\phi}$, as one
approaches the classical limit $k/m_{\phi}\to\infty$ Tulin  _et al._ (2013a,
b).

The different regimes in this problem are delineated by two dimensionless
parameters

$\kappa=k/m_{\phi}\,,\quad\beta=\frac{2\alpha_{\chi}m_{\phi}}{m_{\chi}v^{2}}\,,$
(2)

which correspond to the dimensionless momentum and the strength of the
potential relative to the kinetic energy, respectively. In the classical
scattering problem, the cross section is parametrized in terms of $\beta$
alone, yielding Coulomb-like scattering for $\beta\ll 1$, while $\beta\gg 1$
corresponds to the case of a strong potential Khrapak  _et al._ (2003, 2004).
We can summarize the different regimes as follows:

weakly-coupled ($\beta\ll 1$)
$\displaystyle\textnormal{vs.}\quad\textnormal{strongly-coupled ($\beta\gg
1$)}\,,$ (3) Born ($2\beta\kappa^{2}\ll 1$)
$\displaystyle\textnormal{vs.}\quad\textnormal{non-perturbative regime
($2\beta\kappa^{2}\gg 1$)}\,,$ quantum ($\kappa\ll 1$)
$\displaystyle\textnormal{vs.}\quad\textnormal{semi-classical ($\kappa\gtrsim
1$)}\;.$

Fig. 1 illustrates the landscape of these various cases, each of which
involves different approximations for computing the scattering cross section
analytically.

Figure 1: Sketch of the different regimes relevant for calculating self-
interaction cross sections for a Yukawa potential in terms of the
dimensionless parameters $\beta$ and $\kappa$. The aim of the present work is
to obtain analytical expressions for the scattering cross section in the semi-
classical and non-perturbative regime.

Generally, quoting a single figure-of-merit $\sigma/m$ is an over-
simplification since the differential cross section
$\mathrm{d}\sigma/\mathrm{d}\Omega$ depends nontrivially on both the relative
scattering velocity $v$ and scattering angle $\theta$. The former is treated
by phase-space averaging, while for the latter, one uses a suitable proxy for
the cross section, as opposed to the total cross section
$\sigma=\int\mathrm{d}\Omega(\mathrm{d}\sigma/\mathrm{d}\Omega)$. In the
literature, the most widely-used proxy for $\sigma$ is the momentum transfer
cross section Mohapatra  _et al._ (2002); Buckley and Fox (2010); Feng  _et
al._ (2010a)

$\sigma_{\mathrm{T}}\equiv\int\mathrm{d}\Omega\,(1-\cos\theta)\,\frac{\mathrm{d}\sigma}{\mathrm{d}\Omega}\,,$
(4)

where $(1-\cos\theta)$ is the fractional change in longitudinal momentum.
Analytic formulas for $\sigma_{\mathrm{T}}$ for the Yukawa potential in the
classical limit have been obtained by solving the classical scattering problem
for distinguishable particles, originally motivated in the context of
transport theory for plasmas Khrapak  _et al._ (2003, 2004) and subsequently
imported into a DM context Feng  _et al._ (2010a, b). In the kinetic theory of
gases, $\sigma_{\mathrm{T}}$ arises as the relevant quantity for diffusion
rates Vincenti and Kruger (1965). For DM, this quantity governs the effective
drag force and evaporation rate felt by a subhalo merging into a static
background of DM particles approximating a larger halo Kahlhoefer  _et al._
(2014). However, $\sigma_{\mathrm{T}}$ as defined in Eq. (4) is problematic
since it preferentially weights backward scattering, which leaves DM
distributions unchanged, and moreover is ill-defined for identical
particles.222Ref. Kahlhoefer  _et al._ (2014) proposed modifying
$\sigma_{\mathrm{T}}$ to $\tilde{\sigma}_{\mathrm{T}}=\int
d\Omega\,(1-|\cos\theta|)\frac{d\sigma}{d\Omega}$ to ameliorate these issues.
Since $|\cos\theta|$ has no simple decomposition in terms of Legendre
polynomials, this approach is difficult to incorporate into our quantum
treatment to follow and does not lead to simple expressions in terms of phase
shifts.

As an alternative, there is the viscosity cross section Tulin  _et al._
(2013b)

$\sigma_{\mathrm{V}}\equiv\int\mathrm{d}\Omega\,\sin^{2}\theta\,\frac{\mathrm{d}\sigma}{\mathrm{d}\Omega}\,,$
(5)

which arises from kinetic theory as the relevant quantity for viscosity and
heat conductivity Vincenti and Kruger (1965). This quantity may be a better
figure-of-merit for DM halo dynamics since in fluid formulations of self-
interacting DM, the effect of DM scattering is described by heat conduction
Gnedin and Ostriker (2001); Balberg  _et al._ (2002). Moreover,
$\sigma_{\mathrm{V}}$ is well-defined for identical particles and
preferentially weights perpendicular scattering, which is the maximal
modification of particle trajectories within halos.

The goal of the present work is to provide an analytic and fully quantum
mechanical description of the self-interaction cross sections
$\sigma_{\mathrm{T}}$ and $\sigma_{\mathrm{V}}$ in the semi-classical regime,
which is lacking in the literature. This has two advantages over the purely
classical scattering problem. First, we are able to retain the leading terms
involving factors of $1/\kappa$, which necessarily arise due to the
uncertainty principle but are neglected in the classical limit
$\kappa\to\infty$. This allows us to accurately approximate the entire semi-
classical regime, $\kappa\gtrsim 1$, which we demonstrate by comparing to
exact numerical solutions to the Schrödinger equation. Second, we consider the
case of identical particles for $\sigma_{\mathrm{V}}$, which cannot be treated
classically. Our analytic formulas for $\sigma_{\mathrm{V}}$, both for
distinguishable and identical particles, are new.

The remainder of this work is organized as follows. Sec. II describes the
classical and quantum calculations. For distinguishable particles, we show
that there is a direct correspondence between the two treatments in the limit
$\kappa\to\infty$. We also consider the case of identical particles for which
there is no classical analog. In Sec. III, we derive analytic formulas for
$\sigma_{\mathrm{T}}$ and $\sigma_{\mathrm{V}}$. We obtain separate results in
the weakly-coupled ($\beta\ll 1)$ and strongly-coupled ($\beta\gg 1$) regimes,
which are joined together at $\beta\sim 1$ with an empirical fit function. We
consider cases for both attractive and repulsive Yukawa potentials, as well as
both distinguishable and identical particles. In Sec. IV, we present our main
results, comparing our analytic formulas with numerical results from solving
the Schrödinger equation. We also consider the phenomenological implications
for self-interacting DM in the semi-classical regime (and beyond). Lastly,
Sec. V summarizes our conclusions. Additional details are provided in
Appendices A–C.

We make our results available through the new code CalcuLAtionS of Self-
Interaction Cross Sections (CLASSICS). This code implements analytical
expressions for the various cross sections in the semi-classical and quantum
regimes—covering the full parameter space for self-interactions with a Yukawa
potential—as well as provides tabulated values for velocity-averaged results.
It is available at https://github.com/kahlhoefer/CLASSICS.

##  II General formalism

###  II.1 Classical calculation

Classically, cross sections are calculated by solving for trajectories to
determine the resulting scattering angle $\theta(b)$ for a given impact
parameter $b$ (for a review see Ref. Heer (1972)). For a central potential
$U(r)$, this is given by the integral

$\theta(b)=\pi-\int_{r_{0}}^{\infty}\mathrm{d}r\,\frac{2b}{r^{2}\sqrt{1-U(r)/E-b^{2}/r^{2}}}\;,$
(6)

where $r_{0}$ is the classical turning point, which is the largest root where
the term under the square root vanishes. We also have the usual definitions
for energy $E=\mu v^{2}/2$, momentum $k=\mu v$, reduced mass $\mu=m_{\chi}/2$,
and relative velocity $v$. After solving Eq. (6), the differential cross
section is expressed as
$\mathrm{d}\sigma/\mathrm{d}\Omega=\mathrm{d}^{2}b/\mathrm{d}\Omega$ and Eqs.
(4) and (5) are

$\displaystyle\sigma_{\mathrm{T}}$ $\displaystyle=$
$\displaystyle\int\mathrm{d}\Omega\,(1-\cos\theta)\,\frac{\mathrm{d}^{2}b}{\mathrm{d}\Omega}=2\pi\int_{0}^{\infty}\mathrm{d}b\,b\left(1-\cos\theta(b)\right)$
(7a) $\displaystyle\sigma_{\mathrm{V}}$ $\displaystyle=$
$\displaystyle\int\mathrm{d}\Omega\,\sin^{2}\theta\,\frac{\mathrm{d}^{2}b}{\mathrm{d}\Omega}=2\pi\int_{0}^{\infty}\mathrm{d}b\,b\sin^{2}\theta(b)\,.$
(7b)

In the classical calculation, the lower limit of integration in Eqs. (7) is
simply $b=0$. However, as we show below, the corresponding cross sections for
quantum mechanical scattering (of non-identical particles) are given by
exactly the same formulas in the semi-classical regime, with the simple
modification that the lower limit of integration is $b_{\rm
min}=\mathcal{O}(\frac{1}{2k})>0$.

There is a heuristic argument for $b_{\rm min}$ from the uncertainty
principle. If we assume the initial state has a sufficiently well-defined
momentum $k$, then the corresponding uncertainty must satisfy $\Delta
k\lesssim k$. Similarly, the impact parameter $b$ cannot be smaller than the
uncertainty in position, $\Delta r$. By the uncertainty principle, we have

$b\gtrsim\Delta r\gtrsim\frac{1}{2\Delta k}\gtrsim\frac{1}{2k}.$ (8)

This same result follows from the partial wave analysis below: we find $b_{\rm
min}=\frac{1}{2k}$ for $\sigma_{\mathrm{T}}$ and $b_{\rm min}=\frac{1}{k}$ for
$\sigma_{\mathrm{V}}$.

For the Yukawa potential (1), it is useful to re-express the previous
equations in terms of dimensionless parameters

$R=rm_{\phi}\,,\quad\rho=bm_{\phi}\,.$ (9)

Then, including the appropriate lower limits $b_{\rm min}$, Eqs. (7) become

$\sigma_{\mathrm{T}}m_{\phi}^{2}=2\pi\int_{1/2\kappa}^{\infty}\mathrm{d}\rho\,\rho\left(1-\cos\theta(\rho)\right)\,,\qquad\sigma_{\mathrm{V}}m_{\phi}^{2}=2\pi\int_{1/\kappa}^{\infty}\mathrm{d}\rho\,\rho\sin^{2}\theta(\rho)\,,$
(10)

where

$\theta(\rho)=\pi-\int_{R_{0}}^{\infty}\mathrm{d}R\,\frac{2\rho}{R^{2}\sqrt{1\mp
2\beta e^{-R}/R-\rho^{2}/R^{2}}}$ (11)

with the upper (lower) sign for a repulsive (attractive) potential and the
classical turning point $R_{0}=r_{0}m_{\phi}$. The classical calculation
corresponds to the limit $\kappa\to\infty$, in which the lower limits of Eqs.
(10) are replaced by zero, yielding cross sections that are a function of a
single variable $\beta$ Khrapak  _et al._ (2003, 2004). This approach has been
pursued in detail in the plasma physics literature and we briefly summarize
the central results in Appendix A. However, these results are only valid in
the limit $\kappa\to\infty$, while more generally in the semi-classical
regime, the cross sections now depend on two parameters, $\beta$ and $\kappa$.

###  II.2 Quantum calculation

In the usual partial wave analysis, the asymptotic form of the radial
wavefunction $\mathcal{R}_{\ell}(r)$ is determined by solving the Schrödinger
equation

$\frac{1}{r^{2}}\frac{\partial}{\partial
r}\left(r^{2}\frac{\partial\mathcal{R}_{\ell}}{\partial
r}\right)+\left(E-U(r)-\frac{\ell(\ell+1)}{2\mu
r^{2}}\right)\mathcal{R}_{\ell}(r)=0\,.$ (12)

Rescaling the wavefunction as $u_{\ell}(R)=r\mathcal{R}_{\ell}(r)$, with the
definitions in Eq. (9), the Schrödinger equation becomes

$\frac{d^{2}u_{\ell}}{dR^{2}}+K^{2}(R)\,u_{\ell}(R)=0\,,$ (13)

where

$K^{2}(R)=\kappa^{2}\left(1-\frac{\ell(\ell+1)}{\kappa^{2}R^{2}}\mp\frac{2\beta}{R}e^{-R}\right)\,.$
(14)

The phase shifts $\delta_{\ell}$ are obtained from the asymptotic solutions
for the wavefunction,

$u_{\ell}(R)\xrightarrow[R\to\infty]{}\sin\left(\kappa
R-\frac{\ell\pi}{2}+\delta_{\ell}\right)\,,$ (15)

up to an overall (irrelevant) normalization. Given the phase shifts, the
differential cross section is given by the usual formula

$\frac{\mathrm{d}\sigma}{\mathrm{d}\Omega}=\frac{1}{k^{2}}\left|\sum_{\ell=0}^{\infty}(2\ell+1)e^{i\delta_{\ell}}P_{\ell}(\cos\theta)\sin\delta_{\ell}\right|^{2}\;.$
(16)

By integrating over angle, the transfer and viscosity cross sections can be
expressed as a sum over partial wave modes

$\displaystyle\sigma_{\mathrm{T}}m_{\phi}^{2}$ $\displaystyle=$
$\displaystyle\frac{4\pi}{\kappa^{2}}\sum_{\ell=0}^{\infty}(\ell+1)\sin^{2}(\delta_{\ell+1}-\delta_{\ell})\,,$
(17a) $\displaystyle\sigma_{\mathrm{V}}m_{\phi}^{2}$ $\displaystyle=$
$\displaystyle\frac{4\pi}{\kappa^{2}}\sum_{\ell=0}^{\infty}\frac{(\ell+1)(\ell+2)}{2\ell+3}\sin^{2}(\delta_{\ell+2}-\delta_{\ell})\;.$
(17b)

From the preceding formulas, it is clear that the quantum calculations of Eqs.
(17) depend in general on the two independent parameters, $\kappa$ and
$\beta$, while the classical calculation only depends on $\beta$.

In the semi-classical approximation Berry and Mount (1972), the wavefunction
is approximated using the WKB form

$u_{\ell}^{\rm
WKB}(R)=\frac{2}{K(R)^{1/2}}\cos\left(\int_{R_{0}}^{R}dR^{\prime}K(R^{\prime})-\frac{\pi}{4}\right)\,.$
(18)

The phase shift $\delta_{\ell}$ is simply the difference between the phase of
Eq. (18) and the phase when $\beta=0$, as $R\to\infty$:

$\delta_{\ell}=\lim_{R\to\infty}\left(\int_{R_{0}}^{R}dR^{\prime}K(R^{\prime})-\frac{\pi}{4}\right)-\left(\int_{R_{0}}^{R}dR^{\prime}K(R^{\prime})-\frac{\pi}{4}\right)_{\beta=0}\,.$
(19)

However, there is a well-known problem with Eq. (18) as is, which is made
manifest by considering $\beta=0$. In this case, the free WKB wavefunction is
asymptotically (up to an overall constant)

$u_{\ell}^{\rm WKB}(R)\xrightarrow[R\to\infty]{}\sin\left(\kappa
R-\frac{\pi}{2}\sqrt{\ell(\ell+1)}+\frac{\pi}{4}\right)\,.$ (20)

This does not agree with the expectation for the free wavefunction,
$u_{\ell}(R)\to\sin(\kappa R-\tfrac{\ell\pi}{2})$, unless one makes the
approximation $\ell\gg 1$.

This issue is solved by taking an alternative form for the centrifugal
potential, replacing $\ell(\ell+1)\to(\ell+\tfrac{1}{2})^{2}$. The WKB
wavefunction is thereby rescued provided one substitutes

$K^{2}(R)\longrightarrow K_{\rm
eff}^{2}(R)=\kappa^{2}\left(1-\frac{(\ell+\tfrac{1}{2})^{2}}{\kappa^{2}R^{2}}\mp\frac{2\beta}{R}e^{-R}\right)$
(21)

in Eq. (18), which yields the correct asymptotic behavior for any $\ell$.
Originally discovered by Kramers Kramers (1926) and others as a fix to
reproduce various known exact results (see Ref. Berry and Mount (1972)), it
was put on firm theoretical footing by Langer Langer (1937) and is known in
the literature as the Langer modification. The argument is that the WKB
condition breaks down where $K^{2}(R)$ has singular points, namely at $R=0$
due to the centrifugal term. Remarkably, however, through a change of
variables, one can transform the problem into one for which the WKB condition
is satisfied over the entire domain; upon transforming back to the original
variables, this simply amounts to the substitution
$\ell(\ell+1)\to(\ell+\tfrac{1}{2})^{2}$ Langer (1937). The semi-classical
phase shift, after substituting Eq. (21), is now

$\delta_{\ell}=\kappa\int^{\infty}_{R_{0}}dR\,\left(\sqrt{1\mp\frac{2\beta}{R}e^{-R}-\frac{(\ell+\tfrac{1}{2})^{2}}{\kappa^{2}R^{2}}}-1\right)+\frac{\pi(\ell+\tfrac{1}{2})}{2}-\kappa
R_{0}\,,$ (22)

where $R_{0}$ is also determined from $K_{\rm eff}^{2}$.

Next, we turn to the cross sections in Eqs. (17). Since a large number of
$\ell$ modes are summed over, it is possible to interpret $\delta_{\ell}$ as a
continuous function $\delta(\ell)$ and approximate

$\displaystyle\delta_{\ell+1}-\delta_{\ell}$
$\displaystyle\approx\delta^{\prime}(\ell+1/2)\,,$ (23a)
$\displaystyle\delta_{\ell+2}-\delta_{\ell}$ $\displaystyle\approx
2\delta^{\prime}(\ell+1)\;,$ (23b)

where

$\delta^{\prime}(\ell)=\frac{\pi}{2}-\int_{R_{0}}^{\infty}dR\,\frac{\ell+\tfrac{1}{2}}{\kappa
R^{2}\sqrt{1\mp\frac{2\beta}{R}e^{-R}-\frac{(\ell+1/2)^{2}}{\kappa^{2}R^{2}}}}\;.$
(24)

Using this approximation, we can turn the discrete sums into integrals. While
there is no unique way to do this, we find that numerically obtained results
(see below) can be accurately reproduced when defining the integrals in such a
way that the discrete sums are recovered when approximating the integrals by a
Riemann sum with partition size $\Delta\ell=1$ using the midpoint
rule:333According to this approximation, an integral of the form
$I=\int_{\ell_{\text{min}}}^{\infty}f(\ell)\mathrm{d}\ell$ can be written as
$I\approx\sum_{i=0}^{\infty}f\left(\ell_{\text{min}}+\left(i+\tfrac{1}{2}\right)\Delta\ell\right)\Delta\ell=\sum_{\ell=0}^{\infty}f\left(\ell_{\text{min}}+\ell+\tfrac{1}{2}\right)\;,$
where in the second step we have taken $\Delta\ell=1$. Taking, for example,
$\ell_{\text{min}}=0$ and $f(\ell)=(\ell+1/2)\sin^{2}\delta^{\prime}(\ell)$ as
in Eq. (25a) then yields Eq. (17a) with the approximation from Eq. (23a).
While the midpoint rule yields a sufficient level of accuracy for the present
study, more sophisticated treatments were discussed in Ref. Berry and Mount
(1972).

$\displaystyle\sigma_{\mathrm{T}}m_{\phi}^{2}$ $\displaystyle\approx$
$\displaystyle\frac{4\pi}{\kappa^{2}}\int_{0}^{\infty}\mathrm{d}\ell\,(\ell+\tfrac{1}{2})\,\sin^{2}\delta^{\prime}(\ell)$
(25a) $\displaystyle=$
$\displaystyle\frac{4\pi}{\kappa^{2}}\int_{1/2}^{\infty}\mathrm{d}\ell\,\ell\,\sin^{2}\delta^{\prime}(\ell-1/2)\,,$
$\displaystyle\sigma_{\mathrm{V}}m_{\phi}^{2}$ $\displaystyle\approx$
$\displaystyle\frac{4\pi}{\kappa^{2}}\int_{0}^{\infty}\mathrm{d}\ell\,\frac{(\ell+1/2)(\ell+3/2)}{2\ell+2}\,\sin^{2}2\delta^{\prime}(\ell+1/2)$
(25b) $\displaystyle\approx$
$\displaystyle\frac{2\pi}{\kappa^{2}}\int_{1}^{\infty}\mathrm{d}\ell\,\ell\,\sin^{2}2\delta^{\prime}(\ell-1/2)\;,$

where we have used the approximation
$(\ell+1/2)(\ell+3/2)/(2\ell+2)\approx(\ell+1)/2$ in the final step. Equating
$\ell+\tfrac{1}{2}=\kappa\rho$ and comparing Eqs. (11) and (24), it is clear
that $\delta^{\prime}(\ell)$ is simply $\theta(\rho)/2$ and therefore Eqs.
(25) are identical to Eqs. (7) where $b_{\rm min}=\tfrac{1}{2\kappa}$
($\tfrac{1}{\kappa}$) for $\sigma_{\mathrm{T}}$ ($\sigma_{\mathrm{V}}$). This
comes from the extra term from the Langer modification.

###  II.3 Identical particles

A crucial advantage of the quantum mechanical treatment is that we can
directly extend our discussion to the scattering of identical particles. For
identical bosons, the total wave function needs to be symmetric under particle
exchange, so the spatial part needs to be even (odd) if the spin part is
symmetric (antisymmetric). For identical fermions, a symmetric (antisymmetric)
spin part implies an odd (even) spatial part of the wave function. The
differential cross section is then given by

$\frac{\mathrm{d}\sigma}{\mathrm{d}\Omega}=\frac{1}{k^{2}}\left\lvert\sum_{\ell=0}^{\infty}(2\ell+1)e^{i\delta_{\ell}}\left[P_{\ell}(\cos\theta)\pm
P_{\ell}(-\cos\theta)\right]\sin\delta_{\ell}\right\rvert^{2}\;,$ (26)

where the positive sign corresponds to an even spatial wave function and the
negative sign to an odd one. Since $P_{\ell}(-x)=(-1)^{\ell}P_{\ell}(x)$ it
follows that the contribution from all odd phase shifts vanishes for the even
case, while the contribution from even phase shifts vanishes for the odd case.

Hence we obtain in both cases

$\frac{\mathrm{d}\sigma}{\mathrm{d}\Omega}(\theta)=\frac{\mathrm{d}\sigma}{\mathrm{d}\Omega}(\pi-\theta)\;.$
(27)

It follows immediately that
$\int\mathrm{d}\cos\theta\tfrac{\mathrm{d}\sigma}{\mathrm{d}\Omega}\cos\theta=0$
and hence $\sigma_{\mathrm{T}}=\sigma$. In other words, the momentum transfer
cross section is no longer useful for the scattering of identical particles.
The most relevant quantity for the scattering of identical particles is hence
the viscosity cross section, which is given by

$\displaystyle\sigma_{\mathrm{V}}^{\text{even}}m_{\phi}^{2}$
$\displaystyle=\frac{8\pi}{\kappa^{2}}\sum_{\ell=0}^{\infty}\frac{(2\ell+1)(2\ell+2)}{4\ell+3}\sin^{2}(\delta_{2\ell+2}-\delta_{2\ell})\,,$
(28) $\displaystyle\sigma_{\mathrm{V}}^{\text{odd}}m_{\phi}^{2}$
$\displaystyle=\frac{8\pi}{\kappa^{2}}\sum_{\ell=0}^{\infty}\frac{(2\ell+2)(2\ell+3)}{4\ell+5}\sin^{2}(\delta_{2\ell+3}-\delta_{2\ell+1})\;,$
(29)

where an overall symmetry factor $\tfrac{1}{2}$ has been introduced to avoid
double-counting. Note that it follows directly that

$\sigma_{\mathrm{V}}^{\text{even}}+\sigma_{\mathrm{V}}^{\text{odd}}=2\sigma_{\mathrm{V}}\;.$
(30)

If a large number of phase shifts give a relevant contribution to the
viscosity cross section, one finds

$\sigma_{\mathrm{V}}^{\text{even}}\approx\sigma_{\mathrm{V}}^{\text{odd}}\approx\sigma_{\mathrm{V}}\;.$
(31)

If, on the other hand, the cross section is dominated by the first few phase
shifts, we expect
$\sigma_{\mathrm{V}}^{\text{even}}\gg\sigma_{\mathrm{V}}^{\text{odd}}$.

Again we can approximate the cross sections for the scattering of identical
particles through integrals, making use of the same prescription as above:

$\displaystyle\sigma_{\mathrm{V}}^{\text{even}}\,m_{\phi}^{2}$
$\displaystyle\approx\frac{8\pi}{\kappa^{2}}\int_{0}^{\infty}\mathrm{d}\ell\,\frac{(2\ell)(2\ell+1)}{4\ell+1}\sin^{2}2\delta^{\prime}(2\ell)$
$\displaystyle\approx\frac{2\pi}{\kappa^{2}}\int_{1/2}^{\infty}\mathrm{d}\ell\,\ell\sin^{2}2\delta^{\prime}(\ell-1/2)\,,$
(32) $\displaystyle\sigma_{\mathrm{V}}^{\text{odd}}\,m_{\phi}^{2}$
$\displaystyle\approx\frac{8\pi}{\kappa^{2}}\int_{0}^{\infty}\mathrm{d}\ell\,\frac{(2\ell+1)(2\ell+2)}{4\ell+3}\sin^{2}2\delta^{\prime}(2\ell+1)$
$\displaystyle\approx\frac{2\pi}{\kappa^{2}}\int_{3/2}^{\infty}\mathrm{d}\ell\,\ell\sin^{2}2\delta^{\prime}(\ell-1/2)\;.$
(33)

As we will see below, in certain situations the argument of the sine is
sufficiently small to allow for the approximation $\sin^{2}(x)\approx x^{2}$.
Whenever this is the case, it follows that
$\sigma_{\mathrm{V}}^{\text{even}}\approx 2\sigma_{\mathrm{T}}$, i.e. the
viscosity cross section for the even case is directly related to the momentum
transfer cross section for non-identical particles.

Finally, we note that for DM particles with non-zero spin the more realistic
situation is that the scattering particles are unpolarized and hence the total
spin wave function may be either symmetric or anti-symmetric. Averaging over
spins then amounts to the following replacement

$\sigma_{\mathrm{V}}\to\begin{cases}\sigma_{\mathrm{V}}^{\rm
even}&\text{scalar DM}\,,\\\ \frac{1}{4}\sigma_{\mathrm{V}}^{\rm
even}+\frac{3}{4}\sigma_{\mathrm{V}}^{\rm odd}&\text{fermion DM}\;,\\\
\frac{2}{3}\sigma_{\mathrm{V}}^{\rm even}+\frac{1}{3}\sigma_{\mathrm{V}}^{\rm
odd}&\text{vector DM}\;.\end{cases}$ (34)

##  III Analytical approximations for the semi-classical regime

In this section, we derive analytical approximations for the semi-classical
regime in a weakly- and strongly-coupled theory, corresponding to $\beta\ll 1$
and $\beta\gg 1$, respectively.

###  III.1 Weak potential

Let us first consider a weak dimensionless potential with $\beta\ll 1$. We
begin by considering the case that not only is $\beta$ small but also
$\kappa\beta\ll 1$. In this case, we can expand the square root in Eq. (22) in
$\beta$ and approximate $R_{0}\approx(\ell+\tfrac{1}{2})/\kappa$. The phase
shift is then given by

$\displaystyle\delta_{\ell}\approx\mp\int_{R_{0}}^{\infty}\mathrm{d}{R}\frac{\beta\kappa}{R}e^{-R}\frac{1}{\sqrt{1-\frac{(\ell+\tfrac{1}{2})^{2}}{\kappa^{2}R^{2}}}}=\mp\beta\kappa
K_{0}\left(\frac{\ell+\tfrac{1}{2}}{\kappa}\right)\,,$ (35)

where $K_{i}$ denote the modified Bessel functions of the second kind and the
negative (positive) sign corresponds to an attractive (repulsive) potential.
For the difference between two consecutive phase shifts, we then obtain

$\delta_{\ell}-\delta_{\ell-1}\approx\delta^{\prime}(\ell-1/2)\approx\pm\beta
K_{1}\left(\frac{\ell}{\kappa}\right)\,.$ (36)

Substituting this expression into Eq. (25a), we obtain for both attractive and
repulsive potentials

$\sigma_{\mathrm{T}}\approx\frac{4\pi}{\kappa^{2}m_{\phi}^{2}}\int_{1/2}^{\infty}\ell\sin^{2}\left[\beta
K_{1}\left(\frac{\ell}{\kappa}\right)\right]\mathrm{d}\ell\;.$ (37)

For $\beta\kappa<1/2$ and $\ell>1/2$ we can make use of the fact that
$K_{1}(x)$ is a monotonically decreasing function to show that

$\beta K_{1}\left(\frac{\ell}{\kappa}\right)<\beta
K_{1}\left(\frac{1}{2\kappa}\right)<\beta K_{1}(\beta)<1\;.$ (38)

Hence the argument of the sine is small across the entire integration range
and we obtain approximately

$\displaystyle\sigma_{\mathrm{T}}$
$\displaystyle\approx\frac{\pi\beta^{2}}{2\kappa^{2}m_{\phi}^{2}}\left[-K_{1}\left(\frac{1}{2\kappa}\right)^{2}+K_{0}\left(\frac{1}{2\kappa}\right)K_{2}\left(\frac{1}{2\kappa}\right)\right]\equiv\frac{2\pi\beta^{2}}{m_{\phi}^{2}}\eta\left(\frac{1}{2\kappa}\right)\;,$
(39)

where the second step defines the function $\eta(x)$. For $\kappa\gg 1$ this
expression can be approximated as

$\eta\left(\frac{1}{2\kappa}\right)\approx 2\log
4\kappa-1-2\gamma_{\mathrm{E}}+\frac{1-\gamma_{\mathrm{E}}+\log
4\kappa}{4\kappa^{2}}\;,$ (40)

where $\gamma_{\mathrm{E}}$ is Euler’s constant. We emphasize that the
explicit dependence on $\kappa$ is a direct consequence of the fact that the
integration range in Eq. (37) does not extend to $\ell=0$.

Starting from Eq. (25b), a completely analogous calculation for the viscosity
cross section yields

$\displaystyle\sigma_{\mathrm{V}}$
$\displaystyle\approx\frac{2\pi}{\kappa^{2}m_{\phi}^{2}}\int_{1}^{\infty}\ell\sin^{2}\left[2\beta
K_{1}\left(\frac{\ell}{\kappa}\right)\right]\mathrm{d}\ell\approx\frac{4\pi\beta^{2}}{m_{\phi}^{2}}\eta\left(\frac{1}{\kappa}\right)\;.$
(41)

Let us now consider the case that $\beta\ll 1$ but $\kappa\beta>1/2$. In this
case the expansion in Eq. (35) is only a good approximation for
$\ell>\ell_{\text{min}}=\kappa\beta$. Indeed, for small $\ell$ the argument of
the sine function becomes large and hence the integrand rapidly oscillates
between 0 and 1. We can make a rough estimate of the resulting contribution to
the total cross section by approximating $\sin^{2}(x)\approx 1/2$ for
$\ell<\ell_{\text{min}}$, which gives

$\frac{4\pi}{\kappa^{2}m_{\phi}^{2}}\int_{1/2}^{\ell_{\text{min}}}\ell\sin^{2}\left(\delta_{\ell}-\delta_{\ell-1}\right)\mathrm{d}\ell\approx\frac{\pi}{\kappa^{2}m_{\phi}^{2}}\left(\ell_{\text{min}}^{2}-\frac{1}{4}\right)\;.$
(42)

For $\ell>\ell_{\text{min}}$ we can use the same approximations as before to
obtain

$\displaystyle\frac{4\pi}{\kappa^{2}m_{\phi}^{2}}\int_{\ell_{\text{min}}}^{\infty}\ell\sin^{2}\left[\beta
K_{1}\left(\frac{\ell}{\kappa}\right)\right]\mathrm{d}\ell\approx\frac{2\pi\beta^{2}}{m_{\phi}^{2}}\eta\left(\beta\right)\;,$
(43)

and hence

$\displaystyle\sigma_{\mathrm{T}}\approx\frac{2\pi\beta^{2}}{m_{\phi}^{2}}\left[\frac{\ell_{\text{min}}^{2}-\tfrac{1}{4}}{2\kappa^{2}\beta^{2}}+\eta\left(\beta\right)\right]\;.$
(44)

If $\beta\ll 1$ while $\kappa\beta\gg 1$, the momentum transfer cross section
becomes a function of $\beta$ only:

$\displaystyle\sigma_{\mathrm{T}}$
$\displaystyle\approx\frac{2\pi\beta^{2}}{m_{\phi}^{2}}\left[2\log\frac{2}{\beta}-\frac{1}{2}-2\gamma_{\mathrm{E}}+\beta^{2}\left(1-\gamma_{\mathrm{E}}+\log\frac{2}{\beta}\right)\right]\;.$
(45)

When $\beta$ is small enough that we can approximate the square bracket by
$2\log\beta^{-1}$, this result agrees with the one obtained in the classical
approach. However, for constant $\kappa$ the assumption $\kappa\beta>1$
necessarily breaks down as $\beta\to 0$ and $\sigma_{\mathrm{T}}$ starts to
decrease more steeply, i.e. it scales proportional to $\beta^{2}$ rather than
proportional to $\beta^{2}\log\beta^{-1}$. This observation agrees with what
is found by solving the Schrödinger equation numerically.

We note that the cases $\kappa\beta<1/2$ and $\kappa\beta>1/2$ can be combined
by defining

$\zeta_{n}(\kappa,\beta)\equiv\frac{\text{max}(n,\beta\kappa)^{2}-n^{2}}{2\kappa^{2}\beta^{2}}+\eta\left(\frac{\text{max}(n,\beta\kappa)}{\kappa}\right)\;,$
(46)

which leads to

$\displaystyle\sigma_{\mathrm{T}}\approx\frac{2\pi\beta^{2}}{m_{\phi}^{2}}\zeta_{1/2}\left(\kappa,\beta\right)\;.$
(47)

For the viscosity cross section the integration starts at $\ell=1$ and we
require $2\beta K_{1}(\ell/\kappa)<1$ in order for the argument of the sine to
be small. Hence we obtain

$\displaystyle\sigma_{\mathrm{V}}\approx\frac{4\pi\beta^{2}}{m_{\phi}^{2}}\zeta_{1}\left(\kappa,2\beta\right)\;.$
(48)

Finally, for the case of identical particles, we find

$\displaystyle\sigma_{V}^{\text{even}}$
$\displaystyle\approx\frac{4\pi\beta^{2}}{m_{\phi}^{2}}\zeta_{1/2}\left(\kappa,2\beta\right)\;,$
(49) $\displaystyle\sigma_{V}^{\text{odd}}$
$\displaystyle\approx\frac{4\pi\beta^{2}}{m_{\phi}^{2}}\zeta_{3/2}\left(\kappa,2\beta\right)\;.$
(50)

For $\kappa\beta\ll 1$ (and $n$ of order unity) one finds
$\zeta_{n}\left(\kappa,\beta\right)\approx\eta(n/\kappa)$ independent of
$\beta$ and hence $\sigma_{\mathrm{V}}^{\text{even}}\approx
2\sigma_{\mathrm{T}}\gg\sigma_{\mathrm{V}}^{\text{odd}}$, while for
$\kappa\beta\gg 1$ one finds
$\zeta_{n}\left(\kappa,\beta\right)\approx\tfrac{1}{2}+\eta(\beta)$
independent of $\kappa$ and $n$ and therefore
$\sigma_{\mathrm{V}}^{\text{even}}\approx\sigma_{\mathrm{V}}^{\text{odd}}\approx\sigma_{\mathrm{V}}$.

###  III.2 Strong potential

Let us now turn to the case of large $\beta$, corresponding to a strong
dimensionless potential. We define an effective potential

$U_{\mathrm{eff}}(R)\equiv\left(\frac{\ell+1/2}{\kappa}\right)^{2}\frac{1}{R^{2}}\mp
2\beta\frac{e^{-R}}{R}$ (51)

where the upper (lower) sign is for an attractive (repulsive) interaction. The
largest root of $U_{\mathrm{eff}}(R)=1$ determines the distance of closest
approach $R_{0}$. For large $\beta$, this quantity is decisive for the
calculation of the phase shifts and therefore the cross sections, as we show
below. Since $R_{0}$ depends on the sign of the potential, we obtain different
results for attractive and repulsive cases and therefore need to discuss them
separately. This is in contrast to the weak potential regime, where
$R_{0}\approx(\ell+\tfrac{1}{2})/\kappa$ and our results were independent of
sign.

The calculations in the remainder of this subsection proceed as follows. For
each potential (attractive or repulsive), we determine an analytic form for
the phase shift derivative by estimating $R_{0}$ and specifying the range of
$\ell$ that contributes the most to the cross-section. The momentum-transfer
and viscosity cross sections are then simply the integrals of these phase
shifts over the $\ell$ range of interest, which is given by $0<\ell<\ell_{\rm
max}$ where the upper limit $\ell_{\rm max}$ is defined below. (For the
repulsive viscosity cross section, it is necessary to integrate beyond
$\ell_{\rm max}$, as we discuss below.)

The lower bound of integration $\ell=0$ is taken from the following argument.
Numerically evaluating the integrals in Eqs. (25a) and (25b), we find that the
dependence on $\kappa$ is much weaker for large $\beta$ than for small
$\beta$. Therefore, we will neglect this dependence and assume $\kappa\gg 1$,
which implies $\ell\gg 1$. Since we can approximate
$\ell-1/2\approx\ell+1/2\approx\ell$, we take the lower limit of integration
to be $\ell=0$.

####  III.2.1 Attractive potential

Figure 2:  Effective potential as described in Eq. (51) for two values of
$\beta$. For $\beta\gtrsim 13.2$, a secondary potential barrier emerges for
sufficiently large values of $\ell$, corresponding to a jump in the distance
of closest approach. See text for details.

In the attractive case the effective potential $U_{\text{eff}}(R)$ is
monotonically decreasing only for a sufficiently large angular momentum
barrier (i.e., large $\ell$). For small $\ell$, on the other hand, the
effective potential develops a maximum at finite $R$ which can lead to a
secondary potential barrier (see Fig. 2). For $\beta<13.2$ this effect is
sufficiently small that the equation $U_{\text{eff}}(R)=1$ still has a unique
solution for all values of $\ell$. For larger $\beta$, however, there will be
multiple roots for sufficiently large values of $\ell$. For a given $\beta$ we
define $\ell_{\text{max}}$ as the largest value of $\ell$ for which there is a
single root. We show in Appendix B.1 that

$\displaystyle\ell_{\text{max}}\approx\kappa\,\left(1+\log\beta-\frac{1}{2\,\log\beta}\right)\;.$
(52)

Further, we find that for $\ell>\ell_{\text{max}}$, the distance of closest
approach, $R_{0}$, is so large that the resulting phase shifts are tiny and
give a negligible contribution to the cross section (see Fig. 2). We can
therefore approximate

$\sigma_{\mathrm{T}}^{\text{att.}}m_{\phi}^{2}\approx\frac{4\pi}{\kappa^{2}}\int_{0}^{\ell_{\text{max}}}\ell\sin^{2}\delta^{\prime}(\ell)\mathrm{d}\ell\;.$
(53)

In order to calculate the derivative of the phase shift, we first estimate
$R_{0}$. For $\ell<\ell_{\text{max}}$, one finds $\ell/\kappa<2\beta$ and
hence the equation $U_{\text{eff}}(R)=1$ can only be satisfied for $R<1$. In
this case we can approximate $e^{-R}\approx 1$, i.e., the Yukawa potential can
be approximated by a Coulomb potential. We then find

$R_{0}\approx\frac{\ell^{2}}{2\beta\kappa^{2}}\qquad(\ell<\ell_{\text{max}})\,.$
(54)

For completeness, we also calculate $R_{0}$ in the opposite case
($\ell>\ell_{\text{max}}$). Here, the (largest) root corresponds to the case
that the exponential suppression of the Yukawa potential becomes relevant,
which implies $R\gg 1$, as expected. In this case we can ignore the term
proportional to $\beta$ in Eq. (51) to get

$R_{0}\approx\frac{\ell}{\kappa}\qquad(\ell>\ell_{\text{max}})\,.$ (55)

With these expressions for $R_{0}$, we can now calculate the phase shifts and
their differences.

For $\ell\ll\ell_{\text{max}}$, we show in Appendix B.2 that the phase shift
derivative can be written as

$\delta^{\prime}(\ell)\approx-\frac{\pi}{2}-\frac{\ell}{\kappa}\gamma(\beta)\;,$
(56)

where

$\gamma(\beta)\equiv\frac{W(2\beta)+2}{W(2\beta)^{2}}$ (57)

with $W(x)$ denoting the product logarithm. In practice, we find that Eq. (56)
gives a good approximation for $\tfrac{\ell}{\kappa}\gamma(\beta)<\pi/2$. By
defining the rescaled angular momentum
$\ell^{\prime}\equiv\tfrac{\ell}{\kappa}\gamma(\beta)$, we can then write

$\sigma_{\mathrm{T}}^{\text{att.}}m_{\phi}^{2}\approx\frac{4\pi}{\gamma(\beta)^{2}}\int_{0}^{\pi/2}\ell^{\prime}\sin(\pi/2+\ell^{\prime})^{2}\mathrm{d}\ell^{\prime}+\frac{4\pi}{\kappa^{2}}\int_{\pi\kappa/(2\gamma(\beta))}^{\ell_{\text{max}}}\ell\sin^{2}\delta^{\prime}(\ell)\mathrm{d}\ell\;.$
(58)

The first integral gives $(\pi^{2}-4)/16$. In the second integral there is no
simple expression for the derivative of the phase shifts, but since the
argument of the sine is large we can approximately replace $\sin^{2}(x)\approx
1/2$, giving
$\ell_{\text{max}}^{2}/4-\pi^{2}\kappa^{2}/(16\gamma(\beta)^{2})$. Hence,

$\sigma_{\mathrm{T}}^{\text{att.}}m_{\phi}^{2}\approx\frac{\pi}{\kappa^{2}}\left(\ell_{\text{max}}^{2}-\frac{\kappa^{2}}{\gamma(\beta)^{2}}\right)\approx\pi\left[\left(1+\log\beta-\frac{1}{2\,\log\beta}\right)^{2}-\frac{1}{\gamma(\beta)^{2}}\right]\;.$
(59)

For $\beta\gg 1$ we can approximate $\gamma(\beta)\approx 1/W(2\beta)$ and
$W(2\beta)\approx\log\beta-\log\log\beta$. Hence, the leading term
proportional to $\log^{2}\beta$ cancels and we find

$\sigma_{\mathrm{T}}^{\text{att.}}m_{\phi}^{2}\approx
2\pi\log\beta(\log\log\beta+1)\;.$ (60)

We emphasize that this asymptotic behavior differs from the one found
previously for the classical limit
($\sigma_{\mathrm{T}}^{\text{att.}}\propto\log^{2}\beta$, cf. Appendix A) and
that in contrast to previous calculations it requires no empirical constant of
proportionality. Our new result shows better agreement with the numerically
calculated cross section and therefore supersedes the existing expressions.

In order to calculate the viscosity cross section, we follow an analogous
calculation to the one above to find

$\displaystyle\sigma_{\mathrm{V}}^{\text{att.}}m_{\phi}^{2}\approx\frac{\pi}{2\kappa^{2}}\ell_{\text{max}}^{2}\approx\frac{\pi}{2}\left(1+\log\beta-\frac{1}{2\,\log\beta}\right)^{2}\,,$
(61)

which agrees well with the numerical result.

####  III.2.2 Repulsive potential

In the repulsive case, for $\beta>1$ the equation $U_{\mathrm{eff}}(R)=1$
implies $R>1$. For $\ell/\kappa\ll 1$ it is clear that the second term in Eq.
(51) dominates over the first one in the relevant range of $R$ and we find

$R_{0}\approx W(2\beta)\;.$ (62)

For $\ell/\kappa\gg 1$ the first term dominates and we find

$R_{0}\approx\frac{\ell}{\kappa}\;.$ (63)

As before, we use $\ell_{\text{max}}$ to denote the transition between the two
solutions, which occurs when $W(2\beta)\approx R_{0}$, leading to

$\ell_{\text{max}}\approx\kappa W(2\beta)\;.$ (64)

For $\ell>\ell_{\text{max}}$ the phase shifts are very small and their
contribution to the momentum transfer cross section can be neglected. For
$\ell<\ell_{\text{max}}$ we can use a similar calculation as in Appendix B.2
to show that

$\delta^{\prime}(\ell)\approx\frac{\pi}{2}+\frac{\ell}{\kappa}\int_{R_{0}}^{\infty}\frac{\mathrm{d}R}{R^{2}}\approx\frac{\pi}{2}-\frac{\ell}{\ell_{\text{max}}}\;.$
(65)

Substituting this expression into the formula for the momentum transfer cross
section, we find

$\sigma_{\mathrm{T}}^{\text{rep.}}m_{\phi}^{2}\approx\pi\lambda_{\mathrm{T}}W(2\beta)^{2}\,,$
(66)

with $\lambda_{\mathrm{T}}\equiv(1+\cos 2+2\sin 2)/2\approx 1.20$. For large
$\beta$ this expression becomes approximately equal to

$\sigma_{\mathrm{T}}^{\text{rep.}}m_{\phi}^{2}\approx\pi\lambda_{\mathrm{T}}(\log
2\beta-\log\log 2\beta)^{2}\;,$ (67)

which differs from the result in the literature by the factor $\lambda_{T}$.
We have confirmed that our new result improves the agreement with the
numerically calculated cross section and therefore supersedes existing
expressions.

For the viscosity cross section we find that for $\ell<\ell_{\text{max}}$

$\displaystyle
2\delta^{\prime}(\ell)\approx\pi-\frac{2\ell}{\ell_{\text{max}}}\;.$ (68)

Since this expression is close to $\pi$ for small $\ell$, the resulting
contribution to the viscosity cross section is suppressed and we need to
extend the integration limit beyond $\ell_{\text{max}}$ to better approximate
the cross section. For $\ell>\ell_{\text{max}}$ we can use a calculation
analogous to the case of small $\beta$ to show that

$\displaystyle 2\delta^{\prime}(\ell)\approx 2\beta
K_{1}\left(\frac{\ell}{\kappa}\right)\;.$ (69)

Combining these two expressions yields

$\sigma_{\mathrm{V}}^{\text{rep.}}\approx\frac{2\pi}{\kappa^{2}m_{\phi}^{2}}\left[\int_{0}^{\ell_{\text{max}}}\ell\sin^{2}\left(\pi-\frac{2\ell}{\ell_{\text{max}}}\right)\mathrm{d}\ell+\int_{\ell_{\text{max}}}^{\infty}\ell\sin^{2}\left(2\beta
K_{1}\left(\frac{\ell}{\kappa}\right)\right)\mathrm{d}\ell\right]\;.$ (70)

The first term can be integrated to yield
$\pi\lambda_{\mathrm{V}}\,W(2\beta)^{2}$ with $\lambda_{\mathrm{V}}=(9-\cos
4-4\sin 4)/16\approx 0.793$. For the second term we note that the argument of
the sine is large for $\ell<\kappa\log 2\beta$ and quickly becomes negligible
for larger $\ell$. Hence, we obtain

$\displaystyle\sigma_{\mathrm{V}}^{\text{rep.}}m_{\phi}^{2}$
$\displaystyle\approx\pi\lambda_{\mathrm{V}}\,W(2\beta)^{2}+\frac{\pi}{2}\left(\log^{2}2\beta-W(2\beta)^{2}\right)$
(71) $\displaystyle\approx\pi\log 2\beta\left(\lambda_{\mathrm{V}}\log
2\beta-(2\lambda_{V}-1)\log\log 2\beta\right)\;.$ (72)

To conclude this discussion, we note that for large $\beta$ the scattering of
identical particles becomes trivial, as we always find
$\sigma_{\mathrm{V}}^{\text{even}}\approx\sigma_{\mathrm{V}}^{\text{odd}}\approx\sigma_{\mathrm{V}}$.

###  III.3 Intermediate regime

So far we have focused on the two limiting cases $\beta\ll 1$ and $\beta\gg
1$, where it is possible to derive approximate analytical expressions for the
momentum transfer and viscosity cross sections. In the intermediate regime,
however, the approximations made above do not apply. Although it is still
possible to derive approximate expressions for the phase shifts for
$\ell\ll\kappa$ and $\ell\gg\kappa$, the final cross sections turn out to
depend sensitively on contributions with $\ell\approx\kappa$.

In particular, for $\beta\gtrsim 0.2$ the numerical results for the momentum
transfer cross section differ between the attractive potential and repulsive
potential, which cannot be reproduced using the approximations made above for
$\beta\ll 1$. Conversely, for $\beta\lesssim 50$ the momentum transfer cross
section for an attractive potential exhibits a number of oscillations, which
cannot be reproduced in our treatment of the large $\beta$
regime.444Qualitatively, these oscillations can be understood in terms of the
maximal value of $|\delta^{\prime}(\ell)|$. If
$|\delta^{\prime}(\ell)|_{\text{max}}\approx n\pi$ for some integer $n$ the
momentum transfer cross section exhibits a dip, whereas for
$|\delta^{\prime}(\ell)|_{\text{max}}\approx\left(n-\tfrac{1}{2}\right)\pi$
the momentum transfer cross section exhibits a peak. However, we note that
these peaks and dips are smeared out when averaging over a range of velocities
(see Sec. IV.3) and therefore lose relevance in realistic applications. Rather
than attempting to refine our treatment to reproduce these effects in the
intermediate regime, we derive an improved empirical fitting formula that can
be used for $0.2<\beta<50$.

Specifically, we numerically calculate the momentum transfer cross section for
$\beta=1$ and $\kappa\gg 1$, which is found to be
$\sigma_{\mathrm{T}}^{\text{att.}}m_{\phi}^{2}/\pi=2.8$
($\sigma_{\mathrm{T}}^{\text{rep.}}m_{\phi}^{2}/\pi=1.1$) for the attractive
(repulsive) case. We then assume that $\sigma_{\mathrm{T}}m_{\phi}^{2}/\pi$
can be approximated by $a\log(\beta+b)$ for $1<\beta<50$ and determine $a$ and
$b$ such that the numerical result at $\beta=1$ is reproduced and the function
connects continuously to the analytical expressions for large $\beta$ at
$\beta=50$. This approach yields

$\displaystyle\sigma_{\mathrm{T}}^{\text{att.}}$
$\displaystyle=4.7\log(\beta+0.82)\frac{\pi}{m_{\phi}^{2}}\qquad(1<\beta<50)\,,$
(73) $\displaystyle\sigma_{\mathrm{T}}^{\text{rep.}}$
$\displaystyle=2.9\log(\beta+0.47)\frac{\pi}{m_{\phi}^{2}}\qquad(1<\beta<50)\;.$
(74)

For $0.2<\beta<1$ the dependence of the momentum transfer cross section on
$\kappa$ is non-negligible. To ensure continuity at $\beta=0.2$, we take the
analytical expression obtained for small $\beta$ and multiply by
$e^{c(\beta-0.2)}$, where $c$ is determined such that the momentum transfer
cross section is continuous at $\beta=1$ for $\kappa\gg 1$. This approach
yields

$\displaystyle\sigma_{\mathrm{T}}^{\text{att.}}$
$\displaystyle=\frac{2\pi\beta^{2}}{m_{\phi}^{2}}\zeta_{1/2}\left(\kappa,\beta\right)e^{0.64(\beta-0.2)}\qquad(0.2<\beta\leq
1)\,,$ (75) $\displaystyle\sigma_{\mathrm{T}}^{\text{rep.}}$
$\displaystyle=\frac{2\pi\beta^{2}}{m_{\phi}^{2}}\zeta_{1/2}\left(\kappa,\beta\right)e^{-0.53(\beta-0.2)}\qquad(0.2<\beta\leq
1)\;.$ (76)

Note that for small $\kappa$ this approach leads to a slight discontinuity at
$\beta=1$, which however has no practical consequences.

We follow the exact same approach for the viscosity cross section, with the
only difference being that the intermediate regime is shifted to slightly
smaller values of $\beta$. We numerically evaluate the cross sections at
$\beta=0.5$, which yields
$\sigma_{\mathrm{V}}^{\text{att.}}m_{\phi}^{2}/\pi=1.1$
($\sigma_{\mathrm{V}}^{\text{rep.}}m_{\phi}^{2}/\pi=0.73$) for the attractive
(repulsive) potential, leading to

$\displaystyle\sigma_{\mathrm{V}}^{\text{att.}}$
$\displaystyle=2.5\log(\beta+1.05)\frac{\pi}{m_{\phi}^{2}}\qquad(0.5<\beta<25)\,,$
(77) $\displaystyle\sigma_{\mathrm{V}}^{\text{rep.}}$
$\displaystyle=2.8\log(\beta+0.80)\frac{\pi}{m_{\phi}^{2}}\qquad(0.5<\beta<25)\;,$
(78)

and

$\displaystyle\sigma_{\mathrm{V}}^{\text{att.}}$
$\displaystyle=\frac{4\pi\beta^{2}}{m_{\phi}^{2}}\zeta_{n}\left(\kappa,2\beta\right)e^{0.67(\beta-0.1)}\qquad(0.1<\beta\leq
0.5)\,,$ (79) $\displaystyle\sigma_{\mathrm{V}}^{\text{rep.}}$
$\displaystyle=\frac{4\pi\beta^{2}}{m_{\phi}^{2}}\zeta_{n}\left(\kappa,2\beta\right)e^{-0.37(\beta-0.1)}\qquad(0.1<\beta\leq
0.5)\;,$ (80)

where $n=1$ for distinguishable particles and $n=\tfrac{1}{2}$
($n=\tfrac{3}{2}$) for identical particles with even (odd) spatial wave
function.

###  III.4 Summary

In summary, we obtain the following expressions for the momentum transfer
cross section $\sigma_{\mathrm{T}}$ for the scattering of distinguishable
particles in an attractive or repulsive Yukawa potential:

$\displaystyle\sigma_{\mathrm{T}}^{\mathrm{att.}}$
$\displaystyle=\frac{\pi}{m_{\phi}^{2}}\times\begin{cases}2\beta^{2}\zeta_{1/2}\left(\kappa,\beta\right)&\beta\leq
0.2\,,\\\ \hskip 199.16928pt\ &\\\\[-11.38109pt]
2\beta^{2}\zeta_{1/2}\left(\kappa,\beta\right)e^{0.64(\beta-0.2)}&0.2<\beta\leq
1\,,\\\ \hskip 199.16928pt\ &\\\\[-11.38109pt]
4.7\log(\beta+0.82)&1<\beta<50\,,\\\ \hskip 199.16928pt\ &\\\\[-11.38109pt]
2\log\beta(\log\log\beta+1)&\beta\geq 50\;,\end{cases}$ (81)
$\displaystyle\sigma_{\mathrm{T}}^{\mathrm{rep.}}$
$\displaystyle=\frac{\pi}{m_{\phi}^{2}}\times\begin{cases}2\beta^{2}\zeta_{1/2}\left(\kappa,\beta\right)&\beta\leq
0.2\,,\\\ \hskip 199.16928pt\ &\\\\[-11.38109pt]
2\beta^{2}\zeta_{1/2}\left(\kappa,\beta\right)e^{-0.53(\beta-0.2)}&0.2<\beta\leq
1\,,\\\ \hskip 199.16928pt\ &\\\\[-11.38109pt]
2.9\log(\beta+0.47)&1<\beta<50\,,\\\ \hskip 199.16928pt\ &\\\\[-11.38109pt]
\lambda_{\mathrm{T}}(\log 2\beta-\log\log 2\beta)^{2}&\beta\geq
50\;,\end{cases}$ (82)

with

$\displaystyle\zeta_{n}(\kappa,\beta)$
$\displaystyle=\frac{\text{max}(n,\beta\kappa)^{2}-n^{2}}{2\kappa^{2}\beta^{2}}+\eta\left(\frac{\text{max}(n,\beta\kappa)}{\kappa}\right)\,,$
(83) $\displaystyle\eta(x)$
$\displaystyle=x^{2}\left[-K_{1}\left(x\right)^{2}+K_{0}\left(x\right)K_{2}\left(x\right)\right]$
(84)
$\displaystyle\approx-2\log\left(\frac{x}{2}\right)-1-2\gamma_{\mathrm{E}}+x^{2}\left[1-\gamma_{\mathrm{E}}-\log\left(\frac{x}{2}\right)\right]\;.$
(85)

For the viscosity cross section we obtain

$\displaystyle\sigma_{\mathrm{V}}^{\mathrm{att.}}$
$\displaystyle=\frac{\pi}{m_{\phi}^{2}}\times\begin{cases}4\beta^{2}\zeta_{n}\left(\kappa,2\beta\right)&\beta\leq
0.1\,,\\\ \hskip 199.16928pt\ &\\\\[-11.38109pt]
4\beta^{2}\zeta_{n}\left(\kappa,2\beta\right)e^{0.67(\beta-0.1)}&0.1<\beta\leq
0.5\,,\\\ \hskip 199.16928pt\ &\\\\[-11.38109pt]
2.5\log(\beta+1.05)&0.5<\beta<25\,,\\\ \hskip 199.16928pt\ &\\\\[-11.38109pt]
\frac{1}{2}\left(1+\log{\beta}-\frac{1}{2\log\beta}\right)^{2}&\beta\geq
25\;,\end{cases}$ (86) $\displaystyle\sigma_{\mathrm{V}}^{\mathrm{rep.}}$
$\displaystyle=\frac{\pi}{m_{\phi}^{2}}\times\begin{cases}4\beta^{2}\zeta_{n}\left(\kappa,2\beta\right)&\beta\leq
0.2\,,\\\ \hskip 199.16928pt\ &\\\\[-11.38109pt]
4\beta^{2}\zeta_{n}\left(\kappa,2\beta\right)e^{-0.37(\beta-0.1)}&0.1<\beta\leq
0.5\,,\\\ \hskip 199.16928pt\ &\\\\[-11.38109pt]
2.8\log(\beta+0.80)&0.5<\beta<25\,,\\\ \hskip 199.16928pt\ &\\\\[-11.38109pt]
\log 2\beta\left(\lambda_{\mathrm{V}}\log 2\beta-(2\lambda_{V}-1)\log\log
2\beta\right)&\beta\geq 25\;,\end{cases}$ (87)

with

$\displaystyle\lambda_{\mathrm{T}}$ $\displaystyle=(1+\cos 2+2\sin 2)/2\,,$
(88) $\displaystyle\lambda_{\mathrm{V}}$ $\displaystyle=(9-\cos 4-4\sin
4)/16\;,$ (89)

where $n=1$ for distinguishable particles and $n=\tfrac{1}{2}$
($n=\tfrac{3}{2}$) for identical particles with even (odd) spatial wave
function.

##  IV Results

###  IV.1 Comparison with numerical results

Figure 3: Dimensionless cross sections $\sigma_{\mathrm{T}}m_{\phi}^{2}$ and
$\sigma_{\mathrm{V}}m_{\phi}^{2}$ as functions of $\beta$, for attractive and
repulsive Yukawa potentials and for various values of $\kappa$. Solid lines
show our analytic results, derived in Sec. III, which are in good agreement
with exact numerical results from solving the Schrödinger equation (dots).
Dashed lines correspond to purely classical scattering.

We compare our analytic formulas for the semi-classical regime, given in Eqs.
(81–87), to exact results obtained by numerically solving the Schrödinger
equation. In Fig. 3, the solid lines show our analytic approximations for the
dimensionless cross sections $\sigma_{\mathrm{T}}m_{\phi}^{2}$ (top) and
$\sigma_{\mathrm{V}}m_{\phi}^{2}$ (bottom) as functions of $\beta$ for an
attractive (left) and repulsive (right) potential, each for different values
of $\kappa$. These results are in better agreement with the exact numerical
results (dots) compared to the classical scattering cross section (dashed
line) that is strictly valid in the limit $\kappa\to\infty$. Outside this
limit, however, the cross section is clearly dependent on $\kappa$ in the
Coulomb regime ($\beta\lesssim 1$), but independent of $\kappa$ in the
strongly-coupled regime ($\beta\gtrsim 1$).

Figure 4: Dimensionless cross sections $\sigma_{\mathrm{T}}m_{\phi}^{2}$ and
$\sigma_{\mathrm{V}}m_{\phi}^{2}$ as functions of $\kappa$, for attractive and
repulsive Yukawa potentials and for various values of $\beta$. Solid lines
show our analytic results for the semi-classical regime $\kappa>1$, which are
in good agreement with exact numerical results from solving the Schrödinger
equation (dots). Dotted lines correspond to purely classical scattering.
Dashed lines represent analytic approximations for scattering in the quantum
regime $\kappa<1$ using the Hulthén potential.

Fig. 4 further illustrates the $\kappa$ dependence of our results, as well as
the complementarity with other treatments for Yukawa scattering. As above, we
show the attractive (left) and repulsive (right) dimensionless cross sections
$\sigma_{\mathrm{T}}m_{\phi}^{2}$ (top) and $\sigma_{\mathrm{V}}m_{\phi}^{2}$
(bottom), now as functions of $\kappa$ and each for several values of $\beta$.
Our exact numerical results from solving the Schrödinger equation (dots) span
both the semi-classical ($\kappa>1$) and quantum ($\kappa<1$) regimes. For the
former, our analytic formulas (solid lines) approximately capture the true
$\kappa$ dependence of these cross sections for small values of $\beta$ down
to $\kappa\sim 1$, which is neglected for the classical scattering formula
(dotted lines). On the other hand, the quantum regime is typically dominated
by $S$-wave scattering which can be approximated using the Hulthén potential
(dashed lines), including the effect of resonances and anti-resonances Tulin
_et al._ (2013b).555In this approximation, one replaces the Yukawa potential
by the Hulthén potential $U(r)=\pm\frac{\alpha_{\chi}\delta e^{-\delta
r}}{1-e^{-\delta r}}$, which shares a similar Coulomb-like behavior $\sim 1/r$
at small $r$ and is screened as $e^{-\delta r}$ at large $r$, with screening
parameter $\delta\sim\mathcal{O}(m_{\phi})$ Cassel (2010). The $S$-wave
Schrödinger equation can be solved exactly in terms of hypergeometric
functions, which yields the phase shift
$\delta_{0}=\arg\left(\frac{i\Gamma(\lambda_{+}+\lambda_{-}-2)}{\Gamma(\lambda_{+})\Gamma(\lambda_{-})}\right)\,,\qquad\lambda_{\pm}=1+\frac{i\kappa}{n}\pm\frac{\kappa}{n}\times\left\\{\begin{array}[]{cc}i\sqrt{2\beta
n+1}&{\rm repulsive}\\\ \sqrt{2\beta n-1}&{\rm
attractive}\end{array}\right.\,,$ where $\Gamma$ is the usual gamma function
and $n=\delta/m_{\phi}\approx 1.6$ is a numerical constant Tulin  _et al._
(2013b). Across the two regimes, our new semi-classical formulas combined with
the Hulthén potential approximation provide nearly complete analytic coverage
of the entire parameter space for Yukawa scattering, with the exception of the
region $\kappa\sim 1$ where both approximations break down. In practice, a
good approximation for $\kappa\approx 1$ can be obtained by simply
interpolating between the two different regimes.

Figure 5: Cross sections for distinguishable vs. identical particles for an
attractive Yukawa potential, with $\kappa=5$ (left) and $\kappa=20$ (right).
Distinguishable cross sections $\sigma_{\mathrm{T}}$ (short-dashed) and
$\sigma_{\mathrm{V}}$ (long-dashed) are indicated in gray, while colored
curves (solid) correspond to identical scattering, depending on the DM
particle spin $s=0,\tfrac{1}{2},1$. Continuous lines are our analytic results
and dots are numerical results from solving the Schrödinger equation.

Finally, we consider in Fig. 5 the scattering of identical particles. To
highlight the differences between the various cases, we focus on $\beta<1$ and
plot the cross sections multiplied by $1/\beta^{2}$, which approaches a
constant for $\beta\to 0$. The largest differences are found for small
$\kappa$ (left panel), where only a small number of phase shifts contribute to
the cross section. Since
$\sigma_{\mathrm{V}}^{\text{even}}>\sigma_{\mathrm{V}}^{\text{odd}}$ the cross
section is largest for $\sigma_{\mathrm{V}}^{\text{scalar}}$ (where only the
even part contributes) and smallest for $\sigma_{\mathrm{V}}^{\text{fermion}}$
(where the odd part dominates). For $\kappa=5$, using the momentum transfer
cross section instead of the appropriate viscosity cross section for identical
particles can lead to a difference as large as a factor of 2.

###  IV.2 Velocity averaging

Let us now apply our results above to the description of DM self-scattering in
astrophysical systems. For this purpose we need to account for the
distribution of DM velocities in such systems. Here we assume that the DM
velocity distribution is approximately isothermal, i.e. given by a Maxwell-
Boltzmann (MB) distribution with one-dimensional velocity dispersion $v_{0}$
independent of radius. The relative velocity $v_{\text{rel}}$ between two DM
particles then follows a MB distribution with velocity dispersion
$\sqrt{2}v_{0}$, such that the typical relative velocity is

$\langle v_{\text{rel}}\rangle=\frac{4}{\sqrt{\pi}}v_{0}\;,$ (90)

where the brackets denote the expectation value with respect to the MB
distribution.

The expected rate $\Gamma$ of DM self-scattering is then proportional to
$\langle\sigma v_{\text{rel}}\rangle$. Here we are interested not in the rate
of scattering, but in the rate at which DM self-interactions transfer momentum
or energy. In the literature, this is commonly done by simply replacing
$\sigma$ by either $\sigma_{\mathrm{T}}$ or $\sigma_{\mathrm{V}}$ and
calculating $\langle\sigma_{\mathrm{T,V}}v_{\text{rel}}\rangle$. However, as
shown in Appendix C, the momentum and energy transfer rates are in fact
proportional to $\langle\sigma_{\mathrm{T}}v_{\text{rel}}^{2}\rangle$ and
$\langle\sigma_{\mathrm{V}}v_{\text{rel}}^{3}\rangle$, respectively. By
defining

$\displaystyle\overline{\sigma_{\mathrm{T}}}=\frac{\langle\sigma_{\mathrm{T}}v_{\text{rel}}^{2}\rangle}{16\sqrt{2}v_{0}^{2}/\pi}\,,\qquad\overline{\sigma_{\mathrm{V}}}=\frac{\langle\sigma_{\mathrm{V}}v_{\text{rel}}^{3}\rangle}{24/\sqrt{\pi}v_{0}^{3}}\;,$
(91)

we can write these rates as

$\displaystyle\Gamma_{p}\equiv\frac{\langle\dot{p}\rangle}{\langle
p\rangle}=\frac{\rho}{m_{\chi}}\,\overline{\sigma_{\mathrm{T}}}\,\langle
v_{\text{rel}}\rangle\,,\qquad\Gamma_{E}\equiv\frac{\langle\dot{E_{\perp}}\rangle}{\langle
E\rangle}=\frac{\rho}{m_{\chi}}\,\overline{\sigma_{\mathrm{V}}}\,\langle
v_{\text{rel}}\rangle\;.$ (92)

These rates govern the evolution of DM halos under the influence of DM self-
interactions. For example, we expect core formation to occur whenever
$\Gamma_{E}\,t_{0}>1$, where $t_{0}$ is the age of the system Kaplinghat  _et
al._ (2016); Sokolenko  _et al._ (2018); Robertson  _et al._ (2020).

The rate of momentum and energy transfer place more weight on collisions with
larger velocity than the scattering rate, while the effect of collisions with
smaller velocity is suppressed. This implies in particular that a finite
result is obtained for a cross section that scales proportional to
$v_{\text{rel}}^{-4}$, which is not the case for the naive average
$\langle\sigma_{\mathrm{T,V}}v_{\text{rel}}\rangle$. For a velocity-
independent cross section, on the other hand, one finds
$\overline{\sigma_{\mathrm{T}}}\approx 0.83\sigma_{\mathrm{T}}$ and
$\overline{\sigma_{\mathrm{V}}}\approx 1.34\sigma_{\mathrm{V}}$.

Figure 6: Averaged momentum transfer cross section (top) and viscosity cross
section (bottom) for the attractive (left) and repulsive (right) Yukawa
potential following different averaging prescriptions. Solid lines correspond
to the averaging proposed in this work, while dashed and dotted lines indicate
alternative approaches as explained in the text. The different colors
illustrate the dependence on $\kappa_{0}$.

As shown in Appendix C, in the semi-classical regime
$m_{\phi}^{2}\overline{\sigma_{\mathrm{T}}}$ and
$m_{\phi}^{2}\overline{\sigma_{\mathrm{V}}}$ are functions of only $\beta_{0}$
and $\kappa_{0}$, which are defined as the values of $\beta$ and $\kappa$
obtained for $v=v_{0}$. We show these functions in Fig. 6 and compare to
existing approaches where $\overline{\sigma_{\mathrm{T}}}$ is obtained either
by calculating $\langle\sigma_{\mathrm{T}}v_{\text{rel}}\rangle/\langle
v_{\text{rel}}\rangle$ or simply by evaluating $\sigma_{\mathrm{T}}$ at
$v=\langle v_{\text{rel}}\rangle$. We point out that the wiggles for
intermediate $\beta$ disappear when the averaging is performed and that our
analytical expressions give very similar results to the full numerical
expressions.

###  IV.3 Phenomenological implications

In practice we are interested in astrophysical systems ranging from dwarf
galaxies with $v_{0}\sim 20\,\mathrm{km\,s^{-1}}$ to galaxy clusters with
$v_{0}\sim 1000\,\mathrm{km\,s^{-1}}$. For all of these systems to fall into
the semi-classical regime, we require
$\kappa_{\text{min}}\equiv\kappa(v_{\text{min}})>1$, which implies
$m_{\chi}v_{\text{min}}>2m_{\phi}$ and therefore

$\frac{m_{\phi}}{m_{\chi}}\lesssim 10^{-4}\;.$ (93)

In other words, the semi-classical regime requires a large hierarchy between
the mediator mass and the DM mass.

As we have seen above, for $\beta<1$ the momentum transfer and viscosity cross
sections exhibit a strong velocity dependence:
$\sigma_{\mathrm{T,V}}\propto\beta^{2}\propto v^{-4}$. Given typical upper
bounds at dwarf galaxy scales of $\sigma_{\mathrm{T}}/m_{\chi}\lesssim
50\,\mathrm{cm^{2}\,g^{-1}}$ we then predict unobservably small effects from
DM self-interactions in larger astrophysical systems like groups or clusters.
The much more interesting case is therefore that $\beta>1$ in the
astrophysically interesting range, such that the velocity dependence is much
weaker and $\sigma_{\mathrm{T,V}}/m_{\chi}$ only changes by one or two orders
of magnitude between dwarf and cluster scales.

To get a rough sense of scale we note that for $\beta\gg 1$ we can
approximately write

$m_{\phi}^{2}\,\sigma_{\mathrm{T}}\approx c\,,$ (94)

where $c$ is a number of the order of $100$ that depends only logarithmically
on $\beta$. It then follows that in order to obtain a momentum transfer cross
section in the observationally interesting range
($\sigma_{\mathrm{T}}/m_{\chi}\sim 1\,\mathrm{cm^{2}\,g^{-1}}$) one must
require $m_{\chi}m_{\phi}^{2}\sim 0.02\,\mathrm{GeV^{3}}$. Substituting this
expression into Eq. (93), we obtain $m_{\phi}\lesssim 10\,\mathrm{MeV}$. We
conclude that rather light mediators are necessary in order to obtain
phenomenologically interesting cross sections in the semi-classical
regime.666Note that such light mediators cannot be in thermal equilibrium with
the Standard Model, because they would spoil the agreement between the
observed element abundances and predictions from primordial nucleosynthesis.
These constraints are evaded if the dark sector is (somewhat) colder than the
Standard Model. In such a setup, the DM abundance can be set, for example, via
the freeze-in mechanism Bernal  _et al._ (2016).

Table 1: Velocity averaged self-interaction cross sections for $m_{\chi}=190\,\mathrm{GeV}$, $m_{\phi}=3\,\mathrm{MeV}$, $\alpha_{\chi}=0.5$ at different astrophysical scales. System |  $\langle v\rangle$ |  $\beta_{0}$ |  $\kappa_{0}$ |  $\overline{\sigma_{\mathrm{T}}^{\text{att.}}}/m_{\chi}$ |  $\overline{\sigma_{\mathrm{T}}^{\text{rep.}}}/m_{\chi}$ |  $\overline{\sigma_{\mathrm{V}}^{\text{att.}}}/m_{\chi}$ |  $\overline{\sigma_{\mathrm{V}}^{\text{rep.}}}/m_{\chi}$
---|---|---|---|---|---|---|---
|  |  |  |  $[\mathrm{cm^{2}\,g^{-1}}]$ |  $[\mathrm{cm^{2}\,g^{-1}}]$ |  $[\mathrm{cm^{2}\,g^{-1}}]$ |  $[\mathrm{cm^{2}\,g^{-1}}]$
Dwarf galaxy |  50 |  2890 |  2.34 |  10.9 |  9.0 |  11.7 |  13.6
Galaxy |  250 |  116 |  11.7 |  4.3 |  2.6 |  3.5 |  3.8
Galaxy group |  1150 |  5.46 |  53.8 |  0.66 |  0.36 |  0.64 |  0.54
Galaxy cluster |  1900 |  2.00 |  88.9 |  0.20 |  0.14 |  0.23 |  0.19
Figure 7: Averaged momentum transfer cross section (blue) and viscosity cross
section (green) for $m_{\chi}=190\,\mathrm{GeV}$, $m_{\phi}=3\,\mathrm{MeV}$,
$\alpha_{\chi}=0.5$. For comparison we show $\langle\sigma v\rangle/m_{\chi}$
inferred from the observation of core sizes from Ref. Sagunski  _et al._
(2020) and the shaded region indicates the range of cross sections relevant
for the core-cusp problem.

As an example, we provide in Tab. 1 and Fig. 7 the cross sections
$\sigma_{\mathrm{T}}$ and $\sigma_{\mathrm{V}}$ as functions of $\langle
v_{\text{rel}}\rangle$ for the benchmark point $m_{\chi}=190\,\mathrm{GeV}$,
$m_{\phi}=3\,\mathrm{MeV}$, $\alpha_{\chi}=0.5$. This benchmark point has been
chosen in such a way that the self-interaction cross sections at the scales of
groups and clusters agree with what has recently been inferred from the
observation of core sizes Sagunski  _et al._ (2020).

To more comprehensively explore the allowed parameter space, we show in Fig. 8
how the predicted self-interaction cross sections at dwarf and cluster scales
depend on the underlying parameters. For concreteness, we consider the
viscosity cross section for a repulsive potential, but very similar plots are
obtained in the other cases. In the left panel, we fix
$m_{\phi}=3\,\mathrm{MeV}$ and vary $\alpha_{\chi}$ and $m_{\chi}$. We find
that on dwarf scales (where $\beta_{0}\gg 1$) $\sigma_{\mathrm{V}}$ depends
only very weakly on $\alpha_{\chi}$ and $m_{\chi}$, so that the dominant
dependence stems from the overall factor $m_{\chi}^{-1}$. On cluster scales
(where $\beta_{0}\approx 1$) the dependence of $\sigma_{\mathrm{V}}$ on the
model parameters is more pronounced, such that it is possible to vary the
constraints on the two different scales independently. The grey vertical line
corresponds to $\kappa_{0}=1$ for $v_{0}=50\,\mathrm{km\,s^{-1}}$ and
indicates the transition between the semi-classical regime (to the right) and
the quantum regime (to the left). In the quantum regime, we calculate the
self-interaction cross section using the Hulthén approximation, interpolating
linearly between $\kappa=0.4$ and $\kappa=1$ to ensure a smooth transition.

Figure 8: Self-interaction cross section at cluster scales (orange contours)
and at dwarf scales (blue contours) as a function of the DM mass $m_{\chi}$
and the coupling $\alpha_{\chi}$ (left) and as a function of the DM mass and
the mediator mass $m_{\phi}$ (right). Shaded regions represent approximate
bounds from astrophysical observations. The gray line indicates the transition
between the quantum regime (left) and the semi-classical regime (right) for
dwarf galaxies. At cluster scales self-interactions are always in the semi-
classical regime. For concreteness, we consider the viscosity cross section
for a repulsive potential, but very similar plots are obtained in the other
cases. The red dot indicates the benchmark point considered in Fig. 7 and Tab.
1.

In the right panel of Fig. 8 we fix $\alpha_{\chi}=0.5$ and vary $m_{\phi}$.
As expected from Eq. (94), we find that the cross sections depend strongly on
$m_{\phi}$ and it is not possible to obtain large self-interaction cross
sections for $m_{\phi}>10\,\mathrm{MeV}$ within the semi-classical regime (to
the bottom right of the diagonal grey line). Nevertheless, for smaller
mediator masses it is possible to obtain phenomenologically interesting cross
sections on both dwarf and cluster scales. The benchmark point considered as
an example in Fig. 7 and Tab. 1 is indicated by the red dot in both panels.

##  V Conclusions

Self-interacting DM has long been advocated to alleviate tensions between
astrophysical observations and $N$-body simulations of collisionless cold DM.
For many models, self-interactions are represented by a Yukawa potential and
fall within the semi-classical regime, where the de Broglie wavelengths of DM
particles are small compared to the characteristic length scales involved in
the interaction. However, exploring the parameter space of this regime is
numerically intensive using the standard partial wave calculation, which
requires summing over a large number of $\ell$ modes. Many previous studies
have relied on analytic formulas for the momentum transfer cross section
derived for classical scattering of distinguishable particles. However, this
description necessarily breaks down close to the boundary of the semi-
classical regime and also for the scattering of identical particles.

In this work, we provided a fully quantum mechanical description of DM self-
interactions in the semi-classical approximation, deriving analytic
expressions for the associated cross sections. We demonstrated that the
momentum transfer cross section $\sigma_{\mathrm{T}}$ receives relevant
quantum corrections for weak potentials, and derived improved asymptotic
expressions for strong potentials. We also derived, for the first time,
analytic expressions for the viscosity cross section, $\sigma_{\mathrm{V}}$,
which captures the rate of energy transfer and turns out to be a more useful
quantity than $\sigma_{\mathrm{T}}$ in the case of identical particles. We
established the different formulas for $\sigma_{\mathrm{V}}$ that arise when
considering scattering of identical DM particles with spin $s\leq 1$. These
formulas are summarized in Sec. III.4.

To validate our results, we compared them with the cross section obtained by
numerically solving the Schrödinger equation and found excellent agreement
(see Figs. 3 and 4). We pointed out that with the addition of our results,
almost the entire parameter space of self-interactions in a Yukawa potential
can now be described analytically. We have implemented our results in the new
code CLASSICS, available at https://github.com/kahlhoefer/CLASSICS.

As an application of our results, we considered the case of DM self-scattering
in astrophysical systems. We argued that the velocity-averaged cross section,
$\langle\sigma_{\mathrm{T,V}}v\rangle$ currently used to derive bounds on
$\sigma_{\mathrm{T,V}}$ is imprecise. We showed that the rate of momentum
(energy) transfer is in fact proportional to
$\langle\sigma_{\mathrm{T}}v^{2}\rangle$
($\langle\sigma_{\mathrm{V}}v^{3}\rangle$), which can lead to sizeable
differences for velocity-dependent scattering.

Using this updated treatment for velocity averaging and our analytic results
for the cross sections, we demonstrated that for sufficiently small mediator
masses phenomenologically interesting cross sections at dwarf and cluster
scales can be accommodated within the semi-classical regime. It will be
exciting to implement these cross sections into numerical simulations in order
to study their observational effects in more detail.

An interesting direction for future research is to apply our methodology to
models that generate interactions different from the Yukawa potential
considered here. Our results suggest that even in situations where it is
impossible to calculate the phase shifts analytically, solving the semi-
classical integrals numerically may yield an accurate approximation to the
full solution of the Schrödinger equation. This approach opens up a new avenue
for classifying models of self-interacting DM, which is complementary to the
recently proposed effective-range formalism applicable to $S$-wave scattering
Chu  _et al._ (2020).

## Acknowledgements

We thank Camilo Garcia-Cely, Sophia Gad-Nasr, Andrew Robertson and Kai
Schmidt-Hoberg for discussions. This work has been funded by the DFG Emmy
Noether Grant No. KA 4662/1-1, the DFG grant CRC-TR 211 “Strong-interaction
matter under extreme conditions” and the Natural Sciences and Engineering
Research Council of Canada.

##  Appendix A Review of existing results for the classical limit

So far, analytic formulas exist only for the momentum transfer cross section
$\sigma_{\mathrm{T}}$ and in the limit $\kappa\to\infty$. In the weak coupling
regime $(\beta\ll 1)$, the theory of Coulomb scattering becomes applicable,
and $\sigma_{\mathrm{T}}$ is given by the Coulomb scattering cross section
Fortov  _et al._ (2005)

$\sigma_{\mathrm{T}}\simeq\frac{2\pi}{m_{\phi}^{2}}\beta^{2}\log(1+\beta^{-2})$
(95)

for both attractive and repulsive potentials. In the strong coupling regime
$(\beta\gg 1)$, the scattering resembles that of a hard sphere potential and
the momentum transfer cross section is a geometric cross section of the form
$\sigma_{\mathrm{T}}\sim\pi R^{2}$. For an attractive potential it is
approximately given by Khrapak  _et al._ (2003); Fortov  _et al._ (2005)

$\sigma_{\mathrm{T}}\simeq\frac{0.81\pi}{m_{\phi}^{2}}(1+\log\beta-(2\log\beta)^{-1})^{2}\;,$
(96)

whereas for a repulsive potential one finds Khrapak  _et al._ (2004)

$\sigma_{\mathrm{T}}\simeq\frac{\pi}{m_{\phi}^{2}}(\log 2\beta-\log\log
2\beta)^{2}\,.$ (97)

In the intermediate regime, only empirical fit formulas for
$\sigma_{\mathrm{T}}$ exist. Following Ref. Cyr-Racine  _et al._ (2016), we
can parametrize $\sigma_{\mathrm{T}}$ for an attractive Yukawa potential as

$\sigma_{\mathrm{T}}=\begin{cases}\frac{2\pi}{m_{\phi}^{2}}\beta^{2}\log(1+\beta^{-2})&\beta\lesssim
10^{-2}\,,\\\
\frac{7\pi}{m_{\phi}^{2}}\frac{\beta^{1.8}+280(\beta/10)^{10.3}}{1+1.4\beta+0.006\beta^{4}+160(\beta/10)^{10}}&10^{-2}\lesssim\beta\lesssim
10^{2}\,,\\\
\frac{0.81\pi}{m_{\phi}^{2}}(1+\log\beta-(2\log\beta)^{-1})^{2}&\beta\gtrsim
10^{2}\;.\end{cases}$ (98)

For a repulsive Yukawa potential, the corresponding fitting formula is Cyr-
Racine  _et al._ (2016)

$\sigma_{\mathrm{T}}=\begin{cases}\frac{2\pi}{m_{\phi}^{2}}\beta^{2}\log(1+\beta^{-2})&\beta\lesssim
10^{-2}\,,\\\
\frac{8\pi}{m_{\phi}^{2}}\frac{\beta^{1.8}}{1+5\beta^{0.9}+0.85\beta^{1.6}}&10^{-2}\lesssim\beta\lesssim
10^{4}\,,\\\ \frac{\pi}{m_{\phi}^{2}}(\log 2\beta-\log\log
2\beta)^{2}&\beta\gtrsim 10^{4}\;.\end{cases}$ (99)

To our knowledge, no closed expressions of this form exist for
$\sigma_{\mathrm{V}}$.

##  Appendix B Detailed calculations for an attractive potential and large
$\beta$

###  B.1 Distance of closest approach

As mentioned in Sec. III.2.1, the equation

$\displaystyle
U_{\mathrm{eff}}(R)\equiv\left(\frac{\ell+1/2}{\kappa}\right)^{2}\frac{1}{R^{2}}-2\beta\frac{e^{-R}}{R}=1$
(100)

can have multiple roots, the largest of which determines the distance of
closest approach, $R_{0}$. The existence of multiple roots implies the
presence of a local maximum at $R>0$, for which
$U_{\mathrm{eff}}^{\prime}(R)=0$ or

$e^{-R}=\frac{1}{\beta}\left(\frac{\ell+1/2}{\kappa}\right)^{2}\frac{1}{R(R+1)}\,.$
(101)

For a given $\beta$ we can find the smallest $\ell$ for which multiple roots
exist, by simultaneously solving Eqs. (100) and (101), which yields

$\left(\frac{\ell+1/2}{\kappa}\right)=R_{0}\sqrt{\frac{R_{0}+1}{R_{0}-1}}\,.$
(102)

For $R_{0}\gtrsim 1$, one can expand the equation above to get

$\displaystyle\frac{\ell+1/2}{\kappa}=1+R_{0}+\frac{1}{2R_{0}}\;,$ (103)

which implies

$\displaystyle
R_{0}\approx\frac{1}{2}\left(-1+\frac{\ell+1/2}{\kappa}+\sqrt{-1-\frac{2\ell+1}{\kappa}+\left(\frac{\ell+1/2}{\kappa}\right)^{2}}\right)\,.$
(104)

Substituting this value of $R_{0}$ back in Eq. (101), we get

$\ell_{\mathrm{max}}=\kappa\,\left(1+\log\beta-\frac{1}{2\,\log\beta}\right)-\frac{1}{2}\;.$
(105)

###  B.2 Derivation of phase shifts

We start from the integral

$\displaystyle I^{\ell}$
$\displaystyle=\kappa\int_{R_{0}(\ell)}^{\infty}\mathrm{d}{R}\sqrt{1+\frac{2\beta}{R}e^{-R}-\frac{(\ell+\tfrac{1}{2})^{2}}{\kappa^{2}R^{2}}}$
(106)
$\displaystyle=\kappa\int_{R_{0}(\ell)}^{R_{\text{cut}}}\mathrm{d}{R}\sqrt{1+\frac{2\beta}{R}e^{-R}-\frac{(\ell+\tfrac{1}{2})^{2}}{\kappa^{2}R^{2}}}+\kappa\int_{R_{\text{cut}}}^{\infty}\mathrm{d}{R}\sqrt{1+\frac{2\beta}{R}e^{-R}-\frac{(\ell+\tfrac{1}{2})^{2}}{\kappa^{2}R^{2}}}$
(107) $\displaystyle\equiv I_{1}^{\ell}+I_{2}^{\ell}\;,$ (108)

where $R_{0}=(\ell+1/2)^{2}/(2\beta\kappa^{2})$ and $R_{\text{cut}}$ is chosen
such that $R_{0}\ll R_{\text{cut}}\ll 1$.

In the first integral we can neglect the first term in the square root and
approximate $e^{-R}\approx 1$, leading to

$\displaystyle I^{\ell}_{1}$
$\displaystyle=\kappa\int_{(\ell+\tfrac{1}{2})^{2}/(2\beta\kappa^{2})}^{R_{\text{cut}}}\mathrm{d}{R}\sqrt{\frac{2\beta}{R}-\frac{(\ell+\tfrac{1}{2})^{2}}{\kappa^{2}R^{2}}}$
(109)
$\displaystyle=(\ell+\tfrac{1}{2})\int_{1}^{S_{\text{cut}}}\mathrm{d}{S}\frac{1}{S}\sqrt{S-1}$
(110)
$\displaystyle=(2\ell+1)\left(\sqrt{S_{\text{cut}}-1}-\text{asec}(\sqrt{S_{\text{cut}}})\right)\;,$
(111)

with
$S_{\text{cut}}=R_{\text{cut}}(2\beta\kappa^{2})/(\ell+\tfrac{1}{2})^{2}$. For
$S_{\text{cut}}\gg 1$ we then find

$I^{\ell}_{1}-I^{\ell-1}_{1}=-\pi+\sqrt{\frac{2}{\beta
R_{\text{cut}}}}\frac{\ell}{\kappa}\;.$ (112)

In the second integral, we have $(\ell+\tfrac{1}{2})^{2}\ll\kappa^{2}R^{2}$,
such that we can approximate

$I^{\ell}_{2}\approx\kappa\int_{R_{\text{cut}}}^{\infty}\mathrm{d}{R}\sqrt{1+\frac{2\beta}{R}e^{-R}}-\frac{(\ell+1/2)^{2}}{2\kappa^{2}}\frac{1}{\sqrt{1+\tfrac{2\beta}{R}e^{-R}}R^{2}}\;.$
(113)

We then find

$I^{\ell}_{2}-I^{\ell-1}_{2}\approx-\frac{\ell}{\kappa}\int_{R_{\text{cut}}}^{\infty}\mathrm{d}{R}\frac{1}{\sqrt{1+2\beta
e^{-R}/R}R^{2}}\;.$ (114)

To solve this integral, we again split it into two parts, separated at
$R=R_{\text{t}}$ defined by $2\beta e^{-R_{\text{t}}}/R_{\text{t}}=1$, which
can equivalently be written as $R_{\text{t}}=W(2\beta)$:

$\displaystyle I^{\ell}_{2}-I^{\ell-1}_{2}$
$\displaystyle\approx-\frac{\ell}{\kappa}\int_{R_{\text{cut}}}^{R_{\text{t}}}\mathrm{d}{R}\frac{1}{\sqrt{1+\tfrac{2\beta}{R}e^{-R}}R^{2}}-\frac{\ell}{\kappa}\int_{R_{\text{t}}}^{\infty}\mathrm{d}{R}\frac{1}{\sqrt{1+\tfrac{2\beta}{R}e^{-R}}R^{2}}$
(115)
$\displaystyle\approx-\frac{\ell}{\kappa}\int_{R_{\text{cut}}}^{R_{\text{t}}}\mathrm{d}{R}\frac{1}{\sqrt{2\beta
e^{-R}}R^{3/2}}-\frac{\ell}{\kappa}\int_{R_{\text{t}}}^{\infty}\mathrm{d}{R}\frac{1}{R^{2}}$
(116)
$\displaystyle\approx-\frac{\ell}{\kappa}\left[\sqrt{\frac{2}{R_{\text{cut}}\beta}}-\sqrt{\frac{2e^{R_{\text{t}}}}{R_{\text{t}}\beta}}+\sqrt{\frac{\pi}{\beta}}\text{erfi}(\sqrt{R_{\text{t}}/2})+\frac{1}{R_{\text{t}}}\right]\;,$
(117)

where $\text{erfi}(x)$ denotes the imaginary error function. For
$R_{\mathrm{t}}\gg 1$ we can approximate

$\sqrt{\pi}\text{erfi}\left(\sqrt{R_{\mathrm{t}}/2}\right)\approx\sqrt{\frac{2}{R_{\mathrm{t}}}}e^{R_{\mathrm{t}}/2}\left(1+\frac{1}{R_{\mathrm{t}}}\right)\;.$
(118)

Now we make use of the defining property of $R_{\mathrm{t}}$ to substitute
$e^{R_{\mathrm{t}}}=2\beta/R_{\mathrm{t}}$ and obtain

$I^{\ell}_{2}-I^{\ell-1}_{2}\approx-\frac{\ell}{\kappa}\left[\sqrt{\frac{2}{R_{\text{cut}}\beta}}+\frac{R_{\mathrm{t}}+2}{R_{\mathrm{t}}^{2}}\right]\;.$
(119)

As expected, the dependence on $R_{\text{cut}}$ drops out when combining
$I_{1}$ and $I_{2}$, giving

$I^{\ell}-I^{\ell-1}=-\pi-\frac{\ell}{\kappa}\frac{W(2\beta)+2}{W(2\beta)^{2}}\equiv-\pi-\frac{\ell}{\kappa}\gamma(\beta)\;.$
(120)

Adding $\pi/2$ following Eq. (22) yields Eq. (56).

##  Appendix C Velocity averaging

In this appendix we discuss the role of the momentum transfer cross section
and the viscosity cross section for describing the effects of DM self-
interactions and derive the appropriate expressions for evaluating these cross
sections for a given distribution of DM velocities.

We first consider a DM particle moving with velocity $v$ through a background
density $\rho$ of DM particles at rest. The particle encounters a flux of DM
particles given by $\rho\,v/m_{\chi}$. The probability for scattering to occur
is given by $\sigma\rho\,v/m_{\chi}$. In the center-of mass (cms) frame (which
moves with velocity $v/2$) the distribution of scattering angles is given by
$\sigma^{-1}\mathrm{d}\sigma/\mathrm{d}\theta_{\text{cms}}$. For a given
scattering angle $\theta_{\text{cms}}$, the velocity transfer is given by
$\Delta\mathbf{v}=\mathbf{v}_{\text{cms}}-\mathbf{v}^{\prime}_{\text{cms}}$,
where $\mathbf{v}_{\text{cms}}$ and $\mathbf{v}^{\prime}_{\text{cms}}$ denote
the velocity of the incoming DM particle in the cms frame before and after the
collision, respectively. We decompose $\Delta\mathbf{v}$ into a component
parallel and a component perpendicular to $\mathbf{v}$: $\Delta
v_{\parallel}=v_{\text{cms}}(1-\cos\theta_{\text{cms}})$,
$\Delta\mathbf{v}_{\perp}=-v_{\text{cms}}(\sin\theta_{\text{cms}}\cos\phi,\sin\theta_{\text{cms}}\sin\phi)$,
where $\phi$ is the azimuthal angle. The expectation value of $\Delta
v_{\parallel}$ is then given by

$\langle\Delta
v_{\parallel}\rangle=\frac{v_{\text{cms}}}{\sigma}\int\frac{\mathrm{d}\sigma}{\mathrm{d}\theta_{\text{cms}}}(1-\cos\theta_{\text{cms}})\mathrm{d}\theta_{\text{cms}}=\frac{\sigma_{\mathrm{T}}v}{2\,\sigma}\;,$
(121)

where in the last step we have made use of the fact that $v_{\text{cms}}=v/2$.
Since $\phi$ is distributed uniformly, the expectation value of
$\Delta\mathbf{v}_{\perp}$ vanishes, but we can calculate the expectation
value of $\Delta v_{\perp}^{2}$:

$\langle\Delta
v_{\perp}^{2}\rangle=\frac{v_{\text{cms}}^{2}}{\sigma}\int\frac{\mathrm{d}\sigma}{\mathrm{d}\theta_{\text{cms}}}\sin^{2}\theta_{\text{cms}}\mathrm{d}\theta_{\text{cms}}=\frac{\sigma_{\mathrm{V}}v^{2}}{4\,\sigma}\;.$
(122)

We can now calculate the relative rate of change of the momentum $p$,

$\frac{\dot{p}}{p}=\frac{\rho}{m_{\chi}}\frac{v\sigma_{\mathrm{T}}}{2}\;,$
(123)

and of the energy in the transverse direction $E_{\perp}$,

$\frac{\dot{E_{\perp}}}{E}=\frac{\rho}{m_{\chi}}\frac{v\sigma_{\mathrm{V}}}{4}\;.$
(124)

Let us now consider the case that the background DM particles are not at rest
but have a velocity distribution $g(\mathbf{w})$. We assume that
$g(\mathbf{w})$ is isotropic and hence we can take $\mathbf{v}$ to point in
the $z$ direction. We begin by considering only particles with a specific
velocity $\mathbf{w}$. The rate at which the incoming particle encounters such
particles is given by $\rho v_{\text{rel}}/m_{\chi}$, where
$\mathbf{v}_{\text{rel}}=\mathbf{v}-\mathbf{w}$. Following the same argument
as above (and noting that now $v_{\text{cms}}=v_{\text{rel}}/2$), scattering
will on average lead to a velocity transfer of $\langle\Delta
v_{\parallel}\rangle=\frac{\sigma_{\mathrm{T}}}{\sigma}\frac{v_{\text{rel}}}{2}$,
pointing in the direction of $\mathbf{v}_{\text{rel}}$. The relative change in
momentum is therefore given by

$\frac{\dot{p}}{p}=\frac{\rho}{m_{\chi}}\frac{v_{\text{rel}}^{2}\sigma_{\mathrm{T}}(v_{\text{rel}})}{2v}\;,$
(125)

which differs from the result above by a factor of $v_{\text{rel}}/v$. This
additional factor is a direct consequence of the fact that the change in
velocity in a single collision cannot exceed $v_{\text{rel}}$. The smaller
$v_{\text{rel}}$, the more collisions are necessary to change the velocity of
the incoming particle by a relevant amount. For the case of the perpendicular
energy, one obtains

$\frac{\dot{E_{\perp}}}{E}=\frac{\rho}{m_{\chi}}\frac{v_{\text{rel}}^{3}\sigma_{\mathrm{V}}}{4v^{2}}\;.$
(126)

We now apply these results to the case where both scattering DM particles are
bound to the same DM halo. In this case both $v$ and $w$ approximately follow
a Maxwell-Boltzmann distribution with velocity dispersion $v_{0}$:

$f(v)=\sqrt{\frac{2}{\pi}}\frac{v^{2}e^{-v^{2}/(2v_{0}^{2})}}{v_{0}^{3}}\;.$
(127)

The typical momentum is then given by $\langle
p\rangle=2\sqrt{2/\pi}m_{\chi}v_{0}$ and the typical energy is $\langle
E\rangle=\tfrac{3}{2}m_{\chi}v_{0}^{2}$. The relative velocity
$v_{\text{rel}}$ follows a Maxwell-Boltzmann distribution with dispersion
parameter $\sqrt{2}v_{0}$. The expected relative change in momentum is then

$\Gamma_{p}\equiv\frac{\langle\dot{p}\rangle}{\langle
p\rangle}=\frac{\rho}{m_{\chi}}\frac{\langle\sigma_{\mathrm{T}}v_{\text{rel}}^{2}\rangle}{4\sqrt{2/\pi}v_{0}}=\frac{\rho}{m_{\chi}}\int\mathrm{d}v_{\mathrm{rel}}\sigma_{\mathrm{T}}(v_{\text{rel}})\frac{v_{\text{rel}}^{4}}{8\sqrt{2}v_{0}^{4}}e^{-v_{\text{rel}}^{2}/(4v_{0}^{2})}\;,$
(128)

while the expected relative change in energy in the transverse direction is

$\Gamma_{E}\equiv\frac{\langle\dot{E_{\perp}}\rangle}{\langle
E\rangle}=\frac{\rho}{m_{\chi}}\frac{\langle\sigma_{\mathrm{V}}v_{\text{rel}}^{3}\rangle}{6v_{0}^{2}}=\frac{\rho}{m_{\chi}}\int\mathrm{d}v_{\mathrm{rel}}\sigma_{\mathrm{V}}(v_{\text{rel}})\frac{v_{\text{rel}}^{5}}{12\sqrt{\pi}v_{0}^{5}}e^{-v_{\text{rel}}^{2}/(4v_{0}^{2})}\;.$
(129)

By defining

$\displaystyle\overline{\sigma_{\mathrm{T}}}=\frac{\langle\sigma_{\mathrm{T}}v_{\text{rel}}^{2}\rangle}{16\sqrt{2}v_{0}^{2}/\pi}\,,\qquad\overline{\sigma_{\mathrm{V}}}=\frac{\langle\sigma_{\mathrm{V}}v_{\text{rel}}^{3}\rangle}{24/\sqrt{\pi}v_{0}^{3}}\;,$
(130)

we can write these rates as

$\displaystyle\Gamma_{p}=\frac{\rho}{m_{\chi}}\overline{\sigma_{\mathrm{T}}}\,\langle
v_{\text{rel}}\rangle\,,\qquad\Gamma_{E}=\frac{\rho}{m_{\chi}}\overline{\sigma_{\mathrm{V}}}\,\langle
v_{\text{rel}}\rangle\;.$ (131)

Finally, let us consider the case in which the cross sections can be written
as

$\sigma_{\mathrm{T,V}}=\frac{\pi}{m_{\phi}^{2}}f_{\mathrm{T,V}}(\beta,\kappa)\;.$
(132)

The averaged cross sections are then given by

$\displaystyle\overline{\sigma_{\mathrm{T}}}$
$\displaystyle=\frac{\pi}{m_{\phi}^{2}}\int\mathrm{d}xe^{-x^{2}/4}\frac{x^{4}}{32\sqrt{2/\pi}}f_{\mathrm{T}}(\beta_{0}/x^{2},\kappa_{0}x)\,,$
$\displaystyle\overline{\sigma_{\mathrm{V}}}$
$\displaystyle=\frac{\pi}{m_{\phi}^{2}}\int\mathrm{d}xe^{-x^{2}/4}\frac{x^{5}}{48}f_{\mathrm{V}}(\beta_{0}/x^{2},\kappa_{0}x)\;,$
(133)

where $x=v/v_{0}$, and $\beta_{0}$ and $\kappa_{0}$ are the effective
parameters obtained when setting $v=v_{0}$. For $\kappa_{0}\gg 1$ the velocity
averaged cross section hence depends only on $\beta_{0}$ and (trivially) on
$m_{\phi}$.

## References

  * Spergel and Steinhardt (2000) D. N. Spergel and P. J. Steinhardt, Phys. Rev. Lett. 84, 3760 (2000), arXiv:astro-ph/9909386 [astro-ph] .
  * Tulin and Yu (2018) S. Tulin and H.-B. Yu, Phys. Rept. 730, 1 (2018), arXiv:1705.02358 [hep-ph] .
  * Bullock and Boylan-Kolchin (2017) J. S. Bullock and M. Boylan-Kolchin, Ann. Rev. Astron. Astrophys. 55, 343 (2017), arXiv:1707.04256 [astro-ph.CO] .
  * Salucci (2019) P. Salucci, Astron. Astrophys. Rev. 27, 2 (2019), arXiv:1811.08843 [astro-ph.GA] .
  * Essig  _et al._ (2013) R. Essig _et al._ , in  _Community Summer Study 2013: Snowmass on the Mississippi_ (2013) arXiv:1311.0029 [hep-ph] .
  * Alexander  _et al._ (2016) J. Alexander _et al._ (2016) arXiv:1608.08632 [hep-ph] .
  * Beacham  _et al._ (2020) J. Beacham _et al._ , J. Phys. G 47, 010501 (2020), arXiv:1901.09966 [hep-ex] .
  * Buckley and Fox (2010) M. R. Buckley and P. J. Fox, Phys. Rev. D 81, 083522 (2010), arXiv:0911.3898 [hep-ph] .
  * Tulin  _et al._ (2013a) S. Tulin, H.-B. Yu, and K. M. Zurek, Phys. Rev. Lett. 110, 111301 (2013a), arXiv:1210.0900 [hep-ph] .
  * Tulin  _et al._ (2013b) S. Tulin, H.-B. Yu, and K. M. Zurek, Phys. Rev. D87, 115007 (2013b), arXiv:1302.3898 [hep-ph] .
  * Feng  _et al._ (2010a) J. L. Feng, M. Kaplinghat, and H.-B. Yu, Phys. Rev. Lett. 104, 151301 (2010a), arXiv:0911.0422 [hep-ph] .
  * Feng  _et al._ (2010b) J. L. Feng, M. Kaplinghat, and H.-B. Yu, Phys. Rev. D 82, 083525 (2010b), arXiv:1005.4678 [hep-ph] .
  * Loeb and Weiner (2011) A. Loeb and N. Weiner, Phys. Rev. Lett. 106, 171302 (2011), arXiv:1011.6374 [astro-ph.CO] .
  * Boddy  _et al._ (2014) K. K. Boddy, J. L. Feng, M. Kaplinghat, and T. M. P. Tait, Phys. Rev. D 89, 115017 (2014), arXiv:1402.3629 [hep-ph] .
  * Chu  _et al._ (2020) X. Chu, C. Garcia-Cely, and H. Murayama, JCAP 06, 043 (2020), arXiv:1908.06067 [hep-ph] .
  * Khrapak  _et al._ (2003) S. A. Khrapak, A. V. Ivlev, G. E. Morfill, and S. K. Zhdanov, Phys. Rev. Lett. 90, 225002 (2003).
  * Khrapak  _et al._ (2004) S. A. Khrapak, A. V. Ivlev, and G. E. Morfill, Phys. Rev. E 70, 056405 (2004).
  * Mohapatra  _et al._ (2002) R. Mohapatra, S. Nussinov, and V. Teplitz, Phys. Rev. D 66, 063002 (2002), arXiv:hep-ph/0111381 .
  * Vincenti and Kruger (1965) W. G. Vincenti and C. H. Kruger,  _Introduction to physical gas dynamics_ (Krieger, 1965).
  * Kahlhoefer  _et al._ (2014) F. Kahlhoefer, K. Schmidt-Hoberg, M. T. Frandsen, and S. Sarkar, Mon. Not. Roy. Astron. Soc. 437, 2865 (2014), arXiv:1308.3419 [astro-ph.CO] .
  * Gnedin and Ostriker (2001) O. Y. Gnedin and J. P. Ostriker, Astrophys. J. 561, 61 (2001), arXiv:astro-ph/0010436 .
  * Balberg  _et al._ (2002) S. Balberg, S. L. Shapiro, and S. Inagaki, Astrophys. J. 568, 475 (2002), arXiv:astro-ph/0110561 .
  * Heer (1972) C. V. Heer,  _Statistical mechanics, kinetic theory, and stochastic processes_ (Academic Press Inc, 1972).
  * Berry and Mount (1972) M. V. Berry and K. Mount, Rept. Prog. Phys. 35, 315 (1972).
  * Kramers (1926) H. Kramers, Z. Phys. 39, 828 (1926).
  * Langer (1937) R. E. Langer, Phys. Rev. 51, 669 (1937).
  * Cassel (2010) S. Cassel, J. Phys. G 37, 105009 (2010), arXiv:0903.5307 [hep-ph] .
  * Kaplinghat  _et al._ (2016) M. Kaplinghat, S. Tulin, and H.-B. Yu, Phys. Rev. Lett. 116, 041302 (2016), arXiv:1508.03339 [astro-ph.CO] .
  * Sokolenko  _et al._ (2018) A. Sokolenko, K. Bondarenko, T. Brinckmann, J. Zavala, M. Vogelsberger, T. Bringmann, and A. Boyarsky, JCAP 12, 038 (2018), arXiv:1806.11539 [astro-ph.CO] .
  * Robertson  _et al._ (2020) A. Robertson, R. Massey, V. Eke, J. Schaye, and T. Theuns, (2020), arXiv:2009.07844 [astro-ph.CO] .
  * Bernal  _et al._ (2016) N. Bernal, X. Chu, C. Garcia-Cely, T. Hambye, and B. Zaldivar, JCAP 03, 018 (2016), arXiv:1510.08063 [hep-ph] .
  * Sagunski  _et al._ (2020) L. Sagunski, S. Gad-Nasr, B. Colquhoun, A. Robertson, and S. Tulin, (2020), arXiv:2006.12515 [astro-ph.CO] .
  * Fortov  _et al._ (2005) V. Fortov, A. Ivlev, S. Khrapak, A. Khrapak, and G. Morfill, Physics Reports 421, 1 (2005).
  * Cyr-Racine  _et al._ (2016) F.-Y. Cyr-Racine, K. Sigurdson, J. Zavala, T. Bringmann, M. Vogelsberger, and C. Pfrommer, Phys. Rev. D93, 123527 (2016), arXiv:1512.05344 [astro-ph.CO] .