the semi-annihilation and the pair-annihilation processes, the $\phi$ particle
is in thermal equilibrium with the SM plasma with a vanishing chemical
potential.

####  3.3.1 Estimate of relic abundance

Before proceeding to the discussion of the numerical solutions for the coupled
Boltzmann equations, we first provide a rough estimate of the relation between
the DM relic density ($\Omega_{\chi}$), its mass ($m_{\chi}$), and the CP-
violation parameter $\epsilon$. For this estimate, we shall assume that there
is complete asymmetry between the dark matter and anti-matter states in the
current Universe, i.e., either only the particle or the anti-particle states
survive. It then follows that the present DM relic density $\Omega_{\rm
DM}=m_{\chi}s_{0}\left(Y_{\chi}^{\infty}+Y_{\chi^{\dagger}}^{\infty}\right)/\rho_{c}=m_{\chi}s_{0}Y_{\chi}^{\infty}/\rho_{c}$,
in the scenario when only the $\chi$ particles survive today, where $s_{0}$
and $\rho_{c}$ are the present entropy density and the critical density
respectively.

After the freeze-out of the semi-annihilation process, in the absence of
subsequent pair-annihilations, both the $\chi$ and $\chi^{\dagger}$ co-moving
number densities ($Y_{\chi}$ and $Y_{\chi^{\dagger}}$) remain constants.
However, in the presence of subsequent pair annihilations, namely, the process
$\chi\chi^{\dagger}\rightarrow\phi\phi$, at temperatures below $m_{\chi}$
(when the backward process is not active), each reaction reduces both $\chi$
and $\chi^{\dagger}$ numbers by one unit. Therefore, in this latter case, only
$Y_{\chi}-Y_{\chi^{\dagger}}$ remains a constant, which we can, therefore,
equate to $Y_{\chi}^{\infty}$, assuming the symmetric part is completely
annihilated, and only $\chi$ particles survive today. We now define the net
co-moving charge density in the dark matter sector at the temperature $T$ to
be $\Delta B(T)=Q\left(n_{\chi}(T)-n_{\chi^{\dagger}}(T)\right)$, where $Q$ is
the charge assigned to one DM particle. We can then express the present relic
abundance of DM as

$\Omega_{\rm DM}=\frac{m_{\chi}s_{0}\Delta B(T_{S})}{\rho_{c}\,s(T_{S})Q},$
(94)

where $T_{S}$ is the freeze-out temperature for the semi-annihilation process.

In the semi-annihilation reaction, $\chi\chi\rightarrow\chi^{\dagger}\phi$,
the net change in $\chi$ charge per reaction is negative ($\Delta Q=-3Q$),
while in the CP-conjugate process
$\chi^{\dagger}\chi^{\dagger}\rightarrow\chi\phi$, the net change in $\chi$
charge per reaction is positive ($\Delta Q=3Q$). Hence, the probability of
having a positive change is
$P_{+}=\sigma_{\bar{S}}/\left(\sigma_{S}+\sigma_{\bar{S}}\right)$, while the
probability for a negative change is
$P_{-}=\sigma_{S}/\left(\sigma_{S}+\sigma_{\bar{S}}\right)$. Therefore,
$\Delta Q$ produced per semi-annihilation and its CP-conjugate reaction is
$(3QP_{+}-3QP_{-})=-3Q\epsilon$, where $\epsilon$ is defined as in Eq. 84.
Here, we have used the fact that the final state phase space elements are the
same for the two CP-conjugate processes. Finally, the net DM charge density
produced is $\Delta B=-3\epsilon Qn_{\chi}^{\rm eq}(T_{S})$, assuming that the
near-equilibrium distribution with zero chemical potential, $n_{\chi}^{\rm
eq}(T_{S})$, is being maintained by fast pair-annihilation reactions, and
therefore, $n_{\chi}^{\rm eq}(T_{S})\simeq n_{\chi^{\dagger}}^{\rm
eq}(T_{S})$. Since this assumption is invalid for the scenario discussed in
Sec. 3.2, our estimate of the relic abundance in this section does not apply
for that scenario. In particular, with only the CP-violating semi-annihilation
reaction active in the thermal bath, for large CP-violation (which is
necessary to get a complete asymmetry with only semi-annihilation) the DM and
the anti-DM particles have large and different pseudo-chemical potentials, and
therefore do not follow the equilibrium distribution.

Plugging in the expression for $\Delta B$ as obtained above, we can now write
the relic abundance of DM particles today as follows

$\displaystyle\Omega_{\rm DM}$
$\displaystyle=\frac{3~{}m_{\chi}s_{0}|\epsilon|~{}n_{\chi}^{\rm
eq}(T_{S})}{\rho_{c}~{}s(T_{S})}~{}~{}~{}{\rm(completely~{}asymmetric~{}scenario)}.$
(95)

Since we assumed the DM state $\chi$ to survive in the present Universe,
$\epsilon<0$ in this case, while if the anti-DM state $\chi^{\dagger}$
survives, $\epsilon>0$, as can be seen from Eq. 84. We can re-write the above
expression in terms of a set of particular choices of the parameters as

$\Omega_{\rm DM}h^{2}=0.12\times~{}\bigg{(}\frac{m_{\chi}}{100~{}\rm
GeV}\bigg{)}\bigg{(}\frac{100}{g_{*S}(x_{F})}\bigg{)}~{}\bigg{(}\frac{|\epsilon|~{}x_{S}^{3/2}~{}e^{-x_{S}}}{10^{-9}}\bigg{)}.$
(96)

This shows that apart from the implicit dependence of $\Omega_{\rm DM}$ on
$m_{\chi}$ and $\epsilon$ through the value of $x_{S}(=m_{\chi}/T_{S})$, there
is an explicit linear proportionality with both these parameters expected.
This is to be contrasted with the simple scenarios of asymmetry generation
through the out-of-equilibrium decay of a heavy particle, where the resulting
particle density today is proportional to $\epsilon$ only, and not to the mass
of the decaying heavy particle Kolb:1990vq . Furthermore, in the decay
scenario, the asymmetry parameter $\epsilon$ is independent of the particle
momenta, unlike in the annihilation scenario Kolb:1990vq . For a typical value
of $x_{S}=20$, we see that $|\epsilon|\simeq 5.4\times 10^{-3}$ can reproduce
the present DM abundance, for $m_{\chi}=100$ GeV. In contrast to the scenario
with only semi-annihilation discussed in Sec. 3.2, we see that the CP-
violation required to generate complete asymmetry here is very small.

Unlike in the previous case, for pair-annihilation cross-sections that are not
sufficient to completely remove the symmetric component, there is an explicit
dependence of the DM relic density on the pair-annihilation rate
$\langle\sigma v\rangle_{A}$. In this case, the coupled Boltzmann equations
can be integrated piecewise in different temperature regimes, firstly near the
freeze-out of the semi-annihilation process, in which the pair-annihilation
rate is not relevant, and then near the freeze-out of the pair-annihilation
process, but now with an initial asymmetry in the DM sector generated by the
earlier freeze-out of the semi-annihilation. The resulting relic abundance
already calculated in Eq.36 as

$\Omega_{\rm
DM}=\frac{s_{0}}{\rho_{c}}m_{\chi}C\coth\left(\frac{C\lambda\langle\sigma
v\rangle_{A}}{2x_{A}}\right)~{}~{}~{}~{}{\rm(partially~{}asymmetric~{}scenario),}$
(97)

where, $x_{A}=m_{\chi}/T_{A}$, with $T_{A}$ being the freeze-out temperature
of the pair-annihilation process, $C=Y_{\chi}(T)-Y_{\chi^{\dagger}}(T)$, for
all $T<T_{S}$, and $\lambda=1.32M_{\rm Pl}m_{\chi}g^{1/2}_{*}$. In the limit
$C\rightarrow 0$, the above expression reduces to the well-known result for
symmetric WIMP scenario

$\Omega_{\rm DM}=\frac{2s_{0}m_{\chi}x_{A}}{\rho_{c}\lambda\langle\sigma
v\rangle_{A}}~{}~{}~{}~{}~{}~{}{\rm(symmetric~{}WIMP~{}scenario)}.$ (98)

####  3.3.2 Numerical results

We shall now solve the coupled Boltzmann equations 93 numerically, with four
free parameters. The three parameters $m_{\chi}$, $\langle\sigma v\rangle_{S}$
and $\epsilon_{\rm eff}$ are the same as in Sec. 3.2, with the additional
parameter being the pair-annihilation rate $\langle\sigma v\rangle_{A}$. Since
we have already discussed the role of the first three parameters in
determining the DM properties in the previous section, the primary aim of this
section is to understand the impact of pair-annihilation, in particular its
interplay with the semi-annihilation process. Following our general discussion
above, therefore, the relevant temperature hierarchy is $T_{S}>T_{A}$, in
which the semi-annihilation freezes out earlier. The opposite hierarchy,
$T_{S}<T_{A}$, is exactly equivalent to the scenario in Sec. 3.2, as far as
the DM asymmetry and relic density today are concerned.

Figure 6: Variation of the ratio of dark matter and anti-dark matter yields,
$Y_{\chi}/Y_{\chi^{\dagger}}$, as a function of $m_{\chi}/T$, for different
values of the pair-annihilation rate, $\langle\sigma
v\rangle_{A}=10^{-10}{~{}\rm GeV}^{-2}$ (red solid line) and $\langle\sigma
v\rangle_{A}=10^{-8}{~{}\rm GeV}^{-2}$ (blue solid line), with fixed values of
$m_{\chi}$, $\langle\sigma v\rangle_{S}$ and $\epsilon$.

For $T_{S}>T_{A}$, the essential role of the pair-annihilation process is to
remove the symmetric component of dark matter, as illustrated in Fig. 6. As we
can see from this figure, with fixed values of $m_{\chi}$, $\langle\sigma
v\rangle_{S}$ and $\epsilon$, if we increase the value of the pair-
annihilation rate from $\langle\sigma v\rangle_{A}=10^{-10}{~{}\rm GeV}^{-2}$
(red solid line) to $\langle\sigma v\rangle_{A}=10^{-8}{~{}\rm GeV}^{-2}$
(blue solid line), the ratio of dark matter and anti-dark matter yields
$Y_{\chi}/Y_{\chi^{\dagger}}$ decreases rapidly.

In order to understand the typical values of the cross-sections required to
reproduce the observed relic abundance today, we show in Fig. 7 the regions in
the $\langle\sigma v\rangle_{A}$ and $m_{\chi}$ parameter space in which the
central value of the DM relic density $\Omega h^{2}=0.12$ is reproduced. For
both the plots in this figure (left and right), the values of $\epsilon$ and
$\langle\sigma v\rangle_{S}$ have been kept fixed. We show the results for
$\epsilon=0.01$ and $\langle\sigma v\rangle_{S}=10^{-10}{~{}\rm GeV}^{-2}$ in
the left figure, and for $\epsilon=10^{-4}$ and $\langle\sigma
v\rangle_{S}=10^{-13}{~{}\rm GeV}^{-2}$ in the right figure. We also show
contours in the $m_{\chi}-\langle\sigma v\rangle_{A}$ parameter space for
constant values of the present relative abundance of DM and anti-DM,
parametrized by $\eta$, as defined in Eq. 90.

Figure 7: Required values of the pair-annihilation cross-section
$\langle\sigma v\rangle_{A}$ as a function of the DM mass $m_{\chi}$ that can
reproduce the observed DM relic abundance (red solid line). The results are
shown for different values of the CP-violation parameter $\epsilon$ and the
semi-annihilation rate $\langle\sigma v\rangle_{S}$ (left and right plots).
Corresponding results for the symmetric case ($\epsilon=0$) are also shown for
comparison, with semi-annihilation (black dashed line) and without it (pink
solid line), the latter case being the pure WIMP scenario. Also shown are the
contours for the present DM relative abundance parameter $\eta$. See text for
details.

As expected from our discussion in Sec. 3.3.1, in particular Eq. 96, as we
increase $\epsilon$, the value of $\langle\sigma v\rangle_{S}$ required to
reproduce the relic density is also correspondingly increased. This is
primarily due to the exponential suppression of $\Omega h^{2}$ from $x_{S}$,
which is increased for larger $\langle\sigma v\rangle_{S}$, thereby requiring
larger $\epsilon$. This estimate is applicable only in the case when the
symmetric component today is negligible, i.e., $\eta\simeq 1$. As we see in
Fig. 7, in the entire parameter space under consideration, $\eta$ is close to
$1$. We also show the contour for $\Omega h^{2}=0.12$, in the $\epsilon=0$
limit (black dashed line), which is found to overlap with the corresponding
contour (pink solid line) in the case in which only the pair-annihilation is
active (i.e., $\langle\sigma v\rangle_{S}=0$ as well). This is not surprising,
since for such small values of $\langle\sigma v\rangle_{S}$, which is at least
an order of magnitude below the values of $\langle\sigma v\rangle_{A}$, semi-
annihilation is essentially not relevant in determining the present DM
abundance as long as $\epsilon=0$.

The scenario, however, changes dramatically with the introduction of a small
CP-violation with a non-zero $\epsilon$, when semi-annihilation becomes the
key process in determining the present density. The role of $\langle\sigma
v\rangle_{A}$ for non-zero $\epsilon$ is then to eliminate the symmetric
component of DM that is left over at the freeze-out of the semi-annihilation
process. As we have already seen in Sec. 3.2, for large $\mathcal{O}(1)$
values of $\epsilon$, no other number changing process plays any role in
determining the relic abundance. This is because such a scenario leads to a
large violation of CP in the DM sector, thereby producing an almost completely
asymmetric DM already at the freeze-out of the semi-annihilation process at
temperature $T_{S}$. Since almost no symmetric component is left out in this
case at $T=T_{S}$, the pair-annihilation process is not relevant.

In the limit $\eta\rightarrow 1$, we see from Eq. 96 that for a fixed value of
$x_{S}$ (which in turn is obtained for a fixed value of $\langle\sigma
v\rangle_{S}$ in this limit) and $\epsilon$, the dark matter mass is also
fixed. In particular, as we see from Fig. 7, with $\epsilon=0.01$ and
$\langle\sigma v\rangle_{S}=10^{-10}{~{}\rm GeV}^{-2}$, we obtain
$m_{\chi}\sim 4600$ GeV, while for $\epsilon=10^{-4}$ and $\langle\sigma
v\rangle_{S}=10^{-13}{~{}\rm GeV}^{-2}$ , $m_{\chi}\sim 5$ GeV. Away from the
region in the parameter space for which $\eta\rightarrow 1$, we find it non-
trivial to obtain a semi-analytic solution to the Boltzmann equations.
However, it is clear from Fig. 7 that the DM mass is no longer uniquely fixed
for such a case, but varies with $\langle\sigma v\rangle_{A}$. This is
essentially because the symmetric component is not completely removed in such
scenarios.

We note in passing that the parameter values $\epsilon=10^{-4}$ and
$\langle\sigma v\rangle_{S}=10^{-13}{~{}\rm GeV}^{-2}$ predict a DM mass of
around $5$ GeV in the completely asymmetric DM limit. Since this value of the
DM mass is around five times the proton mass, we expect the current number
densities of the surviving DM particle and protons to be similar in this
scenario. As is well-known, such a DM mass is also expected in scenarios which
dynamically relate the DM and baryon number densities in the current Universe
Petraki:2013wwa ; Zurek:2013wia . Such a mechanism to relate the two
asymmetries might be possible through semi-annihilation.

In the pure WIMP scenario, with $\epsilon=0$ and $\langle\sigma
v\rangle_{S}=0$, in the freeze-out approximation, the dependence of $\Omega
h^{2}$ on the DM mass is logarithmic, while it is inversely proportional to
$\langle\sigma v\rangle_{A}$. Therefore, we see in Fig. 7 that the value of
$\langle\sigma v\rangle_{A}$ required (around $3.5\times 10^{-9}{~{}\rm
GeV}^{-2}$) to reproduce $\Omega h^{2}=0.12$ is largely independent of
$m_{\chi}$ (pink solid line in both figures). As discussed above, this value
remains unchanged with the introduction of a small $\langle\sigma
v\rangle_{S}$, when the CP-violation is zero ($\epsilon=0$). In the
$\eta\rightarrow 1$ limit, for non-zero $\epsilon$, the requirement of
$\langle\sigma v\rangle_{A}$ is larger, and it increases with increasing
$\epsilon$.

Figure 8: Contours in the $m_{\chi}-\epsilon$ plane in which the central value
of the DM relic density $\Omega h^{2}=0.12$ is reproduced, with the pair-
annihilation rate fixed at its s-wave upper bound implied by S-matrix
unitarity, $\langle\sigma v\rangle_{A}=\langle\sigma v\rangle_{\rm
uni}=(4\pi/m_{\chi}^{2})(x_{F}/\pi)^{1/2}$. The semi-annihilation cross-
section is fixed to ensure $\langle\sigma v\rangle_{S}<\langle\sigma
v\rangle_{A}$, for all values of $m_{\chi}$ considered in this figure, such
that the freeze-out temperature hierarchy $T_{S}>T_{A}$ is satisfied.

As in Sec. 3.2, we can obtain an upper bound for possible values of the dark
matter mass by using partial-wave unitarity to bound the annihilation (or
semi-annihilation) cross-sections from above. For the scenario in which
$T_{S}>T_{A}$, the annihilation cross-section must be larger than the semi-
annihilation cross-section, and therefore, we impose the unitarity bound on
$\langle\sigma v\rangle_{A}=\langle\sigma v\rangle_{\rm uni}$, where,
$\langle\sigma v\rangle_{\rm uni}$ is as given in Eq. 91. In this case, we
show the resulting upper bound on the dark matter mass as a function of the
CP-violation parameter $\epsilon$ in Fig. 8. We have fixed the value of the
semi-annihilation cross-section to be $\langle\sigma
v\rangle_{S}=10^{-13}{~{}\rm GeV}^{-2}$, which is chosen to ensure that
$\langle\sigma v\rangle_{S}<\langle\sigma v\rangle_{A}=\langle\sigma
v\rangle_{\rm uni}$, for all values of $m_{\chi}$ considered in this figure.

In Fig. 8, the observed relic abundance $\Omega h^{2}=0.12$ is satisfied along
the solid blue line. As in Fig. 4, we see that as the CP-violation parameter
$\epsilon$ decreases, the resulting mass bound becomes stronger. Furthermore,
higher values of $\epsilon$ lead to larger present asymmetry in the dark
matter sector, and therefore a value of $\eta$ closer to $1$. The general
result obtained in Sec. 3.2 that the bound on $m_{\chi}$ for asymmetric DM is
stronger compared to the symmetric DM scenario, continues to hold in this
scenario as well. In the complete asymmetric limit, i.e., $\eta\rightarrow 1$,
the upper bound on the DM mass is found to be around $25$ TeV (numerically for
$\eta=1.000001$), while for $\eta\rightarrow 2$ it’s around $90$ TeV, assuming
s-wave annihilation. For the opposite hierarchy of the freeze-out
temperatures, i.e., $T_{S}<T_{A}$, the semi-annihilation cross-section must be
larger than the pair-annihilation cross-section, and therefore, the unitarity
bound must be imposed on $\langle\sigma v\rangle_{S}$, which has already been
discussed in Sec. 3.2, in particular in Fig. 4.

Figure 9: Phase diagram showing the interplay between the semi-annihilation
and pair-annihilation rates in determining the asymmetric DM properties. The
observed relic abundance $\Omega h^{2}=0.12$ is satisfied, with and without
CP-violation, along the blue solid and black dashed lines, respectively. The
relative abundance parameter $\eta\rightarrow 2$ represents the symmetric
phase, while $\eta\rightarrow 1$ represents the asymmetric phase. Here, ADM,
Semi DM and WIMP DM denote the dominantly asymmetric dark matter, symmetric
semi-annihilating DM and symmetric pair-annihilating DM phases, respectively.
See text for details.

We can summarize our discussion of the interplay between the semi-annihilation
and pair-annihilation rates in determining the asymmetric DM properties using
an instructive phase diagram, as shown in Fig. 9. In this figure, we study the
values of $\langle\sigma v\rangle_{S}$ and $\langle\sigma v\rangle_{A}$ for
which the observed relic abundance $\Omega h^{2}=0.12$ is satisfied, with and
without CP-violation. When the CP-violation vanishes, i.e., with $\epsilon=0$,
the relic abundance is satisfied along the black dashed contour DEramo:2010keq
. Since for $\epsilon=0$, both semi-annihilation and pair-annihilation can
reproduce the observed relic abundance, with either or both of them
contributing, we obtain an approximate upper bound of $10^{-8}{~{}\rm
GeV}^{-2}$ for both the rates, for a fixed DM mass of $m_{\chi}=100$ GeV. In
contrast, when CP-violation is turned on, i.e., for $\epsilon=10^{-2}$ in Fig.
9, a symmetric phase and an asymmetric phase appear in which the relic density
is satisfied, as seen in the solid blue line. The two phases can be
distinguished by constant values of the DM relative abundance parameter
$\eta$. The symmetric phase, with $\eta\rightarrow 2$ is identical to the
$\epsilon=0$ scenario, and hence the blue solid line and the black dashed
lines overlap. In this phase, the pair-annihilation rate is not large enough
to remove the symmetric component efficiently. On the other hand, the
asymmetric phase appears when $\langle\sigma v\rangle_{A}$ is larger than the
previously obtained upper bound of around $10^{-8}{~{}\rm GeV}^{-2}$, for
$m_{\chi}=100$ GeV. In contrast, $\langle\sigma v\rangle_{S}$ is much smaller
in this phase. Thus, to summarize, there are two ways to produce asymmetric DM
in the absence of any wash-out processes, namely,

  1. 1.

have a large CP-violation $\epsilon$ as in Sec. 3.2, in which case semi-
annihilation is sufficient to create a complete DM asymmetry, and no
subsequent number changing process is necessary, or,

  2. 2.

produce a small asymmetry through a small CP-violation $\epsilon$, and then
have a sufficiently large pair-annihilation rate to remove the symmetric
component, as shown in this section, and as is clear from Fig. 9.

###  3.4 Complex scalar DM with cubic self-interaction

We now discuss a simple toy model in which the generic scenario described in
Sec. 3.3 with both the semi- and pair-annihilation processes can be realized.
The minimal new field content that can lead to a particle-antiparticle
asymmetry through the semi-annihilation process include a complex scalar
$\chi$ which is charged under a $Z_{3}$ symmetry (we assign the charge
$\omega$ to $\chi$, where $\omega^{3}=1$) and a real scalar $\phi$, which is a
singlet under this symmetry, as well as the SM gauge interactions. The SM
fields are also singlets under the discrete $Z_{3}$ symmetry. The $Z_{3}$
symmetry ensures the stability of $\chi$, making it the DM candidate. For
earlier studies involving different aspects of $Z_{3}$ symmetric DM, see, for
example, Refs. Belanger:2012vp ; Belanger:2012zr ; Hochberg:2014dra ;
Bernal:2015bla ; Hektor:2019ote ; Choi:2015bya ; Choi:2016tkj . The effective
low-energy interaction Lagrangian involving the $\chi$ and the $\phi$
particles is given by

$\mathcal{L}\supset\frac{1}{3!}\left(\mu\chi^{3}+{\rm
h.c.}\right)+\frac{1}{3!}\left(\lambda\chi^{3}\phi+{\rm
h.c.}\right)+\frac{\lambda_{1}}{4}\left(\chi^{\dagger}\chi\right)^{2}+\frac{\lambda_{2}}{2}\phi^{2}\chi^{\dagger}\chi+\mu_{1}\phi\chi^{\dagger}\chi+\frac{\mu_{2}}{3!}\phi^{3}+\frac{\lambda_{3}}{4!}\phi^{4}.$
(99)

Here, the couplings $\mu$ and $\lambda$ can be complex in general. However,
one of the phases can be rotated away by an appropriate re-definition of the
field $\chi$. Therefore, in this general effective low-energy theory, there is
one residual complex phase, which is necessary to generate a CP-asymmetry in
the DM sector. We take $\mu$ to be real, and $\lambda$ to have a non-zero
imaginary part, with a phase $\theta$. The parameters in the scalar potential
in Eq. 99 can be suitably chosen to ensure that the $\chi$ field does not
develop a VEV, thereby ensuring that the $Z_{3}$ symmetry is not spontaneously
broken.

In addition to the interaction terms involving the $\chi$ and the $\phi$
fields in Eq. 99, there can be two dimension-four and one dimension-three
couplings to the SM Higgs doublet $H$ as well, namely, $\lambda_{\chi
H}~{}(\chi^{\dagger}\chi|H|^{2})+\lambda_{\phi
H}~{}(\phi^{2}|H|^{2})+\mu_{\phi H}~{}(\phi|H|^{2})$. For $m_{\chi}>m_{H}$,
the $\lambda_{H\chi}$ term contributes in exactly the same way as the
$\lambda_{2}$ term in Eq. 99, and therefore we do not consider it separately
here. Furthermore, the $\lambda_{H\phi}$ and $\mu_{H\phi}$ terms lead to
interactions of the $\phi$ field with the $H$ field, which will thermalize the
$\phi$ field with the SM plasma. Since we assume the $\phi$ particles to be in
equilibrium with the SM bath with zero chemical potential, the effect of these
terms are also included. To note, $\phi$ mixes with the Higgs field via
$\mu_{H_{\phi}}$ coupling, in turn it introduces interaction between $\chi$
and physical Higgs ($h$), e.g. $\lambda\sin\beta~{}h\chi^{3}$,
$\mu_{1}\sin\beta~{}h\chi^{\dagger}\chi$, where $\beta$ is the mixing angle
between $\phi$ and $H$ after the electroweak symmetry breaking. Consequently,
Higgs can decay to a DM pair via $\mu_{1}$ coupling for DM masses,
$m_{\chi}\leq m_{h}/2$, where $m_{h}$ is the mass of the physical Higgs.
However, we find that the decay is well suppressed for the benchmark point,
$m_{\chi}=5~{}\rm GeV$ used in the right panel of Fig.11. Detailed analysison
this can be found in Appendix C.

The above interaction Lagrangian in Eq. 99 leads to several class of
$2\rightarrow 2$, $2\rightarrow 3$ and $3\rightarrow 2$ processes. We find
that in different regions of the multi-dimensional parameter space, different
class of diagrams (or combinations thereof) may dominate. Since in this
section we are presenting a toy model that realises the general features of
the model-independent setup discussed in the previous section, we shall focus
on a restricted region of the parameter space in which a subset of the
$2\rightarrow 2$ diagrams dominate. In particular, we shall consider the
values of the dimensionful parameters to be small compared to the DM mass
scale, i.e., $\mu/m_{\chi}<<1$ and $\mu_{1}/m_{\chi}<<1$. We shall also take
the cubic and quartic self-couplings of the $\phi$ field to be small, which
does not alter the qualitative features of the scenario.

Figure 10: Relevant $2\rightarrow 2$ tree-level and one-loop Feynman diagrams
for the semi-annihilation process $\chi\chi\rightarrow\chi^{\dagger}\phi$.

The relevant tree-level and one-loop Feynman diagrams for the semi-
annihilation process $\chi\chi\rightarrow\chi^{\dagger}\phi$ are shown in Fig.
10. At tree-level there are two Feynman diagrams contributing to this process:
one involving a contact interaction (diagram $T_{1}$), and the other with an
intermediate $\chi$ propagator (diagram $T_{2}$). The second diagram gives a
contribution to the matrix element proportional to
$\left(\mu\mu_{1}/m_{\chi}^{2}\right)$, in the non-relativistic limit for the
$\chi\chi$ initial state, with the centre of mass energy squared $s\simeq
4m_{\chi}^{2}$. Therefore, for $\mu/m_{\chi}<<1$ and $\mu_{1}/m_{\chi}<<1$,
the contact interaction dominates.

In order to determine the CP-asymmetry generated by the semi-annihilation
process, we compute the interference between the tree-level and loop-level
diagrams shown in Fig. 10. In general the CP-asymmetry is proportional to
$\imaginary\left({M}_{\rm tree}(g_{i})^{*}{M}_{\rm loop}(g_{j})\right)$, which
in turn is proportional to
$\imaginary\left(\prod_{i,j}g_{i}^{*}g_{j}\right)\times\imaginary(I)$, where
$I$ is the loop factor which acquires an imaginary part when the particles in
the loop go on-shell. The latter requirement is ensured by the condition
$m_{\phi}<m_{\chi}$. We find that diagram $T_{2}$ gives a non-zero
contribution to the CP-asymmetry, resulting from its interference with the
loop diagram $L_{2}$, while diagram $T_{1}$ leads to a non-zero contribution
from its interference with $L_{1}$ and $L_{3}$. Furthermore, the contributions
from the interference of $T_{2}$ and $L_{2}$, and that from $T_{1}$ and
$L_{3}$ cancel identically. Therefore, the only relevant contribution is from
the interference of $T_{1}$ and $L_{1}$. The resulting difference in matrix
elements squared that contribute to $\epsilon$ as defined in Eq. 84, is given
as:

$|{M}|^{2}_{\chi\chi\rightarrow\chi^{\dagger}\phi}-|{M}|^{2}_{\chi^{\dagger}\chi^{\dagger}\rightarrow\chi\phi}=\frac{4|\lambda|\mu\mu_{1}\lambda_{2}\sin{\theta}}{16\pi\sqrt{s(s-4m^{2}_{\chi})}}\log\left[\frac{m_{\chi}^{2}+m^{2}_{\phi}-s+\beta_{1}}{m_{\chi}^{2}+m^{2}_{\phi}-s-\beta_{1}}\right]$
(100)

where, $s$ is the centre of mass energy squared, and

$\beta_{1}=\sqrt{\frac{(s-4m_{\chi}^{2})(m_{\chi}^{4}+(m^{2}_{\phi}-s)^{2}-2m_{\chi}^{2}(s+m^{2}_{\phi}))}{s}}.$
(101)

Figure 11: Contours of fixed effective CP-violation parameter $|\epsilon_{\rm
eff}|$ (red solid lines), as a function of the complex phase
$\theta=\arg(\lambda)$ and the effective pair-annihilation coupling
$\lambda_{2}$. The results are shown for two different DM mass values
$m_{\chi}=1$ TeV (left panel) and $5$ GeV (right panel). The required values
of the annihilation rates and $\epsilon_{\rm eff}$ are reproduced, as
indicated. See text for details, and Fig. 7 for comparison with the results
obtained in the model-independent analysis.

In the model-independent setup discussed in Sec. 3.2 and Sec. 3.3, the
different annihilation rates and the effective CP-violation parameter were
treated as independent free parameters. However, in a model in which such
processes are realized, these parameters are often correlated, and are
determined in terms of the common set of couplings and masses. Therefore, in
order to understand whether the simple model described by Eq. 99 can
accommodate the required values of the relevant physical parameters found in
the previous section, we study in Fig. 11 the correlation between the
effective CP-violation $\epsilon_{\rm eff}$, and the annihilation rates, as a
function of the CP-violating phase $\theta$, and the relevant couplings
$\lambda_{2}$ and $|\lambda|$.

In Fig. 11, we show contours of fixed effective CP-violation parameter
$|\epsilon_{\rm eff}|$ (red solid lines), as a function of the complex phase
$\theta=\arg(\lambda)$ and the effective pair-annihilation coupling
$\lambda_{2}$. We have also shown the corresponding values of the annihilation
rate $\langle\sigma v\rangle_{A}$ in both panels. The results are shown for
two different DM mass values $m_{\chi}=1$ TeV (left panel) and $5$ GeV (right
panel). As we can see from this figure, the values of the annihilation rates
and $\epsilon$ required to satisfy the DM relic abundance can be obtained in
this model, as indicated by the dashed horizontal and vertical lines. This can
be observed by comparison with the $\Omega h^{2}=0.12$ contour in Fig. 7,
where the results were obtained in the model-independent analysis.

A few comments are in order. First of all, as mentioned earlier, in Fig. 11 we
ensure $\mu/m_{\chi}<<1$ and $\mu_{1}/m_{\chi}<<1$, for which our restriction
to the class of $2\rightarrow 2$ diagrams in Fig. 10 remains valid. Since the
loop amplitudes in this model depend upon the coupling $\lambda_{2}$, the
pair-annihilation process is necessarily present whenever the CP-violation in
the semi-annihilation process is sufficiently large. Thus the first scenario
with only the semi-annihilation process discussed in Sec. 3.2 is not obtained
in this model, while the second scenario in Sec. 3.3 with both semi- and pair-
annihilations can be easily realized. Additional structures are therefore
necessary to have loop graphs with sufficiently large imaginary parts, which
do not induce significant tree-level pair annihilation.

Figure 12: The effective CP-violation parameter $|\epsilon_{\rm eff}|$, as a
function of the pair-annihilation coupling $\lambda_{2}$. The results are
shown for fixed values of all other parameters, including the DM mass value of
$m_{\chi}=5$ GeV. The triangle loop gives the dominant contribution, as shown
by comparison of $|\epsilon_{\rm eff}|$ computed only with the triangle loop
(red dashed line), and all the three loops (blue solid line).

We find that a scenario where the effective CP-violation parameter
$|\epsilon_{\rm eff}|$ is close to unity, thereby leading to the present DM
asymmetry $\eta\rightarrow 1$, can be realized in the model described by Eq.
99, for values of the model parameters within perturbative limits. We show in
Fig. 12 the variation of $|\epsilon_{\rm eff}|$ as a function of the coupling
$\lambda_{2}$, which appears in all the relevant loop graphs in Fig. 10. The
results are shown for fixed values of all the other parameters, including the
DM mass value of $m_{\chi}=5$ GeV. Even though here the dimensionful
parameters $\mu$ and $\mu_{1}$ are taken to be only a factor of two smaller
than $m_{\chi}$, we have checked that only the $2\rightarrow 2$ processes in
Fig. 10 dominate. We find that the triangle loop gives the dominant
contribution, as shown by comparison of $|\epsilon_{\rm eff}|$ computed only
with the triangle loop (red dashed line), and all the three loops (blue solid
line). While the triangle loop ($L_{1}$ in Fig. 10) does not lead to any
ultraviolet (UV) divergence, the other two loops (namely, $L_{2}$ and $L_{3}$)
are divergent in the UV, and therefore $|\epsilon_{\rm eff}|$ computed with
all three loops has a renormalization scale dependence, which is found to be
rather weak. For our computations, we have set the renormalization scale to be
$\Lambda=m_{\chi}$. We emphasize that even though in this example, the value
of the pair-annihilation coupling $\lambda_{2}$ is large, the choice of
parameters for which $|\epsilon_{\rm eff}|\rightarrow 1$ essentially belongs
to the first scenario we studied, in which the semi-annihilation process
almost entirely determines the present DM properties, including its asymmetry.
This is because, for $|\epsilon_{\rm eff}|\rightarrow 1$, at the decoupling of
the semi-annihilation process, the symmetric component is already negligible,
and hence the subsequent pair-annihilation is largely irrelevant. Therefore,
as is clear from Fig. 12, we can achieve $|\epsilon_{\rm eff}|$ close to $1$
in this scenario, being within perturbative limit of all the relevant
parameters including $\lambda_{2}$. Furthermore, we can also essentially
realize, albeit in the presence of the coupling $\lambda_{2}$, the first
scenario in which the semi-annihilation process almost entirely determines the
present DM properties.

##  4 Cosmology of complex scalar dark matter: Interplay of self-scattering
and annihilation

###  4.1 Introduction

Perhaps the simplest studied scenario for thermal dark matter (DM) is that of
a real scalar field ($S$), which is a singlet under the standard model (SM)
gauge interactions of $SU(3)\times SU(2)\times U(1)$ zee ; McDonald . The new
degree of freedom added to the SM field content here is at an absolute minimum
of one. Such a DM particle can be stable due to the existence of an effective
reflection symmetry ($Z_{2}$ symmetry), under which the DM field is odd
($S\rightarrow-S$) and the SM particles are even. A renormalizable interaction
of the real scalar singlet with the SM Higgs doublet ($H$) exists, namely,
$S^{2}|H|^{2}$, which is allowed by the reflection symmetry zee ; McDonald ;
Burgess ; Patt . The cosmology of this scenario is also very simple –
depending upon the mass of the singlet $S$, it may go through pair-
annihilations to different SM particles, achieve and stay in thermodynamic
equilibrium with the SM sector in the early Universe, and eventually freeze-
out with a thermal relic abundance, which is determined by the two unknown
parameters of the singlet mass and coupling to the Higgs. Such a scenario also
presents with different signals at direct, indirect and collider searches –
and is strongly constrained by them in a large region of the parameter space
providing the required DM abundance cline ; Athron:2017kgt ; Athron:2018ipf .

We have already noticed in the earlier chapter that the DM particle may or may
not be self-conjugate and it may be charged under a global symmetry of DM
number. What new phenomena do we expect in such a case in general? Here, there
are two broad possibilities. The first is that of a restricted scenario in
which we impose the strict conservation of this global DM number symmetry,
while the second possibility is of a more general scenario in which such a
symmetry can be broken by different interaction terms. In the first case, the
phenomena are essentially same as that of the self-conjugate DM, albeit with
twice the new degrees of freedom. However, there is no strong argument behind
the conservation of such global continuous symmetries. We know, for example,
that it is very likely that the global $U(1)_{\rm B}$ symmetry of baryon
number must be broken in order to dynamically generate the baryon asymmetry of
the Universe Weinberg:1979sa . Similarly, the SM gauge quantum numbers of
neutrinos allow for a Majorana mass term – whose existence is being searched
for – and such a term would violate the global $U(1)_{\rm L}$ symmetry of
lepton number. Such global symmetry breaking effects are generically found in
unified theories of the strong, weak and electromagnetic interactions
Pati:1974yy ; Georgi:1974sy . It is also generally argued that global
continuous symmetries may not be conserved due to quantum gravitational
effects Witten:2017hdv . Discrete symmetries, on the other hand, may remain in
effective low energy theories, in some cases as a remnant of larger continuous
symmetries Witten:2017hdv , thereby make the DM candidate absolutely stable.

Thus the second general possibility of including interactions that do not
conserve the global symmetry of DM number should be explored in detail. And we
expect to encounter new phenomena in studying the cosmology of such a
scenario. A complex scalar singlet DM constitutes the version of the most
minimal thermal DM scenario, with now a non-self-conjugate DM particle instead
of a self-conjugate one. In this chapter, we carry out a detailed study of
this scenario, and find several interesting effects in its cosmological
history, some of them novel. No complete study of such a general scenario for
a complex scalar DM has been carried out so far, to our knowledge, and the
previous studies of the complex scalar DM are restricted to symmetric weakly
interacting massive particle (WIMP)-like behaviour Barger:2008jx ;
Chiang:2017nmu . The different interaction terms present in such a scenario
and their role in the various scattering processes are discussed in Sec. 4.2
in the following.

The primary effect observed is that of the close interplay between the self-
scattering and annihilation processes in determining the present DM density
and composition. In particular, the fact that DM number may now be violated
with all renormalizable interactions included, leads to the possibility of
creation of a particle-antiparticle asymmetry in the DM sector. Such an
asymmetry generation of course crucially depends on the CP-violation in the
relevant scattering rates, which is non-zero only when all of the self-
scattering and annihilation processes are simultaneously present. The absence
of any one of them leads to the vanishing of the CP-violation, and hence the
asymmetry – by very general arguments of unitarity sum rules. This close
interplay of the self-scatterings and annihilations is a very novel
phenomenon, which we discuss in Sec. 4.3, after setting up the Boltzmann
kinetic equations for the DM and anti-DM number densities. We then present the
approximate analytic solutions to these equations (Sec. 4.3.1), which clearly
demonstrate the role of different type of scattering processes in determining
the net DM yields as well as any possible asymmetries, which are then compared
with the general numerical solutions (Sec. 4.3.2).

In Sec. 4.4 we compute the CP-violation (at one-loop level) and the thermally
averaged symmetric and asymmetric scattering rates in the complex singlet
scenario, that are inputs to the kinetic equations, to understand the typical
values of these quantities that one might expect in this model. We then go on
to discuss the close interplay of the coupling parameters that determine the
self-scattering and annihilation rates in fixing the DM relic density and
asymmetry. We summarize our results in Sec. 4.5. The details of the CP-
violation calculation for the sub-leading coversion process
$\chi+\phi\rightarrow\chi^{\dagger}+\phi$ are provided in Appendix B.

###  4.2 Gauge singlet complex scalar dark matter: general scenario

The most general low-energy effective Lagrangian of a SM gauge singlet complex
scalar ($\chi$), that is odd under a $Z_{2}$ reflection symmetry (with
$\chi\rightarrow-\chi$), is given as follows:

$\mathcal{L}\supset(\partial_{\mu}\chi)^{\dagger}\partial^{\mu}\chi-
m_{\chi}^{2}\chi^{\dagger}\chi-\frac{1}{2}\left(\tilde{\mu}^{2}\chi^{2}+{\rm
h.c.}\right)-\lambda_{\chi H}|\chi|^{2}|H|^{2},$ (102)

where $H$ is the SM Higgs doublet. The interaction with the Higgs boson,
however, is severely constrained by spin-independent direct detection probes
for a large range of the DM mass, as well as by the search for invisible Higgs
decay for a DM that is lighter than half of the Higgs mass. With such
constrained values of the Higgs portal coupling, it is difficult to obtain the
observed DM abundance in a large range of the DM mass. Therefore, we shall
discuss the DM cosmology, assuming, for all practical purposes, that the
coupling $\lambda_{\chi H}\simeq 0$. As we shall see in the following, the
effects of the Higgs coupling-like term in the DM cosmology will be captured
by a new interaction that is then necessary for DM thermalization. The
$\tilde{\mu}^{2}\chi^{2}$ term and its hermitian conjugate both break the
larger global $U(1)_{\chi}$ 101010We replace previously used suffix $B_{\chi}$
by $\chi$ for simplicity. symmetry of DM number under which the DM particle
(denoted by $\chi$) has charge $+1$ and its antiparticle (denoted by
$\chi^{\dagger}$) has a charge of $-1$.

We shall assume the $U(1)_{\chi}$ breaking $\tilde{\mu}^{2}$ term to be much
smaller than the corresponding $U(1)_{\chi}$ conserving mass term
$m_{\chi}^{2}$. In such a case, the DM mass eigenstates will essentially
correspond to the particle states with $U(1)_{\chi}$ charge of $+1$ and $-1$,
throughout the cosmological evolution of interest in this study. In late
epochs, after the chemical decoupling of the DM species, particle-antiparticle
oscillations can be caused by the presence of the $U(1)_{\chi}$ breaking mass
term, whose effects are discussed in Refs. Buckley:2011ye ; Cirelli:2011ac ;
Tulin:2012re . In particular, the oscillation probability is obtained to be
$P(\chi\rightarrow\chi^{\dagger})=\sin^{2}\left({\mid\tilde{\mu}\mid^{2}t}/{m_{\chi}}\right)$.
In order for the oscillation dynamics to not affect the freeze-out process,
the characteristic time scale of oscillations
($\tau=\left({\mid\tilde{\mu}\mid^{2}}/{m_{\chi}}\right)^{-1}$) should be
larger than the freeze-out time scale ($t_{F}\sim H^{-1}(T=T_{F})$), where
$H(T=T_{F})$ is the Hubble parameter at the freeze-out temperature. With
$T_{F}\sim m_{\chi}$, requiring $\tau>t_{F}$ implies that as long as
$|\tilde{\mu}|^{2}\lesssim m_{\chi}^{3}/M_{\rm Pl}$, the mass and charge
eigenstates can be taken to be essentially the same during the cosmological
evolution time scale of the DM species.

For example, with a typical order of the DM mass of $m_{\chi}=10\,\rm TeV$,
which is found to be relevant in the subsequent analyses, this upper bound on
the possible size of $|\tilde{\mu}|$ is around $80$ keV. The upper bound on
$|\tilde{\mu}|$ is modified appreciably in the presence of additional
scattering processes which delay the onset of oscillations (see
Ref.Cirelli:2011ac for details). In our model, we find that for
$|\tilde{\mu}|=10\,\rm GeV$ and $|\tilde{\mu}|=100\,\rm GeV$, the oscillation
starts at temperatures $T\sim 1\,\rm GeV$ and $T\sim 10\,\rm GeV$,
respectively, for $m_{\chi}=10\,\rm TeV$, whereas the corresponding freeze-out
dynamics takes place at temperature $T\sim\mathcal{O}(500\,\rm GeV)$.
Therefore, we can safely carry out the subsequent analysis of the Boltzmann
system in the charge basis, as the oscillation phenomenon is not relevant
before or during the freeze-out process, as long as $|\tilde{\mu}|$ is around
two orders of magnitude smaller than the DM mass.

With the condition that $\lambda_{\chi H}\simeq 0$, as discussed above, in
order to thermalize the DM state with the SM sector, so that a viable thermal
production mechanism for the DM density is obtained, we are forced to
introduce a mediator degree of freedom in the scenario. A minimal new addition
in such a case would be a real scalar field $\phi$, which is also a singlet
under the SM gauge interactions, but is even under the $Z_{2}$ reflection
symmetry. Since $\phi$ can couple to both the DM state as well as the SM
sector, it can effectively bring the two sectors to thermal equilibrium. The
additional interaction terms in the Lagrangian density are now as follows:

$\displaystyle-\mathcal{L_{\rm
int}}\supset\mu\chi^{\dagger}\chi\phi+\left(\frac{\mu_{1}}{2}\chi^{2}\phi+{\rm
h.c.}\right)+\frac{\lambda_{1}}{4}\left(\chi^{\dagger}\chi\right)^{2}+\left(\frac{\lambda_{2}}{4!}\chi^{4}+{\rm
h.c.}\right)+\left(\frac{\lambda_{3}}{4}\chi^{2}\phi^{2}+{\rm h.c.}\right)$
$\displaystyle+\left(\frac{\lambda_{4}}{3!}\chi^{3}\chi^{\dagger}+{\rm
h.c.}\right)+\frac{\lambda_{5}}{2}\phi^{2}\chi^{\dagger}\chi+\frac{\mu_{\phi}}{3!}\phi^{3}+\frac{\lambda_{\phi}}{4!}\phi^{4}+\frac{\lambda_{\phi
H}}{2}\phi^{2}|H|^{2}+\mu_{\phi H}\phi|H|^{2}\,\,\,$ (103)

In this most general renormalizable interaction Lagrangian involving the
$\chi$ and $\phi$ fields, the interaction terms with couplings
$\mu_{1},\lambda_{2},\lambda_{3}$ and $\lambda_{4}$ break the global
$U(1)_{\chi}$ symmetry, in addition to the $\tilde{\mu}^{2}$ term in Eq. 102
above. All of these terms, in general, can also have complex coupling
parameters, one of which can be chosen to be real by an appropriate
redefinition of the $\chi$ field.

The scalar mass and quartic terms are chosen appropriately to ensure that the
$\chi$ and $\phi$ fields do not obtain any non-zero vacuum expectation values,
where the former condition also implies that the $Z_{2}$ reflection symmetry
remains unbroken, rendering the DM stable. With this condition, the couplings
between the $\phi$ and the SM Higgs fields will eventually lead to $\phi$
mixing with the Higgs particle after electroweak symmetry breaking through the
$\phi|H|^{2}$ term only. Therefore, the current Higgs coupling measurements
will strongly constrain the $\mu_{\phi H}$ coupling. On the other hand, the
$\lambda_{\phi H}$ coupling is not as strongly bounded, even if $\phi$ is much
lighter than the Higgs. This is because, if the Higgs boson decays to two
$\phi$ particles, each of the resulting $\phi$ will further decay to visible
SM final states through its small mixing with the Higgs. If the decay width of
$\phi$ is so small that the $\phi$ particles decay outside the LHC detectors,
there will be an upper bound on $\lambda_{\phi H}$ from the search for
invisible decays of the Higgs boson. Using the $95\%$ C.L. upper limit on the
Higgs invisible branching ratio obtained by a combination of the $7,8$ and
$13$ TeV data from the CMS collaboration CMS:2018yfx , we find $\lambda_{\phi
H}\lesssim 10^{-2}$. Such couplings are however sufficient for the
$\lambda_{\phi H}$ and $\mu_{\phi H}$ terms to keep the $\phi$ particles in
thermal equilibrium with the SM sector.

The small $\phi$ mixing with the Higgs however can induce signals in DM direct
detection probes through the $\chi^{\dagger}\chi\phi$ and $\chi^{2}\phi$
terms, which will induce $\chi$ scattering with nucleons. Therefore, to a
first approximation, we also set $\mu\simeq 0$ and $\mu_{1}\simeq 0$. As we
shall see in the following, the essential aspects of the cosmology of $\chi$
particles are effectively captured even in such a case. With these parameter
choices, we have checked that the $U(1)_{\chi}$ symmetry breaking interaction
terms do not generate a large $\tilde{\mu}^{2}$ term through loop
corrections.111111As is well-known from the example of scalar field theory
with a $\phi^{4}$ interaction, at one-loop level, the quartic couplings do not
generate any quantum corrections to the $\tilde{\mu}^{2}$ term. Furthermore,
since we have also set the trilinear couplings to be vanishing to begin with,
the leading quantum correction to the $\tilde{\mu}^{2}$ term will be generated
only at the two loop level. Such two-loop corrections to $|\tilde{\mu}|$ are,
however, expected to be at least two orders of magnitude smaller than
$m_{\chi}$, which is the typical upper bound required for oscillation
phenomenon to occur only much after the DM freeze-out.. We can thus
consistently treat the $U(1)_{\chi}$ charge eigenstates as the DM mass
eigenstates for the cosmological evolution until the freeze-out time scale,
which is the subject matter of this study. For the astrophysical dynamics in
later epochs, the effect of these terms should be included.

Through their interactions with $\phi$, the DM particles and anti-particles
are assumed to remain in kinetic equilibrium with the SM thermal bath
throughout the evolution of the chemical processes that determine its density.
The chemical reactions that affect the DM density and composition, following
from the interactions in Eq. 103 are as follows:

  1. 1.

Self-scattering reaction leading to particle-antiparticle conversion:
$\chi+\chi\rightarrow\chi^{\dagger}+\chi^{\dagger}$. Due to the presence of a
complex coupling parameter, this reaction can be CP-violating and can create a
particle-antiparticle asymmetry in the DM sector. Moreover, it violates the DM
number of $U(1)_{\chi}$ by four units, but cannot change the total DM and
anti-DM number.

  2. 2.

Self-scattering reaction leading to particle-antiparticle conversion:
$\chi+\chi\rightarrow\chi+\chi^{\dagger}$. Similar to the previous process,
this can also lead to CP-violation and asymmetry in the DM sector. This
process violates the DM number by two units, but does not modify the total DM
number.

  3. 3.

DM number violating annihilation: $\chi+\chi\rightarrow\phi+\phi$. This
process can also violate CP-symmetry and create asymmetric DM, due to the
complex coupling of $\lambda_{3}$. It violates DM number by two units, and
also changes the total DM and anti-DM number by the same amount.

  4. 4.

DM number conserving annihilation: $\chi+\chi^{\dagger}\rightarrow\phi+\phi$.
This process is CP-conserving, and cannot by itself create a DM-anti-DM
asymmetry. However, as shown in Ref. Ghosh:2021ivn , it can indirectly affect
the generated asymmetry by controlling the out-of-equilibrium number density
of relevant particles in the thermal bath. This process conserves the
$U(1)_{\chi}$ DM number symmetry, but changes the total DM and anti-DM number
by two units.

In addition to the processes included in the DM and anti-DM number-density
evolution equations, the process, $\chi+\phi\rightarrow\chi^{\dagger}+\phi$
can also change the DM number by $-2$ units, whereas its CP-conjugate process,
$\chi^{\dagger}+\phi\rightarrow\chi+\phi$ changes the DM number by $+2$ units.
Therefore, if the CP-violation in this process is substantial, it can lead to
particle-antiparticle conversions, thereby affecting the DM asymmetry.
However, as discussed in Sec. 4.4, and detailed in Appendix B, we find the
contribution of this process to the CP-violation to be negligible in the
parameter region of interest in this study, and therefore, we have not
included the effect of this scattering process in generating the DM asymmetry.

###  4.3 Boltzmann equations and unitarity sum rules

We shall compute, in Sec. 4.4, the reaction rates for the processes described
in the previous section, and the possible CP-violation in them in terms of the
coupling and mass parameters appearing in Eqs. 102 and 103. However, to
understand the evolution dynamics of the number densities of the $\chi$ and
$\chi^{\dagger}$ particles in the thermal bath, it is useful to first study
the kinetic equations with the thermally averaged reaction rates as inputs.
Through this process, we shall arrive at an understanding of the relationships
between these reaction rates and their consequences — in the DM density, the
CP-violation in the above reactions, and the resulting possible particle-
antiparticle asymmetry in the DM sector. We take up this study in this
section.

The Boltzmann kinetic equations Kolb:1990vq governing the number densities of
DM particle and antiparticle are given as the following, appropriately
accounting for distinct processes along with their conjugate ones.

$\displaystyle\frac{dn_{\chi}}{dt}+3Hn_{\chi}=-\int\prod^{4}_{i=1}\frac{d^{3}p_{i}}{(2\pi)^{3}2E_{p_{i}}}g^{2}_{\chi}(2\pi)^{4}\delta^{(4)}(p_{1}+p_{2}-p_{3}-p_{4})\bigg{[}2\bigg{(}|M|^{2}_{\chi\chi\rightarrow\chi^{\dagger}\chi^{\dagger}}f_{\chi}(p_{1})f_{\chi}(p_{2})$
$\displaystyle-|M|^{2}_{\chi^{\dagger}\chi^{\dagger}\rightarrow\chi\chi}f_{\chi^{\dagger}}(p_{3})f_{\chi^{\dagger}}(p_{4})\bigg{)}+\bigg{(}|M|^{2}_{\chi\chi\rightarrow\chi^{\dagger}\chi}f_{\chi}(p_{1})f_{\chi}(p_{2})-|M|^{2}_{\chi^{\dagger}\chi\rightarrow\chi\chi}f_{\chi^{\dagger}}(p_{3})f_{\chi}(p_{4})\bigg{)}$
$\displaystyle-\bigg{(}|M|^{2}_{\chi^{\dagger}\chi^{\dagger}\rightarrow\chi^{\dagger}\chi}f_{\chi^{\dagger}}(p_{1})f_{\chi^{\dagger}}(p_{2})-|M|^{2}_{\chi^{\dagger}\chi\rightarrow\chi^{\dagger}\chi^{\dagger}}f_{\chi^{\dagger}}(p_{3})f_{\chi}(p_{4})\bigg{)}$
$\displaystyle+2\bigg{(}|M|^{2}_{\chi\chi\rightarrow\phi\phi}f_{\chi}(p_{1})f_{\chi}(p_{2})-|M|^{2}_{\phi\phi\rightarrow\chi\chi}f_{\phi}(p_{3})f_{\phi}(p_{4})\bigg{)}$
$\displaystyle+\bigg{(}|M|^{2}_{\chi^{\dagger}\chi\rightarrow\phi\phi}f_{\chi^{\dagger}}(p_{1})f_{\chi}(p_{2})-|M|^{2}_{\phi\phi\rightarrow\chi^{\dagger}\chi}f_{\phi}(p_{3})f_{\phi}(p_{4})\bigg{)}\bigg{]}.$
(104)

Here, $H$ denotes the Hubble expansion parameter, $f_{i}(p_{i})$ are the
distribution functions of the particle type $i$, and $g_{\chi}$ is the number
of internal degrees of freedom of the $\chi$ field in general, which is equal
to $1$ in the present case. The corresponding equation for
$n_{\chi^{\dagger}}$ can be obtained by replacing $\chi$ by $\chi^{\dagger}$
everywhere in the above equation, and vice versa. We have not explicitly shown
the appropriate symmetry factors for identical particles in the initial and
final states, which have, however, been incorporated in our computations in
the following.

In terms of the standard dimensionless variables $Y_{\chi}=n_{\chi}/s$ and
$x=m_{\chi}/T$, where $s$ and $T$ are the entropy density and temperature of
the radiation bath, respectively, the Boltzmann equations can be re-written as

$\displaystyle\frac{dY_{\chi}}{dx}$
$\displaystyle=-\frac{s}{2Hx}\bigg{[}\left(\expectationvalue{\sigma
v}_{1}+\frac{\expectationvalue{\sigma v}_{2}}{2}+\expectationvalue{\sigma
v}_{3}\right)\left(Y^{2}_{\chi}-Y^{2}_{\chi^{\dagger}}\right)+\expectationvalue{\sigma
v}_{3}\left(Y^{2}_{\chi^{\dagger}}-Y^{2}_{0}\right)$
$\displaystyle+\expectationvalue{\epsilon\sigma
v}_{1}\left(Y^{2}_{\chi^{\dagger}}-Y^{2}_{0}\right)+\expectationvalue{\epsilon\sigma
v}_{2}\left(Y_{\chi^{\dagger}}Y_{\chi}+\frac{Y^{2}_{\chi^{\dagger}}}{2}-\frac{Y^{2}_{\chi}}{2}-Y^{2}_{0}\right)+2\expectationvalue{\sigma
v}_{A}\left(Y_{\chi^{\dagger}}Y_{\chi}-Y^{2}_{0}\right)\bigg{]}$
$\displaystyle\frac{dY_{\chi^{\dagger}}}{dx}$
$\displaystyle=-\frac{s}{2Hx}\bigg{[}\left(\expectationvalue{\sigma
v}_{1}+\frac{\expectationvalue{\sigma v}_{2}}{2}+\expectationvalue{\sigma
v}_{3}\right)\left(Y^{2}_{\chi^{\dagger}}-Y^{2}_{\chi}\right)+\expectationvalue{\sigma
v}_{3}\left(Y^{2}_{\chi}-Y^{2}_{0}\right)$
$\displaystyle-\expectationvalue{\epsilon\sigma
v}_{1}\left(Y^{2}_{\chi}-Y^{2}_{0}\right)-\expectationvalue{\epsilon\sigma
v}_{2}\left(Y_{\chi^{\dagger}}Y_{\chi}+\frac{Y^{2}_{\chi}}{2}-\frac{Y^{2}_{\chi^{\dagger}}}{2}-Y^{2}_{0}\right)+2\expectationvalue{\sigma
v}_{A}\left(Y_{\chi^{\dagger}}Y_{\chi}-Y^{2}_{0}\right)\bigg{]}.$

Here, $Y_{0}=\frac{1}{s}\int\frac{d^{3}p}{(2\pi)^{3}}g_{\chi}f_{0}(p)$, with
$f_{0}(p)=e^{-\frac{E(p)}{T}}$ being the equilibrium distribution function
with zero chemical potential. We have also introduced the thermally averaged
symmetric and asymmetric reaction rates $\expectationvalue{\sigma v}_{f}$ and
$\expectationvalue{\epsilon\sigma v}_{f}$ for a particular final state $f$,
respectively, where the latter is defined as follows:

$\expectationvalue{\epsilon\sigma
v}_{f}=\dfrac{\int\prod^{4}_{i=1}\frac{d^{3}p_{i}}{(2\pi)^{3}2E_{p_{i}}}(2\pi)^{4}\delta^{(4)}(p_{1}+p_{2}-p_{3}-p_{4})\,\epsilon_{f}(p_{i})|M_{0}|^{2}_{f}f_{0}(p_{1})f_{0}(p_{2})}{\int\dfrac{d^{3}p_{1}}{(2\pi)^{3}}\dfrac{d^{3}p_{2}}{(2\pi)^{3}}f_{0}(p_{1})f_{0}(p_{2})}\hskip
14.22636pt,$ (106)

with $|M_{0}|^{2}_{f}=|M|^{2}_{\chi\chi\rightarrow
f}+|M|^{2}_{\chi^{\dagger}\chi^{\dagger}\rightarrow f^{\dagger}}$, and
$\epsilon_{f}(p_{i})$ is given by

$\displaystyle\epsilon_{f}(p_{i})=\frac{|M|^{2}_{\chi\chi\rightarrow
f}-|M|^{2}_{\chi^{\dagger}\chi^{\dagger}\rightarrow
f^{\dagger}}}{|M|^{2}_{\chi\chi\rightarrow
f}+|M|^{2}_{\chi^{\dagger}\chi^{\dagger}\rightarrow f^{\dagger}}}.$ (107)

The corresponding thermally averaged symmetric reaction rates
$\expectationvalue{\sigma v}_{f}$ can be computed using Eq. 144 by removing
the $\epsilon_{f}(p_{i})$ factors. Finally, we denote the reaction rates for
$\chi+\chi\rightarrow\chi^{\dagger}+\chi^{\dagger}$ by
$\expectationvalue{\sigma v}_{1}$ and $\expectationvalue{\epsilon\sigma
v}_{1}$, for $\chi+\chi\rightarrow\chi+\chi^{\dagger}$ by
$\expectationvalue{\sigma v}_{2}$ and $\expectationvalue{\epsilon\sigma
v}_{2}$ and for $\chi+\chi\rightarrow\phi+\phi$ by $\expectationvalue{\sigma
v}_{3}$ and $\expectationvalue{\epsilon\sigma v}_{3}$. For the CP-conserving
reaction $\chi+\chi^{\dagger}\rightarrow\phi+\phi$ we denote the average
reaction rate by $\expectationvalue{\sigma v}_{A}$.

We have parametrized the CP-violation resulting from the first three reactions
discussed in Sec. 4.2 in terms of $\expectationvalue{\epsilon\sigma v}_{1}$
and $\expectationvalue{\epsilon\sigma v}_{2}$ in Eqs. LABEL:eq:boltz_p, and
have eliminated $\expectationvalue{\epsilon\sigma v}_{3}$ from the Boltzmann
equations using unitarity sum rules. This stems from the fact that the
amplitudes for the CP-violating processes are related by CPT and S-matrix
unitarity as follows Kolb:1979qa ; Baldes:2014gca ; Baldes:2015lka :

$\displaystyle\sum_{f}\int dPS_{f}|M|^{2}_{\chi\chi\rightarrow f}=\sum_{f}\int
dPS_{f}|M|^{2}_{f\rightarrow\chi\chi}=\sum_{f}\int
dPS_{f}|M|^{2}_{\chi^{\dagger}\chi^{\dagger}\rightarrow f^{\dagger}},$ (108)

where we have used the S-matrix unitarity in the first equality, and CPT
conservation in the second one. Here, the integral over $dPS_{f}$ sum over the
momenta of the particles in the state $f$, as well as any discrete label that
may be carried by $f$. There is a further sum over all possible final states
$f$ that may be obtained starting from the initial state of $\chi\chi$, which
has been explicitly indicated. This unitarity sum rule in Eq. 108, together
with the definition of the CP-violation parameter $\epsilon_{f}$ in Eq. 107,
imply the following sum rule relating the CP-violation in all the channels

$\displaystyle\sum_{f}\int dPS_{f}\,\,\epsilon_{f}\,|M_{0}|^{2}_{f}=0,$ (109)

where, the quantity $|M_{0}|^{2}_{f}$ has been defined above. Using this sum
rule in Eq. 109 for the three relevant CP-violating reactions with the initial
state of $\chi\chi$ and final states in the set
$f=\\{\chi^{\dagger}+\chi^{\dagger},\chi+\chi^{\dagger},\phi+\phi\\}$, we can
express one of the asymmetric rates $\expectationvalue{\epsilon\sigma v}_{f}$
in terms of the other two, and therefore, can write the Boltzmann equations in
terms of only two asymmetric reaction rates as in Eq. LABEL:eq:boltz_p.

####  4.3.1 Analytic solutions

We now discuss approximate analytic solutions to the Boltzmann equations
LABEL:eq:boltz_p. For this, it is convenient to define the symmetric and
asymmetric yields, $Y_{S}=Y_{\chi}+Y_{\chi^{\dagger}}$ and
$Y_{\Delta\chi}=Y_{\chi}-Y_{\chi^{\dagger}}$, respectively, and rewrite the
Boltzmann equations in terms of these variables as

$\displaystyle\dfrac{dY_{S}}{dx}$
$\displaystyle=-\dfrac{s}{2Hx}\left[\expectationvalue{\sigma
v}_{A}\left(Y^{2}_{S}-Y^{2}_{\Delta\chi}-4Y^{2}_{0}\right)+\expectationvalue{\sigma
v}_{3}\bigg{(}\dfrac{Y^{2}_{S}+Y^{2}_{\Delta\chi}-4Y^{2}_{0}}{2}\bigg{)}-\expectationvalue{\epsilon\sigma
v}_{S}Y_{S}Y_{\Delta\chi}\right]$ $\displaystyle\dfrac{dY_{\Delta\chi}}{dx}$
$\displaystyle=-\dfrac{s}{2Hx}\left[\expectationvalue{\epsilon\sigma
v}_{S}\left(\dfrac{Y^{2}_{S}-4Y^{2}_{0}}{2}\right)+\expectationvalue{\epsilon\sigma
v}_{D}\dfrac{Y^{2}_{\Delta\chi}}{2}+\expectationvalue{\sigma
v}_{all}\,Y_{S}Y_{\Delta\chi}\right].$ (110)

In order to write these equations in a compact form, we have further defined
the following quantities: $\expectationvalue{\epsilon\sigma
v}_{S}=\expectationvalue{\epsilon\sigma
v}_{1}+\expectationvalue{\epsilon\sigma v}_{2}$,
$\expectationvalue{\epsilon\sigma v}_{D}=\expectationvalue{\epsilon\sigma
v}_{1}-\expectationvalue{\epsilon\sigma v}_{2}$ and $\expectationvalue{\sigma
v}_{all}=2\expectationvalue{\sigma v}_{1}+\expectationvalue{\sigma
v}_{2}+\expectationvalue{\sigma v}_{3}$. Let us discuss the analytic solutions
in two different regions of the scaled temperature variable $x$. In the first
region, we consider values of $x$ upto its freeze-out value of
$x_{F}=m_{\chi}/T_{F}$, i.e., $1\leq x\leq x_{F}$. Here, at the temperature
$T_{F}$, all the relevant chemical reactions involving the DM particle freeze-
out. In the second region, we consider values $x>x_{F}$.

For $1\leq x\leq x_{F}$, we can parametrize the number densities of the DM and
anti-DM particles as

$Y_{\chi(\chi^{\dagger})}=Y_{0}(1+\delta_{1(2)}),$ (111)

where, $\delta_{1}$ and $\delta_{2}$ parametrize the small deviations away
from their common equilibrium value with zero chemical potential, $Y_{0}$. We
expect that in this region, the deviations will satisfy the condition
$|\delta_{1,2}|<<1$. In the presence of CP-violating scatterings, in general,
$|\delta_{1}|\neq|\delta_{2}|$. In terms of this parametrization, we have
$Y_{S}=2Y_{0}(1+\delta)$ and $Y_{\Delta\chi}=Y_{0}\bar{\delta}$, where
$\delta=(\delta_{1}+\delta_{2})/2$ and $\bar{\delta}=\delta_{1}-\delta_{2}$.

Dropping terms quadratic in the small deviations, i.e., terms proportional to
$\delta^{2}$, $\bar{\delta}^{2}$ and $\delta\bar{\delta}$, we can solve Eqs.
110 algebraically. The total DM and anti-DM yield is obtained to be as
follows:

$Y_{S}(x)=2Y_{0}\left[1+\dfrac{Hx}{sY_{0}}\dfrac{\expectationvalue{\sigma
v}_{all}}{2\expectationvalue{\sigma v}_{ann}\expectationvalue{\sigma
v}_{all}+\expectationvalue{\epsilon\sigma
v}^{2}_{S}}\right]~{}~{}~{}~{}{\rm(for~{}}1\leq x\leq x_{F}),$ (112)

whereas, the yield for the DM-anti-DM asymmetry is given by

$|Y_{\Delta\chi}(x)|=\dfrac{2Hx}{s}\,\dfrac{\expectationvalue{\epsilon\sigma
v}_{S}}{2\expectationvalue{\sigma v}_{ann}\expectationvalue{\sigma
v}_{all}+\expectationvalue{\epsilon\sigma
v}^{2}_{S}}~{}~{}~{}~{}{\rm(for~{}}1\leq x\leq x_{F}),$ (113)

where, we have defined $\expectationvalue{\sigma
v}_{ann}=\expectationvalue{\sigma v}_{A}+\expectationvalue{\sigma v}_{3}/2$.
In deriving these expressions, we have taken the non-relativistic form of
$Y_{0}$, which is applicable near the freeze-out temperature. We can draw
several important conclusions from these solutions:

  1. 1.

The CP-conserving DM and anti-DM annihilation process,
$\chi+\chi^{\dagger}\rightarrow\phi+\phi$, can indirectly control the
asymmetric yield. This can be seen in Eq. 113, where we find that
$|Y_{\Delta\chi}(x)|$ is inversely proportional to $\expectationvalue{\sigma
v}_{A}$. This is because the out-of-equilibrium number density of the relevant
species can be controlled by this CP-conserving reaction, which, in turn,
controls the asymmetric yield. We have discussed the role of CP-conserving
processes in producing cosmological particle-antiparticle asymmetries in
detail in Ref. Ghosh:2021ivn .

  2. 2.

In our parametrization of CP-violation in the Boltzmann equations
LABEL:eq:boltz_p, $\expectationvalue{\epsilon\sigma
v}_{S}=\expectationvalue{\epsilon\sigma
v}_{1}+\expectationvalue{\epsilon\sigma v}_{2}$ acts as a source term for the
asymmetry, as seen in Eq. 113. This is because, as explained earlier,
$\expectationvalue{\epsilon\sigma v}_{3}$ cannot be independently varied, and
is determined in terms of $\expectationvalue{\epsilon\sigma v}_{1}$ and
$\expectationvalue{\epsilon\sigma v}_{2}$ through the unitarity sum rules in
Eq. 109.

  3. 3.

We see in Eq. 112 that the the total DM yield is significantly affected by the
pair annihilation process $\chi+\chi^{\dagger}\rightarrow\phi+\phi$. This is
because, we expect $\expectationvalue{\epsilon\sigma
v}^{2}_{S}<<\expectationvalue{\sigma v}_{ann}\expectationvalue{\sigma
v}_{all}$, as the asymmetric scattering rates stem from the interference of
tree and loop level amplitudes, and are therefore suppressed compared to the
symmetric scattering rates. With this approximation, we have

$Y_{S}\sim 2Y_{0}\left(1+\dfrac{Hx}{sY_{0}(2\expectationvalue{\sigma
v}_{A}+\expectationvalue{\sigma v}_{3})}\right)$ (114)

In the second region with $x>x_{F}$, we can no longer ignore the terms
quadratic in the deviations $\delta$ and $\bar{\delta}$, and the analytic
solutions in closed form become cumbersome. However, the results are tractable
under certain simplifying assumptions. We note that for $x>x_{F}$, the self-
scattering reactions that can generate the asymmetry are mostly decoupled
since the annihilation rate is proportional to the DM velocity, as detailed in
the next section. Thus we see that in the Boltzmann equation for
$Y_{\Delta\chi}$ in Eq. 110 the terms in the right hand side (RHS)
proportional to $\expectationvalue{\epsilon\sigma v}_{S}$ and
$\expectationvalue{\epsilon\sigma v}_{D}$ will be very small as well.
Furthermore, since $Y_{S}\propto 1/\expectationvalue{\sigma v}_{A}$, the last
term in the RHS will be proportional to the ratio $\expectationvalue{\sigma
v}_{all}/\expectationvalue{\sigma v}_{A}$. Therefore, in the regime
$\expectationvalue{\sigma v}_{all}<<\expectationvalue{\sigma v}_{A}$, i.e.,
when the dominant scattering process is the CP-even annihilation,
$Y_{\Delta\chi}$ essentially remains a constant, being frozen at its value
around $x=x_{F}$. With this as input, we can solve the equation for $Y_{S}$ in
Eq. 110, and for dominantly s-wave contribution to $\expectationvalue{\sigma
v}_{A}$, the solution is given by

$\displaystyle Y_{S}(x>x_{F})$
$\displaystyle=|Y^{F}_{\Delta\chi}|\,\,\dfrac{1+r_{F}\exp[\lambda\,|Y^{F}_{\Delta\chi}|\,(x^{-1}-x^{-1}_{F})]}{1-r_{F}\exp[\lambda\,|Y^{F}_{\Delta\chi}|\,(x^{-1}-x^{-1}_{F})]},$
(115)

where,

$r_{F}=\dfrac{Y^{F}_{S}-|Y^{F}_{\Delta\chi}|}{Y^{F}_{S}+|Y^{F}_{\Delta\chi}|}.$
(116)

Here, $\lambda=1.32\,m_{\chi}M_{Pl}\,g^{1/2}_{*}\expectationvalue{\sigma
v}_{A}$, and $Y^{F}_{S}$ and $Y^{F}_{\Delta\chi}$ represent the symmetric and
asymmetric yields of DM obtained at $x=x_{F}$ using Eqs. 112 and 113. We shall
compare these approximate analytical results with the numerical solutions in
the next sub-section.

####  4.3.2 Numerical results

Figure 13: Evolution of the symmetric (left panel, with
$Y_{S}=Y_{\chi}+Y_{\chi^{\dagger}}$) and the asymmetric yields (right panel,
with $|Y_{\Delta\chi}|=|Y_{\chi}-Y_{\chi^{\dagger}}|$), as a function of the
scaled inverse temperature variable $m_{\chi}/T$. The solid lines in both the
figures represent the exact numerical solution of Eqs. LABEL:eq:boltz_p. The
dashed lines show the predictions based on the approximate analytic solutions,
namely, Eqs. 112 and 113. The dark matter mass and CP-violating symmetric and
asymmetric reaction rates have been kept fixed. We have shown the results for
two different choices of the CP-conserving pair annihilation rate
$\expectationvalue{\sigma v}_{A}$ for the process
$\chi+\chi^{\dagger}\rightarrow\phi+\phi$ (by the blue and pink lines) to
demonstrate its impact not only on the total yield, but also in indirectly
controlling the particle anti-particle asymmetry as well.

Having obtained the approximate analytic solutions to the sum and difference
of DM and anti-DM yields for different ranges of the temperature in the
thermal bath, we now go on to solve Eqs LABEL:eq:boltz_p numerically to
understand the exact relationship between the DM properties at freeze-out and
the microscopic parameters. In Fig. 13 we have shown the evolution of the
symmetric (left panel, with $Y_{S}=Y_{\chi}+Y_{\chi^{\dagger}}$) and the
asymmetric yields (right panel, with
$|Y_{\Delta\chi}|=|Y_{\chi}-Y_{\chi^{\dagger}}|$), as a function of the scaled
inverse temperature variable $m_{\chi}/T$. The solid lines in both the figures
represent the exact numerical solution of Eqs. LABEL:eq:boltz_p. The dashed
lines show the predictions based on the approximate analytic solutions,
namely, Eqs. 112 and 113. The dark matter mass and CP-violating symmetric and
asymmetric reaction rates have been kept fixed. We have shown the results for
two different choices of the CP-conserving pair annihilation rate
$\expectationvalue{\sigma v}_{A}$ for the process
$\chi+\chi^{\dagger}\rightarrow\phi+\phi$, by the blue and pink lines, to
demonstrate its impact not only on the total yield, but also in the particle
anti-particle asymmetry as well. As we can see, by increasing
$\expectationvalue{\sigma v}_{A}$, one obtains a lower value of
$|Y_{\Delta\chi}|$ throughout the evolution. We also see from Fig. 13 (right
panel) that a small out-of-equilibrium asymmetry $|Y_{\Delta\chi}|$ is
generated from around $x\sim 1$, which eventually grows and freezes at around
$x\sim x_{F}$.

The analytic approximation is found to be in good agreement with the exact
numerical solutions, the maximum difference between the two being of the order
of $10\%$. The larger differences are found for smaller values of $x$. This is
because in order to obtain an approximate closed form analytic solution to
Eqs. LABEL:eq:boltz_p, we have taken $dY_{0}/dx\sim-Y_{0}$, and have dropped
an additional term proportional to $1/x$. This approximation is justified for
$x>1$, which is the main region of interest for determining the relic
abundance. However, it is no longer a valid approximation for $x\sim 1$, where
the additional $x-$dependent term also contributes, and an accurate (better
than the $10\%$ accuracy obtained here) closed form solution cannot be
obtained for $x\sim 1$ while retaining such a term.

We note that the patching of the analytic solutions in the regions $1\leq
x\leq x_{F}$ and $x>x_{F}$ requires an input of the value of $x_{F}$ itself,
and is therefore sensitive to how precisely we can estimate $x_{F}$. However,
the asymptotic values of the yields $Y_{S}$ and $|Y_{\Delta\chi}|$ are found
to be well approximated by the analytic solutions. Here, we have estimated
$x_{F}$ as follows. Since we have $Y_{S}=2Y_{0}(1+\delta)$, and in the first
patch the net deviation from equilibrium $\delta<1$, we can use this condition
in the expression for $Y_{S}$ in Eq. 114, and obtain a relation which
determines the boundary of the two regions $x_{F}$ as

$\frac{H(x_{F})x_{F}}{s(x_{F})Y_{0}(x_{F})(2\expectationvalue{\sigma
v}_{A}+\expectationvalue{\sigma v}_{3})}\sim 1.$ (117)

An iterative solution of this equation gives a good estimate of the value of
the scaled freeze-out temperature $x_{F}$.

###  4.4 CP-violation and scattering rates in the complex singlet model

We now compute the relevant CP-violation parameters and the thermally averaged
symmetric and asymmetric scattering rates in the complex scalar singlet model,
as a function of the model parameters. We then go on to study the dependence
of the DM relic abundance and the possible particle-antiparticle asymmetry in
the DM sector on such parameters, with a particular emphasis on the interplay
between the DM self-scattering and annihilation in controlling the DM density
and composition.

We first take up the three CP-violating processes one by one, and discuss the
corresponding results at next-to-leading order in perturbation theory,
following which we show the leading order results for the CP-conserving
process.

####  4.4.1 Process 1: Self-scattering
$\chi+\chi\rightarrow\chi^{\dagger}+\chi^{\dagger}$

The relevant tree and one-loop level Feynman diagrams for the self-scattering
process $\chi+\chi\rightarrow\chi^{\dagger}+\chi^{\dagger}$ are shown in Fig.
14.

Figure 14: Relevant tree and one-loop level Feynman diagrams for the CP-
violating self-scattering process
$\chi+\chi\rightarrow\chi^{\dagger}+\chi^{\dagger}$.

The CP-violation in this process arises from the interference of the tree
level and one-loop graphs shown in Fig. 14. There are additional diagrams for
this process at the one-loop level at the same order in the couplings.
However, those diagrams do not give rise to any CP-violating imaginary parts.
Therefore, we have restricted ourselves to the diagrams shown in Fig. 14. The
resulting difference in the matrix elements squared of the two CP-conjugate
processes is given by

$\displaystyle|M|^{2}_{\chi\chi\rightarrow\chi^{\dagger}\chi^{\dagger}}-|M|^{2}_{\chi^{\dagger}\chi^{\dagger}\rightarrow\chi\chi}=4\,\left[\rm
Im(\lambda^{*}_{2}\lambda^{2}_{3})\,Im\,I_{\phi\phi}+\,\rm
Im(\lambda^{*}_{2}\lambda^{2}_{4})\,Im\,I_{\chi^{\dagger}\chi}\right].$ (118)

The imaginary part of the loop amplitudes are non-zero when internal lines in
the loop go onshell, and using the Cutkosky rules Peskin:1995ev we obtain them
to be

$\displaystyle\rm Im\,I_{\phi\phi}=\dfrac{\beta_{\phi}}{32\pi},\hskip
14.22636ptIm\,I_{\chi^{\dagger}\chi}=\dfrac{\beta_{\chi}}{16\pi}.$ (119)

Here, the factor of $\beta_{f}=(1-4m^{2}_{f}/s)^{1/2}$ comes from the two-body
phase-space integral, with the usual Mandelstam variable
$s=(p_{1}+p_{2})^{2}$, where $p_{1}$ and $p_{2}$ are the four-momenta of the
two initial state particles. Since we take $m_{\phi}<<m_{\chi}$, we can
approximate $\beta_{\phi}\sim 1$.

With the amplitudes in hand, we can now compute the thermally averaged
symmetric rate of this self-scattering process as follows:

$\displaystyle\expectationvalue{\sigma v}_{1}=\dfrac{|\lambda_{2}|^{2}}{16\pi
m^{2}_{\chi}}\dfrac{1}{\sqrt{\pi x}},$ (120)

where we have used the partial wave expansion of the annihilation rate,
keeping only the leading term in the non-relativistic limit. As mentioned in
the previous sections, and as we can see from this expression, the leading
term in the self-scattering rate is proportional to the DM velocity in the
thermal bath. Therefore, in the non-relativistic regime, this rate may be
suppressed. It is sufficient to consider only the tree-level amplitude for
computing the leading contribution to the symmetric annihilation rate. In
obtaining the thermal average, we have included the final state symmetry
factors here, while the initial state symmetry factors have already been taken
into account in the Boltzmann equations.

We can, similarly, obtain the thermal average of the asymmetric reaction rate,
which stems from the interference of the tree level and one-loop level
amplitudes. The resulting expression is given as

$\displaystyle\expectationvalue{\epsilon\sigma v}_{1}$
$\displaystyle=\dfrac{1}{(16\pi m_{\chi})^{2}}\left[\dfrac{1}{\sqrt{\pi
x}}\text{Im}(\lambda^{*}_{2}\lambda^{2}_{3})+\dfrac{3}{2x}\text{Im}(\lambda^{*}_{2}\lambda^{2}_{4})\right].$
(121)

Here again, the leading term is proportional to the DM velocity, whereas the
sub-leading term is proportional to the square of the DM velocity. With this,
the effective CP-violation parameter can be obtained as follows:

$\epsilon^{1}_{eff}=\expectationvalue{\epsilon\sigma
v}_{1}/\expectationvalue{\sigma v}_{1}.$ (122)

For $\mathcal{O}(1)$ values of the absolute values of $\lambda_{i}$, and the
phase of the complex coupling combination $\lambda_{2}^{*}\lambda_{3}^{2}$
(denoted by $\theta_{23}$) set to be $\pi/2$, we obtain for $x>>1$:

$\displaystyle\epsilon^{1}_{eff}\simeq\dfrac{1}{16\pi},\,\,\,\,\text{for}\,\,x>>1.$
(123)

Thus for $\mathcal{O}(1)$ couplings, $\epsilon^{1}_{eff}$ is expected to be
small, around $0.02$, but can of course be made larger with larger couplings.

####  4.4.2 Process 2: Self-scattering
$\chi+\chi\rightarrow\chi^{\dagger}+\chi$

The second CP-violating self-scattering process,
$\chi+\chi\rightarrow\chi^{\dagger}+\chi$ shows very similar behaviour as far
as the reaction rates are concerned, except the fact that different coupling
combinations appear. In particular, the coupling $\lambda_{5}$, which controls
the rate of CP-conserving $\chi+\chi^{\dagger}\rightarrow\phi+\phi$
annihilation, now appears in the dominant loop contribution to the
$\chi+\chi\rightarrow\chi^{\dagger}+\chi$ process, as can be seen from the
Feynman diagrams in Fig. 15. This makes the rate of CP-violation in this
process and the rate of CP-conserving annihilations related.

Figure 15: Relevant tree and one-loop level Feynman diagrams for the CP-
violating self-scattering process $\chi+\chi\rightarrow\chi^{\dagger}+\chi$.

We can now similarly compute the relevant difference between the squared
matrix elements from the interference of the graphs in Fig. 15

$\displaystyle|M|^{2}_{\chi\chi\rightarrow\chi^{\dagger}\chi}-|M|^{2}_{\chi^{\dagger}\chi^{\dagger}\rightarrow\chi^{\dagger}\chi}=4\,\left[\rm
Im(\lambda_{2}\lambda^{*2}_{4})\,Im\,I_{\chi^{\dagger}\chi^{\dagger}}+\,\rm
Im(\lambda^{*}_{4}\lambda_{3})\lambda_{5}\,Im\,I_{\phi\phi}\right],$ (124)

with the following imaginary part of the new loop amplitude, in addition to
the ones shown in Eq. 119:

$\displaystyle\rm
Im\,I_{\chi^{\dagger}\chi^{\dagger}}=\dfrac{\beta_{\chi}}{32\pi}.$ (125)

The thermal averaged symmetric annihilation rate shows the same velocity
dependence as before, and is given by

$\displaystyle\expectationvalue{\sigma v}_{2}=\dfrac{|\lambda_{4}|^{2}}{8\pi
m^{2}_{\chi}}\dfrac{1}{\sqrt{\pi x}}\hskip 28.45274pt.$ (126)

Similarly the $x-$dependence of the asymmetric annihilation rate is also
similar, but now with the explicit appearance of the $\lambda_{5}$ coupling

$\displaystyle\expectationvalue{\epsilon\sigma v}_{2}$
$\displaystyle=\dfrac{1}{(16\pi
m_{\chi})^{2}}\left[\dfrac{3}{2x}\text{Im}(\lambda_{2}\lambda^{*2}_{4})+\dfrac{2}{\sqrt{\pi
x}}\text{Im}(\lambda^{*}_{4}\lambda_{3})\lambda_{5}\right].$ (127)

Thus, the CP-violation will proportionally increase if we increase the pair-
annihilation rate $\expectationvalue{\sigma v}_{A}$. We can also write the
effective CP-violation parameter as

$\displaystyle\epsilon^{2}_{eff}\simeq\dfrac{\lambda_{5}}{16\pi},\,\,\,\,\text{for}\,\,x>>1.$
(128)

Here, as before, we have set $|\lambda_{i}|=\mathcal{O}(1)$, except for
$\lambda_{5}$ which is kept variable, and all the effective phases to be
$\pi/2$. We shall see in the following that to obtain a large asymmetry in the
DM sector, one generally requires a large rate of $\expectationvalue{\sigma
v}_{A}$, and hence a large $\lambda_{5}$. In such cases, we may have
$\epsilon^{2}_{eff}\simeq 0.25$, for $\lambda_{5}\sim\mathcal{O}{(4\pi)}$.

####  4.4.3 Process 3: CP-violating Annihilation
$\chi+\chi\rightarrow\phi+\phi$

Figure 16: Relevant tree and one-loop level Feynman diagrams for the CP-
violating annihilation process $\chi+\chi\rightarrow\phi+\phi$.

Finally, we come to the third CP-violating process, which is an annihilation
that reduces the total DM and anti-DM number as well, unlike the previous two
self-scattering processes. The relevant Feynman diagrams are shown in Fig. 16.
One of the loop diagrams here is again proportional to the CP-conserving
$\lambda_{5}$ coupling. The difference between the matrix elements squared of
the CP-conjugate processes is

$\displaystyle|M|^{2}_{\chi\chi\rightarrow\phi\phi}-|M|^{2}_{\chi^{\dagger}\chi^{\dagger}\rightarrow\phi\phi}=4\,\left[\rm
Im(\lambda^{*}_{3}\lambda_{4})\lambda_{5}\,Im\,I_{\chi^{\dagger}\chi}+\,\rm
Im(\lambda_{2}\lambda^{*2}_{3})\,Im\,I_{\chi^{\dagger}\chi^{\dagger}}\right]$
(129)

Unlike in the self-scatterings, the thermally averaged symmetric scattering
rate is no longer velocity suppressed in the leading term, and is given as

$\displaystyle\expectationvalue{\sigma v}_{3}=\dfrac{|\lambda_{3}|^{2}}{32\pi
m^{2}_{\chi}}\left(1+\dfrac{3}{4x}\right).$ (130)

The asymmetric scattering rate also shows a somewhat different behaviour, with
both terms in the thermal average now suppressed by a single power of the DM
velocity:

$\displaystyle\expectationvalue{\epsilon\sigma v}_{3}$
$\displaystyle=\dfrac{1}{(16\pi m_{\chi})^{2}}\dfrac{1}{\sqrt{\pi
x}}\bigg{[}\text{Im}(\lambda_{2}\lambda^{*2}_{3})+2\,\text{Im}(\lambda^{*}_{3}\lambda_{4})\lambda_{5}\bigg{]}.$
(131)

Thus, the effective CP-violation parameter, for
$|\lambda_{i}|\sim\mathcal{O}(1)$ and the phases in the relevant coupling
combinations set to $\pi/2$, now carries an explicit $x-$dependence even in
the leading term:

$\displaystyle\epsilon^{3}_{eff}\simeq-\dfrac{1}{8\pi}\dfrac{1+2\lambda_{5}}{\sqrt{\pi
x}},\,\,\,\,\text{for}\,\,x>>1.$ (132)

With all our computations in place, we can check the unitarity sum rules
explicitly upto the next-to-leading order in perturbation theory. First of
all, using Eqs. 118, 124 and 129, we see that the sum rule in Eq. 109 holds,
namely

$\int dPS_{2}\,\epsilon_{1}\,|M_{0}|^{2}_{1}+\int
dPS_{2}\,\epsilon_{2}\,|M_{0}|^{2}_{2}+\int
dPS_{2}\,\epsilon_{3}\,|M_{0}|^{2}_{3}=0.$ (133)

The same unitarity sum rule can also be written in terms of the thermally
averaged asymmetric reaction rates:

$\displaystyle\expectationvalue{\epsilon\sigma
v}_{1}+\expectationvalue{\epsilon\sigma
v}_{2}+\expectationvalue{\epsilon\sigma v}_{3}=0,$ (134)

which we can again easily check at this order in perturbation theory using the
expressions in Eqs. 121, 127 and 131.

We now point out a key observation of our study. If the CP-violating
annihilation process is absent, i.e., if we set $\lambda_{3}=0$, then we can
see from Eq. 134 that $\expectationvalue{\epsilon\sigma
v}_{1}+\expectationvalue{\epsilon\sigma v}_{2}=0$. Equivalently, this can be
verified also by setting $\lambda_{3}=0$ in Eqs. 121 and 127. And therefore,
by Eq. 110, the asymmetric yield will not be generated, starting from a
symmetric initial condition. This is because $\expectationvalue{\epsilon\sigma
v}_{S}(=\expectationvalue{\epsilon\sigma
v}_{1}+\expectationvalue{\epsilon\sigma v}_{2})$ acts as the source term for
the asymmetry in our parametrization of the Boltzmann equations, as seen from
both Eq. 110, and the approximate analytic solution in Eq. 113. Thus we see
that the presence of only the CP-violating self-scatterings cannot generate
the asymmetry. Similarly, only the presence of the CP-violating annihilation
cannot generate an asymmetry either, by the unitarity sum rule. It is the
simultaneous presence of both the self-scatterings and the annihilation that
can lead to a generation of particle-antiparticle asymmetries.

A remark about the number of independent phases in the coupling combinations
that appear in the $\expectationvalue{\epsilon\sigma v}_{i}$ is in order.
Defining each coupling as $\lambda_{i}=|\lambda_{i}|e^{i\theta_{i}}$, we can
write the three rates in terms of two relative phase angles, namely
$\theta_{23}=2\theta_{3}-\theta_{2}$ and $\theta_{24}=2\theta_{4}-\theta_{2}$.
The third relevant angle is
$\theta_{4}-\theta_{3}=(\theta_{24}-\theta_{23})/2$, and hence is not
independent. In what follows, we shall set $\theta_{23}=-{\theta}_{24}=\pi/2$,
to maximize the CP-violating effects.

####  4.4.4 Process 4: CP-conserving Annihilation
$\chi+\chi^{\dagger}\rightarrow\phi+\phi$

We have already mentioned the role of the CP-conserving annihilation in
indirectly controlling the particle-antiparticle asymmetry in Sec. 4.3. In
addition to the asymmetry, such a pair annihilation process, if present, also
strongly affects the final relic abundance of the DM and anti-DM particles.
The thermally averaged annihilation rate for this process in the complex
singlet scenario, keeping the first two terms in the partial-wave expansion,
is given as follows:

$\displaystyle\expectationvalue{\sigma v}_{A}=\dfrac{|\lambda_{5}|^{2}}{64\pi
m^{2}_{\chi}}\left(1+\dfrac{3}{4x}\right).$ (135)

####  4.4.5 The $\chi+\phi\rightarrow\chi^{\dagger}+\phi$ process

As mentioned in Sec. 4.2, we can also have the CP-violating DM anti-DM
conversion process $\chi+\phi\rightarrow\chi^{\dagger}+\phi$. As shown by the
explicit computation described in Appendix B, the net contribution to CP-
violation from all the Feynman diagrams contributing to this process is

$\displaystyle\int
dPS_{2}\left(|M|^{2}_{\chi\phi\rightarrow\chi^{\dagger}\phi}-|M|^{2}_{\chi^{\dagger}\phi\rightarrow\chi\phi}\right)\propto{\rm
Im(\lambda^{*}_{3}\mu_{1})}\times{\rm~{}Terms~{}of}~{}\mathcal{O}({\hat{\mu}^{3}/m_{\chi}^{4}}),$
(136)

where, $\hat{\mu}$ is either $\mu$, $\mu_{1}$ or $\mu_{\phi}$. This is in
contrast to the CP-violation stemming from the processes discussed in Sec.
4.2, which is independent of the dimensionful couplings. Since we have assumed
$\mu_{1}\simeq 0$ in order to avoid the stringent constraints from spin-
independent DM direct detection probes, clearly, the contribution to CP-
violation from the $\chi\phi\rightarrow\chi^{\dagger}\phi$ process is much
smaller compared to the ones we have included in the Boltzmann equations for
the (anti-)DM number densities.

####  4.4.6 Dark matter density and asymmetry: the interplay of different
processes

Figure 17: Contour in the $\lambda_{3}-m_{\chi}$ plane for which the DM and
anti-DM densities saturate the DM density of the Universe, with $\Omega
h^{2}\simeq 0.12$ (red solid line). Also shown are the contours of the
constant final asymmetry parameter $r_{\infty}=\lvert
Y_{\Delta\chi}\rvert/Y_{S}$ (blue dashed lines). Here, the coupling
$\lambda_{3}$ determines the symmetric annihilation rate for the
$\chi+\chi\rightarrow\phi+\phi$ process, and also the CP-violation through
loop effects. For this figure, we have set the CP-conserving pair-annihilation
coupling $\lambda_{5}=0$.

Having determined the CP-violation and the symmetric and asymmetric
annihilation rates in the complex scalar singlet model, we can now compute the
relic abundance of the DM and anti-DM system. Due to the possible CP-
violation, there can also be a resulting particle-antiparticle asymmetry in
the DM sector. We have defined the measure of the final particle-antiparticle
asymmetry parameter as $r_{\infty}=\lvert Y_{\Delta\chi}\rvert/Y_{S}$, where
the asymptotic values of the yields are considered. Clearly, we have $0\leq
r_{\infty}\leq 1$. Here, $r_{\infty}=0$ corresponds to the completely
symmetric limit, in which the asymptotic yields of the DM and anti-DM are the
same. On the other hand, $r_{\infty}=1$ corresponds to the completely
asymmetric limit, in which only either the DM or the anti-DM species survives.
As we shall see in the following, depending upon the dominance of either the
CP-conserving or the CP-violating pair-annihilation couplings, we obtain two
different regimes for the DM $-$ a nearly symmetric regime, and an asymmetric
regime.

To begin with, let us consider the range of DM mass that is allowed in this
scenario. We find that the DM relic abundance required for saturating the
observed DM density of $\Omega h^{2}\simeq 0.12$ Planck:2018vyg can be
obtained for $m_{\chi}$ in the range from $\mathcal{O}({\rm GeV})$ to
$\mathcal{O}(10{~{}\rm TeV})$, with the relevant couplings kept within their
perturbative limits. Since both $\lambda_{3}$ and $\lambda_{5}$ can play a
dominant role in determining the DM relic density, as seen in Sec. 4.3 and in
particular in Eqs. 114 and 115, we show the allowed DM mass range as a
function of these two couplings separately in Figs. 17 and 19.

Figure 18: Contour in the $\lambda_{3}-\lambda_{2,4}$ plane for which the DM
and anti-DM densities saturate the DM density of the Universe, with $\Omega
h^{2}\simeq 0.12$ (red solid line). Also shown are the contours of the
constant final asymmetry parameter $r_{\infty}=\lvert
Y_{\Delta\chi}\rvert/Y_{S}$ (blue dashed lines). We have set the two coupling
parameters determining the DM self-scattering rates to be equal
($\lambda_{2}=\lambda_{4}$) for simplicity. The CP-conserving coupling has
been fixed at $\lambda_{5}=0$, see text for details.

In Fig. 17, we show the contour in the $\lambda_{3}-m_{\chi}$ plane for which
the DM and anti-DM densities furnish $\Omega h^{2}\simeq 0.12$ (red solid
line). We also show the contours of the constant final asymmetry parameter
$r_{\infty}$ defined above (blue dashed lines). For this figure, we have fixed
the DM self-couplings as $\lambda_{2}=\lambda_{4}=1$, and have set the CP-
conserving pair-annihilation coupling $\lambda_{5}=0$. The reason for setting
$\lambda_{5}=0$ is to demonstrate the role of $\lambda_{3}$ in determining the
density. The effective CP-violating phases are set to $\pi/2$. As we can see
from this figure, increasing $\lambda_{3}$ implies a corresponding increase in
the mass $m_{\chi}$ that saturates the DM abundance. The final asymmetry
parameter $r_{\infty}$ is found to be rather small, of the order of $10^{-4}$
to $10^{-3}$, indicating a nearly symmetric DM. This is primarily because with
these choices of parameters the CP-violation is not large, and with the CP-
conserving pair-annihilation switched off, the symmetric part is not
subsequently removed as well. In addition to that, in this scenario, the
asymmetric reaction rate for the CP-violating processes are also reduced, as
seen in, for example, Eq. 127.

How do the DM density and the asymmetry parameter vary as a function of the
pair-annihilation ($\lambda_{3}$) and self-scattering couplings
($\lambda_{2},\lambda_{4}$)? We show this correlation in Fig. 18. For this
figure, we have kept the DM mass fixed at $m_{\chi}=100$ GeV, and continue to
consider $\lambda_{5}=0$. As we can see, the DM asymmetry continues to be
small, and in the parameter range of interest can vary from $10^{-5}$ to
$10^{-4}$. Furthermore, with such a small asymmetry, the DM self-scatterings
play no significant role in deciding the DM density, which is fixed by
$\lambda_{3}$. However, the DM composition varies with the variation of the
self-interaction couplings.

Figure 19: Contours in the $\lambda_{5}-m_{\chi}$ plane for which the DM and
anti-DM densities saturate the DM density of the Universe, with $\Omega
h^{2}\simeq 0.12$ (blue and red solid lines), for two different choices of
$\lambda_{2},\lambda_{3}$ and $\lambda_{4}$, as indicated against the lines.
Also shown are the colour-coded contours of the constant final asymmetry
parameter $r_{\infty}=\lvert Y_{\Delta\chi}\rvert/Y_{S}$ (with the blue and
red dashed lines corresponding to the scenarios with
$\lambda_{2}=\lambda_{3}=\lambda_{4}=1$ and
$\lambda_{2}=\lambda_{3}=\lambda_{4}=10^{-3}$, respectively). Here, the
coupling $\lambda_{5}$ determines the CP-conserving pair annihilation rate for
the $\chi+\chi^{\dagger}\rightarrow\phi+\phi$ process, and also the CP-
violation through loop effects.

We now move to the scenario in which the CP-conserving pair-annihilation
coupling is non-zero, and potentially large. We find that in such a case, a
larger asymmetry is obtained. Furthermore, we find a novel and interesting
effect that the self-scattering couplings now play an equally dominant role in
deciding both the DM density and asymmetry, as do the annihilation couplings.
We first show the contours in the $\lambda_{5}-m_{\chi}$ plane for which the
DM and anti-DM densities saturate the DM density of the Universe in Fig. 19.
The mass range obtained is again similar to the one obtained in Fig. 17. For
the blue solid line, we have fixed $\lambda_{2}=\lambda_{3}=\lambda_{4}=1$,
while for the red solid line we have set
$\lambda_{2}=\lambda_{3}=\lambda_{4}=10^{-3}$, for showing two illustrative
cases. We see that when $\lambda_{3}=1$, there is no significant variation in
the DM density for $\lambda_{5}<1$, since $\lambda_{3}$ also plays a dominant
role in determining it. However, for $\lambda_{5}>\lambda_{3}$, the density
changes rapidly with varying $\lambda_{5}$, as seen from the $\lambda_{5}>1$
region along the blue line, and the entire region along the red line.

Figure 20: (Left Panel:) Contours in the $\lambda_{3}-\lambda_{2,4}$ plane for
which the DM and anti-DM densities saturate the DM density of the Universe,
with $\Omega h^{2}\simeq 0.12$, for different values of the CP-even coupling
$\lambda_{5}$ and DM mass $m_{\chi}$. (Right Panel:) Contours of the constant
final asymmetry parameter $r_{\infty}=\lvert Y_{\Delta\chi}\rvert/Y_{S}$ (blue
dashed lines), with a representative value of $\lambda_{5}=4\pi$ and DM mass
$m_{\chi}=14$ TeV. We have set the two coupling parameters determining the DM
self-scattering rates to be equal ($\lambda_{2}=\lambda_{4}$) for simplicity.
In both the figures, we clearly see the strong interplay of the DM
annihilation and self-scattering processes in determining both the DM density
and asymmetry.

With the introduction of the CP-conserving pair annihilation, the asymmetry
parameter $r_{\infty}$ is seen to take values of the order of $10^{-1}$ (or
higher, as we shall see in the following). Thus we can obtain a large
asymmetry in the DM sector in this scenario. As discussed earlier, this is due
to the fact that after the CP-violating processes decouple, thereby freezing
out the difference in the DM and anti-DM yields $Y_{\Delta\chi}$, the CP-
conserving pair annihilation processes with larger rates can still remain
active. These pair annihilations at later epochs reduce the sum of the DM and
anti-DM yields $Y_{S}$. This leads to the enhancement of the final asymmetry
parameter $r_{\infty}=\lvert Y_{\Delta\chi}\rvert/Y_{S}$. The $r_{\infty}$
contours shown in Fig. 19 are colour coded, with the blue dashed contours
corresponding to the $\lambda_{2}=\lambda_{3}=\lambda_{4}=1$ scenario, while
the red dashed ones corresponding to the
$\lambda_{2}=\lambda_{3}=\lambda_{4}=10^{-3}$ scenario.

We now study the variation of the DM density and the asymmetry parameter as a
function of the pair-annihilation ($\lambda_{3}$) and self-scattering
couplings ($\lambda_{2},\lambda_{4}$), in the presence of the $\lambda_{5}$
coupling. As shown in the left panel of Fig. 20, there are strong correlations
between the annihilation and self-scattering in such a scenario. We have shown
the contours in this plane where the relic abundance is satisfied, with
different sets of values for the CP-conserving coupling $\lambda_{5}$ and the
DM mass $m_{\chi}$. For each choice of $\lambda_{5}$, we have chosen
$m_{\chi}$ such that the relic density condition can be satisfied in some
region in the coupling plane. The correlations are especially pronounced for
larger values of $\lambda_{5}$, as seen from the shown representative values
of $\lambda_{5}=10$ (pink dot-dashed line) and $\lambda_{5}=4\pi$ (red solid
line), both of which are within the limits of perturbation theory, such that
our computations can be reliable. There are small correlations for
$\lambda_{5}=5$ as well (blue dashed line), while no observable correlations
are found for $\lambda_{5}=2$ (violet dotted line). The $\lambda_{5}=2$
scenario is thus similar to the previously discussed case of no CP-conserving
annihilation, as both lead to a rather small final asymmetry.

In order to understand the physics behind these strong correlations, we pick
out one representative value of $\lambda_{5}=4\pi$, and study the variation of
the asymmetry parameter $r_{\infty}$ in the right panel of Fig. 20. We see
that $r_{\infty}$, and hence the asymmetry in the DM sector can be large in
this scenario, with the contours upto $r_{\infty}=0.45$ spanning a large range
of the parameter space. This explains the reason behind the strong
correlations being observed. For with such a large asymmetry, the total DM and
anti-DM yield strongly depends upon the asymmetric yield $Y_{\Delta\chi}$
itself, as can be seen from the analytic approximation in Eq. 115, i.e., the
asymmetry plays a dominant role in fixing the relic density. This is the
reason we observe the strong interplay between the DM self-scatterings and the
annihilations in fixing not only the DM asymmetry, but the DM relic density
itself. This role of the DM self-scatterings is a novel observation, which
should be of considerable interest in the study of DM cosmology. We would like
to emphasize again here that within this scenario of complex singlet scalar
DM, we cannot switch off the self-scattering processes, but have only the CP-
violating annihilations, or vice versa, and generate an asymmetry in the DM
sector. The simultaneous presence of both these types of processes is
necessary in order to have a non-zero CP-violation, as dictated by the
unitarity sum rules.

Figure 21: Role of DM self-scattering in determining the total DM relic
abundance $\Omega_{\chi}h^{2}$. (Left panel:) $\Omega_{\chi}h^{2}$ as a
function of the parameters $\lambda_{2}$ and $\lambda_{4}$ which determine the
self-scattering rate. The maximum deviation in the relic density from the case
with no self-scattering, for the choice of parameters in this figure, is
$\mathcal{O}(15\%)$, where we have defined the deviation as
$\Delta\Omega_{\chi}(\lambda_{2},\lambda_{4})=(\Omega_{\chi}(\lambda_{2},\lambda_{4})-\Omega_{\chi}(\lambda_{2}=0,\lambda_{4}=0))/\Omega_{\chi}(\lambda_{2}=0,\lambda_{4}=0)$.
(Right panel:) Contours of fixed values of $\Delta\Omega_{\chi}$ in the
$\lambda_{2}-\lambda_{4}$ plane, with $\Delta\Omega_{\chi}=15\%,10\%$ and
$5\%$.

In order to further understand the impact of the self-scattering in
determining the relic density, we show in Fig. 21 (left panel) the variation
in the total DM density $\Omega_{\chi}h^{2}$ as a function of the parameters
determining the self-scattering rate, namely $\lambda_{2}$ and $\lambda_{4}$.
For this figure, we have used a fixed ratio of these two parameters, with
$2\lambda_{2}=\lambda_{4}$. All other parameters have been kept fixed. We
observe that the relic abundance can vary significantly as a function of
$\lambda_{2}$ and $\lambda_{4}$. For these choice of parameters, the deviation
in $\Omega_{\chi}h^{2}$ from its value with no self-scattering, defined as,
$\Delta\Omega_{\chi}(\lambda_{2},\lambda_{4})=(\Omega_{\chi}(\lambda_{2},\lambda_{4})-\Omega_{\chi}(\lambda_{2}=0,\lambda_{4}=0))/\Omega_{\chi}(\lambda_{2}=0,\lambda_{4}=0)$
takes a maximum value of around $\mathcal{O}(15\%)$.

Such significant deviations in $\Omega_{\chi}h^{2}$ are obtained for a wide
range of values for the self-scattering parameters. This is shown in Fig. 21
(right panel), where we plot contours of fixed values of $\Delta\Omega_{\chi}$
in the $\lambda_{2}-\lambda_{4}$ plane. The contours are shown for three
values of $\Delta\Omega_{\chi}$, $15\%$ (blue solid line), $10\%$ (red dashed
line) and $5\%$ (green dotted line). All other parameters are kept fixed as
before. As we can see from this figure, for a fixed value of
$\Delta\Omega_{\chi}$, the values of $\lambda_{2}$ and $\lambda_{4}$ lie on a
half-circle, and $\mathcal{O}(15\%)$ deviations are observed for a wide range
of values of these parameters. Larger deviations in $\Omega_{\chi}h^{2}$ are
also possible, in circles of smaller radii. Thus, these figures clearly
demonstrate the significant role of the self-scatterings in determining the
total relic density itself, by comparing the relic densities in the scenarios
with and without DM self-scatterings.

###  4.5 Summary and discussion

To summarize, we studied the scenario of a complex scalar dark matter, which
is a singlet under the SM gauge interactions, and is stabilized by an
effective $Z_{2}$ reflection symmetry. We considered all the renormalizable
interactions of the singlet with the SM Higgs doublet, as well as an SM
singlet real scalar, including terms that preserve the reflection symmetry,
but break the larger global $U(1)_{\chi}$ symmetry of DM number. Since the
interactions of the DM with the Higgs boson are already strongly constrained,
we are forced to introduce the minimal scenario with the additional real
scalar which is even under the $Z_{2}$ symmetry $-$ in order to thermalize the
DM sector with the SM sector, and thereby obtain a thermal mechanism for
producing the DM density.

We find that both CP-violating and CP-conserving scattering reactions
involving the DM and the real scalar field ($\phi$) can take place in the
thermal bath. The CP-violating scatterings include both DM self-scattering
processes as well as DM annihilations to a pair of $\phi$ particles. Since
these CP-violating processes involve the $U(1)_{\chi}$ breaking terms, they
can violate DM number as well, and thereby create a possible asymmetry in the
DM sector.

In order to understand the role of the various scattering processes in
determining the DM density and composition, we set up the Boltzmann kinetic
equations for the (anti-)DM yields, and study them first in terms of the
thermally averaged symmetric and asymmetric scattering rates. We obtain
approximate analytic solutions to the system of Boltzmann equations, which are
found to be in good agreement with the exact numerical results, with a maximum
difference of upto $10\%$. We utilize the unitarity sum rules in relating the
CP-violation in the three different scattering processes. These sum rules
imply that the CP-violation is non-zero only in the simultaneous presence of
both the self-scatterings and the pair-annihilation, and would vanish if
either one of them is absent.

We then go on to compute the relevant CP-violations, and thermally averaged
reaction rates in the complex singlet scenario, including relevant Feynman
graphs upto one-loop level, with the CP-violation and asymmetric reaction
rates ensuing from the interference of the tree level graphs with the one-loop
graphs. With these, we compute the relic abundance and asymmetry in the DM
sector as a function of the coupling and mass parameters. There are broadly
two different regimes observed: one in which the CP-violating pair-
annihilation dominantly decides the DM relic abundance, and the other in which
the CP-conserving pair-annihilation plays the major role. In the first
scenario, the resulting asymmetries are found to be small, so are the
correlations between the self-scattering and annihilation couplings in
determining the DM density.

In the second scenario, the asymmetries can be large, with the asymmetry
parameter taking values upto $\mathcal{O}(0.5)$. Furthermore, in this
scenario, we observe strong correlations between the self-scattering and
annihilation processes in obtaining the required DM abundance. This role of DM
self-scatterings in determining the DM density is a novel observation, which
is the primary result of this chapter. The reason behind this effect is that
when the asymmetry in the DM sector becomes large, the asymmetric yield, which
is the difference between the particle and antiparticle yields, plays a
dominant role in determining the total DM density, as we also show through our
analytic solutions. And this asymmetric yield is affected by both the self-
scattering couplings as well as the annihilation ones that violate CP. Thus,
DM self-scatterings can become important in deciding both the DM density and
composition, being strongly correlated with the annihilation. This is further
strengthened by the unitarity sum rules mentioned above $-$ we cannot have a
non-zero asymmetry unless both these types of processes are simultaneously
present.

Although the cosmological, and the related particle physics aspects of the
complex scalar singlet scenario were the focus of this chapter, there are
important astrophysical consequences of this scenario which should be explored
in future studies. The primary direction would be to look into the indirect
detection prospects in the present Universe, which requires a detailed
analysis depending upon the mass of the DM $\chi$ and the real singlet scalar
$\phi$. Although such an analysis is beyond the scope of this work, a few
relevant comments are in order. As we have already seen, the asymmetry
generated in this scenario is not maximal, and therefore, we shall have a
density of both DM and anti-DM particles existing in the galaxies and
clusters. The exact ratio of the densities would be dependent upon both the
asymmetry parameter at the generation level, and any possible oscillation that
may happen at post-freeze out epochs due to the $U(1)_{\chi}$ breaking mass
terms, if these are present with sufficient strength.

All the four relevant annihilation processes involving the $\chi$ and
$\chi^{\dagger}$ particles will therefore take place, with
$\chi\chi\rightarrow\phi\phi$ (and its conjugate
$\chi^{\dagger}\chi^{\dagger}\rightarrow\phi\phi$) and
$\chi\chi^{\dagger}\rightarrow\phi\phi$ leading to the production of $\phi$
particles in the final state. Since the $\phi$ particles can have a small
mixing with the SM Higgs boson, they will eventually decay to SM particles
through this mixing. The exact decay branching ratio to different final states
will depend upon the $\phi$ mass, while the energy spectra of the decay
products will depend both on the decay mode, as well as on the $\chi$ mass,
thus leading to a wide range of possibilities. Given the nature of the Higgs
coupling to SM particles, if kinematically allowed, the possible annihilation
modes for one pair of (anti-)DM particles are then to four final state
particles, drawn in two pairs primarily from the following list
$-\\{\gamma\gamma$, $\tau^{+}\tau^{-}$, $b\bar{b}$, $W^{+}W^{-}$, $ZZ$,
$t\bar{t}$ and $hh\\}$. If $\phi$ is very light, for example, it may decay to
$\gamma\gamma$, thereby leading to four-photon final states in (anti-)DM
annihilations.

Such a process, with
$\\{\chi\chi,\chi^{\dagger}\chi^{\dagger},\chi\chi^{\dagger}\\}\rightarrow\phi\phi\rightarrow
4\gamma$ leads to an interesting feature in the spectra of the gamma ray
signal. Each of the produced photons will have an energy in the band
$E^{min}_{\gamma}\leq E_{\gamma}\leq E^{max}_{\gamma}$ and therefore the
corresponding gamma-ray spectra will be of a box shape of width $\Delta
E=E^{max}_{\gamma}-E^{min}_{\gamma}=\sqrt{m^{2}_{\chi}-m^{2}_{\phi}}$, see,
for example Ibarra:2012dw . Apart from the DM mass $m_{\chi}$, the photon flux
will depend upon the parameters $r_{\infty}$, $\langle\sigma v\rangle_{3}$ and
$\langle\sigma v\rangle_{A}$. Therefore, compared to the standard WIMP
scenario, the correspondence between the observed flux and the pair-
annihilation rate is expected to be different in this model. In particular, if
$\mathcal{F}_{\rm ADM}$ is the photon flux expected in our scenario from
$\\{\chi\chi,\chi^{\dagger}\chi^{\dagger},\chi\chi^{\dagger}\\}\rightarrow\phi\phi$
processes and $\mathcal{F}_{\rm WIMP}$ is the corresponding flux in case of
WIMPs from the same set of
$\\{\chi\chi,\chi^{\dagger}\chi^{\dagger},\chi\chi^{\dagger}\\}\rightarrow\phi\phi$
processes (here, WIMPs represent the symmetric limit of the ADM in which the
final particle and anti-particle number densities are the same), then we
obtain the ratio between the two fluxes to be:

$\frac{\mathcal{F}_{\rm ADM}}{\mathcal{F}_{\rm
WIMP}}=\frac{1}{\expectationvalue{\sigma v}_{A}+\expectationvalue{\sigma
v}_{3}}\bigg{[}(1-r^{2}_{\infty})\expectationvalue{\sigma
v}_{A}+(1+r^{2}_{\infty})\expectationvalue{\sigma
v}_{3}+2r_{\infty}\expectationvalue{\epsilon\sigma v}_{3}\bigg{]}$ (137)

We see that while $\mathcal{F}_{\rm ADM}=\mathcal{F}_{\rm WIMP}$ for the
completely symmetric case of $r_{\infty}=0$, $\mathcal{F}_{\rm
ADM}\neq\mathcal{F}_{\rm WIMP}$ for $r_{\infty}\neq 0$, where the difference
stems from the different number densities of the $\chi$ and $\chi^{\dagger}$
particles in the present epoch, while keeping their sum fixed.

Such a spectrum of photons will be discernible above the astrophysical
background, for example in the Fermi-LAT satellite looking either into the
galactic center or into dwarf-spheroidal galaxies Ibarra:2012dw . However, the
relevant constraints will strongly depend upon the DM mass, and for very heavy
DM particles, only a future experiment such as the CTA probing a higher photon
energy range will be promising Ibarra:2015tya . The other possible final
states do not lead to any distinct spectral feature in the gamma ray or
charged particle energy distribution, and therefore would have to be searched
for in the corresponding diffuse flux. As it is clear from this brief
discussion, given the wide possible range of DM and $\phi$ masses, and thus
the varied possible final states, there are multiple possible indirect
detection probes that would require a detailed independent study.

The direct detection prospect of the present scenario relies also on the
mixing of $\phi$ and SM Higgs in the absence of $\lambda_{\chi H}$ coupling.
In particular, through the mixing $\phi$ couples to the SM quarks which
eventually facilitates the elastic collision between DM particles and target
nuclei. To note, there is no qualitative difference between an ADM and a WIMP
scenario in the context of direct detection experiments albeit the mass of DM
and relevant coupling can be very different from WIMP scenario. Nevertheless,
the current bound on the scalar dark matter particle from direct detection
probes is very much applicable to our scenario. For a comprehensive study see
Refs.He:2008qm ; Barger:2010yn ; Schumann:2019eaa and references therein.

The complex singlet scalar constitutes one of the simplest scenarios for non-
self-conjugate thermal DM. Our study shows that in its full generality, this
scenario offers many novel phenomena as far as the DM cosmology is concerned.
The role of DM self-scatterings, and their interplay with the other processes
in driving the cosmological evolution of the DM properties is a novel effect
of considerable interest which should be explored in other scenarios of
thermal DM as well.

##  5 Revisiting the role of CP-conserving processes in cosmological
particle–antiparticle asymmetries

###  5.1 Introduction

In earlier chapters, we have primarily learned about different CP-violating
scatterings in deciding possible particle-antiparticle asymmetry in the DM
sector, simultaneously generating the present-day density of the same. In our
example models, the CP-conserving processes are naturally present, which may
or may not be relevant in deciding the particle-antiparticle asymmetry. In
this chapter, we focus on the role of CP-conserving processes in deciding the
particle-antiparticle asymmetry in the context of the observed baryon-
antibaryon asymmetry of the Universe (BAU) as well as a probable asymmetric DM
(ADM) sector.

Two of the most well studied BAU generation mechanisms are baryogenesis in
grand unified theories (GUT) Yoshimura:1978ex ; Weinberg:1979bt ;
Nanopoulos:1979gx ; Yoshimura:1979gy , and baryogenesis through leptogenesis
Fukugita:1986hr . In both scenarios, most of the implementations considered
involve the CP-violating out-of-equilibrium decay of a heavy particle in the
early Universe. Thus, the primary quantities that determine the net asymmetry
generated are the rates of the CP-violating decay and its inverse process. If
in addition CP-violating scattering processes are present, they are found to
affect both the generated asymmetry, and its subsequent wash-out, depending
upon the hierarchy of the decoupling temperatures for the decay and scattering
processes; for reviews, see, for example, Refs. Kolb:1990vq ;
Buchmuller:2005eh ; Davidson:2008bu , and references therein.

The primary source of the CP-violation in a given scenario may also turn out
to be scattering processes, with the heavy particle decay processes either
sub-dominant or absent Bento:2001rc ; Nardi:2007jp ; Gu:2009yx . Such
scattering mechanisms have been studied in contexts involving the dark matter
(DM) particles as well, in achieving baryogenesis through DM annihilations
Cui:2011ab ; Bernal:2012gv ; Bernal:2013bga ; Kumar:2013uca , in relating the
baryon and dark matter sector asymmetries Baldes:2014gca ; Baldes:2015lka , in
realizing the ADM scenario through scatterings Ghosh:2021qbo , and through
semi-annihilations Ghosh:2020lma ; DEramo:2010keq .

The role of CP-conserving processes have also been studied in the context of
leptogenesis in see-saw models of neutrino mass. In leptogenesis from type-I
see-saw, CP-conserving scatterings can arise from additional new interactions
of the SM gauge singlets Gu:2009hn ; Sierra:2014sta . As discussed in Ref.
Sierra:2014sta , these additional interactions are CP-conserving by
assumption, that is in general they can violate CP if the relevant couplings
have complex phases. Furthermore, it is a specific dependence of the reaction
rates on the temperature and masses that leads to this competing effect, in
spite of one process being a decay and the other a scattering, as analyzed in
detail in Sierra:2014sta . In leptogenesis from type-II and type-III see-saw,
there are additional gauge interactions, which lead to CP-conserving
scatterings, see, for example Hambye:2003rt ; Hambye:2005tk ; Hambye:2012fh .
The rate of these gauge scatterings are usually smaller than the CP-violating
decay rates, and therefore as shown in Ref. Hambye:2005tk , the role of the
gauge scatterings in determining the asymmetry is small, and the efficiency
factor remains close to maximal.

In this chapter, we revisit the role of CP-conserving reactions in determining
cosmological particle-antiparticle asymmetries, and point out qualitatively
novel scenarios in which their role is significantly exemplified. In
particular, to begin with, we may classify the possible scenarios into three
broad categories:

  1. A.

both the CP-violating and CP-conserving processes are decays

  2. B.

both the CP-violating and CP-conserving processes are scatterings

  3. C.

one of the processes is a decay and the other is a scattering

In scenarios A and B, the two processes can naturally have comparable rates,
and therefore the CP-conserving reactions can in principle play a significant
role in generating the particle-antiparticle asymmetries. In scenario C, which
is the case so far considered in the literature, it is expected that the CP-
conserving scatterings will have a suppressed reaction rate compared to the
CP-violating decays, due to the Boltzmann suppression stemming from the
presence of an additional non-relativistic particle in the initial state. In
this chapter, we shall discuss examples of both scenarios A and B.

In each of these scenarios, one can further have two distinct possibilities.
For processes of the form $M...\rightarrow D...$, where $M$ is heavier than
$D$, we can have a scenario in which

  1. I.

the asymmetry is generated in the mother sector, i.e., in the $M$ sector,

  2. II.

the asymmetry is generated in the daughter sector, i.e., in the $D$ sector.

In conventional scenarios of baryogenesis, we usually find examples of type
II. In this chapter, we shall show examples of both types, belonging to
scenarios A and B above. In particular, we shall point out examples of type I,
in which the CP-conserving processes play a dual role. Until the decoupling of
the CP-violating scatterings, the CP-conserving processes tend to suppress the
asymmetry generation by controlling the out-of-equilibrium number densities of
the bath ($M$) particles. However, subsequently, once the net particle
antiparticle yield difference has been frozen, the same CP-conserving pair-
annihilation reactions modify the ratio of particle anti-particle yields by
eliminating the symmetric component of the mother particles 121212Since either
the DM particle ($\chi$) or the DM anti-particle ($\chi^{\dagger}$) may
dominate the current DM density, we quantify this ratio by the asymptotic
value of the asymmetry parameter $r_{\infty}=\lvert
Y_{\chi}-Y_{\chi^{\dagger}}\rvert/(Y_{\chi}+Y_{\chi^{\dagger}})$ in the
present epoch, where $Y_{i}$ are the respective yields; see Sec. 5.3 for
details.. This leads to a competing effect shown by the same scattering
process at different epochs, since, unlike in the scenarios of type II, in
scenarios of type I the asymmetry is generated in the sector which initiates
both the CP-conserving and violating scatterings in the first place. This dual
role is highlighted for the first time in this thesis.

We shall now demonstrate the above scenarios through illustrative examples. In
Sec. 5.2 we discuss a leptogenesis scenario of type A-II, in which both the
CP-conserving and violating processes are decays, and the asymmetry is
generated in the daughter sector. In Sec. 5.3 we discuss two scenarios of
asymmetric dark matter production from scattering of type B-I, in which both
the processes are scatterings, and the asymmetry is generated in the mother
sector. We summarize our findings in Sec. 5.4.

###  5.2 CP-violating and conserving decays and daughter sector asymmetry

We first discuss an illustrative model which belongs to the type A-II in the
classification described in the introduction, i.e., both the CP-conserving and
violating processes are two-body decays, and the asymmetry is generated in the
daughter sector. To this end, consider a leptogenesis model involving two
standard model (SM) singlet heavy Majorana neutrinos $N_{1}$ and $N_{2}$, with
their mass values satisfying $M_{N_{1}}>M_{N_{2}}$. For the cosmology of
$N_{1}$, the following decays of $N_{1}$ to SM charged lepton ($\ell^{\pm}$),
charged scalar boson ($H^{\pm}$), neutral scalar boson ($h$) and $N_{2}$ are
important:

$\displaystyle N_{1}$
$\displaystyle\rightarrow\ell^{\pm}H^{\mp}~{}~{}~{}~{}~{}{\rm(CP-violating)}$
$\displaystyle N_{1}$ $\displaystyle\rightarrow N_{2}h~{}~{}~{}~{}~{}{\rm(CP-
conserving)}.$

In order to realize these processes obeying the SM gauge symmetries, we need
to extend the SM field content in the scalar sector. In addition to the
dominantly SM-like scalar doublet $\Phi_{1}$ which gives mass to the SM
fermions, we introduce a second Higgs doublet $\Phi_{2}$, and an SM singlet
scalar field $S$. The relevant interaction Lagrangian terms before electroweak
symmetry breaking include the following:

$\mathcal{L}_{\rm
int}\supset-\frac{g}{2}\overline{N_{1}}N_{2}S-y_{i}\overline{L}\tilde{\Phi}_{1}N_{i}-\mu_{i}|\Phi_{i}|^{2}S-\frac{\lambda^{S}_{i}}{2}|\Phi_{i}|^{2}S^{2}-\lambda(\Phi_{1}^{\dagger}\Phi_{1})(\Phi_{2}^{\dagger}\Phi_{2})$
(138)

where $\tilde{\Phi}_{1}=i\sigma_{2}\Phi_{1}^{*}$. Here, we have written the
interaction terms involving the SM singlet Majorana neutrinos using the mass
eigenstates $N_{1}$ and $N_{2}$. In order to avoid tree-level flavour-changing
neutral currents, we have also assumed an additional $Z_{2}$ symmetry, under
which $\Phi_{2}\rightarrow-\Phi_{2}$, that is softly broken by a
$m_{12}\Phi_{1}^{\dagger}\Phi_{2}$ term, as in the type-I two Higgs doublet
model Branco . This symmetry prohibits the
$y^{\prime}_{i}\overline{L}\tilde{\Phi}_{2}N_{i}$ terms. After electroweak
symmetry breaking, the singlet scalar mixes with the neutral components of the
scalar doublets, giving rise to the decay mode $N_{1}\rightarrow N_{2}h$,
while the mixing of the charged scalars leads to the decay
$N_{1}\rightarrow\ell^{\pm}H^{\mp}$ through the Yukawa interaction.

Such a model was proposed in Ref. Bhattacharya:2011sy as a low-scale model for
leptogenesis that utilizes the quartic scalar couplings Kayser:2010fc . In
particular, the lepton number conserving decay mode $N_{1}\rightarrow N_{2}h$
helps in satisfying the requirements of generating a non-zero CP-violation at
the one-loop level, being consistent with the Nanopoulous-Weinberg theorem
Nanopoulos:1979gx ; Bhattacharya:2011sy . However, the role of the CP-
conserving decay mode $N_{1}\rightarrow N_{2}h$ in the cosmology of $N_{1}$ —
and therefore in the generated lepton asymmetry — has not been studied
earlier, and is the focus of the present study.

The Boltzmann equations determining the number density of $N_{1}$, and the
lepton asymmetry produced are given by131313Since $M_{N_{1}}>M_{N_{2}}$,
during the out-of-equilibrium decay of $N_{1}$ at a temperature $T\simeq
M_{N_{1}}$, the $N_{2}$ particles are in thermal equilibrium through rapid
scatterings in the thermal bath, and hence follow a Maxwell-Boltzmann
distribution. Furthermore, for simplicity, we assume that $N_{2}$ decays after
the freeze-out of $N_{1}$ decay do not wash-out the generated lepton
asymmetry, which can be ensured, for example, with $M_{N_{2}}<M_{H^{\pm}}$.

$\displaystyle\dfrac{dY_{N_{1}}}{dx}$
$\displaystyle\simeq-\dfrac{\expectationvalue{\Gamma}_{0}+\expectationvalue{\Gamma}_{A}}{Hx}\,(Y_{N_{1}}-Y_{N_{1},0})$
$\displaystyle\dfrac{dY_{L}}{dx}$
$\displaystyle\simeq-\dfrac{\epsilon\expectationvalue{\Gamma}_{0}}{Hx}\,(Y_{N_{1}}-Y_{N_{1},0}).$
(139)

Here, $Y_{i}=n_{i}/s$ is the yield of the species $i$, with $n_{i}$ being its
number density and $s$ the entropy density in the radiation bath. We also have
$Y_{L}=Y_{\ell^{-}}-Y_{\ell^{+}}$ as the lepton asymmetry produced, and
$Y_{N_{1},0}$ the equilibrium yield of $N_{1}$. Here it is assumed that at
high temperatures $T>M_{N_{1}}$, $N_{1}$ achieves a thermal distribution
through rapid scattering processes in the thermal plasma. The thermally
averaged symmetric decay width is given by $\expectationvalue{\Gamma}_{0}$,
where,
$\Gamma_{0}=\Gamma(N_{1}\rightarrow\ell^{+}H^{-})+\Gamma(N_{1}\rightarrow\ell^{-}H^{+})$,
and $\Gamma_{A}$ is the decay width of the CP-conserving mode
$N_{1}\rightarrow N_{2}h$. Finally, the CP-violation parameter is defined as

$\epsilon=\frac{|M(N_{1}\rightarrow\ell^{-}H^{+})|^{2}-|M(N_{1}\rightarrow\ell^{+}H^{-})|^{2}}{|M(N_{1}\rightarrow\ell^{-}H^{+})|^{2}+|M(N_{1}\rightarrow\ell^{+}H^{-})|^{2}}$
(140)

Here, $|M|^{2}$ denotes the matrix element squared for the corresponding
process. In writing the Boltzmann equation for $Y_{L}$, we have made the
approximation that at the epoch of the generation of the asymmetry, the two-
body scattering reactions have decoupled, which is as expected due to their
lower rates. We have also dropped terms in the right hand side proportional to
$Y_{L}$, as at the epoch when $Y_{L}$ is being generated, those are
subdominant.

Figure 22: The dependence of the lepton asymmetry $Y_{L}$ (scaled by
$Y_{L,0})$, as a function of the ratio of CP-conserving and violating decay
widths, $\expectationvalue{\Gamma}_{A}/\expectationvalue{\Gamma}_{0}$. See
text for details on the choice of the other parameters.

We solve Eqs. 139 for the parameter choices $M_{N_{1}}=500$ GeV,
$\epsilon=10^{-6}$ and $\Gamma_{0}=10^{-13}$ GeV, where the parameters are
chosen such that the required baryon asymmetry of the Universe may be
reproduced. We show the dependence of $Y_{L}$ (scaled by $Y_{L}^{0}$, which is
the asymmetric yield with $\Gamma_{A}=0$) as a function of
$\expectationvalue{\Gamma}_{A}/\expectationvalue{\Gamma}_{0}$ in Fig. 22. As
we can clearly see from this figure, the lepton asymmetry $Y_{L}$ changes by
one order of magnitude if $\Gamma_{A}\sim 10\Gamma_{0}$, compared to the
scenario where $\Gamma_{A}$ is negligible. This simply stems from the fact
that the longer the CP-conserving process is in equilibrium, the smaller the
number density of the mother particle $N_{1}$, and hence the lepton asymmetry
generated from its decays.

Figure 23: The dependence of the lepton asymmetry $|Y_{L}|$ in the model for
low-scale leptogenesis, as a function of the ratio of CP-conserving and
violating decay widths,
$\expectationvalue{\Gamma}_{A}/\expectationvalue{\Gamma}_{0}$. In this model,
the CP-violation parameter $\epsilon$ can be expressed in terms of
$\Gamma_{A}$, for fixed values of the scalar quartic coupling $\lambda$ and
$\tan\beta$. See text for details.

One can simplify the analysis further in the particular low-scale leptogenesis
model discussed above. In this scenario, we can express the CP-violation
parameter $\epsilon$ defined in Eq. 140, stemming from the interference of the
tree and loop level amplitudes, as follows:

$\epsilon\simeq-2\lambda\,v\,\sin\beta\,\tilde{g}\,{\rm Im}(L).$ (141)

Here, $v=246$ GeV is the electroweak symmetry breaking scale and
$\tan\beta=v_{1}/v_{2}$ is the ratio of the vacuum expectation values of the
neutral CP-even components of the two Higgs doublets $\Phi_{1}$ and
$\Phi_{2}$. We have also defined $\tilde{g}=g\,\sin\alpha$, where $\alpha$ is
the mixing angle of the scalar singlet $S$ with the lighter CP-even Higgs
boson. For simplicity, here we have assumed that $S$ dominantly mixes with the
SM-like lighter Higgs state. Finally, ${\rm Im}(L)$ is the imaginary part of
the loop-factor, which is found to be:

${\rm Im}(L)\simeq-\frac{M_{N_{2}}}{8\pi
M^{2}_{N_{1}}}\frac{1}{1-\xi}\log\left(\frac{1}{\xi}\right),$ (142)

where, $\xi=\left(M_{H^{\pm}}/M_{N_{1}}\right)^{2}$. For example, with
$M_{N_{1}}=500$ GeV, $M_{N_{2}}=300$ GeV and $M_{H^{\pm}}=350$ GeV, we obtain
${\rm Im}(L)\simeq-6.7\times 10^{-5}$.

Since the width of the CP-conserving decay $N_{1}\rightarrow N_{2}h$ is
$\Gamma_{A}=\frac{\tilde{g}^{2}}{8\pi}M_{N_{1}}$, we can trade the coupling
$\tilde{g}$ in Eq. 141 with the square-root of the CP-conserving decay width
$\Gamma_{A}$. In addition, if we fix the scalar quartic coupling $\lambda$ and
$\tan\beta$, $\epsilon$ is determined in terms of $\Gamma_{A}$. The residual
dependence of the lepton asymmetry $|Y_{L}|$ in this model as a function of
$\expectationvalue{\Gamma}_{A}/\expectationvalue{\Gamma}_{0}$ is shown in Fig.
23, where $\expectationvalue{\Gamma}_{0}$ is chosen to be $10^{-13}$ GeV,
$\sin\beta\sim\mathcal{O}(1)$ and the quartic coupling
$\lambda\sim\mathcal{O}(0.1)$. As we can see from this figure, for
$\expectationvalue{\Gamma}_{A}<\expectationvalue{\Gamma}_{0}$, the out-of-
equilibrium condition for $N_{1}$ is determined by the CP-violating decay.
Therefore, $|Y_{L}|$ increases with increasing
$\expectationvalue{\Gamma}_{A}$, simply because the CP-violation $\epsilon$
increases as $\sqrt{\Gamma_{A}}$, for a fixed scalar quartic. On the other
hand, for $\expectationvalue{\Gamma}_{A}>\expectationvalue{\Gamma}_{0}$,
$|Y_{L}|$ decreases rapidly with increasing $\expectationvalue{\Gamma}_{A}$,
as $N_{1}$ remains in equilibrium for a longer period, thereby leading to a
reduction in its number density. We would like to emphasize that
$\Gamma_{A}>>\Gamma_{0}$ is easily achieved in the model described above for
natural choices of the parameters.

Thus, CP-conserving processes, when at the same footing as the CP-violating
ones (in this case both of them being decays of $N_{1}$), can play a major
role in deciding the lepton asymmetry production in the early Universe. As
discussed in the introduction, this is in contrast to the role of such
processes observed earlier in the leptogenesis models from different see-saw
mechanisms, as in those cases the CP-conserving modes played a subdominant
role, mostly because they were scattering reactions, whereas the violating
ones were decays, and hence of a higher rate.

###  5.3 CP-violating and conserving scatterings and mother sector asymmetry

We shall now discuss two scenarios of asymmetric dark matter production from
scattering. These two examples belong to the type B-I explained in the
Introduction, in which both the CP-violating and conserving processes are
scatterings, and the asymmetry is generated in the mother sector. We shall see
that due to the latter feature, there is a dual role played by the CP-
conserving processes. The ADM models under discussion have been proposed in
Ref. Ghosh:2021qbo , which discusses ADM from scattering, and in Ref.
Ghosh:2020lma on ADM from semi-annihilation.

Both the ADM models share some common features. There is a complex scalar DM
$\chi$, which is stabilized by a $Z_{N}$ symmetry, interacting with itself and
a $Z_{N}$ even real scalar $\phi$. The scalar $\phi$, taken to be lighter than
$\chi$, can mix with or decay to SM states, thereby maintaining kinetic
equilibrium of the DM sector with the SM bath. In particular, for a GeV DM
particle, the suitable choice for the mass of $\phi$ is around a MeV, as
explained in Chapter 1. This is motivated from the constraints on the
effective relativistic degrees of freedom at the time of BBN ($T_{\gamma}\sim
1~{}{\rm MeV}$). In both cases, the existence of a CP-conserving scattering
$\chi+\chi^{\dagger}\rightarrow\phi+\phi$ immediately follows, which is the
focus of this discussion. In the ADM model from scattering, the stabilizing
symmetry is $Z_{2}$, while for the ADM from semi-annihilation scenario, it is
$Z_{3}$. Let us briefly discuss each model in the following.

####  5.3.1 Asymmetric DM from scattering

In this case, we have the following interaction Lagrangian (detailed in
chapter 4) consistent with the $Z_{2}$ symmetry:

$\displaystyle-\mathcal{L_{\rm
int}}\supset\mu\chi^{\dagger}\chi\phi+\left(\frac{\mu_{1}}{2}\chi^{2}\phi+{\rm