aainstitutetext: CERN, Theoretical Physics Department, CH-1211 Geneva 23,
Switzerlandbbinstitutetext: LAPTh, Université Savoie Mont Blanc, CNRS, BP 110,
F-74941 Annecy-le-Vieux, Franceccinstitutetext: Centre for High Energy
Physics, Indian Institute of Science, C.V. Raman Avenue, Bangalore 560012,
Indiaddinstitutetext: Department of Physics, Indian Institute of Technology
Kanpur, Kanpur, Uttar Pradesh 208016, Indiaeeinstitutetext: Institute of
Theoretical Physics, School of Physics, Dalian University of Technology,
Dalian 116024, People’s Republic of China

# Relic density of dark matter in the inert doublet model beyond leading order
for the low mass region: 2. Co-annihilation

Shankha Banerjee b , Fawzi Boudjema c, d , Nabarun Chakrabarty e , Hao
Sun<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
(August 27, 2024)

###### Abstract

We examine the relic density of the light mass dark matter region in the inert
doublet model (IDM) when the dominant process is due to co-annihilation
between the lightest neutral scalars of the model. The full one-loop
electroweak corrections are computed in an on-shell scheme and are found to be
well approximated as an effective cross-section expressed in terms of
$Z$-observables. The electroweak corrections to the subdominant process which
consists of an annihilation into an on-shell $W$ and an off-shell $W$, that is
calculated as a annihilation into a 3-body final state, is also performed. The
latter reveals an important dependence on a parameter that describes the self-
interaction of the new scalars (solely within the dark sector), a parameter
which is not accessible in tree-level calculations of standard model (SM)-IDM
interactions.

††preprint: LAPTH-002/21, CERN-TH-2021-002

##  1 Introduction

In the very thorough analysis we have conducted in Ref. OurPaper1_2020  on the
available parameter space of the inert ouble model (IDM) Deshpande:1977rw ;
Barbieri:2006dq ; Hambye:2007vf ; LopezHonorez:2006gr ; Cao:2007rm ;
Gustafsson:2007pc ; Agrawal:2008xz ; Hambye:2009pw ; Lundstrom:2008ai ;
Andreas:2009hj ; Arina:2009um ; Dolle:2009ft ; Nezri:2009jd ; Miao:2010rg ;
Gong:2012ri ; Gustafsson:2012aj ; Swiezewska:2012eh ; Arhrib:2012ia ;
Wang:2012zv ; Goudelis:2013uca ; Arhrib:2013ela ; Krawczyk:2013jta ;
Osland:2013sla ; Abe:2015rja ; Blinov:2015qva ; Diaz:2015pyv ; Ilnicka:2015jba
; Belanger:2015kga ; Carmona:2015haa ; Kanemura:2016sos ; Queiroz:2015utg ;
Belyaev:2016lok ; Arcadi:2019lka ; Eiteneuer:2017hoh ; Ilnicka:2018def ;
Kalinowski:2018ylg ; Ferreira:2009jb ; Ferreira:2015pfi ; Kanemura:2002vm ;
Senaha:2018xek ; Braathen:2019pxr ; Arhrib:2015hoa ; Garcia-Cely:2015khw ;
Banerjee:2016vrp ; Basu:2020qoe ; Abouabid:2020eik ; Kalinowski:2020rmb , for
the low mass Dark Matter (DM) (below the $W$ mass, $M_{W}$), we have confirmed
the survival of a very small region with a DM mass of $55-60$ GeV which has
recently been unraveled Datta:2016nfz ; Belyaev:2016lok . A good value of the
relic density Ade:2015xua  is obtained. Within the freeze-out mechanism this
is possible, mainly, through efficient co-annihilation. The two additional
(lightest) neutral scalars of the IDM, $X$ and $A$, need to be
thermodynamically close to one another for this co-annihilation to take place.
In other words, this requires a certain degree of mass degeneracy which, in
the context of the IDM, is not necessarily fine-tuned and unnatural. The
$Z$-mediated co-annihilation is into the (light) fermion pairs, not including
the top quark, of the Standard Model (SM). The process is a gauge interaction
induced process and as such it is quite efficient. To obtain a value of the
relic density around $0.12$, the co-annihilation cross-sections must be
dampened by the Boltzmann factor, $e^{-m_{-}/T}$, where $m_{-}=M_{A}-M_{X}$ is
the mass splitting and $T$ is the temperature. The Boltzmann factor in the
thermally averaged cross-sections should furnish enough reduction to
counterbalance the large cross-sections. This in turn requires the mass
splitting to be not too small. In our analyses we found that $m_{-}=8$ GeV is
an optimal value. This value is at the edge of the LEPII constraint on the
IDM. Higher values of the mass splitting would lead to visible tracks at LEP
that disqualify the model while lower values do not give enough Boltzmann
reduction leading to too small $\Omega h^{2}$. For our one-loop analyses, we
have retained two benchmarks points with mass $M_{X}=58,60$ GeV and $m_{-}=8$
GeV, see Table 1.

| P58 | P60
---|---|---
$M_{X}$ | 58 | 60
$\lambda_{L}$ | 0.0 | 0.0
$M_{A}$,$M_{H^{\pm}}$ | 66, 110 | 68,150
$(\lambda_{3},\lambda_{4},\lambda_{5})$ | (0.28,-0.26,-0.02) | (0.60,-0.58,-0.02)
$\Omega h^{2}$
$\alpha(0)$ | 0.113 | 0.116
$\Omega_{WW^{\star}}(\%)$ | 5 | 9
$\Omega_{AX\to\sum f\bar{f}}(\%)$ | 95 | 91
Table 1: Characteristics of the co-annihilation benchmarks points P58 and P60.
All masses are in GeV. We also show tree-level (calculated with $(\alpha(0)$)
relic density and the weight in percent of each channel contributing to the
relic density. $f$ stands for all light fermions in the SM (the top-quark
channel is closed). For more precise numbers, especially of the underlying
parameters $\lambda_{3,4,5}$, please refer to OurPaper1_2020 .

As we will see, this 2 GeV difference in the DM mass between the two
benchmarks is enough to give a $20\%$ reduction in the (co)-annihilating
cross-sections in P60 compared to P58. While the Boltzmann factor is
(slightly) larger for the $M_{X}=60$ GeV case to somehow compensate for the
drop in the $AX\to f\bar{f}$ cross-sections, what really benefits P60 is a
larger cross-section of the annihilating process $XX\to WW^{\star}$, thanks to
a larger threshold than in the case of $M_{X}=58$ GeV. The masses of all the
scalars of the IDM have been set to define the model, an extra parameter that
is needed to fully describe the interaction between these scalar and the SM
scalar sector is $\lambda_{L}$. In fact, $\lambda_{L}$ describes the
interaction strength of the SM Higgs to the DM, $X$, see OurPaper1_2020  for
details. We find that in order to keep the cross-sections small, including
$XX$ annihilations, $\lambda_{L}$ must be as small as possible. This is the
reason that for the surviving points we take $\lambda_{L}=0$. The latter
suppresses SM Higgs exchange in $XX\to WW^{\star}$ and avoids annihilation
into $b\bar{b}$ through Higgs exchange. We must note that for sufficient
relative velocity the SM Higgs resonance is accessed for this range of DM
masses and even if $\lambda_{L}=0$ SM Higgs exchange may kick in at one-loop.
The Higgs resonance scenarios deserve a dedicated analysis. Of course, the
selection of the benchmark points was based on tree-level cross-sections. It
is interesting to know how these cross-sections are affected by loop
calculations. $AX\to f\bar{f}$ accounts for more than $95\%$ ($90\%$) of the
contribution for the relic density for P58 (P60). In this section we therefore
concentrate on the electroweak radiative corrections for such processes. The
rest of the annihilation contributions are from $XX\to WW^{\star}$. In the
present paper we include the results of the one-loop contribution to $XX\to
WW^{\star}\equiv Wf\bar{f}^{\prime}$ but we leave many of the technical
details covering both $2\to 3$ processes at one-loop and the treatment of the
SM Higgs resonance to more detailed presentations in Refs. OurPaper3_2020 ;
OurPaper4_2020 .

##  2 The co-annihilation cross-section $AX\to f\bar{f}$

###  2.1 $AX\to f\bar{f}$: tree-level considerations

Figure 1: Tree-level Feynman diagrams (in the non-linear Feynman
gaugeBoudjema:1995cb ; OurPaper1_2020 ) for $AX\to b\bar{b}$ as a
representative of $AX\to f\bar{f}$.

The co-annihilation cross-sections are completely determined by the gauge
coupling, given the masses of $X$ and $A$. The annihilations are essentially
$P$-wave with a very small $S$-wave contribution that is noticeable only for
extremely small velocities in the case of the heaviest final state fermion,
the $b$-quark. For such $2\to 2$ processes, the cross-section can be described
by a transparent analytical formula. With $m_{f}$ the mass of the final
fermion, the tree-level cross-section can be written as

$\displaystyle\sigma_{AX\to
f\bar{f}}\;v=\sigma_{v^{2}}^{f\bar{f}}v^{2}+\frac{m_{f}^{2}}{s}\sigma_{0}^{f\bar{f}}.$
(1)

The $P$-wave, $v^{2}$ term, dominates by far while the very small $S$-wave
contribution, $\sigma_{0}$, is suppressed by the chiral factor $m_{f}^{2}$.
The latter is further suppressed by the mass splitting factor, $m_{-}^{2}$, as
shown explicitly below. With the “running" partial width

$\displaystyle\Gamma_{Z\to f\bar{f}}(s)$ $\displaystyle=$ $\displaystyle
N_{c}^{f}\frac{g^{2}\sqrt{s}}{12\pi
c_{W}^{2}}\beta_{f}\Bigg{(}g_{V}^{2}\frac{3-\beta_{f}^{2}}{2}+g_{A}^{2}\beta_{f}^{2}\Bigg{)}\stackrel{{\scriptstyle[}}{{s}}\gg
m_{f}^{2}]{}{\simeq}\frac{\sqrt{s}}{M_{Z}}\Gamma_{Z\to f\bar{f}},\quad
N_{c}\;\text{is the colour factor}$ (2) and
$\displaystyle\quad\beta_{f}=\sqrt{1-\frac{4m_{f}^{2}}{s}},\quad
g_{V}=\frac{T_{3f}}{2}-Q_{f}s^{2}_{W},\quad
g_{A}=\frac{T_{3f}}{2}\quad(T_{3f}=\pm 1/2),$

($Q_{f}$ is the charge of the fermion), we have

$\displaystyle\sigma_{v^{2}}^{f\bar{f}}$ $\displaystyle=$
$\displaystyle\frac{g^{2}}{c_{W}^{2}}\frac{1}{8s}\frac{1-m_{-}^{2}m_{+}^{2}/s^{2}}{(1-M_{Z}^{2}/s)^{2}}\frac{\Gamma_{Z\to
f\bar{f}}(s)}{\sqrt{s}}\simeq\frac{g^{2}}{c_{W}^{2}}\frac{1}{8s}\frac{1-m_{-}^{2}m_{+}^{2}/s^{2}}{(1-M_{Z}^{2}/s)^{2}}\frac{\Gamma_{Z\to
f\bar{f}}}{M_{Z}}$ $\displaystyle\sigma_{0}^{f\bar{f}}$ $\displaystyle=$
$\displaystyle\frac{m_{-}^{2}m_{+}^{2}}{s^{2}-m_{-}^{2}m_{+}^{2}}\frac{g^{2}}{c_{W}^{2}}\frac{3}{2s}\frac{1}{(1-M_{Z}^{2}/s)^{2}}\frac{\beta_{f}\Gamma_{Z\to\nu\bar{\nu}}}{M_{Z}}\simeq\frac{m_{-}^{2}m_{+}^{2}}{s^{2}-m_{-}^{2}m_{+}^{2}}\frac{g^{2}}{c_{W}^{2}}\frac{3}{2s}\frac{1}{(1-M_{Z}^{2}/s)^{2}}\frac{\Gamma_{Z\to\nu\bar{\nu}}}{M_{Z}}.$

The approximation ($\simeq$) in equation 2.1 which amounts to $m_{f}\to 0$, is
excellent for all fermions including the bottom-quark (for which the
approximation is off by about 3 per-mille for the range of velocities relevant
for the calculation of the relic density contributed by these channels).
Despite our $P$-wave appellation, note that the $s$ (and hence $v$) dependence
contained in $\sigma_{v^{2}}^{f\bar{f}}$ is not small for the 2 scenarios that
we are studying, see Figure 2. This $s$-dependent factor in the expression of
$\sigma_{v^{2}}^{f\bar{f}}$ explains that for the same value of the relative
velocity, the co-annihilation processes are about 20% larger for P58 as
compared to P60.

Figure 2: Left panel: The tree-level cross-section for
$\sigma(XX\to\nu\bar{\nu})$ for P58 and P60 as a function of the relative
velocity. The dashed curves are the pure $P$-wave ($\propto v^{2}$)
approximation. We see that this pure $v^{2}$ approximation is only valid for
$v<0.4$. Note that the cross-sections are quite large, however, remember that
a Boltzmann suppression will be applied to these co-annihilations cross-
sections when they are converted to effective velocity/temperature averaged
cross-sections.
Right panel: The $b\bar{b}$ cross-section normalised to the $d\bar{d}$ cross-
section and the $d\bar{d}$ cross-section normalised to the $\nu\bar{\nu}$
cross-section. See text for more detailed comments on these plots.

Figure 2, which is the result of a fully automated computation with our code
SloopS Boudjema_2005 ; Baro:2007em ; Baro:2008bg ; Baro:2009na ;
Boudjema:2011ig ; Boudjema:2014gza ; Belanger:2016tqb ; Belanger:2017rgu ;
Banerjee:2019luv , exhibits the $P$-wave nature of the co-annihilation and
confirms the analytical results. Departure from a pure $v^{2}$ dependence is
clearly seen. The ratio $\sigma(b\bar{b})/\sigma(d\bar{d})$ clearly shows the
very small contribution of the $S$-wave for small $v$ and the tiny effect of
the mass of the fermion for larger $v$. In the case of massless fermions,
Figure 2 confirms that $\sigma(\nu\bar{\nu})/\sigma(d\bar{d})$ is given by the
ratio of the respective partial widths of the $Z$ into these fermions.

###  2.2 $AX\to f\bar{f}$: cross-sections at one-loop

Figure 3: Some one-loop diagrams (in the non-linear Feynman gauge) for $AX\to
b\bar{b}$. We only show a subset of the box corrections and the triangles.
Note the last two diagrams involve the quartic coupling within the dark
sector, $\lambda_{2}$. Unfortunately, the one mediated by the neutral
Goldstone is proportional to the fermion mass. The one mediated by the $Z$
will also give a contribution proportional to the fermion mass, since the one-
loop integration will give, unlike the tree-level structure involving the
difference between the 2 incoming momenta, a new Lorentz structure
proportional to the $Z$ 4-momentum which ends up giving a contribution
proportional to the final fermion mass. Therefore the $\lambda_{2}$ dependence
only affects the $S$-wave.

Since these co-annihilation processes do not depend on $\lambda_{L}$, a fully
OS renormalisation exactly along the lines of the renormalisation of the SM
can be carried out. The one-loop corrections will have no scale dependence.
These processes allow to test whether the use of a running $\alpha$ is a good
approximation for the full one-loop corrections. We will show results for all
light flavours $f=\nu,l,u,d,b$. For definiteness, we take $\nu=\nu_{\tau}$,
and $l=\mu$. We look at both $d$ and $b$ since the latter is affected by the
heavy top-loop, as is the case for the $Z\to b\bar{b}$ decay, and also because
it may be more sensitive to the $\lambda_{2}$ dependence which appears in the
$S$-channel. Recall that $\lambda_{2}$ measures the self-interaction
exclusively within the dark sector, see OurPaper1_2020 . We find however that
this $\lambda_{2}$ dependence is very much suppressed by the mass of the
(very) fermions, see Figure 3, as we explain below.

We perform the full one-loop electroweak corrections for both benchmarks P58
and P60. The electroweak corrections, expressed in terms of relative
corrections, are practically exactly the same for all channels for both
benchmark points. We therefore only show the results for P58. The computation
of the radiative corrections for the charged fermions includes the QED final
state radiation that we obtain through the slicing technique, see Baro:2007em
for example. For a fermion of charge $Q_{f}$, these final state QED
corrections amount to a relative correction $3Q_{f}^{2}\alpha/4\pi$, which are
very small, $<0.2\%$ (for charged leptons). The QCD final state correction
$\sim\alpha_{s}(M_{Z}^{2})/\pi$ amounts to about $3.6\%$. We have not included
these in the figures we show below, Figure 4.

Figure 4: Full results for all channels. The dashed curves corresponds to the
effective cross-sections in Equation 2.2. While a close inspection of the
$b\bar{b}$ channel reveals a very small presence of the $S$-wave contribution
at $v\sim 0$, the $\lambda_{2}$ dependence is too tiny to be seen on this
plot.The $\lambda_{2}$ dependence is magnified in Figure 5.

Apart from the $b$ quarks, the relative corrections for all fermion final
states are practically a modest rescaling of the tree-level cross-section.
Indeed the relative corrections are velocity independent. For the $b$ quark,
this is also the case for all $v$ but $v\sim 0$ where the $S$-wave contributes
negligibly. We will come back to this feature. The flavour dependence of the
correction is also very small, all corrections being within $2\%$ of each
other. Apart from the behaviour at $v\sim 0$, the corrections at all $v$ for
the bottom-quark final states are slightly different due to the contribution
of the heavy top-quark, a feature known from $Z$-physics. On average, the
corrections are about $6\%$ and are therefore a factor 2 lower than if we had
used $\alpha(M_{Z}^{2})$. Genuine box and triangle contributions are therefore
not negligible.

Inspired by equation 2.1, and in view that the $S$-wave contribution is tiny
even in the case of $b\bar{b}$ we suggest to improve the tree-level cross-
sections by using effective cross-sections defined solely in terms of physical
observables,

$\displaystyle\sigma_{v^{2}}^{f\bar{f},\text{eff.}}$ $\displaystyle\simeq$
$\displaystyle\frac{12\pi}{s}\frac{1-m_{-}^{2}m_{+}^{2}/s^{2}}{(1-M_{Z}^{2}/s)^{2}}\frac{\Gamma_{Z\to
f\bar{f}}\;\Gamma_{Z\to\nu\bar{\nu}}}{M_{Z}^{2}}$
$\displaystyle\sigma_{0}^{f\bar{f},\text{eff.}}$ $\displaystyle\simeq$
$\displaystyle\frac{144\pi}{s}\frac{m_{-}^{2}m_{+}^{2}}{s^{2}-m_{-}^{2}m_{+}^{2}}\frac{1}{(1-M_{Z}^{2}/s)^{2}}\frac{\Gamma_{Z\to\nu\bar{\nu}}^{2}}{M_{Z}^{2}}.$
(4)

First $g^{2}/c_{W}^{2}$ in 2.1 is traded for $\Gamma(Z\to\nu\bar{\nu})$. Then
all partial widths in equation 2.2 are one-loop corrected partial widths. The
results of the almost velocity independent corrections are shown in figure 4.
The effective corrective factor misses only about $-0.5\%$ for the benchmark
point P58 (almost independent of the nature of the fermion) of the full one-
loop corrections and about $-1\%$ for P60. These small corrections are the
effect of the box contributions. These effective approximations are therefore
excellent.

Figure 5: The tiny $\lambda_{2}$ dependence in the $b\bar{b}$ co-annihilation
channel. The $\lambda_{2}$ dependence is measured relative to the $d\bar{d}$
cross-section where the $\lambda_{2}$ dependence is vanishing. Observe that
the units in the graph indicate a variation which, at its highest, does not
cross a couple of per-mille.

The very tiny contribution of the $S$-wave at very small velocity (where the
$P$-wave vanishes) is theoretically interesting even if it is without any
phenomenological impact. It could have been larger if the fermion masses were
larger. Theoretically, the $S$-wave is sensitive to $\lambda_{2}$ as explained
in the caption of Figure 3. We verified this property directly in the co-
annihilation into neutrinos where no $\lambda_{2}$ dependence was detected
numerically and we confirmed that the largest $\lambda_{2}$ dependence occurs
with the $b$ final state. Unfortunately, the maximum correction from the dark
sector is at the level of $2$ per-mille for $v=0$ where the cross-section is
smallest. As soon as the $P$-wave contribution kicks in, the tiny
$\lambda_{2}$ dependence in the $S$-wave contribution is totally swamped such
that the one-loop corrected cross-section is, for all purposes, insensitive to
$\lambda_{2}$. This very tiny $\lambda_{2}$ dependence is shown in Figure 5
where we track the $\lambda_{2}$ dependence by considering the ratio in the
$b\bar{b}$ cross-section between $\lambda_{2}=0.01$ and $\lambda_{2}=2$.

##  3 The $XX\to WW^{\star}$ in the co-annihilation region

###  3.1 Tree-level considerations

For P60, $XX$ annihilation to $WW$ below threshold with one $W$ decaying into
a fermion pair, accounts at tree-level, for $10\%$ of the relic density. This
cross-section does not suffer from the Boltzmann suppression. A non-zero value
of $\lambda_{L}$ would have added a Higgs exchange contribution as we show in
Figure 6 in the $WW$ channel, and in the $b\bar{b}$ channel
($XX\xrightarrow{h}b\bar{b}$).

Figure 6: A selection of diagrams for the tree-level cross-section $XX\to
Wf\bar{f}^{\prime}$. Since $\lambda_{L}$=0 for the benchmark point P60, Higgs
exchange does not take place for P60.

As $v$ increases, the phase space for $WW^{\star}$ increases and therefore the
cross-section increases as Figure 7 shows. Note that this cross-section when
compared to $AX\to f\bar{f}$ is far smaller (the latter will be reduced
considerably by the Boltzmann factor). This increase with velocity is smooth
across the Higgs resonance, $v=0.56$, a threshold that, as expected, is not
felt since $\lambda_{L}=0$.

Figure 7: The tree-level cross-section $XX\to Wf\bar{f}^{\prime}$ for the
benchmark points P58 and P60. The arrow at $v\sim 0.56(0.745)$ indicates the
position corresponding to the Higgs resonance (if crossed) for
$M_{X}=60\;(58)$ GeV.

###  3.2 One-loop results

Figure 8: A very small selection of the diagrams that enter the one-loop
calculations of $XX\to WW^{\star}$. We highlight in particular Higgs exchange
through the induced $hXX$ vertex at one-loop. We have also picked
configurations where a $\lambda_{2}$ contribution shows up. These are part of
what constitutes rescattering in the dark sector before annihilation into SM
particles. An example of a pentagon diagram which can not be considered as a
factorised $XX\to WW^{\star}\to Wff^{\prime}$ is also singled out (5-point
function).

While $\lambda_{L}$=0 at tree-level for $XX\to Wff^{\prime}$ with no SM Higgs
exchange contributing, a renormalisation of $\lambda_{L}$ is still called for
in this process. This is because in general the $XX\to WW$ amplitude does
depend parametrically on $\lambda_{L}$. As Figure 8 clearly shows, an induced
one-loop $hXX$ is generated which entails a one-loop Higgs exchange
contribution. The critical question is whether this will lead to an
instability when the pole at $s=M_{h}^{2}$ is reached? Do we then have to, and
how to, include a width? casting doubt on the organisation of the perturbation
series? This same crucial point is dealt with in detail when we study cases
where the resonance is present already at “tree-level". In the present case,
$\lambda_{L}=0$, there is no need to include a width. Since $h\to XX$ is open,
the OS scheme is most appropriate for a definition and renormalisation of
$\lambda_{L}$. The OS scheme means that we trade $\lambda_{L}$ with
$\Gamma_{h\to XX}$. In this particular case, the input translates into
$\Gamma_{h\to XX}=0$. But $\Gamma_{h\to XX}=0$ is maintained at all orders
entails, in fact, that the renormalised amplitude for ${{\cal M}}_{h\to
XX}(s)=0$ at the renormalisation point $s=M_{h}^{2}$, $s$ is the invariant
mass of the $XX$ system. For a selection of one-loop contributions that make
the $h\to XX$ vertex amplitude see Figure 8. Consequently, ${{\cal M}}_{h\to
XX}(s)\frac{1}{s-M_{h}^{2}}|_{s=M_{h}^{2}}$ is continuous and the Laurent’s
series is well defined without a regulator, the width. There is then no pole
structure as such and away from $s=M_{h}^{2}$ other (non-resonant) structures
in the amplitude contribute as importantly. What is very interesting is that
the one-loop amplitude is now $\lambda_{2}$ dependent, see Figure 8. Note that
$XX\to WW^{\star}$ is an abuse of language since such an off-shell amplitude
is not an element of the S-matrix. At one-loop, this statement is supported by
the appearance of essential pentagon contributions where a split into a
$WW^{\star}$ part is not possible. Nonetheless, the fact that the final state
fermions are massless, we find no flavour dependence in the normalised (with
respect to the tree-level) loop corrections when the full one-loop
contribution is taken into account. In fine, this is reminiscent and strongly
related to the fact that (normalised) electroweak corrections to $W$ decay
into fermion pairs are flavour independent Bardin:1986fi . Our results for the
electroweak corrections in this channel are displayed in Figure 9. The
behaviour with respect to velocity is very smooth, for all values of
$\lambda_{2}$, confirming that the location of the Higgs is crossed
continuously.

Figure 9: The relative one-loop contribution to $XX\to Wff^{\prime}$ as a
function of the relative velocity $v$ for $\lambda_{2}=0.01,1,2$. These
corrections are compared to the effect of using an effective tree-level
correction with $\alpha(M_{Z}^{2})$.

The $\lambda_{2}$ dependence at one-loop is important. The full one-loop
correction is about $10\%$ for the smallest of $\lambda_{2}$,
$\lambda_{2}=0.01$, it gives an almost vanishing total correction for
$\lambda_{2}=1$ and then decreases to about $-10\%$ for $\lambda_{2}=2$. For
this value, $\lambda_{2}=2$, there is a $30\%$ difference with the use of an
effective $\alpha(M_{Z}^{2})$. Therefore, $\alpha(M_{Z}^{2})$ does not perform
well as an approximation. The velocity dependence of the $\lambda_{2}$
correction is very weak. To summarise, at the level of the cross-sections the
$\lambda_{2}$ contribution is important and discriminating. Will this
conclusion still hold when we call these cross-sections to compute the relic
density considering that the weight of this channel is only $10\%$?

##  4 Effect on the relic density

|  | $\Omega h^{2}$ | $AX\to f\bar{f}$ | $WW^{*}$
---|---|---|---|---
P58 | tree | 0.113 | 95% | 5%
One-loop $AX\to f\bar{f}$ and tree $XX\to WW^{\star}$ | 0.108 (-4.05%) | 95% | 5%
Effective $AX\to f\bar{f}$ and tree $XX\to WW^{\star}$ | 0.108 (-4.45%) | 95% | 5%
$\alpha(M_{Z}^{2})$ | 0.101 (-10.62%) | 95% | 5%
P60 | tree | 0.116 | 91% | 9%
Full one-loop, $\lambda_{2}=0.01$ | 0.111 (-4.04%) | 91% | 9%
Full one-loop, $\lambda_{2}=1$ | 0.112 (-3.43%) | 91% | 9%
Full one-loop, $\lambda_{2}=2$ | 0.113 (-2.81%) | 91% | 9%
Effective $AX\to f\bar{f}$ and tree $XX\to WW^{\star}$ | 0.111 (-4.36%) | 91% | 8%
Effective $AX\to f\bar{f}$ and loop $XX\to WW^{\star},\lambda_{2}=0.01$ | 0.110 (-4.92%) | 91% | 9%
Effective $AX\to f\bar{f}$ and loop $XX\to WW^{\star},\lambda_{2}=1$ | 0.111 (-4.32%) | 91% | 9%
Effective $AX\to f\bar{f}$ and loop $XX\to WW^{\star},\lambda_{2}=2$ | 0.112 (-3.71%) | 91% | 9%
$\alpha(M_{Z}^{2})$ | 0.103 (-11.21%) | 91% | 9%
Table 2: The relic density for P58 and P60 with cross-sections computed at
tree-level and after including the full electroweak corrections, as well as
implementing the effective cross-sections for $AX\to f\bar{f}$ and the cross-
sections calculated with $\alpha(M_{Z}^{2})$. The percentage correction to the
relic density is given in parenthesis. The percentage weight of each channel
is computed for each implementation of the corrected cross-section.

In order to translate the cross-sections into a prediction on the relic
density, we interface our calculations of the one-loop corrected cross-section
with microMEGAs Belanger:2001fz ; Belanger:2004yn ; Belanger:2006is ;
Belanger:2013oya ; Belanger:2018mqt  to perform the convolution of the
velocity and temperature and solving the Boltzmann equation. Table 2 shows
that the correction to the relic density for these low mass co-annihilation
configurations are small with a correction in line with what we found for the
dominant cross-sections $AX\to f\bar{f}$ (remember that $\Omega\propto
1/\sigma$) which are practically $v$-independent. As expected, the corrections
are much smaller than those found with the approximation of using a running
$\alpha$. Since in P58 the contribution of the $WW^{\star}$ channel is less
than about $5\%$, we do not include the effect of the electroweak corrections
in this channel. However, we include these corrections for P60 since this
channel accounts for almost $10\%$ of the total relic density (at tree-level).
While any small $\lambda_{2}$ dependence (which occurs at the very small $v$
in $AX\to f\bar{f}$) is washed out when the corrections are translated into
the relic density calculation, the $\lambda_{2}$ dependence observed in the
cross-sections $XX\to Wf\bar{f}^{\prime}$, make their way into the relic
density even if the $WW^{\star}$ channel account for only $\sim 10\%$. For
$\lambda_{2}=2$, the correction in the $WW^{\star}$ channel counterbalances
part of the correction from the contributions $AX\to f\bar{f}$ leading to an
overall correction on the relic density less than $3\%$. As a summary, in the
co-annihilation region, if one is satisfied with a calculation of the relic
density with an accuracy no better than $5\%$, one should content oneself with
a tree-level calculation. An “improved" ($\alpha(M_{Z}^{2})$) gives a
correction of about $10\%$. The effective cross-sections that are based on re-
expressing the $AX\to f\bar{f}^{\prime}$ in terms of the one-loop partial
decay widths of the $Z\to f\bar{f}$ reproduce the full one-loop results within
$1\%$.

##  5 Conclusions

We have studied two IDM low mass DM benchmark points where co-annihilation is
the dominant contribution to the relic density. The co-annihilation into
fermions is driven essentially by the SM gauge coupling. The relative
corrections are small. We show that they can be parameterised by simple
effective cross-sections involving $Z$-observables. The fermion flavour
dependence is small. The small contribution from $WW^{\star}$ to the relic
density is interesting since at one-loop it reveals an important $\lambda_{2}$
dependence that accesses the purely dark sector. However, this $\lambda_{2}$
dependence gets diluted because of the small weight of $WW^{\star}$ to the
relic density. Nonetheless, our calculations show that performing a full one-
loop calculation gives smaller corrections than using an effective
$\alpha(M_{Z}^{2})$ into tree-level cross-sections.

###### Acknowledgements.

We thank Alexander Pukhov for helpful discussions. HS is supported by the
National Natural Science Foundation of China (Grant No.12075043, No.11675033).
He warmly thanks the CPTGA and LAPTh for support during his visit to France in
2019 when this work was initiated. SB is grateful for the support received
from IPPP, Durham, UK, where most of this work was performed and is also
grateful to the support received from LAPTh, in 2019, where this work was
initiated. NC is financially supported by IISc (Indian Institute of Science,
Bangalore, India) through the C.V. Raman postdoctoral fellowship. He also
acknowledges the support received from DST, India, under grant number
IFA19-PH237 (INSPIRE Faculty Award).

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