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: 4. The Higgs resonance region

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

###### Abstract

One-loop electroweak corrections to the annihilation cross-sections of dark
matter in the Higgs resonance region of the inert doublet model (IDM) are
investigated. The procedure of how to implement the width of the Higgs in
order to regularise the amplitude both at tree-level and at one-loop together
with the renormalisation of a key parameter of the model, are thoroughly
scrutinised. The discussions go beyond the application to the relic density
calculation and also beyond the IDM so that addressing these technical issues
can help in a wider context. We look in particular at the dominant channels
with the $b\bar{b}$ final state and the more involved 3-body final state,
$Wf\bar{f}^{\prime}$, where both a resonance and an anti-resonance, due to
interference effects, are present. We also discuss how to integrate over such
configurations when converting the cross-sections into a calculation of the
relic density.

††preprint: LAPTH-004/21, CERN-TH-2021-004

##  1 Introduction

There is a fine-tuned region in the parameter space of the inert doublet 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  where if the mass of the lightest
neutral scalar of the model $X$ is such that $2M_{X}\simeq M_{h}=125$GeV ($h$
is the Standard Model (SM) Higgs), a good value of the relic density is
obtained. This is possible, thanks to the efficient annihilation of the DM
pair through, essentially, the SM Higgs resonance. The latter couples to the
DM pair with a coupling of strength, $\lambda_{L}$. In Ref. OurPaper1_2020
where an extensive analysis of the parameter space of the model is conducted,
taking into account various experimental and theoretical constraints, two
benchmarks points for the Higgs resonance region are selected. We list the
characteristics of these 2 points in Table 1.

| P57 | P59
---|---|---
$M_{X}$ | 57 | 59
$\lambda_{L}\times 10^{3}$ | 2.4 | 1.0
$M_{A}$,$M_{H^{\pm}}$ | 113,123 | 113,123
$(\lambda_{3},\lambda_{4},\lambda_{5})$ | (0.382, -0.228,-0.152) | (0.373,-0.224,-0.148)
$\tilde{\beta}_{\lambda_{L}}$ | $1.123+2.157\;\lambda_{2}$ | $1.097+2.095\;\lambda_{2}$
$\Omega h^{2}$
$\alpha(0)$ | 0.113 | 0.108
$\alpha(M_{Z}^{2})$ | 0.118 | 0.113
$\Omega_{b\bar{b}}(\%)$ | 58 | 57
$\Omega_{WW^{\star}}(\%)$ | 22 | 24
$\Omega_{gg}(\%)$ | 8 | 7
$\alpha(M_{Z}^{2}),\lambda_{L}=0$ | 1.97 | 1.97
Table 1: Characteristics of the benchmark points for the Higgs resonance
region, P57 and P59. $H^{\pm}$ are the charged scalars of the IDM. Beside the
DM scalar, $X$, $A$ is the second neutral scalar often referred to as the
pseudo-scalar of the IDM. All masses are in GeV. $\lambda_{2,3,4,5}$ are the
parameters of the scalar potential of the IDM. The tree-level relic density
(calculated with $\alpha(0)$, the electromagnetic coupling in the Thomson
limit) and the weight in percent for all channels contributing to more than
$5\%$ of the relic density, are shown. These are the dominant $b\bar{b}$
bottom quark pair final state, $WW^{\star}$ and gluon pair, $gg$. The last
line gives the value of the relic density when $\lambda_{L}=0$, the Higgs
resonance being thus switched off. We also list the values of the underlying
parameters $\lambda_{3,4,5}$ and $\tilde{\beta}_{\lambda_{L}}$ (the one-loop
$\beta$ constant associated to $\lambda_{L}$) OurPaper1_2020 . We use
micrOMEGAs 5.0.7 Belanger:2001fz ; Belanger:2004yn ; Belanger:2006is ;
Belanger:2013oya ; Belanger:2018mqt  to compute the relic density.

The prominence of the Higgs resonance is made obvious upon switching off Higgs
exchange by setting $\lambda_{L}=0$. With $\lambda_{L}=0$, the relic density
is almost as much as $20$ times larger. This huge increase is due to the
extremely small residual gauge contribution from $XX\to
WW^{\star}(ZZ^{\star})$ through the $t$-channel exchange of $H^{\pm}(A)$,
which we study at length in OurPaper1_2020 . Barring this residual
contribution, the percentages of the participating cross-sections to the relic
density in Table 1 correspond to, a very good approximation 111With the tiny
values of $\lambda_{L}$ for both P57 and P59, ${\rm Br}(h\to XX)<0.3\%$ (see
Ref. OurPaper1_2020 ). This very small branching into invisibles does not
affect much the SM values of the Higgs partial widths. the branching fraction
of the SM Higgs to the corresponding channels including the $gg$ (gluon pair
final state). Only the $WW^{\star}=Wf\bar{f}^{\prime}$ channel for the relic
density is $2\%$ larger than the partial decay fraction of the Higgs to
$WW^{\star}$ due to the small continuum gauge contribution, $XX\to
WW^{\star}$, for the relic. The total width and the branching fractions are
therefore key parameters for describing the Higgs lineshape and the
computation of the relic density in the Higgs resonance region. Small changes
in or very close to the vicinity of the resonance could lead to large
corrections to the cross-sections and hence to the relic density. If one
observes that the non resonant $WW$ channel contributes even in the absence of
Higgs exchange, one could ask, for instance, if other contributions to the
same channels could be induced at one-loop, away from the resonance, impacting
the cross-sections. Because of the inert nature of the model, annihilation to
fermions only takes place through the Higgs exchange at tree-level. However,
at electroweak one-loop corrections these annihilations proceed even when the
Higgs exchange is absent. Nonetheless, because of the small width of the SM
Higgs, the properties of the Higgs resonance are crucial. Apart from the mass
of the Higgs, they rely on the total width and the branching fractions of the
SM Higgs.
The plan of the paper is as follows. We first review the key parameters, in
particular the SM parameters that enter the description of the Higgs
resonance. Since the paper deals with the electroweak corrections, the
discussion about the parameters, in particular the bottom-quark mass, is
covered in the next section which aims to describe how the important QCD
corrections are included. Since, as known and as we will argue, the width is a
higher order effect, its introduction at tree-level and at one-loop should be
treated with care, in particular in order to avoid double counting of
corrections and also, in general, and how to avoid breaking gauge invariance.
In this spirit, section 3 presents the tree-level calculation of the
$b\bar{b}$ final state through the Higgs resonance. The Breit-Wigner
parametrisation will prove to be most useful when we deal with the one-loop
electroweak correction which will be covered in Section 4. We will see how the
perturbation series needs to be reorganised in order to avoid the issue of
double counting. The next 2 sections, Section 5 and Section 6, will follow
similar steps in the case of $WW^{\star}=Wf\bar{f}^{\prime}$. Some important
technical issues need to be tackled here. The $WW^{\star}=Wf\bar{f}^{\prime}$
final state features a resonance contribution and also an interference with a
"continuum" part. The whole effect will exhibit a resonance and "anti-
resonance" behaviour over a small range of energy. Section 7 turns the
corrected cross-sections into the improved relic density. We show how we must
be extra careful in integrating the cross-sections over the velocity
distribution needed in this conversion. Our conclusions are presented in
Section 8.

##  2 The inputs and the parameters to study the resonance region

All the partial widths and total widths can be derived from theory. The latest
state-of-the art calculations deFlorian:2016spz ; 10.1093/ptep/ptaa104  for
the total SM width of the Higgs gives the value

$\displaystyle\Gamma_{h}=4.07\;{\rm MeV}\;\;({\pm 4\%}).$ (1)

Considering the theoretical error on the total width (much larger than that
contribution due to $\Gamma(h\to XX)$ after imposing the direct detection
constraint on $\lambda_{L}$OurPaper1_2020 ), we equate the total width with
the SM total width. These most precise calculations of the SM Higgs width
include crucial higher order QCD corrections. They also lead to the
theoretical prediction for the branching ratio into $b\bar{b}$ to be $58.2\%$,
which corresponds to $\Gamma(h\to b\bar{b})=2.37$ MeV. In order to incorporate
these important QCD higher order predictions in an electroweak calculation at
tree-level we will use, in this paper, an effective $m_{b,\rm eff.}$ for the
Higgs coupling such that the correct value of the branching width into
$b\bar{b}$ is recovered. With the tree-level prediction of the $b\bar{b}$
partial width given by

$\displaystyle\Gamma(h\to b\bar{b})=M_{h}\frac{N_{c}}{8\pi}\frac{m_{b,\rm
eff.}^{2}}{v^{2}}\beta_{f},\quad\beta_{f}=\sqrt{1-4m_{b,\rm
eff.}^{2}/M_{h}^{2}},\quad N_{c}=3,\quad M_{W}=\frac{gv}{2}.$ (2)

$M_{W}$ is the $W$-boson mass ($g$ is the $SU(2)$ coupling and $v$ the vacuum
expectation value). We therefore take for the Higgs resonance channel
$m_{b}=3.15375$ GeV as the effective mass to incorporate the important QCD
corrections 222Note that this effective mass is not to be used as the input
mass in micrOMEGAs. For micrOMEGAs, a default $b$-pole mass is used as input.
Internally, the code calls different routines for running masses and QCD
corrections to the $hbb$ vertex.. In our calculation of the cross-sections for
these benchmark points, we therefore only consider the electroweak
corrections, that unlike the QCD correction, affect both the initial and final
states. We should also keep in mind that $\Gamma(h\to b\bar{b})$ in Equation 2
does not include the electroweak radiative corrections which as we will see
are quite small.

Apart from the numerical value of the Higgs width, we have more to say about
how we implement the Higgs width in our calculations. It is sufficient to say
in this preamble that we take the total SM Higgs width as an input (at all
orders), the same way as we treat the Higgs mass as an input (at all orders)
in perturbation. As Table 1 shows, the most important channels are by far the
$b\bar{b}$ and the $WW$. Although the sum of the rest of the individual
channels is not at all negligible ($20\%$), all fermion pair final states
proceed through the Higgs exchange similarly to the $b\bar{b}$ channel and
hence the procedure of applying the electroweak radiative corrections to them
is the same as in the $b\bar{b}$ channel. The contribution of the $ZZ$ final
state is below $3\%$, which is too small a contribution for any correction to
make an impact on the relic density calculation, especially in view of the
results we find in the $WW$ channel. Technically, $ZZ$ involves the same
treatment as the annihilation to $WW$. Lastly, at the order we are performing
the corrections, the Higgs exchange induced annihilation into gluon pairs will
not be subject to electroweak corrections.

##  3 $XX\stackrel{{\scriptstyle h}}{{\longrightarrow}}b\bar{b}$ at tree-level

Figure 1: A single diagram, Higgs mediated, contributes to $XX\to b\bar{b}$ at
tree-level.

In the IDM, the tree-level annihilation to $b\bar{b}$ only occurs through the
SM Higgs. In this case, it is relatively easy to parameterise the tree-level
cross-section $XX\to b\bar{b}$ in a compact form as

$\displaystyle\sigma^{0}{(XX\stackrel{{\scriptstyle
h}}{{\longrightarrow}}b\bar{b}})\;v=\frac{64\;\pi\frac{\Gamma(h\rightarrow
XX)}{\sqrt{1-4M_{X}^{2}/M_{h}^{2}}}\Gamma(h\rightarrow
b\bar{b})}{(s-M_{h}^{2})^{2}+\Gamma_{h}^{2}M_{h}^{2}}.$ (3)

Here and in the following 0 refers to the tree-level cross-section. $s$, the
invariant mass of the system, can be expressed in terms of the relative
velocity $v$, $\sqrt{s}=2M_{X}/\sqrt{1-\frac{v^{2}}{4}}$. Note that this tree-
level cross-section is written solely in terms of physical observables, the
total width and the partial widths into $XX$ and $b\bar{b}$. Observe also that
in this Breit-Wigner cross-section, the total width and the partial widths are
physical (on-shell) widths that do not run with $s$. It is therefore a totally
gauge invariant cross-section. We take the partial width $\Gamma(h\to XX)$ as
an input parameter in lieu of $\lambda_{L}$. We use the total width of the
Higgs, $\Gamma_{h}$, as an input, making sure that the induced width that is
generated at one-loop (the imaginary part of the Higgs two-point self-energy
function) is properly subtracted (see later). For $\Gamma(h\rightarrow
b\bar{b})$, we stress again that we use the effective $b$-quark mass which is
consistent with the value of $\Gamma(h\rightarrow b\bar{b})$ (Equation 2),
therefore both $\Gamma_{h}$ and $\Gamma(h\rightarrow b\bar{b})$ include
important resummed QCD corrections which we do not have to calculate when we
calculate the electroweak corrections. The latter will also include, at one-
loop, non resonant Higgs contributions. For $\Gamma(h\rightarrow XX)$, we use
Ref. OurPaper1_2020

$\displaystyle\Gamma_{h\to XX}$ $\displaystyle=$
$\displaystyle\frac{\lambda_{L}^{2}v^{2}}{32\pi
M_{h}}\sqrt{1-\frac{4M_{X}^{2}}{M_{h}^{2}}}.$ (4)

It is very important to observe that in the product $\Gamma(h\rightarrow
XX)\Gamma(h\rightarrow b\bar{b})$, the value of the vacuum expectation value,
$v$, drops out. Therefore, the dependence on the electromagnetic coupling that
separately enters these partial widths through $v$ cancels out. As a result,
with $\Gamma_{h}$ as an input parameter, the cross-section, $XX\to b\bar{b}$,
does not depend on the value of the electromagnetic coupling $\alpha$. This is
the reason the values of the relic density in Table 1 for the resonance region
show quite negligible difference between the use of $\alpha(0)$ and
$\alpha(M_{Z}^{2})$ (the small difference comes from the small gauge
contribution to the continuum $WW^{\star}$).
In our analysis, the cross-sections are exactly computed with the help of our
automated code SloopS Baro:2007em ; Baro:2008bg ; Baro:2009na . We verify that
for the tree-level annihilation into $b\bar{b}$, the parameterisation of
Equation 3 reproduces the results of the full calculation over the whole $v$
range with an agreement better than $5\times 10^{-4}$ around the resonance
region, which is the region that matters most for the relic density
calculation, as shown in Figure 2.

Figure 2:  The velocity dependence of the ratio of the full tree-level
computation relative to the fixed width Breit-Wigner calculation based on
Equation 3 for Points P57 and P59. The horizontal line, exact agreement, meets
the curves at exactly the resonance points.

This verification was in order not only to understand the features of the
resonance processes but also to seek effective implementation at one-loop,
later. We have more to say about equation 3 after discussing the set up of the
loop calculation to this observable.

The annihilation cross-section and its velocity dependence is shown in Figure
3. The peaks are extremely sharp because of the extremely narrow width of the
SM Higgs with $\Gamma_{h}/M_{h}=3.26\times 10^{-5}$.

Figure 3:  The relative velocity dependence of the tree-level $XX\to b\bar{b}$
cross-section times $v$ for Points P57 and P59 in units of $10^{-26}{\rm
cm}^{3}{\rm s}^{-1}$ (note the $\log$ scale). We also show the weighted cross-
sections with the velocity distribution OurPaper1_2020  with the freeze-out
parameter $x_{F}=25$.

We now understand that the P57 contribution to the relic density occurs at the
edge of the velocity function, $v\sim 0.82$, in order to catch the Higgs
resonance. It therefore needs a much larger value of $\lambda_{L}$ than P59
that captures the Higgs peak at $v\sim 0.66$. While converting the cross-
sections into a contribution to the relic density, thermal averaging and
convolution with the velocity distribution is performed. The weight of the
velocity distribution is important. This convolution reduces somehow the
weight of the resonance that occurs at large $v$ compared to the off resonance
contributions that occur, say, at $v\sim 0.4$, see Figure 3. Therefore,
possible changes around the peak and possible non-resonant loop-induced
contributions are important to study OurPaper1_2020 . In any case, we need to
very carefully scrutinise the peak. Another very important technical point is
that we need to scan the Higgs resonance extremely carefully, both for the
$b\bar{b}$ and the $WW^{\star}$ channels. This is the reason why will be
generating a very large set of data points around the resonance.

##  4 The $b\bar{b}$ final state in the Higgs resonance region: one-loop
electroweak corrections

Electroweak loop calculations with width effects are very complicated, not
only because of the numerical integration over the very narrow width region
but also conceptually. First of all, the width is a resummation of a loop
effect. Analyticity relates the width to the imaginary part of the self-energy
of the corresponding particle. Introducing the width at the tree-level means
that we are including a part of a higher order loop calculation (imaginary
part of the two-point function). To be consistent and not double count
contributions, at one-loop, the self-energy contribution (included already at
tree-level as a part of $\Gamma_{h}$) must be subtracted. Moreover, the mixing
of different orders in perturbation should be excised with care so as not to
introduce a breaking of gauge invariance. In the case at hand, for the
$b\bar{b}$ channel, the latter issue is easy to deal with.
Renormalisation is a coherent definition of the independent parameters of the
model. Taking as input parameters all the masses of the scalars of the IDM,
two extra independent parameters remain to be defined. The first one is
$\lambda_{L}$. With a spectrum like the one in P57 and P59, $h\to XX$ is open.
$\lambda_{L}$ is defined most easily (parametrically also) from $\Gamma(h\to
XX)$ through Equation 4. We derive the associated counterterm in this full On-
Shell (OS) scheme in OurPaper1_2020 . Naturally, since $\Gamma(h\rightarrow
XX)$ is used as an input, it has the same value at all orders of perturbation.
By construction, no loop correction affects the observable
$\Gamma(h\rightarrow XX)$. Another parameter, $\lambda_{2}$, describes the
self-interaction between the scalars, i.e., it resides solely within the
scalar sector. For the annihilation of DM into the SM particles, the cross-
sections do not depend on $\lambda_{2}$ and therefore a counterterm for
$\lambda_{2}$ for the annihilation cross-sections is not called for. However,
$\lambda_{2}$ can indirectly affect the value of the cross-section, an effect
akin to the top quark dependence of LEPII observables.

###  4.1 $XX\to b\bar{b}$ at one-loop: The theoretical set-up

Figure 4:  A selection of one-loop electroweak corrections diagrams
contributing to $XX\to b\bar{b}$. The first 3 diagrams of the first row are
genuine boxes which are non-resonant. The last 2 diagrams in the first row
show examples where $\lambda_{2}$, rescattering in the dark sector,
contributions can be induced at one-loop. The second row includes vertex
corrections and self-energy contributions to the SM Higgs that can become
resonant.

While at tree-level the structure of the $XX\to b\bar{b}$ is simple since it
proceeds through the unique contribution of Higgs exchange, at one-loop a
variety of contributions are at play, in particular non-resonant, non-Higgs
exchange contributions, as shown by the selection in Figure 4. The question
is, how can one then introduce the Higgs width to regulate the cross-section
at $s=M_{h}^{2}$ in a coherent gauge invariant manner now that we have many
contributions?

Let us first start by reviewing the amplitude at tree-level. In our case it is
not only a simple structure but it is also a trivially gauge invariant
amplitude. Prior to the introduction of the width, the amplitude can be
written as

$\displaystyle{\cal A}_{h}^{0}=\frac{{{\cal M}_{hXX}}\;{{\cal
M}_{hbb}}}{s-M_{h}^{2}},\quad{{\cal M}_{hXX}}={{\cal M}(h\to XX)},\quad{{\cal
M}_{hbb}}={{\cal M}(h\to b\bar{b})}.$ (5)

${{\cal M}(h\to XX)}$ and ${{\cal M}_{hbb}}={{\cal M}(h\to b\bar{b})}$ are the
respective amplitudes of the transitions $XX\to h$ and $h\to b\bar{b}$. The
pole structure requires the introduction of the width

$\displaystyle{\cal A}_{h}^{0,w}=\frac{{{\cal M}_{hXX}}\;{{\cal
M}_{hbb}}}{s-M_{h}^{2}+i\Gamma_{h}M_{h}}=\underbrace{\frac{s-M_{h}^{2}}{s-M_{h}^{2}+i\Gamma_{h}M_{h}}}_{F_{h}}{\cal
A}_{h}^{0},\quad F_{h}=\frac{s-M_{h}^{2}}{s-M_{h}^{2}+i\Gamma_{h}M_{h}}.$ (6)

The last equality shows that the genuinely tree-level amplitude has been
rescaled by a kinematics factor, $F_{h}$. Being a kinematical factor, it is
gauge invariant. In what follows, the regulated amplitude is gauge invariant.
In the code, for such a simple process, the introduction of the width is
trivial. We either correct the amplitude of equation 5 (or for that matter the
cross-section) directly by adding the width to the Higgs propagator or, which
amounts to exactly the same, by multiplying the non-width expression of
equation 5 by the rescaling factor, $F_{h}=F_{h}(s)$. It is important to
observe that we are using a fixed width that corresponds to the observable
total width or the physical width that can be perturbatively computed through
an all-order calculation and is gauge invariant. In particular, it contains
the dominant leading order that appears at one-loop in the self-energy of the
Higgs. The tree-level amplitude therefore includes a contribution from the
one-loop calculation. In turn, in performing the full one-loop correction, we
therefore need to subtract this one-loop contribution since it is already put
in the width-improved tree-level expression, in order to avoid double counting
this specific contribution both in the width-improved tree-level cross-section
and in the one-loop calculation. Note that the introduction of the width, at
tree-level, can be interpreted as a correction that amounts to

$\displaystyle\frac{-i\Gamma_{h}M_{h}}{s-M_{h}^{2}}{\cal A}_{h}^{0}.$ (7)

At one-loop, the full correction consists of i) the self energy insertion,
$\Pi_{hh}(s)$, ii) vertex corrections and iii) box corrections, see figure 4.
The self-energy corrections, from the Higgs two-point function, and its
counterterm, $\hat{\Pi}_{hh}(s)$, add up to

$\displaystyle\Pi_{hh}(s)+\hat{\Pi}_{hh}(s)$ $\displaystyle=$
$\displaystyle\Pi_{hh}(s)+(s-M_{h}^{2})\delta Z_{h}-\delta
M_{h}^{2}+\frac{3\delta T}{v}$ (8) $\displaystyle=$
$\displaystyle\Pi_{hh}(s)+(s-M_{h}^{2})\delta Z_{h}-\hat{\delta}M_{h}^{2},$

where $\delta T$ is the tadpole counterterm and $\delta Z_{h}$ is the SM Higgs
wave function renormalisation, see FawziPhysRepGrace .

The Higgs mass counterterm is defined from the real part of the self-energy of
the Higgs calculated at the Higgs mass

$\displaystyle\hat{\delta}M_{h}^{2}={{\cal{R}}}e\Pi_{hh}(M_{h}^{2}).$ (9)

At one-loop, prior to the introduction of the (total) width (in order to keep
gauge invariance explicit), the total amplitude writes 333Strictly speaking,
the counterterms $\delta{{\cal M}_{hbb}}$ and $\delta{{\cal M}_{hXX}}$ here
refer to the counterterms of the parameters defining the corresponding
vertices.

$\displaystyle\delta{\cal A}_{h}^{1}=\underbrace{\frac{{{\cal
M}_{hXX}}\;{{\cal
M}_{hbb}}}{s-M_{h}^{2}}\;\left(-\frac{\Pi_{hh}(s)+\hat{\Pi}_{hh}(s)}{s-M_{h}^{2}}\right)}_{\text{Higgs
self-energy}}+\underbrace{\Bigg{(}\frac{\delta{{\cal M}_{hXX}}{{\cal
M}_{hbb}}+{{\cal M}_{hXX}}\delta{{\cal
M}_{hbb}}}{s-M_{h}^{2}}\Bigg{)}}_{\text{vertex,Higgs
exch.}}+\underbrace{\Box_{XXbb}.}_{\text{box corr., non-res. cont.}}$ (10)

What we have collected in $\Box_{XXbb}$ includes not only box diagrams but
also vertex ($Z,\gamma$) exchanges which are not Higgs-resonant, examples of
which are shown in figure 4.

From the Higgs two-point function contribution, we have

$\displaystyle\Pi_{hh}(s)+\hat{\Pi}_{hh}(s)=i{\cal{I}}m\Pi_{hh}(M_{h}^{2})+(s-M_{h}^{2})\delta
Z_{h}+(\Pi_{hh}(s)-\Pi_{hh}(M_{h}^{2})).$ (11)

$i{\cal{I}}m\Pi_{hh}(M_{h}^{2})$ is the one-loop term that should be
subtracted for the one-loop contribution because part of it is included in
${\cal A}_{h}^{0,w}$ (the tree-level width inserted cross-section). Indeed,

$\displaystyle{\cal{I}}m\Pi_{hh}(M_{h}^{2})=\Gamma_{h}^{0}M_{h}$ (12)

$\Gamma_{h}^{0}$ is the tree-level calculated width of the Higgs. The easiest
way to implement this subtraction is to define a modified on-shell
renormalisation for the Higgs mass such that

$\displaystyle\hat{\delta}M_{h}^{2}=\Pi_{hh}(M_{h}^{2}),$ (13)

in other words the Higgs mass counterterm is defined from the full two-point
function of the Higgs and not only from its real part. Then the imaginary part
(width at tree-level) is subtracted automatically. Note that since
$M_{h}^{2}\propto\lambda_{h}v^{2}$, this means that we need to introduce
imaginary counterterm for the Higgs self-coupling. Since the processes we
study do not call for this coupling apart from the Higgs mass, this is not
required for this simple process.

The one-loop contribution (before regularisation with the total width) is

$\displaystyle\delta{\cal A}_{h}^{1}$ $\displaystyle=$
$\displaystyle\frac{{{\cal M}_{hXX}}\;{{\cal
M}_{hbb}}}{s-M_{h}^{2}}\;\left(-\delta
Z_{h}-\frac{\Pi_{hh}(s)-\Pi_{hh}(M_{h}^{2})}{s-M_{h}^{2}}\right)$ (14)
$\displaystyle+$ $\displaystyle\Bigg{(}\frac{\delta{{\cal M}_{hXX}}{{\cal
M}_{hbb}}+{{\cal M}_{hXX}}\delta{{\cal
M}_{hbb}}}{s-M_{h}^{2}}\Bigg{)}+\Box_{XXbb}.$

Maintaining gauge invariance of the above amplitude while introducing a Higgs
width, our scheme consists of applying the factorising factor, $F_{h}$, to the
whole amplitude of Equation 14.

The regularised one-loop corrected, gauge invariant, amplitude is then

$\displaystyle\delta{\cal A}_{h}^{1,w}$ $\displaystyle=$
$\displaystyle\frac{s-M_{h}^{2}}{s-M_{h}^{2}+i\Gamma_{h}M_{h}}\delta{\cal
A}_{h}^{1}$ (15) $\displaystyle=$ $\displaystyle\frac{{{\cal M}_{hXX}}\;{{\cal
M}_{hbb}}}{s-M_{h}^{2}+i\Gamma_{h}M_{h}}\;\left(-\delta
Z_{h}-\frac{\Pi_{hh}(s)-\Pi_{hh}(M_{h}^{2})}{s-M_{h}^{2}}\right)$
$\displaystyle+$ $\displaystyle\Bigg{(}\frac{\delta{{\cal M}_{hXX}}{{\cal
M}_{hbb}}+{{\cal M}_{hXX}}\delta{{\cal
M}_{hbb}}}{s-M_{h}^{2}+i\Gamma_{h}M_{h}}\Bigg{)}$ $\displaystyle+$
$\displaystyle\frac{s-M_{h}^{2}}{s-M_{h}^{2}+i\Gamma_{h}M_{h}}\Box_{XXbb}.$

The drawback of the scheme is that the non-resonant contributions gathered in
$\Box_{XXbb}$ vanish at the resonance, $s=M_{h}^{2}$. By default, in the code
we first calculate the one-loop amplitude without the width within the
modified OS scheme (Equation 13) to avoid double counting. In a second step,
we apply the $F_{h}$ factor to the full amplitude to obtain Equation 15.
Observe that the factor containing $\delta Z_{h}$ in Equation 15 (first line)
also has a term with a denominator $(s-M_{h}^{2})$. For this term, no width
can be introduced but the expression is fully regularised because of the
difference appearing in the accompanying numerator.

As promised, let us get back to equation 3 and see how it compares with a full
one-loop calculation. Since we have implemented an OS shell scheme where
$\Gamma(h\to XX)$ is used in lieu of $\lambda_{L}$ and where we are making use
of the total width $\Gamma_{h}$ as input, at the peak $s=M_{h}^{2}$,
$\Gamma(h\to XX)$ is unchanged but we expect a very small electroweak
correction to $\Gamma(h\to b\bar{b})$, such that $\Gamma(h\to
b\bar{b})\equiv\Gamma(h\to b\bar{b})^{\text{tree}}\to\Gamma(h\to
b\bar{b})^{\text{1-loop, EW}}$(which, as we argued before is insensitive to
the dark sector at one-loop). But, more importantly, loop effects bring in new
contributions that are not resonant and that are not described by the Breit-
Wigner approximation. But, for the sake of comparison, at $s\simeq M_{h}^{2}$,
we can seek an improved tree-level electroweak cross-section by an effective
rescaling of $\Gamma(h\to b\bar{b})^{\text{1-loop, EW}}$. This is a very small
correction that we have calculated to be

$\displaystyle\frac{\Gamma(h\to b\bar{b})^{\text{1-loop, EW}}}{\Gamma(h\to
b\bar{b})}=1.0166.$ (16)

Hence the improved $b\bar{b}$ cross-section (that will miss one-loop non-
resonant corrections) writes as

$\displaystyle\sigma^{\text{imp.}}{(XX\stackrel{{\scriptstyle
h}}{{\longrightarrow}}b\bar{b}})\;v=\frac{64\;\pi\frac{\Gamma(h\rightarrow
XX)}{\sqrt{1-4M_{X}^{2}/M_{h}^{2}}}\Gamma(h\rightarrow
b\bar{b})^{\text{1-loop, EW}}}{(s-M_{h}^{2})^{2}+\Gamma_{h}^{2}M_{h}^{2}}.$
(17)

###  4.2 Results for $XX\to b\bar{b}$ at one-loop

Figure 5:  The relative velocity dependence of the one-loop $XX\to b\bar{b}$
cross-section times $v$ for Points P57 (left panels) and P59 (right panels) in
units of $10^{-26}{\rm cm}^{3}{\rm s}^{-1}$ (note the $\log$ scale). The lower
panels help better see the effect of the loop corrections by giving the
relative corrections (with respect to the tree-level) over the full velocity
range as well as a zoom around the peak. The $\lambda_{2}$ dependence of the
one-loop results is given. Also shown is the result of using the full
electroweak correction to $\Gamma(h\to b\bar{b})$ (the improved cross-section
given by Equation 17). We also give the result of using, at tree-level,
$\alpha(M_{Z}^{2})$ instead of $\alpha$.

Our numerical results for the full one-loop electroweak corrections for P57
and P59 are shown in figure 5. Apart from the location of the Higgs peak in
terms of the relative velocity, the results for both benchmarks are very
similar. On the whole, the dependence of the velocity and the peak structure
are maintained when we move from the tree-level to the one-loop corrected
results. Because of the very narrow-peaked structure, the effect of the
radiative corrections at the level of the cross-sections is hardly
distinguishable. To see the subtle details, which amount nonetheless to
important corrections, we look at the normalised corrections with respect to
the tree-level results. First, the numerical computation of the tree-level
cross-section based on the use of $\alpha(M_{Z}^{2})$ confirms that we obtain
exactly the same results, for all $v$, as the use of $\alpha(0)$ (the ratio is
equal to $1$ throughout all $v$). This quantitatively substantiates the remark
we made earlier that the $b\bar{b}$ cross-section is insensitive to $\alpha$.
Away from $s=M_{h}^{2}$, the $\lambda_{2}$ dependence of the cross-section is
important and grows as we further move away from the peak. The $\lambda_{2}$
dependence, though qualitatively similar between the two models, is rather
different numerically. For P57, the difference in the relative correction
between $\lambda_{2}=0.01$ and $\lambda_{2}=2$ reaches $40\%$ while for P59
this difference amounts to more than $60\%$. As expected, at exactly the Higgs
resonance, our computation confirms that the $\lambda_{2}$ dependence drops
out. This is very understandable since the renormalisation of $\lambda_{L}$ is
based on the OS scheme where $\Gamma(h\to XX)$ is used as input, so any
$\lambda_{2}$ dependence has been absorbed in the OS value of $\lambda_{L}$ at
$s=M_{h}^{2}$. Moreover, the value of the correction at the resonance is
reproduced exactly by the use of $\Gamma(h\to b\bar{b})^{\text{1-loop, EW}}$
rather than the pure tree-level which defines what we termed the improved
cross-section. It rests that within $2\Gamma_{h}$ from the peak at $M_{h}$,
the electroweak correction to the this annihilation channel is of the order of
$2\%$. The most important lesson is that this small relative correction is in
large part due to the OS scheme we have used that takes $\Gamma(h\to XX)$ as
input and that the implementation of the width is such that all one-loop
induced contributions are multiplied by $s-M_{h}^{2}$, which leaves then only
the Higgs resonance. The remaining small correction, which is a result of the
electroweak correction $\Gamma(h\to b\bar{b})$, is very small. Important
corrections occur away from the resonance. The interesting observation is that
they reveal the indirect $\lambda_{2}$ contribution hidden in the dark sector.

##  5 $XX\stackrel{{\scriptstyle h}}{{\longrightarrow}}WW^{\star}$ at tree-
level

Figure 6:  Tree-level diagrams contributing to $XX\to Wl\nu_{l}$. Apart from
the Higgs exchange contribution with $h^{\star}\to Wl\nu_{l}$ there are other
non-Higgs resonant contributions. There are other diagrams with $h\to
ll^{\star}\to lW\nu_{l}$ that gives effects proportional to the lepton mass
which we do not show here but which are included in our full calculation.

What we call $XX\to WW^{\star}$ is in fact $XX\to Wf\bar{f}^{\prime}$, a $2\to
3$ amplitude. These cross-sections are studied in Ref. OurPaper3_2020  but in
the case where the DM mass was large enough so that the SM Higgs resonance is
not reached and there was no need to include the Higgs width. In
OurPaper2_2020 , $XX\to WW^{\star}$ contributes subdominantly to the relic
density and while the mass spectrum is such that the Higgs resonance is hit,
we have $\lambda_{L}=0$ so that the Higgs exchange is absent at tree-level and
at one-loop and the induced Higgs exchange crosses the resonances smoothly.
The scenarios in the present paper is that the SM Higgs resonance is crossed
with $\lambda_{L}\neq 0$. The inclusion of the Higgs resonance must be
implemented both at tree-level and at one-loop. The issue is more involved
than in the case of the annihilation into $b\bar{b}$ we just studied. Here,
the total contribution involves a resonant Higgs exchange contribution and a
gauged, non-resonant, amplitude. Moreover, the $\lambda_{L}$ dependence of the
$XX\to WW$ cross-section stems not only from the pure SM Higgs (resonant)
contribution but also from the quartic coupling $XXWW$. Converting (or
regularising) only one of these contributions (the Higgs exchange) to include
the Higgs width, the subtle $\lambda_{L}$ dependence is broken in the sense
that we would be providing a width for the Higgs for only a part of the
$\lambda_{L}$ dependent amplitude. This is akin to a breaking of gauge
invariance when only a part of the amplitude is endowed with a width dependent
propagator. One needs to be careful at how the width is introduced to regulate
the full amplitude.
Here again it is best to start from the full zero-width tree-level amplitude.
This amplitude can be written in terms of its pole structure as a Laurent
series in $1/(s-M_{h}^{2})$. The full amplitude at tree-level can then be
written as

$\displaystyle{\cal{A}}_{XXWW}^{0}={\cal{A}}_{XXWW}(g^{2},\lambda_{L})+\frac{{{\cal
M}_{hXX}}\;{{\cal M}_{hWff}}}{(s-M_{h}^{2})},$ (18)

where ${{\cal M}_{hXX}}\sim\lambda_{L}$ and ${{\cal M}_{hWff}}$ represent,
respectively, the transition amplitude of the SM Higgs to the DM particles
($h\to XX$) and the transition amplitude, $h\to Wf\bar{f^{\prime}}$.
As we did with the $b\bar{b}$ channel, we introduce the width at tree-level
through the overall kinematical fudge factor $F_{h}$ (see Equation 6) to the
full amplitude, such that

$\displaystyle{\cal{A}}_{XXWW}^{0,w}$ $\displaystyle=$ $\displaystyle
F_{h}\Bigg{(}{\cal{A}}_{XXWW}(g^{2},\lambda_{L})+\frac{{{\cal
M}_{hXX}}\;{{\cal M}_{hWff}}}{(s-M_{h}^{2})}\Bigg{)}$ (19) $\displaystyle=$
$\displaystyle\frac{(s-M_{h}^{2})}{(s-M_{h}^{2})+i\Gamma_{h}M_{h}}{\cal{A}}_{XXWW}(g^{2},\lambda_{L})+\frac{{{\cal
M}_{hXX}}\;{{\cal M}_{hWff}}}{(s-M_{h}^{2})+i\Gamma_{h}M_{h}}.$

This treatment is the same as what we apply at tree and one-loop levels for
$XX\to b\bar{b}$. Again, at $s=M_{h}^{2}$, the non resonant contribution is
effectively set to zero with this procedure. The ensuing cross-section can be
parametrised in terms of this pole structure as

$\displaystyle\sigma{(XX\to Wf^{\prime}\bar{f}})^{0,w}$ $\displaystyle=$
$\displaystyle\frac{\;64\;\pi\frac{\Gamma(h\rightarrow
XX)}{\sqrt{1-4M_{X}^{2}/M_{h}^{2}}}\Gamma(h\rightarrow
Wf^{\prime}\bar{f})}{(s-M_{h}^{2})^{2}+\Gamma_{h}^{2}M_{h}^{2}}+F_{h}^{2}\sigma_{XXWW}(g^{2},\lambda_{L})$
(20) $\displaystyle+$
$\displaystyle\frac{s-M_{h}^{2}}{(s-M_{h}^{2})^{2}+\Gamma_{h}^{2}M_{h}^{2}}\text{Ps}{\cal{R}}e\Bigg{(}{\cal{A}}_{XXWW}(g^{2},\lambda_{L})({{\cal
M}_{hXX}}\;{{\cal M}_{hWff}})^{*}\Bigg{)},$

where Ps represents the phase space factor. This is the tree-level cross-
section provided by the code. In the above parametrisation, in the first term
of Equation 20, the width $\Gamma(h\rightarrow Wf^{\prime}\bar{f})$ should
represent the tree-level value, a result similar to what was presented for the
$b\bar{b}$ channel. Figure 7 shows the cross-section as a function of the
relative velocity confirming the Higgs resonance and also the effect of the
interference leading to an anti-resonance structure, with a dip followed by a
spike at exactly the resonance around a very small region of $v(s)$.

Figure 7:  As in Figure 3 but for $XX\to W\tau\nu_{\tau}$. Note the behaviour
of the peak structure that is a characteristic of an interference of a smooth
contribution with a resonance. The panel on the right represents the
convolution of the cross-sections with the velocity distribution with the same
parameters as in Figure 3.

In accordance with the analysis for the $b\bar{b}$ final state, we anticipate
and suggest an improved tree-level cross-section obtained as

$\displaystyle\sigma{(XX\to Wf^{\prime}\bar{f}})^{\text{improved}}$
$\displaystyle=$ $\displaystyle\frac{\Gamma(h\rightarrow
Wf^{\prime}\bar{f})_{\text{1-loop}}}{\Gamma(h\rightarrow
Wf^{\prime}\bar{f})_{\text{tree}}}\sigma{(XX\to Wf^{\prime}\bar{f}})^{0,w},$
(21)

so that the resonant contribution at the peak contains the full one-loop
correction. This constant factor applied to the full tree-level cross-section
will not change the behaviour of the velocity ($s$) dependence from what is
shown in Figure 7. We calculate

$\displaystyle\frac{\Gamma(h\rightarrow
Wl\nu_{l})_{\text{1-loop}}}{\Gamma(h\rightarrow
Wl\nu_{l})_{\text{tree}}}\simeq\frac{\Gamma(h\rightarrow
Wq\bar{q}^{\prime})_{\text{1-loop}}}{\Gamma(h\rightarrow
Wq\bar{q}^{\prime})_{\text{tree}}}\simeq 1.087.$ (23)

This result on the relative correction to the partial decay width is, to a
precision of $0.02\%$, flavour blind and stems from the fact that the
electroweak corrections to $W\to f\bar{f}^{\prime}$ are practically flavour
blind Bardin:1986fi ; OurPaper3_2020 . The $8.7\%$ relative correction is the
combination of the full one-loop correction to $W\to f\bar{f}^{\prime}$
($2.4\%$) and an effective correction at the level of the $hWW$ vertex
($6.3\%$) that is absorbed by using $\alpha(M_{Z}^{2})$. This observation is
important because it points to the fact that the full electroweak correction
in these channels should be flavour blind and that the relative correction
over the whole $v$ range is also flavour blind, a property that we already
found in our study for this process away from the Higgs resonance
OurPaper2_2020 ; OurPaper3_2020 . We directly verify this property in this
case. It is yet another verification of our automatic calculation and the
performance of our code 444For massless fermions, the partial width of the
charged $W$, including QCD corrections, can be written as $\Gamma(W\to
q\bar{q}^{\prime})=N_{C}\;\frac{\alpha
M_{W}}{12(1-M_{W}^{2}/M_{Z}^{2})}|V_{qq^{\prime}}|^{2}(1+\frac{3}{8}\frac{N_{C}^{2}-1}{N_{C}}\frac{\alpha_{s}}{\pi})$,
where $V_{qq^{\prime}}$ is the element of the KM matrix, and $\alpha_{s}$ is
the QCD fine structure constant..

##  6 $XX\stackrel{{\scriptstyle h}}{{\longrightarrow}}Wf^{\prime}\bar{f}$ at
one-loop

The one-loop electroweak corrections to this $2\to 3$ process are challenging
especially because we have to deal with the Higgs resonance. A large number of
one-loop topologies are involved. A subset of these topologies can be found in
Ref. OurPaper2_2020 ; OurPaper3_2020 . The set up of the one-loop calculation
with the insertion of the width goes through the same steps as those that we
presented in the case of $b\bar{b}$. In a nutshell, the width is introduced
through the $F_{h}$ factor for the entire amplitude. Therefore, we can write
for the correction

$\displaystyle\delta{\cal A}_{h_{WWff}}^{1,W}$ $\displaystyle=$
$\displaystyle\frac{{{\cal M}_{hXX}}\;{{\cal
M}_{hWff}}}{s-M_{h}^{2}+i\Gamma_{h}M_{h}}\;\left(-\delta
Z_{h}-\frac{\Pi_{hh}(s)-\Pi_{hh}(M_{h}^{2})}{s-M_{h}^{2}}\right)$ (24)
$\displaystyle+$ $\displaystyle\Bigg{(}\frac{\delta{{\cal M}_{hXX}}{{\cal
M}_{hWff}}+{{\cal M}_{hXX}}\delta{{\cal
M}_{hWff}}}{s-M_{h}^{2}+i\Gamma_{h}M_{h}}\Bigg{)}$ $\displaystyle+$
$\displaystyle\frac{s-M_{h}^{2}}{s-M_{h}^{2}+i\Gamma_{h}M_{h}}\Box_{XXWWf}^{(5)}.$

$\delta{{\cal M}_{hWff}}$ pertains to the correction on the final state of the
Higgs decay. It includes the vertex correction to $hWW^{\star}$, 2-point
function for the $W$ propagator and the $4$ point function.
$\Box_{XXWWf}^{(5)}$ relates to corrections that connect initial/final states
which are 5-point function corrections. These contributions are not
factorisable, with our implementation of the width, these contributions are
effectively put to zero at exactly the Higgs resonance.

Figure 8:  As in Figure 5 but for the $XX\to Wl\nu_{l}$ channel.

First of all, we confirm that the relative correction to the full cross-
section over the whole range of $v$, is within $0.04\%$, independent of the
fermion flavour. The weight of each channel equals the branching ratio of the
$W$ into the corresponding fermion. We therefore only show our results for the
lepton channel. These results are shown in Figure 8. Were we to look at the
total correction, we would see that the corrected cross-sections follow the
tree-level cross-section, the very peaked resonance structure does not show
the details of the corrections. The details of the corrections are most
clearly revealed in the normalised (with respect to the tree-level result)
corrections. Again, as in the case of $b\bar{b}$, apart from the location of
the Higgs peak in terms of the relative velocity, $v\sim 0.82$ for P57, and
$v\sim 0.66$ for P59, the characteristics for the 2 benchmark points are quite
similar. At the Higgs peak the correction is very small. In fact, at this
point, the improved tree-level agrees perfectly with the full one-loop
correction, as expected. As explained previously, by construction, at
$s=M_{h}^{2}$, the fact that $\Gamma(h\to XX)$ is used as an OS input, the
$\lambda_{2}$ dependence of the correction is absorbed in the renormalisation
and therefore the $\lambda_{2}$ dependence does not show up at the peak.
Moreover, at the peak, our implementation of the width makes any non-resonant
contribution vanish. Outside the peak, the $\lambda_{2}$ dependence is
important and is magnified just around the peak in the interference region.
For $\lambda_{2}=0.01$, the variation with $v$ is modest. This variation
amplifies as $\lambda_{2}$ is increased. We also note that the sign of the
interference changes with increasing positive $\lambda_{2}$ for both benchmark
points. Observe also that even far away from the resonance, $v\sim 0$, the
$\lambda_{2}$ dependence is quite different from the one we found in the
$b\bar{b}$ channel. For $b\bar{b}$, $\lambda_{2}=0.01$ amounts to a reduction
of the tree-level cross-section by about $10\%$ while here the same value of
$\lambda_{2}$ gives an enhancement of about $20\%$ for small $v$. Also, the
corrections for P59 are slightly smaller than for P57, again contrary to the
$b\bar{b}$ channel. Another important difference from the $b\bar{b}$ channel
is that the effective $\alpha(M_{Z}^{2})$ does a good job only very close to
the peak. In the region between the dip (anti-resonance) and the peak, where
the interference between the Higgs channel and the gauge continuum
contribution is important, the $\alpha(M_{Z}^{2})$ is not a good
approximation. It affects the Higgs contribution and the $WW^{\star}$
contribution differently. Moreover, for small $v$ the "$\alpha(M_{Z}^{2})$"
cross-section gives corrections which are about a factor 2 larger than the
full loop correction for $\lambda_{2}=0.01$. The discrepancy worsens as
$\lambda_{2}$ increases, confirming that $\lambda_{2}\sim 0$ is what best
approaches the "$\alpha(M_{Z}^{2})$" approximation. It is clear that the
approximations do not reproduce the full one-loop correction (which we can
equate with the result of $\lambda_{2}=0.01$) nor do they catch the
$\lambda_{2}$ dependence. Will these important features have an important
impact on the relic density, considering that the cross-sections are dominated
by the peak contribution which in our implementation, these features are by
construction set to zero? They may also be diluted because of the weight of
this channel (compared to the $b\bar{b}$ channel) to the total effective
annihilation cross-section and hence to the relic density.

##  7 Application to the relic density

One of the steps in the calculation of the relic density, given the velocity
dependent annihilation cross-sections, requires integrating the convolution of
those cross-sections with the velocity distribution. With very narrow
resonance structures, the integrator must carefully pick up the peaked region.
For the case of a resonance with a very nearby anti-resonance like the case of
the $Wf\bar{f}^{\prime}$ final state, the integrator must pick up the entirety
of this rich structure. This is the reason we generate a very large set of
points around the resonance, carefully including the peak and the dip as well
as taking into account the contributions away from the resonance. This is
important because our renormalisation condition together with a consistent
implementation of the Higgs width show that the peak region is practically not
corrected while the (relative) corrections get larger as soon as we move away
from the exact resonance point. The integration over the velocity, for some
values of $x_{F}=M_{X}/T_{F}$ ($T_{F}$ is the freeze-out temperature) is
performed through two dedicated integrators, one calling Python libraries and
the other with Mathematica. Moreover, analytical formulae when the test
(cross-section) function is of the form
$1/((s-M_{h}^{2})^{2}+(\Gamma_{h}M_{h})^{2}),(s-M_{h}^{2})/((s-M_{h}^{2})^{2}+(\Gamma_{h}M_{h})^{2}),1$
555These functions correspond, respectively, to the resonance, the
interference term and the continuum. with the convolution over the velocity
distribution $v^{2}e^{(-x_{F}v^{2})}$ exist when the integration is carried
over the range $v={0,\infty}$. The analytical formulae are then compared with
the numerical evaluation of the dedicated integrators. Only then do we first
feed micrOMEGAs these test functions to conduct sanity checks and then feed in
the exact one-loop cross-sections resulting from our code, SloopS. Of course
micrOMEGAs solves also the Boltzmann equation and returns the effective
$x_{F}$ which we use for the integrators to conduct our checks.

| Tree |  $\lambda_{2}$ = 0.01 |  $\lambda_{2}$ = 1 |  $\lambda_{2}$ = 2 | Improved | $\alpha(M_{Z}^{2})$
---|---|---|---|---|---|---
P57 | 0.113 | 0.110 (-2.42%) | 0.111 (-2.21%) | 0.111 (-2.00%) | 0.110 (-2.36%)) | 0.118 (4.43%)
P59 | 0.108 | 0.105 (-2.73%) | 0.105 (-2.37%) | 0.106 (-2.00%) | 0.105 (-2.62%) | 0.113 (4.63%)
Table 2: Correction to the relic density for the Higgs resonance Points P57
and P59.

The corrections to the relic density for P57 and P59 are shown in Table 2. The
corrections are very small, of the order $-(2-3)\%$. These small corrections
are a consequence of the fact that the cross-sections are overwhelmingly
dominated by the cross-section at the resonance, $s=M_{h}^{2}$, for which our
implementation of the loop corrections suppresses all non-Higgs mediated
contributions and gives a correction to the $b\bar{b}$ cross-section of order
$1\%$ (only slightly more for $WW$). The corrections away from the peak,
though large in relative terms, are small in absolute terms. Their effect is
detectable and in principle (at the per-mille level) the $\lambda_{2}$
dependence is traceable. Note that the use of $\alpha(M_{Z}^{2})$ gives larger
(but small positive) corrections. This is due to the fact that the $b\bar{b}$
channel is independent of the choice of $\alpha$ as explained in the previous
section whereas while its effect is large in $WW$, its effects are much
reduced because of the much smaller weight of this channel in the total
effective cross-section.

##  8 Conclusions

We presented a very thorough procedure for the implementation of the width at
tree-level and at one-loop that avoids the problem of double counting and
eventually of the breakdown of unitarity especially in a situation like the
$WW^{\star}$ final state where interference between the resonant contribution
and other amplitudes is important. It is crucial to stress that the fact that
the one-loop corrections are very small at the resonance is due to, on the one
hand, the implementation of the fudge factor to maintain gauge invariance (for
$WW^{\star}$ this maintains the correct $\lambda_{L}$ dependence) but sets to
zero any non resonant amplitude, and on the other hand, the use of the OS
renormalisation scheme which entails that the correction to the vertex $hXX$
is zero at the resonance. Away from the resonance, these corrections, in
relative terms, are not small. We hope that our analysis of the resonance and
the procedure we detailed proves helpful for many applications, not
necessarily related to dark matter. It rests that for our applications to the
Higgs resonance, an extremely narrow resonance, the very small correction at
the resonance within our scheme and procedure means a very small correction of
the velocity integrated cross-sections and hence the relic density. Because
the integration over the resonance needs to pick up all the very subtle
changes in the cross-sections, we performed sanity checks that ensure that the
numerical integration is performed correctly.

###### Acknowledgements.

We thank Alexander Pukhov for several helpful discussions. HS is supported by
the National Natural Science Foundation of China (Grant No.12075043,
No.11675033). He 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. SB also
acknowledges the support received from LAPTh, during his visit, when this work
was started. NC is financially supported by IISc (Indian Institute of Science,
Bangalore, India) through the C.V. Raman postdoctoral fellowship. NC also
thanks the support received from DST, India, under grant number IFA19-PH237
(INSPIRE Faculty Award).

## References

  * (1) N. G. Deshpande and E. Ma, _Pattern of Symmetry Breaking with Two Higgs Doublets_ , _Phys. Rev._ D18 (1978) 2574.
  * (2) R. Barbieri, L. J. Hall and V. S. Rychkov, _Improved naturalness with a heavy Higgs: An Alternative road to LHC physics_ , _Phys. Rev._ D74 (2006) 015007, [hep-ph/0603188].
  * (3) T. Hambye and M. H. G. Tytgat, _Electroweak symmetry breaking induced by dark matter_ , _Phys. Lett._ B659 (2008) 651–655, [0707.0633].
  * (4) L. Lopez Honorez, E. Nezri, J. F. Oliver and M. H. G. Tytgat, _The Inert Doublet Model: An Archetype for Dark Matter_ , _JCAP_ 0702 (2007) 028, [hep-ph/0612275].
  * (5) Q.-H. Cao, E. Ma and G. Rajasekaran, _Observing the Dark Scalar Doublet and its Impact on the Standard-Model Higgs Boson at Colliders_ , _Phys. Rev._ D76 (2007) 095011, [0708.2939].
  * (6) M. Gustafsson, E. Lundstrom, L. Bergstrom and J. Edsjo, _Significant Gamma Lines from Inert Higgs Dark Matter_ , _Phys. Rev. Lett._ 99 (2007) 041301, [astro-ph/0703512].
  * (7) P. Agrawal, E. M. Dolle and C. A. Krenke, _Signals of Inert Doublet Dark Matter in Neutrino Telescopes_ , _Phys. Rev._ D79 (2009) 015015, [0811.1798].
  * (8) T. Hambye, F. S. Ling, L. Lopez Honorez and J. Rocher, _Scalar Multiplet Dark Matter_ , _JHEP_ 07 (2009) 090, [0903.4010].
  * (9) E. Lundstrom, M. Gustafsson and J. Edsjo, _The Inert Doublet Model and LEP II Limits_ , _Phys. Rev._ D79 (2009) 035013, [0810.3924].
  * (10) S. Andreas, M. H. G. Tytgat and Q. Swillens, _Neutrinos from Inert Doublet Dark Matter_ , _JCAP_ 0904 (2009) 004, [0901.1750].
  * (11) C. Arina, F.-S. Ling and M. H. G. Tytgat, _IDM and iDM or The Inert Doublet Model and Inelastic Dark Matter_ , _JCAP_ 0910 (2009) 018, [0907.0430].
  * (12) E. Dolle, X. Miao, S. Su and B. Thomas, _Dilepton Signals in the Inert Doublet Model_ , _Phys. Rev._ D81 (2010) 035003, [0909.3094].
  * (13) E. Nezri, M. H. G. Tytgat and G. Vertongen, _e+ and anti-p from inert doublet model dark matter_ , _JCAP_ 0904 (2009) 014, [0901.2556].
  * (14) X. Miao, S. Su and B. Thomas, _Trilepton Signals in the Inert Doublet Model_ , _Phys. Rev._ D82 (2010) 035009, [1005.0090].
  * (15) J.-O. Gong, H. M. Lee and S. K. Kang, _Inflation and dark matter in two Higgs doublet models_ , _JHEP_ 04 (2012) 128, [1202.0288].
  * (16) M. Gustafsson, S. Rydbeck, L. Lopez-Honorez and E. Lundstrom, _Status of the Inert Doublet Model and the Role of multileptons at the LHC_ , _Phys. Rev._ D86 (2012) 075019, [1206.6316].
  * (17) B. Swiezewska and M. Krawczyk, _Diphoton rate in the inert doublet model with a 125 GeV Higgs boson_ , _Phys. Rev._ D88 (2013) 035019, [1212.4100].
  * (18) A. Arhrib, R. Benbrik and N. Gaur, _$H\to\gamma\gamma$ in Inert Higgs Doublet Model_, _Phys. Rev._ D85 (2012) 095021, [1201.2644].
  * (19) L. Wang and X.-F. Han, _LHC diphoton Higgs signal and top quark forward-backward asymmetry in quasi-inert Higgs doublet model_ , _JHEP_ 05 (2012) 088, [1203.4477].
  * (20) A. Goudelis, B. Herrmann and O. Stal, _Dark matter in the Inert Doublet Model after the discovery of a Higgs-like boson at the LHC_ , _JHEP_ 09 (2013) 106, [1303.3010].
  * (21) A. Arhrib, Y.-L. S. Tsai, Q. Yuan and T.-C. Yuan, _An Updated Analysis of Inert Higgs Doublet Model in light of the Recent Results from LUX, PLANCK, AMS-02 and LHC_ , _JCAP_ 1406 (2014) 030, [1310.0358].
  * (22) M. Krawczyk, D. Sokolowska, P. Swaczyna and B. Swiezewska, _Constraining Inert Dark Matter by $R_{\gamma\gamma}$ and WMAP data_, _JHEP_ 09 (2013) 055, [1305.6266].
  * (23) P. Osland, A. Pukhov, G. M. Pruna and M. Purmohammadi, _Phenomenology of charged scalars in the CP-Violating Inert-Doublet Model_ , _JHEP_ 04 (2013) 040, [1302.3713].
  * (24) T. Abe and R. Sato, _Quantum corrections to the spin-independent cross section of the inert doublet dark matter_ , _JHEP_ 03 (2015) 109, [1501.04161].
  * (25) N. Blinov, J. Kozaczuk, D. E. Morrissey and A. de la Puente, _Compressing the Inert Doublet Model_ , _Phys. Rev._ D93 (2016) 035020, [1510.08069].
  * (26) M. A. Díaz, B. Koch and S. Urrutia-Quiroga, _Constraints to Dark Matter from Inert Higgs Doublet Model_ , _Adv. High Energy Phys._ 2016 (2016) 8278375, [1511.04429].
  * (27) A. Ilnicka, M. Krawczyk and T. Robens, _Inert Doublet Model in light of LHC Run I and astrophysical data_ , _Phys. Rev._ D93 (2016) 055026, [1508.01671].
  * (28) G. Belanger, B. Dumont, A. Goudelis, B. Herrmann, S. Kraml and D. Sengupta, _Dilepton constraints in the Inert Doublet Model from Run1 of the LHC_ , _Phys. Rev._ D91 (2015) 115011, [1503.07367].
  * (29) A. Carmona and M. Chala, _Composite Dark Sectors_ , _JHEP_ 06 (2015) 105, [1504.00332].
  * (30) S. Kanemura, M. Kikuchi and K. Sakurai, _Testing the dark matter scenario in the inert doublet model by future precision measurements of the Higgs boson couplings_ , _Phys. Rev._ D94 (2016) 115011, [1605.08520].
  * (31) F. S. Queiroz and C. E. Yaguna, _The CTA aims at the Inert Doublet Model_ , _JCAP_ 1602 (2016) 038, [1511.05967].
  * (32) A. Belyaev, G. Cacciapaglia, I. P. Ivanov, F. Rojas-Abatte and M. Thomas, _Anatomy of the Inert Two Higgs Doublet Model in the light of the LHC and non-LHC Dark Matter Searches_ , _Phys. Rev._ D97 (2018) 035011, [1612.00511].
  * (33) G. Arcadi, A. Djouadi and M. Raidal, _Dark Matter through the Higgs portal_ , _Phys. Rept._ 842 (2020) 1–180, [1903.03616].
  * (34) B. Eiteneuer, A. Goudelis and J. Heisig, _The inert doublet model in the light of Fermi-LAT gamma-ray data: a global fit analysis_ , _Eur. Phys. J._ C77 (2017) 624, [1705.01458].
  * (35) A. Ilnicka, T. Robens and T. Stefaniak, _Constraining Extended Scalar Sectors at the LHC and beyond_ , _Mod. Phys. Lett._ A33 (2018) 1830007, [1803.03594].
  * (36) J. Kalinowski, W. Kotlarski, T. Robens, D. Sokolowska and A. F. Zarnecki, _Benchmarking the Inert Doublet Model for $e^{+}e^{-}$ colliders_, _JHEP_ 12 (2018) 081, [1809.07712].
  * (37) P. M. Ferreira and D. R. T. Jones, _Bounds on scalar masses in two Higgs doublet models_ , _JHEP_ 08 (2009) 069, [0903.2856].
  * (38) P. M. Ferreira and B. Swiezewska, _One-loop contributions to neutral minima in the inert doublet model_ , _JHEP_ 04 (2016) 099, [1511.02879].
  * (39) S. Kanemura, S. Kiyoura, Y. Okada, E. Senaha and C. P. Yuan, _New physics effect on the Higgs selfcoupling_ , _Phys. Lett._ B558 (2003) 157–164, [hep-ph/0211308].
  * (40) E. Senaha, _Radiative Corrections to Triple Higgs Coupling and Electroweak Phase Transition: Beyond One-loop Analysis_ , _Phys. Rev. D_ 100 (2019) 055034, [1811.00336].
  * (41) J. Braathen and S. Kanemura, _On two-loop corrections to the Higgs trilinear coupling in models with extended scalar sectors_ , _Phys. Lett. B_ 796 (2019) 38–46, [1903.05417].
  * (42) A. Arhrib, R. Benbrik, J. El Falaki and A. Jueid, _Radiative corrections to the Triple Higgs Coupling in the Inert Higgs Doublet Model_ , _JHEP_ 12 (2015) 007, [1507.03630].
  * (43) C. Garcia-Cely, M. Gustafsson and A. Ibarra, _Probing the Inert Doublet Dark Matter Model with Cherenkov Telescopes_ , _JCAP_ 1602 (2016) 043, [1512.02801].
  * (44) S. Banerjee and N. Chakrabarty, _A revisit to scalar dark matter with radiative corrections_ , _JHEP_ 05 (2019) 150, [1612.01973].
  * (45) R. Basu, S. Banerjee, M. Pandey and D. Majumdar, _Lower bounds on dark matter annihilation cross-sections by studying the fluctuations of 21-cm line with dark matter candidate in inert doublet model (IDM) with the combined effects of dark matter scattering and annihilation_ , 2010.11007.
  * (46) H. Abouabid, A. Arhrib, R. Benbrik, J. E. Falaki, B. Gong, W. Xie et al., _One-loop radiative corrections to $e^{+}e^{-}\to Zh^{0}/H^{0}A^{0}$ in the Inert Higgs Doublet Model_, 2009.03250.
  * (47) J. Kalinowski, T. Robens, D. Sokolowska and A. F. Zarnecki, _IDM benchmarks for the LHC and future colliders_ , 2012.14818.
  * (48) S. Banerjee, F. Boudjema, N. Chakrabarty and H. Sun, _Relic density of dark matter in the inert doublet model beyond leading order: Renormalisation and Constraints for the low mass region_ , .
  * (49) G. Belanger, F. Boudjema, A. Pukhov and A. Semenov, _MicrOMEGAs: A Program for calculating the relic density in the MSSM_ , _Comput. Phys. Commun._ 149 (2002) 103–120, [hep-ph/0112278].
  * (50) G. Belanger, F. Boudjema, A. Pukhov and A. Semenov, _micrOMEGAs: Version 1.3_ , _Comput. Phys. Commun._ 174 (2006) 577–604, [hep-ph/0405253].
  * (51) G. Belanger, F. Boudjema, A. Pukhov and A. Semenov, _MicrOMEGAs 2.0: A Program to calculate the relic density of dark matter in a generic model_ , _Comput. Phys. Commun._ 176 (2007) 367–382, [hep-ph/0607059].
  * (52) G. Belanger, F. Boudjema, A. Pukhov and A. Semenov, _micrOMEGAs $\\_$3: A program for calculating dark matter observables_, _Comput. Phys. Commun._ 185 (2014) 960–985, [1305.0237].
  * (53) G. Belanger, F. Boudjema, A. Goudelis, A. Pukhov and B. Zaldivar, _micrOMEGAs5.0 : Freeze-in_ , _Comput. Phys. Commun._ 231 (2018) 173–186, [1801.03509].
  * (54) LHC Higgs Cross Section Working Group collaboration, D. de Florian et al., _Handbook of LHC Higgs Cross Sections: 4. Deciphering the Nature of the Higgs Sector. arXiv: 1610.07922_ , 1610.07922.
  * (55) P. D. Group, _Review of Particle Physics_ , _Progress of Theoretical and Experimental Physics_ 2020 (08, 2020) , [See in particular, pages 13 and 14 online http://pdg.lbl.gov/2019/reviews/rpp2019-rev-higgs-boson.pdf].
  * (56) N. Baro, F. Boudjema and A. Semenov, _Full one-loop corrections to the relic density in the MSSM: A Few examples_ , _Phys. Lett._ B660 (2008) 550–560, [0710.1821].
  * (57) N. Baro, F. Boudjema and A. Semenov, _Automatised full one-loop renormalisation of the MSSM. I. The Higgs sector, the issue of tan(beta) and gauge invariance_ , _Phys. Rev._ D78 (2008) 115003, [0807.4668].
  * (58) N. Baro, F. Boudjema, G. Chalons and S. Hao, _Relic density at one-loop with gauge boson pair production_ , _Phys. Rev._ D81 (2010) 015005, [0910.3293].
  * (59) G. Belanger, F. Boudjema, J. Fujimoto, T. Ishikawa, T. Kaneko, K. Kato et al., _Automatic calculations in high energy physics and Grace at one-loop_ , _Phys. Rept._ 430 (2006) 117–209, [hep-ph/0308080].
  * (60) S. Banerjee, F. Boudjema, N. Chakrabarty and H. Sun, _Relic density of dark matter in the inert doublet model beyond leading order: Annihilation in 3-body final state for the low mass region_ , .
  * (61) S. Banerjee, F. Boudjema, N. Chakrabarty and H. Sun, _Relic density of dark matter in the inert doublet model beyond leading order: Co-annihilation in the low mass region_ , .
  * (62) D. Bardin, S. Riemann and T. Riemann, _Electroweak One Loop Corrections to the Decay of the Charged Vector Boson_ , _Z. Phys. C_ 32 (1986) 121–125.