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: 3. Annihilation in 3-body final state

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 perform the first one-loop electroweak corrections for $2\to 3$ processes
for dark matter annihilation. These are the dominant processes that enter the
computation of the relic density for the low mass region of the inert doublet
model (IDM) when annihilations to two on-shell vector bosons are closed. The
impact of the one-loop corrections are important as they involve, through
rescattering effects, not only a dependence on the parameter controlling the
dark sector, not present if a calculation at tree-level is conducted, but also
on the renormalisation scale. These combined effects should be taken into
account in analyses based on tree-level cross-sections of the relic density
calculations, as a theoretical uncertainty which we find to be much larger
than the cursory $\pm 10\%$ uncertainty that is routinely assumed,
independently of the model parameters.

††preprint: LAPTH-003/21, CERN-TH-2021-003

##  1 Introduction

The fermions of the the standard model (SM) do not directly couple to the
scalars in 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 . The
annihilation of the dark matter (DM) candidate in the IDM, the lightest
neutral scalar $X$, occurs most naturally through annihilation into the SM
vector bosons. These processes are triggered by the gauge coupling and also by
the interactions stemming from the scalar sector of the model. The latter can
be parametrised by the coupling of the SM Higgs to the pair of DM,
$\lambda_{L}$, once the masses of all the scalars of the IDM are derived
OurPaper1_2020 . For these annihilations into a pair of $WW$ and $ZZ$ to be
possible, the mass of the DM, $M_{X}$, must be larger than $M_{W}$, the mass
of the $W$-boson. Even in this case, these annihilations are so efficient, see
OurPaper1_2020 , that the obtained relic density is too small, unless one
considers very high DM masses Banerjee:2019luv . For the low mass,
$M_{X}<M_{W}$, DM region of the IDM OurPaper1_2020 , the annihilations are
into $WW^{\star}$ and $ZZ^{\star}$ where one of the vector bosons is off-shell
and is materialised by a fermion pair. The cross-sections are then smaller,
bringing the relic density in accord with present measurement of the relic
density. For $WW^{\star}$ and $ZZ^{\star}$, one is then faced with the
calculation of a $2\to 3$ process which has never been attempted at one-loop
for the calculation of the relic density. Unlike the newly discovered co-
annihilation region and the Higgs resonance region, this continuum region does
not require much adjustment of the parameters in order to achieve a good value
of the relic density within the freeze-out mechanism. This explains why a scan
on the parameters of the IDM returns quite a few points with this topology for
the relic density. Following the in-depth preliminary study, $XX\to
W^{+}W^{-},ZZ$, of all the possible benchmarks in this region that pass all
the experimental (and theoretical) constraints, we retain, in the present
analysis of these channels, only those benchmarks which satisfy the one-loop
perturbativity requirement OurPaper1_2020 . This requirement was enunciated in
the preparatory study $XX\to W^{+}W^{-},ZZ$ OurPaper1_2020 . In a nutshell,
only models that return small enough $\tilde{\beta}_{\lambda_{L}}$ (the
$\beta$-function parameter that controls the running of the coupling
$\lambda_{L}$) are perturbative OurPaper1_2020 . On that basis, we keep three
benchmark points defined in OurPaper1_2020 : (points A, F and G) to illustrate
our computations of the one-loop electroweak corrections to $XX\to Zf\bar{f}$
and $XX\to Wf\bar{f}^{\prime}$. Let us therefore recall the characteristics of
these three benchmark points in Table 1.

| A | F | G
---|---|---|---
$M_{X}$ | 70 | 72 | 72
$\lambda_{L}\times 10^{3}$ | 5.0 | 3.8 | 0.1
$M_{A}$,$M_{H^{\pm}}$ | 170,200 | 138,138 | 158,158
$(\lambda_{3},\lambda_{4},\lambda_{5})$ | (1.127, -0.738,-0.384) | (0.447,-0.222,-0.222) | (0.632,-0.316,-0.316)
$\tilde{\beta}_{\lambda_{L}}$ | $4.343+6.082\lambda_{2}$ | $1.402+2.704\lambda_{2}$ | $2.142+3.795\lambda_{2}$
$\Omega h^{2}$
$\alpha(0)$ | 0.156 | 0.119 | 0.121
$\Omega_{WW^{\star}}(\%)$ | 90 | 88 | 88
$\Omega_{ZZ^{\star}}(\%)$ | 10 | 12 | 12
Table 1: Characteristics of the benchmark points A,F and G. All masses are in
GeV. The tree-level (calculated with $(\alpha(0)$) relic density and the
weight in percent of each channel contributing to the relic density, are
given. We also list the values of the underlying parameters $\lambda_{3,4,5}$
and $\tilde{\beta}_{\lambda_{L}}$ OurPaper1_2020 .

The paper is organised as follows. In the next section, we review the $2\to 3$
cross-sections and seek a factorisation where the flavour dependence is
carried by the vector bosons’ partial widths. Section 3 is a general
presentation of the one-loop calculation. Since the bulk of the corrections
are contained in the purely virtual correction in the neutral channel
$Z\nu\bar{\nu}$, section 4 is dedicated to this channel before studying in
section 5 all the other channels where final state radiation (tree-level $2\to
4$ processes are needed) is considered. Section 6 summarises all the one-loop
results on the cross-sections leading the way to the impact of the corrections
and the scale uncertainty on the relic density which we present in section 7.
Our conclusions are presented in section 8.

##  2 Tree-level considerations

Figure 1: A selection of tree-level Feynman diagrams for $XX\to Zf\bar{f}$ in
the Feynman gauge. The displayed diagrams can be built up from $XX\to
ZZ^{\star}\to Zf\bar{f}$ but note also the $Z$-"bremsstrahlung" contribution
triggered from $XX\to ff^{\star}\to f\bar{f}Z$ (last diagram in the first
row).

Like for the $2\to 2$ processes, $XX\to WW$ and $XX\to ZZ$, beside the masses
of the dark sector particles, the cross-sections depend not only on the gauge
coupling but also on $\lambda_{L}$ (because of the SM Higgs exchange and the
quartic $XXVV$ couplings, $V=W,Z$). The massive fermions’ Yukawa couplings may
also play a role, but we will see that they are negligible. As a subset,
contributions to the full $Zf\bar{f}$ is displayed in Figure 1.
It is completely unwise to try to compute such cross-sections, even at tree-
level, by splitting them into a $2\to 2$ process followed by the “decay" of
one of the vector bosons into fermions even if the $Z/W$ current is conserved
in the limit $m_{f}\to 0$. For starters, $XX\to VV^{\star}$ is ill-defined
since it does not correspond to an element of the $S$-matrix. Therefore, a
complete $2\to 3,XX\to Zf\bar{f}$ and $XX\to Wf\bar{f}^{\prime}$, calculation
is in order. Nonetheless, because the mass of the final fermions is very small
compared to the energies involved and the fact that the fermions do not couple
to the dark sector, we expect that the complete calculation of the cross-
sections are arranged such that

$\displaystyle\frac{\sigma(XX\to Zf\bar{f})}{\sigma(XX\to Z\nu\bar{\nu})}$
$\displaystyle\simeq$ $\displaystyle\frac{\Gamma_{Z\to
f\bar{f}}}{\Gamma_{Z\to\nu\bar{\nu}}}=\frac{{\rm Br}_{Z\to f\bar{f}}}{{\rm
Br}_{Z\to\nu\bar{\nu}}},$ (1) $\displaystyle\frac{\sigma(XX\to
Wf\bar{f}^{\prime})}{\sigma(XX\to W\nu_{e}\bar{e})}$ $\displaystyle\simeq$
$\displaystyle\frac{\Gamma_{W\to
f\bar{f}^{\prime}}}{\Gamma_{W\to\nu_{e}\bar{e}}}\simeq
N_{c}^{f},\quad(N_{c}^{f}=1,3\;\;\textrm{is the colour factor}).$ (2)

Figure 2: Benchmark Point G. The tree-level cross-section times the relative
velocity for $XX\to Z\nu\bar{\nu}$ (upper-left panel), $R^{XX}_{Zf\bar{f}}-1$
for $f=\tau,b$ (upper-right panel). The lower panel displays
$R^{XX}_{Z\tau\bar{\tau}}\simeq 1$ and the ratio of the tree-level cross-
sections $R^{XX}_{W/Z}=\sigma(XX\to W\tau\bar{\nu}_{\tau})/\sigma(XX\to
Z\nu_{\tau}\bar{\nu}_{\tau})$ as a function of the relative velocity, $v$.

The ratios of the partial physical widths act as a normalisation of the cross-
sections with respect to the neutrino channels. If we introduce

$\displaystyle R^{XX}_{Zf\bar{f}}$ $\displaystyle=$
$\displaystyle\frac{\sigma(XX\to Zf\bar{f})}{\sigma(XX\to
Z\nu\bar{\nu})}\;\frac{\Gamma_{Z\to\nu\bar{\nu}}}{\Gamma_{Z\to f\bar{f}}}$
$\displaystyle R^{XX}_{Wf\bar{f}^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{\sigma(XX\to Wf\bar{f}^{\prime})}{\sigma(XX\to
W\nu_{e}\bar{e})}\;\frac{\Gamma_{W\to\nu_{e}\bar{e}}}{\Gamma_{W\to
f\bar{f}^{\prime}}},$ $\displaystyle R^{XX}_{W/Z}$ $\displaystyle=$
$\displaystyle\frac{\sigma(XX\to W\nu_{e}\bar{e})}{\sigma(XX\to
Z\nu\bar{\nu})},$ (3)

it follows from the arguments that lead to Equations 1\- 2 that
$R^{XX}_{Zf\bar{f}}\sim R^{XX}_{Wf\bar{f}^{\prime}}\sim 1$. We verify these
approximations and behaviour by carrying a full calculation with fermion mass
effect for the different channels. For Point G, the results are displayed in
Figure 2.

First of all, the velocity dependence of the tree-level $XX\to Zf\bar{f}$ and
$XX\to Wf\bar{f}^{\prime}$ is strong. This is shown in Figure 2 for the
$Z\nu\bar{\nu}$ and consequently for $W\bar{\nu}_{\tau}\tau$. The latter grows
faster past $v\sim 0.5$. This is understandable since as $v$ increases one
gets closer to the opening of the threshold for on-shell pair production of
vector bosons, the $WW$ threshold occurring first. The important observation
though is that below $v\sim 0.5$ (the most important range for the relic
density calculation), when the threshold effect is small, the ratio between
these two cross-sections is almost constant. This is due to the global $SU(2)$
symmetry with the important consequence that the $\lambda_{L}$ dependence is
the same in the neutral and charged channels. This is the same property that
is explicitly confirmed in $XX\to WW$ and $XX\to ZZ$ past the $ZZ$ threshold
in our study in Ref. OurPaper1_2020 . The same $\lambda_{L}$ dependence will
mean that both channels will exhibit the same scale uncertainty in the one-
loop corrected cross-sections.
We confirm this feature at all $v$, and independently of the flavour
$R^{XX}_{Wf\bar{f}^{\prime}}=1$, at better than the per-mille level, to the
point where we can not display this difference in Figure 2.
In the neutral channel, $R^{XX}_{Zf\bar{f}}\simeq 1$ is also nicely confirmed.
Departure from unity are largest for the $Zb\bar{b}$ final state where the
maximal value is below $3\%$ across all values of $v$. The effect is smaller
still for the $Z\tau\tau$ channel as seen in Figure 2. For all other channels,
the mass effects are unnoticeable and are therefore not shown in the figure.
The effect of the fermion masses/Yukawa couplings through Higgs exchange,
$XX\xrightarrow{h}ff^{\star}\to f\bar{f}Z$, is therefore small.
Taken together, these observations lead us to conclude that the cross-section
into the neutrinos can be taken as a representative of the channels, neutral
and charged. We also look at $R^{XX}_{W/Z}$. For the moment we keep these
observations in mind before attempting the one-loop analyses. This first
exploration confirms that the neutrino channels carry the bulk of the $v$
dependence and are the prime channels against which we will measure all other
channels. We also confirm that the same conclusions, with the same level of
accuracy, apply for the other benchmark points (A and F).

##  3 $XX\to Wf\bar{f}$ and $XX\to Zf\bar{f}$ at one-loop: General issues

Figure 3: A small selection of Feynman diagrams for $XX\to Zf\bar{f}$ at one-
loop. We only show a very small subset of pentagons, boxes and triangles but
not self-energy corrections and counterterms. $f^{\prime}$ stands for the
$SU(2)$ partner of $f$ ($\nu_{l}$ for $f=l$). $F$ stands for the sum of all SM
fermions. Although some diagrams may not look like boxes, they fall under the
box category because of the 4-particle vertices they involve. The same applies
to triangles. We see rescattering effects within the dark sector, $XX\to
AA,H^{+}H^{-},XX$ that explicitly involve the $\lambda_{2}$ parameter that
does not show up at tree-level. Note also that because of the off-shell $Z$,
charged fermions $ff$ pairs from $\gamma^{\star}$ must also be taken into
account.

At one-loop, a large number of topologies appears for these $2\to 3$
processes. For $XX\to Zf\bar{f}$, a set of the contributing one-loop diagrams
is shown in Figure 3. A subset of diagrams at one-loop for $XX\to Wf\bar{f}$
is found in our analysis of the co-annihilation region where this cross-
section was a subdominant contribution to the relic density OurPaper2_2020 .
Technically, it is not (only) the sheer number of diagrams that adds to the
complexity of the calculation but also the fact that the calculation,
especially the reduction of the $n-$point integrals, is very much computer-
time consuming. This is particularly the case for the $5-$point functions, the
pentagons, that need to be evaluated, for instance, in configurations of phase
space where $v\sim 0$, dangerously close to the appearance of very small Gram
determinants, see Boudjema_2005 . For the charged final state, tree-level
$XX\to Zf\bar{f}\gamma$ and $XX\to Wf\bar{f}\gamma$ must be considered
together with the virtual one-loop corrections. We find that the phase space
slicing method, as applied in Banerjee:2019luv , converges relatively quickly.
Gauge parameter independence OurPaper1_2020  is carried at some random point
in phase space. This is a check not only on the model implementation
(including counterterms) but also the tensorial reduction of the loop
integrals.

An on-shell scheme for $\lambda_{L}$ based on $h\to XX$ is not possible for
this mass range since this decay is closed. The radiative correction will
therefore be sensitive to the renormalisation scale, $\mu$, associated with
the $\overline{\text{MS}}$ scheme associated with a definition of
$\lambda_{L}$, see OurPaper1_2020 ,

$\displaystyle\delta^{\overline{{\rm
MS}}}\lambda_{L}=\frac{1}{32\pi^{2}}\tilde{\beta}_{\lambda_{L}}C_{{\rm
UV}},\quad C_{{\rm UV}}=-\frac{2}{\varepsilon}-1+\gamma_{E}-\ln(4\pi),$ (4)

where $\varepsilon=4-d$ with $d$ being the number of dimensions in dimensional
regularisation and $\gamma_{E}$ being the Euler-Mascheroni constant. With
$\mu_{\text{dim}}$, the scale introduced by dimensional regularisation, the
scale, $Q^{2}$, dependence of $\lambda_{L}$ is

$\displaystyle
32\pi^{2}\frac{\partial\lambda_{L}}{\partial\ln(Q^{2})}=-32\pi^{2}\frac{\partial\lambda_{L}}{\partial\ln(\mu_{\text{dim}}^{2})}=\tilde{\beta}_{\lambda_{L}}.$
(5)

We will, for the three benchmark points in this mass range, study the scale
dependence. Beside the scale dependence, we also investigate the interesting
$\lambda_{2}$ dependence which, as we showed in $XX\to WW,ZZ$ above threshold,
is not totally contained in $\beta_{\lambda_{L}}$, see OurPaper1_2020 . The
scale dependence, for a fixed $v$, is easily extracted from the $\lambda_{L}$
dependence of the tree-level cross-section combined with the expression of
$\beta_{\lambda_{L}}$ which is known analytically. Barring very small mass
effects, we confirm that at tree-level $R^{XX}_{Zf\bar{f}}\simeq
1,R^{XX}_{Wf\bar{f}^{\prime}}=1$, while $R^{XX}_{W/Z}$ has a slight $v$
dependence above $v>0.5$. This is an indication that the $\lambda_{L}$
dependence of the cross-section is essentially the same for all fermion
channels in $XX\to Zf\bar{f}$ and $XX\to Wf\bar{f}^{\prime}$.

Before displaying the numerical results of the full one-loop computation, we
present the analytical scale variation for a chosen relative velocity in order
to weigh how strong the scale dependence can be. From what we argued and will
confirm shortly through a full calculation, the bulk of the scale variation is
almost flavour independent. We therefore first concentrate on the neutral
channel $XX\to Z\nu\bar{\nu}$. The most important features that are present in
all other channels, are revealed in this channel. From the computation point
of view, the neutrino channel is somehow the easiest since we do not need to
deal with the infrared singularities that require the inclusion of the
radiative $2\to 4(3+\gamma)$ tree-level contribution.

##  4 $XX\to ZZ^{\star}$ at one-loop: $XX\to Z\nu\bar{\nu}$

The large number of diagrams and the appearance of 5-point function loop
integrals makes these computations challenging but SloopS Boudjema_2005 ;
Baro:2007em ; Baro:2008bg ; Baro:2009na ; Boudjema:2011ig ; Boudjema:2014gza ;
Belanger:2016tqb ; Belanger:2017rgu ; Banerjee:2019luv , our automated code,
has been optimised to deal with many of the technicalities that are involved
in these calculations. Another sort of technicality is the renormalisation and
in particular the scheme dependence. All parameters but, in this case,
$\lambda_{L}$ are defined on-shell. $\lambda_{L}$ is here taken
$\overline{{\rm MS}}$, and at the end the one-loop result carries a scale
dependence. As shown in details in the accompanying paper OurPaper1_2020 , the
scale dependence in this mixed scheme only originates from the $\lambda_{L}$
counterterm. One can even exactly determine the scale dependence of the one-
loop cross-section from the parametric $\lambda_{L}$ dependence of the tree-
level cross-section and the knowledge of the corresponding $\beta$ function
for $\lambda_{L}$, $\tilde{\beta}_{\lambda_{L}}$, which can be derived
analytically. That such an approach agrees with the result of a direct
calculation for an arduous calculation such as this $2\to 3$ process, is a
further strong indication of the correctness of the calculation beside the
tests of ultra-violet (UV) finiteness and gauge parameter independence.
Moreover, such an approach which allows an analytical parametrisation of the
scale, is very useful. The first step is to seek the $\lambda_{L}$ parametric
dependence of the tree-level cross-section at a given $v$. To extract this, we
maintain all parameters of the model, namely the masses and the SM parameters
fixed apart from $\lambda_{L}$. Since the dependence is a quadratic
polynomial, the $\lambda_{L}$ dependence is reconstructed numerically by
generating the cross-sections for $\lambda_{L}=0,1,2$. We check the goodness
of the parameterisation by taking a random value of $\lambda_{L}$ and
comparing the cross-section obtained from the reconstructed polynomial against
a direct calculation using the code. We always find excellent agreement for
this check. One can then derive the infinitesimal change of the cross-section
due to an infinitesimal change of $\lambda_{L}$. The latter will then be
turned to a change due to the counterterm for $\lambda_{L}$ through
$\tilde{\beta}_{\lambda_{L}}$ which quantifies the scale dependence as we will
see next for our three benchmark points.

###  4.1 $XX\to Z\nu\bar{\nu}$ at Point G

The $\lambda_{L}$ dependence of the cross-section, for $v=0.4$, is found to be

$\displaystyle\sigma_{XX\to
Z\nu\bar{\nu},G}^{0}(v=0.4)\;v=0.013+0.300\lambda_{L}+1.750\lambda_{L}^{2}\simeq
0.013(1+23\lambda_{L}+135\lambda_{L}^{2}).$ (6)

Observe that the $\lambda_{L}$ dependence is quite strong. The (relative)
coefficient of $\lambda_{L}$ is about $23$. Note also that the
$\lambda_{L}^{2}$ is even larger. The latter will not be so important since
the constraints posed on $\lambda_{L}$ give very small values of
$\lambda_{L}$. One can then relate the one-loop correction ${\rm
d}\sigma(\mu_{2})$ at scale $\mu_{2}$ to the one at scale $\mu_{1}$ according
to

$\displaystyle{\rm d}\sigma_{XX\to Z\nu\bar{\nu}}^{G,v=0.4}(\mu_{2})\;v$
$\displaystyle=$ $\displaystyle{\rm d}\sigma_{XX\to
Z\nu\bar{\nu}}^{G,v=0.4}(\mu_{1})v-\bigg{(}0.300+2\times
1.750\lambda_{L}\bigg{)}\frac{\beta_{\lambda_{L}}\ln(\mu_{2}/\mu_{1})}{16\pi^{2}},$
$\displaystyle\sim$ $\displaystyle{\rm d}\sigma_{XX\to
Z\nu\bar{\nu}}^{I,v=0.4}(\mu_{1})v\;-\;0.300\;\bigg{(}2.141+3.795\lambda_{2}\bigg{)}\frac{\ln(\mu_{2}/\mu_{1})}{16\pi^{2}},\quad\text{leading
to}$ $\displaystyle\left.\frac{{\rm d}\sigma_{XX\to
Z\nu\bar{\nu}}(2\mu_{1})}{\sigma^{0}_{XX\to
Z\nu\bar{\nu}}}\right\rvert_{G,v=0.4}$ $\displaystyle\sim$
$\displaystyle\Bigg{(}\left.\frac{{\rm d}\sigma_{XX\to
Z\nu\bar{\nu}}(\mu_{1})}{\sigma^{0}_{XX\to
Z\nu\bar{\nu}}}\right\rvert_{I,v=0.4}\;-\;21.7\;(1+1.8\lambda_{2})\Bigg{)}\%.$
(7)

The last line gives the difference when the scale is doubled from $\mu_{1}$ to
$2\mu_{1}$.
We verify these formulae against the results of a direct computation of the
full one-loop correction. We obtain a 5 digit agreement for three values of
$\lambda_{2}$ ($\lambda_{2}=0.01,1,2$) and different combinations of the scale
$\mu$. We note that the scale dependence is quite large. This is due to the
strong $\lambda_{L}$ dependence of the cross-section and also to the fact that
$\tilde{\beta}_{\lambda_{L}}$ is not so small. For this benchmark point and
for $v=0.4$ we learn from Equation 4.1 that, in the range
$M_{X}/2<\mu<2M_{X}$, the uncertainty introduced by the scale variation is
about $44\%$ for $\lambda_{2}=0$, it increases to about $200\%$ for
$\lambda_{2}=2$, which should be quoted as the overall theoretical uncertainty
if we allow both the scale $\mu$ to span the range $M_{X}/2-2M_{X}$ and a
variation $0<\lambda_{2}<2$.

Figure 4: Point G. Comparing the relative correction in $\%$ as a function of
the relative velocity for $Z\nu_{\tau}\bar{\nu}_{\tau}$ for
$\lambda_{2}=0.01,1,2$ and for 3 scales $\mu=M_{X}/,M_{X},2M_{X}$. We also
display the improved tree-level based on the use of $\alpha(M_{Z}^{2})$. The
panel on the right is a zoom on the choice $\mu=2M_{X}$ for better
readability.

The results of the full one-loop corrections for three values of
$\lambda_{2}=0.01,1,2$ and for different scales, $\mu=M_{X}/2,M_{X},2M_{X}$
are displayed in Figure 4 for the range of relative velocities of interest for
the relic density calculation. The so-called improved tree-level based on the
use of $\alpha(M_{Z}^{2})$ gives a constant correction of about $21\%$.
$\mu=M_{X}/2$ gives not only the largest correction but shows also a
significant velocity dependence. $\mu=M_{X}/2$ is not an appropriate scale,
this scale is quite removed from the (largest) scales that enter the loop
integrals: the invariant mass of the $XX$ system $\sqrt{s_{XX}}\simeq
2M_{X}(1+v^{2}/8)\sim 2M_{X}$ or $M_{A}$ that enter in the $t$-channel
exchange. As discussed in OurPaper1_2020 , the appropriate scale should be,
$\text{max}(2M_{X},M_{A})=158$ GeV. For point G, there is a small difference
of 14 GeV between $2M_{X}$ and $M_{A}$. We will come back to the choice
$\mu=M_{A}$ whose results will be close to what we obtain for $2M_{X}$. Our
results show that for $\mu=2M_{X}$ and $\lambda_{2}=0.01$, the correction is
about $11\%$. It decreases slowly as $\lambda_{2}$ increases. With this choice
of the scale, the corrections range from $11\%$ to $-3\%$ for $\lambda_{2}$
ranging from $0.01$ to $2$. Observe that while for $\lambda_{2}=0.01$ the
corrections are closest to the value obtained with the improved tree-level
cross-section ($\alpha(M_{Z}^{2})$), there is still as much as $10\%$
difference between the two corrections. An important lesson is that the
$\lambda_{2}$ dependence of the full one-loop correction is clearly important.
This $\lambda_{2}$ dependence is not all contained in
$\tilde{\beta}_{\lambda_{L}}$.

###  4.2 $XX\to Z\nu\bar{\nu}$ at Points F and A

Benchmark points A and F show similar trends to what we just saw for Point G
despite the fact that both benchmarks points A and F have a much larger value
of $\lambda_{L}$. This is understandable since the crucial property that
explains the scale dependence is on the one-hand the relative $\lambda_{L}$
dependence on the cross-section and on the other the value of
$\tilde{\beta}_{\lambda_{L}}$ and its $\lambda_{2}$ dependence. The
$\lambda_{L}$ dependence of the tree-level cross-section is not very different
from that of point G. What is quite different is the magnitude of $\beta_{L}$
and its $\lambda_{2}$ dependence. Point F (A) has a smaller (larger)
$\beta_{L}$ (about a factor 2 for the same value of $\lambda_{2}$) than Point
G. When selecting the most appropriate scale we observe that for point A,
$M_{A}$ is the most appropriate scale while for point F, $2M_{X}$ is quite
close to $M_{A}$.

Figure 5: As in 4 but for Point F (upper panels) and Point A (lower panels).

From the $\lambda_{L}$ dependence of the tree-level cross-section of point F,
at $v=0.4$, we have

$\displaystyle\sigma_{XX\to Z\nu\bar{\nu},F}^{0}(v=0.4)\;v\simeq
0.012\;(1+24\lambda_{L}+143\lambda_{L}^{2}),$ (8)

so that the percentage correction is

$\displaystyle\left.\frac{{\rm d}\sigma_{XX\to
Z\nu\bar{\nu}}(\mu_{2})}{\sigma^{0}_{XX\to
Z\nu\bar{\nu}}}\right\rvert_{F,v=0.4}\simeq\Bigg{(}\left.\frac{{\rm
d}\sigma_{XX\to Z\nu\bar{\nu}}(\mu_{1})}{\sigma^{0}_{XX\to
Z\nu\bar{\nu}}}\right\rvert_{F,v=0.4}\;-\;20.1\;(1+1.9\lambda_{2})\;\ln(\mu_{2}/\mu_{1})\Bigg{)}\%$
(9)

while for point A, we have

$\displaystyle\sigma_{XX\to Z\nu\bar{\nu},A}^{0}(v=0.4)\;v\simeq
0.009\;(1+28\lambda_{L}+201\lambda_{L}^{2}),$ (10)

giving

$\displaystyle\left.\frac{{\rm d}\sigma_{XX\to
Z\nu\bar{\nu}}(\mu_{2})}{\sigma^{0}_{XX\to
Z\nu\bar{\nu}}}\right\rvert_{A,v=0.4}\simeq\Bigg{(}\left.\frac{{\rm
d}\sigma_{XX\to Z\nu\bar{\nu}}(\mu_{1})}{\sigma^{0}_{XX\to
Z\nu\bar{\nu}}}\right\rvert_{A,v=0.4}\;-\;72.1\;(1+1.4\lambda_{2})\;\ln(\mu_{2}/\mu_{1})\Bigg{)}\%$
(11)

Our results for points F and A are shown in Figure 5. They confirm the general
trend observed for point G. The full one-loop results displaying the scale
dependence and $\lambda_{2}$ dependence are also in excellent agreement with
the analytical formulae of Equations 9-11. For $\mu=M_{X}/2$, the corrections
are very large with strong velocity dependence. Of the three scales,
$\mu=M_{X}/2,M_{X},2M_{X}$, the largest scale, $\mu=2M_{X}$, is where the
corrections are the smallest. Yet, for point A, even $\mu=2M_{X}$ gives large
correction. For point A, $M_{A}$ is the largest scale for the process. It is
quite different from the choice $\mu=2M_{X}$, considering the large value of
$\tilde{\beta}_{\lambda_{L}}$. We therefore show the results of taking
$\mu=M_{A}$ as the optimised scale in Figure 6. For Point F, the difference
with $\mu=2M_{X}$ is very small and changes (slightly) mainly the results for
$\lambda_{2}=2$ (there is an upward shift of $-5\%$ when moving from
$\mu=2M_{X}$ to $\mu=M_{A}$). This is due to the fact that for point F,
$2M_{X}$ and $M_{A}$ are very close ($M_{A}=138$ GeV and $2M_{X}=144$ GeV) and
$\tilde{\beta}_{\lambda_{L}}$ is not large. The effect of the change of $\mu$
for point A when moving between the scale $2M_{X}$ and $M_{A}$ is substantial.
In particular, for $\lambda_{2}=0.01$, the correction of $10\%$ is in line
with the correction found for the 2 other benchmarks points while the
corrections for $\lambda_{2}=1,2$ are sensibly smaller. Nonetheless, we warn
the reader again that a large value of $\beta_{\lambda_{L}}$ is sensitive to
large scale variations. Independently of the scale choice, a common feature is
that $\lambda_{2}=0.01$ gives the smallest correction often approaching the
tree-level improved $\alpha(M_{Z}^{2})$ approximation.

Figure 6: The relative corrections for $\sigma v(XX\to
Z\nu_{\tau}\bar{\nu}_{\tau})$ with $\mu=M_{A}$ for the benchmark Point A (left
panel) and Point F (right panel).

##  5 $XX\to Zf\bar{f}$ and $XX\to Wf\bar{f}^{\prime}$: one-loop results

We already saw in our study of the tree-level cross-sections in section 2 that
the flavour dependence of $\sigma(XX\to Wf\bar{f}^{\prime})$ is, to an
excellent accuracy, contained in and represented by the flavour dependence of
$\Gamma(W\to f\bar{f}^{\prime})$. We also learnt that the velocity dependence
of the neutral and charged channels cancels out in the ratio of the cross-
sections $\sigma(XX\to Wf\bar{f}^{\prime})/\sigma(XX\to Z\nu\bar{\nu})$ for
velocities up to $v=0.5$. Above these velocities, the $WW$ channel starts
experiencing the onset of the $WW$ threshold while the $ZZ$ is still not
experiencing the $ZZ$ threshold. The underlying global $SU(2)$ symmetry would
also suggest that, particularly below $v<0.5$, the $\lambda_{L}$ dependence of
$\sigma(XX\to Wf\bar{f}^{\prime})/\sigma(XX\to Z\nu\bar{\nu})$ cancels out. To
wit, we find that the $\lambda_{L}$ dependence for $XX\to
W\nu_{\tau}\bar{\tau}$ for point G and velocity $v=0.4$ writes as

$\displaystyle\sigma_{XX\to W\nu_{\tau}\bar{\nu}_{\tau},G}^{0}(v=0.4)$
$\displaystyle=$ $\displaystyle
0.086+1.960\lambda_{L}+11.343\lambda_{L}^{2}\simeq
0.086\;(1+23\lambda_{L}+133\lambda_{L}^{2})$ (12) $\displaystyle\simeq$
$\displaystyle 6.585\;\sigma_{XX\to Z\nu\bar{\nu},G}^{0}(v=0.4).$

The latter very good approximation means, especially for the very small values
of $\lambda_{L}$ we are permitted, that the $\mu$ dependence, Equation 4.1, is
within machine precision and fitting procedure, identical for $ZZ^{\star}$ and
$WW^{\star}$. Since these two channels carry almost the same relative
$\lambda_{L}$ dependence and that $\lambda_{L}\ll 1$, the scale dependence for
the normalised cross-sections is confirmed to be almost the same. A small
expected departure above $v>0.5$ is confirmed numerically. The flavour
independence of $\sigma(XX\to Wf\bar{f}^{\prime})/\Gamma(W\to
f\bar{f}^{\prime})$ continues to hold true at one-loop. The latter stems from
the fact that the electroweak radiative corrections (relative to the tree-
level) of $\Gamma(W\to f\bar{f}^{\prime})$ are known to be the same for all
flavours Bardin:1986fi , therefore $R^{XX}_{Wf\bar{f}^{\prime}}=1$ at one-loop
also. We also verify all these properties by a direct full one-loop
computation to the different channels. First of all, we confirm that the
relative electroweak correction to the annihilation into $Wl\bar{\nu}_{l}$ is,
within the per-mille level, the same as that of the annihilation into
$Wq\bar{q}^{\prime}$, we therefore only show the leptonic (charged) final
state, $W\tau\bar{\nu}_{\tau}$.

Figure 7: Point G. The relative correction, ${\rm
d}\sigma(Vf\bar{f}^{\prime})/\sigma^{0}(Vf\bar{f}^{\prime}),V=W,Z$ in $\%$ as
a function of the relative velocity for $XX\to
Z\tau^{+}\tau^{-},Z\nu_{\tau}\bar{\nu}_{\tau},W\tau\bar{\nu}_{\tau}$ for
$\mu=2M_{X}$ and $\lambda_{2}=0.01,1,2$. The right panel shows the difference
${\rm d}\sigma(Vf\bar{f}^{\prime})/\sigma(Vf\bar{f}^{\prime})-{\rm
d}\sigma(Z\nu\bar{\nu})/\sigma(Z\nu\bar{\nu})$.

As Figure 7 for $\mu=2M_{X}$ in the benchmark point G shows, the behaviour of
all the annihilation channels follow $XX\to Z\nu_{\tau}\bar{\nu}_{\tau}$. The
$\lambda_{2}$ dependence is indeed represented by the channel we discussed in
detail in the previous section. The $-0.5\%$ difference between the
$(Z)\tau\bar{\tau}$ and $(Z)\nu_{\tau}\bar{\nu}_{\tau}$ is indeed due mostly
to the one-loop relative correction between the $Z$ decay width into
$\tau\bar{\tau}$ and $\nu_{\tau}\bar{\nu}_{\tau}$ ($-0.9\%$) with a smaller
contribution from the $\tau$ Yukawa mass ($h$ exchange for example) as
identified in Figure 2. The deviation observed in $W\tau\bar{\nu}_{\tau}$
increases for larger $v$ (as expected from the tree-level comparison of these
two channels). Nonetheless, the difference between the relative corrections
remains below $1.5\%$ (below $0.5\%$ for $v<0.5$). To an excellent
approximation, the relative one-loop corrections, ${\rm d}\sigma$, between the
different channels, represent the difference between the one-loop electroweak
corrections to the corresponding partial widths, ${\rm d}\Gamma$,

$\displaystyle\frac{{\rm d}\sigma(XX\to Zf\bar{f})v}{\sigma(XX\to
Zf\bar{f})v}-\frac{{\rm d}\sigma(XX\to Z\nu\bar{\nu})v}{\sigma(XX\to
Z\nu\bar{\nu})v}\simeq\frac{{\rm d}\Gamma(Z\to f\bar{f})}{\Gamma(Z\to
f\bar{f})}-\frac{{\rm d}\Gamma(Z\to\nu\bar{\nu})}{\Gamma(Z\to\nu\bar{\nu})}$
(13)

The mass effect is obviously more important for the $b$-quark final state
channel, as we already saw at tree-level (Figure 2). In the radiative
corrections to $\sigma v(XX\to Zb\bar{b})$, there is about a $-6\%$ difference
with the $Z\nu\bar{\nu}$ channel, see Figure 8. Almost half of this correction
is due to the difference in the relative electroweak correction between the
partial width $Z\to b\bar{b}$ and $Z\to\nu\bar{\nu}$.

Figure 8: Point G. Percentage corrections for $\mu=M_{A}$ for the
$Z\nu_{\tau}\bar{\nu}_{\tau}$ and $Zb\bar{b}$ final states. The latter are
about $6\%$ lower than the neutrino case for all values of $v$ and
$\lambda_{2}$.

##  6 Intermediate Summary

Let us take stock.

  * •

An important feature common to all three benchmarks is that whatever the
values of the renormalisation scale $\mu$ and of the parameter $\lambda_{2}$
are, $XX\to Z\nu\bar{\nu}$ encapsulates practically all of the radiative
corrections contained not only in the neutral channels $XX\to Zf\bar{f}$ but
also in the charged channels $XX\to Wf\bar{f}^{\prime}$, in the sense that the
normalised one-loop corrections are, to a high degree of accuracy, equal for
all channels for the same choice of $\lambda_{2}$ and the renormalisation
scale $\mu$

$\displaystyle\frac{\delta\sigma(XX\to Z\nu\bar{\nu})}{\sigma(XX\to
Z\nu\bar{\nu})}{\Huge|}_{\mu,\lambda_{2}}\sim\frac{\delta\sigma(XX\to
Vf\bar{f}^{\prime})}{\sigma(XX\to
Vf\bar{f}^{\prime})}{\Huge|}_{\mu,\lambda_{2}}\quad V=W,Z$ (14)

\- Equation 14 is valid at all $v$ (in the range of interest for the relic
density) for $\sigma(XX\to Zf\bar{f})$. There is a slight $v$ dependence for
$\sigma(XX\to Wf\bar{f}^{\prime})$ when compared to $\sigma(XX\to Vf\bar{f})$.
This difference is small and does not exceed more than $1.5\%$.

-There is no flavour dependence for the charged channels. The flavour dependence in the neutral channel is largest for $Zb\bar{b}$ where the largest difference amounts to $6\%$, more than half of this difference is accounted for by a correction given by $\Gamma(Z\to f\bar{f})_{{\rm 1-loopEW}}/\Gamma(Z\to f\bar{f})_{{\rm tree}}$. We expect these small flavour effects to be diluted when we consider the correction to the relic density, considering that the $ZZ$ channel accounts for about $10\%$ to the relic density and that the $b\bar{b}$ channel is $15\%$ of the whole $ZZ$.

  * •

As expected, the scale dependence is largest for point A which has the largest
$\tilde{\beta}_{\lambda_{L}}$. Our conjecture (based nonetheless on the study
of the scales involved in the loop functions) seems to be a very good one. The
appropriate scale is ${\rm max}(2M_{X},M_{A})$. In particular, we find that
for $\lambda_{2}=0.01$, all 3 benchmark points give very similar corrections
of about $10\%-12\%$. For this choice of scale, points F and G (which have
small $\tilde{\beta}_{\lambda_{L}}$) give very similar corrections for
$\lambda_{2}=1\;(-5\%),2\;(-20\%)$ while for point A, the corresponding
corrections are $v$ dependent with values for $v\sim 0$ similar for all
$\lambda_{2}$. We observe that in our study of $XX\to Wf\bar{f}^{\prime}$
(P60) in the co-annihilation region OurPaper2_2020 , where an on-shell
renormalisation for $\lambda_{L}$ was possible, the electroweak correction for
$\lambda_{2}=0.01,1,2$ were quantitatively very similar to the results we
obtain here, especially for the benchmark points F and G. This validates
further our conjecture about the choice of scale. For the calculation of the
relic density, we consider that the appropriate scale is ${\rm
max}(\mu=2M_{X},M_{A})$ and that theoretical uncertainty can be estimated by
the difference within the range $(2M_{X},M_{A})$.

##  7 Effect on the relic density

| tree | $\alpha(M_{Z}^{2})$ | $\lambda_{2}=0.01$ | $\lambda_{2}=1$ | $\lambda_{2}=2$
---|---|---|---|---|---
Point G (I) | 0.121 | 0.101 |  |  |
|  | (-16.53%) |  |  |
Full, $\mu=M_{X}$ |  |  | 0.093 (-23.43%) | 0.076 (-36.73%) | 0.065 (-46.11%)
Full, $\mu=2M_{X}$ |  |  | 0.109 (-9.53%) | 0.117 (-3.47%) | 0.125 (3.57%)
Full, $\mu=M_{A}$ |  |  | 0.112 (-7.27%) | 0.126 (3.94%) | 0.143 (18.51%)
Simplified, $\mu=M_{X}$ |  |  | 0.093 (-23.48%) | 0.076 (-36.76%) | 0.065 (-46.12%)
Simplified, $\mu=2M_{X}$ |  |  | 0.109 (-9.60%) | 0.117 (-3.55%) | 0.125 (3.47%)
Simplified, $\mu=M_{A}$ |  |  | 0.112 (-7.35%) | 0.126 (3.84%) | 0.143 (18.37%)
Point A (B) | 0.156 | 0.130 |  |  |
|  | (-16.67%) |  |  |
Simplified, $\mu=M_{X}$ |  |  | 0.092 (-41.35%) | 0.063 (-59.57%) | 0.048 (-69.12%)
Simplified, $\mu=2M_{X}$ |  |  | 0.125 (-19.59%) | 0.112 (-28.28%) | 0.101 (-35.31%)
Simplified, $\mu=M_{A}$ |  |  | 0.140 (-10.16%) | 0.144 (-7.91%) | 0.147 (-5.52%)
Point F (H) | 0.119 | 0.099 |  |  |
|  | (-16.81%) |  |  |
Simplified, $\mu=M_{X}$ |  |  | 0.098 (-17.92%) | 0.089 (-25.23%) | 0.082 (-31.38 %)
Simplified, $\mu=2M_{X}$ |  |  | 0.109 (-8.15%) | 0.123 (3.62%) | 0.142 (19.17%)
Simplified, $\mu=M_{A}$ |  |  | 0.108 (-8.82%) | 0.120 (1.21%) | 0.136 (13.94%)
Table 2: The relic density for points G, A and F at tree-level and after
including the one-loop corrections. The percentage changes are given in
parenthesis. The percentage correction corresponding to the use of
$\alpha(M_{Z}^{2})$ at tree-level is also indicated. The full corrections for
point G differ from the simplified one-loop in that the full one-loop
$Zb\bar{b}$ final state is fully taken into account while in the simplified
version all fermion final states are rescaled from the full one-loop in the
$Z\nu\bar{\nu}$ cross-section through the added one-loop flavour correction,
$\Gamma_{Z\to f\bar{f}}/\Gamma_{Z\to\nu\bar{\nu}}$, where the partial widths
are computed at one-loop, see text for details.

We just learnt that the $v$ dependence of the cross-sections, that contributes
to the relic density calculation, is rather smooth. Moreover, the $\mu$
(scale) dependence is sensibly the same in all channels (neutral and charged).
We therefore expect the $\mu$ dependence of the relic density to follow that
of the cross-section $\sigma(XX\to Z\nu\bar{\nu})$, since $\Omega h^{2}\sim
1/<\sigma v>$ ($<\sigma v>$ is the total thermodynamically averaged cross-
section). The difference between the values of the relic density for
$\mu=M_{X}$ and $\mu=2M_{X}$ follows this trend as shown in Table 2 of the
relic densities obtained after passing all our tree and one-loop $v$ dependent
cross-sections to micrOMEGAs. The table shows (as expected) large corrections
for the inappropriate choice $\mu=M_{X}$, particularly for point A. We derive
the relic density by taking into account the Yukawa couplings of the
$b$-quarks (full calculation) beyond the effect of the flavour dependence
contained in the partial decay $\Gamma_{Z\to b\bar{b}}$, see 13, that allows a
nice factorisation of the total cross-section in terms of $\sigma(XX\to
Z\nu\bar{\nu})$, which we call simplified. The difference between the full and
simplified implementations is very small since the overall contribution of the
$Zb\bar{b}$ final state to the total annihilation cross-section is small. An
important feature seen for all scales and benchmark points is that the impact
of $\lambda_{2}$ is large, this parameter is not taken into account when tree-
level analyses are conducted.

The appropriate scale is ${\rm max}(\mu=2M_{X},M_{A})$. For
$\lambda_{2}=0.01$, the 3 benchmark points give very similar results with
small corrections contained in the range $-7\%$ to $-10\%$. These corrections
are smaller than those found through the naive use of a running $\alpha$ at
scale $M_{Z}^{2}$. Even for these scales, the $\lambda_{2}$ dependence is not
negligible at all. To give a quantitative estimate of the theory uncertainty
that a tree-level evaluation of the relic density should incorporate, one
needs to look at the one-loop results by varying both $\mu$ and $\lambda_{2}$.
For instance, take the benchmark point F (very similar results are obtained
for point G) which has a small $\tilde{\beta}_{\lambda_{L}}$. While at tree-
level, $\Omega h^{2}=0.119$ (obtained with $\alpha(0)$), the theory
uncertainty now is $0.108<\Omega h^{2}<0.142$ ($\Omega
h^{2}=0.119^{19.2\%}_{-8.8\%}$), this is more than the uncertainty of $\pm
10\%$ applied routinely in some analyses. Note that the uncertainty/error is
much larger if based on the usage of $\alpha(M_{Z}^{2})$ which is the default
value of micrOMEGAs. For point A, where $\tilde{\beta}_{\lambda_{L}}$ is
larger, the tree-level result is turned into the range $\Omega
h^{2}=0.156^{-5.5\%}_{-35.3\%}$ with the conclusion that a value of $0.156$
that could be dismissed on the basis of the present experimental constraint on
the relic density can in fact be easily brought in line with the measured
value if loop corrections were taken into account.

##  8 Conclusions

This is the first time a calculation of $2\to 3$ processes for the
annihilation of DM has been performed at the one-loop level and the results of
the corrected cross-sections turned into a prediction of the relic density.
While this calculation is within the IDM, the tools at our disposal are now
powerful enough to tackle such calculations for any model of DM provided a
coherent renormalisation programme has been devised and implemented. In the
particular case of the IDM, the reconstruction of the model parameters in
order to fully define the model leaves two underlying parameters not fully
determined in terms of physical parameters. One parameter, $\lambda_{2}$,
describes, at tree-level, the interaction solely within the dark sector of the
IDM. It is therefore difficult to extract directly from observables involving
SM particles. Yet, this parameter contributes significantly to dark matter
annihilation processes such as those we studied here. This indirect one-loop
effect could in principle be extracted from the precise measurement of the
relic density, a situation akin to the extraction of the top mass from LEP
observables provided all other parameters of the model, masses of the
additional scalars and their coupling to the SM, $\lambda_{L}$. $\lambda_{L}$,
in fact measures the strength of the coupling of the SM Higgs to the pair of
DM, there is in fact a one-to-one mapping between Higgs decay to $XX$ and
$\lambda_{L}$, which suggests an extraction of $\lambda_{L}$ from the partial
width of the Higgs into $XX$. While difficult in general, it is impossible
when this Higgs decay is closed. The allowed parameter space for the $2\to 3$
processes we studied is when this Higgs decay is closed. In this case we
suggested an $\overline{\text{MS}}$ scheme for $\lambda_{L}$. The
$\overline{\text{MS}}$ introduces a scale dependence on the one-loop cross-
sections. We showed that the scale dependence can be determined from the
$\lambda_{L}$ parametric dependence of the tree-level cross-section and the
knowledge of the 1-loop $\beta$ constant for $\lambda_{L}$,
$\tilde{\beta}_{\lambda_{L}}$. Despite the fact that the experimentally
allowed values of $\lambda_{L}$ are small, the parametric dependence of the
cross-sections on $\lambda_{L}$ are large for all the benchmarks that we
studied. Combined with not so small $\tilde{\beta}_{\lambda_{L}}$, the scale
dependence can be very large if an inappropriate scale is chosen. Based on a
few other analyses in the framework of the IDM OurPaper2_2020  and also in
supersymmetric scenarios Belanger:2016tqb ; Belanger:2017rgu  regarding the
issue of the choice of the optimal scale, we suggest to restrict the choice of
the scale to values around the maximal scale involved in the process. The
present one-loop analysis is yet another warning to practitioners of the IDM
and other BSM models in respect of the relic density of DM in these models.
The one-loop analyses give an important quantitative estimate of the (often)
large theoretical uncertainty that should be taken into account before
allowing or dismissing scenarios based on a tree-level derivation of relic
density. The latter for instance is not sensitive to the value of
$\lambda_{2}$. $\lambda_{2}$ should be taken into account along side the
uncertainty from the scale variation. We find that the combined theoretical
uncertainty is model-dependent and in many cases is much larger than the
cursory ($\pm$)symmetric $10\%$ theoretical uncertainty that is included in
many analyses.

###### Acknowledgements.

We thank Alexander Pukhov for several insightful 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 started. SB is grateful for the support
received from IPPP, Durham, UK, where most of this work was performed. SB is
also grateful to the support received from LAPTh during his visit, when this
work started. NC is financially supported by IISc (Indian Institute of
Science, Bangalore, India) through the C.V. Raman postdoctoral fellowship. NC
also acknowledges the support received from DST, India, under grant number
IFA19-PH237 (INSPIRE Faculty Award).

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K. Bunkowski, A. Byszuk37, K. Doroba, A. Kalinowski, M. Konecki, J.
Krolikowski, M. Misiura, M. Olszewski, A. Pyskir, M. Walczak Laboratório de
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M. Araujo, P. Bargassa, C. Beirão Da Cruz E Silva, A. Di Francesco, P.
Faccioli, B. Galinhas, M. Gallinaro, J. Hollar, N. Leonardo, J. Seixas, G.
Strong, O. Toldaiev, J. Varela Joint Institute for Nuclear Research, Dubna,
Russia
S. Afanasiev, P. Bunin, M. Gavrilenko, I. Golutvin, I. Gorbunov, A. Kamenev,
V. Karjavine, A. Lanev, A. Malakhov, V. Matveev38,39, P. Moisenz, V. Palichik,
V. Perelygin, S. Shmatov, S. Shulha, N. Skatchkov, V. Smirnov, N. Voytishin,
A. Zarubin Petersburg Nuclear Physics Institute, Gatchina (St. Petersburg),
Russia
V. Golovtsov, Y. Ivanov, V. Kim40, E. Kuznetsova41, P. Levchenko, V. Murzin,
V. Oreshkin, I. Smirnov, D. Sosnov, V. Sulimov, L. Uvarov, S. Vavilov, A.
Vorobyev Institute for Nuclear Research, Moscow, Russia
Yu. Andreev, A. Dermenev, S. Gninenko, N. Golubev, A. Karneyeu, M. Kirsanov,
N. Krasnikov, A. Pashenkov, A. Shabanov, D. Tlisov, A. Toropin Institute for
Theoretical and Experimental Physics, Moscow, Russia
V. Epshteyn, V. Gavrilov, N. Lychkovskaya, V. Popov, I. Pozdnyakov, G.
Safronov, A. Spiridonov, A. Stepennov, V. Stolin, M. Toms, E. Vlasov, A.
Zhokin Moscow Institute of Physics and Technology, Moscow, Russia
T. Aushev National Research Nuclear University ’Moscow Engineering Physics
Institute’ (MEPhI), Moscow, Russia
R. Chistov42, M. Danilov42, S. Polikarpov42, E. Tarkovskii P.N. Lebedev
Physical Institute, Moscow, Russia
V. Andreev, M. Azarkin, I. Dremin39, M. Kirakosyan, A. Terkulov Skobeltsyn
Institute of Nuclear Physics, Lomonosov Moscow State University, Moscow,
Russia
A. Baskakov, A. Belyaev, E. Boos, M. Dubinin43, L. Dudko, A. Ershov, A.
Gribushin, V. Klyukhin, O. Kodolova, I. Lokhtin, I. Miagkov, S. Obraztsov, S.
Petrushanko, V. Savrin, A. Snigirev Novosibirsk State University (NSU),
Novosibirsk, Russia
A. Barnyakov44, V. Blinov44, T. Dimova44, L. Kardapoltsev44, Y. Skovpen44
Institute for High Energy Physics of National Research Centre ’Kurchatov
Institute’, Protvino, Russia
I. Azhgirey, I. Bayshev, S. Bitioukov, V. Kachanov, A. Kalinin, D.
Konstantinov, P. Mandrik, V. Petrov, R. Ryutin, S. Slabospitskii, A. Sobol, S.
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A. Babaev, S. Baidali, V. Okhotnikov University of Belgrade: Faculty of
Physics and VINCA Institute of Nuclear Sciences
P. Adzic45, P. Cirkovic, D. Devetak, M. Dordevic, J. Milosevic Centro de
Investigaciones Energéticas Medioambientales y Tecnológicas (CIEMAT), Madrid,
Spain
J. Alcaraz Maestre, A. Álvarez Fernández, I. Bachiller, M. Barrio Luna, J.A.
Brochero Cifuentes, M. Cerrada, N. Colino, B. De La Cruz, A. Delgado Peris, C.
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J. Puerta Pelayo, I. Redondo, L. Romero, S. Sánchez Navas, M.S. Soares, A.
Triossi Universidad Autónoma de Madrid, Madrid, Spain
C. Albajar, J.F. de Trocóniz Universidad de Oviedo, Oviedo, Spain
J. Cuevas, C. Erice, J. Fernandez Menendez, S. Folgueras, I. Gonzalez
Caballero, J.R. González Fernández, E. Palencia Cortezon, V. Rodríguez Bouza,
S. Sanchez Cruz, J.M. Vizan Garcia Instituto de Física de Cantabria (IFCA),
CSIC-Universidad de Cantabria, Santander, Spain
I.J. Cabrillo, A. Calderon, B. Chazin Quero, J. Duarte Campderros, M.
Fernandez, P.J. Fernández Manteca, A. García Alonso, J. Garcia-Ferrero, G.
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Arbol, F. Matorras, J. Piedra Gomez, C. Prieels, T. Rodrigo, A. Ruiz-Jimeno,
L. Scodellaro, N. Trevisani, I. Vila, R. Vilar Cortabitarte University of
Ruhuna, Department of Physics, Matara, Sri Lanka
N. Wickramage CERN, European Organization for Nuclear Research, Geneva,
Switzerland
D. Abbaneo, B. Akgun, E. Auffray, G. Auzinger, P. Baillon, A.H. Ball, D.
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Camporesi, M. Cepeda, G. Cerminara, E. Chapon, Y. Chen, G. Cucciati, D.
d’Enterria, A. Dabrowski, N. Daci, V. Daponte, A. David, A. De Roeck, N.
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Gilbert, K. Gill, F. Glege, M. Gruchala, M. Guilbaud, D. Gulhan, J. Hegeman,
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Institut, Villigen, Switzerland
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Kaestli, D. Kotlinski, U. Langenegger, T. Rohe, S.A. Wiederkehr ETH Zurich -
Institute for Particle Physics and Astrophysics (IPA), Zurich, Switzerland
M. Backhaus, L. Bäni, P. Berger, N. Chernyavskaya, G. Dissertori, M. Dittmar,
M. Donegà, C. Dorfer, T.A. Gómez Espinosa, C. Grab, D. Hits, T. Klijnsma, W.
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Neutelings, G. Rauco, P. Robmann, D. Salerno, K. Schweiger, C. Seitz, Y.
Takahashi, A. Zucchetta National Central University, Chung-Li, Taiwan
T.H. Doan, R. Khurana, C.M. Kuo, W. Lin, A. Pozdnyakov, S.S. Yu National
Taiwan University (NTU), Taipei, Taiwan
P. Chang, Y. Chao, K.F. Chen, P.H. Chen, W.-S. Hou, Y.F. Liu, R.-S. Lu, E.
Paganis, A. Psallidas, A. Steen Chulalongkorn University, Faculty of Science,
Department of Physics, Bangkok, Thailand
B. Asavapibhop, N. Srimanobhas, N. Suwonjandee Çukurova University, Physics
Department, Science and Art Faculty, Adana, Turkey
A. Bat, F. Boran, S. Cerci52, S. Damarseckin, Z.S. Demiroglu, F. Dolek, C.
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I.S. Zorbakir, C. Zorbilmez Middle East Technical University, Physics
Department, Ankara, Turkey
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Istanbul, Turkey
I.O. Atakisi, E. Gülmez, M. Kaya58, O. Kaya59, S. Ozkorucuklu60, S. Tekten,
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M.N. Agaras, A. Cakir, K. Cankocak, Y. Komurcu, S. Sen62 Institute for
Scintillation Materials of National Academy of Science of Ukraine, Kharkov,
Ukraine
B. Grynyov National Scientific Center, Kharkov Institute of Physics and
Technology, Kharkov, Ukraine
L. Levchuk University of Bristol, Bristol, United Kingdom
F. Ball, J.J. Brooke, D. Burns, E. Clement, D. Cussans, O. Davignon, H.
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Titterton Rutherford Appleton Laboratory, Didcot, United Kingdom
K.W. Bell, A. Belyaev64, C. Brew, R.M. Brown, D. Cieri, D.J.A. Cockerill, J.A.
Coughlan, K. Harder, S. Harper, J. Linacre, K. Manolopoulos, E. Olaiya, D.
Petyt, C.H. Shepherd-Themistocleous, A. Thea, I.R. Tomalin, T. Williams, W.J.
Womersley Imperial College, London, United Kingdom
R. Bainbridge, P. Bloch, J. Borg, S. Breeze, O. Buchmuller, A. Bundock, D.
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Zenz Brunel University, Uxbridge, United Kingdom
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of California, Los Angeles, USA
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University of California, Riverside, Riverside, USA
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Carnegie Mellon University, Pittsburgh, USA
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Accelerator Laboratory, Batavia, USA
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Whitbeck University of Florida, Gainesville, USA
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Florida Institute of Technology, Melbourne, USA
M.M. Baarmand, V. Bhopatkar, S. Colafranceschi, M. Hohlmann, D. Noonan, M.
Rahmani, T. Roy, M. Saunders, F. Yumiceva University of Illinois at Chicago
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Gandrajula, M. Haytmyradov, V. Khristenko, J.-P. Merlo, A. Mestvirishvili, A.
Moeller, J. Nachtman, H. Ogul71, Y. Onel, F. Ozok72, A. Penzo, C. Snyder, E.
Tiras, J. Wetzel Johns Hopkins University, Baltimore, USA
B. Blumenfeld, A. Cocoros, N. Eminizer, D. Fehling, L. Feng, A.V. Gritsan,
W.T. Hung, P. Maksimovic, J. Roskes, U. Sarica, M. Swartz, M. Xiao The
University of Kansas, Lawrence, USA
A. Al-bataineh, P. Baringer, A. Bean, S. Boren, J. Bowen, A. Bylinkin, J.
Castle, S. Khalil, A. Kropivnitskaya, D. Majumder, W. Mcbrayer, M. Murray, C.
Rogan, S. Sanders, E. Schmitz, J.D. Tapia Takaki, Q. Wang Kansas State
University, Manhattan, USA
S. Duric, A. Ivanov, K. Kaadze, D. Kim, Y. Maravin, D.R. Mendis, T. Mitchell,
A. Modak, A. Mohammadi Lawrence Livermore National Laboratory, Livermore, USA
F. Rebassoo, D. Wright University of Maryland, College Park, USA
A. Baden, O. Baron, A. Belloni, S.C. Eno, Y. Feng, C. Ferraioli, N.J. Hadley,
S. Jabeen, G.Y. Jeng, R.G. Kellogg, J. Kunkle, A.C. Mignerey, S. Nabili, F.
Ricci-Tam, M. Seidel, Y.H. Shin, A. Skuja, S.C. Tonwar, K. Wong Massachusetts
Institute of Technology, Cambridge, USA
D. Abercrombie, B. Allen, V. Azzolini, A. Baty, G. Bauer, R. Bi, S. Brandt, W.
Busza, I.A. Cali, M. D’Alfonso, Z. Demiragli, G. Gomez Ceballos, M. Goncharov,
P. Harris, D. Hsu, M. Hu, Y. Iiyama, G.M. Innocenti, M. Klute, D. Kovalskyi,
Y.-J. Lee, P.D. Luckey, B. Maier, A.C. Marini, C. Mcginn, C. Mironov, S.
Narayanan, X. Niu, C. Paus, C. Roland, G. Roland, Z. Shi, G.S.F. Stephans, K.
Sumorok, K. Tatar, D. Velicanu, J. Wang, T.W. Wang, B. Wyslouch University of
Minnesota, Minneapolis, USA
A.C. Benvenuti${}^{\textrm{\textdagger}}$, R.M. Chatterjee, A. Evans, P.
Hansen, J. Hiltbrand, Sh. Jain, S. Kalafut, M. Krohn, Y. Kubota, Z. Lesko, J.
Mans, N. Ruckstuhl, R. Rusack, M.A. Wadud University of Mississippi, Oxford,
USA
J.G. Acosta, S. Oliveros University of Nebraska-Lincoln, Lincoln, USA
E. Avdeeva, K. Bloom, D.R. Claes, C. Fangmeier, F. Golf, R. Gonzalez Suarez,
R. Kamalieddin, I. Kravchenko, J. Monroy, J.E. Siado, G.R. Snow, B. Stieger
State University of New York at Buffalo, Buffalo, USA
A. Godshalk, C. Harrington, I. Iashvili, A. Kharchilava, C. Mclean, D. Nguyen,
A. Parker, S. Rappoccio, B. Roozbahani Northeastern University, Boston, USA
G. Alverson, E. Barberis, C. Freer, Y. Haddad, A. Hortiangtham, G. Madigan,
D.M. Morse, T. Orimoto, A. Tishelman-charny, T. Wamorkar, B. Wang, A.
Wisecarver, D. Wood Northwestern University, Evanston, USA
S. Bhattacharya, J. Bueghly, O. Charaf, T. Gunter, K.A. Hahn, N. Odell, M.H.
Schmitt, K. Sung, M. Trovato, M. Velasco University of Notre Dame, Notre Dame,
USA
R. Bucci, N. Dev, M. Hildreth, K. Hurtado Anampa, C. Jessop, D.J. Karmgard, K.
Lannon, W. Li, N. Loukas, N. Marinelli, F. Meng, C. Mueller, Y. Musienko38, M.
Planer, R. Ruchti, P. Siddireddy, G. Smith, S. Taroni, M. Wayne, A. Wightman,
M. Wolf, A. Woodard The Ohio State University, Columbus, USA
J. Alimena, L. Antonelli, B. Bylsma, L.S. Durkin, S. Flowers, B. Francis, C.
Hill, W. Ji, T.Y. Ling, W. Luo, B.L. Winer Princeton University, Princeton,
USA
S. Cooperstein, P. Elmer, J. Hardenbrook, N. Haubrich, S. Higginbotham, A.
Kalogeropoulos, S. Kwan, D. Lange, M.T. Lucchini, J. Luo, D. Marlow, K. Mei,
I. Ojalvo, J. Olsen, C. Palmer, P. Piroué, J. Salfeld-Nebgen, D. Stickland, C.
Tully University of Puerto Rico, Mayaguez, USA
S. Malik, S. Norberg Purdue University, West Lafayette, USA
A. Barker, V.E. Barnes, S. Das, L. Gutay, M. Jones, A.W. Jung, A. Khatiwada,
B. Mahakud, D.H. Miller, N. Neumeister, C.C. Peng, S. Piperov, H. Qiu, J.F.
Schulte, J. Sun, F. Wang, R. Xiao, W. Xie Purdue University Northwest,
Hammond, USA
T. Cheng, J. Dolen, N. Parashar Rice University, Houston, USA
Z. Chen, K.M. Ecklund, S. Freed, F.J.M. Geurts, M. Kilpatrick, Arun Kumar, W.
Li, B.P. Padley, R. Redjimi, J. Roberts, J. Rorie, W. Shi, Z. Tu, A. Zhang
University of Rochester, Rochester, USA
A. Bodek, P. de Barbaro, R. Demina, Y.t. Duh, J.L. Dulemba, C. Fallon, T.
Ferbel, M. Galanti, A. Garcia-Bellido, J. Han, O. Hindrichs, A.
Khukhunaishvili, E. Ranken, P. Tan, R. Taus Rutgers, The State University of
New Jersey, Piscataway, USA
B. Chiarito, J.P. Chou, Y. Gershtein, E. Halkiadakis, A. Hart, M. Heindl, E.
Hughes, S. Kaplan, R. Kunnawalkam Elayavalli, S. Kyriacou, I. Laflotte, A.
Lath, R. Montalvo, K. Nash, M. Osherson, H. Saka, S. Salur, S. Schnetzer, D.
Sheffield, S. Somalwar, R. Stone, S. Thomas, P. Thomassen University of
Tennessee, Knoxville, USA
A.G. Delannoy, J. Heideman, G. Riley, S. Spanier Texas A&M University, College
Station, USA
O. Bouhali73, A. Celik, M. Dalchenko, M. De Mattia, A. Delgado, S. Dildick, R.
Eusebi, J. Gilmore, T. Huang, T. Kamon74, S. Luo, D. Marley, R. Mueller, D.
Overton, L. Perniè, D. Rathjens, A. Safonov Texas Tech University, Lubbock,
USA
N. Akchurin, J. Damgov, F. De Guio, P.R. Dudero, S. Kunori, K. Lamichhane,
S.W. Lee, T. Mengke, S. Muthumuni, T. Peltola, S. Undleeb, I. Volobouev, Z.
Wang Vanderbilt University, Nashville, USA
S. Greene, A. Gurrola, R. Janjam, W. Johns, C. Maguire, A. Melo, H. Ni, K.
Padeken, F. Romeo, J.D. Ruiz Alvarez, P. Sheldon, S. Tuo, J. Velkovska, M.
Verweij, Q. Xu University of Virginia, Charlottesville, USA
M.W. Arenton, P. Barria, B. Cox, R. Hirosky, M. Joyce, A. Ledovskoy, H. Li, C.
Neu, T. Sinthuprasith, Y. Wang, E. Wolfe, F. Xia Wayne State University,
Detroit, USA
R. Harr, P.E. Karchin, N. Poudyal, J. Sturdy, P. Thapa, S. Zaleski University
of Wisconsin - Madison, Madison, WI, USA
J. Buchanan, C. Caillol, D. Carlsmith, S. Dasu, I. De Bruyn, L. Dodd, B.
Gomber75, M. Grothe, M. Herndon, A. Hervé, U. Hussain, P. Klabbers, A. Lanaro,
K. Long, R. Loveless, T. Ruggles, A. Savin, V. Sharma, N. Smith, W.H. Smith,
N. Woods †: Deceased
1: Also at Vienna University of Technology, Vienna, Austria
2: Also at IRFU, CEA, Université Paris-Saclay, Gif-sur-Yvette, France
3: Also at Universidade Estadual de Campinas, Campinas, Brazil
4: Also at Federal University of Rio Grande do Sul, Porto Alegre, Brazil
5: Also at Université Libre de Bruxelles, Bruxelles, Belgium
6: Also at University of Chinese Academy of Sciences, Beijing, China
7: Also at Institute for Theoretical and Experimental Physics, Moscow, Russia
8: Also at Joint Institute for Nuclear Research, Dubna, Russia
9: Now at Helwan University, Cairo, Egypt
10: Now at Fayoum University, El-Fayoum, Egypt
11: Also at British University in Egypt, Cairo, Egypt
12: Now at Ain Shams University, Cairo, Egypt
13: Also at Department of Physics, King Abdulaziz University, Jeddah, Saudi
Arabia
14: Also at Université de Haute Alsace, Mulhouse, France
15: Also at Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State
University, Moscow, Russia
16: Also at Tbilisi State University, Tbilisi, Georgia
17: Also at CERN, European Organization for Nuclear Research, Geneva,
Switzerland
18: Also at RWTH Aachen University, III. Physikalisches Institut A, Aachen,
Germany
19: Also at University of Hamburg, Hamburg, Germany
20: Also at Brandenburg University of Technology, Cottbus, Germany
21: Also at Institute of Physics, University of Debrecen, Debrecen, Hungary
22: Also at Institute of Nuclear Research ATOMKI, Debrecen, Hungary
23: Also at MTA-ELTE Lendület CMS Particle and Nuclear Physics Group, Eötvös
Loránd University, Budapest, Hungary
24: Also at Indian Institute of Technology Bhubaneswar, Bhubaneswar, India
25: Also at Institute of Physics, Bhubaneswar, India
26: Also at Shoolini University, Solan, India
27: Also at University of Visva-Bharati, Santiniketan, India
28: Also at Isfahan University of Technology, Isfahan, Iran
29: Also at Plasma Physics Research Center, Science and Research Branch,
Islamic Azad University, Tehran, Iran
30: Also at ITALIAN NATIONAL AGENCY FOR NEW TECHNOLOGIES, ENERGY AND
SUSTAINABLE ECONOMIC DEVELOPMENT, Bologna, Italy
31: Also at Università degli Studi di Siena, Siena, Italy
32: Also at Scuola Normale e Sezione dell’INFN, Pisa, Italy
33: Also at Kyung Hee University, Department of Physics, Seoul, Korea
34: Also at International Islamic University of Malaysia, Kuala Lumpur,
Malaysia
35: Also at Malaysian Nuclear Agency, MOSTI, Kajang, Malaysia
36: Also at Consejo Nacional de Ciencia y Tecnología, Mexico City, Mexico
37: Also at Warsaw University of Technology, Institute of Electronic Systems,
Warsaw, Poland
38: Also at Institute for Nuclear Research, Moscow, Russia
39: Now at National Research Nuclear University ’Moscow Engineering Physics
Institute’ (MEPhI), Moscow, Russia
40: Also at St. Petersburg State Polytechnical University, St. Petersburg,
Russia
41: Also at University of Florida, Gainesville, USA
42: Also at P.N. Lebedev Physical Institute, Moscow, Russia
43: Also at California Institute of Technology, Pasadena, USA
44: Also at Budker Institute of Nuclear Physics, Novosibirsk, Russia
45: Also at Faculty of Physics, University of Belgrade, Belgrade, Serbia
46: Also at INFN Sezione di Pavia a, Università di Pavia b, Pavia, Italy
47: Also at University of Belgrade, Belgrade, Serbia
48: Also at National and Kapodistrian University of Athens, Athens, Greece
49: Also at Riga Technical University, Riga, Latvia
50: Also at Universität Zürich, Zurich, Switzerland
51: Also at Stefan Meyer Institute for Subatomic Physics (SMI), Vienna,
Austria
52: Also at Adiyaman University, Adiyaman, Turkey
53: Also at Istanbul Aydin University, Istanbul, Turkey
54: Also at Mersin University, Mersin, Turkey
55: Also at Piri Reis University, Istanbul, Turkey
56: Also at Ozyegin University, Istanbul, Turkey
57: Also at Izmir Institute of Technology, Izmir, Turkey
58: Also at Marmara University, Istanbul, Turkey
59: Also at Kafkas University, Kars, Turkey
60: Also at Istanbul University, Istanbul, Turkey
61: Also at Istanbul Bilgi University, Istanbul, Turkey
62: Also at Hacettepe University, Ankara, Turkey
63: Also at Rutherford Appleton Laboratory, Didcot, United Kingdom
64: Also at School of Physics and Astronomy, University of Southampton,
Southampton, United Kingdom
65: Also at Monash University, Faculty of Science, Clayton, Australia
66: Also at Bethel University, St. Paul, USA
67: Also at Karamanoğlu Mehmetbey University, Karaman, Turkey
68: Also at Purdue University, West Lafayette, USA
69: Also at Beykent University, Istanbul, Turkey
70: Also at Bingol University, Bingol, Turkey
71: Also at Sinop University, Sinop, Turkey
72: Also at Mimar Sinan University, Istanbul, Istanbul, Turkey
73: Also at Texas A&M University at Qatar, Doha, Qatar
74: Also at Kyungpook National University, Daegu, Korea
75: Also at University of Hyderabad, Hyderabad, India