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

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


Introduction
In the very thorough analysis we have conducted in Ref. [1] on the available parameter space of the inert ouble model (IDM) , for the low mass Dark Matter (DM) (below the W mass, M W ), we have confirmed the survival of a very small region with a DM mass of 55 − 60 GeV which has recently been unraveled [33,49]. A good value of the relic density [50] is obtained. Within the freeze-out mechanism this is possible, mainly, through efficient co-annihilation. The two additional (lightest) neutral scalars of the IDM, X and A, need to be thermodynamically close to one another for this coannihilation to take place. In other words, this requires a certain degree of mass degeneracy which, in the context of the IDM, is not necessarily fine-tuned and unnatural. The Z-mediated co-annihilation is into the (light) fermion pairs, not including the top quark, of the Standard Model (SM). The process is a gauge interaction induced process and as such it is quite efficient. To obtain a value of the relic density around 0.12, the co-annihilation cross-sections must be dampened by the Boltzmann factor, e −m − /T , where m − = M A − M X is the mass splitting and T is the temperature. The Boltzmann factor in the thermally averaged cross-sections should furnish enough reduction to counterbalance the large cross-sections. This in turn requires the mass splitting to be not too small. In our analyses we found that m − = 8 GeV is an optimal value. This value is at the edge of the LEPII constraint on the IDM. Higher values of the mass splitting would lead to visible tracks at LEP that disqualify the model while lower values do not give enough Boltzmann reduction leading to too small Ωh 2 . For our one-loop analyses, we have retained two benchmarks points with mass M X = 58, 60 GeV and m − = 8 GeV, see Table 1.  Table 1: Characteristics of the co-annihilation benchmarks points P58 and P60. All masses are in GeV. We also show tree-level (calculated with (α(0)) relic density and the weight in percent of each channel contributing to the relic density. f stands for all light fermions in the SM (the top-quark channel is closed). For more precise numbers, especially of the underlying parameters λ 3,4,5 , please refer to [1].
As we will see, this 2 GeV difference in the DM mass between the two benchmarks is enough to give a 20% reduction in the (co)-annihilating cross-sections in P60 compared to P58. While the Boltzmann factor is (slightly) larger for the M X = 60 GeV case to somehow compensate for the drop in the AX → ff cross-sections, what really benefits P60 is a larger cross-section of the annihilating process XX → W W , thanks to a larger threshold than in the case of M X = 58 GeV. The masses of all the scalars of the IDM have been set to define the model, an extra parameter that is needed to fully describe the interaction between these scalar and the SM scalar sector is λ L . In fact, λ L describes the interaction strength of the SM Higgs to the DM, X, see [1] for details. We find that in order to keep the cross-sections small, including XX annihilations, λ L must be as small as possible. This is the reason that for the surviving points we take λ L = 0. The latter suppresses SM Higgs exchange in XX → W W and avoids annihilation into bb through Higgs exchange. We must note that for sufficient relative velocity the SM Higgs resonance is accessed for this range of DM masses and even if λ L = 0 SM Higgs exchange may kick in at one-loop. The Higgs resonance scenarios deserve a dedicated analysis. Of course, the selection of the benchmark points was based on tree-level cross-sections. It is interesting to know how these cross-sections are affected by loop calculations. AX → ff accounts for more than 95% (90%) of the contribution for the relic density for P58 (P60). In this section we therefore concentrate on the electroweak radiative corrections for such processes. The rest of the annihilation contributions are from XX → W W . In the present paper we include the results of the one-loop contribution to XX → W W ≡ W ff but we leave many of the technical details covering both 2 → 3 processes at one-loop and the treatment of the SM Higgs resonance to more detailed presentations in Refs. [51,52].
2 The co-annihilation cross-section AX → ff 2.1 AX → ff : tree-level considerations Figure 1: Tree-level Feynman diagrams (in the non-linear Feynman gauge [1,53]) for AX → bb as a representative of AX → ff .
The co-annihilation cross-sections are completely determined by the gauge coupling, given the masses of X and A. The annihilations are essentially P -wave with a very small S-wave contribution that is noticeable only for extremely small velocities in the case of the heaviest final state fermion, the b-quark. For such 2 → 2 processes, the cross-section can be described by a transparent analytical formula. With m f the mass of the final fermion, the tree-level cross-section can be written as The P -wave, v 2 term, dominates by far while the very small S-wave contribution, σ 0 , is suppressed by the chiral factor m 2 f . The latter is further suppressed by the mass splitting factor, m 2 − , as shown explicitly below. With the "running" partial width The approximation ( ) in equation 2.3 which amounts to m f → 0, is excellent for all fermions including the bottom-quark (for which the approximation is off by about 3 per-mille for the range of velocities relevant for the calculation of the relic density contributed by these channels). Despite our P -wave appellation, note that the s (and hence v) dependence contained in σ ff v 2 is not small for the 2 scenarios that we are studying, see Figure 2. This s-dependent factor in the expression of σ ff v 2 explains that for the same value of the relative velocity, the co-annihilation processes are about 20% larger for P58 as compared to P60. Figure 2, which is the result of a fully automated computation with our Figure 2: Left panel: The tree-level cross-section for σ(XX → νν) for P58 and P60 as a function of the relative velocity. The dashed curves are the pure P -wave (∝ v 2 ) approximation. We see that this pure v 2 approximation is only valid for v < 0.4. Note that the cross-sections are quite large, however, remember that a Boltzmann suppression will be applied to these co-annihilations cross-sections when they are converted to effective velocity/temperature averaged cross-sections. Right panel: The bb cross-section normalised to the dd cross-section and the dd cross-section normalised to the νν cross-section. See text for more detailed comments on these plots.
code SloopS [54][55][56][57][58][59][60][61][62], exhibits the P -wave nature of the co-annihilation and confirms the analytical results. Departure from a pure v 2 dependence is clearly seen. The ratio σ(bb)/σ(dd) clearly shows the very small contribution of the S-wave for small v and the tiny effect of the mass of the fermion for larger v. In the case of massless fermions, Figure 2 confirms that σ(νν)/σ(dd) is given by the ratio of the respective partial widths of the Z into these fermions.

AX → ff : cross-sections at one-loop
Since these co-annihilation processes do not depend on λ L , a fully OS renormalisation exactly along the lines of the renormalisation of the SM can be carried out. The one-loop corrections will have no scale dependence. These processes allow to test whether the use of a running α is a good approximation for the full one-loop corrections. We will show results for all light flavours f = ν, l, u, d, b. For definiteness, we take ν = ν τ , and l = µ. We look at both d and b since the latter is affected by the heavy top-loop, as is the case for the Z → bb decay, and also because it may be more sensitive to the λ 2 dependence which appears in the S-channel. Recall that λ 2 measures the self-interaction exclusively within the dark sector, see [1]. We find however that this λ 2 dependence is very much suppressed by the mass of the (very) fermions, see Figure 3, as we explain below.
We perform the full one-loop electroweak corrections for both benchmarks P58 and P60. The electroweak corrections, expressed in terms of relative corrections, are practically exactly the same for Figure 3: Some one-loop diagrams (in the non-linear Feynman gauge) for AX → bb. We only show a subset of the box corrections and the triangles. Note the last two diagrams involve the quartic coupling within the dark sector, λ 2 . Unfortunately, the one mediated by the neutral Goldstone is proportional to the fermion mass. The one mediated by the Z will also give a contribution proportional to the fermion mass, since the one-loop integration will give, unlike the tree-level structure involving the difference between the 2 incoming momenta, a new Lorentz structure proportional to the Z 4-momentum which ends up giving a contribution proportional to the final fermion mass. Therefore the λ 2 dependence only affects the S-wave.
all channels for both benchmark points. We therefore only show the results for P58. The computation of the radiative corrections for the charged fermions includes the QED final state radiation that we obtain through the slicing technique, see [55] for example. For a fermion of charge Q f , these final state QED corrections amount to a relative correction 3Q 2 f α/4π, which are very small, < 0.2% (for charged leptons). The QCD final state correction ∼ α s (M 2 Z )/π amounts to about 3.6%. We have not included these in the figures we show below, Figure 4. Apart from the b quarks, the relative corrections for all fermion final states are practically a modest rescaling of the tree-level cross-section. Indeed the relative corrections are velocity independent. For the b quark, this is also the case for all v but v ∼ 0 where the S-wave contributes negligibly. We will come back to this feature. The flavour dependence of the correction is also very small, all corrections being within 2% of each other. Apart from the behaviour at v ∼ 0, the corrections at all v for the bottom-quark final states are slightly different due to the contribution of the heavy top-quark, a feature known from Z-physics. On average, the corrections are about 6% and are therefore a factor 2 lower than if we had used α(M 2 Z ). Genuine box and triangle contributions are therefore not negligible.
Inspired by equation 2.3, and in view that the S-wave contribution is tiny even in the case of bb we suggest to improve the tree-level cross-sections by using effective cross-sections defined solely in terms of physical observables,   The tiny λ 2 dependence in the bb co-annihilation channel. The λ 2 dependence is measured relative to the dd cross-section where the λ 2 dependence is vanishing. Observe that the units in the graph indicate a variation which, at its highest, does not cross a couple of per-mille.
The very tiny contribution of the S-wave at very small velocity (where the P -wave vanishes) is theoretically interesting even if it is without any phenomenological impact. It could have been larger if the fermion masses were larger. Theoretically, the S-wave is sensitive to λ 2 as explained in the caption of Figure 3. We verified this property directly in the co-annihilation into neutrinos where no λ 2 dependence was detected numerically and we confirmed that the largest λ 2 dependence occurs with the b final state. Unfortunately, the maximum correction from the dark sector is at the level of 2 per-mille for v = 0 where the cross-section is smallest. As soon as the P -wave contribution kicks in, the tiny λ 2 dependence in the S-wave contribution is totally swamped such that the one-loop corrected cross-section is, for all purposes, insensitive to λ 2 . This very tiny λ 2 dependence is shown in Figure 5 where we track the λ 2 dependence by considering the ratio in the bb cross-section between λ 2 = 0.01 and λ 2 = 2.
3 The XX → W W in the co-annihilation region

Tree-level considerations
For P60, XX annihilation to W W below threshold with one W decaying into a fermion pair, accounts at tree-level, for 10% of the relic density. This cross-section does not suffer from the Boltzmann suppression. A non-zero value of λ L would have added a Higgs exchange contribution as we show in Figure 6 in the W W channel, and in the bb channel (XX h − → bb).
As v increases, the phase space for W W increases and therefore the cross-section increases as Figure 6: A selection of diagrams for the tree-level cross-section XX → W ff . Since λ L =0 for the benchmark point P60, Higgs exchange does not take place for P60. Figure 7 shows. Note that this cross-section when compared to AX → ff is far smaller (the latter will be reduced considerably by the Boltzmann factor). This increase with velocity is smooth across the Higgs resonance, v = 0.56, a threshold that, as expected, is not felt since λ L = 0.  While λ L =0 at tree-level for XX → W f f with no SM Higgs exchange contributing, a renormalisation of λ L is still called for in this process. This is because in general the XX → W W amplitude does depend parametrically on λ L . As Figure 8 clearly shows, an induced one-loop hXX is generated which entails a one-loop Higgs exchange contribution. The critical question is whether this will lead to an instability when the pole at s = M 2 h is reached? Do we then have to, and how to, include a width? casting doubt on the organisation of the perturbation series? This same crucial point is dealt with in detail when we study cases where the resonance is present already at "tree-level". In the present case, λ L = 0, there is no need to include a width. Since h → XX is open, the OS scheme is most appropriate for a definition and renormalisation of λ L . The OS scheme means that we trade λ L with Γ h→XX . In this particular case, the input translates into Γ h→XX = 0. But Γ h→XX = 0 is maintained at all orders entails, in fact, that the renormalised amplitude for M h→XX (s) = 0 at the renormalisation point s = M 2 h , s is the invariant mass of the XX system. For a selection of one-loop contributions that make the h → XX vertex amplitude see Figure 8. Consequently, M h→XX (s) 1

One-loop results
is continuous and the Laurent's series is well defined without a regulator, the width. There is then no pole structure as such and away from s = M 2 h other (non-resonant) structures in the amplitude contribute as importantly. What is very interesting is that the one-loop amplitude is now λ 2 dependent, see Figure 8. Note that XX → W W is an abuse of language since such an off-shell amplitude is not an element of the S-matrix. At one-loop, this statement is supported by the appearance of essential pentagon contributions where a split into a W W part is not possible. Nonetheless, the fact that the final state fermions are massless, we find no flavour dependence in the normalised (with respect to the tree-level) loop corrections when the full one-loop contribution is taken into account. In fine, this is reminiscent and strongly related to the fact that (normalised) electroweak corrections to W decay into fermion pairs are flavour independent [63]. Our results for the electroweak corrections in this channel are displayed in Figure 9. The behaviour with respect to velocity is very smooth, for all values of λ 2 , confirming that the location of the Higgs is crossed continuously. The λ 2 dependence at one-loop is important. The full one-loop correction is about 10% for the smallest of λ 2 , λ 2 = 0.01, it gives an almost vanishing total correction for λ 2 = 1 and then decreases to about −10% for λ 2 = 2. For this value, λ 2 = 2, there is a 30% difference with the use of an effective α(M 2 Z ). Therefore, α(M 2 Z ) does not perform well as an approximation. The velocity dependence of the λ 2 correction is very weak. To summarise, at the level of the cross-sections the λ 2 contribution is important and discriminating. Will this conclusion still hold when we call these cross-sections to compute the relic density considering that the weight of this channel is only 10%?

Effect on the relic density
In order to translate the cross-sections into a prediction on the relic density, we interface our calculations of the one-loop corrected cross-section with microMEGAs [64][65][66][67][68] to perform the convolution of the velocity and temperature and solving the Boltzmann equation. Table 2 shows that the correction to the relic density for these low mass co-annihilation configurations are small with a correction in line with what we found for the dominant cross-sections AX → ff (remember that Ω ∝ 1/σ) which are practically v-independent. As expected, the corrections are much smaller than those found with the approximation of using a running α. Since in P58 the contribution of the W W channel is less than about 5%, we do not include the effect of the electroweak corrections in this channel. However, we include these corrections for P60 since this channel accounts for almost 10% of the total relic density (at tree-level). While any small λ 2 dependence (which occurs at the very small v in AX → ff ) is washed out when the corrections are translated into the relic density calculation, the λ 2 dependence observed in the cross-sections XX → W ff , make their way into the relic density even if the W W channel account for only ∼ 10%. For λ 2 = 2, the correction in the W W channel counterbalances part of the correction from the contributions AX → ff leading to an overall correction on the relic density less than 3%. As a summary, in the co-annihilation region, if one is satisfied with a calculation  Table 2: The relic density for P58 and P60 with cross-sections computed at tree-level and after including the full electroweak corrections, as well as implementing the effective cross-sections for AX → ff and the cross-sections calculated with α(M 2 Z ). The percentage correction to the relic density is given in parenthesis. The percentage weight of each channel is computed for each implementation of the corrected cross-section.
of the relic density with an accuracy no better than 5%, one should content oneself with a tree-level calculation. An "improved" (α(M 2 Z )) gives a correction of about 10%. The effective cross-sections that are based on re-expressing the AX → ff in terms of the one-loop partial decay widths of the Z → ff reproduce the full one-loop results within 1%.

Conclusions
We have studied two IDM low mass DM benchmark points where co-annihilation is the dominant contribution to the relic density. The co-annihilation into fermions is driven essentially by the SM gauge coupling. The relative corrections are small. We show that they can be parameterised by simple effective cross-sections involving Z-observables. The fermion flavour dependence is small. The small contribution from W W to the relic density is interesting since at one-loop it reveals an important λ 2 dependence that accesses the purely dark sector. However, this λ 2 dependence gets diluted because of the small weight of W W to the relic density. Nonetheless, our calculations show that performing a full one-loop calculation gives smaller corrections than using an effective α(M 2 Z ) into tree-level crosssections.
fellowship. He also acknowledges the support received from DST, India, under grant number IFA19-PH237 (INSPIRE Faculty Award).