Relic density of dark matter in the inert doublet model beyond leading order for the low mass region. IV. The Higgs resonance region

One-loop electroweak corrections to the annihilation cross sections of dark matter in the Higgs resonance region of the inert doublet model are investigated. The procedure of how to implement the width of the Higgs in order to regularize the amplitude both at tree level and at one loop together with the renormalization of a key parameter of the model, are thoroughly scrutinized. The discussions go beyond the application to the relic density calculation and also beyond the inert doublet model so that addressing these technical issues can help in a wider context. We look in particular at the dominant channels with the $b\overline{b}$ final state and the more involved three-body final state, $Wf{\overline{f}}^{\prime }$, where both a resonance and an antiresonance, 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.

Figure 1

A single diagram, Higgs mediated, contributes to $XX\to b\overline{b}$ at tree level.

Figure 2

The velocity dependence of the ratio of the full tree-level computation relative to the fixed width Breit-Wigner calculation based on Eq. (3.1) for points P57 and P59. The horizontal line, exact agreement, meets the curves at exactly the resonance points.

Figure 3

The relative velocity dependence of the tree-level $XX\to b\overline{b}$ cross section times $v$ for points P57 and P59 in units of ${10}^{-26}{\mathrm{cm}}^3{\mathrm{s}}^{-1}$ (note the log scale). We also show the weighted cross sections with the velocity distribution [1] with the freeze-out parameter $x_F=25$.

Figure 4

A selection of one-loop electroweak correction diagrams contributing to $XX\to b\overline{b}$. The first three diagrams of the first row are genuine boxes which are nonresonant. The last two 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.

Figure 5

The relative velocity dependence of the one-loop $XX\to b\overline{b}$ cross section times $v$ for points P57 (left panels) and P59 (right panels) in units of ${10}^{-26}{\mathrm{cm}}^3{\mathrm{s}}^{-1}$ (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 an enlargement 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 $\mathrm{\Gamma }(h\to b\overline{b})$ [the improved cross section given by Eq. (4.13)]. We also give the result of using, at tree level, $\alpha (M_Z^2)$ instead of $\alpha$.

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 $h\to l{l}^{\star }\to lW{\nu }_l$ that gives effects proportional to the lepton mass, which we do not show here but which are included in our full calculation.

Figure 7

As in Fig. 3 but for $XX\to W\tau {\nu }_{\tau }$. Note the behavior 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 Fig. 3.

Figure 8

As in Fig. 5 but for the $XX\to Wl{\nu }_l$ channel.